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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 15 — Jul. 16, 2012
  • pp: 17160–17173
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Nonparaxial propagation of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal

Yongzhou Ni and Guoquan Zhou  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 17160-17173 (2012)
http://dx.doi.org/10.1364/OE.20.017160


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Abstract

Analytical expressions of the nonparaxial propagation of even and odd elegant Laguerre-Gaussian beams orthogonal to the optical axis of a uniaxial crystal are derived. The intensity distributions of even and odd elegant Laguerre-Gaussian beams and their three components propagating in uniaxial crystals orthogonal to the optical axis are illustrated by numerical examples. Even though one of the two transversal components of even and odd elegant Laguerre-Gaussian beams in the input plane is set to be zero, this transversal and the longitudinal components have nonzero intensities and cannot be neglected upon propagation inside the uniaxial crystal. The evolution laws of even and odd elegant Laguerre-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis are also demonstrated. The intensity distributions of even and odd elegant Laguerre-Gaussian beams can be modulated by the uniaxial crystal, which is beneficial to the some applications involving in the special beam profile.

© 2012 OSA

1. Introduction

The cylindrically symmetric higher-order modes of laser cavities with spherical mirrors are the standard Laguerre-Gaussian beams. As an extension of the standard Laguerre-Gaussian beams, the elegant Laguerre-Gaussian beams, which are also the eigenmodes of the paraxial wave equation, have been introduced by T. Takenaka et al [1

1. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]

]. The elegant Laguerre-Gaussian beams differ from the standard Laguerre-Gaussian beams in that the former contains polynomials with a complex argument, while the argument in the latter is real. Within the paraxial and the non-paraxial frameworks, the properties of the elegant Laguerre-Gaussian beams propagating in free space, through a paraxial optical system, in turbulent atmosphere, and at a dielectric interface have been extensively investigated [2

2. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986). [CrossRef]

17

17. H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express 19(22), 21163–21173 (2011). [CrossRef] [PubMed]

]. Also, the elegant Laguerre-Gaussian beam has been extended to the partially coherent case [18

18. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009). [CrossRef] [PubMed]

20

20. Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011). [CrossRef]

]. The relationship between the elegant Laguerre-Gaussian and the Bessel-Gaussian beams has been also examined [21

21. M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18(1), 177–184 (2001). [CrossRef] [PubMed]

]. Elegant Laguerre-Gaussian beams can be used as a new tool to describe the axisymmetric flattened Gaussian beam [22

22. R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18(7), 1627–1633 (2001). [CrossRef] [PubMed]

].

Research in the propagation of laser beams in a uniaxial crystal plays an important role in designing polarizing prism, amplitude- and phase-modulation devices. The propagation of laser beams in a uniaxial crystal can be treated by solving Maxwell’s equation in crystals. The propagation of various kinds of laser beams in uniaxial crystals has been widely examined [23

23. 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]

32

32. G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012). [CrossRef] [PubMed]

]. In the remainder of this paper, therefore, the propagation of an elegant Laguerre-Gaussian beam in uniaxial crystals orthogonal to the optical axis is to be investigated. Moreover, here we consider the nonparaxial propagation of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal. Analytical propagation expressions of even and odd elegant Laguerre-Gaussian beams orthogonal to the optical axis of a uniaxial crystal are derived, and the corresponding figures are demonstrated.

2. Theoretical derivations

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 dielectric tensor of the uniaxial crystal is described by
ε=(ne2000no2000no2),
(1)
where no and ne are the ordinary and the extraordinary refractive indices, respectively. The elegant Laguerre-Gaussian beam can be divided into two types: the even and the odd modes. First, we consider the even elegant Laguerre-Gaussian beam. The even elegant Laguerre-Gaussian beam considered here is linearly polarized in the x-direction and is incident on a uniaxial crystal in the plane z = 0. The even elegant Laguerre-Gaussian beam in the input plane z = 0 takes the form [1

1. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]

