Abstract

This paper considers diffusers characterized by random variations of optical path, such as ground-glass-surface diffusers. The theoretical limits on the light distributions realizable with random phase diffusers are derived, and the important parameters controlling these light distributions are identified. Methods for generating controlled random signals that, when converted to optical-path variations, have the correct parameters to synthesize diffusers with any desired, realizable light distribution are described. Some goniophotometric data taken from diffusers synthesized by this method are given in support of the theory. These results represent a basic solution of the random-phase-diffuser synthesis problem.

© 1973 Optical Society of America

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References

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  1. J. Upatneiks, Appl. Opt. 6, 1905 (1967).
    [Crossref]
  2. J. S. Chandler and J. J. DePalma, J. Soc. Motion Pict. Telev. Eng. 77, 1012 (1968).
  3. C. N. Kurtz, J. Opt. Soc. Am. 62, 982 (1972).
    [Crossref]
  4. H. J. Caulfield, in Proceedings of the SPIE Seminar on Developments in Holography, Vol. 25 (Society of Photo-Optical Instrumentation Engineers, Redondo Beach, Calif., 1971), p. 111.
    [Crossref]
  5. R. C. Waag and K. T. Knox, J. Opt. Soc. Am. 62, 877 (1972).
    [Crossref]
  6. F. Bestenreiner and W. Weiershausen, Optik 32, 446 (1971).
  7. H. F. Langworthy and J. J. DePalma, J. Opt. Soc. Am. 63, 488A (1973).
  8. D. O. Muhleman, Astron. J. 69, 34 (1964).
    [Crossref]
  9. D. E. Barrick, IEEE Trans. Antennas Propag. 16, 449 (1968).
    [Crossref]
  10. M. Bora and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 402.
  11. Reference 10, p. 752.
  12. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw–Hill, New York, 1958), pp. 32–35.
  13. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun. Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55 (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965), p. 297.
  14. A. Papoulis, Probability, Random Signals and Stochastic Processes (McGraw–Hill, New York, 1965), p. 226.
  15. Reference 10, p. 513, Prob. No. 14-12.
  16. Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 332.

1973 (1)

H. F. Langworthy and J. J. DePalma, J. Opt. Soc. Am. 63, 488A (1973).

1972 (2)

1971 (1)

F. Bestenreiner and W. Weiershausen, Optik 32, 446 (1971).

1968 (2)

J. S. Chandler and J. J. DePalma, J. Soc. Motion Pict. Telev. Eng. 77, 1012 (1968).

D. E. Barrick, IEEE Trans. Antennas Propag. 16, 449 (1968).
[Crossref]

1967 (1)

1964 (1)

D. O. Muhleman, Astron. J. 69, 34 (1964).
[Crossref]

Abramowitz, M.

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun. Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55 (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965), p. 297.

Barrick, D. E.

D. E. Barrick, IEEE Trans. Antennas Propag. 16, 449 (1968).
[Crossref]

Bestenreiner, F.

F. Bestenreiner and W. Weiershausen, Optik 32, 446 (1971).

Bora, M.

M. Bora and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 402.

Caulfield, H. J.

H. J. Caulfield, in Proceedings of the SPIE Seminar on Developments in Holography, Vol. 25 (Society of Photo-Optical Instrumentation Engineers, Redondo Beach, Calif., 1971), p. 111.
[Crossref]

Chandler, J. S.

J. S. Chandler and J. J. DePalma, J. Soc. Motion Pict. Telev. Eng. 77, 1012 (1968).

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw–Hill, New York, 1958), pp. 32–35.

DePalma, J. J.

H. F. Langworthy and J. J. DePalma, J. Opt. Soc. Am. 63, 488A (1973).

J. S. Chandler and J. J. DePalma, J. Soc. Motion Pict. Telev. Eng. 77, 1012 (1968).

Knox, K. T.

Kurtz, C. N.

Langworthy, H. F.

H. F. Langworthy and J. J. DePalma, J. Opt. Soc. Am. 63, 488A (1973).

Lee, Y. W.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 332.

Muhleman, D. O.

D. O. Muhleman, Astron. J. 69, 34 (1964).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Signals and Stochastic Processes (McGraw–Hill, New York, 1965), p. 226.

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw–Hill, New York, 1958), pp. 32–35.

Stegun, I. A.

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun. Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55 (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965), p. 297.

Upatneiks, J.

Waag, R. C.

Weiershausen, W.

