Critical angle refractometry with optically isotropic attenuating media

Abstract

We report on a refractometry method for determining the complex refractive index of isotropic attenuating media, which relies on the measurement of the two critical angles at the s and p polarization states. Real and imaginary indices are retrieved by numerically solving a well-posed system of algebraic equations. Data regression is avoided and accurately calibrated reflectance values become superfluous. Monte Carlo simulations confirm that the method’s relative error is proportional to the error in locating the critical-angles, which can be made sufficiently small.

Appendix

All algebraic expressions presented in Sect. 2 appear in Ref. [25] with the exception Eq. 6b, which is presented here for the first time and derives from the derivative condition for the p-polarization [25]

$$\begin{aligned} \rho _p'' = \frac{2}{1+\rho _p}(\rho _p')^2. \end{aligned}$$

(A.1)

Differentiation of $\rho _p$ is with respect to the angle of incidence on the reflectance equation. The derivatives of $\alpha _p$ and $\gamma _p$ with respect to $\theta$ are needed and the recursive relations

$$\begin{aligned} \alpha _p' = -2t_p(1-\alpha _p), \quad \gamma _p' = 2t_p\left( \gamma _p - \frac{\alpha _p}{\gamma _p}\right) \end{aligned}$$

(A.2)

hold [25]. Furthermore, it is $t_p' = 1+t_p^2$. The derivatives of $\rho _p$ calculated by differentiation of the reflectance equation are substituted in the derivative condition and the resulting first order equation for $\sqrt{\alpha _p+\gamma _p}$ is solved, while simplifying along the way, to obtain Eq. 6b.

It is finally worth noting that computation speed seems to improve when Eq. 6a is replaced by its solution for $\sqrt{\alpha _p+\gamma _p}$, according to the perfectly equivalent form:

$$\begin{aligned}&2\sqrt{2}\, t_p^2 \sqrt{\alpha _p+\gamma _p} = \rho _p (1+t_p^2)(t_p^2+\gamma _p) \nonumber \\&\quad -\sqrt{\rho _p^2 (1+t_p^2)^2 (t_p^2+\gamma _p)^2- 4t_p^2(1+\gamma _p)(t_p^4+\gamma _p)}, \end{aligned}$$

(A.3)