Abstract

Theoretical computations are presented for the far-zone radiation patterns from strong diffusers having a normally distributed height profile and different autocorrelation functions for the surface height. Diffusers having two scales of roughness are also analyzed. Data are presented from ground-glass and acid-etched-glass diffusers using a scatterometer that permits measurements over an entire hemisphere with a dynamic range of 6–8 orders of magnitude. For the ground glass, excellent agreement is obtained using an autocorrelation function that is conical for small spatial offsets; this is consistent with our physical expectation based on the need for a rapid fall-off in surface correlation, due to the jagged nature of the surface relief. For the acid-etched case, excellent agreement is found using an autocorrelation function that is paraboloidal for small spatial offsets. Remote sensing of surface roughness and surface correlation are seen to be practical and accurate.

© 1988 Optical Society of America

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References

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  1. C. N. Kurtz, “Transmittance Characteristics of Surface Diffusers and the Design of Nearly Bandlimited Binary Diffusers,” J. Opt. Soc. Am. 62, 982 (1972).
    [CrossRef]
  2. C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and Synthesis of Random Phase Diffusers,” J. Opt. Soc. Am. 63, 1080 (1973).
    [CrossRef]
  3. R. C. Waag, K. T. Knox, “Power-Spectrum Analysis of Exponential Diffusers,” J. Opt. Soc. Am. 62, 877 (1972).
    [CrossRef]
  4. Y. Nakayama, M. Kato, “Diffuser with Pseudorandom Phase Sequence,” J. Opt. Soc. Am. 69, 1367 (1979).
    [CrossRef]
  5. M. Kowalczyk, “Spectral and Imaging Properties of Uniform Diffusers,” J. Opt. Soc. Am. A 1, 192 (1984).
    [CrossRef]
  6. L. G. Shirley, N. George, “Wide-Angle Diffuser Transmission Functions and Far-Zone Speckle,” J. Opt. Soc. Am. A 4, 734 (1987).
    [CrossRef]
  7. C. Pask, “Derivation of Source-Field Coherence Properties from Radiation Angular Distribution,” Opt. Acta. 24, 235 (1977).
    [CrossRef]
  8. E. Wolf, W. H. Carter, “Fields Generated by Homogeneous and by Quasi-Homogeneous Planar Secondary Sources,” Opt. Commun. 50, 131 (1984).
    [CrossRef]
  9. M. V. Berry, “Diffractals,” J. Phys. A 12, 781 (1979).
    [CrossRef]
  10. D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental Measurements of Non-Gaussian Scattering by a Fractal Diffuser,” Appl. Phys. B 31, 179 (1983).
    [CrossRef]
  11. E. Jakeman, “Fraunhofer Scattering by a Sub-Fractal Diffuser,” Opt. Acta 30, 1207 (1983).
    [CrossRef]
  12. D. L. Jaggard, Y. Kim, “Diffraction by Band-Limited Fractal Screens,” J. Opt. Soc. Am. A 4, 1055 (1987).
    [CrossRef]
  13. W. T. Welford, “Review-Optical Estimation of Statistics of Surface Roughness from Light Scattering Measurements,” Opt. Quantum Electron. 9, 269 (1977).
    [CrossRef]
  14. J. M. Elson, J. M. Bennett, “Relation Between the Angular Dependence of Scattering and the Statistical Properties of Optical Surfaces,” J. Opt. Soc. Am. 69, 31 (1979).
    [CrossRef]
  15. Y. Wang, W. L. Wolfe, “Scattering from Microrough Surfaces: Comparison of Theory and Experiment,” J. Opt. Soc. Am. 73, 1596 (1983).
    [CrossRef]
