Scaling Helmholtz equation

[Tuuba]

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

∇×1/μ∇×E-k²E=0

in 2D, where k²=k²₀-β² 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=ε_rε₀ and μ=μ_rμ0 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

 

[jvicens]

Hi Tuuba,

Thank you for sharing your approach and question. It seems like you are on the right track with your scaling method. Using dimensionless units can make the calculations more manageable and can help avoid numerical issues.

In order to accurately solve the eigenvalue problem for the Helmholtz equation, it is important to use consistent units throughout all the calculations. This means that all physical dimensions, such as a and λ0, should be in the same unit (e.g. meters or micrometers). In your case, if you are using micrometers as your unit of length, then both a and λ0 should be in micrometers.

Your equation for neff looks correct, but it is important to note that k0 should also be in the same unit (micrometers) as a and λ0. So in your case, k0=2π/1.55μm=4.05μm^-1.

I hope this helps. Let me know if you have any further questions.