EMSolve Selected Results

Coaxial Waveguide and Numerical Dispersion

Computation of Eigenfrequencies and Eigenmodes of Three-Dimensional, Inhomogeneous Resonant Cavities

Eigenmodes of a Stanford Linear Accelerator Center (SLAC) Induction Cell

Full-Wave Simulation of Substrate Coupling Effects in Integrated Circuits

High-Order Spatial Discretization

Low-Frequency Signal Propagation in a Mountain Environment

Optical Trapping of Dielectric Microspheres

Photonic Band-Gap Waveguides

Radar Propagation in Buildings: A 10-Billion-Element Simulation

RF Signal Propagation in a Cave

Simulation of Accelerator Wakefields

Single-Mode Optical Fiber

TITAN Petawatt Facility EMP Modeling

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Photonic Band-Gap Waveguides

In these examples, we use EMSolve to simulate two different wave-guiding structures based on the notion of a photonic band-gap (PBG) structure (also known as a photonic crystal). PBG materials allow for the engineering of devices that can prohibit the propagation of light, allow propagation only in certain directions at certain frequencies, or localize light in specified areas [1] ,[2].

A PBG structure works very similarly to a currently well-known and widely used optical device: the dielectric mirror (or "quarter-wave stack"), which consists of alternating layers of different dielectric materials. Light of the proper wavelength, when incident on such a layered material, is completely reflected. This is because the light wave is scattered at the layer interfaces, and if the spacing is just right, the multiplied scattered waves interfere destructively inside the material. This effect is well known—however, while such mirrors are tremendously useful, they only reflect light at normal incidence or near-normal incidence to the layered material. A PBG structure is the generalization of this notion for periodic arrays of dielectric material in two and three dimensions. In general, a PBG structure is defined to be a periodic array of dielectric materials with a characteristic dimension (or lattice constant) a, such that incident light of wavelength ~2a is forbidden to propagate in the structure.

If, for some frequency range, a PBG structure reflects light of any polarization incident at any angle, then the crystal is said to have a "complete photonic band gap" [3]. In such a crystal, no light modes can propagate if they have a frequency within that range. The simple dielectric mirror cannot have a complete band gap, because scattering occurs along only one axis. In order to create a material with a complete photonic band gap, the contrasting dielectrics must be arranged in a lattice that is periodic along three axes. Due to the nature of Maxwell's equations, a PBG structure can be scaled to any operating wavelength (or frequency). Below are two examples of three-dimensional PBG structures with a complete photonic band gap; one designed to operate at optical wavelengths (left) and one designed for the RF regime (right).

2-D Slab Optical Waveguide

Here we simulate the propagation of an optical signal around a sharp 90-degree bend in a two-dimensional slab (i.e., one element thick in the z-direction) PBG waveguide using the high order spatial discretization capabilities of EMSolve [4]. The PBG structure consists of a 9 × 9 array of Gallium-Arsenide (GaAs) cylinders oriented normal to the x-y propagation plane, with a relative dielectric permittivity of 12.0, surrounded by a square section of air. The rod spacing (or lattice constant) for the crystal is a = 0.62 micrometers, yielding a band-gap in the optical regime. A waveguide can be constructed by removing some of the cylinders and introducing a defect, allowing a small range of wavelengths of light around a central defect frequency to propagate through the structure.

The computational mesh for this problem consists of only 4,432 hexahedral elements. However, we use high order p=3 interpolatory polynomial basis functions to represent the electric and magnetic fields, resulting in a total of 441,123 electric field unknowns and 399,888 magnetic flux density unknowns. Because of its large size, this problem must be solved in a parallel computational environment. The mesh and its corresponding linear system are partitioned over 16 processors as shown in the image below.

To account for the open region boundary at the top of the mesh, a single-element-thick artificial absorbing layer (or PML) is added to terminate the guide. Below is an animation of the computed electric field of a 1.55-micrometer optical signal traversing the bend of the PBG waveguide. Note how the electric field is guided around the sharp bend with minimal loss.

