Complex source point theory of paraxial and nonparaxial elliptical Gaussian beams

Jie Zhu; Taofen Wang; Kaicheng Zhu

doi:10.1364/OE.467428

1. Introduction

Optical beams with elliptical geometry can have great superior features during propagation and have become increasingly interesting and demanding in numerous applications [1–4]. For instance, considerable research activity has been directed toward the fundamental properties of the elliptical Gaussian beam (EGB) which are often encountered in laser optics, material processing, as well as nonlinear optics, for it possesses high orbital angular momentum (1000ћ per photon), although without an optical vortex, and can play a unique role for rotating micrometer-sized particles and cold atoms [2–4]. Such novel beams are known to be solutions of the paraxial propagation equation [1] and can be generated via different methods, such as from a mode-locked laser beam, a frequency-double laser beam or a semiconductor laser beam by focusing a Gaussian beam with a cylindrical lens. Their propagation properties have also been extensively studied. It is found that a vectorial nonparaxial EGB has anomalous behavior during propagation near focus [5]. The self-focusing and rotating properties of an EGB in a nonlinear medium have further been analyzed, implying that upon proper choice of the beam parameters, the difficulties which arise from bulk damage due to self-focusing in high-power optical systems can be avoided or ameliorated effectively [6,7]. More recently, the propagation properties of an EGB in turbulent atmosphere were reported, and remarkably found to have smaller scintillation index compared with that of a circular symmetric Gaussian beam in a weakly turbulent atmosphere, which may be particularly useful for applications with long-distance free-space optical communications [8–11].

On the other hand, we note that beams with large divergence angle or small spot size that is of the order of light wavelength are usually confronted in practical application. It is obvious that the theory of optical propagation and transformation based on paraxial approximation is no longer valid in this case [12–16], and, in order to resolve the problem of beam’s nonparaxial propagation, various more accurate methods have been developed, which includes vectorial Rayleigh-Sommerfeld diffraction integral method [13], perturbation power series method [14], transition operators [15], angular spectrum representation [16], virtual source point technique [17], and so on. Among them, virtual source method which was firstly introduced by Deschamps [18] and systematically extended by Felsen [19,20] has been widely applied to investigate the characterization and propagation of beams within and beyond the paraxial regime. Later in a series of papers researchers anticipated this method to obtain Hermite-, Bessel– and Laguerre–Gaussian waves [17–27]. A higher-order complex point source was also constructed to produce the elegant Laguerre–Gaussian waves and the elegant Hermite-Laguerre–Gaussian waves [28–30]. Note that this method was further extended to four or many complex point sources to obtain other beam models such as cosh-Gaussian waves [31–33]. Recently, the propagation characteristics of Airy–Gaussian beam [34], cosine-Gaussian and Bessel-Gaussian beam [35], Pearcey beam [36], Mathieu–Gauss beam [37], Lommel–Gauss beam [38], asymmetric Bessel–Gauss beam [39] and fractional-order Bessel–Gauss beam [40] were extensively and precisely studied by using the virtual source technique. Additionally, other types of shaped beams like the propagation-invariant beam or pulsed Gaussian beam were also investigated using this method [41–45].

However, to the best of our knowledge, the complex point sources have not been used to explore the beams with elliptical geometry symmetry. The EGBs have extraordinary optical characteristics comparing to Gaussian beams with circular symmetry spot. Considering that these unique optical topology structure and mechanical property of EGBs are expected to open up new avenues in related scientific fields, accurately studying their propagation characteristics, especially nonparaxial propagation or nearfield propagation, is of great scientific and practical interest. In this paper, our aim is to investigate the free-space propagation features of an EGB with planar or cylindrical wavefront in both the paraxial and the nonparaxial regimes using the virtual source technique. We analytically derive a closed-form expression of the composite wave through the operator transformation method, and numerically confirm the influence of the nonparaxial corrections on the propagation of EGBs in free space. Our analysis may help for better understanding and application of the EGBs with cylindrical wavefront propagating beyond the paraxial limit.

2. Theoretical modeling

In free space the propagation of a monochromatic scalar light field E(x, y, z) is described by the homogeneous Helmholtz equation for z > 0:

(1)$${\nabla ^2}E + {k^2}E = 0,$$

where ${\nabla ^2} = \partial _x^2 + \partial _y^2 + \partial _z^2$ with ${\partial _p} = {\partial / {\partial p}}$ (p = x, y, z) and $k = {{2\pi } / \lambda }$ is the wavenumber in free space. Throughout this paper the time dependence of factor exp(iωt) is assumed and will be canceled.

Let $E({x,y,z} )= \textrm{exp} ({ - ikz} )E_G^p({x,y,z} )$, while $E_G^p({x,y,z} )$ obeys the paraxial equation $(\partial_x^2 + \partial_x^2 - 2ik{\partial_z} )E_G^p({x,y,z} )= 0$ and its solutions can be the well-known circular or elliptical Gaussian beam [1]:

(2)$$E_G^p({x,y,z} )= \frac{1}{q}\textrm{exp} \left( { - ikz - \frac{{{u^2} + {v^2}}}{q}} \right),$$

where u = x/w₀ and v = y/w₀ are the scaled coordinates with the unit of w₀, q =1 − iz_s, z_s = z/z_R is the scaled propagation distance with the unit of the Rayleigh range of the beam ${z_R} = {{kw_0^2} / 2}$, and w₀ is an arbitrary transverse scale which characterizes the beam width.

