Nonlinear polarization creates new optical frequencies

Polarization density is electric dipole moment per volume, measured in coulombs per square metre. It is distinct from the direction or ellipse called the polarization of a light beam. In a weakly nonlinear instantaneous scalar model, expand this density in powers of the local electric field. The linear susceptibility is dimensionless, the quadratic susceptibility has units metres per volt, and each higher order supplies another inverse field factor. The expansion is useful only while omitted terms remain sufficiently small.

Real optical materials have memory, dispersion, and vector response. More generally the quadratic susceptibility is a tensor with an output index and two input indices, and it depends on the participating frequencies. A scalar coefficient represents a specified projection for a chosen geometry and polarization. Near resonances or for broad pulses, one must use the frequency-dependent response or a time-domain convolution rather than assume one instantaneous constant describes every frequency component.

Choose the real driving field E(t)=E₀ cos(ωt), where E₀ is its peak amplitude. Squaring the cosine gives one half plus one half cosine of twice the angular frequency. The quadratic polarization therefore contains a constant optical-rectification term and a source oscillating at the second harmonic. Its peak second-harmonic polarization amplitude is ε₀χ⁽²⁾E₀²/2, with χ⁽²⁾ here meaning the second-order susceptibility, not the square of the linear one.

For two real driving tones, the cross product of their cosines produces both sum and difference frequencies. Their separate squares also produce doubled frequencies and DC. The relative coefficients depend on whether one uses real peak fields or complex positive-frequency amplitudes; switching conventions midway changes apparent factors of two. Most importantly, identifying a nonlinear source does not calculate the outgoing optical power. Boundary conditions, absorption, phase matching, and any pump depletion still have to be considered.

In a centrosymmetric bulk material under the local electric-dipole approximation, reversing every electric-field component reverses the induced polarization: P(−E)=−P(E). Even powers do not change sign under that operation. Their coefficients must therefore vanish in this bulk response expansion, including every second-order tensor component. Third-order effects remain possible. The argument is a symmetry constraint on this constitutive response, not a claim that all nonlinear optical phenomena vanish in an inversion-symmetric material.

Surfaces, interfaces, applied static fields, magnetic-dipole or electric-quadrupole effects, and broken structural symmetry can change the scope of that conclusion. State those mechanisms explicitly rather than declaring an observed second harmonic impossible. The function lab can isolate quadratic scaling: Reset and set both controls to coefficient one and position one, predicting output one. Reset again and set coefficient one and position two, predicting output four. Its curve y=ax² is dimensionless here; it does not contain an optical carrier, a susceptibility tensor, or emitted radiation.