Broadband meta-converters for multiple Laguerre-Gaussian modes

Huade Mao; Yu-Xuan Ren; Yue Yu; Zejie Yu; Xiankai Sun; Shuang Zhang; Kenneth K. Y. Wong

doi:10.1364/PRJ.423344

Abstract

Metasurface provides miniaturized devices for integrated optics. Here, we design and realize a meta-converter to transform a plane-wave beam into multiple Laguerre-Gaussian (LG) modes of different orders at various diffraction angles. The metasurface is fabricated with Au nano-antennas, which vary in length and orientation angle for modulation of both the phase and the amplitude of a scattered wave, on a silica substrate. Our error analysis suggests that the metasurface design is robust over a 400 nm wavelength range. This work presents the manipulation of LG beams through controlling both radial and azimuthal orders, which paves the way in expanding the communication channels by one more dimension (i.e., radial order) and demultiplexing different modes.

1. INTRODUCTION

The Laguerre-Gaussian (LG) mode, a solution of the Helmholtz equation in cylindrical coordinates, characterized by the radial index [1], $p$ , and the azimuthal index [2], $l$ , has attracted tremendous attention recently owing to the ability to encode information [3 –6]. The azimuthal index, known as the orbital angular momentum (OAM) [2], has shown applications in object detection [7,8], optical communication [9], holography imaging [10], etc., mainly counting on the momentum conservation during propagation [11]. Meanwhile, the LG mode, as a complete orthogonal basis [7], has been demonstrated to increase the communication speed [12,13], e.g., multiplexing and demultiplexing in multiple orthogonal OAM channels [13]. However, the radial index has been largely underexplored by the community as it is not supported by the single-mode fiber used in many optical setups. However, this is not a concern in the free space. Since the radial index is a valid quantum number [14] and could be transmitted through a graded-index fiber [1] or in free space, it has the potential to further increase the capacity of communication system [1] by one more dimension (i.e., radial order).

However, many studies primarily focused on the generation [3,6,15] or the observation [16,17] of LG modes. For example, a mode sorter [4] was proposed to generate up to 325 LG modes by utilizing seven phase masks to transform an array of Gaussian beams into the quasi-complete set of LG mode beams. Despite its technical brilliance, it still transforms each Gaussian beam into a single LG mode but not multiple ones, not to mention the setup’s complexity.

Here, we report a complex-modulated metasurface to simultaneously tailor multiple LG modes. Many previous optical devices for LG modes generation [4,6,18] feature phase-only modulation, which need an iterative algorithm [19] to minimize the error between the output and target field. In recent years, complex modulation [20,21] was proposed to generate LG modes. However, they were only able to generate one LG mode at a time. By comparison, our metasurface could (1) generate the field in a faster way (no need for iteration) and (2) achieve more sophisticated goals (simultaneous conversion and demultiplexing for multiple LG modes). Our metasurface consists of a 2D array of Au nano-antennas on a glass substrate coated with indium tin oxide. Our experiments suggest that the designed metasurface performs in a broadband wavelength spanning over 400 nm, which is also corroborated by simulation. As the metasurface is a promising miniaturization technique [22 –25] and many high-order LG modes are achieved through a single chip [6,26], our work paves the way to future LG modes application and communication channel expansion.

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2. COMPLEX MODULATION

The building block of the metasurface is the nano-antenna made of Au [Fig. 1(a)], with the height $h$ and width $w_{x}$ fixed respectively at 80 nm and 200 nm throughout the paper. The orientation of the nano-antenna (“atom”) defines the Pancharatnam–Berry phase modulation [27 –30].

Figure 1.(a) Configuration of a unit block. $P$ , period; $w_{x}$ , width; $w_{y}$ , length; $h$ , height; $α$ , orientation angle. (b) Configurations of Au block to accomplish complex modulation. The color bars are amplitude range of [0, 0.5] and phase range of [ $- π, π$ ], which are the same as in (c) and (d). (c), (d) Amplitude and phase conversion over 500–1500 nm range for the 10 configurations specified in the red rectangle in (b). (e) Experimental setup. PBS, polarization beam splitter; LCP, LCP generator, consisting of a polarizer and a $λ / 4$ wave plate; RCP, RCP filter, composed of a polarizer and a $λ / 4$ wave plate.