To explore new solution of Helmholtz Eq. (1), we introduce the following transformation operator $T({{\partial_x}} )$ as

(3)$$\hat{T}({{\partial_x}} )= \textrm{exp} ({\alpha w_0^2\partial_x^2} ),$$

where α = α_r + iα_i is an adjustable complex parameter with the real part α_r ≥ 0 being always assumed in this paper. Because the operators $\hat{T}$, $\partial _x^m$ and $\partial _y^m$ are always commutable, it follows that $E({x,y,z} )$ and ${E_e}({x,y,z} )= T({{\partial_x}} )E({x,y,z} )$ satisfy the same homogeneous Helmholtz Eq. (1). In principle, a large number of new solutions E´(x, y, z) of Eq. (1) can be constructed based on the existed solutions E(x, y, z) of the homogeneous Helmholtz Eq. (1) or its corresponding paraxial equation through carefully designing the transformation operator $\hat{T}$.

In this paper the new paraxial solution $E_{e - G}^p({x,y,z} )$ is constructed as

(4)$$E_{e - G}^p({u,v,{z_s}} )= \hat{T}E_G^p({u,v,{z_s}} )= \frac{1}{q}\textrm{exp} ({\alpha \partial_u^2} )\textrm{exp} \left( { - ikz - \frac{{{u^2} + {v^2}}}{q}} \right).$$

Making use of the relations (Seeing Eq. (A.33) of Ref. [46])

(5)$$f(x )= \int_{ - \infty }^\infty {f(u )} \delta ({x - u} )du,$$

(6)$$\textrm{exp} \left( {\alpha \frac{{{\partial^2}}}{{\partial {x^2}}}} \right)\delta ({x - u} )= \frac{1}{{\sqrt {4\pi \alpha } }}\textrm{exp} \left[ { - \frac{{{{({x - u} )}^2}}}{{4\alpha }}} \right],$$

we obtain

(7)$$E_{e - G}^p({u,v,z} )= \textrm{exp} ({i{\alpha_i}\partial_x^2} )E_G^p({u,v,z} )= \frac{1}{{\sqrt {4\pi \alpha } }}\int_{ - \infty }^\infty {E_G^p({x,v,z} )} \textrm{exp} \left[ { - \frac{{{{({x - u} )}^2}}}{{4\alpha }}} \right]dx.$$

In practice, for a pure imaginary parameter α = iα_i, Eq. (7) is the well-known Fresnel integration (transformation). In this sense, we point out that the operator transformation represents an additional diffraction along a specified direction or an anisotropy diffraction, for example, along x-direction here.

Substituting the paraxial solution (2) into Eq. (7) and performing the Gaussian integral, we get

(8)$$\begin{array}{c} E_{e - G}^p({u,v,z} )= \frac{1}{{q\sqrt {4\pi \alpha } }}\textrm{exp} \left( { - ikz - \frac{{{v^2}}}{q}} \right)\int_{ - \infty }^\infty {\textrm{exp} \left[ { - \frac{{{x^2}}}{q} - \frac{{{{({x - u} )}^2}}}{{4\alpha }}} \right]} dx\\ = \frac{1}{{\sqrt {q({q + 4\alpha } )} }}\textrm{exp} \left( { - ikz - \frac{{{u^2}}}{{q + 4\alpha }} - \frac{{{v^2}}}{q}} \right). \end{array}$$

It should be pointed out that the expression (8) can also be directly obtained using the Gleisher identity [47].

In fact, on the initial plane of z = 0, Eq. (8) reduces to

(9)$$E_{e - G}^p({u,v,0} )= \frac{1}{{\sqrt {1 + 4\alpha } }}\textrm{exp} \left( { - \frac{{{u^2}}}{{1 + 4\alpha }} - {v^2}} \right),$$

which, depending on the parameter α, represents an elliptical Gaussian source model with planar wavefront (for real α) or with cylindrical wavefront (for complex α). According to the above equations, we obtain the scaled lengths of the major and the minor axes of the elliptical optical spot on the initial plane respectively as $w_{sx}^{(0 )} = \frac{1}{2}\sqrt {1 + 4{\alpha _r} + \frac{{16\alpha _i^2}}{{1 + 4{\alpha _r}}}}$ and $w_{sy}^{(0 )} = \frac{1}{2}$.

