Optical on-axis imperfect phase-only correlator using liquid-crystal television
Yunlong Sheng and Gilles Paul-Hus
We present a VanderLugt-type optical phase-only correlator that uses the currently available liquid-crystal television. Theoretical and experimental results show that phase mismatching and phase and amplitude cross coupling of the liquid-crystal television can reduce the peak intensity to approximately 60%-70%. The imperfect phase-only filter yields on-axis correlation with high light efficiency and utilizes all the available space-bandwidth product of the liquid-crystal television.
The phase-only filter (POF) is a kinoform in the Fourier plane of an optical correlator. Its correlation output is on axis so that the POF has 100% theoretical light efficiency and utilizes the full available space—bandwidth product of the spatial light modulator (SLM). The POF with unit amplitude amplifies high-frequency components of the input object and produces a sharp autocorrelation peak with good discrimination capability.1 Because of difficulties in realization of continuous phase modulation by real-time SLM's, the binary POF and other filters have been proposed. These filters produce the POF response with reduced light efficiency and a reduced signal-to-noise ratio (SNR).
Recent progress in technology has made the high-contrast high-resolution low-cost liquid-crystal television (LCTV) available.2 Continuous phase modulation by the LCTV has been obtained3 and used in the optical joint transform correlator and in the electro-optical zoom lens. The major drawback of implemen-tion of the POF with the LCTV is that the maximum phase retardation of the LCTV could be less than the required 2π (phase mismatching) and that there is undesired amplitude modulation cross coupled with the phase modulation. Amako and Sonehara4 fabricated a LC SLM that is capable of pure phase modulation of 0 to 2π; however, their LCTV is not commercially available. Optimal filters that use the SLM's with the phase amplitude cross coupling or with arbitrary constraint have been proposed.5-6 In
The authors are with the Centre d'Optique, Photonique et Laser, Département de Physique, Université Laval, Quebec G1K 7P4, Canada.
this Note we optically implement the on-axis POF with the currently available LCTV in the Fourier plane of a VanderLugt-type correlator. We present theoretical and experimental analysis to show that, although the imperfections of the LCTV reduce intensity of the correlation peak, the POF coded on the LCTV yields on-axis correlation with still high light efficiency and utilizes the whole space-bandwidth product of the LCTV.
The LCTV used in our optical correlator is a liquid-crystal panel in the Epson Crystal Image video projector. The panel has an aperture roughly 1 in. × 1 in. (2.54 cm × 2.54 cm) and 220 × 320 pixels, each measuring 60 μm × 55 μm with a center-to-center spacing of 90 μm × 80 μm. There are basically two light-modulation mechanisms: rotation of the polarization caused by twisting of the liquid-crystal molecules and phase retardation caused by the birefringence. Kirsch et al. presented measurements on the modulation characteristics of this device in detail.3
We tested our LCTV by using similar methods. The modulation characteristics vary from one LCTV panel to other. The maximum phase retardation Φmax of 1.3π to 2.4π was obtained when the brightness control voltage was turned to the minimum and the linear polarization of the incident laser beam was parallel to the director of the LCTV molecules. Under this condition the maximum polarization rotation angle was Θmax = 60° for the tested panel.
The LCTV was immersed in a liquid gate to reduce random-phase errors caused by defects in the flatness of the panel. The LCTV was placed in the filter plane in a conventional 4-ƒ optical correlator. The two Fourier lenses were of focal length f = 300 mm. Neither a polarizer nor an analyzer was used in the correlator. The input image on a photographic film was illuminated by a parallel He-Ne laser beam of λ =
632.8 nm. The image was then reduced by an optical system so that the size of its Fourier transform matched the size of a square of 200 × 225 pixels on the LCTV. The filter was generated in a computer as a 200 × 200 matrix and then converted into a 200 × 225 matrix before being fed to the LCTV input by a frame grabber.
The LCTV permits continuous phase modulation. However, in practice we used a limited number of phase levels, because when the number of the phase levels N ≥ 16, no significant improvement in the filter performance can be obtained by a further increase of N.
