Abstract

A general theory is presented for the characteristics of speckle when electromagnetic waves are scattered by a rotating, rough surface which has underlying shape features. An expression is derived for the cross-correlation function of the electric field scattered to two different observation points, with the two states including wavelength changes as well as object motion. This function is the basic building block from which to study remote sensing of object features using FM-laser systems or to study the spectral broadening caused by object rotation. Illustrative calculations are presented for rotating cylinders and for roughened spheres.

© 1976 Optical Society of America

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References

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  1. These are discussed and referenced in the topics volume edited by J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
  2. N. George, “The wavelength sensitivity of back-scattering,” Opt. Commun. 16, 328–333 (1976).
    [Crossref]
  3. N. George, A. C. Livanos, J. A. Roth, and C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
    [Crossref]
  4. J. V. Evans and Tor Hagfors, Radar Astronomy (McGraw-Hill, New York, 1968).
  5. R. M. Goldstein, Radar Exploration of Venus (Ph. D. thesis, California Institute of Technology, Pasadena, 1962).
  6. R. F. Broderick and H. S. Hayre, “Doppler Return From a Random Rough Surface,” IEEE Trans. Aerosp. Electron. Sys. AES-5, 441–449 (1969).
    [Crossref]
  7. J. van Bladel, “Electromagnetic Fields in the Presence of Rotating Bodies,” Proc. IEEE 64, 301–318 (1976).
    [Crossref]
  8. T. C. Mo, “Theory of electrodynamics in media in noninertial frames and applications,” J. Math. Phys. 11, 2589–2610 (1970).
    [Crossref]
  9. C. Yeh, “Scattering of obliquely incident microwaves by a moving plasma column” J. Appl. Phys. 40, 5066–5075 (1969).
    [Crossref]
  10. H. S. Cabayan and R. C. Murphy, “Scattering of Electromagnetic Waves by Rough Perfectly Conducting Circular Cylinders,” IEEE Trans. Antennas Propag. AP21893–895 (1973).
    [Crossref]
  11. M. C. Teich, “Homodyne Detection of Infrared Radiation from a Moving Diffuse Target,” Proc. IEEE 57, 786–792 (1969).
    [Crossref]
  12. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
    [Crossref]
  13. J. W. Goodman, “Degredation of image quality and velocity measurement accuracy by speckle phenomena,” Stanford University, Stanford, Calif., unpublished memorandum.
  14. C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965).
  15. J. Van Bladel, “Relativistic Theory of Rotating Disks,” Proc. IEEE 61, 260–268 (1973).
    [Crossref]
  16. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  17. B. Crosignani, P. DiPorto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).
  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964).
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  20. H. E. Rowe, Signals and Noise in Communication Systems (Van Nostrand Reinhold, New York, 1965).
  21. W. R. Symthe, Static and Dynamic Electricity, 3rd ed. (McGraw-Hill, New York, 1968).

1976 (3)

N. George, “The wavelength sensitivity of back-scattering,” Opt. Commun. 16, 328–333 (1976).
[Crossref]

N. George, A. C. Livanos, J. A. Roth, and C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[Crossref]

J. van Bladel, “Electromagnetic Fields in the Presence of Rotating Bodies,” Proc. IEEE 64, 301–318 (1976).
[Crossref]

1973 (2)

H. S. Cabayan and R. C. Murphy, “Scattering of Electromagnetic Waves by Rough Perfectly Conducting Circular Cylinders,” IEEE Trans. Antennas Propag. AP21893–895 (1973).
[Crossref]

J. Van Bladel, “Relativistic Theory of Rotating Disks,” Proc. IEEE 61, 260–268 (1973).
[Crossref]

1970 (1)

T. C. Mo, “Theory of electrodynamics in media in noninertial frames and applications,” J. Math. Phys. 11, 2589–2610 (1970).
[Crossref]

1969 (3)

C. Yeh, “Scattering of obliquely incident microwaves by a moving plasma column” J. Appl. Phys. 40, 5066–5075 (1969).
[Crossref]

R. F. Broderick and H. S. Hayre, “Doppler Return From a Random Rough Surface,” IEEE Trans. Aerosp. Electron. Sys. AES-5, 441–449 (1969).
[Crossref]

M. C. Teich, “Homodyne Detection of Infrared Radiation from a Moving Diffuse Target,” Proc. IEEE 57, 786–792 (1969).
[Crossref]

1965 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Bertolotti, M.

