Bayesian estimation of optical constants using mixtures of Gaussian process experts

Abstract

We propose modeling absorption spectrum measurements as mixtures of Gaussian process experts. This enables us to construct a flexible statistical model for interpolating and extrapolating measurements, facilitating statistical integration of Kramers-Kronig relations to estimate the whole complex refractive index. Additionally, we statistically model the anchoring points used in subtractive Kramers-Kronig relations to account for possible measurement errors of the anchor point. In addition to flexible statistical modeling, the mixtures of Gaussian process formulation enables automatic selection of measurement points to use for extrapolation. We apply the method to experimental absorption spectrum measurements of gallium arsenide, potassium chloride, and transparent wood.

Figures (6)

• Figure 1: An example partition with $K = 5$ experts and 10 realizations drawn from the mixture of Gaussian process experts model estimated with nested sequential Monte Carlo sampling denoted by $(\boldsymbol{\omega}_k, \boldsymbol{\kappa}_k)$ and $\widetilde{\boldsymbol{\kappa}}^*$, respectively. Each realization can be used to compute a corresponding real part of the complex refractive index through Kramers-Kronig relations. The realizations closely follow the low-noise measurements. Outside the measurement range, $\omega \geq 6\text{ eV}$, the realizations show varied, slowly-decaying behavior encoded by the covariance function of the Gaussian process experts. The extrapolation is also performed for $\omega \leq 1.5\text{ eV}$, however, this part of the realizations is left out for visual clarity.
• Figure 2: Example realizations $\widetilde{\boldsymbol{\eta}}^*$ corresponding to realizations $\widetilde{ \boldsymbol{\kappa} }^*$ in Figure \ref{['im:examplePartitionRealization']} and anchor points $\left( \widetilde{\omega}_\text{a}, \widetilde{\eta}_\text{a} \right)$ sampled from $\pi_0( \omega_\text{a}, \eta_\text{a} )$. With a large ensemble of realizations, we can construct a numerical approximation for the complex refractive index distribution, $\pi( \boldsymbol{n}^* \mid \boldsymbol{\omega}^*, \boldsymbol{\omega}, \boldsymbol{\kappa} )$.
• Figure 3: Results for the gallium arsenide dataset. At the top, attenuation measurement data $\boldsymbol{\kappa}$, together with estimated predictive mean and 95% marginal predictive intervals denoted by $\rm{E} \left[ \boldsymbol{\kappa}^* \right]$ and $\rm{Var}_{95\%} \left[ \boldsymbol{\kappa}^* \right]$, respectively. In the middle and bottom, predictive means and 95% marginal predictive intervals $\rm{E} \left[ \boldsymbol{\eta}^* \right]$ and $\rm{Var}_{95\%} \left[ \boldsymbol{\eta}^* \right]$ for the real, refractive component estimated with and without extrapolation, respectively. The reference measured refractive index is denoted by $\boldsymbol{\eta}$ and the 95% intervals of the anchor point distribution $\pi_0( \omega_\text{a}, \eta_\text{a} )$ are illustrated in red.
• Figure 4: Results for the potassium chloride dataset. At the top, attenuation measurement data $\boldsymbol{\kappa}$, together with estimated predictive mean and 95% marginal predictive intervals denoted by $\rm{E} \left[ \boldsymbol{\kappa}^* \right]$ and $\rm{Var}_{95\%} \left[ \boldsymbol{\kappa}^* \right]$, respectively. In the middle and bottom, predictive means and 95% marginal predictive intervals $\rm{E} \left[ \boldsymbol{\eta}^* \right]$ and $\rm{Var}_{95\%} \left[ \boldsymbol{\eta}^* \right]$ for the real, refractive component estimated with and without extrapolation, respectively. The reference measured refractive index is denoted by $\boldsymbol{\eta}$ and the 95% intervals of the anchor point distribution $\pi_0( \omega_\text{a}, \eta_\text{a} )$ are illustrated in red.
• Figure 5: Results for the transparent wood dataset without background correction. At the top, attenuation measurement data $\boldsymbol{\kappa}$, together with estimated predictive mean and 95% marginal predictive intervals denoted by $\rm{E} \left[ \boldsymbol{\kappa}^* \right]$ and $\rm{Var}_{95\%} \left[ \boldsymbol{\kappa}^* \right]$, respectively. At the bottom, predictive mean and 95% marginal predictive interval $\rm{E} \left[ \boldsymbol{\eta}^* \right]$ and $\rm{Var}_{95\%} \left[ \boldsymbol{\eta}^* \right]$ for the real, refractive component estimated with extrapolation. The fixed anchor point $( \omega_\text{a}, \eta_\text{a} )$ is illustrated in red.