We experimentally study the thickness dependence of the terahertz (THz) response in 〈110〉-oriented GaAs crystals for free space electro-optic sampling at 1.55 µm. The THz response bandwidths are analyzed and simulated under phase-matching condition with a model frequency response function. The results indicate that the detection bandwidth increases from 2 THz to 3 THz when the thickness of GaAs is reduced from 2 mm to 1 mm. Below 1 mm, the detected bandwidth is increasingly limited by the emitter characteristics and the finite probe pulse duration. The broadest bandwidth in experiment reaches 3.3 THz when using a 0.2 mm thick crystal, while it exceeds 5 THz in theory. The THz response sensitivity was studied experimentally and modeled taking into account the absorption of the THz radiation in the GaAs crystal. While absorption was found to be negligible for the crystal thickness range studied here, strong saturation is predicted theoretically for crystal thicknesses exceeding 5 mm.
© 2010 OSA
The rapidly growing demand for low-cost compact THz spectrometers has given rise to a strong interest in the development of coherent detection schemes with 1.55 µm femtosecond fiber lasers [1, 2]. Photoconductive antennas based on In0.53Ga0.47As compounds [3–5] have been studied since their intrinsic bandgap (E gap = 0.74 eV) suits this optical excitation wavelength. Free space electro-optic sampling (FSEOS) is an alternative detection scheme that is based on the Pockels effect and allows for measuring transient electromagnetic radiation with extremely high temporal resolution . The 〈110〉-oriented ZnTe crystal possesses a large electro-optic coefficient and excellent phase-matching at around 0.8 µm probe wavelength so that it is successfully used in THz time domain spectroscopy driven by Ti:sapphire lasers [7, 8]. However, the detection efficiency of ZnTe deteriorates at a probe wavelength of 1.55 µm due to the phase mismatch between the optical sampling pulse and the THz radiation . A possible solution is the conversion of the electro-optic probe wavelength from 1.55 µm to 0.8 µm via second harmonic generation in nonlinear optical crystals . However, the limited conversion efficiency reduces the optical intensity available for THz sampling. Such an approach also increases the complexity of the setup. Nagai et al.  proposed 〈110〉-oriented GaAs as a substitution for ZnTe in THz spectrometers driven by fiber lasers, since the optical group velocity at 1.55 µm is closer to the THz phase velocity in GaAs than in ZnTe. Pradarutti et al.  studied the THz response in 〈110〉-oriented GaAs at 1.06 µm probe wavelength. Vossebürger et al.  studied the THz response in GaAs crystals of different thicknesses at 0.9 µm optical sampling wavelength. However, there is no detailed study of the thickness dependence of the THz response in 〈110〉-oriented GaAs using electro-optic sampling at 1.55 µm wavelength, which could be helpful to design commercial THz spectrometers driven by fiber lasers.
In this paper, we present a study of the THz response of 〈110〉-oriented GaAs for different crystal thicknesses in the range from 0.2 mm to 2.0 mm. To this end, we perform FSEOS in 〈110〉-oriented GaAs using a 1.55 µm femtosecond fiber laser for the sampling of a THz pulse form. The experimental results for the bandwidth response are compared with theory by modeling the frequency response function in order to identify signal features that depend on the thickness of the detection crystal. The response bandwidth is limited by the propagation velocity difference between the probe beam and the frequency components of the THz pulse. In addition to this group velocity mismatch, absorption of the THz radiation in the electro-optic crystal can have detrimental effects on the response sensitivity of FSEOS. Therefore, we also study the influence of the crystal thickness on the sampling sensitivity. We find that absorption has a negligible effect on the response sensitivity in the crystal thickness range studied here, while our theoretical model predicts strong saturation for crystal thicknesses exceeding 5 mm.
