Abstract

Paraxial propagation of a partially coherent Hermite–Gaussian beam through aligned and misaligned ABCD optical systems is investigated based on the generalized Collins formula for treating the propagation of a partially coherent beam through such optical systems. Analytical formulas for the cross-spectral density of a partially coherent Hermite–Gaussian beam propagating through such optical systems are derived. As an application example, we derive the propagation formulas for a partially coherent flattened Gaussian beam by expressing it as a superposition of a series of partially coherent Hermite–Gaussian beams by using polynomial expansion. The focusing properties of a partially coherent Hermite–Gaussian beam focused by a thin lens are studied as a numerical example.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  37. Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
    [CrossRef]
  38. Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
    [CrossRef]
  39. Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A, Pure Appl. Opt. 8, 537-545 (2006).
    [CrossRef]
  40. X. Lu and Y. Cai, "Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system," Opt. Commun. 269, 39-46 (2007).
    [CrossRef]
  41. Y. Cai and S. He, "Partially coherent flattened Gaussian beam and its paraxial propagation properties," J. Opt. Soc. Am. A 23, 2623-2628 (2006).
    [CrossRef]
  42. D. W. Coutts, "Double-pass copper vapor laser master-oscillator power-amplifier systems: generation of flat-top focused beams for fiber coupling and percussion drilling," IEEE J. Quantum Electron. 38, 1217-1224 (2002).
    [CrossRef]
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2007 (1)

X. Lu and Y. Cai, "Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system," Opt. Commun. 269, 39-46 (2007).
[CrossRef]

2006 (4)

2005 (4)

B. Lu, X. Tao, and Y. Nie, "Effect of quartic-phase aberrations on the focal switch of Hermite-Gaussian beams," Optik (Stuttgart) 116, 454-458 (2005).
[CrossRef]

Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
[CrossRef]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C. S. Chuu, and M. G. Raizen, "A high frequency optical trap for atoms using Hermite-Gaussian beams," Opt. Express 13, 2843-2851 (2005).
[CrossRef]

K. Duan, B. Wang, and B. Lu, "Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 22, 1976-1982 (2005).
[CrossRef]

2004 (1)

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

2003 (4)

Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

B. Lu and P. Peng, "Focal shift in Hermite-Gaussian beams based on the encircled-power criterion," Opt. Laser Technol. 35, 435-440 (2003).

S. Yu and W. Gu, "Generation of elegant Hermite-Gaussian beams using the graded-phase mirror," Opt. Laser Technol. 5, 460-463 (2003).

Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gaussian beam," J. Opt. Soc. Am. A 20, 1111-1119 (2003).
[CrossRef]

2002 (5)

Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002).
[CrossRef]

Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through misaligned optical systems," Opt. Commun. 211, 1-8 (2002).
[CrossRef]

D. W. Coutts, "Double-pass copper vapor laser master-oscillator power-amplifier systems: generation of flat-top focused beams for fiber coupling and percussion drilling," IEEE J. Quantum Electron. 38, 1217-1224 (2002).
[CrossRef]

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, "Turbulence-induced beam spreading of higher-order mode optical waves," Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).

2001 (3)

2000 (1)

B. Lu and H. Ma, "Beam combination of a radial laser array: Hermite-Gaussian model," Opt. Commun. 178, 395-403 (2000).
[CrossRef]

1998 (5)

H. Laabs, "Propagation of Hermite-Gaussian-beams beyond the paraxial approximation," Opt. Commun. 147, 1-4 (1998).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Opt. 45, 1999-2009 (1998).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, "The beam propagation factor for higher-order Gaussian beams," Opt. Commun. 153, 207-210 (1998).
[CrossRef]

A. N. Alekseev, K. N. Alekseev, O. S. Borodavka, A. V. Volyar, and Yu. A. Fridman, "Conversion of Hermite-Gaussian and Laguerre-Gaussian beams in an astigmatic optical system. 1. Experiment," Tech. Phys. Lett. 24, 694-696 (1998).
[CrossRef]

O. Mata-Mendez and F. Chavez-Rivas, "New property in the diffraction of Hermite-Gaussian beams by a finite grating in the scalar diffraction regime: constant-intensity angles in the far field when the beam center is displaced through the grating," J. Opt. Soc. Am. A 15, 2698-2704 (1998).
[CrossRef]

1997 (1)

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers," IEEE J. Quantum Electron. 33, 1025-1030 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

1990 (1)

1988 (1)

R. Simon, N. Mukunda, and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
[CrossRef]

1986 (1)

1985 (1)

1980 (1)

1970 (1)

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

Alekseev, A. N.