  16. P. Roche, E. Pelletier, “Characterizations of Optical Surfaces by Measurement of Scattering Distribution,” Appl. Opt. 23, 3561 (1984).
    [CrossRef] [PubMed]
  17. K. J. Allardyce, N. George, “Diffraction Analysis of Rough Reflective Surfaces,” Appl. Opt. 26, 2364 (1987).
    [CrossRef] [PubMed]
  18. H. E. Bennett, “Specular Reflectance of Aluminized Ground Glass and the Height Distribution of Surface Irregularities,” J. Opt. Soc. Am. 53, 1389 (1963).
    [CrossRef]
  19. M. R. Latta, “The Scattering of 10.6 Micron Radiation from Ground Glass Surfaces,” M.S. Thesis, U. Rochester, (1968).
  20. R. I. Hamaguchi, “Transmission Characteristics of Ground Glass,” M.S. Thesis, U. Rochester, (1970).
  21. P. J. Chandley, “Surface Roughness Measurements from Coherent Light Scattering,” Opt. Quantum Electron. 8, 323 (1976).
    [CrossRef]
  22. P. J. Chandley, “Determination of the Probability Density Function of Height on a Rough Surface from Far-Field Coherent Light Scattering,” Opt. Quantum Electron. 11, 413 (1979).
    [CrossRef]
  23. N. G. Gaggioli, M. L. Roblin, “Etudes des etats de surface par les properties de diffusion a l’infini en lumiere transmise,” Opt. Commun. 32, 209 (1980).
    [CrossRef]
  24. P. Croce, L. Prod’Homme, “Contribution of Immersion Technique to Light Scattering Analysis of very Rough Surfaces,” J. Opt. 11, 319 (1980).
    [CrossRef]
  25. L. G. Shirley, N. George, “Diffuser Transmission Functions and Far-Zone Speckle Patterns,” Proc. Soc. Photo-Opt. Instrum. Eng. 556, 63 (1985).
  26. P. F. Gray, “A Method of Forming Optical Diffusers of Simple Known Statistical Properties,” Opt. Acta 25, 765 (1978).
    [CrossRef]
  27. B. M. Levine, J. C. Dainty, “Non-Gaussian Image Plane Speckle: Measurements from Diffusers of Known Statistics,” Opt. Commun. 45, 252 (1983).
    [CrossRef]
  28. E. R. Mendez, K. A. O’Donnell, “Observation of Depolarization and Backscattering Enhancement in Light Scattering from Gaussian Random Surfaces,” Opt. Commun. 61, 91 (1987).
    [CrossRef]
  29. We use Ien instead of the BRDF because Ien is proportional to the power at the detector. The BRDF can be obtained from Ien by dividing Ien by cosθ. See F. E. Nicodemus, “Reflectance Nomenclature and Directional Reflectance and Emissivity,” Appl. Opt. 9, 1474 (1970).
    [CrossRef] [PubMed]
  30. Equation (7) is obtained from Ref. 6 as follows: Equation (56) for 〈|ν|2〉 is multiplied by (R0/A0)2, Eq. (79) for 〈U〉 is then substituted into Eq. (56) with Eq. (64) for the aperture factor, and spatial frequencies are converted to angles through Eqs. (18) and (19). The cos2θ obliquity factor is also generalized to cosnθ.
  31. E. Lukacs, Characteristic Functions (C. Griffin, London, 1970), p. 68.
  32. P. Beckmann, “Scattering by Composite Rough Surfaces,” Proc. IEEE 53, 1012 (1965).
    [CrossRef]
  33. J. Dyson, “Optical Diffusing Screens of High Efficiency,” J. Opt. Soc. Am. 50, 519 (1960).
    [CrossRef]
  34. E. G. Rawson, A. B. Nafarrate, R. E. Norton, “Speckle-Free Rear-Projection Screen Using Two Close Screens in Slow Relative Motion,” J. Opt. Soc. Am. 66, 1290 (1976).
    [CrossRef]
  35. Armour Products, Midland Park, NJ 07432.
  36. Allied Chemical, Allied Corp., Morristown, NJ 07960.

1987 (4)

1985 (1)

L. G. Shirley, N. George, “Diffuser Transmission Functions and Far-Zone Speckle Patterns,” Proc. Soc. Photo-Opt. Instrum. Eng. 556, 63 (1985).

