Hermite–Gaussian beams with sinusoidal vortex phase modulation

Guanxue Wang; Yue Li; Xinzhi Shan; Yu Miao; Xiumin Gao

doi:doi:10.3788/COL202018.042601

Chinese Optics Letters, 2020, 18 (4): 042601, Published Online: Apr. 14, 2020

Hermite–Gaussian beams with sinusoidal vortex phase modulation Download： 708次

论文大纲

Guanxue Wang Yue Li Xinzhi Shan Yu Miao Xiumin Gao ^*

Author Affiliations

School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

Hermite–Gaussian beams phase modulation sinusoidal vortex phase modulation spatial light modulator focus mode

Abstract

In this Letter, vortex phase and sinusoidal phase modulations of Hermite–Gaussian beams are studied theoretically and experimentally. The coding method of the experiment is introduced in detail, and the evolution law of focus under different beam order (m, n) and topological charge (l) is given. In order to verify the accuracy of the generation experiment, the optical field distribution under sinusoidal vortex modulation is analyzed deeply. The relevant analysis and methods provided in this Letter have certain practical significance for the development of laser mode analysis, optical communication, and other fields.

Hermite–Gaussian (HG) beams are one of many high-order beams. Because of its wide application in optical communication, laser mode analysis, super-resolution imaging, and other fields, it attracts an increasingly growing number of researchers to study it^{[1–7" target="_self" style="display: inline;">–7]}. Among them, the propagation characteristics of HG beams in atmospheric turbulence have been widely studied^{[8–11" target="_self" style="display: inline;">–11]}. Seshadri reviews the methods of processing real-argument HG beams by Fourier transform^[12]. Reflected in a dielectric/vacuum interface, it has been proved that a group of high-order HG beams can generate well-defined evanescent light modes, where the light intensity distribution of the light modes is limited (confined) to a subwavelength region near the interface outside the dielectric^[13]. In addition, the combination of HG modes and other modes such as Laguerre–Gaussian (LG) is widely studied as well^[14,15].

In recent years, there is a hot topic about researches on the vortex-modulated HG beams. Hebri et al. conducted a comprehensive analysis of the diffraction of the structural aperture vortex beam^[16]. Chen et al. derived the analytical wave expressions used to represent asymmetric elliptical beams, which are generated by the conversion of asymmetric HG modes through an astigmatic converter^[17]. Zhou et al. proposed an implementation of a robust mode sorter that can order a large number of HG patterns based on the relationship between LG modes and HG modes^[18]. Shen et al. proposed a dual off-axis pumping scheme for generating wavelength-tunable high-order HG modes, in which the mode and wavelength can be actively controlled by off-axis displacement and pump power in the Yb:CaGdAlO₄ lasers^[19]. Some scientists have found that the LG beams and the HG beams carry a cross phase $\exp (i θ x y)$ , where the two modes can be converted to each other^[20]. The elliptical vortex HG beams are found to be described by the complex amplitude proportional to the $n$ th-order Hermite polynomial, whose parameter is a function of a real parameter, $a$ . Moreover, the parameter $a$ and the ellipticity parameter of the Gaussian beam together determine the orbital angular momentum (OAM) of the vortex HG beams^[21]. The beam with OAM^{[2224" target="_self" style="display: inline;">–24]} and its beam profiles are also fascinating^{[2527" target="_self" style="display: inline;">–27]}, and the light propagation process with a Gaussian-modulated mode profile in the few-photon regime has been studied.

In this Letter, two different modulations of the HG beams are carried out, and the focus evolution of light field is deeply studied. Lasers are developed on the basis of previous research and have been promoted in the current market, but these lasers produce a single laser mode that cannot meet the needs of many applications, especially in high-end and sophisticated industries. Therefore, it is very important to develop a specific beam pattern. In recent years, there have been many studies on such beam patterns, but most of them have only stayed in the stage of formula derivation and theoretical simulation. Few researchers have carried out corresponding experimental verification. On the other hand, it remains an active question of how to improve the encoding efficiency using beam spatial modes. The introduction of additional OAM degrees of freedom by modulating the laser beam pattern is envisioned as cutting-edge technology for quantum manipulation, quantum communication, and quantum measurement. Our recent work is not only the calculation and simulation of the modulation of the HG beams, but also the experimental verification using a programmable spatial light modulator (SLM). The use of SLM to perform two different modulations of the Hermite beam is of great interest for the study of specific beam modes.

