Abstract

Multi-transmitter aperture synthesis increases the effective aperture in coherent imaging by shifting the backscattered speckle field across a physical aperture or set of apertures. Through proper arrangement of the transmitter locations, it is possible to obtain speckle fields with overlapping regions, which allows fast computation of optical aberrations from wavefront differences. In this paper, we present a method where Zernike polynomials are used to model the aberrations and high-order aberrations are estimated without the need to do phase unwrapping of the difference fronts.

© 2012 OSA

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References

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2012 (2)

2011 (1)

2010 (2)

2008 (1)

2007 (1)

2003 (1)

2000 (1)

1996 (1)

1988 (1)

R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” in Statistical Optics, Proc. SPIE 976, 37–47 (1988).

1976 (1)

1975 (1)

1974 (1)

Buffington, A.

Dai, F.

Dierking, M. P.

Duncan, B. D.

Feng, P.

Fienup, J. R.

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

Goodman, J. W.

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

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2010).

Gunturk, B. K.

Harbers, G.

Jameson, D.

Jameson, D. F.

Kamiya, K.

Kunst, P. J.

Leibbrandt, G. W. R.

Marron, J. C.

R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” in Statistical Optics, Proc. SPIE 976, 37–47 (1988).

Miller, J. J.

Miller, N. J.

Miyashiro, H.

Muller, R. A.

Noll, R. J.

Nomura, T.

Okuda, S.

Paxman, R. G.

R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” in Statistical Optics, Proc. SPIE 976, 37–47 (1988).

Rabb, D.

Rabb, D. J.

Rimmer, M. P.

Sasaki, O.

Schmidt, J. D.

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

Stafford, J.

Stafford, J. W.

Stokes, A.

Stokes, A. J.

Tang, F.

Thurman, S. T.

Wang, X.

Watson, E. A.

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

Wyant, J. C.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Express (5)

Statistical Optics (1)

R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” in Statistical Optics, Proc. SPIE 976, 37–47 (1988).

Other (4)

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

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

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2010).

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Figures (8)

Fig. 1
Fig. 1

Multi-transmitter system. The scene is illuminated with one transmitter at a time. A shift of Δx in the transmitter location results in a shift of −Δx in the backscattered field.

Fig. 2
Fig. 2

(a) Wavefront difference between two overlapping apertures. (b) Sub-regions with uniform phase offset.

Fig. 3
Fig. 3

(a) Actual multi-transmitter system. (b) Equivalent multi-aperture system.

Fig. 4
Fig. 4

(a) to (c) show the phase difference in overlap regions. (d) to (e) show the subregions within each overlap region.

Fig. 5
Fig. 5

(a) Image formed at an aperture by averaging 30 speckle realizations. (b) Estimated phase error using the proposed algorithm. (Units are in waves; up to fifth order Zernike polynomials are used.) (c) Aberration corrected first aperture, corresponding to (a). (d) Composite formed by all three aberration-corrected apertures.

Fig. 6
Fig. 6

Root mean square error in the wavefront reconstruction as a function of root mean square wavefront aberration.

Fig. 7
Fig. 7

Root mean square error in the wavefront reconstruction as a function of signal to noise ratio in the pupil field.

Fig. 8
Fig. 8

A sample restoration, where the RMS wavefront aberration is 0.4951 and the signal to noise ratio is 100. (a)–(c) The difference fronts and the corresponding sub-regions are displayed. (d) Actual wavefront aberration. (e) Estimated wavefront aberration. (f)–(g) The difference between the actual and estimated wavefront aberrations. The RMS wavefront reconstruction error is 0.0403. Note that in (d)–(f), the same colormap is used for comparison purposes.

Equations (6)

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U i ( x , y ) = P ( x , y ) exp ( j 2 π W e ( x , y ) ) U b ( x x i , y y i ) , i = 1 , 2 ,
W i ( x , y ) = W e ( x , y ) + W b ( x x i , y y i ) , i = 1 , 2 .
Δ W ( x , y ) = W 1 ( x + x 1 , y + y 1 ) W 2 ( x + x 2 , y + y 2 ) = ( W e ( x + x 1 , y + y 1 ) + W b ( x , y ) ) ( W e ( x + x 2 , y + y 2 ) + W b ( x , y ) ) = W e ( x + x 1 , y + y 1 ) W e ( x + x 2 , y + y 2 ) .
Δ W ( x , y ) = k a k Z k ( x + x 1 , y + y 1 ) k a k Z k ( x + x 2 , y + y 2 ) = k a k ( Z k ( x + x 1 , y + y 1 ) Z k ( x + x 2 , y + y 2 ) ) = k a k Δ Z k ( x , y ) ,
Δ W ( m ) ( x i ( m ) , y i ( m ) ) = α ( m ) + k a k Δ Z k ( m ) ( x i ( m ) , y i ( m ) ) ,
[ Δ W ( 1 ) ( x 1 ( 1 ) , y 1 ( 1 ) ) Δ W ( 1 ) ( x N 1 ( 1 ) , y N 1 ( 1 ) ) Δ W ( 2 ) ( x 1 ( 2 ) , y 1 ( 2 ) ) Δ W ( 2 ) ( x N 2 ( 2 ) , y N 2 ( 2 ) ) Δ W ( M ) ( x 1 ( M ) , y 1 ( M ) ) Δ W ( M ) ( x N M ( M ) , y N M ( M ) ) ] = [ 1 0 0 Δ Z 1 ( 1 ) ( x 1 ( 1 ) , y 1 ( 1 ) ) Δ Z K ( 1 ) ( x 1 ( 1 ) , y 1 ( 1 ) ) 1 0 0 Δ Z 1 ( 1 ) ( x N 1 ( 1 ) , y N 1 ( 1 ) ) Δ Z K ( 1 ) ( x N 1 ( 1 ) , y N 1 ( 1 ) ) 0 1 0 Δ Z 1 ( 2 ) ( x 1 ( 2 ) , y 1 ( 2 ) ) Δ Z K ( 2 ) ( x 1 ( 2 ) , y 1 ( 2 ) ) 0 1 0 Δ Z 1 ( 2 ) ( x N 2 ( 2 ) , y N 2 ( 2 ) ) Δ Z K ( 2 ) ( x N 2 ( 2 ) , y N 2 ( 2 ) ) 0 0 1 Δ Z 1 ( M ) ( x 1 ( M ) , y 1 ( M ) ) Δ Z K ( M ) ( x 1 ( M ) , y 1 ( M ) ) 0 0 1 Δ Z 1 ( M ) ( x N M ( M ) , y N M ( M ) ) Δ Z K ( M ) ( x N M ( M ) , y N M ( M ) ) ] [ α ( 1 ) α ( 2 ) α ( M ) a 1 a K ]

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