B. Crosignani, P. DiPorto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Broderick, R. F.

R. F. Broderick and H. S. Hayre, “Doppler Return From a Random Rough Surface,” IEEE Trans. Aerosp. Electron. Sys. AES-5, 441–449 (1969).
[Crossref]

Cabayan, H. S.

H. S. Cabayan and R. C. Murphy, “Scattering of Electromagnetic Waves by Rough Perfectly Conducting Circular Cylinders,” IEEE Trans. Antennas Propag. AP21893–895 (1973).
[Crossref]

Crosignani, B.

B. Crosignani, P. DiPorto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

DiPorto, P.

B. Crosignani, P. DiPorto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Evans, J. V.

J. V. Evans and Tor Hagfors, Radar Astronomy (McGraw-Hill, New York, 1968).

George, N.

N. George, “The wavelength sensitivity of back-scattering,” Opt. Commun. 16, 328–333 (1976).
[Crossref]

N. George, A. C. Livanos, J. A. Roth, and C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[Crossref]

Goldstein, R. M.

R. M. Goldstein, Radar Exploration of Venus (Ph. D. thesis, California Institute of Technology, Pasadena, 1962).

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

J. W. Goodman, “Degredation of image quality and velocity measurement accuracy by speckle phenomena,” Stanford University, Stanford, Calif., unpublished memorandum.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Hagfors, Tor

J. V. Evans and Tor Hagfors, Radar Astronomy (McGraw-Hill, New York, 1968).

Hayre, H. S.

R. F. Broderick and H. S. Hayre, “Doppler Return From a Random Rough Surface,” IEEE Trans. Aerosp. Electron. Sys. AES-5, 441–449 (1969).
[Crossref]

Livanos, A. C.

N. George, A. C. Livanos, J. A. Roth, and C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[Crossref]

Mo, T. C.

T. C. Mo, “Theory of electrodynamics in media in noninertial frames and applications,” J. Math. Phys. 11, 2589–2610 (1970).
[Crossref]

Murphy, R. C.

H. S. Cabayan and R. C. Murphy, “Scattering of Electromagnetic Waves by Rough Perfectly Conducting Circular Cylinders,” IEEE Trans. Antennas Propag. AP21893–895 (1973).
[Crossref]

Papas, C. H.

N. George, A. C. Livanos, J. A. Roth, and C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[Crossref]

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965).

Roth, J. A.

N. George, A. C. Livanos, J. A. Roth, and C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[Crossref]

Rowe, H. E.

H. E. Rowe, Signals and Noise in Communication Systems (Van Nostrand Reinhold, New York, 1965).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964).

Symthe, W. R.

W. R. Symthe, Static and Dynamic Electricity, 3rd ed. (McGraw-Hill, New York, 1968).

Teich, M. C.

M. C. Teich, “Homodyne Detection of Infrared Radiation from a Moving Diffuse Target,” Proc. IEEE 57, 786–792 (1969).
[Crossref]

van Bladel, J.

J. van Bladel, “Electromagnetic Fields in the Presence of Rotating Bodies,” Proc. IEEE 64, 301–318 (1976).
[Crossref]

J. Van Bladel, “Relativistic Theory of Rotating Disks,” Proc. IEEE 61, 260–268 (1973).
[Crossref]

Yeh, C.

C. Yeh, “Scattering of obliquely incident microwaves by a moving plasma column” J. Appl. Phys. 40, 5066–5075 (1969).
[Crossref]

IEEE Trans. Aerosp. Electron. Sys. (1)

R. F. Broderick and H. S. Hayre, “Doppler Return From a Random Rough Surface,” IEEE Trans. Aerosp. Electron. Sys. AES-5, 441–449 (1969).
[Crossref]

IEEE Trans. Antennas Propag. (1)

H. S. Cabayan and R. C. Murphy, “Scattering of Electromagnetic Waves by Rough Perfectly Conducting Circular Cylinders,” IEEE Trans. Antennas Propag. AP21893–895 (1973).
[Crossref]

