### Integral Targets

 OptiLayer allows you to specify integral targets. An integral characteristic $$F$$ has the form: $F=\frac{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda) S(\lambda)d\lambda}{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda) d\lambda},$ where $$\lambda_d$$ and $$\lambda_u$$ are boundaries of the wavelength interval of interest, $$W(\lambda)$$ is a given weight function, $$S(\lambda)$$ is a spectral characteristic of a coating. For example, $$S(\lambda)$$ can be reflectance for s-polarization, or absorptance for p-polarization, and so on. In the example presented on the right panel transmittance (TA, no polarization) is integrated with three commonly used spectral weight functions. It is required that the integral values are to be more than 60%. In OptiLayer integral targets are represented as finite sums and can be considered as approximations of an integral expression with the help of rectangle rule formula. It is necessary to specify the spectral weight function $$W(\lambda)$$  at some wavelength grid. Up to 128 different spectral weight functions can be defined and used in OptiLayer. Some commonly used spectral weight functions are available in the Catalog and can be used almost instantly. You can specify your own spectral weight functions as well. Specified integral functions can be used in "Integral column" (see Figure above). Final expression for the corresponding term in the merit function looks as follows: $MF_{int}^2=\frac 1M\sum\limits_{i=1}M\left[\frac{F^{(i)}-\hat{F}^{(i)}}{\Delta F^{(i)}}\right]^2,$ where $$M$$ is the number of different integrals in the resulting target, $$F^{(i)}$$ are target values,  $$\Delta F^{(i)}$$ are corresponding tolerances. As in conventional targets it is possible to specify Angle of incidence (Angle); Integral type (Integral); Spectral characteristic (Char); Value of the integral expression (Target); Tolerances for target values (Tol); Qualifiers for the target values in the case of range targets  (Q). OptiLayer allows you to monitor integral and averaged characteristics: Analysis --> Integral and averaged characteristics These characteristics is not necessary the target integral characteristics. You can monitor:  Merit Function value and its components, Integral values for spectral distributions defined in the current Integral Target or predefined spectral distributions, Various Color characteristics of the current coating, Averaged values over a range of wavelengths, Averaged values over a range of angles of incidence, Averaged values over a 2D region of [wavelengths x angles of incidence].

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