If I have a look at the Fresnel equations on Wikipedia, I get the following expressions for reflection and transmission at perpendicular light incidence
$$ R = \left|\frac{\hat n_1 - \hat n_2}{\hat n_1 + \hat n_2}\right|^2 ,\qquad T = \frac {\hat n_2}{\hat n_1}\cdot \left|\frac{2\hat n_1}{\hat n_1 + \hat n_2}\right|^2 , $$ where $\hat n_1 = n_1 + ik_1$ and $\hat n_2 = n_2 + ik_2$ are the complex refractive indices on the left and right side, respectively.

Normally, $R$ and $T$ should sum up to a total of 1, but if I add them up, I get the result:

$$R+T = 1 + \frac{4(n_1k_2 - n_2k_1)}{(n_1+n_2)^2+(k_1+k_2)^2}i$$

which is due to $T$ in fact a complex number. Why is the transmission coefficient $T$ a complex number in the first place, if it only is the ratio of two real powers?

Hence, $R+T$ is only equal to 1, if $k_1$ and $k_2$ are both exactly zero (no absorption/dissipation) or if I only take the real part of $R+T$.

What is the physically correct way of getting $R+T$ to exact 1 for complex refractive indexes?


2 Answers 2


The Given coefficient :

$$R^2=\left|\frac{\hat n_1-\hat n_2}{\hat n_1+\hat n_2}\right|$$ $$T=\frac{n_2\cos\theta_t}{n_1\cos\theta_i}|t|^2$$

Note that in the cases of an interface into an absorbing material (where $n$ is complex) or total internal reflection, the angle of transmission might not evaluate to a real number.

Light propagation in absorbing materials can be described using a complex-valued refractive index. The imaginary part then handles the attenuation, while the real part accounts for refraction. For most materials the refractive index changes with wavelength by several percent across the visible spectrum.

When light passes through a medium, some part of it will always be attenuated. This can be conveniently taken into account by defining a complex refractive index,

$$\hat n= n+i\kappa$$

Here, the real part $n$ is the refractive index and indicates the phase velocity, while the imaginary part $κ$ is called the extinction coefficient — although $κ$ can also refer to the mass attenuation coefficient— and indicates the amount of attenuation when the electromagnetic wave propagates through the material.

Now You asked why we work with complex numbers? The answer is simple. Because It makes things much more simple. Working with exponential is much simple than with trigonometric functions. Just see the derivative of them. More on this here.

  • $\begingroup$ thanks for your answer! For $R$ this concept works out, but in $T$, there are complex refractive indexes without modulus $\endgroup$
    – Pixel_95
    Dec 2, 2020 at 10:06
  • $\begingroup$ Can you point heading or line? I'm unable to find your formula on wikipedia. $\endgroup$ Dec 2, 2020 at 10:20
  • $\begingroup$ R ist directly after [8] and T after [9] $\endgroup$
    – Pixel_95
    Dec 2, 2020 at 12:22
  • $\begingroup$ I found $R=|r|^2$ and $T$ to be what I have written above. There is no issue with these two. $\endgroup$ Dec 2, 2020 at 15:15
  • 1
    $\begingroup$ @Pixel_95 I edited the answer. Please give a look. $\endgroup$ Dec 3, 2020 at 15:33

I've found an interesting paper of Steve Byrnes on arXiv: https://arxiv.org/abs/1603.02720

There are some details about cases, where $R+T \neq 1$. In short: $R+T=1$ does not hold, if the medium of the superstrate of your transfer-matrix-stack is a complex number ($k \neq 0$)


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