Sum-frequency generation
Sum-frequency generation (SFG) is a second-order nonlinear optical process in which two input optical fields at frequencies \omega_1 and \omega_2 interact within a nonlinear medium to produce an output field at the sum frequency \omega_3 = \omega_1 + \omega_2.[1] This interaction arises from the quadratic term in the medium's polarization response, governed by the second-order nonlinear susceptibility tensor \chi^{(2)}, which is nonzero only in materials lacking inversion symmetry, such as certain crystals like beta barium borate (BBO) or lithium niobate.[1] For efficient SFG, phase matching must be achieved, ensuring the wave vectors of the input and output fields satisfy \mathbf{k_3} = \mathbf{k_1} + \mathbf{k_2} to prevent destructive interference over the interaction length.[2] This condition is typically met through birefringence in anisotropic crystals or quasi-phase matching using periodic poling, allowing conversion efficiencies up to several percent in optimized setups.[1] The first experimental observations of nonlinear optical effects, including sum-frequency generation, were reported in 1961 by Franken et al. using a ruby laser on quartz. The process was first theoretically described in 1962 by Armstrong, Bloembergen, Ducuing, and Pershan, building on foundational work in nonlinear optics following the invention of the laser.[1] SFG finds broad applications in frequency conversion to access wavelengths not directly available from lasers, such as generating mid-infrared light for gas spectroscopy by mixing near-infrared and visible beams, or ultraviolet radiation for photochemical studies.[2] In bulk media, it enables tunable sources. At interfaces, SFG serves as a surface-selective probe in vibrational spectroscopy, providing insights into molecular orientation, chirality, and dynamics at buried interfaces like liquid-solid or air-water boundaries, with sub-monolayer sensitivity due to the symmetry-breaking at surfaces.[3] These capabilities have made SFG indispensable in fields ranging from materials science and electrochemistry to biological interface analysis.[4]Fundamentals
Definition and basic principles
Sum-frequency generation (SFG) is a second-order nonlinear optical process wherein two input photons, possessing frequencies \omega_1 and \omega_2, interact within a nonlinear medium to generate a single output photon at frequency \omega_3 = \omega_1 + \omega_2. This interaction conserves energy, as the energy of the emergent photon equals the combined energies of the incident photons, \hbar \omega_3 = \hbar \omega_1 + \hbar \omega_2.[2] SFG belongs to the class of parametric nonlinear processes, in which the medium acts as a catalyst without undergoing net absorption or permanent change, relying instead on virtual excitations of the material's electrons to facilitate the frequency mixing.[2] The fundamental mechanism of SFG stems from the second-order nonlinear polarization induced in the medium by the oscillating electric fields of the input beams. In the electric dipole approximation, this polarization arises from the quadratic response of the material's electrons to the applied fields, proportional to the product of the two input field strengths. However, such a second-order response vanishes in centrosymmetric media due to inversion symmetry, which forbids a permanent dipole moment; thus, SFG requires non-centrosymmetric materials, such as certain crystals lacking a center of symmetry, to exhibit a nonzero second-order nonlinear susceptibility. Qualitatively, the photon interaction can be visualized as the two input photons "colliding" via the medium's nonlinear response, effectively annihilating to produce the higher-energy output photon, with the medium providing the necessary momentum transfer without real energy exchange.[5] In addition to energy conservation, SFG obeys momentum conservation, requiring the wave vectors of the photons to satisfy \mathbf{k_3} = \mathbf{k_1} + \mathbf{k_2} vectorially, where \mathbf{k_i} = n_i (\omega_i / c) \hat{\mathbf{d_i}} and n_i is the refractive index at each frequency. This condition often necessitates careful alignment to achieve efficient generation, though detailed phase-matching strategies are essential for practical implementation. The related process of second-harmonic generation (SHG), a degenerate case of SFG, was first observed in 1961 by Franken et al., who irradiated a quartz crystal with a ruby laser beam. Non-degenerate SFG was first experimentally observed in 1962 by Maker et al., establishing these parametric processes as pioneering demonstrations of nonlinear optics shortly after the invention of the laser.[5][1]Relation to other nonlinear processes
