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.
© 2021 Chinese Laser Press
The Laguerre-Gaussian (LG) mode, a solution of the Helmholtz equation in cylindrical coordinates, characterized by the radial index , , and the azimuthal index , , has attracted tremendous attention recently owing to the ability to encode information [3–6]. The azimuthal index, known as the orbital angular momentum (OAM) , has shown applications in object detection [7,8], optical communication , holography imaging , etc., mainly counting on the momentum conservation during propagation . Meanwhile, the LG mode, as a complete orthogonal basis , has been demonstrated to increase the communication speed [12,13], e.g., multiplexing and demultiplexing in multiple orthogonal OAM channels . 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  and could be transmitted through a graded-index fiber  or in free space, it has the potential to further increase the capacity of communication system  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  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  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.
2. COMPLEX MODULATION
The building block of the metasurface is the nano-antenna made of Au [Fig. 1(a)], with the height and width 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].
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  from transmitted light is1(a); and 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 of the unit as 500 nm, and orientation from 0 to 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 6] or diffraction filter , 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 as1(e)]. The pixel vector 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 and modes with their deflection direction being respectively and , with deflection angles 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 unit blocks. Since each unit block has a size of , the metasurface has an area of .
We decompose the complex pattern to determine the coefficient for each mode as (Appendix C)34]. The result [Fig. 2(d)] suggests a ratio of 6:4 between the modes of and , which is well consistent with our design.
Here, we analyzed the effect of fabrication error, detailed in Section 4. Under our assumption, 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 .
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 here, in our consideration, is [190,210] nm for width and nm for length. We conduct full-wave FDTD simulation over this region, and get the error map:
The average error could be calculated as
Since the two variables, width and length are uncorrelated, we just draw one variable at a time in Fig. 3. If the standard deviation is small enough, the most distribution will be inside our consideration region. Here we assign three levels to different standard deviations, ranging from “pessimistic,” “neutral,” to “optimistic” in Fig. 3 and Table 1. When the standard deviation is set to 3.33 nm, nearly 99.4% of the whole distribution, which is the yellow area in Fig. 3, is within our consideration region. When the standard deviation goes up to 10 nm, only 46.5% of the distribution falls into our assumption region, which is termed “pessimistic” in this case. We will calculate the total error according to Eq. (7) under these three scenarios.
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.
The average error , 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.
5. SIMULATION AND EXPERIMENTS8); is the wavenumber; denotes the distance between and the position on the holographic image . The final normalized intensity is calculated as .
We next evaluate the broadband performance of the metasurface. The complex pattern 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 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 . 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 . The and 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).
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 . After normalization, the desired coefficients vector is . After decomposition at 1 μm, the decomposed vector is . 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 . 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.
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. 1(a). The configuration of the Au block is determined by three parameters, namely, the width , the length , and the orientation angle .
The Au nano-block, considered as a birefringent crystal, has its ordinary and extraordinary refractive index as and along the and axes given no orientation angle. If the incident light is LCP, the output field after the Au block’s tuning is
In the end, we shall extract the RCP component of the output field , which is
From above, we could have the output field specially modulated in amplitude and phase as
In phase modulation, the dynamic phase is determined by the Au block’s own parameter, and the Pancharatnam–Berry phase is dependent on the orientation angle, which means if the Au block is rotated from 0 to , the output phase could have a modulation range, which is consistent with the phase conversion in Fig. 6(b).
The FDTD simulation results are presented in Figs. 6(a) and 6(b). We set the width and height to 200 nm and 80 nm, respectively, and alter the length from 200 to 400 nm. Such a range could reach the maximum amplitude conversion range, e.g., from 0 to 0.464, and is within our fabrication capacity. If equals , the Au block will show no birefringence. It means no LCP will be converted to RCP, which is consistent with the dark area in Fig. 6(a) where is roughly equivalent to .
