Applications of the Wigner distribution function in signal processing

Eurasip Journal on Applied Signal Processing

Volumn 2005, Issue 10, 2005, Pages 1520-1534

  Dragoman, Daniela ^a  

^a UNIVERSITY OF BUCHAREST   (Romania)

Author keywords

Phase space; Signal processing; Wigner distribution function

Indexed keywords

ARBITRARY SIGNALS; PHASE SPACE; SIGNAL RECOGNITION; WEIGNER DISTRIBUTION FUNCTION;

MATHEMATICAL MODELS; OPTICAL DEVICES; OPTICAL SYSTEMS; PATTERN RECOGNITION; THROUGHPUT;

SIGNAL PROCESSING;

EID: 27844542553     PISSN: 11108657     EISSN: None     Source Type: Journal    
DOI: 10.1155/ASP.2005.1520     Document Type: Review

References (127)

1. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, "The Wigner distribution - a tool for time-frequency signal analysis. Part I. Continuous-time signals," Philips Journal of Research, vol. 35, no. 3, pp. 217-250, 1980.
2. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, "The Wigner distribution - a tool for time-frequency signal analysis. Part II: Discrete-time signals," Philips Journal of Research, vol. 35, pp. 276-300, 1980.
3. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, "The Wigner distribution - a tool for time-frequency signal analysis. Part III: Relations with other time-frequency transformations," Philips Journal of Research, vol. 35, pp. 372-389, 1980.
4. L. Cohen, "Time-frequency distributions - a review," Proc. IEEE, vol. 77, no. 7, pp. 941-981, 1989.
5. P. Flandrin, Temps-Fréquence, Hermès, Paris, France, 1993.
6. D. Dragoman, "The Wigner distribution function in optics and optoelectronics," Progress in Optics, vol. 37, pp. 1-56, 1997.
7. G. Matz and F. Hlawatsch, "Wigner distributions (nearly) everywhere: time-frequency analysis of signals, systems, random processes, signal spaces, and frames," Signal Processing, vol. 83, no. 7, pp. 1355-1378, 2003.
8. B. Boashash, Ed., Time Frequency Signal Analysis and Processing - A Comprehensive Review, Elsevier, Amsterdam, The Netherlands, 2003.
9. F. Hlawatsch and G. F. Boudreaux-Bartels, "Linear and quadratic time-frequency signal representations," IEEE Signal Processing Mag., vol. 9, no. 2, pp. 21-67, 1992.
10. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Physical Review, vol. 40, pp. 749-759, 1932.
11. A. Walther, "Radiometry and coherence," Journal of the Optical Society of America, vol. 58, pp. 1256-1259, 1968.
12. M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Optics Communications, vol. 25, pp. 26-30, 1978.
13. D. S. Bugnolo and H. Bremmer, "The Wigner distribution matrix for the electric field in a stochastic dielectric with computer simulation," Advances in Electronics and Electron Physics, vol. 61, pp. 299-389, 1983.
14. R. Martínez-Herrero, P. M. Mejías, and J. M. Movilla, "Spatial characterization of general partially polarized beams," Optics Letters, vol. 22, no. 4, pp. 206-208, 1997.
15. J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron., vol. 28, no. 10, pp. 2262-2273, 1992.
16. M. J. Bastiaans, "Application of the Wigner distribution function to partially coherent light," Journal of the Optical Society of America {A}, vol. 3, pp. 1227-1237, 1986.
17. B. Boashash and P. J. Black, "An efficient real-time implementation of the Wigner-Ville distribution," IEEE Trans. Acoust., Speech, Signal Processing, vol. 35, no. 11, pp. 1611-1618, 1987.
18. F. Peyrin and R. Prost, "A unified definition for the discretetime, discrete-frequency, and discrete-time/frequency Wigner distributions," IEEE Trans. Acoust., Speech, Signal Processing, vol. 34, no. 4, pp. 858-867, 1986.
