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Abstract
We demonstrate that Laguerre-Gauss (LG) beams can have a polygonal distribution with the radial and azimuthal components, the so-called polygonal Laguerre-Gauss beams. These light beams are generated with a linear and trigonometric phase function of the angular coordinate which transform the Perfect LG beams with a lens. The polygonal Laguerre-Gauss beams preserve their distribution within a maximum propagation distance without diffraction similar to that used in the propagation-invariant light fields. We expect that these polygonal Laguerre-Gauss beams will be a method for creating any distribution controlling the radial and azimuthal components; moreover, they can be a proposal to control the number of the ring structures of the light beams.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Structured light beams represent a valuable way to tailor the properties or the freedom grades within a system in itself [1,2], for instance, by the amplitude, phase, and polarization among some of their properties. Since the demonstration that Laguerre-Gauss beams have orbital angular momentum (OAM) due to their helical phase $\exp (im\phi )$ [3], the structured light beams have gained more importance and have increased their applications [4–6], such as in optical manipulation [7,8], in entanglement of low and high dimensional quantum state of photons [9,10], in nonlinear optics [11–13], and in optics communications under free-space and turbulent media [14–16].
The intensity distribution of the structured light, for instance, circular, elliptical, and parabolic shapes are obtained by the solutions of the scalar Helmholtz equation as the Bessel, Mathieu, and Weber beams, [17–20], which are the propagation-invariant light fields (PILFs). In the paraxial approximation, the solutions of the paraxial wave equation correspond to Hermite-Gauss, Laguerre-Gauss, and Ince-Gauss beams [21,22]. Recently, in the PILFs, a method has been developed to design any desired curves by concentrating the intensity with caustics [23], and the spectral phases are also used to construct the required PILFs [24].
Resonators with astigmatism have been used to generate polygonal distributions with triangular, square, and parallelogram shapes carrying OAM [25], and the spiral petal-shaped zonal plates [26] allow to modify the polygonal shapes in their size and OAM. Also, the optical elements are calculated to generate the beams with multi-contour plane curves [27], and the superimposed Bessel beams are used to generate dark-hollow optical beams with a controllable cross-sectional intensity distribution [28]. A general method for tailoring the shape of light beams is based on a tunable phase gradient defined along parametric curves [29–31], but these light beams have limited propagation. As not to limit their propagation, methods are used which are based in multi-singularity manipulation with the high-order cross-phase [32,33], and the corresponding caustics [23].
Previous works customize the light beam distributions for one or two radial concentric rings. To achieve the modulation of beams with more than two concentric rings and control their propagation distance, it is known that the Bessel beams and the Laguerre-Gauss beams can be comparable in their circular distribution and free-space propagation [34]. These structured light beams can adapt their intensity distribution by modulating their Fourier spectrum or seed beam with a phase [35] in the same experimental setup where only the seed beams and a lens are used, known as the seed-lens setup. This process can be seen as the application of a differential operator on a solution of the paraxial wave equation to generate a new distribution of the light beams; in Fourier space the differential operator is an algebraic function applied to the Fourier spectrum of the beams [36,37].
Thus, if the seed beams are defined by the perfect Laguerre-Gauss beams [35,37], the radial and azimuthal components are controlled, as well as the number of concentric rings in the beams. The structured beam in radial and angular parts is applied to the propagation properties in the beams by the radial part [38], and the changes in the transverse distributions can be achieved by modifications in the angular components [39].
This paper demonstrates that Laguerre-Gauss beams can have a polygonal distribution, the light beams that we call polygonal Laguerre-Gauss beams. They are generated with linear and trigonometric phase functions of the angular coordinate, which acts on perfect Laguerre-Gauss beams in Fourier space [35]. In these polygonal Laguerre-Gauss beams, their concentric rings, and the amount of OAM are controlled by the radial index $n$ and the azimuthal index $m$, respectively.
The polygonal LG beams preserve their distribution at the maximum distance without diffraction used in the PILFs, which is determined by the seed-lens setup with similar radii in their seeds. The Dirac Delta function and the perfect LG beams are the seed beams for the PILFs and the polygonal LG beams respectively.
