174 APPLIED OPTICS / Vol. 30, No. 2 / 10 January 1991

Real time self-pumped optical phase-conjugator-based Wigner distribution processor for complex signals Yao Li and Berlin Ha

CUNY-City College, Department of Electrical Engineer­ing, New York, New York 10031. Received 20 June 1990. 0003-6935/91/020174-03$05.00/0. © 1991 Optical Society of America.

Based on a combination of linear shift-invariant optical elements and a nonlinear photorefractive crystal serving as a self-pumped optical phase-conjugator, the real time opti­cal generation of an auto- and cross- Wigner distribution for complex signals is demonstrated.

Although in many applications an analysis or synthesis of a signal's global power spectrum can provide useful informa­tion, it is often of interest to analyze and manipulate the signal's local distribution of energy as a function of time or space. A way to display a signal's [f(x)] local frequency spectrum uses the so-called auto-Wigner or Wigner distribu­tion (WD) defined as1

where ξ denotes its spatial frequency. For two dissimilar signals, the results is the cross-Wigner distribution. While the WD was first introduced as a thermodynamic description for quantum mechanics, recently it has found applications in many other areas of science and engineering.2-5 Because the WD is a Fourier transform of the Hermitian function,5

many optical implementations have been proposed. In the early optical WD (OWD) implementation approaches,6-9 be­cause of the difficulty associated with the generation of the complex conjugate wavefront, only real signals were consid­ered. For a real time complex signal OWD generation, one method requires an interferometric technique10 that pro­duces, in addition to the OWD, three additional optical quantities, while another method works only under the small

Fig. 1. Schematic OWD setup for 1-D complex signals: E, beam expander; M, mirrors; BS, beam splitter; T, input transparencies; L, lenses; CL, cylindrical lens; D, detector. As a self-pumped OPC

generator, a BaTiO3 crystal was used.

signal condition.11 Recently, the use of a four-wave-mixing optical-phase-conjugation (OPC) based system for the real time OWD display of complex signals was proposed.12 The purpose of this Letter is to report on a simplified version of this OPC-based approach, and its experimental demonstra­tion using a BaTiO3 photorefractive crystal.

In Fig. 1, for a complex 1-D signal, a schematic is shown of the new OPC-based OWD device. A single-mode λ = 514.5-nm Ar+-ion laser was used. The source beam, after spatial filtering and collimation, was guided through mirror M1 into the OWD system. The main optical components were two spherical lenses L1,L2, a cylindrical lens CL, a beam splitter BS, and a 0° cut BaTiO3 photorefractive crystal. Two iden­tical 1-D signal transparencies T1(T2) were placed at the front focal plane of the lenses L1(L2) and were rotated -26.5° (+26.5°) about the ;y-axis. To generate the self-pumped OPC wavefront of the input beam, a BaTiO3 crystal was placed at the L1 back focal plane.13-14 Although fixed trans­parencies were employed in our experiment, because of the BaTiO3 crystal, their replacement with real time input de­vices, such as spatial light modulators, is feasible. The mul­tiplication of the two wavefronts reproduces the terms in Eq. (2). To generate a 1 -D image (1 -D Fourier transform) of this signal along its x(y) direction, the lenses L2 and CL had an identical focal length ƒ with the CL acting along the system's x-axis. The thus formed lens combination was placed at a

focal length ƒ behind T2. At the output plane D, the OWDs of the complex 1-D signals were generated. To generate the self-pumped OPC signal, ~100 mW of optical power were used. The BaTiO3 crystal was a 4- × 4- × 5-mm3 cube. The input light was polarized linearly along the crystal's extraor­dinary polarization direction.13 The average angle between the input beam and the c-axis was ~40°. To generate the image (Fourier transform) along the x(y) direction, both the spherical L2 and cylindrical CL lenses had an identical focal length of ƑF = 100 mm. With these parameters, the following experiments were performed.

In the first experiment, a 1-D prism wedge object de­scribed as the complex signal

where θ denotes the wedge's deflection angle, was used. With this input, the corresponding Hermitian function is

Thus its spatially displayed OWD is

Fig. 2. Experimental results of an OWD of 1-D phase objects: (a) the OWD of a collimated plane wave and of a 1.5° prism wedge; (b)

OWD of a ƒ = 150-mm cylindrical lens.

where yF = λƒFξ and Cg is a constant, respectively. Since in our experiment θ was 1.5°, to display this horizontal light bar with a vertical translation of 2.6 mm [see Fig. 2(a)] on the x-axis a reference light bar was also produced. In a second experiment, as the input signal, a quadratic phase object, i.e., a cylindrical lens, was used. For an ideal quadratic lens with a focal length ƒh, its transmittance is15

Correspondingly, its Hermitian function is

The corresponding OWD, neglecting an aperture effect, is

To verify this result, as the inputs T1 and T2, two identical focal length ƒh = 150-mm cylindrical lenses were used. The obtained OWD experimental result [see Fig. 2(b)] indicates a linear chirp function with a slope of a = tan-1[100/150] = 33.6°.

