1. INTRODUCTION

The use of optically sampled crystals for the detection of terahertz (THz) frequency radiation has become widespread within applications such as THz time-domain spectroscopy (THz-TDS) [1–5]. The most commonly adopted detection scheme exploits the linear electro-optic (EO) (or Pockels) effect in noncentrosymmetric crystals whereby the THz field induces a birefringence in the crystal (typically ZnTe) that can be probed optically [6–9]. In addition to THz-TDS, EO sampling has similarly been applied to THz-pulsed imaging schemes employing both raster-scanning of the sample [10] and full-field CCD detection [10–12]. In each of these cases, and by virtue of the ultrafast timescales of the EO mechanism, the THz field can be sampled coherently using fs laser pulses that are synchronized with the THz emission [7].

EO crystals have more recently been applied to incoherent [13,14] and coherent sampling of THz fields generated using a quantum cascade laser (QCL) source. In the latter case, synchronization of the optical probe with the QCL has been accomplished by phase-seeding the QCL emission with THz pulses generated both externally [15] and internally [16] to the QCL cavity, as well as by electrical stabilization of the QCL using a phase-locked loop [17]. The use of these coherent EO sampling schemes has led to the study of active mode-locking [18], gain clamping [15], and more generally, the sampling coherence [19] of THz QCLs.

However, an alternative incoherent interaction mechanism between a ZnTe crystal and radiation from a THz QCL source has recently been demonstrated in a standard EO sampling arrangement [20]. Based on the dependence of the measured signal on the modulation frequency of the THz source, this previous study identified that this interaction has a thermal origin. As such, this incoherent response was reported to be much slower than the picoseconds time response indicative of an EO mechanism such as that reported in [19]. Nevertheless, the specific mechanism of this thermal response was not established unequivocally in this previous study.

In this paper, we further investigate this interaction in ZnTe and GaP crystals illuminated by QCL sources in the frequency range 2.2–2.9 THz. We show that the measured response cannot be accounted for using the simple thermo-optic description proposed in [20], whereby the crystal refractive indices vary proportionally to the local temperature change. Rather, our results indicate a photothermoelastic origin whereby the stress distribution established through localized heating of the crystal induces a change in optical birefringence via the photoelastic response of the crystal. In addition to being of fundamental interest, an understanding of this incoherent interaction mechanism and its implications for EO sampling schemes is important for future studies involving THz QCLs as well as other THz sources.

This paper is organized as follows. In Section 2.A, we describe the experimental arrangement for optical sampling of the thermally induced birefringence in noncentrosymmetric crystals. Measurements of the response obtained from ZnTe crystals, under illumination by QCL sources emitting in the range 2.2–2.9 THz, are then presented in Section 2.B. In Section 2.C, we develop a simple model to describe the thermal response of these crystals to the absorption of THz radiation, and relate this to the measurements obtained. It is shown that the measured response cannot be fully accounted for using a simple thermo-optic model. In Section 3.A, a full model of the photothermoelastic response of zincblende crystals to THz illumination is presented. Measurements of the magnitude and spatial dependence of the sampled signal are shown to agree well with the predictions of this model for ZnTe and GaP crystals, in Sections 3.B and 3.C, respectively. Finally, the possible presence of a thermally induced EO response in zincblende crystals is briefly discussed in Section 4, before conclusions are drawn in Section 5.

