I'm studying the effects of absorption in a 1D photonic structure with repeating alternate layers (a Distributed Bragg Reflector). To include absoprtion effects, the refractive index is defined as $bar{n} = n +ik$ where $n$ is the real refractive index and $k$ is the extinction coefficient describing the absorption due to free charge carriers.

My question is whether there is an analytical equation for the reflectivity of such a system? For a system without absorption, the reflectivity is given by:

$$R = \Bigg(\frac{\big(\frac{n_b}{n_a}\big)^{2m} - 1}{\big(\frac{n_b}{n_a}\big)^{2m} + 1}\Bigg)^2$$ where $m$ is the number of periods (repeated pairs of layers) and $n_a$ and $n_b$ are the real refractive indices of the two materials.

To develop a similar equation which accounts for absorption, would it be just a simple substitution of e.g. $n_a$ for $n_a + ik$ (assuming $k$ is the same for both $a$ and $b$)?

This would give: $$R = \Bigg(\frac{\big(\frac{n_b+ik}{n_a+ik}\big)^{2m} - 1}{\big(\frac{n_b+ik}{n_a+ik}\big)^{2m} + 1}\Bigg)^2$$

Is this correct? I can't seem to find any resources describing this.


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