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We use our recently developed beam-deflection technique to measure the dispersion of the nondegenerate nonlinear refraction (NLR) of direct-gap semiconductors. The magnitude and sign of the NLR coefficient n2(ωa; ωb) are determined over a broad spectral range for different values of nondegeneracy. In the extremely nondegenerate case, n2(ωa; ωb) is positively enhanced near the two-photon absorption (2PA) edge and is significantly larger than its degenerate counterpart, suggesting applications for nondegenerate all-optical switching. At higher photon energies within the 2PA regime, n2(ωa; ωb) switches sign to negative over a narrow wavelength range. This strong anomalous nonlinear dispersion provides large phase modulation of a femtosecond pulse with bandwidth centered near the zero-crossing frequency. The measured nondegenerate dispersion closely follows our earlier predictions based on nonlinear Kramers-Kronig relations [Sheik-Bahae et. al, IEEE J. Quant. Electron. 30, 249 (1994)].
© 2016 Optical Society of America
1. Introduction
In semiconductors, bound-electronic nonlinear refraction (NLR), also known as the optical Kerr effect, is one of the dominant optical nonlinearities in the sub-gap regime, resulting in a refractive index change, Δn, directly proportional to the irradiance with coefficient n2 [1–3]. Owing to the fast response of bound valence electrons, the Kerr effect has been exploited for various ultrafast applications including Kerr-lens mode locking [4], optical limiting [5,6] and soliton propagation [7,8]. Another application of n2 of semiconductors is all-optical switching (AOS) [9–14], particularly in waveguide geometries such as nonlinear directional couplers [15], Mach-Zehnder interferometers [10,16] or resonant cavities [17–19], in which the transmission of the signal beam can be modulated by a control beam. Precise knowledge of the magnitude, sign, and dispersion of n2 of semiconductors is needed for the design and prediction of AOS devices. In general, NLR is related to nonlinear absorption (NLA) via a Kramers-Kronig (KK) transformation of the nondegenerate (ND) NLA spectrum [20,21]. This theory has been extensively used for prediction of n2 of various solids [22–24]. For a single beam at frequency ω, Z-scan measurements of this nonlinear refractive index, denoted by n2(ω; ω), have shown excellent agreement with theoretical predictions [25–28]. However, the ND-NLR, namely the refractive index change at frequency ωa due to the presence of a beam at frequency ωb, of coefficient n2(ωa; ωb), is much less explored, particularly for the extremely nondegenerate case (i.e. ħωa ħωb) and for spectral regions where 2PA is present.
We have observed orders of magnitude enhancement of two-photon absorption (2PA), of coefficient α2(ωa; ωb), using extremely nondegenerate photon pairs. Such enhancement has been utilized in applications including mid-infrared pulse detection [29], 3D IR imaging [30] and observation of two-photon gain [31]. Linked by KK relations, the nondegenerate enhancement of 2PA translates into enhancement of n2(ωa; ωb) [22,32], which suggests the use of semiconductors for applications such as nondegenerate AOS. The enhanced n2(ωa; ωb) lowers the switching energy required for various device geometries [33]. Traditional AOS devices with degenerate or nearly degenerate signal and control beams operate best below or near half the bandgap energy, Eg, to avoid loss from 2PA [15,19,34]. This excludes certain materials to be used for efficient Kerr effect based AOS at wavelengths of interest such as the telecommunication band. Nondegenerate operating schemes for AOS have the potential to overcome such limitations, as it tailors the nonlinearities, and, in principle, allows any signal wavelength below the bandgap to be used without the presence of 2PA. In the extremely nondegenerate case, the theory predicts that when ћωa → Eg, the positively enhanced n2(ωa; ωb) becomes anomalously dispersive and rapidly switches sign from a large positive value to a large negative value over a very narrow spectral range, i.e., ~10 nm [32]. This rapid anomalous nonlinear dispersion provides a large modulation for a femtosecond pulse with bandwidth centered near the zero crossing frequency. This may enable other applications such as nonlinear pulse shaping. However, it should be noted that the ND-2PA is also greatly enhanced in this same spectral region.
