Progress in Nonlinear Optics Research

PROGRESS IN NONLINEAR OPTICS RESEARCH

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PROGRESS IN NONLINEAR OPTICS RESEARCH

MIYU TAKAHASHI AND

HINA GOTÔ EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter cover herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal, medical or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Library of Congress Cataloging-in-Publication Data Takahashi, Miyu. Progress in nonlinear optics research / Miyu Takahashi and Hina Gotô. p. cm. ISBN 978-1-60741-929-7 (E-Book) 1. Nonlinear optics. I. Goto, Hina. II. Title. QC446.2.T35 2008 535’.2—dc22

Published by Nova Science Publishers, Inc.

New York

2008015606

CONTENTS Preface Chapter 1

Chapter 2

Chapter 3

Chapter 4

vii Nonlinear Photonic Fibre Ring Lasers: Stability, Harmonic Detuning, Temporal Diffraction and Bound States Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

1

Maneuvering Atoms for Lithography Using Near Resonant Spatially Varying Laser Field Kamlesh Alti and Alika Khare

63

Highly Efficient Nonlinear Frequency Conversion Schemes for Compact Femtosecond Erbium Fiber Lasers: From the Near Ultraviolet through the Entire Visible into the Infrared Konstantinos Moutzouris, Florian Adler, Florian Sotier, Daniel Träutlein, Alexander Sell, Elisa Ferrando-May and Alfred Leitenstorfer The New Process of Adaptive Optics Based on Nonlinear Control Algorithms for Applications in Solid-State Lasers and ICF System Ping Yang, Mingwu Ao, Bing Xu and Wenhan Jiang

85

115

Chapter 5

High-Order Harmonic Generation in Laser-Produced Plasma R. A. Ganeev

Chapter 6

Sum-Frequency Generation and Multiphoton Ionization in Rare Gases with Non-Gaussian Laser Beams V. Peet

197

Real-Time Electroholography Using FPGA Technology, and Color Electroholography by the Time Division Switching Method Tomoyoshi Shimobaba and Tomoyoshi Ito

261

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement in Artificial Nanostructures Guohong Ma, Jielong Shi and Qi Wang

281

Chapter 7

Chapter 8

149

vi Chapter 9

Chapter 10 Index

Contents Monochromatic Wavefield Evolution in Waveguide Arrays with Gain and Nonlinearity Anatoly P. Napartovich and Dmitry V. Vysotsky Vortices of Light: Generation, Characterization and Applications R. P. Singh, Ashok Kumar and Jitendra Bhatt

327 359 383

PREFACE Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light. This nonlinearity is typically only observed at very high light intensities such as those provided by pulsed lasers. Nonlinear optics is of considerable research interest since it includes: Free space and guided wave nonlinear optics; Weak (second and third order etc. effects) and strong (non-perturbative) nonlinearities; Fast (electronics fs timescale) and slow (thermal ms timescale) nonlinearities; Novel nonlinear materials; Numerical simulation of nonlinear optical propagation; Applications of nonlinear optics in fields such as signal processing, optical communications, holographic memory and soliton phenomena. This new book provides the latest research from around the globe in this fast-moving field. Chapter 1 - We present the operational principals and implementation of mode-locked fibre lasers operating in the nonlinear region, whether via the saturation optical saturated amplification or photonic interactions of the pump sources and generated lightwaves. Harmonic and regenerative mode locked type for 10 and 40 G-pules/sec., harmonic detuning type for up to 200 G-pulses/sec., the harmonic repetition multiplication via temporal diffraction, and the multi-wavelength type are reported. An ultra-stable mode-locked laser operating at 10 GHz repetition rate has been designed, constructed and tested. The laser generates optical pulse train of 4.5 ps pulse width when the modulator is biased at the phase quadrature quiescent region. Long term stability of amplitude and phase noise that indicates that the optical pulse source can produce an error-free pattern in a self-locking mode for more than 20 hours, the most stable photonic fibre ring laser reported to date. The rep-rate of the mode locked fibre ring laser is demonstrated up to 200 Gpulses/sec. using harmonic detuning mechanism in a ring laser. In this system, we investigate the system behaviour of rational harmonic mode-locking in the fibre ring laser using phase plane technique. Furthermore, we examine the harmonic distortion contribution to this system performance. We also demonstrate 660x and 1230x repetition rate multiplications on 100MHz pulse train generated from an active harmonically mode-locked fibre ring laser, hence achieve 66 GHz and 123 GHz pulse operations, which is the highest rational harmonic order reported to date. The system behaviour of group velocity dispersion repetition rate multiplication is also demonstrated. The stability and the transient response of the multiplied pulses are studied using the phase plane technique. Furthermore, we demonstrate, for the first time, experimental generation of multi-soliton bound states in an active FM mode-locked

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fiber laser. Not only bound soliton pairs but also triple- and quadruple-soliton bound states are generated. Chapter 2 - Two kinds of forces are experienced by an atom in a laser radiation field, viz; radiation pressure resulting form absorption followed by the random spontaneous emission of photons and the conservative or dipole force originating from interaction of neutral atom with a near resonant non uniformly distributed laser field. The magnitude of radiation pressure is limited by the rate of spontaneous emission and saturates as the laser intensity increases. This dissipative spontaneous force is responsible for laser cooling. On the other hand, the magnitude of dipole force depends on the intensity gradient and amount of detuning and the direction of force depend on the sign of detuning. With the proper configuration of the atomic beam and the laser beam, the trajectories of the atoms can be manipulated so as to focus down the atoms in the periodic nano-size structures of desired geometry. The intensity gradient acts as the atomic lens as well as mask and thereby obviating the need of material mask for writing periodic patterns. The complete writing using dipole force can be performed in a single step. This lithography technique is very general and is applicable to any atomic and molecular species provided the tunable laser for the particular transition frequency of atoms/molecules is available. This article primarily reviews simulation studies of atomic trajectories under dipole force in various spatially varying laser light field for the lithographical applications. A novel concept of multiple atomic beams traveling in TEM00 mode of laser is also discussed to generate the periodic patterns of periodicity less than /2, where  is the wavelength of light used, and sizes in range of tens of nanometer. Chapter 3 - Recent advances in fiber lasers unlock the potential for novel nonlinear devices and generation of new wavelengths. In this review article we examine the application of five nonlinear frequency conversion schemes to a femtosecond fiber laser system. We discuss in detail particular issues concerning the design, choice of material and performance characteristics for each converter. Exploiting single-pass and cascaded harmonic generation as well as sum frequency mixing, a nearly continuous spectral coverage from the near ultraviolet to the near-infrared is achieved. This widely tunable radiation appears in the form of ultrashort pulses with a repetition rate in the range of 100 MHz and average power levels ranging between 1 mW and more than 100 mW. We also report on a concept for broadband-to-narrowband frequency doubling of femtosecond pulses. Unexpectedly high efficiencies are predicted owing to intra-pulse sumfrequency mixing. Experimental proof is provided via demonstration of frequency doubling of sub-30-femtosecond near infrared pulses, with spectral bandwidths exceeding 100 nm, into the visible. Efficiencies higher than 30% are achieved. Spectral narrowing by a factor of up to more than 50 and an increase in spectral intensity by one order of magnitude are observed simultaneously without gain saturation. The performance characteristics of this relatively simple device, such as the wide coverage of the visible spectrum, as well as the coexistence of several perfectly synchronized outputs, have attracted attention for its immediate use in various scientific and technological fields, including bioimaging, ultrafast spectroscopy and optical frequency metrology. Chapter 4 - A new adaptive optics (AO) system for optimizing the output laser mode of a diode-laser pumped Nd:YAG solid-state laser has been built in our laboratory. A piezoelectric deformable mirror (DM) which is taken as the rear mirror of the solid-state laser is controlled by a real encoding genetic algorithm. To improve the AO system convergence rate, a group of

Preface

ix

Zernike mode coefficients is taken as the optimizing basis instead of the voltages on the DM. The transform matrix between voltages and Zernike mode coefficients has been deduced. A series of comparative numerical results show that the convergence speed and the correction performance of the AO system based on optimizing Zernike mode coefficients is far better than that of optimizing voltages. Moreover, Experimental results also showed that this AO system could change higher transverse modes into TEM00 mode successfully. In another way, so as to detect the entire beam aberrations of an inertial confinement fusion (ICF) system᧨an amendatory phase-retrieval method is presented, simulative and experimental results shown that using this method, the aberrations of entire beam path can be reconstructed just from a few pairs of DM surface shapes and their corresponding focal spots intensity. Chapter 5 - The studies of the high-order harmonic generation (HHG) in laser plasma are reviewed. We discuss the harmonic generation conditions in various laser plumes, which allowed achieving the HHG in plasma up to 7.9 nm. New approach for harmonic enhancement through the resonance-induced growth of HHG conversion efficiency in some low-excited plasmas is offered, which allowed achieving the 10-4 conversion efficiency for single harmonic. Various other experimental schemes and approches are reviewed as well, such as the harmonic generation in nanoparticles-contained laser plumes, excitation of laser plasma by the prepulses of different duration, variations of the chirp of femtosecond radiation for improving the brightness of harmonics, application of short-wavelength radiation for harmonic generation, etc. Chapter 6 - In recent years the generation, properties, and interaction with matter of new types of non-Gaussian laser beams have been of great interest. Such novel beams like Bessel, Bessel-Gauss, Mathieu, segmented conical beams and others show a number of interesting effects in the field of nonlinear optics. In many cases, the nonlinear effects, which are well known in ordinary laser beams, show significant differences under excitation by nonGaussian beams. The present paper gives an overview of some gas-phase nonlinear effects observed under excitation by several new types of laser beams. Considered is the generation of resonance-enhanced third and fifth harmonics under excitation by different conical beams (Bessel, Bessel-Gauss, and segmented beams) and comparison of these processes with ordinary Gaussian beams. A similar comparison is made for sum-frequency generation processes under two-color excitation by spatially coherent and incoherent non-Gaussian and Gaussian beams. Under intense excitation, the multiphoton excitation and ionization of the target gas atoms become important. A large number of experimental and theoretical works have shown the importance of harmonics of the fundamental laser light in these processes. Here again the use of non-Gaussian laser beams and different beam configurations results in several interesting observations. Some of the effects of internally-generated harmonic fields on the excitation of atomic resonances in non-Gaussian laser beams are considered and discussed. Chapter 7 - In this chapter, we describe two topics for electroholographic threedimensional (3D) display. The first topic is real-time electroholography using the fieldprogrammable gate array (FPGA) technology. We developed an electroholographic display unit, which consists of a special-purpose computational chip (SPC) for holography and a reflective liquid-crystal display (LCD) panel, for a 3D display. We used a FPGA chip for the SPC, and we designed the SPC by adopting our proposed method, which can calculate the phase on a computer-generated hologram (CGH) using two recurrence formulas. The SPC

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Miyu Takahashi and Hina Gotô

can compute a computer-generated hologram (CGH) of 800 x 600 grids in size from a 3D object consisting of approximately 400 points in approximately 0.15 seconds. We implemented the SPC and LCD panel on a printed circuit board. After the calculation, the CGHs produced by the SPC are displayed on the LCD panel. When we illuminate a reference light to the LCD panel, we can observe a real-time 3D animation of approximately 3cm x 3cm x 3cm in size. The second topic is color electroholography using the time division switching method. We used a reflective LCD panel with a high refresh rate as a displaying device for a CGH. A color 3D object data is divided into red, green and blue components, from which we compute three CGHs. The LCD panel displays the CGHs in sequence at a refresh rate of about 100Hz. The LCD panel also outputs synchronized signals, indicating that one of the CGHs is currently displayed on the LCD panel. Red, green and blue light-emitting diodes (LEDs) used for reference lights, are switched by the synchronized signals. As a result of the afterimage effect on human eyes, we can clearly observe a colored 3D object. Chapter 8 - Materials with large optical nonlinearity and ultrafast response are the fundamental requirements for fabricating photonic devices such as optical switching and modulators. In general, two key factors are often used to evaluate the merit of electronic and photonic materials for high speed communications-modulation depth and response speed. The modulation depth is referred to the value of nonlinear susceptibility, and the response speed is related to the optical response. In the past several decades, a large number of research work was carried out on the search for and synthesis of this kind of materials, the candidate materials include semiconductor materials (especially for semiconductor quantum wells and quantum dots), photorefractive materials, metal-dielectric composite and some organic materials such as conjugated polymer, phthalocyanine and other organic materials with conjugated S-electron. From another point of view, if the incident optical electric field was confined into a small region inside a target material (in other words, electric field localization), the final result is equal to the materials possessed a large optical nonlinearity. Following this idea, by designing a certain composite structures, a large optical nonlinearity and fast response can be realized. In this article, we will address the ultrafast optical nonlinearity response enhancement for two-type of artificial structures including 1. noble metal/dielectrics and metal/semiconductor nanocomposite structure; 2. one dimensional photonic crystal with defect which is consisted of nonlinear optical materials. This article includes two chapters, chapter 1 will present the fundamental optical properties and advanced nonlinear optical response of some novel metal nanoparticles. First we will touch on the size and shape-dependence of surface plasmon resonance of metal particle. Then we will discuss the ultrafast nonlinear optical response following the ultrafast laser pulse on-resonant excitation. The last section of the chapter will focus on our findings, size- and dielectric-dependence of nonlinear optical enhancement with off-resonance excitation in Au-dielectric nanocomposite structure. In chapter 2 we will discuss another kind of artificial structures which can realize light localization, i.e. one dimensional defective photonic crystal. One-dimensional photonic crystal (1D PC) may be regarded as a waveguide consisting of a sequence of dielectric mirrors with periodically modulated dielectric constants. A defective 1D PC is constructed by inserting a defect layer into such a multilayer structure. Introduction of defect permits

Preface

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localized modes to exit in the range of the frequencies of bandgap. Owing to the highly localized field, strong enhancement of optical nonlinearity by several orders of magnitude can be expected in a defect layer. If the defect layer includes nonlinear optical materials, it is envisaged that the nonlinearity may be substantially enhanced by the presence of such a strong electric field. In this chapter, we will first give briefly introduction to nonlinear photonic crystal, following that we will focus on discussion on how to optimize optical nonlinearity, at last, we will confine our discussion on defect modes interaction as well as metallodielectric structure for realization of nonlinear optical enhancement. Chapter 9 - Wavefield propagation in an array of parallel waveguides exhibits a wide variety of nonlinear phenomena. The arrays of passive waveguides with the refractive index nonlinearity in cores were analyzed by the well developed instruments of the perturbation theory. The effect of capturing the wave field in one of the waveguides was suggested for design of optical switches. Modeling the wavefield propagation in active waveguides in the strong nonlinearity condition attracts special attention last years due to the development of the high power fiber lasers and amplifiers. Such devices contain the lattice of active cores, so the wavefield can be expanded on a small number of guided modes. Both the refractive index and gain depend on the wavefield intensity. Doping of the core material by rare earth ions can increase the nonlinear coefficient of the refractive index by several orders due to polarizability difference of ions in different states. The gain disposition in the cores changes dramatically the amplification of monochromatic wave field. The laser radiation self-synchronization at pump increase was demonstrated experimentally with 7-core fiber laser. Numerical modeling by 3D diffraction code has shown the crucial role of the gain nonuniformity for this phenomenon. The evolution of linear combination of two optical modes in the fiber amplifier has been analyzed. Due to crossmodal gain power of the mode with greater start power grows linearly and can limit the lower-power modes. The single mode lasing establishes if distributed losses are supposed for laser simulation. The mode selection in the ring lasers with strong cross saturation or in the random lasers with strong scattering are the nearest analogies to this effect. The behavior of the monochromatic wavefield is studied for different constructions of the multicore fiber amplifier. The conditions for a given mode domination are found for the twin waveguides amplifier. Possible usage of the developed approach is discussed for the single mode output ensuring in single core fiber amplifier. Chapter 10 - Waves that possess a phase singularity and a rotational flow around the singular point are called vortices. In the light wave, such structures are called optical vortices. These are generated as natural structures when light passes through a rough surface or due to phase modification while propagating through a medium. However, these can be generated in a controlled manner as well. We will discuss the method used to generate optical vortices in the laboratory and how to characterize their topological charge using interferometry. In recent years optical vortices have got applications in variety of fields starting from biological physics to quantum information and computation. Therefore study of their coherence properties becomes very important. Study of the Wigner distribution function (WDF), originally discovered in quantum mechanics, can be quite useful for this purpose since it can provide coherence information in terms of the joint position and momentum phase-space distribution of the optical field. We will present experimental as well as theoretical results for the WDF of an optical vortex. Being vortices they posses helical wavefront and consequently each photon in the vortex beam carries an orbital angular momentum (OAM) l  depending

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Miyu Takahashi and Hina Gotô

on the order l of the vortex. Most of the applications with vortices of light use this property of orbital angular momentum. However, OAM per photon in the beam will be integer or noninteger (in units of ) that depends on if the vortex is axial (centered at origin) or non-axial (shifted from the origin). Thus, axial nature of the vortex can affect intrinsic property of a vortex beam. We show that the WDF can also be used to discriminate between an axial and a non-axial vortex. We will end the discussion by describing our work on second order coherence properties, based on intensity correlation studies of these novel light structures and some exciting applications of vortices of light.

In: Progress in Nonlinear Optics Research Editors: Miyu Takahashi and Hina Goto, pp.1-62

ISBN 978-1-60456-668-0 © 2008 Nova Science Publishers, Inc.

Chapter 1

NONLINEAR PHOTONIC FIBRE RING LASERS: STABILITY, HARMONIC DETUNING, TEMPORAL DIFFRACTION AND BOUND STATES Le Nguyen Binh 1, Nhan Duc Nguyen1 and Wenn Jing Lai2 1

Department of Electrical and Computer Systems Engineering, Monash University, Clayton Victoria 3168 Australia 2 Network Technology Research Centre, Techno-Plaza, Nanyang Ave., Nanyang Technological University, Singapore

Abstract We present the operational principals and implementation of mode-locked fibre lasers operating in the nonlinear region, whether via the saturation optical saturated amplification or photonic interactions of the pump sources and generated lightwaves. Harmonic and regenerative mode locked type for 10 and 40 G-pules/sec., harmonic detuning type for up to 200 G-pulses/sec., the harmonic repetition multiplication via temporal diffraction, and the multi-wavelength type are reported. An ultra-stable mode-locked laser operating at 10 GHz repetition rate has been designed, constructed and tested. The laser generates optical pulse train of 4.5 ps pulse width when the modulator is biased at the phase quadrature quiescent region. Long term stability of amplitude and phase noise that indicates that the optical pulse source can produce an error-free pattern in a self-locking mode for more than 20 hours, the most stable photonic fibre ring laser reported to date. The rep-rate of the mode locked fibre ring laser is demonstrated up to 200 Gpulses/sec. using harmonic detuning mechanism in a ring laser. In this system, we investigate the system behaviour of rational harmonic mode-locking in the fibre ring laser using phase plane technique. Furthermore, we examine the harmonic distortion contribution to this system performance. We also demonstrate 660x and 1230x repetition rate multiplications on 100MHz pulse train generated from an active harmonically mode-locked fibre ring laser, hence achieve 66 GHz and 123 GHz pulse operations, which is the highest rational harmonic order reported to date. The system behaviour of group velocity dispersion repetition rate multiplication is also demonstrated. The stability and the transient response of the multiplied pulses are studied using the phase plane technique. Furthermore, we demonstrate, for the first time, experimental

E-mail: [email protected]

2

Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai generation of multi-soliton bound states in an active FM mode-locked fiber laser. Not only bound soliton pairs but also triple- and quadruple-soliton bound states are generated.

1. Introduction Generation of ultra-short optical pulses with multiple gigabits repetition rate is critical for ultra-high bit rate optical communications, particularly for the next generation of terabits/sec. optical fibres systems. As the demand for the bandwidth of the optical communication systems increases, the generation of short pulses with ultra-high repetition rate becomes increasingly important in the coming decades. The mode locked fibres laser offers a potential source of such pulse train. Although the generation of ultra-short pulses by mode locking of a multi-modal ring laser is well known, the applications of such short pulse trains in multi-gigabits/sec optical communications challenges its designers on its stability and spectral properties. Recent reports on the generation of short pulse trains at repetition rates in order of 40 Gb/s, possibly higher in the near future[1], motivates us to design and experiment with these sources in order to evaluate whether they can be employed in practical optical communications systems. Further the interest of multiplexed transmission at 160 Gb/s and higher in the foreseeable future, requires us to experiment with optical pulse source having a short pulse duration and high repetition rates. This report describes laboratory experiments of a mode-locked fibres ring laser (MLFRL), initially with a repetition rate of 10 GHz and preliminary results of higher multiple repetition rates up to 40 GHz. The mode locked ring lasers reported hereunder adopt an active mode locking scheme whereby partial optical power of the output optical waves is detected, filtered and a clock signal is recovered at the desired repetition rate. It is then used as a RF drive signal to the intensity modulator incorporated in the ring laser. A brief description on the principle of operation of the MLFRL is given in the next section followed by a description of the mode-locked laser experimental set up and characterisation. Active mode-locked fibre lasers remain as a potential candidate for the generation of such pulse trains. However, the pulse repetition rate is often limited by the bandwidth of the modulator used or the radiofrequency (RF) oscillator that generates the modulation signal. Hence, some techniques have been proposed to increase the repetition frequency of the generated pulse trains. Rational harmonic mode-locking is widely used to increase the system repetition frequency [1-3]. 40GHz repetition frequency has been obtained with 4th order rational harmonic mode locking at 10GHz base band modulation frequency [2]. [3] has reported 22nd order rational harmonic detuning in the active mode-locked fibre laser, with 1GHz base frequency, leading to 22GHz pulse operation. This technique is simple and achieved by applying a slight deviated frequency from the multiple of fundamental cavity frequency. Nevertheless, it is well known that it suffers from inherent pulse amplitude instability as well as poor long-term stability. Therefore, pulse amplitude equalization techniques are often applied to achieve better system performance [3], [4, 5]. Other than this rational harmonic detuning, there are some other techniques have been reported and used to achieve the same objective. Fractional temporal Talbot based repetition rate multiplication technique [6, 7] uses the interference effect between the dispersed pulses to achieve the repetition rate multiplication. The essential element of this technique is the dispersive medium, such as linearly chirped fibre grating (LCFG) [6, 8] and dispersive fibre

Nonlinear Photonic Fibre Ring Lasers

3

[9-11]. Intracavity optical filtering [12, 13] uses modulators and a high finesse Fabry-Perot filter (FFP) within the laser cavity to achieve higher repetition rate by filtering out certain lasing modes in the mode-locked laser. Other techniques used in repetition rate multiplication include higher order FM mode-locking [14], optical time domain multiplexing [15], etc. The stability of high repetition rate pulse train generated is one of the main concerns for practical multi-Giga bits/sec optical communications system. Qualitatively, a laser pulse source is considered as stable if it is operating at a state where any perturbations or deviations from this operating point is not increased but suppressed. Conventionally the stability analyses of such laser systems are based on the linear behavior of the laser in which we can analytically analyze the system behavior in both time and frequency domains. However, when the mode-locked fibre laser is operating under nonlinear regime, none of these standard approaches can be used, since direct solution of nonlinear different equation is generally impossible, hence frequency domain transformation is not applicable. Some inherent nonlinearities in the fibre laser may affect its stability and performance, such as the saturation of the embedded gain medium, non-quadrature biasing of the modulator, nonlinearities in the fibre, etc., hence, nonlinear stability approach should be used in any laser stability analysis. In section 2, we focus on the stability and transient analyses of the rational harmonic mode-locking in the fibre ring laser system using phase plane method, which is commonly used in nonlinear control system. This technique has been previously used in [11] to study the system performance of the fractional temporal Talbot repetition rate multiplication systems. It has been shown that it is an attractive tool in system behaviour analysis. However, it has not been used in the rational harmonic mode-locking fibre laser system. In Section 3.1, the rational harmonic detuning technique is briefly discussed. Section 3.2 describes the experimental setup for the repetition rate multiplication used. Section 3.3 investigates the dynamic behaviour of the phase plane of the fibre laser system, followed by some simulation results. Section 3.4 presents and discusses the results obtained from the experiment and simulation. Finally, some concluding remarks and possible future developments for this type of ring laser are given. Rational harmonic detuning [3, 16] is achieved by applying a slight deviated frequency from the multiple of fundamental cavity frequency. 40GHz repetition frequency has been obtained by [3] using 10GHz base band modulation frequency with 4th order rational harmonic mode locking. This technique is simple in nature. However, this technique suffers from inherent pulse amplitude instability, which includes both amplitude noise and inequality in pulse amplitude, furthermore, it gives poor long-term stability. Hence, pulse amplitude equalization techniques are often applied to achieve better system performance [2], [4, 5]. Fractional temporal Talbot based repetition rate multiplication technique [4-8] uses the interference effect between the dispersed pulses to achieve the repetition rate multiplication. The essential element of this technique is the dispersive medium, such as linearly chirped fibre grating (LCFG) [8, 16] and single mode fibre [8, 9]. This technique will be discussed further in Section II. Intracavity optical filtering [13, 14] uses modulators and a high finesse Fabry-Perot filter (FFP) within the laser cavity to achieve higher repetition rate by filtering out certain lasing modes in the mode-locked laser. Other techniques used in repetition rate multiplication include higher order FM mode-locking [13], optical time domain multiplexing, etc. Although Talbot based repetition rate multiplication systems are based on the linear behaviour of the laser, there are still some inherent nonlinearities affecting its stability, such

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

as the saturation of the embedded gain medium, non-quadrature biasing of the modulator, nonlinearities in the fibre, etc., hence, nonlinear stability approach must be adopted. In Section 4, we focus on the stability and transient analyses of the group-velocity-dispersion (GVD) multiplied pulse train using the phase plane analysis of nonlinear control analytical technique [2]. This is the first time, to the best of our knowledge that the phase plane analysis is being used to study the stability and transient performances of the GVD repetition rate multiplication systems. In Section 4.1, the GVD repetition rate multiplication technique is briefly given. Section 4.2 describes the experimental setup for the repetition rate multiplication. Section 4.3 investigates the dynamic behaviour of the phase plane of GVD multiplication system, followed by some simulation results. Section 4.4 presents and discusses the results obtained from the experiment. Finally, some concluding remarks and possible future developments for this type of lasers are given.

2. Ultra-High Rep-Rate Fibre Mode-Locked Lasers This section gives a detailed account of the design, construction and characterisation of a mode-locked (ML) fibres ring laser. The ML laser structure employs in-line optical fibres amplifiers, a guided-wave optical intensity Mach-Zehnder interferometric modulator (MZIM) and associate optics to form a ring resonator structure generating optical pulse trains of several GHz repetition rate with pulse duration in order of pico-seconds. Long term stability of amplitude and phase noise has been achieved that indicates that the optical pulse source can produce an error-free PRBS pattern in a self-locking mode for more than 20 hours. A mode-locked laser operating at 10 GHz repetition rate has been designed, constructed, tested and packaged. The laser generates optical pulse train of 4.5 ps pulse width when the modulator is biased at the phase quadrature quiescent region. Preliminary experiment of a 40 GHz repetition rate mode-locked laser has also been demonstrated. Although it is still unstable in long term, without an O/E feedback loop, optical pulse trains have been observed.

2.1. Mode-Locking Techniques and Conditions for Generation of Transform Limited Pulses from a Mode Locked Laser 2.1.1. Schematic Structure of MLRL Figure 1 show the composition of a MLRL with an optical-electronic feedback loop. It consists principally, for a non-feedback ring, an optical close loop with an optical gain medium, an optical modulator (intensity or phase type) an optical fibres coupler and associated optics. An O/E feedback loop detecting and repetition-rate signal and generating RF sinusoidal waves to electro-optically drive the intensity modulator is necessary for the regenerative configuration as shown in Figure 1.

Nonlinear Photonic Fibre Ring Lasers

5

10 GHz repetition rate - ps pulse width EFA-MLL -10 or -25 dB RF dir.coupler Basic fibre 2-STAGE EDFA ring

10GHz BP Amp

Fibre connection delay T

DC bias

Adjustable phase shifter

RF Amp

EDFA Section 1

10GHz BPF clock recovery

Loop MZI

Re-circulating port

Linear a mplifier

3nm OF

PC

90%

2 1

Opt.att

3 10:90 FC

4 10%

SYNC

LD det.

EDFA Section 2

3dB FC

OUTPUT PULSES

HP-det

High Speed Digital Sampling CRO

Figure 1. Schematic arrangement of a mode-locked ring laser with an O/E-RF electronic active feedback loop.

2.1.2. Mode-Locking Conditions The basic conditions for MLRL to operate in pulse oscillation are: For Non-Feedback Optical Mode-Locking Condition 1: The total optical loop gain must be greater than unity when the modulator is ON-state, i.e. when the optical waves transmitting through the MZIM is propagating in phase[28]; Condition 2: The optical lightwaves must be depleted when the optical modulator is in the OFF-state, i.e. when the lightwaves of the two branches of the MZIM is out of phase or in destructive interference mode[28]; Condition 3: The frequency repetition rate at a locking state must be a multiple number of the fundamental ring resonant frequency[29]. For Optical-RF Feedback Mode Locking - Regenerative Mode-Locking Condition 4: Under an O/E-RF feedback to control modulation of the intensity modulator the optical noise at the output of the laser must be significantly greater than that of the electronic noise for the start-up of the mode locking and lasing. In other words the loop gain of the optical-electronic feedback loop must be greater than unity. Thus it is necessary that the EDF amplifiers are operated in saturation mode and the total average optical power of the lightwaves circulating in the loop must be sufficiently adequate for the detection at the photo-detector and electronic preamplifier. Under this condition the optical quantum shot noise dominates the electronic shot noise.

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2.1.3. Factors Influencing the Design and Performance of Mode-Locking and Generation of Optical Pulse Train The locking frequency is a multiple of the fundamental harmonic frequency of the ring defined as the inverse of the travelling time around the loop and is given by

Nc neff L

f RF

(1)

where fRF is the RF frequency required for locking and the required generation rate, N is an integer and indicates mode number order, c is the velocity of light in vacuum, neff is effective index of the guided propagating mode and L is loop length including that of the optical amplifiers. Under the requirement of the OC-192 standard bit rate the locking frequency must be in the region of 9.95 Giga-pulses per second. That is the laser must be locked to a very high order of the fundamental loop frequency that is in the region of 1 MHz to 10 MHz depending on the total ring length. For an optical ring of length about 30 metres and a pulse repetition rate of 10 GHz, the locking occurs on approximately the 1400th harmonic mode. It is noted also that the effective refractive index n can be varied in different sections of the optical components forming the laser ring. Furthermore the two polarised states of propagating lightwaves in the ring, if the fibres is not a polarisation maintaining type, would form two simultaneously propagating rings, and they could interfere or hop between these dual polarised rings. The pulse width, denoted as 'W of the generated optical pulse trains can be found to be given by[30] 1/ 4

'W

§D G · 0.45 ¨ t t ¸ © 'm ¹

1 ( f RF 'X )1/ 2

(2)

with DtGt is the round trip gain coefficient as a product of all the loss and gain coefficients of all optical components including their corresponding fluctuation factor, 'm is the modulation index and 'Q is the overall optical bandwidth of the laser. Hence the modulation index and the bandwidth of the optical filter influence the generated pulse width of the pulse train. However the optical characteristics of the optical filters and optical gain must be flattened over the optical bandwidth of the transform limit for which a transform limited pulse must satisfy, for a sech2 pulse intensity profile, the relationship 'W .'X

0.315

(3)

Similarly, for Gaussian pulse shape the constant becomes 0.441. The fluctuation of the gain or loss coefficients over the optical flattened region can also influence the generated optical pulse width and mode locking condition. In the case for regenerative mode-locking case as illustrated in Figure 1, the optical output intensity is split and opto-electronically (O/E) detected, we must consider the sensitivity and noises generated at the photo-detector (PD). Two major sources of noises are

Nonlinear Photonic Fibre Ring Lasers

7

generated at the in put of the PD, firstly the optical quantum shot noises generated by the detection of the optical pulse trains and the random thermal electronic noises of the small signal electronic amplifier following the detector. Usually the electronic amplifier would have a 50 : equivalent input impedance referred to the input of the optical preamplifier as evaluated at the operating repetition frequency, this gives a thermal noise spectral density of SR

4kT A2/Hz R

(4)

with k the Boltzmann's constant. This equals to 3.312 x 10-22 A2/Hz at 300 oK. Depending on the electronic bandwidth Be of the electronic pre-amplifier, i.e. wideband or narrow band type, 2

the total equivalent electronic noise (square of noise 'current') is given by, iNT

S R Be .

Under the worst case when a wide-band amplifier of a 3-dB electrical bandwidth of 10 GHz, the equivalent electronic noise at the input of the electronic amplifier is 3.312 x 10-11 A2, that is an equivalent noise current of 5.755 PA is present at the input of the 'clock' recovery circuit. If a narrow band-pass amplifier of 50 MHz 3-dB bandwidth centred at 10 GHz is employed this equivalent electronic noise current is 0.181 PA. Now considering the total quantum shot noise generated at the input of the 'clock' recovery circuit, suppose that a 1.0 mW (or 0 dBm) average optical power is generated at the output of the MLRL, then a quantum shot noise1 of approximately 2.56 x 10-22 A2 /Hz (i.e. an equivalent electronic noise current of 16 nA) is present at the input of the clock recovery circuit. This quantum shot noise current is substantially smaller than that of the electronic noise. In order for the detected signal at the optical receiver incorporated in the 'clock' recovery circuit to generate a high signal-to-noise ratio the optical average power of the generated pulse trains must be high, at least at a ratio of 10. We estimate that this optical power must be at least 0 dBm at the PD in order for the MLRL to lock efficiently to generate a stable pulse train. Given that a 10% fibres coupler is used at the optical output and an estimate optical loss of about 12 dB due to coupling, connector loss and attenuation of all optical components employed in the ring, the total optical power generated by the amplifiers must be about 30 dBm. To achieve this we employ two EDF amplifiers of 16.5 dBm output power each positioned before and after the optical coupler, one is used to compensate for the optical losses and one for generating sufficient optical gain and power to dominate the electronic noise in the regenerative loop.

2.2. Experimental Set-Up and Results The experimental set ups for MLL and RMLL are as shown in Figure 1. Associate equipment used for monitoring of the mode locking and measurement of the lasers are also included. However we note the followings: (i) In order to lock the lasing mode of the MLL to a certain repetition rate or multiple harmonic of the fundamental ring frequency, a synthesiser is required to generate the required sinusoidal waves for modulating the optical intensity 1

By using the relationship of the quantum noise spectral density of 2qRPav with Pav the average optical power, q the electronic charge and R the responsivity of the detector.

8

Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

modulator and tuned to a harmonic of the cavity fundamental frequency. (ii) A signal must be created for the purpose of triggering the digital oscilloscope to observe the locking of the detected optical pulse train. For the HP-54118A amplitude of this signal must be greater than 200 mV. This is also critical for the RMLL set up as the RF signal detected and phase locked via the clock recovery circuitry must be spit to generate this triggering signal. Typical experimental procedures are: (i) After the connection of all optical components with the ring path broken, ideally at the output of the fibres coupler, a CW optical source can be used to inject optical waves at a specific wavelength to monitor the optical loss of the ring; (ii) Close the optical ring and monitor the average optical power at the output of the 90:10 fibres coupler and hence estimate the optical power available at the PD is about -3 dBm after a 50:50 fibres coupler; (iii) Determine whether an optical amplifier is required for detecting the optical pulse train or whether this optical power is sufficient for O/E RF feedback condition as stated above; (iv) Set the biasing condition and hence the bias voltage of the optical modulator (v) Tune the synthesiser or the electrical phase to synchronise the generation and locking of the optical pulse train. The following results are obtained: (i) The optical pulse train generated at the output of the MLL or RMML. Experimental set up is shown in Figure 2; (ii) Synthesised modulating sinusoidal waveforms can be monitored as shown in Figure 4 and Figure 5 and Figure 6. Figure 4 illustrates the mode locking of a MLL operating at around 2 GHz repetition rate with the modulator driven from a pattern generator while in Figure 5and Figure 6 show the sinusoidal waveforms generating when the MLRL is operating at the self-mode-locking state. (iii) The interference of other super-modes of the MLL without RF feedback for self locking is indicated in Figure 4; (iv) Observed optical spectrum (not available in electronic form); (v) Electrical spectrum of the generated pulse trains was observed showing a -70dB super-mode suppression under the locked state of the regenerative MLRL; (vi) Figure 5 and Figure 7 show that the regenerative MLRL can be operating under the cases when the modulator is biased either at the positive or at the negative going slope of the optical transfer characteristics of the Mach-Zehnder modulator; (vii) Optical pulse width is measured using an optical auto-correlator (slow or fast scan mode). Typical pulse width obtained with the slow scan auto-correlator is shown in Figure 9. Minimum pulse duration obtained was 4.5 ps with a time-bandwidth product of about 3.8 showing that the generated pulse is near transform limited; (viii) BER measurement was used to monitor the stability of the regenerative MLRL. The BER error detector was then programmed to detect all '1' at the decision level at a tuned amplitude level and phase delay. The clock source used is that produced by the laser itself. This set up is shown in Figure 3 and an error-free has been achieved for over 20 hours. The O/E detected waveform of the output pulse train for testing the BER shown in Figure 8 after 20 hours operation, the recorded waveform is obtained under infinite persistence mode of the digital oscilloscope; (ix) A drift of clock frequency of about 20 kHz over one hour in open laboratory environment is observed. This is acceptable for a 10 GHz repetition rate. (x) The 'clock' recovered waveforms were also monitored at the initial locked state and after the long-term test as shown in Figure 5 and Figure 6 respectively. Figure 6 obtained under the infinite persistence mode of the digital oscilloscope.

Nonlinear Photonic Fibre Ring Lasers

9

Clock signals generated from RMLL

Trig input

HP-54118A 0.5 - 18 GHz OPTICAL PULSE TRAINS GENERATED FROM R-MLL

HP-54123A (DC-34 GHz)

Channel inputs Opt. Att. If req. dep. on detector

Output trig pulses

input

HP-34GHz pin DECTECTOR or Fermionics HSD-30 HP-54118A sampling head and display unit

Figure 2. Experimental set up for monitoring the locking of the photonic pulse train.

BER MEASUREMENT OF RMLL-EDF BER circuitry and equipment set-up

FIGURE 4

To CLK recovery and feedback to drive MZIM

Optical signals from output port of RMLL 3 dB FC

PIN bias (9V) HBT TI amp supply (11.5V)

DC blocking cap

Linear power amplifier to provide > 0.2 V signals for BER input port

RMLL Clock

delay

Set @ ALL ‘1’ 16-bit length periodic

Nortel PP-10G 500 :TI optical receiver

RF amp AL-7 MA Ltd

BER detector ANRITSU MP1764A

RF amp ERA 10GHz RZ sinusoidal waveform

Data IN CLK

Clk freq: 9.947303 GHz var. 100 MHz

Figure 3. Experimental set up for monitoring the BER of the photonic pulse train.

10

Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai Modulation frequency = 2,143,500 kHz

SCOPE 2

Competition from other supermodes

2 GHz ML_EDFA_L Neg CLK PULSES

Generated -Detected Laser puls es

0.000501

Generated output pulses 0.000401 @ coupler port 4 0.000301

0.000201

Clock for modulating MZI Modulator in-loop 0.000101 9.99996E-07

-9.9E-05 17508.0000 17708.0000 17908.0000 18108.0000 18308.0000 18508.0000 18708.0000 18908.0000 19108.0000 19308.0000 19508.0000 Time (picoseconds)

Figure 4. Detected pulse train at the MLRL output tested at a multiple frequency in the range of 2 GHz repetition frequency.

Regenerative mode-locked generated pulse trains: SCOPE 3 10 G rep rates - detected by 34 GHz (3dB BWHP-detector into 50 :) 31.375

21.375

Generated pulse trains 100 ps spacing - pulse width ~ 12 ps - limited by PD bandwidth

11.375

1.375

-8.625

-18.625

Clock recovery signals for driving MZIM

-28.625

-38.625

-48.625 16199.0010

16299.0010

16399.0010

16499.0010

16599.0010

16699.0010

Time (picoseconds)

Figure 5. Output pulse trains of the regenerative MLRL and the RF signals as recovered for modulating the MZIM for self-locking.

Nonlinear Photonic Fibre Ring Lasers

11

SCOPE 4 RML laser with V(MZIM bias)=1.9V (more sensitive to competition of supermodes than biased @0 V - maximum transmission bias) 43

Generated pulse train 33

23

13

3

-7

Clocked regen for modulating MZIM

-17

-27

-37 0.0162

0.0162

0.0163

0.0163

0.0164

0.0164

0.0165

0.0165

0.0166

0.0166

0.0167

Time (microseconds)

Figure 6. Detected output pulse trains of the regenerative MLRL and recovered clock signal when the MZIM is biased at a negative going slope of the operating characteristics of the modulator.

RMLLL-09-May-2000 V(bias)= 9.34 Volts

10G ps-pulses generated

09-May-2000 R egen MLL

RF pulses to drive MZIM

Generated optical output pulses

c lk & RF signals to M ZIM

43

33

23

13

3

-7

-17

-27

-37 16.1990

16.2490

16.2990

16.3490

16.3990

16.4490

16.4990

16.5490

16.5990

16.6490

16.6990

Tim e (na nose conds)

Figure 7. Output pulse trains and clock recovered signals of the 10 G regenerative MLRL when the modulator is biased at the positive going slope of the modulator operating transfer curve.

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

BER measurement BER=0 x 10-15 measured for over 20 hours RF (CLK) frequency varied from 9.954 to 9.952 GHz gradually over measurement period BER m e as ure m e nt - clock s ignal - infinite pe rs is te nce non- pulses

clock RF signals w hile BER measurement

280

180

80

-20

-120

-220

-320 16.1990

16.2490

16.2990

16.3490

16.3990

16.4490 16.4990

16.5490

16.5990

16.6490

16.6990

Tim e (nanos e conds )

Figure 8. BER measurement – O/E detected signals from the generated output pulse trains for BER test set measurement. The waveform is obtained after 20 hours persistence.

Auto-correlated pulse V(bias) = 1.55 volt - phase quadrature neg slope FWHM 'W  ps autocorrelator set thumbwheel 6 and 0 (100ps range @10ps/s delay rate)

Figure 9. Auto-correlation trace of output pulse trains of 9.95 GHz regenerative MLRL.

We note the following factors which are related to the above measurements (Figure 4 to Figure 6): (i) All the above measurements have been conducted with two distributed optical

Nonlinear Photonic Fibre Ring Lasers

13

amplifiers (GTi EDF optical amplifiers) driven at 180 mA and a specified output optical power of 16.5 dBm; and (ii) Optical pulse trains are detected with 34 GHz 3dB bandwidth HP pin detector directly coupled to the digital oscilloscope without using any optical preamplifier.

Figure 10. Stability of output pulse trains of 9.95 GHz regenerative MLRL, FWHM = 4.48 ps: pulse with and period versus average optical pulse power.

Figure 11. Proposed set up of a THz regenerative MLRL using nonlinear effects such as parametric amplification sequence incorporating the proposed CONTROLLER.

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

2.3. 40 GHz and Tera-Hz Regenerative Mode-Locked Laser We have also been successful in the construction and testing of a regenerative MLRL at 40 GHz repetition rate regenerative mode locked laser using appropriate modulator and electronic feedback circuitry. However a nonlinear optical guided wave photonic component can be inserted in the fibre ring so as to obtain an ultra-wideband amplification via parametric amplification whose bandband reaches several hundreds of nm. Initial observation of the locking and generation of the laser has been observed and progress of this laser design and experiments will be reported in the near future.

2.4. Remarks In this section we have demonstrated a mode locked laser operating under an open loop condition and with O/E RF feedback providing regenerative mode locking. The O/E feedback can certainly provide a self-locking mechanism under the condition that the polarisation characteristics of the ring laser are manageable. This is done by ensuring that all fibres path is under constant operating condition. The regenerative MLRL can self-lock even under the DC drifting effect of the modulator bias voltage (over 20 hours)2. The generated pulse trains of 4.5 ps duration can be, without difficulty, compressed further to less than 3 ps for 160 Gb/s optical communication systems. The regenerative MLRL can be an important source for all-optical switching of an optical packet switching system. We recommend the following for future works of ultra-high repetition rate (up to few tens of THz) regenerative MLRL: (i) Eliminating polarisation drift through the use of Faraday mirror or all polarisation maintaining (PM) optical components, for example polarised Er-doped fibre amplifiers, PM fibres at the input and output ports of the intensity modulator and employment of nonlinear guided wave element, for example the parametric amplification via nonlinear pumping; (ii) Stabilising the ring cavity length with appropriate packaging and via piezo/thermal control to improve long term frequency drift, filtering of the stimulated Brillouin scattering (SBS) ;(iii) Control and automatic tuning of the DC bias voltage of the intensity modulator; (iv) Developing electronic RF 'clock' recovery circuit for regenerative MLRL operating at 40 GHz or higher via heterodyne detection, repetition rate together with appropriate polarisation control strategy; (iv) Study of the dependence of the optical power circulating in the ring laser by varying the output average optical power of the optical amplifiers under different pump power conditions; (v) Incorporating a phase modulator, in lieu of the intensity modulator, to reduce the complexity of polarisation dependence of the optical waves propagating in the ring cavity, thus minimising the bias drift problem of the intensity modulator.

2

Typically the DC bias voltage of a LiNbO3 intensity modulator is drifted by 1.5 volts after 15 hours of continuous operation.

Nonlinear Photonic Fibre Ring Lasers

15

3. Active Mode-Locked Fibre Ring Laser by Rational Harmonic Detuning In this section we investigate the system behaviour of rational harmonic mode-locking in the fibre ring laser using phase plane technique of the nonlinear control engineering. Furthermore, we examine the harmonic distortion contribution to this system performance. We also demonstrate 660x and 1230x repetition rate multiplications on 100MHz pulse train generated from an active harmonically mode-locked fibre ring laser, hence achieve 66GHz and 123GHz pulse operations by using rational harmonic detuning, which is the highest rational harmonic order reported to date.

3.1. Rational Harmonic Mode-Locking In an active harmonically mode-lock fibre ring laser, the repetition frequency of the generated pulses is determined by the modulation frequency of the modulator, fm=qfc, where q is the qth harmonic of the fundamental cavity frequency, fc, which is determined by the cavity length of the laser, fc=c/nL, where c is the speed of light, n is the refractive index of the fibre and L is the cavity length. Typically, fc is in the range of kHz or MHz. Hence, in order to generate GHz pulse train, mode-locking is normally performed by modulation in the states of q >>1, i.e. q pulses circulating within the cavity, which is known as harmonic mode-locking. By applying a slight deviation or a fraction of the fundamental cavity frequency, 'f=fc/m, where m is the integer, the modulation frequency becomes fm

qf c r

fc m

(5)

This leads to m-times increase in the system repetition rate, fr=mfm, where fr is the repetition frequency of the system [2]. When the modulation frequency is detuned by a m fraction, the contributions of the detuned neighbouring modes are weakened, only every mth lasing mode oscillates in phase and the oscillation waveform maximums accumulate, hence achieving in m times higher repetition frequency. However, the small but not negligible detuned neighbouring modes affect the resultant pulse train, which leads to uneven pulse amplitude distribution and poor long term stability. This is considered as harmonic distortion in our modelling, and it depends on the laser linewidth and amount of detuned, i.e. a fraction m. The amount of the allowable detuneable range or rather the obtainable increase in the system repetition rate by this technique is very much limited by the amount harmonic distortion. When the amount of frequency detuned is too small relative to the modulation frequency, that is very high m, contributions of the neighbouring lasing modes become prominent, thus reduce the repetition rate multiplication capability significantly. In other words, no repetition frequency multiplication is achieved when the detuned frequency is unnoticeably small. Often the case, it is considered as the system noise due to improper modulation frequency tuning. In addition, the pulse amplitude fluctuation is also determined by this harmonic distortion.

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

3.2. Experiment Setup In this case the experimental setup of the active harmonically mode-locked fibre ring laser is similar to Figure 1 but without the feedback loop. The principal element of the laser is an optical open loop with an optical gain medium, a Mach-Zehnder amplitude modulator (MZM), an optical polarization controller (PC), an optical bandpass filter (BPF), optical couplers and other associated optics. The gain medium used in our fibre laser system is an erbium doped fibre amplifier (EDFA) with saturation power of 16dBm. A polarization independent optical isolator is used to ensure unidirectional lightwave propagation as well as to eliminate back reflections from the fibre splices and optical connectors. A free space filter with 3dB bandwidth of 4 nm at 1555 nm is inserted into the cavity to select the operating wavelength of the generated signal and to reduce the noise in the system. In addition, it is responsible for the longitudinal modes selection in the mode-locking process. The birefringence of the fibre is compensated by a polarization controller, which is also used for the polarization alignment of the linearly polarized lightwave before entering the planar structure modulator for better output efficiency. Pulse operation is achieved by introducing an asymmetric coplanar travelling wave 10Gb/s lithium niobate, Ti:LiNbO3 Mach-Zehnder amplitude modulator into the cavity with half wave voltage, VS of 5.8 V and insertion loss of d 7dB. The modulator is DC biased near the quadrature point and not more than the VS such that it operates around the linear region of its characteristic curve. The modulator is driven by a 100MHz, 100ps step recovery diode (SRD), which is in turn driven by a RF amplifier (RFA) a RF signal generator. The modulating signal generated by the step recovery diode is a ~1% duty cycle Gaussian pulse train. The output coupling of the laser is optimized using a 10/90 coupler. 90% of the optical field power is coupled back into the cavity ring loop, while the remaining portion is taken out as the output of the laser and analysed.

3.3. Phase Plane Analysis Nonlinear system frequently has more than one equilibrium point. It can also oscillate at fixed amplitude and fixed period without external excitation. This oscillation is called limit cycle. However, limit cycles in nonlinear systems are different from linear oscillations. First, the amplitude of self-sustained excitation is independent of the initial condition, while the oscillation of a marginally stable linear system has its amplitude determined by the initial conditions. Second, marginally stable linear systems are very sensitive to changes, while limit cycles are not easily affected by parameter changes [31]. Phase plane analysis is a graphical method of studying second-order nonlinear systems. The result is a family of system motion of trajectories on a two-dimensional plane, which allows us to visually observe the motion patterns of the system. Nonlinear systems can display more complicated patterns in the phase plane, such as multiple equilibrium points and limit cycles. In the phase plane, a limit cycle is defined as an isolated closed curve. The trajectory has to be both closed, indicating the periodic nature of the motion, and isolated, indicating the limiting nature of the cycle [31]. The system modelling of the rational harmonic mode-locked fibre ring laser system is done based on the following assumptions: (i) detuned frequency is perfectly adjusted

Nonlinear Photonic Fibre Ring Lasers

17

according to the fraction number required, (ii) small harmonic distortion, (iii) no fibre nonlinearity is included in the analysis, (iv) no other noise sources are involved in the system, and (v) Gaussian lasing mode amplitude distribution analysis. The phase plane of a perfect 10GHz mode-locked pulse train without any frequency detune is shown Figure 12 and the corresponding pulse train is shown in Figure 13a. The shape of the phase plane exposes the phase between the displacement and its derivative. From the phase plane obtained, one can easily observe that the origin is a stable node and the limit cycle around that vicinity is a stable limit cycle, hence leading to stable system trajectory. 4x multiplication pulse trains, i.e. m = 4, without and with 5% harmonic distortion are shown in Figure 13b and c. Their corresponding phase planes are shown in Figure 14a and b. For the case of zero harmonic distortion, which is the ideal case, the generated pulse train is perfectly multiplied with equal amplitude and the phase plane has stable symmetry periodic trajectories around the origin too. However, for the practical case, i.e. with 5% harmonic distortion, it is obvious that the pulse amplitude is unevenly distributed, which can be easily verified with the experimental results obtained in [3]. Its corresponding phase plane shows more complex asymmetry system trajectories. One may naively think that the detuning fraction, m, could be increased to a very large number, so a very small frequency deviated, 'f, so as to obtain a very high repetition frequency. This is only true in the ideal world, if no harmonic distortion is present in the system. However, this is unreasonable for a practical mode-locked laser system. We define the percentage fluctuation, %F as follows: %F

Emax  Emin u 100% Emax

(6)

where Emax and Emin are the maximum and minimum peak amplitude of the generated pulse train. For any practical mode-locked laser system, fluctuations above 50% should be considered as poor laser system design. Therefore, this is one of the limiting factors in a rational harmonic mode-locking fibre laser system. The relationships between the percentage fluctuation and harmonic distortion for three multipliers (m=2, 4 and 8) are shown in the Figure 14. Thus, the obtainable rational harmonic mode-locking is very much limited by the harmonic distortion of the system. For 100% fluctuation, it means no repetition rate multiplication, but with additional noise components; a typical pulse train and its corresponding phase plane are shown in Figure 16 (lower plot) and Figure 17 with m=8 and 20% harmonic distortion. The asymmetric trajectories of the phase graph explain the amplitude unevenness of the pulse train. Furthermore, it shows a more complex pulse formation system. Thus, it is clear that for any harmonic mode-locked laser system, the small side pulses generated are largely due to improper or not exact tuning of the modulation frequency of the system. An experiment result is depicted in Figure 20 for a comparison.

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

Figure 12. Phase plane of a 10GHz mode-locked pulse train. (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E’(t)).

Figure 13. Normalised pulse propagation of original pulse (a) detuning fraction of 4 , with 0% (b) 5% (c) harmonic distortion noise.

Nonlinear Photonic Fibre Ring Lasers

a

19

b

Figure 14. Phase plane of detuned pulse train, m=4, 0% harmonic distortion (a), and 5% harmonic distortion (b) ; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E’(t)).

1.2

% Fluctuation

1

0.8

0.6

0.4

0.2

0 0

0.16

0.32

0.48

0.64

0.8

0.96

% Harmonic Distortion m=2

m=4

m=8

Figure 15. Relationship between the amplitude fluctuation and the percentage harmonic distortion (diamond – m=2, square – m=4, triangle – m=8).

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

Figure 16. 10GHz pulse train (upper plot), pulse train with m =8 and 20% harmonic distortion (lower plot).

Figure 17. Phase plane of the pulse train with m =8 and 20% harmonic distortion.

Nonlinear Photonic Fibre Ring Lasers

a

21

b

c

d

Figure 18. Autocorrelation traces of 66 GHz (a) and 123 GHz (c) pulse operation; optical spectrums of 66GHz (b) and 123 GHz (d).

Figure 19. Phase plane of the 66 GHz (a) and 123 GHz (b) pulse train with 0.001% harmonic distortion noise.

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

Figure 20. Autocorrelation trace (a) and optical spectrum (b) of slight frequency detune in the modelocked fibre ring laser.

3.4. Results and Discussions By careful adjustment of the modulation frequency, polarization, gain level and other parameters of the fibre ring laser, we are managed to obtain the 660th and 1230th order of rational harmonic detuning in the mode-locked fibre ring laser with base frequency of 100MHz, hence achieving 66 GHz and 123 GHz repetition frequency pulse operation. The auto-correlation traces and optical spectrums of the pulse operations are shown in Figure 27. With Gaussian pulse assumption, the obtained pulse widths of the operations are 2.5456ps and 2.2853ps respectively. For the 100MHz pulse operation, i.e. without any frequency detune, the generated pulse width is about 91ps. Thus, not only we achieved an increase in the pulse repetition frequency, but also a decrease in the generated pulse widths. This pulse narrowing effect is partly due to the self phase modulation effect of the system, as observed in the optical spectrums. Another reason for this pulse shortening is stated by Haus in [18], where the pulse width is inversely proportional to the modulation frequency, as follow:

W4

2g M Zm2 Zg2

(7)

where W is the pulse width of the mode-locked pulse, Zm is the modulation frequency, g is the gain coefficient, M is the modulation index, and Zg is the gain bandwidth of the system. In addition, the duty cycle of our Gaussian modulation signal is ~1%, which is very much less than 50%, this leads to a narrow pulse width too. Besides the uneven pulse amplitude distribution, high level of pedestal noise is also observed in the obtained results.

Nonlinear Photonic Fibre Ring Lasers

23

For 66GHz pulse operation, 4nm bandwidth filter is in used in the setup, but it is removed for the 123 GHz operation. It is done so to allow more modes to be locked during the operation, thus, to achieve better pulse quality. In contrast, this increases the level of difficulty significantly in the system tuning and adjustment. As a result, the operation is very much determined by the gain bandwidth of the EDFA used in the laser setup. The simulated phase planes for the above pulse operation are shown in Figure 19. They are simulated based on the 100MHz base frequency, 10 round trips condition and 0.001% of harmonic distortion contribution. There is no stable limit cycle in the phase graphs obtained; hence the system stability is hardly achievable, which is a known fact in the rational harmonic mode-locking. Asymmetric system trajectories are observed in the phase planes of the pulse operations. This reflects the unevenness of the amplitude of the pulses generated. Furthermore, more complex pulse formation process is also revealed in the phase graphs obtained. By a very small amount of frequency deviation, or improper modulation frequency tuning in the general context, we obtain a pulse train with ~100 MHz with small side pulses in between as shown in Figure 10. It is rather similar to the Figure 6 (lower plot) shown in the earlier section despite the level of pedestal noise in the actual case. This is mainly because we do not consider other sources of noise in our modelling, except the harmonic distortion.

3.5. Remarks We have demonstrated 660th and 1230th order of rational harmonic mode locking from a base modulation frequency of 100MHz in the erbium doped fibre ring laser, hence achieving 66GHz and 123GHz pulse repetition frequency. To the best of our knowledge, this is the highest rational harmonic order obtained to date. Besides the repetition rate multiplication, we also obtain high pulse compression factor in the system, ~35x and 40x relative to the nonmultiplied laser system. In addition, we use phase plane analysis to study the laser system behaviour. From the analysis model, the amplitude stability of the detuned pulse train can only be achieved under negligible or no harmonic distortion condition, which is the ideal situation. The phase plane analysis also reveals the pulse forming complexity of the laser system.

4. Rep-Rate Multiplication Ring Laser Using Temporal Diffraction Effects The pulse repetition rate of a mode-locked ring laser is usually limited by the bandwidth of the intra-cavity modulator. Hence, a number of techniques have to be used to increase the repetition frequency of the generated pulse train. Rational harmonic detuning [3, 16] is achieved by applying a slight deviated frequency from the multiple of fundamental cavity frequency. 40GHz repetition frequency has been obtained by [3] using 10GHz base band modulation frequency with 4th order rational harmonic mode locking. This technique is simple in nature. However, this technique suffers from inherent pulse amplitude instability, which includes both amplitude noise and inequality in pulse amplitude, furthermore, it gives poor long-term stability. Hence, pulse amplitude equalization techniques are often applied to

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

achieve better system performance [2], [4, 5]. Fractional temporal Talbot based repetition rate multiplication technique [4-8] uses the interference effect between the dispersed pulses to achieve the repetition rate multiplication. The essential element of this technique is the dispersive medium, such as linearly chirped fibre grating (LCFG) [8, 16] and single mode fibre [8, 9]. This technique will be discussed further in Section II. Intra-cavity optical filtering [13, 14] uses modulators and a high finesse Fabry-Perot filter (FFP) within the laser cavity to achieve higher repetition rate by filtering out certain lasing modes in the mode-locked laser. Other techniques used in repetition rate multiplication include higher order FM mode-locking [13], optical time domain multiplexing, etc. The stability of high repetition rate pulse train generated is one of the main concerns for practical multi-Giga bits/sec optical communications system. Qualitatively, a laser pulse source is considered as stable if it is operating at a state where any perturbations or deviations from this operating point is not increased but suppressed. Conventionally the stability analyses of such laser systems are based on the linear behaviour of the laser in which we can analytically analyse the system behaviour in both time and frequency domains. However, when the mode-locked fibre laser is operating under nonlinear regime, none of these standard approaches can be used, since direct solution of nonlinear different equation is generally impossible, hence frequency domain transformation is not applicable. Although Talbot based repetition rate multiplication systems are based on the linear evolution of the laser, there are still some inherent nonlinearities affecting its stability, such as the saturation of the embedded gain medium, non-quadrature biasing of the modulator, nonlinearities in the fibre, etc., hence, nonlinear stability approach must be adopted. The stability and transient analyses of the Group Velocity Dispersion (GVD) multiplied pulse train can be conducted using the phase plane analysis of nonlinear control engineering [2]. Furthermore it allows the study the stability and transient performances of the GVD repetition rate multiplication systems. The stability and the transient response of the multiplied pulses are studied using the phase plane technique of nonlinear control engineering. We also demonstrated four times repetition rate multiplication on 10Gbits/s pulse train generated from the active harmonically mode-locked fibre ring laser, hence achieving 40Gbits/s pulse train by using fibre GVD effect. It has been found that the stability of the GVD multiplied pulse train, based on the phase plane analysis is hardly achievable even under the perfect multiplication conditions. Furthermore, uneven pulse amplitude distribution is observed in the multiplied pulse train. In addition to that, the influences of the filter bandwidth in the laser cavity, nonlinear effect and the noise performance are also studied in our analyses. In Section 4.1, the GVD repetition rate multiplication technique is briefly given. Section 4.2 describes the experimental setup for the repetition rate multiplication. Section 4.4 investigates the dynamic behaviour of the phase plane of GVD multiplication system, followed by simulation and experimental results. Finally, some concluding remarks and possible future developments are given.

4.1. GVD Repetition Rate Multiplication Technique When a pulse train is transmitted through an optical fibre, the phase shift of kth individual lasing mode due to group velocity dispersion (GVD) is

Nonlinear Photonic Fibre Ring Lasers

Mk

SO 2 Dzk 2 f r2 c

25

(8)

where O is the centre wavelength of the mode-locked pulses, D is the fibre’s GVD factor, z is the fibre length, fr is the repetition frequency and c is the speed of light in vacuum. This phase shift induces pulse broadening and distortion. At Talbot distance, zT=2/'Ofr~D~ [6] the initial pulse shape is restored, where 'O = frO2/c is the spacing between Fourier-transformed spectrum of the pulse train. When the fibre length is equal to zT/(2m), (where m = 2,3,4, …), every mth lasing modes oscillates in phase and the oscillation waveform maximums accumulate. However, when the phases of other modes become mismatched, this weakens their contributions to pulse waveform formation. This leads to the generation of a pulse train with a multiplied repetition frequency with m-times. The pulse duration does not change that much even after the multiplication, because every mth lasing mode dominates in pulse waveform formation of m-times multiplied pulses. The pulse waveform therefore becomes identical to that generated from the mode-locked laser, with the same spectral property. Optical spectrum does not change after the multiplication process, because this technique utilizes only the change of phase relationship between lasing modes and does not use fibre’s nonlinearity. The effect of higher order dispersion might degrade the quality of the multiplied pulses, i.e. pulse broadening, appearance of pulse wings and pulse-to-pulse intensity fluctuation. In this case, any dispersive media to compensate the fibre’s higher order dispersion would be required in order to complete the multiplication process. To achieve higher multiplications the input pulses must have a broad spectrum and the fractional Talbot length must be very precise in order to receive high quality pulses. If the average power of the pulse train induces the nonlinear suppression and experience anomalous dispersion along the fibre, solitonic dynamics would occur and prevent the linear Talbot effect from occurring. The highest repetition rate obtainable is limited by the duration of the individual pulses, as pulses start to overlap when the pulse duration becomes comparable to the pulse train period, i.e. mmax = 'T/'t, where 'T is the pulse train period and 't is the pulse duration.

4.2. Experiment Setup GVD repetition rate multiplication is used to achieve 40Gbits/s operation. The input to the GVD multiplier is a 10.217993Gbits/s laser pulse source, obtained from active harmonically mode-locked fibre ring laser, operating at 1550.2nm. The principle element of the active harmonically mode-locked fibre ring laser is an optical closed loop with an optical gain medium, that is the Erbium doped fibre under 980nm pump source, an optical 10GHz amplitude modulator, optical bandpass filter, optical fibre couplers and other associated optics. The schematic construction of the active mode-locked fibre ring laser is shown in the Figure 1. In this case the active mode-locked fibre laser design is based on a fibre ring cavity where the 25 meter EDF with Er3+ ion concentration of 4.7x1024 ions/m3 is pumped by two diode lasers at 980 nm: SDLO-27-8000-300 and CosetK1116 with maximum forward pump power of 280mW and backward pump power of 120mW. The pump lights are coupled into the cavity by the 980/1550 nm WDM couplers;

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

with insertion loss for 980 nm and 1550 nm signals are about 0.48 dB and 0.35 dB respectively. A polarization independent optical isolator ensures the unidirectional lasing. The birefringence of the fibre is compensated by a polarization controller (PC). A tunable FP filter with 3dB bandwidth of 1 nm and wavelength tuning range from 1530 nm to 1560 nm is inserted into the cavity to select the center wavelength of the generated signal as well as to reduce the noise in the system. In addition, it is used for the longitudinal modes selection in the mode-locking process. Pulse operation is achieved by introducing a JDS Uniphase 10Gb/s lithium niobate, Ti:LiNbO3 Mach-Zehnder amplitude modulator into the cavity with half wave voltage, VS of 5.8 V. The modulator is DC biased near the quadrature point and not more than the VS such that it operates on the linear region of its characteristic curve and driven by the sinusoidal signal derived from an Anritsu 68347C Synthesizer Signal Generator. The modulating depth should be less than unity to avoid signal distortion. The modulator has an insertion loss of d 7dB. The output coupling of the laser is optimized using a 10/90 coupler. 90% of the optical field power is coupled back into the cavity ring loop, while the remaining portion is taken out as the output of the laser and is analyzed using a New Focus 1014B 40 GHz photo-detector, Ando AQ6317B Optical Spectrum Analyzer, Textronix CSA 8000 80E01 50GHz Communications Signal Analyzer or Agilent E4407B RF Spectrum Analyzer. One rim of about 3.042km of dispersion compensating fibre (DCF), with a dispersion value of -98ps/nm/km was used in the experiment; the schematic of the experimental setup is shown in Figure 21. The variable optical attenuator used in the setup is to reduce the optical power of the pulse train generated by the mode-locked fibre ring laser, hence to remove the nonlinear effect of the pulse. An DCF length for 4x multiplication factor on the ~10 GHz signal is required and estimated to be 3.048173 km. The output of the multiplier (i.e. at the end of DCF) is then observed using Textronix CSA 8000 80E01 50GHz Communications Signal Analyzer.

Figure 21. Experiment setup for GVD repetition rate multiplication system.

4.3. Phase Plane Analysis Nonlinear system frequently has more than one equilibrium point. It can also oscillate at fixed amplitude and fixed period without external excitation. This oscillation is called limit cycle. However, limit cycles in nonlinear systems are different from linear oscillations. First, the amplitude of self-sustained excitation is independent of the initial condition, while the oscillation of a marginally stable linear system has its amplitude determined by the initial

Nonlinear Photonic Fibre Ring Lasers

27

conditions. Second, marginally stable linear systems are very sensitive to changes, while limit cycles are not easily affected by parameter changes. Phase plane analysis is a graphical method of studying second-order nonlinear systems. The result is a family of system motion of trajectories on a two-dimensional plane, which allows us to visually observe the motion patterns of the system. Nonlinear systems can display more complicated patterns in the phase plane, such as multiple equilibrium points and limit cycles. In the phase plane, a limit cycle is defined as an isolated closed curve. The trajectory has to be both closed, indicating the periodic nature of the motion, and isolated, indicating the limiting nature of the cycle. The system modelling for the GVD multiplier is done based on the following assumptions: (i) perfect output pulse from the mode-locked fibre ring laser without any timing jitter, (ii) the multiplication is achieved under ideal conditions (i.e. exact fibre length for a certain dispersion value), (iii) no fibre nonlinearity is included in the analysis of the multiplied pulse, (iv) no other noise sources are involved in the system, and (v) uniform or Gaussian lasing mode amplitude distribution.

4.3.1. Uniform Lasing Mode Amplitude Distribution Uniform lasing mode amplitude distribution is assumed at the first instance, i.e. ideal modelocking condition. The simulation is done based on the 10Gbits/s pulse train, centred at 1550 nm, with fibre dispersion value of –98ps/km/nm, 1 nm flat-top passband filter is used in the cavity of mode-locked fibre laser. The estimated Talbot distance is 25.484km. The original pulse (direct from the mode-locked laser) propagation behaviour and its phase plane are shown in Figure 22(a) and Figure 23(a). From the phase plane obtained, one can observe that the origin is a stable node and the limit cycle around that vicinity is a stable limit cycle. This agrees very well to our first assumption: ideal pulse train at the input of the multiplier. Also, we present the pulse propagation behaviour and phase plane for 2-times, 4times and 8-times GVD multiplication system in Figure 22 and Figure 23. The shape of the phase graph exposes the phase between the displacement and its derivative. As the multiplication factor increases, the system trajectories are moving away from the origin. As for the 4-times and 8-times multiplications, there is neither stable limit cycle nor stable node on the phase planes even with the ideal multiplication parameters. Here we see the system trajectories spiral out to an outer radius and back to inner radius again. The change in the radius of the spiral is the transient response of the system. Hence, with the increase in multiplication factor, the system trajectories become more sophisticated. Although GVD repetition rate multiplication uses only the phase change effect in multiplication process, the inherent nonlinearities still affect its stability indirectly. Despite the reduction in the pulse amplitude, we observe uneven pulse amplitude distribution in the multiplied pulse train. The percentage of unevenness increases with the multiplication factor in the system.

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

Figure 22. Pulse propagation of (a) original pulse, (b) 2x multiplication , (c) 4x multiplication, and d)8x multiplication with 1nm filter bandwidth and equal lasing mode amplitude analysis.

Figure 23. Phase plane of (a)original pulse, (b) 2x multiplication , (c) 4x multiplication, and d)8x multiplication with 1nm filter bandwidth and equal lasing mode amplitude analysis; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E’(t)).

Nonlinear Photonic Fibre Ring Lasers

29

Figure 24. Pulse propagation of (a)original pulse, (b) 2x multiplication , (c) 4x multiplication, , and (d)8x multiplication with 1nm filter bandwidth and Gaussian lasing mode amplitude analysis.

Figure 25. Phase plane of (a) original pulse, (b) 2x multiplication , (c) 4x multiplication, and (d) 8x multiplication with 1nm filter bandwidth and Gaussian lasing mode amplitude analysis; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E’(t)).

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

Figure 26. Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3nm filter bandwidth and Gaussian lasing mode amplitude analysis.

Figure 27. Phase plane of (a)original pulse, (b) 2x multiplication , (c) 4x multiplication, and (d) 8x multiplication with 3nm filter bandwidth and Gaussian lasing mode amplitude analysis; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E’(t)).

Nonlinear Photonic Fibre Ring Lasers

31

Figure 28. Pulse propagation of (a) original pulse, (b) 2x multiplication , (c) 4x multiplication, and (d)8x multiplication with 3nm filter bandwidth, Gaussian lasing mode amplitude analysis and input power = 1W.

Figure 29. Phase plane of (a) original pulse,(b) 2x multiplication , (c) 4x multiplication, and (d) 8x multiplication with 3nm filter bandwidth, Gaussian lasing mode amplitude analysis and input power = 1W; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and yaxes – E’(t)).

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

Figure 30. Pulse propagation of (a) original pulse, (b)2x multiplication , (c) 4x multiplication, and (d) 8x multiplication with 3nm filter bandwidth, Gaussian lasing mode amplitude analysis and 0dB signal to noise ratio.

Figure 21. Phase plane of (a) original pulse, (b)2x multiplication , (c) 4x multiplication, and (d)8x multiplication with 3nm filter bandwidth, Gaussian lasing mode amplitude analysis and 0dB signal to noise ratio; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E’(t)).

Nonlinear Photonic Fibre Ring Lasers

33

4.3.2. Gaussian Lasing Mode Amplitude Distribution This set of the simulation models the practical filter used in the system. It gives us a better insight on the GVD repetition rate multiplication system behaviour. The parameters used in the simulation are exactly the same except the filter of the laser has been changed to 1 nm (125GHz @ 1550nm) Gaussian-profile passband filter. The spirals of the system trajectories and uneven pulse amplitude distribution are more severe than those in the uniform lasing mode amplitude analysis.

4.3.3. Effects of Filter Bandwidth Filter bandwidth used in the mode-locked fibre ring laser will affect the system stability of the GVD repetition rate multiplication system as well. The analysis done above is based on 1 nm filter bandwidth. The number of modes locked in the laser system increases with the bandwidth of the filter used, which gives us a better quality of the mode-locked pulse train. The simulation results shown below are based on the Gaussian lasing mode amplitude distribution, 3nm filter bandwidth used in the laser cavity, and other parameters remain unchanged. With wider filter bandwidth, the pulse width and the percentage pulse amplitude fluctuation decreases. This suggests a better stability condition. Instead of spiralling away from the origin, the system trajectories move inward to the stable node. However, this leads to a more complex pulse formation system. 4.3.4. Nonlinear Effects When the input power of the pulse train enters the nonlinear region, the GVD multiplier loses its multiplication capability as predicted. The additional nonlinear phase shift due to the high input power is added to the total pulse phase shift and destroys the phase change condition of the lasing modes required by the multiplication condition. Furthermore, this additional nonlinear phase shift also changes the pulse shape and the phase plane of the multiplied pulses. 4.3.5. Noise Effects The above simulations are all based on the noiseless situation. However, in the practical optical communication systems, noises are always sources of nuisance which can cause system instability, therefore it must be taken into the consideration for the system stability studies. Since the optical intensity of the m-times multiplied pulse is m-times less than the original pulse, it is more vulnerable to noise. The signal is difficult to differentiate from the noise within the system if the power of multiplied pulse is too small. The phase plane the multiplied pulse is distorted due to the presence of the noise, which leads to poor stability performance.

4.4. Demonstration The obtained 10 GHz output pulse train from the mode-locked fibre ring laser is shown in Figure 32. Its spectrum is shown in Figure 33. This output was then used as the input to the

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

dispersion compensating fibre, which acts as the GVD multiplier in our experiment. The obtained 4-times multiplication by the GVD effect and its spectrum are shown in Figure 35 and Figure 35.

Figure 32. 10 GHz pulse train from mode-locked fibre ring laser (100ps/div, 50mV/div).

Figure 33. 10 GHz pulse spectrum from mode-locked fibre ring laser.

Nonlinear Photonic Fibre Ring Lasers

35

Figure 34. 40GHz multiplied pulse train (20ps/div, 1mV/div).

Figure 35. 40 GHz pulse spectrum from GVD multiplier.

The spectrums for both cases (original and multiplied pulse) are exactly the same since this repetition rate multiplication technique utilizes only the change of phase relationship between lasing modes and does not use fibre’s nonlinearity. The multiplied pulse suffers an amplitude reduction in the output pulse train; however, the pulse characteristics should remain the same. The instability of the multiplied pulse train is mainly due to the slight deviation from the required DCF length (0.2% deviation). Another reason for the pulse instability, which derived from our analysis; is the divergence of the pulse energy variation in the vicinity around the origin, as the multiplication factor gets

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

higher. The pulse amplitude decreases with the increase in multiplication factor, as the fact of energy conservation, when it reaches certain energy level, which is indistinguishable from the noise level in the system, the whole system will become unstable and noisy.

4.5. Remarks In this section we have demonstrated 4-times repetition rate multiplication by using fibre GVD effect; hence, 40GHz pulse train is obtained from 10GHz mode-locked fibre laser source. However, its stability is of great concern for the practical use in the optical communications systems. Although the GVD repetition rate multiplication technique is linear in nature, the inherent nonlinear effects in such system may disturb the stability of the system. Hence any linear approach may not be suitable in deriving the system stability. Stability analysis for this multiplied pulse train has been studied by using the nonlinear control stability theory, which is the first time, to the best of our knowledge, that phase plane analysis is being used to study the transient and stability performance of the GVD repetition rate multiplication system. Surprisingly, from the analysis model, the stability of the multiplied pulse train can hardly be achieved even under perfect multiplication conditions. Furthermore, we observed uneven pulse amplitude distribution in the GVD multiplied pulse train, which is due to the energy variations between the pulses that cause some energy beating between them. Another possibility is the divergence of the pulse energy variation in the vicinity around the equilibrium point that leads to instability. The pulse amplitude fluctuation increases with the multiplication factor. Also, with wider filter bandwidth used in the laser cavity, better stability condition can be achieved. The nonlinear phase shift and noises in the system challenge the system stability of the multiplied pulses. They not only change the pulse shape of the multiplied pulses, they also distort the phase plane of the system. Hence, the system stability is greatly affected by the self phase modulation as well as the system noises. This stability analysis model can further be extended to include some system nonlinearities, such as the gain saturation effect, non-quadrature biasing of the modulator, fibre nonlinearities, etc. The chaotic behaviour of the system may also be studied by applying different initial phase and injected energy conditions to the model.

5. Generation of Bound Solitons Using FM Modulation ModeLocked Fiber Ring Resonators Mode-locked fiber lasers described in the above sections can offer possible uses as important laser source for generating ultrashort soliton pulses. Recently, soliton fiber lasers have attracted significant research interests with experimentally demonstration of bound states of solitons as predicted in some theoretical works [32]-[33]. These bound-soliton states have been observed mostly in passive mode-locked fiber lasers [34]-[38]. The very short “solitonlike” pulses are generated through passive mode locking mechanisms such as nonlinear polarization rotation and saturable absorption. There are, however, few reports on bound solitons in active mode-locked fiber lasers. The active mode locking offers significant advantage in the control of the repetition rate that would be critical for optical transmission

Nonlinear Photonic Fibre Ring Lasers

37

systems. Observation of bound soliton pairs was first reported in a hybrid FM mode-locked fiber laser [39], in which a regime of bound-soliton pairs harmonic mode locking at 10 GHz could be generated. There are, however no report on multiple bound soliton states. Depending on the strength of soliton interation, the bound solitons can be classified into two categories: loosely bound solitons and tightly bound solitons with different the relative phase difference between adjacent solitons. The phase difference may take the value of S or S/2 or any value depending on the fiber laser structures and mode locking conditions. In this section, the bound states of multiple solitons in an active mode-locked fiber laser using FM mechanism under nonlinear saturated optical power. By tuning the parameters of the fiber loop, not only that we could observe the dual-soliton bound state but also the tripleand quadruple-soliton bound states. Relative phase difference and chirping caused by phase modulation of LiNbO3 modulator in the fiber loop have significantly influenced the interaction between solitons and hence their stability as they circulate in the anomalous pathaveraged dispersion fiber loop.

5.1. Formation of Bound States in a FM Mode-Locked Fiber Laser Although the formation of the stable bound soliton states which determined by the Kerr effect and anomalous averaged dispersion regime has been discussed in some configurations of passive mode-locked fiber lasers [35]-[38], it can be quite distinct in our active mode-locked fiber laser with the contribution of phase modulation of the LiNbO3 modulator in stabilization of bound states. The formation of bound soliton states in a FM mode-locked laser can experience through two stages which include a process of pulse splitting and stabilization of muti-soliton bound states in presence of a phase modulator in the cavity of mode-locked laser. The first stage is splitting of a single pulse into multi pulses which occurs when the power in the fiber loop increases above a certain mode locking threshold [39], [40]. At higher power, higher order solitons can be excited and in addition the accumulated nonlinear phase shift in the loop is so high that a single pulse breaks up into many pulses [41]. Number of splitted pulses depends on the optical power in the loop, so there is a specific range of power for each splitting level. The fluctuation of pulses may occur at region of power where there is a transition from the lower splitting level to the higher. Moreover, the chirping caused by phase modulator in the loop also makes the process of pulse conversion from a chirped single pulse into multi-pulses taking place more easily [42], [43]. After splitting into multi pulses, multi-pulse bound states are stabilized subsequently through the balance of the repulsive and attractive forces between neighboring splitted pulses during circulating in a fiber loop of anomalous-averaged dispersion. The repulsive force comes from direct soliton interaction depending on the relative phase difference between neighboring pulses [44] and the effectively attractive force comes from the variation of group velocity of soliton pulse caused by the frequency chirping [45], [46]. Thus, in an anomalous average dispersion regime, the locked pulses should be located symmetrically around the extreme of positive phase modulation half-cycle, in other words the bound soliton pulses acquire an up chirping when passing the phase modulator. In a specific mode-locked fiber laser setup, beside the optical power level and dispersion of the fiber cavity, the modulatorinduced chirp or the phase modulation index determine not only the pulsewidth but also the time separation of bound-soliton pulses at which the interactive effects cancel each other.

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

The presence of a phase modulator in the cavity to balance the effective interactions among bound-soliton pulses is similar to the use of this device in a long haul soliton transmission system to reduce the timing jitter, for this reason the simple perturbation theory can be applied to understand the role of phase modulation on mechanism of bound solitons formation. A multi-soliton bound state can be described as following: N

ubs

¦ u ( z, t ) i

i 1

(9)

and

Ai sec h^Ai >(t  Ti ) / T0 @`exp( jT i  jZi t )

ui

(10)

where N is number of solitons in the bound state, T0 is pulsewidth of soliton and Ai, Ti, Ti, Zi represent the amplitude, position, phase and frequency of soliton respectively. In the simplest case of multi-soliton bound state, N is equal 2 or we consider the dual-soliton bound state with the identical amplitude of pulse and the phase difference of S value ('T = Ti+1 - Ti = S), the ordinary differential equations for the frequency difference and the pulse separation can be derived by using the perturbation method [3]

dZ dz

d'T dz



ª 'T º 4E 2 exp « »  2D m 'T 3 T0 ¬ T0 ¼

(11)

E 2Z (12)

where E2 is the averaged group-velocity dispersion of the fiber loop, 'T is pulse separation between two adjacent solitons (Ti+1 – Ti = 'T) and D m

mZm2 (2 Lcav ) , Lcav is the total

length of the loop, m is the phase modulation index. Eq. (11) and (12) show the evolution of frequency difference and position of bound solitons in the fiber loop in which the first term on the right hand side represents the accumulated frequency difference of two adjacent pulses during a round trip of the fiber loop and the second one represents the relative frequency difference of these pulses when passing through the phase modulator. At steady state, the pulse separation is constant and the induced frequency differences cancel each other. On the other hand, if Eq. (11) is equated to zero, we have



ª 'T º 4E 2 exp « »  2D m 'T 3 T0 T 0 ¬ ¼

0

ª 'T º 'T exp « » ¬ T0 ¼ or



4 E 2 Lcav T03 mZm2

(13)

From Eq. (13) we can see the effect of phase modulation to the pulse separation and E2 and Dm must have opposite signs which mean that in an anomalous dispersion fiber loop with

Nonlinear Photonic Fibre Ring Lasers

39

negative value of E2, the pulses should be up chirped. With a specific setup of FM fiber laser, when the magnitude of chirping increases, the bound pulse separation decreases subsequently. The pulsewidth also reduce according to the increase in the phase modulation index and modulation frequency, so that the ratio 'T/T0 can change not much. Thus, the binding of solitons in the FM mode-locked fiber laser is assisted by the phase modulator. Bound solitons in the loop experience periodically the frequency shift and hence their velocity in response to changes in their temporal positions by the interactive forces in equilibrium state.

Figure 36. Experimental setup of the FM mode-locked fiber laser. PM, phase modulator; PC, polarization controller; OSA, optical spectrum analyzer.

5.2. Experimental Setup and Results [-] Shows the experimental setup of the FM mode locked fiber laser. Two erbium doped fiber amplifiers (EDFA) pumped at 980 nm are used in the fiber loop to control the optical power in the loop for mode locking. Both are operating in saturated mode. A phase modulator driven in the region of 1 GHz modulation frequency assumes the role as a mode locker and controls the states of locking in the fiber ring. At input of the phase modulator, a polarization controller (PC) consisting of two quarter-wave plates and one half-wave plate is used to control the polarization of light which relates to the nonlinear polarization evolution and influences to multi-pulse operation in formation of bound soliton states. A 50 m Corning SMF-28 fiber is inserted after the phase modulator to ensure that the average dispersion in the loop is anomalous. The fundamental frequency of the fiber loop is 1.7827 MHz that is equivalent to the 114 m total loop length. The outputs of the mode locked laser from the 90:10 coupler are monitored by an optical spectrum analyzer (HP 70952B) and an oscilloscope (Agilent DCA-J 86100C) of an optical bandwidth of 65 GHz.

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Figure 37. (a) The oscilloscope trace and (b) optical spectrum of dual-soliton bound state.

Under normal conditions, the single pulse mode locking operation is performed at the average optical power of 5 dBm with the harmonic mode locking at the 560th order as shown in [-]. The narrow pulses of 8 to 14 ps width (the correct pulsewidth can be much smaller if excluding the effect of the oscilloscope’s risetime) depending on the RF driving power of the phase modulator are observed on the oscilloscope. The measured pulse spectrum has spectral shape of a soliton rather than a Gaussian pulse. By adjusting the polarization states of the PC wave plates at higher optical power, the dual-bound solitons or bound soliton pairs can be observed at average optical power circulating inside the fiber loop of about 10 dBm.[-] (a) shows the typical time-domain waveform and (b) the corresponding spectrum of the dualbound soliton state. The estimated FWHM pulsewidth is about 9.5 ps and the temporal separation between two bound pulses is 24.5 ps which are correlated exactly to the distance between two spectral main lobes of 0.32 nm of the observed spectrum. When the average power inside the loop is increased to about 11.3 dBm and a slight adjustment of the polarization controller is performed, the triple-bound soliton state occurs as shown in Figure 39 with that the FWHM pulsewidth and the temporal separation of two close pulses are slightly less than 9.5 ps and 22.5 ps respectively. Insets in [-] (a) and [-] (a) show the periodic sequence of bound solitons at the repetition rate equal exactly to the modulation frequency.

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41

This feature is quite different from that in a passive mode-locked fiber laser in which the positions of bound solitons is not stable and the direct soliton interaction causes a random movement and phase shift of bound soliton pairs [36]. On the other hand, it is more advantageous to perform a stable periodic bound soliton sequence in a FM mode-locked fiber laser. The symmetrical shapes of optical modulated spectrum in (b) and (b) indicate clearly that the relative phase difference between two neighboring bound solitons is of S value. At the center of spectrum is there a dip due to the suppression of the carrier with S phase difference in case of dual-soliton bound state.

Figure 38. (a) The oscilloscope trace and (b) optical spectrum of triple-soliton bound state.

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Figure 39. Calculated spectrums of double- soliton and triple-soliton bound states respectively with assumption of the S phase difference between neighboring pulses.

While a small hump is there in case of triple-soliton bound state because three soliton pulses are bound together with the first and the last pulses in phase and out of phase with the middle pulse. The shape of spectrum will change which can be symmetrical or asymmetrical when this phase relationship varies. We believe that the bound state with the relative phase relationship of S between solitons is the most stable in our experimental setup and this observation also agrees with the theoretical prediction of stability of bound soliton pairs relating to photon-number fluctuation in different regimes of phase difference [36]. Similar to a single soliton, the phase coherence of bound solitons is still maintained as a unit when propagating through a dispersive medium shows the dual-soliton bound state and triple-soliton state waveforms on the oscilloscope after they propagate through 1 km SMF fiber. There is also no change in their observed spectral shapes in both cases. Clearly, the multi-soliton bound operation can be formed in an FM mode-locked fiber by operating at a critical optical power level and phase modulator-induced chirp in a specific fiber loop of anomalous average dispersion.

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Figure 40. (a) The oscilloscope trace and (b) optical spectrum of quadruple-soliton bound state.

Figure 41. A circulating model for simulating the FM mode-locked fiber ring laser, SMF: Standard single mode fiber, EDFA: Erbium doped fiber amplifier.

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Figure 42. (a) Numerically simulated dual-soliton bound state formation from noise, (b) the evolution of the formation process in contour plot.

0.11 0.1 0.09 Intensity (a.u)

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 -200

-100

0 Time (ps)

100

200

0

Power spectral density (a.u)

-20

-40

-60

-80

-100

-120 -4

-3

-2

-1

0 1 Bandwidth (nm)

2

3

4

Figure 43. Dual-soliton bound state (a) waveform, and (b) corresponding spectrum of simulated at the 2000th roundtrip.

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Figure 44. (a) Numerically simulated triple-soliton bound state formation from noise, (b) evolution of the formation process by contour plot.

0 -10

Power spectral density (a.u)

-20 -30 -40 -50 -60 -70 -80 -90 -100 -110 -4

-3

-2

-1

0 1 Bandwidth (nm)

2

3

4

Figure 35. Simulated triple-soliton bound state (a) Waveform, and (b) corresponding spectrum at 2000th roundtrip.

Figure 46. (Continued).

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

Figure 46. Simulated evolution of dual-soliton bound state over 2000 roundtrips in the fiber loop at different phase modulation indexes (a) m = 0.1S, (b) m = 0.4S, (c) m = 0.8S, (d) m = 1S.

Figure 47. Simulated evolution of triple-soliton bound state over 2000 roundtrips at the low phase modulation index m = 0.1S.

(a)

(b)

Figure 48. Simulated evolutions over 2000 roundtrips in the fiber loop of (a) triple-soliton bound state at m = 1S and (b) quadruple-soliton bound state at m = 0.7S (Insets: contour views).

Nonlinear Photonic Fibre Ring Lasers

47

(a)

(b) Figure 49. Simulated evolution of (a) triple-soliton bound state at Psat = 9 dBm, m = 0.7S and (b) quadruple-soliton bound state at Psat = 10 dBm, m = 0.4S in the fiber loop (Insets: contour views).

It is really amazing when we increase the average optical power in the loop to maximum of 12.6 dBm and decrease the RF driving power of 15 dBm, the quadruple-soliton state is generated as observed in. The bound state occurring at lower phase modulation index is due to maintaining a small enough frequency shifting in a wider temporal duration of bound solitons would be holding the balance between interactions of the group of four solitons in the fiber loop. However the optical power can still not be sufficient to stabilize the bound state, the time-domain waveform of quadruple-soliton bound state is much noisier and its spectrum only can be seen clearly two main lobes being inversely proportional to the temporal separation of pulses which is about 20.5 ps in our experimental setup. The results show the pulse separation reduces when number of pulses in the bound states is larger. Thus both the phase modulation index and the cavity’s optical power influence the existence of the multi-soliton bound states in the FM mode-locked fiber laser. The optical power determines number of initially split pulses and maintains the pulse shape in the loop.

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

Observations from the experiment shows that there is a threshold of optical power for each bound level.

(a)

(b)

(c)

Figure 50. Simulated unstable evolution of the bound soliton states (a) dual-soliton bound state at Psat = 10 dBm and m = 0.2S, (b) triple-soliton bound state at Psat = 13 dBm and m = 0.6S, and (c) quadruplesoliton bound state at Psat = 14 dBm and m = 0.8S in the fiber loop (Insets: contour views).

Nonlinear Photonic Fibre Ring Lasers

49

At threshold value, the bound soliton shows strong fluctuations in amplitude and oscillations in position considering as a transition state between different bound levels and the collisions of adjacent pulses even occur as shown in [-]. The phase difference of adjacent pulses can also change in these unstable states which mean that the neighboring pulses is not out of phase anymore but in phase as seen through the measured spectrum in [-]. Although the decrease in phase modulation index is required to maintain the stability at higher bound soliton level, it increases the pulsewidth and reduces the peak power. So the waveform seen is noisier and its spectrum is not strongly modulated as shown in [-].

5.3. Simulation of Dynamics of Bound States in a FM Mode-Locked Fiber Laser 5.3.1. Numerical Model of a FM Mode-Locked Fiber Laser To understand the dynamic of FM bound soliton fiber laser, simulation technique is used to see what is going on in the FM mode-locked fiber loop. A recirculation model of the fiber loop is used to simulate propagation of the bound solitons in the fiber cavity. The simple model consists of basic components of a FM mode-locked fiber laser as shown in Figure 52, in other words, the cavity of FM mode-locked fiber laser is modeled as a sequence of different elements. The optical filter has a Gaussian transfer function with 2.4 nm bandwidth. The transfer function of phase modulator is uout uin exp[ jm cos(Zm t )] , m is the phase modulation index and Zm = 2Sfm is angular modulation frequency, assumed to be a harmonic of the fundamental frequency of the fiber loop. The pulse propagation in the optical fibers is governed by the nonlinear Schrodinger equation in general [44].

wu E w 2u E 3 w 3u D j 2   u wz 2 wT 2 6 wT 3 2

jJ u u 2

(14)

where u is the complex envelop of optical signal, E2 and E3 account for the second- and thirdorder fiber dispersions, D and J are the loss and nonlinear parameters of the fiber respectively. Amplification of signal including the saturation of the EDFA is modelled as following [44]:

uout

Guin

(15)

and

G

§ G  1 Pout G0 exp¨¨  G Ps ©

· ¸¸ ¹

where G is the amplification factor, G0 is unsaturated amplifier gain, Pout and Psat power and saturation power respectively.

(16) are output

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

(a)

(b) Figure 51. The time-domain evolution of dual- and triple-soliton bound states respectively with inphase pulses.

The difference of dispersion between the SMF fiber and the Er-doped fiber in the cavity arranges a certain dispersion map and the fiber loop gets a positive net dispersion or an anomalous average dispersion which is important in forming the “soliton – like” pulses in a FM fiber laser. Basic parameter values used in our simulations are listed in Table 5.1.

