Multidimensional spectroscopy at visible and infrared frequencies has opened a window into the transfer of energy and quantum coherences at ultrafast time scales. For these measurements to be performed in a manageable amount of time, one spectral axis is typically recorded in a single laser shot. An analogous rapid-scanning capability for THz measurements will unlock the multidimensional toolkit in this frequency range. Here, we first review the merits of existing single-shot THz schemes and discuss their potential in multidimensional THz spectroscopy. We then introduce improved experimental designs and noise suppression techniques for the two most promising methods: frequency-to-time encoding with linear spectral interferometry and angle-to-time encoding with dual echelons. Both methods, each using electro-optic detection in the linear regime, were able to reproduce the THz temporal waveform acquired with a traditional scanning delay line. Although spectral interferometry had mediocre performance in terms of signal-to-noise, the dual echelon method was easily implemented and achieved the same level of signal-to-noise as the scanning delay line in only 4.5% of the laser pulses otherwise required (or 22 times faster). This reduction in acquisition time will compress day-long scans to hours and hence provides a practical technique for multidimensional THz measurements.

The advancement of terahertz (THz) spectroscopy has relied on the increasing sensitivity and resolution of optical detection methods for ultrafast THz waveforms. In this regime, THz time domain spectroscopy (THz-TDS) is used to measure the full time-dependent waveform of the THz electric-field (E-field).1 A common technique employs electro-optic (EO) sampling with a mechanical scanning delay line.2 This classic approach has its temporal resolution given by the readout pulse duration, which is typically much shorter than a THz period, and its time window set by etaloning effects in the EO detection crystal or the sample. Early THz-TDS focused on measuring the linear THz response of a system at equilibrium, but has since advanced to more complex pump-probe geometries capable of measuring dynamics of reversible processes following photoexcitation.

In the THz regime, many studies only measure the average THz probe response (the peak or integrated power) at various pump-probe time delays. This approach has proven very insightful, for example, in measuring the carrier dynamics using an optical pump in silicon on sapphire,3 microcrystalline silicon,4 graphene,5 and InAs/GaAs quantum dots;6 and similarly using a THz pump in semiconductors.7,8 It also lends itself to many other experiments, including optically pumped liquid solvation dynamics9,10 and an optically induced phase transition in vanadium dioxide.11 An important distinction is made between these measurements and ones in which the THz probe is spectrally resolved for a limited number of fixed pump-probe time delays such as in the photoconductivity study of liquid n-hexane using an UV pump12 and the determination of relaxation dynamics of carriers and excitons in GaAs-AlGaAs multiple quantum wells.13 

While these THz probe experiments are sufficient for observing dynamics, they lack the ability to deduce the extent of coupling between THz-resonant degrees of freedom; this would require a comprehensive study of a spectrally resolved THz probe as a function of a near-continuous sweep of pump-probe time delays. In multidimensional THz spectroscopy, the pump-induced response is measured as the time-evolution of the change in the THz probe spectrum. Multidimensional studies have the potential to reveal the couplings between different degrees of freedom in a system (nuclear, vibrational, rotational, or electronic) that are otherwise difficult to isolate unambiguously from the vast number of possible connections between all states.

Multidimensional spectroscopy has yielded major insights in other frequency regimes, for example, in studying spin-spin couplings in complex molecular systems using 2D nuclear magnetic resonance spectroscopy,14 coherence transfer using 2D infrared vibrational spectroscopy in organic solutions,15 or 2D visible electronic spectroscopy for studying electronic couplings in photosynthetic systems16 and GaAs quantum wells,17 and exciton-phonon coupling in molecular aggregates.18 Recently, there has also been an endeavor to perform true 2D THz spectroscopy to elucidate the coupling between electronic degrees of freedom in quantum wells and graphene.19–22 However, the technological barriers in THz generation and detection have limited the measurement of THz nonlinear signals to this small subset of systems which have relatively large transition dipole moments that arise from strong electronic resonances.