J. Appl. Phys. (1)

C. Yeh, “Scattering of obliquely incident microwaves by a moving plasma column” J. Appl. Phys. 40, 5066–5075 (1969).
[Crossref]

J. Math. Phys. (1)

T. C. Mo, “Theory of electrodynamics in media in noninertial frames and applications,” J. Math. Phys. 11, 2589–2610 (1970).
[Crossref]

Opt. Acta (1)

N. George, A. C. Livanos, J. A. Roth, and C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[Crossref]

Opt. Commun. (1)

N. George, “The wavelength sensitivity of back-scattering,” Opt. Commun. 16, 328–333 (1976).
[Crossref]

Proc. IEEE (4)

J. van Bladel, “Electromagnetic Fields in the Presence of Rotating Bodies,” Proc. IEEE 64, 301–318 (1976).
[Crossref]

M. C. Teich, “Homodyne Detection of Infrared Radiation from a Moving Diffuse Target,” Proc. IEEE 57, 786–792 (1969).
[Crossref]

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

J. Van Bladel, “Relativistic Theory of Rotating Disks,” Proc. IEEE 61, 260–268 (1973).
[Crossref]

Other (11)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

B. Crosignani, P. DiPorto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D. C., 1964).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

H. E. Rowe, Signals and Noise in Communication Systems (Van Nostrand Reinhold, New York, 1965).

W. R. Symthe, Static and Dynamic Electricity, 3rd ed. (McGraw-Hill, New York, 1968).

J. W. Goodman, “Degredation of image quality and velocity measurement accuracy by speckle phenomena,” Stanford University, Stanford, Calif., unpublished memorandum.

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965).

These are discussed and referenced in the topics volume edited by J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

J. V. Evans and Tor Hagfors, Radar Astronomy (McGraw-Hill, New York, 1968).

R. M. Goldstein, Radar Exploration of Venus (Ph. D. thesis, California Institute of Technology, Pasadena, 1962).

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Figures (5)

FIG. 1
FIG. 1

Rough surface S illuminated by a monochromatic source at A with the scattered field monitored at B.

FIG. 2
FIG. 2

Configurations for (a) the wavelength sensitivity of backscatter from a stationary sphere, and (b) the cylinder is rotating about the z axis at Ω rad/s.

FIG. 3
FIG. 3

The amplitude of the real part of the autocorrelation function | Re R e a | vs the offset wavelength parameter, 2a0Δk. Plotted from Eq. (33) for c0 of (○) 0.63, (△) 1, (+) 2.

FIG. 4
FIG. 4

The wavelength sensitivity of backscatter from roughened spheres is shown in plots of the amplitude | R e a | vs 2a0Δk for the labeled values of the roughness parameter c0 = α1/(2σ).

FIG. 5
FIG. 5

Curves of | R e a | vs 2a0Δk for roughened spheres but using the function B = cos2ψ (for comparison to the biased slope case in Fig. 4 where B = 1/cos2ψ).

Tables (1)

Tables Icon

TABLE I Relative values for terms in the cross correlation of the electric field, computed from Eqs. (B22)–(B26).

Equations (88)

