Abstract

The speckle pattern produced by laser light scattered from a moving diffuse surface is described with linear system analysis. The mathematical procedures are greatly simplified by operator algebra. Results cited in the literature are shown to be special cases, derived under simplifying assumptions, from the general expressions obtained in the present work.

© 1981 Optical Society of America

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References

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  1. J. W. Goodman, Proc. IEEE 53, 1688 (1965); Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86 (1979).
    [CrossRef]
  2. E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970); A. E. Ennos, “Laser Speckle and Related Phenomena,” in Topics in Applied Physics, J. C. Dainty, Ed. (Springer, Heidelberg, 1975), Vol. 9.
    [CrossRef]
  3. C. B. Burckhardt, Bell Syst. Tech. J. 49, 309 (1970).
  4. J. C. Dainty, Opt. Acta 17, 761 (1970).
    [CrossRef]
  5. S. Lowenthal, H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
    [CrossRef]
  6. D. A. Gregory, “Speckle Photography in Engineering Applications,” in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., London, 1975), pp. 263–282; Opt. Acta 27, 481 (1980).
  7. N. Takai, T. Asakura, Appl. Opt. 17, 3785 (1978); N. Takai, T. Iwai, T. Ushizaka, T. Asakura, Opt. Commun. 30, 287 (1979); N. Takai, X. Sutanto, T. Asakura, J. Opt. Soc. Am. 70, 827 (1980).
    [CrossRef] [PubMed]
  8. I. Yamaguchi, S. Komatsu, Opt. Acta 24, 705 (1977); I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301 (1975); S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715 (1976).
    [CrossRef]
  9. J. C. Leader, Opt. Eng. 19, 593 (1980).
    [CrossRef]
  10. J. Ohtsubo, Opt. Commun. 34, 147 (1980); J. Opt. Paris 11, 323 (1980).
    [CrossRef]
  11. B. Stoffregen, Optik, 52, 305, 385 (1978/79); Optik, 55, 261 (1980).
  12. M. Nazarathy, J. Shamir, J. Opt. Soc. Am. 70, 150 (1980).
    [CrossRef]
  13. M. Nazarathy, J. Shamir, J. Opt. Soc. Am. 71, 529 (1981).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  15. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  16. E. Ingelstam, S. Ragnaksson, Vision Res. 12, 411 (1972).
    [CrossRef] [PubMed]
  17. W. N. Charman, Am. J. Optom. Physiol. Opt. 51, 832 (1974).
    [CrossRef] [PubMed]
  18. J. Shamir, Appl. Opt. 18, 4195 (1979).
    [CrossRef] [PubMed]
  19. G. B. Smith, K. A. Stetson, Appl. Opt. 19, 3031 (1980).
    [CrossRef] [PubMed]
  20. B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).
  21. Y. Fainman, E. Lenz, J. Shamir, Appl. Opt. 20, 158 (1981).
    [CrossRef] [PubMed]
  22. M. Nazarathy, J. Shamir, Isr. J. Technol. 18, 229 (1980).

1981

1980

G. B. Smith, K. A. Stetson, Appl. Opt. 19, 3031 (1980).
[CrossRef] [PubMed]

M. Nazarathy, J. Shamir, J. Opt. Soc. Am. 70, 150 (1980).
[CrossRef]

J. C. Leader, Opt. Eng. 19, 593 (1980).
[CrossRef]

J. Ohtsubo, Opt. Commun. 34, 147 (1980); J. Opt. Paris 11, 323 (1980).
[CrossRef]

M. Nazarathy, J. Shamir, Isr. J. Technol. 18, 229 (1980).

1979

1978

1977

I. Yamaguchi, S. Komatsu, Opt. Acta 24, 705 (1977); I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301 (1975); S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715 (1976).
[CrossRef]

1974

W. N. Charman, Am. J. Optom. Physiol. Opt. 51, 832 (1974).
[CrossRef] [PubMed]

1972

E. Ingelstam, S. Ragnaksson, Vision Res. 12, 411 (1972).
[CrossRef] [PubMed]

1970

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970); A. E. Ennos, “Laser Speckle and Related Phenomena,” in Topics in Applied Physics, J. C. Dainty, Ed. (Springer, Heidelberg, 1975), Vol. 9.
[CrossRef]

C. B. Burckhardt, Bell Syst. Tech. J. 49, 309 (1970).

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

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

1965

J. W. Goodman, Proc. IEEE 53, 1688 (1965); Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86 (1979).
[CrossRef]

Archbold, E.

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970); A. E. Ennos, “Laser Speckle and Related Phenomena,” in Topics in Applied Physics, J. C. Dainty, Ed. (Springer, Heidelberg, 1975), Vol. 9.
[CrossRef]

Arsenault, H.