In this paper, we perform an extensive experimental study of the dispersion of ND-NLR for three direct-gap semiconductors (ZnO, ZnSe, and CdS) using our recently developed beam-deflection (BD) technique [35]. BD is a highly sensitive excite-probe technique capable of simultaneously measuring the absolute magnitude, sign, and temporal dynamics of both NLR and NLA. We previously applied BD to measure the nonlinearity of fused silica [35], the response functions of various liquid solvents [36,37] and NLR transients of molecular gases [38]. Here, using very different excitation and probe wavelengths, the dispersion of the ND-NLR is measured over the sub-gap region, along with ND-2PA or nondegenerate three-photon absorption (ND-3PA). We observe the resonance enhancement of NLR and the rapid anomalous dispersion near the bandgap. The results are compared to theoretical predictions from the KK transformation of [22,23]. We also define a nondegenerate figure of merit (FOM) for AOS in both ND-2PA and ND-3PA regimes and compare theory and experiment showing potential advantages with respect to the degenerate case.
2. Theory
As a consequence of causality in any linear system, a dispersion relation governs the real and imaginary parts of the frequency-domain complex response function via Hilbert transform pairs [39]. In optical systems, this principle results in the Kramers-Kronig transformation, which relates the real and imagery parts of the linear susceptibility [20]. To include nonlinear effects, we introduce nondegenerate perturbations on the refractive index ∆n = 2n2(ωa; ωb)Ib and absorption ∆α = 2αNL(ωa; ωb)Ib due to the presence of an excitation optical field at frequency ωb with irradiance Ib. After subtracting the linear components and dividing out Ib, we derive the KK transformation that relates ND-NLR with ND-NLA [22,23].
where αNL is the total NLA coefficient, c is the speed of light and P donates the Cauchy principle value of the complex integral. Note that the degenerate NLR n2(ωb; ωb) can be evaluated by performing the integral and then taking ωa = ωb. As discussed in [20–24], this method of using KK with the ND-NLA linearizes the problem since the change in absorption is linear in the probe irradiance.In semiconductors, the major physical mechanisms of αNL(ωa; ωb) consist of 2PA, electronic Raman and the optical (AC) Stark effect, which together can be viewed as the overall change of absorption of a material due to the presence of an excitation beam at ωb. As graphically illustrated in Fig. 1(a), ND-2PA describes simultaneous absorption of both photons of frequency ωa and ωb, which requires ћωa + ћωb ≥ Eg. Stimulated Raman scattering leads to simultaneous absorption and emission of photons at ωa and ωb, respectively, where |ћωa – ћωb| ≥ Eg is required. The optical Stark effect, also referred to as virtual band blocking [23], can be viewed as saturation of linear absorption for ћωa ≥ Eg, caused by virtual carriers generated by a strong beam at ωb. The nondegenerate NLA spectra of 2PA, Raman and AC Stark effect were formulated previously based on a two-parabolic band model using perturbation theory [22,23], from which the nondegenerate n2(ωa; ωb) dispersion is calculated via the KK transformation resulting in;
where Ep is the Kane energy parameter, and na and nb are the linear refractive indices at the respective frequencies. G2 is the dimensionless dispersion function as explicitly defined for each corresponding NLA mechanism in [22,23]. K is a material-dependent parameter, which is ~3100 cm GW−1 eV5/2 from an experimental best fit in [6,40], but may vary by a factor of ~2 from one semiconductor to another due to the simplicity in the assumption of the band structure [6,22,23]. This simple theory results in a generality, making it possible to predict NLR of various solids given the bandgap, Kane energy, linear index and optical frequencies.
Fig. 1 (a) Illustrations of nondegenerate NLA processes contributing to NLR in the sub-gap regime; (b) calculated degenerate NLR dispersion function G2 with total contribution (solid line) decomposed into 2PA, Raman and AC Stark contributions (dashed lines) [22]; (c) calculated ND-NLR (solid lines) and ND-2PA (dashed lines) coefficients of ZnO with excitation photon energy ћωb of 20% (2), 15% (3) and 10% (4) of the bandgap, as compared to the degenerate case (1).
In Fig. 1(b), the dispersion function G2 for degenerate (ωa = ωb) NLR is calculated from [22,23], which is independent of materials and therefore represents the universal nonlinear dispersion of direct-gap semiconductors. The total G2 is decomposed into contributions from 2PA, Raman, and AC Stark effect. Of these, 2PA contributes the most in the spectral region well below the bandgap, which grows from low frequencies to a maximum near Eg/2 (onset of 2PA) and then becomes anomalously dispersive and ultimately turns negative as ћω approaches the bandgap.