We cut a 〈110〉-oriented semi-insulating GaAs crystal with a thickness of 2 mm into 10 mm × 10 mm sized pieces. Each piece is polished mechanically to different thicknesses from 2.0 mm to 0.2 mm in 0.2 mm steps. The THz pulse forms are recorded in a THz time domain spectroscopy setup. A mode-locked Er-doped fiber laser (TC-1550, MenloSystems GmbH) is used as light source. It emits pulses with 100 fs duration and 1.55 µm central wavelength at a repetition rate of 100 MHz. In our setup, the THz emitter is triggered with laser pulses of 33 mW average power, while a probe beam of 5 mW is used for FSEOS. The THz emitter consists of a photoconductive switch patterned on a 500 nm thick ErAs:In0.53Ga0.47As superlattice with 10 nm period grown on top of an InAlAs buffer layer and a semi-insulating InP substrate. The ErAs:In0.53Ga0.47As superlattice is a high performance photoconductor at 1.55 µm, with a short carrier lifetime (0.3 ps)  and a high electron Hall mobility (906 cm2/Vs) . The antenna consists of a bow-tie geometry with an opening angle of 30° and a gap of 10 µm × 10 µm. In order to improve the signal-to-noise ratio (SNR), the photoconductive ErAs:In0.53Ga0.47As is etched away everywhere except at the center of the bow-tie antenna, which is illuminated, in order to expose the well insulating substrate and reduce the dark current . Our emitter is biased at 20 kV/cm field using a block function voltage at a frequency of 33 KHz for lock-in amplification. A hemispherical high-resistivity silicon lens with 6 mm diameter is used to collimate the THz emission in order to improve the detection efficiency. Two off-axis parabolic mirrors focus the THz radiation on the GaAs crystal where it overlaps with the 1.55 µm probe pulse train. The electric field of the THz radiation induces birefringence in the GaAs crystal, which causes a change of the polarization state of the 1.55 µm probe beam. This change is analyzed with a quarter-wave plate, a Wollaston prism and a balanced photodetector. The THz propagation path is enclosed in a nitrogen atmosphere to avoid spectral dips induced by aqueous vapor absorption. The SNR exceeds 40 dB in our experiment.
3. Results and Discussion
Figure 1(a) shows the normalized THz pulse forms detected by GaAs crystals with thicknesses of 2.0 mm, 1.0 mm and 0.2 mm, respectively. Internal reflections at the surfaces of the GaAs crystals cause the signal echoes in the traces recorded with the 1.0 mm and 0.2 mm thick crystals. These echoes lead to frequency ripples in the THz spectra, which are obtained by Fast Fourier Transform (FFT). In order to avoid this rippling, we transform the time domain signals into frequency spectra in a selected time window excluding all the echoes. This time window is fixed for all crystal thicknesses and limited by the positions of the echoes for the thinnest crystal. The peak SNR of the spectra remains at about 40 dB. The time domain response is estimated from the full width at half maximum (FWHM) of the main positive peak of the pulse form as indicated by arrows in Fig. 1(a). When the thickness of GaAs is reduced from 2.0 mm to 0.2 mm, the FWHM decreases from 0.5 ps to 0.3 ps.
The change of the pulse form in the time domain for different crystal thicknesses is also reflected in the frequency domain data. Figure 1(b) shows the THz spectra corresponding to the pulse forms in the time domain. The detected bandwidth is approximately equal to the reciprocal temporal FWHM and increases from 2 THz to 3.3 THz when the thickness of the GaAs crystal is reduced from 2.0 mm to 0.2 mm. However, the detected bandwidth is limited both by the emitter and the finite duration of the probe pulse as shown below.
The efficiency of the electro-optic sampling process critically depends on the fulfillment of the phase-matching condition
For the frequencies contained in the optical pulse (νopt) and the terahertz pulse (νTHz) in our experiments, νTHz ≪ νopt, which justifies the approximation
where n gr is the group refractive index and n ph the phase refractive index. Optimal sampling of a specific THz frequency is therefore achieved when this frequency component of the THz pulse form traverses the GaAs crystal with the same phase velocity as the group velocity of the infrared probe pulse. Velocity mismatch leads to a temporal walk-off δ after propagation through the crystal of
where c is the speed of light in vacuum and l is the crystal thickness.