Asakura, T.

Bertolotti, M.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Burch, J. M.

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970); A. E. Ennos, “Laser Speckle and Related Phenomena,” in Topics in Applied Physics, J. C. Dainty, Ed. (Springer, Heidelberg, 1975), Vol. 9.
[CrossRef]

Burckhardt, C. B.

C. B. Burckhardt, Bell Syst. Tech. J. 49, 309 (1970).

Charman, W. N.

W. N. Charman, Am. J. Optom. Physiol. Opt. 51, 832 (1974).
[CrossRef] [PubMed]

Crosignani, B.

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

Dainty, J. C.

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

Di Porto, P.

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

Ennos, A. E.

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970); A. E. Ennos, “Laser Speckle and Related Phenomena,” in Topics in Applied Physics, J. C. Dainty, Ed. (Springer, Heidelberg, 1975), Vol. 9.
[CrossRef]

Fainman, Y.

Goodman, J. W.

J. W. Goodman, Proc. IEEE 53, 1688 (1965); Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86 (1979).
[CrossRef]

Gregory, D. A.

D. A. Gregory, “Speckle Photography in Engineering Applications,” in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., London, 1975), pp. 263–282; Opt. Acta 27, 481 (1980).

Ingelstam, E.

E. Ingelstam, S. Ragnaksson, Vision Res. 12, 411 (1972).
[CrossRef] [PubMed]

Komatsu, S.

I. Yamaguchi, S. Komatsu, Opt. Acta 24, 705 (1977); I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301 (1975); S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715 (1976).
[CrossRef]

Leader, J. C.

J. C. Leader, Opt. Eng. 19, 593 (1980).
[CrossRef]

Lenz, E.

Lowenthal, S.

Nazarathy, M.

Ohtsubo, J.

J. Ohtsubo, Opt. Commun. 34, 147 (1980); J. Opt. Paris 11, 323 (1980).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Ragnaksson, S.

E. Ingelstam, S. Ragnaksson, Vision Res. 12, 411 (1972).
[CrossRef] [PubMed]

Shamir, J.

Smith, G. B.

Stetson, K. A.

Stoffregen, B.

B. Stoffregen, Optik, 52, 305, 385 (1978/79); Optik, 55, 261 (1980).

Takai, N.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Yamaguchi, I.

I. Yamaguchi, S. Komatsu, Opt. Acta 24, 705 (1977); I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301 (1975); S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715 (1976).
[CrossRef]

Am. J. Optom. Physiol. Opt.

W. N. Charman, Am. J. Optom. Physiol. Opt. 51, 832 (1974).
[CrossRef] [PubMed]

Appl. Opt.

Bell Syst. Tech. J.

C. B. Burckhardt, Bell Syst. Tech. J. 49, 309 (1970).

Isr. J. Technol.

M. Nazarathy, J. Shamir, Isr. J. Technol. 18, 229 (1980).

J. Opt. Soc. Am.

Opt. Acta

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970); A. E. Ennos, “Laser Speckle and Related Phenomena,” in Topics in Applied Physics, J. C. Dainty, Ed. (Springer, Heidelberg, 1975), Vol. 9.
[CrossRef]

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

I. Yamaguchi, S. Komatsu, Opt. Acta 24, 705 (1977); I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301 (1975); S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715 (1976).
[CrossRef]

Opt. Commun.

J. Ohtsubo, Opt. Commun. 34, 147 (1980); J. Opt. Paris 11, 323 (1980).
[CrossRef]

Opt. Eng.

J. C. Leader, Opt. Eng. 19, 593 (1980).
[CrossRef]

Optik

B. Stoffregen, Optik, 52, 305, 385 (1978/79); Optik, 55, 261 (1980).

Proc. IEEE

J. W. Goodman, Proc. IEEE 53, 1688 (1965); Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86 (1979).
[CrossRef]

Vision Res.

E. Ingelstam, S. Ragnaksson, Vision Res. 12, 411 (1972).
[CrossRef] [PubMed]

Other

D. A. Gregory, “Speckle Photography in Engineering Applications,” in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., London, 1975), pp. 263–282; Opt. Acta 27, 481 (1980).

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

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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Figures (6)

Fig. 1
Fig. 1

General outline of the optical system: S(S′), illumination source for opaque (transparent) object; T (ρ,r), optical system.

Fig. 2
Fig. 2

(a) Speckle pattern with large aperture; (b) aperture with small round holes; (c) large aperture superposed by a coarse grating [see Fig. 5(a)]; (d) pattern modulated by the aperture having FT shown in (e).

Fig. 3
Fig. 3

(a) Pattern produced by an isotropically rough stainless steel surface; (b) same surface after anisotropic surface treatment.