Since the fabrication of a certain Au block has an error of about 10 nm, there is no point to pick in Figs. 6(a) and 6(b) to match the exact amplitude and phase needed. Because later fabrication may deviate the selected point thus the desired complex modulation could not be achieved. To solve this issue, we conduct a parameter approximation. We select 10 points, which are 10 different values, to round the continuous 0–1 amplitude to the nearest decimal, viz., from 0.1 to 1 in 10 discrete values. The continuous phase modulation is reached through the Pancharatnam–Berry phase by designing the orientation angle of each individual gold nano-block, so there is no need to round the phase. The selected 10 different configurations are elaborated in Table 2 and are the same as in Fig. 1(c) in the red box.
Due to the fabrication limit, the maximum amplitude conversion we could obtain is 0.464. Therefore, a projection mapping from to is adopted to yield quasi amplitude modulation. And the approximation mentioned above is taken to round the desired amplitude to the nearest one decimal. For example, if the desired amplitude conversion is 0.34, it will be rounded to 0.3 denoted as , which after approximation could be written as . Next, we search over the simulated conversion map in Figs. 6(a) and 6(b) to find the appropriate parameters, including the length and initial orientation , on which the recomputed actual amplitude and phase are based. Last, the relative error is calculated as2.
APPENDIX B: PHASE-ONLY MODULATION COULD NOT ACHIEVE ZERO-ERROR
The electric field of an LG mode is detailed in Eq. (2). Here we simplify it as
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 , where is a constant. The power is . Naturally, we will have
The above equation is not true for any arbitrary combination of more than two LG modes. Given the unimodular property , 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. , where the phase-only modulation for OAM combination is proved impossible.
APPENDIX C: LG MODE DECOMPOSITION
We use to denote the Laguerre-Gaussian mode with radial index and azimuthal index . The detailed expression is detailed in Eq. (2). The field of the LG mode is normalized throughout the paper, which means the highest amplitude is 1 for all the different modes.
Consider an electric field , which could be decomposed into the combination of different LG modes as follows:
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. , 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.
For our metasurface , detailed in Eq. (3), we first introduce the complex field , which is equivalent with but without deflection.
For the field , the decomposition is
The relative intensity of each mode is written as
From Eq. (C1), we simplify the electric field as
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 is4.
Figure 7 reveals the complex pattern and LG decomposition results with regard to a metasurface, whose complex field is , with the weight ratio being but without deflection compared to the metasurface field. The decomposition result is also shown in Fig. 2(e). Apparently, the error has little effect on our LG mode decomposition.
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. . We assume the orientation does not change, so the Pancharatnam–Berry phase remains the same. The width and length deviation will affect the dynamic phase change, which is negligible if only considering change. Therefore, the width and length deviation will only affect amplitude. Given that the fabrication machine is unbiased, it is intuitive to take the mean as 0. Under optimistic distribution in Table 2, the expectation of error ranges from 3.96% to 9.38%. It would be reasonable to choose a standard deviation in this scale. In Fig. , we assign a value of 0.05. The Gaussian noise is applied to the amplitude part of the combined LG mode pattern.
For demonstration, we write the LG mode as , where and are both real numbers. The decomposition for the electric field after fabrication is
Since the normalization factor is constant, we do not show it explicitly in Eq. (C8) for simplicity.
Given that is a random variable, we take the expectation to get the final coefficient:C9), we could conclude that as long as the mean of amplitude error is 0, the noise term, which is the second term in Eq. (C9), could be waived. Even if the amplitude error has some bias, which means is not 0, the decomposition coefficient will be somewhere between complex modulation and phase modulation . If the bias is infinitely large (which is quite impossible in the sense), which means the amplitude error is large, the coefficient will degenerate into phase modulation .
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 , while the coefficient after decomposition is , where and means the desired and decomposed coefficient for the mode, respectively, and means all the modes desired.
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 is low.
National Natural Science Foundation of China (N_HKU712/16); Research Grants Council, University Grants Committee (CityU T42-103/16-N, E-HKU701/17, HKU 17200219, HKU 17209018, HKU C7047-16G).
The authors declare no conflicts of interest.
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