]
[Ex(ρ0,0)Ey(ρ0,0)]=[(ρ0w0)mLnm(ρ02w02)exp(ρ02w02)cos(mφ0)0],
(2)
where ρ0 = x0ex + y0ey. Vectors ex and ey are the unit vectors in the x- and y-directions, respectively. ρ0=(x02+y02)1/2 and φ0=tan1(y0/x0). Integers n and m are the radial and the angular mode numbers, respectively. The symbol w0 is the waist radius of the Gaussian part. Lnm is the associated Laguerre polynomial. We recall the following relation [33

33. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]

]:
ρ0mLnm(ρ02)cos(mφ0)=(1)n22n+mn!l=0ns=0[m/2](1)s(nl)(m2s)HM1(x0)HN1(y0),
(3)
where M1 = m + 2l-2s and N1 = 2n-2l + 2s. The expression [m/2] denotes the integer part of m/2, and HM1 is the Hermite polynomial of order M1. The expressions (nl) and (ms)are binomial coefficients. Therefore, Eq. (2) can be rewritten as
[Ex(ρ0,0)Ey(ρ0,0)]=[(1)n22n+mn!l=0ns=0[m/2](1)s(nl)(m2s)HM1(x0w0)HN1(y0w0)exp(ρ02w02)0].
(4)
The propagation of the even elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal obeys the following equation [34

34. 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]

]:
E(ρ,z)=d2kexp(ikρ)exp(ikezz)(E˜x(k)kxkyk02no2kx2E˜x(k)kezkxk02no2kx2E˜x(k))+d2kexp(ikρ)exp(ikozz)(0kxkyk02no2kx2E˜x(k)+E˜y(k)kykoz[kxkyk02no2kx2E˜x(k)+E˜y(k)]),
(5)
where k = kxex + kyey, ρ = xex + yey, and k0 = 2π/λ is the wave number with λ being the optical wavelength. E˜j(k) is the two-dimensional Fourier transform of the transverse components of the optical field in the input plane z = 0 and is given by
E˜j(k)=1(2π)2Ej(ρ0,0)exp[i(kxx0+kyy0)]dx0dy0,
(6)
where j = x or y (hereafter). kez and koz are defined by
kez=[ko2ne2(ne2/no2)kx2ky2]1/2,koz=(ko2n02kx2ky2)1/2.
(7)
When the beam waist radius of the even elegant Laguerre-Gaussian beam is comparable with the wavelength, the paraxial approach is not accurate enough. A nonparaxial laser beam can be produced from a paraxial laser beam by its focusing with a high numerical aperture, and is commonly encountered in holography microscopy and microlithography. When the field in a uniaxial crystal is nonparaxial, Eq. (5) can be expressed as a sum of the dominant paraxial result and a nonparaxial correction term [34

34. 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]

]:
E(ρ,z)=exp(ik0nez)d2kexp(ikρ)exp(ine2kx2+no2ky22k0neno2z)(E˜x(k)kxkyk02no2E˜x(k)nekxk0no2E˜x(k))+exp(ik0noz)d2kexp(ikρ)exp(ikx2+ky22k0noz)(0kxkyk02no2E˜x(k)+E˜y(k)kykonoE˜y(k)).
(8)
Ex(ρ, z) is just the paraxial result. Ey(ρ, z) and Ez(ρ, z) are the first nonparaxial correction to the paraxial result. The nonparaxial correction contains all the nonparaxial features of the beam and can be easily evaluated from the knowledge of the paraxial result [34

34. 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]

]. The nonparaxiality couples the Cartesian components of the field. The resultant longitudinal component is greater than the correction to the transverse component orthogonal to the optical axis. By using the property of the Fourier transforms [35

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

], the three components of the nonparaxial field in the uniaxial crystal orthogonal to the optical axis can also be rewritten as
Ex(ρ,z)=k0no2πizEx(ρ0,0)Λe(ρ,ρ0)dx0dy0,
(9)
Ey(ρ,z)=ik0no2πz3Ex(ρ0,0)(xx0)(yy0)[Λe(ρ,ρ0)Λo(ρ,ρ0)]dx0dy0+k0no2πizEy(ρ0,0)Λo(ρ,ρ0)dx0dy0,
(10)
Ez(ρ,z)=ik0no2πz2[Ex(ρ0,0)(xx0)Λe(ρ,ρ0)+Ey(ρ0,0)(yy0)Λo(ρ,ρ0)]dx0dy0,
(11)
with Λe(ρ,ρ0)and Λo(ρ,ρ0)being given by
Λe(ρ,ρ0)=exp(ik0nez)exp{k02izne[no2(xx0)2+ne2(yy0)2]},
(12)
Λo(ρ,ρ0)=exp(ik0noz)exp{k0no2iz[(xx0)2+(yy0)2]}.
(13)
Using the following mathematical formulae [35

