MAXIM: Metasurfaces-oriented electromagnetic wave simulation software with intuitive graphical user interfaces,☆☆

https://doi.org/10.1016/j.cpc.2021.107846Get rights and content

Abstract

We develop MAXIM which is electromagnetic wave simulation software based on rigorous coupled-wave analysis. The principal advantage of MAXIM is an intuitive graphical user interface drastically improving the accessibility of the software to who are not familiar with computer programming. Here, we present the basic formulation and computation methods that are incorporated in MAXIM. The computation performance is also evaluated for several didactic examples of dielectric metasurfaces which are the main application of MAXIM. The comparison of the calculation results with commercial software based on a finite-difference time-domain method confirms that the computation results of two programs coincide closely with each other within 1% difference. Considering the easy accessibility, wide availability and high reliability, MAXIM will serve the development of related research fields of metasurfaces and nanophotonics.

Program summary

Program Title: MAXIM

CPC Library link to program files: https://doi.org/10.17632/352jpd593h.1

Licensing provisions: LGPL

Programming language: Python 3.8

Supplementary material: User guide and tutorial movie

Nature of problem: Time-harmonic electromagnetic wave simulations on multilayered periodic structures composed of isotropic materials. Diffraction occurs as a propagating electromagnetic wave meets periodically-structured arrays of materials that have different refractive indices. Wave parameters such as amplitude and phase of diffracted waves are modulated depending on the structure configuration. The main purpose of this software is to calculate complex transmission and reflection coefficients of diffracted waves from dielectric metasurfaces that are composed of arrays of subwavelength antennas.

Solution method: Rigorous coupled-wave analysis associated with an extended scattering matrix method. According to Bloch’s theorem, diffracted waves from periodic structures can be represented as a truncated Fourier series in which the primitive reciprocal vector is the same as that of periodic structures. Appropriate boundary conditions at the interface of a single structure layer enable development of an eigenvalue equation for which the solution gives a scattering matrix of the layer, and the coupling coefficients within it. In the case of a multilayered structure, a total scattering matrix can be obtained using the Redheffer star product, which interconnects scattering matrices of each layer. Total coupling coefficients can be also calculated using the extended Redheffer star product; this ability is a major advantage of the extended scattering matrix method. Diffraction from periodic structures can be fully described by the total scattering matrix and total coupling coefficients.

Additional comments including restrictions and unusual features: MAXIM builds on several open-source Python packages including PySide2 [1], Numpy [2], Scipy [3], Pandas [4], Matplotlib [5] and Pyinstaller [6].

References

[1] PySide2, https://wiki.qt.io/Qt_for_Python

[2] Numpy, https://numpy.org/

[3] Scipy, https://www.scipy.org/

[4] Pandas, https://pandas.pydata.org/

[5] Matplotlib, https://matplotlib.org/

[6] Pyinstaller, https://www.pyinstaller.org/index.html

Introduction

Metasurfaces that consist of subwavelength optical antenna arrays have shown exceptional capabilities as ultrathin flat optical devices [1]. Conventional optical components such as lenses control light by exploiting refraction, so the final devices are usually bulky and heavy. In contrast, metasurfaces in which thickness is comparable to the operating wavelength can perform the same functions because tiny antennas of metasurfaces can efficiently interact with incident light. Due to those advantages, many promising applications have been demonstrated using metasurfaces such as ultrathin lenses [2], holograms [3], [4], [5], structural colors [6], [7], [8] and skin cloaks [9].

Metasurfaces can be classified as plasmonic or dielectric, according to the nature of the material [10]. Plasmonic metasurfaces exploit surface plasmons, which are collective electron oscillations at the interface between conductors and dielectrics. Reflective plasmonic metasurfaces of metal–insulator–metal configurations exploit Fabry–Perot cavity effects, and therefore can achieve very high modulation efficiency in the visible spectrum [11]; however, the modulation efficiency of transmissive plasmonic metasurfaces is usually very low compared with conventional refractive optical elements.

Dielectric metasurfaces have high modulation efficiency for both reflective and transmissive types. Scattering properties of dielectric antennas are determined by multipole resonances that incident light induces inside antennas; the resonances can be tailored by modifying the geometry of antennas [12]. Many types of antennas have been developed to control wave parameters such as amplitude, phase and frequency. Simultaneous control of multiple parameters is also possible [13], [14]. Recently efforts are focused on active tunability [15] as well as largescale manufacturing of dielectric metasurfaces for practical applications[16], [17].

Appropriate design of metasurfaces requires understanding of transmission or reflection characteristics of individual antennas. For this purpose, several computational electromagnetic (EM) wave simulation methods can be used, such as the finite element method (FEM) and finite-difference time-domain method (FDTD). Such methods are well developed and benefit from high degrees of freedom (DoF) in simulation, including boundary conditions, light sources and structure configurations.