2. INVESTIGATION OF THERMALLY INDUCED BIREFRINGENCE IN ZINC TELLURIDE

A. Experimental Arrangement

Figure 1(a) shows a schematic diagram of the experimental apparatus, which is based on a standard balanced electro-optic sampling arrangement [7]. In this study, three separate QCL devices were used: two bound-to-continuum (BTC) QCLs [21] tailored to emit at 2.2 and 2.6 THz, and a 2.9 THz QCL based on BTC active region with a phonon extraction/injection stage [22]. Each of the three devices was processed into a 150-μm-wide semi-insulating surface plasmon ridge waveguide with lengths 4.5, 3, and 4.6 mm, respectively. The devices were cooled to $\sim 25\mathrm{K}$ using a helium-cooled continuous-flow cryostat and the THz radiation was collimated, passed through a rotatable wire-grid polarizer, and focused onto the detector crystal using parabolic reflectors. The focused spot sizes on the crystal were measured to be $\sim 395$, $\sim 330$, and $\sim 340\mathrm{\mu m}$ for the 2.2, 2.6, and 2.9 THz devices, respectively. Three crystals were investigated in this work, namely a wedged (110) ZnTe crystal with thickness $L\sim 1.9\mathrm{mm}$, a 0.5-mm-thick (110) ZnTe crystal, and a 1-mm-thick (110) GaP crystal. QCL devices were driven either in continuous-wave, employing mechanical modulation, or in pulsed mode with 100 kHz pulse trains electrically modulated in the range 10–4 kHz, with lock-in detection of the photodiode output in both cases used to improve the detection sensitivity.

 

Fig. 1. (a) Experimental apparatus for optical sampling of the birefringence induced in a crystal by THz radiation. C, crystal; $\lambda /4$, quarter-wave plate; WP, Wollaston prism; P, optical polarizer; TP, THz polarizer. (b) Geometry of the (110) zincblende crystal showing the crystallographic axes ($x^{\prime }–y^{\prime }–z^{\prime }$), the Cartesian ($y–z$), and cylindrical $(r,\phi )$ axes in the (110) plane of the crystal, and the principal axes of the index ellipsoid ($Y–Z$). The $x^{\prime }$ and $y^{\prime }$ directions are bisected by the normal to the page. All other axes shown are in the plane of the page. Also shown are the angles $\psi$ and $\theta$ of the polarization directions of the THz and probe beams relative to the [$-1$, 1, 0] direction.

Download Full Size | PDF

The crystal was optically sampled using the combined beams from two external cavity diode lasers at 778.3 nm, providing a combined power of $\sim 20\mathrm{mW}$. The use of two lasers in this experiment is solely to provide greater optical power. Control of the polarization direction of the combined beams was achieved using a rotatable Glan–Thompson polarizer, with a rotatable half-wave plate being used in each of the individual beams to maximize the power delivered to the crystal. The diode lasers were focused onto the crystal collinearly with the THz beam and a balanced sampling arrangement was used. Initially, the polarization of the sampling beam and THz beam were set to be parallel to the [$-1$, 1, 0] direction of the crystal, giving $\theta =0$ [see Fig. 1(b)]. The sampling position of the probe beam relative to the THz beam was adjusted to give the maximum response on the balanced photodiodes.

B. Optical Sampling of Birefringence

In a standard EO sampling arrangement [7], the ellipticity of a linearly polarized near-infrared probe beam is modified by the optical anisotropy induced in a detector crystal by THz radiation. The relative phase delay $\delta$ between the orthogonal polarization components of the probe beam that result from this birefringence is obtained by integrating along the beam propagation axis,

For the experimental arrangement described in Section 2.A, the phase delay can be inferred from the measured photodiode signal using Eq. (2) with $\theta =0$ and $\alpha =\pi /4$. Figure 2(a) shows the phase delays obtained using each of the three QCL devices with the $\sim 1.9\mathrm{mm}$ ZnTe crystal, and also for the 0.5 mm ZnTe crystal with the 2.2 THz laser. For these measurements the QCL powers were set to be approximately equal and were measured using a THz power meter to be 845, 800, and 805 μW for the 2.2, 2.6, and 2.9 THz lasers, respectively. As can be seen, the measured response decreases with modulation frequency and tends toward an inverse relationship at higher frequencies. A similar behavior has been observed previously [20] using a 3.2 THz QCL, and is symptomatic of a thermal mechanism [25]. By rotation of the wire grid polarizer it has also been confirmed that, after correction for the variation of the transmitted THz power arising from the elliptically polarized QCL output, the measured phase delay is independent of the polarization direction $\psi$ of the THz field. This is as expected for a thermal mechanism, and in agreement with observations reported elsewhere [20].

