Third-harmonic generation

Third-harmonic generation is a nonlinear optical process in which three photons at an angular frequency (\omega) are converted into one photon at the angular frequency (3\omega). The process is governed directly by the third-order electric susceptibility of a medium and therefore belongs to the broader class of four-wave mixing interactions. In the absence of absorption or energy transfer to other degrees of freedom, the generated photon carries the combined energy of the three incident photons,

[ 3\hbar\omega=\hbar(3\omega). ]

The output wavelength in vacuum is consequently one third of the pump wavelength. A pump at (1.5\ \mu\text{m}), for example, produces third-harmonic radiation near (500\ \text{nm}). This frequency conversion differs from third-harmonic distortion in electronic systems because it is a coherent electromagnetic interaction described by the nonlinear polarization of matter.

Nonlinear polarization

The macroscopic polarization of a dielectric exposed to an optical electric field can be expanded as a power series,

[ P_i(t)=\varepsilon_0\left( \chi^{(1)}{ij}E_j+ \chi^{(2)}{ijk}E_jE_k+ \chi^{(3)}_{ijkl}E_jE_kE_l+\cdots \right), ]

where repeated Cartesian indices denote summation and the susceptibility tensors encode the response of the material. For a monochromatic field written in complex notation as

[ \mathbf E(t)=\frac{1}{2}\mathbf E_\omega e^{-i\omega t}+\text{c.c.}, ]

the cubic term contains components oscillating at both (\omega) and (3\omega). The component responsible for direct third-harmonic generation is

[ P_i^{(3)}(3\omega)= \frac{\varepsilon_0}{8} \chi^{(3)}{ijkl}(3\omega;\omega,\omega,\omega) E{\omega,j}E_{\omega,k}E_{\omega,l}. ]

Numerical prefactors vary with the convention used for complex field amplitudes, whereas the cubic dependence and tensor contraction remain invariant. The same third-order response also contributes to the optical Kerr effect, self-phase modulation, and other frequency-mixing processes, although those phenomena correspond to different frequency arguments of (\chi^{(3)}).

Unlike second-harmonic generation, electric-dipole third-harmonic generation is not forbidden in media possessing inversion symmetry. Under spatial inversion, both the electric field and polarization change sign, while the cubic field product also changes sign. A nonzero third-order susceptibility is therefore compatible with centrosymmetric crystals, glasses, gases, liquids, and plasmas. Crystal symmetry nevertheless constrains the independent components of (\chi^{(3)}), making the generated polarization dependent on propagation direction and input polarization.

Propagation and phase matching

Energy conservation alone does not ensure sustained harmonic growth. Contributions generated at different positions interfere constructively only when the pump and harmonic fields retain the required phase relationship. For collinear propagation, the wave-vector mismatch is

[ \Delta k=k(3\omega)-3k(\omega). ]

Perfect phase matching corresponds to (\Delta k=0). Under the plane-wave, slowly varying envelope, and undepleted-pump approximations, the third-harmonic intensity follows the characteristic dependence

[ I_{3\omega}(L)\propto \left|\chi_{\mathrm{eff}}^{(3)}\right|^2 I_\omega^3L^2 \operatorname{sinc}^2\left(\frac{\Delta kL}{2}\right), ]

where (L) is the interaction length and (\chi_{\mathrm{eff}}^{(3)}) is the tensor component selected by the optical geometry. The cubic intensity scaling applies when pump depletion, nonlinear absorption, and competing nonlinear processes remain negligible.

Ordinary material dispersion usually gives (k(3\omega)\neq3k(\omega)), so the generated field reverses from constructive to destructive interference over the coherence length

[ L_{\mathrm c}=\frac{\pi}{|\Delta k|}. ]

This cancellation limits the net output of a uniform bulk medium. Birefringence can compensate for the dispersion when different polarization eigenmodes provide suitable refractive indices at the two frequencies. Structured media instead alter modal propagation constants, allowing phase matching in optical waveguides, microresonators, and photonic-crystal structures. Periodic modulation of the nonlinear response can supply an additional reciprocal-lattice momentum, producing a third-order form of quasi-phase matching.

Focused beams introduce longitudinal phase shifts that are absent from the plane-wave description. In particular, the Gouy phase changes rapidly near a focus and can cause harmonic radiation generated on opposite sides of the focal plane to interfere destructively. The observable field is therefore determined by an overlap integral involving the spatial modes, the nonlinear polarization, and the longitudinal phase mismatch. Confinement in a waveguide changes this balance because its eigenmodes replace freely diffracting Gaussian beams.

Direct and cascaded generation

Direct third-harmonic generation is mediated by (\chi^{(3)}(3\omega;\omega,\omega,\omega)). A field at (3\omega) can also arise through two sequential second-order interactions in a non-centrosymmetric medium. The first interaction converts two pump photons into a field at (2\omega), after which sum-frequency generation combines that field with another pump photon:

[ \omega+\omega\rightarrow2\omega, \qquad 2\omega+\omega\rightarrow3\omega. ]

The cascaded mechanism depends on two second-order susceptibilities and on the phase matching of both conversion stages. It can coexist coherently with the direct third-order contribution, in which case the total harmonic amplitude includes the relative phase of the two pathways. The distinction is physical rather than merely notational because suppressing the intermediate second harmonic removes the cascaded contribution without eliminating the direct process.