Sum-frequency generation (SFG) is closely related to second-harmonic generation (SHG), which represents a specific instance of the SFG process. In SHG, the two input photons possess identical frequencies, denoted as \omega_1 = \omega_2, leading to an output photon at \omega_3 = 2\omega_1. This equivalence arises because SHG can be viewed as the degenerate case of SFG, where the nonlinear interaction simplifies due to frequency degeneracy. However, SFG extends this capability by allowing inputs at distinct frequencies \omega_1 \neq \omega_2, enabling the production of tunable output wavelengths across a broader spectral range, which is particularly useful for applications requiring flexibility in frequency selection.[2] Another key related process is difference-frequency generation (DFG), which serves as the inverse of SFG in the context of second-order nonlinear optics. While SFG produces an output frequency \omega_3 = \omega_1 + \omega_2 through up-conversion, combining lower-frequency inputs to yield a higher-frequency output, DFG generates \omega_3 = |\omega_1 - \omega_2|, facilitating down-conversion from higher to lower frequencies. This duality allows DFG and SFG to complement each other in frequency manipulation schemes, such as in tunable laser systems, where one process can generate signals for the other. Both rely on the same second-order nonlinear susceptibility \chi^{(2)} in non-centrosymmetric media.[2][6] SFG differs fundamentally from third-order nonlinear processes, such as third-harmonic generation (THG), in terms of the governing material response. SFG depends on the second-order nonlinear susceptibility \chi^{(2)}, which is absent in centrosymmetric materials and drives three-wave mixing interactions. In contrast, THG involves \chi^{(3)}, enabling four-wave mixing and occurring even in materials lacking \chi^{(2)} symmetry, such as glasses or isotropic media. This distinction underscores SFG's requirement for non-centrosymmetric crystals, while THG offers broader material compatibility but typically lower efficiency for harmonic generation without additional cascading.[7] As a member of the parametric family of nonlinear optical processes, SFG functions as a frequency mixer within a nonlinear medium, converting input frequencies without the need for population inversion, unlike laser gain media. In parametric amplification contexts, SFG can contribute to the generation of idler waves or signal amplification through coupled wave interactions, maintaining energy conservation via Manley-Rowe relations. This parametric nature highlights SFG's role in efficient, coherent frequency translation for photonics applications.[8][9]Theoretical framework
Mathematical description
Sum-frequency generation (SFG) arises from the second-order nonlinear polarization induced in a non-centrosymmetric medium by interacting electric fields. The nonlinear polarization \mathbf{P}^{(2)}(t) is expressed as \mathbf{P}^{(2)}(t) = \epsilon_0 \boldsymbol{\chi}^{(2)} : \mathbf{E}(t) \mathbf{E}(t), where \boldsymbol{\chi}^{(2)} is the second-order nonlinear susceptibility tensor and \epsilon_0 is the vacuum permittivity. This quadratic response generates new frequencies when the input field \mathbf{E}(t) comprises multiple frequency components. For input fields at frequencies \omega_1 and \omega_2, the polarization is expanded in the frequency domain, yielding a term at the sum frequency \omega_3 = \omega_1 + \omega_2:P_i^{(2)}(\omega_3) = 2 \epsilon_0 \sum_{j,k} \chi_{ijk}^{(2)}(\omega_3; \omega_1, \omega_2) E_j(\omega_1) E_k(\omega_2),
where the indices i, j, k denote Cartesian components, and the factor of 2 accounts for time-dependent permutations in the Fourier expansion. This polarization acts as a source for the generated field at \omega_3. The propagation of the fields is described by the nonlinear wave equation:
\nabla^2 \mathbf{E} - \frac{n^2}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = \frac{1}{\epsilon_0 c^2} \frac{\partial^2 \mathbf{P}^{(2)}}{\partial t^2},
where n is the refractive index and c is the speed of light in vacuum. For collinear plane waves propagating along z, assuming slowly varying envelopes A_j(z) for each field E_j(z, t) = A_j(z) e^{i(k_j z - \omega_j t)} + \mathrm{c.c.}, the equation yields coupled differential equations. Neglecting pump depletion, the equation for the SFG amplitude is
\frac{d A_3}{dz} = i \frac{\omega_3 \chi^{(2)}}{n_3 c} A_1 A_2 e^{i \Delta k z},
with \Delta k = k_3 - k_1 - k_2 the phase mismatch and n_3 the refractive index at \omega_3. Integrating over interaction length L gives the SFG field amplitude proportional to A_1 A_2 L \mathrm{sinc}(\Delta k L / 2). The generated SFG intensity I_3 follows from I_3 \propto |A_3|^2 and scales as
I_3 \propto I_1 I_2 |\chi^{(2)}|^2 L^2 \mathrm{sinc}^2(\Delta k L / 2),
where I_1 and I_2 are the input intensities; efficiency is maximized when \Delta k = 0. This relation highlights the quadratic dependence on input powers and the role of phase matching in enhancing conversion. The susceptibility \boldsymbol{\chi}^{(2)} is a third-rank tensor with 27 components, constrained by the medium's point group symmetry to fewer independent elements. In contracted notation, it is often expressed via the second-order nonlinear coefficient tensor d_{il} = \frac{1}{2} \chi_{ijk}^{(2)}, where l runs over 1 to 6 corresponding to strain-like indices (e.g., l=1 for xx, l=4 for yz). The effective nonlinearity d_{\mathrm{eff}} in the coupled equations depends on the polarization directions of the fields and crystal orientation, determining the SFG response for specific input configurations. For many crystals, Kleinman symmetry further simplifies the tensor by assuming invariance under index permutation (\chi_{ijk}^{(2)}(\omega_3; \omega_1, \omega_2) = \chi_{ikj}^{(2)}(\omega_3; \omega_2, \omega_1), etc.), valid in the absence of strong dispersion; this reduces the number of independent components, as seen in classes like 3m (e.g., LiNbO_3) where only a few d_{ij} elements dominate.