Optical systems for combined 1-D image–orthogonal Fourier transform processing

B. J. Pernick

doi:10.1364/AO.19.000754

Applied Optics
Vol. 19,
Issue 5,
pp. 754-760
(1980)
•https://doi.org/10.1364/AO.19.000754

Optical systems for combined 1-D image–orthogonal Fourier transform processing

B. J. Pernick

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B. J. Pernick, "Optical systems for combined 1-D image–orthogonal Fourier transform processing," Appl. Opt. 19, 754-760 (1980)

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Abstract

A number of spherical and cylindrical lens combinations are described to achieve a 1-D image and an optical Fourier transform normal to the imaging direction. Both output distributions must be coplanar. Optical features dealt with are the image and transform magnification, lens separation, and input-to-output plane distance. Variable magnification setups with fixed focal length combinations are given.

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Fig. No.	Range of Magnification M	Input–Output Distance D	Lens Separation S	Transform Scale Factor	Zero Quadratic Phase Factor in Transform	Comments

1	M = − f_s/f_c < 0	f_c + f_s + S	-	f_s	No	Arbitrary S

	M = − f_s/f_c < 0	f_c + f_s	0	f_s	No	Zero Lens Spacing

	M = −1	2 f_s	0	f_s	Yes	f_c = f_s

2	M = − f_s/f_c ≤ − 1	2 f_s	f_s − f_c	f_s	Yes	f_s ≥ f_c

	M = − f_c/f_s	2 f_s	f_s − f_c	f_s	Yes	f_s ≥ f_c
	− 1 ≤ M < 0	2 f_s	f_s − f_c	f_s	Yes	f_s ≥ f_c

3	$- \frac{f_{s}}{f_{c}} or - \frac{f_{c}}{f_{s}} < M < 0$	$2 f_{s} - \frac{f_{s} (f_{s} + f_{c} / M)}{f_{c} + \sqrt{(- M) f_{s} f_{c}}}$	$f_{s} - \sqrt{(- M) f_{s} f_{c}}$	f_s	No	D > 2 f_s
3	$- \frac{f_{s}}{f_{c}} or - \frac{f_{c}}{f_{s}} < M < 0$		$f_{s} - \sqrt{(- M) f_{s} f_{c}}$	f_s	No	f_c > f_c − S

4	$- \frac{f_{s}}{f_{c}} < M < - \frac{f_{c}}{f_{s}}$	$2 f_{s} - \frac{f_{s} (f_{s} + f_{c} / M)}{f_{c} + \sqrt{(- M) f_{s} f_{c}}}$	$f_{s} - \sqrt{(- M) f_{s} f_{c}}$	f_s	No	D < 2 f_s
4	$- \frac{f_{s}}{f_{c}} < M < - \frac{f_{c}}{f_{s}}$		$f_{s} - \sqrt{(- M) f_{s} f_{c}}$	f_s	No	f_s > f_c

5	M < − 4 f_c/f_s	f_s(2 − M − 1/M)	$f_{s} + f_{c} - \frac{f_{c}}{M_{T}}$	\|M_T\| f_s	No	Two value for M_T from Eq. (27), with M_T < 0

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Applied Optics