Area-Modulated Signal Recordings for Coherent Optical Correlators B. J. Pernick

Grumman Aerospace Corporation, Bethpage, New York 11714. Received 24 June 1971.

In many optical signal processing systems the input signal to be analyzed is usually recorded as an amplitude transmittance varia­tion within an aperture area of fixed dimensions. The use of a different recording technique, a variable area-modulation input format, has been demonstrated for optical Fourier transforma­tions by Felstead.1 With an area-modulated format, only the aperture width is varied in accordance with the input signal. The amplitude transmittance within the aperture is not altered or related to the input signal. Felstead has also described several types of area-modulated recordings and indicated an important advantage common to all his recording methods, i.e., the non-linearities of the recording material do not influence the system performance. It will be shown in this note that area-modulated recording techniques can also be adapted with further advantage to coherent optical correlators.

In general, offset or bias levels in both input and reference signal recordings of bipolar signals are needed when the signals are written as unipolar amplitude transmittance variations. This requirement introduces a major source of error in the correlation measurement from a conventional three-lens correlator.2 The desired correlation and measured quantity differ by additive terms that depend upon the bias levels. Consider an optical

correlator in which both bipolar signals are contained as unipolar, variable transmittance functions. Now the influence of the bias level contained in one signal record (e.g., input signal) can be removed from the output correlation measurement by spatially filtering its optical Fourier transform with a dc stop.2 The transform spectrum and dc stop are located in the back focal plane of the first lens in the correlator. The filtered optical spectrum is then retransformed and projected with a second imaging lens onto the reference function. The correlation of these superposed signals is measured on axis in the back focal plane of the transform lens following the reference function. However, this correlation value still contains a bias error intro­duced by a unipolar recording of the bipolar reference signal.

One of the several types of area-modulated recordings can be used to store the reference function and eliminate the remaining error. Let the reference function a(x) be stored in the correlator as an unbiased-with-phase-plate recording. (The notation of Ref. 1 is used herein.) The input signal f (x – x0 ) , recorded as a variable transmittance function, is spatially filtered with a dc stop and projected onto this reference as above. The parameter x0 denotes an aribitrary shift in the x direction. The frequency spectrum of the product f(x — x0)a(x) is realized as the system output after the transform lens. The output amplitude distribu­tion along one axis in the back focal plane of the transform lens is now

The desired correlation, free of bias errors, is measured on axis, i.e., at the spatial frequency fx = 0.

As an alternate method, one can also record the bipolar input f(x – x0) as an unbiased-with-phase-plate recording. In this case no intermediate dc spatial filtering is required. Again, the measured correlation does not include bias error terms.

A simplified correlator system utilizing a hybrid combination of area-modulation and amplitude transmittance recording methods is of value if unipolar signals are to be processed. In this case both signals are recorded in different formats onto a single input device rather than on two separate input units placed in close proximity.3 The need to align carefully and register two input devices is thereby avoided. For example, the input signal f(x – x0), is recorded as a variable transmittance pattern in the x-direction and a constant transmittance in the y direction. The reference function, a(x), is recorded as a variable area-modulation signal superposed upon this transmittance pattern. Two of the four forms of area-modulation described in Ref. 1 are considered, namely, biased bilateral and biased unilateral modulation. Since the signals to be correlated are assumed to be unipolar, pro­cessing with the unbiased-with-phase-plate format is not needed. Use of the fourth mentioned area-modulation scheme is not con­sidered here. {The reader has pointed out that a correlator of the form described by C. Atzeni and L. Pantani, Proc. IEEE 57, 344 (1969), has been operated using this method of area-modulation [E. B. Felstead, "Some Optical Correlations," Research Report 70-2, Electrical Engineering Department, Queens University, Kingston, Ontario, April (1970)].} The amplitude transmittance for each of the first two forms is now given by

where both f (x) and a(x) are ≥ 0. If the record containing either t1(x,y) or t2(x,y) is used in an

optical Fourier transform system, the desired cross correlation is obtained from a measure of the light intensity in the back focal plane of the transform lens, i.e., at spatial frequencies fx = 0 = fy one has

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Thus a single lens transform system with one mixed format data record can be used in place of a conventional three lens optical correlator. Unfortunately, linearity problems associated with recording one of the signals as an amplitude transmittance varia­tion still remain. Nevertheless, in conjunction with a real time transducer to record the signals, such a single lens system offers significant and practical component simplification.

One method to obtain dual signal recordings onto a single input device is suggested. Simultaneous mixed format recording in a low- to mid-frequency range is achieved with a modulated cathode ray tube display4 and at high frequencies with a modulated laser or electron beam recording technique.5 For example, the input signal to be inscribed as a transmittance pattern is applied to the z axis of a CRT for beam intensity modulation. The reference signal that provides biased-bilateral area modulation is first used to amplitude modulate a high frequency carrier. This amplitude modulation signal in turn drives the vertical input or y deflection circuitry of the CRT. For recording fidelity one chooses a carrier frequency greater than the maximum frequency compo­nent contained in either signal. The varying vertical motion of the modulated CRT beam is then projected onto the recording transducer. The CRT beam can be moved horizontally (in the x direction) to record a length of the signals.

References

1. E. B. Felstead, Appl. Opt. 10, 2468 (1971). 2. A. R. Shulman, Optical Data Processing (Wiley, New York, 1970, pp. 223-230. 3. G. Meltz and W. T. Maloney, Appl. Opt. 7, 2091 (1968). 4. B. J. Pernick, D. Yustein, and C. Bartolotta, Appl. Opt. 8, 65 (1969). 5. C. J. Peters, Proc. IEEE 58, 886 (June 1970).

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