19. K.-H. Brenner, "A discrete version of the Wigner distribution function," in Signal Processing II: Theories and Applications, H. W. Schüssler, Ed., Elsevier (North-Holland), Amsterdam, The Netherlands, 1983.
20. K. B. Wolf, M. A. Alonso, and G. W. Forbes, "Wigner functions for Helmholtz wave fields," Journal of the Optical Society of America {A}, vol. 16, no. 10, pp. 2476-2487, 1999.
21. D. Dragoman, "Phase space interference as the source of negative values of the Wigner distribution function," Journal of the Optical Society of America {A}, vol. 17, no. 12, pp. 2481-2485, 2000.
22. D. Dragoman and M. Dragoman, "Quantum coherent versus classical coherent light," Optical and Quantum Electronics, vol. 33, no. 3, pp. 239-252, 2001.
23. F. Hlawatsch and P. Flandrin, "The interference structure of the Wigner distribution and related time-frequency signal representations," in The Wigner Distribution - Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, Eds., pp. 59-132, Elsevier, Amsterdam, The Netherlands, 1997.
24. M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," Journal of the Optical Society of America, vol. 69, pp. 1710-1716, 1979.
25. M. J. Bastiaans, "The Wigner distribution function and Hamilton's characteristics of a geometric-optical system," Optics Communications, vol. 30, pp. 321-326, 1979.
26. R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, Calif, USA, 1966.
27. A. A. Godil, B. A. Auld, and D. M. Bloom, "Picosecond timelenses," IEEE J. Quantum Electron., vol. 30, no. 3, pp. 827-837, 1994.
28. B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron., vol. 30, no. 8, pp. 1951-1963, 1994.
29. A. G. Kostenbauder, "Ray-pulse matrices: a rational treatment for dispersive optical systems," IEEE J. Quantum Electron., vol. 26, no. 6, pp. 1148-1157, 1990.
30. J. Yang and D. Fan, "Intensity-moments characterization of general pulsed paraxial beams with the Wigner distribution function," Journal of the Optical Society of America {A}, vol. 16, no. 10, pp. 2488-2493, 1999.
31. K. B. Wolf, "Wigner distribution function for paraxial polychromatic optics," Optics Communications, vol. 132, no. 3-4, pp. 343-352, 1996.
32. D. Onciul, "Characterization of the propagation of light beams through misaligned linear optical systems," Optik, vol. 93, pp. 23-26, 1993.
33. D. Dragoman, "Wigner distribution function for Gaussian-Schell beams in complex matrix optical systems," Applied Optics, vol. 34, pp. 3352-3357, 1995.
34. M. J. Bastiaans, "Transport equations for the Wigner distribution function," Optica Acta, vol. 26, pp. 1265-1272, 1979.
35. M. J. Bastiaans, "Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium," Optica Acta, vol. 26, pp. 1333-1344, 1979.
36. J. T. Mendonça and N. L. Tsintsadze, "Analog of the Wigner-Moyal equation for the electromagnetic field," Physical Review E, vol. 62, no. 3, pp. 4276-4282, 2000.
37. D. Dragoman, "Wigner distribution function for a complex matrix optical system," Optik, vol. 100, pp. 137-139, 1995.
38. J. Oz and E. Heyman, "Wigner-Ville distribution for pulse propagation in random media: pulsed plane wave," Waves in Random Media, vol. 8, no. 2, pp. 175-191, 1998.
39. D. Dragoman, "The Wigner distribution function and the energy conservation of a light beam," Journal of Modern Optics, vol. 43, pp. 1127-1134, 1996.
40. M. Born and E. Wolf, Principles of Optics, Pergamon, Oxford, UK, 1970.
41. D. Dragoman, "Wigner-transform approach to paraxial light propagation in stratified media," Journal of the Optical Society of America {A}, vol. 14, no. 5, pp. 1103-1109, 1997.
42. D. Dragoman, "Higher-order moments of the Wigner distribution function in first-order optical systems," Journal of the Optical Society of America {A}, vol. 11, no. 10, pp. 2643-2646, 1994.
