Abstract

A moving diffuser destroys the spatial coherence of laser light only partially when the integration time is finite; this can be expressed by the signal-to-noise (S/N) ratio in the observed illuminance, due to the residual speckle. The method can be improved by the use of a two-diffuser system of which one diffuser is motionless, the other moving. In this case, the integration time (or the displacement of the diffuser) required to obtain a given S/N ratio can be greatly reduced, allowing the use of a slowly moving diffuser. Moreover, the S/N ratio does not depend on the optical-system parameters, whereas it depends on these parameters when a single diffuser is used.

© 1971 Optical Society of America

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References

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  1. R. F. van Ligten, Nouv. Rev. Opt. Appl. 1, Suppl. n°2, 17 (1970).
    [Crossref]
  2. A. W. Lohmann, J. Opt. Soc. Am. 55, 1030 (1965).
    [Crossref]
  3. H. Arsenault and S. Lowenthal, Opt. Commun. 1, 451 (1970).
    [Crossref]
  4. S. Lowenthal, D. Joyeux, and H. Arsenault, Opt. Commun. 2, 184 (1970).
    [Crossref]
  5. L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).
    [Crossref]
  6. H. H. Hopkins, Nouv. Rev. Opt. Appl. 1, Suppl. n°2, 3 (1970).
  7. S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
    [Crossref]
  8. L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
    [Crossref]
  9. W. Martiensen and E. Spiller, Am. J. Phys. 32, 919 (1964).
    [Crossref]
  10. J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
    [Crossref] [PubMed]
  11. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw–Hill, New York, 1965).
  12. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces (Pergamon, New York, 1963).

1970 (5)

R. F. van Ligten, Nouv. Rev. Opt. Appl. 1, Suppl. n°2, 17 (1970).
[Crossref]

H. Arsenault and S. Lowenthal, Opt. Commun. 1, 451 (1970).
[Crossref]

S. Lowenthal, D. Joyeux, and H. Arsenault, Opt. Commun. 2, 184 (1970).
[Crossref]

H. H. Hopkins, Nouv. Rev. Opt. Appl. 1, Suppl. n°2, 3 (1970).

S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
[Crossref]

1967 (2)

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

J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
[Crossref] [PubMed]

1965 (2)

1964 (1)

W. Martiensen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[Crossref]

Arsenault, H.

S. Lowenthal, D. Joyeux, and H. Arsenault, Opt. Commun. 2, 184 (1970).
[Crossref]

H. Arsenault and S. Lowenthal, Opt. Commun. 1, 451 (1970).
[Crossref]

S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
[Crossref]

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces (Pergamon, New York, 1963).

Enloe, L. H.

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

Goldfischer, L. I.

Goodman, J. W.

Hopkins, H. H.

H. H. Hopkins, Nouv. Rev. Opt. Appl. 1, Suppl. n°2, 3 (1970).

Joyeux, D.

S. Lowenthal, D. Joyeux, and H. Arsenault, Opt. Commun. 2, 184 (1970).
[Crossref]

Lohmann, A. W.

Lowenthal, S.

S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
[Crossref]

H. Arsenault and S. Lowenthal, Opt. Commun. 1, 451 (1970).
[Crossref]

S. Lowenthal, D. Joyeux, and H. Arsenault, Opt. Commun. 2, 184 (1970).
[Crossref]

Martiensen, W.

W. Martiensen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[Crossref]

Papoulis, A.

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

Spiller, E.

W. Martiensen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[Crossref]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces (Pergamon, New York, 1963).

van Ligten, R. F.

R. F. van Ligten, Nouv. Rev. Opt. Appl. 1, Suppl. n°2, 17 (1970).
[Crossref]

Am. J. Phys. (1)

W. Martiensen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[Crossref]

Bell System Tech. J. (1)

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

J. Opt. Soc. Am. (4)

Nouv. Rev. Opt. Appl. (2)

R. F. van Ligten, Nouv. Rev. Opt. Appl. 1, Suppl. n°2, 17 (1970).
[Crossref]

H. H. Hopkins, Nouv. Rev. Opt. Appl. 1, Suppl. n°2, 3 (1970).

Opt. Commun. (2)

H. Arsenault and S. Lowenthal, Opt. Commun. 1, 451 (1970).
[Crossref]

S. Lowenthal, D. Joyeux, and H. Arsenault, Opt. Commun. 2, 184 (1970).
[Crossref]

Other (2)

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

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces (Pergamon, New York, 1963).

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Figures (7)

F. 1
F. 1

One possible optical configuration. The object plane Π0 consists of two diffusers. The diffuser d1 is motionless and d2 is moving with constant velocity υ. The image plane Π is formed by a linear and shift-invariant (in amplitude) optical system whose magnification is normalized to +1. The pupil P is in the Fourier plane of the system; the aperture angle is α.

