Abstract

The combination of broad-band illumination and coherent (synchronous) detection techniques provides a means by which imaging systems can derive added object information and consequently enhance their resolution. In this paper a unified theory of coherent imaging or holography with broad-band illumination is developed in terms of the temporal impulse response, the spectral frequency response, and cross-correlation imaging with noise illumination of the object. The equivalence of these three methods is established and their resolution capabilities in the imaging of one- and two-dimensional nondispersive objects is determined. The resolution is shown to be dependent on bandwidth and geometry. The advantages of broad-band illumination over monochromatic illumination are also discussed.

© 1977 Optical Society of America

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References

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  1. H. F. Harmuth, Proceedings of the 1975 IEEE International Radar Conference, New York,1975 (IEEE, New York, 1975), p. 256, IEEE Order No. 75CH0938–1AES.
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 46.
  3. H. F. Harmuth and et al., Applications of Walsh Functions and Sequency Theory (IEEE, New York, 1974), p. 275, IEEE Order No.74CH861–5EMC.
  4. J. C. Vienot and et al., Proceedings of the Third IEEE International Conference on Optical Computing, 1975 (IEEE, New York, 1975).
  5. K. Iizuka and L. G. Gregoris, Appl. Phys. Lett. 17, 509 (1970).
    [Crossref]
  6. W. Lukosz, J. Opt. Soc. Am. 56, 1463 (1966).
    [Crossref]
  7. W. Lukosz, J. Opt. Soc. Am. 57, 932 (1967).
    [Crossref]
  8. See, for example, A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 334;also, M. Gupta, Proc. IEEE 63, 997 (1975).
  9. B. Chatterjee and A. B. Bhattacharyya, IEEE Trans. Automat. Contr. AC–8, 186 (1963).
    [Crossref]
  10. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), p. 61.
  11. W. E. Kock, Proc. IEEE Lett. 61, 1518 (1973).
    [Crossref]
  12. T. Sato and S. Wadaka, J. Acoust. Soc. Am. 58, 1013 (1975).
    [Crossref]
  13. S. Ueda and et al., Appl. Opt. 14, 1478 (1975).
    [Crossref]
  14. In practice, only a positive frequency spectral window is available. The impulse response [eq. (20)] becomes then complex. This can be shown to lead to enhancement of edges and other fine detail in the image with resolution remaining essentially unchanged. The same effect occurs in conventional holography of nondiffusely illuminated planar objects when the hologram aperture is not centered in front of the object.
  15. N. H. Farhat, Proc. IEEE Lett. 64, 379 (1976).
    [Crossref]
  16. N. H. Farhat, Proceedings of the International Optical Computing Conference (IEEE, New York, 1976), IEEE Cat. No. 76CH1100-7C, p. 19.
  17. N. H. Farhat and et al., Proceedings of the 1976 Ultrasonics Symposium (IEEE, New York, 1976), IEEE Cat. No. 76 CH1120–5SU, p. 168.
    [Crossref]
  18. R. M. Lewis, IEEE Trans. Ant. Prop. AP–17, 308 (1969).
    [Crossref]
  19. S. Rosenbaum-Raz, IEEE Trans. Ant. Prop. AP–24, 66 (1976).
    [Crossref]

1976 (2)

N. H. Farhat, Proc. IEEE Lett. 64, 379 (1976).
[Crossref]

S. Rosenbaum-Raz, IEEE Trans. Ant. Prop. AP–24, 66 (1976).
[Crossref]

1975 (2)

T. Sato and S. Wadaka, J. Acoust. Soc. Am. 58, 1013 (1975).
[Crossref]

S. Ueda and et al., Appl. Opt. 14, 1478 (1975).
[Crossref]

1973 (1)

W. E. Kock, Proc. IEEE Lett. 61, 1518 (1973).
[Crossref]

1970 (1)

K. Iizuka and L. G. Gregoris, Appl. Phys. Lett. 17, 509 (1970).
[Crossref]

1969 (1)

R. M. Lewis, IEEE Trans. Ant. Prop. AP–17, 308 (1969).
[Crossref]

1967 (1)

1966 (1)

1963 (1)

B. Chatterjee and A. B. Bhattacharyya, IEEE Trans. Automat. Contr. AC–8, 186 (1963).
[Crossref]

Bhattacharyya, A. B.

