The problem

At the interface with a slab material, a monochromatic wave experiments a reflection and a refraction as well described by the well-known Snell-Descartes law as follows for air environment \[ \sin \theta_i = n \sin \theta_r, \] where $\theta_i \in \mathcal{R}$ is the incident angle, and $\theta_{r}$ is the refracted angle.

However, as soon as we consider an absorbing material, then, the optical index (and thus the dielectric permittivity $ \varepsilon=n^2 $) is a complex-valued quantity \[ n \in \mathcal{C} \] and the angles of refraction also become a complex-valued quantity. As a consequence, the Snell law can remains valid, but only under certain conditions and an extended physical picture has to be considered.

 

The meaning of a complex angle

First, a complex angle has not a clear physical meaning. In the outer region of the sample, the incidence angle is well defined and real-valued. As soon as the wave gets into the material, the angles (both refraction and reflection) become complex-valued. Two interpretations can be given to a complex angle.

(i) Born and Wolf considered it as a change of optical index as the angle is changing, which can be understood as anisotropy. However, nothing tells that simply using complex angles inside the sample is a sufficient formalism to properly consider this anisotropy.

(ii) A physical picture was proposed describing a shift between the equal amplitude wave front and the equal phase front.

In both cases, the angle of refraction cannot be considered as a well-defined (peak) value anymore and becomes a distribution. A change of angle of refraction is nothing else than a change of the wave momentum, which is actually a scattering event. The laser wave is scattered by matter and interacts (at optical frequencies) with the electronic band structure. The case of an absorbing matearial requires to consider a rigorous approach to consider properly the wave scattering by the sample, for example by using transmission matrix formulation.

 

How to solve the problem ?

Before considering any advanced theories, it is proposed to consider the Snell Law using a multilayered reflection model to calculate the sample reflectivity of a stratified material.

By comparison with a FDTD Maxwell 3D calculation, it will be shown that Snell-Descartes law can remain valid for low absorbing materials, but fails in the case of strongly absorbing materials. A simple criterion identifying the validity limit of the complex-valued Snell-Descartes law can be also proposed.

 

Could an experiment measure that ?

About measuring the angle

Not really. Actually, the refracted angle can be measured if you can measure inside matter. Due to the backward light principle, the angle that is measured by reflection or by tranmission through a thin-film is the same as the incident one, whatever happens inside the sample!

The way to produce such a measurement is thus to make the problem time-dependent and to observe a change of the refraction angle as a function of time. This can be performed by an intense and ultrashort laser pulse which changes the optical properties of the sample as a function of time, and observe the transmitted wave direction evolution (how ?).

About measuring the reflected beam phase

However, by considering a multilayer model to calculate the surface reflectivity, it would be possible to obtain the phase of the reflected signal theoretically. The formula involves the tranmission angles inside the material, and complex optical indices. As a consequence, it is possible to compare the phase of the reflected beam with the phase of the calculated one. A comparison with FDTD calculation is also possible, despite that I focused on obtained the angle instead of the phase of the reflected beam.

So, is it so bad, doctor ?

So it seems that this theoretical limit of the Snell law draws a real physical problem, and that the concept of refractive angle inside matter is not well defined for all materials. However, in the theoretical world (so in some simulation codes), this angle is required, and is still a complex-valued quantity. So this problem is important and must be solved in a clear way.

 

Theoretical limit of Snell-Descartes law: mathematical and physical reasons

Let's investigate now why Snell-Descartes low is limited to low-absorbing materials.

The Snell law is expressed by the following formula, using the dielectric permittivity

\[ \sqrt{\varepsilon_i} \sin \theta_{i} = \sqrt{\varepsilon_{i+1}} \sin \theta_{i+1} \] where $i$ is the i-th layer of the sample. The complex-valued refractive angle $\theta_{i+1}$ is given by

\[ \theta_{i+1} = \sin^{-1} \left[ \sin \theta_i \sqrt{\frac{\varepsilon_i}{\varepsilon_{i+1}}} \right] \]

The real-valued $\arcsin$ function applied to $x \in \mathcal{R}$ has the following property [1] \[ \left| \arcsin(x) \right| \le \frac{\pi}{2} \]

However, the complex-valued number $\arcsin(z)$, where $z \in \mathcal{C}$ has a modulus which looses its trigonometric properties as soon as \[ \left| \sqrt{\frac{\varepsilon_i}{\varepsilon_{i+1}}}\right| > 1 \] This situation actually leads to a loss of the angle definition and constitutes a criterion which allows to check if the Snell-Descartes law can give an accurate description in an absorbing medium. This is the mathematical reason.

The physical reason is that the angle of refraction does not stand any longer since scattering occurs and that the angle becomes a broad distribution, which should be taken into account to conserve the energy and momentum.

What if I just take the real part of the angle ?

Because it is not describing what happens physically. Taking the real part or the imaginary part is simply wrong and is not linked with any physical description in that case.

References

[1]: http://mathworld.wolfram.com/InverseSine.html