There are many potential practical applications of optical nonlinearities in condensed matter, including all-optical switching [1-4], optical limiting [5, 6], sensing [7, 8], and imaging [9, 10]. Additionally, these same nonlinearities limit the performance, particularly in optical telecommunication, and can lead to unwanted absorption or even damage of solid-state devices . The study of two-photon absorption (2PA) is instrumental in investigating material properties, since two-photon transitions are subject to a set of selection rules that are distinct from those governing one-photon transitions . Furthermore, nonlinear refraction (NLR) is related to nondegenerate 2PA though Kramers-Kronig relations , and NLR causes effects such as self-focusing, self- and cross-phase modulation and four-wave mixing [14, 15], each of them having a number of practical applications. Therefore, precise knowledge of the nonlinear optical properties of semiconductor materials is of great practical importance. While there has been much theoretical and experimental work done on degenerate, or self-, nonlinearities, where only a single beam of a particular wavelength interacts with a material, nondegenerate nonlinear interactions are much less explored. In this work, we discuss recent theoretical and experimental progress on the understanding of nondegenerate nonlinearities in both bulk semiconductor and semiconductor quantum wells.
The principle nonlinear interactions of interest are nonlinear refraction and nonlinear absorption (namely two- and three-photon absorption), which result in changes in refractive index and absorption coefficient due the presence of a strong optical field. The material’s response depends on its electronic energy levels, as well as the transition dipole moments between them. Knowledge of the wave functions and their energy levels thus allows prediction of the two-photon absorption coefficient α2 and the nonlinear refractive index n2. Alternatively, measurement of n2 and α2 provide information about the electronic energy levels and wave functions of a material.
Two-photon absorption, for example, depends on the initial, final, and intermediate states involved in the interactions. Figure 1(a) shows a simple three level model of the electronic states of a material, including an initial state i, final state f, and intermediate state j. The 2PA coefficient may occur for two photons when ħωa + ħωb = Efi, and depends on the detuning with respect to the intermediate state ΔEa = Eji – ħωa 
where the frequency arguments indicate irradiance loss of field a due to the presence of field b, and is the matrix element coupling states n and m by the field c = a, b. As either photon energy becomes close to the energy difference between initial and intermediate states Eji, α2(ωa; ωb) is greatly enhanced, an effect known as intermediate state resonance enhancement (ISRE). In semiconductors, the intermediate states involved in the transition are within the conduction and valence bands themselves, meaning Eji is either near the band-gap energy or near zero, respectively (see Figure 1(b)). In this case, ISRE occurs when one photon energy becomes close to the band-gap energy, ħωa → Eg, and the other goes towards zero, ħωb → 0. Thus, the largest enhancement occurs when the photon energies are very different from one another, so called extremely nondegenerate (END) enhancement. Similar enhancement is investigated here theoretically for semiconductor quantum wells , as well as for three-photon absorption in bulk semiconductors.
This nondegenerate enhancement also applies to nonlinear refraction (NLR). ND-NLR is the refractive index change at frequency ωa caused by the presence of a beam at frequency ωb. The knowledge of the magnitude, sign and dispersion of the Kerr index n2(ωa; ωb) is critical for applications utilizing cross-phase modulation, such as optical switching . Compared to degenerate NLR, ND-NLR is less often investigated experimentally, particularly for the END case and the spectral region where 2PA is present . The theory is based on Kramers-Kronig transformation of the nondegenerate nonlinear absorption spectrum, where two-photon absorption (2PA), electronic Raman and optical Stark effect are taken into account [16, 18]. In this work, we have studied several direct-gap semiconductors using our nonlinear beam deflection technique . The magnitudes and nonlinear dispersion of n2(ωa; ωb) are resolved over a broad spectral range with high nondegeneracy. We found agreement between our experimental results and theoretical predictions.
