Abstract

A fundamental limitation to the quality of wavefront reconstruction images is noise generated by the granular structure of the recording medium. Predictions of the signal-to-noise ratios that can be achieved in wavefront-reconstruction imaging are based on the checkerboard and overlapping circular-grain models of the recording medium. When the object consists of a multitude of resolvable point sources, the signal-to-noise ratio is found to be proportional to the space-bandwidth product of the recording medium; when the object is a diffuse surface, the signal-to-noise ratio is found to be independent of that space-bandwidth product. The quantum limit to signal-to-noise ratio is approachable only with a judicious choice of reference exposure and a recording medium free of other classical noise sources.

© 1967 Optical Society of America

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References

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  1. D. Gabor, Nature 161, 777 (1948).
    [CrossRef]
  2. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
  3. D. Gabor, Proc. Phys. Soc. B64, 449 (1951).
  4. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [CrossRef]
  5. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).
    [CrossRef]
  6. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [CrossRef]
  7. E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).
    [CrossRef]
  8. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]
  9. J. Armstrong, IBM J. Res. Dev. 9, 171 (1965).
    [CrossRef]
  10. D. Gabor, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), Vol. I, pp. 122–124.
  11. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [CrossRef]
  12. R. F. van Ligten, J. Opt. Soc. Am. 56, 1 (1966).
    [CrossRef]
  13. C. W. Helstrom, J. Opt. Soc. Am. 56, 433 (1966).
    [CrossRef]
  14. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963).
  15. L. Silberstein, Phil. Mag. 44, 257 (1922).
  16. L. Silberstein, J. Opt. Soc. Am. 31, 343 (1941).
    [CrossRef]
  17. G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966).
  18. While a planar object is assumed here, the results can readily be extended to three-dimensional objects.
  19. The linearity of the process is proved by applying two point-source objects and noting that they generate two real (and two virtual) images, and that the relative amplitudes of the images are the same as the relative amplitudes of the objects.
  20. That the imaging process is space invariant when the film records all incident spatial structure is implied by the results of R. F. van Ligten, J. Opt. Soc. Am. 56, 1009 (1966).
    [CrossRef]
  21. Note that while Es(x,y) refers to the exposure due to the entire object, Eσ refers to the constant exposure contributed by a single resolution cell on the object. Eσ is, of course, different for different resolution cells; or equivalently, Eσ depends on the image coordinates (α0,β0).
  22. This result follows directly from the Fresnel-Kirchhoff diffraction formula, as found, for example, in M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2d ed., p. 340.
  23. Ref. 14, Sec. 7–3.
  24. Ref. 14, Sec. 7–4.
  25. An analogous result is known in the theory of optical heterodyne detection. If a strong local oscillator drives a detector well above its sensitivity threshold, the signal-to-noise ratio is limited solely by the quantum efficiency of the detector and the number of signal photons incident per resolution period.
  26. This restriction is a necessary (but in general not sufficient) condition if the incident spatial structure is to be fully recorded.
  27. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill Book Company, New York, 1960), p. 356.

1966 (4)

1965 (3)

1964 (1)

1963 (1)

1962 (1)

1951 (1)

D. Gabor, Proc. Phys. Soc. B64, 449 (1951).

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

1948 (1)

D. Gabor, Nature 161, 777 (1948).
[CrossRef]

1941 (1)

1922 (1)

L. Silberstein, Phil. Mag. 44, 257 (1922).

Armstrong, J.

J. Armstrong, IBM J. Res. Dev. 9, 171 (1965).
[CrossRef]

Born, M.

This result follows directly from the Fresnel-Kirchhoff diffraction formula, as found, for example, in M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2d ed., p. 340.

Gabor, D.

D. Gabor, Proc. Phys. Soc. B64, 449 (1951).

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

D. Gabor, Nature 161, 777 (1948).
[CrossRef]

D. Gabor, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), Vol. I, pp. 122–124.

Haines, K. A.

Helstrom, C. W.

Kozma, A.

Leith, E. N.

Meier, R. W.

Middleton, D.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill Book Company, New York, 1960), p. 356.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963).

Silberstein, L.

L. Silberstein, J. Opt. Soc. Am. 31, 343 (1941).
[CrossRef]

L. Silberstein, Phil. Mag. 44, 257 (1922).

Stroke, G. W.

