Abstract

The statistical properties of dynamic speckles produced by a moving diffuse object were reviewed by providing the space–time correlation function and the power spectrum of speckle-intensity fluctuation for five combined cases of both the optical configuration and the illumination light. In the optical configuration, three kinds of geometry (free-space, single-lens, and double-lens) were taken, and three kinds of illumination light (a Gaussian beam, a plane-wave beam, and a Gaussian Schell-model beam) were used. Consequently, it was shown that the cross-correlation function and the power spectrum are both Gaussian under some assumptions. From the dynamic properties, two types of speckle motion, boiling and translation, were also evaluated for various conditions of object motion, optical configuration, and illumination light.

© 1986 Optical Society of America

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1986

1985

1984

T. W. Darling, A. G. Klein, G. I. Opat, S. N. Tovey, Opt. Acta 31, 813–821 (1984).
[CrossRef]

T. Yoshimura, H. Doi, N. Wakabayashi, J. Opt. Soc. Am. A 1, 1078–1084 (1984).
[CrossRef]

C. A. Schmidt-Harms, Appl. Opt. 23, 2353–2358 (1984).
[CrossRef]

T. Yoshimura, H. Saito, N. Wakabayashi, Opt. Commun. 52, 265–268 (1984).
[CrossRef]

1983

1982

A. T. Friberg, R. J. Sudol, Opt. Commun. 41, 383–387 (1982).
[CrossRef]

K. A. O’Donnell, B. J. Brames, J. C. Dainty, Opt. Commun. 41, 79–82 (1982).
[CrossRef]

A. Hayashi, Y. Kitagawa, Opt. Commun. 43, 161–163 (1982).
[CrossRef]

K. A. O’Donnell, J. Opt. Soc. Am. 72, 191–197 (1982).
[CrossRef]

A. Starikov, E. Wolf, J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

J. H. Churnside, J. Opt. Soc. Am. 72, 1464–1469 (1982).
[CrossRef]

1981

N. Takai, T. Iwai, T. Asakura, Opt. Eng. 20, 320–324 (1981).
[CrossRef]

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 857–870 (1981).
[CrossRef]

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 1425–1437 (1981).
[CrossRef]

J. Ohtsubo, J. Opt. 12, 129–142 (1981).
[CrossRef]

T. Asakura, N. Takai, Appl. Phys. 25B, 179–194 (1981).
[CrossRef]

C. T. Stansberg, Opt. Acta 28, 917–932 (1981).
[CrossRef]

M. Giglio, S. Musazzi, U. Perini, Appl. Opt. 20, 721–722 (1981).
[CrossRef] [PubMed]

J. H. Churnside, H. T. Yura, Appl. Opt. 20, 3539–3541 (1981).
[CrossRef] [PubMed]

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

1980

A. F. Fercher, Opt. Commun. 33, 129–135 (1980).
[CrossRef]

J. Ohtsubo, Opt. Commun. 34, 147–152 (1980).
[CrossRef]

J. C. Leader, Opt. Eng. 19, 593–601 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

1979

R. Vehrenkamp, K. Schatzel, G. Pfister, E. O. Schulz-DuBois, J. Phys. E 12, 119–125 (1979).
[CrossRef]

J. W. Goodman, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).

J. Ohtsubo, T. Asakura, Optik 52, 413–420 (1979).

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, Opt. Commun. 30, 287–292 (1979).
[CrossRef]

1978

1977

J. C. Leader, J. Opt. Soc. Am. 67, 1091–1098 (1977).
[CrossRef]

E. Jakeman, W. T. Welford, Opt. Commun. 21, 72–79 (1977).
[CrossRef]

I. Yamaguchi, S. Komatsu, Opt. Acta 24, 705–724 (1977).
[CrossRef]

1976

J. Ohtsubo, T. Asakura, Opt. Quantum Electron. 8, 523–529 (1976).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715–1724 (1976).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314–316 (1976).
[CrossRef]

P. N. Pusey, J. Phys. D 9, 1399–1409 (1976).
[CrossRef]

E. Jakeman, J. G. Mcwhirter, J. Phys. A 9, 785–797 (1976).
[CrossRef]

N. George, J. Opt. Soc. Am. 66, 1182–1194 (1976).
[CrossRef]

J. C. Erdmann, R. I. Gellert, J. Opt. Soc. Am. 66, 1194–1204 (1976).
[CrossRef]

H. M. Pedersen, J. Opt. Soc. Am. 66, 1204–1210 (1976).
[CrossRef]

1975

B. E. A. Saleh, Appl. Opt. 14, 2344–2346 (1975).
[CrossRef] [PubMed]

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301–306 (1975).
[CrossRef]

H. Ogiwara, H. Ukita, Jpn. J. Appl. Phys. 14, 307–310 (1975).
[CrossRef]

E. Jakeman, J. Phys. A 8, L23–L28 (1975).
[CrossRef]

1974

T. S. Mckechnie, Optik 39, 258–267 (1974).

N. Takai, Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[CrossRef]

A. A. Scribot, Opt. Commun. 11, 238–241 (1974).
[CrossRef]

1973

1971

M. Rousseau, J. Opt. Soc. Am. 61, 1307–1316 (1971).
[CrossRef]

B. Crosignani, B. Daino, P. D. Porto, J. Appl. Phys. 42, 399–403 (1971).
[CrossRef]

1970

J. A. Leendertz, J. Phys, E. Sci. Instrum. 3, 214–218 (1970).
[CrossRef]

1969

T. M. Sporton, Brit. J. Appl. Phys. (J. Phys. D) Ser. 2, 2, 1027–1034 (1969).
[CrossRef]

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, Opt. Spectrosc. (USSR) 27, 258–294 (1969).

1966

G. Stavis, Instrum. Control Syst. 39, 99–102 (1966).

H. Kogelnik, T. Li, Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1965

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

J. W. Goodman, Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

1963

R. V. Langmuir, Appl. Phys. Lett. 2, 29–30 (1963).
[CrossRef]

B. M. Oliver, Proc. IEEE 51, 220–221 (1963).
[CrossRef]

1962

J. D. Rigden, E. I. Gordon, Proc. IRE 50, 2367–2368 (1962).

Anisimov, V. V.

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, Opt. Spectrosc. (USSR) 27, 258–294 (1969).

Asakura, T.

