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Abstract
We investigate the effects of nonlinear absorption of the pump beam on second-harmonic generation in GaAs nanowires. Our model includes nonlinear absorption of the pump and allows obtaining a self-consistent solution of the nonlinear Maxwell equations. First, we observe that SHG conversion efficiency can be limited from two-photon absorption and generated free-carriers depending on the pump intensity. Second, we show a method to modulate the SHG response by varying the pump beam intensity. We find that varying the pump intensity from 1 GW/cm2 up to 15 GW/cm2 can red-shift the SH peak wavelength up to 5 nm and modulate the conversion efficiency at a fixed pump wavelength up to 60%. Our results enable new applications of dielectric nanoresonators for nonlinear applications such as harmonic generation, optical switching, and all-optical ultrafast modulation.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nanophotonics based on high refractive index nanoparticles has gained huge momentum in this decade, enabled by the possibility to have both electric and magnetic Mie-type resonances in a single element [1,2]. Among others, one promising application of all-dielectric nanoresonators is nonlinear nanophotonics [2–5]. The combination of low or negligible optical absorption, high second- and third-order bulk nonlinear response, and strong enhancement of electric field in the nanoresonator volume has enabled a new class of nonlinear nanodevices with efficiencies that outclass those of their plasmonic counterpart [3–5]. For example, the conversion efficiency of second-harmonic generation (SHG) in AlGaAs nanodisks, [6–8], can be enhanced by more than 5 orders of magnitude compared to plasmonic nanoantennas [9].
In the undepleted pump approximation, the conversion efficiency in, e.g., second-order nonlinear processes as SHG, increases proportionally to the intensity of the pump beam. This is true as long as the materials involved are transparent at high intensities. However, at high intensities, nonlinear absorption processes such as two-photon absorption (TPA) may become significant in the semiconductors that are typically used for the nanoresonator [10]. On one hand, TPA limits conversion efficiency [11], on the other hand, it induces free carriers (FC) that may significantly modify the optical response of the system [12,13]. Recently, the optical generation of FCs has also been proposed, both theoretically and experimentally, as an efficient strategy to enable ultrafast all-optical control of the optical response of all-dielectric metasurfaces [13–18]. However, current demonstrations and numerical models are focused only on the control over the response of the nanoresonators at the pump frequency.
In this work, we investigate the modulation of SHG via variation of the pump beam intensity. We develop a full-wave model to account for complex permittivity variation in SHG induced by nonlinear absorption. In our numerical simulations, the responses at both pump and SH frequencies are coupled, in the assumption of undepleted pump approximation, to provide a self-consistent solution of the nonlinear Maxwell equations [19–21]. In contrast with previous models, we consider the effects of nonlinear absorption due to TPA and subsequent free-carrier phenomena. This allows describing and understanding the impact of TPA on the conversion efficiency and its ability to modulate the signal at the second harmonic (SH) frequency.
2. Second harmonic generation in GaAs nanowires
As an example, we take a single, infinitely long, GaAs nanowire in air. Here we consider a two-dimensional structure for simplicity but our model is rather general and it can be used with other materials (e.g. AlGaAs, InP, GaP, and halide perovskites) and structures such as semiconductor resonators [7,8,19,22–24], metasurfaces [25,26], hybrid metal-dielectric structures [27–30], and semiconductor gratings [31,32]. Our choice of GaAs is motivated by the fact that this III-V semiconductor exhibits strong second-order nonlinear response, strong TPA in the near-infrared, and a direct bandgap [10,33]. For our analysis, we adjust the nanowire radius to enhance SHG efficiency from a near-infrared pump. We evaluate the SHG conversion efficiency as PSH/PFF2 where PSH is the power emitted at the SH and PFF is the pump power incident on the nanowire. Since the analyzed structure is two-dimensional, all power values, e.g. PSH and PFF, are power per unit length (i.e. W/m). In our numerical simulations, the pump is a transverse-magnetic (TM) polarized plane wave with intensity I0, thus the pump power incident on the nanowire is PFF=I0d where d is the nanowire diameter. The procedure to solve the nonlinear scattering problem at the SH wavelength is described in the Appendix B. As shown in Fig. 1(a), for nanowires with a radius of 200 nm we observe a sharp SHG peak when the pump wavelength is tuned close to 1800 nm. As we can see from the E-field maps at the pump and SH wavelengths, we obtain enhanced SHG because the magnetic dipole (MD) resonance at the pump wavelength (see Fig. 1(b)) couples efficiently to a multipolar mode at the SH (see Fig. 1(c)). The peculiar feature of this SH mode is that the SH conversion efficiency peak has a narrow full width at half-maximum (FWHM) of about 12 nm.