Figure 1 shows some typical numerical evaluating results on the intensity pattern and the corresponding phase distribution evolutions of the EGBs paraxial propagating in free space at various distances z_s for several sets of α. It is clearly seen that, depending on the values of α, the intensity patterns experience different variations upon propagation. For a pure imaginary parameter α (α_r = 0) as shown in Fig. 1(a)-(d), the intensity pattern first rotates and finally evolves into a circular spot in the far field. It is interesting to note that when the beam propagates at z_s = 8 (i.e., z_s = 4α_i which will be discussed in detail below), the intensity distribution restores the shape back to its initial field (z_s = 0) but with the orientation being rotated by 90 degrees. Meanwhile, for other cases of parameter α (α_r ≠ 0) as shown in Fig. 1(e)-(l), the intensity pattern only rotates its orientation but retains its elliptical configuration during the whole propagation. The phase distribution evolutions are also checked to understand the propagation features of the EGBs comprehensively. We point out that for all the values of α, the wavefront always evolves into a spherical wave in the far field (for example, z_s = 30). However, the wavefront in the near field is highly sensitive to the values of α. Specifically, for a pure real parameter α (α_i = 0), the wavefront of the propagating beam is observed to evolve from a planar configuration to an elliptical shape, and finally to a spherical structure (seeing Fig. 1(i)-(l)), while for a complex parameter α (α_i ≠ 0), the wavefront of the propagating beam is confirmed to transform from a focusing cylindrical shape with the cylindrical axis located along x-axis at the initial plane into a defocusing cylindrical configuration with the cylindrical axis located along y-axis at z_s = 4α_i plane (seeing Fig. 1(a)-(h)). Obviously, the phase wavefront exhibits a profound astigmatism during propagation between z_s = 0 and z_s = 8. In short, the results confirm that for the operator transformation parameter α, its imaginary part α_i mainly influences the wavefront configuration evolutions while its real part α_r primarily affects the intensity pattern variations.

Fig. 1. Evolutions of the intensity patterns (Upper) and the corresponding phase distributions (Bottom) of the EGB paraxial propagating in free space at various distances z_s with α = 2i (a-d); α = 2 + 2i (e-h); α = 2 (i-l).

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We further investigate the on-axial intensity of the paraxial propagating EGBs based on the expression (8) as

(10)$$I_{e - G}^p(z )= {|{E_{e - G}^p({0,0,z} )} |^2} = \frac{1}{{\sqrt {({1 + z_s^2} )[{{{({{z_s} - 4{\alpha_i}} )}^2} + {{({4{\alpha_r} + 1} )}^2}} ]} }}.$$

Obviously, at z_s = 4α_i, the on-axial intensity of the beam arrives its maximum ${I_{\max }} =\{ {|{E_{e - G}^p({0,0,z} )} |}^2 \}_{\max } = \frac{1}{{({4{\alpha_r} + 1} )\sqrt {1 + 16\alpha _i^2} }}$.

For the case of a pure real parameter α (α_i = 0), the on-axial intensity $\textrm{I}_{e - G}^{On - axis}(z )$ monotonously decreases with increasing the propagation distance. However, for other cases of complex parameter α (α_i ≠ 0), the behavior of the on-axial intensity to propagation distance is quite different and exhibits more complicated variations. These results are further demonstrated in Fig. 2 where the on-axial intensity evolutions against the propagation distance for the EGBs paraxial propagating in free space with different values of α are illustrated. Note that when the parameter α takes a complex value, instead of monotonously decreasing during propagation, the on-axial intensity is observed to experience a rapid increase when the beam travels over certain distances, and after reaching its peak at some specified distances, starts to decrease again in the far field. Due to this abnormal growth behavior, the EGBs restore their initial on-axial intensity during propagation which can be termed as self-focusing. Comparing curves (a) to (d) shown in Fig. 2(A) indicates that decreasing values of the real part α_r will effectively reinforce the self-focusing effect. The most extreme example is that for a pure imaginary α = iα_i, the beam completely recovers the original on-axial intensity of the input plane at the self-focusing plane. Besides, we find that the self-focusing position directly relates to the parameter with z_s = 4α_i, which is also seen from Fig. 2(B).

Fig. 2. On-axial intensity distributions versus the propagation distances z_s for EGBs paraxial propagating in free space with different values of α: (A): α = 2i (a), 0.25 + 2i (b), 1 + 2i (c), 2 (d); (B): α = 0.5i (a), 1i (b), 2i (c), 3i (d).

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Mathematically, for a pure imaginary parameter α = iα_i, Eq. (9) can be rewritten as

(11)$$\begin{aligned}{c} E_{e - G}^p({u,v,z = 0} )&= \frac{1}{{\sqrt {1 + 16\alpha _i^2} }}\textrm{exp} \left[ { - \frac{i}{2}\arctan ({4{\alpha_i}} )- \frac{{{u^2}}}{{1 + 16\alpha_i^2}} - {v^2} - \frac{{4i{\alpha_i}{u^2}}}{{1 + 16\alpha_i^2}}} \right]\\ &= |{E_{e - G}^p({u,v,z = 0} )} |\textrm{exp} \left[ { - \frac{i}{2}\arctan ({4{\alpha_i}} )- \frac{{4i{\alpha_i}{u^2}}}{{1 + 16\alpha_i^2}}} \right], \end{aligned}$$

where

(12)$$|{E_{e - G}^p({u,v,z = 0} )} |= \frac{1}{{\sqrt {1 + 16\alpha _i^2} }}\textrm{exp} \left( { - \frac{{{u^2}}}{{1 + 16\alpha_i^2}} - {v^2}} \right),$$

and from Eq. (8) we can obtain

(13)$$\begin{aligned}{c} E_{e - G}^p({u,v,{z_s} = 4{\alpha_i}} )= \frac{1}{{\sqrt[4]{{1 + 16\alpha _i^2}}}}\textrm{exp} \left[ { - 4ik{\alpha_i} - \frac{i}{2}\arctan ({4{\alpha_i}} )- {u^2} - \frac{{{v^2}}}{{1 - 4i{\alpha_i}}}} \right]\\ = |{E_{e - G}^p({v,u,{z_s} = 0} )} |\textrm{exp} \left[ { - 4ik{\alpha_i} - \frac{i}{2}\arctan ({4{\alpha_i}} )+ \frac{{4i{\alpha_i}{v^2}}}{{1 + 16\alpha_i^2}}} \right]. \end{aligned}$$

