1. INTRODUCTION

Electro-optic (EO) sampling, hereafter reported as conventional EO sampling (CEOS), is a widely used technique for the measurement of the electric field of a terahertz (THz) pulse with subpicosecond time resolution [1–3]: in a zinc-blende crystal, the polarization state of a linearly polarized ultrashort optical pulse is modulated by the THz pulse through the Pockels effect. The corresponding phase retardation $\mathrm{\Gamma }$, proportional to THz electric field amplitude $E_{\mathrm{THz}}$, is then measured by ellipsometry for different time delays $\tau$ between the two pulses. The ellipsometer, consisting of a $\lambda /4$ plate followed by a suitably oriented Wollaston polarizer, converts the phase modulation into an amplitude modulation of the optical probe pulse measured by a balanced photodetector. Different alternatives to this technique have been proposed, but they suffer from some detection nonlinearities [4,5]. Even though these nonlinearities can be corrected, they increase the complexity of the measurement process with respect to conventional EO. Very recently, new methods based on a technique commonly used in optically heterodyne detected optical Kerr effect spectroscopy have been introduced [6,7]. These distortion-free techniques are based on two measurements at opposite optical biases near the zero transmission point in a crossed polarizers detection geometry. They lead to an increase, by 1 order of magnitude, of the signal enhancement factor, defined as the ratio of the normalized detected balanced difference signal to $\mathrm{\Gamma }$.

Hereafter, we propose two novel and simple schemes, based on a special geometry of the polarization states of the optical and THz beams. The first one leads to an amplitude modulation of the optical pulse at the output of the zinc-blende crystal and makes it possible to separate the optical pulse generated by the Pockels effect, hereafter called the signal pulse, from the incident pulse [8]. Thanks to an optical heterodyne detection of these signal pulses, one can perform a distortion-free sampling of THz pulses within a single measurement. In the second one, we take advantage of this geometry where the Pockels nonlinearity couples to the intrinsic second-order nonlinearity of the EO crystal, leading to the generation of a second-harmonic signal [the Pockels-induced second-harmonic (PISH) effect], which can be used, through an optical heterodyne detection, to sample terahertz pulses [9]. Since the latter detection scheme, which shows some similarities with the first one, offers interesting perspectives for the sampling of THz pulses generated by near-infrared pulses, we will present it here with more details than in [9].

This paper is organized as follows. In Section 2, we present the experimental setup of our optically heterodyne-detected EO sampling and derive the expression of the detected signal. The latter expression is then analyzed and confronted with the experiment. Next, in Section 3, we give a comprehensive analysis of the optically heterodyne-detected PISH signal and compare it, experimentally, with conventional EO sampling. Finally, in Section 4, we compare both detection schemes.

2. OPTICALLY HETERODYNE-DETECTED ELECTRO-OPTIC SAMPLING

A. Experimental Setup

The experiment is the following [Fig. 1(a)]: a linearly polarized THz pulse is emitted from air ionized by a two-color, namely 400 and 800 nm, femtosecond laser field [10]. This THz pump field is then collimated and focused onto a $⟨110⟩$ EO crystal (ZnTe or GaP), with a thickness $L$ ($L=300\mathrm{\mu m}$ for ZnTe and 200 μm for GaP), by two off-axis paraboloidal mirrors with a 150 mm focal length. Inside the crystal, the $\widehat{y}$-polarized THz beam propagates collinearly with a weak 800 nm probe beam, initially $\widehat{z}$-polarized by a half-wave plate and a polarizer, the $\widehat{z}$-axis coinciding with the $⟨001⟩$-axis of the crystal [Fig. 1(b)]. After the crystal, the transmitted optical probe beam is split by a 50/50 nonpolarizing beam splitter. Both beams are then sent through a quarter-wave plate and a polarizer, and their intensity is recorded by photodiodes. The fast axis of the quarter-wave plate in front of the photodiode PD1 (resp. PD2) is oriented at an angle $\beta$ (resp. $-\beta$) with respect to the $\widehat{z}$-axis. The axes of the polarizers are set along the $\widehat{y}$-axis, so that the incident probe beam is blocked in the absence of THz and $\beta =0°$. However, when the probe and THz pulses temporally overlap in the crystal, their interactions within the crystal generates a weak optical signal polarized along the $\widehat{y}$-axis [8]. The latter is therefore transmitted by the polarizers. The THz beam is chopped and the signal, $\mathrm{\Delta }I(\mathrm{\Gamma },\beta )=I_1(\mathrm{\Gamma },\beta )-I_2(\mathrm{\Gamma },\beta )$, measured by the two photodiodes versus the time delay $\tau$ between the probe and THz pulses, is recorded by a lock-in amplifier.

