Abstract

An imaging scheme is described that is based on the transmission of image-forming information encoded within optical coherence functions. The scheme makes use of dynamic random-valued encoding–decoding masks placed in the input–output planes of any linear optical system. The mask transmittance functions are complex conjugates of each other, as opposed to a similar coherence encoding scheme proposed earlier by two of this paper’s authors that used identical masks. [ Rhodes and Welch, in Euro-American Workshop on Optoelectronic Information Processing, SPIE Critical Review Series (SPIE, 1999), Vol. CR74, p. 1 ]. General analyses of the two coherence encoding schemes are performed by using the more general mutual coherence function as opposed to the mutual intensity function used in the earlier scheme. The capabilities and limitations of both encoding schemes are discussed by using simple examples that combine the encoding–decoding masks with free-space propagation, passage through a four-f system, and a single-lens imaging system.

© 2005 Optical Society of America

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References

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2000 (1)

1999 (1)

1997 (2)

1996 (4)

1992 (1)

1967 (1)

1966 (1)

Brady, D. J.

Brown, M.

Chen, C.

Goodman, J. W.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, 1985), Chap. 5.

James, J. C.

J. C. James, “Imaging systems based on the encoding of optical coherence functions,” Ph.D. dissertation (Georgia Institute of Technology, 2003).

Kiryuschev, I.

Konforti, N.

Leith, E.

Leith, E. N.

Lohmann, A. W.

Lukosz, W.

Marks, D. L.

Marks, D. M.

Mendlovic, D.

Naulleau, P.

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. TheoryIT-8, 194–195 (1962).

Rhodes, W. T.

G. Welch, W. T. Rhodes, “Image reconstruction by spatio-temporal coherence transfer,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE4489, 60–65 (2001).

W. T. Rhodes, G. Welch, “Remote imaging by transfer of coherence functions through optical fibers,” in Euro-American Workshop on Optoelectronic Information ProcessingP. Réfrégier and B. Javidi, eds. Critical Review Series (SPIE, 1999), Vol. CR74, p. 1–9.

Rosen, J.

Stack, R. A.

Sun, P. C.

Welch, G.

G. Welch, W. T. Rhodes, “Image reconstruction by spatio-temporal coherence transfer,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE4489, 60–65 (2001).

W. T. Rhodes, G. Welch, “Remote imaging by transfer of coherence functions through optical fibers,” in Euro-American Workshop on Optoelectronic Information ProcessingP. Réfrégier and B. Javidi, eds. Critical Review Series (SPIE, 1999), Vol. CR74, p. 1–9.

G. Welch, “Application of coherence theory to enhanced backscatter and superresolving optical imaging systems,” Ph.D. dissertation (Georgia Institute of Technology, 1995).

Yariv, A.

Zalevsky, Z.

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

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

Opt. Lett. (2)

Other (6)

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, 1985), Chap. 5.

G. Welch, W. T. Rhodes, “Image reconstruction by spatio-temporal coherence transfer,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE4489, 60–65 (2001).

G. Welch, “Application of coherence theory to enhanced backscatter and superresolving optical imaging systems,” Ph.D. dissertation (Georgia Institute of Technology, 1995).

W. T. Rhodes, G. Welch, “Remote imaging by transfer of coherence functions through optical fibers,” in Euro-American Workshop on Optoelectronic Information ProcessingP. Réfrégier and B. Javidi, eds. Critical Review Series (SPIE, 1999), Vol. CR74, p. 1–9.

J. C. James, “Imaging systems based on the encoding of optical coherence functions,” Ph.D. dissertation (Georgia Institute of Technology, 2003).

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. TheoryIT-8, 194–195 (1962).

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Figures (6)

Fig. 1
Fig. 1

Schematic representation of an optical coherence encoding system. U in ( x , t ) and U out ( x , t ) denote the complex amplitudes of input and output waves, m 1 ( x , t ) and m 2 ( x , t ) the complex transmittances of the encoding and decoding masks, and h ( x , ξ ) a linear optical system.

Fig. 2
Fig. 2

Three examples of coherence encoding systems: (a) masks separated by free space, (b) masks placed at input and output planes of a four-f imaging system, and (c) masks on either side of a single finite-diameter lens, where d 1 , d 2 , and f satisfy the Gaussian imaging formula.

Fig. 3
Fig. 3

Illustration of the response of a simple encoding system to illumination by light from two mutually incoherent point sources. (a) With identical masks, the light exiting the decoding mask produces a real image of the two sources plus an additional uniform incoherent bias field. (b) With conjugate masks, the light exiting the decoding mask appears to be identical to the light that entered the system except for the presence of the bias field (the field’s coherence has been replicated, not the field itself). (Note: lines representing the coherent wavefield’s phase fronts were drawn for illustration only.)

