ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 99
EDITOR-IN-CHEF
PETERW. HAWKES CEMESILaboratoire d 'OptiqueElectronique du Centre National de la Recherche Scient$que Toulouse. France
ASSOCIATE EDITORS
BENJAMIN W A N Xerox Corporation Alto Research Center Palo Alto, California
TOM MULVEY Department OfElectronic Engineering and Applied Physics Aston University Birmingham, United Kingdom
Advances in
Imaging and Electron Physics EDITEDBY PETER W. HAWKES CEMES/Lnhomtoire d’Optique Electroriique du Centre N d o i i a l de In Recherche Scientifique Toulouse.Froiice
VOLUME 99
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CONTENTS . . . . . . . . . . . . . . . . . . . . . . CONTRIBUTORS PREFACE . . . . . . . . . . . . . . . . . . . . . . . .
I. I1. I11. IV. V. V1. VII . V111.
I. I1. 111.
1V. V. VI . V11. VIII .
Morphological Scale-Spaces PAULT. JACKWAY Introduction . . . . . . . . . . . . . Multiscale Morphology . . . . . . . . . Multiscale Dilation-Erosion Scale-Space . . . Multiscale Closing-Opening Scale-Space . . Fingerprints in Morphological Scale-Space . . Structuring Functions for Scale-Space . . . . A Scale-Space for Regions . . . . . . . . Summary. Limitations, and Future Work . . . Appendix . . . . . . . . . . . . . . References . . . . . . . . . . . . . .
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Characterization and Modeling of SAGCM InPlInGaAs Avalanche Photodiodes for Multigigabit Optical Fiber Communications C . L . F. MA. M . J . DEENA N D L . E . TAROF Introduction . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . Planar SAGCM lnP/InGaAs APD . . . . . . . . . . . Critical Device Parameters Extraction . . . . . . . . . . Photogain . . . . . . . . . . . . . . . . . . . . Temperature Dependence of Breakdown Voltage and Photogain Dark Current Noise . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . List of Acronyms . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . Appendix A: Electric Field in SAGCM APD . . . . . . . References . . . . . . . . . . . . . . . . . . . .
ix xi
2 8 16 22 29 37 46 53 57 61
66 74 96 102 120 135 150
153 157 157 158
161 164
CONTENTS
1. I1. 111.
IV. V. V1. VII .
Electron Holography of Long-Range Electrostatic Fields G . MATTEUCCI. G. F. MISSIKOLI A N D G . POZZl Introduction . . . . . . . . . . . . . . . . . . . . Electron-Specimen Interaction . . . . . . . . . . . . . . Recording and Processing of Electron Holograms . . . . . . Charged Dielectric Spheres . . . . . . . . . . . . . . . P-N Junctions . . . . . . . . . . . . . . . . . . . . Investigation of Charged Microtips . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
171 174 192 207 216 229 235 236 237
The Imaging Plate and Its Applications NOBUFUMI MORIA N D %TSUO OIKAWA I. I1. 111. IV. V. VI . VII . VIII .
I. I1. 111. IV. V. VI .
Introduction . . . . . . . . . . . . . . . . . . . . Mechanism of Photostimulated Luminescence (PSL) . . . . . Imaging Plate (IP) . . . . . . . . . . . . . . . . . . Elements of the Imaging Plate (IP) Systetn . . . . . . . . . Characteristics of the Imaging Plate (IP) System . . . . . . . Practical Systems . . . . . . . . . . . . . . . . . . Applications of the Imaging Plate . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
241 242 248 250 254 262 265 288 288 288
Space-Variant Image Restoration ALREKTO DE SANTIS. ALFREDO G E R M A NA IN D LEOPOLDO JETTO Introduction . . . . . . . . . . . . . . . . . . . . Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Image Model Image Restoration . . . . . . . . . . . . . . . . . . Numerical Results . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . . Appendix C . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
292 295 300 308 317 320 321 322 324 326
CONTENTS
RAFAFI
INDEX
Erratum and Addendum for Iniage Representation with Gabor Wavelets and Its Applications NAVAIZIIO, AN I O N 1 0 T A H L R N ~ A NKDO GAHIULL clil5 IOI3Al
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CONTRIBUTORS Nuiiihcrs iii parciillicws iridicale tlic page\
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~it11110r~‘ conlributions
begin
M. JAMALDEEN(6 S ), School of Engineering Science, Simon Fraser University, Burnaby, British Columbia VSA IS6, Canada ALHERTO DE SANTIS (291), Dipartimento di Informatica e Sistemistica, Universiti degli Studi “La Sapienza” di Roma, via Eudossiano 18, Roma 00184, Italy ALFREDOGERMANI(291), Dipartimento di Ingegneria Elettrica, Universita dell’Aquila, 67100 Monteluco (L‘Aquila), Italy, and Istituto di Analisi dei Sistemi ed Informatica del CNR, Viale Manzoni 30, 00185 Roma, Italy PAULT. JACKWAY ( I ) , Centre for Sensor Signal and Information Processing (CSSIP), Department of Electrical and Computer Engineering, The University of Queensland, St. Lucia, Brisbane 4072, Australia LEOPOLDO JETTO (29 1 ), Dipartimento di Elettronica e Automatica, Universita di Ancona, via Breccie Bianche, Ancona 601 3 1, Italy
C. L. F. MA (65), School of Engineering Science, Simon Fraser University, Burnaby, British Columbia VSA 1S6, Canada G. MATrEUCCl (171), Department of Physics, and lstituto Nazionale per la Fisica della Materia, University of Bologna, viale B. Pichat 6/2, Bologna 40127. Italy G. F. MISSIROL.~ (171), Department of Physics, and Istituto Nazionale per la Fisica della Materia, University of Bologna, viale B. Pichat 6/2, Bologna 40127, Italy
NOHUFUMIMORI(241), FUJI Photo Film Co., Ltd., 798, Miyanodai, Kaisei, Ashigarakami, Kanagawa 258, Japan
TETSLJO OIKAWA (241), JEOL, LTD., 1-2 Musashino 3-chome, Akishima, Tokyo 196, Japan G. POZZI (171), Department of Physics, and Lstituto Nazionale per la Fisica della Materia, University of Bologna, viale B. Pichat 6/2, Bologna 40127, Italy
L. E. TAROF ( 6 5 ) ,Bell-Northern Research Ltd., P.O. Box 35 1 1, Station C, Ottawa, Ontario K 1Y 4H7, Canada
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PREFACE
Mathematical morphology, avalanche photodiodes, electron holography, the properties of imaging plates (IPS) and space-variant image processing: the range of themes covered in this volume is wide and all of them are active research topics. The opening chapter is concerned with morphological scale-spaces. Paul Jackway has made major contributions to this difticult and important branch of mathematical morphology, which addresses the question of extracting reliable information from images across a range of scales. This description is deceptively simple and, as the author shows, sophisticated mathematical tools are needed to answer it. The procedure proposed by Jackway is certainly not the last word on the subject and, as in so much of digital image processing, the effect of noise is still incompletely understood. Nevertheless, this full account of a very promising approach is extremely exciting and 1 am very glad to include it here. In the second chapter we have a full study of an avalanche photodiode to be used in present and future optical fiber communication systems. This is a planar, separate absorption, grading, charge and multiplication InPflnGaAs photodiode and C. L. F. Ma, M. J. Deen and L. E. Tarof take us through the design and the physics of such devices in great detail. There are sections on the device itself, on measurement of its parameters, on the photogain, on the temperature dependence ofthe working properties and on the dark-current noise. This extremely full account of the physics and engineering aspects of these components will surely be heavily used. Anyone who has participated in a recent major electron microscopy conference will know that electron holography is in a stage of explosive growth and that the physicists of Bologna have made important contributions, especially in the study of electric tields by holography. The third chapter provides a connected account of all this Italian work and related activity elsewhere, with prospects for future investigations. G. Matteucci, G. F. Missiroli, and Giulio Pozzi first discuss electron-specimen interactions, with a section on the less well known electrostatic Aharonov-Bohm effect. They then describe how electron holograms are formed and recorded before turning to real specimens, charged spheres, p-n junctions and charged microtips. This survey of the holography of electric tields usefully complements the magisterial publications of Akira Tonomura, which have been concentrated more on magnetic tields, and those of Hannes Lichte, in the area of very high resolution. It is not often that some completely new recording medium is introduced but the imaging plate, or IP, has already had considerable impact in x-ray diffraction
xi
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and in autoradiography and is gradually coming into use in electron microscopy. The authors, Nobufumi Mori and Tetsuo Oikawa have been largely instrumental in extending the applications of the IP to electron microscopy and here they discuss cveiy aspect of it, from the physics of its mode of action to its many applications. The characteristics of typical IPS are described and practical systems for radiography, radio luminography and microscopy are presented. This first long account of these new recording and digitizing devices is likely to be frequently consulted. Finally, we have a chapter by Alberto de Santis, Alfredo Germani, and Leopoldo Jetto on image restoration in situations in which the relation between object and image cannot be represented by a convolution: space-variant image restoration. The authors have developed a new approach to this rebarbative problem and here they explain their model and their procedure in full. The basic assumptions are clearly stated and the state-space approach extensively used in signal processing is adapted to the present problem. Their method is applicable to a wide range of image types and this presentation of it should attract many more potential users. In conclusion, I thank all the authors for sharing their scholarship and inventiveness with readers of this series and list articles to be found in the next few volumes. Volume 100, a cumulative index, will appear shortly after volume 102. Peter W. Hawkes
FORTHCOMING CONTRIBUTIONS
Nanofabrication Finite-element methods for eddy-current problems Mathematical models for natural images Use of the hypermatrix Image processing with signal-dependent noise The Wigner distribution Hexagon-based image processing Microscopic imaging with mass-selected secondary ions Modern map methods for particle optics ODE methods Electron microscopy in mineralogy and geology Microwave tubes in space
H. Ahmed and W. Chen (vol. 102) R. Albanese and G. Rubinacci (vol. 102) L. Alvarez Leon and J.-M. Morel D. Antzoulatos H. H. Arsenault M. J. Bastiaans S. B. M. Bell M. T. Bernius M. Berz and colleagues J. C. Butcher P. E. Champness (vol. 101) J. A. Dayton
...
PREFACE
Fuzzy morphology The study of dynamic phenomena in solids using field emission Gabor filters and texture analysis Miniaturization in electron optics Liquid metal ion sources X-ray optics The critical-voltage effect Stack tiltering Median tilters The development of electron microscopy in Spain Space-time representation of ultra-wideband signals Structural analysis of quasicrystals Formal polynomials for image processing Contrast transfer and crystal images Optical interconnects
Numerical methods in particle optics Surface relief Spin-polarized SEM Sideband imaging Vector transformation SEM image processing The dual de Broglie wave Electronic tools in parapsychology Z-contrast in the STEM and its applications Phase-space treatment of photon beams Aspects of mirror electron microscopy Image processing and the scanning electron microscope Representation of image operators
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E. R. Dougherty and D. Sinha M. Drechsler J. M. H. Du Buf A. Feinerman (vol. 102) R. G. Forbes E. Forster and F. N. Chukhovsky A. Fox M. Gabbouj N. C. Gallagher and E. Coyle M. I. Herrera and L. Bru E. Heyman and T. Melamed K. Hiraga (vol. 101) A. Imiya (vol. 101) K. Ishizuka M. A . Karim and K. M. Iftekharuddin (vol. 102) E. Kasper J. J . Koenderink and A. J. van Doorn K. Koike W. Krakow W. Li N. C. MacDonald M. Molski (vol. 101) R. L. Morris P. D. Nellist and S. J . Pennycook G. Nemes S. Nepijko (vol. 102) E. Oho
B. Olstad
xiv
PREFACE
Fractional Fourier transforms HDTV Scattering and recoil imaging and spectrometry The wave-particle dualism Digital analysis of lattice images (DALI) Electron holography X-ray microscopy Accelerator mass spectroscopy Applications of mathematical morphology Set-theoretic methods in image processing Focus-deflection systems and their applications Electron gun system for color cathode-ray tubes Study of complex fluids by transmission electron microscopy New developments in ferroelectrics Electron gun optics Very high resolution electron microscopy Morphology on graphs Analytical perturbation methods in charged-particle optics
H. M. Ozaktas E. Petajan J. W. Rabalais H. Rauch A. Rosenauer D. Saldin G. Schmahl J . P. F. Sellschop J . Serra M. 1. Sezan T. Soma H. Suzuki
J. Talmon J . Toulouse Y. Uchikawa D. van Dyck L. Vincent M. 1. Yavor (vol. 103)
ADVANCES IN IMAGINGAND ELECTRON PHYSICS VOL 99
Morphological Scale-Spaces PAUL T. JACKWAY Cooperative Research Centrefor Sensor Signal and Information Processing. Department of Electrical and Computer Engineering. University of Queensland. Brisbane. Queensland 4072. Australia
I. Introduction . . . . . . . . . . . . . . . . . . . A. Gaussian Scale-Space . . . . . . . . . . . . . . B . Related Work and Extensions . . . . . . . . . . . . 11. Multiscale Morphology , , , , , , , , , , , , , , , A. Scale-Dependent Morphology . . . . . . . . . . . B . Semi-Group and General Properties of the Structuring Function 111. Multiscale Dilation-Erosion Scale-Space . . . . . . . . . A. Continuity and Order Properties ofthe Scale-Space Image . B . Signal Extrema in Scale-Space . . . . . . . . . . . IV. Multiscale Closing-Opening Scale-Space . . . . . . . . . A. Properties of the Multiscale Closing-Opening . . . . . .
. . . . . . . . . VI. Structuring Functions for Scale-Space . . . . . . . . . A . Semi-Group Properties . . . . . . . . . . . . . . B . A More General Umbra . . . . . . . . . . . . . . C. Dimensionality . . . . . . . . . . . . . . . . . D. The Poweroid Structuring Functions . . . . . . . . VII. A Scale-Space for Regions . . . . . . . . . . . . . . A. The Watershed Transform . . . . . . . . . . . . . B . Homotopy Modification of Gradient Functions . . . . C . A Scale-Space Gradient Watershed Region . . . . . . VIII. Summary, Limitations. and Future Work . . . . . . . . A . Summary . . . . . . . . . . . . . . . . . . . B . Limitations . . . . . . . . . . . . . . . . . . C. Futurework . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . B . Monotone Theorem forthe Multiscale Closing-Opening V. Fingerprints in Morphological Scale-Space . . . . . A. Equivalence of Fingerprints . . . . . . . . . B. Reduced Fingerprints . . . . . . . . . . . . C . Computation of the Reduced Fingerprint . . . . .
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I.
~NTRODIJCTION
The term scale-space is a very successful one: From its modest launch in thc title of a four-page conference paper by Witkin (19831, it has grown to denote a whole subfield of study and a raft of generalizations, extensions, and theories (unfortunately, not all of them compatible!). As an indication of the enduring nature of this contribution, we note that at the prescnt time (over a decade since its publication), Witkin’s papcr still rcceives over 20 citations per annum in the Scierice Ciruriori Index, which covers only the majorjournals in the field of computer vision. What then is scale-space about? Let’s follow Witkin’s introduction: Any sophisticated signal understanding task must rely on a description of the signal which extracts meaningful objects or events. The problem of “scale” has emerged consistently as a fundamental source of difficulty in finding a good signal descriptor, as we need to separate events at different scales arising from distinct physical processes (Marr, 1982). It is possible to introduce a “parameter of scale” by smoothing the signal with a mask of variable size. but every setting of the scale parameter yields a different description! How can we decide which if any of this continuum o f descriptions is “right”’? For many tasks it has become apparent that no one scale of description is categorically correct so there has been considerable interest in multi-scale descriptions (Ballard and Brown, 1982; Rosenfeld and Thurston. 1971; MaiTand Poggio, 1979; Marr and Hildreth, 1980). However, merely computing signal descriptions at multiple scales does not solve the problem; if anything it exacerbates it by increasing the volume of data. Some means must be found to organize the description by relating one scale to provides a means for managing the ambiguity of scale another. SL.Li/e-spuce~/tcrinR in an organized and natural way. (Witkin. 1983)
From the above passage we see that scale-space filtering concerns signals, in particular, signal understanding or analysis. It proceeds by dealing with dcscriptions of the signal smoothed by masks of varying sizes corresponding to multiple scales. Important too is the idea of dealing with all the resulting descriptions as a whole-we arc not trying to determine which is the single “best” scale for analysis. As we will sec in the next section. the “stack” of signal descriptions is organized by relying on continuity properties of signal features across the scale dimension. Indeed, scale-space is most useful if we demand that the signal rcprcsentation gets simpler with increasing smoothing. This turns out to lead to very interesting theoretical questions, such as: For what signal features. and €or which smoothcrs, and which class of signals do the required properties exist’? And given the signal/smoother/€eature combination, what are thc stability, uniqueness, invertability, and differential properties of the resulting signal representation. These theoretical questions have exercised the minds and pens of many researchers, starting from Witkin (1983) who speculated that (under some restrictions) the Gaussian filter was the unique filter in ID that possessed the requircd properties, and there is quite a large body of work on thcsc topics (for examples,
MOKPHOLOGICAI. SCAl .E-SPACI:.S
3
see Yuille and Poggio, 1985; Babaud et al., 1986; Hummel, 1986; Humnicl and Moniot, 1989; Wu and Xie, 1990; ter Haar Romeny et a/., 1991; Alvarez and Morel, 1994; Jackway and Deriche, 1996). A. Gaussiari Scale-Space
To start, we need to review Witkin’s (1983) approach. Suppose we have a signal, , f ( x - ) : R” -+ R and a smoothing kernel g(x, (T) : R” x R. -+ R.. The scale-space image F(x, a) : R” x R + R of the signal is obtained by smoothing the signal at all possible scales and is a function o n the ( n + 1)-dimensional spacc called scalc-space: F ( x .a ) = f(x)
* g(x. a),
(1)
where * denotes a smoothing operation. F is known as the scale-spuce iiricige of the signal. The ideas behind scale-space first appeared in a report on expert systems by Stansficld (1980) who was looking at ways to extract features from graps of commodity prices. The scale-space concept was named, formalized, and brought to image analysis by Witkin (1984). Both these authors used the linear convolution IS the smoothing operation: (2)
und Gaussian functions as the scale-dependent smoothing kernel:
With this smoother, the Marr-Hildreth edge detector (Marr and Hildreth, 1980) (zero-crossings of the second derivative of the signal) is the appropriate feature dc tector. Witkin’s idea is elegant: If scalc is considered as a co~zrirz~~ous variable rather than a parameter, then a signal feature at one scale is identified with that at another scale if they lie on the same feature path in the resulting scale-space. A central idea in Witkin’s work is that important signal features would persist through to relatively coarse scales even though their location may be distorted by thc filtering process. However, by course-to-jrie tmckitig they could be tracked back down a path in scale-space to zero-scale to be located exactly on the original signal. In this way the benefit of large smoothing t o detect the major features could be comined with precise localization. In a way thrcse linkages across scale are used to overcome thc uncertainty principle, which states that spatial localization and frequency domain localization are conflicting requirements (Wilson and Granlund, 1984).
4
I'Alll. T. JACKWAY
A defining feature of scale-space theory, in contrast to other multiscalc approaches, is the property that a signal feature, once present at some scale, must persist all the way through scalc-space to zero-scalc (otherwise the fcaturc would bc spurious: being caused by the filtcr and not thc original signal). This is callcd a moiiototie proper5 since thc number of features must necessarily be a monotone dccrcasing function of scale. If 2, I'O denotes the point set of the positions o f fcatures in a signal , f ( x ) , and if C[Z] denotes the nuinbcr of fcatures in the set. thcn we require,
A corititrnity property is also implied since the feature paths should be continuous across scalc to cnable tracking. The plot of signal feature positions versus scale has been tcrmed thc,fingerprint of the signal (Yuillc and Poggio, 1985). Figure 1 shows the Gaussian scalc-space analysis ofa 1D signal. Note that the monotone and continuity properties cnsurc that all thc fingerprint lines of a signal form continous
i+iLJIW. I . The Gaussian scale-\p;icc analysis o f a I Ll signal ( n \ciiii 111ie11.oiii the "Lena" image). Froiii Icl't to right. top to boitoiii: thc signal: tlic scale-space imagc a s a s u r t x e plot: ihe scale-\pace iiiiage iis a grnysc;ilc image: the Iingcrprini (the plot o ~ ~ e I o - e ~ - o ~ sof i i ithe g s s x w i d yuttnl tIcrivati\c).
MOKPHOI.OGICAL SCAIJ-SPACES
5
paths in scale-space. All the paths start at zero-scale and continue upward until [hey stop at some scale (possibly infinite) which is characteristic for that feature. I t was Witkin's plan to understand the signal by using its fingerprint. Because of problems with signals in higher dimensions, the chief applications of Gaussian scale-space have been those involving 1D signals, for example, the description and recognition of planar curves (Asada and Brady, 1986; Mokhtarian and Mackworth. 1986), histogram analysis (Carlotto, 1987), signal matching (Witkin et al., 1987). ECG signal analysis (Tsui et al., 1988), the pattern matching of 2D shapes (Morita etal., 1991), boundary contour refinement in images (Raman et al., 1991), the analysis of facial profiles (Campos et al.. 1993), and the matching of motion trajectories (Rangarajan et al., 1993). To motivate the use of scale-space and to illustrate the general favor of the above scale-space applications, consider the following example. Suppose we want to represent the shape of an object in a binary image. First, we express the boundary curve of the shape C as a pair of functions of path length t along the curve:
Then the curvature at each point can be computed, K(t)
=
x v - )'t + $) 1 / 2 '
(x'
where .i and j;. denote the first and second derivatives with respect to t . Now we smooth the curvature function with a scaled Gaussian [Eqs. ( 2 ) and ( 3 ) ] ,directly (Asada and Brady, 1986) or via smoothing x ( t ) and y(t) (Mokhtarian and Mackworth, 1986) to give a scale-space curvature image K ( t , cr), and then plot the zcro-crossings of this image (to detect points of inflection of the curve) to give a fingerprint diagram representing the shape. Now, to recognize or match this shape we can usc the fingerprints. The idea is that noise and minor features are confincd to small scales in the fingerprint while thc most important features persist through to larger scales in the representation. Thus, it makes sense and is very efficient to match fingerprints in a hierarchical fashion starting at the larger scales (Mokhtarian and Mackworth, 1986; Jackway et (/I., 1994). Other ways of dealing with fingerprints are also possible; for example. they can be rcprcsented as a ternary tree (Witkin, 1984) and the stability of various branches of this tree considered (Bischof and Caelli, 1988).
B. Kelcrtcd Work arid Exterisions Unfortunately, it is generally impossible to find smoothing filters which would satisfy all the desired propcrtics on images and higher dimensional signals. Thcrefore, various authors. i n an effort to extend and generalize the theory to higher dinicn-
6
PAUI. T. JACKWAY
sions and nonlinear smoothing operators, have emphasized certain properties and sacrificed others. Koenderink (1984) emphasized the differential structure of scale-space; that is, what are the laws governing the shape of the surface surrounding an arbitrary point F(xo, (TO) in scale-space‘? Koenderink showed that Gaussian filtering is the Green’s function (DuChateau and Zachniann, 1986) of the differential equation known as the heat equation. That is, the Gaussian scale-space image F ( x , ( T ) given by Eqs. (2) and ( 3 ) is a solution of
With this approach, the original signal is the initial condition 0 = 0, which propagates into scale-space under control of Eq. (7). The solutions to Eq. (7) obey a maximum principle (Protter and Weinbcrger, 1967), which states that if F is a solution to Eq. (7) on the open and bounded region with F of class C’ and continuous on the closure of the region, then F assumes its maximum at some point on the boundary of the region or for (T = 0. It has been shown that the maximum principle implies an evolution property for zerocrossings of the solution of the heat equation (Hummcl and Moniot, 1989): Let C be a connected component of the set of zero-crossings in the domain: { (x, CJ) : x E R ” , SI 5 CT 5 S ? ) , where 0 5 SI iS2. Then C f l ((x,a): (T = SI 1 f. !A. This property ensures that a new zero-crossing component cannot begin at nonzero-scale, and that all zero-crossing components can be traced to features on the original signal. This evolution property, called causali9 (Koenderink, 1984), under certain conditions leads uniquely to the scaled Laplacian-of-Gaussian filter (Babaud ef id., 1986). Important later work generalizing the heat equation has shown that space variant anisotropic operators can also satisfy causality while not degrading image edges with increasing scale (Perona and Malik, 1990). The maximum principle and its evolution or causality properties arc indeed one way to extend Witkin’s 1D results to images and higher dimensions. However, part of the elegance of the 1D result is lost since, “a closed zero-crossing contour can split into two as the scale increases, just as the trunk of a tree may split into two branches” (Yuille and Poggio, 1986). This may be a problem as two separate contours at a coarse scale may in fact be caused by the same signal feature (see, for cxaniple. the diagrams in Lifshitz and Pizer (1990)). The monotone property in the sense of our Eq. (4) is therefore not valid, which is a disadvantage with the linear scale-space formulations using zero-crossings in 2D and higher dimensions. The recently developed field of mathematical morphology (Scrra, 1982; Haralick et al., 1987) deals in its own right with the analysis of images but also provides quite general nonlinear operators, which can be used to remove structure from a signal. Therefore, scaled morphological operations have been used as scale-space smoothers. Chen and Yan (Chen and Yan, 1989) have used a scalcd disk for the
MOIIPIHOLOGICAI~SCALE-SPACES
7
moiyholigical opening of objects in binary images to create a scale-space theorem for zero-crossings of object boundary curvature. These results have since been extended to general compact and convex structuring elements (Jang and Chin, 1991). Unfortunately, these results only apply to zero-crossings of boundary curvature of objects in binary images, although they can also be applied to 1D functions, through the use of umbras (Sternbcrg, 1986). However, the extension to higher dimensions seems problematic (Jackway, 199521). Recent work has also considered to construction of a scale-space through scaled morphological operations (van den Boomgaard and Smeulders, 1994). Van den Boomgaard’s approach proceeds by considering the nonlinear differential cquation, which governs the propagation of points on a signal into scale-space under morphological operations with a scaled convex structuring function. However, this work mainly examined the d rential structure of the scale-space itself and did not explicitly emphasize a monotone property. All the points on a signal propagate into scale-space and van den Boomgaard did not explicitly consider special signal points (features) except for singular points which do not obey a monotone property. Alvarez and Morel have recently presented an excellent theoretical unification and axiomization of many niultiscale image analysis theories including most of those mentioned above (Alvarez and Morel, 1994). Once again this approach emphasizes the partial differential equations governing the propagation of the image into scale-space. Here the causality principle is essentially the maximum principle already discussed. The importance of image features and the monotone property is not stressed in this work. The way we have chosen to extend Witkin’s work is to seek to return to first principles. If the aim is to cxamine the deep structure of images, then we should seek to relate signal features across differing scales of image blurring. To be a scale-space theory, we require a monotone property that ensures that increasing scale removes features from the image. In Witkin’s original work, extrema of the signal and its first derivative are seen as fundamental signal features (Witkin, 1984). However, as discussed in Lifshitz and Pizer (1990), there is no convolution kernel with the property that it does not introduce new extrema with increasing scale in 2D, so the monotone property does not hold for linear filters and signal extrema. Therefore, we must turn to nonlinear filters. We have found that scaled operations from mathematical morphology can act as signal smoothers and allow a monotone property for signal extrema, and indeed this result holds for signals on arbitrary dimensional space (Jackway and Dcrichc, 1996). We have also found a way to combine the dilation and erosion to give meaning to negative values of the scale parameter, thereby creating a full-plane (a E R) scale-space. The emphasis in our work is on the monotone property for signal features. We use signal cxtrcma rather than zero-crossings as the signal feature of importance. We no longer use a linear smoothing operator, and we do
8
PAUI, T. JACKWAY
not restrict the scale parameter to nonnegative values. In the remainder of this article we discuss these developments. Multiscale dilation-erosion is introduced and its scale-space properties are discussed in Section 11, followed in Section 111 by a discussion of the multiscale closing-opening scale-space and its relation to the dilation-erosion. Section 1V considers dirnensio~ialityand the selection of the structuring function, and Section V extends the scale-space theory to regions via thc watershed transform. Finally, Section VI is a summary and conclusion. Where possible, the reader is directed to previously publishcd papers for proofs of the various mathematical results, proofs of the new material are placed in an appendix. Parts of this work can be found in earlier conference papers (Jackway, 1992; Jackway e f a l . , 1994) and more fully later(Jackway, 1995a; Jackway, 1995b; Jackway and Deriche, 1996; Jackway, 1996). 11. MULTISCALE MORPHOLOGY
Mathematical morphology grew out of theoretical investigations of a geometrical or probabilistic nature needed in the analysis of spatial data from geology. The work was carried out by a team at the Fontaincbleau research centre of the Paris School of Mines from 1964. This theoretical work was first released widely with the publication of the book by Matheron (1975). A more practical book related to image analysis was later published by Serra (1982) followed by a second volume on theoretical advances (Serra, 1988). Mathematical morphology, developed originally for sets, can be applied to numerical functions either via umbras (Sternberg, 1986) or directly and prcfcrably via the complete lattice approach (Heijmans and Ronsc, 1990). However, since we will consider only functions, we can skip the preliminaries and merely defiric the required operations directly on functions. Also, since notation varies between sources, we need to state that we will follow that of Haralick ef al., 1987). Denoting the functions, f : D c R” + R and g : G c R.” + R , the two fundamental operations of grayscale morphology arc:
Definition I (Dilution): The dilarion of the function , f ( x )by the function g(x) is denoted by (,f @ g)(x) : D c H.” -+ H., and is defined by
(f @ g) ( x) =
v {.f@
-
ttmfj
t)
+ g(t)l.
,
Definition 2 (Erosion): The erosion of the function f ( x ) by the function g(x) is denoted by ( f 8 g)(x) : D c R” + R, and is defined by
cf e g m ) = A td;rlll
+
{ f ( x t) ,
-
g ( t ~
+
Where D, is the translate of D ,D,= (x t : t E D), D is the reflection of L). D = (.t : -x # D). and V ( f ] and A [ f } refer to the mprerriutti (least upper bound) and ir!firriim (greatest lowcr bound) of .f' (DcPrec and Swarlz, 1988). In the discrete case (and for computation) where the function is a countable set ofpoints, max(,f) and min(f') are used for V ( , f ]and A(J').The above definitions arc general, in practice one function, say ,f.denotes the signal and thc other g is a compact shape called the st,.rrc.frir.iiigJ'~ric'tiori.Note also that we have taken particular care with thc edge c
Dejnition 3 (Opening): The opeiiitig ofthe function ,f(x)by the function g(x) is denoted by (J' o g ) ( x ) : D c R" + 13, and is dcfincd by
(J'
0
g)(x) = ( ( f ' 0 g ) a3 g)(x);
Definition 4 (Closing): The closirig of the function J'(x) by the function g(x) is denoted by (,f' 0 g)(x) : D c R" + R. and is dcfincd by
(,f e
m )= CCJ'a3 g ) e m).
These four basic morphological operations possess many interesting relationships and properties. which can be fou~idin the standard sources (Matheron. 1975; Serra, 1982; 1988) and in the tutorial (Haralick et cil., 1987). We will list here just thosc few which are used in this article. 1. All the morphological operations arc i~nrzlinenr, +(ti,f
+ hh) # +(cr,f) + +(hh)
for all functions ,f, h ,
(8)
where + ( . f )= (J' a3 g ) , (J' 8 g). (J' 0 g ) , or (,f 0 g ) . 2. Dilation and erosion arc dircrl.s.
(.f a3 8 ) = - c c - , f ) where the reflection.
3. Closing and opening are duals,
4. Closing and opening arc icletripoferzt,
e s).
(9)
10
PAUL T. JACKWAY
5. Dilation is commutative and associative and erosion admits a chuin rule,
6. Some further identities also prove useful,
f
@ g = (.f @ g )
.f @ R = ( f e g )
0g
= (f * g ) @ g :
.g = ( f o g ) eg.
(17) (18)
Now we need to define the following partial order relation on the functions f ,h :
D c R” -+ R : f I h is defined as f ( x ) I h ( x ) for all x
E D.
7. All the morphological opcrations are increasing,
f 5 h =+ I l r ( f 1 I Ilr(h),
(19)
where Ilr(f1 = (f @ g ) , ( f e g ) . (.f g ) , or (.f 0 g ) . 8. If the structuring function contains the origin g(0) 2 0, thc dilation is extensive and erosion anti-extensive,
g ( 0 ) ? 0 =+ f I (f @g): g(0) 2 0 =+ cf e R ) i f .
(20) (21)
9. Closing is extensive and opening anti-extensive,
Putting the previous two properties together, we have the order relation:
10. If the structuring function contains the origin g ( 0 ) > 0, then:
(.f e g ) I (J’
0g)
if I (.f g ) I cf CB8)
(24)
Finally, we say that function g is open with respect to 11 if, ( g 0 h ) =
(25)
11. The opening and closing obey a sieving property, for if g is opcn with respcct to h , then
12. and also an ordering property,
MORPHOLOGICAL SCALE-SPACES
11
A. Scale-Dependent Morphology We will now proceed to review a scale parameterized signal smoothing operation we have developed for the purpose of scale-space using the basic morphological operations of dilation and erosion (Jackway and Deriche, 1996). Note, from the aspect of morphology, there is little new in the following sections, multiscale morphology has been used since the beginning, for example, in the “Granulometrics” as introduced in Matheron ( 1975). Granulometries enable the distribution of particle sizes in an image to be found by measuring the residue following morphological openings with increasingly large structuring elements. The emphasis in the following work is not on the morphological operations but on the resulting scale-space properties; the fact that the operations used come from morphology is incidental-(but fortunate!) due to the large volume of well-developed theory on which to draw. The simplest structuring function might be that representing an n+ 1-dimensional ball of radius r . this can be written as:
4.+
+ +
where 11 . II= x ’i .xi . . . x: is the norm in the n-dimensional Euclidean space. I n general, the result of dilation or erosion depends on the position of the origin g(0) of the structuring function. We can see this by performing the dilation and erosion of the constant function f(x) = 0, Vx E R ” by the structuring function given by (29),
These level-shifting effects are easy to fix; we simply require:
hence, the structuring function should be everywhere nonpositive with a global maximum value of zero. g(t) 5 0 for all t E G additionally, to avoid horizontal translation effects, we require this maximum to occur at the origin,
(33)
12
PAllI, T. SACKWAY
+
We can make the 11 1-dimensional ball function satisfy thc foregoing conditions by shifting down by the radius:
” .J
=
gh;illl(X)
11 x 11 5 r .
-
(34)
We henceforth assume that all structuring functions discussed in this article satisfy Conditions (32) and (33). The morphological dilation, erosion, closing, and opening can be made scale dependent by the use of a scaled striictirrirzg,f~irilnctiori,g, : G , c R” + R . The most natural way is to equate scale to the radius of the ball in Eq. (34). For reasons which will become clear in the next section, we will in fact equate radius to the rriagriitiide of the scale parameter o # 0, so the scaled ball becomes: =
d
m
-
IIXII
I01
5
(35)
101.
Wc can see how this relates to a prototype ball of unit radius by noting:
d
m
-
la1 = l =
a
l
(
J
a
-
I ~ l I R u , , l , l ~ a l l ( I ~ I - ~ X0 )
1)
# 0,
(36)
and for the support region,
This suggests that given any prototype structuring function g : G use the scaled structuring functions given by,
c It”
4
R, we
Now, if G is bounded, say, G C {x : 11 x 11 < R ) for some R , then g, is defined on G , C {x : II x II < IcrlR) so the support region of the structuring function scales correctly, in particular:
G , + (0) as lo1 + 0.
(40)
However, further conditions need to be imposed on the structuring function to ensure reasonable scaling behavior in all cases. Consider the threshold set of the structuring function G,(t) = {x : g,(x) 2 t ) for any t < 0. To incorporate the idea of scaling, we wish to ensure that for all t < 0: (41)
l m l + 0 =+ G,(t) +
I < I n 2 1 =+ G,, ( f ) c G,,(t), =+ G, ( t ) 1 (x : 11 x 11 5 R ) for all R
(42)
1 0 1
(oI +
30
> 0.
(43)
MORPI-IOI.OGICAI. SCALE-SPACES
1.3
In terms of the functions involved this is equivalent to, lo1 -+
0 =+go@) -+
0
ifx=0;
-cc
i f x # 0,
(44)
This relation indicates that a chord from the origin to any point on the structuring function should lie on or below the structuring function (see Fig. 2). This, togcthcr with the nonpositivity condition (32), nieans that the structuring function should bc monotone decreasing along any radial direction from the origin. It also means the function is coiivex.' In fact in almost all practical cases we use functions from the slightly smaller class of continuous convcx structuring functions. Convex shapes are widely used
t
Flcime 2. A chord troin the origin to any point o i l the structtiring tuiictioii should lie on o r helow the structuring function.
I By convention. in matht.nintica~niorphohgy. a < o m w ttinction h a s any chord entirely o i i o r hclow the function. Note, thi\ would he called a w ! t < ' o \ ' eliinclion i n ;innlysi\.
14
PAUL T. JACKWAY
as structuring elements in morphology as they possess many useful properties (Serra, 1982). In image analysis it is often desirable to ensure isotropic propertics in any filtering; this translates directly to the morphological structuring functions being circularly symmctric. A useful family of isotropic structurin functions is iven by powcr functions of thc vector norm 11 x It= = d ! e . So we define the poweroid family of scaled structuring functions:
JX'AX
Definition 5 (Poweroid Structuring Functions): The scaled poweroid structuring functions are given by, = - l ~ l ~ l l x l l / l o l ) ua 1.0,
fl
f0.
Some commonly used structuring functions are presented in Fig. 3 including representative members of the 2D poweroid functions g ( x , y ) = -(\/x2+y?)u.
(49)
This family includes cones (a= I ) , paraboloids (a = 2 ) , and cylinders (a = co).
((u
-(.\I
FlciLJl
+ \.?)a/?,
ID: (:I) sphere; ( h ) colic ( a = I ) ;(c) paraholoid hclong to the circular poweroid Ihinily, %g(.\,y ) =
15
MORPHOLOGICAL SCALE-SPACES
To cater for nonisotropic (directional) filtering we can define a elliptic poweroid family of scaled structuring functions: &(X)
=
-
lal(JXTAX/(oI)u
(Y
> 0,
CT
# 0,
(50)
where A is a symmetric positive definite matrix. In H’ the contours gn(x) = constant are ellipses of various orientation and eccentricity. If A is the unit matrix thcn (50) reduces to the isotropic case (Definition 5 ) . This idea is an extension of van den Boomgaard’s ( 1992) nonisotropic quadratic .structuring,function. The elliptic powcroid structuring functions have some special properties and arc discussed further in Section V1.
B. Semi-Group and General Properties qf the Structuring Function Scrra ( 1982) has presented the following proposition: A family B h ( h 2 0 ) of nonempty compact sets is a one-parameter continuous scmigroup ( i t . , Bh $B,, = B h + / , , h , 0) if and only ifBh = hB whereB is a convex coinpact set.
We will present a related result for our scaled structuring function, which helps to explain our choice of the scaling Eq. (38), and why the convex property of structuring functions is necessary. Wc could obtain the umbras of our functions (Stcrnberg, 1986) and directly use Serra’s result above but it is more informative to work from first principles to show how the definition of convexity comes into play. We introduce the following proposition:
Proposition 1. A farnily g, (a > 0) of scaled structirring functions given by Ey. (38), which is convex, is a one-parameter continuous semi-group. That is, K n @ g/, = go+/, .for fl, P ? 0. Proof.
A proof of this proposition can be found in the appendix.
As our scaled structuring functions are dependent only on the magnitude of the scale parameter, we have the further result, go @ g/, = glf1 ti/,[.
0,P
E
It.
(51)
The concept of being morphologically open, Eq. (25), places an order on the structuring functions. We have the following proposition:
Proposition 2. If g, (x) denotes a conve.~scaled structuring junction given by Eq. (38), and if I ? 101 1, then the scaled structuring function go: (x)is morphologically open with respect to g,, (x).
Proof. A proof of this proposition can be found in the appendix.
16
PAIJI- T. JACKWAY
111. MULTISCALE. DILATION-EROSION SCAIE-SPACE
Using the scaled structuring functions just defined, we can join dilation and crosion at zero-scale to form a single multiscale operation, which unifies the two morphological operations as follows:
Dejnition 6 (Multiscale Dilation-Erosion): The multiscale dilation-erosion of the signal f ( x ) by the scaled structuring function g,(x) is denoted’ by f @ g , and is defined by ( f @ g , ) ( x )=
{
( f @ g n ) ( x ) if (T > 0; f(x)
i f a = 0;
(.f e g , ) ( x )
i f a < 0.
That is, for positive scales we perform a dilation, for negative scales an erosion. With this method, scale may be negative; it is 1 0 I which corresponds to the intuitive notion of scale. Unlike linear operators, dilation and erosion and “non-self-clual” (Serra, 1988); therefore, positive and negative scales in scale-space contain differing aspects of the information in a signal. As we shall see, positive scales pertain to local maxima in the signal, whereas negative scales pertain to local minima. Other authors (e.g., van den Boomgaard, 1992) have considered scaled dilations and erosions separately, and it is well known from mathematical morphology that if f is sufficiently smooth, then both lim,+o(f @ g,) + f , and, lim,,o(f 8 g,) -+ f , but we explicitly combine these operations into a sirigle operation. We specifically wish to consider the scale-space fingerprint for positive and negative scales as a whole since the information content o f a signal is expanded into this entire region. This approach is consistent with the scale-space philosophy of treating the scale-space image as a whole for the purposes of analysis. Having defined a suitable operator, we now define the associated scale-space image F : D C R f x R + Rdefined by (cf. Eq. ( I ) ) :
+
where the ( n 1)-dimensional space given by D x R is known as the triultisccrle diltrtiotz-erosion sccrle-space. Since we arc using the operations of mathematical morphology to smooth a signal, the well-known geometric visualizations of dilation and erosion are intuitively helpful: For the moment, take thc scaled structuring function to be a ball with the radius as a scale parameter with a positive radius corresponding to rolling the ball along the top of the “surface” of the signal, and a negative radius to rolling the ball along the underneath. The smoothed signal can be visualized as the surface traced
’
Thc symhol @ has previously hccn iiscd by Serra (1987) to rcler which tlocs not appcar i n this article.
10 h e
hi1 01 mi.\\
mfii.\/&wi.
MOIZPIIOI.O~~ICAI. SC‘AI I:-SPACES
17
out by the ccntrc of the ball when it is traced over the top (dilation) or underneath (crosion)of the surface of the signal. We illustrate this operation for a ID signal in Fig. 4. The multiscalc dilation-erosion smoothing of the “Lena” image is shown it1 Fig. 5 for scvcral scalcs, positive and negative. Intuitively, this new surfacc is smoother (in the sense of having smoother and less hills) than the original signal, and furthermore the largcr the radius the smoother the filtered surfacc becomes. In the limits, as the radius approaches zero the original image is recovered, and as thc radius approaches infinity the output becomes flat. It should bc apparent that if the ball touchcs the top of a hill (local maxima), then a hill will appcar on the output at exactly that point. If, however, the radius is such that the ball is prevented from touching that hill by nearby hills, then no hill will appcar at that point on the output, and more importantly that hill cannot rcappcar for any increased value of radius, Y . Thus, i t would seem that the number of local maxima may be a monotone tlccrcasing function of Y . This turns out to be so-but it is not quite so easy to prove! The scale-space properties of the multiscalc dilation-erosion will be revicwcd formally in the form of propositions building to a theorem and corollaries expressing the scale-space monotone principlc. Thc proofs of Propositions 3 to 7 will not be presented here as they can bc found in Jackway and Deriche (1996). In outlining the results o f this section, we will often make use of the following tluality between morphological dilation and erosion, Eq. (9). Therefore, many rcsults on , f @ gn for (T 0, which correspond to dilation, can be immediately applied to (T < 0, corresponding to erosion. I n practice most structuring functions arc symmetrical about the origin (g(x) = g ( - x ) ) so that g(x) = g ( x ) . I n this case the erosion of a signal by a structuring function may be obtained by negating
PAUI. T. JACKWAY
F'l(;Lu
the signal, perfornung a dilation with that same structuring function, and negating thc rcsult.
A. Continuits urid Order Properties cf the Scale-Spce Image
Thc definition of the multiscale dilation-erosion (Definition 6) consists of three parts corresponding to positive, zero, or negative valucs of the scale parameter. It is thcreforc natural to consider the behavior of the scale-space image F ( x , a ) across this scam at CT = 0 in scalc-space. Since the structuring function is zero at the origin [see Eq. (33)], the dilation is extensive and the erosion anti-extensive [Eqs. (20) and ( 2 I)]; therefore, we have
19
MORPWOLOGICAL SCALE-SPACES
the result,
( f ogn)(X)n
E
(53)
D.
This order property looks promising and indeed a further continuity property applies that shows that at continuous points of ,f (x), ( f ag,)(x) approaches j ( x ) as the scale parameter approaches zero from either above or below. Wc have the following proposition.
Proposition 3. Ifthe bounded signal f (x) is continuous at some x E D , then the .scale-space image F(x, a) is continuous with respect to (T at a = 0. That is, at points x where ,f (x) is continuous. F(x, a) + j ( x ) U S a +. 0. In f a d a slightly stronger result holds; full continuity of the signal is not necessary for the one-sided limits to converge to the signal. At points x = x,, wherc ,f(x,,) is upper semi-continuous (u.s.c.) F(x,,, a ) --+ j ( x , , )as (T -+ O’, and at points x = X I whcre f (x/)islowersemi-continuous(I.s.c)F(x/,a) +. f ( X I ) a s a + OW. We recall that a function f (x) is said to be upper semi-continuous at x,, if the nondeleted limit superior Lim SUP^+^,, f ( x ) = f(x,,), see, for example, Bartle (1964). Further f is lower semi-continuous at c iff -f ( c ) is U.S.C. Upper semicontinuous functions are often used to model pictures since thcir threshold sets, F ( t ) = (x : f(x) L I ) , are closed scts (Serra, 1982). If the structuring function is sufficiently smooth, this property transfers to the scalc-space image and we have the following proposition:
Proposition 4. I f the structuring junction g(t) is c1 continuous function on R”, then the scale-space image of the bounded signrrl f (x) is continuous on R ’ x R f o r all x E D , a # 0. This applies for any signal ,f (x) (as long as it is bounded) and shows that the scalcspace image is much better behaved than the signal itself. This is to be cxpectcd since the signal has been smoothed to give the scale-space image. The following point-wise order properties of the scale-space image follow directly from the extensivity, and increasing propcrtics of the morphological dilation and erosion [Eqs. (19)-(21)], and the order properties of the scalcd structuring function [Eqs. (44)-(46)].
Proposition 5. The scale-space image F(x. a) = (f@ g,)(x) possesses the,following properties: F(x, 0) = f(x) F(X,
00) =
for all x E D:
V ( j ( t ) l for a11 x
E
D;
ItD
~ ( x-00) , = A{f(t)l
for a11 x E D ;
tell
a(, < a,>=+ F(x, (T~,)i F(x,a,’) foralla,,, a(/E R.;
x ED.
Propositions 3 t o 5 show that the scale-space image has good continuity antl order properties but we have yet to show the essential scale-space monotone property. The major rcsult of this section is a theorem, which shows in a precise way how , f @ go bccomes smoother with increasing la 1. Furthermore. we show that the monotone property holds for local extrcnia of the signal so this is the signal feature appropriate to the niultiscalc dilation-crosion scale space. Prior to presenting this theorem soinc ncccssary partial rcsults arc obtained. The lirst result relates the position antl aniplitudc of a local maximum ( o r minimum) in the filtered signal to that i n thc original signal.
Proposition 6. Let the strrrctirr-irig~~rictiori I I L I Vcr~sitigle r ~ ~ c i . x i i i i r(it ~ ~ the i i origiri: /litit i s . g(x) cr Incul I ~ L I . Y ~ I ~ ~ Iiriiplirs IIII x = 0,tlicn: ff'a > 0 L
i d (,f@g,)(x,,,,,,)is cr local t t ~ ( r . ~ i i i z ~ i trl/iic. t i . ,f(xl,,~,~) is L I loccrl of .f(x)Nlld ( f a g , = j(X,,,'l,): is LI loccrl rriiiiiriiirrri. theri. ,f'~~,,,,,,)i s cr local ( h ) I f n < 0 miel (,f8gn)(xllllll) /liirlirrllrtH ofj(x) L I l i d (,f8,qn )(x,,,,,,) = .f'(X,,,,,,). (61)
tIlNX.iII?CIII7
We arc now able to rclatc a signal feature at nonzero-scale to the original signal (zero-scale). However. to obtain a nionotonc result we nced the next proposition.
Proposition 7. Let tlic .str-irctirriri~,fiiric~tioti liervc ti siiigle lncril origin: tlitrt is, g(x) is cr lorrrl I I T C ~ X - ~ I ~ I Mirriplies III x = 0,t h c ~ i :
r t i r r s i r t ~ r m( i t
tJic
(fa,)> a > 0 trrid
(.f'@gs,)(xI,,,,,) is rr l o c d ~ t i t ~ x i t t t u ttheti. n, (,f'@gn)(x,,,,,,) (,f'8g,)(XIll,,\) = ( f a g n , ,)(XIll;,\); (/>) (fa,, < a < 0 llild (,f@g,,,)(x,,,,,,) is Cl local rrlirlirll~rm, tl1et1, (,f@g, )(XI,,,,,) is tr loco1 rIlinirr7lrun Lllld, (.fagn)(XI,,,,,) = (,feg,,,)(x,,,,,,).
((I)
/ S (I lOCN/ / 1 7 ~ / . ~ ~ I 7 l l[lJ/ l/ l?d ,
These propositions provide very important scale-space results because they cnablc coarse-to-fine tracking in the scale-space image. If a signal feature (extrerna) appcars at some scale DO,i t also appears at zero-scale and all scales in between. Stated as a monotone property, we can state that the numbcr of features may not decrease as scale approaches zero. This property is now encapsulated in a theorem.
Theorem 1 (Scale-Space Monotone Property for Extrema): Lrr ,f' : D 5 13'' + H clenote (I borriiderl~filtictiori,g, : G H" + R N sc~mlec/st~iic~turirig,fiirictiori srnti.sf.+iiig tlie cotic1itiori.s offProposition 7, trizdthe/,oitzt.sets. El,,,,, ( , f ) = {x : j is CI lncul tnaxir~iion),and, E , , , ~ , , (= j )(x : j is rr loccil riiiriiriiiirri] ck.iiote t l i r locrrl c~rrriicrof ,f. Tlieri, ,for mi). s(z1e.s (TI < n2 < 0 < 03 < n ~ , ((I)
~lll,,l(,f@g~,~ G El,,l,l(.f'~g~,)
c E,,,,,,(f');
trr1cl.
(h)
~ l , , ' l , ( f s s f T , )
c ~ 1 , , , \ ( . f Q S n ; )c ~,ll,\(f').
MORPHOLOGICAL SCALE-SPACES
21
Proof. Suppose the theorem is false andE,,,;,,(,f@g,,) 9 E,,,,,(,f@g,,) for some 0 < ‘53 < 04, then there exists some xlll~lk E D such that F(x,,,,~, 0-1)is a local maximum but F(xIll~,~, 0 3 ) is not, which contradicts Proposition 6(a). The case for E,,,,,,is proved similarly using Proposition 6(b). This theorem is actually stronger than required since i t governs the positions of the extrema as well as thcir number. To obtain a monotone property of the form o f Eq. (4), we need some functional # : R i + R such that
For the practical case where E c Z” we simply choose #[El = the number cf points in E . We have the following corollary to Theorem 1 :
Corollary 1.1 (Scale-Space Monotone Property for the Number of Local Extrema): For # : R” -+ R, such that El E? c R” =+ # ( E l ) 5 #(E?)then, ,for LIny 0 1 < CT? < 0 < C I T < rrJ,
We can further extend Theorem 1 from local to regional extrema, which makes it more useful when dealing with operations that rely on the number of regional cxtrema, such as the watershed transform (to be discussed later). First, we recall the definitions of the various types of extrcma: (a) f is said to have a strict local muxirnum at x = xo if there exists a neighborhood N ( x ( , )such that f (x) < f(x0) for all x E N(xo). (b) ,f is said to have a local maximum at x = xo if there exists a neighborhood N(xo)such that f(x) I f(xO) for all x E N ( x g ) . (c) ,f is said to have a regional maximum of value h on the connected component M if there exists neighborhood of M , N ( M ) , such that f (x) = h for all x E M , and f ( x ) < h for all x E N ( M ) . (d) The corresponding definitions for minima follow directly with the inequalities reversed. Since we will be counting regional cxtrema. we will assume that ,f has a finite number of connected components in all upper and lower thresholds. We can now present the following corollary to Theorem 1:
Corollary 1.2 (Scale-Space Monotone Property for the Number of Regional ( f ) ] denote the number of connected comExtrema): Let C[R,,,,,( f ) ] und C[R,,,,,, ponents in the point sets of the regiorml extrerna of LI signal f . then for any scales, CI] < (T? < 0 < (Tj < 0 4 ,
22
PAUL T. JACKWAY (0)
C[Riwn(f~g,,11 I ClRniiii(fQ~rr~)l I C[Riiiin(f11;
and. ( b ) CLRlll,, (f
s g,, )I I C[Rl,It,X( f Q go,) 1 I C[Rlll,,, ( f ) l .
Proof. A proof of this corollary is given in Jackway (1996). Corollaries 1.1 and 1.2 are monotone properties of the form of Eq. (4) and we can thereforc claim the production of a scale-space. This scale-space allows all input signals in any dimensionality as long as they are bounded (infinite amplitudes upset the morphological operations!). The signal is expanded into a scale-space image by smoothing with the multiscale morphological dilation-erosion. The features in this scale-space are the signal local extrema (maxima for positive scales, minima for negative scales). We have given a meaning to the concept of negative scale through the use of the morphological erosion. We have shown that the number of features may not increase with increasing scale but we have not shown that they decrease! However, if a signal contains information at different scales, this will generally be reflected as a decrease in the number of features with increasing scale magnitude. If the signal has a single unique global maximum (minimum), thcn for sufficiently large positive (negative) scale, there remains only a single feature in the scale-space image. IV. MULTISCALE CLOSING-OPENING SCAI~E-SPACE We have developed a scale-space theory based on the morphological dilationerosion, but to some readers it may seem strange that we did not use the opening or closing operations. First, the morphological dilation and erosion are not true morphologicaljilters (as they are are not idempotent) like the opening and closing (Serra, 1988). Second, Chcn and Yan (1989) have published a well-known paper titled “A Multiscaling Approach Based on Morphological Filtering” in which they demonstrate a scalespace causality property for the zero-crossings of curvature on the boundaries of objects in binary images when opened by multiscale disks. This work has since been generalized by Jang and Chin (199 I ) to show that convexity and compactness of the structuring element are the necessary and sufficient conditions for the monotonic property of the multiscalc morphological opening filter. Their theorem is:
Theorem 2 (Monotonic Property of the Multiscale Opening: Jang and Chin, 1991): Suppose X is a compact set in R’. Z [ a X ]denotes thejinite number of zero-crossings of curvature,function along the contour ax, and C N [ X ] is the
MORPHOLOGICAL SCALE-SPACES
number of connected components
X o B(r)#
and
of X . For any r
23
> 0,
C N [ X ]= C N [ X o B ( r ) ] = 1,
w e have,
crrid Z ( a [ X o B ( r ) ] )is monotonic decreasing as r increasing ifand only if B ( r ) is CI compact convex set.
A review and comparison of Gaussian and morphological opening scale-spaces for shape analysis have recently appeared in the literature (Jang and Chin, 1992). Interestingly this review stresses the signal ,feature-smoothing jilter aspects o f scale-space and the importance of the scale-space causality or monotone property as we do here. Note some technical problems have been found in the approaches of both Chen and Yan (1989) and Jang and Chin (1991), which limit the generality of their results (Nacken, 1994; Jackway, 1995a). However, the paper by Chen and Yan (1989) is noteworthy in being the first attempt to use nonlinear operations to create a scale-space. An advantage of the above approach is that in using zero-crossings of boundary curvature as the feature there is an obvious close connection with the Gaussian approach which uses zero-crossings of the second derivative. When applied to functions, zero-crossings of curvature are equivalent to zero-crossings of the second derivative, since,
K/ ( X )
<0
,f”(X)
< 0.
(56)
In unpublished work we have extended Theorem 2 to functions and obtained a scale-space monotone property for zero-crossings of the second derivative of ID functions smoothed with a multiscale closing-opening operation (Jackway, 1995a). In common with all the other zero-crossing approaches, including Gaussian scalespace, the big disadvantage with using zero-crossings as the scale-space feature is that there is no obvious extension to functions on higher dimensional spaces (e.g., images). Our previous success with using signal extrema as the feature with morphological dilation-erosion scale-space suggests that using extrema rather than zerocrossings with the morphological opening may be a way to achieve a scale-space for higher dimensions. This is indeed so, and we will now present the development of a closing-opening scale-space and then we will examine its relation to the dilation-erosion scale-space.
24
PAUL. T. JACKWAY
A. Properties cfthe Multiscale Closing-Opening Throughout this section we will work with the multiscale closing-opening, which can be defined in terms of the closing and openings with scaled structuring functions:
Definition 7 (Multiscale Closing-Opening): The multiscale closing-opening of the signal .f (x) by the scaled structuring function g,(x) is denoted3 by ,f 0g,, and is defined by
Note: A similar multiscale operation has also been defined by van den Boomgaard (1992). The multiscale dilation-erosion smoothing of the “Lena” image is shown in Fig. 6 for several scales, positive and negative. In this section we will often obtain results for scale 0 < 0 since this corresponds to the opening operation which is used in the literature. In these cases we then appcal to the duality principle of the opening and closing (Eq. ( I 1)) to extend the results to the morphological closing and hence to the combined closing-opening operation. Since both the closing and opening are idempotent, Eqs. (12) and (13), the closing-opening is as well,
Since the closing is extensive and the opening anti-extensive [Eqs. (22) and (23)], we have the following result,
and we have the following order properties with respect to scale,
Proposition 8. I f
01
< 02 < 0 < ~3 < a ~then ,
(f 0so,)i ( f 0g,:) Proof.
5f 5
(f0
SITl)
5
(f 0XnJ.
(59)
A proof of this proposition is given in Appendix A.l.
The symbol 0 has previously been tiscd by Serra ( 19x2) to refer to the thickrrriii~,which docs not appear in this article.
The rnultiscale closing-opening filter also satisfies thc lollowing scale-relatcd conditions (cl. Chcn and Yan, 1989): 1. 11 is scale invariant, that is,
(1
1
f ( f ) O g , ( t ) = (7 - . f ( r J f ) o g , ( t ):
( 60)
2 . The filter recovers the input signal for zero-scale (by definition!),
( f 0g o i ( r ) = f u ) :
(61)
26
PAUL T JACKWAY
3. As scale approaches positive (negative) infinity, the output approaches the global maximum (manimum) of the input signal, that is,
Therefore, the multiscale closing-opening appears suited to the formation of a scale-space, similar to that formed by the multiscale dilation-erosion. This is, we should consider the multiscale closing-opening scale-space F : D 2 R ’ x R -+ R. defined by,
F(x, 0 )= ( f 0gn)(x).
(64)
Now we need to obtain monotonic properties for signal features within this scalespace. The results of Chen and Yan (1989) and Jang and Chin (1991) depend on partitioning the result of the opening operation into arcs ofthe original set and arcs of the translated structuring element; we will extend this partitioning idea to work with functions on multidimensional spaces and, therefore, with multidimensional arcs orputches of the structuring function. The first step is a basic morphological result, outlined in Haralick et ul. (1987), which provides a geometrical interpretation to the opening and closing: To obtain the opening off’ by a paraboloid structuring element, for example, take the paraboloid, apex up, and slide it under all the surface o f f pushing it hard up against the surface. The apex of the paraboloid may not be able to touch all points of f . For example, if ,f has a spike narrower than the paraboloid, the top of the the apex may only reach as far as the mouth of the spike. The opening is the surface of the highest points reached by any part of the paraboloid as it slides under all the surface of f . (. . .) To close ,f with a paraboloid structuring element, we take the reflection of the paraboloid in the sense of (Eq. (lo)), turn it upside down (apex down), and slide it all over the top of the surface of f . The closing is the surface of all the lowest points reached by the sliding paraboloid.
In terms of the opening we have thc following proposition.
Proposition 9.
f
08 = T
[
u
UkL]
UlrlcUI/Il
(1
Where, U [ g ]is the umbra of g, that is, U[gl = ((x,y ) : y 5 g(x)l. T[U[gll : R” + Risthe“topsu~ace”oftheumbra;fhatis, T [ U [ g ] ] ( x = ) max(y : (x,y) E U [ g ] ) .U [ g ] ,indicutes the translate o f U [ g ] by z E R x R, U [ g ] ,= (U z : u E UkII.
+
27
MORPHOLOGICAL SCA1,E-SPACES
Prooj
This result is proved in Proposition 71 of Haralick et al. (1987).
From this geometrical interpretation of the opening (or closing), we see that thc output signal can be partitioned; that is, with f : D E R” + R,
.f(x) i f x = S ’ ( ~ T ) ;
( f 0 gn)(x) =
if x = $’(a).
s(x)
with,
S ’ ( O )U S ” ( ( T )= D
S ’ ( O ) n s”(a) = fl x = S”(0.)
s(x) < f ( x ) S(X) =
u I
PATCH[(g,):,]
(69)
d
PATCH[(g,);,] n PATCH[(g,);,] = fl f o r i
#j
(70)
where PATCH[(g,),,] is a patch on the structuring function g,, which has the origin translated to z , E Ulf]. Note that I is afinife index family (Jang and Chin, 1991). In words, the opening of a signal consists of patches of the original signal combined with patches of translated structuring functions. By duality, the closing can be partitioned in a similar way. This geometrical interpretation of the opening and closing (on a ID function) is illustrated in Fig. 7. Now we examine how this partitioning varies with scale and we obtain the following proposition:
Proposition 10. Given that 01
J
O
g,, =
< a2 < 0 , then,
f(X)IxeSln,)
u
u
PATCH[(sn,);,]
(71)
u u PATCH[(g,?):,].
(72)
I
f o g , ? = f(X)IxFs,n?)
I
(a)
(1))
d
tl
(C)
(11)
FIOIJRE7. Geometrical interpretation of thc opcning and closing with partitioning. ( a ) parabolic structuring function ,q(.\ 1: (h) signal , f ( . r ) ; (c) opcning / i 8: (d) closing , / ,g.
PAUIA'I. JAC'KWAY
28
Proof. A proof of this proposition is given in Appendix A . 1. The above proposition states that with increasing scale, the opening replaces more and more of the original function with patches from the structuring element. Likewise, from the duality of opening and closing, we see that a corresponding result applies for the closing, with the function being replaced with patches from the inverted structuring function. B. Moriotorie Theorern f o r the Multiscde Clo.sirzg-Operiirig
The patches of the upright (inverted) structuring function posscss several stnoothness propcrtics, namely. that they are convex (concave). cannot contain a local minimum (local maximum), and contain at most one local maximum (local minimum). Therefore, we have the following scale-space monotone theorem.
Theorem 3 (Monotone Property of Local Extrema of the Multiscale Closing-Opening): Let f : D c R" + R. denote N boltridcd jiiiirtiort, go : G c R" + R a sccrlecl strurtiiriiig ,firrictiotz wtisfi~iiigthe coriditioizs o j P r o p siiiori 10, aiid the poiiit sets, E,,,,,(f) = (x : f is a loctrl rnnxiniirrii), a r i d . E,lllll(,f) = ( x : ,f is a local i~iinimirin),denote the local e.rtreina of j . 'l'heri. f o r any sccrles ( T I < ff? < 0 < ( 7 3 < (74, ([I)
Ei1iiiic.f
O gol) C Eiiiiii(.f O ~
C Eiiiin(f);
n : )
crrzd, (17)
~ t l l a \ ( f
(3 So,)
c
~ l l l i l Y ( f
O Sn;) c EillJX(.f).
Proof. The patches for the opening contain no local minima,
(U PATCH[(Rn.),,]
~111111
)
= u;
(75)
The occurrence of a local minimum is a local property of a function, therefore. the rcplaccment of part of ,f with a patch from go cannot affect the existence of local extrcma outside the patch except possibly at the patch boundary. However, from Eq. (68), the patch is everywhere less than the function it replaces, and no new minima can be created at the patch boundaries; therefore,
(.f O gm1)
~llllll
Eiiiiti(f
o
~ m , )
(mI x tS' ( nl
= ~llllll
));
= Eti,i,i(f(x)l~~~s,(m~,).
(76) (77)
MORPHOLOGICAL SCALE-SPACES
29
But, by relation (73), S’(a1) C S’(cr2) C D , so,
and therefore, ~ i i i i i i ( f
o
gml)
c
Eiiiiii(f
o
SO?)
c Eiiiin(f).
(79)
And, as usual, by duality,
E,,,;u( f 0)8,
G El11,lX ( f 0SOl) C E,ll,X ( f ).
(80)
This theorem is the direct analog of Theorem 1. Corollaries 1.1 and 1.2 can also be applied to Theorem 3 and we will state them for completeness without proof.
Corollary 1.3 (Scale-Space Monotone Property for the Number of Local Extrema): For # : R” + R., such that El C El c R ” =+ # ( E l ) 5 #(I??) then, f.r uny a1 < 0 2 < 0 < (Tj < (TJ, (a)
#[Eiiiiii(f
O gml)l i
#[~lll,X(f
08 , ) l 5 #[EIll,X(f O gm)I I #[~111;1x(.f)l.
#[Eiiiin(.f
O g o ? ) ] i #[Eiiiiii(f)l;
und,
(0)
Corollary 1.4 (Scale-Space Monotone Property for the Number of Regional Extrema): Let C[RnlilX(f)] and C[R,,,,( f ) ] denote the number of connected components in the point sets of the regional extremu of u signal f , then,for any scales, CII < a2 < 0 < CI,< < (TJ, (0)
c[Riiiiii(f
O gul)l F C[Riiiin(.f O g o , ) ] I C[Riiiiii(,f11;
und, ( h ) C[Rlll,,(f 0gn,)l I C[Rl,l‘,X(f O gm)I I ~l~Ill~lX(f)l.
Having presented another scale-space, the question now is how are the two scale spaces, F,(x, a ) = (f@g,)(x) and F,(x, (T) = (.f 0 g,)(x) related? We can answer this in the next section using fingerprints.
V. FINGERI’IIINTS IN MORPHOI,OGICAl SCA1.E-SPACE
As mentioned in the introduction, the central idea of scale-space filtering is that we trace the paths of signal features through the scale-spacc. These paths are termed the scale-.spuce~nger~~riiit and we have previously seen in Fig. 1 a typical Gaussian scale-space fingerprint. We have already defined the sets of feature positions, EIlli,,
30
PAUL>.r. JACKWAY
and E,,,,,,for use in Theorems 1 and 3 , now we start with some further definitions and notations concerning fingerprints.
Definition 8 (Multiscale Dilation-Erosion Scale-Space Fingerprint): The tnultiscale dilation-erosion scale-space~ngrrprintis a plot, versus scale, of the scaledependent point-set:
Likewise, for the multiscale closing-opening, we define:
Definition 9 (Multiscale Closing-Opening Scale-Space Fingerprint): The multiscale closing-opening scale-space~ngrrprintis a plot, versus scale, of the scaledependent point-set:
To illustrate these definitions we show in Fig. 8 fingerprints for the same random 1D signal from the zero-crossings of second derivative in Gaussian scale-space, local extrema in multiscale dilation-erosion scale-space, and local extrema in multiscale closing-opening scale-space.
A. Equivalence of Fingerprints It is clear that the smoothed signal formed by the dilation-erosion is in general different from the smoothed signal formed by the closing-opening. However, the fingerprints are concerned only with the local extrema of these signals and it is not immediately apparent what the relationship is (if any) between fingerprints of the two scale-spaces. Let us compare the dilation and the closing. From Proposition 6 we see that (for CI > 0) a local maximum on the dilated signal corresponds directly in height and position to a local maximum on the underlying signal. Then from the geometrical interpretation of the closing we can also see that the same should apply for the closing operation. Furthermore, an identical condition of the structuring function “touching” the signal at a local maximum is responsible for the local maximum in both the dilated and the closed signals. A minima in the smoothed signal can be formed in slightly different ways between the dilation and the closing but Proposition 12 will show that these too are equivalent. To formalize these ideas wc have the following propositions. Note that we havc split the results between two propositions as the method of proof differs considerably between them.
Proposition 11. Let the structuring function have a single local maximum at the origin. The,following two statements are equivalent:
MORPHOI,OGICAL SCALE-SPACES
31
FIGUIU;X. A comparison of lingerprin~s-(n) iero-ci-ossinga of second derivative in Gatisinn \cale-space:(h) lociil extrema in multiscalc ililation-crosion scale-space; (c) local cxtrciiia i n multiscnle closingopening scale-space. Figtirc X(h) rcpriiited li-om (Jackway CG Deriche 1996. @ IEEE).
1.
2.
(f @ g,)(x) hus a local maximum at x = x,,,;,, ( , f 0 g,)(x) has a local maximum at x = x,,,;,,
Likewise, ,for the erosion and opening, thefollowing two statements are equivalent: 1. (f 8 g,)(x) has a locul rniriirmun ofx = x ~ , ~u ~,, 2. ( f o g,)(x) has CI local minimum at x = x,,,,,,
32
PAUL T. JACKWAY
Proof
A proof of this proposition is given in the appendix.
Proposition 12. Let the structuringfunction have a single local maximum at the origin. The following two statements are equivalent. I. 2.
( f @ gn)(x) has a local minimum at x = x,,,,,, (,f g,)(x) has a local minimum at x = X,lljn
Likewise, for the erosion and opening, tlie,following two statements are equivalent:
2.
( f 8 g,)(x) has a local maximum at x = x,,,,,u ( f o gn)(x) has a local maximum at x = x,,,,,
Proof
A proof of this proposition is given in the appendix.
1.