NONDEGENERATE TWO- AND THREE-PHOTON ABSORPTION
Nondegenerate Enhancement of 2PA
Two-photon absorption in bulk semiconductors can be greatly enhanced by ISRE using END photon pairs, which has been shown both theoretically  and experimentally . For a two parabolic band model, the 2PA coefficient has been shown to follow
and K = 3100 cm GW-1 eV5/2, Ep is the Kane energy parameter, na and nb are the linear refractive indices, and Eg is the bandgap energy. This enhancement has been measured for nondegeneracies as large as ħωa/ħωb ~ 10 showing enhancement factors of up to 270 in ZnO over the degenerate case . Such large enhancement factors suggests the use of END 2PA may be of practical use in applications. Gated detection of mid-IR pulses using END-2PA in p-i-n photodiodes have demonstrated greater sensitivity than liquid nitrogen cooled HgCdTe detectors . This detection scheme has recently been used for 3D IR imaging , where the (nearly) instantaneous nature of the process ensures detection only when the two pulses are incident on the diode simultaneously. Depth information from reflective objects is encoded in the arrival time of the pulse to the detector, which only provides a signal when the gate pulse is synchronized, allowing for precise surface profile measurements.
Additionally, two-photon gain (2PG), which is the inverse process of 2PA, is likewise enhanced when using nondegenerate photons. 2PG is the stimulated emission of two photons, which is the two-photon analog to traditional stimulated emission. The 2PG coefficient is related to the 2PA coefficient by the inversion factor
where α2,0 is the 2PA coefficient at equilibrium, and fc and fv are the Fermi-Dirac distributions of conduction and valence bands, respectively. Thus two-photon gain is enhanced whenever 2PA is enhanced, as shown in Figure 2. ND-2PG has recently been experimentally demonstrated in optically excited GaAs, using pump-probe techniques, demonstrating such enhancement . END-2PG holds potential for the creation of a two-photon semiconductor laser.
Nondegenerate 2PA in Quantum Wells
While degenerate (D) 2PA of quantum wells (QW’s) has been studied by many groups [22-24], the nondegenerate regime has not been explored. The ND-2PA in QW’s is expected to be enhanced more near the band edge as the density of states and the transition matrix elements exceed those of bulk semiconductors . Since QW’s are anisotropic, the 2PA selection rules differ for the electric field polarized in the plane of the QW’s (TE) and light polarized perpendicular to the plane of the QW’s (TM). Figure 3(a) shows a finite QW structure with the electric vector polarization for TE and TM cases. The transition paths for the 2PA in the TE-TE case is shown in Figure 3(b). The selection rules in the TE-TE case transitions are necessarily interband-intraband where the initial and final states are within valence and conductions bands of the same subband index n, respectively, i.e., Δn = 0. For the TM-TM case (i.e., both photons TM polarized) the selection rules are different and Δn must be odd. Thus the “interband” transition goes from a valence subband to a conduction subband, and the “intraband” transition occurs between two subbands within either the conduction or valence bands themselves. Thus, TM-TM 2PA transitions are interband-intersubband, as shown in Figure 3(c).
For the ND-2PA, an analytic expression is derived using second order perturbation theory both for TE-TE and TM-TM polarizations , which assumes an infinite QW. The ND-2PA coefficient for the TE-TE polarized case is
where ζν = (ħωa + ħωb – Eg)/Ev,11 defines how far the two-photon transition energy is above the bandgap normalized to the linear absorption edge, Ev,11 = Ec,1 + Ev,1 = ħ2π2/(2μν,⊥d2), Nv = Int(√ζν) is the number of two-photon transitions between the valence and conduction subbands, α is the fine structure constant, m0 is the electron mass, μν are the reduced effective masses of the bands, and d is the well width. For TM-TM polarized light,
The ND-2PA coefficients may be measured experimentally by pump-probe where the transmission of the weak probe beam at ωa is monitored in the presence of the strong pump beam at ωa. Figure 4 shows the calculated ND-2PA coefficient of bulk GaAs and a 10 nm wide QW for TE-TE and TM-TM polarized light with a pump photon energy ħωb = 0.12Eg (λb = 7.5 μm) and varying the probe photon energy ħωa. For bulk, α2 is plotted against the sum of photon energies normalized to the bandgap (Eg), whereas for the QW, the energy is normalized to the respective one-photon absorption threshold (Eg + Ehh,11 for TE-TE and Eg + Elh,11 for TM-TM). The photon energy of the pump is chosen to have a long wavelength to avoid D-2PA or three photon absorption (3PA) from the pump itself. This allows comparison of the ND-2PA coefficient of the bulk and QW on the same scale. At a normalized energy of 1.02, Figure 4(a), α2,|| in a 10 nm QW is ~ 2× that of the bulk, and in a 5 nm QW is ~ 3.4× that of the bulk. The continuous increase of α2,|| from nth valence band to nth conduction band is due to the linear dependence of the intraband transition matrix elements on the inplane wave vector (k||), and the signature step-like features of QW’s, as seen in linear absorption spectra, is not observed.