Fig. 4
Fig. 4

Plot of replicated coherence field’s sinc-squared attenuation function arising from the use of a four-f system with a rectangular pupil and identical masks. The plot domain covers the decoding mask’s area. The arrows indicate directions of increasing values of measurement point separations x 1 x 2 . The system parameters used to produce this plot satisfy the relationship w p w m λ f = 4 .

Fig. 5
Fig. 5

Plot of replicated coherence field’s attenuation function in the case of a single-lens imaging system with identical masks. The plot domain covers the decoding mask’s area. The system parameters used to produce this plot satisfy the relationship w p w m λ d 2 = 4 , and the system magnification factor was 4.

Fig. 6
Fig. 6

Plot of replicated coherence field’s attenuation function for the case of a single-lens imaging system and conjugate masks. The plot domain covers the decoding mask’s area. The system parameters used to produce this plot satisfy the relationship w p w m λ d 2 = 4 , and the system magnification factor was 4.

Equations (33)

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U out ( x , t ) = m 2 ( x , t ) + m 1 ( ξ , t ) U in ( ξ , t ) h ( x , ξ ) d ξ .
Γ out ( x 1 , x 2 ; τ ) = U out ( x 1 , t ) U out * ( x 2 , t + τ ) ,
Γ out ( x 1 , x 2 ; τ ) = U in ( ξ 1 , t ) U in * ( ξ 2 , t + τ ) m 1 ( ξ 1 , t ) m 1 * ( ξ 2 , t + τ ) m 2 ( x 1 , t ) m 2 * ( x 2 , t + τ ) × h ( x 1 , ξ 1 ) h * ( x 2 , ξ 2 ) d ξ 1 d ξ 2 .
m ( x 1 , t 1 ) m ( x 2 , t 2 ) m * ( x 3 , t 3 ) m * ( x 4 , t 4 ) = κ 2 [ δ ( x 3 x 1 ) δ ( x 4 x 2 ) R ( t 3 t 1 ) R ( t 4 t 2 ) + δ ( x 3 x 2 ) δ ( x 4 x 1 ) R ( t 3 t 2 ) R ( t 4 t 1 ) ] ,
m ( ξ 1 , t ) m * ( ξ 2 , t + τ ) m ( x 1 , t ) m * ( x 2 , t + τ ) = κ 2 R 2 ( τ ) [ δ ( x 2 ξ 1 ) δ ( ξ 2 x 1 ) + δ ( x 2 x 1 ) δ ( ξ 2 ξ 1 ) ] .
m ( ξ 1 , t ) m * ( ξ 2 , t + τ ) m * ( x 1 , t ) m ( x 2 , t + τ ) = κ 2 [ δ ( x 1 ξ 1 ) δ ( x 2 ξ 2 ) R 2 ( 0 ) + δ ( ξ 1 ξ 2 ) δ ( x 2 x 1 ) R ( τ ) 2 ] .
Γ out ( x 1 , x 2 ; τ ) = κ 2 R 2 ( τ ) [ h ( x 1 , x 2 ) h * ( x 2 , x 1 ) Γ in ( x 2 , x 1 ; τ ) + δ ( x 1 x 2 ) I incoh ( x 1 ; τ ) ] .
Γ out ( x 1 , x 2 ; τ ) = κ 2 [ h ( x 1 , x 1 ) h * ( x 2 , x 2 ) R 2 ( 0 ) Γ in ( x 1 , x 2 ; τ ) + δ ( x 1 x 2 ) R ( τ ) 2 I incoh ( x 1 ; τ ) ] .
I incoh ( x ; τ ) = + Γ in ( ξ , ξ ; τ ) h ( x , ξ ) 2 d ξ .
I incoh ( x ) = + I in ( ξ ) h ( x , ξ ) 2 d ξ .
Γ ( x 1 , x 2 ; τ ) = J ( x 1 , x 2 ) exp ( j 2 π ν ¯ τ ) ,
J ( x 1 , x 2 ) = U ( x 1 , t ) U * ( x 2 , t ) .