1984 (3)

1983 (4)

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental Measurements of Non-Gaussian Scattering by a Fractal Diffuser,” Appl. Phys. B 31, 179 (1983).
[CrossRef]

E. Jakeman, “Fraunhofer Scattering by a Sub-Fractal Diffuser,” Opt. Acta 30, 1207 (1983).
[CrossRef]

B. M. Levine, J. C. Dainty, “Non-Gaussian Image Plane Speckle: Measurements from Diffusers of Known Statistics,” Opt. Commun. 45, 252 (1983).
[CrossRef]

Y. Wang, W. L. Wolfe, “Scattering from Microrough Surfaces: Comparison of Theory and Experiment,” J. Opt. Soc. Am. 73, 1596 (1983).
[CrossRef]

1980 (2)

N. G. Gaggioli, M. L. Roblin, “Etudes des etats de surface par les properties de diffusion a l’infini en lumiere transmise,” Opt. Commun. 32, 209 (1980).
[CrossRef]

P. Croce, L. Prod’Homme, “Contribution of Immersion Technique to Light Scattering Analysis of very Rough Surfaces,” J. Opt. 11, 319 (1980).
[CrossRef]

1979 (4)

P. J. Chandley, “Determination of the Probability Density Function of Height on a Rough Surface from Far-Field Coherent Light Scattering,” Opt. Quantum Electron. 11, 413 (1979).
[CrossRef]

M. V. Berry, “Diffractals,” J. Phys. A 12, 781 (1979).
[CrossRef]

Y. Nakayama, M. Kato, “Diffuser with Pseudorandom Phase Sequence,” J. Opt. Soc. Am. 69, 1367 (1979).
[CrossRef]

J. M. Elson, J. M. Bennett, “Relation Between the Angular Dependence of Scattering and the Statistical Properties of Optical Surfaces,” J. Opt. Soc. Am. 69, 31 (1979).
[CrossRef]

1978 (1)

P. F. Gray, “A Method of Forming Optical Diffusers of Simple Known Statistical Properties,” Opt. Acta 25, 765 (1978).
[CrossRef]

1977 (2)

W. T. Welford, “Review-Optical Estimation of Statistics of Surface Roughness from Light Scattering Measurements,” Opt. Quantum Electron. 9, 269 (1977).
[CrossRef]

C. Pask, “Derivation of Source-Field Coherence Properties from Radiation Angular Distribution,” Opt. Acta. 24, 235 (1977).
[CrossRef]

1976 (2)

1973 (1)

1972 (2)

1970 (1)

1965 (1)

P. Beckmann, “Scattering by Composite Rough Surfaces,” Proc. IEEE 53, 1012 (1965).
[CrossRef]

1963 (1)

1960 (1)

Allardyce, K. J.

Beckmann, P.

P. Beckmann, “Scattering by Composite Rough Surfaces,” Proc. IEEE 53, 1012 (1965).
[CrossRef]

Bennett, H. E.

Bennett, J. M.

Berry, M. V.

M. V. Berry, “Diffractals,” J. Phys. A 12, 781 (1979).
[CrossRef]

Carter, W. H.

E. Wolf, W. H. Carter, “Fields Generated by Homogeneous and by Quasi-Homogeneous Planar Secondary Sources,” Opt. Commun. 50, 131 (1984).
[CrossRef]

Chandley, P. J.

P. J. Chandley, “Determination of the Probability Density Function of Height on a Rough Surface from Far-Field Coherent Light Scattering,” Opt. Quantum Electron. 11, 413 (1979).
[CrossRef]

P. J. Chandley, “Surface Roughness Measurements from Coherent Light Scattering,” Opt. Quantum Electron. 8, 323 (1976).
[CrossRef]

Croce, P.

P. Croce, L. Prod’Homme, “Contribution of Immersion Technique to Light Scattering Analysis of very Rough Surfaces,” J. Opt. 11, 319 (1980).
[CrossRef]

Dainty, J. C.

