Abstract

All gradient-index optical filter calculations may be made using the characteristic matrix approach by dividing the index profile into many small homogeneous index layers. This attests to the versatility of the characteristic matrix technique. However, a disadvantage is the computation time required for thick filters and possibly the lack of insight for certain types of gradient-index structure. The coupled-wave theory (variation of constants) has been used to derive a closed-form expression for the reflection of filters having periodic index variation. Boundary conditions are used to account for arbitrary substrate and incident media. The total complex electric field is resolved into left- and right-traveling components. Such an approach makes it convenient to derive the amplitude reflection coefficient as well as the electric field standing waves inside the structure. The results are also used to predict an optical monitor response during deposition of such gradient-index structures. The computational time for the spectral response is independent of the number of periods or filter thickness. Consequently, for thick filters the computer computation time may be orders of magnitude faster than the characteristic matrix approach.

© 1987 Optical Society of America