Effects of two-photon absorption on terahertz radiation generated by femtosecond-laser excited photoconductive antennas

Terahertz (THz) radiation can be generated more efficiently from a low-temperature-grown GaAs (LT-GaAs) photoconductive (PC) antenna by considering the two-photon absorption (TPA) induced photo-carrier in the photoconductor. A rate-equation-based approach using the DrudeLorentz model taking into account the band-diagram of LT-GaAs is used for the theoretical analysis. The use of transform-limited pulses at the PC antenna is critical experimentally. Previously unnoticed THz pulse features and anomalously increasing THz radiation power rather than saturation were observed. These are in good agreement with the theoretical predictions. The interplay of intensity dependence and dynamics of generation of photoexcited carriers by single-photon absorption and TPA for THz emission is discussed. ©2011 Optical Society of America OCIS codes: (160.5140) Photoconductive materials; (300.6495) Spectroscopy, terahertz; (320.0320) Ultrafast optics References and links 1. D. H. Auston, K. P. 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Introduction
Since Auston's group generated picosecond electromagnetic pulses from photoconductive (PC) antennas by ultrafast laser pulses several decades ago [1], such coherent terahertz (THz) radiation has been studied extensively and generated by various methods, e.g., ultrafast switching of PC antennas, optical rectification in nonlinear crystals, surge-current at the semiconductor surface, charge oscillations in semiconductor quantum wells, coherent excitation of optical phonons, etc.Among them, the technique of generation and detection of THz radiation using the PC antenna is well-established and widely used for various applications [2].
The output THz power from a PC antenna is strongly correlated with the photoexcited carrier density and saturated by when it is pumped harder.It is widely accepted that the saturation phenomena is due to the screening effect by photoexcited carriers accelerated in the PC gap [3][4][5].However, this model only considered the single source of generated photocarriers.Some experimental data hinted a more complicated picture, however, as peak THz pulse amplitudes were showing a decreasing trend at higher excitation densities [6][7][8].It is well-known that extra photoexcited carriers can be generated by two-photon absorption (TPA) process [9].To our knowledge, there is no previous work taking into account both single-photon absorption (SPA) and TPA processes for generation of photoexcited carriers and the radiation of THz pulses from PC antennas.
In this paper, we consider low-temperature-grown GaAs (LT-GaAs), of which the band diagram has been thoroughly investigated [10][11][12] as the PC.Using the rate-equation approach and the Drude-Lorentz model [13][14][15], we present detailed calculations of THz radiation generated from LT-GaAs antennas and compare with experiments.It will be shown that the THz power radiated from PC antennas will depend on the competition between TPA and SPA.Because TPA effect converts two photons to electron-hole pairs, this process has lower conversion efficiency for photoexcited carriers, comparing with SPA effect.By virtue of this mechanism, the peak amplitude of the THz field and total power of the THz radiation will begin to saturate.However, optically induced carriers still can be excited to the conduction band by TPA for sufficiently optical excitation fluence.We propose and demonstrate that this mechanism can help alleviate the saturation of THz radiation power if SPA only is considered.Besides, possibility of enhancement of higher-frequency spectral components of THz radiation by TPA is also discussed.