We use the theory of Goodman and Silvestri7 and Dallas8 to analyze the multilevel POF with phase mismatching and coupled amplitude modulation. Assume that a POF, exp[j(u, v)], with 0 ≤ ≤ 2π, is quantized in the computer and fed to the LCTV. The optical filter displayed on the LCTV would have phase modulation exp[j'(u, v)] and coupled amplitude modulation cos[θ'(w, υ)]. Both the phase and amplitude modulations are controlled through a single video signal. Therefore the optical filter is piecewise constant over the intervals of quantization of (u, v) as
when
where k = 0, 1, . . . , N - 1 and N is the number of phase quantization levels. Because of the phase mismatching, we have 0 ≤ '(u, v) ≤ Φmax, where Φmax = 2πc(N - 1)/N and c < 1 is the phase-matching factor. The coupled rotation of polarization is 0 ≤ θ'(u, v) ≤ θmax, where Θmax = Θ(N - 1)/N is the maximum polarization rotation angle. Here, we assume that both the phase retardation and the rotation angle of polarization are approximately linear to the video signal. According to Eq. (1) the optical filter on the LCTV is a function of (u, υ):
The function is periodic in (u, υ) of period 2π. We expand F'(u, υ) into a Fourier series of exp[jm(u, υ)]. First, the rect function may be expanded as
Substituting Eq. (3) into Eq. (2), we obtain the optical filter on the LCTV as
When there is no phase mismatching and coupled amplitude modulation, c = 1 and Θ = 0, respectively, Eq. (4) describes a classical two-dimensional multilevel phase hologram. Its component of order m = 1,
is the ideal POF with the light efficiency [sinc(l/N)]2. The response of the ideal POF,
is usually the contour of the image because of the high-pass-filter nature of the POF. The zero-order component with m = 0 is equal to zero.
When the phase-matching factor c ≠ 1 and the coupled amplitude modulation Θ = 0, the first-order component with m = 1 in Eq. (4) is
which is also the ideal POF with reduced light efficiency. When c ≠ 0 and Θ ≠ 0, the optical filter still contains the component of m = 1 corresponding to the ideal POF, but the light efficiency would be further reduced. Other components in Eq. (4) with m ≠ 1 would not be zero. The zero order yields a central spot, the order m = - 1 corresponds to the conjugate image, and the orders m = ±2 correspond to the convolution images ƒ * f. The optical filter on the LCTV is an on-axis hologram with many superimposed diffraction orders.
The above theory is based on the assumption of linear phase and amplitude modulation. If the phase and amplitude modulations are not linear to (u, v), the analytical expression of Eq. (4) cannot be obtained. But the filter F'(u,v) is still a periodic function of (w, υ). Its Fourier series still contains the term m = 1 corresponding to the ideal POF. This filter can be studied by computer simulation and optical experiments.
To evaluate the noise associated with the orders m ≠ 1 in Eq. (4), we consider a one-dimensional (1-D) multilevel phase grating with the phase mismatching and the coupled amplitude modulation. The 1-D grating yields separated diffraction orders whose intensities may be optically measured. We calculated the theoretical coefficients in Eq. (4) and the
intensities of the diffraction orders of the 1-D grating by computer simulation. We measured the intensities of the diffraction orders of the 1-D grating optically implemented on the LCTV. The results are given in Table 1. The theory and the computer simulation yielded the same results. When phase level N = 16, phase-matching factor c = 0.7, and coupled rotation of polarization Θmax = 60°, ~68% of the energy goes to the required first-order POF. The zero order takes ~21% of the energy, and the rest (11%) goes to other orders. The optical intensity distribution among the diffraction orders roughly agreed with that given by the theory and the computer simulation. The optical intensities were measured in the on-axis diffraction pattern. In fact, the grid structure of the LCTV yields multiple separated diffraction patterns, reducing the light efficiency of the on-axis diffraction pattern. The dead interpixel region reduces the light passing through the LCTV to roughly (55 μm/80 μm)2. This additional light loss is not taken into account in the above theory and optical experiment.
We obtain 60%-70% energy efficiency of the first order of diffraction of the LCTV, which is the response of the POF. This efficiency is much higher than that of the binary POF [-1,1] (40%) and that of the unipolar binary POF [0, 1] (10%).9 The optical binary POF also suffers from an important additional loss of energy when it is implemented on a SLM.