Obviously, the spatial structure of the optical field intensity at z_s = 4α_i denoted by Eq. (13) is exactly the same as that at z = 0 depicted by Eq. (11), except a rotation of 90 degrees with respect to the original pattern. Besides, we also find from Eq. (11) that the optical field at z = 0 has a focusing cylindrical wavefront with the cylindrical axis located along x-axis, which can be used to well explain the self-reappearance and self-focusing effects near the self-focusing plane. Equation (13) further confirms that the wavefront of the EGB turns into a defocusing cylindrical one with cylindrical axis positioned along y-axis at z_s = 4α_i, which is in accordance with that shown in Fig. 1.

We now proceed to study the evolution behavior of the beam width of the EGBs paraxial propagating in free space. It is straightforward to obtain

(14)$${w_{sx}} = \sqrt {\left\langle {{u^2}} \right\rangle } = {\left[ {\frac{{\int_{ - \infty }^\infty {{u^2}{{|{E_{e - G}^p({x,y,z} )} |}^2}dudv} }}{{\int_{ - \infty }^\infty {{{|{E_{e - G}^p({x,y,z} )} |}^2}dudv} }}} \right]^{{1 / 2}}} = \frac{1}{2}\sqrt {1 + 4{\alpha _r} + \frac{{{{({{z_s} - 4{\alpha_i}} )}^2}}}{{1 + 4{\alpha _r}}}} ,$$

(15)$${w_{sy}} = \sqrt {\left\langle {{v^2}} \right\rangle } = \frac{{\sqrt {1 + z_s^2} }}{2},$$

and the ratio

(16)$$e = \frac{{{w_{sx}}}}{{{w_{sy}}}} = {{\sqrt {1 + 4{\alpha _r} + \frac{{{{({{z_s} - 4{\alpha_i}} )}^2}}}{{1 + 4{\alpha _r}}}} } / {\sqrt {1 + z_s^2} }} = \sqrt {\frac{{1 + 4{\alpha _r}}}{{1 + z_s^2}} + \frac{{{{({{z_s} - 4{\alpha_i}} )}^2}}}{{({1 + 4{\alpha_r}} )({1 + z_s^2} )}}} ,$$

which directly relates to the ellipticity of the transverse beam spot.

Equations (14) and (15) demonstrate that the scaled beam width w_sy always monotonously increases with the propagation distance, while the scaled beam width w_sx displays a more complex behavior depending on the values of the parameter α, i.e., it monotonously increases with increasing propagation distance z_s for pure positive real α (α_i = 0) but shows a decreasing first and then increasing trend with reaching minimum ${\left\langle {{u^2}} \right\rangle _{\min }} = {\alpha _r} + {1 / 4}$ at z_s = 4α_i for complex α (α_i ≠ 0). These analytical predictions are in good agreement with the results from numerical evaluations as shown in Fig. 3 where the variations of the parameters w_sx and w_sy against the propagation distance for the EGBs with different values of α are graphed in detail.

Fig. 3. Scaled beam width w_sx (w_sy) versus propagation distances z_s for EGBs paraxial propagating in free space with different values of α: (A): α = 2i (a), 1 + 2i (b), 2 (c); (B): α = 0.5i (a), 1i (b), 2i (c).

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According to Eq. (16), we plot in Fig. 4 the variations of the ratio of the beam width e against the propagation distance z_s for the EGBs travelling in free space with different values of α. It is found that for a pure imaginary parameter α (α_r = 0), the ratio e falls quickly to the minimum value and then rises gradually to reach a final stable value of 1 with increasing propagation distance, implying that the ellipticity of the EGB varies remarkably in the near field and a circular Gaussian beam profile without elliptical symmetry is eventually formed in the far field. However, note that for other values of parameter α (α_r ≠ 0) the evolution behavior of the ratio e is quite different: it remains stable all the way to z_s = 60 after decreasing to the minimum value in the near field, indicating that under such conditions, the EGB can preserve an almost constant ellipticity when propagating in the far field. Obviously, the ellipticity of the propagating EGB is associated with the real part α_r, and our results suggest that a stable elliptical beam structure can be realized in the far field when the real part α_r is taken as nonzero values.

Fig. 4. Ratio of the beam width e versus the propagation distances z_s for EGBs paraxial propagating in free space with different values of α denoted on the corresponding curves.

