1. Introduction

Hermite-Gaussian (HG) beams and Laguerre-Gaussian (LG) beams are the two main beam families in the theory of laser optics [1]. The HG beams have rectangular symmetry and the LG beams have rotational symmetry. It is well known that sinusoidal functions represent the modes of rectangular metal waveguides and closed resonators, the HG functions represent the modes of complex lenslike media and open resonators. Hermite-sinusoidal-Gaussian (HsG) beams are obtained as rectangular-symmetric solutions to the paraxial wave equation [2–4]. While in the cylindrically symmetric systems, Bessel functions represent the modes of metal waveguides and closed resonators, and Laguerre-Gaussian functions represent the modes of complex lenslike media and open resonators. The existence of rectangular-symmetry HsG function solutions implies the existence of the cylindrically symmetry Laguerre-Bessel-Gaussian (LBG) function solutions. Recently, Tovar introduced new LBG beams as solutions of the wave equation in cylindrical coordinates [5]. By changing parameter in the LBG beams it can reduce to the LG beam modes and Bessel-Gaussian (BG) modes. The new LBG solutions are valid for electromagnetic waves traveling through free space, and they are valid for propagation through rotational symmetric paraxial optical systems represented by ABCD matrices as well. The above modes are restricted within the framework of the paraxial approximation. However, the paraxial approximation is not relevant for the beams with a large divergence angle or a small spot size that is comparable with the wavelength. Our purpose in the present paper is to extend the paraxial LBG beams to the nonparaxial LBG modes. Up to now, a variety of nonparaxial approaches have been developed to treat beam propagation beyond the paraxial approximation [6–10]. In this paper, we use the vectorial Rayleigh-Sommerfeld formulas to deal with the nonparaxial propagation of vectorial LBG beams. In Section 2, the nonparaxial propagation equation of vectorial LBG beams is derived from which the far-field and paraxial expressions are obtained by making use of the far-field and paraxial approximations, respectively. The obtained results can reduce to those of the cases for the LG beams and the BG beams under certain conditions. Detailed numerical results are presented in Section 3 to illustrate the nonparaxial propagation properties of LBG beams and to compare results with those for the paraxial cases. Finally, the main results obtained in this paper are summarized in Section 4.

2. Nonparaxial propagation equation of vectorial Laguerre-Bessel-Gaussian beams

Consider a LBG beam that is linearly polarized in the x direction, whose field E=E _x(x ₀, y ₀, 0)i+E _y (x ₀, y ₀, 0)j at the source plane z=0 reads as [5]

where i and j are the unit vectors in the x and y directions, respectively, L ^|m| _n denotes the Laguerre polynomial with the radial and angular mode orders of n and m, w ₀ is the waist width, J _q is the qth-order Bessel function of the first kind, (x ₀, y ₀) and (ρ ₀, θ ₀) are the transverse coordinates in the Cartesian (x, y, z) and cylindrical coordinates (ρ, θ, z) system, respectively. ρ ₀=(x ² ₀+y ² ₀)^1/2 and θ ₀=tan^-1(y ₀/x ₀). Figure 1 shows the three-dimensional intensity distributions and corresponding contour graphs of LBG beams with n=3, m=3, q=0 for various values of the coefficient α at the plane z=0. When α=0 and q=0, it is clear from the Fig. 1(a), as from Eq. (1), that the result reduces to the pure LG mode. With the increasing of α, the modulated effect of Bessel factor becomes more and more evident, as Figs. 1(b), 1(c) and 1(d). For the larger values of α, such as Figs. 1(e) and 1(f), the beam is largely a Bessel beam.

