Abstract

Analytical propagation expression of an Airy beam in uniaxial crystals orthogonal to the optical axis is derived. The ballistic dynamics of an Airy beam in uniaxial crystals is also investigated. The Airy beam propagating in uniaxial crystals orthogonal to the optical axis mainly depends on the ratio of the extraordinary refractive index to the ordinary refractive index. As an example, the propagation of an Airy beam in the positive uniaxial crystals orthogonal to the optical axis is demonstrated. The acceleration of an Airy beam in the transversal direction along the optical axis is more rapidly than that in the other transversal direction. With increasing the ratio of the extraordinary refractive index to the ordinary refractive index, the acceleration of the Airy beam in the transversal direction along the optical axis speeds up and the acceleration of the Airy beam in the other transversal direction slows down. The Airy beam propagating in uniaxial crystals orthogonal to the optical axis follows a ballistic trajectory. The effective beam size of the Airy beam in the transversal direction along the optical axis is always larger than that in the other transversal direction.

© 2012 OSA

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References

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  1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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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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2011 (3)

2010 (5)

Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010).
[CrossRef] [PubMed]

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82(4), 043832 (2010).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57(5), 375–384 (2010).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

2009 (4)

D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 11(6), 065710 (2009).
[CrossRef]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[CrossRef] [PubMed]

B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(12), 2480–2487 (2009).
[CrossRef] [PubMed]

2008 (5)

2007 (3)

2004 (1)

B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

2003 (1)

2002 (2)

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[CrossRef]

Alfano, R. R.

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[CrossRef]

Broky, J.

Cai, Y.

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57(5), 375–384 (2010).
[CrossRef]

Chen, C. G.

Chen, R.

R. Chen and C. Ying, “Beam propagation factor of an Airy beam,” J. Opt. 13(8), 085704 (2011).
[CrossRef]

R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82(4), 043832 (2010).
[CrossRef]

Chen, R. P.

Chen, Y.

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

Christodoulides, D. N.

Chu, X.

X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36(14), 2701–2703 (2011).
[CrossRef] [PubMed]

R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82(4), 043832 (2010).
[CrossRef]

Ciattoni, A.

Cincotti, G.

Dai, C. Q.

Deng, D.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[CrossRef]

Dogariu, A.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Fan, Z.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Ferrera, J.

Fleischer, J. W.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Gutiérrez-Vega, J. C.

Heilmann, R. K.

Heyman, E.

Jia, S.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

Kaganovsky, Y.

Kolesik, M.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[CrossRef] [PubMed]

Konkola, P. T.

Lee, J.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

Li, J.

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

Liu, D.

D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 11(6), 065710 (2009).
[CrossRef]

Lü, B.

B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

Luo, S.

B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[CrossRef]

Moloney, J.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[CrossRef] [PubMed]

Palma, C.

Polynkin, P.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[CrossRef] [PubMed]

Schattenburg, M. L.

Shao, J.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Siviloglou, G. A.

Sztul, H. I.

Tang, B.

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Wang, H.

R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82(4), 043832 (2010).
[CrossRef]

Xin, Y.

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

Xu, S.

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Yin, C.

R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82(4), 043832 (2010).
[CrossRef]

Ying, C.

R. Chen and C. Ying, “Beam propagation factor of an Airy beam,” J. Opt. 13(8), 085704 (2011).
[CrossRef]

Yu, H.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Zhao, C.

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57(5), 375–384 (2010).
[CrossRef]

Zheng, H. P.

Zhou, Z.

D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 11(6), 065710 (2009).
[CrossRef]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[CrossRef]

Eur. Phys. J. D (1)

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

J. Mod. Opt. (1)

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57(5), 375–384 (2010).
[CrossRef]

J. Opt. (1)

R. Chen and C. Ying, “Beam propagation factor of an Airy beam,” J. Opt. 13(8), 085704 (2011).
[CrossRef]

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

D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 11(6), 065710 (2009).
[CrossRef]

J. Opt. Soc. Am. A (5)

Nat. Photonics (2)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[CrossRef]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Opt. Commun. (1)

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Opt. Express (4)

Opt. Laser Technol. (1)

B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82(4), 043832 (2010).
[CrossRef]

Phys. Rev. Lett. (3)

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[CrossRef] [PubMed]

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[CrossRef] [PubMed]

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, Oxford, 1999).

