Calculate Dispersion Relation Using COMSOL

Analyze wave propagation, photonic bands, and waveguide characteristics.

k-point (Normalized)	Frequency (Normalized)	Effective Index (n_eff)	Wavelength (nm)

Table 1: Discrete dispersion data points for COMSOL validation.

What is the Dispersion Relation in COMSOL?

To calculate dispersion relation using comsol is to determine the relationship between the angular frequency (ω) and the wavevector (k) of an electromagnetic or acoustic wave within a specific medium or structure. This is a fundamental task for photonics engineers, acoustic researchers, and materials scientists.

Who should use this? Anyone working with photonic crystals, surface plasmon resonance, or fiber optic waveguides. A common misconception is that dispersion is only about material properties; however, “geometric dispersion” caused by the structure’s physical boundaries often dominates at the micro and nano scales. By using COMSOL Multiphysics, users can numerically solve Maxwell’s equations to find these relations where analytical solutions do not exist.

Calculate Dispersion Relation Using COMSOL: Formula and Mathematical Explanation

The mathematical backbone involves solving the Eigenvalue problem derived from the wave equation. For a periodic structure, we apply Bloch-Floquet boundary conditions.

The simplified linear relation used in this calculator is:

f = (c * k) / (2π * n_eff)

Variable	Meaning	Unit	Typical Range
ω (Omega)	Angular Frequency	rad/s	10^12 – 10^15
k	Wavevector	1/m	0 – 2π/a
a	Lattice Constant	nm	100 – 2000
n_eff	Effective Index	Unitless	1.0 – 4.5

Practical Examples of Dispersion Calculations

Example 1: Silicon-on-Insulator (SOI) Waveguide

Suppose you are using COMSOL to model a standard SOI waveguide with a width of 500nm. You run a Boundary Mode Analysis and find an effective index (n_eff) of 2.45 at a wavelength of 1550nm. To calculate dispersion relation using comsol in this scenario, you would vary the frequency and observe how n_eff changes, allowing you to plot the group velocity dispersion (GVD).

Example 2: 1D Photonic Crystal Bandgap

In a periodic stack of TiO2 and SiO2 layers (period a = 400nm), you seek the bandgap. By sweeping the Bloch wavevector k from 0 to π/a in COMSOL and extracting the eigenfrequencies, you can identify the forbidden frequency ranges where no modes exist. This tool helps you normalize those frequencies to compare against literature values.

How to Use This Dispersion Relation Calculator

1. Enter Periodicity (a): Input the physical size of your unit cell in nanometers.
2. Input Effective Index: Retrieve the “n_eff” value from your COMSOL Mode Solver results.
3. Set Wavevector: Choose the k-point in the Brillouin zone you are analyzing (0.5 represents the zone edge).
4. Review Results: The calculator instantly provides the normalized frequency and physical wavelength.
5. Analyze Chart: Use the SVG/Canvas chart to see how your guided mode deviates from the vacuum light line.

Key Factors That Affect Dispersion Results

• Material Dispersion: The refractive index of materials like Silicon or Silica changes with frequency (Sellmeier equations).
• Geometric Constraints: The width and height of a waveguide force the wave to “see” a different environment, altering the dispersion curve.
• Meshing Accuracy: When you calculate dispersion relation using comsol, a coarse mesh can lead to numerical artifacts and incorrect band positions.
• Boundary Conditions: Using Perfect Electric Conductors (PEC) vs. Scattering Boundary Conditions changes the mode leakage and dispersion.
• Temperature: Thermal expansion and the thermo-optic effect modify both the geometry and the refractive index.
• Symmetry: Exploiting symmetry in COMSOL can reduce calculation time but requires careful selection of symmetric vs. anti-symmetric modes.

Frequently Asked Questions (FAQ)

How do I extract k-vectors in COMSOL?

Usually, you define a parameter for ‘k’ and use it in the Bloch-Floquet boundary conditions, then run a Parametric Sweep over that variable.

Why is my dispersion curve folded?

This happens in periodic structures due to the Brillouin Zone boundaries. The curve is periodic with 2π/a.

What is the difference between phase and group velocity?

Phase velocity is ω/k, while group velocity is dω/dk. Group velocity is what carries information and energy.

Can COMSOL handle non-linear dispersion?

Yes, by incorporating non-linear material properties in the Physics settings, but the Eigenfrequency solver remains linear; you may need Transient analysis.

What is a normalized frequency?

It is often expressed as ωa/2πc or a/λ, making the results scalable to any size of the same geometry.

Why use Boundary Mode Analysis?

It allows you to find the cross-sectional modes of a waveguide at a specific frequency, providing the initial n_eff for the dispersion curve.

Does the tool account for loss?

This specific calculator assumes a lossless medium. In COMSOL, the imaginary part of the refractive index represents loss.

How accurate is the linear approximation?

It is accurate for simple waveguides but fails near bandgaps or resonances where the curve highly bends.