
- Photonics Research
- Vol. 9, Issue 9, 1689 (2021)
Abstract
1. INTRODUCTION
The Laguerre-Gaussian (LG) mode, a solution of the Helmholtz equation in cylindrical coordinates, characterized by the radial index [1],
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
Figure 1.(a) Configuration of a unit block.
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
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
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]
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
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
Figure 2.(a) Rounded complex pattern to convert the fundamental Gaussian beam into a combination of
We decompose the complex pattern to determine the coefficient for each mode as (Appendix C)
Here, we analyzed the effect of fabrication error, detailed in Section 4. Under our assumption,
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:
The total possible range of
The average error
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
The average error
Parameters Approximation
0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0.0464 | 0.0928 | 0.1392 | 0.1856 | 0.2320 | 0.2784 | 0.3248 | 0.3712 | 0.4176 | 0.4640 | |
0 | 220 | 225 | 230 | 250 | 255 | 265 | 270 | 300 | 330 | 400 | |
0 | 0 | 0 | |||||||||
0 | 0.0452 | 0.103 | 0.125 | 0.181 | 0.236 | 0.275 | 0.326 | 0.378 | 0.424 | 0.450 | |
0 | −1.129 | −1.112 | −1.177 | −1.080 | −1.154 | −1.208 | −1.212 | −1.151 | −1.112 | −1.100 | |
0 | 0.258% | 2.23% | 3.39% | 2.17% | 1.53% | 4.77% | 5.82% | 2.30% | 2.07% | 2.90% | |
0 | 11.17% | 9.29% | 8.47% | 10.68% | 8.39% | 9.85% | 8.20% | 8.62% | 6.83% | 5.18% | |
0 | 9.48% | 7.87% | 7.28% | 9.95% | 6.91% | 9.01% | 7.00% | 7.92% | 6.30% | 4.50% | |
0 | 7.49% | 6.04% | 6.10% | 9.38% | 5.58% | 8.23% | 5.89% | 7.19% | 5.75% | 3.96% |
5. SIMULATION AND EXPERIMENTS
The holographic image of the metasurface could be calculated through a complex transmission function [32,35],
We next evaluate the broadband performance of the metasurface. The complex pattern
The supplementary visualization shows how the complex pattern and the LG decomposition results change over the broadband wavelength range.
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
Figure 5.Experimental results: broadband performance of meta-converters. (a)–(d) Diffraction patterns with metasurface designed for
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
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
We set two basic orthogonal electric field units as
Then the light field could be written in Jones vector form:
As LCP and RCP could be decomposed into two linear polarization, and LCP and RCP could be written as follows:
In our metasurface, each periodic unit block contains the glass substrate and Au nano-block as in Fig.
The Au nano-block, considered as a birefringent crystal, has its ordinary and extraordinary refractive index as
In the end, we shall extract the RCP component of the output field
From above, we could have the output field specially modulated in amplitude and phase as
In phase modulation, the dynamic phase
Figure 6.(a) Amplitude conversion and (b) phase conversion under different
The FDTD simulation results are presented in Figs.
Since the fabrication of a certain Au block has an error of about 10 nm, there is no point to pick
Due to the fabrication limit, the maximum amplitude conversion we could obtain is 0.464. Therefore, a projection mapping from
APPENDIX B: PHASE-ONLY MODULATION COULD NOT ACHIEVE ZERO-ERROR
The electric field of an LG mode is detailed in Eq. (
The ideal complex pattern containing exactly the desired LG modes could be written as
If the metasurface could feature phase-only modulation, the complex pattern could be rewritten as
The above equation is not true for any arbitrary combination of more than two LG modes. Given the unimodular property [
APPENDIX C: LG MODE DECOMPOSITION
We use
Consider an electric field
Since LG modes are orthogonal to each other, obviously we have
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. [
For our metasurface
For the field
The relative intensity of each mode is written as
From Eq. (
Given the assumption that the fabrication error of the width and length of the Au block is Gaussian distributed, it is reasonable to assume that it will lead to amplitude error being Gaussian distributed, which means the electric field of the fabricated metasurface is
Figure
Figure 7.First column: amplitude pattern. Second column: phase pattern. Third column: LG decomposition results. (a) Rounded complex pattern, featuring
The deviation of width and length of the Au block will affect the amplitude and phase output. The error in this regard has been obtained through FDTD in Fig.
For demonstration, we write the LG mode as
Since the normalization factor
Given that
In order to quantify the deviation caused by the fabrication noise, here we set a criterion to judge the error in LG mode expansion.
The “error” in LG mode expansion here is defined as the sum of unwanted modes and the deviation from desired modes. Our desired coefficient vector of different LG modes is denoted as
The error could be calculated as
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.
Here we define another term called efficiency, which is
The efficiency is the sum of our desired modes. If a large ratio of power is split into unwanted modes, the metasurface will not generate desired modes even if the error
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