 

Fig. 2. (a) Phase delays measured as a function of modulation frequency for a $\sim 1.9\mathrm{mm}$-thick ZnTe crystal, using QCL devices emitting at 2.2 THz (red), 2.6 THz (blue), and 2.9 THz (black), and also for a 0.5 mm ZnTe crystal at 2.2 THz (green). The error bars have been calculated based on three repeated measurements. The solid lines are fits to Eq. (6). (b) Corresponding temperature modulation at the crystal surface predicted using Eq. (7).

Download Full Size | PDF

C. Simple Thermal Model and Analysis

In order to characterize the optically sampled signal it is necessary first to evaluate the temperature increase arising from local heating of a crystal under THz illumination.

We consider the case of a THz beam of radius $w_0$ incident on a crystal of density $\rho$ and specific heat capacity $C$. The modulated THz power at a depth $x$ below the crystal surface is assumed to take the form $P=(1-R)P_0{e}^{-\alpha x}{e}^{i\omega t}$, where $\alpha$ is the absorption coefficient, $R$ is the Fresnel reflection coefficient, $P_0$ is the incident power and $\omega$ is the modulation frequency. From conservation of energy we arrive at the heat equation for the temperature rise $T(x,t)$

In the simple thermo-optic model [20,26] it was assumed that this temperature modulation gives rise to a modulation of the optical birefringence $\mathrm{\Delta }n$ through the linear relation

In the present case of a thermally induced birefringence, the phase delay $\delta$ can be obtained by applying Eqs. (1), (4), and (5) to give the result

Figure 2(a) shows fits of the data to Eq. (6), in which the values $\beta$ and $G$ are treated as free parameters. The absorption coefficients and Fresnel reflection coefficients used in these fits were obtained using THz-TDS, with $\alpha$ being found to increase from $15{\mathrm{cm}}^{-1}$ at 2.2 THz to $31{\mathrm{cm}}^{-1}$ at 2.9 THz. As such, the factor $(1-e^{-\alpha L})\approx 1$ for the crystal with $L\sim 1.9\mathrm{mm}$, indicating almost complete absorption of the THz radiation at all three frequencies. Similarly, only $\sim 47\%$ is absorbed in the case of the 0.5 mm crystal at 2.2 THz, which gives rise to the smaller optical phase delay observed. For ZnTe the value $\rho C=1.54{\mathrm{Jcm}}^{-3}{\mathrm{K}}^{-1}$ was also used [27]. The fits are seen to agree well with the measured data and yield the average values $\beta =1.1\times {10}^{-6}{\mathrm{K}}^{-1}$ and $G=77{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$. As a matter of interest, the data can also be expressed in terms of the amplitude of temperature modulation at the crystal surface predicted via Eq. (7). This is shown in Fig. 2(b) and reveals temperature rises on the order of $\sim 10\mathrm{mK}$ at low modulation frequencies for the experimental conditions described here.

D. Analysis Based on Simple Thermo-Optic Model

One mechanism proposed to explain the origin of the observed modulation of optical birefringence is through a simple temperature dependence of the refractive indices [20]. Under this model the modulation of the birefringence can be expressed in terms of the temperature modulation according to Eq. (5) in which the coefficient $\beta$ is given by

Briefly, and following the analysis in [23], the signal obtained in a crossed sampling arrangement with no THz radiation, and with the wave plate adjusted (in the absence of the crystal) to give minimum signal transmitted through the crossed-polarizer, is given by