Table 5.1 Parameter values used in the simulations

E 2SMF = -21 ps2/km,

E 2ErF = 6.43 ps2/km,

= 0.0019 W /m, J SMF D = 0.2 dB/km Lcav = 115 m

J = 0.003 W /m, Psat = 7 ÷ 13 dBm NF = 6 dB

SMF

-1

ErF

-1

'Ofilter = 2.4 nm, fm | 1GHz, m = 0.1S ÷ 1S O = 1558 nm

SMF – Standard single mode fiber, ErF – Erbium doped fiber, NF – noise figure of the EDFA.

Nonlinear Photonic Fibre Ring Lasers

51

Figure 52. A circulating model for simulating the FM mode-locked fiber ring laser, SMF: Standard single mode fiber, EDFA: Erbium doped fiber amplifier.

5.3.2. Simulation of the Formation Process of the Bound Soliton States Firstly, we simulate the formation process of bound states in the FM mode-locked fiber laser and the parameters used in the simulation are similar to the parameters of components in our experiments above. The lengths of the Er-doped fiber and SMF fiber are chosen to get the cavity’s average dispersion

E 2 = -10.7 ps2/km. Figure 53 shows a simulated dual-soliton

bound state building up from initial Gaussian-distributed noise as a input seed over the first 2000 round-trips with Psat value of 10 dBm and G0 of 16 dB and m of 0.6S. The built-up pulse experiences transitions with large fluctuations of intensity, position and pulse width during around the first 1000 roundtrips before reaching to the final bound state. Figure 54 shows the time-domain waveform and spectrum of the output signal at the 2000th round trip. When the gain G0 value is increased to 18 dB which enhances the average optical power in the loop, the triple-soliton bound state is formed from the noise seed via simulation as shown in Figure 55. In the case of higher optical power, the fluctuation of signal at initial transitions is stronger and it needs more round trips to reach to a more stable three pulses bound state. The waveform and spectrum of the output signal from the FM mode-locked fiber laser at the 2000th round trip are shown in the Figure 56 (a) and (b) respectively.

(a)

(b)

Figure 53. (a) Numerically simulated dual-soliton bound state formation from noise, (b) the evolution of the formation process in contour plot view.

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

0.11 0.1 0.09 Intensity (a.u)

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 -200

-100

0 Time (ps)

100

200

0

Power spectral density (a.u)

-20

-40

-60

-80

-100

-120 -4

-3

-2

-1

0 1 Bandwidth (nm)

2

3

4

Figure 54. (a) The waveform, and (b) the corresponding spectrum of simulated dual-soliton bound state at the 2000th roundtrip.

(a)

(b)

Figure 55. (a) Numerically simulated triple-soliton bound state formation from noise, (b) the evolution of the formation process in contour plot view.

Nonlinear Photonic Fibre Ring Lasers

53

0.14 0.12

Intensity (a.u)

0.1 0.08 0.06 0.04 0.02

-200

-100

0 Time (ps)

100

200

0 -10

Power spectral density (a.u)

-20 -30 -40 -50 -60 -70 -80 -90 -100 -110 -4

-3

-2

-1

0 1 Bandwidth (nm)

2

3

4

Figure 56. (a) The waveform, and (b) the corresponding spectrum of simulated triple-soliton bound state at the 2000th roundtrip.

Although the amplitude of pulses is not equal which indicates the bound state can require a larger number of round trips before the effects in the loop balance, the phase difference of pulses accumulated during circulating in the fiber loop is approximately of S value that is indicated by strongly modulated spectra. In particular from the simulation result, the phase difference between adjacent pulses is 0.98S in case of the dual-pulse bound state and 0.89S in case of the triple-pulse bound state. These simulation results agree with the experimental results (shown in Figure 56 (a) and (b)) discussed above to confirm the existence of multisoliton bound states in a FM mode-locked fiber laser.

5.3.3. Simulation of the Evolution of the Bound Soliton States in a FM Fiber Loop Stability of bound states in the FM mode-locked fiber laser strongly depends on the parameters of the fiber loop which also determine the formation of these states. Beside the phase modulation and group velocity dispersion (GVD) as mentioned in 5.1, the cavity’s optical power also influence to existence of the bound states. Using the same model above, the effects of active phase modulation and optical power can be simulated to see the dynamics of bound solitons. Instead of the noise seed, the multi-soliton waveform following the Eq. (9)

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

is used for simplification of our simulation processes. Initial solitons is assumed to be identical with the phase difference between adjacent pulses of S value. Figure 57 shows the evolution of dual-soliton bound state over 2000 roundtrips in the loop at different phase modulation indexes with the same saturation optical power of 9 dBm. The simulation results also indicate that the pulse separation decreases corresponding to the increase in the modulation index. In the first roundtrips, there is a periodic oscillation of bound solitons that is considered as a transition of solitons to adjust their own parameters to match to the parameters of the cavity before reaching to a stable state. Simulations in other multi-soliton bound states also manifest this similar tendency. The periodic phase modulation in the fiber loop is not only to balance the interactive forces between solitons but also to remain the phase difference of S between them. At too small modulation index, the phase difference changes or reduces slightly after many round trips or in other words, the phase coherence is looser this leads to the amplitude oscillation due to the alternatively periodic exchange of energy between solitons as shown in Figure 58. The higher the number of solitons in bound state is, the more sensitive it is to the change in phase modulation index. At too high modulation index, it is more difficult to balance the effectively attractive forces between solitons especially when the number of solitons in the bound state is larger. The increase in chirping leads to faster variation in group velocity of pulses when passing the phase modulator which can create the periodic oscillation of pulse position in the time domain evolution. Psat = 9 dBm m = 0.4%

Psat = 9 dBm m = 0.1%

(a)

(b) Psat = 9 dBm m = 0.8%

(c)

Psat = 9 dBm m = 1%

(d)

Figure 57. The simulated evolution of dual-soliton bound state over 2000 roundtrips in the fiber loop at different phase modulation indexes (a) m = 0.1S, (b) m = 0.4S, (c) m = 0.8S, (d) m = 1S.

Nonlinear Photonic Fibre Ring Lasers

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Figure 58. The simulated evolution of triple-soliton bound state over 2000 roundtrips at the low phase modulation index m = 0.1S.

Figure 59 shows the evolution of the triple-soliton bound state at the m value of 1S and the quadruple-soliton bound state at the m value of 0.7S. In case of quadruple-soliton bound state, solitons oscillate strongly and tend to collide together.

(a)

(b)

Figure 59. The simulated evolutions over 2000 roundtrips in the fiber loop of (a) triple-soliton bound state at m = 1S and (b) quadruple-soliton bound state at m = 0.7S (Insets: contour views).

The optical power of the fiber loop plays an important role not only in determination of multi-pulse bound states as in simulation of previous section, but also in stabilization of the bound states circulating in the loop. As mentioned in section 5.2, each bound state has a specific range of operational optical power. In our simulations, the dual- and triple- and quadruple- soliton bound states behave stably in the loop at Psat of about 9 dBm, 11 dBm and 12 dBm respectively. When the optical power of the loop is set outside this range, the bound states become unstable and they are more sensitive to the change of phase modulation index. At the power lower than threshold, the bound states tend to be destroyed and switched to the lower level of bound state as shown in Figure 60. While at the too high power level, the

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

pulses can be broken into many pulses or decay into radiation. Figure 61 shows unstable behaviours of the bound soliton states in different conditions of the loop.

(a)

(b) Figure 60. The simulated behaviors of (a) triple-soliton bound state at Psat = 9 dBm, m = 0.7S and (b) quadruple-soliton bound state at Psat = 10 dBm, m = 0.4S in the fiber loop (Insets: contour views).

The cases of other phase difference between bound solitons are also simulated and the simulation results also show the S phase difference is the most stable state in the FM modelocked fiber ring laser. The states of not S phase difference often behave unstably in the loop and easily to be destroyed as displayed in Figure 62. The simulation results agree totally with the experimental observations discussed in 5.2.

Nonlinear Photonic Fibre Ring Lasers

57

(a)

(b)

(c) Figure 61. Simulated unstable behaviors of the bound soliton states (a) dual-soliton bound state at Psat = 10 dBm and m = 0.2S, (b) triple-soliton bound state at Psat = 13 dBm and m = 0.6S, and (c) quadruplesoliton bound state at Psat = 14 dBm and m = 0.8S in the fiber loop (Insets: contour views).

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Le Nguyen Binh, Nhan Duc Nguyen and Wenn Jing Lai

(a)

(b) Figure 62. The time-domain evolution of dual- and triple-soliton bound states respectively with in phase pulses.

6. Conclusions We have successfully demonstrated a mode locked laser operating under the open loop condition and with O/E RF feedback providing regenerative mode locking. The O/E feedback can certainly provide a self-locking mechanism under the condition that the polarisation characteristics of the ring laser are manageable. The regenerative MLRL can self-lock even under the DC drifting effect of the modulator bias voltage (over 20 hours)3. The generated pulse trains of 4.5 ps duration can be, with minimum difficulty, compressed further to less than 3 ps for 160 Gb/s optical communication systems. We have also demonstrated 660th and 1230th order of rational harmonic mode locking from a base modulation frequency of 100 MHz in the optically amplified fibre ring laser, 3

Typically the DC bias voltage of a LiNbO3 intensity modulator is drifted by 1.5 volts after 15 hours of continuous operation.

Nonlinear Photonic Fibre Ring Lasers

59

hence achieving 66GHz and 123GHz pulse repetition frequency. To the best of our knowledge, this is the highest rational harmonic order obtained to date. Besides the repetition rate multiplication, we also obtain high pulse compression factor in the system, ~35x and 40x relative to the non-multiplied laser system. In addition, we use phase plane analysis to study the laser system behaviour. From the analysis model, the amplitude stability of the detuned pulse train can only be achieved under negligible or no harmonic distortion condition, which is the ideal situation. The phase plane analysis also reveals the pulse forming complexity of the laser system. We have demonstrated 4-times repetition rate multiplication by using fibre GVD effect; hence, 40GHz pulse train is obtained from 10 GHz mode-locked fibre laser source. Stability analysis for this multiplied pulse train has been studied by using the nonlinear control stability theory that phase plane analysis is being used to study the transient and stability performance of the GVD repetition rate multiplication system. Surprisingly, from the analysis model, the stability of the multiplied pulse train can hardly be achieved even under perfect multiplication conditions. Furthermore, we observed uneven pulse amplitude distribution in the GVD multiplied pulse train, which is due to the energy variations between the pulses that cause some energy beating between them. Another possibility is the divergence of the pulse energy variation in the vicinity around the equilibrium point that leads to instability. The pulse amplitude fluctuation increases with the multiplication factor. Also, with wider filter bandwidth used in the laser cavity, better stability condition can be achieved. The nonlinear phase shift and noises in the system challenge the system stability of the multiplied pulses. They not only change the pulse shape of the multiplied pulses, they also distort the phase plane of the system. Hence, the system stability is greatly affected by the self phase modulation as well as the system noises. This stability analysis model can further be extended to include some system nonlinearities, such as the gain saturation effect, non-quadrature biasing of the modulator, fibre nonlinearities, etc. The chaotic behaviour of the system may also be studied by applying different initial phase and injected energy conditions to the model. Currently the design and demonstration of multi-wavelength mode-locked lasers to generate ultra-short and ultra-high rep-rate pulse sequences are under considerations by employing a multi-spectral filter demultiplexers and multiplexers within the photonic fibre ring. Furthermore the locking in the THz region will also be reported. The principal challenge is the conversion from the THz photonic to electronic domain for stabilisation. This can be implemented by either electronic or photonic sampling. Finally it is experimentally demonstrated that stable multi-soliton bound states can be formed in a nonlinear FM mode-locked fiber laser. This stable existence of multi-soliton bound states is effectively supported by the phase modulation in an anomalous-dispersion fiber loop under nonlinear saturation of optical power gain in optical amplifiers and in the circulating loop. Simulation results have confirmed the existence of multi-soliton bound states in the FM mode-locked fiber laser. Created bound states can be easily harmonic mode locked to generate periodically multi-soliton bound sequence at high repetition rate in this type of fiber laser rendering more prominent than that by passively mode-locked fiber lasers.

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References [1] [2]

[3]

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K. Kuroda, and H. Takakura, "Mode-locked ring laser with output pulse width of 0.4 ps," IEEE Trans. Inst. Meas., vol. 48, pp. 1018-1022, 1999. G. Zhu, H. Chen, and N. Dutta, "Time Domain Analysis of A Rational Harmonic Mode-locked Ring Fibre Laser," Journal of Applied Physics, vol. 90, pp. 2143-2147, 2001. C. Wu, and N.K. Dutta, "High repetition rate optical pulse generation using a rational harmonic mode-locked fibre laser," IEEE Journal of Quantum Electronics, vol. 36, pp. 145-150, 2000. K. K. Gupta, N. Onodera and M. Hyodo, "Technique to generate equal amplitude, higher-order optical pulses in rational harmonically mode locked fibre ring lasers," Electronics Letters, vol. 37, pp. 948-950, 2001. Y. Shiquan, L. Zhaohui, Z. Chunliu, D. Xiaoyi, Y. Shuzhong, K. Guiyun and Z. Qida, "Pulse-Amplitude Equalization in a Rational Harmonic Mode-Locked Fibre Ring Laser by Using Modulator as Both Mode Locker and Equalizer," IEEE Photonics Technology Letters, vol. 15, pp. 389-391, 2003. J. Azana, and M.A. Muriel, "Technique for Multiplying the Repetition Rates of Periodic Trains of Pulses by Means of a Temporal Self-Imaging Effect in Chirped Fibre Gratings," Optics Letters, vol. 24, pp. 1672-1674, 1999. S. Atkins, and B. Fischer, "All optical pulse rate multiplication using fractional Talbot effect and field-to-intensity conversion with cross gain modulation", IEEE Photonics Technology Letters, vol. 15, pp. 132-134, 2003. J. Azana, and M.A. Muriel, "Temporal Self-Imaging Effects: Theory and Application for Multiplying Pulse Repetition Rates," IEEE Journal of Quantum Electronics, vol. 7, pp. 728-744, 2001. D. A. Chestnut, C. J. S. de Matos and J. R. Taylor, "4x Repetition Rate Multiplication and Raman Compression of Pulses in the Same Optical Fibre," Optics Letters, vol. 27, pp. 1262-1264, 2002. S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, "Repetition Frequency Multiplication of Mode-locked Using Fibre Dispersion," Journal of Lightwave Technology, vol. 16, pp. 405-410, 1998. W. J. Lai, P. Shum, and L.N. Binh, "Stability and Transient Analyses of Temporal Talbot-Effect-Based Repetition-Rate Multiplication Mode-Locked Laser Systems," IEEE Photonics Technology Letters, vol. 16, pp. 437-439, 2004. K. K. Gupta, N. Onodera, K.S. Abedin and M. Hyodo,, "Pulse Repetition Frequency Multiplication via Intracavity Optical Filtering in AM Mode-locked Fibre Ring Lasers," IEEE Photonics Technology Letters, vol. 14, pp. 284-286, 2002. K. S. Abedin, N. Onodera and M. Hyodo, "Repetition-rate Multiplication in Actively Mode-locked Fibre Lasers by Higher-order FM Mode-locking Using a High Finesse Fabry Perot Filter," Applied Physics Letters, vol. 73, pp. 1311-1313, 1998. K. S. Abedin, N. Onodera and M. Hyodo, "Higher Order FM Mode-locking for PulseRepetition-Rate Enhancement in Actively Mode-locked Lasers: Theory and Experiment," IEEE Journal of Quantum Electronics, vol. 35, pp. 875-890, 1999.

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[15] W. Daoping, Z. Yucheng, L. Tangjun and J. Shuisheng, "20 Gb/s Optical Time Division Multiplexing Signal Generation by Fibre Coupler Loop-Connecting Configuration," presented at 4th Optoelctronics and Communications Conference,, 1999. [16] D. L. A. Seixasn, and M.C.R. Carvalho, "50 GHz Fibre Ring Laser Using Rational Harmonic Mode-locking," presented at Microwave and Optoelectronics Conference, 2001. [17] R. Y. Kim, "Fibre lasers and their applications," presented at Laser and Electro-Optics, CLEO/Pacific Rim'95, 1995. [18] H. Zmuda, R.A. Soref, P. Payson, S. Johns, and E. N. Toughlian, "Photonic beamformer for phased array antennas using a fibre grating prism," IEEE Photon. Technol. Lett., vol. 9, pp. 241 -243, 1997. [19] G. A. Ball, W. W. Morey, and W. H. Glenn, "Standing-wave monomode erbium fibre laser," IEEE Photon. Technol. Lett., vol. 3, pp. 613-115, 1991. [20] D. Wei, T. Li, Y. Zhao, et alia, "Multi-wavelength erbium-doped fibre ring laser with overlap-written fibre Bragg gratings," Opt. Lett., vol. 25, pp. 1150-1152, (2000). [21] S. K. Kim, M. J. Chu, and J. H. Lee, "Wideband multi-wavelength erbium-doped fibre ring laser with frequency shifted feedback," Opt. Commun., vol. 190, pp. 291-302, 2001. [22] Z. Li, L. Caiyun, and G.Yizhi, "A polarization controlled multi-wavelength Er-doped fibre laser," presented at APCC/OECC99, 1999. [23] R. M. Sova, C. S. Kim, and J. U. Kang, "Tunable dual-wavelength all-PM fibre ring laser," IEEE Photon. Technol. Lett., vol. 14, pp. 287-289, 2002. [24] I. D. Miller, D. B. Mortimore, P. Urquhart, et al., " A Nd3+-doped cw fibre laser using all-fibre reflectors," Appl. Opt., vol. 26, pp. 2197-2201, 1987. [25] X. Fang, and R. O. Claus, "Polarization-independent all-fibre wavelength-division multiplexer based on a Sagnac interferometer," Opt. Lett., vol. 20, pp. 2146-2148, 1995. [26] X. Fang, H. Ji, C. T. Aleen, et al., "A compound high-order polarization-independent birefringence filter," IEEE Photon. Technol. Lett., vol. 19, pp. 458-460, 1997. [27] X. P. Dong, Li, S., K. S. Chiang et al., "Multi-wavelength erbium-doped fibre laser based on a high-birefringence fibre loop," Electron. Lett., vol. 36, pp. 1609-1610, 2000. [28] D. Jones, H. Haus, and E. Ippen, "Subpicosecond solitons in an actively mode locked fibre laser," Opt. Lett., pp. 1818-1820., 1996. [29] X. Zhang, M. Karlson and P. Andrekson, "Design guideline for actively mode locked fibre ring lasers," IEEE Photonic tech. Lett., pp. 1103-1105, 1998. [30] A. E. Siegman, Laser. Mill Valley, C.A.:: University Press, 1986. [31] J. J. E. Slotine, and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: PrenticeHall, 1991. [32] B. A. Malomed, “Bound solitons in coupled nonlinear Schrodinger equation,” J. Phys. Rev. A 45, R8321-R8323 (1991). [33] B. A. Malomed, “Bound solitons in the nonlinear Schrodinger- Ginzburg-Landau equation,” J. Phys. Rev. A 44, 6954-6957 (1991). [34] N. H. Seong and Dug Y. Kim, “Experimental observation of stable bound solitons in a figure-eight fiber laser,” Opt. Lett., Vol. 27, No. 15, August 2002. [35] D. Y. Tang, B. Zhao, D. Y. Shen, and C. Lu, “Bound-soliton fiber laser,” J. Phys. Rev. A 66, 033806 (2002).

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[36] Y. D. Gong, D. Y. Tang, P. Shum, C. Lu, T. H. Cheng, W. S. Man, and H. Y. Tam, “Mechanism of bound soliton pulse formation in a passively mode locked fiber ring laser”, Opt. Eng., 41(11), 2778-2782 (2002) [37] P. Grelu, F. Belhache, and F. Gutty, “Relative phase locking of pulses in a passively mode-locked fiber laser,” J. Opt. Soc. Am. B 20, 863-870 (2003). [38] L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, C. Lu, “Bound states of dispersionmanaged solitons in a fiber laser at near zero dispersion,” App. Opt. 46, 4768-4773 (2007). [39] W. W. Hsiang, C. Y. Lin, and Y. Lai, “Stable new bound soliton pairs in a 10 GHz hybrid frequency modulation mode locked Er-fiber laser,” Opt. Lett. 31, 1627-1629 (2006). [40] C. R. Doerr, H. A. Hauss, E. P. Ippen, M. Shirasaki, and K. Tamura, “Additive-pulse limiting,” Opt. Lett. 19, 31-33 (1994). [41] R. Davey, N. Langford, A. Ferguson, “Interacting solitons in erbium fiber laser,” Electr. Lett. 27, 1257-1259 (1991). [42] D. Krylov, L. Leng, K. Bergman, J. C. Bronski, and J. N. Kutz, “Observation of the breakup of a prechirped N-soliton in an optical fiber”, Opt. Lett., 24, 1191-1193 (1999). [43] J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasi-lossless and lossy optical fiber spans”, J. Opt. Soc. Am. B, 24, 1254-1261 (2007). [44] G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001). [45] T. Georges and F. Favre, “Modulation, filtering, and initial phase control of interacting solitons,” J. Opt. Soc. Am. B 10, 1880-1889 (1993). [46] N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, “Suppression of soliton interactions by periodic phase modulation”, Opt. Lett., 19, 16-18 (1994). [47] R. K. Lee, Y. Lai, and B. A. Malomed, “Photon-number fluctuation and correlation of bound soliton pairs in mode-locked fiber lasers,” Opt. Lett. 30, 3084-3086 (2005).

In: Progress in Nonlinear Optics Research Editors: Miyu Takahashi and Hina Goto, pp. 63-84

ISBN 978-1-60456-668-0 © 2008 Nova Science Publishers, Inc.

Chapter 2

MANEUVERING ATOMS FOR LITHOGRAPHY USING NEAR RESONANT SPATIALLY VARYING LASER FIELD Kamlesh Alti1,2 and Alika Khare1, 1

Department of Physics, Indian Institute of Technology Guwahati Guwahati-781039, India 2 Division of Laser Spectroscopy, Manipal Life Sciences Centre, Manipal University Manipal-576104, India

Abstract Two kinds of forces are experienced by an atom in a laser radiation field, viz; radiation pressure resulting form absorption followed by the random spontaneous emission of photons and the conservative or dipole force originating from interaction of neutral atom with a near resonant non uniformly distributed laser field. The magnitude of radiation pressure is limited by the rate of spontaneous emission and saturates as the laser intensity increases. This dissipative spontaneous force is responsible for laser cooling. On the other hand, the magnitude of dipole force depends on the intensity gradient and amount of detuning and the direction of force depend on the sign of detuning. With the proper configuration of the atomic beam and the laser beam, the trajectories of the atoms can be manipulated so as to focus down the atoms in the periodic nano-size structures of desired geometry. The intensity gradient acts as the atomic lens as well as mask and thereby obviating the need of material mask for writing periodic patterns. The complete writing using dipole force can be performed in a single step. This lithography technique is very general and is applicable to any atomic and molecular species provided the tunable laser for the particular transition frequency of atoms/molecules is available. This article primarily reviews simulation studies of atomic trajectories under dipole force in various spatially varying laser light field for the lithographical applications. A novel concept of multiple atomic beams traveling in TEM00 mode of laser is also discussed to generate the periodic patterns of periodicity less than /2, where  is the wavelength of light used, and sizes in range of tens of nanometer.

Keywords: Atom lithography, Dipole force, Atomic beam.

Corresponding Author E-mail: [email protected]

Kamlesh Alti and Alika Khare

64

1. Introduction Atom lithography using dipole force is a new upcoming technique. This technique uses the dipole or gradient force1 for focusing the atomic beam down to tens of nanometer in a periodic one-, two- and three-dimensional structure. It offers massive parallelism without using any material mask. The low energy atomic beam used in this lithography does not damage the substrate. The dipole force, originating from the interaction of neutral atom with a near resonant, non-uniformly distributed laser field can be expressed as below (Eq. 1) using semi classical treatment [1]

JG F

JG I =* 2 '’( ) 2 I sat  I * 2 (1  )  4' 2 I sat

(1)

where * is the natural line width , ' is the detuning from the resonance, I is the intensity of laser and Isat is the saturation intensity for the corresponding transition. FromEq 1, it is clear that the magnitude of this force depends on the intensity gradient and amount of detuning and the direction of force depend on the sign of detuning. With the proper configuration of the atomic beam and the laser beam, the trajectories of the atoms can be manipulated so as to focus down the atoms in the periodic nano size structures of desired geometry. The intensity gradient act as the atomic lens as well as mask and so no additional material mask is required. Bjorkholm et al.[2] reported the focusing of sodium atomic beam in presence of TEM00 mode laser in 1978 for the first time experimentally. Subsequently, they had reported [3] a minimum spot size of 28 Pm using the same configuration. Aberration problem due to large thickness of atomic lens in their work was overcome by Sleator et al.[4]. They used large period standing wave produced by bouncing the laser beam off a glass substrate under very small angle of incidence. This large period standing wave acts as a cylindrical atomic lens. Using this, beam of metastable helium atoms were successfully focused down to 4 Pm. The idea of using the standing wave was further developed by Timp et al.[5], where the atomic beam of sodium was launched perpendicular to the standing wave of near resonant radiation field of wavelength O. Each period of standing wave acts as the cylindrical lens to focus the atomic beam. This results into grating structures of sodium on the substrate with periodicity of O/2 (294.3r0.3 nm). Since then, there has been a number of experimental reports on the focusing of atomic beams of chromium [6], cesium [7], aluminium [8], ytterbium [9] and iron [10, 11] using dipole force generated by continuous wave optical standing wave. This paper contains our simulation studies [12-14] of atomic trajectories under dipole force for various light fields (standing wave and TEM00 mode of laser) and atomic beam configurations. A novel configuration of multiple periodic atomic beams (MPAB) traveling in TEM00 mode of laser is explored for the first time to generate the structure via dipole force with periodicity below /2. A new configuration of using square arrays of multiple atomic lenses produced by interference of four nearly collinear optical beams for atom lithography using dipole force is proposed. The rubidium atomic beams are considered for simulation of

Maneuvering Atoms for Lithography…

65

atomic trajectories via dipole force using Eq.1 for rubidium transition 5S21/2- 5P23/2 (Please see Figure 1). The natural line width of this transition is 6 MHz.

Figure 1. Energy Level diagram of Rubidium.

2. Gaussian and Standing Wave Configuration 2.1. Gaussian Wave Configuration Simplest light configuration is TEM00 mode of a laser. Gaussian variation of intensity in the transverse plane of TEM00 laser beam acts as a single atomic lens for the focusing of atomic beam to a single spot. Intensity distribution of TEM00 mode laser traveling along y direction is defined by

I

I 0 w0 2 2( x 2  z 2 )  exp[ ] w( y ) 2 w( y ) 2

(2)

where

w( y )

w0 (1  (

y 2 12 ) ) y0

is the beam size at a distance y from the location of beam waist (2w0) as depicted in Figure 2 and y0 is the Rayleigh length given by

y0

w0 2S

O

where  is the wavelength of laser.

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66

2.1.1. Boundary Conditions The collimated as well as divergent one-dimensional cold and thermal atomic beams of rubidium were considered for simulations of trajectories via dipole force using Eq. 1 in the counter propagating TEM00 mode of laser. The direction of propagation of rubidium atomic beam is taken along y axis and that of laser beam along -y axis as shown in Figure 2. The atoms were assumed to enter in the field at a Rayleigh distance with respect to the location of beam waist (w0=50 m) of laser (please see Figure 2). From Eq. 1, equations of motion of atoms in x and y direction are given by

m

2 x='

w2x wt 2

and m

y w( y ) (1  ( ) 2 ) y0 *2  2 I sat (* 2  4'2 ) I o exp[ 2 x 2] w( y )

(3a)

2

w2 y wt 2

0

(3b)

Eq. 3a and b are computed simultaneously for rubidium atomic beams (collimated as well as divergent) for following initial boundary conditions. For collimated atomic beams two initial boundary conditions at t=0 are

wx wt

vx

0 and

wy wt

vy

17 or 1700 m / s

(4)

Other two boundary conditions are x and y coordinates at t=0, depicting the distribution of atoms in the atomic beam at the launching position, given by

x

NG x ,

y

y0 5

where N=-10,-9,-8,....,8,9,10 and x=1 m, giving 21 atoms in the beam with 1 m spacing between the adjacent atom. For divergent atomic beam, instead of Eq. 4, two initial boundary conditions at t=0 are

vx

v sin( X u 0.0025), v y

v cos( X u 0.0025) 6

where X=-2,-1, 1, 2 and v=17 or 1700 m/s. The other two boundary conditions are same as given by Eq. 5 apart from the difference that now from each location 5 atoms are launched in coaxial cones; therefore the total number of atoms in the beam is now 5 times more with a maximum cone angle dmax of 5 mrad.

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67

Figure 2. Launching position of atoms with respect to Gaussian laser beam.

2.1.2. Simulation Results for Gaussian Wave Configuration The rubidium trajectories are computed for cold as well as thermal beams traveling in TEM00 mode of laser with peak intensity, I0, equal to 16.5 W/m2, which corresponds to saturation intensity of rubidium transition (Figure 1), for red detuning of L=-200 MHz. The computed rubidium trajectories of cold rubidium atomic beam in TEM00 mode of laser for red detuning are shown in Figure 3a for first focal spot. Interaction time required for focusing the atomic beam is 35.88 s. Figure 3b shows the effect of longer interaction time of atoms with laser beyond the first focus. Atomic trajectories in this case show multiple focusing and defocusing depending on the interaction time. The collimated atomic beam is an ideal case; therefore effect of divergence of atomic beam was studied for the same interaction time. Figure 4a shows the first focal spot of the initially divergent (dmax = 5 mrad) cold rubidium atomic trajectories in TEM00 mode of laser. The focus spot size is increased in this case as compared to collimated atomic beam due to the initial divergence of the atom. Multiple focusing effects for longer interaction time of atoms with the laser are shown in Figure 4b.

Figure 3. Trajectories of initially collimated cold rubidium atomic beam in TEM00 mode of laser showing (a). First focal spot and (b). Multiple focusing due to longer interaction time of atomic beam with laser.

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Kamlesh Alti and Alika Khare

Figure 4. Trajectories of initially divergent cold rubidium atomic beam in TEM00 mode of laser showing (a). First focal spot and (b). Multiple focusing due to longer interaction time of atomic beam with laser.

Figure 5. Trajectories of initially collimated thermal rubidium atomic beam showing first focal spot for the same interaction time as with the cold atomic beam.

Figure 5 shows the first focal spot of thermal (1700 m/s) rubidium atoms for the same interaction time (35.88 s) as that required for cold rubidium atoms. It seems that the interaction time for the focusing of the atomic beam is independent of its axial velocity. The TEM00 mode of laser beam gives only one single focus spot of atoms of micron dimensions. Therefore a light configuration which gives multiple periodic atomic lenses is needed for periodic writing.

2.2. Standing Wave Configuration Standing wave of light is one such configuration which gives series of periodic micro lenses with period /2, as shown in Figure 6. The atomic beam is launched perpendicular to standing wave and focus down into the grating structure by each period of standing wave, which acts as a cylindrical atomic lens. Intensity distribution of a one-dimensional standing wave can be taken as

Maneuvering Atoms for Lithography…

I 0 cos 2 (kx)

I

where k

69 (7)

2S

O

Figure 6. Launching position of atoms with respect to standing wave.

2.2.1. Boundary Conditions The collimated as well as divergent cold and thermal one-dimensional atomic beam of rubidium were considered for simulations of trajectories via dipole force. The direction of propagation of rubidium atomic beam is taken along -y axis while the standing wave is along x axis as shown in Figure 6. From Eq. 1, equations of motion of atoms in x and y directions are given by

wx wt 2

2

m

I cos(kx) sin( kx) I0 I cos 2 ( kx) * 2 (1  0 )  4'2 I sat =* 2 '

m

,

w2x wt 2

0

(8)

Eqs. 8 are computed simultaneously for rubidium atomic beams (collimated as well as divergent). For collimated atomic beams two initial boundary conditions at t=0 are

wx wt

vx

wy wt

0

vy

17 or 1700 m / s

(9)

Other two boundary conditions are x and y coordinates at t=0, which are given by

x

NG x

y

y0

(10)

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70

where N=1,2,3,....,90 and x =13 nm. For divergent atoms two initial boundary conditions for t=0 modified to

vx

v sin( X u 0.0025), v y

v cos( X u 0.0025)

(11)

where X=-2, -1, 1, 2 and v=17 or 1700 m/s. The other two boundary conditions are same as given by Eq. 7 apart from the difference that now from each location 5 atoms are launched in coaxial cones, such that the total number of atoms is now 450 with a maximum cone angle (or dmax) of 5 mrad.

Figure 7. Trajectories of initially collimated cold rubidium atomic beam in optical standiing wave at intensity 16.5 W/m2 showing (a). First focal spot and (b). Multiple focusing due to longer interaction time of atomic beam with laser. Intensity distribution is shown in the bottom of the figure.

2.2.2. Simulation Results for Standing Wave Configuration The computed trajectories of cold (17 m/s) rubidium atomic beam in standing wave for blue detuning (L=200 MHz) for first focal spot is shown in Figure 7a. Interaction time required for focusing the atomic beam is 0.35 s. Figure 7b shows the effect of multiple focusing for longer interaction time of atoms with laser. Figure 8a shows the initially divergent (dmax = 5 mrad) cold rubidium atomic trajectories in standing wave for first focal spot. The focus spot size is increased in this case as compared to collimated atomic beam due to divergence. Figure 8b shows multiple focusing due to longer interaction time of atoms with the standing wave. The computed trajectories of cold beam of rubidium in standing wave at intensities 1 and 10 W/m2 for blue detuning are shown in Figure 9a and b, respectively. At higher intensities atoms are tightly focused with better contrast. From Figure 7a and Figure 9a and b, it is obvious that at lower intensity, the atomic beam has to travel a larger distance, in other words interaction time with the field are required to be large. Interaction times required for the atomic beam for focusing at various intensities are listed in Table. 1.

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71

Figure 8. Trajectories of initially divergent cold atomic beam in optical standing wave at intensity 16.5 W/m2 showing (a). First focal spot and (b). Multiple focusing due to longer interaction time of atomic beam with laser. Intensity distribution is shown in the bottom of the figure.

Figure 9. Trajectories of initially collimated cold rubidium atomic beam in optical standing wave at intensity (a). I0=1 W/m2 and (b). I0=10 W/m2. Intensity field is shown in the bottom of the figure.

Table 1. Focusing time for various intensities of laser Intensity (W/m2) 1 10 16.5

Focusing time (s) 1.35 0.47 0.35

Results for a thermal beam at atomic velocity 1700 m/s are shown in Figure 10. Results are similar to that of cold beam. Thermal beam has to travel a distance longer by two orders of magnitude before being focused compared to that of cold atomic beam. This confirms that interaction time required for the focusing at any given intensity is constant and is independent of velocity of the atom.

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Figure 10. Trajectories of rubidium thermal atomic beam at Intensity, I0= 1W/m2 and I0=10W/m2.

3. Discrete Arrays of Atomic Beams for Sub-/2 Lithography The periodicity of the standing wave configuration is limited to O/2. To overcome this limitation, a new configuration is being proposed in this section. The possibility of compressing the discrete regular arrays of atomic beams to generate sub O/2 structures via dipole force using TEM00 mode of laser is discussed below. This configuration is the reverse analogy of the standing wave configuration. One can also draw analogy of this configuration with optics, as shown in Figure 11.

Figure 11. Analogy of focusing of multiple atomic beams using TEM00 mode of laser with optics.

In conventional optics, large number of periodic light beams can be focused using a convex lens. In the present system, light beams are replaced by atoms and lens by TEM00 mode of laser. In optics minimum size of focal spot is restricted by wavelength of light. But in case of atoms, the de Broglie wavelength is quite small ( 0.1 nm for thermal atoms), so theoretically it is possible to get sub-nanometer atomic spots with this configuration.

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73

3.1. Configuration Details Intensity distribution of TEM00 mode of laser traveling along y direction is given in section 2. The collimated [cold (17 m/s) and thermal (1700 m/s)] atomic beams were considered for simulations of trajectories via dipole force using Eq.1. The cross section (z-x plane) of square arrays of atomic beam launched is shown in Figure 12.

Figure 12. (a). Cross sectional view of initial positions of rubidium atomic beam arranged in a square geometry. (b). Arrangements of atoms in each individual atomic beam.

The initial positions of atomic beams correspond to the corner of squares of size 2 Pm. The diameter of each atomic beam is considered to be 1 Pm with 91 atoms placed symmetrically in it (Figure 12). The direction of propagation of atomic beam is taken along y axis and that of laser beam along –y axis as shown in Figure 2. The atoms were assumed to enter in the field at a Rayleigh distance with respect to the location of beam waist (w0= 50 Pm) of laser (as shown in Figure 2).

3.2. Boundary Conditions From Eq.1, equations of motion of atoms in x, y and z direction are given by

w2 x m( 2 ) wt

m(

2 x=' y 2 ) ] y0 *2  2( x 2  z 2 ) I sat (* 2  4' 2 ) I 0 exp[ ] w( y ) 2

w2 y ) 0 wt 2

(12)

w( y ) 2 [1  (

(13)

Kamlesh Alti and Alika Khare

74

m(

w2 z ) wt 2

2 z=' y w( y ) [1  ( ) 2 ] y0 *2  2( x 2  z 2 ) I sat (* 2  4' 2 ) I 0 exp[ ] w( y ) 2

(14)

2

Eq. 12, 13 and 14 are computed simultaneously for the proposed configuration of atomic beams (collimated as well as divergent) for the following initial boundary conditions. For collimated atomic beams three initial boundary conditions at t=0 are

wx wt wy wt wz wt

vx

0

vy

17or1700m / s

vz

0

(15)

Other three boundary conditions are x, y and z coordinates at t=0, which are given by

x

ab  cd cos(T )

y

y0

z

eb  cd cos(T )

(16)

where a=-10, -8……., 8, 10, b=1 Pm, c=0, 1, 2, 3, 4, 5, T=0, 0.348, 0.696, 1.044,…., 6.28 rad, d=0.1 Pm, e=-10, -8,……, 8, 10. Pictorially x-z coordinates are shown in Figure 12. For divergent atomic beams three initial boundary conditions at t=0 modified to

vx vy

v sin(T d ) 1.41 v cos(T d )

vz

v sin(T d ) 1.41

wr wt

v

(17)

vx  v y  vz 2

2

2

17 or 1700 m / s

where d is the divergence angle. The other three boundary conditions are same as shown in Figure 12 and (and given by Eq. 16) apart from the difference that now from each location 41 atoms are launched in coaxial cones (as shown in Figure 13), such that the total number of atoms in each beam are now 91 41 with a maximum cone angle (or dmax) of 1 mrad and 4.5 mrad.

Maneuvering Atoms for Lithography…

75

Figure 13. Launching coaxial cones of atomic beams for divergence angle d.

The peak intensity of laser at the center of the beam waist is taken to be 16.5 W/m2 corresponding to the saturation intensity of rubidium transition (Figure 1). The laser wavelength is red detuned ('=-200 MHz) from the resonance line of 780 nm of rubidium.

3.3. Results The sequence of computed lithographic patterns for collimated arrays of cold atomic beams for interaction time of 41.17 Ps and 44.11 Ps (corresponding interaction length of 700 Pm and 750 Pm respectively) are shown in Figure 14a and b respectively. Overall pattern compression factor as compared to initial configuration of Figure 12 is ~ 5.8 in Figure 14a. The minimum periodicity of this pattern is 380 nm ( O/2) and the individual atomic beams compressed to 200 nm in the center of the laser beam. Figure 14b shows the pattern after an interaction length of 750 Pm. The compression factor in this case is ~ 3.6. The spot size at the center is expanded to 300 nm with periodicity 580nm (>O/2). The astigmatism in the pattern is because of non-uniform gradient of intensity distribution of TEM00 mode. The well collimated atomic beams for such lithography is an ideal choice but practically not possible. Therefore we have considered the effect of divergence of the atomic beam. The starting configuration of atomic beam is shown in Figure 12 and Figure 13. Figure 15 and Figure 16 shows the lithographic patterns for such configuration for interaction length of 700 m and 750 m, corresponding to interaction time of 41.17 s and 44.11 s for cold divergent rubidium atomic beams having initial cone angle of 1 and 4.5 mrad. The lithographic pattern of atoms after an interaction time of 41.17 s with the field is shown in Figure 15a and Figure 16a. These patterns shows the line structure of atoms with periodicities 320 nm and 300 nm (both /2) for maximum divergence of 1 mrad and 4.5 mrad respectively at the center of the pattern. The corresponding line thicknesses, at the center are nearly equal to 220 nm and 210 nm respectively. The overall compression factor in this case is  5 and 3.2 respectively. The pattern further expands after an interaction time of 44.11 s, as shown in Figure 15b and Figure 16b with overall compression factor  3.3 and 2.4 respectively. The periodicities of these patterns are 490 nm and 500 nm (both > /2) with line thickness 330 nm and 340 nm

76

Kamlesh Alti and Alika Khare

respectively. The reason for observed line structure can be symmetric atomic distribution along x and z direction of the atomic beams at the launching position.

Figure 14. Final position of cold rubidium atoms after interacting with TEM00 mode of laser for initially collimated set of atomic beams for time (a). 41.17 s and (b). 44.11s.

Figure 15. Final position of cold rubidium atoms after interacting with TEM00 mode laser for set of atomic beams having initial cone angle of 1 mrad for time (a). 41.17 s and (b). 44.11s.

Figure 16. Final positions of cold rubidium atoms after interacting with TEM00 mode of laser for set of atomic beams having intial cone angle of 4.5 mrad for time (a). 41.17 s and (b). 44.11s.

Maneuvering Atoms for Lithography…

77

3.4. Dependency of Uniformity and Compression Factor of Final Lithographic Pattern on Beam Waist of Laser The intensity gradient in the gaussian beam (TEM00) is non uniform which results into the non uniformity in the periodicity of the focused pattern. A careful choice of the beam waist (w0) of TEM00 beam may improve upon the uniformity of the finally focused pattern of atomic beams via dipole force. Figure 17 represents series of pictures which show final position of initially collimated cold rubidium atoms for the interaction time of 41.17 s with TEM00 mode of laser having beam waist of 40 m, 50 m, 60 m, 70m, 80m and 90 m 5 1.5

z(microns)

z(microns)

1

0

0.5 0 -0.5 -1

(a) 5

0 x(microns)

-1.5

5

z(microns)

z(microns)

0

(c) 0 x(microns)

0.8

-2.5 -2.5

0 0.5 x(microns)

1

1.0

(d) 0 x(microns)

2.5

0 x(microns)

5

5

z(microns)

z(microns)

-0.5

0

4

0

-4 -4

-1

2.5

0.8

-0.8 -0.8

(b)

-1.5

(e) 0 x(microns)

4

0

-5 -5

(f)

Figure 17. Final position of cold rubidium atoms for initially collimated set of atomic beams for interaction time of 41.17 s after interacting with TEM00 mode of laser having beam waist (a). 40 m (b). 50 m (c). 60 m (d). 70 m (e). 80 m and (f) 90 m. Figures show how the change in the beam waist of laser can affect the uniformity and compressibility of multiple atomic beam configuration.

respectively. These figures show a very systematic change in uniformity and compression of the lithographic pattern with increasing beam waist of laser. This is expected because of the

Kamlesh Alti and Alika Khare

78

slow variation of the spatial distribution of intensity gradient with the increase in the beam waist of the laser. Approximately compression factors for Figure 17 (a)-(f) with respect to initial configuration (Figure 12 (a)) are 2.5, 5.88, 12.5, 4, 2.5 and 2 respectively. Figure 17 (f) which corresponds to beam waist of 90 m, looks most uniform as compared to others. The compression factor for this figure is 50%, which may improve with higher intensities. The computed results yield interesting patterns with periodicities much less than O/2 in presence of single potential from a TEM00 mode of laser. The proposed configuration is relatively simple because it doesn't require any conditioning of light potential as directly TEM00 mode is used. The only requirement of this configuration is generation of multiple periodic atomic beams with micron or sub-micron range periodicity [15-16].

4. Inter Ferometric Configuration In this section, a new configuration of multiple atomic lenses formed via interference of four nearly collinear optical beams [12, 17] for writing the square arrays of structures is proposed. The interference of four beams results into the square arrays of equally illuminating light spots. These light spots act as the two dimensional periodic potential for the focusing of the atomic beam via dipole force.

Figure 18. Experimental set up for the production of four beam interferometric pattern.

4.1. Configuration Details Experimental set up used to generate square arrays of equal illuminating light spots is shown in Figure 18. It comprises of a MachZhander interferometer coupled to Michelson interferometer. Beam splitters BS1, BS2 and mirrors M1, M2 forms the MachZhander interferometer giving the two interfering beams. These two beams can be launched into the second stage, which is a Michelson interferometer comprising of beam splitter BS3 and mirrors M3 and M4. The output of BS3 consists of four nearly collinear interfering beams. The mirror tilts can be adjusted in such a way that the interference patterns of individual

Maneuvering Atoms for Lithography…

79

stages get oriented perpendicular to each other. This gives the arrays of equal illuminating light spots in the square geometry as shown in Figure 19b.

Figure 19. Three dimensional plot of intensity distribution of four beam interferometric pattern (a). Recorded four beam interferometric pattern using a He-Ne laser (b.)

Figure 20. Launching position of rubidium atoms in presence of interfering optical beams.

The resultant intensity distribution of interference pattern generated by using such set up is given by

I

I 0 [4  4 cos(kx)  4 cos( ky )  2 cos(kx  ky )  2 cos( kx  ky )]

(18)

Kamlesh Alti and Alika Khare

80 where k

2SP and spatial frequency P

sin(T ) ,  is the angular separation between two

O interfering beams of either of the interferometer and O is the wavelength of the laser light used. Figure 19a shows the computed plot of intensity distribution of Eq. 18 and Figure 19b shows the recorded pattern from the set up of Figure 18 using a He-Ne laser. The output of the interferometer (from the BS3) of Figure 18 can be compared as if the parallel light pipes arranged in square geometry as coming out. Atomic beam launched in the counter propagating way as shown in Figure 20. Thus each light pipes acts as an atomic lens for the focusing of the atoms. The collimated as well as divergent cold (v=17 m/s) rubidium atomic beams traveling in counter propagating interfering beams from the set up of Figure 18 are considered (as shown in Figure 19) for simulation of atomic trajectories via dipole force using Eq.1. 4.2. Boundary Conditions Equations of motion of rubidium atoms traveling in this optical configuration (from Eq.1) are given by

m

w2 x wt 2

w2 y m 2 wt

m

w2 z wt 2

='I 0 k[2sin( kx)  sin(kx  ky )  sin( kx  ky)]  I 0 * [4  4 cos(kx)  4 cos( ky)  2 cos( kx  ky)  2 cos( kx  ky)] 1 I sat (* 2  4' 2 ) 2

='I 0 k[2sin(ky )  sin(kx  ky )  sin( kx  ky)]  I 0* [4  4 cos(kx)  4 cos(ky )  2 cos(kx  ky )  2 cos(kx  ky )] 1 I sat (* 2  4' 2 ) 2

0

(19a)

(19b)

(19c)

Eq. 19a, b and c are computed simultaneously for the proposed configuration of atomic beams (collimated as well as divergent) for following initial boundary conditions. For the collimated rubidium atomic beams the three boundary conditions at t=0 are

wx wt wy wt wz wt

vx

0

vy

0

vz

17m / s

Other three boundary conditions are x, y and z coordinates at t=0, which are given by

(20)

Maneuvering Atoms for Lithography…

x

N H cos(I )

y

N H sin(I )

z

z0

81

(21)

where N=-2000, -1900, -1800, ........, 1800, 1900, 2000; H=10-9, I=0,0.174,0.348,...., 2S. Pictorially initial arrangement of the atoms in the transverse plane x-y is shown in Figure 21. For divergent atomic beams three boundary conditions at t=0 are

vx vy vz v

v sin(T d ) 1.41 v sin(T d ) 1.41 v cos(T d )

(22)

(vx 2  v y 2  vz 2 )1/ 2 17m / s

where Td is the divergence angle. The other three boundary conditions are same as shown in Figure 21 (and given by Eq. 21) apart from the difference that now from each location 5 atoms are launched in coaxial cones (as shown in Figure 22) with a maximum cone angle (or Tdmax ) of 5 mrad.

Figure 21. Cross sectional view of initial positions of rubidium atoms in the beam.

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Kamlesh Alti and Alika Khare

Figure 22. Launching coaxial cones of atomic beams for maximum divergence angle dmax.

4.3. Results and Discussions The sequence of computed lithographic patterns for collimated cold rubidium atomic beams for interaction time of 82.3 Ps and 88.2 Ps (corresponding interaction length of 1400 Pm and 1500 Pm respectively) are shown in Figure 23.

Figure 23. Final positions of initially collimated rubidium atoms after interacting with interferometric beams for an interaction length of a. 1400 m and b. 1500 m.

It can be seen from the both the lithographic patterns that the atoms have relocated themselves towards the intensity peaks. The effect of divergence of the atomic beam is also considered . For this, the starting configuration of atomic beam is shown in Figure 21 and Figure 22. Figure 24 shows the lithographic patterns for such configuration for interaction length of 1400 Pm and 1500 Pm, corresponding to interaction time of 82.3 Ps and 88.2 Ps for cold rubidium atoms. These lithographic patterns look quite similar to as that with the case of collimated atomic beams. In this case also one can see the relocation of atoms around the intensity peaks. The maximum density in the center of Figure 23 – 24 is because of the maximum density of the atom in the center of Figure 21. If a uniform distribution of atoms in

Maneuvering Atoms for Lithography…

83

the beam is considered then the focused pattern will also show the same density of atoms in each spot.

Figure 24. Final positions of divergent rubidium atoms after interacting with interferometric beams for an interaction length of a. 1400 m and b. 1500 m.

5. Conclusion The computed results for the Gaussian and Standing wave configuration shows Multiple focusing of the atomic beam for the longer interaction time with the light field Interaction time required for the focusing at any given intensity is constant and is independent of velocity of the atom. At higher intensities of light field, atoms are tightly focused with better contrast. The computed results for the proposed new configuration of matrix of micro-ovens yield interesting patterns with periodicities much less than /2 in presence of single potential from a TEM00 mode laser. The proposed configuration is relatively simple because it doesn't require any conditioning of light potential as directly TEM00 mode is used. The idea of using arrays of atomic beams in TEM00 mode appears to be a promising configuration for the generation of periodic nano structures. The computed lithographic patterns for a single atomic beam traveling in counter propagating two dimensional periodic potentials generated via interference of four optical beams shows the accumulation of atoms towards the center of the potentials. Thus the proposed configuration may find its application in micro fabrication. The trajectories simulated in this paper are for rubidium atoms but this configuration is very much applicable to other atoms also.

References [1]

Claude Cohen-Tannoudji. Atom-Photon Interaction, John Wiley and Sons, New York, (1992).

84 [2]

[3]

[4] [5] [6] [7] [8] [9] [10]

[11]

[12]

[13]

[14]

[15]

[16] [17]

Kamlesh Alti and Alika Khare J. E. Bjorkholm, R. R. Freeman, A. Ashkin, D. B. Pearson, “Observation of Focusing of Neutral Atoms by the Dipole Forces of Resonance-Radiation Pressure,” Phys. Rev. Lett., 41, 1361-1364 (1978). J. E. Bjorkholm, R. R. Freeman, A. Ashkin, D. B. Pearson, “Experimental observation of the influence of the quantum fluctuations of resonance-radiation pressure,” Opt. Lett., 5, 111-113 (1980). T. Sleator, T. Pfau, V. Balykin, J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B, 54, 375-379 (1992). G. Timp, R. E. Behringer, D. M. Tennant, J. E Cunningham, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett., 69, 1636-1639 (1992). J. J. McCelland, R. E. Scholten, E. C. Palm, R. J. Celotta, “Laser-focused atomic deposition,” Science, 262, 877-880 (1993). F. Lison, H.-J. Adams, D. Haubrich, M. Kreis, S. Nowak, D. Meschede, “Nanoscale atomic lithography with a cesium atomic beam,” Appl. Phys. B, 65, 419-421 (1997). R. W. McGowan, D. M. Giltner, S. A. Lee, “Light force cooling, focusing, and nanometer-scale deposition of aluminum atoms,“ Opt. Lett., 20, 2535-2537 (1995). R. Ohmukai, S. Urabe, M. Watanabe, “Atom lithography with ytterbium beam,” Appl. Phys. B, B77, 415-419 (2003). E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, K. A. H. van Leeuwen, “Atom lithography of Fe,“ Appl. Phys. Lett., 85, 4493-4495 (2004). G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett., 85, 3842-3844 (2004). Kamlesh Alti and Alika Khare, “Simulated lithographic pattern for periodic arrays of atomic beams focused with a single atomic lens,” International Journal of Nanoscience, 5 (1), 145 (2006). Alika Khare, Kamlesh Alti, Susanta Das, Ardhendu Sekhar Patra and Monisha Sharma, “Application of laser matter interaction for generation of small sized materials,” J. Radiation and Chemistry, 70 (4-5), 553 (2004). Kamlesh Alti, Ardhendu Sekhar Patra and Alika Khare, “Two dimensional periodic potentials via multiple beam interferometry for atom lithography,” Journal of Microlithography Microfabrication and Microsystem, 5, 023005 (2006). Kamlesh Alti and Alika Khare, “Arrays of discrete atomic beams for sub-/2 lithography via dipole force,” Microelectronics Engineering, 83 (10), 1975-1980 (2006). Kamlesh Alti and Alika Khare, “Sculpted pulsed indium atomic beams via selective laser ablation of thin film,” Laser and Particle beams, 24, 469-473 (2006). A.S Patra and A Khare, “Interferometric Array Generation,” J. of Optics and Laser Technology, 38, 37-45 (2006).

In: Progress in Nonlinear Optics Research Editors: Miyu Takahashi and Hina Goto, pp. 85-114

ISBN 978-1-60456-668-0 © 2008 Nova Science Publishers, Inc.

Chapter 3

HIGHLY EFFICIENT NONLINEAR FREQUENCY CONVERSION SCHEMES FOR COMPACT FEMTOSECOND ERBIUM FIBER LASERS: FROM THE NEAR ULTRAVIOLET THROUGH THE ENTIRE VISIBLE INTO THE INFRARED Konstantinos Moutzouris 1, Florian Adler1, Florian Sotier1, Daniel Träutlein1, Alexander Sell1, Elisa Ferrando-May2 and Alfred Leitenstorfer1 1

Department of Physics and Center for Applied Photonics, University of Konstanz, Universit tsstr. 10, 78464 Konstanz, Germany 2 Department of Biology and Center for Applied Photonics, University of Konstanz, 78457 Konstanz, Germany

Abstract Recent advances in fiber lasers unlock the potential for novel nonlinear devices and generation of new wavelengths. In this review article we examine the application of five nonlinear frequency conversion schemes to a femtosecond fiber laser system. We discuss in detail particular issues concerning the design, choice of material and performance characteristics for each converter. Exploiting single-pass and cascaded harmonic generation as well as sum frequency mixing, a nearly continuous spectral coverage from the near ultraviolet to the near-infrared is achieved. This widely tunable radiation appears in the form of ultrashort pulses with a repetition rate in the range of 100 MHz and average power levels ranging between 1 mW and more than 100 mW. We also report on a concept for broadband-to-narrowband frequency doubling of femtosecond pulses. Unexpectedly high efficiencies are predicted owing to intra-pulse sumfrequency mixing. Experimental proof is provided via demonstration of frequency doubling of sub-30-femtosecond near infrared pulses, with spectral bandwidths exceeding 100 nm, into the visible. Efficiencies higher than 30% are achieved. Spectral narrowing by a factor of up to

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Konstantinos Moutzouris, Florian Adler, Florian Sotier et al. more than 50 and an increase in spectral intensity by one order of magnitude are observed simultaneously without gain saturation. The performance characteristics of this relatively simple device, such as the wide coverage of the visible spectrum, as well as the coexistence of several perfectly synchronized outputs, have attracted attention for its immediate use in various scientific and technological fields, including bioimaging, ultrafast spectroscopy and optical frequency metrology.

1. Introduction The first demonstration of a prototype fiber resonator [1] followed shortly after the operation of the first free-space lasers [2]. It was soon realized that fiber laser geometries bear significant advantages over other alternatives, such as conventional solid state or gas lasers. These advantages include a higher degree of compactness and reliability, higher pump conversion efficiencies, better spatial beam profiles and avoidance of thermal effects that are common with laser crystals. For several decades, however, fiber lasers met applications nearly exclusively in the telecommunications sector. he restricted use of fiber lasers was due to the limited output power (in the range of few milliwatts) that could be generated. This problem associated with early fiber sources was addressed successfully in recent years, to such an extent that today these devices are competing thin disc and other high-power solid state lasers. Several technological breakthroughs helped in this direction. On one hand, the implementation of optical fiber amplifiers [3] allowed the external enhancement of the weak signals from fiber oscillators. Additionally, the initial side-pumping design (which compromised the efficiency and leaded to multi-mode operation), was gradually replaced by longitudinal [4] and cladding-pumping [5] geometries. Increasing the fiber laser output was also made possible due to significant developments in laser-diodes. These devices, which deliver up to hundreds of Watts, are almost exclusively used for pumping fiber lasers either through air, or directly via fiber. Fiber lasers typically exploit the waveguiding effect in an optical fiber, in which the active core is doped with rear-earth ions [6]. One common dopant is neodymium (Nd3+) [7]. This system is a four-level active medium, thus associated with low threshold, absorbing light at 808 nm and exhibiting a primary emission wavelength near 1060 nm. More recently, ytterbium (Yb3+) has attracted a lot of attention [8]. Ytterbium has a broad emission bandwidth between 1000 nm and 1100 nm, it may be pumped at 975 nm, and allows for an unprecedented scaling of output power. Yet, the most widely used rear-earth material in fiber technologies is erbium (Er3+). Erbium-doped fiber lasers (EDFL) are less efficient than ytterbium based systems. However, they emit radiation at the eye-safe wavelengths near 1540 nm, corresponding to minimum transmission loss for glasses. The extensive applications of EDFL’s in the telecommunications industry establish them as the maturest fiber sources. Longer infrared wavelengths may also be accessed by use of other dopants, such as Thulium (Tm3+), which emits in the spectral range between 1800 nm and 2100 nm [9]. Fiber lasers are proven excellent sources for the production of pulsed radiation. Highenergy pulses with nanosecond time durations have been generated via passive Q-switching by use of semiconductor saturable-absorber mirrors [10]. Actively Q-switched fiber lasers based in acousto-optic modulators (AOM) [11], as well as hybrid configurations exploiting simultaneously AOM and fiber backscattering effect have been reported [12]. In recent years, ultrashort pulses with durations in the pico- and femtosecond range are produced

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conveniently by mode-locked fiber lasers. Saturable absorbers [13], nonlinear fiber loop mirrors [14], and electro-optic modulators [15] have been employed for mode-locking fiber sources. Nevertheless, the shortest pulses from a fiber laser have been produced by use of nonlinear polarization rotation [16], a technique introduced by Tamura et al in 1993, also referred to as additive-pulse mode-locking. The continuously increasing output power levels, the variable output wavelengths provided by different dopants and the ability to generate short pulses explain the key role of fiber devices in current laser technology. Yet, the modest wavelength tunability and the nearly exclusive operation in the near infrared set a limit to the practical use of these sources. Wide spectral coverage at shorter wavelengths (most interestingly in the visible) has recently been reported via exploitation of third-order nonlinearities in photonic crystal [17] and tapered fibers [18]. However, this class of devices is associated with modest spectral brightness, relatively high noise and low temporal coherence. A promising alternative involves the use of (2) nonlinear frequency conversion schemes. Efforts in this direction include frequency doubling and tripling (via sum frequency of two independent sources) of nanosecond pulses from erbium fiber lasers [19]. Femtosecond EDFL’s have also been efficiently frequency doubled to generate ultrashort pulses with up to 8.7 mW average power at a wavelength of 771 nm [20-21]. In the first part of this article, we review a complete set of recent experiments on upconversion of a mode-locked erbium fiber laser [22-24]. By use of second harmonic generation and sum frequency mixing in several stages, we obtain an overall spectral coverage from the near ultraviolet through the entire visible to the near infrared. Ultrashort pulses with average power levels ranging between 1 mW, and more than 100 mW, are collected in this wide spectral band. Our compact, stable, and relatively inexpensive fiberbased system competes with other nonlinear devices operating in the same spectral region, such as frequency up-converted infrared lasers [25-30], as well as parametric devices pumped in the ultraviolet [31-33]. In the second section, we introduce a novel concept for highly efficient broadband-tonarrowband frequency doubling of ultrashort pulses. This technique allows for simultaneous wavelength conversion and spectral filtering under conditions of large group-velocitymismatch (GVM), i.e. in “thick” nonlinear crystals. This goal is achieved via an additional phase-matched sum-frequency interaction between the wavelength components coexisting in the fundamental pulse. Contrary to simple linear filtering techniques, like those employing slits [34], our approach utilizes the entire spectral content of the fundamental pulses, thus leading to high conversion efficiencies. Today, infrared mode-locked lasers (including fiber sources) providing femtosecond pulses (typically, 10 fs to 100 fs) are commonplace. The proposed method permits efficient doubling of these sources for the generation of tunable visible radiation with narrower spectra and longer durations (in the femto- and picosecond range). Such visible pulses, not available directly from laser oscillators, are of great importance for numerous applications, ranging from condensed-matter spectroscopy via biological imaging to industrial test devices.

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2. Erbium Fiber Laser Pumped Nonlinear Converters

a) The Pump Fiber Source The pump fiber system (oscillator – twin amplifiers – highly nonlinear fiber) is schematically illustrated in Figure 1. The basic design concepts of the device have been discussed in detail previously [35-38]. It comprises a diode-pumped erbium-fiber stretched-pulse ring oscillator. The pump diode emits 175 mW of continuous-wave (cw) radiation at 980 nm. Mode-locking is achieved via nonlinear polarization rotation [16]. To enable this scheme, two fiber-to-freespace coupling lenses are employed, permitting the introduction of an intracavity sequence of polarizing elements (quarter-and half-wave plates, and polarizing beam splitter). Er:fiber oscillator

st

1 Aplifier

Diode #1

Diode #4

WDM

M

WDM Er:f

FI PBS O O

CL

WDM 80/20 coupler

Er:f

CL O O

Diode #5

CL

50/50 splitter

M M‘

2

nd

Si-prism pair

Aplifier

Diode #2

WDM

Glass-prism pair

M

WDM

M

f

Er:f

v

M‘ Diode #3 CL

M‘

M‘ HNLF

f

M

Figure 1. The fiber pump source. WDM: Wavelength Division Multiplexer; /2 and /4: Half and Quarter Wave Plates; FI: Faraday Isolator; PBS: Polarizing Beam Splitter; CL: Coupling Lens; M: Gold-coated mirrors; M’: Pick-off or flipper mirrors. HNLF: Highly nonlinear fiber; Er:f: Erbiumdoped fiber.

In the free space region a Faraday isolator is also inserted to ensure unidirectional operation. Typically, the fiber laser provides a sub-3 mW average power in the form of 60 fs pulses centered at a wavelength of 1.55 m and operates at a repetition rate of approximately 100 MHz. The laser radiation is split into two identical components, which are then used to seed two parallel erbium-doped fiber amplifiers (EDFAs) [17]. Both EDFAs are bi-directionally

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pumped by a total of four diode lasers. Each pump diode emits 850 mW of cw power at 980 nm. After leaving the oscillator, the seed pulses are temporally stretched in a standard single-mode fiber with a group-velocity-dispersion (GVD) of = -0.023 ps /m. The fiber used in the amplifiers is highly doped with erbium ions and has a length of less than 2 m. Owing to the positive GVD of the erbium fiber, the pre-chirped pulses shorten during amplification. The synchronous and mutually coherent outputs from the two amplifiers (thereafter, referred to as fundamental frequency f) are passed through silicon prism pairs for dispersion compensation. Each branch of the pump source delivers sub-80 fs pulses centered at 1.55 m with a spectral full-width-at-half-maximum (FWHM) bandwidth of approximately 60 nm. These values correspond to a time-bandwidth product of   0.6 indicating nearly transform limited pulses. Average power levels of 300 mW are obtained from each arm. In combination with a repetition rate around 100 MHz, this value translates into pulse energies of 3 nJ. The pulse train from the second of the twin EDFAs is coupled into a highly-nonlinear fiber (HNLF). This device is a single-mode fiber with a reduced mode-field diameter of 3.7 m and a zero dispersion wavelength of 1.52 m, with a length of approximately 8 cm. The HNLF generates an octave-spanning supercontinuum [39]. Most notably, the output of the HNLF comprises a continuously and independently tunable near-infrared component with center wavelengths between 1.05 m and 1.4 m (thereafter, denoted as variable frequency v). The peak wavelength is selected by adjusting the fundamental pulse prechirp with the Si prism sequence. This tunable component amounts to an average power of up to 30 mW and exhibits spectral FWHM bandwidths in excess of 100 nm. After recompression in an additional glass prism pair the temporal duration of the tunable near-infrared pulses is measured to range between 30 fs and 13 fs, indicating nearly transform-limited pulses. Typical spectra and temporal profiles of the two components of interest (f and v), collected, respectively, by use of an infrared spectrometer and frequency-resolved-opticalgating (FROG), are shown in Figures 2 and 3. It is worth mentioning that the specifications of these two perfectly synchronized outputs may be compared to the signal and idler waves from a synchronously pumped optical parametric oscillator (OPOs) [40]. This observation reveals the potential of our source for the construction of several nonlinear frequency converters, in analogy to the great number of schemes available for mixing the signal and idler outputs from OPOs.

b) The Nonlinear Converters Figure 4 provides a layout of the five mixing configurations employed for up converting the fiber source wavelengths. The fundamental harmonic (f) (FH) from the first amplifier branch is passed through a frequency doubler to generate the second harmonic (SH) output (2f). The SH is then directed towards a cascaded doubler, to lead to the fourth harmonic component (4f). Alternatively, the SH may be mixed with the FH from the second amplifier branch in a sum-frequency crystal, to give the third harmonic component (3f). The tunable infrared output (v) of the HNLF may also be doubled to provide tunable visible pulses at frequencies (2v).

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Intensity (arb. units)

Intensity (arb. units)

0.8 0.6

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2

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0

0.4 56 fs

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-2

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1.0

200

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0.4 0.2 0.0 1400

1450

1500

1550 1600

1650

1700

1750

Wavelength (nm)

Figure 2. A typical fundamental spectrum from a nonlinear Er:amplifier with a central wavelength near 1550 nm. It exhibits a spectral bandwidth of approximately 60 nm. The inset shows a characteristic FROG trace, indicating a pulse duration as short as 56 fs.

Intensity (arb. units)

500 400 300

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50

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600

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200 100 0

1000

1100

1200

1300

1400

1500

1600

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Figure 3. A selection of typical spectra at the output of the nonlinear fiber. This infrared component (v) has a variable central wavelength from 1050 nm to 1400 nm. Spectral bandwidths from 80 nm to 130 nm are observed. Inset: FROG trace at an indicative wavelength in the middle of the band near 1250 nm. Solid line: Temporal intensity envelope of sub-30 fs pulses. Dashed line: Temporal phase profile.

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 = 780 nm PUMP LASER

Oscillator

Amplifier 1

f

f + f

 = 390 nm

2f + 2f

 = 520 nm

f

2f + f

Amplifier 2

HNLF

 = 520 nm – 700 nm

v  v+  v

 = 460 nm – 500 nm

2f + v

Figure 4. A map of the five nonlinear frequency converters. From top: The fundamental frequency from the first arm is doubled, and quadrupled in a cascaded process in the following. The second harmonic from one arm is then mixed with the fundamental from the second arm to generate the third harmonic. One more converter doubles the tunable infrared component at the output of the HNLF. Finally, the second harmonic and the tunable infrared components are mixed to generate tunable blue pulses. For simplicity, some optical elements used in our apparatus are not shown here. These include: half-wave plates used for control of the polarization of the input beams; input focusing, and output collimating, lenses; flipper mirrors to enable operation of the desired converter; dichroic mirrors, and computercontrol translation stages, for spatial and temporal overlapping of different input components, respectively; Various filters for isolating the outputs of the converters.

Finally, another sum-frequency crystal may be used to mix the tunable component (v) from the second branch, with the SH from the first one, to provide blue wavelengths at frequencies (2f + v). For simplicity, some optical elements of our apparatus are not shown in Figure 4. These include: half-wave plates used for control of the polarization of the input beams; input focusing, and output collimating, lenses; flipper mirrors to enable operation of the desired converter; dichroic mirrors, and computer controlled translation stages for spatial and temporal, respectively, overlapping of different input components; Various filters for isolating the outputs of the converters. More explicit details on the geometry may be found on references [22-24]. Four out of five nonlinear converters comprise electrically poled, MgO-doped LiNbO3 (MgO:LN) crystals with gradually varying grating periods  (fan-out design) [41,42]. This material offers a large nonlinear coefficient and a wide optical transparency in the wavelength range between 350 nm and 4500 nm. The orientation of the nonlinear crystals was chosen for the conventional e + e o e interaction with all waves polarized along the optical axis in the z direction. This geometry utilizes the maximum nonlinear coefficient d33 and avoids spatial walk-off effects (which are common with birefringent crystals) leading to nearly perfect Gaussian beams. The facets of the crystals were optically polished. The fan-out design

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ensures that the required period for a quasi-phase-matched (QPM) process is readily available by linear translation of the crystals, without stringent and time wasting requirements in temperature control. All crystals were mounted on a copper plate connected to a heating resistor; elevating the temperature to 60 C ensured that photorefractive damage is avoided. Crystals with typical lengths of 2 mm and 1 mm were employed. These values are larger than corresponding values of the group-velocity-mismatch characteristic length. This fact leads to higher conversion efficiencies and will be discussed in more detail in the following section. The QPM structures had a duty cycle of 0.5 and an order m = 1, with the exception of the quadrupling crystal which used m = 0.5 Avoiding the latter efficiency-limiting factor was difficult in the case of the quadrupling crystal, since it would require a sub-2.5 m grating period. Short grating period resulting in increased cost, drove us to opt a 2-mm-long -barium borate (BBO) crystal for the last sum frequency mixer used for tunable blue light generation. This relatively cheap material is particularly suitable for the blue spectral region. The nonlinear crystal was antireflection coated for the near-infrared. It was cut at angles  = 240 and  = 00. This geometry allows type-I phase matching of sum frequency mixing between the tunable near-infrared (v) and the SH (2f) input components with only a 20 (internal) angle tuning of the crystal. All important specifications of the crystals in use are summarized in Table 1.

Table 1. Specifications of nonlinear crystals employed in the five different converters Converter

Nonlinear medium

Available gratings / angles

Crystal length

f + f 2f + f 2f +2f

MgO:LiNbO3 MgO:LiNbO3 MgO:LiNbO3

v + v

MgO:LiNbO3

2f + v

-BaB2O4

 = 14.9– 21.0 m  = 5.5 – 6.8 m  = 5.5 – 6.8 m  = 6.45–7.9 m  = 7.25–11.72 m  = 9.73–15.41 m  = 240,  = 00

2 mm 1 mm 1 mm 1 mm 2 mm 1 mm 2 mm

c) Second Harmonic Output Figure 5 illustrates a selection of typical spectra along with FROG temporal traces, obtained at the output of the first frequency doubler. A maximum power of 120 mW was collected at a central wavelength of 770 nm. Combined with an available fundamental average power of 300 mW, this value leads to an overall conversion efficiency (defined as P2 / P) of 40%. By linear translation of the crystal the SH output was tunable in a 50 nm wide band between 750 nm and 800 nm. The FWHM bandwidth of the SH signal varied from 2.5 nm to 6 nm. These values are in reasonable agreement with theoretical calculations in the plane wave approximation.

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20 Intensity (arb. units)

15

10

40

1.0 0.8 0.6 0.2 0.0

-150

0

150

0

time (fs)

51 mW 45 mW

5

0 740

20

142.7 fs

0.4

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Spectral Intensity (mW/nm)

120 mW

760

780

800

820

840

Wavelength (nm) Figure 5. Typical spectra at the second harmonic wavelength. The tuning range is between 750 and 800 nm. Spectral power is shown in absolute values. Inset: FROG measurement for the second harmonic output indicating 143 fs chirp-free pulses. Solid line: Temporal intensity envelope. Dashed line: Temporal phase profile.

The narrow generated bandwidths imply that the doubling process utilizes only a small fraction of the broad fundamental spectrum, an observation contradicting the large conversion efficiencies. This effect, which will be discussed in the next section, can be understood via sum-frequency-mixing (SFM) contributions from input frequency components which are positioned symmetrically with respect to the fundamental central frequency. The dependence of the generated average power on the fundamental average power was studied, revealing significant gain saturation when input power exceeded ~150 mW. Durations ranging between ~140 fs and ~250 fs were measured at the SH without any external elements for dispersion compensation (see inset in Figure 5). The measured pulse durations are significantly shorter than theoretical estimates considering unfocused beams and group-velocity-mismatch. We attribute the pulse shortening in our experimental results to focusing effects.

d) Third Harmonic Output Figure 6 shows a typical spectrum collected at the output of the frequency tripler stage, as well as the respective temporal trace observed via second-harmonic autocorrelation. Pulse durations of 285 fs (assuming sech2 profiles) are determined at a central wavelength of 520 nm. Combined with a spectral FWHM bandwidth of 2.18 nm, this value leads to a timebandwidth product of  ~ 0.69. Temporal broadening of the third harmonic output is predominantly due to large GVM between the fundamental and third harmonic components (3 ps/mm).

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53mW

12 8

Intensity (arb. units)

Spectral power (mW/nm)

20

FWHM 2.18 nm

3 2

440 fs

1 0

-0.4

0.0

0.4

Dealy (ps)

4 0 510

515

520

525

530

535

540

545

Wavelength (nm) Figure 6. Typical spectrum at the third harmonic wavelength of 520 nm. It exhibits a FWHM bandwidth of 2.18 nm. Spectral power is shown in absolute values. Inset: Autocorrelation trace with 440 fs width. Assuming sech2 pulses, this value reveals pulse durations of 285 fs.

The average power in the green reaches 55 mW. In this case both amplifiers are employed, and thus the total initial power at the fundamental wavelength is 600 mW. Therefore, we calculate an overall conversion efficiency (defined as P3 / P) of 9.2%. However, the quantum efficiency with which the second harmonic pump beam is converted (defined as N3 / N2, where N indicates the number of photons at the corresponding frequency) exceeds 30%. Accounting for losses in the uncoated facets of the crystal, this value indicates that average power levels as high as 75 mW may be generated at the third harmonic by further optimization. A linear output power scaling was observed (with increasing power on each one of the pump beams), demonstrating the overall quadratic nature of the process and lack of gain saturation.

e) Fourth Harmonic Output Figure 7 shows a typical output spectrum at the fourth harmonic. It is centered at 390 nm and exhibits a FWHM bandwidth of 1.5 nm. Te corresponding output average power is 6 mW. This amount of generated power in the ultraviolet, in combination with 120 mW of available input power at the SH, translates into an overall efficiency for the frequency quadrupler (defined as P4 / P2) of 5%. Due to lack of crystal coatings, we believe that conversion efficiency may be further increased up to 7%. This value constitutes our estimate for the current internal efficiency (i.e., the ratio of the generated to the input power inside the crystal).

Highly Efficient Nonlinear Frequency Conversion Schemes…

Spectral power (mW/nm)

3

95

6mW

2 FWHM 1.53 nm

1

0 384

387

390

393

396

Wavelength (nm) Figure 7. Typical spectrum at the fourth harmonic. It is located at a wavelength of 390 nm and exhibits a FWHM bandwidth of 1.53 nm.

However, the most significant performance-limiting factor in this process is the use of third order QPM. Evidently, a first-order QPM interaction would increase the conversion efficiency by a factor of 9 resulting in the generation of up to 50 mW of UV radiation. Although the fabrication of first order QPM gratings for this particular interaction involves a great amount of technical complexity, recent developments in short-period poling technologies [43, 44] indicate that improvement of our current set-up is a realistic possibility. Temporal measurements in the ultraviolet at this power levels were proven impossible with available laboratory equipment. However, accounting for a GVM as large as 2.5 ps/mm, we estimate that the generated fourth harmonic pulses are well within the 1-picosecond range.

f) Tunable Green-to-Red Output By frequency doubling of the tunable near-infrared component from the HNLF (employing one amplifier only) widely tunable radiation in the visible from green to red is obtained. Two nonlinear crystals, with lengths of 2 mm and 1 mm, respectively, were used. Figure 8 shows a selection of typical visible spectra throughout the tuning range of our device. The observed FWHM bandwidths (e.g., 3.55 nm bandwidth at 680 nm for a 1 mm crystal and corresponding input bandwidths exceeding 100 nm) indicate a spectral filtering by a factor of up to 50. A maximum average visible power of up to 10 mW (7.6 mW) was collected at 630 nm (620 nm) for the 2 mm (1 mm) long crystal and less than 30 mW of pump power. These values translate into to a maximum overall efficiency larger than 30%.

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Pulse duration (fs)

Spectral Power (mW/nm)

5 4 3

L = 2 mm L = 1 mm

1200 800 400 0

520

560

600

640

Wavelength (nm)

2 1 0 510

540

570

600

630

660

690

720

Wavelength (nm) Figure 8. A selection of spectra after frequency doubling of the tunable near infrared component from the HNLF. A tuning range from 520 nm in the green to 700 nm in the red is observed. Spectral intensity is shown in absolute values. Three nonlinear crystals were used to cover the entire band, with lengths of 1 mm and 2 mm. Inset: Pulse duration for various visible wavelengths and different crystal lengths. Temporal measurements were carried out via second-order nonlinear autocorrelation.

The high conversion efficiencies combined with the strong spectral filtering effect, manifest the strong presence of a more-general mixing process. The dependence of the visible power on the input power was also examined. The quadratic character of the interaction was verified, and no saturation was observed. A typical set of temporal measurements, based on autocorrelation and the assumption of Gaussian pulses, for different wavelengths and crystal lengths is depicted as inset in Figure 8. The pulses broaden as the wavelength decreases (owing to an increase in GVM) and, as expected, exhibit a nearly linear dependence on interaction length. Durations of 1000 fs (600 fs) are observed for the 2 mm (1 mm) crystal in the green. These values are approximately one half of our estimates based on plane wavefronts and GVM considerations alone. The discrepancy may be attributed to focusing effects. The time-bandwidth product assumes values from 0.36 to 1.4 (0.7 to 0.9) for the 2 mm (1 mm) long crystal.

g) Tunable Blue Output A free-space-coupled sub-nm resolution spectrometer is used to collect spectra at the output of the last nonlinear converter. A selection of typical spectra in the blue is shown in Figure 9. The central wavelength is tunable between 460 nm and 500 nm, while FWHM bandwidths take values between 2.6 nm and 4.9 nm. At least 1.5 mW of average output power was collected throughout the entire blue band, and a maximum power of 1.8 mW was observed at a central wavelength of 480 nm.

Normalised intensity (arb. units)

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1.0 0.8 0.6 0.4 0.2 0.0

460

470

480

490

500

510

Wavelength (nm) Figure 9. A selection of typical spectra at blue wavelengths from 480 nm to 500 nm. Corresponding average power exceed 1.5 mW. Spectral FWHM bandwidths vary from 2.6 nm to 4.9 nm.

The corresponding input power levels, measured before the crystal input facet, were 23 mW and 80 mW for the tunable near-infrared (v) and SH (2f) components, respectively. Taking into account an additional 6% loss of blue power due to reflection at the crystal output facet, these values indicate an overall photon efficiency (defined as the percentage of photons at a frequency (v) converted into the blue) of 21%. The output power was also measured for variable input power levels. The observed quadratic scaling of the blue light intensity was a clear confirmation of lack of gain saturation. Direct temporal measurements of the blue pulses were proven unattainable. However, a combination of theoretical estimates based on GVM considerations and indirect experimental measurements of blue power as a function of input pulse delay, provided strong arguments towards sub-300 fs temporal durations.

h) Summary of Device Performance The performance characteristics of the device are depicted in Table 2. Wavelength ranges, power levels and temporal durations (either measured or theoretically estimated) are specified. All outputs have reference to the same seed pulses from the fiber oscillator, and thus are perfectly synchronized with a repetition rate of approximately 100 MHz. An overall spectral coverage from 390 nm in the ultraviolet to 770 nm in the near infrared is obtained, with average power levels well in the mW range (up to more than 100 mW). The last columns of the table include the specifications of available output pulses from the fiber source itself, providing a further extended spectral coverage up to 1.55 m and containing four consequent harmonics.

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Table 2. Performance characteristics of the fiber pump source and the nonlinear converters. Spectral coverage, average power and pulse durations are indicated for all output components of our device, including the two laser outputs shown in the last columns Converter

Wavelength range

Average power

Pulse duration

f + f 2f + f 2f +2f v + v 2f + v f v

770 nm 520 nm 390 nm 520 nm – 700 nm 460 nm – 500 nm 1550 nm 1050 nm – 1400 nm

120 mW 55 mW 6 mW 10 mW 1.5 mW 300 mW 30 mW

> 150 fs 285 fs > 1000 fs 300 – 1000 fs 300 fs 80 fs 30 fs

3. Highly Efficient Broadband-to-Narrowband Frequency Doubling a) Review on Second Harmonic Generation in Thick Media Second harmonic generation (SHG) is a widely employed technique for extending the spectral coverage of available sources. As a tool, SHG is particularly appropriate in combination with mode-locked lasers emitting radiation in the form of ultrashort pulses, and offering high peakintensities that are advantageous for F(2) interactions. Ultrashort pulses, however, with large spectral bandwidth impose stringent limitation in choice of nonlinear material thickness. This well-known effect is due to the linear decrease in phase-matching acceptance bandwidth with increasing interaction length. In the time domain, this problem is dealt with considering the group-velocity-mismatch. Therefore, an optimum crystal thickness L exists, corresponding to the distance over which the crystal acceptance equals the pump spectral bandwidth. For most nonlinear materials, femtosecond pulses and within the optical frequencies, L lies in the range ~ 100 m to ~ 1000 m. Exceeding this characteristic value results in a compromise in conversion efficiency with simultaneous temporal broadening and spectral narrowing of the SHG field. Unfortunately, the use of thin crystals necessitates tight focusing and gives rise to various complications, the most important of which are optical damage, excitation of higher order nonlinearities, and chromatic aberrations. To avoid such problems, numerous schemes have been proposed for ultrashort pulse frequency conversion in media with thickness larger than the characteristic acceptance (or GVM) length L. In the most direct approach, known as spectrally non-critical phasematching, pairs of fundamental and second harmonic wavelengths are identified for which GVM vanishes naturally due to material dispersion [20], [45-47]. Despite its simplicity, this method lacks wavelength flexibility. A more versatile, yet alignment sensitive, route to cancel GVM involves the controlled localization of the interaction inside the nonlinear medium for the different frequency components. This goal has been achieved by either inducing a tilted wave front in the pump beam with use of dispersive elements such as gratings and prisms [4855], or by engineering an axial variation of material properties such as temperature in case of

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birefringent phase-matching (BPM) [56] and longitudinal patterning of gratings in quasiphase-matching [57-66]. A conceptually analogous scheme exploits a series of independently angle-tuned crystals, each converting complementary parts of the broad input spectrum [67, 68]. Group-velocity matched SHG has also been demonstrated in a non-conventional type-I QPM geometry, exploiting the off-diagonal nonlinear coefficient d31 of MgO-doped periodically-poled lithium-niobate (MgO:PPLN) by proper choice of doping concentration [69]. Recently, several works have indicated that broadband conversion is possible in thick nonlinear media under conditions of large GVM and extreme focusing [70-73]. In this case, acceptance increase is due to the strong divergence inside the nonlinear medium, introducing an effective interaction length far shorter than the physical thickness of the crystal. Other special configurations include type II processes with temporally-pre-delayed and orthogonally polarized FH pulses [74, 75], Cerenkov SHG in QPM waveguides [76], as well as multimode field interaction [77]. In general, the spectral bandwidth of the SH output is twice (half) that of the FH acceptance bandwidth in frequency (wavelength) units. Hence, all the schemes for artificial enhancement of crystal acceptance summarized above result in broadband SH output fields. These broadband-to-broadband frequency-doubling configurations are of interest for all applications requiring short pulses. There exist, however, scientific and technological areas such as atomic physics, biophotonics and optical metrology, in which narrow-line laser radiation is often preferred. (For example, confocal microscopy requires pump sources delivering small bandwidths in order to ensure than the excitation and fluorescent signals can be distinguished at the detection stage). In this section, we discuss a novel concept for highly efficient generation of narrow output spectra by frequency doubling a broadband input signal in a thick medium. This technique exploits sum frequency mixing of spectral components coexisting in a single input pulse. The role of SFM contributions in the spectral bandwidths of nonlinear frequency doubling was realized as early as 1977 by Liu [78]. He employed a tunable, two-color, linenarrowed laser with variable spacing between the two fundamental frequencies. By use of a cesium - dihydroarsenate (CDA) crystal, he was able to generate a particular SH component via either SFM of two laser colors or direct SHG of only one. Broad side-peaks in the temperature-tuning curve were observed for SFM up-conversion, which vanished for direct SHG. In 1985, Kwok and Chiu [79] reported on frequency doubling of 1.4 nm broadband 500-picosecond pulses at a wavelength of 585 nm from a dye laser into 0.21 nm narrowband ultraviolet radiation. The employment of a thick potassium-dihydrogen-phosphate (KDP) crystal resulted in spectral narrowing of the SH output (compared to the FH) by a factor of approximately 7, and the achieved 10% conversion efficiency was interpreted as evidence of the SFM effect. Other occasions in which unexpected SHG efficiencies were observed in thick nonlinear crystals include that of Ref. [80].

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Table 3. Materials and corresponding phase-matching configurations under study. Acronyms: BBO, E-BaB2O4 (negative uniaxial); LBO, LiB3O5 (negative biaxial); KTP, KTiOPO4 (positive biaxial); PPKTP, Periodically poled KTP; MgO:PPLN, Periodically poled MgO-doped-LiNbO3 (negative uniaxial). Column with non-accessible wavelengths (lack of phase-matching) accounts only for the 0.8 Pm to 2.5 Pm spectral band of interest. The dotted horizontal line separates the three BPM (upper), from the two QPM (lower), geometries. Source of dispersion data was Ref. [82] and [83] Crystal BBO LBO KTP PPKTP MgO:PPLN

Phase-matching ooe ooe, XY plane oeo, XZ plane zzz eee

Non-accessible wavelengths ~ 1.2– 1.4 m ~ 0.8 – 1.1 m -

Tuning parameter  = 20 - 29 deg  = 0 - 49 deg = 51 - 90 deg  = 3.3 – 42.1 m  = 2.2 – 30.0 m

Our recent results (in particular, the tunable green-to-red output), constitute an unambiguous demonstration of SFM-dominated frequency doubling in MgO:LN. Nearinfrared pulses as short as 13 to 30 femtoseconds with corresponding FWHM spectral bandwidths of up to 120 nm are frequency doubled into a less than 3 nm broad spectrum in the visible. Contrary to the known performance of standard SHG, this spectral narrowing by a factor of up to 50 is accompanied by one order of magnitude increase in spectral brightness and an unprecedented conversion efficiency of higher than 30%. Following our original report of these experimental results, along with the recognition of the generalized nature of the doubling process [23], Marangoni et al [81] observed similar behaviour in periodically poled lithium tantalite. In their article a theoretical treatment of spectral and temporal effects is also included. In the following, we provide an introductory analysis of the proposed scheme, with different nonlinear materials being examined in the wavelength range between 0.8 µm and 2.5 µm. This spectral region has a two-fold significance: First, it corresponds to the emission of numerous femtosecond sources, including Ti:Sapphire, Cr:Forsterite/LiSAF, YAG, and fiber lasers, as well as various synchronously-pumped optical parametric oscillators. Second, up-conversion of near infrared light leads to the generation of visible radiation, which represents a wavelength band of particular interest for current laser research.

b) Intra-Pulse Sum Frequency Mixing We begin with assuming a broad fundamental spectrum of central frequency Z o and FWHM bandwidth GZ o , consisting of a number of longitudinal laser modes, as shown in Figure 10a. It is evident that the SH component of the central frequency may be generated via either direct SHG ( Z o  Z o 2Z o ) or SFM of two symmetric frequency components that co-exist in the pulse:

Z1  Z 2 2Z o

(1)

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where: Z1 Z o  f /2 , and Z 2 Z o  f /2 , with 0 d f d GZ 0 being the frequency spacing between Z1 and Z 2 (f also in angular frequency units). We first show that such intrapulse SFM exhibits a fundamental acceptance bandwidth ( 'Z 0

SFM

) which may be

significantly larger than that of direct SHG ( 'Z We then extend the concept of SFM-dependent frequency-doubling to any pair of fundamental components Z1 and Z 2 that coexist in the input pulse (Figure 10b). We establish that, even in the presence of wide parametric SFM, the SH output is limited to a SHG narrow spectral width nearly identical to that for exclusive SHG ( 2 ˜ 'Z 0 ). This analysis reveals the potential for highly-efficient broadband-to-narrowband frequency-doubling in thick media. SHG ). 0

Z0

a Z0-f/2

Z0 Z0+ f/2

f Z0

b

Z0 Z0+f

Z0-f1+f Z0-f1

Z0+f1 f

f1

f

Figure 10. Schematic of the generalized frequency doubling process at a frequency Z0. (a) Sumfrequency-mixing of symmetric components separated by a frequency f. This situation leads to generation of the second harmonic at 2Z0. (b) Sum-frequency-mixing of arbitrary components Z1 = Z0 + f1 and Z2 = Z0 - f1 + f. This process leads to generation of a frequency Z3 detuned by a factor f from the second harmonic. In this case, f is the distance between Z2 and the the symetric of Z1.

c) Fundamental Acceptance Bandwidth The gain of the SFM interaction described by Eq. (1), in the low-depletion plane-wave approximation, is given by the well-known expression

G 2Z 0 where

I 2Z 0 I Z1

§ l ˜ 'k · * 2 ˜ l 2 ˜ sinc 2 ¨ ¸ © 2 ¹

(2)

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102

*

ª 2 ˜ (2Z 0 ) 2 d eff2 º « » ˜ I Z2 3 ¬« H 0 c nZ1 nZ2 n 2Z0 »¼

2

(3)

is the gain coefficient. Ii is the intensity at the respective frequency ( i 2Z 0 ,Z1,Z 2 ) and l is the thickness of the nonlinear medium. deff denotes the effective nonlinear coefficient and c

H0 is the dielectric constant in vacuum, n i the refractive indices at the respective frequencies and k is the wave number. For scalar interaction, the phase the speed of light in vacuum.

mismatch parameter 'k may be written as:

'k

'k(Z 0, f ,m p )

(2Z 0 ) ˜

n 2Z 0 (m p ) c

 (Z 0  f /2) ˜

nZ 0  f / 2 (m p ) c

 (Z 0  f /2) ˜

nZ 0  f / 2 (m p ) c

 KQPM (m p )

(4)

It can be recognized that the phase-mismatch parameter depends upon the central frequency Z o , the characteristic spacing f , and a number of material-related variables, which are collectively symbolized by m p . These quantities include the angles I and T defining the propagation direction with respect to optical axis, the temperature T and the grating period / . It is to be noted that the added grating

KQPM

in Eq. 4 is only relevant for

quasi-phase-matching interactions, and vanishes in birefringent-phase-matching: kQPM 2 ˜ S / / , equivalent to first order QPM with 50% duty cycle. For a given fundamental frequency Z 0 and a suitable nonlinear material, specific values (denoted by a stress) of the material variables m p can be trivially determined for phasematching the direct SHG process (operation at degeneracy, with f

mp

m pc

c : 'k (Ic,Tc, T c, /)

'k(Z 0, f

0,m p c ) 0

0 and Z1 Z 2

Z 0 ): (5)

Having set these values, one can define the normalized conversion efficiency of the SFM process with respect to direct SHG via:

nnorm

nnorm (l , Z 0 , f , m p c )

ª l ˜ 'k (Z , f , m c ) º 0 p » sinc 2 « « 2 » ¬ ¼

(6)

In a next step, an effective acceptance bandwidth for the SFM interaction is defined as the value of the frequency spacing f f c, for which the normalized efficiency n norm acquires a value of 0.5:

'Z 0SFM

'Z (l,Z 0 )

f c: n norm (l,Z 0 , f c,m p c ) 0.5

(7)

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The preceding equation indicates that this effective SFM acceptance bandwidth depends upon crystal length and fundamental frequency. It may be readily calculated along the line described above and directly compared with the standard fundamental acceptance bandwidth for direct SHG given by:

'Z 0SHG

5.57 ˜ c w ˜ Z 0 ˜ nZ 0  n2Z 0 2 ˜ l wZ 0

> 

@

(8)

It is interesting to point out that effective acceptance for SFM depends in a more complicated fashion upon crystal length compared to the usual inverse scaling for direct SHG. Later on this section, we apply the above analysis to different materials and show that, in some cases, SFM is significantly more broadband than SHG.

d) Second Harmonic Bandwidth In the standard case of frequency doubling via direct SHG, the generated radiation exhibits a spectral bandwidth equal to twice the fundamental acceptance given by Eq. (8) (in wavelength units, the SHG acceptance is one half of the fundamental). In presence of strong SFM contributions, however, this behavior is not a priori maintained. In fact, one has to take into consideration that any two frequency components Z1 and Z 2 within the input pulse may interact to generate a third frequency at Z 3 which, in general, is different to 2Z 0 . To analyse the spectral characteristics of the SH field, we now assume a generalized SFM process without restricting contributions to symmetric components, such that:

Z1  Z 2 Z 3

(9)

with:

Z1 Z 0  f1 Z 2 Z 0  f  f1

(10)

where, as shown in Figure 10b, f1 locates the first of the interacting components Z1 with respect to the central frequency Z 0 , and f is the frequency spacing between Z 2 and the symmetric of Z1 ( 0 d f d GZ 0 ). Combining the last two equations, we can obtain the following expression for the generated field:

Z 3 2Z 0  f

(11)

By use of the same basic principles as in the previous, we are able to write the intensity increase at Z 3 due to a single mixing process as: I Z3

I Z3 (l , Z 0 , f , f 1 , m p c )

ª 2 ˜ (2Z 0  f ) 2 d eff2 º 2 2 § l ˜ k · ¸ « » ˜ I Z0  f ˜ I Z0  f  f1 ˜ l ˜ sinc ¨ 3 c n n n H © 2 ¹ ¬« 0 Z0  f Z0  f  f1 2Z0  f »¼

(12)

Konstantinos Moutzouris, Florian Adler, Florian Sotier et al.