With highly nondegenerate photon pairs, the two-photon transition rate is greatly enhanced due to intermediate-state resonance enhancement (ISRE) of the small photon energy with the intraband self-transition and the larger photon energy with the interband transition [32,41,42]. This directly results in the enhancement of 2PA, e.g., 270 × enhancement measured in ZnSe using photons with a ratio of energies of 12 [32]. Based on the ND-2PA spectrum derived from a two-band model, the ND-NLR dispersion can be calculated via a KK transformation from Eq. (2). Figure 1(c) shows the calculated n2(ωa; ωb) and α2(ωa; ωb) of ZnO by varying ћωa from 0 to Eg with the excitation photon energy, ћωb, fixed at 20% (λb = 1.9 μm), 15% (2.5 μm) and 10% (3.8 μm) of the bandgap. The degenerate spectra of n2(ωa; ωa) and α2(ωa; ωa) are also shown for comparison. Linked by KK relation, n2(ωa; ωb) is positive for low ωa and grows to a maximum near the ND-2PA resonance (ћωa = Eg – ћωb). Then n2(ωa; ωb) becomes anomalously dispersive and ultimately switches to negative as ћωa approaches the bandgap. Associated with ISRE of ND-2PA, n2(ωa; ωb) is positively enhanced near ћωa = Eg – ћωb, and the increase of nondegeneracy (ћωa/ћωb) results in larger enhancement. In the extremely nondegenerate case, with ћωb only ~10% of Eg, the maximum n2(ωa; ωb) is enhanced by ~30 × over its zero-frequency limit (ћωa = 0), and ~9 × larger than the maximum for the degenerate case. However, the nondegenerate NLR changes sign and is not as enhanced as the ND-2PA. Also, the anomalous dispersion near the bandgap becomes very steep for extremely ND-NLR, and n2(ωa; ωb) can switch sign over a very narrow spectral range. For example, with ћωb = 0.1Eg, n2(ωa; ωb) changes from 11 × 10−5 cm2/GW to –11 × 10−5 cm2/GW by varying ℏωa from 3.1 eV (400 nm) to 3.2 eV (388 nm). Picosecond pulses with narrow bandwidth might be required to resolve such rapid anomalous dispersion experimentally. However, one can also directly observe this nonlinear dispersion using a linearly chirped femtosecond pulse with its bandwidth centered near the zero crossing frequency of n2(ωa; ωb). This is similar to time-wavelength spectroscopy [43–46], as discussed in detail in Section 4.
3. Nondegenerate beam deflection
To experimentally measure ND-NLR of semiconductors, we utilize our recently developed beam-deflection technique [35–38]. BD is an excite-probe technique where we use fs pulses along with a temporal delay line, as shown in Fig. 2(a). The strong excitation pulse at ωb creates an index change Δn(r) = 2n2(ωa; ωb)Ib(r) that follows its Gaussian spatial irradiance distribution, shown in Fig. 2(b). The probe pulse at ωa is focused to a spot size ~3-5 × smaller than that of the excitation beam and is spatially displaced to the Gaussian wings off the excitation’s center where the index gradient is nearly constant and maximized. This deflects the probe by a small angle which is measured using a position sensitive 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, ΔE/E, is directly proportional to n2(ωa; ωb), and the transmission change in E is proportional to the NLA (here 2PA). By choosing different wavelengths of excitation and probe, BD measures the magnitudes and signs of both nondegenerate NLR and NLA as well as their temporal dynamics. These measurements make possible the determination of the material FOM.
Fig. 2 (a) Illustration of the nondegenerate BD experiment [35]; (b) spatial irradiance distribution of the excitation beam (red) and overlapping geometry with the probe beam (blue) at the sample plane.