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966).

Upatnieks, J.

van Ligten, R. F.

Wolf, E.

This result follows directly from the Fresnel-Kirchhoff diffraction formula, as found, for example, in M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2d ed., p. 340.

IBM J. Res. Dev. (1)

J. Armstrong, IBM J. Res. Dev. 9, 171 (1965).
[CrossRef]

J. Opt. Soc. Am. (10)

Nature (1)

D. Gabor, Nature 161, 777 (1948).
[CrossRef]

Phil. Mag. (1)

L. Silberstein, Phil. Mag. 44, 257 (1922).

Proc. Phys. Soc. (1)

D. Gabor, Proc. Phys. Soc. B64, 449 (1951).

Proc. Roy. Soc. (London) (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Other (12)

D. Gabor, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), Vol. I, pp. 122–124.

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966).

While a planar object is assumed here, the results can readily be extended to three-dimensional objects.

The linearity of the process is proved by applying two point-source objects and noting that they generate two real (and two virtual) images, and that the relative amplitudes of the images are the same as the relative amplitudes of the objects.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963).

Note that while Es(x,y) refers to the exposure due to the entire object, Eσ refers to the constant exposure contributed by a single resolution cell on the object. Eσ is, of course, different for different resolution cells; or equivalently, Eσ depends on the image coordinates (α0,β0).

This result follows directly from the Fresnel-Kirchhoff diffraction formula, as found, for example, in M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2d ed., p. 340.

Ref. 14, Sec. 7–3.

Ref. 14, Sec. 7–4.

An analogous result is known in the theory of optical heterodyne detection. If a strong local oscillator drives a detector well above its sensitivity threshold, the signal-to-noise ratio is limited solely by the quantum efficiency of the detector and the number of signal photons incident per resolution period.

This restriction is a necessary (but in general not sufficient) condition if the incident spatial structure is to be fully recorded.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill Book Company, New York, 1960), p. 356.

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Figures (9)

Fig. 1
Fig. 1

Hologram-recording geometry. The reference and object waves interfere at the photographic plate P.

Fig. 2
Fig. 2

Hologram-reconstruction geometry.

Fig. 3
Fig. 3

Checkerboard model of photographic plate.

Fig. 4
Fig. 4

Overlapping circular-grain model of photographic plate.

Fig. 5
Fig. 5

Dependence of normalized average noise intensity, G(tb), on bias transmittance tb. The result for the checkerboard model is shown by the solid curve, while that for the overlapping circular-grain model is shown by the dotted curve.

Fig. 6
Fig. 6

Signal-to-noise ratio Ii/σ as a function of signal-to-average-noise Ii/〈IN〉.

Fig. 7
Fig. 7

Determination of optimum reference exposure. (a) the transmittance-exposure curve, and (b) the dependence of Ii/〈IN〉 on reference exposure, checkerboard model shown solid, overlapping circular-grain model shown dotted.

Fig. 8
Fig. 8

Transmittance vs. exposure, as predicted by the Silberstein model.

Fig. 9
Fig. 9

Dependence of the factor Qm on reference exposure.

Equations (67)