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where, following the usual fashion, the material parameters m p have been set to the values

m p c that phase match the direct SHG of the central component Z 0 . The phase mismatch parameter 'k is properly modified compared to Eq. (4). Therefore, for any given central frequency Z 0 Z 0c and crystal length l lc, the overall growth of a particular frequency component Z 3 (exclusively defined by a unique value of f , say f f c) can be calculated by integrating Eq. (12) for all allowed values of f1 , such that: mp

IZtot3

1 N

GZ 0

³ IZ (lc,Z c, f c, f ,m c) ˜ df

f1 GZ 0

0

3

1

p

1

(13)

where N is a normalizing factor with respect to the total intensity of the central SH frequency ( Z 3 2Z 0 , with f 0 ): GZ 0

N

I2totZ 0

³ IZ (lc,Z c, f

f1 GZ 0

3

0

0, f1,m p c ) ˜ df1

(14)

Solving Eq. (13) for different values of f allows for direct calculation of the normalized spectral intensity at any component Z 3 . This process may be used for reconstructing the generated spectrum at the second harmonic.

e) Material Evaluation We now apply the preceding analysis to various materials and phase-matching schemes that are commonly employed for frequency doubling the infrared spectrum between 0.8 m and 2.5 m. Details on the examined crystals and configurations are presented in Table 1. These include birefringent phase-matching in BBO, LBO, and KTP, as well as quasi-phasematching in PPKTP and MgO:PPLN (definitions of acronyms are provided in Table 1). Figure 11 illustrates simulation results for the effective acceptance bandwidth of the SFM process and comparison with the corresponding values for direct SHG. Studies were carried out for interaction lengths of 1 mm, 2 mm and 5 mm, and for BBO, LBO, PPKTP and MgO:PPLN crystals (Figures 11a to 11d, respectively). It is revealed that in all cases and throughout nearly the entire spectral region of interest, the SFM contributions are broader by one order of magnitude or more. Exceptions are the wavelengths near 2.5 m for the two QPM crystals, as well as the telecommunications-window neighborhood for BBO. It is to be recognized that two distinguishable patterns of acceptance scaling are formed, one for the QPM and one for the BPM crystals. Typical values of several tens of nanometers (hundreds of nanometers) of SFM doubling bandwidth are calculated at the short (long) wavelength end of the band under examination, reducing by a factor of more than 10 for direct SHG. Combined with a corresponding spectral width of ~250 nm (~750 nm) for a transform-limited single-cycle optical pulse, this observation leads to the unexpected conclusion that millimeter-

Highly Efficient Nonlinear Frequency Conversion Schemes…

105

100

10

(B)

1 0.8

BBO

1.2 1.6 2.0 2.4 Fundamental wavelength (P m)

1000

(A) 100

10

1 0.8

(B)

PPKTP

1.2 1.6 2.0 2.4 Fundamental wavelength (Pm)

Fundamental acceptance (nm)

(A)

1000

Fundamental acceptance (nm)

1000

Fundamental acceptance (nm)

Fundamental acceptance (nm)

thick crystals are well-suited for efficient doubling of infrared radiation as short as 10 cycles. An unavoidable limitation for the efficiency of the SFM contributions, which should become prominent at this temporal regime, involves dispersion effects and will be discussed further in the following. (A)

100

10

(B)

LBO

1 0.8

1.2 1.6 2.0 2.4 Fundamental wavelength (Pm)

1000

100

(A)

10

1 0.8

(B)

MgO:PPLN

1.2 1.6 2.0 2.4 Fundamental wavelength (Pm)

Figure 11. Scaling of spectral acceptance for frequency doubling via sum-frequency-mixing (A) and second-harmonic-generation (B), for: (a) BBO, (b) LBO, (c) PPKTP, and (d) MgO:PPLN. Calculations assume fundamental wavelengths in the 0.8 m to 2.5 m range and crystal thicknesses of 1 mm, 2 mm, and 5 mm. Arrows indicate direction of increasing crystal thickness.

Figure 12 presents theoretical calculations of SFM acceptance for bulk and periodically poled KTP. It is evident that, despite the use of the same material, the two different configurations result in largely different spectral tolerances, with the QPM option offering the wider SFM contributions. The strong dependence of spectral acceptance on the type of interaction (for a particular nonlinear medium) has been observed previously [69]. It is worth mentioning that bulk KTP was found to have the narrowest SFM acceptance from all the investigated materials, with respective values that do not exceed notably those for direct SHG.

Konstantinos Moutzouris, Florian Adler, Florian Sotier et al.

Fundamental acceptance (nm)

106

1000

(A) 100

10

(B) KTP / PPKTP

1 0.8

1.2

1.6

2.0

2.4

Fundamental wavelength ( m)

1000 (a)

100

100 Op = 1550 nm

10

10 (b)

1

1 2 4 6 8 Crystal thickness (mm)

10

Normalised acceptance (nm/mm)

Fundamental acceptance (nm)

Figure 12. Spectral acceptance for generalised sum-frequency-mixing doubling, for periodically poled (A) and bulk (B) KTP. Arrows indicate direction of increasing thickness, for values of 1 mm, 2 mm, and 5 mm.

Figure 13. Spectral acceptance (solid lines) and normalised spectral acceptance (dotted lines) for SFM (a) and SHG (b) interactions, as a function of crystal thickness. Calculations assume a MgO:PPLN crystal designed for frequency doubling of 1.55 m radiation.

Figure 13 shows the dependence of the spectral acceptance on material thickness for the indicative case of 1.55 m doubling in MgO:PPLN. As expected, the calculated bandwidth decreases with increasing interaction length for both SFM and direct SHG process. However, the normalized crystal acceptance (in units of nm per mm, shown in dotted lines) varies significantly with changing thickness for SFM, contrary to the case of direct SHG in which, according to the well-known behavior, it remains constant. Figure 14 illustrates modelling results for red, green and blue generation at 650 nm (right column), 550 nm (middle column) and 450 nm (left column), respectively, obtained via the

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concept described in section 3d. We have chosen 1-mm-thick MgO:PPLN as a nonlinear medium. The upper (middle) row assumes a Gaussian input spectrum with 100 nm (150 nm) FWHM width and accounts for the presence of generalized SFM contributions. The lower row presents the usual sinc2-bandwidth for direct SHG. It is an impressive outcome that, even in presence of ultra-broad SFM contributions, the generated fields exhibit practically the same narrow bandwidth as that for exclusive SHG. A small blue-shift in the output spectra, along with a small asymmetry, is observed for the generalized SFM interaction, while vanishing for the perfectly symmetric sinc2-shaped output for direct SHG. This effect should be attributed to the fact that the symmetry of the SFM contributions indicated in Figure 10a (i.e., mixing of equidistant components) breaks during transformation of units from frequency to wavelength.

0,8

A-1

'O = 0.4 nm

A- 2

'O = 1.1 nm

A-3

'O = 2.3 nm

B-1

'O = 0.5 nm

B-2

'O = 1.2 nm

B-3

'O = 2.4 nm

C-1

'O = 0.4 nm

C-2

'O = 1.1 nm

C-3

'O = 2.2 nm

0,4

Intensity (arb. units)

0,0 0,8

0,4

0,0 0,8

0,4

0,0

448

450

452

548

550

552

648

650

652

SH wavelength (nm)

Figure 14. Modeling of generated blue (1), green (2) and red (3) spectra. Graphs (A) account for SFM from a 100 nm broad Gaussian spectrum. Row (B) accounts for SFM from a 150 nm broad Gaussian spectrum and (C) shows the usual sinc2 result for direct SHG. Calculations assume a 1 mm thick MgO:PPLN crystal. 'O is the corresponding FWHM width.

Our theoretical results indicate that efficient frequency doubling of broadband infrared pulses is possible in thick media. In such geometries, the drawback arising from narrow SHG spectral acceptance can be naturally detoured due to the presence of ultra-wide intra-pulse SFM contributions. More interestingly, despite the large tolerances of the additional SFM interaction, the spectral properties of the generated radiation are predominantly determined by the narrow bandwidths of exclusive SHG. Based on these conclusions, the design criteria for a broadband-to-narrowband frequency doubler are evident: This device should employ a

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crystal with the largest possible SFM and smallest SHG spectral acceptance. As an example, we concentrate in the case of fundamental wavelengths in the ~ 1 m to 1.4 m range. From Figure 11 it is evident that an excellent candidate for this situation is MgO:PPLN (with PPKTP coming a close second). Additional advantages of this material include large effective coefficient and avoidance of spatial walk-off. Finally, it is worth pointing to the fact that acceptance in QPM geometries does not depend upon the order of interaction. This observation might be of interest for extending the operation of this device to shorter wavelengths, requiring technologically challenging short periods for first order QPM.

f) Experimental Proof Frequency doubling of the tunable infrared component form the HNLF in thick MgO:LN crystal provide clear evidence of the proposed scheme. Illumination of the crystals at normal incidence ensured that the fan-out-poling pattern does not result in chirping of gratings. Therefore, the phase-matching geometry is identical to a standard periodically poled structure and our theoretical discussion is applicable to the experimental materials. For a wavelength near the center of the tuning range of our pump source, say 1.3 m, the standard SHG fundamental acceptance bandwidth is 4.5 nm (2.25 nm), for the 1 mm (2 mm) crystals, respectively. Compared with available pump bandwidths in the order of 100 nm, this indicates that less than 10% of the input spectral content is useful for an exclusive SHG interaction. Therefore, the absolute power limit (100% conversion at full laser power) for frequency doubling in absence of broadband SFM contributions is ~3 mW. We measure up to 10 mW of average power in the visible, corresponding to conversion efficiencies exceeding 30%. These values alone indicate the strong presence of SFM contributions from the entire pump spectrum. Our argumentation is strengthened by additional measurements that revealed strictly quadratic scaling of the generated power and thus, lack of gain saturation. A selection of visible spectra has been presented in Figure 8 (shown in absolute power scale). The measured spectral intensities of several mW per nm translate into an increase in spectral brightness of approximately one order of magnitude compared to the pump radiation (typically sub-0.5 mW/nm, as shown in Figure 3). In good agreement with our theoretical prediction, visible spectra exhibit narrow FWHM spectral widths ( 3.5 nm), limited by standard SHG acceptance. Further experimental proof of a more general parametric SFM interaction was provided by comparison of fundamental spectra recorded before and after the nonlinear crystal, shown in Figure 15. No partial spectral depletion was observed in the transmitted spectra. Combined with lack of gain saturation and high conversion efficiencies, this observation contradicts the expected behavior of an exclusive SHG interaction.

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Pump spectral power (PW/nm)

600

650

700

160

1000

120

750

80

500

40

250

0

1200

1300

1400

0

Visible spectral power (PW/nm)

Visible wavelength (nm)

Fundamental wavelength (nm) Figure 15. Fundamental spectra collected before (black) and after (grey) the nonlinear crystal. Corresponding visible spectra, measured experimentally (blue) and calculated theoretically (red).

Figure 15 also illustrates the corresponding experimental visible spectrum, as well as the theoretically predicted emission. It is interesting to point out that, although our model reconstruction is in reasonable agreement with measurements, the actual experimental spectrum exhibits a stronger blue-shift with respect to the central wavelength. This finding may be attributed to slight deviations of the laser output from an ideal Gaussian shape. Up to now, dispersion related effects on the magnitude of SFM have been neglected. Although a quantitative analysis of such effects is beyond the scope of the present article, it has to be acknowledged that SFM contributions are expected to drastically reduce in presence of pulse chirp. This argument may be understood via considerations of temporal and spatial overlap between the mixing frequency components. As a first evaluation step, we varied the input pulse chirp by controlling the optical path length inside the SF10 glass of a prism compressor and recorded its influence on: a) the generated visible power, shown in Figure 16, and b) the transmitted fundamental spectrum at the output of the nonlinear crystal, shown in Figure 17. Not only did we observe a decrease in efficiency with increasing chirp, but most importantly, we were able to verify the build-up of a partial spectral depletion of the pump, as expected for direct SHG. These results are strong evidence of highly-efficient, wide-band doubling via parametric SFM in absence of chirp, reducing to inefficient, narrowband conversion via common SHG in presence of pulse chirp. Finally, we wish to report on additional experimental tests by use of the same pump source, in various type I BBO and type II KTP crystals. In both cases the direct SHG acceptance is larger than for MgO:PPLN, an effect that leads to generation of significantly broader spectra than these of Figure 8. Compared to MgO:LN, generated power levels are reduced by one order and three orders of magnitude for BBO and KTP, respectively.

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6 SHG power (mW)

5 4 3 2 1 0

0

250

500

750 1000 1250 2

Fundamental pulse chirp (fs ) Figure 16. Scaling of visible power as a function of the input pulse chirp. Pump power was kept constant at ~ 20 mW.

2

E = 0 fs 2 E = 1000 fs

Intensity (arb. units)

10 8 6 4 2 0

1200

1300

1400

Fundamental wavelength (nm) Figure 17. Transmitted fundamental spectra for input pulse chirp E = 0 fs2 (black line) and E = 1000 fs2 (red line). Partial spectral depletion is observed in the latter case.

In the first case, broad parametric-SFM is possible, and the power reduction is predominantly due to lower effective nonlinearity. In the latter case, efficiency is limited by both lower nonlinear coefficients and a notably narrower SFM acceptance. Further measurements indicated that the transmitted fundamental spectrum for KTP crystals does not vary with input pulse chirp. This finding originates in a combined effect of absence of parametric SFM and modest SHG gain.

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Conclusion This article reviewed the application of various nonlinear wavelength converters to an extended mode-locked erbium-fiber laser. The pump source comprised a single fiber oscillator and two identical fiber amplifiers, one of which was equipped with a highlynonlinear fiber to generate tunable radiation in the near infrared. The fundamental component from one arm and the variable infrared wavelength from the second arm were perfectly synchronized with each other. Mixing the two outputs in media exhibiting strong secondorder nonlinearities was the straightforward root to expand the coverage of the laser system from the ultraviolet through the entire visible to the near infrared. More specific, the second, third and fourth harmonic of the fundamental laser wavelength at 1.55 m were generated. Remarkable conversion efficiencies (up to 40% at the first stage) allowed light generation of 120 mW, 55 mW and 6 mW in average power at respective wavelengths of 770 nm, 520 nm and 390 nm. This radiation came in the form of ultrashort pulses with durations ranging between 140 fs (at the lower) and more than 1 ps (at the higher) harmonic. The tunable near infrared component (100 nm broad spectra, 30 fs pulses) from the second amplifier branch was also frequency doubled in a single-pass configuration. Tunable sub-picosecond pulses from green (520 nm) to red (700 nm) with average power levels reaching 10 mW were collected. High conversion efficiencies were achieved with simultaneous strong spectral filtering. It was recognized that the performance of this converter depends upon a generalized intra-pulse sum-frequency mixing process, which frequency doubles the fundamental radiation in a different way to standard second harmonic generation. Accessing the only part of the visible spectrum yet inapproachable was made possible by a final wavelength converter. This geometry was designed to mix the second harmonic of the fundamental laser wavelength with the tunable near infrared component. The sum-frequency interaction led to generation of tunable blue pulses in the spectral range between 460 nm and 500 nm. More than 1.5 mW of average power was collected in the entire blue band, with corresponding photon efficiency exceeding 20%. Pulses with sub-5 nm spectral widths and sub-300 fs durations were determined. In the last section of the present article, we discussed in more detail the novel concept for broadband-to-narrowband frequency doubling by means of generalized sum-frequencyinteraction. Theoretical investigations for various phase-matching geometries and nonlinear materials indicated that the collateral sum-frequency interaction introduces an effective acceptance that can exceed significantly the well-known wavelength tolerances of direct second harmonic generation. Theoretical reconstructions of the output radiation suggested that the additional wideband nonlinearity influences only to a negligible extend the generated spectral characteristics. In accordance with our experimental results, MgO:PPLN was found to be an excellent candidate for the implementation of such a broadband-to-narrowband doubler in the near infrared. The new nonlinear mechanism is to a great amount responsible for the current specifications of our multi-stage nonlinear converter. It is now clear to us that this fiber pumped device is of great interest to numerous scientific and technological situations, with recent work already revealing its immediate applicability for confocal microscopy and optical metrology.

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In: Progress in Nonlinear Optics Research Editors: Miyu Takahashi and Hina Goto, pp. 115-148

ISBN 978-1-60456-668-0 2008 Nova Science Publishers, Inc.

Chapter 4

THE NEW PROCESS OF ADAPTIVE OPTICS BASED ON NONLINEAR CONTROL ALGORITHMS FOR APPLICATIONS IN SOLID-STATE LASERS AND ICF SYSTEM Ping Yang 1,2, Mingwu Ao1,2, Bing Xu1 and Wenhan Jiang1 1

Institute of Optics and Electronics, Chinese Academy of Sciences, P.O.Box 350 Chengdu 610209, China 2 Graduate School of Chinese Academy of Sciences, Bei ing 10084, China

Abstract A new adaptive optics (AO) system for optimizing the output laser mode of a diode-laser pumped Nd:YAG solid-state laser has been built in our laboratory. A piezoelectric deformable mirror (DM) which is taken as the rear mirror of the solid-state laser is controlled by a real encoding genetic algorithm. To improve the AO system convergence rate, a group of Zernike mode coefficients is taken as the optimizing basis instead of the voltages on the DM. The transform matrix between voltages and Zernike mode coefficients has been deduced. A series of comparative numerical results show that the convergence speed and the correction performance of the AO system based on optimizing Zernike mode coefficients is far better than that of optimizing voltages. Moreover, Experimental results also showed that this AO system could change higher transverse modes into TEM00 mode successfully. In another way, so as to detect the entire beam aberrations of an inertial confinement fusion (ICF) system᧨an amendatory phase-retrieval method is presented, simulative and experimental results shown that using this method, the aberrations of entire beam path can be reconstructed ust from a few pairs of DM surface shapes and their corresponding focal spots intensity.

E-mail: pingyang2516 163.com

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1. Introduction Solid-state lasers have been used more and more widely in industrial fields and scientific fields. Solid-state lasers which can generate high quality fundamental mode (TEM00) output are indispensable in more and more scientific application fields, such as laser communication, high precision laser machining and so on. Therefore, there is an increasing demand for solidstate lasers to generate TEM00 mode. nfortunately, since thermal lens and thermally induced birefringence are the main thermal effects in solid-state laser resonators and can destroy the output laser beam quality greatly, thus, for obtaining fundamental mode output, the thermal effects must be compensated. It is known that adaptive optics technology has been used widely in many fields since it emerges as an independent optics technology. Adaptive optics can be used to compensate static or dynamic aberrations of a light beam after propagation through a distorting medium. A typical adaptive optical system is implemented with a wave-front sensor to measure the wave-front error, a DM to correct the wave-front, and a feedback control algorithm (more often than not, the direct-slope algorithm) to link these two elements in real time. However, these adaptive optical systems are maybe too large and too expensive for applications in solid-state lasers. In view of this case, we present a compact adaptive optical system based on piezoelectricity deformable mirrors controlled by the feedback of a single-performance metric to compensate phase aberrations and optimize laser modes in a solid-state laser. A global genetic algorithm is applied to control the deformable mirror by maximizing a performance metric inversely related to the wave-front error. A series of experiments have been done and the experimental results will be presented and discussed in this chapter. It is also known that in an ICF system, the wave-front aberrations that exist in laser beam and brought in by optical elements will enlarge the focal spot size and decrease power density at the target. Fortunately, an AO system can be employed in ICF system to correct the beam aberrations. As a powerful wave-front detector, Hartmann-Shack (H-S) sensor is often utilized as the wave-front sensor in AO system, however, when it is applied for ICF system, it can ust detect a part of the aberrations of entire beam path due to the spatial limitation of target chamber. For detecting and correcting the phase aberrations of entire beam path completely, a new method is presented to measure the aberrations of entire beam path in this chapter. A CCD camera is located in target chamber instead of H-S sensor and a series of different surface shapes of DM are produced which can generate their corresponding focal spots, and then these surface shapes and the intensity signals of their corresponding focal spots could be recorded by a H-S sensor(which is located in front of the chamber) and the CCD camera respectively. An amendatory phase-retrieval method is introduced to reconstruct the aberrations of entire beam path from a few pairs of DM surface shapes and their corresponding focal spots intensity. Simulative and experimental results show that the AO system can correct the aberrations of entire beam path of ICF successfully based on this method.

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2. Phase Aberrations in Solid-State Lasers Solid-state lasers are one of the most important and promising lasers during many kinds of lasers, nowadays, they are applied widespread in more and more fields. Make a solid-state laser to generate high quality and high power output beam simultaneously is one of the most interesting research orientations of researchers. However, many bad factors will affect the quality and power of the output beam. It is known that defocus is the main phase aberration in the solid-state laser beam (excluding tip-tilt), besides, astigmatism, spherical, lower order comas and so on also occupy a great large percentage in the phase aberrations of the laser beam[1]. As a result, for obtaining high quality output, phase aberrations, especially those induced by thermal effect in the laser cavity have to be compensated. Thermal lenses and thermally induced birefringence within the gain material can greatly affect the stability and efficiency of the laser. The birefringence could be eliminated through selecting a naturally birefringence laser crystal or adopt a polarization insensitive laser cavity configuration [2]. The spherical parts of the thermal lens can be eliminated by designing the resonator cavity well [3], whereas the nonspherical aberrations couldn t be removed in the same way [4]. Generally, one of the most conventional ways to make solid-state laser generate high quality TEM00 mode output is to locate a pinhole in its resonators. However, this way will abate the output power greatly and may cause the resonator to be misaligned once the pumping conditions changes. Since the thermal effects in the resonators change with the varying pumping conditions, thereby the promising ways for compensating the thermally induced phase aberrations and thermally lenses are to adopt adaptive methods. It is known that AO technique is a powerful technique that allows dynamic correction of phase aberrations. Although it is initially developed for astronomy, it can also be used in the laser fields [5]. A 37 element AO system has been used intracavity to optimize the laser modes [3]. In this chapter, we will study the possibility of using a 19-element piezoelectric DM as the rear mirror of a laser resonator and in con unction with a genetic algorithm (GA) to control the beam mode of a diode-laser pumped Nd:YAG solid laser[6].

3. Genetic Algorithm and Its Application in Solid-State System GA is a stochastic parallel algorithm based on natural selection and biological genetics. It is often used to search the global optimum value of some multi-ob ect problem[7]. In past a few years, the GA community has turned much of its attention to the optimization problems of industrial engineering. Literature [8] firstly reported that a kind of GA was successfully used in an adaptive optical system to control a DM to optimize the laser intensity distribution. That paper shows that the optimum laser intensity distribution could be searched by using this GA as long as the amplitude of the phase aberration is in the stroke range of the DM. However, the GA introduced in literature [8] is a simple GA based on binary encoding. There are some disadvantages in this binary encoding GA. Because the binary string is not a natural encoding, thus when the binary string is too short, GA may not satisfy the precision request of given problems, moreover, when the binary string is too long, the search space will become too large for GA to search rapidly. These drawbacks of the binary encoding indicate that in many GA application fields, especially for those problems from industrial engineering world, the simple GA based on binary encoding may be not the most promising ways.

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So as to overcome the shortage of the GA based on binary encoding, a real number encoding GA is presented. This algorithm is applied to control a 19-element intracavity DM to optimize the output modes of a solid-state laser. The following paragraphs will demonstrate the basic steps of typical GA and the flowchart of GA is shown in Figure 1.

Generate initial population

Fitness evaluation

Selection

No

Crossover

Mutation

Satisfy the end condition Yes end

Figure 1. The flowchart of a typical genetic algorithm.

3.1. Overview of a Real Number Encoding GA Compared to the binary encoding GA, the real number encoding GA whose chromosomes comprises real numbers is proved to give better performance [9]. The implementation of the GA based on real number encoding strategy is as follows:

3.1.1. Encoding Strategy The real number encoding is the natural description of the solution to a certain problem, for example, consider a problem to be optimized has three variable x1, x2, x3, then X=[x1, x2, x3] is a chromosome which is also a encoding form for the variables of the this problem. Variable x1, x2 or x3 is not only a gene of the chromosome respectively, but also stands for a real value within its own range. In comparison with binary encoding, real number encoding can improve calculation complexity and increase the solution precision without encoding or decoding error. 3.1.2. Fitness Evaluation and Selection Operator Before selection operation, the fitness of every individual in a population must be evaluated firstly. A roulette wheel approach is adopted as the selection procedure, which is one of the fitness-proportional selections and can select a new population with respect to the probability distribution based on fitness values. The roulette wheel can be constructed with as following steps: firstly, suppose that there is a population which includes M individuals (chromosomes), the fitness of the kth individual is Fk, then the selection probability Pk can be calculated as follow

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Fk

Pk

M

¦F

(K=1, 2,…M)

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(1)

k

j 1

The second step is to calculate cumulative probability qk K

¦P

qk

(K=1,2 ,…M)

(2)

j

step 3 is to generate a random number r from the range [0,1] the last step is to estimate: If r q1, then select the first individual V1 otherwise select the kth chromosome Vk in terms of qk1ื

rื qk (2kM).

3.1.3. Crossover Operator Crossover operator is the main manner to generate the new individuals from the old individuals, there are several crossover methods such as Simple crossover, Direction-based crossover, Non-uniform, Arithmetical crossover and so on. In this chapter, we adopted the non-uniform Arithmetical crossover. The operation processes are as follow: assume that YAt᧨YBt are two individuals which are selected from an old population by a crossover probability Pc, then the two new individuals YAt+1 ,YBt+1 are generated as follows YAt+1᧹mhYAt + (1-m) YBt

(3)

YBt+1᧹m YBt + (1-m) YAt

(4)

The variable m that is decided by the evolution generation can be described as following equation

m=r

au (1-t/T)

(5)

In equation (5), r is a random number from the range [0, 1] T is the maximum evolution generation t stands for the generation which is at the range of [1,T] a is weight factor from 1.0 to 4.0. What can be seen from equation (5) is that m decreases as t approaches T, in this way, the variable can be ad usted adaptively.

3.1.4. Mutation Operator The local search ability of GA is determined by mutation operation, crossover operation and mutation operation are two indispensable part of GA. There are also kinds of mutation operation ways, the mutation used here is nonuniform mutation and the mutation course is as follows: choose the gene of a parent chromosome randomly, then alters this gene with a mutation rate Pm. Assume that the kth gene of a parent chromosome X=[x1, x2,…xk,…xn] is

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selected for a mutation, the new offspring chromosome after mutation is X =[x1 ,x2,…x k,…xn] and the new kth gene x k can be obtained from following formula x k᧹xk+(t, xk - xk)

(6)

where the xk is the upper bounds for xk and the funcion (t, y) can be described as (t, y)=yr(1-t/T)b

(7)

where r is a random uniform distributed number from [0, 1] t is the current generation number T is the maximal generation number parameter b, which is set to 3 in this paper, determines the degree of nonuniformity. From (7) we can see (t, y) returns a value in the range of [0, y] such that the value of(t, y) approaches to 0 as t increases. When a new generation is generated, the previous generation is eliminated. The process of selection, crossover and mutation is repeated until the maximum generation T is reached or fitness value reaches a preset value.

3.2. The Improved Genetic Algorithm Based On Optimizing The Zernike Mode Coefficients This section will present an improved GA, for comparing the convergence speed and the correction performance of this GA with that of the traditional one, some numerical simulations have been accomplished in the following several sections Figure 2. is the photograph of the DM used in our solid-state laser system. The DM which was fabricated in our laboratory has continuous face plate with stacked PZT actuators[10] and with effective diameter 32mm, maximum deflection 2m, maximum voltage 300V, nonlinearity and hysteresis 4%, resonance frequency 10kHz. According to its principle, the DM can deform its surface shape by applying voltages on the actuators: n

M (x,y)= ¦ v V (x,y) =1

(8)

where v is the voltage applied on the th actuator, V (x,y) is the influence function of the th actuator on the wave-front, n is the number of actuator.

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Figure 2. The photograph of the DM.

The influence function of the DM can be written in following form:

V (x,y)=exp[ln(w) ( (x-x )2 +(y-y )2 / d))p

(9)

where w is the coupling coefficients of DM and set to 0.082 (x , y ) is the space position of the ith actuator, p is set to 2 and d is the distance between every two neighbor actuators, x and y represent the value in x-coordinate and y-coordinate respectively of the orthogonal coordinate plane. The voltages applied on the 19 actuators of the DM are in the range of 300V and can vary with a step of 1 V, therefore, the possible DM surface shapes number on basis of voltages can reach 60019 [11] . Since the GA is used to find the optimum shape for optimizing the laser mode intracavity in such a vast DM surface shapes space, thus, to speed the system convergence, and to improve the calculation efficiency of GA, one promising approach is to decrease the search space.

3.2.1. The Transform Matrix between Zernike Mode Coefficients and Voltages We have known that most of the phase aberrations in the Nd:YAG solid-state lasers are low order aberrations(such as defocus, astigmatism and coma) and can be described well by the First 10 orders Zernike polynomial [1], in another way , to ascertain the performance of our DM, we investigate the correction capability of the 19 element DM through simulation firstly. Figure 3 demonstrate the correct ability of this DM where the red line represents the RMS value of the original wave-front and the green curve stands for the RMS value of the wave-front generated by DM (which is used to correct the original wave-fronts that correspond to each Zernike order) while the blue one represents the RMS value of the residual wave-front. These results are obtained through 35 times simulative correction calculations. What we can know from Figure 3 is that the DM employed is very suitable for correcting the first 10 orders Zernike aberrations although it can also correct the 11th to the 14th and the 18th to20th orders Zernike aberrations to some extent.

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Figure 3. The correction capability of the 19 element piezoelectric DM.

From discussion above, we choose the first 10 order Zernike mode coefficients on actuators as the ob ect function of GA to optimize rather than the voltages, as a result, the whole search space of GA are reduced to60010. The following paragraphs will deduce the transform matrix between 19 voltages and first 10 Zernike mode coefficients. It is known that any given wave-front in a unit circle can be described by a series of Zernike polynomial: m

M ( x, y ) b0  ¦ bk Z k ( x, y ) k 1

(10)

where b0 is the piston coefficient, Z k ( x, y ) and bk (k=1, 2… m) are the kth Zernike polynomial and their corresponding coefficients respectively, m is the Zernike order. Considering that whether optimizing the voltages or the Zernike mode coefficients by GA, the ultimate aim is to obtain the needed DM surface shapes, moreover, as to our DM, voltages are the key parameters that affect its surface shape directly, thereby we have to change the Zernike mode coefficients into voltages after the optimization sequence. As a result, the relationship between mode coefficients and voltages should be ascertained. From Eq.(8) and Eq.(10), we can obtain: n

m

=1

k 1

¦ v V (x,y) = b0  ¦ bk Z k ( x, y)

(11)

Because in most applications, the piston coefficient is often omitted, and then Eq.(11) can be rewritten as:

The New Process of Adaptive Optics Based on Nonlinear Control Algorithms… n

m

=1

k 1

¦ v V (x,y)  ¦ bk Z k ( x, y)

123

(12)

Now, our ob ect is to build the transform matrix between Zernike coefficients and actuators voltages which can be calculated according to:

ªv 1 º «v » « 2» «... » « » ¬ vn ¼

ªu11 , u12 ...u1m º ªb1 º «u , u ...u » « » « 21 22 2 m » u «b 2 » «................ » «... » « » « » ¬un1 , un 2 ...unm ¼ ¬bm ¼

(13)

Equation (13) can be rewritten:

V

UB

(14)

Limited by the DM deflection range( 2m), we set the value of each Zernike coefficient to 0.1. The simulative target grid is set to 100x100, and then the size of each influence function matrix ᧤ =1, 2…19᧥is also 100x100. For calculation convenience, we transform the form of 19 influence function matrixes into a 10000x19 matrix and using V (x,y) to describe. We using the first 10 order Zernike polynomials and their corresponding mode coefficients (each is set to 0.1) in sequence to produce a surface shape matrix\ ( x, y ) with a size of 10000x1, u represents the voltage vector. According to Eq.(10)and Eq.(12) we can obtain:

V (x,y) 10000x19u19u1 \ ( x, y )10000u1

(15)

System stability and the capability of resistance to interference is evaluated by the condition number of matrix V (x,y) : Cond( V (x,y) )=max/min

(16)

where max andmin are the maximum and minimum singular values respectively. In fact, the condition number is the measurement of matrix ill-condition. Traditionally, this value should be smaller than 20. Since the condition number is 4.47 calculated by Eq.(16), thus, the capability of V (x,y) is very good. Once \ ( x, y ) and V (x,y) are ascertained, the transform vector between Zernike mode coefficients and actuators voltages can be calculated:

u19u1

V (x,y) 19u10000\ ( x, y )10000u1

(17)

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where V (x,y) is the generalized inverse matrix of V (x,y) . Finally, the transform matrix

U between the first 10 orders Zernike mode coefficients and 19 actuators voltages on the DM could obtained by calculating Eq.(17) ten times:

U

>u1 , u2 ...ui ...u10 @

(18)

where ui ᧤i=1, 2...10᧥is a 19x1 vector. When U has been calculated, GA will optimize the first 10 order Zernike mode coefficients rather than 19 voltages on actuators. It should be noticed that since each coefficient is set to 0.1, therefore, the voltages calculated by (18) can not be applied on actuators directly before they are multiplied by a factor of 10. So as to evaluate the effectiveness of the transform matrix, we firstly use the fist 10 orders Zernike polynomial coefficients to produce a wave-front, and then using the transform matrix to figure out 19 voltages which will drive DM to produce another wave-front. Compare the similarity of the two wave-fronts and then subtract the Zernike generating wavefront from the DM generating wave-front at their corresponding position to obtain the residual wave-front. What we can know from Figure 4 is that although there is a little difference about the peak-to-valley (PV) between Zernike generating wave-front and that of DM generating wavefront, their RMS approach each other, besides, their shapes are also similar. Because RMS value is conventionally regarded as one of the most important parameters to evaluate the whole surface quality of a wave-front in applications, therefore, we can think the results is able to testify the availability of the transform matrix because the RMS of the residual wavefront is small as a result, the transform matrix can be used to describe the relationship of the first 10 order Zernike mode coefficients and 19 voltages.

Figure 4. The left figure is the wave-front generated by the first 10 order Zernike polynomial coefficients and the center figure is the wave-front reconstructed by DM while the right one is the corresponding residual wave-front.

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3.2.2. The Comparative Simulations To compare the performance of our AO system based on optimizing voltages and the Zernike mode coefficients, we have accomplished some comparative numerical simulations. On two bases, the initial populations of GA both consist of 20 individuals (DM surface shape). The far-field light spot are obtained through executing FFT algorithm. The initial phase aberrations of both cases are generated randomly by a group of Zernike polynomial coefficients. 3.2.2.1. Correcting the Defocus Aberration We firstly test the comparative ability of GA on both bases through correcting the defocus aberration generated by the program in the computer. As for the Zernike mode coefficients base, for speeding the convergence of GA , we ust set each value of Zernike mode coefficients to 0 except the defocus coefficients, and then using this group of coefficients to correct the defocus aberration generated by computer. Figure 5(I) is the near-field wave-front before and after the phase aberrations are corrected by DM on basis of optimizing the voltages. Figure 5(I)A and Figure 5(I)B are the planar and three-dimensional wave-fronts respectively before phase aberrations are corrected whereas Figure 5(I)C and Figure 5(I)D correspond the corrected cases. We can known that the PV value and RMS value of the wave-front are reduced to 2.113᧤=1064nm᧥and 0.263 from 2.126and 0.606respectively after 200 times iterative calculation, thus, we can t say the correction effect is promising. Similarly, Figure 5(II) is the near-field wave-front before and after the phase aberrations are corrected by DM on basis of optimizing the defocus Zernike mode coefficient. Figure 5(II)A and Figure 5(II)B show the planar and threedimensional wave-fronts respectively before phase aberrations are corrected whereas Figure 5(II)C and Figure 5(II)D correspond the corrected cases.

I

II

Figure 5. The near-field distribution before and after correction based on optimizing voltages. (II) The near-field distribution before and after correction based on optimizing the defocus Zernike mode coefficients.

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We can know that the PV value and RMS value are reduced to 0.52and 0.085from 2.126and 0.606respectively after 200 generations iterative calculations, therefore, the correction ability based on optimizing the defocus Zerinike polynomial coefficients is powerful and the wave-front improvement is obvious. Figure 6(I) and Figure 6(II) are the far-field focal spot distribution on basis of optimizing the voltages and mode coefficients respectively.

I

II

Figure 6. The far-field distribution before and after correction based on optimizing voltages. (II) The far-field distribution before and after correction based on optimizing the defocus Zernike mode coefficient.



Figure 7. The fitness value based on two different bases: the bottom curve is the fitness value based on optimizing actuator voltages of DM whereas the top curve is the fitness value based on optimizing defocus Zernike mode coefficient.

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Figure 6(I) demonstrates that most energy of the far-field focal spot converges to the center after phase aberrations are corrected, and the Strehl ratio is raised from 0.12 to 0.62. Compared with Figure 6(I), after correction, more energy are converged towards the spot center in Figure 6(II), and the Strehl ratio is increased from 0.12 to 0.86. Figure 7 shows the comparative fitness function curves Y-coordinate represents the fitness value which is normalized to 1 whereas X-coordinate is the number of iterative generations. What we can know is that the GA based on optimizing the Defocus Zernike mode coefficients is nearly converged after 200 generations, whereas based on voltages is far from convergence. Furthermore, after about 30 iterative generations, the fitness value of GA based on defocus coefficients can reach about 0.75 which will take about 200 generations for the GA based on voltages to reach. 3.2.2.2. Correcting the Astigmatism Aberration We then compare the correction performance of GA on two bases through correcting the single astigmatism aberration, the way of bringing in and correcting the astigmatism aberration is similar as that of defocus. The correction performance is as following pictures. Figure 8 and Figure 9 are the correction performance for astigmatism based on two bases, from this two figure, We can also draw a conclusion that the correction ability and convergence of GA based on Voltages is not as good as that of GA based on astigmatism coefficient.

I

II

Figure 8. The near-field distribution before and after correction based on optimizing voltages. (II) The near-field distribution before and after correction based on optimizing the astigmatism Zernike mode coefficients.

Figure 10 shows the comparative fitness function curves Y-coordinate represents the fitness value which is normalized to 1 whereas X-coordinate is the number of iterative generations. What we can know is that the GA based on optimizing the Astigmatism Zernike mode coefficients is nearly converged after 200 generations. However, although the GA based on optimizing the voltages is also seemed to convergence, we can know it is far from reaching the global maximum even after 200 generations calculation.

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I

II

Figure 9. The far-field distribution before and after correction based on optimizing voltages. (II) The far-field distribution before and after correction based on optimizing the astigmatism Zernike mode coefficient.

Figure 10. The fitness value based on two different bases: the bottom curve is the fitness value based on optimizing actuator voltages of DM whereas the top curve is the fitness value based on optimizing astigmatism Zernike mode coefficient.

3.2.2.3. Correcting the Combination Aberrations In this section, we evaluate the correction capability of GA based on two bases through correcting combination aberrations which are generated by the first 10 order Zernike mode coefficients.

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Figure 11 is the near-field wave-front before and after the phase aberrations are corrected by DM on basis of optimizing the voltages. Figure 11(I)A and Figure 11(I)B are the planar and three- dimensional wave-fronts respectively before phase aberrations are corrected whereas Figure 11(I)C and Figure 11(I)D correspond the corrected cases. We can known that the PV value and RMS value of the wave-front are reduced to 1.45᧤=1064nm᧥and 0.30 from 3.38and 0.64respectively even if after 400 times iterative calculation, thus, we can t say the correction effect is promising. Similarly, Figure 11(II) is the near-field wave-front before and after the phase aberrations are corrected by DM on basis of optimizing the Zernike mode coefficients. Figure 11(II) is the plot according to the mode coefficients optimizing case. Figure 11(II)A and Figure 11(II)B show the planar and three-dimensional wave-fronts respectively before phase aberrations are corrected whereas Figure 11(II)C and Figure 11(II)D correspond the corrected cases. Figure 11(II) are corrugated to some extent after correction, there are two possible reasons can explain this: one is due to the reduced basis, another is due to the deformable mirror (DM) employed. Because there is nearly no such a DM that can generate complete precise surface shape needed to compensate the phase aberrations generated and represented by Zernike polynomials. However, we can know that the PV value and RMS value are reduced to 0.52and 0.06from 3.38and 0.64respectively after 200 generations iterative calculations, therefore, the correction ability based on optimizing the Zerinike polynomial coefficients is powerful and the wave-front improvement is obvious. Figure 12(I) and Figure 12(II) are the far-field focal spot distribution on basis of optimizing the voltages and mode coefficients respectively. Figure 12(I) demonstrates that most energy of the far-field focal spot converges to the center after phase aberrations are corrected, and the Strehl ratio is raised from 0.06 to 0.64.





Figure 11. The near-field distribution before and after correction based on optimizing voltages (the iterative number is 400). (II) The near-field distribution before and after correction based on optimizing the Zernike mode coefficients (the iterative number is set at 100).

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Figure 12. (I) The far-field distribution before and after correction based on optimizing voltages(the iterative number is 400). (II) The far-field distribution before and after correction based on optimizing the Zernike mode coefficients (the iterative number is 200).

Figure 13. The fitness value based on two different bases: the four curves at the bottom of the picture are the fitness value based on optimizing actuator voltages of DM whereas the top four curves are the fitness value based on optimizing the first 10 order Zernike mode coefficients.

Compared with Figure 12(I), after correction, more energy are converged towards the spot center in Figure 12(II), and the Strehl ratio is increased from 0.06 to 0.95. Figure13 shows the comparative fitness function curves Y-coordinate represents the fitness value which is normalized to 1 whereas X-coordinate is the number of iterative generations. What we can know is that the GA based on optimizing the first 10 order Zernike mode coefficients is nearly converged after 200 generations, whereas based on voltages is far from convergence. Furthermore, after about 20 iterative generations, the fitness value of GA

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based on coefficients can reach 0.5 which will take about 200 generations for the GA based on voltages to reach.

3.2.3. The Mode Control Experiments on the Basis of Optimizing Mode Coefficients We have known that most of the phase aberrations in the Nd:YAG solid-state lasers are low order aberrations(such as defocus, astigmatism and coma) and can be described well by the First 10 orders Zernike polynomial [1]. Consequently, GA based on optimizing the first 10 order Zernike mode coefficients is suitable for application in the solid-state lasers. To investigate the availability of the 19 element DM as the rear mirror to optimize the Nd:YAG solid laser, a Nd:YAG laser resonator of Figure 8 is configured. A 5x telescope is used intracavity to expand the laser beam (6.2 mm) to about 30mm diameter so that the beam could match the DM aperture (32mm) and cover as many actuators as possible. In this way the resolution of the beam/mirror interaction could be enhanced and allowing a greater degree of optimization to be performed [1]. The active medium is a 85mm long Nd:YAG rod with a diameter of 7mm. The Nd3+ doping concentration of the crystal is 0.8%, the rod is surrounded by the antireflection-coated cooling sleeve. The largest repetition rate, pumping current of the pump head are 100Hz, 70A respectively. A 70% reflective mirror is used as the output coupler (OC) and the output power could be ad usted from 0 to50W. After attenuated by an Attenuator, the output beam is firstly reflected by a beam splitter (BS1), and then passes through a 1064nm narrow filter before it is focused by a lens onto an infrared CCD camera. The intensity information of the focus light spot is acquired with a frequency of 25 Hz by a frame grabber. sing one part of (within a selected center area) the intensity information as the ob ect function to maximize, the industrial computer calculates the 10 element Zernike mode coefficients which would finally be transformed into 19 voltages by the transform matrix U obtained in section 3.2.1. At last, these voltages are amplified by a high voltage amplifier (HVA) before applied on DM actuators. A monitor which is placed behind another beam splitter (BS2) was used to watch the laser mode profile. A power meter was also employed to detect the output laser power in real time. DM

Telescope

Laser head

Attenuator BS1

BS2

Nd Rod

Monitor OC Pinhole Filter

Industrial computer

HVA

D/A

Power meter

Lens

Frame grabber

CCD

Figure 14. Experimental schematic layout of the adaptive mode control system.

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We firstly tested the optimization performance of the solid-state laser at a pump current of 40 A. Before optimization, the output power was 5.6W the initial transverse mode which is shown in Figure 15(a) was a TEM10 mode. During the course of optimization, we found that the TEM10 mode converged towards a fundamental TEM00 mode which is shown in Figure 15 (b) after about 60 seconds. This phenomenon may be explained as follows: as the DM changed its surface shape in the course of optimization, its curvature radius also changed which resulted in a change of the resonator configuration, thereby could establish conditions for generating the TEM00 mode more efficiently and suppression of the higher order modes to a great extent. The output power was reduced to 5.3W after mode optimization has been accomplished.

(a)

(b)

Figure 15. (a) The far-field beam profiles distribution (TEM10) when AO system is off, whereas (b) according to the beam profiles distribution (TEM00) when AO system is on.

We slowly increased the pump current and observed the output laser mode simultaneously. It is found that the beam mode structure became more and more complex as the pumping current increased. Experimental results shown that it was even impossible for the DM to select the TEM00 mode successfully when the mode structure became too complex. In order to control the mode structure efficiently in a relative high pumping current status, a variable size pinhole was placed near to the OC of the resonator to coarsely restrict the complex high order modes. Figure16 is the Far-field laser beam profiles recorded at various intervals during an optimization sequence when the pump current was 50A with a 3 mm pinhole in the resonator. At this case, the aperture of the laser beam on the DM was restricted to about 15 mm however, experimental results demonstrated that it was still effective to optimize the mode in spite of only the central part of the DM is covered by laser. Figure 16 shows that the TEM20 mode distribution was changed into TEM00 mode after about 85 seconds. The relative power in the selective region of the CCD camera was increased from 1 to 5.6. The selective region could be altered by program built in computer in this case, the size of the selected region was as large as one diffraction-limit. As a consequence of optimization, the output power was increased from 3.2 W to 4.7 W. Figure 17 is the Far-field beam profiles from the Nd:YAG laser recorded at various intervals during another optimization sequence where the pump current was increased to 59 A with a 2.5 mm pinhole in the resonator.

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Relative Power in the selective region of the CCD 5.6 4.5 3.5 2.25 1

10

43

60

73

85

Time/s

Figure 16. Far-field beam profiles from the Nd:YAG laser recorded at various intervals during an optimization sequence(the pump current is 50A). The X-coordinate represents the optimization time the Y-coordinate represents the relative power in a selective region of the CCD camera.

Relative Power in the selective region of the CCD 5.2 5

3 2.4 1.3 1

15

40

72

100

125

140

Time/s

Figure 17. Far-field beam profiles from the Nd:YAG laser recorded at various intervals during another optimization sequence. The X-coordinate represents the optimization time the Y-coordinate represents the relative power in a selective region of the CCD camera. (The pump current is 59A).

Figure 17 shows that the TEM11 mode distribution was also changed into TEM00 mode after optimization was finished, and the relative power in the selective region of the CCD camera was increased from 1 to 5.2. In this case, the size of the selected region was also set as

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large as one diffraction-limit. Output power was brought down from 7.2 W to 5.8 W during this course. It took about 140 seconds to accomplish this optimization course.

4. ICF System with New AO System As a new ICF system, SG-ċ Prototype [12-16] has been constructing in China. A block diagram of the SG-ċ Prototype main laser optical system is shown in Figure 18. It consists of the front-end, the master amplifier (Amp1), the booster amplifier (Amp2), the frequency converter, and the target chamber. The front-end is the laser pulse generation and preamplifying unit, which generates the spatially and temporally shaped 1 (1.053 m) pulse. This laser pulse was in ected into the Amp1 which is a four-pass cavity amplifier. After passing the Amp1, this laser pulse passes through Amp2 and heads towards the target chamber. There beam is frequency-converted to 3 (351 nm) and focused onto the target. nfortunately, wave-front aberrations existed in laser beam will increase the focal spot size and decrease power density at the target. Adaptive Optics technique is often used to control the wave-front aberrations. sually, two kinds of Adaptive Optics techniques are employed in ICF system to correct wave-front aberrations [12-16]. One technique is Hill-climbing wave-front correcting technique6. The technique takes the focal spot intensity distribution as a goal to optimize, drive the actuators of DM to compensate phase aberrations in the beam and then decrease the focal spot size. The most advantage of this technique is that the focal spot can be controlled directly, however, low efficiency to drive actuators of DM ust according to focal spot intensity information is its drawback. This technique had been used in SG-I ICF system of which more than one hundred times needed to find a proper DM surface for compensating wave-front aberrations, as a result, the speed of this technique is very slow. Another technique is to adopt a wave-front sensor to detect the wave-front aberrations and then use a DM to correct them by generating a con ugate wave-front of the phase aberrations. In this way, the DM can control the focal spot directly. Most ICF systems adopted this technique to control the focal spot, such as NIF, SG-ċ Prototype, LM and so on. In order to meet the spot-size requirement and goal, in this chapter, an Adaptive optics system is employed in SG-ċ Prototype to restrict wave-front aberrations. As shown in Figure 18, a Hartmann-Shack sensor (H-S) and a DM constitute an Adaptive optics system (not including the CCD which is placed in the far-field) the DM is yielded to correct the wavefront aberrations which are detected by the H-S. nfortunately, H-S can ust be used to detect a part of the aberrations of entire beam path (before the beam-splitter) the aberrations that after the beam-splitter can not be detected. Thereby, the DM can only correct aberrations detected by sensor. However, the aberrations of optical elements after the beam-splitter are considerable which will increase the focal spot size and decrease power density greatly at the target. It is difficult to compensate these aberrations ust depends on optical fabrication due to the high cost of manufacture, and the best way to solve the problem is to detect and correct the whole aberrations of entire ICF beam path.

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Figure 18. Configuration of SG㧙III Prototype.

In order to measure the whole aberrations of ICF beam path, one may think to put another H-S in the chamber, however, it is unpractical to install H-S sensor in target chamber for detecting the whole wave-front aberrations of ICF system. Because the high power laser beam must pass through a frequency converter crystal before reaching the target chamber [17]. The frequency converter crystal is nonlinear medium that would disturb the linear propagation of laser beam therefore, the measurement of the phase aberration in the chamber can not represent the actual entire phase aberration precisely. In the following paragraphs, a new technique will be recommended this technique is capable of detecting and correcting the aberrations of entire ICF beam path, furthermore, it is more efficient to drive the actuators of DM to correct wave-front aberrations than that of Hillclimbing wave-front correcting technique.

4.1. The Phase Aberrations of Entire Beam Path In the SG-ċ prototype ICF system, the main factors that affect the wave-front quality of entire beam path include the static aberrations of the optical surfaces, the dynamic aberrations induced by nonlinear phenomena of the high power laser, pump-induced thermal aberrations in the amplifier and so on. As shown in Figure 18, when high power laser pass through the ICF beam path, since most of the dynamic aberrations occur in the optical units before the beam splitter and we can deem that the H-S can detect the dynamic aberration of entire ICF beam path. However, each optical element has contribution to the static aberrations of entire ICF beam path, moreover , the H-S which is located before the Beam-splitter isn t able to

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detect the static aberrations of optical elements after the beam splitter. In the following paragraph, a new technique about the static aberrations measurement of entire ICF beam path will be presented.

4.2. Static Aberrations Measurement of Entire Beam Path As shown in Figure 18, an extra CCD is installed in the far-field to detect the focal spot intensity distribution. The entire ICF beam path static aberrations can be measured by the cooperation of the H-S, DM and CCD. Let the DM generate different surfaces when the ICF system shoot the static pulse beam, and then these different DM surfaces could modulate different wave-front aberrations which will be added to the entire ICF beam path static aberrations to create their corresponding focal spot in the far-field. At last, the different wavefront aberrations modulated by DM and the different intensity distribution of focal spots could be recorded simultaneously by H-S sensor and CCD respectively. An amendatory phase-retrieval method [18-19] can reconstruct the static aberrations of entire ICF beam path from these different data pairs that recorded by H-S sensor and CCD. nder the condition of static pulse shoot, the near-field of the laser wave-front in the entire ICF beam path can be expressed as

A᧤ 0 x, y᧥ A0 ( x, y) exp i 2SI0 ( x, y) where

(19)

I0 ( x, y ) is the wave-front that contains static aberrations of the entire ICF beam path.

Our purpose is to figure out them through the following steps: The corresponding far-field can be expressed as

I 0 ( x1 , y1 )

I 0 ( x1 , y1 ) exp i 2S\ 0 ( x1 , y1 )

(20)

Let the DM generate different surfaces, and then the different wave-front aberrations are modulated into ICF beam path the near-field can now be described as:

An ( x, y )

An ( x, y ) exp[i 2S (I 0 ( x, y )  I n ( x, y))] n=1, 2

(21)

where the In means that the nth aberration has been modulated into the ICF beam path by DM. The corresponding far-field can be expressed as

I n ( x1 , y1 )

I n ( x1 , y1 ) exp(i 2S\ n ( x1 , y1 )) n=1,2…

The DM can not change the intensity distribution of near-field, therefore

(22)

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An ( x, y )

2

137

2

A0 ( x, y ) n=1,2… (23) 2

In equation (19)~(22)᧨the I 0 ( x1 , y1 ) and I n ( x1 , y1 )

2

are obtained by the CCD

installed in the far-field The In obtained by H-S sensor, at the same time the H-S sensor can 2

also detect the A0 ( x, y ) . According to Fourier optics principle, the relation between the near-field and the far-field is expressed as

I 0 exp(i 2S\ 0 )᧹F A0 exp(i 2SI0 )

(24)

A0 exp(i 2SI0 )᧹F-1 I 0 exp(i 2S\ 0 )

(25)

I1 exp(i 2S\ 1 )᧹F A1 exp[i 2S (I0  I1 )]

(26)

A1 exp[i 2S (I0  I1᧥ ]᧹F-1 I1 exp(i 2S\ 1 )

(27)

I n exp(i 2S\ n )᧹F An exp[i 2S (I0  In )]

(28)

An exp[i 2S (I0  In᧥ ]᧹F-1 I n exp(i 2S\ n )

(29)

4.3. The Amendatory Phase-Retrieval Method The entire ICF wave-front static aberrations I0 can be retrieved by iteration which is an amendatory phase-retrieval method based on the classical G-S iteration the flowchart is shown in Figure 19. The detailed steps of iteration can be demonstrated in the following paragraphs. (1) Input the initial phase estimate value I0 , the I0 and the known A0 make up of the near field expression A0 exp(i 2SI0 ) , and then the near-field expression is transformed to the

expression I 0c exp(i 2S\ 0 ) by Fourier transform let the known I 0 replace the I 0c and make up of the amendatory far field expression I 0 exp(i 2S\ 0 ) .The expression I 0 exp(i 2S\ 0 ) is (1) then transform to A0c exp(i 2SI0 ) by executing inverse Fourier transform, and then obtain

I0(1)

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Figure 19. Flowchart of the iterative reconstruction processing. (1) (2) The I 0 ,

(1) I1 and A1 make up of the near-field expression A1 exp[i 2S (I0  I1)]) ,

and the near field expression is transformed to the expression I1c exp(i 2S\1 ) by Fourier transform let the known I1 replace the expression I1 exp(i 2S\1 ) .

The

I1c and make up of the amendatory far field

expression I1 exp(i 2S\1 ) is

transformed

to

A1c exp[i 2S (I0(2)  I1)] by inverse Fourier transform, and then obtain I0( 2) ( n 1) by calculating step (2) iteratively (3) Working out I0 ( n 1)

(4) Let I0 = I0

, go to step (1) and continue the iteration until the condition error

H d c (a very small constant ) , and then the calculation output I0 can be deemed as the static wave-front aberration of the entire ICF beam path. The condition error H is defined as

H᧹¦ > A0  A0c @ / ¦ A0 . 2

¦ >I

or H᧹

2

 I 0c @ / ¦ I 0 2

0

2

(30)

(31)

It is advantageous for the iteration to reach convergence when In are mainly represented by low orders Zernike aberrations. The following simulative part imply that it is enough to retrieve the static aberrations of entire beam path when the first 5 low orders Zernike aberrations are modulated into ICF system.

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4.4. Numerical Simulation Suppose the near-field intensity distribution is a 8 order Gauss distribution. The entire beam path static aberrations

2

I0 and their corresponding far-field intensity distributions I 0 are

shown in Figure 20 (a). Now suppose the static aberration

I0 is unknown, and then retrieve

and correct I0 according to the technique described above. The configuration of the 41actuator DM [16] and the corresponding 25 25 sub apertures H-S sensor are shown in Figure 21 (a), the round pads are actuators and the grids are sub apertures. The unit voltage response function of actuators are second-order super-Gauss functions, and the inter-actuator coupling coefficient =12%. The DM modulates extra low order Zernike aberrations respectively,

the

corresponding

near-field

static

wave-front

are

I1 ᨺ I5 into I0 changed 2

to 2

( I1  I0 )ᨺ( I5  I0 ) and their corresponding far-field intensity distributions are I1 ᨺ I 5 . The five near-field wave-fronts and far-field distributions are shown in Figure 20(b)~ (f).

a) I0 is static aberrations of entire beam path of ICF, I 0 2 is corresponding intensity distribution of far-field.

I1

2

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I2

2

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Figure 22. The relationship between  and iteration times.

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143

Figure 25. Far-field encircled energy vs DL (diffraction limit).

4.5. Experiments for Entire Beam Static Aberrations Measurement For evaluating the effectiveness of the amendatory phase-retrieval method based on the classical G-S, We have accomplished the principle experiments. Figure 26 is the experimental layout of entire beam phase aberrations measurement. sing a super-Gaussian laser as the light source, a phase plate I0 is located ust before the second focus lens, and our aim is to ascertain the information of

I0 . At first, put no other phase plates in the entire optics beam 2

and recorded the far-field focal intensity distribution I 0 by a 8 bit CCD camera, and then put phase plate

I1 ~ I5 (which are shown in Figure 27) into the laser system( before the beam 2

splitter) respectively, and five corresponding far-field focal intensity distribution I1 ~ I 5 (which can be seen in Figure 28) is also recoded by CCD respectively. At last, the

2

I0 can be

figured out by executing the iterative steps described in Figure 19. From Figure 29 we can know that after about 100 times iterative calculation, H reach convergence. Figure 30 show The experimental reconstruction results of interferometer and᧤b᧥is the

I0 , of which᧤a᧥is the I0 measured by

I0 reconstructed by the amendatory phase-retrieval method

based on the classical G-S᧨whereas᧤c᧥is the residual wave-front aberrations, many factors that can induce the residual aberrations, for example, the non-linear response of the CCD to the focal intensity can affect the measurement greatly. What should also be noted is that since we using the theoretical focus length for calculation, and the difference between the actual focus length and theoretical focus length can also affect the measurement precision, therefore, there are some defocus aberration in the residual aberrations which can be seen in

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Figure 30᧤c᧥. For eliminating the defocus aberration, we should calibrate the difference in proper way.

Figure 26. The experimental layout of entire beam phase aberrations measurement.

Figure 27. The far-field focal intensity distribution of phase plate

2

I0 ~ I5

2

.

The New Process of Adaptive Optics Based on Nonlinear Control Algorithms…

Figure 28. The near-field wave-fronts of phase plates

Figure 29. The relation between

H

I1 ~ I5

and iterative times.

145

146

Ping Yang, Mingwu Ao, Bing Xu et al. Figure 30. The experimental reconstruction results of

I0 (a) is the I0 measured by

I

interferometer (b) is the 0 reconstructed by the amendatory phase-retrieval method based on the classical G-S, (c) is the residual wave-front aberrations.

Figure 30. The experimental reconstruction results of ᧤b᧥is the

I0 ᧤a᧥is the I0 measured by interferometer

I0 reconstructed by the amendatory phase-retrieval method based on the classical G-S, (c) is the residual wave-front aberrations.

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147

5. Conclusions In this chapter, An adaptive optics system based on genetic algorithms for optimizing a solidstate laser output modes has been presented, the principle of genetic algorithm on base of optimizing the Zernike mode coefficients has been introduced. Simulative and experimental results proved that this method can be used to control the laser mode well. We have also introduced a new adaptive optics technique for detecting and correcting the whole aberrations of entire ICF beam path. The DM and H-S sensor cooperate to modulate extra aberrations which are added on the ICF beam path, and the corresponding focal spots intensity distribution are detected by the CCD located on far-field. An amendatory phase retrieval algorithm that based on the G-S algorithm can reconstruct the whole aberrations of ICF. The numerical simulation shows that this technique is capable of detecting and correcting the whole aberrations of entire ICF beam path. Principle experimental results also show that this technique can well detecting the whole phase aberration of the entire laser system.

References [1]

Ping Yang, Shi ie Hu, Xiaodong Yang et al. (2005) Test and analysis of the time and space characteristics of phase aberrations in a diode-side-pumped Nd:YAG laser. Proc. SPIE, 6108, 182-91. [2] W.A.Clarkson, N.S.Felgate, D,C.Hanna,(1999) Simple method for reducing the depolarization loss resulting from thermally induced birefringence in solid-state lasers, Opt. Lett., 24,.820-822. [3] W. Lubeigt, G. Valentine, . Girkin, E. Bente , and D. Burns, (2002) Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptiveoptic mirror, Opt. Express. 10, 550-555. [4] D. Burns, G. . Valentine, W. Lubeigt, E. Bente, and A.I. Ferguson, (2002) Development of high average power picosecond laser systems, Proc. SPIE 4629, 4629-18. .W. Hardy, Adaptive Optics for Astronomical Telescope. (Oxford niversity Press, [5] 1998). [6] W H. iang, H G.Li, Hartmann-Shack wave-front sensing and wave-front control algorithm, Proc. SPIE. 1237, 64-67. [7] Goldberg D E. Genetic algorithms in search, optimization and machine learning [M]. Reading M A, SA: Addison-Wesley Publishing Company, Inc, 1989. [8] Oshichi Nemoto, Takuya Nayuki, Takashi Fu ii et al. (1997) Optimum control of the laser beam intensity profile with a deformable mirror . Appl.Opt,. 36 (30) 7689-7695. [9] A.H.Wright,Genetic algorithms for real parameter optimization. In Foundation of Genetic Algorithms, G. .E.Rawlines,ed. (Morgan Kaufman, San Mateo, Calif.,1991), . 205-218 [10] H. iang , N. Ling , X B. Wu, C H. Wang , H. Xian, S F. iang, Z . Rong, C L. Guan, L T. iang, Z B. Gong, Y. Wu , and Y . Wang, (1996) 37-element adaptive optics experimental system and turbulence compensation experiments , Proc. SPIE 2828, 312-320

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[11] Yang, S . Hu, S Q. Chen , W. Yang, B. Xu , and W H. iang, (2006) Research on the phase aberration correction with a deformable mirror controlled by a genetic algorithm, Journal of Physics: Conference series 48: 1017-1024 [12] W. H. iang, S. F. Huang, Ning Ling, X. B. Wu, Hill᧩climbing wave front correcting [13] [14]

[15]

[16]

[17] [18] [19]

system for large laser engineering, Proc. of SPIE. 1988, 1965 H. iang, Adaptive Optics Techniques Investigations in Institute of Optics and Electronics, Opto-Electronics Enginering, 1995, 22(1):1-13. Wang Fang, Zhu Qi-hua, iang Dong-bin, et al. Optimization of optical design of the master amplifier in multi-pass off-axis amplification system, Acta Physica Sinica, 2006, 55 (10): 5277-5282. D. Zhang, Z. P.Yang, H. F. Duan. Characteristics of wavefront aberration in the single beam principle prototype of the next generation ICF system, Proc. of SPIE. 2002, 4825:249-265. M. W. Ao , Z.P.Yang, Yang Ping, et al, Research on aberration correction for multipass amplification system with beam rotate 90 , High Power Laser and Particle Beams, 2007, 19(7):1167-1171. P. . Wegner , . M. Auerbach, C. E. Barker , et al, Frequency converter development for the National Ignition Facility, SPIE .1999, 3492 : 392- 405. R. W. Gerchberg, W. O. Saxton, Phase determination from imagine and diffraction plane pictures in the election microscope, Optik, 1971, 34(2):275-283. Z. Yang, B. Z. Dong , B. Y. Gu ,et al. Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a ninunitary transform system: a comparison, Applied Optics, Vol. 33, No. 2, 1994:209-218.

In: Progress in Nonlinear Optics Research Editors: Miyu Takahashi and Hina Goto, pp. 149-196

ISBN 978-1-60456-668-0 2008 Nova Science Publishers, Inc.

Chapter 5

HIGH-ORDER HARMONIC GENERATION IN LASER-PRODUCED PLASMA R. A. Ganeev Scientific Association Akadempribor, Academy of Sciences of zbekistan, Akademgorodok, Tashkent 100125, zbekistan

Abstract The studies of the high-order harmonic generation (HHG) in laser plasma are reviewed. We discuss the harmonic generation conditions in various laser plumes, which allowed achieving the HHG in plasma up to 7.9 nm. New approach for harmonic enhancement through the resonance-induced growth of HHG conversion efficiency in some low-excited plasmas is offered, which allowed achieving the 10-4 conversion efficiency for single harmonic. Various other experimental schemes and approches are reviewed as well, such as the harmonic generation in nanoparticles-contained laser plumes, excitation of laser plasma by the prepulses of different duration, variations of the chirp of femtosecond radiation for improving the brightness of harmonics, application of short-wavelength radiation for harmonic generation, etc.

1. Introduction High-order harmonic generation (HHG) can be considered as most effective method of coherent expreme ultraviolet (X V) radiation generation in a wide spectral range [1-9]. An alternative methods include the x-ray lasers [10,11] and free electron lasers. The achievements of x-ray lasers cannot pretend on the generation in a broad range compared with the sources based on the HHG. Free electron lasers remain to be the exotic sources due to their extremely high price and presently can be used in a few centres. Both of these methods suffer from the worse spatial coherence and divergence with respect to the harmonic radiation. Harmonic generation became a robust technique since the availability of moderate and compact laser systems possessing the high pulse energy, high pulse repetition rate, and short

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pulse duration. HHG is mostly realized through the two mechanisms, harmonic generation in gases [1,3-7] and harmonic generation from the surfaces [2,8,9]. A notable achievements of these sources allowed to extend the coherent radiation into the spectral range, where it can penetrate through the water-containing media (so called water-window range, 2.3-4.6 nm) [6,7]. This spectral range attracts the attention due to the interest to the biologican ob ects, which can be probed using the coherent radiation. However, the presently reported data on the conversion efficiency of these methods are still insufficient (10-5 and less), which restricts the application of such a radiation. A search of the methods for the improvement of the HHG in X V region during long time remained (and still remains) the important goal of the nonlinear optics. Currently, the harmonic yield remains below the applicable level for biology, plasma diagnostics, medicine, microscopy, photolithography, etc. Probably this led to switching the interests to the process, which was developed during the gas HHG studies, – attosecond pulses generation. The growth of harmonic yield through various processes was the cornerstone of many studies in this very competitive field. In particular, the search of the harmonic enhancement through the resonances with the atomic and ionic transitions of gaseous media was reported, though mostly by the theoretical methods [13,14]. Some calculations show that, at resonance conditions, one can achieve a considerable growth of harmonic intensity. This methods can be considered as an alternative (or additional) way compared with the phase matching technique using the waveguides with variable thickness of the gas-filled tubes [6,7]. Harmonic generation in the laser plasmas produced on the surfaces of solid targets initially was considered as a considerably less successful method compared with the two above-mentioning HHG techniques. The studies using highly excited laser plasma containing multiply charged ions [16-21] did not allowed to achieve both high harmonic cutoffs and high conversion efficiencies. Besides, the harmonic distribution in that case did not obeyed the three-step HHG model [22], which predicts the appearance of plateau (i.e. the approximately equal intensities of high-order harmonics). These studies were carried out at the mid of nineties and have demonstrated relatively low cutoffs (ranging from 11th to 27th order, in some of best cases, see last illustration). This led to the drop of interest in such method of the HHG. At the same time, the laser plasma can be used for the efficient HHG and shorterwavelength cutoffs. It was noted in Ref. 16 that, irrelevant of the physical and chemical properties of the targets, the plasma produced on those materials could be an efficient medium for the HHG. No principal restrictions appear here, and one has to find the optimal conditions for the laser plume, which could be used as an efficient nonlinear medium. For the creation of such conditions, one has to analyze the plasma parameters for different cases of excitation and define those of them when the influence of restricting factors (i.e., phase mismatch, self-defocusing) would be minimized [16,19,21]. Among the peculiarities of plasma HHG one has to mention a broad range of nonlinear medium s characteristics, which should be varied to optimize the harmonic generation. Among them are the plasma length, ion and electron concentration, excitation state of the plasma particles, etc. The application of various solid-state elements of the periodic table considerably broadens the range of the materials, which could be used as the nonlinear media, contrary to the gas HHG, where only few noble gases are used. This allowed for achieving in some cases the quasi-resonance conditions for the considerable growth of the single harmonic efficiency due to the influence of ionic transitions with strong oscillator strengths on the

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nonlinear susceptibility of those harmonics. This process unlikely could be realized in gas HHG due to the rare opportunity for the coincidence of the frequencies of harmonics and transitions in the gaseous media. The advatages of plasma HHG can be realized using the low-excited, low-ionized plumes due to the insignificance of the restricting factors in those conditions. This approach has been confirmed during recent studies of the HHG in such plumes [23-28]. Among the advantages of using such a plasma are the considerable growth of the maximum harmonic order, appearance of the plateau, resonance enhancement of single harmonics and other peculiarities (see also [29-32], where the above processes were further improved). As a results of recent plasma HHG studies, the harmonics in the range of sixtees and seventees orders were reported [24,26,33,34]. The highest harmonic cutoff (101st harmonic, O = 7.9 nm) was observed from the manganese plasma [35]. The conversion efficiency as high as 10-5 in the range of plateau-like distribtion of harmonics was reported [36]. Alongside with these studies, the resonace-induced enhancement of single harmonic allowed achieving the 10-4 conversion efficiency [25,26]. Some of these achievements are in the same range or higher compared with the parameters reported in the cases of gas HHG. However, most important difference between the plasma and other HHG techniques is the very high cutoffs achieved in the latter cases. Presently, the harmonics exceeding the 300th orders were reported in the gas HHG studies [1,6]. Moreover, recently, the harmonics over the 3200th orders were achieved during the harmonic generation at the specular reflection of ultraintense pulses from the surfaces [37]. To overcome this gap, one has to study some new possible approaches in the case of plasma HHG, in particular the application of doubly-charged ions for the harmonic generation. As it was mentioned, the resonance enhancement of single harmonic (but not the group of harmonics at the palteau region) allows us the practical application of such sources of coherent X V radiation. There are few new reports indicating that such an opportunity can be realized when one uses the tuning of harmonic wavelength toward the ionic transitions with strong oscillator strengths [25,31,32], or uses the chirp-induced ad ustment between the frequencies of harmonic wavelength and appropriate transitions of the atoms and ions [26,28,30]. The theoretical basis of HHG in isotropic media has detailly been described during the early studies of this phenomenon in gaseous media [22,38,39], as well as in recent publications [40-43]. Since the goal of this review is to acquaint the reader with new approaches of the plasma HHG, we mostly concentrate on the experimental findings, reported in this field. The Chapter is organized as follows. In Sections 2 – 5, the peculiarities of harmonic generation from the B, Ag, Au, Mn, and V low-charged plasmas are presented. In Section 6, the resonance-induced enhancement of single harmonic obtained in various plumes (In, Cr, InSb, GaAs) is discussed. The application of nanoparticles-contained plumes for the enhancement of harmonic yield is presented in Section 7. The HHG using the second harmonic of Ti:sapphire laser radiation is analyzed in Section 8. The application of the laser prepulses of different duration for plasma formation is analyzed in Section 9. The selfdefocusing properties of laser plumes and the optimization of plasma conditions are discussed in Section 10. We summarize these studies in Section 11.

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2. Boron Plasma Two schemes of the interaction of the femtosecond radiation with the low-excited plasma are used in the studies of the HHG in laser plasmas. In the first (orthogonal) scheme, a portion of the uncompressed radiation of Ti:sapphire laser was split from the main beam by a beam splitter and used as a prepulse. This sub-nanosecond prepulse was focused by a spherical lens L2 on a solid target T located in the vacuum chamber and produced a plume predominantly consisting on the neutrals and singly charged ions (Figure 1a). After some delay, the femtosecond main pulse was focused on the target plasma from the orthogonal direction. In the second (longitudinal) scheme, the sub-nanosecond prepulse was focused by a lens L1 at the edges of the hole drilled in the target T (Figure 1b). The divergence of this radiation was increased using the additional lens pair L2 to achieve a broad beam size (700 Pm) at a focal plane of the focusing lens L1. The second beam was used as a main radiation interacting with the plasma at different delays with regard to the sub-nanosecond prepulse. The femtosecond main pulse was focused on the plasma by a lens L1 from the longitudinal direction.

Figure 1. Experimental arrangements of the (a) orthogonal and (b) longitudinal pump schemes of harmonic generation. T: target L1, L2: focusing lenses TFP: thin film polarizer PSP: picosecond prepulse FSP: femtosecond pulse FSP+HH: femtosecond pulse and high-order harmonics.

In these studies, a commercial, chirped-pulse amplification Ti:sapphire laser system was used, whose output was further amplified by a homemade three-pass amplifier. A portion of the uncompressed radiation (24 m , 210 ps, 796 nm center wavelength) was split from the main beam by a beam splitter and used as the prepulse. This prepulse was focused on a 5 mm thick boron slab target by a spherical lens (Figure 1a). The focal spot diameter of the prepulse beam at the target surface was ad usted to be 600 Pm. The intensity (Ipp) of the prepulse on the target surface was varied between 1u1010 to 1u1011 W cm2. After some delay, the femtosecond main pulse (10 m , 150 fs) was focused by a 200 mm focal length lens on the boron plasma from the orthogonal direction. The maximum intensity (Ifp) of the main

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femtosecond pulse was 2u1015 W cm2. The confocal parameter (b) of the femtosecond laser beam was 4.8 mm. The generated harmonics were analyzed by a flat-field grazing incidence X V spectrometer with a Hitachi 1200 groove/mm grating. X V spectrum was detected by a micro-channel plate (MCP) with a phosphor screen, and its image was recorded by a chargecoupled device (CCD) camera. The high-order harmonics up to the 63rd order (O = 12.6 nm) were observed in these experiments [23]. The HHG from B plume appeared to be similar to those observed in the gas- et studies, with a characteristic shape of the plateau for the harmonics exceeding the 21st order (Figure 2). The conversion efficiencies varied in the range from 104 (for the 3rd harmonic) to 107 (for the harmonics in the plateau region). The plateau disappeared when the prepulse energy was increased above certain level, which generated a considerable amount of free electrons and continuum emission from the B plume. In this case, the highest harmonic observed was the 19th order. Further growth of prepulse energy led to the appearance of the strong spectral lines from the multiply ionized boron, which prevented observing any harmonics below 65 nm.

Figure 2. Harmonic spectra generated in the boron plasma obtained in the case of orthogonal scheme.

The optimization of the HHG in the boron plasma was carried out by changing the focusing geometry of the femtosecond beam to generate the harmonics at the tight focusing conditions. In that case, the confocal parameter of the focused radiation (b = 1.2 mm) was close to the plasma sizes (Lp | 0.7 mm). High-order harmonics up to the 65th order (O = 12.24 nm) were observed in these experiments [26]. The conversion efficiency at the plateau range was measured to be 5u10-7. The harmonics disappeared at the high intensities of prepulse radiation (Ipp 5u1010 W cm-2). Further studies of the HHG in the boron plasma were carried out using the longitudinal pump scheme. The 210 ps, 15 m , 796 nm prepulse from the uncompressed radiation of Ti:sapphire laser was used for the excitation of boron plasma. This radiation was focused by a

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200 mm focal length lens into the hole of the boron target placed in the vacuum chamber (Figure 1b). The femtosecond main pulse was focused by a 200 mm focal length lens from the longitudinal direction on the plasma volume with the maximum intensity at the plasma area of Ifp = 2u1015 W cm-2. The beam waist diameter and the confocal parameter of the femtosecond radiation were measured to be 60 Pm and 6 mm, respectively. This radiation and generated harmonics propagated through the plasma and the 0.2 mm hole drilled in the 4-mmthick boron. The heating picosecond radiation played a crucial role in the creation of the optimal conditions of harmonic generation in this scheme. At small prepulse energies, the IqZ (Ipp) dependences were obeyed to the scale low with a slope in the range of 3 to 4. The high-order harmonics up to the harmonic cutoff Hc = 57 (O = 13.96 nm, Figure 3) were observed in these experiments [44].

Figure 3. Harmonic intensity as a function of the wavelength of generated X V radiation in the case of longitudinal scheme of the HHG in boron plasma.

3. Silver Plasma In the case of silver plasma, a plateau pattern starting from the 11th harmonic was observed (Figure 4). A conversion efficiency at the plateau region was measured to be 1u10-6, taking into account the absolute measurements of the conversion efficiency of low-order harmonics described in Ref. [36].

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Figure 4. Part of the harmonic spectra generated in the Ag plasma. Inset: the Hc(Ifp) dependence.

The harmonic output at the plateau region was further improved by using the optimal prepulse energy, delay between the 210 ps prepulse and 150 fs main pulse, distance between the target and main beam, etc. The increase of harmonic yield at loose focusing of the femtosecond pulse was observed. A conversion efficiency in that case was improved up to 8u10-6. The Hc (Ifp) dependence was saturated at Ifp = 3.5u1014 W cm-2 (see inset in Figure 4). This intensity of 150 fs main pulse is well above the barrier suppression intensity at which neutral Ag atoms are ionized. The questions arise (a) whether the free electrons generating during ionization of neutral atoms affect the Hc (Ifp) dependence, and (b) what the physical process governs the saturation of this dependence In the case when this free electron concentration is not sufficient for the phase mismatching, self-defocusing, and suppression of peak intensity, then the saturation arises due to further ionization of singly charged ions and growth of free electron concentration. The calculations of the barrier suppression intensity of singly charged Ag ions (2.1u1014 W cm-2), which was close to the observed saturation of Hc (Ifp) dependence, confirm the consideration of singly charged ions as the main source of harmonics in these experiments. Further studies of silver plasma were carried out using the 48 fs, 795 nm laser pulses. In these experiments, the Ag plasma predominantly consisted on the neutrals and small amount of singly charged ions, which was confirmed by the spectral analysis in the visible, V, and X V ranges. However, during the interaction of this plasma with the femtosecond pulses, an increase in the concentration of singly charged ions as well as the generation of multiply charged ions were registered. The harmonics up to the 61st order (O = 13 nm) and the prolonged plateau pattern were observed in these studies (Figure 5). A systematic study was performed to maximize the harmonic efficiency, cutoff energy, and sharpness of harmonics. The optimal conditions for harmonic generation were found to be at a distance 100 - 150 Pm above the target surface.

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The optimal distance, at which the maximum harmonic yield was observed, depended on the delay between the prepulse and the main pulse. A decrease of the conversion efficiency was observed with the increase of the prepulse intensity above 3u1010 W cm-2 [33].

Figure 5. High-order harmonics generated in the silver plume. The satellites appearing from the both sides of the 21st to 29th harmonics are the second-order diffractions of the higher-order harmonics.

The high-order harmonic yield as a function of the focus position of the driving laser radiation with regard to the plume [Ih(z)] depended on the main pulse intensity. At low laser intensity, a single maximum of Ih(z) dependence was observed when the femtosecond pulse was focused inside the plasma, while at high intensity, a two-peak pattern of Ih(z) dependence with distinct difference between peak intensities was appeared. In latter case, no harmonics were observed when the laser beam was focused inside the plume. The high-order harmonic yield saturated at the laser intensity above the barrier suppression intensity of singly charged Ag ions. Further growth of laser intensity in the vicinity of the plume led to the abrupt decrease of the HHG. A key point of the next set of Ag plasma studies was a variable energy controller, which allowed independently change the energies of the two pulses [45]. Initially, with the increase of the prepulse intensity, the peak spectral intensity of the harmonics also increases. This process saturates at relatively low prepulse intensities for the low-order harmonics and at higher prepulse intensities for the high-order harmonics. When one continues to further increase the prepulse intensity, the bandwidth of the low-order harmonics significantly increases comparing with the higher-order harmonics. Figure 6 shows the spectral bandwidths of the 17th, 25th, and 43rd harmonics as the functions of the prepulse intensity [46]. The calculations of plasma characteristics were carried out using the HYADES code. This one-dimensional code has been developed to simulate laboratory experiments on plasmas driven by intense sources of energy. The code was constructed with the ob ectives of providing a tool that is (1) easy to use by an experimentalist (2) formulated using simple yet accurate numerical approximations to the physics models, (3) easy to modify and extend with new models, and (4) run on a variety of computers. Recent additions and modifications have been made to allow more realistic simulations of materials at temperatures below a few eV. Initially elaborated by Larson and Lane [47], it currently applied for simulation of energy deposition and modeling of various plasma instabilities [48], underdense plasma optimization [49], etc. The following conclusion could be drawn based on the analysis of the simulations by using HYADES code (Table 1). At the prepulse intensities below 0.9u1010 W cm-2, the ionization level of Ag plasma remains lower than 1 but higher than 0.5. The ablation contains singly ionized and neutral atoms, which generate the harmonic spectrum.

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Figure 6. Bandwidths of the 17th, 25th, and 43rd harmonics as the functions of prepulse intensity.

Table 1. Results of the calculations of silver plasma characteristics at different prepulse intensities Prepulse intensity, u1010 W cm-2

0.3

0.59

0.9

1.3

1.95

2.6

3.25

0.89

2.56

3.2

4.0

5.79

6.33

8.0

Ion density, u10 cm

0.89

2.56

3.17

3.41

4.45

4.55

5.06

Ionization level

0.5

0.7

1.01

1.17

1.3

1.39

1.58

17

-3

Electron density, u10 cm 17

-3

At the prepulse intensities between 0.9u1010 W cm-2 and 3u1010 W cm-2, the ionization level becomes higher than 1. The plasma contains a considerable amount of singly ionized atoms and small amount of doubly ionized atoms. At the intensities exceeding 3u1010 W cm-2, the ionization level becomes higher than 1.5, thus creating a significant amount of doubly charged ions. The growth of free electron concentration at this case prevents the efficient harmonic generation due to the phase mismatch, self-defocusing, and self-modulation of femtosecond pulse.

4. Gold Plasma The characterization of laser plasma emission plays important role in the optimization of the HHG from the plasma plumes. Previous analysis of the plasma spectra was carried out at the time-integrated mode, so it was impossible to define precisely the plasma conditions existing

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at the moment of the propagation of the femtosecond pulse through the plume. In this Section, the optimization of the HHG using the time-resolved spectral studies of the gold plasma in the V range is presented. The time gate for each spectral measurement was 20 ns. The measurements were carried out each 10 ns after the beginning of the irradiation of Au target by the prepulse until to 150 ns. The time-resolved studies of the gold plume were carried out at both the optimal and nonoptimal conditions of plasma excitation. The variation of Au plasma emission in the vicinity of the V spectral lines related with the excitation of neutral atoms and singly charged ions (266-295 nm) was measured. The optimal plasma, at which both the plateau-like distribution and maximum harmonic cutoff were observed (at Ipp | 1u1010 W cm-2), was generated for these purposes. The variations of Au plasma spectra in the case of the interaction of the main pulse with the laser plasma at the 100 ns delay are presented in Figure 7.

Figure 7. Time-resolved V spectra of the optimal Au plasma. Ipp= 1.0u1010 W cm-2.

The decay times of the excited neutral lines considerably exceeded those of the ionic lines. One can see a slow decrease of the intensities of the neutral Au lines (267.59 and 274.82 nm) compared to the Au+ lines (280.20, 282.25, and 291.18 nm). The laser radiation (t = 35 fs) arriving 100 ns after the beginning of plasma formation excites exclusively the ionic lines, while no considerable changes in the intensities of neutral lines occur after the propagation of the main pulse. Another pattern of V spectrum dynamics was observed in the case of relatively small increase of the prepulse intensity at the surface of Au target. The increase of Ipp from 1u1010 to 2u1010 W cm-2 led to a considerable excitation of the gold plume (Figure 8).

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Figure 8. Time-resolved V spectra of the non-optimal Au plasma. Ipp= 2.0u1010 W cm-2.

Apart from an approximately ten-fold growth of the intensities of above-mentioned neutral and ionic lines, some other ionic lines appeared in plasma spectra. At these conditions, the excitation of the ionic and neutral lines by the femtosecond pulse was insignificant and barely seen, when one compares the spectra at 80 and 100 ns timescales. The slices of the time-resolved V spectra for 90 and 100 ns timescales showed a considerable difference in the excitation of ionic and neutral lines by the femtosecond radiation at the conditions of the formation of optimal plasma. Analogous pattern could be expected in the X V range as well, though no measurements of the time-resolved spectra of plasma lines in the range of plateau-like harmonics were carried out. However, the appearance of strong ionic lines in the time-integrated X V spectra coincided with a decrease or even almost disappearance of the high-order harmonics. In particular, the increase of prepulse intensity from 1u1010 to 2.5u1010 W cm-2 led to a decrease of the 13th harmonic intensity with the factor of 2.5. The optimization of plasma parameters allowed further improving the harmonic generation from the Au plasma (Figure 9). The harmonics up to the 53rd order (O = 15.09 nm) were generated at the plasma conditions chosen after the analysis of time-resolved spectra [50]. The comparison of harmonic generation efficiencies in the gold and silver plumes showed that, in the former case, the intensities of plateau-like harmonics were approximately a few times lower compared with the intensities of the harmonics generated in silver plasma at the same spectral region. The conversion efficiency in the case of gold plume was estimated to be 2u10-6.

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Figure 9. Harmonics obtained from the optimally prepared gold plume at a 100 ns delay between the prepulse and main pulse.

There are several advantages of time-resolved plasma spectroscopy studies in attaining the goal of harmonic efficiency enhancement. 1. Time-resolved analysis of plasma spectra reveals the detailed physics of harmonic generation from plasma plume. 2. Temporal characterization of the plasma emission allows improving the enhanced intensity of single harmonic in resonance-induced HHG experiments. 3. Time-resolved analysis is essential when one studies the use of new approaches (such as application of double-target schemes of plasma excitation, doubly charged ions, etc), which have recently emerged for further enhancement of the output characteristics of plasma harmonics. The above-presented results are the first optimization of plasma HHG using the timeresolved plasma spectroscopy. This technique allows defining the optimal delays between the prepulse and main pulse for specific plumes, at which the maximum conversion efficiency can be achieved. In particular, the recombination times and dynamics of plasma expansion for low-Z plumes show considerably different pattern with regard to the high-Z plumes. For such cases, the time-resolved plasma spectroscopy data allows for optimizing the interaction of the femtosecond pulse with the plume to produce best conditions for the HHG.

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5. Manganese and Vanadium Plasmas A strong correlation between the harmonic cutoffs and the ionization potentials (Ii) of the particles participating in the HHG in different plasmas was reported in Refs. 28 and 33. In most cases, the singly charged ions were responsible for the harmonic generation from laser ablation, so one can use a second ionization potential of target atoms as a parameter for the prediction of maximally observed cutoff energy. This relation has found a confirmation in the measurements using three Ti:sapphire lasers operated at different pulse duration (35, 48, and 150 fs) (Figure 10). The experimental dependence can be approximated by the following empirical relation Hc [harmonic cutoff] | 4 Ii [eV] – 32.1.

(1)

The conclusion that could be drawn from this relation is as follows. In the case of the involvement of singly charged ions in the HHG from laser ablation, shortest wavelength harmonics are generated from the targets that have higher second ionization potentials. The appearance of additional free electrons (due to the ionization of singly charged ions by further increasing the fundamental intensity) leads to the saturation of the Hc (Ifp) dependence, thus restricting the generation of higher-order harmonics. In most cases, there is no need to include doubly charged ions for explaining the cutoff of the high-order harmonics from the ablated medium, since their appearance coincides with the growth of the free electron concentration in the plume. The studies of the HHG from the manganese ablation using the 800 nm, 35 fs pulses have revealed new features of this process. As in previous cases with other targets, the harmonic generation from this medium up to the harmonic cutoff Hc = 29 was observed, which well coincides with the empirical relation (1), taking into account the second ionization potential of Mn (I2i = 15.64 eV) (see Figure 10). However, with further variation of plasma conditions (when the manganese surface was irradiated by the subnanosecond prepulse radiation of higher intensity than the one corresponded to the optimal plasma conditions for most of targets, Ipp | 1u1010 W cm-2), a considerable increase in the harmonic cutoff was obtained from this plume. Harmonics as high as the 101st order were clearly identified in this case, though with less conversion efficiency compared with the case of smaller prepulse intensities (Figure 11). Instead of a plateau-like harmonic distribution in the range of 15th to 29th orders (for moderate irradiation of the Mn target by the subnanosecond prepulse), another plateau pattern was appeared at higher orders (from the 33rd to 93rd harmonic) with further steep drop of harmonic intensity up to 101st order (O = 7.9 nm) [35]. The observed harmonic cutoff well coincided with the empirical relation (1), taking into account the involvement of doubly charged ions and the third ionization potential of manganese (Isi = 33.67 eV) (Figure 10). During the studies of other plasma plumes, the opposite feature of the influence of prepulse intensity on the preparation of optimal plasma was observed, when both highest harmonic order and conversion efficiency were achieved. In particular, in the case of the Au plume, the harmonic spectra worsened when the prepulse energy was increased above 17 m , which corresponded to the intensity of Ipp | 3u1010 W cm-2 at the target surface.

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Figure 10. Maximum harmonic order observed in plasma HHG experiments as a function of the ionization potentials of the singly charged and doubly charged ions participating in harmonic generation. Dotted line shows the empirical relation Hc [harmonic cutoff] | 4Ii[eV] – 32.1. Filled squares are the results obtained in Institut National de la Recherche Scientifique, Canada, open circles are the results obtained in the Institute for Solid State Physics, apan, and open triangles are the results obtained in Ra a Ramanna Centre for Advanced Technology, India.

Figure 11. A lineout of the high-order harmonic spectrum obtained at Ifp = 2u1015 W cm-2 and Ipp = 3u1010 W cm-2. Inset: Part of harmonic distribution between the 67th and 101st orders.

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The strong spectral lines from multiply charged gold ions appeared in the X V spectrum when the prepulse energy was further increased, and no harmonics were distinguished in the region below 33 nm. The simulations using the HYADES code were performed to analyze the phenomenon of different behavior of the HHG for two different plasma plumes. The expansion of the heated manganese and gold surfaces interacting with the laser prepulse was simulated, and the electron density, ionization level, and ion density of these plumes as a function of the prepulse intensity, at a distance of 300 Pm from the target surface, were determined. These data for 100 ns delay are summarized in Table 2. One can see from the data that, already at 1u1010 W cm-2, a considerable difference in the ionization states of the two plasmas can lead to the difference in nonlinear optical response of Au and Mn plumes. At this intensity, the ionization level of Au plume becomes higher than 1, which leads to the appearance of additional free electrons, due to the ionization of singly charged ions. The ratio between the electron density and ion density in gold plasma continues to increase at higher prepulse intensities, and at Ipp = 5u1010 W cm-2, their ratio becomes higher than 4. The growth of free electrons concentration prevents efficient harmonic generation and extension of harmonic cutoff, due to the self-defocusing of femtosecond pulse and growing phase mismatch between the harmonic and pump laser waves. The characteristics of manganese plasma under the same conditions are considerably different from those of Au plume. The appearance of doubly charged ions in the preformed plasma can only be expected at Ipp 3u1010 W cm-2, thus restricting the growth of free electrons concentration at smaller intensity. This means less influence of the free electrons on the self-interaction of main femtosecond pulse inside the Mn plume at higher prepulse intensity compared to the gold plume. These features of manganese plasma allowed for achieving the highest observed harmonics.

