## Abstract

On the basis of angular spectrum representation and the stationary-phase method, a far-field expression for nonparaxial Gaussian beams diffracted at a circular aperture is derived, which permits us to study the far-field nonparaxial properties of apertured Gaussian beams both analytically and numerically. It is shown that for the apertured case, the *f*-parameter and the truncation parameter affect the beam’s far-field properties. The *f*-parameter plays the more important role in determining the beam nonparaxiality than does the truncation parameter, whereas the truncation parameter additionally influences the beam diffraction. A comparison with the paraxial case is made. For the unapertured case our results reduce to the previous ones.

©2003 Optical Society of America

## 1. Introduction

Thus far most of the problems involving beam propagation have been studied successfully within the framework of the paraxial approximation. However, the paraxial approximation is not relevant for beams with a large divergence angle or a small spot size that is comparable with the wavelength. For these cases a rigorous nonparaxial theory is necessary. Various nonparaxial approaches—for example, the perturbation method [1], the power-series expansion [2,3], transition operators [4,5], angular spectrum representation [6–8], and the virtual source method [9]—have been developed to treat beam propagation beyond the paraxial approximation, but the results have been restricted to the unapertured case. In practice, the aperture effect exists, more-or-less. Our purpose in the present paper is to deal with nonparaxial Gaussian beams diffracted at a circular aperture. In Section 2, by use of angular spectrum representation and the method of stationary phase, a far-field expression for apertured nonparaxial Gaussian beams is derived and analyzed physically. Detailed numerical results are presented in Section 3 to illustrate the nonparaxial properties of apertured Gaussian beams in the far field and to compare results with those for the paraxial and unapertured cases, respectively. Finally, Section 4 summarizes the main results obtained in this paper.

## 2. Far-field expression for apertured nonparaxial Gaussian beams

In the Cartesian coordinate system the field of an initial Gaussian beam at the plane *z*=0 is given by

where *w*
_{0} is the waist width. Assume that, as shown in Fig. 1, a circular aperture of radius *a* is located at the plane *z*=0. The field just behind the aperture reads as

where the window function of the aperture is

By using the angular spectrum representation [10] and Eqs. (2) and (3), and by performing a coordinate transformation from the Cartesian coordinate system to the cylindrical one, we can express the field at the *z*-plane as

where

$$={\left(\frac{k}{2\pi}\right)}^{2}{\iint}_{{x}^{2}+{y}^{2}\le {a}^{2}}{E}_{0}(x,y,0)\mathrm{exp}[-ik\left(\mathit{px}+\mathit{qy}\right)]dxdy$$

$$={\left(\frac{k}{2\pi}\right)}^{2}{\int}_{0}^{a}{\int}_{0}^{2\pi}\mathrm{exp}(-\frac{{\rho}^{2}}{{w}_{0}^{2}})\mathrm{exp}[-ik\rho \left(p\phantom{\rule{.2em}{0ex}}\mathrm{cos}\theta +q\phantom{\rule{.2em}{0ex}}\mathrm{sin}\theta \right)]\mathrm{d\theta}\mathrm{d\rho}$$

$$=\frac{{k}^{2}}{2\pi}{\int}_{0}^{a}\mathrm{exp}(-\frac{{\rho}^{2}}{{w}_{0}^{2}})\rho {J}_{0}\left[k\rho \sqrt{{p}^{2}+{q}^{2}}\right]\mathrm{d\rho},$$

where *k* is the wave number related to the wavelength λ by *k*=2π/λ and *J*
_{0}(·) denotes the zeroth-order Bessel function. As usual, the contribution of evanescent waves can be ignored; thus Eq. (4) is written as

The substitution from Eq. (5) into Eq. (7) yields

where

The triple integration of Eq. (8) is quite time consuming. To obtain the analytical expression and gain intrinsic physical insight, we perform the far-field approximation. As pointed out in Refs. [6] and [10], Eq. (9) can be evaluated asymptotically by the method of stationary phase when *k*(*x*
^{2}+*y*
^{2}+*z*
^{2})^{1/2}→∞. Thus the integrals over *p* and *q* can be calculated analytically, and the final result is arranged as

