1. What is a beam splitter

Beam Splitter is used to split incident beam power into multiple beams in different spatial directions, reflection and transmission in most cases, via multilayer dielectric stack coating in most cases. The splitting is by design, for example 50:50 non-polarizing beam splitter would divide incident beam into 50% reflectance and 50% transmittance, and the multilayer stack coating is designed to achieve that specific ratio under design parameters (spectral and angular). A polarizing beam splitter (PBS) would want all the s-polarized light to reflect while all the p-polarized light to transmit through.

2. How to model a beam splitter

Since here we are talking about splitting via multilayer coating, this can be modelled using Transfer Matrix Method (TMM). Please click on the link to go through the step by step derivation of reflectance and transmittance from a multilayer stack.

The coating that needs to be optimized for beam splitting is mostly made of alternating stacks of high and low refractive index dielectric layers (e.g. high refractive index example being TiO2, Ta2O5 etc while low refractive index example being SiO2 etc).

3. Design Problem: Broadband 50:50 Non-Polarizing Plate Beam Splitter

Design a multilayer dielectric coating for a plate beam splitter that divides incident optical power equally between reflected and transmitted beams over the wavelength range of 450–550 nm, while minimizing polarization dependence.

Assume you can use TiO2, SiO2 as your layers.

What do we have: We have a multilayer stack sandwiched between air and glass (N-BK7). We know the spectral range of deisgn (450-500 nm). We also know the design angle of incidence (AOI) to be 45 degrees - which is normally the case in case of beam splitters. We can optimize the number of layers, but for now we would start with 10 alternating stacks of TiO2 and SiO2.

Design Objective: Our target is to get reflectance and transmittance of 50 % in the design wavelength irrespective of polarization (s or p or unpolarized)

Optimization variables (variable we are optmizing to reach our target) : thickness of each layer.

We will use TMM and optimization methods to minimize the following error function then:

\[ \mathrm{Error}_{type} = \frac{1}{N}\sum_{\lambda_i \in B}\left[type(\lambda_i)-0.5\right]^2 \]

where type is R_s, R_p, T_s, T_p because we want all of them to be as close to 0.5 as possible.

The final error function we would want to minimize is:

\[ \mathrm{Error}_{\mathrm{total}} = \mathrm{Error}_{R_s} + \mathrm{Error}_{T_s} + \mathrm{Error}_{R_p} + \mathrm{Error}_{T_p} \]

The Reflectance of 8 alternately layered TiO2|SiO2 stack is shown below. Within our waveleght of interest, we get closer to 50% reflectance for s and p polarized light. For s polarized light, R over 450,550 nm is 50% ±12.25 % while p polarized light is 50% ±14.24 %. This percentage is calculated using the maximum variation of R from target of 50%.

Please note that till now we have only modelled the dielectric stack and the interface it has with air and glass. We have not yet modelled the rear interface of this spiltter which it shares with air.

We assumed a fixed number layers. So here we run optimization over different number of layers. The following example is only for normal incidence and therefore s and p polarized results would be exactly equal.

This shows that even numver of layers as low as 4 can lead to target reflectance for 0 AOI. 4 here means 4 total layers (i.e., 2 alternating stacks).

4. Plate and Cube beam splitters

If the beam splitter is a plate of glass substrate with a specialized coating, it is called as a plate beam splitter. A plate beam splitter will be modelled as above.

If instead the beam splitteris two right angle prisms cemeted together along their diagonal face, with the coating on the diagonal face - it is called as a cube beam splitter. To model a cube beam splitter, the first medium would also be the glass (instead of air as in case of plate splitter).