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H inc ( x , y , z , t ) = e y H 0 exp { i ( ω t - k R a p ) } ,
E inc ( x , y , z , t ) = e z ( μ 0 / 0 ) 1 / 2 H 0 exp { i ( ω t - k R a p ) } ,
K = 2 n × H inc ,
A ( x , y , z , t ) = μ 0 4 π K ( x , y , z , t - R / c ) R d s ,
A ( x , y , z , t ) = μ 0 2 π n × ( e y H 0 ) R p b exp { i ω ( t - R a p + R p b c ) } d s ,
A b ( t ) = μ 0 2 π [ n ] × ( e y H 0 ) [ R p b ] exp { i ω ( t - [ R a p + R p b ] c ) } d s ,
[ R a p + R p b ] = 2 r 0 - w sin θ { cos ( ϕ ) + cos ( ϕ - ϕ 1 ) } - 2 ω w r 0 c sin θ { sin ϕ + sin ( ϕ - ϕ 1 ) } + 2 Ω w 2 c sin 2 θ { sin ϕ cos ϕ + sin ( ϕ - ϕ 1 ) cos ( ϕ - ϕ 1 ) } ,
ϕ = ϕ r + Ω t .
8 Ω w r 0 / λ c < 1 ,
A b ( t ) = μ H 0 exp { i ( ω t - 2 k r 0 ) } 2 π r 0 × n × e y exp ( i k w sin θ { cos ϕ + cos ( ϕ - ϕ 1 ) } ) d s ,
E z ( ω 1 , ϕ 1 , t ) = - i λ 1 ( μ ) 1 / 2 H 0 exp [ i ( ω 1 t - 2 k 1 r 0 ) ] r 0 × exp ( i k 1 w sin θ { cos ϕ + cos ( ϕ - ϕ 1 ) } ) ( n · e x ) d s .
w ( θ r , ϕ r ) = w 0 ( θ r , ϕ r ) + h ( θ r , ϕ r ) ,
w ( θ , ϕ - Ω t ) = w 0 ( θ , ϕ - Ω t ) + h ( θ , ϕ - Ω t ) .
R e = E z ( ω 1 , ϕ 1 , t 1 ) E z * ( ω 2 , ϕ 2 , t 2 ) ,
R e = I 0 exp { i k 1 w 0 sin θ [ cos ϕ + cos ( ϕ - ϕ 1 ) ] - i k 2 w 0 sin θ [ cos ϕ + cos ( ϕ - ϕ 2 ) ] } × F ( η 1 , - η 2 ; r 12 ) B ( n · e x , n · e x ) d s d s ,
I 0 = [ H 0 2 μ / ( λ 1 λ 2 r 0 2 ) ] exp { i [ ω 1 t 1 - ω 2 t 2 - 2 r 0 ( k 1 - k 2 ) ] } , k 1 , 2 = 2 π / λ 1 , 2 , w 0 = w 0 ( θ , ϕ - Ω t 1 ) , w 0 = w 0 ( θ , ϕ - Ω t 2 ) .
F ( η 1 , - η 2 ; r 12 ) = exp ( i η 1 h 1 - i η 2 h 2 ) .
h 1 = h ( θ , ϕ - Ω t 1 ) , h 2 = h ( θ , ϕ - Ω t 2 ) .
η 1 = k 1 sin θ [ cos ϕ + cos ( ϕ - ϕ 1 ) ] , η 2 = k 2 sin θ [ cos ϕ + cos ( ϕ - ϕ 2 ) ] .
B ( ψ ) = 1 / [ cos ψ ] 2 ,
f ( h 1 , h 2 ; r 12 ) = 1 2 π σ 2 ( 1 - r 12 2 ) 1 / 2 × exp ( - h 1 2 - 2 r 12 h 1 h 2 + h 2 2 2 σ 2 ( 1 - r 12 2 ) ) ,
F ( η 1 , - η 2 ; r 12 ) = exp [ - 1 2 σ 2 ( η 1 2 - 2 r 12 η 1 η 2 + η 2 2 ) ] .
r 12 ( h 1 , h 2 ) = { 1 - ( ρ s / α 1 ) 2 when ( ρ s / α 1 ) 2 1 0 otherwise ,
R e = R e a + E z 1 E z 2 * ,