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

]:
HM(x)exp[(xy)2/α]dx=πα(1α)M/2HM[y(1α)1/2],
(14)
xHM(x)=12HM+1(x)+MHM1(x),
(15)
the three components of the even elegant Laguerre-Gaussian beam in the uniaxial crystal orthogonal to the optical axis are found to be
Ex(ρ,z)=(1)nk0no22n+m+1n!iπzexp(iek0noz)l=0ns=0[m/2](1)s(nl)(m2s)TM1(x)UN1(y),
(16)
Ey(ρ,z)=(1)nik0no22n+m+1n!πz3l=0ns=0[m/2](1)s(nl)(m2s){exp(iek0noz){xyTM1(x)UN1(y)w0x×TM1(x)[0.5UN1+1(y)+N1UN11(y)]w0yUN1(y)[0.5TM1+1(x)+M1TM11(x)]+w02[0.5TM1+1(x)+M1TM11(x)][0.5UN1+1(y)+N1UN11(y)]}exp(ik0noz){xy×VM1(x)VN1(y)w0xVM1(x)[0.5VN1+1(y)+N1VN11(y)]w0yVN1(y)[0.5VM1+1(x)+M1VM11(x)]+w02[0.5VM1+1(x)+M1VM11(x)][0.5VN1+1(y)+N1VN11(y)]}},
(17)
Ez(ρ,z)=(1)nik0no22n+m+1n!πz2exp(iek0noz)l=0ns=0[m/2](1)s(nl)(m2s){xTM1(x)UN1(y)w0UN1(y)×[0.5TM1+1(x)+M1TM11(x)]},
(18)
with Tμ(x),Uμ(y), Vμ(x), and Vμ(y)being given by
Tμ(x)=w0πα(1α)μ/2exp(ax2)Hμ[αzrnoxiezw0(1α)1/2],
(19)
Uμ(y)=w0πβ(1β)μ/2exp(by2)Hμ[βzrenoyizw0(1β)1/2],
(20)
Vμ(j)=w0πγ(1γ)μ/2exp(cj2)Hμ[γzrn0jizw0(1γ)1/2],
(21)
where µ is an arbitrary integer. The auxiliary parameters are defined as
e=neno,zr=k0w022,α=(1izrnoez)1,β=(1iezrnoz)1,γ=(1izrn0z)1,
(22)
a=α(zrnoiezw0)2+k0no2iez,b=β(ezrnoizw0)2+ek0no2iz,c=γ(zrnoizw0)2+k0no2iz.
(23)
where zr is the Rayleigh length of the Gaussian part. As shown in Eqs. (19)-(21), the polynomials in three components of the even elegant Laguerre-Gaussian beam have complex argument, which can be compared with those of the standard elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis by using the method suggested in Ref [36

36. S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001). [CrossRef]

].

The odd elegant Laguerre-Gaussian beam in the input plane z = 0 is described by
[Ex(ρ0,0)Ey(ρ0,0)]=[(ρ0w0)mLnm(ρ02w02)exp(ρ02w02)sin(mφ0)0].
(24)
Based on the following expansion [33

33. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]

]:
ρ0mLnm(ρ02)sin(mφ0)=(1)n22n+mn!l=0ns=0[(m1)/2](1)s(nl)(m2s+1)HM11(x0)HN1+1(y0),
(25)
Equation (25) can be rewritten as
[Ex(ρ0,0)Ey(ρ0,0)]=[(1)n22n+mn!l=0ns=0[(m1)/2](1)s(nl)(m2s+1)HM11(x0w0)HN1+1(y0w0)exp(ρ02w02)0].
(26)
The three components of the odd elegant Laguerre-Gaussian beam in the uniaxial crystal orthogonal to the optical axis turn out to be

