The general nonlinear susceptibility expression for sum frequency generation process χsum(2) is given by (27)χsum(2)=1ε0VkmnvμmnμnvμvmΔEmn+EkEsumiΓ×(ρmρvΔEvm+EkENIRiΓ+ρnρvΔEnv+EkENIRiΓ)(1)where m, n, and v are the various confined electron and hole states in the band structure and k is the in-plane wave vector (which is conserved in the dipole approximation). ΔEijij) refers to the transition energy (dipole matrix element) between states i and j. Ek is the kinetic energy for the relative electron-hole motion, and ρi represents the population of state i. The broadening coefficient Γ was set to 2 meV corresponding to a QCL operating temperature of 10 K (28). ENIR and Esum refer to the NIR pump energy and the sum energy, respectively. The occupancy ρi is taken to be 1 for states in the valence band and, owing to the low-level doping and weak photoexcitation, is negligible for states in the conduction band. The susceptibility (related to either LH or HH states) can then be split into two terms, χc(2) and χv(2), refering to the conduction and valence bands, respectivelyχsum(2)=χc(2)+χv(2)=1ε0Vk(mnnμmnμnnμnm(ΔEmn+EkEsumiΓ)(ΔEnm+EkENIRiΓ)+mmnμmnμnmμmm(ΔEnm+EkEsumiΓ)(ΔEnm+EkENIRiΓ))(2)

Indices n and n″ refer to electron states in the conduction band and indices m and m″ refer to hole states in the valence band. The total second-order susceptibility of the QCL can then be noted as χ(2) = χ(2)sum(HH) + χ(2)sum(LH).

Note that |χ(2)|2, calculated at 10 K, was red shifted by 13 meV to account for a bandgap shift, observed in the experiments, due to a local temperature increase by the electrical power dissipated in the QCL (29). This shift appears clearly when comparing (see fig. S2) the photoluminescence of the biased QCL with the calculated energies (at 10 K) of the first electron-hole transitions of the structure.

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