1. Introduction

Electro-optic (EO) mixing [1] of an optical probe pulse and a terahertz (THz) pulse is the favored method for detection of THz fields in single-shot experiments. Single-shot THz detection is becoming increasingly important in a number of applications including ultrafast relativistic electron beam diagnostics [2, 3, 4], shock induced phase transformations and laser driven melting [5]. Additionally, the characterization of ultrashort electron bunches are of growing interest at large facilities, such as the Stanford Linear Accelerator Center [6], Brookhaven National Laboratory [7], and FELIX [2], as well as at laser based accelerators - for example, experiments at the Lawrence Berkeley National Laboratory [4]. In many of these experiments, the sources have either low repetition rates or they have inherent shot-to-shot fluctuations, making single-shot schemes necessary. Several schemes exist for single-shot detection of THz fields and, as stated above, most of these exploit the EO effect with a chirped probe pulse [8, 9, 10].

The EO effect with a chirped probe, E _opt(t), is conventionally described as the product of the THz field, E _THz(t), and the probe beam as [8, 9, 11], E _out(t)=E _opt(t)[1+aE _THz(t)], where a is a constant. This conventional equation has been experimentally verified [8, 10] using the data from a multi-shot scan. However, in a recent Letter [12], the equation describing the electric field of a chirped optical probe pulse was reported to be, $E_{\mathrm{out}}\left(t\right)=E_{\mathrm{opt}}\left(t\right)+a\frac{d}{\mathrm{dt}}\left[E_{\mathrm{opt}}\left(t\right)E_{\mathrm{THz}}\left(t\right)\right]$ , which is in contradiction with the conventionally used equation. Furthermore, these authors also reported agreement with experiment [12]. In this letter, we resolve this conflict and show that the correct equation depends on various factors such as the residual birefringence of the EO crystal, the analyzer detuning, and the phase retardation from waveplates. In the small bandwidth limit, the correct equation is E _out(t)=E _opt(t)[1+aexp(iθ)E _THz(t)], in agreement with both the conventional equation and the recently reported result [12]. We also experimentally verify our result.

2. Equations governing EO effect with a chirped pulse

A typical single-shot EO detection set-up consists of an EO crystal placed between crossed polarizers. The THz induced leakage of the probe is either detected in a spectrometer [8, 10] or by performing a cross-correlation [9]. Our goal is to derive the equation for the optical probe exiting the analyzer. For coherent detection, some background probe light is necessary which is obtained by the inherent residual birefringence of the EO crystal or analyzer detuning, or through an additional waveplate after the EO crystal (see Fig. 1(a).). For definiteness, we assume that the optical probe is y-polarized as it enters the crystal, and interacts with a co-polarized THz field. The electric field at the output of the crystal is described by equations similar to Eq. (6) of Ref. [12]. The EO crystal generates both the x and y -polarized optical fields due to the effective second order crystal susceptibility coefficients χ^x _eff and χ^y _eff along x and y directions [13]. In the small signal limit [13], the field exiting a thin birefringence free EO crystal can be written as [12],

The constants B^x,B^y account for the efficiencies of sum- and difference-frequency mixing along x and y due to χ^x _eff and χ^y _eff, which are considered frequency independent within the detection bandwidth. Here we restrict our derivation to isotropic EO crystals. Typically, even isotropic EO crystals have some strain induced residual birefringence [14]. We account for the small residual birefringence as a separate waveplate having a small phase shift Γ₁ with crystal axes oriented at an angle ϕ ₁ with respect to x -axis. For situations when the residual birefringence is not sufficient for coherent detection, a wave plate can be used. Let the orientation of the waveplate be ϕ ₂ and the corresponding phase shift be Γ₂. The field exits the analyzer detuned by a small angle, θ_d. Using the Jones calculus, this output field is given by

where M_1,2 are the Jones matrices corresponding to the residual birefringence and the wave-plate.