3. PHOTOTHERMOELASTIC EFFECT IN ZINCBLENDE CRYSTALS

A. Theory of Photothermoelastic Effect

We next consider the presence of a photothermoelastic modulation of the optical birefringence in zincblende crystals under THz illumination. Such a response may be present owing to the thermal stresses induced in the crystal by localized heating. To evaluate this stress distribution we make use of the model developed for the case of an isotropic thin disk subject to a radial temperature distribution $\mathrm{\Delta }T(r,x)$, where $r$ is the radial coordinate. The radial ${\sigma }_{rr}$ and tangential ${\sigma }_{\phi \phi }$ components of the thermally induced stress are given as [30]

For a crystal of any symmetry a stress field will induce a perturbation to the refractive indices via the photoelastic mechanism [31]. This effect can be described in terms of the perturbation to the optical indicatrix according to

The optical indicatrix ellipsoid under the applied stress field thus becomes

Thus, we may model the balanced photodiode signal accordingly. The stress field tensor ${\sigma }_i$ can be obtained from Eq. (12), making use of Eqs. (10), (11), and (24). These tensor components thereby enable calculation of the modulation of the birefringence via Eq. (23), and by applying Eqs. (14) and (16). The total phase delay can then be obtained from Eq. (1), from which the balanced photodiode signal can be calculated using Eq. (2). An important characteristic of this signal is that it exhibits spatial inhomogeneity originating from both the radial stress distribution, arising from localized heating in the crystal, and also through the $\alpha$-dependence of Eq. (2), which accounts for the orientation of the indicatrix ellipsoid relative to the probe beam polarization. This spatial dependence is confirmed experimentally in the next section.

B. Experimental Results and Analysis for ZnTe

The spatial inhomogeneity of the photothermoelastic signal was investigated experimentally using the experimental arrangement described in Section 2.C, with the 2.2 THz QCL and the $\sim 1.9\mathrm{mm}$ ZnTe crystal. For these measurements the QCL was driven in pulsed mode and electrically modulated at a frequency of 40 Hz. In order to adjust the position of the probe beam focus relative to the focused THz beam, the focusing lens was manually scanned in the plane perpendicular to the beam axes using translation stages, with the balanced photodiodes being realigned for each measurement. Figure 3(a) shows a contour plot of the photodiode signal $\mathrm{\Delta }I/I_p$ measured across an area of $1.4\mathrm{mm}\times 1.4\mathrm{mm}$ using a step size of 0.2 mm in both $y$ and $z$ directions. For comparison the spatial variation of the photodiode signal, calculated using Eq. (2) as described in Section 3.A, for the same experimental conditions is shown in Fig. 3(b). For this calculation the following values of the photoelastic constants [33] and stiffness matrix components [27] have been used for ZnTe: $p_{11}^{\prime }=0.144$, $p_{12}^{\prime }=0.094$, $p_{44}^{\prime }=0.046$, and $c_{11}^{\prime }=7.1\times {10}^{10}\mathrm{Pa}$, $c_{12}^{\prime }=4.1\times {10}^{10}\mathrm{Pa}$, $c_{44}^{\prime }=3.1\times {10}^{10}\mathrm{Pa}$. As can be seen, for both the theoretical prediction and experimental results, the sign of the signal alternates in adjacent quadrants of the crystal surface and reaches a maximum magnitude at a critical radius along the $\phi =\pm \pi /4$ diagonals. The intensity is also seen to vanish toward zero at the origin. This radial variation of the signal originates from the radial dependence of the factor (${\sigma }_{rr}-{\sigma }_{\phi \phi }$) appearing in Eq. (12), which exhibits its maximum magnitude at some critical radius beyond that at which the tangential component of the stress tensor becomes tensile (${\sigma }_{\phi \phi }>0$), as described by Eq. (11). It should be noted that such a spatial dependence of the measured response would not arise through a simple temperature-dependence of the refractive indices, as described by the simple thermo-optic model.

