Abstract

The autocorrelation of the amplitude transmittance of surface diffusers and their scattering characteristics are related to the optical-path autocorrelation for several sets of optical-path statistics. Simple relationships are found for the gaussian case and experimental evidence is given to support the view of a ground-glass surface being accurately modeled by a gaussian process. Some closed-form results are also obtained for binary diffusers, including the design of a quasi-band-limited binary diffuser with a much improved power spectrum.

© 1972 Optical Society of America

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References

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  1. J. Upatnieks, Appl. Opt. 6, 1905 (1967).
    [Crossref] [PubMed]
  2. S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
    [Crossref]
  3. L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).
    [Crossref]
  4. See P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963). Chapters 4 and 5 discuss several alternative models of a rough surface and provide many pertinent references.
  5. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw–Hill, New York, 1965), p. 306.
  6. Reference 5, p. 213.
  7. R. B. Crane, J. Opt. Soc. Am. 60, 1658 (1970).
    [Crossref]
  8. L.I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
    [Crossref]
  9. Reference 5, p. 289.
  10. Reference 5, p. 290.
  11. Reference 5, p. 486.
  12. A. Y. Khintchine, Mathematical Methods in the Theory of Queueing (Hafner, New York, 1960).

1970 (2)

1967 (2)

L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).
[Crossref]

J. Upatnieks, Appl. Opt. 6, 1905 (1967).
[Crossref] [PubMed]

1965 (1)

Arsenault, H.

Beckmann, P.

See P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963). Chapters 4 and 5 discuss several alternative models of a rough surface and provide many pertinent references.

Crane, R. B.

Enloe, L. H.

L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).
[Crossref]

Goldfischer, L.I.

Khintchine, A. Y.

A. Y. Khintchine, Mathematical Methods in the Theory of Queueing (Hafner, New York, 1960).

Lowenthal, S.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw–Hill, New York, 1965), p. 306.

Spizzichino, A.

See P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963). Chapters 4 and 5 discuss several alternative models of a rough surface and provide many pertinent references.

Upatnieks, J.

Appl. Opt. (1)

Bell System Tech. J. (1)

L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).
[Crossref]

J. Opt. Soc. Am. (3)

Other (7)

Reference 5, p. 289.

Reference 5, p. 290.

Reference 5, p. 486.

A. Y. Khintchine, Mathematical Methods in the Theory of Queueing (Hafner, New York, 1960).

See P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963). Chapters 4 and 5 discuss several alternative models of a rough surface and provide many pertinent references.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw–Hill, New York, 1965), p. 306.

Reference 5, p. 213.

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Figures (10)

Fig. 1
Fig. 1

Unity-magnification diffuser imaging system. Dashed lines show the contribution of the v0 spatial-frequency component of the diffuser transmittance T(x) to the spectrum display in the aperture plane. Lenses L1 and L2 are identical, with focal length f. The dashed lines are located in the aperture plane at y = ±fλv0. U1 is the amplitude at the diffuser surface and U2 is the amplitude in the aperture plane.

Fig. 2
Fig. 2

Comparison of RT(τ) and ρΦ(τ) for the gaussian case, when ρΦ(τ) = sinτ/τ. For the small deviation of only σΦ ≐ λ/4 assumed, RT(∞) = 0.1, resulting in a significant specular component.

Fig. 3
Fig. 3

Plot of relative specular component from ground glass as the surface variance σΦ2 is reduced by liquid gating with liquids of different indices. The variance is proportional to (Δn)2, where Δn is the difference of the liquid and glass refractive indices. The curve is an exponential fit to the data.

Fig. 4
Fig. 4

Plot showing RT(τ) and ρΦ(τ) when ρΦ is gaussian with correlation distance τΦ The transmittance correlation distance τT is always less than the surface correlation distance τΦ by the factor 1/Φ when σΦ is large enough to eliminate the specular component. In the illustration, σΦ ≐ 0.8λ, so Φ = 5.

Fig. 5
Fig. 5

Relative exitance vs scattering angle for ground-glass diffusers ground with various grits. The back surface of the glasses was index matched to eliminate multiple reflections.

Fig. 6
Fig. 6

General character of one-dimensional binary surface. The total optical-path difference is 2μ, and the points where the optical path changes between +μ and −μ are randomly located.

Fig. 7
Fig. 7

Illustration of the method of generation of the hard-clipped gaussian binary wave. The limiter provides an output of +μ or −μ when the gaussian input is positive or negative, respectively.

Fig. 8
Fig. 8

A plot of the transform of Eq. (34). Since the transform goes negative it cannot represent a power spectrum.

Fig. 9
Fig. 9

Comparison of the ideal, sincτ, and the function ρT given by Eq. (35). The value at the origin and all the zeros correspond, along with fairly similar shapes. The largest differences occur near the origin; particularly note the difference of slope at τ = 0.

Fig. 10
Fig. 10

Comparison of the actual power spectra computed by Fourier transforming Eq. (35) and the ideal, rect(v). A quite ideal spectrum is achieved.