The optical group refractive index is defined as and can be approximated with the Sellmeier equation . In the following, we assume the probe pulse to be temporally and spectrally δ-like at a wavelength λ 0 = 1.55 µm and neglect dispersive broadening. This yields for the probe pulse a group refractive index n gr(λ 0) = 3.5. We also do not take into account absorption of the probe pulse because its photon energy is smaller than the bandgap in GaAs (E gap = 1.43 eV). In the THz range, the phase refractive index can be calculated as , where ν TO = 8.05 THz is the TO-phonon resonance frequency and ε the frequency dependent permittivity . The features in our experiments and simulations appear at frequencies ν THz < 6 THz, so that the phonon absorption is irrelevant here. The values for the GaAs high frequency permittivity limit ε (∞) = 10.8 and the DC permittivity ε(0) = 12.7 are best-fit values from the comparison of the distinct features in the response function and the recorded spectra. They lie within the range of measured values reported in the literature . The amplitude of the electro-optic modulation of the probe beam by an electric field of frequency νTHz is proportional to the average of the field over the mismatch time δ(ν THz), which can be expressed by the frequency response function [19, 20]
The most striking feature of Eq. (5) is the periodical occurrence of zero response whenever ν THz·δ(ν THz) takes on an integer value. This corresponds to the sampling of an integer multiple of a full THz field oscillation so that the electro-optic modulations of the probe beam induced by the two THz half-waves cancel each other out. For our analysis, we take the frequency of the first minimum as the detectable bandwidth. This frequency will be referred to as the cut-off frequency. Furthermore, we notice that the response decreases with increasing frequency, as can be seen from the appearance of ν THz in the denominator. Physically, this reflects the fact that the effective interaction length for maximum response is half a wavelength. One should be aware that this simplified model neither takes into account the frequency dependence of the electro-optic characteristics of the GaAs crystal nor the convolution of the response function with the incident THz pulse. It is however sufficient to model the positions of the cut-off frequencies in real data, which is demonstrated in Fig. 2. The panels (a) to (f) display the frequency response function of Eq. (5) in the upper part and the corresponding measurement in the lower part of each panel for electro-optic crystal thicknesses ranging over one order of magnitude. For the thickest crystal [2 mm, panel (a)], the position of the first signal minimum in the measurement coincides very well with the first zero point in the response function. Even the spectral tail beyond this cut-off frequency resembles the model, at least up to the next minimum. When the GaAs thickness is reduced towards 1 mm [panels (b) to (d)], the correlation between the cut-off frequency in the measurement and the bandwidth of the model function persists. The detectable bandwidth increases as the crystal thickness is reduced, since the temporal walk-off δ of Eq. (4) decreases with l. For crystal thicknesses of less than 1 mm [panels (e) and (f)], the theoretical and the measured bandwidths differ. While the measured cut-off frequency does not rise beyond 3.3 THz, the first zero of the frequency response function for 0.2 mm thickness, which corresponds to our thinnest crystal, is located above 5 THz.
Figure 3 plots the measured and calculated bandwidths for all crystal thicknesses considered in this work. It shows the excellent congruence for l ≥ 1 mm as well as the increasing discrepancy between theory and experiment for smaller l, for which the experimental bandwidth saturates. We attribute this mainly to the following two aspects: (1) the limited output bandwidth of our emitter ; and (2) the deterioration of the SNR due to the limited interaction length in thinner crystals.