We use a commercial Ti:sapphire 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 generator/amplifier (TOPAS-800, Light Conversion) to generate the excitation pulses from the idler beam at a wavelength of λb = 2.3 μm with ~150 nm bandwidth (FWHM), which is then focused to a beam waist of wb = 210 μm (HW1/e2M) at the sample. To obtain the probe pulses with wavelength tunability, another beam from the Ti:sapphire laser system is focused into a 1 cm quartz cuvette filled with water to produce a white-light continuum (WLC), which is then spectrally filtered using narrow bandpass interference filters (10-25 nm, FWHM) to select wavelengths in the range of λa = 430-750 nm. The measured beam waist of the probe for different wavelengths at the sample varies from wa = 30-50 μm (HW1/e2M). The beam crossing angle is kept < 2°. The linear polarization of the excitation and probe beams was maintained using polarizers and for most experiments was set to be parallel to each other; however, we also used crossed polarizations for some cases where noted. The deflection of the probe induced by the excitation pulse is detected using a quad-segmented Si photodiode (OSI QD50-0-SD) placed in the far field after the sample. Both ΔE and E are detected via a lock-in amplifier (Stanford Research Systems, SR830) at the 285 Hz modulation frequency of an optical chopper in the excitation beam. While this technique is in principle absolutely calibrated [35,37], the relative errors between different wavelengths and samples can be reduced by comparing to a standard reference. In this work, all BD measurements of the semiconductors studied were conducted relative to the results obtained from a 1 mm thick fused silica sample using n2 = 2.5 × 10−16 cm2/W [47].
The bound-electronic nonlinearity leads to a nearly instantaneous response for NLR, which only occurs when the excitation and probe pulses are temporally overlapped within the sample. In extremely nondegenerate experiments, significant group velocity mismatch (GVM) between excitation and probe pulses causes temporal walk-off within the sample. Following Negres et. al. [48], we have developed a methodology to account for GVM in the BD analysis [37], which analytically solves the electric field distribution of the probe at the rear surface of the sample including the effects of NLR induced by the excitation. This allows us to fit BD data and extract n2(ωa; ωb). To do this, we first define the probe electric field to be Ea(r, z, t) = ½Ea(r, z, t)exp[i(kaz – ωat)]ê + c.c., where Ea is the complex field amplitude, r is the radial vector, ka = naωa/c is the wave number, and ê is the unit vector in the probe field direction. Within the slowly-varying-envelope approximation [2,3], the nonlinear wave equation is derived to describe the evolution of a weak probe propagating through the sample in z;
where ng,a is the group index of the sample at the probe frequency ωa. Here, the group velocity dispersion (GVD) and linear absorption are neglected. The thin sample approximation is also applied (external self-action [49]), meaning the sample thickness L is less than the Rayleigh ranges of both beams and the nonlinear induced phase distortion does not propagate within the sample to produce irradiance changes on either beam. In our experiments with fixed ћωb, α2(ωa; ωb) is nonzero only if ћωa ≥ (Eg – ћωb). For (Eg – ћωb) > ћωa ≥ (Eg – 2ћωb), the dominant NLA becomes ND-3PA, of coefficient α3(ωa; ωb, ωb), with two photons from the excitation and one from the probe absorbed [50]. Assuming temporally Gaussian pulses following [37,48,50] and analytically solving for Ea at the back surface of the sample, we obtain:where E0,a is the input probe field, T = (t – ng,az/c)/τb and Td are the normalized time and delayin the frame of the excitation pulse, respectively, and τa and τb are pulse durations (HW1/eM) for probe and excitation, respectively. ρ = (ng,a – ng,b)L/(cτb) is the GVM parameter, related to the temporal walk-off between excitation and probe pulses. Treating n2(ωa; ωb), α2(ωa; ωb) and α3(ωa; ωb, ωb) as invariant w.r.t. T, Eq. (4) can be further simplified to:The probe field at the detector can be calculated by numerically propagating Ea from Eq. (5) via Fresnel diffraction. Then ΔE and E are calculated by spatial and temporal integration of the probe irradiance profile at the detector, Ia(r,t) = cε0|Ea(r,t)|2/2, which we fit to the experimental results to extract both nondegenerate NLR and NLA coefficients.4. Results
Examples of measured BD signals (ΔE/E) with fits are shown in Fig. 3(a), where ZnO, ZnSe, and CdS, along with a 1 mm thick fused silica sample, are measured under identical beam overlapping geometries with excitation (ћωb = 0.54 eV) and filtered WLC probe (ћωa = 1.9 eV). The respective beam radii are 210 μm and 35 μm. For fused silica, the GVM between excitation and probe pulses is negligible (ρ = 0.03 [52]), therefore ΔE/E simply follows the cross correlation of the two pulses. For semiconductor samples, GVM becomes a significant factor in interpreting the measured signals. Particularly in ZnSe and CdS, the temporal walk-off between excitation and probe within the sample results in a broadened ΔE/E towards negative delays where the probe comes prior to excitation but travels at a slower group velocity such that the faster excitation pulse catches up and walks through the probe over a distance less than L. The corresponding fits (solid lines) for each material take into account GVM using Eqs. (4, 5) where ρ = 3.7, 8.9 and 10.2 for ZnO [53], ZnSe [54], and CdS [55], respectively.
