Abstract

Paraxial propagation of a general-type beam through a truncated fractional Fourier transform (FRT) optical system is investigated. Analytical formulas for the electric field and effective beam width of a general-type beam in the FRT plane are derived based on the Collins formula. Our formulas can be used to study the propagation of a variety of laser beams—such as Gaussian, cos-Gaussian, cosh-Gaussian, sine-Gaussian, sinh-Gaussian, flat-topped, Hermite-cosh-Gaussian, Hermite-sine-Gaussian, higher-order annular Gaussian, Hermite-sinh-Gaussian and Hermite-cos-Gaussian beams—through a FRT optical system with or without truncation. The propagation properties of a Hermite-cos-Gaussian beam passing through a rectangularly truncated FRT optical system are studied as a numerical example. Our results clearly show that the truncated FRT optical system provides a convenient way for laser beam shaping.

© 2010 Optical Society of America

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2010 (1)

F. Wang, Y. Cai, and Y. Ma, “Sixth-order coincidence fractional Fourier transform implemented with partially coherent light,” Appl. Phys. B 98, 187-193 (2010).
[CrossRef]

2009 (3)

G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56, 886-892 (2009).
[CrossRef]

B. Tang and M. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276-1282 (2009).
[CrossRef]

G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express 17, 10529-10534 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (7)

2006 (3)

2005 (4)

2004 (3)

2003 (5)

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation properties,” Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

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

2002 (2)

1999 (1)

D. Ding and X. Liu, “Approximate description of Bessel, Bessel-Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1296-1293 (1999).
[CrossRef]

1994 (1)

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

1993 (3)

1988 (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

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).

Arpali, Ç.

Arpali, S. A.

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227-239 (2008).
[CrossRef]

Ç. Arpali, C. Yazıcıoğlu, H. T. Eyyuboğlu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14, 8918-8928 (2006).
[CrossRef] [PubMed]

Baykal, Y.

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Cai, Y.

F. Wang, Y. Cai, and Y. Ma, “Sixth-order coincidence fractional Fourier transform implemented with partially coherent light,” Appl. Phys. B 98, 187-193 (2010).
[CrossRef]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25, 2001-2010 (2008).
[CrossRef]

Y. Cai and F. Wang, “Lensless optical implementation of the coincidence fractional Fourier transform,” Opt. Lett. 31, 2278-2280 (2007).
[CrossRef]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937-1944 (2007).
[CrossRef]

F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14, 6999-7004 (2006).
[CrossRef] [PubMed]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798-1804 (2005).
[CrossRef]

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

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

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation properties,” Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian Schell-model beams,” Opt. Lett. 27, 1672-1674 (2002).
[CrossRef]

Carter, W. H.

Chu, X.

Collins, S. A.

Deng, D.

Ding, D.

D. Ding and X. Liu, “Approximate description of Bessel, Bessel-Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1296-1293 (1999).
[CrossRef]

Erdelyi, A.

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

Eyyuboglu, H. T.

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227-239 (2008).
[CrossRef]

Y. Baykal and H. T. Eyyuboğlu, “Intensity fluctuations of focused general-type beams in atmospheric optics links,” Proc. SPIE 6603, 60320-1-60320-8 (2007).

H. T. Eyyuboğlu and Y. Baykal, “Generalized beams in ABCD GH systems,” Opt. Commun. 272, 22-31 (2007).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Angle-of-arrival fluctuations for general-type beams,” Opt. Eng. (Bellingham) 46, 096001 (2007).
[CrossRef]

Ç. Arpali, C. Yazıcıoğlu, H. T. Eyyuboğlu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14, 8918-8928 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709-2718 (2005).
[CrossRef]

H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 1527-1535 (2005).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

Ge, D.

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Gori, F.

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

Guo, Q.

He, S.

Ji, G.

Ji, X.

Jing, F.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Li, X.

Li, Y.

Lin, Q.

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25, 2001-2010 (2008).
[CrossRef]

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

Y. Cai and Q. Lin, “Properties of 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, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beam and its propagation properties,” Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian Schell-model beams,” Opt. Lett. 27, 1672-1674 (2002).
[CrossRef]

Liu, H.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

Liu, X.

D. Ding and X. Liu, “Approximate description of Bessel, Bessel-Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1296-1293 (1999).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181-2186 (1993).
[CrossRef]

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

Lu, X.