The thickness l of the electro-optic crystal does not only influence the THz response bandwidth of the FSEOS process, but it also constitutes the maximum interaction length of THz and probe pulse and thus determines the THz response sensitivity. To model this sampling sensitivity, we again assume δ-like characteristics of the optical probe pulse. While the group velocity mismatch is straightforward to account for in the frequency domain as demonstrated in the previous paragraphs, this effect is not easily modeled in the time domain due to the broadband characteristics of the THz radiation. For this reason, we neglect this effect for the modeling of the THz response sensitivity and assume a coherent co-propagation of the optical probe pulse and the THz pulse form. Under these assumptions, the THz response sensitivity, which we define as the ratio of the electro-optic signal ΔI and the incident optical intensity I, can be written as follows :
Here, γ 41 = 1.5 pm/V is the electro-optic coefficient, n gr(λ 0) = 3.5 is the group refractive index of GaAs at the optical wavelength λ 0 = 1.55 µm, and E THz is the electric field of the THz pulse form. Since γ 41 is very small, the sine can be approximated by its argument. Therefore, for a fixed electric field, the THz response sensitivity ΔI/I is proportional to the crystal thickness l. We would like to point out that this simple model so far neglects absorption of both the optical probe pulse and the THz radiation. We use near infrared (NIR) transmission spectroscopy and Fourier transform infrared (FT-IR) spectroscopy to investigate the transmission properties of GaAs in the optical and THz region, respectively. While we experimentally find its optical absorption coefficient to be negligible (α = 3 × 10−3 cm−1), the THz absorption coefficient is considerable (e.g., α THz = 5.6 cm−1 at 3 THz). These results are in good agreement with previously published data [23, 24]. In order to account for the absorption of the THz radiation in the GaAs crystal, we replace the static term E THz · l in Eq. (6) by
Here, the attenuation of E 0, the THz electric field incident on the electro-optic crystal, in dependence of the penetration depth z, is described with the Lambert-Beer law (see also the inset of Fig. 4). For the sake of completeness, note that we also lump the partial reflection of the THz field at the crystal surface into E 0, which will be used as a fit parameter in the corrected THz response sensitivity
We would like to point out that this equation reduces to Eq. (6) when α THz approaches zero. The inclusion of the absorption of the THz radiation in the model leads to a sublinear dependence of the THz response sensitivity on l, as intuitively expected.
Figure 4 plots the measured and fitted THz response sensitivity as a function of thickness. We use the incident THz electric field E 0 = 20.6 V/cm as a fit parameter and the experimentally obtained THz absorption coefficient of α THz = 5.6 cm−1 as mean value in our range of THz frequencies. The fitted values match our experimental results within the error bars. However, the sublinearity due to absorption in the thickness range of up to 2 mm is found to be negligible. The simulation of the THz electro-optic sampling sensitivity predicts stronger saturation effects for a crystal thickness range beyond the experimentally studied thicknesses. For example, at l = 5 mm, already 75% of the theoretical maximum sensitivity are reached, and a 10 mm thick crystal would exhibit 94% sensitivity.
In summary, we have studied the THz response bandwidth and sensitivity of FSEOS in 〈110〉-oriented GaAs at a probe wavelength of 1.55 µm for different crystal thicknesses. The measured THz response bandwidths are compared to a basic frequency response function. The cut-off frequencies are in very good agreement for crystal thicknesses l ≥ 1 mm, supporting the usability of the model. A crystal with a thickness of 0.2 mm was successfully employed for FSEOS. Limitations in the emitter bandwidth and the finite optical pulse duration prevented to experimentally demonstrate the theoretically predicted response bandwidth of up to 5 THz. Moreover, the effects of the absorption of the THz radiation in the electro-optic crystal on the detection sensitivity were studied. While in our crystal thickness range absorption was found to have no considerable detrimental effect on the response sensitivity, the model predicts strong saturation of the sensitivity for crystal thicknesses exceeding 5 mm.
The authors would like to thank M. Theuer and C. Petermann for helpful discussions regarding the setup and acknowledge financial support from the TEKZAS project (Grant No. 13N9471) and the NanoFutur Young Investigator Award of the German Ministry of Science and Education (Grant No. 03X5503A). A. S. is grateful for support from the Thomas Gessmann-Foundation.
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