When a left-handed circularly polarized (LCP) light impinges onto this unit, the Au block, functioning as a locally defined birefringent crystal, could alter the amplitude and phase for the orthogonally decomposed light respectively along the fast axis and slow axis, thus transforming a certain amount of LCP into right-handed circular polarized (RCP) light [28,31]. The RCP component [32] from transmitted light is $S_{out} = ⟨ R | Γ (- α) \hat{Q} Γ (α) | L ⟩,$ (1)where $R$ and $L$ denote RCP and LCP, respectively; $Γ (α)$ is the rotation matrix; $α$ is the orientation angle of the block in Fig. 1(a); and $\hat{Q}$ is the transformation matrix.

We perform finite-difference time-domain (FDTD) simulation to calculate the full field of the transmitted RCP light through nano-blocks with various orientations and geometries (Appendix A). We select 10 different configurations with length ranging from 220 nm to 400 nm while keeping the period $P$ of the unit as 500 nm, and orientation from 0 to $π$ to encode the full phase and amplitude of the light wave. In order to offset dynamic phase’s difference, each configuration has a unique initial orientation angle as shown in the red block [Fig. 1(b) and Appendix A]. The performance of 10 configurations was evaluated in the wavelength range from 500 to 1500 nm using FDTD simulation. The complete conversion maps of both the amplitude and phase are revealed in Figs. 1(c) and 1(d).

The laser beam is converted to LCP before shining on the metasurface under test [Fig. 1(e)]. The RCP component of the forward scattering light will carry the information encoded with multiple LG modes deflected into different angles (demultiplexing).

3. METASURFACE FIELD

The electric field of a certain Laguerre-Gaussian beam can be written as [33] ${LG}_{p}^{l} = \frac{\sqrt{2} r^{| l |}}{w_{0}^{| l | + 1}} \cdot \exp (- \frac{r^{2}}{w_{0}^{2}}) \cdot L P_{p}^{| l |} (\frac{2 r^{2}}{w_{0}^{2}}) \cdot \exp (i l ϕ),$ (2)where $p$ is the radial index; $l$ is the azimuthal index that represents the magnitude of OAM; $r$ and $ϕ$ are the cylindrical coordinates; $w_{0}$ is the beam waist; and $LP$ is the Laguerre polynomial. In the past, devices featuring only phase modulation, whether a metasurface [6] or diffraction filter [18], have been proposed for generating LG modes. However, for more than two LG modes’ encoding, phase-only modulation could not achieve zero-error (Appendix B). Complex modulation [20,21], proposed in recent years, has not been explored in multiple LG modes encoding.

Here, we show that an incident plane-wave beam can be converted into multiple LG modes diffracted into arbitrary directions in transmission by a metasurface, whose complex modulation is expressed as $M = \sum_{s} σ_{s} {({LG}_{p}^{l})}_{s} \cdot \exp (i \cdot \sin θ \cdot \vec{g_{s}} \cdot \vec{ρ}) = A_{M} \exp (i ϕ_{M}),$ (3)where $σ_{s}$ represents different weight assigned to the specified LG mode; ${LG}_{p}^{l}$ denotes the normalized complex distribution of each mode; and $θ$ is the deflection angle while $g_{s}$ and $ρ$ are the deflection direction and pixel vector, respectively [inset of Fig. 1(e)]. The pixel vector $\vec{ρ}$ points from the center point to one pixel on the metasurface. The amplitude and phase parts of the output field in Eq. (3) could be achieved by the metasurface with complex modulation which is fabricated by electron-beam lithography.

The meta-convertor allocates the incident light energy into different decomposition modes according to the designed information magnitude. As an example, we convert the plane-wave beam into two LG modes with non-identical deflection angles. Figure 2(a) shows the complex modulation of the combination of ${LG}_{4}^{2}$ and ${LG}_{1}^{1}$ modes with their deflection direction being respectively $\vec{g_{1}} = (1,0)$ and $\vec{g_{2}} = (1,0)$ , with deflection angles $θ$ of 5° and 10°, and an amplitude ratio of 6:4. We fabricate the metasurface by a mapping from the complex field [Fig. 2(a)] to the exact configuration in Fig. 1(b). Figure 2(b) shows the scanning electron microscope image of the fabricated metasurface with a zoom-in view shown in Fig. 2(c), which elaborates the building blocks of various angles. The metasurface is composed of $400 \times 400$ unit blocks. Since each unit block has a size of $500 nm \times 500 nm$ , the metasurface has an area of $200 μm \times 200 μm$ .

