Are Fresnel Equations with Complex Indices and Angles always valid? This question-answer pair came after i was asked the following question and realized i had to do some research of my own to answer it fully, and to be sure that the answer is „yes“, in the sense described in my answer.
One fairly often sees the assertion that the Fresnel equations are unconditionally valid for linear isotropic mediums, even when the refractive indices are complex, so that they can be freely used for calculations even with metal-metal and metal-dielectric interfaces. For example, a well known „Photonics Knowledge“ site (RP Photonics) makes this claim on its „Fresnel Equations“ entry in its „Encyclopedia of Photonics“.
I have been unable to find a justification for the truth of this statement; can someone please point me to a reference, or give a concise, rigorous justification of its truth?
 A: First let’s define „unconditionally“. The answer is yes in the sense that, as long as the mediums concerned and (a) linear and (b) isotropic the standard derivation of the Fresnel equations works, that is, (1) Assume a plane wave solutions to Maxwell’s equations for (a) an incident wave, (b) a reflected wave on the incoming wave side of the interface and (c) a transmitted plane wave beyond the interface then (2) derive the complex ratios of the amplitudes of these three waves  by imposing the continuity conditions for the electromagnetic fields across the interface.
One can even relax the isotropy condition and appropriate simple (but extremely messy) generalizations of following reasoning still works perfectly for complex component dielectric tensor, or even the full blown rank 4 tensor that maps the Faraday tensor to the Displacement tensor, but the equations analogous to the wonted Fresnel equations are vastly more complicated tensor equations, albeit linear. 
Another way of putting this statement is that when we are dealing with linear materials with general complex-component wavevectors, the Fresnel equations always describe a valid solution of Maxwell’s equations comprising an incident plane wave, a reflected plane wave on the  incoming wave side of the interface and a lone transmitted plane wave in the medium beyond the interface.
Ok, let’s move onto justification. As always, if you want to know whether a certain set of equations is valid in certain circumstances, one must look at the derivation of those equations and check whether all the assumptions used to make any and all inferences hold give the said circumstances.
Let us take, as a canonical derivation of the Fresnel equations, that to be found in Section 1.5.2 of Born and Wolf entitled „Fresnel Formulae“; in my (the Seventh) edition, it begins on page 40:
Max Born and Emil Wolf, "Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light"
To make this derivation work, one needs the following steps:


*

*The  spatial variation of three assumed plane wave fields (incident (I), reflected (R), transmitted) (T) can all be written in the form $\left(\mathbf{E},\,\mathbf{H}\right) = \left(\mathbf{E}_0,\,\mathbf{H}_0\right)\,\exp\left(i\,\left<\mathbf{k},\,\mathbf{r} \right>\right)$. That is, the spatial variation of each plane is encoded by the wavevector $\mathbf{k}$. The components of this wavevector can naturally be complex to account for absorptive materials. So, so far, nothing in Born and Wolf derivation changes. We can either assume arbitrary directions for these three (I, R, T)  fields and find that their wavevectors are forced to be coplanar in step 3 below, or we can make this assumption at the outset and find that Maxwell’s equations can be successfully solved with the further assumption in place;

*Given this assumptions, Maxwell’s Equations reduce to:
$$\begin{array}{lcl}
\mathbf{k} \times \mathbf{E}_0 &=& \omega\,\mu\,\mathbf{H}_0\\
\mathbf{k} \times \mathbf{H}_0 &=& -\left(\omega\,\epsilon+i\,\sigma\right)\,\mathbf{E}_0\\
\left<\mathbf{k},\,\mathbf{E}_0\right>&=& \left<\mathbf{k},\,\mathbf{H}_0\right>=0
\end{array}\tag{1}$$
and this is not changed by the components of $\mathbf{k}$’s being complex rather than real;


*To complete the derivation, one writes down the continuity conditions for the tangential electromagnetic fields and uses Gaussian elimination to find the required amplitude ratios for the incident, reflected and transmitted fields. Both $\mathbb{R}$ and $\mathbb{C}$ are commutative fields, thus both equally allow Gaussian elimination and so nothing about this step in the derivation is changed by a complex component $\mathbf{k}$. With electromagnetic waves, one must work through the derivation twice, once for each of a basis pair of polarization states. The wonted and easiest choice for the latter are linear polarization states, firstly (a) the Transverse Magnetic (TM) or Senkrecht state (s) where the magnetic field lies everywhere parallel to the interface and normal to the plane of incidence (the plane spanned by the incident wavevector and the unit normal to the interface) and (b) the Transverse Electric, or Parallel state (p), where the electric field is everywhere parallel to the interface plane and lies normal to the plane of incidence.