Fig. 1. (a) SH conversion efficiency as a function of pump wavelength calculated for a nanowire with radius 200 nm. (b) and (c) show the normalized electric field amplitude at pump and second harmonic wavelengths of 1807nm and 903.5 nm, respectively. Panel (b) shows also the in-plane electric field vector at the pump wavelength represented by the white arrows inside the GaAs nanowire.
2.1 Permittivity modulation at the pump wavelength
We begin our analysis from the study of the effect of nonlinear absorption of the pump beam on the SHG conversion efficiency. In this step, the material properties at the SH wavelength are those of the unperturbed GaAs. As the pump wavelength is far from the bandgap energy, we consider two nonlinear absorption phenomena that are TPA and the subsequent free-carrier absorption (FCA). The energy that is absorbed from TPA is transferred to electrons in the semiconductor band structure and it generates free-carriers (FC). If the density of the generated FC is sufficiently high, this can perturb the semiconductor optical properties. The FC concentration can be calculated solving the continuity equation for electrons [34]
where β is the TPA coefficient, I is the intensity at the pump wavelength, τr is the carrier relaxation time, ωFF is the pump angular frequency, and ħ is the reduced Planck’s constant. At steady-state condition, the FC density is given by N=βI2τr/(2ħωFF). This condition implies that the FC generation is balanced by the FC relaxation, and it occurs when the pump pulse duration is comparable to the FC relaxation time, τr. The case of optical pulses with duration smaller or comparable to the FC relaxation time is discussed in the Appendix A. In the calculations, β is estimated from the dispersion of the GaAs permittivity, and it assumes a value of 6×10−11 m/W at the wavelength of 1800 nm, τr is assumed to be equal to 1.9 ps [16]. In Fig. 2(a) we show the generated FC density at steady-state from Eq. (1) in the nanowire as a function of the pump intensity. The absorption coefficient relative to FCA is given by [12,16,35]Fig. 2. (a) Absorption efficiency (left y-axis, continuous lines), free-carrier density (right y-axis, purple dashed line), and (b) Peak SHG conversion efficiency as a function of pump beam intensity. In both (a) and (b), the results shown by the blue line are obtained considering only TPA effect, with orange line TPA and FCA are considered, and with yellow line TPA, FCA, and FCD are considered.
We include both TPA and FCA at the pump wavelength in our model as an intensity-dependent increase in the GaAs extinction coefficient: k’=k0+ΔkTPA+ΔkFCA, where k0 is the extinction coefficient of GaAs at ground state [36], ΔkTPA is the TPA induced extinction and ΔkFCA is the FCA induced extinction. In our model ΔkTPA=βIλ/4π, where β is the TPA coefficient of GaAs, I is the pump intensity and λ is the wavelength. The FCA coefficient is obtained from Eq. (2) with the relation ΔkTPA=ΔαFCA 4π/λ.
The inclusion of the nonlinear absorption at the pump wavelength is accounted for in our model by following a two-step procedure. First, we solve the problem of the linear scattering of the nanowire at the pump wavelength for an incident plane wave of intensity I0. In this step, we use the unperturbed GaAs optical properties. The electric field inside the nanowire is used to estimate the TPA absorption and density of generated free-carriers, N, by applying Eqs. (1) and (2). Thus, to account for the spatial distribution of the mode in the nanowire, we map the variations in the material extinction as k’(x,y)=k0+(ΔkTPA+ΔkFCA)I(x,y)/IM, where k'(x,y) is the perturbed extinction, ΔkTPA and ΔkFCA are the extinction modulations associated with the maximum intensity IM in the nanowire and I(x,y) is the local intensity at the pump wavelength. Finally, with the perturbed material properties of the GaAs, we solve the nonlinear scattering problem at both pump and SH frequency.
Figure 2(a) shows the absorption efficiency, Qabs = Pabs/PFF, at different pump intensities from 1 GW/cm2 up to 100 GW/cm2. As can be seen, there is an increase in absorption efficiency as the intensity increases. This can be understood due to increased absorption because of TPA and FCA. Nonlinear absorption is mainly due to TPA at low pump intensities, while at high intensities, FCA becomes the most important contribution. This occurs because the induced absorption from TPA and FCA scales proportionally to I and I2, respectively.