N. Takai, T. Asakura, J. Opt. Soc. Am. A 2, 1282–1290 (1985).
[CrossRef]

T. Ushizaka, T. Asakura, Appl. Opt. 22, 1870–1874 (1983).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, Appl. Opt. 22, 170–177 (1983).
[CrossRef] [PubMed]

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

T. Asakura, N. Takai, Appl. Phys. 25B, 179–194 (1981).
[CrossRef]

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 1425–1437 (1981).
[CrossRef]

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 857–870 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, Opt. Eng. 20, 320–324 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

J. Ohtsubo, T. Asakura, Optik 52, 413–420 (1979).

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, Opt. Commun. 30, 287–292 (1979).
[CrossRef]

J. Ohtsubo, T. Asakura, Opt. Quantum Electron. 8, 523–529 (1976).
[CrossRef]

Barakat, R.

R. Barakat, Opt. Acta 20, 729–740 (1973).
[CrossRef]

Battin, R. H.

J. H. Laning, R. H. Battin, Random Processes in Automatic Control (McGraw–Hill, New York, 1956).

Beckmann, P.

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

Born, M.

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

Brames, B. J.

K. A. O’Donnell, B. J. Brames, J. C. Dainty, Opt. Commun. 41, 79–82 (1982).
[CrossRef]

Churnside, J. H.

Collet, E.

Crosignani, B.

B. Crosignani, B. Daino, P. D. Porto, J. Appl. Phys. 42, 399–403 (1971).
[CrossRef]

Daino, B.

B. Crosignani, B. Daino, P. D. Porto, J. Appl. Phys. 42, 399–403 (1971).
[CrossRef]

Dainty, J. C.

T. Gonsiorowski, J. C. Dainty, J. Opt. Soc. Am. 73, 234–237 (1983).
[CrossRef]

K. A. O’Donnell, B. J. Brames, J. C. Dainty, Opt. Commun. 41, 79–82 (1982).
[CrossRef]

J. C. Dainty, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 14, pp. 1–46.
[CrossRef]

Darling, T. W.

T. W. Darling, A. G. Klein, G. I. Opat, S. N. Tovey, Opt. Acta 31, 813–821 (1984).
[CrossRef]

Doi, H.

T. Yoshimura, H. Doi, N. Wakabayashi, J. Opt. Soc. Am. A 1, 1078–1084 (1984).
[CrossRef]

T. Yoshimura, H. Doi, N. Wakabayashi, Opt. Commun. 48, 17–20 (1983).
[CrossRef]

Elek, A. C.

Erdmann, J. C.

Fercher, A. F.

A. F. Fercher, Opt. Commun. 33, 129–135 (1980).
[CrossRef]

Friberg, A. T.

A. T. Friberg, R. J. Sudol, Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Gellert, R. I.

George, N.

Giglio, M.

Gonsiorowski, T.

Goodman, J. W.

J. W. Goodman, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).

J. W. Goodman, Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968).

Gordon, E. I.

J. D. Rigden, E. I. Gordon, Proc. IRE 50, 2367–2368 (1962).

Hayashi, A.

A. Hayashi, Y. Kitagawa, Appl. Opt. 22, 3520–3525 (1983).
[CrossRef] [PubMed]

A. Hayashi, Y. Kitagawa, Opt. Commun. 43, 161–163 (1982).
[CrossRef]

Ishimura, Y.

T. Yoshimura, H. Saito, Y. Ishimura, N. Wakabayashi, Kogaku 14, 198–205 (1985).

Iwai, T.

N. Takai, T. Iwai, T. Asakura, Appl. Opt. 22, 170–177 (1983).
[CrossRef] [PubMed]

N. Takai, T. Iwai, T. Asakura, Opt. Eng. 20, 320–324 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 857–870 (1981).
[CrossRef]

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 1425–1437 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, Opt. Commun. 30, 287–292 (1979).
[CrossRef]

Jakeman, E.

E. Jakeman, W. T. Welford, Opt. Commun. 21, 72–79 (1977).
[CrossRef]

E. Jakeman, J. G. Mcwhirter, J. Phys. A 9, 785–797 (1976).
[CrossRef]

E. Jakeman, J. Phys. A 8, L23–L28 (1975).
[CrossRef]

E. Jakeman, P. N. Pusey, J. Phys. A 6, L88–L92 (1973).
[CrossRef]

E. Jakeman, in Photon Correlation and Light-Beating Spectroscopy, H. Z. Cummins, H. R. Pike, eds. (Plenum, New York, 1974), pp. 75–149.

Kitagawa, Y.

A. Hayashi, Y. Kitagawa, Appl. Opt. 22, 3520–3525 (1983).
[CrossRef] [PubMed]

A. Hayashi, Y. Kitagawa, Opt. Commun. 43, 161–163 (1982).
[CrossRef]

Klein, A. G.

T. W. Darling, A. G. Klein, G. I. Opat, S. N. Tovey, Opt. Acta 31, 813–821 (1984).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Komatsu, S.

I. Yamaguchi, S. Komatsu, Opt. Acta 24, 705–724 (1977).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715–1724 (1976).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314–316 (1976).
[CrossRef]

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301–306 (1975).
[CrossRef]

Kozel, S. M.

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, Opt. Spectrosc. (USSR) 27, 258–294 (1969).

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R. V. Langmuir, Appl. Phys. Lett. 2, 29–30 (1963).
[CrossRef]

Laning, J. H.

J. H. Laning, R. H. Battin, Random Processes in Automatic Control (McGraw–Hill, New York, 1956).

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Leendertz, J. A.

J. A. Leendertz, J. Phys, E. Sci. Instrum. 3, 214–218 (1970).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lokshin, G. R.

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, Opt. Spectrosc. (USSR) 27, 258–294 (1969).

Mandel, L.

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Marron, J.

Mckechnie, T. S.

T. S. Mckechnie, Optik 39, 258–267 (1974).

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E. Jakeman, J. G. Mcwhirter, J. Phys. A 9, 785–797 (1976).
[CrossRef]

Mehta, C. L.

C. L. Mehta, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 373–440.
[CrossRef]

Mohon, N.

Morris, G. M.

Musazzi, S.