Ex(ρ,z)=(1)nk0no22n+m+1n!iπzexp(iek0noz)l=0ns=0[(m1)/2](1)s(nl)(m2s+1)TM11(x)UN1+1(y),
(27)
Ey(ρ,z)=(1)nik0no22n+m+1n!πz3l=0ns=0[(m1)/2](1)s(nl)(m2s+1){exp(iek0noz){xyTM11(x)UN1+1(y)w0xTM11(x)[0.5UN1+2(y)+(N1+1)UN1(y)]w0yUN1+1(y)[0.5TM1(x)+(M11)×TM12(x)]+w02[0.5TM1(x)+(M11)TM12(x)][0.5UN1+2(y)+(N1+1)UN1(y)]}exp(ik0noz){xyVM11(x)VN1+1(y)w0xVM11(x)[0.5VN1+2(y)+(N1+1)VN1(y)]w0yVN1+1(y)[0.5VM1(x)+(M11)VM12(x)]+w02[0.5VM1(x)+(M11)VM12(x)]×[0.5VN1+2(y)+(N1+1)VN1(y)]}}, (28) Ez(ρ,z)=(1)nik0no22n+m+1n!πz2exp(iek0noz)l=0ns=0[(m1)/2](1)s(nl)(m2s+1){xTM11(x)UN1+1(y)w0UN1+1(y)[0.5TM1(x)+(M11)TM12(x)]}.
(29)

3. Numerical calculations and analyses

Based on the analytical formulae derived in the last section, now we investigate the propagation properties of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal. Here we mainly pay attention to the influence of the uniaxial crystal on the propagation of an elegant Laguerre-Gaussian beam. The calculation parameters are chosen as follow. m = 3, n = 2, w0 = λ, and the ordinary refractive index is fixed to be 2.616. The extraordinary refractive index is variable. Figures 1
Fig. 1 Contour graph of the intensity of the x-component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.
3
Fig. 2 Contour graph of the intensity of the y-component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.
Fig. 3 Contour graph of the intensity of the longitudinal component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.
represent the contour graphs of the intensity distribution of the three components of an even elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes. e is set to be 1.5 in Figs. 13. The observation planes are z = 0.1zr, z = zr, z = 3zr, and z = 5zr, respectively. The intensity is proportional to the scale of 1/λ2, which is not marked in the figures. The beam profile of the x-component of an even elegant Laguerre-Gaussian beam has six lobes. Upon propagation orthogonal to the optical axis of the uniaxial crystal, the spreading of the beam profile in the y-direction is far slower than that in the x-direction, which is caused by the anisotropic effect of the crystal. As a result, the beam profile in the uniaxial crystal is elongated in the x-direction. The magnitude of the intensity of the six lobes close to the input plane is equivalent. However, the equality of magnitude of the intensity of the six lobes is broken upon propagation in the uniaxial crystal. At first the magnitude of the intensity of the two middle lobes is smaller than that of the other four lobes. When the observation plane is far enough, the magnitude of the intensity of the four top and bottom lobes is smaller than that of the two middle lobes. Though the y-component of an even elegant Laguerre-Gaussian beam in the input plane is set to be zero, it is no longer equal to zero upon propagation in the uniaxial crystal, which can be interpreted as follow. The y-component is affected by the x-component because of the dependence on E˜x(k), and this corresponds to a change of polarization state of the radiation that occurs during propagation in the uniaxial crystal [32

