Difference-frequency generation (DFG)

Difference-frequency generation (DFG) is the second-order nonlinear process in which a strong beam at frequency $\omega_1$ and a weaker beam at $\omega_2$ drive a polarization at $\omega_3 = \omega_1 - \omega_2$, which radiates a new beam at that difference frequency. It is the standard way to turn two convenient near-infrared lasers into mid-infrared light: mixing 1064 nm from an Nd:YAG laser with 1550 nm from a telecom source produces 3.39 µm, and two lines near 1550 nm separated by 8 nm produce 1 THz. Like sum-frequency generation, it requires a medium without inversion symmetry and a nonzero nonlinear coefficient.

Energy conservation and the wavelength formula

In photon terms, one photon at $\omega_1$ (usually called the pump) is annihilated, and one photon at $\omega_2$ (the signal) and one at $\omega_3$ (the idler) are created. Energy conservation $\hbar\omega_1 = \hbar\omega_2 + \hbar\omega_3$ gives, in wavelengths,

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

For $\lambda_1 = 1064$ nm and $\lambda_2 = 1550$ nm the frequencies are 281.76 THz and 193.41 THz, so the idler sits at 88.35 THz: $\lambda_3 = 3.393$ µm, or 2947 cm⁻¹, within the C–H stretching band used in hydrocarbon sensing. Because every idler photon is accompanied by a new signal photon, DFG also amplifies the signal beam; when that amplification is the goal, the same interaction is called an optical parametric amplifier, and placing it inside a resonator makes an optical parametric oscillator.

The photon picture also sets a ceiling on power conversion. Each pump photon of energy $\hbar\omega_1$ yields at most one idler photon of energy $\hbar\omega_3$, so the largest fraction of pump power that can appear at the idler is $\omega_3/\omega_1 = \lambda_1/\lambda_3$: 31% for the 1064/1550 nm example. This is the Manley–Rowe limit, and real DFG sources operate well below it.

Phase matching and the low-conversion limit

The generated idler adds up coherently along the crystal only if the wave vectors satisfy $\Delta k = k_1 - k_2 - k_3 = 0$, the phase-matching condition. It is met either by birefringence or, more often in modern sources, by quasi-phase matching in periodically poled lithium niobate or KTP, where the poling period is chosen for the desired idler. In the undepleted-pump limit with plane waves, the idler intensity scales as

$$I_3 \propto \frac{\omega_3^2\, d_\text{eff}^2\, I_1 I_2\, L^2}{n_1 n_2 n_3},$$

multiplied by $\mathrm{sinc}^2(\Delta k L/2)$ when the phases are not matched. The $\omega_3^2$ factor penalizes long idler wavelengths. At fixed input intensities and crystal parameters, moving the idler from 3.39 µm to 300 µm (1 THz) lowers $\omega_3^2$ by a factor of about 7800, which is one reason optical-to-terahertz conversion efficiencies are small.

How it is observed in the lab

A typical setup combines the two beams on a dichroic mirror, overlaps them spatially and, for pulsed sources, in time, and focuses them into the crystal. The idler is separated with a long-pass filter or a germanium window that blocks the near-infrared inputs, and detected with a mid-infrared detector such as a thermopile, a cooled InSb or HgCdTe photodiode, or a spectrometer. The idler wavelength is tuned by tuning either input laser and then retuning the crystal temperature or poling period to restore phase matching. A quick check that the signal is genuine DFG: the idler power should scale linearly with each input power separately.

Where DFG is used

• Mid-infrared spectroscopy. DFG sources cover the molecular fingerprint region with narrow linewidths inherited from the near-infrared lasers, complementing quantum cascade lasers.
• Offset-free combs. When both inputs come from the same frequency comb, the carrier-envelope offset frequency cancels in the difference, giving a mid-infrared comb with zero offset.
• Terahertz sources. Mixing two nearby lasers, or the spectral components within one femtosecond pulse, gives tunable terahertz generation in crystals such as GaSe, DAST or lithium niobate.

Pitfalls

Lithium niobate absorbs strongly beyond roughly 5 µm, so longer idlers need crystals such as AgGaS₂, GaSe, ZnGeP₂ or orientation-patterned GaAs, several of which require pump wavelengths beyond about 2 µm to avoid two-photon or residual absorption. The idler beam diverges faster than the inputs because its wavelength is longer, so collection optics must be sized for it. Thermal effects from pump absorption and photorefractive damage in undoped lithium niobate can shift the phase-matching temperature over time; MgO-doped crystals and elevated operating temperatures are the usual remedies.

Common questions

What is the difference between DFG and an OPA?

The interaction is the same three-wave mixing. The name DFG is used when the goal is the new difference-frequency beam and the input signal is supplied by a laser; OPA is used when the goal is to amplify the signal, often starting from a weak seed. Both create idler photons and signal photons in equal numbers.

Can DFG generate wavelengths shorter than both inputs?

No. The output frequency is the difference of the input frequencies, so its wavelength is always longer than that of the higher-frequency input. Shorter wavelengths come from sum-frequency or harmonic generation.

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; J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962).