Abstract

An orthonormal family of super Lorentz–Gauss (SLG) modes is proposed to describe the highly divergent higher-order modes. The first-order and the second-order SLG modes SLG01 and SLG11 are illustrated as examples. Analytical propagation formulas of the SLG01 and SLG11 modes through a paraxial ABCD optical system are derived, and analytical beam propagation factors of the SLG01 and SLG11 modes are presented. The paraxial propagation properties of the SLG01 and SLG11 modes in free space are also compared with those of the corresponding Hermite–Gaussian (HG) HG01 and HG11 modes, respectively. This research indicates that SLG modes are more appropriate than HG modes to describe the highly divergent higher-order modes.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2009 (5)

2008 (4)

G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25, 2594-2599 (2008).
[CrossRef]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93, 891-899 (2008).
[CrossRef]

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55, 3571-3577 (2008).
[CrossRef]

2007 (2)

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
[CrossRef]

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32, 3179-3181 (2007).
[CrossRef] [PubMed]

2006 (1)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

2005 (3)

Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edge diffracted flattened Gaussian beams,” Appl. Opt. 44, 1381-1386 (2005).
[CrossRef] [PubMed]

J. Gu and D. Zhao, “Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture,” J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341, 352-356 (2005).
[CrossRef]

2004 (3)

2003 (1)

2000 (1)

B. Lü and H. Ma, “Beam propagation factor of decentred Gaussian and cosine-Gaussian beams,” J. Mod. Opt. 47, 719-723 (2000).

1995 (1)

1993 (1)

1990 (1)

1975 (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” J. Quantum Electron. 11, 400-402 (1975).
[CrossRef]

Alonso, M. A.

Arias, M.

Bandres, M. A.

Cai, Y.

Chávez-Cerda, S.

Deng, D.

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341, 352-356 (2005).
[CrossRef]

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” J. Quantum Electron. 11, 400-402 (1975).
[CrossRef]

Durst, F.

Erdelyi, A.

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

Gawhary, O. E.

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Gu, J.

J. Gu and D. Zhao, “Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture,” J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

Gutiérrez-Vega, J. C.

Lin, Q.

Lu, X.

Lü, B.

B. Lü and H. Ma, “Beam propagation factor of decentred Gaussian and cosine-Gaussian beams,” J. Mod. Opt. 47, 719-723 (2000).

Ma, H.

B. Lü and H. Ma, “Beam propagation factor of decentred Gaussian and cosine-Gaussian beams,” J. Mod. Opt. 47, 719-723 (2000).

Magnus, W.

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

Martínez-Herrero, R.

Mei, Z.

Mejías, P. M.

Moore, N. J.

Naqwi, A.

Nemes, G.

G. Nemes and J. Serna, “Laser beam characterization with use of second-order moments: an overview,” in DPSS Lasers: Applications and Issues, M.W.Dowley, ed., Vol.17 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1998), pp. 200-207.

Oberhettinger, F.

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

Rodríguez-Morales, G.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Schwarz, U. T.

Serna, J.

G. Nemes and J. Serna, “Laser beam characterization with use of second-order moments: an overview,” in DPSS Lasers: Applications and Issues, M.W.Dowley, ed., Vol.17 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1998), pp. 200-207.

Severini, S.

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

Xu, Y.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

Zhao, D.

J. Gu and D. Zhao, “Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture,” J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edge diffracted flattened Gaussian beams,” Appl. Opt. 44, 1381-1386 (2005).
[CrossRef] [PubMed]

Zheng, J.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

Zhou, G.

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41, 953-955 (2009).
[CrossRef]

G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26, 350-355 (2009).
[CrossRef]

G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26, 141-147 (2009).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96, 149-153 (2009).
[CrossRef]

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93, 891-899 (2008).
[CrossRef]

G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25, 2594-2599 (2008).
[CrossRef]

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55, 3571-3577 (2008).
[CrossRef]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (2)

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93, 891-899 (2008).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96, 149-153 (2009).
[CrossRef]

J. Mod. Opt. (4)

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55, 3571-3577 (2008).
[CrossRef]

J. Gu and D. Zhao, “Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture,” J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

B. Lü and H. Ma, “Beam propagation factor of decentred Gaussian and cosine-Gaussian beams,” J. Mod. Opt. 47, 719-723 (2000).