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K. A. O’Donnell, B. J. Brames, J. C. Dainty, Opt. Commun. 41, 79–82 (1982).
[CrossRef]

K. A. O’Donnell, J. Opt. Soc. Am. 72, 191–197 (1982).
[CrossRef]

Ogiwara, H.

H. Ogiwara, H. Ukita, Jpn. J. Appl. Phys. 14, 307–310 (1975).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, J. Opt. 12, 129–142 (1981).
[CrossRef]

J. Ohtsubo, Opt. Commun. 34, 147–152 (1980).
[CrossRef]

J. Ohtsubo, T. Asakura, Optik 52, 413–420 (1979).

J. Ohtsubo, T. Asakura, Opt. Quantum Electron. 8, 523–529 (1976).
[CrossRef]

Oliver, B. M.

B. M. Oliver, Proc. IEEE 51, 220–221 (1963).
[CrossRef]

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T. W. Darling, A. G. Klein, G. I. Opat, S. N. Tovey, Opt. Acta 31, 813–821 (1984).
[CrossRef]

Parry, G.

G. Parry, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 77–121.
[CrossRef]

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Perini, U.

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Phys, J.

J. A. Leendertz, J. Phys, E. Sci. Instrum. 3, 214–218 (1970).
[CrossRef]

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B. Crosignani, B. Daino, P. D. Porto, J. Appl. Phys. 42, 399–403 (1971).
[CrossRef]

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P. N. Pusey, J. Phys. D 9, 1399–1409 (1976).
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J. D. Rigden, E. I. Gordon, Proc. IRE 50, 2367–2368 (1962).

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T. Yoshimura, H. Saito, Y. Ishimura, N. Wakabayashi, Kogaku 14, 198–205 (1985).

T. Yoshimura, H. Saito, N. Wakabayashi, Opt. Commun. 52, 265–268 (1984).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715–1724 (1976).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314–316 (1976).
[CrossRef]

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301–306 (1975).
[CrossRef]

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B. E. A. Saleh, Appl. Opt. 14, 2344–2346 (1975).
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R. Vehrenkamp, K. Schatzel, G. Pfister, E. O. Schulz-DuBois, J. Phys. E 12, 119–125 (1979).
[CrossRef]

Schmidt-Harms, C. A.

Schulz-DuBois, E. O.

R. Vehrenkamp, K. Schatzel, G. Pfister, E. O. Schulz-DuBois, J. Phys. E 12, 119–125 (1979).
[CrossRef]

Scribot, A. A.

A. A. Scribot, Opt. Commun. 11, 238–241 (1974).
[CrossRef]

Spizzichino, A.

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

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T. M. Sporton, Brit. J. Appl. Phys. (J. Phys. D) Ser. 2, 2, 1027–1034 (1969).
[CrossRef]

Stansberg, C. T.

C. T. Stansberg, Opt. Acta 28, 917–932 (1981).
[CrossRef]

Starikov, A.

Stavis, G.

G. Stavis, Instrum. Control Syst. 39, 99–102 (1966).

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A. T. Friberg, R. J. Sudol, Opt. Commun. 41, 383–387 (1982).
[CrossRef]

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N. Takai, T. Asakura, J. Opt. Soc. Am. A 2, 1282–1290 (1985).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, Appl. Opt. 22, 170–177 (1983).
[CrossRef] [PubMed]

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

T. Asakura, N. Takai, Appl. Phys. 25B, 179–194 (1981).
[CrossRef]

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 857–870 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, Opt. Eng. 20, 320–324 (1981).
[CrossRef]

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 1425–1437 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, Opt. Commun. 30, 287–292 (1979).
[CrossRef]

N. Takai, Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[CrossRef]

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T. W. Darling, A. G. Klein, G. I. Opat, S. N. Tovey, Opt. Acta 31, 813–821 (1984).
[CrossRef]

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H. Ogiwara, H. Ukita, Jpn. J. Appl. Phys. 14, 307–310 (1975).
[CrossRef]

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T. Ushizaka, T. Asakura, Appl. Opt. 22, 1870–1874 (1983).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, Opt. Commun. 30, 287–292 (1979).
[CrossRef]

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R. Vehrenkamp, K. Schatzel, G. Pfister, E. O. Schulz-DuBois, J. Phys. E 12, 119–125 (1979).
[CrossRef]

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T. Yoshimura, K. Nakagawa, N. Wakabayashi, J. Opt. Soc. Am. A 3, 1018–1022 (1986).
[CrossRef]

T. Yoshimura, H. Saito, Y. Ishimura, N. Wakabayashi, Kogaku 14, 198–205 (1985).

T. Yoshimura, H. Doi, N. Wakabayashi, J. Opt. Soc. Am. A 1, 1078–1084 (1984).
[CrossRef]

T. Yoshimura, H. Saito, N. Wakabayashi, Opt. Commun. 52, 265–268 (1984).
[CrossRef]

T. Yoshimura, H. Doi, N. Wakabayashi, Opt. Commun. 48, 17–20 (1983).
[CrossRef]

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E. Jakeman, W. T. Welford, Opt. Commun. 21, 72–79 (1977).
[CrossRef]

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A. Starikov, E. Wolf, J. Opt. Soc. Am. 72, 923–928 (1982).
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[CrossRef]

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

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

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I. Yamaguchi, S. Komatsu, Opt. Acta 24, 705–724 (1977).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314–316 (1976).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715–1724 (1976).
[CrossRef]

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301–306 (1975).
[CrossRef]

I. Yamaguchi, in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1985), pp. 271–340.

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T. Yoshimura, K. Nakagawa, N. Wakabayashi, J. Opt. Soc. Am. A 3, 1018–1022 (1986).
[CrossRef]

T. Yoshimura, H. Saito, Y. Ishimura, N. Wakabayashi, Kogaku 14, 198–205 (1985).

T. Yoshimura, H. Doi, N. Wakabayashi, J. Opt. Soc. Am. A 1, 1078–1084 (1984).
[CrossRef]

T. Yoshimura, H. Saito, N. Wakabayashi, Opt. Commun. 52, 265–268 (1984).
[CrossRef]

T. Yoshimura, H. Doi, N. Wakabayashi, Opt. Commun. 48, 17–20 (1983).
[CrossRef]

Yura, H. T.

Appl. Opt.

Appl. Phys.

T. Asakura, N. Takai, Appl. Phys. 25B, 179–194 (1981).
[CrossRef]

Appl. Phys. B

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

Appl. Phys. Lett.