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3. Complex-source-point spherical waves for EGBs

To extend our studies beyond the paraxial limit to achieve large angle trajectories desirable for various applications, in this section we are going to use the virtual source method to discuss the nonparaxial propagation characteristics of the EGBs travelling in free space. According to the method, the beam is assumed to be generated by a source of strength S_ext located at z = z_ext exterior to the physical space z > 0. Note that although the source strength S_ext and the (complex-valued) source location z_ext are undetermined for the present, the requirement of yielding the desired input field distribution will specify them in the subsequent stage. With the source included, the wave function E(x, y, z) obeys the inhomogeneous Helmholtz equation

(17)$$({\partial_x^2 + \partial_y^2 + \partial_z^2 + {k^2}} ){E_G}({x,y,z} )={-} {S_{cs}}\delta (x )\delta (y )\delta ({z - i{z_R}} ),$$

with ${S_{cs}} ={-} i{z_R}\textrm{exp} ({ - k{z_R}} )$ and the solution reads as [17,22,27,28]

(18)$${E_G}({x,y,z} )= {S_{cs}}{{\textrm{exp} ({ikR} )} / {4\pi R}},$$

where $R = \sqrt {{x^2} + {y^2} + {{({z + i{z_R}} )}^2}} = {z_R}\sqrt {4{\varepsilon ^2}({{u^2} + {v^2}} )+ {{({{z_s} - i} )}^2}} $ with ɛ⁻¹ = kw₀. Equation (18) is called as a complex-source-point spherical wave which satisfies exactly the scalar imhomogeneous Helmholtz Eq. (17) [21]. Applying the operator $\hat{T}$ on both sides of Eq. (17), we have

(19)$$\begin{array}{c} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over T} ({\partial_x^2 + \partial_y^2 + \partial_z^2 + {k^2}} ){E_G}({x,y,z} )= ({\partial_x^2 + \partial_y^2 + \partial_z^2 + {k^2}} ){E_{e - G}}({x,y,z} )\\ ={-} {S_{cs}}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over T} \delta (x )\delta (y )\delta ({z - i{z_R}} ). \end{array}$$

Then let us construct a new wave solution as

(20)$${E_{e - G}}({u,v,z} )= \hat{T}{E_G}({x,y,z} )= \frac{{{S_{cs}}}}{{4\pi }}\textrm{exp} ({\alpha \partial_u^2} )\frac{{\textrm{exp} ({ikR} )}}{R},$$

Obviously, the wave E_e-G also exactly meets the scalar Helmholtz Eq. (17).

As pointed out by Couture and Belanger [21], the paraxial Gaussian beams can become the complex-source-point spherical waves (18) when all-order corrections are included. Based on their results, the complex amplitude of a propagating Gaussian wave can be expressed as

(21)$${E_G}({u,v,{z_s}} )= E_G^p({u,v,{z_s}} )+ \sum\limits_{n = 1}^\infty {{\varepsilon ^{2n}}E_G^{({2n} )}({u,v,{z_s}} )} ,$$

where $E_G^p({u,v,{z_s}} )$ is the paraxial Gaussian beam given by Eq. (2), and $E_G^{({2n} )}({u,v,{z_s}} )$ (n = 1, 2, 3, …) are the higher order corrections taking the form as (seeing Eq. (16) in [21])

(22)$$E_G^{({2n} )}({u,v,{z_s}} )= {\left( {\frac{{{u^2} + {v^2}}}{{{q^2}}}} \right)^n}L_n^n\left( {\frac{{{u^2} + {v^2}}}{q}} \right)E_G^p({u,v,{z_s}} ),$$

with $L_n^n({\cdot} )$ being the associated Laguerre polynomial

(23)$$L_n^n(x )= ({2n} )!\sum\limits_{m = 0}^n {\frac{{{{({ - x} )}^m}}}{{m!({n - m} )!({n + m} )!}}} .$$

Similarly, according to Eq. (21), for an EGB, we arrive

(24)$${E_{e - G}}({u,v,{z_s}} )= \textrm{exp} ({\alpha \partial_u^2} ){E_G}({u,v,{z_s}} )= E_{e - G}^p({u,v,{z_s}} )+ \sum\limits_{n = 1}^\infty {{\varepsilon ^{2n}}E_{e - G}^{({2n} )}({u,v,{z_s}} )} ,$$

where $E_{e - G}^p({u,v,{z_s}} )= \textrm{exp} ({\alpha \partial_u^2} )E_G^p({u,v,{z_s}} )$ is just the paraxial approximation solution of the EGB determined by Eq. (8). According to Eq. (7) we can generally express the correction terms as

(25)$$E_{e - G}^{({2n} )}({u,v,{z_s}} )= \textrm{exp} ({\alpha \partial_u^2} )E_G^{({2n} )}({u,v,{z_s}} )= \frac{1}{{\sqrt {4\pi \alpha } }}\int_{ - \infty }^\infty {E_G^{({2n} )}({\eta ,v,{z_s}} )} \textrm{exp} \left[ { - \frac{{{{({u - \eta } )}^2}}}{{4\alpha }}} \right]d\eta$$

for n = 1, 2, 3, ….

