Abstract

The statistical properties (i.e., the probability density function and the average contrast) of laser speckle produced by a weak diffuse object in the diffraction field have been theoretically and experimentally studied with the assumption of the Gaussian statistics for the formation of speckles. The general formulas for the probability density function and the average contrast, which are valid for an entire range of object surface and for the whole diffraction field, are introduced, and their special cases, which have been studied in the past, are derived and discussed. These formulas for the probability density function and the averagae contrast are actually evaluated in the diffraction field of a weak diffuse object illuminated by the Gaussian laser beam. The experimental study has been conducted to verify the above theoretical study, and its results are in good agreement with the theoretical ones. The circularity and noncircularity of the speckle statistics in the diffraction field are discussed on the basis of the theoretical study.

© 1977 Optical Society of America

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References

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  1. J. C. Dainty, Opt. Acta 17, 761 (1970).
    [CrossRef]
  2. J. C. Dainty, J. Opt. Soc. Am. 62, 595 (1972).
    [CrossRef]
  3. R. A. Sprague, Appl. Opt. 11, 2811 (1972).
    [CrossRef] [PubMed]
  4. H. Fujii, T. Asakura, Opt. Commun. 11, 35 (1974).
    [CrossRef]
  5. H. Fujii, T. Asakura, Opt. Commun. 12, 32 (1974).
    [CrossRef]
  6. H. Fujii, T. Asakura, Nouv. Rev. Opt. 6, 5 (1975).
    [CrossRef]
  7. J. Ohtsubo, H. Fujii, T. Asakura, Jpn. J. Appl. Phys. Suppl. 14-1, 293 (1975).
  8. J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
    [CrossRef]
  9. J. Ohtsubo, T. Asakura, Nouv. Rev. Opt. 6, 189 (1975).
    [CrossRef]
  10. J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).
  11. H. Fujii, T. Asakura, Y. Shindo, Opt. Commun. 16, 68 (1976).
    [CrossRef]
  12. H. M. Pedersen, Opt. Commun. 12, 156 (1974).
    [CrossRef]
  13. H. M. Pedersen, Opt. Commun. 16, 63 (1976).
    [CrossRef]
  14. H. M. Pedersen, J. Opt. Soc. Am. 66, 1204 (1976).
    [CrossRef]
  15. N. George, A. Jain, Appl. Phys. 4, 201 (1974).
    [CrossRef]
  16. N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
    [CrossRef]
  17. J. W. Goodman, Opt. Commun. 14, 324 (1975).
    [CrossRef]
  18. J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, Berlin, 1975), p. 9.
    [CrossRef]
  19. W. T. Welford, Opt. Quant. Electron. 7, 413 (1975).
    [CrossRef]
  20. N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
    [CrossRef]
  21. N. Takai, Opt. Commun. 14, 24 (1975).
    [CrossRef]
  22. E. Jakeman, P. N. Pusey, J. Phys. A 8, 369 (1975).
    [CrossRef]
  23. P. N. Pusey, E. Jakeman, J. Phys. A 8, 392 (1975).
    [CrossRef]
  24. E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
    [CrossRef]
  25. J. Ohtsubo, T. Asakura, Opt. Quant. Electron. 8, 523 (1976).
    [CrossRef]

1976 (6)

J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).

H. Fujii, T. Asakura, Y. Shindo, Opt. Commun. 16, 68 (1976).
[CrossRef]

H. M. Pedersen, Opt. Commun. 16, 63 (1976).
[CrossRef]

H. M. Pedersen, J. Opt. Soc. Am. 66, 1204 (1976).
[CrossRef]

E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
[CrossRef]

J. Ohtsubo, T. Asakura, Opt. Quant. Electron. 8, 523 (1976).
[CrossRef]

1975 (10)

N. Takai, Opt. Commun. 14, 24 (1975).
[CrossRef]

E. Jakeman, P. N. Pusey, J. Phys. A 8, 369 (1975).
[CrossRef]

P. N. Pusey, E. Jakeman, J. Phys. A 8, 392 (1975).
[CrossRef]

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

J. W. Goodman, Opt. Commun. 14, 324 (1975).
[CrossRef]

W. T. Welford, Opt. Quant. Electron. 7, 413 (1975).
[CrossRef]

H. Fujii, T. Asakura, Nouv. Rev. Opt. 6, 5 (1975).
[CrossRef]

J. Ohtsubo, H. Fujii, T. Asakura, Jpn. J. Appl. Phys. Suppl. 14-1, 293 (1975).

J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
[CrossRef]

J. Ohtsubo, T. Asakura, Nouv. Rev. Opt. 6, 189 (1975).
[CrossRef]

1974 (5)

H. Fujii, T. Asakura, Opt. Commun. 11, 35 (1974).
[CrossRef]

H. Fujii, T. Asakura, Opt. Commun. 12, 32 (1974).
[CrossRef]

N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
[CrossRef]

N. George, A. Jain, Appl. Phys. 4, 201 (1974).
[CrossRef]

H. M. Pedersen, Opt. Commun. 12, 156 (1974).
[CrossRef]

1972 (2)

1970 (1)

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

Asakura, T.