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

J. Quantum Electron. (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” J. Quantum Electron. 11, 400-402 (1975).
[CrossRef]

Opt. Commun. (1)

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
[CrossRef]

Opt. Laser Technol. (1)

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41, 953-955 (2009).
[CrossRef]

Opt. Lett. (7)

Phys. Lett. A (1)

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341, 352-356 (2005).
[CrossRef]

Other (3)

G. Nemes and J. Serna, “Laser beam characterization with use of second-order moments: an overview,” in DPSS Lasers: Applications and Issues, M.W.Dowley, ed., Vol.17 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1998), pp. 200-207.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

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

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Figures (5)

Fig. 1
Fig. 1

Contour graphs of normalized irradiance distributions of the SLG 01 and HG 01 modes in the source plane z = 0 . (a) SLG 01 mode with α 0 x = α 0 y = 10 λ . (b) SLG 01 mode with α 0 x = α 0 y = 30 λ . (c) SLG 01 mode with α 0 x = 10 λ and α 0 y = 30 λ . (d) HG 01 mode.

Fig. 2
Fig. 2

Contour graphs of normalized irradiance distributions of the SLG 11 and HG 11 modes in the source plane z = 0 . (a) SLG 11 mode with α 0 x = α 0 y = 10 λ . (b) SLG 11 mode with α 0 x = α 0 y = 30 λ . (c) SLG 11 mode with α 0 x = 10 λ and α 0 y = 30 λ . (d) HG 11 mode.

Fig. 3
Fig. 3

Contour graphs of normalized irradiance distributions of the SLG 01 and HG 01 modes in the plane z = 1000 λ . (a) SLG 01 mode with α 0 x = α 0 y = 10 λ . (b) SLG 01 mode with α 0 x = α 0 y = 30 λ . (c) SLG 01 mode with α 0 x = 10 λ and α 0 y = 30 λ . (d) HG 01 mode.

Fig. 4
Fig. 4

Contour graphs of normalized irradiance distributions of the SLG 11 and HG 11 modes in the plane z = 1000 λ . (a) SLG 11 mode with α 0 x = α 0 y = 10 λ . (b) SLG 11 mode with α 0 x = α 0 y = 30 λ . (c) SLG 11 mode with α 0 x = 10 λ and α 0 y = 30 λ . (d) HG 11 mode.

Fig. 5
Fig. 5

(a) M x 2 factor of a SLG 01 mode as a function of the parameter a. (b) M y 2 factor of a SLG 01 mode as a function of the parameter b.

Equations (53)