Ma, Y.

F. Wang, Y. Cai, and Y. Ma, “Sixth-order coincidence fractional Fourier transform implemented with partially coherent light,” Appl. Phys. B 98, 187-193 (2010).
[CrossRef]

Magnus, W.

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

Mao, H.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

Medlovic, D.

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

Mei, Z.

Mendlovic, D.

Oberhettinger, F.

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

Ozaktas, H. M.

Shen, M.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

Stegun, I.

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

Tang, B.

B. Tang and M. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276-1282 (2009).
[CrossRef]

Wang, F.

Wei, X.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Xu, M.

B. Tang and M. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276-1282 (2009).
[CrossRef]

Yazicioglu, C.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

Zhao, D.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21, 2375-2381 (2004).
[CrossRef]

Zhou, G.

Zhu, Q.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

Zhu, S.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798-1804 (2005).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

F. Wang, Y. Cai, and Y. Ma, “Sixth-order coincidence fractional Fourier transform implemented with partially coherent light,” Appl. Phys. B 98, 187-193 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

J. Mod. Opt. (3)

G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56, 886-892 (2009).
[CrossRef]

B. Tang and M. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276-1282 (2009).
[CrossRef]

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227-239 (2008).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (3)

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

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (13)

D. Ding and X. Liu, “Approximate description of Bessel, Bessel-Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1296-1293 (1999).
[CrossRef]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21, 2375-2381 (2004).
[CrossRef]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937-1944 (2007).
[CrossRef]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25, 2001-2010 (2008).
[CrossRef]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798-1804 (2005).
[CrossRef]

H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 1527-1535 (2005).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709-2718 (2005).
[CrossRef]

Y. Baykal, “Formulation of correlations for general-type beams in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 889-893 (2006).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Optical system for performing the truncated FRT of a light beam.

Fig. 2
Fig. 2

Normalized intensity distribution and cross line ( y = 0 ) of a Hermite-cos-Gaussian beam in the FRT plane for different values of the fractional order p. The solid curves are calculated by using the analytical formula Eq. (13) and the dotted curves are calculated by direct integration of Eq. (3) numerically.

Fig. 3
Fig. 3

Normalized intensity distribution and cross line ( y = 0 ) of a Hermite-cos-Gaussian beam in the FRT plane for different values of aperture widths a, b with a = b .

Fig. 4
Fig. 4

Effective beam width W x of a Hermite-cos-Gaussian beam in the FRT plane versus the fractional order p.

Fig. 5
Fig. 5

Effective beam width W x of a Hermite-cos-Gaussian beam in the FRT plane versus the aperture widths a, b with a = b .

Tables (1)

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Table 1 List of Specific Beam Parameters for Characterizing Various Laser Beams

Equations (32)

