Abstract

When a diffuse surface is illuminated by a coherent monochromatic source such as a laser, the illuminated area appears speckled. Exposing photographic film directly to the backscattered radiation confirms the independent existence of the speckles. The autocorrelation function of the speckle pattern so recorded is shown to be proportional to the diffraction pattern corresponding to the illumination function plus a constant. The power spectral density of the pattern is shown to be the convolution of the illumination function against itself displaced by an amount proportional to the space frequency of interest. The cases of single beam and double beam uniform circular illumination are treated explicitly and experimental verification is offered.

© 1965 Optical Society of America

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References

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  1. B. M. Oliver, Proc. IEEE 51, 220 (1963).
    [Crossref]
  2. J. D. Rigden and E. I. Gordon, Proc. IRE 50, 2367 (1962).
  3. E. A. Guillemin, The Mathematics of Circuit Analysis (John Wiley & Sons, Inc., New York, 1949), Chap. 7, Sec. 23.
  4. S. O. Rice, Selected Papers on Noise and Stochastic Processes (Dover Publications, Inc., New York, 1954), pp. 263–264.
  5. F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Co., Inc., New York, 1937), p. 180.

1963 (1)

B. M. Oliver, Proc. IEEE 51, 220 (1963).
[Crossref]

1962 (1)

J. D. Rigden and E. I. Gordon, Proc. IRE 50, 2367 (1962).

Gordon, E. I.

J. D. Rigden and E. I. Gordon, Proc. IRE 50, 2367 (1962).

Guillemin, E. A.

E. A. Guillemin, The Mathematics of Circuit Analysis (John Wiley & Sons, Inc., New York, 1949), Chap. 7, Sec. 23.

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Co., Inc., New York, 1937), p. 180.

Oliver, B. M.

B. M. Oliver, Proc. IEEE 51, 220 (1963).
[Crossref]

Rice, S. O.

S. O. Rice, Selected Papers on Noise and Stochastic Processes (Dover Publications, Inc., New York, 1954), pp. 263–264.

Rigden, J. D.

J. D. Rigden and E. I. Gordon, Proc. IRE 50, 2367 (1962).

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Co., Inc., New York, 1937), p. 180.

Proc. IEEE (1)

B. M. Oliver, Proc. IEEE 51, 220 (1963).
[Crossref]

Proc. IRE (1)

J. D. Rigden and E. I. Gordon, Proc. IRE 50, 2367 (1962).

Other (3)

E. A. Guillemin, The Mathematics of Circuit Analysis (John Wiley & Sons, Inc., New York, 1949), Chap. 7, Sec. 23.

S. O. Rice, Selected Papers on Noise and Stochastic Processes (Dover Publications, Inc., New York, 1954), pp. 263–264.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Co., Inc., New York, 1937), p. 180.

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Figures (8)

Fig. 1
Fig. 1

Speckle pattern produced by laser illumination.

Fig. 2
Fig. 2

Geometry pertinent to speckle pattern analysis.

Fig. 3
Fig. 3

Experimental arrangement.

Fig. 4
Fig. 4

Optical correlator.

Fig. 5
Fig. 5

(a) Speckle pattern for a 1-mm aperture; (b) autocorrelation function of Fig. 5(a); (c) diffraction pattern for a 1-mm aperture.

Fig. 6
Fig. 6

(a) Speckle pattern for a 8-mm aperture, taken at ×8, shown here at ×3.84; (b) autocorrelation function of Fig. 6(a); (c) diffraction pattern for a 8-mm aperture.

Fig. 7
Fig. 7

(a) Speckle pattern for two 1-mm apertures separated 2 mm on centers; (b) autocorrelation function of Figure 7(a); (c) diffraction pattern for two 1-mm apertures separated 2 mm on centers.

Fig. 8
Fig. 8

(a) Speckle pattern for two 8-mm apertures separated 16 mm on centers, taken at ×8 and shown here at ×3.84; (b) autocorrelation function of Fig. 8(a); (c) diffraction pattern for two 8-mm apertures separated 16 mm on centers.