Γ out ( x 1 , x 2 ; τ ) = κ 2 R 2 ( τ ) [ h ( x 1 , x 2 ) h * ( x 2 , x 1 ) exp ( j 2 π ν τ ) J in * ( x 1 , x 2 ) + δ ( x 1 x 2 ) exp ( j 2 π ν τ ) I incoh ( x 1 ) ] ,
Γ out ( x 1 , x 2 ; τ ) = κ 2 exp ( j 2 π ν τ ) [ h ( x 1 , x 1 ) h * ( x 2 , x 2 ) R 2 ( 0 ) J in ( x 1 , x 2 ) + δ ( x 1 x 2 ) R ( τ ) 2 I incoh ( x 1 ) ] ,
I incoh ( x 1 ) = + I in ( ξ 1 ) h ( x 1 , ξ 1 ) 2 d ξ 1 .
h ( x , ξ ) = 1 j λ d exp ( j π d λ ) exp [ j π λ d ( x ξ ) 2 ] rect ( ξ w m ) ,
h ( x 1 , x 2 ) h * ( x 2 , x 1 ) = h ( x 1 , x 1 ) h * ( x 2 , x 2 ) = 1 λ d rect ( x 1 w m , x 2 w m ) .
I incoh ( x ; τ ) = 1 λ d w m 2 + w m 2 Γ in ( ξ , ξ ; τ ) d ξ .
J in ( x 1 , x 2 ) = I a λ d a exp { j π λ d a [ ( x 1 x a ) 2 ( x 2 x a ) 2 ] } + I b λ d b exp { j π λ d b [ ( x 1 x b ) 2 ( x 2 x b ) 2 ] } .
J out ( x 1 , x 2 ) = 1 λ d rect ( x 1 w m , x 2 w m ) ( I a λ d a exp { j π λ d a [ ( x 1 x a ) 2 ( x 2 x a ) 2 ] } + I b λ d b exp { j π λ d b [ ( x 1 x b ) 2 ( x 2 x b ) 2 ] } + w m { I a λ d a + I b λ d b } δ ( x 1 x 2 ) )
h ( x , ξ ) = w p λ f sinc [ w p λ f ( x + ξ ) ] rect ( ξ w m ) .
h ( x 1 , x 2 ) h * ( x 2 , x 1 ) = w p 2 λ 2 f 2 sinc 2 [ w p λ f ( x 1 x 2 ) ] rect ( x 1 w m , x 2 w m )
h ( x 1 , x 1 ) h * ( x 2 , x 2 ) = w p 2 λ 2 f 2 rect ( x 1 w m , x 2 w m )
I incoh ( x ; τ ) = w p 2 λ 2 f 2 w m 2 + w m 2 Γ in ( ξ , ξ ; τ ) sinc 2 [ w p λ f ( x + ξ ) ] d ξ .
h ( x , ξ ) = w l λ 2 d 1 d 2 exp [ j π λ ( d 1 + d 2 ) ] exp [ j π λ ( x 2 d 2 + ξ 2 d 1 ) ] × sinc [ w l λ d 2 ( x + d 2 d 1 ξ ) ] rect ( ξ w m ) .
h ( x 1 , x 2 ) h * ( x 2 , x 1 ) = w l 2 λ 4 d 1 2 d 2 2 exp [ j π λ d 2 ( M 1 ) ( x 1 2 x 2 2 ) ] sinc [ w l λ d 2 ( M x 2 x 1 ) ] sinc [ w l λ d 2 ( M x 1 x 2 ) ] rect ( x 1 w m , x 2 w m )
h ( x 1 , x 1 ) h * ( x 2 , x 2 ) = w l 2 λ 4 d 1 2 d 2 2 exp [ j π λ f lens ( x 1 2 x 2 2 ) ] rect ( x 1 w m , x 2 w m ) sinc [ w l λ d 2 ( M 1 ) x 1 , w l λ d 2 ( M 1 ) x 2 ]
I incoh ( x ; τ ) = w l 2 λ 4 d 1 2 d 2 2 w m 2 + w m 2 Γ in ( ξ , ξ ; τ ) sinc 2 [ w l λ d 2 ( x + M ξ ) ] d ξ .
E [ x r ( t m ) ] = E [ x i ( t m ) ] = 0 ,
E [ x r ( t m ) x r ( t n ) ] = E [ x i ( t m ) x i ( t n ) ] ,
E [ x r ( t m ) x i ( t n ) ] = E [ x r ( t n ) x i ( t m ) ] ,
E [ X ( t 1 ) X ( t 2 ) X ( t m ) X * ( t m + 1 ) X * ( t m + 2 ) X * ( t N ) ]
E [ X ( t 1 ) X ( t 2 ) X ( t m ) X * ( t m + 1 ) X * ( t m + 2 ) X * ( t N ) ] = π E [ X ( t ρ ( 1 ) ) X * ( t m + 1 ) ] E [ X ( t ρ ( 2 ) ) X * ( t m + 2 ) ] E [ X ( t ρ ( m ) ) X * ( t N ) ] ,

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