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:Update equationsalong with the coefficients:coefficientsI am trying to implement this system but the numerical result I obtain is different than the analytical one. For reference:Results from my current implementationFor 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
Now 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:
#Initialisation #Units ps = 1e-12 #sμm = 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/psλₘᵢₙ = c/(nₘₐₓ*fₘₐₓ) #μm Nₗ = 20 #number of points per wavelengthΔₗ = λₘᵢₙ/Nₗ #μm #Grid Resolution (Structure) Nₛ = 4Δₛ = 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 Gridμᵣ = 1 #relative permeability of materialμₛ = 1 #relative permeability of sourceεₛ = 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 boundaryΔt = (nb*Δz)/(2c) #psτ = 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 Functionsω = 26*2π #THz t = (0:Steps-1)*Δt #psδt = (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 Motionμ₀ = 4π*1e-7*H_per_m #H/mϵ₀ = 1/(c^2*μ₀) #F/mε₀, εₓ = ones(Nᵣ), ones(Nᵣ)Γ, Ω, α, β = zeros(Nᵣ), zeros(Nᵣ), zeros(Nᵣ), zeros(Nᵣ)ε₀[n₁:n₂].= 9.66εₓ[n₁:n₂].= 6.52Γ[n₁:n₂].= 0.2 #THzΩ[n₁:n₂].= 24.9 #THzΩ₀ = 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).^2ε₂ = 2*real(n).*imag(n) #Determine optical properties from analytical solution if matεₜₑₛₜ = εₓ[n₁].+(Ω[n₁]^2*(ε₀[n₁]-εₓ[n₁]))./(Ω[n₁]^2 .-Freq.^2 .-im.*Γ[n₁]*Freq) end if !matεₜₑₛₜ = fill(εₓ[n₁],Steps) end 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.