R. V. Langmuir, Appl. Phys. Lett. 2, 29–30 (1963).
[CrossRef]

Brit. J. Appl. Phys. (J. Phys. D) Ser. 2

T. M. Sporton, Brit. J. Appl. Phys. (J. Phys. D) Ser. 2, 2, 1027–1034 (1969).
[CrossRef]

E. Sci. Instrum.

J. A. Leendertz, J. Phys, E. Sci. Instrum. 3, 214–218 (1970).
[CrossRef]

Instrum. Control Syst.

G. Stavis, Instrum. Control Syst. 39, 99–102 (1966).

J. Appl. Phys.

B. Crosignani, B. Daino, P. D. Porto, J. Appl. Phys. 42, 399–403 (1971).
[CrossRef]

J. Opt.

J. Ohtsubo, J. Opt. 12, 129–142 (1981).
[CrossRef]

J. Opt. (Paris)

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

E. Jakeman, J. Phys. A 8, L23–L28 (1975).
[CrossRef]

E. Jakeman, J. G. Mcwhirter, J. Phys. A 9, 785–797 (1976).
[CrossRef]

E. Jakeman, P. N. Pusey, J. Phys. A 6, L88–L92 (1973).
[CrossRef]

J. Phys. D

P. N. Pusey, J. Phys. D 9, 1399–1409 (1976).
[CrossRef]

J. Phys. E

R. Vehrenkamp, K. Schatzel, G. Pfister, E. O. Schulz-DuBois, J. Phys. E 12, 119–125 (1979).
[CrossRef]

Jpn. J. Appl. Phys.

N. Takai, Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[CrossRef]

H. Ogiwara, H. Ukita, Jpn. J. Appl. Phys. 14, 307–310 (1975).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Jpn. J. Appl. Phys. 15, 1715–1724 (1976).
[CrossRef]

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. 14, 301–306 (1975).
[CrossRef]

Kogaku

T. Yoshimura, H. Saito, Y. Ishimura, N. Wakabayashi, Kogaku 14, 198–205 (1985).

Opt. Act.

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 857–870 (1981).
[CrossRef]

T. Iwai, N. Takai, T. Asakura, Opt. Act. 28, 1425–1437 (1981).
[CrossRef]

Opt. Acta

I. Yamaguchi, S. Komatsu, Opt. Acta 24, 705–724 (1977).
[CrossRef]

C. T. Stansberg, Opt. Acta 28, 917–932 (1981).
[CrossRef]

R. Barakat, Opt. Acta 20, 729–740 (1973).
[CrossRef]

T. W. Darling, A. G. Klein, G. I. Opat, S. N. Tovey, Opt. Acta 31, 813–821 (1984).
[CrossRef]

Opt. Commun.

A. T. Friberg, R. J. Sudol, Opt. Commun. 41, 383–387 (1982).
[CrossRef]

E. Jakeman, W. T. Welford, Opt. Commun. 21, 72–79 (1977).
[CrossRef]

T. Yoshimura, H. Doi, N. Wakabayashi, Opt. Commun. 48, 17–20 (1983).
[CrossRef]

A. A. Scribot, Opt. Commun. 11, 238–241 (1974).
[CrossRef]

S. Komatsu, I. Yamaguchi, H. Saito, Opt. Commun. 18, 314–316 (1976).
[CrossRef]

A. Hayashi, Y. Kitagawa, Opt. Commun. 43, 161–163 (1982).
[CrossRef]

T. Yoshimura, H. Saito, N. Wakabayashi, Opt. Commun. 52, 265–268 (1984).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, Opt. Commun. 30, 287–292 (1979).
[CrossRef]

A. F. Fercher, Opt. Commun. 33, 129–135 (1980).
[CrossRef]

J. Ohtsubo, Opt. Commun. 34, 147–152 (1980).
[CrossRef]

K. A. O’Donnell, B. J. Brames, J. C. Dainty, Opt. Commun. 41, 79–82 (1982).
[CrossRef]

Opt. Eng.

N. Takai, T. Iwai, T. Asakura, Opt. Eng. 20, 320–324 (1981).
[CrossRef]

J. C. Leader, Opt. Eng. 19, 593–601 (1980).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

J. Ohtsubo, T. Asakura, Opt. Quantum Electron. 8, 523–529 (1976).
[CrossRef]

Opt. Spectrosc. (USSR)

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, Opt. Spectrosc. (USSR) 27, 258–294 (1969).

Optik

T. S. Mckechnie, Optik 39, 258–267 (1974).

J. Ohtsubo, T. Asakura, Optik 52, 413–420 (1979).

Proc. IEEE

B. M. Oliver, Proc. IEEE 51, 220–221 (1963).
[CrossRef]

J. W. Goodman, Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

H. Kogelnik, T. Li, Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Proc. IRE

J. D. Rigden, E. I. Gordon, Proc. IRE 50, 2367–2368 (1962).

Proc. Soc. Photo-Opt. Instrum. Eng.

J. W. Goodman, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).

Rev. Mod. Phys.

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Other

C. L. Mehta, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 373–440.
[CrossRef]

E. Jakeman, in Photon Correlation and Light-Beating Spectroscopy, H. Z. Cummins, H. R. Pike, eds. (Plenum, New York, 1974), pp. 75–149.

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[CrossRef]

E. O. Schulz-DuBois, ed., Photon Correlation Techniques (Springer-Verlag, Berlin, 1983).
[CrossRef]

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

J. C. Dainty, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 14, pp. 1–46.
[CrossRef]

R. K. Erf, ed., Speckle Metrology (Academic, New York, 1978).

I. Yamaguchi, in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1985), pp. 271–340.

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

G. Parry, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 77–121.
[CrossRef]

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

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968).

J. H. Laning, R. H. Battin, Random Processes in Automatic Control (McGraw–Hill, New York, 1956).

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Figures (18)

Fig. 1
Fig. 1

Physical representation forming a speckle pattern.

Fig. 2
Fig. 2

Dependence of the beam-spot radius w and the wave-front curvature radius ρ of the Gaussian-beam on the propagating distance.

Fig. 3
Fig. 3

Free-space geometry. (a) Gaussian-beam illumination, (b) Gaussian Schell-model source illumination.

