1. Introduction

Free space electro-optic (EO) sampling of terahertz waveforms by femtosecond laser pulses [1] is a technique that is commonly used in terahertz time-domain spectroscopy [2], along with photoconductive sampling [3]. In the standard EO sampling scheme, the probe optical pulse co-propagates collinearly with the terahertz pulse in an EO crystal and accumulates a variation of polarization caused by the Pockels effect. By measuring the polarization variation as a function of delay between the optical and terahertz pulses, the terahertz electric field can be mapped. To achieve efficient EO sampling for a given wavelength λ of the probe optical beam, one should use a specific EO crystal that provides optical-terahertz velocity matching. For example, ZnTe is routinely used to perform EO sampling with a Ti:sapphire laser (λ ≈ 0.8 μm). Unfortunately, no crystal is known to provide velocity matching at the wavelength (λ ≈ 1.55 μm) of femtosecond fiber lasers, which are an appropriate light source for low-cost compact terahertz spectrometers.

In noncollinear extensions of the ellipsometric technique [4,5], the probe optical beam propagates in an EO crystal at the Cherenkov angle to the terahertz beam. This allows one to achieve synchronous propagation of the probe optical pulse and a terahertz wavefront for an arbitrary optical wavelength in crystals with an optical-terahertz velocity mismatch, such as, LiNbO₃ [4] and GaAs (at the wavelengths larger than 1.33 μm) [5].

Recently, a nonellipsometric method of EO sampling has been proposed [6]. In this method, direct modulation of the probe optical beam intensity, induced by terahertz field in an EO crystal, is measured without any polarization optics. The mechanism of the nonellipsometric sampling has been elucidated as spatial separation of the sum-frequency-generation (SFG) and difference-frequency-generation (DFG) contributions of opposite polarities to the optical intensity modulation [7]. The separation is achieved due to noncollinear propagation of the optical and terahertz beams. By using a Si-prism-coupled LiNbO₃ crystal, it was shown that noncollinear nonellipsometric sampling could provide efficiency comparable to that of the standard ellipsometric technique [6]. Later, spectral filtering was proposed as another way to frustrate mutual compensation of the SFG and DFG contributions in the collinear scheme [8].

In this paper, we demonstrate experimentally noncollinear nonellipsometric EO sampling with a femtosecond fiber laser (λ = 1.55 μm) in a GaAs crystal. Due to a smaller, than in LiNbO₃, velocities mismatch between optical and terahertz waves in GaAs, we implemented noncollinear nonellipsometric scheme without a Si-prism coupler, unlike Ref. [6]. Such simplification of the scheme facilitates its practical applications in compact terahertz spectrometers. We obtained detection efficiency of the same order of magnitude as that of the noncollinear ellipsometric technique [5]. Using a 1-cm thick crystal ensured higher, as compared to mm- or sub-mm-thick crystals [9,10], spectral resolution (as high as a few GHz [5]). We derived theoretically and confirmed experimentally optimal orientation of the terahertz and optical polarizations with respect to the crystallographic axes of the crystal. This orientation provides a reasonable tradeoff between high modulation depth and practicality of implementation of the noncollinear nonellipsometric EO sampling in GaAs. We also verified experimentally the theory developed in Ref. [7].

2. Experimental

The experimental setup is shown in Fig. 1(a). A femtosecond Er³⁺–doped fiber laser with 1.55 μm central wavelength, 70 fs pulse duration (FWHM), and 100 MHz repetition rate was used as an optical source. The optical beam was split into the pump beam (35 mW average power), which triggered a photoconductive antenna (PCA) on an InGaAs/InAlAs substrate, and probe beam (30 mW) whose intensity was varied by using a half-wave plate (λ/2) and Glan prism (GP). PCA was biased with a ± 20 V, 10 kHz square wave voltage. By using a TPX lens and a parabolic mirror, the terahertz beam was collimated and focused onto the 1-cm thick <110>-cut GaAs crystal (with 5 × 5 mm² cross-section). From knife-edge measurements with use of a Golay cell, the 1/e beam width was estimated to be ~1.2 mm. The corresponding Rayleigh length is as high as ~1.5 cm (at the frequency ~0.5 THz of the spectrum maximum). Therefore, diffraction broadening of the terahertz beam was negligible.

 

Fig. 1 (a) Schematic of the experimental setup. (b) The polarizations of the optical and terahertz beams that optimize the efficiency of the nonellipsometric sampling.

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The probe beam was collimated by a 25.4 mm focal length lens f1 and focused onto the GaAs crystal by a lens f2, whose focal length was varied to provide different beam widths in the crystal. We used the 1/e-widths of 80, 120, 180, and 500 µm measured at the entrance surface of the crystal. The corresponding Rayleigh lengths (>2 cm) substantially exceed the crystal thickness. The incident angle of the probe beam on the entrance surface of the crystal was taken equal to 45° to ensure the propagation of the probe beam in the crystal at the Cherenkov angle β ≈ 12° to the terahertz beam [5].

The setup could be switched from the nonellipsometric scheme to the ellipsometric one [5] by using a flipped mirror (it also required rotating the GaAs crystal by 90°). In the ellipsometric scheme, a standard combination of a quarter-wave plate (λ/4) and Wollaston prism (WP) was used to separate orthogonal polarizations of the probe beam. To improve the signal-to-noise ratio in the nonellipsometric scheme, we used a D-shape mirror, which divided the probe beam into two halves, which correspond to the SFG and DFG processes, and directed them to a balanced detector. Due to opposite polarities of the SFG and DFG signals they are added constructively whereas the background signals from the two photodiodes are subtracted thus improving the signal-to-noise ratio. To minimize water vapor absorption, the measurements were made in dry air conditions.

