Naturally Phase-Matched Second-Harmonic Generation in a Whispering-Gallery-Mode Resonator

We demonstrate for the first time natural phase matching for optical frequency doubling in a high-$Q$ whispering-gallery-mode resonator made of lithium niobate. A conversion efficiency of 9% is achieved at $30\mu \mathrm{W}$ in-coupled continuous wave pump power. The observed saturation pump power of 3.2 mW is almost 2 orders of magnitude lower than the state-of-the-art value. This suggests an application of our frequency doubler as a source of nonclassical light requiring only a low-power pump, which easily can be quantum noise limited. Our theoretical analysis of the three-wave mixing in a whispering-gallery-mode resonator provides the relative conversion efficiencies for frequency doubling in various modes.

Figure 1

Normalized Clebsch-Gordan coefficients $⟨m+{\nu }_p,m+{\nu }_p;m,m|2m+{\nu }_s,2m⟩/⟨m,m;m,m|2m,2m⟩$.

Figure 2

Normalized absolute-value coupling coefficients for frequency doubling $\{{\omega }_p,q_p\}\to \{2{\omega }_p,q_s\}$.

Figure 3

Experimental setup.

Figure 4

Transmitted pump power and the SH signal. The coupling is optimized for high SH output, which corresponds to overcoupled fundamental modes. On the inset: a typical WGM spectrum of a critically coupled resonator.

Figure 5

SH conversion efficiency as a function of the in-coupled power: experimental data and theoretical fit [10] corresponding to the saturation pump power $W_0=3.2\mathrm{mW}$. On the inset: transmitted pump power and the second harmonic signal above the “self-limiting” threshold which arises at a much lower power than $W_0$.