Together these propositions show that the full scale-space fingerprints for the dilation-erosion and the closing-opening are identical,
E @ ( a )= Ea(a) for all a E R,
(83)
so we can drop the superscript and simply write E ( o ) . Thus, the surfaces formed by the multiscalc dilation-erosion and multiscale closing-opening of a multidimensional function, although different almost everywhere, have local extrema of the same height and at the same points. This has computational implications, since the dilation is usually quicker to compute than the closing. Therefore, if it is desired to cxtract the morphological scale-space fingerprints from a signal, then the dilation-erosion scale-space is the most efficient way to do this.
B. Reduced Fingerprints
Proposition 6 shows that fingerprint paths corresponding to signal maxima (minima) do not change spatial position as scale is varied above (below) zero. Therefore, this subset of the full fingerprint consists of straight lines only, and so can be represented vary compactly. This leads us to define a subset (maxima only for positive scales, minima only for ncgative scales) of the full fingerprint called the reduced fingerprints and denoted by the subscript r : Definition 10 (Morphological Scale-Space Reduced Fingerprint): phologicul scale-space reduced fingerprint is defined as:
The rnor-
33
Now, from Theorem 1 (or alternately, Theorem 3j, we find that in the reduced fingerprint, (TI
<
02
04
>
03
C El (0); > 0 =+ E,-(oJ)C El ( ( ~ 3 ) C E,.(O). < 0 =+ E , - ( a l )5 El ( 0 1 )
(85)
(86)
Since at any scale, El ( o ) is a point-set, the above set inclusions confirm that the reduced fingerprint consists only o f vertical lines. beginning at zero-scale and cxtending into positive and negative scale-space until they end. This behavior can hc seen in the example of a reduced morphological scale-space fingerprint shown in Fig. 9. This property makes the reduced fingerprint particularly easy to represent; all we need is to specify the position ofcach fingerprint line and the scale (positive or negative) at which it ends. This is easily stored as a list: Suppose we have a signal ,f : R ” + €3. with k local extrema at ,f(x,),i = I , 2, , k , thcn, since each local cxtrcma is the origin of a fingerprint line, we can represent the whole reduced fingerprint as a list of k ( n I)-tuples, (xl.o,), i = 1 , 2, . . . , k , where O, is the scale associated with fingerprint line i.
+
C. Coinputcitiori of tlic Hetluccd Fingerprint The practical importance o f the rcduccd fingerprint is enhanced by the fact that we have developed an efficient method for computing the reduced morphological scale-space fingerprint, which docs not involve actually computing the scale-space image or indeed any signal smoothing. We can consider the ii-tuple (x,,0-1) as assigning a scale (T, to the local cxtrema at position (x,)in the original signal. where (T, rcpresents the scale value at which the fingerprint line at (x,)ends. From the geometrical interpretation of
34
PAUL T. JACKWAY
the morphological operations, we see that this of is the value such that structuring functions of m 5 ml “touch” the signal at (x,)while those of scale m > nf do not. This is the property that is used in the algorithm which follows-for each extrcma in the signal, we search for a structuring element satisfying this property and then assign its scale value to the fingerprint line. It may help to visualize this operation as inflating a balloon resting on top of a local maxima (see Fig. 10). If the local maxima is lower than the global maximum, then eventually the balloon gets so big that it touches the surface elsewhere and, from thcn on, no longer rests on the local maximum. We equate the radius of the balloon (structuring function) at “lift-off“ as the critical scale of the local maximum. Of course, if the signal local maximum is equal to the global maximum, we can immediately assign the scale to be m. To formally present the algorithm, we consider a bounded discrete signal f : Z“ + [O, 1, . . . , M ] with k local maxima at f ( x , ) i = 1,2, . . . , k . To translate the description of the previous paragraph into a practical algorithm, we note that at the critical scale (T, for the local maximum at position x,,the structuring function with origin at f ( x ; ) and of scale (T; passes through a point in common with the surface at ut least one other point. Let’s denote any one of these other points by ,f(x,,). We need to find f ( x , , ) , for by knowing this point we can immediately determine crl by solving (fog of) the equation,
Since go is convex, we can always find a function g-’(x) such that if gn(x) = z then, (T = g-l(x, z ) . Using this function the solution to Eq. (87) is,
MORPHOLOGICAL SCALE-SPACES
35
As an example, for spherical structuring functions given by:
we have,
giving,
The remaining step is to find the point x,’. We could conduct an exhaustive search by applying Eq. (88) to all the points x/,j # i in the signal and taking the minimum of all the resulting scales,
Fortunately, there is generally no need to search all the points of the signal. For a start, if we are calculating the scale for the local maximum at xi, we can ignore all other points x,, in the signal for which ,f(x,) < f ( x , ) , as the structuring function cannot touch there. Second, as we progressively compute Eq. (92), if we denote the minimum scale found so far by 6 then we only have to search the points in the region defined by,
{x : R;;;(X- X I ) 1 f ( X , )
-
M}
(93)
where M is the global maximum of the signal f . This is because, outside of this region, the magnitude of the structuring function is such that it cannot touch the signal. Note that as the search in Eq. (92) progresses and the minimum scale found so far decreases, the region defined by Eq. (93) also decreases, ensuring that the algorithm terminates. For this reason it is mot efficient to search over points x, in order of increasing radius from x,. In practice, it easier to search along the sides of an expanding square around x,. We can now write the algorithm:
Algorithm 1 (To find the scale of the local maximum at x,): Let M denote t h e g l o b a l maximum of s i g n a l f ( x ) .
ENTRYPOINTx; i s a l o c a l maximum of .f Step 1 Step2 Step3 Step4 Step5
c, t co
IF ( f ( x , ) = M ) , RETURN R t0 K t R+ 1 FOR a l l p o i n t s xI of r a d i u s K from x, DO
36
PAlJI. 1’. IACKWAY
Step6 Step7 Step 8 Step 9 Step 10 Step 11
H + .f(x,)- f@,) IF ( H 2 0) GOTO S t e p 10 0 , = ~ - ‘ ( x, x,.H ) IF ( 0 , < ~ ~ 0,1 + . cr, ENDFOR
IF ( g n , ( R )2 f ( x , ) - M I , GOTO S t e p
RETURN:value
in
4
0,
A similar algorithm finds the scale associated with the local minima of , f . An examination of Algorithm 1 shows that for each local cxtrcma i n the signal we scarch all the points in an n-dimensional volume given by
cf
I Therefore, the algorithm is approximately of order O(a,“).If we let 3 = 0,” then the computation for the extraction of the complete reduced morphological scale-space fingerprint is O(kcr“).An example code fragment in the coinpiitcr language C for the implementation of this algorithm for a 2D function (a rangc image) is presented in Appcndix 2. Before we leave the computational details of fingerprint extraction, one further point deserves mention. Thc input to the fingerprint algorithm is a list of the positions of the local cxtrcnia in the signal. For computational purposes, where we deal with a discrete signal, two difficulties may occu~’. First, because the range of the signal is discrete, it is possible (and common for some signals!) to have plateaus or areas of equal level. This presents difficulties as all the internal points of the platcau are both local maxima and local minima by definition-which is somewhat counter-intuitive. We can avoid this problem by using only the strict local extrema in the fingerprint; that is, the central point must be higher (lower) than its neighborhood. Now, another problem occurs: For a flattopped hill, there is no point higher than all its neighborhood, whereas, intuitively, a hill should possess a single local maximum. One solution. is to first find all the regional extrema, and then to follow this step by a procedure that represents each cxtreina by a single point near the center of its region. This center point can be found by a shririkirig algorithm to shrink the connected regions down to single points (Rosenfeld, 1970). The outcome is that we have a list of isolated points dcscribing the extrema in the signal, which seems an adequate solution to the problem. Second, the definition of extrema is closcly related to that of “neighborhood” of a point. For the square 2D lattice, there are two usiial possibilities, the four horizontal and vertical neighbors, or the eight neighbors which include the diagonal points as well. The definition of neighborhoods in digital spaces is closely related to issues of connectivity (Rosenfeld, 1970; Lee and Roscnfcld, 1986); in particular
MORPHOLOGICAL SCALE-SPACES
37
it is well known that square lattices on 2D suffer from a number of deficiencies and that for issues such as neighborhoods the hexagonal lattice has a number of advantages (Serra and Lay, 1985; Bell e f al., 1989). To summarize the process of extracting the reduced fingerprint: We first find all the regional extrema in the signal, then we reduce connected regions in this extrema-map to isolated single points. This results in a list of coordinates x,, i = 1 , 2 , . . . , k . This list is passed to the finger-print extraction step, which adds the scalc cntries to the list and returns, (x,, c,), i = 1, 2 , . . . , k . This list is the morphological scale-space reduced$ngerprint. A number of examples of the use of the reduced fingerprint for object recognition in range images have been presented in Jackway (1 995a).
VI. STRUCTURING FUNCTIONS FOR SCALE-SPACE I n Section 11, we have treated the structuring function in a general way, introducing constraints only where mathematically needed to obtain desired theoretical results. From those results we may conclude so far that, depending on which of the properties of Sections 111 and IV we require, the structuring function should be continuous, convex, and have a single local maximum at the origin. In this section we will collect together some further results which impact on the choice of structuring function. A. Semi-Group Properties We begin our discussion of structuring functions by looking at the semi-group property of dilation-erosion scale space. We may view the multiscale dilation and erosion as operators on a signal, that is,
We have already seen in Seclion B and Proposition 1 that the convex scaled structuring functions form a one-parameter continuous semi-group undcr thc morphological dilation. Now using the chain rules for dilations and erosions Eqs. (15) and (16) lead directly to the semi-group property (Butzer and Berens, 1967) for the scale parameterized morphological operations.
38
PAUL T. JACKWAY
A word of caution is needed here. We have dealt with the erosions and dilations separately, as we do not have the full group structure, which would require a negative scale operation to cancel out the effect of a positive scale operation. The erosion comes the closest to canceling the effect of a dilation, but erosion following dilation does not give the identity operator-it gives the closing!
We should also note in passing that the closings and openings do not enjoy any such semi-group property; the sieving properties (26) and (27) ensure that instead of achieving an additive effect, successive openings or closings merely preserve the effect of the strongest structuring function. Many authors consider the semi-group property of scale-space one of the fundamental foundations (Koenderink, 1984; Lindeberg, 1990). Lindeberg (1988) considers the semi-group property to be very important in his construction of a (linear) discrete scale-space. In fact he makes it one of his scale-space axioms from which the whole structure of the scale-space is developed. The semi-group property of dilation-erosion scale space enables the signal at scale a p to be obtained directly from the previous signal at scale a by repeated dilation (or erosion) and specifies how the global structure of the scale-space is related to the local structure. In practical terms the semi-group property ensures that the result of smoothing a signal at some scale is independent of the path taken to arrive at that smoothing; that is, a signal smoothed at scale a3 may be obtained by smoothing the original signal ,fn3 = f o g n , or by smoothing an already smoothed signal f n q = f m , og,: where ( ~ 3= (TI 02. Subject to the propagation of numerical errors, the construction of a set of smoothed signals by incremental smoothing with a small scale structuring function may be computationally more efficient than using larger and larger -scaled structuring functions.
+
+
B. A More Gerieral Urnbra
In applications, due to the physical nature of the problem to be addressed, more constraints on the desired behavior of any scale-dependent operator may need to be imposed. The morphological operations depend on shape (from the Greek root morphe = shape). We continue our discussions of structuring functions by showing how we require the introduction of an additional parameter for the notion of the shape of afunction to be properly defined. This leads to a generalization of the umbra concept of Sternberg. When working with functions (rather than sets) we need to be careful with the concept of shape, which is central to morphology. Binary images are readily represented as sets (for example, the set of all white points) and mathematical morphology was originally set based (Matheron, 1975;
39
MORPHOLOGICAL SCA1.E-SPACES
Serra, 1982). However, grayscalc images are more naturally represented as functions where the value of the function at a point represents the image intensity. The umbra (Sternberg, 1986) provides a natural way to associate a set U with a function f (x, y ) and was the first way in which thc opcrations of mathematical morphology were extended to grayscale functions,
U [ f1 = ((x, y, z ) : z 5 f (x, )))I.
(101)
Since morphological opcrations depend on shape, we can see a short-coming of the umbra approach, as defined above, for physical signals. This is best illustrated with a more common function-to-set mapping-the drawing of a graph. Consider the process of drawing the graph of a signal voltage u ( r ) as a function of time. First, a scale is chosen for the x-axis to display the required timc interval. Then a scale is chosen for the y-axis (relative to that of the x-axis) to cause thc shape of the resulting graph to convey the rcquired information to the viewer. Mathematically, we can view this operation as choosing appropriate scaling constants a! and /? to relate a physical timc-varying, signal voltage to a set of points G in the plane (the graph). In symbols:
G = ((x, E’) : x = t / a ! , ?‘ = U ( f , / / ! ? )
( 102)
In the digilization of a physical signal (by sampling in time and quantizing the samples) thc signal is mapped into the digital space Z2. The scaling factors appear in the selection of sampling rate ( a ! ) and the gain of the A-to-D converter (B). We wish to emphasize here that furicrion shape depends on the values of a! and /!? (see Fig. 11). More precisely, we can see that scale depends on a! and shape on the ratio h = a!//?. Since morphological operations are scale invariant, a ( G @ B ) = a!G @ a!B, we can, without loss of generality, take the scale to be unity. This suggests that (103) be reparameterized as,
G = ((x, y) : x = I , y = h U ( f ) I Thus, in applying mathematical morphology to gray scale images we consider Sternberg’s ( 1986) umbra to be underparametcrized. A more gcneral formulation would be: A grayscale image is a gray level function f ( x . y) on the points of Euclidean 2space. A gray level function can be thought of in Euclidean 3-space as a set ofpoints [ x . y , h f ( x . y ) ] ,imagined as a thin, undulating, not necessarily connected sheet. A grayscale image f ( r , y ) is represented in the mathematical morphology by an umbra U [ , f ,h ] in Euclidean 3-space, where a point p = (-1,y, z ) belongs to the umbra if and only i f z 5 h f ’ ( x , y ) . Where h is a shape parameter (cf. Sternberg, 1986).
This cxtra shape parameter, A, however presents a problem due to the dimensional inhomogeneity between the spatial dimensions and the intensity dimension of the grayscale image. The value of h is thereforc undefined by the physical problem. It
40
PAUL T. JACKWAY 10 9.
e '' 7
.'
6 .. 5 4
<
..
3 ' 2. 1 , 0.
::I,,
,
,
:
,
15
2
25
3
02 0
0
05
1
A
i:4il
LLL 135
1 1 105 0
FIGIIRE
1
(.\
1 I . Shape depends
0.5
011
1
1.5
2
2.5
3
X , Y scaling. Thrce graphs of the function y =
4.1'- 3 r + 2
(2.
would be advantageous if the morphological operations were invariant with respect to the implicit value of A. This type of shape invariance had received some attention in Verbeek and Verwer (1989) but the most thorough and general treatment is to be found in the dimensionality property of Rivest et al. (1992).
C. Dimensionality
Rivest et al. ( 1 992) have recently shown the importance in image processing and analysis of a concept they call dimensional consistency or dimensionality. Dimensional measurements on grayscale images have a physical signijcance. Suppose that an intensity image is mathematically modeled as a function f ( x ) , x E R' into the closed segment Z = [0, I]. The intensity axis I represcnts the irradiancc (light intensity) at the image plane and is, therefore, not dimensionally
homogeneous with the spatial dimensions. Any measurement with physical signilicance should not couple these physically different dimensions. Scaling in the spatial dimensions is known as homorheq and scaling in the intensity direction as ufliiify, indicating the fundamental physical difference bctwccn magnifying an image and brightening it (Rivest cf d . , 1992). Making a measurement o n an image consists in applying a functional on the image, where a jirnctional is a global parameter associated with a function. The property of dimensionality applies to functionals; that is:
De$nition 1 I (Dimensional Functionals): The functional W is detined to be clirmtisiori~ilif there exists constants k l k7 such that for all h l h l > 0:
where h I is the affinity and h? the homothety (Rivest c f d . , 1992). This relation restricts the way in which affinities and hotnotheties of an image affect dimensional measurements o n it, and results in a decoupling between affinity and homothcty measurements. For example, the durn(’ of ,f. V ( f ) = f(x)dx, is dimensional since, V ( f ’ ) = hlhT’V(,f) with f ’ = h l , f ( h ? x ) . If f(x) is the irradiancc at x then V ( f )has physical significance as the total radiant power. Rivest ct al. (1992) indicate that useful image processing operators should conserve dimensionality. It is commonly thought that volurnic (nonfat) morphological structuring elements lead to the breakdown of dimensionality in morphological opcrations (Rivcsteral., 1992; Stcrnberg, 1986).This is true for fixed scale structuring elements. However, we will show in the next section that if the morphological dilation-erosion scale-space is formed by using scaled elliptic poweroid structuring functions, which are in general volumic, any dimensional functional o f the scale-space image is also a dimensional functional of the underlying image. In scale-space filtering we work with the scale-space image, which is of higher dimensionality than the signal (Witkin, 1983). We can be careful to ensure that any operations used preserve dimensionality in the scale-space image. As an illustrative (somewhat nonpractical!) example, suppose we were to try matching 1D signals by counting the number of closed loops in their morphological scale-space fingerprints. Since the existence and relative position of local extrema are invariant under homothety and affinity, the number of closed fingerprint loops is a ditnensionul firnctiorial on the scale-space image. Now, a moment’s rellection shows that this is not the important point; the real question is whether or not our dimensional functional on the scale-space image F (which after all is a construct) is a dimcnsional functional on the origiriul sigtirrlJ’and thus has a phy.ricu/ significance. So we need to find out how to construct dimensional functionals on , f . First, we have extended the definition of dimensionality to scale-space images:
42
PAUL T. JACKWAY
Dejinition 12 (Scale-Space Dimensionality): W is a dimensional functional on F if given h l , h2, h3 0 there exists constants k l , kZ, k3 such that: k~
k-. k
W ( ~F(h2x, I h 3 ~ )= ) hi h,-h,'(F(x, a)) We will show that if the dilation-erosion scale-space is constructed with the elliptic poweroid structuring functions, all dimensional functionals on F are also dimensional functions on f . Consider the scale-space formed by the multiscale dilation F ( x , u ) = ( f @I g,) ( x ) ,u > 0.
Proposition 13. With elliptic poweroid structuring functions, go ( x ) = - I D I x ICT I)", all dimensional functionals W ( F ) are also dimensional functionals o f f :
(m/
Proof. A proof of this proposition can be found in Jackway (1995b). As a point of interest, as a + 00 the circular poweroid structuring functions approach the flat (nonvolumic) cylindrical structuring element well known in grayscale morphology and image processing (Nakagawa and Rosenfeld, 1978; Sternberg, 1986; Haralick et al., 1987). It is rather difficult to demonstrate the effect of nondimensionality in scale spaces as the effects are likely to be small. One difference, however, can be seen in the connectivity of fingerprints. As an example of the dimensionality property in ID, we present Fig. 12, which shows the differing effects of the use of dimensional (parabolic) and nondimensional (spherical) structuring functions in the computation of the multiscale dilation-erosion scale-space fingerprint of a certain signal and that same signal with an affinity. A ver?, close examination of Fig. 12 shows that with a nondimensional structuring function, the fingerprint of the stretched signal may be differently connected, whereas with the parabolic structuring function the fingerprint is merely compressed in the scale direction. We can obtain a functional by counting the closed loops of the fingerprints. In this case we have Fig. 12c with 19 loops, and Fig. 12d with 20 loops (the difference is in the 8th loop from the right) indicating the breakdown of dimensionality. In contrast (with a parabolic structuring function) both Figs. 12e and 12f contain 19 loops. If the signal was an intensity image and some image analysis operation, such as pattern recognition, was sensitive to the connectivity of the fingerprint, then with the nondimensional structuring function, the output would depend on the arbitrary scale chosen to represent the intensity dimension relative to the spatial dimensions of the image.
D. The Poweroid Structuring Functions We make the general observation that (for any fixed scale) the more pointed (low a ) structuring functions tend to give more emphasis to the local shape near a signal
MORPHOLOGICAL SCALE-SPACES
43
I: FIGIJRE12. A n example of dimensionality i n scale-spncc. ( a ) A rnndom signal. ( b ) T h i \ signal with an aftinity ol siLe 4.0. ( c ) and (tl) The multiscale tliI;itioii-ei-osion lingerprints of the prcccding signals with ;I ii(iiidiiiiciisioiial (spherical) structuring function. Note the connectivity and structure of the fingerprints differ because o f the allinity: (c) has I 9 closcd l o o p and (d) 20 closed loops (the diflerence i s i n thc 8th loop from the right) indicating the breakdown of dimcnsionality. ( c ) and (tj Thc nidtiscale dilation-erosion lingerprints of the preceding signals with a tlinicnaional (pirabolic) structuring function. Note the structure o f thc fingerprints reniaiiis similar (with 19 closed loops) indicating the conservation of dimensionality.
44
PAlJL 1'.J A C K W A Y
feature. Other constraints or requirements may dictate the choice of (Y and, hence. the structuring function. The flat structuring function ( a = m ) is commonly used because the inorphological operations reduce to simply taking the maximum or minimum of the signal over some neighborhood (Nakagawa and Roscnfeld, 1978). That is, for flat structuring functions,
The use of flat structuring functions is most appropriate for binary images where the morphological operations are identical to those on point-sets. However, on grayscalc images, the use of flat structuring functions leads to flat regions in the output signal around the local extrema, and the local extrema are no longer exactly localized in position. This is certainly a disadvantage in multiscale dilation-erosion scale-space. since it is the local extrema of the output signal which arc our scalespace features and exact localization is absolutely necessary. There is a computational reason for the importance of the paraboloid structuring functions in particular. For a fixed scale, the 2D morphological dilation can be computed most directly by the rrior-phological c o m o l i ~ i o ~ ~
where G is a square neighborhood of (i, , j ) . If the size of this neighborhood is r 7 , then the computational burden of the direct implementation of Eq. (105) is B,/ = U ( r 2 ) ,The Landau symbol 0 is often used to indicate computational complcxity: R ( $ ) = U(q5')means that R($)/$' is bounded as $ + 30 (Lipschutz, 1969). However, for the 2D paraboloid structuring function, g,(s. y ) = -~u~(.xr' ??)/a', we have a separability property:
+
g (.I.. y) = g:'l(.I.)
' 0
where the ID structuring function g,!,"(x) a similar property,
+ g,
= -10
(I!
0.).
[ x 2 / u ' . The "max" function has
where G'" and G'"'are the projections of G on the these properties we get the desired result ( f @ g , ) ( i , .i) = inax { v ( i- s,j) 1E
G
'
(106)
I-and
y-axes. Combining
+s ~ ' ' ~ - x ) }
(108)
MORPHOLOGICAL SCALE-SPACES
45
where
The computation has been reduced to a sequence of two 1D morphological convolutions with a computational burden, B I = O ( r ) . The cost is that additional storage is required for the intermediate result h. This result has recently become known in the literature as the separable decomposition of structuring elements (Shih and Mitchell, 1991; Gader, 1991; van den Boomgaard, 1992; Yang and Chen, 1993). There are actually two kinds of separability involved here. First, that is additive separability where g, (x, y ) = gX’(x) gA”(y); second, that is morphological separability, g,(x, y ) = g,!,l’(x) @ gh’)(p). Note in the result (108-109) we have used additive separability to obtain a morphological separability result. Recent work has in fact shown that for square morphological templates (i.e., discrete structuring elements) the two kinds of separability are in fact equivalent (Yang and Chcn, 1993). Yang and Chen (1993) show in a theorem that if g ( x , y ) is additively separable of size (2r 1) x (2r I), and it is convex, then it can be expressed as g(x, p) = ( k f @ k t @ . . . @ k l ! ) C B ( k ; ’ @ k ; ’ @ . . . @ k : ’wherekf ), isahorizontal 1Dstructuring element of size 3, and k:’ is a vertical I D structuring element of size 3. The importance of this result is that by the chain rules for dilations (end erosions), if g = k l @ k2 @ . . . k r , then f CB g = ( ( ( , f CB k2) @ k2) @ . . . ) k , . . So the whole opcration can be performed as a sequence of 1D three-point operations. The point to stress here is that to obtain all these nice results we need additive separability of the structuring function. Writing the 2D elliptic poweroids as
+
+
+
where.
The conditions necessary are therefore that (a) N I Z = 0 ( A is a diagonal matrix), and (b) a = 2. Therefore g(x, y) must be a circular paraboloid (a1I = a??), or an elliptic paraboloid with the major and minor axes of the ellipse aligned with thc coordinate system x , y-axes (for a1 I # a?*). In practical terms this is a very favorable property of the paraboloids. Van den Boomgaard (1992) has shown that the elliptic paraboloid structuring functions (callcd the quadratic structuring ~functions,QSF) are closed with respect to morphological dilation (and erosion). This result is an extension of the
PAUL ‘r,JACKWAY
46
semi-group property of Section B to arbitrary QSF kernel matrices. In fact van den Boomgaard (1992) argues that the elliptic paraboloids can be considered to be the morphological equivalent of the Gaussian convolution kernels because this class is dimensionally separable and closed with respect to dilation (and erosion), thereby establishing an equivalence between the parabolic structuring function in mathematical morphology and the Gaussian kernel in convolution.
v11. A
SCALE-SPACE FOR
REGIONS
In this section we will show how we can change the signal feature involved and still maintain the monotone property of scale-space. We will first extend the morphological scale-spaces from signal regional extrema (Corollaries 1.2 and 3.2) to signal watershed regions via the watershed transform. Then, via homoropy mod$cation of the gradient, we will further extend the scale-space property to watershed regions of the gradient function where we will demonstrate its application to multiscale segmentation. The idea in both cases is that if we can find some transform of a signal that gives a new feature that is 1: 1 to the signal regional extrema, then Corollaries I .2 and 3.2 ensure that this new feature also possesses a scale-space monotone property. Let’s make this a proposition:
Proposition 14. If C[R(f ) J denotes the number of connected componerits in the point-sets of the regional extrema of a signal f , and there exists some measure # on the transforms @I ( f ) and @*( f ) such that:
and.
(c) # [ @ 2 ( f 0ggn,)l 5 # [ @ 2 ( f 0gnz)l I #[@z(f)l: fd) #1@1 ( f 0gnJl 5 #[@I ( f 0go,)] I #[@I (f)l.
Proof. A proof of this proposition is given in Appendix 1. Wc will discuss two such transforms, the Watershed transform and a certain homotopy modijication.
MORPHOILOGICAL SCALE-SPACES
47
A . The Wutershed Trunsfiwm Wtrtershed trunsforrns arc used primarily for image segmentation and arc part of the tools of mathematical morphology (Lantuijoul, 1978; Serra, 1982; Vincent and Beucher, 1989; Beucher, 1990). Thc recent dcvelopmcnt of powerful and fast algorithms (Vincent and Soille, 199 I ) has further served to popularizc the method. For segmentation (edge detection) the idea is that the Watershed lines of a surface tend to follow the “high ground’ so that if we find the watershed transform of the gradient image, the watershed lincs will follow the edges (regions of high gradient) in the image, thereby performing a useful segmentation of the image into regions of low-intensity gradient, which are regions without edges. However, although watershed image segmentation methods are very powerful and general, in many applications they tend to oversegmcnt (Vincent and Bcucher, 1989). We can decomposc the watcrshcd transform into catchment basins (watershed regions) U: i = 1, 2, . . . , q and the watershed lines themselves L:
W S ( f ) = WI
u wz u . . u w, u L. ’
(1 12)
Suppose a function ,f : D c R’ + R possesses q regional minima; that is, C[R,,,,,,(f)l = q . We can, therefore, write: Riiiin(f) = N I UNz U . ’ . U N q ,
(113)
where we have identified the individual regional minima N, i = I , 2, . . . , q . The watershed transform of a surface possesses the following properties (Bcuchcr, 1990):
I . The watershed lines delineatc open connected regions, W ; . 2. All points of the surface cither belong to a region or fall on a watershed line x E D + (x E W ; for some i = 1 , 2 , . . . , q ) OR (x E L ) . 3. Each watershed region contains a single regional minimum and each regional minimum belongs to a single watershed region (its catchmcnt basin). So we can make the correspondences,
c U:
foralli = 1 , 2, . . . , q.
(114)
From the watcrshcd propcrties above, we can write
C [ W S ( f ) l= ~ [ ~ , l , l l l c f ) l where C [ W S ( f ) counts ] the number of catchment basins in W S ( f ) .This cquation is of the form required by Proposition 14 so it is possible to create a monotone scale-space property for watcrshcd regions.
Theorem 4 (Scale-Space Monotone Property for the Number of Watershed Regions): Let C[WS(f )] denote the number of watershed regions ojun irnage f ,
48
PAUL T. JACKWAY
then for uny scales 01 > 02 > 0,
and, fb)
Prooj
C [ W S ( f 0 g,, 11 5 C[WS(f
0
go?)] I C[WS(f,l.
This theorem follows directly from Eq. (115) and Proposition 14.
Although theoretically correct, unfortunately, this theorem is less useful in practice than it might at first appear. We have glossed over the role played by the function f in the foregoing treatment but we must consider it now. To correspond to something useful (such as edges in the image), the watershed should be applied to the gradient of the original image. This suggests that f should be the gradient imagc. However, the gradient surface is quite unlike the original image and smoothing the gradient with the multiscale morphological operations is not equivalent to smoothing the original image. For instance, small scale features in the original can have arbitrarily high gradients and thus dominate the gradient image. Smoothing the gradient image will leave these dominant features until last but they should be the first to disappear. Moreover, since the gradient image we use is actually the magnitude of the gradient function, and the magnitude operation removes any symmetry between the shapes of the maxima and minima, many minima occur as narrow cusps at zero, and applying an erosion or opening to this image will not really help to analyze the signal. No! To make sense, it is necessary to perform the multiscale smoothing on the original image; however, the watershed operation must still be performed on the gradient image. We therefore need a link between the two, which maintains the scale-space monotone property. We can construct the necessary link by using Vincent’s (1993) grayscale reconstruction to modify the homotopy of the gradient image.
B. Homotopy Modijication of Gradient Functions Loosely speaking, two functions (i.e., surfaces) are said to be homotoj>ic-if their hills, channels, and divides have the same relationship to each other in both the functions; that is, their watershed transforms will make the same pattern (NB: for a more precise definition see Serra, 1982, Def. XII-3). Vincent’s (1993) grayscale reconstruction provides a way to modify the homotopy of a function based on the values of another function called the murker function. Basically, the grayscale reconstructions can remove designated ( i t . , marked) extrema from a function while leaving the remainder of the function unchanged. The use of these reconstructions is now the standard way to apply marker functions to watershed segmentation (Beucher, 1990). A full discussion of the grayscale reconstruction is beyond thc scope of the present article but interested
readers are urged to consult Vincent (1993), which gives full details, examples, and fast algorithms. We will employ the d i r i 1 1 gruyscale r.eC.oIi.strIICtiorip i ( g ) t o remove from ,f’ the regional minima “designated” by g. To set this up, suppose we have a bounded function ,f’(x),which has r regional minima which we label arbitrarily, R,,,,,,(,f) = N I U N2 U . . U N, . Suppose wc thcn select (mark) s < r of these regional minima by choosing s sets A, so that A , c N , i = I, 2 , . . . , s . Let the union of these sets be denoted by A = -I A , . Now we form the marker function,
u:
K(X) =
0.
vf.
ifx
E
A:
othcrwisc.
Wc then have the following proposition:
Proposition 15. I f ,f is a contiriuoirs Dorrrirlrdfirric.tiori f : D c R ” + (0. B ] . w t l g : D c R” --f [ 0, B 1. is c.ori.strirctcdas oirtlirietlrnrlier, then if w’e rec~)ristruct the,
liotnotopy riindi~eu’,firri(,tioti ,f’K irsirig tlie dirirl grnysctrle r~~c,otistrrrc’tioti,
.f
= P;
(s).
(117)
tl?ot I ,
.
LIP1 tl,
R l l l l , l ( j K ) =U NN ~? U . . . U N ,
(118)
C[~llilll(.f’K)] = s.
( I 19)
therqf o r - 0 ,
Prooj
A proof of this result can be found i n Jackway ( 1996).
To proceed we must now consider the relationship between a function f ( . r , y ) and thc magnitude of its gradient l V ( , f ) l ( . ~y). , Assume that .f‘ is of class C 1 (that is, its first dcrivative exists and is continuous), then the gradient is zero on any regional extrenia of f . Note thal if ,f is not of class C ’ (or in the discrete case) lV(,f’)l can simply be defined ( i c , forced) to equal zero on the regional extrema of , f . We have,
(x,.Y)
u
E (R,,l,l,(fj R,l,,,l(,f)) =+ IV(fjl(.r, y j = 0.
( 120)
Since l V ( f ) I is a nonnegative function, all its zero points belong to its regional minima, so,
Equations ( 120) and ( 12 I ) imply,
50
PAUL T, JACKWAY ...
.,
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..
,
.
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f-
n
FIGURE13. Modifying the homotopy of the gradient function. From lop to hottoni: a ILinctioii. i t ) gradient. the homolopy modified gradient function retaining only minima corresponding to maxima in the original ftiiiction, a n d the homotopy modified gradient function retaining only illiniillii corresponding to minima in the original function. Keprinktl froni (Jackway I996 @ IEEE).
which shows that the regional extrema of f are subsets of the regional minima of I V ( f ) l . Therefore, we can use Proposition 15. By choosing appropriate marker functions related to selected regional maxima or minima o f f and using the dual reconstruction on the gradient function, we can modify the homotopy of the gradient function to possess regional minima corresponding only to the selected regional extreina of f ,thereby providing the necessary link between the homotopy of image and its gradient image. This idea is illustrated in Fig. 13. We formalize the above modification of gradient homotopy in a proposition.
Proposition 16. I f f (x, y ) isa boundedfunction ofclass C' f :D c R' -+ 10, B ] with C[R,,,,,(f ) ] = p , with C[R,,,( f ) ] = r, and with gradienr lV( f )I, and suppose we select q 5 p regional maxima M , , i = 1 , 2 , . ... q and s 5 r regional minima NJ j = I , 2, .... s and form the marker function
.
MORPHOLOGICAI. SCALE-SPACES
Riiiin(IV(,f)I’)
c (Riiim(f) U Riiiiii ( f ) )
51
(125)
and, CIRiiiiii(IV(f)I’)I = q
ProoJ
+ s.
(126)
A proof of this result is found in Jackway (1996).
Examples of similar homotopy modifications of gradient functions are presented in a more ad hoc manner in Beucher (1990), Vincent and Beucher (1989), and Vincent (1993). If we set q = p and s = 0 in Proposition 16, denoting the corresponding reconstructed magnitude-of-gradient by lV(,f) I+, wc obtain, ~ [ ~ , , ,(lV(f ,,,
11’- )1 = C[Rlll,IX(f)l,
(127)
and, since the regional minima of lV(f)l’ by construction coincide with the regional maxima of ,f, we get the stronger result,
Rlllln(I V ( f ) )1’
= Rll,;lx( f ) .
( 128)
and, conversely, if q = 0 and s = r , denoting the corresponding reconstructed magnitudc-of-gradient by lV(f)l-, C[Riiiiii (I V ( f ) I
)I
= CIRiiiiii (.f)
1 3
(129)
The effects of the above homotopy modifications of the gradient and the resulting watersheds are shown in Fig. 14.
C. A Scale-Space Crudient Wutershed Region In the sequcl we will use only the values q = p , s = 0 to make the minima of the gradient corrcspond to all the maxima of the image (for positive scales), or q = 0, s = r to make the minima ofthe gradient corrcspond to all the minima of the image (for negative scales). lnstcad of the original image f we can use the above rcsults on the smoothed image f @ g,. In particular, for cr 2 0, wc can find the watersheds of l V ( f ) l i , and for negative scales, the watcrshcds of I V(f ) I -.
52
I'AlJl.
I. IACKWAY
Iioiiiokipy iiiotlilicd graeiient rcraining only iiiiiiiiiiii ct,rrc.;poiidttig l o
5 ) we get,
' )I: clws(lv(f)l-) I
= ~ l ~ l , l l l , ( l)I. ~~,f~l ~
(131) ( 132)
ButusingEqs. (127)and(129)tosubstituteforC[R,,,,,,(lV(f)l-' 11 andC[R,,,,,, x (IV (,f)l- )I. gives Cl WS(I V ( f )I -I- ) 1 = C[R,,,;,, ( f) I : ClwS(lV(,f)l
)I
= C[Riiiin(.f)l.
(133) ( 134)
which arc transforms of the form required by Proposition 14. Therefore, we have a scale-space monotone property for gradient watershed regions.
MORPHOL.OGICAL. SCALE-SPACES
53
Theorem 5 (Scale-Space Monotonicity Property for the Number of Gradient Watershed Regions): Let C [WS( ,f)] denote the nurnber qf wutershed regions of an imagc f, I V ( ,f) I and I V (,f ) I the homotopy modified grudirnt images as defined earlier: Then for any scules 01 < cr2 < 0 < crj < crd, C[WS(lV(f @gfT,)I+)J5 C[WS(lV(f' 63 snz)l+)l5 C ~ W S ~ l ~ C f ~ l S ~ l ; el?, ClWS(lV(.fosfT4)l-)l I C[WS(lV(.f 63 g f n I - ) l I c[Ws(lv(f>l-)l> (0)
and, ccj
en,
C[WS(lV(f Qgn,)I'+)l I C[WS(lV(f 0gn,)lt)l I C[WS(lV(,f)I+)l;
C[WS(lV(.f 0g,,)l-)l I C[WS(lV(f 0gn3)l-)1I c[Ws(lV(f)l--)l,
Prooj This theorem follows directly from Eqs. (133) and (134) and Proposition 14. With this theorem we have completed the formal development of a scale-space monotonicity theory for gradient watershed regions. The full algorithm for obtaining a multiscale set of gradient watersheds is:
Algorithm 2 (Gradient Watersheds): I. S e l e c t a s e t of s c a l e s of i n t e r e s t J { a k ) ; For each s c a l e c r k DO: 2. smooth f t o o b t a i n fog,, u s i n g Eq. (52); 3. f i n d t h e r e g i o n a l minima, N ; , ( f o r ffk 1 o), o r maxima, M i ,( f o r f!, 5 01, of fog,,, and compute a s u i t a b l e marker f u n c t i o n g(x,y) (Eq.(123)); 4. compute t h e magnitude of g r a d i e n t image lV(f@g,)I; 5. modify t h e homotopy of t h i s image (Eq.(125)); 6. f i n d t h e watershed r e g i o n s Ws(lv(fSg,,)I*). ENDDO : As an example of the scale-space properties of the gradient watershed, we present Fig. 15.
VIII. SUMMARY, LIMITATIONS, AND FLJTLJRE WORK A. Sunzmaty
Two scaled morphological operations, the multiscale dilation-erosion and the multiscale closing-opening, have been introduced for the scale-space smoothing of
'
Note: The inleresfing rcales may he prescrihed hy the application. o r perhaps sumpled (linearly or logarilhinically) over soiiie range. A set ot \tale\ al which regions vanish can be found wilhoul smciothing hy Algorilhm I .
54
PAUL T. JACKWAY
FK;LIIU:IS. The closing-opening \cale-space for gradient watershed regions. Hoinotopy niodiliccl gradient watersheds for the "Lena" image smoothed by inultiscale closing-opening. From Id1 to right, top 10 bottom. thc scales arc: -2.5. -1.6. -0.9, -0.4, -0.1. 0.0-. 0.0+. 0.1, 0.4. 0.9. 1.6. 2.5. A c i irc ti I nr paraboloid s t ruc luri ng ftinct i oti was 11 scd .
MORPIIOLOGICAL SCALE-SPACES
55
signals. These multiscale operations arc translation invariant, nonlinear, incrcasing, and dependent on a real scale paramctcr, which can be ncgativc. The smoothed signals across all scales can bc considercd as a function on the so-callcd scalcspace. This scale-space image cxists for negative as well as positive scalc and, thus, the information in the signal is more cxpanded than in the linear (Gaussian) scale-space image, which only exists for nonncgative scale. The scalc-space image has good continuity and order properties. The position and hcight of extrema in thc signal are preserved with increasing scale (maxima for positive scales and minima for negative scales), until they vanish at thcir characteristic scale. A monotone propcrty for these signal features has been demonstrated. Fingerprint diagrams from these scale-spaces are identical and may be used to rcprcsent signals. To summarize, the morphological scale-space diffcrs from the Gaussian scalcspace in that it: possesses a monotone property in two and higher dimensions represents local extrcma instead of zero-crossings cxists for negative as well as positivc scale With the morphological scale-spaces subsets of the full fingerprints, which are called reduced fingerprints, have been defined. The reduced fingcrprint consists of local signal maxima for positive scale and local signal minima for negative scale. This reduced fingerprint: consists of vertical lines only (since the position of signal features in not altcrcd by the smoothing); is equivalent to a set of ( n 1)-tuples, where 17 is the dimension of the signal; can be efficicntly computed without signal smoothing by the algorithm presented.
+
The scale-spaces have been extended from point-set features-the extrcma, to regions through the watershed transform. Through honiotopy modification of the gradient function, the rnonotonc has then been extended to gradient watershcd regions.
It is important to note the limitations and restrictions on the proposed theory. In essencc we return to thc carly days of scale-space theory by placing emphasis on the importance of signal features, and the tracking of these fcatures through scale. Featurc tracking is an idea which wc bclievc is as yct underexplored. Modern scale-space theory seems to concentrate on powerful mathcrnatical results dcscribing the axiomatic bases and the diffcrcntial structurc and invariants of thc various scale-spaces. While important theoretically, causality in the form of the maximum principle on partial diffcrential cquations has taken prccedencc over
56
PAUL T. JACKWAY
a monotone principle for signal features. We have emphasized such a monotonic principle in this article. Since we do not use an averaging filter for signal smoothing, the question of the sensitivity of this method to signal noise naturally arises. As the dilation and erosion depend on the extreme values of the signal in the neighborhood of a point, impulse noise in particular will upset the method. A high amplitude impulse will be seen as a large-scale feature, especially if it is on a relatively flat region on the signal. In some applications this may be reasonable behavior; in others this may be unacceptable. In the end it is the application which determincs if the definition of scale embodied in the proposed method is useful or otherwise. Additionally, almost any high-frequency noise will introduce many new local extrcma into the signal, causing many spurious features in the analysis. The glib answer is to say that if noise is present in the signal it should be filtered out before the signal is analyzed. This may indeed be appropriate in many cases, but realistically this sensitivity to noise may be one of the limitations of the approach. C. Future Work
The practical utility of our approach has yet to be demonstrated; in particular, work is needed on the computation, stability, inversion, and application of the full fingerprint and the watershed transform. In particular, which classes of signals are well represented by these quantities, and are these methods of use for signal compression? In ID, the noncreation of maxima (minima) implies the noncreation of minima (maxima) due to the interleaving of maxima and minima. However, in higher dimensions this does not necessarily hold. We have as yet no results on these other possible monotonic properties in higher dimensions. In common with Gaussian scale-space, the theory of morphological scale-space has been developed in the continuous domain. Digital signal and image processing, however, involves discrete signals. The various results need to be formally obtained for digital signals. Thinking out loud, we can make the following suggestions for future directions:
1. The formal extension of the theory to real and discrete functions on discrete spaces. 2. The inversion of the fingerprint and watersheds to reconstruct the original signal. 3. The use of multiscale morphology on the derivatives or integrals of multidimensional signals. 4. The use of transforms on the signal, before or after (or both!) scale-space analysis to alter the quantity represented by the scale dimension in the analysis.
57
MORPHOI .OCI(‘AI. SCALE SPACES
5. A soft morphological hcale-spacc based on the kth and statistics. 6. . . . !
12
- kth order
Part of thc material containctl in this article was P1i.D. work under the supervision of V. V Anh, W. Boles. and M. Dcrichc lroni the Quccnsland University of Tcchnology. The work on watersheds was suplmrted by a grant from thc University of Quccnsland.
P r o ( ? f ’ ~ f P I . ~ p o s i l i o1 i i
Sincc we consider only 0,p 1 0, lor clarity. we imnicdiatcly drop the 1. 1 signs in (38). We start from the definition 01 convexity: hg(a)+(l-h)g(b)(g(ha+(l-h)b).
Nowniakethefollowingsubstitutions: h =
A:( I - h )
( O ( h 5 1) = &;a = $;
(135)
b = +,
so that,
This implies, ( 137)
And wc also note that,
SO,
( 139)
58
PAUL T. JACKWAY
Thus, the semi-group property (Hille and Phillips, 1957) of convex structuring functions is proved. Proof of Proposition 2 From the property of the opening of the dilation, Eq. (17), we have, (f@g)og =.f@g,
(140)
and from the semi-group property for scaled structuring functions, Eq. (5 l), we have, gv? = glnrl-lnll @g1.
(141)
Combining these equations we get the required result, g,,
O
gv, =
(Rin.l-ln,l
- gl.zl-l..II - goz.
@ s n , ) 0 gm,
@go,
(142)
Proof of Corollav 1.2 Consider the case for positive scales, 0 < 03 < 0 4 . From the order properties for grayscale dilation, Eq. (20), we have,
(f@g,)(x) ?
(fog&)
? . f ( x ) forall (XI,
(143)
so with increasing scale the value of any fixed point (f@gu,)(xo) can never decrease. A regional maximum is a connected component of the point-set of local maxima. A necessary condition for a regional maximum to exist is that all its points are local maxima. We may associate with each point of this set a scale (T; being the scale at which this point ceases to be a local maximum because one of its neighbors has exceeded its value. Then the whole regional maximum ccases to be a regional minimum at a scale of min{cr,]. Thus, each regional maximum exists for a rangc of scales of 0 5 (T 5 min(a;). Part (a) of the corollary follows. Part (b) follows from duality. Proof of Proposition 8 From Proposition 2, we find g,, is open with respect to g,:, and g,, is open with respect to g,, . Then, from Relation (28), the result follows. Proof of' Proposition 10 Equations (71) and (72) follow directly from Eq. (65). Relation (73) follows from Proposition 8, on the order (anti-extensive) properties of thc opening.
59
MORPHOLOGICAL SCALE-SPACES
Proof of Proposition 11 From Proposition 6: (f @ gn)(xlnay) is a local maximum =+ f(x,,,,,,)is a local maximum, and (f@ g,)(x,,,,,,) = f(x,,,,,). However, from Property (24) we have a sandwich result:
f ( x ) 5 (.f g,)(x) i ( f @ g,)(x),
Vx
E
D,
(144)
is also a local maximum and (.f 0 gn)(xnlax) =f(~,,,~,~). therefore, ( f 0 gn)(xlll.lx) To show the reverse relation, we appeal to the geometric interpretation of the closing. If ( f 0 g,)(x,,,,,) is a local maximum, then the origin of the translated (reflected) structuring element at x,,,,, must be greater than the origin for the structuring element at all x in some t-neighborhood of x,,,,,,. Since the locii of this origin form the surface of the dilation operation, we have,
which shows that (f@g,,)(~,,,~,,) is a local maximum. This completes the proof of the first part of the proposition. Again, the second part follows from the morphological duality properties.
Proof of Proposition 11 If ( f @ g,)(x,,,,,) is a local minimum, thcn the origin of the translated (negated) structuring function is lower than in the surrounding neighborhood. Then, since the negated structuring function has a local minimum at the origin, and is convex, the union of the structuring functions in the neighborhood of x,,,,,has a minimum at x,,,,,,and, since this union is the closing, (f0 ,gn)(xllllll), is also a local minimum. To show the reverse relation, we note that if ( f 0 g,)(x,,,,,,)is a local minimum, then, since the negated structuring function has a local minimum at the origin, and is convex. we have
(f 0 gn)(Xnlln) = (f @ Rn)(XIIIIII).
(146)
Appeal to Property (24): (fog,)(X)
i (f@g,)(x), VXED.
(147)
l n ) also be a local minimum. This completes the we see that ( f @ g n ) ( ~ , r rmust proof of the first part of the proposition. Once again, the second part follows from the morphological duality properties.
60
P A U I . T. JACKWAY
and,
Then thc proposition parts (a) and (b) follow fi-om Corollary 1.2 parts (a) and (b), and the proposition parts (c) and (d) follow from Corollary 3.2 parts ( a ) and (b).
Appcndh 2: Computer Code A fragment of C code to extract the reduced morphological scale-space fingerprint from local inaxiina of a 2D function:
/ * typedef s t r u c t
/* /* /* /* /* /*
{int x; i n t y ; f l o a t scale;} scaleitemtype; f l o a t f [N,N] h o l d s t h e s i g n a l . s c a l e i t e m t y p e FP[K] h o l d s t h e reduced f i n g e r p r i n t , on i n p u t FP[] c o n t a i n s t h e c o - o r d i n a t e s of t h e l o c a l maxima; on o u t p u t a l s o c o n t a i n s t h e associated s c a l e s . M i s t h e g l o b a l maximum of f [I
*/ */ */ */ */ */ */
f o r ( i = l ; i<=K; i + + ) { x i = FP[i] . x ; y i = FPLiI . y ; s i g i = MAXFLOAT; i f (f [ x i ] [yi] < M) { R = 0; do { R++ ; x j = x i + R; y j = y i + R ; f o r ( j = i ; j<=2*R; j + + ) { H = f [xi] [yi] - f [ x j l [ y j l ; i f (H < 0 . 0 ) { s i g j = - ( h y p o t ( x i - x j , y i - y j ) + H*H) / ( 2 . 0 * H); i f (sigj < sigi) sigi = sigj; } /* i f * / xj--; } / * f o r */ f o r ( j = l ; j<=2*R; j + + >1 H = f [xi] [ y i l - f [ x j l [ y j l ; i f (H < 0 . 0 ) { s i g j = -(hypot ( x i - x j , y i - y j ) + H*H) / ( 2 . 0 * H) ; if (sigj < sigi) sigi = sigj; 1 /* i f */
MORPHOLOGICAL SCALE-SPACES
61
yj--; f o r */ f o r ( j = l ; j<=2*R; j + + >{ H = f [xi] [yi] - f [ x j l [ y j l ; i f (H < 0 . 0 ) { s i g j = - ( h y p o t ( x i - x j , y i - y j ) + H*H) / ( 2 . 0 * HI; i f ( s i g j < sigi) s i g i = s i g j ; 1 / * i f */ xj++; 1 / * f o r */ f o r ( j = l ; j<=2*R; j + + ) { H = f [xi] [ y i ] - f [ x j l [ y j l ; i f (H < 0.0) s i g j = - ( h y p o t ( x i - x j , y i - y j ) + H*H) / ( 2 . 0 * HI; if (sigj < sigi) sigi = sigj; 1 /* i f */ xj++; 1 / * f o r */ 1 w h i l e ( ( R <= s i g i ) && (R*R <= -H*(2.0*sigi + H I ) ) ; 1 / * i f */ FP[i] . s c a l e = s i g i ; /* f o r i */
1 /*
1
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Lindeberg, T. (1990). Scale-space for discrete signals, ZEEE Z Putt. Anal. Mach. Zntell. 12(3): 234-254. Lipschutz, M. M. (1969). Theory and Problems d Geometry, Schaum's Outline Series, New York. Morphological filters -part I: Their set-theoretic analysis Maragos, P., and Schafer, R. W. and relations to linear shift-invariant filters, ZEEE 7: Acoust. Speech Sig. Process. 169. Maragos, and Schafer, R. W. Morphological filters -part Their relations to median, order statistic, and stack filters, ZEEE Z Acoust. Speech Sig. Process. ASSP-35(8): 1 170-1 184. Man; D. (1982). Vision,Freeman, San Francisco. Marr, D., and Hildreth, E. (1980). Theory of edge detection, Proc. Royal SOC.Lonu'. B 207: 187-2 17. Marr, D., and Poggio, T. (1979). A computational theory of human stereo vision, Proc. Royal SOC. Lond. B 204: 301-328. Matheron, G. (1975). Random Sets and Integral Geometry, Wiley, New York. Mokhtarian, F., and Mackworth, A. (1986). Scale-based description of planar curves and twodimensional shapes, ZEEE Pad Anal. Mach. Zntell. Morita, S.,Kawashima, and Aoki, Y. (1991). Patt. matching of 2-D shape using hierarchical descriptions, Comput. Nacken, E M. (1994). Openings can introduce zerocrossings in boundary curvature, ZEEE Putt. Anal. Much. Intell. Nakagawa, Y., and Rosenfeld, A. (1978). A note on the use of local and max operations in digital picture processing, ZEEE Sysv.Man 632-635. Perona, P., and Malik, J. (1990). Scale-space and edge detection using anisotropic diffusion, ZEEE Z Pad. Anal. Mach. Zntell. 12(7): 629-639. Protter, M., and Weinberger, H. (1967). Maximum Principles in Diflerenrinl Equations, Prentice-Hall, Englewood Cliffs, NJ. Rarnan, S . V., Sarkar, S., and Boyer, K. L. (1991). Tissue boundary refinement inmagnetic resonance images using contour-based scale space matching, ZEEE Z Med Inzag. lO(2): 109-121. Rangarajan, K., Allen, W., and Shah, M. (1993). Matching motiontrajectories using scale-space,Putt. Recog. J.-F.,Serra, J., and (1992). Dimensionality in image analysis, Represent. 137-146. 146- 160. Rosenfeld, A. (1970). Connectivity in digital pictures, Assoc. Comput.Mach. Rosenfeld, A,, and Thurston, M. (1971). Edge and curve detection for visual scene analysis, ZEEE 7: comput. Serra, J. 982). ZmageAnalysis and MathematicalMorphology, Academic Press, London. Serra, J. (1988). Image Analysis and Mathematical Morphology. Volume 2: TheoreticalAdvances, Academic Press, London. Serra, J., and Lay, B. (1985). Square to hexagonal lattices conversion, Sig. Process. 9: 1-13. Shih, F. Y., and Mitchell, 0.R. (1991). Decomposition of gray-scale morphological structuring elements, Putt. Recog. 24(3): 195-203. Stansfield, J. L. (1980). Conclusions from the commodity expert project, A.Z. Lab Memo No. 601, Massachusetts Institute of Technology. Sternberg, R. (1986). Grayscale morphology, Comput. Graph.Image Process. Scale space: M. A. (1991). Itsnatural ter Haar Romeny, B. M., Florack, L. M. J., Koenderink J. J., and operators and differential invariants, Proc. 12th Image Process. Lecture Notes in Computer Science Wye, pp. H.-T., Choy, T.-C., and Ho, (1988). Biomedical signal analysis by scale space technique, Proceedings d the 5th Znt. Biomed. Eng.
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van dcn Boonigaard. R. ( 1992). Mtrthrrricrriccrl Mor./hdofi~E.x/cvi.\iorr.\ 7iwtrnlr Corirpidter. Vi.\iorr. Ph.11. thcsis, University of Amsterdam. van den Boonigaard, R.. and Smculders, A. ( 1994). The morphological structure of images: The differential equations of morphological scale-space, IEEE 7: PNII.A r i d . A4uc.h. I J J I ~ 16( / . I I ): I I O I - I 113. Verbcck. P. W.. and Vcrwer. B. J . H. (1989). 2-D adaptive smoothing by 3-D distance transformation. Air/. Recog. Lett. 9: 53-65. Vincent, L. (1993). Morphological grayscale reconstruclion in image analysis: Applications and cfficicnt algorithms, IEEE 7: Irrrcrge Prot Vincent, L., and Beucher, S. ( 1989). The morphological approach to segmentation: An inlroduclion. /ii/i,rriir/ Report C-OK/b”UMM, School of Mines, Paris. Vincent. I.., and Soille, P. (1991). Watersheds in digital spaces: an efficient algorithm bascd o n immersion siinulations, IEEE 7: Prrrr. Aiitrl. M d z . h e / / . 13(6):583-598. Wilson. R.. and Granlund. G. H. (1984). The uncertainty principle in iinagc processing. IEEE Trcirw iw/iori,s o r r Pott. A r i d . M a c k . /rr/e//.PAM1-6(6): 758-767. Witkin. A. P. ( 1983). Scale-space filtering. Pro(.. I r i t . Joirit Cor!f:Arr. / r i / t 4 / . , Kaufmann, Palo Alto. CA. pp. 1019-1022. Witkin, A. P. (1984). Scale-space filtering: a new approach to multi-scale description. irr S. Ullnian and W. Richards (eds.), I i i i q e Urrdrr,stirridrn,q19x4, Ablcx. Norwood, NJ. pp. 79-95, Witkin, A,. TcIropoulos.D., and Kass. M. (1987). Signal matching through scale space. Irir. J . Coriiput. vis. pp. 133-144. Wu, L.. and Xie. Z. ( 1990). Scaling theorems for zcro-crossings. IEEE 7: Ptrtt. A i r d . Moch. Iritcdl. 12( 1 1 ): 46-54. Yang. J.-Y.. and Chen. C.-C. ( 1993). Decomposition ofrtdditively separable \tructui-ing elements with applications. Port. K c ~ o g 2. 6 ( h ) :867-X7S. ~ . A ~ JAL 2(5): Yuille, A. L.. and Poggio, T. ( 1985). Fingerprints theorenis for zero crossings. J. O / J So<,. 683-692. Yuille. A. L., and Poggio. T. A. (1986). Scaling theorcm forzerocrossings, IEEEK Port. A r i d . M o d r . I J I I P / / PAMI-& . I ): 15-25,
Characterization and Modeling of SAGCM InPDnGaAs Avalanche Photodiodes for Multigigabit Optical Fiber Communications C. L. F. M A
M. J. DEE"
L. E. TAROF
I. Introcluclion . . . . . . . . . . . . . . . A . Opticid Fihcr C ~ i ~ i i i ~ i t i i i i e ~ i t.t ~ .~ i .i ~ . . . . 13. Optical I
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C . L . F. MA. M . J . DEEN. AND I. . E . TAROF
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A . Introduction
B . Empirical Formula
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C . Physical Model D . Summary
VI . Temperature Dependence of Breakdown Voltage and Photogain A . Theory
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B . Experimental Rcsults C . Discussion
D . InP-Based APDs
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E . Temperature Dependence o i Photopin F. Summary
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V11. Dark Current Noise
A . Low-Frequency Noise Measurements
B . Multiplication Shot Noise C . Flicker Noise
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C . Temperature Dependence of Breakdown Voltage and Photogain
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D . Dark Current Noise
VI11. Conclu .;ions
A . Device Parameter Extraction B . Photopain
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Acknowledgment
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List of Acronyms
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Appendix A: Electric Field in SAGCM APD Relcrcnces
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I . INTRODUCTION In this paper. we describe extensive characterization and modeling of a specific type of state-of-the-art InGaAdInP avalanche photodiode (APD). utilized in current and future multigigabit optical fiber communication systems . In this section. a short introduction to optical fiber communications (Senior. 1992; Palais. 1992; Gowar. 1993) is given by considering the general system architecture. the major advantages over other technologies. and the historical developments of the technology . The general design requirements for optical receivers are discusscd. Finally. thc motivations of this investigation and the outline of this papcr arc presented .
CHARACTERIZATION AND MODELING 01; SAGCM
Information signal
Electrical * processing
Optical transmitter
processing
Fiber
67
+ Destination
) . Optical receiver
A. Optical Fiher Corrimunicatioizs 1. Gerieral System
An optical fiber communication system as shown in Fig. 1 is similar in its basic concept to any type of communication systems (i.e., its purpose is to convey the signal from the information source to the destination). In an optical fiber communication system, the information signal is converted from electrical form to optical form (E/O) by modulating an optical carrier of an optical source, such as laser or light-emitting diode (LED). Then the signal is transmitted through an optical fiber that can be as long as a few hundred kilometers; finally it is converted back into electrical form (O/E) by an optical receiver. I n an electrical communication system, the optical transmitter and receiver in the optical fiber communication system are replaced with an electrical transmitter (modulator) and receiver (demodulator), respcctivcly. The modulator converts the information signal into a form of propagation suitable to the transmission medium, such as a pair of wires, coaxial cable, or RF/microwave link. The demodulator transforms the modulated signal into the original information signal. Optical fiber communications use the superb advantages of optical fiber, mainly low attenuation and huge bandwidth. The optical fiber is made of silica glass with the optical refractive index being slightly lower in cladding than in core, resulting in light propagation in the core through total internal reflection. Generally speaking, an optical fiber can support many propagation modes. With reduced core diameter and other optimizations, monomode fibers, which have better characteristics than multimode fibers, can be fabricated, and they are the most popular fibers for long-haul optical fiber communications. In monomode optical fibers, the dominant mechanism of attenuation is impurity scattering within the I .O-1.6 p m
68
C . 1.. F. M A . M. J . D E N . A N D I.. E. TAROF
wavelength range. With advancements in impurity control of optical fibers, today's commercial monomode optical fibers have almost achieved the theoretical attenuation limit of -0.16 dB/km at 1.55 p m . Optical fiber communications benefit from an inherently huge bandwidth since thc frcqucncies of the optical carriers are in the order of 10" Hz. Since the wavelengths for zero dispersion and lowest attenuation in regular (standard) optical fibers are 1.3 and 1.55 pin, respectively, then most optical fiber communication systems operate at eithcr of these two Wavelengths. Optical fiber communication may be either analog or digital (binary), as in the case of electrical communication. Analog optical fiber communication is limited to applications of short distance and lower bandwidth, such as cable television. Compared to digital optical fiber communication, analog optical fiber communication is less efficient; that is, a far higher signal-to-noise ratio (SNR) is required and thus is difficult to achieve due to the inherent quantum noise associated with photon statistics. For example, the required SNR for cable television is 40 dB. In addition, the linearity required for analog modulation is not readily provided by semiconductor optical sources, especially at higher frcqucncies. The unavoidably high quantum noise and very wide bandwidth available in fiber optics make the digital format a natural choice. Historically, the installation of optical fibers followed the process of digitizing telephone systems. Optical carrier modulation is in the form of intensity modulation (IM), in which the intensity of the optical carrier is modulated by the power of the information signal pulse, and accordingly, optical detection is in the form of direct detection (DD). The more sensitive method of coherent modulation and detection still faces some technical difficulties, and at present it is not economical. In this paper, unless otherwise specified, a digital, IM/DD, and long-haul (with repeater spacing LIPto 100 km) optical fiber communication system is assumcd.
2 . System Considerutions In long-haul digital optical fiber communication systems, repeaters are used along the communication route to regenerate optical pulses in order to extend the communication distance. Similar to digital clcctrical communications, the major system considerations for optical fiber communications are bit rate B and repeater spacing L . Depending on the operating Wavelengths (0.85, 1.3, or 1. S S pm), B and L are limited either by attenuation or dispersion. An optical pulse propagating in an optical fiber loses power because of attenuation. Optical output power from a fiber of length L (km) can be relatcd to input power Pi,,by p ,,,,, = p,,,1o--~ ~ c l l I l/ ~10(1) where (Y,IH is the attenuation coefficient in decibels (dB/km). Considering an optical transmitter with maximum average power Pt,. (mW) and an optical receiver with
CHARACTERIZATION AND MODELING OF SAGCM
69
sensitivity P,, (mW), which is proportional to B, the maximum repeater distance L limited by attenuation is given by 10 L = -log UdB
P,,
Prcc
An optical pulse, which is unavoidably chromatic, also spreads out in transmission since the spectral power of the optical pulse associated with different wavelengths travels with different speeds (medium dispersion). In regular monomode optical fibers, one of the dominant dispersion mechanisms is material dispersion caused by the variation of refractive index n with wavelength h, and this is quantified by the material dispersion parameter D,,, = - where c is the speed of light. Another dispersion mechanism is waveguide dispersion caused by the propagation of a significant fraction of the optical power in the cladding. The combination of these two dispersions is called chromatic dispersion D,..For an optical source with root-mean-square (rms) spectral width C T ~the , rms pulse broadening cr after transmitting a distance L is given by
9I 1
u = CTALD,..
(3)
The maximum bit rate B due to chromatic dispersion can be estimated from B = 1/4a; therefore, the repeater distance L is limited by dispersion, and it is given by
L = 1/40iBD,.
(4)
It is obvious that the bit-rate 0 distance product, B 0 L , is independent of both B and L. If cri is so small that the limiting rms spectral width is due to finite time duration of the optical pulse, then the minimum value of cr is given by (Gowar, 1993) (T
=Apz
2 nc
Thcreforc,
and in this case B 2 0 I, rather than B 0 L is independent of both B and L. Using typical values of previously described parameters in Table 1, and Eq. ( 2 ) , Eq. (4), or Eq. (6), the attenuation limits and dispersion limits on the repeater spacing L as a function of the bit rate B for regular monomode optical fibers at 1.3 and 1.55 prn wavelength are displayed in Fig. 2. The attenuation and dispersion are limiting factors at bit rates lower and higher than 5 Gb/s, respectively. If the laser is directly modulated at 1.55 pin, I), will be ten times larger than when it is externally modulatcd, and the attenuation would be the limiting factor cvcn
70
C. L. F. M A , M . J . DEEN. AND L. E. TAROF
I .3 1.55
04 0.25
2
0.25 0.03"
-40 -40
0 0
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"At I Gh/s. hExicmalniodulatcd.
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lo-'
100
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Bit rate (Gb/s) FI(;uw 2. Repcatcr spacing versuz hit rate tising distributed fccdhack lasers and rcgulaimonoinode optical fihers. The solid lines are for I .3 pin. and the dashed lines for 1.S.5 pin.
at 0.5 Gb/s. Note that the preceding calculations are estimates only, and power budget and rise time analysis are required for detailed system design.
3. Historical Perspective Kao and Hockham (Kao and Hockham, 1966) in 1966 may be credited for having laid the foundation for modern optical fiber communications, even though the attenuation of optical fibers was 1000 dB/km at visible wavelengths at that time. The attenuation was successfully reduced to 20 dB/km by 1970, and to 2 dB/km by 1975 for visible wavelengths (0.85 pm). By the mid-I980s, optical fibers having attenuation less than 0.4 dB/km at 1.3 p m and less than 0.25 dB/km at 1.55 p m were commercially available. These three wavelengths (0.85, 1.3, and 1.55 p m ) with lower attenuation are so-called first, second, and third transmission windows, and they were used by first, second, and third generations of optical fiber communication systems, respectively. The first generation optical fiber systems used multimode fibers with core dianieters of50 pm, GaAlAs-GaAs lasers as sources operating at 0.85 p m and Si APDs
CHARACTERIZATION AND MODELING OF SAGCM
71
as detectors. Such systems have been installed since 1980, but were obsolete by the mid-1980s. The bit rate used was up to 150 Mb/s with the repeater spacing up to 10 km. The second and third generation systems used monomode fibers, InCaAsPbased lasers as sources operating at 1.3 p m or I .5S p m , respectively, and InGaAs/ InP PINS or APDs as detectors. Such systems have been used exclusively since 1985 in all high-capacity, long-haul, trunk route ovcrcontinental and underwater communication systems. The bit rate is up to 2.5 Gb/s with the repeater spacing up to 80 km. A fiber amplifier uses rare-earth ions to dope the fiber core as gain medium. Erbium-doped fiber amplifiers (EDFAs) with high gain and low noise at I .55 p m were demonstrated in 1987 (Mears et ul., 1987; Desurvire et a]., 1987). This development has revolutionized the field of optical fiber communications. The attenuation at 1.55 p m is not a serious design problem where EDFAs are utilized, even for underwater multigigabit communication systems, and the repeater spacing is dispersion limited. One expensive option for reducing dispersion at 1.55 p m is to use dispersion shifted fiber (DSF), which has both minimum dispersion and minimum attenuation at 1.55 pm. Unfortunately, reliable fiber amplifiers with high gain and low noise operating at 1.3 p m rcmain to be demonstrated.
4. Major Advaiitages of Fibers over Metallic Transmission Media Herc, the major advantages of optical fibers over metallic media are briefly discussed. 1. Huge bandwidth and low attenuution. Light is a much higher-frequency electromagnetic wave ( lo5 GHz) than RF/Microwave (up to 10 GHz in coaxial cablcs). The attenuation is 10 to 20 dB lower than the popular RG-19/U coaxial cable at 500 MHz. It is easy to transmit over 100 km at S GHz, or over 300 km at 500 MHz in optical fibers without rcpcatcrs. By comparison, broadband coaxial cable systems restrict the transmission distance without repeaters to only a few kilometers at a few hundred MHz. 2. Srnall size and weight. Common optical fiber cables have cladding diameters of 125 p m (fiber core diameter from 10 p m to 100 pin) enclosed in a sheath with outer diameter of 2.5 mm, and the weight is 6 kg/kni. By comparison, the RG19/U coaxial cables have outer diameter 28.4 mm, and the weight is 11 10 kg/km. Furthermore, the transmission capacity of the fiber cables is at least 100 times that of the coaxial cables. 3. l m m u r z i ~to interfererzcc and crosstulk. Optical fibers are dielectric waveguides and are, therefore, free from electromagnetic interferences. Since they have extremely small optical leakages, crosstalk between fibers is negligible even when many fibers are cabled together. The immunity to interference and crosstalk rcpresents a major design advantage for optical fiber communication systems.
C. L. E MA, M. J. DEEN, AND L. E. TAROF
12
Additional advantages are electrical isolation, signal security, ruggedness and flexibility, reliability, and potential low cost. Optical fiber systems have totally dominated long-haul telecommunications since the early 1980s. Shorter-distance communication systems are being gradually replaced by optical fibers. The major disadvantages of optical fiber communication should also be mentioned. Due to the difficulty and cost of splicing fibers, and the high cost of O/E and E/O conversions, it is presently not economically viable to build local area networks and short-distancecommunication systems with optical fibers. Because of the higher photon quantumnoise and nonlinearity of optical sources, the applications of optical fibers in analog systems are limited at present. B. Optical Receivers
1. Design Considerations
The most important design considerationsfor a receiver in (electrical or optical) digital communications are sensitivity and bandwidth. The sensitivity is conventionally defined as the minimum average received power pr,, required by the receiver to operate at a BER of lom9, where BER (bit-error rate) is defined as the probability of incorrect identification of a bit.