Fig. 4
Fig. 4

Single-lens geometry. (a) Plane-wave light illumination, (b) Gaussian-beam illumination.

Fig. 5
Fig. 5

Two-lens geometry.

Fig. 6
Fig. 6

Schematic diagram for analyzing the Doppler shift frequency. In this two-lens system, the source placed at the origin of the l 1 plane is imaged on the origin of the l 4 plane.

Fig. 7
Fig. 7

Schematic diagram of the space–time correlation function.

Fig. 8
Fig. 8

Second-order moment of the two-lens imaging system.

Fig. 9
Fig. 9

Diagram indicating the type of motion of dynamic speckles as a function of the object position z under Gaussian-beam illumination. The symbols B and T indicate the boiling and translation regions, respectively, of the speckle motion.

Fig. 10
Fig. 10

The type of motion of dynamic speckles and the translation direction in free space under Gaussian-beam illumination. −ρ shows the position indicating pure boiling.

Fig. 11
Fig. 11

Diagram indicating the type of motion of dynamic speckles as a function of the lens pupil q under plane-wave illumination. The symbols B and T indicate the boiling and translation regions, respectively, of the speckle motion.

Fig. 12
Fig. 12

The type of motion of dynamic speckles and the translation direction in the image space under plane-wave illumination.

Fig. 13
Fig. 13

Diagram indicating the type of motion of dynamic speckles as a function of the position lb at which speckles are in pure boiling motion.

Fig. 14
Fig. 14

Dependence of the position Lb of the imaged pure boiling plane on the position Lw of the beam waist under Gaussian-beam illumination. Here, the conditions q = 10 mm, F = 50 mm, λ = 0.6328 μm, w0 = 500 μm, and l1 = 2F were used. The dashed lines (Lρ) show the position of the pure boiling plane in free space without the imaging lens.

Fig. 15
Fig. 15

The type of motion of dynamic speckles and the translation direction in the image space under Gaussian-beam illumination with four different beam-waist positions. Here lb shows the position of the imaged pure boiling plane under the condition indicated in Fig. 14.

Fig. 16
Fig. 16

The coordinate system for (a) the optical arrangement and (b) the observation plane used for analysis of the dynamic speckles produced by a rotating object.

Fig. 17
Fig. 17

The coordinate system for the optical arrangement used for analysis of the dynamic speckles produced by a rotating object.

Fig. 18
Fig. 18

The spectral linewidth Γ2 versus the point spread ri for the incident angle α = 5.77°, Ω = 0.544 rad/sec, R = 20 mm, and F1 = F2.

Tables (1)

Tables Icon

Table 1 Parameters Indicating the Speckle-Intensity Fluctuation for Various Conditions of Both the Optical Geometry and the Illuminationa

Equations (173)