HG beams are a set of solutions of the paraxial wave equation in the Cartesian coordinate system. The expression modulated by adding the sinusoidal-vortex-phase-modulated (SVPM) function $[\exp (i l \sin ϕ)]$ is $HG (x, y, z) = \frac{1}{ω (z)} {(\frac{2^{1 - n - m}}{π n! m!})}^{1 / 2} H_{n} [\frac{\sqrt{x}}{ω (z)}] H_{m} [\frac{\sqrt{y}}{ω (z)}] \times \exp [i (n + m + 1) ξ (z)] \exp {- {[\frac{p}{ω (z)}]}^{2}} \times \exp (- \frac{i k p^{2}}{2 R}) \exp (- i k z) \exp (i l \sin ϕ) .$ (1)

The SVPM function is $\exp (i l \sin ϕ)$ . $k$ is the wave number related to the wavelength, $k = 2 π / λ$ , $m$ and $n$ are the number of modes in the $x$ and $y$ directions, and the number of modes is a positive integer starting with zero. $l$ is the number of the topological charge, and $ϕ$ is the phase angle.

$ω_{0}$ is the waist radius, $ω_{0} = z {(λ z_{R} / π)}^{1 / 2}$ , $ω (z)$ is the fundamental mode spot radius at the $z$ point along the propagation axis, $ω (z) = ω_{0} {[1 + {(z / z_{R})}^{2}]}^{1 / 2}$ , and $R (z)$ is the radius of curvature of the phase plane of the beam intersecting the propagation axis with the $z$ point, $R (z) = z \sqrt{1 + {(\frac{z_{R}}{z})}^{2}} .$ (2) $p = \sqrt{x^{2} + y^{2}},$ (3) $ξ (z) = \arctan (\frac{z}{z_{R}}) .$ (4)

According to Eq. (1), the detailed derivation of the beam generation experiment is carried out. The SVM HG beams are expressed in the form of complex amplitudes, and the expression of the amplitude term is $A_{HG} (x, y, z) = \frac{1}{ω (z)} \sqrt{\frac{2^{1 - n - m}}{π n! m!}} H_{n} [\frac{\sqrt{x}}{ω (z)}] H_{m} [\frac{\sqrt{y}}{ω (z)}] \exp [{- \frac{p}{ω (z)}]}^{2} .$ (5)

The phase expression is $ϕ_{HG} (x, y, z) = \exp [i (1 + n + m) ξ (z)] \exp (- \frac{i k p^{2}}{2 R}) \times \exp (- i k z) \exp (i m \sin ϕ) .$ (6)

The expression after combining the phase and amplitude part is $h (x, y) = \exp {i ψ [A_{HG} (x, y, z), ϕ_{HG} (x, y, z)]},$ (7)where $ψ [A_{HG} (x, y, z), ϕ_{HG} (x, y, z)]$ represents the covariant domain of phase and amplitude. In this domain, the Fourier series expression for $h (x, y)$ is $h (x, y) = \sum_{- \infty}^{\infty} c_{q}^{A_{HG} (x, y, z)} \exp [i q ϕ_{HG} (x, y, z)],$ (8)where $q$ is a positive integer. The first-order diffraction beam of HG beams is $c_{1}^{A_{HG} (x, y, z)} = A_{HG} (x, y, z) a$ , in which $a$ is the position parameter. In addition, since $ψ [A_{HG} (x, y, z), ϕ_{HG} (x, y, z)]$ is odd, it can be converted into $ψ [A_{HG} (x, y, z), ϕ_{HG} (x, y, z)] = f_{HG} \sin [ϕ_{HG} (x, y, z)] .$ (9)

So, the complex amplitude $h (x, y)$ is $\exp {i \cdot f_{HG} \cdot \sin [ϕ_{HG} (x, y, z)]}$ , and the Fourier expansion is $\exp {{if}_{HG} \sin [ϕ_{HG} (x, y, z)]} = \sum_{m = - \infty}^{\infty} J_{m} f_{HG} \exp [im ϕ_{HG} (x, y, z)] .$ (10)

$J_{m}$ represents the $m$ -order Bessel function.

According to Eqs. (8), (10), and $c_{1}^{A_{HG} (x, y, z)} = A_{HG} (x, y, z) a (a = 1)$ , it can be obtained as $A_{HG} (x, y, z) = J_{1} f_{HG} .$ (11)

The coding hologram used by SLM is given by the formula $Φ_{SLM} = f_{HG} \sin (J_{HG} + G_{X} X + G_{Y} Y),$ (12)where $J_{HG}$ is the amplitude phase function, and $G_{X}$ and $G_{Y}$ are the grating constants.