Fig. 4
Fig. 4

Optical system with lens.

Fig. 5
Fig. 5

Observation of speckle pattern at the image plane (μ = 0) with a grating attached to the object; (a) large imaging lens; (b) a grating serving as a lens aperture placed with lines parallel to the object grating; (c) as (b) but with grating lines perpendicular to the object grating; (d) lens aperture similar to that used to produce Fig. 2(d).

Fig. 6
Fig. 6

Explanation of notation for a point source illumination.

Equations (128)

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Γ I I ( r 1 , r 2 ; t 1 , t 2 ) = I ( r 1 , t 1 ) I ( r 2 , t 2 ) ,
r 2 - r 1 = V ( t 2 - t 1 ) ,
V ( r , t ) = u ( r , t ) exp ( j ω t ) ,
I ( r , t ) = V ( r , t ) 2 = u ( r , t ) 2 .
Γ I I ( r 1 , r 2 ; t 1 ; t 2 ) = Γ u u ( r 1 , r 2 ; t 1 , t 2 ) 2 + Γ u u ( r 1 , r 2 ; t 1 ; t 2 ) 2 + I ( r 1 , t 1 ) I ( r 2 , t 2 ) - 2 u ( r 1 , t 1 ) 2 u ( r 2 , t 2 ) 2 ,
Γ u u ( r 1 , r 2 ; t 1 , t 2 ) = u ( r 1 , t 1 ) u * ( r 2 , t 2 )
Γ u u ( r 1 , r 2 ; t 1 , t 2 ) = u ( r 1 , t 1 ) u ( r 2 , t 2 ) .
u 0 ( r , t ) = L r ρ u i ( ρ , t ) ,
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = u 0 ( r 1 , t 1 ) u 0 * ( r 2 , t 2 )
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = L 1 L 2 * Γ i ( ρ 1 , ρ 2 ; t 1 , t 2 ) ,
L 1 = L r 1 ρ 1             L 2 = L r 2 ρ 2
Γ i ( ρ 1 , ρ 2 ; t 1 , t 2 ) = u i ( ρ 1 , t 1 ) u i * ( ρ 2 , t 2 ) .
u e = G ρ [ s ] Q ρ [ 1 - ( ρ ^ · s ) 2 R ] ,
u i ( ρ , t ) = a ( ρ ) G [ s ] Q ρ [ 1 - ( ρ ^ · s ) 2 R ] γ ( ρ - v t ) × exp [ - j ϕ ( ρ - v t ) ] .
u 0 ( ρ , t ) = L exp [ - j ϕ ( ρ - v t ) ,
L = T a ( ρ ) G ρ [ s ] Q ρ [ 1 - ( ρ ^ · s ) 2 R ] γ ( ρ - v t ) ,
Γ ϕ ( ρ 1 , ρ 2 ; t 1 , t 2 ) exp [ j ϕ ( ρ 2 - v t 2 ) - j ϕ ( ρ 1 - v t 1 ) ] .
Γ ϕ ( ρ 1 , ρ 2 ; t 1 , t 2 ) = S ρ 1 ( v t 1 ) S ρ 2 ( v t 2 ) Γ ϕ ( ρ 1 , ρ 2 ) ,
Γ ϕ ( ρ 1 , ρ 2 ) = exp [ j ϕ ( ρ 2 ) - j ϕ ( ρ 1 ) ] ,
Γ 0 [ r 1 , r 2 ; t 1 , t 2 ] = L 1 L 2 * S ρ 1 [ v t 1 ] S ρ 2 [ v t 2 ] Γ ϕ ( ρ 1 , ρ 2 ) .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = L 1 L 2 S ρ 1 [ v t 1 ] S ρ 2 [ v t 2 ] Γ ϕ ( ρ 1 , ρ 2 ) ,
Γ ϕ ( ρ 1 , ρ 2 ) = exp [ - j ϕ ( ρ 1 ) - j ϕ ( ρ 2 ) ] ,
I ( r 1 , t 1 ) = Γ 0 ( r 1 , r 2 = r 1 ; t 1 , t 2 = t 1 ) ,
I ( r 2 , t 2 ) = Γ 0 ( r 1 = r 2 , r 2 ; t 1 = t 2 ; t 2 ) .
T = R r ρ [ d ]