2. Components A digital optical receiver consists of three sections as shown in Fig. 3. The frontend section,which is the criticalpart of the three sections,includes aphotodetector, followedby apreamplifier.The photodetector convertsthe opticalbit stream into an electrical bit stream. A photodetector with internal gain is beneficial in increasing
Front end Optical input
Decision circuit
Photodetector
amplifier
Clock recovery
-
CHARACTERIZATION AND MODtLlNG OF SAGCM
73
the overall sensitivity of the optical receiver. A transimpedance amplifier is the best choice for the preamplifier since the transimpedance amplifier can provide both high gain and high bandwidth, and in this case, the stability of feedback loop at highcr gains is the niajor design challengc. In contrast, the design of the
preamplifier using a voltage amplifier forces a trade-off between band-width and gain and is, therefore, less attractive. The linear channel section consists of a high-gain main amplifier and a low-pass filter. The main amplifier includes automatic gain control (AGC), which limits its avcragc output power to a fixed level regardless ofthe optical input power at the photodetector. The low-pass filter shapes the electrical pulses to reduce noise and intersymbol interference (ISI). The data recovery section is made up of a decision circuit and a clock recovery circuit. The decision circuit compares the output from the linear channel to a Lhrcshold level at sampling times determined by the extracted clock, and decides if thc signal correspond to bit “ I ” or “0.”
C. Motivatioti Long-haul optical fiber communication systems necessitate high-performance optical receivers. One of thc most critical components in the receiver is a highperformance photodetector. In this paper, an extensive investigation is presented o n such a state-of-the-art, high-performance photodetector, planar separate ahsorption, grading, charge, and multiplication (SAGCM) I n P h C a A s avalanche photodiode (APD), utilized today and in the near future commercial 1.3or 1.55p m optical fiber communication systems. In this paper, comprehensive characterization and modeling, particularly DC and noise characteristics and their temperature dependence, which have never been done bcforc, are important parts of overall efforts to design, fabricate, characterize, model, and test the devices. The knowledge discovered here can also be used for other devices. D. Outline
In Section 11, a review of the fundamentals of photodetectors is presented, including difierent types of photodctcctors and the reasons why APD was selected. The fundamentals of APD are reviewed, followed by a brief account of its historical devclopment. This leads to a discussion of the reasons why SAGCM InP/InGaAs APDs arc the best choice as the photodetector in the intended optical fiber communication systems. In Section 111, specific details of the devices are presented, including their structure, fabrication, and calibration. In Section IV, a simple, innovative, and nondestructive technique for extracting of two critical device parameters, multiplication layer thickness x,/ and integrated areal charge density uct,a,.gc in the charge layer, is described. The systematic uncer-
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C. L. F. MA. M. J . DEEN. AND L. E. TAKOF
tainties, arising from the neglect of ionizations in the absorption and charge layers, and from different ionization rates in InP reported in the literature, are considered in detail. The random errors caused by uncertainties from experiments and other device parameters are also investigated. In Section V, the photogain ( M ) versus bias voltage ( V ) characteristic is studied both experimentally and theoretically. An empirical formula for Si and Ge is tested on the SAGCM InPAnGaAs APDs. A physics-based model is developed to theoretically calculate the M - V characteristics at room temperature for all the SAGCM InP/InGaAs APDs with different device parameters. In Section VI, a general theoretical model for the temperature dependence of breakdown voltage V,, in any type of InP-based APD (including the SAGCM InP/InGaAs APDs) is developed. The temperature dependence of breakdown voltage Vb, from -4O’C to 110°C in the SAGCM InP/InGaAs APDs with a range of device parameters is investigated. The temperature dependence of the M - V characteristics is also discussed. In Section VII, low, frequency noise (LFN) in dark currents is also investigated experimentally. The dark current multiplication shot noise is determined experimentally as a function of dark current and multiplication gain M in a few SAGCM InP/InGaAs APDs. Flicker noise is also observed in some SAGCM InP/InGaAs APDs, and it can be related to surface and interface imperfections and defects.
11. BACKGROUND In this section, different types of photodetcctors and their general operating principles are first briefly reviewed. This review highlights the reasons why APDs are a good choice for optical fiber communications. A detailed review of opcrating principles, structures, and calculations of major performance characteristics of general APDs follows. The section concludes with a brief description of the historical evolution of APDs, and the reasons why a specific structure APD, separate absorption, grading, charge, and multiplication (SAGCM) InP/InGaAs APD, is required for optical fiber communications.
A. Fundamentals of Photodetectors The most popular photodetectors, except avalanche photodiode (APD), are discussed in detail in the next three sections. Excellent general discussions of photodetectors can be found in many references, for examples, (Senior, 1992; Saleh and Teich, 1991). The comparisons of photodetectors for optical fiber communications are discussed in the references (Beneking, 1982; Forrest, 1986; Rolland et al., 1992).
CHARACTERIZATION AND MODELING OF SAGCM
0
Electron
0
Hole
75
FIGLIRI:. 4.Intelnal photoeniission eltec~--ahsorplion of a photon.
1. Photodetection There are two distinct photodetection mechanisms. The first mechanism is the external photoemission effect, typified by the photomultiplier tube (PMT). When the primary photocathode o f a PMT is bombarded by photons, electrons are emitted if the energy of the incoming photons is larger than the work function of the photocathode. The emitted electrons are accelerated toward secondary electrodes (dynodes), where more electrons can be emitted by the impact from each incoming electron. This process is repeated for many stages, resulting in gains as high as lo6. The PMT is not suitable for optical fiber communications because it is incompatible with microelectronic monolithic integration in these systems, and there are no photocathode metals with a suitable work function for 1.3-1.6 Krn wavelength light. The second mechanism is the internal photoemission effect (absorption) as shown in Fig. 4. A photon with energy equal or larger than the bandgap energy of the semiconductor E,s can excite an electron and hole pair. Photodetectors based on internal photoemission effect are named solid-state detectors since they are made of solid-state materials, predominantly semiconductors. With good performance characteristics as well as simplicity, small size, ruggedness, and compatibility to integration, semiconductor photodetectors are undoubtedly better choices than PMT or any other phototubes for visible and near-infrared (IR) optical fiber communications. Semiconductor photodetectors can be classificd further according to one of the following criteria: semiconductor material, device structure, and internal gain.
2. Pe'erforrnance Chamcteristirs The major performance characteristics appropriate for all photodetectors are the following. Quantum efficiency and responsivity. The quantum efficiency is defined as the ratio of the number of collected electron-hole pairs (without gain) over the number of incident photons, and is always less than 100%. q is determined by the absorption coefficient a,the surface reflection coefficient, and any other carrier
76
C . 1. I:. MA. M. J. DEIIN. A N D 1.. E. TAROF
recombination mechanisms occurring before collection. In common with a. rl is a function of the photon wavelength, A related but more often used figure of merit is responsivity !N. It is defined as the ratio oT generated photocurrent, I,,, (without internal gain) over incidcnt optical power, P(,, and it can be related to q using
where y is the electron charge. Bandwidth. Electrical bandwidth is defined as the frequency of the electrical signal at which the power of the electrical signal is 3 dB lower than its DC power. Gain. Gain is defined as the internal gain of the photodetector ( i t . , the ratio of number of collected electron-hole pairs Lo the number of primary photogencrated pairs). Another popular name is multiplication gain (photogain), and they arc used interchangeably in this paper. Noise. Noise is defined as the fluctuation of the generatcd electrical signal and is described with corresponding rms values of current or voltage. Long cutoff wavelength, The photon energy with wavclcngth h must be larger than the bandgap energy, E,, (direct bandgap); that is, h c / h 2 E,,. Thus. the long cutoff wavclcngth, A(.,is hc
h, = - .
E, 3. Device Reyuiretnent.\
Being one of the crucial components which dictates the overall system performance in optical fiber communication systems, photodetectors must satisfy very stringent requirements for performance, compatibility, cost, and reliability. High sensitivity. The photodetectors are designed to detect weak optical signals of a specific wavelength, This demands high responsivity (quantum efficiency), low internal noise, and high multiplication gain (if applicable). Fast speed. The time delay in the conversion from electrical to optical forms should be minimal. In frcqucncy domain, it means the electrical bandwidth should be adequate for the operating bit rate. High reliability and stability. The photodetector must maintain dcsigncd pcrformance characteristics for many ycars in sometimes demanding environmcnts. This requires better design and stringent fabrication quality for eliminating potential sources of degradation. Low cost. This normally means high yield in fabrication. High fidelity and dynamic range. For analog transmission, the photodetectors should be able to reproduce faithfully the received optical waveform over a widc range of optical power. This is not critical in digital applications.
CHARACTERIZATION AND MODELING OF SAGCM
77
A suitable photodetector must satisfy these requirements. Various types of photodetectors are examined in the following sections in order to select a suitable one. B. Semiconductor Photodetectors without Internal Gain I . p-n Photodiode
A p-n photodiode normally operates under reverse bias, and both the depletion and diffusion regions may absorb photons, which generate electron-hole (e-h) pairs in proportion to the optical power, as shown in Fig. 5. In the depletion region, the e-h pairs are separated and drift in the electric field. In the diffusion region, the pairs within a few diffusion lengths may slowly diffuse to the depletion region, and eventually generate some terminal current. The other pairs in the diffusion region recombine and generate no terminal current. Therefore, the depletion region should be wide enough to achieve high quantum efficiency. This also makes the p-n capacitance smaller to meet the high bandwidth requirement. However, the depletion region cannot be too wide since it would increase the drift time of the carriers through the depletion region, which results in reduced bandwidth. This exemplifies the compromise between quantum efficiency (gain) and speed (bandwidth) in all photodetectors.
2 . p i - n Photodiode
.
p-i-n photodiodes (PINS) as shown in Fig. 6 have superseded p-n photodiodes, and are the most popular photodetectors without internal gain in optical fiber communications. p-i-n photodiodes are better than p-n photodiodes since they can be easily tailored for optimum quantum efficiency and bandwidth, and are relatively easy to fabricate, highly rcliable, and of low noise. I n G a A s h P p-i-n Optical Power
-vel
Depletion
................................................................................
Y
Diffusion
..................................................................................
78
++
C. L. F. MA, M. J . DEEN, AND L. E. TAROF
/ I
TdAucontact
n intrinsic InCaAs absorption
\
bf InP buffer
I
@ InP substract
Back contact FI(;IJKL: 6. A front-illuminated InGaAs homojunction p-i-n photodiode.
photodiodes arc most widely used, and have gained even more popularity after the breakthrough of the EDFAs. Since all the generated carriers are in the depletion region (i region), the collection process is fast and efficient. Therefore, the intrinsic bandwidth is very high, and the overall bandwidth is usually limited by extrinsic effects up to a fcw tens of GHz. The p+ InGaAs layer in homojunction p-i-n photodiodes, as shown in Fig. 6, has to be thin to allow light to penetrate without severe loss. However, in this case, the strong surface recombinations impair the quantum efficiency. One solution is to open an optical window at the bottom of the substrate, and the p-i-n photodiode is back-illuminated rather than front-illuminated in this case (heterojunction), since InP is transparent to light with its wavelength longer than 0.92 pm. Another solution is to replace the thin p+ InGaAs layer with a thick p+ InP laycr (double heterojunction). Heterojunction p-i-n photodiodes with up to 30 GHz bandwidth and about 70% quantum efficiency are fabricated commcrcially. 3. Schottky-Barrier Photodiode
A Schottky-barrier photodiode as shown in Fig. 7 is made of metal-semiconductor rectifying junction rather than the p-n junction in the case of p-n and p-i-n photodiodes. The Schottky-barrier photodiodes inherently have a very narrow active rcgion, and are particularly useful in the visible and ultraviolet wavclcngths due to their large absorption coefficients. Because of the short transit time, Schottkybarrier photodiodes can operate at frequencies as high as 100 GHz. One technical
79
CHARACTLRIZATION AND MODELING OF SAGCM FET
Metal
n GaAs
n+ InGaAs Semi-insulating aubWate
I
I t
I
t;lGllKs
I
Ohmiccontact
7. Schottky-harrier photodiodc.
I I
FIGLIHI; 8. MSM pholotletector ink+vated with MESFET.
challenge is how to avoid surface traps and recombination centers, which cause substantial loss of generated carriers at the surface. In addition, the quantum efficiency is low, and there are no suitable mctals li,r fabricating Schottky-barrier photodiodes operating in near-infrared wavelengths. Therefore, Schottky-barrier photodiodes are not suitable for optical fiber communications for wavelengths of current interest.
4. Metal-Serni~onductor-MetalPhotodetector A metal-semiconductor-metal (MSM) photodetector is a variation of the Schottkybarrier photodiode (i.e., two Schottky contacts are made at the same side of the substrate). This results in easy fabrication and monolithic integration with othcr clcctronic devices, such as the mctal-semiconductor field effect transistor (MESFET), as demonstrated in Fig. 8. The capacitance of an MSM photodetector is extremely small. However, it suffers similar limitations as a Schottky-barrier photodiode, and it is not widely used in optical fiber communications. As for optical receivers using optoelectronic integrated circuits (OEICs), it seems that the more popular choices are the combination of a p-i-n photodiode and a high electron mobility transistor (HEMT) or a heterojunction bipolar transistor (HBT) (Akahori, 1992). C. Semiconductor Photodetectors with ltiternul Guiri 1, Photorotiductive Detector
A photoconductive delector is made of some absorptive semiconductor, such as InGaAs, with two electrical terminals. In the dark, the semiconductor behaves like a resistor with a finite dark current at a constant voltage bias. Electrical conductivity increases when it is illuminatcd because of photo-gcncrated carriers and the current increases. The simplicity makes i t attractive in OEICs. The internal gain arises from the space charge neutrality rcquirement. The photo-generated electron-hole pairs move toward their respcctive collecting terniinals with different velocities. The faster carriers are collected first, which results in
80
C. L. F. MA, M. J. DEEN, AND L. E. TAROF
excessivecharge in the photoconductor. The excessivecharge draws another faster carrier into the conducting layer until the slower one is recombined or collected. Therefore, the internal gain is equal to r/r,, where r is the slower carrier lifetime (transit time), and r, the fast carrier transit time. The bandwidth is inversely proportional to the relatively large t;therefore, the bandwidth is normally limited to 100 MHz. Thus, an increase in gain can only be obtained at the expense of the speed of response. This is clearly another example of the trade-off between gain (or quantum efficiency)and bandwidth. As for optical fiber communications,the relative large dark current (from lack of a p-n junction and required narrow bandgap energies) and its associated noise are the major problems. Quantum efficiency would be compromised significantly if the dark current is reduced to an acceptablelevel with current technology. Because of the limitedbandwidth and large leakagecurrent, photoconductorsare rarely used in optical fiber communications.
2. Phototransistor A phototransistor is similar to a bipolar transistor except normally there is no electrical contact to the base as shown in Fig. 9.The base and the base-collector junction regions are used as the photosensitive region. The holes generated in these regions accumulatein the base. This excessive charge causes electronsto be injected from the emitter, and the current gain is achieved as in a normal bipolar transistor. Phototransistorshave yet to find use in major optical fiber communication systems,although these devices have been investigatedintensively in the late 1970s. The problems are quantum efficiency, gain and bandwidth, and maybe, the more critical one is that a steady-stateilluminationis required to bias the phototransistors to achieve linear amplification and fast response time.
FIGURE9. n-p-n InGaAdlnP phototransistor,
CHARACTERIZATION A N D MODELING OF SAGCM
D. Fundmientnls
81
of APD
From the above discussions, it is clear that a p-i-n photodiode is by far the better choice among the photodetectors without internal gain. However, the photodetectors with inteinal gain arc advantageous ovcr the photodetectors without internal gain, especially when it comes to dctccting weak signals. The counterpart ofp-i-n photodiode, avalanche photodiode (APD), is a good choice, and is widely used. The optical receivers with APDs are normally about 10 d B more sensitive than the optical receivers with p-i-n photodioclcs, which may translate into an extra 40 kiii repeater spacing if 0.25 dB/kni attenuation i n optical fibers is assumed. The combination of p-i-ti photodiode and EDFA is adcqiiate in sensitivity. However, for sufficiently short distances or high launched powers, it may be advantageous to use a high-speed APD over an EDFA/PIN configuration for significant cost reduction (Rolland et nl., 1992). More importantly, no reliable optical amplifiers operating at I .3 pum wavelength arc available y c ~ .For 1.3 pin optical fiber comniunication systems, APDs arc far more scnsitivc than PINS without using optical aniplificrs. APDs suffer several drawbacks compared to PINs: more complex structure and complicated fabrication. therefore, more expensive and less reliablc; of inherently lowcr bandwidth at higher inultiplication gains because the avalanchc multiplication process takes extra time; and a higher bias voltage (30-70 V) is required, compared to 5 to 10 V required for PINs. 1 . Av(ilanche Multipliccitioii
l h e APD’s internal gain is realized by the avalanche multiplication process, which is achicvcd through impact ionization. The impact ionization phenomenon has been extensively investigated both theoretically and expcrinicntally (Chynoweth, 1968; Stillman and W o k , 1977; Capasso, 1985). Electrons and holes are accelerated by ; Ihigh electric field until they gain sufficient energy to excite an electron from the valence band into the conduction band. as shown i n Fig. 10. This process is a thrccbody collision process. The required iuiniinum carrier energy for causing impact ionization is the ionization threshold energy E , , which is obviously larger than the handgap energy. Electrons and holes nornially have different ionization energies. The impact ionization process is quantified by (impact) ionization rates (Y and /3 for clcctrons and holes, respectively. They are defined as the reciprocal of the average distance, measured along the direction of the electric field, traveled by an electron or a hole to create an electron-hole pair. In thc process of acceleration.
82
C. L. F. MA, M . J. DEEN, AND L. E. TAROF
Electron
0 Hole
the electrons and holes inevitably lose energy to nonionization collision processes, such as phonon scattering. One carrier undergoing the impact ionization process creates a pair of free carriers. All three carriers get reaccelerated, and then continue to undergo impact ionization events and generate more free carriers. This process is only terminated if all the free carriers are swept out of the high electric field region. In the end, one initial electron (hole) generates M extra electron-hole pairs. M is therefore the APD multiplication gain (photogain).
2. Rate Equutions and Photoguin The mathematical descriptions of avalanche multiplication are well documented (Stillman and Wolfe, 1977; Gowar, 1993). Suppose that the avalanche multiplication occurs only within the multiplication region (the high electric field region) as shown in Fig. 1 l a (x between 0 and w).The elcctron current, J,, (x), increases and the hole current, J [ , ( x ) , decreases along positive x direction. The rate equation for J,, ( x ) can be written as
and for J , , ( x )
where G ( x ) is the space charge generation rate (optically or thermally). The total current J = J , , ( x ) J , ] ( x ) is a constant throughout the structure (current
+
CHARACTERIZATION AND MODELING OF SAGCM
0
W
Y
83
) X
+ x
FIGtIRE I I . ( a ) Avalanche mulLiplication within ii multiplication region, a n d gain along .t direction lor the case of' (b) LY > 6 or (c) (Y < 6 .
continuity). Substituting this condition into each of the above expressions, and integrating both sides from 0 to w by using integrating factor exp(- yi,'(a D)d.w') = exp(-p(x)), the total current can be obtained in two equivalent forms:
84
C L F MA. M J D E N . AND I. b TAKOE
Pure electron and hole injection conditions are represented by J,,(w) = 0 and J l l ( 0 )= 0, respectively. Therefore, the gains with pure electron injection. Mil, and with pure hole injection, M,,, can be respectively derived as
J M -I' - J,,(O) I
1 exp[cp(uj)l 1 ,A:' a exp[-cp(x)l dx - J;,'" /3 exp[cp(u~) - cp(x-)]d x
'
(13) M,, =
J
I
-
~
J,,(uI)
I
-
-
J;' p exp[cp(w) - cp(x)] d x
I
-
exp I0(U' 11 J:' a exp[-cp(.r)l
dx ' (14)
The avalanche breakdown occurs when the multiplication gain is infinite, and it can be shown that for both pure electron and hole injections, the breakdown condition is the same; that is, 1
-
l"'
a exp[-cp(x) I d x
=I
-
1:
/3 exp[cp(w) - cp(x)] d x
= 0.
(15)
It can also be shown that if a > ,!I, then MI, > M,,, and vice versa. Thus, a higher gain is obtained if the carriers with larger ionization rate are injected. It will be demonstrated later that this is also the condition for lower excess noise and higher bandwidth. In the case of no carrier injections, the carriers can still be photogenerated within the multiplication region. Suppose that the carriers are photogencrated at x,, between 0 and W, and let G ( x ) = G,,S(x then the gain M ( x , , ) is given by .I-()),
J = q ~ , , I
cxpIcp(llJ)- cp(x0) I J;;pexplcp(u~) - cp(x)ldx I
exp[-cp(x,,)l J(1'aexpI-cp(.I-)]dx-' (16) It is obvious that MI, = M ( 0 ) and M,, = M ( w ) . The variations of M ( s ) along x direction are displayed in Fig. 1 Ib and c. The current can be written in a more physical expression M(&) =
~
-
-
3. Multiplication Excess Noise The avalanche multiplication process is a random process, and the multiplication gain described in Eq. ( 16) represents only the average gain. The statistical variation of the multiplication gain is responsible for the multiplication excess noise, which is additional to the inultiplication shot noise associated with the current. The total mean square noise spectral density (Mclntyre, 1966; McIntyrc, 1972) can be
85
CHARACTERIZATION AND MODELING OF SAGCM
described by
I)
+ I [ Z ~ " o M ? ( x ) d x MI', -
,
where I is the total current. In the case of pure electron or hole injection, ( t M )= ? ~~I,,,,M'F,
(19)
where I,), is the primary photocurrent, M is either M,, or MI,, and F is the excess noise factor. F is always greater than or equal to 1, and F = 1 only if the multiplication process does not cause additional noise. It has been shown to a very good approximation (McIntyre, 1966; McIntyre, 1972) that
F = M { 1 - (1
-
k)
(y)'},
where k is either replaced by k,ll or kA,l for the respective case of pure electron or hole injection, and
(21) For the case of uniform electric field in the multiplication region, k,ll = B/a and k i t , = o/B. In Fig. 12, F is plotted for different k,,, (or kLll) values. It can be seen that for lower excess noise, the carriers with higher ionization rate should be
1
10
1W
Gain M 1:IGtlRF.
12. Excess noise Iaclors l o r various values ofk:.,, and M in thc caw of pure hole injection.
86
C. L. F. MA, M . J. D E N , AND I,. E. TAKOF
injected and the ionization rates for electrons and holes should be as different as possible. For the case of mixed carrier injections without charge space carrier generation (Webb et al., 1974), together with the average gain
the effective excess noise factor is given by
where f is the fraction of the total injection current due to electron injection I ] , ( W ) ) , and F,, and F,, are the corresponding excess noise factor for respective pure electron and hole injection. F,tf is always larger than F at the same average gain since in this case MI,or M , is larger than M . (= I,((O)/(I,,(O)
+
4. Bandwidth In addition to the limiting factors to the response speed in p-i-n photodiodes, such as the parasitic effect (RC) and the time required for the primary carriers to transit the depletion region, two additional limiting factors in the APDs are the avalanche buildup time and the transit time of the secondary carriers. Practically, two more mechanisms are important: the hole trapping at the heterointerface and the diffusion tail (Campbell ef a]., 1988; Tarof, 1990; Tarof et al., 1993; Webb etal., 1974). In general, to analyze the time and frequency response of APDs, a pair of coupled time-dependent transport equations must be solved (Emmons, 1967; Kuvas and Lee, 1970). These solutions are generally very complicated and only rccently have more rigorous analytical and numerical solutions been attempted (Hollenhorst, 1990; Kahraman et al., 1992; Roy and Chakrabarti, 1992). Different limiting factors become dominant at different gains. At high enough gains, the bandwidth is inversely proportional to the avalanche buildup time. Since the gain is proportional to the buildup time, gain-bandwidth product GBW is independent of the gain. It can be proved that GBW (hole injection) is equal to [2nNs(a//l)]-' with N between 1/3 and 2, and 5 is the transit time of the carricrs in the multiplication region (Emmons, 1967; Kuvas and Lee, 1970). It is clear that GBW is higher if kl,, is smaller. In summary, to achieve lower excess noise and higher GBW product, the carriers with higher ionization rate should be injected and the difference between the ionization rates for electrons and holes should be as great as possible.
CHARACTERIZATION A N D MODELING OF SAGCM
87
(1-R)Pi
X-
~
0
xabs
FIGURE13. Optical power attenuation in an ahsorption layer.
5 . Quantum EfJiciency As shown in Fig. 13, suppose that the width of absorption layer is X&,\ with the absorption coefficient y , and the reflection coefficient at the incident surface is R ,
then the optical power P ( x ) loss d P at x is given by
d P = -yPdx,
(24)
with solution
P ( x ) = (I - R)P,e-Y'.
(25)
Therefore, the quantum efficiency 17 without any carrier recombination is given by
6. Photocurrent The photocurrent I , is related to the photogain M by
Ip = I p o M , where I/,(, is the primary photocurrent. Without any carrier recombinations,
Using Eq. (7), the responsivity is then given by
88
C. L. F. MA, M. J. DEEN. AND L. E. TAROF
7. Dark Current and its Noise The dark current Ill of APDs consists of two components-multiplied tiplied dark currents; that is,
I(/ = Id,, M
+ Ill,,
1
and unmul(30)
where I,/,, is the primary dark current, and Illlr is the unmultiplied leakage dark current via shunt paths. The important conduction mechanisms for the primary dark current are diffusion, generation-recombination (GR), and tunneling. The diffusion current in p-n diodes at reverse bias voltages greater than a few k T / q is given by (Forrest, 1981)
where, r,,(r/,),D,,(D/,),A / , ( A , , ) ,and NA(ND) are the minority carrier diffusion lifetime, the minority carrier diffusion constant, the area of the depletion region boundary, and the doping concentration in the p (n) region, respectively. n , is the intrinsic carrier concentration, and it is given by n , = 2(4n 2mpmll kZ/h4)1/4T ' I 2exp( - E, /2kT),
(32)
where m e ( m / , )is the effective mass of electrons (holes), h is Planck's constant, and T is the absolute temperature. The GR current at reverse bias V is approximately given by
where A and W are the area and width of the depletion region, respectively, and r,tt is the effective carrier lifetime (Grove, 1967). GR is assumed to occur via traps near the middle of the bandgap (Moll, 1964). Since I,J~,, cx nf cx exp(- E,/kT) and I,?,. C( a; cx exp(-E,/2kT), it is expected that the GR current would be dominant at lower temperatures, and the diffusion current dominant at higher temperatures. For band-to-band tunneling in a direct bandgap semiconductor, the tunneling current is given by (Zemel and Gallant, 1988; Moll, 1964).
where F,,, is the maximum electric field, and the parameter 0 depends on the detailed shape of the tunneling barrier. 0 is 1.11 for a triangular barrier and 1.88 for a parabolic barrier. The only temperature-dependent parameter is the bandgap energy E,, and therefore, the tunneling current has a considerably weaker temperature dependence than either the diffusion or GR currents.
89
CHARACTERIZATION A N D MODI-.I.ING 01;SAGCM
I
p+ substrate
metal 1;i(mi11:1.1. A \ilicon rcacli-through aval;iiichc photodiotlc.
E. APD Evolution 1. Silicori APD
Because of the large ratio of electron-to-hole ionization rate and maturc processing technology, silicon is the most suitable material for the 0.8-0.9 p m wavelength first generation optical fiber communication systems. The absorption coefficient dictates that absorption width is a few tens of p m . A simple Si p+-n structure is riot a satisfactory design for Si APDs, since it would require morc than 500 V bias voltage to deplcte such a long absorption layer. In addition, the high clcctric fields at operating bias voltages degrade the noisc performance considerably. The reach-through structure consisting of a pi -n-p-n'- structure is the standard design of Si APDs, and it is shown in Fig. 14 (Kaneda, 1985). Most of the photons are absorbcd in the n region (-SOptn) since both p and n+ regions are thin (a fcw pm). At operating bias voltages, the electric ticlds i n the n region are low enough not to cause impact ionization, but high enough (20 Vlpm) to ensure the generated carriers being swept at their scattering-limited velocity (IO'purdns) to the p region, where the impact ionization occurs because of higher electric ficld. This type of Si APD can achieve breakdown voltage as low as 200 V, nearly 100% quantum efficiency, photogain as high as loJ, excess noisc with kc,-( 0.04 for photogains up to 500, and 500 MHz bandwidth for photogains up to 100.
-
2. G~wznnirrmAPD
Germanium is not the idcal matcrial for APDs because it has almost equal electron and hole ionization ratcs. To achieve better noise pcrformance, the n ~-n-p ' structure shown in Fig. 15 is thc preferred one for the wavelength 1.3 pin (Kaneda, 1985). Almost all incident light is absorbcd in the thick n region (a few pni), and pure hole injection is achieved. It can achieve -30 V breakdown voltage, -200 nA lcakagc
90
C. L. F. MA, M. J . DEEN, AND L. E. TAROF
n-guard ring
metal
'
FIGLIRE. 15. Germaniuni 11 -n-p APD
currcnt, >80% quantum efficiency, photogain 30 to 40, excess noise with kill 0.6 - 0.7 for photogain up to 50, and larger than 1 GHz bandwidth. For 1.55 pm, the absorption in Ge is weak. A reach-through structure as in the case of Si APD is dcsired. Therefore, germanium APDs may be used reasonably well for the whole interested wavelength range 0.8-1.6 pm. However, ideally, the selected semiconductor should have only a marginally smaller bandgap energy than the energy of photons. This would ensurc a compact structure with sufficient response, as well as a minimized tunneling leakage current. Large leakage current and its associated noise in germanium photodetectors are detrimental to their performance as sensitive photodetectors. In addition, germanium APDs suffer from some technical fabrication difficulties, and a large excess noise factor. Direct bandgap 111-V compound semiconductor alloys are superior to germanium because their bandgap energies can be tailored to suit a particular application by changing the relative concentrations of their constituents. In addition, a number of alloys with different bandgap energies can be utilized to fabricate a device because of their lattice match possibilities. This additional flexibility has been critical in designing photodetectors which must meet many conflicting design requirements. Therefore, 111-V compound APDs almost completely replace Ge APDs in optical fiber communications at 1.3 and 1.55 p m wavelengths. Tcrnary alloys such as InGaAs and GaAlSb have been used to fabricate photodetectors for longer wavelengths. In particular, the alloy Ino.53Gao.47As ( E , = 0.75 eV), which is lattice matched to InP ( E , s = 1.35 eV), has been extensively employed to dcsign and fabricatc photodetectors for 1.3 and 1.55 p m wavelengths. InGaAs represents In0,53Gq),47Ashereafter, unless stated.
3. SAM and SAGM InPAnGaAs APDs Thcre are some excellent reviews on the evolution of InP/InGaAs heterojunction APDs up to the late 1980s (Forrest, 1988; Campbell, 1989). Because of the relatively narrow bandgap and light masses of electrons and holes, carriers in InGaAs undergo band-to-band tunncling breakdown at an electric field of about
91
CHARACTERIZATION AND MODELING OF SAGCM
Metal
u
Metal
I
I
P+ InP
P+ InP n InP multiplication
I
n- InGaAs absorption
I
n InP buffer n+InPsubstrate Metal
1
1
I
I
n- InGaAs absorption n InP buffer
n+InPsubsuate
I
I
Metal
15-20 V l p m , which is below the threshold electric field for avalanche multiplication. Homojunction InGaAs APDs (with a structure like p-i-n) were reported with the problem of large leakage currents (Nishida el al., 1979). This problem was clearly understood at the beginning, and a separate absorption and multiplication (SAM) structure shown in Fig. 16a was proposed (Takanishi et al., 1980) to circumvent this tunneling problem. The objectives are to make absorption occur in a narrower bandgap layer, such as InGaAs, while avalanche multiplication occurs in a wider bandgap layer, such as InP. For this structure to operate properly as an APD, the following conditions must be satisfied. The absorption layer width is -2 p m . The goals are to obtain good quantum efficiency by ensuring adequate absorption, and to ensure adequatc bandwidth by not unnecessarily increasing the carrier transit time. The electric field at the InGaAslInP heterointerface is smallcr than 15 V/pm at operating bias voltages to avoid significant tunneling currents there. The absorption layer is completely depleted with minimum electric field 10 V l p m to ensure the photogeneratcd carriers being swept with the saturated velocities to thc InP multiplication layer at operating bias voltages. The conflicting rcquirements for the clectric fields in the absorption layer imply that the background doping concentration in the InGaAs absorption layer must be smaller than 2 x 10” cmP3 if it is uniformly doped. The maximum electric field in the InP multiplication layer is larger than 45 VIpm to achieve significant avalanche multiplication. The doping concentration in thc InP multiplication layer is smaller than 2 x 10” cmP3 to avoid the large tunneling current there (Takanishi rt 01.. 1980).
92
C L E M A . M J D E k N A N D 1 L rAAKOE
InP
InGaAsP InGaAs
InP
InPDnGaAs
InGaAs
SAM InP/InGaAs APDs were fabricated successfully (Susa et al., 1980; Diudiak et NI., 1981; Kim et al., 1981) with useful gain -10, leakage current -25 nA before breakdown, and the primary multiplied dark current smaller than 1 nA. A severe limitation of SAM InP/InGaAs APDs is that the photo-generated holes are trapped at the InP/InGaAs heterointerface due to the valence band discontinuity (0.4 eV). The trapping results in a slow component of the photoresponse (Forrest et d . ,1982), which limits the bandwidth to a few hundred MHz (Campbell et al., 1983). As shown in Fig. 16b, this problem can be alleviated by incorporating a thin grading layer of InGaAsP between the InP multiplication and the InGaAs absorption layer (Forrest et al., 1982; Kobayashi et al., 1984; Taguchi eta]., 1988; Cainpbell et ul., 1983). The quaternary alloy InGaAsP can be tailored compositionally to have a bandgap energy anywhere between the bandgap energy of InGaAs (0.75 eV) and InP (1.35 eV), while simultaneously maintaining a fixed lattice parameter (5.869 A). The gain-bandwidth products GBW for such devices are normally bctween 10 t o 30 GHz (Taguchi et 01.. 1988; Campbell et al., 1985). An improved technique for increasing thc bandwidth is to provide up to three InGaAsP transition layers as the grading layer, shown in Fig. 17a (Campbell et a / . , 1988; Hollenhorst, 1990). Such devices have shown gain-bandwidth products up to 70 GHz, and are thcrefore adequate for 2.5 Gb/s optical fiber communications. This type of grading is the most common form in separate absorption, grading, and multiplication (SAGM) lnP/LnGaAs APDs. Alternatively, a sophisticated approach (Capasso et al., 1984) is to grow a variable-period superlattice of InP and InGaAs, where the ratio of widths of thin layers of InP to InGaAs varies from very large at the InP side to very small at the InGaAs side, as demonstrated in Fig. 17b. In this way, a “pseudo-quaternary” compound is produced, whose effective bandgap is determined by the local ratio and is shown as the dashed line in Fig. 17b. 4. SAGCM IizP/InGaAs APL) The GBW of SAGM InP/lnGaAs APDs can be increased by reducing the effective multiplication width (Shiba et NI.,1988; Kuchibhotla and Campbell, 1991). The
CHARACTERIZATION AND MODELING OF SAGCM
93
electric field profile in the multiplication layer is triangular, with the maximum electric field near the p+-n junction. Most of the avalanche multiplication occurs around the location of maximum electric field. To reduce this width, the doping concentration in the InP multiplication layer should be increased. However, as discussed above, there is an upper limit of 2 x lOI7 cm-3 to the doping concentration, which is almost impossible to fabricate. In addition, the theoretically possible achievement of GBW (about 90 GHz with a reasonable concentration of 5 x 10" cm-3) is at the price of higher excess noise (Osaka and Mikawa, 1986; Kuchibhotla and Campbell, 1991) because of the higher maximum electric field in the multiplication layer. This results in higher kcf,since it is well known that the ionization rates for electrons and holes in InP converge rapidly at electric fields higher than 60 Vlpm (Capasso, 1985). In SAGM APDs, because of the strict requirements of the electric field at the InPlInGaAs heterointerface and in the absorption InGaAs layer, it can be shown that the product of the doping concentration and the width of the InP multiplication layer (charge requirement) must be within a very small range. Thus, an increase in the doping concentration must be accompanied by a reduction in the multiplication width, and the accuracy of the multiplication layer width becomes critical to satisfy this charge requirement. Therefore, the increase in the doping concentration results in a narrow multiplication width with a small tolerance. For example (Hollenhorst, 1990), the control of a multiplication width of 0.5 p m must be within 0.02 pm for the doping concentration of 6 x 10lhcm-3. The tolerance would be 0.01 wm if the doping concentration is 2 x lOI7 ern-%, an improbable task with today's fabrication technology for planar diffusion structure APDs. To circumvent these problems, a high-low AlInGdInGaAs was fabricated (Capasso et al., 1984), and a 8-doped SAGM InGaAslInP APD was proposed and analyzed (Webb ef al., 1988; Ito ef al., 1990; Kuchibhotla and Campbell,
Metal
p+ InP
I
n- InGaAs absorption n- InGaAsP grading
n- InP multiplication
I
n+ InP
I
Metal FI
94
C. 1.. E MA, M . J . DEEN. AND L. E. TAROF
Bandwidth Hole trapping, diffusion tail
TI
parasitic, carrier transit
FIGURE 19. Various contributions to the overall bandwidth of SAGCM InP/InGaAs APDs. Both gain and bandwidth arc i n log scale.
1991) as shown in Fig. 18. The primary advantages are that it decouples the doping concentration and width of the multiplication layer, and the flat electric field profile (because the multiplication layer is undoped) reduces the maximum elcctric field. It was predicted that GBW as high as 140 GHz could be achieved with a 0.2 p m thick multiplication layer, but with a concomitant increase in excess noise (Kuchibhotla and Campbell, 1991). The high-low planar InP/InGaAs structure devices (which are named SAGCM) have been fabricated (Tarof et al., 1990; Tarof, 1990; Tarof, 1991; Kuchibhotla etal., 1991) with record-setting GBW of 122 GHz with a 0.1 p n i wide multiplication layer (Tarof et al., 1993), and without severe excess noise penalty (Yu et al., 1994). The planar SAGCM I n P h G a A s APDs is discussed in detail in the next section. A properly designed SAG(C)M I n G a A s h P APD normally has bandwidth as a function of gain as shown in Fig. 19 (Tarof et al., 1993). At low gains ( 2 4 ) , thc overall bandwidth is limited by hole trapping due to an insufficient heterointcrface electric field, and by the diffusion tail due to incomplete depletion of the absorption layer. At medium gains (5-20), the bandwidth is limited by the various transit times (particularly the transit of the secondary electrons) and thc parasitic effects. At higher gains, the bandwidth is limited by the avalanche buildup time and GBW is a constant. 5 . Low-Noisearid Fast-Speed Heterojunction APD
The excess noise and bandwidth of APDs depend strongly on the ratio of impact ionization rates for electrons and holes. Therefore, thcrc is a great incentive to reduce this ratio since it can improve both noise and bandwidth performance characteristics simultaneously. The bulk 111-V compounds are not very favorable in this regard. Popular schemes have been proposed in the early 1980s ( i s . ,
CHARACTERIZATION AND MODELING OF SAGCM
0
Elecuon
0
Hole
95
\ (a)
(b)
Fr[iLiI<[-.20. (a) MQW and (h) staircac APDs energy band diagranis under biasing.
niultiquantum well (MQW)/superlattice (SL) (Chin et al., 1980), and staircase (Capasso et al., 1983) APDs), and both of them use of a few tens of thin layers with the thickness of each layer a few hundred A as the multiplication layer. The stcplike MQW energy band diagram is shown in Fig. 20a. When an electron enters the well, it abruptly gains an energy equal to the conduction band discontinuity. This effectively reduces the electron ionization threshold energy, which greatly enhances the electron ionization rate. When the electron enters the next barrier, thc opposite occurs. However, since the ionization rate in the well material is larger than the one in the barrier and the ionization rates depend exponentially on the threshold energy the average ionization rate for electrons is largely enhanced. The holes undergo a similar process but with less enhancement due to the smaller valence band discontinuity. Such devices have been demonstrated successfully in the AIGaAs/GaAs material (Capasso rt al., 1982; Kagawa et d., 1989). Unfortunatcly, the structure does not provide the same favorable enhancement when using InGaAs(P)/InP material systems (Osaka et al., 1986), and any potential noise improvement is impaired by large tunneling leakage current in the lnGaAs layers. For this reason, the more complex staircase scheme is devised as shown in Fig. 20b. The energy band structure is obtained by compositional grading. In this case, the impact ionization can occur at lower electric fields so that the tunneling current problem is eliminated. However, this structure presents substantial fabrication difficulty and has not been fabricated successfully. The only commercially fabricated MQW/SL APDs for 1.3 and 1.55 p m wavelength optical fiber communications is the InGaAs(P)/lnA 1As superlattice (Mohammed et NI., 1985). in which the electron ionization rate is enhanced by a factor of 20 over the hole ionization rate (Kagawa et al., 1989). Separate absorption, charge, and multiplication InGaAs/InAlAs superlattice multiplication APDs have been fabricated (Kagawa et d., 1990), and such devices have achieved gain-bandwidth product of 90 GHz and k , ~between 0.1 and 0.2 (about half of bulk InP) (Kagawa et d., 1991). However, the improvement is impaired by a large tunneling current in InGaAs. To eliminate this tunneling problem, a larger bandgap semiconductor is required. For example, InGaAsP/InAlAs superlattice
96
C. L. F. MA. M . J . DEEN. AND L. E. TAROF
-
-
APDs have been fabricated with GBW 110 GHz and kell 0.2. The superior advantage is best illustrated by a improved flip-chip structure with 17 GHz bandwidth (Kagawa ef al., 1993), and by the successful 10 GB/s optical receiver with -27.4 dBm sensitivity (NRZ) (Miyamo et al., 1994), using this type of APDs. 111. PLANARSAGCM InP/InGaAs APD
In this section, the detailed structure, fabrication, and calibration of the planar SAGCM InP/InGaAs APDs are described. A. Device Structure
The APDs investigated in this research were designed to operate at 2.5 Gb/s, and possibly 10 Gb/s reliably with reasonable cost. As has been demonstrated in the last section, both the SAGM and SAGCM APDs can operate at data rates up to 10 Gb/s with a gain of 10. However, the yield of the SAGM APDs is low because of the tight tolerance in the thickness of the multiplication layer. Therefore, the SAGCM structure is selected. Another consideration is that the devices should be mesa or planar. The mesa structure is simpler to design and fabricate, but its reliability is poor due to the requirement of passivation on crystal surfaces other than (100) surface. On the other hand, the planar structure is complicated to design and fabricate, but is of high reliability because the passivation is required on (100) surface only. Since reliability is crucial, planar structure is selected, similar to the structures of most commercial electronic devices. One of the challenging problems in designing planar diodes is premature edgc breakdown (Grove, 1967). The elcctric field at thc cdge is always higher than thc central rcgion duc to its small curvature. The standard solution is to usc a guard ring shown in Fig. 21 (Taguchi et al., 1988; Hollenhorst, 1990; Liu et d., 1992). The guard ring modifies the doping profile at the edge region from abrupt into graded, which has a lower maximum electric field. An additional benefit is that thc guard ring increases the curvature. which also results in a lower maximum clcctric field. Guard ring
n
CHARACTEIZIZATION AND MODELING OF SAGCM
97
However, the design and fabrication of a reliable (non-ion-implanted) guard ring constitutes a complex task. Because a technique for fabricating a linear graded impurity profile in ti-type InP is not available yet. double guard rings must be used. Furthermore, the diffusion process used in fabricating the guard rings may limit thc accurate control of the depth of the pi- region. Another solution is to enhance the electric field in the central active region of the p-n junction by selectively increasing the charge density under that region
SiO,
AR S O 2
I
I P + InP
x.i
,,‘ InP multiplication
xd
,,-InGaAs absorption tabsorp tun doped
1
,,- InP buffer
98
C. L. F. MA. M. J . DEEN, AND L. E. TAROF
(Webb et al., 1988). This approach is extended further (Tarof ef al., 1990) as shown in Fig. 22. Here, two lateral charge sheets, the full-charge sheet o;~~,,,,~ in the central active region and the partial charge sheet r ~ , , , . , - i ~ in ~ , ~the ~ periphery rcgion, are incorporated. This extra degree of freedom allows independent control of the electric fields in both the central active region and the periphery region. The full charge sheet ~ ~ ~ ~ is, iused , ) ~ to meet APD design requirements, and lhc partial charge sheet o ~ , ~ ~is , , used ,~,~ to~reach ~ a balance between the electric ficld near the pt region edge and the electric tield in the InGaAs absorption layer i n the periphery region. Although the periphery charge sheet somewhat degrades the edge breakdown in the device, it is required to keep the electric field low in the InGaAs absorption layer and, hence, to limit tunneling currents. B. Fabrication
The fabrication and calibration of the planar SAGCM InPAnGaAs APDs with partial charge sheet in device periphery are described in detail in (Tarof, 1993). The metal-organic chemical vapor deposition (MOCVD) technique including an overgrowth is used to grow the wafers. The maximum lattice mismatching among all the laycrs is about 0.07%. The upper limit of the unintentional background doping concentration is 1.0 x 10" cnip3 in both InP and InGaAs. An unintentionally doped InP buffer layer with a thickness of 0.5 1.0 p m is grown on a 2"nt-InP substrate. An unintentionally doped InGaAs absorption layer with thickness 2.5 3.3 p m follows. At thc very end of it, a "boost" layer o f thickness about 0.02 p m is formed by doping with Si (2.0 x 10'' cm- j). Thc boost layer might increase the bandwidth since it helps to prevent hole trapping due to thc higher heterointerface electric field. The grading layer consists of nine quaternary layers, compositionally graded from InGaAs to InP with the total thickness of -0.09 pm. The dopant is Si and the doping concentration is -1.0 x l O ' " ~ n i - . ~ . The InP charge layer is grown onto the wafer with thickness about 0.2 p m , and the integrated areal charge density ~ ~ ~ ~ (including , i , , ~ the areal charge density in the boost and grading layers) is -3.0 x 10" cm-2 with Si as the dopant. The top 0.025 p m of the charge layer (sacrificial layer) as shown in Fig. 23 is undoped to avoid the following two problems. First, there is always a thin Si layer incorporated at the top of the charge layer before the overgrowth, and this causes substantial offset of the integrated areal charge density. Second, the cleaning procedure before the overgrowth removes a thin (less than 0.025 p m ) uncontrollable thickness of the charge layer. The mesas are patterned by standard photolithographic techniques with wavelengths of 4050 or 2900 Wand a dark field mask set (ZOI). The etching is performed with either wet chemical ctching (WCE) or reactive ion ctching (RIE). To facilitatc bctter control of the etching, more complicated charge layer structures arc fabricated in the first growth. For RIE, a low-high doping profile is adopted as
-
-
99
CHARACTERIZATION A N D MODELING OF SAGCM
n sacrifice
I
high doping
I I n InP low doping
nlnP
I
n InP
FI(,LJI
shown in Fig. 2321. In this way, the uncertainty and nonuniformity of RIE have a minimal impact on o.,utlvc. For WCE, an etch stop (InGaAs) within the charge layer is incorporated as shown in Fig. 23b. The thickness of the etch stop is chosen as 0.005 p m to balance the requirement of stopping etching and minimizing the tunneling current due to the narrowcr bandgap of the etch stop layer. Unless otherwise spccificd, WCE with an etch stop is selected. The wafer is then put back into the MOCVD reactor and 2.5 pm of unintentionally doped InP cap layer is grown. A dielectric mask consisting of either of SiOl or SiN, is then deposited. After the overgrowth, the mesa is still visible and the alignment for p+ diffusion mask can be done without further complications. The Zn diffusion is monitored by a reference calibration samplc and the depth x , can be controlled within 0.1 p m . To precisely control the multiplication layer thickness -I(,/ the InP cap layer thickness is measured before the diffusion, and this is described in the next section. An antireflection (AR) coating of SiOl or SIN, is applied to the wafer with a thickness about a quarter of the wavelength (- quarter of 1.3 or 1.55 p m ) in the material. The p-metal contact window is etched into the dielectric coating by RIE. The p-metal, which consists of 0.04 pm Cr and 0.25 p m Au, is evaporated using an e-beam evaporator. A standard lift-off'technique using a bilayer of polymethyl-methacrylate (PMMA) and photoresist is used to pattern the p-metal. The wafer is quartered and thinned down to a 150-200 p m thickness. Cr/Au n-metal is evaporated onto the polished substrate. The finished wafer is diced into 500 x 500 pm' dies and the dies are mounted onto 2 x 2 x 4 mm3 ceramic carriers. The n-contact is formed by thermally compressing Au/Sn solder between the die and an Au pad on the ceramic carrier. The p contact is formed with I .0 or 1.4 mil diameter bonding wire of length -2 mm. C. Calibration
Thc Hall measurement technique is used to calibrate the doping concentrations. It uses a van der Pauw measurement sample, about 4 x 4 mni' in size that is cleaved
100
C I F MA M J I1PI-N A N D 1 F IAKOP
1.1
1.o
0.9
0.8
' 0
I
I
I
I
10
20
30
40
50
Distance from leading edge (mm) FIc;LII
MOCVD gas Ilow.
from a calibration wafer. The Hall measurements determine that thc upper limit of the unintentional background doping concentration is 1.0 x 10" cm-j. The intentional doping concentration is calibrated on the whole calibration wafers with layer thickness about 2 3 p m . It is found that the doping concentration is reduced by about 20% from the leading edge to the trailing edge (major flat) of the wafer, as shown in Fig. 24. This is due to the gas flowing along that direction relative to the wafer in the MOCVD reactor. Along the direction perpendicular to the gas flow direction, the doping concentration variation is minimal. Reflectance spectroscopy has been introduced for measuring the epitaxial layer thicknesses, such as the InP charge layer before the overgrowth and the InP cap layer after the overgrowth at both central and periphery regions. Reflectance is measurcd as a function of wavelength from 0.9 to 1.5 pm. Since in this spectral range, the InP layers are transparent while the InGaAs layers are opaque, the measurement is similar to that of a single film thickness (Tarof rt a/., 1989). It is generally found that the variation of the InP cap layer thickness after the overgrowth over a wafer is nearly radial, and it is thicker at the center and about 0.1 p m thinner at 8 mm from the edge along the direction of the gas flow. For measuring the InGaAs layer thicknesses, the technique of measuring the epitaxial layer thickness of A-B-A double heterostructure using reflectance spectroscopy is employed (Tarof et d., 1990). At present, the thickness measurement is only applied to wafers since the spot size of the spectrometer is 6 mm in diameter. Secondary ion mass spectroscopy (SIMS) is used to determine the thickness of the multiplication layer and the doping profiles of the p+, charge, and grading
-
101
CHARACTERIZATION AND MODELING OF SAGCM
Ga
7
1019
10’8 1017
10’6 1015
0
1
2
3
4
layers, as shown in Fig. 25. SIMS relies on removal of material from a solid by sputtering and on analysis of the sputtered ionized species. The depth axis of the SIMS profile is converted from the elapsed time by a subsequent stylus profilonietry measurement. It is found that the doping profile of the p+-n junction obeys a “modified erfc” function, which is quite close to an abrupt junction. The fall-off of the Zn concentration is a good indicator of the diffusion depth, which is found to be uniform across wafers within the SIMS uncertainty (0.1 pni). The abrupt increase of Si concentration provides an unambiguous location of the top of the charge layer. The separation between the Zn and Si profile edges is the thickness of the multiplication layer.
D. Devices Investigated Thc planar SAGCM I n P h G a A s APDs with partial charge sheet in device periphery investigated in this paper were designed and fabricated by Bell Northern Research, Ottawa, Canada. The devices investigated in the following sections are from two wafers-P555 and P623. The nominal fabrication parameters are listed in the Table 2. The various thicknesses are already defined in Fig. 22. ng,,lcling, and (Tp,.,,l)hc.I-y are the integrated areal charge density in the charge, the grading, and the boost layers, respectively, and n;,,,ive= nctl;,,pe D ~ ( T [, , , C ; ~~ ,~~ in the active central region. cT,lcr,ptlcry is the corresponding value in the periphery region. Unless otherwise specified, all the devices investigated here have GBW product in 50 100 GHz range. The highest is 122 GHz, a record-setting value (Tarof et al., 1993).
+
-
+
~
~
102
C. I.. F. MA, M J. DEEN, AND L. E. TAROF TABLE 2 THE NOMIN,\L DEVICE FABRICATION PARAMETERS t O R WAFERS 1'555 AND
P623 THE THICKNESS IS IN fin1 AN11 THE INrEGRATED ARBAI. C H A R G E D E N S I T Y 10"
IS I N
PSSS
30
Clll
',
2.2
0.3-0.4
0.17
fu~Ic,pcd
0ch;irgc
flgrxluig
'TlnCdA\
fldclivu
3.S-3.8
l,9-2,6 2.2-2.9
0.09
0.4
2.4-3.1 2.7-3.4
S
0.11
0.09
0.02
P623 fahcurp
PSSS 2,s-2.8 Ph23
3.0-3.3
~,""l'hCr)
I .?-I .'l I .3-1 .s
LED Power
LED
@ 'I
1.3 pm
Fibre
FIcimE
26. Photocurrent nieasurcment setup.
IV. CRITICAL DEVICEPARAMETERS EXTRACTION The multiplication layer thickness x,/and the integrated areal charge density o,,,,\.~ are crucial parameters for SAGCM APDs to meet design requirements. For our APDs, both device parameters vary significantly over a wafer when the devices are fabricated with current MOCVD technology. Even with the inherent advantages of SAGCM InP/InGaAs APDs, it is still very important to control the variation of xc/ when it is less than 0.5 p m (note that this is crucial for SAGM APDs). In our APDs, x,/ is determincd by two processes: growth of the epitaxial InP layer of thickness xj +x,/, and subsequent p + diffusion of depth x,. Because of the nonuniformity of the InP cap layer thickness and the uncertainty of the diffusion depth, it is difficult to control x,)to better than 0.1 p m across a wafer for devices intended for bandwidths greater than 5 GHz. Due to the gas flow direction in the MOCVD reactor and the uncertainty in mesa etching, C T ~ ~ ,can , , ~ vary more than 20% across a wafer. Typical direct measurements of x,/ and o ~ ~ ,such , , ~ ,as with secondary ion mass spectroscopy (SIMS) and Hall analysis. have limitations in that they are destructive andor they must be performed on separate calibration wafers. Another common tcchnique, capacitance-voltage (C-V) measurement, is not suitable for these de-
CHARACTERIZATION AND MODELING OF SAGCM
103
vices because the capacitances are too small due to the small device size, resulting in a very noisy signal. In addition, the mesa structure makes the extraction of x,f and a,,,i,, from C-V measurements a fairly complicated process, if not entirely impossible. Larger-area devices were not readily available during the period of this investigation. Therefore, a new nondestructive technique to determine device parameters at device level is critical for ongoing and further characterizing and modeling of our SAGCM InP/lnGaAs APDs. In addition, determination of device parameters at the wafer level is very helpful to monitor wafer fabrication, and to reduce unnecessary fabrication and processing steps to reduce device cost. In this section, a simple, innovative, and nondestructive method for determining both xd and a,,,,,on , a wafer and on an individual APD is presented (Ma et al., 1995; Ma et a/., 1994). The method relies on punchthrough voltage V,,,,,;, and breakdown voltage Vh, easily obtained from photocurrent-voltage measurements. First, the experimental setup is briefly described, and then the detailed device parameters extraction technique is presented. Finally, this technique is applied to two wafers, and all possible errors, including both systematic and random ones, are discussed.
A. Experimental Setup The light is generated with a pigtail 1.3 pin LED, and is transmitted over a monomode optical fiber to the APD. The incident optical power is controlled hy an HP optical attenuator. The end of the optical fiber is cleaved with a core diameter of about 10 p m , and it is pointed to thc APD from a distance less than 50 p m away the surface. The incident light is directed at the central active region of the APD by maximizing the photocurrent at a bias voltage 0.9. Vhr while moving the end of the optical fiber along the APD surface. The DC photocurrent is measured with a Kcithley 617 Electrometer with the APD reverse-biased. The sensitivity is about 100 nA due to environmental noise. The normally employed phase-sensitive measurement of photocurrent is not necessary since both dark current and its associated noise are much smaller than the photocurrent. All instruments are controlled with a personal computer through an lEEE 488 bus. B. Extmctiori Technique
All properly designed APDs including SAGCM InP/InGaAs APDs show impact ionization breakdown at some reverse bias voltage v h , . However, as shown in Fig. 27, an additional characteristic voltage in our planar SAGCM InP/lnGaAs APDq, which can be unambiguously determined from photocurrcnt-voltage characteristics, is the punchthrough voltage V,,,,,,,. The punchthrough voltagc is that at which the electric field starts to penetrate the InGaAs layers in the central active re-
I04
C. L. F. M A , M. J . DEEN, AND 1,. E. TAROF
10”
a
n
102
10-6
I
Y U
c
E
10-7
L
3 0
c n
0 0
c
c
n
lo-’ 100
10-9
0
10
20
30
40
50
60
Bias Voltage (V) FKXJRE27. A typical (APD32. wafer P623) phoiocurrent vcrsus bias voltage and its derived photogain.
gion (Tarof eta!., 1990). The corresponding punchthrough voltage in the periphery region is Van,,. The detailed photo I-V characteristic in Fig. 27 is now explained. When V < Van,,, the InGaAs absorption layers in both the active and the periphery regions are not depleted. The photocurrent is very small since the photogenerated holes in the absorption region are blocked by the energy barrier at the InPDnGaAs heterointerface, and eventually, the photogencrated electrons and holes recombine. Whcn Van,, < V < V,,,,,,, the periphery region of the InGaAs absorption layer is depleted whereas the central active region is not. The energy barrier at the I n P h G a A s heterointerface in the central active region still blocks the photogenerated holes. However, the energy barrier in the periphery region is diminishcd. Because of the very high quality of the InGaAs semiconductor, the hole lifetime is about 18.5 ps and the diffusion length is about 140 p m (Gallant and Zemel, 1988). Therefore, the long-livcd photogenerated electron-hole pairs in the central active region can diffuse to the depleted periphery region and are collected by the terminals. The photocurrent is constant over this range of voltage, and the gain is unity since the gain is due to impact ionization in the periphery region, which is approximately unity at this range of low-bias voltages. This is one of the unique features of planar SAGCM APDs with partial charge shcct i n device periphery as the gain is calibrated automatically in this bias range. When V > V,,,,,;,, the energy barrier in the active region is diminished, and the photogeneratcd holes are passing through the central active multiplication region.
CHARACTERIZATION AND MODELING OF SAGCM
105
Therefore, the photocurrent and its derivative with respect to V starts to increase monotonically due to the increased impact ionization multiplication until the bias voltage reaches the avalanche breakdown voltagc v h r . The existence of V,,,,,, is common in both planar and mesa SA(G)M InP/InGaAs APDs (Susa et al., 1981; Kobayashi e f al., 1984; Taguchi e f al., 1986), but the existence of the unity gain bias voltage region is only observed in our high-quality planar SAGCM APDs with partial charge sheet in device periphery or in high-quality planar SAGM InP/lnGaAs APDs with preferential lateral extended guard (PLEG) ring (Taguchi c ~al., t 1986). Also, the cxislencc of the unity gain in our devices makes it possible lo derive directly the photogain. as shown in Fig. 27. With these two experimental characteristic voltages, V,,,,,;, and VI,,, two device parameters may be extracted if all other device parameters and the ionization rates are known. I t is clear from the above discussions that it is most beneficial to extract . Y ~ / and the two most critical and uncertain device parameters. To relate the breakdown voltage v h r , the punchthrough voltage V1llc,;l,and the breakdown electric field Fh,. in the multiplication layer to the device parameters, the following assumptions and simplifications are made.
I . The p ~ + - InP n junction is abrupt. 2 . The voltage drop at the InP/InGaAs hcterointerface ( 0 . 1 4 . 4 V) is ignored. 3 . The doping concentrations in the multiplication, charge, and grading layers are uniform. 4. The thickness of the thin highly doped lnGaAs layer is ignored. 5. The relative dielectric constant of'the InGaAsP of the grading layer is assumed to be equal to the relative dielectric constant of InP. 6. The InP buffer layer is treated as part of the absorption layer. 7. The multiplication layer background doping concentration is assumed to be zero. 8. The doping concentration in the absorption layer is uniform. 9. Thc absorption layer is completely depleted at breakdown. From Poisson's equation and appropriate boundaiy conditions as shown in Fig. 28, from Eqs. 80 and 82 in Appendix A, it can be shown that
106
C. L. F. MA, M. J . DEEN. AND L. E. TAROF
Xd
'1-'d
-
'InP
x2
x1
x2-x
1
=
1
grading
x3-x2
=
t
u ndop e d
FIG~JRF2X. The electric lieldj in SAGCM InP/lnGaAs APD
In the above equations, E I and ~2 are the relative dielectric constants for InP and InGaAs, and are 12.3 and 12.9, respectively. E ( ) is the dielectric constant of vacuum and q is the electron charge. Nc; and NlI,(ja~\ are the respective doping concentrations in the grading and highly doped thin InGaAs layers. N D is the unintentional background doping concentration in the multiplication and absorption layers. c ~ ~ ~ c , ~rlnGaA~NII,GaA, A~(= = 0.4 x 10" cm-') is the integrated areal charge density ofthe thin highly doped InGaAs layer. ( ~ , isl the ~ integrated ~ ~ ~ ~ areal ~ charge density o f t h e charge layer, and aLwlive = (T,tIargc N G t g r a d l n g ( ~ l ~ ~ ~vh ~ ,, is 4 ,the . zero bias built-in voltage across the APD structure and it is approximately 1 V. Two more simplifications are assumed for breakdown condition.
+
+
10. Impact ionization in the InGaAs absorption layer is ignored. 1 1. Impact ionizations in the charge and the grading layers are ignorcd. The breakdown condition can be found from
which can be obtained from the more general breakdown condition in Eq. (15) when the electric field in the multiplication layer is constant. a1 and P I are impact ionization rates for electrons and holes in InP, respectively. There are a few reported experimental values of a1 and BI in the literature, and the particular set of values
CHARACTERIZATION AND MODELING OF SAGCM
107
used results in differences in the extracted x,!and (rac,ive.This point is discussed in detail later. For now, the most commonly cited values (Osaka and Mikawa, 1986) for a1 and are used, which are
a l ( F ) = 555 .exp(-310/F),
BI(F)= 198 .exp(-229/F),
(38)
with the electric field F in Vlpm, and the ionization rates in p m - ' . The three unknown variables F h r , (rchargc. and x,/ can be found by iteratively solving Eqs. (35), (361, and (37), using values of (YI and from Eq. (38), and with experimental V,,,,,, and V,, as sole input parameters. Two more characteristic voltages are useful to calculate: the voltages V,:,,,, and VLlcpleled at which the electric field starts and finishes depleting the absorption layer (not including thc thin highly doped InGaAs layer), respectively. From Eqs. (91) and (86) in Appendix A, it can be shown that
Another two interesting values are the heterointerface electric field Fhclcr and F,~,,,, (Fig. 28), the respective electric fields at the boundary of the thin highly doped InGaAs layer and the absorption layer, which are given by (Eq. 81 in Appendix A).
C. Kesults trnd Discussions
The device parameter extraction technique is tested on P555 and P623 wafers. The experimental values of V,,,,,, and Vb, from the leading edge to the trailing edge of the wafers are displaycd in Fig. 29. The values of the dcvice parameters used for modeling are listed in Table 2. Thc extracted values of x,/ and cT,ic,l,,c are shown in Fig. 30a for wafer P55.5 and 30(b) for P623. The extracted rcsults agree quite well with the independent measurements (SIMS, reflectance spectroscopy, and Hall analysis), for which the results have been presented in (Tarof, 1993; Tarof et d., 1994) and in Section 111. These independent measurements establish that for both wafers, x,/ is greatest in the center of the wafer with a value about 0.4 pm, and falls off toward the edges to a value of -0.3 p m . The trend is consistent with what has been calculated, but the calculated values are systematically 0.1-0.15 p m larger than the independently measured x , ~
108
C. I-. P. M A , M. J. DEEN, A N D I-. E TAKOI-
10 .
0
1
0
2
0
3
0
4
0
1
Distance from leading edpe (mm)
0
1
0
0
2
0
3
0
4
o
M
Distance from leading edge (mm)
(a)
(h)
P[(;L1tw29. The cxpcriniental valucs 0 1 V,,,,,,, and Vl:, . and thc calculated V,:,,,,, m d Vtl+,L,c~a c r o u ( a ) PSSS atid (h) PO23 wafers.
I
o'a n
0.a '
E1
W
x"
0.40
'
I
values. However, the error of these independent measurements is about 0.1 p n i and the error of this extraction technique is about 0.05 p m (shown later). Therefore, it can be concluded that the extracted and experimentally measured x,! values agree within the errors. The small differences in x , between ~ wafer P555 and P623 reflects (he fabrication uncertainty of -0.1 pm. The small differences between wafer P555 and P623 reflect the fabrication uncertainty (0.I p m ) stemming from p i - diffusion and the InP cap layer MOCVD growth.
CHARACTERIZATION AND MODELING OF SAGCM
109
u ~ ~ ,for , , ,wafer ~ P623 is designed to be (2.7-3.4) x 10l2cm-2 (Table 2), which is in good agreement with the extracted values. Wafer P555 is designed to have oac,ive (0.2 - 0.3) x 10’’ cm-2 smaller than the wafer P623, which agrees with the cxtracted values for the APDs near the leading edge. The variation of the extracted cr21ctivc over wafer in Fig. 30 is -29% for wafer P555 and -22% for wafer P623. The Hall measurements for calibrating this particular MOCVD reactor, which was used to grow the two wafers, established that aaclive increases by 15% to 20% from the location 8 mm from the leading edge to the location 8 mm from the trailing edge (Fig. 24). This confirms that this extraction technique is successful in determining both the absolute value of u;,,,,~,and its variation over wafer for wafer P623. However, the extracted variation of 29% for wafer P555 is larger than the expected 20% variation. One possible reason for this discrepancy is that this fabricated wafer is slightly different from its design due to the very delicate fabrication processes. The calculated V,:,,,;, and Vcicpletcdare shown in Fig. 29, and the calculated F h r , Fl:c,,.r,and F h c r c r in Fig. 31. The calculated Vclci,lctcd is always lower than v h r , therefore, the assumption that the absorption layer at breakdown is completely depleted is satisfied. The calculated V,hCSais systematically about 4 V higher than V,,,,,,, which is not surprising since is assumed unchanged, and xo changes very little. The calculated Fhr is between 52 to 57 V/pm. This small variation of F h r is consistent with the small variation of the extracted xd. since F b r depends only on x,~.Also FL,,,, is about 5.6 Vlpm lower than filclrr. The extraction for individual APDs cannot be independently examined since the uncertainty of the independent measurements for individual APDs using SIMS is
110
C. L. F. MA. M. J. DEEN. AND L. E. TAROF
too large to reach meaningful conclusions. The validity of this new extraction technique is ultimately decided by future modeling of other characteristics. The extraction technique describe earlier (which will be called standard technique) relies on some assumptions and simplifications. Some of them are fairly obvious (items 1-6) and no further discussions are warranted. Some of the assumptions represent the reality of the devices (items 8, 9). The rest of the assumptions and simplifications (items 7, 10, I I ) are examined in the next section, and the resulting variations are treated as the errors of this extraction technique.
D. Error Analysis One aspect of error analysis is to determine the absolute uncertainty of the extracted values, which is very important for further investigations at the individual device level. Another aspect is that the errors should be small so that the general trends across the wafer are preserved. The errors can be generally classified into two types: systematic and random. The systematic errors cause the extracted values to change systematically in a predicted direction, whereas the random errors cause the extracted values to change in an unpredictable direction. The general trend across the wafer of the extracted values is generally not affected by the systematic errors, whereas the absolute uncertainties of the extracted values depend on both the systematic and random errors. The systematic errors are contributed by ignoring the ionizations in the absorption layer, and the ionization in the charge and grading layers, the assumed zero doping concentration in the multiplication layer, and the selection of InP ionization rates. The random errors are contributed by the experimental errors of V,,,,,,,and v h , , and the fabrication and calibration errors of the device parameters listed in Table 2. 1. Ionization in the Absorption h y e r
Ionization in the InGaAs absorption layer has long been suspected to be responsible for some unexplained experimental results, such as a lower than expected gainbandwidth product (GBW) (Hollenhorst, 1990; Tarof et al., 1993; Tarof et al., 1994), and also a lower than expected excess noise (Osaka and Mikawa, 1986; Yu et al., 1994). Ionization in the InGaAs absorption layer will cause systematic errors to the extracted x,/and a,,,,,,, and in principle this effect should be included. However, in many cases, it can be ignored when extracting xd and adcllve. In this section, ionization in the InGaAs absorption layer is examined in detail so as to determine ( I ) the error due to the neglect of this effect; and (2) under what conditions this ionization can be ignored. To include the ionization in the absorption layer, the right-hand side of Eq. (37)
CHARACTERIZATION AND MODELING OF SAGCM
111
should be modified
Fend is the electric field at the end of the absorption layer at breakdown, and
rfF/dx is the spatial gradient of the electric field in the absorption layer, and it is ( q N L ) / E Z E O ) here. a2 and 82 in pm-' are the ionization rates for electrons and holes in InGaAs with electric field F in V/pm, and their experimental values have been reported by Pearsall (Pearsall, 1980) as az(F) = 5130. exp(-195/F),
P2(F) = 7300. exp(-220/F),
(45)
and by Osaka (Osaka and Mikawa, 1986) as a z ( F ) = 227. exp(-l13/F),
B z ( F ) = 395 exp(-145/F),
(46)
Their values are plotted with F between 15 and 25 V/pm in Fig. 32. A cautionary note is that for both formulas the valid F range is between 20 to 25 Vlpm. Also
Inverse electric lleld 1/F (wN) (:I)
Inverse electrlc fleld 1/F (wN)
(h)
32. The cxperiiiiental values for the ionintion rates i n InGaAs lor (a) eleclrons. and (b) holes. The solid lines are from Pearsall. (lie clashed lines from Osaka. FIGURE
C. .L F. MA, M . J. DEEN. AND I-. E. TAROF
112
t El
L
... .
o
i
o
m
5
~
0
(
0
Distance from leading edge (mm)
i
o
0
1
0
2
0
3
0
4
0
5
Distance from leading edge (mm)
(a) (b) lor wafer P555 with the ionimtion in the 33. The extracted values of (a) .vL, and (b) absorption layer. The solid lines are extractcd with the standard technique. The dotted lines with solid squares are calculated with Pearsall’s InGaAs ionization rates. and with open squares with Osaka’s InGaAs ioniialion rates. FIGURE
note that unlike InP, the ionization rate for electrons is larger than the ionization rate for holes in InGaAs. The extracted x,/and oactive for wafer P555 with both sets of the ionization rates are shown in Fig. 33. The inclusion of the ionization in the absorption layer clearly makes some difference in the extracted values when the breakdown voltages are high, and makes no difference when the breakdown voltages are low. This is exactly as expected, since at lower breakdown voltages, o‘acrivc are higher, which results in lower Fie,,, (lower than 12 Vlpm). Therefore, the ionization in the absorption layer is too small, as seen from Fig. 32. When breakdown voltages are high, crLlcllbc are lower, which results in higher Fk,,,,(higher than 15 Vlpm). Therefore, the ionization in the absorption layer is significant. However, the extracted x , ~and oacl,,, from the calculation with Osaka’s InGaAs ionization rates are contradicted by other independent measurements. The trend of x,/and the 52% variation of oL,,.,,,,,over the wafer cannot be explained. It seems that Osaka’s cr? and 82 arc not correct when F is below 20 Vlpm. However, a? and /?r are determined to be between 20 to 25 Vlpm, and the only conclusion from this investigation is that extrapolation of Osaka’s a2 and 82 to electric fields below 20 Vlpm is not accurate. Therefore, only the results calculated with Pearsall’s a2 and 8 2 will be discussed further. From the extraction for wafer P555, it seems that when Fi,,,,.is higher than 17 Vlpm, the ionization in the absorption layer is significant, and when FL,,,, is lower than 15 Vlpm, the ionization is negligible. For the purpose of a rule-of-
~
CHARACT1:RIZATION AND MODELING OF SAGCM
t I
o.60
0.30
L P
I13
10
I
mtl C T , , ~ , ,which ,~. are overlapping with the vulucs FlGlJlct 34. The extracted vtiIuc\ 01' (a) cxtractcd l'roiii the s1and;it-d technique and. ( h ) Fl~c,,c,, iiiicl Fc,,,l for walcr P623. including the ioiiiiiitioii i n the ahsorption layer.
thumb estimation, F,:,,,, of 17 and 15 Vlpm correspond to experimental Vh, of 90 and 80 V, if xc/is 0.4-0.5 pm. It also reveals that this inclusion causes an increase in ,q/ and a decrease in o-~,,~,,,, when its contribution is important. The inclusion of the ionization in the absorption layer reduces the necessity of the ionizaiton contribution from the multiplication layer, which causes a rcduction of F,,,.. However, this reduction of fi,r will reduce the electric fields in the absorption layer, which will result in a reduction of the ionization contribution from the absorption layer. To compensate for the recduction of thc electric fields in the absorption layer, o - ~ , is~ ~reduced. ~ ~ , ~ The increase of x,~.which has a much smaller cffect on the ionization contribution from the multiplication layer than from the reduction of F h r , is for maintaining V,,,,,;,. Wafer P623 is also extracted as shown in Fig. 34a. The results are almost identical to the values extracted with the standard technique, since cl;lcr,\.eis higher for this wafcr than wafer P5.55. This higher ~ , ~ c lresults l ~ c in the values of F,~,,,,. lower than 15 Vlpm, as can be seen from Fig. 3413.