Theoretical analysis
The approach we adopted is based on the rate-equation analysis using the Drude-Lorentz model [13][14][15] of semiconductors.Specifically, we consider the LT-GaAs system [10][11][12], which has many interesting properties, including short carrier lifetime, high mobility, and high breakdown voltage.As a result, LT-GaAs are widely in ultrafast photonic devices.Previously, Benjamin and associates observed TPA effect in LT-GaAs and determined key parameters such as TPA coefficient and carrier trapping time by the optical pump-probe techniques and the Z-scan measurements [10][11][12].
Following Benjamin et al. [10], the band diagram of LT-GaAs is shown in Fig. 1.The symbol in Fig. 1 will be explained in the next paragraph.Optical pump pulses with energies above the band-gap will excite carriers to the bottom of the conduction band.These photoexcited carriers can be trapped in the mid-gap states and then relax back to the valance band or be re-excited to upper states in the conduction band.Meanwhile, the valence electrons can also be excited to the upper states of the conduction band by the TPA effect.The carriers in the upper states can relax to the bottom of the conduction band and then are trapped by the mid-gap states.
(1 ) (1 ) where N is the number of carriers at the bottom of the conduction band, N 0 is the saturation carrier density, N T is the number of carriers in the mid-gap states contributing to induced absorption, N T0 is saturated number density of trapped carrier in the mid-gap states, n is the number of carries excited to upper states in the conduction band, hν is the photon energy of the incident light, σ BB is the cross section for band-to-band transitions, σ TB is the cross section for the band-to-trap transitions, β is the TPA coefficient, τ 1 is the carrier trapping time from the bottom of the conduction band to the mid-gap states, τ 2 is the trap emptying time, τ 3 is the hot carrier relaxation time from the upper states of the conduction band to the bottom of the where v(t) is the average velocity of the carrier, E loc (t) is the local electric field, τ s is the momentum relaxation time, e is the charge of an electron, and m* is the effective mass of the electron.The local electric field is screened by the polarization induced by the space charges, ( ) ( ) , where E b is the applied bias electric field on the PC antenna, P sc is the polarization induced by the separation of the hole and electron, η is the geometrical factor for the PC antenna, and ε is the dielectric constant of the PC.The temporal evolution of polarization is then where τ r is the recombination time of carrier, and J(t) is the current density induced by photoexcited carriers of which the time derivative gives rise to the radiated THz field for the Hertzian dipole.Most of the parameters used in the calculation were taken from results of Benjamin's group [10][11][12].Specifically, N 0 = 5×10 18 cm −3 , N T0 = 5×10 19 cm 2 , σ BB = 2.5×10 −14 cm 2 , σ TB = 5×10 −16 cm 2 , and β = 35 cm/GW, as well as the relevant carrier relaxation times given by τ 1 = 1 ps, τ 2 = 1.5 ps, τ 3 = 100 ps, τ 4 = 300 fs.According to Jepsen and associates, τ r is much longer than the carrier lifetime and ranging from 1 to 100 ps [13].To simplify the calculation, we set τ r ≈10 ps, which is larger than both τ 1 (1 ps) and τ 4 (300 fs).The other parameters of the PC antenna needed include τ s = 30 fs,, E b = 10 4 V/cm, η = 900, refractive index of LT-GaAs = 3.6, and m* = 0.067m 0 .The excitation wavelength is 800 nm and the pulse duration is 35 fs.By changing the focused laser spot size on the PC antenna, we consider two regimes of photo-excitation.In the first regime, carriers in the valance band can only be excited to the bottom of the conduction band (SPA effect only).In the second regime, carriers can also be excited to the upper states of the conduction band, i.e., both SPA and TPA effects are considered.In Fig. 2(a) and Fig. 2(b), we show simulation results for relatively low excitation powers, corresponding to the case of 10μm-spot size in our experimental configuration.It can be seen that peak amplitudes of the THz fields and total radiated powers first increase linearly with pump power initially at lower power densities and then exhibit saturation behavior at higher excitation densities.If the TPA effect is considered, THz peak amplitude and power saturate faster.Because the conversion of photons into carriers by the TPA effect is much less efficiently than the SPA effect, the simulated result is reasonable.Figure 2(c) and Fig. 2(d) show the peak amplitude and total power as the spot size is further focused down to 5μm.Significantly, if the TPA effect is included, the THz peak amplitudes and total powers are expected to increase and exceed values predicted for photoexcitation by SPA only at higher pump powers.This anomalous increase can be attributed to that the carriers still can be excited to the conduction band by TPA effect.Further, we find a similar tendency of nonsaturation or anomalous increase in THz power for various values of geometrical factor η from 1 to 1000.Increasing geometric factor η corresponds to decreasing screening field or higher saturation intensity [13].This observation further corroborates our contention that the anomalous increase of THz is due to the TPA effect.