B. Chatterjee and A. B. Bhattacharyya, IEEE Trans. Automat. Contr. AC–8, 186 (1963).
[Crossref]

Chatterjee, B.

B. Chatterjee and A. B. Bhattacharyya, IEEE Trans. Automat. Contr. AC–8, 186 (1963).
[Crossref]

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), p. 61.

Farhat, N. H.

N. H. Farhat, Proc. IEEE Lett. 64, 379 (1976).
[Crossref]

N. H. Farhat, Proceedings of the International Optical Computing Conference (IEEE, New York, 1976), IEEE Cat. No. 76CH1100-7C, p. 19.

N. H. Farhat and et al., Proceedings of the 1976 Ultrasonics Symposium (IEEE, New York, 1976), IEEE Cat. No. 76 CH1120–5SU, p. 168.
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 46.

Gregoris, L. G.

K. Iizuka and L. G. Gregoris, Appl. Phys. Lett. 17, 509 (1970).
[Crossref]

Harmuth, H. F.

H. F. Harmuth and et al., Applications of Walsh Functions and Sequency Theory (IEEE, New York, 1974), p. 275, IEEE Order No.74CH861–5EMC.

H. F. Harmuth, Proceedings of the 1975 IEEE International Radar Conference, New York,1975 (IEEE, New York, 1975), p. 256, IEEE Order No. 75CH0938–1AES.

Iizuka, K.

K. Iizuka and L. G. Gregoris, Appl. Phys. Lett. 17, 509 (1970).
[Crossref]

Kock, W. E.

W. E. Kock, Proc. IEEE Lett. 61, 1518 (1973).
[Crossref]

Lewis, R. M.

R. M. Lewis, IEEE Trans. Ant. Prop. AP–17, 308 (1969).
[Crossref]

Lukosz, W.

Papoulis, A.

See, for example, A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 334;also, M. Gupta, Proc. IEEE 63, 997 (1975).

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), p. 61.

Rosenbaum-Raz, S.

S. Rosenbaum-Raz, IEEE Trans. Ant. Prop. AP–24, 66 (1976).
[Crossref]

Sato, T.

T. Sato and S. Wadaka, J. Acoust. Soc. Am. 58, 1013 (1975).
[Crossref]

Ueda, S.

Vienot, J. C.

J. C. Vienot and et al., Proceedings of the Third IEEE International Conference on Optical Computing, 1975 (IEEE, New York, 1975).

Wadaka, S.

T. Sato and S. Wadaka, J. Acoust. Soc. Am. 58, 1013 (1975).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. Iizuka and L. G. Gregoris, Appl. Phys. Lett. 17, 509 (1970).
[Crossref]

IEEE Trans. Ant. Prop. (2)

R. M. Lewis, IEEE Trans. Ant. Prop. AP–17, 308 (1969).
[Crossref]

S. Rosenbaum-Raz, IEEE Trans. Ant. Prop. AP–24, 66 (1976).
[Crossref]

IEEE Trans. Automat. Contr. (1)

B. Chatterjee and A. B. Bhattacharyya, IEEE Trans. Automat. Contr. AC–8, 186 (1963).
[Crossref]

J. Acoust. Soc. Am. (1)

T. Sato and S. Wadaka, J. Acoust. Soc. Am. 58, 1013 (1975).
[Crossref]

J. Opt. Soc. Am. (2)

Proc. IEEE Lett. (2)

W. E. Kock, Proc. IEEE Lett. 61, 1518 (1973).
[Crossref]

N. H. Farhat, Proc. IEEE Lett. 64, 379 (1976).
[Crossref]

Other (9)

N. H. Farhat, Proceedings of the International Optical Computing Conference (IEEE, New York, 1976), IEEE Cat. No. 76CH1100-7C, p. 19.

