Calculate Dispersion Relation Using Comsol

Analytic Verification and Brillouin Zone Mapping Tool

What is Calculate Dispersion Relation Using COMSOL?

To calculate dispersion relation using COMSOL is a fundamental procedure for physicists and engineers working with periodic media. This process involves determining the relationship between the wave vector (k) and the eigenfrequency (ω or f). In COMSOL Multiphysics, this is typically achieved through an “Eigenfrequency Study” applied to a single unit cell utilizing Bloch-Floquet periodic boundary conditions.

Researchers use this to identify bandgaps in phononic or photonic crystals, analyze group velocity dispersion in waveguides, and characterize the refractive index of metamaterials. A common misconception is that COMSOL provides a single button to “plot dispersion.” In reality, it requires a parametric sweep over the wave vector across the irreducible Brillouin zone.

Calculate Dispersion Relation Using COMSOL Formula and Mathematical Explanation

The mathematical foundation relies on Bloch’s Theorem, which states that waves in a periodic medium take the form of a plane wave modulated by a periodic function. When you calculate dispersion relation using COMSOL, you are solving the Helmholtz equation subject to:

u(x + a) = u(x) ⋅ exp(-i ⋅ k ⋅ a)

Variable	Meaning	Unit	Typical Range
k	Wavenumber / Wave Vector	rad/m	0 to π/a
f	Eigenfrequency	Hz	1 Hz – 1 THz
a	Lattice Constant	m	10^-9 to 1 m
vg	Group Velocity (dω/dk)	m/s	Material dependent

Practical Examples (Real-World Use Cases)

Example 1: Acoustic Metamaterial Design

A researcher wants to design a 1D phononic crystal with a lattice constant of 0.1m in air. By using an periodic structure simulation, they sweep k from 0 to π/0.1. The resulting plot shows a “flat band” at 1500 Hz, indicating zero group velocity and the potential for energy localization.

Example 2: Photonic Crystal Fiber

An engineer calculates the dispersion for a silica fiber with hexagonal air holes. They use the electromagnetic dispersion model in COMSOL to find the Zero Dispersion Wavelength (ZDW), crucial for nonlinear optics applications. The inputs involve the refractive index of silica and the hole-to-pitch ratio.

How to Use This Calculate Dispersion Relation Using COMSOL Calculator

• Step 1: Enter your Lattice Constant (a). This should match the width of your geometry in the COMSOL Graphics window.
• Step 2: Input the Phase Velocity (c). For acoustics, use the speed of sound; for optics, use the speed of light divided by the refractive index.
• Step 3: Adjust the Wavenumber (k). This allows you to verify if your COMSOL eigenfrequency at a specific point (like the X or M point) matches the theoretical expectation for a homogeneous medium.
• Step 4: Observe the Brillouin Zone Dispersion Curve. If your COMSOL results deviate significantly from this linear line, you have likely identified a bandgap or dispersion effect caused by periodicity.

Key Factors That Affect Calculate Dispersion Relation Using COMSOL Results

• Mesh Density: In COMSOL, the accuracy of high-frequency bands depends heavily on having enough elements per wavelength.
• Periodic Boundary Conditions: Ensuring the Floquet-Bloch vector is correctly parameterized using a “Global Definition” variable (e.g., k_vector).
• Material Dispersion: If the material properties themselves are frequency-dependent, the dispersion relation becomes non-linear even without periodic effects.
• Unit Cell Symmetry: The shape of the Brillouin zone changes based on whether the lattice is square, hexagonal, or rectangular.
• Study Type: Using an “Eigenfrequency Study” vs. a “Frequency Domain Study” — eigenfrequency is the standard for dispersion mapping.
• Number of Eigenfrequencies: COMSOL must be instructed to find more than just the first few modes to capture higher-order bands.

Frequently Asked Questions (FAQ)

Why is my dispersion curve folded?

This is due to the Bloch-Floquet periodicity. When you calculate dispersion relation using COMSOL, any frequency above the first Brillouin zone is folded back into the 0 to π/a range.

What is a bandgap?

A frequency range where no eigenfrequencies exist for any value of k. It signifies that waves cannot propagate through the structure at those frequencies.

How do I define ‘k’ in COMSOL?

Create a ‘Parameter’ named k_val and use it in the Periodic Boundary Condition settings under the Floquet Wave Vector components.

Does this calculator handle 2D or 3D lattices?

This tool provides a 1D analytic baseline. For 2D/3D, the Brillouin zone mapping follows paths like Γ-X-M-Γ.

Can I use this for surface waves?

Yes, by applying Bloch theorem calculator principles to a unit cell with a finite thickness and vacuum/perfectly matched layers (PML).

What is the difference between phase and group velocity?

Phase velocity is ω/k, while group velocity is dω/dk, representing the speed of energy/information transport.

Why does COMSOL show complex frequencies?

Usually, this indicates damping or leaky modes. For standard dispersion curves, look for the real part of the eigenfrequency.

How do I export the data for plotting?

In COMSOL, use a ‘1D Plot Group’ with a ‘Global’ feature, then use the ‘Export’ function to save a CSV for external tools.

Related Tools and Internal Resources

• COMSOL Eigenfrequency Guide – Master the solver settings for modal analysis.
• Wave Propagation Physics – Understanding the fundamentals of elastic and EM waves.
• Periodic Structure Simulation – Tips for setting up unit cells efficiently.
• Bloch Theorem Calculator – Quick verification of periodic phase shifts.
• Acoustic Bandgap Analysis – Specific workflows for noise control structures.
• Electromagnetic Dispersion Model – Dealing with dielectric and metallic periodicity.