<title>Light Propagation In An Inhomogeneous Optical Medium</title>

Abstract

AbstractConsider the light propagation in a weak -inhomogeneous medium ( 1 An << 1 for dis-tances on the order of a, where n is the refractive index of the ediumnand a is thelight wavelength). It is known that for the conditions of this problem, the Maxwell equa-tions for monochromatic electric field component in Fock- Leontovich paraxial (parabolic)approximation can be reduced to the equivalent Schrodinger equation (1 -3). Thus, if Z -axis of a rectangular coordinate system X, Z is chosen in the dire tion of light propagation,then for the reduced field 4(X, Z) = no /2E (X,Z) exp { -i K nodZ}we have i - 1 ( aGA) 1 k ( n2 n2)11) 2K2 aX2 2 0 (1) where E (X, Z) is the monochromatic electric field component, n = n(0,Z), K- 2-ff and ?Zd . Therefore, as was noted by Man'ko (4), such methods °of quantum mechanics as the r.o integrals -of- motion method, the coherent states method and the method of dynamical symmetrygroup may be used to find solutions of equation (1). Such an approach was discussed in de-tail and realized in papers (5 --8). In the present paper, the main results are formulated.It turns out that the modes of the medium correspond to the Fock states of quantum me-chanics and the solution of (1) in the coherent state representation is the generatingfunction for these modes. The coherent states for homogeneous in Z- direction medium withparabolic transverse index distribution minimize uncertainty relation,and average meaningof coordinate operators in the coherent states representation defines the geometrical raytrajectories. The axis ray corresponds to the ground state. The uncertainty relation isnot minimized for inhomogeneous in Z- direction medium with quadratic transverse index dist-ribution (the ray width is changing), the average meaning of coordinate operators defines,however, the ray trajectories as well as in the previous case. This property is not validwhen an arbitrary inhomogeneous medium is discussed and the coherent states may be onlysaid to be eigenstates of the invariant operator whose eigenvalues set an initial point inphase space of geometrical rays. These properties of the coherent states, as well as theircompleteness condition seem to be helpful in calculations and permit the connection betweenthe ray and mode descriptions to be observed in a more clear form.Consider,as an example, the inhomogeneous parabolic -index mediumn2(x,C)