N. H. Farhat and et al., Proceedings of the 1976 Ultrasonics Symposium (IEEE, New York, 1976), IEEE Cat. No. 76 CH1120–5SU, p. 168.
[Crossref]

In practice, only a positive frequency spectral window is available. The impulse response [eq. (20)] becomes then complex. This can be shown to lead to enhancement of edges and other fine detail in the image with resolution remaining essentially unchanged. The same effect occurs in conventional holography of nondiffusely illuminated planar objects when the hologram aperture is not centered in front of the object.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), p. 61.

See, for example, A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 334;also, M. Gupta, Proc. IEEE 63, 997 (1975).

H. F. Harmuth, Proceedings of the 1975 IEEE International Radar Conference, New York,1975 (IEEE, New York, 1975), p. 256, IEEE Order No. 75CH0938–1AES.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 46.

H. F. Harmuth and et al., Applications of Walsh Functions and Sequency Theory (IEEE, New York, 1974), p. 275, IEEE Order No.74CH861–5EMC.

J. C. Vienot and et al., Proceedings of the Third IEEE International Conference on Optical Computing, 1975 (IEEE, New York, 1975).

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Figures (3)

FIG. 1
FIG. 1

Geometry referred to in determining the impulse response.

FIG. 2
FIG. 2

Two-dimensional object geometry.

FIG. 3
FIG. 3

Frequency synthesized scan format for a linear array of receivers.

Equations (44)

