Periodic Boundary Conditions in Comsol

Example: Propagation of Plane Waves in Layered Media

Periodic Boundary condition

• BCs that are used to model a large system by simulating a small part that is far away from the edges.

Ex1. Diffusion of particles in a large container

0 0.1 0.2

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

X (m.)

Z (

m.)

Real part of the solution

-1

-0.5

0

0.5

1

Ex2. Propagation of Plane Waves

Information obtained from one boundary is put as a boundary condition to the related boundary

Plane Waves

• Not found in nature– Infinitely large space + infinitely large source

• Used to analyze antennas, small particles in nano photonics (structures work in the far-field).

• Used to build up the physical sources mathematically:– Sommerfeld Integration– Fourier analysis in EM etc…

Plane Waves in Free Space

• Solution of Homogenous Helmholtz equation:

• Must have a magnetic part that obeys:

zjkzjkx

xx

eEeEE

EkE00

00

20

2 0

Propagating in +z direction Propagating in -z direction

zjky e

EH

HjE

0

0

0

Directions: Right hand rule (E x H=direction of prop.)Magnitudes: H=E/impedance

0

00

Plane Waves in Layered Media

• In layered media reflection and transmission occur.

1: incident2: reflected

3: transmitted-14: transmitted-2

5: transmitted-3The magnitudes and the angles of plane waves in all the media can be solved analytically.

Q: How to solve in FEM?A: FEM needs some help! (Use periodic BCs to truncate the system)

Plane Waves in Layered Media by FEM

1: incident

FEM BOX

Needed for FEM with Per. BCs:The direction of propagation of the waves other than the incident,Analytically!! (Snell’s Law)

2: reflected

3: transmitted-1

ri

ir

t

2

1

sin

sin

n

n

i

t

iririn ,,

Idea: Get the information from right boundaryimport it as a new boundary condition to the left

Comsol Implementation

• Draw the geometry

Comsol Implementation

• Initialize constants needed for the simulation– Options – Constants…

Comsol Implementation

• Set the Boundary conditions

Incident field

Comsol Implementation

• Set the boundary conditions - 2

The wave propagating outside

Comsol Implementation

• Boundary Conditions - 3

Periodic Boundary conditions

Need to define different index for each set

The x and y component of wave number the field in medium 3

Comsol Implementation

• Set the sub domain settings

Comsol Implementation

• Set the frequency value to the value in constants:– In Physics – Scalar Variables…

• Press Solve