Abstract

An implementation of an acousto-optic heterodyning image processor was reported earlier [ Opt. Lett. 4, 317 ( 1979)]. However, details of that report are limited only to the confirmation of the basic principle of operation. In this paper, we present and emphasize the mathematical structure of the processor. We also compare this processor with other kinds of scanning and nonscanning processors. The analysis is then extended to the defocused case, where the optical transfer function is derived. A potential application of this in scanning holography is discussed.

© 1985 Optical Society of America

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References

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  1. W. Lukosz, “Properties of linear low-pass filters for nonnegative signals,”J. Opt. Soc. Am. 52, 827–829 (1962).
    [Crossref]
  2. A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1141–1150 (1978).
    [Crossref] [PubMed]
  3. D. Goerlitz, F. Lanzl, “Methods of zero-order non-coherent filtering,” Opt. Commun. 20, 68–72 (1977).
    [Crossref]
  4. A. W. Lohmann, “Incoherent optical processing of complex data,” Appl. Opt. 16, 261–263 (1977).
    [Crossref] [PubMed]
  5. W. Stoner, “Incoherent optical processing via spatially offset pupil masks,” Appl. Opt. 17, 2454–2466 (1978).
    [Crossref] [PubMed]
  6. W. T. Rhodes, “Bipolar pointspread function synthesis by phase switching,” Appl. Opt. 16, 265–267 (1977).
    [Crossref] [PubMed]
  7. T. C. Poon, A. Korpel, “Optical transfer function of an acousto-optic heterodying image processor,” Opt. Lett. 4, 317–319 (1979).
    [Crossref] [PubMed]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed., Vol. 48 of Topics in Applied Physics (Springer-Verlag, New York, 1981), pp. 69–110.
    [Crossref]
  10. W. T. Rhodes, “Noncoherent spatial filtering with temporal frequency carriers,” presented at the Electro-Optics/Laser 1978 Conference, Boston, Mass., September 1978.
  11. P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,”J. Opt. Soc., Am. 66, 14–23 (1976).
    [Crossref]
  12. P. Chavel, S. Lowenthal, “Implementation of an incoherent optical image restoration method: limitations related to optical subtraction,” Appl. Opt. 20, 1438–1449 (1981).
    [Crossref] [PubMed]
  13. J. M. Florence, “Color coding in incoherent spatial filtering,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 19–23 (1980).
  14. W. T. Rhodes, “Temporal frequency carriers in noncoherent optical processing,” in Proceedings of the 1978 Interfational Optical Computing Conference London (Institute of Electrical and Electronics Engineers, New York, 1979).
  15. A. Korpel, “Acousto-optics,” in, Applied Solid State Science, R. Wolfe, ed. (Academic, New York, 1972), Vol. 3.
  16. W. Stoner, “Edge enhancement with incoherent optics,” Appl. Opt. 16, 1451–1452 (1977).
    [Crossref] [PubMed]
  17. N. Konforti, E. Marom, “Low frequency de-emphasis of the MTF. II. Two-dimensional case,” presented at the International Optical Computing Conference, Cambridge, Mass., April 1983.

1981 (1)

1980 (1)

J. M. Florence, “Color coding in incoherent spatial filtering,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 19–23 (1980).

1979 (1)

1978 (2)

1977 (4)

1976 (1)

P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,”J. Opt. Soc., Am. 66, 14–23 (1976).
[Crossref]

1962 (1)

Chavel, P.

P. Chavel, S. Lowenthal, “Implementation of an incoherent optical image restoration method: limitations related to optical subtraction,” Appl. Opt. 20, 1438–1449 (1981).
[Crossref] [PubMed]

P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,”J. Opt. Soc., Am. 66, 14–23 (1976).
[Crossref]

Florence, J. M.

J. M. Florence, “Color coding in incoherent spatial filtering,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 19–23 (1980).

Goerlitz, D.

D. Goerlitz, F. Lanzl, “Methods of zero-order non-coherent filtering,” Opt. Commun. 20, 68–72 (1977).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Konforti, N.

N. Konforti, E. Marom, “Low frequency de-emphasis of the MTF. II. Two-dimensional case,” presented at the International Optical Computing Conference, Cambridge, Mass., April 1983.