2. Basic principles
To customize the structured light beams, we consider that a solution of paraxial wave equation $U(\mathbf {r},z)$ can be modified with the following transformation [35,36,40]:
where $\mathbf {r}{=}(x,y){=}(\rho \cos \phi,\rho \sin \phi )$ is the transverse radius vector in Cartesian and polar coordinates, and $\hat {\mathcal {A}}$ is a linear differential operator that transforms a beam $U_{0}(r,z)$, a solution of the paraxial wave equation (PWE), the seed beam with purely radial dependence. The differential operator can be expressed in general as a sum of products of derivatives $\hat {\mathcal {A}}(D_{x}, D_{y})\equiv h_{a,s}D^{a}_{x}D^{s}_{y}$, and $D^{a}_{x}$ and $D^{s}_{y}$ are the derivatives of $a$th-order and $s$th-order with respect to $x$ and $y$ respectively, and $h_{a,s}$ is an amplitude factor [35].In Fourier space, $\mathbf {k}{=}(k_x,k_y){=}(r\cos \phi,r\sin \phi )$ represents the Cartesian and polar coordinates, $\mathcal {F}^{-1}$ and $\mathcal {F}$ are the Fourier inverse transform and Fourier transform respectively. The transformation of Eq. (1) is evaluated as follows:
The algebraic operator can be considered with different forms of phase, and it can obtain many different beam shapes, whereas the differential operator does not have easy to find forms. The phase function $\mathcal {A}(r,\phi )$ can be seen as an astigmatism aberration [25,41–43], as well as some phase plates [26], the low and high phase-crosses [33], and some optical elements calculated to the generation of multi-contour plane curves [27]. Nevertheless, the phase functions which transform the seed beams with $n$ rings have been scarcely explored [40,44].
The phase function to generate the polygonal shapes comes from the ideas of previous works [35,40] where the sides in an intensity distribution can be controlled; nevertheless, the specific differential operator form is difficult to find. The beams with astigmatism or astroid distribution [35], guide us to define the phase function $\mathcal {A}(r,\phi )=\mathcal {A}(r)$ with a linear and sinusoidal dependence of the angular coordinate to find the polygonal Laguerre-Gauss beams as follow:
where $m_{1}$ is the topological charge, $q$ gives the sides of the polygon, and $m_{2}$ emphasizes the sides. The phase function is determined by the Eq. (3) acts on the seed perfect Laguerre-Gauss (PLGbs) [37]:To establish a possible experimental generation of the polygonal LG beams, we follow the scheme of Fig. 1, which describes the transformation of seed beams in the Fourier space of Eq. (2). The scheme shows the phase function $\mathcal {A}$, Eq. (3), which acts on the seed beams $\tilde {U}_{0}$ with the radius $b$, and these distributions are transformed with a converging thin lens of focal length $F$ of radius $R$. The polygonal LG beams are found at $z{=}2F$, whose spot radius is $W(z{=}F){\approx }R$.
Fig. 1. A scheme to transformation process described in Eq. (2), and a possible experimental generation to the polygonal Laguerre-Gauss (LG) beams, the seed-lens setup. Left and center: Triangular LG beams with a ring. Right: Pentagonal LG beams with three rings. At $z{=}0$ is the seed perfect LG beams with the same radii $b$, which is modulated with a phase function $\mathcal {A}$, and a lens of focal length $F$ of radius $R$. The polygonal LG beams are generated at $z{=}2F$, and they are propagated at a distance $Z_{max}{=}FR/b$. The perfect LG beams have the radial indexes $n{=}0$, $n{=}1$ and $n{=}4$, and the azimuthal index $m{=}0$.
The seed-lens setup of Fig. 1 is similar to the setup for the generation of the PILFs where the seed beam is defined by the Dirac Delta function [35]. The characteristics in this setup make it possible for the polygonal LG beams to have a propagation distance $Z_{max}{=}FR/b$ analog to the maximum propagation distance without diffraction of the PILFs [17,43].