On substituting Eq. (10) into Eq. (8) and recalling the formula

we obtain

$$=i\frac{z}{2k{r}^{2}}{e}^{\mathit{ikr}}\sum _{m=0}^{\infty}\frac{{\left(-1\right)}^{m}}{{2}^{2m}{f}^{2m+2}{\left(m!\right)}^{2}}{\left(\frac{{x}^{2}+{y}^{2}}{{r}^{2}}\right)}^{m}\left[\Gamma (1+m,{\delta}^{2})-m!\right],$$

where

and Γ(·) denotes the incomplete Gamma function.

Equation (12) is the basic analytical result obtained in this paper, which provides an approximate asymptotic form for evaluation of the far-field of apertured nonparaxial Gaussian beams. Equation (12) indicates that the axial symmetry and spherical wave front remain unchanged for apertured nonparaxial Gaussian beams in the far field. Moreover, from Eq. (12) it follows that *E*(*x,y,z*) depends mainly on two key parameters: *f* and δ. *f* is a measure of the relative spot size *w*
_{0}/λ. If *w*
_{0}/λ is much larger than 1, the paraxial approximation is allowable; otherwise the nonparaxial correction should be taken into consideration.

Expanding *r* into a series and keeping the first and second terms,

When we replace *r* of the exponential part in Eq. (12) with Eq. (16), and other terms with *z*, Eq. (12) simplifies to

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times \sum _{m=0}^{\infty}\frac{{\left(-1\right)}^{m}}{{2}^{2m}{f}^{2m+2}{(m!)}^{2}}{h}^{m}\left[\Gamma (1+m,{\delta}^{2})-m!\right],$$

where

It can be readily shown that Eq. (17) is the Fraunhofer diffraction formula of apertured paraxial Gaussian beams in the paraxial regime. In fact, the Fraunhofer diffraction for the apertured case reads as

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times {\iint}_{{x}_{0}^{2}+{y}_{0}^{2}\le {a}^{2}}E({x}_{0},{y}_{0},0)\mathrm{exp}\{-\frac{\mathit{ik}}{2z}\left(x{x}_{0}+y{y}_{0}\right)\}d{x}_{0}d{y}_{0}.$$

The substitution from Eq. (2) into Eq. (19) and straightforward integral calculations lead to Eq. (17).

The diffraction effect introduced by an aperture in Eq. (12) is described by the truncation parameter δ. The smaller the truncation parameter, the more strongly the field is diffracted by the aperture. For the unapertured case δ→∞, Eq. (12) reduces to

Apart from an unimportant factor, Eq. (20) is consistent with Eq. (12) in Ref. [11], which is the far-field expression for unapertured nonparaxial Gaussian beams.

## 3. Numerical calculation results and analysis

Numerical calculations were made with Eq. (12), which contains factorial and incomplete Gamma functions whose numerical processing is not easy but can be performed by means of computer. In addition, the series in Eq. (12) is fast convergent; thus usually the use of ten terms of the series is sufficient to achieve satisfactory numerical results in comparison with the direct numerical integral of Eq. (12). Figure 2 gives irradiance distributions |*E*(*x*,0,10*z*_{R}
)|^{2} of a nonparaxial Gaussian beam (solid curve) at the plane *z*=10*z*_{R}
(*z*_{R}
=π${w}_{0}^{2}$/λ-Rayleigh length) for different values of *f* and δ. For convenience of comparison, the paraxial results (dotted curve) calculated by use of Eq. (17) are compiled together. From Fig. 2 we find that the difference between the nonparaxial and the paraxial results is negligible for *f*=0.08 and δ≥1 [see Figs. 2(e) and 2(f)] A comparison of Fig. 2(e) with Figs. 2(b) and 2(d) shows that the difference between them becomes evident as *f* increases and δ decreases. Therefore both parameters *f* and δ affect the beam’s far-field properties. The *f*-parameter plays a more important role in determining the beam nonparaxiality than does the truncation parameter δ [for example, compare Figs. 2(b) and 2(e); Figs. 2(a) and 2(c), respectively], whereas, as shown in Figs. 2(a)–2(c), the truncation parameter δ additionally influences the beam diffraction.