Figure 2.(a) Rounded complex pattern to convert the fundamental Gaussian beam into a combination of ${LG}_{4}^{2}$ and ${LG}_{1}^{1}$ directed to ${\vec{g}}_{1} = (1,0)$ , with angles ${θ = 5}^{°}$ and ${θ = 10}^{°}$ . The assigned weight has a ratio of ${LG}_{4}^{2} ∶ {LG}_{1}^{1} = 6 ∶ 4$ . (b) Overall scanning electron microscope (SEM) image. (c) Zoomed SEM image. (d) LG decomposition results for the complex pattern in (a) without deflection. (e) LG decomposition results for the modes combination of ${LG}_{4}^{2} ∶ {LG}_{1}^{1} = 1 ∶ 1$ . Scale bar: 20 μm in (b); 1 μm in (c).

We decompose the complex pattern to determine the coefficient for each mode as (Appendix C) $C_{p}^{l} = \frac{⟨ {LG}_{p}^{l}, U ⟩}{⟨ {LG}_{p}^{l}, {LG}_{p}^{l} ⟩},$ (4)where $U$ denotes the complex field without deflection to maintain central symmetry and the denominator is the normalization term. This process is one kind of Fourier decomposition with the Fourier basis being the LG modes which form the complete set [34]. The result [Fig. 2(d)] suggests a ratio of 6:4 between the modes of ${LG}_{4}^{2}$ and ${LG}_{1}^{1}$ , which is well consistent with our design.

Here, we analyzed the effect of fabrication error, detailed in Section 4. Under our assumption, $\pm 10 nm$ deviation of width and length of the Au block will cause an average error of mostly lower than 10%, which is acceptable. Then, we also evaluate the influence of fabrication error on the decomposition results (Appendix C). The decomposition would have no significant change provided that the fabrication error has a mean of 0. The decomposition error deviates only 4.5% for a mean of 0.05, which suggests the robust tolerance to the fabrication error. This has also been validated with another design [Fig. 2(e)] with a combination of ${LG}_{4}^{2} ∶ {LG}_{1}^{1} = 1 ∶ 1$ .

4. ERROR ANALYSIS

The tolerance of the size of the nano-block is critical for the broadband performance. The metasurface is fabricated with the electron-beam lithography. But there is no statistical data reflecting the fabrication error for our metasurface. Empirically, the fabrication has an error of 10 nm. So here we assume the machine will have a 10 nm deviation on both the width and length of the Au block and the deviation is Gaussian distributed. Therefore, the probability density function of the distribution could be represented as follows: $p (Z; μ, Σ) = \frac{1}{{(2 π)}^{\frac{d_{i}}{2}} {| Σ |}^{\frac{1}{2}}} \cdot \exp [- \frac{1}{2} {(Z - μ)}^{T} Σ^{- 1} (Z - μ)],$ (5)where $Z = {(w_{x}, w_{y})}^{T}$ is the vector of the length and width of the Au block; $μ = {(μ_{x}, μ_{y})}^{T}$ is the mean of the length and width; $Σ = (σ_{1}^{2} 0; 0 σ_{2}^{2})$ is the covariance matrix of the Gaussian distribution, where two variables are uncorrelated under our assumption; and $d_{i}$ denotes the dimension of the variable.

The total possible range of $Z$ here, in our consideration, is [190,210] nm for width and $[μ_{y} - 10, μ_{y} + 10]$ nm for length. We conduct full-wave FDTD simulation over this region, and get the error map: $ϵ (Z) = \frac{| C (w_{x}, w_{y}; α_{0}) - C (μ_{x}, μ_{y}; α_{0}) |}{0.464},$ (6)where $C (x, y; α)$ denotes the complex modulation under the configuration $x$ and $y$ given orientation $α$ .

The average error $ξ$ could be calculated as $ξ = \int_{μ_{x} - 10}^{μ_{x} + 10} \int_{μ_{y} - 10}^{μ_{y} + 10} ϵ (Z) \cdot p (Z; μ, Σ) d w_{x} d w_{y},$ (7)where the integral region is the total possible range of $Z$ .

Figure 3.Gaussian distribution under three scenarios: “pessimistic,” “neutral,” “optimistic.”

The error map is revealed in Fig. 4 for four nano-block configurations. The first and third columns, captioned “Error” in the figure, are the absolute error among the fabrication regions. The second and fourth columns show the error with Gaussian distribution under “optimistic” assumption.

