1. What is dichroic mirror

A mirror reflects most of the light that incidents on it. This is the optical – electronic response of the atoms and bonds that make the mirror. A Dichroic Mirror on the other hand has a wavelength dependent (and angle and polarization) reflection response. This spectral selectivity is a resultant of constructive/destructive interference of the reflected light from the multiple interfaces of a multilayer structure. optical-electronic-phononic response of the material. Thus, while the former is just what it is by nature, the latter is engineered.

2. Wavelength selectivity

The reflection (R and T) at each interface is decided by the Fresnel reflection coefficients, which is a function of the layer’s refractive index. These reflected or transmitted waves have to propagate through the surrounding films – the optical phase they pick there is also function of the refractive index. However, after reflections and transmissions through multiple layers, the interference between these waves is engineered via the material and geometric properties of this multilayer structure (e.g. thickness, material of layers, number of layers), and this leads to wavelength selectivity of the net reflection and transmission spectra. Depending on whether the transmission zone is at higher wavelengths in comparison to reflection dominant wavelength range, the dichroic mirrors are called long pass and short pass dichroic mirrors.

3. How to model dichroic mirrors

Since dichroic mirrors are uniform dielectric multilayer coatings in most cases, they are modelled using Transfer Matrix Method. You can refer to the article linked for a proper derivation of this method. In this method, each interface contributes an interface matrix, and each layer contributes a propagation matrix. All these matrices together lead to the net Transfer matrix which can then be used to calculate the net reflected and transmitted fields and thus, the net R and T of the entire multilayer structure.

4. Design Challenge: Design a long pass dichroic

Assume a design challenge, design a visible-spectrum long-pass dichroic mirror that has Reflection 100% in the wavelength range 400 -500 nm and 100% transmission in the wavelength range 600-750 nm. 

Our design is a multilayer structure of alternating high refractive index layer (TiO2) and low refractive index layer (SiO2) sandwiched between air and glass substrate. The main design variables are thickness of all the layers. Here we assume that the number of layers is fixed and the refractive index of the materials are lossless (real refractive index only) and non-dispersive (do not change with wavelengths) within the range of wavelengths being studied. We also assume normal incidence and therefore no polarization (electric field and magnetic field are in the interface plane perpendicular to each other).

TMM runs on many starting configurations using stochastic optimization. The objective is to minimize the sum of two errors:

\[ \mathrm{Error}_R = \frac{1}{N_R}\sum_{\lambda_i \in B_R}\left[R(\lambda_i)-1\right]^2 \]

where

\[ B_R = \{\lambda \mid 0.40\,\mu\mathrm{m} \leq \lambda \leq 0.50\,\mu\mathrm{m}\} \]

\[ \mathrm{Error}_T = \frac{1}{N_T}\sum_{\lambda_i \in B_T}\left[T(\lambda_i)-1\right]^2 \]

where

\[ B_T = \{\lambda \mid 0.60\,\mu\mathrm{m} \leq \lambda \leq 0.75\,\mu\mathrm{m}\} \]

When we run this for a fixed number of layers (total = 20 or 10 alternating stacks), we get an optimized stack. Note that it is optimized for specific conditions (eg, specific reflectivity and transmissivity criteria under normal incidence).

Note that this design was optimized for normal incidence only. Angle of incidence ranges (depending on the FOV requirements) and hence s and p polarization needs to be considered as well. 

The above design of total 20 layers of alternating TiO2/SiO2 fulfills our engineering requirement of long pass dichroic. We could however question our initial choices of materials or number of layers. The following figure shows an optimization study over number of layers. It shows that a total of 12 layers could also have been used. Optical engineering design is a compromise between cost and performance. If by letting go of some performance, costs can be reduced, that should be considered as well

5. Uses of Dichroic Mirrors

Dichroic mirrors are used where spectral ranges need to have different directions. For example, dichroic mirrors can be used in Wavelenght Division Multiplexing (WDM) used in optical networks. Optical networks carry infomration through various wavelenght channels. In order to direct them towards separate sensors to get back the information, the different channels need to directed towards their own sensors. Dichroic mirrors can do that by reflecting a specific spectral channel while tranmitting another.

Similarly dichroic mirrors can be used in fluorescence microscopy where the excitation and emission wavelengths need to be separated so that emission wavelenghts can be studied.

6. Tolerances

In addtion to the design criteria, fabrication tolerances i.e., how much the performance would change as a function of slight errors in layer thicknesses is important. Surface problems like localized scratches and dig can also impact performance and they need to be considered as well. In addition, chemical deposition issues due to deposition techniques need to be kept in mind as well.