Generation of super-high-order petal-like laser beam using Theon-sieve resonator

Xianghe Guan; Junyong Zhang; Yanli Zhang; Jianqiang Zhu

doi:10.3788/COL202018.041404

The Laguerre–Gaussian (LG) beam can be obtained by solving the Helmholtz equation in the condition of cylindrical symmetry. Different from the fundamental mode in a resonator, the LG beam carries orbital angular momentum. High-order LG transverse electromagnetic ( ${TEM}_{p, l}$ )mode, where $p$ and $l$ are the radial and azimuthal order, respectively, exhibits some interesting and unique physical properties^[1–4] and has been widely used in the field of precise manipulation and optical information transfer^[5–8]. The petal-like beam that can be generated by coherently superimposing two high-order LG beams with a zero order in the radial direction and conjugate orders in the azimuthal direction^[9–11] has better performance in the field of controlling the rotation of trapped microparticles^[12,13]. Due to its special intensity distribution and great applications, the petal-like laser gradually catches people’s attention. The method of generation of the petal-like laser can be divided into two types, intra-cavity^[9,14–16] and extra-cavity shaping techniques^[17–19]. Note that the intra-cavity method cannot output high-energy and high-power laser beams, while the extra-cavity method is more complex and cannot have simple operation. In addition, the order of the petal-like beam generated by previous methods is usually not very high.

The Theon sieve (TS)^[20,21] is a kind of diffractive optical element with two axial focal spots, and the transmissive rings of TS are replaced by a large number of isolated pinholes. In this paper, we propose a simple, stable, and adjustable structure by inserting a TS into a resonator to generate a super-high-order petal-like laser. The model of the proposed resonator is established and simulated by the optical software GLAD, and the per round trip energy loss of the resonator and the intensity distribution of the output beam are obtained. The influence of the location of TS on the output beam and the stability of the optical resonator cavity are also analyzed.

The electric field of high-order LG modes can be written as^[22] $u_{p, l} = \sqrt{\frac{2 p!}{π (p + | l |)!}} \frac{1}{ω (z)} {[\frac{\sqrt{2} r}{ω (z)}]}^{| l |} L_{p}^{| l |} [\frac{2 r^{2}}{ω {(z)}^{2}}] \exp [\frac{- r^{2}}{ω {(z)}^{2}} - \frac{i k r^{2}}{2 R (z)}] \times \exp [- i (2 p + | l | + 1) \arctan (\frac{z}{z_{R}})] \exp (- i l Φ),$ (1)where $p$ and $l$ are the radial and azimuthal order, respectively, $r$ is the radius, $Φ$ is the azimuthal angle, $z$ is the distance from the beam waist, $z_{R}$ is the Rayleigh range, $ω (z)$ and $R (z)$ are the radius of curvature and beam width of a Gaussian beam, and $L_{p}^{| l |}$ is the generalized Laguerre polynomial. The superposition field of the petal-like mode can be expressed as^[9] $u_{0, \pm l} = u_{0, + l} + u_{0, - l} = \sqrt{\frac{2 p!}{π (p + | l |)!}} \frac{1}{ω (z)} {[\frac{\sqrt{2} r}{ω (z)}]}^{| l |} \exp [\frac{- r^{2}}{ω {(z)}^{2}} - \frac{i k r^{2}}{2 R (z)}] \times \exp [- i (| l | + 1) \arctan (\frac{z}{z_{R}})] \exp (- i l Φ) .$ (2)

The intensity profile of this structure will result in $2 l$ petals around a circle.

Based on the parallel plane mirror resonator with two condenser lenses inserted inside, we replaced one of the lenses with a TS. The focal length of the TS needs to be confirmed by the stability conditions of the resonator. The structure of the proposed resonator is shown in Fig. 1. $M_{1}$ and $M_{2}$ are a pair of parallel planar mirrors. $L_{1}$ is a condenser lens that is placed a distance $z_{1}$ away from $M_{1}$ , and $C_{1}$ is a pinhole close to $M_{1}$ . $S_{1}$ is a TS that is placed a distance $z_{2}$ away from $M_{2}$ . The distance between $S_{1}$ and $L_{1}$ is $d$ . The ${LG}_{0 l}$ mode is nearly circularly symmetric in the resonator. Two high-order LG beams with zero order in the radial direction and conjugate orders in the azimuthal direction have identical loss and coherently superimpose in the same position.

Figure 1.Theon-sieve (TS) resonator.

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GLAD software is used to simulate the properties of the resonator by the Fox–Li iteration method^[23]. Although the calculation process of the Fox–Li iteration method is tedious, it has universal applicability. The most important point is that GLAD is suitable for the modeling of the physical process, such as various diffraction effects that cannot be ignored in the design of the optical system. In fact, it is possible to calculate the self-reproducing mode in any resonator.

In the numerical calculation, the focal length of $L_{1}$ is 80 mm, the radius of pinhole $C_{1}$ is 0.45 mm, $z_{1}$ and $z_{2}$ are both 100 mm, $d$ is 95 mm, and the initial optical field is set to random noise. A TS is designed with a diameter of 8.2 mm and focal lengths of 55.07 mm and 73.03 mm, respectively, as indicated in Fig. 2.

Figure 2.Structure of the TS.

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The simulation starts from the left plane mirror, and the light goes through $L_{1}$ , $S_{1}$ , $M_{2}$ , $S_{1}$ , $L_{1}$ , $C_{1}$ , and $M_{1}$ in turns. In the end, the normalized light field intensity distribution on the left surface of $M_{1}$ and the per round trip energy loss percentage are obtained after 800 times propagation in the resonator. From these results, as shown in Fig. 3, we know that the output light is a petal-like laser beam, and the TS resonator mode becomes stable when the beam propagates for about 200 cycles. Figure 3(a) shows the normalized intensity distribution on the left surface of $M_{1}$ . The number of the petals is 36, and the radius from the center of petal-like laser beam is 0.3 mm.

Figure 3.TS resonator: (a) normalized intensity distribution; (b) per round trip energy loss percentage.

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Changing the distance between the TS and $L_{1}$ , the radii and the number of the petals versus the distance $d$ are described in Table 1. The normalized intensity distributions of the petal-like beam are presented in Figs. 4(a)–4(f), and the per round trip energy loss percentage is shown in Fig. 5. In Figs. 4(a)–4(c), the distance between the TS and $L_{1}$ is 87 mm, 90 mm, and 102 mm, respectively, and the number of the petals is 36, the same as that of $d = 95 mm$ . A distance of at least 15 mm can generate the same transverse mode with 36 petals. In Figs. 4(d)–4(f), it is seen that the numbers of the petals are 26, 46, and 54, when the distances between TS and $L_{1}$ are 65 mm, 120 mm, and 140 mm, respectively. Obviously, the number of the petals increases with the increase of the distance. Besides, higher-order petal-like beams can also be generated by adjusting the parameters of the TS resonator appropriately. On the basis of the data in Table 1, the radius of the petal-like laser decreases with the increase of distance in the condition of the number of the petals being invariable. Although the size of the diffraction pattern, which is generated by TS, at the plane $L_{1}$ varies with the distance between the TS $L_{1}$ , the clear aperture of the $L_{1}$ is invariable. On the other hand, different distances correspond to the different ABCD matrices of the resonator. Therefore, there is a big difference in the per round trip energy loss, just as shown in Fig. 5.

Table 1. Radii and Number of Petals at Different Distances

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Table 1. Radii and Number of Petals at Different Distances

d (mm) 87 90 102 65 120 140
Petal number 36 36 36 26 46 54
Radius (mm) 0.32 0.31 0.30 0.30 0.31 0.31

Figure 4.Simulation of petal-like modes at different distances between TS and L₁. (a) $d = 87 mm$ ; (b) $d = 90 mm$ ; (c) $d = 102 mm$ ; (d) $d = 65 mm$ ; (e) $d = 120 mm$ ; (f) $d = 140 mm$ .

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Figure 5.Per round trip energy loss percentage versus different distances.

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Figure 6 shows the relationship between the number of petals and the distance between the TS and $L_{1}$ . Obviously, the ABCD matrix of the laser resonator will change according to the variation of the distance, meaning that different losses exist in the cavity. In this case, different resonator modes will be generated. Note that for a stable resonator, a slight disturbance of the system parameters has no influence on the mode of the output laser.

Figure 6.Number of petals versus different distances.

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In conclusion, we propose a stable and adjustable structure to generate super-high-order petal-like laser beams by inserting a TS into the resonator, and the effectiveness is verified by GLAD software. The output normalized intensity distribution and the per round trip energy loss percentage are obtained successfully by the Fox–Li iteration method. The relationship among the radius, the number of petals, and the distance between the TS and lens is also analyzed in detail. The proposed method can enlighten us in constructing more complex light fields so as to be beneficial to interference diagnostics and interferometric imaging.

Category: Lasers and Laser Optics

Received: Oct. 22, 2019

Accepted: Jan. 3, 2020

Posted: Jan. 3, 2020

Published Online: Apr. 14, 2020

The Author Email: Junyong Zhang (zhangjy829@siom.ac.cn), Jianqiang Zhu (jqzhu@siom.ac.cn)

DOI:10.3788/COL202018.041404

Table 1. Radii and Number of Petals at Different Distances

Table 1. Radii and Number of Petals at Different Distances