Here $a_{\mathrm{1,2}}=e^{-i\frac{{\Gamma }_{\mathrm{1,2}}}{2}}{\cos }^2\left({\varphi }_{\mathrm{1,2}}\right)+e^{i\frac{{\Gamma }_{\mathrm{1,2}}}{2}}{\sin }^2\left({\varphi }_{\mathrm{1,2}}\right),\mathrm{and}{b}_{\mathrm{1,2}}=\sin \left(\frac{{\Gamma }_{\mathrm{1,2}}}{2}\right)\sin \left(2{\varphi }_{\mathrm{1,2}}\right)$ are the Jones matrix coefficients [15]. Since the residual birefringence is small, we can approximate a ₁≈1 and $b_1\approx \frac{{\Gamma }_1}{2}\sin \left(2{\varphi }_1\right)$ . After simplification, we can write the output field after the analyzer as,

In these equations, $\alpha \exp \left(\mathrm{i\theta }\right)=\frac{B^x\left(a_2-b_1{b}_2\right)-i{B}^y\left(b_2+b_1{a}_2\right)}{\left(b_2+b_1{a}_2\right)}$ and the proportionality constant, -i, is dropped.

For most practical scenarios these equations can be simplified further. Typically, the THz bandwidth is much narrower than the optical bandwidth, which in turn is much less than the optical carrier frequency, ω ₀. Rewriting ω=ω ₀+Δω and making the approximation Δω<<ω ₀, the first ω in the second term of Eq. (5) can be approximated to be ω ₀. With this approximation, Eqs. (5) and (6) are used to derive the following equations,

Here a=αω ₀, is a real constant and t is the time in the group velocity frame of the optical pulse. We note that THz pulse bandwidths, ΔΩ, are always less than the optical frequency, ω ₀, by at least an order of magnitude. In most chirped probe pulse detection schemes, the THz pulse can be correctly resolved, provided, the probe has a slightly greater bandwidth than the THz pulse, that is ΔΩ≤Δω. This suggests that for the detection, we can always choose an optical chirped probe that easily satisfies the small bandwidth condition Δω<<ω ₀. Therefore, the small bandwidth approximation can always be satisfied during the THz pulse detection. However, if for some reason, the probe is derived from a few cycle pulse then it is necessary to use Eq. (5) and Eq. (6).

Alternatively, Eqs. (7) and (8) can also be derived using the time domain Eq. (6) under the slowly varying amplitude approximation. We write the fields in Eq. (6) in terms of their complex envelopes and carrier frequencies as E ^eff _THz(t)=A _THz(t), and E _opt(t)=A _opt(t)exp(iω ₀ t). Here we have assumed that the THz frequency is much less than the optical bandwidth. The derivative in Eq. (6) can be expressed as,

 

Fig. 1. (a) Schematic of experimental scheme for the verification of EO effect with chirped pulse. (b) Set-up for direct characterization of THz field.

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Using the slowly varying amplitude approximation,

in Eq. (9), Eq. (6) can be written as Eq. (8). This approximation is valid when the total bandwidth of the optical field and THz field is much smaller than the optical carrier frequency.

Equations (7) and (8) represent the field in the general case when the background field is derived from a combination of methods. In practice the background needed for coherent detection is obtained from only one method. In most experiments, the residual birefringence is sufficiently large to enable coherent detection eliminating the need for a waveplate or analyzer detuning. For such cases, it can be shown that θ=0, and the commonly used description of the EO effect is applicable. However, when the EO crystal is thin and the residual birefringence cannot provide necessary background, an additional waveplate can be used. For the case of a quarter wave plate (QWP) at small angle, $\theta =\frac{\pi }{4}$ . For a half wave plate (HWP) at small angle, $\theta =\frac{\pi }{2}$ . Alternatively, the background needed for coherent detection can be obtained by detuning the analyzer. When no residual birefringence or waveplates are present, analyzer detuning results in $\theta =\frac{\pi }{2}$ .

3. Experimental verification

We verified these derivations with an experiment based on spectral in-line interferometry. The experimental verification involves a comparison of the direct measurement of spectral difference, ΔĨ=|Ẽ_out|²-|Ẽ_opt|², with the calculated spectral difference using Eq. (7) (or Eq. (8)). In order to calculate this difference, we experimentally measure the optical probe pulse and the THz pulse separately and use them with Eq. (8).

