Cone Angle
When a sample is inserted at the focus of a convergent beam (at the focal plane), then biconical transmittance should be calculated: \[T_b=\frac{total \; flux\; in \;output \;beam}{total\; flux\; in\; input \;beam}\]. The biconical transmittance \(T_b\) can be calculated as follows: \[ T_b=\frac{\int_{\varphi=0}^{2\pi}\int_{\xi=0}^{\xi_max} I(\xi,\varphi)T(\xi,\varphi)\sin\xi d\xi d\varphi}{\int_{\varphi=0}^{2\pi}\int_{\xi=0}^{\xi_max} I(\xi,\varphi)\sin\xi d\xi d\varphi}, \] where the angles \(\xi\) and \(\varphi\) are polar and azimuthal angles defined in the spherical coordinate system. The \(\xi_{\max}\) is the maximum semiangle of the convergent cone. 

In the case of Lambertian source \(I(\xi,\varphi)=I_0\cos\xi\) and equations can be simplified:
\[ T_b=(\pi \sin^2\xi_{\max})_{\varphi=0}^{1}\int_{\varphi=0}^{2\pi}\int_{\xi=0}^{\xi_{\max}}T(\xi,\varphi)\sin\xi\cos\xi d\xi d\varphi.\] In the case when the principal ray of the cone is perpendicular to the multilayer sample the formula can be further simplified: \[T_b=2(\sin\xi_{\max})^{2}\int_{\xi=0}^{\xi_{\max}} T(\xi)\sin\xi\cos\xi d\xi\] 
Cone Angle database of OptiLayer allows you to create cone angles averaging specifications. When Cone Angle is loaded to the memory, all computations of Transmittance, Reflectance, Back Reflectance, and Absorptance take into account cone angle averaging. All synthesis procedures also will take into account cone averaging. The only exclusion from this rule is a target with UDT characteristics.

Cone can be specified with Halfangle (in degrees), f/number or Numerical aperture. Computations are performed on angular grid, with growing number of points the accuracy is improving, but computational time is growing proportionally. OptiLayer uses highprecision sophisticated integration procedures, so for most cases 1020 points are quite sufficient.  Type of distribution can be Uniform Intensity, Lambertian, or UserDefined. In the last case Cone Angle Editor allows to specify the number of Angular points for Cone Angle Intensity distribution and the spreadsheet itself. The number of angular points should not coincide with Cone averaging grid parameter. OptiLayer will perform all necessary interpolation procedures automatically. 
Example: Comparison of a narrow bandpass filter without taking into account cone angle (solid curve) and with taking into account cone angle (dashed curve)

Example: Comparison of a twoline filter without taking into account cone angle (solid curve) and with taking into account cone angle (dashed curve) 