Table 2. Simulations of Mn and Au plasma characteristics at different prepulse intensities Intensity (1010 W cm-2)

Target Mn Au

1 1.3 7.32

2 3.25 14.2

3 3.77 18.5

5 6.13 25.2

Mn

1.3

3.25

3.77

4.50

Ion density (10 cm )

Au

4.7

6.03

6.8

7.52

Ionization level

Mn Au

0.62 1.56

1.0 2.35

1.0 2.72

1.36 3.35

Electron density (1017 cm-3) 17

-3

One can conclude from the above analysis that the search of optimal conditions for the plasma generation using other targets through simulation of plasma characteristics and the time-resolved analysis of the spectral dynamics of expanded plume can reveal new findings in the extension of highest harmonic toward the soft x-ray region. Some indirect confirmation of this approach was recently found for harmonic generation from vanadium plasma [34]. The harmonic cutoff of this medium (I2i = 14.65 eV) could be estimated in the range of about the thirties harmonics, according to the empirical relation (1), assuming that the singly charged ions are the highest ionization states participating in HHG. In fact, the 39th harmonic of 35 fs,

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800 nm radiation was observed from this plume, which is in the range of uncertainty of the empirical relation presented in Figure 10. However, recent studies using the 796 nm, 150 fs pulses have shown the harmonic generation from this plume at considerably shorter wavelength (with harmonics generating up to the 71st order). The consideration of the HHG from vanadium plasma at specific conditions of ablated plume using the same approach as the one, which was presented in above discussion, predicts that, with the involvement of doubly charged ions (I3i = 29.31 eV), the maximum expected cutoff would be about the 79th harmonic, which is in a reasonable agreement with the experimentally obtained cutoff (see Figure 10). Note that the simulations revealed the closeness of V and Mn plasma characteristics. Recently, analogous observation of the enhancement of harmonic cutoff by using doubly charged ions was reported in the case of titanium plasma [51].

6. Resonance-Induced Processes Influencing the Single Harmonic Intensity A bright, monochromatic, coherent source in the X V range could be useful for many applications. X-ray lasers and high-order harmonics are the coherent sources of radiation in this region. Although the latter sources have a better coherence than the x-ray lasers, they are not monochromatic, unless one selects some harmonic from the plateau using a monochromator. The use of phase matching conditions to replace the well-known plateau of the harmonic distribution by the group of intense harmonics is a step toward this direction [7]. Though the achievement of an intense single harmonic looks unrealistic at the moment, there are some techniques, which can considerably enhance a single harmonic with regard to the neighboring harmonic orders. The possibility of the enhancement of the high-order harmonics in gaseous media using the atomic and ionic resonances has been extensively studied theoretically [13,14]. This approach can be an alternative to the phase matching of the pump and harmonic waves using the gas-filled waveguides [6]. Furthermore, for the plumes generated at the surfaces of some solid targets, the resonance conditions between the harmonic wavelength and the excited states of neutrals and singly charged ions can lead to the enhancement of the yield for some specific harmonic orders [31,32]. The availability of a much wider range of target materials for this purpose compared to a few gases increases the possibility of the resonance of an ionic transition with a harmonic order during plasma HHG studies. The origin of the strong yield of the single harmonic in the plateau region could be associated with the resonance-induced growth of conversion efficiency. The role of atomic resonances in harmonic generation was an important sub ect of discussion in the early studies of low-order harmonic generation ([52] and references therein). The first observation of such enhancement was related with the harmonic at the beginning of the plateau region [24]. In this Section, the observation of the enhancement of a single harmonic intensity in different parts of plateau in the case of plasma HHG is discussed. Such a finding paves the way for the creation of almost single wavelength source of X V radiation through the HHG from different plumes.

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6.1. Giant Enhancement of the 13th Harmonic in the Indium Plasma A typical harmonic spectrum obtained from the In plume using the 796 nm, 150 fs laser pulses is shown in Figure 12. High-order harmonics up to the 39th order (O = 20.4 nm) were observed in these experiments and showed a plateau pattern (Figure 12, curve 1). The harmonic spectrum generated from silver plasma under the same experimental conditions is presented here for the comparison, which showed an analogous characteristic plateau region for the harmonics exceeding the 11th order (Figure 12, curve 2). The conversion efficiency at the plateau region in the case of the In plasma was measured to be 8u10-7. The most intriguing feature observed in these studies was a very strong 13th harmonic, whose intensity was almost two orders of magnitude higher than those of its neighbors. The conversion efficiency of the 13th harmonic was 8u10-5, and for the pump energy of 10 m , this corresponded to 0.8 P [24]. After the observation of such an unusual harmonic distribution, the question arises as to whether the strong emission associated with the 13th harmonic (O = 61.2 nm) originates from amplified spontaneous emission, re-excitation of plasma by a femtosecond beam, or nonlinear optical process related to enhancement of a separate harmonic due to its spectral proximity to resonance transitions. The pump laser polarization was varied to analyze strong emission near 61 nm in the harmonic spectrum. Small deviation from linear polarization led to a considerable decrease of the 61.2 nm pulse intensity, which is a typical behavior for high-order harmonics. The application of circularly polarized laser pulses led to the complete disappearance of 61.2 nm emission (Figure 13). At the same time, the excited lines of the plasma spectrum observed at different polarizations of the main beam remained unchanged, which clearly shows that the strong emission at 61.2 nm was of nonlinear optical origin. The wavelength of main pump beam was tuned to analyze whether the excited ionic transitions from indium plasma influence the plateau pattern of harmonic distribution. The central wavelength of the output radiation of Ti:sapphire laser was tuned from 770 to 796 nm. The 13th harmonic output was considerably decreased with the detuning of the fundamental wavelength from 796 nm toward the shorter wavelength region. At the same time, a strong enhancement of the 15th harmonic of the 782 nm radiation was observed, while the intensities of other harmonics remained relatively unchanged (Figure 14). These observations show the influence of ionic transitions on the intensity of individual harmonics. These studies have demonstrated that the 26-nm shift in the central wavelength of the main pulse (which corresponded to a 2-nm shift in the wavelength of the 13th harmonic) considerably changed the overall pattern of harmonic distribution at the plateau region. The question arises as to why one can achieve a strong enhancement of a single harmonic within the harmonic spectrum in the case of indium plasma. Comparison with a past study on In plasma emission [53] showed that the emission in the range of 40 – 65 nm is due to radiative transitions to the ground state (4d10 5s2 1S0) and the low-lying state (4d10 5s 5p) of In+.

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Figure 12. High-order harmonic spectra obtained in the (1) indium and (2) silver plumes.

Figure 13. 61.2 nm emission yield as a function of the rotation of quarter-wavelength plate.

Previous work [53] revealed an exceptionally strong line at 62.1 nm (19.92 eV), corresponding to the 4d10 5s2 1S0 o 4d9 5s2 5p (2D) 1P1 transition of In II. The oscillator strength gf of this transition has been calculated to be 1.11, which is more than 12 times larger than other transitions in this spectral range.

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Figure 14. Harmonic spectra from indium plasma using (a) 796 nm, (b) 782 nm, and (c) 775 nm main pulse. (d) Spectrum of In plasma generated at high intensity of prepulse radiation.

This transition can be driven into resonance with the 13th harmonic (O = 61.2 nm, Eph = 20.26 eV) by the AC-Stark shift, thereby resonantly enhancing its intensity. Such intensity enhancement can be attributed to the existence of oscillating electron tra ectories that revisit the ionic core twice per laser cycle [14]. Since such tra ectories start from the resonantly populated excited state, with a nonzero initial kinetic energy, they still have nonzero instantaneous kinetic energies when they return to the origin. As usual, recombination results in the emission of harmonics, but due to the relatively low probabilities, the population in the laser-driven wave packets increases continuously and the probability for harmonic emission grows with the number of allowed recollisions. This multiple recollision is predicted to enhance harmonics in the spectral ranges close to the atomic and ionic resonances. Such an assumption has recently been confirmed in the case of further studies of harmonics from the indium plasma [25,54].

6.2. Single Harmonic Enhancement in Cr, GaAs, and InSb Plasmas High-order harmonic generation from the plumes of a number of targets was studied prior to choosing these samples as suitable ones for the observation of a single harmonic enhancement in the short-wavelength range. The harmonic distribution from most of plasmas showed a featureless plateau-like shape in the X V range, while some plumes demonstrated a steady or even steep decrease of conversion efficiency for each next harmonic order. The harmonics of 48 fs, 795 nm radiation from the chromium, gallium arsenate, and indium antimonite plumes were studied in more details due to the observation of abnormal harmonic distribution in the plateau region. These observations were compared with the harmonics generating in the indium plasma. The harmonics up to the 29th (27.3 nm), 43rd (18.4 nm), and 47th orders (16.9 nm) were observed in these experiments with InSb, GaAs, and Cr plumes, respectively. These targets showed some enhancement (or decrease) of specific harmonic order. These studies were performed using the chirp-free 48-fs pulses. In further studies, the chirp of the driving laser pulse was varied by ad usting the distance between the gratings in the pulse compressor. The

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change of laser chirp resulted in a considerable variation of the harmonic distribution in the mid- and end-plateau ranges, due to the tuning of some harmonics toward the ionic transitions [55]. The HHG in Cr plasma showed a considerable variation of the 27th harmonic intensity at different chirps of the driving radiation [27,56]. At some chirps, the 27th harmonic almost disappeared from the harmonic spectrum. At the same time, a strong 29th harmonic (27.3 nm) was observed in the case of chirp-free pulses (Figure 15). The chirp variation led to a change in the 29th harmonic yield compared to the neighboring harmonics. The maximum ratio of the 29th and 31st harmonic intensities was measured to be 23. It should be noted that, for negatively chirped pulses, the harmonic lines became sharper [Figure 15(b)]. At high negative chirp, the harmonic spectrum was considerably detuned from the resonances causing the absorption of the 27th harmonic and the enhancement of the 29th harmonic. In that case, the intensities of theses harmonics became comparable to each other.

a

b Figure 15. Harmonic distribution in the case of Cr plume at different chirps of the driving pulse. (a) chirp-free 48 fs pulses, (b) negatively chirped 160 fs pulses.

Analogous variations of harmonic distribution at the end- and mid-plateau ranges were observed for GaAs and InSb plumes. The change of laser chirp resulted in a tuning of the harmonics generating in the GaAs plasma [29]. At the chirp-free case and for negatively chirped pulses, a featureless plateau-like distribution of high-order harmonics with a gradual decrease of the harmonic intensity was observed [Figure 16(a)]. However, for positively chirped laser pulses, an enhanced 27th harmonic (29.4 nm) was appeared [Figure 16(b)]. The intensity of this harmonic was six times higher then the intensities of neighboring harmonics. In the case of InSb plume, a strong 21st harmonic (37.8 nm) of chirp-free driving radiation was observed (Figure 17). This figure shows a 20u enhancement of the 21st harmonic with regard to the neighboring harmonics [25]. The 21st harmonic intensity varied at different chirps of pump laser. In particular, in the case of positively chirped 140-fs pulses, the 21st harmonic exceeded the neighboring ones by a factor of 10. The enhancement of this harmonic considerably decreased in the case of negatively chirped pulses.

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a

b Figure 16. Harmonic distribution in the case of GaAs plume using the (a) chirp-free 48 fs pulses, and (b) positively chirped 130 fs pulses.

Figure 17. Harmonic distribution in the case of InSb plasma and chirp-free 48 fs driving pulses.

The origin of the enhanced emission in the vicinity of 27.3 nm (the 29th harmonic from the Cr plume), 29.4 nm (the 27th harmonic from the GaAs plume), and 37.8 nm (the 21st harmonic from the InSb plume) was analyzed by inserting a quarter-wave plate in the path of the femtosecond laser beam. No harmonics appeared for circularly polarized laser pulses, as could be expected assuming the nonlinear optical origin of observed spectra. The enhancement of the single harmonics belonging to the mid- and end-plateau regions was smaller comparing with the enhancement of the 13th harmonic generated from the indium plume, which showed a 10-4 conversion efficiency [25]. The enhancement of the 13th harmonic (200u) generated from the In plasma, considerably exceeded the enhancements for the 21st (20u), 27th (6u), and 29th (23u) harmonics generated from the InSb, GaAs, and Cr plasmas, respectively. Such a difference was attributed to the different oscillator strengths of the ionic transitions involved in the resonance enhancement of harmonics [54]. It follows from preceding results that the origin of the strong yield of the single harmonics in the plateau region is associated with the resonance-induced growth of nonlinear optical frequency conversion. Therefore, let us examine the resonance-induced growth mechanism in a little more detail. Some experimental observations (in particular, the dependences of the harmonic yield on the beam waist position, plasma sizes, and laser radiation intensity) point out effects related to a collective character of the HHG from laser plumes. Among the factors enhancing harmonic output are effects related to the difference in the phase conditions for different harmonics. The phase mismatch ('k=nk1-ki, where k1 and ki are the wave numbers of the laser radiation and ith harmonic) changes due to the ionisation caused by propagation of the driving pulse through the plume. According to calculations

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[16,17], in the plateau region the phase mismatch caused by the influence of free electrons is about one to two orders higher than those caused by the influence of atoms and ions. At the resonance conditions, when the harmonic frequency is close to the frequency of the atomic transition, the variation of the wave number of a single harmonic could be considerable [52], and the influence of free-electron-induced mismatch can be compensated for by the atomic dispersion for specific harmonic order. In that case, improvement of the phase conditions for single-harmonic generation can be achieved. Such a mechanism probably was responsible for the enhancement of above-described nonlinear optical processes, as well as the single harmonic enhancement observed in Sn [31], Sb [32], Te [57] and Mn [35,54], plasmas. Recently, the theoretical analysis of resonance-induced enhancement of the HHG in laser plasma was presented and compared with the experimental data described in this Section [40]. It was shown that this strong intensity enhancement of single harmonic can be explained using the theory presented in [41], generalized to the case of a superposition of states having different parity. The excited state is embedded into the continuum and is metastable. The crucial point of this consideration is that the transition probability between the ground and the excited state is large and that the excitation energy is resonant with integer multiple of the laser photon energy. A small Stark shift of the ionic state levels can be used to drive the system into the exact resonance.

7. Nanoparticle-Contained Plumes An interesting and important aspect of using nanoparticles as the nonlinear medium for HHG is that one can tune the surface plasmon resonance (SPR) of such structures to a multiple harmonic wavelength of the pump laser. In this way, one can enhance the nonlinear susceptibility of HHG without relying on coincidental overlap between a strong radiative transition and a harmonic. Therefore, the results of such investigations could open a new method of increasing the efficiency of high-order harmonics, which will have a strong impact on the application of these unique x-ray sources in areas such as attoscience and coherent xray nonlinear optics. Nanoparticles are known to enhance second and third harmonic generation [58,59], which has been used to improve the performance of microscopes. In this section, the demonstration of using nanoparticles to enhance phenomenon in high-field physics, namely HHG, is presented. HHG has had huge impacts in exploring new sciences in the attosecond time regime [60]. One ma or goal for developing HHG sources is to increase its efficiency and intensity, to further advance the field of coherent x-ray science. One feature of nanoparticles with strong interests in photonics is its peculiar absorption characteristics. Like all materials, nanoparticles absorb light at characteristic frequencies. Yet, the efficiency of this absorption, as determined by an optical absorption cross-section, can be enormous. Such SPR excited in metallic nanoparticles exhibit selective photo-absorption, scattering, and local electromagnetic field enhancement. An interesting characteristic of SPR with many potential applications is that the wavelength of its peak absorption can be tuned. This can be achieved by controlling the particle size, shape, particle-to-particle distance, and surrounding dielectric medium of the SPR [61-63]. Important issue here is the integrity of nanoparticles after ablation. The laser intensity used to create the plume is very sensitive factor that has to be carefully investigated before one can make statements about the integrity of nanoparticles in the plasma area after the

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ablation. One could expect the aggregation or melting of nanoparticles during the interaction of the laser prepulse with the target surface. In described experiments, the presence of silver nanoparticles in the plumes was confirmed by analyzing the spatial characteristics of the deposited material. However, further studies of plasma components during ablation of the nanoparticles-contained targets will benefit the statement about the presence and integrity of nanoclusters in the plumes at the moment when femtosecond pulse arrives in the area of interaction. The experiments on harmonic generation were performed using the silver nanoparticles glued on various substrates. The size of these nanoparticles was analyzed using a scanning electron microscope, which confirmed that they varied between 80 to 250 nm with mean size of 110 nm. It was initially verified that harmonics generated from the substrates itself (drop of glue, tape, and glass), without nanoparticles, were negligible compared with those from silver nanoparticles-contained plasma. The targets were fabricated so that a slab silver target was ad acent to the nanoparticle target, with the two target surfaces at the same height. These targets were placed onto the target holder, so that they interacted with both the prepulse and main pump laser at the same intensities. First, the prepulse (t = 210 ps) and main pulse (t = 35 fs) were aligned using a solid slab target of silver, to obtain conditions for maximum harmonic intensity within the plateau. Next, the target was translated so that the prepulse beam irradiated the Ag nanoparticle target. The harmonic yield for silver nanoparticle targets was compared with that from bulk silver target, under the same prepulse and main pulse conditions, as well as at the short pulse delay. Figure 18 shows the lineouts of the harmonic spectra between the 21st and 29th harmonics within the plateau. One can clearly see that the HHG intensity from nanoparticle target was six times higher compared with that from bulk silver target. The energy of these harmonics was estimated based on the calibrations performed using longer pulses. As it was shown in Section 3, for 150 fs pump lasers, a conversion efficiency of 8u10-6 for bulk silver target was achieved. This would be a conservative estimate of the conversion efficiency for bulk silver targets in the described study, which used the 35 fs pulses. Therefore one can estimate a minimum harmonic conversion efficiency of 4u10-5 from silver nanoparticles within the plateau region. For the maximum main pump laser energy of 25 m , the energy of the 21st to the 29th harmonics was evaluated to be in the range of 1 P . When one compare the cutoff observed for harmonics from nanoparticle and slab silver targets, one can also observe a slight extension of the harmonic cutoff for nanoparticles. Harmonics as high as the 67th order (Eph = 103 eV) were obtained in these studies with silver nanoparticles, while, for bulk silver target, the cutoff was at the 61st order (Eph = 94 eV) under the same conditions (Figure 19). This slight extension of harmonic cutoff is in agreement with past work, which observed similar extension in the cutoff for argon clusters, as compared to isolated atoms [64]. This difference has been explained by the increase in the effective binding energy of electrons in the cluster. The higher binding energy will allow the cluster to interact with laser intensities that are higher than in the case of isolated atoms, resulting in the extended cutoff for the former medium. In past work [64], for Ar, the cutoff for clusters was at the 33rd order, compared to the 29th order cutoff for monomer harmonics.

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Figure 18. Harmonic distribution in the mid-plateau region in the cases of the plasma produced on the surface of bulk Ag target (thin lineout) and the Ag nanoparticles-contained plasma (thick lineout).

Figure 19. High-order harmonic spectra generated from (1) silver nanoparticle-containing plasma, (2) plasma produced on the bulk silver target, and (3) plasma of colloidal silver.

Next, the dependence of the harmonic yield as a function of the pump intensity was studied. However, the measurement was made difficult by the rapid shot-to-shot change in the intensity of harmonics from Ag nanoparticle target. For experiments with solid slab targets, stable harmonic generation can be obtained for a long time at 10 Hz repetition rates, without translation for a new target surface. However, for nanoparticle targets, the harmonics were extremely strong for the first few shots, which were followed by a rapid decrease in harmonic yield when the plume was created at the same target position, due to evaporation of the thin

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layer of nanoparticles. Figure 20 shows this shot-to-shot change in the harmonic spectrum, when the target was not moved. The first shot results in a strong harmonic spectrum, with the typical plateau-like structure starting from the 17th order. Then, for the second and third shots, the intensity of the harmonics decreased drastically, and, for the fourth shot and after, the harmonics almost disappeared. The above observations give a rough picture of the ablation process for nanoparticle targets. The material directly surrounding the nanoparticles is polymer (epoxy glue), which has a lower ablation threshold than metallic materials. Therefore the polymer starts to ablate at relatively low intensities, carrying the nanoparticle with it, resulting in the lower prepulse intensity. Polymer also has a lower melting temperature than metals. Therefore, repetitive irradiation of the target leads to melting and change in the properties of the target. This leads to the change in conditions of the plasma plume, resulting in a rapid reduction in the harmonic intensity with increased shots. Due to such rapid change in the conditions for harmonic generation with nanoparticle targets, it was difficult to precisely define the dependence of harmonic yield on prepulse and main pulse intensities. Nevertheless, rough measurements of the dependence of harmonic yield as the function of main pulse intensity for Ag nanoparticles have shown a saturation of this process at relatively moderate intensities (Ifp | 8u1014 W cm-2). Harmonics from plasma nanoparticles also exhibited several characteristics similar to gas harmonics. First, the harmonic intensity decreased exponentially for the lower orders, followed by a plateau, and finally a cutoff. Next, the harmonic intensity was strongly influenced by the focus position of the main pump laser, along the direction parallel to the harmonic emission. The strongest harmonic yield was obtained when the main pump laser was focused 4 to 5 mm after the nonlinear medium. The same tendency was observed for harmonics using bulk silver target. The typical intensity of the pump laser for maximum harmonic yield was between 5u1014 to 1u1015 W cm-2. The harmonic generation experiments using colloidal silver targets, with blocks of silver 100 to 1000 nm in size, were performed to study the size effect of nanoparticles. The size of the silver blocks was confirmed by observations using a scanning tunneling microscope. The results showed that the harmonic yield for these sub-Pm-sized silver blocks was much smaller that that from nanoparticles, and was comparable to those from bulk silver targets. A tendency in the harmonic cutoff to be slightly extended with smaller particle sizes was observed. The cutoff for harmonics from sub-Pm-sized silver blocks was at the 63rd order, whereas that for nanoparticles was at the 67th order, and that for bulk silver at the 61st (Figure 19). These studies have shown that the increase of nanoparticles sizes over some limit is undesirable due to the disappearance of enhancement-inducing processes. The observed enhancement of harmonic yield in the case of 110 nm nanoparticles-containing plume can be probably further improved by using the smaller sizes of particles. To obtain maximum HHG conversion efficiency from such nanostructured media, it is essential to know the maximum tolerable particle size. On one hand, increasing the size of the particles increases its polarizability, and large polarizability of a medium is critical for efficient harmonic generation [65]. On the other hand, the increase in the size of the particles leads to the phenomena that reduce harmonic yield (such as the involvement of only surface atoms for HHG [66], reabsorption of

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harmonics, recombination of electrons from inner parts of clusters). Presented experimental results show that this limit lies between the largest size of the nanoparticles used, at 110 nm, to the median size of the sub-Pm silver blocks, at 500 nm.

Figure 20. Low-order harmonic spectra obtained in the Ag nanoparticles-contained plasma for different shots (from 1 to 4) of the prepulse on the same spot of the target.

The described results with nanoparticles are the first experimental demonstrations of increasing the harmonic yield, resulting from resonance with nanotailored material. Theoretical investigations have shown that, in the strong-field regime, a single multiphoton resonance can increase the intensity of many harmonics, using alkali-metal atom driven by an intense, ultrafast mid-infrared pulse [15]. This regime is characterized by the nonperturbative response of a medium to an intense driving field, when the nonlinear polarization, corresponding to the qth harmonic, increases with the laser intensity Ifp at a much slower rate than the Iq rule. However, as it was mentioned in previous section, there has not been many reports on resonance enhancement of harmonic intensity using Ti:sapphire lasers. This would be a disadvantage, since Ti:sapphire lasers provide one of the most intense femtosecond pump lasers. Although these lasers are widely used for harmonic generation, significant atomic resonance effects in gas HHG have not been observed, except for a few cases, in which a single harmonic was enhanced [66,67]. Even in these studies, the enhancement was relatively small. One can achieve the enhancement of harmonic yield from the nanoparticle-containing plumes by tuning the harmonic wavelength toward the ionic transition with strong oscillator strength, analogous to the resonance-induced enhancement of a single harmonic described in Section 6. Tuning of harmonic wavelength could be achieved by (a) tuning the fundamental wavelength of laser pulse [24,68], (b) chirping the laser radiation [27,54,69], (c) altering the laser intensity, which leads to the control of ionization rate of nonlinear medium [70-72], and

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(d) adaptive control using shaped laser pulses [73]. Note that effective tuning can be achieved only in the case of a broadband radiation. In that case only the leading part of the fundamental pulse consisting of either blue or red components participates in the HHG. In the case of narrowband radiation, the difference in the components at the leading and trailing parts of the chirped pulse is not so pronounced. However, the interplay between the SPM-induced chirp and the artificial chirp of narrowband radiation induced by variation of the grating position can lead to an ad ustment of some specific harmonic orders to quasi-resonance conditions, which can alter the harmonic output. This process was observed in recent studies in the case of the plasma consisting on GaN nanoparticles [74]. GaN powder consisting of 20-nm nanoparticles was glued on the tape and a drop of Superglue and then irradiated by the prepulse. We analyzed the harmonics from this plume in the range of 20 - 65 nm. Figure 21 presents a few lineouts of the harmonic spectra at different chirps of the driving radiation. Harmonic generation was followed by plasma emission from GaN nanoparticles in the X V region, which dominated by the strong 50.3 and 53.8 nm ionic transitions marked by two black lines. In the case of the negatively chirped 280 fs pulses, the 15th harmonic wavelength (52.7 nm) was ust between these two ionic transitions (curve 1). Application of chirp-free pulses at these conditions led to a redshift-induced tuning of the 15th harmonic toward the longer-wavelength transition (curve 2). This redshift is consistent with a previously reported discussion of the role of nanoparticles and clusters in a shift of fundamental radiation wavelength [75]. Simultaneously, the appearance of both the enhanced 15th harmonic and higher-order harmonics was the main feature of this spectral pattern.

Figure 21. Harmonic spectra obtained in the GaN nanoparticles-contained plasma using the radiation of different chirp and pulse duration.

These experiments with GaN-nanoparticle-contained plasma revealed the opportunity of the resonance-induced modification of harmonic spectra using the intensity-induced shift of the harmonic wavelength. Further analysis of resonance-induced enhancement of single

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harmonic is out of the scope of this section. One have to note that the presented result of single harmonic enhancement in the case of a GaN nanoparticle-contained plume is based on principles other than those that were used for enhancement of harmonics in previous plasma HHG studies. The narrow bandwidth of 790 nm, 120 fs radiation (a10 nm) allowed for the tuning of, for example, the 15th harmonic only within a narrow spectral range (0.3 nm) that was insufficient to detune harmonic wavelength far from the initial position. Note that, in the case of ultrashort 800-nm pulses, the variation of laser chirp allowed for a considerable change in the enhancement of specifics harmonic due to the broad bandwidth of fundamental radiation (40 - 50 nm), which gave the opportunity of harmonic wavelength tuning in a much broader range (1.3 nm [27,54]) compared with the narrow-bandwidth radiation.

8. Application of 400 Nm Radiation for the Harmonic Generation in Laser Plasma The application of shorter wavelength sources of driving radiation for the HHG has some attractions. The important peculiarity of using the shorter wavelength photons for the HHG is a less mismatch between the driving radiation and harmonic waves due to the less influence of the free electrons on the phase matching conditions. For the high-order harmonics, the phase mismatch in plasma determined mostly by free electrons ['k=(n2-1)(Zp)2/2ncZ, where Zp is the plasma frequency, n is the order of harmonics, c is the speed of light, and Z is the laser frequency]. In our case, at free electron density of 2u1017 cm-3 (i.e., at ionization rate of 1), the phase mismatch is of order of 90 cm-1 (for 400 nm driving pulses). Thus, the coherence length for harmonics up to the 21st order (~ 19 nm) is approximately 0.7 mm, which is comparable with the plasma sizes (0.6 mm). At the same time, for 800 nm radiation, the same free electron density creates the acceptable phase mismatch only for harmonics above 38 nm. These estimations show that the use of shorter-wavelength sources is advantageous to satisfy the phase matching conditions for shorter wavelengths of harmonic radiation in the case of abundance of the free electrons (at the same concentration of electrons). At the same time, the three-step model describing HHG in gaseous and plasma media predicts that the cutoff energy Ec would drop with the decrease of the wavelength of fundamental radiation with a factor of O2. Therefore, the cutoff harmonic order should scale as O3. It is important analyzing the application of shorter wavelength sources compared to longer-wavelength ones to define the advantages and disadvantages of conversion efficiency optimization through the wavelength control. First observations of the HHG from the laser surface plasma excited by a subpicosecond short-wavelength radiation (KrF excimer laser, O = 248nm) were reported in Refs. 16 and 17. They observed the harmonics up to the 21st order (O = 11.8 nm) generated in a Pb plasma [17]. They also analyzed the positive phase mismatch caused by free electrons in the cases of infrared and V driving radiation and found a considerable difference in these two cases showing the significantly lower phase mismatch in X V range in the case of using the V pulses. The same conclusions were reported in the early studies of the HHG in gas ets [76]. Some time ago, a two-color (fundamental and second harmonic) scheme was proposed for gas HHG to increase the harmonic yield and to generate single attosecond pulses. For these purposes one has to add a weak second-order field to the fundamental beam [77].

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Recently, various schemes using second harmonic pump for phase matching and improvement of electron tra ectories were proposed [78,79]. Similar studies could be performed using the plasma HHG to achieve the attosecond timescale. For these purposes one has to learn more about the properties of the harmonics generated from the 400 nm pump laser itself. In this Section, the application of 400 nm radiation for the HHG from various plasma plumes created on the surfaces of solid-state targets is discussed [80]. The experimental scheme was described in details in previous sections. To create the plasma plume, a prepulse from the chirped radiation of the Ti:sapphire laser (t = 210 ps, Opp = 800 nm) was focused on a target placed in the vacuum chamber. After some delay (40 to 100 ns), the femtosecond main pulse (E = 5 m , t = 35 fs, Ofp = 400 nm central wavelength, 9 nm FWHM bandwidth) was focused on the plasma from the orthogonal direction. This radiation was produced during second harmonic generation of the 800 nm radiation of Ti:sapphire laser in the nonlinear crystals. The generated harmonics were analyzed by the X V spectrometer. The measurements of harmonic cutoff (Hc) for the 400 nm laser pumping silver plasma showed a four-fold decrease compared with the case of the 800 nm pump, instead of the expected eight-fold decrease (Hc v O3). The comparison with previous studies of harmonic efficiency in the plateau range in the case of the HHG of the 800 nm radiation in Ag plasma (8u10-6) showed a ten-fold decrease of conversion efficiency in the case of the 400 nm radiation. For the 800 nm pump, the cutoff order from the Ag plasma was measured to be in the range of sixties harmonics [33,36,45]. Analogous tendency was observed for the Al plasma (Figure 22), when the harmonic cutoff (15th harmonic) considerably exceeded the predictions of three-step model. The Hc in Al plasma obtained from the 800 nm laser (43rd harmonic) was only three times higher than in the case of doubled frequency pump radiation. The discrepancy between the theory and experiment was also observed in the case of the ablation of Al powder, when the signs of the 17th and higher orders appeared in the harmonic spectrum. The reason for such discrepancy between the three-step model and the experiments using shorter-wavelength radiation can be attributed to the different species (atoms, singly- and doubly-charged ions) that contribute to the harmonic generation of the 800 nm and 400 nm pump lasers. One should also keep in mind that, since the harmonic cutoff energy scales linearly with laser intensity, it is important that the pump laser intensity in the medium was kept the same for the two pump lasers when one compares these two cases. Another reason is that this scaling neglects the ionization potentials of particular species, since the scaling is only valid for high-order harmonics when the ionization potential becomes less important. Also, the difference in the ionization-induced defocusing in the cases of the 800 nm and 400 nm pulses could play a role in the unexpected scaling of the harmonics. Less influence of free electrons on the phase matching conditions for the 400 nm pump laser can also be a reason of departing from the O3 scaling.

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Figure 22. Harmonic spectra obtained in the Ag, Al, Al powder, and Be plasmas in the case of 400 nm driving radiation.

The plateau pattern was not so pronounced in the case of the 400 nm driving radiation compared to the 800 laser pulses when the harmonic distribution from most plumes showed almost equal intensities for the harmonics exceeding the 13th order. The only sample, where the plateau pattern appeared in the case of the 400 nm driving radiation, was the beryllium plasma. The harmonics between the 17th to 31st orders showed a clear plateau-like pattern (Figure 22). Some of plasma samples (Sn, Mn, Cr, Sb) demonstrated the resonance-induced enhancement of single harmonics, which was discussed in Section 6 for the 800 nm driving radiation. Such enhancement occurred when the harmonic wavelength was in the vicinity of a radiative transition possessing strong oscillator strength. Figure 23(a) shows the enhanced yield of the 9th harmonic (O = 44.4 nm, Eph = 28 eV) generated from the tin plasma. The intensity of enhanced harmonic eight to ten times exceeded those of neighboring orders. Analogous enhancement in this spectral region has previously been reported in the case of harmonic generation from the tin plasma using the fundamental radiation of Ti:sapphire laser [31]. Such an enhancement using the 800 nm radiation has also been observed in described studies [see inset in Figure 23(a)]. The closeness of the wavelengths of the 17th harmonic of 800 nm radiation (47 nm, Eph = 26.45 eV) and the 9th harmonic of 400 nm radiation (44.4 nm) in the area of strong transitions of singly charged Sn ions (25 - 27 eV) showed identical resonance-induced enhancement. The influence of ionic 4d105s5p ൺ 4d95s5p2 transitions (28.33 – 28.71 eV) of Sn III on the enhancement of the 9th harmonic of 400 nm radiation could be even more pronounced compared with the Sn II transitions due to the high gf values of the former transitions [81].

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a

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c Figure 23. Single harmonic enhancement obtained in the case of the 400 nm radiation propagating through the (a) tin, (b) chromium, and (c) manganese plasmas. Insets in Figs. 23(a) – 19(c) show the enhancement of single and multiple harmonics obtained in the case of the 800 nm pump radiation.

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The chromium plasma has been studied during last time from the point of view of its ability on both the suppression and enhancement of some harmonics belonging to the plateau region in the case of fundamental radiation of Ti:sapphire lasers [27,54]. In particular, the suppression of the 27th harmonic (29.5 nm) and the 23-fold enhancement of the 29th harmonic (27.4 nm) of 793 nm radiation were discussed in Section 6.2 and attributed to the influence of the short-wavelength wing of the strong spectral band of the 3p o 3d transitions of Cr II ions [27]. The studies using the 400 nm radiation confirmed previously reported peculiarities of the harmonic spectra in above-mentioned region [Figure 23(b)]. The observed three-fold enhancement of the 15th harmonic of 400 nm driving radiation can be attributed to the enhancement of the nonlinear susceptibility of this harmonic induced by the influence of the same transitions, though not so pronounced compared with the 29th harmonic of 800 nm radiation [see inset in Figure 23(b)]. In the case of Mn plasma, the maximum harmonic order (21st harmonic) was considerably lower compared with the case of the 800 nm pump (101st harmonic). The sixfold enhancement of the 17th harmonic was observed in these studies [Figure 23(c)]. The single harmonic enhancement in the case of the 400 nm pump was in a stark contrast with the multiple harmonic enhancement observed in the case of the 800 nm driving pulses [see inset in Figure 23(c)]. In the latter case, the harmonics between the 33rd and 41st orders demonstrated the three-fold enhancement, though not so strong as in the former case (6u). The observation of the enhancement of a group of harmonics in the case of manganese plasma is the first experimental evidence of the theory proposed by Taieb et al [14]. In their theory, it has been suggested that electrons that are e ected from an excited bound state, can have nonzero initial velocities and undergo an elastic recollision, thus being available for subsequent recollisions, twice per laser cycle. The theory cannot exactly say which bound states participate in this process, but the observations show that these states should be multiphoton resonant with the main frequency (considering AC-Stark shift as well) and should possess the strong oscillator strength. Their theory predicts, that only harmonics with energies of Zn, where

1.4U p  I p n d =Z n d 2U p  I p n ,

(2)

could be resonance enhanced. Here Up is the ponderomotive energy of electron in the electromagnetic field, and Ip¨n is the ionization potential of nth excited bound state (n=0 is the ground state), in which the electrons recombine. Taking into account the optimal intensity for HHG in Mn plasma and the second ionization potential of Mn, one can obtain from the above relation that, for Ofp = 800 nm, the harmonics between the 33rd and 41st orders can be, in principle, resonance enhanced, while, for Ofp = 400 nm, only a single (17th) harmonic can be resonance enhanced. These predictions were confirmed during the experiments with Mn plasma, where we observed the enhancement of the multiple harmonics of the 800 nm pump, and the single harmonic enhancement in the case of the 400 nm driving radiation [Figure 23(c)]. Note that there are strong radiative transitions in Mn in the range between the 33rd and 41st harmonic wavelengths [82].

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As it was shown in [13], if the accelerated electron recombines to the core, it can fall into either the ground state, or into the excited state, from which it originates. The probability of recombining into these states is determined by their oscillator strength. That is why, in case of excited states with weak gf, a weak multiple harmonic enhancement attributed to the recombination into the ground state could be observed, while, in case of strong gf of a single excited state, one can notice the prevailing recombination into this state with subsequent transition into ground state emitting a single harmonic. The HHG studies at different pump laser chirps were performed to tune some harmonic wavelengths in the proximity of the ionic transitions possessing strong oscillator strengths. The chirp of 800 nm laser pulse was varied by ad usting the distance between the two gratings of the pulse compressor. Varying the laser chirp resulted in a considerable change in the harmonic distribution from laser plasma in the case of the HHG using the 800 nm chirped pulses. However, in the case of the 400 nm radiation, the control of chirp conditions appeared to be not so efficient for manipulating the single harmonic conversion efficiency. The enhancement of single harmonics of the 400 nm radiation in the case of the chirp variations of the 800 nm radiation did not change with the variation in the pulse duration and sign of the chirp. Figure 24 presents the summary of the studies of the enhancement factor at different chirps and pulse durations of the driving radiation. In the case of Mn and Sb plasmas, the enhancement factors [i.e., the ratios between the enhanced harmonic and neighboring harmonic intensities (IEH/INH)] remained approximately the same in a broad range of the variations of chirp and pulse duration, contrary to the variations of the enhancement factors for different chirps and pulse durations in the case of the 800 nm pump (see the inset in Figure 20). The intensity of the 17th harmonic of 400 nm radiation generated in Mn plasma remained strong in a broad range of chirp and pulse duration variations and it was difficult to detune this harmonic from the quasi-resonance conditions [84]. Such a feature can be explained by the relatively narrow bandwidth of the 400 nm radiation (a9 nm), which allowed the tuning of the 17th harmonic within a narrow spectral range (0.3 nm) that was insufficient to detune the harmonic wavelength far from the resonance line responsible for the enhancement of this harmonic. The same can be said about the experiments with the antimony plasma (Figure 24). Note that, in the case of the 800 nm radiation, the variation of laser chirp allowed for a considerable change in the enhancement of specifics harmonic due to the broad bandwidth of pump radiation (40 nm). In the meantime, the tuning of harmonic wavelength in the case of the 400 nm pump could be achieved by tuning the phase matching conditions in the KDP crystal used for the second harmonic generation of the 800 nm radiation. As it was mentioned above, the coherence length (Lc = S/'k) for shorter wavelength pump is expected to be few times longer for the same spectral range, which can be considered as an advantage of using the 400 nm pump instead of 800 nm. This allows increasing the plasma length (Lp) in the former case, thus enhancing the harmonic yield, which is proportional to the (Lp)2. Some other advantages of shorter wavelength source include higher harmonic conversion efficiency at the specific spectral ranges. In particular, the low-order harmonics from the shorter wavelength sources correspond to the mid-order harmonic of the 800 nm pulses. This means that, in the range of 100 – 200 nm, the harmonic energy from the 400 nm driving pulses will be higher compared with those from the 800 nm pump.

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Figure 24. Ratio of the enhanced harmonic and neighboring harmonic intensities obtained in the manganese and antimony plasmas at different chirps and pulse durations of the 400 nm driving radiation. Inset: Single harmonic variations at different chirps of the 800 nm radiation in the cases of indium (filled triangles), chromium (filled squares), and antimony (filled circles) plasmas. Positive and negative signs of the pulse duration correspond to the positive and negative chirps of driving radiation.

Shorter pulse duration and better coherence properties of the second harmonic of 800 nm radiation could also be advantageous for the HHG. This is because the improved phase conditions of the pump radiation result in improving the phase matching condition between the harmonic and driving waves. This can lead to further shortening of the pulse duration for harmonics generated by the 400 nm pump. However, the relatively small second harmonic conversion efficiency in the thin nonlinear crystals (~20%) still remains an obstacle for the application of the 400 nm pump in the HHG studies.

9. High-Order Harmonic Generation from Laser Plasma Produced by Pulses of Different Duration Above-presented studies have shown a crucial role of plasma formation for achieving the maximum conversion efficiency and cutoff of generated harmonics. Various techniques were applied for the analysis of the ionization level, electron and ion concentration, spectral characteristics, and nonlinear optical parameters of the laser plasma (e.g., the calculations using HYADES code [45], time-integrated [33] and time-resolved [83] spectral analysis of plasma in the visible and V ranges, analysis of the divergence of the femtosecond beam propagated through the laser plasma [84], z-scan studies of laser ablation [84], etc.). Meanwhile, the important parameter that has never been explored during plasma HHG studies is a prepulse duration. Since almost all of the HHG studies from plasma plumes were carried out by using a few hundreds picosecond prepulses, the conclusion was drawn about

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the optimal intensity of the prepulse radiation at the target surface [(1-4)u1010 W cm-2, depending on the absorptive properties of targets] [25,28,45]. It would be interesting to compare the HHG from the plasmas produced by the prepulses of different duration, since it can clarify which parameter (energy or intensity) of the prepulse plays a crucial role in the formation of optimal plasma. Another interesting issue is the influence of the atomic number (Z) of target materials on the HHG at different delays between the prepulse and main pulse. It would be worth analyzing whether the optimal delay between the two pulses is sensitive to the target properties. One can expect a difference in the HHG from the low- and high-Z targets, since the heating as well as the expansion and the subsequent cooling and recombination of the plasma depend on the atomic weight and density of the target material. Different velocities and recombination properties of the low- and high-Z atoms and ions can considerably change the concentration characteristics of plasma at different delays. The optimization of the delay between the prepulses for specific plasma samples allows for achieving the maximum harmonic intensities and highest harmonic cutoffs. In this Section, the studies of the HHG from the laser plumes produced by the interaction of the prepulse radiation of different pulse duration (160 fs, 1.5 ps, 210 ps, and 20 ns) with the silver, manganese, barium, lithium, zinc, nickel, boron, and carbon targets are discussed [85]. These studies showed that plasma formation plays a crucial role for achieving the efficient harmonic generation and the optimization of this process mostly depends on the prepulse energy rather than its intensity at the target surface. It was also shown the difference in harmonic generation in the cases of the low- and high-Z targets, when the delay between the prepulse and femtosecond pulse becomes a crucial factor for achieving the efficient HHG from laser ablation. The experimental set-up of the HHG from laser plasma was described in previous sections. The initial prepulse duration was 210 ps. The pulse duration of this radiation was varied by compressing it in an additional compressor stage. In particular, the 160 fs and 1.5 ps prepulses were used in these experiments by ad usting the distance between the gratings of this compressor. The pulse energy of these three prepulses was ad usted to be approximately 10 m . A spherical or cylindrical lens focused the prepulse on a solid target placed within a vacuum chamber to generate a laser ablation plume. High-Z (barium and silver), medium-Z (zinc, nickel, and manganese), and low-Z (carbon, boron, and lithium) materials were used as the targets in these studies. The width of the line focus at the target surface in the case of the focusing by a cylindrical lens was ad usted between 50 and 200 Pm at 2 mm length of the plasma, and the intensity of 210 fs prepulse at the target surface was varied in the range of Ipp = (0.7 - 7.0)u1010 W cm-2. In the case of spherical focusing, the area of ablation was ad usted in the range of 0.6 mm and the 210 ps prepulse intensity at the target surface was varied in the range of Ipp = (1 - 5)u1010 W cm-2. In the case of 160 fs and 1.5 ps prepulses, the intensities at the target surface were considerably higher, since the geometrical characteristics of these studies were similar for three prepulse durations, while keeping the same pulse energy. A main pump pulse at a center wavelength of 795 nm had the energy of 12 m with pulse duration of 115 fs. After the proper delay with regard to the prepulse irradiation, this pulse was focused by a spherical lens (f/10) on the ablation plume from the orthogonal direction. The maximum intensity of the femtosecond laser beam at the focal spot was 6u1016 W cm-2.

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Since this intensity considerably exceeded the barrier suppression intensity of singly charged ions, the position of laser focus was ad usted to be either before or after the laser plume to maximize the high-order harmonic yield. The intensity of the main pump pulse at the plasma area was varied between 7u1014 to 3u1015 W cm-2. A magnesium fluoride window possessing small nonlinear refractive index was used in a vacuum chamber containing targets to exclude the self phase modulation and white-light generation during the propagation of femtosecond radiation. The grating compressor was ad usted to achieve the shortest duration of the main pulse at the plasma area after the propagation through the focusing lens and chamber window. These experiments were carried out at loose focusing conditions [b Lp, b is the confocal parameter of the focused radiation, and Lp is the plasma length b = 3 mm, Lp = 0.6 mm (in the case of the spherical focusing of prepulse radiation), and Lp = 2 mm (in the case of the cylindrical focusing of prepulse radiation)]. Figures 25-27 show the influence of the prepulse duration on the harmonic cutoff in the cases of relatively heavy [Ba (Z=56) and Ag (Z=47)], mid-weight [Zn (Z=30), Ni (Z=28), and Mn (Z=25)], and light [C (Z=6), B (Z=5), and Li (Z=3)] targets. One can see the different behavior of the Hc (tpp) dependences for these three groups of targets. In the case of the highZ targets (and in most cases of the medium-Z targets), no considerable change in the harmonic cutoffs was observed at the 160 fs, 1.5 ps, and 210 ps pulse durations of prepulse radiation. In some cases, the second harmonic of Nd:YAG laser was used as a prepulse radiation of longer pulse duration (tpp = 20 ns). These conditions also did not lead to a considerable difference (for heavy targets) in harmonic cutoffs with regard to shorter prepulses. 70

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Figure 25. Harmonic cutoffs obtained in the high-Z plasmas prepared using the prepulses of different duration in the cases of short (17 ns) and long (88 ns) delays. ( a ) Ba, ( b ) Ag.

In the case of the low-Z targets, a more pronounced Hc (tpp) dependence was found compared to the high-Z samples. However, no restricting factors were observed in that case, which stopped the HHG at significantly different intensities of heating radiation. The only exception was the lithium plasma, where the high harmonics were generated only using the relatively long (tpp = 210 ps) prepulses and short delays. The main differences in the HHG properties in that case were related with the delay between the prepulse and main pulse, rather than the prepulse intensities.

High-Order Harmonic Generation in Laser-Produced Plasma

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Figure 26. Harmonic cutoffs obtained in the medium-Z plasmas prepared using the prepulses of different duration in the cases of short (17 ns) and long (88 ns) delays. (a) Zn, (b) Ni, (c) Mn.

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Figure 27. Harmonic cutoffs obtained in the low-Z plasmas prepared using the prepulses of different duration in the cases of short (17 ns) and long (88 ns) delays. ( a ) C, ( b ) B and Li.

These observations showed the main parameter of laser prepulse, which is responsible for the creation of the optimal plasma conditions for the HHG. This is a pulse energy, which is responsible for the ablation of targets and preparation of the plasma where the efficient HHG can be realized. Since the most important parameter of plasma (from the point of view of harmonic generation ability) is a concentration of singly charged (or in some cases doubly charged) ions, the conditions of plasma preparation should be maintained with the appropriate manner to achieve the best characteristics of harmonic generation. The time-integrated spectra from laser plasmas were analyzed in the visible and V ranges in the cases of different pulse durations of prepulse radiation. Figure 28 shows some of these spectra obtained from the Ag and Mn plasmas prepared for the efficient harmonic generation. Though some of these spectra varied from each other, the overall pattern remained the same at different prepulse durations. These studies showed that the pulse duration of heating radiation does not considerably change the emission spectra of targets. The same can be said about the emission in the X V region that confirmed the existence of approximately equal plasma conditions at considerably different intensities of target excitation.

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Ag Intensity (arb. units)

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Figure 28. Plasma spectra emitted from the (a) Ag and (b) Mn plumes in the cases of the excitation by the 160 fs, 1.5 ps, and 210 ps prepulses.

The harmonic generation strongly depended on the delay between the prepulse and main pulse. At very small delays (less than 5 ns), no harmonics were generated from all plasma samples due to insufficient concentration of the particles at the axis of the femtosecond main beam. However, the harmonic properties of different plasma plumes considerably distinguished from each other at longer delays depending on the atomic number of the targets. In the case of 17 ns delay, the harmonic generation was observed from all of the samples, while in the case of 88 ns delay, only the high- and medium-Z plumes produced harmonics (Figures 25 and 26). No harmonics were obtained at long delays in the case of the low-Z plumes (excluding carbon, Figure 27). At the same time, the harmonic cutoffs from the highand most of medium-Z plumes remained approximately unchanged for both the short and long delays. Such a behavior can be attributed to the dynamic characteristics of the plasmas in the cases of the low- and high-Z target materials. Light particles possess higher velocity and this can lead to the depletion of the particles concentration in the laser plasma after a few tens nanosecond. At the same time, the high- and medium-Z particles for a longer time remain close to the target surface where the HHG occurs. For heavy targets, the dependence of the

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high-order harmonic yield on the delay is not so pronounced compared to the low-Z materials. The difference between these dependences was clearly seen in the case of longer delays due to different concentration of the particles in the interaction area of a laser plume for the low- and high-Z targets. Based on this, the search of the optimal plasma conditions should take into account the role of the atomic characteristics of the targets [87]. It was discussed in Section 5 that the second ionization potential of target materials was assumed as playing an important role in the definition of the harmonic cutoffs from different laser plumes [28,33]. It was found that the harmonic cutoffs observed from different plumes directly depended on the second ionization potentials of materials. This finding underlined the role of singly charged ions in harmonic generation. The studies reported in this Section revealed the additional parameters of target material and prepulse, which should be taken into account during further search of higher harmonic cutoffs and harmonic yields from laser ablation [83]. From the empirical rule Hc | 4Ii  32.1 [35], one can assume that the highest cutoff can be achieved in the case of highest ionization potentials of the particles participating in the HHG. Indeed, in the case of singly charged ions (which are proven to be the main source of harmonics in most plasma HHG experiments), highest second ionization potentials of used targets corresponded to the highest cutoffs ever observed. At the same time, as it was mentioned in previous Sections, the plasma HHG studies have also revealed in some cases the role of doubly charged ions in achieving highest harmonic cutoff energies [34,35]. The ionization potentials of V III and Mn III (i.e., the third ionization potentials of vanadium and manganese) and the observed harmonic cutoffs (71st and 101st orders, respectively) obeyed the above-mentioned empirical relation (Figure 10). One could expect to achieve higher harmonics by applying the materials with higher third ionization potentials. The targets under investigation in the studies described in this Section were chosen from the point of view of their highest third ionization potentials among other solid materials. However, no higher harmonic cutoffs with regard to previously reported data were achieved. The reason for this fault could be related with the specific phase-matching conditions in the case of a few plasma samples, which led to the efficient harmonic generation through the interaction of the radiation with the doubly charged ions, while in the case of other plumes, the influence of the enhanced free electron concentration led to the growth of phase mismatch and self-defocusing. Laser-induced plasma itself is a complex phenomenon, especially from the point of view of its nonlinear optical properties. The behavior of laser ablation changes considerably, depending on the equation of state, ionization potential, and cohesive energy of the material. The processes that determine harmonic generation from the plasma plume are complex and may involve various factors that are not considered for gas harmonics. For example, the nonlinear medium is already weakly ionized for plasma harmonics, whose level depends on the prepulse characteristics. If the free electron density is too high (which is the case when doubly charged particles exist in the plasma), it can induce the phase mismatch between the pump and harmonic waves, or defocus the pump beam. Both of these processes can reduce or even stop the harmonic generation.

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10. Analysis of Plasma Properties Rapidly ionizing plasma medium modifies the temporal structure of femtosecond laser pulse due to the self-phase modulation. In addition, the spectral structure of high-order harmonics depends critically on the frequency modulation, or chirp, of the driving laser pulses. The propagation of intense chirped laser pulses through the ionized gas ets has previously been studied, and the spatial, spectral, and temporal characteristics of the laser radiation and generated harmonics were defined (see Ref. 86 and references therein). Those and other studies underlined the influence of the self-defocusing of the intense femtosecond pulse in the ionized medium on the phase matching conditions between the pump and harmonic waves. Since the self-defocusing seems a common limiting factor for both HHG schemes, it is important to analyze this process and define the experimental conditions when its influence becomes less significant. The shape of the main beam (t = 150 fs) propagating through the laser plume at different plasma densities varying from 5u1016 to 2u1017 cm3 was registered in the far field using a CCD camera. The nonlinear refractive properties of indium and molybdenum plasmas were analyzed by the z-scan technique with a 2-mm aperture placed in the far field and a detector behind it. In the conventional z-scan configuration, the medium under investigation moves with regard to the focal plane of the focused radiation. However, in these experiments, a focal plane of femtosecond radiation was moved with regard to the plasma area by changing the position of focusing lens. The beam shape remained unchanged at small intensities of the femtosecond radiation even at the conditions of the ionization of neutral atoms. This observation showed that the concentration of the free electrons generating during ionization of the neutrals was insufficient for the self-defocusing of the laser beam. Another beam profile appeared after the propagation through the plume in the case when the pulse intensity exceeded a barrier suppression intensity of singly and doubly charged ions. The barrier suppression intensities of the Mo I, Mo II, and Mo III particles were calculated to be 1.2u1013, 7u1013, and 2.4u1014 W cm2, respectively. The amount of the free electrons appearing in the focal area at Ifp 3u1014W cm2 became sufficient for the self-defocusing of the laser beam. In that case, the ring profile of the pump radiation propagated through the plume was observed indicating the variation of refractive index in the vicinity of the axis at the focal plane. The closed aperture z-scans of various plumes were carried out using the femtosecond pulse propagating through the plasma. These studies were performed at the optimal plasma conditions corresponding to the efficient HHG. At small laser intensities, no variations of the normalized transmittance of the femtosecond radiation through the far field aperture at the optimal plasma conditions were observed. With further growth of laser intensity, a characteristic peak–valley shape of normalized transmittance appeared indicating the negative nonlinear refraction inside the plasma. At these intensities, a strong nonlinear absorption of femtosecond beam caused by the multiphoton ionization of neutrals and singly charged ions was observed as well. These studies showed that the concentration of the free electrons produced by laser ablation was insignificant for the self-defocusing of low-intensity femtosecond beam. The free electrons led to the self-defocusing of laser beam at the intensities exceeding the barrier suppression intensities of singly and doubly charged ions.

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The closed aperture scheme was capable determining both the sign and magnitude of the nonlinear refractive index (J) and nonlinear absorption coefficient (E) of the plasma plume. In particular, in the case of indium plasma, the J and E were defined to be 2u1018 cm2 W1 and 5u1013 cm W1, respectively [84]. Note that the latter parameter strongly depended on the intensity of the femtosecond pulse. The four-photon absorption was found as a dominant one in the experiments with the indium plasma. The calculated four-photon absorption coefficient of In plume was found to be 5u1042 cm5 W3. In the case of the Mo plume (Ii = 7.1 eV), the suitable mechanism of nonlinear absorption is a five-photon absorption. The studies of the shape variation of the femtosecond beam propagating through the optimal plasma showed the influence of the free electrons produced during the ionization of singly charged particles on the spatial distribution of laser radiation in the focal area. The free-electron-induced self-defocusing prevents achieving the higher intensity available in the case of vacuum conditions, thus limiting the conversion efficiency and cutoff energy of the high-order harmonics. The z-scan studies of laser plasma confirmed that the combined influence of self-defocusing and nonlinear absorption could restrict the maximum observed harmonic order. The abundance of free electrons also causes the phase mismatch between the harmonic and main pulse waves. The value of nonlinear susceptibility can be enhanced up to several orders in the case of the coincidence of the pump or harmonic wavelength with the ionic or neutral transitions of the medium. In the case of nanostructured medium, such resonances could originate from the absorption peaks associated with the appearance of SPR. The resonance amplification is of especial importance for the frequency conversion of laser radiation since the resonanceinduced enhanced susceptibilities allow realizing various interactions using relatively lowpower tunable lasers. Besides this, resonance enhancement defines the maximum efficiency of the HHG, which could be achieved in the nonlinear medium. Several conditions have to be fulfilled for optimizing the HHG during the interaction of laser radiation with plasma. Since the expansion time of laser plume ranges from tens to hundreds nanosecond, one has to synchronize the propagation of main pulse through the optimally prepared plasma . This term refers to the conditions when maximum conversion efficiency and harmonic cutoff can be achieved. The above-described studies revealed that the main parameters for the preparation of such plasma are the density, excitation, and charge state of the plasma. The presence of excited atoms and ions can considerably increase the nonlinear optical response of laser-produced plasma. The delay between the prepulse and main pulse and the location of the focusing area of main beam in the appropriate regions of plasma are also of considerable importance. And finally, depending on the experimental geometry, it is possible to produce a point-like plasma as well as a prolonged one. This would affect the conversion efficiency due to the competition between the growth of the length of nonlinear medium and the re-absorption of generated harmonics in the prolonged plasma. Such a peculiarity was confirmed in the studies of the HHG from boron plasma using the spherical and cylindrical focusing geometries for prepulse beam, as well as in the studies of C [87], Mg [88], Al [89], and Mo [90] plasmas. An enhancement of HHG efficiency should include the optimization of both the macroscopic and microscopic responses. From the macroscopic point of view, such an optimization dominantly means the achievement of phase matching conditions. Microscopic optimization includes the methods of the growth of the harmonic polarization, which depends

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on the nonlinear susceptibility of the medium. In particular, the enhancement of single harmonics from specific plasma plumes was induced by the increase of nonlinear susceptibility of the medium at quasi-resonance conditions [91]. One of the options to enhance the harmonic yield is the generation of high-order harmonics using the two-component laser-ablation plumes [92]. The harmonic spectra from double target schemes comprise those obtained from both the targets. In particular, for the indium/chromium plumes, the enhancement of two harmonics (13th and 29th orders) was obtained at the wavelengths of 61.15 and 27.41 nm, respectively. The conversion efficiencies of these two harmonics were estimated to be 10-5. This demonstration of the two-color enhancement in the short-wavelength range can open new opportunities in the applications of such sources for the X V nonlinear optics and pump-probe X V spectroscopy.

11. Summary and Future Prospective This review is based on the description of the HHG experiments using the low-excited plasma, which show quite different approaches compared to the conventional gas HHG. At the same time, it is clear that the gas HHG presently shows better results compared with the reviewed studies (from the point of view of higher harmonic cutoffs). To not be very optimistic we did not compete with the conventional technique in this field but instead described various new approaches, which are difficult to implement in gas HHG. The characteristic range of conversion efficiencies achieved in the plateau region in the case of the HHG from gas ets did not considerably exceed 10-6 in early experiments. The conversion efficiency then has been improved up to 10-5 by proper phase matching. In the case of laser-surface experiments, when both the odd and even harmonics generated at specular direction from solid surfaces, the conversion efficiency is higher and comparable to those of saturated collisional X V lasers, and is much higher than other X V sources. The HHG from laser plasma (in particular from silver plume) obtained in conventional conditions shows the conversion efficiency comparable with the one achieved in the case of the HHG from gas ets. However, as it was shown in this review, the resonance conditions can considerably increase the conversion efficiency of single harmonic at the plateau region that exceeds the one achieved during laser-surface HHG experiments in the X V range. The application of nanoparticle-contained plumes can further enhance the harmonic yield. The studies presented in this review considerably improved both the harmonic cutoff and the conversion efficiency from the laser plasma produced on the surfaces of various solidstate targets. Figure 29 presents the comparison of the best results obtained in previous studies (curves 1-8) and recent studies described in this review, where the low-excited, lowionized plasma was used as a nonlinear medium (curves 9,10). The generation of high harmonics (up to the 101st order, O = 7.9 nm) after the propagation of femtosecond laser pulses of different duration through the low-excited, low-charged plasma was demonstrated during recent studies. The high-order harmonics generating from most plumes showed a plateau pattern. The harmonic generation in these conditions assumed to occur due to the interaction of femtosecond pulses with singly and doubly charged ions. The conversion efficiency for the harmonics generated in the plateau region was varied in the range of 10-7 to 8u10-6 depending on the target material. The saturation of harmonic generation efficiency was

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caused by the free-electron-induced phase mismatching and the self-defocusing of the main beam.

Figure 29. Harmonic spectra from the laser ablation of different materials obtained (1-8) in previous studies and (9,10) recent studies using the low-excited, low-charged plasma for the HHG from the laser plumes produced on the surfaces of various solid-state targets.1: K II, 2: LiF II, 3: Al II, 4: C II, 5: C II, 6: Pb III, 7: K II, 8: Al II, 9: Ag II, 10: Mn III.

The observation of the considerable resonance-induced enhancement of a single harmonic (O = 61.2 nm) at the plateau region with the efficiency of 10-4 in the case of In plume offered some expectation that analogous processes can be realized in other plasma samples even in the shorter wavelength range where the highest harmonics were achieved. It is difficult to realize this approach in gas ets due to the necessity of the preparation of the specific conditions for the excitation of the appropriate transitions of nonlinear medium prior to laser-matter interaction. We reviewed the studies of resonance-induced enhancement of single harmonic from other plumes as well as analyzed the harmonic output from different targets. We discussed the enhancement of high-order harmonic yield in the case of nanoparticles-contained plasmas.

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Among the future potential improvements of plasma HHG, one can propose the double excitation of laser plasma by different prepulses, improvements in the longitudinal HHG scheme, optimization of the nanostructures sizes in the plasma, creation of the guiding regime of pulse propagation through the plasma, etc. One can expect that the HHG experiments with specially prepared plasmas can demonstrate further enhancement of harmonic yield and the extension of the harmonic cutoff toward the shorter-wavelength range.

Acknowledgments Author thanks H. Kuroda, P. D. Gupta, P. A. Naik, and T. Ozaki for fruitful discussions at various stages of these studies. Author is indebted to M. Suzuki, H. Singhal, L. B. Elouga Bom, . A. Chakera, M. Baba, . Chakravarty, I. A. Kulagin, P. V. Redkin, R. A. Khan, M. Raghuramaiah, and V. Arora for their contribution during these studies.

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In: Progress in Nonlinear Optics Research Editors: M. Takahashi and H. Goto, pp. 197-259

ISBN 978-1-60456-668-0 c 2008 Nova Science Publishers, Inc. 

Chapter 6

S UM -F REQUENCY G ENERATION AND M ULTIPHOTON I ONIZATION IN R ARE G ASES WITH N ON -G AUSSIAN L ASER B EAMS V. Peet∗ Institute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia

Abstract In recent years the generation, properties, and interaction with matter of new types of non-Gaussian laser beams have been of great interest. Such novel beams like Bessel, Bessel-Gauss, Mathieu, segmented conical beams and others show a number of interesting effects in the field of nonlinear optics. In many cases, the nonlinear effects, which are well known in ordinary laser beams, show significant differences under excitation by non-Gaussian beams. The present paper gives an overview of some gasphase nonlinear effects observed under excitation by several new types of laser beams. Considered is the generation of resonance-enhanced third and fifth harmonics under excitation by different conical beams (Bessel, Bessel-Gauss, and segmented beams) and comparison of these processes with ordinary Gaussian beams. A similar comparison is made for sum-frequency generation processes under two-color excitation by spatially coherent and incoherent non-Gaussian and Gaussian beams. Under intense excitation, the multiphoton excitation and ionization of the target gas atoms become important. A large number of experimental and theoretical works have shown the importance of harmonics of the fundamental laser light in these processes. Here again the use of non-Gaussian laser beams and different beam configurations results in several interesting observations. Some of the effects of internally-generated harmonic fields on the excitation of atomic resonances in non-Gaussian laser beams are considered and discussed.

1.

Introduction

Generation of low-order harmonics in gases is a well-known phenomenon of nonlinear optics. Under proper excitation conditions, the third- and fifth harmonics of the fundamental ∗

E-mail address: vikp@fi.tartu.ee

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laser light may achieve a relatively high intensity and serve thus as an efficient source of coherent emission in the VUV spectral region where the number of direct laser sources is very limited. The efficiency of the harmonic generation process is strongly enhanced near intermediate atomic resonances due to the resonance character of nonlinear susceptibilities and a good phase matching of the focused fundamental beam and harmonics generated in the negatively-dispersive side of the resonance. For strong isolated atomic resonances the negatively-dispersive region can be rather broad and, with tunable pump lasers, an efficient generation of tunable narrowband VUV light can be obtained. The output beam of convenient pump lasers is usually close to a Gaussian profile of order zero and until the late 1980’s the vast majority of all investigations on gas-phase harmonic generation was carried out with focused Gaussian beams. In the limit of a large confocal parameter, such Gaussian beams can be considered in terms of plane waves. In 1987, a new type of beam as a solution to the homogeneous Helmholtz equation was introduced by Durnin [1] and soon afterwards was realized experimentally [2]. These beams have become known as Bessel beams owing to their amplitude profile given by Bessel function J0 of the first kind and of order zero. Actually, the light field profiles described by Bessel functions are well known in classical optics, but less attention was paid until recently to the depth of such fields. Durnin and co-workers have demonstrated a very unusual feature of Bessel light fields: namely, the transverse profile of a Bessel beam does not depend on the coordinate along propagation direction and the beam keeps its intensity profile unchanged under propagation. The Bessel beams are capable of giving a high concentration of light energy into a small spot, and the spot size remains nearly constant over a distance of propagation, which can be made much longer than the Rayleigh length of a focused Gaussian beam. This feature contrasts drastically ”standard” laser beams, where diffraction leads to an inevitable divergence of the field profile. With this, Bessel beams introduced into optics the concept of propagation-invariant light fields. The interesting and unusual features of Bessel beams have, for many years, inspired further research on the nondiffracting aspect of these beams, but it took much longer to appreciate the possibility of using the Bessel beams in nonlinear optics. Actually, experimental realization and application of intense Bessel beams in the creation of long unbroken laser sparks in gases [3, 4, 5] were made well before the concept of Bessel beams was introduced, but interest in a regular study of Bessel beams was initiated after the works of Durnin and co-workers. The first demonstration of Bessel beams in the generation of second harmonics was reported in [6]. It was shown that with Bessel beams the phase-matching condition for optimum harmonic generation is not limited to one direction (that is, the propagation direction) but can be tuned continuously by varying the inclination angle of the Bessel beam. Concurrently, a similar result was obtained using excitation by Bessel-like conical beams in generation of third harmonics in metal vapors [7]. It was demonstrated that with the use of conical excitation geometry of a Bessel-like beam, the harmonic output can be increased by several times together with certain immunity of harmonic output to variations of excitation conditions. In the following years, different nonlinear optical effects with Bessel beams have been under extensive consideration (see, for example, [6, 7, 8, 9, 10, 11, 12, 13, 14] and references therein). The Bessel beam can be viewed as a superposition of infinitely many plane waves whose

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wave vectors form a conical surface. For a J0 Bessel beam all these plane waves have equal amplitudes and phases. Besides zero-order J0 Bessel beams, there is a family of higherorder Jm beams (m = 1,2,3, etc.), where phase of partial plane waves changes in azimuthal direction [15, 16, 17]. All these Bessel beams are propagation-invariant like J0 beams, but all higher-order Bessel beams have zero intensity on their axes. Other examples of novel laser beams are segmented beams [18, 19], Mathieu beams [20, 21, 22], and truncated Bessel beams [23]. These conical beams share the propagation properties of Bessel beams, but they have amplitude profiles different from the Bessel pattern. Segmented conical beams [18, 19] remain propagation-invariant in the same sense as the initial Bessel beam. It was shown [20, 21, 22] that some of such segmented conical beams represent a new type of propagation-invariant light fields, where the field distribution is described by radial and angular Mathieu functions. These Mathieu beams appear as solutions to the Helmholtz equation in elliptic coordinates and the transverse profiles of Mathieu beams consist of several lobes, where the light is concentrated [20, 21, 22]. In all above examples the conical light field is characterized by a complete spatial coherence when all partial waves maintain their mutual correlation under propagation. Another class of propagation-invariant light fields includes partially coherent conical beams. A special case of such beams is represented by the so-called Bessel-correlated beams [24], where correlation between individual plane waves vanishes in azimuthal direction. In general, both azimuthal and radial incoherence can be introduced into conical superposition of multiple plane waves and it allows one to synthesize a variety of specific field configurations like, for example, light strings and light capillary beams [25, 26]. Under excitation by intense laser beams the processes of harmonic generation are accompanied by multiphoton excitation and ionization of the target gas atoms. These processes change the properties of gaseous media and the onset of intense ionization is often one of the limiting factors for conversion efficiency. On the other hand, gas-phase multiphoton excitation and resonance-enhanced multiphoton ionization (REMPI) are effective and widespread techniques of modern laser spectroscopy. A large number of experimental and theoretical works on REMPI have shown the importance of harmonics of the fundamental light in the response of atomic system under excitation (see, for example, [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] and references therein). The harmonic field, once generated, becomes an additional source of coherent excitation of the target gas atoms. The excitation pathways, where the harmonic and fundamental photons are involved, may interfere constructively or destructively, leading to strong enhancement or suppression of multiphoton excitation and ionization processes. Until recently, these complex processes were studied with Gaussian beams, but much less was known about the peculiarities of these processes under excitation by non-Gaussian laser beams. Interesting and unusual properties of novel laser beams make them very attractive for nonlinear optics, where such beams often demonstrate significant difference, compared with ordinary Gaussian beams. The present paper gives an overview of low-order nonlinear optical effects such as generation of resonance-enhanced third and fifth harmonics, sumfrequency generation, and accompanying multiphoton excitation and ionization of the target gas atoms under excitation by some non-Gaussian laser beams and beam configurations. For the standard Gaussian beams, these processes have been well studied and analyzed in numerous experimental and theoretical works and it gives a natural basis for comparison

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lens

MgF2 window

Ta cathode

gas cell

wire collectors Figure 1. Schematic of basic experimental arrangement. of Gaussian and different non-Gaussian beams. The study was not aimed toward any optimization of conversion efficiency, but special attention was paid to the understanding of peculiarities of novel laser beams and the underlying physics in the excitation of nonlinear optical processes.

2.

Experiment

Figure 1 shows a schematic diagram of basic experimental arrangement. The output of an excimer-pumped tunable dye laser was focused into a static gas cell. The cell was made of stainless steel and contained xenon at a pressure ranging from 1 mbar to 10 bar. The dye laser pulses had energies of a few mJ, pulse duration of 8-10 ns, and spectral widths of about 10 pm (FWHM). To enhance the available laser pulse energy, a tandem pumping system with two excimer lasers could be used, where the first excimer laser pumped a dye laser and another excimer laser pumped an additional dye amplifier cell. Both excimer laser were synchronized with a ± 2 ns time jitter. The maximum laser pulse energy in the tandem system was increased by 4-5 times. Photoelectrons resulting from the multiphoton ionization process were monitored with a wire collector biased at a positive voltage from 15 to 250 V. Ionization signal was amplified, digitized and stored on a computer. To detect the generated VUV light, a VUV monochromator and a solar-blind phototube could be used. In many cases, however, the generated VUV emission was strong enough to be measured by a simple ionisation cell [28, 38]. The VUV emission was passed through a MgF2 window and was measured in a second vacuumized cell, where VUV photons impinged a tantalum photocathode. Ejected photoelectrons were collected by a wire biased at +200 V and detected by an electrometer. Such a double-cell experimental arrangement with two ionization detectors has allowed both the harmonic output and accompanying multiphoton ionization to be measured simultaneously. The optical arrangement shown in Fig. 1 was used mainly in reference experiments with ordinary Gaussian beams, when the laser beam was focused into the cell by a spherical lens. This basic arrangement was easily transformed and modified depending on particular tasks. In experiments with J0 Bessel and other conical beams, the focusing lens was removed and

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J02

J12

Figure 2. Theoretical and experimental profiles of J0 and J1 Bessel beams. a conical lens (axicon) was mounted as an input window of the gas cell. With the aid of different glass and quartz axicons the conical beams with inclination angles up to 33◦ could be obtained. To produce a higher-order Jm Bessel beam, the pumping laser beam was passed through a special zone plate, which converted the lowest-order Gaussian mode into a family of annular Laguerre-Gauss modes with helical phase surface. Being focused by axicon, these Laguerre-Gauss modes formed the corresponding Jm Bessel beams. Fig. 2 demonstrates theoretical and experimental intensity profiles for J0 and J1 Bessel beams. The experimental profiles were measured with the aid of a microscope and a CCD camera. With the aid of special amplitude masks, a full-aperture Bessel beam was transformed into different segmented conical beams (see Section IV). A disk-shaped mask was used in experiments with Bessel-like conical beams. Such beams were formed when an annular input beam was focused by spherical lenses. In multibeam excitation geometries the initial laser beam was split into two or more parallel sub-beams and focused by a lens or by an on-axis spherical mirror. Besides of single-color excitation, different two-color excitation modes were employed. In experiments with incoherent beams, the excimer laser served as a source of such spatially-incoherent UV emission synchronized with the pulses of dye laser. Another mode of two-color excitation was obtained with the use of two synchronously pumped dye lasers where the parameters of each laser beam (wavelength, intensity, polarization, etc.) could be controlled independently. Among different gaseous media, the rare gases are especially convenient for genera-

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Xe

+

1

2

3

4

Xe 1

S0

Figure 3. Schematic energy level diagram of xenon showing relevant excitation and ionization channels: 1 - three-photon resonance excitation; 2 - resonance-enhanced third harmonic; 3 - five-photon resonance excitation; 4 - resonance-enhanced fifth harmonic.

tion of tunable VUV light. The excited states of rare gas atoms have rather high excitation energy and the resonance-enhanced harmonics of the fundamental light are generated with photon energies above 8.44 eV. Rare gases are chemically inert, photostable, and transparent in a broad spectral range. The frequency converters with rare gases operate usually at a room temperature and do not require any additional heating like gas cell with metal vapors. It allows one to design very compact gas cell to be used with convenient pump lasers. In the present work, xenon gas was used as a nonlinear media for generation of resonance-enhanced harmonics and REMPI. Fig. 3 demonstrates a schematic energy level diagram of xenon together with relevant excitation and ionization pathways. In the simplest single-color excitation mode, the resonance-enhanced third harmonic (TH) was generated in the negatively-dispersive side of the 6s or 6s states of xenon with excitation energies of 8.44 and 9.57 eV, respectively. The same excited states were used in two-color experiments on the resonance-enhanced sum-frequency generation. The resonance-enhanced fifth harmonic (FH) was generated near the 5d[3/2] state with excitation energy of 10.4 eV. For a resonance-enhanced nonlinear process the absorption of generated VUV emission on the resonance atomic transitions is of critical importance. This absorption leads to formation of excited xenon atoms, which are easily ionized through absorption of one or two additional laser photons. In intense laser fields the ionization probability is close to unity and nearly each act of a VUV photon absorption leads to appearance of a photoelectron in the ionization continuum. It is well known [27, 39] that in many cases the near-resonance ionization profiles closely follow the intensity and tuning curves of the generated sum-frequency field. Thus, the measurements of the ionization yield present a simple way to monitor the production of VUV photons in the near-resonance region. This method

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becomes especially important for optically thick media, where the VUV absorption is very strong and no VUV emission exits the gas cell. In addition to direct measurement of the ionization yield and VUV light intensity, there is another quite interesting method to monitor the excitation and ionization processes in the target gas. The initial photoelectrons created during the laser pulse may gain energy from the laser field and produce an avalanche ionization leading to a complete ionization of the gas in the excitation region. This laser-induced breakdown of the gas (”laser spark”) is a well-known phenomenon associated with high-power lasers. With tunable lasers, the onset of breakdown can have very good selectivity with respect to the excitation wavelength [40, 41, 42]. It allows one to monitor the multiphoton excitation of different excited atomic states including higher-lying Rydberd states, generation of near-resonance harmonics, various effects of internally-generated harmonics on excitation of atomic states, and others [40, 41, 42]. This method is especially convenient for the study of multiphoton excitation processes in conditions when it is difficult to measure directly the VUV emission or the ionization yield (laser-induced breakdown of the gas, high gas pressure, electronegative gases, etc.). Selective laser-induced breakdown was used in early studies on harmonic generation with Bessel beams [43].

3.

Bessel and Bessel-Gauss Beams

In general, the main difference in initiation of nonlinear processes by Gaussian and Bessel beams arises from the excitation geometry of these beams. In the beam waist of a Gaussian beam, where the intensity is maximum, the nonlinear processes are driven in a nearly collinear excitation geometry. On the contrary, the excitation geometry of a conical beam is essentially noncollinear and the cone angle may serve as an additional tunable parameter [6, 9]. In early experiments on the TH generation with Bessel beams in high-density xenon [43] an intense ionization band was detected near the three-photon 6s atomic resonance of xenon. That band was produced by TH photons, generated in the Bessel beam [43]. In further experiments on the TH generation and REMPI [10] compared were a tightly focussed Gaussian beam with a confocal parameter of an order of 0.1 mm and a Bessel beam with inclination angle α = 17◦ . Such a choice of tightly focused Bessel beam was dictated by the light intensity of 109 − 1010 W/cm2 necessary to drive the TH generation and REMPI. The length of the high-intensity central lobe of the Bessel beams was about 2 mm. Additionally, the excitation processes were studied for a Bessel-like focussed annular beam, when the central part of the initial Gaussian beam was blocked by a disk-shaped mask. This excitation mode reproduced the experimental arrangement of Ref. [7] on the TH generation in metal vapors. Both REMPI and tunable TH generation were studied near the three-photon 6s resonance of xenon. The general picture of excitation processes for this system has been studied well in a broad range of excitation conditions [27, 28, 29, 30, 31, 32, 33, 34, 44]. An example of simultaneously recorded REMPI spectra and TH is shown in Fig. 4. For a Gaussian beam the ionization signal at a moderate xenon pressure closely follows the TH excitation profile. For the used xenon pressures the three-photon 6s resonance in ionization spectra is absent as a result of well-known cancellation effect (see Section VII). Resonanceenhanced TH with a pressure-dependent excitation profile is generated near λ =147 nm. When TH overlaps the four-photon 4f resonances, a distinct reabsorption dip appears on

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Figure 4. Wavelength scans of REMPI and TH signals. Left column - xenon pressure 20 mbar; right column - 100 mbar. From top to bottom: ionization signal for the Gaussian beam; TH output for the Gaussian beam; ionization signal for the Bessel beam (from Ref. [10]).

the TH excitation profile due to the onset of two-photon absorption of a TH photon and a laser photon. In REMPI spectra it results in the appearance of an intense ionization peak at the resonance position. Further increase of pressure shifts TH band off the 4f resonance and the corresponding ionization peak is diminished. Ionization spectra underwent significant changes when the laser beam was focused by axicon (see Fig. 4). For the Bessel beam, a prominent ionization peak appeared very close to the 6s resonance. Instead of an intense 4f resonance a weak peak was registered at 20 mbar. At an increased xenon pressure the situation was inverted and an intense 4f resonance was developed for the Bessel beam, whereas a very weak peak, if any, could be registered with the Gaussian beam. The peak near the 6s resonance is similar to the ionization band registered in early highpressure ionization experiments with Bessel beams [43]. Keeping in mind such an intense ionization feature, an efficient TH generation could be expected for the Bessel beams. However, all the attempts to detect the VUV emission from the gas cell failed if the axicon was used to focus the laser beam. For comparison, for the Gaussian and focused annular beams the VUV signal was intense enough to be measured easily by a simple ionization cell. The

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absence of VUV signal did not allow one to carry out direct measurements of TH excitation profiles for Bessel beams. Such information was obtained from the ionization spectra and from experiments with annular beams. One of the reasons why the VUV emission was not detected in the Bessel beams is obvious: an increased interaction length of a Bessel beam is obtained at the expense of power. An axicon gives the light intensity by about an order of magnitude lower than the corresponding spherical lens [45]. It reduces the tripling efficiency because of the I 3 power dependence for the TH generation in a perturbative regime. Another reason is the reabsorption of the TH. As it will be discussed below, for a tightly focused Bessel beam the maximum of generated TH always is located very close to the atomic resonance where the TH absorption is very strong. This absorption produces an intense resonance-enhanced ionization but no measurable VUV light exits the gas cell. To get a deeper insight into the TH generation in Bessel beams, a detailed numerical simulation was carried out. A theory of the TH generation in conical beams was developed in [46, 47]. In a simplified manner the results of [46, 47] were used for interpretation of experimental observations of [43, 48]. The goal of present numerical work was to get the spectral dependence of the tunable TH generation and to compare the calculation results with experimental observations. Intensity of TH field is given by the expression [46, 47]     2 |E3| = 2π ψ3+  ρ dρ , 2

(1)

where the amplitude of the generated TH is 

ψ3+ ∼



sin θ ω32 (3) χ (ω1 , ω1, ω1) tan β IJ0(k3 sin βρ) , k3 θ θ=

I = 2π

 ∞ 0

(2)

L (k3 cos β − 3k1 cos α) , 2

(3)

ρJ0(k3 sin βρ)[J0(k1 sin αρ)]3 dρ ,

(4)

where χ(3) (ω1, ω1, ω1 , ) is the nonlinear susceptibility, ρ is the radial coordinate, L is the length of the medium, α (β), k1 (k3) and ω1 (ω3 ) are the inclination angle, the wave vector and the angular frequency of the fundamental (TH) light, respectively. Some terms in expression (2), which are independent on the wavelength, are omitted. In numerical calculations a simplification with the treatment of Bessel functions was made. In experiments, the Bessel beams are realized for a finite aperture of the pump beam. Therefore, the J0 amplitude profile is actual for some limited spatial volume but vanishes elsewhere. This feature is even more enhanced in a nonlinear optical process, where the main contribution comes from the central part of the beam with maximum light intensity. It allows one to treat the fundamental beam as a Bessel-Gauss beam [49] rather than a Bessel one. For a Bessel-Gauss beam the function J0 is multiplied by the Gaussian term exp[−(ρ/w)2]. This term suppresses outer oscillations of J0 but, at a proper choice of w, makes no influence at small ρ. Such an approach simplifies the treatment of the transversephase-matching integral (4), since for the Bessel-Gauss beam the integral can be solved

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Figure 5. Calculated spectral dependences of the TH generation: 1 - Bessel beam with α = 17◦; 2 - Gaussian beam with b=100 μm. Xenon pressure 50 mbar (from Ref. [10]). over a finite volume without oscillations of solution. Besides, it eliminates the divergence of the integral (4) at T = 1, where T = (k3sinβ)/(k1sinα) [46, 47]. In calculations, the integration was carried over the first 20 rings of the fundamental beam and the Gaussian parameter w was arbitrarily chosen to suppress the 20th maximum of |J0 | below 0.01 of its value. For the nonlinear susceptibility χ(3) the calculation results of [50] were used and only the resonant part of χ(3) was taken into account. The refraction indices for the fundamental beam and for the TH light were calculated with the use of Sellmeir formula and the data of [51] about the oscillator strengths for xenon transitions. Curve 1 in Fig. 5 shows the TH excitation profile calculated for the Bessel beam. This curve represents the result of direct numerical treatment of (1)-(4) without any fitting parameter. A close similarity of the calculated curve and the ionization peaks in REMPI spectra (Fig. 4) is obvious. The calculated dependence of TH generation reproduces well the sharp maximum near the atomic resonance, the long tail toward the shorter wavelength and the cutoff at the red edge. The maximum of TH corresponds to the maximum spatial overlap of the cube of the fundamental Bessel beam with the Bessel beam of generated TH. In this case T = 1 and the cone of TH light has the inclination angle β0 given by [46, 47] 1 tan α (5) 3 In the case here β0 = 5.8◦. The long tail of the TH tuning curve corresponds to 1 < T < 3 and in this region β0 < β < α. Near the long-wavelength edge 0 < T < 1 and tan β0 =

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λ

λ

Sum-Frequency Generation and Multiphoton Ionization in Rare Gases...

Figure 6. Calculated shift of the TH maximum with pressure: 1 - Gaussian beam with b =100 μm ; 2 - Bessel beam with α = 3.3◦; 3 - Bessel beam with α = 17◦. Full circles show the measured positions of the TH ionization peak in REMPI spectra (from Ref. [10]). 0 < β < β0. For comparison, Fig. 5 demonstrates the TH excitation profile calculated for the Gaussian beam with the confocal parameter b = 100 μm. This dependence was obtained by numerical calculation of the well-known phase-matching integral for a tightly focused Gaussian beam [44]  2

|F (bΔk)| =

π 2(bΔk)2exp(bΔk) , 0,

Δk < 0 Δk > 0 ,

(6)

where Δk = k3 − 3k1. It is interesting to note, that for the Bessel beam the spectral dependence of TH can be reproduced quite well by the phase-matching integral (6) if an extremely short (3-4 μm) confocal parameter b is assumed. In this case, the calculated curve shows very similar sharp peak near the resonance but without a long tail. Such a formal similarity may be useful for qualitative comparison of nonlinear optical effects in Gaussian and in Bessel beams. Figure 6 shows the calculated shift of the TH maximum from the 6s resonance as a function of xenon pressure. For a tightly focused Gaussian beam the maximum of TH is achieved at bΔk = −2 and with an increased gas pressure the maximum shifts rapidly off the resonance (curve 1 in Fig. 6). For the Bessel beams (curves 2 and 3 in Fig. 6) the position of the TH maximum is more tolerant against the pressure and for beams with large α the TH bands remains in the vicinity to the atomic resonance for the whole pressure range. Note a good agreement of the calculated dependence 3 in Fig. 6 and the data of experiments.