43. J. B. Murphy, "Phase space conservation and selection rules for astigmatic mode converters," Optics Communications, vol. 165, no. 1-3, pp. 11-18, 1999.
44. R. Martínez-Herrero and P. M. Mejías, "Second-order spatial characterization of hard-edge diffracted beams," Optics Letters, vol. 18, no. 19, pp. 1669-1671, 1993.
45. G. Piquero, P. M. Mejías, and R. Martínez-Herrero, "Propagation of laser beam parameters through pure phase transmittances," Optics Communications, vol. 129, no. 3-4, pp. 161-166, 1996.
46. K.-H. Brenner and A. W. Lohmann, "Wigner distribution function display of complex ID signals," Optics Communications, vol. 42, no. 5, pp. 310-314, 1982.
47. G. Shabtay, D. Mendlovic, and Z. Zalevsky, "Proposal for optical implementation of the Wigner distribution function," Applied Optics, vol. 37, pp. 2142-2144, 1998.
48. Y. Li, G. Eichmann, and M. Conner, "Optical Wigner distribution and ambiguity function for complex signals and images," Optics Communications, vol. 67, no. 3, pp. 177-179, 1988.
49. J. Lancis, E. E. Sicre, E. Tajahuerce, and P. Andrés, "White-light implementation of the Wigner-distribution function with an achromatic processor," Applied Optics, vol. 34, pp.8209-8212, 1995.
50. R. A. Athale, J. N. Lee, E. L. Robinson, and H. H. Szu, "Acousto-optic processors for real-time generation of time-frequency representations," Optics Letters, vol. 8, no. 3, pp. 166-168, 1983.
51. D. Dragoman, "Can the Wigner transform of a two-dimensional rotationally symmetric beam be fully recovered from the Wigner transform of its one-dimensional approximation?" Optics Letters, vol. 25, no. 5, pp. 281-283, 2000.
52. M. Conner and Y. Li, "Optical generation of the Wigner distribution of 2-D real signals," Applied Optics, vol. 24, pp. 3825-3829, 1985.
53. R. Bamler and H. Glünder, "The Wigner distribution function of two-dimensional signals. Coherent-optical generation and display," Optica Acta, vol. 30, pp. 1789-1803, 1983.
54. T. Iwai, A. K. Gupta, and T. Asakura, "Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object," Optics Communications, vol. 58, no. 1, pp. 15-19, 1986.
55. G. T. Hermann, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, NY, USA, 1980.
56. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, "Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms," Optics Letters, vol. 20, no. 10, pp. 1181-1183, 1995.
57. M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Physical Review Letters, vol. 72, no. 8, pp. 1137-1140, 1994.
58. K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, "Heterodyne measurement of Wigner distributions for classical optical fields," Optics Letters, vol. 24, no. 19, pp. 1370-1372, 1999.
59. A. Wax, S. Bali, and J. E. Thomas, "Optical phase-space distributions for low-coherence light," Optics Letters, vol. 24, no. 17, pp. 1188-1190, 1999.
60. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, "Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses," Optics Letters, vol. 18, no. 23, pp. 2041-2043, 1993.
61. J. W. Wu, P. K. Lam, M. B. Gray, and H. A. Bachor, "Optical homodyne tomography of information carrying laser beams," Optics Express, vol. 3, no. 4, pp. 154-161, 1998.
62. D. Dragoman, M. Dragoman, and K.-H. Brenner, "Amplitude and phase recovery of rotationally symmetric beams," Applied Optics, vol. 41, pp. 5512-5518, 2002.
63. T. Alieva and M. J. Bastiaans, "Wigner distribution and fractional Fourier transform for two-dimensional symmetric optical beams," Journal of the Optical Society of America {A}, vol. 17, no. 12, pp. 2319-2323, 2000.
64. G. F. Boudreaux-Bartels and T. W. Parks, "Time-varying filtering and signal estimation using Wigner distribution syn thesis techniques," IEEE Trans. Acoust., Speech, Signal Processing, vol. 34, no. 3, pp. 442-451, 1986.