Experimental methods and results
Experimentally, we employ a typical antenna-based THz time-domain spectroscopy (THz-TDS) which has been described in detail previously [16].The femtosecond laser is a modelocked Ti:sapphire laser (Spectra Physics, Tsunami) with an average power of ~450mW at a repetition rate of 82 MHz producing ~35fs.Its central wavelength is 800nm and the bandwidth (full-width-at-half maximum; FWHM) is around 43nm.To agree with the assumptions of our simulations, we would like to ensure that the optical pulse incident on the PC antenna is transform-limited.This is realized by adopting an adaptive pulse shaper consisting of a pair of gratings (600 lines/mm), two concave reflectors with a focal length of f = 20cm, and a liquid crystal spatial light modulator (SLM) (Cambridge Research and Instrumentation Inc, SLM-128).The SLM consists of 128 phase-modulating elements 100µm-wide with a 3µm gap between adjacent pixels.The Freezing algorithm is used to adaptively feedback-control phase of the optical pulses [17].We used second harmonic generation by a 300μm thick BBO crystal located at the antenna position as the feedback signal.The shaped femtosecond laser output is then split into two beams and guided to the emitting and receiving PC antennas, respectively.The PC dipole antenna is fabricated on the LT-GaAs sample with a 30 μm antenna length, 5 μm gap, 5 μm antenna width, and 10 μm transmission line width.Both antennas are mounted with Si hemispherical lenses.Apparently, the THz waveform exhibits oscillations in the rising edge.The ringing disappeared when the laser spot size is defocused to 10 μm.This is shown in the inset of the Fig. 3(a).As the same shaped pulses were employed for both cases, ringing could not be an artifact contributed by multiple reflections or pulse multiplexing in the shaper, but rather related to excitation power densities.various excitation powers.Clearly, the positive parts of the profile have two peaks.The relative delay and peak amplitude change as the excitation power was increased from 5 mW to 55 mW (or 2.5×10 4 to 2.8×10 5 W/cm 2 ).In general, the heights of negative peaks of the waveforms increase with excitation powers, and the positive peaks grow as well but with relative changes in heights and delay of the two peaks.Figure 3(c) plots the relative delay of the two positive peaks as a function of the excitation intensity.The relative delay time decreases rapidly with increasing excitation power density and then settles down to around 0.45 ps for excitation intensity greater than 1.0 × 10 5 W/cm 2 .Concurrently, the relative peak amplitude initially decreases and then monotonically increases from low to high excitation intensities.The amplitude of the first positive peak is initially smaller and then starts to exceed the second one as the excitation power is increased beyond 45 mW.This behavior can depend on whether TPA or SPA is dominant in generation of photoexcited carriers.We expect TPA to exhibit stronger power dependence compared to SPA due to the former's dependence on square of the laser intensity.Further, shorter lifetime (~300fs) of carriers from TPA should result in the corresponding THz signal taking place before the one from SPA whose lifetime is longer (~ps).Based on the above arguments, we tentatively attributed the first and second positive peaks in the THz waveforms mainly to photocarriers generated by TPA and SPA, respectively.The second peak for each trace in Fig. 3(b) actually should contain contributions by both the SPA and TPA effects.Indeed, it increases with power.To check the above picture, we performed the following simulation as shown in Fig. 3(e).Two THz waveforms such as the one plotted in the inset of Fig. 3(a) with a relative delay of 1 ps are superimposed.The first pulse is assumed to be due to photocarriers generated by the TPA effect with squared intensity dependence.The following pulse, on the other hand, is generated through SPA and exhibits a linear dependence on intensity.By varying the excitation power from 20mW to 50mW and fixing the spot size at 5 μm, almost the same trend of relative delay time and relative amplitude of two positive peaks with increasing excitation power like Fig. 3(b) can be demonstrated.Changing the relative delay does not affect the general behavior shown in Fig. 3(e).This further confirms our picture that THz pulse generated by a PC antenna could contain at least two parts with different excitation power dependences.As for the actual value and origin of relative delay between SPA and TPA, more detailed investigations, such as chirp controlled pump-probe measurement, will be done.
We employed a PC antenna for sampling.It is well known that this technique can distort the detected temporal profile of the THz pulse due to its relatively narrow spectral bandwidth [18,19].Nevertheless, trends such as excitation power dependence of THz peak amplitude and total power should be preserved.Hence, we compare the total THz power and peak amplitude of simulated and experimental results in this work.Figure 4 shows the pump power density dependencies of the total radiated THz powers as a function of pump intensity.These are obtained by integrating the corresponding Fourier-transformed spectra.For the lower power density condition (spot size is 10 μm), the experimental results shown in Fig. 4(a) agrees with the simulation results in Fig. 2(b) with or without consideration of TPA effect.This is reasonable because the excitation is not high enough to generate sufficient carriers through TPA.In contrast, the THz emission from PC antennas in the tight focus geometry should consider both SPA and TPA effects.Figure 4(b) depicts the excitation power dependency of THz total power for spot size of 5μm.Clearly, simulation without taking into account the TPA effect departs drastically from the experimental data.