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Different pressure dependences result from different phase-matching conditions for the TH generation in Bessel and Gaussian beams. In a Bessel beam, a ring-type phase matching is realized, when different combinations of individual waves from the fundamental light cone contribute the overall TH production (see below). This feature of conical beams was termed in [7] as self-phase-matching (SPM). Formally, the SPM should facilitate the TH production in a more broad range of excitation wavelengths and gas pressure as compared with the Gaussian beams. However, there are other important factors which can spoil such a formal advantage of the Bessel beams for frequency tripling. For a tightly focused Bessel beam the maximum of TH always is located near the atomic resonance, where the absorption of TH photons is very strong. For an excitation wavelength far enough from the resonance the efficiency of TH generation is much lower (the tail of the TH excitation profile). To shift the maximum of TH off the resonance the inclination angle α must be reduced. Such Bessel beams, however, lose intensity in the central lobe. In this case, the I 3 power dependence will lead to significant decrease of the overall efficiency in spite of reduced TH reabsorption. Thus, for the Bessel beams two contradictory conditions are encountered. To provide necessary light intensity, which would be comparable with that in the reference Gaussian beam, the Bessel beams with large α should be used, whereas small angles α are required to avoid the TH reabsorption. Instead of using axicons, there is another way to realize conical excitation geometry similar to that in a Bessel beam. With a disk-shaped mask, the input Gaussian beam can be transformed into annular one, where the intensity distribution has the form of a ring. Focused annular beam forms a cone of waves like a Bessel beam and the intensity profile in the focal spot is very similar to the central part of a Bessel beam (see insets in Fig. 8). This type of beam is termed as Bessel-Gauss beams, where the outer rings of the Bessel profile are suppressed by the exponentially decaying Gaussian envelope. The Bessel-Gauss beam appears as a solution to the Helmholz equation and both the Bessel and the Gaussian beams emerge as special cases of the Bessel-Gauss beam. Excitation by focused annular beams does not require special conical optics and is thus more flexible in the choice of focusing conditions. Fig. 7 shows the spectral dependences of the TH output measured with focused Gaussian and annular laser beams. In both cases, the laser pulse energy was kept constant and the loss of energy for the masked annular beam was compensated by attenuating the dye laser output in the measurements with the Gaussian beam. Again, as it was observed with the Bessel beams, the conical excitation geometry shifts the TH excitation profile toward the longer wavelength and the maximum of generated TH is achieved closer to the atomic resonance. As a result, within some spectral range the VUV output for the annular beam exceeds that for the Gaussian beam. This observation agrees with the results of [7], where the efficiency of TH generation was increased by 4-5 times when a ring-shaped beam was used instead of a disk-shaped one. In the case here, the maximum VUV output for the Gaussian beam was four times higher than that for the annular beam of the same energy. However, for the used annular beam the light intensity in the focal region was about two times lower than in the corresponding Gaussian beam. Thus, being normalized on light intensity and taking into account the I 3 dependence of non-saturated TH output, the internal efficiency of the annular beam is, actually, about two times higher. Figure 8 demonstrates another example of the TH generation with focused annular

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Figure 7. Spectral dependences of the TH output: 1 - Gaussian beam ; 2 - annular beam. Xenon pressure 50 mbar (from Ref. [10]). beams [52]. In this experiment, a disk-shaped mask was shifted through the input laser beam. When the mask was located off the beam, the TH generation was driven by a pure Gaussian beam. With the mask inserted, the Gaussian excitation profile was transformed into a Bessel-Gauss one (see inserts in Fig. 8). If the excitation wavelength was set to the maximum of the TH tuning curve for the annular beam, a significant increase of the TH output was obtained despite of the loss of energy in masked input beam. This experiment demonstrates again a possibility of increasing the VUV output through the change of excitation geometry. Note, however, that such an increase is obtained for a rather narrow range of excitation wavelength. Outside this range the Gaussian beam is more effective. Experimental dependences of the TH output in Fig. 7 were simulated by numerical calculations. Again, the theory of SPM [46, 47] was applied, where the annular beam was considered as a set of thin-ring slits [47]. Each slit produces an individual Bessel beam and these Bessel beams overlap in the nonlinear medium. The generated TH photons come from all combinations of the three photons of the fundamental beams and the output TH radiation results from a superposition of multiple Bessel beams with different angles β. In the case here, the TH output was calculated as a superposition of ten Bessel beams. Such an approach corresponds to an annular beam, represented by three ring slits [47]. Figure 9 plots the calculated phase-matching curves for the Gaussian beam with b = 100 μm, for the annular beam and for a single Bessel beam. The last corresponds to a single ring slit having mean diameter of the annular beam and α = 3.3◦. For this single Bessel beam the position of the TH maximum coincides with that registered in experiments,

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mask

laser beam

laser pulse energy

VUV output

0

2

4

6

8

mask position (mm) Figure 8. Enhancement of the TH output in focused annular beams. Xenon pressure 30 mbar, laser wavelength 440.4 nm, TH wavelength 146.8 nm. (from Ref. [52]). Insets show the focused beam profiles. but the calculated TH excitation profile is too sharp and narrow. Superposition of multiple Bessel beams increases the width of TH profile and the calculated curve reproduces much better the shape of experimental spectral dependence. Note again, that the TH output for the annular beam can be approximated quite well by the phase-matching integral (6). In the case here, the phase-matching curve 2 in Fig. 9 formally corresponds to that for the initial Gaussian beams (curve 1) but with b reduced by 5-6 times. The analysis of the harmonics generation process in conical beams [53, 54] has shown, that under proper excitation conditions the conversion efficiency for an optimized configuration of a Bessel-Gauss beam can be significantly higher than for the Gaussian beam of the same focal intensity and confocal parameter [54]. However, at an equal pulse energy content the focal intensity in a Bessel-Gauss beam always is lower than in the reference Gaussian beam. Thus, in the perturbative regime a higher intrinsic conversion efficiency of a Bessel-Gauss beam does not mean necessarily a higher overall efficiency because of the I q power law, where q is the harmonic order. For q = 3 the intrinsic efficiency of a Bessel-Gauss beam can be high enough to overcome lower focal intensity and to increase the TH output in some spectral region, where the reference Gaussian beam has lower TH output because of a poorer phase matching. However, such an advantage of Bessel-Gauss beams becomes questionable for higher-order harmonics. In [55], the generation of third and fifth harmonics has been studied for a positively dispersive medium. It was shown that for q = 3 the use of Bessel-Gauss beams can give an increase by about 2 times in the TH

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Figure 9. Numerical simulation of experimental TH dependences in Fig. 7: 1 - Gaussian beam ; 2 - annular beam as a superposition of ten Bessel beams ; 3 - single Bessel beams α = 3.3◦ (from Ref. [10]). output, but for q = 5 the Gaussian beam always is more effective. Experiments on generation of resonance-enhanced fifth harmonic (FH) [56] were conducted in the negatively dispersive side of the 5d[3/2] resonance of xenon, and the tunable FH was generated in the range of λ=117-119 nm. When the Gaussian beam was transformed into Bessel-Gauss one, the fifth-harmonic excitation profile was shifted to the longer wavelength and was located closer to the 5d[3/2] resonance [56]. This behavior is very similar to the red shift of the TH excitation profiles, considered above. Due to this shift, the maximum of the FH output for the Bessel-Gauss beam at any gas pressure is achieved at a longer wavelength, than that for the reference Gaussian beam. Due to the differences in the FH excitation profiles, the relative conversion efficiency for the Bessel-Gauss and the Gaussian beams in an essential manner is wavelength-dependent. The maximum FH output for the Bessel-Gauss beam was about 5 times lower than for the Gaussian beam. In this sense, the ordinary Gaussian beam is more effective in the FH generation than the BesselGauss beam with the same pulse energy. However, such a comparison of the FH output relates to different excitation wavelengths. Within some spectral region, the Bessel-Gauss beam could have a comparable or even higher efficiency of the FH generation. At an equal energy content the focal intensity in the used Bessel-Gauss beam was about 1.5 times lower than in the reference Gaussian beam. In non-saturated perturbative regime it should result in a decrease by about an order of magnitude in the FH output. Nevertheless, at the red wings of the FH profiles the FH output for the Bessel-Gauss beam was nearly the same or

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even exceeded slightly the FH output for the Gaussian beam [56]. It means that for this particular spectral region the Bessel-Gauss beam has comparable or even higher efficiency in the FH generation. However, the FH output for these excitation wavelengths was rather weak being only a few percent of the FH maximum in the Gaussian beam. For any other spectral region within the FH tuning curve the Gaussian beam was always more effective. The loss of intensity in Bessel and other non-Gaussian beams is one of the most important factors which determine the overall efficiency of these beams for harmonic generation in comparison with the reference Gaussian beam. Even though the internal efficiency of some non-Gaussian beams can be increased substantially, the overall efficiency of these beams is competitive with Gaussian beams only for some specially chosen excitation conditions (excitation wavelength, gas pressure, beam geometry, etc.) With tunable lasers, the harmonic output can be increased for some spectral region, but this increase is obtained at the expense of tunability. With Gaussian beams, a relatively high harmonic output is maintained in a much broader spectral range. Laguerre-Gauss laser modes (optical vortices) were another type of annular beams checked for TH generation in conical excitation geometry. Frequency tripling of LaguerreGauss laser modes is a reliable way of generating optical vortices in VUV. In this case, a fundamental Laguerre-Gauss mode with the azimuthal index l is transformed into a mode with the index 3l and the orbital angular momentum per photon is tripled. The TH generation process was studied for a Laguerre-Gauss beam with l = 1 and the TH efficiency and excitation profiles were compared with the ordinary Hermite-Gaussian beam [52]. Experiments have shown that the maximum TH efficiency for the Laguerre-Gauss beam is about three times lower than for the ordinary Gaussian beam of the same energy. Here again a ring-shaped focal spot of the Laguerre-Gauss beam leads to reduced light intensity and, thus, a reduced overall efficiency of the TH generation. In a sharp contrast to other annular beams, the TH excitation profile remained unchanged when the Hermite-Gaussian laser beam was transformed into the Laguerre-Gauss mode. It means that the phase-matching conditions for the Laguerre-Gauss beam are nearly the same as for the ordinary HermiteGaussian beam. A similar conclusion about phase matching has been stated from experiments on second-harmonic generation in the Laguerre-Gauss beams [57]. Being focused by axicon, the Laguerre-Gauss beams form higher-order Jm Bessel beams where the maximum intensity is achieved within the first interference ring (see Fig. 2). The light intensity in this ring, however, is much lower than in the sharp onaxis peak of the lowest-order J0 Bessel beams. Therefore, no ionization signal nor VUV emission was registered with Jm beams with the available laser pulse energy. Numerical simulation of experimental TH dependences (Figs. 5 and 9) was based on the SPM theory of Refs. [46, 47], where the nonlinear response was derived from direct treatment of the scalar wave equation with the conical source field represented as a Bessel beam. Similar approach was used in other studies on generation of low-order harmonics with Bessel beams [53, 54]. Detailed comparison of experimental observations and numerical results, however, reveals several disadvantages of such an approach. First, the SPM theory fails to explain the appearance of the on-axis TH component in Bessel beams. For a Bessel beam having an inclination angle α the TH is generated along a cone with an inclination angle β. Depending on excitation conditions, the angle β can be tuned from 0 to α. According to Eq. (2), the amplitude of generated TH is proportional to tan β and it

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yields a sharp cutoff of the calculated dependences of the TH output at β → 0. In other words, the theory predicts the absence of any TH component directed along the propagation axis of the Bessel beam. However, significant on-axis TH was observed in [7] in conditions near the maximum conversion efficiency. Similarly, in ionization profiles measured with Bessel beams, the on-axis TH is responsible for a discernible broadening of the spectral TH excitation profiles toward the atomic resonance [13], whereas the calculated TH profiles are always cut to zero at a point where β = 0. Similar problem arises with the analysis of the TH angular spectrum. If the generated TH field is idealized as a Bessel beam, the far-field TH profile for a finite beam length should be a ring as a cross-section of the outgoing TH cone. In experiments of [7], it was the case for conditions far enough from the maximum of conversion efficiency, but near the maximum significant transformations of spatial profiles were observed. Namely, the increase of vapor density was followed by filling in the empty part of the TH ring without significant decrease in the ring diameter. At some pressure the ring was transformed into a disk. In [7] these observations were explained as due to the onset of Kerr nonlinearity. However, as it will be shown below, the presence of an intense on-axis TH component is a general feature of the TH generation with realizable Bessel or Bessel-Gauss beams independent on the Kerr effect. In [13, 18, 19] another way to evaluate the TH generation process in conical light beams was suggested. The Bessel and other conical beams can be viewed as a superposition of plane waves with their wave vectors all inclined by a constant angle α toward the propagation axis. It has been shown in [13, 18, 19] that the TH excitation profiles for Bessel beams can be analysed from a simple picture of a few angled plane waves constituting the Bessel beam. In the TH generation process, every TH photon is made from three photons picked from the fundamental light cone. The source field can thus be decomposed into elementary spatial configurations of three sub-beams having some azimuth angles φ1 , φ2 , and φ3 on the fundamental cone. For a given spatial arrangement of sub-beams, the nonlinear response builds-up according to the phase-matching requirements. Since every partial field configuration contributes the overall TH production, the TH output can be found as a superposition of contributions from all possible elementary configurations of the source field. The problem of TH generation with a few plane waves is a well-studied subject of nonlinear optics and it gives a simple and reliable way to evaluate the TH generation in conical light beams. Consider first the TH generation in an optically thick medium. This situation is realized when tightly-focussed conical beams are used for the TH generation and the TH maximum is achieved close to the atomic resonance. The TH excitation profiles in this case are monitored through resonance-enhanced multiphoton ionization. In these ionization processes the generated TH field plays a significant role in determining the total transition probability. For a certain threshold product of number density and oscillator strength, the TH field generated by two crossed plane waves evolves in amplitude and phase to interfere destructively with direct three-photon excitation of dipole-allowed atomic transition everywhere except for a region on the blue side of the resonance position [58, 59, 60]. At this region the interference becomes constructive and it produces an ionization profile which is identical to the expected pressure-broadened atomic line[58, 59, 60, 61]. This Lorentzian profile or cooperative line is located essentially at the frequency where the phase-matched TH is produced by crossed beams. The shift of the Lorentzian line from the resonance position

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is given by the frequency expression for the so-called cooperative line shift [58, 59, 60]. With this picture, the TH excitation profiles for Bessel beams in an optically thick medium were shown to be principally describable from a simple picture of a few angled plane waves together with the characteristic feature of the cooperative line shifting associated with noncollinear three-photon excitation [13, 18, 19]. At any gas pressure the location of the main pressure-dependent TH peak in a Bessel beam with an inclination angle α matches exactly the value of the cooperative shift for two plain waves angled by 2 α [13]. It was shown in [13, 18, 19] that the overall TH envelope for a Bessel beam can be viewed as a superposition of ”homogeneous” Lorentzian profiles variously displaced by the cooperative shift. These Lorentzians correspond to different spatial configurations of two and three plain waves from the conical wave front. In a description of the TH generation process in a Bessel beam, all combinations of partial waves having any azimuth angle φ from 0 to 360◦ on the light cone are superimposed. For every configuration of two or three crossing plane waves the concept of the cooperative line shift predicts the ionization profile as a Lorentzian peak. The location of this peak in the spectrum is determined by the applicable value of the cooperative shift. Remarkable feature of these elementary excitation profiles is that for a fixed interaction length, the magnitude of the Lorentzian line remains nearly constant over rather large range of detunings from the atomic resonance [59, 61]. This is because the TH gets stronger as the detuning gets larger, while the absorption gets weaker at the same rate. Thus, the product of the TH intensity and the atomic absorption remains nearly constant (see Fig. 3 in Ref. [61]). In an optically thick medium the propagation distance of the TH light is very short and all the generated TH photons are absorbed entirely within the central high-intensity core of the Bessel beam. It means that for all sub-beam combinations an equal excitation volume can be assumed. Hence, the TH excitation profile in a Bessel beam can be evaluated from a simple picture of equally-weighted Lorentzians, where any partial combination of the fundamental sub-beams yields the same ”brightness” of the corresponding elementary TH peak. For the general case of three interacting waves from a conical wave front, the location of the cooperative line is given by the cooperative shift δc written as [18]

δc = Δ0(−1.11 +

18 ), sin α[3 − cos(φ1 − φ2 ) − cos(φ1 − φ3 ) − cos(φ2 − φ3)] 2

(7)

where Δ0 = πN F01e2 /2mω, N is the gas density, F01 is the oscillator strength, ω is the TH angular frequency, and φ1, φ2 , and φ3 are the azimuth angles of three fundamental waves on the light cone. The angles φi have all possible choices from 0 to 2 π. Neglecting the small constant term, (7) can be written as δc =

4δ0 , 3 − cos(φ1 − φ2) − cos(φ1 − φ3 ) − cos(φ2 − φ3 )

(8)

9Δ0 where δ0 = 2 sin 2 α determines the location of the ”main” TH peak. This peak corresponds to the excitation geometry when φ2 = φ1 ± 180◦ and φ3 is arbitrary [13, 18]. The minimum value of the cooperative shift is δc = 89 δ0 and it is realized for symmetrical configuration of

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three fundamental sub-beams separated by 120◦. In this case, β = 0 and the generated TH is directed along the propagation axis of the Bessel beam. The inclination angle β of the TH wave vector is given by

tan β =

1 tan α · 3







3 + 2 cos(φ1 − φ2 ) + cos(φ1 − φ3 ) + cos(φ2 − φ3 ) .

(9)

Thus, (8) can be written as δc =

8δ0 9(1 −

tan2 β ) tan2 α

.

(10)

Equation (10) couples spectral variable δc and angular variables α and β. The angle β changes from 0 to α when δc changes from its minimum value of 89 δ0 to infinity. The overall TH excitation profile in a Bessel beam builds up from all elementary TH components. This profile is to be found as a squared sum of all partial TH amplitudes. For an optically thick medium, however, all the generated TH photons are absorbed entirely within a very short propagation distance. One can suppose that the integral excitation in such conditions can be derived as simple incoherent sum of intensities rather than coherent sum of amplitudes. It simplifies the calculation procedure very much and gives very good agreement with experimental observations. Every elementary TH profile, monitored in an optically thick medium through atomic excitation and subsequent ionization, is a Lorentzian cooperative line with the pressuredependent width γ and the cooperative shift δc . For an excitation frequency ω, the contribution of a Lorentzian line into the overall TH excitation envelope is given by I(ω) =

γ/π , (ω − ω0 − δc )2 + γ 2

(11)

where ω0 is the resonance frequency, ω0 + δc is the central frequency of the shifted Lorentzian line, and δc is given by (8). Cylindrical symmetry of the Bessel beam allows one to fix φ1 = 0 and the overall signal S(ω) from the TH generation is then given by an integral over azimuth angles φ2 and φ3 : S(ω) =

2π2π

I[ω, δc(φ2 , φ3)] dφ2dφ3 .

(12)

0 0

Figure 10 shows the TH excitation profile calculated using (12) for δ0 = 18γ. Such δ0 corresponds to the experimental conditions of [13, 18]. Calculated profile in Fig. 10 shows sharp near-resonance peak followed by a tail toward the blue side of the spectrum. Such shape of the profile has an excellent agreement with the TH excitation bands registered at a moderate gas pressure in ionization experiments [10, 13]. For a higher gas pressure, the experimental profiles show similar shape but with less pronounced peak and more intense tail [43, 13] because of additional molecular absorption in a dense gas. In agreement with experimental data [13, 18], the location of the TH peak coincides the location of the cooperative line at δ0 (curve 2 in Fig. 10). Note distinct red wing of the calculated profile, where according to the SPM approach the TH should vanish. Red wing covers the region toward

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(ω−ω0)/γ

Figure 10. Calculated near-resonance excitation profiles associated with the phase-matched TH in an optically thick medium: 1 - superposition of multiple Lorentzians in a Bessel beam; 2 - single Lorentzian peak (not scaled to 1) located at δ0 (from Ref. [19]). and beyond the resonance position at ω0 . Again, it agrees very well with experimental observations [43, 10, 13]. The peak of the TH profile occurs at a wavelength where δc = δ0 . In this case, Eq. (10) gives the known condition (5) for the angle β. In experiments of [7] on the TH generation with conical beams the excitation wavelength was fixed and the phase matching was controlled by changing the vapor density. In order to analyze experimental observations of Ref. [7] the source field is again decomposed into elementary sub-beam combinations and the overall response is constructed from all partial TH components. Observable quantity now is the TH emission and evaluated is the angular spectrum of generated TH. Generation of resonance-enhanced TH with crossed plane waves has been analyzed in [61]. It has been shown that for two beams crossed at some angle, both the maximum of TH radiation and the atomic excitation show frequency shift Δω. This shift is written in the present notations as Δω = 4Δ0/(1 − μ2 ), where μ = κc/ω. Parameter κ is determined ˆ 2 ) so that κ2 = (ω 2/c2)[1 − 89 sin2 α], where 2α is the crossing n1 + n as κ= ( 13 ω/c)(2ˆ angle between two beams. Frequency shift Δω can be written as Δω = 9Δ0/2 sin2 α and it is just the value of the cooperative shift δ0 (see above). Thus, for any pair of sub-beams from conical wave front the generated TH emission can be characterized by parameter μ and corresponding frequency shift Δω = δc . The general case of the TH generation with conical beams involves interaction of three pumping waves. For three-beam configurations the parameter κ is generalized as ˆ2 + n ˆ 3 ) and, using (9) and (10), one has κ= ( 13 ω/c)(ˆ n1 + n 1 μ= 3



9−8

 δ0 sin2 α sin2 α = 1 − sin2 α(1 − ρ2 )  1 − (1 − ρ2 ) , δc 2

(13)

where ρ = tan β/ tan α. For a Bessel beam, μ changes from its minimum value of cos α for the on-axis TH (β = 0, ρ = 0) to 1 for collinear TH (β = α, ρ = 1). Any TH

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component, regardless of its origin in one-, two- or three-beam pumping process, can thus be characterized by parameter μ. Intensity of the TH emission in crossed beams is given by a sync function [61] I(ω) ∼

sin2( 12 uω ) , u2ω

(14)

where uω = ωL c (n3 − μ), and n3 is the refraction index at the TH frequency (for the fundamental frequency the refraction index n1 =1 is assumed). For a two-level system the refraction index n3 can be approximated as n3 = 1 − 2Δ0 /(ω − ω0 ) [13]. Following terminology of Ref. [7], N0 is determined as the gas density when the TH is peaked along 2 the propagation axis, i.e. β = 0, ρ = 0 and Δ0(N0) = (ω−ω04) sin α . It is easy to see that the density N0 corresponds to the frequency shift δc = 89 δ0 . The TH output is then written as I(ω) ∼ (N/N0)2 ·

sin2( 12 uω ) , u2ω

(15)

2 2 where uω = ωL 2c (1 − ρ − N/N0 ) sin α. Circular symmetry of the pumping beam results in a ring-shaped far-field profile of emerging TH. If the far-field radii of the fundamental and the TH lights are R and r, respectively, then r/R = tan β/ tan α = ρ. The expression (15) determines the TH intensity as a function of ρ and N/N0 and, being integrated over radial coordinate ρ, (15) gives the overall TH output as a function of N/N0 . Figure 11 shows angular spectrum of the TH output calculated for different values of N/N0. These numerical results were obtained with the use of (15) and experimental parameters of Ref. [7], namely L=4 cm, α=10 mrad, and λ=1.064 μm. At a low pressure, the TH is generated along a cone surface close to the fundamental beam. In this case, there are fewer atoms in the cell and the TH intensity is small. With the increased vapor density the TH is increased gradually in intensity and decreased in cone angle. This evolution is followed by filling in the inner part of the TH cone. At N = N0 the far-field TH profile is transformed into a disk. Further, the TH becomes peaked along the beam axis and decreases rapidly. All these transformations follow exactly the experimental observations of Ref. [7]. Note, that the calculations results show the presence of an intense on-axis TH, whereas within the frames of the SPM approach the TH vanishes at β → 0. This on-axis TH has much of a character of a plane wave driven by symmetrical three-beam combinations of the source field. A few concluding remarks concerning the TH generation in conical beams, should be added. Non-collinear interaction of individual waves in conical beams induces nonlinear polarization wave which travels along the beam axis with the phase velocity vexc = c/n1 cos α. Momentum conservation imposes the axial phase-matching condition for the wave vectors k1 of the fundamental and k3 of the generated TH as

k3 cos β = 3k1 cos α ,

(16)

or cos β =

n1 cos α c = , n3 vexc n3

(17)

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ρ Figure 11. Evolution of the TH angular spectrum with gas pressure (labeled in units of N/N0; from Ref. [19]).

which is the known Cherenkov condition, where β is the cone angle of the Cherenkov emission. In other words, the generation of conical TH proceeds as a Cherenkov-type process [8, 62, 64] when phase velocity of the driven polarization vexc = c/n1 cos α exceeds the phase velocity c/n3 of the generated radiation. At β → 0 the Cherenkov-type conical emission disappears and this threshold condition is achieved when n3 = n1 cos α  cos α. It is easy to see that it occurs when N = N0 and δc = 89 δ0 . If δc > 89 δ0 (N < N0 ), the phase velocity of the driven nonlinear polarization vexc exceeds the phase velocity c/n3 of the TH emission. In this case, a Cherenkov-type process is established and the TH output is identical to the Cherenkov cone. When δc < 89 δ0 (N > N0 ), the phase velocity of TH exceeds vexc. Phase-matched TH can still be generated and resulting TH emission is directed along the propagation axis. These two regimes of the TH generation may be termed as superluminal (vexc > c/n3) and subluminal (vexc< c/n3) ones. The evolution of TH profiles in Fig. 11 can then be interpreted as a gradual transition from superluminal to subluminal regimes of the TH generation. When gas pressure is low, the TH is produced in a Cherenkov-type superluminal process and the TH output is identical to the Cherenkov cone. This TH cone is squeezed toward the beam axis when gas pressure is increased. At N > N0 the cone disappears and the TH output, driven by a subluminal excitation process,

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becomes peaked along the beam axis.

4.

Segmented Conical Beams

The field distribution in a Bessel beam results from a superposition of infinitely many plane waves all inclined by a constant angle α toward the propagation axis. In addition to the angle α, which is the same for all components of a conical beam, there is also the azimuthal angle φ of a particular fundamental wave on the light cone. In a three-photon-induced process, the azimuthal angles of each of three interacting fundamental waves determine a given component of the nonlinear polarization of the medium and, in the present instance, the resultant TH field. In the description of TH generation process, all combinations of particular waves having any azimuthal angle φ from 0 to 360◦ on the light cone are superimposed. Segmented [18] and Mathieu [20, 21, 22] beams share the propagation properties of a Bessel beam, but their amplitude profiles differ significantly from the Bessel pattern. In these beams the light energy also propagates along a cone surface, but the light field is comprised of sub-beams from a limited range of azimuth angles. Thus, some sub-beam combinations and the correspondent TH components of a full-aperture Bessel beam disappear in segmented and Mathieu beams. In this sense, a Bessel beam represents a particular case of the full-aperture conical excitation geometry when the azimuth angles φi have any value from 0 to 360◦. (a)

(b)

(c)

(d)

Figure 12. CCD pictures of the focal regions: (a) full-aperture conical beam (Bessel beam); (b)-(d) segmented conical beams. The insets show the used slit masks (from Ref. [18]). In experiments with segmented conical beams, the geometry of the driving field within

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the excitation volume was controlled through the azimuthal selection of interacting waves [18]. Laser beam was focused into the gas cell by axicon. Different slit masks were placed close to the entrance surface of the axicon (see masks depicted in the corner of each separate component of Fig. 12). The width of the slits was 0.6 - 0.8 mm. Selected radial slices of the laser beam were focused by the axicon forming segmented conical beams. As in the case of a full-aperture Bessel beam, all the components of the segmented beam were inclined by an angle α toward the propagation axis and crossed along this axis, but the composition as a function of azimuthal angle was controlled. Fig. 12 shows the focal regions of different conical beams used in experiments (in all cases the central part of the axicon was blocked by a 1 - 1.5 mm obstacle). The full-aperture laser beam focused by the axicon produced the known pattern of a zero-order Bessel beam. For segmented beams, this interference pattern was reduced to a pattern of a few crossed plane waves selected by the masks as shown in Fig. 12.

Figure 13. TH excitation profiles: (a) full-aperture Bessel beam; (b) linear slit geometry; (c) two slits 90◦; (d) three slits 120◦; (e) two slits 120◦ (from Ref. [18]).

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Results of experiments with segmented conical beams are summarized in Fig. 13. The upper trace in Fig. 13 shows the TH ionization profile for the full Bessel beam (no mask). This profile was considered and analyzed in the previous section. The other four traces in Fig. 13 show results with the geometrical masks indicated in the figure. For a linearslit mask (Fig. 13b) the ionization profile is reduced to a single Lorentzian-like peak at the position of the TH maximum in the initial unmasked Bessel beam. For other masks allowing more than one spatial combination of sub-beams, additional spectral components can be recognized in the TH profiles. For two crossed slits (Fig. 13c), the same Lorentzian peak was registered together with an additional weak band on the short-wavelength side of the main peak. For the mask with three slits separated by 120◦ (Fig. 13d), the TH profile again has two components - a sharp Lorentzian peak and a blue band. The Lorentzian peak is located closer to the 6s resonance than the peak for one slit or for two slits at right angles. When any one of the three slits was blocked, the Lorentzian peak disappeared and only the broad blue band remained in the spectrum (Fig. 13e). Again, a description of TH generation in segmented conical beams requires consideration of two geometries of interacting sub-beams. The first is the general case of photons from each of three interacting fundamental plane waves having different azimuthal angles φ1 , φ2 , and φ3 . The second is the two-beam case, where two fundamental photons come with the same azimuthal angles φ1 = φ2 and the third photon comes from another part of the light cone. The collinear (single-beam) case φ1 = φ2 = φ3 is excluded from consideration since no resonant three-photon excitation and no phase-matched TH generation is produced by a single plane wave in the negatively-dispersive side of the atomic resonance. The analysis of the TH excitation profiles was similar to that for a full-aperture Bessel beam, where the concept of the cooperative line shift was used. For two angled plane waves the excitation profile has the shape of a Lorentzian line shifted to the blue side of the expected atomic line position [59, 61]. The shift of this atomic resonance line profile is given by the analytic expression for the interference-based cooperative shift, and the Lorentzian shape of the profile is identical to the unshifted pressure-broadened atomic line [59, 58]. For the full-aperture conical beam there is an infinite number of pairs of interacting waves having different crossing angles and, thus, different values of the cooperative shift. This leads to a large spreading of individual Lorentzians in the full Bessel-beam spectrum. The overall TH envelope for a Bessel beam builds up as a superposition of such Lorentzians (see previous section). This superposition of shifted lines has an interesting formal similarity with inhomogeneous broadening of spectral response in a system where individual homogeneous components get spectral spreading and superposition of these components produces the overall inhomogeneous spectral profile. With this analogy, the whole TH profile for a Bessel beam can be viewed as the result of ”beam-geometry broadening” of the cooperative line, analogous to inhomogeneous broadening of an atomic line profile. This broadening arises due to an infinite number of possible combinations of interacting waves in Bessel beams, when every particular combination gives its own ”homogeneous” contribution to the resulting excitation profile. Segmentation of the input beam reduces the number of available combinations of interacting waves. Some combinations of the fundamental waves in segmented beams are blocked and the corresponding parts of the TH profile disappear. Again, this situation is in some sense analogous to hole burning in an inhomogeneous spectral profile. Here one can

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extract a selected ”homogeneous” component from the profile of superimposed Lorentzians. But, because of axial symmetry of a full-aperture conical beam it is impossible to form a ”homogeneous dip” in the TH profile as none of the masks is able to block all spatial configurations of a given geometry. Nevertheless, for conical beams it is possible to separate some single configurations while blocking all others. In this case the overall TH profile should be reduced to a single ”homogeneous” spectral component. This is just the case illustrated in Fig. 13b, where a slit mask selected a single configuration of beam components entering the excitation zone from opposite sides of the light cone. For this single-slit mask the TH generation proceeds via a planar combination of two interacting waves, where two photons of the fundamental are taken from one sub-beam and the third photon comes from another sub-beam (φ1 = φ2 and φ3 = φ1 ± 180◦). Fig. 13b shows that for a single-slit mask a Lorentzian-like peak appears at the position of the TH maximum from the full Bessel beam of Fig. 13a. The width of this very Lorentzian profile is proportional to xenon pressure (0.035 nm/bar FWHM) and matches exactly the theoretical value for pressure-induced broadening of the atomic 6s resonance of xenon [63] and the shift is in agreement with [59, 58]. For two crossed slits (Fig. 13c) the same Lorentzian peak was observed together with an additional weak band on the blue side of the main peak. For this case there are three possible combinations of sub-beams. The first combination is realized for each set of two slits similar to the case of a single slit. These combinations produce the same Lorentzian peak as in Fig. 13b. In a second possible combination, two photons come from the opposite sides of a slit and the third photon comes from another slit ( φ2 = φ1 ±180◦ , φ3 = φ1 ±90◦ ). As it was shown in [13], such non-planar three-beam combination is equivalent to a planar two-beam combination giving the main TH peak. The last is a two-beam combination, where two photons come from one side of a slit and the third photon comes from another slit (φ1 = φ2 , φ3 = φ1 ±90◦ ). This particular combination is responsible for the appearance of the weak TH band to the blue of the main peak. For two crossed unfocussed beams the location of the pure Lorentzian component in the spectrum is determined by the applicable value of the frequency shift given by Eq. (8). From simple considerations of the conical interaction geometry, it is easy to see that two particular waves with azimuth angles φ1 and φ2 will cross at an angle θ given by θ φ = sin sin α , 2 2 where φ = φ2 − φ1 . Using this expression, Eq. (8) can be written as sin

δc =

δ0 9Δ0 9Δ0 = = 2 θ φ 2 2 2 sin ( 2 ) 2 sin 2 sin α sin2 φ2

(18)

(19)

9Δ0 where δ0 = 2 sin 2 α determines the position of the main peak with θ = 2α. Thus, the shifted position of the Lorentzian peak for any given pair of sub-beams can be determined in units of δ0 . For instance, for two beams crossing at φ = 90◦ the shift is δc = 2δ0 . This point corresponds exactly to the maximum of the blue band registered with two crossed perpendicular slits (Fig. 13c). Similarly, for φ = 120◦ δc = 43 δ0 . Again, this point can be recognized in REMPI spectra as the maximum of the band registered with two or three slits separated by 120◦ (Fig. 13d and Fig. 13e).

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For any two-beam combination the TH profile should have a Lorentzian shape with the width independent of crossing angle. However, the spectral profiles produced by beams crossing at 90◦ and 120◦ were registered as broad bands instead of narrow Lorentzians. This effect results from the angular spreading of the individual beams which was up to ±20◦ in the case considered. The sensitivity of δc to the angular spreading dφ can be estimated by d(δc )/dφ as d(δc ) = −

δ0 sin2 φ2

tan

φ 2

dφ = −

δc tan φ2

dφ

(20)

This expression shows that the larger the value of φ the less sensitive is d(δc ) to dφ, and that d(δc ) → 0 at φ → 180◦. For the pure Lorentzian peak produced from a single slit geometry, where δc = δ0 , the angular spreading of ±20◦ leads to a negligible (about 3%) broadening of this peak. By contrast, for beams crossing at 120◦ the same spreading leads to an asymmetrical broadening of the corresponding Lorentzian by 0.2 δ0 to the red and by 0.37δ0 to the blue. For beams crossing at 90◦ the broadening is further increased, being about 0.5δ0 and δ0 for the red and the blue sides, respectively. As a result, instead of narrow Lorentzians, broader bands were registered for crossing angles 90◦ and 120◦. For the general case of three non-planar interacting waves from the axicon, the cooperative shift is given by Eq. (8). Assuming φ1 = 0 this equation is written as δc =

4δ0 3 − cos φ2 − cos φ3 − cos(φ2 − φ3 )

(21)

For three beams separated by 120◦ this expression gives the value δc = 89 δ0 for the shift. Among all possible two- and three-beam combinations from segmented conical beams this is the minimum value of the cooperative shift. The corresponding peak was registered in experiments with the three-slit mask (see Fig. 13d). As expected, this peak had the Lorentzian profile with the same width of 0.035 nm/bar FWHM. The sensitivity of the three-beam δc to the angular spreading dφ can again be estimated by d(δc )/dφ: d(δc ) = −

4δ0 [(sin φ2 + sin(φ2 − φ3 ))dφ2 + (sin φ3 − sin(φ2 − φ3 ))dφ3] [3 − cos φ2 − cos φ3 − cos(φ2 − φ3)]2

(22)

For the symmetrical three-beam configuration d(δc ) → 0 when φ2 → 120◦ and φ3 → 240 . In our case of ±20◦ spreading, the corresponding Lorentzian peak gets only a minor (about 5%) broadening toward the shorter wavelength. Figure 14 shows the TH excitation profile calculated for a conical beam segmented by symmetrical three-slit mask. This profile was obtained by numerical evaluation of the integral ◦

  

S(ω) =

I(ω, δc) dφ1dφ2dφ3 ,

(23)

φ1 φ2 φ3

where I(ω, δc) is given by (11) and δc is given by (8). The simulation of the TH profile in Fig. 14 was again based on the approach of equally-weighted Lorentzians. Being

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Figure 14. Near-resonance TH excitation profiles for segmented conical beam. Upper trace, experiment; lower trace, numerical simulation. The inset shows the used slit mask (from Ref. [19]). segmented by a mask, the range of possible azimuth angles is determined by two factors. First, this range is given by geometry of the used mask. For the case shown in Fig. 14 the mask selected three sub-beams separated by 120◦. Second, a finite width of mask slits and diffraction on mask edges give some spreading ±Δφ of the azimuth angles for every selected sub-beam. Thus, for the case considered all the azimuth angles φi in (23) run over three ranges of ±Δφ, 120◦ ±Δφ, and 240◦ ±Δφ. Best calculation results were obtained for Δφ = 15◦ − 20◦, which agrees well with the value of spreading estimated in experiment. The TH profiles in Fig. 14 have two distinct spectral components. Sharp near-resonance peak results from three-beam excitation, when all three sub-beams enter the excitation zone from different slits. Spectral spreading of individual Lorentzians in this case is small and the envelope is formed as a sharp and narrow peak located at δc = 89 δ0 . If any pair of sub-beams enter the excitation zone from the same slit (two-beam excitation), the range of δc is larger and individual Lorentzians are spread over a more broad spectral range. It yields a second component of the profile as a broad band located at the blue side of the peak. Finally, when all three sub-beams enter the excitation zone from the same slit (single-beam excitation), the range of shift is δc > 15δ0 and the corresponding TH components are located off the range of interest. Such far wing of the TH envelope is very weak since the spectral density of Lorentzians is reduced rapidly with an increased δc .

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225

Crossed Focused Beams

Noncollinear interaction of multiple beam components is the main feature of conical light beams in initiation of nonlinear processes. The simplest noncollinear excitation mode of any conical beam is realized with two sub-beams selected from the light cone. This excitation mode produces an elementary nonlinear response of the medium and shares main properties of conical beams in nonlinear optics. Such a two-beam excitation geometry can be arranged through segmentation of the conical wave front with the aid of slit masks as it was shown in the previous section. Fig. 15 demonstrates another way of producing such an elementary noncollinear excitation mode. The initial laser beam is split into two parallel sub-beams. Being focused by a lens, these sub-beams form noncollinear configuration of two crossed focused beams, where the crossing angle can be changed by changing the focal length of the lens or by changing the distance between the two parallel sub-beams. With the splitter mirror removed and minor readjustment of the focusing lens, the two-beam excitation mode was transformed easily into a reference single-beam excitation geometry. mirror R=1

lens gas cell

mirror R=0.5

λ/2 plate λ/4 plate

Figure 15. Schematic of setup arrangement in experiments with crossed focused beams. In the TH generation process, two angled beams produce four spatial TH components. First, there are two collinear TH components generated by each of sub-beams. For this collinear TH a spectral selection is needed to separate the intense fundamental and the weak TH lights. Second, the two sub-beams drive two additional components of nonlinear polarization and the TH photons are generated in a process when two fundamental photons are taken from one sub-beam and the third photon comes from another sub-beam. It gives two noncollinear TH components generated along some directions different from the propagation direction of the fundamental sub-beams. Therefore, the noncollinear TH is free of fundamental background and can easily be selected spatially. An additional flexibility in the choice of excitation conditions is provided through the control of polarization of the two sub-beams. With the aid of half-wave and quarter-wave plates (see Fig. 15), the polarization of either single or both sub-beams could be controlled and any combination of linearly or circularly polarized sub-beams could be obtained. Figure 16 shows an example of TH excitation profiles measured with single linearly polarized beam and two crossed beams polarized normally to the common propagation plane(s-polarization) [65]. The laser pulse energy was the same in both cases. For a single beam, the TH excitation profile has the well-known shape of a broad band in the negatively-

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6s TH INTENSITY (arb. units)

4f 1

2

439.5

440.0

440.5

LASER WAVELENGTH (nm) Figure 16. Wavelength scans of TH output: 1 - single beam; 2 - two focused sub-beams with crossing angle 8◦. Xenon pressure 10 mbar (from Ref. [65]). dispersive side of the atomic resonance. This band has a reabsorption dip at the position of the four-photon 4f resonance of xenon. With an increased xenon pressure the TH band is broadened and shifted rapidly toward shorter wavelength. The maximum of the TH output is achieved at a pressure of about 30 mbar. Above 30 mbar the TH output is reduced gradually. Excitation by two crossed beams changes the TH excitation profiles. In this case, the collinear TH band is reduced and an intense peak of noncollinear TH appears near the atomic resonance. Intensity of this noncollinear TH was very sensitive to the overlap of both beams. At any gas pressure, however, it was quite easy to get the same or even higher peak amplitude as compared with the TH band for the initial single beam. Note, that the gain length for noncollinear TH generation is only a fraction of the confocal parameter of initial single beam because of an oblique propagation of crossed sub-beams. Besides, crossed focused beams produce a specific interference pattern within the excitation volume [66]. The TH excitation profiles in Fig. 16 demonstrate again the main peculiarities of the TH generation in collinear (single beam) and noncollinear (crossed beams) excitation modes. The maximum conversion efficiency is comparable in both cases, but for a given gas pressure the maxima of the TH output are achieved at different wavelengths. The noncollinear excitation mode is able to increase significantly the TH output within some spectral range, but this range is relatively narrow. For a single-beam, the TH output may be weaker, but the

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TH generation process is phase matched within a much broader spectral range. All these observations agree well with experimental results described in previous sections. TH

REMPI

1

1

2

2 6s

3

440.4

440.6

440.8

3

440.4

440.6

6s

440.8

laser wavelength (nm) Figure 17. Wavelength scans of TH output (left) and REMPI signal (right) near the 6s resonance of xenon in crossed beams: 1 - s-polarization; 2 - opposite circular polarizations; 3 - orthogonal polarizations. Crossing angle 7◦; xenon pressure 20 mbar (from Ref. [67]). Figure 17 demonstrates excitation profiles of the resonance-enhanced TH and REMPI for different polarizations of pumping sub-beams [67]. Again, the intense TH peak near the resonance position belongs to the non-collinear TH generated by crossed beams. For s-polarization of pump beams this TH peak is followed by a long weak tail of collinear TH. For crossed polarizations of sub-beams the tail remains unchanged but the non-collinear TH peak is reduced by 7-8 times. Under excitation by beams with opposite circularity the collinear TH band disappears since no TH is generated along each of circularly polarized sub-beams. However, the non-collinear TH does not disappear and quite an intense TH peak is registered. The ionization spectra closely follow the TH excitation profiles for all used polarizations (see Fig. 17). This is a well-known observation when the absorption of TH photons on the wing of the atomic resonance and subsequent ionization of excited gas atoms by laser light dominate REMPI process and the ionization profiles look exactly like the phase-matching curve for TH generation [27, 39]. The peaks of REMPI profiles are shifted slightly toward the resonance position with respect to the TH peaks, but this shift is very small. For spectra shown in Fig. 17 the shift between TH and REMPI peaks is about 20 pm. A deeper insight into the process of TH generation in crossed laser beams with different polarizations can be obtained from the general approach to the frequency tripling in isotropic media [44, 68, 69]. The induced nonlinear polarization at 3ω is P(3)(3ω) = . χ(3) ..E(ω)E(ω)E(ω), where χ(3) is the third-order nonlinear susceptibility and E(ω) is the fundamental field at frequency ω. For crossed beams the fundamental field can be written as a sum of two plane waves with wave vectors k1 and k2 : E = E1 eik1 ·r + E2 eik2 ·r . Assume that |E1| = |E2|=E. For the simplest case of s-polarization both beams are polarized linearly normal to the common propagation plane and, for the coordinate system shown in

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Fig. 18, the field components are E1x = E2x = 0, E1y = E2y = E, and E1z = E2z = 0. The components of induced nonlinear polarization are Px(3) = Pz(3) = 0,





3 i3k1 ·r + 3ei(2k1 +k2 )·r + 3ei(k1 +2k2 )·r + ei3k2 ·r . Py(3) = χ(3) yyyy E e

k2

x α α

y z

k1 Figure 18. Two crossed plane waves with wave vectors k1 and k2 in the chosen coordinate system. For s-polarized crossed waves the induced nonlinear polarization has four non-zero terms and, under phase-matching conditions, there are four TH emission components. Two of them are generated collinearly along k1 and k2 of pump beams. Other two non-collinear components are generated along (2k1 + k2 ) and (k1 + 2k2). All these four TH components are linearly polarized in y direction. For cross-polarized beams E1x = E, E2x = 0, E1y = 0, E2y = E, and E1z = E2z = 0. The nonlinear polarization components are Pz(3) = 0,



i3k1 ·r (3) (3) i(k1 +2k2 )·r + (χ(3) Px(3) = E 3 χ(3) xxxx e xxyy + χxyxy + χxyyx )e







i3k1 ·r = E 3χ(3) + ei(k1 +2k2 )·r , xxxx e



i3k2 ·r (3) (3) i(2k1 +k2 )·r Py(3) = E 3 χ(3) + (χ(3) yyyy e yyxx + χyxyx + χyxxy )e







i3k2 ·r = E 3χ(3) + ei(2k1 +k2 )·r . yyyy e (3)

(3)

In the case here, Px = Py . Again, four TH components can be generated under phase-matching conditions. Collinear components follow x and y polarizations of the corresponding pump beams. The intensity of these two TH beams remains the same as in s-polarization. Non-collinear TH components have orthogonal polarization planes and an intensity reduced by 9 times as compared with s-polarization. This ratio follows from the ratio of squared coefficients at corresponding terms. The component propagating along (k1 +2k2 ) is polarized in x−z plane, and another component propagating along (2k1 +k2) is polarized in y direction. Contrary to the case of s-polarization, the intensity of noncollinear TH components in cross-polarized beams is, in general, different. It arises from

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(3)

229

(3)

the inequality of radiative parts of Px and Py , where Py is perpendicular to the prop(3) agation direction of TH generated along (2k1 + k2 ), but Px is not perpendicular to the corresponding propagation direction along (k1 +2k2). Therefore, the TH component polarized in x − z plane has a lower intensity. For relatively small crossing angles the difference of non-collinear TH intensities, however, is rather small. The TH generation process is not allowed in isotropic media under single-beam excitation by circularly polarized light. Therefore, in crossed circularly polarized sub-beams there is no collinear TH emission. Non-collinear TH arises from a combination of two fundamental fields from one beam and one from the other. In this case, the conservation of angular momentum allows the generation of two non-collinear TH beams with opposite circularity. It gives a possibility for the direct generation of circularly polarized VUV light in a frequency tripling process. Experiments have shown that such an excitation mode provides a rather high TH output which is comparable with the maximum TH output for linear s-polarization (see Fig. 17). Actually, the TH emission under circularly polarized pumping may have some ellipticity where the major axis of TH polarization ellipse is directed perpendicular to the propagation plane. It results from the inequality of two orthogonal nonlinear polarization components with respect to the corresponding TH propagation directions exactly as it takes place for cross-polarized beams. For relatively small crossing angles this ellipticity is small and the polarization of TH emission is close to circular. Experimental results and analysis of the frequency tripling process in crossed laser beam indicate a possibility of the polarization control over the process of TH generation. For an ordinary single-beam excitation the TH output is maximum for linearly polarized light but vanishes under circular pumping. By contrast, for two crossed beams the TH generation process is much more flexible with respect to the polarizations of pump beams. The non-collinear TH components, which are well separated from the fundamental beams, can be made polarized linearly with parallel or crossed polarization planes, or they can be made circularly polarized with opposite circularity. Such a polarization control may be a useful feature of non-collinear excitation geometry for direct generation of coherent VUV light with a desired polarization.

6.

Spatially Incoherent Conical Beams

All quasi-monochromatic conical beams considered in previous sections result from a superposition of plane waves whose wave vectors lie on a conical surface. For a J0 Bessel beam all these plane waves have equal amplitudes and phases. Similarly, the higher-order Jm Bessel beams have equal amplitudes of individual plane waves but their phases change in azimuthal direction. Other conical beams like segmented conical or Mathieu beams are generated with the aid of corresponding azimuthal modulation of the amplitudes of partial plane waves. In all these examples the conical field is characterized by a complete spatial coherence when all partial waves maintain their mutual correlation under propagation. Another class of propagation-invariant light fields includes partially-coherent conical beams. For the so-called Bessel-correlated beams [24] the correlation between individual plane waves vanishes in azimuthal direction. Conical superposition of such mutually uncorrelated plane waves produces a propagation-invariant intensity profile, as well. In a

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stationary case of partial azimuthal incoherence the field profile is decomposed into multiple irregular speckles which preserve their location and shape under propagation (see Fig. 4 in Ref. [24]). In other words, such partially coherent beam is comprised of multiple parallel light filaments distributed irregularly within the beam profile and extended along the beam axis. In general, both the azimuthal and radial incoherence can be introduced into conical superposition of multiple plane waves and a variety of specific light field configurations can be obtained [25, 26]. In most studies with propagation-invariant light fields in nonlinear optics the simplest coherent conical beams like J0 Bessel or Bessel-Gauss beams were used, but much less is known about nonlinear optical processes driven by incoherent conical beams. In the process of nonlinear optical conversion the incoherent beams are usually considered as less attractive since the presence of wave-front irregularities reduces the length of coherent nonlinear interaction and leads to significant destructive interference of secondary emission from multiple uncorrelated regions. However, it has been demonstrated recently that conical excitation geometry allows one to preserve a relatively high conversion efficiency in the process of optical parametric oscillation even for pumping beams with significantly degraded spatial coherence [70]. Such a tolerance may be very interesting and useful feature of conical beams in the excitation of nonlinear optical processes. A variety of excitation schemes can be arranged to drive nonlinear processes with conical laser beams. The simplest is a single-beam configuration with the use of either coherent or incoherent conical beam. More complex though more flexible are two-beam combinations of coherent-coherent, coherent-incoherent, or incoherent-incoherent beam pairs. In this section, considered are the experimental observations and analysis of sum-frequency generation in conical beams under coherent single-beam excitation and mixed two-color excitation by coherent and incoherent beams. The two-color combination of coherent and incoherent excitation was used in a study of sum-frequency generation in ordinary geometry of focused beams [72] and it allows one to make a reasonable comparison of these processes in different excitation conditions. The general experimental procedure was similar to that used in experiments with coherent conical beams. The resonance-enhanced sum-frequency generation was driven by three-photon excitation at the negatively-dispersive side of the 6s resonance of xenon. The excitation energy of this resonance is 9.57 eV and the generated sum-frequency field had the wavelength near 129 nm. Sum-frequency photons were registered through their nearresonance absorption and subsequent ionization of excited xenon atoms by laser light. In single-color excitation mode the generated sum-frequency field was the third harmonic 3ω of the fundamental laser light (see Fig. 3). It was the reference excitation mode, where the laser source had a high degree of spatial coherence. The second scheme employed a two-color 2ω1 + ω2 excitation of the same transition, but one of the laser sources ( ω2 ) had a low degree of spatial coherence. The ionization profiles measured in such a combined coherent-incoherent two-color excitation mode were compared with the profiles measured in the same conditions under coherent one-color excitation. Additionally, the results of experiments with conical beams were compared with previous results obtained in ordinary excitation geometry of focussed Gaussian beams [72]. Figure 19 demonstrates the arrangement of main optical elements for one- and twocolor excitation by conical beams. In both cases, a tunable dye laser was used as a source of

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splitter

ω1 ω2

axicon

Figure 19. Schematic of two-color excitation geometry in conical beams.

coherent radiation ω1 . In one-color experiments the dye laser operated in the 388-389 nm spectral range (QUI dye in dioxane). In two-color experiments the laser was tuned in the 446-448 nm spectral range (Coumarin 120 dye in ethanol) and the pumping XeCl excimer laser (λ =308 nm) served as a source of incoherent UV radiation ω2 synchronized with pulses of the dye laser. For this purpose, a small portion of the XeCl laser output was directed to a quartz splitter and reflected toward the ionization cell together with the dye laser beam. Laser beams were focused to the ionization cell by an axicon. Two quartz axicons with apex angles 120◦ and 140◦ were used. Depending on the laser wavelength, these axicons produced conical beams with inclination angles α ranging within 17.2◦ − 17.9◦ and 10.1◦ − 10.5◦, respectively. In most cases, small variations of inclination angles could be neglected and further the corresponding conical beams will be considered as beams with fixed inclination angles α = 17◦ and α = 10◦. Divergence of the dye laser beam (about 0.5 mrad full angle 2 θ) was close to the diffraction limit. It means a high degree of the beam spatial coherence. Divergence of the excimer laser beam (2-2.5 mrad full angle 2θ) exceeded the diffraction limit by about two orders of magnitude. Such beam contained multiple wave-front irregularities with the characteristic size of lateral correlations d0 ∼ λ/θ  0.3 mm. The temporal coherence length of the excimer laser beam was measured with the aid of Michelson interferometer. This length was determined as the difference of optical paths when visibility of interference fringes was reduced to a half of maximum. The value obtained was 1.0±0.2 mm which corresponds to a bandwidth of about 4 cm−1 FWHM for a Gaussian spectral profile. The presence of two emission lines in the XeCl laser spectrum modulated the fringe visibility function with a period of 0.36-0.38 mm, which corresponds to a spectral distance of 26-28 cm−1 between emission lines. The temporal coherence length of the dye laser beam was 10-12 mm. Important feature of conical beams is their extended excitation region. In coherent J0 Bessel beams, a nonlinear process is confined to the central lobe of the Bessel profile where light intensity is maximum. This high-intensity focal line is usually much longer than the Rayleigh length of the reference Gaussian beam. Spatial incoherence destroys regular Bessel or Gaussian patterns and the excitation volume in both cases is decomposed into multiple regions where some degree of coherence persists, but these coherent domains are mutually uncorrelated. The general beam geometry, however, is preserved and the focal region of conical beams remains extended along the beam axis. It means that the domain structure of incoherent conical beam differs significantly from that in the reference Gaussian

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beam. For any coherent nonlinear process the structure and characteristic dimensions of coherent domains in the focal region are of crucial importance. Therefore, a special study was arranged aiming to a detailed characterization of the used coherent and incoherent beams.

without lens

with lens

laser beam

microscope diffuser lens

axicon

CCD camera

Figure 20. Schematic of experimental setup for characterization of partially coherent laser beams. CCD pictures (each picture size is 2x2 mm2 ) show the speckled patterns of light beam on focusing element with and without the collimating lens (from Ref. [71]). The characterization of partially coherent nondiffracting light beams was the subject of several theoretical studies (see, for example, [24, 25, 26, 49, 73, 74] and references therein). In the case here, a simple empirical approach was used where necessary parameters of the focal coherent domains were obtained from model experiments with laser beams having variable spatial coherence. To have a basis for comparison, the same experimental procedure was applied also to the Gaussian beams. Fig. 20 illustrates a schematic of experimental arrangement for characterization of different laser beams. The focal region of beams was imaged by a microscope and a CCD camera. To match tightly focused conical beams with microscope optics, the inclination angle of beams was reduced with the aid of axicon immersion. The axicon with the apex angle 120◦ was placed into an optical cuvette filled by distilled water. Such immersion reduced the inclination angle of conical beams to about 4◦ . Fig. 21 demonstrates the intensity distributions in the focal region of the dye laser beam λ=447 nm and the UV beam λ=308 nm from the XeCl laser. The last profile is shown in a scale of intensity reduced to 8 gradations. Spatially coherent beam of the dye laser produced the known pattern of a J0 Bessel beam, where the diameter of the central maximum was 4.81λ/(2π sin α)  5 μm. Incoherent beam from the XeCl laser produced a smooth profile peaked on the beam axis. The width of profile was 30 - 40 μm FWHM. This profile had very slow evolution along the propagation direction and, to some extent, such incoherent conical beam can be considered as a particular case of propagation-invariant light fields. In model experiments with spatially incoherent beams examined was the pattern of

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Figure 21. CCD pictures of the focal regions of conical beams: top, dye laser beam; bottom, XeCl laser beam. Each picture size is 70x70 μm2 (from Ref. [71]).

stationary speckles formed with the aid of a He-Ne laser and a holographic diffuser (see Fig. 20). A small part of scattered speckled beam was passed through a diaphragm and was focused by the axicon. The mean size d0 of speckles on the axicon was 0.3-0.4 mm. Additionally, a lens of focal length f =27 cm could be placed behind the diffuser as shown in Fig. 20. This lens collimated the scattered beam and reduced d0 by 3-4 times. The same elements were used in experiments with Gaussian beams, where a lens of focal length f =35 mm was placed instead of the cuvette with axicon. This lens gave the focussing angle α = D/2f  4◦ , where D is the beam diameter. In all model experiments the condition α  θ was satisfied, where θ is the divergence angle of speckled beams on focussing elements. Figure 22 shows the focal regions of the beams in model experiments. Without the diffuser, the axicon focusing of the coherent He-Ne beam produced again the known pattern of a J0 Bessel beam with the central peak diameter of about 7 μm (Fig. 22a). When the same beam was focused by the lens, a Gaussian focal spot of about 10 μm was formed (Fig. 22b). With the diffuser inserted, these Bessel and Gaussian profiles were decomposed into irregular patterns of multiple speckles (Figs. 22c,d). Evolution of these patterns under propagation was examined by moving the microscope along the beam axis.

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(b)

(c)

(d)

d0

(a)

Δz Figure 22. CCD pictures of the focal regions: (a) Bessel beam; (b) Gaussian beam; (c) spatially incoherent conical beam; (d) spatially incoherent Gaussian beam. Each picture size is 70x70 μm2 . Bottom, schematic of formation of coherent focal domains in conical and Gaussian beams (from Ref. [71]). The general structure of volume speckles in focused incoherent Gaussian beams is well known. In the case here, the measured transverse size d and the length ΔzG of focal speckles were d=5-10 μm and ΔzG =0.2-0.3 mm in good agreement with simple estimates d ∼ λ/α and ΔzG ∼ λ/α2 . Both d and ΔzG were insensitive to the transverse size d0 of input speckles since in all cases the condition α  θ was valid and the characteristic dimensions of focal speckles were determined by the focusing angle α. Fig. 22 shows that the focal speckles in incoherent conical beam have nearly the same transverse size d as in the reference Gaussian beam. It means that the same estimate d ∼ λ/α remains valid for conical beams. The length of speckles Δz, however, differed significantly from the Gaussian case. For large input speckles the length Δz=5-8 mm was measured. This length was reduced to 1.2-1.8 mm when input speckles were reduced by the collimating lens f =27 cm. These results imply that Δz in conical beams is proportional to the transverse size of input speckles or, more generally, Δz is proportional to the characteristic size of lateral wavefront correlations in the input beam. The focal speckles in conical beams are extended significantly along the beam axis so that Δz  d and Δz  d0. It is fairly easy to see that such an extended Δz is formed as a projection of input correlation d0 toward the beam

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axis, when Δz  d0/ sin α  d0/α (see Fig. 22). It means that conical excitation geometry is able to produce very long correlated regions even for relatively small input correlations. Since d0 ∼ λ/θ and α  θ, the ratio Δz/ΔzG ∼ α/θ  1. It indicates significant advantage of conical beams in the formation of very long coherent domains in the focal region as a key feature of these beams in the excitation of coherent nonlinear processes. With analytical guidance of model experiments, all necessary parameters of the dye and the XeCl laser conical beams are readily obtained. In the gas cell the coherent dye laser beam produces a J0 Bessel beam with the central peak diameter of about 2 μm and 1 μm for the beams α = 10◦ and α = 17◦, respectively. The length L of this high-intensity region is proportional to 1/ tan α and it was 10-12 mm for the beam α = 10◦ and 5-6 mm for the beam α = 17◦ . The incoherent UV beam from XeCl laser forms a conical beam, where the transient coherent domains have characteristic dimensions d  2 μm, Δz  2mm and d  1 μm, Δz  1mm for α = 10◦ and α = 17◦, respectively. Some general features of near-resonance ionization processes under mixed two-color excitation by coherent and incoherent beams are known from [72]. Briefly, a tightly focused dye laser beam produced several peaks in the ionization spectra due to multiphoton excitation of some intermediate resonances of xenon. With the addition of incoherent UV beam from XeCl laser, an intense ionization band appeared in spectra. That band was associated with the phase-matched sum-frequency field 2ω1 + ω2 generated near the 6s resonance of xenon. With conical beams, the light intensity considerations again become important since in conical excitation geometry the laser light is concentrated along an extended focal line instead of a much more compact excitation region of focused Gaussian beams. In the case here, the light intensity in conical beams was by at least an order of magnitude lower than in the focal region of beams in Ref. [72]. Therefore, no multiphoton atomic resonances of xenon were detected with conical beams even with maximum pulse energy available from the dye laser. However, with addition of the XeCl laser beam an intense ionization band appeared in spectra similar to the observations of [72] and quite a moderate laser pulse energy produced the same ionization signal in conical excitation geometry as with focused Gaussian beams. Figure 23 demonstrates the evolution of the ionization band with xenon pressure in conical beams α = 10◦. The ionization profiles have two distinct peaks. Each of these peaks results from the same 2ω1 + ω2 process, where the field ω2 contained two spectral components - the two emission lines from the XeCl laser spectrum (see inset in Fig. 23). Note a distinct dip on the blue ionization peak. This spectral feature corresponds to the dip on the red peak of the XeCl laser spectrum, where it appears due to absorption by OH radicals in the XeCl laser active medium [75]. With conical beams, the ionization dip could be traced up to a pressure of about 500 mbar, but no such feature was observed in experiments of [72]. It is again a consequence of relatively low light intensity in conical beams. The high light intensity in tightly-focused Gaussian beams induced remarkable ac Stark broadening of the ionization peaks. In conical beams the light intensity is much lower. It diminished significantly the ac Stark broadening and has allowed the observation of fine spectral features. With an increased xenon pressure, the ionization peaks were broadened and shifted toward shorter wavelengths. The shift was linear with pressure and, in terms of the dye laser wavelength, the slope was 0.88 nm/bar for the beam α = 10◦ and 0.29 nm/bar for the

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α

Figure 23. Wavelength scans of ionization signal in two-color conical beams α = 10◦. The inset shows XeCl laser emission spectrum (from Ref. [71]). beam α = 17◦. Figure 24 demonstrates another example of ionization profiles for conical beams α = ◦ 17 in two- and one-color excitation modes. The profiles are shown in the scale of sumfrequency detuning from the 6s resonance position. For the two-color case this detuning was determined with respect to the red peak of the ionization band. The one-color coherent excitation mode produced a J0 Bessel beam, where the general features of near-resonance ionization profile were considered above. With an increased xenon pressure the TH peak is broadened and shifted off the resonance position. The same behavior is demonstrated in

Sum-Frequency Generation and Multiphoton Ionization in Rare Gases...

3ω1

3ω1

2ω1+ω2

2ω1+ω2

237

Figure 24. Sum-frequency 3ω1 and 2ω1 + ω2 excitation profiles for one- and two-color conical beams α = 17◦. Left, xenon pressure 100 mbar; right, xenon pressure 600 mbar (from Ref. [71]).

the two-color excitation profiles, but with smaller pressure-induced shift. The coherent onecolor excitation gave the shift of 30 cm−1 /bar and 95 cm−1 /bar for the beams α = 17◦ and α = 10◦, respectively. For the two-color case, the corresponding values were 29 cm−1 /bar and 88 cm−1 /bar. Experiments with conical beams have shown the ability of conical excitation geometry to support effectively sum-frequency generation process in conditions of relatively low light intensity and a poor spatial coherence of one of the pumping fields. Low conversion efficiency is a usual feature of incoherent beams in nonlinear frequency conversion processes. Moreover, the use of spatially incoherent laser beams may serve as a tool for suppression the coherent transfer of energy from the pumping field(s) toward undesirable nonlinear processes [72]. However, there are distinct indications that sum-frequency generation process is much less degraded in conical excitation geometry and, in agreement with previous observations [70], conical beams demonstrate their better tolerance in the wave-front aberrations. Wave-front irregularities in the input beams decompose the excitation volume into multiple mutually uncorrelated domains. The generated sum-frequency field gains intensity along the beam axis as long as an appreciable correlation of the pumping field components and phase matching between the fundamental and the generated sum-frequency fields are maintained. Decay of correlation destroys this coherent process and evolution of the sum-frequency field in amplitude is limited by the length of coherent focal domains. In tightly-focussed incoherent Gaussian beams this gain length ΔzG can be very short (a few microns) and it reduces significantly the overall conversion efficiency. Axicon focussing gives a much lower light intensity, but the correlation length Δz becomes extended significantly so that Δz  ΔzG . In two-color conical beams the ionization signal originates from

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the region, where the central lobe of the coherent Bessel beam ω1 overlaps the transient coherent domains from the incoherent beam ω2 . Nonlinear process is driven within these embedded domains and the extended gain length Δz facilitates the build-up of coherent sum-frequency field. A general analysis of sum-frequency generation in conical beams requires consideration of all possible combinations of three fundamental photons with wave vectors k1 , k2, and k3 at azimuthal angles φ1, φ2 , and φ3 on the light cones. For two-color conical excitation geometry, the 2ω1 + ω2 process is driven by two superimposed light cones. Elementary configurations of pump fields in this case involve two photons from one conical beam and one photon from another beam. Here again all possible combinations of fundamental photons contribute the excitation profile. Cylindrical symmetry of conical beams allows one to fix the azimuth angle φ3 = 0 for the wave ω2 and consider further only the variations of φ1 and φ2 for two waves at ω1 . When three crossed waves drive a nonlinear polarization at ω = 2ω1 + ω2 within the negatively dispersive side of an atomic resonance at ω0 , the phase mismatch between the incident fields and generated sum-frequency field vanishes at some detuning Δω = ω − ω0 from the resonance position. For a two-level system and far enough from the resonance position this detuning for any configuration of three crossed waves can be found from simple wave-vector diagrams. It gives Δω =

2Δ0 (2 + r)2 , sin2 α (1 + 2r − cos(φ1 − φ2 ) − r cos φ1 − r cos φ2 )

(24)

where r = ω2 /ω1 , Δ0 = πN F01e2 /2mω0 , N is the gas density, and F01 is the oscillator strength. In the present experiments the ratio r was 1.45. The general expression (24) is valid also for some other excitation geometries. For example, for a one-color conical beam r =1 and (24) gives the frequency shift as Δω =

18Δ0 , sin α (3 − cos(φ1 − φ2) − cos φ1 − cos φ2 ) 2

(25)

which is the known expression for TH generation in Bessel beams [18] (see Eq.(8)). Another example is the generation of sum-frequency field by two crossed beams [59, 60]. In this case φ1 = φ2 = π and (24) gives the detuning Δω as Δω =

Δ0 (2ω1 + ω2 )2 Δ0 (λ1 + 2λ2) = , 2 λmix(1 − cos 2α) 2ω1 ω2 sin α

(26)

where λmix = 2πc/(2ω1 + ω2 ), and 2α is the crossing angle between two waves. Eq. (26) is again the known expression for near-resonance two-color excitation profiles in crossed laser beams [58, 59, 60]. Owing to the noncollinear phase matching and circular symmetry of pumping fields, the sum-frequency emission forms an outgoing light cone with inclination angle β < α. For a given spatial configuration of fundamental waves from the two light cones, the angle β is found from wave-vector diagram as

tan β =

2 + r2 + 2 cos(φ1 − φ2) + 2r cos φ1 + 2r cos φ2 tan α . 2+r

(27)

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For r ≤ 2 the angle β changes from 0 to α. For r = 1 (single-color Bessel beam) the minimum value β = 0 corresponds to symmetrical configuration of three fundamental waves separated by the azimuth angle 120◦ from each other. This configuration gives also the minimum value of detuning Δωmin = 4Δ0/ sin2 α. For r = 2 the same values of Δωmin and β = 0 correspond to planar configurations of pumping waves, when φ1 = φ2 = π. Further increase of r increases the gap between the resonance position and the closest phase-matching point. For r > 2 the angle β > 0 for any configuration of the fundamental photons. For an optically thick medium, the ionization profile for a single-color Bessel beam can be simulated as a superposition of equally-weighted Lorentzian profiles (see Section III). To some extent, the same approach can be applied also for two-color excitation by conical beams with large inclination angles α, when the ionization peaks are located close to the atomic resonance and the absorption of sum-frequency field is very strong. The general features of two-color ionization profiles were evaluated in another way. From (24) it follows, that any elementary configuration of pump fields with given azimuth angles φ1 and φ2 can be characterized by a phase-matching point located at some detuning Δω from the atomic resonance. Since angles φ1 and φ2 have all possible choices from 0 to 2 π, these phase-matching points are spread within a broad spectral region starting from minimum detuning Δωmin . This spectral distribution can be characterized by spectral density function ρ(ω) which can be written in the following form: 2π2π

ρ(ω) =

δ[ω − Δω(φ1, φ2) − ω0 ] dφ1dφ2 ,

(28)

0 0

where δ is the Dirac delta function, and Δω(φ1, φ2) is given by (24). Fig. 25 shows the calculated spectral density ρ(ω) for conical beams α = 10◦ . In numerical work, the width Γ (FWHM) of pressure-broadened atomic line and Δ0 were taken from Ref. [76] as Γ/Δ0 = 4.66. Calculation results demonstrate the general structure of near-resonance excitation profiles in different excitation conditions. The case r = 1 corresponds to the process of TH generation in Bessel beams. Note distinct similarity of ρ(ω) at r = 1 and the transverse phase-matching integral [46, 47] derived as the overlap of the TH Bessel beam and the cube of the fundamental Bessel beam. The maximum of ρ(ω) occurs when φ1 = π and φ2 is arbitrary, or φ2 = π and φ1 is arbitrary, or |φ1 − φ2 | = π. According to (27) this maximum corresponds to tan β/ tan α = 1/3 which matches exactly Eq. (5) for the maximum TH output in Bessel beams. The maximum of ρ(ω) is infinite similar to the maximum of the transverse phase-matching integral (4). In the case here, the divergence arises from the use of the δ function instead of an elementary excitation profile with a finite spectral width. The calculation results have shown that the peak of the excitation profile is split into two maxima if r > 1. The blue maximum corresponds to configurations of the pumping field when |φ1 − φ2 | = π. A particular case here are planar configurations φ1 = 0 and φ2 = π, or φ1 = π and φ2 = 0 (recall, that the field ω2 is fixed at the azimuth angle φ3 = 0). The blue maximum is shifted toward shorter wavelengths with an increased r. The red maximum has an opposite shift toward the red edge of the excitation profile. This maximum corresponds to configurations of the pumping fields when

V. Peet

ρ(ω)

240

(ω−ω )/Γ Figure 25. Spectral density ρ(ω) of phase-matching points as a function of detuning from the resonance position for one- and two-color conical beams α = 10◦ (from Ref. [71]).

cos(φ1 − φ2 ) + r cos φ1 + r cos φ2 = 1 − 2r .

(29)

The ratio of detunings Δω for the red and the blue maximum is (r + 1)/2r and is thus independent on the inclination angle α and the gas density. For given r and α, the spectral distance between two peaks is proportional to Δ0/sin2 α and is increased with gas pressure since Δ0 is proportional to the gas number density N . The calculation results have shown that two-color excitation by conical beams produces the excitation profiles with two maxima in contrast to a single maximum for single-color Bessel beams. One of these two peaks is located closer to the resonance and has thus a smaller pressure-induced shift than the peak of the corresponding one-color Bessel beam. It agrees with experimental observations, where the ionization peak under two-color excitation had smaller shift than the peak of the reference Bessel beam. Besides, the calculation results predict the existence of a second peak located at a shorter wavelength. Since the pumping field ω2 had two components, the ionization profiles should actually contain two

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additional spectral components located at the blue sides of the two main peaks. The spectral distance between each pair of peaks is relatively small and, from above consideration, the additional peaks could be detected at an elevated gas pressure with beams of small α. Fig. 26 shows the excitation profiles measured under two-color excitation by beams α = 10◦ in dense xenon. Despite of a relatively poor spectral resolution, distinct blue satellites can be recognized in spectra for each of the two main peaks. The location of these satellites is in very good agreement with calculation results.

Figure 26. Wavelength scans of ionization signal in two-color conical beams α = 10◦ . Upper trace - xenon pressure 500 mbar; lower trace - xenon pressure 800 mbar. Paired bars indicate the calculated positions of the two maxima of sum-frequency excitation profile. Vertical dashed lines indicate the location of origins for two main peaks (from Ref. [71]).

Small magnitude of blue satellites can be understood from the following consideration. In experiments, the generated sum-frequency field was registered through ionization of the target gas, but in conical beams the efficiency of such method is reduced gradually if the excitation wavelength is tuned off the resonance position. Sum-frequency photons propagate, in general, at some angle with respect to the axis of pumping conical beams. It is easy to see from (24) and (27) that the inclination angle β for the phase-matched sum-frequency field is increased together with increased detuning Δω:

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tan β = tan α

1−

4Δ0 . Δω sin2 α

(30)

Near the red edge of the excitation profiles the angle β is small and the generated sumfrequency photons propagate close to the axis of conical beams. These photons are absorbed entirely within the central lobe of the Bessel beam, where the excited xenon atoms are effectively ionized by intense laser light. With an increased Δω the absorption of sumfrequency photons gets weaker and the cone angle β is increased. As a result, a part of the generated sum-frequency emission is absorbed outside the high-intensity core and the corresponding ionization signal is weaker.

7.

Effects of Internally-Generated Sum-Frequency Field on Atomic Resonance Excitation

In experiments on generation of resonance-enhanced sum-frequency fields, the target gas is excited by intense laser light at a frequency which is near-resonant in three or five photons with a J = 1 atomic state. In this condition, the multiphoton transition amplitude should be very large and an intense excitation of atomic states followed by ionization to continuum could be expected. However, in all REMPI spectra shown above there was no ionization peaks at the location of the driven J = 1 atomic resonance. Such an anomaly was first noted in [77] and it turned out to be a very general phenomenon studied in the next years in numerous experimental and theoretical works. For a detailed review of these phenomenon see [37] and references therein. Briefly, when three-photon excitation of a dipole-allowed atomic transition is attempted at elevated concentrations, a coherent sum-frequency field is generated at the resonant transition frequency. The overall transition amplitude becomes determined by two coherent processes of the three-photon excitation by fundamental field and the one-photon excitation by sum-frequency field. In conditions of strong resonant absorption of sum-frequency field these excitation processes become equal in magnitude but opposite in phase and the two excitation pathways interfere destructively. As a result, no excitation of the atomic state occurs above some certain threshold product of number density and oscillator strength. In REMPI experiments this destructive interference leads to the cancellation of odd-photon atomic resonances, which are diminished and disappear into background at an increased gas pressure. The wave-mixing interference that suppresses resonance excitation in unidirectional beams maintains a strong influence even for counterpropagating beam geometry, when resonance ionization is restored [27, 28, 30, 31, 32]. This point was not recognized in those early studies where the wave-mixing interference in counterpropagating beams was assumed to be negligible. However, the restored atomic resonance in counterpropagating beams was later found to have large pressure-induced shift [58, 63], and it was realized that the wave-mixing field is responsible for this shift [59, 61]. Thus the wave-mixing interference is not negligible in any excitation geometry. In all cases, the resonant wave-mixing field evolves to destructively interfere with direct excitation by the laser photon except for a region on the blue side of the resonance position, where the interference becomes constructive. The shifted position where constructive interference occurs is closely associated

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with the point where the real part of the phase mismatch between the incident and generated fields vanishes [59, 61]. The underlying physics of the cancellation effect is rather well understood and a very complete theoretical treatment of the problem has been developed. In a sharp contrast, much less attention was paid to another subset of relevant experimental observations. These observations have been well documented in several REMPI studies with rare gases but have remained without any satisfactory explanation so far. Namely, the cancellation of atomic resonances does not mean necessarily an entire disappearance of any resonant ionization signal at the corresponding spectral region. In many cases, distinct residual ionization feature was registered close to the canceled atomic peak. Such a weak ionization feature near the canceled three-photon 6s atomic resonance of xenon was first reported in [77], where it was assigned to the four-photon excitation of the intermediate 8s[3/2]◦J=2 resonance of xenon. Such a transition, however, represents a parity change and is not allowed in four photons. Similar residual peaks were registered in several further studies near the canceled three-photon 6s and the 6s resonances of xenon [28, 29, 78], and near the 5s resonance of krypton [79]. However, the main attention in those studies was focused on the dramatic cancellation effect, but other features like weak residual ionization peaks remained unappreciated. The interest to near-resonance ionization features arose after REMPI experiments in rare gases with Bessel beams [10, 43]. As it was shown above, in Bessel beams an intense TH ionization band appears close to the canceled atomic resonance. The red wing of the TH excitation profile is extended toward the resonance position and the ionization signal in near-resonance region is increased significantly as compared with Gaussian beams. The experiments of [43] have indicated the presence of a distinct ionization anomaly near the cancelled atomic resonance, where a specific interference structure with a peak and a dip was formed. These observations stimulated a further detailed study of near-resonance ionization in different excitation conditions. In [13], utilizing both Bessel and Gaussian laser beams the ionization profiles near the 6s resonance of xenon were studied over a broad range of gas pressure. In unidirectional beams, either Bessel or Gaussian, an ionization structure was easily registered close to the canceled resonance. This structure could have the shape of a weak resonance-like peak. In many cases this peak was accompanied from the blue side by an ionization dip where significant suppression of ionization occurred. In some experiments with unidirectional Bessel beams even more complex structures were observed, when the peak was surrounded by two ionization dips [13]. Fig. 27 demonstrates an example of near-resonance ionization anomaly on the red wing of the TH band registered near the 6s resonance of xenon under excitation by Bessel beam. In unidirectional Gaussian beams the excitation of atomic resonance is canceled and the residual ionization peak is relatively weak. Experiments with Bessel beams have shown that similar anomaly persists also in conditions of a much stronger near-resonance ionization when the resonant atomic state is populated through absorption of TH photons. Similarly, the experiments of [13] with counterpropagating beams have shown significant deformations of the restored 6s ionization profile at an elevated xenon pressure and it was revealed that these deformations follow exactly the pattern of ionization anomaly in unidirectional beams. For example, significant suppression of the blue wing of pressure-broadened atomic profile was observed with counterpropagating beams in the region where the ionization dip

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Figure 27. Ionization anomaly on the TH ionization band registered with Bessel beam near the canceled 6s resonance of xenon. was registered with unidirectional beams. Similar observations can be found in the results of early REMPI study [29]. Figure 28 demonstrates REMPI spectra measured with two crossed focussed beam near the canceled 6s resonance. The crossing angle between two focussed beams was relatively large and the TH profile was located close to the 6s resonance. Again, distinct nearresonance ionization anomaly was registered over the whole range of gas pressure. At a low pressure, the TH band is located close to the resonance position and the TH profile demonstrates an unusual asymmetric shape with a suppressed blue wing. If the pressure is increased the TH band moves off the resonance position and distinct ionization dip is registered on the TH red wing. This dip, where significant suppression of ionization occurs, can be registered even at xenon pressure of 10 bar, when all other spectral features almost entirely disappear in background ionization [13]. The principal features of near-resonance ionization anomalies can be summarized as follows. First, the ionization yield in anomaly region experiences significant modulations. These modulations produce a variable ionization profiles with distinct regions of enhanced

Sum-Frequency Generation and Multiphoton Ionization in Rare Gases...

6s

IONIZATION SIGNAL (arb. units )

α

245

Figure 28. Ionization profiles measured with crossed focused beams near the 6s resonance of xenon. Arrows mark the ionization dip.

or suppressed ionization. Second, the pattern of ionization anomaly is identical in unidirectional and counterpropagating excitation geometries. It means that the ionization anomaly is quite a strong effect, when the ionization peaks and dips may have significant magnitudes. In some experiments, the ionization peak in unidirectional beams had intensity up to several percents of that for the most prominent spectral components like the atomic peak in countrepropagating beams or the TH ionization band in Bessel beams. Similarly, a strong suppression of ionization could be obtained within the ionization dip.

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The residual peak and the ionization dip demonstrated very different evolution with pressure [13]. The dip shifted linearly toward shorter wavelength when the gas pressure was increased and the slope was close to that for the cooperative line in counterpropagating beams. In a sharp contrast, the shift of the peak was negligibly small. It may point to a very different nature of these two ionization features. The cooperative line registered with counterpropagating beams in an essential manner is produced due to the wave-mixing field and the shift of this line is much larger than the actual pressure-induced shift of the probed atomic resonance. Even though the peak registered with counterpropagating beams looks like the expected atomic line, it is not the ”real” atomic profile but a cooperative line shifted from the resonance position. The ionization dip follows the evolution of the cooperative line with pressure, but it is not the case for the residual ionization peak. Hence, the appearance of this peak may point to a process, where the interference of two coherent excitation pathways is perturbed and the ”real” atomic resonance manifests itself as a weak ionization peak. The pressure dependence of the TH ionization profiles in all experiments was extrapolated at zero pressure to the position of atomic resonance, as expected. By contrast, for both the ionization dip and the peak the plots of pressure-induced shifts were extrapolated at zero pressure to some points on the blue wing of the atomic line, but not to the line center [13]. Further experiments have shown that the location of this zero-pressure point is not a constant but is dependent on excitation conditions. In the same manner, the excitation conditions influenced significantly the magnitude and shape of near-resonance ionization features and very different ionization profiles were registered in different sets of experiments. In [13] it was concluded that the near-resonance ionization anomaly is not readily explained within the framework of existing theory of resonant three-photon excitation and associated wave mixing. No adequate explanation has been suggested to date and the origin of these mysterious anomalies has remained unknown for more than two decades. Detailed experiment with counterpropagating beams have shown that the nearresonance dip and peak are located entirely within the range of the ac Stark shifts for the probed atomic resonance. This fact was not recognized in previous study [13], where intensity of retroreflected beam was too weak to reveal the whole ac Stark atomic profile. The ac Stark effect results in a specific asymmetric broadening of the probed atomic resonance toward shorter wavelengths. This profile builds up from all space-time variations of light intensity in the focal region [80, 81, 82]. The blue wing of the profile is determined by space-time domains with maximum light intensity and it is just the region where the dip and the peak are registered. Fig. 29 demonstrates ionization profiles measured near the 6s resonance in unidirectional and counterpropagating Gaussian beams. In these experiments the laser pulse energy was fixed but the light intensity in focal region was changed through variations of the input beam diameter. For upper traces in Fig. 29 the beam diameter was relatively small and, under excitation by counterpropagating beams, an asymmetric ac Stark atomic profile was revealed (curve 1 in Fig. 29). Under excitation by unidirectional beam, a well-pronounced dip-peak structure is registered on the ionization background (curve 2 in Fig. 29). For lower two traces in Fig. 29 the input beam diameter was increased. It resulted in a more tight focus and an increased light intensity (note increased ionization background). The ac Stark profile becomes extended significantly toward shorter wavelengths (curve 3 in Fig. 29). In the same manner, the dip-peak structure becomes extended

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Figure 29. Ionization profiles near the 6s resonance of xenon: 1,3 - excitation by counterpropagating beams; 2,4 - excitation by unidirectional beams.

along the atomic profile and is much less pronounced (curve 4 in Fig. 29). These experiments explain qualitatively the variability of the ionization pattern observed in different excitation conditions. This variability, when a well-pronounced anomaly may almost entirely disappear after apparently minor changes of excitation conditions, complicated very much any regular study of this effect. One can suppose that the ionization anomaly results from the onset of some transient higher-order nonlinear processes which modify the character of near-resonance excitation and ionization in high-intensity domains. It explains both the blue shift of registered ionization structures and their variability in different excitation conditions. A more detailed understanding of the problem, however, is absent. Other striking and unexpected effects were observed in REMPI experiments with Bessel beams. In unidirectional Bessel beams the ionization signal near the canceled three-photon 6s resonance of xenon demonstrated distinct sensitivity to the polarization purity of the incident laser beam. The emission of pulsed dye laser used in experiments had a degree of natural polarization P=0.1-0.3 depending on laser adjustment and pulse energy. Usually, the output emission of the laser was purified by a polarizing prism and most experiments were carried out with linearly polarized laser light. In this case, a known pattern of the cancelation effect was observed as expected. In some experiments, however, the polarizing prism

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was removed and the beam with its natural polarization was used. Such excitation always resulted in a distinct ionization enhancement near the resonance. In control experiments, when the beam was focused by an ordinary spherical lens, such polarization effect was absent and, except for the weak residual peak, no other resonant ionization features were detected with or without the polarizing prism. Such a difference means that there are at least two general conditions which should be satisfied to observe polarization effects. First, the three-photon excitation should be driven in an essentially noncollinear geometry, which is the principal difference of Bessel and Gaussian beams. Second, some of combining fundamental waves should have orthogonal polarization planes. Such field components with orthogonal polarizations are present in nonpolarized light, but they are absent in linearly polarized light. These two conditions were checked with Bessel beams for the 6s resonance of xenon. With linearly polarized light, when the resonance ionization peak was absent, a half-wave plate was used to cover nearly a half of the incident laser beam. For this beam part, the plate turned the polarization plane by 90◦ while the rest of the beam preserved its initial polarization and the whole beam became comprised of two half-beams with orthogonal polarization planes. Such excitation mode resulted in the appearance of a distinct resonant ionization peak exactly as it was observed with nonpolarized light. As expected, the peak disappeared if the half-wave plate was removed or covered the whole beam cross-section. Similarly, the effect of cross-polarized sub-beams was absent if spherical lens was used to focus the laser beam. Exactly the same behavior was observed in the UV spectral region, when the same experimental procedure was applied for the excitation of the 6s resonance of xenon. Fig. 30 shows ionizations profiles measured with a full-aperture Bessel beam near this atomic resonance. The intense ionization band is produced due to the generation of resonance-enhanced TH. In the UV region, the dye laser output had a high polarization degree P >0.8 and, except for a weak near-resonance structure, no ionization enhancement was observed under resonant excitation. However, if a half-wave plate was used to cover a half of the incident laser beam, a distinct ionization peak appeared at the 6s resonance position exactly as it was observed for the 6s resonance. As it was considered above, the simplest noncollinear excitation geometry can be obtained by either focusing two parallel sub-beams or by selecting of two radial slices from the conical wave front with the aid of slit masks. Both of these methods were checked in experiments [83]. The polarization of the incident laser beam was purified by a Glan prism and the sub-beams were polarized initially normal to the common plane of propagation (spolarization). With the aid of half-wave and quarter-wave plates several combinations of linearly or circularly polarized sub-beams was checked. For linearly polarized light, the sub-beams had parallel s-polarization, p-polarization when polarization planes of both subbeams were turned by 90◦ , or crossed polarizations when the polarization plane for one of sub-beams was turned by 90◦. With circularly polarized light the sub-beams had either opposite circularity or co-circular polarizations. As before, care was undertaken in all experiments to avoid any parasitic retroreflection of laser light from the exit window of the cell since even weak retroreflected light could give a resonance ionization enhancement. Figure 31 demonstrates an example of REMPI spectra measured under axicon focussing of two radial beam slices. The intense peak in spectra is again due to the resonance-

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Figure 30. Wavelength scans of ionization signal near the 6s resonance of xenon in conical beams: top - linearly polarized beam (Bessel beam); bottom - cross-polarized beam (from Ref. [83]). enhanced TH. For given excitation geometry, the TH profile appears as a Lorentzian-like peak which is identical to the expected pressure-broadened atomic line (see Section IV). In the case here, the width of TH peak was 35 pm/bar FWHM in good agreement with theoretical estimates of the 6s resonance broadening [63]. The position of the TH peak in spectra is given by δ0 in Eq. (8). In s- and p-polarization, a weak residual ionization peak is registered as a hump near the canceled 6s resonance. Again, a well-pronounced resonance ionization peak appears in spectra under excitation by cross-polarized beams. The background hump gives some asymmetry and broadening of this peak toward shorter wavelength. Figure 32 demonstrates ionization profiles near the canceled 6s resonance. These spectra were measured for four different polarizations of crossed beams with a fixed laser pulse energy. With co-circular beams the three-photon excitation of the 6s resonance is forbidden. The case of s-polarization is usually considered in theory of the odd-photon interference effect in crossed beams [58, 59, 61], where a very complete cancellation of the resonant excitation process is predicted. However, experiments have shown the existence of a distinct near-resonance ionization structure. This structure has a dispersion-like shape with a peak followed by an ionization dip from the blue side. A distinct ionization peak appears under excitation by cross-polarized beams. Very similar peak appears also for beams with opposite circularity. The last is especially important since with circularly polarized beams all single-beam excitation processes are eliminated.

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Figure 31. Wavelength scans of ionization signal near the 6s resonance of xenon in angled beams: top - s-polarization; middle - p-polarization; bottom - orthogonal polarization. Excitation energy in near-resonance region was increased (from Ref. [83]). The profiles in cross-polarized and circularly polarized beams have nearly equal amplitudes, while the amplitudes of corresponding off-resonance TH peaks differed by 7-8 times (see Fig. 17). Note distinct shift between the dispersion-like ionization feature in s-polarization of pump beams and the peak registered with cross-polarized or circularly polarized beams. All these observations indicate once more that even though the general picture of resonance excitation and associated wave mixing is apparently well understood, there are several specific effects which are not readily explained within the frames of existing models. Even for the simplest case of s-polarization the observed ionization profile differs significantly from the expected picture of cancellation effect. With cross-polarized or circularly polarized beams a rather strong additional ionization peak appears under three-photon resonance excitation. Pressure-induced width of the 6s and the 6s ionization peaks in crossed-polarized or circularly polarized beams is in reasonable agreement with the expected width of corresponding atomic lines [63]. Besides, for each of these peaks the observed blue shift is very close to the blue shift of the resonance ionization peaks in counterpropagating beams, where the shift achieves its minimum value of δ0 = 4.5Δ0 (see Eq. (8) and δ0 at the crossing angle 2α = 180◦). So, the peak in angled cross-polarized beams appears as a peak in counterpropagating beams even though there is no obvious origin of such light.

ionization signal (arb. units)

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4 3 2 1 440.7

440.8

440.9

laser wavelength (nm) Figure 32. Ionization profiles in crossed laser beams near the 6s resonance of xenon: 1 co-circular polarizations; 2 - s-polarization; 3 - crossed linear polarizations; 4 - opposite circular polarizations (from Ref. [67]).

A close similarity of the resonance ionization peak in cross-polarized beams and the peak produced by weak counterpropagating light may point to a possible reason of the resonance-enhanced ionization in cross-polarized beams. Namely, such resonance ionization could be induced by the onset of a process where the weak counterpropagating light is generated within the excitation region as a reflected, conjugated or back-scattered wave at pump frequency. The nature of this process, driven by cross-polarized beams but not by beams with parallel polarization planes, is unknown. This internally-generated light, if present, is rather weak and has an intensity of 10−3 − 10−5 of the pump [83]. In ionization measurements, the weak counterpropagating light, if present, is easily registered on canceled atomic resonances which serve as a sensitive internal detector of such light. Direct detection of weak backward emission from the gas cell, however, is much more complicated since parasitic reflections and scattering of the input beam on focusing elements produce a strong background which limits the detection threshold in optical measurements. In the case here, the parasitic background exceeded the expected intensity of internally-generated light by 1-2 orders of magnitude. Therefore, all attempts to detect possible backward emission have failed and such a tentative reason for the resonance ionization enhancement has neither been confirmed nor ruled out.

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In one-color experiments with crossed beams the one-photon coupling of the ground and excited atomic states is provided by an internally-generated TH of the fundamental laser light. A more general case is the two-color excitation mode, when the resonant transition is driven by two independent laser sources of frequencies ω1 and ω2 . If these lasers are tuned to three-photon resonance at 2ω1 + ω2 or ω1 + 2ω2 , then the generated sum-frequency field should interfere with three-photon excitation at resonance. The general pattern of a suppressed resonance ionization for such two-color excitation by crossed pump beams is known from previous studies [58, 59, 84]. The observed polarization anomalies were checked for two-color excitation [85]. For this purpose, the output of the pumping excimer laser was split and used to pump two dye lasers. The parallel propagating laser beams were focused by a lens f =75 mm into the gas cell. The crossing angle between two focused beams was about 15◦. The output emission of dye lasers was polarized linearly in the direction normal to the common plane of propagation (s polarization). With the aid of zero-order quarter-wave plates the linear polarization of two beams was transformed into circular polarizations with opposite circularity. The cancellation effect was probed for the three-photon 6s resonance (λ0=440.86 nm). To monitor the canceled 6s resonance, one-color reference experiments with unidirectional and counterpropagating laser beams were arranged. For two beams with arbitrary frequencies ω1 and ω2 , the three-photon excitation of J = 1 atomic states occurs when ω1 + 2ω2 = ω0 or 2ω1 + ω2 = ω0 , where ω0 is the resonance frequency. In experiments, the wavelength of one laser was fixed at λ2 so that Δλ = λ2 −λ0 << λ0 . When the wavelength of another laser λ1 was tuned the excitation of the probed resonance occurred at two wavelengths λ1 = λ0 − 0.5Δλ and λ1 = λ0 − 2Δλ. Figure 33 shows REMPI spectra measured in different one- and two-color excitation modes. Fig. 33a,b demonstrate the cancellation of three-photon atomic resonance under one-color excitation, when the resonance ionization is absent under single-beam excitation (Fig. 33a), but strong resonance ionization peak appears in the presence of weak counterpropagating light (Fig. 33b). A very similar pattern of the cancellation effect persists also for two-color excitation by crossed beams in s polarization, where the resonance ionization peak is absent and only a weak and broad dispersion-like feature is registered near the canceled resonance (Fig. 33c). The ionization spectra undergo significant changes if the excitation process is driven by beams with opposite circular polarizations (Fig. 33d,e). In this case, two distinct ionization peaks appear in spectra. If the excitation wavelength λ2 is set to the red from the resonance position, the two ionization peaks appear at the blue side of the resonance (Fig. 33d). If the excitation wavelength λ2 is tuned to the blue from the resonance position, the two ionization peaks are registered at the red side of the resonance (Fig. 33e). The relative magnitude of these peaks is determined mainly by relative intensity and mutual overlap of pump beams in the interaction region. The position of two peaks in spectra is determined entirely by the detuning Δλ exactly as it was considered above. Very similar peaks were registered also if the two pump beams had linear polarization with orthogonal polarization planes. Figure 34 demonstrates evolution of the ionization peaks with pressure. Within the limits of experimental uncertainty, the shift Δ and the width Γ (FWHM) were linear with pressure (see the inset in Fig. 34) and a slope of 30-40 pm/bar for the shift and 70-80 pm/bar for the width were measured for the most intense peak at 2ω1 + ω2 . Such a ratio of Δ/Γ 

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2ω1+ω2

λ2

ω1+2ω2

Ionization signal (arb. units)

(e) 2ω1+ω2

λ2

ω1+2ω2

(d) λ2 (c) (b) (a) 440.4

440.8

441.2

Laser wavelength λ1 (nm) Figure 33. Wavelength scans of the ionization signal near the 6s resonance of xenon: (a) - single-beam excitation; (b) - single-color excitation by counterpropagating beams; (c) two-color excitation by crossed s-polarized beams; (d,e) - two-color excitation by crossed beams with opposite circular polarizations. The excitation wavelength λ2 for two-color profiles is marked by arrows (from Ref. [85]). 0.5 is again exactly the same as for the ionization peaks registered in counterpropagating beams and the peaks in angled beams look exactly as it would be in the presence of weak counterpropagating light. So far, the physical interpretation and an adequate theoretical model of these polarization anomalies are absent. Some general background of the unusual interference character can be derived from the following qualitative consideration. In previous theoretical studies the quantum interference effect was considered for the simplest case of crossed beams with linear s polarization. It corresponds to the usual choice of polarization in two-beam experiments on interference effect. For s-polarized beams both one- and three-photon couplings

Δ, Γ (pm)

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Γ

120 80

Δ

40

0

0.5

1.0

1.5

Xe pressure (bar) 1.5 bar

1.0 bar

0.5 bar 0.2 bar 440.4

440.6

440.8

441.0

Laser wavelength λ1 (nm) Figure 34. Evolution of the resonance ionization peaks with pressure under excitation by crossed beams with opposite circular polarizations. Wavelength λ2=441 nm. The intense band on the blue side of the profile at 0.2 bar is due to the phase-matched sum-frequency field 2ω1 + ω2 generation in the negatively-dispersive side of the 6s resonance. The inset shows plots of the blue shift Δ and width Γ of the ionization peak vs xenon pressure (from Ref. [85]). for an atomic J = 0 to 1 transition follow the selection rule ΔmJ = 0 for the magnetic quantum number mJ , where the quantization axis is along the light polarization direction. In this case, the magnetic degeneracy of the upper J = 1 state can be neglected [60] and, independent of the crossing angle between two beams, the whole excitation problem can be framed by a simple two-level model. Such an approach, however, becomes questionable for a more general case of crossed beams with different polarizations. If pump beams are cross-polarized or they have opposite circular polarizations, the angular momentum considerations become important in analyzing the response of atomic system. Under three-photon

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excitation by angled circularly polarized beams the atom acquires angular momenta from two photons taken from one beam and a photon with opposite circularity from another beam. Non-collinear addition of angular momenta ±2 and ∓1 gives the total-momentum eigenstates |1, 0 , |1, +1 , and |1, −1 for the upper J = 1 atomic state, and this state enters the excitation problem as a coherent superposition of three degenerated sub-levels mJ = 0, ±1, where amplitudes of this superposition depend on the crossing angle between pump beams. Sum-frequency generation adds coherent one-photon couplings between the ground state J = 0 and each of these sub-levels. As a result, a multilevel excitation scheme with multiple interfering processes is established. Experiments have shown that with circularly polarized beams the resonant excitation still remains suppressed but, in sharp contrast to the parallel polarizations of pump beams, the overall transition amplitude does not vanish and the resonance excitation is easily registered through resonance-enhanced multiphoton ionization.

8.