2. Ioni:ritions in the Charge miid Glzrdirig Loycrs 111 principle, the ionizations in both the charge and grading layers ought to be considered together. In this section, it will be demonstrated that the inclusion of the ionization in the charge layer will cause acceptable systematic errors in the extracted x,/ and crdc,iVc. The systematic errors caused by the inclusion of the ionization in the grading laycr arc even smaller. To include the ionization in the charge layer, the right-hand sidc of Eq. (37)
114
C. L. F. MA, M. J. DEEN, AND L. E. TAROF
should be modified to
F2 is the electric field at the boundary between the charge and grading layers, and
d F f d x is the spatialgradientofthe electricfieldinthe chargelayer, andis (qccharge/ E ~ E O ~ I , here. ,~)
The inclusion of the ionization in the charge layer causes a systematic increase of 0.011-0.024 ,urn in the extractedxd, and a systematicdecrease of (0.05-0.1) x 10l2cm-* in the exracted craactive, as shown in Fig. 35 for both wafers. This phenomenon is quite similar to the inclusion of the ionization in the absorption layer as discussed earlier, except that the effect of the ionization in the charge layer will be important for all the values of craCtive,since the electric field in the charge layer is not a strong function of aactive. It shouldbe noted that even if the ionization in the charge layer canbe treated as a systematicerror in the deviceparameters extraction,
Distance from leading edge (mm)
(4
Distance from leading edge (mm) (b)
FIGURE35. The extracted values of xd and tractive (the dotted lines with open symbols) with the ionization in the charge layer for wafers (a) P555 and (b) P623. The solid lines with closed symbols are extracted from the standard technique.
CHARACTERIZATION AND MODELING OF SAGCM
115
it does not necessarily mean that the effect can be treated as a sedcond-order effect in modeling other characteristics.
3. Non-Zero Doping Coiicentrrrtion in the Multiplication Layer Principally, the electric field in the multiplication layer is not constant due its nonzero background doping concentration. It will be demonstrated that this effect will cause negligible systematic errors in the extracted x,/and (T:,,~~~~. To include this effect, Eqs. (35) and (36) should be modified to (Appendix A):
The breakdown condition becomes
where d F / d x is the spatial gradient of the electric field in the multiplication layer, and it is ( qN / ) / E ~ here. E ( ~ )Eqs. (41) and (42) should be modified to
The inclusion of the nonzero doping concentration in the multiplication layer causes a negligible systematic increase of less than 0.0006 p m in the extracted x ~ ,
116
C L I- MA. M J DbEN. A N D I. b TAKOF
1)istaiire from leading edgc (inin)
Distance from leading edge (mni)
(hi
(a)
FIGLIRE 36. The cx[racLcd values of .v,, and ~ l ~ , (domd ~ ~ , , line\ ~ ~ with open rymbols) hflilli Ihc noniero doping in the multiplication layer lor wulci-s ( a ) PSSS and ( b ) P623. The d i d liiics with cIo\cd symbols arc extracted lroin the slandard bxhniquc.
and a decrease of less than 0.018 x 10” cin-’ in the extracted Fig. 36 for both wafers.
mac,ibe, as
shown in
4. InP ioriization Rates The selection of the InP ionization rates is of potential importance for the absolute accuracy ofthe extracted x,/and c ~ ~ , ~ ,Thcrc ~ \ ~ . are substantial diffcrcnccs in reported cxperimental values a1 and PI from Osaka (Osaka and Mikawa, 1986) used so far and othcrs, such as from Uinebu (Umebu et ul., 1980), (YI
( F ’ ) = 736. exp(-345/F),
PI( F ) = 204. cxp(-242/F),
(54)
from Cook (Cook et nl., 1982) for F between 36 and 56 V l ~ m ,
a l ( F ) = 293 .cxp(-264/F),
P I ( F ) = 162,exp(-211/F),
(55)
B 1 ( F ) = 321 .exp(-256/F).
(56)
and from Armicnto (Rolland et al., 1992) a l ( F ) = 555.cxp(-310/F),
Thcy are plotted with F between 50 and 60 Vlpm in Fig. 37 in order to both imprcss upon the reader the differences and to guide the reader through the discussion that follows. Note that Armiento’s values, a ~arc , idcntifcal to Osaka’s and Armiento’s values, P I , are quite similar to Osaka’s. The extracted x , ~and ( T ~ , for ~ ~ both ~ \ ~ wafers with Umebu’s and Cook’s InP ionization rates are plotted in Fig. 38. Note that all the extracted Fhl- are within the valid F range of the corresponding set of the InP ionization rates. Sincc Armiento’s values of a1 and arc quite similar to Osaka’s, there is little difference
117
CHARACTERIZATION AND MOIIE1,ING OF SAGCM
&
r
.
In
0.016
~
0.017
Osaka's
.
Cook's
0.019
0.01s
o.o?o
0.016
0.017
0.018
0.020
0.019
Inverse electric field 1IF ( p W )
Inverse electric field 1/F (pm/V)
(a)
(b) FIGLIKE 37. The experimental values for the ionization rates in InP for (a) electrons, and (b) holes froin the literatui-e.
4 0
I
d b
o t o m l o r o l l o
Distance from leading edge (inin) (21)
o
~
o
m
a
a
m
Distance from leading edge (mm) (b)
F I ( ~ u 38. K ~ The extracted values o h < / and (T.,~,,,~ l o r wafers (a) PSSS and (b) Ph23 with Ihe InP ioniiation rates lroni Umcbu (the dotted lines with open circles) iiiid Cooh (the dotted lines with opcn sc~ii;ires).Thc solid lines with closed symbols are extracted from thc rtandard technique.
if Armiento's values of (YI and PI are used. Compared to the standard technique, usign Umebu's values of a1 and PI causes a typical decrease of 0.06 p m of the extracted x,! and 0.3 x 10" cm-' increase of the extracted a~,,,,,, while , using Cook's values of ( Y I and PI causes a typical increase of 0.05 p m and a decreasc of 0.2 x 10I2cm-?, rcspectively. There arc a few reasons to discard both the extractions using Umebu's and Cook's values of a ]and P I . One of the reasons is that the two calculations result in exactly opposite extracted values of x,, and cr,,,,,,,
118
C. L. F. MA, M. J . DEEN, AND L. E. TAROF
TABLE 3 THEVALUES OF T H E DEVICE PARAMETERS U S E D FOR MODELIN(;.
THET H I C K N t S S
IS IN pm, THE I)OPING CONCENTRATION IS IN
clnP3, ANL) I H F . INTEGRATE11 AREALCIIARGE DENSITY IS IN 10'2 cl11-2.
P5.5.5 P623
0.17
0.09
10
0.4
3.8
0.7
TABLE 4 T H E THICKNESS IS IN p111. THE THERANDOMERRORSO F THE EXTRACTED .x,/ AND DOPING CONCENTKAI'ION IS IN 10" Gin-', AND THE INTk;(iRATtD AREAL. CHARGE DENSITY IS IN IO" c l l l ~ 2 .
"The doping in the grading layer is not independently monitored
and one of the two must be wrong. Osaka's values of a1 and P I are more up-to-date and most frequently used in the recent literature. Finally, the extracted values are most important in the sense of self-consistency, and the selection of different InP ionization rates may not be very critical in modeling other characteristics of the APDs. Therefore, Osaka's InP ionization rates are used for the calculations in this investigation. 5 . Raridorn Errors
The random errors are contributed by the experimental errors of V,,,,,, and Vb,, and the fabrication and the calibration errors of the device parameters used in the extraction, such as tlIlp, tgratling, f u l l d o l , c ~ , N G , N L ) ,and c ~ ~ G ~that A ~are , listed in Table 3. The maximum errors corresponding to each contributing source are listed in Table 4. The uncertainties in V,,,,, and tunc~opcdare the dominant contributing sources to the total random errors, and the value for x d is approximately f 0 . 0 5 p m , and for c ~ ~ approximately , ~ ~ i ~ ~ f0.09 x 10" ern-?. The total random error for auc,l,rc is clearly small enough that the trend for both wafers is preserved. The total random error for X ~ I which , is calculated in the worst case, principally is large enough to smear out the trend of both wafers. However, the reflectance spectroscopy clearly established that the trend shown is correct, and thus the worse case is not likely
CHARACTERIZATION AND MODELING OF SAGCM
119
occuring. In addition, the uncertainties of V,,,,,,, and r,,,,~lol,,~iare most likely to be on one side rather than random, which will result in systematic rather than random uncertainties, thus preserving the trend over the wafer. However, it is not certain on which sidc these two errors are and, therefore, they are still treated as random errors here.
6. 6-dopedSA(G)CMAPD It would be rewarding to use an even simpler model to conduct the extraction, such as the simplified &doping structure model (Tarof, 1993; Tarof et al., 1994). The formula is similar to the standard technique if it is assumed that rlnP = fgraLling = 0 and N u = 0. The extracted values with this simple model are surprisingly close to the extracted values with the standard technique. x,/ is increased by less than 0.04 p m and aaCrlve is decreased by less than 0.08 x 10” cm-’. However, this obvious success does not necessarily qualify this simple model as appropriate for more general cases. To examine its validity, a better model with N n f 0 is used to do the extraction. Not surprisingly, this “better” model causes an unacceptable 0.06 pin increase of x,/ and 0.23 x 10” cmp2 decrease of oaaclive. Obviously, the two simplifications in the simple model cause similar but opposite systematic errors to x , ~and a,,,iVz. Physically, the 6-doping simplification reduces the voltage drop in the charge and grading layers because of the zero thicknesses, and fortuitously, the N,) = 0 simplification increases the voltage drop in the absorption layer because the increased electric fields in the absorption layer. For quantitative modeling, caution should be exercised when the &doping structure model is used.
E. Sumtnary In this section, a novel, fast, accurate, and nondestructive technique has been invented to extract the two most critical device parameters in the SAGCM APDs-x,/ and To the authors’ best knowledge, the innovative use of V,,,,,, obtained from simple DC photocurrent measurements in this technique is novel. The technique is tested on two wafers, and the extracted values agree very well with the independent measurements within both the experimental and extraction errors. The systematic errors have been considered in detail. The ionization in the absorption layer is found to cause significant errors only when F,!,,,,, is higher than 17 Vlpm; that is, x,/ is increased by -0.03 p m and a,,,,,is, decreased by -0.1 x 10” cmP2. The corrcsponding Vhr is about 90 V ifx,! is 0.4-0.5 p m . This effect is negligible for 90% of the APDs from a wafer. Also, it is found that for further modeling, Pearsall’s rather than Osaka’s InGaAs ionization rates are more consistent with this model and the experimental results. The simplification of thc zero doping concentration in the multiplication layer is found to cause very small systematic errors. The selection of either Umebu’s
I20
C. L. F. MA, M. J . DEEN. A N D L. E. TAROF
or Cook’s 1nP ionization rates is found to cause large systematic errors, but in opposite directions. It has been argued that the best choice for the extraction and further investigations are Osaka’s values. The largest systematic error is contributed by the simplification of no ionization in the charge layer. x“ is increased by less than 0.03 p m and oncrivc is decreased by less than 0.1 x 101’cmp’. The combination of this simplification and the simplification of no ionization in the absorption layer will cause an unacceptable error of 0.2 x 10”cmp’ in oactivr when Vhr is larger than 90 v. However, for the overwhelming majority of the APDs, the systematic errors are acceptable, and are dominated by the simplification of no ionization in the charge layer. The random errors are contributed by the experimental errors of V,,,,,, and vh,, and by the fabrication and calibration errors of the device parameters used in the extraction. The uncertainties in V,,,,,, and fund,,ped are the dominant contributing sources of the random errors, and the overall random error for xd is about fO.OSprn, and for (T,~,,,,~ about f0.09 x 10” cmp2. This technique is applicable to SAM and SAGM APDs to some extent with the modification that the multiplication layer is heavily doped. The critical device parameters in those devices are the doping concentration and the thickness of the multiplication layer. However, it will be less accurate since the electric fields in the multiplication layer change significantly over the layer. The physical model established here is very useful for further investigations on the planar SAGCM InP/InGaAs APDs, and the extraction technique makes it possible to conduct detailed quantitative modeling. V. PHOTOGAIN
In the previous section, a detailed description of a new technique to extract some critical device parameters was given. The next step in the process is to use the extracted and known device parameters of the APDs for prediction of important performance characteristics, such as DC multiplication gain (photogain), breakdown voltage, bandwidth, and noise. In this section, the photogain versus bias voltage ( M - V ) characteristics for a number of devices are investigated (Ma er al., 1995). The Miller empirical formula (Miller, 195.5; Miller, 1957) invented for Si and Ge M - V characteristics is applied for the first time to the M - V characteristics of the SAGCM InP/InGaAs APDs. A physical model is developed to interpret the experimental M - V characteristics, and possible improvements and confirmations to the dcvice parameter extraction technique are also discussed. A . Introduction
Analytical expressions of the M - V characteristics are very useful when estimating the bias voltage required for a particular operating M . They also make possible
CHARACTERIZATION A N D MOIIkLING OF SAGCM
121
the derivation of analytical expressions for the differentiation of M with respect to V required to extract the ionization rates, since numerical differentiation may introduce unwanted spikes and noise. With an analytical expression for the M - V cliaracteristics, it is also possible to investigate gain saturation effects, such as those due to parasitic and load resistances, space charge, and heating. The wcllknown Miller empirical formula for the M - V characteristics in APDs has been used successfully in the cases of Si (Miller, 1957) and Ge (Miller, 1955) APDs. However, this formula has never been investigated systematically for InP-based APDs. In addition, the Miller formula makes it possiblc to extract the true experimental breakdown (at which the current through the device is infinite). In somc cases, i t is possible to determine if the breakdown is truly duc to impact ionization or to some other mechanism such as tunneling, as shown soon. It is relatively simple to physically modcl the M - V characteristics for an arbitrary electric field variation and distribution of injected carriers with Eq. (16) when the elcctron and hole ionization rates are known, as in the case o f Si (Conradi, 1974). As a matter offact, the electron and hole ionization rates arc determined by the inverse of this procedure under the condition of pure electron and hole injections (Stillman and Wolfe. 1977). The determination of thc elcctron and hole ionization rates in I n P was performed from a differentiation of M - V curve (Umebu et ul., 1980; Cook et al., 1982; Armiento and Groves, 1983; Osaka et nl., 1985). Therefore, they cannot be translated directly into an M - V curve without some assumptions regarding quantum efficiency and unity gain (Osaka ef al., 1985). Photogain versus bias voltagc. along with breakdown voltage, bandwidth, and noise, are the most critical periormance characteristics of APDs. Thus, it is important to theoretically predict the M - V characteristics. However, physical modeling of the M - V characteristics in InP-based SA(GC)M APDs has never been done before, except that the M - V characteristics of SAGM InP/InGaAs APD have been calculated as a consistency check (Campbell et al., 1983; Campbell et al., 1988) with the doping concentration in the multiplication layer takcn as an adjustable paramctcr to provide a better fit. Thcrcfore, calculation ofthc M - V characteristics will be a good test o f the device parameter extraction tcchniquc discussed above, and it may even offer some insights into possible improvements to the extraction technique. Planar SAGCM InP/InGaAs APDs are almost ideal for studying the M - V characteristics as it is shown soon. However, two obstacles must be overcome when dealing with M - V characteristics i n any type of APDs. One is how to define unity gain, and the other is how to maintain a constant primary photocurrcnt (quantum efficiency) throughout the bias voltages. or how to calculate i t when it is voltage dependent. The photogain M in planar SAGCM InP/lnGaAs APDs is conveniently self-calibrated and the quantum efticicncy does not change with bias voltage. The gain of other types of InP/lnGaAs APDs must bc calibrated with another photodiode, or unity gain is defined at an arbitrary low voltage, or at punchthrough voltage V,,,c\L, in mesa SAM InP/InCaAs APDs (Susa et al., 19x1; Kobayashi ef ui., 1984). which may not be necessarily correct (Kuchibhotla r t al., 1991 ). For
122
C. 1.. F. MA, M . J. DEEN, AND I-. E. TAROF
voltage dependent quantum efficiency in APDs, complicated baseline fitting must be employed to account for this effect (Osaka er al., 1985). The calibration of M and baseline fitting not only make the theoretical model more complicated, but more importantly, they introduce extra large uncertainties. As discussed later, the primary photocurrent (quantum efficiency) for the planar SAGCM InPAnGaAs APDs is independent of the bias voltage for the voltage range V > V.,,,, and M is indeed unity when Van, < V < V,,,,,;,, as shown in Fig. 27. When Vi,,, < V iV,,,,,,, the periphery region of the InGaAs absorption layer is depleted whereas the central active region is not. The energy barrier at the I n P h G a A s heterointcrface in the central active region still blocks the holes photogenerated in the absorption layer from passing. Because of the very high quality of the InGaAs semiconductor, the hole lifetime is about 18.5 ~s and the diffusion length is about 140 k m (Gallant and Zemel, 1988). Therefore, the long-lived, photogenerated electron-hole pairs in the central active region can diffuse laterally to the periphery region, and are collected by the terminals since the energy barrier in the periphery region is diminished. The photocurrent is constant over this range, and the gain is unity since at these low bias voltages the gain is due to impact ionization in the periphery region. This is strongly supported by the experimental photocurrent. Should the gain not be unity within this bias range, the photocurrents would not be constant. It should be noted that the multiplication gain due to the impact ionization in the central region is not necessarily unity. However, this multiplication gain is irrelevant to the photocurrent since the photocurrent does not flow through the central active region at all. This is one of the very unique features of planar SAGCM APDs with partial charge sheet in the periphery as the photogain is calibrated automatically in this bias range. The existence of the unity gain bias voltage region is only observed in our high-quality planar SAGCM APDs with partial charge sheet in device periphery, or high-quality planar SAGM I n P h G a A s APDs with preferential lateral extended guard (PLEG) ring (Taguchi rt al., 1988). When V > V,,,,,,, the energy barrier in the active central region is diminished, and the photogenerated holes can pass through the central active multiplication region. In the case of a completely depleted absorption layer ( V > Vdc,,lrtccl, Vdcplc,,.~ is about 10-15 V higher than V,,,,,,), the holes photogenerated in the absorption layer drift through the central region. In the case of an incompletely depleted absorption layer, the holes photogenerated in the undepleted absorption region diffuse vertically along the central regions, since it is much shorter than to diffuse to the periphery region. Both the holes diffusing to the depleted absorption region and the holes photogenerated in the depleted absorption region are collected through the central region. Therefore, when V z Vl,,,,i,, all the holes photogenerated in the absorption layer are collected through the central region, and the primary current is the same as when Vi,,,,,, < V < V,,,,,,. However, the energy barrier at the I n P h G a A s interface in the central region does not disappear at V = V,,,,,,. The energy barrier
123
CHARACTERIZATION AND MODELING OF SAGCM
at the heterointerface may prevent the hole current flow through the central region. In this case, the blocked holes will diffuse laterally to the periphery and are collected through the periphery region. Let’s estimate this extra voltage above V,ll,,, required to eliminate this hole blockage completely. The valence band discontinuity of the InP/InGaAs junction is about 0.36 V, which spreads over the thickness of the grading layer. Therefore, thc electric field at the heterointerface Fl,,,,, of 0.36 V/O. 1 p m = 3.6 V/pm is approximately strong enough toeliminate this energy barrier. This means that an extra 3 V above V,,,,,i, will eliminate this hole blockage caused by this energy barrier. It should be noted that this phenomenon is different from the hole-trapping phenomena, which is a high-frequency phenomenon. Therefore, when V z ( Vlll,si, 3 V), thc primary current through the central active region does not change with bias voltage and is the same as the primary current (through the periphery region) when Val,,, < V < V,,,,,i,. It should be also noted that when V > (V,,,,,;, 3 V) the multiplication gain in the periphery region does not contribute to the photocurrent since the photocurrent does not flow through it and the gain may not be unity. The photocurrent and its differentiation with respect to V start to increase monotonically due to the increased impact ionization multiplication until the bias voltage reaches the avalanche breakdown voltage Vl,. The existence of the unity gain and the voltage-independent quantum efficiency in our devices make it possible to calculate the photogain directly, as shown in Fig. 2, and more importantly, they make empirical and physical modeling more reliable. Another important advantage of SAGCM APDs is that the electric field in the multiplication layer is uniform. This significantly simplifies the mathematics involved. More importantly, it eliminates the complicated dead space correction encountered in conventional APDs with a heavily doped multiplication layer (Woods et ul., 1973). All these advantages allow for a more accurate modeling of the M - V characteristics in planar SAGCM InP/InGaAs APDs.
+
+
B. Empirical Forrnula 1. Theoty The Miller empirical formula is given by M=
1
I
-
( V / V,)’. ’
(57)
where VH is an adjustable breakdown voltage, and r is an adjustable positive power coefficient. The formula is numerically displayed in Fig. 39 for different r values. V/{is different than the experimental V b , , which is defined as the voltage at which the photocurrent is -10 p A . Certainly, V , is the most appropriate breakdown voltage from the physical point of view. However, V,,, is usually only 0.5 V smaller
124
C. I,. F. MA. M. J. D E N , A N D I-. E. T A R O F
0.00
0.20
0.40
0.60
0.80
1.00
vml FIGURE.
39. Eimpirical formula for M - V characteristics with different
I’
than VB, and both of them may be used interchangeably. To fit the experimental data, it is better to fit
which can be performed conveniently as a linear regression.
2 . Results arid Discussioiis Photocurrents (photogains) versus bias voltage for nine APDs from wafer P623 were measured. These nine APDs showed different v,r and can be divided into four groups according their V,, and V,,,,,;,, besides APD4. In fact, two APDs in each group are from the same area on the wafer. Each experimental M-V characteristic is fitted to the Miller empirical formula with 95% confidence, and the results are given in Table 2. The uncertainty for V,] is 0.02 V, and for I’, it is less than 0.006 for all the APDs. One ofthe typical fits (APD32) is illustrated in Fig. 40. The fit is very well from V,,,,,:, f 3 V to V,,,.and VR is only about 1 V higher than V,,,.. The extra 3 V is required to completely eliminate the hole blockage, and it confirms the estimate as given above. However, the estimation is done for the most optimized condition; that is, there are no other photocurrent shunt flow paths, and no other hole recombinations at the heterointerface (for example, due to interface defects). As a matter of fact, in some devices, up to 5 V extra bias voltage is required to eliminate the hole blockage. Therefore, to be consistent, all the fittings are done to data points for M > 5 . In addition, the higher photogain region is more relevant to the APD’s design requirement. Even though the fit of log(1 - I / M ) does not agree with
-
125
CHARACTERIZATlON AND MODELING OF SAGCM TABLE 5 THEF17TE[) PAIIAMETURS FOII 8 APDS FROMWAI:t:R P623. ALL V0l:rAGES A R E I N VOl TS. APD# 4 V,,,,,, Vu t.
14
37
25.8 25.4 21.2 37.41 46.50 45.84 0.640 0.880 0.732
32
31
13
35
34
24.8 25.0 14.6 15.0 20.6 20.2 56.62 55.50 70.00 68.45 70.44 71.22 0.993 1.020 1.343 1.372 1.443 1.403
1oc
L
36
1
5c
la
c
1
20
30
40
50
60
0.1 20
30
Bias V
Bias V (V)
40
50
60
(V)
F I C X J R40. ~ : A typical lilted M - V charactcristics (APD32) to the Miller empirical iorinula Ihr 5 data. The solid and dashcd lincs arc the fitted and experimental charactcristics, respectively. (a) M-V. (h) ( l - l / M ) - V . M
>
the experimental data at lower gains from the graph, it does not undermine the integrity of empirically fitting M - V characteristics, as can be seen from the M - V curves. This fitting is also performed o n some APDs from wafer P624 (APD27, APD28, APD29, and APD30), and very satisfactory results are obtained. One of the ten APDs examined (APD 5 , which is not listed in Table 5 ) has an abnormal value of extracted r (0.41) as shown in Fig. 41 value much smaller than the value of r for the other APDs ( r I ) with comparable breakdown voltage. In this case, a nonzero gain will be extrapolated at zero bias voltage for this experimental M - V characteristic. The reason remains elusive. This implies that fitting to the Miller empirical formula may even be useful for monitoring the meas urements. Because r does not have clear physical meaning, an empirical r - F,, relation between r and M,,. is useful for device screening and modeling. Vb, is chosen since it is probably the most easily and frequently measured quantity of APDs. Plotting r versus V,,, the APDs studied result in the graph shown in Fig. 42, and this includes the APDs from both wafers P623 and P624. From the plot, an empirical linear
-
126
C. L. F. MA, M. J. DEEN, A N D L. E. TAROF
1
10
0 7
20
30
40
50
60
30
20
40
50
60
A fitted M - V characteristics (APDS) to the Miller empirical forinula with unreasonable I' (0.41). The solid and dashed lines are the litted and experiniental characteristics, respectively. ( a ) M - V . (b) (l-i/M)- V . E'IGIIRE 41,
1S O
1.30
1.10
0.90
0.70
0.50 30
40
60
50
70
80
Vb,(V) Fioul
I'
antl
w,r. The symbols are from the APDs, antl the solid
relation between r and Vb, can be obtained as r = (-0.06 f 0.13)
+ (0.020 f0.002)Vhr.
(59)
To cxaminc the merit of this empirical V - R , relation, ~ two APDs with extracted r values that deviated most from the empirical relation r-Vh, shown in Fig. 42 were selected, and the new M-V characteristics were recalculated using r values obtained from the empirical r-Vh, relation with v h , as input parameters, and the results are shown in Fig. 43a and b respectively. The difference between r obtained from thc empirical r-Vbr relation and the experimentally extracted r is -0.1. It
CHARACTERIZATION AND MODELING OF SAGCM
I27
(?I) it)) F I G U R43. ~ . The recalculated M - V characteristics with Miller ciiipirical lormula, using I- ohtninetl lrom r-Vh, empirical relalion (EL]. S O ) . (a) APD37. r = O.XS7. (b) APD34. I’ = 1.349.
causes little change for the APD (APD34) with higher v h r (70 V), while it causcs visible change for the APD (APD37) with lower Vt,r(46 V). This occurs because the M-V characteristics for APDs with lower r values (lower F,,)are more sensitive to a variation of r than for APDs with higher r values (higher &,,-). Therefore, r obtained from the empirical r-v,,. relation can reproduce well the experimental M-V characteristics for the APDs with v h , larger than 60 V. As for APD37, thc discrepancy in bias voltagc corrcsponding to M = 5 and 10 is 0.8 and 0.6 V, respectively, and thc discrepancy in M for V = 0.9Vhl. is 2 ( M = 13), which is caused by the steep slope there. Therefore, it is still reasonably accurate to cstimate the bias voltage for M = 10 even for APDs with Vh, less than 60 V. One of the design objectives is to have lower operating voltages. Thus, it is important to design APDs with low brcakdown voltages. Since r generally increases with V,,,.,the normalized voltage V / is lower for thc APDs with lowcr values of &,, than for the APDs with higher values of Vhr for achieving thc same photogain M , as can be seen from Fig. 39. Therefore, it is more beneficial to design APDs with lower breakdown voltages since this translates to even lowcr operating voltages. For Si APDs, r has been found to rangc bctween 1.9 and 4.0 (with vh,. between 34 and 48 V), which is significantly higher than the r values found here. This means that M is lower in Si APDs than in SAGCM InP/InGaAs APDs at a same normalized voltage, and this is mainly due to different ionization rates in Si and InP. It appears that gain saturation effccts (Kagawa et ul.. 1992) such as parasitic and load resistances, spacc charge effect, and heating are not important, at least to thc maximum gain of 100 with light input power of approximately 0.1 pW. These gain saturation effects can be combined analytically as a series resistance, but the calculation discussed above docs not need this series resistance to reproduce the experimental M-V characteristics. The reason for this is that the photocurrent is 10 p A at breakdown and, thcreforc, a 50 kR serics rcsistancc is needed to make some difference, but such a large resistancc is highly unlikely.
v,,-
128
C. L. F. MA. M. J . DEEN, AND L. E. TAROF
Even though the Miller empirical formula can account for the general features of the M-V characteristics, deeper insight of the M-V characteristics requires a physical model. In addition, the empirical r-Vbr relation has only been tested on APDs with a relatively smaller range of x,1(0.4-0.6 Fm), and it may not be adequate to describe the M- V characteristics of APDs with significantly different values of xd. A physical model will not suffer from such limitations. C. Physical Model
1. Theory Physical modeling of M-V characteristics can be derived easily from the general formulas presented in Section 11. With the identical assumptions and simplifications as in the standard technique for device parameter extraction, the electric field in the InP multiplication layer at bias voltage is given by
with
The depletion depth t (V) at bias voltage V is given by
(62) The photogain M(V) at bias voltage V can be derived from the more general formula Eq. (14), when the electric field in the multiplication layer is constant, and it is given by
where x,) and
mLtl,l,ge
are extracted with V,,,,,, and Vn, and V, is extracted by
CNAKACTl:RIZATION A N D MOI)EI.ING O F SAGCM
I29
fitting the experimental M - V characteristics t o the Miller empirical formula as ju\t discussed. V,i is better than Vh, in reproducing the cxpcrimental M - V characteristics.
2 . Kcsirlts ~ i i Discirssioris d To compare the physical modcl predictions t o experiments, all 14 APDs discussed previously are investigated again. A typical good match and a typical fair match between the theoretical and expcriincntal M - V characteristics are shown in Fig. 44a and b, respcctively, and even the agreement in the fair match is quite acceptable. There arc about an equal number of APDs with good and fair matches, as well as one APD which did not give good agreement (APDS) as expected. It can be concluded that the physical model is successful, considering that many parametcrs are only estimated from the MOCVD growth conditions, rather than from dircct measurements. The success indicates that both the physical model and the arguments in converting I,]-V into M - V are correct. Similar to what has been found in the Miller empirical section, the physical model predicts that the normalized voltage V / V h r is lower for the APDs with lower values of Vt,r than for the APDs with higher values of V I , for ~ achieving same photogain M . Recall that C T ~ ~ ~(which , ; , ~ is proportional to the electric field discontinuity A F between the multiplication and absorption layers) is higher in the APDs with lower vh,.,and the bias voltage is mainly dropped across the absorption layer. Then denoting the electric field in the multiplication layer F.,v under the bias voltage V , at photogain M , it can be shown that V,v/ v h , X (F,L,- A F ) / ( F h , A F ) = I - (Fhl-- Fbf)/(FI,, - A F ) . Therefore, Vbt/Vl,, is smaller when A F is larger. 1t is very interesting to note that the theoretical M is not unity at the bias voltage V,,,,,;,, as has been discussed in the introduction. This nonunity phenomenon is more severe for the APDs with lower VI,~’sthan Cor the APDs with higher Vhr’s. For the examples in Fig. 44, assuming unity gain at V,,,,,,i would cause more than
130
C. L. F. M A , M . J. DEEN. A N D 1. E. TAKOF
r 50
h
E
-
.
3
45-
l
i
.
La
’ . 40
-’
35 20
30
40
50
60
Bias V (V) FIGLJR 45. ~ , The electric field F[,,p versus the bins voltage V (or APD32.
v,,.,
50% error in M for the APD with lower and 10%for the APD with higher Vb,.. The electric field in the central multiplication layer has been built up before the bias voltage reaches V,,,,,, and the multiplication gain is larger than unity, but it docs not contribute to the photocurrent because there is no injection of carriers to the central active multiplication layer. This is why the Miller empirical formula is adequate to interpret p-n APDs as well as these types of complicated APDs, since it predicts nonzero multiplication gain as long as the bias voltage is nonzero. The curvatures of the theoretical M-V characteristics when V,,,,,, < V < V,:,,,,, are quite different than the curvatures when V > V,:,, and this is in general agreement to the experimental observations. This general agreement is due to the highly doped InGaAs layer, which modifies the slope of the electric field &p against the bias voltage as shown in Fig. 45. Another interesting point confirmed is that the absorption layer is not completely depleted until at a bias voltage much higher than Vine,;,is reached, as shown in Fig. 44a. This effect is particularly important for the APDs with lower Vhr. It is obvious that a slow diffusion tail in the photoresponse will be observed at higher frequencies. However, this will not occur for the DC photogain described here. More importantly, this would not cause the normally observed variation of the quantum efficiency over a voltage bias range, as explained in the introduction. This phenomenon is confirmed by the good agreements between the theoretical and experimental M-V characteristics. The agreements between the experimental data and the Miller empirical M-V characteristics are always good, while the agreements between the experimental data and the physical modeling are only good for half the APDs examined. This is not surprising since the empirical formula is also a good curve-fitting scheme. The discrepancy in bias voltage between the experiment and the physical modcl predictions for M = 5 and 10 is 2.6 and 1.6 V, respectively, and the discrepancy
CHARACTERIZATION AND MODELING OF SAGCM
131
in M for V = 0.9 vh, is 2 ( M = 7), which is caused by the steep slope there. Therefore, this physical model is adequate to physically model M - V characteristics exccpt for the bias range close to breakdown, and tolally adequate to model in logarithm scale as usually required. 3. Iniproved Physical Model
I n the device parameter extraction discusscd in the last section, the device parameters, such as flnp, tgrilciillg, t,ln~ol,lcc~. N G , Nl,, and c ~ ~ ~ Gcan ~ ~ be A ~adjusted , within thcir respective experimental uncertainties. In addition, the physical model may be modified to include the effects of the ionizations in thc charge and absorption layers. However, any modifications in the above device parameters, or any modification of the physical model, will cause corresponding changes in the extracted values of x,/ and ~ ~ l ~ ~ ~ ~ ~ ~ . If similar modifications are made in the physical modeling of M - V charackristics, then x , ~and cfchargu will have to be recxtractcd. As a matter of fact, the theoretical M - V characteristics calculated using extracted x,/ and Cchargc will ensure a perfect match of two unique points in M-V characteristics-VI,,,,:, and Vh,. It is clear that this physical model of M-V characteristics is not truly a6 inifio but is rather self-consistent. The inclusion of the ionization in the absorption layer modifies M ( V )of Eq. (63) into M(V)=
(64) a I and BI are calculated with the electric field in the multiplication layer [Eq. (60)]. It is easy to check that if the ionization in the absorption layer is ignored, the above equation becomes Eq. (63). The inclusion of this effect makes very little diffcrcnce i n the theoretical M-V characteristics. Since all the APDs examined have a v h , of 70 V or lower, then as expected, ionization in the absorption layer contributes little to the photogain. The inclusion of the ionization in the charge layer modifies M ( V ) of Eq. (63) into
I32
C I F M A , M J DFEN A N D L F, TAROF
I
20
30
40
so
00
Bias V (V) The theoretical M - V characteristics COI- APD37 calculated with Osaka‘s InP iiiiii/atioti ( d i d line). with Umebu’h (dottcd line). and with Cooh‘s (dashed linc).
FlcillRr; 46. rates
Similarly, the inclusion of this effect makes very little difference in the theoretical M - V characteristics. The self-consistency is largely responsiblc for this ncgligible differcnce. Perhaps better examples to demonstrate the self-consistency are the theoretical calculations with the other reported InP ionization rates, such as Umebu’s (Umcbu, 1980) or Cook’s (Cook et al., 1982). This makes very little difference in the theoretical M - V characteristics as shown in Fig. 46, cven though it makes a significant diffcrence in the extracted device parameters. Independent and accurate measuremcnts of x,/ and Dcharge charge are the only way to decide which InP ionization ratcs are better. Other device parametcrs uscd in calculating the thcoretical M - V characteristics are also examined to seek any improvement within their respcctive uncertainties, and they are listed in Table 6. All the parameters make little difference to the theoretical M - V characteristics cxcept the device paramctcr tLinr~opcl~. Clearly, flinl~~,,,c~~ is the only parameter which may improve thc agreement bctween the thcoretical and experimental M - V characteristics. For the particular wafers investigated here (wafers P623 and P624), fll,,~~~ll,,cl~is not directly measured. The default value of 3.8 p m is estimated from the argument that t,l,,,l~lpcclis equal to its value from calibration wafers with identical nominal MOCVD growth conditions. The unccrtainty of ~llil~~lll,cl~ (+/ - 0.2 p m ) refers to the variation among APDs from the same wafer. It is found that it is always possible to find onc particular value between 3.6 to 2.9 p m for each of the APDs with which the agreement of the theoretical
133
CHARACTERIZATION AND MODELING OF SAGCM TABLE 6 T H EEFFECTSTO THE THEORETICALM-V C M ~ R ~ C I ~ B M K S DTOU THE E -OF DEVICE -P THE THICKNESS IS IN p m , THE C I I - ~ , AND THE INTEGRATED DOPING -m IS IN DENSITY IS I N 1012cm-2.
Default Uncertainty
f 1
0.17 f0.009
V
Bias V (V)
0.09
10
f0.005 f 2 . 5 Little effect
0.4 f0.05
0.7 f0.07
3.8
3.8 -0.2
Worse
ktter
Bias V (V)
(a)
FIGURE47. Typical almost perfect matchedM-Y characteristics. The solid and dashed lines are Y characteristicspspectively. (a) with = 3.4
the theoreticaland expenmentalMand (b) with = 2.9
and esperiiiieiital M-V cliaracteristicsis best. as sliown 111 Fig. 47. A best value of fundo@ is deteniillied by iiillilliiizllig the sum of the square of the differences between tlie esperiiiieiitaland tlie theoretical M-V Characteristics at all measured and2V lowerthan Vbr. Certallily.foreac1i bias voltage points5 Vliiglierthan ,V individual APD, a range of values for rundoped caii be found to obtain a reasonable agreement. which is much better than the typical good fit as sliown in Fig. 44a. This range is typically f 0 . 3 pm. This process offers an lliiproveiiieiitover tlie previous described parameter estractions. Basically. by fitting tlie entire M-V Characteristics (including and t h e e device parameters caii be obtained siiiiultaiieously ccharge. rundoped). Measuring photocurrent versus bias voltage V at rooiii temperature should be mandatory for device characterization. modeling. and screening. since much infonilation caii be obtained from th~ssimple measurement. From a physical iiiodeliiig pollit of view. the best coiiiproiiiisedvalue of rundoped for all tlie APDs from wafer P623 is 3.2 pm. This iiew default value gives a better fit than the previous estimated value (3.8 pm). Tlie iiew theoretical M- V cliaracteristicsare displayedlli Fig. 48. Tlie shift 111 bias voltage (tlie worse case 111 the graph) for M = 5 and 10 is 1.0 and 0.7 V, respectively. and tlie shift in M for
134
C. L. F. MA, M. .I.DEEN. AND I*. E. TAROF
100
100
I
/
I
I
-4
.4 0
0
-D -
10
//
Vdeple'ed
I
-
10
a
1
+. -
./' I >mv
1
"
" " ' ~
" '
"
" "
" "
"
"
" '
=, . / 2 1
,
,
, ,
,
V = 0.9Vhl is 0.7 (M = 7. l), which is caused by the steep slope there. Therefore, the physical model with this new default value is improved, and it is adequate to theoretically calculate the M- V characteristics in SAGCM InP/InGaAs APDs. However, it should be cautioned that the curvatures of theoretical M-V characteristics also strongly depend on the slope of (YI and @, with respect to 1/F in addition to the value of t L I n d o p c ~ It . will be demonstrated in the next section that the curvatures of the theoretical M-V characteristics are significantly different when they are calculated using ( Y I and /I1 with different slopes. However, Osaka's ionization values are most trustworthy, and it is supported by four more experimental data (Umebu's (Umebu, 1980),Cook's (Cook et al., 1982). Armiento's (Armiento et al., 1983),and Taguchi (Taguchi et al., 1986)')as far as the slope is concerned. In other words, the selection of (YI and /I1 is critical to both the absolute values of the extracted device parameters (x,,,cqldlgC, and tundopl.d) and the absolute correctness of the theoretical M-V characteristics. However, the consistency is much more important and this does not depend on the selection of a1 and @ I . D. Surnmary
In this section, we have demonstrated both experimentally and physically that for our planar SAGCM InP/InGaAs APDs the photogain is unity when V,,,,, < V < V,,,,,, and the quantum efficiency does not depend on the bias voltage. The Miller empirical formula for M-V characteristics is shown to be very appropriate to our APDs with r between 0.6 and 1.5, and an empirical relation between the power coefficient r in the Miller empirical formula and experimental Vhr is derived. Finally, a physical model of the M-V characteristics is developed and proven to be successful in interpreting the experimental data from our APDs. It is also found that fu,,do,,cd can be extracted from the experimental M- V characteristics-
'
At rooiii temperature,
(YI
( F ) = 9 2 0 . exp(-344/F).
Dl
( F ) = 430 ' exp(-272/F)
135
CHARACTERIZATION AND MODELING OF SAGCM
an improvement to the device parameter technique developed in the last section. The inclusion of the ionizations in the charge and absorption layers is not important in modeling the M - V characteristics.
VI. TEMPERATURE DEPENDENCE OF BREAKDOWN VOLTAGE. A N D PHOTOGAIN
Temperature dependence of breakdown voltage is one of the critical performance parameters of APDs for coolerless applications. However, there has not been any theoretical modeling of temperature dependence of breakdown voltage in any types of InP-based APDs, even though a few early experimental works were published (Takanashi and Horikoshi, 1981; Susa e f a l . , 1981; Forrest etal., 1983; Chau and Pavlidis, 1992). Maybe one of the reasons is that a large discrepancy can be found bctwcen the temperature dependence of a1 and /l1 reported by different authors (Kao and Crowell, 1980; Takanashi and Horikoshi, 1981; Taguchi et al., 1986; Chau and Pavlidis, 1992). Thcreforc, it is important to examine this discrepancy for future modeling. In this section, the temperature dependence of breakdown voltage in SAGCM I n P h G a A s is investigated (Chau and Pavlidis, 1992; Ma et al., 1995). Detailed experimental results of the breakdown voltages for temperature from -40°C to 1 1O’C of the SAGCM APDs having a range of device parameters, including a small sublinearity at higher temperaturcs arc rcported. A physical model of temperature dependence of breakdown voltage as well as numcrical results are presented with a full account of all important device details. Then the model is tested on other types of InP-based APDs from the published literature. Finally, the temperature dependence of the M - V characteristics is discussed.
v,,.
A. Theory
The device parameters extraction technique ensures a perfect match of Vbr at room temperature since x , ~and (T,I,;,,.~~ arc cxtracted using V,, at room temperature. Vh,. at any temperature can be calculated from Eq. (36) once crct,,,pc and x,/ are determined, and F h r , calculated from Eq. (37), is the only variable which is a strong function of tempcraturc for the tempcraturc range investigated here. The tcrnpcrature dependence of F h r is mainly due to the temperature dependence of a! I and PI, as can be seen from Eq. (37). The ionization in the InGaAs absorption layer can be ignored for a first-order approximation, and this is discussed in detail later. A generalized theory (Baraff theory) for lhe impact ionization process in semiconductors has been developed (Baraff, 1962) in terms of the threshold energy E , . the average energy loss per phonon scattered E r , and the carrier mean free path A. from optical-phonon scattering. Unfortunately, this formula is not very
136
C. L. F. M A , M . J . DEEN. AND L. E. TAROF
convenient for modeling since it can only be numerically calculated. Later, an analytic expression (Okuto-Crowell theory) applicable for a wide range of electric fields was given in (Okuto and Crowell, 1972) and it is a,B = -exp qE,F
{
0.217 (::)I -
l4
-
,}d-
(66)
where F is the electric field. Assuming that the energy loss per unit path length is independent of temperature, then the temperature dependence of E, and h was obtained (Crowell and Sze, 1966) as
A = h,tanh(Er,/2kT).
E, = Er,tanh(E,.,/2kT),
(67)
and E,, and h, are the values of E,. and h at 0 K. Furthermore, the temperature dependence of El can be assumed to be the same as the temperature dependence of the bandgap energy E,(T) (eV), and for InP (Casey and Panish, 1978) it is given by E,(T) = .421 -
3.63.10-4~2 (T 162) '
(68)
+
where T is in degrees Kelvin (K) In principle, 011 and can be completely determined once E,, E , , and h at one particular temperature (room temperature) are known. In general, some or all of the parameters in the Okuto-Crowell theory are adjusted to fit experimentally measured a I and BI at room temperature. However, a large discrepancy can be found between E,, E,, and h reported by different authors (Kao and Crowell, 1980; Takanashi and Horikoshi, 1981; Taguchi ef al., 1986; Chao and Pavlidis, 1972) (Table 7). Different parameters used for fitting and the discrepancy between experimentally TABLE 7 E,. E,- AND h A T 300 K FOR InP FROMR E I . W E N C ~E,.,, S . AND A,, ARE CA1.C'LlLATk.U F R O M EQ.(67) E X C ~ ~IN P T( T A K A N I S H IAND HORIKOSHI, 198 1 ), W H E R E THEY WERE DETERMINED BY FITTING EXPERIMENTAL. ( Y I AND PI A T A FEW TEMPERATURES. Fitted E, ( e v ) E,.(meV) h ( i ) E,-,,(nieV) &,(A) parameter 011
1.99
,91
1.65
13 26
18.0 30.1
27 40
37.6 46.4
011
pI
1.57 I .hl
33.7 42.9
27.0 33.1
46.9 54.7
37.5 42.2
(YI
1.84
61 1.65
32.7 21.6
29.6 24.8
46 36
41.7 41.3
2.37 2.00
39.9 39.9
4.5.2 45.2
52.2 52.2
59.1 59.1
(YI
frl
Source 0 1 n1casured da1a
(Kao and Crowell. (Kao and Crowell. 1980) 1980)
I I
(Takanashi and Umebu's (Umcbu Horikoshi. 19x1) efu/..1980) (Taguchi et d.. 1986)
Er"""
6 .E,
Ref.
1
(Taguchi el o/.. 1986)
(Chau and Pavliclia. Cooh'b (Cook C I ti/.. 1992) 19x2)
137
CHARACTEKI%ATION AND MOIIEI,ING OF SAGCM
determined a1 and are the main reasons [or this largc discrcpancy. Note that, to the author's best knowledge. (Taguchi et a/., 1986) is the only work where the temperature dependence of a1 and is experimentally determined, and its a1 and PI at room temperature are close to Osaka's values (Osaka and Mikawa, 1986) for electric fields from 50 to 60 V/pm.
B. Experinzentrrl Results The temperature dependence of our APDs was measured by placing devices in a variable-temperature oven (-80 to 200'.C). The temperature uncertainty of the devices is -1;C. The dark current versus (reverse) bias voltage was measured with an HP 4145B Semiconductor Parameter Analyzer (SPA), and the accuracy of measuring DC currents is 1 PA. Typical (APD32) dark currents versus bias voltages at different temperatures arc: shown in Fig. 49. The breakdown is impact ionization breakdown since it shows a hard breakdown rather than a soft breakdown, as in thc case of Zener tunneling breakdown (Forrest et al.. 1983). The experimental Vhr is defined as the voltage corresponding to 10 p A through the APD. Experimental Vhr as a function of temperature from -40 to 1 10 C are shown as symbols in Fig. 50 for three APDs (APD14, APD32 and APD34) with different room temperature Vt,, from the same wafer (P623). Experimental Vh,at temperature T can be approximated as a linear function of T
-
10-5
lom6 108 109
10-9 10-10 10-11 10-12
0
10
20
30
40
50
60
70
Bias (V) FIGLIRE 49. Typical (APD32) dark ciirreiits verstis hias voltages for dit'fcrenr temperatures. The hrcakdown voltage is clelinecl its the voltage at which the dark current is 10 p A .
138
C. I,. F. MA. M. J . DEEN. A N D L. E. TAROF TABLE 8 THELINEARLY FITTED )Iexp AND Vhr ( 0 )FOR T H E APDS. FOR ALL THE TEMPEKATURES (AVE). BETWI;~:N-40 1 0 0 C (LT), A N D BETWEEN50 TO 105‘c (HT). t]c,,, IS I N THE UNIT OF v / C A N D Vhr (0) I N v.
APD14 APD32 APDM
0.156 0.146 0.128
0.184
0.170 0.146
31.7 52.5 66.4
0.131 0.128 0.10s
42.0 52.8 66.6
43.5 53.7 68.0
80
70
60 50
40
30 -50
-30
-10
10
30
50
70
90
110
T (“C) The Vhr as a fiinction of ternperature from 4 0 to I 10 C for three APDs. The ayinhols arc cxperimcntal data. and the solid line\ arc linear fits. FICiURb S O .
v,,
where vh,(o) is at O”c,T is in ” c , and vex,, is the experimental temperature coefficient of Vhr(T). For a first-order approximation, the experimental Vhr(T) is well represented by Eq. (69). However, a close examination reveals that qcxpis different at lower and higher temperatures for each APD. By linearly fitting the experimental data for all temperatures (-40 to 105”C,AVE), for lower temperatures (-40 to OC, LT), and for higher temperatures (50 to 105‘C, HT), respectively, different qexphave been found and they are listed in Table 8. In addition, there is a small variation of qcxp(AVE)among the APDs ranged from 0.16 to 0.13 V/-’C. To confirm the existence of these second-order variations, photocurrents versus bias voltages were also measured for different temperatures for each APD, and the experimental v , r ( T ) is extracted from this measurement. A similar trend can be confirmed from these experimental vh,-(T).A very generous estimation of measurement error in vex,, is f0.05 V / C .
139
CHARACTERIZATION AND MODELING OF SAGCM
C. Discussion The extracted x,! and q l l a r g c for the three APDs (with Taguchi's (Taguchi et a]., 1986)a1 and PI at 2 0 C ) are listed in Table 9. The extractions are performed with the parameter extraction standard technique, using V,llcsLl,which hardly change with temperature, and vh, at 20°C. The extracted x,/ does not change significantly among the APDs with different vh,, whereas the extracted crcllargefor the APDs with large Vh, is noticeably smaller than for the APDs with smaller Vbr. This extraction procedure is repeated whenever a different set o f a l and is used for calculating V , J T ) . Note that 3.2 wm rather than 3.8 pm is used as the value of f l l , l ~ , ~ l ,toe ~be consistent with the results presented in Section V. The theoretical Vh,.(T) for the three APDs calculated with each set of ( Y I and in Table 7 are displayed in Fig. 51a-d, respectively. These figures show that it does not matter which a1 and is used; that is, despite which set of parameters is used in the Okuto-Crowell theory, the theoretical Vhr(T) is very close to a linear function of T , Vhr(T) = Vhr(0)
+
%he
'
T,
(70)
and the theoretical temperature coefficient q,llc, calculated with a particular set of a1 and P I does not change from one APD to another. For example, the linearly fitted q [ h c is 0.154 f 0.001 V / C , calculated with a1 and B1 from (Taguchi et al., 1986) and 0.105 f 0.001 VPC from (Chau and Pavlidis, 1992). The theoretical vh,.(T), which agree best with the experimental data for all the APDs, are the ones calculated with a1 and from Taguchi's (Taguchi etal., 1986), as shown in Fig. 5 Ic. Because of this good agreement, and the fact that they are the only experimentally determined (YI and PI at a few different temperatures, this set ofal and BI (or this set of paraincters in the Okuto-Crowell theory) is considered most reliable, at least as far as the temperature dependence is concerned, and this set will be used in the rest of this paper. A typical (APD32) calculated Fh,(T) from Eqs. (37),(661, and (67) is shown in Fig. 52. It can be proved generally that F h r ( T ) is a linear function of temperature T , which is not easy to see from the equations. Since F h r depends only weakly on x,~,its temperature coefficient aFh,/aT does not vary much among the APDs investigated here, and it is approximatcly 3.84. lo-' (V/(pm .' C)). Therefore, TABLE 9 TtIE INPlJT PARAMEThRS AND Tllli EXTRACT~IIIIFiVlCE
APDI4 APD32 APD34
25.8 14.8 20.6
34.X
55.4 69.0
0.562 0.590 0.544
PARAMETIxS I4)H 'TIIIJ SAGCM
2.72 x 10" 2.50 x 1 0 ' 2 2.23 x 10"
51.6 51.2 51.9
APDS.
4.8 7.6 12.2
140
C . L. F. MA. M. J . DEEN. AND
NO
NO
70
70 h
n
Lao
Lao
:so
>’
so 40
40
30
30 -SO -30 -10 10
30 SO
70
-SO -30 -10
90 110
to
to
70
70
n
t u
3
E. TAROF
10 30 SO
70
90 110
n
f
50
40
>
50
I 8 8
30 -50 -30 -10 10 30 50 70 90 110
30 -SO -30 -10 10 30 SO 70 90 110
FIGLIIW. 5 I . Thc Ilicorctical vh,(r)(solid lines) calculated with O(I and 61 from ref’s. fa) (Cnsey and Panish. 1978). (h) (Kao and Crowell. 1980).( c )(Taknnnshi and Horikadii. 1981 ).and ( d ) ( T a y d i i e/ ol.. 1980). The symhols are experimenrnl daln.
t ~ , l l Ccan be related to the teinperaturc coefficient of and it is given by
FhI
from Eq. (36) analytically,
The numerical value is obtained if it is assumed that x,) 0.54 pin. Thc variation of x,/ among the APDs is too small to significantly change this numerical value.
141
CHARACTERIZATION AND MODELING OF SAGCM
Ba
2B a
-50
-30 -10
30
10
50
70
90
110
T ("C) FIGURE 52. The theoretical Fhr(T) for APD32 calculated with LY, and
from Taguchi
Thus, it can be analytically calculated that ~ ~ = 1 0.154 , ~ VPC, which is identical to what has been numercially calculated. This analytical expression offers some physical insights which are not obvious from the numerical calculations. For example, it is now easy to explain the small but discernible variation of qcxpamong the APDs (Table 8), which cannot be accounted for, regardless of the choice for the parameters in Okuto-Crowell theory. From Eq. (71), it is clear that only a variation of t,ll,c~c,,,cd is possible to cause this small variation of vexp since the variations of the other thicknesses there arc too small. For example, if t,,,lopecl = 3.8 p m , then the theoretical value of Vhr/aT becomes approximately 0.177 VPC. At higher temperatures, the theoretical and experimental Vhr(T)has a visible discrepancy for all the APDs, but particularly for APD34 (Fig. 5 Ic). The neglect of ionization in the InGaAs absorption layer may be the possible reason. F;,,,, at room temperature for APD34 is much larger than for other APDs (Table 9), and it is expected that the contribution from the ionization in the InGaAs absorption layer is largest for APD34. The only reported parameters for the Okuto-Crowell theory for ionization in InGaAs from the literature (Chau and Pavlidis, 1992) is listed in Table 10, and this was fitted to Pearsall's experimental data at room temperature (Pearsall, 1980). Thc bandgap energy ( E , y for ) InGaAs is given by (Casey and Panish, 1978)
4.5. IOP"T' T +327 ' where T is in degrees Kelvin. It must be emphasized that there are no reports about the temperature dependence of a? and 02, and the parameters for OkutoCrowell theory from Ref. (Chau and Pavlidis, 1992) were obtained by fitting at room temperature only. E,q(T) = 0.822
-
I42
C. L. F. MA. M. J. DEEN, AND L. E. TAROF TABLE 10 E , , E , , A N D h AT 300 K FOR InCaAs. E,,, A N D h,, AKk CALCULATED FROMEQ. (67).
Fitted E , ( e ~ E,. ) (mev) A. (A) E,.,, (mev) A,, (A) parameter cT2
1.05
pz
1.26
25.1 25.1
44.7 44.7
-50
39.24 39.24
Source of measured data
Ref.
69.9 E , , E , , I (Chau and Pavlidia, 1992) (Pearsall. 1980) 69.9
-30 -10
10
30
50
70
90
110
TW) FIGURE53. The theoretical vhl(T) with (solid line) or without (dashed line) the ionir.ation in the InCaAs absorption layer for APD34.
The re-calculated Vhr(T) is shown in Fig. 53 for APD34. The theoretical Vbr( T ) including InGaAs ionization makes a small but not sufficient improvement with respect to the theoretical Vbr( T ) without including InGaAs ionization. This failure is due to the fact that a2 and 8 2 decrease comparatively as fast as a1 and 8, do with increasing temperature, and the ratio between the contribution to ionization from the InGaAs absorption layer and the InP multiplication layer remains comparable at all temperatures. Should the ionization in the InGaAs layer be responsible for the discrepancy at higher temperatures, then a2 and 82 would not decrease as much as a ]and 81do, when the temperature increases. There are no independent measurements to support this claim at present time.
D. InP-Based APDs It is important that the theoretical modeling of Vhr(T) developed here be applicable to other InP-based APDs. Two p+-n InP APDs listed in Table 11 (Samples A and B) in the literature are discussed, and their experimental values are taken from the graphs/values (Susa et al., 1981; Takanashi and Horikoshi, 1981).
CHARACTERIZATION AND MODELING OF SAGCM
143
The experimental Vhr(T)indicates that Vhr is a linear function of T, and qexpcan be significantly different from one APD to another. Nlnp and tlnp are the respective doping concentration and thickness of the n region in a p+-n InP diode. For a reachthrough structure like Sample A (n region is completely depleted at breakdown), the relevant formulas are given by
( is the relative temperature coefficient of Vhr(T). For a long pf-n diode like Sample B (n region is not completely depleted at breakdown), the relevant formulas are given by
v h , has been ignored for the last expression ( 5 ) in each of the last two equations. Thc breakdown condition is very similar to Eq. (5 l), and it is given by,
is the electric field at x = tlnp, and it is zero in the case of a long p f - n diode. Fbr(T) is found numerically to be a linear function of temperaturc (Fig. 54), and its value depends on Nlnp only. This can bc readily understood from Eq. (75) that Fclld
144
C. L. F. MA, M. J . DEEN, AND L. E. TAROF
70 68
*
n
66
v
64 62
60
.o
FIcimri S4. The theoretical V h , ( T ) and F b , ( T ) (solid lines) for p+-n InP diode saiuple H . Thc dotted line is the experimental data.
the most important contribution to ionization is from the narrow region with highest electric field. Therefore, it is clear from the analytical expressions in Eqs. (73) or (74) that for a first-order approximation, q(he is indeed independent of temperature. From Fig. 54 and Table I I , it is clear that the theoretical modeling is very satisfactory, even without considering the fact that the input parameters (Nlnp and rInp) are subject to some uncertainties. It also indicates that the valid range of electric field for the temperature dependcnce of and ,81 is from 50 to 65 V/pm at room temperature, and there are no reasons why the valid range of electric field cannot be extended even further. The model is also applied to thrce SAM InP/InGaAs APDs in the litcraturc (Susa et al., 1981; Forrest et al., 1983) listed in Table 12, where Nlnp and flnp are the respective doping concentration and the thickness of the multiplication layer, and N l l 1 ~ :and , ~ \ tlnca~, are the respective doping concentration and the thickness of the absorption layer. Since the InGaAs layer is not completely depleted at breakdown, then the relevant formulas can be given by
CHAKACTEKI%ATION A N D M O D I ~ L I N GOF SAGCM
it
I
I3
0.x 3.2 0.92
(.
3
77
0.15
1
100
I
80
0.18 0.15
0.67
95.9 I I0.h 102.2
0.147 0.153 0.146
52.93 51.51 52.25
145
0.0360 (Fot-[w\t('/(i/..1983) 0.0359 ( S u s a ~ ' l t r / . .1981) 0.0361 ( S u s a r r O l . . 1981)
The breakdown condition is unchanged from the case of a pf-n diode as discussed above. Not surprisingly, the thcorctical calculations agree well with the experimental data as listed in Table 12. Another widely used expression to describe the experimental Vhl-(T) in InPbased APDs (Takanashi and Horikoshi, 1981; Susa et d., 1981; Forrest c't al., 1983) is given by
where T is in 'C, and t is the relative temperature coefficient of Vl,,.. This expression is mathematically equivalent to the cxpression of VI,,(T ) with temperature coefficient 77 used so far. However, physically the expression of Eq. (77) implicitly states that t depends strongly on the properties of semiconductor, but weakly on thc specific details ofdevices, which has bcen proven to be correct both experinientally and theoretically for Si pi--" APDs (Chau and Pavlidis, 1992). However, this cannot be generalized to other structures. For example, the experimental values of as listed in Fig. 55 are very close among the SAM APDs, but change by more than 100% among the SAGCM APDs. It can be demonstrated that the insensitivity of 5' to device dctails for Si p'-n APDs is also true for long InP pi--n APDs. The numerical results for different Nlnp calculated from Eqs. (74) and (7.5) are displayed in Fig. 56. These results show that aFtl,./aT is not a strong function of NlI,p; therefore, neither is <. However, since Vh,. at 0 C is a strong function of Nlnp,then 17 will also be a strong function of NlI,p. Note that the ps -n Sample A is a reach-through diode rathcr than a long diode; thus, it does not agree with the values calculated for long InP pi -n APDs. As stated above, this phenomcnon that t is insensitive to device details of long InP p+-n APDs cannot be generalized, and the agreement of the values of < between the SAM APDs in Fig. 55 is coincidental rather than generally correct, since their structures and doping concentrations are similar. For example, if Nltlp keeps increasing, thcn SAM APDs become p+-n APDs; thereffore, the value of r will become 1.1 rathcr than 1.8. The SAGCM InP/InGaAs APDs are
<
146
C. L. F. M A . M. J. DEEN, AND I>. E. TAKOF
0.2
,........................
I
..._
K'-/'
Q343J -................................... SAGCM 0.0'
"
30 40
'
' 50
"
60
,/ ~
"
70
\ ................................ SAM 1-
"
"
80
'
"
'
90 100 110
Breakdown voltage at O°C (v) p+-n
V,,(O"C) (V)
A 80
SAM B 36
A 77
SAGCM B 100
C 80
0.15 0.05 0.15 0.18 0.15 Tl (V/'c) r(lo-' "c-') 1.9 1.4 1.94 1.8 1.88
APDl 4
APD3 2
APD3 4
41.7 0.156 3.74
52.5 0.146 2.78
66.4 0.128 1.93
E. Temperature Dependence of Photogairi
It is relatively straightforward to extend the theoretical model of the M - V characteristics at room temperature to any temperatures, combining the model of photogain at room temperature and the model of temperature dependence of breakdown voltage. Typical (APD32) experimental M - V characteristics for different temperatures are shown in Fig. 57. Theoretical M - V characteristics for different temperatures can be calculated using Eqs. (63) and (60) with Taguchi's (YI and ,B1,and the results
CHARACTERIZATIONAND MODELING OF SAGCM
FIGURE 56. Theoreticalresults for long
147
with differentN1"p. They are also tabulated.
are displayed in Fig. 58a. The unsatisfactory agreement between the theoretical and experimentalM-V characteristicscan be improved dramatically if fundoped 3.8 pm is used, as shown in Fig. 58b. It should be noted that in principle, the values of x d and cfcharge used in the calculationsof M-V characteristicsshould be extracted each time a different value of fundoped is used. However, it is found that t h ~ is s not necessary as long as M versus V/ Vbr is calculated. In other words, the and Ccharge, shape of the M-V characteristicsis not sensitive to the values of whereas the values of vbr are. = 3.2 has been found that is the better choice for modeling the M-V characteristics at room temperature as long as it is calculated with one of the ionizationrates reportedby Osaka,Umebu, Cook, or Note that are very similar among these the slopes of the ionizationrates with respect to reported InP ionization rates. Formodeling the temperaturedependenceofthe M-V characteristics,rundo,d 3.8 Fm is the better choice. The obvious contradiction is due to the significant difference in the slopes between Osaka's and Taguchi's InP ionization rates that are shown in Fig. 59 and that are used in the calculations. The absolute values of the ionization rates are certainly very important for calculating Vbr, but only the
148
C. I. F. MA. M. J. DEEN. A N D 1. E. TAROF
2 d
.d
d
20
30
40
50
60
Bias voItage (V)
slope of the ionization rates with respect to I I F is important for calculating the M - V characteristics, as already demonstrated in Section V. Most likely 3.2 p m rather than 3.8 pm is the true value of run~~,,,,c~~r since modeling the M - V characteristics at room temperatures is very reliable, and supported by four independently reported InP ionization rates. It is even supported by the experimental room temperature InP ionization rates reported by Taguchi in the same publication, which arc different from the calculated ones at 20 C. This discrepancy can be due to a few reasons. The values of the parameters used in [he Okuto-Crowell theory for the temperature dependence of InP ionization rates are
149
CHARACTERIZATION AND MODELING OF SAGCM
I
t Tagucht’s , , 1
1
.
0.024
Inverse electric field l / F (pm/V)
(a)
0025
Inverse electric field 1/F (pm/V) (b)
FKiLIRE 59. Comparison of InP ioni/ations rates (solid lines). Taguchi’s ionimtion rates are cnlculatcd iit 7 0 C, and the dashed lincs are the expcrimental rooni teniperature InP ionization rates reported
in the saint: puhlication.
obtained by fitting to the experimental data for a number of temperatures. In addition, the range of the electric field in Fig. 59 is much smaller than the range fitted. More importantly, all the experimental errors of the InP ionization rates are fairly large. Therefore, the temperature dependence of Taguchi’s InP ionization rates are much more reliable than their absolute values at a particular temperature. In conclusion, tLlnt~~,lle(~ = 3.2 p m with Osaka’s InP ionization rates are the better combination for modeling the characteristics at room temperature, whereas tunt~,,pe~ = 3.8 p m with Taguchi’s InP ionization rates are the better combination for modeling the temperature dependcnce of the characteristics. This is the best compromise unless a better expression for the temperature dependence of thc InP ionization rates can be found.
E Summan)
In summary, the temperature dependcnce of breakdown voltage Vhr in the SAGCM InP/InGaAs APDs with a range of device parameters from -40 to 1 1 0 C is investigated (Yu et al., 1994; Ma et al., 1994; Ma et al., 1995). The experimental data show that V,,,.is approximately a linear function of temperature, with a temperature coefficient qcxpbetwecn 0.13 and 0.16 V/ C. A physical model is developed, and it demonstrates that V,, indeed varies linearly with temperature with a temperaturc cocfficient qlhC approximately 0.155 V/’ C. It is also shown that the electric field in the multiplication layer at breakdown is a linear function of temperature. It can explain successfully the small variation of q,..,, among the APDs. This physical model is also successfully applied to pi-n and SAM InP-based APDs (Ma et al., 1994; Ma et al., 1995). Good agreement between the physical model predictions and experimental data of published InP-based APDs is obtaincd. This good agreement demonstrates that the proposed physical model is appropriate to modcl the tempcraturc depcndence of characteristics in any InP-based APDs. It is also demonstrated that the widely used expression for Vh,.(T)with a rclative
C. L. F. M A . M . J . DEEN, A N D L. E. TAROF
150
temperature coefficient is only correct for long pt-n InP-based diodes. The good agreement between the experimental and calculated Vb,( T )for the InP-based APDs implies that the temperature dependence of the impact ionization coefficients in InP is vcrified independently for thc first time, and Taguchi’s InP ionization ratcs are the best. The temperature dependence of the M - V characteristics also is examined (Ma et al., 1995), and the results are satisfied within the uncertainties. This work is also very useful for further modeling the temperature dependence of the charactcristics of InP-based APDs. It demonstrates that tundc,,,ed = 3.2 p m with Osaka’s InP ionization rates are better for modeling the characteristics at room temperature, whereas t,,,l~,pcd = 3.8 p m with Taguchi’s InP ionization rates are better for modeling the temperature dependence of the characteristics.