The vectorial Rayleigh-Sommerfeld formulas deliver the exact propagation of the optical field in the whole half-space z>0 once the transverse component at the initial plane z=0 is given, i.e. [10, 11],

where r ₀=x ₀ i+y ₀ j and r=x i+y j+z k, i, j and k are the unit vectors in the x, y and z directions, respectively, and

k is the wave number related to the wavelength λ by k=2π/λ. Making use of the following approximation [10, 12]

where r=(x ²+y ²+z ²)^1/2. On substituting from Eqs. (1) and (4) into Eq. (2), transforming Cartesian to cylindrical coordinates and integrating with respect to the coordinate θ, we get

where

 

Fig. 1. Three-dimensional intensity distributions and corresponding contour graphs of LBG beams with n=3, m=3, q=0 for various values of the coefficient α at the plane z=0. (a) α=0, (b) α=0.5, (c) α=1, (d) α=2, (e) α=3.5, (f) α=5.

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After tedious but straightforward integral calculations to Eq. (5), we obtain

where

and

is the f parameter, Γ(·) is the Gamma function.

Equation (7) is the basic result obtained in this paper, which provides a general expression for the field distribution of nonparaxial vectorial LBG beams and can reduce to the far-field and paraxial expressions under the far-field and paraxial approximation conditions. It can be found from Eq. (7) that the field distribution of nonparaxial vectorial LBG beams is dependent on the parameters of m, n, q and α, but also dependent on the parameter of f.

In the far field, approximation (4) can be further approximated as [10, 12]

From Eq. (7), we obtain the far-field expressions for the nonparaxial LBG beams:

Expanding r into series and only keeping the first and second terms

and replacing r of the exponential part in Eq. (7a) with Eq. (13) and other terms with z, we obtain the field distribution of paraxial LBG beams

where

When q=0 and α=0, Eq. (7) reduces to

where

Equation (16) is the nonparaxial propagation expression for the vectorial LG beams. Although Eq. (16) and the results in Ref. [10] are different in form, their numerical calculation results are consistent.

In addition, letting n=m=0 in Eq. (7), we get

where

Equation (18) is the nonparaxial propagation equation for the vectorial BG beams, which is a new result also obtained in this paper.

Similarly, the far-field expressions of nonparaxial LG and BG beams along with the field distribution of paraxial LG and BG beams can also be obtained from Eqs. (16) and (18) under the far-field and paraxial approximation conditions, respectively.

 

Fig. 2. Normalized intensity distributions of nonparaxial vectorial LBG beams at various values of the propagation distance z for f=0.02.

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3. Numerical calculation results and analyses

Some numerical calculations are carried out by using the analytical formulas derived in the previous section. In the calculation, the series l ₁ in Eq. (7) is fast convergent, but the series l ₂ is more slowly convergent, thus usually the use of fifteen terms of the series l ₁ and eighty terms of the series l ₂ is sufficient to achieve satisfactory numerical results in comparison with the direct numerical integral results. The normalized transversal irradiance distributions of nonparaxial vectorial LBG beams at various values of the propagation distance z are given in Figs. 2 and 3 (solid curve), where I=|E _x|²+|E y|²+|E _z|² and z _R=πw ² ₀/λ is the Rayleigh length. For convenience of comparison, the corresponding paraxial results I _p (dotted curve) calculated by using Eq. (14) are compiled together. The calculation parameters are n=3, m=3, q=0, α=1, f=0.02 in Fig. 2, and f=0.2 in Fig. 3.

 

Fig. 3. Normalized intensity distributions of nonparaxial vectorial LBG beams at various values of the propagation distance z for f=0.2.

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Comparing Figs. 2 and 3, we find that the f parameter is not relevant for the paraxial propagation of LBG beams, whereas it plays an important role in determining the nonparaxial propagation properties of LBG beams. It can be seen from Fig. 2 that there is little difference between nonparaxial and paraxial results, which means the paraxial approximate is valid for the case of smaller value of f=0.02. For large value f, for example f=0.2 as showing in Fig. 3, the propagation behavior of nonparaxial LBG beams is notably different from that of paraxial LBG beams.

 

Fig. 4. Three-dimensional intensity distributions and corresponding contour graphs of nonparaxial (a) and paraxial (b) LBG beams with n=3, m=3, q=0 for f=0.2 at the plane z=15z _R.

