Four-wave mixing (FWM)

Four-wave mixing (FWM) is the third-order nonlinear process in which light at frequencies $f_i$, $f_j$ and $f_k$ interacts through the Kerr nonlinearity of a medium to generate light at

$$f_{ijk}=f_i+f_j-f_k\mathrm{.}$$

In terms of photons, two are annihilated and two created, with energy conserved. When two of the input frequencies coincide the process is called degenerate: a strong pump at $f_p$ and a signal at $f_s$ produce an idler at $2f_p-f_s$, as far on the other side of the pump as the signal is on its side. The generated power grows as the product of the three input powers, so four-wave mixing matters at the milliwatt-per-channel powers of long fiber links, where it is an impairment, and at the watt-level pumps of wavelength converters and parametric amplifiers, where it is the working principle.

Phase matching

A generated wave builds up along the medium only while it stays in step with the nonlinear polarization that drives it, which requires the propagation constants of the four waves to match (phase matching). In a fiber, for degenerate mixing with the pump and signal $\mathrm{\Delta }f$ apart, the mismatch is

$$\mathrm{\Delta }\beta ={\beta }_2(2\pi \mathrm{\Delta }f{)}^2=\frac{2\pi {\lambda }^{2}D\mathrm{\Delta }{f}^2}{c},$$

where ${\beta }_2$ is the group-velocity dispersion and $D$ the dispersion parameter. In standard single-mode fiber at 1550 nm, $D$ = 17 ps/(nm·km), channels 100 GHz apart have $\mathrm{\Delta }\beta$ = 8.56 km⁻¹, so the generated wave slips out of step with its source within $2\pi \mathrm{/}\mathrm{\Delta }\beta$ = 0.73 km and never builds up. Over a fiber of loss $\alpha$ and length $L$, the efficiency relative to perfect phase matching is

$$\eta =\frac{{\alpha }^2}{{\alpha }^2+\mathrm{\Delta }{\beta }^2}[1+\frac{4e^{-\alpha L}{\sin }^2(\mathrm{\Delta }\beta L\mathrm{/}2)}{{(1-e^{-\alpha L})}^2}]$$

(Shibata, Braun and Waarts; Inoue). For an 80 km span with a loss of 0.2 dB/km:

Halving the spacing reduces $\mathrm{\Delta }\beta$ fourfold, so dense grids mix more strongly. Measuring the idler as the pump is tuned through the zero-dispersion wavelength locates that wavelength, since the efficiency peaks there (Inoue).

Crosstalk in WDM links

With $N$ channels, the number of mixing products is $N^2(N-1)\mathrm{/}2$: 9 for three channels, 224 for eight and 1,920 for sixteen. On an equally spaced grid many of them fall exactly on other channels, where they beat with the signal and cannot be filtered out. A degenerate product has a power, relative to the transmitted channel, of $(\gamma L_{\text{eff}}P{)}^2\eta$ for a channel power $P$, a nonlinear coefficient $\gamma$ and an effective length $L_{\text{eff}}$; a non-degenerate product, from three distinct channels, is four times (6 dB) stronger (Agrawal). In an 80 km span of a phase-matched fiber with an effective area of 50 µm² ($\gamma$ = 2.1 W⁻¹km⁻¹, $L_{\text{eff}}$ = 21.2 km), a single degenerate product lies 27 dB below its channel at 0 dBm per channel and 21 dB below at +3 dBm, since it grows as the square of the channel power, and many products add on each channel. In standard single-mode fiber at 100 GHz spacing the same product is 45 dB weaker still.

That difference is why dispersion-shifted fiber, whose zero-dispersion wavelength lies in the C-band, proved unsuitable for dense WDM, and why non-zero dispersion-shifted fiber keeps a few ps/(nm·km) of dispersion throughout the band (Tkach and colleagues). Unequal channel spacing, which moves the products off the channels, is another remedy. Four-wave mixing belongs to the same Kerr nonlinearity as self-phase modulation and cross-phase modulation, which change the phase of the existing waves rather than create new frequencies.