F. Bestenreiner and W. Weiershausen, Optik 32, 446 (1971).

Wolf, E.

M. Bora and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 402.

Appl. Opt. (1)

Astron. J. (1)

D. O. Muhleman, Astron. J. 69, 34 (1964).
[Crossref]

IEEE Trans. Antennas Propag. (1)

D. E. Barrick, IEEE Trans. Antennas Propag. 16, 449 (1968).
[Crossref]

J. Opt. Soc. Am. (3)

J. Soc. Motion Pict. Telev. Eng. (1)

J. S. Chandler and J. J. DePalma, J. Soc. Motion Pict. Telev. Eng. 77, 1012 (1968).

Optik (1)

F. Bestenreiner and W. Weiershausen, Optik 32, 446 (1971).

Other (8)

H. J. Caulfield, in Proceedings of the SPIE Seminar on Developments in Holography, Vol. 25 (Society of Photo-Optical Instrumentation Engineers, Redondo Beach, Calif., 1971), p. 111.
[Crossref]

M. Bora and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 402.

Reference 10, p. 752.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw–Hill, New York, 1958), pp. 32–35.

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun. Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55 (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965), p. 297.

A. Papoulis, Probability, Random Signals and Stochastic Processes (McGraw–Hill, New York, 1965), p. 226.

Reference 10, p. 513, Prob. No. 14-12.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 332.

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Figures (15)

Fig. 1
Fig. 1

Optical-path variation of an equivalent microelement. For the case of an element consisting of a surface-to-air interface, the optical path is proportional to the actual surface contour.

Fig. 2
Fig. 2

Prismatic-equivalent dement designed to redirect light equally into each of two directions.

Fig. 3
Fig. 3

Equivalent element Φe(x) and its corresponding derivative, or slope Ψe(x). The inverse curve of Ψe(x) is denoted xe) and is simply the lower curve plotted with the abscissa and ordinate interchanged.

Fig. 4
Fig. 4

(a) Quadratic microelement given by Eq. (14). (b) Slope Ψe, calculated from Eq. (15). (c) Density of slopes pe), calculated for this microelement from Eq. (16). A quadratic element has uniformly distributed slopes.

Fig. 5
Fig. 5

Interpretation of the envelope concept. The top illustration shows the irradiance distribution I2(y) in the back focal plane of lens L when a single microelement is in the front focal plane. Regular interference effects from an array of microelements are shown in the center; random interference effects from a random surface are shown at the bottom. In the latter two cases the dotted line represents the envelope of the irradiance distribution.

Fig. 6
Fig. 6

Noise generator NG provides random zero-mean signal to frequency filter FF. Its output V(t) is fed through nonlinear element NE to produce random-slope signal Ψ(t), which is integrated by integrator I to produce the optical-path signal Φ(t).

Fig. 7
Fig. 7

Plot of the nonlinearity g(V) required, according to Eq. (48), to produce a random signal having the same slope probability density p(Ψ) as a quadratic microelement.

Fig. 8
Fig. 8

Sinusoidal equivalent element with maximum slope of magnitude πδ/2a.

Fig. 9
Fig. 9

Plot of the nonlinearity g(V) required, according to Eq. (51), to produce a random signal having the same slope probability density p(Ψ) as a sinusoidal microelement.

Fig. 10
Fig. 10

Density of slopes of a sinusoidal microelement. The signal out of the nonlinear element shown in Fig. 9 also has this same density of slopes.

Fig. 11
Fig. 11

Prismatic-equivalent element. Two directions and the relative amount of light directed into each of them depend upon parameters b and m.

Fig. 12
Fig. 12

Plot of the nonlinear element needed to generate a random signal with the same slope probability density as the prismatic element of Fig. 11. The nonlinearity is essentially a biased hard limiter, the bias level depending on parameters b and a. The bias level is determined by b according to the abscissa, which corresponds to an ordinate of b/a on the upper error-function curve.

Fig. 13
Fig. 13

Required nonlinearity for a combination prismatic- and quadratic-equivalent element is given by the heavy line, D. It is made up of sections of B and C, given by Eqs. 57(b) and 57(a), respectively. The location of the discontinuity in curve D is determined from the error-function curve A by the abscissa corresponding to the ordinate value b/a which, in this case, was chosen as b/a = 0.5. For the plot, m/a was taken as 0.2 and πδ/2a was taken as 0.4.

