Finite Element Anisotropic Beam propagation method (FEAB)

Introduction

We have developed a finite element beam propagation method that is suitable to simulate the light propagation in liquid crystal devices such as liquid crystal (micro)displays, spatial light modulators (SLMs) and tunable photonic components based on liquid crystals. Typically, the liquid crystal orientation is simulated first and the resulting director profile is then used in the optical simulation. The combination of the efficiency of the beam propagation algorithm, the flexibility of the finite element implementation and the wide applicability to general anisotropic materials makes our method an interesting tool for the optical analysis of photonic components.

A compiled version of FEAB is available for free download (see below).
NOTE: Please cite our paper in Optics Express vol. 17 (13) in your future publications when reporting on results that have been (partly) obtained by using FEAB.

The beam propagation algorithm

The algorithm of our beam propagation algorithm is described and illustrated in a paper in Optics Express, vol. 17 (13). The basic principle is to derive a recurrence scheme from the wave equation for the electric field which relates the electric field at two positions z0 and z0+dz. In order to represent the inhomogeneous orientation of the anisotropic liquid crystal, the full dielectric tensor is considered in the wave equation. Once the (analytical) input optical field is discretized, the light can be propagated through the structure of interest by iterative application of this recurrence relation. Therefore, the BPM scheme is a very fast and efficient algorithm to simulate the propagation of light. Furthermore, the method is very versatile because there are no restrictions on the shape of the input optical field that is considered for propagation. Employing a finite element scheme offers a high accuracy in combination with the flexibility to model arbitrarily curved structures.

Demonstration

Below we present three illustrations of the use of FEAB to simulate the propagation of light in photonic devices. Other more detailed applications are presented in the paper in Optics Express.

  • Polarization rotation of a gaussian beam in a uniaxial material

    Explanation: The movie illustrates the polarization rotation of a light beam with a gaussian intensity profile upon propagation in a homogeneous uniaxial medium as simulated with FEAB. The Ex and Ey field components of the input optical field with wavelength lambda=1um are in phase at z = 0, leading to a linear polarization state parallel to the first bisector. The input beam shown propagates through a homogeneous uniaxial material with refractive indices nx = 1.5, ny = 1.6 and nz = 1.5. Due to light diffraction, there is a weak broadening of the gaussian intensity profile and the transverse field components Ex and Ey decrease slowly during propagation. Furthermore, Ex and Ey will propagate at a different phase velocity c/nx resp. c/ny through the medium, which leads to a phase delay Gamma = 2pi/lambda(ny - nx)d between these components after propagation over a distance d. The change in the phase delay Gamma will alter the polarization state of the light as illustrated in the movie. After propagation over a distance d = 1.25um, the phase difference between Ex and Ey becomes Gamma = pi/4 which leads to an elliptical polarization state with long axis parallel to the first bisector of the transverse plane. For d = 2.5um, the Ex and Ey components are in quadrature with Gamma = pi/2, which leads to a circular polarization. After d = 3.75um the phase difference Gamma = 3pi/4 leads to an elliptic polarization state with long axis parallel to the -45 degrees bisector. Finally, after propagation over a distance d = 5um, Ex and Ey become opposite in sign and a linear polarization state is obtained which is perpendicular to the initial polarization. A similar rotation towards the original linear polarization state occurs if the beam if further propagated to d=10um.

  • Waveguide coupling in a directional coupler with a liquid crystal cladding layer

    Explanation: If two waveguides are sufficiently close to each other, light can be coupled from one to the other. The interchange of optical power between the two waveguides can be used to make optical couplers and switches. In the illustration, a laser beam with wavelength 1.5um is launched into the left waveguide of the coupler. The light couples to the right waveguide upon propagation through the coupler, as shown in the movie. To realize a tunable coupler, the waveguides can be surrounded by a layer of uniaxial liquid crystal. The orientation of the liquid crystal molecules can be controlled because of their dielectric anisotropy by applying a voltage over the liquid crystal layer. As the coupling characteristics of the directional coupler are determined by the effective indices of the supermodes propagating in the waveguides, changing the optical properties of the surrounding liquid crystal layer will alter the coupling behavior. Therefore, the presented structure will behave as a voltage controllable tunable directional coupler. The simulation of the tunability of such a directional coupler (first the liquid crystal director is calculated, afterwards an optical simulation is performed) is described in Optics Express.

  • Light diffraction in 4um pixels of a microdisplay or spatial light modulator (SLM)

    Explanation:
    As well as being able to model continuous liquid crystal layers, such as the tunable directional coupler possesses, the BPM can also be used to simulate light propagation through structures consisting of pixels as in liquid crystal displays (LCDs) or spatial light modulators (SLMs). Applying the BPM becomes relevant for the simulation of structures in which the pixel dimensions approach the light wavelength. In this case, a rigorous vector method is required to accurately model light diffraction, which plays a crucial role. Recently, small pixel SLMs are gaining interest for holographic display applications as phase modulating components. Application of FEAB can provide detailed information on the near-field profile of light after propagation.

    In the movie, the propagation in the z-direction of a plane wave with vertical linear polarization and wavelength 1um is illustrated through a structure consisting of 4 square pixels of 4um by 4um. To illustrate the influence of the pixel size only on light diffraction whilst excluding boundary effects due to the elastic distortion of the liquid crystal between neighboring pixels, ideal pixels are considered which are modeled as homogeneous uniaxial materials. The top left and bottom right pixel are in the ON state (so polarization rotation will occur) while the top right and bottom left pixel are in the OFF state (no polarization rotation will occur). The movie shows the evolution of Ex upon propagation of the input optical field with vertical linear polarization (Ex = 0) through the structure. It can be seen that the polarization is indeed rotated over 90 degrees after propagation through the ON pixels over 5um to obtain a horizontal linear polarization. As expected, the initial vertical polarization largely remains unchanged in the OFF pixels (Ex = 0). For the 4um by 4um pixels, the amplitude profile of Ex consists of several peaks due to elementary Fresnel diffraction of a plane wave. It can be seen that due to light diffraction of Ex in the ON pixels, this component spreads out to partially cover the OFF pixels. This effect degrades the contrast of the pixels and puts a limit on the pixel size for high quality amplitude modulation SLMs. It is clear from this example that FEAB can provide detailed information on the near-field pattern to aid the design of advanced structures consisting of several pixels such as LCDs or SLMs.

Download

A compiled version of FEAB is available for free download. This version is intended for educational and academic purposes only. Commercial use, including commercial research, is strictly prohibited. This evaluation version is limited to homogeneous material layers and the number of elements is restricted to 400 elements. Contact us if commercial use or additional program functionality (e.g. more elements, other input optical fields, inhomogeneous or bianisotropic material layers) is desired.

NOTE: Please cite our paper in Optics Express vol. 17 (13) in your future publications when reporting on results that have been (partly) obtained by using FEAB.

  • Download the paper in Optics Express.
  • Download the short user manual.
  • Download FEAB without Matlab component runtime (322KB).
  • Download FEAB with Matlab component runtime (228MB).

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