# Scaling Helmholtz equation

1. Apr 16, 2013

### Tuuba

Hi, I have tried to solve eigenvalue problem of the Helmholtz equation

∇×1/μ∇×E-k2E=0

in 2D, where k2=k202 is the eigenvalue and k0=2*π/λ0 is the wavenumber in vacuum. Also β=neff*2*π/λ0 where λ0 is the wavelength in vacuum.

Because constants ε=εrε0 and μ=μrμ0 are not very convinient numerically I have tried to scale my equations so that I don't have to deal with very large or very small numbers. So I have tried to use dimensionless units

ε0=μ0=1 => c0=1 (1)

where c0 is speed of light in vacuum. Let's assume that my physical dimensions are order of micrometers

a=1μm

and wavelength λ0 is for example 1.55μm. Now I want to do all the calcutions in a mesh where coordinates of the grid points are in microns not in meters and I want that equations (1) hold. Thus a=1 and λ0=1.55 am I right? If I want to solve effective refractive index neff, is this correct?

neff=(1-k2/k20)1/2

where k0=(2*π)/1.55 Now neff should correspond to the same value if everything is calculated in SI-units.

-Tuuba