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

1. Introduction

As the solution of the force-free Schrödinger equation, an Airy beam contains infinite energy and exhibits a nonspreading property in vacuum [1]. The free acceleration [2, 3], the self-healing properties [4], the evolution of the Poynting vector and angular momentum [5], the ballistic dynamics [6], and the beam propagation factor [7] of an Airy beam have been widely investigated. The methods of the Wigner distribution function and the geometrical optics have been used to interpret the intriguing features of an Airy beam, respectively [8, 9]. The Airy beam has vital applications in optical micromanipulation and all-optical switching [10, 11]. The propagation of an Airy beam through an ABCD optical system [12], in water [13], in a nonlinear medium [14, 15], and in turbulence [16] has been examined, respectively.

Besides in free space, in a nonlinear medium, and in turbulent atmosphere, laser beams also propagate in anisotropic medium such as uniaxial crystals, which is treated by solving Maxwell’s equations. Some applications, e. g., the design of the polarizer and the compensator, are involved in laser beams propagating in uniaxial crystals. The propagation of various kinds of laser beams in uniaxial crystals has been reported [1723]. To the best of our knowledge, however, no literature has been reported on the propagation of Airy beams in uniaxial crystals. In the remainder of this paper, therefore, the paraxial propagation of Airy beams in uniaxial crystals orthogonal to the optical axis is to be investigated.

2. Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis

In the Cartesian coordinate system, the z-axis is taken to be the propagation axis. The optical axis of the uniaxial crystal coincides with the x-axis. The input plane is z=0 and the observation plane is z. The ordinary and extraordinary refractive indices of the uniaxial crystal are no and ne, respectively. The relative dielectric tensor of the uniaxial crystal reads as

ε=(ne2000no2000no2),

The Airy beam considered here is linearly polarized in the x-direction and is incident on a uniaxial crystal in the plane z=0. The Airy beam in the input plane z=0 takes the form as [2, 3]

[Ex(x0,y0,0)Ey(x0,y0,0)]=[Ai(x0w0)Ai(y0w0)exp(ax0w0+ay0w0)0],
where w0 is the transverse scale. Ai(⋅) is the Airy function. a is the modulation parameter. Within the framework of the paraxial approximation, the propagation of the Airy beam in uniaxial crystals orthogonal to the optical axis obeys the following equations [24, 25]:
Ex(x,y,z)=exp(iknez)E˜x(kx,ky)exp[i(kxx+kyy)ine2kx2+no2ky22kno2nez]dkxdky,
Ey(x,y,z)=exp(iknoz)E˜y(kx,ky)exp[i(kxx+kyy)ikx2+ky22knoz]dkxdky,
with the two-dimensional Fourier transform E˜x(kx,ky) and E˜y(kx,ky) being given by
E˜j(kx,ky)=1(2π)2Ej(x0,y0,0)exp[i(kxx+kyy)]dkxdky,
where j = x or y (hereafter). In the paraxial approximation, the longitudinal component of the field can be neglected as long as the transversal beam radius w0 is larger compared with the wavelength [26]. Equations (3) and (4) indicate that the x component of the optical field is only a superposition of extraordinary plane waves and the y component uniquely contains ordinary plane waves [24]. Equations (3) and (4) can also be rewritten as [24, 25]
Ex(x,y,z)=kno2πizexp(iknez)Ex(x0,y0,0)exp{ik2zne[no2(xx0)2+ne2(yy0)2]}dx0dy0,
Ey(x,y,z)=kno2πizexp(iknoz)Ey(x0,y0,0)exp{ikno2z[(xx0)2+(yy0)2]}dx0dy0,
where k=2π/λ is the wave number with λ being the optical wavelength. Inserting Eq. (2) into Eq. (6), we can obtain
Ex(x,y,z)=kno2πizexp(iknez)U(x,z)U(y,z),
withU(x,z)andU(y,z)being given by

U(x,z)=Ai(x0w0)exp(ax0w0)exp[kno22izne(xx0)2]dx0,
U(y,z)=Ai(y0w0)exp(ay0w0)exp[kne2iz(yy0)2]dy0.