VII. DARKCURRENT NOISE Dark currents and their associated multiplication shot noises are potential sources of degradation to the sensitivity of the APDs (Smith and Forrest, 1982). In addition, flicker noises may be a powerful tool for finding sources of degradation in many elcctronic devices, including the APDs. In this section, the dark current low-frequency noise characteristics (Ma et al.. 1993; Ma et al., 1994; Kanbe et al., 1981) of the planar SAGCM I n P h G a A s APDs are investigated, including both multiplication shot and flicker noises.
A. Low-Frequency Noise Measurements The low-frequency noise spectra from I Hz to 100 KHz were measured with an HP3561A Dynamic Signal Analyzer (Fig. 60) (Deen, 1993; Deen el al., 1995; Li et al., 1990; Zhu et al., 1992). A low-noise Ithaco 564 current amplifier was used to extend the sensitivity up to -270 dBA2/Hz. APDl4, APD32, APD34 and APD37 are investigated in this section, and all of them are from one wafer (P623).
+ Voltage source
Dynamic Signal Analyzer
F1c;uizt: 60. APD dark current low-frequency noise measurement setup.
CHARACTERIZATION A N D MODELING OF SAGCM
151
B. Multiplication Shot Noise The multiplication shot noise spectra S/ is of a white-noise type, and visible in all the APDs but one, APD37, which has a dominant flicker noise spectra. The multiplication shot noise spectra are measured at some I[/'s with bias voltages V > V,,,,,;, so that M > 1. Also I,/ are selected to avoid the breakdown region since the very low dynamic resistance there would make measurements much less sensitive. To understand quantitatively the multiplication shot noise, the results from APD32 and APD 14 of SIversus Id arc shown in Fig. 61a. The corresponding noise versus M (derived from the photocurrent measurement) are plotted in Fig. 61b. The linear relation in Fig. 61b implies that the multiplication shot noises are proportional to some power of M , but not to I(,. Therefore, if we assume I,/ = I,/,,M , where I(/()is the primary dark current, then I(/(,varies with M . There are up to 10 dB differences between the two APDs at the same I(,, whereas they are very close at the same M . This shows that the shot noise sources are directly related to M rather than I',, as expected. At a small range of M ,S, can be approximated to be proportinal to M Y , and for APD32 and APD14 they are fitted with y's of 2.75 f 0.05 and 2.95 0.1, respectively. The corresponding ke,, is about 0.5. The error of mcasuring the low-frequency noise spectra ( 1 dB) prevents accurate calculations of F directly. The observed dark current multiplication shot noise is similar to what has been found in Ge APDs (Kanbe et al., 1981).
*
C. Flicker Noise Flicker noises S, are found in two APDs (APD37 and APD34). All the flicker noise spectra (Deen, 1993; Deen et al., 1995; Ng ef al., 1992) are fitted with S,(.f) = hlf" = K&fU
(78)
and K are constants independent of both f and I,,, whcre f is the frequency, a,/l, and b = K I ~ A typical . flicker noise spectrum of APD37 at I , = 5 nA is shown in Fig. 62 with a = 1.0 and b = 1.5 x 10-"A'. As seen in Fig. 62, it is not a perfect fit to the flicker noise spectra. Hence, the fitted a's are distributed between 0.80 and 1.10 at different I , for both APDs without clear correlation to either 4 or M . However, it is still very meaningful and interesting to compare the flicker noise at 10 Hz at different I(/as shown in Fig. 63a for both APDs, or at different M in Figure 63b. It is very important to note that the flicker noise spectra are found only in some of the APDs, whereas the multiplication shot noise spectra are found in all the APDs (except in the case where the flicker noise is too dominant), cven though there arc no fundamental differences between their dark current characteristics. This occurs because the dominant I
152
C. I>. F. MA, M. J . DEEN. AND L. E. TAROF r.
N
F 2
3-
-230
-240 APD14
v)
2 .-0 C
-250
2 -270
-.$.-
APD32
-260
.*
.
~
.:
*
0
-
-280
I
5
Multiplication Gain M
(b)
FIGURE 61, The multiplication shot noises vcrsus ( a ) the dark CLirrents. and (b) the multiplication gains M lor APD.32 and APD 14. The noises arc proportional t o MJ'with y2.75 10.05 and 2.95 ?cO. 10 lor APD3Z and APD 14. i-cspcctively.
the InGaAs layers for all the APDs. In addition, there are substantial differences in S , for APD34 and APD37 at the same A possible reason for this is that only a small portion of I(,, if any, contributes to the observed flicker noises. It will be useful to understand what happens ifa current with flicker noise spectra goes through an impact multiplication, defining S, (I(/(,)and S , (I(/)as the flicker noises from I(/,)and I(/,respectively. In addition, it is assumed that 10 = I < / , , M . Since the multiplication process does not contribute an excess flicker noise ( F = 1) as in the case of multiplication shot noise, then,
s, ( 4 )= s, ( I d 0 ) M 2 .
(79)
For APD34 at higher gains and APD37 at lower gains, the flicker noises do tend to follow Eq. (79). For APD34 at lower gains, it appears that a constant leakage
current is the major contributor to the flicker noise. For APD37 at higher gains, the flicker noise increases faster than M', and a reasonable explanation is that I(/,, increases with M , too.
153
CHARACTERIZATION AN11 M O D t L I N G 01: SAGCM
10-22
10-a
10-24
10-25
10.26 10’
102
103
Frequency (Hz) !, A rypical (AP1137, I,, = 5 IIA)Ilickci- tioiw yicctra. 7‘he solid l i n e is tittctl to ELI.(78) / J = 1.5 x l0-”,4’.
with rr = I 0 iiiid
I n conclusion, we find that the dark current multiplication shot noises are proportional to MI’ with y about 2.75 to 2.95. The flicker noisc is observed in some APDs and is believed to come from w n c lcakagc current sources.
V111. CONCI~USIONS
In this chapter, an extensive investigation on a state-of-art photodetector, namely, planar separate absorption, grading, chargc, and multiplication (SAGCM) InP/lnGaAs avalanche photodiode (APD) with partial chargc sheet in device periphery has been conducted. Specifically, the following was accomplished.
A . Device t’urunieter Extructiori A simple, innovative, and nondestructive technique has been invented to extract the two most crilical device parameters in the SAGCM APDs, xd and oac,lbc. To thc authors’ best knowledge, the innovative use of V,,,,,;,obtained from simple DC photocurrent measurements in this technique is novel. The technique was tested on two wafers, and the extracted values agreed very well with the independent measurements, such as Hall analysis, SIMS, and reflectance spectroscopy. within both the experimental and cxtraction errors.
154
C . L. F. MA, M. J. DEEN. AND L. E. TAROF
lo-'
10-9
Dark Current Id (A) (a)
Q -220 X 3 4
a
a, $ -230
2 6D
!.
-240
;
4 -250 t bd
.r(
,/,
, , , ,
,
10'
100
Multiplication M (b) FIGURE6 3 . The flicker noises at 10 Hr versus (a) the dark currents, and (b) the multiplication gains for APD34 and APD37. The solid line is from Eq. (79).
CHARACTERIZATION AND MODELING OF SAGCM
155
The systematic errors have been considered in details. The ionization in the absorption layer was found to cause significant errors only when F,:,,,, was higher than 17 V/pm. That is, x , was ~ increased by -0.03 p m and crac,ivc was decreased by -0.1 x 10” cm-’. The corresponding Vh,.was about 90 V if xd is 0.4 to 0.5 p m . This effect was negligible for 90% of the APDs from a wafer. Also, it was found that for further modeling, Pearsall’s rather than Osaka’s lnGaAs ionization rates were more consistent with this model and the experimental results. The simplification of the zero doping concentration in the multiplication layer was found to cause very small Systematic errors. The selection of either Umebu’s or Cook’s InP ionization rates was found to cause large systematic errors, but in opposite directions. It has been argued that the best choice for the extraction and further investigations were Osaka’s values. The largest systematic error was contributed by the simplification of no ionization in the charge layer. x , ~was increased by less than 0.03 pin and (T,,,,,~~ was decreased by less than 0.1 x 10” cm-2. The combination of this simplification, and the simplification of no ionization in the absorption layer would cause an unacceptable error of 0.2 x 10” cm-2 in plc,lbc when Vh, was larger than 90 V. However, for the overwhelming majority of the APDs, the systematic errors were acceptable, and were dominated by the simplification of no ionization in the charge layer. The random errors were contributed by the experimental errors of V,,,,,, and Vh,, and by the fabrication and calibration errors of the device parameters used in the extraction. The uncertainties in V,,,,,L,and ~,,,,,I,,~,,~I were the dominant contributing sources of the random errors, and the overall random error for x,/was about 2~0.05p m , and for oac.ivc about f0.09 x 10” cmP2.
B. Photogain It has been demonstrated both experimentally and physically that for our planar SAGCM InP/lnGaAs APDs that the photogain is unity when V,,,, < V < V,,,,,,,, and the quantum efficiency did not depend on the bias voltage. The Miller empirical formula for M - V characteristics was shown to be very appropriate to our APDs with r between 0.6 and 1.5,and an empirical relation between the power coefficient r in the Miller empirical formula and experimental h,, was obtained. Finally, a physical model of the M-V characteristics was developed and proven to be successful in interpreting the experimental data from our APDs. It was also found that tundop,.d could be extracted from the experimental M - V characteristics-an improvement to the device parameter technique developed. The inclusions of the ionizations in the charge and absorption layers wcre not important in modeling thc M - V characteristics.
156
C. L. F. MA. M. J . DEEN. AND I,. E. TAROF
C. Terripercitur-eDependence o j Bt-eukdowii Voltcige arid Photogciiri The temperature dependence of the breakdown voltage Vbr from -40 to 110 C in the SAGCM I n P h G a A s APDs with a range of device parameters was investigated. The experimental data show that V,,,.was approximately a linear function of temperature, with a temperature coefficient 17ex,, between 0.13 and 0.16 V/ C. A physical model was developed and it demonstrated that Vh,indeed varied linearly with temperature with a temperature coefficient q , ~about , ~ 0.155 V/’ C. It was also shown that the electric field in the multiplication layer at breakdown was a linear function of temperature. The model could successfully explain the small variation of o~.,,,among the APDs. This physical model was also successfully applied to p+-n and SAM InP-based APDs. Good agreement between the physical model predictions and experimental data of published InP-based APDs was obtained. This good agreement demonstrated that the proposed physical model was appropriate to model the temperature dependence of characteristics in any InP-based APDs. It was also demonstrated that the widely used expression for N , r ( T )with a relative temperature coefficient was only correct for long p’-n InP-based diodes. The good agreement between the experimental and calculated T ) for the InP-based APDs implied that the temperature dependence of the impact ionization coefficients in InP was verified independently for the first time, and Taguchi’s InP ionization rates were the best. The temperature dependence of the M - V characteristics was also examined, and the results were satisfactory.
v,,.(
D. Durk Current Noise Low-frequency noise (LFN) in dark currents was investigated. It was found that for the SAGCM APDs investigated, the dark current multiplication shot noise was proportional to MI’ with y from 2.75 to 2.95, which was in general agreement with theory. Flicker noise was also observed in some of the SAGCM APDs and it was believed to be due to leakage currents from surfacehterface imperfections or defects. The major contributions from this paper make it possible (1) to extract critical device parameters fast and nondestructively, which is critical for monitoring fabrication and further detailed modeling; (2) to develop and verify a general theory of breakdown voltages and its temperature dependence; (3) to accurately calculate photogain versus bias voltage and its temperature dependence; (4) to understand dark current noise; ( 5 ) to help to understand the degradation sources from lowfrequency noise spectra; and (6) to lay a solid foundation to characterize and model more operating (high-frequency) characteristics of APDs, such as bandwidth and multiplication noise.
CHARACTERIZATION AND MODEI.ING OF SAGCM
157
ACKNOWLEDGMENTS We are indebted to the help received from members of Advanced Technology Laboratory of Bell-Northern Research (BNR), especially to J. Yu, R. Bruce, Dr. D. G. Knight, T. Baird, D. McGhan, K. Fox, and Dr. M. Gallant. We also thank the BNR organization, and in particular the management, Drs. G. Chik, G. Ribakovs, and C. Rolland, for encouragement. We are also grateful to several members of the lntegrated Devices and Circuits Research Group, School of Engineering Science, Simon Fraser UniversityZhixin Yan, Arya Raychaudhuri, Mihai Margarit, Plamen Kolev, Xiaojun Zhao, Chihhung Chen, Tim Hardy, Wing Suen Kwan, and Joseph Liang, for their suggestions and valuable comments. The financial support from Centre of System Sciences of Simon Fraser Univcrsity, Natural Science and Engineering Research Council of Canada, and Micronet is acknowledged.
LISTOF ACRONYMS AGC APD BER C-V DD EDFA E/O GBW GR HBT HEMT IM 1R LED LFN MESFET MOCVD MQW MSM NRZ
automatic gain control avalanche photodiode bit-error rate capacitance-voltage direct detection Eribium-doped fiber amplifier electrical to optical gain-bandwidth product generation-recombination heterojunction bipolar transistor high electron mobility transislor intensity modulation infrared light-emitting diode low-frequency noise metal-semiconductor field effect transistor metal-organic chemical vapor deposition multiquantum well metal-semiconductor-metal nonreturn to zero
158 O/E OEIC PIN PLEG PMMA PMT RF RIE SAM SAGCM
C. L.F. MA,M.J. DEEN. AND L.E.TAROP
optical to electrical optoelectronic integrated circuit p-i-n photodiode preferential lateral extended guard polymethyl-methacrylate photomultiplier tube radio frequency reactive ion etching separate absorption and multiplication separate absorption, grading, charge, and multiplication separate absorption, grading, and multiplication secondary ion mass spectroscopy superlattice signal-to-noise ratio wet chemical etching
LISTOF SYMBOLS
bit rate chromatic dispersion of optical fiber material dispersion parameter electron diffusion coefficient hole diffusion coefficient bandgap energy threshold energy for impact ionization threshold energy for impact ionization at 0 Kelvin average optical phonon energy optical phonon cnergy at 0 Kelvin cxccss noise factor (Section 11) electric field electric field in the multiplication layer at breakdown electric field at the interface between the buffer and substrate layers at breakdown electric field at the heterointerface between the grading and boost layers at breakdown electric field at the heterointerface between the boost and absorption layers at breakdown excess noise factor in the case of pure electron injection
CHARACTERIZATION AND MODELING OF SAGCM
159
excess noise factor in the case of pure hole injection dark current diffusion dark current primary dark current unmultiplied leakage dark current GR dark current photocurrent primary photocurrent electron current density hole current density k-factor in the excess noise formula k-factor in the excess noise formula in the case of electron injection k-factor in the excess noise formula in the case of hole injection repeater spacing multiplication (gain, photogain) average multiplication multiplication in case of pure electron injection multiplication in case of pure hole injection refraction index intrinsic carrier density in semiconductor doping concentration in the absorption and multiplication layers doping concentration in the grading layer N I ! , c ~ ,doping ~, concentration in thc boost layer average receiving power of an optical receiver for a certain BER average transmitting power of an optical transmitter power coefficient in the Miller formula responsivi ty noise power in A'IHz thickness of the absorption layer thickness of the grading layer thickness of the boost layer thickness of the charge layer etching depth in the periphery region thickness of the unintentionally doped layers (the buffer and absorption laycrs) temperature bias voltage bias voltage at which the electric field starts to penetrate the boost layer in the periphery region theoretical breakdown voltage
I60 Vh,
vh, Vde,,lcceLl
C. L. F. MA. M . J . DEEN, A N D 1.. E. T A R O F
built-in voltage experimental breakdown voltage bias voltage at which the buffer layer is completely depleted bias voltage at which the multiplication is M bias voltage at which the electric field starts to penetrate the boost layer in the central active region bias voltage a1 which the electric field starts to penetrate the absorption layer in the central active region pf diffusion depth thickness of the multiplication layer the diameter of the active region in our APDs impact ionization for electrons impact ionization for electrons in InP impact ionization for elcctrons in InGaAs impact ionization for holes impact ionization for holes in InP impact ionization for holes in InGaAs attenuation coefficient in decibels in optical fiber power coefficient in the multiplication noise formula (Section VlI) optical wavelength (Sections I and 11) carrier free path (Okuto-Crowell theory) optical long cutoff wavelength rms spectral width of an optical pulse rms spectral width of an optical source quantum efficiency experimental absolute temperature coefficient of breakdown voltage theoretical absolute temperature coefficient of breakdown voltage integrated areal charge density in the charge, grading, and boost layers in the active region integrated areal charge density in the charge, grading, and boost layers i n thc periphery region integrated areal charge density in the charge layer integrated areal charge density in the grading layer integrated areal charge density in the boost layer carrier transit time electron lifetime in n-type semiconductor electron lifetime in p-type semiconductor relative temperaiurc cocfficient of breakdown voltage
161
CIHARACT1,RIZATION AN11 MOI)EI.ING OF SAGCM
E: FInF
:tnc f
:Id
\ b
Xd
x1
x2
xo
x3
FIGLIRI: 64. Thc electric liclils iii SAGCM Iiil’/liiG;iAs A P D .
APPENDIX A: EL.6CTIUC F1f.d2I>IN SAGCM APD The electric field F ( x ) is related to the net doping concentration N ( x ) by the Poisson equation
where E is the relative dielcctric constant and is the dielectric constant of vacuum. I n the following. the one-dimensional Poisson equation is applied to the central region. Before deriving the clcctric field in SAGCM InP/lnGaAs APD, a few assumptions and simplifications are madc: p-‘-n is an abrupt junction. The voltage drop at thc InP/lnGaAs heterointerface (0.1-0.4) is ignored. The doping concentrations in the multiplication, charge, and grading layers are uniform. The thickness of the boost layer is ignored. The relative dielectric constant of thc InGaAsP grading layer is equal to the relative dielectric constant of InP. The InP buffer layer is treated as part of the absorption layer. All the symbols are defined in Figs. 22, 64, and Table 2. The electric field as a function of bias voltage V is derived according to the bias range. Let us define V,,,,,;, as the bias voltage at which the electric field starts to penetrate the boost layer, V,~,,,;,as the bias voltage at which the electric field starts to penetrate the absorption layer, and V,~,,,,,,,,~as the bias voltage at which thc electric field reaches the I I - ~InP substrate. Obviously, V,,,,,,, 5 V,~,,,;,5 Vc~cl,~c,c~~.
162
C. 1.. E M A . M . J. DEEN. AND L. E. TAROF
When the electric field is nonzero in the absorption layer, the electric field F ( x ) is
F(x)= <
with E I and ~ 2 the , relative dielectric constants for InP and InGaAs, and having values of 12.3 and 12.9, respectively. The bias voltage is given by
where Vb, is the zero bias potential and is about 1.O V, and in two cases:
has to be found out
I . V 2 Vc~clcplctcd, (i.e., the absorption layer is completely depleted). In this case, to = t , l , l ~ ~ , p c dwhich r can be substituted into Eq. (82) to derive the bias voltage. 2. V 5 Vdc,,leted, the electric field is zero at the location xo. Thus, to is related to Flnp from Eq. (81).
CHARACTERIZATION AND MODELING OF SAGCM
Substituting Eq. (83) into Eq. (82), to is related to the bias voltage by
V = Vl~l,,, when to = 0. Thus
163
164
C. L. F. MA. M. J. DEEN. A N D L. E. TAKOF
The electric field in the absorption layer is zero but nonzero in the grading layer. The electric fields are
The two punchthrough voltages are related by
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Kagawa. T.. Kawamura. Y.. and Iwamura, H. ( 1992). “lnGaAsP/InAIAs superlattice avalanche phoJ. L ) L { ~ i i i .E / f ~ c t m i QE-28. i. I’p. 141‘)-1423. Kagawa. T., Kawamura, Y., and Iwamura, H. ( 1993). ”A wide-bandwidth low-noise InGaAsP/ InAlAs superlattice avalanche photodiodc with ii Hiychip structure for wavelength 0 1 I.3 and I.5 pm,” IEEE J. Qiiuti. E/ecmnti. QE-29. pp. 1387-1 392. Kanbe, H.. Grosskopf, G., Mikami, 0..and Machida. S. (1981). “Dark current noise characteristics and their teinperature dependence in GermaniLim avalanche photodiodes,” IEEE J. Q i i c i i i . E/rc,/mri. QE-17. pp. 1534-1 539. Kahraman, G., Saleh. R. E. A,, Sargcant, W. L., and Teich.M. C. (1992). ”Time and frequency response of avalanche photodiodes with arbitrary structure,” Trms. E/rc/roti. I h i c ~ r sED-39. pp. 553-560. Kaneda. T. ( 1985). “Silicon and gerinaniuni avalanche photodiodes,” in W. T. Tsang (etl.).Sernic~~tid/trror,soiid srtnitrirto/.\ 22D, Academic Press. Kno, C. W., and Crowell, C. R. ( 1980). “Iinpact ionimlion hy electrons and holes i n InP,” So/ic/-S/u/[, E/rct,niii~~.s 23. pp. 881-891. Kao, K. C.. and Hockhani, G. A. ( 1966). “Dielectric fibre surlace waveguides for optical frequencies,” PMJC.IEE 113, pp. 1151-1 158. Kim. 0. K., Forrest, S. R.. Bonncr, W. A,, and Smith. K. G. (1981). ’ A high gain InGaAs/InP avalanche photodiodes with no tunneling lcakagc current,” A p p / . Phvs. Lett. 39, pp. 4 0 2 4 0 4 . Kobayashi. M.. Yamazaki, S . . and Kaneda, T. ( 1984). “Planar InP/InGaAsP/lnGaAs buried structure APD,” A/@. Phys. Le/t. 45, pp. 759-761. Kuchibhotla. R., and Campbell, J. C. (19911. “Delta-doped avalanche photodiodes for high bit-rate lightwave receivers.” J. Lighrir,tiiv Trc/iti[~/. 9, pp. 900-905. Kuchihhotla, R.. Camphell, .I.C., Tsai, C., Tsang. W. T., and Choa, F. S. ( I 99 I ). “InP/lnGaAsP/lnGaAa SAGM avalanche photodiode with dclta-doped multiplication region,” E/cc./rmi.Lett. 27, pp. I36 I1363. Kuvas. R., and Lee. C. A. (1970). “Qu itic approximation for scniiconductor avalanches.” J. A/J/J/. Pkvs. 41, pp. 1743-1755. 1.1, X. M., Deen, M. J., Stapleton. S., Hardy. K. H. S.. and Berolo. 0. (December 1990). Ct:vogeriI“~.\ 30(12). pp. 1140-1 145. I i u . Y., Forrest. S . . Hladky, J . Langc. M. J.. Olaen, G. H.. and Ackley, D. E. (1992). “A Planar InP/ InGaAs avalanche photodiode with Iloating guard ring and double diffused junction,” J. Lightwrive TP<’htl(J/.10, I’P. 182-192. hki, C. L. F., Dcen. M. J . . and Tarof, L. (14-17 Scptembcr 1993). “D.C. and Noise Characteristics of InP-Based Avalanche Photodiodes for Optical Communication Applications”, Pmcwcling.t of /lie Cmcidiuti Coi!fi.rc.ric.c,oti E/rc,/r.iu/K. Coiiipiirer Eiigitiwritig, Vancouver, pp. 1270- 1273. Ma, C. I.. F, Deen, M. 1.. and Tarof, L. ( I 1-15 Septcmbcr 1994). “Dark Current NoiseCliaractcristics of Separate Absorption, Grading, Charge. rind Multiplication in InP/lnGaAs Avalanche Photodi.’Yo/i&s/~i/r I)evic? Ke.srnrr/i Coi!fi.rc.iicr(ESSIIERC ‘Y4J. odes.” P rncrrditig.~oftlie 24th Ei/f’o[~rm Edinburgh, Scotland, pp. 727-730. Ma, C. L. F., Deen. M . J., and Tarof. L. ( I I - 15 September 1994). “Fast and Accurate Melhod of Extracting Two Critical Device Paraiiieters 0 1 SAGCM InP/InGaAs Avalanche Photodiodes”. 24th Eiiroprciii Solid-Stritc, l h i c . r R~.scwn.h C<~t!f<,f?ii[,~ (ESSDERC ’Y4). Edinburgh, Scotland. pp. 459462. Ma. C. L. F.. Decn. M. J . , and Tarof, L. (Novrmher 1995). “Multiplication in Separate Absorption. Grading, Charge and Multiplication InP/InCnAs Avalanche Phototliodes,” IEEE J o i i r i i r i l o f ’ Q i u i i i t i ~ i i ~ E/rc~troiiic.s31( 1 l ) , pp. 2078-2080. Ma. C. L. E, Dcen. M. J.. and Tarof. I.. (25-27 Septemher 1995). “Multiplicarion in InP/lnGaAs r EffrcJ/Je(lllS~J/id-Sttrtr /)n,ic.r ~ f ~ . s C ‘ r r , r C~Jrrfi.,z~ilc~r ~/I Avalanche PhOlndiOlkS.” Pt-oc.rer/irfg.\o / / / f 2.j//I (ESSDERC ’YSJ, The Hague, Ncdcrland. pp. 765-768.
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ADVANCES IN IMAGING AND ELECTRON PHYSICS VOL. 99
Electron Holography of Long-Range ElectrostaticFields G .MATTEUCCI, G .F. MISSIROLI. AND G . POZZI Department d Physics and lstituto Nazionale per la Fisica della Mareriu University d Bologna. viale B .Pichnt 612.40127Bologna.Italy
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Electron-Specimen Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. the phase-Object Approximation B. Wave-Optical Analysis of the Electron Biprism . . . . . . . . . . . . . C. Effect of the Biprism on the Image Wavefunction . . . . . . . . . . . . . . . . . . . . . . . D. On the Validity of the Phase-Object Approximation E . The Electrostatic Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Recording and Processing of Electron Holograms A. Hologram Recording . . . . . . . . . . . . . . . . . . . . . . B. Hologram Reconstruction and Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Double-Exposure Electron Holography IV. Charged Dielectric Spheres . . . . . . . . . . . . . . . . . . . . . A. Recording and Processing of Electron Holograms . . . . . . . . . . . . . . . . . . . . . . . . . . B . Interpretation of the Experimental Results C . Numerical Simulations of Contour Maps . . . . . . . . . . . . . . . V. P-N Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . A. Experimental Results . . . . . . . . . . . . . . . . . . . . . . B . Theoretical Interpretation . . . . . . . . . . . . . . . . . . . . VI . Investigation of Charged Microtips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Field Model B . Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Conclusions Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
171 174 175 176 179 184 187 192 192 200 205 207 207 210 214 216 216 222 229 230 233 23 5 236 237
I . INTRODUCTION About fifteen years ago it was foreseen that. thanks to the introduction of high brightness and coherence sources such as field emission guns (FEG), electron holography would play an increasing role in electronmicroscopy (Missiroli et al., 1981). Reality has shown itself to be greater than expectations as witnessed by the dedicated sessions at international meetings and by the success of the first international workshop on the subject (Tonomura et al., 1995). 171
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Roughly speaking, we may divide the field of electron holography into two main streams: one, pursued mainly by H. Lichte and his coworkers (Lichte, 1991, 1995), aims at the realization of the original Gabor dream, that is, to correct the spherical and other residual aberrations in order to improve the resolution limit in electron microscopy down to the physical limit set by the chromatic coherence of the electron beam. Unfortunately, the main obstacle is represented by the extremely high accuracy, a fraction of a percent, by which the aberration coefficients need to be known in order to carry out effectively the reconstruction process; in fact, other experimental troubles linked to the instrumentation are about to be solved (Lichte, 1995). The other main stream is related to the application to problems in the medium-low resolution range, where, with respect to the standard phase contrast methods in electron microscopy (Wade, 1973; Chapman, 1984), holography allows the extraction of quantitative information with increased sensitivity limits owing to the use of image processing and phase amplification methods which have no counterpart in electron microscopy (Hanszen, 1982, 1986; Tonomura, 1986, 1987a, 1987b. 1992, 1993). Among these problems, a prominent place is held by the application of electron holography to a basic issue in quantum physics, that is, the significance of electromagnetic potentials, also known as Aharonov-Bohm effect (Aharonov and Bohm, 1959). The lively theoretical and experimental debate concerning this effect (Olariu and Popescu, 1985: Peshkin and Tonomura, 1989) culminated in the outstanding experiments by Tonomura and his group (Tonomura et ul., 1986), who reached an almost ideal shield of the magnetic field by surrounding a micro toroidal magnet by means of superconducting niobium. The offspring of these researches eventually led to the first successful observation of quantized flux lines in superconductors by means of Lorentz (Harada et ul., 1992) and holographic (Bonevich et al., 1993) methods. The interest of our group in electron holography was motivated by its many applications for the investigation of materials science problems. The developments in magnetic information storage technology and microelectronics require the characterization of the magnetic recording media, junction devices, and interfaces in terms of magnetic and electric field distributions. The study of magnetic fields (Matteucci et ul., 1984b3 developed from the first experiments aimed at demonstrating the possibility to display magnetic lines of force in a thin tilm (Lau and Pozzi, 1978) to more recent experiments regarding the magnetic probes used in magnetic force microscopy (Matteucci et ul., 1994; Matteucci and Muccini, 1994; Frost et ul., 1996). As far as electric fields are concerned, following the pioneering work of Titchmarsh and Booker (1972) and in collaboration with the Lamel-CNR laboratory, we started in the mid-1970s to investigate reverse-biased p-n junctions, first by standard Lorentz methods (Merli et al., 1973; 1975), then by means of interference electron microscopy (Merli et ul., 1974) and finally by
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mapping the electric field across a reverse-biased p-n junction by means of off-axis image electron holography (Frabboni et al., 1985; 1987). The problems encountered in the reconstruction of holograms of reverse-biased p-n junctions (Frabboni et al., 1987) demonstrated unambiguously that the longrange field perturbed the so-called reference wave. A basic assumption of holography was thus manifestly violated and in order to assess the consequences of this fact we started to investigate other specimens with long-range electric fields, namely charged dielectric particles (Chen et al., 1989; Matteucci et al., 1991) or biased tips (Matteucci et al., 1992b), having, with respect to p-n junctions, the advantages of an easier specimen preparation and of a simpler theoretical description. Recently, we have reviewed the work done by our group, with main emphasis on the experimental results (Matteucci et ul., 1996); however, for the particular class of problems investigated, the theoretical interpretation is equally important, since on one hand it helps to clarify the obtained results with their sometimes puzzling aspects, and on the other hand it allows the extraction of maximum useful information from the recorded patterns. The aim of this paper is, therefore, to give a balanced review of the work done so far emphasizing that the interest of some issues is not limited to the particular field of investigation, but could be potentially useful to everyone engaged or wishing to engage in the exciting field of electron holography. Section I1 deals with the fundamental theoretical considerations underlying the observation of electric fields. The basic tool, that is, the phase-object approximation, is applied to the case of the electron biprism, in order to obtain its transmission function and to analyze its effect on the image wavefunction within an interferometric or holographic setup. The validity range of the phase-object approximation is then considered, showing that it can be safely applied to the electric fields investigated in this work. Last but not least, the fact that the potentials and not the fields enter the basic equations has the consequence that quantum nonlocal effects can be detected also with electrostatic fields, in close analogy with the more striking and fundamental effects linked to magnetic fluxes. A short account of the basic ideas and experiments is therefore given, as a reminder that sometimes, starting from an applicative problem, we may touch a fundamental issue. Section 111recalls the basic principles of holographic recording and processing, with the modifications introduced by the long-range behavior of a particular class of electric fields, namely the perturbation of the reference wave. Although the main emphasis is on image electron holography by means of an electron biprism, also Fresnel holography using a single crystal film as an amplitude beam splitter is briefly described, since this method can be carried out even if the microscope is not equipped with a field emission gun. Moreover, in the analysis of the reconstruction process we have limited considerations to our experience in the optical realm. Today, with the introduction of digital image recording devices like the CCD cameras (de Ruijter and Weiss, 1992), the way is paved for carrying out in-line the
174
G. MATTEUCCI, G. F. MISSIROLI, AND G. POZLI
whole process of image recording and processing by digital methods. The interested reader is referred to the recent papers of the pioneers in this field (Lichte, 1995; Lehmann and Lichte, 1995; Volkl et al., 1995; Ade, 1994, 1995). However, owing to the complete analogy between Fourier optics and Fourier analysis, the optical considerations reported in this paper can be profitably transferred to the digital realm, although the reverse may not always be true, as demonstrated by the brilliant alternative method to extend the spatial resolution of an electron hologram, realized by Ade and Lauer ( 1994). The last three sections deal with applications of the formerly developed ideas and methods. First, the case of charged dielectric spheres is treated in Section IV. This specimen can be considered as an ideal test object, since it is easy to prepare and is described by a simple theoretical model, which gives an analytical expression for the associated phase shift. Theoretical modeling is considerably more complicated for the case of reversebiased p-n junctions (Section V). Also the specimen preparation is challenging, especially if the p-n junction needs to be biased: According to our experience, this is a mandatory requirement for an unambiguous interpretation of the experimental data. Last but not least, charged microtips are analyzed in Section VI. Fortunately, in this case too an analytical model for the phase shift is available, which allows the interpretation of the puzzling features of the experimental data. In fact, contrary to the naive expectations, according to which the equiphase lines are a good representation of the equipotential surfaces of the field, in this case there is a striking difference which stresses how cautious the interpretation should be. The conclusions complete the chapter.
11. ELECTRON-SPECIMEN INTERACTION
The aim of this section is to introduce the phase-object approximation (POA), which gives the basic theoretical tool for the interpretation of the effects associated with the interaction of the electron beam with electrostatic fields at the mesoscopic level. This approximation will then be applied to the case of the electron biprism which, being the most diffused type of electron interferometer, has been extensively investigated both from a theoretical and experimental point of view. It turns out that the biprism can be described by a very simple and useful transmission function, which is at the basis of the theoretical analysis regarding the effects of the biprism on the image wavefunction. The validity limits of the POA are then considered with particular reference for the case of reverse-biased p-n junctions where some authors claimed that the POA would be no longer valid. However, we will recall and discuss how this conclusion is superseded by new calculations based on the multislice approach.
ELECTRON HOLOCRAI'HY OF LONG-RANGE ELECTROSTATIC FIELDS
175
Finally, the electrostatic Aharonov-Bohm effect is briefly reviewed, and it is shown how nonlocal quantum effects can arise with a particular configuration of electrostatic fields.
A . The Phuse-Object Apprmimcrtioiz Let us recall some fundamental theoretical aspects concerning the interaction of electrons with static electric (Glauber, 1959; Landau and Lifshits, 1965) and magnetic (Wohlleben, 197 1) fields. By considering only elastic scattering events, the solution of the time-independent, nonrelativistic Schrodinger equation in the high energy approximation gives the transmission function (i.e., the ratio between the complex amplitudes of the ingoing and outgoing wavefunctions) as:
T(r) = e'4'"
(1)
where r = (x,y) is a bidimensional vector perpendicular to the optic axis z , which is parallel and in the same direction as the electron beam, and @(r) is the phase term, given by:
The integrals in Eq. ( 2 ) are taken along a trajectory
t parallel to the optic axis
z ; V ( x , y, ,-) and A , ( x , y, z ) are the electrostatic potential and the ,--component
of the magnetic vector potential A(x, y , 2 ) ; e , h , E and h are the absolute value of the electron charge, the de Broglie electron wavelength, the accelerating voltage of' the electron microscope in the nonrelativistic approximation, and the Planck constant divided by 2x,respectively. Moreover, the fact that electrons can be either stopped by a thick specimen or scattered by the specimen atoms at large angles until they are cut off by the aperture of the objective lens can be accounted for by introducing a real amplitude term C(r) in the object wavefunction, so that the Eq. (1) becomes:
In order to understand better the validity range of the above approximation, it is worthwhile reconsidering how it is derived. The starting point is the timeindependent Schrodinger equation:
where the additional constraint div A = 0 has been imposed on the vector potential. In the purely electrostatic case, (A = 0 ), the crystal potential energy eV ( A , y, z ) is considered as a small perturbation with respect to the kinetic energy e E of the incident electron beam (Glauber, 1959; Landau and Lifshits, 1965). Therefore, if the plane wave solution of the unperturbed Schrodinger equation propagating
176
G. MATTEUCCI, G. F. MISSIROLI, AND G. PO%%I
parallel to the optic axis z , is given by: +o
= exp
(T ) 2niz
then the solution of the perturbed Schrodinger equation is looked for in the form: $ = $,ox The resulting equation for
(6)
x is given by: ,
v-x
4n’ + _4niax _+ -VX h 8: Eh’
= 0.
(7)
The phase-object approximation is obtained when the V’x term is neglected; in this case, the equation for x results as:
which can be immediately integrated along the trajectory C to give the first phase term in Eq. (2). Let us introduce, for the sake of completeness, in the same approximation, also the effect of a magnetic field. The equation for x,once the V 2 x term is neglected, turns out to be:
As can be seen, with respect to Eq. (8) we have three additional terms. The first:
can be simply added to the electrostatic term in Eq. (8) and the integration can be carried out in the same manner as before. It ensues that the resulting phase shift is the magnetic contribution reported in Eq. ( 2 ) .Therefore, the standard phase-object approximation for magnetic fields amounts to neglecting the other two terms of Eq. (9), as shown by Wohlleben (1971). This approximation is no longer valid for the magnetic lens case, where the main shift, Eq. (lo), vanishes identically and, as shown by Pozzi (1995), the other two terms are responsible for the focusing and image rotation effects. B . Wave-Optical Analysis of the Election Biprism
The above considerations can be applied to the study of a particular configuration of electrostatic field such as that produced by the electrostatic biprism invented by Mollenstedt and Duker (1956). It consists of a thin charged wire W whose axis is coincident with the y direction and placed between two earthed plates as shown
ELECTRON I-IOLOGRAI’HY OF LONG-RANGE ELECTROSTATIC FIEl.IX
177
in Fig. la. When the observations are carried out near the central region of the wire, far from the edge of the supporting apertures, a bidimensional approach for the electrostatic field can be usefully employed. Two equivalent models have been proposed for the electric field associated to an electron biprism, differing only in their boundary conditions. The first starts from thc field of a line charge lying along the y-axis and placed between two earthed plates as shown in Fig. l b (Septier, 1959); the second considers the field as that arising from a cylindrical condenser as sketched in Fig. l c (Komrska, 197 I ) . Let us extend the Septier approach to the case of an asymmetric line charge placed between two earthed plates as shown i n Fig. 2 (Matteucci et al., 19923). If a is the distance between the plates and h that of the line charge from the left one, by considering a coordinate system as in Fig. 2a it turns out that the potential is given by (Durand, 1966):
I;“ I;“
U
(1
-cos[
n(a - h)
n(s+ h)
] ]
}
(11)
178
G. MATTEUCCI, G. F. MISSIROLI, AND G. PO%%I
(b)
(3)
Flc;lilux 2. Potential (solid) and field (dashed) distribution\ of the electrostatic field due to ;I charged line between two earthed pluleh. (a) Large scale: L I , distance between the twn plates; h , di\tancc e between Ihe l e l t plate and the line charge. (h) Sinall scale iieiir the charged line: R radiw o l ~ h charged cylinder (dashed region).
where o is the line charge density and EO is the vacuum dielectric constant. The trend of the equipotential surfaces (solid lines) and of the lines of force (dashed lines) are reported in Fig. 2a. It can be easily ascertained that near the charged line, the field is negligibly influenced by the boundaries, (as shown in Fig. 2b, which reports an enlarged view of the region around the line charge), and is identical to the logarithmic field of a single charged line without boundaries. The potential is given by:
(12)
r being the radial distance from the charged wire. Therefore, this model describes also the field of a macroscopic charged cylinder of radius R , (dashed central region in Fig. 2b), brought to the potential VW with respect to the earthed plates. The relation between the wire potential, line charge density, and radius R is given by:
2rr EO
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
I79
The phase shift suffered by the electron wave impinging on the biased biprism can be calculated by introducing in the Eq. (2) the potential of Eq. ( 1 1) and by considering that for the electrostatic field of the wire the vector potential is zero. It turns out that the resulting integral can be reduced to a tabulated definite integral (Gradshteyn and Ryzhik, 1980) by means of the substitution cosh(ni/u) = I / r . After some calculations, the following result is obtained for the phase shift:
n a
@(x)= ---A(u
AE El) n u
-
h ) / u for 0 5 x 5 h
( 14a)
(u - x ) h / u for h 5 x 5 a . LIZ E o By taking into account that the electrons impinging on the wire are stopped, and by considering the symmetric case in which h = u / 2 (i.e., the wire is centered between the two earthed plates), the following transmission function adequately describes the effect of the biprism on the electron beam:
@(x)=
--
Tfj(rR)= 0 for
1.1~1
(1%)
where rlj = (,I 8 , y l j ) is the coordinate vector i n the biprism plane (r = z / j ) centered on the wire axis. (Y = c r / 4 E ~is ~the ~ classical angular deflection due to the biprism which, according to Eq. ( 1 3 ) , is directly proportional to the wire potential.
The basic features of an electron microscope modified for interferometry or holography experiments are schematically shown in Fig. 3 . The electron beam emitted by the microscope filament crosses the specimen S under investigation and then, after being focused near the back focal plane of the objective lens Oh, enters in the biprism region. The biprism is usually inserted at the selected area aperture plane and its central wire W splits the incoming wave into two parts, which travel on the left- and right-hand sides of the wire ihelf. The specimen wavefunction in the presence of the biprism can be calculated, according to Glaser’s paraxial theory (Glaser, 1952, 1956), by following its propagation through the microscope in two steps. The first one is that from the object to the biprism plane, the second from the biprism, described by the transmission function of Eq. (15), to the observation plane OP. The remaining lenses PS of the microscope (which includes the intermediate lens and the projector system) provide a further magnification of the intermediate image in the final imaging plane IP. The resulting wavefunction is complicated. However, the main features of the image, including diffraction effects, can be derived using the asymptotic approximation (Pozzi, 1975, 1980a) assuming that the electron wavelength is the small parameter.
180
G. MATTEIJCCI. G. F. MISSIKOLI. A N D G. I’O%%I
Here the main results of this analysis will be briefly summarized. If
represents the wavefunction which is formed at the first intermediate image plane (: = : I ) in the absence of the biprism, the image wavefunction taking into account the effect of the biprism is given by the following expression:
ELECTRON HOLOGRAPHY OF LONGRANGE ELECTROSTATIC FIELDS
181
where XI, y~ are the coordinates in the intermediate image plane, ,I- and y are dummy integration variables, and y is an unessential phase factor. It should be stressed that this expression holds also for : I < :u and can also be obtained by applying the Kirchhoff integral twice between the planes : I and -8. By inserting Eqs. ( IS) and ( 16) in Eq. ( 17) it results after some calculations:
where:
with
and
with wfj = -4-1
+ (:I
- Z8)ff.
(184
$11 and @H represent the wavefunctions to the left and right respectively of the biprism wire. Owing to the oscillatory behavior of the phase, it can be assumed that the main contributions to the values of the integrals arise from small intervals surrounding the singular points s = u,1and s = W A and the points of stationary phase s,,, and S R , which are solutions to the implicit equations: ( Z / - : H ) ( P : (,S,.\,
y / ) - I.(,
(I/ - :/<1(P:
y/)
(SH,
+
J',,
- s/1 -
=0
s/< = 0.
( 194
(1%)
,\I = s,\and .\-/ = so as functions of the parameter y/ define curves in the (,I-, , y l ) plane, which do not depend on the angle and represent the shadow images of the wire edges when the biprism is held at zero volt applied potential.
182
G . MATTEUCCI, G . F. MISSIROLI. AND G . PO%%I
For points far from the unfolded edge, the intervals are disjointed and the asymptotic approximation to the @A can be written as:
whereas across the edge, the asymptotic wavefunction is given by:
where
with 6 = w,\
- SA.
and
F(u,
=
1’‘ [f] exp
iTlEAt-
dr
In the case of no object present, Eqs. (18), (20), and (21) are reduced to the exact wavefunction of a defocused opaque half-plane and to its asymptotic approximations, respectively. Then the two expressions join together because, as long as Iu I > 1.8, no relevant error is made in taking either of the two wavefunctions. If this criterion is extended to the general case (object present), we should assume that the object phase can be approximated by its second-order Taylor expansion over an interval having the same width as the first Fresnel fringe. When this condition is not fulfilled, the asymptotic image presents features such as unexpected
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
183
maxima and discontinuities at the joining points, which indicate the failure of the approximation. Some calculations carried out for the case of ferromagnetic domain walls (Pozzi, 1980b) or reverse-biased p-n junctions (Pozzi and Vanzi, 1982) suggest that a reasonable upper limit for the phase error is about 0 . 1 ~ . The above equations allow the interpretation of the main features of the observed interferograms. When the potential of the wire VU = 0, on the imaging plane IP an in-focus image of the specimen can be observed although part of it is hidden by the shadow of the wire as shown in Fig. 3. The portions of the image on the left and on the right of the shadow are of the same extension, provided that the wire is centered on the optical axis. The shadow of the wire may be distorted if the first partial derivative of the object phase in the direction normal to the wire axis is different from zero; similarly, the second derivative affects the spacing of the diffraction fringes at the edge of the shadow of the wire, as impressively shown in Fig. 4a in the case of a magnetic specimen. These effects are independent of the voltage applied to the wire and may be useful to detect local electric and magnetic fields and thickness variations in the specimen.
FIGIIRE 4. Ellect of the magnctic field of a thin perinalloy film on the hiprism wire shadow. (a) Hiprisni wire at 0 V. The wire h i d o w is allcctctl by the magnetic field of ii thin perinalloy lilni. Convergent C nnd divergent D tlomnin walls can he seen since the specimen image is recovered slightly out of focus. (b) Interference pattern ohtninctl with the hiprism wire zit 4.2 V. The specimen IS the smie iis i n (a). (c) As in (h) hut with the spccimen recorclcd at a larger opposite dcfocus.
184
G. MAI’TELJCCI, G. F MISSIROLI. A N D G. POLLI
On applying a bias voltage to the biprism, the shadow will widen or shrink depending on the overall electron optical conditions. In the case of shrinkage, by increasing VIVthe left-hand side and right-hand side partial images, corresponding to two different regions of the specimen, move toward each other in a direction perpendicular to the wire by an amount proportional to V w , until the shadow disappears. When the voltage is further increased, the two images overlap producing an interference pattern where both amplitude and phase differences between the two waves are encoded in the appearance of modulated interference fringes. It should be noted that the boundaries of the interference field, together with diffraction effects (Fig. 4b), have the same profile as that of the distorted shadow (Fig. 4a), although with left-right inversion. These effects could be better appreciated by Fig. 4c, where the specimen features at the right and left of the biprism distorted shadow are evidenced by the larger opposite defocus, so that the divergent wall D in the upper part of the figure becomes a convergent one, whose bright contrast line can be easily seen also within the interference field. When the interference field is much larger than the first Fresnel fringe width, the diffraction effects due to the wire edge do not affect appreciably the trend of the interference fringes in the central part of the overall interference field. In this case the interferogram can be considered as a hologram, although the borderline between interferograms and holograms is not well defined. As a rule of thumb, the fringe number should be larger than 100,but the larger the better, although the upper limit is set by the finite lateral coherence of the illuminating beam. D . On the Validity of‘the Phase-Object Approximation A question arises as to whether or not the use of the phase-object approximation is always justifed to describe the interaction between the electron beam and electric fields. The answer is based on the knowledge accumulated in the interpretation of a few case studies. As described in Sections IIB, and IIC, from the POA the following simple model of the biprism effect on the electron wavefunction can be justified. The wavefunction is the sum of two waves, each wave describing the diffraction of the electrons, originating from a virtual point source, by an opaque half-plane. A careful analysis made by Komrska et al. (1967) shows a fairly good agreement between the theoretical predictions and the experimental data. Therefore, several efforts have been made in order to justify this simple model and the POA on conceptually more satisfactory grounds. The scattering of electrons by the electrostatic tield of the biprism has been investigated within the framework of the scalar diffraction theory developed by Komrska (1971) for the case of weak electrostatic fields. The wavefunction in the observation plane can be expressed in this case in terms of a diffraction integral. Numerical calculations are, however, necessary in order
ELECTRON HOLOGRAPHY O F 1 A ) N G I I A N G E ELECTROSTATIC FIELDS
185
to derive the intensity distribution of the out-of-focus pattern. It turns out that the intensities calculated according to the POA and to the diffraction integral agree at least to four decimal places. This fact prompted Komrska and Vlachova (1973) to investigate and successfully demonstrate, by means of the method of stationary phase, the equivalence of the two descriptions. For the case of p-n junctions calculations based on the Komrska diffraction integral were carried out by Lo Vecchio and Morandi ( 1979). who, on the contrary. found a striking disagreement with the results calculated according to the POA. Their results led them to state that the POA is hardly tenable for the interpretation of experimental data. The experience recently gained in the analysis of electromagnetic lenses by means of the multislice method (Pozzi, 1995), as well as the continuing interest toward the observation of p-n junctions by TEM methods (Capiluppi c't d., 1995), for which the POA is an invaluable tool, stimulated a reconsideration of the whole issue (Pozzi, 1996a) whose main results are reported here. Let us recall that the basic idea of the multislice method is to divide the electric field under investigation into thin slices perpendicular to the direction of the incident beam and to project each slice into the entrance plane, which acts as a two-dimensional phase object. The propagation of the electron wavefunction between two neighboring slices is then calculated according to the Huygens-Fresnel principle in the paraxial (Fresnel) approxiination (Goodman, 1968). I n the computer-oriented versions of this method, Fourier transforms are used to reduce the convolution between the object wavefunction after the slice and the Fresnel propagator between the two slices to a multiplication in the Fourier space, taking advantage of the existence of appropriate numerical algorithms like that of the Fast Fourier Transform (FFT). The availability of a high-level language like Matheniatica (Wolfram, 1994)allows the writing of transparent codes and the analysis and display of the results with outstanding graphic capabilities. Let us consider the case of a one-dimensional reverse-biased p-n junction parallel to the y-axis, present in a specimen of thickness t , whose internal field is described by the Spivak model (Spivak et ul., 1968),given by:
where V/;.and d are the junction reverse bias and half-width respectively. This topography has the advantages that both the external potential and the phase shift associated to each slice can be calculated in an analytical form (Capiluppi et ol., 1976; Lo Vecchio and Morandi, 1979). Therefore, numerical calculations have been carried out with the multislice method for the same data as those used by Lo Vecchio and Morandi (1979). that is, assuming an illuminating spherical wave originating from a point source at 10 cm from the specimen, of thickness t = 0.3 p m , in which a junction is present with reverse bias \/,( = -4 V and
186
G. MATTEIJCCI, G . F. MISSIROLI, AND G. POZZI
d = 0.4 pm. The observation plane has been placed to out-of-focus distance up to 20 mm. Larger distances were not allowed because, owing to the periodic continuation inherent in the numerical FFT methods, leakage from neighboring intervals introduces severe artefacts in the out-of-focus images. In order to minimize these effects, calculations were made over an interval of width 16 p m across the junction with N = 1024 sampling points, introducing also a modified cosine window. Fortunately, the practical upper limit to the defocus values is not a limiting factor, since, according to Lo Vecchio and Morandi (1979), the larger discrepancies between the two approximations should be more evident for comparatively small values of the defocus distance. Thus, the difference between the rectilinear paths used in the Komrska approximation and the L parallel path used in the POA is more marked. Another important parameter, the cutoff distance, should be taken as low as possible, in order to improve the accuracy of the multislice calculations. By means of the POA this value has been chosen equal to 2Sd, for which it has been checked that the intensity approximates that obtained for larger values of the cutoff with an error lower than 0.001. The upper and lower external fields from the specimen surfaces up to the cutoff have been divided into N.7 slices and the specimen internal tield has been taken as a single slice, for a total of 2 N,s + I slices. Figure Sa reports the results of the numerical calculations, over an interval of 4 p m in width across the junction, for h and E corresponding to 100 kV electrons and the following values of the defocus: 5 mm, curve 1; 10 mm, curve 2; and 20 mni, curve 3. All the images calculated for the POA and multislice case, with N,y = 10 and N s = 20 are indistinguishable. Therefore, Figs. 5b, c, d report on an expanded scale the differences A1 between the POA case and the multislice calculation with N s = 10, curves (1) whereas curves (2) report the differences between two multislice calculations with N s = 10 and N s = 20. It can be seen that, as expected, these differences increase with larger defocus distance: (b) 5 mm, (c) 10 mm, and (d) 20 mm. It is also worthwhile to note that curves ( 1 ) are offset by a roughly constant amount with respect to curves (2), indicating that the main effect of the transition between the POA and multislice approximation is the taking into account of a quadratic term in the phase, corresponding to a weak lens effect. From the data it can be ascertained that the maximum difference between the POA and each of the multislice calculations never exceeds 5 . lopJ; that is, all the calculations agree to three decimal places, as found by Komrska and Vlachova (1973) for the case of an electron biprism. As the multislice approach is an improvement with respect to the Komrska diffraction integral, since in the limit of very large N s the action is calculated along a piecewise rectilinear path approaching more and more the classical electron trajectory, we may conclude that the POA is validated by this approach.
187
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
I
0.0005
A1
-
i(h)
I I
I
1
2
-0.0005
~
1 c
2 2
I
'd)
FIGLNE5. (a) Intensity distrihu~ion\I iici-o\s the out-ol-l'ocus image ol the p-n junction for the following values of the dcCocus: curve ( I 5 iiini: curve (2), 10 nini: curve ( 3 ) . 20 nim. (h. c, tl) Diffrrences A / between the POA C I I S C and tlic inultislice calculation with N.7 = 10. CUI-VCS ( I ) nncl hetween two multislice calculations with N y = 10 and N.5 = 20. curves (2) for the ahovc v:iIues oithc delocus: (b) 5 nini: ( c ) 10 mm: (d) 20 iiini.
E. The Electrostatic Akuroiiov-Bohni Eflect
In 1959, in a famous paper, Aharonov and Bohm (1959), hereafter referred to as AB, called attention to the significance of the electromagnetic potentials in quantum theory. They proposed two different electron interference experiments in order to test their conclusions. The first of them, concerning the effect of the vector potential associated to a static magnetic field, has stimulated a wealth of experimental work (see for reviews Olariu and Popescu, 1985; Peshkin and Tonomura, 1989), which shows the attempts of the experimentalists to satisfy the ever increasingly stringent conditions required by the theoreticians, especially by those who do not believe in the effect. Much less attention has been paid to the second experiment regarding the electric scalar potential: In this case a coherent electron beam is split into two parts and chopped. Subsequently, each part is allowed to enter a long cylindrical metal tube, the electric potential of which is varied only when the electron wave packets are well inside. The beams are then recombined to give an interference pattern. This experiment (which so far has never been carried out) should show a phase difference due to the time-dependent scalar potential even though no force is ever exerted on the electron wave packets.
I 88
G MATTELJCCI. (i t- MISSIROLI. A N D G I’OLLI
In 1973 Boyer, in his considerations on the AB effect, noticed that, if the experiment involving time-dependent electric fields is carried out by static potentials, its result will be very similar to that produced in the magnetic AB effect. When electrons enter and leave the tubes, they experience classical electrostatic forces, which produce no net change of momentum or energy but only a classical time lag. This can be related to the phase difference A+ calculated in the WKB approximation (Boyer, 1973) through the de Broghe wavelength h:
where AV is the potential difference between the two tubes of length I , and E , in the nonrelativistic case, is the accelerating potential. A different point of view in considering these experiments has been expressed by Aharonov (1984) who, in addition to the “true” AB effects (which are defined as type-1 nonlocal phenomena), introduced a new kind of quantum nonlocal phenomenon (referred to as type 2). In the type-2 phenomena the particles experience local interactions with fields (or other forces), which result in a change in their semiclassical action independent of the trajectory, and hence a change of phase for the quantum state of the particle. The electrostatic AB experiment proposed by Boyer (1973) can therefore be regarded as a nonlocal type-2 phenomenon. The use of the two tubes proposed by Boyer (1973) requires a highly sophisticated experimental setup. A modified version of Boyer’s experiment was realized with a single tube by Schmid (1984). To overcome the difficulties inherent in the realization of two microtubes, we conceived a different and simpler method (Matteucci et ul., 1 9 8 2 ~ ) . The central wire of the electrostatic biprism was evaporated on one side with a thin layer of a different metal, (Fig. 6a). In this way, the contact potential difference A V between the two metals produces an electrostatic potential distribution around
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROS'I'A'rIC FIELDS
189
the wire given by:
V ( x ,3 ) =
~
n
arctan
2 Rs
The following facts should be noted: If the radius R of the wire tends to zero, while R A V = const, the potential distribution becomes that of a line of dipoles and it can be easily verified in this case that the electrons do not suffer any lateral deflection and leave the bimetallic wire with the same energy as when they enter its field (Matteucci and Pozzi, 1987; Boyer, 1987). Moreover, a constant phase difference given by:
is introduced between the two split parts of the electron beam, which travel 011 both sides of the wire. These same conclusions hold also for the wire with finite diameter (Matteucci et al., 1 9 x 2 ~ )By . comparing Eq. (25) with Eq. (23), it turns out that the device is equivalent to two tubes of length 4a. For electrons accelerated at 100 kV, with a wire radius o f 0 3 p m and a contact potential difference of 0.5 V, the phase difference amounts to 1.617. This effect can be explained also on the basis of the following heuristic considerations: Roughly speaking, the contact potential difference between the two metals causes a charge redistribution, Fig. 6b, in such a way that the resulting field is equivalent to that produced by two parallel linear charge densities of opposite sign (no net charge on the bimetallic wire), which are laterally displaced, one with respect to the other, Fig. 6c. Therefore, the bimetallic wire can be modeled by a system of two biprisms of opposite power. In the region between the two lines of charge the phase shift varies linearly; this effect cannot be observed for the case of the bimetallic wire because the wire itself acts as an impenetrable barrier for electrons, but can be revealed in the case of two wires carrying opposite charges, allowing thus the measurement of' the total phase difference. This consideration has led us to develop a further shifting device (Matteucci and Pozzi, 1987). it consists of two parallel conducting wires held at opposite potentials by an external voltage supply. They act as a macroscopic line of dipoles with the additional advantage, with respect to the bimetallic wire, of controlling its strength. 1. E , ~ p ~ ~ i r i i e i t Mctkods tuI utrd Rewlts
u. Binietallic Wire. A schematic drawing of the whole setup of our first experiment (Matteucci et ul., 1 9 8 2 ~is) shown in Fig. 7a. The coherent electron beam coming from a field emission source S propagates to the biprism plane. located at the level of the selected area aperture of a Philips EM 400T electron microscope. The wire W was coated laterally for half of its length with a thin layer of gold
190
G. MATTEUCCI, G. F. MISSIROLI, AND G. I’OTLI
FIGLIKE 7. (a) Schematic drawing for the electron interference experiment. S , electron source; u’ and P . wire and earthed plates of the biprim: OP observation plane. (b), (c) interference patterns corrcspontling to the uncoated (h) and coated (c) part of thc biprism wire.
(black region), thus becoming a bimetallic biprism. The biprism wire splits the wavefront of the incoming beam and its electrostatic field produces a deflection and a subsequent overlapping in the plane OP below the wire, where a system of interference fringes will be observed. The interference fringe systems of the wire recorded in correspondence of uncoated (Fig. 7b) and coated (Fig. 7c) regions are shown in the right part of Fig. 7. The displacement of the interference fringe system due to the constant phase difference with respect to the unperturbed diffraction envelope is clearly visible through a change of symmetry of the pattern: The central maximum corresponding to the uncoated part becomes nearly a minimum in the coated part, thus indicating a phase difference of about n. In this experiment, the phase difference cannot be varied since the rotation of the wire around its axis is not allowed. The dependence of the effect on the angle 6’ cannot be revealed by this setup. Different experiments were made to display this dependence using interference electron microscopy (Matteucci et al., 1984a) and diffraction methods (Matteucci and Pozzi, 1985). h. Macroscopic Dipole. This experiment has been realized by inserting in a special specimen holder, provided with electrical contacts, a macroscopic electric dipole D , (Fig. 8a). Two platinum wires soldered on platinum apertures are superimposed and electrically insulated (Matteucci and Pozzi, 1987). The wires were oriented in parallel and the whole assembly inserted in the microscope. The lower wire was earthed, whereas the upper one could be biased by means of an external voltage supply. Observations were carried out by means of interference electron microscopy, the instrument being equipped also with a conventional electron biprism, Fig. 8a. Quite
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
191
FIGIIRE X. (a) Schematic setup for the inlerlcrcncc fringe formation in the case o f a macroscopic tlipolc D acting as shifting device. The C S W I I I I ~ cleiiients are (lie siiine iis i n Fig. 721. ( b ) Interlcrencc pallern nccording to (a). D I and 111 pro,jectcd dipole sliudows: N region of the interference pattern wlicrc the pIin\c shift is revealed: M unnfl'eclcd interference region.
surprisingly, by applying 24 V to the upper wire, the condition could be achieved when the two wires carried opposite charges. This fact was easily detectable by observing the relative alignment of the outer system of interference fringes. In this arrangement the interference region in the OP plane due to the biprism is divided into three parts by the projected shadows D of the macroscopic dipole. In the outer regions M , provided the wires are carrying opposite charges, two wavefronts with the same phase shift overlap and, therefore, a symmetric fringe system is expected, equal to that observable with the biprism alone. In the region N between the shadows, the waves coming from the opposite sides of the dipole overlap. Thus, electrons experience different phase shifts and a resulting phase difference arises. The net effect is a lateral displacement of the interference fringes with respect to the unperturbed system or to the diffraction envelope. Owing to the different geometry and to the value of the electrostatic potential of' the dipole, the effect is much larger than in the case of the bimetallic wire. In our arrangement, by tilting the assembly by a11 angle of 1 degree, it is possible to observe directly on the screen the changes in the interference fringe system between the two shadows, whereas the outer fringes are stationary. The static recording on a photographic plate does not adequately render the observed phenomena, as shown in Fig. Xb. However, by tilting the macroscopic dipole in order to obtain an equivalent configuration of the two lines of charge as in Fig. 6c, electrons are allowed to travel also in the region between the two wires.
192
I)?
G. MATTEIJCCI, G. F. MISSIIIOLI. A N D G. PO%%I
Flciont 9. ( a ) Interferencc pallern with the dipolc rolii~ccls o tliiil h e two wire h i d o w . ; /Iand l iio longer overlq?piiig. (h) Magnitied view of (lie region hclwccn D I and D z .
;IIK
it is expected to observe not a single shadow D as in Fig. Sb, but the projected shadows of both the two wires of the dipole. In the interference pattern shown in Fig. 9a the shadows D I and L)? of the two wires are clearly distinguishable together with the region between them, shown enlarged in Fig. 9b. The linear increase of the phase is evident and it can be estimated that the phase difference is larger than 407s. These results show that a quantum phase shift can be introduced by a macroscopic dipole and detected in an interference experiment. This phase shift arises through a local interaction of electrons with the electrostatic field of the dipole and can be interpreted either as a classical lag effect or as a local effect of the electric scalar potential. A deeper analysis has also demonstrated that this effect is due to the dipole field and cannot be interpreted as arising from the asymmetric section of the bimetallic wire (Matteucci et al., 1992a).
111. RECORDING AND
PROCESSING OF
ELECTRON HOLOGRAMS
The close similarity between light optics and electron optics is used to classify interferometry devices into two categories: ( 1) division of amplitude (e.g., Michelson's
ELECTIION HOLOGRAI’I-IY OF LONG-RANGE: ELEC~ROSTATICFIELDS
I93
interferometer in light optics) and (2) division of wavefront (e.g., Fresnel biprism). For a review see Missiroli et ul. (198 I ) . The electrostatic biprism (Section IIB), which belongs to the second category. is by far the most used device in electron optics and its perforniances will be discussed further in Section IIIA2. Moreover, for sake of completeness, in Section IIlA 1, we present the operating principle of an amplitude division interferometry, origitially realized for holography purposes by Matteucci et ul. (198 I , 19X2a) and Pozzi (1983).
I . Awiplitirdc Di\i.sion Itir~~t:fi.r.otiictr:\. The electron optical setup of the interferometer is shown in Fig. 10. A thin single crystal C, used as amplitude division beam splitter and inserted in the standard specimen holder, is oriented in such a way that one first-order Bragg reflection is strongly excited. The objective lens Oh forms the Fraunhofer diffraction pattern in its back focal plane. where the two spots 1 and 2 (the zero order and one of the tirst order, respectively) are selected by the objective aperture A . The illumination is tilted so that the microscope optic axis bisects the angle formed by the direct and the Bragg-reflected beams. The specimen S under investigation, assumed to be a thin film of constant thickness, is inserted below the objective lens, in our case at the level of the selected aperture plane. A lattice fringe system is formed i n the observation plane OP, which is cotijugate to the final imaging plane through the
194
G. MATTEUCCI, G. F. MISSIROLI. AND G. POZZI
remaining lenses of the microscope. As depicted in Fig. 10, the plane OP does not coincide with the specimen plane so that an out-of-focus image of S is observed in the plane OP, superimposed on the lattice fringes of the single crystal C . These fringes will no longer be straight and parallel as they are in the absence of the object but are modulated by the phase shift introduced by the specimen. This interferogram is called the Fresnel hologram. The performance regarding intensity, coherence, and versatility of an electron microscope equipped with an amplitude division interferometer compared with one which uses an electrostatic biprism will be briefly summarized, although some basic differences in the two setups render this comparison rather difficult. In both cases it is necessary to insert at the level of the diffraction aperture plane either the object or the electrostatic biprism. In the case of the electrostatic biprism the main difficulty is the production of the thin conducting wire; however, once mounted, the interferometry device can operate with specimens which can be easily changed through the standard specimen air-lock. Moreover, it is possible to observe the specimen both out of focus and in focus. The fringe spacing can be varied simply by changing the biprism voltage. However, if interferograms with a large number of fringes are required, it is necessary to use a field emission gun. In any case diffraction effects modulate the intensity of the interference fringe system near the edges of the interference pattern. In the present case, unless an additional air-lock is built, the specimen change requires a break of the vacuum in the column. However, as the diffraction aperture holder can carry up to three apertures, three specimens can be observed on each run. The limitation, due to the fact that the fringe spacing can be varied only by changing the crystal orientation or the crystal itself, is compensated by the much less experimental effort required in setting up the beam-splitting device. Moreover, with respect to the electrostatic biprism, a much wider interference field, limited only by the lateral dimension of the crystal, is available; edge diffraction effects are negligible since the crystal is observed nearly focused. Other points to consider are the brightness and the coherence. In the amplitude division interferometer the weakening of the beam, due to the crystal thickness, is compensated by the greater intensity available and by the less stringent coherence condition for illumination. Recently Ru et al. (1994) have adopted the amplitude division interferometer in a modified form to record electron holograms of latex spheres and charged microtips (Ru, 1995a, 1995b). Problems related to the formation and reconstruction of electron holograms in a non-FEG, nonbiprism TEM using an amplitude division interferometer have been recently analyzed by Wang (1995).
2. Wuvejkont Division Iiiterferonietry
a. Theoretical Considerations. As we have demonstrated in Section 11, the interferometric technique is a very powerful method to reveal the phase difference
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATlC FIELDS
195
introduced by electrostatic fields. Such a phase difference is measured directly i n the interferogram, for example, as a deviation from the rectilinear trend of the central brightest fringe. The achievement of a complete map of the phase variations related to electromagnetic fields through a thin foil or of the stray fields outside it can be obtained i n a much more appealing fashion using electron holography. As already discussed in Section IIC, by applying a suitable voltage to the wire, an overlapping region can be obtained, as the two object wavefunctions, each one passing on either side of the biprism filament, are shifted respectively by +D/2 and -D/2 in the direction normal to the wire, D being the vector that connects the points brought to interfere. Henceforth, the modulus of D will be called the interference distance. In this overlapping region the total wavefunction, referred back to the object plane, is described by the following equation, which neglects diffraction effects due to the biprism edges and unessential multiplying phase factors (see Section IIC):
where f, parallel to D and perpendicular to the biprism axis, is the spatial frequency wave vector corresponding to the interference fringes, referred to the object plane. In absence of the object, it is found that the image displays an interference pattern in which the fringes are parallel to the biprism axis and spaced s = l/lfl. The trend of these fringes is modified by the object wavefunction in such a way that information on it can be obtained either by the direct analysis of the fringes, when these are few and refer to one-dimensional objects (interference electron microscopy), or by analogic or digital processing of the interferogram when the fringe number is increased and the interferogram can be considered a hologram. Let us analyze first the ideal situation, reported in Fig. 1 l a in which a plane wave PW illuminates a specimen S . Only that part of the wave 0 which has passed through S suffers a phase modulation. The reference wave R travels outside the specimen rim through a field-free region and is not affected by any field (Matteucci c t al., 1991). The biased biprism provides the superposition of the object and reference wave. In this condition the reference wave can be written as:
+
+
From Eq. ( 3 ) it follows that C(r D/2) = 1 and @(r D/2) = constant = so that the interferogram results as the superposition of the object wave +,l,,,(r)= T ( r - D/2) with a reference wave +,c, (r) = T ( r D/2) = e'@".
+
196
G. MATTEUCCI, G. F. MISSIROLI. A N D G. I’O%%I
LZc7/
PW
(h)
(it)
FIOUKF:1 1 . Sketch of clcciron hologram lomiation with ( a ) n relei-cncc planc w:ivc iind ( b ) ii perturbed reference MWC. P W . incident pliine wave; .S. \pccinicn: 0 and K . ob.ieci and relcrcncc wnve: CV. biprism wire.
The intensity distribution is therefore given by:
[(
x c o s Cp r - -
9
1
-Cpo+27rf.r.