J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).

H. Fujii, T. Asakura, Y. Shindo, Opt. Commun. 16, 68 (1976).
[CrossRef]

J. Ohtsubo, T. Asakura, Opt. Quant. Electron. 8, 523 (1976).
[CrossRef]

H. Fujii, T. Asakura, Nouv. Rev. Opt. 6, 5 (1975).
[CrossRef]

J. Ohtsubo, H. Fujii, T. Asakura, Jpn. J. Appl. Phys. Suppl. 14-1, 293 (1975).

J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
[CrossRef]

J. Ohtsubo, T. Asakura, Nouv. Rev. Opt. 6, 189 (1975).
[CrossRef]

H. Fujii, T. Asakura, Opt. Commun. 11, 35 (1974).
[CrossRef]

H. Fujii, T. Asakura, Opt. Commun. 12, 32 (1974).
[CrossRef]

Dainty, J. C.

Fujii, H.

H. Fujii, T. Asakura, Y. Shindo, Opt. Commun. 16, 68 (1976).
[CrossRef]

J. Ohtsubo, H. Fujii, T. Asakura, Jpn. J. Appl. Phys. Suppl. 14-1, 293 (1975).

H. Fujii, T. Asakura, Nouv. Rev. Opt. 6, 5 (1975).
[CrossRef]

H. Fujii, T. Asakura, Opt. Commun. 12, 32 (1974).
[CrossRef]

H. Fujii, T. Asakura, Opt. Commun. 11, 35 (1974).
[CrossRef]

George, N.

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

N. George, A. Jain, Appl. Phys. 4, 201 (1974).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Opt. Commun. 14, 324 (1975).
[CrossRef]

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, Berlin, 1975), p. 9.
[CrossRef]

Jain, A.

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

N. George, A. Jain, Appl. Phys. 4, 201 (1974).
[CrossRef]

Jakeman, E.

E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
[CrossRef]

E. Jakeman, P. N. Pusey, J. Phys. A 8, 369 (1975).
[CrossRef]

P. N. Pusey, E. Jakeman, J. Phys. A 8, 392 (1975).
[CrossRef]

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
[CrossRef]

Melville, R. D. S.

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).

J. Ohtsubo, T. Asakura, Opt. Quant. Electron. 8, 523 (1976).
[CrossRef]

J. Ohtsubo, T. Asakura, Nouv. Rev. Opt. 6, 189 (1975).
[CrossRef]

J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
[CrossRef]

J. Ohtsubo, H. Fujii, T. Asakura, Jpn. J. Appl. Phys. Suppl. 14-1, 293 (1975).

Pedersen, H. M.

H. M. Pedersen, Opt. Commun. 16, 63 (1976).
[CrossRef]

H. M. Pedersen, J. Opt. Soc. Am. 66, 1204 (1976).
[CrossRef]

H. M. Pedersen, Opt. Commun. 12, 156 (1974).
[CrossRef]

Pusey, P. N.

E. Jakeman, P. N. Pusey, J. Phys. A 8, 369 (1975).
[CrossRef]

P. N. Pusey, E. Jakeman, J. Phys. A 8, 392 (1975).
[CrossRef]

Shindo, Y.

H. Fujii, T. Asakura, Y. Shindo, Opt. Commun. 16, 68 (1976).
[CrossRef]

Sprague, R. A.

Takai, N.

N. Takai, Opt. Commun. 14, 24 (1975).
[CrossRef]

N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
[CrossRef]

Welford, W. T.