A. N. Alekseev, K. N. Alekseev, O. S. Borodavka, A. V. Volyar, and Yu. A. Fridman, "Conversion of Hermite-Gaussian and Laguerre-Gaussian beams in an astigmatic optical system. 1. Experiment," Tech. Phys. Lett. 24, 694-696 (1998).
[CrossRef]

Alekseev, K. N.

A. N. Alekseev, K. N. Alekseev, O. S. Borodavka, A. V. Volyar, and Yu. A. Fridman, "Conversion of Hermite-Gaussian and Laguerre-Gaussian beams in an astigmatic optical system. 1. Experiment," Tech. Phys. Lett. 24, 694-696 (1998).
[CrossRef]

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Arpali, C.

Arpali, S. A.

Bagini, V.

Basov, Y. G.

Y. G. Basov, "Divergence of excimer laser beams: a review," J. Commun. Technol. Electron. 46, 1-6 (2001).

Baykal, Y.

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Borghi, R.

Borodavka, O. S.

A. N. Alekseev, K. N. Alekseev, O. S. Borodavka, A. V. Volyar, and Yu. A. Fridman, "Conversion of Hermite-Gaussian and Laguerre-Gaussian beams in an astigmatic optical system. 1. Experiment," Tech. Phys. Lett. 24, 694-696 (1998).
[CrossRef]

Cai, Y.

X. Lu and Y. Cai, "Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system," Opt. Commun. 269, 39-46 (2007).
[CrossRef]

Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A, Pure Appl. Opt. 8, 537-545 (2006).
[CrossRef]

Y. Cai and S. He, "Partially coherent flattened Gaussian beam and its paraxial propagation properties," J. Opt. Soc. Am. A 23, 2623-2628 (2006).
[CrossRef]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gaussian beam," J. Opt. Soc. Am. A 20, 1111-1119 (2003).
[CrossRef]

Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002).
[CrossRef]

Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through misaligned optical systems," Opt. Commun. 211, 1-8 (2002).
[CrossRef]

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).

Carter, W. H.

Chavez-Rivas, F.

Chen, Y. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers," IEEE J. Quantum Electron. 33, 1025-1030 (1997).
[CrossRef]

Chen, Z.

Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
[CrossRef]

Chuu, C. S.

Collins, S. A.

Coutts, D. W.

D. W. Coutts, "Double-pass copper vapor laser master-oscillator power-amplifier systems: generation of flat-top focused beams for fiber coupling and percussion drilling," IEEE J. Quantum Electron. 38, 1217-1224 (2002).
[CrossRef]

Duan, K.

Erdelyi, A.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Eyyuboglu, H. T.

Fridman, Yu. A.

A. N. Alekseev, K. N. Alekseev, O. S. Borodavka, A. V. Volyar, and Yu. A. Fridman, "Conversion of Hermite-Gaussian and Laguerre-Gaussian beams in an astigmatic optical system. 1. Experiment," Tech. Phys. Lett. 24, 694-696 (1998).
[CrossRef]

Fukumitsu, O.

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, "Turbulence-induced beam spreading of higher-order mode optical waves," Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Gori, F.

Gu, W.

S. Yu and W. Gu, "Generation of elegant Hermite-Gaussian beams using the graded-phase mirror," Opt. Laser Technol. 5, 460-463 (2003).

Guo, H.

Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
[CrossRef]

Hanssen, J. L.

He, S.

Huang, T. M.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers," IEEE J. Quantum Electron. 33, 1025-1030 (1997).
[CrossRef]

Kao, C. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers," IEEE J. Quantum Electron. 33, 1025-1030 (1997).
[CrossRef]

Kojima, T.

Laabs, H.

H. Laabs, "Propagation of Hermite-Gaussian-beams beyond the paraxial approximation," Opt. Commun. 147, 1-4 (1998).
[CrossRef]

Lin, Q.

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gaussian beam," J. Opt. Soc. Am. A 20, 1111-1119 (2003).
[CrossRef]

Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002).
[CrossRef]

Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through misaligned optical systems," Opt. Commun. 211, 1-8 (2002).
[CrossRef]

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).

Lu, B.

