An exact theory for two coupled modes traveling in the same direction with white random propagation constants is developed. Applications include tolerance studies for integrated optical directional couplers. The expected fields and the expected powers of the two modes are determined versus distance along the coupler, with the ratio of propagation-constant spectral density B0 to coupling C as a parameter. For small values of B0/C the powers in the two modes behave in a damped oscillatory manner, with distance along the coupler. For large values of B0/C the behavior is monotonic. In all cases, equal power division occurs for sufficiently large distance along the coupler when heat loss is absent. For an optical switch the maximum B0/C is given as a function of the required switch performance by the present results. The dividing line between oscillatory and monotonic behavior of the powers in the two modes versus distance occurs for B0/C = 4. This choice of parameters yields a power divider of minimum length whose performance is insensitive to the dimensions.
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