 

Fig. 3. (a) Contour plot showing the spatial variation of the photodiode signal $\mathrm{\Delta }I/I_p$ in the $y–z$ plane of the ZnTe crystal. The origin (0,0) corresponds to the center of the THz beam. For these measurements $\theta =0$ and the polarization direction of the THz field $\psi =0$. (b) Spatial variation of the (normalized) photodiode signal calculated using the photothermoelastic model for the same experimental conditions.

Download Full Size | PDF

We may further validate the photothermoelastic model through comparison of the magnitude of the predicted modulation of the birefringence with that observed experimentally. As noted previously, the optical birefringence arising through the photothermoelastic mechanism cannot be described fully using the simple model described by Eqs. (4) and (5). Nevertheless, to aid comparison with the experimental results presented in Section 2.C, it is convenient to cast the birefringence $\mathrm{\Delta }n(r,x)$ in terms of $\mathrm{\Delta }T(x)$ defined by Eq. (4). In this case spatial-dependence of $\mathrm{\Delta }n(r,x)$ becomes absorbed into the coefficient $\beta$. Figure 4(a) shows the simulated modulation amplitude calculated using Eq. (23), as described in Section 3.A, and cast in terms of the spatially dependent coefficient $\beta$. Here, the value of the linear thermal expansion coefficient ${\alpha }_{\text{th}}=8\times {10}^{-6}{\mathrm{K}}^{-1}$ [27] has been used, and the Young’s modulus for ZnTe has been obtained from the stiffness constants using the relation $Y=(c_{11}^{\prime }+2c_{12}^{\prime })(c_{11}^{\prime }-c_{12}^{\prime })/(c_{11}^{\prime }+c_{12}^{\prime })$ [31]. It can be seen that the modulation is greatest away from the center of the THz beam and exhibits local maxima at positions $\phi =m\pi /2$ on the crystal surface (i.e., along the $y$ and $z$ axes). It should thus be noted that, by suitable rotation of the probe beam polarization away from $\theta =0$, the photodiode signal could attain a maximum at these positions of maximum birefringence. Nevertheless, for these positions the indicatrix component $\mathrm{\Delta }{B}_4=0$ and hence the main axes of the indicatrix ellipsoid become aligned with the probe beam polarization [i.e., $\alpha =m\pi /2$, see Fig. 4(b)] for the case when $\theta =0$, giving zero sampled signal. Conversely, at the position of maximum signal [see Fig. 3(b)], a value $\beta =3.2\times {10}^{-6}{\mathrm{K}}^{-1}$ is predicted by the model. This agrees well with the value $\beta =1.1\times {10}^{-6}{\mathrm{K}}^{-1}$ obtained experimentally at the position of maximum response, which further confirms the proposed photothermoelastic origin of the signal. The discrepancy between simulated and measured values could be attributed to a number of assumptions made in the theoretical model. First, the approximation was made that the THz beam and radial temperature distribution follow a Gaussian distribution. We note that any asymmetry in the THz beam would cause deviation from the radial and tangential stress components described by Eqs. (10) and (11), and therefore to the optically sampled birefringence. Furthermore, a simple proportional form was adopted in our thermal model to describe the radial conduction of heat away from the irradiated volume, and axial heat conduction was neglected. The consequences of the latter assumption may be negligible, however, by virtue of the fact that the measured signal arises from the integration of the phase delay accumulated along the probe beam axis. A full numerical model of the temperature distribution within the crystal would nevertheless be feasible. Further discrepancies of the simulated response will have arisen due to uncertainties in the material parameters used, for example the photoelastic matrix components, which were taken from measurements at a slightly different optical wavelength of 633 nm.

 

Fig. 4. (a) Spatial variation of the modulation of the birefringence in the $y–z$ plane of the ZnTe crystal, calculated using the photothermoelastic model and cast in terms of the coefficient $\beta$ defined by Eq. (5). (b) Spatial variation of the angle $\alpha$ obtained via Eq. (16). The origin (0,0) corresponds to the center of the THz beam. 