W. T. Welford, Opt. Quant. Electron. 7, 413 (1975).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (2)

N. George, A. Jain, Appl. Phys. 4, 201 (1974).
[CrossRef]

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. A (3)

E. Jakeman, P. N. Pusey, J. Phys. A 8, 369 (1975).
[CrossRef]

P. N. Pusey, E. Jakeman, J. Phys. A 8, 392 (1975).
[CrossRef]

E. Jakeman, J. G. McWhirter, J. Phys. A 9, 785 (1976).
[CrossRef]

Jpn. J. Appl. Phys. (1)

N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
[CrossRef]

Jpn. J. Appl. Phys. Suppl. (1)

J. Ohtsubo, H. Fujii, T. Asakura, Jpn. J. Appl. Phys. Suppl. 14-1, 293 (1975).

Nouv. Rev. Opt. (2)

J. Ohtsubo, T. Asakura, Nouv. Rev. Opt. 6, 189 (1975).
[CrossRef]

H. Fujii, T. Asakura, Nouv. Rev. Opt. 6, 5 (1975).
[CrossRef]

Opt. Acta (1)

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

Opt. Commun. (8)

H. Fujii, T. Asakura, Opt. Commun. 11, 35 (1974).
[CrossRef]

H. Fujii, T. Asakura, Opt. Commun. 12, 32 (1974).
[CrossRef]

J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
[CrossRef]

H. Fujii, T. Asakura, Y. Shindo, Opt. Commun. 16, 68 (1976).
[CrossRef]

H. M. Pedersen, Opt. Commun. 12, 156 (1974).
[CrossRef]

H. M. Pedersen, Opt. Commun. 16, 63 (1976).
[CrossRef]

J. W. Goodman, Opt. Commun. 14, 324 (1975).
[CrossRef]

N. Takai, Opt. Commun. 14, 24 (1975).
[CrossRef]

Opt. Quant. Electron. (2)

W. T. Welford, Opt. Quant. Electron. 7, 413 (1975).
[CrossRef]

J. Ohtsubo, T. Asakura, Opt. Quant. Electron. 8, 523 (1976).
[CrossRef]

Optik (1)

J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).

Other (1)

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, Berlin, 1975), p. 9.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Experimental arrangement for producing speckle patterns in the diffraction field of a transmitting diffuse object.

Fig. 2
Fig. 2

Probability density functions P(I/〈I〉) of the speckle intensity I for the diffuse object, having σφ = 1.86 and α = 6.0 μm, at four detecting points of r/W = 0, 0.66, 0.99, and 1.65 in the receiving plane of R = 80 cm. (a) The experimental results and (b) the theoretical ones corresponding to (a).

Fig. 3
Fig. 3

Probability density functions P(I/〈I〉) of the speckle intensity I for the diffuse object, having σφ = 2.53 and α = 65 μm, at three detecting points of R = 10 cm, and 80 cm, along the optical axis in the diffraction field. (a) The experimental result and (b) the theoretical ones corresponding to (a).

Fig. 4
Fig. 4

A variation of the contrast V of speckle intensity variations for the diffuse object, having σφ = 1.86 and α = 6.0 μm, as a function of r/W at three receiving planes of R = 25 cm, 80 cm, and 400 cm under three different conditions of illumination characterized by N and W0. Three different symbols represent the experimental values while the solid curves denote the theoretical results.

Fig. 5
Fig. 5

A variation of the contrast V of speckle intensity variations along the optical axis with the distance R from the object for three different objects having σφ = 2.53, 1.86, and 0.92 and α = 6.5 μm, 6.0 μm, and 6.5 μm. Three different symbols represent the experimental values for three objects, and the solid curves denote the theoretical results.

Fig. 6
Fig. 6

Specular and diffuse components of the speckle field as a function of r/W at the receiving plane R = 400 cm for the diffuse object having σφ = 1.86 and α = 6.0 μm. (a) The specular component IS and (b) the diffuse component 〈IN〉 together with the variances σr2 and σi2 of the real and imaginary parts of the speckle field.

Fig. 7
Fig. 7

Specular and diffuse components of the speckle field as a function of R along the optical axis for the diffuse object having σφ = 1.86 and α = 6.0 μm. (a) The specular component Is together with the squared values (cr2 and ci2) of its real and imaginary parts and (b) the diffuse component 〈IN〉 together with the variances (σr2 and σi2) of the real and imaginary parts of the speckle field.

Fig. 8
Fig. 8

Coefficients of h+ and h in the logarithmic scale as a function of the standard phase deviation σs.

Fig. 9
Fig. 9

Schematic outline of the two regions in the diffraction field in which two different types of the circularity and noncircularity of the speckle statistics are established. The noncircular statistics are kept in the hatched area, while the circular one is held in the dotted area. The broken lines indicate the radius of a propagating beam, centered on the optical axis, when the diffuse object is not placed at the object position.