Figure 4.Error distribution for four different configurations with 10 nm deviation along the width and length of the Au block. The first and third columns are the absolute error over different width and length. The second and fourth columns are the Gaussian-distributed error from the desired configuration. For the four configurations we selected here, the width $w_{x}$ is fixed at 200 nm. The lengths $w_{y}$ are (a) 220 nm, (b) 225 nm, (c) 230 nm, and (d) 250 nm.

The average error $ξ$ , elaborated in Eq. (7), is calculated under all the three scenarios and listed in Table 2. Even under pessimistic assumption, the average error is about or lower than 10%, which is acceptable. And in Appendix C, we will see that the deviation will be fully waived under LG mode decomposition, which shows the robustness of our design.Table 2.

Parameters Approximation^a

$η_{N}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
$η_{P}$	0.0464	0.0928	0.1392	0.1856	0.2320	0.2784	0.3248	0.3712	0.4176	0.4640
$w_{y} (nm)$	220	225	230	250	255	265	270	300	330	400
$α_{0}$	0	0	$- \frac{1}{60} π$	$- \frac{1}{18} π$	$- \frac{1}{12} π$	$- \frac{1}{9} π$	$- \frac{5}{36} π$	$- \frac{1}{6} π$	$- \frac{7}{36} π$	$- \frac{41}{180} π$
$η_{ac}$	0.0452	0.103	0.125	0.181	0.236	0.275	0.326	0.378	0.424	0.450
$ϕ_{ac}$	−1.129	−1.112	−1.177	−1.080	−1.154	−1.208	−1.212	−1.151	−1.112	−1.100
$ξ$	0.258%	2.23%	3.39%	2.17%	1.53%	4.77%	5.82%	2.30%	2.07%	2.90%
$ξ_{1}$	11.17%	9.29%	8.47%	10.68%	8.39%	9.85%	8.20%	8.62%	6.83%	5.18%
$ξ_{2}$	9.48%	7.87%	7.28%	9.95%	6.91%	9.01%	7.00%	7.92%	6.30%	4.50%
$ξ_{3}$	7.49%	6.04%	6.10%	9.38%	5.58%	8.23%	5.89%	7.19%	5.75%	3.96%

$η_{N}$ : desired quasi-amplitude conversion. $η_{P}$ : desired actual amplitude conversion. $w_{y}$ : length of the nano-antenna along the $y$ axis. $α_{0}$ : initial orientation angle. Given the predetermined length $w_{y}$ and orientation $α_{0}$ , the actual amplitude and phase conversion are denoted as $η_{ac}$ and $ϕ_{ac}$ . $ξ$ : error between the actual and desired complex conversion. Error under three different Gaussian distributions: $ξ_{1}$ : pessimistic; $ξ_{2}$ : neutral; $ξ_{3}$ : optimistic.

5. SIMULATION AND EXPERIMENTS

The holographic image of the metasurface could be calculated through a complex transmission function [32,35], $H (x, y) = \sum_{m, n} M_{m n} \cdot \frac{\exp [i k R_{m n} (x, y)]}{R_{m n} (x, y)},$ (8)where $M_{m n}$ represents a pixel on the metasurface determined by Eq. (8); $k$ is the wavenumber; $R_{m n} (x, y)$ denotes the distance between $M_{m n}$ and the position $(x, y)$ on the holographic image $H$ . The final normalized intensity is calculated as $I_{norm} = ⟨ H, H ⟩ / {⟨ H, H ⟩}_{\max}$ .

We next evaluate the broadband performance of the metasurface. The complex pattern $M_{mn}$ of a certain wavelength is converted using the broadband conversion map for 10 configurations [Figs. 1(c) and 1(d)]. The metasurfaces are designed at the wavelength of 1000 nm. The complex pattern varies slightly from 500 nm to 1500 nm (Visualization 1). Then we alter the wavenumber $k$ in Eq. (8) to simulate the holographic image.

The supplementary visualization shows how the complex pattern and the LG decomposition results change over the broadband wavelength range. Visualization 1 is about the metasurface featuring modes of ${LG}_{4}^{2} ∶ {LG}_{1}^{1} = 6 ∶ 4$ . In the video, the first row contains the amplitude and phase map of the complex field without deflection over the wavelength range from 500 nm to 1500 nm. The third figure of the first row is the bar chart of the LG decomposition results. The second row contains the complex field of the metasurface with deflection, which is the fabricated complex field we use for experiments.