Conclusion

Excitation of low-order nonlinear optical processes with non-Gaussian laser beams offers several extensions beyond the well-known patterns of these effects in ordinary Gaussian beams. It is because any variations of amplitude and/or phase profile of the driving pump beam(s) influence the nonlinear response of a target atomic system under excitation. Therefore, significant variations in the driven nonlinear processes can be obtained in both micro and macro levels. Theoretical and experimental study of novel light fields is a rapidly developing field of modern optics, and the implications of these novel fields in nonlinear optics provide valuable information about their interesting and potentially useful properties. Even though non-Gaussian laser beams will hardly be competitive with standard beams in their conversion efficiency in a sum-frequency generation process, the non-traditional beams and different beam configurations may serve as a useful additional tool for several specific tasks of nonlinear optics like enhancement of harmonic output at some excitation conditions, spatial separation of fundamental and harmonic lights, improvement of immunity of a nonlinear process to variations of excitation conditions, and others. Several additional controls and tunable parameters become available with non-Gaussian beams, adding flexibility in excitation of nonlinear processes. With non-Gaussian laser beams, some old and well-known light-matter interaction effects may be opened to several new insights. For example, for standard Gaussian beams or plane waves, there are well-defined circumstances when a multi-photon-resonant response of a target system leads to the generation of new electromagnetic fields changing drastically the overall system response because of the constructive or destructive interference between multiple excitation pathways. The general picture of these phenomena has rather well been understood but there still are some observations that have remained without satisfactory explanation thus far. For three-photon atomic excitation and associated wave-mixing there are at least three of such effects. First, a strong ionization suppression occurs at the blue side of canceled three-photon atomic resonances. Second, a residual ionization peak is registered near the canceled resonances. Third, the pattern of resonance ionization suppression is essentially dependent on polarization of pump beams. All these experimental observations have remained unexplained and present a challenge for theory before a com-

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plete understanding of the problem can be claimed. Some of these effects are especially well pronounced with non-Gaussian beams which may serve thus as an efficient tool for the study of these puzzling anomalies.

Acknowledgements This work was supported by the Estonian Science Foundation.

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In: Progress in Nonlinear Optics Research Editors: M. Takahashi and H. Goto, pp. 261-280

ISBN 978-1-60456-668-0 c 2008 Nova Science Publishers, Inc. 

Chapter 7

R EAL -T IME E LECTROHOLOGRAPHY U SING FPGA T ECHNOLOGY, AND C OLOR E LECTROHOLOGRAPHY BY THE T IME D IVISION S WITCHING M ETHOD Tomoyoshi Shimobaba1 and Tomoyoshi Ito2 1 Graduate School of Science and Engineering, Yamagata University, Jonan 4-3-16, Yonezawa, Yamagata 992-8510 Japan 2 Graduate School of Engineering, Chiba University, Yayoi 1-33, Inage, Chiba, Chiba 263-8522 Japan

Abstract In this chapter, we describe two topics for electroholographic three-dimensional (3D) display. The first topic is real-time electroholography using the fieldprogrammable gate array (FPGA) technology. We developed an electroholographic display unit, which consists of a special-purpose computational chip (SPC) for holography and a reflective liquid-crystal display (LCD) panel, for a 3D display. We used a FPGA chip for the SPC, and we designed the SPC by adopting our proposed method, which can calculate the phase on a computer-generated hologram (CGH) using two recurrence formulas. The SPC can compute a computer-generated hologram (CGH) of 800 × 600 grids in size from a 3D object consisting of approximately 400 points in approximately 0.15 seconds. We implemented the SPC and LCD panel on a printed circuit board. After the calculation, the CGHs produced by the SPC are displayed on the LCD panel. When we illuminate a reference light to the LCD panel, we can observe a real-time 3D animation of approximately 3cm × 3cm × 3cm in size. The second topic is color electroholography using the time division switching method. We used a reflective LCD panel with a high refresh rate as a displaying device for a CGH. A color 3D object data is divided into red, green and blue components, from which we compute three CGHs. The LCD panel displays the CGHs in sequence at a refresh rate of about 100Hz. The LCD panel also outputs synchronized signals, indicating that one of the CGHs is currently displayed on the LCD panel. Red, green and blue light-emitting diodes (LEDs) used for reference lights, are switched by the synchronized signals. As a result of the afterimage effect on human eyes, we can clearly observe a colored 3D object.

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Tomoyoshi Shimobaba and Tomoyoshi Ito

Introduction

Optical holography, which utilizes the phenomena of interference and diffraction of light, can record and reconstruct the light wave of an existing object correctly and, as such, has been referred to as the ultimate three-dimensional (3D) display technique since the birth of holography [1]. However, optical holography cannot deal with a virtual 3D object, which is made on a computer. To simulate the phenomena of interference and diffraction of light from a virtual 3D object on a computer, we can compute a hologram. This hologram is referred to ”computer-generated-hologram (CGH)”. If we display a CGH on a displaying device such as a liquid crystal display (LCD) or digital micro mirror (DMD) device and illuminate a reference light to the device, we can observe a reconstructed 3D object from the CGH. An electroholographic 3D display using the CGH technique has been implemented in the research field of 3D display because the CGH can correctly reconstruct the light wave of a virtual 3D object.

Figure 1. Arrangement of a CGH and a 3D object. Here, we describe how to calculate a CGH. First, we prepare a virtual 3D object’s data on a computer, and we consider that the virtual 3D object is composed of many point light sources. Let’s suppose that a virtual 3D object consists of N point light sources and, the coordinate for j-th point light source is expressed as (ˆ xj , yˆj , zˆj ). An object’s light xα, yˆα ) on a CGH is O(ˆ xα, yˆα ) propagating from j-th point light source to the coordinate (ˆ expressed as, 2π  (ˆ xα − x ˆj )2 + (ˆ yα − yˆj )2 + zˆj2 ), (1) λ where λ is the wave length of a reference light and Aj is the amplitude of the j-th point light source. In order to record the 3D information of a 3D object on a hologram, holography needs O(ˆ xα, yˆα ) = Aj exp(j

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the interference between an object’s light and a reference light. A reference light on a CGH is expressed as, R(ˆ xα, yˆα ) = Ar exp(jφr (ˆ xα, yˆα)),

(2)

xα , yˆα) are the amplitude and phase of the reference light. So, the intenwhere Ar and φr (ˆ sity of the interference is expressed as, I(ˆ xα, yˆα ) = |O(ˆ xα, yˆα) + R(ˆ xα, yˆα)|2

(3)

xα, yˆα)| + = |O(ˆ xα, yˆα)| + |R(ˆ 2

2

O(ˆ xα, yˆα )R∗(ˆ xα, yˆα ) + O∗(ˆ xα , yˆα)R(ˆ xα, yˆα),

where * denotes complex conjugate. In CGH calculation, we can omit the first and second terms because these terms do not include the 3D information of a 3D object. Therefore, we can rewrite the above equation as follows: I(ˆ xα, yˆα ) = O(ˆ xα , yˆα)R∗(ˆ xα, yˆα ) + O∗ (ˆ xα, yˆα)R(ˆ xα, yˆα).

(4)

When we use a planer wave as the reference light (namely, the parameters for the reference light are Ar = 1 and φr (ˆ xα, yˆα ) = 0), eq.(4) is expressed as, 2π  (ˆ xα − x ˆj )2 + (ˆ yα − yˆj )2 + zˆj2 ). (5) λ Using the above equation, we can calculate a CGH from the j-th point light sources. The constant value 2 in front of Aj can be omitted because it changes only the amplitude of a virtual 3D object. And also, we can use the following equation to calculate a CGH from a virtual 3D object composed of N point light source. I(ˆ xα, yˆα ) = 2Aj cos(

I(ˆ xα, yˆα ) =

N

Aj cos(

j

2π  (ˆ xα − x ˆj )2 + (ˆ yα − yˆj )2 + zˆj2 ). λ

(6)

In terms of the computational cost of eq.(6), the calculation of the square root in the equation takes much computational time. In optics, the distance calculation in eq.(6) is approximated by Fresnel approximation. If zˆj2  (ˆ xα − x ˆj )2 + (ˆ yα − yˆj )2, the distance calculation is expressed as



(ˆ xα − x ˆj )2 + (ˆ yα − yˆj )2 + zˆj2 ≈

xα − x ˆj )2 + (ˆ yα − yˆj )2)/2ˆ zj . zˆj + ((ˆ Therefore, we can rewrite eq.(6) as follows. I(ˆ xα, yˆα ) =

N

j

Aj cos(

2π (ˆ xα − x ˆj )2 + (ˆ yα − yˆj )2 (ˆ zj + )). λ 2ˆ zj

(7)

If we compute a CGH using eq.(6) or eq.(7) on a computer, the coordinates (ˆ xj , yˆj , zˆj ) and (ˆ xα , yˆα) must be sampled by the sampling spacing p, that is, the coordinates are exxα , yˆα) = (pxα, pyα), where xα, yα , xj , yj pressed as (ˆ xj , yˆj , zˆj ) = (pxj , pyj , pzj ) and (ˆ and zj are integer values.

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Tomoyoshi Shimobaba and Tomoyoshi Ito Therefore, we can rewrite eq.(7) as follows. I(xα, yα ) =

N

j

2.

Aj cos(

2π (xα − xj )2 + (yα − yj )2 )). (pzj + p2 λ 2pzj

(8)

Real-Time Electroholography Using FPGA Technology

An electroholographic 3D display using the CGH technique has the potential to realize an ideal 3D display [2]. However, due to two significant problems, no practical electroholographic 3D display has been developed. First problem is the enormous computational time required for CGH calculation, and the second problem is an optical system for reconstructing a 3D object from a CGH. In this section, we introduce software and hardware approaches in order to overcome the first problem. We will discuss the second problem in the next section. The complexity of eq.(8) is O(M N ), where M is the total grid number of a CGH and N is the total number of point light sources of a virtual 3D object. For example, for the case in which a CGH (M = 1, 000 × 1, 000 grids) is computed from a 3D object with a simple structure (N = 1, 000 points), the computation requires approximately 10 seconds using a personal computer with a 1.8GHz Pentium-4 processor. Thus, even when using a computer with general computational ability, the CGH cannot be computed at video rate, i.e., 30 CGHs per second. Furthermore, in order to develop a practical electroholographic display, we must calculate a more high-resolution CGH from a 3D object consisting of many point light sources. Increasing the resolution of a CGH and the number of points of a virtual 3D object requires increased computational ability. In order to solve this problem, several software approaches have been proposed. For example, there is the approach of the Look-Up table method [3] suggested by M. Lucente et al. The method accomplishes a fast computation of CGH by using the cosine function table. And, H. Yoshikawa et al. [4] and K. Matsushima et al. [5] have proposed fast computation methods of CGH by translating the expression of CGH into recurrence formulas. We have, also, proposed another fast computation method of CGH by two recurrence formulas [6]. In the next subsection, we describe our proposed method briefly.

2.1.

Recurrence Formulas Algorithm for Rapid CGH Calculation

Our recurrence formulas algorithm can compute the phase component of the cosine function in eq.(8) by two recurrence formulas. Figure 2 shows the arrangement of a virtual 3D object and a CGH. We rewrite eq. (8) as follows. I(xα, yα ) =

N

Aj cos(2πθH ).

(9)

j

At the coordinate (xα , yα ) on a CGH, the phase θH of the cosine function in eq.(9) is defined by, p2 pzj 2 θH (xαj , yαj , zj ) = + (x2 + yαj ) = θZ + θXY , (10) λ 2pλzj αj

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Figure 2. Arrangement of a CGH and a 3D object. where xαj and yαj denote (xα − xj ) and (yα − yj ), respectively. Here, θXY and θZ are defined as, 2 ), (11) θXY (xαj , yαj , zj ) = Pj (x2αj + yαj θZ (zj ) = where, Pj is expressed as, Pj =

pzj , λ

p2 . 2pλzj

(12)

(13)

Note that Pj involves the depth information for a virtual 3D object. We can compute a CGH using eq.(9) and the phase θH , which is computed by θXY and θZ . Here, we consider the phase θXY (xαj + d, yαj , zj ) at the position (xα + d, yα ) on the CGH. The phase θXY (xαj + d, yαj , zj ) is expressed by, 2 ) θXY (xαj + d, yαj , zj ) = Pj ((xαj + d)2 + yαj 2 = Pj (x2αj + yαj ) + Pj (2dxαj + d2) = θXY (xαj , yαj , zj ) + Γd .

Here, Γd is defined as,

Γd (xαj , zj ) = Pj (2dxαj + d2 ).

(14) (15)

We substitute 1, 2, 3, · · ·, n into d on Γd . For d = 1, Γ1 is expressed by, Γ1 = Pj (2xαj + 1).

(16)

For d = 2, Γ2 is expressed by, Γ2 = Pj (4xαj + 4) = Pj (2xαj + 1) + Pj (2xαj + 1) + Pj × 2 = Γ1 + Γ1 + Δ.

(17)

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Here, Δ is defined as, Δ(zj ) = Pj × 2.

(18)

For d = 3, Γ3 is expressed by, Γ3 = Pj (6xαj + 9) = Pj (4xαj + 4) + Pj (2xαj + 1) + Pj × 4 = Γ2 + Γ1 + 2Δ.

(19)

For d = n, we can commonly express it as, Γn = Γn−1 + Γ1 + (n − 1)Δ.

(20)

Here, we put the second and third terms as δn−1 . Therefore, we rewrite eq.(20) as follows.

Here, δn−1 is expressed by,

Γn = Γn−1 + δn−1 .

(21)

δn−1 = Γ1 + (n − 1)Δ.

(22)

And, δn at the next coordinate is expressed by, δn = Γ1 + nΔ.

(23)

When we subtract eq.(22) from eq.(23), δn is expressed by, δn = δn−1 + Δ.

(24)

Eventually, we can compute the phase Γn at the next coordinate by the two recurrence formulas [eq.(21] and eq.(24)). Therefore, we can compute the phase θH at the position (xα + n, yα ) on a hologram by adding Γn and θZ to θXY . In the case that we compute a CGH to use this algorithm with a fixed-point operation, we can ignore the overflow bit caused by addition in the process of the all phase computations, namely, which are from eq. (10) to eq. (24). Because the overflow bit is regarded as one period of the cosine function. Then, the dynamic ranges of θZ , θXY , Γ1 , Δ, Γn and δn are always constant, respectively. The computation time on a general purpose computer of the recurrence formulas algorithm is about 24 times faster than that of the direct computation by eq.(8).

2.2.

Special-Purpose Computer for Holography

The recurrence formulas algorithm can compute a CGH at high speed, however, even a general computer on which the recurrence formulas algorithm is implemented is difficult to compute a CGH at video rate. In order to obtain more computational speed, we need to use a hardware approach together with software approaches, such as the recurrence formulas algorithm. Fortunately, CGH calculation is suitable to implement it into hardware. So far, we have designed and built special-purpose computers for holography, called HORN (HOlographic ReconstructioN), in order to overcome the computational cost of CGH calculation. HORN computers designed using a pipeline architecture [7] can calculate a CGH at high speed [8–10, 13–16]. Thus far, we have constructed several HORN

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computers, which attained computational speeds several times higher than contemporary personal computers and workstations. All HORN computers except our second machine, HORN-2, execute all calculations in the pipelines with a fixed-point format. We designed the pipelines of all HORN computers whose computational precision for hologram showed no difference between a fixed-point format and a floating-point format. Therefore, we use, ”flops (floating point number operations per second) equivalent” as the unit of the computational performance in the following discussion. The first machine, HORN-1 [8], was constructed in March 1993. HORN-1 had some restrictions since we developed the hardware easily. HORN-1 can deal with only a fixed size CGH of 400 × 400 grids. HORN-1 consists of mainly TTL(Transistor-TransistorLogic) chips and ROM(Read Only Memory) chips. The total number of chips is 26. The computational speed is about 0.3Gflops equivalent. The second machine, HORN-2 [9], was constructed in April 1994. HORN-2 was improved over HORN-1. HORN-2 can deal with any size of CGH. The total chip number of HORN-2 board is 76. The computational speed of one HORN-2 board is about 0.3Gflops equivalent. We made same three boards of HORN-2. Operating them in parallel, we got about 1Gflops equivalent. The third machine, HORN-3 [10], was constructed in July 1999. After HORN-3 machine, we used a FPGA (Field Programmable Gate Array) chip as a pipeline circuit for calculating a CGH, and, we adopted PCI (Peripheral Component Interconnect) bus as the interface between a host computer and the HORN computers. A FPGA chip is a semiconductor device containing programmable logic blocks, and programmable interconnects [11, 12]. Using HDL (Hardware Description Language), a user can design programmable logic blocks and interconnects to perform arbitrary logic functions. When a compiled HDL code is downloaded to a FPGA chip, the FPGA chip can perform arbitrary logical functions. HORN-3 can calculate only eq.(8). We could integrate one pipeline for computing eq.(8) into one FPGA chip which is equivalent to about 70,000 gates. We call it HORN3 chip. On a HORN-3 board, we mounted two HORN-3 chips and eight RAM (Random Access Memory) chips for storing virtual 3D object information. The number of operations in the summation of eq.(8) is about 30. HORN-3 chips are activated at 20M Hz. Therefore, the HORN-3 machine can calculate eq.(8) about 1.2Gflops equivalent ( 30operations × 20M Hz × 1pipeline × 2chips). The fourth machine, HORN-4, is our fourth HORN computer [13]. HORN-4 was constructed in January 2001. HORN-4 was also built with FPGA technology. After the HORN-4 machine, we designed the pipeline by adopting our recurrence algorithm [6, 13] which can calculate the phase on a CGH by the two recurrence formulas, as described in subsection 2.1. As a result, we could integrate 21 units for calculating a CGH into one FPGA chip, which is equivalent to 300,000 gates. We call it HORN-4 chip. On a HORN-4 board, we mounted two HORN-4 chips and five RAM chips. HORN-4 chips are activated at 35M Hz. We could obtain a fast computational speed of about 42Gflops equivalent (30operations × 35M Hz × 20pipelines × 2chips). The fifth machine, HORN-5, is our fifth HORN computer [14]. On a HORN-5 board, we mounted four large-scale FPGA chips. The number of pipelines for calculating a CGH implemented to the board was 1,408. The board calculated a hologram at a speed 360 times higher than a personal computer with Pentium4 processor. A personal computer connected

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with four HORN-5 boards calculated a CGH of 1, 408 × 1, 050 made from a virtual 3D object consisting of 10,000 points at 0.0023 s.

2.3.

Electroholographic Display Unit

Figure 3. Photograph of the electroholographic display unit.

Table 1. Specification of reflective LCD panel (CMD8X6D) Resolution 800 × 600

Pixel Pitch 12μm

Active Area 9.6mm × 7.2mm

Maximum Refresh Rate 360 Hz

We developed an electroholographic display unit, which has a special-purpose chip for holography based on the HORN-4 machine and a reflective LCD panel on a printed circuit board, for a 3D display system [15, 16]. Figure 3 shows a photograph of the electroholographic display unit. The unit consists of mainly four modules, namely, a universal serial bus (USB) controller, a special-purpose chip (SPC) for holography, an LCD controller, and a reflective LCD panel. We mounted these modules on a printed circuit board of approximately 28cm × 13cm in size. The USB controller is used for communication, which is the datum (xj , yj ) and Pj of 3D objects, between a host computer and the electroholographic display unit. We used an FT8U245AM manufactured by the FTDI Corporation as a USB controller. The datum is stored in Static RAM chips (SRAM). The SPC automatically starts CGH calculation by the recurrence algorithm after receiving the datum. The SPC can perform CGH calculation faster than general-purpose computers, such as personal computers. We adopted an FPGA chip (EP20K300EQC240-1X, ALTERA Corporation), which was equivalent to approximately 300,000 gates, for the SPC. After finishing the computation, the SPC sends the CGH data to a frame buffer of the LCD controller. We implemented the LCD controller on a FPGA chip (EP1K100QC2081, ALTERA Corporation), which is equivalent to approximately 100,000 gates. The chip

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was activated at 60M Hz. We adopted Synchronous DRAM chips (SDRAM) as the frame buffer. The LCD controller controls the reflective LCD panel, and the fringe pattern of the CGH is displayed on the reflective LCD panel. We used the reflective LCD panel of CMD8X6D made by Colorado Microdisplay (The LCD panel is currently made by Brillian Corporation). The specification of the reflective LCD panel is shown in table.1. The reflective LCD has a resolution of 800×600, a pixel pitch of 12μm, an active area of 9.6mm×7.2mm, and a maximum refresh rate of 360Hz. When we illuminate a reference light into the reflective LCD panel, we can observe a reconstructed 3D animation.

2.4.

Special-Purpose Chip for Holography

Figure 4. Outline of special-purpose chip for holography. We designed a pipeline in the SPC by adopting the recurrence formulas algorithm, as described in subsection 2.1. Figure 4 shows the outline of the SPC. The SPC consists of two modules: the BPU(Basic Phase Unit) and the CU(Cascade Unit). We implemented these modules into a FPGA chip. The SPC can calculate 60 light intensities on a CGH in parallel. In the figure, the symbols “MUX1” and “MUX2” mean a multiplexer circuit which can select one of the input signals. “MUX1” supplies alternately two coordinates (xα, yα ) and (xα + CN , yα) to the BPU. These coordinates are stored on registers in the FPGA chips. Here, CN is the total number of CUs in the FPGA chip. This detail will be described in subsection 2.6. After finishing the CGH calculation, “MUX2” selects sequentially one of the 60 light intensities in order to display the CGH on the reflective LCD panel. After sending the 60 light intensities to the reflective LCD panel, the SPC updates automatically new coordinates (xα, yα) and (xα + CN , yα) to add xα to 60, then, the SPC automatically starts calculating the next 60 light intensities on a CGH.

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Figure 5. The module BPU.

2.5.

The Module BPU

The BPU is used to compute three parameters, θXY (eq.(11)), Γ1 (eq.(16)) and Δ(eq.(18)), which are expressed as,

θXY = Pj ((xα − xj )2 + (yα − yj )2), Γ1 = Pj (2(xα − xj ) + 1), Δ = Pj × 2, (25) where Pj (eq.(13)) is pre-computed by a host computer. Pj , xj and yj are stored in the SRAM chips on the electroholographic display unit. Figure 5 is the block diagram of BPU. In the figure, the symbols “ABS” and “SHL” mean the circuit of the absolute value and that of left-shift operation. Figure 5 is divided into four steps as follows. (1) (2a) (2b) (3) (4a) (4b) (4c)

Subtraction Absolute value and Square Shift and Addition Addition Multiplication and Addition Multiplication Shift

Dx = xα − xj and Dy = yα − yj D2x = |Dx| × |Dx| and D2y = |Dy | × |Dy | Θ = 2Dx + 1 D2xy = D2x + D2y θXY = D2xy × Pj Γ 1 = Θ × Pj Δ = Pj × 2

In step (1), each calculation is executed in parallel. In steps (2a) and (2b), D2x, D2y and Θ are executed in parallel. In step (2b), we can obtain 2xαj + 1 to shift xαj to the left and add it to 1. In step (3), we can obtain D2xy to add D2x to D2y . In steps (4a), (4b) and (4c), θXY , Γ1 and Δ are executed in parallel. In step (4b), we can obtain Γ1 by multiplying Θ by Pj . In step (4c), we can obtain Δ by shifting Pj to the left with 1 bit.

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Figure 6. The module CU.

2.6. The Module CU The CU can compute the following two recurrence formulas and two light intensities I(xα + n, yα ) and I(xα + CN + n, yα ) on a CGH: Γn = Γn−1 + δn−1 , δn = δn−1 + Δ, I(xα + n, yα ) =

N

(26)

Aj cos(2πθH (xα + n, yα )).

(27)

Aj cos(2πθH (xα + n + CN , yα )).

(28)

j

I(xα + n + CN , yα ) =

N

j

The parameters θXY , Γ1 and Δ computed by the BPU are sent to the inputs Γn−1 , δn−1 and Δ of the first CU. Therefore, the first CU (n = 1) calculates θH (xα + 1, yα) = θXY (xα +1, yα )+Γ1 (xα +1, yα ) at the coordinates (xα +1, yα ), and θH (xα +1+CN , yα) = θXY (xα + 1 + CN , yα ) + Γ1(xα + 1 + CN , yα) at the coordinates (xα + 1 + CN , yα). Note that we can omit θz from θH because θz is not important when reconstructing a 3D object. The cascade connection of CUs, as shown in fig.4, can compute the two recurrence formulas and two intensities. Figure 6 is the block diagram of CU. It is divided into one step as follows. (1a) (1b)

Addition Addition

Γn = Γn−1 + δn−1 δn = δn−1 + Δ

In steps (1a) and (1b), Γn and δn are executed in parallel, which are used by next CU. Although all calculations may occur an overflow bit, we can ignore it since it is regarded as one period of the cosine function. Figure 7 is the block diagram of IU(Intensity Unit), which calculates eq.(27) and eq.(28). The upper 8 bit of Γn is used as θH .

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Figure 7. The module IU.

In this figure, the symbols “DFF1” and “DFF2” denote D flip-flop 1 and D flip-flop 2. The symbol “MUX” denotes a multiplexer that selects the outputs of “DFF1” and “DFF2” by the vp sw signal. The symbol “ACC” denotes an accumulator that accumulates the outputs of “DFF1” or “DFF2”. In the electroholographic display unit, the clock frequency for reading 3D object information from the SRAM chips is 40M Hz. In addition, we used a phase-locked loop (PLL) in the FPGA chip to generate 80M Hz clock signal from the 40M Hz clock signal. The 80M Hz clock signal is supplied to circuits in the BPU, the CUs and the IUs. During CGH calculation, the vp sw signal is periodically toggled between 1 and 0 at 40M Hz. The parameters xj , yj and Pj are supplied to the BPU from the SRAM chips at 40M Hz. In fig.4, the coordinates (xα, yα ) and (xα +CN , yα ) on a CGH are automatically set on the registers by a controller in the FPGA chip. Here, CN is the total number of CUs in the FPGA chip. The two coordinates are alternately supplied to the BPU at 80M Hz. Therefore, we can calculate eq.(25), eq. (26), eq. (27) and eq. (28) alternately for the two coordinates at 80M Hz. Namely, the IU of n-th CU can calculate the intensities I(xα + n, yα ) and I(xα + n + CN , yα ) at 40M Hz because the output of the look-up table for the cosine function is selected and accumulated by “MUX” and “ACC” by the vp sw signal. We implemented one BPU and 30 CUs (CN = 30) into the SPC. For example, the first CU can calculate two intensities I(xα + 1, yα ) and I(xα + 31, yα ). Similarly, the second CU can calculate two intensities I(xα + 2, yα ) and I(xα + 32, yα ), and the last CU can calculate two intensities I(xα + 30, yα ) and I(xα + 60, yα ). Thus, we can compute the two intensities by introducing the multiplexer and two DFFs to the IU. A pipeline implemented by the above method is referred to as ”Virtual Multiple Pipeline” architecture [17]. The SPC can compute 60 light intensities on a CGH at one clock cycle of 40M Hz because we implemented one BPU and 30 CUs into the SPC.

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2.7.

273

Performance of the Electroholographic Display Unit

The communication speed of the USB controller was approximately 53 KByte/s. The communication time from a host computer to the SRAM chips for a virtual 3D object consisting of approximately 400 point light sources is approximately 0.06 seconds. The computation time by the SPC for computing a CGH of 800 × 600 grids from the 3D object is approximately 0.08 seconds. The communication time for the CGH data from the SPC to the LCD controller is approximately 0.012 seconds. Therefore, the total time, i.e., the sum of the communication times and the computation time, is approximately 0.15 seconds. Namely, we can obtain CGHs and can reconstruct 3D animation from the CGHs at about 7 flames per second.

2.8. Reconstructed 3D Animation Using the Electroholographic Display Unit

Figure 8. Outline of an optical system with the electroholographic display unit.

Figure 9. Snapshot of a reconstructed 3D animation.

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Figure 8 shows the outline of an optical system in order to reconstruct a 3D animation from CGHs generated by the electroholographic display unit. In the figure, a green light emitting diodes (LED) is used as the reference light for the CGH, and the combination of a pinhole and lens L1 are used to form the plane wave of the reference light. For CGH calculation using the electroholographic display unit, we set the wave length λ in Pj to 532nm. The beam splitter leads the plane wave to the reflective LCD and leads light diffracted by the reflective LCD to lens L2. L2 and L3 are convex lenses. L2 is used to shorten the viewing distance of a reconstructed 3D object, and L3 is used to expand the viewing size of the reconstructed 3D object. We can obtain a CGH of 800 × 600 grids in size from a 3D object consisting of approximately 400 points in approximately 0.15 seconds by the electroholographic display unit, i.e., approximately 7 CGHs per second. An observer can perform operations, such as rotation and movement, to reconstructed 3D objects by use of a keyboard, while viewing the reconstructed 3D object. Figures 9 shows an example of observer’s operations on reconstructed 3D animations which consists of a circle and cone. This photograph shows the scene in which a 3D object consisting of a circle and cone is rotating from (a) to (h) by observer’s operations.

3.

Color Electroholography by the Time Division Switching Method

An electroholographic 3D display has two significant problems, that is, the first problem is the enormous computational time required for CGH calculation, and the second problem is an optical system for an electroholographic 3D display. We discussed already the first problem and its solution in section 2. In this section, we discuss the second problem, which has the following problems. (1) The viewing angle for a reconstructed 3D object is narrow. (2) The size of a reconstructed 3D object is small. (3) A reconstructed 3D object has only monochrome color. We need a device, that can display a high resolution CGH, in order to solve problems (1) and (2). In order to enlarge the viewing angle for a reconstructed 3D object, a displaying device for a CGH must have a large displaying area because the viewing angle is proportional to the displaying area. And, in order to enlarge the size of a reconstructed 3D object, a displaying device for a CGH must have a minute pixel pitch because the size is inversely proportional to the pixel pitch. Therefore, an ideal displaying device for a CGH must have a minute pixel pitch and large displaying area simultaneously. It is difficult to develop such a displaying device. Here, we introduce several methods in order to solve the problems (1) and (2). Benton et al. used an acoustic optical modulator (AOM) as a displaying device. The size of a reconstructed 3D object from an electroholographic display system with an AOM

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is large, because the diffraction angle of an AOM is wide. However, the system has only horizontal parallax in principle, and becomes a complex because it includes mechanism in the system [18]. On the other hand, an electroholographic display system with some LCDs as a displaying device has also been developed [19]. Generally, an electroholographic display with an LCD can realize full parallax, however, it is difficult to reconstruct a 3D object with a wide viewing area and large size because an LCD does not have enough minute pixel pitch and large displaying area. Using the combination of some LCDs and a reduction optical system for reducing the pixel pitch could equivalently realize a LCD which has a large displaying area and minute pixel pitch. Although transmissive LCDs have been primarily used in this field, we use a reflective LCD, which may be referred to as a Liquid Crystal On Silicon (LCOS). The reflective LCD has a high contrast and a minute pixel pitch, compared to transmissive LCDs. Therefore, we can obtain a bright and large 3D object [20].

3.1.

Color Reconstruction Using Time Division Switching Method

In this subsection, we discuss problem (3), namely, color reconstruction. So far, the research field has been studied primarily for monochrome reconstruction, while several pieces of research have also studied color reconstruction.

Figure 10. Optical system for color reconstruction using three LCD panels. For example, K.Sato developed a color electroholographic display system in 1994 [21]. Figure 10 shows the outline of the optical system. The system used three LCD panels to display CGHs, upon which we recorded the red, green and blue components of a 3D object. When illuminating red, green and blue laser lights to each LCD, the system can reconstruct a color 3D object by compounding the diffracted light from each LCD, using half mirrors. In case we expand the viewing zone of an electroholographic 3D display, we need some LCD panels as mentioned above. If we develop a wide-viewing color electroholographic 3D display system by the color reconstruction method using three LCD panels, we will

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need three times the number of LCD panels needed for a monochrome system. This causes an increase in the cost and the scale of the system. Therefore, a color reconstruction method is desirable to be able to reconstruct a color 3D object by using only an LCD panel. We developed a new color reconstruction method, which can reconstruct a color 3D object by using only an LCD panel. We call it ”the time division switching method” [22,23]. The method uses red, green and blue light emitting diodes (LEDs) as the reference lights, and one reflective LCD panel, with a high refresh rate, as a displaying device. Of course, we can also use lasers as the reference lights. We divide a color 3D object into red, green and blue components and compute three CGHs corresponding to the components. The LCD panel displays one of the CGHs in sequence and outputs synchronized signals, indicating that one of the CGHs is currently displayed on the LCD panel. Red, green and blue LEDs are switched by the synchronized signals and due to the afterimage effect on human eyes, we can observe a color 3D object. Despite the simplicity of the time division switching method, it can reconstruct a clear color 3D object. The CGH calculation for the time division switching method uses the following formula: I(xα, yα ) =

N

Aj cos(

j

2π (xα − xj )2 + (yα − yj )2 (pzj + p2 )). λt 2pzj

(29)

This equation is the same as eq.(8) expect the wave lenght in eq.(8). λt is the wavelength of the reference lights. Of course, we can also use the recurrence formulas algorithm discussed in section 2 to change λ to λt. The wavelength is generally constant with regard to monochrome reconstruction, however, in the time division switching method λt becomes variable for a time, since we need to switch the red, green and blue reference lights at certain intervals.

3.2.

Computational Process

Firstly, we describe how CGHs for the time division switching method are produced. The process is as follows: (1) Using a 3D graphics library, such as OpenGL and DirectX, we prepare a color 3D object on a computer. (2) We divide the color 3D object into red, green and blue components. (3) We compute three CGHs from the components using eq.(29). Namely, CGHs corresponding to the red, blue and green are computed with the wavelengths λt1 = 633 nm, λt2 = 432 nm and λt3 = 533 nm of the reference lights, respectively.

3.3.

Reconstruction Process

Next, we describe the reconstruction process of the time division switching method. Figures 11 and 12 show the outline and photograph of an optical system for the time division switching method.

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Figure 11. Outline of optical system for the time division switching method. In Fig.11, red, green and blue LEDs are used as the reference lights for the CGHs. These LEDs are high power LEDs made by Lumileds. The pinholes P1, P2 and P3, with a diameter of about 0.8mm, are used for making spherical waves from the LEDs, respectively. Determining the diameter is based on experience. The dichroic mirrors DM1, DM2 are used for combining the lights from three LEDs. The dichroic mirror DM1 can reflect red light, and allow other lights to pass through it. Likewise, the dichroic mirror DM2 can reflect blue light, and allow other lights to pass through it. The beam splitter BS channels the spherical waves onto the reflective LCD panel and then channels the diffracted light to lens L1 and L2 via the reflective LCD panel. L1 and L2 are convex lenses. L1 is used to shorten the viewing distance of a reconstructed color 3D object, while L2 is used to expand the viewing size of the reconstructed color 3D object. The specification of the reflective LCD panel is shown in Table 1. The LCD panel outputs three synchronized signals, indicating that one of the red, green and blue CGHs is currently displayed on the LCD panel. The synchronized signals are connected to each LED via the buffer circuit in order to amplify the drive capacity of the LEDs. The timing chart of the synchronized signals is shown in Fig. 13. Red, green and blue LEDs are switched via the synchronized signals in order to illuminate the CGHs. When the synchronized signal of red is high, the red LED lights up and the LCD panel displays the CGH computed with the wavelength of λt1 . Therefore, the red 3D object is reconstructed. When the synchronized signal of blue is high, the blue LED lights up and the LCD panel displays the CGH computed with the wavelength of λt2 . Therefore, the blue 3D object is reconstructed. Likewise, when the synchronized signal of green is high, the green LED lights up and the LCD panel displays the CGH computed with the wavelength of λt3 . Therefore, the green 3D object is reconstructed. We repeat the above process at intervals of 100Hz. As a result of the

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Figure 12. Photograph of optical system for the time division switching method.

Figure 13. Synchronized signals for time division switching of reference lights. afterimage effect on human eyes, we can clearly observe a colored 3D object. In order to adjust the brightness of the reconstructed 3D objects from the corresponding CGHs, we experimentally controlled the electric currents of the LEDs. Figure 14 shows the results of reconstructed color 3D animation using the time division switching method. The original 3D object of fig.14 consists of a circle, corn and torus with blue, green and red. The size of the reconstructed 3D object is approximately 30mm × 30mm × 30mm.

4.

Conclusion

In this chapter, we described two topics for electroholographic 3D display, namely the electroholographic display unit using the FPGA technology and color electroholography using the time division switching method. We developed the electroholographic display unit, which consists of the SPC and the reflective LCD panel. We used a FPGA chip for the SPC, and we designed the SPC by adopting our recurrence formulas algorithm, which can calculate the phase on a CGH using two recurrence formulas. The SPC can compute a CGH of 800 × 600 grids in size from a

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Figure 14. Reconstructed color 3D objects using the time division switching method. 3D object consisting of approximately 400 points in approximately 0.15 seconds. Using the electroholographic display unit, we could obtain a reconstructed 3D animation faster than by using current personal computers. However, the viewing zone of a reconstructed 3D animation from the electroholographic display unit was narrow because the reflective LCD panel does not have enough of a displaying area. A well-known method for expanding the viewing zone of an electroholographic 3D display is to enlarge the display area of a CGH, since the area of a CGH is proportional to the viewing zone. We can easily add multiple electroholographic display units to the optical system because we have implemented the electroholographic display unit on a small printed circuit board. Despite the simplicity of the time division switching method, it can reconstruct a color 3D object clearly with one LCD panel, which has a high refresh rate. The advantage of the time division switching method is the following: In case we expand the viewing zone of an electroholographic 3D display, we need to use some LCD panels. For example, the researchers studied a large-scale electroholographic 3D display with regard to monochrome reconstruction. The system used some LCDs in order to expand the viewing zone [19]. In case we develop a color electroholographic 3D display system with wide viewing zone, we will realize the system by using the time division switching method. The applications of the electroholographic display unit and the time division switching method will include the visualization of numerical simulations, entertainment, medical imagery, computer aided design, and so on.

References [1] R.J.Collier; C.B.Burckhardt; L.H.Lin. Optical Holography; Academic Press, 1971. [2] Ting-Chung Poon. Digital holography and three-dimensional display: principles and applications; Springer, 2006. [3] M.Lucente. J. Electronic Imaging 1993, 2-1, pp.28–34.

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[4] H.Yoshikawa; S.Iwase; T.Oneda. Proc.SPIE Practical Holography XIV 2000, 3956, pp.48–55. [5] K.Matsushima; M.Takai. Appl.Opt. 2000, 39, pp.6587–6594. [6] T.Shimobaba; T.Ito. Comput. Phys. Commun. 2001, 138, pp.44–52. [7] D.A.Patterson; J.L.Hennessy. Computer Organization and Design: The Hardware/Software Interface; Morgan Kaufmann Pub., 2007. [8] T.Ito; T.Yabe; M.Okazaki; M. Yanagi. Comput.Phys.Commun. 1994, 82, pp.104–110. [9] T.Ito; H.Eldeib; K.Yoshida; S.Takahashi; T.Yabe; T.Kunugi. Comput.Phys.Commun. 1996, 93, pp.13–20. [10] T.Shimobaba; N.Masuda; T.Sugie; put.Phys.Commun. 2000, 130, pp.75–82.

S.Hosono;

S.Tsukui;

T.Ito.

Com-

[11] http://www.altera.com/ [12] http://www.xilinx.com/ [13] T.Shimobaba; S.Hishinuma; T.Ito. Comput.Phys.Commun. 2002, 148, pp.160-170. [14] T.Ito; N.Masuda; K.Yoshimura; A.Shiraki; T.Shimobaba; T.Sugie. Opt.Express 2005, 13, pp.1923-1932. [15] T.Ito; T.Shimobaba. Opt.Express 2004, 12, pp.1788–1793. [16] T.Shimobaba; A.Shiraki; N.Masuda; T.Ito. Opt.Express 2005, 13, pp.4196-4201. [17] J.Makino; M.Taiji; T.Ebisuzaki; D.Sugimoto. ApJ. 1997, 480, pp.432–446. [18] S.A.Benton. Proc.SPIE 1991, 8, pp.247–267. [19] K.Maeno; N.Fukaya; O.Nishikawa; K.Sato; T.Honda. Proc.SPIE 1996, 2652, pp.15– 13. [20] T.Ito; T.Shimobaba; H.Godo; M.Horiuchi. Opt.Lett. 2002, 27, pp.1406–1408. [21] K.Sato. J.Inst.Telev.Eng.Jpn. 1994, 48, pp.1261–1266. [22] T.Shimobaba; T.Ito. Opt.Rev. 2003, 10, pp.339–341. [23] T.Shimobaba; A.Shiraki; N.Masuda; T.Ito. J.Opt.A-Pure.Appl.Op. 2007, 9, pp.757– 760.

In: Progress in Nonlinear Optics Research Editors: Miyu Takahashi and Hina Goto, pp. 281-326

ISBN 978-1-60456-668-0 © 2008 Nova Science Publishers, Inc.

Chapter 8

ELECTRIC FIELD LOCALIZATION AND ULTRAFAST OPTICAL NONLINEARITY ENHANCEMENT IN ARTIFICIAL NANOSTRUCTURES Guohong Ma, Jielong Shi and Qi Wang Department of Physics, Shanghai University Shanghai 200444, P. R. China

Abstract Materials with large optical nonlinearity and ultrafast response are the fundamental requirements for fabricating photonic devices such as optical switching and modulators. In general, two key factors are often used to evaluate the merit of electronic and photonic materials for high speed communications-modulation depth and response speed. The modulation depth is referred to the value of nonlinear susceptibility, and the response speed is related to the optical response. In the past several decades, a large number of research work was carried out on the search for and synthesis of this kind of materials, the candidate materials include semiconductor materials (especially for semiconductor quantum wells and quantum dots), photorefractive materials, metal-dielectric composite and some organic materials such as conjugated polymer, phthalocyanine and other organic materials with conjugated S-electron. From another point of view, if the incident optical electric field was confined into a small region inside a target material (in other words, electric field localization), the final result is equal to the materials possessed a large optical nonlinearity. Following this idea, by designing a certain composite structures, a large optical nonlinearity and fast response can be realized. In this article, we will address the ultrafast optical nonlinearity response enhancement for two-type of artificial structures including 1. noble metal/dielectrics and metal/semiconductor nanocomposite structure; 2. one dimensional photonic crystal with defect which is consisted of nonlinear optical materials. This article includes two chapters, chapter 1 will present the fundamental optical properties and advanced nonlinear optical response of some novel metal nanoparticles. First we will touch on the size and shape-dependence of surface plasmon resonance of metal particle. Then we will discuss the ultrafast nonlinear optical response following the ultrafast laser pulse on-resonant excitation. The last section of the chapter will focus on our findings,

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Guohong Ma, Jielong Shi and Qi Wang size- and dielectric-dependence of nonlinear optical enhancement with off-resonance excitation in Au-dielectric nanocomposite structure. In chapter 2 we will discuss another kind of artificial structures which can realize light localization, i.e. one dimensional defective photonic crystal. One-dimensional photonic crystal (1D PC) may be regarded as a waveguide consisting of a sequence of dielectric mirrors with periodically modulated dielectric constants. A defective 1D PC is constructed by inserting a defect layer into such a multilayer structure. Introduction of defect permits localized modes to exit in the range of the frequencies of bandgap. Owing to the highly localized field, strong enhancement of optical nonlinearity by several orders of magnitude can be expected in a defect layer. If the defect layer includes nonlinear optical materials, it is envisaged that the nonlinearity may be substantially enhanced by the presence of such a strong electric field. In this chapter, we will first give briefly introduction to nonlinear photonic crystal, following that we will focus on discussion on how to optimize optical nonlinearity, at last, we will confine our discussion on defect modes interaction as well as metallodielectric structure for realization of nonlinear optical enhancement.

1. Ultrafast Nonlinear Optical Response of Metal/Dielectric Nanocomposite Structures

1.1. Surface Plasmon Resonance and Linear Optical Properties of Metal Nanocomposite 1.1.1. Metal Nanospheres Optical properties of metallic systems are mainly dominated by conduction electron and bond electron responses. The conduction electrons of bulk noble metals, such as Au, Ag and Cu etc, follow a quasi-free electron behavior and their contribution to the dielectric constant at the frequency Z is well described by a Drude formula [1]

H f Z 1 

Z p2

Z >Z  i / W (Z )@

with the plasma frequency

Zp

(1)

ne e 2 / H 0 m , ne being the conduction band electron density

and m the electron effective mass. W is the electron optical relaxation time which is frequency dependent. In fact, the bond electrons will also make contributions to the dielectric constant. In most of noble metallic systems, the bond electrons contribution is dominated by the interband transition from the fully occupied d-bands below the Fermi energy to the half filled s-p conduction band. After considering the contributions of bond electrons, Hb(Z), the complete dielectric constant can be expressed as [2] H(Z)=Hf(Z)+Hb(Z) Electron optical relaxation time W(Z) in bulk metal is determined by electron-phonon and electron-electron scatterings:

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement…

1 W (Z )

1



1

283

(2)

W e e Z W e ph Z

In thermal equilibrium at room temperature, We-ph is much smaller than We-e and yields maim contributions to 1/W [3, 4]. The linear optical properties of small spherical metal particles (nanoparticles) accounting for surface plamson resonance can be understood with Mie scattering theory [5]. Mie theory is the solution of Maxwell`s equation for an electromagnetic wave interacting with small metal particle having the same macroscopic, frequency-dependent material dielectric constant as the bulk metal. In most of optical experiments one is often dealing with an ensemble of metal nanoparticle dispersed in a dielectric matrix. For a low volume fraction p<<1 of small spherical nanocrystal (normally, less than 20 nm in diameter), the effective dielectric constant for the nanocomposite can be expressed as: [6]

H Z H 1 (Z )  iH 2 (Z ) H d  3 pH d

H Z  H d H (Z )  2H d

(3)

Hd is the frequency-independence of dielectric constant of the dielectric matrix. The absorption coefficient of the nanocomposite is related to the imagery part H2(Z) which can be expressed as [6]

D (Z )

9 pH d3 / 2 ZH 2 (Z ) c >H 1 Z  2H d @2  H 22 (Z )

(4)

This expression corresponds to remaining only the dipolar terms in Mie`s theory, and thus is valid only for small particles. It is seen from Eq. (4), the absorption coefficient can be resonantly enhanced when the term H1(Z)+2Hd approaches zero, which is the condition for the surface plasmon resonance (SPR). Also, the SPR frequency :R can be determined by the term H1(Z)+2Hd =0. It is seen the :R is particle size independent according to Mie theory (dipole approximation), this situation is on agreement with experimental findings when metal particle is not too small, normally the particle size is larger than 2 nm [7, 8]. When the particle size is smaller than 2 nm, the SPR frequency shows particle size dependence [9]. When the laser wavelength is on resonance with the SPR frequency, the local electric field inside the metal nanoparticle can be greatly enhanced due to a dielectric confinement effect. And the nonlinearity of the composite is expected to be greatly improved, which will be discussed in details later.

1.1.2. Metal Nanorods SPR frequency also shows particle shape dependence. For simplicity, we focus on discussion linear optical properties of randomly oriented ellipsoids dispersed in a dielectric matrix. As done by Gans [10], by introducing a geometrical factor Pj, the absorbance of light by ellipsoids can be calculated. The geometrical factor corresponds to each of the axes A, B, and

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C of the particle as indicated in Figure 1.1 Gans' formula for randomly oriented elongated ellipsoids in the dipole approximation is

2SH 3O

J

3/ 2 d

N pV

§ 1 ¨ ¨ P2 © j

C

¦

j A

ª § 1  Pj «H 1  ¨¨ «¬ © Pj

· ¸H 2 ¸ ¹

(5) 2

· º ¸H d »  H 22 ¸ » ¹ ¼

Figure 1.1. schematic of geometrical factors for an elongated ellipsoid metal particle.

For elongated ellipsoids (or rods) the B and C axes are equal and correspond to the particle diameter (d) while the A axis represents the particle length (L). The geometrical factors Pj for elongated ellipsoids along the A and B/C axes are respectively given by [11]

ª 1 §1 e · º « 2e ln¨ 1  e ¸  1» © ¹ ¼ ¬ (1  PA ) / 2

PA

1  e2 e2

PB

PC

with e

§ L2  d 2 ¨¨ 2 © L

· ¸¸ ¹

(6)

1/ 2

In analogy with the absorbance of spheres, the SPR frequency is then given by

H1  s jH d

0

(7a)

The sj is called screening parameter, and sj=(1-Pj)/Pj, which is strongly depend on the anisotropy of the nanoparticle. RB=RC corresponds to the transverse Plasmon oscillation, and RA represents the longitudinal oscillation, with increasing aspect ratio the screening parameters of longitudinal oscillation shifts towards infinity, where in the case of the transverse oscillation, the screening parameter reaches unity. Figure 2 shows the aspect ratio dependence of SPR wavelength in silver nanorod.

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement… aspect ratio d/L 1/2 1/3 1/4

1.0

Normalized absorbance

285

0.8 0.6 0.4 0.2 0.0 500

1000

1500

2000

2500

3000

3500

Wavelength (nm)

Figure 1.2. Aspect ratio (d/L) dependence of SPR wavelength of silver nanorod with d/L=1/2, 1/3, and 1/4, respectively. The arrow indicated peak is corresponding to transverse mode which shows aspect ration independence.

1.1.3. Metal Nanoshells Core-shell structure in metallic system can also be used to tune the SPR frequency [12]. The geometry of this structure is shown in Figure 3. Region 1 is core and is characterized by a radius r1 and dielectric function H1. the shell has a thickness r2-r1 and a dielectric function H2. The embedding medium has a dielectric function H3. Among them H1 and H3 are dielectric matrix with zero imaginary part, and H2 is metal nanoshell.

Figure 1.3. Nanoshell geometry: H1, H2, and H3 are the dielectric constants for core (with radius of r1), shell (with thickness of r2-r1) and embedding regions, respectively.

According to Mie theory and proper boundary condition, the absorption cross section is obtained as follows: [13, 14]

V

8S 2 H 3

O

§ H H  H 3H b r23 Im¨¨ 2 a © H 2 H a  2H 3H b

· ¸¸ ¹

(7b)

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by defining the ratio of the shell volume to the total particle volume as P, we have P=1-(r3 1/r2) , and the parameters Ha and Hb are, respectively

Ha Hb

H 1 (2  3P)  2H 2 P H 1 P  H 2 (3  P)

with Eq. 7 the calculations of the extinction cross section for gold nanshells are shown in Figure 4. The results of these calculations are for a constant shell thickness of 2 nm and r2=4, 10, and 17 nm. The core dielectric constant was determined by use an estimate of the calculated bandgap of Ag2S and empirical expression H2Eg=77, [15] where H is the dielectric constant and Eg is the bandgap.

Figure 1.4. Normalized cross section versus wavelength for various Au2S (core) capped with 2-nmthicness-Au.

1.2. Nonlinear Optical Response of Nanocomposite with Resonance Excitation 1.2.1. Introduction Optical materials with large nonlinear optical susceptibility and fast response time are essential requirements for future photonic devices such as all-optical switches and modulators [16-18]. In the past few years, considerable attention has been devoted to metallic nanoparticles embedded into transparent dielectric matrix [19-21]. Motivation for this effort resides in the nanoparticles-dielectrics composite system exhibition of a large third-order optical nonlinearity F(3), ultrafast response time and tunable surface plasmon resonance frequency. Linear and nonlinear optical properties of the composite system have been extensively investigated both in basic and applied research fields. A strong enhancement of

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement…

287

the F(3) (~10-7 esu) has been observed around the peak of the SPR [19]. It is well known that the considerable enhancement of F(3) in such a composite system arises from the local field enhancement near and inside the metal particle at the SPR [18, 22-23], which actually originates from an inhomogeneous distribution of the matrix. Many methods have been employed to determine the F(3) of the composite systems, such as Z-scan [21, 24], optical Kerr effect (OKE) [18, 25] and four wave mixing (FWM) [17, 26] etc. Single beam Z-scan method can provide both the real and imaginary parts of F(3), it can not give dynamical information which is very important for designing all optical switching devices. OKE and FWM methods can provide both dynamic information and the modulus of F(3), but the real and imaginary components are not separatable. In homedyne OKE setup, by introducing a local oscillator field with or without 900 optical phase biasing (optically heterodyne) [27], i.e. optically heterodyne optical Kerr effect (OHD-OKE), we can separately investigate the sign, magnitude and response of both real and imaginary parts of F(3). Furthermore, the signal-tonoise ratio of experimental data can be greatly improved by this method. Before discussing the experimental results, I will discuss experiment setup of ultrafast OKE and OHD-OKE used in our lab.

1.2.2. Ultrafast Experimental Technique Refractive index of optical medium depends upon the intensity of light through the simple relation n=no+n2I, with no linear refractive index, n2 Kerr coefficient and I for light intensity. In other words, optical field induced birefringence can modify the polarization of a beam in the medium. Conversely, measurement of the change of the beam polarization should allow us to deduce values of the induced birefringence. In this part, we will discuss the principle of optical Kerr effect and the ultrafast experimental setup, so that we can characterize the magnitude and dynamic information for both real and imaginary components of metal nanocomposites. For simplicity, we assume that both pump and probe beam are linearly polarized, and the polarizations are at 450 degree with respect to each other. As shown in Figure 1(a), both probe field E1(Z1) and pump field E2(Z2) propagate along z direction. The polarization of E1(Z1) is along x-direction, and the polarization of E2(Z2) has a 450 degree with that of E1(Z1). Suppose that the angular frequencies and wave vectors for intense pump field and weak probe field are Z1, k1 and Z2, k2, respectively. By defining, Probe field: E1 ( z , Z1 )

G E1x x

Pump field: E 2 ( z , Z 2 )

G G E2 x x  E2 y y

(1 / 2) A1 ( z ) exp[i (k1 z  Z1t )]  c.c. (1 / 2) A2 ( z ) exp[i (k 2 z  Z 2 t )]  c.c.

where, the c.c. stands for complex conjugation, and A1x

A2 y

A1 , A2 x

A2 cos 45 0 ,

A2 sin 45 0 . For normal OKE arrangement, P1 and P2 are set with crossed polarization

[28]. When pump field is on, the detected signal is referring as Kerr signal, Is.

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Figure 1.5. (a) Schematic showing the direction of pump (E1) and probe (E2) fields; (b) Geometrical layout of the propagation of pump and probe beams in pump-probe setup, P1 and P2 are polarizers, D is the angle between the propagating pump and probe beams.

In order to separate the real and imaginary components of nonlinear coefficient, a local field ALO is introduced, which is mixed with Kerr signal. For doing that, we can rotate P2 slightly from the crossed point with an angle ) (<50), or by inserting a quarter-wave plate between P1 and sample, the optical axis of the quarter-wave plate has a small angle ) (<50) with the polarization of probe beam. The former configuration can be used to obtain the information of imaginary part of F(3), and the real part can be determined by using the later case [27]. As is referring as Kerr signal, the total signal detected by detector is

(8) where, |AS| stands for Kerr signal, and |ALO| for local signal, IH=ASA*LO+ALOA*S for heterodyne signal. The local signal ILO is dc signal that can be removed by using special experimental arrangement. The detected signal I is

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement… I=IH+IS

289 (9a)

We have three cases: Case 1, )=0 in absence of O/4, This is normal OKE arrangement, we have IH=0, so the total detected signal is I=Is, with

I

H 0 nc

Is

2

( Asig )1 y v i

* ( Asig )1 y ( Asig )1 y , with

Z1 2 H 0 k1 c 2

>

( 3) ( 3) L F yxyx A2 x A2* y A1x  F yxxy A2*x A2 y A1x

@

for degenerated case, k1=k2 and Z1=Z2, we have

Z4

L2 2 ( 3) ( 3) I 2 I 1 F yxyx  F yxxy H 04 k 2 c 6 n 2

IS

2

(9b)

This is homodyne OKE, it is seen that the measured signal is proportional to the square of the modulus of

( 3) ( 3) F yxyx  F yxxy

It is seen that the OKE signal is proportional to I12, (F(3))2 and L2. By taking a reference (CS2, for example), the OKE signal of sample is given by

(10) R is absorption correction factor, with Case 2, )z0 (<50) in absence of O/4, Under this situation, the local field ALO is proportional to )A1x

IH

H 0 nc 2

* ( AS ALO  AS* ALO )

Replacing ALO with )A1x, we have

IH v i i

>

nZ 2 2 ( 3) ( 3) ( 3) ( 3) * )L 2 2 2 I 2 I 1 u F yxyx  F yxxy  F yxyx  F yxxy kc H0 n c

4Z 2 )L ( 3) ( 3) I 2 I 1 i ImF yxyx  F yxxy 2 3 H 0 k1 c n

>

@

@

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Guohong Ma, Jielong Shi and Qi Wang



4Z 2 )L ( 3) ( 3) I 2 I 1 ImF yxyx  F yxxy H 02 k1c 3 n

(11)

Obviously, the heterodyned signal is proportional to the imaginary component of sample. Case 3, ) z0 in presence of O/4, Under this condition, local field ALO is proportional to i)A1x

IH v ( AS ) ALO I1

H 0 nc 2

* u ( AS* ALO  AS ALO )

( ASig )1 y v i

Z2 H 0 k1 c

2

L

2 1 ( 3) ( 3) A2 A1 ( F yxyx )  F yxxy 2

i)A1x i)A1 H 0 nc 2 H 0 nc 2 A1 ; I 2 A2 ; 2 2

IH ~

Ai

2

2Ii H 0 nc

4Z 2 )L ( 3) ( 3) I 1 I 2 Re( F yxyx  F yxxy ) 2 3 H 0 kc n

(12)

It is seen that the imaginary (case 2) and real (case 3) components of the material can be determined by introducing a local oscillator. The total detected signal I for case 2 and 3 can be written as I=Z1+)Z2

with Z 2, r ( im )

(13)

4Z 2 LI1 I 2 ( F yxyx  F yxxy ) r ( im ) H 02 nkc 3

Subscript r and im represent for real and imaginary parts of F(3). Z2 can be determined from I ~ ) linear relation, where Z2 is the slop of the linear. Compared with standard sample (for example. CS2) at identical condition, we have s

( F ( 3) eff ,s ) r ,im R

(

ns 1/ 2 Lref Z 2 ) ( F ( 3) eff ,ref ) r ,im u DLs / R ref nref Ls Z 2

(14)

exp(DLs / 2)(1  exp(DLs ))

Here, subscripts s and ref denote sample and reference respectively, and n represents refractive index. L ref stands for overlapped length of pump and probe beams in reference sample, and Ls is the thickness of the sample, while D is the linear absorption coefficient of the sample. With introducing a heterodyne angle in case 2 and 3, the real and imaginary components of nonlinear optical susceptibility can be measured separately, normally, case 2

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement…

291

and case are called optical heterodyne optical Kerr effect (OHD-OKE). Figure 6 shows experimental setup, where W is denoted for quarter wave plate (O/4).

Figure 1.6. Ultrafast OKE experimental setup. Ch, M, P, BS, f, S, A and D are denoted for chopper, mirror, polarizer, bean splitter, lens, sample, aperture, and detector, respectively. W is quarter wave plate.

In our experimental, the ultrashort laser pulse was generated from a Ti: sapphire laser (Spectra-Physics, Tsunami). The laser beam was divided into pump beam and probe beam by a beam splitter, and the ratio of pump-to-probe power was set to be larger than 10:1. The pump beam traveled through a chopper, a computer-controlled delay line, and the reflected probe beam went through a polarizer P1 with the polarization direction set at 45q with respect to that of the pump beam. Two beams were focused by a lens of 5 cm focal length and overlapped on the same spot of the sample with a spot size of about 50 Pm. An analyzer P2 was placed behind the sample with cross polarization to P1. In order to separate the imaginary and real parts of F(3), a local oscillator has to be introduced. For the F(3) imaginary part measurement, polarization of P2 was tuned slightly with an angle ) (where defined as the heterodyning angle, is the deviated angle of P2 from the cross polarization of P1). As for obtaining the F(3) real component, a quarter-wave plate (dz/4) with its optical axis parallel to the polarization direction of probe beam was inserted between P1 and the sample. By slightly rotating the analyzer with an angle, ). The generated local oscillator field ()Ax for imaginary part and i)Ax for real part of F(3)) and the Kerr signal was mixed in the photodiode leading to a heterodyned detection. It should be mentioned that the OKE setup above is applicable for ultrafast pump and probe measurement just by taking off P2 as well as changing the polarization of P1 to be parallel or perpendicular with that of pump pulses.

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Guohong Ma, Jielong Shi and Qi Wang

1.2.3. Nonlinearity Enhancement in Au Nanocomposite with Resonant Excitation In order to investigate nonlinearity enhancement in metal composite, Au:TiO2 composite films fabricated by alternating rf-sputtering were chose for studying. The deposition rates of Au and TiO2 were calibrated by separate sputtering of Au and TiO2 prior to the alternating sputtering process under the same deposition condition. The relative composition of Au in the composite films was determined by adjusting the deposition time and power supply of the Au target. All the samples were deposited on glass substrate without intentional heating in the presence of Ar. The detailed deposition process is as follows: (1) under high vacuum (~ 5 pa) condition, the pure TiO2 (purity 99.98%) was deposited onto the glass substrate with a deposition rate of 1 nm/min for 10 min, with the Au target off (2) TiO2 target was turned off and the Au target was turned on to deposit Au target under the same vacuum condition with a deposition rate of 2 nm/min for 1 min. Steps (1) and (2) were repeated 7 times. The sample is then annealed at a temperature of 400 0C for 3 hours in the presence of Ar atmosphere. The resulting film has a thickness of 85 nm measured on D-step profiler. The mean diameter d was estimated to be 32 nm using Scherrer`s equation from X-ray-diffraction measurement [29]. The UV spectra of the Au:TiO2 film fabricated by the alternating sputtering method was shown in Figure 7. One can recognize an absorption peak at 770 nm along with increasing background. The origin of the peak is assigned to a surface plasmon resonance. The SPR peak position shows a large red shift compared with other reports in which the sample was fabricated by cosputtering method [17, 26]. In order to make a comparison, we also fabricated Au:TiO2 composite film by cosputtering method as reported in ref. 17. The SPR peak at 590~600 nm was observed for cosputtered films although the Au concentration and annealing temperatures were kept the same as that of alternating sputtering films. In gold nanoparticles, the interband transition of the d band to sp bands lie in the higher energy region (larger than 2 eV), the SP resonance is far from the interband transition in our sample. Thus we can separately investigate free electron contribution to the dynamical properties of the nonlinear response. Additionally, SP resonance peak is at 770 nm in the alternating sputtering films, which is close to the wavelength of the laser.

Figure 1.7. UV spectra of alternating sputtering (solid line) and cosputtering (dash line) composite film annealed at 400 0C for 3 hours in Argon environment. The arrow indicates the laser wavelength used in the OHD-OKE measurement.

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement…

Figure 1.8. OHD-OKE response of Au:TiO2 film at heterodyne angle ˓ 

s30 with ˨

293

/4 plate (Figure

8a) and without ˨ /4 plate (Figure 8b). The long time behavior is shown in the inset of Figure 8a.

As mentioned in the experimental section, the real and imaginary components of F(3) in Au:TiO2 composite film can be determined based on the presence or absence of O/4 plate in OKE experiment. Figure 8 shows the OKE response at )= r 30 of the Au:TiO2 film for the presence (a) and absence (b) of O/4 plate, a long-time-scale behavior is shown in the inset of Figure 8 (a). It is seen that the OKE temporal response is also decomposed into two decay components, the fast decay component (about 3 ps) and the slow decay component (about 150 ps), which agrees very well with the pump-probe result. Figure 9 presents the OKE signals with and without O/4 plate at zero delay time vs heterodyning angle ) for both Au:TiO2 and ZnSe, it is seen that the slope of Au:TiO2 composite film has an opposite sign to that of reference, ZnSe, for both real and imaginary parts. This means that both of the real and imaginary components of Au:TiO2 films are negative. The negative real part of F(3) has selfdefocused characteristics, while the negative imaginary part of F(3) exhibits saturable absorption. Based on formula (14), the Feff(3) of Au:TiO2 film was determined to be –2.3u10-8i*6.7u10-8 esu. Here, we use F(3)xxxx =1u10-11 + i* 3.5u10-11 esu at 780 nm with 130 fs pulse for ZnSe sample (F(3)xyxy+F(3)xyyx=2/3 F(3)xxxx), and take nref=1.7 and ns=2.4 for ZnSe and the sample, respectively [30].

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Guohong Ma, Jielong Shi and Qi Wang

Figure 1.9. The OKE signals with and without O/4 plate at zero delay time was plotted as heterodyning angle ) for both Au:TiO2 and ZnSe, respectively.

At the surface plasmon resonance, large enhancement of local field near and inside the metal particle can be realized according to the well-known “effective medium theory”. In the metal: dielectrics nano-composite structure, local field factor f takes the form of f=3Hd/(Hm+2Hd), where Hm = H1+iH2 and Hd are dielectric constants of metal particle and dielectrics matrix, respectively. The F(3) of the composite system is proportional to the fourth power of local field factor, f4. Under the resonance condition, H1+2Hd ~ 0, f is reduced to 3Hd/iH2. Therefore, at the SPR, the giant local field enhancement can induce large nonlinear optical response. The origin of the large nonlinear optical susceptibility in the composite film can be attributed to the hot electrons contribution [31, 23]. In the Au:TiO2 film fabricated by alternating sputtering method, the SPR peak is at 770 nm, which is far from the interband transition of the d band to sp bands peaked at 560~600 nm, the contribution of interband transition to the nonlinearity can be neglected. The high intensity pump pulse energizes the conduction electrons resulting in a high electron temperature while the lattice remains cool because the heat capacity of electron is much smaller than that of phonon. The heated electrons lead to Fermi smearing, which rearranges the occupancy distribution of conduction electrons near the Fermi surface, resulting in an increase in the occupancy above the Fermi energy and a decrease in the occupancy below Fermi energy. The value of F(3) in our experiment is one order smaller than the value reported by other groups [17, 19]. When excitation energy is on resonant with both SP and band-to-band transition, the contributions of band-to-band transition are comparable to or even larger than that of Drude part contribution. In addition, the value of F(3) in nanoseocnd region is much larger than that of femtosecond region because the nonlinearity in nanosecond is mainly governed by thermal effect [17]. Therefore, the value of F(3) obtained in our experiment should mainly come from the contribution of free electrons.

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement…

295

In order to study the excited electron dynamics, pump-probe measurement was carried out with femtosecond laser pulse. Figure 10 shows the time-evolution of transmittance change ('T) obtained from pump-probe experiment with parallel polarization configuration. It is clearly seen that the temporal behavior is decomposed into two components, a fast rise photobleaching followed by its recovery process. The recovery process is consisted of two decay processes, a fast process (about 3 ps) and a slow one (150 ps). With increasing pump intensity, the fast decay process retarded while the slow components remains almost unchanged.

Figure 1.10. Time evolution of transmittance changes at 780 nm with different pump intensity.

It has been suggested that fast decay reflects the thermal equilibrium process between the electron system and the lattice system in the metal nanoparticle and the slow decay arises from the cooling process by a thermal diffusion from the metal nanoparticle to the host matrix [23, 27]. For the whole relaxation process in a metal nanoparticle system with ultrafast light excitation, the widely accepted model is the two-temperature model [33] which is described in short as follows: A pump pulse with short pulse duration strikes nanoparticles, exciting the surface plasmons. Plasmons lose their coherence instantaneously through an electron-electron scattering process and change into a quasiequilibrated hot electron system within 100 fs. As the hot electron system loses its energy through an electron-phonon coupling process, the phonon system becomes hot. After several ps, a quasiequilibrium state is formed between the electron system and the phonon system in a nanoparticle. In a long time scale over 150 ps, the heat transferred from the nanoparticle to the host matrix should be a dominant process. From Figure 10, one can recognize that the electron system cools down with two characteristic time constants of a few picoseconds.

1.2.4. Size-Dependence of the Nonlinearity Enhancement in Au Nanocomposite Film with Off-Resonance Excitation As discussed in last section, metal nanoparticles (such as Au, Ag and Cu) embedded in various dielectric show a large value of fast optical Kerr susceptibility, F(3), and hence possible applications in all-optical switching devices [33-37]. In general, the considerable

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enhancement of F(3) in such a composite film is a consequence of local field enhancement near the surface plasmon resonance [33,35,38] which actually originates from inhomogeneity of the matrix with embedding metal nanoparitlces. As a result of the sharp difference in indices between the metal particles and the surrounding matrix, the electromagnetic field distribution in the composite system will be very different from that of a homogenous matrix without embedding particles. It can be expected that the local field enhancement also play an important role in off-resonant excitation. Furthermore, it is envisaged that the enhancement factor will be essentially determined by the particle diameter and the surrounding matrix by ways of the quantum confinement and dielectric confinement effects, respectively [17, 35]. In fact, in order to evaluate the nonlinear optical characteristics of materials, one often considers the figure-of-merit F(3)/D, where D is the absorption coefficient of the material, instead of F(3). Furthermore, SPR excitation will induce both a loss in pump beam and an increase in sample heating, both of which can be critical disadvantages in designing photonics devices. It is therefore equally important to search for nonlinear optical of materials with large effect and fast response under off-resonance condition. Here, we focused on studying the nonlinearities of gold nanoparticles embedded in BaTiO3 and ZrO2 matrices by femtosecond optical Kerr effect. Under off-resonant excitation, experimental results show that F(3) of the composite film is strongly dependent on both particle sizes and host matrices. By using Lorentz-Mie scattering theory, we calculated the average third-order nonlinear optical susceptibility under both resonant and off-resonant excitation. The simulation confirms that the dependence of F(3) on particle size and dielectric constant stems from local field enhancement. Here, 260-nm-thick Au:BaTiO3 and ZrO2 films were fabricated by sol-gel and dipcoating method[39-40]. Briefly, HAuCl4·4H2O was added into barium titanium methacrylate (the precursor for preparation BaTiO3) and ZrOCl2·8H2O, respectively. The fixed molar ratio of Au/Ba and Au/Zr was kept to be 15%. The as-prepared films were dried at about 60 0C, and finally were densified at 300-600 0C in air. The average sizes are about 12, 18.5, 31 and 43 nm for Au nanoparticles in barium titanate films at annealing temperatures of 300, 400, 500 and 600 0C respectively, and the average size is about 11.5 nm for Au:ZrO2 films at annealing temperature of 450 0C. The standard deviation of particle distribution was 30% as determined by high-resolution TEM. The optical absorption spectra of Au:BaTiO3 composite film with different Au particle sizes are shown in Figure 11, and the peak positions vs. Au nanopartice diameter are also plotted in the inset. For the 12 nm Au particles, the surface plasmon resonance absorption is located at about 690 nm as shown in Figure 11. The resonant peak shifts towards a longer wavelength with increasing particle size, which agrees well with the other reports [17, 41]. For Au:ZrO2 film annealed at 450 0C, the surface plasmon resonance absorption was found to be 574 nm (not shown in the figure).

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0.5 610

Absorption

0.4

0.3

605

peak position

12 nm 18.5 nm 31 nm 43 nm

600 595 590 585 580 575

0.2

10 15 20 25 30 35 40 45 particle diameter (nm)

0.1

0.0 300

400

500

600

700

800

Wavelength (nm)

Figure 1.11. The absorption spectra of Au:BaTiO3 composite films with gold nanoparticles of diameter d=12, 18.5, 31 and 43 nm. The dependence of surface plasmon resonance wavelength on particle diameter plotted in the inset.

The third-order nonlinear optical susceptibility of the films was measured by means of femtosecond optical Kerr effect (OKE). The third-order nonlinear optical susceptibility, F(3), of the Au:BaTiO3 films is shown in Figure 12 as a function of Au particle size. The inset of Figure 12 shows the OKE response of the films with particle diameter dAu=31 nm and pure BaTiO3 (d=0 nm). The films show an ultrafast OKE response with FWHM about 200 fs, which is just a little longer than that of the laser pulse duration. As the excitation wavelength is far from any absorption resonance of composite material, the ultrafast OKE response arises from the photo-induced anisotropy contributed mainly by free electrons near the Fermi surface. For nanparticle sizes ranging from 12 nm to 31 nm, the OKE signal is seen to increase with Au particle size. However, as the nanoparticle size further increases to 43 nm, the OKE signal is seen to decrease. According to corresponding formula, the third-order nonlinear optical susceptibilities F(3) of these composite films were calculated to be 6.1u10-11, 7.8u10-11, 1.3u10-10 and 5.6u10-11 esu for diameters d=12, 18.5, 31 and 43 nm, respectively.

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24

Au:BaTiO3 (d=32 nm) pure BaTiO3 (d=0)

14

OKE signal (a.u.)

16

(a)

12 10 8 6 4 2

12



F X10

-11

(esu)

20

16

0 -400

-200

0

200

400

Delay time (fs)

8 Au:BaTiO3 Au:ZrO2

4 0 0

10

20

30

40

50

Particle diameter d (nm)

OKE Signak (a.u.)

(b)

Au:BaTiO3 Au:ZrO2

4 3 2 1 0 -400

-200

0

200

400

Delay time (fs) Figure 1.12. (a) The dependence of third-order nonlinear optical susceptibility (F(3)) on gold nanoparticle diameter (d) in Au:BaTiO3 films, d=0 was supposed for pure BaTiO3 film. The OKE responses of composite film (with dAu=31 nm) and pure BaTiO3 film were shown in the inset. The calculation of F(3) is based on equation (2) with parameters: FCS2(3)=1u10-13 esu., nref=1.69 (CS2) and ns=2.37 (BaTiO3) and 2.14 (ZrO2). Figure 2 (b) the OKE response of Au:BaTiO3 (dAu=12 nm) and Au:ZrO2 (dAu=11.5 nm) films, the corresponding value of F(3) for Au: ZrO2 was plotted in Figure 2(b).

By comparison with the pristine BaTiO3 film (F(3) = 1.1 u10-12 esu, here we supposed that d=0 nm), nonlinear enhancement is clearly demonstrated in the Au:BaTiO3 composite films, and is strongly dependent on the nanoparticle size. The value of F(3) for Au:ZrO2 composite film with dAu=11.5 nm is also shown in the Figure 12. Comparing with that of Au:BaTiO3 film with identical Au particle, the difference of F(3) mainly comes from different matrices, i.e. BaTiO3 and ZrO2. In general, size dependence of third-order nonlinear optical response of metal nanoparticles doped dielectric matrices arises from local field enhancement when the excitation energy is on resonant with the frequency of surface plasmon of the sample, and F(3)

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has been found to increase nearly as d-3[34,35].Thus our experimental observation represents a significant departure from the results reported in Ref. 35. It is surmised that this departure may have originated from off-resonance excitation employed in our experiment. In order to confirm that the effect is indeed due to off-resonance excitation, we calculated the electric field in the region surrounding the gold sphere through Lorenz-Mie scattering theory. In the calculation, we invoked Eqn. (15) for the effective F(3) of the matrix,

³

F 

2 3 effective

F 3 E 3 dV 2

shell

E 03

2

(15)

³ dV

shell

where F(3) is susceptibility of the matrix, E the electric field calculated through Lorentz-Mie scattering theory for an incident electromagnetic wave E0. The dielectric constants of gold at 600nm and 800 nm are taken from Ref.42. Eqn. (15) can be easily understood by the following consideration. As the observed electric field of third harmonics is proportional to F (3) E 3 , the light intensity contributed by a small volume 'V is seen to be proportional 2

to F (3) E 3 'V . The total contribution from the whole shell can be expressed as

³



3 E 03 F 3 E 3 dV , which should be equal to F effective 2

shell

³ dV . In the calculation, a constant 2

shell

volume ratio of the shell to the gold sphere is being kept for different sizes of gold spheres. It should be noted that interference arising from neighboring volumes has been neglected in the integration, which is reasonable as all the spherical gold particles are much smaller than the wavelengths. Our simulation provides a good qualitative understanding for the size and dielectric dependence of nonlinear optical response short of an exact calculation. Figure 13 gives the dependence of calculated

F effective (3) on the Au diameter at

wavelength 800 nm, in BaTiO3 (n=2.37), ZrO2 (n=2.1) and SiO2 (n=1.5) media. It is seen that for small Au particles, the calculated

F effective (3) shows an initial increase with metal particle

size, but the curve is not monotonously increasing. The maximal points are localized at d=32 nm for BaTiO3, d=40 nm for ZrO2 and d=66 nm for SiO2. It is obvious that the surrounding matrices with different refractive indices induce different local fields near the metal surface. More exactly, the local field intensity strongly depends on the refractive index difference between the matrix and the metal particle, and it is seen to increase with the refractive index difference. Basing on the above analysis, and considering both the particle size and the difference in refractive indices of particle and surrounding matrix, optimization of F(3) in a composite system can be obtained. We also calculated the

F effective (3) dependence on Au

nanoparticle diameter in Au:BaTiO3 films under resonant excitation (at 600 nm), which is the solid line shown in Figure 13. The result closely indicates that although increase with a decrease in particle size.

F effective (3) does

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Guohong Ma, Jielong Shi and Qi Wang

Figure 1.13. The calculated effective F(3),

F effective (3)

, plotted as a function of gold nanoparticle

diameter at resonant (600 nm) and off-resonant (800 nm) excitation. Curve (a) is d-dependence of

F effective (3)

for Au:BaTiO3 film at 600 nm excitation; curves (b), (c) and (d) are d-dependence of

F effective (3)

for Au:BaTiO3, Au:ZrO2 and Au:SiO2 films with 800 nm excitation. For comparison,

curve (a) and (d) are multiplied by factors of 1/15 and 2, respectively; The arrows indicate the gold nanoparticle diameters corresponding to Figure 12 (a).

The dependence of optical nonlinearity on particle size in our case can be explained by local field enhancement arises from morphology resonance (MR). According to Lorentz-Mie theory, the scattering field distributions are strongly modulated by particle sizes. Such modulation of the scattering field distribution could in turns lead to a maximum in the local field intensity for selected particle size. An applied field with frequency far from the surface plasmon resonance may exert its influence by exciting the eigenmodes of the selected particles, and induces a local field enhancement. It is envisaged that such a fluctuating strong local fields will lead to an enhancement of optical nonlinearity. Optical nonlinearity in small metal clusters (diameter below 10 nm) has been attributed to intraband transition [35], interband transition [17] and hot electrons [33] contribution due to real optical excitation. Among these contributions, only the intraband transition is expected to be size dependent ( v 1 / d 3 ) because of quantum confinement as shown in Ref.35. For particle diameter above 10 nm, the effect of quantum confinement is progressively weak and the size dependence of nonlinearity will gradually depart from 1 / d 3 . When the excitation wavelength is far from any absorption resonance of the nanocomposite material, the nonlinear optical enhancement can only be derived from electrical-field-induced free electron anisotropy of metal nanoparticles. This fact can be further verified from the ultrafast and symmetric OKE response (Figure 2). More concisely, the ultrafast OKE signal originates from the contribution of free electrons, and is indirectly particle size dependent through local field dependence.

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301

In summary, surface plansmon resonance of metal nanocrystal takes an important role for describing the optical properties of metal-dielectric nanocomposite. When the working wavelength of laser is on resonance with SPR frequency, the great optical nonlinearity enhancement occurs. The large optical nonlinearity comes from the local field enhancement due to the sharp refractive index difference between the metal particle and dielectrics. When the working wavelength is far away from the SPR frequency of metal particles, the local field enhancement also works although the enhancement factor is smaller than that in the onresonance excitation. Materials working in off-resonance condition are essentially important for most of modern photonic devices. Our studies shows that optimization of nonlinear optical response can be attained by monitoring local field distribution. One special case in our present study is that optimized nonlinearity enhancement can be realized with particular particle size and matrices refractive index. In conclusion, metal-dielectric nanocomposite film is a strong candidate for fabricating various photonic devices.

2. Light Localization and Optical Nonlinearity Enhancement In Defective One-Dimensional Photonic Crystal Photonic crystals (PCs) and periodic structures in general, have one-, two- or threedimensional ordered structure with a periodicity comparable to the optical wavelength; have been studied extensively from both fundamental and practical points of view [43-48]. An important feature of these structures is the formation of a photonic bandgap (PBG) which strongly forbids the propagation of electromagnetic waves of certain frequencies [43]. Such a regular periodic structure may be disrupted by introducing some defects in the PCs [49-50]. The resulting composite system has attracted a fair amount of attention recently because defects in PCs can be manipulated to modify the optical properties of devices [45-48, 50-55]. In the past decades, various applications of PCs in electromagnetics and photonics have been proposed, such as enhancement of nonlinear optics [45-48], quasi-phase-matching technique [56,57], photonic-crystal fiber [58,59], photonic-crystal lasers and waveguides et al [53,54,60-63]. However, complete 3D PCs with a periodicity equivalent to the visible wavelength remain a technical challenge, therefore, 1D PCs have been given special attention because of their well known properties and ease of fabrication. One-dimensional photonic crystal (1D PC) may be regarded as a waveguide consisting of a sequence of dielectric mirrors with periodically modulated dielectric constants [43]. A defective 1D PC is constructed by inserting a defect layer into the center of such a multilayer structure. Introduction of defect permits localized modes to exit in the range of the frequencies of bandgap. [64] Owing to the highly localized field, strong enhancement of optical nonlinearity by several orders of magnitude can be expected in a defect layer [65,66]. If the defect layer includes nonlinear optical materials, it is envisaged that the nonlinearity may be substantially enhanced by the presence of such a strong electric field. In particular, defect modes that lie in the vicinity of the mid-gap of the PBG structure will have stronger effects on light localization and its associated optical nonlinearity [67,68]. As the incident field is intricately coupled to the local defect modes [66], the structure has emerged to be a promising candidate for many optical applications such as ultrafast all-optical switches and optical modulators [69-71]. In fact, since the first theoretical predictions of optical switching

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Guohong Ma, Jielong Shi and Qi Wang

and limiting in nonlinear periodic structure in the late 1970s [72], a lot of theoretical and experimental studies had been carried out [69-76]. For example, Larochelle et al first investigated optical response of nonlinear periodic structure by employing an optical Kerreffect cross-phase modulation in optical fiber grating [74]. Sankey et al first reported experimental observation of all-optical switching in a silicon-on-insulator periodical waveguide structure [75]. Scalora et al demonstrated optical limiting and switching can be realized in 1D PC doped with F(3) type of nonlinearity [72]. The Southhampton group first demonstrated switching at the optical communication wavelength of 1.55 Pm [76]. Many references can be found in a review paper by Brown and Eggleton [77]. In this chapter, I will focus on discussion on light localization, linear and nonlinear optical properties of defect mode in one-dimensional photonic crystal. We start from introducing the linear optical transmittance/reflection of 1D PC with proper defects, and then, we studied the nonlinear optical enhancement in defect state of 1D PC, and the optimization of the nonlinearity enhancement. Following that, I will touch on the discussion on the defect mode interaction in multi-defective 1D PC structure. Finally, but not end, I will extend our research interests on the optical properties of a metal-dielectric structure, a new kind of 1D PC.

2.1. One-Dimensional Photonic Crystal and Its Optical Properties One dimensional photonic crystal (1D PC), as shown in Figure 2.1, consists of alternating layers of materials with different dielectric constants. The well known Bragg mirrors widely used in modern technology is one example of 1D PC.

Figure 2.1. Schematic of the composition of 1D PC. n denotes for refractive index, the subscript H, L, D and S represent the high, low, defect and substrate, respectively.

This periodical structure is known to exhibit interesting light transmission and reflection characteristics. For example, in the stop band or more commonly the photonic band gap (PBG), certain frequencies of light are unable to penetrate these crystals because photon modes corresponding to those frequencies do not exist in the structure [43,44]. This is in analogy to the forbidden movements of electrons in the energy gap of an electronic band

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303

structure. Defective photonic crystal can be constructed by disrupting the periodical structure. By inserting a dielectric layer with different the dielectric constant from the intrinsic dielectric constant, for example, by introducing CdS layer in TiO2/SiO2 multilayer structure, defective 1D PC can be constructed. Figure 1 shows the schematic of the composition of 1D PC. The centered gray box denotes for defect layer. The periodical structures on both side of the defect layer represent pure 1d PC, the right hand part is substrate for supporting the 1D PC structure.

2.1.1. Transfer-Matrix Method In this section, we will employ transfer-matrix theory to simulate the optical transmission and reflection spectra of 1D PC, and the local field distribution in PC structure will also be discussed with the theory [79]. Light (plane wave, for simplicity) was incident from left side, the electric and magnetic field in air was E0 and H0, respectively. The transmitted electric and magnetic fields in the substrate matrix are ES and HS, respectively. We have [80]

ª E0 º «H » ¬ 0¼

ª cos G j « – j 1 « ¬iK j sin G j k

iK j 1 sin G j º ª Es º » cos G j »¼ «¬ H s »¼

(1)

the corresponding jth film has a 2u2 matrix Mj

Mj

ª cos G j « ¬«iK j sin G j

where, G j

iK j 1 sin G j º » cos G j ¼»

(2)

2S n j d j cos T j / O is the phase thickness of the layer, O is the wavelength of the

incident light, dj is it’s the physical thickness, Kj is the optical admittance of the jth film, with

Kj

­°n j cos T j ® °¯n j / cos T j

TE

polarization

TM

(3)

polarization

nj and Tj represent the refractive index and refraction angle in the jth film. As defining in Eq. (3), by introducing the optical admittance of air and substrate, K0 and Ks, the Eq.(1) can be written as

ªB º «C » ¬ ¼

§ k ¨ ¨– © j1

ªcos G j « «¬iK j sin G j

iK j 1 sn G j º ·¸ ª1 º » « » cos G j »¼ ¸¹ ¬K s ¼

>M 1 @>M 2 @ ˜ ˜ ˜ >M k @ª«

1 º » ¬K s ¼

(4)

M1 indicates the matrix associated with layer 1, and so on. If M1 to Mk are known, then the elements of B and C in left hand matrix can be determined. Then, the transmittance (T),

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Guohong Ma, Jielong Shi and Qi Wang

reflectance (R) and absorbance (A) of the multilayer stacks can be calculated with the following formula [80]

R T A

§ K 0 B  C ·§ K 0 C  C · ¸¸ ¸¸¨¨ ¨¨ © K 0 B  C ¹© K 0 B  C ¹ 4K 0 ReK s

*

(5)

(6)

K 0 B  C K 0 B  C *



4K 0 Re BC *  K s



(7)

K 0 B  C K 0 B  C *

It is known that the basic unit of a 1D PC is composed two dielectric layers with high and low refractive index, respectively. Supposed the matrix of the two layers is M1 and M2, the matrix of the basic unit of the 1D PC is

M

M 1M 2

ª cos G1 iK11 sin G1 º ª cos G 2 « »« cos G1 ¼ ¬iK 2 sin G 2 ¬iK1 sin G1

iK 21 sin G 2 º » cos G 2 ¼

ª M 11 «M ¬ 21

M 12 º M 22 »¼

(8)

the elements of the matrix are

M 11

cos G1 cos G 2  K11K2 sin G1 sin G 2

M 12

i (K sin G1 cos G 2  K cos G1 sin G 2 )

(10)

M 21

i (K1 sin G1 cos G 2  K2 cos G1 sin G 2 )

(11)

M 22

cos G1 cos G 2  K K sin G1 sin G 2

(12)

1 1

Assume

(9)

1 2

1 1 2

x

( M 11  M 22 ) / 2 , we have

M s U s1 ( x)M  U s2 ( x) I ( s

2 , 3," )

(13)

where, I is the unit matrix, and

U N ( x)

sin[( N  1) arccos x ] 1  x 2

(14)

According to the absolute value of the x=(M11+M22)/2, the bandgap width and cut off wavelength of the PC can be determined. Whenµ(M11+M22)/2<1, all wavelengths are falling beyond the bandgap, andµ(M11+M22)/2>1, all wavelengths are falling inside the bandgap. By using transfer matrix method, the electric field inside the 1D PC can be calculated directly. Without loss of generality, we suppose the 1D PC has the structure as shown in Figure 1, denoted as (HL)kD(LH)k, where H and L represent the high and low refractive index

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement…

305

layer, and D stands for defect layer, k is repeat unit. The calculation was carried out from right-hand (substrate, as shown in Figure 1), incident light with normalized intensity, E0=1, t is the transmission coefficient of the 1D PC. KS is the optical admittance of the substrate, then

ª t º » .The electric field in the PC is ¬K s t ¼

the electric field at the substrate can be written as «

ª Ek º «H » ¬ k¼

ª t º M k « » , after substituting into Eq.(13), we have ¬K s t ¼

ª Ek º «H » ¬ k¼

ª [U k 1 ( x)( M 11  M 12K s )  U k  2 ( x)]t º « » ¬«[U k 1 ( x)( M 21  M 22K s )  U k  2 ( x)K s ]t ¼»

(15)

the electric field at the defect layer can be expressed as

ª Ez º «H » ¬ z¼ with

ª cos G z « ¬iK D sin G z

Gz

iK D1 sin G z º ª Ek º »« » cos G z ¼ ¬ H k ¼

(16)

2S nD z cos T D / O , z is the distance from the point to the right interface, nD and TD

are refractive index and refraction angle at defect layer, KD is the optical admittance of defect layer. The average electric field for the whole defect layer can be expressed as

| E |2

1 dD

³

dD

0

| Ez |2 dz

(17)

with dD is the thickness of the defect layer.

2.2. Defect Mode Dependence of Optical Nonlinearity Enhancement in 1D PC Structure As discussed in previous part, defective photonic crystal can be realized by inserting a defect layer in the center of a multilayer stacks (Figure 2.1). It is expected that a localized optical mode appears in the photonic band gap (PBG) as a defect state, light can be strongly confined within the defect layer of the PBG when the light excitation is on resonance with the defect mode. The localization of light leads to an increase in the optical electric field in the defect layer. If the defect layer includes a nonlinear optical material, it is envisaged that the nonlinearity may be substantially enhanced by the presence of such a strong electric field. Many interesting optical effects, both linear and nonlinear, can be realized [48]. Optical devices, such as filters, optical switches and optical modulator, can be fabricated under this scheme. The occurrence of such resonating defect mode is strongly dependent on the optical thickness of defect layer for a fixed PBG structure. In general, the resonant frequency of defect mode exhibits a blue- or red-shift with decreasing or increasing optical thickness of

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Guohong Ma, Jielong Shi and Qi Wang

defect layer [64]. As the magnitude of electric field in the defect layer can be very sensitive to the optical thickness of defect layer, it can be expected that the nonlinearity of defect layer will show defect-thickness dependence. In this section, with femtosecond pump-probe method, we studied the optical thickness dependence of two-photon-absorption (TPA) coefficient of a CdS defect layer inserted into a four-period PBG structure. The results show that the TPA coefficient of CdS defect layer strongly depends on the defect mode of the photonic crystal, in consistent with the predicted dependence. By designing proper PBG structure and defect mode, Optimization of TPA enhancement can be realized.

Figure 2.2. (a) Optical transmission spectrum of 1D PC with a CdS defect layer thickness of 324 nm: the dotted line represents experimental results while the solid line is the simulation calculated according to the transfer matrix method. The schematic of the composition of 1D PC with a defect layer is shown in the inset; (b) Measured transmission spectra of 1D PC with a defect layer thickness at 273, 318, 324, 336 and 368 nm, respectively; (c) On-resonance defect mode vs defect layer thickness.

In order to study the defect-mode dependence of optical nonlinearity enhancement in a 1D PBG, a 1D PC with various defect layers was fabricated. All the photonic crystals used in this study consist of nine layers of dielectric thin films stack with a CdS layer in the center, as shown in the inset of Figure 2.2(a). TiO2 and SiO2 were chosen as the high and low refractive index dielectric materials in the PBG structure, respectively. The odd layers (1, 3, 7, 9) were TiO2 and the even layers (2, 4, 6, 8) were SiO2. The films were fabricated on a glass substrate by electron beam evaporation under oxygen atmosphere (2h10-2 Pa). The thicknesses of TiO2 layer and SiO2 layer were 99 nm and 151 nm, respectively. Refractive indices were determined to be 2.21 and 1.45 at 800 nm for TiO2 and SiO2, respectively. A CdS layer (layer

Electric Field Localization and Ultrafast Optical Nonlinearity Enhancement…

307

5, with different thickness) was deposited by thermal evaporation. For the TPA measurement, a single CdS layer with the same thickness was deposited on the glass substrate for comparison. The refractive index of CdS was found to be 2.26 at 800 nm. XRD results show that the CdS film has a cubic zinc blend structure. Fig 2-2 (a) shows the transmission spectrum in the range from 650 nm to 1050 nm for the photonic structure with a CdS layer thickness of 324 nm. The dashed line is the fitted curve which was calculated with transfer matrix formalism under the assumption of plane wave propagation [79]. The spectrum in Figure 2.2 (a) exhibits a broad rejection band ranging from about 750 nm to about 1010 nm, and a resonance transmission band peaking at 800 nm. The transmission peak is seen to correspond to the defect mode in the photonic structure. By increasing the thickness of CdS layer, the defect mode and hence the transmission peak is seen to shift correspondingly toward a lower frequency. Fig 2-2 (b) shows the measured transmission peaks for different thicknesses of the CdS layers at 274, 318, 324, 336 and 368 nm, which corresponds to the resonance defect modes at wavelength of 744, 792, 800, 812 and 847 nm, respectively. It is seen that the resonance defect mode wavelength is proportional to the thickness of CdS layer with a slope close to unity, as shown in Figure 2.2 (c). The series of transmission spectra further shows that when the thickness of the defect layer was varied, there is no significant change in the position of the photonic band gap. The observation is in line with the expectation that the bandgap is essentially defined by the periodical structure of the photonic crystal, and that the frequency of the localized mode is essentially determined by the optical thickness of the defect layer. Pump and probe experiments were carried out by using a Ti: Sapphire laser (Tsunami, Spectra-Physics) with a pulse duration of 200 fs, repetition rate of 82 MHz, and a center wavelength of 800 nm. The detailed description of pump-probe experimental arrangement has been shown in last chapter. The peak intensity of the pump beam at the sample position was about 2 GW/cm2. For comparison, a pump-probe measurement was also performed to determine the TPA coefficient for a single layer CdS film with thickness of 385 nm deposited on the blank glass substrate. The TPA coefficient E of the CdS film was found to be 5.8 cm/GW. The calibration was carried out with reference to a well-characterized sample. Here we used bulk CdS (thickness of 0.5 mm) as the reference crystal, with E reported to be 6.4 cm/GW [30]. It is seen that the CdS film has a slightly smaller E relative to that of bulk CdS. The difference may arise from the lower density of the film deposited by this method. Similar pump-probe measurement was carried out for samples with a single CdS defect layer. Figure 2.3 (a) shows the dependence of the change in transmission ('T) of the probe beam on the intensity of pump beam at zero delay time (t=0) for the photonic structure with 324 nm-thickness of CdS layer (corresponding resonant mode at 800 nm). It is seen that the change in transmission ('T) due to TPA process is proportional to the intensity of pump (Ipump) and probe (Iprobe) beams as well as the thickness (L) of film, i.e. [81]

'T ~ (1  R) 3 EI pp I pr L

(18)

where 'T is the TPA signal at zero delay time, E the TPA coefficient, R the reflectance of the sample, L the interaction length of pump and probe beam over the sample, and Ipp and Ipr the pump and probe beam intensities, respectively.