As a typical example, we consider a propagating EGB with the lowest correction term (n = 1) and express $E_G^{(2 )}({u,v,s} )$ as follows

(26)$$E_G^{(2 )}({u,v,{z_s}} )= \frac{{{u^2} + {v^2}}}{{{q^3}}}\left( {2 - \frac{{{u^2} + {v^2}}}{q}} \right)\textrm{exp} \left( { - ikz - \frac{{{u^2} + {v^2}}}{q}} \right).$$

Upon inserting Eq. (26) into Eq. (25) we have

(27)$$\begin{aligned}{c} E_{e - G}^{(2 )}({u,v,{z_s}} )&= \textrm{exp} ({\alpha \partial_u^2} )E_G^{(2 )}({u,v,{z_s}} )\\ &= \frac{1}{{{q^3}\sqrt {4\pi \alpha } }}\textrm{exp} \left( { - ikz - \frac{{{v^2}}}{q}} \right)\int_{ - \infty }^\infty {({{\eta^2} + {v^2}} )} \left( {2 - \frac{{{\eta^2} + {v^2}}}{q}} \right)\textrm{exp}\!\left[ { - \frac{{{\eta^2}}}{q} - \frac{{{{({u - \eta } )}^2}}}{{4\alpha }}} \right]\!d\eta . \end{aligned}$$

After making use of the following relation

(28)$$\int_{ - \infty }^\infty {} {x^n}\textrm{exp} ({ - \alpha {x^2} + \beta x} )dx = \frac{{\sqrt \pi }}{{{{({2i} )}^n}{\alpha ^{{{({n + 1} )} / 2}}}}}\textrm{exp} \left( {\frac{{{\beta^2}}}{{4\alpha }}} \right){H_n}\left( {\frac{{i\beta }}{{2\sqrt \alpha }}} \right),$$

and completing straightforward calculations, Eq. (27) which represents the lowest-order correction can be evaluated and turns out to be

(29)$$\begin{aligned}{c} E_{e - G}^{(2 )}({u,v,{z_s}} )&= \frac{{E_{e - G}^p({u,v,{z_s}} )}}{{{q^2}}}\left\{ {2{v^2} - \frac{{2\alpha q}}{{q + 4\alpha }}{H_2}\left( {\frac{{iu}}{2}\sqrt {\frac{q}{{\alpha ({q + 4\alpha } )}}} } \right)} \right.\\ &\left. { - \frac{1}{q}\left[ {{v^4} - \frac{{2\alpha q{v^2}}}{{q + 4\alpha }}{H_2}\left( {\frac{{iu}}{2}\sqrt {\frac{q}{{\alpha ({q + 4\alpha } )}}} } \right) + \frac{{{\alpha^2}{q^2}}}{{{{({q + 4\alpha } )}^2}}}{H_4}\left( {\frac{{iu}}{2}\sqrt {\frac{q}{{\alpha ({q + 4\alpha } )}}} } \right)} \right]} \right\}. \end{aligned}$$

Actually, note that other higher-order correction terms can also be checked using this same approach.

Furthermore, setting x = y = 0 and using ${H_{2n}}(0 )= {({ - 1} )^n}\frac{{({2n} )!}}{{n!}} \ne 0$, Eq. (29) reduces to

(30)$$E_{e - G}^{(2 )}({0,0,{z_s}} )= E_{e - G}^p({0,0,{z_s}} )\frac{{\alpha q}}{{q + 4\alpha }}\left( {4 - \frac{{12\alpha }}{{q + 4\alpha }}} \right) = \frac{{{\alpha ^2}}}{{{q^{{3 / 2}}}{{({q + 4\alpha } )}^{{5 / 2}}}}}\textrm{exp} ( - ikz).$$

Equation (30) represents the nonparaxial corrections to the propagating EGBs on the z-axis, indicating that the on-axial optical field amplitude of an EGB does not vanish although its counterpart of a circular Gaussian beam does always vanish during their non-paraxial propagation. In fact, for other higher-order correction terms the conclusion is also true because $E_G^{({2n} )}({u,v,s} )$ are the even functions of u which will always lead to even-order ${H_{2n}}(u )$ occurrence in the $E_{e - G}^{({2n} )}({u,v,s} )$ expressions. We can take $E_G^{(4 )}({0,0,s} )$ and $E_G^{(6 )}({0,0,s} )$ for example, and, according to Eq. (25), they can be obtained respectively as

(31)$$\begin{aligned}{c} E_{e - G}^{(4 )}({0,0,{z_s}} )&= \frac{1}{{\sqrt {4\pi \alpha } }}\int_{ - \infty }^\infty {E_G^{(4 )}({x,0,{z_s}} )\textrm{exp} \left( { - \frac{{{x^2}}}{{4\alpha }}} \right)} dx\\ &= \frac{1}{{q\sqrt {4\pi \alpha } }}\int_{ - \infty }^\infty {\frac{{{x^4}}}{{{q^4}}}L_2^2\left( {\frac{{{x^2}}}{q}} \right)\textrm{exp} ({ - B{x^2}} )} dx = \frac{1}{{4{q^5}{B^2}\sqrt {\alpha B} }}\left( {9 - \frac{{15}}{{qB}} + \frac{{105}}{{16{q^2}{B^2}}}} \right),\end{aligned}$$

(32)$$\begin{aligned}{c} E_{e - G}^{(\textrm{6} )}({0,0,{z_s}} )&= \frac{1}{{\sqrt {4\pi \alpha } }}\int_{ - \infty }^\infty {E_G^{(6 )}({x,0,{z_s}} )} \textrm{exp} \left( { - \frac{{{x^2}}}{{4\alpha }}} \right)dx = \frac{1}{{q\sqrt {4\pi \alpha } }}\int_{ - \infty }^\infty {\frac{{{x^6}}}{{{q^6}}}L_3^3\left( {\frac{{{x^2}}}{q}} \right)\textrm{exp} ({ - B{x^2}} )} dx\\ &= \frac{1}{{4{q^7}{B^3}\sqrt {\alpha B} }}\left( {75 - \frac{{1575}}{{8qB}} + \frac{{2835}}{{16{q^2}{B^2}}} - \frac{{3465}}{{64{q^3}{B^3}}}} \right), \end{aligned}$$

where $B = \frac{{q + 4\alpha }}{{4\alpha q}}$.