3. Results and discussions
For the numerical generation of the light beams, the transverse radial coordinate is normalized with the Gaussian waist $w_0$. The longitudinal coordinate is normalized with respect to $L_{D}{=}kw^{2}_{0}/2$. The setup of Fig. 1 has the seeds with the same radii $b{=}1w_{0}$, $r_{0}{=}2w_{0}$, and the focal length of the lens is considered as $F{=}0.5L_{D}$. We numerically observe the propagation under free-space of the polygonal LG beams solving the paraxial wave equation $\nabla ^2_{T} U +2ik\partial U/\partial z {=} 0$ and simulate the experimental setup of Fig. 1 [43]. Figure 1 shows two cases of triangular and pentagonal LG beams with one and three rings at the position $z{=}2F$ and $Z_{max}{=}FR/b$ respectively. The polygonal shape with a Gaussian seed is not well defined at $z=2F$; however, their propagation evolution produces well-defined sides at $z{=}Z_{max}$, and the polygonal distributions, with more than two concentric rings, need the PLGbs seeds necessarily. In the rest of the work, we use the PLGbs with a radial index greater than zero due to its qualitatively better definition at $z{=}2F$ and $z{=}Z_{max}$.
Figure 2 shows the polygonal Laguerre-Gauss beams. These light beams are generated with the phase function $\mathcal {A}(\phi ){=}exp\left [i(m_{1}\phi +m_{2}sin(q\phi ))\right ]$, which acts on the seed perfect LG beams at the seed-lens setup. The seed Perfect LG beams have radial indexes of $n{=}1$ and $n{=}4$, and an azimuthal index of $m{=}0$, and the same radii $b{=}1w_{0}$. We consider that $m_{1}{=}6$, $m_{2}{=}0.2$, and $q{=}1,2,3,4, 8$ to generate the polygonal distributions, and the circular and elliptical shape. To have a remarkable elliptical shapes $m_{2}{=}0.7$. The parameters $q{=}3,4$, and $8$ determine the sides of the polygon, and $m_{2}$ is used to distinguish their sides. The elliptical shape vortex shown in Fig. 2(b) is similar to the elliptical vortex generated with the Ince-Gauss beams [22], where the radial number of rings can be controlled with the index $n$.
Fig. 2. Polygonal Laguerre-Gauss beams. The first and third rows show the polygonal intensity distribution with one and four rings and their corresponding phase distributions at $z{=}2F$ at the seed-lens setup. The intensity of the fourth ring is less than 1${\%}$ of its maximum intensity. The first and second columns show the circular and elliptical shapes. The columns $c$, $d$, and $e$ show the polygonal distributions with three, four, and eight sides.
Figure 2 shows the polygonal distribution with one and four rings and their corresponding phase in the first and third rows. The intensity of the fourth ring is less than $0.1$ with respect to maximum intensity. The first and second columns show the circular and elliptical shapes. The columns $c$, $d$, and $e$ show the polygonal shape with three, four, and eight sides. Figure 2 shows the phase, and the central part corresponds to a topological charge of $m_{1}{=}6$ for the five distributions; these phases have a polygonal shape. Additionally, these light beams have from three to eight well-defined sides in the four-ring polygonal distribution, see Fig. 2.
Figure 3 shows the polygonal LG beams propagation with one ring in Fig. 3(a), and with four rings in Fig. 3(b), both beams with eight sides. We can see that the beams have a broadening at $Z_{max}$, such as the regular Laguerre-Gauss beams where their intensity is reduced by half [34]. At this position, we consider that their distributions maintain their structure, and they have only a broadening or scaling in their cross section, for example, the beams with four rings in Fig. 3(b). Figure 3 shows the light beams at position $Z_{max}{=}0.9L_{D}$, and $Z_{max}{=}2.3L_{D}$ for one and four rings respectively, and the spots radii of the beams are $W(z{=}F){=}1.8w_{0},$ and $W(z{=}F){=}4.6w_{0}$.