B. Lu, X. Tao, and Y. Nie, "Effect of quartic-phase aberrations on the focal switch of Hermite-Gaussian beams," Optik (Stuttgart) 116, 454-458 (2005).
[CrossRef]

K. Duan, B. Wang, and B. Lu, "Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 22, 1976-1982 (2005).
[CrossRef]

B. Lu and P. Peng, "Focal shift in Hermite-Gaussian beams based on the encircled-power criterion," Opt. Laser Technol. 35, 435-440 (2003).

B. Lu and H. Ma, "Beam combination of a radial laser array: Hermite-Gaussian model," Opt. Commun. 178, 395-403 (2000).
[CrossRef]

Lu, X.

X. Lu and Y. Cai, "Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system," Opt. Commun. 269, 39-46 (2007).
[CrossRef]

Luk, K. M.

Ma, H.

B. Lu and H. Ma, "Beam combination of a radial laser array: Hermite-Gaussian model," Opt. Commun. 178, 395-403 (2000).
[CrossRef]

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, "Turbulence-induced beam spreading of higher-order mode optical waves," Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

Mata-Mendez, O.

Meyrath, T. P.

Mukunda, N.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Nie, Y.

B. Lu, X. Tao, and Y. Nie, "Effect of quartic-phase aberrations on the focal switch of Hermite-Gaussian beams," Optik (Stuttgart) 116, 454-458 (2005).
[CrossRef]

Oberhettinger, F.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Pacileo, A. M.

Peng, P.

B. Lu and P. Peng, "Focal shift in Hermite-Gaussian beams based on the encircled-power criterion," Opt. Laser Technol. 35, 435-440 (2003).

Qiu, Y.

Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
[CrossRef]

Raizen, M. G.

Ronchi, L.

S. Wang and L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, "The beam propagation factor for higher-order Gaussian beams," Opt. Commun. 153, 207-210 (1998).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Opt. 45, 1999-2009 (1998).
[CrossRef]

Santarsiero, M.

Schreck, F.

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Opt. 45, 1999-2009 (1998).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, "The beam propagation factor for higher-order Gaussian beams," Opt. Commun. 153, 207-210 (1998).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simon, R.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

Sudarshan, E. C. G.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Takenaka, T.

Tao, X.

B. Lu, X. Tao, and Y. Nie, "Effect of quartic-phase aberrations on the focal switch of Hermite-Gaussian beams," Optik (Stuttgart) 116, 454-458 (2005).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Volyar, A. V.

A. N. Alekseev, K. N. Alekseev, O. S. Borodavka, A. V. Volyar, and Yu. A. Fridman, "Conversion of Hermite-Gaussian and Laguerre-Gaussian beams in an astigmatic optical system. 1. Experiment," Tech. Phys. Lett. 24, 694-696 (1998).
[CrossRef]

Wang, B.

Wang, C. L.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers," IEEE J. Quantum Electron. 33, 1025-1030 (1997).
[CrossRef]

Wang, S.

S. Wang and L. Ronchi, "Principles and design of optical arrays," in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

Wang, S. C.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, "Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers," IEEE J. Quantum Electron. 33, 1025-1030 (1997).
[CrossRef]

Woerdman, J. P.

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Figures (1)

Fig. 1
Fig. 1

Normalized irradiance distribution of partially coherent Hermite–Gaussian beams ( m = 0 , 2 , 5 ) with different initial transverse coherence widths in the geometrical focal plane ( z = f ) .

Equations (26)