 

Fig. 1. (a) Experimental setup. BS, beam splitter; nBS, nonpolarizing beam splitter; P, polarizer; PD, photodiode. (b) Geometry of the experiment. $(\widehat{\mathbf{X}},\widehat{\mathbf{Y}},\widehat{\mathbf{Z}})$, Cartesian frame of the crystal; $(\widehat{\mathbf{x}},\widehat{\mathbf{y}},\widehat{\mathbf{z}})$, Cartesian frame of the laboratory. ${\mathbf{k}}_{\mathrm{THz}}$ (resp. ${\mathbf{k}}_{\mathrm{pr}}$) is the wave vector of the THz (resp. probe) pulse.

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B. Electro-Optic Sampling by Optical Heterodyne Detection

Let us see what is the expression of the signal $\mathrm{\Delta }I(\mathrm{\Gamma },\beta )$ provided by the setup of Fig. 1. In the $(\widehat{\mathbf{y}},\widehat{\mathbf{z}})$ basis, the electric field of the incident probe beam writes

At the output of the EO crystal, after interaction with the THz pulse, this field is given by [8]

Using the Jones matrix formalism [11], one can derive the expression of the electric fields $E_1$ and $E_1$ transmitted by polarizers $P_1$ and $P_2$, respectively, and their intensities $I_1$ and $I_2$ detected by photodiodes PD1 and PD2, respectively. All calculations done, one has

Now, if one considers the modulation depth $M(\mathrm{\Gamma },\beta )=[I_1(\mathrm{\Gamma },\beta )-I_1(0,\beta )]/I_1(0,\beta )$ of the signal detected by PD1 [6], it is then possible to express the enhancement factor $g$ and the distortion parameter $d$ associated with our method. They are related to the modulation depth, in the weak field limit, by

From Eqs. (4) and (5), one has

Equation (8) agrees with the expression of the enhancement factor found for EO sampling expressed in the terminology of optical heterodyne detection [6], where the heterodyning of the signal is performed in front of the EO crystal. On the other hand, the distortion parameter we found is multiplied by a factor ${\cos }^2(2\beta )$ compared to [6], meaning that our detection scheme is less sensitive to distortion as $\beta$ increases. However, since the enhancement factor is better for small values of $\beta$, $d\simeq 1/2$ over the optimal operating range of our detection scheme.

Finally, this distortion is removed by measuring the subtraction of the signals detected by both photodiodes,

C. Results and Discussion

Figure 2 displays the THz electric fields and their spectra, detected with our setup when ZnTe [Figs. 2(a) and 2(b)] or GaP [Figs. 2(c) and 2(d)] crystals are used, for values of the angle $\beta$ ranging from 5° to 29°. For comparison, all these fields are normalized to one and superimposed to the normalized THz fields measured with a CEOS setup, as described in [1], using the same EO crystal. At first sight, there is a very good agreement between both detection schemes, especially in terms of signal-to-noise ratio, which is about ${10}^3$. Discrepancies between the signals, if any, might be due to some small mismatch between the heterodyning angles of the quarter-wave plates, leading to a nonvanishing contribution from the homodyne signal, which would distort the measured signal shape. We kept such a source of distortion as small as possible by tuning the angle of the quarter-wave plates so that the intensities of the local oscillator on each photodiode, $I_1(0,\beta )$ and $I_2(0,\beta )$, respectively, were equal, the signals delivered by PD1 and PD2 being measured by means of a voltmeter.