(28)
It should be noted that, apart from the unessential constant phase factor 40. the object phase Cp is stored in the hologram. The situation is completely different when the specimen gives rise to long-range electric and/or magnetic fields as sketched in Fig. 1 l b where two electrostatic charges (black dots) generate a field, which extends all around the object and will perturb electron motion (Matteucci Pt ul., 199 1 ). The resulting phase-modulated reference wave can be written as:
ELECTRON HOI,OCRAI'HY OF LONG-RANGE ELbCTROSTATlC FIELDS
I97
The intensity distribution in the hologram becomes:
In this case the hologram stores the information due to a fictitious specimen, whose amplitude and phase are given by: (31a)
(31b) h. E,tperinientul Setup. Since the specimens are rather coarse phase objects, to record electron holograms of long-range electric or magnetic fields, it is necessary either to use low-magnification objective pole pieces or to switch off the objective lens in order to obtain useful holograms with a large enough interference tield. This operating mode has been chosen involving a minimum modification of the instrument. A Philips EM400Tequipped with an FEG was used in the experiments presented in this review. The essential electron optical arrangement for hologram formation of electric and magnetic fields is sketched in Fig. 12. A coherent beam illuminates the specimen S placed off-axis so that the reference beam travels outside the object rim. The objective lens is switched off and the microscope operates in the diffraction mode. The electron interferometer operates at the selected area plane. The intermediate lens, included in the projector lens system P S , is used to focus the specimen plane in the imaging plane IP. The final magnification is in the range of ( 1000-2500) x . In this case the biprisin should be negatively biased, giving a virtual hologram on the specimen plane, which becomes a real one in the image plane. Condenser lenses (not shown in Fig. 12) are usually strongly excited, in order to have the highest possible lateral coherence on the specimen plane: This means that often it can be safely assumed that the specimen is illuminated by a plane wave. However, especially for electron microscopes equipped with field emission sources, the demanding lateral coherence requirements can be met also with the effective electron source located at a smaller distance from the biprism and object planes. In this case illumination can be better modeled by means of a spherical (instead of a plane) wave. The divergence of the illumination introduces the following modifications of the main features of the interferogram, which can be accounted for by simple geometric optical considerations (Missiroli et ul., 1981). Let us denote by u the
198
G. MATTEUCCI, G. F. MISSIROLI, AND G. PO%%1
F l c ; w t 12. Schematic ray diagram for hologram formalion. S,specimcn; W , biprism wire: P.S, pro,jector ICII\ systcm; IP. imaging pliine.
distance between the effective source plane and the biprism plane, u being positive if, along the optical axis, the source plane precedes the biprism plane, and negative in the opposite case. Similarly, b is the distance between the biprism and specimen planes, b being negative when the specimen plane precedes that of the biprism.
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
199
The interference distance D is then given by:
D = 21ha1,
(32)
where a is the biprism deflection angle. D is unaffected by the beam divergence, which modifies both the fringe spacing s and the interference field width Wr;,given respectively by,
where .so and 2R are, respectively, the fringe spacing and the biprism wire shadow for parallel illumination. It should be noted that the same projection factor ( a h ) / u enters the fringe spacing and the projected shadow of the biprism plane, both referred to the specimen plane. The same consideration holds if diffraction effects are taken into account by means of the asymptotic approximation, Section IIC, (Pozzi, 1975; 1980a). It ensues that both the shadow effect (i.e., the deformation of the biprism edges caused by the presence of the specimen) and the defocus distance entering in the Fresnel diffraction fringe spacing around the edges follow the same similarity relation with respect to the plane wave illumination. Therefore, the possibility of varying the divergence of the beam offers an additional degree of freedom to the experimenter: In fact, keeping (Y fixed, and hence D , because h is dictated by the experimental setup, it is possible to vary the fringe spacing and the interference field width according to Eqs. (33) and (34). These considerations are illustrated by our experimental observations on an array of parallel reverse-biased p-n junctions with a spacing of 8 p m and an expected depletion layer width of 1.4 pm (Frabboni et ul., 1987). Figure 13a shows a hologram taken with a reverse bias of 2 V and a voltage applied to the electron biprism of 1 8 V. The first condenser of the FEG was at its maximum excitation, and the second condenser formed the image of the source above the specimen, inserted in the normal eucentric position. The interferogram contains about 200 fringes, of spacing 45 p m , and the interfering distance D is 5 p m . The electron-optical magnification was 1 8 O O x . In order to increase the interfering distance, the objective lens was weakly excited (0.6 A); this allowed the formation of the effective source image below the specimen and hence to suitably vary u and the associated projection factor. The hologram obtained under these new operating conditions is shown in Fig. 13b. The biprism potential has been raised to 40 V, corresponding to a doubled interference distance with respect to Fig. 13a. The electron-optical magnification was 1200x and the fringe spacing 70 pin. It can be ascertained that the projection factor has been increased by a factor of four, which corresponds to the doubling of the
+
200
G . MATTEUCCI. G. F MISSIROLI, A N D G. I'OLL1
FIOIIKE13. Electron Iiolograms laken with: ( a ) t l i r ohjectivc lens switchcd off, and (1)) :I \ ~ c ; ~ h l y excited oh,iectivc lens. After Frahhoni ('I o/.. 1987; reprinted with kind pcrmi\ion of Elhevier Science.
Fresnel fringe spacing at the biprism edges. Moreover, in Fig. 13b, we also have a stronger deformation of the edges (shadow effect) due to the same amplification effect of the projection factor, and possibly to the rotation between specimen and biprism axis introduced by the objective lens. Finally, the interference field is not doubled because part of the increased interference distance is lost in the enlarged projected shadow of the biprism wire. The exposure times were, in both cases, about 10 s. Although our instrument was equipped with an FEG, with an expected brightness of about 10' Acni-'sr-', the measured brightness was, in our case, an order of magnitude lower, in agreement with other measurements (Hanszen et ul., 1985).
B. Hologram Reconstiuction und Processing 1. Theoretical Considerutiom
As is well known, Gabor's idea (Gabor, 1948, 1949, 1951) was to recover the information contained in a hologram by optical means. This is the second step of the
ELECTRON HOL0012APMY OF LONG-RANGE ELECTROSTATIC FIELDS
201
holographic method consisting in the reconstruction and processing of the object wave stored in the hologram. In other words, the electron hologram, registered on a photographic emulsion, is tirst developed so that its amplitude transmittance becomes a linear function of the recorded intensity given by Eqs. (28) and (30). The plate is then inserted in an optical bench and illuminated by a coherent plane laser wave $1, given by:
where kn is the spatial frequency vector of the optical wave. The wavefunction after the hologram is given by: @(r) = e ~ 2 ~ kr I(r) 0 = @(O)
+
@(+I)
+
@(-I)
The three terms in Eq. (36) correspond respectively to: ( I ) the intermodulation wave @((,), which propagates in the same direction as the laser beam and is therefore spread around the spatial frequency ko; ( 2 ) the primary wave +(+ 1 ) and its twin wave @(-I), separated by the term $(o,. These two waves spread around the spatial frequencies (ko f) and (ko - f).
+
The object phase information we want to decode is contained in the wave term which can be isolated from the others, since the beams travel in different directions and achieve their maximum spatial separation in the back focal plane of the reconstructing lens, where an aperture can be inserted (see Section IIIB2). In order to display the phase information stored in the optically reconstructed beam, it is possible to overlap it to a plane optical interferometric wave @,((I-): @K(r)= ~
~ ( ~ l rO+ d nJ f ok ~
(37)
where C K is the amplitude, kK the spatial frequency wave vector, and 4~ the longitudinal phase, so that the observed intensity in the image plane is given by:
+ = [~'(r)]+ ' [c,1' + 2C'(r)CK x cos[A@(r)+ 2 ~ r k l. r
I(r) = [+(&I)(r)@l((r)]'
-
(38)
where kl = (ko - kK +f) is the resulting spatial frequency wave vector. In the wave vector kl unknown contributions are also included, being due to the misalignment of the optical bench or of the hologram plate, not always under the
202
G. MATTEUCCI, G. F. MISSIROLI, AND G POL7.1
FIC;LiKE 14. Recoii\truclioii of an electron hologram using an in-line optical bench. L . laser: HE. heirin expandel-; H , hologram; RL, reconstruction lens: F. filter; S , observation \creen.
experimenter's control. In this optical interferogram the trend of the interference fringes indicates the local phase distribution of the fictitious object. Contrary to electron interferometry experiments, azimuth and spacing of the fringes can now be adjusted to meet the actual needs. In principle, by putting k, = 0, a particular interferogram called a contour map is obtained. This is very useful because a direct physical meaning can be attached to these fringes (Wahl and Lau, 1979; Tonomura, 1986, 1987a, 1987b, 1992). In fact, when no leakage fields are present, according to Eq. (2), in the case of ferromagnetic specimens the optical contour fringes represent the lines of force of the magnetic field (averaged along the electron trajectory), whereas in the electrostatic case they represent the equipotential lines of the projected potential. 2. Optical Reconstruction and Phase Detection N. In-Line Optical Bench. The conventional reconstruction of the electron hologram can be carried out with an in-line optical bench (Fig. 14) equipped with an He-Ne laser source L and a beam expander BE so that a plane wave illuminates the hologram H . In the back focal plane the reconstruction lens RL performs the Fourier transform of the intensity distribution of the hologram H . A spatial filter F intercepts all the other beams except the (+1) allowing a free propagation of the (+1) beam. In this way the inverse Fourier transform is performed and in the observation screen S the intensity distribution replica of the object wavefunction is displayed. If the screen location can be freely varied along the optical axis, its position can be chosen to correct the defocus aberration present in the electron hologram (Hanszen, 1982, 1986; Tonomura, 1986, 1987a, 1987b, 1992; Matteucci et al., 1984b). In order to obtain the phase information from the hologram in the form of a contour map, a parallel coherent optical interferometric wave is superimposed on the reconstructed image wave. This can be achieved by inserting in the inline bench of Fig. 15 the hologram H2 of the object to be investigated in contact with the hologram H I recorded without the object (Wahl and Lau, 1979). The
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
203
FIGLIKE 15. Linage reconstruction a n d phase detection with an in-line optical bench. H I elccti-on interference pattern without the objcct; H: ol>ject Hologram: the otlicr elenients arc the S a m as in Fig. 14.
interference pattern H I gives an additional plane wave, which superimposes on the reconstructed wavefront. The two waves are then filtered and imaged as discussed before with reference to Fig. 14. In the observation screen the parallelism condition is realized when the interferometric wave is made parallel to the background wave (i.e., a plane wave) of the reconstructed image (Hanszen, 1982) in such a way that this region presents a contrast as uniform as possible, which may be bright or dark according to the longitudinal phase difference I$R between the waves (Hanszen, 1982, 1986). It is clear that, for objects whose fringing field affects the electron reference wave (Matteucci et a/., 1991, 1994),the contour maps cannot be pursued because:
I . there is no unequivocal criterion for the parallel superposition of the “presumed” reconstructed object wave with the optical reference one; 2. the object phase is “buried” in the phase difference registered in the hologram and there is no way to “unearth” it with optical processing. In any case even if point (1) is overcome, at the end of the process only the fictitious and not the original wavefunction is reconstructed. b. Mach-Zehnder. Intet-jfernmeter and Phuse Aniplijication. A more versatile optical setup used for the reconstruction of the hologram is the Mach-Zehnder interferometer, as it allows many different processing schemes to be performed in addition to the standard ones used with the in-line bench (for reviews, see Hanszen, 1982, 1986). In particular, we used the Mach-Zehnder interferometer reported in Fig. 16 to carry out experiments of phase difference holography. By processing two holograms H I and H2 of the same object in two different states (in our case a p-n junction at two different reverse applied potentials), the change of the phase corresponding to the change of the state can be displayed. In this case phase changes correspond to different projected equipotential configurations. A coherent light wave, coming from a laser L and a beam expander BE, is divided into two beams by the beam splitter A. The two beams are deflected by mirrors M to illuminate the two holograms H I and H2. A second splitter B allows the parallel recombination
204
G. MATTEUCCI, G. I;. MISSIROLI. A N D G. PO%%I
FIC~OIW 16. Mach-Zehiidcr intcrfei-oiiictei- for hologram I-ccoiistructioii iiiid pi-occshing. A mid 11. heum splitters: M . mirrors: the other clcnient\ arc the snme ;I\ i n Fig. 14.
of the two beams and the formation of images and contour maps on the final screen S as described before for the in-line bench. The versatility of this bench is demonstrated by its use for obtaining phase difference amplification maps, which give more detailed information concerning the trend of equiphase lines (see Tonomura, 1986, 1987a, 1987b, 1992). In Fig. 17 is sketched the basic phase difference amplification setup using a Mach-Zehnder interferometer. The electron hologram H is illuminated by the two coherent beams coming from the two interferometer arms and each of them forms a set of ( + I ) , (0) and (-1) diffracted beams. The mirrors are adjusted to superimpose on the focal plane of the lens RL the ( + I ) order of the beam, which comes from one arm to the (- 1) generated by the other one. The spatial filter F prevents the propagation of all the other beams. In this way the optically reconstructed object wave ( ( + I )
ELECTRON IIOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
205
beam) and its coiljugate wave ( ( - 1 ) beam) interfere, thus giving the desired two times amplification. More generally, a 2n-time amplification factor can be performed by developing the electron hologram nonlinearly and using higher-order diffracted beams (+n) and ( - n ) (Matsumoto and Yakahashi, 1970), or by using the iterative method suggested by Bryngdahl ( 1969).
C. Double-Exposure Electron Holography In the foregoing section we have briefly outlined the problems related to achieving the contour map condition. In fact, it is impossible to determine unequivocally the parallelism between the optically reconstructed object wave and the reference wave (see Section 11IB) used to extract phase information. This ambiguity can be removed by the electron double-exposure method (Wahl, 1975; Matteucci ef al., 1988). An off-axis image hologram is first recorded as described in Section IIIA2 and, after having removed the specimen from the beam, the interference pattern between two unperturbed waves is recorded on the same plate (this last recording provides the most reliable reference wavefront). This recording process guarantees the parallelism between the object and reference wave vectors, since these waves are registered with the same tilt angle (which is related to the biprism potential). The whole procedure is analytically described by considering that the intensity function of the first hologram is given by: Il(r) = 1
+ coslA@(r)
-
2 n f . r].
(39)
The second hologram is simply a linear grating, whose intensity is: Il(r) = I
+ c o s ( 2 n f . r),
(40)
so that the total intensity stored in the double-exposure hologram is, apart from an unessential multiplying factor:
flol(r)= 2
+ cos(2nf. r) + cos[A$(r)
-
2 n f . r].
These phase-dependent intensity variations will appear directly on the plate once it has undergone a linear photographic development. Figure 18 shows a double-
exposure electron hologram of charged latex spheres. The most interesting feature of this micrograph is the contrast variation of the interference fringe system, which presents regions of a strongly reduced contrast in the form of bands. These map the in-plane projected potential distribution of the electric field directly (Matteucci etal., 1988, 1991; Chen el a/., 1989). If the double-exposure hologram is inserted in the optical bench of Fig. 14 and is illuminated by a plane laser wave, the carrier fringe system is removed in the reconstructed image, thus leaving an optical interferogram alone which
206
G. MATTEUCCI. G. F. MISSIROLI, AND G. I‘OLZI
FIGLIKE18. Double-exposure electron hologram of electrostatic charged spherical particles 21ssernhled o n thc rim of a carbon film. The regions where the interference fringes are bluri-ed map llie projccted potential distribution.
displays the map of the phase difference A$. Moreover, contrary to the optical interferometry techniques, it should be emphasized that double-exposure electron holography also eliminates any possible additional longitudinal phase term. As experimentally demonstrated in the following section, the importance of recording double-exposure holograms is due to the fact that in the reconstruction and processing of a standard hologram taken with a perturbed reference wave, no objective criterion exists for determining the correct phase-difference map condition (i.e., for recovering the object phase unambiguously). These problems become more serious when phase-difference amplification techniques are employed (Tonomura, 1986, 1987a, 1987b, 1992; Tonomura et al., 1985; Hasegawa e l al., 1989). With the lack of a double-exposure hologram which could be used as a reference to compare the phase-difference maps, the physical interpretation of the final amplified map suffers the same shortcomings as for those obtained with the optical arrangement shown in Figs. 14 and IS. These considerations suggest a standard procedure for mapping and amplifying the stored phase-difference. When possible, it is worth taking a set of three electron micrographs of the same specimen: 1. a single exposure hologram; 2. an image of the interference field without the object; this fringe system is used to generate the interferometric wave to extract the phase-difference map; 3 . a double-exposure hologram; this furnishes directly the map of the phasedifference between the object and the perturbed reference wave and can be used also as a “guide” hologram when the phase amplification methods are applied.
However, it should be borne in mind that even if one uses the double-exposure method, it will not be possible to avoid the distortion of the contour map, which is caused by the perturbation of the reference wave in the presence of long-range fields, and whose effect can be investigated only by computer simulation.
ELECTRON HOLOGRAI’HY OF LONG-RANGE ELECTROSTATIC FIELDS
207
Moreover, an additional merit of the double-exposure method is that the mapping of electric or magnetic fields on a larger area than that recorded on a single micrograph can be obtained by a suitable montage of different holograms. A number of double-exposure holograms are taken of successive adjacent areas of the field under investigation and then pasted together. Careful attention must be paid when the object is shifted with respect to the biprism. Reliable results can be obtained particularly when a theoretical model is available to take into account the phase differences recorded on each hologram.
IV. CHARGED DIELECTRIC SPHERES
A. Recording arid Processing of Electron Holograms The specimens were prepared by depositing on a conducting carbon film, of about 30 nm in thickness, a little drop of an aqueous suspension of spherical polystyrene latex particles of diameter 2a = 0.31 p m . Under the action of the electron beam the randomly distributed latex spheres acquire a stationary positive charge Q as a result of a dynamical equilibrium of charging up by secondary emission, and neutralizing by field emission (Drahos er al., 1969). The magnitude and the distribution of the stationary charge depend on the material, on the geometry of the scattering object, and on the energy and current density of the electron beam. As regards the field outside the sphere, it can be fairly well approximated by the field of a point charge Q located in the sphere center at a distance a from the conducting grounded plane (Drahos et al., 1969). The off-axis electron holograms were recorded by the setup schematically shown in Fig. 12. The holograms have an interference field of about 120 fringes of 85 pm spacing, which allows for a resolution referred to the specimen of about 0.1 p m . Exposure times were of the order of a few seconds. The conventional reconstruction of electron holograms has been carried out in an in-line optical bench. In Fig. 19 is shown the reconstructed specimen image. Only the amplitude information is present in this image, showing a set of opaque dielectric spheres deposited on the thin carbon film F and the edge of the hole H . Note the presence of two single spheres: A near and B far from the edge of the carbon film F . Phase-difference maps were performed by two different experimental methods: The first made use of the optical arrangement represented in Fig. 15; the second by using the double-exposure technique which allows the recording of the correct trend of the contour fringes (Wahl, 1975). In Fig. 20 is shown the phase-difference map of the same region of Fig. 19, obtained by the optical reconstruction of the double-exposure electron hologram.
208
G. MATTEUCCI. G. F. MISSIROLI, AND G. 1'0%%1
FiciLliu: 19. Optical reconstruclion of an electron hologram showing latex porticles. F. cni-hon filni: H . hole in tlie carbon film; A and B cliargcd spheres near ( A ) cind far ( B ) horn the edge of tlic carhon film. After Clien et NI.. 1989: reprinted with kind permission of The American I'hy\ical Society.
FI(;I:KE20. Optical rcconslruction of n double-cxpohurc hologruni aliowing a mnp o l thc phn\ctlilfcrcncc o f tlic same region as Fig. 19. After C'licn ('I ( I / . , I 9 W reprinted with kind pcimi\\ion of The Amcricon Physical Society.
Note that the contour fringes suffer an abrupt discontinuity when crossing the edge of the film. The importance of this map is better shown by considering that if the optical experimental procedure is repeated, the wave vector k, [see Eq. (38)] slightly changes and the interferogram displayed shows different trends of the field under investigation. These considerations are demonstrated by the experimental results reported in Figs. 2 la, b, c, d as compared with the optical reconstructionof the double-exposure
ELECTRON HOLOGRAI’I IY O F I.ONG-RANGE EL6CIKOSTATIC FIELDS
209
electron hologram of Fig. 20. Each of the parts of Fig. 2 1 reports an optical interferogram obtained with small angular variations of k, (of the order of lop3rad) with respect to the interferometric plane wave. The different trend of the various maps is immediately perceived. For this reason we regard the double-exposure hologram as a “guide hologram” with which the trend of the phase-difference fringes obtained by the reconstruction of a single-exposure hologram should be tested and the phase-difference map selected. In conclusion, the experimental results reported in Fig. 21 show that the optical processing of an electron hologram can produce a large number of different interferograms, which do not represent the most reliable phase-difference map. This latter can only be reached by comparing the optical interferograms with the double-exposure hologram. More detailed infomation about the trend of equipotential lines can be obtained with the technique of the phase-difference amplification as discussed in Fig. 17, Section IIIB2. Figure 22a shows a two-time amplification contour map of the same region reported in Fig. 20, whereas Fig. 22b shows a four-time amplification contour map. In these figures two neighboring equipotential fringes differ in phase of il and of n / 2 , respectively.
210
G. MATTEUCCI, G. F. MISSIROLI, AND G. 1'0%%1
FIGIIIIII 22. ( a ) Two-timc and
(11)
lour-time phasc-difference amplilicd contour maps ohtaincd C/ u / . , 1989: rcpi-intcd will1 kind pcrnii\sioii 01' The
from tlic miic hologram of Fig. 20. After Chcn
Ainerican Pliysicul Society.
Figures 23a and 23b show a magnified image of the phase distribution around the two single dielectric spheres A and B of Fig. 19, amplified by a factor of four. It can be seen that whereas the trend of the fringes has a circular symmetry in the vicinity of the particle B , which is far from the edge of the carbon film, this symmetry is no longer present for the sphere A located near the edge. That is, the presence of the edge has a detectable effect on the phase distribution.
B. Interpretation of the Experimental Results
In order to interpret the results of the foregoing paragraph, the electric field around the spherical particles has been modeled by that of a point charge Q localized at the center of the sphere of radius a and placed in front of a conducting plane.
ELECTRON HOLOGRAI’NY OF LONG-RANGE ELECTROSTATIC FIELDS
2 11
For a particle far from the edge, whose center is positioned at the point (q, yo, a ) , the potential in the half-space z > 0 can be calculated by means of the image charge method (Feynman, 1967) and is given by the expression:
212
G . MATIEUCCI. Ci. F. MISSIIZOLI, A N D G . I’O%%I
while i n the half-space :< 0, V ( x , y , :) = 0. The phase shift 4 can be calculated analytically from Eq. (42). It follows that:
4(x,?’) =
~
[2&1
arcsinh
[
U
[ ( x - .Yo)?
+ (y - ?‘,)-I
7
I/?
1
.
(4.3)
Simulations of the contour map images produced by the phase distribution [Eq. (43)], in which the edge effects of the carbon film and of the perturbed reference wave have been momentarily neglected, have been carried out using an IBM PC/AT equipped with a video board able to display 5 12 x 5 12 pixels at 256 grey levels. Figure 24 shows the results for our point charge distribution of Fig. 20 assuming Q =40Oe = 6.4 . lOP”C and ~i=0.155 pm. This value of the charge is in agreement with the findings of Komrska ( 1971), who measured a charge of I 100r on spheres of radius 0.28 p m by studying their diffraction patterns. The overall similarity between calculated and experimental images is satisfying f,: ‘it trom the edge. thus confirming the main hypotheses made and allowing, within this framework, the determination of the charge Q with an accuracy of 20%. Of course, near the edge, differences are detectable due to the fact that the simple model does not take into account the discontinuity of the conducting film. If the actual shape of the edge is replaced by a straight line, the electrostatic problem still has an analytical solution (Durand, 1966). Refelring to Fig. 25 for the definition of (Y and 29, the solution reads:
ELECTRON HOLOGIIAI'HY OF L O N G R A N G E I~LECTIIOSTATI(' FIELDS
2 13
Flcii IKE 2.5. Coordinate sy\tern lor the potential and phaw colcul:ition in the contlucling hall-plane prohlem. ( I < , .Y,~.( I ) , position of the point cliarge: ( t , Y , :). observation point; (0. \'(,. (1 1. projection 0 1 tlic position 01' the point charge on plnne \:: (0. v. :). projection ol the observalion point on pl;ine \':.
where
hi =
[
1 [r2
+ ( z fa)']
1
"I
and
Yt =
{
n -PI. [n- P , 2x1,
I,
l9 E [O,
-1,
l9 E
1,
19 E [O, 71 +PI,
-1,
19 E [ T f
P , 2x1.
Since no analytical expression for the phase shift was found, it has been calculated by numerical integration of Eq. (2).
214
G. MATTEUCCI, G. F. MISSIROLI, AND G. POZLI
FIGIJKE 26. Comparison of simu1;itctl contour maps of a single charged sphere located on the wliolc conducting plane W P (upper part of each picture) and near the edge (lowor part) 01' ii halfconducting plane H P . The vertical line E marks the edge position. Distance between the sphei-e x n d the edge: (a) 5 pm: (bj 2 pni; (c) 0.155 pni. Atter Chen efd.,1089; reprinted with kind permission of The American Physical Society.
Figure 26 shows the results of the contour map simulation for the charged sphere on the whole plane WP or on the half-plane H P . Here again it can be ascertained that the effect of the edge E is to break the circular symmetry of the contour lines around the particle and that their distortion increases as the particle is located at diminishing distances from the edge. However, it is not responsible for the step observed experimentally, which is entirely due to, and can be properly accounted for, the introduction of a constant phase shift in the half-plane due to the mean inner potential of the carbon film. C. Numerical Simulations of Contour Maps
In order to evaluate the influence of the perturbing phase term +(r - D/2) which appears in Eq. (30), computer-simulated contour maps, relative to the two experimental cases considered in this work, are here presented and discussed.
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
2 15
FIGLIKE27. Simulatetl conIot~ri n a p ofthe potential distribution generaled by the charges located Fig. 20 for different viilues of the interfering distance D : ( a ) / I = 3 pni; (h) D = S ~ 1 1 1 ;(c) I ) = X p m and (d) I ) = infinite. Aftei- Matteucci C I o/., 1991: reprinted with kind permission of
as
iii
American Institute of Physics.
The function I (r) which describes intensity variations through the contour map is given, after a translation of D/2, by:
in which the term of the phase perturbation due to the reference wave is evident; see also Eq. (30). The intensity variation of Eq. (SO) was displayed by means of an IBM-PC-AT personal computer equipped with a Matrox PIP 1024-B video board. The set of images in Fig. 27 shows four simulations of the interference maps of the field associated with a system of charges like that in Fig. 22 for increasing values of the interfering distance D. The sphere diameter and charge are 0.31 pm and Q = 400e. The values of D for each micrograph of Fig. 27 are: (a) D = 3 p m ; (b) D = S pm; (c) D = 8 pm;(d) D = co;this latter value corresponds to the case of an unperturbed reference wave. It can be ascertained that differences arise between the patterns, which are more detectable in the regions far from the particle centers. According to theoretical expectations, such differences become smaller as the distance D between interfering points is increased. I n practice, further simulations show that the actual contour map is indistinguishable from the unperturbed one (Fig. 27d) only when D is greater than 15 p m . However, near the particles, where the phase shift is
2 16
G MATTEIICCI, G. F MISSIKOL.1, A N D G. P(>%/,I
larger, the effect of the perturbation is less and, for values of L) wider than 5 p m , the overall trend of the fringes is not substantially affected. Taking into account that the upper value of the interference distance of our experimental setup is of about 10 p m , we can see that the ideal condition can be almost reached. For lower values the effect of the perturbed reference wave becomes important. Since latex spheres samples can be easily prepared and can be used to check the holographic method, our experiments have been repeated by Frost et al. (199%). Their results show the possibility to reveal a small amount of charge of the order of magnitude of ten electrons.
V. P-N JUNCTIONS A. Experinwntal Results 1 . Specimen Prepurutiori urid Electron Microscopy Ohsetvations
In this section we present and discuss the experimental results obtained by applying electron holography techniques to the observation of reverse biased p-n junctions. The procedure to obtain a specimen suitable for transmission electron microscopy from an n-type silicon wafer is the following. The surface layer of the sample was preamorphized by the implanting of Si' ions in order to minimize channeling phenomena. Boron ions with a dose of 1OIs cm-' were successively implanted at 10 keV. The wafer was then annealed at 900' C for 30 niin in a nitrogen atmosphere. The implantation was carried out through a SiO? mask consisting of parallel slits 8 p m wide and of 8 F m spacing in order to produce a set of parallel diodes. The resulting p-n junctions have a depletion-layer width W = 1.4 p m , and a built-in potential of V = 0.76 V. In order to bias the junctions, one end of the structure made up with parallel slits was electrically shorted by vacuum deposition of a TiAg layer; this continuous layer connects the p regions together and is isolated from the n regions by the SiOz layer used for ion implantation. Far from the TiAg layer, the SiOz mask was removed by photolithography (Fig. 28). A flat rectangular region was obtained, which consists of a set of p-n .junctions. The junctions can be biased with an external voltage source applied between the TiAg layer and the back of the wafer. The implanted wafer was subsequently chemically thinned, from the side of the backing material, in correspondence to the region where the SiO? mask was removed. By protracting the thinning up to the formation of a hole H , it was possible to obtain around it a thin area containing several parallel junctions side by side. Since the lower portion of the junctions has been removed by the thinning process, the remaining part of them is perpendicular to the wafer surface.
ELECTRON I-IOLOGI
2 17
. . .
In order to carry out the observations i n the electron microscope, the thinned specimen was mounted on a special specimen holder equipped with electrical contacts for biasing the junctions. When a reverse bias is applied to the array, an enhancement of the electrostatic field of the junctions is produced both inside and outside the specimen. The inner field lines of the junctions are parallel to the upper specimen surface and perpendicular to the beam. Figure 29 is intended to show schematically the trend ofthe equipotential surfaces around the specimen by means of their intersection with ( 1 ) the plane inside the specimen parallel to the direction :ofthe electron beam, and perpendicular to the junction boundary, and (2)the xy plane at the edge of the hole H . The equipotential surfaces spread out from the upper and lower surfaces ofthe specimen where thejunctions are localized and extend as far as the edge of the hole where they rotate. Figure 30 shows an out-of-focus electron image of the specimen area where two p-ti junctions are present. The reverse bias was 4 V, the defocus distance - 1.25 mm, and the electron optical magnification I700 x . The micrograph shows sharp contrast lines (see arrows), with a fine wave-optical structure, which arise in correspondence with the depletion layer, or more precisely, in correspondence with the regions having the highest phase gradient (i.e., the highest electric field). Unfortunately. it is rather difficult to extract quantitative information from these patterns, since the relation between image intensity and the phase shift of the object wavefunction is highly nonlinear and depends on other parameters such as defocus, spherical aberration, and intensity distribution of the electron source, which limit the accuracy of the obtained rcsults. Similar criticism can be applied to the other methods of Lorentz microscopy (Chapman, 1984).
218
G. MATTEUCCI, G. F. MISSIROLI. AND G. PO%%I
H 20. Sketch of the intersections of the equipotential aurfiices associated with the cxtcrnd licltl of p-ti junctions with planes parallel nnd perpcntlicular to the optic axis :. After Frahhoni (I/., 19x5: reprinted with kind permiasion of The Anicrican I'hysical Society. E'I(itIw
(11
FIGIME 30. Out-of-focus ininge of a region whcre two p-n junctions iirc prescnt. A n o w s in;rrk contra\( lines associated with depletion layers. Reverse hias. 4 V. After Frnbhoni C I nl.. 1987; rcprintctl with k i n d permission of Elsevier Science.
Electron holography, with its unique capability of reconstructing the two-dimensional object wavefunction, both in amplitude and phase, could be the answer to the problems posed by the standard methods of observation. As shown in Section IIIA2b holograms containing about 200 fringes at two interfering distances, 5 p m and 10 p m , were taken with exposure times of about 10 s.
2. Optical Recotistructioti and Processing of Electron Ho1ogrum.c In this section we report the results concerning the optical processing of the image holograms recorded under the experimental conditions of Fig. 13 (Section IIIA2),
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
2 I9
FIGIiKE 3 1. Contour i m p oflhc specimen a10 V reverse hias. Aficr Frahboni ('I ol., 1987: reprinted with kind pcmiission of Elsevier Scicncc.
with the largest interference distance, for which the effect of the fringing field spreading outside the specimen should be minimized. The p-n junctions were reverse biased at 0 , 2 V, and 4 V in addition, also the hologram of the interference tield without the object was recorded. The optical setup used for the reconstruction of the holograms is a MachZehnder interferometer, whose versatility allows us a choice of different processing schemes. Figure 31 shows the contour map obtained with the hologram of the specimen at 0 V reverse bias and a plane wave obtained from a hologram recorded without the specimen. Two regions S and H can be observed: The lower part of the figure, S, refers to the in-focus image of the specimen, whereas the upper part, H , refers to the hole. The reconstructed specimen area, which extends laterally over a distance of about 20 p m , contains the same two junctions as reported in the out-of-focus image of Fig. 30. The interference fringes superimposed on the specimen do not reveal any distortion attributable to the electrical field generated by the contact potential (0.76 V) between the areas n and p, although, as demonstrated in the following, the sensitivity of the method is high enough to detect such potential difference. The fringe behavior in region S, however, is related to the variations in the specimen thickness and reflects the wedge shape around the edge due to the chemical thinning. In area H a bright field can be observed, with some shadows whose origin can be sought in the weak perturbing electrostatic fields originated by impurities on the biprism wire or in the photographic plates used for the holograms whose thickness varies within the reconstruction field. Figure 32 shows an interferogram obtained by the simultaneous reconstruction of one hologram at 2 V reverse bias and another without the specimen. First of all, we verified the parallelism between the two transmitted beams by checking the absence of optical interference fringes within the field of view when the filter was removed. On examination of the interferogram obtained by reinserting the
220
C . MATTELICCI. G. F. MISSIIIOLI. A N D G. i'O%%I
tilter, it was impossible to decide the correct condition of the contour map, since we could choose among a large number of possible superpositions between the diffracted spot generated by the hologram with the specimen and the one generated by the reference hologram as demonstrated in Fig. 21. The former spot, in fact, is rather broadened because of the electric field while the latter is a dotlike spot. The superimposing of the interference fringes of the two holograms in unperturbed regions is not applicable in this case, since the carrier fringes of the holograms recorded at 2 V are strongly distorted all over by the electric tield. All these drawbacks could be eliminated if the double exposure of the holograms gave reliable results when recorded directly in the electron microscope. In spite of the uncertain optical conditions, the interferogram shows that the optical fringes propagate in the whole interference field and fan out through the hole connecting the neighboring junctions, whereas within the specimen they are influenced also by its thickness variations and topographical features. Considerable improvement of information about the trend of the electric field distribution within the specimen can be obtained if two holograms of the specimen, recorded at different reverse bias, are simultaneously reconstructed. With respect to the foregoing case, once the parallelism is achieved as before, the coincidence of in images of the specimen eliminates that degree of freedom due to the relative rotation of the holograms. so that reproducible and reliable results can be obtained. Figure 33 shows the differential contour map obtained by superimposing the reconstructed images relative to the holograms of the specimen at 0 and 4 V reverse bias. As the fringes represent the positions of those points where the difference between the phases of the reconstructed waves is constant, since it is the same for both waves, the thickness contribution to the phase is eliminated. The improvement of Fig. 33 with respect to Fig. 32 is evident, and in this case the contour fringes
ELECTRON HOLOGRAI'I IY Ol-LONG-RANGE LLECIROCTATIC FIELDS
22 I
directly display the trend of the in-plane projected equipotential surfaces resulting from the total electric field associated with the junctions. In the depletion layer areas, the contour lines are parallel to each other and to the junction and show a different spacing on both sides of the p-n junctions, reflecting the local asymmetry of the electric field, usually described by one-sided step models. Outside the edge of the specimen, the contour lines fan out again to join the nearest junction, as before. Finally, Fig. 34 shows the differential contour map obtained with two holograms of the specimen held at a reverse bias of 2 V and 4 V, respectively. The density of the contour fringes is diminished roughly by a factor of two when compared to Fig. 33: that is, it is linear with the reverse bias, confirming that the sensitivity of the technique is adequate enough to clearly detect a potential difference of 0.76 V (i.e., the built-in potential).
222
G. MATTEUCCI, G. F. MISSIROLI, AND G. 1’0%%1
B. Theoretical Interpretation
I . The Electrostatic Field Model In this section it will be shown how the analytical solution can be obtained for the electrostatic field associated to a periodic array of alternating p and n regions lying in a semi-infinite plane. From this solution the phase shift can be calculated and used for the interpretation of the main features of the holographic contour maps. Let us consider the following boundary value problem (Wendt, 1958), that is, finding the electrostatic potential produced by a parallel array of stripes having pitch 6, which lie in the half-plane L = 0, x > 0, and are biased at alternate potential, namely +Vo for 2nb < y 5 ( 2 n 1)h and -Vo for (2n 1)b < y 5 (2n 2)h, n being an arbitrary positive or negative integer (see Fig. 35). On the half-plane, the potential is given by: s i n 2k ( 7 . y1) VO V ( x ,y, 0 ) = (51) 4 k=O 2k 1
+
+
c
+
+
+
so that, considering each Fourier component separately, we have to find a solution of Laplace’s equation in the form:
X
FICiLlr<E
3.5. Schcmatic drawing of h e specimen ;mi coordinate system
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
It follows that
@h(x, i ) should
223
satisfy the following differential equation
with the boundary condition:
2 0,O) = 1
@*(X
(54)
As the half-plane := 0, x 1 0 can be considered as the limiting case of an infinite parabolic cylinder, it is convenient to solve Eq. (53) by the method of separation of variables in parabolic coordinates (, r j related to the Cartesian coordinates x, z by the relations:
z
= (11
(56)
where -00 < ( < 00,and 0 5 11 < co. The orthogonal system of surfaces consists of parabolic cylinders with foci at the origin (Fig. 36) and the specimen half-plane corresponds to the value r j = 0. In this coordinate system, Eq. (53) for the Fourier component of the electrostatic
FICiIJKE 36. Cartesian and parabolic cylinder coordillnte\. After Capiluppi (’I (,/., 1995; reprinted with kind prnni.~sionof Les Etliiionc tle I’hysiquc.
224
G . MA1'TEUCCI. G. r-'. MISSIIIOLI, A N D G I'O%%I
potential V takes the form:
for which a solution is sought by the method of separation of the variables (Lebedev. 1965). Writing: @ L ,
= A(4)BOl)
the following equations result:
where /L is a constant. By introducing the new variables eter u related to p by the formula:
M =
J2hc and u = a
p = -h(2u
+ 1)
i l , and a new parani-
(60)
Eqs. (58) and (59) can be reduced to the form:
cl'A + (u + - 1- dlC'
L;)
=o
(61)
and
respectively. The solutions D , , ( u )of Eq. (61),called Weber's equation lor of the otherwise identical Eq. (62) with index --v - I ] are called parabolic cylinder functions or Weber-Hermite functions and their properties are sutnniarized in the book edited by Erdelyi (1953), whose notation will be hereafter strictly followed. hi addition to D , , ( u ) also Dl,(-i{), 11l,-i(iu) and D,,+i(-iu) satisfy Weber's equation. As there are only two linearly independent solutions, the above solutions are connected. In particular, if u = 11 is a nonnegative integer, then:
where H , , ( u ) is the Hermite polynomial of degree 1 1 , being Ho(ir) = I , whereas if u is a negative integer, then D,,( u ) can be expressed in terns of the complementary
ELbC‘I RON kIOLOGRAI’HY OF LONO-RANGF E L E C T R 0 9 T A T I C FIELDS
225
error function Erfc defined by:
that is D-,,, I ( u ) = 2z
(65)
~
tn!
so that
Unfortunately, the solution of Eq. (53) with the boundary condition of Eq. (54) cannot be found by using Hermite polynomials, as outlined by Lebedev (1965), since the value of the function at the boundary does not satisfy the conditions for its expansion in series of Hermite polynomials. Therefore, we should use the more general Hermite functions. When u is not an integer, D , , ( u )and D,,(-u) are linearly independent, so that a particular solution of Eqs. (61) and (62) can be written as:
+ QD,,(-u)
A ( u ) = PD,,(u)
(67)
and
respectively. Noting that the required solution of Eq. ( 5 3 ) should be symmetric with respect to the plane (x,y), then A ( u ) should be unchanged by the substitution of u + -it, which implies that P = Q. It follows that the most general solution for the Fourier component of the potential can be put in the form:
where the coefficients R and S are functions of v, and should be determined in such a way that @,- satisties the boundary condition, which in parabolic coordinates becomes:
Putting S = 0 for simplicity, [his leaves iis to determine only the function K , and this can be done by comparing our expression with the one which can be deduced from Erdelyi’s Eq. ( 1 1) in Section 8,5,2, (ErdClyi, I953), specifying there c = - 1/2,
226
G. MATTEUCCI, G. F. MISSIROLI, AND G. POLL1
t = I,andy=O
(71) It results that: I
= 2fisinvn
so that our solution for the field in the whole plane can be written:
(73) Our final solution can be changed into a more manageable form, if it is compared with Cherry’s Eq. (8) in the same Section 8,5,2, (ErdClyi, 1953), which, rewritten by putting 4 = n / 2 , h( = u and h v = u , becomes:
so that it finally ensues:
It should be noted that Cherry’s formula has been derived with complex arguments in relation to the solution of the Helmoltz equation to find Sommerfeld’s secondary wave, while in our case we are dealing with real arguments and the solution of Laplace’s equation. Nonetheless, it can be easily ascertained that our final solution is the right one, being the correct combination of parabolic cylinder functions and satisfying the boundary condition, but with the peculiarity that its orientation has been rotated by n / 2 with respect to the original one (Fig. 36). By using the inverse transform between parabolic and Cartesian coordinates:
ELECTRON HOLOGRAPHY OF LONG-IIANGE ELECTROSTATIC FIELDS
Eq. (75) can be transformed, in the half-space z 2 0, into:
+ exp(-h:)
Erfc
(-hdm)]
for x 2 0, whereas for x 5 0 it results:
+ exp(-hz)
Erfc (Ad-)]
The symmetric form holds for the half-space L 5 0. Finally, the expression for the electrostatic potential in the whole space satisfying the required boundary conditions is given by: sin
V ( x , J’,
vo
z) = 4
c
(
2k
L =o
+
k2kn y )1
+I
c D u r r ( x z, ) .
(80)
2. Numericul Siniulutions of Hologruphic Contour Maps As shown in Section IIA once the phase shift 4 (r)is calculated inserting into Eq. (2) the electrostatic potential given by Eq. (go), the ideal contour map is given by:
Z(r) = 1
+ cos[4(r)]
(81)
whereas the real contour map, taking into account the perturbed reference wave, Eq. (31b), is given by:
I(r) = 1
+ cos[A$(r)].
(82)
Figure 37 shows the ideal contour map due to the external field alone of our p-n junctions array, calculated for a pitch h = 8 p m and a potential V, = 0.5 V (Fig. 37a) and Vo = 1 V (Fig. 37b), the potential difference being the double of this value ( A V = 2Vo). It should be remembered that the ideal contour map is the optical interferogram which would be obtained when the reference wave in the recording step is an unperturbed plane wave, and the optical wave in the recording step is plane and parallel to the object wave. It can be seen that the trend of the contour fringes, giving the configuration of the projected potential distribution, is in good agreement with what is expected on the basis of naive assumptions regarding the potential distribution in Fig. 29 (Frabboni e f u l . , 1985). The fringes are running in parallel to the junctions far from the specimen edge, and at the edge they fan out and connect neighboring regions.
228
G. MATTEUCCI, G F.. MISSIKOLI. A N D G. I’O%%I
However, it should be remarked that experimental results show unexpected features, like closed loops of the equiphase lines between the junctions in Figs. 32. 33, and 34 (Frabboni et a/., 1987). This effect can be accounted for by considering the real contour map, Eq. (82), where the phase difference is inserted instead of the phase itself. Figure 38 shows the real contour maps obtained for different values of the interference distance D whose direction has been taken parallel to the junctions. Figure 3Xa refers to D = 4 pin: Fig. 38b to D = 6 p m ; Fig. 3Xc to D = 8 p m and Fig. 3Xd to D = 10 p m ; Vo has been taken equal to I V and, hence, the potential difference is 2 V. It can be seen that, as expected, the closed contour lines become increasingly elongated along the junctions as the interference distance D increases, corresponding to a lessening of the effect of fringing field on the reference wave; it is also interesting to note that the radius of curvature of the loops is larger near the edge toward the vacuum region and smaller far from the edge within the specimen, as observed also experimentally. Finally, the figures refer to a very large area, side 2 h = 16 p m , whereas the actual reconstructed differential contour maps display only a strip, parallel to the biprisin wire, a few microns in width. However, an image like those reported could be obtained by reconstructing several double-exposure holograms and pasting them together, as recently demonstrated for the mapping of the electrostatic field around charged microtips (Matteucci et al., 1992b). In view of their importance for semiconductor devices, further attempts to observe the field distribution of unbiased p-n junctions with electron holography have been undertaken by McCartney et a/. (1994), and Frost et al. (1995b).
ELECTRON €IOLOGRAI’HY OF LONG-RANGE ELECTROSTATIC FIELDS
229
As a further example of the capability of electron holography, we present the study of the electrostatic tield around a charged niicrotip (Matteucci et ul., 1992b), which has so far been studied only by means of interference electron microscopy (Kulyupin et ul., 1978-79) with the tinal aim of investigating the newly developed monoatomic point sources (Fink, 19x8). This kind of source is meeting an ever increasing interest in low-voltage electron holography (Fink et ul., 1995; Spence C ’ t d . , 19%).
With the aid of a theoretical model for the field around the tip, which gives an analytical result for the projected potential, it has been possible to predict the general trend of the experimental contour map. Moreover, the comparison between theoretical and experimental results is made more accurate by making a montage of reconstructed contour maps of ad.jacent regions, using the double-exposure technique. In this way, it has been possible to partly overcome the experimental limitations due to the reduced width of the interference field of a single hologram.
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G. MATTEUCCI, G. F. MISSIROLI, AND G. POZZI
\
i'p \
39 Theoretical iiiodel to cdlculate the lield iiedr d charged microtip The lree pdr'inickm are \hewn together with the equipotcntt,il wrt'icer ncdr each chargctl segment ot length 2~
FIGURE
h dnd
A. The Field Model The theoretical analysis of the electrostatic field in the outer space of a charged tip first considers the simple model made by two linear segments, (see Fig. 39) each of length 2c and whose centers are 2h distant, placed along the y-axis in a symmetric position with respect to the x; plane of an xy; coordinate system. Each segment has a constant and opposite charge density CT.
ELECTRON HOLOGRAPHY OF LONG-RANGE ELECTROSTATIC FIELDS
23 1
The analytical expression of the potential distribution V ( x , y , z ) can be obtained (Durand, 1966) by integrating the formula:
V ( x ,y , z ) =
0 ~
di
The integration leads to:
and it can be seen that the potential distribution has a rotational symmetry and is zero when y = 0. Near and around the extremities of the two charged lines, the equipotential surfaces behave approximately as a family of hyperboloids of rotation. Therefore, it is reasonable to assume that the field described by Eqs. (83) and (84) may be used to represent, at least in a first approximation, the field produced by a charged tip in front of a conducting plane ( y = 0). The distance between the tip vertex and the conducting plane ( y = 0) was 15 km. The charge density 0 was chosen in order to obtain the equipotential surface that represents the tip shape at about 10 V. In Fig. 40 are shown the simulated equipotential lines around the tip in the specimen plane ( z = 0 ) . In order to display such a distribution, a set of equipotential surfaces was chosen with a constant potential difference. The region T inside the
FIGLIKE 40. Computer simulation 01' ecpipotential lines, in the .vy plane, of a charged microtip. Alter Matteucci cr u/., 1992b; reprinted with kind permission 0 1 Elsevier Science.
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G MATTEIJCCI. Ci f. MISSIKOLI, AND G 1'07/1
equipotential surface (which more closely resembles the tip) was darkened. From the analytical expression of the potential, the phase #(x, y ) can be calculated by performing the integral of Eq. (2). The integration leads again to an analytical expression:
+ [c
-
(y
+ h ) ] I n Jx' + Ic
-
(?I
+ h)l'
The holographic method reveals the loci of points with constant phase shift as a set of curves with a phase difference of 2n between two successive dark and white ones. Figure 41 shows the computer simulation of the equiphase lines obtained by the coherent superposition of the object wave, Eq. ( 8 5 ) , and a plane reference wave. While the trend of the potential in the (A-, y , := 0) plane is easy to guess (Fig. 40), the interpretation of Fig. 41, where the equiphase lines seem to enter the tip shadow T , is less intuitive because the phase shift, suffered by electrons along their trajectories, is related to the potential distribution around the tip integrated along the :-axis. However, when experimental observations are made of the tield close to the tip apex, we must consider that also the reference beam is modulated by the tield of the tip which extends microns away from the tip itself. Therefore, the final contour map will show the loci of constant phase difference between the perturbed reference wave and the object wave and does not exactly represent the object phase variations. In our case by using Eq. (85) and by taking into account the distance between the interfering points (in the electron microscope),
ELECTRON HOLOGRAI'HY OF LONG-RANGE ELEC'TIIOSTATIC FIELIX
2.33
the perturbed reference wave can be calculated and the resulting interferogram displayed directly in the computer. In the following section this theoretical result will be compared with the experimental one.
B. E.rperimrntal Kesults Using a standard electrolytical thinning process, the tips were obtained from a polycrystalline tungsten wire (0.25 mm diameter), in a cell with 2% NaOH solution and by applying 2 V, SO Hz alternating voltage (Dyke and Dolan, 1956). One was mounted in the center of a 2 mm aperture, which was inserted on a special specimen holder equipped with electrical contacts connected to an external voltage supply. The aperture and the tip, electrically insulated from the microscope, could then be biased and, by rotating the aperture, it was possible to arrange the tip and the biprism axis i n a mutually perpendicular position. A voltage of the order of 10 volts was applied to the tip. Holograms were recorded according to the electron optical arrangment of Fig. 12. Double-exposure holograms were recorded with an interference distance of about 5 Fm. Figure 42 shows a double-exposure electron hologram in which the reference wave is perturbed by the near-apex electric tield. The dark regions represent the equiphase lines in the area near the tip 7 when it was held at 7.5 V. I n the previous example dealing with charged latex spheres, we showed that the eqiiiphase lines were strictly related to and also displayed the trend of the projected equipotential surfaces. On the contrary, in the present case, the equiphase lines observed in the tinal interferogram cannot be related simply to the equipotential surface shape. Since the investigated area around the tip is fairly limited to about 5 pin, the overall trend of these lines cannot be displayed i n a large
234
G. MATTEUCCI, G. F. MISSIROLI, AND G. POLL1
FIGLIKE 42. Double-exposure hologram displaying equiphase difference lincs near the apex of chnrged microtip T . Alter Matwucci ef ( I / . . 1992h: reprinted with kind pemiission of Elscvicr Science.
ii
enough region. In order to follow their trend around the tip on a wider area, three double-exposure holograms were taken from parallel and adjacent regions and then mounted together. It is important to note that the success of this procedure is linked to the fact that double-exposure holograms are recorded, so that the interferometric wave for the contour mapping is provided by the hologram without object. Figure 43a shows a montage of these three regions (labeled as 1, 2 and 3) in which the useful interference field extending along the tip axis is about IS wm. The three strips are of different width since the overlapping regions were removed. It can be noted that in this overall map the equiphase difference lines circle around the vertex of the tip T and then join the tip itself, behavior that could not be inferred previously. Figure 43b reports the computer simulation obtained by the coherent superposition of the object wave and the perturbed reference wave and adjusting the parameters h , c and the charge density 0 in order to fit with the experimental data. The number of equiphase difference lines is the same as what would be obtained by a double-exposure electron hologram performed with a perturbed reference wave passing 5 p m distant from the object wave and with the same relative orientation of the biprism and the tip as that shown by the electron holograms of Fig. 43a. The satisfactory agreement between experimental and theoretical results is evident. The comparison between Figs. 41 and 43b clearly shows the difference between the trend of the phase distribution displayed by a hologram recorded with an unaffected reference wave instead of a modulated one. Our results regarding the mapping of electric field distribution around charged tips have been confirmed by Ru el al. (1994) and Ru ( 1 9 9 % ~199Sb) who used an amplitude-division interferometer, Section IIIA1 (Matteucci et al., 1982b; Pozzi, 1983) instead of a biprism.
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FIGIJKE 43. (a) Collnge of three ( I, 2, 3 ) double-exposure 1iologr;ims Inkcn from iidjilcent regions showing lhc lrend of lhc cquiphase difference line iii :I witlei- iircit. (17) Coiiipuler simulation of the cquiphase lines obt:iined with a pcrlurhctl rcfci-ence wave. Ailer Matlcucci PI o/,,lW2h; reprinted with kind permission o f Elsevier Science.
VII. CONCLUSIONS The main experimental and theoretical results obtained by our group in the investigation by holographic techniques of long-range electrostatic fields (i.e., not strictly localized in the sample from which they arise) have been reviewed in this work. We have demonstrated how the phase-object approximation can be reliably adopted for the calculation of the phase shift associated with these electric fields (obtaining in most cases an analytical expression for it) and how this phase shift (proportional to the projected electrostatic potential) can be strikingly different from the shape of the equipotential surfaces. Moreover, the external fringing field, usually treated as a perturbation with respect to the internal one, may also be the main responsible cause of the observed effects, as shown especially by the case of reverse-biased p-n junctions. As a consequence, when dealing with electrostatic fields, extreme care should always be paid to the interpretation of the experimental data. In fact, another disturbing feature of the fringing field is that it can affect the reference wave, which can no longer be considered an unperturbed plane wave, as is customary in electron holography.
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G. MATTEUCCI, G. F MISSIROLI, A N D G. 1’0%%1
It turns out that the artifacts introduced in the reconstructed images have been thoroughly investigated in this work, and our conclusion is that it is highly recommended to have a good model for the field under investigation. Preliminary attempts have been carried out to cope with this central issue i n a general way (Kou and Chen, 1995) from the theoretical point of view, but we are still far from a satisfying solution. From the experimental point of view, the solution is to increase the interference distance far above the presently obtainable values, in the range of about 10 p m . This could be done, in principle, by using a multiple biprism setup, as developed in Tubingen for the experiments on the magnetic Aharonov-Bohm effect (Schaal et al., 1966/67) or by mixed-type arrangements, combining amplitude and wavefront division beam splitters (Matteucci and Pozzi, 1980). However, for this, radical changes in the basic instrument may be
In conclusion, since electron holography is a very powerful technique, able to solve problems at the frontier of modem technology, we hope that our work, in addition to clarifying the basic elements involved in this technique and to giving warning of some of the pitfalls when searching for a reliable interpretation of the data, will pave the way to further developments. Recent examples are represented by the study of ferroelectric domain walls (Zhang et al., 1992; Spence eta!., 1993) and by the observations of charged grain boundaries in Mn-doped strontium titanate (Ravikumar et al., 199.5; Lin ct al., 199.5), a case where some of the knowledge accumulated in the investigation of reverse biased p-n junctions can be protitably applied (Pozzi, 1996b). There are also prospects that the potential distribution of charged dislocations could be detected by holographic methods (Cavalcoli cr ul., 1995; Matteucci cr al., 1995). The commercial availability of holographic electron microscopes up to 300 kV and their increasing diffusion in university and research laboratories will open a wide range of new applications. We hope that some of our ideas described in this work will be useful for a better understanding of the problems at hand and will help to develop a critical attitude to extract the maximum useful information from holograms. ACKNOWLEDGMENTS We are deeply indebted to our collaborators in the tield, C. Capiluppi, J. W. Chen, S. Frabboni, F. F. Medina, P. G. Merli. A. Migliori, M. Muccini, E. Nichelatti, and M. Vanzi. The critical reading of the manuscript and the helpful comments of M. Beleggia and R. Patti are gratefully acknowledged. Finally, the skillful technical assistance in preparing the drawings of Stefan0 Patuelli has been highly appreciated.
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M;ittcucci, G., Muccini, M., and Cavalcoli, D. ( 1905). In Elec/rvri Holo,qrriplig ( A . Tonoinura, L. F. Allard. G. Poiri, L). C. Joy, and Y. A . Ono, etls.). pp. 1.59-168. North-Holland. Elsevicr Science. Amstrrdam. Molteucci, G., Muccini, M., and Hurtniann. CI. (1094). P/iv.s. Kei: B 50, 6823-6828. Matteucci, G., and Pozzi. C. ( I 9x0). Ul/,-ririiic.rv.\c.opg5. 2 19-22?, Mattcucci. G., and I’oLzi. G. (1985). P/ig.\. Kei: Lcrr. 54, 346‘1-2472. Mattcucci, G.. nnd Porzi, G. ( 19x7). In Q i m i ~ i u r iU ~ i w r / ~ i i r i / i eRwrrir .s; m r l Future E.tperiiiirvi/.s ciritl /,i/rr/~rc.fn/i~Jri.\ (W. M. Honig, D. V. Kralt, mid E. l’anarella, eds.), pp. 297-3 12. Plcnuiii Press, New York. McCartney. M. R., Smith, D. J., Hull,R., Hcan. J. C., Viilkl. E., and Frost. B. ( 1904). A/>/>/.Pliys. Lr,//. 6.5, 2603-2605. f r i f . S o l . (n) 20, KX7TK89. Mci-li, I? G., Missiroli. G. F.. arid Pozri. G. (1973). H I Mcrli, P. G.. Missiroli, G. F,, nnd l’omi, G. (1974). Joi(rrio1 rlr Mic,rn.sc,opir21, 11-20. Mcrli. I? G.. Missiroli. G. F.. inid I’oui, G. (1975). Pliw. Srrit. S o l . ( ( I ) 30. 699-7 I I. Mi\\iroli, G. F., P o u i , G.. iincl Valdri.. U. ( 19X I). .I. Pliv.s. E: S r i . /ri,sfrirtii.14, 649-67 I. Mtillcnstedt G.. and Diiker 1H. (1956). Z. Pliys. 145. 377-307. Olariu, S., and Popcscu 1. I . ( 19x5). Re\: M i ~ t l P/ig.\. . 45, 339436. Pc\kin, M., and Tonoinura A. ( 19x9). 7 l i r A l i t i m r i r , ~ ~ - H o / i , r&ffk/. i Springer-Verlag, Berlin. I’wri, G. (1975). Oprik ( S / i i f r g r i r . / j42, 07-102. G. ( 19XOa). O p i k (Stiittpirt) 56, 243-2.50. l’wri, G. (1980h). In Elec./ro,i M i ~ ~ r o . s ( ~ iVol. p g . I (P. Hroderoo and G. Boom, cds.), pp. 32-33. Seventh European Congress on Electron Microscopy Foundation I’ublisher. G. (19X3). Oprik (Stirrrgtirrj 66, 91-100. G. ( 1995). InArh~criice.sit1 /niri,qir~,qt i r i d Elccfnui f-’/iy.sirx, Vol. 93, pp. 173-218 (I? W. Hawkcs. L L . , Academic Press, New York. P o r i G. (1996a). Pligs. Ski/. Sol. ( ( 1 ) 156. KILK4. Porri, G. (1996h). J o i i i : Phr..c I): A/>/>/.P/~g.s.29, 1 X07-I X I I . G., antl VanLi, M. (1982). Opfik (.S/iu/,qrir/) 60, 175-1 X0. Ravikuinar, V., Rodrigues, R. I?. antl Druvid V. 1’. ( 1 9 9 5 ) . f’ligs. Rev. I,e//er,s75, 40634066. RU,0. (199Sa). .I A/>/>/.f‘/ig.s. 77, 1421-1426. Ru, Q.(1995h). In E k r r o r i Ho/o,qrriphy ( A . Tonomura, L. F. Allard. G. PozLi. U. C. Joy, nnd Y. A. Ono, cds.), pp. 343-353. Noi-tli-Holland. Elsevier Science, Arnsterdani. Ru, Q., Osakahc. N.. Entlo, J., and ‘Tnnoniur;~,A . ( 1494). Ulfrriiriic.,n.sc.(>/~~~ 53. 1-7. Scliaal, G., Jniis\on. C., and tiriminel, E. F. ( l966/67). Opfik (S/u//,yrir/j 5, 52Y-538. Sclimid. 1-1. ( 19x4). In Prrrc.ectli,i,q.\ of / / i c Eigli//i E i o r ~ / x w rCoiigrc,s.r on E l e r w m Mi~~ro.sc~opg (A. Csanritly, I? Riihlicli. a d D. S n h l i , elk.), pp. 285-286. PI-ogramni Committee of tlic Congress, Hudapest.
Septier, A. (1959). C. K. Acrid. 5’r.i.. P t i r i . ~ 249, 662-664. Spcncc, J . C. H.. Cowley, J. M.,and Zuo, 1. M. ( 1993). A/>/>/.P/iy.s. L m . 62. 2446-2447. Spencc, J . C. H.. %hang, X . . and Qian. W. ( 1095). In E/cc/rr~riHolo,qrciphy (A. Tonornura, L. F. Allaril, <;. I’iuzi. U. C. Ioy. and Y. A. O n o , cds.), pp. 267-276. North-Holland. Elsevicr Science. Ainslerdani. Spivak, G. V., Saporin, G. V.. Scdov. N. N.. and Komolov;~,L. F. (IYhX). H i i l l . Acvid. S<,i. USSK. Srr: /’li,v.\. (USA) 32, 1036-105 I . Titchiiiiii-sli. J . M., and Booker, G . K. (1972). In Pro<, i r i g s of / l i e FI/i/i Eirropwti Cor7,qrc.s~o r i Elrctrori M k r o . v r w p y , pp. 540-541 . The Institute of Physics Publisher, London mid Hristol. Toiioniura. A. ( 19x0). Pro,q. @ I . 23, 1XS-220. Tononiura, A . ( I9X701. Kc.1: M d . /’/ig,\. 59. 639-669. Tonomura, A . (I9X7h). ./. Appl. Phvs. 61. 4 2 0 7 4 3 0 2 . x ~ l ~ o ~ l l A~ .r (iow). a, Ad. ir1 ~/jg.S.41, s - 1 0 3 . Tonomura, A. ( 1993). E / ~ / r o tffo/o,qnip/i,v, i Springer-Verlag, Rei-lln.
240
G . MATTEUCCI. C. F. MISSIROLI, AND G. I’o%%I
Tonomura, A,. Allard, L. F.. I’oui. G.. Ioy, D. C.. and Ono, Y.A . (1995). E / w / r o t / H o / o g / ~ c / p / ~(A. y. Tonomura. L. F. Allard. G . P o u i , 0. C. Joy, and Y. A. Ono. eds.). North-Holland. Elwvier Science. Ainstei-dnm.
‘Ionomurn, A,, Malsuda. T.. Kawasaki. T.. Entlo, J.. and Osakahe, N. (1985). Ph,~.s.K o : Lrrr. 51.60(12. Tonomul.a. A,. Osnknhe. N., Mat\uda, T., Kn ilii. T., Endo J., Yano, S., and Ximnda. 11. ( I‘186). P/IY.S. Re),. Lrrt. 56. 792-795. Vdkl, E., Allard. L. F., and Frost, H. ( 19YS). I n Elec.rroti Hologrtrphy ( A . Tononiura. L. F. AII;irtl. G . I’ozzi. 0. C. Joy, and Y. A. Ono, eds.), pp. 103-1 16. Norlli-Holland. Elsevier Science. Ain\lcrdnin. Wude, R. H. (1973). I n At/i,crr/cas itr Op/iccrl oiid Ekrrori Mic.ro.ccopv (R. Hnrer iind V. E. Co\\le~. eds.), Vol. 5, pp. 239-296. Academic Press, New York. WahI, H. (1975). Ph.D. Thesis. University of Tiihingen. WahI. H., and Lxi, H. (1979). Opfik ( S / i t t t g ~ i r r54. ) 27-36. Wang, %. L. (1995). OpriX ( S / i i / I g c i r / J101. 24-28. Wendr, G. ( 1958). 111 H m d / n d i rlor Plryxik (S. Fliiggc, 4.).Vol. 16, pp. 1-164. Springer-Verlng,
Berlin. Wohllchen, 0 . (1971 ). In E / K ~ R ) JMic,rn.scq>y /
iri Mderitrl Sc.icwc.r ( U . ValdrL:, etl.). Vol. 2. pp. 71 2-757. Academic Press, Ncw York. Wolfram. S. ( 1994). Mu//ieriici/ic; (I .sy.v/rnr jijr tloiug iiru//ietmrric.s by c.ornpirrrr. Se.c.ot/t/ d i r i o t / Addicon-Wesley, Redwood City, Illinois. %Iimg. X., Hashinioto, T., and Joy. D. C. ( 1992). A/)/>/.Plry.v. L P ~60. . 784-786.
The Imaging Plate and Its Applications NOBUFUMI MORl
I. Iii1rotluctiori
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L) . Imllge I’rocc\\or . . . . . . . . . . . . . . . . . V . CIiai-ack~htic\ of tlie linagiiig Plate (11’) System . . . . . . A . Scmitivity . . . . . . . . . . . . . . . . . . . H . Rcwlutioii . . . . . . . . . . . . . . . . . . . C . Fudiiig . . . . . . . . . . . . . . . . . . . . . L) . Granularity arid Uniloriiiity . . . . . . . . . . . . VI . 1’1-acliciil Syslcim . . . . . . . . . . . . . . . . . . A . Triinsniishioii Electron Micro\copc ( ‘ I E M ) Syslcm . . . . H . Coiiiptilctl Ratliogi-aptiy iiiid Radio Luiiiinogr~iphySyhteiii . VII . Applicntion 01 tlic Imaging I’latc . . . . . . . . . . . A . High Sciisitivity . . . . . . . . . . . . . . . . . H . Wide Ilynaiiiic Rangc . . . . . . . . . . . . . . C . Qunntitativc Image Analy\i\ . . . . . . . . . . . . L) . Image I’roccshlng . . . . . . . . . . . . . . . . . E . Anolhcr Field 01‘ Applicatioii o t t h c Imaging I’lalc . . . . V I I I . Conclwion . . . . . . . . . . . . . . . . . . . AcLnowledpmeiil\ . . . . . . . . . . . . . . . . . . Rc lcrc rice s . . . . . . . . . . . . . . . . . . .
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253 253 251 257 258 259 262 262 263 265 265 269
271 285 286
288 288 288
1. INTROD~JCTION
The Imaging Plate (IP) system was first developed for diagnostic X-ray radiography (Sonoda et a/., 1983). Recently the 1P system has been used to obtain quantitative analysis in X-ray diffraction experiments and i n autoradiography. that is. an image
242
NOBUFUMI MORI A N D m m u o OIKAWA
by the radioactive nuclei (Amemiya et al., 1988). It is useful for detecting neutron images (Niimura et al., 1994). The image of Transmission Electron Microscope (TEM) also can be recorded on the IP (Mori et al., 1988). The results showed that the IP system is much more useful than the conventional photographic film system or other image sensors, television cameras, solid-state image devices, and gas-type area detectors. The IP is approximately 0.5 mm thick, and it is composed of flexible plastic film coated with photostimulable phosphor powder (BaFX : Eu", X = CI, Br, I) together with an organic binder. In the 1P system, a pattern of radiation is temporarily stored in the IP. The data on the 1P are read out by an image reader, which scans the IP with a focused beam of red-light laser and causes photostimulated luminescence (PSL). The emitted light falls on a photomultiplier tube (PMT), the output of PMT then becomes the time series of the digital signal. We call this new technology of radiography rudio luniinogt-uphy, along with coniputed r.adiogr-up h y in diagnostic X-ray imaging. Advantages of the IP system are wide dynamic range over five decades of magnitude, linearity of response to the intensity of radiation throughout the dynamic range, and 100 or 1,000 times higher sensitivity than photographic films. We will discuss the mechanism of PSL. The mechanism has attracted many researchers because it is interesting not only as the study of phosphor but also as that of the relation between luminescent centers and defects. There were some different opinions, and we will discuss the latest interpretations. We will deal with the system including the IP, scanner (reader), printer, and processor. The structure of the IP will be discussed in relation to image quality. There are various ways of scanning the IP, which have merits and drawbacks. We must discuss the characteristics of sensitivity, linearity, resolution, noise, and uniformity. The numerous practical data will show the reader the usefulness of the IP system and become a good reference for new applications of the IP.
11. MECHANISM OF PHOTOS'TIMULATED LUMINESCENCE (PSL) Phosphor is a material that emits light excited by energy such as another light, ionizing radiation like electrons or X-rays, electric power, force, and chemical reaction (Blasse and Grabmaier, 1994). In these types of luminescence, first the excitation energy is converted to photon energy. However, photostimulated or thermally stimulated luminescence is caused by a second excitation of light or heat, after storage of the energy of the first excitation. We can use this second type phosphor as memory for images. BaFX(X = C1, Br, I) : Eu" (Sonoda eta/., 1983), RbBr : TI+ (Amitani ct a/., 1986), BaSSiOjBrb : Eu" and BasGeOJBr6 : Eu" (Meijerink and Blasse, 1991), Ba3(P04)? : Eu2+ (Schipper et ul., 1993),
243
THE IMAGING PLATE A N D ITS AI’PLICAI‘IONS
A
f
8
F
0
Ba
0 0
........
.......
...... .......
X X
0
Ba
0
F
I . The lallicc sirticlure or HaFX ( X = CI. HI.. 1). The lattice pal-amcler dcpentlr 011 tlic larger Iiaiogeii coinponelit. ELI” ions replace HLI’ ions wlieii ELI’ ion are inrroiIuccti as I U iiiiiicscciit centers in thc crystal. The layer is chown on [tic righ-hand side (Licbicli ant1 Nicollin. 1977). FlCillRE
’
’
Y2SiOS : Ce3’ (Meijerink et al., 1991), (Ca, Sr)S : Eu2’-. Sm3+ (Gasiot et al., 1982, Keller and Pettit. 1958) were reported as suitable second-type phosphors. Among these, the most popular phosphor for photostimulated luminescence (PSL) is BaFX(X = CI, Br, 1): Eu” phosphor. In this chapter we will discuss the mechanism of PSL in BaFBr : Eu” phosphor. The crystal structure of BaFX is shown in Fig. 1 (Liebich and Nicollin, 1977). This crystal has the layer structure F-Ba-X-X-Ba-F and belongs to the D:,, space group. Since the X-X plane has weaker ionic force, the crystal tends to be cleaved along this surface. BaFX is colored under irradiation of ionizing radiation or ultraviolet light, the origin of the color being a defect called an F-center. The characteristics of F-centers have been studied in alkali halide and it was revealed that halogen vacancies (F+)in the structure capture electrons (Fowler, 1968). Yuste et ril. (1976) observed two types of F-centers corresponding to the two kinds of halogen atoms F(CI-) and F(F-) in BaFCI; Takahashi et al. (1985) observed F(Br-) and F(F-) in BaFBr. BaFX : ELI?’ phosphor emits blue light as shown in Fig. 2 (Sonoda Ptal., 1983). The origin of the blue light at about 390 nm is Eu” ions that are found in the BaFX crystal in small numbers as luminescent centers. Direct optical excitation of Eu” ions is about 4.6 eV (270 nm), not shown in the figure. The excitation spectrum of PSL in BaFBr : Eu’~‘ phosphor spreads over the visible light region (Fig. 3, Umemoto et al., 1988). In particular, i n the red-light region we can get good efficiency, since this material is well suited to the red-light laser.
