Hi, I have tried to solve eigenvalue problem of the Helmholtz equation(adsbygoogle = window.adsbygoogle || []).push({});

∇×1/μ∇×E-k^{2}E=0

in 2D, wherekis the eigenvalue and^{2}=k^{2}_{0}-β^{2}kis the wavenumber in vacuum. Also_{0}=2*π/λ_{0}β=neff*2*π/λwhere_{0}λis the wavelength in vacuum._{0}

Because constantsε=εand_{r}ε_{0}μ=μare 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_{r}_{μ0}

ε_{0}=_{μ0}=1 => c_{0}=1 (1)

wherecis speed of light in vacuum. Let's assume that my physical dimensions are order of micrometers_{0}

a=1μm

and wavelengthλis for example 1.55μm. Now I want to do all the calcutions in a mesh where coordinates of the_{0}grid points are in microns not in meters and I want that equations (1) hold.Thusa=1andλam I right? If I want to solve effective refractive index neff, is this correct?_{0}=1.55

neff=(1-k^{2}/k^{2}_{0})^{1/2}

wherekNow neff should correspond to the same value if everything is calculated in SI-units._{0}=(2*π)/1.55

-Tuuba

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# Scaling Helmholtz equation

Can you offer guidance or do you also need help?

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