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Abstract
Metasurfaces, which possess unprecedented capabilities in manipulating electromagnetic wavefronts, are promising for accurate complex amplitude modulation with a compact device. However, current strategy of complex amplitude modulation based on metasurfaces focuses on anisotropic unit design which is intrinsically constrained of polarization states. In this study, we propose a design methodology of polarization-independent metasurface which comprises an array of nanocylinders with various radii and heights. The effectiveness of the proposed scheme is verified using an optical vortex generator and a complex-amplitude hologram device. The straightforward, cost-effective, and polarization-independent design can provide robust and reliable solutions for wavefronts modulation in various optical applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The amplitude and phase are fundamental properties of electromagnetic (EM) waves. Achieving simultaneous and independent manipulation of wavefront amplitude and phase, also called complex amplitude modulation, is necessary to achieve accurate wavefront modulation; however, the various limitations of conventional devices make this goal difficult to be achieved. Metasurfaces are two-dimensional (2D) metamaterials comprising an array of subwavelength-scale nanostructures. These ultrathin planar devices exhibit unprecedented capabilities in the manipulation of the EM wavefront; they can introduce abrupt changes in one or more of the amplitude, phase, and polarization states [1]. Based on this property, remarkable progress has been achieved with metamaterials in several critical applications that are difficult or even impossible to achieve using typical EM wave modulation devices such as anomalous reflection / refraction [1], metalens [2–4], holograms [5], structured light [6], imaging [7], and quantum photonics [8].
To date, the most common metasurfaces have been designed to modify only one aspect of light, such as via phase to modulate the light field. Although phase-only modulation has yielded numerous astonishing results, the lack of amplitude modulation prevents the realization of perfect wavefront modulation. More advanced and complex wavefront control can benefit a variety of EM wave modulation-based applications such as high-fidelity holograms, structured light generation, multichannel optical communication, and optical simulation operations.
With the development of metasurfaces, recent studies have achieved significant progress towards realizing complex-amplitude modulation of EM waves. The methodologies employed are typically based on the geometric phase principle, and the representative design schemes use a wide range of creative anisotropic unit structures, such as C-shape [9–11], V-shape [12], X-shape [13,14], and their variants, such as the single rectangular structure [15,16], composite structures with multiple structures [17,18], multilayer vertical structure [19–21], and single rectangles of different heights [22]. By altering their geometric parameters and azimuth angle, these anisotropic unit structures can be made to yield various conversion efficiencies and phase delays for light in different polarization states, thereby enabling both amplitude and phase modulations. These studies considerably broaden the scope of metasurfaces in the field of complex amplitude and even multidimensional property modulation domains. However, devices based on anisotropic unit structures function effectively only if the incident light has one or more specified polarization states. For practical applications, polarization states are not easily achievable. Moreover, numerous applications exist in which induces a specific polarization state is difficult or even impossible. Consequently, polarization-dependent designs are severely limited, which prevents them from becoming a widespread and universal solution. Therefore, modern advanced wavefront engineering must consider the functional diversification of devices in a polarization-independent manner.
In this study, we propose a polarization-independent dielectric metasurface design that can realize complex amplitude modulation of light. The unit structure is designed based on an isotropic nanopillar to achieve polarization insensitivity. In the proposed method, complex amplitude modulation is achieved by altering the radius and height of the nanopillar with a circular cross-section, which is significantly different from most common metasurface devices, as shown in Fig. 1. We develop a pure optical vortex generator and metasurface holographic device to demonstrate the effectiveness of our strategy in modulating complex-amplitudes. An optical vortex generator can directly produce an optical vortex from a single metasurface, while the holographic device can display distinct holographic patterns at various propagation distances. These two adequate scenarios enable us to emphasize the conceptual simplicity and effectiveness of our strategy for achieving complex amplitude modulation. This new scheme can accomplish perfect wavefront modulation, and which has a wide range of application prospects.
Fig. 1. Schematic illustration of polarization-independent metasurface operating in transmission mode for achieving complex amplitude modulation.