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = R r 1 ρ 1 [ d ] G ρ 1 [ s ] Q ρ 1 [ 1 - ( ρ ^ 1 · s ) 2 R ] · a ( ρ 1 ) γ ( ρ 1 - t 1 v ) · R r 2 ρ 2 [ - d ] G ρ 2 [ - s ] Q ρ 2 [ - 1 - ( ρ ^ 2 · s ) 2 R ] · a * ( ρ 2 ) γ * ( ρ 2 - t 2 v ) S ρ 1 ( t 1 v ) · S ρ 2 ( t 2 v ) Γ ϕ ( ρ 1 , ρ 2 ) .
R r ρ [ d ] = Q r [ 1 / d ] r V [ 1 / ( λ d ) ] r ρ Q ρ [ 1 / d ] ,
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q r 1 [ 1 d ] V r 1 [ 1 λ d ] r 1 ρ 1 Q ρ 1 [ α 1 ] G ρ 1 [ s ] a ( ρ 1 ) · γ ( ρ 1 - t 1 v ) Q r 2 [ - 1 d ] V r 2 [ - 1 λ d ] r 2 ρ 2 Q ρ 2 [ - α 2 ] · G ρ 2 [ - s ] a * ( ρ 2 ) γ * ( ρ 2 - t 2 v ) S ρ 1 [ t 1 v ] · S ρ 2 [ t 2 v ] Γ ϕ ( ρ 1 , ρ 2 ) ,
α = 1 / d + 1 - ( ρ ^ . s ) 2 R .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q r 1 [ 1 d ] V r 1 [ 1 λ d ] r 1 ρ 1 G ρ 1 [ s ] S ρ 1 [ t 1 v ] · Q t 1 v [ β ] G ρ 1 [ t 1 d V ] Q ρ 1 [ α 1 ] a ( ρ 1 + t 1 v ) γ ( ρ 1 ) · Q r 2 [ - 1 d ] V r 2 [ - 1 λ d ] r 2 ρ 2 G ρ 2 [ - s ] S ρ 2 [ t 2 v ] Q t 2 v [ - β ] · G ρ 2 [ - t 2 d V ] Q ρ 2 [ - α 2 ] a * ( ρ 2 + t 2 v ) γ * ( ρ 2 ) Γ ϕ ( ρ 1 , ρ 2 ) ,
β = 1 - ( v ^ . s ) 2 R + 1 d ,
V = v + d R [ v - s ( s · v ) ] .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q r 1 [ 1 d ] Q r 2 [ - 1 d ] Q t 1 v [ β ] Q t 2 v [ - β ] · S r 1 [ s d ] G r 1 [ - t 1 v d ] S r 1 [ t 1 V ] S r 2 [ + s d ] G r 2 [ t 2 v d ] · S r 2 [ t 2 V ] M ( r 1 , r 2 ; t 1 , t 2 ) ,
M ( r 1 , r 2 ; t 1 , t 2 ) = V r 1 [ 1 λ d ] r 1 ρ 1 Q ρ 1 [ α 1 ] a ( ρ 1 + t 1 v ) γ ( ρ 1 ) · V r 2 [ - 1 λ d ] r 2 ρ 2 Q ρ 2 [ - α 2 ] · a * ( ρ 2 + t 2 v ) γ * ( ρ 2 ) Γ ϕ ( ρ 1 , ρ 2 ) .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q r 1 [ 1 d ] Q r 2 [ - 1 d ] Q t 1 v [ β ] Q t 2 v [ - β ] · G s d [ t 1 v d ] G r 1 [ - t 1 v d ] S r 1 [ s d + t 1 V ] · G s d [ - t 2 v d ] G r 2 [ t 2 v d ] · S r 2 [ s d + t 2 V ] M ( r 1 , r 2 ; t 1 , t 2 ) .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q r 1 [ 1 d ] Q r 2 [ - 1 d ] Q t 1 v [ β ] Q t 2 v [ - β ] · G t 1 [ s · v - r 1 · v d ] G t 2 [ - s · v + r 2 · v d ] · S r 1 [ s d + t 1 V ] S r 2 [ s d + t 2 V ] M ( r 1 , r 2 ; t 1 , t 2 ) .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q r 1 [ 1 d ] Q r 2 [ 1 d ] Q t 1 v [ β ] Q t 2 v [ β ] · G t 1 [ s · v - r 1 · v d ] · G t 2 [ s · v - r 2 · v d ] S r 1 [ s d + t 1 V ] · S r 2 [ s d + t 1 V ] M ( r 1 , r 2 ; t 1 , t 2 ) ,