The generation of structured laser beams can be achieved by using an astigmatic mode converter (AMC)^[28] or SLM^[29,30]. The AMC has obvious advantages in beam mode conversion^[31]; Yoshikawa et al.^[32] demonstrated a simple and versatile method for generating various configurations of optical vortices from a Gaussian light beam by using glass plates and an AMC. Compared with an AMC, the SLM is more suitable and has better flexibility in this Letter.

In accordance with the above principles, Fig. 1 depicts an experimental setup for generating vortex-phase-modulated (VPM) HG beams.

Fig. 1. Experimental device. $λ / 2$ , half-wave plate; $λ / 4$ , 1/4-wave plate; lens with focal length of 100 mm; SLM, space light modulator; CCD, charge coupled device.

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The linearly polarized light required for the experiment is produced by a He–Ne laser with a power of 2 mW (wavelength of 632.8 nm, polarization ratio of 1000:1) and a polarizer (extinction ratio of 500:1) and then is incident into the beam expanding collimation system (expanded by a spatial filter consisting of a $40 \times$ objective lens and a 25 μm pinhole, collimated by a collimating lens with a focal length of 100 mm). The beams exiting in parallel pass through the aperture, the filter, and the beam splitter (BS) and are incident on the effective area of the reflective SLM (Holoeye SLM with resolution 1920 × 1200). After the beam reflected by the SLM is collimated by the lens (focal length is 150 mm), the beam is received and recognized by the CCD. Phase encoding for the HG beam with an SLM could produce vortex-modulated HG beams.

First, we use MATLAB to simulate the HG_mn beam. Figure 2 is a simulation diagram of different m and $n$ values and corresponding experimental results.

Fig. 2. Simulation and experimental comparison. (a)–(d) are simulation plots, and (e)–(h) are experimental plots. (a), (e) $m = 3$ , $n = 3$ ; (b), (f) $m = 4$ , $n = 6$ ; (c), (g) $m = 5$ , $n = 5$ ; (d), (h) $m = 5$ , $n = 8$ .

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In Figs. 2(a)–2(h), obviously, the focal regions of the beams are distributed in a matrix, and the values of $m$ and $n$ change the number of focal spots and the distribution of beam intensity. In Figs. 2(e)–2(h), the focal spot on the outermost side is larger. Among them, the four vertices are the largest, and the light intensity is the strongest. Moreover, the central focal spot is evenly distributed. Compared with the light intensity distribution in Figs. 2(a)–2(d) obtained by MATLAB simulation with the experimental results in Figs. 2(e)–2(h), the focal distributions of the same $m$ and $n$ values were completely identical. According to the above comparison, the focal spot shape and the light intensity distribution obtained by the simulation and the experiment are completely consistent. This uniformity is ubiquitous for the HG beam of any order, which fully demonstrates the accuracy of the experiment. However, the experimentally obtained focal spot has many burrs, which may be related to the interference of light.

The different intensity distributions of VPM HG beams, where the VPM function is $\exp (i l ϕ)$ , with the HG beam focal region are shown in Fig. 3.

Fig. 3. When the vortex parameter $l = 5$ , the HG beam pattern is vortex modulated by different orders. (a) $m = 1$ , $n = 1$ ; (b) $m = 2$ , $n = 2$ ; (c) $m = 3$ , $n = 4$ ; (d) $m = 4$ , $n = 5$ ; (e) $m = 5$ , $n = 5$ ; (f) $m = 6$ , $n = 4$ ; (g) $m = 7$ , $n = 2$ ; (h) $m = 8$ , $n = 3$ .

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In terms of focal spot distribution, we can conclude that the VPM-HG beams have a square distribution of focal spots when the values of $m$ and $n$ are the same in Figs. 3(a), 3(b), and 3(e). The VPM-HG beams are symmetrically distributed in a cross shape as the values of $m$ and $n$ change. Moreover, with the change of $m$ and $n$ , the focus mode evolves around spatially. The spot distribution of the focus is related to $m$ and $n$ , which affect the vertical and horizontal direction of the focusing mode, respectively. The complex evolution of the spot in space can carry more information. In terms of intensity distribution, the HG beams with a lower order have stronger intensity. As the order increases, the focal spot with the strongest intensity gradually shifts to both sides, and the central focal spot is a little weaker, but the matrix is distributed symmetrically.