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E 0 ( x , t ) = E 0 ( x , t ) exp [ i ϕ ( x v t ) ] ,
E ( X , t ) = E 0 ( x , t ) K ( x , X ) d x ,
E 1 E 2 * = E ( X 1 , t 1 ) E * ( X 2 , t 2 ) = E 0 ( x 1 , t 1 ) E 0 * ( x 2 , t 2 ) K ( x 1 , X 1 ) K * ( x 2 , X 2 ) × exp { i [ ϕ ( x 1 ) ϕ ( x 2 ) ] } d x 1 d x 2 ,
x j = x j + v t j
E 1 E 2 * = E 0 ( x 1 + v t 1 , t 1 ) E 0 * ( x 2 + v t 2 , t 2 ) × exp { i [ ϕ ( x 1 ) ϕ ( x 2 ) ] } × K ( x 1 + v t 1 , X 1 ) K * ( x 2 + v t 2 , X 2 ) d x 1 d x 2 .
ϕ ( x ) = j a j Φ j ( x ) .
Φ i ( x ) Φ j ( x ) d x = δ i j , a i a j = a j 2 δ i j .
exp { i [ ϕ ( x ) ϕ ( x ) ] } = j P j ( a j ) exp { i a j [ Φ j ( x ) Φ j ( x ) ] } d a j .
P j ( a j ) = ( 2 π a j 2 ) 1 / 2 exp ( a j 2 2 a j 2 ) .
P j ( a j ) exp ( i m a j ) d a j = exp ( 1 2 m 2 a j 2 ) ,
exp { i [ ϕ ( x ) ϕ ( x ) ] } = exp { 1 2 j a j 2 [ Φ j ( x ) Φ j ( x ) ] 2 } = exp { 1 2 [ ϕ ( x ) ϕ ( x ) ] 2 } = exp { Φ 2 [ 1 ϕ ( x ) ϕ ( x ) ϕ 2 ] } .
ϕ ( x ) ϕ ( x ) ϕ 2 = exp ( ( x x ) 2 r h 2 ) ,
circ ( x x r h ) = { 1 for | x x | r h 0 for | x x | > r h .
ϕ 2 1
exp { i [ ϕ ( x ) ϕ ( x ) ] } = { 1 for | x x | r h 0 for | x x | > r h .
E 1 E 2 * = π r h 2 E 0 ( x 1 + v t 1 , t 1 ) E 0 * ( x 1 + v t 2 , t 2 ) × K ( x 1 + v t 1 , X 1 ) K * ( x 1 + v t 2 , X 2 ) d x 1 .
I 1 I 2 = E 1 E 1 * E 2 E 2 * + | E 1 E 2 * | 2 ,
r = X 2 X 1 , τ = t 2 t 1 ,
g ( 2 ) ( r , τ ) = I 1 I 2 I 1 I 2 = 1 + | g ( 1 ) ( r , τ ) | 2 ,
g ( 1 ) ( r , τ ) = E 1 E 2 * ( | E 1 | 2 | E 2 | 2 ) 1 / 2
E 0 ( x , t ) = w 0 w exp [ | x | 2 w 2 i ( ω 0 t k 0 z π λ ρ | x | 2 φ 0 ) ] ,
w = w 0 [ 1 + ( z a ) 2 ] 1 / 2 , ρ = z [ 1 + ( a z ) 2 ] ,
a = π λ w 0 2 .
E ( x , t ) = E 0 exp { i [ w 0 t k 0 x φ 0 ] } ,
E s ( ξ 1 ) E s * ( ξ 2 ) = I 0 exp ( | ξ 1 + ξ 2 | 2 2 h 0 2 ) exp ( | ξ 1 ξ 2 | 2 2 ζ 0 2 ) ,
E j + 1 ( x j + 1 , t ) = k i 2 π l j exp ( i k l j ) E j ( x j , t ) × exp ( i k | x j x j + 1 | 2 2 l j ) d x j ,
E j + 1 ( x j + 1 , t ) G ( x j ) E j ( x j , t ) exp ( i k | x j x j + 1 | 2 2 l j ) d x j .
K ( x , X ) = k i 2 π l exp ( i k | x X | 2 2 l ) exp ( i k l ) .
K ( x , X ) = k 2 q 2 4 π l 1 l 2 exp [ k 2 q 2 4 ( 1 + θ 1 2 ) | x l 1 X l 2 | 2 ] × exp { i [ k ( l 1 + l 2 + | x | 2 2 l 1 + | X | 2 2 l 2 ) + tan 1 θ 1 k 2 q 2 θ 1 4 ( 1 + θ 1 2 ) | x l 1 X l 2 | 2 ] } ,
θ 1 = k q 2 2 ( 1 l 1 + 1 l 2 1 F ) .
K ( x , X ) = k 2 q 2 4 π l 1 l 2 l 3 l 4 A 1 A 2 ( 1 + θ 2 2 ) 1 / 2 × exp [ k 2 q 2 4 ( 1 + θ 2 2 ) | x A 1 l 1 l 2 X A 2 l 3 l 4 | 2 ] × exp { i k ( l 1 + l 2 + l 3 + l 4 ) + i tan 1 θ 2 + i k 2 [ 1 l 1 ( 1 1 A 1 l 1 ) | x | 2 + 1 l 4 ( 1 1 A 2 l 4 ) | X | 2 ] i k 2 q 2 θ 2 2 4 ( 1 + θ 2 2 ) | x A 1 l 1 l 2 X A 2 l 3 l 4 | 2 } ,
A 1 = 1 l 1 + 1 l 2 1 F 1 , A 2 = 1 l 3 + 1 l 4 1 F 2 , θ 2 = k q 2 2 ( 1 l 2 1 A 1 l 2 2 + 1 l 3 1 A 2 l 3 2 ) .
I X ( ω ) = ( 1 2 π ) 1 / 2 g X ( 1 ) ( 0 , τ ) e i ω τ d τ ,
g ( 1 ) ( X 1 , X 2 , τ ) = exp [ | v | 2 2 w 2 τ 2 w 2 k 0 2 8 l 2 | X 2 X 1 ( 1 + l ρ ) v τ | 2 ] × exp [ i ω 0 τ i k 0 2 l ( X 1 + X 2 ) ( X 2 X 1 v τ ) ] .
g ( 2 ) ( r , τ ) 1 = exp ( | r | 2 r s 2 + τ d 2 τ c 2 ) exp [ 1 τ c 2 ( τ τ d ) 2 ] ,
1 τ c = | v | [ 1 r s 2 ( 1 + l ρ ) 2 + 1 w 2 ] 1 / 2
τ d = τ c 2 r s 2 ( 1 + l ρ ) vr .
r s = 2 l / ( k 0 w ) ,
I X ( ω ) = τ c exp [ τ c 2 2 ( ω ω 0 k 0 l vX ) 2 ] ,
ω D = k ( û v ) = k ( X l υ x + Y l υ y ) = k 0 l ( vX ) ,
E 0 ( x 1 ) E 0 * ( x 2 ) = E s ( ξ 1 ) E s * ( ξ 2 ) K ( ξ 1 , x 1 ) K * ( ξ 2 , x 2 ) d ξ 1 d ξ 2 = 2 I 0 ( π h 0 ζ 0 λ l 0 ) 2 exp [ i ( π λ l 0 ) ( | x 1 | 2 | x 2 | 2 ) ] × exp ( | x 1 + x 2 | 2 2 q 2 ) exp ( | x 2 x 1 | 2 2 ζ 1 2 ) ,