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Ψ ( u , ν ) = j ( c Z 0 ) 1 / 2 ν x 0 a ( ν ) D ( x 0 ) × e j 2 π ( u / c ) ν x 0 d x 0 ,
h s ( u , t ) = j ( c Z 0 ) 1 / 2 ν x 0 ν δ ( ν ν 0 ) D ( x 0 ) × e j 2 π ( u / c ) ν x 0 e j 2 π ν t d x 0 d ν
= j ( c Z 0 ) 1 / 2 ν 0 e j 2 π ν 0 t D ( u c ν 0 ) ,
D ( u c ν 0 ) = x 0 D ( x 0 ) e j 2 π ( u / c ) ν 0 x 0 d x 0
h ( u , t ) = Ψ ( u , ν ) e j 2 π ν t d ν ,
Ψ ( u , ν ) = j ( c Z 0 ) 1 / 2 ν x 0 D ( x 0 ) e j 2 π ( u / c ) ν x 0 d x 0 .
h ( u , t ) = j ( c Z 0 ) 1 / 2 ν D ( u c ν ) e j 2 π ν t d ν
h ( u , t ) = j ( c Z 0 ) 1 / 2 c u 1 t * d d t D ( c u t )
h ( u , t ) = j ( c Z 0 ) 1 / 2 c u d d t ( 1 t ) * D ( c u t ) = j ( c Z 0 ) 1 / 2 c u t 3 / 2 * D ( c u t ) .
h ( u , t ) = [ j / ( c Z 0 ) 1 / 2 ] ( c / u ) D [ ( c / u ) t ] .
Ψ ( u , ν ) = [ j / ( c Z 0 ) 1 / 2 ] D [ ( u / c ) ν ] .
h ( u , t ) = [ j / ( c Z 0 ) 1 / 2 ] ( c / u ) D [ ( c / u ) t ] ,
H ( u , ν ) = [ j / ( c Z 0 ) 1 / 2 ] ν D [ ( u / c ) ν ] ,
D ( u c ν ) = j ( c Z 0 ) 1 / 2 H ( u / ν ) / ν .
ϕ x y ( τ ) = h ( t ) ϕ x x ( τ t ) d t .
ϕ x y ( τ ) = h ( τ ) ,
h ( u , τ ) = ϕ x y ( u , τ ) .
D [ ( u / c ) ν ] = j ( c Z 0 ) 1 / 2 ϕ x y ( u / ν ) / ν ,
h ( u , t ) = j ( c Z 0 ) 1 / 2 c u Δ ν 2 D ( c u t ) * sinc ( π Δ ν t ) .
h ( u , ξ ) = j ( c Z 0 ) 1 / 2 c u Δ ν 2 D ( ξ ) * sinc ( π u c Δ ν ξ ) .
h s ( u , ξ ) = j ( c Z 0 ) 1 / 2 c u Δ ν 2 sinc ( π u c Δ ν ξ ) ,
Δ ξ = 2 c / u Δ ν .
Δ ξ = 2 c u ( c / λ 2 c / λ 1 ) = 2 λ 1 λ 2 u ( λ 1 λ 2 )
Δ ξ = 2 λ eff / u ,
λ eff = λ 1 λ 2 λ 1 λ 2 = λ 1 λ 1 / λ 2 1 .
Δ ξ = ( 2 λ eff / B ) Z 0 .
Ψ ( u , υ , ν ) = j c Z 0 ν x 0 y 0 a ( ν ) D ( x 0 , y 0 ) × e j ( 2 π ν / c ) ( u x 0 + υ y 0 ) d x 0 d y 0 ,
Ψ ( u , υ , t ) = j c Z 0 ν x 0 y 0 ν δ ( ν ν 0 ) D ( x 0 , y 0 ) × e j ( 2 π ν / c ) ( u x 0 + υ y 0 ) e j 2 π ν t d x 0 d y 0 d ν = j c Z 0 ν 0 e j 2 π ν 0 t x 0 y 0 D ( x 0 , y 0 ) × e j ( 2 π ν 0 / c ) ( u x 0 + υ y 0 ) d x 0 d y 0 , Ψ ( u , υ , t ) = j c Z 0 ν 0 D ( ν 0 c u , ν 0 c υ , ) e j 2 π ν 0 t .
h ( u , υ , t ) = j c Z 0 ν x 0 y 0 ν D ( x 0 , y 0 ) × e j ( 2 π ν / c ) ( u x 0 + υ y 0 ) e j 2 π ν t d x 0 d y 0 d ν = j Z 0 u d d t D ( c u t , c υ t ) ,
D ( c u t , c u t ) = j u Z 0 h ( u , υ , t ) d t .
D ( u c ν , υ c ν ) = j c Z 0 H ( u , υ , t ) ν ,
H ( u , υ , ν ) = ( ν / j c Z 0 ) D [ ξ ( ν ) , η ( ν ) ] ,
ξ ( ν ) = ( u / c ) ν , η ( ν ) = ( υ / c ) ν .
ξ = ( u / υ ) η .
u / υ = sin θ x / sin θ y x / y ,
S ξ , η = { [ ξ ( ν 2 ) ξ ( ν 1 ) ] 2 + [ η ( ν 2 ) η ( ν 1 ) ] 2 } 1 / 2 = ( 1 / c ) ( ν 2 ν 1 ) ( u 2 + υ 2 ) 1 / 2 ,
S ξ , η ν 2 ν 1 c Z 0 ( x 2 + y 2 ) 1 / 2 = ( 1 λ 2 1 λ 1 ) B Z 0 ,
ξ = ( x / c Z 0 ) ν m , η = ( y / c Z 0 ) ν m .
d S ξ , η = ( d ξ 2 + d η 2 ) 1 / 2 = ( ν m / c Z 0 ) d S x , y ,
S x , y = λ m Z 0 S ξ , η ,
S x , y = λ m ( 1 / λ 2 1 / λ 1 ) B
= 1 2 ( λ 1 / λ 2 + 1 ) ( λ 1 / λ 2 1 ) λ 1 λ 2 B .
Δ θ = λ m / S x , y .
Δ θ = λ eff / B ,