We re-elaborate on the basic properties of lossless multilayers by using bilinear transformations. We study some interesting properties of the multilayer transfer function in the unit disk, showing that hyperbolic geometry turns out to be an essential tool for understanding multilayer action. We use a simple trace criterion to separate multilayers into three classes that represent rotations, translations, or parallel displacements. Moreover, we show that these three actions can be decomposed as a product of two reflections in hyperbolic lines. Therefore, we conclude that hyperbolic reflections can be considered as the basic pieces for a deeper understanding of multilayer optics.
© 2003 Optical Society of America
(000.3860) General : Mathematical methods in physics
(120.5700) Instrumentation, measurement, and metrology : Reflection
(120.7000) Instrumentation, measurement, and metrology : Transmission
(230.4170) Optical devices : Multilayers
Alberto G. Barriuso, Juan J. Monzón, Luis L. Sánchez-Soto, and José F. Cariñena, "Hyperbolic reflections as fundamental building blocks for multilayer optics," J. Opt. Soc. Am. A 20, 1812-1817 (2003)