Fig. 3 (a) Examples of measured BD signals (circles) from ZnO, ZnSe, CdS, and fused silica, along with fits (lines) using Eqs. (3-5) considering GVM; Measured n2(ωa; ωb) dispersion (red circles) of (b) ZnO, (c) ZnSe, and (d) CdS, compared to theoretical calculations for nondegenerate (solid lines) and degenerate (dashed lines) n2 using Eq. (2); Shaded region represents errors from the bandwidth of the excitation pulse; degenerate n2 data is from [23] (open squares) and [51] (black squares).
The measured dispersion of n2(ωa; ωb) for ZnO, ZnSe, and CdS is shown in Fig. 3(b-d) as red circles along with the theoretical calculations for both nondegenerate (red curve) and degenerate (black curve) cases. Errors in n2(ωa; ωb) are mainly from irradiance uncertainties as well as experimental noise in ΔE/E, and those in ћωa/Eg originate from the bandwidth of the probe. In the measurements of ZnO, shown in Fig. 3(b), ℏωb is fixed to ~16% of the bandgap, and the WLC probe is filtered at several wavelengths from 750 nm to 440 nm to map out the dispersion of n2(ωa; ωb). With ћωa/Eg < 0.68 (λa > 600 nm), both 2PA and 3PA are not possible, since ћωa + 2ћωb < Eg, and no NLA is observed, giving values of n2(ωa; ωb) close to the zero-frequency limit. For ћωa/Eg = 0.69 and 0.72 (570 nm and 550 nm), ND-3PA occurs, and we measure n2(ωa; ωb) along with α3(ωa; ωb, ωb). Significantly larger n2(ωa; ωb) is measured in the 2PA spectral region of ћωa/Eg = 0.82 – 0.90 (λa = 480 – 440 nm) owing to the ISRE. Near the onset of 2PA (i.e., at λa = 480 nm), n2(ωa; ωb) is measured ~6 × larger than the degenerate n2 (black squares) at 1.06 μm measured via Z-scan near its zero-frequency limit [23]. The greatest nondegenerate enhancement in ZnO is measured at λa = 440 nm, where n2(ωa; ωb) is more than one order-of-magnitude larger than the degenerate n2 (black squares) at 1.06 μm measured via Z-scan near its zero-frequency limit [23] and ~3 × larger than the maximum of the calculated degenerate n2. But it should be noted that at the wavelength where n2(ωa; ωb) reaches its maximum enhancement, ND-2PA is also greatly enhanced. Using K = 2400 cm GW−1 eV5/2 in Eq. (2), gives good agreement with the measured ND-NLR. Additionally, the dependence of n2(ωa; ωb) on the polarization of the probe with respect to that of the excitation is also investigated for selected probe wavelengths. For λa = 550 and 600 nm, we measure that with perpendicularly polarized waves, n2(ωa; ωb) becomes 3.2 × and 2.7 × smaller respectively than for the parallel polarization case.