ζ 1 = λ l 0 π h 0
q = λ l 0 π ζ 0 .
g ( 1 ) ( X 1 , X 2 , τ ) = exp [ i k 0 2 l ( X 1 + X 2 ) ( X 2 X 1 v τ ) + i ω 0 τ ] × exp [ | v | 2 2 ζ 1 2 τ 2 | ( 1 + l / l 0 ) v τ ( X 2 X 1 ) | 2 2 r s 2 ] .
r s = l l 0 ζ 0 .
g ( 2 ) ( r , τ ) 1 = exp ( | r | 2 r s 2 + τ d 2 τ c 2 ) exp [ 1 τ c 2 ( τ τ d ) 2 ] .
1 τ c = | v | [ ( 1 + l / l 0 ) 2 r s 2 + 1 ζ 1 2 ] 1 / 2
τ d = τ c 2 r s 2 ( 1 + l l 0 ) vr .
I X ( ω ) = τ c exp [ τ c 2 2 ( ω ω 0 k 0 vX l ) 2 ] ,
g ( 1 ) ( r , τ ) = exp [ i ( ω 0 k 0 v ) τ ] × exp [ i k 0 2 l 2 ( X 1 + X 2 ) ( X 2 X 1 + v τ ) ] × exp { 1 2 [ k 0 q 2 l 2 ( X 2 X 1 l 2 F F v τ ) ] 2 | v | 2 2 q 2 τ 2 } .
g ( 2 ) ( r , τ ) 1 = exp ( | r | 2 r s 2 + τ d 2 τ c 2 ) exp [ 1 τ c 2 ( τ τ d ) 2 ] ,
1 τ c = | v | [ 1 r s 2 ( l 2 F 1 ) 2 + 1 q 2 ] 1 / 2 ,
τ d = τ c 2 r s 2 ( l 2 F 1 ) vr ,
r s = 2 l 2 k 0 q .
I X ( ω ) = τ c exp [ τ c 2 2 ( ω ω 0 + k 0 v + k 0 vX l 2 ) 2 ] .
ω D = ( k u ˆ k 0 ) v = k l 2 vX k 0 v ,
w 2 l 1 k 0 q ( 1 + θ 1 2 ) 1 / 2 = 2 r i ,
g ( 1 ) ( X 1 , X 2 , τ ) = exp { i [ ω 0 τ k 0 2 l 2 ( X 1 + X 2 ) × ( X 2 X 1 + μ v τ ) ] } × exp ( 1 2 | k 0 q 2 l 2 { ( X 2 X 1 ) [ l 2 l 1 ( 1 μ ) + μ ( l 2 F 1 ) ] v τ } | 2 μ 2 2 q 2 | v | 2 τ 2 ) ,
μ = ( 1 + l 1 ρ ) .
g ( 2 ) ( r , τ ) 1 = exp [ | r | 2 r s 2 + τ d 2 τ c 2 ] exp [ 1 τ c 2 ( τ τ d ) 2 ] ,
1 τ c = | v | ( B 1 2 r s 2 + μ 2 q 2 ) 1 / 2 ,
τ d = τ c 2 r s 2 B 1 vr ,
r s = 2 l 2 k 0 q ,
B 1 = μ ( l 2 F 1 ) l 2 l 1 ( μ 1 ) .
I X ( ω ) = τ c exp [ τ c 2 2 ( ω ω 0 + k 0 μ l 2 vX ) 2 ] .
g ( 1 ) ( X 1 X 2 , τ ) = exp [ i ( ω 0 k 0 v ) τ ] × exp [ i k 0 2 ( l 4 + l 3 F 2 F 2 l 3 ) ( X 1 + X 2 ) × ( X 2 X 1 + 1 l 2 / F 1 1 l 3 / F 2 v τ ) ] × exp [ k 0 2 q 2 8 A 2 2 l 3 2 l 4 2 ( X 2 X 1 B 2 v τ ) 2 1 2 ( 1 l 2 F 1 ) 2 | v | 2 q 2 τ 2 ] ,
B 2 = A 2 l 3 l 4 [ 1 A 1 l 1 l 2 ( 1 l 2 F 1 ) ( 1 l 2 1 A 1 l 2 2 + 1 l 3 1 A 2 l 3 2 ) ] .
g ( 2 ) ( r , τ ) = exp [ | r | 2 r s 2 + τ d 2 τ c 2 ] exp [ 1 τ c 2 ( τ τ d ) 2 ] ,
1 τ c = | v | [ B 2 2 r s 2 + ( l 2 F 1 1 ) 2 1 q 2 ] 1 / 2 ,
τ d = τ c 2 r s 2 B 2 vr ,
r s = 2 k 0 q ( l 3 + l 4 l 3 l 4 F 2 ) ,
I X ( ω ) = τ c exp [ τ c 2 2 ( ω ω 0 + k 0 v l 2 / F 1 1 A 2 l 3 l 4 k 0 vX ) 2 ] .
ω D = ( k u ˆ 1 k 0 ) v .
u 1 = a l 2 ( l 2 F 1 1 ) = u 2 ( l 2 F 1 )
u 2 = b l 3 = X A 2 l 3 l 4 ,
u ˆ 1 = X A 2 l 3 l 4 ( l 2 F 1 )
I X ( ω ) = τ c exp [ τ c 2 2 ( ω ω 0 ω D ) 2 ] ,
g ( 2 ) ( r , τ ) 1 = exp [ | r | 2 r s 2 + ( τ d τ c ) 2 ] exp [ 1 τ c 2 ( τ τ d ) 2 ] ,
g ( 2 ) ( r , τ ) 1 = exp ( r s 2 τ c 2 | V s | 2 τ c 2 r s 2 τ 2 ) × exp [ 1 r s 2 ( r V s τ ) 2 ] ,
W ( X j , t j ) = H D j H T j I ( X , t ) d X d t ,
g ( 2 ) ( r , τ ; D , T ) = W ( X 1 , t 1 ) W ( X 2 , t 2 ) W ( X 1 , t 1 ) W ( X 2 , t 2 ) ,
g ( 2 ) ( r ; τ ; D , T ) = H D 1 H D 2 H T 1 H T 2 I 1 I 2 d X 1 d X 2 d t 1 d t 2 H D 1 H T 1 I 1 d X 1 d t 1 H D 2 H T 2 I 2 d X 2 d t 2
H D j = circ ( X X j D ) .
H T j = rect ( t t j T ) .
H D j = exp [ 2 ( X X j ) 2 D 2 ]
H T j = exp [ 2 ( t t j ) 2 ( T / 2 ) 2 ] .
g ( 2 ) ( r , τ ; D , T ) 1 = f D f T exp [ ( f T τ c D ) 2 ( τ τ d ) 2 + ( τ d D τ c D ) 2 f D | r | 2 r s 2 ] ,
f D = 1 1 + ( D / r s ) 2 , f T = { 1 + [ ( T / 2 ) / τ c D ] 2 } 1 / 2 .
W ( X ) 2 W 2 = g ( 2 ) ( 0 , 0 ; D , T )
W ( X ) 2 W 2 1 = f D f T = f D [ 1 + ( T 2 τ c D ) 2 ] 1 / 2 .
W ( X ) 2 W 2 1 = W 2 W 2 W 2 = C W 2 ( X ) .
W 2 W 2 1 = f D [ 1 + f D N 2 ] 1 / 2 ,
N = M | v | T / ( 2 r s ) .
τ d = ( τ c r s ) 2 V s r ,
g ( 2 ) ( r , τ ) 1 = exp [ r 2 r c 2 ] exp [ 1 τ c 2 ( τ τ d ) 2 ] ,
