Nonlinear susceptibility

A light wave drives the electrons of a material, and the resulting dipole moment per unit volume, the polarization $P$, radiates the transmitted and reflected fields. At low intensity $P$ is proportional to the field $E$, and the proportionality constant $\chi^{(1)}$ sets the refractive index through $n^2 = 1 + \chi^{(1)}$. At higher intensity the response contains terms in higher powers of the field:

$$P = \varepsilon_0\left(\chi^{(1)}E + \chi^{(2)}E^2 + \chi^{(3)}E^3 + \cdots\right).$$

The coefficients $\chi^{(2)}$ and $\chi^{(3)}$ are the second- and third-order nonlinear susceptibilities. The processes each one drives are surveyed in the overview of second- and third-order effects. In SI units $\chi^{(2)}$ is in m/V and $\chi^{(3)}$ in m²/V². Typical values are tens of pm/V for $\chi^{(2)}$ in ferroelectric crystals such as lithium niobate, and about $2 \times 10^{-22}$ m²/V² for $\chi^{(3)}$ in fused silica. Because both are small, nonlinear effects become noticeable only at the high intensities of focused laser beams, pulses, or long, tightly confining waveguides.

Tensor form and symmetry

For real materials each susceptibility is a tensor that connects field components along different axes: $P_i = \varepsilon_0 \sum_{jk} \chi^{(2)}_{ijk} E_j E_k$, and similarly for $\chi^{(3)}_{ijkl}$. The susceptibilities also depend on the frequencies involved, since the nonlinear response is enhanced near material resonances.

Symmetry determines which components survive. In a centrosymmetric medium, reversing the field must reverse the polarization, which forces every even-order term to zero. Glass, liquids, gases, and crystals such as silicon therefore have $\chi^{(2)} = 0$ in the bulk, and second-order processes require non-centrosymmetric crystals such as lithium niobate, KTP, BBO or GaAs. Surfaces and interfaces break inversion symmetry locally, which makes weak second-order signals from them useful as surface probes. $\chi^{(3)}$ is allowed in every material. For second-order crystals the tensor is usually quoted in the contracted form of the nonlinear coefficient, $d = \chi^{(2)}/2$.

Processes each order produces

The $\chi^{(2)}$ term mixes two fields: second-harmonic generation, sum-frequency generation, difference-frequency generation, optical rectification, and parametric amplification all come from it, as does the linear electro-optic Pockels effect when one of the fields is static. The $\chi^{(3)}$ term mixes three: third-harmonic generation, four-wave mixing, and the intensity-dependent index of the Kerr effect, which underlies self-phase modulation, self-focusing and solitons. Raman and Brillouin scattering also appear as the imaginary, delayed parts of $\chi^{(3)}$.

Magnitudes and a worked number

A useful estimate compares the optical field with the field that binds an electron in an atom, $E_{at} = e/(4\pi\varepsilon_0 a_0^2) \approx 5.1 \times 10^{11}$ V/m. Each additional order is then smaller by roughly a factor $E_{at}$, giving $\chi^{(2)} \sim 2$ pm/V and $\chi^{(3)} \sim 4 \times 10^{-24}$ m²/V² for nonresonant materials; real values scatter by an order of magnitude or more around these.

For fused silica, the measured nonlinear index $n_2 \approx 2.6 \times 10^{-20}$ m²/W relates to $\chi^{(3)}$ through

$$n_2 = \frac{3\,\chi^{(3)}}{4\,n^2\,\varepsilon_0\,c},$$

a relation that assumes the intensity convention $I = \tfrac{1}{2} n\varepsilon_0 c|E|^2$. With $n = 1.45$ this gives $\chi^{(3)} \approx 1.9 \times 10^{-22}$ m²/V². At an intensity of 1 GW/cm² the field amplitude in silica is about $7.2 \times 10^{7}$ V/m and the index change $n_2 I$ is $2.6 \times 10^{-7}$, which at 1550 nm amounts to about 1 rad of nonlinear phase per metre of propagation. In lithium niobate at the same intensity, with $\chi^{(2)} \approx 50$ pm/V, the second-order polarization is of order $10^{-3}$ of the linear one.

How it is measured

Second-order susceptibilities are measured by second-harmonic generation, often by the Maker-fringe method: a plate is rotated in the beam, the harmonic power oscillates as the path length changes, and the envelope and fringe spacing give the coefficient relative to a reference crystal such as quartz. Third-order susceptibilities are measured by the Z-scan technique, in which a sample is translated through a focus and the far-field transmission through an aperture reveals the nonlinear phase; by third-harmonic generation; or, for fibers, from the spectral broadening of self-phase modulation at known power.

Pitfalls

Conventions differ by factors of 2, 3/4 and $n$ depending on whether fields are written as real amplitudes or complex envelopes, and on how intensity is defined; values from different sources can be compared only after checking these. Older literature uses Gaussian (esu) units: $\chi^{(3)}$ in m²/V² equals $4\pi/(3\times 10^4)^2 \approx 1.40 \times 10^{-8}$ times the esu value, and $\chi^{(2)}$ in m/V equals $4.19 \times 10^{-4}$ times the esu value. Measurements with nanosecond pulses can include thermal and electrostrictive contributions that femtosecond measurements exclude, so $n_2$ values for the same glass differ with pulse duration.

Common questions

Why is χ(2) zero in glass and silicon?

Both are centrosymmetric: the glass on average and silicon by crystal structure. Inversion symmetry makes the polarization an odd function of the field, so even-order terms vanish. Strained silicon, poled glass and interfaces show weak effective $\chi^{(2)}$ where that symmetry is broken.

What is the difference between χ(3) and n2?

$\chi^{(3)}$ is the material coefficient in the polarization expansion; $n_2$ is the derived, directly measurable change of refractive index per unit intensity. They are proportional through the relation above, with a factor that depends on $n$.

References: R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020); Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984); M. Sheik-Bahae et al., IEEE J. Quantum Electron. 26, 760 (1990); G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. (Academic Press, 2019).