Nonlinear optical susceptibility (${\chi }^{(2)}$)

Choose one driving electric field component and one responding electric polarization component. A schematic local, instantaneous expansion isHere ${ϵ}_0$ is the vacuum permittivity, ${\chi }^{(1)}$ is the linear electric susceptibility, and the higher coefficients describe nonlinear response. In SI, $P$ has units $\mathrm{C}{\mathrm{m}}^{-2}$, $E$ has units $\mathrm{V}{\mathrm{m}}^{-1}$, and ${\chi }^{(2)}$ has units $\mathrm{m}{\mathrm{V}}^{-1}$. The selected scalar coefficient depends on polarization of waves and crystal orientation; it is not a universal material number independent of geometry.

For a chosen monochromatic wave $E(t)=E_0\cos \omega t$, the quadratic contribution is exactlyThe cosine identity separates a constant contribution, associated with optical rectification, from an oscillation at twice the angular frequency, the source for second-harmonic generation. A term oscillating at $2\omega$ is a local driving source; efficient outgoing light additionally requires propagation and phase matching.

For a material with optical dispersion, the scalar instantaneous expansion is only a teaching model. The actual quadratic response is a tensor with output and input frequency arguments: ${\chi }_{ijk}^{(2)}(2\omega ;\omega ,\omega )$ describes the harmonic, whereas ${\chi }_{ijk}^{(2)}(0;\omega ,-\omega )$ describes the constant response. Their coefficients need not coincide. The cosine calculation establishes which frequencies are available, without assigning measured coefficients to any crystal.

The order-$n$ response has one output vector index and $n$ input vector indices. Material point-group invariance, permutation of input index-frequency pairs, and reality of the electric polarization impose different constraints. They must not be conflated with the stronger dispersion-neglecting Kleinman symmetry.

For identical input frequencies, intrinsic permutation symmetry of nonlinear susceptibility makes the last two Cartesian indices symmetric. Store their six independent pairs as one index, so the quadratic response uses a three-by-six array $d_{iJ}$. In the convention $d_{ijk}={\chi }_{ijk}^{(2)}/2$, the contracted input vector is $(A_x^2,A_y^2,A_z^2,2A_y{A}_z,2A_z{A}_x,2A_x{A}_y{)}^T$. The factors of two preserve the sum over both off-diagonal orderings. This bookkeeping does not assert the stronger Kleinman symmetry and does not change the tensor's physical order.

An isotropic fourth-rank polar tensor has the displayed form. If its three input slots are fully interchangeable at the same relevant frequency arguments, then $a=b=c$, giving ${\chi }_{xxxx}=3{\chi }_{xxyy}$. For mixed-sign or unequal frequencies, intrinsic permutation exchanges the arguments too; it need not make all coefficients at one fixed ordering equal.

Far from relevant resonances, when frequency dispersion of the nonlinear response can be neglected, a susceptibility can acquire full permutation symmetry among spatial indices. For second order this reduces a general symmetric-input tensor's 18 components to 10 before material point-group reductions. Mere transparency does not establish this approximation. The name refers to D. A. Kleinman.^[ref]

Commuting input field factors permit the response kernel to be symmetrized under interchange of input index-frequency pairs. For example ${\chi }_{ijk}^{(2)}({\omega }_s;{\omega }_1,{\omega }_2)={\chi }_{ikj}^{(2)}({\omega }_s;{\omega }_2,{\omega }_1)$. This does not authorize exchanging indices while keeping unequal frequency arguments in their old slots, nor exchanging the output index without further assumptions.

Suppose a homogeneous bulk material is unchanged when all spatial coordinates are reversed. Both electric polarization and electric field reverse under this inversion, so a local electric dipole response must obey $P(-E)=-P(E)$. Substituting the nonlinear optical susceptibility expansion givesFor every sufficiently small $E$, all even coefficients vanish. In the full tensor relation, two reversed input electric fields leave their product unchanged while the output electric polarization changes sign, forcing every bulk electric-dipole component of ${\chi }^{(2)}$ to vanish.

The qualification matters: the argument assumes inversion symmetry, a local bulk electric dipole response, and no applied bias that breaks that symmetry. A surface breaks bulk inversion symmetry; spatially varying or magnetic responses also lie outside this local electric-dipole model. Thus detecting second-harmonic generation at a surface or through a weaker magnetic contribution does not contradict the bulk selection rule. Conversely, absence of inversion symmetry permits some quadratic components but does not guarantee a nonzero component in every beam geometry.