In a final experiment, an optical cross-WD (OCWD) of two 1-D complex quadratic phase functions p(x) and q(x) was generated. As the two inputs, different focal length (ƒp = 300-mm, fq = 75-mm) cylindrical lenses were used. Now the complex Hermitian function is

Fig. 3. Experimental results of an OCWD of 1-D phase objects: (a) the OCWD of two cylindrical lenses with fp = 300 mm and fq = 75 mm, respectively; (b) the output generated from the same inputs but

with the detector shifted from a distance of 100-134 mm.

= 49° was obtained [see Fig. 3 (b)]. This result was due to the diffraction effect. In Eq. (9), for an OCWD generation, the last two terms require a Fourier transformation along the y-direction. To generate an output similar to Fig. 2(b), it is essential to eliminate the effect of the second term in Eq. (9). To accomplish this task, the constraint

where needs to be satisfied.15 Thus, we have a new output location d0 where

In Fig. 3(a), the obtained OCWD pattern, i.e., the Fourier transform of Eq. (9b), is shown. It is interesting to note that when the output plane was axially shifted from d = ƑF = 100 mm to d = d0 = 134 mm, where d is the distance between the CL and output screen, a linear slope function with an angle a

is required. For the case of fp = fq ,d0 is equal to ƑF, in agreement with the result of the previous example. At this new output plane d0 the output pattern is

10 January 1991 / Vol. 30, No. 2 / APPLIED OPTICS 175

176 APPLIED OPTICS / Vol. 30, No. 2 / 10 January 1991

where yd = λd0ξ and Cpq is a constant. For the above given parameters, Eq. (11) and (12) predict the result observed in our experiment. The Fig. 3 (a) pattern can now be interpret­ed as a convolution of an out-of-focused signal with a linear slope δ-function, i.e.,

where • denotes a convolution, CPQ is a phase function, and

is a lens aperture function. While in our experiment two input transparencies were

needed, however, with slight modification, the use of a single input transparency is also possible. The use of the OPC device allows a real time spatial light modulator input. This new real time OWD method can also be extended to display the OWD and OCWD of 2-D signals.9 In addition, since the WD is also closely related to the ambiguity function (AF),16 a function with many applications in radar technology, this new method can also be modified to display the AF. The use of a photorefractive effect promises a high speed operation. Recently, a self-pumped OPC with a nanosecond response was demonstrated.17 With the advent of more efficient non­linear materials, a high speed OWD display for various scien­tific and industrial applications will be feasible. This work is supported in part by a grant from the Air Force Office of Scientific Research (AFOSR-90-0346). References 1. E. Wigner, "On the Quantum Correction for Thermodynamic

Equilibrium," Phys. Rev. 40, 749 (1932). 2. C. P. Jansse and J. M. Kaizer, "Time-Frequency Distributions of

Loudspeakers: The Application of the Wigner Distribution Function," J. Audio Eng. Soc. 31, 198 (1983).

3. W. Martin and P. Flandrin, "Detection of Changes of Signal Structure by Using the Wigner-Ville Spectrum," Signal Process. 8, 215 (1985).

4. G. Eichmann and N. M. Marinovic, "Scale-Invariant Wigner Distribution and Ambiguity Functions," Proc. Soc. Photo-Opt. Instrum. Eng. 519, 18 (1984).

5. G. Cristobal, J. Bescos, and J. Santamaria, "Image Analysis Through the Wigner Distribution Function," Appl. Opt. 28, 262-271 (1989).

6. R. J. Marks II, M. I. Jones, E. I. Kral, and J. F. Walkup, "One-Dimensional Linear Coherent Processing Using a Single Optical Element," Appl. Opt. 18, 2783-2786 (1979).

7. M. J. Bastiaans, "Wigner Distribution Function and Hamilton's Characteristic of a Geometric-Optical System," Opt. Commun. 30, 321 (1979).

8. G. Eichmann and B. Z. Dong, "Two-Dimensional Filtering of 1-D Signals," Appl. Opt. 21, 3152-3156 (1982).

9. M. Conner and Y. Li, "Optical Generation of the Wigner Distri­bution of 2-D Real Signals," Appl. Opt. 24, 3825-3829 (1985).

10. K.-H. Brenner and A. W. Lohmann, "Wigner Distribution Func­tion Display of Complex 1-D Signals," Opt. Commun. 42, 310-(1982).

11. T. Mateeva and P. Sharlandjiev, "The Generation of a Wigner Distribution Function of Complex Signals by Spatial Filtering," Opt. Commun. 57, 153 (1986).

12. Y. Li, G. Eichmann, and M. Conner, "Optical Wigner Distribu­tion and Ambiguity Function for Complex Signals and Images," Opt. Commun. 67, 177 (1988).

13. J. Feinberg, "Self-Pumped, Continuous-Wave Phase Conjuga­tion Using Internal Reflection," Opt. Lett. 7, 486-488 (1982).

14. A. E. Chiou and P. Yeh, "Parallel Image Subtraction Using a Phase-Conjugate Michelson Interferometer," Opt. Lett. 11, 306-308 (1986).

15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

16. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, London, 1953).

17. B. Monson et al., "Self-Pumped Phase-Conjugation with Nano­second Pulses in Strotium Barium Niobate," Opt. Lett. 15, 12-14 (1990).