244
NOBUFUMI MORI A N D TETSUO OIKAWA
400
500
Wave Length (nm) 2. ~uminesccncc~pcciraof H ~ F X : EU’ ( X o/.. 1983).
FIGLIRE (Sollcltla C f
CI. ~ r I ). ptw\pIlol- ;II
1-00111icmpcr3iut.c
Takahashi et al. (1984) proposed the following model as the mechanism of PSL in BaFBr : Eu2+. Ionizing radiation creates electrons and holes in the crystal. Electrons trapped by halogen vacancies from F-centers. Holes are trapped by Eu” ions. Irradiation by light liberates electrons from F-centers to the conduction band, after which electrons recombine with holes of ELI‘-‘ions. The recombination energy is converted to light from Eu”. ions. These processes are represented as an energy-level diagram (Fig. 4, Iwabuchi er al., 1994). The following experimental findings support this model. Measurement of electron spin resonance spectra revealed that F-centers are formed by X-ray radiation (Takahashi et ul., 198.5). The optical absorption spectrum due to F-centers, the excitation spectrum of photostimulated photoconductivity, and the excitation spectrum of PSL were measured in an X-irradiated BaFBr : Eu” single crystal (Fig. 5 ) . Good agreement of the absorption spectrum of F-centers with the PSL excitation spectrum meant that an electron from the F-center is related to the PSL mechanism. Simultaneously observed photostimulated photoconductivity meant that the electron moves through the conduction band. Temperature dependence of the intensity
T H E IMAGING PLATE AND ITS AI'I'LICATIONS
245
1 .o
0.5
500
600
700
800
Wave Length (nm)
of the photoconductivity revealed that the thermal energy difference between the first excited state and the conduction band is 37 meV for F(Br-) and 1.3 meV for F(F-) respectively (Iwabuchi rt a/., 1990). The change of luminescence intensity of Eu" ions before and after exposure to ionizing radiation or light stimulation supported the hypothesis that the Eu'~'-ion traps a hole (Takahashi ct ul., 1984). In the above discussion, trapping an electron by a preexisting halogen vacancy is the origin of the F-center in the BaFX system. However, on the formation mechanism of color centers, there have been many arguments concerning the alkali halide system (Williams and Song, 1990). Two different processes of Fcenter formation have been discussed. ( I ) The lattice originally has defects such as halogen vacancies. When a free electron is introduced by ionizing radiation or an electrode, F-centers can be easily created by trapping electrons at halogen
246
NOBUFUMI MORl A N D TETSUO OIKAWA
Conduction Band
T-r Valence Band Memory
4 - 1 - 1 -
Read Out
FIGLIRE 4. The energy diagram that illustrates the photostimulated luminescence mechiinisin in BaFBr : Eu” . The band gap is 8.2 eV and the optical absorption energy ofthe F-center that is fornied hy irradiation with ionizing radiation is 2.1 eV for F(Br- ) and 2.5 eV forF(F - ) . Holes are trapped by EuIi ions. 6.2 CV is the ionization energy ofEu”- ions. which means the direct creation of holes at Eu” ions, and releascd electrons are trapped by F+ centers to form F-centers. The memory process is represented hy the solid line, and the readout process is represented by the dashed line (Lwahuchi c’t uI.. 1094).
vacancies. This is the same process as that mentioned above in the proposal of Takahashi et al. ( 2 ) High-energy ionizing radiation distorts the lattice and pushes out an atom from a lattice site. Thus, defects including F-centers are introduced into the crystal. The growth curve of F-centers as a function of radiation dose enables us to distinguish which process is dominant. In alkali halide, Rabin and Klick (1960) found linear growth, which means new creation of F-centers; thus, the second process is dominant in alkali halide. However, in BaFBr Kondo et al. (1994) found that the F(Br-) band grew and became saturated after 60 min of Xirradiation. The saturation of growth suggested that filling preexisting vacancies with electrons is the main process of F-center creation in BaFBr (the first process). The possibility of a complex center of an F-center (a hole center and an Eu” ion) was another candidate discussed. The point in question was which process, single or bimolecular, accounted better for the linear characteristic of the Imaging
I BaFBr:Eu*+
I
I
Optical Absorption
I
I
I
1
Photo Stimulation
I
500
I
I
600
I
I
700
Wave Length (nm) FIGLJKE 5. Comparison of the three spectra o f a IinFHr : ELI’+ single crystal (optical ahsorption, photoconductivity, ;ind pliotostiniulntioii spectra). The sample was heavily exposed 10 X-rays: measurcnicnt w i i h exccukd at room temperature with conliguration of E 1 c. (Repi-inted froin Tahahashi r f uI., J . Lumin., 3 1 CG 32 ( 1984). 266. “Mcclianism of Photo5timulated Luminescence in IhFX:Eu?+(X = CI. €31.) I’hosphors,” with kind permission from Elsevicr Science-NL, Sara Burgerhartstraut 25, 1055 KV Anisterdnni, The Nether1;inds.)
Plate (IP) system. Following the proposal of Takahashi et al., photostimulated luminescence occurs by the meeting of electrons with holes; the probability of meeting depends on the product of their concentrations. This is essentially a bimolecular process. On the other hand, in the case of complex centers of Fcenters and hole centers, the recombination probability of an electron and a hole should be some fixed value, not depending on their concentration (single molecular process). In 1988 von Seggern ef al. calculated that the bimolecular reaction led to a quadratic relationship with dose whereas the single molecular reaction led to a
248
NOHIIFCIMI MORI A N D ‘ETSUO OIKAWA
linear relationship. However, lwabuchi c>t01. ( 1994)repeated this calculation much more precisely, and found that linearity also occurs in the bimolecular process. the quadratic relationship occurring in only restricted conditions. Thus, there is no problem with the model of Takahashi et ul. Only in the low-temperature region, where an electron cannot escape from F-center without thermal activation energy, must we consider complex centers and electron hole recombination that occurs when electrons move by the tunneling process. Hangleiter ct ul. ( 1990) found this tunneling process at LHeT. As a summary of this section, there have been many discussions on the process of photostimulation in BaFBr : Eu’ The model proposed by Takahashi (it (11. explained most of the experimental results. The study of BaFBr : ELI’+ has still been attractive. Ohnishi rt ul. ( 1994) and Radzhabov and Egranov (1994) revealed luminescence of self-trapped excitons. Koshnick et ul. (1992) and Kondo et (11. (1994) discussed stability of F-centers, which is important for fading characteristics discussed later. These studies will contribute to the improvement of the characteristics of the IP system.
‘.
111. IMAGING PLATE(IP) The IP is essentially composed of a protective layer, a phosphor layer, and a support layer (Fig. 6). The phosphor layer has the structure o f a dispersion of phosphor particles and organic binder. Since the amount of binder is of the order of 1/ 10 that of phosphor by weight, the phosphor particle is not fully buried in the organic binder, but bound to a neighboring phosphor particle only by contact with it (not shown in the figure). The phosphor layer is sensitive to ionizing radiation. and thickness is about 50-300 pm. The protective layer, 3-10 p m thick, protects the phosphor layer from dust, stain, or damage by external forces. The support layer gives the appropriate rigidity to the IP for transportation by the mechanical system. Flexibility is also useful in autoradiography for close contact between samples and the plate. The luminescent intensity of the 1P is closely related to the reading system in respect to excitation wavelength and energy density; however, the characteristics of the phosphor are the most important. Though BaFBr : Eu”- phosphor was originally used combined with He-Ne laser (633 nm) as a stimulation light source, recently BaFBro,&15 : Eu” has been used much more together with a high-power semiconductor laser of 680 nm. This is because the semiconductor laser is preferable for reliability and reducing the system size. Figure 3 shows a comparison of the stimulation spectra of the two phosphors. The advantage of the BaFBr,),gsIo,ls: Eu’+ is apparent for the 680 nm laser. The particle size of the phosphor affects resolution and noise of the image. The IP in the 1980s was made from about 7 p m phosphor particles. but in the
THE IMAGING PLATE AND ITS AI’I’LICATIONS
249
199Os, 4 p m phosphor particles were made. Using a smaller particle size improves resolution and reduces the noise of the grain of phosphor. Therefore, the direction of phosphor development is to get the phosphor of higher luminescent intensity and smaller particle size. Figure 6 also illustrates schematically how the reading light scatters in the phosphor layer. The degree of scattering determines the resolution in position and detection intensity of the luminescence. The thicker phosphor layer increases the absorption efticiency of ionizing radiation and luminescent intensity, especially for X-rays, however, scattering degrades resolution. To improve resolution, a blue pigment, which absorbs only reading light, is sometimes useful. As the phosphor particle scatters light, it may be useful to use a transparent phosphor layer such as a single crystal; however, the reading light is
250
NOHLIFUMI MORI AND TETSUO OIKAWA
reflected at the other side of the layer, and this reflection occurs back and forth many times. Thus, resolution should become much worse, and another technology is needed to reduce the broadening of the laser beam. The protective layer is important for the durability of using it many times; however, the thickness of the layer affects the resolution of the image. Generally, a thicker protective layer is better for durability but worse for resolution. Another point concerning the protective layer is the attenuation effect for ionizing radiation. The penetration depth of the electron at the protective layer estimated by the equation of Katz-Penfold (1952) is 70 p m for 100 keV, 2 p m for 10 keV and 0.07 p m for 1 keV electrons. The maximum energy of an electron of tritium is about 10 keV; more than half of its energy will be dissipated even for the I p m protective layer. Thus, the IP for tritium has no protective layer. The phosphor does not degrade in normal humidity, but does decompose on contact with water. In the field of autoradiography, where the surface is in contact with the sample, water contained in the sample often permeates the phosphor through the protective layer. Thus, sample dryness is important for durability of the 1P. These features give a survey of the structure; however, there are many commercially available types of the IP similar in outward appearance; size, thickness, and flexibility, may vary. There are high-resolution types and high-sensitivity types. In practice; some are better for X-rays, others for Transmission Electron Microscopy (TEM) and autoradiography. Each of them combines almost exclusively with a particular reading system. Thus, in selecting an IP for a specific purpose, it is important to study all the characteristics of the system.
Iv. ELEMENTS OF THE IMAGING PLATE (IP) SYSTEM Figure 7 shows the typical configuration of the Imaging Plate (IP) system. The reader reads out the IP after exposure to ionizing radiation. Luminescence from the IP is photoelectrically detected, converted to a digitized electrical signal to be processed by the computer system. The eraser then exposes the IP to visible light to erase stored data and the IP becomes reusable. The details of this procedure are as follows.
A. Exposure The IP is a two-dimensional sensor for ionizing radiation. When the IP is exposed to ionizing radiation, the IP stores the radiation energy as a latent image. Since stored energy on the IP disappears with light exposure, we must ensure that the ionizing radiation falls on the 1P in the dark. For diagnostic X-ray imaging, it is convenient to use the light shield case known as the cassette. This cassette is
TNE IMAGING PLATE AND ITS AI’PLICATIONS
25 1
I
I
1111 Reader Exposure
L 7
FIGLIKE 7. The conliguration of the Imaging Plate (11’) system. The II’ system is comprised o t the 11’. the reader. tlie eraser, and the procesor. After exposure to ionizing radiation, the IP is fed into the reader where Ihe II’ is scanned with the visihle laser. The Il’einits blue light, the intensity of which ia proportional to tlie dose of ionizing radiation. The luminescence i\ dekcted by ;I photoniultiplier tuhc and converted to an eleclrical signal. In the eraser, &ila are processed to he enhanced or analyzed to nieasure intensity and so on. After rending, the 11’ is irradiated with light to erase data stored in thc IP.
almost the same as that used in a conventional film screen system. For autoradiography, when the sample is very thin, like a membrane, we can expose the 1P by contact with the sample in the cassette. However, when the sample is too thick to use the cassette, we need a box to shield light. The IP system is so sensitive a detector of environmental radioactivity, that the latter causes a noiselike fog level of photographic film. In order to avoid this, it is preferable to erase just before an exposure to eliminate any prior stored activity and, furthermore, to use a shield box made of lead for long exposures as in autoradiography. After exposure, it is better to read the plate as soon as possible because the stored energy gradually escapes even in a dark place. We call this phenomenon fading. B . Reading
In the reader, the 1P is scanned with a red-light beam that is focused on the surface of the IP. Luminescence is about 390 nm for BaFX : Eu” and comes to the
252
NOHLlFUMl MORI A N D ’IETSLIO OIKAWA
(4 Laser
PMT
PMT FK~CIK X. FVarious types of waimei-. ( a ) Flat-bed-type scanner: the Iiiiiiging I’liite (11’) I\ lielcl on the flal bed. The h e r hearn rellcctctl by a rotnting or turning iiiii-ror \cans the IP. The luminescence from 11ic IP i \ guided In thc phntomultiplier ttihe (PMT) through the light guide. ( b j Spinner-type scnnner: the 11’ i s held on Ihe mncr side 01‘ Ihe cylinder and move5 along lhe direction oi the axis o i the cylinder while the reading tiend (spinner) r o ~ a t e b .(c) Drum-type scanner: the It‘ i \ put 011 the I-otating cylintlcr (drumj. Thc reading herd iiinves along the direction parallel to the axih of the cylinder. (ti) Disk-type scanner: the 11’ turns uround. The reading head. which irrudiatch it with laser hcam light and collects Iuniiiiescciice. iiioves along the radinl direction.
detector with excitation light. Thus, the optical filter in front of the photodetector is used for cutting off laser light. There are many possible ways of scanning (Fig. 8). In case of a flat-bed scanner, the IP is held on the flat bed and transported. A rotating mirror reflects laser beam light and focused beam spots move in a straight line on the IP. An F-8 lens is used to achieve uniform velocity scanning on the 1P. The plate moves along the perpendicular direction to that of the spot mot ion.
‘THE IMAGING I’LA’I‘E A N D ITS APPLICATIONS
253
In the spinner-type scanner, the IP is tixed along the inner surface of the cylinder. Laser light passes through the dichromatic mirror ( A ) , is reflected by the mirror (B), and is then focused on the 1P surface. Luminescence of the IP is collected by a lens; lens and mirror rotate as indicated by the dashed line in the figure. The pair of lenses forms a confocal configuration, which is used for PIXsysTEM. However, in the FDL-5000 system, lens (C) is not used. In the drum-type scanner, the IP is tixed on the cylindrical drum and the reading head moves parallel to the axis of the drum. In the disk-type scanner, the 1P rotates and the reading head moves along the radial direction. In this type of scanner, the spatial density of reading must be kept the same between inner and outer positions of the plate. The flat-bed type is the most popular for medical applications or biotechnology; the spinner type is used for Transmission Electron Microscope (TEM).
After reading, exposing the 1P to visible light erases the data stored. The light source is an ordinary fluorescent lamp or sodium lamp, chosen for its electrical power efficiency. The erasing level restricts the lowest detected level of the next measurement. High sensitivity means essentially competition with detection of unwanted environmental activity. Though the film has a one-way characteristic of storing information, the 1P has a reset procedure by erasing. This is one of the reasons that the IP system can achieve high-sensitivity detection.
The latest technology of computer and memory devices makes it possible to execute complicated image processing tasks much more rapidly. Image processing is useful to distinguish patterns or to measure the quantity of activity and pattern shape, gradation processing, narrowing the range, and enhancement contrast of the image. By broadening the range, we can easily observe whole patterns of large dynamic range such as diffraction patterns. Since the IP system has good linearity. direct reliable quantization is possible with image data, and display the protiles of image data is also useful to compare activity. FFT or contour map processing is useful to improve distinguishing power. The “superimpose” function allows one to write letters or arrow marks on the recorded image, and this is useful for presentation. In the field of diagnosis, image processing may indicate the point that doctors should note. These types of processing enhance the value of the image and this is one of the merits of the IP system. Image data can be stored on a large-capacity memory device such as an optical magnetic disk; we can archive image data and retrieve images quickly.
254
NOBUFUMI MORI AND TETSlJO OIKAWA
v.
CHARACTERISTICS OF THE IMAGING PLATE
(IP) SYSTEM
In the following, we will discuss mainly the sensitivity and dynamic range, resolution, fading, and noise. The noise characteristic is important to assess the efficiency of the detector, though it is difficult to calculate. The characteristics will be discussed by using the data of the Transmission Electron Microscope (TEM) system. However, this discussion should be applicable to other fields, taking into consideration any differences of ionizing radiation. A. Sensitivity
Sensitivity is the luminescent intensity detected. Thus, the flow of image data is important for any discussion of the sensitivity factor (Fig. 9). Let N be the initial number of quanta of ionizing radiation. This number is multiplied by efficiency factors. ~ ( 1 is) the efficiency of the 1P. a! includes the absorption efficiency of ionizing radiation, electron- and hole-creating efficiency and readout efficiency. a! depends on the intensity I of reading light; the dependence of a! on laser intensity I is gradually saturated in a practical system. In the case of X-rays of 80 kVp, a! is estimated to be about 10-200 in practical systems. /Iis the light-collecting efficiency, including the transmission characteristics of optical elements such as filters, light-collecting guides, or lenses. This is normally 0.1-0.5. x is the quantum
Step of Process
Number of Quanta
(1) Ionizing Raditation
N
(2) Imaging Plate
a(I>N
am
P
(3) Light Gu'ide(lens, Filter)
(4) Photo Electrical Conversion
(5) Amplifier
Efficiency
xN
x
Px6N
6
a(1)p
FIGLIKE 9. The How of image carriers. N quanta of ionizing radiation fall on the Iinaging Plate (IP), which absorbs ionizing radiation and emits photons with efliciency a ( I ) ,when stimulated with light of intensity I.Photons from the 11' reach the photodetector with efficiency b , which is delinetl hy the light-collccting efliciency and ti-ansmission coeflicient of the optics. Photons arc convei-led l o electrons hy the pholodetector with efficiency x and the number of electrons increahes both in the photodeteclor and electrical ainplifier by a factor 8. The linal nurnher is the product of these efficiencies.
255
THE IMAGING PLATE A N D ITS API’LICATIONS
efficiency of the photodetector. As for the photomultiplier tube (PMT), it is the quantum efficiency of photocathode, typically 0.14.3. 6 is the amplifying factor of the PMT or electrical circuit, normally 102-107.By using this notation, detected luminescent intensity becomes Luminescent Intensity = N c u ( I ) B x 6 The signal intensity of the reader of PIXsysTEM as a function of electron dose is shown in Fig. 10 with the data of photographic film (Mori et ul., 1990). The figure shows good linearity of signal intensity to electron dose over 5 decades. The IP is used for many other types of ionizing radiation, and linearity of the PSL intensity to the dose of radiation is generally observed. This is because ionizing radiation creates electrons and holes in the phosphor without any nonlinear process irrespective of the kind of radiation, though the efficiency will be different. The vertical axis for film is optical density. Though it may be possible to get a straight
I 10 5
5
104
4
103 10 2
10 1
1
100
1 0
Electron Dose (C/Cm2) FIGURE10. Sensitivity chiiriiclerislics o f t h e Imaging Plate (IP) system (PIXsysTEM). The signal intensity of the 11’ is plotted. The dcnsily ci~ilieol FG film, developed by 0-19 for 2 inin is :ilso plolled a h a reference (Mori r i o/., 1990).
256
NORUFLJMI MOKl A N D TETSUO OIKAWA
10
5
2 50
100
500
Accelerating Voltage (kV)
line by using another unit or calibrated data, the drawbacks of using film are its narrow dynamic range of about 2 decades and slightly poorer reproducibility, since the density changes by chemical condition like the concentration or the temperature of the developer. Thus, using the IP improves the precision compared with the photographic film method. Figure 1 I shows the dependence of the sensitivity of PIXsysTEM on accelerating voltage (Mori et ul., 1990). The IP system shows its maximum intensity at about 150 keV. Ogura and Nishioka (1995) measured the dependence of the sensitivity for 40-200 keV for FDL-5000, and obtained similar results to those of Fig. 1 1 . The origin of the decrease below 100 keV is thought to be due to electron absorption by the protective layer. The interpretation of the decrease in the higher-energy region is as follows. As the energy of the electrons goes up, the penetration depth of the incident electrons increases and electron energy is mainly dissipated in the deeper part of the phosphor layer. However, the intensity of the light for reading becomes weaker in the deeper part of the phosphor layer because of the absorption and diffusion of light. Luminescence from the deeper
TIHE IMAGING PLATE A N D ITS AI’PLICATIONS
257
part of the phosphor layer is also diffused and weakened. As a result, the detected intensity of luminescence becomes weakened. Electrons of much higher energy will pass right through the phosphor layer, and the intensity will then decrease substantially.
B. Resoliitiori The IP itself does not have discrete pixels, but a pixel is created as the electrical signal by the reader. Thus, signal response is very important for resolution. One of the factors determining resolution is the scattering of the laser beam in the phosphor layer as discussed in the IP section. Another factor is the time response of the luminescence and the photodetecting system. The decay characteristic of the luminescence, the time i n which the luminescence declines to I/e intensity, is about 0.6 ps in case of BaFBro.~sIO.I~ : Eu”; the reading time for one pixel should be longer than this time. The response of the electrical system, which converts luminescence to a digital electrical signal, should be shorter than the time for one pixel. Of the many ways of evaluating resolution, some select the method in which the lattice image of a gold crystal of graphitized carbon is used. This way is very practical, but the result is affected by the characteristics of the TEM and the operating condition when taking images. A method using a metal wire has been examined (Isoda e t a / . , 1992, Burmester et a/., 1994). The wire was directly fixed on the IP, uniform electron radiation creates a shadow of the wire on the 1P. The resolution as MTF (modular transfer function) was determined by the frequency analysis of the difference between the theoretical image and the observed image; MTF(q) = F ~ , , , , ( L / ) / F , ~ , where ~ ~ ~ ( Yq) . was spatial frequency, F(,,,,(q) was the amplitude of the Fourier spectrum of the observed shadow profile, F,l,?(,(q)was the amplitude of the spectrum of theoretical square-well profile. Instead of Fourier analysis, one may use a metal mask that has a pattern of openings of various spatial frequencies (Mori et u/., 1990). Uniform exposure made a square wave pattern on the IP; readout amplitude of the wave pattern declined at a higher spatial frequency. Thus, resolution was expressed by Response ( 4 ) = A ( q ) / A ( O ) , where A ( 4 ) was the amplitude of image profile at spatial frequency q . This corresponds to the contrast transfer function (CTF). MTF and CTF give almost the same result; however, MTF is more suitable for treatment of theoretical analysis. Figure 12 shows the improvement of the resolution by comparing the resolution of the three systems, measured with the method of metal mask: (a) is the result of flat-bed type, with a pixel size of 100 p m , (b) is that of PIXsysTEM, with a pixel size of 50 p m , (c) is that of FDL-5000, with a pixel size of 25 p m . The improvement of resolution is important for TEM systems, since we can take images covering a wider area and at lower TEM, magnification.
25 8
NOHUFUMI MORl AND TETSUO OIKAWA
1 .o
a,
tn c
0
a . 0.5 v) Q)
[II
0
2
4
6
8
10
12
14
16
Spatial Frequency (Ip/mm) FIGUKE12. The resolution of the Imaging Plate (IP) system. The resulth of response measured with the metal plate method is summarized. Square is HR-I1 IP and CR-IOI system. closed circle; 1JR-I11 IP and PIXsysTEM (Mori ef u/., 1990). Open circlc: FDL-URV and FDL-5000 (Ogur;~cf ( I / . . 1994).
C. Fading The intensity of the stored image on the IP decreases with the passage of time. Figure 13 shows the fading characteristics of PIXsysTEM (Oikawa et al., 1994). The degree of fading depends on temperature, however, and is generally larger as the temperature is higher. This characteristic depends on the phosphor itself and on the wavelength of the reading light. There is no precise comparison of dependence on various types of the IP, but there is not much difference. Oikawa et a/. proposed empirical equations of fading as a function of temperature. This is useful for estimating the degree of fading. The fading characteristic does not depend on the dose; this is very important as it is possible to compare the intensity even after fading. The fading is negligible provided that the IP is kept in cool
THE IMAGING PLATE AND ITS API'LICATIONS
259
10 5
to
4
10 3
102
10
P IXsysTEM
' 10 -1
100
10 1
102
10 3
Time (hr) F I G UKE13. Fading characteristics. lnteiisity change with thc passage of time is plotted a1 0 and 25 degrees Centigrade. TIKn i e a s t m m e n t was made will1 doses of I x 10-I", IO-", l o - ' * C/crn' (Oikawa ( ' r o / . , 1994).
conditions, but this is not practical for TEM use. For autoradiography, however, it should help to increase sensitivity because of its long time exposure. D . Grani.rlar.ityarid Uniformity
Image noise (granularity) is directly related to perceptivity by image. In this sense, noise is another aspect of sensitivity of the system. Granularity is the deviation of the intensity of each pixel, composed of mainly two components. One is dependent on the number of image carriers, while the other is not, and has a fixed value. The former follows a statistical deviation, the Poisson distribution: (Noise)2 = I / / I , where / I is the number of image carriers (Dainty and Shaw, 1974). This number of image carriers changes as the detection process proceeds (Fig. 9). The fixed noise is electrical noise or the fixed noise of the IP. Total noise (reciprocal of
260
N O H U F U M I M O R I A N D TETSUO OIKAWA
signal-to-noise ratio (SIN))is expressed as the sum of the individual types of noise.
On the basis of the above equation, the fixed noise appears at high dose ( N is large) and determines the lower limit of the signal-to-noise ratio of the system. On the other hand, the 1/N term appears at low dose (N is small). The multiplier is composed of (Y, /?, x, and 6. a, efficiency of the IP, is contained except for first term. B and x are important, since they are less than unity and may become the dominant part of the noise at low dose. Figure 14 shows noise characteristics of the FDL-5000 system (Ogura and Nishioka, 1995). The noise becomes better as the electron dose increases. The noise power is inversely proportional to the number of electrons exposed; however, improvement saturates because of the fixed noise. This figure shows that noise follows the relation above. The efficiency of detectors is often discussed using a term called dPtective quantunr c ~ c i e n i . y(DQE).related to the noise characteristics, because it does not
1 100 keV
10-1
FDL-5000
10 -2
10 -3
10-4
Electron Dose (C/cm2)
THE IMAGING PLATE A N D ITS API'I.ICATIONS
26 1
depend on the method of detection. The DQE is expressed as DQE=
($)'I(;)'
where as usual S is signal and N is noise. Subscripts o and i are output and input, respectively. The denominator is the number of quanta of ionizing radiation. The numerator is generally expressed by the equation discussed in the last paragraph. However, to calculate the numerator for detective quantum efficiency, it is false to take the noise of granularity directly because the resolution characteristic reduces the noise in appearance. With compensation for this factor, frequency analysis of noise becomes important together with resolution. In the case of 80 kVp X-rays at 1 mR, Ogawa et ul. reported a DQE of 0.2 at 1 line-pair per mm for the FCR 9OOO/ST-V system. This is an accurate way of characterizing a system, but is difficult because it needs resolution data (MTF) and the data must be processed by FFT (Dainty and Shaw, 1974). A convenient way of calculating DQE with larger pixels such as 3 x 3 is sometimes used for minimizing the effect of the response, though information about the frequency dependence is lost. Thus, in this case, the value should be discussed together with resolution. Burmester et al. (1994) estimated by this convenient way that the DQE of their IP system is about 0.9 for 120 keV electrons at about lo-" C/cm'. Ogura and Nishioka (1995) also calculated the DQE of the FDL-5000 using the data of Fig. 14, and found a value of almost unity for 100 keV electrons at about the same dose region, taking care to measure the electron dose accurately. The difference of DQE is large between the result of Ogawa et d.and Burmester el a/. We suppose that this is due to the difference of ionizing radiation (X-rays and electrons). Electrons in this energy region will all be stopped as is predicted by Katz-Penfold equation; for X-rays, however, above SO% of the X-ray photons escape from the phosphor layer. Thus, the efliciency (Y will be very different between the two sources. Another factor is involved when the continuous signal is converted to a digital signal: the density resolution of the signal. When the density resolution is not as small as the noise level of the image data, the pattern will have artifacts such as contours, or the precision of quantitative analysis becomes degraded. However, too small a density resolution leads to a waste of memory resources or time for image processing. Normally, data are logarithmically transformed, as expressed by the following equation:
where L is the dynamic range of the image, 111 is the density resolution expressed by the bit number : ni = 2'' and Q denotes the digital data. The change of fraction between Q and Q 1 is D = L/ni (Inlo), which is sometimes called the error of quantization. The value D should be almost the same as that of the
+
262
NOBUFUMI MORI A N D TETSlJO OIKAWA
image noise. For example, in the case of L = 4 and noise = 0.4%, then w7 should be 1000, which means that the density resolution should be 10 bit (1024). This density resolution should be selected depending on the application field because the necessary signal-to-noise ratio depends on the application field. Uniformity of sensitivity is important for quantitative analysis. In the flat-bed scanner, the uniformity of laser light intensity and the light-collective efficiency govern the uniformity characteristics. Uniformity is always the same and can in principle be calibrated in the system. In some systems, the calibration is executed automatically and the user does not need to recognize this factor. The uniformity of the IP originates mainly from the uniformity of thickness of the phosphor layer. Amemiya et al. (1988) reported that uniformity error is about 1.3%. They concluded that this degree of uniformity is sufficient for X-ray diffraction analysis for their purpose.
VI. PRACTICAL SYSTEMS In the previous section, we discussed the principle of the system and dealt with the basic ideas. In this chapter we will consider the practical system.
A . Transmission Electron Microscope (TEM)System Figure 15 shows the layout and components of the TEM system of the FDL5000 (Ogura et al., 1994). We can use the IP in the TEM just like photographic film because we can use an ordinary film cassette for the IP, together with the film magazine of the TEM. After removal from the cassette, exposed IPS are put into the magazine for reading the system. After the information such as operation conditions of TEM, sample names, and reading parameters are set, the reader reads all the IPS automatically. The data of the IP are stored on the Digital Data Storage (DDS) unit simultaneously while reading. When the printer is connected to the 1P reader, the image hard copies are also available at the same time. The image data in the DDS are transferred to the processor and processed and displayed. Since the processor is independent of the reader, image capture and image analysis can be performed separately. The size of the IP used is about 94 x 75 mm. The pixel size is 25 pm. The data volume is about 23 M bytes. In the TEM, photographic film or a TV camera system has been used (Reimer, 1984). Burmester et a/. (1994) summarized the DQE of image devices: less than 0.35 for photographic film and 0.4-0.7 for slow-scan charge-coupled device (SSCCD) ( Kujawa and Krahl, 1992). They also reported that the DQE of the IP system of their own was about 0.9, as discussed in Section IV. This high efficiency is one of the merits of the IP system. The high sensitivity is useful not only for saving the
263
THE IMAGING PLATE AND ITS APPLICATIONS
DDS Cartridge IP
TEM Cassette
IP Magazine IP
Reader Printer
IP Eraser
FIGLIREIS. Transmission Elcctron Microscope (TEM) system (FDL-5000). In this configuration, the Imaging Plate (11’) is used with the TEM cassetkc i n thc TEM and with an 1P niagazine in the reader. Data from the IP are transferred to the computer system hy the data storage media of DDS. This is because the quantity of data in the system is several 10 M bytes. 50 the data transfer time is not ncgligihle. The separation of data processing and reading make thc bcst of the independent operation of each step (Ogura ~f ( I / . , 1994).
sample from damage by the electron beam, but also for making it possible to use a high-speed shutter, which is helpful for avoiding the deterioration of the image quality by the vibration of the sample. The pixel size is 25 p m ; thus, the image enlarged 16 times by area is not unnatural because the resolution limit of the naked eye is 100 p m . This digital enlargement contains no distortion factor caused by the optical system of enlarging equipment, as in the case of photographic systems.
B. Computed Radiography arid Radio Luniinography Systenr The IP system was first used in the medical field for X-ray imaging. In this field the technique was called computed radiography (Tateno et al., 1987). High sensitivity is good for reducing the dose to the patient. The digital image enables us to make a Picture Archiving and Communication System (PACS) and allows comparative diagnosis between isolated hospitals by the transmission of digital images. The IP system is widely used in this field and various systems are now available. A built-in system, where the system circulates the 1P and exposure and the reading and erasing process is executed in one system, is very convenient for examination. TEM application, autoradiography, X-ray diffraction, and so on are called radio luminography. In these fields the scanner most popularly used is the flat-bed type and for high resolution, the spinner type. The IP system was evaluated in 19x6 in the field of X-ray diffraction (Miyahara et u/.). The high Detective Quantum Efficiency (DQE) and wide dynamic range of the system, together with its absence of
264
NOKUFIJMI MORI A N D mrsiio OIKAWA
count-rate limitation, resulted in a significant reduction of exposure time. Thus, the IP has helped protein crystallographers to obtain accurate measurements in a shorter time. This saves the sample from beam damage and so full data can be obtained with the use of only one sample. In the case of photographic film. many samples are needed to get full data and this degrades the accuracy of the data. This is the reason why the 1P system has led to much progress in this field (Amemiya et al., 1988; Sakabe, 1991). In the field of X-ray diffraction, the combination with a synchrotron-radiation source is most successful; in addition, the 1P system should be promising for use with a conventional laboratory-scale X-ray source (Sato et al., 1993). In the biotechnology industry, autoradiography is commonly used to analyze gene and protein sequences. Since the exposure time ranges from a day to a month in the conventional way of using photographic film, a reduction of exposure time by a factor of more than ten by the IP system is very useful (Amemiya and Miyahara, 1988). In addition, the radioactivity of part of the sample can be measured by image processing, without taking off the part of the sample and measuring by liquid scintillation counter. These merits raise the importance of the autoradiography method. Neutron radiography is used for nondestructive testing, such as inspection of organic material in a metal vessel, or neutron diffraction analysis to investigate the position of hydrogen in the protein. However, the conventional IP is not sensitive to neutrons. Niimura et al. (1994) developed an IP that contains a Gd or Li compound in the phosphor layer. Since Gd or Li atoms have large cross sections for neutron, absorb neutrons, and emit gamma rays or electrons, these can be detected by the phosphor. They justified the merits of this system and demonstrated neutron radiography with the IP. Katto et ul. (1993) measured the beam profile of a UV laser with the IP for tritium. Since BaFX : Eu” phosphor is sensitive to UV-VUV light (Iwabuchi er al., 1994), the IP is a valuable image device in this region. Nishikawa et ul. (1994) examined field emission and field ion microscopies with the IP, that is, images of Het or Ne.’.. They showed the possibility of a quantitative analysis of electron tunneling and a field ionization probability over individual surface atoms. It is the combination of all these characteristics-sensitivity, dynamic range, resolution, and large effective area-that generates the superiority of the IP system. In some characteristics, another image system is better than the 1P system. For example, the film system has good resolution and a wide effective area, but its sensitivity and dynamic range are not sufficient. The TV camera system has good sensitivity, spatial resolution, and time resolution; however, the effective area is small. The IP system does not suffer from the drawbacks of the film system and is suitable for the detection for images of ionizing radiation. Furthermore, it is important to comment on the easiness of handling of the IP system. The IP itself does not need any electric power. It is merely a thin plate and the only essential precaution is to exclude stray light. On reading, we need a large precision system;
THE IMAGING PLATE A N D ITS AI’PLICATIONS
265
however, this is not an obstacle at exposure. This easiness is another merit of the 1P system. Thus, we can apply the IP to many fields of imaging--electromagnetic waves from the UV region to the gamma-ray region, electrons, ion beams, neutrons. Its characteristics overcome the drawbacks of conventional image sensors. With the development of new types of 1P like those for neutron imaging, this new technology called radio Iumimgraphy will expand the field and make itself more valuable.
v11. AITLICATIONS OF THE IMAGING PLATE
I n this chapter, application data obtained by many researchers are introduced, which illustrate the advantages of the Imaging Plate (IP). The application fields where the IP is expected to exhibit its performance are listed in Table 1. In these fields, there have been limitations to observation with conventional photographic tilm. Use of the 1P is expected to break through those limitations. A . High Sensitivity
In this section, application data, illustrating the high-sensitivity performance of the Imaging Plate (IP), are introduced. The IP was applied to TEM observation of silver bromide microcrystals, which are typical of the electron-sensitive materials, by Ayato et al. (Ayato rt al., 1990). Silver bromide (AgBr) microcrystals are so susceptible to beam irradiation damage TAHLE I L I K E L Y FIEI.L)S OF APPI.ICATION 01; ‘THE IMA(iINC P L A I E
No.
Advantages of the 11’
1
High scrisirivity
Applicalion lields
oi heam-sen\itive specinien b) Data acquisition with high-speed shutiers (low- and high-tcmpcrnturc a) Observation
stages. etc.) c ) Dark lieltl iiiicl weak beam method
2
Wide dynamic range
3
Linear sensitivily
4
High-precision
5
DI-y systcm and
digital im;igc
others
d ) High-contriist images c ) Electron diffraction mid CBED pattcrn5 lj Electron intensiiy measureinent g) Quantit;itive image annly\is ti) Iinagc proccssing i ) Image coiitrast en1i;iiiccmcnt i) linage liling and retrieving k ) Reductioii of manpower
266
NOBlJFUMI MORI AND TETSUO OIKAWA
FIGLIKE16. Electron micrographs of silver hroinicle microcrystals taken at room temperature. (Direct magnification: x 15.000) (a) Recorded with conventional photographic tilm (Fuji FG). Electron dose: 700 elecrrons/nrn’. (b) Rocorded with the Imaging Plate. Electron dose: 7 electrons/nm’.
that they are destroyed during room temperature recording using conventional photographic film, making recording difficult. Figure 16a shows AgBr microcrystals destroyed during exposure with conventional photographic film. The authors therefore reduced the electron dose by a factor of 100 by using the IP and thus succeeded in recording AgBr microcrystals without destroying them (Fig. 16b). The high sensitivity of the IP allowed us to record images of the silver bromide microcrystals at room temperature with very little irradiation damage by reducing the electron dose at the specimen. In low-dose observation, the IP is of great use for recording an image with good image contrast even at low-electron intensity. This is because the IP has a linear response to exposure even at low-exposure levels. Another example is a measurement of electron irradiation damage to a polyethylene single crystal (Oikawa et al., 1992). The degree of specimen damage was evaluated from the degree of intensity fading of an electron diffraction spot from the specimen (Kobayashi and Sakaoku, 1964.) Figure 17 shows electron diffraction patterns of a polyethylene single crystal. These diffraction patterns were obtained at an accelerating voltage of 200 kV and an extremely low-electron dose rate, 1 electron/(nm’. sec). Moreover, the exposure time was set to 0. I sec in order to improve the time resolution per image during the exposure. Figure 17a shows an electron diffraction pattern taken by irradiating a fresh field of view with an electron beam. The image clearly shows even higher-order diffraction spots.
THE IMAGING PLATE AND ITS APPLICATIONS
267
FIG"? 17. Electron diffraction patreriis of a polyethylene single crystal with thickness about 10 nm, and their intensity tlistrihutions obtained hy the tinaging Hate (Oikawa et u/.. 1092). (200 kV. rooin lempcrature). (a) Electron dose: 0. I clectrons/nm' (fresh licld o l view). (b) Electron dose: 600 eIectrons/nm'.
Figure 17b shows a pattern taken after a dose of 600 electrons/nm'. The logarithms of the intensity distributions of the two patterns are shown along the horizontal lines in the figures. Figure 18 shows background subtraction of the intensity of an electron diffraction with three-dimensional distributions. The spots are (200) irradiated with 200 kV electrons at doses of 250 electrons/nm' and480 electrons/nm'. Figures 18a and d show the original intensity distributions of the diffraction spots (200). Figures 18b and e show background intensity distributions obtained by a background fitting method (Shindo rt a/., 1993). Figures I8c and f show the net intensity distributions of spot (200) after background intensity subtraction. Figure 19 shows the net intensity distribution changes after background subtraction of diffraction spots (200) after irradiation with 0.1, 250 and 480 electrons/nm'. By integrating the spot intensity, the change of the diffraction intensity with electron irradiation was measured. Figure 20 shows the change of the integrated (200) reflection spots for
268
NOHUFUMI MORI A N D TETSUO OIKAWA
FIGLIKE 18. Background suhtraclion process ofthe intensily ol ail elcctron diffraction s p o ~(700) irradiated with 200 kV e I e c t r o ~ ~ats closes 250 electrons/nm' (ax) and 480 elecii-undnni' (d-f).
FIGL'RE 19. Chmge of intensity distrihution ofthe ciiffraction spot (200) ofpolyethyIene irI.atllntcd with 200 kV clcclrons. ( a ) 0.1 elcctrons/nm'. ( h ) 250 clcctrons/nm'. (c) 4x0 electi-ons/nni2.
200 kV and 100 kV. Here, the incident electron intensity was obtained as the whole intensity of the diffraction pattern, and the integrated (200) reflection intensities were normalized relative to the incident intensity. In the same electron irradiation condition, the reflection intensity at 100 kV fades more rapidly than that at 200 kV. At 200 kV, the reflection intensity at 730 electrons/nm' irradiation faded to 1/20 of the original value, and at 100 kV, the intensity at 480 electrons/nm' faded to 1 / 10 of the original value.
THE IMAGING PLATE AND ITS APPLICATIONS
0
200
400
GOO
Irradiated Electrons FI(;IIHE20. Change (11 ilillraclion intensity circles) and I00 kV ( o p circles) elrclron\.
o l ii
269
800 (e-/nm')
( 2 0 0 ) rellection irl.ncliatcd with 200 hV (lull
Because of its wide dynamic range, the 1P records both high intensities (diffraction spots) and weak intensities (halo rings) in a single image. In addition, its linear response characteristic allows quantitative measurement of the beam intensity. Furthermore, by using the high sensitivity of the IP, the exposure was carried out with a very low dose, using a high-speed shutter. The intensity fading of the diffraction spots of polyethylene with electron irradiation had already been measured by the X-ray diffraction method (Kawaguchi. 1970.) However, the electron diffraction method is more useful than the X-ray diffraction method because the electron diffraction intensity is recorded simultaneously from the same specimen field of view, during electron irradiation.
B . Wide Dytiuniic Rurige I n this section, application data illustrating the wide dynamic range performance of the Imaging Plate (1P) are introduced. Since a convergent-beam electron diffraction (CBED) pattern has an intensity range covering about three orders of magnitude, the entire pattern cannot be recorded in a single image with conventional photographic film. With the IP, the dynamic range covers four orders of magnitude on a single image, allowing all the intensities of a CBED pattern to be covered. Figure 21a shows a CBED pattern recorded using the IP. Figure 21b shows a line profile (intensity distribution)
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NOBUFUMI MORI A N D TETSUO OIKAWA
FIGLIKE 21. CHED patterns taken with the Imaging Plale and a JEM-2000FX I1 TEM :it 100 kV. Thc specimen was a silicon ( I 1 I ) aingle ci-yslal. ( a ) Low-contrast print. (b) Line profile (the intensity distribution) of part a.
along the center position of Fig. 2 1a (indicated by the horizontal line). This profile shows that the pattern was recorded without saturation or loss, from the center to the periphery of the CBED pattern, indicating the large width of the dynamic range. Figure 22 is a kind of a contour map presentation, obtained by dividing the intensity range of the image of Figure 21 a into 16 parts and rendering the intensity steps of each part white and black alternately (Oikawa et al., 1990). It is seen that this presentation not only allows the pattern of the entire image to be recognized, but also is effective for extracting the features of the fine structures. With the IP, which has high-intensity resolution (4096 gray levels), contrast enhancement and image analysis applications can be carried out with high precision. Electron diffraction patterns of a Cu3Pd alloy were quantitatively analyzed by making good use of the wide dynamic range and good linearity of the IP by Shindo et al. (1990). Intensities of both fundamental and superlattice reflections of the alloy having a one-dimensional, long-period superstructure were measured in situ as a function of the temperature. The intensity changes of the superlattice reflections quantitatively evaluated clearly show the characteristic disordering process of the Cu3Pd alloy. It was demonstrated that quantitative structure analysis by electron diffraction patterns is possible with the use of the IP if the dynamical diffraction effect is taken into acount. In this study, by measuring the intensities
THE IMAGING PLATE A N D ITS AI’PLICATIONS
27 1
FIGLIKE22. A contour map of llic data iii Fig. 21a. showing that the intensity is recorded well over the whole pattern (Oikawa (’f u / . , 1990).
of the superlattice reflections and short-range-order diffuse scattering, the orderdisorder transition of the Cu3Pd alloy was quantitatively investigated using the advantages of the IP, that is, a wide dynamic range and good linearity for the electron beam. In Fig. 23, an electron diffraction pattern of Cu3Pd obtained with the IP is shown. The original signal intensities of 4096 gray levels were simply converted to 256 gray levels for the output; that is, each of the 16 gray levels of the original data were converted into one gray level in the output print of a diffraction pattern. The electron diffraction pattern shows sharp superlattice reflections, labeled A ] , A?, B I , Bz, and C. These superlattice reflections indicate a one-dimensional, long-period superstructure. In the single-crystal film, superlattice reflections from three variants are usually observed. The spots A, B, and C indicated in the pattern correspond to the three variants. The reflections A , and B I correspond to the periodicity of the basic ordered structure of Llz-type whereas A?, Aj, Bz, and B j correspond to the periodicity of long-period superstructure along each direction. By measuring the separation of superlattice reflections such as A? and Aj, the period of the one-dimensional, long-period superstructure was obtained as M = 3.6.
272
NOHUFUMI MORI AND E T S L J O OIKAWA
FIG( R E 23. Electron diffraction pattern of a single crystal Cu3Pd observed by tht Imaging Plate. ( I / = 1-3) nnd C indicate the \uperlattice reflections corrcsponding to three viiri:ints.
A,,. B,,
F I G I I K24. ~ (a) Electi-on diltrnction pattein of CuqPd after heating to X23 K in :in elccti-on iiiicroThe conversion o T tlic originnl intensity into tlic output is the same a s i n Fig. 2 3 . ( b ) Thc w i l e eltctron tlifffiiction puttem a s that i n part a, hut only the gray levels less t1i;iii level I .400 ol tlic origiixil intensity were converted into 2.56 gray levels in the output print. scope.
Figures 24a and b are electron diffraction patterns observed with the 1P after heating the alloy in the electron microscope at 823 K. Figure 24a is a pattern output in the same manner as in Fig. 23, whereas in Fig. 24b, only the g d y levels below gray level 1400 in the original signal intensity were converted into 256 gray levels; the gray levels above gray level 1400 were set to the value 256 for the output. It should be noted that in Fig. 24a the superlattice reflections sharply observed i n Fig. 23 become faint. However, in Fig. 24b, the diffuse scattering broadening at
THE IMAGING PLATE AND ITS AI'PLICATIONS
273
Flc;ol
the positions of reflections such as A2 and A3 is clearly observed suggesting the existence of a short-range-ordered state, although the intensity of the transmitted beam and the fundamental reflections are saurated in this case. In Fig. 25, the intensity distribution of the electron diffraction patterns was plotted as a contour map in order to make clear the change of the intensity distribution with the increase of temperature. It should be noted that even the intensity of the fundamental reflection is not saturated owing to the wide dynamic range of the IP. Although reflections such as A? and B I correspond to the different regions with different variants, it was possible to compare these two reflections quantitatively to examine the disordering process, assuming that the thicknesses of these regions in each of these two variants are almost equal. This is because, during heating of the sample, a small drift of the sample was noticed and so the intensity variations due to the change of the excitation errors may be considerable when the
274
NORUFUMI MORI A N D E T S U O OIKAWA
intensities of superlattice reflections A? and A1 situated relatively far from each other are compared. It is interesting to point out, that the intensity of superlattice reflections such as the one indicated by A>, which corresponds to the periodicity of the one-dimensional, long-period superstructure, decreases first, and above 790 K, the intensity of the superlattice reflections B1 decrease next. The different rates of decrease of the intensities in these superlattice reflections with the increase of temperature are consistent with the another report (Hirabayashi and Ogawa, 1957), indicating that the disordering process preferentially occurs at the antiphase boundary of the long-period superlattice, leaving a fairly highly ordered state between the boundaries below 790 K . By utilizing the IP, the disordering process of CuiPd was quantitatively analyzed by measuring the intensities of both superlattice reflections and fundamental reflections. The characteristic disordering process and the transition to the short-range-order state were quantified from the in situ experiment by using the IP. It was demonstrated that the 1P can be used for quantitative analysis by taking account of the dynamical factor.
C . Quantitutixv Inluge Analysis In this section, the application data of quantitative image analysis illustrating the linear response of the Imaging Plate (IP) are introduced. High-resolution electron microscope (HREM) images of W8Ta2020 were observed quantitatively by using the 1P with a 400 kV electron microscope by Shindo etul. (1991). Figure 26 is an example of an HREM image taken with the 1P. The specimen used was W-Ta-0; the image was recorded with an HREM, the JEM-4000EX, at an accelerating voltage of 400 kV, direct magnification of x 1,500,000,current density of 10 pA/cm', and an exposure time of 2 seconds. The image data were subjected to contrast adjustment and x 2 magnification, using the image processing software of the IP processor (Oikawa et ul., 1990). An original print that was magnified x 1.8 (finally x3.6) with the IP-printer was used directly for printing. Figure 27 shows a three-dimensional presentation of the electron intensity distributions in areas A and B in Fig. 26, which were measured from the IP. In area A (where the specimen is thin), the measured intensity is least at heavy atomic columns (indicated by arrows H in Fig. 27a), showing a good agreement with the projected potential of the atoms in the structure model (the inset in Fig. 26). In area B (where the specimen is a little thicker), on the other hand, the intensity is greatest in the low potential region (indicated by arrows L in Fig. 27). It was thus clear from this quantitative measurement that the region was subjected to a strong dynamical diffraction effect. An HREM image of the high-Tc superconductor T12BazCul0, was quantitatively observed by using the IP by Shindo et al. (1994). In order to evaluate quantitatively the difference between the intensity of the observed image and that
275
THE IMAGING PLATE A N D ITS AI’I’LICATIONS
Flc;LIKI(
2 6 . Exmiple of 11IZEM iin;ipc (Shilldo cf o/.. I%) I ). Spccimcil L\’;IY
w-Fri~-() iiiiil
iiccc1-
crating volliipc 400 hV.
of calculated images. a residual index KIII
276
NOHIIFIIMI MORI A N D TETSIJO OIKAWA
HREM image of T1.Ba,Cul O,., some requirements for quantitative HREM were pointed out, and were briefly discussed in comparison with those for the standard X-ray and neutron diffraction methods. An HREM study was carried out with a JEM-4000EX electron microscope. HREM images were recorded on the 1P and were converted into digital data (2048 x 1536 pixels, 4096 gray levels) at the JEOL Laboratory. After investigating the image intensity i n the image processing system (PIXsysTEM) (Oikawa ct al., 1090). the digital data were transferred to Tohoku University on magnetic tapes and were there analyzed with an engineering workstation (Sun: Argoss 5230) and a mainframe (NEC: ACOS-2020). An HREM image ofTI2Ba.Cul0, is shown i n Fig. 2X. The incident electron beam w a s parallel to the [ O l O ] direction. The image was taken with a ?-second exposure and a direct magnification of x 1,500,000. It was noted that the image was observed with a defocus value which was rather smaller than the so-called Scherzer locus value (i.e., -48 nm). Although the image was recorded with 2048 x 1536 pixels and 4096 gray levels, only a part of the 1024 x 1024 pixels was output with 256 gray levels in Fig. 2X. In the image, small dark dots show heavy atom positions projected along the incident electron beam. In Fig. 29. the number of pixels used for recording this HREM image is shown as a function of the gray level. Although the number of gray levels needed for recording HREM images seems to be much smaller than that for electron diffraction patterns, it is seen that about 1000 gray levels were used for recording the HREM image. A model of the atomic arrangment of TI?Ba?CulO, is presented i n Fig. 30a. which was proposed earlier by an X-ray diffraction study (Parkin Ct "I., 1988). I n
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277
Fig. 30b, the intensity distribution of a part of the image near the crystal edge is shown as a contour map. The squares in the model of Fig. 30a and in the intensity distribution of Fig. 30b indicate unit cells of TIzBalCulO,, which has a tetragonal structure with the lattice constants u = 0.3866 nm, L’ = 2.324 nm. In order to renove the noise such as quantum noise, the contour map was produced by smoothing the data with 2 x 2 sampling points and averaging the intensity after displacing the image by +u and - ( I . Even after the averaging process, there is a small asymmetry around metal atom positions in the contour map. The asymmetry is considered to come from the crystal thickness change. The observed intensity of the HREM image was divided by the intensity of the incident electron beam, which was measured at the vacuum region near the crystal edge. Thus, the normalized observed intensity can be directly compared with the calculated intensity without any scaling
NOBUFUMI MORl A N D TETSUO OIKAWA
0
1024
2048
3072
4096
Gray Level FIGURE29. Number of pixels as a function of the gray level uscd for recording the HREM iinage of Fig. 28.
factor. Although the contour map reveals the detailed intensity distribution of the HREM image, it is not easy to distinguish the intensity maxima from the minima, since both intensity maxima and minima appear as similar dense contour lines. In order to make a detailed investigation of both high intensity and low intensity, which may correspond to low and high potential regions, respectively, the contour map of Fig. 30b was separated into two contour maps as shown in Figs. 30c and d. In Fig. 30c, the grid indicates the positions where the observed intensities were measured with the IP. The number of sampling points on the grid in the unit cell was 743. The observed intensities at these sampling points were compared with the calculated ones. In the contour map of Fig. 30d, which shows low intensity, the heavier atomic columns of TI and Ba can be easily distinguished from those of Cu. It should be noted that there is no marked difference between the density of the contour lines at the TI site and those at the Ba site, although the potential of
THE IMAGING PLATE AND ITS APPLICATIONS
279
T1
Ba
Ba *
(a)
FIGURE30. (a) Structure: model of TlzBazCulOy. (b) Contour map showing the intensity distribution of the HREM image of T12Ba~CuiOyin Fig. 28. (c) High-intensity region of the contour map (h). The grid corresponds to the sampling points ;it which thc observed and calculated intensities were coinpared to evaluate a rcsidual indcx RIIRI..M.(d) Low-intensity regions of the contour map (h).
TI atoms is much larger than that of Ba atoms. This will be taken into account for the refinement of the computer simulation below. An image calculation based on a structure model suggested by an X-ray diffraction study was camed out, which is shown in Fig. 30a. In order to evaluate the difference between the observed intensity and the calculated one, a residual index RC~KE was M calculated.
RHREM is the index for the observed and calculated image intensity, and is basically different from the so-called R-factor or the residual index generally used in diffraction studies, where the factor or the index is evaluated for the absolute value of the structure factor. C in Eq. ( I ) indicates the summation for the sampling points in the unit cell, which number 743 in this study and correspond to the grids of Fig. 30c. In order to get smaller values of RHREM, parameters, which depended
3x0
NOHCIFLIMI MOlil ANL) 'IE 1'$110 OIKAWA
on the experimental conditions, i.e., crystal thickness, defocus and chromatic aberration. were changed. With the structure model of Fig. 30a, R l l ~ < l ;=~ ,0.0506 was obtained with the experimental parameters shown in Table 2. where the parame~ are indicated ters which were changed to get a smaller R ~ I KinEthe~ calculation M are shown in Fig. 3 1 , with an asterisk. Images simulated with R I ~ K ~=; 0.0506 where three types of contour maps (i.e., whole intensity, higher intensity, and lower intensity) are presented i n Figs. 31a, b. and c in a similar manner to the observed images shown in Figs. 30b, c, and d, respectively.
TlHE IMAGING I’LATE A N D I I’S AI’I’LI(’A7‘1ONS
28 I
x 10-2
0
0
I n order to see the variation of R H K I : with \ ~ the change of the parameters in the calculation, Rlll
282
NOBUFUMI MORI AND TETSUO OIKAWA x 10-2
0
20
40
00
UCfOCllS Value ( 11111 )
(a)
XI1
I0
?I)
10
1)Cf'UCllS
v:lluo (
1111l
40 )
(h)
33. Variation of R I I R I . M a s ;I function ol'tlclocus. The other parameters cxcept thc tlelocus value arc the \aiiie ;IS IIiose in T~ihlc1.( a ) Thc i-anyc 0 1 dcl'ocus values is 5-75 nm. (13) The ra~lgcis 14-35 nm. Fi(;IIi
intensity deviates widely from the observed intensity. Region A corresponds to the positions around the TI atomic columns, As pointed out in the observed image of Fig. 30d, the contrast of T1 atoms is similar to that of Ba atoms despite its much larger atomic number. It is thus reasonable to say that the discrepancy may be attributed to the fact that the concentration of TI atoms is lower than the nominal concentration. This was noticed in their previous HREM experiment of T 1 ~ B a ~ C u l O by, Shindo et al. (1991). They therefore took into account the partial occupancy of TI atoms, and made new image calculations. It was found that R H K E Mbecame smaller if the partial occupancy of TI atoms was taken into account. As a result, R H K E M = 0.0473 was obtained with an 87% occupancy of TI atoms, as shown in Table 3. The parameters with an asterisk indicate those changed to get a small value of R H Kin~the ~ calculation. Figure 35 indicates the variation of R l j as ~ a ~function ~ of occupancy of TI atoms. In Fig. 34b, the low intensity of the calculated image with RI,I<~:M = 0.0473 was plotted as a contour map. It is noted that the density of the contour lines at the TI position is slightly lower that in Fig. 31c, which was calculated with full occupancy of TI atoms.
THE IMAGING PLATE AND ITS APPLICATIONS
(a)
(b)
283
(c)
FIGURE 34. (a) Difference hetween observed and cnlculatetl intensities of HREM iinngcs with R~~RE =M 0.0506. (h) Lower-intcnsily distribution of llie calculated images taking into accwnt 87%) occupancy of TI atoms. (c) Difference between obseived and calculated intcnsities o l HREM images with R ~ I R ~= :M 0.0473. Note tliiif there are s l i l l sonic peak\ at positions indicated by H.
In Fig. 34c, the difference between the observed and the calculated images was shown as a contour map. Some of the contour lines around the TI atom positions observed in Fig. 30a disappear. However, there is still a fairly large difference at the positions indicated by B. These positions correspond to the interstices among oxygen atoms and Ba atoms in Fig. 30a. As pointed out above, there is some oxygen deficiency in the quenched samples. Thus, the difference between the observed and calculated intensities in the above refinement may be attributed to some oxygen deficiency. In the analysis of an HREM image of T12Ba2Cu10,, a residual index R ~ ~ K E M of 0.0473 was obtained by changing the experimental parameters and introducing the partial occupancy of TI atoms. By the refinement of the computer simulation, deficient oxygen positions were also detected. It was pointed out that a smaller
284
NOBUFUMI MOM AND TETSUO OIKAWA TABLE 3 PARAMETERS USED FOR THE CALCULATION OF FINAL REFINEMENTCORRESPONDING TO THE CONTOUR MAPO F FIGURE34B.
Wavelength Spherical aberration constant Thickness of one slice Number of beams *Defocus of objective lens *Defocus due to chromatic aberration *Crystal thickness *Occupancy of TI atoms
x
0.00164 nm l.Onm 0.3866nm 3 2 x 128 24.5 nm 24 nm 5 slice (= 1.93nm) 87%
10-2
5.3
5.2
5.1
5 .o
4.9
4.8
4.7
70
80
90
100
Occupancy of TI Atoms ( % ) FIGURE35. Variations of RHREMas a function of occupancy of T1 atoms. The other parameters except the occupancy of T1 atoms are the same as those in Table 3.
THE IMAGING l’l.ATE A N D ITS AI’I’LI(‘AT1ONS
285
residual index K I I I < C M and a higher resolution limit are indispensible to get more accurate atomic arrangements from HREM images observed with the 1P.
Since the Imaging Plate (1P) generates digital image data, it i s convenient for digital image processing. In this section, two types of application data of the image processing are introduced. One is the simple contrast enhancement of an image. Figure 36 shows an example of the image contrast enhancement of a biological specimen (a thin section of dragonfly). The image contrast was enhanced by the look-up-table (LUT) as shown in Fig. 37. Here, the image contrast y is defined as i n Eq. (2). y = - W,,
w, W , :dynamic range of input data W,,: dynamic range of output data The other is spatial frequency filtering. Figure 38 shows an example of the Fourier transfoimation of an HREM image (SijN4 single crystal taken with the JEM-2010F 200 kV field-emission HREM). Figure 3Xa shows an original image, Fig. 38b shows the Fourier transfoiined two-dimensional power spectrum pattern (difffractogram), 38c a reconstructed image by selecting periodic spots in the spectrum, as indicated by the circles in 38b.
286
NORUFUMI MORI A N D TETSUO OIKAWA
16384
0
Input data FItiuKE 37. Look-Up-table (LUT) used for contraat enhancement in Fig. 36. A gray-level hi+ togram ofthe original image data is also shown in lhe figure.
The IP has a wide dynamic range and high-intensity resolution (16,384 gray levels); contrast enhancement and image analysis applications can hence be carried out with high precision.
E. Another Field of Application of the Imaging Plate
Recently, the Imaging Plate (IP) had been begun to be used in the RHEED field (Miura et ul., 1995). In this field as well as in electron diffraction, the superior
THE IMAGING PLATE AND ITS AI-’I’L21CATIONS
287
FIGLIRE 38. Image proccsaing of spatial frequency filkring. (a) HREM iningc oi Si ~ N \ingle J crystal taken with the JEM-70IOF FE-T’EM and the 1m;iging I’latc. (b) Fourier 1i-anst’ormed twodimensional powei- spectruin pattern of ( a ) . (c) Rcconstructcd (spatial lrcquency liltered) image by selecting periodic spectrol spots indicated hy tlic circles in (b).
Timc-rcsporisc
Rcwlution
Width of fidd-of-view
Slmsitivity
Dynamic-range
Linear-r e s p nse
Photo-Film
TV Camera
Slow Scan CCD
Imaging Plate
FIG(I R E 39. Comp:irison 01‘ some characteristics lor the image tlekclion devices widely used today.
characteristics of the IP are valuable. Originally, the IP was developed as a highly sensitive image recording device for X-ray images. The IP is widely used today in the field of clinical medical science (Sonoda et a/., 1983) and medicine and bioscience (Nakajima, 1993). In the field of X-ray crystallography, the IP has been began to be used (Fuji and Kozaki, 1993). Since the IP has good sensitivity
288
NOBUFUMI MORI AND TETSUO OIKAWA
for ultraviolet rays and ions (Nishikawa et al., 1995), applications in these fields have also been started.
VIII. CONCLUSION The transmissionelectronmicroscope (TEM) is an instrument for observing magnified images of microscopic objects and outputs experimental results in the form of images. Moreover, the TEM outputs not only the morphology of the specimen but also the result of interaction between the incident electron and the specimen. In this sense, the TEM image is not a mere “picture,”but a “message from the microscopic world.” Of course, imaging with the TEM is modulatedby instrumental factors such as lens aberrations. Image detection devices also have specific characteristics. Figure 39 shows comparisons of some characteristicsfor the image detection devices widely used today. These devices have both advantages and disadvantages, and have very different characteristics. Among these devices, it is hoped that the Imaging Plate, which has high sensitivity and high quantitative precision for beam intensity and which is also suited for image processing, will be widely used and assist in new research using the TEM.
ACKNOWLEDGMENTS Among the application data introduced in t h ~ sarticle, Fig. 1 was obtained in a joint research project by Dr. Hiroshi Ayato of the Ashigara Research Laboratory of Fuji Photo Film Co., Ltd. and the authors. Many of the application data in this article were obtained in ajoint research project by Professor Daisuke Shindo of Institute for Advanced Materials Processing, Tohoku University and one of the authors (T. 0 .).We hereby express our gratitude to them for allowing us to use the data included in t h ~ article. s
REFERENCES Amemiya, Y., and Miyahara, J. (1988). Nature 336, 89-90. Amemiya, Y., Satow, Y., Matsushita, T., Chikawa, J., Wakabayashi, K., and Miyahara, J. (1988). In “Topics in Current Chemistly,” Vol. 147, pp. 121-144. Springer-Verlag, Berlin Heidelberg. Amitani, K., Kano, A., Tsuchino, H., and Shimada, F. (1986). SPSE’sConference and Exhibirion on Electronic- Imaging, 26rh. A Fall Symposium, Advance Printing of Paper Summaries 180. Ayato, H., Mori, N., Miyahara, J., and Oikawa, T. (1990). .I. Electron Microsc. 39,444448,
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A1)VANCES IN IMA(;ING ANI) LI.I.("l R O N I'HYSICS . VOI . 49
Space-Variant Image Restoration ALBERT0 DE SANTIS
. . . . . . . . . . . . . . . . . . . . A . The State of the Art . . . . . . . . . . . . . . . . B . The Image Signal: Basic Assumptions . . . . . . . . Kalnian Filtering . . . . . . . . . . . . . . . . . . A . Slate-Space Representation . . . . . . . . . . . . . . . . . . . . . . . . . . B . The Estimation Algorithm C . The Steady-State Solution . . . . . . . . . . . . . The Image Model . . . . . . . . . . . . . . . . . . A . The Homogeneous Iniage Equation . . . . . . . . . . . . . B. The Component Equations of the Sampled Imagc C . Modeling the State Noise . . . . . . . . . . . . . . D . The Constitutive Equation . . . . . . . . . . . . . Image Restoration . . . . . . . . . . . . . . . . . A . Space-Variant Realization of the linage . . . . . . . . I3 . The Edge Problem . . . . . . . . . . . . . . . . C . The Filtering Algorithm . . . . . . . . . . . . . . L) . Deblurring . . . . . . . . . . . . . . . . . . . Numerical Results . . . . . . . . . . . . . . . . . Concltisions . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . Appendix C . . . . . . . . . . . . . . . . . . . . Rcl'erences . . . . . . . . . . . . . . . . . . . .
1. Introduction
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292 292 294 29.5 295 297 299 300 300 302 304 305 308 308 312 314 315 317 320 321 322 324 326
In this work a substantial part was relxoduced and adapted. with pcrniission. lroni "Space-Variant Recur\ivc Restoration of Noisy Iniages. by A . Dc Sanliu C I i l l . , IEEE Tt-irii.srrc.rionso t i Circui/.\ mid Sy.crcni~-ll:Atrrrlog rrrztl Digirtrl Sigtitil Prnc~.s.sirig.Vol . 41. No . 4. 249-261. April 1994 @ 1994.
..
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292
AI.RERTO 116 SANTIS. ALFRED0 GERMANI. AND I.EOPOI,DO IEl"K1
I. INTRODUCTION A. The State qf the Art The considerable attention that the image restoration problem has been rccciving in the literature has motivated the interest in extending optimal one-dimensional (ID) filtering procedures to two-dimensional (2D) data ficlds. In particular most authors investigated the applicability of Kalman filtering techniques to the restoration of images corrupted by additive noise; as a consequence many efforts have bccn made toward the synthesis of image models suitable to recursive restoration techniques. In the literature (Nahi, 1972; Nahi and Assefi, 1972; Nahi and Franco, 1973; Jain and Angel, 1974; Powell and Silverman, 1974; Woods and Radewan, 1977; Murphy and Silverman, 1978; Katayama, 1980; Suresh and Shenoi, 1981; AzimiSajadi and Khorasani, 1990) a 2D image is transformed into ID scalar or vector stochastic process using a line-by-line scan or a vector-scanning scheme. Othcr approaches that use a 2D model can be found (Habibi, 1972; Attasi, 1976; Jain, 1977; Jain and Jain, 1978; Katayama and Kosaka, 1978; Kalayama and Kosaka, 1979; Suresh and Shenoi, 1979). All these papers arc based on the common assumption that the image is the realization of a wide-sense stationary random field. By this simplifying hypothesis, an image model suitable to a state-space rcprcscntation can bc derived; nevertheless, the corresponding space-invariant filters arc insensitive to abrupt changes in thc image signal and give restored images with reduced contrast and blurred edges. Actually, a real image is composed of an ensemble of several different regions and, in general, no correlation among them may be assumed save that of casually being clemcnts of the samc picture. Thus, the stationarity assumption may fit for the statistics of each single region, but not for thc whole image; consequently, blurring and oversmoothing phenomena occur at edge locations. Adaptive space-variant filters based on identification-estimation algorithms havc been proposed (Keshavan and Srinath, 1977; Keshavan and Srinath, 1978; Katayama, 1979; Kaufman et al., 1983; Yum and Park, 1983; Wcllstead and Caldas Pinto, 1985a and 1985b; Azimi-Sajadi and Bannour, 1991). These niethods allow the parameters describing the image model to vary inside the image itself according to the local statistics. The specific problem of reducing the numerical complexity involved in the adaptive parameter estimation procedures for a 2D image model is considered (Zou et al., 1994). Other space-variant filters have been proposed (Rajala and De Figueiredo, 198 I ; Woods et al., 1987; Tckalp et ul., 1989; Jeng and Woods, 1988; Jcng and Woods, 1991; Wu and Kundu, 1992). In Rajala and De Figueiredo (1981) the imagc is partitioned into disjoint subregions with similar spatial activity; the so called riirrskirzg,fcirzc.tiorz, first introduced in Anderson and Netravali ( 1976). is then used
SPACf-VARIANT IMAGh I
293
to properly weigh the response of human eye to additive noise. In Woods et d. image is inodeled as a globally homogeneous random field with a local structure created by a 2D hidden Markov chain. This hidden chain controls the switching o f the coefficients of a conditionally Gaussian autoregressive model to take into account the prescncc of image edges. The method proposed in Woods et ( I / . (1987) is extended i n Tekalp et id. (1989) to dcconvolution-type problems. In Jeng and Woods (1988) each image pixel is processed by a switched linear filfer with the switching govcrncd by a visibility function; in Jeng and Woods ( 1991) images arc described as a compound o f several submodels having different chxacteristics. an underlying structure model governs the transition between these submodels. I n Wu and Kundu (1992) the image is modeled as a nonstationary mean and stationary variance autoregressive Gaussian process, some modifications of thc reduced update Kalman filler described in Woods and Radewan (1977) are also proposed to obtain a filtering algorithm with a reduced numerical complexity. 'The method proposed in Bieniond and Jerbrands (1979) deals with the edge problem using the output o f a linear edge detector as an additional input to iniprovc thc step response of a space-invariant Kalman filter. All the above mentioned papers arc based on a description of the image in terms of its statistical properties. The self-tuning methods attempt to draw this information starting from noisy data. Their main drawbacks are the computational cost, that in many cases may be unacceptable, and/or the increased structural complexity of the algorithm. Moreover, it seems difficult to obtain a fast switching of the image model parameters in correspondence to a sudden change in the image statistics, such as at edge points. The other methods assume that the information on the image model is available a priori or that it can be obtained from the noisy frcc image or by a sample of similar pictures. In inany practical situations it is unrcalistic to assume that these data are available. Based on the results of Germani and Jetto (1988) and Bedini and Jetto (1991), thc method proposed in De Santis et n/. ( 1994) starts from a completely different point of view, based on very physical assumptions on the structure of the stochastic image model (see Section IB). As shown in De Santis et a/ . (1994). such assumptions allow one to construct a space-variant iinagc model where the problem of image parameter identification is greatly simplified and where the presence of image edges is intrinsically taken into account, so that edge oversmoothing is automatically avoided. The resulting filtering algorithm is suitable for irnplcmentation as a strip processor (Woods and Radcwan, 1977). Other 2D recursive filtering algorithms paralleling the ID Kalman filter and not requiring strip processing have been proposed (Habibi, 1972; Panda and Kak, 1976; Katayaina and Kosaka, 1979; Biemond and Gerbrands, 1980). These filters are based on the quartcr-plane system, first introduced in Habibi (1972), and their nonoptimnlity was proved in Striiitzis (1976) and Murphy (1980). As shown in Barry ( 1976), thcrc is no optimal finite-dimensional causal filter for ( 1987) the
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ALBERT0 DE SANTIS. ALFRED0 GERMAN], AND LEOPOLDO JETTO
the quarter-plane system, while a finite dimensional approximation to the optimal half-plane filter has been presented (Attasi, 1976). B. The Image Signal: Basic Assumptions
The new approach first proposed in De Santis a/. (1994) allows one to define an image model incorporating a priori structural information about edge locations. In general, such information can be more reliably obtained than a complete statistical description of the image process. The model is derived from the following assumptions 1. Smoothness assumption: The image is modeled by the union of open disjoint subregions whose interior is regular enough to be well described by a 2D surface of class C’. 2. Stochastic assumption: All the derivatives of order ti 1 of the 2D signal are modeled by means of zero-mean independent Gaussian random fields. 3. Inhomogeneity assumption: The random fields representing the image process relative to different subregions are independent.