 

Fig. 2. EO sampling by optical heterodyne detection performed in ZnTe [(a) and (b)] and GaP [(c) and (d)]. (a),(c) Normalized THz electric field, for different $\beta$ angles, versus the time delay $\tau$ compared to the field measured with a CEOS setup (dotted line). (c),(d) Corresponding spectra.

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However, a closer look at the detected waveforms, especially their dependence with respect to the heterodyning angle $\beta$ [Figs. 3(a) and 3(c)], reveals some discrepancies. This is clearly evidenced by Figs. 3(b) and 3(c), which display the evolution of the peak value of the nonnormalized signal $\mathrm{\Delta }I$ with respect to the angle $\beta$. According to Eq. (10), this amplitude is expected to scale like $\sin (2\beta )$. For $\beta \le 20°$, the experimental results are in agreement with this angular dependence, as shown by the fits on Figs. 3(b) and 3(d). But, for higher values of $\beta$, the peak value of $\mathrm{\Delta }I$ saturates and no longer obeys to Eq. (10). We attribute this behavior, and the subsequent distortion of the detected THz waveforms, to the saturation of PD1 and PD2 by the local oscillator signal, $I_{\mathrm{LO}}$, once $\beta >20°$. Nevertheless, this saturation of the photodetectors is not detrimental since, according to Eq. (8), it is better to work at small values of $\beta$. Indeed, the smaller the heterodyning angle $\beta$, the better is the enhancement factor $2g$ of EO sampling by optical heterodyne detection. For instance, for $\beta =5°$, the enhancement factor is $2g\simeq 1/\beta \sim 10$, a value identical to the enhancement factor obtained in [7].

 

Fig. 3. EO sampling by optical heterodyne detection performed in ZnTe [(a) and (b)] and GaP [(c) and (d)]. (a),(c) THz electric field, for different $\beta$ angles, versus the time delay $\tau$. (c),(d) Evolution of the amplitude of the signal displayed in (a) and (c) with respect to the heterodyning angle $\beta$. The solid line curve corresponds to a fit of the data with the function $\sin (2\beta )$.

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3. TERAHERTZ PULSES SAMPLING THROUGH OPTICALLY HETERODYNE-DETECTED PISH SIGNAL

In a recent paper [9], we have shown that it was possible to detect THz pulses via the detection of the second harmonic (SH) of an optical pulse induced through its interaction with an intense THz pulse in a zinc-blende crystal (the so-called PISH effect). Briefly, with the geometry of Fig. 1, an optical signal pulse is induced by the THz pulse through the Pockels effect. For the sake of simplicity, assuming monochromatic plane waves at angular frequencies $\omega$ in the optical range and $\mathrm{\Omega }$ in the THz range, the electric field of the signal is $E_{\mathrm{sig}}=E_y^{(\omega \pm \mathrm{\Omega })}\simeq E_y^{\omega }$, since $\mathrm{\Omega }\ll \omega$. This field then couples to the fundamental electric field, $E_z^{\omega }$, through the second-order nonlinearity of the crystal, ${\chi }_{XYZ}^{(2)}$, to generate a second-harmonic signal with an electric field given by [9]: $E_{\mathrm{PISH}}\propto {\chi }_{XYZ}^{(2)}{r}_{41}{(E_z^{\omega })}^2{E}_{\mathrm{THz}}L/\lambda$. Since the PISH signal is proportional to the THz electric field, it has been used to detect the latter. However, such a detection requires to heterodyne the PISH signal in order to avoid losing the phase of the THz waveform. This was done by inserting a quarter-wave plate on the optical beam pathway in front of the crystal.

Recently, to circumvent some limitations of lithium niobate or ZnTe (two- or three-photon absorption in the spectral range of the pump, absorption in the THz range, phase mismatch), new materials have been developed to generate intense THz pulses by optical rectification: orientation-patterned GaAs [12] or stilbazolium-salt single crystals [13], for example. They share the common characteristic of being pumped in the near-infrared, and may require some frequency doubling of the pump pulse for further characterization of the emitted THz wave by EO sampling [13]. One can avoid the insertion of a nonlinear crystal for SH generation of the pump beam prior to EO sampling, or the use of a THz Michelson interferometer to retrieve the THz waveform by first-order autocorrelation [14], by resorting to the heterodyne detection of the THz wave via THz-field-induced second-harmonic generation in a plasma [15] or in a suitable EO crystal as we did.