For ZnSe, where ћωb ~20% of the bandgap, the increase of n2(ωa; ωb) is also observed as ћωa increases towards the 2PA resonance as shown in Fig. 3(c). For longer wavelengths (λa = 750 – 600 nm), ND-NLR is accompanied by ND-3PA. The data of degenerate n2 dispersion (black squares) of ZnSe is taken from previous measurements using WLC Z-scan [51] for comparison. A remarkable agreement is found between theoretical calculations and experimental results from both BD and WLC Z-scan measurements when using the same K = 3900 cm GW−1 eV5/2, indicating the consistency of the theory in both degenerate and nondegenerate cases. This K value is similar to the previously reported K = ~4000 as the experimental best fit for ZnSe [22,23]. For λa = 600 and 750 nm, with perpendicularly polarized waves, n2(ωa; ωb) is measured to be 3.7 × and 2.4 × smaller respectively than the values measured with parallel polarizations. Note that the theoretical model assuming a two-parabolic band structure cannot account for this effect [22], but these ratios may be used to compare to the theory considering a more realistic band structure such as Kane’s model [42,50].
As shown in Fig. 3(d), n2(ωa; ωb) of CdS is measured with ℏωb ~23% of the bandgap. Besides the nondegenerate enhancement of n2, the anomalous dispersion after the maximum, as well as the associated sign change, are also resolved with a probe wavelength very close to the bandgap (λa = 570 nm and 550nm). Particularly at λa = 570 nm near the zero-crossing frequency, a small value of n2 (2.2 × 10−5 cm2/GW) is measured in the presence of a large 2PA (13 cm/GW) background, indicating the excellent ability of BD for separately measuring NLR and NLA. Such a separation would be very difficult using other techniques, including Z-scan. All the measured nondegenerate NLR and NLA coefficients for ZnO, ZnSe and CdS are summarized in Table 1.
Table 1. Measured nondegenerate NLR and NLA coefficients for semiconductors with λb = 2.3 μma
Limited by the large bandwidth of femtosecond pulses, the frequency resolution is insufficient to directly measure the absolute value of n2 near the bandgap in the extremely nondegenerate case (e.g. ZnO in Fig. 3(b)), as the theory predicts a rapid anomalous nonlinear dispersion followed by a sign change in a very narrow spectral range. Here we utilize a methodology similar to time-wavelength spectroscopy [43–46] to resolve this strongly dispersive NLR. In ZnO measurements with the same excitation of 2.3 μm, the WLC probe is filtered at a center wavelength of 430 nm to 10 nm (FWHM) bandwidth, which covers the zero-crossing frequency of n2(ωa; ωb) predicted by theory (422 nm). Higher and lower frequency components will see a negative and positive n2, respectively, as shown in the inset of Fig. 4(a). In this experiment, the probe pulse is assumed to be linearly up-chirped, with lower frequencies arriving before higher frequencies, and the pulsewidth τa is determined to be ~500 fs from the cross-correlation in fused silica using the same BD setup. As shown in Fig. 4(b), the measured ΔE/E of ZnO is greatly different from the signals shown in Fig. 3(a), where the dispersion of n2(ωa; ωb) over the bandwidth of the probe is negligible. Around zero delay, the excitation temporally overlaps the rising edge of the probe pulse (lower frequency) at the front surface of the sample, giving a positive signal. A negative signal is observed at a delay of ~– 2.5 ps, where the excitation pulse overlaps with the falling edge of the probe (higher frequency) at the back surface of the sample. For delays in between, the excitation and probe completely walk through each other within the sample, giving an averaged small positive signal from all frequency components.
Fig. 4 (a) Converted n2(T) assuming linear chirp at the front (black) and back (red) surface of the sample compared to the initial probe pulsewidth (blue dashed). Inset is the theoretical n2(ωa; ωb) dispersion (black solid) of ZnO (same as Fig. 3(b)) relative to the probe bandwidth (blue solid) at λa = 430 nm. (b) measured ΔE/E (black circles), compared to theoretical predictions based on chirping conditions at the front (red dashed) and back (red solid) surface of the sample along with the averaged curve (blue solid).
