Derivation of Coupled Power Equations

T h e interaction of the modes of a multimode waveguide can be described by coupled wave equations. 1,2,3 The coupling between the waves is caused by imperfections of the waveguide structure. These imperfections are either deviations of the refractive index from the index distribution of the perfect waveguide or they are departures of the waveguide geometry from its nominal value. Changes of the core diameter of an optical fiber causes coupling between the guided modes and also coupling of the guided modes to the radiation modes. Solutions of the coupled wave equations are hard to obtain for many modes since not only the wave amplitudes but also their relative phases enter into the description. In most problems of practical interest the coupling coefficient is a random function of distance and only the exchange of power between the modes is of interest. A description of this problem in terms of coupled wave equations yields more information (phase information) than is required and consequently is quite complicated. One might expect intuitively that a description in terms of power exchange between the modes should exist. 1 If it were permissible to add power instead of amplitude one would be tempted to write down power rate equations that account for the incremental loss of power of one mode in terms of the power that is transferred per unit length from this mode to all the other guided modes while an increase in power can be expressed 229