# Physics Cup – TalTech 2021 – Problem 5

by Jaan Kalda (TalTech).

A circular resonator in fibre optics is made from one circular single-mode fibre coupled to two straight single-mode fibres as shown in the figure. In single-mode fibres, light propagation is essentially one-dimensional: the phase of the electric and magnetic fields in each cross-section of the fibre is constant. The couplings $A$ and $C$ between the circular fibre and the straight fibres are such that a very small fraction $\alpha\ll 1$ of light energy propagating in one of the fibres is transmitted through the coupling into the other fibre. Assume that the both couplings are identical. As a wave of amplitude $E$ propagates in the circular fibre, a double-frequency light wave of amplitude $\mathcal E$ is generated due to nonlinearity of the fibre (however, the double-frequency wave propagates linearly without generating a quadruple-frequency wave). We assume that the dispersion is negligible, i.e. the coefficient of refraction is the same for both the main wave, and for the double-frequency wave so that the increase $\Delta \mathcal E$ in the amplitude of the double-frequency wave is proportional to the travel distance. Here we quantify the nonlinearity of the circular fibre with the constant $\delta$, such that if a wave of amplitude $E$ travels through a semicircle $ABC$, the amplitude $\mathcal E$ of the co-propagating double-frequency wave is increased by $\Delta \mathcal E=\delta E^2$; this expression is valid assuming that $\delta \mathcal E\ll 1$ (you can use this assumption). The coupling constant $\alpha$ is a function of the frequency of the wave; here we denote the coupling constant for the main wave by $\alpha$, and assume that the coupling constant for the double-frequency wave equals to $\alpha^2$. The length of the circular fibre equals to an integer multiple of the wavelength of the main-frequency wave.

Laser light of intensity $I_0=E_0^2$ (with $E_0$ denoting the amplitude of the wave) and of a very long coherence length is directed into the bottom fibre at the input $O$; at the output $D$, both the main-frequency and double-frequency light will appear. For fixed values of $I_0$ and $\delta$, the intensity of the double-frequency light $\mathcal I$ at the output depends on the coupling strength $\alpha$ which can be easily controlled through the manufacturing process. Find the maximal value $\mathcal I_m$ of $\mathcal I$, and the corresponding coupling strength $\alpha_m$ (express in terms of $I_0$ and $\delta$; only closed-form expressions are accepted).

Please submit the solution of this problem via e-mail to physcs.cup@gmail.com.

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