Fig. 3. Polygonal Laguerre-Gauss beams under free-space propagation: a) Polygonal beam with one ring, and b) polygonal beam with four rings, both beams with eight sides. The free-space propagation at plane $x{-}z$ for the polygonal Laguerre-Gauss beams, whose propagation distance without diffraction is $Z_{max}{=}0.9L_{D}$ a), and $Z_{max}{=}2.3L_{D}$ b).
For the distributions, with two, three, and four sides ($q=2,3,4$) and one ring, are notable that at $Z_{max}$ the distributions have rotated slightly because the phase has nonlinear dependence [45], see Fig. 1. In Fig. 3 the changes are less notable because of the number of their sides. The helical phase $exp(im\phi )$ in the beams has a linear dependence in $\phi$ and a nonlinear dependence can be periodic in $\phi$, for example, with the phase $exp\left [im sin(q\phi )\right ]$ [45], where the angular rotation in the beams is constant or non-constant associated with the linear phase and the nonlinear phase respectively, and $m$ and $q$ establish the magnitude of the rotation. In addition, there are some other cases where the phase can be dependent $\phi$ and $r$ [44].
Figure 4 shows the polygonal LG beams behavior modifying parameter $m_{1}$ in the phase function $A(\phi )$. The three columns show the polygonal LG beams with four sides when $m_{1}{=}3, 6, 9$. The first column corresponds to one, four, and ten rings with $n{=}1,4,10$ in the radial component, and $m_{1}{=}3$. The light beams have a broadening when the topological charge increases. For $m_{1}{\geq }6$ the last ring starts to disappear when the radial index is $n{\geq }2$, see the second and third columns of Fig. 4. For $5 \leq n \leq 10$ the sides of the polygonal LG beams are less notable, where it can be necessary to look for other phase functions $A(\phi )$.
Fig. 4. Polygonal Laguerre-Gauss beams with different topological charge $m_{1}$. The three columns correspond to $m_{1}{=}3, 6, 9$, and the three rows correspond to different radial index components with $n=1,4, 10$.
In order to avoid the broadening of the cross section in polygonal LG beams when the topological charge increases, some articles have recently been published on the perfection property in polygonal distributions [46], i.e. the intensity distributions and the cross sections are independent of their topological charges [47]. For the polygonal LG beams, the possibility that they maintain the size independently of their topological charge can be explored, for instance, with the generation of the perfect LG beams [37,48].
Meanwhile, the polygonal LG beams experimental generation can be made according to some techniques implemented in the spatial light modulator (SLM) to create the light beams [40,49]. It knows that an SLM needs a field defined analytically or numerically then we can numerically simulate the two-stage of the scheme of Fig. 1 [43], and the polygonal LG beams obtained at $z=2F$ in the simulation can be the distribution numerical necessary to implement an SLM with a technique to recover the beams required.
4. Conclusions
In this paper, we demonstrate that Laguerre-Gauss beams have a polygonal distribution with the radial and azimuthal components, which we call polygonal Laguerre-Gauss beams. These beams are generated with a linear and trigonometric phase function of the angular coordinate which transform the perfect LG beams with a lens.
The polygonal LG beams preserve their distribution at a maximum propagation distance without diffraction determined by the seed-lens setup, which is used as a reference in the propagation-invariant light fields. We consider that the phase function shown in this work can be a first function that inspires a generatrix function to create any distribution which will control the radial and azimuthal components in the light beams. Additionally, we expect that these light beams can be applied in optical manipulation and optical communications contexts.
Acknowledgment
Thanks to the Consejo Nacional de Humanidades, Ciencias y Tecnologias de Mexico (CONAHCYT), the Sistema Nacional de Investigadoras e Investigadores (SNII), and the Vicerrectoria de Investigación y Estudios de Posgrado (VIEP) of the Benémerita Universidad Autónoma de Puebla (BUAP). Also, thanks to Adriana Inclán-Ladino for her valuable comments, and the author thanks the reviewers for their helpful insights.
Disclosures
The author declares no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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