Optical sparse aperture imaging

Nicholas J. Miller; Matthew P. Dierking; Bradley D. Duncan

doi:10.1364/AO.46.005933

Applied Optics
Vol. 46,
Issue 23,
pp. 5933-5943
(2007)
•https://doi.org/10.1364/AO.46.005933

Optical sparse aperture imaging

Nicholas J. Miller, Matthew P. Dierking, and Bradley D. Duncan

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Nicholas J. Miller, Matthew P. Dierking, and Bradley D. Duncan, "Optical sparse aperture imaging," Appl. Opt. 46, 5933-5943 (2007)

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Abstract

The resolution of a conventional diffraction-limited imaging system is proportional to its pupil diameter. A primary goal of sparse aperture imaging is to enhance resolution while minimizing the total light collection area; the latter being desirable, in part, because of the cost of large, monolithic apertures. Performance metrics are defined and used to evaluate several sparse aperture arrays constructed from multiple, identical, circular subapertures. Subaperture piston and∕or tilt effects on image quality are also considered. We selected arrays with compact nonredundant autocorrelations first described by Golay. We vary both the number of subapertures and their relative spacings to arrive at an optimized array. We report the results of an experiment in which we synthesized an image from multiple subaperture pupil fields by masking a large lens with a Golay array. For this experiment we imaged a slant edge feature of an ISO12233 resolution target in order to measure the modulation transfer function. We note the contrast reduction inherent in images formed through sparse aperture arrays and demonstrate the use of a Wiener–Helstrom filter to restore contrast in our experimental images. Finally, we describe a method to synthesize images from multiple subaperture focal plane intensity images using a phase retrieval algorithm to obtain estimates of subaperture pupil fields. Experimental results from synthesizing an image of a point object from multiple subaperture images are presented, and weaknesses of the phase retrieval method for this application are discussed.

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Golay-3	$(- \frac{1}{2}, - \frac{\sqrt{3}}{6})$	$(\frac{1}{2}, - \frac{\sqrt{3}}{6})$	$(0, \frac{\sqrt{3}}{3})$
Golay-6	$(0, - \frac{2 \sqrt{3}}{3})$	$(1, - \frac{2 \sqrt{3}}{3})$	$(1, \frac{\sqrt{3}}{3})$
	$(\frac{1}{2}, - \frac{5 \sqrt{3}}{6})$	$(- 1, - \frac{\sqrt{3}}{3})$	$(- \frac{3}{2}, \frac{- \sqrt{3}}{6})$
Golay-9	$(- \frac{1}{2}, - \frac{\sqrt{3}}{6})$	$(\frac{1}{2}, - \frac{\sqrt{3}}{6})$	$(0, \frac{\sqrt{3}}{3})$
	$(\frac{7}{2}, - \frac{\sqrt{3}}{6})$	$(\frac{3}{2}, \frac{5 \sqrt{3}}{6})$	$(- \frac{3}{2}, \frac{11 \sqrt{3}}{6})$
	$(- 2, \frac{\sqrt{3}}{3})$	$(- 2, - \frac{5 \sqrt{3}}{3})$	$(\frac{1}{2}, - \frac{7 \sqrt{3}}{6})$
Golay-12	$(0, - \sqrt{3})$	$(- 2, - \sqrt{3})$	$(- \frac{1}{2}, - \frac{3 \sqrt{3}}{2})$
	$(- 4, - 2 \sqrt{3})$	$(5, - \sqrt{3})$	$(\frac{5}{2}, - \frac{\sqrt{3}}{2})$
	$(\frac{5}{2}, \frac{\sqrt{3}}{2})$	$(\frac{3}{2}, \frac{\sqrt{3}}{2})$	$(- \frac{1}{2}, \frac{3 \sqrt{3}}{2})$
	$(- \frac{3}{2}, \frac{\sqrt{3}}{2})$	$(- 2, \sqrt{3})$	$(- 1, 3 \sqrt{3})$

Array	Expansion Factor (s)	D _{circumscribed}	FWHM_PSF	PISLR [dB]	ρ_min(λf)	ρ_max(λf)	MTF_midfreq	Fill Factor (α) (%)
Golay-9	1.0	8.02	$δ_{0}$ ∕6.98	−6.14	4.36	7.08	0.089	47.2
Golay-9	1.1	8.73	$δ_{0}$ ∕7.58	−7.09	4.71	7.68	0.076	40.6
Golay-9	1.2	9.43	$δ_{0}$ ∕8.16	−7.94	5.02	8.29	0.067	35.7
Golay-9	1.3	10.1	$δ_{0}$ ∕8.77	−8.70	5.32	8.90	0.059	31.9
Golay-9	1.4	10.8	$δ_{0}$ ∕9.38	−9.39	5.68	9.50	0.052	27.9
Golay-9	1.5	11.5	$δ_{0}$ ∕9.98	−10.0	5.92	10.1	0.046	25.7
Golay-9	1.6	12.2	$δ_{0}$ ∕10.6	−10.6	6.23	10.8	0.041	23.2
Golay-9	1.7	12.9	$δ_{0}$ ∕12.7	−11.4	2.12	11.4	Voids	—
Golay-9	1.8	13.6	$δ_{0}$ ∕13.4	−11.9	1.66	12.2	Voids	—
Golay-9	1.9	14.3	$δ_{0}$ ∕14.1	−12.4	1.20	12.7	Voids	—
Golay-9	2.0	15.0	$δ_{0}$ ∕14.8	−12.8	1.18	13.2	Voids	—

