2.1

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 [16] and experimentally [20]. For a two parabolic band model, the 2PA coefficient has been shown to follow

where

and K = 3100 cm GW^-1 eV^5/2, E_p is the Kane energy parameter, n_a and n_b are the linear refractive indices, and E_g is the bandgap energy. This enhancement has been measured for nondegeneracies as large as ħω_a/ħω_b ~ 10 showing enhancement factors of up to 270 in ZnO over the degenerate case [20]. 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 [7]. This detection scheme has recently been used for 3D IR imaging [9], 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 f_c and f_v 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 [21]. END-2PG holds potential for the creation of a two-photon semiconductor laser.

Figure 2.

(left) E-k diagram showing 2PG. (right) 2PG spectra in GaAs for various nondegeneracy factors for GaAs at T = 20 K, N = 2×10¹⁸ cm^-3 for (black) degenerate, (red) ħω_b = 0.2E_g, and (blue) ħω_b = 0.1E_g.

2.2

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 [17]. 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).

Figure 3.

(a) Sketch of a QW structure with polarization for TE and TM cases. (b) Interband-intraband transitions for two TE polarized photons, and (c) interband-intersubband transitions for two TM polarized photons.

For the ND-2PA, an analytic expression is derived using second order perturbation theory both for TE-TE and TM-TM polarizations [17], which assumes an infinite QW. The ND-2PA coefficient for the TE-TE polarized case is

with

where ζ_ν = (ħω_a + ħω_b – E_g)/E_v,11 defines how far the two-photon transition energy is above the bandgap normalized to the linear absorption edge, E_v,11 = E_c,1 + E_v,1 = ħ²π²/(2μ_ν,_⊥d²), Nv = Int(√ζ_ν) is the number of two-photon transitions between the valence and conduction subbands, α is the fine structure constant, m₀ is the electron mass, μ_ν are the reduced effective masses of the bands, and d is the well width. For TM-TM polarized light,

where

and

The ND-2PA coefficients may be measured experimentally by pump-probe where the transmission of the weak probe beam at ω_a is monitored in the presence of the strong pump beam at ω_a. 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 ħω_b = 0.12E_g (λ_b = 7.5 μm) and varying the probe photon energy ħω_a. For bulk, α₂ is plotted against the sum of photon energies normalized to the bandgap (E_g), whereas for the QW, the energy is normalized to the respective one-photon absorption threshold (E_g + E_hh,11 for TE-TE and E_g + E_lh,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), α_2,|| 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 α_2,|| from n^th valence band to n^th 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.

Figure 4.

ND-2PA coefficient in bulk GaAs and GaAs QW’s of different widths: (a) TE-TE case and (b) TM-TM case.

At the same normalized energy, α_2,⊥ in a 10 nm QW is ~ 36× that of the bulk. In the TM-TM case, the onset of 2PA occurs at ħω_a + ħω_b = E_g + E_lh,12, rather than ħω_a + ħω_b = E_g + E_lh,11 as in the TE-TE case, so the sum of the two-photon energies is normalized to E_g + E_lh,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 E_g + E_lh,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

and

This limitation in pump and probe photon energies is depicted in Figure 5, which describes different scenarios for α₂,⊥.

Figure 5.

ND-2PA coefficient α_2,⊥ in a GaAs QW (a) of width 10 nm for various pump photon energies, and (b) of different widths for a constant pump photon energy ħω_b corresponding to a wavelength of 5.5 μm.

As observed in Figure 5, as we decrease the photon energy ħω_b, we observe a strong increase in α_2,⊥ at the C1LH2 transition. The C2LH1 transitions are not observed for λ_b = 7 μm and 8 μm because ħω_a + ħω_b = E_g + E_lh,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 α_2,⊥ 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 ħω_a nears E_g + E_{lh, 11} (interband) and when ħω_b approaches E_c,2 – E_c,1 or E_lh,2 – E_lh,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.

2.3

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 ħω_a/ħω_b = 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 [28]. We observe significant enhancement using a pump photon energy of 0.16 eV (0.11E_g). 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.

Figure 6.

(a) E-k diagram of Kane’s (spin-degenerate) 4-band model showing three examples of quantum pathways for 3PA. (b) Comparison of (circles) measured data of GaAs to (solid curves) theory (×3) for both (black) degenerate and (red) nondegenerate (with ħω_b = 0.16 eV) [28].

3.1

Nonlinear Kramers-Kronig relations

The NLR dispersion can be calculated from the nonlinear absorption spectrum using Kramers-Kronig (KK) transformations by

where α_NL 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 [16]. The nondegenerate dispersion of n₂(ω_a; ω_b) can also be predicted from the ND-2PA spectrum via KK calculations. Fig. 3(b) shows the predicted n₂(ω_a; ω_b) dispersion by fixing ħω_b at a certain percentage of E_g and varying ħω_a from 0 to E_g. Owing to the large enhancement of α₂(ω_a; ω_b), n₂(ω_a; ω_b) is also positively enhanced near the 2PA edge with increasing nondegeneracy and becomes significantly larger than its degenerate counterpart. Particularly for the END case, ħω_b → 0.1E_g, n₂(ω_a; ω_b) becomes anomalously dispersive near the bandgap and rapidly switches sign from positive to negative over a very narrow spectral range.

Figure 7.

(a) Theory of degenerate n₂ of ZnO with total NLR (solid line) decomposed into 2PA, Raman and AC Stark mechanisms (dash line) [16]; (b) calculated nondegenerate enhancement of NLR of ZnO (solid lines), for various pump energies, compared to the degenerate case (dash line) [18].

3.2

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 = E_left – E_right and normalizing to the total energy E. The BD signal, ΔΕ/Ε, is proportional to n₂(ω_a; ω_b).

Figure 8(b) shows agreement between the measured and theoretical nondegenerate n₂(ω_a; ω_b) 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(ω_a; ω_b) approaching the 2PA edge. Particularly, at a probe wavelength of 480 nm (0.81Eg) where 2PA starts, n₂(ω_a; ω_b) is enhanced 7× over the zero frequency limit and 1.8× over its degenerate counterpart with this moderate nondegeneracy (ħω_b = 0.26E_g). We have measured a much larger enhancement with excitation photon energy ħω_b = 0.16E_g. 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 [16].

Figure 8.

(a) The nondegenerate beam deflection measurement setup including excitation and probe overlap geometry and position sensitive quad-segmented photodetector. (b) Experimental data of n₂(ω_a; ω_b) of ZnO (square), along with a theoretical calculations of degenerate (dash curve) and nondegenerate (solid curve) NLR.

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.