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U a ( x , y ) = U r ( x , y ) + U s ( x , y ) .
I a ( x , y ) = U r 2 + U s ( x , y ) 2 + U r * ( x , y ) U s ( x , y ) + U r ( x , y ) U s * ( x , y ) ,
I a ( x , y ) U r 2 + U r * ( x , y ) U 0 ( x , y ) + U r ( x , y ) U 0 * ( x , y ) .
t t b - χ ( E - E r ) ,
t ( x , y ) t b - χ [ E r E s ( x , y ) ] 1 2 exp { j [ ϕ r ( x , y ) - ϕ s ( x , y ) ] } - χ [ E r E s ( x , y ) ] 1 2 exp { - j [ ϕ r ( x , y ) - ϕ s ( x , y ) ] }
t ( x , y ) t b - χ [ E r E σ ] 1 2 exp { j [ ϕ r ( x , y ) - ϕ σ ( x , y ) ] } - χ [ E r E σ ] 1 2 exp { - j [ ϕ r ( x , y ) - ϕ σ ( x , y ) ] } ,
U t ( x , y ) = - χ [ E r E σ I p ] 1 2 exp { j [ ϕ r ( x , y ) - ϕ σ ( x , y ) + ϕ p ( x , y ) ] } .
U i ( α 0 , β 0 ) = ( χ [ E r E σ I p ] 1 2 / λ 2 d i ) A t ,
I i ( α 0 , β 0 ) = ( χ 2 ( E r A t ) ( E σ A t ) / λ 2 2 d i 2 ) I p .
t ( x , y ) = τ ( x , y ) ,
χ [ E r E s ( x , y ) ] 1 2 t b
τ ( x , y ) = t b + [ τ ( x , y ) - t b ] = t b + τ ( x , y ) ,
I N ( α , β ) = I p A t λ 2 2 d i 2 - + ϕ τ τ ( Δ x , Δ y ) d Δ x d Δ y ,
ϕ τ τ ( Δ x , Δ y ) = τ ( x , y ) τ ( x - Δ x , y - Δ y ) .
ϕ τ τ ( Δ x , Δ y ) = { t b ( 1 - t b ) [ 1 - ( Δ x / l ) ] × [ 1 - ( Δ y / l ) ] Δ x , Δ y l 0 otherwise .
ϕ τ τ ( Δ x , Δ y ) = { t b 2 [ t b - F ( ρ / l ) - 1 ] 0 ρ l 0 otherwise ,
ρ = [ ( Δ x ) 2 + ( Δ y ) 2 ] 1 2 ,
F ( ρ / l ) = 2 / π { cos - 1 ρ / l - ρ / l [ 1 - ( ρ / l ) 2 ] 1 2 } .
I N ( α , β ) = I p A t A g t b ( 1 - t b ) / λ 2 2 d i 2 .
I N ( α , β ) = I p A t A g λ 2 2 d i 2 { 8 0 1 [ t b 2 - F ( ξ ) - t b 2 ] ξ d ξ } .
I N ( α , β ) = ( I p A t A g / λ 2 2 d i 2 ) G ( t b ) ,
G ( t b ) = { t b ( 1 - t b ) checkerboard model 8 0 1 [ t b 2 - F ( ξ ) - t b 2 ] ξ d ξ overlapping circular-grain model .
I i / I N = χ 2 E r ( E σ A t ) / A g G ( t b ) .
I i / σ = ( I i / I N ) / ( [ 1 + 2 I i / I N ] ) 1 2 .
I i σ = { I i / I N I i / I N 1 [ I i / 2 I N ] 1 2 I i / I N 1.
σ = I N [ 1 + 2 I i / I N ] 1 2 .
( I i / I N ) ( A t / A g ) ( L / l ) 2 .
f max = ( 2 κ l ) - 1 ,
I i / I N ( 2 κ L f max ) 2 .
Prob { τ = 0 } = k = m ( E T A g / ω ) k k ! exp [ - E T A g ω ] ,
Prob { τ = 1 } = k = 0 m - 1 ( E T A g / ω ) k k ! exp [ - E T A g ω ] .
t = 0 · Prob { τ = 0 } + 1 · Prob { τ = 1 } = k = 0 m - 1 ( E T A g / ω ) k k ! exp [ - E T A g ω ] .
t b = k = 0 m - 1 ( E r A g / ω ) k k ! exp [ - E r A g ω ] .
χ = A g ω ( E r A g / ω ) m - 1 ( m - 1 ) ! exp [ - E r A g ω ] .
I i I N = ( n r ) 2 m - 1 exp [ - n r ] ( N σ ) [ ( m - 1 ) ! ] 2 { k = 0 m - 1 [ ( n r ) k / k ! ] } { 1 - k = 0 m - 1 [ ( n r ) k / k ! ] exp [ - n r ] } ,