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E ( x 1 , y 1 ) = i λ B 1 exp [ i k f tan ( ϕ 2 ) ] E ( x , y ) exp { i k 2 B 1 [ A 1 ( x 2 + y 2 ) 2 ( x 1 x + y 1 y ) + D 1 ( x 1 2 + y 1 2 ) ] } d x d y ,
A 1 = 1 , B 1 = f tan ( ϕ 2 ) , C 1 = 0 , D 1 = 1 .
E ( x 2 , y 2 ) = i λ B 2 exp [ i k f tan ( ϕ 2 ) ] H ( x 1 , y 1 ) E ( x 1 , y 1 , 0 ) exp { i k 2 B 2 [ A 2 ( x 1 2 + y 1 2 ) 2 ( x 2 x 1 + y 2 y 1 ) + D 2 ( x 2 2 + y 2 2 ) ] } d x 1 d y 1 ,
A 2 = cos ϕ , B 2 = f tan ( ϕ 2 ) , C 2 = sin ϕ f , D 2 = 1 .
E ( x , y , 0 ) = l = 1 N A l exp ( i θ l ) H n l ( a x l x + b x l ) H m l ( a y l y + b y l ) × exp [ 0.5 k ( α x l x 2 + α y l y 2 ) ] exp [ i ( V x l x + V y l y ) ] ,
E ( x 1 , y 1 ) = i λ B 1 exp [ i k f tan ( ϕ 2 ) ] l = 1 N X l exp ( d x 2 4 c x ) exp ( d y 2 4 c y ) exp [ i k D 1 2 B 1 ( x 1 2 + y 1 2 ) ] H n l [ b x l + a x l d x ( 2 c x ) [ 1 ( a x l c x ) 2 ] 1 2 ] H m l [ b y l + a y l d y ( 2 c y ) [ 1 ( a y l c y ) 2 ] 1 2 ] ,
H n ( x ) = n ! m = 0 [ n 2 ] ( 1 ) m 1 m ! ( n 2 m ) ! ( 2 x ) n 2 m ,
H n ( x + y ) = 1 2 n 2 k = 0 n ( n k ) H k ( 2 x ) H n k ( 2 y ) ,
x n exp [ ( x β ) 2 ] d x = ( 2 i ) n π H n ( i β ) .
H ( x 1 , y 1 ) = { 1 , | x 1 2 + y 1 2 | a 2 0 , | x 1 2 + y 1 2 | > 0 } .
H ( x 1 , y 1 , z ) = t = 1 T α t exp [ β t ( x 1 2 + y 1 2 ) a 2 ] ,
E ( x 2 , y 2 ) = 1 λ 2 B 1 B 2 t = 1 T α t exp [ 2 i k f tan ( ϕ 2 ) ] l = 1 N X l exp [ i k D 2 2 B 2 ( x 2 2 + y 2 2 ) ] π e x e y exp [ ( V x l 2 4 c x + V y l 2 4 c y ) ] exp [ f x 2 4 e x + f y 2 4 e y ] [ 1 ( Ω x e x ) 2 ] n l 2 [ 1 ( Ω y e y ) 2 ] m l 2 H n l [ T x + Ω x f x ( 2 e x ) [ 1 ( Ω x e x ) 2 ] 1 2 ] H m l [ T y + Ω y f y ( 2 e y ) [ 1 ( Ω y e y ) 2 ] 1 2 ] ,
W s = 2 s 2 | E ( x 2 , y 2 , z 2 ) | 2 d x 2 d y 2 | E ( x 2 , y 2 , z 2 ) | 2 d x 2 d y 2 ( s = x 2 , y 2 ) .
W x = 2 S 1 S 2
S 1 = Ψ S x S y ,
S x = Υ 4 ( 1 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) 2 H g + q 2 h 2 w + 2 ( i ( i V x l k 2 4 e x c x B 1 B 2 i V x l * k 2 4 e x * c x * B 1 * B 2 * ) 2 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) ,
S y = g = 0 m l h = 0 g 2 q = 0 m l π 2 m l w = 0 q 2 g ! ( 1 ) h h ! ( g 2 h ) ! q ! ( 1 ) w w ! ( q 2 w ) ! ( m l g ) ( m l q ) { i 2 2 Ω y k e y B 2 [ 1 ( Ω y e y ) 2 ] 1 2 } g 2 h { i 2 2 Ω y * k 2 e y * B 2 * [ 1 ( Ω y * e y * ) 2 ] 1 2 } q 2 w ( 2 i ) ( g + q 2 h 2 w ) ( 1 k 2 4 e y B 2 2 + k 2 4 e y * B 2 * 2 ) g + q 2 h 2 w + 1 H m l g [ 2 ( T y + Ω y V y l k 4 e y B 1 c y ) [ 1 ( Ω y e y ) 2 ] 1 2 ] H m l q [ 2 ( T y * + Ω y * V y l * k 4 e y * B 1 * c y * ) [ 1 ( Ω y * e y * ) 2 ] 1 2 ] exp [ V y l 2 k 2 16 e y B 1 2 c y 2 + V y l * 2 k 2 16 e y * B 1 * 2 c y * 2 + ( ( i V y l k 2 4 e y c y B 1 B 2 i V y l * k 2 4 e y * c y * B 1 * B 2 * ) 2 k 2 4 e y B 2 2 + k 2 4 e y * B 2 * 2 ) 2 ] H g + q 2 h 2 w ( i ( i V y l k 2 4 e y c y B 1 B 2 i V y l * k 2 4 e y * c y * B 1 * B 2 * ) 2 k 2 4 e y B 2 2 + k 2 4 e y * B 2 * 2 ) ,
S 2 = Ψ S x 1 S y ,
S x 1 = Υ H g + q 2 h 2 w ( i ( i V x l k 2 4 e x c x B 1 B 2 i V x l * k 2 4 e x * c x * B 1 * B 2 * ) 2 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) ,
Ψ = ( 1 λ 2 B 1 B 2 ) ( 1 λ 2 B 1 * B 2 * ) t 1 = 1 T α t 1 t 2 = 1 T α t 2 * l 1 = 1 N A l 1 l 2 = 1 N A l 2 * π 2 c x c y e x e y π 2 c x * c y * e x * e y * [ 1 ( a x l c x ) 2 ] n l 2 [ 1 ( a x l * c x * ) 2 ] n l 2 [ 1 ( a y l c y ) 2 ] m l 2 [ 1 ( a y l * c y * ) 2 ] m l 2 exp [ ( V x l 2 4 c x + V y l 2 4 c y ) ] exp [ ( V * x l 2 4 c x * + V * y l 2 4 c y * ) ] [ 1 ( Ω x e x ) 2 ] n l 2 [ 1 ( Ω x * e x * ) 2 ] n l 2 [ 1 ( Ω y e y ) 2 ] m l 2 [ 1 ( Ω y * e y * ) 2 ] m l 2 ,
Υ = g = 0 n l h = 0 g 2 q = 0 n l w = 0 q 2 π 2 n l g ! ( 1 ) h h ! ( g 2 h ) ! q ! ( 1 ) w w ! ( q 2 w ) ! ( n l g ) ( n l q ) { i 2 2 Ω x k e x B 2 [ 1 ( Ω x e x ) 2 ] 1 2 } g 2 h ( 2 i ) ( g + q 2 h 2 w ) × { i 2 2 Ω x * k 2 e x * B 2 * [ 1 ( Ω x * e x * ) 2 ] 1 2 } q 2 w H n l g [ 2 ( T x + Ω x V x l k 4 e x B 1 c x ) [ 1 ( Ω x e x ) 2 ] 1 2 ] H n l q [ 2 ( T x * + Ω x * V x l * k 4 e x * B 1 * c x * ) [ 1 ( Ω x * e x * ) 2 ] 1 2 ] × ( 1 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) g + q 2 h 2 w + 1 exp [ V x l 2 k 2 16 e x B 1 2 c x 2 + V x l * 2 k 2 16 e x * B 1 * 2 c x * 2 + ( ( i V x l k 2 4 e x c x B 1 B 2 i V x l * k 2 4 e x * c x * B 1 * B 2 * ) 2 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) 2 ] ,
H ( r 1 ) = { 1 , | x 1 | a , | y 1 | b 0 , | x 1 | > a , | y 1 | > b } ,
H ( x 1 , y 1 ) = t = 1 T α t exp ( β t x 1 2 a 2 ) u = 1 U α u exp ( β u y 1 2 b 2 ) ,
E ( x 2 , y 2 ) = 1 λ 2 B 1 B 2 exp [ i k ( z 1 + z 2 ) ] exp ( i θ l ) π c x c y π e x e y t = 1 T α t u = 1 U α u l = 1 N A l [ 1 ( a x l c x ) 2 ] n l 2 [ 1 ( a y l c y ) 2 ] m l 2 [ 1 ( Ω x e x ) 2 ] n l 2 [ 1 ( Ω y e y ) 2 ] m l 2 exp [ i k D 2 2 B 2 ( x 2 2 + y 2 2 ) ] exp [ ( V x l 2 4 c x + V y l 2 4 c y ) + f x 2 4 e x + f y 2 4 e y ] H n l [ T x + Ω x f x 2 e x [ 1 ( Ω x e x ) 2 ] 1 2 ] H m l [ T y + Ω y f y 2 e y [ 1 ( Ω y e y ) 2 ] 1 2 ] ,
W x = 2 P 1 P 2 ,
P 1 = ζ P x P y ,