At the same normalized energy, α2,⊥ in a 10 nm QW is ~ 36× that of the bulk. In the TM-TM case, the onset of 2PA occurs at ħωa + ħωb = Eg + Elh,12, rather than ħωa + ħωb = Eg + Elh,11 as in the TE-TE case, so the sum of the two-photon energies is normalized to Eg + Elh,12. The signature step like features of the TM-TM case are due to the selection rules and transition paths that allow intersubband transitions and the fact that the intraband transition matrix elements are independent of k||.
Since in the TM-TM case the first two-photon transition occurs at Eg + Elh,12, the range over which the pump and probe photon energies can vary depends on the QW width, and is limited to a range given by
This limitation in pump and probe photon energies is depicted in Figure 5, which describes different scenarios for α2,⊥.
As observed in Figure 5, as we decrease the photon energy ħωb, we observe a strong increase in α2,⊥ at the C1LH2 transition. The C2LH1 transitions are not observed for λb = 7 μm and 8 μm because ħωa + ħωb = Eg + Elh,12. This is also observed in Figure 5(b) as we decrease the QW width from 10 nm to 8 nm. As the confinement increases there is a strong enhancement of α2,⊥ at the C1LH2 transition, but the C2LH1 transition is not observed for 5.5 μm.
In ND-2PA for bulk semiconductors the enhancement is limited by the density of states and the intraband matrix element, both of which approach zero towards the band edge. In QW’s, however, for the TM-TM case, the enhancement is due to interband and intersubband resonances. These occur when ħωa nears Eg + Elh, 11 (interband) and when ħωb approaches Ec,2 – Ec,1 or Elh,2 – Elh,1 (intersubband). But as we approach these resonances, both the transition matrix elements and the density of states remain finite. The density of states is also large near the band edge, which leads to the enhancement of the ND-2PA over that for the bulk semiconductor.
The large enhancement of ND-2PA in QW’s is useful for determining the potential of QW’s for nonlinear optical devices. One of the applications is mid-IR detection using the strong enhancement of END-2PA in the TM-TM polarization case with the QW’s acting as the active region in a photodiode. In addition, inverted QW’s may provide two-photon gain and a possible path to a semiconductor two-photon laser.
Nondegenerate Enhancement of 3PA
Enhancement of three-photon absorption (3PA) is even larger than the orders of magnitude enhancement observed in 2PA since there is an additional photon to give intermediate state resonance enhancement. This provides an extra energy term in the denominator of the 3PA equivalent of Eq. (1). 3PA in semiconductors is well described by application of third-order perturbation theory to Kane’s (spin degenerate) four-band model [25-27]. The presence of the various bands, as compared to the simpler two parabolic band model, gives rise to quantum interference between the various pathways, yielding a rich structure to the D-3PA spectra of various zinc-blende semiconductors. Figure 6(a) shows the energy-momentum diagram of Kane’s band model, which includes conduction, heavy- and light-hole, split-off bands, along with examples of three different quantum pathways for 3PA. Using such a model at a nondegeneracy of ħωa/ħωb = 10 yields a predicted enhancement for ND-3PA of 2410× its degenerate value (for the same energy sum), compared to only 150× enhancement for ND-2PA. Experimentally, we have measured enhancement in GaAs via pump-probe techniques, the results of which are shown in Figure 6(b), compared to measurements of D-3PA . We observe significant enhancement using a pump photon energy of 0.16 eV (0.11Eg). Such large 3PA coefficients may be important to consider in the design of semiconductor based IR optical parametric devices, and will play a limiting factor in a two-photon semiconductor laser.