+
Hypotheses 1 and 2 are based on the consideration that most images are composed of open disjoint subregions whose interior is regular enough to be well described as a finite support restriction of a smooth two-dimensional Gaussian process. The boundary of each subregion is constituted by the image edges, which represent sharp discontinuities in the distribution of the gray level. Assumption 3 means that no correlation can be assumed among pixels belonging to different subregions. Hypotheses 1 and 2 were exploited in Germani and Jetto (1988) to derive a space-invariant image model which does not take into account the edges’ presence. To reduce the consequent blurring phenomenon, a heuristic restoration procedure was defined by forcing the filter with the output of an edge detector. In this case the detector performance is crucial in estimating both edges’ location and amplitude. The inhomogeneity assumption introduced in this paper allows the edges to be directly taken into account by the image model. A stochastic image-generating process is so obtained describing the gray level discontinuities by a space-varying model, where only the information on edge location is needed. Consequently, the optimal restoration procedure is guaranteed by the corresponding nonstationary Kalman filter. Blurring is intrinsically avoided because at any pixel the estimate is obtained by using the information carried by the neighboring pixels belonging to a convex set contained in the same subregion. The information on edge location can be obtained by an edge-detector operator whose main feature should be robustness with respect to noise (see, e.g., Agile, 1971; From and Deutsch, 1975; From and Deutsch, 1978; Rosenfield and Kak, 1982; Lunscher and Beddoes, 1986; Nalva and Rinford. 1986).
SPACE-VARIANT IMAGE RESTORATION
295
In the sequel we will use the following definition.
Dejinition 1. Let P I , Pr be two points in a given subregion R, . We say P I , Pz are adjacent if and only if they belong to a convex set contained in R, . Remark 1. Note that P I , P2 are not adjacent if and only if either they belong to different subregions or their convex combination [ P E IR2 : P = U P 1
+(I
-
a ) P 2 , a E [O, l]},
contains at least a boundary point of R, .
11. KALMANFILTERING
In this section a brief account on the Kalman approach to the filtering of noisy signals is given. While the theory is well established even for signals defined in the continuum (the independent variable can be indifferently time or space), we shall restrict the attention to sampled signals since all the data processing is nowadays performed by means of computers; moreover, the mathematics involved is greatly simplified.
A. State-Space Representation In any practical situation data are collected by periodically sampling the quantities of interest obtaining, at any step, a vector Sk E IR“of observations. Yet, the physical determination of Sk is invariably corrupted by a random quantity representing thc measurement noise Nf E IRy;its influence on the observed data Yk E IR‘I may vary at any step, so that we can write Yk=ShfGI;N;‘,
k = l , 2 ,...,
(2.1)
with Gk E R q X ‘ / . The sequence [ sh } of the “true signal” samples may be a random sequence as well, owing to the randomness of the generating process. A full statistical description of [Yk}would require the knowledge of the finite distributions p ( Y , ,, . . . , Y I Lfor ) any set of indexes i f , . . . , i!, . Such an information can be efficiently obtained from the knowledge of a limited number of parameters once a model for the mechanism that generates the sequence IS,} is adopted. Then, in addition to the measurement Eq. ( 2 .l), the following signal “state-space model” may be considered
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ALBERT0 DE SANTIS. A I I R E D O GERMANI. A N D I~EOPOI.IX)JETTO
with Ak E IR””” , Fk E R‘”/’, Ck E IR4””.Vcctor Xk E If7” is the “system’s memory,” since it accounts for thc proccss past history up to thc kth step; in this respect the statc “initial condition” X o is correctly modeled as a random variable. Vcctor N i E IR/’is a random excitation so that, according to (2.2), the new systcm outcome X i + I is computed as linear combination of the available information X,: and the occurring input N ; . Then the signal samplc S, + 1 is obtained as partial obscrvation of X k + I through (2.3). Equations (2.2) and (2.3) dcfinc a linear nonslationary modcl. This schcmc is sufficiently gcncral to dcscribc a grcat dcal of practical situations, or can be considered as approximation of more sophisticated nonlinear processcs. Then wc see that the statistics of sequences { Xk }, { S, 1, and { Yk } arc completely dctcrniincd by those of X o , { N ; } and { N , “ } .Now, our intuition about the behavior of the system’s evolution suggests that thc probabilistic structure induced by Eqs. (2.1), ( 2 . 2 ) ,and (2.3) should be such that the information at any step k dcpended only on the past information and not on the futurc onc. In other words, we want to modcl the causality usually featured by the physical processcs wc dcal with. Indeed, this would be accomplished by the causal form of Relations (2.2) and (2.3) in the deterministic case. Causality is retained at thc statistical lcvcl providing some assumptions are made on thc random variables involved. Thus, I N ; ) and { N t ] are assumed to be independent zero-mcan white scqucnccs, i.c., sequences of independent identically distributcd (i.i.d.) random variables EIN;N;l”] = O , V h , k , where I,,
Et/’‘/’,I,,
E[N,”NA’] = ~ ‘ L , / ~ I , ,E, [ N ; N ; : ’ ] = h’k,/,I,>.
E IRL’xy
and
I
k=/1
.
is the Kronecker delta. The i’ { random variable X O is assumcd indcpcndcnt of { N,”} and {PI;’}as well, i.c., E
h’k,/l
=
I)
Then, according to (2.l ) , (2.2), (2.3), and the just stated hypothesis, the sequences ( Xk }, { Sk ] and { Yk } arc “nonanticipative;” i.c., XL , Sk, and Yk arc independent of Xk,~,,, Sk+/, and Yk+,, for any k and 1’ 1 1. Moreover { X k } featurcs thc so-callcd Markov property P(Xk I Xk-1, . .
. 1
x/<-/J) = P(Xk I Xk-l),
(2.4)
with ~ ( I I Iu ) dcnoting the probability density function of II conditioned t o the knowledge of u . Relation (2.4) states that, for any k , Xk-1 conveys all the past information. Property (2.4) need not to hold for the sequences {Sk) and ( Y k ) . For signal processing purposes we shall be concerned only with second-ordcr theory, so that only mean and covariance do matter about sequences { Xk }, { Sk }, and {Yk]. Moreover, if the Gaussian assumption is made it turns out that the sccond-order statistics suffices for a signal full description. Then, suppose that X O is a Gaussian random variable with mean X o and covariance matrix PO,and that { N ; ) and IN,“} are Gaussian as well. From (2. I ) , (2.2), and (2.3) we readily ohlain
SPACE-VARIANT IMAGF RESTORATION
297
that
/=(I
Yktl = S L + I + G k ~ ~ N ~ ' - ~ l ~
(2.7)
with @,,,,, = A , . A , - l . . . . . A , , , . We see that X I , + ] ,S I , , ~ ,Y,+,, k = 0, 1,. . . , arc linear transformations of the random variables Xo,N ( ; , . . . , N i , which, because of independence, are jointly Gaussian. Consequently X I , , Sk, and Yk are Gaussian sequences, The mean evolution is obtained as follows ElXk+I I = @L,(IX(I = XI,+, = AI,X/,,
Els/,+~l= CLtiXk.+l =SI,,.I, ElYI,+II = & + I
=YL+I,
(2.8) (2.9) (2.10)
which hold since the noise sequences have zero mean. For the covariance matrices
it rcadily follows that 1
(2. I I ) = AkR;A[
+ FkF,',
q+1= G i l R ; + l c ; , R ; , I = R;+l
I>
(2.12)
+ GktlG:,,.
(2.13)
In particular (2.8) and (2.11) provide recursive rclations to compute the state mean and covariance matrix. B. The Estiriiutiori Algorithni
Our goal consists in estimating the valuc of Sk, k = 1, 2, . . . , given the measurement sample Y I, Y?, . . . , Y,\. This is the filtering problem; when a measurement sample Y I , Y2, . . . , Y,,, is used we talk about prediction if m < k and smoothing if 117 > k . Thc estimate is then correctly intended as a measurable
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A I B E R T O DE SANTIS, ALFRED0 GERMANI. AND LEOPOLDO JETTO
s,,!,
function of the available data, = $ ( Y l , . . . , Yk). Such a function is usually selected according to an optimization critcrion. For instance, the optimal estimate gklkshould be unbiased, i.e., [.?klk] = E[&] and may be chosen in order to minimize the trace E[(& - , ? k ~ k ) ’ ( S k - , ? k i k ) ] of the estimation error covariance matrix. A standard result in estimation theory states that such an optimal estimate coincide with the conditional mean expectation E[ Sk I Yk, . . . , Y I] of the signal Sk given the measurement sample Yk , . . . , Y I. Now, from (2.3) we see that
The celebrated Kalman filter is a linear system that recursively provides the optimal state estimate .?,I, on real time with data acquisition. With the assumption that for any k matrix Gk G; is full rank, the following scheme is obtained
According to Eq. (2.15), the optimal estimate is composed of two terms. The first one represents the “one-step prediction,” that is, the best estimate at kth step (in the minimum variance sense) that we can obtain without processing further data than Y I, . . . Yk-1. The second term is a “correction” based on the innovation carried by the kth measurement with respect to the information contained in the past measurements already accounted in the prediction term. The II x q matrix Kk is called “Kalman gain” and, according to (2.16), (2.17), and (2.18), can be computed on the base of the signal model coefficients Ah, FA, Ck, G!, and the initial condition covariance matrix Po. The choice 2,) = 81,yields an unbiased estimate for any k. The matrix Rh E R””” is the state estimation error covariance and Eq. (2.17) is known as (dynamical) “Riccati equation.” Matrix f f - 1 E R“’” is the state one-step prediction error covariance; moreover, it can be shown that matrix ( 1 4- Hh- 1 c,/(Gk c;)-lck)is always invertible. The remarkable feature of the Kalman filter is that it allows for a nonstationary data processing as opposite to the frequency domain methods for filtering which are ultimately a steady-state approach.
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SPACE-VARIANT IMAGE RESTORATION
C. The Steady-State Solution
In thecase that the signal model is stationary, it is interesting to study the asymptotic behavior of the Kalman filter, once Eqs. (2.15)-(2.18) are updated with constant system matrices A, F , C , and G . Here the notion of system stability, as well as the structural properties of system controllability and observability, play an essential role. We briefly recall them. A state X E IR" is "A-stable" if IIA'XII tends to zero as k increases. We say that X is "(C, A ) unobservable" if C A' X = 0 for any k 2 0, whereas it is "(A, F ) uncontrollable'' if F T A T k X= 0 for any k 2 0. The states featuring either one of the stated properties form a linear subspace of the state-space R". Concerning the system behavior, the notions of stability, observability, and controllability can be considered together to give the properties usually referred to as detectubilit?, and stahilirubility. Then we say that a system is detectable if all the ( A , C ) unobservable states arc A-stable, and that it is stabilizablc if all the ( A , F ) uncontrollable states are A7'-stable. Besides the inherent interest, the steady-state investigation is needed since no Kalman filter can be optimal unless the initial condition covariance Po is known exactly. To overcome this difficulty consider Eqs. (2.17) and (2.18) in the stationary case RI, = (I
+ H'-IC~(GG'')-'C)-'H~-I,
Hk-l = A R k - j A 7 ' + F F ' . (2.19)
Then we want to check if the Riccati equation admits a steady-state solution K , = Iinik+% Ra. The advantage would be twofold: first, if the limit exists it is unique regardless of the way the sequence Ra is started up according to (2. I9), thus obtaining a filtering algorithm robust with respect to initial data knowledge inaccuracies; then, from a practical point of view, we would not need to update matrix Rk for k sufficiently large, saving computational resources (time and memory). It can be shown that if thc signal model is stabilizablc and detectable, then the Riccati equation (2.19) admits a unique steady-state nonnegative definite and selfadjoint solution R,. This matrix solves the so-called steady-state Riccati equation (SSRE): R,
=(I
+ H,C"(GG')-'C)-~H>,
H,
=AR,A~
+ FF".
(2.20)
Then the following suboptimal filter can be designed
Nevertheless it can be shown that the prcvious algorithm is asymptotically optimal; that is, the stateestimation crrorcovariance matrix l?k=El(Xk - z k l ~ ) ( X k- 2 ~ , ' ) ~ . ] approaches R, a s k increases.
300
A L B E R T 0 DE SANTIS. ALFRLDO GhKMANI. A N D LEOPOLDO JETIO
111. THE IMAGE MODEIL
In this section we describe an image by means of the gray-level signal together with its partial derivatives with respect to the spatial coordinates, up to a certain order. The vector so obtained is assumed to be the state vector of the image model. Moreover, by assumptions 1 and 2, a stochastic relation between the states evaluated at two different points in the same subregion is obtained. Let us indicate by x ( r , s ) the value of the original monochromatic image at spatial coordinate ( r , s ) inside a smooth subregion. The continuous variables tand s denote the vertical and horizontal positions respectively. For simplicity, but without loss of generality, we assume ( r . s) E [O, I]'. Because of the smoothness assumption, it is possible to define a state vector composed of the signal x ( r , s) and its partial derivatives with respect to r and s X(r,s ) =
8'l.a ( r . s ) ar,l-
i)su
'
I1
=o,
1, . . . , n;
cy
=o,
I , . . . , I7
I'
.
(3.1)
If ti is the maximum order of derivation used, the dimension of X ( r , s ) is N = [ (17 I)(n 2 ) ] / 2 . The kth component of X ( r . .s), denoted by Xk ( r , .s), is given by
+
+
(3.2) with
where [ . ] stands for the integer part. Let I'
+ yu. + p14,
= r ( u ) = r()
s = s ( 1 1 ) = S,)
denote a parametric representation in ~r of a straight line passing through the point (ri~, sI)) belonging to a hoinogencous subregion KI c 10, I]'. As a direct conscquencc of the state vector definition ( 3 .I ) , the following equation can be written for any I / E [O, I I , , , , , , ] such that the point (r(i4).s ( i 4 ) ) is adjacent to ( Q ~ s, o ) according to Definition I
SPACt-VARIANTI M A G E RESTORATION
30 1
the dot denoting the derivative with respect to u . Moreovcr, by direct computation we have i)
-X(/.(/O, il I'
J(CI))
+
= A X ( I ' ( I ~~ )( ,1 0 ) BW, ( I ' ( I ~ ). ,\ ( l o ) ,
(3.4)
where A and A' arc ( N x N ) cominuting inatrices (Germani and Jetto, 1988) (see Appendix A), whosc elements I l l and t i ; I l l arc such that Nl
Ill
=
I = r +;
1, 0,
ifrri
otherwise,
1
+ [ (--I1
5N
(3.6) (3.7)
+ 1 and
where the vectors W, ( Y ( M ) , J ( I I ) ) and W , ( I ' ( I ~~ )( , u )have ) dimension ii are given by
(3.8)
(3.9)
+
where the ( N x (I? 1) ) matrix B has the form B = I 0' I 1' . The dimension of the null block and the identity matrix are ( ( N - ( r i ( N I ) } and (ri 1) x (ii I ) , respectively. Using (3.4) and (3.5),Eq. (3.3) can be rewritten in the following form:
+
+
+ I )) x
+
+ B A ' ) X ( r ( u ) ~, ( u ) ) f B [ y W , ( r ( u ) . s ( u ) )+ B w , ( r ( u ) . s ( u ) ) ] .
X ( r ( l r ) .\(U)) , =(yA
(3.10)
Formal integration of (3.10) with respect t o 14 between u0 and u 1 allows us to derive a relation between the state vectors evaluated at two generic points ( r o v"o, .so Brro) and ( r ~ ym I , s[l pu 1 ). Exploiting the commutativity of A and A', we obtain:
+
+
+
+
X(ro+yrii..s(~+pu,) - c, ( 1'
\ i - /:\J' I ( (1 1
~
11,) 1
X(I'c1
+
I/l'o. S I I
+ Bull)
302
ALBERT0 DE SANTIS, ALFRED0 GERMANI. AND LEOPOLDO JETTO
By the stochastic assumption, W,.(.,.), lVS(.,.) are white Gaussian vector fields, so that the integral term in Eq. (3.11) is intended as a stochastic Wiener integral.
B. The Component Equations of the Sampled Image In this section we use the results of Section IIIA in order to get a semicausal statistical model for the sampled image, which takes into account each possible edge configuration at any pixel. We denote by xi,j
= x ( i A , . ,j A , s ) , i, j = 1 , . . . , m ,
the true value of the sampled image at the pixel with vertical coordinate i A,. and horizontal coordinate j A,,, where A,. and A,, denote respectively the vertical and the horizontal spatial sampling steps. If the image is sampled with an equal number m of pixels on each row and on each column, the normalized value of A,. and A, are both equal to l/(m - 1). We consider the situation where the image is observed under additive white Gaussian noise v,,; N ( O ,
-
Y i J = xi.;
+ Ui.1.
(3.12)
We assume that the image state vector at pixel coordinates ( i , j ) depends on the state vectors at neighboring pixels according to the scheme shown in Fig. I, where causality is assumed only for the first coordinate (semicausal model). The coefficients c:!] ( l = 1, . . . , 5 ) may be zero or one; they are one if the corresponding pixels are adjacent, and zero otherwise. This implies that 25 different configurations of the image model can be obtained depending on the edge shape at pixel (i, j ) .
\ c;;
0
X ii-1
‘3)
0 i,i
X i,j+i
FIGURE I . Spatial structure of seinicausal dependence model.
SPACE-VARIANT IMAGE RESTORATION
303
The relations between the state vector X I , / at the pixel ( i , j ) and the state evaluated at neighboring pixels for which c!:; = 1 can be obtained by applying (3.1I), with a suitable choice of y and ,B. The following component equations are derived
where
1 I
w,!.:.) = eAAtll-r)A,.BW ((i,
- I)A,
+ t A l ,j A , ) d s ,
. [A, W,((i - l ) A , + A , T , ( j + l ) A , - A , t ) - A , W,((i - l)A, + A , r , ( j + l)A, - A , r ) ] d t ,
(3.20)
(3.21)
Note that only the component equations for which c,'.'J'= 1 give significant information. If some c;! = 0, only a trivial relation is obtained because (3.10) cannot be integrated along that direction. A pixel is said to be internal, boundary, or isolated if, for it, the number pi,j of component equations which rcally hold is pi,j = 5, 1 5 p , . j < 5 , p I . / = 0, respectively.
304
AL.BLRTO DE SANTIS, ALFRED0 GERMANI. A N D LEOPOL.DO JETTC)
C. Modelivig the State Noise From the stochastic assumption and using (3.18)-(3 22), it is possible to show that Wl"/' ( l = I . . . . , 5 ) are zero-mean white Gaussian random fields with the following properties (Germani and Jetto, 1988): = 6,
E[W,':'W:',;,']
( 3 23)
lllQ\,
E [ W,';'W:::,']61
111 Q l
( 3 24)
,
E[W,';'W,'j;,'] = 0,
( 3 25)
with
Q,-
=
1 I
e/\Atll-Ti
B q,B re.l ' A, I I --Ti
ds,
where q ,and q,arc diagonal matrices such that E[W,(r..s)W;'(F,F)] = q 5 6 ( l 1 ( r , . s )- ( F , S ) I I ) , E [ W , ( ~ , S ) W , ~ . ( F , S= )] ~ 1 6 ( ~ ~ ( r( F, ,LS )~( I ) .
Estimates of q,\and qr can be obtained as functions of the image spectrum (Germani and Jetto, 1988) (see Appendix B). Moreover, from (3.13)-(3.17) the following identities can be proven to hold: W'5' = -e-, 1.
/
''A\
Wl(;)t, ,
Wl" I . / = e;\'A. Wl(,;\,
W""/ = e - A ' A , I.
+ wl',;',
( W'3' l . / + l - Y:;;,).
(3.26) (3.27) (3.28)
The previous identities, obtained by assuming c)(li = 1, (e = 1, . . . , 5 ) . imply the following relations among the covariance matrices of the white Gaussian random fields W;"i), ( l = 1, . . . , 5 ) E I W l ! ~ i W / ~=~ ~l ~l , ' ];
~
~
,
l
l
l
Q,e-""'\., ~ ~ ~ ~ " " ~
E I W l ( ~ i W / ~ ~= l : '6] , , / 6 / , l l , ( e ' " AQ,-e'"'"~ ~ + Q,), E[W"iU;'J''] I./ ,111 = 8I . /, 6/ . I l l e ~ - i \ ' A )( Q \ + Qr)0-,-"'
(3.29) (3.30) (3.31)
The statistics of the infinitc two-dimensional Gaussian process corresponding to a generic smooth subregion are completely defined by (3.23)-(3.25) and (3.29)(3.3 I ). As a consequcncc these equations definc the statistics for any pixel of each subregion. Of course, some of the r.h.s.'s of (3.26)-(3.28) might loose thcir
SPACk-VARIANTIMAGE RES'IORATION
305
physical meaning in the presence of an edge, but the statistical meaning is retained because it is related to the infinite two-dimensional Gaussian random field. Therefore, Relations (3.29)-(3.31) are considered true even if sonic ri,': = 0.
Assuming pr.,> 0, we now exploit the component equations in order to derive a unique relation among XI,I and the state evaluated at its pI,,/neighboring pixels. Tlic case p , / = 0 (isolated pixels) will bc considered separately. For convcnicnce thc following notation is introduced HI := e,''ls, HJ :=
H2 :=
p,\''\.-l
c,-,\'ll,+,\L\,
, \ 1,
H3 := H5 := e - , \ ' - j s
e,'A$
Defining:
1
W l . , :=
whcrc I,, is the N x N identity matrix. Eqs. (3.13)-(3.17) can be rewritten as Ci./Xi.j=
Ci./Zi,,
+ Ci,/Wi,/.
cl,
(3.32)
Note that / X I , belongs to IR'", being composed of five N x 1 dimensional blocks. We stress that only pl,/ ( 1 5 p , / 5 5 ) of them are nonzero blocks c o t ~ c sponding to the 11,./ component equations for which c,':; = 1 . Now we define an operator nI,/: IR'" + IR'" selecting the tionzero block entries of C, I XI,/ . It is given by a p , , / N x 5 N matrix whose generic block entry Ill / ( k ,/ I ) , k = 1 , . . . pl,,, / I = I . , . . 5. is l N
Note that definitions of C, and
n, I
imply that
306
ALBERT0 DE
AND
Hence, applying & , j to both sides of (3.32) one has Y
c:jxi,j
= zij
+
-
(3.33)
wij 3
1
I
where CCj : = l l i , j C i , j , Z i ? , :=lli,jZi,j and Wt., J -=ll. 1 3 J.W. 1,J. Equabon (3.33) soobtamed is ageneral compact formto expressthe pi,, component equations whch actually hold at pixel (i, j ) . We want to combine such equations in order to obtain a unique relation with a minimum variance stochastic term. It is well established (see, 1979,p. 403) that such an equation is of the type
=~
(3.34)
[ ~ i , j / z i +, wj i~, j ,
/...
h
where E [ X i : j / & j ] denotesthe conditionalexpectation givenZi,j, andW,,j is the menboned stochasticterm In the present linear Gaussian case, it is known that (SoderstrBm and Stoica, 1989): EIXi,j / z i , j l = (C:,Tqc] CFj)-lCgj~ P GTlZ, I , -2, j and
9
w. = (CPJ.mc)c&)-1 c:j 5;; Fi,j , h
1.J '
where 6 i , j is the Gi,j
jN x
E[Gi,jE'Tj]
Using (3.23)-(3.25) and
'3
jN covariance l-Ii,jW,,,A-I~j,
of
givenby
With 9, :=E[wi.jwTj]-
it is found that
SPACE-VARIANT IMAGE RESTORATION
307
n,
and that, by (3.35) and definition of I , the following general expression for A , , := nT,(n, ,q,L,IIT,)-lll,., is found
A,.,=
where
(3.39)
as it can be directly verified for any possible configuration assumed by the general semicausal dependence scheme of Fig. 1. By definition of ci,;one has that the N x N matrix
is given by the sum of all the block entries of the matrix A;,;,while the N x 5N matrix
(c:;n:,
( ~ , , , q w n ~ni.;)? j)
is given by the sum of the block rows of A,,,. Hence (3.37) results in Xj.1
+ A 3 ( 1 . j+) A4ii.l) 2A4.5(,.1) + . [Alii,jl - A l , l ( i . , ) ; - ~ l , r c , . j ) + Azit,/); A.t(J.,) &.s(,.,): - A 4 . 5 i i . j ) + A ~ c i . ( Z i . j + W,.
= ( A I ~ J ,, ) 2Al.?(i,;J-I- A2ii.j)
-
I AS(i.jJ)-
Ajil.;):
-
j)]
j).
(3.40)
Equation (3.40) is the unique relation with minimum variance stochastic term we were looking for. It does not hold for isolated pixels. In fact, in this case Eq. (3.10) cannot be integrated along any one of the five directions of Fig. 1. Hence, p i , , = 0 and the pseudoinverse cannot be defined. It seems natural to consider such pixels
308
ALBERT0 DE SANTIS. ALFRED0 GERMANI, AND LEOPOLDO JETTO
(3.41) where p , . j is P1.J =
0, if pI.J = 0.
Equation (3.41) is referred to as the Constitutive Equation (CE) of the sampled image. For internal or boundary pixels it provides a relation between the state evaluated at spatial point (i, j ) and the state evaluated at neighboring points. For isolated pixels Eq. (3.41) resets the state. The form of the CE is identical for each image pixel, but its actual expression depends on the spatial position ( i , J ) through the coefficients cl;;. From Eq. (3.39) we note that c1.1 ! ' )= c"! 1.1 = 1
A I , ~ (=~ A, ~Z () ~ , / ) ,
c l .!J ~=) c!'! 1 . 1 = 1 ===+ A 4 , 5 ( , . j ) = A d ( ; , j ) .
+
In this case the matrices - A , , ~ ( , , J )A 2 ( r , jand ) A 4 ( ; , j )- A ~ , s ( , . jthat ) , in Eq. (3.41) weigh the contribution of the diagonal directions (corresponding to l = 2 and l = 4, respectively), are null. This means that for every "internal" pixel all the information necessary to estimate X l ~ isj contained in the horizontal and vertical directions. Diagonal directions are useful to estimate boundary pixels for which c"! and/or C"! = 0. '.J
1.J
IV. IMAGERESTORATION A. Space-Variant Realization of the Image
In this section we exploit the CE (3.41) to derive a state-space representation of the sampled image suitable for the Kalman filter implementation as a strip processor (Woods and Radewan, 1977). For this purpose consider the following ensemble of pixels composed of L columns and two contiguous rows in Fig. 2
309
SPACE-VARIANT IMAGE. RESTORATION
4
J 2
0
0
0
....
i-1
0
0
lXiXi - iXIXi ..*.
0 -0
FICXiRt
i
0-0-0
2. Enseinble of pixels consitlercd
where
j ? := j l
+ 1,
J,
= jl
+ 2, ...,j ,
-
I := j l + L - 2 ,
J ’L
= jl
+L
-
1.
We derive the desired image state-space representation by expressing in a suitable compact form the complete set of CE (3.41) that can be written for each pixel at , j , . By defining the following matrices coordinates ( i , j ) with j = j l ,
Fi.1 := pi,j(A1(t,jl- 2A1.2it./l+ Arc;.;]+ A . ~ t . j+ l AAii.;]
+
- 2 ~ 3 . 5 ( i . j i As(i.jl+ (1 - P ~ . / ) I N ) ~ ’ > F/l.” ._ 1.1
(2.11
.- Fi,/ ( A l ( i . j )- At,r(i,jl); F,.;
+
,
.= Fi.,j( - A I , ~ ( ~ . ,A/ 2I( i . / , ) *
Fl!;,’:= Fi.,A3(i~j)? (1.5)._ F/”.“’ ._ .- Fi,j( - &.s(l.j] + A ~ i . , j ] ) , Fl,, .- Ft,j(A3(;.,1- AJ,s(i,j)); the CE (3.41) can be rewritten in a shorter notation as
X IJ .
- [F-“..21. 1. J ’ FI (. . /2 : 1 )F!’.); ;1 . / F-“”; 1. / F,!:’](Zl,/
+ W l , J+) ( I
-
pl,j)Xjl,.
(4.1)
For each of the ( L - 2 ) pixels at coordinates (i, j ) , j = j 2 , j j , . . . j L - 1, the CE has the general form of Eq. (4.1). The extremal pixels ( i , j l ) and (i, j L ) are considered as boundary pixels, so that the relative boundary conditions must be taken into account. In particular for the pixel at coordinates (i, j l ) , the relative CE is derived by putting c:,yl = c)!: = 0. By definition of F;!:’” and F,!?” and taking into account (3.39). Eq. (4.1) gives
+
+
X ; , ; , = [O; 0; F“.’; I.JI F.’3’; I.J F i ~ ~ ~ ] ( Z IW, jl l, J , ) ( I
-
p 1 , / , ) X ~ ~ , , . (4.2)
Analogously, for the pixel at coordinates ( i , j l , ) , the relative CE is obtained by assuming c:!jl = c:l = 0. In this case (4.1) assumes the following form
Xi,,( = [F(’;’’; I.,//. F.“”’; !./I. F l ! ~ ~ ; O ; O ] ( Z+ lW , , ,~, ,j , ) + ( l - p l , J 1 ) X ~ ~ , l .(4.3)
3 10
ALBERT0 DE SANTIS, ALFRED0 GERMANI, AND LEOPOLDO JETTO
To express in a suitable compact form the complete set of CE composed of the L - 2 Eqs (4. l), (4.2), and (4.3),it is convenient to define the following vectors
31 I
SPACE-VARIANT IMAGE RESTORATION
From (4.9, when ( I - @:ji'L))is nonsingular, the following linear dynamical state equation of the image state-space representation is finally obtained xjjl.L)
= (I
)- 1 ri(j1.L)X,-i L )
- @;jl.L)
+ ( 1 - @jjl,'-)
(/I
)-I * ( J I . L ) + 0;-l
( l - Q;jl.L))-lx;(j13L)
= 1, 2 , . . . .
(4.6) We assume to know the mean value Xy"',) and the covariance matrix Po of the initial condition
X:l'L) := E[Xbjl%'2)],
PO := E[(XfI.[,) - X:/l'r.)) (Xj/l,L)- X ; ' . L ) ) T ] To define
@? L , and riJ.L) a reasonable a priori choice of the coefficients ci!:,
j =
j , , . . . j [ < ,is the following (3)
=I,
(2)
= C1.j = 0,
c1.j CI , ,
(4)
c(I) -
I.jl
j=A,...j[,,
-
(5) CI,,
j=h,...jL,
=O,
while the coefficients CI.jt (I)
j = j z , ..., j L ,
c (,5 .) ~ , j = j l , . . . , j L
-
1,
depend on the actual edge configuration of the first row. is singular (4.6) cannot be derived. This particular case When ( I corresponds to the existence at ith row of at least one "isolated ensemble" of pixels as shown on Fig. 3. Here the pixels of coordinates
(i, jl,), . . . , (i, j r ) , j,, < j r , are isolated from the other pixels of the same strip belonging both to the same row and to the previous one. In this situation the entries of the state vector X:'l'L) corresponding to the isolated pixels need to be reinitialized according to the same
FIGURE3. Edge configuration giving rise to an ensemble of isolated pixels
3 12
ALBERT0 DE SANTIS, ALFRED0 GERMANI. AND LEOPOLDO JETTO
procedure used to define the initial conditions. By virtue of this reinitialization the matrix cP"i'"'' is modified in such a way that ( I - cP~.""~') is nonsingular. An observation equation can be associated to (4.6) by writing Eq. (3.12) for each pixel at coordinates (i, j ) , j = j l , . . . , j l ~ To . this aim define the following vectors
and matrices
M :=
-M' 0 . . . 0 M' . . .
-0
0 0
.
.
.
.
0
".
M
,
-
M'=[1
0
. . . 01.
N elemr.nls
Taking into account that M ' X i , , = xi,,,, the complete set of scalar observation Eqs. (3.12) can be written in the following compact form y;~i.L)
= M X ; ~ ~ 3+[ v,(jl.L) ,)
(4.7)
Since the state noise @:J"',' is a white-noise sequence (see Appendix C), the state-space representation given by Eqs. (4.6) and (4.7) have a form amenable to the Kalman filtering implementation as a strip processor. The resulting algorithm behaves like the classical Kalman filter if no isolated point is met in the image scanning. At isolated points, where no correlation with neighboring pixels exists, the dynamical model simply assigns the observed value, i.e.
B. The Edge Problem The proposed image model requires the preliminary identification of coefficients c:'; associated to each pixel; therefore, edge locations need to be estimated. This problem has been widely investigated in the literature and many methods have been proposed depending on the various definitions of edge and image models, either deterministic or stochastic. The deterministic approach describes edge points as locations of suitable order discontinuities (Marr and Hildreth, 1980; Nalva and Binford, 1986; Torre and Poggio, 1986; Huertas and Mcdioni, 1986; Asada and Brady, 1986; Mallat and
SPACE-VARIANT IMAGE RESTORATION
313
Hwang, 1992) so that the presence of a contour is associated to local extrema of the signal derivatives. For steplike edges a classical method for detecting such extrema consists in determining zero crossing points of the convolution of the data with a Laplacian of Gaussian masks (Huertas and Medioni, 1986). A more recent approach uses multiscale wavelet analysis for identifying higher-order discontinuity points (Mallat and Hwang, 1992). The stochastic approach is based on a probabilistic description of signals obtained either by defining the signal a priori distribution (Geman and Geman, 1984) or defining the signal generation model (Basseville and Benveniste, 1983). In the first case stochastic relaxation procedures are used to generate a sequence of images converging to the maximum a posteriori estimate of the true image. In the second case, sudden variations in the signal model parameters are detected by statistical hypothesis tests on the output innovation process. We chose an edge-detection algorithm based on the gradient method (Rosenfield and Kak, 1982). The motivation for such a choice is twofold. The gradient method is relatively simple to implement and is based on the definition of edge points as locations where abrupt changes occur in the image gray-level distribution. Such definition well agrees with the image model here proposed. According to (Rosenfield and Kak, 1982), we estimated the gradient magnitude M f , / and direction di,,as
where f,.,,,and f S l , represent , the rate of change of the gray-level distribution along the vertical and horizontal directions respectively. We thresholded Mi,, with the mean value of the gradient computed over all the image pixels. As suggested in Rosenfield and Kak (1982), f , - i , J and f y i j were estimated by means of noisesmoothing difference operators. Besides the‘differentiation procedure, these linear operators perform some smoothing action in order to reduce noise effects. From a practical point of view, the implementation of these operators requires a 2D discrete convolution of the noisy image with a matrix operator. As in Germani and Jetto (1988), to computef,.;,, we used the (6 x 3) edge-detector operator, most sensitive to horizontal edges, given by -1 -1 -2 -1.5 -2 --2.5 2 2.5 2 1.5 I I
-1
-
-.1.5
-2 2 1.5 I-
The matrix K , = KT was used to estimate f \ l , / . The entries have decreasing numerical values to weaken the influence on f ,, J , f S f . , of pixels lying on neighboring
314
ALBERT0 DE SANTIS, ALFRED0 GERMANI, AND LEOPOLDO JETTO
edges. The size (6 x 3) is a reasonable compromise in recognizing small objects while retaining a sufficient noise smoothing. Arguments about shape and dimensions of convolution operators are extensively discussed in the literature (Argile, 1971; From and Deutsch, 1975; From and Deutsch, 1978; Rosenfield and Kak, 1982). The response of the edge detectors was improved by a thinning procedure based on the nonmaximum suppression technique (Rosenfield and Kak, 1982). This facilitated the subsequent operation of determining the coefficients c,(!j e = I , . . . , S for each image pixel, by exploiting the on-off information about edge locations.
C. The Filtering Algorithm Equations (4.6) and (4.7) provide us the sought image signal-state space representation to use in the Kalman filter design, according to the theory described in Section 11. The particular filter implementation is named strip processor since Eq. (4.6) describes the gray-level spatial evolution for pixels lining up in a row of width L so that, as the row index increases from 1 tom, a image strip is scanned and processed. To simplify the notation, let us rename some of the matrices appearing in Eq. (4.6)
(4.1 1)
H k -i jl 1 . L )
- A (kj l . l - ) R i ki i-,lL )
(j1.L)'
+
F;j,.L)F;.il.L~7
(4.13) cjl.L)T
Equations (4.11) and (4.12) are obtained considering that EIV,'J1'L)Vk ] = -0ijl.L) . uiZ. The term FLJ1'L'Xk in Eq. (4.10) accounts for the singular cases discussed in Section IVA; it resets the filter when isolated ensembles of pixels are met.
SPACE-VARIANT IMAGE RESTORATION
315
We finally note that a space-variant filter is obtained, so that no steady-state argument can be applied. As a consequence, Eqs. (4.1 I), (4.12), and (4.13) must be recursively solved on line.
D. Deblurring In the image model devised through Sections I1 and 111, only the degradation due to additive measurement noise was considered. The signal recording process usually introduces other deterministic kinds of perturbations, whose overall effect on the detected image is known as blur(Hwang, 197 I). It mainly depends on the low-pass filter behavior of the measurement equipment, as well as on typical aberration of the optical components of the imaging system. Furthermore, the relative motion between source and sensor results in a defocused signal. The deblurring problem has been widely considered in the literature (see, cg., Cannon, 1976; Tekalp er al., 1986; Tckalp and Kaufman, 1988; Lagendijk et al., 1990). The purpose of this section is to give an outline of how the image modeling previously described can be extended to deal with this problem. Referring directly to the discretized model, blurring is commonly described by a convolution of the original signal x , , ~(i, , j ) E [l,fi]’, with a 2D linear spaceinvariant system generally characterized by a “Point Spread Function” (PSF) hk.1 with a rectangular support. Then the noisy blurred image is given by
(4.14)
The blurred image can be considered as a unique smooth domain, so that any information about edge locations is lost. Models (4.6) and (4.7) can be well adapted to deal with this case by removing the inhomogeneity assumption. This implies c i,t,), ~= 1 and el,/ = I for every L and ( i , j ) . As a consequence system matrites @(;I./-) and rj’l’L) in (4.6) become constant arrays @ and r, depending only on the Atrip width L ; moreover X{/”’~)= (0) and the state noise vector sequence ((i3:!;L))becomes a stationary one with covariance matrix Q. To obtain matrices @, r, and Q substitute the stationary version of Eqs. (3.13)-(3.17) = I, L = I , . . . 5 ) in formulas defining @:””,), rlcil’’*)in Section IV, and Ql!;”’ in Appendix C. Defining the following vectors and matrices
ii+
1 clcmrnls
A+ I elements
316
ALBERT0 DE SANTIS, ALFRED0 GERMANI, AND LEOPOLDO JETTO
......
...
hl.-L2 ...
h. 0 4
......
0
......
...
. . . . . . . . .
0
-
0
h
'..
hk.LZ
(I
I
- @)-I
0
0
(I
A=
-
@)-lr I 0
0 0 I
......
0
...
h --k.-L.?
kk.0
b,,&
0
0
'
..
2L1
'
"
h -k.O
-
-
+ 1 blocks,
...... 0 ...... 0 0
...
0
. . .. 0
...
0
1
0
it can be shown that the following state-space representation of the blurred image is obtained
Equations (4.15) and (4.16) define a suitable model for Kalman filter design as a strip processor for blurred images. Moreover, a space-invariant scheme is obtained so that a steady-state implementation can be adopted.
SPACE-VARIANT IMAGE RESTORATION
317
V. NUMERICAL RESLJLTS Two 256 x 256 pixels eight-bit images were used to test the proposed restoration method. The first one is a simulated image consisting of concentric rhombi with constant gray level in each homogeneous subregion (see Fig. 4). The second one is the real image shown in Fig. 5 (particularly of Susan, courtesy of IBM scientific center of Rome). The synthesized image has been chosen because it contains sharp edges, while the real image has been chosen to evaluate the filter performance on an actual image. For each of the original test images two different noisy versions were generated with an SNR (signal variancehoke variance) equal to 4 and 16. See Figs. 6a-7a and 6b-7b. This experimental situation was considered to test the method capability in restoring noisy images of heavily different characteristics.
F I G ~ J R5 F. Original Susati image.
3 18
ALBERT0 DE SANTIS, ALFRED0 GERMANI, AND LEOPOLDO JETTO
FI(iIJRE
(a) (b) 6. Noisy versions of rhombi corresponding to (a) SNR = 4 and (b) SNR = 16.
FIGURE 7.
(a) (b) Noisy versions of Susan corresponding to (a) SNR = 4 and (b) SNR = 16.
Once the parameters crc(:(t?= 1, . . . , 5 ) have been obtained, according to the procedure described in Section IVB, the Kalman filter was implemented as a strip processor (Woods and Radewan, 1977) according to the image representation of Eqs. (4.6) and (4.7). The images were partitioned into strips 15 pixels wide; these strips were overlapped and only the 11 middle pixels were retained as final estimates to avoid strip-edge effects. For each strip, the Kalman filter estimate equation was initialized by assuming
SPACE-VARIANT IMAGE RESTORATION
3 19
The Riccati equation was implemented starting from an initial value of the ( L N ) x ( LN ) error covariance matrix Po given by Po‘”
0
.........
0
......
0
0
Pi;2) 0
0
. . . ... 0 . . . . . . . . . . . . P,;”’
Pi) =
0
’.
0
-a,:
o... . . . . . .
0... 0
0
0
0
0
0
... . . . 0
0
......
p p= 0 0 -
The image of concentric rhombi was processed with a model order corresponding to the choice ii = 0 because it can be considered a piecewise constant image, while the valueii = 1 seemed to be more appropriate for processing a shaded image such as Susan. To measure the improvement in SNR introduced by the filter, the following performance parameter q , expressed in decibel (db), was defined
where y;,, is the noisy signal observed at pixel (i, J ) , xi,,, is the corresponding true signal value and Zi,; is the Kalman estimate of x I , ,. Filtered images are reported in Figs. 8a-8b and Figs. 9a-9b. The values of q for SNR = 4 (SNR = 16) were 5.78 and 3.55 (4.35 and 2.01) for rhombi and for Susan respectively. Figures 8 and 9 reveal an effective reduction of the observation noise; edges are clearly demarcated and the original image contrast is well preserved. As final comments to the numerical simulation, it is worth underlining the following. Our filtering algorithm requires only the on-off information on edge location; no edge amplitude estimate is needed. This simplifies the edge-detection procedure and fixes the two following limiting situations in the filter performance:
I . No edge is detected; the filter behaves according to the corresponding space invariant structurc. 2. All image pixels are marked as edge points (all p,,,’s are zero); the noisy image is reproduced.
320
ALBERT0 IIE SANTIS, ALFRED0 GERMANI, AND LEOPOIAXI JETTO
(a)
(b)
FIwku: 8. Filtered rhonibi imagcs coiwspondinp to (a) SNK = 4 and (b) SNR = 16.
(a)
(b)
Flc;u#t: 9. Filtered Sum1 imagcs concymiiding 10 (a) SNR = 4 aiid ( h ) SNR = I6
This means that even in the theoretically worst possible cases, unrealistic images are not produced. Moreover, we mention that numerical experiments performed by varying entries and size of K,. and K , produced filtered images very similar to those reported here. Hence, the overall filtering algorithm can be considered robust enough with respect to the edge-detection procedure. The presented numerical results show improved filter performances with respect to the other existing methods similarly based on the information drawn from real data (noisy picture). Furthermore, they are even comparable with the best ones obtained by using noise-free image statistics, information which is not always available in practice.
The necessity of processing real images calls for algorithms where a rapid switching of the filter characteristics is allowed. The method proposed in this work seems
SPACE-VARIANT IMAGE RESTORATION
32 1
to be a simple and efficient way to meet this requirement. It should be emphasized that the main feature of the proposed approach is the analytical construction of the image model. Starting from the smoothness, stochastic, and inhomogeneity assumptions, a nonstationary state-space representation is obtained without the necessity of onerous identification procedures, or the a priori knowledge of the image autocorrelation function. The obtained model is space varying according to the presence of image edges. In this way the edge defocusing phenomenon is greatly reduced. The adaptive behavior of the proposed restoration method is obtained by including the information on edge locations into the image model. In this way, the filter transitions in correspondence of edge locations are not the result of heuristic procedures, but are justified on a theoretical basis since they are strictly related to the image model. It is stressed that the choice, here adopted, of detecting edges by the gradient method is just one among many existing possibilities. Other reliable edge detectors for noisy environments proposed in the literature can be used (see, e.g., references of Section IVB). These characteristics make the method amenable to be applied to a large class of images; moreover, the experimental results presented in the previous section confirmed the merit of the approach by showing that high filter performances are really attainable.
APPENDIX A
In order to show that A and A' commute, we need to prove that N
N
According to Eqs. (3.6) and (3.7), we obtain N
(A.2) Ill=
I
N
(A.3) 111 = I
where
322
A12BERT0 DE SANTIS. ALFRED0 GERMANI. AND LEOPOLDO JETTO
Then, from (3.7) and (A.2), it follows that all the entries of each row of the product matrix AA‘ are zero except for the kth entry which is equal to 1, with k given by
In the same way, from Eqs. (3.6) and (A.3), each row of A’A has null entries but the hth one, which is equal to 1, with h given by
According to (A.4), is easy to verify that (AS) and (A.6) are equivalent, thus proving (A.1).
B APPENDIX Here we propose a method to obtain feasible estimates of Wf and W5 under the hypothesis of a finite limited and isotropic spectrum in each subregion. As both these matrices can be estimated in a fully analogous manner, the calculations are developed only for one of them, say W,. If, for convenience, we indicate the vector composed of the last ii 1 derivatives of order ii with Xi’) ( r , s), the last ti+ 1 equations of system (3.5) can be rewritten as
+
a
- X ( ! i ) ( r , s) = W,(r, s).
(B.1)
as
By integrating this equation with respect to r between two adjacent pixels of coordinates (i 1, j ) and (i I , j I), we only obtain the last ii 1 equations (I) of system (3.13) (c,+ I J + l = 1)
+
+
+
+
+
The vector W,(l)c,/+ I is composed of the last ii 1 elements of W,!!)], / + and, under the hypothesis (3.23), it is a discrete white noise -N (0, Q,:.”)).Taking into account that from (3.23), (B.I), and (B.2), we obtain Q:’) = an estimate of W scan be obtained through a feasible estimate of (3:”). To this purpose, let us rewrite system (3.13) as
(B.3) n = O , l , . . . ,i i ,
c t = O , I , . . . ,n ,
where Wk(,ylI L I is the kth component of W:k\,J+l.
k = 1 , 2 ,...,N ,
323
SPACE-VARIANT IMAGE RESTORATION
Equation (B.3) is formally identical to the Taylor series expansion for the signal and its derivatives. Our idea is to get an estimate of QF)through a feasible estimate of the Taylor remainder. Let us indicate by G(w,-,w,) the spectrum of the image as a function of the spatial frequencies w,. and w,$and let us assume the following hypotheses: there exist O,vand W,such that
G ( O , ~w,.) , = 0 if Ic(w,s, wr)I 5 K ,
w,
1
and/or w,- > W,,
(B.4)
V(w5 w,.) E [0,0,l x [0,0,1.
(B.5)
1
By consequence of (B.4), the signal is of class C": (Papoulis, 1977). Using the two-dimensional Fourier transform, we can now rewrite Eq. (B.3) in the following way arlx(r,s) ri -11
I
where
kwl
j&l+a+l-rl-n+I
< -
J
n2(1+a
where 0 = max[W, ,4
+
w,. l)(n -a
+I1
+2
<
+ 1) - n ? ( l +n + 2 ) '
1 and the condition a 5 n has been used.
(B.9)
324
ALBERT0 DE SANTIS, ALFRED0 GERMANI, AND LEOPOLDO JETTO
(B.lO)
+
Inequality (B.lO) represents an upper bound for all the n 1 components of W,(l’,, , which corresponds to the same order n of derivation for x ( r ,s). Hence, for n = r? we have that all the elements of l V ~ ~ ~satisfy , + l the following inequality &)li+3A\ ii(E 1) (1) ,‘uA, =: qE, for k = 1, . . . , N , ( B . l l ) IWk,+l, * I 5 2x2 2 which states an upper bound for the remainder relative to the Taylor series expansion of order 0 for the derivatives of order f i . It seems reasonable to estimate Qi”)as
I
~
~
+ +
(B.12) 3 ’ where q,?/3 is the variance of a random variable uniformly distributed between [-qfi, 4,i]. Therefore, Q:<)
=
(B. 13) In a fully analogous manner the following estimate for \vris found
(B.14) where
APPENDIXC In this section we show that the noise sequence @:’I”,) satisfies the whiteness condition o;.il .jd)@:J, ./. I - 8, ~ ( j 1 . L ) (C.1) - 1.k I
[
{I
3
where 8,.k is the Kronecker delta, and we compute the covariance matrix Ql””-’. Whiteness of OIJ”L’ is a straightforward consequence of (3.23)-(3.25) and (3.29)-( 3.31).
SPACE-VARIANT IMAGE RESTORATION
325
,
For convenience denote by 19;., j = j1 , . . . , j,, thc cntries of the vector 0;’;i‘,) defined in (4.4). Then the covariance matrix Q;!;;”’can be computed as follows. Taking intoaccount (3.23)-(3.25) and (3.29)-(3.31) we have, form, k = ,I2,. . . , .iL -
1
(
+ F l ~ ~( H; ”IQ r HT + Q \ ) F,!:’”’ + Fll~l~ Q r F,::”
E[8,,,,18i7k] = Fi:;,;’)Q\Fl:L”17
+ F,l~;”Hs(Q,-+ Q.\)H5 Fj,k’ + F;!:;4)Hs Qs H< F 1 ~ ~ J“l l”l , k) 7’
( 3 5)‘
326
ALBERT0 DE SANTIS, ALFRED0 GERMANI, AND LEOPOLDO JETTO
the matrix
i L, has the following five band structure
(J1.L)
Q,-I
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AIIVANCES IN IMAGING AND EI.ECTKON PHYSICS. VOI. 99
Image Representation with Gabor Wavelets and Its Applications RAFAEL NAVARRO lii.stituro dr 0pric.n "l>rrztr tlr Kilrl6.s (CSIC). Sermiin 12 I , 28006 Mtidrid, Spcriii "
ANTON10 TABERNERO f i i c i i l / c r r l rlr
/iiforniti/iccr, Uiiiver.sirlnd PolirL:c~iiicridr Mtidrid. Rotrdilki del Moilre, 28660 Mtrdrid, Sprriii
and
GABRIEL CRISTOBAL I m t i t i i t o tlo Opricri "Driztr rle KildPs" (CSIC),Serrcirio 121, 28006 Mmlricl. .'$xiit1
After publication of the article by Navarro, Tahernero. and Cristcihal in Volume 97 (1996). i t was found that Figure 30(b) was omitted. Thc complete Figure 30 is as follows:
b 0"
f, f, f, f,
I
0.076
45"
90"
135"
0.028 0.021 0.031
0.137 0.036 0.023 0.034 0.240 0.037 0.026 0.034 0.263 0.042 0.036 0.041
~ l G ~ J l
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Index
rate cqudtions dnd photogdin, 82-84 SAGCM InP/lnGaA\ APDs, 77.92-94, 99- I56 SAM and SAGM InP/InGnA\ APDs, 90-92 Fillcon APDs, 89 rtaircase avalanche photodiode\, 94-95
Affinity. 41 AGC. See Autoniatic gain control Aharonov-Bohm effect electron holography, 172, 187-192 electrostatic, 187- I92 Amplitude division interferometry, 193-194 Analog optical fiber coiiirnunication, 68 APDs. See Avalanche photodiodes Automatic gain control (AGC), 73 Auloradiography, 264 Avalanche buildup time, 86 Avalanche multiplication, 8 1-82. 84 Avalanche photodiodes (APDs) avalanche multiplication, 8 1-82, 84 bandwidth, 86 critical device parameters extraction, 102-120,153,155 erroranalysis. 110-1 13, 118-1 19, 155 dark current. 74, 88. 150-153, 154. 156 germanium APDs, 89-90 indium phospide-based, 142-146, 156 low-noise and fast-speed heterojunction APDs, 94-96 for inultigigabit optical fiber communications, 65- I56 multiplication excess noise, 84-86 multiquanturn well/superlattice (MQW/SL) photodiodes, 94-96 photocurrent. 87 photogain, 74. 120-1 35, 155 temperature dependence, 135- I SO planar, 96-102. 121 quantum efficiency, 87
B Bandgap semiconductor, tunneling. 88 Bandwidth avalanche photodiodes, 86,92-94 defined, 76 Bimetallic wire. biprism, 188-190 Binary images. representation, 38 Biprisin electron, 173. 176-1 84 electrostatic, 188-190, 193 Breakdown voltage. temperature dependence. 135-146
C Causality, 6 CRED. S w Convergent-beam electron diffraction Charged dielecti-ic spheres, electron holograms, 174,207-2 I6 Charged microtips, electron holography. 229-235 Closing function, 9-10 inultiscale closing-opening. 22-29
33 I
332
INDEX
Computed radiography, imaging plate with, 242, 263-265 Constitutive equation (CE). space-variant image restoration. 305-308 Continuity property, 4, 6 Contour maps. numerical simulation, 2 14-2 16, 227 Contrast enhancement, imaging plate. 285 Convcrgent-beam electron diffraction (CBED), imaging plate and, 269-270 Convex structuring functions, 13-14 Crystal structure and, luminescence, 243
Dark currents, 74 avalanche photodiodes. 88. 150-1 53, IS4. I56 DD. S P Direct ~ detection Dehlurring, space-variant image restoration, 3 15-31 8 Detective quantum efficiency (DQE). 260-26 I Dielectric spheres, charged, electron holograms. 174,207-2 Ih Digital optical receiver, 72 Dilation function, 8 multiscale dilation-erosion scale-space, 16-22 Dimensional lunctionals, 41 Dimensionality, 8, 4 W 2 Direct detection (DD), 68 Dispersion shifted fiber (DSF), 7 I Double-exposure electron holography, 205-207, 207-2 I0 DQE. See Detective quantum efficiency Dual gray-scale reconstruction. 49
Aharonov-Bohm effect, 172, 187- I 9 2 charged dielectric spheres, 174, 207-216 charged microtips, 229-235 double-exposure electron holography, 205-207,207-2 I0 electron biprism, 173 image wavefunction and, 179-1 84 wave-optical analysis. 176-1 79 electron-specimen interaction, 174- I76 phase-object approximation. 174, 175-1 76, 184-186 reverse-biased p-n junctions, 172-173, 174. 185.2 16-220 Electrostatic Aharonov-Bohni effect, 187- I92 Electrostatic biprism, 188-1 90, I93 Erbium-doped fiber amplifiers (EDFAs), 7 I Erosion function, 8 niultiscale dilation-erosion scale-space. 16-22
F Fading. imaging plate system, 258-259 Filtering, scale-space filtering, 2 Filtering algorithm, space-variant image restoration, 314 Fingerprints, morphological scale-space, 4-5, 29-30,SS equivalence. 30-32 reduced, 32-37 Flat structuring function, 44 Flicker noise, avalanche photodiodes ( APDs), 74, 151-153, 154 Fresnel holography, 173 Functional (noun), 41
E
G
EDFAs. See Erbium-doped fiber amplifiers Edge detection, space-variant image restoration, 3 12-34 Electrical bandwidth, defined, 76 Electron biprism. 173 image wavefunction and, 179-1 84 wave-optical analysis, 176-1 79 Electron diffraction, imaging plate, 270-274 Electron holography, 171-173,235-236
Gain, defined, 76 Gain-bandwidth product (GRW), 86. 110 Gaussian filter, 2-3 Gaussian scale-space, 3-5,55 Germanium avalanche photodiodes, 89-90 Gradient functions, hornotopy modification, 46, 48-5 I , 54 Gradient watershed region, scale-space, 5 1-53 Granularity, imaging plate system, 259-262
INDEX Granulometries, I 1 Gray-scale images. mathematical morphology, 39 Gray-scale morphology. 8 Gray-scale reconstruction, 48-49
Hcat equation. 6 Helerojunction bipolar transistor (HBT), 79 High electron mobility transistor (HEMT), 79 High-rcsolution electron microscope (HREM), imaging platc with, 274-285 Holograms, electronic, 173-1 74 iniage reconstruction. 200-205. 207-2 10. 2 18-221 double-exposure electron hologram. 207-2 10 in-line optical bench. 200-203, 207 Mach-Zcnder interferometer, 203-2 IS. 2 I9 rccording. 192-200, 207-2 10 Holography electron holography. 171-173. 235-236 charged diclectric spheres, 174, 207-2 I6 charged microtips. 229-235 double-cxposure electron holography, 205-207 electron hiprism, 173, 176-184 electron-specimen interaction. 174- 176 electrostatic Aharonov-Bohm effect, 187-192 phase-object approximation, 174-1 75. 184-186 revcrsc-biased p-n junctions. 172- 173. 174, 185.216-229 Fresnel holography, 173 Honiothcty, defined. 4 I Hoiiiotopy modification, gradient functions. 46, 48-5 I . 54 HREM. See High-resolution electron inicroacopc
Idcnipotence, 9 IM. Srr lntcnsity modulation Image analysis. quantitative. imaging plate systcrn, 274-285
333
Image processing contrast enhancement. 285 imaging plate. 253. 285-286 spatial frequency filtcring. 285 liuage reconstruction double-exposure electron holography, 207-2 10 pray-scale reconstruction. 4 8 4 9 holograms, 200-205, 207-2 10, 2 18-22 I double-exposure electron holoram. 207-2 I 0 in-line optical bench, 200-203. 207 Mach-Zender interferometer, 203-205, 219 Image restoration, 292-293 space variant. S w Space-variant image restoration space-variant realization. 308-3 12 Image signal. 294-295 space-variant image restoration, 294-295 Imaging plate, 241-242, 242, 248-250 CBED pattern with. 269-270 computed radiography and, 242. 263-265 dynamic range, 269-274 electron diffraction with. 270-274 erasing. 253 exposurc, 25@?51 lading. 258-259 granularity. 259-262 high-resolution electron microscopy with. 274-285 image processing, 253, 285-286 qiiantitative image analysis, 274-285 radio luminography and, 242, 263-265 reading, 25 1-253 resolution, 257 RHEED and. 2 8 6 2 8 8 sensitivity. 254-257, 262, 265-269 transmission electron microscopy and. 262-263.26.5-269.288 Impact ionization, 8 1-82 Indium phosphide avalanche phosphodiodcs bascd on. 142-146. I56 ioniration rates, 1 16- I 18 Indium phosphidehndium gallium arsenitic avalanche photodiodes SACGM, 73.92-94. 96- 1 % SAM and SAGM, 9&92 Intcnsity modulation (IM), 68
334
INDEX
Interferometry amplitude division interferometry, 193-194 Mach-Zender interferometer, 203-205, 2 I9 wavefront division interferometry, 194-200 Ionization photodiode absorption layer, 110-1 13 in semiconductors, charge and grading layers, 113-1 15 Ionization rates, indium phosphide, I 16-1 18
Kalman filtering, space-variant image restoration, 292, 293, 295 estimation algorithm, 297-298 state-space representation, 295-297 steady-state solution, 299
Morphological convolution, 44 Morphological scale-space, 55 fingerprints, 4-5, 29-37,55 future work. 56-57 limitations, 55-56 multiscale closing-opening, 22-29. 53, 55 multiscale dilation-erosion, 8. 16-22. 53, 55 MQW/SL photodiodes. See Multiquantum well/superlattice photodiodes MSM photodetector. See Metal-semiconductor-metal photodetector Multimode fibers, history, 70-71 Multiplication shot noise, avalanche photodiodes, IS I Multiquantum well/superlattice (MQWKL) photodiodes, 94-96 Multiscale closing-opening, 22-29, 53. 55 Multiscale dilation-erosion, 8, 16-22, 53, 55
L Long cutoff wavelength, defined, 76 Low-frequency noise, avalanche photodiodes, 150, 156 Luminescence crystal structure and, 243 phosphostimulated. See Phosphostimulated luminescence
Mach-Zender interferometer, 203-205, 2 19 Marker function, 4 8 4 9 Masking function, 292-293 Mathematical morphology, 6, 8 convex function, 13-14 scale-dependent morphology, 1 1-1 5 watershed transform, 46, 4 7 4 8 Metal-semiconductor field-effect transistor (MESFET), 79 Metal-semiconductor-metal (MSM) photodetector, 79 Microtips, charged, 229-235 Miller empirical formula, 120, 121, 123-1 24 Modulator, defined. 67 Monomode fibers, history, 7 I Monotone property, 4, 20-22, 22-23.28-29, 4748
Neutron radiography, 264 Noise analog optical fiber communication. 68 avalanche photodiodes, 88, 94-96. 150-1 53, 156 flicker noise, 151-153, 154 low-frequency noise measurements, 1 SO, IS6 multiplication shot noise, 15 I defined, 76 imaging plate system, 2.59-260 multiplication excess noise, 84-86 Nonlocal type 2 phenomenon, 188 Numerical simulation, contour maps, 214-2 16, 227
OEICs. See Optoelectronic integrated circuits Okuto-Crowell theory, 136, 141. 148 Opening function, 9-10 multiscale closing-opening, 22-29 Optical fiber cables, 71 Optical fiber communications, 67-70 advantages, 71-72
INDEX avalanche photodiodes ( APDs), 8 1 gennaniurn APDs. 89-90 InP-based, 142-146 low-noise and fast-speed heterojunction APDs, 94-96 SACGM InP/lnGaAs APDs, 73,92-94, 99-156 SAM and SAGM InP/lnGaAs APDs, 90-92 silicon APDs, 89 theory, 8 1-88 critical device parameters extraction, 102-120, 153, 155 dark current, 74, 88, 150-153, 154. 156 disadvantages, 72 erroranalysis, 110-113, 118-119. 155 history, 70-7 I metal-semiconductor-metal photodetector, 79 optical receivers. 72-73 photoconductive detector. 79-80 photodetectors, 74-89 photogain, 74, 120-150, 155, 156 phototransistor. 80 p-i-n photodiode. 77-78 p-n photodiode, 77 Schottky-barrier photodiode, 78-79 temperature-dependence, photogain and breakdown voltage. 135-150, 156 Optical fibers, 67-88 advantages, 7 1-72 history, 70-71 Optical pulses, 68-69 Optical receivers, 72-73 Optoelectronic integrated circuits (OEICs), 79
P Paraxial theory, 179 Phase-difference maps, 207-208 Phase-object approximation (POA), 174. 175- 176, 184-1 86 Phosphor, defined, 242 Phosphostimulated luminescence (PSL), 242 applications CBED pattern, 269-270 computed radiography, 242, 263-265 electron diffraction patterns, 270-274 high-resolution electron microscopy, 274-285 image processing, 285-286
335
quantitative images analysis, 274-285 radio luminography. 242, 263-265 RHEED, 286-288 transmission electron microscopy. 262-263,265260,288 imaging plate erasing. 253 exposure, 250-25 I fading. 2.58-2.59 granularity. 259-262 reading. 25 1-253 resolution, 257 sensitivity, 262, 265-267 mechanisms, 242-248 Photoconductive detector, 79-80 Photocurrent. avalanche photodiodes, 87 Photodetectors, 74-75 device requirements, 76-77 performance characteristics, 75-76 semiconductor photodetectors avalanche photodiodes. See Avalanche photodiodes metal-semiconductor-metal photodetectors. 79 photoconductive detector, 79-80 phototransistors, 80 p-i-n photodiode, 77-78 11-n photodiode, 77 Schottky-barrier photodiodes, 78-79 Photodiodes avalanche photodiodes, See Avalanche photodiodes InP/lnGaAs avalanche photodiodes, 73, 90-94,99-156 ionization in absorption layer, I 10-1 13 in charge and grading layer, I 13-1 15 multiquantum well/superlattice (MQW/SL) photodiodes, 94-96 photogain, 74, 120-1 50, I55 p-i-ti photodiode, 77-78 p-ii photodiode, 77 Schottky-barier photodiode, 78-79 Photogain SAGCM InP/lnCaAs avalanche photodiodes, 74, 120-150, 155 temperature dependence, 146-1 50 Photomultiplier tube (PMT), 75 Phototransistor, 80 nhotodiodes. 77-78 io-i-n .
336
INDEX
Pixels space-variant image restoration. 309-1 2 types, 303 Planar SAGCM InPllnGaAs avalanche photodiodes, 96- 102, I2 I /)-ti photodiode, 77 POA. See Phase-object approximation Poweroid structuring functions. 14-15. 41, 3246 PSL. See Pliosphostiinulated luminescence
Quadratic structuring functions. 15. 4 5 4 6 Quantitative image analysis, imaging plate system, 274-285 Quantum efficiency avalanche photodiodes, 87 photodetectors. 75
R Radio luniinography. imaging plate with, 242, 263-265 Receiver. defined, 67 Reduced fingerprints, mathematical morphology, 32-37 Reflectance spectroscopy, photodiode calibration, 100 Reflection high-energy electron diffraction (RHEED), imaging plate system with, 286288 Resolution, imaging plate system, 257 Responsivity, photodetectors, 76 Reverse-biased p n junction, 172- 173. 174. 185,216-229 RHEED. Stw Reflection high-energy electron diffraction
SAGCM avalanche photodiodes, 73,92-94 critical device parameters cxtraction, 102-120, 153, 155 error analysis, 110-1 13, 118-1 19, IS5 photogain. 74, 12C-135,155 teriiperature dependence, 135-1 50 planar. 96-1 02. I 2 1
SAGM InP/lnCaAs avalanche photodiode. 90-92 SAM InP/lnGaAs avalanche photodiodes. 90-92 Scale-space delined. 2 Gaussian scale-space, 3-5, 55 gradient watershed region. 5 1-53 morphology. 55 tingerprints, 4-5, 29-37, 55 future work, 56-57 limitations, 55-56 inultiscale closing-opening scale-space, 22-29 multiscale dilation-erosion scale-space. 16-22 multiscale morphology, 8-1 5 for regions, 46-53 signal extrema. 20-22 structuring functions. 3 7 4 6 Scale-space filtering. 2 Schottky-barrier photodiode, 78-79 Secondary ion niass spectroscopy (SIMS). photodiode calibration, 100-101 Semiconductor photodetectors avalanche photodiodes. See Avalanche phototiodes metal-semiconduc tor-metal photodetectors. 79 photoconductive detector. 79-80 phototransistors, X0 />-i-ti photodiode, 77-78 p r i photodiode, 77 Schottky-barrier photodiodes, 78-79 Senii-group propertics. structuring functions. 15, 37-38 Sensitivity. imaging plate system, 254-257. 262 Signal density resolution, 261-262 fingerprint. 4-5 Signal-to-noise ratio (SNR), analog optical fiber cornmunication. 68 Silicon avalanche photodiodes. 89 SIMS. See Secondary ion mass spectroscopy Space-variant image restoration, 293-294. 320-32 I image model, 30C309 image restoration, 308-3 17 image signal, 294-295 Kalman filtering. 292. 293, 295 estimation algorithm. 297-298
INDEX state-space representation, 295-297 steady-state solution, 299 numerical results, 3 18-320 Spatial frequency filtering. imaging plate, 285 Staircase avalanche photodiodes, 94-95 State-space, Kalinan filtering, 295-297 Strip processor, 3 14 Sti-ucturing functions, 9, 13-1 5, 18-20 coilvex structuring functions, 13- 14 dimensionality, 40-42 Hat structuring function, 44 poweroid structuring functions, 14-1 5, 41. 4246 quadratic structuring functions, IS, 45-46 scale-space, 37-46 bemi-group properties, IS, 37-38 signal extrerna. 20-22
331
T Transimpedance amplifier, 73 Transmission electron microscope (TEM), imaging plate with, 262-263, 265-269, 288 Tunneling, bandgap semiconductor, 88
Uncertainty principle, scale-spaces and, 3
Watershed transform, 46, 4 7 4 8 Wavefront divirion interferometry, I944200
