# Effect of complex refractive index on reflectivity in a periodic structure?

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.