NONDEGENERATE NONLINEAR REFRACTION
Nonlinear Kramers-Kronig relations
The NLR dispersion can be calculated from the nonlinear absorption spectrum using Kramers-Kronig (KK) transformations by
where αNL includes 2PA as well as the stimulated Raman and optical Stark effects. Figure 7(a) illustrates these contributions for the degenerate case of ZnO. In spectral regions well below linear absorption resonances, the 2PA contribution dominates the dispersion of n2 which grows from low frequencies to a maximum near the onset of 2PA and then turns negative as the photon energy approaches the bandgap. The Raman and Stark effects give overall minor contributions in positive and negative ways, respectively, and only become resonant at the bandgap . The nondegenerate dispersion of n2(ωa; ωb) can also be predicted from the ND-2PA spectrum via KK calculations. Fig. 3(b) shows the predicted n2(ωa; ωb) dispersion by fixing ħωb at a certain percentage of Eg and varying ħωa from 0 to Eg. Owing to the large enhancement of α2(ωa; ωb), n2(ωa; ωb) is also positively enhanced near the 2PA edge with increasing nondegeneracy and becomes significantly larger than its degenerate counterpart. Particularly for the END case, ħωb → 0.1Eg, n2(ωa; ωb) becomes anomalously dispersive near the bandgap and rapidly switches sign from positive to negative over a very narrow spectral range.
Experiment and discussion
We apply our recently developed ultrafast beam deflection (BD) technique [19, 29, 30] to measure the ND-NLR of several direct-gap semiconductors. The experimental setup and beam geometries are shown in Figure 8(a). Beam deflection is an excite-probe technique using fs pulses, where an infrared excitation pulse creates an index gradient that follows its spatial Gaussian profile. The probe beam at another wavelength is focused ~ 5× smaller than the excitation and is spatially displaced to the Gaussian wings off the excitation’s center. The index gradient is nearly linear near where the gradient is maximized. Therefore, the probe is deflected by a small angle which can be measured using a quad-segmented detector by taking the difference of the energy falling on the left and right halves ΔE = Eleft – Eright and normalizing to the total energy E. The BD signal, ΔΕ/Ε, is proportional to n2(ωa; ωb).
Figure 8(b) shows agreement between the measured and theoretical nondegenerate n2(ωa; ωb) for ZnO, as compared to the calculated degenerate counterpart. We use a Ti:sapphire amplified laser system (Coherent Legend Elite Duo HE+) producing 12 mJ, ~40 fs (FWHM) pulses at a 1 kHz repetition rate to pump an optical parametric amplifier (TOPAS-HE) to produce the excitation pulses at 1440 nm. A portion of the excitation is then used to produce a white-light continuum (WLC) in a 5 mm thick sapphire crystal, from which narrow bandpass filters select the desired wavelength. A 1 mm thick fused silica sample is measured for calibration at all wavelength combinations. We observe an increase of n2(ωa; ωb) approaching the 2PA edge. Particularly, at a probe wavelength of 480 nm (0.81Eg) where 2PA starts, n2(ωa; ωb) is enhanced 7× over the zero frequency limit and 1.8× over its degenerate counterpart with this moderate nondegeneracy (ħωb = 0.26Eg). We have measured a much larger enhancement with excitation photon energy ħωb = 0.16Eg. The theoretical prediction using the KK transformation agrees with both the dispersion relation and the magnitude of nondegenerate NLR when using experimental best fit parameter of the two-band model .
In the END case, this enhanced n2 may impact numerous applications such as all-optical switching, and the rapid anomalous nonlinear dispersion will provide large modulation of a femtosecond pulse with narrow bandwidth centered near the zero crossing frequency which may enable other new applications such as nonlinear pulse shaping.
Nondegenerate nonlinearities in semiconductors are orders of magnitude greater than those in the degenerate case, and yet remain largely underexplored. This enhancement is due to intermediate state resonance enhancement, coupled with the semiconductor band structure, which provides the largest enhancements for extremely different photon energies. Such enhancements offer great potential for a host of applications, from gated detection of mid-IR pulses and 3D-IR imaging, to all-optical switching for communication, to the potential for two-photon semiconductor lasers.