So I am trying to model a dispersive dielectric using the Lorentz model in Julia, more specifically I am trying to obtain the frequency-dependent reflectivity of the material. I am ultimately trying to replicate the results in this paper.
I have the set of update equations:
along with the coefficients:
I am trying to implement this system but the numerical result I obtain is different than the analytical one. Results from my current implementation:
For some context, my source is a Gaussian pulse with central frequency 26 THz and bandwidth 6 THz. I also use the parameters fmax, nmax, d (the thickness of the slab) to determine the minimum wavelength I want to be able to resolve with my cell size through the relations lambda=c/(fmax*nmax), dzi=lambda/20,N = ceil(d/dzi), dz = d/N. Now I obtain the result above when I put in fmax=10,nmax=1, and d=304.8 which are not the expected values. When I put in the actual values fmax=32 (as the maximum freqeuncy in the source), nmax=16 (since the analtical result gives n=16+16i near the resonance), d=38.1 I get the following:
Wrong? result for the right values
The solution for the actual values results in a sin^2 pattern just as in the case with no dispersion. The distance between the peaks for a linear material can be found analytically to be c/(2nmaxd) and here the peculiar thing is that the distance between the peaks is the same as the mean of this expression for all the different frequencies (as nmax is a function of frequency now since the material is dispersive). I remain dumbfounded by these results and can't figure out why my implementation is not working.
I have my code written in julia below (note: I am also taking D/epsilon_0 Q/sqrt(epsilon_0) and J/sqrt(epsilon_0)):
#Initialisation
#Units
ps = 1e-12 #s
μm = 1e-6 #m
THz = 1e12 #Hz
F_per_m = ps^-4*μm^3 #(ps⁴*A²)/(kg*μm³)
H_per_m = ps^2/μm #(kg*μm)/(ps²*A²)
#Grid Resolution (Wavelength)
c = 299792458*ps/μm #μm/ps
λₘᵢₙ = c/(nₘₐₓ*fₘₐₓ) #μm
Nₗ = 20 #number of points per wavelength
Δₗ = λₘᵢₙ/Nₗ #μm
#Grid Resolution (Structure)
Nₛ = 4
Δₛ = d/Nₛ #μm
#Initial Grid Resolution (Overall)
Δzᵢ = min(Δₗ,Δₛ) #μm
#Snap Grid to Critical Dimension(s)
dᵣ = d #μm
N = ceil(dᵣ/Δzᵢ)
Δz = dᵣ/N #μm
#Determine Size of the Grid
Nᵣ = Int64(N+23) #number of total cells
z = (1:Nᵣ)*Δz
#Compute Position of Materials
n₁ = 13 #first cell of the material
n₂ = Int64(n₁+round(d/Δz)-1) #last cell of the material
#Add Materials to the Grid
μᵣ = 1 #relative permeability of material
μₛ = 1 #relative permeability of source
εₛ = 1 #relative permittivity of source
nₛ = sqrt(μₛ*εₛ) #refractive index of source
UR=fill(μₛ,Nᵣ) #relative permeability across the grid
UR[n₁:n₂].=μᵣ
#Calculate Time Step
nb = 1 #refractive index of the boundary
Δt = (nb*Δz)/(2c) #ps
τ = 1/(2*6) #1/(2fₘₐₓ) #ps
t₀ = 6τ #ps