Equations (58)

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U 2 ( y ) = T ˆ ( y / f λ ) ,
T ˆ ( v ) = - T ( x ) e 2 π i v x d x ,
I 2 ( y ) = U 2 ( y ) 2 = T ˆ ( y / λ f ) 2 .
E { I 2 ( y ) } = E { T ˆ ( y / λ f ) 2 } S ( y / λ f ) .
sin θ = ± y / f = ± λ v 0 .
T ( x ) = e i k Φ ( x ) ,
S ( v ) R ˆ T ( v ) .
R T ( τ ) = - e i k Φ 1 e - i k Φ 2 p 2 ( Φ 1 , Φ 2 , τ ) d Φ 1 d Φ 2 .
C ( ω 1 , ω 2 , τ ) = - e i ω 1 Φ 1 + i ω 2 Φ 2 p 2 ( Φ 1 , Φ 2 , τ ) d Φ 1 d Φ 2 ,
R T ( τ ) = C ( k , - k , τ ) .
R T ( τ ) = e - k 2 σ Φ 2 [ 1 - ρ Φ ( τ ) ] ,
R T ( 0 ) = 1 ,
R T ( ) = e - k 2 σ Φ 2 ,
e - 2 k 2 σ Φ 2 R T ( τ ) 1.
R T ( τ ) = R T ( ) + R T ( τ ) ,
S ( v ) = R T ( ) δ ( v ) + - R T ( v ) e 2 π i v τ d τ .
R T ( τ ) = e - k 2 σ Φ 2 [ 1 - exp ( - τ 2 / τ Φ 2 ) ] .
τ T τ Φ / k σ Φ .
p ( μ / μ ) = p ( - μ / - μ ) = p e ( τ ) ,
p ( - μ / μ ) = p ( μ / - μ ) = p 0 ( τ ) ,
p μ , μ = p ( Φ = + μ ) p ( μ / μ ) = 0.5 p e ( τ ) ,
p μ , - μ = p ( Φ = + μ ) p ( - μ / μ ) = 0.5 p 0 ( τ ) ,
p - μ , μ = p ( Φ = - μ ) p ( μ / - μ ) = 0.5 p 0 ( τ ) ,
p - μ , - μ = p ( Φ = - μ ) p ( - μ / - μ ) = 0.5 p e ( τ ) .
R T ( τ ) = - - e i k Φ 1 e - i k Φ 2 { 0.5 p e ( τ ) δ ( Φ 1 - μ ) δ ( Φ 2 - μ ) + 0.5 p e ( τ ) δ ( Φ 1 + μ ) δ ( Φ 2 + μ ) + 0.5 p 0 ( τ ) δ ( Φ 1 - μ ) δ ( Φ 2 + μ ) + 0.5 p 0 ( τ ) δ ( Φ 1 + μ ) δ ( Φ 2 - μ ) } d Φ 1 d Φ 2 ,
R T ( τ ) = p e ( τ ) + p 0 ( τ ) cos ( 2 μ k ) .
R T ( τ ) = 1 - p 0 ( τ ) [ 1 - cos ( 2 μ k ) ] .
R T ( 0 ) = 1
R T ( ) = 0.5 [ 1 + cos ( 2 μ k ) ] ,
2 μ = ( m + 1 2 ) λ ,
R T ( τ ) = 1 - 2 p 0 ( τ ) .
p 0 ( τ ) = 0.5 ( 1 - e - 2 β τ ) ,
R T ( τ ) = e - 2 β τ
p 0 ( τ ) = ( 1 / π ) cos - 1 [ ρ g ( τ ) ] ,
cos - 1 ( ρ ) = π / 2 - sin - 1 ( ρ )
p 0 ( τ ) = ( 1 2 ) - ( 1 / π ) sin - 1 [ ρ g ( τ ) ] .
R T ( τ ) = ( 2 / π ) sin - 1 [ ρ g ( τ ) ] ,
- 1 R T ( τ ) 1.
ρ g ( τ ) = - H ( v ) 2 e - 2 π i τ v d v / - H ( v ) 2 d v .
ρ g ( τ ) = sin [ ( π / 2 ) ρ T ( τ ) ] ,
ρ g ( τ ) = sin [ ( π / 2 ) sinc ( τ ) ] .
ρ T ( τ ) = ( 2 / π ) sin - 1 [ sinc τ ]
ρ T ( τ ) ~ ( 2 / π ) sinc τ .
p 0 ( τ ) = 0.5 - 0.5 sinc τ .
p { N ( A i + y ) = n i ,             i = 1 , , k }
v k ( τ ) = p { exactly k zeros in length τ } ,
m τ = k = 1 k v k ( τ ) k = 1 v k ( τ ) = ω ( τ ) = p { more than one zero in length τ } ,
m ω ( τ ) τ             for every τ .
lim τ 0 + ω ( τ ) τ = γ > 0 ,
m = E { N ( 0 l ] } = ,
v k ( τ ) = γ 0 τ [ ψ k - 1 ( u ) - ψ k ( u ) ] d u ,             k = 1 , 2 , .
p 0 ( τ ) = k odd v k ( τ ) = γ k odd 0 τ [ ψ k ( u ) - ψ k - 1 ( u ) ] d u .
k odd ψ k - 1 ( u ) = ψ 0 ( u ) + k odd , > 1 ψ k - 1 ( u ) = ψ 0 ( u ) + E { u } ,
p 0 ( τ ) = γ 0 τ ψ 0 ( u ) d u + γ 0 τ E { u } d u - γ 0 τ θ ( u ) d u .
d + d τ p 0 ( τ ) = lim 0 + 1 [ p 0 ( τ + ) - p 0 ( τ ) ] = γ ψ 0 ( τ ) + γ E { τ } + γ θ ( τ ) .
lim τ 0 + E { τ } = lim τ 0 + θ ( τ ) = 0 ,
lim τ 0 + d + d τ p 0 ( τ ) = γ > 0.
( d + / d τ ) sinc τ = ( d / d τ ) sinc τ 0             as τ 0 ,