Abstract

Imaging of Gaussian Schell-model sources by general lossless systems is analyzed with an extended ray-transfermatrix method. Algebraic expressions are derived for the location, size, and coherence area of the image waist and for the depth of focus and the far-field diffraction angle. These results are shown to provide a continuous transformation between laser-beam optics and geometrical optics. They also lead naturally to several equivalence and invariance relations pertaining to isotropic and anisotropic Gaussian Schell-model sources. As an application, the importance of effects due to partial spatial coherence in beam focusing is examined.

© 1988 Optical Society of America

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  1. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
    [CrossRef]
  2. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  3. H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
    [CrossRef]
  4. W. H. Carter, “Statistical radiometry,” Radio Sci. 18, 149–158 (1983).
    [CrossRef]
  5. E. Collett, E. Wolf, “Beams generated by Gaussian quasihomogeneous sources,” Opt. Commun. 32, 27–31 (1980).
    [CrossRef]
  6. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [CrossRef]
  7. E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [CrossRef]
  8. B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
    [CrossRef]
  9. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  10. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  11. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  12. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
    [CrossRef]
  13. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
    [CrossRef]
  14. C. Pask, “Application of Wolf’s theory of coherence,” J. Opt. Soc. Am. A 3, 1097–1101 (1986).
    [CrossRef]
  15. P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]
  16. J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
    [CrossRef]
  17. J. Deschamps, D. Courjon, J. Bulabois, “Gaussian Schellmodel sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256–261 (1983).
    [CrossRef]
  18. A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
    [CrossRef]
  19. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  20. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  21. Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schellmodel sources,” Opt. Lett. 7, 256–258 (1982).
    [CrossRef] [PubMed]
  22. R. Simon, “A new class of anisotropic Gaussian beams,” Opt. Commun. 55, 381–385 (1985).
    [CrossRef]
  23. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  24. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  25. J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
    [CrossRef]
  26. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  27. J. Turunen, A. T. Friberg, “Propagation of Gaussian Schellmodel beams: a matrix method,” in Optical System Design, Analysis, and Production for Advanced Technology Systems, R. E. Fischer, P. J. Rogers, eds. Proc. Soc. Photo-Opt. Instrum. Eng.655, 60–66 (1986).
    [CrossRef]
  28. P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
    [CrossRef]
  29. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).
  30. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
    [CrossRef]

1986 (5)

C. Pask, “Application of Wolf’s theory of coherence,” J. Opt. Soc. Am. A 3, 1097–1101 (1986).
[CrossRef]

A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
[CrossRef]

J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[CrossRef]

1985 (1)

R. Simon, “A new class of anisotropic Gaussian beams,” Opt. Commun. 55, 381–385 (1985).
[CrossRef]

1984 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1983 (3)

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

J. Deschamps, D. Courjon, J. Bulabois, “Gaussian Schellmodel sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256–261 (1983).
[CrossRef]

W. H. Carter, “Statistical radiometry,” Radio Sci. 18, 149–158 (1983).
[CrossRef]

1982 (3)

1981 (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1980 (3)

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

E. Collett, E. Wolf, “Beams generated by Gaussian quasihomogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1979 (2)

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

1978 (3)

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
[CrossRef]

1976 (1)

1970 (1)

1967 (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[CrossRef]

1965 (2)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
[CrossRef]

Baltes, H. P.

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
[CrossRef]

Bulabois, J.

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Carter, W. H.

W. H. Carter, “Statistical radiometry,” Radio Sci. 18, 149–158 (1983).
[CrossRef]

Collett, E.

E. Collett, E. Wolf, “Beams generated by Gaussian quasihomogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

Collins, S. A.

Courjon, D.

DeSantis, P.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Deschamps, J.

Farina, J. D.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

Foley, J. T.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

Friberg, A. T.

A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
[CrossRef]

J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
[CrossRef]

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

J. Turunen, A. T. Friberg, “Propagation of Gaussian Schellmodel beams: a matrix method,” in Optical System Design, Analysis, and Production for Advanced Technology Systems, R. E. Fischer, P. J. Rogers, eds. Proc. Soc. Photo-Opt. Instrum. Eng.655, 60–66 (1986).
[CrossRef]

Geist, J.

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
[CrossRef]

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Gori, F.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Guattari, G.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
[CrossRef]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Li, Y.

Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schellmodel sources,” Opt. Lett. 7, 256–258 (1982).
[CrossRef] [PubMed]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Mandel, L.

Mukunda, N.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Narducci, L. M.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

Palma, C.

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Pask, C.

Saleh, B. E. A.

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[CrossRef]

Schell, A. C.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[CrossRef]

Simon, R.

R. Simon, “A new class of anisotropic Gaussian beams,” Opt. Commun. 55, 381–385 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Starikov, A.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Turunen, J.

A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
[CrossRef]

J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
[CrossRef]

J. Turunen, A. T. Friberg, “Propagation of Gaussian Schellmodel beams: a matrix method,” in Optical System Design, Analysis, and Production for Advanced Technology Systems, R. E. Fischer, P. J. Rogers, eds. Proc. Soc. Photo-Opt. Instrum. Eng.655, 60–66 (1986).
[CrossRef]

Walther, A.

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
[CrossRef]

Wolf, E.