65. G. F. Boudreaux-Bartels, "Time-varying signal processing using Wigner distribution synthesis techniques," in The Wigner Distribution-Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, Eds., pp. 269-317, Elsevier, Amsterdam, The Netherlands, 1997.
66. M. R. Teague, "Image analysis via the general theory of moments," Journal of the Optical Society of America, vol. 70, no. 8, pp. 920-930, 1980.
67. M. R. Teague, "Optical calculation of irradiance moments," Applied Optics, vol. 19, no. 8, pp. 1353-1356, 1980.
68. M. O. Freeman and B. E. A. Saleh, "Moment invariants in the space and frequency domains," Journal of the Optical Society of America {A}, vol. 5, no. 7, pp. 1073-1084, 1988.
69. M. J. Bastiaans and T. Alieva, "Wigner distribution moments measured as fractional Fourier transform intensity moments," in Proc. 19th Congress of the International Commission for Optics: Optics for the Quality of Life, vol. 4829 of Proceedings of SPIE, pp. 245-246, Firenze, Italy, August 2002.
70. G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics," Journal of the Optical Society of America {A}, vol. 11, no. 8, pp. 2257-2264, 1994.
71. B. Eppich, C. Gao, and H. Weber, "Determination of the ten second order intensity moments," Optics & Laser Technology, vol. 30, no. 5, pp. 337-340, 1998.
72. N. Hodgson, T. Haase, R. Kostka, and H. Weber, "Determination of laser beam parameters with the phase space beam analyzer," Optical and Quantum Electronics, vol. 24, no. 9, pp. S927-S949, 1992.
73. J. Serna, R. Martínez-Herrero, and P. M. Mejías, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," Journal of the Optical Society of America {A}, vol. 8, no. 7, pp. 1094-1098, 1991.
74. M. A. Alonso and G. W. Forbes, "Uncertainty products for nonparaxial wave fields," Journal of the Optical Society of America {A}, vol. 17, no. 12, pp. 2391-2402, 2000.
75. S. Lavi, R. Prochaska, and E. Keren, "Generalized beam parameters and transformation laws for partially coherent light," Applied Optics, vol. 27, no. 17, pp. 3696-3703, 1988.
76. A. E. Siegman, "Denning the effective radius of curvature for a nonideal optical beam," IEEE J. Quantum Electron., vol. 27, no. 5, pp. 1146-1148, 1991.
77. P. A. Bélanger, "Beam propagation and the ABCD ray matrices," Optics Letters, vol. 16, no. 4, pp. 196-198, 1991.
78. M. A. Porras, J. Alda, and E. Bernabeu, "Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams," Applied Optics, vol. 31, no. 30, pp. 6389-6402, 1992.
79. M. J. Bastiaans, "Wigner distribution function applied to twisted Gaussian light propagating in first-order optical systems," Journal of the Optical Society of America {A}, vol. 17, no. 12, pp. 2475-2480, 2000.
80. J. Serna, P. M. Mejías, and R. Martínez-Herrero, "Rotation of partially coherent beams propagating through free space," Optical and Quantum Electronics, vol. 24, no. 9, pp. S873-S880, 1992.
81. R. Martínez-Herrero, P. M. Mejías, M. Sánchez, and J. L. H. Neira, "Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems," Optical and Quantum Electronics, vol. 24, no. 9, pp. S1021-S1026, 1992.
82. G. Piquero, P. M. Mejías, and R. Martínez-Herrero, "Sharpness changes of Gaussian beams induced by spherically aberrated lenses," Optics Communications, vol. 107, no. 3-4, pp. 179-183, 1994.
83. R. Martínez-Herrero, G. Piquero, and P. M. Mejías, "On the propagation of the kurtosis parameter of general beams," Optics Communications, vol. 115, no. 3-4, pp. 225-232, 1995.
84. D. Dragoman, "The Wigner distribution function of self-Fourier functions," Journal of Modern Optics, vol. 43, no. 9, pp. 1933-1938, 1996.