B. M. Levine, J. C. Dainty, “Non-Gaussian Image Plane Speckle: Measurements from Diffusers of Known Statistics,” Opt. Commun. 45, 252 (1983).
[CrossRef]

DePalma, J. J.

Dyson, J.

Elson, J. M.

Gaggioli, N. G.

N. G. Gaggioli, M. L. Roblin, “Etudes des etats de surface par les properties de diffusion a l’infini en lumiere transmise,” Opt. Commun. 32, 209 (1980).
[CrossRef]

George, N.

Gray, P. F.

P. F. Gray, “A Method of Forming Optical Diffusers of Simple Known Statistical Properties,” Opt. Acta 25, 765 (1978).
[CrossRef]

Hamaguchi, R. I.

R. I. Hamaguchi, “Transmission Characteristics of Ground Glass,” M.S. Thesis, U. Rochester, (1970).

Hoadley, H. O.

Hollins, R. C.

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental Measurements of Non-Gaussian Scattering by a Fractal Diffuser,” Appl. Phys. B 31, 179 (1983).
[CrossRef]

Jaggard, D. L.

Jakeman, E.

E. Jakeman, “Fraunhofer Scattering by a Sub-Fractal Diffuser,” Opt. Acta 30, 1207 (1983).
[CrossRef]

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental Measurements of Non-Gaussian Scattering by a Fractal Diffuser,” Appl. Phys. B 31, 179 (1983).
[CrossRef]

Jordan, D. L.

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental Measurements of Non-Gaussian Scattering by a Fractal Diffuser,” Appl. Phys. B 31, 179 (1983).
[CrossRef]

Kato, M.

Kim, Y.

Knox, K. T.

Kowalczyk, M.

Kurtz, C. N.

Latta, M. R.

M. R. Latta, “The Scattering of 10.6 Micron Radiation from Ground Glass Surfaces,” M.S. Thesis, U. Rochester, (1968).

Levine, B. M.

B. M. Levine, J. C. Dainty, “Non-Gaussian Image Plane Speckle: Measurements from Diffusers of Known Statistics,” Opt. Commun. 45, 252 (1983).
[CrossRef]

Lukacs, E.

E. Lukacs, Characteristic Functions (C. Griffin, London, 1970), p. 68.

Mendez, E. R.

E. R. Mendez, K. A. O’Donnell, “Observation of Depolarization and Backscattering Enhancement in Light Scattering from Gaussian Random Surfaces,” Opt. Commun. 61, 91 (1987).
[CrossRef]

Nafarrate, A. B.

Nakayama, Y.

Nicodemus, F. E.

Norton, R. E.

O’Donnell, K. A.

E. R. Mendez, K. A. O’Donnell, “Observation of Depolarization and Backscattering Enhancement in Light Scattering from Gaussian Random Surfaces,” Opt. Commun. 61, 91 (1987).
[CrossRef]

Pask, C.

C. Pask, “Derivation of Source-Field Coherence Properties from Radiation Angular Distribution,” Opt. Acta. 24, 235 (1977).
[CrossRef]

Pelletier, E.

Prod’Homme, L.

P. Croce, L. Prod’Homme, “Contribution of Immersion Technique to Light Scattering Analysis of very Rough Surfaces,” J. Opt. 11, 319 (1980).
[CrossRef]

Rawson, E. G.

Roblin, M. L.

N. G. Gaggioli, M. L. Roblin, “Etudes des etats de surface par les properties de diffusion a l’infini en lumiere transmise,” Opt. Commun. 32, 209 (1980).
[CrossRef]

Roche, P.

Shirley, L. G.

L. G. Shirley, N. George, “Wide-Angle Diffuser Transmission Functions and Far-Zone Speckle,” J. Opt. Soc. Am. A 4, 734 (1987).
[CrossRef]

L. G. Shirley, N. George, “Diffuser Transmission Functions and Far-Zone Speckle Patterns,” Proc. Soc. Photo-Opt. Instrum. Eng. 556, 63 (1985).

