1. What are Anti Reflective Coatings

Anti-reflective coatings are layers deposited on a substrate that interact with light in certain wavelength ranges to lead to negligible reflectance. This happens due to the destructive interference between various fresnel reflections from all the interfaces of the (multi)layer stack that makes the coating.

2. How do they work?

All the reflected waves from the multilayer coating stack undergo destructie interference. In order for anti-reflective coatings to truly work ie., for there to be a true destructive interference, two things need to happen - amplitude of the two waves about to destructively interfere has to exactly match and they have to be out of phase i.e., the phase difference have to be odd multiples of pi.

2. How to model Anti-reflective coating

Anti-reflective coating is a multilayer stack on a substrate. Therefore they are modelled using Transfer Matrix Method. To revise TMM, refer this article in optical methods.

Using the transfer matrix method, the net reflectance of any multilayer stack is:

\[ R = |r|^2 = \left|\frac{M_{21}}{M_{11}}\right|^2 \]

Single layer anti reflective coating is theoretically possible. For a single layer coating to be anti-reflective - we need two things:

1. The phase difference has to be odd multiples of pi.

2. R has to be zero

Lets see if we can achieve that. Quarter Wave optical thickness (d = lamda/4) leads to the extra optical phase acquired by the second reflected wave in comparison to the first reflected wave of pi. Thus if n_layer > n_air and n_substrate > n_layer then, the two reflected waves will be out of phase, and hence destructively interfere.

Using the QW optical thickness, and using R = 0 in the above equation, we would get:

\[ n_2 = \sqrt{n_1n_3} \]

Thus if we have a layer that has QW optical thickness and has a very specific refractive index, we can get zero reflectance. But this will happen only at a very narrow wavelength range (design wavelength at which QW optical thickness was designed).

An example could be a coating sandwhiched between air and glass (n=1.5). Thus,

\[ n_{\text{coating}} = \sqrt{1 \times 1.5} \approx 1.225 \].

A real close example would be MgF2 with a refractive index of 1.38. The following figure shows the R vs wavelength for a single layer AR coating of MgF2 (QW at deisgn wavelengthof 550 nm), a fictional material with the theoretically correct refractive index at QW thickness and at 20% off thickness.

The figure above shows that MgF2 can give low reflectance but not completely zero. This result also shows that change in thickness can also lead to change in intended performance.

3. Multilayer Coatings

In most cases, to have anti-reflective performance across bigger spectral ranges and angular field of view, we need multilayer stacks of alternating low and high refractive index materials.

One such example is SiO2/TiO2 alternating layers. To design such multilayer stacks, optimization over layer thickness needs to be done.