Experimentally, the performance of the fabricated metasurface is evaluated at various wavelengths (1030 nm, 1200 nm, and 808 nm) to demonstrate the broadband modulation ability. The diffractive images are collected at a distance of 1.5 mm or 2.0 mm from the metasurface (Fig. 5). Figure 5(a) shows the diffraction pattern for $λ = 1030 nm$ . The ${LG}_{4}^{2}$ and ${LG}_{1}^{1}$ deflect to roughly 130 μm and 260 μm at a propagation distance of 1.5 mm, which correspond to the diffraction angles of 5° and 10°, respectively. As seen from both the hologram simulation and the experiment [Fig. 5(a)], there are apparent interference fringes in the region where the two LG modes overlap, which is owing to the insufficient separation in the near field. However, the fringes will disappear when the paired beams propagate further (2 mm away from the metasurface), as shown in Fig. 5(b).

$Experimental results: broadband performance of meta-converters. (a)–(d) Diffraction patterns with metasurface designed for LG42∶LG11=6∶4. The test wavelength and the measurement distance from the metasurface are all labeled with the beam profile. First row: simulation. Second row: experiments. (e) Simulated broadband performance for the meta-converter of LG42∶LG11=6∶4 in terms of error and efficiency. (f) Metasurface contains modes of LG11∶LG22∶LG23=3∶4∶5. The left edge corresponds to zero deflection for all figures. Scale bar for all: 200 μm.$

Figure 5.Experimental results: broadband performance of meta-converters. (a)–(d) Diffraction patterns with metasurface designed for ${LG}_{4}^{2} ∶ {LG}_{1}^{1} = 6 ∶ 4$ . The test wavelength and the measurement distance from the metasurface are all labeled with the beam profile. First row: simulation. Second row: experiments. (e) Simulated broadband performance for the meta-converter of ${LG}_{4}^{2} ∶ {LG}_{1}^{1} = 6 ∶ 4$ in terms of error and efficiency. (f) Metasurface contains modes of ${LG}_{1}^{1} ∶ {LG}_{2}^{2} ∶ {LG}_{2}^{3} = 3 ∶ 4 ∶ 5$ . The left edge corresponds to zero deflection for all figures. Scale bar for all: 200 μm.

In Figs. 5(c) and 5(d), we test the performance of the meta-converter at two other wavelengths of 1200 nm and 808 nm, and the experiments (bottom) agree well with the simulation results (top). We analyze its error and efficiency from 700 nm to 1500 nm [Fig. 5(e)] through simulation. The decomposition error measures the deviation of energy allocation from design and the efficiency measures the ratio of energy flowing into desired modes defined in Eq. (C10). Take the metasurface in Fig. 2(b) for example. The amplitude ratio of the desired modes is ${LG}_{4}^{2} ∶ {LG}_{1}^{1} = 6 ∶ 4$ . After normalization, the desired coefficients vector is $\vec{ω} = (0.6,0.4)$ . After decomposition at 1 μm, the decomposed vector is $\vec{σ} = (0.504,0.354)$ . The error is calculated as 0.076 and the efficiency is 0.858, which are both acceptable. We repeat the procedure from 700 to 1500 nm with a step of 50 nm to get Fig. 5(e). It shows that from 800 nm and up to 1500 nm, the metasurface has an error about or below 10% mostly (except 28% at 850 nm). Although the above analysis adopts two LG modes, in general, the design and analysis procedure could apply to a combination of multiple LG modes. For demonstration, we show the three LG modes encoding with an energy ratio of ${LG}_{1}^{1} ∶ {LG}_{2}^{2} ∶ {LG}_{2}^{3} = 3 ∶ 4 ∶ 5$ . Since the RCP filter is not at 100% efficiency, the unconverted LCP passing the metasurface will form an interference pattern resembling its arrangement, which will cause a concentric ring interference at the center in Figs. 5(a)–5(d).

Although the result is decent for three LG modes here, the metasurface has an issue of energy transmission efficiency. The maximum amplitude modulation is 0.464. The transmitted power is 21.5% at most. Given the metasurface is complex modulated, the whole energy transmission would be even lower. If we increase the amount of LG modes, there must exist a limit where the energy allocated to each mode is undetectable in experiment. Future work about dielectric metasurface to improve the energy transmission is prospected.

6. CONCLUSION

In conclusion, we designed a metal metasurface to generate multiple LG modes and separate them simultaneously. The metasurface features a complex modulation and has a decent performance over the 400 nm wavelength range. Although we demonstrate the energy allocation into a limited number of LG modes, the meta-convertor could in principle distribute the light energy into an arbitrary number of modes. Since the radial index of the LG mode could be transmitted through a graded-index fiber [1,36] or in free space, it is envisaged it could potentially increase the communication bandwidth in the optical communication systems with integrated optical devices.