Equation (9) can be rewritten as

U(x,z)=exp(axw0+ia2nez2no2z0)Ai(x0w0)exp[kno22izne(x+iaw0nezno2z0x0)2]dx0,
where z0=kw02. We note that the convolution of two functions,f1(τ)andf2(τ), is defined by [27]
f1(τ)f2(τ)=f1(x0)f2(τx0)dx0,
where ⊗ denotes the convolution. Therefore, Eq. (11) can be expressed in the form of the convolution:
U(x,z)=exp(axw0+ia2nez2no2z0)[f1(x+iaw0nezno2z0)f2(x+iaw0nezno2z0)],
where the auxiliary functionsf1(τ)andf2(τ)are given by

f1(τ)=Ai(τw0),
f2(τ)=exp(kno22izneτ2).

The convolution theorem of the Fourier transform has the following property [27]

f1(τ)f2(τ)=f1(ξ)f2(ξ)exp(iξτ)dξ,
where the auxiliary functionsf1(ξ)andf2(ξ)are the Fourier transform of f1(τ)andf2(τ), respectively. f1(ξ) and f2(ξ) are given by [27]

f1(ξ)=12πAi(τw0)exp(iξτ)dτ=w02πexp(iw033ξ3),
f2(ξ)=12πexp(kno22izneτ2)exp(iξτ)dτ=iznekno2exp(izne2kno2ξ2).

Therefore, Eq. (13) turns out to be

U(x,z)=w0iez2πknoexp(axw0+ia2ez2noz0)exp[iw033ξ3iez2knoξ2+(aw0eznoz0ix)ξ]dξ=w0iez2πknoexp[axw0a2(eznoz0)2i12(eznoz0)3+(a2+xw0)iez2noz0]×exp[i3(w0ξ+ez2noz0)3]exp{i[xw0(ez2noz0)2+iaeznoz0](w0ξ+ez2noz0)}dξ,
where e = ne/no is the ratio of the extraordinary refractive index to the ordinary refractive index. We recall that the Airy function can be written in terms of the integral representation:

Ai(u)=12πexp(i3x3)exp(iux)dx.

Therefore, Eq. (19) can be analytically expressed as

U(x,z)=i2πezknoAi[xw0(ez2noz0)2+iaeznoz0]exp[axw0a2(eznoz0)2i12(eznoz0)3+(a2+xw0)iez2noz0].

Similarly, Eq. (10) is found to be

U(y,z)=i2πzkenoAi[yw0(z2enoz0)2+iazenoz0]exp[ayw0a2(zenoz0)2i12(zenoz0)3+(a2+yw0)iz2enoz0].

As a result, the analytical paraxial propagation equations of an Airy beam in uniaxial crystals orthogonal to the optical axis yield

Ex(x,y,z)=exp(iknez)Ai[xw0(ez2noz0)2+iaeznoz0]exp[axw0a2(eznoz0)2i12(eznoz0)3+(a2+xw0)iez2noz0]×Ai[yw0(z2enoz0)2+iazenoz0]exp[ayw0a2(zenoz0)2i12(zenoz0)3+(a2+yw0)iz2enoz0],
Ey(x,y,z)=0.

Under the condition of ne = no = 1, Eq. (23) reduces to the familiar propagation formula of Airy beams in free space.

As shown in Eq. (23), the Airy beam in uniaxial crystals orthogonal to the optical axis follows a ballistic trajectory in the x-z and y-z planes that are described by the following parabolas

x=e24no2k2w03z2,
y=14e2no2k2w03z2.

The corresponding Newtonian equations describing theses ballistics read as

gx=d2xdz2=e22no2k2w03,
gy=d2ydz2=12e2no2k2w03,
where gx and gy play the role of “gravity”. As e>1 holds for the positive uniaxial crystals, the displacement in the x-direction is always larger than that in the y-direction, and gx is always larger than gy. When ne = no = 1, Eqs. (25)-(28) reduce to the ballistic dynamics in free space, which are same as those in Ref [6]. The effective beam size of the Airy beam in the x- and y-directions of the observation plane is defined as [28]:
Wjz(z)={2(jjC)2|Ex(x,y,z)|2dxdy|Ex(x,y,z)|2dxdy}1/2,
where jC is the centre of gravity of the Airy beam in uniaxial crystals orthogonal to the optical axis and is given by

jC=j|Ex(x,y,z)|2dxdy|Ex(x,y,z)|2dxdy.

According to Ref [6], the centres of gravity of the Airy beam in uniaxial crystals orthogonal to the optical axis yield

xC=yC=(a214a)w0.