32. G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012). [CrossRef] [PubMed]

]. The beam profile of the y-component of an even elegant Laguerre-Gaussian beam apparently varies upon propagation in the uniaxial crystal. Upon propagation in the uniaxial crystal, the two central lobes including the top and bottom side lobes will diminish until disappear. Instead, the four bilateral lobes gradually become the dominant lobes. Compared with the magnitude of the x-component, the magnitude of the y-component is small. When propagating from the observation plane z = 0.1zr to the observation plane z = zr, the beam profile of the longitudinal component undergoes a process of the split of the lobes. When propagating from the observation plane z = zr to the observation plane z = 3zr, the two lobes in the y-axis gradually diminish until disappear, and the two lobes in the x-axis evolve from the accessory lobes into the dominant lobes. When the observation plane is larger than 3zr, the beam profile of the longitudinal component keeps stable. The magnitude of the longitudinal component is smaller than that of the x-component but larger than that of the y-component. The contour graph of the intensity distribution of an even elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes is shown in Fig. 4
Fig. 4 Contour graph of the intensity of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.
where e = 1.5. As |Ex|2>|Ez|2>|Ey|2, the beam profile of an even elegant Laguerre-Gaussian beam is close to that of the x-component. However, the details in the beam profile between the even elegant Laguerre-Gaussian beam and its x-component are different. Moreover, the magnitude of the intensity of the even elegant Laguerre-Gaussian beam is larger than that of its x-component, which sometimes cannot be distinguished through the comparison of the corresponding figures as the label of the intensity is marked in the rank.

Figure 5
Fig. 5 Contour graph of the intensity of an even elegant Laguerre-Gaussian beam propagating in the observation plane z = 3zr of different uniaxial crystals. (a) e = 0.6. (b) e = 0.8. (c) e = 1.0. (d) e = 1.2.
represents the contour graph of the intensity of an even elegant Laguerre-Gaussian beam in the observation plane z = 3zr of different uniaxial crystals. The uniaxial crystals can be divided into two kinds. One is the positive uniaxial crystal, which corresponds to e>1. The other is the negative uniaxial crystal, which corresponds to e<1. As to a uniaxial crystal, the value of e is fixed. For simplicity, here a model which is the combination of two kinds of uniaxial crystals is used to illustrate the effect of the uniaxial crystals. e = 1 corresponds to the isotropic mediums. When e = 1, the beam profile of the even elegant Laguerre-Gaussian beam is symmetric and the magnitude of the intensity of the six lobes is equivalent. When e is not equal to unity, the beam profile becomes an astigmatic beam. When e<1, the beam profile is elongated in the y-direction, and the magnitude of the intensity of the top and bottom lobes is larger than that of the two middle lobes. When e>1, the beam profile is elongated in the x-direction, and the magnitude of the intensity of the top and bottom lobes is smaller than that of the two middle lobes. With increasing the deviation of e from unity, the elongation of the beam spot augments, and the difference of the magnitude of the intensity between the middle lobes and the other lobes also increases.

The contour graphs of the intensity distribution of the three components of an odd elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes are shown in Figs. 6
Fig. 6 Contour graph of the intensity of the x-component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.
8
Fig. 7 Contour graph of the intensity of the y-component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.
Fig. 8 Contour graph of the intensity of the longitudinal component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.
where e = 1.5. When z = 0.1zr, the beam profile of the x-component of the odd elegant Laguerre-Gaussian beam is that of the even elegant Laguerre-Gaussian beam rotating 90° from the y-axis. Upon propagation in the uniaxial crystal, the magnitude of the intensity of the lobes in the y-axis quickly decreases and is smaller than that of the four bilateral lobes. When z = 0.1zr, the beam profile of the y-component of the odd elegant Laguerre-Gaussian beam is somewhat similar to that of the even elegant Laguerre-Gaussian beam rotating 90° from the y-axis. However, the magnitude of the intensity slightly decreases. When z = zr or z = 3zr, the beam profile of the y-component of the odd elegant Laguerre-Gaussian beam is completely different from that of the even elegant Laguerre-Gaussian beam. In these cases, the beam profile of the y-component of the odd elegant Laguerre-Gaussian beam takes on an annular shape. When z = 5zr, the beam profile of the y-component of the odd elegant Laguerre-Gaussian beam is somewhat similar to that of the even elegant Laguerre-Gaussian beam. The beam profile of the longitudinal component of the odd elegant Laguerre-Gaussian beam is always completely different from that of the even elegant Laguerre-Gaussian beam. The beam profile of the longitudinal component of the odd elegant Laguerre-Gaussian beam has eight lobes. When z = 3zr, each two lobes is mutually connected to form a group. Upon propagation in the uniaxial crystal, the magnitude of the intensity of the four central lobes diminishes and that of the four bilateral lobes increases. Figure 9
Fig. 9 Contour graph of the intensity of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.
represents the contour graph of the intensity distribution of an odd elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes. Comparing Fig. 9(d) with Fig. 6(d), the magnitude of the intensity denotes that the y- and z-components cannot be ignored. Figure 10
Fig. 10 Contour graph of the intensity of an odd elegant Laguerre-Gaussian beam propagating in the observation plane z = 3zr of different uniaxial crystals. (a) e = 0.6. (b) e = 0.8. (c) e = 1.0. (d) e = 1.2.
shows the contour graph of the intensity of an odd elegant Laguerre-Gaussian beam in the observation plane z = 3zr of different uniaxial crystals. When e<1, the magnitude of the intensity of the lobes in the y-axis is larger than that of the four bilateral lobes. When e>1, the magnitude of the intensity of the lobes in the y-axis is smaller than that of the four bilateral lobes. When e<1, the beam profile of the odd elegant Laguerre-Gaussian beam and its components is elongated in the y-direction. When e>1, the beam profile of the odd elegant Laguerre-Gaussian beam and its components is elongated in the x-direction. Comparing Figs. 15 with Figs. 610, one can find that the even and odd elegant Laguerre-Gaussian beams have their respective evolution laws.

