Analytical paraxial and nonparaxial propagation expressions for vectorial elegant Laguerre-Gaussian (eLG) beam together with its even and odd modes are introduced by use of the vectorial Rayleigh-Sommerfeld formulas and the relations between eLG and elegant Hermite-Gaussian (eHG) modes. The propagation features of vectorial eLG beams are studied and analyzed comparatively in the paraxial and nonparaxial regimes with vivid illustration. It is shown that the propagation behavior of nonparaxial vectorial eLG beams is notably different from that of paraxial cases.
© 2009 OSA
The concept of elegant beams was introduced by Siegman to describe the eHG beams , and then was extended to study the eLG beams by Takenaka et al [2–4]. The elegant solutions are also eigenmodes of the paraxial wave equation, but they differ from the standard solutions in that the former contain polynomials with a complex argument, but in the latter the argument is real. Some authors have tried to extend paraxial eLG solutions to the nonparaxial generalization of eLG beams. Takenaka et al obtained the correction terms of eLG beams in scalar theory by using the perturbation method proposed by Lax et al . Seshadri et al gave the first three orders of nonparaxial corrections for the corresponding paraxial eLG beam [5,6]. Kim and Lee obtained the power-series solutions of eLG beams by use of Lommel’s lemma in vector theory . Recently, April gave an explicit closed form for nonparaxial eLG beams, which is an exact solution of the Helmholtz equation and can be expressed as linear combination of a finite number of analytic functions . In this paper, by the representations of eLG modes in terms of the Hermite polynomials and the vectorial Rayleigh-Sommerfeld formulas [9–11], we obtain the analytical propagation expressions of vectorial eLG beam along with its even and odd modes in the nonparaxial and paraxial regime, which could enlighten the main features of nonparaxial propagation of eLG beams, and then investigate and analyze comparatively their propagation properties with vivid numerical examples.
2. Theoretical analyses
In the following we shall consider the free-space propagation of a quasi-monochromatic electromagnetic beam whose mean propagation axis coincides with the z axis of a cylindrical Cartesian reference frame , where with as the unit vector along the α axis. The quantities and will denote the transverse and longitudinal components, respectively. We consider a beam that, across the plane z = 0, is characterized by the transverse electric field . The total electric field in the half-space z>0 can be derived by the vectorial Rayleigh-Sommerfeld formulas 12].Eq. (1), the polar coordinates (ρ, θ, z) in Eq. (2) need to be transform into rectangular coordinates (x, y, z). Based on the following relation expressing Laguerre in terms of Hermite polynomials Eq. (2) can be expressed as sum of eHG modes:Eq. (1), and recalling the following integral formulaEquation (6) provides a general propagation expression for the field of nonparaxial vectorial eLG beams.
Within the paraxial regime, we often neglect the longitudinal component of the beam field. Expanding r into series and keeping the first and second terms, i.e., r≈z + (x 2 + y 2)/2z, Eq. (6) reduces to the field distribution of paraxial eLG beams
The eLG modes can alternatively defined with non-negative values of m as the following two types. Superscript “e” in the field distribution E ( e )(ρ,z) stands for even mode with the even function cos(mθ) for the azimuthal dependence and superscript “o” in E ( o )(ρ,z) stands for odd mode with the odd function sin(mθ). Based on the following relations Eqs. (6), the nonparaxial propagation expression for the even modes is given byEq. (10) reduces to13]Eqs. (14) reduces to
From the above paraxial results, the paraxial fundamental Gaussian mode yields
3. Numerical calculation results and comparative analyses
For illustrating the propagation properties of vectorial nonparaxial eLG beams in free space and comparing the results with that of the paraxial case, numerical calculations were carried out by use of the formulas derived in the previous section. Figure 1 gives the normalized strength distribution of a vectorial nonparaxial eLG beam with n = 2, m = 3 for different value f, as functions of the normalized transverse coordinate x/λ, evaluated at the transverse plane z = 15zR, where I = |Ex|2 + |Ey|2+|Ez|2. For convenience of comparison, the corresponding paraxial results Ip (dotted curve) and the contribution z longitudinal component to the strength of the electric field expressed Iz (dashdotted curve) are compiled together. One can see from Fig. 1 that when the value of parameter f is very small, the nonparaxial results by using Eqs. (6) coincide with the paraxial results by using Eqs. (7) quite well, so that for this case the paraxial approximate holds true and the z component is very small and can be negligible. However, with the increasing f, the difference between the paraxial and nonparaxial transverse components becomes obvious, and that the longitudinal component Iz become large and cannot be neglected. It means that for the large values of parameter f the vectorial nonparaxial approach instead of the scalar paraxial one should employed, and the vectorial nonparaxial behavior of eLG beams should be taken into consideration. Figure 2 shows the normalized strength distribution of a fundamental Gaussian beam at the transverse plane z = 15zR for different f value. It is worth noting that the fundamental Gaussian beam compared to the higher-order eLG beams with n = 2, m = 3, paraxial approximation provides better accuracy for values of f as high as 0.25, which is consistent with the previous results. However, toward the eLG as shown in Fig. 1, for values of f as low as 0.05, the difference between the nonparaxial results I and paraxial results Ip have been large, and that the condition is change with the radial number n and azimuthal parameter m.