There are a few points that need special care:
Magnetic Constants
Most derivations of the Fresnel formulas assume that the magnetic constant for both mediums is the same. This doesn’t have to be so for the derivation to hold, but the final equations are subtly different if there is a magnetic constant difference between the two media and indeed i can’t actually find a convenient reference for them right now. So you may need to be careful of some exotic materials and include the magnetic constant if need be;
Unconjugated Dispersion Condition and Complex Wavenumber
The wavevector magnitude condition is no longer a condition on the magnitude but a condition on the unconjugated inner product of $\mathbf{k}$ with itself:
$$\left<\mathbf{k},\,\mathbf{k}\right> = \omega\,\mu\,\left(\omega\,\epsilon+i\,\sigma\right)\tag{2}$$
a condition which follows simply from the vector wave equation that results from substituting one of the curl equations in (1) into the other. I have in the past made some considerably strife for myself in making this mistake.
One then defines the wavenumber of the medium in question as the complex square root 
$$k = \stackrel{def}{=}\sqrt{\left<\mathbf{k},\,\mathbf{k}\right>}\tag{3}$$ 
of the medium in question. The branch of the square root is such that the imaginary part is of  consistent sign. In the convention  (as used in (1)) that takes $\exp(-i\,\omega\,t)$ to be the positive frequency variation, the imaginary part of k is chosen to be positive.
The index of the medium is then defined as:
$$n \stackrel{def}{=} \frac{\lambda\,k}{2\,\pi} = \frac{\lambda\,\sqrt{\left<\mathbf{k},\,\mathbf{k}\right>} }{2\,\pi}\tag{4}$$
Snell’s Law
Lastly, Snell’s law needs some care to interpret, for it implies some quite nonintuitive things.  As i explain in my answer here, all three of:


*

*Snell’s law;

*The law of reflexion and;

*The principle that the wavevectors of all three plane waves must be coplanar are the direct result of equating the components of all wave vectors when the wavevectors are projected into the interface plane. 


The only way that the components of $\mathbf{E},\, \mathbf{H}$ parallel to the interface and of $\mathbf{D},\,\mathbf{B}$ normal to the plane can be continuous is if the components of the wavevector $\mathbf{k}$ that defines the variation in the spatial variation $\exp\left(i\,\left<\mathbf{k},\,\mathbf{r}\right>\right)$ are the same for all three incident, transmitted and reflected fields. This assertion immediately constrains all three wavevectors to be coplanar and contained in the plane spanned by the incident wavevector and the interface normal, i.e. the plane normal to $\mathbf{k}_i\times \hat{\mathbf{n}}$, the so called plane of incidence and it constrains components of all three vectors along the interface in this plane of incidence to be equal. Thus the analysis is always simplified by aligning our co-ordinates with the the plane of incidence, and so wavevectors all have only two nonzero Cartesian components $k_x,\,k_y$.
The above constraint on exponential variation in the direction of the interface in the plane of incidence also means that the the angle of incidence must always equal the angle of reflexion and, across the interface it also gives us:
$$n_i\, \sin\theta_i = n_r\,\sin\theta_r\tag{5}$$
where $n_i$ and $n_r$ are the indices defined by (4) for the two mediums on either side of the interface the brach convention mentioned after (3). When the values of $n_i$ and $n_r$ are complex, to make (5) hold we now in general require a complex refraction angle $\theta_r$. In multilayer problems the incidence angle can also be complex. This means that the Cartesian vector components of $\mathbf{k}$ are out of phase. It betokens evanescence. Clearly in general general a complex angle is needed to match the variation for different values of $\arg(n_i)$ and $\arg(n_r)$ in (5). This phenomenon is the presence of oscillating stores of energy at the interface that arise to match the EM field variations across the interface. This is a very like phenomenon to the oscillator reactive power in inductive / capacitive lumped circuit elements. The purely imaginary angle, for example, arising when $k_i, k_r$ and $\theta_i$ are all real but leave $\sin\theta_r > 1$, can corresponds to the purely evanescent situation that arises in total internal reflexion as discussed in my answer here.
Complex refraction angles are extremely hard if not impossible to visualize, but it is evident that with such angles (5) can be fulfilled and then the normal derivation of the Fresnel equations runs to completion.
One last word of warning. Snell’s law is not always as powerful and useful as it seems. Strictly speaking, it always holds when it refers to wavevector components as in (5), but it may not always correspond to the correct relationship between directions of propagation of the fields. The wavevector concept defines phase fronts of the wave fields, and in evanescent and anisotropic cases this can be radically different from the direction of power flow. So phase front stops being a useful notion in defining the direction of propagation of the electromagnetic field: if we trace the path of the field with our power meters and detectors, we can in general end up following directions that are quite different from those defined by the wavevector.