At surfaces and interfaces, translational symmetry changes discontinuously and the bulk-generated waves encounter abrupt variations in refractive index and nonlinear polarization. These changes can produce localized third-harmonic emission even when propagation through the surrounding bulk material gives strong cancellation. The resulting sensitivity to boundaries underlies the use of third-harmonic signals in nonlinear optical microscopy, where contrast frequently originates from interfaces between regions having different optical properties.

Historical development

The experimental basis of modern nonlinear optics followed the development of the laser. In 1961, Peter Franken, A. E. Hill, C. W. Peters, and Gabriel Weinreich observed optical second-harmonic generation in crystalline quartz. Their experiment established that sufficiently intense coherent light could drive a detectable polarization at an integer multiple of the incident frequency.

The theoretical treatment of wave propagation in nonlinear dielectrics was developed during the same period. Nicolaas Bloembergen and Peter S. Pershan formulated boundary and propagation analyses that connected nonlinear polarization sources with measurable harmonic fields. Peter D. Maker and Robert W. Terhune subsequently investigated optical effects arising from polarization terms of third order in the electric field, clarifying the relation between material susceptibility, coherence length, and nonlinear propagation.

The first spectrally resolved optical third-harmonic experiment in a gas was reported in 1967 by G. H. C. New, J. F. Ward, and You Watanabe. Their measurements used intense laser radiation to generate a coherent output at three times the incident optical frequency and established the characteristic pressure and focusing dependence of gas-phase generation. The work placed third-harmonic generation within the experimentally accessible family of third-order optical interactions rather than treating it solely as a formal term in the polarization expansion.

Later studies extended the process to condensed matter, guided-wave systems, resonant nanostructures, and ultrashort pulses. These developments retained the same underlying conservation laws while modifying field confinement, dispersion, and the spatial overlap between the pump and harmonic modes.

Material and resonant response

The third-order susceptibility is generally complex and frequency dependent. Its real part contributes to coherent polarization, while its imaginary part is associated with nonlinear absorption and other dissipative channels. Near an electronic or vibrational resonance, the magnitude of the susceptibility can increase, although linear absorption at either (\omega) or (3\omega) can reduce the distance over which a measurable harmonic field accumulates.

In gases, the nonlinear response scales with particle number density when interparticle interactions are negligible. Increasing pressure therefore increases the local nonlinear polarization, while simultaneously changing dispersion and phase mismatch. Atomic and molecular resonances further determine the tensor structure and spectral dependence of the response.

In solids, third-harmonic emission depends on band structure, crystal symmetry, and the polarization of the incident field. Semiconductor resonances can produce strong spectral variation near interband transitions, whereas transparent dielectrics commonly support propagation over longer distances with a smaller resonant response. Metals and nanostructured composites exhibit highly localized fields near plasmonic resonances, but their interpretation requires inclusion of absorption, spatial inhomogeneity, and surface contributions.

Pulsed and microscopic regimes

Ultrashort pulses contain a finite bandwidth rather than a single pump frequency. Their third-order polarization combines frequency components satisfying

[ \omega_1+\omega_2+\omega_3=\Omega, ]

with output frequencies (\Omega) concentrated near three times the pump carrier frequency. Group-velocity mismatch causes the pump and harmonic envelopes to separate during propagation, limiting their effective temporal overlap. Higher-order dispersion also changes the phase relation among spectral components and thereby alters the duration and shape of the emitted harmonic pulse.

At the quantum level, direct third-harmonic generation is represented by an interaction that annihilates three pump photons while creating one photon in the harmonic mode, together with the Hermitian-conjugate reverse process. A strongly occupied pump mode permits the classical-field approximation used in most macroscopic treatments. The generated field then remains phase coherent with the pump, and its phase is three times the pump phase apart from the phase contributed by the nonlinear susceptibility and propagation.

Measurement and interpretation

Third-harmonic radiation is identified through its frequency, polarization, angular distribution, and dependence on pump intensity. Under perturbative conditions, a cubic relationship between harmonic power and incident intensity distinguishes the direct third-order process from linear fluorescence. Deviations from this relationship can result from pump depletion, saturation, multiphoton absorption, thermal modification, or a cascaded second-order pathway.

Because the detected signal represents the coherent sum of radiation generated throughout the illuminated volume, intensity alone does not directly measure the local magnitude of (\chi^{(3)}). Quantitative interpretation incorporates phase mismatch, absorption, collection geometry, and the spatial profiles of the participating fields. In heterogeneous samples, boundary emission and bulk emission can possess comparable amplitudes but different phases, producing either enhancement or suppression in the measured direction.