In fact, we can also express the exact elliptical Gaussian wave (20) as

(33)$$\begin{aligned}{c} {E_{e - G}}({u,v,{z_s}} )&= \textrm{exp} ({\alpha \partial_u^2} ){E_G}({u,v,{z_s}} )= \frac{1}{{\sqrt {4\pi \alpha } }}\int_{ - \infty }^\infty {{E_G}({\eta ,v,{z_s}} )} \textrm{exp} \left[ { - \frac{{{{({u - \eta } )}^2}}}{{4\alpha }}} \right]d\eta \\ &= \frac{1}{{\sqrt {4\pi i\alpha } }}\frac{{{S_{cs}}}}{{4\pi }}\textrm{exp} \left( { - \frac{{{u^2}}}{{4\alpha }}} \right)\\ &\times \int_{ - \infty }^\infty {\frac{1}{{R(\eta )}}} \textrm{exp} \left[ {ikR(\eta )- \frac{{{\eta^2}}}{{4\alpha }} + Re \left( {\frac{1}{{2\alpha }}} \right)u\eta - i2\pi {\mathop{\rm Im}\nolimits} \left( {\frac{1}{{4\pi \alpha }}} \right)u\eta } \right]d\eta , \end{aligned}$$

In this sense, Eq. (33) can be considered as the one-dimensional Fourier transformation on the function $f(x )= \frac{1}{{R(x )}}\textrm{exp} \left[ {ikR(x )+ \frac{{i{x^2}}}{{4\alpha }} + Re \left( {\frac{1}{{2\alpha }}} \right)ux} \right]$ with $R(x )= {z_R}\sqrt {4{\varepsilon ^2}({{x^2} + {v^2}} )+ {{({{z_s} - i} )}^2}} $ for complex α, and can directly reduce to

(34)$${E_{e - G}}({0,0,{z_s}} )= \frac{1}{{\sqrt {4\pi \alpha } }}\frac{{{S_{cs}}}}{{4\pi {z_R}}}\int_{ - \infty }^\infty {\frac{1}{{\sqrt {4{\varepsilon ^2}{x^2} + {{({{z_s} - i} )}^2}} }}} \textrm{exp} \left( {\frac{i}{{2{\varepsilon^2}}}\sqrt {4{\varepsilon^2}{x^2} + {{({{z_s} - i} )}^2}} - \frac{{{x^2}}}{{4\alpha }}} \right)dx.$$

which represents the on-axial field amplitude of the elliptical Gaussian wave.

Figure 5 shows the evolutions of the intensity distribution patterns ${I_{e - G}}({u,v,{z_s}} )= |E_{e - G}^p({u,v,{z_s}} )+ {\varepsilon^2}E_{e - G}^{(2 )}({u,v,{z_s}} ) |^2$ and the corresponding phase distributions of the EGBs at various distances z_s for several sets of α where the lowest non-paraxial correction is taken into consideration. Compared with the paraxial approximation results shown in Fig. 1, it is interesting to note that although the transverse intensity patterns taken at different propagation distances present a quite similar tendency, the corresponding phase configurations exhibit some remarkable variations especially for the EGB with pure imaginary parameter α (α_r = 0) propagating near the focusing position at z_s = 4α_i (seeing Fig. 5(c)).

Fig. 5. Same as Fig. 1 but including the lowest-order correction term.

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We then investigate the nonparaxial effect on the on-axial intensities $I_{e - G}^{p + (n )}({0,0,{z_s}} )= {\left|{E_{e - G}^p({0,0,{z_s}} )+ \sum\limits_{l = 1}^n {{\varepsilon^{2l}}} E_{e - G}^{({2l} )}({0,0,{z_s}} )} \right|^\textrm{2}}$ of the EGBs in free space. Figure 6 comparatively presents the on-axial intensities against the propagation distance for the paraxial approximation solution and the ɛ²ⁿ-order (n = 1, 2, 3) non-paraxial correction solutions of the EGBs with different values of α. It is easy to find that for a parameter α with α_r ≠ 0, the effect of the nonparaxial correction is inconspicuous during the whole propagation (seeing curves (B) in Fig. 6). However, note that for a pure imaginary parameter α = iα_i (seeing curves (A) in Fig. 6), the on-axial intensities of the nonparaxial correction solutions are greatly enhanced compared with that of the paraxial approximation solution particularly near the focusing position. The right figure of Fig. 6 highlights the intensity variations near the self-focusing position, in which a shift of the self-focusing position toward the input plane can also be found by adding the higher-order nonparaxial correction.

Fig. 6. On-axial intensity distributions versus the propagation distances z_s for EGBs in free space with α =2i (A) and 0.5 + 2i (B). The right figure is the local part of that shown in the left figure for $I_{e - G}^p({0,0,{z_s}} )$ (a), $I_{e - G}^{p + (1 )}({0,0,{z_s}} )$ (b), $I_{e - G}^{p + (2 )}({0,0,{z_s}} )$ (c), and $I_{e - G}^{p + (3 )}({0,0,{z_s}} )$ (d).