Figure 1 shows an optically reconstructed image from a POF for an image of the space shuttle with N = 16 phase levels, coded on the LCTV with phase mismatching and phase and amplitude cross coupling. We see the response of the ideal POF, the zero-order spot, and other random noise. The conjugate image and ±2 orders are weak with respect to the first and zero orders. This is in agreement with the above results of the 1-D grating.
After we analyzed the response of the imperfect POF, our LCTV was ready for the optical correlator. The results of a three-dimensional (3-D) plot of the optical correlation output is presented in Fig. 2. The input was an image of the space shuttle. A sharp autocorrelation peak was obtained in a rather uniform noise background in the output plane. The peak intensity was equal to 62. The noise background intensity was ~25. We used the SNR de-
Table 1. Diffraction Efficiency of the Liquid-Crystal-Television-Encoded Phase-Only Filter
Fig. 1. Optical impulse response of a POF encoded in a LCTV.
fined by Horner and Bartelt as the correlation-peak value divided by the rms noise outside the full width at half-maximum of the correlation peak. This SNR was equal to 4.1. We considered the energy on the pixels with the intensities above the half-maximum of correlation-peak intensity as the energy of the correlation peak. This peak energy was ~ 80% of the total energy in the correlation output. The on-axis correlation peak was so bright that a neutral filter with density of 5 was used to avoid saturation of the CCD camera when the laser source was 10 mW.
The correlation peak was wider than the theoretical one-pixel-wide correlation peak of the POF owing to the noise in the response of the optical POF, shown in Fig. 1. Other sources of noise in the LCTV could be nonuniformity and nonlinearity of the phase modulation over the LCTV screen, noise of the electronic addressing, and optical misalignment.
The second experiment was with an input scene containing an image of the space shuttle and an image of an aircraft to demonstrate the discrimination of the optical POF. Figure 3 shows the optical correlation output. The autocorrelation-peak intensity for the space shuttle was equal to 54 and the cross-correlation maximum for the aircraft was equal to 41, which can be eliminated by thresholding. Note that the image of the aircraft was similar to that of the space shuttle.
In summary, we have built an on-axis phase-only VanderLugt-type correlator that uses a commercial LCTV with phase mismatching and phase and amplitude cross coupling. The theory, the computer simulation, and the optical experiment show that the
Fig. 2. 3-D plot of optical autocorrelation output; the input image was the space shuttle.
References
Fig. 3. 3-D plot of optical autocorrelation output; the input image was an image of the space shuttle plus an image of an aircraft.
optical filter coded on the LCTV contains the ideal POF component that yields a sharp autocorrelation peak with high light efficiency and utilizes all the available space-bandwidth product of the LCTV. The phase mismatching, the coupled amplitude modulation, and other source of noise can reduce the SNR of the correlation output and widen the correlation peak. This simple and low-cost optical correlator will have many interesting applications for optical pattern recognition and optical computing.
1. J. L. Horner and P. D. Gianino, "Phase-only matched filtering," Appl. Opt. 23, 812-816 (1984).
2. H. K. Liu, J. A. Davis, and R. A. Lilly, "Optical-data-processing properties of a liquid-crystal television spatial light modulator," Opt. Lett. 10, 635-637 (1985).
3. J. C. Kirsch, D. A. Gregory, M. W. Thie, and B. K. Jones, "Modulation characteristics of the Epson liquid crystal television," Opt. Eng. 31, 963-969 (1992).
4. J. Amako and T. Sonehara, "Kinoform using an electrically controlled birefringent liquid-crystal spatial light modulator," Appl. Opt. 30, 4622-4628 (1991).
5. R. Juday, "Correlation with a spatial light modulator having phase and amplitude cross coupling," Appl. Opt. 28, 4865-4869 (1989).
6. M. W. Farm and J. W. Goodman, "Optimal maximum correlation filter for arbitrarily constrained devices," Appl. Opt. 28, 3362-3366(1989).
7. J. W. Goodman and A. M. Silvestri, "Some effects of Fourier-domain phase quantization," IBM J. Res. Dev. 478-484 (1970).
8. W. J. Dallas, "Phase-quantization: a compact derivation," Appl. Opt. 10, 673-674 (1971).
9. L. Leclerc, Y. Sheng, and H. H. Arsenault, "Optical binary phase-only filters for circular harmonic correlations," Appl. Opt. 30, 4643-4649 (1991).