Consequently, since THz detection through optically heterodyne-detected PISH signal deserves some attention, we are going to present it in detail compared to the elementary treatment of [9] where, for instance, no derivation of the analytical expression for the measured signal or evaluation of the enhancement factor was given.

A. Experimental Setup and Analytical Expression of the Optically Heterodyne-Detected PISH Signal

The experimental setup is the same as in [9]. THz pulses are generated in ${\mathrm{LiNbO}}_3$ by optical rectification of tilted-pulse-front, 800 nm, femtosecond pulses, and the EO crystal is a 300 μm thick ZnTe. A quarter-wave plate, with its fast axis oriented at an angle $\pm \beta$ with respect to the $\widehat{z}$-axis, is inserted on the optical probe beam pathway in front of the EO crystal (Fig. 4). The THz beam is modulated by a mechanical chopper. The THz-induced SH signal, filtered by a 400 nm bandpass filter and measured by a photomultiplier tube with a monochromator, is recorded by a lock-in amplifier as a function of the time delay $\tau$ between the probe and THz pulses. Note that, contrary to usual heterodyne detection schemes, this method does not require any polarizer prior to the detector.

 

Fig. 4. Experimental setup. PMT, photomultiplier tube; $(\widehat{\mathbf{x}},\widehat{\mathbf{y}},\widehat{\mathbf{z}})$, Cartesian frame of the laboratory.

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Without any loss of generality, we model the probe and THz beams by monochromatic nondepleted waves [9]:

The $⟨110⟩$ cut ZnTe crystal, due to the applied THz field, is birefringent with the following principal axes [16]: ${\widehat{\mathbf{e}}}_1=\widehat{\mathbf{x}}$, ${\widehat{\mathbf{e}}}_2=(\widehat{\mathbf{y}}+\widehat{\mathbf{z}})/\sqrt{2}$, and ${\widehat{\mathbf{e}}}_3=(-\widehat{\mathbf{y}}+\widehat{\mathbf{z}})/\sqrt{2}$, whose refractive indices are $n_1=n_{\omega }$, $n_2=n_{\omega }+\frac{1}{2}{n}_{\omega }^3{r}_{41}{E}_{\mathrm{THz}}$, and $n_3=n_{\omega }-\frac{1}{2}{n}_{\omega }^3{r}_{41}{E}_{\mathrm{THz}}$, respectively. As a consequence, in the frame of the crystal [see Fig. 1(b)], the probe field amplitude at each point $x$ of the crystal is given by

The expansion of $f_{\beta }^2$, $g_{\beta }^2$, and ${\mathcal{E}}_z^{2\omega }(\beta ,L)$, assuming that $\mathrm{\Gamma },\beta \ll 1$, makes it possible to calculate the SH intensity for an heterodyning angle $+\beta$, $I_{2\omega }(\mathrm{\Gamma },\beta )\propto {|{\mathcal{E}}_y^{2\omega }(\beta ,L)|}^2+{|{\mathcal{E}}_z^{2\omega }(\beta ,L)|}^2$. Neglecting irrelevant terms of order 3 and more in $\mathrm{\Gamma }$ and $\beta$, one has

The first term in Eq. (26) corresponds to the intensity of the SH local oscillator,

In the terminology of optical heterodyne detection, one has to consider the modulation depth, $M(\mathrm{\Gamma },\beta )=[I_{2\omega }(\mathrm{\Gamma },\beta )-I_{\mathrm{LO}}^{2\omega }]/I_{\mathrm{LO}}^{2\omega }$, of the PISH signal detected for a positive heterodyning angle $+\beta$. As $\mathrm{\Gamma },\beta \to 0$, we found that

Finally, the heterodyned PISH signal is

As expected, the heterodyned PISH signal is proportional to the THz electric field and makes it possible to measure the latter. Note that, in the equation given in [9] for $\mathrm{\Delta }{I}_{2\omega }(\mathrm{\Gamma },\beta )$, the square power law dependence regarding $I_{\omega }$, ${\chi }^{(2)}$, and $r_{41}$ is missing.