Fig. 14
Fig. 14

Relative spectra of V, Ψ, and Φ. The upper curve is the ideal pass band |H(f)|2 corresponding to the power spectrum of V, and the middle curve is the power spectrum of Ψ. Both were normalized to unity; they differ only slightly. The bottom curve indicates the spectrum of the optical-path signal Φ; it shows the effect of the integration of Ψ.

Fig. 15
Fig. 15

Light-distribution characteristic caused by the random undulations of the optical-path variations Φ(x). It is square because the signal had a uniform density of slopes, and it is smooth because the measurement was made with white light to average out any irradiance fluctuations.

Equations (78)

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U 1 ( x ) = T ( x ) ,
U 2 ( y ) = T ˆ ( y / f λ ) ,
T ˆ ( ν ) = T ( x ) e 2 π i ν 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 T ( y / λ f ) .
sin θ = ± λ / f = ± λ ν 0 .
T ( x ) = e i k Φ ( x ) ,
Ψ e ( x ) = d Φ e ( x ) / d x ,
F ( Ψ ) = prob { slope Ψ } = Ψ p ( Ψ ) d Ψ ,
p ( Ψ e ) = d F ( Ψ e ) / d Ψ e .
F ( Ψ e ) = m / n = [ a x ( Ψ e ) ] / 2 a .
F ( Ψ e ) = m / n = % Flux refracted by slopes Ψ e ,
p ( Ψ e ) = d F / d Ψ e = ( 1 / 2 a ) d [ x ( Ψ e ) ] / d Ψ e .
Φ e ( x ) = ( π / 4 ) [ 1 δ ( x / a ) 2 ] , δ < 2 a / π
Ψ e ( x ) = d Φ e / d x = π δ x / 2 a 2 ,
x ( Ψ e ) = 2 a 2 Ψ e / δ π ;
p ( Ψ e ) = ( 1 / 2 a ) d x / d Ψ e = a / δ π ,
T ˆ ( ν ) = e i k Φ ( x ) 2 π i ν x d x .
T ˆ ( ν ) = e i k [ Φ ( x ) 2 π ν x / k ] d x .
Φ ( x 0 ) = 2 π ν / k ,
T ˆ ( ν ) = e i k Φ ( x 0 ) e i 2 π ν x 0 e i k ( x x 0 ) 2 Φ ( x 0 ) / 2 d x .
e i k ( x x 0 ) 2 Φ ( x 0 ) / 2 d x = { ( 2 π / k Φ ) 1 2 e i π / 4 , Φ > 0 ( 2 π / k | Φ | ) 1 2 e i π / 4 Φ < 0 .
| T ˆ ( ν ) | 2 2 π / k | Φ ( x 0 ) | .
Ψ ( x 0 ) = 2 π ν / k = λ ν ;
| T ˆ ( ν ) | 2 λ / | d Ψ / d x | x 0 .
| T ˆ ( ν ) | 2 | d x / d Ψ | Ψ = λ ν ,
S T ( ν ) E { | d x / d Ψ | Ψ = λ ν } .
p ( Ψ e ) = ( 1 / 2 a ) d x / d Ψ e
p ( λ ν ) = ( λ / 2 a ) | d x / d Ψ e | Ψ e = λ ν ,
S T ( ν ) p ( λ ν )
S T n ( ν ) = λ p ( λ ν ) ,
Ψ = g ( V ) ,
d h / d Ψ = p ( Ψ ) / p υ ( h ) ,
V = h ( Ψ ) .
h p υ ( h ) d h = Ψ p ( Ψ ) d Ψ ,
F υ ( h ) = F ( Ψ ) .
F υ ( h ) = 0.5 + 0.5 f υ ( h )
F ( Ψ ) = 0.5 + 0.5 f ( Ψ )
f ( Ψ ) = f υ ( h ) .
Ψ ( f ) = { f ( Ψ ) } 1 ,
Ψ { f } = Ψ { f υ ( h ) } ,