Equation (31) demonstrates again that the center of gravity in uniaxial crystals is propagating on a straight line. After lengthy integral, the effective beam sizes of the Airy beam in the x- and y-directions of the observation plane turn out to be

Wxz(z)=Wxz(0)[1+(z/zrx)2]1/2,
Wyz(z)=Wyz(0)[1+(z/zry)2]1/2,
where
Wxz(0)=Wyz(0)=w0(1+8a3)1/2/2a
are the effective beam sizes in the input plane. zrx and zry represent the Rayleigh length in the x- and y-directions of the Airy beam in uniaxial crystals and are given by

zrx=1+8a32anoz0e,
zry=1+8a32aenoz0.

3. Numerical calculations and analyses

Here we mainly pay attention to the influence of the uniaxial crystals on the propagation of an Airy beam, and the uniaxial crystals considered here are positive. Calculation parameters are chosen as λ = 0.53μm, w0 = 100μm, a = 0.1, and no = 2.616. Figure 1 represents the contour graph of the normalized intensity distribution of an Airy beam propagating in the uniaxial crystals at several observation planes. The normalized intensity is given by |Ex(x,y,z)|2/|Ex(x,y,z)|2max where the subscript means taking the maximum value. The top and bottom rows denote e = 1.1 and e = 1.5, respectively. The two observation planes are z = 0.1z0 and z = 5z0. Upon propagation in the uniaxial crystals orthogonal to the optical axis, the acceleration of the Airy beam in the y-direction is far slower than that in the x-direction, which is caused by the anisotropic effect of the crystals. In free space, the Airy beam remains almost invariant up to a certain distance, and it accelerates in the same manner along the 45° axis in the x-y plane [3]. While in the uniaxial crystal, the Airy beam no longer accelerates along the 45° axis in the x-y plane. The angle along which the Airy beam accelerates in the x-y plane of the uniaxial crystal is equal to θ=tan1(e4). With increasing the value of e, we can find that the acceleration of the Airy beam in the x-direction speeds up and the acceleration of the Airy beam in the y-direction slows down. When e = 1.1, θ = 34.33°. θ decreases with increasing the value of e.

 

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.

Download Full Size | PPT Slide | PDF

Equation (23) denotes that the x and y variables can be separated from each other. To further reveal the accelerating properties of the Airy beam in uniaxial crystals, the normalized intensity distribution in x- and y-directions of an Airy beam propagating in the uniaxial crystals are shown in Figs. 2 and 3 . When propagating in the uniaxial crystals orthogonal to the optical axis, the Airy beam in the x- and y-directions accelerates, which is more evident than that in Fig. 1. With increasing the value of e, the acceleration of Airy beam in the x-direction quickens up, and the acceleration of Airy beam in the y-direction retards. The above exotic features only belong to the Airy beam. Equations (25) and (26) denote that the displacements in the x- and y-directions are proportional to the square of the axial propagation distance. When the observation plane is far from the input plane, therefore, the displacement deflection augments with increasing the value of e.

 

Fig. 2 The normalized intensity distribution in the x-direction of an Airy beam propagating in the uniaxial crystals.

Download Full Size | PPT Slide | PDF

 

Fig. 3 The normalized intensity distribution in the y-direction of an Airy beam propagating in the uniaxial crystals.

Download Full Size | PPT Slide | PDF

The contour graphs of the normalized intensity distribution of an Airy beam propagating in the uniaxial crystals at x-z and y-z planes are shown in Fig. 4 . The top and bottom rows correspond to e = 1.1 and e = 1.5, respectively. The Airy beam in uniaxial crystals orthogonal to the optical axis follows a ballistic trajectory. With increasing the value of e, gx increases and gy decreases, which is indicated by Eqs. (27) and (28). Therefore, the parabolic deflection of the normalized intensity in the x-z plane augments with increasing the value of e. However, the parabolic deflection of the normalized intensity in the y-z plane decreases with increasing the value of e, which is open and shut in Fig. 4. Comparison between the right and left parts of Fig. 4 denotes that the acceleration of the Airy beam in the x-direction is far faster than that in the y-direction under the same condition.

 

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.