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Guohong Ma, Jielong Shi and Qi Wang

Figure 2.3. (a) Incident intensity (I) dependence of the transmission changes ('T) of probe beam at zero delay time in 1D PC with CdS defect mode at 800 nm (it was plotted as log'T ~ logI); (b) Defect mode dependence of TPA coefficient; dotted line denotes the TPA coefficient of pure CdS film. The inset is the transient time evolution of the three samples with resonant mode at 800, 812 and 847 nm, respectively.

The expected square dependence is indicated as the solid fitted line. Figure 2.3 (b) shows the dependence of TPA coefficient on defect mode for a pump beam incident irradiance of 2 GW/cm2, while the inset of Figure 2.3 (b) shows the transient time evolution of the three samples with resonant modes at 800, 812 and 847 nm, respectively. It is seen that the magnitude of E is substantially enhanced relative to that of the single layer CdS film when defect mode comes into resonant with the laser wavelength, while E remains small when the laser wavelength involved is far from the defect mode. In order to analyses the resonant wavelength-dependence of the TPA coefficient, matrix transfer method is employed for the calculation of the electric field distribution in the 1D PC structure. Figure 2.4 illustrates the field intensity distribution (_E_2) inside the 1D PC with different thicknesses of defect layer. It is seen that the electric field in the defect layer exhibits

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appreciable enhancement when the defect mode is in resonant or near-resonant with the laser wavelength. It should be mentioned that the electrical field in near-resonant mode at 812 nm is slightly larger than that of the on-resonant mode at 800 nm. This is attributed to the fact that the mode at 812 nm closer to the midgap of the 1D PC structure. Thus the total magnitude of electric field in the defect layer is determined by two factors, i.e. the resonant transmittance and the position in the rejection band. It is envisaged that the magnitude of the electric field can attain its maximum when defect mode is localized in the midgap of the photonic bandgap structure. It is seen that optical nonlinearity in a defective PBG structure is strongly defect mode dependent. The reason for the amplification of electrical field in the defect layer is entirely due to light localization. Therefore, optimization of optical properties (both linear and nonlinear) can be attained by designing proper resonant defect mode and proper PBG structure. For doing that, two types of defective 1D PC were fabricated.

Figure 2.4. Calculated defect mode dependence of electric field distribution within 1D PC structure with incidence wavelength at 800 nm. Light gray and white blocks represent TiO2 and SiO2 stacks, respectively. The dark gray block, from z=500 nm extending to 774 nm (minimum) and 868 nm (maximum), represents CdS defect layer.

The two types of PC structure are similar as shown in Figure 2.1. Also, TiO2 and SiO2 were chosen as the high and low refractive index dielectric materials in the PBG structure, respectively. The oxide films were deposited on a glass substrate by electron beam evaporation under oxygen atmosphere of 2u10-2 Pa, whereas the CdS defect layer by thermal evaporation. The four samples thus fabricated can be classified into two groups. Group A consists of two samples that have the same PBG structure but with different number of repeat unit (NOR). Sample PA-4 has a total of 9 dielectric layers comprising of 4 repeat units of TiO2 (thickness 90 nm) and SiO2 (thickness 138 nm) layers with a CdS (thickness 355 nm) defect layer in the center. Sample PA-8 has a total of 17 dielectric layers with 8 repeat units instead of 4. The structure of Group A samples was designed such that the defect mode is on resonant with the laser wavelength at 800 nm, and that the defect mode is positioned nearly at

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the mid-gap of the PBG structure. The two samples in Group B have different PBG structure from those of Group A in terms of thicknesses of layers. The thicknesses of TiO2 and SiO2 in Group B are 99 and 151 nm, respectively, while that of the CdS layer is 324 nm. Samples PB4 and PB-8 have the same stacking configurations as their group A counterparts. The defect mode is designed to remain on resonant with laser wavelength at 800 nm, but deviates from the mid-gap of the PBG. The refractive indices were determined to be 2.21, 1.45 and 2.26 at 800 nm for TiO2 , SiO2 and CdS respectively. The transmission spectra in the range from 600 nm to 1100 nm are shown in Figure 2.5(a) for PA and Figure 2.5(b) for PB. The fitting curves calculated with transfer matrix formulation under the assumption of plane wave propagation are also shown in the figures. The parameters for best-fittings are given in the caption of Figure 2.5. The spectra in Figure 2-5(a) exhibit a broad rejection band ranging from about 680 nm to about 930 nm, and a resonant transmission band is peaking at 800 nm which falls on the mid-gap of PA structure. The transmission peak within the PBG is seen to correspond closely to the defect mode in the photonic structure. By increasing the repeating unit from 4 to 8, it is clearly seen that the band edge becomes steeper and the bandwidth of the defect mode becomes narrower although the position of resonant mode remains unchanged. The defect resonant mode is determined solely by the optical thickness of the defect layer. The defect mode is expected to be blue-shifted with increasing thickness of the defect layer [68]. This behavior, due to multiple Bragg scatterings inside the 1D PC, is consistent with the expectation of transfer matrix formulation. By modifying the thickness of the dielectric mirrors in samples PB, the mid-gap frequency in the PBG is seen to be shifted correspondingly to 870 nm, as shown in Fig 2-5(b). By adjusting the thickness of defect layer to 324 nm, the defect transmission mode can be tuned to peak at around 800 nm so that it remains on resonant with the laser wavelength employed and provides a comparison with samples in group A. The TPA coefficient of defect CdS layer was characterized by pumpprobe technique as described before, (pulse duration of 200 fs, repetition rate of 82 MHz, and a center wavelength of 800 nm). Figure 2.6 (a) shows the transient time evolution of the four samples with a peak pump beam intensity of 2 GW/cm2. For comparison, a pump-probe signal of 0.5 mm-thick bulk CdS is also shown in Figure 6 (a), where the signal of the bulk CdS was multiplied by a factor of 0.1 for comparison. The ultrafast negative transmission change of the probe beam is seen to have arisen from the two-photon absorption in the CdS layer. In order to rule out any contributions from the TiO2 and SiO2 films, the pumpprobe measurement was repeated for homogeneous films of TiO2 and SiO2 as well as multilayer films of alternating TiO2/SiO2 with 4 and 8 periods respectively. No TPA signal was observed in all these films.

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Figure 2.5. The measured (solid line) and simulated (dashed line) optical transmission spectra of PA (Figure 5a) and PB (Figure 5b). The simulated curves are calculated based on transfer matrix formulation with fitting parameters: (a) dH=90, dL=138 and dD=355 nm; (b) dH=99, dL=151 and dD=324 nm. The refractive indices are nH=2.21-0.002i, nL=1.45-0.002i and nD=2.26-0.004i. The subscripts H, L and D represent TiO2, SiO2 and CdS, respectively.

It is seen from Figure 2.6 (a) that the TPA signal has increased by about threefold with increasing number of unit from 4 to 8 layers. It is also noted that the TPA signal depends critically on the 1D PC structure; it is significantly stronger for samples in PA than samples in PB. The pump irradiance dependence of pumpprobe signal at zero delay time is shown in Figure 2.6 (b) for both PA-4 and PA-8. Good fitting by straight line indicates that the signal increases linearly with the pump irradiance. The result is consistent with the prediction given by Eq. (18) [81]. In order to evaluate the TPA coefficient, E, for the CdS layer in the 1D PC, we compared it against the E for bulk CdS [30], the values of E for PA-4, PA-8, PB-4 and PB-8 were determined to be 60.1, 160.3, 19.3 and 74.6 cm/GW and the corresponding enhancements relative to the bulk CdS are 9.4, 25.0, 3.0 and 11.7 times, respectively. From these experimental data, we can conclude that the optical nonlinearity increases with the NOP of the 1D PC structure; and that the large optical nonlinearity can be reached when the defect mode falls in the center of the photonic bandgap.

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Figure 2.6. (a) Transient transmittance changes of the probe beam for samples PA and PB as well as 0.5 mm-thick bulk CdS (the signal of the bulk CdS was multiplied by a factor of 0.1 for comparison). (b) Pump intensity dependence of the transmittance change of the probe beam at zero delay time for PA-8 (square) and PA-4 (circle).

Enhancement of optical nonlinearity in 1D PC structure with defect is understood to originate from electric field localization within the defect layer [48, 65]. The intensity of local field increases with NOP because the number of trips of a beam of light trespassing the defect layer increases with the NOP of the dielectric mirrors. Similarly, for the same reason the defect mode localized at the mid-gap of PBG will experience a stronger local field than that for a defect mode departed from the mid-gap. In order to explain the nonlinear enhancement in TPA qualitatively, transfer matrix method was employed again to calculate the steady-state electric field distribution within these 1D PC samples at 800 nm (defect mode wavelength). In the calculation, for simplicity the extinction coefficients of TiO2, SiO2 and CdS were assumed to be zero, and the electric field amplitude of the incident light was set to be unity. The calculation shows that the amplitude of the electric field at defect layer increases with the repeating unit of the PC structure. On comparing the field distributions, between PA-8 and PB-8 as well as between PA-4 and PB-4, it is seen that large local field appears when the defect mode is localized at the mid-gap position. These results are consistent with the experimental observations. This strongly enhanced electric field is caused by the localization

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of incident light in the defect layer, and leads to an enhancement in the TPA coefficient as observed in our experiment. To evaluate the TPA signal enhancement relative to pure CdS film, based on Eq. (17) we can introduce an enhancement factor G defined as

G

1

d

E defect ( z )

d defect

0

Eincident

³

2

2

dz

(19)

where Edefect(z) is amplitude of field in the defect layer expressed as a function of position z, ddefect is the thickness of the defect layer, and Eincident is the amplitude of incident field. Figure 2.7(a) shows the repeating unit dependence of electric field intensity for both PA and PB structures. It can be seen from Figure 2.7(a) that two factors, namely the number of repeating unit and the PC structure, have significant influence on the enhancement of electric field intensity. The enhancement factor G in the defect layer shows an exponential increase with the NOP, and furthermore this dependence is seen to be more pronounced in PA structure where the factor G increases faster than that in PB structure. In fact, the factor G for PA structure increases exponentially with NOP at a gain rate of (nL/nH)2=0.418, while that for PB structure has a smaller gain rate of 0.315. The difference in gain rate can be attributed to a deviation of the defect mode from the mid-gap. With defect modes nearly fixed at 800 nm, Figure 2.7(b) shows the dependence of the factor G on the positions of mid-gap with 8 periods PC structure. The calculation was carried out for the configurations of quarter-wave stack where nHdH=nLdL=Omid/4 and for wavelengths Omid at mid-gap ranging from 720 to 880 nm. PA structure PB structure

15

(a)

(b)

14

100

13

G

G

12 10

11 10 9 8

1

7 0

2

4

6

8

10

12

Number of period

14

16

18

720

750

780

810

840

870

900

Midgap (nm)

Figure 2.7. (a) Number of repeating unit dependence of G factor for both PA and PB structures, these samples, PA-4, PA-8 as well as PB-4, PB-8 were indicated as arrows in the figure; (b) Calculated enhancement factor G at defect layer as a function of mid-gap position for 8 periods PC structure.

During the calculation, defect mode was remained at 800 nm, and wavelength of mid-gap was set to be Omid=4nHdH=4nLdL ranging from 720 to 880 nm with step of 10 nm. While n and d represents refractive index and thickness of dielectric stacks, subscript H and L represent TiO2 and SiO2, respectively.

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It is clearly seen that the factor G decreases with the deviation of defect mode from the mid-gap, and reaches a maximum when the defect mode is located precisely at the mid-gap. These experimental observations are consistent with the hypothesis that the intensely enhanced electric field is caused by the localization of incident light in the defect layer, which leads in turns to an enhancement of the TPA coefficient. In summary, we have successfully demonstrated an enhancement of the TPA coefficient for a single CdS defect layer imbedded in 1D PC with different defect modes, number of repeating unit and structures. The enhanced TPA coefficient is seen to have arisen from the localization of electric field within the defect layer. Our results clearly show that: (i) linear transmittance and nonlinear coefficient due to the defect strongly depends on the defect mode in the photonic band gap (ii) the larger number of repeat unit the greater the enhancement of the TPA signal; and (iii) the optimization of nonlinear coefficient of the defect layer reaches its peak when the defect mode falls on the center of the photonic bandgap. The experimental observation is further shown to be consistent with the calculation based on transfer matrix formulation.

2.3. Multiple Defect Modes and Their Interaction in 1D PC Structure Although photonic crystals (PCs) have been studied extensively from both fundamental and application points of view, some technical challenges still remain [43, 82-84]. One such challenge is the design of the controllable defect mode in the bandgap structure of two- and three-dimensional PCs with a periodicity equivalent to the visible wavelength. 1D PC system retains all the features of the PBG concept but may be implemented more easily than two dimensional (2D) and three dimensional (3D) systems. A 1D PC structure with proper defect mode is a promising candidate since the incident field is fully coupled to the local modes. Many photonic devices, such as modulators, filters, switches and reflectors et al can be fabricated quite readily in 1D PC system. In the previous section, we have discussed the defect-mode dependence of optical properties in 1D PC system. In fact, the ability to control the defect modes has been given special attention in the application of 1D PC system. In the present section, we will extend our studies on a 1D PC system with multiple defect states. Similar to a single-mode defective PC, a multimode defective 1D PC is constructed by inserting multiple defect layers into a 1D PC structure. For the sake of simplicity, our pure 1D PC (with no defects) structure consists of 8 units of alternating high (H) and low (L) refractive index layers, and is denoted as (HL)8. A defective PC can be constructed by inserting a multidefect layer (D) in the center of the pure PC structure. For instance, (HL)4D(LD)m(LH)4, here, m=0, 1, 2, 3… corresponds to single, double, triple, … defect modes in the PC structure, respectively. On the other hand, with a fixed value for m, the separation of the two defect states is dependent on the separation between the two defect layers. In the present work, we focus on two-defect mode PCs with the structure defined by (HL)4D(LH)nLD(LH)4, where n=0,1,2…. By adjusting the parameter n, the position as well as the separation of the two defect modes can be modified correspondingly. As done before, the transmission spectra of the multi-defect 1D PC were calculated with the transfer matrix method [79]. Without loss of generality, we assumed that the light falls at a normal incidence and propagates in the PC structure as a plane wave, and that the dielectrics are nondispersive and lossless. TiO2 and SiO2 were chosen as the high and low refractive index media and CdS

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was chosen as defect layer. The schematic of the composition of 1D PC with two defect layers is shown in Figure 2.8. The grey and white block represent TiO2 and SiO2 layers, and the dark blocks D1 and D2 represent CdS defect layers, respectively.

Figure 2.8. The schematic of the composition of 1D PC with two defect layers.

Figure 2.9 shows the simulated transmission spectra of 1D PC structures of (HL)4D(LD)m(LH)4 with m = 1 and 7. It is seen that the number of the defect modes in the photonic bandgap is equal to the number of defect layers. The split defect modes have approximately the same frequency intervals as the property of a comb filter [85]. It should be emphasized that there is no one-to-one correspondence of split modes to the defect layers in a PC. Each split mode, in fact, consists of contributions from all of the defect layers. Figure 2.10 shows the transmission spectra of the PC structures of (HL)4D(LH)nLD(LH)4, with n=0, 1, 2, 3, and 7. This is the case with a 1D PC structure with two-defect modes. It is seen that the mode separation can be modulated by changing the separation of the two defect layers. The frequency difference between two split modes diminishes with the increasing separation between the two defect layers. This is because the coupling coefficients of the two localized states are weaker when their separation becomes larger. When the separation of the two defect layers is sufficiently large, mode degeneracy is observed where the defect states merge into a single state as shown in Figure 2.10 with n=7 [85].

Figure 2.9. The simulated transmission spectra of 1D PC structures of (HL)4D(LD)m(LH)4 with m=1 [Figure 1(a)] and m=7 [Figure 1(b)], respectively. The refractive indices are nH=2.21 for TiO2, nL=1.45 for SiO2, and nD=2.26 for CdS around the wavelength of 760 nm. The respective thicknesses of the dielectric layers dH (TiO2) and dL(SiO2) are 86 and 131 nm, and the thickness of each defect layer dD (CdS) was set to 352 nm.

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Figure 2.10. The simulated transmission spectra of 1D PC structures with two defect layers, i.e., (HL)4DL(HL)nD(LH)4 with n=0, 1, 2, 3, and 7, respectively.

In the experimental study, two types of 1D PC samples each with two defects were fabricated, where one sample was designed with n=2 and the other n=3. These samples were fabricated with similar procedure as described in previous section. Characterization of an ultrafast nonlinear optical response was performed using a pump-probe setup consisting of a Ti: Sapphire laser (Tsunami, Spectra-Physics) with a pulse duration of 200 fs, repetition rate of 82 MHz, and a center wavelength of 800 nm. Figure 2.11(a) shows the measured transmission spectra of two-defect structure (HL)4D(LH)nLD(LH)4 with n=2 and 3, respectively. The two resonant modes are located at 765 and 805 nm for n=2, and at 772 and 800 nm for n=3, respectively. In the simulation the resonant modes are respectively located at 761 and 798 nm for n=2 (Figure 2.10, green line) and 767 and 792 nm for n=3 (Figure 2.10, blue line). The difference between the measured and simulated results may have originated from a deviation of the thickness of defect layers in the sample from the value 352 nm used in simulation. Because the defect layer was deposited with thermal evaporation, it is difficult to control the deposition rate precisely with this technique. The optical spectroscopy analysis indicates that the two defect layers have different thicknesses; one has a thickness of about 352 nm, and the other has a thickness of 362 nm. Ultrafast nonlinear optical response characterization was carried out on the sample with n=3, where the defect mode localized at 800 nm is on resonance with the wavelength of the laser output. Figure 2.11(b) shows the transient time evolution of the PC sample with the pump intensity of 1.1 GW/cm2 at wavelength of 800 nm. A pump-probe signal of 0.5-mmthick bulk ZnSe crystal is also plotted in Figure 2.11 with a red line, where the signal of the bulk ZnSe was multiplied by a factor of 0.1 for comparison. The ultrafast negative transmission change of the probe beam is seen to have arisen from the two-photon absorption (TPA) in the CdS layer. In order to rule out any contributions from the TiO2 and SiO2 films, the pump-probe measurement was repeated for homogeneous films of TiO2 and SiO2 as well as multilayer films of alternating TiO2/SiO2 with 8 periods. No TPA signal was observed in all these films. As a reference, TPA coefficient E of bulk ZnSe is reported to be 3.5 cm/GW at wavelength of 780 nm [30]. The E value of CdS layers in the 1D PC structure was calibrated

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as 307 cm/GW. Thus the enhancement relative to the bulk CdS is about 48 times (E=6.4 cm/GW is reported at wavelength of 780 nm [30] for bulk CdS). It is well known that such enhancement of optical nonlinearity in 1D PC with defect originated from the high electric field localization within the defect layers.

Figure 2.11. (a) The measured transmission spectra of 1D PC structures of (HL)4DL(HL)nD(LH)4 with n=2 (solid) and 3 (dashed), respectively. The arrow indicates the wavelength of the pump beam in the experiment. (b) Transient transmission changes of probe beam for (HL)4DL(HL)3D(LH)4 PC structure (solid) and bulk ZnSe (dashed) with the incident wavelength of 800 nm. The signal of ZnSe was multiplied by a factor of 0.1 for comparison. The pump intensity is about 1.1 GW/cm2 at the wavelength of 800 nm.

Taking the linear refractive index n0 = 2.26 and the thickness of CdS layers in the PC structure to be 704 nm (352u2 nm), n2 can be calculated from n2='n/I0=(n-n0)/I0. Here, I0 denotes the light intensity of the incident pump beam, which has a value of about 1.1 GW/cm2. The value of n2 of the CdS layers is calculated to be 3.9u10-3 cm2/GW. Compared with the reported value of n2 in bulk CdS of 7.9u10-5 cm2/GW [30], n2 of CdS layers inside

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the PC have achieved an enhancement greater than 48 times. This enhancement factor is almost the same as that in the TPA measurement mentioned previously. Again, such a large nonlinear refractive coefficient inside the defect layers is seen to have originated from light localization in the defect modes [86]. It should be pointed out that the large value of n2 of the CdS layer is obtained from the formulation n2='n/I0=(n-n0)/I0, where I0 denotes the light intensity of incident pump beam instead of light intensity I inside the CdS layer [86-87]. Due to the local field enhancement, the light intensity inside the CdS layer, I, is much stronger than that of incident light, I0. In fact, if we define an effective nonlinear refractive index as (n2)eff=(n-n0)/I, we can find that (n2)eff remains exactly the same value as in bulk status, i.e., 9u10-5 cm2/GW . The same conclusion can also be applied to the discussion on the enhancement of the TPA process mentioned before.

2.5. Optical Properties in a Metal-Dielectric Multilayer Structure In recent year, it has been realized that metal-dielectric multilayer structures behave a high transmission within a certain controllable spectral range due to multiple Bragg reflections, even when the total thickness of metal significantly exceeds the conventional skin depth [8889]. The transmission window can be tuned by adjusting the thickness of metal and dielectric layers, or by changing the number of metal layers [88-91]. This artificial structure provides another candidate material for fabricating all-optical devices. In this section, we will focus our interests on the linear and nonlinear optical response of a metal-dielectric multilayer structure. By designing the thickness of metal and dielectric layers, the transmission window can be tuned to be on-resonance with the laser wavelength, 800 nm. By employing femotsocond optical Kerr effect measurement, enhanced nonlinear optical response was investigated. Our results demonstrate that metal-dielectric composite structure also shows a suitable candidate for fabricating ultrafast photonic devices. The alternative Ag/TiO2 multilayer films were fabricated by alternating sputtering of Ag and TiO2 in a multitarget rf-sputtering system. The detailed processes are described before. The final sample is consisted of alternative TiO2 (5-layer) and Ag (4-layer) films. The thickness of each TiO2 and Ag layer were determined to 310 and 25 nm, respectively, by second ion mass spectrometer. The sample is then annealed at a temperature of 400 0C for 3 hours in the presence of Ar atmosphere. The annealed film shows a better stability and the damage threshold can be improved as high as 5-6 GW/cm2. The measured transmission spectrum in the range from 500 to 1100 nm is shown in Figure 2.12. It is seen that the transmission spectra remain almost same in the range of 500 to 1100 nm before and after annealing, indicating the annealing process at temperature of 400 0C does not affect the multilayer structure. But after annealing, the damage threshold of the composite films can be improved greatly compared to the pristine film. It can be seen from Figure 2.12 that the transmission at 800 nm is about 35% for annealed film, and two side peaks with relatively low transmittance peaked at 745 nm and 875 nm, respectively. In order to understand the wavelength-dependence of transmission, a transfer matrix formulation is employed to simulate the transmission spectrum of the multilayer structures [93]. During the calculation, plane wave propagation was assumed for simplicity. The calculated spectrum is also shown in the figures.

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Figure 2.12 Transmittance spectra of alternative 5-layer TiO2 and 4-layer Ag multilayer structure, with the thickness of TiO2 and Ag being 310 and 25 nm, respectively. The solid and dashed lines represent the pristine multilayer film and its annealing at 400 0C under Ar environment for 3 hours, respectively. The theoretical simulation spectrum is shown in dotted line. Duration calculation, real, n(O), and, imaginary, k(O), components of refractive index for Ag film were referred to ref. [92], and refractive indices of TiO2 films and substrate glass were assumed to be 2.21 and 1.52, respectively.

It is seen that the numerical calculation can reproduce the transmission spectrum around 800 nm. But the side peaks on both sides of 800 nm are not reproducible, especially large discrepancy occurs at the short wavelength region. The reason of the discrepancy between the simulation and experimental measurement may be caused by i) the blur interfaces between Ag and TiO2, ii) variations of deposited layers from the nominal thickness, and/or 3) light scattering due to the non-uniform distribution of the film. It should be mentioned that if all the four-layer Ag films are stacked together to form a 100 nm-thickness film, the transmission at 800 nm is about 1.5%, even for 25 nm-thickness-Ag film, the linear transmission at 800 nm is as low as 7 %. The relative high transmission in the multilayer structure indicates that a good fraction of light can still penetrate into the inside of the metal film, and it is expected that the contribution to nonlinear optical response from metal film can be greatly enhanced. Nonlinear optical response of the multilayer films is characterized on OKE setup with laser wavelength centered at 800 nm. As shown in Figure 2.13 (a), the OKE signal shows an ultrafast rise followed by a slow recovery process. In order to confirm the originality of the OKE signal, the experiment was repeated on the pure TiO2 film with thickness of about 1.5 Pm, the corresponding OKE signal shows an almost symmetric response with FWHM is same order of the laser pulse duration. And the magnitude of the OKE signal in pure TiO2 film is more than one order smaller than that in the multilayer films. Under same experimental condition, we also performed OKE measurement on 25-nm-thickness Ag film, as shown in Figure 2.13(a) in dashed line, although the thin film show quite good signal-to-noise ratio, the corresponding signal is more than one order smaller than that in the composite film.

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Figure 2.13. (a) Optical Kerr responses of the multilayer structure (solid line) and 25-nm-Ag-film (dashed line), and (b) 1-mm-thickness bulk ZnSe single crystal.

Consider effective thickness of Ag film in multilayer structure, the average signal in the multilayer film is seen to improve by a factor of about 3 on comparing with the single layer Ag film. Furthermore, the experiment was also repeated on the pure Ag film with thickness of 100 nm, and no clear signal was detected. Since OKE signal comes from the pump-field induced refractive index change, we believed that the large OKE response in the multilayer film is contributed by the specific structure. By referring to a reference sample, ZnSe crystal [30], the magnitude of F(3) in the composite film was calculated to be 3.2u10-9 esu, and that of the 25-nm-thickness Ag film was determined to be 1.1u10-9 esu. It is seen that the OKE recovery processes have two components, a fast process with time constant of ~2 ps and a slow process with lifetime more than 100 ps. These processes reflect the photo-induced refractive index change and its relaxation. The excitation at 800 nm in our experiment is far away from the surface plasmon resonance in Ag film, which is localized around 400 nm according to previous reports [94]. Pump-probe experiment with parallel polarization configuration was employed to study the electron behavior in the composite

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films. An ultrafast photobleach followed by a biexponential recovery process was detected, which shows a very similar temporal profile as in OKE measurement. We can conclude that the fast component in OKE response reflects the thermal equilibrium process between the electron system and the lattice system in the metal film, and the slow component of OKE profile arises from the cooling process by a thermal diffusion from the metal to the host matrix [4]. In order to analyze originality of nonlinearity enhancement in the composite multilayer structure, matrix transfer method is employed again for the calculation of the steady-state electric field distribution in the metal layer at wavelength of 800 nm. In the calculation, for simplicity the electric field amplitude of the incident light was set to be unity. The square of electric field amplitude (|E|2) distribution within the multilayer structure is shown in Figure 2.14 (a). On comparing the field distributions, we also calculate the electric field distribution in 100-nm-silver-film as well as 25-nm-silver-film (Figure 2.14 (c)). It is clearly seen that the amplitude of the electric field of silver layer in the multilayer structure is much stronger than that in pure silver film. This strongly enhanced electric field is caused by the large penetration depth of incident light in metallic layer, and leads to large Kerr response as observed in our experiment. In conclusion, metal-dielectric multilayer structure with proper structure also shows an enhanced optical nonlinearity. The originality of the large nonlinearity arises from the reason that the light can penetrate into the highly nonlinear metallic layers. Our studies show that metal-dielectric multilayer structure is a strong candidate for fabricating nonlinear photonic devices. Finally, we would mention that material requirements for the fabrication of all-optical switching devices based on waveguide structures with the exploration of nonlinear phase changes. To evaluate the material requirement for all-optical switching devices, one often introduces one-photon and two-photon figures of merit, W and T, respectively. The two figures of merit are defined as W=n2I0/(DO) and T=EO/n2 [95-97], where I0 is the light intensity outside the material, D is the one-photon absorption coefficient due to linear absorption and scattering, E is two-photon absorption coefficient, and O is the working wavelength in air. For realization of all-optical switching devices, materials with W>1 and T<1 have to be met. Obviously, in order to fabricate nonlinear photonics, materials with large n2 and small D and E are required. For the case of metal-dielectric nanostructure, two-photon figure of merit with T<1 can be reached in most of metal nanostructure. When the laser wavelength is on resonance with SPR absorption of the metal composite, large linear optical absorption occurs even for very thin composite film. Although magnitude of n2 of the composite film is as large as ~1 cm2/GW due to large local field enhancement, the one-photon figure of merit W is still less then one, i.e. W<1. For off-resonance excitation, the linear absorbance in a thin film can be as low as several cm-1, it is easy to realize the condition W>1. In the case of 1D PC with defect layers, let’s consider the case for the Figure 2.11 with two defect layers. Considering ~50% transmittance at defect mode of 800 nm and the total thickness of 3232 nm for PC films (Figure 2.11(a)), the value of D was calculated to be 2.14u103 cm-1. With other data, n2=3.9u10-3 cm2/GW, E=307 cm/GW, I0=1.1 GW/cm2, and O=800 nm, T and W were calculated to be 6.3 and 0.025, respectively. Although our data show that our samples are not yet met to the requirements of all-optical switching devices, it is expected that this target can be achieved by choosing proper materials.

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Figure 2.14. Calculated square of electric field amplitude (|E|2) distribution within the multilayer film (a) and |E|2 distribution within the 100 nm- (solid) and 25 nm- (dashed) Ag-film (c) with incidence wavelength at 800 nm; (b) is the expansion of the circled region of (a). The incident electric field amplitude is set to unity during the calculation.

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The condition of W>1 can be met by reducing light scattering in the PC structure. In a high quality PC structure, the transmittance at a defect mode can be as high as 90%, and then the W>1 can be achieved. The condition T<1 can be met by choosing a Kerr material in which the value of E is as small as possible at the working wavelength; for example, some polymers or wide gap semiconductors such as ZnO and ZnS etc.

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In: Progress in Nonlinear Optics Research Editors: M. Takahashi and H. Goto, pp. 327-358

ISBN 978-1-60456-668-0 c 2008 Nova Science Publishers, Inc. 

Chapter 9

M ONOCHROMATIC WAVEFIELD E VOLUTION IN WAVEGUIDE A RRAYS WITH G AIN AND N ONLINEARITY Anatoly P. Napartovich and Dmitry V. Vysotsky State Research Center for Russian Federation Troitsk, Institute for Innovation and Fusion Research, Troitsk, Russia

Abstract Wavefield propagation in an array of parallel waveguides exhibits a wide variety of nonlinear phenomena. The arrays of passive waveguides with the refractive index nonlinearity in cores were analyzed by the well developed instruments of the perturbation theory. The effect of capturing the wave field in one of the waveguides was suggested for design of optical switches. Modeling the wavefield propagation in active waveguides in the strong nonlinearity condition attracts special attention last years due to the development of the high power fiber lasers and amplifiers. Such devices contain the lattice of active cores, so the wavefield can be expanded on a small number of guided modes. Both the refractive index and gain depend on the wavefield intensity. Doping of the core material by rare earth ions can increase the nonlinear coefficient of the refractive index by several orders due to polarizability difference of ions in different states. The gain disposition in the cores changes dramatically the amplification of monochromatic wave field. The laser radiation self-synchronization at pump increase was demonstrated experimentally with 7-core fiber laser. Numerical modeling by 3D diffraction code has shown the crucial role of the gain nonuniformity for this phenomenon. The evolution of linear combination of two optical modes in the fiber amplifier has been analyzed. Due to cross-modal gain power of the mode with greater start power grows linearly and can limit the lower-power modes. The single mode lasing establishes if distributed losses are supposed for laser simulation. The mode selection in the ring lasers with strong cross saturation or in the random lasers with strong scattering are the nearest analogies to this effect. The behavior of the monochromatic wavefield is studied for different constructions of the multicore fiber amplifier. The conditions for a given mode domination are found for the twin waveguides amplifier. Possible usage of the developed approach is discussed for the single mode output ensuring in single core fiber amplifier.

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1.

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Introduction

Optical waveguides provide unique environments for nonlinear optics, because of the combination of high intensities, long interaction lengths, and control of the propagation constants. In designing the waveguide one has freedom to engineer the optical response over a broad parameter space, so as to enhance or suppress such optical effects as stimulated Raman scattering (SRS), four-wave-mixing, self-focusing or soliton formation. An excellent introduction to nonlinear optical processes in optical fibers is the textbook by Agrawal [1]. Latest studies of nonlinear effects in optical waveguides are reviewed in [2]. Wave field propagation in an array of parallel waveguides leads to a wide variety of nontrivial physical phenomena, thus making them very appealing for research. Hystorically, passive evanescently coupled optical waveguides and multicore fibers (MCF) were in the center of attention as important photonic devices capable of manipulating the lightwave. Discrete character of the media in the waveguide arrays results in diffraction effects, which are quite different from the light behaviour in homogeneous media. Among such phenomena are diffraction reversal and cancellation [3] and lattice solitons [4]. The wellknown effect Talbot effect: self-reproduction of the spatially periodical wavefield at integer multiples of so-called Talbot distances, occurs in discrete lattice only for distributions with periods {1, 2, 3, 4, 6} lattice elements [5]. Bending of the waveguides allows to manage the frequency dependence of the coupling constant. As a result, an array construction was calculated and manufactured [6], in which the Talbot distances were the same in large spectral length and multicolor Talbot effect was demonstrated. The method commonly used in describing guided field evolution in such passive systems is the coupled-mode theory [7] (CMT), in which the wave field is presented as a sum of sole waveguide modes with coefficients depending on time and propagation distance. The equations for these coefficients are derived by the perturbation method. CMT is intuitively evident and can be easily extended to new physical effects [8]. The simplest evanescently coupled array is a pair of collinear waveguides, which was analyzed by the CMT for the case of Kerr nonlinearity of the core index in [9, 10]. This theory was generalized by Hardy and Streifer [11], who introduced nonidentical waveguides and vector field effects. Recently twisting the pair of waveguides as a means to manipulate the coupling strength was analyzed in [12] by the CMT. Optical switch development is the practical result of these studies. This device is the 2x2 fiber coupler with long twin-core  section. One of the inputs was seeded by powerful beam, which provides high intensity ∼ 1 GW/cm2 in the cores resulting notable distortions of the core refractive index. If the second input is idle then the wavefield oscillates between the cores with constant period. The core index distortions due to nonlinearity can enlarge the beating length up to capturing the total power in one of the cores. A signal in the second input channels in this situation can change dramatically the power distribution between two output channels for given coupler length. Thus, the high power beam is switched between two outputs by small governing signal. Theoretical and experimental studies of optical switches including interaction of wave fields with different polarizations and frequencies in systems of two and three evanescently coupled waveguides were summarized in [13]. The CMT approach was recently implemented for analysis of dynamic stability for an array of fiber lasers with a separate coupler [14]. Using a model coupling matrix, Peles et

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al. [14] explained the robustness of the phase synchronization in such systems, studied in the experiment [15], provided special asymmetry is introduced into the construction. The CMT is analogues in quantum mechanics to tight-binding approximation. More general approach, which considers the Schroedinger equation for periodic potential, reveals many phenomena analogous to that known in solid-state physics. For example, adding a transverse index gradient in the array has led to the observation of the optical analogue of the Bloch oscillations of a charged particle in a periodic potential subjected to a dc electric field [16]. Periodic axis bending of the array results in the phenomenon of dynamic localization in the presence of an ac field [17]. All-optical Landau-Zener tunneling [18] has been demonstrated in a one dimensional array of liquid crystalline waveguide and explained on the basis of Floquet-Bloch modes. Another example, which needs the true array modes for analyzing, is the array of active cores, which has been suggested in [19] for multicore fiber (MCF) lasers and amplifiers. Different types of MCF constructions are considered in sec. 2.. It is shown in sec. 3., that the major factor governing the modes competition of the monochromatic wavefield in the MCF amplifier is the cross-gain, which is proportional to the product of the mode field profiles with the gain distribution integrated over the fiber cross section area. The results of two mode competition in fiber amplifier are discussed in sec.4. in an arbitrary fiber amplifier and in sec. 5. in the model constructions with two or three ultrathin planar waveguides. It is shown in sec 6., that the cross gain effect ensure the amplification with single mode output in the planar waveguide and the possible effects destroying the single spatial mode propagation are briefly discussed in sec. ??. The main results are reviewed in conclusion.

2.

Waveguide Arrays in Fiber Lasers and Amplifiers

Rare-earth-doped single transverse mode fiber lasers pumped by low cost diodes have attracted considerable attention in commercial and military applications. With the development of high-power laser diodes and pumping technologies, the output power of doubleclad fiber lasers with multi-stage amplifiers is able to reach 2 kilowatts in continuos-wave (CW) regime at good quality output beam [20]. The physical processes in different materials, which provides the rare-earth-doped fibers lasing, are deeply studied in [21]. The Raman and thermal effects restrict laser performance in a kilowatt power domain. The SRS in fibers can convert part of the signal into the undesirable Raman-Stokes waves at longer wavelengths [1], and hence degrades the laser efficiency. The pump-induced heating can cause such problems as degradation of laser beam quality due to thermal lensing, formation of thermal cracks due to thermal expansion, shortening of fiber lifetime, even melting of the glass. Increasing the effective mode area and decreasing the fiber length increase the threshold power for these processes in the fiber (A theoretical analysis of Raman and thermal effects in kilowatt ytterbium-doped double-clad fiber lasers is presented in [22]). The core size is limited by the condition the core to guide single mode: V = 2πaλ−1(2n0 Δn)1/2 ≤ 2.4, where V is the waveguide parameter, a is the core radius, λ is the lasing wavelength in vacuum, n0 is the cladding refractive index, and Δn is the core-cladding refractive index difference, which cannot safely be made less than 0.001. This limits the single mode core size by a < 7λ. Various techniques have been implemented to preserve single-mode output in large-core fibers: selective bend-loss in coiled multimode

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refractive index (a.u.)

330

cores 'n

cladding

n0 coating

radius (a.u.) Figure 1. Schematic of the multicore fiber index profile. (MM) fibers [23], engineering of radial index and dopant profiles in MM fibers [24], use of photonic crystal fibers [25], and selective bend-loss in helical-core MM fibers [26]. Recently, close to diffraction limited output beam after propagation through a 20 cm length of 300 μm core MM fibre (V = 236.8) was achieved by the seed conditions control [27]. The MCF laser, in which several single mode cores are placed into the common pump cladding, is an alternative way to increase the effective mode area.

2.1.

Weakly Coupled Waveguides

MCF lasers first suggested in [19] contain a number (18–61) single mode cores placed on the circle inside the MM cladding (see fig. 1 ). More simple constructions contain only two [28] or four [29] microcores. Hexagonal lattices of cores inside the common cladding are now studied due to well developed technology of manufacturing [30]. The active lattices up to 37 waveguides in common fibers are now available [31]. The numerical code [32] based on the three-dimensional (3D) beam propagation method (BPM) has been developed for modeling the MCF laser used in [19]. This code solves Cauchy problem for scalar parabolic equation of difraction optics using the split-step Fast Fourier Transform technique [33]. A gain-refraction step of the split-step scheme represents a set of transcendental equations which can be solved iteratively. The wavefield backreflection from the cladding/coating boundary was neglected and the absorbing boundary conditions [34] were used at the boundaries of the numerical domain. The results of 3D BPM calculations have confirmed [35] the applicability of the CMT for the MCF amplifier with weak coupling. The MCF design has an advantage over single-core large mode area constructions only if the fields in the cores are phase locked, i. e. the fields are coherent between each other and the phase differences between them are stable (the best situation is these difference to be zeroed). Due to scattering of properties of individual cores, the propagation constants in different cores are different, which results in optical lengths spread for the radiation in different cores much larger than the laser wavelength. The weak coupling or practically

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Figure 2. Cross-section of the ribbon fiber and profiles of effective index and gain. 1 is the cladding; 2 is the ribbon waveguide, 3 is the core. [44]. independent propagation of the wavefields in the cores prevents highly spatial coherent radiation. In order to select a single array mode an external strong optical coupling is required. A number of methods has been proposed to stabilize the single-array-mode operation of the MCF laser by virtue an external optics: plane or spherical mirror [36, 37], sectoral mirror [38,39] or 2x2 couplers [28]. The combination of plane mirror with annular waveguide, which allows to reduce diffraction losses for the inphase mode, was suggested in [40]. In all these methods the in-phase mode for identical core array has a smallest losses in external filter in comparison with other modes and independent lasing. As a result, the frequency of the real MCF laser self-tunes inside the gain spectral range to minimize these losses [41] and the field distribution over the array is close to the in-phase mode. Last years the methods to couple cores in all-fiber construction have been suggested and checked experimentally [29–31]. The drawbacks of using external coupling is design complications, temporal unstability and problem with quality of output beam. Phase locking can be provided also by diffractive exchange of fields between the cores if the coupling between them is strong. In this case the CMT is not applicable and it is necessary to analyze the competition of the array modes.

2.2.

Antiguided Arrays

Numerous studies of phase-locking the waveguide arrays in semiconductor lasers [42] have shown that the evanescent wave coupling between the neighboring waveguides is not sufficient for stable phase locking. The better way is to use leaky wave systems, in which the refractive index is uniform or the gain is localized in the periodically spaced channels with smaller refractive index (antiguides). The fiber laser containing inside the cladding the ribbon waveguide with periodically modulated gain was suggested in [43] and is schematically depicted in fig. 2. Authors of ref. [43] have concluded that the construction with the refractive index modulation less 0.001 is the most effective one. However, such small index modulation amplitude is vulnerable to random variations of the refractive index and to

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*

*res *adj

1-V -2

Figure 3. Gain overlap integral of the resonant (inphase) mode and all the other modes vs. the construction parameter 1 − σ −2 for m = 3/2. thermal lensing. The analysis of the ribbon construction by the methods used for the antiguiding arrays of diode lasers has shown [44] the possibility for single mode lasing in the 1D antiguided array with large number of elements if the resonant condition is satisfied. for the index step inside the ribbon Δn = n1 − n0 and the widths of guiding (s) and antiguiding (d) regions: (1) 4m2s−2 − d−2 = 8n0 Δnλ−2 , where 2m = 1, 2, 3 . . .. Condition (1) means ceasing the band gap in the spectrum of the Bloch waves propagating across the array. The corresponding array mode (called resonant) has the fields in nearest antiguides in-phase if m = 1/2, 3/2, . . ., and out-of-phase if m = 1, 2, . . .. In the construction considered there is a non-zero difference in the field overlap with the gain between the resonant mode (in-phase or out-of-phase) and all other array modes. This difference does not depend on the number of antiguiding gain regions, N, until this number is less than critical one 





−1

Nc = 4πλ (2m + σ) (2m + σ 3 ) n0 gsd σ 2 − 1

,

where σ = 2md/s is the ratio of the maximum field amplitudes in the guiding and antiguiding regions for the resonant mode. Typical values of Nc are ∼ 103 for fiber laser constructions while being less than 100 for typical diode laser arrays. The dependences of the gain overlap are illustrated in fig. 3 for m = 3/2. The ribbon fiber laser with uniform index profile and five Nd-doped cores has demonstrated two-mode operation in the experiment [45]. Coherent propagation of radiation was observed in 20 cm length fiber with linear array of 16 antiguided cores [46]. Practically, the ribbon fiber construction turn out to be difficult for manufacturing due to large difference of size in two transverse dimensions. Additionally, the antiguide construction is very sensitive to distortion of the resonant condition (1) and relies on three types of glass material. Disposition of the cores with axial symmetry as in [19] and, especially, in hexagonal lattice as

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in [30] is more suitable in fiber technology. Besides, evanescent wave coupling does not use resonant wave propagation or reflection, which is vulnerable for core lattice deformations produced by the fiber amplifier manufacturing.

2.3.

Evanescently Coupled Arrays: Strong Coupling

The coupling strength reduces exponentially with distance between the cores in the case of evanescently coupled array. However, the coupling of the field in the nearest cores can be strong if the field outside the cores is comparable with the field inside them. For this reason the best synchronization is expected in hexagonal lattices containing 7 or 19 elements, in which the cores are packed in the densest structure and maximum distance between cores in the array is minimal. The fiber laser containing 7-core hexagonal lattice of strongly coupled cores has exhibited in the experiment [47] the far-field patterns typical for phase-locked operation at power levels more than 100 W. An analogous 19-core fiber amplifier [48] has demonstrated 20 dB gain at near to diffraction limit beam quality. Single-mode cores can be arranged into the hexagonal lattice also in the form of defects doped by rare-earth in photonic crystal fiber lasers [49]. The fibers used in the experiments [47, 48] contain, respectively, 7 and 19 evanescently coupled cores doped by Yb and having diameter 7 μm, the distance between the cores is 10.5 μm. The core-cladding index step is Δn = 2.57 × 10−3 in [47] and Δn = 1.54 × 10−3 in the construction [48]. The respective values of waveguide parameter are V = 1.73 and 1.4. The observed self-synchronization has been ascribed in [50] to the influence of the resonance refractive index non-linearity. The calculation relied on the CMT and on the assumption about self-focusing nature of non-linear refractive index associated with an intensity dependence of the Yb3+ ions population in the excited states. It was shown later [51] in terms of the CMT too, that the self-organization occurs even if the nonlinear refractive index is negative. It should be mentioned, that the actual dependence of the refractive index on the light intensity is determined in Yb-doped glasses by many factors. It is known from the studies of the rare-earth-doped crystals, that besides refractive index changes induced by the thermal heating, there is comparable or even prevailing distortion of the refractive index due to the polarizability difference between excited and ground states of the Yb3+ ions [52–54]. The measurements of nonlinear refractive index Yb-doped phosphate glass fibers [55] show that n2 (λ) is nearly constant far away from the resonance at λ = 975 μm. Bochove et al. [50] have found in modelling the Yb-doped silica glass in frame of a simplified twolevel model that n2 for the signal wavelength 1.1 μm changes a sign at the pump intensity level 40 kW/cm2. They also derived numerically estimation of the maximum value of nonlinear refractive index for the experiment condition |n2 | = 10−16m2 W−1. However, in calculating the index variation in visible and near infrared (IR) regions it is necessary [56] to take into account the ultraviolet (UV) transitions, although these transitions are never actually excited by the pump wavelength. The reason for the large contributions of the UV transitions is that the IR bands represent transitions within the same 4f electron shell, which are only weakly allowed quantum mechanically, and the corresponding oscillator strengths are ∼ 10−5. The UV bands represent transitions between 4d and 5f shells, which are strongly allowed, with oscillator strengths near unity. Besides, Arkwright et al. have mentioned in [56], that variation of the host fiber composition can significally modify and

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even change the sign of the induced index change. For this reason the value of the refractive index quadratic nonlinearity, which was calculated in [50], was considered rather as the maximum value in subsequent numerical modelling. Direct numerical modeling of the wavefield propagation in the seven-core amplifier has been made in [57] using the BPM program developed in [32]. The initial distribution was taken as a combination of sole core modes injected in the cores with user-defined complex coefficients. The gain saturation was described by the Rigrod formula g = g0/(1+I/ISat). Waveguide parameters were taken from [50, 53]: the small signal gain g0 = 0.26 cm−1, that roughly corresponds to ions concentration 7.7 × 1018 cm−3 and the pump intensity 48 kW/cm2 . The pump depletion along the fiber was neglected. The saturation intensity and quadratic nonlinearity of the refractive index inside the cores were defined as ISat = 64.4 kW/cm2 and n2 = 2 × 10−12 cm2W−1 . In addition, calculations were performed, where the gain or the refractive index nonlinearity were turned off. The initial distribution was taken with phase dispersion 0.3 radian among the microcores. It was demonstrated in [57] that in presence of both the gain and nonlinearity the phase dispersion in the array diminishes with the decay length 120±10 cm. If the nonlinearity was turned off (n2 = 0) the decay length enlarges to 146 ± 15 cm. In the absence of the gain and nonlinearity n2 = 2 × 10−12 cm2 W−1 the decay rate diminishes to the values beyond the model accuracy limit. Thus, the analysis [57] has shown that the gain nonuniformity in the system is the major factor responsible for the effect, while the role of refractive index nonlinearity is subsidiary. If the refractive index is diminished to Δn = 1.27 × 10−3, that corresponds to the waveguide parameter V = 1.22, then the decay length decreases to 46.9 ± 1 cm if n2 = 0, and 41.4 ± 1.3 cm if n2 = 2 × 10−12 cm2 W−1 . Later ( [58]), the self-synchronization was demonstrated to preserve also in the case of random variations of the core parameters. The core-cladding refractive index difference across the array was taken as the random parameter in numerical modelling [58]. The critical variation of the core-cladding index step normalized to the mean value δn/Δn was found to be equal 5-6% for the construction studied in the experiment [47] and up to 15% for the construction with Δn = 1.27 × 10−3 . This tolerance limit is not restrictive for up-to-date technology. This behaviour was observed if the input wavefield distribution has the phase spread among the cores 0.3 rad and the same intensity in all cores. If the initial phase spread was 3.1 rad, then the wavefield tends to converge to combination of in-phase and out-of-phase modes [59]. An important feature of the MCF amplifier is that the field distribution in a transverse plane is strictly determined by the index profile, with distortions induced by gain being negligible. This fact supports an approach based on wave field expansion over passive structure modes, which can be found easily by a standard solver. In particular, such modes for the seven-core fiber design in the case of identical cores were calculated in [60] by a finite-element solver. The mode solver for the structure with complex-valued refractive index, which can model the fibers with gain, was developed [61] on the base of Krylov’s subspace method. The wavefield evolution in the fiber obeys Schroedinger equation with a Hamiltonian H accounting the processes of diffraction, refraction and amplification. The guided modes determination is mathematically equivalent to calculating the eigenvalues and eigenfunctions of the corresponding Hamiltonian. A finite-difference approximation of this problem leads to an algebraic eigenvalue problem for a complex non-hermitian matrix of high dimension. But only several eigenfunctions with propagation constant larger than kn0 ,

Monochromatic Wavefield Evolution...

1

2

3

335

4

In-phase mode 5

6

7

Out-of-phase mode

Figure 4. The near field distributions of the seven core fiber amplifier guided modes Δn = 2.57 × 10−3. which corresponds to guided modes, are of practical interest. The shift-and-invert Arnoldi’s method [63] was used in [61], in which the eigenvalue problem for the operator (H−μI)−∞ has to be solved, where μ is the shift parameter and I is the identity matrix. This problem has the same eigenfunctions as the finite difference approximation the original problem and eigenvalues of both problems are expressed through each other. The Berenger’s Perfectly Matched Layer boundary conditions [62] were used at the numerical domain boundaries. The shift parameter μ was fitted to reach optimal convergence of the Arnoldi process. The chosen method for the case of passive fiber is equivalent to the corresponding shift-andinvert Lanczos method. The eigenvectors were calculated by the method [64], combining an incomplete LU decomposition with generalized conjugate gradient method. For passive system the incomplete Cholesky – conjugate gradient algorithm [64] was used. The guided modes of passive and active fibers were compared in [61] for the model construction with two cores. The results of calculation have shown that if the gain is equal 0.2 cm−1 the relative error for eigenvalues due to gain neglect was ∼ 0.0005 for the real part of the first eigenvalue and ∼ 0.0009 for the second one. The construction [47] was found to guide 7 array modes, of which two are axially symmetric. They are shown in fig. 4. If small spread in core refractive indices is added, then the number of guided modes increases to 8. The modal gain of the in-phase mode (1) is minimal, while the highest angular mode possesses maximal modal gain. If the refractive index is diminished to Δn = 1.27 × 10−3, then the system guide only one axially symmetric mode and two angular modes. The in-phase mode has the maximum gain in this case for small signal regim, i. e. without gain saturation. This behaviour is typical fo evanescently coupled arrays: at large distances between the cores the modes with zeroes

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Figure 5. The ratio of mode 1 and mode 7 powers as a function of the amplifier length: a) the wavefields of the modes are incoherent; b) the wavefields are coherent.

between the cores has larger modal gain, but when the distances are small, the in-phase mode has the best overlap with the gain. The in-phase mode modal gain becomes lower that the high angular mode gain for high intensities of the in-phase mode, but this effect can be compensated by tailoring the core-cladding index step [59]. Comparison between results of 3D BPM calculations [57] and of modal analysis revealed [65] a seeming contradiction: an in-phase mode according to the mode solver possesses the lowest gain, while according to the 3D BPM approach the wave field converges to this mode, if the differences in phases of the launched beams are within a few tenths of a radian. Combination of the mode solver with 3D beam propagation program allows to study in detail the propagation of linear combination of two-modes. The fig. 5 illustrates the importance of the coherent interaction of the modes on the example of linear combination in-phase mode and out-of-phase mode. The seven-core fiber amplifier construction corresponds with [47]. The input power of the in-phase mode power is equal to 87 mW, the out-of-phase power is 24 mW. In the case of incoherent wavefields (a), when the total intensity is the sum of modal intensities, the mode with larger modal gain (7) amplifies more effectively. In the case when the wavefields are coherent, i. e. the total field amplitude is the sum of the modal amplitudes, the in-phase mode strongly dominates. It was shown in ref. [65] that the convergence to one or another mode depends also from the input power. To understand this complicated behaviour, the theory of light amplification in a system of parallel waveguides with saturable gain should be reconsidered.

Monochromatic Wavefield Evolution...

3.

337

Wavefield Propagation in Fiber Amplifier

The built-in index profile of a multicore fiber amplifier is a step wise function with higherindex regions, called cores, each guiding a single mode. To approach the goal of having a large mode area, small values ∼ 10−3 of the core-cladding index difference Δn are required in experiments. In addition, we do not consider birefringence effects. It is then possible [76] to use the scalar approximation instead of solving Maxwell’s equations for the elecromagnetic field in the fibre. The wave field can be characterized in this approximation by a scalar function ψ(x, y) exp(iβz − iωt), where z is the propagation distance, ω = kc is the oscillation frequency, k is the vacuum wave number, c is the speed of light, and β is the propagation constant. The function ψ could be any transverse component of the electrical or magnetic field, obeying the 2D Helmholtz equation, which formally coincides with the stationary Schroedinger equation in quantum mechanics: 

Hψ = −β ψ, Hψ = − 2

∂ 2ψ ∂ 2ψ + ∂x2 ∂y 2



+ U ψ,

(2)

where H is the Hamiltonian, β 2 plays the role of energy, x and y are the transverse spatial variables, the potential function U = −k2 n2 (x, y), and n is the refractive index. Equation (2) is to be solved with corresponding conditions at the core-cladding boundary and at infinity. This problem has a continuous spectrum of the leaky modes ψQ and a discrete spectrum of the guided modes ψj , so any field injected into the MCF can be presented as the sum over the spectrum of ψj plus the integral over the spectrum of ψQ with some coefficients [76]. The field amplitudes of guided modes ψj (x, y) are real-valued functions for no-loss fiber. Additionally, the guided and leaky modes are mutually orthogonal to each other. The wave field Ψ(x, y, z) in an active fiber with gain g (x, y, Ψ) satisfies the 3D Helmholtz equation ∂ 2Ψ ∂ 2Ψ ∂ 2 Ψ  2 2 + + + k n − ikn g Ψ = 0, 0 ∂z 2 ∂x2 ∂y 2

(3)

where n0 is the cladding index. Thus, in contrast to quantum mechanics the potential in laser optics is a complex-valued function U + ikn0 g (x, y, Ψ) . The modes of equation (2) can be taken as a basis for expansion of the wave field: Ψ(x, y, z) =



cj (z)ψj e

iβj z

+

cQ(z)ψQeiβj z dQ.

(4)

j

The nonuniform gain leads to an interaction between guided and radiating (leaky) modes, thus producing additional losses for the guided modes due to the radiating modes carrying away the energy. However, transformation of guided modes into leaky ones is a negligible effect for typical parameters of fiber amplifiers, as it was confirmed by 3D BPM calculations [77]. From the other side, leaky modes see the gain much lower than do the guided modes because the cladding area is much larger than the area of active cores. For these reasons we can substitute expansion (4) into (3) and neglect the radiating modes. The gain in the cores is usually of the order of 10−1 cm−1 , while a typical spectral separation between

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the propagation constants of guided modes is ∼ 10 cm−1 . Thus, variation of the guided mode amplitudes cj due to amplification is a slow process and it is possible to use the approximation, in which the terms d2 cj /dz 2 are neglected. Then the following system of equations for the expansion coefficients cj can be derived by multiplying the resulting equation by ψj exp(−iβj z) and integrating over the fiber aperture: 1 dcj 1 Gjl cl ei(βl−βj )z , = cj Gjj + dz 2 2 l=j

(5)

where the summation is made over N guided modes. The optical modes introduced are normalized for convenience as s ψj2 dx dy = 1. The difference of propagation constants is a small parameter in comparison with kn0 , so the factors βj / (kn0 ) in (5) have been replaced by 1. The applicability of this approximation was verified by 3D modelling of the MCF aplifier. The matrix elements Gjl describe an interaction between the wave field and the gain: Gjl = s g(x, y)ψj ψl dx dy, (6) where integration is made over the fiber aperture. The coefficients Gjj are the modal gains of the j-th mode. The nonuniform spatial gain distribution results in coupling of the orthoghonal modes of equation (3). This cross-modal gain effect is described by the coefficients Gjl , j = l in the sum in (5). In passive fiber this effect can be produced by the refractive index nonlinearity, but it can be neglected for intensity values typical for cw fiber amplifiers. If the non-diagonal terms are abscent (G12 = 0), then the modes are still coupled due to the gain saturation by the total radiation intesity. The system (5) of complex equations can serve as the basis for analyzing the mode competition in a wide variety of MCF amplifiers. To give an idea of the features possessed by the theory based on the equation system (5), let us consider the simplest situation when a waveguide array supports only two guided modes. In this case, system (5) can be reduced to three ordinary equations for real-valued functions dP1 = G11 P1 + P1 P2 G12 cos φ, dz dP2 = G22 P2 + P1 P2 G12 cos φ, dz dφ P1 + P2 G12 sin φ. = δβ − √ dz P1 P2

(7) (8) (9)

Here the modal powers Pj , j = 1, 2, are introduced by the expression cj = Pj exp(iφj − iβj z), and φ = φ2 − φ1 is the phase difference between the modal fields, δβ = β2 − β1 . Pj is the fraction of the wave field power carried by the j-th mode. Equation (9) describes the behavior of the modal phase difference. Since gain coefficients are much smaller than δβ, this phase difference grows along the propagation axis almost linearly, dφ/dz ≈ δβ. In this approximation, the problem is reduced to solving equations (7) and (8) only. Mode competition for gain has been studied since the early times of laser research. First terms at the right-hand sides of eqs. (7)–(8) include the mode competition due to the total intensity oscillations. Differing spatial profiles of modal intensities result in the gain spatial hole burning effect, which, in turn, induces instability of single-mode lasing

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[66–68]. Spatial beating between two modes leads to oscillatory modulation of the total field intensity along the axis with a characteristic spatial frequency of a few tens of cm−1 . This modulation induces through the gain saturation similar oscillations in the material gain, the resulting gain grating in different cores correlates with the modal powers. It is known [69], that the gain non-uniformity in lasers is a principal reason for destroying single mode oscillation and spectral broadening, which is concerned with the longitudinal modes. In the case of the field evolution in the fiber amplifier we have a tangled interaction between the wave field and transverse and longitudinal gain non-uniformities. This dynamics is traditionally described by taking into account the effect of the modal gain saturation by the intensity of another mode (so-called cross-saturation, which was studied mainly for counterpropagating modes of ring lasers [70]). The dependency of the modal gains ( G11 and G22) on the wavefield intensity results in known as the cross saturation effect [70, 71], while the additional coupling through the cross-gain being new effect, which is special for the active waveguides. Our purpose is to make this interaction clearer by using the simplest waveguide configuration.

4.

Two-Mode Competition in the MCF

To complete the system of equations (7)–(9) it is necessary to specify how the material gain coefficient depends on the total intensity. Generally, the kinetics of population inversion depends on the material used in fiber laser. The common feature for all cw systems is so-called gain saturation, i.e. reduction of inversion and gain induced by stimulated emission. This effect can be qualitatively well described by the simplest formula g = g0/ (1 + I/Isat) , where g0(x, y) is the small signal gain, I is the wave field intensity, and Isat is the saturation intensity. We will normalize the intensity to the saturation intensity value. Even with such a simple model for gain saturation and taking the mode phase difference as φ ≈ δβ · z, the system (7)–(9) is still rather difficult to analyze. The point is that the Gjl terms defined by (6) are complicated functions of P1,2, which cannot be found explicitly for realistic waveguide structures. Nevertheless, some general properties of solutions to (7)–(9) can be identified. It should be noted, that the studies of two-mode laser with high-Q cavity including polarization and population densities dynamics [72–74] results in the same sort of mathematical structure as eqs. (7)–(9) in the limit of small saturation. However, the physical content of the problem studied is essentially different. The authors of [72–74] analyzed the stability of two-mode regimes in a two-wavelength laser, but the complete dynamics had not been modeled. The equations similar to two-mode laser dynamics are recently have been derived for the random laser with streong scattering [75]. The wavefield in the problem, which we consider, evolve in space, not in time. Additionally the total power rise in our system is not limited along the fiber length. An important distinction of these equations is the identity of the second terms in the right-hand sides of equations (7) and (8). For this reason we can deduce from eqs. (7) and (8) for the evolution of the mode power ratio d dz



P2 P1







P2 P1 − P2 = G22 − G11 + G12 cos φ · √ . P1 P1 P2

(10)

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If G12 cos φ is negative, then the last term in parentheses on the right-hand side of (10) supports the trend of the power ratio to grow when this ratio is greater than 1. The dimensionless total wave field intensity can be expressed in the form I = P1 ψ12 + √ P2 ψ22 + 2 P1 P2 ψ1ψ2 cos φ. The cross-gain variation as a function of z can be understood from the expression G12 cos φ = s

g0ψ1 ψ2 cos φ dx dy √ , 1 + P1 ψ12 + P2 ψ22 + 2 P1 P2 cos φ

which is obtained from (6). The mode amplitude product necessarily changes sign within the cores to provide orthogonality of modes. It is seen that the integrand generally takes on a negative value with larger absolute magnitude when ψ1 ψ2 cos φ < 0. Therefore the quantity G12 cos φ is preferably negative. It can be rigorously proved that G12 cos φ is nonpositive for a system of two parallel waveguides possessing mirror symmetry Δn(x, y) = Δn(−x, y) and supporting two guided modes. One of the modes is symmetric (j = 1) and another mode is antisymmetric (j = 2) . Taking into account the symmetry properties, the cross-gain coefficient can be expressed as

G12 cos φ = −4 P1 P2 cos2 φ s

S1

g0 ψ12ψ22 dx dy, Cψ

(11)

where the integration is made over one of the waveguides, and the saturation factor Cψ is 

Cψ = 1 + P1 ψ12 + P2 ψ22

2

− 4P1P2 ψ12ψ2 cos2 φ.

It is evident from (11) that the term G12 cos φ in this case is always non-positive and it turns to zero at P1 = 0 or P2 = 0. Since the cross-gain term G12 cos φ diminishes the amount of energy extracted by stimulated emission equally for both modes, the increase of the mode power is favored for the mode with higher power. This is a rather important conclusion, radically differing from the intuitive suggestion that the mode possessing higher saturated gain is dominant at the output of a sufficiently long amplifier. It is shown below that the mode can possess higher modal gain throughout the entire amplifier but the part of the power carried by this mode in the total power diminishes in the course of amplification. The idea of using the saturated modal gains [78] stems from consideration of the incoherent fields of competing modes. Such a situation can be thought of as the competition of two signals launched in the fiber amplifier from independent sources at the same frequency. The wave field intensity is expressed in the incoherent case as I = P1 ψ12 + P2 ψ22, and the equations for the two-mode evolution are dP2 dP1 (12) = G11P1 , = G22P2 , dz dz where G11 and G22 are the modal gains of modes 1 and 2, respectively. These equations include so-called cross-saturation of gain associated with the fact that both modes saturate gain. In the limit of weak saturation, equations (12) are reduced to a form closely resembling the known dynamic equations for a two-mode ring laser ( [71]) X˙ = 2X (η1 − β1X − θ12 Y ) , Y˙ = 2Y (η2 − β2Y − θ21 X) ,

(13)

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341

Figure 6. Schematic of the system of two parallel ultrathin waveguides. The profiles of the symmetric (solid line) and antisymmetric (dashed line) modes correspond to κ = 1.2. where X and Y are the dimensionless intensities of modes 1 and 2, ηj , and βj j = 1, 2, are the above-threshold small signal gains and self-saturation coefficients, respectively, θ12 and θ21 are the cross-saturation coefficients. If there is a difference in the modal gains, then one mode may suppress the growth of the other. If the modal gains are equal, both modes lase the same power for the case of an inhomogeneously broadened (θ12 θ21 ≤ β1 β2) ring laser [71]. For a homogeneously broadened (θ12 θ21 > β1β2 ) ring laser [79] equations (13) predict that of two modes starting at t = 0, the mode with higher power completely suppresses the second mode. In practice, the operation regime of a laser in the last case is random due to spontaneous emission effects [70, 80]. Actually, the analogy between the weak-saturation limit of (12) and the laser equations (13) is valid only for low-power wave fields, i. e. for lasers starting from small signals.

5. 5.1.

Wavefield Evolution in Two and Three Ultra-thin Active Planar Waveguides Guided Modes

The delta-function [δ(x)] is a favorite potential well in quantum mechanics. In application to optics, the δ-function well is the mathematical limit of a high-contrast thin planar waveguide with arbitrarily small waveguide width d and a high refractive index difference Δn, characterized by a single parameter dΔn. In this limit, the waveguide supports a single mode, the wave field of which is almost constant within the core and extends far outside the

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core. It should be mentioned that the core-cladding difference is restricted by the condition Δn 1 for applicability of the scalar model (2) . The δ-function as a model waveguide is very fruitful for analytical studying the waveguide arrays [81, 82]. Figure 6 shows the system of two ultra-thin waveguides situated at locations x = ±a. It is convenient for this particular system to introduce a dimensionless transverse coordinate ξ = x/a. Equation (2) for guided modes of the waveguide array reads  d2 ψ  2 + −η + 2κ [δ (ξ − 1) + δ (ξ + 1)] ψ = 0, dξ 2

(14)

where κ = k2 adn0 Δn is the δ-function amplitude, η is the wave field attenuation rate outside the waveguides, and η 2 is an eigenvalue characteristic for a given mode. As is well known [83], such a system supports the two guided modes shown in fig. 6, provided the condition 2κ ≥ 1 is satisfied. The fundamental mode is symmetric ψ1 (ξ) (in-phase mode), and the second mode is antisymmetric, ψ2 (ξ) (out-of-phase mode). The amplitudes of these modes attenuate exponentially at ξ → ±∞ with corresponding rates η1 and η2 , while in the space between the waveguides ψ1 ∼ cosh(η1ξ), and ψ2 ∼ sinh(η2ξ). The attenuation rates satisfy the transcendental equations 



η1 = κ 1 + e−2η1 , 

(15)



η2 = κ 1 − e−2η2 .

(16)

The modal shift of the propagation constant δβj = βj − kn0 is expressed as δβj = ηj2/ (4LR) , where LR = kn0 a2/2 is the Rayleigh length. Gain in the waveguides can be included in the model by adding an imaginary part into κ: κ = κ + iκ, where κ = gκ/(2kΔn). The modal gains in the small signal limit read 

2η  κ sinh 2η1 + (sin 2η1) η1 /η1 G1 = 1 1+ LR sinh 2η1 + cos 2η1 

−1

2η  κ sinh 2η2 − (sin 2η2) η2 /η2 1+ G2 = 2 LR sinh 2η2 − cos 2η2

, −1

,

where ηj and ηj are the real and imaginary parts of the corresponding parameters satisfying (15). For the symmetric system under consideration, it is convenient to introduce a confinement factor of the mode, Γj , as being proportional to the overlap of the mode intensity and gain in one waveguide. For ultra thin waveguides the confinement factor is equal to the squared amplitude of the mode in the waveguide. The modal gain is expressed as Gj = 2Γj gd/a. As long as κ  κ (2kΔn  g) , the expressions for modal gains and confinement factors can be simplified. The confinement factors of the symmetric and antisymmetric modes can be expressed as Γ1 = Γ2 =

κ [1 + exp (−2η1 )]2 , 2 1 + 2κ exp (−2η1 )

κ [1 − exp (−2η2 )]2 . 2 1 − 2κ exp (−2η2 )

(17) (18)

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Figure 7. Confinement factors for the symmetric (solid line) and antisymmetric (dashed) modes vs. the coupling strength parameter κ in the system of two thin planar waveguides. The quantities are dimensionless. [82] ηj (j = 1, 2) satisfy equations (15) with κ replaced by κ . In the limit of weak coupling between waveguides (κ  1) , the antisymmetric mode has a higher confinement factor, than the symmetric one:  −1 κ  1 + 2κ exp −2η1 , 2  −1 κ  1 − 2κ exp −2η2 . Γ2 ≈ 2

Γ1 ≈

As coupling between waveguides increases the antisymmetric mode tends to transform into a leaky mode, (η2 → 0) , and its modal gain diminishes proportionally to η2 , while the symmetric mode gain remains of finite value. Thus, there is a critical distance between the waveguides, at which both gains are equalized. The confinement factors are shown in fig. 7 as functions of κ . The curves intersect at κcr = 0.900126 . . .. This value corresponds to a ≈ 3.9 μm for a system of two waveguides with Δn = 2 · 10−3, d = 2 μm, n0 = 1.456, and the radiation wavelength 2π/k = 1 μm. The Rayleigh length LR in this case is 70 μm. The seven-channel fiber amplifier, which was studied experimentally [47] and numerically [57], consisted of one waveguide in the center of array and six waveguides placed symmetrically around. Thus the central waveguide had an exceptional position. The guided modes of the array can be dividied into the axially symmetrical modes and high anglular index harmonics. The wavefields of the latter modes are very small in the central core and the former modes take almost all gain in this core. Thus the axially symmetrical modes including the in-phase mode of the array see the gain, which is not available fotr the high angular indices modes. This situation can be modeled in the system of three ultra-thin planar waveguides (see the fig. 8). We will suppose below that the refractive indices in all waveguides are the same. The guided modes of the array are symmetrical or antisymmetrical as functions of the transverse coordinate ξ. Asymmetrical modes are governed by

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Figure 8. Schematic of the system of tree parallel ultrathin waveguides. The normalized profiles of the symmetric (solid line) and antisymmetric (dashed line) modes correspond to κ = 1.2. equation 16, while the dispersion equation for symmetrical modes reads (η1 − κ)2 = κ exp (−2η1) η1 + κ

(19)

There is the one (symmetrical) guided mode in the considered system if the condition 2κ < 1 is satisfied. We restrict our analyses by the case 1 < 2κ < 3, in which there are two guided modes: symmetrical (1) and antisymmetrical (2). The small signal gain in the lateral waveguides are supposed to be equal. The small signal gain coefficient in the central waveguides is μ times larger the small signal gain coefficients in the lateral cores. The wavefield of the antisymmetrical mode (2) is equal zero in the central waveguide. Due to dispersion equation 19 the ratio of the intensity in the central waveguide to the intensity in the lateral waveguide can be expressed for the symmetrical mode through the formula ρ = 14 |1 − η1/κ|2 exp(2η1). It is useful to re-define the coefficient Γ1 as the overlap of the symmetrical mode wavefield with a one lateral waveguide. It is evident, that the confinement factor of the antisymmetric mode, Γ2 obeys eq.(18) The expression for the confinement factor Γ1 can be derived in the case κ  κ in the form Γ1 =

κ , 1 + Qκ/ρ

where the quantity Q introduced is Q =





1 + |κ/η1 |2 sinh 2η1 −

  κ  1 + cosh 2η1 + η1 1 − |κ/η1|2 .  η1

(20)

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345

The total confinement factor for the symmetrical mode can be expressed as Γtot = Γ1 (2 + 1 μ/ρ).

5.2.

Amplification of Two Incoherent Wavefields

It is instructive to analyze wave field amplification in the array of δ-function-type coupled waveguides, taking as a reference case the standard equations applicable for description of two incoherent modes with gain cross-saturation taken into account. In this case the total wave field intensity is the sum of the two mode intensities. For definiteness, the simplest gain saturation model is adopted g = g0/ (1 + I/Isat) , and the field intensity in the following is measured in Isat units. For the incoherent fields of two modes in the two waveguide system shown in fig. 6, the equations for the modal powers (12) read dP1 2κ0 P1 Γ1 = , dζ 1 + P1 Γ1 + P2 Γ2 dP2 2κ0 P2 Γ2 = , dζ 1 + PS 1Γ1 + P2 Γ2 where ζ = z/LR and κ0 = g0 (LRd/a) . This system of equations can be easily integrated: ln P1 /P10 + Γ1 (P1 − P10 ) + Γ1 (P2 − P20) = 2Γ1 κ0 ζ, where P2 /P20 = (P1 /P10)γ , γ = Γ2 /Γ1 , and P10 and P20 are the mode powers at the amplifier entrance. The asymptotic (ζ → ∞) behavior of the modes depends on the value of γ, which is equal to the ratio of modal gains. That is, at γ = 1 both modes grow with equal rates. This case is illustrated in fig. 9, which shows the diagram in dimensionless variables I1 = P1 Γ1 and I2 = P2 Γ2 . It is clearly seen that the proportion P2 /P1 remains constant at any distance. If the modes have different overlaps with the gain, the mode with higher confinement factor wins, and its power increases linearly with length, while the power of the second mode grows as a fractional power of the length. This means that the powers of both modes increase with no limit. This result does not depend on the initial proportion of the powers of the modes, if the amplification is large enough. The propagation of two incoherent waves in the system of three waveguides is described by the system of equations 



dP1 μP1 Γ1 /ρ 2P1 Γ1 = κ0 + , dζ 1 + P1 Γ1 + P2 Γ2 1 + P1 Γ1 /ρ dP2 2κ0 P2 Γ2 . = dζ 1 + P1 Γ1 + P2 Γ2

(21) (22)

Results of numerical integration of equations is 22 presented in fig. 10 as dependencies P2 (P1) for the conditions κ = 0.0001, μ = 0.7 κ = 1.139. In the limit P1 Γ1 , P2Γ2  1, the system 22 can be integrated so the asymptotic dependencies of powers in the modes are P1 ≈ θP2 + P2 μ(1 − γ)−1, γ = 1, P2 ≈ P2 μ (ln P2 + θ) , γ = 1, γ

(23)

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Figure 9. Powers carried with the modes in the two waveguide system for different values of initial power ratio for the incoherent wave fields of the modes, Γ1 = Γ2 . Axes are in ISata.

Figure 10. Powers carried with the modes in the three waveguide system for different values of initial power ratio for the incoherent wave fields of the modes, κ = 1.139, μ = 0.7 (Γtot 1 = Γ2 ). Axes are in ISata.

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where γ = (1 + μ) Γ1 /Γ2 , θ is a constant. Thus if the fields of the modes are incoherent the output powers ratio tends to μ/ (1 − γ) if γ < 1. In the case γ > 1 the power carried with the fundamental mode raises faster than the power in the antisymmetric mode. The main result is that the final domination of one of the modes in the output is not determined by the initial ratio of powers in the modes but it is produced by the parameters of the construction if the amplifier length is sufficiently large.

5.3.

Amplification of Two Coherent Wavefields

If the wave fields of two competing modes in the system with two ultra-thin planar waveguides are coherent, the gain coefficients entering into the system of equations (7)–(9) can be found explicitly from eqns. (6) and (11) as Gjj = 2κ0 Γj (1 +√P1 Γ1 + P2 Γ2 ) / (CLR ) , G12 = −4κ0 Γ1 Γ2 P1 P2 cos2 φ/ (CLR ) ;

(24)

here j = 1, 2 and the denominator contains C = (1 + P1 Γ1 + P2 Γ2 )2 − 4Γ1 Γ2 P1 P2 cos2 φ. In the specific system under consideration, the ratio of the modal gains of the two modes is independent of the field in the amplifier: G11 /G22 = Γ1 /Γ2 . The reason is that both the modes are equally distributed over the two waveguides and the mode profiles inside the waveguides are considered to be identical due to the small waveguide thickness. Equations (7)–(9) read dP1 2κP1 Γ1 = 0 (1 + P1 Γ1 − P2Γ2 cos 2φ) , dζ C dP2 2κP2 Γ2 = 0 (1 + P2 Γ2 − P1Γ1 cos 2φ) , dζ C dφ ≈ LR δβ. dζ

(25) (26) (27)

Direct integration of (25)–(27) was performed using a Mathcad software solver based on 4th-order Runge-Kutta method. The calculations were made for the cases (a) κ = 0.900126 (Γ1 = Γ2 ), and (b) κ = 1.2 (Γ2 = 1.112Γ1). For calculations we took κ0 = kn0 g0da/2 = 0.001, that corresponds with the small signal gain g0 ≈ 0.21 cm−1 for a = 5.2 μm. The other parameters of the construction as in the sec. 5.1., so that κ = 1.2. −1 The spectral separation of the modes in this case is δβ = 0.277L−1 R ≈ 22 cm . The results of the calculations are presented in fig. 11. For equal confinement factors Γ2 = Γ1 , the behavior of the curves on the diagram in fig. 11a showing P2 as a function of P1 for different initial conditions is in contrast with their behavior in the case of incoherent modes (fig. 9). In the latter case, both modes increase in power so that the proportion of the powers is constant. In the former case, there is the line P1 = P2 , that is unstable in Lyapunov’s

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Figure 11. Powers carried with the modes normalized to Isat a for different values of initial power ratio: (a) Γ1 = Γ2 ; (b) Γ2 = 1.112Γ1. Dashed line in (b) is the bisector I1 = I2. sense: an infinitesimal deviation from this trajectory results in further divergence with the curves approaching asymptotically either a vertical or a horizontal line depending on the direction of the initial deviation. This behavior corresponds to dominance of one mode, the power of which grows linearly with the length, while the power carried by the other mode is stabilized at a certain level. This behavior is similar to that observed in Lamb’s model [71] for a two-mode laser in the case when the gain cross-saturation coefficient is greater than the self-saturation coefficient. In the case of different confinement factors (see fig. 11b), the diagram on the whole is nearly the same. The straight line P2 = IP1 in the diagram P2(P1 ) in fig. 11a transforms to a curve (not shown in 11b) dividing the (P1 , P2) plane into two parts. In the upper part all curves approach vertical asymptotes (the asymmetric mode dominates), while in the lower part all curves approach horizontal asymptotes (the symmetric mode dominates). The shape of this curve can be found numerically. We illustrate the general arguments adduced in sec. 4. by the results of analysis of the specific construction under consideration in order to get better insight into the mechanism leading to depression of one of the modes in two-mode amplification, . Figure 12 illustrates the behavior of the terms in the right-hand sides of eqs. (7)–(8) Gjj Pj (j = 1, 2) and √ P1 P2 G12 cos φ with modal gain and cross gain coefficients defined by (24). P10 = 0.27 and P20 = 0.13. Small-scale oscillations of these terms associated with mode beating are shown in fig. 12a. It is seen that the power increase G22P2 in the antisymmetric mode is lower than that √ of the symmetric mode even though its modal gain is higher. The cross-gain component P1 P2 G12 cos φ is nonpositive and antiphase to the power increments associated with modal stimulated emission. The same terms averaged over oscillations are shown in fig. 12b. The cross gain-term is multiplied by (−1) for convenience of presentation. It

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√ Figure 12. Dimensionless terms G12 cos φ P1 P2 LR (solid line), G22P2 LR (dashed line), and G11P1 LR (dotted line) as functions of propagation distance normalized to LR : (a) axial oscillations √shown within a short propagation interval; (b) values averaged over oscillations; G12 cos φ P1 P2 LR (solid line) is taken with minus sign. P10 = 0.27 and P20 = 0.13; Γ2 = 1.112Γ1. is clearly seen in fig. 12b, that the decrease in emitted power caused by the cross-gain term tends to equilibrate the term G22P2 at a sufficiently long amplification length. In dimensional variables the Rayleigh length at the parameters taken is 123 μm. This phenomenon leads to stabilization of the antisymmetric mode power on the level reached at that moment. The trend to domination of one of the modes in simultaneous amplification of two coherent modes can be illustrated by the behavior of the mode power ratio P2 /P1 . The proportion P2 /P1 calculated for κ = 1.2 is shown in fig. 13 as a function of propagation distance for various input proportions at constant total input power P20 + P10 = 0.4. It is seen that increasing P20/P10 results in a change of amplification regime from dominance of the symmetric mode to dominance of the asymmetric one. It is worth noting that the sign of the derivative at ζ = 0 of the proportion P2 /P1 cannot serve as a criterion for the change of

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Figure 13. Proportion of the modal powers for varied inputs at constant total input power as a function of the amplifier length. The quantities are dimensionless, Γ2 = 1.112Γ1. [82] the amplification regime. As long as P20 /P10 grows, the curve appears, for which growth at small distances changes to decrease at longer distances. Figure 13 proves that there exists a critical value of P20 /P10, which separates the regimes of dominance of the symmetric or asymmetric mode. Actually, the critical value of P20/P10 is a function of the total power and depends parametrically on the confinement factor values. A series of calculations allows us to find the critical ratio of powers of the symmetric (in-phase) and antisymmetric modes in the launched signal shown in fig. 14 as a function of the total power for confinement factor values Γ2 = 1.112Γ1. At such values of Γ1 and Γ2 the modal gain of the antisymmetric mode is greater than of the symmetric one. Nevertheless, the in-phase mode dominates at the output of the amplifier when its initial power ratio to the antisymmetric mode power is above the curve shown in fig. 14. The higher the total launched power, the lower the critical fraction of the in-phase mode. It follows from this figure that 40% excess of in-phase mode power in the input signal is sufficient to suppress the out-of-phase mode power for P > 0.3 In the case of te system of three ultra-thin waveguides the coefficients G12 and G22 are still defined by the equations (24). For the coefficient G11 it can be derived the expression: G11 =

2κΓ1 kn0





1 + P1 Γ1 + P2 Γ2 μ/ρ + . C 1 + P1 Γ1 /ρ

(28)

It should be noted, that the ratio of the modal gains in the case of three waveguides can be characterized by the confinement factors only for small signal operation. Substituting the expressions for Gij into eqs. (7)–(9) the system of ordinary differential equations for modal power can be derived, which is analogous to eqs. (25)–(27) except additional term in the equation for P1 due to gain in the central waveguide. The dependencies P2 (P1) are presented in fig. 15 for the condition κ = 0.0001, μ = 0.7 and two values of κ : (a)

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Figure 14. Critical ratio of the in-phase mode power to the antisymmetric mode power, above which the in-phase mode dominates vs. total initial power P = P20 + P10. Γ2 = 1.112Γ1. κ = 1.139, when the small signal gains for both modes are the same, and (b) κ = 1.45, when the confinement factor of the antisymmetric mode is larger. As for the case of two thin waveguides, the diagrams in fig. 15 possess the separatrices, dividing the plane P2 (P1 ) into two areas with different asymptotic behaviour of the powers. However in the case of two waveguides the areas higher and lower the separatrix mean the different type of the dominating mode, while in fig. 15a powers of both the modes increase at high values of P2 . The antisymmteric mode power rises with higher speed. This behaviour is evident because the gain in the central waveguide (where the asymmetrical mode intensity is zero) can be taken by the symmetrical mode, only and its power rises unlimited. At small values of P2 and high values of P1 , i. e. in the situation of initial dominance of the symmetrical mode, the gain supports the tendency for symmetrical mode to dominate, while the amplification of antycimmetrical mode practically ceases. This behaviour is the same as in the case of two waveguides. If the small signal gain of the antisymmetric mode is larger than that of the symmetrical mode, the behaviour of dependencies P2 (P1) is qualitatively the same. Only the form of the separatix dividing the two amplification regime is changed. The symmetrical mode dominates in the output signal only if the trajectrory P2(P1 ) is below the separatix in fig. 15b. This means, that the input wavefield distribution is closely approximated by the symmetrical mode, i. e. the initial spread of phases over the array is small. This result is qualitatively agrees with the results of 3D modelling of the seven-core fiber amplifier [59].

6.

Cross-Gain Effect in Single Waveguide

The analysis performed above is strictly valid for the model system under consideration. However, the mechanism of weak mode suppression by a strong mode is of quite general nature. Thus, it is expected that this mechanism will work in any fiber amplifier pro-

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Figure 15. Normalized to Isata powers carried with the modes in the system of three waveguides for different values of initial power ratio: (a) κ = 1.139; (b) κ = 1.45. I1 = I2 . vided the input signal has a sufficiently narrow spectral width in order that results found for amonochromatic wave field are applicable for a real signal. 2D planar waveguide (see fig. 16) is one of the simplest construction for modelling the guided wavefield propagation. Guided modes spectrum calculation is in solving 1D Schroedinger equation with when the potential energy profile is rectangular well of width 2a. Due to the symmetry of the potential the modes are or symmetrical, or antisymmetrical. The lateral component of the modal wavevectors govern by the trancendent equations tan qa = η/q, tan−1 qa = −η/q,

(29) 

for the symmetrical and antisymmetrical modes, respectively. Here η = 2k2nΔn − q 2

1/2

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Figure 16. Schematic of the planar waveguide. The normalized profiles of the symmetric (solid line) and antisymmetric (dashed line) modes correspond to κ = 1.2. is the wavefield decrement outside the waveguide, q is the lateral komponent of the wavevector inside the waveguide. The modal gain for the case of small signal decreases steadily as a function of q: 

GSS = g0



q2 1− 2 . 2k nΔn(1 + ηa)

The waveguide confines two transverse modes if the condition π 2 < 2k2 a2 nΔ < 4π 2 is satisfied. These modes possess transverse profiles ψ1 = A1 cos q1 x and ψ = A1 sin q2 x respectively, where q1 and q2 are the corresponding eigenvalues of the transverse wavevector inside the waveguide, A1 , A2 are the coefficients normalizing the eigefunctions Ψj , so  2 that |Ψ| dx = 1. If the gain is taken in the Rigrod’s form g = g0/(1 + I), then the coefficients Gij due to the symmetry of the modal wavefields can be expressed as Gjj =

 a 0





2g0ψj2Cψ−1 1 + P1ψ12 + P2 ψ22 dxdy

G12 = − 4 P1 P2 cos (δβ · z)

 a 0

j = 1, 2,

g0ψ12ψ22Cψ−1 dxdy,

(30) (31)

It can be seen from the formula for G12, that the cross-gain impact into the modal power increase is always non-positive. The reason of diminishing the powers in both the modes is clear: the beating of the total intensity in the waveguide results in the gain to be loosed in the places of destructive interference of the modes. It should be noted, that the modal gains increase in these places. Because the cross-gain terms are the same as for the symmetrical modes as for the antisymmetrical ones the result is more critical for the mode carrying smaller power, thus suppressing the amplification of this mode. For this reason if the input wavefield profile is well-fitted with the fundamental mode, then the cross gain results in extension of the area of parameters where the fundamental mode is dominated.

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Practically, the maximum power of the single mode amplification is restricted by the spectral transformation. As it was discussed above the refractive index nonlinearity plays an auxilliary role in mode competition at the given frequency. However, this nonlinearity is the main reason of the four wave mixing (FWM), which can result in spectral broadening of the amplifier [84, 85]. The FWM in amplifiers is radically differs from this effect in passive fibers. The reason is that the wavefields with different frequencies have different propagation constants, and if we consider the FWM process of transferring the energy from the laser field into s noise wavefield: 2ωlas −→ ωnoise + ωidle, then the amplitude of the noise wavefield oscillates due to the phase mismatch. If the power of the laser field rises, as it is in the amplifier, then the energy received by the noise field in the oscillation period is not equal to the energy it return to the laser field [84]. This result to the amplification of the waves with frequencies in the very wide range, which was observed in experiment with fiber amplifier in short-pulse regime [85]. The intensity values, at which the laser signal amplification was stopped, are not used in cw sytems. However we should mention, that if the amplifier guides several transverse modes, as it was in the systems considered above, then the propagation constant difference can be arbitrary small for different transverse modes with different frequencies. In this case the role of the refractive index nonlinearity can not be neglected.

7.

Conclusion

The multicore fibers reveals a wide variety of physical effects making them perspective instruments for wavefields manipulating and amplifying. Several types of active fibers with independent, leaky-wave or evanescent-wave coupled cores have been explored as the fiber lasers and amplifiers components. The most developed technology now is now to insert hexagonal lattices comprising up to 37 single mode cores into the common cladding. The wavefields in the cores can be synchronized by an external Talbot filter or by diffractional exchange between the cores. A theory of monochromatic wave field amplification in a waveguide array is developed. An approach based on expansion of the wave field in terms of guided array modes leads to the appearance of additional terms in the system of ordinary evolution equations for the mode amplitudes. These terms have the meaning of cross-modal gain and, as shown, completely change the behavior of the amplified wave field. Analysis of two-mode amplification reveals that instead of unlimited growth of both modes for incoherent fields, the effect of weak-mode suppression by a strong one takes place. A detailed analysis is made for an amplifier composed of a pair of ultrathin waveguides. The critical values of waveguides and input signal parameters are found at which the in-phase mode dominates at the amplifier output. The model developed is applicable for studies on transverse modes competition in a single-core fiber amplifier. It is shown, that this effect expands the area of parameters, where the single-mode lasing is stable, and the factors limiting the maximal power in cw fiber amplifier are discussed. Authors acknowledge help in calculations from professor N. N. Elkin and V. N. Troshchieva. The work was partially supported by RFBR projects no. 07-02-01112-a, 07-02-12166-ofi

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[64] D. S. Kershaw J. Comput. Phys. 26, 43 (1978). [65] N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, J. Lightwave Technol. 25 3072(2007). [66] T. I. Kuznetsova and S. G. Rautian, Sov. Phys. Solid State 5, 1535 (1964). [67] H. Statz and C. L. Tang, J. Appl. Phys. 35, 1377 (1964). [68] C. L. Tang, H. Statz, and G. A. de Mars, J Appl. Phys. 34, 2289 (1963). [69] Yu. A. Anan’ev , Sov. Phys. Tech. Phys. 12, 97 (1967). [70] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, U. K. 1995), chap. 19. [71] W. E. Lamb, Jr., Phys. Rev. 134, A1429 (1964). [72] L. A. Ostrovskii, Sov. Phys. JETP. 21, 727 (1965). [73] N. G. Basov, V. N. Morozov, and A. N. Oraevskii, Sov. Phys.-Dokl. 10, 516 (1965). [74] J. A. Fleck, Jr. and R. E. Kidder, J. Appl. Phys. 36, 2327 (1965). [75] L. I. Deych, Phys. Rev. Lett. 95, 043902 (2005). [76] A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), chap. 13, 33. [77] N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, in Proc. of IVth Conf. on Finite Difference Methods: Theory and Application, Lozenetz, 2006 , edited by I. Farago, P. Vabishchevich, and L. Vulkov (Rousse University, Rousse, Bulgaria, 2007), p. 167. [78] K. M. Gundu, M. Kolesik, and J. M. Moloney, Opt. Lett 32, 763 (2007). [79] S. Singh and L. Mandel, Phys. Rev. A 20, 2459 (1979). [80] S. Singh, Phys. Rep. 108, 217 (1984). [81] A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. E 65, 036609 (2002). [82] A. P. Napartovich and D. V. Vysotsky, Phys. Rev. A 76, 063801 (2007). [83] V. M. Galitsky, B. M. Karnakov, and V. I. Kogan, Zadachi po kvantovoi mekhanike [in Russian] (Problems in Quantum Mechanics) (Nauka, Moscow, 1992), p. 27. [84] J. P. Feve Opt. Express 15, 577 (2007). [85] J. P. Feve, P. E. Schrader, R. L. Farrow, and D. V. Kliner, Opt. Express 15, 4647 (2007).