R e a = I 0 A 1 exp ( - i w 0 ( θ , ϕ - Ω t 1 ) Δ η - ( σ Δ η ) 2 2 ) × P ( θ , ϕ , t 1 - t 2 ) rect ( ϕ - Ω ( t 1 - t 2 ) π ) B ( ψ ) d s .
Δ η = k 2 sin θ { cos [ ϕ - Ω ( t 1 - t 2 ) ] + cos [ ϕ - Ω ( t 1 - t 2 ) - ϕ 2 ] } - k 2 sin θ [ cos ϕ + cos ( ϕ - ϕ 1 ) ] .
Δ η = 2 Δ k sin θ cos [ ϕ - Ω ( t 1 - t 2 ) - 1 2 ϕ 2 ] cos ( 1 2 ϕ 2 ) + 2 k 1 sin θ ( sin [ 1 2 Ω ( t 1 - t 2 ) ] sin ( ϕ - 1 2 Ω ( t 1 - t 2 ) ] + sin { 1 2 [ Ω ( t 1 - t 2 ) + ϕ 2 - ϕ 1 ] } sin { ϕ - 1 2 [ Ω ( t 1 - t 2 ) + ϕ 1 + ϕ 2 ] } ) ,
P 1 = exp [ - ( 1 2 α 1 b ) 2 / μ ] / μ .
b - 2 k 2 cos ( 1 2 ϕ 2 ) { 1 - sin 2 θ cos 2 [ ϕ - Ω ( t 1 - t 2 ) - 1 2 ϕ 2 ] } 1 / 2 .
μ = σ 2 η 1 η 20 , μ = σ 2 k 1 k 2 sin 2 θ [ cos ϕ + cos ( ϕ - ϕ 1 ) ] × { cos [ ϕ + Ω τ ] + cos [ ϕ + Ω τ - ϕ 2 ] } .
R e = E z ( ω 1 , ϕ 1 , t 1 ) E z * ( ω 2 , ϕ 1 , t 1 ) .
R e a = I 0 A 0 A 1 4 k 2 σ 2 × 0 π / 2 exp [ - i 2 a 0 Δ k cos ψ - 2 ( σ Δ k cos ψ ) 2 - ( α 1 2 σ tan ψ ) 2 ] × sin ψ d ψ [ cos ψ ] 4 .
R e a = I 0 A 0 A 1 4 k 2 σ 2 m = 0 ( - i 2 a 0 Δ k ) m c 0 m - 3 exp ( c 0 2 ) m !                         2 Γ ( - m + 3 2 , c 0 2 ) ,
R u = u ( ω 1 , ϕ 1 , t 1 ) u ( ω 1 , ϕ 2 , t 2 )
R u = E z 1 E z 1 * E z 2 E z 2 * ,
R e a = E z ( ω 1 , 0 , t 1 ) E z * ( ω 1 , 2 Ω τ , t 1 + τ ) .
Δ η = - 4 k sin 2 ( Ω τ / 2 ) cos ϕ , b = 2 k sin ϕ cos ( Ω τ ) , μ = 4 ( k σ cos ϕ ) 2 cos ( Ω τ ) .
R e a = I 0 A 1 A 2 4 k 2 σ 2 × - π / 2 π / 2 exp [ i 4 k a 0 sin 2 ( Ω τ 2 ) cos ϕ - ( α 1 2 σ ) 2 tan 2 ϕ ] d ϕ [ cos 4 ϕ ] .
R e a = I 0 A 1 A 2 π 1 / 2 exp [ 1 2 c 0 2 ] 4 k 2 σ 2 × m = 0 ( i c 1 ) m c 0 ( m - 5 ) / 2 m ! W ( - m + 3 ) / 4 , ( m - 3 ) / 4 ( c 0 2 ) ,
2 a 0 Δ k cos ψ and k a 0 ( Ω τ ) 2 cos ϕ ,
R e a I 0 A 1 A 2 4 k 2 σ 2 - π / 2 π / 2 exp ( i c 1 cos ϕ ) d ϕ , R e a = I 0 A 1 A 2 4 k 2 σ 2 [ π J 0 ( c 1 ) - i π E 0 ( c 1 ) ] ,
( Ω τ ) 0 [ λ / a 0 ] 1 / 2 .
N = 4 ( a 0 / λ ) 1 / 2 .
R e = E z ( ω 1 , 0 , t 1 ) E z * ( ω 1 , 0 , t 2 ) .
P 1 B = exp { - c 0 2 sin 2 ( ϕ + Ω τ ) / [ cos ϕ cos ( ϕ + Ω τ ) ] } 4 k 2 σ 2 [ cos ϕ cos ( ϕ + Ω τ ) ] 2 .