Waag, R. C.

Wang, Y.

Welford, W. T.

W. T. Welford, “Review-Optical Estimation of Statistics of Surface Roughness from Light Scattering Measurements,” Opt. Quantum Electron. 9, 269 (1977).
[CrossRef]

Wolf, E.

E. Wolf, W. H. Carter, “Fields Generated by Homogeneous and by Quasi-Homogeneous Planar Secondary Sources,” Opt. Commun. 50, 131 (1984).
[CrossRef]

Wolfe, W. L.

Appl. Opt. (3)

Appl. Phys. B (1)

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental Measurements of Non-Gaussian Scattering by a Fractal Diffuser,” Appl. Phys. B 31, 179 (1983).
[CrossRef]

J. Opt. (1)

P. Croce, L. Prod’Homme, “Contribution of Immersion Technique to Light Scattering Analysis of very Rough Surfaces,” J. Opt. 11, 319 (1980).
[CrossRef]

J. Opt. Soc. Am. (9)

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781 (1979).
[CrossRef]

Opt. Acta (2)

P. F. Gray, “A Method of Forming Optical Diffusers of Simple Known Statistical Properties,” Opt. Acta 25, 765 (1978).
[CrossRef]

E. Jakeman, “Fraunhofer Scattering by a Sub-Fractal Diffuser,” Opt. Acta 30, 1207 (1983).
[CrossRef]

Opt. Acta. (1)

C. Pask, “Derivation of Source-Field Coherence Properties from Radiation Angular Distribution,” Opt. Acta. 24, 235 (1977).
[CrossRef]

Opt. Commun. (4)

E. Wolf, W. H. Carter, “Fields Generated by Homogeneous and by Quasi-Homogeneous Planar Secondary Sources,” Opt. Commun. 50, 131 (1984).
[CrossRef]

B. M. Levine, J. C. Dainty, “Non-Gaussian Image Plane Speckle: Measurements from Diffusers of Known Statistics,” Opt. Commun. 45, 252 (1983).
[CrossRef]

E. R. Mendez, K. A. O’Donnell, “Observation of Depolarization and Backscattering Enhancement in Light Scattering from Gaussian Random Surfaces,” Opt. Commun. 61, 91 (1987).
[CrossRef]

N. G. Gaggioli, M. L. Roblin, “Etudes des etats de surface par les properties de diffusion a l’infini en lumiere transmise,” Opt. Commun. 32, 209 (1980).
[CrossRef]

Opt. Quantum Electron. (3)

W. T. Welford, “Review-Optical Estimation of Statistics of Surface Roughness from Light Scattering Measurements,” Opt. Quantum Electron. 9, 269 (1977).
[CrossRef]

P. J. Chandley, “Surface Roughness Measurements from Coherent Light Scattering,” Opt. Quantum Electron. 8, 323 (1976).
[CrossRef]

P. J. Chandley, “Determination of the Probability Density Function of Height on a Rough Surface from Far-Field Coherent Light Scattering,” Opt. Quantum Electron. 11, 413 (1979).
[CrossRef]

Proc. IEEE (1)

P. Beckmann, “Scattering by Composite Rough Surfaces,” Proc. IEEE 53, 1012 (1965).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

L. G. Shirley, N. George, “Diffuser Transmission Functions and Far-Zone Speckle Patterns,” Proc. Soc. Photo-Opt. Instrum. Eng. 556, 63 (1985).

Other (6)

Equation (7) is obtained from Ref. 6 as follows: Equation (56) for 〈|ν|2〉 is multiplied by (R0/A0)2, Eq. (79) for 〈U〉 is then substituted into Eq. (56) with Eq. (64) for the aperture factor, and spatial frequencies are converted to angles through Eqs. (18) and (19). The cos2θ obliquity factor is also generalized to cosnθ.