2. Mechanism of the modulation of the complex amplitude
The use of nanocylinders to spatially vary the phase without depending on a specific polarization state is a well-known approach. For dielectric metasurfaces, equivalent refractive index theory is an effective theoretical model for explaining this phase modulation mechanism. Only the current design scheme considers the amount of height variation as a variable, with different heights inducing significantly different effects on phase accumulation. The entire phase accumulation can be defined as follows:
where neff is the equivalent refractive index, k = 2π/λ is the wave vector, and h is the height of the nanopillar.In contrast to the conventional approach using nanocylinders, our design aims to produce responses with high and low transmittances to incident EM waves by varying the geometry of the nanocylinder. The mechanism of the targeted transmittance manipulation can be explained using the Mie resonance theory of subwavelength structures. By changing the geometrical parameters of the dielectric scattering particles, which can independently tune the spectral positions of electric and magnetic dipole resonances, unique electric and magnetic resonances are optically induced [23]. Several studies have already been conducted in the field of phase metasurfaces for high transmittance responses [24–28]; therefore, we would like to focus on the mechanism and possibility of achieving both high and low transmittance responses.
Related literature [29] reported that when the incident wave is incident parallel to the symmetry axis of the cylinder, varying the height and radius of the dielectric cylindrical resonator can cause the electric and magnetic dipole resonances to overlap at the same frequency, thereby resulting in a strong absorption. Therefore, based on the above theory, we can alter the radius and height of the nanocylinder over a wide range, allowing us to obtain both higher and lower transmittance responses. However, in the process of changing the parameters, special phenomena such as high reflectivity and the interaction between the mode fields leaking from the structure inevitably occur, which can complicate the interactions with EM waves.
3. Methods and results
The effectiveness of this principle can be proved by applying it to the design of a transmission-mode dielectric metasurface operating in the visible light band. The incident light wave was considered to be a general plane wave that can be in any polarized or unpolarized state, and the wavelength of the light source is set to 633 nm. The wave vector of the incident light is parallel to the nanocylinder axis and normal to the substrate surface. We selected a silicon nitride nanopillar with a circular cross section located on a silica substrate. Each nanocylinder had a different radius and height parameter and is in a square lattice (in the Cartesian coordinate system) with a period of 600 nm. The design of the unit structure is illustrated in Fig. 2(a). The normalized amplitude and phase difference imparted by different geometric parameters (radius and height) of the nanocylinder are calculated using the finite-difference time-domain (FDTD) algorithm (Lumerical, Inc.). Importantly. all parameters are not limited to those currently selected for demonstration purposes, and our scheme does not lose generality.
Fig. 2. Design scheme of cylindric nanopillar with varying radius and height. (a) Geometric parameters of nanocylinder located in the center of square lattice, W is the side length of the square lattice, R and H are the radius and height of the nanocylinder, respectively. (b) Results of orthogonal parameter sweep for the parameter range of R from 94 nm to 280 nm and H from 134 nm to 1000 nm for W = 600 nm and λ = 633 nm. The top and bottom maps represent the normalized amplitude and phase modulation values provided by all nanocylinders with different geometric parameters, respectively. (c) and (d) A series of unit structures selected from the sweep results provide 10 discrete complex amplitude modulation values. While providing 10 discrete amplitude modulation values, each discrete value of amplitude can also provide 10 discrete phase modulation values. The i-th color bar represents the level number of discrete values, and the ascending ordered number corresponds to the ascending discrete values in the target modulation value. (c) denotes the amplitude distribution, with each row representing a level of amplitude modulation value, and (d) denotes the phase distribution, with each column representing a level of phase modulation value.
As previously mentioned, owing to the complex interaction of subwavelength metamaterials with EM waves, using the forward structure design becomes extremely difficult; therefore, we use an inverse design approach to find the appropriate structural parameters. The amount of complex amplitude modulation provided by the unit structure with different parameters is obtained by progressively varying the radius from 94 nm to 280 nm, and simultaneously, the height value from 134 nm to 1000 nm, in an orthogonal sweep of the two sets of parameters. Figure 2(b) shows the normalized amplitude and phase-difference maps corresponding to the different height and radius profiles. In Supplement 1, the sweep results are further discussed to corroborate that the above results for the amplitude and phase are consistent with the theoretical model that we analyzed earlier.
For the transmission phase structure, achieving continuous complex amplitude modulation is almost impossible. Therefore, we need to find an appropriate unit structure to achieve discrete complex amplitude modulation. The challenge is to find as many unit structures as possible with proper geometric parameters in these two maps so that M uniformly distributed discrete values can be found in the interval 0–1 as the amplitude modulation value, and for each amplitude discrete value, N uniformly distributed discrete values can be found in the 2π interval as the phase modulation value. In other words, a parametric search must be performed under the simultaneous constraints of both amplitude and phase values.
Inspired by the Euler sieve algorithm, we employed a sieve algorithm with tolerance error to find the maximum amplitude and phase discrete values that can be obtained from the sweep data. The algorithm first assumes the number of discrete target values and subsequently calculates these for the ideal case (i.e., the search was performed on a continuous interval). In the next step, by combining the tolerance error for each discrete value, a range of values can be obtained, which is used to filter all the data. Finally, among the filtered values, the intermediate value is taken as the actual searched discrete value.