M ( r 1 , r 2 ; t 1 , t 2 ) = V r 1 [ 1 λ d ] r 1 ρ 1 Q ρ 1 [ α 1 ] a ( ρ 1 + t 1 v ) γ ( ρ 1 ) · V r 2 [ 1 λ d ] × r 2 ρ 2 Q ρ 2 [ α 2 ] a ( ρ 2 + t 2 v ) γ ( ρ 2 ) Γ ϕ ( ρ 1 , ρ 2 ) ,
I ( r 1 , t 1 ) = S r 1 [ s d + t 1 V ] M ( r 1 , r 2 = r 1 ; t 1 , t 2 = t 1 ) ,
I ( r 2 , t 2 ) = S r 2 [ s d + t 2 V ] M ( r 1 = r 2 , r 2 ; t 1 = t 2 , t 2 ) .
u ( r , t ) = Q r [ 1 d ] Q t v [ β ] G t [ s · v - r · v d ] × S r [ s d + t V ] m ( r , t ) ,
m ( r , t ) = V r [ 1 λ d ] r ρ Q ρ [ α ] a ( ρ + t v ) γ ( ρ ) exp [ - j ϕ ( ρ ) ] .
Γ I I ( r 1 , r 2 ; t 1 , t 2 ) = S r 1 [ s d + t 1 V ] S r 2 [ s d + t 2 V ] · { M ( r 1 , r 2 ; t 1 , t 2 ) 2 + M ( r 1 , r 2 ; t 1 , t 2 ) 2 + M ( r 1 , r 2 = r 1 ; t 1 , t 2 = t 1 ) · M ( r 1 = r 2 , r 2 ; t 1 = t 2 , t 2 ) - 2 M ( r 1 , t 1 ) 2 m ( r 2 , t 2 ) 2 } .
M ( r 1 , r 2 ; t 1 , t 2 ) = V r 1 [ 1 λ d ] V r 2 [ - 1 λ d ] r 1 ρ 1 r 2 ρ 2 · Q ρ 1 [ α 1 ] a ( ρ 1 + t 1 v ) γ ( ρ 1 ) · Q ρ 2 [ - α 2 ] a * ( ρ 2 + t 2 v ) γ * ( ρ 2 ) Γ ϕ ( ρ 1 , ρ 2 ) .
M ( r 1 , r 2 ; t 1 , t 2 ) = [ A ( r 1 , t 1 ) A * ( r 2 , t 2 ) } * * C ( r 1 , r 2 ) ,
A ( r 1 , t 1 ) = V r 1 [ 1 λ d ] r 1 ρ 1 Q ρ 1 [ α 1 ] a ( ρ 1 + t 1 v ) γ ( ρ 1 ) ,
A * ( r 2 , t 2 ) = V r 2 [ - 1 λ d ] r 2 ρ 2 Q ρ 2 [ - α 2 ] a * ( ρ 2 + t 2 v ) γ * ( ρ 2 ) ,
C ( r 1 , r 2 ) = V r 1 [ 1 λ d ] V r 2 [ - 1 λ d ] r 1 ρ 1 r 2 ρ 2 Γ ϕ ( ρ 1 , ρ 2 ) .
M ( r 1 , r 2 ; t 1 , t 2 ) = { A ( r 1 , t 1 ) A ( r 2 , t 2 ) } * * C ( r 1 , r 2 ) ,
m ( r , t ) = A ( r , t ) * Φ ( r ) ,
Φ ( r ) = V r [ 1 λ d ] r ρ exp [ - j ϕ ( ρ ) ] .
A Q * ( a ) * ( γ ) .
r 1 - s d - t 1 V = r 2 - s d - t 2 V .
V = v + d R ( v - v s 2 ) = v ( 1 + d R cos 2 θ ) ,
V = v ( 1 + d / R ) .
T s = ( t 2 - t 1 ) max = a ¯ / v ,
u ( r , t ) = G t [ s · v - r · v d ] Q r [ 1 d ] Q t v [ β ] S r [ s d + t V ] m ( r , t ) ,
u ( r , ω ) δ ω [ k s · v - k r · v d ] * ω t Q r [ 1 d ] × Q t v [ β ] S r [ s d + t V ] m ( r , t ) ,
Δ ω = k ( s - r / d ) · v .
Γ ϕ ( ρ 1 , ρ 2 ; t 1 , t 2 ) = Γ ϕ ( ρ 2 - ρ 1 - τ v ) = S ρ 2 [ τ v 2 ] S ρ 1 [ - τ v 2 ] S ρ 2 [ ρ 1 ] Γ ϕ ( ρ 2 ) ,
τ = t 2 - t 1 or t 1 = t - τ / 2 ;             t 2 = t + τ / 2.
Γ 0 ( r 1 , r 2 ; τ , t ) = Q r 1 [ 1 d ] Q r 2 [ - 1 d ] G τ [ s · v - v · ( r 1 + r 2 ) 2 d ] · S r 1 [ s d - τ 2 V ] S r 2 [ s d + τ 2 V ] M ( r 1 , r 2 ; τ , t ) ,
