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Filter plane resolution requirements for coherent optical correlators David Casasent and Alan Furman Carnegie-Mellon University, Department of Electrical Engineering, Pittsburgh, Pennsylvania 15213. Received 19 July 1976. The joint transform correlator 1 has recently received con- siderable attention because it is compatible with reflex mode readout real-time devices. 2 This system and the classic fre- quency plane (or Vander Lugt) correlator 3 have recently been compared. We report here on the fact that the resolution requirements of the transform plane material in the frequency plant correlator (FPC) are always less than or equal to those of the joint transform correlator (JTC). In the FTC, the interference of the Fourier transform of a reference function g (χ) (1-D functions are used only for simplicity) and a plane wave at angle θ is recorded at P 1 . With f(x) present at the input plane P 0 , its transform F(u) formed by lens L 1 is incident on P 1 ; and the output pattern at P 2 formed by L 2 is where * denotes convolution, ® denotes correlation, a = (sinθ)/λ, and fl, is the focal length of L 1 or L 2 . From Eq. (1), the on-axis terms at P 2 have width 2 W g + W f , where W g and W f are the widths of g(x) and f(x). The terms at χ = ±aλfL have width W f + W g . For the case of W f = W g = W, sepa- ration of the terms at χ = ±aλfL from the on-axis terms re- quires a ≥ 2.5Wλ fL . When this criterion is followed, the total width of the output correlation plane pattern is 7 W. In the JTC the two functions f(x) and g(x) are placed side-by-side at P 0 with a center-to-center spacing of 2b, and the interference of their overlapping transforms (formed by L 1 ) is recorded at P 1 . When P 1 is subsequently illuminated with a plane wave, the Fourier transform of its transmittance formed by L 2 at P 2 is From Eq. (2) we see that the width of the on-axis term is now 2W g or 2W f or 2W for W g = W f , and the width of the corre- lation terms at χ ±26 is W g + W f = 2W for W g = W f . To separate these terms, we require b ≥ W, from which the maximum spatial frequency recorded at P 1 must be 2Wλfz, and the total width of the pattern in the correlation plane is 6W. If one assumes that the bandwidth required in the Fourier transform plane is proportional to the width of the pattern in the correlation plane, 2 then the resolution required at P 1 in the FPC appears to be higher than in the JTC. However this is not true. The difference is due to the fact that P 1 in the JTC is illuminated by a plane wave, and the correlation pat- tern is thus the impulse response of the filter at P 1 . Hence the total width of the correlation pattern reflects the resolu- tion required at P 1 . However, P 1 in the FPC is illuminated by the transform F(u) of a function f(x) of width W rather than a plane wave. The pattern recorded at P 1 in the FPC is equivalent to the joint transform of a delta function and g(x) with a center-to-center spacing of 2.5W. The impulse re- sponse for the filter recorded at P 1 consists of an on-axis term of width 2W and first-order terms of width W separated from the on-axis terms by W, for a total width of 6W for the impulse response. Since the widths of the impulse responses of the recordings at P 1 are the same for both systems, the resolutions February 1977 / Vol. 16, No. 2 / APPLIED OPTICS 285
