Here, the math involved with the calculations of the wave amplitudes can easily become complicated, but there are several ways for avoiding excessive algebra. One (possibly the best) way is to parametrize the mode conversion rate through the fraction of energy lost by the main wave while travelling over one semiarc inside the circular fibre; let us denote this fraction, for instance, by \kappa. Then you can start with expressing the amplitude of the main wave inside the circular fibre in terms of \kappa and \alpha. This expression can be used to find the optimal value of \kappa. After that, it is already very easy to express the amplitudes of the outgoing second-harmonic-waves in terms of the amplitude of the main wave inside the fibre, and the value of \kappa. With all these results, finding which value of \alpha yields the desired value of \kappa isn’t too difficult, either.
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