For the sake of comparison with the previous research [11], the beam width *w*(*z*) of the axially symmetric beams is defined as one at which the irradiance is reduced to 1/2 its maximum value. Thus

The far-field divergence angle reads as

Figure 3 shows the variation of the far-field divergence angle of a Gaussian beam versus parameters δ and *f* for the nonparaxial (solid curve) and paraxial (dotted curve) cases, which gives some interesting physical results. First, let us consider Fig. 3(a) for a fixed value of δ; the difference between the solid and the dotted curves becomes small as 1/*f* increases. For δ=2.2, where the aperture effect can be neglected, there is no difference between the nonparaxial and paraxial cases as *f*≤0.18, which is consistent with the result in Ref. [12], if the difference of the initial field expression Eq. (1) in this paper and Eq. (7) in Ref. [12] is considered, which gives rise to the factor 1/√2. Depending on the truncation parameter δ, the nonparaxial and paraxial results are coincident for *f*≤0.12, δ=1 and for *f*≤0.07, δ=0.5.

Furthermore, our calculations indicate that the far-field divergence angle approaches zero as the value of 1/*f* becomes large enough, because for this case the Gaussian beam given in Eq. (1) approaches an infinite plane wave as the waist width becomes large enough. Then Fig. 3(b) shows additionally that the far-field divergence angle reaches the value corresponding to the unapertured case of nonparaxial and paraxial Gaussian beams, respectively, as the truncation parameter δ≥2.2. For our calculation parameters, θ (for the nonparaxial case)=18.5°, θ_{0}(for the paraxial case)=20.8° for *f*=0.32, θ=12.7°, θ_{0}=13.3° for *f*=0.20, and θ=θ_{0}=5.4° for *f*=0.08. Finally, Figs. 4(a) and 4(b) indicate that the far-field divergence angle θ of nonparaxial Gaussian beams asymptotically reaches the same value θ=32.77° as the parameter δ or *w*
_{0}/λ approaches zero. The physical reason for this is evident, because for both cases the source can be regarded as one point described by a δ-function, whose divergence angle approaches the upper limit of Eq. (22), which is expressed as [11]
${\theta}_{max}={\mathrm{tan}}^{-1}{\left(\sqrt{2}-1\right)}^{\frac{1}{2}}=32.77\xb0.$

## 4. Concluding remarks

In this paper the nonparaxial propagation of Gaussian beams through a circular aperture has been studied. The stationary-phase method has been used to solve Eq. (8), resulting in the analytical field expression Eq. (12), which is valid when the propagation distance *z* is much larger than the Rayleigh length *z*_{R}
. We have found that if there is an aperture, both the *f*-parameter and the truncation parameter δ affect the far-field behavior of nonparaxial Gaussian beams. The *f*-parameter plays the more important role in determining the beam nonparaxiality than does the truncation parameter δ, whereas the truncation parameter additionally influences the beam diffraction. For the two limiting cases, our results reduce to those for the paraxial case and the unapertured cases, respectively. Finally, for clarifying the main physical aspect, our treatment is limited to the scalar case in the cylindrical coordinate system. For an arbitrarily shaped aperture without rotational symmetry (e.g., for a rectangular or polygonal aperture) and vectorial nonparaxial beams, the Cartesian coordinate system has to be used, and the relevant results will be published elsewhere.

## Acknowledgments

This research was supported by the Foundation of State Key Laboratory of Laser Technology and the National Hi-Tech Project of China. The authors are grateful to the reviewers for insightful comments, which were useful for improving the paper.

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