It worth making a more farther comparison between the nonparaxial eLG beams and its paraxial approximation, the three-dimensional intensity distributions and corresponding contour graphs of nonparaxial and paraxial eLG beam with n = 2, m = 3 for f = 0.2 at the plane z = 15zR are plotted by using Eqs. (6) and (7), as shown in Fig. 3 . Figure 3(a) and (b) express the nonparaxial eLG beams, (c) and (d) are the paraxial cases. We can see from Fig. 3 that the paraxial eLG beam is a circular symmetrical dark hollow beam and the nonparaxial eLG beam is an elliptical symmetrical dark hollow beam. Figure 4 is plotted by using Eqs. (10) and (11) for even modes and Fig. 5 is plotted by using Eqs. (14) and (15) for odd modes, the calculation parameters are the same as those in Fig. 3. From Figs. 4-5, it can be seen that the intensity patterns of even and odd modes are notable different from those of the paraxial cases. The paraxial intensity patterns of even and odd modes are circular symmetry and approximately similar except for the difference of spatial orientation of beam lobes. While for the nonparaxial case, the intensity distributions of eLG beams lose their circular symmetry and become elliptical symmetry, and that the intensity patterns of even and odd modes also have obvious difference, due to the longitudinal component to the strength of the electric field become large and cannot be neglected when the parameter f is large value.
In the present paper the free-space propagation features of vectorial eLG beams have been investigated and analyzed comparatively in the paraxial and in the nonparaxial regimes. Based on the vectorial Rayleigh-Sommerfeld formulas and the relations between eLG and eHG modes, the analytical propagation expressions of a vectorial eLG beam along with its even and odd modes have been derived. The corresponding paraxial approximation expressions have been dealt with as special case of our general results. With the help of numerical calculations, the propagation properties of the nonparaxial vectorial eLG beams have been studied and analyzed comparatively with the paraxial results. The numerical results have shown that the parameter f plays an important role in determining the nonparaxial propagation properties of vectorial eLG beams. Under the condition that f<0.05 the paraxial approximation for eLG beams with n = 2, m = 3 is allowable. Otherwise, one must take into account the vector structure of eLG beams. The condition is different for different order eLG modes, also different from the case of fundamental Gaussian beam. Moreover, the intensity patterns of a nonparaxial vevtorial eLG beam along with its even and odd modes are notable different from those of the conventional transverse-mode patterns in the paraxial regime.
This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Y6090247, Y606320), Huzhou Civic Natural Science Fund of Zhejiang Province of China (2009C50064) and National Natural Science Foundation of China (10874150).
References and links
1. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973). [CrossRef]
2. 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]
3. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]
4. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998). [CrossRef]
5. S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 (2002). [CrossRef]
7. H. C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169(1-6), 9–16 (1999). [CrossRef]
9. R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004). [CrossRef]
10. K. Duan, B. Wang, and B. Lü, “Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 22(9), 1976–1980 (2005). [CrossRef]
12. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002). [CrossRef]
13. I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]