Equations (60)

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A ( r , t ) = A = a ( r , t ) + c ( r , t ) ,
Re [ A ( r , t ) ] = A r = a r + c r ,
Im [ A ( r , t ) ] = A i = a i + c i .
A r = a r + c r = c r ,
A i = a i + c i = c i ,
σ r 2 = A r 2 - A r 2 = a r 2 ,
σ i 2 = A i 2 - A i 2 = a i 2 .
I N = A 2 - A 2 = a r 2 + a i 2 = σ r 2 + σ i 2 ,
I S = A 2 = c r 2 + c i 2 .
P ( A r , A i ) = 1 2 π σ r σ i exp { - [ ( A r - c r ) 2 2 σ r 2 + ( A i - c i ) 2 2 σ i 2 ] } .
V = ( I 2 - I 2 ) 1 / 2 I [ 2 ( σ r 4 + σ i 4 ) + 4 ( c r 2 σ r 2 + c i 2 σ i 2 ) ] 1 / 2 σ r 2 + σ i 2 + c r 2 + c i 2 .
P ( I ) = 1 4 π σ r σ i 0 2 π exp { - [ ( cos 2 ϕ 2 σ r 2 + sin 2 ϕ 2 σ i 2 ) I - ( c r σ r 2 cos ϕ + c i σ i 2 sin ϕ ) I 1 / 2 + c r 2 2 σ r 2 + c i 2 2 σ i 2 ] } d ϕ .
V = [ 2 ( σ r 4 + σ i 4 ) + 4 I S σ r 2 ] 1 / 2 σ r 2 + σ i 2 + I S ,
P ( I ) = 1 2 σ r σ i exp { - 1 4 [ ( 1 σ r 2 + 1 σ i 2 ) I + 2 σ r 2 I S ] } · [ I o ( s 1 ) I o ( s 2 ) + 2 n = 1 I n ( s 1 ) I n ( s 2 ) ] ,
s 1 = 1 4 ( 1 σ r 2 - 1 σ i 2 ) I ,
s 2 = ( I S I ) 1 / 2 σ r 2 ,
V = ( I N 2 + 2 I S I N ) 1 / 2 I N + I S ,
P ( I ) = 1 I N exp ( - I + I S I N ) I o [ 2 ( I S I ) 1 / 2 I N ]
A ( r , t ) = - E ( r ) T ( r , t ) K ( r , r ) d 2 r ,
E ( r ) = exp ( - r 2 W 0 2 ) ,
T ( r , t ) = exp [ i φ ( r , t ) ] ,
K ( r , r ) = exp [ i H ( r , r ) ] = exp [ - i ( 2 π λ R r · r - π λ R r 2 ) ] ,
A r = - E ( r ) cos [ φ ( r , t ) + H ( r , r ) ] d 2 r
A i = - E ( r ) sin [ φ ( r , t ) + H ( r , r ) ] d 2 r .
A r = exp ( - σ φ 2 / 2 ) - E ( r ) cos H ( r , r ) d 2 r = 2 π exp ( - σ φ 2 / 2 ) × 0 r J 0 ( 2 π λ R r r ) exp ( - r 2 / W 0 2 ) cos ( π λ R r 2 ) d r ,
A i = exp ( - σ φ 2 / 2 ) - E ( r ) sin H ( r , r ) d 2 r = 2 π exp ( - σ φ 2 / 2 ) × 0 r J 0 ( 2 π λ R r r ) exp ( - r 2 / W 0 2 ) sin ( π λ R r 2 ) d r ,
A r 2 = 1 2 exp ( - σ φ 2 ) × - E ( r 1 ) E ( r 2 ) { C + ( r 1 - r 2 ) cos [ H ( r 1 , r ) - H ( r 2 , r ) ] + C 1 ( r 1 - r 2 ) cos [ H ( r 1 , r ) + H ( r 2 , r ) ] } d 2 r 1 d 2 r 2 ,
A i 2 = 1 2 exp ( - σ φ 2 ) × - E ( r 1 ) E ( r 2 ) { C + ( r 1 - r 2 ) cos [ H ( r 1 , r ) - H ( r 2 , r ) ] - C - ( r 1 - r 2 ) cos [ H ( r 1 , r ) + H ( r 2 , r ) ] } d 2 r 1 d 2 r 2 ,
A r A i = 1 2 exp ( - σ φ 2 ) × E ( r 1 ) E ( r 2 ) { C + ( r 1 - r 2 ) sin [ H ( r 2 , r ) - H ( r 1 , r ) ] + C - ( r 1 - r 2 ) sin [ H ( r 2 , r ) + H ( r 1 , r ) ] } d 2 r 1 d 2 r 2 ,