R e a = I 0 A 1 A 2 × - π / 2 π / 2 exp ( - i 2 k 1 a 0 Φ - 2 ( k 1 σ Φ ) 2 ) rect ( ϕ + Ω τ π ) P 1 B d ϕ
Φ = - 2 sin ( 1 2 Ω τ ) sin ( ϕ + 1 2 Ω τ ) .
sin Ω τ Ω τ 1.
R e a = I 0 A 1 A 2 - π / 2 π / 2 exp [ i 2 k 1 a 0 Ω τ sin ϕ - 2 ( k 1 σ Ω τ sin ϕ ) 2 ] P 1 B d ϕ .
S e ( ω ) = - R e a ( τ ) e - i ω τ d τ .
S e ( ω ) = I 1 A 1 A 2 - π / 2 π / 2 ( 1 2 π ) 1 / 2 k 1 σ Ω sin ϕ × exp { - ( ω - ω c + 2 k 1 a 0 Ω sin ϕ ) 2 8 ( k 1 σ Ω sin ϕ ) 2 } P 1 B d ϕ ,
δ ( ω ) = lim Δ 0 exp ( - π ω 2 / Δ 2 ) Δ .
S e ( ω ) = I 1 A 1 A 2 - π / 2 π / 2 δ ( ω - ω c + 2 k 1 a 0 Ω sin ϕ ) P 1 B d ϕ ,
[ R a p + R p b ] = R a p { 1 + [ β a p ] 2 - 2 [ β a p ] cos [ ψ a p ] } 1 / 2 + R p b { 1 + [ β p b ] 2 - 2 [ β p b ] cos [ ψ p b ] } 1 / 2 ,
R p b = { r 0 2 + w 2 - 2 w r 0 sin θ cos ( ϕ - ϕ 1 ) } 1 / 2 ,
R p b r 0 - w sin θ cos ( ϕ - ϕ 1 ) ,
w ( θ r , ϕ r ) = w ( θ , ϕ - Ω t ) ,
cos ψ p b = - sin ( ϕ - ϕ 1 ) ( 1 + ( w r 0 ) 2 - 2 w r 0 sin θ cos ( ϕ - ϕ 1 ) ) 1 / 2
cos ψ p b - sin ( ϕ - ϕ 1 ) .
[ R p b ] = r 0 - w sin θ cos ( ϕ - ϕ 1 ) - 2 Ω w r 0 c sin θ sin ( ϕ - ϕ 1 ) + 2 Ω w 2 c sin 2 θ sin ( ϕ - ϕ 1 ) cos ( ϕ - ϕ 1 ) .
R e = I 0 d s d s G 1 G 2 * N 1 N 2 P 12 B ,
G 1 = exp { i k 1 w 0 ( θ , ϕ - Ω t 1 ) sin θ [ cos ϕ + cos ( ϕ - ϕ 1 ) ] } , G 1 = exp { i k 2 w 0 ( θ , ϕ - Ω t 2 ) sin θ [ cos ϕ + cos ( ϕ - ϕ 2 ) ] } .
F ( η 1 , - η 2 ; r 12 ) = N 1 N 2 P 12 ,
N 1 = exp [ - 1 2 ( σ η 1 ) 2 ] , N 2 = exp [ - 1 2 ( σ η 2 ) 2 ] , P 12 = exp ( σ 2 r 12 η 1 η 2 ) ,
F ( η 1 , - η 2 ; r 12 ) = exp [ - 1 2 σ 2 ( η 1 - η 1 ) 2 - σ 2 η 1 η 1 ( 1 - r 12 ) ] .
R e = I 0 d s B G 1 N 1 ( r 12 0 d s G 2 * N 2 P 12 ) + I 0 d s B 1 G 1 N 1 ( r 12 = 0 d s B 2 G 2 * N 2 ) ,
R e = I 0 d s B G 1 N 1 [ d s G 2 * N 2 ( P 12 - 1 ) ] × E z 1 E z 2 * ,
E z 1 = - i λ 1 ( μ 0 0 ) 1 / 2 H 0 exp [ i ( ω 1 t 1 - 2 k 1 r 0 ) ] r 0 d s B 1 G 1 N 1 , E z 2 * = i λ 2 ( μ 0 0 ) 1 / 2 H 0 exp [ - i ( ω 2 t 2 - 2 k 2 r 0 ) ] r 0 d s B 2 G 2 * N 2
θ = θ - θ , θ = ϕ - ϕ + Ω ( t 1 - t 2 ) .
R e a = E z 1 E z 2 * - E z 1 E z 2 * , R e a = I 0 d s B G 1 G 20 * N 1 N 20 exp ( + μ ) × [ d s exp ( - i w 0 d η 2 ) { exp [ μ ( r 12 - 1 ) ] - exp ( - μ ) } ] .