F. 2
F. 2

Static diffuser: the speckle-size correlation radius L depends only on the optical system, i.e., the image aperture sinα, through the relation L = L0= 1.22/(2 sinα) for a circular pupil. Here L0 = 0.18 mm.

F. 3
F. 3

Single moving diffuser, same optical system as in Fig. 2; the residual speckling is described by the S/N ratio ρ, which is independent of the nature of the diffuser, but depends on the total diffuser displacement X and on the optical system. Here, with X = 0.13 mm, L = 0.18 mm, the averaging is hardly noticeable, i.e., ρ∼1.

F. 4
F. 4

Double diffuser (Kodatrace sheet), same optical system as in Fig. 3, same displacement X of the diffuser. The S/N ratio does not depend on the optical system but on the diffuser only through ρ = ( X / L ) 1 2, where L is the correlation length of the diffuser. Here L = 6 μm, X = 0.13 mm. The averaging yields ρ∼5.

F. 5
F. 5

Double diffuser (milk glass). The conditions are the same as in Fig. 4 except for the correlation length L, which is much smaller. Here L∼0.4μ and X = 0.13 mm yield ρ∼18. Comparison with Fig. 4 shows clearly the influence of the diffuser properties in the double-diffuser method.

F. 6
F. 6

Single moving diffuser. A S/N ratio ρ∼5.3 (for L = 0.18 mm) is achieved with a displacement of the diffuser X = 5 mm, whereas with the double Kodatrace the same ρ is obtained for a smaller (1/30) displacement, i.e., X = 0.17 mm. With the double milk glass, the displacement would be only X = 11 μm.

F. 7
F. 7

Double diffuser rotating around the optical axis. It can be seen how the averaging increases with the linear displacement (i.e., with increasing radius). Comparison with Fig. 6 shows that the averaging corresponds to a different physical phenomenon than when a single diffuser is used. The displacement here is about 0.05 mm at the edge of the field; with L = 6 μ (Kodatrace), this corresponds to ρ = 3.

Equations (25)

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J T ( r , t ) = t ( T / 2 ) t + ( T / 2 ) I ( r , t ) d t .
ρ = J T / σ .
I ( r , t ) = | A ( r , t ) | 2 = | d 1 ( r ) d 2 ( r v t ) * h ( r ) | 2 ,
J T ( r , t ) = I ( r , t ) * g ( t )
g ( t ) = { 1 / T for | t | < T / 2 0 elsewhere
J T ( r , t ) = C ,
σ 2 = R J ( 0 , 0 ) [ J T ( r , t ) ] 2 ,
R J ( r , τ ) = J T ( r 1 , t 1 ) J T ( r 2 , t 2 ) ,
R J ( r , τ ) = R I ( r , τ ) * g ( τ ) * g ( τ ) .
R I ( r , τ ) = | A ( r 1 , t 1 ) | 2 | A ( r 2 , t 2 ) | 2
X 1 2 X 2 2 = X 1 2 X 2 2 + 2 X 1 X 2 2
| Z 1 | 2 | Z 2 | 2 = | Z 1 | 2 | Z 2 | 2 + | Z 1 Z 2 * | 2 ,
R I ( r , τ ) = | A ( r 1 , t 1 ) | 2 | A ( r 2 , t 2 ) | 2 + | A ( r 1 , t 1 ) A * ( r 2 , t 2 ) | 2 .
σ 2 = [ | R d ( r ) R d ( r v τ ) * h ( r ) * h * ( r ) | 2 * g ( τ ) * g ( τ ) ] τ = 0 , τ = 0
r = r 2 r 1 ,
R d ( r ) = d 1 ( r 1 ) d 1 * ( r 2 ) = d 2 ( r 1 ) d 2 * ( r 2 )
f ( v τ ) = f ( v τ , 0 ) = | R d ( r ) R d ( r v τ ) * h ( r ) * h * ( r ) | 2 r = 0 ,
σ 2 = [ C | S d ( v τ ) | 2 * g ( τ ) * g ( τ ) ] τ = 0 ,
S d ( r ) = [ R d ( r ) * R d ( r ) ] / [ R d ( r ) * R d ( r ) ] r = 0
ρ = [ 1 T T + T ( 1 τ T ) | S d ( v τ ) | 2 d τ ] 1 2 .
ρ [ T / + | S d ( υ τ , 0 ) | 2 d τ ] 1 2 = ( υ T L ) 1 2 ,
L = + | S d ( υ τ , 0 ) | 2 d ( υ τ ) ,
R d ( x , y ) = exp [ π ( x 2 + y 2 ) / a 2 ] ,
ρ = N 1 2 ,
ρ = N 1 2 with N = X / L .