Sum-frequency generation (SFG)

Sum-frequency generation combines two light beams at frequencies $\omega_1$ and $\omega_2$ in a crystal with a nonzero second-order nonlinear susceptibility, producing a third beam at $\omega_3 = \omega_1 + \omega_2$. In wavelength terms,

$$\frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2}.$$

Mixing 1064 nm from an Nd:YAG laser with 1550 nm gives 630.9 nm, red light that neither source emits. Second-harmonic generation is the degenerate case with $\lambda_1 = \lambda_2$, and 1064 nm doubles to 532 nm. SFG is used to reach visible and ultraviolet wavelengths, to convert infrared signals to wavelengths where silicon detectors work, and to measure ultrashort pulses.

Energy and momentum conservation

In photon terms, one photon from each input beam is annihilated and one photon at the sum frequency is created. Energy conservation is the frequency relation above: for the example, $1.165\ \mathrm{eV} + 0.800\ \mathrm{eV} = 1.965$ eV. The process is efficient only when momentum is also conserved, which requires the wavevectors to satisfy

$$\Delta k = k_3 - k_1 - k_2 = 0,$$

with $k = 2\pi n/\lambda$ for each wave. Normal dispersion makes $n_3$ larger than $n_1$ and $n_2$, so $\Delta k$ is positive in an isotropic medium and the generated field drifts out of phase with its driving polarization after a coherence length $\pi/\Delta k$, typically micrometres. Phase matching cancels the mismatch, either with birefringence, by choosing polarizations and a propagation angle, or by quasi-phase matching with a poling period $\Lambda = 2\pi/\Delta k$.

Because each output photon consumes one photon from each input, the photon flux converted from beam 1 equals the photon flux generated at $\omega_3$ (the Manley–Rowe relations). The converted power therefore grows by the frequency ratio: each watt of 1550 nm light fully converted to 630.9 nm would emerge as 2.46 W, with the additional energy supplied by the 1064 nm beam.

Efficiency and practical conditions

In the low-conversion limit, the generated power is proportional to the product of the input powers, to $d_{\mathrm{eff}}^2$ (the effective nonlinear coefficient), and, for a phase-matched waveguide, to the square of the length. Efficiencies are therefore quoted, as for SHG, in %/W/cm². With a strong pump and a weak signal, the signal can in principle be converted almost completely, after which further propagation converts it back; the optimum length depends on pump power.

Both beams must overlap in space and, for pulses, in time. Group-velocity mismatch between the inputs separates short pulses over the crystal length, and the finite acceptance bandwidth in wavelength, angle and temperature narrows as the crystal gets longer. Each input also generates its own second harmonic if it is phase matched nearby, which appears as a parasitic output.

Applications

SFG is a standard route to short wavelengths. Mixing 1064 nm with its own second harmonic at 532 nm gives the 354.7 nm third harmonic of Nd:YAG lasers, the usual cascaded form of third-harmonic generation; sodium-guide-star lasers at 589 nm have been made by summing 1064 nm and 1319 nm Nd:YAG lines.

Upconversion detection shifts infrared light into the range of silicon detectors, which have lower noise and higher efficiency than InGaAs infrared detectors. A 1550 nm single-photon signal mixed with a strong 1950 nm pump in periodically poled lithium niobate produces 863.6 nm, where silicon single-photon avalanche diodes work well; choosing a pump at longer wavelength than the signal keeps pump-induced Raman and parametric noise out of the signal band. The same principle, with preserved quantum states, is used for quantum frequency conversion between memories and telecom fiber.

In ultrafast optics, SFG between a pulse and a delayed replica or reference gives cross-correlations, and the SFG and SHG versions of frequency-resolved optical gating recover pulse shape and phase. Vibrational SFG spectroscopy mixes a visible and a tunable infrared beam at a surface; because bulk centrosymmetric media give no signal, it isolates the molecular layer at an interface.

Common questions

How is the SFG wavelength calculated?

Add the inverse wavelengths: $\lambda_3 = 1/(1/\lambda_1 + 1/\lambda_2)$. For 1064 nm and 1550 nm this is 630.9 nm; for 1550 nm and 1950 nm it is 863.6 nm. Frequencies, or photon energies, simply add.

What is the difference between SFG and SHG?

SHG takes both photons from the same beam; SFG takes them from two beams at different frequencies, which gives more freedom in choosing the output wavelength but requires spatial and temporal overlap of the two inputs. The phase-matching condition generalizes accordingly.

How is SFG related to difference-frequency generation?

They are the same $\chi^{(2)}$ interaction run in opposite directions. Difference-frequency generation subtracts frequencies, $\omega_3 - \omega_1 = \omega_2$, and amplifies the lower-frequency wave; which process dominates depends on the relative phases and the fields present at the input.

References: R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).