4. Conclusions

The analytical nonparaxial formulae of even and odd elegant Laguerre-Gaussian beams propagating in a uniaxial crystal orthogonal to the optical axis have been derived, respectively. The intensity distributions of even and odd elegant Laguerre-Gaussian beams and their three components propagating in a uniaxials crystal orthogonal to the optical axis have been demonstrated by numerical examples. Though the y-component of even and odd elegant Laguerre-Gaussian beams in the input plane is set to be zero, the y-component emerges upon propagation in the uniaxial crystal. Moreover, the y- and z-components cannot be ignored. The even and odd elegant Laguerre-Gaussian beams propagating in uniaxial crystal become astigmatic beams. The beam profile of even and odd elegant Laguerre-Gaussian beams in the uniaxial crystal is elongated in the x- or y-direction, which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index. With the increase of the deviation of the ratio of the extraordinary refractive index to the ordinary refractive index from unity, the elongation of the beam profile also increases. The even and odd elegant Laguerre-Gaussian beams propagating in the uniaxial crystal have their respective evolution laws. As the beam profile of even and odd elegant Laguerre-Gaussian beams can be modulated by the uniaxial crystal, this research is beneficial to the some applications involving in the special beam profile.

Acknowledgments

This research was supported by National Natural Science Foundation of China under Grant Nos. 10974179 and 61178016. The authors are indebted to the reviewer for valuable comments and suggestions.

References and links

1.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]

2.

E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986). [CrossRef]

3.

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45(10), 1999–2009 (1998). [CrossRef]

4.

S. Luo and B. Lü, “Propagation of the kurtosis parameter of elegant Hermite-Gaussian and Laguerre-Gaussian beams passing through ABCD systems,” Optik (Stuttg.) 113(5), 227–231 (2002). [CrossRef]

5.

M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 (2004). [CrossRef] [PubMed]

6.

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(12), 2375–2381 (2004). [CrossRef] [PubMed]

7.

D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE 5639, 149–158 (2004). [CrossRef]

8.

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004). [CrossRef]

9.

Y. Pagani and W. Nasalski, “Diagonal relations between elegant Hermite-Gaussian and Laguerre-Gaussian beam fields,” Opto-Electron. Rev. 13, 51–60 (2005).

10.

J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express 15(10), 6300–6313 (2007). [CrossRef] [PubMed]

11.

D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24(3), 636–643 (2007). [CrossRef]

12.

Z. Mei, “The elliptical elegant Laguerre-Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” Optik (Stuttg.) 118(8), 361–366 (2007). [CrossRef]

13.

A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008). [CrossRef] [PubMed]

14.

Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 (2009). [CrossRef] [PubMed]

15.

W. Nasalski, “Elegant Hermite-Gaussian and Laguerre-Gaussian beams at a dielectric interface,” Opt. Appl. 40, 615–622 (2010).

16.

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre-Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010). [CrossRef]

17.

H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express 19(22), 21163–21173 (2011). [CrossRef] [PubMed]

18.

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009). [CrossRef] [PubMed]

19.