n r = E r A g / ω ,
N σ = E σ A t / ω .
I i / I N = ( Q m ) N σ ,
Q m = ( n r ) 2 m - 1 exp [ - n r ] [ ( m - 1 ) ! ] 2 { k = 0 m - 1 [ ( n r ) k / k ! ] } { 1 - k = 0 m - 1 [ ( n r ) k / k ! ] exp [ - n r ] } .
I i / I N N σ ,
I i / σ N σ / [ 1 + 2 N σ ] 1 2 .
n r = m - 1 2
( 2 / π ) N σ [ 1 + ( 4 / π ) N σ ] 1 2 < I i σ < N σ [ 1 + 2 N σ ] 1 2 .
U N ( α , β ) = 1 j λ 2 d i - + { U p ( x , y ) τ ( x , y ) rect x L rect y L × exp [ j 2 π r 12 λ 2 ] } d x d y ,
r 12 d i [ 1 + 1 2 ( α - x / d i ) 2 + 1 2 ( β - y / d i ) 2 ] ,
U N ( α , β ) = exp [ j ( π / λ 2 d i ) ( α 2 + β 2 ) ] λ 2 d i - + U p ( x , y ) τ ( x , y ) · rect x L rect y L exp [ j π λ 2 d i ( x 2 + y 2 ) ] exp [ - j 2 π λ 2 d i ( α x + β y ) ] d x d y .
U p ( x , y ) = U p exp [ j ( π / λ 2 d p ) ( x 2 + y 2 ) ] .
I N ( α , β ) = I p λ 2 2 d i 2 - + d x 1 d y 1 - + d x 2 d y 2 rect x 1 L rect y 1 L · rect x 2 L rect y 2 L exp [ j π λ 2 ( 1 d p + 1 d i ) ( x 1 2 - x 2 2 + y 1 2 - y 2 2 ) ] · exp { - j 2 π λ 2 d i [ α ( x 1 - x 2 ) + β ( y 1 - y 2 ) ] } τ ( x 1 , y 1 ) τ ( x 2 , y 2 ) .
x 1 - x 2 = Δ x             and             y 1 - y 2 = Δ y
I N ( α , β ) = I p λ 2 2 d i 2 - + d Δ x d Δ y - + d x 1 d y 1 rect x 1 L rect y 1 L · rect ( x 1 - Δ x ) / L rect ( y 1 - Δ y ) / L exp [ j ( 2 π / λ 2 d i ) ( 1 + ( d i / d p ) ) ( x 1 Δ x + y 1 Δ y ) ] · ϕ τ τ ( Δ x , Δ y ) exp [ j ( π / λ 2 d i ) ( 1 + ( d i / d p ) ) ( Δ x 2 + Δ y 2 ) ] · exp [ - j ( 2 π / λ 2 d i ) ( α Δ x + β Δ y ) ] ,
ϕ τ τ ( Δ x , Δ y ) = τ ( x 1 , y 1 ) τ ( x 1 - Δ x , y 1 - Δ y )
( π l 2 / λ 2 d i ) [ 1 + ( d i / d p ) ] < ( π / 4 ) .
( π L l / λ 2 d i ) [ 1 + ( d i / d p ) ] < ( π / 4 ) .
exp { j ( 2 π / λ 2 d i ) [ 1 + ( d i / d p ) ] ( x 1 Δ x + y 1 Δ y ) } 1 exp { j ( π / λ 2 d i ) [ 1 + ( d i / d p ) ] ( Δ x 2 + Δ y 2 ) } 1 ,
I N ( α , β ) = I p A t λ 2 2 d i 2 - + Λ ( Δ y L ) Λ ( Δ y L ) ϕ τ τ ( Δ x , Δ y ) · exp [ - j ( 2 π / λ 2 d i ) ( α Δ x + β Δ y ) ] d Δ x d Δ y ,
Λ ( x ) = { 1 - x x 1 0 otherwise .
Λ ( Δ x / L ) 1             Λ ( Δ y / L ) 1 ,
I N ( α , β ) = I p A t λ 2 2 d i 2 - + ϕ τ τ ( Δ x , Δ y ) × exp [ - j 2 π λ 2 d i ( α Δ x + β Δ y ) ] d Δ x d Δ y .
I N ( α , β ) = I p A t λ 2 2 d i 2 - + ϕ τ τ ( Δ x , Δ y ) d Δ x d Δ y .
θ > ( L / 2 ) ( d i + d p ) / ( d i d p ) .
θ < λ 2 f max = λ 2 / 2 κ l .
( κ L / λ 2 d i ) [ 1 + ( d i / d p ) ] < 1.
I ( α , β ) = U i ( α , β ) + U N ( α , β ) 2 ,
σ 2 = I 2 - I 2 .
I 2 = I i 2 + 4 I i I N + 2 ( I N ) 2 .
( I ) 2 = I i 2 + 2 I i I N + ( I N ) 2
σ = I N [ 1 + 2 I i / I N ] 1 2 .