M ( r 1 , r 2 ; τ , t ) = V r 1 [ 1 λ d ] r 1 ρ 1 Q ρ 1 [ α 1 ] a ( ρ 1 - τ 2 v ) γ ( ρ 1 - t v ) · V r 2 [ - 1 λ d ] r 2 ρ 2 Q ρ 2 [ - α 2 ] a * ( ρ 2 + τ 2 v ) · γ * ( ρ 2 - t v ) S ρ 2 [ ρ 1 ] Γ ϕ ( ρ 2 ) .
C ( r 1 , r 2 ) = V r 1 [ 1 λ d ] V r 2 [ - 1 λ d ] r 1 ρ 1 r 2 ρ 2 S ρ 2 [ ρ 1 ] Γ ϕ ( ρ 2 ) = C ( r 2 ) δ r 1 [ r 2 ] ,
C ( r 2 ) = V r 2 [ - 1 λ d ] r 2 ρ 2 Γ ϕ ( ρ 2 ) .
M ( r 1 , r 2 ; τ , t ) = A * ( r 2 , τ , t ) r 2 * A ( r 1 - r 2 , τ , t ) C ( r 2 , τ , t ) ,
Γ ϕ ( ρ 1 , ρ 2 ) = e ( ρ 1 ) δ ( ρ 2 - ρ 1 ) = e ( ρ 1 ) S ρ 2 [ ρ 1 ] δ ρ 2 [ 0 ] .
C ( r 1 , r 2 ) = V r 1 [ 1 λ d ] r 1 ρ 1 e ( ρ 1 ) G r 2 [ ρ 1 d ] = E ( r 1 - r 2 ) ,
M ( r 1 , r 2 ; τ , t ) = A * ( ξ ; τ , t ) A ( r 1 - r 2 + ξ ; τ , t ) × E ( R 1 - r 2 + ξ ) d ξ ,
M ( r 1 , r 2 ; τ , t ) = S r 1 [ r 2 ] r 1 V [ 1 λ d ] r 1 ρ 1 a ( ρ 1 - τ 2 v ) × a * ( ρ 1 + τ 2 v ) γ ( ρ 1 - t v ) 2 e ( ρ 1 ) ,
M ( r 1 , r 2 ; τ , t ) = A ( r 1 ; τ , t ) A * ( r 2 ; t , τ ) ,
T = R Q R .
L = L a L b .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = L 1 a L 1 b L 2 a * L 2 b * Γ i ( ρ 1 , ρ 2 ; t 1 , t 2 ) = L 1 a L 2 a * Γ b ( η 1 , η 2 ; t 1 , t 2 ) ,
Γ b ( η 1 , η 2 ; t 1 , t 2 ) = L 1 b L 2 b * Γ i ( ρ 1 , ρ 2 ; t 1 , t 2 )
L a = R r η [ b ] P ( η ) Q η r [ - 1 / f ] ,
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = R r 1 η 1 [ b ] P ( η 1 ) Q η 1 [ - 1 f ] R r 2 η 2 [ - b ] P * ( η 2 ) Q η 2 [ 1 f ] · Q η 1 [ 1 d ] Q η 2 [ - 1 d ] Q t 1 v [ β ] Q t 2 v [ - β ] · G t 1 [ s · v - η 1 · v d ] · G t 2 [ - s · v + η 2 · v d ] S η 1 [ s d + t 1 V ] · S η 2 [ s d + t 2 V ] M ( η 1 , η 2 ; t 1 , t 2 ) .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q r 1 [ 1 b ] V r 1 [ 1 λ b ] r 1 η 1 Q η 1 [ μ ] P ( η 1 ) Q t 1 v [ β ] · G t 1 [ s · v - η 1 · v d ] · S η 1 [ s d + t 1 V ] Q r 2 [ - 1 b ] V r 2 [ 1 λ b ] r 2 η 2 · Q η 2 [ - μ ] P * ( η 2 ) Q t 1 v [ - β ] · Q t 2 [ - s · v + η 2 · v d ] · S η 2 [ s d + t 1 V ] M ( η 1 , η 2 ; t 1 , t 2 ) ,