Next, we investigate the effect of TPA and FCA of the pump on the SHG conversion efficiency. As can be seen in Fig. 2(b), the SHG conversion efficiency decreases as the pump intensity increases. As expected from the results in Fig. 2(a), FCA is the main limiting factor of the conversion efficiency as the intensity increases above 10 GW/cm2. Furthermore, as the intensity increases above 20 GW/cm2 we can observe a decrease in the conversion efficiency of more than 10% with respect to the value at 1 GW/cm2 that is principally due to FCA.
The excitation of free-carriers in the GaAs nanowire induces also a modulation of the refractive index of the material at the pump wavelength via the free-carrier dispersion (FCD) phenomenon [34]. The effect of FCD onto the refractive index modulation is obtained by describing free carriers as a Drude electron gas:
2.2 Permittivity modulation at the SH wavelength
In the proximity of the material bandgap energy, also inter-band mechanisms lead to changes in the material absorption and refractive index [12,16]. In this study, the SH wavelength is tuned in such region and thus two other phenomena need to be included: bandfilling and bandgap shrinkage [12]. The bandfilling effect causes a decrease in inter-band absorption for photon energies slightly above the bandgap. This occurs due to occupancy of the lowest conduction band states by the excited FCs so that electrons in the valence band require more energy than the nominal bandgap energy to be optically excited into the conduction band. Reversely, bandgap shrinkage signature is an increase in inter-band absorption for photons energies slightly below the bandgap. If the density of free-electrons that occupy the lowest energy states in the conduction band is high enough, the interaction between them results into a decrease in their energy. We assume the contribution of each effect to the perturbation of the material properties as independent, thus the perturbed refractive index and extinction of the material at the SH wavelength after optical excitation are given by
In the following, we study the modulation of SHG due to modulation of the material properties at the SH wavelength for pump intensities up to 20 GW/cm2. As observed in Fig. 2(b), this value of intensity results in a slight decrease of the SHG conversion efficiency and we can thus ignore modulation of the material parameter at the pump to better understand the physical mechanisms at play. We estimate the modulation of the SHG response in GaAs nanowires using the following procedure. First, we solve the problem of the linear scattering of the nanowire at the pump wavelength for an incident TM-polarized plane wave of intensity I0. The fields inside the nanowire are used to estimate the density of generated free-carriers, N, generated by TPA by applying Eq. (1). From N, we estimate the change in the GaAs refractive index and extinction at the SH wavelength using Eqs. (2)–(9). We account for the electric field distribution of the mode at the pump wavelength in the nanowire by mapping the modulation in refractive index and extinction coefficient at the SH wavelength with the same approach used previously at the pump wavelength. Finally, with the perturbed material properties of the GaAs, we solve the nonlinear scattering problem at the SH frequency.
Figures 3(a) and 3(b) show the SHG conversion efficiency, η, and SH absorption, αSH, as a function of the pump wavelength and for different levels of intensity. We observe that the position and height of the SHG peak changes as the pump intensity is varied. This is because the variation in the material permittivity moves the nanowire mode at the SH frequency. As we can observe in Fig. 3(a), the SHG peak red-shifts for pump intensities up to 5 GW/cm2 and then, as the pump intensity increases up to 20 GW/cm2, the SHG peak blue-shifts and reduces in amplitude. To understand these dynamics, we analyze the effect of each modulation phenomena to the SHG process. Figure 4(a) shows the refractive index variation as a function of the pump intensity. The single contributions to the total variation are shown in the graph. As we can observe, the bandfilling and the FCD effect cause a reduction of refractive index, while the bandgap shrinkage effect induces an increase of the same quantity. For low pump intensities (below 10 GW/cm2), the overall effect of all these contributions can mostly be attributed to the bandgap shrinkage, whereas at high intensities the bandfilling effect becomes dominant. These variations in refractive index are comparable to those observed in GaAs metasurfaces [16].
Fig. 3. Conversion efficiency (a) and SH absorption (b) as a function of pump beam intensity and wavelength.
Fig. 4. (a) Variation of refractive index as a function of pump beam intensity due to free-carrier dispersion (D), bandfilling (BF), bandgap shrinkage (BS), and all effects (Tot). (b) Optical contrast of the SH at different wavelengths as a function of the pump intensity.
Thus, the dynamics observed in Figs. 3(a) and 3(b) can be attributed to the weight of the different contributions of bandfilling and bandgap shrinkage as seen in Fig. 4(a). For pump intensities up to 5 GW/cm2, the counterbalancing effects of BF and BS cause a slight red-shift of the SHG peak. As the pump intensity increases from 10 GW/cm2 up to 20 GW/cm2, the magnitude of the bandfilling effect increases more rapidly than the bandgap shrinkage effect and thus the SHG peak experience a net blue shift that is in agreement with a decrease in refractive index. Moreover, as the pump intensity increases, the overall SH conversion efficiency decreases. This is due to an increase of the absorption at the SH frequency as it can be seen in Fig. 3(b). The SH absorption is defined here as ${\alpha _{SH}} = P_{SH}^{abs}/{I_0}d$, where $P_{SH}^{abs}$ is the power absorbed in the nanowire at the SH wavelength. This effect is due to the crossing of the SH peak with the nominal bandgap energy of GaAs.