85. R. Martínez-Herrero and P. M. Mejías, "Beam characterization through active media," Optics Communications, vol. 85, no. 2-3, pp. 162-166, 1991.
86. J. R. Martín de Los Santos, R. Martínez-Herrero, and P. M. Mejías, "Parametric characterization of Gaussian beams propagating through active media," Optical and Quantum Electronics, vol. 28, no. 8, pp. 1021-1027, 1996.
87. C. Hirlimann and J.-F. Morhange, "Wavelet analysis of short light pulses," Applied Optics, vol. 31, no. 17, pp. 3263-3266, 1992.
88. L. Helczynski, D. Anderson, R. Fedele, B. Hall, and M. Lisak, "Propagation of partially incoherent light in nonlinear media via the Wigner transform method," IEEE J. Select. Topics Quantum Electron., vol. 8, no. 3, pp. 408-412, 2002.
89. C. Paré and P.-A. Bélanger, "Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments," Optical and Quantum Electronics, vol. 24, no. 9, pp. S1051-S1070, 1992.
90. W. Nasalski, "Scale formulation of first-order nonlinear optics," Optics Communications, vol. 137, no. 1-3, pp. 107-112, 1997.
91. H. Konno and P. S. Lomdahl, "The Wigner transform of soliton solutions for the nonlinear Schrödinger equation," Journal of the Physical Society of Japan, vol. 63, no. 11, pp. 3967-3973, 1994.
92. D. Dragoman and M. Dragoman, "Phase space characterization of solitons with the Wigner transform," Optics Communications, vol. 137, no. 4-6, pp. 437-444, 1997.
93. D. Dragoman and M. Dragoman, "Time-frequency characterization of magnetostatic envelope soliton waves," Applied Physics Letters, vol. 70, no. 6, pp. 714-716, 1997.
94. L. P. J. Kamp, "Local frequency spectra for nonlinear wave equations," Journal of Physics A, vol. 20, no. 11, pp. 3279-3291, 1987.
95. L. Galleani and L. Cohen, "The Wigner distribution for classical systems," Physics Letters A, vol. 302, no. 4, pp. 149-155, 2002.
96. D. Dragoman, "Wigner distribution function in nonlinear optics," Applied Optics, vol. 35, no. 21, pp. 4142-4146, 1996.
97. (2) media," Optik, vol. 111, pp. 20-24, 2000.
98. A. Pe'er, D. Wang, A. W. Lohmann, and A. A. Friesem, "Wigner formulation of optical processing with light of arbitrary coherence," Applied Optics, vol. 40, no. 2, pp. 249-256, 2001.
99. T. Iwai, N. Kawamura, and T. Takemori, "Real-time profiling of a pure phase object using an auto-Wigner distribution function," Optics Communications, vol. 95, no. 4-6, pp. 199-204, 1993.
100. D. Dragoman, "Wigner distribution function applied to third-order aberrations," Applied Optics, vol. 35, no. 1, pp. 161-168, 1996.
101. A. W. Lohmann, J. Ojeda-Castañeda, and N. Streibl, "The influence of wave aberrations on the Wigner distribution," Optica Applicata, vol. 13, pp. 465-471, 1983.
102. R. Martínez-Herrero, G. Piquero, and P. M. Mejías, "Beam quality changes produced by quartic phase transmittances," Optical and Quantum Electronics, vol. 27, pp. 173-183, 1995.
103. J. Serna, P. M. Mejías, and R. Martínez-Herrero, "Beam quality changes of Gaussian Schell-model fields propagating through Gaussian apertures," Applied Optics, vol. 31, no. 22, pp. 4330-4331, 1992.
104. G. Saavedra, W. D. Furlan, E. Silvestre, and E. E. Sicre, "Analysis of the irradiance along different paths in the image space using the Wigner distribution function," Optics Communications, vol. 139, no. 1-3, pp. 11-16, 1997.
105. W. D. Furlan, D. Zalvidea, and G. Saavedra, "Synthesis of filters for specified axial irradiance by use of phase-space tomography," Optics Communications, vol. 189, no. 1-3, pp. 15-19, 2001.
106. J. Lancis, E. E. Sicre, A. Pons, and G. Saavedra, "Achromatic white-light self-imaging phenomenon: an approach using the Wigner distribution function," Journal of Modern Optics, vol. 42, no. 2, pp. 425-434, 1995.