Download Full Size | PPT Slide | PDF

The effective beam sizes of an Airy beam propagating in the uniaxial crystals versus the propagation distance z are shown in Fig. 5 . With increasing the value of e, the effective beam size in the x-direction increases and the effective beam size in the y-direction decreases. The reason is that ne/no of the anisotropic crystal is larger than unity. The effective beam size in the x-direction is always larger than that in the y-direction at any given propagation distance.

 

Fig. 5 The effective beam sizes of an Airy beam propagating in the uniaxial crystals versus the propagation distance z.

Download Full Size | PPT Slide | PDF

4. Conclusions

Analytical propagation expression of the Airy beam in uniaxial crystals orthogonal to the optical axis is derived. The parabolic defection of the normalized intensity features and the corresponding Newtonian equations describing theses ballistics are presented, respectively. The analytical formulae of the effective beam sizes are also derived. The Airy beam propagating in uniaxial crystals orthogonal to the optical axis depends on the ordinary refractive index no, the ratio of the extraordinary refractive index to the ordinary refractive index e, the axial propagation distance z, and the beam parameters w0 and a. As an example, the influence of the positive uniaxial crystals on the propagation properties of an Airy beam is numerically investigated. Upon propagation in the uniaxial crystals, the Airy beam in the two transversal directions accelerates. Moreover, the acceleration of the Airy beam in the transversal direction orthogonal to the optical axis is far slower than that in the transversal direction along the optical axis. The Airy beam in uniaxial crystals orthogonal to the optical axis follows a ballistic trajectory. With increasing the value of e, 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 transversal direction orthogonal to the optical axis slows down. The dependence of the effective beam size of the Airy beam in uniaxial crystals on the value of e is also examined. With increasing the value of e, the effective beam size in the transversal direction along the optical axis increases and that in the other transversal direction orthogonal to the optical axis decreases. At any given propagation distance, the effective beam size in the transversal direction along the optical axis is always larger than that in the other transversal direction.

Acknowledgments

This research was supported by National Natural Science Foundations of China under Grant No. 10974179 and 61178016. The authors are indebted to the reviewer for valuable comments.

References and links

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

2. 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]  

3. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef]   [PubMed]  

4. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef]   [PubMed]  

5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef]   [PubMed]  

6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef]   [PubMed]  

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

8. R. P. Chen, H. P. Zheng, and C. Q. Dai, “Wigner distribution function of an Airy beam,” J. Opt. Soc. Am. A 28(6), 1307–1311 (2011). [CrossRef]   [PubMed]  

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

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

11. 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]  

12. M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15(25), 16719–16728 (2007). [CrossRef]   [PubMed]  

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

14. 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]  

15. 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]  

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

17. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002). [CrossRef]   [PubMed]  

18. 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]  

19. 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]  

20. 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]  

21. 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]  

22. 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]  

23. 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]  

24. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003). [CrossRef]   [PubMed]  

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

26. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19(2), 404–412 (2002). [CrossRef]   [PubMed]  

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

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

References

  • View by:
  • |
  • |
  • |

  1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [Crossref]
  2. 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]
  3. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  4. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
    [Crossref] [PubMed]
  5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008).
    [Crossref] [PubMed]
  6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
    [Crossref] [PubMed]
  7. R. Chen and C. Ying, “Beam propagation factor of an Airy beam,” J. Opt. 13(8), 085704 (2011).
    [Crossref]
  8. R. P. Chen, H. P. Zheng, and C. Q. Dai, “Wigner distribution function of an Airy beam,” J. Opt. Soc. Am. A 28(6), 1307–1311 (2011).
    [Crossref] [PubMed]
  9. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010).
    [Crossref] [PubMed]
  10. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
    [Crossref]
  11. 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]
  12. M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15(25), 16719–16728 (2007).
    [Crossref] [PubMed]
  13. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
    [Crossref] [PubMed]
  14. 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]
  15. 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]
  16. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36(14), 2701–2703 (2011).
    [Crossref] [PubMed]
  17. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002).
    [Crossref] [PubMed]
  18. 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]
  19. 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]
  20. 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]
  21. 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]
  22. 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]
  23. 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]
  24. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003).
    [Crossref] [PubMed]
  25. M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, Oxford, 1999).
  26. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19(2), 404–412 (2002).
    [Crossref] [PubMed]
  27. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, New York, 1980).
  28. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer Press, New York, 2005).

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)

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]

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]

Other (3)

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).

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

Equations on this page are rendered with MathJax. Learn more.

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 .

Metrics