E. Lukacs, Characteristic Functions (C. Griffin, London, 1970), p. 68.

M. R. Latta, “The Scattering of 10.6 Micron Radiation from Ground Glass Surfaces,” M.S. Thesis, U. Rochester, (1968).

R. I. Hamaguchi, “Transmission Characteristics of Ground Glass,” M.S. Thesis, U. Rochester, (1970).

Armour Products, Midland Park, NJ 07432.

Allied Chemical, Allied Corp., Morristown, NJ 07960.

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Figures (10)

Fig. 1
Fig. 1

SEMs of glass surfaces (a) ground with 820 grit and (b) preroughened for 60 min with Armour Etch and etched for 45 min in BOE.

Fig. 2
Fig. 2

Coordinate system with input angles (θ00) and observation angles (θ,Φ). The diffuser is located in the x-y plane of the coordinate system.

Fig. 3
Fig. 3

In〉 vs θ for the conical diffuser of Eq. (9) with θ0 = 0 and for various values of w/(λS2). The dashed line represents an idealized Lambertian diffuser.

Fig. 4
Fig. 4

In〉 vs θ for the paraboloidal diffuser of Eq. (10) with θ0 = 0 and for various values of w/(λS). The dashed line represents an idealized Lambertian diffuser.

Fig. 5
Fig. 5

Universal plots of Eq. (24) with β of Eq. (25) ranging between 0 and 4.

Fig. 6
Fig. 6

In〉 vs θ for the composite diffusers of Eq. (36), negative angles, and of Eq. (37), positive angles.

Fig. 7
Fig. 7

Block diagram of the scatterometer for measuring angular dependence of radiation patterns Ien.

Fig. 8
Fig. 8

Measured radiation patterns Ien, solid lines, and theoretical radiation patterns 〈In〉, dashed lines, for (a) the “Lambertian” surface compared to Eq. (10), (b) the ground-glass diffuser compared to Eq. (9) with wS2) = 1.4, and (c) the etched-glass diffuser compared to Eq. (36) with w1S1) = 6, w2/λ = 0.75, and S2 = 0.14.

Fig. 9
Fig. 9

Measured radiation patterns Ien from the etched-glass diffuser for various angles of incidence θ0.

Fig. 10
Fig. 10

In〉 vs θ as calculated by Eq. (36) for various angles of incidence θ0 and for w1/(λS1) = 6, w2/λ = 0.75, and S2 = 0.14.

Equations (42)