In order to study the influence of vortex parameters, HG beams are with a fixed order of $m = 5$ , $n = 8$ , and the topological charge value is changed to observe the variation law. The experimental results are shown in Fig. 4. It is not difficult to see that VPM HG beams have a great influence on the focal mode. With the increase of topological charge, VPM-HG beams become more and more obvious. In Figs. 4(a)–4(d), the focus mode in the central region retains the original distribution and is unaffected by phase modulation, but changes occurring at the vertex are relatively large. In Figs. 4(e)–4(h), the original mode of HG beams disappears, and the whole spot appears as a ‘cross’ distribution. We can draw a preliminary conclusion that the topological charge can significantly change the distribution of optical field in the focusing region, and the area of the central region is related to the size of the topological charge, presenting an inverse proportion trend.

Fig. 4. Diagram of HG beams with $m = 5$ , $n = 8$ , and different topological charge $l$ . (a) $l = 2$ , (b) $l = 3$ , (c) $l = 4$ , (d) $l = 5$ , (e) $l = 6$ , (f) $l = 7$ , (g) $l = 8$ , and (h) $l = 10$ .

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SVPM process is performed on the HG beam, and the different orders under the same vortex parameters are compared. For the convenience of comparison, we select the same parameters as in Fig. 3, and the experimental results are shown in Fig. 5.

Fig. 5. Sinusoidal vortex modulated HG beam diagram with different-order vortex parameters, $l = 5$ . (a) $m = 1$ , $n = 1$ ; (b) $m = 2$ , $n = 2$ ; (c) $m = 3$ , $n = 4$ ; (d) $m = 4$ , $n = 5$ ; (e) $m = 5$ , $n = 5$ ; (f) $m = 6$ , $n = 4$ ; (g) $m = 7$ , $n = 2$ ; (h) $m = 8$ , $n = 3$ .

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In Fig. 5, the topological charge and beam parameters are the same as in Fig. 3, where the focusing field distribution is obtained by SVPM process. The intensity distribution is only symmetric with respect to the $x$ axis and skewed to the right, which is caused by sinusoidal phase modulation with an asymmetric structure. On the other hand, the beam parameter can be embodied concretely. In Fig. 3, we can observe the symmetric structure with the influence of vortex beams, but cannot observe the rule of $m$ and $n$ . In Fig. 5, the focusing spot is distorted based on the HG beam, and we can clearly see the evolution rules of different $m$ and $n$ . In addition, we can also find that the beam from the outside to inside is less and less affected by modulation. In Figs. 5(e)–5(h), the peripheral light spot is seriously distorted, and the distribution of the HG beam is still maintained inside. In conclusion, through sinusoidal vortex modulation, we obtain a set of focal patterns with symmetrical structure and a special focusing distribution with certain regularity, which is obviously related to $m$ and $n$ .

In order to study the influence of vortex parameters on the SVPM-HG beams, a fixed-order HG beam is selected to change its topological charge value, and the variation law of the beam is observed. For the convenience of comparison, we choose the same order as in Fig. 4, that is $m = 5$ , $n = 8$ . The experimental results are shown in Fig. 6.

Fig. 6. Sinusoidal vortex modulated HG beam with $m = 5$ , $n = 8$ , and different topological charge $l$ . (a) $l = 2$ , (b) $l = 3$ , (c) $l = 4$ , (d) $l = 5$ , (e) $l = 6$ , (f) $l = 7$ , (g) $l = 8$ , and (h) $l = 10$ .

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As can be seen from Fig. 6, the value of vortex parameter $l$ has a great influence on the focal region of SVPM-HG beams. The focused distribution is still symmetric with respect to the $x$ axis and topological charge directly changes the distortion degree of the light field. With the increase of the topological charge, the modulation characteristic is still from the outside to the inside. We expect a quantitative relationship between the number of distorted points and the topological charge with introduction of additional OAM degrees of freedom by modulating the laser beam pattern. This will not only have academic significance, but also have practical value. After preliminary analysis, more experimental data and more experimental schemes are needed, such as fractional topological charge, higher-order, and more accurate light field details. In this Letter, only the qualitative relation is studied, that is, the higher the topological charge is, the greater the degree of distortion of the beam is, and fewer spots are maintained in the HG mode.

It is of great significance to study the quality of the experimental beam. In Fig. 7, the generated beam is analyzed in detail, and a three-dimensional distribution diagram of the beam is given.