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E ( x 0 , y 0 , 0 ) = E m ( x 0 , 0 ) E n ( y 0 , 0 ) ,
E 2 p ( j 0 , 0 ) = exp ( i k j 0 2 2 q 0 ) ( j 0 2 p + l = 1 p 1 c ( 2 p ) ( 2 l ) j 0 2 l ) q = 1 N 1 j 0 2 + α ( q 1 ) j 2 ,
E 2 p + 1 ( j 0 , 0 ) = exp ( i k j 0 2 2 q 0 ) ( j 0 2 p + 1 + l = 1 p 1 c ( 2 p + 1 ) ( 2 l + 1 ) j 0 2 l + 1 ) q = 1 N 1 j 0 2 + α ( q 1 ) j 2 ,
c ( 2 p ) ( 2 l ) = R ( 2 p ) R ( 2 l ) , c ( 2 p + 1 ) ( 2 l + 1 ) = R ( 2 p + 1 ) R ( 2 l + 1 ) ,
R ( 2 p ) = 2 π ( 1 ) p + 1 s = 0 N 1 lim j 0 α s j j 0 ( j 0 2 p ( j 0 + α s j ) 2 q = 1 , q s + 1 N 1 j 0 2 + α ( q 1 ) j 2 ) ,
R ( 2 p + 1 ) = 2 π ( 1 ) p + 1 s = 0 N 1 lim j 0 α s j j 0 ( j 0 2 p + 1 ( j 0 + α s j ) 2 q = 1 , q s + 1 N j 0 j 0 2 + α ( q 1 ) j 2 ) .
E ( x , y , z ) , E ( x , y , z ) = E ( x 0 , y 0 , 0 ) , E ( x 0 , y 0 , 0 ) = E m ( x 0 , 0 ) E m ( x 0 , 0 ) d x 0 E n ( y 0 , 0 ) E n ( y 0 , 0 ) d y 0 = 0 ,
E 00 ( x 0 , y 0 , 0 ) = 2 π α 0 x α 0 y 1 [ 1 + ( x 0 α 0 x ) 2 ] [ 1 + ( y 0 α 0 y ) 2 ] exp [ i k ( x 0 2 + y 0 2 ) 2 q 0 ] ,
E 01 ( x 0 , y 0 , 0 ) = 2 π α 0 x α 0 y y 0 α 0 y [ 1 + ( x 0 α 0 x ) 2 ] [ 1 + ( y 0 α 0 y ) 2 ] exp [ i k ( x 0 2 + y 0 2 ) 2 q 0 ] ,
E 10 ( x 0 , y 0 , 0 ) = 2 π α 0 x α 0 y x 0 α 0 x [ 1 + ( x 0 α 0 x ) 2 ] [ 1 + ( y 0 α 0 y ) 2 ] exp [ i k ( x 0 2 + y 0 2 ) 2 q 0 ] ,
E 11 ( x 0 , y 0 , 0 ) = 2 π α 0 x α 0 y x 0 α 0 x 1 + ( x 0 α 0 x ) 2 y 0 α 0 y 1 + ( y 0 α 0 y ) 2 exp [ i k ( x 0 2 + y 0 2 ) 2 q 0 ] .
E ( x , y , z ) = 1 i λ B exp ( i k z ) E ( x 0 , y 0 , 0 ) exp { i k 2 B [ A ( x 0 2 + y 0 2 ) 2 ( x 0 x + y y 0 ) + D ( x 2 + y 2 ) ] } d x 0 d y 0 ,
E 01 ( x , y , z ) = 1 i λ B exp ( i k z ) T 1 ( x , z ) T 2 ( y , z ) ,
E 11 ( x , y , z ) = 1 i λ B exp ( i k z ) T 2 ( x , z ) T 2 ( y , z ) ,
T 1 ( x , z ) = α 0 x 2 α 0 x π exp ( i k C x 2 2 A ) 1 x 0 2 + α 0 x 2 exp [ i k A 2 B ( x 0 x A ) 2 ] d x 0 ,
T 2 ( j , z ) = 2 α 0 j π exp ( i k C j 2 2 A ) j 0 j 0 2 + α 0 j 2 exp [ i k A 2 B ( j 0 j A ) 2 ] d j 0 ,
T 1 ( x , z ) = α 0 x 2 α 0 x π exp ( i k C x 2 2 A ) [ f 1 ( x A ) f 2 ( x A ) ] ,
T 2 ( j , z ) = 2 α 0 j π exp ( i k C j 2 2 A ) [ f 3 ( j A ) f 2 ( j A ) ] ,
f 1 ( τ ) = 1 α 0 x 2 + τ 2 ,