To investigate this transient NLR quantitatively, the n2(ωa; ωb) dispersion is converted to a time-domain quantity via the instantaneous probe frequency, which is defined as
where ω0,a is the carrier frequency of the probe (here at 430 nm), τG is the pulsewidth of the probe field (HW1⁄eM), a is the chirping parameter, which is estimated as 14 for the initial input pulse based on the measured pulsewidth and spectrum. Due to GVD, the probe pulse is chirped further upon propagating through sample, which gives a = 20.5 at the back surface of sample, as calculated using the linear dispersion from [53]. n2(T) is calculated by substituting Eq. (6) into Eq. (2), which is plotted in Fig. 4(a) for the two different chirps at the front (red dashed) and back (red solid) surface of the sample, where the temporal profile of the probe (blue dashed) is also plotted for comparison. Clearly, Eq. (5) is not applicable for predicting ΔE/E in this experiment because n2(ωa; ωb) in Eq. (4) cannot be assumed to be constant in the integral of T. But the variation of α2(ωa; ωb) is negligible within the same spectral (temporal) range. For extremely ND-NLR, this anomalous dispersion is very steep within the instantaneous frequency (or temporal profile) of the probe, so that we can take a first-order approximation to obtain a linear dependence of n2 on ωa(t) (or T), giving n2(ωa; ωb) = n2(T) = n2,0 + n2,1T. By substituting this into Eq. (4), the output probe field with ND-2PA becomes,where n2,0 and n2,1 are derived from a linear fit of n2(T) at T = 0. Figure 4(b) shows theoretical predictions of ΔE/E calculated from Eq. (4) for an initial probe chirp of a = 14 (red dashed) and that after the sample of a = 20.5 (red solid). The averaged curve between the two cases (blue solid) leads to a better agreement of the shape of the NLR transients. Note an even better agreement can be achieved by varying n2,0 and n2,1 as fitting parameters. Therefore, we confirm the theoretical prediction of the anomalous dispersive sign switching of extremely nondegenerate n2 near the bandgap, which provides a large nonlinear modulation of a femtosecond pulse.5. Nondegenerate FOM of AOS
Owing to the large enhancement of NLR, the nondegenerate operating scheme can be implemented into Kerr-effect based photonic devices such as AOS, where the signal beam, with ℏωa close to the bandgap, can be modulated with an infrared control beam at ℏωb. With a larger n2(ωa; ωb), the switching irradiance of the control beam, Isw, can be significantly reduced for a required nonlinear phase change ∆ϕ = 2kan2(ωa; ωb)IswL. In practice, the nonlinear loss terms imposed by ND-2PA and ND-3PA should satisfy 2α2(ωa; ωb)IswL < 1 and 3α3(ωa; ωb, ωb)Isw2L < 1, respectively [33], from which FOMs are defined by the absolute value of the ratio of ∆ϕ and loss terms as
where a factor of 2 is added to be consistent with other definitions (i.e., FOM2PA = |Re(χ(3))/Im(χ(3))|) [56]. For different AOS geometries, both FOMs need to be larger than 4π for a Mach-Zehnder (MZ) interferometer [33], 8π for a nonlinear directional coupler (NLDC) [21,23] and 2/√3 for a Fabry-Perot (FP) filter [34,57]. Table 2 summarizes measured nondegenerate FOM2PA and FOM3PA for ZnO, ZnSe, and CdS based on the definitions of Eqs. (8) and (9). It should be noted that the most enhanced n2, measured with a probe at 440 nm, 550 nm, and 600 nm in ZnO, ZnSe, and CdS respectively, do not result in the largest FOM. This is because in the spectral region where nondegenerate enhancement of NLR is maximized, the enhancement of 2PA is even larger, leading to a smaller FOM2PA [32]. The largest FOM values measured in the 2PA spectral region are either near the onset of 2PA (i.e. 480 nm in ZnO and 570 nm in ZnSe) or near the bandgap with a negative n2 (i.e. 550 nm in CdS).Table 2. Measured nondegenerate FOM in 2PA and 3PA spectral region a, b
As an example, Fig. 5 shows measured FOM2PA and FOM3PA of ZnSe for both degenerate and nondegenerate cases, along with theoretical curves calculated using Eq. (2) and α2(ωa; ωb) spectra from the same two-band model [22,23]. The degenerate data (black squares) are calculated based on the experimental results of n2 and α2 from [51]. Within the 2PA region, ℏωa/Eg > 0.5, the degenerate FOM2PA is too small to meet the criteria of all the AOS geometries, except ℏωa/Eg > 0.9 where loss from band-tail absorption increases. This essentially limits the operating wavelength of ZnSe for AOS to below half of the bandgap, λa > 928 nm. Such limitations may be overcome with nondegenerate enhancement of NLR. In agreement with theory, the nondegenerate FOM2PA (red circles) near the onset of ND-2PA (ℏωa/Eg ~81%) is measured to be ~7 × larger than the degenerate FOM2PA at the same wavelength of λa = 570 nm. The improvement over the degenerate case can become even larger with higher nondegeneracy (e.g. ~50 × in ZnO at λa = 480 nm). For 0.6 < ℏωa/Eg < 0.8, ND-3PA becomes the dominate loss mechanism in ZnSe as ND-2PA no longer occurs since ℏωa + ℏωb < Eg. Based on Eq. (9), FOM3PA is inversely proportional to Isw. In Fig. 5, nondegenerate FOM3PA of ZnSe is calculated from the measured parameters in Table 2 using two different irradiances, Isw = 10 GW/cm2 (green triangles) and Isw = 1 GW/cm2 (blue stars), where the smaller Isw results in significant improvement of FOM3PA over the degenerate case in the same spectral region (i.e. ~340 × at λa = 700 nm) and satisfies the criteria of all AOS geometries.