Korpel, A.

T. C. Poon, A. Korpel, “Optical transfer function of an acousto-optic heterodying image processor,” Opt. Lett. 4, 317–319 (1979).
[Crossref] [PubMed]

A. Korpel, “Acousto-optics,” in, Applied Solid State Science, R. Wolfe, ed. (Academic, New York, 1972), Vol. 3.

Lanzl, F.

D. Goerlitz, F. Lanzl, “Methods of zero-order non-coherent filtering,” Opt. Commun. 20, 68–72 (1977).
[Crossref]

Lohmann, A. W.

Lowenthal, S.

P. Chavel, S. Lowenthal, “Implementation of an incoherent optical image restoration method: limitations related to optical subtraction,” Appl. Opt. 20, 1438–1449 (1981).
[Crossref] [PubMed]

P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,”J. Opt. Soc., Am. 66, 14–23 (1976).
[Crossref]

Lukosz, W.

Marom, E.

N. Konforti, E. Marom, “Low frequency de-emphasis of the MTF. II. Two-dimensional case,” presented at the International Optical Computing Conference, Cambridge, Mass., April 1983.

Poon, T. C.

Rhodes, W. T.

A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1141–1150 (1978).
[Crossref] [PubMed]

W. T. Rhodes, “Bipolar pointspread function synthesis by phase switching,” Appl. Opt. 16, 265–267 (1977).
[Crossref] [PubMed]

W. T. Rhodes, “Noncoherent spatial filtering with temporal frequency carriers,” presented at the Electro-Optics/Laser 1978 Conference, Boston, Mass., September 1978.

W. T. Rhodes, “Temporal frequency carriers in noncoherent optical processing,” in Proceedings of the 1978 Interfational Optical Computing Conference London (Institute of Electrical and Electronics Engineers, New York, 1979).

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed., Vol. 48 of Topics in Applied Physics (Springer-Verlag, New York, 1981), pp. 69–110.
[Crossref]

Sawchuk, A. A.

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed., Vol. 48 of Topics in Applied Physics (Springer-Verlag, New York, 1981), pp. 69–110.
[Crossref]

Stoner, W.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

J. Opt. Soc., Am. (1)

P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,”J. Opt. Soc., Am. 66, 14–23 (1976).
[Crossref]

Opt. Commun. (1)

D. Goerlitz, F. Lanzl, “Methods of zero-order non-coherent filtering,” Opt. Commun. 20, 68–72 (1977).
[Crossref]

Opt. Lett. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. M. Florence, “Color coding in incoherent spatial filtering,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 19–23 (1980).

Other (6)

W. T. Rhodes, “Temporal frequency carriers in noncoherent optical processing,” in Proceedings of the 1978 Interfational Optical Computing Conference London (Institute of Electrical and Electronics Engineers, New York, 1979).

A. Korpel, “Acousto-optics,” in, Applied Solid State Science, R. Wolfe, ed. (Academic, New York, 1972), Vol. 3.

N. Konforti, E. Marom, “Low frequency de-emphasis of the MTF. II. Two-dimensional case,” presented at the International Optical Computing Conference, Cambridge, Mass., April 1983.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed., Vol. 48 of Topics in Applied Physics (Springer-Verlag, New York, 1981), pp. 69–110.
[Crossref]

W. T. Rhodes, “Noncoherent spatial filtering with temporal frequency carriers,” presented at the Electro-Optics/Laser 1978 Conference, Boston, Mass., September 1978.

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Figures (4)

Fig. 1
Fig. 1

One-pupil optical scanner.

Fig. 2
Fig. 2

Mach–Zehnder configuration.

Fig. 3
Fig. 3

Idealized acousto-optic heterodyning image processor.

Fig. 4
Fig. 4

Interference pattern in out-of-focus plane 2′, created by line source and plane wave.