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, New York, 1980).

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer Press, New York, 2005).

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Figures (5)

Fig. 1
Fig. 1

Contour graph of the normalized intensity distribution of an Airy beam propagating in the uniaxial crystals at several observation planes. The top and bottom rows denote e = 1.1 and e = 1.5, respectively. (a) and (c) z = 0.1z0. (b) and (d) z = 5z0.

Fig. 2
Fig. 2

The normalized intensity distribution in the x-direction of an Airy beam propagating in the uniaxial crystals.

Fig. 3
Fig. 3

The normalized intensity distribution in the y-direction of an Airy beam propagating in the uniaxial crystals.

Fig. 4
Fig. 4

Contour graph of the normalized intensity distribution of an Airy beam propagating in the uniaxial crystals at different observation sections. The top and bottom rows denote e = 1.1 and e = 1.5, respectively. (a) and (c) x-z plane. (b) and (d) y-z plane.

Fig. 5
Fig. 5

The effective beam sizes of an Airy beam propagating in the uniaxial crystals versus the propagation distance z.

Equations (36)

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ε=( n e 2 0 0 0 n o 2 0 0 0 n o 2 ),
[ E x ( x 0 , y 0 ,0) E y ( x 0 , y 0 ,0) ]=[ Ai( x 0 w 0 )Ai( y 0 w 0 )exp( a x 0 w 0 + a y 0 w 0 ) 0 ],
E x (x,y,z)=exp(ik n e z) E ˜ x ( k x , k y )exp[ i( k x x+ k y y)i n e 2 k x 2 + n o 2 k y 2 2k n o 2 n e z ]d k x d k y ,
E y (x,y,z)=exp(ik n o z) E ˜ y ( k x , k y )exp[ i( k x x+ k y y)i k x 2 + k y 2 2k n o z ]d k x d k y ,
E ˜ j ( k x , k y )= 1 (2π) 2 E j ( x 0 , y 0 ,0)exp[ i( k x x+ k y y) ]d k x d k y ,
E x (x,y,z)= k n o 2πiz exp(ik n e z) E x ( x 0 , y 0 ,0)exp{ ik 2z n e [ n o 2 (x x 0 ) 2 + n e 2 (y y 0 ) 2 ] }d x 0 d y 0 ,
E y (x,y,z)= k n o 2πiz exp(ik n o z) E y ( x 0 , y 0 ,0)exp{ ik n o 2z [ (x x 0 ) 2 + (y y 0 ) 2 ] }d x 0 d y 0 ,
E x (x,y,z)= k n o 2πiz exp(ik n e z)U(x,z)U(y,z),
U(x,z)= Ai( x 0 w 0 )exp( a x 0 w 0 )exp[ k n o 2 2iz n e (x x 0 ) 2 ]d x 0 ,
U(y,z)= Ai( y 0 w 0 )exp( a y 0 w 0 )exp[ k n e 2iz (y y 0 ) 2 ]d y 0 .