To evaluate the SHG modulation at a constant pump wavelength, in Fig. 4(b) we show the optical contrast as a function of the pump intensity that is defined as
where η’ and η0 are the SHG conversion efficiencies of modulated and unperturbed GaAs nanowire. As we can see, the SHG conversion efficiency at the pump wavelength of 1808 nm, that corresponds with the peak SHG at the ground state, can be reduced by more than 60% for pump intensities up to 20 GW/cm2. Furthermore, due to the resonance blue-shift caused by the net reduction of refractive index, we can also observe an increase of about 40% in the SHG conversion efficiency at a pump wavelength of 1800 nm.3. Conclusion
In summary, we developed a numerical model for SHG that includes the modulation of the material optical properties (refractive index and absorption) due to nonlinear phenomena when high pump beam intensities are used. Our results unveil, first, that the SHG conversion efficiency may be limited by different phenomena depending on the pump intensity. Second, the induced effects on the SH enable all-optical control of the light emitted at the SH wavelength. We show that, by varying the pump intensity between 1 GW/cm2 and 15 GW/cm2, the SHG conversion efficiency peak wavelength can be red-shifted by about 5 nm and, for a fixed pump wavelength, the SHG conversion efficiency can be modulated by up to 60%. Furthermore, our model is rather general, and it can be extended to other nonlinear phenomena such as third and high-harmonic generation, sum-frequency generation, and four-wave mixing. Our model contributes to the advancement in the full understanding of the optical properties of nonlinear nanoantennas and metasurfaces [3,5] and it opens new perspectives for the development of optoelectronic devices based on tunable metasurfaces [17,18].
Appendix A: Extension of the model to ultrashort optical pulses
In the main paper, for the calculation of the generated free-carrier (FC) density, we assumed a temporal pulse duration comparable to or longer than the FC relaxation time (τr = 1.9 ps). However, our model can also be applied to the case of optical pulses with time duration much shorter than τr. For example, let us assume to have a pulse with a Gaussian temporal intensity profile I(t)=I0exp(-t2/T02) where I0 is the peak intensity and T0 is the 1/e temporal duration of the pulse. In the case when T0 << τr, FCs do not have enough time to recombine over the pulse duration and thus the last term on the right-hand side of the FC rate equation (Eq. (1) in the manuscript) can be ignored. Thus, integration of the continuity equation for electrons yields [21]
Appendix B: Modelling of the second-harmonic generation in GaAs
The nonlinear scattering problem is solved following the procedure developed in Ref. [19]. This is based on solving, first, the scattering problem at the pump wavelength and, then, the scattering problem at the SH wavelength driven by the nonlinear polarization induced by the electric field of the pump. For GaAs, the nonlinear polarization, P, for the SHG process is given by
where ɛ0 is the vacuum permittivity, $\chi _{ijk}^{(2 )}$ is the second-order nonlinear susceptibility, and E is the electric field at the pump wavelength. The value of $\chi _{ijk}^{(2 )}$ in our simulations is 300 pm/V [33]. The simulated nanowires are constituted by [100] GaAs with the longitudinal direction that is coincident with the crystalline z-axis. The incident electric field is linearly polarized and parallel to one of the crystalline axes in the transverse nanowire plane.Funding
Army Research Office (W911NF-18-1-0424); Ministero dell’Istruzione, dell’Università e della Ricerca (Law 232/2016); Office of Naval Research Global; RDECOM-Atlantic; Università degli Studi di Padova (STARS-StG-PULSAr).
Acknowledgments
This research was partially funded by the Rita Levi-Montalcini Program. L.C. acknowledges STARStG project PULSAR. Research of D. d. C. was sponsored by the RDECOM-Atlantic, US Army Research Office, and Office of Naval Research Global, and partly performed within Project “Internet of Things: Sviluppi Metodologici, Tecnologici E Applicativi”, cofunded (2018-2022) by the Italian Ministry of Education, Universities and Research (MIUR) under the aegis of the “Fondo per il finanziamento dei dipartimenti universitari di eccellenza” initiative (Law 232/2016).
Disclosures
The authors declare no conflicts of interest.
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