Download Full Size | PDF

As noted above, for the case of the probe beam polarization being aligned parallel to the [$-1$, 1, 0] direction of the crystal, the maximum measured response is observed at positions for which $\phi \approx \pm \pi /4$, $\pm 3\pi /4$. According to the photothermoelastic model the stress component ${\sigma }_4$, and hence $\mathrm{\Delta }{B}_4$, attain maximum amplitudes in these cases and also $\mathrm{\Delta }{B}_2\approx \mathrm{\Delta }{B}_3$. Therefore, at these positions of maximum response, the orientation of the ellipsoid is predicted to be $\alpha \approx \pm \pi /4$, $\pm 3\pi /4$ from Eq. (16). This can be verified experimentally by monitoring the photodiode signal for different orientations of the probe beam polarization $\theta$, with the quarter-wave plate being adjusted in each case to recover the a balanced signal in the absence of THz radiation. From Eq. (2) the optically sampled signal should then vary according to $\mathrm{\Delta }I\sim {\cos }^2(2\theta )$. Figure 5 shows this measured response, and also a fit to an offset form of this functional dependence. The angular variation of the measured signal can be seen to agree well with that expected for the photothermoelastic response described.

 

Fig. 5. Photodiode signal $\mathrm{\Delta }I/I_p$ (black squares) measured for different polarization directions $\theta$ of the probe beam. Also shown (blue line) is the functional form $\mathrm{\Delta }I\sim {\cos }^2(2\theta )$. These measurements were taken at the position corresponding to the maximum signal attained in Fig. 3(a), and with the polarization direction of the THz field $\psi =0$.

Download Full Size | PDF

C. Experimental Results and Analysis for GaP

In order to evaluate the photothermoelastic model further, measurements were also performed using a 1-mm-thick (110) GaP crystal, which also belongs to the $-43\mathrm{m}$ point group. Figure 6 shows the phase delays measured using the 2.2 THz laser for electrical modulation frequencies in the range 10 Hz–3 kHz. For these measurements the same experimental procedure described in Section 2.C was used, with the sampling position of the probe beam again corresponding to the position of maximum response on the photodiodes. Figure 6 also shows a fit of the data to Eq. (6), in which the values $\beta$ and $G$ are treated as free parameters. In the case of GaP, the following material parameters were used: $\alpha =4{\mathrm{cm}}^{-1}$, $R=0.30$ (calculated from THz–TDS measurements), and $\rho C=1.78{\mathrm{Jcm}}^{-3}{\mathrm{K}}^{-1}$ [34]. The fit is again seen to agree well with the measured data and yields the values $\beta =1.8\times {10}^{-6}{\mathrm{K}}^{-1}$ and $G=79{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$.

 

Fig. 6. Phase delays measured as a function of modulation frequency for a 1-mm-thick GaP crystal, using a QCL device emitting at 2.2 THz.

Download Full Size | PDF

Using the same approach as described for ZnTe, the coefficient $\beta$ can be quantitatively estimated at the position of maximum sampled signal using the photothermoelastic model. In this case a value $\beta =4.9\times {10}^{-6}{\mathrm{K}}^{-1}$ is predicted, which is again in good agreement with that measured. For this calculation the following values were used for GaP: $p_{11}^{\prime }=0.161$, $p_{12}^{\prime }=0.088$, $p_{44}^{\prime }=0.073$; $c_{11}^{\prime }=14.1\times {10}^{10}\mathrm{Pa}$, $c_{12}^{\prime }=6.3\times {10}^{10}\mathrm{Pa}$, $c_{44}^{\prime }=7.0\times {10}^{10}\mathrm{Pa}$ [35]; ${\alpha }_{\text{th}}=4.7\times {10}^{-6}{\mathrm{K}}^{-1}$ [34]; and $n=3.20$ [36]. This predicted value also overestimates the measured result by a similar factor to that found for ZnTe, which can be explained by consideration of the approximations adopted in the photothermoelastic model, as discussed previously.