To search for both amplitude and phase datasets, we apply the above algorithm to both datasets using the corresponding tolerance errors. Subsequently, we solve for the intersection of the indexes of the filtered values to find the parts belonging to the same set of parameters, based on which the search for the target values is performed. Through an automatic iterative process, in which a small initial target search value is incremented while reducing the tolerance error, the entire dataset is searched under a more balanced constraint.
We determine the sets of structural parameters of the nanopillars that could provide 10 discrete amplitude modulation values, each of which can provide 10 discrete phase modulation values, in turn. The selected value maps are shown in Figs. 2(c) and (d) and detailed parameters of these unit structures are illustrated in Supplement 1. Benefiting from the strictly constrained search strategy, the fluctuation between the different discrete values provided by both amplitude and phase modulation is relatively uniform.
3.1 Laguerre-Gaussian beam generator
We first design a metasurface that can generate a pure optical vortex beam, also known as the Laguerre-Gaussian (LG) beam, as proof of concept. Optical vortexes can carry orbital angular momentum and have a ring intensity distribution, which makes them attractive to researchers for use in numerous applications, such as quantum information [30] and super-resolution microscopy [31].
Typically, phase-only modulation devices are used to generate an optical vortex beam. However, the lack of amplitude modulation will be compensated by the spreading of the beam energy during the propagation of the high-order radial mode. This spread of beam energy leads to a superposition of modes, and, finally, the beam enters an impure state [32].
The operation scheme using a polarization-independent complex-amplitude modulation metasurface to generate an LG beam is illustrated in Fig. 3(a). In the case of plane wave incidence, by applying paraxial and scalar wave approximations, the normalized complex-amplitude LGp,l at the beam waist position (we set z = 0 to denote this position) can be expressed in cylindrical coordinates as follows:
Fig. 3. Generation of the optical vortex from a single metasurface. (a) Schematic of the system generating a Laguerre-Gaussian beam using a single complex amplitude modulation metasurface. The incident light is a plane wave source without any polarization state requirements at 633 nm. The normalized intensity and phase maps are obtained by FDTD simulations. (b), (c) and (d) denote the simulation results of the normalized far-field intensity and far-field phase distributions of the device generating different LG modes. In all cases, far-field results are obtained at a distance of 1- m from the device, and the central area is cropped to facilitate viewing of the result. Both the X and Y axes are expressed using direction cosines. (b), (c) and (d) denote the far-field simulation results of LG0,5, LG2,5, and LG4,2 respectively.
The complex-amplitude distribution of the LG beam is solved based on the above equation, with the beam waist set to 16 μm and the radius of the device set to 60 μm. Subsequently, the distribution is mapped to the circular arrangement of the searched unit structure.
Based on the above process, we designed three beam generators to generate beams with LG0,5, LG2,5, and LG4,2 modes. The LG0,5 beam was designed for the common case of pure optical vortex generation, whereas LG2,5 and LG4,2 beams were designed to test the effectiveness of our strategy for other higher-order mode combinations.
The near-field intensities of these three LG beams are obtained by FDTD simulations. Subsequently, the far-field projection was applied to map the near-field results to a hemisphere surface at a distance of 1-m from the device. In this manner, the result of long-distance propagation is calculated and the final distribution of intensity and phase is obtained by mapping the hemisphere result to a plane. The results for these three generators are shown sequentially in Figs. 3(b), (c), and (d), respectively.
The far-field results show that the number of intensity rings with the same value as their radial mode p+1 and phase sectors with the same value as their azimuth mode l are retained. This behavior is expected and indicates that the distribution characteristics of the output beam are consistent with those predicted by the theoretical results. Using a phase-only modulation device can have similar phase distributions in most cases; however, it tends to introduce intensity ring superposition of multiple modes, [33] which becomes more noticeable when beams with higher-order modes are generated. We also calculated the purity of the corresponding modes using the overlap integration method [34], and the results showed that the mode purity of LG0,5, LG2,5, and LG4,2 reached 95.8%, 96.9%, and 95.2%, respectively.
The entire device, as detailed in our strategy, is polarization-independent; therefore, the input light can be in any polarization state, and a consistent output beam can be directly obtained using a single metasurface. The aforementioned results prove that our strategy is an efficient method to achieve complex-amplitude modulation and requires no additional or complex setup. In principle, the field distribution of LG beams with any combination of radial and azimuth modes can be directly mapped to the structural distribution. Furthermore, our simulation results have preliminarily verified that our device can provide a direct, efficient, and simple method for generating LG beams with higher-order modes or other structured light based on compact metasurface devices.
