# Do the Fresnel equations of reflection apply to monochromatic light?

I have a monochromatic light source (wavelength ~ 420 nm), which will be incident on the interface of two different media. Could someone please explain if the Fresnel equations applies with monochromatic light when estimating the reflectance and transmittance?

Fresnel equation:

\begin{align} R_s &= \left| \frac{n_1 \cos(\Theta_\mathrm{in}) - n_2 \cos(\Theta_\mathrm{out})}{n_1 \cos(\Theta_\mathrm{in}) + n_2 \cos(\Theta_\mathrm{out})} \right|^2 = \left| \frac{n_1 \cos(\Theta_\mathrm{in}) - n_2\sqrt{1-\left(\frac{n_1}{n_2}\sin(\Theta_\mathrm{in})\right)}}{n_1 \cos(\Theta_\mathrm{in}) + n_2 \sqrt{1-\left(\frac{n_1}{n_2}\sin(\Theta_\mathrm{in})\right)}} \right|^2 \\ R_p &= \left| \frac{n_1 \cos(\Theta_\mathrm{out}) - n_2 \cos(\Theta_\mathrm{in})}{n_1 \cos(\Theta_\mathrm{out}) + n_2 \cos(\Theta_\mathrm{in})} \right|^2 = \left| \frac{n_1 \sqrt{1-\left(\frac{n_1}{n_2}\sin(\Theta_\mathrm{in})\right)} - n_2 \cos(\Theta_\mathrm{in})}{n_1 \sqrt{1-\left(\frac{n_1}{n_2}\sin(\Theta_\mathrm{in})\right)} + n_2 \cos(\Theta_\mathrm{in})} \right|^2 \end{align}

• Please do not post formulae as screenshots, but use MathJax instead. I have transcribed your images for now (including a mistake in a missing square), but you should use direct MathJax typesetting for future posts. – Emilio Pisanty Sep 3 '19 at 14:46
• I've also removed your final paragraph, as it asks a separate question -- threads here should contain one question per question (or, at most, a tightly-related set of queries). If you're still curious, post a separate thread. – Emilio Pisanty Sep 3 '19 at 14:48

Yes. Fresnel equation as written as valid for some fixed wavelength since the index of refraction $$n$$ is usually wavelength-dependent. If your source is not monochromatic, you need to compute $$R_\lambda$$ for each wavelength (using the appropriate $$n$$ is each case), multiply each $$R_\lambda$$ by its proportion in the incident beam, and sum over all $$\lambda$$ (or integrate if your distribution is continuous).
Usually $$n$$ will vary slowly with $$\lambda$$ most of the time. This results in dispersion, as illustrated (presumably for glass) by this iconic album cover: