The semiconductor industry is getting closer to standardizing photonic systems, so designers will need to include photonic structures in their electrical and all-optical circuits.
One fundamental structure is an optical resonator, which forms the basis of many other structures and devices in photonic circuits.
Simple optical resonator cavities are easy to design and tailor to particular geometries, but complicated cavities need a field solver for visualization and analysis.
Laser light in your photonic ICs is generated from an optical resonator.
Photonic ICs are on the horizon and optical resonator structures are a fundamental part of these components and systems. An optical resonator is deceptively simple, but it can have very complex structures that do not have an obvious set of resonant frequencies. In order to design these structures, an engineer needs to determine the resonant frequencies of optical cavities so that they can support electromagnetic resonances at desired frequencies.
Simple geometries like parallelepipeds, cylinders, and spheres have well-known closed-form solutions for their resonances. Furthermore, simple dispersion relations at the resonator boundary are used to determine the resonant frequencies of the system. This is a standard process in optical resonator design and it extends into numerical techniques for more complex optical structures.
Simple Optical Resonator Eigenfrequencies
The resonant frequencies of an optical resonator, also known as the eigenfrequencies, will determine how the system responds to an arbitrary stimulus. Each eigenmode is a particular electric field distribution in the system, and each mode has an eigenfrequency associated with it. For example, when we think about the possible standing waves that can exist on a string, the various wavelengths that can fit on the string are the eigenmodes, and the oscillation frequency is the eigenfrequency. RF and photonics designers can determine the eigenmodes for the electromagnetic field distribution from a wave equation, and the electromagnetic field is built up from the eigenmodes by exploiting their orthogonality.
It’s a bit more complicated in an optical system, as the system can have absorption, leading to nonlinear effects at high photon flux. However, the mathematical formalism required to determine the eigenmodes of a system and its response to stimuli are well-known and can be found in many textbooks. Below is a short summary of the relevant systems one might encounter in photonic systems design.
The rectangular cavity, as its name suggests, has a rectangular cross-section and may have highly reflective surfaces to confine light. This is a good place to introduce optical eigenmodes, as the process shown here is useful for any other resonator. The goal in this process is to solve the inhomogeneous wave equation for the system with known boundary conditions.
This is normally done by expanding the electric field (or magnetic field) as a series of orthogonal eigenmodes, i.e., we have a Sturm-Liouville problem. Here, we solve the simple Helmholtz problem for the electric or magnetic field and then apply the boundary conditions to the Helmholtz solution to get the optical resonator eigenmodes and their eigenfrequencies.
Solving the inhomogeneous wave equation using terahertz microcavity resonator geometry and eigenfrequencies.
If you apply the boundary conditions you select to a rectangular cavity, you’ll get an equation you can use to determine the eigenfrequency for each eigenmode. This may be as simple as finding the solutions to a trigonometric equation, or it may involve solving a transcendental equation. The latter is required in partially open cavities, where some light is allowed to exit the cavity at one port. The image below shows an example of rectangular geometry and the eigenfrequencies:
Rectangular optical resonator geometry and eigenfrequencies.
For any other non-rectangular geometry, we will have a situation where the eigenfrequencies are not multiples of each other and are not solved from a simple equation at the boundary. Although there will be a simple method for finding the dispersion relation, it often has tabulated numerical solutions or it requires solving a transcendental equation. An example for a ring cavity is shown below:
A ring cavity is a cylindrical cavity with boundary conditions defined at the interior and exterior of the structure. For the ring cavity, we have an internal and external boundary with specified boundary conditions. These are critical for determining the eigenfrequencies from the cylindrical Bessel functions, where the field inside and outside the cavity are matched at the ring boundary. The interior boundary of the ring is normally taken to be totally reflective to simplify the problem.
Ring cavity geometry and eigenfrequencies.
Thankfully, the Bessel functions are tabulated for various orders, or they can be redefined in terms of sines and cosines. This allows complex arguments to be considered, which will account for absorption in the system. However, this means there is no simple closed-form solution for the resonant frequencies in the system, in contrast to the case for a rectangular resonator.
More Complex Optical Resonator Geometries
The two resonators shown above are common, but simple, geometries with closed-form solutions for the set of orthogonal basis functions (solution to the Helmholtz equation). In addition, arbitrary sources in space and time may not have an analytical solution, so determining the resonances in an arbitrary optical cavity with arbitrary sources requires numerical techniques. In common linear semiconductors, this still becomes a complex electromagnetics problem requiring a field solver to generate useful data.
Materials for Optical Resonators
Conventional optical resonators on an optical table use multiple mirrors, filters, polarizers, and a range of other components to manipulate light. At the photonic IC level, this all has to be integrated onto a semiconductor wafer. Silicon is the most desirable platform for photonic ICs, as the manufacturing capacity already exists. However, other materials have proven themselves at SMF and MMF wavelengths. Some available material platforms are shown in the table below.
By carefully designing the resonator geometry and selecting the appropriate materials, the designer can tailor the resonator’s optical response within a desired range of wavelengths. In addition, by setting a resonance at a particular frequency in the resonator, the designer can create a strong coherent field inside the resonator, which can exploit nonlinear effects in the system. This is the basis of forming nonlinear amplifiers, lasers, up/down converters, and other nonlinear components in photonic circuits.
As the semiconductor industry moves closer to commercializing photonic circuits and all-optical systems, optical resonator structures will become increasingly fundamental for manipulating optical signals.