Second-harmonic generation (SHG)

In a material without inversion symmetry the polarization responds to the optical field with a quadratic term, $P^{(2)} = \varepsilon_0 \chi^{(2)} E^2$. A field at frequency $\omega$ therefore drives a polarization at $2\omega$, which radiates a second harmonic: 1550 nm light produces 775 nm, and 1064 nm produces the green 532 nm of doubled Nd:YAG lasers. In photon terms two photons of energy $\hbar\omega$ are annihilated and one of energy $2\hbar\omega$ is created. The same $\chi^{(2)}$ is responsible for the Pockels effect, which is why the materials that make good modulators, lithium niobate above all, also make good frequency doublers.

Efficiency is governed by phase. The driving polarization travels at the phase velocity of the fundamental, the harmonic it radiates at its own, and because of dispersion the two differ. After a coherence length

with $\lambda$ the fundamental wavelength, the newly generated harmonic arrives out of phase with what was generated earlier, and conversion reverses. In most crystals $L_c$ is only micrometers to tens of micrometers, so without correction the harmonic oscillates at a trivial level. Phase matching removes the mismatch. Birefringent phase matching uses a polarization and propagation direction for which the two indices are equal, at the cost of beam walk-off and a restricted choice of crystal cut. Quasi-phase matching reverses the sign of $\chi^{(2)}$ every coherence length, by periodic poling of a ferroelectric crystal with period $\Lambda = 2L_c$, so the conversion is reset before it can reverse; it allows the largest nonlinear coefficient to be used and any wavelength to be designed for by choosing the period.

In the low-conversion limit, harmonic power grows as the square of fundamental power and, for a phase-matched waveguide, as the square of length, which is why efficiencies are quoted normalized as %/W/cm². Confining both waves in a small waveguide raises the intensity and therefore the efficiency: periodically poled waveguides in thin-film lithium niobate, with a poling period of about 4 µm for 1.5 µm light, reached 2600 %/W/cm², more than twenty times conventional diffused waveguides (Wang et al. 2018). The price of a long, efficient device is a narrow acceptance bandwidth in wavelength and temperature.

SHG is used to reach wavelengths lasers do not emit well, in f-2f interferometers that lock the carrier-envelope offset of a frequency comb, and as a measurement: an SHG autocorrelator measures femtosecond pulse widths, and the SHG signal from a surface or interface reveals broken symmetry in materials characterization.

References: R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020); C. Wang et al., Optica 5, 1438 (2018).