Quantum (not frustrated) theory of the total internal reflection as the source of the Goos-H?nchen shift

摘要

The frustrated total internal reflection theories (FTIR) from previous century are thoroughly recalculated from the, so called, monodromy operator's point of view – a theory lunched by Born and Wolf [Principles of Optics (Pergamon Press, 1975), Chap. 1.6] and Arnold [Geometric Methods in the Theory of Ordinary Differential Equations (Springer, 1987)]. Monodromy is a theory of simultaneous solution (for both reflection and transmission amplitudes) of one dimensional Schr?dinger equation (for the wavefunction and its derivative) and the Maxwell equation (for electric and magnetic fields). Introducing new quantities: the dwell distance and the phase distance, we get general Goos-H?nchen (G-H) shift formula for optical tunneling for three layer system with refraction indexes n0, n1, n2. This formula reduces itself to expressions known from the scientific literature for infinite air gap (infinite width of second layer). Extension to many layers is possible.