Equations (46)

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I ( x , y ) = U 2 ( x 2 , y 2 ) U 2 * ( x 2 , y 2 ) × Γ 2 ( x 2 + x , y 2 + y ) 2 d x 2 d y 2
I ( - x , - y ) = U 2 ( x 2 , y 2 ) U 2 * ( x 2 , y 2 ) × Γ 2 ( x 2 - x , y 2 - y ) 2 d x 2 d y 2 = ( U 2 U 2 * ) Γ 2 2 ,
I * ( - x , - y ) = ( U 2 * U 2 ) Γ 2 2 ;
F { I } = F * { I * ( - x , - y ) } = [ F { U 2 * U 2 } F * { Γ 2 2 } ] * = F * { U 2 * U 2 } F { Γ 2 2 } ,
F { I } ( f x , f y ) = I ( x , y ) exp ( - j 2 π f x x - j 2 π f y y ) d x d y ,
U 2 ( x 2 , y 2 ) = F { U 1 } ( x 2 λ f 2 , y 2 λ f 2 ) = U 1 ( x 1 , y 1 ) exp [ - j 2 π λ f 2 ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 ,
OTF ( f x , f y ) = F { I } / F { Γ 2 2 } = F * { U 2 * U 2 } .
OTF = F * { U 2 * U 2 } = [ U 2 * ( x 2 , y 2 ) U 2 ( x 2 , y 2 ) × exp ( - j 2 π f x x 2 - j 2 π f y y 2 ) d x 2 d y 2 ] * = [ U 1 * ( x 1 , y 1 ) × exp [ i 2 π λ f 2 ( x 1 x 2 + y 1 y 2 ) ] U 1 ( x 1 , y 1 ) × exp [ - j 2 π λ f 2 ( x 1 x 2 + y 1 y 2 ) ] × exp ( - j 2 π f x x 2 - j 2 π f y y 2 ) × d x 1 d y 1 d x 1 d y 1 d x 2 d y 2 ] * = [ U 1 * ( x 1 , y 1 ) U 1 ( x 1 , y 1 ) × δ ( 2 π λ f 2 x 1 - 2 π f x - 2 π λ f 2 x 1 ) × δ ( 2 π λ f 2 y 1 - 2 π f y - 2 π λ f 2 y 1 ) d x 1 d y 1 d x 1 d y 1 ] * = [ U 1 * ( x 1 , y 1 ) U 1 ( x 1 - λ f 2 f x , y 1 - λ f 2 f y ) d x 1 d y 1 ] * = U 1 ( x 1 , y 1 ) U 1 * ( x 1 - λ f 2 f x , y 1 - λ f 2 f y ) d x 1 d y 1 = U 1 U 1 ,
U i ( x i , y i ) = U 0 ( x 0 , y 0 ) h ( x i - x 0 , y i - y 0 ) d x 0 d y 0 = U 0 * h ,
I i ( x i , y i ) = U i ( x i , y i ; t ) U i * ( x i , y i ; t ) ,
F { U i } = F { U 0 } F { h } .
H ( f x , f y ) = F { U i } F { U 0 } = F { h } .
h = F { P }
H ( f x , f y ) = F { h } = F { F { P } } = P ( f x , f y ) .
I i ( x i , y i ) = I 0 ( x 0 , y 0 ) h ( x i - x 0 , y i - y 0 ) 2 d x 0 d y 0 ,
F { I i } = F { I 0 } F { h 2 } .
OTF = F { I i } F { I 0 } = F { h 2 } ,
OTF = P P .
OTF tot = F { h 1 2 - h 2 2 } = OTF 1 - OTF 2 = U 1 U 1 - V 1 V 1 ,
U eff = U 1 + V 1 .
PSF = F { U eff } 2 = U 2 + V 2 2 = U 2 2 + V 2 2 + U 2 V 2 * + U 2 * V 2 ,
OTF = F { PSF } = F { U 2 2 } + F { V 2 2 } + F { U 2 V 2 * } + F { U 2 * V 2 } = U 1 U 1 + V 1 V 1 + U 1 V 1 + V 1 U 1 .