C + ( r 1 - r 2 ) = exp [ σ φ 2 ρ ( r 1 - r 2 ) ] ,
C - ( r 1 - r 2 ) = exp [ - σ φ 2 ρ ( r 1 - r 2 ) ] ,
ρ ( r ) = exp ( - r 2 / α 2 ) ,
C + ( r ) = exp [ σ φ 2 ρ ( r ) ] 1 + [ exp ( σ φ 2 ) - 1 ] exp [ - r 2 f ( σ φ 2 ) / α 2 ] ,
C - ( r ) = exp [ - σ φ 2 ρ ( r ) ] 1 + [ exp ( - σ φ 2 ) - 1 ] exp [ - r 2 g ( σ φ 2 ) / α 2 ] ,
R + ( x ) = exp { i [ φ ( r ) - φ ( r + x ) ] } - exp [ i φ ( r ) ] exp [ i φ ( r + x ) ] = exp ( - σ φ 2 ) { exp [ σ φ 2 ρ ( x ) ] - 1 } ,
R - ( x ) = exp { i [ φ ( r ) + φ ( r + x ) ] } - exp [ i φ ( r ) ] exp [ i φ ( r + x ) ] = exp ( - σ φ 2 ) { exp [ - σ φ 2 σ ( x ) ] - 1 } ,
C + 1 + exp ( σ φ 2 ) W + ( 0 ) δ ( x ) ,
C - 1 + exp ( σ φ 2 ) W - ( 0 ) δ ( x ) ,
W + ( 0 ) = π α 2 exp ( - σ φ 2 ) n = 1 σ φ 2 n n ! n = π α 2 exp ( - σ φ 2 ) h + ,
W - ( 0 ) = π α 2 exp ( - σ φ 2 ) n = 1 ( - σ φ 2 ) n ! n = - π α 2 exp ( - σ φ 2 ) h -
h + = n = 1 σ φ 2 n n ! n , h - = - n = 1 ( - σ φ 2 ) n ! n .
A r 2 = A r 2 + π α 2 2 exp ( - σ φ 2 ) { h + - E ( r ) d 2 r - h - - E ( r ) 2 cos [ 2 H ( r , r ) ] d 2 r } ,
A i 2 = A i 2 + π α 2 2 exp ( - σ φ 2 ) { h + - E ( r ) 2 d 2 r + h - - E ( r ) 2 cos [ 2 H ( r , r ) ] d 2 r } .
σ r 2 = π 2 W 0 4 4 N exp ( - σ φ 2 ) [ h + - 4 W 0 2 h - 0 r J 0 ( 4 π λ R r r ) exp ( - 2 r 2 / W 0 2 ) cos ( 2 π r 2 / λ R ) d r ] ,
σ i 2 = π 2 W 0 4 4 N exp ( - σ φ 2 ) [ h + + 4 W 0 2 h - 0 r J 0 ( 4 π λ R r ) exp ( - 2 r 2 / W 0 2 ) cos ( 2 π r 2 / λ R ) d r ] ,
N = ( W 0 / α ) 2 .
A r = c r = π W 0 2 exp ( - σ φ 2 / 2 ) exp [ - ( π W 0 / λ R ) 2 r 2 ] ,
A i = c i = 0 ,
σ r 2 = π 2 W 0 4 4 N { h + - h - exp [ - 2 ( π W 0 / λ R ) 2 r 2 ] } exp ( - σ φ 2 ) ,
σ i 2 = π 2 W 0 4 4 N { h + + h - exp [ - 2 ( π W 0 / λ R ) 2 r 2 ] } exp ( - σ φ 2 ) .
A r = c r = π W 0 2 exp ( - σ φ 2 / 2 ) [ 1 + ( π W 0 2 / λ R ) 2 ] - 1 ,
A i = c i = π W 0 2 exp ( - σ φ 2 / 2 ) [ 1 + ( π W 0 2 / λ R ) 2 ] - 1 ( π W 0 2 / λ R ) ,
σ r 2 = π 2 W 0 4 4 N { h + - h - [ 1 + ( π W 0 2 / λ R ) 2 ] - 1 } exp ( - σ φ 2 ) ,
σ i 2 = π 2 W 0 4 4 N { h + + h - [ 1 + ( π W 0 2 / λ R ) 2 ] - 1 } exp ( - σ φ 2 ) .
W 0 = λ f / π W i ,
σ l = σ h ( n 1 - n 2 ) ,
σ φ 2 = ( 2 π / λ ) 2 σ l 2 .
W = λ R / π W 0 3.0 mm .
R π W 0 2 / λ 10 cm .
σ r 2 = σ i 2 = π 2 W 0 4 4 N h + exp ( - σ φ 2 ) = 1 2 I N .

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