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W m ( x 1 , x 2 , 0 ) = H m ( 2 x 1 w 0 ) H m ( 2 x 2 w 0 ) exp [ x 1 2 + x 2 2 w 0 2 ( x 1 x 2 ) 2 2 σ 0 g 2 ] ,
W m ( ρ 1 , ρ 2 , z ) = 1 λ B W m ( x 1 , x 2 , 0 ) exp [ i k 2 B ( A x 1 2 2 x 1 ρ 1 + D ρ 1 2 ) ] exp [ i k 2 B ( A x 2 2 2 x 2 ρ 2 + D ρ 2 2 ) ] d x 1 d x 2 ,
W m ( ρ 1 , ρ 2 , z ) = 1 A W m ( x 1 , x 2 , 0 ) exp [ i k C 2 A ( ρ 1 2 ρ 2 2 ) ] δ ( x 1 ρ 1 A ) δ ( x 2 ρ 2 A ) d x 1 d x 2 = 1 A exp [ i k C 2 A ( ρ 1 2 ρ 2 2 ) ] W m ( ρ 1 A , ρ 2 A , 0 ) .
W m ( ρ 1 , ρ 2 , z ) = 1 λ B exp [ i k D ρ 1 2 2 B + i k D ρ 2 2 2 B ] H m ( 2 x 1 w 0 ) exp [ x 1 2 ( 1 w 0 2 + 1 2 σ 0 g 2 + i k A 2 B ) ] exp [ i k x 1 ρ 1 B ] d x 1 H m ( 2 x 2 w 0 ) exp [ x 2 2 ( 1 w 0 2 + 1 2 σ 0 g 2 i k A 2 B ) + x 2 ( x 1 σ 0 g 2 i k ρ 2 B ) ] d x 2 .
W m ( ρ 1 , ρ 2 , z ) = 1 λ B π C 1 ( 1 2 w 0 2 C 1 ) m 2 exp [ i k D ρ 1 2 2 B + i k D ρ 2 2 2 B ] H m ( 2 x 1 w 0 ) exp [ x 1 2 ( 1 w 0 2 + 1 2 σ 0 g 2 + i k A 2 B ) ] exp [ i k x 1 ρ 1 B ] exp [ 1 4 C 1 ( x 1 σ 0 g 2 i k ρ 2 B ) 2 ] H m [ 1 2 w 0 C 1 ( x 1 σ 0 g 2 i k ρ 2 B ) 1 2 w 0 2 C 1 ] d x 1 ,
exp [ ( x b ) 2 2 a ] H p ( x ) d x = 2 π a ( 1 2 a ) p 2 H p [ b 1 2 a ] ,
H n ( x + y ) = 1 2 n 2 k = 0 n ( n k ) H k ( 2 x ) H n k ( 2 y ) ,
H n ( x ) = n ! m = 0 [ n 2 ] ( 1 ) m 1 m ! ( n 2 m ) ! ( 2 x ) n 2 m ,
W m ( ρ 1 , ρ 2 , z ) = 1 λ B π C 1 ( 1 2 w 0 2 C 1 ) m 2 exp [ i k D ρ 1 2 2 B + i k D ρ 2 2 2 B ] exp [ k 2 ρ 2 2 4 C 1 B 2 ] m ! j = 0 [ m 2 ] n = 0 m p = 0 [ n 2 ] ( m n ) ( 1 ) j n ! j ! ( m 2 j ) ! 1 2 m 2 ( 1 ) p 1 p ! ( n 2 p ) ! H m n ( i k ρ 2 B w 0 2 C 1 2 2 C 1 ) ( 2 2 x 1 w 0 ) m 2 j ( 2 x 1 σ 0 g 2 w 0 2 C 1 2 2 C 1 ) n 2 p exp [ x 1 2 C 2 ] exp [ x 1 ( i k ρ 1 B i k ρ 2 2 C 1 B σ 0 g 2 ) ] d x 1 ,
x n exp [ ( x β ) 2 ] d x = ( 2 i ) n π H n ( i β ) ,
W m ( ρ 1 , ρ 2 , z ) = π λ B 1 C 1 ( 1 2 w 0 2 C 1 ) m 2 exp [ i k D ρ 1 2 2 B + i k D ρ 2 2 2 B ] exp [ k 2 ρ 2 2 4 C 1 B 2 ] m ! j = 0 [ m 2 ] n = 0 m p = 0 [ n 2 ] ( m n ) ( 1 ) j + p n ! j ! ( m 2 j ) ! 1 2 j i 2 j + 2 p m n p ! ( n 2 p ) ! 1 C 2 m + n 2 j 2 p + 1 2 1 w 0 m 2 j ( 1 σ 0 g 2 w 0 2 C 1 2 2 C 1 ) n 2 p exp [ k 2 4 C 2 B 2 ( ρ 1 ρ 2 2 C 1 σ 0 g 2 ) 2 ] H m n ( i k ρ 2 B w 0 2 C 1 2 2 C 1 ) H m + n 2 j 2 p [ k ρ 2 4 C 2 C 1 B σ 0 g 2 k ρ 1 2 C 2 B ] .