μ = 1 / d + 1 / b - 1 / f .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q t 1 v [ β ] Q t 2 v [ - β ] G t 1 [ s · v ] G t 2 [ - s · v ] · Q r 1 [ 1 b ] Q r 2 [ - 1 b ] S r 1 [ - t 1 v d b ] S r 2 [ - t 2 v d b ] · V r 1 [ 1 λ b ] r 1 η 1 Q η 1 [ μ ] P ( η 1 ) Q η 1 [ s d + t 1 V ] · V r 2 [ - 1 λ b ] r 2 η 2 Q η 2 [ - μ ] · P * ( η 2 ) S η 2 [ s d + t 2 V ] M ( η 1 , η 2 ; t 1 , t 2 ) .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q t 1 v [ β ] Q t 2 v [ - β ] G t 1 [ s · v ] · G t 2 [ - s · v ] Q s d + t 1 v [ μ ] · Q s d + t 2 v [ - μ ] Q r 1 [ 1 b ] Q r 2 [ - 1 b ] · S r 1 [ - t 1 v d b ] S r 2 [ - t 2 v d b ] · G r 1 [ - 1 b ( s d + t 1 V ) ] G r 2 [ 1 b ( s d + t 2 V ) ] · S r 1 [ b μ ( s d + t 1 V ) ] · S r 2 [ b μ ( s d + t 2 V ] N ( r 1 , r 2 ; t 1 , t 2 ) ,
N ( r 1 , r 2 ; t 1 , t 2 ) = V r 1 [ 1 λ b ] r 1 η 1 Q η 1 [ μ ] P ( η 1 + s d + t 1 V ) · V r 2 [ - 1 λ b ] r 2 η 2 · Q η 2 [ - μ ] P * [ η 2 + s d + t 2 V ) M ( η 1 , η 2 ; t 1 , t 2 ) .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = Q t 1 v [ β ] Q t 2 v [ - β ] G t 1 [ s · v ] G t 2 [ - s · v ] Q s d + t 1 V [ μ ] · Q s d + t 2 V [ - μ ] Q r 1 [ 1 b ] Q r 2 [ - 1 b ] G r 1 [ - 1 b ( s d + t 1 V ) ] · G t 1 v d b [ 1 b ( s d + t 1 V ) ] G r 2 [ 1 b ( s d + t 1 V ) ] · G t 2 v d b [ - 1 b ( s d + t 1 V ) ] S r 1 [ b d μ s + t 1 b ( μ V - v / d ) ] · S r 1 [ b d μ s + t 1 b ( μ V - v / d ) ] N ( r 1 , r 2 ; t 1 , t 2 ) .
G b d t v [ 1 b ( s d + t V ) ] = G t [ v · s ] Q t v [ 2 d v 2 V · v ] = G t [ v · s ] Q t v [ 2 β ] ,
1 d v 2 V · v = β .
Q s d + t v [ μ ] = Q s d [ μ ] G t [ d μ s · V ] Q t v [ μ ( V v ) 2 ] .
Γ 0 ( r 1 , r 2 ; t 1 , t 2 ) = G t 1 [ ( d μ s - r 1 / b ) · V ] G t 2 [ - ( d μ s - r 2 / b ) · V ] · Q t 1 v [ - β + μ ( V / v ) 2 ] Q t 2 v [ β - μ ( V / v ) 2 ] · G r 1 [ - d b s ] G r 2 [ d b s ] · Q r 1 [ 1 b ] Q r 2 [ - 1 b ] S r 1 [ b d μ s + t 1 b ( μ V - v / d ) ] · S r 2 [ b d μ s + t 2 b ( μ V - v / d ) ] N ( r 1 , r 2 ; t 1 , t 2 ) .
Γ I I ( r 1 , r 2 ; t 1 , t 2 ) = S r 1 [ b d μ s + t 1 b ( μ V - v / d ) ] · S r 2 [ b d μ s + t 1 b ( μ V - v / d ) ] · { N ( r 1 , r 2 ; t 1 , t 2 ) 2 + N ( r 1 , r 2 ; t 1 , t 2 ) 2 + N ( r 1 , r 2 = r 1 ; t 1 , t 2 = t 1 ) · N ( r 1 = r 2 , r 2 ; t 1 = t 2 , t 2 ) - 2 n ( r 1 , t 1 ) 2 n ( r 2 , t 2 ) 2 } ,
N ( r 1 , r 2 ; t 1 , t 2 ) = V r 1 [ 1 λ b ] r 1 η 1 Q η 1 [ μ ] P ( η 1 + s d + t 1 V ) · V r 2 [ 1 λ b ] r 2 η 2 Q η 2 [ μ ] P ( η 2 + s d + t 1 V ) · M ( η 1 , η 2 ; t 1 , t 2 ) ,
n ( r , t ) = V r [ 1 λ b ] F r η Q ( μ ) P ( η + s d + t V ) m ( r , t ) ,
V = - b d ( v - μ d V ) .