Figure 5(a) describes pump power dependencies of the THz peak amplitudes when the laser beam is defocused to a spot size of 10 μm.Clearly, it is not essential to include contribution by TPA.However, for higher pump power density, our model predict that the peak amplitudes of THz fields to exhibit a plateau with excitation power from 20 to 30 mW, before a monotonic rise with pump intensity above ~30mW.Good qualitative agreement with the experimental trend is obtained, as shown in Fig. 5(b).We note that the pump pulse incident on the generating PC antenna in typical THz time-domain spectroscopy (THz-TDS) system is chirped since most components in the system are dispersive and not precompensated.This implies longer pulses, lower pump intensity and diminishing role played by TPA in conventional PC-Antenna-based THz-TDS.For a THz-TDS system employing a broadband source without pre-compensation, the TPA will play a diminishing role while SPA will totally dominate.That is the reason why anomalous increase rather than saturation of THz radiation is observed for the first time here with transform-limited excitation using the pulse shaping technique.From Benjamin's model [10], relaxed carriers from bottom of the conduction band could be generated by either TPA or SPA.This indicates the complexity of screening field since there are three possible way for carrier relaxation.However, we can consider just the inter-band transitions because the carrier lifetime for intra-band transitions (τ 3 ) is long.Increasing electron-phonon scattering due to the high phonon density caused by the phonon bottleneck effect in LT-GaAs [20] could reduce the τ 3 in LT-GaAs.At the pump intensity employed in this work, this mechanism should be negligible.Therefore, we argue that photo-induced carriers caused by SPA and TPA effects can be considered separately.It follows that the two-photon induced carriers will not enhance the screening effect by the SPA.
In our experiments employing 30fs-wide pulses, maxima excitation power is limited to 50mW due to the damage issue.We find that the peak amplitudes of the generated THz pulses increase by a factor of two when the excitation power increased from 30 to 50mW.We anticipate the use of transform-limited pulses 15 fs wide can be used to generate THz pulses with peak amplitude as high as four times compared to the present case of 30fs-wide pulses.
The pulse shaping system is used to provide transform-limited pulses through compensating not only linear but also high-order chirp.Although we can compensate linear chirp using prism pairs, residual higher-order chirp means broader-than-transform-limited pulses and lower peak power which would reduce the beneficial role played by TPA compared with the pulse compensated by pulse shaper.Nonetheless, TPA will play a role if the peak intensity is still high enough even if the excitation pulse is not transform-limited.In addition, maximum THz emission power is limited by the electrostatic energy stored in the photoconductive gap, the estimated value should be around the order of 10 8 w/cm 2 [3].Apparently, this value is much larger than the excitation density in our case (around 10 5 W/cm 2 ).Therefore, the generated THz power does not exceed the electrostatic energy stored in the photoconductive gap.As for the overall efficiency, it is difficult to measure in our case because the higher spectral component have been filtered out during the detection by PC antenna sampling.However, we estimate the overall efficiency increased by a factor of 2 by transform-limited pulses at an excitation power of 50 mW (see Fig. 2 (d)).Previous theoretical studies show that the local electric field oscillates if the generated carrier densities exceed 10 20 cm −3 in LT-GaAs.Further, the oscillation frequency increases monotonically with the carrier density.This can be explained by the interplay between the bias field and the polarization induced in the PC [14,21].The free carriers are initially accelerated and the polarization is generated to screen the bias field.While the bias field is canceled by the polarization, the momentum of free carriers is not zero and free carriers still drift to increase the polarization.The direction of the local field is reversed and the carriers are accelerated to the backward direction.As a result, the local electric field oscillates.Therefore the central frequency of the THz radiation generated should increase with the carrier density.However, it is difficult to observe this phenomenon experimentally because the carrier density at the bottom of the conduction band is limited to about 5 × 10 18 cm −3 .Further increase of generated carrier density can be achieved by TPA effect as shown in this work.This hints at the possibility of generating higher-frequency THz radiation from PC antennas.