Rigorous coupled-wave analysis (RCWA) is an appropriate method to efficiently calculate the EM responses of metasurfaces [18]. RCWA has distinct advantages over other methods. First, RWCA can provide analytic solutions of diffracted waves, whereas other methods, in principle, provide only numerical electric or magnetic field data. Therefore, RCWA can easily discriminate among diffracted waves by their diffraction orders. In RWCA, the calculation time is not affected by the thickness of structures, so nanostructures combined with multiple thick layers can be considered without difficulty. In particular cases such as one dimensional (1D) gratings on a distributed Bragg reflector, RCWA shows outstanding computation speed and accuracy. Several open-source packages based on RCWA were already developed to boost the advance of nanophotonics [18], [19], [20], [21], [22]. However, those scripting-based programs suffer from limited structural configuration or low accessibility to users who are not familiar with computer programming. There are some open-source RCWA programs with a graphical user interface (GUI), but they must be incorporated in internet connection or other paid software [23], [24].

Here, we develop MAXIM which is open-source RCWA software to calculate the EM response of periodically distributed structures composed of isotropic materials, with the purpose of guiding design of dielectric metasurfaces. MAXIM features a powerful GUI that improves the accessibility of the software to users who do not have expertise in programming languages. MAXIM has four distinct advantages as follows.

  • Highly-intuitive GUI. All simulation variables are tabulated in the main table widget, so users can check their simulation conditions easily. Most open-source RCWA codes require users to manually define each variable and corresponding values; this process usually causes troubles because users may not be aware of all necessary variables and available options in a comprehensive simulation environment. In contrast, MAXIM displays all the required variables in the table, so the whole simulation can be understood without difficulty.

  • High-DoF in structure configurations. MAXIM can consider various kinds of structures including films, rectangles, circles, arbitrary polygons and their combinations. Truncated cones and rectangular horns composed of multiple alternating materials are also available. These objects are essential in the characterization of a side-wall effect, which is one of the most common fabrication defects in dielectric metasurfaces.

  • Convenient parametric sweep. When designing metasurfaces, a parametric sweep is unavoidable because general analytic scattering solutions of non-spherical objects rarely exist. MAXIM contains powerful functions to build a calculation schedule for the parametric sweep. Thus, users can easily construct a large simulation schedule as desired.

  • Portable software. The package of MAXIM incorporates an executable file that can be run in a Windows operating system. MAIXM does not require preloaded packages, compilation processes, or virtual environments, so beginners have real opportunities to access RCWA.

The calculation accuracy of MAXIM is also verified by comparing its calculation results with those of commercial FDTD simulation software (Lumerical). RCWA solves the Maxwell equations in a frequency domain whereas FDTD computes electric and magnetic fields in a time domain. Due to the obvious contrast between two frameworks, FDTD is suitable to evaluate the calculation accuracy of RCWA. We calculate complex transmission and reflection coefficients from several exemplary structures such as Huygens, geometric and chiral metasurfaces. The calculation results of MAXIM agree within 1% with those of Lumerical. Moreover, MAXIM automatically provides the complex coefficients after calculation, so additional computation is not necessary; this attribute is highly convenient for the process of designing metasurfaces.

This article presents the basic formulation and computation methods of MAXIM. Section 2 presents a brief overview of RCWA formulation implemented in MAXIM. Section 3 introduces the computation methods of MAXIM such as the data structures, logic and internal processes of particular functions. Section 4 validates the computation performance of MAXIM through several didactic examples. Section 5 summarizes the capability and competence of MAXIM.

Section snippets

Basic formulation

Since the 1960s, coupled-wave analysis that is a predecessor of current RCWA has been developed to study sinusoidal diffraction gratings [25], [26], and the original RCWA formulation is founded in 1995 [27]. RCWA calculates diffraction from multilayered periodic structures by solving algebraic Maxwell equations in a spatial frequency domain. Bloch’s theorem provides the fundamental basis of RCWA, so EM waves in each layer are expanded to a truncated Fourier series of Bloch eigenmodes that

Computation methods

This section presents specific computation methods that use the formulation presented in the preceding section. A table is a fundamental apparatus of MAXIM. Simulation conditions are entered in the form of a table by users, and this table is converted to another form of a table by internal algorithms. Only the ‘string’ datatype is allowed in the former table, so its contents should be transformed to calculable forms of numbers in the latter table. This conversion process may look inefficient,

Performance verification

In this section, we demonstrate computation tests using MAXIM to verify its calculation accuracy and appropriateness for various applications of dielectric metasurfaces. The calculation results are also compared with the commercial FDTD software of Lumerical FDTD instead of other RCWA schemes. The computation scheme of FDTD is clearly distinct from that of RCWA, so we increase the credibility of the comparison and ensure the capability of MAXIM. To compare the results quantitatively, we use the

Conclusion

We present MAXIM, which is EM wave simulation software that uses RCWA. A highly user-friendly GUI is a primary advantage of MAXIM compared with conventional open-source RCWA programs, which usually suffer from low accessibility, especially for beginners who are not familiar with computer programming. The table-based intuitive GUI of MAXIM is extremely easy and straightforward, so MAXIM can provide a real opportunity to introduce RCWA to the communities of related fields.

The basic formulation of

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was financially supported by the National Research Foundation (NRF), Republic of Korea grants (NRF-2019R1A2C3003129, CAMM-2019M3A6B3030637, NRF-2019R1A5A8080290, NRF-2018M3D1A1058997) funded by the Ministry of Science and ICT of the Korean government.

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