APPENDIX A: COMPLEX MODULATION

E_{x} = \vec{e_{x}} \cdot \exp [i (k z - ω t)],

| x ⟩ = [\begin{matrix} 1 \\ 0 \end{matrix}],

| L ⟩ = \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ i \end{matrix}],

w_{x}

n_{o}

| Ψ (z) ⟩

A_{out} = abs (E_{out}) = \sin [\frac{k d (n_{o} - n_{e})}{2}],

k d (n_{o} + n_{e}) / 2

Figure 6.(a) Amplitude conversion and (b) phase conversion under different $w_{y}$ and orientation angle $α$ when $x$ is set at 200 nm and incident source is set at 1000 nm.

w_{x}

(α, w_{y})

0 − 1

APPENDIX B: PHASE-ONLY MODULATION COULD NOT ACHIEVE ZERO-ERROR

{LG}_{p}^{l} = A_{pl} (x, y) \cdot \exp [i ϕ_{pl} (x, y)],

U = \sum_{p = 0}^{\infty} \sum_{l = - \infty}^{\infty} σ_{p l} \cdot A_{p l} (x, y) \cdot \exp [i ϕ_{p l} (x, y)],

U = C \cdot \exp [i (ϕ (x, y)]

The above equation is not true for any arbitrary combination of more than two LG modes. Given the unimodular property [37], the Fourier basis, here the LG mode basis, should be either one or infinite in Eq. (B3). For finite LG modes, which are more than two LG modes but not infinite, Eq. (B3) is impossible. This is quite similar to Ref. [38], where the phase-only modulation for OAM combination is proved impossible.

APPENDIX C: LG MODE DECOMPOSITION

{LG}_{p}^{l}

U

\frac{\int {LG}_{p}^{l} * \cdot U d S}{\int {LG}_{p}^{l} * \cdot {LG}_{p}^{l} d S} = {\begin{matrix} 1, & p = p^{'} and l = l^{'} \\ 0, & otherwise \end{matrix} .

Here we only consider the LG modes combination without deflection. It could give a basic idea of how the fabrication error will affect the final LG modes. However, if we introduce deflection into the decomposition, the results will cover a wide range of LG modes basis. It is similar to Ref. [39], where the OAM state is fixed but located away from the center, leading to a non-pure singular OAM in the spectrum. Such simplification could help us quantify the error and determine how much the error will influence our results.

M

U

{(I_{p}^{l})}_{k} \frac{σ_{k}^{2}}{⟨ U, U ⟩} = \frac{σ_{k}^{2}}{\sum_{i} ⟨ σ_{i} {(L G_{p}^{l})}_{i}, σ_{i} {(L G_{p}^{l})}_{i} ⟩} = \frac{σ_{k}^{2}}{\sum_{i} σ_{i}^{2}},

U = A (x, y) \cdot \exp [i ϕ (x, y)] = A_{U} \cdot \exp (i ϕ_{U}) .

U_{fab} = [A (x, y) + t] \cdot \exp [i ϕ (x, y)],

U = {LG}_{4}^{2} + {LG}_{1}^{1}

Figure 7.First column: amplitude pattern. Second column: phase pattern. Third column: LG decomposition results. (a) Rounded complex pattern, featuring ${LG}_{4}^{2} ∶ {LG}_{1}^{1} = 1 ∶ 1$ . (b) Complex pattern with a Gaussian noise $N {(0,0.05}^{2})$ applied to the amplitude. (c) Complex pattern with a Gaussian noise $N {(0.05,0.05}^{2})$ applied to the amplitude.

4

{LG}_{p}^{l} = A_{p}^{l} \cdot \exp (i ϕ_{p}^{l})

⟨ {LG}_{p}^{l}, {LG}_{p}^{l} ⟩

t

In order to quantify the deviation caused by the fabrication noise, here we set a criterion to judge the error in LG mode expansion.

\vec{ω} = (ω_{0}, \dots, ω_{i}, \dots, ω_{n})

ϵ = \sum_{i} | \frac{ω_{i}}{ω_{0}} - \frac{σ_{i}}{σ_{0}} | .

The error could be interpreted as the deviation from the desired modes. We check whether the ratio of different modes is close to our predetermined ratio.

γ = \sum_{i} σ_{i} .

ϵ

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