107. R. Castaneda, "On the relationship between the cross-spectral density and the Wigner distribution function," Journal of Modern Optics, vol. 45, no. 3, pp. 587-593, 1998.
108. C. Martínez, R. Martínez-Herrero, and P. M. Mejías, "Parametric characterization of light pulses propagating through single-mode fibres," Journal of Modern Optics, vol. 43, no. 9, pp. 1785-1796, 1996.
109. [ 109] J. Serna, P. M. Mejías, and R. Martínez-Herrero, "Beam quality in monomode diode lasers," Optical and Quantum Electronics, vol. 24, no. 9, pp. S881-S887, 1992.
110. R. Gase, "The multimode laser radiation as a Gaussian Schell model beam," Journal of Modern Optics, vol. 38, no. 6, pp. 1107-1115, 1991.
111. K. B. Wolf and A. L. Rivera, "Holographic information in the Wigner function," Optics Communications, vol. 144, no. 1-3, pp. 36-42, 1997.
112. L. Onural and M. T. Özgen, "Extraction of three-dimensional object-location information directly from in-line holograms using Wigner analysis," Journal of the Optical Society of America {A}, vol. 9, no. 2, pp. 252-260, 1992.
113. M. Testorf, "Analysis of the moiré effect by use of the Wigner distribution function," Journal of the Optical Society of America {A}, vol. 17, no. 12, pp. 2536-2542, 2000.
114. J. Paye and A. Migus, "Space-time Wigner functions and their application to the analysis of a pulsed shaper," Journal of the Optical Society of America B, vol. 12, no. 8, pp. 1480-1490, 1995.
115. K. B. Wolf, D. Mendlovic, and Z. Zalevsky, "Generalized Wigner function for the analysis of super resolution systems," Applied Optics, vol. 37, no. 20, pp. 4374-4379, 1998.
116. D. Mendlovic and Z. Zalevsky, "Definition, properties, and applications of the generalized temporal-spatial Wigner distribution function," Optik, vol. 107, no. 2, pp. 49-56, 1997.
117. V. J. Pinto-Robledo and T. A. Hall, "Chronocyclic description of laser pulse compression," Optics Communications, vol. 125, no. 4-6, pp. 338-348, 1996.
118. M. A. Muriel, J. Azana, and A. Carballar, "Fiber grating synthesis by use of time-frequency representations," Optics Letters, vol. 23, no. 19, pp. 1526-1528, 1998.
119. J. Azaña and M. A. Muriel, "Reconstruction of fiber grating period profiles by use of Wigner-Ville distributions and spectrograms," Journal of the Optical Society of America {A}, vol. 17, no. 12, pp. 2496-2505, 2000.
120. D. Dragoman, M. Dragoman, and J.-P. Meunier, "Recovery of the refractive-index profile from the Wigner distribution of an optical waveguide," Applied Optics, vol. 37, no. 12, pp. 2357-2360, 1998.
121. D. Dragoman and J. P. Meunier, "Recovery of longitudinally variant refractive index profile from the measurement of the Wigner transform," Optics Communications, vol. 153, no. 4-6, pp. 360-367, 1998.
122. R. März, Integrated Optics: Design and Modeling, Artech House, Norwood, Mass, USA, 1995.
123. D. Dragoman, J. P. Meunier, and M. Dragoman, "Beam-propagation method based on the Wigner transform: a new formulation," Optics Letters, vol. 22, no. 14, pp. 1050-1052, 1997.
124. D. Onciul, "Efficiency of light launching into waveguides: a phase space approach," Optik, vol. 96, no. 1, pp. 20-24, 1994.
125. D. Dragoman, "Wigner-distribution-function representation of the coupling coefficient," Applied Optics, vol. 34, no. 29, pp. 6758-6763, 1995.
126. D. Dragoman and M. Dragoman, "Integrated optic-devices characterization with the Wigner transform," IEEE J. Select. Topics Quantum Electron., vol. 2, no. 2, pp. 181-186, 1996.
127. D. Onciul, "Waveguide launching efficiency for multimoded and partially coherent light sources," Optik, vol. 97, no. 2, pp. 75-77, 1994.