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From Figs. 2 and 3, we can see that the propagation of paraxial and nonparaxial LBG beams is all instable in the near field. With the increasing of propagation distance z, the optical field distributions gradually reach stable. For vividly comparing nonparaxiality with paraxiality of LBG beams, the three-dimensional intensity distributions and corresponding contour graphs of nonparaxial and paraxial LBG beams for f=0.2 at the plane z=15z _R are plotted in Fig. 4. It can be seen from Fig. 4 that the difference between nonparaxial result I and paraxial result I _p is very obvious. The energy is mostly concentrated in the outer lobes of the beam for paraxial LBG beams. However, for the nonparaxial case, the energy is mostly concentrated in the inner lobes of the beam, which becomes similar to that of an elegant LG beam.

 

Fig. 5. Normalized intensity distributions of a LBG beam with n=3, m=3, q=0 at the plane z=15z _R for various values of α when f=0.2. The solid curves denote the nonparaxial results I, the dash curves express I _x and the dotted curves are the paraxial results I _p. (a) α=0, (b) α=0.5, (c) α=1, (d) α=2, (e) α=3.5, (f) α=5.

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Fig. 6. Normalized intensity distributions of a LG beam with n=3, m=3 at the plane z=15z _R. (a) f=0.02, (b) f=0.2.

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The normalized intensity distributions at the e plane z=15z _R of a LBG beam with n=3, m=3, q=0 are given in Fig. 5 for various values of α when f=0.2, where I _x=|E _x|². The solid curves denote the nonparaxial results I, the dash curves express I _x and the dotted curves are the paraxial results I _p. As seen from Fig. 5, there is a little difference between I and I _p, which means the paraxial approximation is not applicable. Furthermore, I differs from I _x, the longitudinal component I _z(x,0,15z _R) is not negligible. However, the difference between I, I _x and I _p changes irregularly with the parameter of α. For example, when α=3.5 as shown in Fig. 5(e), the difference between the three results is very small. Especially within the range of main energy of beam, the paraxial approximation is applicable and the longitudinal component is small and negligible. It should be pointed out that the analytical formulas for the nonparaxial propagation of vectorial LBG beams in this paper provide advantages not only of the explicit demonstration of their physical meaning, but also of the reduction of the computation time. Moreover, under certain conditions, the obtained results can be more easily reduced to those of the cases for the nonparaxial LG and BG beams. Figure 6 is plotted by using Eq. (16) for a pure LG beam with n=3, m=3, where z=15z _R, f=0.02 in Fig. 6(a) and f=0.2 in Fig. 6(b). The nonparaxial propagation of vectorial BG beams with α=0, q=3.5 is given in Fig. 7 by using Eq. (18). The other calculation parameters are the same as Fig. 6. It can be seen from Figs. 6 and 7 that the f parameter also plays an important role in determining the nonparaxiality of vectorial LG and BG beams.

 

Fig. 7. Normalized intensity distributions of a BG beam with q=0, α=3.5 at the plane z=15z _R. (a) f=0.02, (b) f=0.2.

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4. Conclusions

In this paper, the nonparaxial propagation of vectorial Laguerre-Bessel-Gaussian beams has been studied in detail. Starting from the vectorial Rayleigh Sommerfeld formulas, the closed-form formulas for the nonparaxial propagation of vectorial LBG beams have been derived, and the far field approaches and paraxial approximation expressions have been dealt with as special cases of our general results. With the help of the derived formulas, the propagation properties of nonparaxial and paraxial LBG beams in free space are illustrated and analyzed comparatively with numerical examples. The numerical results have shown that the propagation of paraxial and nonparaxial LBG beams is all instable in the near field. The f parameter is not relevant for the paraxial propagation of LBG beams, whereas it plays an important role in determining the nonparaxial propagation properties of LBG beams. Furthermore, the beam parameter α also affects the propagation behavior of nonparaxial LBG beams. Under certain conditions, the obtained results reduce to those of the cases for the LG beams and the BG beams.