Wavelength conversion and phase conjugation

A pump at $f_p$ copies a signal at $f_s$ onto an idler at $2f_p-f_s$ that carries the same modulation. The idler's field is proportional to $A_p^2{A}_s^{\ast }$, the complex conjugate of the signal's, so its spectrum is inverted and its phase reversed. Converters use highly nonlinear fiber, with a small effective area ($\gamma$ of about 10 W⁻¹km⁻¹, against 1.3 for standard fiber) and a zero-dispersion wavelength near the pump, or semiconductor optical amplifiers, in which the beating of pump and signal modulates the gain and index and gives strong mixing in a short device. The phase reversal can undo dispersion: a conjugated signal sent through a second span like the first retraces the dispersion it accumulated, which is called optical phase conjugation or mid-span spectral inversion.

Parametric amplification

With a strong pump, four-wave mixing transfers power from the pump to the signal and the idler together, which makes a fiber optical parametric amplifier. When the phases are matched, including the contribution of the pump's own self-phase modulation, the signal gain is

$$G={\cosh }^2(\gamma PL)$$

for a pump power $P$ over a length $L$: 37.4 dB for $\gamma PL$ = 5, which a 1 W pump reaches in 0.5 km of fiber with $\gamma$ = 10 W⁻¹km⁻¹. A fiber amplifier pumped continuously has reached 49 dB of gain (Hansryd and Andrekson). The gain lies on both sides of a pump placed slightly on the anomalous-dispersion side of the zero-dispersion wavelength, and because it involves no energy levels it can be placed wherever the fiber and pump can be phase matched. When signal and idler both enter the amplifier in the right phase relation, it becomes phase-sensitive and can amplify with a noise figure below the 3 dB limit of ordinary optical amplifiers; a fiber phase-sensitive amplifier has shown 1.1 dB (Tong and colleagues).

Combs, photon pairs and waveguides

In a high-Q microresonator, cascaded four-wave mixing from a single continuous-wave pump generates a comb of equally spaced lines (microcomb). Below the threshold of such oscillation, spontaneous four-wave mixing converts pairs of pump photons into signal and idler photon pairs, a standard source of heralded single photons and entangled pairs in silicon and silicon nitride circuits (quantum photonic integrated circuits); the same interaction produces squeezed light.

In silicon waveguides the Kerr coefficient is about 170 times that of silica and the mode about 800 times smaller in area (0.1 against 80 µm²), so a silicon wire has $\gamma$ of about 180 W⁻¹m⁻¹ and useful mixing occurs over millimeters to centimeters. At 1550 nm, two-photon absorption and the free carriers it creates limit the pump power, which is why silicon mixing works better beyond 2.2 µm, where two-photon absorption stops, and why silicon nitride, which has no two-photon absorption in the telecom bands, is widely used for chip-scale combs and parametric amplifiers.

Four-wave mixing is compared with the other third-order effects in Nonlinear Optics: An Overview.

Common questions

What is four-wave mixing?

A nonlinear optical process in which light at two or three frequencies interacts through the Kerr effect to generate light at new frequencies, $f_i+f_j-f_k$. Energy is conserved photon by photon, and the process is efficient only when the waves are phase matched.

Why is four-wave mixing a problem in WDM systems?

Every combination of channels generates mixing products, and on an equally spaced grid many of them land on other channels, where they cannot be filtered out and add crosstalk. The products grow with the square of the channel power relative to the channels, and are strongest where the fiber's dispersion is near zero.

How does dispersion suppress four-wave mixing?

Dispersion makes the waves travel with slightly different phase velocities, so the generated wave falls out of step with the polarization that drives it and cannot build up. In standard single-mode fiber at 1550 nm, mixing between channels 100 GHz apart over an 80 km span is 45 dB weaker than in a fiber with zero dispersion.

What is four-wave mixing used for?

Wavelength conversion and optical phase conjugation, parametric amplification (including phase-sensitive amplification below the usual 3 dB noise figure), Kerr frequency combs in microresonators, and photon pairs and squeezed light for quantum optics. It is also used to measure a fiber's zero-dispersion wavelength.

References: G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2013). R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020). N. Shibata, R. P. Braun and R. G. Waarts, "Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber," IEEE Journal of Quantum Electronics 23, 1205 (1987). K. Inoue, "Four-wave mixing in an optical fiber in the zero-dispersion wavelength region," Journal of Lightwave Technology 10, 1553 (1992). R. W. Tkach et al., "Four-photon mixing and high-speed WDM systems," Journal of Lightwave Technology 13, 841 (1995). J. Hansryd and P. A. Andrekson, "Broad-band continuous-wave-pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency," IEEE Photonics Technology Letters 13, 194 (2001). Z. Tong et al., "Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers," Nature Photonics 5, 430 (2011).