g ( V ) = Ψ { f υ ( V ) } ,
Ψ e ( a u ) = Ψ ( u ) ,
g ( V ) = Ψ e { a f υ ( V ) } .
p υ ( V ) = ( 1 / ( 2 π ) 1 2 V 0 ) e ( V / 2 V 0 ) 2 ,
g ( V ) = Ψ { erf ( V / 2 V 0 ) }
g ( V ) = Ψ e { a erf ( V / 2 V 0 ) } ,
Φ e ( x ) = ( π / 4 ) [ 1 δ ( x / a ) 2 ] , | x | a , δ < 2 a / π
Ψ e ( x ) = δ π x / 2 a 2 , | x | a
g ( V ) = ( π δ / 2 a ) erf ( V / 2 V 0 ) .
Φ e ( x ) = δ cos ( π x / 2 a ) , | x | a , δ < 2 a / π
Ψ e ( x ) = ( π δ / 2 a ) sin ( π x / 2 a ) ,
g ( V ) = ( π δ / 2 a ) sin { ( π / 2 ) erf ( V / 2 V 0 ) } ,
p ( Ψ ) = ( 1 / π ) / [ ( δ π / 2 a ) 2 Ψ 2 ] 1 2 , δ < 2 a / π
Φ e ( x ) = { m ( 1 x / a ) , b x a m [ ( a + b ) / ( a b ) ] ( 1 + x / a ) , a x b
Ψ e ( x ) = { ( m / a ) , b x a ( m / a ) ( a + b ) / ( a b ) , a x b
g ( V ) = { ( m / a ) , 1 erf ( V / 2 V 0 ) b / a ( m / a ) ( a + b ) / ( a b ) , b / a erf ( V / 2 V 0 ) 1 .
Ψ e ( x ) = ( π δ x / 2 a 2 ) + { ( m / a ) , b x a ( m / a ) ( a + b ) / ( a b ) , a x b .
g ( V ) = { ( π δ / 2 a ) erf ( V / 2 V 0 ) ( m / a ) , 1 erf ( V / 2 V 0 ) b / a ( π δ / 2 a ) erf ( V / 2 V 0 ) + ( m / a ) ( a + b ) / ( a b ) , b / a erf ( V / 2 V 0 ) 1
S υ ( f ) | H ( f ) | 2 ,
ρ υ ( τ ) = | H ( f ) | 2 e 2 π i τ f d f / | H ( f ) | 2 d f .
Ψ = g ( V ) = ( π δ / a ) Erf ( V / V 0 ) ,
erf ( x ) = 2 Erf ( 2 x )
R Ψ ( τ ) = E { g [ V ( t ) ] g [ V ( t + τ ) ] } = E { ( π δ / a ) 2 Erf [ V ( t ) / V 0 ] Erf [ V ( t + τ ) / V 0 ] } .
d R Ψ / d R υ = ( 2 / V 1 V 2 ) [ ( π δ / a ) 2 Erf ( V 1 / V 0 ) × Erf ( V 2 / V 0 ) ] p ( V 1 , V 2 ) d V 1 d V 2 ,
p ( V 1 , V 2 ) = { 2 π V 0 [ 1 ρ υ ( τ ) ] 1 2 } 1 × exp { [ 2 V 0 2 ( 1 ρ υ ( τ ) ] 1 × [ V 1 2 + V 1 2 2 ρ υ ( τ ) V 1 V 2 ] } .
d R Ψ / d R υ = ( δ π / 2 a V 0 ) 2 × exp { ( V 1 2 / 2 V 0 2 ) ( V 2 2 / 2 V 0 2 ) } p ( V 1 , V 2 ) d V 1 d V 2 ,
d Erf ( x ) / d x = ( 1 / ( 2 π ) 1 2 ) e x 2 / 2 .
p υ ( V ) = ( [ 2 π ] 1 2 V 0 ) 1 e V 2 / 2 V 0 2
d R Ψ / d R υ = ( π δ / a ) 2 E { p υ ( V 1 ) p υ ( V 2 ) } .
d R Ψ / d R υ = ( π δ / a ) 2 { 2 π [ 4 V 0 4 R υ 2 ( τ ) ] 1 2 } 1 ,
R Ψ ( τ ) = ( π δ 2 / 2 a 2 ) sin 1 [ ρ υ ( τ ) / 2 ] .
ρ Ψ ( τ ) = R Ψ ( τ ) / R Ψ ( 0 ) = ( 6 / π ) sin 1 [ ρ υ ( τ ) / 2 ]
S Ψ ( f ) = R ˆ Ψ ( f ) .
S Φ ( f ) = S Ψ ( f ) / 4 π 2 f 2 ,
Ψ = 2 erf ( V / 2.2 2 ) ,
Φ r ( x ) = 2 Φ ( x ) / ( 1 n ) ,
Φ ( x , z ) = Φ 1 ( x ) + Φ 2 ( z ) .