In: Progress in Nonlinear Optics Research Editors: Miyu Takahashi and Hina Goto, pp. 359-381

ISBN 978-1-60456-668-0 2008 Nova Science Publishers, Inc.

Chapter 10

VORTICES OF LIGHT: GENERATION, CHARACTERIZATION AND APPLICATIONS R. P. Singh, Ashok Kumar and Jitendra Bhatt Physical Research Laboratory, Navrangpura, Ahmedabad - 380009

Abstract Waves that possess a phase singularity and a rotational flow around the singular point are called vortices. In the light wave, such structures are called optical vortices. These are generated as natural structures when light passes through a rough surface or due to phase modification while propagating through a medium. However, these can be generated in a controlled manner as well. We will discuss the method used to generate optical vortices in the laboratory and how to characterize their topological charge using interferometry. In recent years optical vortices have got applications in variety of fields starting from biological physics to quantum information and computation. Therefore study of their coherence properties becomes very important. Study of the Wigner distribution function (WDF), originally discovered in quantum mechanics, can be quite useful for this purpose since it can provide coherence information in terms of the oint position and momentum phase-space distribution of the optical field. We will present experimental as well as theoretical results for the WDF of an optical vortex. Being vortices they posses helical wavefront and consequently each photon in the vortex beam carries an orbital angular momentum (OAM) l  depending on the order l of the vortex. Most of the applications with vortices of light use this property of orbital angular momentum. However, OAM per photon in the beam will be integer or non-integer (in units of ) that depends on if the vortex is axial (centered at origin) or non-axial (shifted from the origin). Thus, axial nature of the vortex can affect intrinsic property of a vortex beam. We show that the WDF can also be used to discriminate between an axial and a non-axial vortex. We will end the discussion by describing our work on second order coherence properties, based on intensity correlation studies of these novel light structures and some exciting applications of vortices of light.

1. Introduction Optical vortex, the beam carrying orbital angular momentum, is a phase singularity in light. The phase of the field in an optical vortex beam varies in a cork-screw like manner along the

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beam s direction of propagation [1, 2]. It was Nye and Berry who pointed out that like crystals, wavefronts – surfaces of constant phase- can also have dislocations [3]. The first experiments were done with ultrasonic wave trains [3], which were followed by few experiments in optical field [4]. However, the name optical vortex for such dislocations was coined by Coullet et al [5] in 1989 only as they found the structures in their solution of wave equation for a laser cavity with large Fresnel number, which have resemblance with superfluid vortex. It is interesting to observe that Nye found the similarity through the route of solids while Coullet found it through the superfluid. It would be relevant to point out that vortices were known on a formal level in hydrodynamics for quite some time [6]. It was almost two decades after Nye and Berry when Allen et al [1] showed that photons in such light beams carry an orbital angular momentum of l, l being the order of the vortex. Since then it has emerged as a potential research field, however, by the end of the chapter one would realize that still the field remains highly unexplored and generating curiosity across the fields. Though in laboratories it is experimentally produced using computer generated hologram [7, 8], astigmatic mode converter [9, 10], spiral phase plate [11, 12], spatial light modulator [13] etc but it is ubiquitous in nature. These are generated as natural structures when light passes through a rough surface, the dark spots in the well known speckles or due to phase modification while propagating through a medium, being dislocations of the wavefront. Ordinary laser beams with planar wavefronts are usually characterized by Hermite-Gaussian modes [14]. These modes have rectangular symmetry and are shown as HG(m,n) where m and n are mode indices. In contrast, optical vortices are the beams having helical wavefronts and are characterized by azimuthal phase term associated with Laguerre-Gaussian modes. These modes have circular symmetry and are shown as LG(l,p) where the l index relates azimuthal phase and it tells the number of 2 cycles of phase in the azimuthal direction around the circumference of the mode, while the p index relates the number of additional concentric rings around the central zone and p+1 gives the number of nodes across the radial field distribution. Though spin angular momentum of a circularly polarized light was first experimentally demonstrated by Beth [15] in 1936 but it was only in 1992 when Allen et al [1] showed that the orbital angular momentum (OAM) into the propagation direction of the beam having helical phase front has a discrete value of OAM l  per photon. The orbital angular momentum carried by an optical vortex enables to trap and rotate microparticles and even living cells so it can act as an optical spanner [16]. In such a way optical vortex makes its importance in biophysics [17-19], micromechanics [20, 21] or microfluidics [22].

2. Generation Though often encountered as natural structures, optical vortices can be generated in laboratory as well. Here we will discuss the most commonly used methods to produce optical vortices.

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2.1. Computer Generated Holograms This is the simplest yet quite efficient method to generate an optical vortex. Computer generated hologram (CGH) is a fork like pattern cast on a holographic sheet [23]. On passing a laser beam through branch point of CGH we get a diffraction pattern consisting of various orders of optical vortex. The intensity of these vortices decreases as the order increases. The dark area at the centre of the cross section of the beam increases with the increase in the order of the vortex. Figure 1 shows the generation of first order vortex using a computer generated hologram and TEM00 mode of a He-Ne laser. The CGH is produced by calculating the interference pattern of a plane beam with an optical vortex. A highly reduced pattern of this is then transferred on a holographic sheet which is the desired hologram. The hologram which consist only black and white areas without any grey scale is called binary hologram. The limitation of binary hologram is that the fraction of power that goes to first diffracted order is very small but this is overcome by using blazed grating [24]. The more sophisticated holograms are also created for generating modes having good purity [25]. To get the required radial profile in the diffracted beam the contrast of the pattern is varied as a function of radius.

Figure 1. Experimental scheme to generate optical vortex: CGH, computer generated hologram A, aperture CCD, charge coupled device camera. Inset shows the diffraction pattern of the forked holographic grating, the bright spot at top is central order that is followed by first order optical vortex.

2.2. Astigmatic Mode Converter We can also convert Hermite Gaussian mode into Laguerre Gaussian mode by using combination of two cylindrical lenses which is called astigmatic mode converter [9, 10]. The conversion is based on the appropriate use of Gouy phase [26]. Guoy phase shift is a phase shift occurring in a Gaussian beam along the direction of propagation during propagation and its value at a distance z from the beam waist (z = 0) is defined as – (z) = - arctan(z/zR),

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where zR is the Rayleigh range. The astigmatic mode converter converts a Hermite Gaussian mode of arbitrary order into the same order of Laguerre Gaussian mode and vice-versa [9, 10].

2.3. Spiral Phase Plate It is also a type of mode converter so it can also generate LG mode from HG beam. A spiral phase plate is a transparent disc whose thickness varies circumferally but is uniform radially. These are constructed from a piece of transparent dielectric material with increasing, spiraling thickness. When a spherical wavefront falls on such plates then due to variable thickness of spiral phase plates the light suffers dense media for different times. The light beams passing through the thicker part suffer longer optical path and hence greater phase shift. Due to having spiraling thickness it generates spiral phase distribution of optical vortex. The traditional spiral phase plate is useful only for one wavelength of light and one topological charge hence a more versatile phase plate is created, known as ad ustable spiral phase plate [27]. This phase plate can be used with multiple wavelengths and can produce a range of topological charges. Such kind of plates are created by making a crack in a parallel sided transparent plexiglass, the crack starts from one edge and terminates near the centre. Now the plate is mounted in a rigid frame and keeping one tab fixed the other tab is twisted. If a laser beam falls perpendicular to one tab then it will not remain perpendicular to other tab. In such a way the phase of incident light is modified and a laser beam directed at the end of crack will produce an optical vortex. Thus by changing the twist we can change the order of vortex. The high power efficient vortices in a single beam can be generated by a kinoform type spiral phase plate [28].

2.4. Spatial Light Modulator Spatial light modulator (SLM) is a device that can modulate light spatially in amplitude and phase. The principle of SLM is based on the properties of liquid crystals. It consists of a liquid cell in which the arrangement of liquid crystal molecules can be altered by applying electric field. SLMs can be electrically addressed or optically addressed. We generate a program which gives a fork type pattern and then we introduce this pattern in SLM through computer. The liquid crystals in SLM orient themselves as a fork type pattern. If now we incident a laser on the SLM it results into the optical vortices as output in reflection or transmission. SLMs are used to generate optical vortex beams in a prompt and efficient manner. We can generate an array of optical vortices using SLMs [29-33], which are used for trapping of microscopic particles. The main advantage of SLM over CGH lies in it being a flexible diffractive optical element and thus can generate optical vortex of any order when required.

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2.5. Other Methods Apart from methods described so far people are generating optical vortices by non linear wave mixing [34], uniaxial crystals [35], interferometry [36, 37], Wollaston prism [38], diffraction from dislocation in two dimensional colloidal crystals [39], nonspiral phase plate [40], single phase wedge [41], left handed materials [42], helical phase spatial filtering [43], an image-rotating optical parametric oscillator [44], 2 phase plate [45], stressed fiber optic waveguide [46] etc.

3. Characterization Once we have produced an optical vortex, a beam having darkness at the centre of intensity profile, how we will confirm that we have got Laguerre Gaussian mode or the optical vortex because only darkness at the centre of the beam does not imply the helical wavefront. To identify the helical wavefront or Laguerre Gaussian mode we do the interferometry. We know the interference fringes resulting from the superposition of two beams with plane wavefront are straight lines with a spacing that depends on both the intersection angle and the wavelength. However, when we interfere a beam having helical wavefront with a beam having plane wavefront we get a complicated fringe pattern. The resulting fringe pattern is similar to fork pattern. The number of prongs in the resulting fork in interference pattern decides the order of vortex. In general for l th order vortex we get l+1 prongs. When we pass laser beam through a hologram having l+1 prongs in its fork, we get lth order vortex as first diffracted order. It is easier to interfere the optical vortex with its own mirror image rather than a plane wave, the resulted interference pattern consists of 2l dark spokes (for lth order vortex and its mirror image) [47]. To characterize optical vortex, in general we look for its following physical properties.

3.1. Topological Charge The order of optical vortex is characterized by its topological charge. Topological charge is defined as the number of twists in wavefront per unit wavelength of light. Higher the topological charge more the twisting of wavefront. The topological charge of an optical vortex is measured by using interferometry. If we have been given an optical vortex and we interfere it with a plane Gaussian beam then we will get a fork pattern as resulting fringes [48]. By counting the number of prongs in fork we can determine the topological charge. In general if we have l+1 number of prongs in interference pattern then the topological charge of vortex will be l. The sign of this topological charge may be positive or negative and it is determined by the orientation of the fork if it is upside or down. Figure 2 shows an experimental arrangement, a Mach-Zhender interferometer, to find out the charge of the vortex. The figure also shows experimentally observed interference fringe pattern.

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Figure 2. Experimental setup to find out topological charge and sign of an optical vortex: M1, M2 mirrors B1, B2 - beam splitters A – aperture to select the vortex L – lens to pro ect the interference pattern onto the Screen that is recorded by a CCD camera. Inset shows interference patterns characteristic of topological charge +1 (left) and -1 (right) recorded by the CCD.

3.2. Orbital Angular Momentum nlike Hermite Gaussian beams, Laguerre Gaussian beams have an orbital angular momentum. The spin angular momentum of light or photon is always intrinsic but the orbital angular momentum of the photon may be either intrinsic or extrinsic [49, 50]. The orbital angular momentum of a light beam depends on the inclination of wavefront [51, 52]. The spin angular momentum of light is always  where degree of polarization  = 1 for right and left handed circularly polarized light respectively and it is zero for linearly polarized light. The spin angular momentum does not depend on the choice of aperture or calculation axis but the orbital angular momentum depends on aperture and calculation axis. The OAM is l per photon only when the aperture or calculation axis coincides with the axis of original beam. Allen et al [1] proposed the measurement of OAM of LG beam by measuring the torque applied to optical system by the transfer of OAM. The direct transfer of OAM to absorptive particles from a laser beam with phase singularity (optical vortex) has been observed [17]. They trapped an absorptive particle by an optical vortex and observed the rotation of particles in the trap. To confirm that this rotation is only due to orbital angular momentum associated with the helical wavefront they took the linearly polarized light having zero spin angular momentum. They also observed that on inverting the sign of optical vortex the direction of rotation of particles was also inverted. Presently, experimental techniques are available which can measure orbital angular momentum of single photon [53] along with spin angular momentum and total angular momentum [54] o f the photon. The transfer of OAM by a fractional optical vortex has also been observed [55]. It was observed that the rotating speed of particles deceases with increase of radial opening width of

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optical vortex even though the fractional charge has increased. nlike the integer order optical vortices, fractional optical vortices have radial opening which resists or even stops the rotation of particles.

3.3. Propagation The propagation of an optical vortex embedded in a host beam depends upon the shape of the host beam and its evolution as well as spatial properties of the wavefront dislocation i.e. the embedded vortex [56-61]. A canonical vortex, a vortex for which lines of zero crossing of the field form a right angle, sitting in a rotationally symmetric beam can propagate without change of its shape and location in free space [62]. However, it is not so for a non-canonical vortex for which lines of zero crossing of the field don t form a right angle [63]. For such a vortex topological charge associated with the vortex changes its sign even for free space propagation [64, 65]. The evolution of such vortices as they propagate in free space has been shown in Figure 3. Propagation of non-canonical vortices can be controlled by the noncanonical parameter associated with these vortices [65, 66]. It is not only the free space, but vortex propagation in nonlinear media has also been studied with interesting results [67-69].

4. Coherence Properties To study coherence properties of the optical vortices, we describe the Wigner function approach. We obtain the Wigner distribution function (WDF) of the vortex experimentally as well as theoretically and use it to extract information about the vortex. We show how the WDF can be used to discriminate an axial vortex from a non-axial vortex and further use transport properties of the WDF to study their propagation, which is found to be different for an axial and non-axial vortex.

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Figure 3. Evolution of canonical and non-canonical vortex as they propagate in free space: 1st row – canonical vortex (theoretical) 2nd row – canonical vortex (experimental) 3rd row – non-canonical vortex (theoretical) 4th row – non-canonical vortex (experimental). Experimental and theoretical plots are for same distances of propagation in each case.

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4.1. Wigner Function Approach In recent years optical vortices have found applications in a variety of fields that we would discuss in the next section. The study of their coherence properties therefore becomes very important. The study of Wigner distribution function (WDF), originally discovered in quantum mechanics [70, 71] can be quite useful for this purpose since it can provide coherence information in terms of the oint position and momentum (direction) phase-space distribution of the optical field [72]. In fact, the WDF has already been applied to study properties of an electromagnetic field in various optical systems [73-81]. For an optical vortex the WDF has been applied to few theoretical studies [75-77], however, no experimental work has been done except the one in our group [82]. Here, we present experimental results for the WDF of an optical vortex along with a closed analytical expression for the WDF of an optical vortex of order m [66]. Theoretical results for the vortex of order one were found in good agreement with our experimental results obtained for a vortex of order one produced in the laboratory. These results are general in nature, the WDF being a quasi-probability distribution can provide almost all the information about the vortex including spatial correlation singularity of the vortex [83]. In contrast to the field of a Gaussian beam and other fields studied earlier [78-81], which could be taken as functions of one transverse coordinate only, the field associated with an optical vortex is essentially a function of two transverse coordinates, making it a difficult proposition. We have used shearing Sagnac interferometer to obtain two-point correlation function or spatial coherence function [78-79] in both the transverse coordinates for an optical vortex embedded in a Gaussian beam. The WDF of the vortex is obtained by the FT of this measured two-point correlation function.

4.2. Wigner Function of an Optical Vortex Let us start with a vortex,

E ( x, y)

[( x  x 0 )  i ( y  y 0 )] m e



 x2  y2



V

2

(1)

that is a vortex of order m centered at (x0, y0) in the Gaussian beam of size V. Since the WDF of an optical field is given by [7]

) ( x, y, px, p y )

f

1 (2S )

2

f

³ ³

f f

E  x  R1 2, y  R 2 2 E

 x  R1

2, y  R 2 2 e



i R1 p x  R 2 p y



dR1 dR 2 ,

(2)

the WDF of the vortex, therefore becomes ) OV  x, y , p x, p y

 2  x 2  y 2 V 2   p x2  p 2y V 2 2 (  1) m m V 2 ( m  1) 2 E0 e e m 1 S2 ª 2  ( x  x 0 ) 2  ( y  y 0 ) 2  p x2  p y2 V 2 º Lm «   2( y  y 0 ) p x  2( x  x 0 ) p y » , 2 V 2 ¬« ¼»

(3)

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where x, y are position and px, py are con ugate momentum variables in the phase space. One can see that quantity written in angle brackets in Eq. (2) is mutual coherence function (MCF) of the field that can be measured experimentally with interferometry [78-79]. Fourier transform of that can provide us with the WDF of the field. An optical vortex of order one nested in a Gaussian beam and centered at origin was produced in the laboratory by passing the TEM00 beam of the He-Ne laser ( = 632.8nm) through the branch point of the CGH [8]. This vortex was introduced into a shearing Sagnac interferometer shown in Figure 4. Interferograms were recorded through a CCD by rotating the glass block in both the transverse directions that produces lateral shifts in respective directions. These interferograms were used to find out complex spatial coherence function as discussed in reference [78-79]. To find out imaginary part of the spatial coherence function, half wave plate (HWP) and quarter wave plate (QWP) were used as in the reference above. A Fourier transform of this two-point correlation function provides us with the Wigner distribution of the vortex [84]. To verify our experimental set up we compared theoretical and observed fringe patterns for a particular shear that is shown in the inset of Figure 4.

Figure 4. Experimental set up of shearing Sagnac interferometer for obtaining two-point correlation function and WDF for the vortex A: Aperture, BS: Beam splitter, M: Mirror, CGH: Computer generated hologram, HWP: Half wave plate, QWP: Quarter wave plate. Inset shows experimental and theoretical fringe patterns for a particular shear.

The WDF for the vortex in our experiment is given in Figure 5a. In this plot and others as well )ov values have been normalized with maximum value of )ov. To get a three dimensional plot of the Wigner function, x and y have been taken as constant (x=0, y=0) as it will give angular distribution across the beam axis. It must be noted that the Wigner function obtained is not a section of the four dimensional WDF, rather it is the WDF of the vortex with origin as the reference point since it has been obtained by introducing lateral shifts in x and y directions with respect to the origin. In the Figure 5a small wiggles away from the region of interest and sliced shape of the 3D-plot for the experimental WDF can be attributed to noise in the interferograms and finite number of data points for discrete Fourier transform.

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(a)

(b) Figure 5. Wigner distribution function of a vortex of order one (a) Experimental (b) Theoretical.

Most of the applications with optical vortices are based on orbital angular momentum (OAM) carried by the beams having vortices. However, OAM per photon in the beam will be integer or non-integer (in units of ) that depends on if the vortex is axial (centered at origin) or non-axial (shifted from the origin) [15]. Thus, axial nature of the vortex can affect intrinsic

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property of a vortex beam. The studies on vortex dynamics also show that an axial vortex will behave differently from a non-axial vortex [85]. We show that the WDF can be used to find out if the vortex is axial or non-axial for a vortex embedded in a Gaussian beam, the most commonly encountered vortex in experiments. Eq. (3) is an analytical expression for the WDF of an optical vortex of order m centered at (x0, y0) in a Gaussian beam that can be used to find the WDF of any order of the vortex. Now we use Eq. (3) to show how the WDF of an axial vortex is different from a nonaxial vortex. The calculations have been done keeping in mind the experimental set up in our laboratory that uses a red He-Ne laser having a beam size of 0.8 mm and the CCD camera with a pixel size of 13 µm. In all the WDF plots x and y have been taken x=0, y=0 as it will give angular distribution across the beam axis. For a complete angular distribution of the beam one will have to integrate the WDF over space variables, theoretically this is very much possible, however, not feasible experimentally. One can see that the WDF of a vortex (m=1) centered at origin (x0=0, y0=0) is symmetric (Figure 6a). Figures 6b – 6d show the WDF of vortices of order m=1, but shifted from the origin in different amounts. It is very much clear that the WDF is no more symmetric rather it is modulated in one direction. The utility of the concept can be seen in Figure 5, where an experimental result (Figure 5a) has been compared with one of the results obtained with Eq. (3) (Figure 5b). The experimental Wigner function is modulated in one direction, quite similar to the theoretical one that has been plotted for a vortex of order one shifted from the origin by a distance of (-65 µm, 39 µm). It shows that although we have tried to generate a vortex centered at the origin, however it is not so, it is slightly away from the origin. One can see that as the vortex moves away from the center, the WDF becomes more asymmetric. This can be interpreted as a single pure vortex state getting transformed to a superposition of many vortex states. In other words, the orbital angular momentum state of a photon in a non-axial vortex is superposition of various orbital angular momentum states unlike in an axial vortex where all the photons belong to the same single pure angular momentum state. The case of a vortex shifted by (260 µm, 273 µm) from the origin or center of the beam is quite interesting. The WDF shows positive as well as negative peaks side by side indicating almost equal contribution of even and odd orbital angular momentum states in the superposition that defines the photons in such a vortex. Please note the term (-1)m in Eq. (3) that makes the Wigner function negative for an odd order and positive for even. One can also make out the shift of the vortex with respect to the origin by looking at the Wigner function, for example see the direction of modulation of the WDF for vortices shifted by (130 µm, 26 µm) and (-130 µm, 26 µm) in Figures 6b and 6c. Once we know the Wigner function of the vortex, their propagation through any optical system can be written in terms of A, B, C, D matrices [86]. However, in the present work we would be dealing with free space propagation only.

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(a)

(b)

(d) Figure 6. Wigner function of the vortex of order (m=1) centered at origin (a), shifted from origin by (130 µm, 26 µm) (b), (-130 µm, 26 µm) (c), (260 µm, 273 µm) (d).

For free space propagation of a distance z = L, the WDF can be obtained by using the transform

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) ov  r  ( L / k ) p, p , z where r

( x, y ) and p

L ) ov  r , p, z

0

(4)

 p , p . Figs. 7a – 7d show the Wigner functions of an axial x

y

and a non-axial vortex for different distances of propagation. One can see that the WDF for an axial vortex remains same except narrowing of the peak in momentum space while for a non-axial vortex there is a marked change in the distribution, clearly visible in contour plots given in Figures 8a-8d. The narrowing of the peak of an axial vortex implies that as the vortex propagates away from the waist the beam size increases, consequently the angular spread i.e. the beam divergence reduces. In case of a non-axial vortex a marked change in distribution implies that with propagation coherence properties of an axial and a non-axial vortex change in different ways.

Figure 7. Wigner function of propagating vortices, (a) and (b) for a vortex at origin after propagating a distance of 1.26 meters, 2.52 meters (c) and (d) for a vortex shifted at (260 µm, 273 µm) after propagating a distance of 1.26 meters, 2.52 meters.

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Figure 8. Contour plots of Wigner function of propagating vortices, (a) and (b) for a vortex at origin after propagating a distance of 1.26 meters, 2.52 meters (c) and (d) for a vortex shifted at (260 µm, 273 µm) after propagating a distance of 1.26 meters, 2.52 meters.

4.3. Second Order Coherence Robert Hanbury Brown and Richard Q. Twiss discovered a new technique of two photon correlation or intensity correlation by performing several experiments in 1950s [87, 88]. They showed that there is a close relationship between the coherence and the correlation of intensity fluctuations in light beams. After their experiments the definition of coherence could also be given in terms of intensity fluctuations of light beams: two light beams are said to be coherent if the intensity fluctuations in them are correlated [89]. The second order coherence is defined in terms of the correlation function for intensity fluctuation and the degree of second order temporal coherence can be given as g(2)( ) =  I(t) I(t + ) ! /  I(t) !2 =  E ( t) E ( t+ ) E( t+ ) E( t) ! /  E (t) E(t) !2 where  I(t) ! is the long time average intensity.

(5)

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For all types of chaotic light the relationship between the degree of 1st and 2nd order coherence is g(2)( ) = 1+ _g(1)( )_2.

(6)

Second order coherence or intensity correlation function can be experimentally determined but the direct measurement of first order coherence function, g(1)( ), is not experimentally possible due to nature of detection process [90]. So with the help of relation (2) we can predict the first order coherence and first order spectral properties. To study second order coherence of optical vortices, we passed a Hermite Gaussian beam HG(0,0), TEM00 mode of a red He-Ne laser and a Laguerre Gaussian beam LG(0,1) (optical vortex of order 1) made out of the same laser through a rotating ground glass. Intensity correlation curves for them were obtained experimentally using a digital correlator (PhotoCor-FC), keeping the experimental conditions e.g. sample time, rotation speed of ground glass, intensity etc. same. We found that the intensity correlation for an optical vortex decays faster than the original TEM00 mode of the laser, which could be attributed to the complex phase structure and intensity profile of the vortex.

5. Applications Because of their specific spatial structure, associated helical wavefront and consequent orbital angular momentum, optical vortices find a variety of applications in the optical trapping of atoms [91], optical tweezers [92, 93], optical spanners [16, 94], micromachining [17, 21], optical communication [95], optical logic gates [96], and also in quantum information and computation [97, 98]. Optical trapping has potential applications in material science since an array of traps can be utilized to form various lattice structures and their properties can be studied by controlling the lattice parameters. Optical vortices have been used to trap particles both having refractive index less than surrounding and more than surroundings [99, 100]. The trapping and spanning has got applications in biological sciences to trap and move around live cells. At the same time optical spanning can be used to implement various gears and motors for micro-machining. The topological charge and its inversion can be made use of in optical communication and implementation of logic gates while orbital angular momentum states of photons in an optical vortex can be used to achieve multidimensional entanglement for quantum information and computation. The OAM carried by an optical vortex have also been used in super high density data storage [101] for imaging and meteorology [102-105]. Optical vortex can also be used as a coronagraph [106, 107] and vortex stellar interferometry is also proposed [108]. Below we will discuss some of the popular applications in detail.

5.1. Optical Tweezing and Spanning Optical tweezers pro ected as one of the top five technologies of the present century were first realized in 1986 [109, 110] at ATandT Bell laboratories. This modern optical tool [18, 19, 111-114] can trap ob ects as small as 5nm [115, 116] and can apply forces in pN range [117-

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119] with resolution as fine as 100aN [120-122]. These length scales and force ranges correspond to many important processes both intra- and inter-cellular like respiration, reproduction, signaling etc. In physics and chemistry these correspond to overlapping region of classical and quantum mechanics which is not yet fully understood. In this size and force regime even the statistical many body theories are generally inapplicable. It is because of these reasons optical tweezers are being used in a variety of areas from life sciences to colloidal sciences, from fluorescence studies to photonics and material sciences. Since the mechanism of optical trapping is such that the particles having refractive index more than the surroundings are attracted towards the region of higher intensity while the particles having refractive index less than the surrounding experience a force towards the region of lower intensity, it leads to a limitation on optical trap that only particles having refractive index more than the surroundings can be trapped by TEM00 mode of a laser. However, optical vortex has a region of lower intensity at the centre of its cross section so the particles having refractive index lower than the surroundings are attracted towards the dark region of the centre and get trapped [99, 100] while the particles having refractive index more than the surroundings are trapped by the higher intensity region at the periphery of the beam. The particle trapped by optical vortex can show rotational effects if it absorbs optical vortex photons. Though rotational effects can also be obtained in absorbing and birefringent particles using circularly or elliptically polarized light but an optical vortex beam can impart greater rotational speed to absorbing particles since it can have orbital angular momentum per photon many times . Valuable information about the quantum nature of helical beams and interplay of spin and orbital angular momentum of photon can be obtained by studying the motion of ob ects trapped by such beams [16, 123-125].

5.2. Micromachining Diffractive optical devices can be used to produce multiple optical traps with arbitrary intensity profile [126-129]. Several computational methods have been developed to compute diffractive elements for producing desired intensity distribution in multiple traps [126, 130, 131]. In general diffractive elements modulate both amplitude and phase of the incident light but there are diffractive optical elements as well that primarily modify the phase but not the amplitude, which are termed as kinoforms [126]. With the development in SLM technology and algorithms for calculating real time holograms, generation of even 400 optical traps, having higher order beams in them is possible [17-19, 132].

5.3. Logic Gates It has been shown that there are various optical elements which can invert the charge of an optical vortex [96]. sing this property of optical elements we implement a CNOT gate. In the schematic diagram of Figure 9 to implement a CNOT gate, the most interesting optical element is the beam splitter BS1. It reflects a part of the vortex beam and acts like a mirror by inverting its charge. At the same time another part that is transmitted remains the same as the input vortex. The transmitted vortex from BS1 is interfered with a reference beam obtained from the He-Ne laser used for generating optical vortex to find out the charge of the vortex. If

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the interferogram recorded on the CCD corresponds to a vortex of charge +1, a trigger is sent to enable OECI2 otherwise it remains disabled. OECI1 is used as and when required to produce the desired input. The input and output clearly show that the scheme can implement a CNOT gate. The two sets of input and the corresponding output form the truth table of the gate.

Figure 9. Schematic diagram to implement CNOT gate: OECI – optical element for charge inversion BS – beam splitter. Please see the text for details.

It s important to note that the charge states are not affected by the environment and at the same time they can be initialized and read-out with high efficiency. Also scaling of operations can be done easily with proper selection of beam splitters. These properties of the present implementation make it a strong candidate on the roadmap to quantum computation. It should be noted that the implementation described above does not use a single photon source or quantum entanglement, which is required for quantum computation. However, methods can be devised to produce entangled charge states [97, 133] for quantum implementation.

Conclusion In conclusion, we can say that the field of optical vortices is quite interesting. Vortices being ubiquitous in nature starting from Bose Einstein condensates [134-137], superfluids [138], superconductors [139] to water reservoirs [140], air columns [141] and cosmos [142], study of optical vortices can be quite useful in understanding some of the phenomena associated with these systems. It is true that the systems mentioned above are quite different, however, topological nature of the vortex being same in all the cases, they could find similarity in some respect. It would be heartening if scientists working on vortices and phase singularities in

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different branches like hydrodynamics, condensed matter, nonlinear dynamics and optics could come together on a common platform which will synergize all the fields. Optical vortices have a peculiar property of having a phase singularity in the coherence function itself [83, 143], although we have not touched upon that, a lot of remain to be explored pertaining to their behavior and more importantly their useful applications. Light scattering by plane wave has been worked out theoretically as well as experimentally and finds lot of applications, however, light scattering with helical wavefront has not been studied either theoretically or experimentally. It would be worthwhile to pursue the field that can supplement the knowledge gained by the plane wave scattering. Since the orbital angular momentum (OAM) carried by the beam has been found even at quantum level which shows things are quite subtle and needs to be thought over deeply. In parallel with the wide spread applications of optical vortices, theoretical and experimental work on fundamental aspects of optical vortices is being actively pursued.

Acknowledgements Authors acknowledge useful discussions with Prof. G. S. Agarwal and . Baner i. They remain thankful to Prof. L. Allen for his constant encouragement.

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INDEX A absorption, viii, 36, 63, 170, 172, 191, 192, 205, 206, 207, 208, 217, 218, 230, 233, 238, 242, 245, 246, 285, 287, 291, 292, 294, 295, 298, 299, 302, 308, 312, 318, 323 absorption coefficient, 192, 285, 292, 298, 323 absorption spectra, 298, 299 AC, 169, 182 accounting, 97, 285 accuracy, 336 achievement, 166, 192 acoustic, 276 active feedback, 5 actuators, 122, 123, 124, 125, 126, 133, 136, 137, 141, 143 Adams, 84 adaptive control, 177 adjustment, 22, 23, 40, 153, 177, 250 Ag, 153, 157, 158, 173, 174, 175, 176, 179, 180, 186, 188, 189, 194, 284, 297, 320, 321, 322, 324 aggregation, 173 aid, 204, 228, 232, 234, 235, 236, 251, 255 AIP, 359 air, 88, 298, 305, 323, 378 algorithm, viii, 117, 118, 119, 120, 127, 149, 150, 266, 268, 269, 270, 271, 278, 280, 337 alkali, 176 alternative, 89, 151, 152, 166, 320, 321, 332 alternatives, 88 alters, 121 aluminium, 64 aluminum, 85 AM, 60, 361, 371, 379 amplitude, vii, 1, 2, 3, 4, 8, 15, 16, 17, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36,

38, 49, 53, 54, 59, 60, 119, 200, 202, 204, 208, 215, 216, 222, 229, 240, 245, 258, 264, 265, 314, 315, 323, 324, 334, 338, 342, 344, 356, 364, 377 Amsterdam, 381 angular momentum, xii, 215, 232, 257, 361, 362, 366, 371, 372, 376, 377, 379 animations, 276 anisotropy, 286, 299, 302 annealing, 294, 298, 320, 321 anomalous, 25, 37, 38, 39, 42, 50, 59 antimony, 183, 184 application, viii, ix, 84, 87, 113, 118, 119, 133, 151, 152, 153, 162, 167, 172, 178, 179, 184, 193, 200, 316, 343 applied mathematics, 359 applied research, 288 argon, 173 argument, 110, 111 artificial, x, 101, 177, 283, 284, 320 AS, 290 aspect ratio, 286, 287 assumptions, 16, 27 astigmatism, 75, 119, 123, 129, 130, 133 astronomy, 119 asymmetry, 17, 109, 252, 331 asymptotic, 347, 353 asymptotically, 350 atmosphere, 294, 308, 311, 320 atomic physics, 101 atoms, viii, ix, 63, 64, 66, 67, 68, 69, 70, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 153, 157, 158, 159, 160, 163, 172, 173, 175, 179, 185, 191, 192, 199, 202, 205, 220, 230, 233, 245, 376 attention, viii, xi, 88, 119, 152, 200, 203, 246, 288, 303, 316, 329, 330, 331 Au nanoparticles, 298

384

Index

Australia, 1 autocorrelation, 95, 98 Autocorrelation, 21, 22, 96 availability, 126, 133, 151, 166 avoidance, 88, 110 azimuthal angle, 222, 223, 224, 241

B backscattering, 88 Bali, 381 band gap, 304, 307, 309, 316, 334 bandgap, xi, 284, 288, 303, 306, 309, 311, 313, 316, 317 bandwidth, 2, 6, 7, 8, 13, 16, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 36, 39, 49, 59, 88, 91, 92, 94, 95, 96, 97, 98, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 158, 178, 179, 183, 234, 312 barium, 94, 185, 298 barrier, 157, 158, 186, 191 beams, viii, ix, 63, 64, 66, 67, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 93, 95, 96, 199, 200, 201, 202, 203, 204, 206, 207, 208, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 290, 292, 293, 309, 338, 362, 364, 365, 366, 371, 375, 377 beating, 36, 59, 330, 341, 350, 355 behavior, xi, 3, 105, 108, 110, 165, 167, 186, 189, 190, 214, 239, 251, 284, 295, 297, 312, 322, 329, 340, 349, 350, 351, 356, 379 behaviours, 56 Beijing, 117 bending, 331 beryllium, 180 Bessel, ix, 199, 200, 201, 202, 203, 204, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 222, 223, 224, 225, 232, 233, 234, 235, 236, 237, 238, 239, 241, 242, 243, 245, 246, 247, 248, 250, 251, 252 bias, 8, 14, 58 biaxial, 101 binding, 39, 173, 331 binding energy, 173 biological, xi, 89, 119, 361, 376 biology, 152 biophysics, 362 birefringence, 16, 26, 61, 118, 119, 149, 289, 339 birth, 264

black, 111, 112, 177, 363 blocks, 175, 176, 269, 311, 317 borate, 94 boron, 154, 155, 156, 185, 192 Boron, 154 Bose, 378 boundary conditions, 66, 69, 70, 74, 75, 81, 82, 332, 337 bounds, 122 BPM, 101, 102, 106, 332, 336, 338, 339 Bragg grating, 61 breakdown, 206 broad spectrum, 25, 102 broadband, viii, 87, 89, 101, 103, 105, 109, 110, 113, 177 Broadband, 100 buffer, 270, 271, 279 Bulgaria, 360 burning, 224, 340

C calibration, 309 Canada, 164 capacity, 279 capillary, 202 carbon, 185, 189 carrier, 41 cast, 363 cathode, 203 Cauchy problem, 332 CDA, 101 cell, 203, 204, 205, 206, 207, 208, 220, 223, 228, 234, 238, 251, 254, 255, 364 cesium, 64, 84, 101 channels, 205, 279, 330, 333 chaotic, 36, 59, 376 charge coupled device, 363 charged particle, 190, 192, 331 chemical, 152 chemical properties, 152 chemistry, 377 Cherenkov, 221 China, 117, 136, 283 Chinese, 117 chromium, 64, 169, 181, 182, 184, 193 chromosome, 120, 121 chromosomes, 120 circularly polarized light, 232, 251, 362, 366 cis, 339 CL, 90 cladding, 88, 331, 332, 333, 335, 336, 338, 339, 344, 356

Index 8classical, 64, 139, 145, 148, 200, 377 classified, 37, 311 clusters, 173, 176, 177, 302, 325 Co, 332, 335 coatings, 96 coherence, xi, 42, 54, 89, 151, 166, 178, 183, 184, 232, 233, 234, 235, 240, 297, 361, 367, 369, 370, 374, 375, 376, 379, 381 collateral, 113 collisions, 49 Colorado, 271 colors, 101 coma, 123, 133 combined effect, 112 commercial, 154, 331 communication, 2, 14, 33, 58, 118, 270, 275, 304, 376 communication systems, 2, 14, 33, 58 community, 119 compensation, 91, 95, 149 competition, 192, 331, 333, 340, 342, 356 complementary, 101 complexity, 14, 23, 59, 97, 120, 266 complications, 100, 333 components, x, 6, 7, 8, 14, 17, 49, 51, 89, 90, 91, 93, 94, 95, 99, 100, 102, 103, 105, 109, 111, 173, 177, 218, 219, 222, 223, 224, 225, 227, 228, 231, 232, 238, 243, 251, 277, 278, 289, 290, 292, 295, 297, 321, 322, 356 composite, x, 283, 285, 288, 294, 295, 296, 298, 299, 300, 301, 303, 320, 321, 322, 323 composition, 4, 223, 294, 304, 305, 308, 317, 335 compressibility, 78 compression, 23, 59, 75, 77 computation, xi, 266, 268, 270, 275, 361, 376, 378 computer, x, 93, 127, 133, 134, 203, 263, 264, 265, 266, 268, 269, 270, 272, 275, 278, 281, 293, 362, 363, 364 computers, 158, 268, 269, 270 computing, 269, 275 concentration, 25, 101, 133, 152, 157, 159, 163, 165, 178, 184, 185, 188, 189, 190, 191, 200, 294, 336 condensed matter, 379 conditioning, 78, 84 conduction, 284, 296 configuration, viii, 4, 63, 64, 65, 68, 72, 73, 74, 75, 76, 77, 78, 79, 81, 83, 84, 113, 119, 134, 141, 191, 217, 225, 226, 228, 233, 241, 242, 290, 297, 322 confinement, 285, 298, 302 Congress, iv

385 conjugate gradient method, 337 conjugation, 289 conservation, 36, 220, 232 construction, 4, 14, 25, 91, 330, 331, 333, 334, 335, 336, 337, 338, 349, 350, 354 control, 3, 4, 5, 14, 15, 24, 36, 39, 59, 62, 93, 94, 118, 119, 120, 133, 134, 136, 149, 176, 178, 183, 228, 232, 251, 316, 318, 330, 332 controlled, viii, xi, 61, 93, 100, 117, 118, 136, 150, 204, 219, 223, 228, 280, 293, 361, 367 convergence, viii, 117, 122, 123, 127, 129, 132, 140, 145, 337, 338 conversion, viii, ix, 37, 59, 60, 87, 88, 89, 94, 95, 96, 97, 98, 100, 101, 102, 104, 110, 111, 113, 151, 152, 153, 155, 156, 157, 158, 161, 162, 163, 166, 167, 169, 171, 173, 175, 178, 179, 183, 184, 192, 193, 202, 203, 213, 214, 216, 229, 233, 240, 258, 363 convex, 72, 276, 279 cooling, viii, 63, 85, 133, 185, 297, 323 cooling process, 297, 323 copper, 94 corn, 280 correlation, xii, 12, 22, 62, 163, 202, 232, 237, 240, 361, 369, 370, 375, 376 correlation function, 369, 370, 375, 376 correlations, 234, 237, 238 cosine, 266, 268, 273, 274 couples, 218 coupling, 7, 16, 26, 90, 123, 141, 255, 317, 330, 332, 333, 335, 340, 341, 345 coverage, viii, 87, 88, 89, 99, 100, 113 crack, 364 critical value, 352, 356 cross-phase modulation, 304 crystal, ix, x, 89, 91, 93, 94, 96, 97, 98, 99, 100, 101, 105, 106, 107, 108, 109, 110, 111, 119, 133, 137, 183, 263, 264, 283, 284, 303, 304, 305, 307, 308, 309, 318, 322, 332, 335, 364 crystalline, 331 crystals, 88, 89, 93, 94, 97, 98, 100, 101, 106, 110, 111, 112, 116, 179, 184, 303, 304, 335, 362, 365 curiosity, 362 cycles, 16, 26, 27, 107, 362

D DCA, 39 DCF, 26, 35 de Broglie, 73 decay, 56, 160, 295, 297, 336 decay times, 160

386 decoding, 120 defects, 303, 304, 316, 318, 335 definition, 190, 375 degradation, 331 degree, 88, 122, 133, 233, 234, 250, 251, 289, 366, 375, 376 delays, 154, 162, 185, 186, 187, 188, 189, 190 delta, 343 demand, 2, 118 density, 7, 83, 118, 136, 159, 165, 178, 185, 192, 206, 216, 217, 219, 220, 227, 241, 242, 243, 245, 309, 376 depolarization, 149 deposition, 84, 85, 158, 294, 318 deposition rate, 294, 318 depression, 350 designers, 2 detection, 5, 7, 14, 101, 254, 293, 376 deviation, 15, 23, 35, 167, 315, 316, 318, 350 DF, xi, 361, 367, 369 diamond, 19 diaphragm, 236 dielectric, vii, x, xi, 104, 172, 283, 284, 285, 287, 288, 296, 297, 298, 300, 301, 303, 304, 305, 306, 308, 311, 312, 314, 315, 317, 320, 323, 327, 364 Dielectric, 284, 320 dielectric constant, xi, 104, 284, 285, 287, 288, 296, 298, 301, 303, 304, 305 dielectric function, 287 dielectric materials, 308, 311 dielectrics, x, 283, 288, 296, 303, 316 diffraction, vii, xi, 1, 134, 136, 143, 145, 150, 200, 227, 234, 264, 277, 294, 329, 330, 332, 333, 335, 336, 363, 365 diffusion, 297, 323 diode laser, 25, 91, 334 diodes, 88, 263, 276, 331 dipole, viii, 63, 64, 66, 69, 72, 73, 77, 79, 81, 85, 216, 245, 285, 286 Dirac delta function, 242 direct measure, 206, 208, 376 disabled, 378 dislocation, 365, 367 dislocations, 362 dispersion, vii, 1, 4, 24, 25, 26, 27, 34, 37, 38, 39, 42, 50, 51, 53, 59, 62, 91, 95, 100, 102, 107, 111, 172, 252, 253, 255, 336, 346 displacement, 17, 27 disposition, xi, 329 distilled water, 235 distortions, 330, 336

Index distribution, xi, 15, 17, 22, 24, 27, 33, 36, 59, 65, 66, 68, 70, 71, 73, 75, 77, 79, 80, 84, 119, 127, 128, 129, 130, 131, 132, 134, 135, 136, 138, 141, 142, 143, 144, 145, 146, 149, 152, 160, 163, 164, 166, 167, 169, 170, 171, 174, 180, 183, 192, 202, 211, 222, 242, 289, 296, 298, 302, 303, 305, 310, 311, 314, 321, 323, 324, 330, 331, 333, 336, 340, 353, 361, 362, 364, 367, 369, 370, 371, 372, 374, 377 distribution function, xi, 361, 367, 369, 371 divergence, 35, 36, 59, 67, 70, 75, 77, 82, 83, 101, 151, 154, 184, 200, 209, 236, 242, 350, 374 division, x, 61, 263, 278, 279, 280, 281 dominance, 350, 351, 352, 353 dopant, 88, 332 dopants, 88, 89 doped, 14, 16, 23, 25, 39, 43, 50, 51, 61, 88, 90, 93, 101, 102, 300, 304, 331, 334, 335, 357 doping, 101, 133 duration, ix, 2, 4, 8, 14, 25, 47, 58, 91, 92, 98, 100, 151, 152, 153, 163, 177, 183, 184, 185, 186, 187, 188, 193, 203, 297, 299, 309, 312, 318, 321 dynamical properties, 294

E earth, 88, 331, 335, 357 eigenvalue, 336, 337, 344 eigenvalues, 336, 337, 355 Einstein, 378 election, 150 electric current, 280 electric field, vii, x, xi, 283, 284, 285, 301, 303, 306, 307, 310, 311, 314, 315, 316, 319, 323, 324, 364 electrical, 7, 8, 302, 311, 339 electromagnetic, 172, 182, 258, 285, 298, 301, 303, 369 electromagnetic fields, 258 electromagnetic wave, 285, 301, 303 electromagnetic waves, 303 electron, x, 151, 152, 157, 159, 163, 165, 169, 172, 173, 178, 179, 182, 183, 184, 190, 192, 194, 283, 284, 294, 296, 297, 302, 308, 311, 322 electron beam, 308, 311 electron density, 165, 178, 190, 284 electronic, iv, x, 4, 5, 7, 8, 14, 59, 283, 304 electronics, vii electron-phonon, 284, 297 electron-phonon coupling, 297

Index electrons, 155, 157, 163, 165, 172, 173, 176, 178, 179, 182, 191, 192, 284, 296, 299, 302, 304 electrostatic, iv emission, viii, 63, 88, 102, 111, 155, 159, 160, 162, 167, 168, 169, 171, 175, 177, 188, 200, 203, 204, 206, 207, 208, 215, 219, 220, 221, 231, 232, 233, 234, 238, 239, 241, 245, 250, 254, 255, 341, 342, 343, 350 employment, 14, 101 encoding, viii, 117, 119, 120 Encoding, 120 encouragement, 379 energy, 18, 19, 28, 29, 30, 31, 32, 35, 36, 54, 59, 64, 88, 129, 131, 132, 143, 145, 151, 155, 157, 158, 163, 165, 167, 172, 173, 178, 179, 182, 183, 185, 188, 190, 192, 200, 203, 205, 206, 211, 212, 213, 214, 215, 222, 228, 233, 238, 240, 249, 250, 252, 253, 294, 296, 297, 300, 304, 339, 342, 356 engineering, 15, 24, 100, 119, 150, 332 Enhancement, v, 60, 167, 169, 213, 283, 294, 297, 303, 307, 314 entanglement, 376 entertainment, 281 envelope, 92, 95, 211, 217, 218, 224, 227 environment, 8, 294, 321, 378 epoxy, 175 equilibrium, 16, 26, 27, 36, 39, 59 equilibrium state, 39 equipment, 7, 97 erbium, 16, 23, 39, 61, 62, 88, 89, 90, 113 ES, 305 Estonia, 199 ethanol, 234 Europe, 357 evaporation, 174, 308, 311 evidence, 101, 110, 111, 182 evolution, xi, 24, 38, 39, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 58, 121, 220, 221, 235, 238, 240, 249, 255, 297, 310, 312, 318, 329, 330, 336, 341, 342, 356, 367 excimer lasers, 203 excitation, ix, x, 16, 26, 100, 101, 151, 152, 155, 160, 161, 162, 167, 172, 188, 189, 192, 194, 195, 199, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 232, 233, 234, 238, 239, 240, 241, 242, 243, 244, 245, 246, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 283, 296, 297, 298, 299, 300, 301, 302, 303, 307, 322, 323 exotic, 151

387 experimental condition, 167, 191, 218, 321, 376 expert, iv exploitation, 89 exponential, 315 extinction, 288, 314 extrinsic, 366 eye, 88 eyes, x, 263, 278, 280

F fabricate, 323 fabrication, 84, 97, 136, 303, 323 family, 16, 27, 202, 204 fax, 87 feedback, 4, 5, 8, 14, 16, 58, 61, 118 Fermi, 284, 296, 299 Fermi energy, 284, 296 Fermi surface, 296, 299 FFP, 3, 24 FFT, 127 fiber, viii, xi, 2, 36, 37, 38, 39, 40, 42, 43, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 87, 88, 89, 90, 91, 92, 99, 100, 102, 113, 303, 365 fibers, 89 fibre laser, vii, 1, 2, 3, 16, 17, 24, 25, 27, 36, 59, 60, 61 film, 294, 295, 296, 298, 300, 302, 303, 305, 309, 310, 315, 320, 321, 322, 323, 324 films, 294, 295, 298, 299, 300, 301, 308, 311, 312, 318, 320, 321, 323 filters, 6, 93, 307, 316 finite volume, 209 fitness, 120, 122, 128, 129, 130, 132 flexibility, 100, 228, 258 floating, 269 flow, xi, 361 fluctuations, 17, 49, 51, 375 fluorescence, 377 fluoride, 186 focusing, 64, 65, 67, 68, 70, 71, 72, 79, 80, 84, 85, 93, 95, 98, 100, 101, 154, 155, 157, 185, 186, 191, 192, 203, 211, 228, 235, 236, 237, 251, 254 Fourier, 25, 139, 140, 332, 370 FP, 26 FPGA, v, ix, 263, 265, 266, 267, 269, 270, 271, 273, 274, 275, 277, 279, 280, 281 freedom, 330 FSP, 154 fusion, ix, 117

388

Index

FWHM, 13, 40, 91, 94, 95, 96, 97, 98, 99, 102, 109, 110, 179, 203, 225, 226, 234, 235, 242, 252, 255, 299, 321

growth mechanism, 171 guidance, 238

H G GaAs, 153, 169, 170, 171 gallium, 169 gas, ix, 88, 152, 153, 155, 166, 175, 176, 178, 190, 191, 193, 194, 199, 200, 202, 203, 204, 205, 206, 207, 208, 210, 211, 214, 215, 217, 218, 220, 221, 223, 228, 229, 230, 238, 241, 243, 244, 245, 246, 247, 249, 254, 255 gas jet, 178, 191, 193, 194 gases, 152, 166, 199, 200, 204, 205, 206, 246 Gaussian, v, ix, 6, 16, 17, 22, 27, 29, 30, 31, 32, 33, 40, 49, 51, 62, 65, 67, 84, 93, 98, 109, 111, 145, 199, 200, 202, 203, 204, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 233, 234, 235, 236, 237, 238, 240, 246, 249, 251, 258, 362, 363, 365, 366, 369, 370, 372, 376 GAUSSIAN, 199 gene, 120, 121 generation, vii, viii, ix, 2, 6, 8, 14, 25, 60, 78, 84, 85, 87, 89, 94, 97, 100, 101, 102, 103, 107, 108, 111, 113, 121, 122, 136, 151, 152, 153, 154, 156, 157, 159, 161, 162, 163, 164, 165, 166, 169, 172, 173, 174, 175, 176, 177, 179, 180, 183, 185, 186, 188, 189, 190, 193, 199, 200, 201, 202, 205, 206, 207, 208, 209, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 224, 225, 228, 229, 230, 232, 233, 240, 241, 242, 245, 251, 257, 258, 363, 377 genetic, viii, 117, 118, 119, 120, 149, 150 genetic algorithms, 149 genetics, 119 Germany, 87, 357 GH, 276 Ginzburg-Landau equation, 61 glass, 64, 91, 111, 173, 204, 294, 308, 309, 311, 321, 331, 334, 335, 370, 376 glasses, 88, 335 gold, 160, 161, 162, 165, 288, 294, 298, 299, 300, 301, 302 gold nanoparticles, 294, 298, 299 graph, 17, 27 gratings, 94, 97, 100, 110, 169, 183, 185 grazing, 155 grids, x, 141, 263, 266, 269, 275, 276, 280 groups, 186, 296, 311 growth, ix, 106, 151, 152, 153, 155, 157, 158, 159, 161, 163, 165, 166, 171, 190, 191, 192, 343, 352, 356

Hamiltonian, 336, 339 harmonics, ix, 99, 151, 152, 153, 154, 155, 156, 157, 158, 159, 161, 162, 163, 165, 166, 167, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 186, 189, 190, 191, 192, 193, 194, 199, 200, 201, 202, 205, 206, 213, 215, 301, 345 HDL, 269 head, 133 heat, 296, 297 heat capacity, 296 heat transfer, 297 heating, 94, 156, 185, 186, 188, 205, 294, 298, 331, 335 height, 173 helium, 64 Helmholtz equation, 200, 339 hexagonal lattice, 334, 335, 356 high resolution, 276 Holland, 381 hologram, x, 263, 264, 268, 269, 362, 363, 365, 370 holograms, 363, 377 homogeneous, 200, 224, 312, 318, 330 homogenous, 298 Honda, 282 host, 269, 270, 272, 275, 297, 298, 323, 335, 367 human, x, 263, 278, 280 hybrid, 37, 62, 88 hydrodynamics, 362, 379 hypothesis, 316 hysteresis, 122

I id, 116 identity, 337, 341 ILO, 290 imagery, 281, 285 images, 381 imaging, 89, 376 immersion, 235 immunity, 201 implementation, vii, 1, 88, 113, 120, 376, 378 incidence, 64, 110, 155, 311, 316, 324 independence, 285, 287 India, 63, 164

Index Indian, 63 indices, 209, 298, 301, 308, 345, 362 indium, 85, 167, 168, 169, 171, 184, 191, 192, 193 industrial, 89, 118, 119, 133 industry, 88 inequality, 3, 23, 232 inert, 205 inertial confinement, ix, 117 infinite, 8, 224, 242 Information Technology, 328 infrared, viii, 87, 88, 89, 91, 92, 93, 94, 97, 98, 99, 102, 106, 107, 109, 110, 113, 133, 176, 178, 335 infrared light, 102 inhomogeneity, 298 initiation, 206, 228 injury, iv Innovation, 329 insertion, 16, 26 insight, 33, 208, 230, 350 instabilities, 158 instability, 2, 3, 23, 33, 35, 36, 59, 340 instruments, xi, 329, 356 integration, 209, 301, 340, 342, 347, 349 integrity, 172 intensity, viii, ix, xi, xii, 2, 4, 5, 6, 7, 14, 25, 33, 51, 58, 60, 63, 64, 65, 67, 70, 71, 72, 75, 77, 79, 80, 83, 84, 88, 92, 95, 98, 99, 104, 105, 106, 117, 118, 119, 133, 136, 138, 141, 142, 143, 145, 146, 149, 152, 154, 156, 157, 158, 159, 160, 161, 162, 163, 165, 166, 167, 169, 170, 171, 172, 173, 174, 175, 176, 177, 179, 180, 182, 183, 185, 191, 192, 200, 202, 204, 205, 206, 208, 211, 213, 214, 215, 217, 220, 231, 232, 234, 235, 238, 240, 245, 248, 249, 250, 254, 255, 289, 296, 297, 301, 302, 307, 309, 310, 312, 314, 315, 318, 319, 323, 329, 330, 335, 336, 338, 340, 341, 342, 344, 346, 347, 353, 355, 356, 361, 363, 365, 375, 376, 377 interaction, viii, ix, xi, 37, 41, 63, 64, 67, 68, 70, 71, 72, 75, 76, 77, 78, 83, 84, 85, 89, 93, 97, 98, 100, 103, 104, 106, 107, 108, 109, 110, 113, 133, 154, 157, 160, 162, 165, 173, 185, 190, 192, 193, 194, 199, 208, 217, 219, 220, 225, 228, 233, 255, 258, 284, 304, 309, 330, 338, 339, 340, 341 Interaction, 67, 70, 71, 84, 316 interactions, vii, 1, 38, 47, 62, 100, 104, 108, 192 interface, 307 interference, 2, 3, 5, 8, 24, 64, 79, 84, 125, 215, 216, 223, 224, 229, 233, 234, 245, 246, 249,

389 252, 256, 258, 264, 265, 301, 355, 363, 365, 366 interpretation, 208, 256 intrinsic, xii, 213, 305, 361, 366, 371 inversion, 341, 376, 378 Investigations, 150 ionic, 152, 153, 160, 161, 166, 167, 169, 170, 171, 172, 176, 177, 180, 183, 192 ionization, ix, 157, 158, 159, 163, 164, 165, 176, 178, 179, 182, 184, 190, 191, 192, 199, 202, 203, 205, 206, 207, 208, 209, 210, 215, 216, 217, 218, 224, 230, 233, 234, 238, 239, 240, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258 ionization potentials, 163, 164, 179, 190 ions, xi, 25, 88, 91, 152, 153, 154, 157, 158, 159, 160, 162, 163, 164, 165, 166, 172, 179, 180, 182, 185, 186, 188, 190, 191, 192, 193, 329, 336 IR, 335 iron, 64, 85 irradiation, 160, 163, 175, 185 IS, 291 isotropic, 153, 230, 232 isotropic media, 153, 230, 232 iteration, 139, 140, 144

J January, 269 Japan, 164, 263

K kinetic energy, 169 kinetics, 341 King, 359 Korean, 262 krypton, 246

L L1, 154, 198, 276, 279 L2, 154, 276, 279, 291 Langmuir, 325 laser, vii, viii, ix, x, xi, 1, 2, 3, 4, 5, 6, 8, 14, 15, 16, 17, 22, 23, 24, 25, 26, 27, 33, 34, 36, 37, 39, 41, 43, 47, 49, 50, 51, 53, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 84, 85, 87, 88, 89, 90, 100, 101, 102, 110, 111, 113, 117, 118, 119, 120, 122, 123, 133, 134, 135, 136, 137, 138,

390 145, 149, 150, 151, 152, 153, 154, 155, 157, 158, 159, 160, 163, 165, 167, 169, 170, 171, 172, 173, 175, 176, 178, 179, 180, 182, 183, 184, 185, 186, 188, 189, 190, 191, 192, 193, 194, 195, 199, 200, 202, 203, 204, 205, 206, 207, 211, 212, 213, 215, 223, 228, 230, 232, 233, 234, 235, 236, 238, 239, 240, 241, 245, 246, 249, 250, 251, 252, 254, 255, 258, 277, 283, 285, 293, 294, 297, 299, 303, 309, 310, 311, 312, 318, 320, 321, 323, 329, 331, 332, 333, 334, 335, 339, 340, 341, 342, 343, 350, 356, 362, 363, 364, 365, 366, 370, 372, 376, 377 laser ablation, 85, 163, 184, 185, 190, 191, 194 laser radiation, viii, xi, 63, 90, 101, 153, 158, 160, 171, 176, 191, 192, 329 lasers, viii, xi, 2, 4, 7, 36, 37, 59, 60, 61, 62, 87, 88, 89, 100, 102, 118, 119, 123, 133, 149, 151, 163, 166, 173, 176, 179, 182, 192, 193, 200, 204, 205, 206, 215, 255, 278, 303, 329, 330, 331, 332, 335, 341, 343, 356 lattice, xi, 296, 297, 323, 329, 330, 335, 376 lattice parameters, 376 lattices, 332 law, 213 LCDs, 277, 281 lead, 91, 165, 166, 177, 184, 186, 189, 211, 296, 302 LED, 276, 279 lens, viii, 63, 64, 65, 68, 72, 80, 84, 85, 133, 145, 154, 156, 185, 191, 203, 204, 208, 228, 235, 236, 237, 251, 255, 276, 279, 293, 366 lenses, 64, 68, 79, 90, 93, 119, 154, 204, 276, 279, 363 LH, 306, 316, 317, 318, 319 life sciences, 377 lifetime, 322, 331 light beam, 72, 118, 216, 235, 362, 364, 366, 375 light emitting diode, 278 light scattering, 321, 325, 379 light transmission, 304 light-emitting diodes, x limitation, 72, 100, 107, 118, 363, 377 linear, xi, 3, 16, 24, 25, 26, 36, 89, 94, 96, 98, 100, 137, 167, 223, 224, 232, 233, 238, 254, 255, 256, 285, 289, 292, 304, 307, 311, 316, 319, 320, 321, 323, 329, 334, 335, 338, 365 linear dependence, 98 linear systems, 16, 27 liquid crystals, 364 literature, 119 lithium, 16, 26, 101, 102, 185, 186 lithography, viii, 63, 64, 75, 84, 85

Index localization, x, 100, 283, 284, 303, 304, 307, 311, 314, 316, 319, 320, 331 location, 65, 66, 70, 73, 75, 82, 192, 217, 218, 225, 233, 244, 245, 249, 367 London, 360 long-term, 2, 3, 8, 23 losses, xi, 7, 96, 329, 333, 339 low-intensity, 191 low-power, 192, 343 lying, 167, 206

M M1, 79, 305, 306, 366 machine learning, 149 Madison, 383 magnesium, 186 magnetic, iv, 257, 305, 339 magnetic field, 305 manganese, 153, 163, 165, 181, 182, 184, 185, 190 Manganese, 163 manipulation, 85 manufacturing, 334, 335 Mars, 360 mask, viii, 63, 64, 204, 206, 211, 212, 213, 224, 225, 226, 227 material sciences, 377 mathematical, 341, 343 Mathieu function, 202 matrix, ix, 84, 117, 124, 125, 126, 133, 285, 287, 288, 296, 297, 298, 301, 305, 306, 308, 309, 310, 312, 313, 314, 316, 320, 323, 330, 336, 337, 340 MCP, 155 measurement, 7, 8, 12, 95, 125, 137, 138, 145, 146, 160, 174, 289, 293, 294, 297, 309, 312, 318, 320, 321, 323, 366 mechanical, iv media, vii, 25, 100, 103, 109, 113, 152, 153, 166, 175, 178, 202, 204, 205, 206, 301, 316, 330, 364, 367 median, 176 medicine, 152 melting, 173, 175, 331 melting temperature, 175 memory, vii metal nanoparticles, x, 283, 297, 300, 302 metals, 175 metric, 118 microparticles, 362 microscope, 150, 173, 204, 235, 236 microscopy, 101, 113, 152

Index Microsystem, 85 military, 331 mirror, viii, 14, 79, 117, 118, 119, 131, 133, 149, 150, 204, 228, 264, 279, 293, 333, 342, 365, 377 mixing, viii, 87, 89, 91, 94, 95, 98, 101, 103, 105, 107, 108, 109, 111, 113, 245, 249, 253, 258, 289, 330, 356, 365 ML, 4 MLL, 7, 8 model system, 353 modeling, xi, 158, 329, 332, 336 models, 33, 158, 253 modulation, x, 2, 3, 5, 6, 15, 17, 22, 23, 36, 37, 38, 39, 40, 46, 47, 49, 53, 54, 55, 58, 59, 60, 62, 159, 186, 191, 232, 283, 302, 333, 334, 341, 372 modules, 270, 271 modulus, 289, 291 molar ratio, 298 molecules, viii, 63, 364 molybdenum, 191 momentum, xi, 258, 361, 362, 366, 369, 370, 372, 374, 377 monochromator, 166, 203 monomer, 173 Moon, 380 morphology, 302 Moscow, 360 motion, 16, 27, 66, 69, 73, 81, 377 motors, 376 movement, 41, 276 multidimensional, 376 multilayer films, 312, 318, 320, 321, 327 multiples, 330 multiplexing, 3, 24 multiplication, vii, 1, 2, 3, 15, 17, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 59, 60 multiplier, 25, 26, 27, 33, 34, 35 mutation, 121, 122 mutation rate, 121

N nanoclusters, 173 nanocomposites, 289 nanocrystal, 285, 303 nanofabrication, 85 nanometer, viii, 63, 64, 73, 85 nanometers, 106 nanoparticles, ix, 151, 153, 172, 173, 174, 175, 176, 177, 194, 285, 288, 297 Nanostructures, v, 283

391 nanparticle, 299 Nanyang Technological University, 1 natural, xi, 64, 65, 119, 120, 202, 250, 251, 361, 362 natural selection, 119 Nd, viii, 117, 119, 123, 133, 134, 135, 149, 186, 334 neglect, 337, 339 neodymium, 88 New York, iii, iv, 84, 197, 261, 326, 327, 357, 379, 382 next generation, 2, 150 Ni, 186, 187 nickel, 185 noble gases, 152 noble metals, 284 nodes, 362 noise, vii, 1, 3, 4, 5, 7, 15, 16, 17, 18, 21, 22, 23, 24, 26, 27, 32, 33, 36, 44, 45, 50, 51, 52, 53, 89, 356, 370 non-Gaussian, 199, 202, 203, 215, 258, 259 nonlinear, iv, vii, viii, ix, x, xi, 1, 3, 4, 13, 14, 15, 16, 24, 25, 26, 27, 33, 36, 37, 39, 49, 59, 61, 87, 89, 90, 91, 92, 93, 94, 97, 98, 100, 101, 102, 104, 107, 109, 110, 111, 112, 113, 116, 137, 152, 165, 167, 171, 172, 175, 176, 179, 182, 184, 186, 190, 191, 192, 193, 194, 199, 200, 201, 202, 203, 205, 206, 208, 209, 210, 212, 215, 216, 221, 222, 228, 230, 231, 232, 233, 234, 235, 238, 240, 241, 250, 258, 283, 284, 288, 290, 292, 294, 296, 298, 299, 300, 301, 302, 303, 304, 307, 311, 314, 316, 318, 320, 321, 323, 325, 329, 330, 335, 367, 379 non-linear, 145 non-linear, 335 nonlinear dynamics, 379 nonlinear optical response, x, 165, 192, 283, 296, 300, 301, 303, 318, 320, 321 nonlinear optics, iv, vii, ix, 152, 172, 193, 199, 200, 202, 216, 228, 233, 258, 303, 330 nonlinear systems, 16, 26, 27 nonlinearities, vii, 3, 24, 27, 36, 59, 89, 100, 113, 298 non-linearity, 335 non-uniform, 64, 75, 121, 321, 341 non-uniformities, 341 non-uniformity, 341 normal, 40, 110, 230, 251, 255, 289, 291, 316 normal conditions, 40

392

Index

O observations, ix, 56, 167, 169, 171, 175, 178, 182, 188, 199, 208, 215, 216, 218, 219, 220, 230, 233, 238, 243, 246, 247, 253, 258, 314, 316 one dimension, x, 283, 284, 331 operator, 121, 337 optical, vii, viii, x, xi, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 16, 21, 22, 24, 25, 26, 33, 36, 37, 39, 40, 41, 42, 43, 47, 49, 51, 53, 54, 55, 58, 59, 60, 62, 64, 70, 71, 79, 80, 81, 84, 88, 91, 93, 100, 101, 102, 104, 106, 111, 113, 116, 118, 119, 136, 137, 150, 167, 171, 172, 184, 190, 201, 202, 203, 208, 210, 215, 233, 234, 235, 254, 258, 264, 266, 275, 276, 277, 278, 279, 280, 281, 283, 284, 285, 288, 289, 290, 292, 293, 296, 297, 298, 299, 300, 302, 303, 304, 305, 307, 308, 309, 311, 312, 313, 314, 316, 318, 319, 320, 321, 323, 325, 329, 330, 331, 332, 333, 340, 361, 362, 363, 364, 365, 366, 367, 369, 370, 371, 372, 376, 377, 378, 379 optical communications, vii, 2, 3, 24, 36 optical fiber, 49, 62, 88, 304 optical gain, 4, 6, 7, 16, 25 optical parameters, 184 optical polarization, 16 optical properties, x, 190, 283, 285, 288, 303, 304, 311, 316 optical pulses, 2, 60 optical systems, 118, 369 Optical Time Division Multiplexing, 61 optical transmission, 36, 305, 313 optical tweezers, 376, 377 optics, iv, vii, viii, 4, 16, 25, 72, 117, 118, 136, 139, 145, 149, 199, 200, 211, 235, 258, 265, 327, 332, 333, 339, 343, 379 optimization, 96, 119, 124, 133, 134, 135, 149, 153, 155, 158, 159, 161, 162, 178, 185, 192, 195, 301, 303, 304, 311, 316 opto-electronic, 6 ordinary differential equations, 38, 352 organic, x, 283 orientation, 93, 365 originality, 321, 323 orthogonality, 342 OSA, 39, 382 oscillation, 5, 15, 16, 25, 26, 54, 233, 286, 339, 341, 356 oscillations, 16, 26, 49, 208, 209, 331, 340, 341, 350 oscillator, 2, 90, 91, 99, 113, 152, 153, 168, 171, 176, 180, 182, 183, 209, 216, 217, 241, 245, 289, 292, 293, 335, 365

oxide, 311 oxygen, 308, 311

P PA, 311, 312, 313, 314, 315 Pacific, 61 packaging, 14 packet switching, 14 paper, ix, 64, 84, 119, 122, 199, 202, 304 parabolic, 332 parallel algorithm, 119 parallelism, 64 parameter, 16, 27, 50, 102, 104, 106, 122, 149, 155, 156, 163, 184, 186, 188, 192, 200, 206, 209, 210, 213, 219, 220, 229, 286, 316, 330, 331, 334, 335, 336, 337, 340, 343, 345, 367 particle shape, 285 particles, 152, 163, 175, 189, 190, 191, 285, 298, 301, 302, 303, 364, 366, 376, 377 passive, xi, 36, 37, 41, 88, 329, 330, 336, 337, 340, 356 pathways, 202, 205, 245, 249, 258 patterning, 101 Pb, 178, 194 PCs, 303, 316 PD, 6, 7, 8 performance, vii, viii, ix, 1, 2, 3, 15, 24, 33, 36, 59, 87, 88, 97, 99, 102, 113, 117, 118, 120, 122, 123, 127, 129, 134, 172, 269 periodic, viii, 16, 17, 27, 40, 54, 62, 63, 64, 68, 72, 78, 79, 84, 85, 152, 303, 304, 331 periodic table, 152 periodicity, viii, 63, 64, 72, 75, 77, 78, 303, 316 Peripheral, 269 personal, 266, 269, 270, 281 personal computers, 269, 270, 281 perturbation, xi, 38, 329, 330 perturbation theory, xi, 38, 329 perturbations, 3, 24 phase space, 370 phased array antennas, 61 phonon, 296, 297 phosphate, 101, 335 phosphor, 155 photobleaching, 297 photolithography, 152 photon, xii, 42, 99, 113, 172, 192, 205, 206, 207, 215, 216, 217, 222, 224, 225, 228, 229, 233, 241, 245, 246, 249, 250, 251, 252, 253, 255, 256, 257, 258, 304, 308, 312, 318, 323, 361, 362, 366, 371, 372, 375, 377, 378

Index photonic, vii, x, 1, 9, 14, 59, 89, 283, 284, 288, 303, 304, 307, 308, 309, 311, 312, 313, 316, 317, 320, 323, 330, 332, 335 photonic crystals, 308, 316 photonic devices, x, 283, 288, 303, 316, 320, 323, 330 photonics, 172, 298, 303, 323, 377 photons, viii, 63, 96, 99, 178, 202, 203, 205, 206, 211, 212, 216, 217, 218, 224, 225, 228, 230, 233, 241, 242, 244, 245, 246, 258, 362, 372, 376, 377 physical properties, 365 physics, xi, 158, 162, 172, 203, 246, 331, 361, 377 piezoelectric, viii, 117, 119, 124 piezoelectricity, 118 pinhole, 119, 134, 276 pipelines, 269 pitch, 271, 276, 277 planar, 16, 127, 131, 225, 226, 242, 331, 343, 345, 349, 354, 362 plane waves, 200, 201, 202, 216, 217, 219, 223, 224, 230, 231, 232, 233, 258 plasma, ix, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 182, 183, 184, 185, 186, 188, 189, 190, 191, 192, 193, 194, 195, 284 plasmons, 297 play, 179, 298 PM, 14, 39, 61, 97 polarizability, xi, 175, 329, 335 polarization, vii, 16, 22, 26, 36, 39, 40, 61, 89, 90, 93, 119, 167, 176, 192, 204, 220, 221, 222, 228, 230, 231, 232, 241, 250, 251, 252, 253, 254, 255, 256, 257, 258, 289, 290, 293, 297, 322, 341, 366 polarization planes, 231, 232, 251, 254, 255 polarized, 16, 93, 101, 167, 171, 228, 230, 231, 232, 250, 251, 252, 253, 254, 255, 256, 257, 258, 289, 366, 377 polarized light, 16, 251, 366, 377 polymer, x, 175, 283 polymers, 325 polynomial, 123, 124, 126, 127, 128, 131, 133 polynomials, 125, 131 poor, 2, 3, 15, 17, 23, 33, 240, 244 population, 120, 121, 169, 335, 341 ports, 14 potassium, 101 potential energy, 354 powder, 177, 179, 180

393 power, viii, xi, 2, 5, 7, 8, 13, 14, 16, 25, 26, 31, 33, 37, 39, 40, 42, 47, 49, 51, 53, 54, 55, 59, 87, 88, 89, 90, 91, 94, 95, 96, 97, 98, 99, 100, 110, 111, 112, 113, 118, 119, 133, 134, 135, 136, 137, 149, 206, 208, 211, 213, 279, 293, 294, 296, 329, 330, 331, 335, 338, 340, 341, 342, 343, 347, 348, 349, 350, 351, 352, 353, 355, 356, 363, 364 powers, 338, 340, 341, 347, 349, 352, 353, 355 prediction, 42, 110, 163, 313 preparation, iv, 163, 188, 192, 194, 298 pressure, viii, 63, 84, 203, 206, 207, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 220, 221, 224, 225, 229, 230, 238, 239, 240, 242, 243, 244, 245, 246, 247, 249, 252, 255, 257 pristine, 300, 320, 321 probability, 120, 121, 169, 172, 183, 205, 369 probability distribution, 120, 369 probe, 193, 289, 290, 292, 293, 295, 297, 308, 309, 310, 312, 313, 314, 318, 319, 322 procedures, 8 production, 79, 88, 205, 211, 216 program, 127, 134, 336, 338, 364 propagation, vii, xi, 16, 18, 27, 28, 29, 30, 31, 32, 49, 66, 69, 73, 104, 118, 137, 160, 171, 186, 191, 192, 193, 195, 200, 201, 202, 216, 217, 218, 220, 221, 222, 223, 228, 229, 230, 232, 233, 235, 236, 251, 255, 290, 303, 329, 330, 331, 332, 333, 334, 336, 338, 340, 344, 347, 351, 354, 356, 362, 363, 367, 368, 372, 373, 374 property, iv, xii, 25, 317, 361, 372, 377, 379 proposition, 369 prototype, 88, 137, 150 PSP, 154 pulse, vii, viii, x, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 47, 49, 51, 53, 54, 55, 58, 59, 60, 62, 87, 89, 90, 91, 92, 95, 96, 99, 100, 101, 102, 103, 105, 106, 109, 111, 112, 113, 136, 138, 151, 154, 156, 157, 158, 159, 160, 161, 162, 163, 165, 167, 169, 170, 171, 173, 175, 176, 177, 179, 183, 184, 185, 186, 188, 189, 191, 192, 195, 203, 206, 211, 213, 214, 215, 228, 238, 249, 250, 252, 283, 293, 295, 296, 297, 299, 309, 312, 318, 321, 356 pulsed laser, vii pulses, vii, viii, 1, 2, 3, 6, 15, 17, 23, 24, 25, 33, 36, 37, 38, 39, 40, 42, 47, 49, 50, 51, 53, 54, 56, 58, 59, 62, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 109, 113, 152, 153, 157, 158, 163, 166, 167, 169, 170, 171,

394

Index

173, 177, 178, 179, 180, 182, 183, 185, 191, 193, 203, 204, 234, 293 pump fields, 241, 242 pumping, 14, 88, 119, 133, 134, 179, 203, 204, 219, 220, 230, 232, 233, 234, 240, 241, 242, 243, 244, 255, 331

Q quantization, 257 quantum, x, xi, 5, 7, 84, 96, 256, 257, 283, 298, 302, 331, 335, 339, 343, 361, 369, 376, 377, 378, 379 quantum confinement, 298, 302 quantum dot, x, 283 quantum dots, x, 283 quantum entanglement, 378 quantum fluctuations, 84 quantum mechanics, xi, 331, 339, 343, 361, 369, 377 quantum well, x, 283 quartz, 204, 234

R RA, 286 radiation, viii, ix, 56, 63, 64, 84, 87, 88, 89, 90, 97, 100, 101, 102, 105, 107, 108, 109, 110, 113, 151, 152, 153, 154, 155, 156, 161, 163, 166, 167, 169, 170, 171, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 188, 190, 191, 212, 219, 221, 234, 332, 333, 334, 340, 345 Radiation, 84, 85, 178 radiofrequency, 2 radius, 27, 134, 287, 331, 363 Raman, 60, 330, 331 Raman scattering, 330 random, viii, xi, 7, 41, 63, 121, 122, 329, 334, 336, 341, 343 range, viii, xi, 10, 15, 26, 37, 55, 63, 78, 87, 88, 89, 91, 93, 95, 97, 98, 99, 100, 102, 107, 110, 113, 119, 120, 121, 122, 123, 125, 151, 152, 153, 155, 156, 160, 161, 163, 165, 166, 167, 168, 169, 173, 177, 178, 179, 182, 183, 185, 193, 194, 195, 205, 206, 210, 211, 212, 214, 215, 217, 222, 227, 229, 230, 234, 246, 247, 249, 284, 303, 309, 312, 320, 333, 356, 364, 376 rare earth, xi, 329 Rayleigh, 65, 66, 73, 200, 234, 344, 345, 351, 364 RB, 286

RC, 286 reading, 274 real numbers, 120 real time, 118, 133, 377 real-time, ix recall, 242 recognition, 102 recombination, 162, 169, 176, 183, 185 reconstruction, 111, 140, 143, 145, 148, 277, 278, 281 recovery, 7, 8, 14, 16, 297, 321, 322 recovery processes, 322 recurrence, x, 263, 266, 268, 269, 270, 271, 273, 278, 280 red light, 279 red shift, 214, 294 redshift, 177 reduction, 27, 35, 112, 175, 277, 341 reflection, 99, 153, 304, 305, 364 refraction index, 220 refractive index, xi, 6, 15, 186, 191, 192, 289, 292, 301, 303, 304, 305, 306, 307, 308, 311, 315, 316, 319, 321, 322, 329, 330, 331, 333, 334, 335, 336, 337, 339, 340, 343, 356, 376, 377 refractive indices, 104, 301, 312, 313, 317, 321, 337, 345 regular, 72, 200, 234, 250, 303 rejection, 309, 311, 312 relationship, 6, 7, 25, 35, 42, 124, 126, 144, 375, 376 relationships, 17 relaxation, 284, 297, 322 relaxation process, 297 relaxation time, 284 reliability, 88 reproduction, 377 research, iv, vii, x, 36, 102, 119, 200, 264, 277, 283, 304, 330, 340, 362 researchers, 119, 281 reservoirs, 378 residual error, 143 resistance, 125 resolution, 98, 133, 244, 266, 271, 298, 377 resonator, 4, 88, 119, 133, 134 respiration, 377 response time, 288 returns, 122 RF, 2, 4, 5, 6, 8, 10, 14, 16, 26, 40, 47, 58 RFA, 16 rings, 6, 209, 211, 362 RMS, 123, 126, 127, 128, 131 roadmap, 378

Index robustness, 331 rods, 286 room temperature, 205, 285 roundtrip, 44, 45, 52, 53 rubidium, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 84 Russia, 329 Russian, 329, 360

S sample, 180, 290, 291, 292, 293, 294, 295, 298, 300, 309, 318, 320, 322, 376 sampling, 59, 265 sapphire, 153, 154, 155, 163, 167, 176, 179, 180, 182, 293 saturation, vii, viii, xi, 1, 3, 4, 5, 16, 24, 36, 49, 54, 59, 64, 67, 75, 88, 95, 96, 98, 99, 110, 157, 163, 175, 193, 329, 336, 337, 340, 341, 342, 343, 347, 350 scalar, 104, 215, 332, 339, 344 scaling, 88, 96, 99, 105, 106, 110, 179, 378 scanning tunneling microscope, 175 scattering, xi, 14, 172, 254, 285, 297, 298, 301, 302, 323, 329, 332, 341, 379 Schrodinger equation, 49, 61 science, 172, 376 scientific, viii, 88, 101, 113, 118 scientists, 378 search, x, 119, 121, 123, 124, 149, 152, 165, 190, 283, 298 seed, 51, 53, 90, 99, 332 segmentation, 228 selecting, 119, 251 selectivity, 206 Self, 60 self-focusing, 330, 335 self-organization, 335 self-phase modulation, 191 self-reproduction, 330 semiconductor, x, 88, 269, 283, 333 semiconductor lasers, 333 semiconductors, 325 sensing, 149 sensitivity, 6, 226, 250 separation, 37, 38, 40, 47, 54, 79, 258, 316, 317, 339, 349 Sequoia, 197 series, ix, 68, 77, 101, 117, 118, 124, 150, 309, 352 services, iv SH, 91, 93, 94, 95, 96, 99, 101, 102, 103, 105, 106

395 Shanghai, 283 shape, x, 6, 17, 25, 27, 33, 36, 40, 42, 47, 59, 111, 122, 123, 124, 125, 127, 131, 134, 155, 169, 172, 191, 192, 213, 218, 224, 226, 228, 233, 246, 247, 249, 252, 283, 350, 367, 370 shares, 228 shear, 370 shoot, 138 short period, 110 shortage, 120 sign, viii, 63, 64, 183, 192, 289, 295, 335, 336, 342, 351, 365, 366, 367 signaling, 377 signals, x, 10, 11, 12, 26, 88, 101, 118, 207, 263, 271, 278, 279, 280, 295, 296, 342, 343 signal-to-noise ratio, 7, 289, 321 signs, 38, 179, 184 silica, 335 silica glass, 335 silicon, 91, 304 silver, 156, 157, 158, 159, 161, 167, 168, 173, 174, 175, 176, 179, 185, 193, 286, 287, 323 similarity, 126, 209, 210, 242, 254, 362, 378 simulation, vii, viii, xi, 3, 4, 24, 27, 33, 49, 51, 53, 54, 55, 56, 63, 64, 81, 106, 123, 149, 158, 165, 208, 214, 215, 226, 227, 298, 301, 308, 318, 321, 329 simulations, 33, 50, 55, 66, 69, 73, 122, 127, 158, 165, 166, 281 Singapore, 1 singular, xi, 125, 361 singularities, 378 SiO2, 301, 302, 305, 308, 311, 312, 313, 314, 315, 316, 317, 318 SiO2 films, 302, 312, 318 skin, 320 sodium, 64 software, 266, 268, 349 solar, 203 sol-gel, 298 solid state, 88 solid surfaces, 193 solid-state, viii, 117, 118, 119, 120, 122, 123, 133, 134, 149, 152, 179, 193, 194, 331 soliton, vii, 2, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 330 solitons, 36, 37, 38, 39, 40, 41, 42, 47, 49, 53, 54, 55, 56, 61, 62, 330 solutions, 202, 341 SP, 294, 296 space-time, 249

396 spatial, 77, 79, 88, 93, 110, 111, 118, 151, 173, 191, 192, 202, 208, 209, 216, 217, 224, 225, 228, 232, 233, 234, 235, 240, 241, 258, 331, 333, 339, 340, 341, 362, 365, 367, 369, 370, 376, 381 spatial frequency, 79, 341 species, viii, 63, 179 spectra, 53, 89, 91, 92, 94, 95, 97, 98, 99, 101, 109, 110, 111, 112, 113, 155, 157, 159, 160, 161, 162, 163, 168, 169, 171, 173, 174, 176, 177, 180, 182, 188, 189, 193, 194, 206, 207, 208, 209, 210, 225, 230, 238, 244, 245, 247, 251, 252, 255, 294, 305, 308, 309, 312, 313, 316, 317, 318, 319, 320, 321 spectral analysis, 157, 184 spectral component, 101, 224, 225, 227, 244, 248 spectroscopy, viii, 88, 89, 162, 193, 202, 318 spectrum, viii, 8, 22, 25, 33, 34, 35, 39, 40, 41, 42, 43, 44, 45, 47, 49, 51, 52, 53, 88, 92, 95, 96, 97, 101, 102, 106, 109, 110, 111, 112, 113, 155, 158, 164, 165, 167, 170, 175, 179, 216, 217, 218, 219, 220, 221, 224, 225, 234, 238, 239, 308, 309, 320, 321, 334, 339, 354 speed, ix, x, 15, 25, 104, 117, 122, 123, 136, 178, 268, 269, 275, 283, 339, 353, 366, 376, 377 speed of light, 15, 25, 104, 178, 339 spheres, 286, 301 spin, 362, 366, 377 SPR, 172, 192, 285, 286, 287, 289, 294, 296, 298, 303, 323 sputtering, 294, 296, 320 SRD, 16 SRS, 330, 331 stability, vii, 1, 2, 3, 4, 8, 15, 23, 24, 27, 33, 36, 37, 42, 49, 59, 119, 125, 320, 330, 341 stabilization, 37, 55, 351 stabilize, 47, 333 stages, 37, 79, 89, 93, 195 stainless steel, 203 standard deviation, 298 Stark effect, 249 steady state, 38 stochastic, 119 storage, 376 strength, 37, 168, 176, 180, 182, 183, 216, 217, 241, 245, 330, 335, 345 stress, 104 strikes, 297 stroke, 119 substrates, 173 Sun, 116, 197, 325, 380 superconductors, 378 superfluid, 362

Index superposition, 172, 201, 202, 212, 214, 216, 217, 219, 222, 224, 232, 233, 242, 258, 365, 372 supply, 294 suppression, 8, 25, 41, 134, 157, 158, 182, 186, 191, 202, 240, 246, 247, 248, 258, 353, 356 susceptibility, x, 153, 172, 182, 192, 193, 208, 209, 230, 283, 288, 292, 296, 297, 298, 299, 300, 301 switching, x, 14, 88, 152, 263, 278, 279, 280, 281, 283, 289, 297, 304, 323 symbols, 271, 272, 274 symmetry, 17, 109, 218, 220, 225, 241, 334, 342, 354, 355, 362 synchronization, xi, 329, 331, 335, 336 synchronous, 91 synthesis, x, 283 systematic, 77, 157 systems, 2, 3, 16, 24, 27, 36, 37, 88, 136, 149, 151, 284, 289, 316, 330, 331, 333, 341, 356, 378

T tantalum, 203 tar, 202 targets, 152, 163, 165, 166, 169, 173, 174, 175, 179, 185, 186, 188, 189, 190, 193, 194 technological, viii, 88, 101, 113 technology, ix, 89, 118, 263, 269, 280, 304, 332, 335, 336, 356, 377 telecommunications, 88, 106 TEM, 298 temperature, 94, 100, 101, 104, 294, 296, 297, 298, 320 temporal, vii, 1, 2, 3, 24, 39, 40, 47, 89, 91, 93, 94, 95, 98, 99, 100, 102, 107, 111, 191, 234, 295, 297, 323, 333, 375 theoretical, ix, xii, 36, 42, 94, 95, 99, 102, 107, 109, 110, 145, 152, 153, 172, 199, 202, 204, 225, 235, 245, 246, 256, 303, 321, 331, 361, 368, 369, 370, 372, 379 theory, 36, 59, 172, 179, 182, 208, 212, 215, 216, 249, 252, 258, 285, 287, 296, 298, 301, 302, 305, 330, 338, 340, 356 thermal, vii, 7, 14, 66, 67, 68, 69, 71, 72, 73, 88, 118, 119, 137, 285, 296, 297, 309, 311, 318, 323, 331, 334, 335 thermal equilibrium, 285, 297, 323 thermal evaporation, 309, 311, 318 thermal expansion, 331 thermal lens, 118, 119, 331, 334 thin film, 85, 154, 308, 321, 323 thin films, 308

Index third order, vii, 97 three-dimensional, ix, 64, 127, 131, 316, 332 threshold, 37, 48, 49, 55, 88, 175, 216, 221, 245, 254, 320, 331, 343 Ti, 16, 26, 38, 102, 153, 154, 155, 163, 167, 176, 179, 180, 182, 293, 309, 318 time, vii, x, 1, 3, 4, 6, 8, 24, 36, 37, 40, 47, 50, 51, 54, 58, 64, 67, 68, 70, 71, 72, 75, 76, 77, 78, 83, 84, 88, 91, 94, 95, 98, 100, 135, 139, 149, 152, 159, 160, 161, 162, 165, 167, 170, 172, 174, 178, 182, 184, 188, 189, 190, 192, 193, 203, 263, 265, 266, 268, 275, 276, 278, 279, 280, 281, 288, 294, 295, 296, 297, 309, 310, 312, 313, 314, 318, 322, 330, 341, 362, 375, 376, 377, 378 timing, 27, 38, 279 tin, 180, 181 TiO2, 294, 295, 296, 305, 308, 311, 312, 313, 314, 315, 316, 317, 318, 320, 321 titanium, 166, 298 tolerance, 233, 240, 336 topological, xi, 361, 364, 365, 366, 367, 376, 378 torque, 366 torus, 280 TPA, 308, 309, 310, 312, 313, 314, 315, 316, 318, 320 trajectory, 16, 17, 27, 350 trans, 91, 96, 214 transfer, 8, 11, 49, 240, 305, 306, 308, 309, 310, 312, 313, 314, 316, 320, 323, 366 transformation, 3, 24, 109, 339, 356 transformations, 216, 220 transition, viii, 37, 49, 54, 63, 64, 65, 67, 75, 166, 168, 169, 172, 176, 177, 180, 183, 216, 221, 233, 245, 246, 255, 257, 258, 284, 294, 296, 302 transitions, 51, 152, 153, 167, 168, 170, 171, 177, 180, 182, 183, 192, 194, 205, 209, 335 translation, 93, 94, 174 transmission, 2, 38, 88, 307, 308, 309, 310, 312, 316, 317, 318, 319, 320, 321, 327, 364 transparency, 93 transparent, 288, 364 transport, 367 traps, 376, 377 travel, 71 trend, 342, 351 Tsunami, 293, 309, 318 tunneling, 331 turbulence, 149 two-dimensional, 16, 27

397

U ubiquitous, 362, 378 UK, 382 ultra-thin, 344, 345, 349, 352 ultraviolet, viii, 87, 89, 96, 97, 99, 101, 113, 151, 335 Ultraviolet, v, 87 uncertainty, 166, 255 uniform, 27, 33, 77, 84, 121, 122, 333, 334, 364 uniformity, 77, 78 UV, 97, 157, 160, 161, 178, 184, 188, 204, 234, 235, 238, 251, 294, 335 UV radiation, 97, 234 UV spectrum, 160 Uzbekistan, 151

V vacuum, 6, 25, 104, 154, 156, 179, 185, 186, 192, 294, 331, 339 values, 50, 91, 94, 95, 96, 97, 98, 99, 104, 106, 107, 108, 110, 120, 125, 180, 220, 224, 240, 242, 265, 289, 313, 334, 335, 336, 339, 340, 348, 352, 353, 356, 370 vanadium, 165, 190 vapor, 216, 219, 220 variability, 250 variable, 26, 89, 91, 92, 99, 101, 113, 120, 121, 134, 152, 158, 218, 235, 247, 278, 364 variables, 104, 120, 218, 339, 347, 351, 370, 372 variation, 35, 36, 37, 54, 59, 65, 77, 100, 160, 163, 170, 172, 177, 178, 183, 191, 192, 335, 336, 340, 342 vector, 125, 126, 208, 241, 330 velocity, vii, 1, 4, 6, 24, 37, 38, 39, 53, 54, 68, 71, 84, 89, 91, 94, 95, 100, 101, 189, 220, 221 Victoria, 1 video, 266, 268 visible, viii, 87, 88, 89, 91, 97, 98, 102, 110, 111, 112, 113, 157, 184, 188, 303, 316, 335, 374 visualization, 281 vortex, xii, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378 vortices, xi, 215, 361, 362, 363, 364, 365, 367, 369, 371, 372, 374, 375, 376, 378, 379

W water, 152, 378 wave number, 104, 171, 339

398 wave packet, 169 wave packets, 169 wave propagation, 309, 312, 320, 335 wave vector, 202, 216, 218, 220, 230, 231, 232, 241, 289 waveguide, x, 284, 303, 323, 330, 331, 333, 335, 336, 340, 341, 343, 344, 345, 346, 347, 348, 349, 352, 353, 354, 355, 356, 365 waveguides, xi, 101, 152, 166, 303, 329, 330, 331, 332, 333, 338, 341, 342, 343, 344, 345, 346, 347, 349, 352, 353, 356 wavelengths, viii, 87, 88, 89, 91, 93, 98, 99, 100, 102, 106, 107, 110, 113, 178, 180, 182, 183, 193, 211, 214, 215, 229, 238, 242, 249, 255, 278, 301, 306, 315, 331, 364 workers, 200 writing, viii, 63, 68, 79

Index

X xenon, 203, 205, 206, 207, 209, 210, 214, 225, 229, 230, 233, 238, 239, 240, 244, 245, 246, 247, 248, 250, 251, 252, 253, 254, 256, 257 x-ray, 151, 165, 166, 172 X-ray, 166, 294 XRD, 309

Y yield, 78, 84, 152, 153, 157, 158, 166, 168, 170, 171, 173, 174, 175, 176, 178, 180, 183, 186, 190, 193, 194, 195, 205, 206, 247 ytterbium, 64, 85, 88, 331

Z Zener, 331