i ( x , y , t ) = | [ U 2 ( x 2 , y 2 ) exp ( - j 2 π Δ ν t 2 ) + V 2 ( x 2 , y 2 ) exp ( j 2 π Δ ν t 2 ) ] × Γ 2 ( x + x 2 , y + y 2 ) | 2 d x 2 d y 2 ,
U 2 ( x 2 , y 2 ) exp ( - j 2 π Δ ν 2 t ) ,             V 2 ( x 2 , y 2 ) exp ( + j 2 π Δ ν 2 t ) ,
A ˜ cos ( 2 π ν t + ϕ ) = Re [ A ˜ exp ( - j 2 π ν t ) ] ,
i ˜ ( x , y , t ) = [ U 2 ( x 2 , y 2 ) V 2 * ( x 2 , y 2 ) exp ( - j 2 π Δ ν t ) + V 2 ( x 2 , y 2 ) U 2 * ( x 2 , y 2 ) exp ( j 2 π Δ ν t ) ] × Γ 2 ( x + x 2 , y + y 2 ) 2 d x 2 d y 2 = Re [ U 2 ( x 2 , y 2 ) V 2 * ( x 2 , y 2 ) × Γ 2 ( x + x 2 , y + y 2 ) 2 d x 2 d y 2 × exp ( - j 2 π Δ ν t ) ]
i ˜ ( x , y , t ) = Re [ I ˜ ( x , y ) exp ( - j 2 π Δ ν t ) ] ,
I ˜ ( x , y ) = U 2 ( x 2 , y 2 ) V 2 * ( x 2 , y 2 ) × Γ 2 ( x + x 2 , y + y 2 ) 2 d x 2 d y 2 .
OTF = F { I ˜ ( x , y ) } / F { Γ 2 2 } ,
OTF = F * { U 2 * V 2 } = U 1 V 1 ,
OTF ( f x , f y ) = U 1 ( x 1 , y 1 ) × V 1 * ( x 1 - λ f 2 f x , y 1 - λ f 2 f y ) d x 1 d y 1 .
OTF ( f x , f y ) = F { I ˜ ( x , y ) } / F { Γ 2 2 } = F * { U 2 * ( x 2 , y 2 ) V 2 ( x 2 , y 2 ) } ,
U 2 ( x 2 , y 2 ) = exp ( j k f 2 ) j λ f 2 U 1 ( x 1 , y 1 ) × exp [ - j 2 π λ f 2 ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 ,
U 2 ( x 2 , y 2 ) = exp ( j k z ) j λ z exp [ j k 2 z ( x 2 2 + y 2 2 ) ] × U 2 ( x 2 , y 2 ) exp [ j k 2 z ( x 2 2 + y 2 2 ) ] × exp [ - j k z ( x 2 x 2 - y 2 y 2 ) ] d x 2 d y 2 ,
U 2 ( x 2 , y 2 ) = { - exp ( j k f 2 ) exp ( j k z ) × exp [ j k 2 z ( x 2 2 + y 2 2 ) ] / λ 2 f 2 z } U 1 ( x 1 , y 1 ) × exp [ - j 2 π λ f 2 ( x 1 x 2 + y 1 y 2 ) ] exp [ j k 2 z ( x 2 2 + y 2 2 ) ] × exp [ - j k z ( x 2 x 2 + y 2 y 2 ) ] d x 1 d y 1 d x 2 d y 2 .
OTF ( f x , f y ) = exp [ j π λ f ( f x 2 + f y 2 ) ] × U 1 ( x 1 , y 1 ) V 1 * ( x 1 - λ f 2 f x , y 1 - λ f 2 f y ) × exp [ - j 2 π z f 2 ( x 1 f x + y 1 f 1 ) ] d x 1 d y 1 .
OTF ( f x , f y ) = exp [ j π λ z ( f x 2 + f y 2 ) ] .
I ˜ ( x , y ; z ) = F - 1 { OTF } = exp [ j π / 2 ] λ z exp [ - j π λ z ( x 2 + y 2 ) ] ,
I ˜ ( x , y ; z ) = lim z 0 I ˜ ( x , y ; z ) = δ ( x , y ) ,