3.2 Complex amplitude hologram
Furthermore, we apply this strategy to design a metasurface that generates a different pattern at different propagation distances to verify its performance in the case of a more complex task.
In conventional phase-only computational holograms, iterative algorithms such as the Gerchberg–Saxton (GS) algorithm [35] are commonly used to generate the phase distribution of the holographic plane according to a target pattern on the observation plane. This approach is time consuming and results in the loss of the amplitude information of the light field. However, with complex amplitude modulation, an accurate complex-amplitude distribution of the holographic plane can be achieved with merely one step of the reverse calculation. Furthermore, the quality of the final reconstructed hologram will be significantly improved.
Here, we select two target patterns: the logo of the Wuhan National Laboratory for Optoelectronics and that of the Huazhong University of Science and Technology.
The design is based on the Cartesian coordinate system, and the positive direction of the z-axis is the propagation direction. The metasurface is located on the X-Y plane with 400 pixels (corresponding to 240 μm) on x axis and 200 pixels (corresponding to 120 μm) on y axis. The target display locations on the propagation direction of the two logos are designed to be 100 μm and 300 μm, respectively. The entire system is illustrated in Fig. 4(a).
Fig. 4. Hologram metasurface. (a) Schematic illustration of the system implementation for generating two holograms at different positions using a single complex amplitude modulation metasurface. The incident light is a plane wave source without any polarization states requirements at 633 nm. (b) and (c) denote the designed patterns (the top image) and the FDTD simulation results (the bottom image) at 100 μm and 300 μm, respectively.
A scalar diffraction algorithm [36] is employed to directly calculate the distribution of the holographic plane. For comparison, we use the classical GS algorithm to calculate the phase hologram. Each of these two images requires approximately 35 iterations to converge. In contrast, when the complex amplitude is used to calculate the hologram, the target field distribution is obtained by performing merely one inverse calculation. Subsequently, we construct the entire metasurface and simulate it using the FDTD algorithm to obtain the near-field result. Finally, an exact far-field projection is applied after the simulation is complete to project the near field to z = 100 μm and z = 300 μm to obtain the holograms on the target z-axis positions.
The results are shown in Figs. 4(b) and (c). The reconstructed patterns display intensity distributions that are almost identical to those of the theoretical pattern, which indicates that our metasurface faithfully record both the amplitude and phase distribution of the light field and accurately reproduced the hologram.
4. Conclusions
We proposed a polarization-independent dielectric metasurface using a transmission mode design strategy. Based on this strategy, isotropic nanopillars are used to simultaneously modulate the amplitude and phase of the wavefront without any polarization conversion. However, changes in the tunable geometries of the nanopillars contribute to changes in both transmittance and phase, which complicates the reverse design process. To solve this problem, we use an automatic iterative sieve algorithm to find target structures within the constraint parameters. The applications of the LG beam generator and hologram metasurface demonstrated above corroborate the effectiveness of the proposed strategy. The LG beam generator can directly produce a high-purity beam with specific LG modes using a single metasurface without any extra setup. The results of the experiment with the hologram metasurface indicates that our strategy enables the encoding and decoding of any light field to instantly record and reproduce a hologram.
Most conventional metasurfaces are fabricated using the complement metal-oxide-semiconductor (CMOS) processes because they are flat devices based on isotropic or anisotropic structures. Our devices have different height distributions, which are neither periodic nor regular, depending on the specific application goal; thus, the proposed devices are difficult to fabricate using the CMOS processes. Three-dimensional processing at the micro/nano scale is an alternative and effective approach to process devices with a three-dimensional (3D) distribution. One particular study [22,37] proposed an approach for processing geometric phase metasurfaces with different heights using two-photon polymerization (TPP) 3D processing technology, which can make our strategy sufficiently feasible.
Overall, this study offers a novel and robust strategy for polarization-independent metasurface design, which provides a simple but effective way to achieve perfect wavefront modulation. Furthermore, the proposed strategy can be widely used in numerous applications with the assistance of micro- and nano-fabrication techniques.
Funding
National Key Research and Development Program of China (2021YFF0502700); National Natural Science Foundation of China (52275429, 62205117); Knowledge Innovation Program of Wuhan-Shuguang; Innovation research project of Optics Valley Laboratory.
Acknowledgments
We appreciate the efforts of all medical workers and volunteers around the world who are fighting against the COVID-19 pandemic.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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