f 2 ( τ ) = exp ( i k A 2 B τ 2 ) ,
f 3 ( τ ) = τ α 0 j 2 + τ 2 .
f 1 ( ξ ) ˜ = 1 2 π 1 α 0 x 2 + τ 2 exp ( i ξ τ ) d τ = π 2 1 α 0 x exp ( α 0 x | ξ | ) ,
f 2 ( ξ ) ˜ = 1 2 π exp ( i k A 2 B τ 2 ) exp ( i ξ τ ) d τ = i B k A exp ( B ξ 2 2 i k A ) ,
f 3 ( ξ ) ˜ = 1 2 π τ α 0 j 2 + τ 2 exp ( i ξ τ ) d τ = i π 2 | ξ | ξ exp ( α 0 j | ξ | ) .
f 1 ( τ ) f 2 ( τ ) = f 1 ( x ) ˜ f 2 ( ξ ) ˜ exp ( i ξ τ ) d ξ .
T 1 ( x , z ) = i α 0 x B k A exp ( i k C x 2 2 A ) { 0 exp [ B ξ 2 2 i k A ( α 0 x + i x A ) ξ ] d ξ + 0 exp [ B ξ 2 2 i k A ( α 0 x i x A ) ξ ] d ξ } ,
T 2 ( j , z ) = i i α 0 j B k A exp ( i k C j 2 2 A ) { 0 exp [ B ξ 2 2 i k A ( α 0 j + i j A ) ξ ] d ξ 0 exp [ B ξ 2 2 i k A ( α 0 j i j A ) ξ ] d ξ } .
T 1 ( x , z ) = π α 0 x 2 exp ( i k C x 2 2 A ) [ E x + ( x , z ) + E x ( x , z ) ] ,
T 2 ( j , z ) = i π α 0 j 2 exp ( i k C j 2 2 A ) [ E j + ( j , z ) E j ( j , z ) ] ,
E j ± ( j , z ) = exp [ k A 2 i B ( α 0 j ± i j A ) 2 ] erfc [ k A 2 i B ( α 0 j ± i j A ) ] ,
where erfc ( x ) = 2 π x exp ( s 2 ) d s
E 01 ( x , y , z ) = π α 0 x α 0 y 2 λ B exp ( i k z ) exp [ i k ( x 2 + y 2 ) 2 q ] [ E x + ( x , z ) + E x ( x , z ) ] [ E y + ( y , z ) E y ( y , z ) ] ,
E 11 ( x , y , z ) = i π α 0 x α 0 y 2 λ B exp ( i k z ) exp [ i k ( x 2 + y 2 ) 2 q ] [ E x + ( x , z ) E x ( x , z ) ] [ E y + ( y , z ) E y ( y , z ) ] ,
q = A C = A q 0 + B C q 0 + D .
E 01 ( x 0 , y 0 , 0 ) = E ( x 0 , 0 ) E ( y 0 , 0 ) ,
E ( x 0 , 0 ) = α 0 x 2 α 0 x π 1 α 0 x 2 + x 0 2 exp ( x 0 2 w 0 2 ) ,
E ( y 0 , 0 ) = 2 α 0 y π y 0 α 0 y 2 + y 0 2 exp ( y 0 2 w 0 2 ) .
x 0 = x 0 | E ( x 0 , 0 ) | 2 d x 0 | E ( x 0 , 0 ) | 2 d x 0 = exp ( 2 x 0 2 w 0 2 ) x 0 ( α 0 x 2 + x 0 2 ) 2 d x 0 exp ( 2 x 0 2 w 0 2 ) 1 ( α 0 x 2 + x 0 2 ) 2 d x 0 = 0 ,
y 0 = y 0 | E ( y 0 , 0 ) | 2 d y 0 | E ( y 0 , 0 ) | 2 d y 0 = exp ( 2 y 0 2 w 0 2 ) y 0 3 ( α 0 y 2 + y 0 2 ) 2 d y 0 exp ( 2 y 0 2 w 0 2 ) y 0 2 ( α 0 y 2 + y 0 2 ) 2 d y 0 = 0 .
x 0 2 = ( x 0 x 0 ) 2 | E ( x 0 , 0 ) | 2 d x 0 | E ( x 0 , 0 ) | 2 d x 0 = w 0 x 2 exp ( a τ 2 ) τ 2 ( 1 + τ 2 ) 2 d τ exp ( a τ 2 ) 1 ( 1 + τ 2 ) 2 d τ = w 0 x 2 π ( 1 + 2 a ) exp ( a ) erfc ( a ) 2 a π ( 1 2 a ) exp ( a ) erfc ( a ) + 2 a ,
