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.

© 2007 Optical Society of America

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References

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2003

J. D. Monnier, "Optical interferometry in astronomy," Rep. Prog. Phy. 66, 789-857 (2003).
[CrossRef]

2002

R. D. Fiete, T. Tantalo, J. R. Calus, and J. A. Mooney, "Image quality of sparse-aperture designs for remote sensing," in Opt. Eng. 41, 1957-1969 (2002).
[CrossRef]

S.-J. Chung, D. W. Miller, and O. L. deWeck, "Design and implementation of sparse aperture imaging systems," in Highly Innovative Space Telescope Concepts, H. A. MacEwen, ed., Proc. SPIE 4849, 181-192 (2002).
[CrossRef]

2000

J. R. Fienup, "MTF and integration time versus fill factor for sparse-aperture imaging systems," in Imaging Technology and Telescopes, J. W. Bilbro, J. B. Breckinridge, R. A. Carreras, S. R. Czyzak, M. J. Eckert, R. D. Fiete, and P. S. Idell, eds., Proc. SPIE 4091, 43-47 (2000).
[CrossRef]

1999

1997

E. Keto, "The shapes of cross-correlation interferometers," Astrophys. J. 475, 843-852 (1997).
[CrossRef]

1996

1989

1988

J. E. Harvey and R. A. Rockwell, "Performance characteristics of phased array and thinned aperture telescopes," Opt. Eng. 27, 762-768 (1988).

S. M. Watson, J. P. Mills, and S. K. Rogers, "Two-point resolution criterion for multiaperture optical telescopes," J. Opt. Soc. Am. A 5, 893-903 (1988).
[CrossRef]

1987

1982

1971

M. J. E. Golay, "Point arrays having compact nonredundant autocorrelations," J. Opt. Soc. America 61, 272-273 (1971).
[CrossRef]

1969

1967

1965

Appl. Opt.

Astrophys. J.

E. Keto, "The shapes of cross-correlation interferometers," Astrophys. J. 475, 843-852 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. America

M. J. E. Golay, "Point arrays having compact nonredundant autocorrelations," J. Opt. Soc. America 61, 272-273 (1971).
[CrossRef]

Opt. Eng.

R. D. Fiete, T. Tantalo, J. R. Calus, and J. A. Mooney, "Image quality of sparse-aperture designs for remote sensing," in Opt. Eng. 41, 1957-1969 (2002).
[CrossRef]

J. E. Harvey and R. A. Rockwell, "Performance characteristics of phased array and thinned aperture telescopes," Opt. Eng. 27, 762-768 (1988).

Proc. SPIE

S.-J. Chung, D. W. Miller, and O. L. deWeck, "Design and implementation of sparse aperture imaging systems," in Highly Innovative Space Telescope Concepts, H. A. MacEwen, ed., Proc. SPIE 4849, 181-192 (2002).
[CrossRef]

J. R. Fienup, "MTF and integration time versus fill factor for sparse-aperture imaging systems," in Imaging Technology and Telescopes, J. W. Bilbro, J. B. Breckinridge, R. A. Carreras, S. R. Czyzak, M. J. Eckert, R. D. Fiete, and P. S. Idell, eds., Proc. SPIE 4091, 43-47 (2000).
[CrossRef]

Rep. Prog. Phy.

J. D. Monnier, "Optical interferometry in astronomy," Rep. Prog. Phy. 66, 789-857 (2003).
[CrossRef]

Other

J. W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985).

A. A. Michelson, Studies in Optics (University of Chicago Press, 1927).

P. Burns, Spatial Frequency Response code written for Matlab, sfrmat2: version 2.1 (International Imaging Industry Association, 2003).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

ISO 12233, "Photography--electronic still-picture cameras--resolution measurements" (International Organization for Standardization, 2000).

N. J. Miller, B. D. Duncan, and M. P. Dierking, "Resolution enhanced sparse aperture imaging," in Proceedings of IEEE Aerospace Conference (IEEE, 2006) IEEEAC paper 1406.

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Figures (17)

Fig. 1
Fig. 1

Golay-4 array with expansion factor of 1.6.