B. Results and Discussion

Figure 5(a) displays the PISH intensities, $I_{2\omega }(\mathrm{\Gamma },\beta )$, measured for $\beta =\pm 10°$. Both signals oscillate around zero thanks to the mechanical chopper and the lock-in amplifier, which make it possible to get rid of the contribution of the local oscillator, $I_{\mathrm{LO}}^{2\omega }$. Note also that, by adding these two signals, one would get the homodyne signal, $I_{\mathrm{sig}}^{2\omega }$. On the other hand, their substraction gives a signal directly proportional to the THz electric field, as evidenced by Fig. 5(b) where $\mathrm{\Delta }{I}_{2\omega }(\mathrm{\Gamma },\beta )=I_{2\omega }(\mathrm{\Gamma },\beta )-I_{2\omega }(\mathrm{\Gamma },-\beta )$ is compared to the THz waveform measured by CEOS. As can be seen, the agreement between our technique and CEOS is excellent. This is also evidenced by the corresponding spectra displayed in Fig. 5(c). The signal-to-noise ratio was found to be $\sim {10}^3$ in both cases. Note that, contrary to Section 2, these experiments were not performed in dry air, explaining why the measured THz waveforms and spectra are less clean than those of Fig. 2.

 

Fig. 5. (a) PISH signals for two opposite heterodyning angles: $\beta =10°$ and $\beta =-10°$. (b) EO signal measured with a CEOS setup (dashed red curve) and heterodyned PISH signal difference, $\mathrm{\Delta }{I}_{2\omega }$, for $\beta =10°$ (black curve). All the signals are plotted as a function of the time delay, $\tau$, between the probe and THz pulses. (c) Corresponding spectra.

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4. COMPARISON OF THE TWO DETECTION SCHEMES

First, the two detection schemes presented above offer the same performances in terms of enhancement factor and distortion parameter. So, the choice of one technique rather than the other will depend on the user’s aim. If one intends to measure THz pulses generated by amplified Ti:sapphire laser systems, the recommended scheme is the optically heterodyne-detected electro-optic sampling (Section 2) for the following reasons: first of all, this setup is very close to the widespread CEOS setup usually dedicated to this task; it makes it possible to acquire the THz waveforms in one scan, limiting the distortions that may arise from any drift of the experimental setup between two consecutive scans for two different heterodyning angles; finally, it does not require the use of a photomultiplier tube, which is much more expensive and fragile than a photodiode. However, this scheme, like the second one, requires good polarizing optics (polarizers, wave plates). Furthermore, these optics must be identical (i.e., from the same manufacturer) on both arms after the EO crystal in order to reduce as much as possible the influence of any intrinsic birefringence. As already mentioned in Section 3, the second detection scheme is interesting if one intends to characterize THz pulses from a setup pumped by near-infrared laser pulses. It is a good alternative to much more complex detection techniques, like THz-field-induced second-harmonic generation in a plasma or the use of a Michelson interferometer, since it is very close to the setup of CEOS and requires very few modifications of the latter to be implemented in the laboratory.

5. CONCLUSION

Two simple and distortion-free methods to sample THz pulses in zinc-blende-type crystals, based on an optical heterodyne detection scheme, have been presented and demonstrated experimentally. The first one makes it possible to measure, in a single measurement and without distortion, a THz waveform by detecting an optical probe beam. Regarding the second one, its relies on the so-called PISH effect and makes it possible to sample a THz wave through THz-induced second-harmonic generation. It might be useful in characterizing THz pulses generated by optical rectification of near-infrared pulsed sources other than Ti:sapphire lasers. Both methods are as good as the more conventional, and widespread, electro-optic sampling technique.