OTF ( f x , f y ) = [ U 2 * ( x 2 , y 2 ) V 2 ( x 2 , y 2 ) × exp [ - j 2 π ( f x x 2 + f y y 2 ) ] d x 2 d y 2 ] * .
OTF ( f x , f y ) = ( 1 λ 2 f 2 z ) 2 { 2 4 U 1 * ( x 1 , y 1 ) × exp [ j 2 π λ f 2 ( x 1 x 2 + y 1 y 2 ) ] d x 1 d y 1 × exp [ - j k 2 z ( x 2 2 + y 2 2 ) ] exp [ j k z ( x 2 x 2 + y 2 y 2 ) ] d x 2 d y 2 × [ 4 V 1 ( x ^ 1 , y ^ 1 ) exp [ - j 2 π λ f 2 ( x ^ 1 x ^ 2 + y ^ 1 y ^ 2 ) ] × d x ^ 1 d y ^ 1 exp [ j k 2 z ( x ^ 2 2 + y ^ 2 2 ) ] × exp [ - j k z ( x ^ 2 x 2 + y ^ 2 y 2 ) ] d x ^ 2 d y ^ 2 ] × exp [ - j 2 π f x x 2 - j 2 π f y y 2 ] d x 2 d y 2 } * ,
2 exp [ j k z ( x 2 x 2 + y 2 y 2 ) ] exp [ - j k z ( x ^ 2 x 2 + y ^ 2 y 2 ) ] × exp [ - j 2 π f x x 2 - j 2 π f y y 2 ] d x 2 d y 2 = ( 2 π ) 2 δ ( k z x 2 - k z x ^ 2 - 2 π f x , k z y 2 - k z y ^ 2 - 2 π f y ) = ( 2 π ) 2 ( z k ) 2 δ ( x 2 - x ^ 2 - 2 π z k f x , y 2 - y ^ 2 - 2 π z k f y ) .
OTF ( f x , f y ) = ( ( 1 λ f 2 ) 2 6 U 1 * ( x 1 , y 1 ) × exp [ j 2 π λ f 2 ( x 1 x ^ 2 + x 1 2 π z k f x + y 1 y ^ 2 + y 1 2 π z k f y ) ] d x 1 d y 2 × exp { - j k 2 z [ x ^ 2 2 + ( 2 π z k f x ) 2 + 2 x ^ 2 2 π z k f x + y ^ 2 2 + ( 2 π z k f y ) 2 + 2 y ^ 2 2 π z k f y ] } V 1 ( x ^ 1 , y ^ 1 ) × exp [ - j 2 π λ f 2 ( x ^ 1 x ^ 2 + y ^ 1 y ^ 2 ) ] × d x ^ 1 d y ^ 1 exp [ j k 2 z ( x ^ 2 2 + y ^ 2 2 ) ] d x ^ 2 d y ^ 2 ) * .
OTF ( f x , f y ) = ( 1 λ f 2 ) 2 exp { j k 2 z [ ( 2 π z k f x ) 2 + ( 2 π z k f y ) 2 ] } 6 U 1 ( x 1 , y 1 ) exp { - j 2 π λ f 2 × [ x 1 x ^ 2 + 2 π z k ( x 1 f x + y 1 f y ) + y 1 y ^ 2 ] } d x 1 d y 1 × exp [ j 2 π x ^ 2 f x + j 2 π y ^ 2 f y ] V 1 * ( x ^ 1 , y ^ 1 ) × exp [ j 2 π λ f 2 ( x ^ 1 x ^ 2 + y ^ 1 y ^ 2 ) ] d x ^ 1 d y ^ 1 d x ^ 2 d y ^ 2 .
2 exp [ - j 2 π λ f 2 ( x 1 x ^ 2 + y 1 y ^ 2 ) ] exp [ j 2 π x ^ 2 f x + j 2 π y ^ 2 f y ] × exp [ j 2 π λ f 2 ( x ^ 1 x ^ 2 + y ^ 1 y ^ 2 ) ] d x ^ 2 d y ^ 2 = ( λ f 2 ) 2 δ ( - x 1 + x ^ 1 - λ f 2 f x , - y 1 + y ^ 1 + λ f 2 f y ) .
OTF ( f x , f y ) = exp [ j π λ z ( f x 2 + f y 2 ) ] × U 1 ( x 1 , y 1 ) V 1 * ( x 1 - λ f 2 f x , y 1 - λ f 2 f y ) × exp [ - j 2 π z f 2 ( x 1 f x + y 1 f y ) ] d x 1 d y 1 .

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