W 0 ( ρ 1 , ρ 2 , z ) = π λ B C 1 C 2 exp [ i k D ρ 1 2 2 B + i k D ρ 2 2 2 B ] exp [ k 2 ρ 2 2 4 C 1 B 2 ] exp [ 1 4 C 2 ( i k ρ 1 B i k ρ 2 2 C 1 σ 0 g 2 B ) 2 ] .
I ( ρ 1 , z ) = W m ( ρ 1 , ρ 1 , z ) = π λ B 1 C 1 ( 1 2 w 0 2 C 1 ) m 2 exp [ k 2 ρ 1 2 4 C 1 B 2 ] m ! j = 0 [ m 2 ] n = 0 m p = 0 [ n 2 ] ( m n ) ( 1 ) j + p n ! j ! ( m 2 j ) ! 1 2 j i 2 j + 2 p m n p ! ( n 2 p ) ! 1 C 2 m + n 2 j 2 p + 1 2 1 w 0 m 2 j ( 1 σ 0 g 2 w 0 2 C 1 2 2 C 1 ) n 2 p exp [ k 2 4 C 2 B 2 ( ρ 1 ρ 1 2 C 1 σ 0 g 2 ) 2 ] H m n ( i k ρ 1 B w 0 2 C 1 2 2 C 1 ) H m + n 2 j 2 p [ k ρ 1 4 C 2 C 1 B σ 0 g 2 k ρ 1 2 C 2 B ] .
W m ( ρ 1 , ρ 2 , z ) = 1 λ B W m ( x 1 , x 2 , 0 ) exp [ i k 2 B ( A x 1 2 2 x 1 ρ 1 + D ρ 1 2 + E x 1 + G ρ 1 ) ] exp [ i k 2 B ( A x 2 2 2 x 2 ρ 2 + D ρ 2 2 + E x 2 + G ρ 2 ) ] d x 1 d x 2 .
E = 2 ( α T ϵ x + β T ϵ x ) ,
G = 2 ( B γ T D α T ) ϵ x + 2 ( B δ T D β T ) ϵ x ,
α T = 1 A , β T = l B , γ T = C , δ T = ± 1 D .
W m ( ρ 1 , ρ 2 , z ) = π λ B 1 C 1 ( 1 2 w 0 2 C 1 ) m 2 exp [ i k D ρ 1 2 2 B + i k D ρ 2 2 2 B i k G 2 B ρ 1 + i k G 2 B ρ 2 ] exp [ k 2 ( ρ 2 E 2 ) 2 4 C 1 B 2 ] m ! j = 0 [ m 2 ] n = 0 m p = 0 [ n 2 ] ( m n ) ( 1 ) j + p n ! j ! ( m 2 j ) ! 1 2 j i 2 j + 2 p m n p ! ( n 2 p ) ! 1 C 2 m + n 2 j 2 p + 1 2 1 w 0 m 2 j ( 1 σ 0 g 2 w 0 2 C 1 2 2 C 1 ) n 2 p exp [ k 2 4 C 2 B 2 ( ρ 1 E 2 ρ 2 E 2 2 C 1 σ 0 g 2 ) 2 ] H m n ( i k ( ρ 2 E 2 ) B w 0 2 C 1 2 2 C 1 ) H m + n 2 j 2 p [ k ( ρ 2 E 2 ) 4 C 2 C 1 B σ 0 g 2 k ( ρ 1 E 2 ) 2 C 2 B ] .
W m ( ρ 1 , ρ 2 , z ) = π λ B 1 C 1 C 2 exp [ i k D ρ 1 2 2 B + i k D ρ 2 2 2 B i k G 2 B ρ 1 + i k G 2 B ρ 2 ] exp [ k 2 ( ρ 2 E 2 ) 2 4 C 1 B 2 ] exp [ k 2 4 C 2 B 2 ( ρ 1 E 2 ρ 2 E 2 2 C 1 σ 0 g 2 ) 2 ] .
