For example, a quarter-wave stack with n pairs of layers with another non-quarter wave stack with m pairs and QWOT thicknesses equal to a and b can be represented as

\[ (HL)^n\; (aHbL)^m\]

We can consider some set of possible values for integers n and m, and for every possible combination of n and m OptiLayer will try to find optimal values of continuous parameters a and b, optimizing the design performance with respect to loaded targets.

Another example may be a stack having period with varying optical thicknesses. Ultra-wide range high reflectors, chirp mirrors and other coatings can be designed in this way.

Example. High reflector operating in the spectral range from 400 nm to 900 nm. Refractive indices of layer materials n_H=2.35 and n_L=1.45, glass substrate.

Width of the first high reflection zone of a quarter wave mirror with the central wavelength can be estimated with the help of a known formula:

\[ \Delta= \lambda_u-\lambda_l, \;\; \frac{\lambda_u}{\lambda_l}=\frac{\pi+\arccos(-\xi)}{\pi-\arccos(-\xi)}\]

\[ \xi=\frac{n_H^2+n_L^2-6n_H n_L}{(n_H+n_L)^2} \]

where \(\lambda_u\) and \(\lambda_l\) are upper and lower boundaries of the high reflection zone.

If the central wavelength is 650 nm then the width of the high reflection zone of a quarter wave mirror is about 200 nm. In our design problem we need to cover a spectral range 400-900 nm, i.e. we need a high reflection zone of 500 nm.

It means that we need to combine several quarter wave or near quarter wave stacks.  

Illustration of the design process of a wide band reflector. In order to observe the design process, put the mouse of the picture.

Assuming that \(\lambda_l=400\)nm and using formulas above, it is possible to estimate that three quarter wave stacks are required. The corresponding width of the high reflection zones are:

\[\Delta_1=140\; nm,\;\Delta_2=200\;nm,\;\Delta_3=260\;nm\]   

Then we need three quarter wave or near quarter wave stacks and matching layers.  The design can be therefore represented by a formula:

m₁, m₂, m₃ are integer parameters indicating the number of pairs 

a₁, b₁,... are continuous parameters expressing the fraction of quarter wave optical thickness.

m₁, m₂, m₃ can, for example, take values from 5 to 14.

The parameters a₁, b₂,... can be varied in the segments [0.8;2]

\[c_1 H d_1 L (a_1H b_1 L)^{m_1}\;\; c_2Hd_2L (a_2Hb_2L)^{m_2} c_3Hd_3L (a_3H b_3 L)^{m_3} c_4 H\]

In OptiLayer design formulas and their parameters can be specified in Synthesis --> Formula Constrained Optimization:

OptiLayer will consider all possible combinations of specified integer parameters m₁, m₂, m₃. Each of these parameters can take 10 values: 5, 6, ..., 14. For each m₁, m₂, m₃ combination, OptiLayer optimizes the design with respect to continuous parameters a₁, b₁, c₁,...

All calculated design are stored in the "Design Collection Window". The designs are ranged with respect to the merit function values.

This design example is illustrated in our video example "High Reflector Part II" on YouTube