b = f ( R + d cos 2 θ ) R + ( d - f ) cos 2 θ
b = f ( r + d ) R + d - f .
Δ ω = k ( d μ s - r / b ) · V .
N ( r 1 , r 2 ; t 1 , t 2 ) = K ( r 1 , t 1 ) K * ( r 2 , t 2 ) * * H ( r 1 , r 2 ; t 1 , t 2 ) ,
K ( r , t ) = V r [ 1 λ b ] r η P ( η + s d + t V ) Q η [ μ ] ,
H ( r 1 , r 2 ; t 1 , t 2 ) = V r 1 [ 1 λ b ] V r 2 [ - 1 λ b ] F r 1 η 1 F r 2 η 2 × { A ( η 1 , t 1 ) A * ( η 2 , t 2 ) * * C ( η 1 , η 2 ) } .
H ( r 1 , r 2 ; t 1 , t 2 ) = { V r 1 [ 1 λ b ] V r 1 [ - 1 λ b ] F r 1 η 1 F r 2 η 2 A ( η 1 , t 1 ) A * ( η 2 , t 2 ) } · { V r 1 + [ 1 λ b ] V r 2 [ - 1 λ b ] F r 1 η 1 F r 2 η 2 C ( η 1 , η 2 ) } .
H ( r 1 , r 2 ; t 1 , t 2 ) = { Q r 1 [ ( d b ) 2 α 1 ] Q r 2 [ - ( d b ) 2 α 2 ] a ( - d b ( r 1 + t 1 v ) ) · γ ( - d b r 1 ) a * [ - d b ( r 2 + t 2 v ) ] γ * ( - d b r 2 ) } · { Γ ϕ ( d b r 1 , - d b r 2 ) } ,
T s = P ¯ / V .
Q ρ [ a ] exp j k 2 a ρ 2 ;
G ρ [ s ] exp j k s · ρ ;
S [ m ] u ( ρ ) = u ( ρ - m ) ;
V [ b ] u ( ρ ) = u ( b ρ ) ;
F u ( ρ ) - d ρ exp ( 2 j π ν · ρ ) u ( ρ ) ;
δ ρ [ m ] = δ ( ρ - m ) .
Q ρ [ a 1 ] Q ρ [ a 2 ] = Q ρ [ a 1 + a 2 ] ;
G ρ [ s 1 ] G ρ [ s 2 ] = G ρ [ s 1 + s 2 ] ;
S ρ [ m 1 ] S ρ [ m 2 ] = S ρ [ m 1 + m 2 ] ;
G ρ [ a ρ ] = G ρ 2 [ a ] = Q [ 2 a ] ;
Q ρ [ a ] S ρ [ m ] = S ρ [ m ] Q ρ [ a ] Q m [ a ] G ρ [ a m ] ;
S ρ [ m ] G ρ [ s ] = G m [ - s ] G ρ [ s ] S ρ [ m ] ;
Q ρ [ a ] G ρ [ s ] = G ρ [ s ] Q ρ [ a ] ;
F r ρ G ρ [ s ] = S r [ s λ ] F r ρ ;
F r ρ S ρ [ m ] = G r [ - λ m ] F r ρ ;
{ F r ρ G ρ [ s ] } = δ r [ s λ ] ;
δ ρ [ m ] * = S ρ [ m ] ;
V ρ [ b ] G ρ [ s ] = G ρ [ b s ] V ρ [ b ] ;
V ρ [ b ] S ρ [ m ] = S ρ [ m b ] V ρ [ b ] ;
V ρ [ b 1 ] V ρ [ b 2 ] = V ρ [ b 1 b 2 ] .
u ( ρ 2 ) = A r 12 exp ( j k r 12 ) ,
r 12 = [ ( ρ 1 - ρ 2 ) 2 + z 12 ] 1 / 2 = ( R 1 2 + ρ 2 2 - 2 ρ 1 · ρ 2 ) 1 / 2 ,
R 1 = ( z 12 2 + ρ 1 2 ) 1 / 2 .
R 1 2 ρ 2 2 - 2 ρ 1 · ρ 2 ,
r 12 R 1 + ρ 2 2 2 R 1 - ρ · ρ 2 R 1 - ( ρ 1 · ρ 2 ) 2 2 R 1 3 ,
u ( ρ 2 ) = A R 1 exp ( j k R 1 ) G ρ 2 [ s ] Q ρ 2 [ 1 - ( ρ ^ 2 · s ) 2 R 1 ] ,
s = - ρ 1 R 1 = - ρ ^ 1 sin θ
Q ρ [ 1 - ( ρ ^ 2 · s ) 2 R 1 ] S ρ [ m ] = S ρ [ m ] exp [ j k 2 R ( ρ + m ) 2 ] · exp [ - j k 2 R { ρ + m ) · s } 2 ] = S ρ [ m ] Q ρ [ 1 - ( ρ ^ · s ) 2 R ] × Q m [ 1 - ( m ^ · s ) 2 R ] G ρ [ m - s ( s · m ) R ]

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