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” PIER 103, 33–56 (2010). [CrossRef]

20.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011). [CrossRef]

21.

M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18(1), 177–184 (2001). [CrossRef] [PubMed]

22.

R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18(7), 1627–1633 (2001). [CrossRef] [PubMed]

23.

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]

24.

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]

25.

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]

26.

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]

27.

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]

28.

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009). [CrossRef]

29.

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]

30.

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]

31.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011). [CrossRef]

32.

G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012). [CrossRef] [PubMed]

33.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]

34.

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]

35.

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

36.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001). [CrossRef]

OCIS Codes
(260.1180) Physical optics : Crystal optics
(260.1960) Physical optics : Diffraction theory
(350.5500) Other areas of optics : Propagation

ToC Category:
Physical Optics

History
Original Manuscript: May 23, 2012
Revised Manuscript: June 28, 2012
Manuscript Accepted: July 9, 2012
Published: July 12, 2012

Citation
Yongzhou Ni and Guoquan Zhou, "Nonparaxial propagation of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal," Opt. Express 20, 17160-17173 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-17160


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References

  1. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A2(6), 826–829 (1985). [CrossRef]
  2. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am.3(4), 465–469 (1986). [CrossRef]
  3. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt.45(10), 1999–2009 (1998). [CrossRef]
  4. S. Luo and B. Lü, “Propagation of the kurtosis parameter of elegant Hermite-Gaussian and Laguerre-Gaussian beams passing through ABCD systems,” Optik (Stuttg.)113(5), 227–231 (2002). [CrossRef]
  5. M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett.29(19), 2213–2215 (2004). [CrossRef] [PubMed]
  6. Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A21(12), 2375–2381 (2004). [CrossRef] [PubMed]
  7. D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE5639, 149–158 (2004). [CrossRef]
  8. Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt.6(11), 1005–1011 (2004). [CrossRef]
  9. Y. Pagani and W. Nasalski, “Diagonal relations between elegant Hermite-Gaussian and Laguerre-Gaussian beam fields,” Opto-Electron. Rev.13, 51–60 (2005).
  10. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express15(10), 6300–6313 (2007). [CrossRef] [PubMed]
  11. D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B24(3), 636–643 (2007). [CrossRef]
  12. Z. Mei, “The elliptical elegant Laguerre-Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” Optik (Stuttg.)118(8), 361–366 (2007). [CrossRef]
  13. A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett.33(12), 1392–1394 (2008). [CrossRef] [PubMed]
  14. Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express17(17), 14865–14871 (2009). [CrossRef] [PubMed]
  15. W. Nasalski, “Elegant Hermite-Gaussian and Laguerre-Gaussian beams at a dielectric interface,” Opt. Appl.40, 615–622 (2010).
  16. J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre-Gaussian beam in a turbulent atmosphere,” Opt. Commun.283(14), 2772–2781 (2010). [CrossRef]
  17. H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express19(22), 21163–21173 (2011). [CrossRef] [PubMed]
  18. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express17(25), 22366–22379 (2009). [CrossRef] [PubMed]
  19. F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” PIER103, 33–56 (2010). [CrossRef]
  20. Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B102(4), 937–944 (2011). [CrossRef]
  21. M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A18(1), 177–184 (2001). [CrossRef] [PubMed]
  22. R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A18(7), 1627–1633 (2001). [CrossRef] [PubMed]
  23. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A19(4), 792–796 (2002). [CrossRef] [PubMed]
  24. 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]
  25. 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]
  26. 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]
  27. B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A26(12), 2480–2487 (2009). [CrossRef] [PubMed]
  28. D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D54(1), 95–101 (2009). [CrossRef]
  29. 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]
  30. 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. D57(3), 419–425 (2010). [CrossRef]
  31. J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53(2), 20701 (2011). [CrossRef]
  32. G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express20(3), 2196–2205 (2012). [CrossRef] [PubMed]
  33. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron.29(9), 2562–2567 (1993). [CrossRef]
  34. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A20(11), 2163–2171 (2003). [CrossRef] [PubMed]
  35. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 1980)
  36. S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun.191(3-6), 173–179 (2001). [CrossRef]

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