Abstract

In this paper, we propose and demonstrate the synthetic aperture imaging by using spatial modulation diversity technology with stochastic parallel gradient descent (SPGD) algorithm. Instead of creating diversity images by means of focus adjustments, the technology, proposed in this paper, creates diversity images by modulating the transmittance of individual sub-aperture of multi-aperture system, respectively. Specifically, spatial modulation is realized by switching off the transmittance of each sub-aperture with electrical shutters, alternately. Based on these multi diversity images, SPGD algorithm is used for adaptively optimizing the coefficients of Zernike polynomials to reconstruct the real phase distortions of multi-aperture system and to restore the near-diffraction-limited image of object. Numerical simulation and experimental results show that this technology can be used for joint estimation of both pupil aberrations and an high resolution image of the object, successfully. The technology proposed in this paper can have wide applications in segmented and multi-aperture imaging systems.

© 2015 Optical Society of America

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References

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2012 (2)

2010 (1)

2009 (1)

2005 (1)

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 58960G (2005).

2003 (1)

1998 (1)

1997 (1)

1992 (1)

1988 (1)

1979 (1)

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

1976 (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976).
[Crossref]

Blanc, A.

Bolcar, M. R.

M. R. Bolcar and J. R. Fienup, “Sub-aperture piston phase diversity for segmented and multi-aperture systems,” Appl. Opt. 48(1), A5–A12 (2009).
[Crossref] [PubMed]

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 58960G (2005).

Chidlaw, R.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Crosby, E. R.

Dente, G. C.

Fienup, J. R.

Gonsalves, R. A.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Gunturk, B. K.

Idier, J.

Jameson, D.

Lee, D. J.

Miller, N. J.

Mugnier, L. M.

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976).
[Crossref]

Paxman, R. G.

Rabb, D.

Roggemann, M. C.

Schulz, T. J.

Sivokon, V. P.

Stafford, J.

Stokes, A.

Tilton, M. L.

Vorontsov, M. A.

Watson, E. A.

Welsh, B. M.

Appl. Opt. (3)

J. Opt. Soc. Am. A (5)

Opt. Express (2)

Proc. SPIE (2)

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 58960G (2005).

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Other (3)

M. R. Bolcar and J. R. Fienup, “Phase diversity with broadband illumination,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JTuA6.
[Crossref]

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2004).

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Figures (16)

Fig. 1
Fig. 1 The configuration of a synthetic aperture imaging system (two sub-apertures).
Fig. 2
Fig. 2 The four-aperture configuration and the USAF 1951 resolution test chart used in the numerical simulation, (a) the four-aperture configuration, (b) the USAF 1951 resolution test chart.
Fig. 3
Fig. 3 Phase distributions of imaging beams and four sub-apertures telescopes (the unit is radian), (a) phase aberration of imaging beams, (b) piston errors of the four sub-apertures telescopes ( 0.3π,0.2π,0.5π,0.9π ) , (c) the whole phase distortion of the synthetic aperture imaging system.
Fig. 4
Fig. 4 Image of the resolution test chart blurred by phase distortion.
Fig. 5
Fig. 5 Images of the resolution test chart obtained by using spatial modulation technology, (a) image formed by four sub-apertures with aperture 1 turned off, (b) image formed by four sub-apertures with aperture 2 turned off, (c) image formed by four sub-apertures with aperture 3 turned off, (d) image formed by four sub-apertures with aperture 4 turned off.
Fig. 6
Fig. 6 The results generated by the spatial modulation diversity technology, (a) corresponding evolutions of the relative value of metric function as the SPGD algorithm proceeds, (b) the restored phase distribution (the unit is radian).
Fig. 7
Fig. 7 The results generated by the spatial modulation diversity technology, (a) the difference between the restored phase distortion and the loaded phase distortion, the RMS value is about 0.3rad, (b) the reconstructed near-diffraction-limited image of the resolution test chart.
Fig. 8
Fig. 8 Defocus phase aberration used in the conventional focus adjustment phase diversity for focus adjustment (the unit is radian.), (a) with positive value, (b) with negative value.
Fig. 9
Fig. 9 The results generated by the conventional focus adjustment phase diversity, (a) the restored phase distribution, (b) the difference between the restored phase distribution and the load phase distortion, the RMS value is about 0.85rad.
Fig. 10
Fig. 10 Proof-of-principle experimental setup for proving the spatial modulation diversity technology.
Fig. 11
Fig. 11 Phase distribution loaded on the two apertures.
Fig. 12
Fig. 12 Images of the resolution test chart formed by two aperture diaphragm, (a) image of the resolution test chart without being blurred by phase distortion of the phase screen, (b) image of the resolution test chart blurred by phase distortion of the phase screen.
Fig. 13
Fig. 13 Images of the resolution test chart obtained by using spatial modulation technology, (a) image formed by two sub-apertures with left sub-aperture turned off, (b) image formed by two sub-apertures with right sub-aperture turned off.
Fig. 14
Fig. 14 Experimental results generated by spatial modulation diversity technology, (a) corresponding evolutions of the relative value of metric function as the SPGD algorithm proceeds, (b) phase distribution of the imaging system (the unit is raidan).
Fig. 15
Fig. 15 The phase errors between the restored phase distortion and loaded phase distortion (the unit is radian), (a) the phase error without excluding the tilted errors, the PV value is about 0.78λ, the RMS value is about 0.2λ, (b) the phase error with excluding the tilted errors, the PV value is about 0.36λ, the RMS value is about 0.13λ.
Fig. 16
Fig. 16 The reconstructed near-diffraction-limited image of the resolution test chart.

Equations (10)

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d k ( x,y )=o( x,y ) h k ( x,y )+ n k ( x,y )
h k ( x,y )= | f k ( x,y ) | 2
f k ( x,y )=exp[ i π D k λ B k ( x 2 + y 2 ) ]× P k ( u,v )exp[ i π A k λ B k ( u 2 + v 2 ) ] ×exp[ i 2π λ B k ( ux+vy ) ]dudv
P k ( u,v )= q=1 Q p q ( u,v )× w q,k div ( u,v )×exp[ iw( u,v ) ]
w q,k div ( u,v )={ 1q=k 0qk
w( u,v )= m=1 M α m Z m ( u,v )
E( f x , f y ,α )= k=1 K f x , f y [ D k ( f x , f y )O( f x , f y )× H k ( f x , f y ) ] 2
E( f x , f y ,α )= f x , f y k=1 K | D k ( f x , f y ) | 2 f x , f y k=1 K | D k ( f x , f y ) H k ( f x , f y ) ¯ | 2 k=1 K | H k ( f x , f y ) | 2
O( f x , f y ,α )= k=1 K D k ( f x , f y ) H k ( f x , f y ) ¯ k=1 K | H k ( f x , f y ) | 2
S= 1 MN i=1 M j=1 N { [ I( i,j )I( i1,j1 ) ] 2 + [ I( i1,j )I( i,j1 ) ] 2 } 1/2

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