Our main experimental geometry is shown in Fig. 1(a), and it is similar to the single-shot THz detection schemes using a spectrometer [8, 10]. A 815 nm, 50 fs pulse from a regenerative amplifier is directed to a beamsplitter, where the intense fraction passes through a 1 mm thick ZnTe crystal generating a THz field, and the remaining fraction passes through a second beam-splitter creating two pulses of equal amplitude. Following the second beamsplitter, one pulse of this pulse-pair is chirped, while the other pulse of this pulse-pair is used as a short-probe useful for scanned detection of THz. These two pulses are recombined and aligned to traverse the same beam path through the polarizers, detector ZnTe crystal and the waveplate. By blocking one of the two pulses, we can realize either the setup shown in Fig. 1(a) or Fig. 1(b). A spectrometer is used with the chirped probe, while a photo-diode is used with the short probe.

 

Fig. 2. (a) Background spectrum, |Ẽ_opt|², of chirped optical probe in absence of THz field. (b) Modulated probe, |Ẽ_out|², in presence of THz field. (c) Temporal intensity profile of the probe pulse calculated using |Ẽ_opt|² and the spectral phase. (d) THz field measured directly using a short optical probe pulse.

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The chirped probe pulse is derived from the 50 fs pulse using a grating stretcher. The stretcher consists of a pair of 1200 lpm grooved gratings in a standard geometry separated by 2.4 cm. This stretcher results in a calculated spectral phase, ϕ, which is characterized by the second and third order dispersion of $\frac{1}{2}\frac{d^2\varphi }{d{\omega }^2}=-3.9\times {10}^4{\mathrm{fs}}^2\mathrm{and}\frac{1}{6}\frac{d^3\varphi }{d{\omega }^3}=4.6\times {10}^4{\mathrm{fs}}^3$ respectively [16]. We also experimentally characterize the second order chirp by delaying the chirped probe with respect to the THz pulse and noting the shift in the difference spectrum, ΔĨ, as detailed in Refs. [8, 10]. The experimental GDD, -3.5×10⁴ fs², is close to the calculated value. The background probe spectrum (|Ẽ_opt|²) measured in absence of the THz field, using a 0.05 nm resolution imaging spectrometer, is plotted in Fig. 2(a). The spatial imaging dimension is used to verify that the probe pulse is free from space-time tilt. The spectrum has modulations due to multiple reflections from the beam splitters in the beam path, which result in some uncharacterized phase. Using the calculated second and third order dispersions and the optical spectrum, the probe temporal pulse shape is calculated and the intensity is plotted in Fig. 2(c).

For the EO mixing experiments, one of two different ZnTe crystals is used. The first crystal, 1 mm thick, has sufficient residual birefringence and does not require the use of an additional waveplate. The second crystal, 200 μm thick, is strain-free and has negligible birefringence. This crystal requires an additional waveplate or detuned analyzer. The THz field is characterized using a short-probe as shown in Fig. 1(b). The THz field measured with the first crystal (with no waveplate) is very similar to that measured using the second crystal and a QWP. The characterized THz field is plotted in Fig. 2(d).

 

Fig. 3. The blue curve is the directly measured spectral difference for a residual birefringence case (θ=0). The red curves are calculated spectral difference using (a) Eq. (8) and (b) using Eq. (6) of Ref [12] for various delays (T).

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We verify the derived equations for four different cases outlined earlier, namely: residual birefringence (θ=0), QWP at small $\left(\theta =\frac{\pi }{4}\right)$ , HWP at small angle $\left(\theta =\frac{\pi }{2}\right)$ , and detuned analyzer $\left(\theta =\frac{\pi }{2}\right)$ . For each of these cases, since θ is known, we can use the measured probe pulse shape and the THz pulse in Eq. (8) to calculate the spectral difference, ΔĨ. The only unknown in this calculation is the relative time delay between the probe pulse and the THz pulse, which is adjusted so that the calculated and measured spectral peaks line up. The calculated spectrum is compared with the direct measurement obtained by subtracting the background spectrum (|Ẽ_opt|²) from the spectrum measured in presence of THz (|Ẽ_out|²). A typical spectrum in presence of the THz is shown in Fig. 2(b). For the case of residual birefringence, we use the first crystal in between a pair of crossed crystal polarizers. The second crystal with negligible birefringence is used in the other cases. For the second and third cases, the QWP and HWP are oriented at an angle of ϕ ₂≈2^o respectively. For the last case, the analyzer is detuned by ≈ 1.5^o from the crossed position. The measured and calculated spectra for all the four cases are summarized in Fig. 4, and these results are discussed in detail later.