Equations (32)

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[ α P ( u , v ) Δ u Δ v / π r 2 ] 1 2 cos [ 2 π ( c t - r ) / λ + φ u v ] ,
r = [ h 2 + ( x - u ) 2 + ( y - v ) 2 ] 1 2 .
r = h [ 1 + ( x - u ) 2 + ( y - v ) 2 h 2 ] 1 2 , h [ 1 + ( x - u ) 2 + ( y - v ) 2 2 h 2 ] , h + x 2 + y 2 - 2 ( x u + y v ) + u 2 + v 2 2 h .
[ α P ( u , v ) Δ u Δ v π h 2 ] 1 2 cos [ 2 π λ { c t - h - x 2 + y 2 - 2 ( x u + y v ) + u 2 + v 2 2 h } + φ u v ] .
[ α P ( u , v ) Δ u Δ v π h 2 ] 1 2 cos [ 2 π λ { c t + x u + y v h } + θ x y + φ u v ] .
A ( x , y ) = [ α Δ u Δ v π h 2 ] 1 2 { cos ( 2 π c t λ + θ x y ) u v [ P ( u , v ) ] 1 2 × cos ( 2 π λ [ x u + y v h ] + φ u v ) - sin ( 2 π c t λ + θ x y ) × a n [ P ( u , v ) ] 1 2 sin ( 2 π λ [ x u + y v h ] + φ u v ) } .
B ( x , y ) = α Δ u Δ v 2 π h 2 { ( u v [ P ( u , v ) ] 1 2 cos [ 2 π λ ( x u + y v h ) + φ u v ] ) 2 + ( u v [ P ( u , v ) ] 1 2 sin [ 2 π λ ( x u + y v h ) + φ u v ] ) 2 } .
B ¯ = α Δ u Δ v 2 π h 2 u v P ( u , v ) { cos 2 [ 2 π λ ( x u + y v h ) + φ u v ] + sin 2 [ 2 π λ ( x u + y v h ) + φ u v ] } , = α 2 π h 2 u v P ( u , v ) Δ u Δ v .
B ¯ = α 2 π h 2 - d u - d v P ( u , v ) .
B ˜ ( x , y ) = α Δ u Δ v 2 π h 2 u v p q [ P ( u , v ) P ( p , q ) ] 1 2 · { cos [ 2 π λ ( x u + y v h ) + φ u v ] cos [ 2 π λ ( x p + y q h ) + φ p q ] + sin [ 2 π λ ( x u + y v h ) + φ u v ] sin [ 2 π λ ( x p + y q h ) + φ p q ] } = α Δ u Δ v 2 π h 2 u v p q [ P ( u , v ) P ( p , q ) ] 1 2 · cos [ 2 π λ ( x [ u - p ] + y [ v - q ] h ) + ( φ u v - φ p q ) ] ; u p             and / or             v q .
2 π ( u - p ) / λ h = ω             and             2 π ( v - q ) / λ h = Ω ,
p = u - λ h ω / 2 π             and             q = v - λ h Ω / 2 π .
B ˜ ( x , y ) = α Δ u Δ v 2 π h 2 u v ω Ω [ P ( u , v ) P ( u - λ h 2 ω ω , v - λ h 2 π Ω ) ] 1 2 × cos ( ω x + Ω y + Ψ ω Ω u v ) ,
B ( x , y ) B ( x + γ , y + δ ) = [ B ¯ ] 2 + B ˜ ( x , y ) B ˜ ( x + γ , y + δ ) ,
Δ ω = ( 2 π / λ h ) Δ u             and             Δ Ω = ( 2 π / λ h ) Δ v .
K a b K c d cos ( a x + b y + Ψ a b ) cos ( c [ x + γ ] + d [ y + δ ] + Ψ c d ) ,