U(x,z)=exp( ax w 0 +i a 2 n e z 2 n o 2 z 0 ) Ai( x 0 w 0 )exp[ k n o 2 2iz n e ( x+i a w 0 n e z n o 2 z 0 x 0 ) 2 ]d x 0 ,
f 1 (τ) f 2 (τ)= f 1 ( x 0 ) f 2 (τ x 0 ) d x 0 ,
U(x,z)=exp( ax w 0 +i a 2 n e z 2 n o 2 z 0 )[ f 1 ( x+ ia w 0 n e z n o 2 z 0 ) f 2 ( x+ ia w 0 n e z n o 2 z 0 ) ],
f 1 (τ)=Ai( τ w 0 ),
f 2 (τ)=exp( k n o 2 2iz n e τ 2 ).
f 1 (τ) f 2 (τ)= f 1 (ξ) f 2 (ξ)exp(iξτ) dξ,
f 1 (ξ)= 1 2π Ai( τ w 0 )exp(iξτ) dτ= w 0 2π exp( i w 0 3 3 ξ 3 ),
f 2 (ξ)= 1 2π exp( k n o 2 2iz n e τ 2 )exp(iξτ) dτ= iz n e k n o 2 exp( iz n e 2k n o 2 ξ 2 ).
U(x,z)= w 0 iez 2πk n o exp( ax w 0 +i a 2 ez 2 n o z 0 ) exp[ i w 0 3 3 ξ 3 iez 2k n o ξ 2 +( a w 0 ez n o z 0 ix )ξ ] dξ = w 0 iez 2πk n o exp[ ax w 0 a 2 ( ez n o z 0 ) 2 i 12 ( ez n o z 0 ) 3 +( a 2 + x w 0 ) iez 2 n o z 0 ] × exp[ i 3 ( w 0 ξ+ ez 2 n o z 0 ) 3 ] exp{ i[ x w 0 ( ez 2 n o z 0 ) 2 + iaez n o z 0 ]( w 0 ξ+ ez 2 n o z 0 ) }dξ,
Ai(u)= 1 2π exp( i 3 x 3 )exp(iux) dx.
U(x,z)= i2πez k n o Ai[ x w 0 ( ez 2 n o z 0 ) 2 + iaez n o z 0 ]exp[ ax w 0 a 2 ( ez n o z 0 ) 2 i 12 ( ez n o z 0 ) 3 +( a 2 + x w 0 ) iez 2 n o z 0 ].
U(y,z)= i2πz ke n o Ai[ y w 0 ( z 2e n o z 0 ) 2 + iaz e n o z 0 ]exp[ ay w 0 a 2 ( z e n o z 0 ) 2 i 12 ( z e n o z 0 ) 3 +( a 2 + y w 0 ) iz 2e n o z 0 ].
E x (x,y,z)=exp(ik n e z)Ai[ x w 0 ( ez 2 n o z 0 ) 2 + iaez n o z 0 ]exp[ ax w 0 a 2 ( ez n o z 0 ) 2 i 12 ( ez n o z 0 ) 3 +( a 2 + x w 0 ) iez 2 n o z 0 ] ×Ai[ y w 0 ( z 2e n o z 0 ) 2 + iaz e n o z 0 ]exp[ ay w 0 a 2 ( z e n o z 0 ) 2 i 12 ( z e n o z 0 ) 3 +( a 2 + y w 0 ) iz 2e n o z 0 ],
E y (x,y,z)=0.
x= e 2 4 n o 2 k 2 w 0 3 z 2 ,
y= 1 4 e 2 n o 2 k 2 w 0 3 z 2 .
g x = d 2 x d z 2 = e 2 2 n o 2 k 2 w 0 3 ,
g y = d 2 y d z 2 = 1 2 e 2 n o 2 k 2 w 0 3 ,
W jz (z)= { 2 (j j C ) 2 | E x (x,y,z) | 2 dxdy | E x (x,y,z) | 2 dxdy } 1/2 ,
j C = j | E x (x,y,z) | 2 dxdy | E x (x,y,z) | 2 dxdy .
x C = y C =( a 2 1 4a ) w 0 .
W xz (z)= W xz (0) [1+ (z/ z rx ) 2 ] 1/2 ,
W yz (z)= W yz (0) [1+ (z/ z ry ) 2 ] 1/2 ,
W xz (0)= W yz (0)= w 0 (1+8 a 3 ) 1/2 /2a
z rx = 1+8 a 3 2a n o z 0 e ,
z ry = 1+8 a 3 2a e n o z 0 .

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