Fig. 2
Fig. 2

Resolution metrics for a sparse array consisting of four identical, circular subapertures.

Fig. 3
Fig. 3

Threefold symmetric Golay arrays with compact nonredundant autocorrelations. Top row: Point array configurations. Bottom row: Associated autocorrelations.

Fig. 4
Fig. 4

PSF and MTF for a Golay-3 array with an optimum expansion factor of 1.6.

Fig. 5
Fig. 5

PSF and MTF for a Golay-6 array with an optimum expansion factor of 1.5.

Fig. 6
Fig. 6

PSF and MTF for a Golay-9 array with an optimum expansion factor of 1.4.

Fig. 7
Fig. 7

PSF and MTF for a Golay-12 array with an optimum expansion factor of 1.3.

Fig. 8
Fig. 8

Golay-9 array geometry for an optimum expansion factor of 1.4.

Fig. 9
Fig. 9

PSF and MTF of the optimized Golay-9 array with λ / 2 piston added to one subaperture.

Fig. 10
Fig. 10

PSF and MTF of the optimized Golay-9 array with λ tilt added to one subaperture.

Fig. 11
Fig. 11

Imaging of an incoherently illuminated resolution target through a Golay aperture array.

Fig. 12
Fig. 12

Measured and theoretical MTF cross-sections for the optimized Golay-9 array ( s = 1.4 ) .

Fig. 13
Fig. 13

A raw Golay-9 image (left) and its Wiener restored image (right).

Fig. 14
Fig. 14

(a) Image synthesis using a single real imaging lens. (b) Postdetection image synthesis using a virtual imaging lens.

Fig. 15
Fig. 15

Imaging of an infinite point object with a fixed Golay aperture array.

Fig. 16
Fig. 16

Pupil plane phase and intensity retrieval for subaperture B of the Golay-3 array shown at left.

Fig. 17
Fig. 17

Postdetection synthesis of Golay-3 intensity images.

Tables (3)

Tables Icon

Table 1 Threefold Symmetric Golay Array Subaperature Center Coordinates ( s = 1)

Tables Icon

Table 2 Golay-9 Array Metrics as a Function of Expansion Factor s

Tables Icon

Table 3 Quality Measures of Optimized N Element Golay Arrays

Equations (12)

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α = N D e f f 2 ,
P a r r a y ( x , y ) = n = 1 N P s u b ( x x n , y y n ) e j ϕ n ( x , y ) ,
PSF a r r a y ( u , v ) = PSF s u b ( u , v ) [ N + 2 k = 1 N ( N 1 ) / 2 × cos [ 2 π λ f ( Δ x k u + Δ y k v ) ] ] ,
D e f f ( P S F ) = δ 0 FWHM a r r a y P S F = 1.03 λ f FWHM a r r a y P S F ,
ω p e a k = 2.44 λ f D e f f .
PISLR = 10   log [ 0 2 π 0 ω p e a k PSF a r r a y ( ρ , ϕ ) ρ d ρ d ϕ 0 2 π ω p e a k PSF a r r a y ( ρ , ϕ ) ρ d ρ d ϕ ] ,
OTF ( f x , f y ) { PSF a r r a y ( u , v ) } PSF a r r a y ( u , v ) d u d v ,
MTF a r r a y ( f x , f y ) = MTF s u b ( f x , f y ) [ δ ( f x , f y ) + 1 N k = 1 N ( N 1 ) / 2 × δ ( f x ± Δ x k λ f , f y ± Δ y k λ f ) ] ,
MTF s u b ( ρ ) = { 2 π [ arccos ( λ f D ρ ) ( λ f D ρ ) 1 ( λ f D ρ ) 2 ] for   ρ D λ f 0 for   ρ > D λ f ,
D e f f ( M T F ) = ρ min ρ o = ρ min λ f ,
MTF m i d f r e q = 1 2 π ( ρ min 2 1 ) 0 2 π 1 ρ min MTF ( ρ , ϕ ) ρ d ρ d ϕ .
W ( f x , f y ) = * ( f x , f y ) | ( f x , f y ) | 2 + K ,

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