Data analysis diagram of sinusoidal vortex modulation with parameters m=5, n=8, and l=8. (a) Light intensity distribution collected by CCD. (b) A cross section perpendicular to the x axis. (c) A cross section perpendicular to the y axis. (d) Three-dimensional intensity distribution.

Fig. 7. Data analysis diagram of sinusoidal vortex modulation with parameters $m = 5$ , $n = 8$ , and $l = 8$ . (a) Light intensity distribution collected by CCD. (b) A cross section perpendicular to the $x$ axis. (c) A cross section perpendicular to the $y$ axis. (d) Three-dimensional intensity distribution.

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We have made a detailed analysis of the intensity distribution at the focal point of a sinusoidal-modulated beam, with the data in Fig. 7 as the original data, without any algorithm modification. Figure 7(a) is the distribution of intensity acquired by CCD. Figures 7(b) and 7(c) show the intensity curves perpendicular to the $x$ -axis and $y$ -axis sections, respectively. From Figs. 7(b) and 7(c), we can see that the number of peaks with the highest intensity values is 9 and 6, respectively (marked with blue points in Fig. 7); this is the same order as that of the HG beam. The small peaks in Figs. 7(b) and 7(c) represent the distortion under the influence of modulation. In order to show the modulated beam more intuitively, we also show the three-dimensional distribution diagram of intensity in Fig. 7(d). Figures 7(b) and 7(c) are obtained from the sections perpendicular to the $x$ and $y$ axes in Fig. 7(d).

In this Letter, the sinusoidal modulation mode of HG beams is deduced by formula calculation. The comparison between simulation and experimental results shows the accuracy of the experiment. The phase distribution of the optical field was analyzed by controlling the parameters $m$ , $n$ , and $l$ , respectively, and the phase distribution of the optical field of ordinary vortex modulation and sinusoidal vortex modulation was observed and analyzed, respectively, by controlling the same parameters $m$ , $n$ , and $l$ . The light field of $m = 5$ , $n = 8$ , and $l = 5$ is quantitatively analyzed. The characteristic of this Letter is the combination of theory and experiment, which is of great significance for studying different modes of HG beams.

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Guanxue Wang, Yue Li, Xinzhi Shan, Yu Miao, Xiumin Gao. Hermite–Gaussian beams with sinusoidal vortex phase modulation[J]. Chinese Optics Letters, 2020, 18(4): 042601.

Hermite–Gaussian beams with sinusoidal vortex phase modulation Download： 708次

Fig. 1. Experimental device. $λ / 2$ , half-wave plate; $λ / 4$ , 1/4-wave plate; lens with focal length of 100 mm; SLM, space light modulator; CCD, charge coupled device.

Fig. 2. Simulation and experimental comparison. (a)–(d) are simulation plots, and (e)–(h) are experimental plots. (a), (e) $m = 3$ , $n = 3$ ; (b), (f) $m = 4$ , $n = 6$ ; (c), (g) $m = 5$ , $n = 5$ ; (d), (h) $m = 5$ , $n = 8$ .

Fig. 3. When the vortex parameter $l = 5$ , the HG beam pattern is vortex modulated by different orders. (a) $m = 1$ , $n = 1$ ; (b) $m = 2$ , $n = 2$ ; (c) $m = 3$ , $n = 4$ ; (d) $m = 4$ , $n = 5$ ; (e) $m = 5$ , $n = 5$ ; (f) $m = 6$ , $n = 4$ ; (g) $m = 7$ , $n = 2$ ; (h) $m = 8$ , $n = 3$ .

Fig. 4. Diagram of HG beams with $m = 5$ , $n = 8$ , and different topological charge $l$ . (a) $l = 2$ , (b) $l = 3$ , (c) $l = 4$ , (d) $l = 5$ , (e) $l = 6$ , (f) $l = 7$ , (g) $l = 8$ , and (h) $l = 10$ .

Fig. 5. Sinusoidal vortex modulated HG beam diagram with different-order vortex parameters, $l = 5$ . (a) $m = 1$ , $n = 1$ ; (b) $m = 2$ , $n = 2$ ; (c) $m = 3$ , $n = 4$ ; (d) $m = 4$ , $n = 5$ ; (e) $m = 5$ , $n = 5$ ; (f) $m = 6$ , $n = 4$ ; (g) $m = 7$ , $n = 2$ ; (h) $m = 8$ , $n = 3$ .