I ( ρ 1 , z ) = W m ( ρ 1 , ρ 1 , z ) = π λ B 1 C 1 ( 1 2 w 0 2 C 1 ) m 2 exp [ k 2 ( ρ 1 E 2 ) 2 4 C 1 B 2 ] m ! j = 0 [ m 2 ] n = 0 m p = 0 [ n 2 ] ( m n ) ( 1 ) j + p n ! j ! ( m 2 j ) ! 1 2 j i 2 j + 2 p m n p ! ( n 2 p ) ! 1 C 2 m + n 2 j 2 p + 1 2 1 w 0 m 2 j ( 1 σ 0 g 2 w 0 2 C 1 2 2 C 1 ) n 2 p exp [ k 2 4 C 2 B 2 ( ρ 1 E 2 ρ 1 E 2 2 C 1 σ 0 g 2 ) 2 ] H m n ( i k ( ρ 1 E 2 ) B w 0 2 C 1 2 2 C 1 ) H m + n 2 j 2 p [ k ( ρ 1 E 2 ) 4 C 2 C 1 B σ 0 g 2 k ( ρ 1 E 2 ) 2 C 2 B ] .
W M ( x 1 , x 2 ; 0 ) = m = 0 M h = 0 M 1 h ! m ! ( x 1 2 w 0 M 2 ) m ( x 2 2 w 0 M 2 ) h exp [ x 1 2 + x 2 2 w 0 M 2 ( x 1 x 2 ) 2 2 σ 0 g 2 ] ,
x 2 m = ( 2 m ) ! 2 3 m p = 0 m 1 ( m p ) ! ( 2 p ) ! H 2 p ( 2 x ) ,
W M ( x 1 , x 2 ; 0 ) = m = 0 M h = 0 M l = 0 m f = 0 h 1 h ! m ! ( 2 m ) ! 2 3 m ( 2 h ) ! 2 3 h 1 ( h f ) ! ( 2 f ) ! 1 ( m l ) ! ( 2 l ) ! H 2 f ( 2 x 2 w 0 M ) H 2 l ( 2 x 1 w 0 M ) exp [ x 1 2 + x 2 2 w 0 M 2 ( x 1 x 2 ) 2 2 σ 0 g 2 ] .
W M ( ρ 1 , ρ 2 , z ) = π λ B exp [ i k D ρ 1 2 2 B + i k D ρ 2 2 2 B ] exp [ k 2 ρ 2 2 4 C 1 B 2 ] m = 0 M h = 0 M l = 0 m f = 0 h j = 0 [ f ] n = 0 2 h p = 0 [ n 2 ] ( 1 ) p + j ( 2 h n ) n ! ( 2 m ) ! ( 2 h ) ! p ! ( n 2 p ) ! i 2 j + 2 p 2 f n j ! ( 2 f 2 j ) ! 1 h ! m ! ( 2 f ) ! ( h f ) ! ( 2 f ) ! 1 ( m l ) ! ( 2 l ) ! 1 w 0 2 f 2 j 2 f j 3 m 4 h C 2 f j p + n 2 + 1 2 1 C 1 ( 1 2 w 0 2 C 1 ) h ( 1 σ 0 g 2 w 0 2 C 1 2 2 C 1 ) n 2 p exp [ k 2 4 C 2 B 2 ( ρ 1 ρ 2 2 C 1 σ 0 g 2 ) 2 ] H 2 h n ( i k ρ 2 B w 0 2 C 1 2 2 C 1 ) H 2 f + n 2 j 2 p ( k ρ 2 4 C 1 C 2 B σ 0 g 2 k ρ 1 2 C 2 B ) .
W M ( ρ 1 , ρ 2 , z ) = π λ B 1 C 1 exp [ i k D ρ 1 2 2 B + i k D ρ 2 2 2 B i k G 2 B ρ 1 + i k G 2 B ρ 2 ] exp [ k 2 ( ρ 2 E 2 ) 2 4 C 1 B 2 ] m = 0 M h = 0 M l = 0 m f = 0 h j = 0 [ f ] n = 0 2 h p = 0 [ n 2 ] ( 1 ) p + j ( 2 h n ) n ! ( 2 m ) ! ( 2 h ) ! p ! ( n 2 p ) ! i 2 j + 2 p 2 f n j ! ( 2 f 2 j ) ! 1 h ! m ! ( 2 f ) ! ( h f ) ! ( 2 f ) ! 1 ( m l ) ! ( 2 l ) ! 1 w 0 2 f 2 j 2 f j 3 m 4 h C 2 f j p + n 2 + 1 2 ( 1 2 w 0 2 C 1 ) h ( 1 σ 0 g 2 w 0 2 C 1 2 2 C 1 ) n 2 p exp [ k 2 4 C 2 B 2 ( ρ 1 E 2 ρ 2 E 2 2 C 1 σ 0 g 2 ) 2 ] H 2 h n ( i k ( ρ 2 E 2 ) B w 0 2 C 1 2 2 C 1 ) H 2 f + n 2 j 2 p ( k ( ρ 2 E 2 ) 4 C 1 C 2 B σ 0 g 2 k ( ρ 1 E 2 ) 2 C 2 B ) .
( A B C D ) = ( 1 z 0 1 ) ( 1 0 1 f 1 ) = ( 1 z f z 1 f 1 ) .

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