Abstract
Rigorous coupled wave analysis (RCWA) has emerged as a versatile and efficient technique for analyzing diffraction gratings, metamaterials, photonic crystals, and related periodic structures. This expanded tutorial provides a comprehensive overview of the theoretical basis, numerical implementation, applications, advancements as well as comparisons with other modeling approaches for RCWA. The discussion ranges from fundamental RCWA concepts and formulations to recent innovations regarding improved algorithms, extensions to complex materials and geometries, and integration with other computational electromagnetics and optics simulation tools.
Introduction
Periodic micro and nanostructures are broadly utilized in photonics applications like optical filters, sensors, and detectors. The ability to accurately model the interaction of electromagnetic waves and light propagation through such complex media is critical for effective analysis and design. Rigorous coupled wave analysis solves Maxwell’s equations in the Fourier domain for periodic structures, providing a balance between physical accuracy and computational efficiency. Since its conception in the 1980s, RCWA has undergone major improvements regarding robustness, flexibility, and performance to become an indispensable simulation platform across nanophotonics, optoelectronics, and electromagnetic devices.
Theoretical Basis
RCWA basically expands electromagnetic fields within periodically structured media as a summation of space harmonics using Floquet theorem. By appropriately truncating the infinite series, Maxwell’s differential equations can be converted into a matrix eigenvalue problem solvable on a computer. The technique divides any general diffraction grating profile into a number of thin uniform layers to which the coupled wave equations obtained from Maxwell’s relations can be matched at the boundaries. The resulting matrix equation has the form:
Ax=λBx
Where x is the state variable representation of the harmonic electric and magnetic amplitudes, while A and B are characteristic matrices deduced systematically from the boundary conditions and refractive indices of the layers. Solving the above matrix eigenvalue system yields the necessary parameters for computing diffraction efficiencies, field intensities, power losses and other critical figures-of-merit.
Working Mechanism
Conceptually, RCWA essentially models an arbitrarily curved periodic profile as a multilayer stair-case grating. By slicing the structure into a number of flat layers and applying boundary condition match at each interface, the problem gets reduced to solving a truncated set of wave equations for the amplitudes of the coupled harmonics expanded using Floquet modes [170]. The number of equations to solve equals twice the number of harmonics retained times the layers used. Though simple in principle, multiple mathematical innovations regarding convergence, stability, and efficiency of the core algorithm have expanded the power of RCWA tremendously over the past few decades.
Key Enabling Developments
Several advances have played a pivotal role in establishing RCWA as one of the most widely used techniques for modeling periodic optical and EM structures:
Improved algorithms for stable eigenmode solutions have increased accuracy and range of validity.
Incorporating higher order Fourier terms has enhanced flexibility for complex material geometries with non-vertical walls.
Anisotropic, bianisotropic and nonlinear materials can now be analyzed with specialized RCWA implementations.
Curved structures can be accurately simulated using stair-case approximations without loss of fidelity.
Diffraction regimes from reflection to transmission can be studied within a single unified framework.
Combining RCWA with methods like FDTD and FEM leverages advantages of both frequency and time domain approaches.
These advances have made RCWA an ideal platform for designing, optimizing and characterizing a wide range of modern photonic components like waveguides, optical filters, sensors, and anti-reflection structures.
Implementing the RCWA Algorithm
The procedural steps involved in implementing the RCWA algorithm are:
Specify relevant geometrical parameters like periodicity, layer materials and thicknesses.
Discretize structure into sufficient slices (typically > 30) for numerical stability.
Identify appropriate Fourier harmonics and field expansions for system complexity.
Construct field transmission matrices at boundaries from Maxwell’s relations.
Formulate and solve matrix eigenvalue equation for eigenmodes.
Calculate measurable quantities like diffraction efficiencies.
Based on the output parameters sought, iterative calculations may be required with adaptive improvisations regarding grid sizes and harmonic orders for acceptable accuracy within reasonable times.
Scope of Applications
Owing to its versatility, RCWA remains unrivaled as an accurate modeling tool and has been leveraged to design and analyze devices across the entire electromagnetic spectrum spanning:
Microwave regimes: Modeling radar and satellite communication reflector antennas.
Millimeter waves: Design of dielectric transmit/receive modules.
Infrared region: Photonic crystals for thermal photovoltaics.
Visible spectrum: Optimization of LEDs and displays.
Ultraviolet: Photolithography masks and filters.
Apart from traditional optics applications, RCWA has proven invaluable in breakthrough research areas as well including:
Metamaterials with negative refractive index.
Anisotropic media like liquid crystals.
Optical sensing elements and biosensors.
Atomically thin 2D materials.
Each class of problem above utilizes specialized RCWA adaptations targeted to the specific geometry, material properties and output metrics sought from the analysis.
Comparison with Beam Propagation Method (BPM)
Both BPM and RCWA constitute useful analytic tools for periodic optical structures. However, some salient differences exist:
BPM relies on paraxial approximations while RCWA imposes no constraints on incident angles.
RCWA incorporates multiple backward reflections unlike BPM beam forward propagation assumption.
RCWA provides better accuracy for boundary condition mismatch problems in distributed Bragg reflectors.
BPM suffices for simple waveguide-like structures, but RCWA works for general diffraction gratings.
BPM constitutes an easier implementation, but RCWA offers higher flexibility and fidelity.
The choice between employing RCWA and BPM is dictated by the specific nature of application and expected insights from the analysis - with RCWA delivering better performance for modeling general diffraction gratings and related structures.
Conclusion
Rigorous coupled wave analysis forms an efficient, accurate and flexible platform for computational characterization of periodic optical media and devices. With extensive toolboxes already available in commercial and open-source EM solvers, and the advent of GPU computing, the future scope of RCWA adoption across nanophotonics appears brighter than ever.
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