Fig. 6. Sinusoidal vortex modulated HG beam with $m = 5$ , $n = 8$ , and different topological charge $l$ . (a) $l = 2$ , (b) $l = 3$ , (c) $l = 4$ , (d) $l = 5$ , (e) $l = 6$ , (f) $l = 7$ , (g) $l = 8$ , and (h) $l = 10$ .

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Hermite–Gaussian beams with sinusoidal vortex phase modulation Download： 708次

Fig. 1. Experimental device. λ/2, half-wave plate; λ/4, 1/4-wave plate; lens with focal length of 100 mm; SLM, space light modulator; CCD, charge coupled device.

Fig. 2. Simulation and experimental comparison. (a)–(d) are simulation plots, and (e)–(h) are experimental plots. (a), (e) m=3, n=3; (b), (f) m=4, n=6; (c), (g) m=5, n=5; (d), (h) m=5, n=8.

Fig. 3. When the vortex parameter l=5, the HG beam pattern is vortex modulated by different orders. (a) m=1, n=1; (b) m=2, n=2; (c) m=3, n=4; (d) m=4, n=5; (e) m=5, n=5; (f) m=6, n=4; (g) m=7, n=2; (h) m=8, n=3.

Fig. 4. Diagram of HG beams with m=5, n=8, and different topological charge l. (a) l=2, (b) l=3, (c) l=4, (d) l=5, (e) l=6, (f) l=7, (g) l=8, and (h) l=10.

Fig. 5. Sinusoidal vortex modulated HG beam diagram with different-order vortex parameters, l=5. (a) m=1, n=1; (b) m=2, n=2; (c) m=3, n=4; (d) m=4, n=5; (e) m=5, n=5; (f) m=6, n=4; (g) m=7, n=2; (h) m=8, n=3.

Fig. 6. Sinusoidal vortex modulated HG beam with m=5, n=8, and different topological charge l. (a) l=2, (b) l=3, (c) l=4, (d) l=5, (e) l=6, (f) l=7, (g) l=8, and (h) l=10.

Fig. 7. Data analysis diagram of sinusoidal vortex modulation with parameters m=5, n=8, and l=8. (a) Light intensity distribution collected by CCD. (b) A cross section perpendicular to the x axis. (c) A cross section perpendicular to the y axis. (d) Three-dimensional intensity distribution.

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Fig. 1. Experimental device. $λ / 2$ , half-wave plate; $λ / 4$ , 1/4-wave plate; lens with focal length of 100 mm; SLM, space light modulator; CCD, charge coupled device.

Fig. 2. Simulation and experimental comparison. (a)–(d) are simulation plots, and (e)–(h) are experimental plots. (a), (e) $m = 3$ , $n = 3$ ; (b), (f) $m = 4$ , $n = 6$ ; (c), (g) $m = 5$ , $n = 5$ ; (d), (h) $m = 5$ , $n = 8$ .

Fig. 3. When the vortex parameter $l = 5$ , the HG beam pattern is vortex modulated by different orders. (a) $m = 1$ , $n = 1$ ; (b) $m = 2$ , $n = 2$ ; (c) $m = 3$ , $n = 4$ ; (d) $m = 4$ , $n = 5$ ; (e) $m = 5$ , $n = 5$ ; (f) $m = 6$ , $n = 4$ ; (g) $m = 7$ , $n = 2$ ; (h) $m = 8$ , $n = 3$ .

Fig. 4. Diagram of HG beams with $m = 5$ , $n = 8$ , and different topological charge $l$ . (a) $l = 2$ , (b) $l = 3$ , (c) $l = 4$ , (d) $l = 5$ , (e) $l = 6$ , (f) $l = 7$ , (g) $l = 8$ , and (h) $l = 10$ .

Fig. 5. Sinusoidal vortex modulated HG beam diagram with different-order vortex parameters, $l = 5$ . (a) $m = 1$ , $n = 1$ ; (b) $m = 2$ , $n = 2$ ; (c) $m = 3$ , $n = 4$ ; (d) $m = 4$ , $n = 5$ ; (e) $m = 5$ , $n = 5$ ; (f) $m = 6$ , $n = 4$ ; (g) $m = 7$ , $n = 2$ ; (h) $m = 8$ , $n = 3$ .

Fig. 6. Sinusoidal vortex modulated HG beam with $m = 5$ , $n = 8$ , and different topological charge $l$ . (a) $l = 2$ , (b) $l = 3$ , (c) $l = 4$ , (d) $l = 5$ , (e) $l = 6$ , (f) $l = 7$ , (g) $l = 8$ , and (h) $l = 10$ .