μ = σ 2 η 1 η 20 η 20 = k 2 sin θ { cos [ ϕ - Ω ( t 1 - t 2 ) ] + cos [ ϕ - Ω ( t 1 - t 2 ) - ϕ 2 ] } , N 20 = exp { - 1 2 ( σ η 20 ) 2 } , G 20 = exp ( i k 2 w 0 ( θ , ϕ - Ω t 1 ) sin θ { cos [ ϕ - Ω ( t 1 - t 2 ) ] + cos [ ϕ - Ω ( t 1 - t 2 ) - ϕ 2 ] } ) ,
R e a = I 0 A 1 d s B G 1 G 20 * N 1 N 20 exp ( + μ ) P ( θ , ϕ , t 1 - t 2 ) ,
P = ( 2 π / A 1 ) 0 α 1 d ρ s ρ s J 0 ( b ρ s ) { exp [ μ ( r 12 - 1 ) ] - exp ( - μ ) } ,
b = 2 k 2 cos ( 1 2 ϕ 2 ) { 1 - sin 2 θ cos 2 [ ϕ - Ω ( t 1 - t 2 ) - 1 2 ϕ 2 ] } 1 / 2 .
b 2 k 2 cos ( 1 2 ϕ 2 ) sin ψ ,
P = m = 0 ( - 1 ) m ( α 1 b ) 2 m 2 2 m     m ! m ! × γ ( m + 1 , μ ) μ m + 1 - exp ( - μ ) J 1 ( α 1 b ) α 1 b / 2 .
P = exp ( - μ ) μ n = 2 μ n J n ( α 1 b ) ( 1 2 α 1 b ) n .
P μ J 2 ( α 1 b ) ( 1 2 α 1 b ) 2             when ψ 1 2 π .
P = P 1 + P 2 + P 3 ,
P 1 = exp [ - ( 1 2 α 1 b ) 2 / μ ] μ , P 2 = - exp ( - μ ) μ m = 0 ( - 1 ) m ( α 1 b 2 μ ) m J m ( α 1 b ) , P 3 = - exp ( - μ ) J 1 ( α 1 b ) / ( 1 2 α 1 b ) .
μ = 1.6 × 10 4 and α 1 b = 0 for ψ = 0 , μ = 172 and α 1 b = 250 for ψ = 84° .
R e 1 = I 0 A 0 A 1 k 1 k 2 ( 2 σ ) 2 × 0 1 d u exp { - i 2 Δ k a 0 u - 2 ( σ Δ k u ) 2 - c 0 2 [ ( 1 - u 2 ) / u 2 ] } ,
R e 1 ( 0 ) = 1 2 c 0 1 / 2 exp ( c 0 ) Γ ( - 1 2 , c 0 ) I 0 A 0 A 1 [ k 1 k 2 ( 2 σ ) 2 ] .
R e 2 = - I 0 A 0 A 1 k 1 k 2 ( 2 σ ) 2 × 0 1 d u exp [ - i 2 Δ k a 0 u - 2 σ 2 ( k 1 2 + k 2 2 ) u ] .
R e 2 = - I 0 A 0 A 1 k 1 k 2 ( 2 σ ) 2 × m = 0 i m ( - 2 Δ k a 0 ) m γ [ 1 2 ( m + 1 ) ,     2 σ 2 ( k 1 2 + k 2 2 ) ] m !                         2 [ 2 σ 2 ( k 1 2 + k 2 2 ) ] ( m + 1 ) / 2 .
R e 2 = - I 0 A 0 A 1 k 1 k 2 ( 2 σ ) 2 × m = 0 i [ 2 σ 2 ( k 1 2 + k 2 2 ) ] m γ ( 2 m + 1 ,     i 2 Δ k a 0 ) m !                         ( 2 Δ k a 0 ) 2 m + 1 .
R e 3 = - I 0 A 0 A 1 × m = 0 i m ( - 2 Δ k a 0 ) m γ [ 1 2 ( m + 3 ) ,     2 σ 2 ( k 1 2 + k 2 2 ) ] m !                         2 [ 2 σ 2 ( k 1 2 + k 2 2 ) ] ( m + 3 ) / 2 .
E z 1 E z 2 * = I 0 A 0 A 0 × | n = 0 - ( 2 k 1 σ ) 2 n γ ( 2 n + 2 ,     i 2 k 1 a 0 ) ( 2 k 1 a 0 ) 2 n + 2 | 2 .