Whereas the coefficient $\beta$ characterizes the photoelastic response of a crystal to a particular temperature distribution (or equivalently a particular stress field), the choice of sampling crystal will also influence the measured response through the magnitude of the temperature modulation induced by the incident THz radiation, as well as the crystal thickness. We may define a figure-of-merit $\mathcal{F}$ that determines the maximum measured response for a general (110)-orientated crystal with point group $-43\mathrm{m}$. Retaining only the crystal physical parameters, and assuming a modulation frequency $\omega \gg G/\rho C\pi w_0^2$, we arrive at the relation

4. THERMALLY INDUCED ELECTRO-OPTIC EFFECT IN ZINCBLENDE CRYSTALS

It is interesting to note that an additional incoherent mechanism for modulation of the optical birefringence under THz illumination may be present in the zincblende crystals considered in this work. It is known that cubic crystals do not exhibit primary pyroelectricity owing to their nonpolar crystal structure [31]. Nevertheless, a secondary pyroelectric response [31] may occur in noncentrosymmetric crystals under localized heating, whereby the thermally induced stress field results in an electric displacement through the piezoelectric response of the material. This in turn can induce a change in the optical birefringence through a linear (Pockels) EO effect, which is also exhibited by noncentrosymmetric crystals.

To estimate the magnitude of this response we start by calculating the electric displacement vector $D_i$ arising from the piezoelectric response to the thermally induced stress field in the form [31,32]

The optical birefringence arising from the EO response to this electric field can be obtained following the analysis presented elsewhere [8,24]. The final modulation of the birefringence is found to be

The balanced photodiode signal can again be obtained from the birefringence through Eqs. (1) and (2). Figure 7 shows the spatial variation of this EO signal $\mathrm{\Delta }I/I_p$ calculated using the values $d_{14}^{\prime }=0.9\times {10}^{-12}{\mathrm{mV}}^{-1}$ and $r_{41}=3.9\times {10}^{-12}{\mathrm{mV}}^{-1}$ [27] for ZnTe, in which the signal amplitude is shown on the same numerical scale as Fig. 3(b). As for the photothermoelastic case, this signal exhibits spatial inhomogeneity originating from both the radial stress distribution in the crystal and also through the geometrical dependence of the optical sampling described by Eq. (2). However, as can be seen, the EO signal magnitude is everywhere $<4\%$ that of the photothermoelastic signal, confirming that the latter is the dominant effect contributing to our measured response.

 

Fig. 7. Spatial variation of the photodiode signal $\mathrm{\Delta }I/I_p$ in the $y–z$ plane of the ZnTe crystal, calculated assuming only a thermally induced EO effect. The signal is displayed on the same numerical scale as Fig. 3(b).

Download Full Size | PDF

5. CONCLUSIONS

We have presented, for the first time, evidence of a photothermoelastic response in zincblende crystals under THz illumination. Using THz QCL sources emitting in the frequency range 2.2–2.9 THz, the optical birefringence induced in ZnTe and GaP crystals have been measured using a balanced sampling arrangement. The dynamic behavior of this response demonstrates a mechanism of thermal origin, although we have shown that the measured response cannot be accounted for using a simple thermo-optic description whereby the induced birefringence is proportional to the temperature rise in the crystal. Rather, our results indicate a photothermoelastic origin whereby the stress distribution established through localized heating of the crystal induces a change in optical birefringence via the photoelastic response of the crystal. A full mathematic model of this photothermoelastic mechanism in (110)-orientated zincblende crystals has been developed, and shown to agree well with experimental data. Specifically, measurements of the magnitude, and the orientational- and spatial-dependence of the sampled signal are found to agree well with the predictions of this model. In addition to being of fundamental interest, an understanding of this incoherent interaction mechanism and its implications for EO sampling schemes is important for future studies involving THz QCLs as well as other THz sources.