Third-harmonic generation (THG)

Third-harmonic generation (THG) converts light at frequency $\omega$ into light at $3\omega$. An Nd:YAG laser at 1064 nm is tripled to 354.7 nm, the familiar "355 nm" of solid-state ultraviolet lasers, and a Yb-doped laser at 1030 nm is tripled to 343.3 nm. In photon terms, the energy of three 1.165 eV photons at 1064 nm is carried away by one 3.496 eV photon. Two routes lead there: a direct third-order process through the nonlinear susceptibility $\chi^{(3)}$, which occurs in every material including glasses and gases, and a cascade of two second-order steps in crystals, which is how practically all commercial solid-state 355 nm and 343 nm lasers work.

Direct THG through χ(3)

A single field $E\cos\omega t$ drives a cubic polarization with a component at $3\omega$,

$$P^{(3)}(3\omega) = \tfrac{1}{4}\,\varepsilon_0\,\chi^{(3)} E^3,$$

so the third-harmonic intensity scales as the cube of the input intensity: doubling the input intensity raises the harmonic eightfold. Because $\chi^{(3)}$ is small, direct THG is weak and becomes practical mainly at the high peak powers of pulsed lasers, especially femtosecond pulses. Dispersion limits it further. In fused silica the indices at 1064 nm and 354.7 nm are 1.4496 and 1.4761 (Malitson Sellmeier fit), and the coherence length

$$L_c = \frac{\lambda}{6\,(n_{3\omega} - n_\omega)}$$

is only 6.7 µm. Direct THG is therefore used where crystals cannot be used or efficiency is secondary: in gas cells and hollow fibers for vacuum-ultraviolet generation, as a diagnostic of pulses (third-harmonic FROG), and in microscopy.

Cascaded THG: SHG followed by SFG

The efficient route uses two $\chi^{(2)}$ crystals, commonly LBO. The first performs second-harmonic generation, 1064 nm to 532 nm; the second performs sum-frequency generation between the leftover fundamental and the harmonic,

$$\frac{1}{1064\ \text{nm}} + \frac{1}{532\ \text{nm}} = \frac{1}{354.7\ \text{nm}}.$$

Each stage is separately phase-matched, with a waveplate or crystal orientation choice to set the polarizations. The SFG step consumes one fundamental photon and one second-harmonic photon, and making one second-harmonic photon consumes two fundamental photons. Equal photon numbers entering the second crystal therefore require converting two thirds of the fundamental power in the first crystal, which is the usual design target; doubling harder leaves too few fundamental photons for the mixer. Q-switched and mode-locked systems reach overall 1064 nm to 355 nm efficiencies of tens of percent this way, far above what direct THG reaches in bulk media. The same cascade can be built inside one periodically poled crystal with two poling sections.

THG microscopy

A tightly focused Gaussian beam passes through a Gouy phase shift of π at the focus. In a homogeneous, normally dispersive medium the third harmonic generated before and after the focus then cancels, and almost no signal leaves the sample. When the focus straddles an interface or a small inhomogeneity, such as a lipid droplet, a membrane or a cell boundary, the cancellation is broken and a harmonic appears. This makes THG a label-free contrast mechanism for structure, demonstrated by Barad et al. in 1997. Excitation near 1.2–1.3 µm is common because tissue scatters less there and the harmonic, 410 nm for 1230 nm excitation, stays in the visible where detectors and objectives work well. It is usually combined with SHG and two-photon fluorescence on the same ultrafast laser scanning microscope.

Pitfalls

UV output degrades optics: tripling crystals and output windows suffer from color-center formation and surface contamination under 355 nm exposure, and crystal spot-shifting is a routine maintenance step in industrial lasers. The walk-off and narrow angular acceptance of the SFG stage make it sensitive to pointing and temperature. In THG microscopy, a signal from a uniform sample usually indicates a surface or a coverslip interface near the focus.

Common questions

Why do 355 nm lasers use two crystals instead of one?

In a crystal without inversion symmetry the second-order response is far stronger than the third-order one at practical intensities. Splitting the process into SHG and SFG uses that stronger response twice, and each step can be phase-matched independently.

Is third-harmonic generation the same as the Kerr effect?

Both come from $\chi^{(3)}$. The Kerr effect is the part of the response at the original frequency, which changes the refractive index; THG is the part at $3\omega$, which radiates new light. They share the same tensor but different frequency arguments, so their magnitudes are related but generally differ.

Why does glass give only odd harmonics?

In an isotropic medium the even-order susceptibilities vanish by symmetry, so the lowest harmonic available is the third. Surfaces break the symmetry locally and can give a weak second harmonic.

References: R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 22; I. H. Malitson, J. Opt. Soc. Am. 55, 1205 (1965); Y. Barad, H. Eisenberg, M. Horowitz, Y. Silberberg, Appl. Phys. Lett. 70, 922 (1997).