Array	Expansion Factor (s)	D _{circumscribed}	FWHM_PSF	PISLR [dB]	ρ_min(λf)	ρ_max(λf)	MTF_midfreq	Fill Factor (α)
Single	NA	1.00	$δ_{0}$	7.13	1.00	1.00	NA	1.00
Golay-3	1.6	2.85	$δ_{0}$ ∕2.92	−2.92	1.99	2.60	0.136	75.8%
Golay-6	1.5	5.58	$δ_{0}$ ∕5.83	−7.52	3.98	4.96	0.067	37.9%
Golay-9	1.4	10.8	$δ_{0}$ ∕9.38	−9.39	5.68	9.50	0.052	27.9%
Golay-12	1.3	14.8	$δ_{0}$ ∕12.7	−10.6	6.59	12.9	0.045	27.6%

Golay-3	$(- \frac{1}{2}, - \frac{\sqrt{3}}{6})$	$(\frac{1}{2}, - \frac{\sqrt{3}}{6})$	$(0, \frac{\sqrt{3}}{3})$
Golay-6	$(0, - \frac{2 \sqrt{3}}{3})$	$(1, - \frac{2 \sqrt{3}}{3})$	$(1, \frac{\sqrt{3}}{3})$
	$(\frac{1}{2}, - \frac{5 \sqrt{3}}{6})$	$(- 1, - \frac{\sqrt{3}}{3})$	$(- \frac{3}{2}, \frac{- \sqrt{3}}{6})$
Golay-9	$(- \frac{1}{2}, - \frac{\sqrt{3}}{6})$	$(\frac{1}{2}, - \frac{\sqrt{3}}{6})$	$(0, \frac{\sqrt{3}}{3})$
	$(\frac{7}{2}, - \frac{\sqrt{3}}{6})$	$(\frac{3}{2}, \frac{5 \sqrt{3}}{6})$	$(- \frac{3}{2}, \frac{11 \sqrt{3}}{6})$
	$(- 2, \frac{\sqrt{3}}{3})$	$(- 2, - \frac{5 \sqrt{3}}{3})$	$(\frac{1}{2}, - \frac{7 \sqrt{3}}{6})$
Golay-12	$(0, - \sqrt{3})$	$(- 2, - \sqrt{3})$	$(- \frac{1}{2}, - \frac{3 \sqrt{3}}{2})$
	$(- 4, - 2 \sqrt{3})$	$(5, - \sqrt{3})$	$(\frac{5}{2}, - \frac{\sqrt{3}}{2})$
	$(\frac{5}{2}, \frac{\sqrt{3}}{2})$	$(\frac{3}{2}, \frac{\sqrt{3}}{2})$	$(- \frac{1}{2}, \frac{3 \sqrt{3}}{2})$
	$(- \frac{3}{2}, \frac{\sqrt{3}}{2})$	$(- 2, \sqrt{3})$	$(- 1, 3 \sqrt{3})$

Array	Expansion Factor (s)	D _{circumscribed}	FWHM_PSF	PISLR [dB]	ρ_min(λf)	ρ_max(λf)	MTF_midfreq	Fill Factor (α) (%)
Golay-9	1.0	8.02	$δ_{0}$ ∕6.98	−6.14	4.36	7.08	0.089	47.2
Golay-9	1.1	8.73	$δ_{0}$ ∕7.58	−7.09	4.71	7.68	0.076	40.6
Golay-9	1.2	9.43	$δ_{0}$ ∕8.16	−7.94	5.02	8.29	0.067	35.7
Golay-9	1.3	10.1	$δ_{0}$ ∕8.77	−8.70	5.32	8.90	0.059	31.9
Golay-9	1.4	10.8	$δ_{0}$ ∕9.38	−9.39	5.68	9.50	0.052	27.9
Golay-9	1.5	11.5	$δ_{0}$ ∕9.98	−10.0	5.92	10.1	0.046	25.7
Golay-9	1.6	12.2	$δ_{0}$ ∕10.6	−10.6	6.23	10.8	0.041	23.2
Golay-9	1.7	12.9	$δ_{0}$ ∕12.7	−11.4	2.12	11.4	Voids	—
Golay-9	1.8	13.6	$δ_{0}$ ∕13.4	−11.9	1.66	12.2	Voids	—
Golay-9	1.9	14.3	$δ_{0}$ ∕14.1	−12.4	1.20	12.7	Voids	—
Golay-9	2.0	15.0	$δ_{0}$ ∕14.8	−12.8	1.18	13.2	Voids	—

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Applied Optics