Just like many other schemes [8, 9, 10] which compare a THz waveform obtained by a multi-shot scan with a single-shot method, there is naturally an unknown time-delay involved in our experiments. The unknown time delay arises because in either a multi-shot scan or a single-shot method, only the scale of the time axis -not the absolute time- is known. Typically, this unknown time delay can be determined by matching the peaks of the THz waveforms [8, 9, 10].

 

Fig. 4. Comparison of directly measured difference spectra (blue lines) with the calculated difference spectra (red lines) using Eq. (8) for four different cases. (a) Residual birefringence (θ=0), (b) QWP at small angle $\left(\theta =\frac{\pi }{4}\right)$ , (c) HWP at small angle $\left(\theta =\frac{\pi }{2}\right)$ , and (d) detuned analyzer $\left(\theta =\frac{\pi }{2}\right)$ .

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In our experiments, we adjust the time delay so that the calculated and measured spectral differences agree both in shape as well as the wavelength position. Fig. 3 (a) shows our approach to determine the time delay for the first case (residual birefringence, θ=0). The red curves represent calculated difference spectra for various time delays (T). We notice that for T=5.8 ps, the measured and calculated spectra match closely. The red curves in Fig. 3(b) show the calculated difference spectra using Eq. (6) of Ref [12]. Obviously none of these curves agree with the measured difference spectra, indicating that Eq. (6) of Ref [12] does not describe the residual birefringence case.

We summarize our results for the four different cases in Fig. 4. For all these plots, the time delay is determined as mentioned before and it is approximately the same. From the figure, we see that there is an excellent agreement between the measured and calculated difference spectra (using Eq. (8)), verifying the validity of Eq. (8). We also used Eq. (6) instead of Eq. (8) and found that for all cases, the calculated spectra overlap, as expected, with the corresponding red curves in the Fig. 4. Next, we compared the calculated difference spectra obtained using the conventional equation [8] and the Eq. (6) of Ref [12], with the measured difference spectra. With the conventional equation, there is a match with Fig. 4(a) alone, indicating that the conventional equation describes only the residual birefringence case. With Eq. (6) of Ref [12], the curves match with Fig. 4(c) and Fig. 4(d). Thus the Eq. (6) of Ref [12] is only valid for the case of a HWP at small angle $\left(\theta =\frac{\pi }{2}\right)$ or a detuned analyzer $\left(\theta =\frac{\pi }{2}\right)$ .

4. THz retrieval with the new equation

Although several schemes exist for single-shot THz detection, the simplest techniques are the ones which use a single chirped probe arranged collinear to the THz field. In these schemes, the probe-bandwidth-limited THz field detection is only possible using a reconstruction technique along with a well characterized probe pulse. We have recently described two approaches [11, 17] for such a THz field reconstruction. Here we show that reconstruction approaches can be adapted to our new equation. The first reconstruction approach [11] is based on matrix inversion, where the difference spectrum ΔĨ=|Ẽ_out|²-|Ẽ_opt|², is expressed as a linear matrix equation, ΔĨ=a B E ^eff _THz. Here B denotes a two dimensional matrix whose elements, using Eq. (6) are given by,

The second reconstruction approach [17] is based on phase-retrieval, where a guess field is iteratively updated while transforming between real and frequency domains. In this scheme, the frequency domain measured background is replaced, while in the time domain an error metric is minimized. For the new equation derived here, the error metric,

is minimized to obtain an update for the THz field, E ^eff _THz(t)

5. Conclusions

We have presented equations that describe the EO effect with chirped pulses in a typical single-shot THz measurement set-up. The conventional equation can be used when the EO crystal has sufficient residual birefringence. In the small signal limit, nearly all experimental schemes are described by the equation, E _out(t)=E _opt(t)[1+aexp(iθ)E ^eff _THz(t)], provided, the optical probe pulse is not derived from a few cycle pulse. We have experimentally verified this equation using spectral in-line interferometry. For an extremely broadband optical probe, derived from a few cycle pulse, the general equation is $E_{\mathrm{out}}\left(t\right)=E_{\mathrm{opt}}\left(t\right)-\mathrm{i\alpha }\exp \left(\mathrm{i\theta }\right)\frac{d}{\mathrm{dt}}\left[E_{\mathrm{opt}}\left(t\right)E_{\mathrm{THz}}^{\mathrm{eff}}\left(t\right)\right].$ .