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S = 2 π σ h λ ( n 1 )
R h ( x 2 x 1 , y 2 y 1 ) = h ( x 1 , y 1 ) h ( x 2 , y 2 ) σ h 2 ,
R h ( r ) = 1 r w + ,
R h ( r ) = 1 ( r w ) 2 + ,
I = d P d Ω .
I e n = Δ P Δ Ω P 0 .
I n = cos n θ 2 π λ 2 0 r × J 0 ( 2 π r λ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ) × exp { S 2 [ 1 R h ( r ) ] } d r .
exp { S 2 [ 1 R h ( r ) ] } = exp ( C 1 S 2 w | r | + C 2 S 2 w 2 r 2 + C 3 S 2 w 3 | r | 3 + ) .
I n = cos θ 2 π ( w λ S 2 ) 2 { 1 + ( 2 π w λ S 2 ) 2 × [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } 3 / 2 .
I n = cos θ π ( w λ S ) 2 exp { ( π w λ S ) 2 × [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } .
I n = cos θ π .
P = P 0 { 1 [ 1 + ( 2 π w λ S 2 ) 2 ] 1 / 2 }
P = P 0 { 1 exp [ ( π w λ S ) 2 ] }
I n max = 2 π ( w λ S 2 ) 2
I n c o s θ ( 2 π ) 2 λ S 2 w sin 3 θ .
I n max = π ( w λ S ) 2
S ( θ 0 ) = 4 π σ h λ cos θ 0 ,
S ( θ 0 ) = 2 π σ h λ ( n 2 sin 2 θ 0 cos θ 0 ) .
h ( x , y ) = h 1 ( x , y ) + h 2 ( x , y ) .
σ h 2 R h ( r ) = σ h 1 2 R h 1 ( r ) + σ h 2 2 R h 2 ( r ) ,
exp { S 2 [ 1 R h ( r ) ] } = exp { S 1 2 [ 1 R h 1 ( r ) ] } × exp { S 2 2 [ 1 R h 2 ( r ) ] } .
w λ S 2 = ( λ S 1 2 w 1 + λ S 2 2 w 2 ) 1 .
w λ S = [ ( λ S 1 w 1 ) 2 + ( λ S 2 w 2 ) 2 ] 1 / 2 .
I n = cos θ 2 π λ 2 0 r × J 0 ( 2 π r λ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ) × exp ( r S 1 2 w 1 2 r 2 S 2 2 w 2 2 ) d r .
β = w 2 / S 2 w 1 / S 1 2 ,
exp { S 2 2 [ 1 R h 2 ( r ) ] } = exp ( S 2 2 ) ( 1 + { exp [ S 2 2 R h 2 ( r ) ] 1 } ) ,
I n = cos θ 2 π exp ( S 2 2 ) [ ( w 1 λ S 1 2 ) 2 { 1 + ( 2 π w 1 λ S 1 2 ) 2 [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } 3 / 2 + 1 λ 2 0 r J 0 [ 2 π r λ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] exp ( r S 1 2 w 1 ) { exp [ S 2 2 R h 2 ( r ) ] 1 } d r ] .
I n = cos θ π exp ( S 2 2 ) [ ( w 1 λ S 1 ) 2 exp { ( π w 1 λ S 1 ) 2 [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } + 2 λ 2 0 r J 0 [ 2 π r λ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] exp [ ( r S 1 w 1 ) 2 ] { exp [ S 2 2 R h 2 ( r ) ] 1 } d r ] .
exp [ S 2 2 R h 2 ( r ) ] 1 = S 2 2 R h 2 ( r ) + .
R h ( r ) = exp ( r w )
R h ( r ) = exp ( r 2 w 2 ) .
I n = cos θ 2 π exp ( S 2 2 ) [ ( w 1 λ S 1 2 ) 2 { 1 + ( 2 π w 1 λ S 1 2 ) 2 [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } 3 / 2 + ( s 2 λ / w 2 + λ S 1 2 / w 1 ) 2 { 1 + 4 π 2 [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 ] + sin 2 θ 0 ( λ / w 2 + λ S 1 / w 1 ) 2 } 3 / 2 ] .
w 2 w 1 / S 1 2 ,
I n = cos θ π exp ( S 2 2 ) [ 2 ( w 1 λ S 1 2 ) 2 { 1 + ( 2 π w 1 λ S 1 2 ) 2 × [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } 3 / 2 + ( S 2 w 2 λ ) 2 exp { ( π w 2 λ ) 2 × [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } ] .
w 2 w 1 / S 1 ,
I n = cos θ π exp ( S 2 2 ) [ ( w 1 λ S 1 ) 2 exp { ( π w 1 λ S 1 ) 2 × [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } + 2 ( S 2 w 2 λ ) 2 { 1 + ( 2 π w 2 λ ) 2 × [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } 3 / 2 ] .
I n = cos θ π exp ( S 2 2 ) [ ( w 1 λ S 1 ) 2 exp { ( π w 1 λ S 1 ) 2 [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] } + S 2 2 ( λ / w 2 ) 2 + ( λ S 1 / w 1 ) 2 exp { π 2 [ sin 2 θ 2 cos ( ϕ ϕ 0 ) sin θ sin θ 0 + sin 2 θ 0 ] ( λ / w 2 ) 2 + ( λ S 1 / w 1 ) 2 } ] .
I n cos θ ( 2 π ) 2 λ S 2 2 w 2 sin 3 θ .
g = V s 0 V r 0 .
P 0 = α g V r ,
Δ P = α V s .
I e n = V s Δ Ω g V r .

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