In the ellipsometric operation mode, the [001] crystallographic axis of the crystal was aligned along the optical beam polarization and orthogonal to the terahertz beam polarization, i.e., at θ = −90° [Fig. 1(b)]. This orientation is optimal for ellipsometric detection in <110>-cut zinc-blende crystals [11–13]. In the nonellipsometric mode, we rotated the crystal by 90° to the position θ = 0° shown in Fig. 1(b). This orientation, as we explain below, may be optimal in terms of high optical intensity modulation depth and practical convenience. To verify experimentally a theoretically predicted dependence of the nonellipsometric EO signal on angle θ, we rotated the crystal around the [110] direction [Fig. 1(b)].

3. Results and discussion

Figure 2 shows the EO signals obtained by using the nonellipsometric method with different 1/e-widths (80, 120, 180, and 500 µm) of the probe beam in the crystal. The signal obtained by the ellipsometric method with an 80-µm wide probe beam is shown for reference. From Fig. 2, it is seen that the efficiencies of both methods are comparable but the shapes of the output signals differ significantly. For a narrow (80-µm wide) beam, the shape of the nonellipsometric signal is close to the time derivative of the ellipsometric one (inset in Fig. 2). With increasing the beam width, the nonellipsometric signal deviates from the time derivative shape and becomes smoother.

 

Fig. 2 EO signals obtained by the nonellipsometric (NEL) method with different widths of the probe beam. EO signal obtained by the ellipsometric (EL) method is shown for reference. The inset compares the normalized shapes of the NEL 80 μm signal (solid) and the time derivative of the EL signal (dashed).

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Figure 3 shows the corresponding amplitude spectra. The experimental results in Fig. 3 confirm the theoretical prediction [7] that noncollinear nonellipsometric scheme operates as a bandpass filter, whose transmission band is defined by the probe beam width. According to Ref. [7], the output signal of the nonellipsometric scheme can be evaluated as

 

Fig. 3 Amplitude spectra of the EO signals shown in Fig. 2.

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Fig. 4 Transfer function of the nonellipsometric scheme for different widths of the probe beam. The spectrum of the ellipsometric EO signal is shown for reference.

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According to Figs. 3 and 4, for the 500-µm wide beam, the scheme operates as a low-pass filter that attenuates high frequencies. Physically, it can be explained by the fact that in noncollinear geometry the pancake-shaped probe pulse is tilted to the terahertz wavefronts, its different parts interact with different phases of the terahertz field. This reduces the EO response, the effect is more pronounced for a wider probe beam and smaller terahertz wavelengths. For the 80-µm wide beam, high frequencies are boosted and low frequencies are filtered out (Figs. 3 and 4). The weakening of the response to low frequencies can be explained by a smaller angular separation of the SFG and DFG contributions to the optical intensity modulation from low frequencies relative to the divergence angle of the probe beam [7]. The optimal width of the probe beam for correctly measuring the spectrum of the used terahertz source (PCA) is ~120 µm (Fig. 3).

The shapes of the nonellipsometric EO signals (Fig. 2) can be explained by using Eq. (1) and Fig. 4. For the 80-µm wide probe beam, the terahertz spectrum lies almost completely at the frequencies significantly lower than the maximum frequency of the transfer function (Fig. 4), where H(Ω) ≈ 2π^-1/2Ω/Ω_c. Taking into account the phase shift of all spectral components of the input spectrum by π/2 in Eq. (1) (because of Im), the output spectrum is proportional to the input spectrum multiplied by iΩ. This corresponds to differentiating the input terahertz signal with respect to time. With increasing the probe beam width, the output signal deviates from the time derivative shape and becomes smoother.

In the standard collinear geometry, a short coherence length of laser pulses at 1.55 μm and terahertz waves in GaAs leads to the necessity of using crystals thinner than ~1 mm. For thin crystals, an echo, originating from internal reflections of the terahertz pulse in the crystal, is separated from the main EO signal by a short time interval – about 12 ps for a 0.5 mm thick crystal [2]. The necessity to filter out the signal echo puts the upper limit on the time window of EO sampling and, as a result, significantly restricts the spectral resolution – by 40 GHz for a 1 mm thick crystal. In the noncollinear geometry, using cm-thick crystals allows one to achieve an order of magnitude increase in the spectral resolution [5]. In particular, the spectral resolution in Fig. 3 is as high as ~3 GHz.

To provide a highest efficiency of the nonellipsometric scheme, the terahertz and optical beam polarizations should be different from those in the ellipsometric scheme. The optimal polarizations, shown in Fig. 1(b), can be found from following consideration. Fourier transform (with respect to time) of the optical electric field after the GaAs crystal can be written as

To corroborate the consideration above, we derived the dependence of the output nonellipsometric signal $\Delta I$ on the crystal rotation angle θ [Fig. 1(b)] for fixed polarizations of the optical and terahertz fields [Fig. 1(b)]. This dependence coincides with the θ-dependence of the P^NL(ω) component parallel to $E^{(0)}(\omega )$. By using the tensor ${\chi }_{ijk}^{\left(2\right)}$ [11], we obtained the dependence $\Delta I\propto \cos \theta \left(2-3{\cos }^2\theta \right)$ that was confirmed experimentally (Fig. 5).

 

Fig. 5 The output nonellipsometric EO signal as a function of θ: experiment (crosses) and theory (solid line).

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4. Conclusion

We have demonstrated efficient broadband EO sampling of terahertz pulses in a GaAs crystal without any polarization and coupling optics. The optimal, in terms of high modulation depth and practical convenience, polarizations of the terahertz and probe optical beams have been theoretically found and experimentally verified. Using a cm-thick crystal provided spectral resolution as high as a few GHz. The measured dependencies of the output terahertz waveform and spectrum on the probe beam width agree well with the theoretical predictions [7].