Conclusions
In this work, feasibility of enhancing THz radiation power due to TPA induced photo-carrier in PC antennas is studied theoretically and experimentally.Features of the THz waveform associated with SPA and TPA are clearly identified for the first time.At low excitation power densities, TPA actually caused radiated THz power to decrease.When photocurrent due to SPA saturated, however, extra carriers can still be excited to the conduction band by TPA, if the exciting photon density is sufficiently high.Experimentally, using the pulse shaping technique, we observed an anomalously increasing radiated power rather than saturation.This is in good agreement with predictions of the theoretical predictions and simulation results.With transform-limited pulses at the PC antenna, TPA effect can be enhanced for higherpower excitation condition.We show that TPA effect can help to break through the limitation of generation of THz radiation power by the PC antenna due to SPA.Besides, possibility of enhancing the high-frequency spectral components of THz radiation by TPA is also discussed.

Fig. 1 .
Fig. 1.Band diagram for low-temperature-grown GaAs describing the main excitation and decay transitions in the rate-equation model The rate-equation for the dynamics of carriers in the PC antenna based on LT-GaAs are

Fig. 2 .
Fig. 2. Simulation results of peak amplitudes of THz fields and total radiated powers for two scenarios: (i) Carriers can only be excited to the bottom of the band gap (SPA effect only, blue traces), or (ii) carriers can also be excited to the upper states of the conduction band (Both SPA and TPA effects are present, red traces).For the pump power considered, the focus spot size is assumed to be 10 μm in (a) and (b) and 5 μm in (c) and (d), respectively.

Fig. 3 .
Fig. 3. (a) Temporal profiles of THz radiation at excitation power of 5 mW when the laser spot size is 5 μm.The inset shows the corresponding THz waveform when the laser beam is defocused to 10μm.(b) Temporal profiles of THz radiation at various laser excitation power.(c) The relative delay time between the first and the second peak fields of (b).(d) The relative peak field between the first and the second peak fields of (b).(e) simulated plot of generated THz radiation including SPA and TPA Figure 3(a) shows the temporal profiles of THz radiation generated with transform-limited pulse of 35 fs focused to a spot size of around 5 μm.The excitation laser power was 5 mW.Apparently, the THz waveform exhibits oscillations in the rising edge.The ringing disappeared when the laser spot size is defocused to 10 μm.This is shown in the inset of the Fig.3(a).As the same shaped pulses were employed for both cases, ringing could not be an artifact contributed by multiple reflections or pulse multiplexing in the shaper, but rather related to excitation power densities.Figure 3(b) shows temporal profiles of THz radiation at Figure 3(a) shows the temporal profiles of THz radiation generated with transform-limited pulse of 35 fs focused to a spot size of around 5 μm.The excitation laser power was 5 mW.Apparently, the THz waveform exhibits oscillations in the rising edge.The ringing disappeared when the laser spot size is defocused to 10 μm.This is shown in the inset of the Fig.3(a).As the same shaped pulses were employed for both cases, ringing could not be an artifact contributed by multiple reflections or pulse multiplexing in the shaper, but rather related to excitation power densities.Figure 3(b) shows temporal profiles of THz radiation at

Fig. 4 .
Fig. 4. Pump power dependencies of the THz total powers.(a) The laser beam spot size is 10 μm.(b) The laser beam spot size is 5 μm.

Fig. 5 .
Fig. 5. Pump power dependencies of the THz peak amplitude.(a) The laser beam spot size is 10 μm.(b) The laser beam spot size is 5 μm.