P x = δ 4 ( 1 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) 2 H g + q 2 h 2 w + 2 ( i ( i V x l k 2 4 e x c x B 1 B 2 i V x l * k 2 4 e x * c x * B 1 * B 2 * ) 2 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) ,
P y = g = 0 m l h = 0 g 2 q = 0 m l w = 0 q 2 π 2 m l g ! ( 1 ) h h ! ( g 2 h ) ! q ! ( 1 ) w w ! ( q 2 w ) ! ( m l g ) ( m l q ) { i 2 2 Ω y k e y B 2 [ 1 ( Ω y e y ) 2 ] 1 2 } g 2 h { i 2 2 Ω y * k 2 e y * B 2 * [ 1 ( Ω y * e y * ) 2 ] 1 2 } q 2 w × H m l g [ 2 ( T y + Ω y V y l k 4 e y B 1 c y ) [ 1 ( Ω y e y ) 2 ] 1 2 ] H m l q [ 2 ( T y * + Ω y * V y l * k 4 e y * B 1 * c y * ) [ 1 ( Ω y * e y * ) 2 ] 1 2 ] ( 1 k 2 4 e y B 2 2 + k 2 4 e y * B 2 * 2 ) g + q 2 h 2 w + 1 × ( 2 i ) ( g + q 2 h 2 w ) exp [ V y l 2 k 2 16 e y B 1 2 c y 2 + V y l * 2 k 2 16 e y * B 1 * 2 c y * 2 + ( ( i V y l k 2 4 e y c y B 1 B 2 i V y l * k 2 4 e y * c y * B 1 * B 2 * ) 2 k 2 4 e y B 2 2 + k 2 4 e y * B 2 * 2 ) 2 ] × H g + q 2 h 2 w ( i ( i V y l k 2 4 e y c y B 1 B 2 i V y l * k 2 4 e y * c y * B 1 * B 2 * ) 2 k 2 4 e y B 2 2 + k 2 4 e y * B 2 * 2 ) ,
P 2 = ζ P x 1 P y ,
P x 1 = δ H g + q 2 h 2 w ( i ( i V x l k 2 4 e x c x B 1 B 2 i V x l * k 2 4 e x * c x * B 1 * B 2 * ) 2 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) ,
ζ = ( 1 λ 2 B 1 B 2 ) ( 1 λ 2 B 1 * B 2 * ) t 1 = 1 T α t 1 t 2 = 1 T α t 2 * u 1 = 1 U α u 1 u 2 = 1 U α u 2 * l 1 = 1 N A l 1 l 2 = 1 N A l 2 * π 2 c x c y e x e y π 2 c x * c y * e x * e y * [ 1 ( a x l c x ) 2 ] n l 2 [ 1 ( a x l * c x * ) 2 ] n l 2 [ 1 ( a y l c y ) 2 ] m l 2 [ 1 ( a y l * c y * ) 2 ] m l 2 exp [ ( V x l 2 4 c x + V y l 2 4 c y ) ] exp [ ( V * x l 2 4 c x * + V * y l 2 4 c y * ) ] [ 1 ( Ω x e x ) 2 ] n l 2 [ 1 ( Ω x * e x * ) 2 ] n l 2 [ 1 ( Ω y e y ) 2 ] m l 2 [ 1 ( Ω y * e y * ) 2 ] m l 2 ,
δ = 1 2 n l g = 0 n l h = 0 g 2 q = 0 n l w = 0 q 2 g ! ( 1 ) h h ! ( g 2 h ) ! q ! ( 1 ) w w ! ( q 2 w ) ! ( n l g ) ( n l q ) { i 2 2 Ω x k e x B 2 [ 1 ( Ω x e x ) 2 ] 1 2 } g 2 h { i 2 2 Ω x * k 2 e x * B 2 * [ 1 ( Ω x * e x * ) 2 ] 1 2 } q 2 w × H n l g [ 2 ( T x + Ω x V x l k 4 e x B 1 c x ) [ 1 ( Ω x e x ) 2 ] 1 2 ] H n l q [ 2 ( T x * + Ω x * V x l * k 4 e x * B 1 * c x * ) [ 1 ( Ω x * e x * ) 2 ] 1 2 ] ( 1 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) g + q 2 h 2 w + 1 ( 2 i ) ( g + q 2 h 2 w ) π exp [ V x l 2 k 2 16 e x B 1 2 c x 2 + V x l * 2 k 2 16 e x * B 1 * 2 c x * 2 + ( ( i V x l k 2 4 e x c x B 1 B 2 i V x l * k 2 4 e x * c x * B 1 * B 2 * ) 2 k 2 4 e x B 2 2 + k 2 4 e x * B 2 * 2 ) 2 ] .

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