Zubairy, M. S.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

IEEE Trans. Antennas Propag. (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[CrossRef]

Opt. Commun. (10)

R. Simon, “A new class of anisotropic Gaussian beams,” Opt. Commun. 55, 381–385 (1985).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

P. DeSantis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

E. Collett, E. Wolf, “Beams generated by Gaussian quasihomogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Opt. Eng. (1)

A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
[CrossRef]

Opt. Laser Technol. (1)

J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Radio Sci. (1)

W. H. Carter, “Statistical radiometry,” Radio Sci. 18, 149–158 (1983).
[CrossRef]

Other (3)

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

J. Turunen, A. T. Friberg, “Propagation of Gaussian Schellmodel beams: a matrix method,” in Optical System Design, Analysis, and Production for Advanced Technology Systems, R. E. Fischer, P. J. Rogers, eds. Proc. Soc. Photo-Opt. Instrum. Eng.655, 60–66 (1986).
[CrossRef]

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Illustration of the quantities and notation associated with the passage of a Gaussian Schell-model beam through a general optical system.

Fig. 2
Fig. 2

Focusing of a Gaussian Schell-model beam by a thin lens: w0, wi, and wG and z0, zi, and zG are the sizes and distances, respectively, of the object, the image, and the Gaussian (geometrical-optics) image.

Fig. 3
Fig. 3

Effect of partial spatial coherence on the relative displacement |(zizG)/zG| of the image-beam waist from the Gaussian (geometrical-optics) image plane for various source distances: (a) z0/F > 1; (b) z0/F < 1. The parameter Nβ (given explicitly in the text) is the coherence-dependent Fresnel number of the imaging setup.

Fig. 4
Fig. 4

Effect of partial spatial coherence on the relative image magnification |m/mG| for various values of the quantity 1 − z0/F; here z0/F is the normalized source distance, and mG is the corresponding geometrical-optics magnification. The parameter Nβ is the coherence-dependent Fresnel number, as before.

Equations (38)

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b ( z ) = π n w 2 ( z ) β / λ ,
b ( z ) = b 0 [ 1 + ( z / b 0 ) 2 ] ,
R ( z ) = z [ 1 + ( b 0 / z ) 2 ] ,
b 2 = b 1 [ ( A + B R 1 ) 2 + ( B b 1 ) 2 ] ,
R 2 = [ ( A + B R 1 ) 2 + ( B b 1 ) 2 ] / [ ( A + B R 1 ) ( C + D R 1 ) + B D b 1 2 ] .
1 q = 1 R i b ,
q 2 = A q 1 + B C q 1 + D ,
b 2 = b 0 [ A 2 + B 2 b 0 2 ] ,
R 2 = [ A 2 + B 2 b 0 2 ] / [ A C + B D b 0 2 ] .
z i = A 2 + B 2 b 0 2 A C + B D b 0 2 + ( A C b 0 2 + B D ) 1 ,
m 2 = b i b 0 A 2 + B 2 b 0 2 1 + ( A C b 0 + B D b 0 1 ) 2 ,
w i = m w 0 .
σ i = m σ 0 .
d = 2 m 2 b 0 ( η 2 1 ) 1 / 2 .
Θ = arctan ( λ / π n m ω 0 β ) ,
z i B / D ,
| m | 1 / | D | .
z i A / C ,
| m | 0 .
1 w L 2 = 1 w 0 2 + 1 σ 0 2 .
1 w 0 x 2 + 1 σ 0 x 2 = 1 w 0 y 2 + 1 σ 0 y 2 ,
Δ z z G = { z 0 F [ 1 + ( 1 z 0 / F N β ) 2 ] } 1 ,
| m m G | = [ 1 + ( N β 1 z 0 / F ) 2 ] 1 / 2 .
Δ F = F ( 1 + N β 2 ) 1 ,
w ( ρ , z ) = ink 2 π B × w ( ρ , z ) exp [ ink 2 B ( D ρ 2 2 ρ ρ + A ρ 2 ) ] d 2 ρ ,
U ( r ) = w ( ρ , z ) exp ( inkz ) .
W 1 ( ρ 1 , ρ 2 ) = w * ( ρ 1 , z 1 ) w ( ρ 2 , z 1 ) ,
I 1 ( ρ ) = W 1 ( ρ , ρ ) ,
μ 1 ( ρ 1 , ρ 2 ) = [ I 1 ( ρ 1 ) I 1 ( ρ 2 ) ] 1 / 2 W 1 ( ρ 1 , ρ 2 ) ,
I 1 ( ρ ) = I 0 exp ( 2 ρ 2 w 1 2 )
μ 1 ( ρ 1 , ρ 2 ) = exp [ ( ρ 1 ρ 2 ) 2 2 σ 1 2 ] exp [ ink 2 R 1 ( ρ 1 2 ρ 2 2 ) ] ,
W 2 ( ρ 1 , ρ 2 ) = I 0 ( w 1 w 2 ) 2 exp [ ( ρ 1 + ρ 2 ) 2 2 w 2 2 ] × exp [ w 1 2 ( ρ 1 ρ 2 ) 2 2 w 2 2 w 2 2 ] × exp [ ink 2 R 2 ( ρ 1 2 ρ 2 2 ) ] ,
w 2 2 = w 1 2 [ ( A + B R 1 ) 2 + ( λ π n w 1 w c ) 2 B 2 ] ,
R 2 = ( A + B R 1 ) 2 + ( λ π n w 1 w c ) 2 B 2 ( A + B R 1 ) ( C + D R 1 ) + ( λ π n w 1 w c ) 2 B D ,
1 w c 2 = 1 w 1 2 + 2 σ 1 2 .
I 2 ( ρ ) = I 0 ( w 1 w 2 ) 2 exp ( 2 ρ 2 w 2 2 )
μ 2 ( ρ 1 , ρ 2 ) = exp [ ( ρ 1 ρ 2 ) 2 2 σ 2 2 ] exp [ ink 2 R 2 ( ρ 1 2 ρ 2 2 ) ] ,
σ 2 = ( w 2 w 1 ) σ 1 .

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