r c = ( 1 r s 2 τ c 2 r s 4 V s 2 ) 1 / 2 .
g ( 2 ) ( r , τ ) 1 = exp [ ( 1 τ c 2 V s 2 r s 2 ) τ 2 ] × exp [ 1 r s 2 ( r V s τ ) 2 ] ,
r = V s τ .
τ = ( 1 τ c 2 V s 2 r s 2 ) 1 / 2 .
r T = V s ( 1 τ c 2 V s 2 r s 2 ) 1 / 2 ,
r c 2 = r T 2 + r s 2 .
η = r T r s = τ d ( r = r c ) τ c .
g ( 2 ) ( r , τ ) 1 = exp ( | v | 2 w 2 τ 2 ) × exp [ 1 r s 2 | r ( 1 + l ρ ) v τ | 2 ] .
τ T = w | v | ,
r T = ( 1 + l ρ ) w
η = { ( l + z ) z a + a } 1 l ,
a 2 + z 2 a z l a 2 + z 2 a + z .
l a 2 + z 2 a + z , l a 2 + z 2 a z .
a 2 + z 2 a z l a 2 + z 2 a + z .
l = ρ = z [ 1 + ( a z ) 2 ] .
V s = r τ = ( 1 + l ρ ) v .
V s = v ,
g ( 2 ) ( r , τ ) 1 = exp [ | v | 2 ζ 1 2 τ 2 ] × exp [ 1 ζ 0 2 ( l 0 l ) 2 | r ( 1 + l l 0 ) v τ | 2 ] .
τ T = ζ 1 | v | ,
r T = ( 1 + l l 0 ) ζ 1 .
η = l 0 l ζ 0 ( 1 + l l 0 ) ζ 1 = ζ 1 ζ 0 ( 1 + l / l 0 l / l 0 ) .
V s = ( 1 + l l 0 ) v .
g ( 2 ) ( r , τ ) 1 = exp [ | v | 2 q 2 τ 2 ] × exp [ 1 r s 2 | r ( l 2 F 1 ) v τ | 2 ] .
τ T = q | v | .
r T = ( l 2 F 1 ) q ,
η = ( l 2 F 1 ) q r s = ( 1 F l 2 ) q r s F .
q q + r s F l 2 F q q r s F .
l 2 F q q + r s F .
V s = ( l 2 F 1 ) v .
V s = l 2 l V v V = ( l 2 F 1 ) v V .
v V = v
g ( 2 ) ( r , τ ) 1 = exp [ μ 2 q 2 | v | 2 τ 2 ] × exp { 1 r s 2 | r [ μ ( l 2 F 1 ) l 2 l 1 ( μ 1 ) ] v τ | 2 } ,
τ T = q μ | v | ,
η = 1 r s [ q ( l 2 F 1 ) l 2 l 1 q ( μ 1 μ ) ] .
1 l 1 + ρ + 1 l b = 1 F .
η = k 0 q 2 2 ( 1 l b 1 l 2 )
1 l 1 + 1 l g = 1 F .
l b 1 2 l b k 0 q 2 l 2 l b 1 + 2 l b k 0 q 2 ,
l 2 l b 1 + 2 l b k 0 q 2 .
l 2 l b 1 + 2 l b / ( k 0 q 2 ) .
l b 1 2 l b / ( k 0 q 2 ) l 2 l b 1 + 2 l b / ( k 0 q 2 )
l 2 l b 1 + 2 l b / ( k 0 q 2 ) .
L b = n ( L w 2 m L w + n m ) L w 2 ( n + m ) L w + n m + 1 ,
L ρ = L w + 1 L w m ,
V s = [ μ ( l 2 F 1 ) l 2 l 1 ( μ 1 ) ] v .
1 l V + 1 l 2 = 1 F
V s = l 2 l V v V .
v V = ( l ρ + 1 ) v = ( l 1 ρ + 1 ) v l V ρ v
l 2 = l b .
g ( 2 ) ( r , τ ) 1 = exp [ ( l 2 F 1 ) 2 | v | 2 q 2 τ 2 ] × exp [ 1 r s 2 | r B 2 v τ | 2 ] ,
τ T = q ( l 2 / F 1 ) | v | ,
r T = B 2 q / ( l 2 / F 1 )
η = B 2 q ( l 2 / F 1 ) r s
V s = B 2 v = A 2 l 3 l 4 [ 1 A 1 l 1 l 2 ( 1 l 2 F 1 ) ( 1 l 2 1 A 1 l 2 2 + 1 l 3 1 A 2 l 3 2 ) ] v .
( 1 τ c ) 2 = ( V s r s ) 2 + ( 1 τ T ) 2 .
η = V s r s τ T = r T r s .
x j = ( cos Ω t j sin Ω t j sin Ω t j cos Ω t j ) ( x j y j ) = x j cos Ω t j + A x j sin Ω t j ,
A = ( 0 1 1 0 ) .
r = X 2 X 1 , R = ( X 1 + X 2 ) / 2 ,
g R ( 2 ) ( r , τ ) 1 = exp ( | r | 2 r s 2 ) exp { ( 1 cos Ω τ ) × [ 2 | d | 2 w 2 + 2 r s 2 ( R l ρ d ) 2 | r | 2 2 r s 2 ] } × exp [ 2 sin ( Ω τ ) 1 r s 2 ( R l ρ d ) Ar ] .
g R ( 2 ) ( 0 , τ ) 1 = exp { 2 [ d 2 w 2 + 1 r s 2 ( R l ρ d ) 2 ] ( 1 cos Ω τ ) } .
g R ( 2 ) ( r , τ ) 1 = exp [ | d | 2 w 2 Ω 2 τ 2 ] × exp { 1 r s 2 [ r A ( R l ρ d ) Ω τ ] 2 } ,
| r | 2 r s 2 r 2 4 r s 2 Ω 2 τ 2
V S R = A ( R l ρ d ) Ω ,
exp [ | d | 2 w 2 Ω 2 τ 2 ] = exp [ | v d | 2 w 2 τ 2 ] ,
τ T = w | v d | ,
r T = | R l ρ d | w | d | .
η = | R l ρ d | k 0 w 2 2 l d .
g ( 2 ) ( r , τ ) 1 = exp { 1 r s 2 [ | r | 2 + ( Ω τ ) 2 ( X 1 X 2 + r s 4 | k 0 | 2 4 M 2 sin 2 α ) + 2 Ω τ X 1 A X 2 ] } ,
| X 2 X 1 | = R Δ θ , X 1 X 2 = R 2 , X 1 A X 2 = R 2 Δ θ
g R ( 2 ) ( r , τ ) 1 = exp [ ( r s 2 | k 0 | sin α 2 M R ) 2 ( V S R τ r s ) 2 ] × exp [ 1 r s 2 | r V S R τ | 2 ] ,
r = R Δ θ , V S R = Ω R .
r T = 2 M R r s | k 0 | sin α ,
R = M d , r s = 2 M r i ,
η = d 2 r i 2 | k 0 | sin α = ω V Δ ω 0 / 2 ,
ω V = d Ω / 2 r i
Δ ω D = 2 r i Ω | k 0 | sin α .
ω D = k 0 v d = d | k 0 | Ω sin α .

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