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To explore the influence of the parameter α on the on-axial intensity evolution of the non-paraxial correction solutions of the EGBs, we plot in Fig. 7 the evolutions of the on-axial intensities $I_{e - G}^{p + (3 )}({0,0,{z_s}} )$ including the ɛ⁶-order corrections (dashed line), and, for comparison, the on-axial intensities $I_{e - G}^p(z )$ associated with the paraxial propagation (solid line) against the propagation distance with various values of α. It is clearly seen that for a pure real parameter α, there is an excellent agreement between $I_{e - G}^p(z )$ and $I_{e - G}^{p + (3 )}({0,0,{z_s}} )$ (seeing (d) in Fig. 7(A)). For a complex parameter α, increasing the values of the real part α_r will greatly reduce the effect of the nonparaxial correction, and $I_{e - G}^p(z )$ is also found to be in good accordance with $I_{e - G}^{p + (3 )}({0,0,{z_s}} )$ (seeing (b) and (c) in Fig. 7(A)). However, when the real part α_r decreases to zero, the on-axial intensity difference between paraxial approximation and nonparaxial correction expression becomes significant (seeing (a) in Fig. 7(A) and Fig. 7(B)) and the higher-order correction solution must be considered. Therefore, the results imply that the nonparaxial correction term can enhance and accelerate the self-focusing behavior of the EGBs with cylindrical wavefront especially for α being pure imaginary, suggesting the important role of the virtual source method in the study of field EGB propagation.

Fig. 7. On-axial intensity distributions versus the propagation distances z_s for EGBs in free space with the values of the parameter α being the same as used in Fig. 2. Solid curves correspond to $I_{e - G}^p({0,0,{z_s}} )$, while dashed curves correspond to $I_{e - G}^{p + (3 )}({0,0,{z_s}} )$.

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According to Eq. (34), we further make a precise numerical simulation for the evolution of the on-axial intensity of an elliptical Gaussian wave. In fact, Eq. (34) represents a high oscillatory integral which can be evaluated with various highly efficient calculation methods [48,49]. Figure 8 shows the on-axial intensities against the propagation distance for the elliptical Gaussian wave as well as the paraxial approximation solution and the ɛ⁶-order non-paraxial correction solution of the EGBs with different values of α. As shown in Fig. 8(A), when the real part α_r ≠ 0, the exact calculation result agrees well with the paraxial approximation solution and ɛ⁶-order correction solution. The difference appears and becomes evident when the real part α_r is small or zero. The right figure of Fig. 8 confirms that for a pure imaginary α, there is a shift of the self-focusing position toward the input plane and a greater on-axial intensity near the focusing position for the elliptical Gaussian wave, which is similar to the results of the EGBs taking into account the nonparaxial correction.

Fig. 8. On-axial intensity distributions versus the propagation distances z_s for EGBs in free space with α =2i (A) and 0.5 + 2i (B). The right figure is the local part of that shown in the left figure for $I_{e - G}^p({0,0,{z_s}} )$(a), the exact intensity based on Eq. (34) (b), and $I_{e - G}^{p + (\textrm{3} )}({0,0,{z_s}} )$ (c).

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Finally, it should be pointed out that, based on Eq. (33), the intensity pattern on the specified plane can also be obtained by making use of the numerical integrals. The performed calculation results indicate that the intensity patterns obtained are very similar to those shown in Fig. 5 where the lowest-order correction term is included. Then here these figured results are omitted.

4. Conclusions

In summary, we have comparatively studied the paraxial and nonparaxial propagation characteristics of the EGBs with cylindrical wavefront in free space. Firstly, we have derived the analytical expression for the field distribution of an EGB paraxial propagating in free space. Self-reappearance and self-focusing behaviors are observed during their propagation and the associated parameter conditions have been analytically derived and numerically proved. Secondly, we have further obtained the strict integral expression of the EGBs under nonparaxial conditions using the virtual (complex) sources and operator transformation method. As an extension from the previous studies about the nonparaxial Gaussian beams with circular symmetric spot, such an EGB also satisfies the same Helmholtz equation obeyed by the corresponding Gaussian beam with circular spot. The paraxial approximation and the nonparaxial corrections of all orders of an EGB have been determined and can be used to accurately reveal some important information about their near field propagation characteristics. Note that the numerical analysis of the on-axial intensity of the EGBs with added nonparaxial correction confirms that the nonparaxial correction term can effectively enhance and accelerate the self-focusing behavior of the EGBs with cylindrical wavefront. Our findings are expected to lay a theoretical foundation for better application of the EGBs in future scientific experiments, especially in the field of near-field optics even though calculating nonparaxial propagation is relatively complex. Finally, it should be pointed out that the operator transformation technology can also be readily extended to treat other beams with elliptical geometry straightforwardly for a variety of applications.

Funding

Science and Technology Program of Guizhou Province ([2020]1Y025); National Natural Science Foundation of China (62163008).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Complex source point theory of paraxial and nonparaxial elliptical Gaussian beams

Abstract

1. Introduction

2. Theoretical modeling

3. Complex-source-point spherical waves for EGBs

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (34)

Optics Express