B ˜ ( x , y ) B ˜ ( x + γ , y + δ ) = α 2 λ 2 2 π 4 h 2 u v P ( u , v ) Δ u Δ v × ω Ω P ( u - λ h 2 π ω , v - λ h 2 π Ω ) Δ ω Δ Ω · cos ( ω x + Ω y + Ψ ω Ω u v ) × cos ( ω [ x + γ ] + Ω [ y + δ ] + Ψ ω Ω u v ) .
cos ( ω x + Ω y + Ψ ω Ω u v ) cos ( ω [ x + γ ] + Ω [ y + δ ] + Ψ ω Ω u v ) = 1 2 cos ( ω γ + Ω δ ) + 1 2 cos ( ω [ 2 x + γ ] + Ω [ 2 y + δ ] + 2 Ψ ω Ω u v ) = 1 2 cos ( ω γ + Ω δ ) .
B ˜ ( x , y ) B ˜ ( x + γ , y + δ ) = α 2 λ 2 32 π 4 h 2 - - d ω d Ω cos ( ω γ + Ω δ ) × - - d u d v P ( u , v ) P ( u - λ h 2 π ω , v - λ h 2 π Ω ) .
< B ˜ ( x , y ) B ˜ ( x + γ , y + δ ) > = ( α 2 / 8 π 2 h 4 ) F ( γ , δ ) 2 ,
F ( γ , δ ) = - - d u d v P ( u , v ) ( j 2 π / λ h ) ( γ u + δ v ) .
S ( ω , Ω ) Δ ω Δ Ω = { α Δ u Δ v 2 π h 2 u v [ P ( u , v ) P ( u - λ h 2 π ω , v - λ h 2 π Ω ) ] 1 2 × cos ( ω x + Ω y + Ψ ω Ω u v ) } 2 .
cos 2 ( ω x + Ω y + Ψ ) = 1 2 [ 1 + cos ( 2 ω x + 2 Ω y + 2 Ψ ) ] ,
S ( ω , Ω ) Δ ω Δ Ω = α 2 ( Δ u Δ v ) 2 8 π 2 h 4 u v P ( u , v ) P ( u - λ h 2 π ω , v - λ h 2 π Ω ) .
S ( ω , Ω ) = α 2 λ 2 32 π 4 h 2 u v Δ u Δ v P ( u , v ) P ( u - λ h 2 π ω , v - λ h 2 π Ω ) .
S ( ω , Ω ) = α 2 λ 2 32 π 4 h 2 - - d u d v P ( u , v ) × P ( u - λ h 2 π ω , v - λ h 2 π Ω ) ,
B ˜ ( x , y ) B ˜ ( x + γ , y + δ ) = α 2 λ 2 32 π 4 h 2 - - d ω d Ω cos ( ω γ + Ω δ ) × - - d u d v P ( u , v ) P ( u - λ h 2 π ω , v - λ h 2 π Ω ) .
B ˜ ( x , y ) B ˜ ( x + γ , y + δ ) = α 2 λ 2 32 π 4 h 2 - - d u d v P ( u , v ) × - - d ω d Ω e - j ( ω γ + Ω δ ) P ( u - λ h 2 π ω , v - λ h 2 π Ω ) .
exp [ - j ( ω γ + Ω δ ) ] = exp [ - j ( 2 π γ u λ h + 2 π δ v λ h ) ] × exp [ j ( [ u - λ h ω 2 π ] 2 π γ λ h + [ v - λ h Ω 2 π ] 2 π δ λ h ) ] ,
B ˜ ( x , y ) B ˜ ( x + γ , y + δ ) = α 2 8 π 2 h 4 - - d u d v P ( u , v ) exp [ - j ( 2 π γ u λ h + 2 π δ v λ h ) ] × - - d ( - λ h ω 2 π ) d ( - λ h Ω 2 π ) P ( u - λ h 2 π ω , v - λ h 2 π Ω ) × exp [ j ( [ u - λ h ω 2 π ] 2 π γ λ h + [ v - λ h Ω 2 π ] 2 π δ λ h ) ] .
F ( 2 π γ λ h , 2 π δ λ h ) = - - d u d v P ( u , v ) × exp [ j ( 2 π γ u λ h + 2 π δ v λ h ) ] ,
B ˜ ( x , y ) B ˜ ( x + γ , y + δ ) = ( α 2 / 8 π 2 h 4 ) | F ( 2 π γ λ h , 2 π δ λ h ) | 2 .