3-D "Multi-Bend" Woodpile RF Waveguide

Here we simulate a 3-D PBG waveguide with a complete photonic band-gap designed to operate in the RF regime. The PBG crystal is based on the "woodpile" structure as investigated by [5]. In particular, we utilize the unit cell originally proposed by [6], which consists of a series of aluminum rods (index of refraction, 3.1) arranged in an alternating, stacked configuration. The lattice constant for this crystal is a = 1.123 cm and the unit cell has dimensions of 1.123 cm × 1.123 cm × 1.272 cm, making it suitable for operation in the radio frequency regime. We construct a 3-D crystal by arranging the unit cell in a 9 × 13 × 7 layer configuration as shown in the image below.

Our goal is to exploit the complete photonic band-gap of this crystal and create a "multi-bend" wave guide, making the radio signal traverse two separate 90-degree bends in three-dimensional space. This can be accomplished by introducing three separate defects into the crystal as shown in the image below. We create a 90-degree bend in the x-y plane by removing a half-portion of two of the rods. While the vast majority of computational research in PBG waveguides has been performed on two-dimensional structures like that described in the previous section, single bends in 3-D crystals have been studied as well. What makes this simulation unique is the introduction of a third z-defect by removing a section of rods 2 lattice constants wide from each of the stacked z layers. Because of the 3-D nature of the multibend, this type of simulation cannot be performed using standard 2-D codes, which are extensively used in the study of PBG devices. In addition, trustworthy simulations of PBG waveguides require that phase velocities of propagating waves be computed as accurately as possible. A high order method is therefore highly desirable for an electrically large waveguide such as this.

The computational mesh of this problem consists of 419,328 hexahedral elements. We excite the problem with a time-dependent voltage source boundary condition applied at the x-z input plane with an operating frequency of 11 GHz. The rest of the mesh is terminated with a PEC boundary condition. We use high order p = 2 basis functions to represent the electric and magnetic fields resulting in a linear system with approximately 10.5 million unknowns. This large linear system requires that the problem be distributed in parallel across 150 processors. We let the simulation run for a total of 6,500 time steps. In the image below, we show a 3-D iso-surface plot of the electric field magnitude in the wave guide at the end of the simulation. Note how the wave has made two complete 90 degree bends with a negligible loss due to radiation.

Below is an animation of the time-dependent electric field plotted over three separate slices into the crystal. The source of the wave is originally polarized along the z-axis of the guide. Note how as the the wave traverses the first bend, it remains polarized along the z-axis but as it traverses the second bend, it becomes polarized along the y-axis. The results of this computation represent the first time a high-order, full-wave simulation of a 3-D PBG waveguide has ever been performed, and to our knowledge, the first time a multibend PBG waveguide has ever been demonstrated.

References:

1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, 1995, Photonic Crystals:  Molding the Flow of Light: Princeton Univ. Press.
2. E. Yablonovitch, 1993, Photonic band-gap structures: J. Opt. Soc. Am. B, vol. 10, no. 2, 283-295.
3. E. Lidorikis, M. Povinelli, S. Johnson, and J. Joannopoulos, 2003, Polarization-independent linear waveguides in 3D photonic crystals: Phys. Rev. Let., vol. 91,no. 2, Art. No. 023902.
4. R. Rieben, D. White, and G. Rodrigue, 2004, Application of novel high order time domain vector finite element method to photonic band-gap waveguides: Proceedings of the 2004 IEEE International Antennas and Propagation Symposium, volume 4, 3469-3472, Monterey, CA.
5. H. S. Sozuer and J. P. Dowling, 1994, Photonic band calculations for woodpile structures: J. Mod. Opt., vol. 41, no. 2, 231-239.
6. E. Ozbay, et. al., 1994, Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods: Phys. Rev. B, vol. 50, no 3, 1945-1948.

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January 20, 2009