tₚᵣₒₚ = (nₘₐₓ*Nᵣ*Δz)/c #ps
T = 12τ+tc*tₚᵣₒₚ #ps
Steps = Int64(ceil(T/Δt)) #number of total time steps
#Compute the Source Functions
ω = 26*2π #THz
t = (0:Steps-1)*Δt #ps
δt = (nₛ*Δz)/(2c)+Δt/2 #ps
Eₛ = exp.(-((t.-t₀)./τ).^2).*exp.(im*ω.*(t.-t₀)) #located at source
Hₛ = -sqrt(εₛ/μₛ).*exp.(-((t.-t₀.+δt)./τ).^2).*exp.(im*ω.*(t.-t₀.+δt)) #located at source-1
#Parameters for Equation of Motion
μ₀ = 4π*1e-7*H_per_m #H/m
ϵ₀ = 1/(c^2*μ₀) #F/m
ε₀, εₓ = ones(Nᵣ), ones(Nᵣ)
Γ, Ω, α, β = zeros(Nᵣ), zeros(Nᵣ), zeros(Nᵣ), zeros(Nᵣ)
ε₀[n₁:n₂].= 9.66
εₓ[n₁:n₂].= 6.52
Γ[n₁:n₂].= 0.2 #THz
Ω[n₁:n₂].= 24.9 #THz
Ω₀ = sqrt(ε₀[n₁]/εₓ[n₁]*Ω[n₁]^2) #THz
#Compute Update Coefficients
#UR *= nb/(2*ri) .*(sin.(π*fₘₐₓ*ri*Δz/c))./(sin.(π*fₘₐₓ*Δt))
#ε₀ *= nb/(2*ri) .*(sin.(π*fₘₐₓ*ri*Δz/c))./(sin.(π*fₘₐₓ*Δt))
#εₓ *= nb/(2*ri) .*(sin.(π*fₘₐₓ*ri*Δz/c))./(sin.(π*fₘₐₓ*Δt))
meh = nb./(2*UR)
mjj = (2 .-Γ.*Δt)./(2 .+Γ.*Δt)
mqj = (2 .*Ω.^2 .*Δt)./(2 .+Γ.*Δt)
mej = (2*Ω.*sqrt.((ε₀.-εₓ)).*Δt)./(2 .+Γ.*Δt)
mhd = nb/2
mde = 1./(εₓ)
mqe = (Ω .*sqrt.((ε₀.-εₓ)))./(εₓ)
#Initialize fields and Boundary Terms
E, H, J, Q, D =zeros(ComplexF64,Nᵣ),zeros(ComplexF64,Nᵣ),zeros(ComplexF64,Nᵣ),zeros(ComplexF64,Nᵣ),zeros(ComplexF64,Nᵣ)
e₁, e₂ = 0,0
h₁, h₂ = 0,0
#Initialize Reflected and Transmitted Fields
Er = zeros(ComplexF64,Steps)
Et = zeros(ComplexF64,Steps)
#Setup Fourier Transforms
Freq = fftshift(fftfreq(Steps,1/Δt))
Esf = fftshift(fft(Eₛ))
#Main FDTD Loop
for T in 1:Steps
#Record H at Boundary
h₂ = h₁
h₁ = H[1]
#Update H from E
for nz in 1:Nᵣ-1
H[nz] = H[nz] + meh[nz]*(E[nz+1] - E[nz])
end
H[Nᵣ] = H[Nᵣ] + meh[Nᵣ]*(e₂-E[Nᵣ])
#H Source
H[1] = H[1] - meh[1]*Eₛ[T]
#Update J from Q and E
for nz in 1:Nᵣ
J[nz] = mjj[nz]*J[nz] - mqj[nz]*Q[nz] + mej[nz]*E[nz]
end
#Update Q from J
for nz in 1:Nᵣ
Q[nz] = Q[nz] + Δt*J[nz]
end
#Update D from H
D[1] = D[1] + mhd*(H[1] - h₂)
for nz in 2:Nᵣ
D[nz] = D[nz] + mhd*(H[nz] - H[nz-1])
end
#D Source
D[2] = D[2] - mhd*Hₛ[T]
#Record E at Boundary
e₂ = e₁
e₁ = E[Nᵣ]
#Update E from D and Q
for nz in 1:Nᵣ
E[nz] = mde[nz]*D[nz]-mqe[nz]*Q[nz]
end
#Update reflected and transmitted fields
Er[T] = E[1]
Et[T] = E[Nᵣ]
end
#Determine optical properties from FDTD values
Erf = fftshift(fft(Er))
Etf = fftshift(fft(Et))
r = Erf./Esf
tr = Etf./Esf
Ref = abs.(r).^2
Trn = abs.(tr).^2
Con = Ref.+Trn
n = (1 .-r)./(1 .+r)
ε₁ = real(n).^2-imag(n).^2
ε₂ = 2*real(n).*imag(n)
#Determine optical properties from analytical solution
εₜₑₛₜ = εₓ[n₁].+(Ω[n₁]^2*(ε₀[n₁]-εₓ[n₁]))./(Ω[n₁]^2 .-Freq.^2 .-im.*Γ[n₁]*Freq)
nₜₑₛₜ = sqrt.(εₜₑₛₜ)
Rₜₑₛₜ=abs.((1 .-nₜₑₛₜ)./(1 .+nₜₑₛₜ)).^2
I would appreciate any help in fixing my code asap as I have not been able to fix this issue for almost a month now.
As mentioned above I tried changing the values of fmax, nmax, and d to get more accurate results and did manage to except at the wrong values.