y 0 2 = ( y 0 y 0 ) 2 | E ( y 0 , 0 ) | 2 d y 0 | E ( y 0 , 0 ) | 2 d y 0 = w 0 y 2 exp ( b τ 2 ) τ 4 ( 1 + τ 2 ) 2 d τ exp ( b τ 2 ) τ 2 ( 1 + τ 2 ) 2 d τ = w 0 y 2 2 ( b + 1 ) π b ( 3 + 2 b ) exp ( b ) erfc ( b ) π b ( 1 + 2 b ) exp ( b ) erfc ( b ) 2 b ,
θ x 2 = 1 k 2 | E ( x 0 , 0 ) | 2 d x 0 | E ( x 0 , 0 ) x 0 | 2 d x 0 = 1 k 2 w 0 x 2 exp ( a τ 2 ) a 2 τ 6 + ( 2 a 2 + 4 a ) τ 4 + ( a + 2 ) 2 τ 2 ( 1 + τ 2 ) 4 d τ exp ( a τ 2 ) 1 ( 1 + τ 2 ) 2 d τ ,
θ y 2 = 1 k 2 | E ( y 0 , 0 ) | 2 d y 0 | E ( y 0 , 0 ) y 0 | 2 d y 0 = 1 k 2 w 0 y 2 exp ( b τ 2 ) b 2 τ 8 + 2 ( b 2 + b ) τ 6 + ( b 2 + 1 ) τ 2 2 ( b + 1 ) τ 2 + 1 ( 1 + τ 2 ) 4 d τ exp ( b τ 2 ) τ 2 ( 1 + τ 2 ) 2 d τ .
θ x 2 = 1 6 k 2 w 0 x 2 π ( 3 6 a 2 4 a 3 ) exp ( a ) erfc ( a ) + ( 6 + 4 a + 4 a 2 ) a π ( 1 2 a ) exp ( a ) erfc ( a ) + 2 a ,
θ y 2 = 1 6 k 2 w 0 y 2 π ( 3 + 18 b 2 + 4 b 3 ) exp ( b ) erfc ( b ) + ( 6 16 b 4 b 2 ) b π ( 1 + 2 b ) exp ( b ) erfc ( b ) 2 b .
x 0 θ x = π i k { x 0 [ E ( x 0 , 0 ) x 0 ] * E ( x 0 , 0 ) x 0 E ( x 0 , 0 ) x 0 [ E ( x 0 , 0 ) ] * } d x 0 | E ( x 0 , 0 ) | 2 d x 0 = 0 ,
y 0 θ y = π i k { y 0 [ E ( y 0 , 0 ) y 0 ] * E ( y 0 , 0 ) y 0 E ( y 0 , 0 ) y 0 [ E ( y 0 , 0 ) ] * } d y 0 | E ( y 0 , 0 ) | 2 d y 0 = 0 ,
M x 2 = 2 k ( x 0 2 θ x 2 x 0 θ x 2 ) 1 2 = 2 3 [ π ( 1 + 2 a ) exp ( a ) erfc ( a ) 2 a ] 1 2 π ( 1 2 a ) exp ( a ) erfc ( a ) + 2 a × [ π ( 3 6 a 2 4 a 3 ) exp ( a ) erfc ( a ) + ( 6 + 4 a + 4 a 2 ) a ] 1 2 ,
M y 2 = 2 k ( y 0 2 θ y 2 y 0 θ y 2 ) 1 2 = 2 3 [ 2 ( b + 1 b ) π ( 3 + 2 b ) exp ( b ) erfc ( b ) ] 1 2 π ( 1 + 2 b ) exp ( b ) erfc ( b ) 2 b × [ π ( 3 + 18 b 2 + 4 b 3 ) exp ( b ) erfc ( b ) + ( 6 16 b 4 b 2 ) b ] 1 2 .
M 2 = ( M x 2 M y 2 ) 1 2 .
M x 2 = 2 3 [ π ( 3 + 18 a 2 + 4 a 3 ) exp ( a ) erfc ( a ) + ( 6 16 a 4 a 2 ) a ] 1 2 [ 2 ( a + 1 a ) π ( 3 + 2 a ) exp ( a ) erfc ( a ) ] 1 2 π ( 1 + 2 a ) exp ( a ) erfc ( a ) 2 a ,
M x 2 = 2 3 [ π ( 3 + 18 b 2 + 4 b 3 ) exp ( b ) erfc ( b ) + ( 6 16 b 4 b 2 ) b ] 1 2 [ 2 ( b + 1 b ) π ( 3 + 2 b ) exp ( b ) erfc ( b ) ] 1 2 π ( 1 + 2 b ) exp ( b ) erfc ( b ) 2 b .
erfc ( b ) = 1 π b exp ( b ) ( 1 + s = 1 ( 1 ) s ( 2 s 1 ) !! 2 s b s ) , b .

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