Nonlinear coefficient

"Nonlinear coefficient" names two different quantities depending on the field. In crystal optics it is the second-order tensor $d$, defined as half the second-order nonlinear susceptibility, $d_{ijk}={\chi }_{ijk}^{(2)}\mathrm{/}2$, in pm/V; it determines how efficiently a crystal performs second-harmonic generation, sum- and difference-frequency generation and parametric amplification. Approximate values are $d_{33}\approx 25$–27 pm/V for lithium niobate, $d_{33}\approx 15$–17 pm/V for KTP, and $d_{22}\approx 2.2$ pm/V for BBO. In fiber optics, "nonlinear coefficient" usually means the parameter $\gamma =2\pi n_2\mathrm{/}(\lambda A_{\mathrm{e}\mathrm{f}\mathrm{f}})$, in 1/(W·km), which gives the Kerr phase per watt per kilometer; for standard single-mode fiber at 1550 nm it is about 1.3 /(W·km).

Contracted notation

Because the two field indices of $d_{ijk}$ can be interchanged, the last pair is contracted into a single index $l$ from 1 to 6: $xx\to 1$, $yy\to 2$, $zz\to 3$, $yz\to 4$, $xz\to 5$, $xy\to 6$. The tensor then becomes a 3 × 6 matrix $d_{il}$, and crystal symmetry leaves only a few independent elements. Far from absorption resonances, Kleinman symmetry makes the tensor symmetric in all three indices and reduces the count further. Lithium niobate (point group 3m) is described mainly by $d_{33}$, $d_{31}$ and $d_{22}$; the large $d_{33}$ couples fields polarized along the crystal $z$ axis.

Effective coefficient

In an experiment, the fields have particular polarizations and directions relative to the crystal axes, and the tensor contracts to a single number, $d_{\mathrm{e}\mathrm{f}\mathrm{f}}$. For birefringent phase matching, $d_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is a trigonometric combination of tensor elements that depends on the propagation angles and the phase-matching type; it is often much smaller than the largest element, because birefringent matching requires orthogonal polarizations, which excludes the diagonal $d_{33}$. In quasi-phase matching, all fields can be polarized along $z$, and with a first-order 50 % duty-cycle grating

$$d_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{2}{\pi }{d}_{33}\mathrm{.}$$

For $d_{33}=25$ pm/V this gives 15.9 pm/V. Birefringently phase-matched lithium niobate uses $d_{31}$, roughly 4–5 pm/V. Since low-conversion efficiency scales as $d_{\mathrm{e}\mathrm{f}\mathrm{f}}^2$, taking $d_{31}=4.5$ pm/V the periodically poled crystal is about 12.5 times more efficient for the same length and focusing, and it avoids walk-off as well.

Measurement

Absolute values of $d$ are hard to measure, because the harmonic power depends on beam quality, focusing and the exact phase-matching condition. Most tables are built from relative Maker-fringe or phase-matched SHG measurements against a reference such as quartz ($d_{11}\approx 0.3$ pm/V), and published values for the same crystal commonly differ by 10–20 %. The coefficients also fall with increasing wavelength (Miller's rule relates them to the linear susceptibilities at the frequencies involved), so a value quoted at 1064 nm overstates $d$ for mid-infrared mixing.

The fiber parameter γ

In fibers and waveguides the relevant quantity combines the Kerr index $n_2$ of the material with how tightly the mode is confined:

$$\gamma =\frac{2\pi n_2}{\lambda A_{\mathrm{e}\mathrm{f}\mathrm{f}}}\mathrm{.}$$

For silica, $n_2=2.6\times {10}^{-20}$ m²/W, $\lambda =1550$ nm and an effective area of 80 µm²,

$$\gamma =1.32\times {10}^{-3}{\mathrm{W}}^{-1}{\mathrm{m}}^{-1}\approx 1.3{\mathrm{W}}^{-1}\mathrm{k}{\mathrm{m}}^{-1}\mathrm{.}$$

The nonlinear phase accumulated by power $P$ over an effective length $L_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is ${\varphi }_{NL}=\gamma P{L}_{\mathrm{e}\mathrm{f}\mathrm{f}}$. For a long fiber with 0.2 dB/km loss, $L_{\mathrm{e}\mathrm{f}\mathrm{f}}\approx 21.7$ km, so 10 mW produces a phase of 0.29 rad; the nonlinear length $1\mathrm{/}(\gamma P)$ at 100 mW is 7.6 km. This phase drives self-phase modulation, cross-phase modulation and four-wave mixing in optical links. Highly nonlinear fibers reach about 10 /(W·km) with small cores, and silicon wire waveguides, with $n_2\approx 4.5\times {10}^{-18}$ m²/W and $A_{\mathrm{e}\mathrm{f}\mathrm{f}}\approx 0.1$ µm², reach about 180 /(W·m), five orders of magnitude above standard fiber.

Pitfalls

The naming collision is the main source of confusion: a crystal's $d$ in pm/V and a fiber's $\gamma$ in 1/(W·km) describe different orders of nonlinearity and cannot be compared. Within crystal optics, check whether a table lists $d$ or ${\chi }^{(2)}=2d$, and whether signs are given; the sign matters for the relative phase in cascaded processes. Quoting $d_{33}$ for a birefringently phase-matched design overstates its efficiency; the relevant figure is $d_{\mathrm{e}\mathrm{f}\mathrm{f}}$ for the actual geometry.

Common questions

What is d33 of lithium niobate?

About 25–27 pm/V near 1064 nm, with MgO-doped material similar. It is the largest element of lithium niobate's tensor and the reason periodically poled lithium niobate, including thin-film x-cut devices, is the most widely used material for quasi-phase-matched frequency conversion.

Why is d_eff smaller than d33?

The effective value projects the tensor onto the actual field polarizations. Birefringent matching uses off-diagonal elements, and a quasi-phase-matching grating contributes only its first Fourier component, the factor $2\mathrm{/}\pi$.

How does γ change with wavelength?

It scales as $1\mathrm{/}\lambda$ explicitly and also through $A_{\mathrm{e}\mathrm{f}\mathrm{f}}$, which grows with wavelength, so $\gamma$ falls faster than $1\mathrm{/}\lambda$ in a given fiber.

References: R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020); V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, 3rd ed. (Springer, 1999); I. Shoji et al., J. Opt. Soc. Am. B 14, 2268 (1997); G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. (Academic Press, 2019).