Fig. 5 Measured FOM of ZnSe in the presence of 2PA for degenerate (black squares) and nondegenerate (red circles) NLR, as compared to theory (solid lines) [22,23]; FOM in the presence of ND-3PA are based on Isw = 10 GW/cm2 (green triangles) and Isw = 1 GW/cm2 (blue stars). The degenerate data is from [51]. The minimum requirements for AOS geometries of a Mach-Zehnder (MZ) interferometer, nonlinear directional coupler (NLDC) and Fabry-Perot (FP) filter are included for comparison.
6. Conclusions
We have presented experimental measurements on the dispersion of the nondegenerate nonlinear refraction of ZnO, ZnSe, and CdS using our recently developed beam-deflection technique. With various values of nondegeneracy, the ND-NLR coefficient n2(ωa; ωb) is measured over a broad spectral range, along with the corresponding ND-2PA and ND-3PA coefficients. To extract the values of these nonlinear coefficients, GVM has been taken into account in the analysis of the data. Using an excitation pulse at ℏωb = 0.54 eV, a positively enhanced n2(ωa; ωb) is measured in all direct-gap semiconductors with a probe ℏωa near the ND-2PA resonance. In ZnO, with ℏωb at 16% of the bandgap, more than one order-of-magnitude enhancement of n2(ωa; ωb) over its zero-frequency limit is measured. In CdS, the sign change of n2(ωa; ωb) near the bandgap is also resolved. We found good agreement between experimental results and our earlier theoretical predictions using nonlinear Kramers-Kronig relations [20–23,25], in both nondegenerate enhancement and spectral dispersion shapes of n2. We therefore confirm the theory over an extremely large range of photon energy ratios, allowing predictions of the best operating wavelengths of semiconductor devices engineered to use cross-phase modulation.
Based on time-wavelength mapping of a linearly up-chirped probe pulse, we also demonstrate the anomalously dispersive behavior of extremely nondegenerate n2(ωa; ωb) near the bandgap in ZnO, which rapidly switches sign from the enhanced positive n2 to a large negative value over a very narrow spectral range. This strongly dispersive NLR has potential to provide not only large but ultrafast modulation of a femtosecond pulse with bandwidth centered near the zero-crossing frequency to enable new applications such as nonlinear pulse shaping.
This enhancement of NLR also suggests the possibility of all-optical switching based on the nondegenerate Kerr effect. A larger n2(ωa; ωb) significantly reduces the switching energy for the required phase changes for all AOS devices. Also, the magnitude, sign, and dispersion of NLR can be significantly tailored with nondegenerate beams by selecting the appropriate excitation wavelength, which can lead to advantages over its degenerate counterpart. Operating in a nondegenerate scheme with a control beam at a (mid-) infrared wavelength, we can essentially avoid 2PA in the signal beam, leaving a relatively smaller nondegenerate 3PA as the dominant nonlinear loss, resulting in a larger FOM3PA. Approximately 340 × improvement of the FOM3PA over the degenerate case at the same signal wavelength has been measured in ZnSe when using < 1 GW/cm2 switching irradiance.
Funding
Air Force Office of Scientific Research (MURI grant FA9550-10-1-0558); National Science Foundation (NSF) (ECCS-1202471 and ECCS-1229563); Army Research Laboratory (W911NF-15-2-0090).
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