ReTraF

Reflectance and Transmittance fitter for arbitrary layered optical systems.

Instalation:

Add src folder to Matlab path.

Suported Models:

• Constant refractive index model: $n(\lambda) = n0$
• Real Cauchy Model: $n(\lambda) = A_{1} + 10^{4}\frac{A_{2}}{\lambda^2} + 10^{9}\frac{A_{3}}{\lambda^4}$
• Imaginary Cauchy Model: $n(\lambda) = A_{1} + 10^{4}\frac{A_{2}}{\lambda^2} + 10^{9}\frac{A_{3}}{\lambda^4} + A_{4}\cdot i + 10^{4}\frac{A_{5}}{\lambda^2}\cdot i + 10^{9}\frac{A_{6}}{\lambda^4}\cdot i$
• Forouhi-Bloomer model: $n(\lambda) = n_0 + \sum_{j=1,2,3,4} \frac{B_j\cdot (E(\lambda)-E_j) + C_j}{(E(\lambda)-E_j)^2 + G_j^2} + \frac{f_j\cdot (E(\lambda)-E_g)^2\cdot \delta(E(\lambda)-E_g)}{(E(\lambda)-E_j)^2+G_j^2}\cdot i$, being $\delta$ the step function.
• Linear gradient model: $n_j(\lambda)= \frac{n_2-n_1}{N-1}\cdot j$, being $N$ the number of sublayers.

Usage:

ReTraF function call takes the following form:

[models_out,N,D,Data_exp,Data_theor,xbest,foptions_out] = ReTraF(wl,theta,models,data_file,foptions)

Input arguments:

• data_file full path to .mat file containing Reflectance and Transmitance data to fit. Variables inside this file must be:
  • wl_exp wavelength in nm.
  • RSample_S reflectance (in %) for S polarization.
  • RSample_P reflectance (in %) for P polarization.
  • TSample_S transmittance (in %) for S polarization.
  • TSample_S transmittance (in %) for P polarization.

Reflectance and transmittance variables are matrices of dimension (m x n) where m is the number of measured wavelengths and n is the number of measurements for different incident angles.

• wl wavelength (in nm) vector
• theta incident angles struct:
  • theta.values vector containing the incident angles (in degrees) of the measurements to fit.
  • theta.index vector containing the indices of the corresponding angles.
• models cell of structs containing the information of each layer.
• foptions options struct.

Models struct:

All the required information of each layer is stored inside a model struct. We make a distinction between known layers and unknown layers.

• Known layers:
  • Forouhi-Bloomer model :
    • model.type = "Fh-N"
    • model.Eg Bandgap in Ev
    • model.n0 low frequency refractive index
    • model.fi fi parameter (length should be equal to the number of oscillators)
    • model.Ei Ei parameter (length should be equal to the number of oscillators)
    • model.Gi Gi parameter (length should be equal to the number of oscillators)
    • model.D layer thickness in nm
  • Real Cauchy model
    • model.type = "Ch-n"
    • model.A vector with real Cauchy parameters [ A1 , A2 , A3 ]
    • model.D layer thickness in nm
  • Complex Cauchy model
    • model.type = "Ch-nk"
    • model.A vector with real Cauchy parameters [ A1 , A2 , A3 , A4 , A5 , A6 ]
    • model.D layer thickness in nm
  • Linear refractive index gradient
    • model.type = "lin-grad"
    • model.n1 refractive index of the first layer
    • model.n2 refractive index of the last layer
    • model.nlayers number of layers
    • model.D total tickness in nm
  • Constant refractive index
    • model.type = "cnst"
    • model.n refractive index
    • model.D layer thickness in nm
  • Load from .mat file
    • model.type = "file"
    • model.filename full path to the nk .mat file. The variables inside this file must be:
      • wl wavelenth in nm
      • n real part of the refractive index
      • k imaginary part of the refractive index
• Unknown layers
  • Forouhi-Bloomer model :
    • model.type = "U-Fh-N"
    • model.l_Eg lower bound for the bandgap in Ev
    • model.u_Eg upper bound for the bandgap in Ev
    • model.l_n0 lower bound for the low frequency refractive index
    • model.u_n0 upper bound for the low frequency refractive index
    • model.l_fi lower bound for the fi parameter (length should be equal to the number of oscillators)
    • model.u_fi upper bound for the fi parameter (length should be equal to the number of oscillators)
    • model.l_Ei lower bound for the Ei parameter (length should be equal to the number of oscillators)
    • model.u_Ei upper bound for the Ei parameter (length should be equal to the number of oscillators)
    • model.l_Gi lower bound for the Gi parameter (length should be equal to the number of oscillators)
    • model.u_Gi upper bound for the Gi parameter (length should be equal to the number of oscillators)
    • model.l_D lower bound for the layer thickness in nm
    • model.u_D upper bound for the layer thickness in nm
  • Real Cauchy model
    • model.type = "U-Ch-n"
    • model.l_A lower bound for the vector with real Cauchy parameters [ A1 , A2 , A3 ]
    • model.u_A upper bound for the vector with real Cauchy parameters [ A1 , A2 , A3 ]
    • model.l_D lower bound for the layer thickness in nm
    • model.u_D layer thickness in nm
  • Complex Cauchy model
    • model.type = "U-Ch-nk"
    • model.l_A lower bound for the vector with complex Cauchy parameters [ A1 , A2 , A3 , A4 , A5 , A6 ]
    • model.u_A upper bound for the vector with complex Cauchy parameters [ A1 , A2 , A3 , A4 , A5 , A6 ]
    • model.l_D lower bound for the layer thickness in nm
    • model.u_D upper bound for the layer thickness in nm
  • Linear refractive index gradient
    • model.type = "U-lin-grad"
    • model.l_n1 lower bound for the refractive index of the first layer
    • model.u_n1 upper bound for the refractive index of the first layer
    • model.l_n2 lower bound for the refractive index of the last layer
    • model.u_n2 upper bound for the refractive index of the last layer
    • model.nlayers number of layers
    • model.l_D lower bound for the total tickness in nm
    • model.u_D upper bound for the total tickness in nm
  • Constant refractive index
    • model.type = "U-cnst"
    • model.l_n lower bound for the refractive index
    • model.u_n upper bound for the refractive index
    • model.l_D lower bound for the layer thickness in nm
    • model.u_D upper bound for the layer thickness in nn
  • Load from .mat file
    • model.type = "U-file"
    • model.l_D lower bound for the layer thickness in nm
    • model.u_D upper bound for the layer thickness in nn
    • model.filename full path to the nk .mat file. The variables inside this file must be:
    • wl wavelenth in nm
    • n real part of the refractive index
    • k imaginary part of the refractive index

Once all the models are properly defined, they must be packed in a cell array: models = {model_1 model_2 model_3 model_4}.

Options struct:

• foptions.method

  • "fmincon" use Matlab fmincon minimization.
  • "genetic" use Matlab genetic algorithm minimization.

• foptions.itermax maximum number of iterations.

• foptions.poppize genetic algorithm population size.

• foptions.parallel use Matlab parallelization (true or false).

• foptions.lcoher coherence length (1e4 is recommended).

• foptions.scatt apply scattering correction (only for R&T of non-absorbing media).