In contrast, we have a particular interest in the study of intermolecular and low-frequency intramolecular motion using multidimensional vibrational THz spectroscopy, a technique that currently exists in an early developmental stage. An example of a class of materials that pose key challenges in the understanding of coherent coupling and energy transfer across the different vibrational degrees of freedom is energetic materials (i.e., explosives). In these systems, mechanical energy from a shock front can be up-converted to high-frequency molecular vibrations that lead to initiation of a chemical reaction.23 Comprehensive studies of the intricate vibrational couplings throughout such a system embody the motivation for development of 2D techniques across the entire electromagnetic spectrum. For low-frequency vibrational couplings, these nonlinear signals are even smaller than those observed in Refs. 19–22 and typically have narrower spectral features that are only resolved with sufficiently large temporal observation windows in these time domain measurements. One study that has started to observe these couplings used 2D Raman-THz spectroscopy to observe the collective intermolecular modes of water molecules,24 but proved difficult to characterize the spectral response due to a limited observation window.

Because THz waveforms are measured in the time domain, multidimensional THz measurements require substantial scaling up of total experimental time with the number of time points (i.e., 1D scan: ∼n time points, 2D scan: ∼n2 time points, etc.), which makes faster acquisition techniques crucial for their practical implementation. Current EO sampling is limited by its inherently multi-shot, scanning approach, which is incompatible with the observation of rapid time dynamics of irreversible phenomena (e.g., material damage, irreversible chemical reactions, or structural phase transitions) and impractical for experiments that use THz sources with large shot-to-shot fluctuations or low-repetition rates such as particle accelerators. The desire to record non-repeating phenomena has previously driven the development of scan-free THz techniques, but single-shot recording, when used in conjunction with averaging, can drastically reduce the time required to record high signal-to-noise (S/N) ratio measurements.

In view of the shortfalls of conventional EO sampling, a proposed detection scheme is one that accurately recovers the entire THz waveform in a single laser shot and has a sufficiently long temporal window to provide the desired resolution in the THz spectrum. Furthermore, under equal averaging times the S/N of the recorded single-shot THz trace should surpass that recorded using the traditional scanning method. In this paper, we review several THz single-shot methods with a focus on their ability to record accurate, low-noise THz traces in a short period of time when directly compared to traditional EO sampling. We first analyzed the different single-shot methods based on their temporal resolution, temporal window, accurate reproduction of the THz E-field, and the anticipated S/N, and then selected two of the most promising techniques to evaluate experimentally: a method that used amplitude encoding spectral interferometry and one based on dual echelon optics. For these two single-shot methods, we constructed experimental setups that implemented balancing and high-frequency modulation of the THz arm for optimal noise suppression, in addition to the averaging of many temporal traces. The designs allowed for rapid switching between single-shot detection and scanning delay line detection without perturbing the THz generation so that we could directly verify the single-shot method’s ability to accurately measure THz waveforms with respect to the standard scanning methodology.

To date, there has been little emphasis on single-shot methods as reliable, quantitative, and quick data acquisition techniques for spectrally resolving a THz probe between 0.1 and 10 THz. Many of these single-shot methods have a tradeoff between fast experimental acquisition and some fundamental capabilities that will be discussed below. In this case, we deviate from the objectives of irreversible single-shot experiments or diagnostic techniques, where while we still require a low noise floor for each temporal THz waveform acquired in a single laser pulse, there is the added possibility of extensive averaging of many of these measurements.

A worthwhile technique will not make major concessions compared to traditional delay line EO sampling: the temporal resolution should be close to the transform-limited pulse duration, the temporal observation window should be long enough to give the requisite spectral resolution, and the experimental design should allow for straightforward incorporation of noise suppression techniques like balanced detection. Optimally, the experimental geometry would allow the THz beam to be focused at the EO detection crystal for higher sensitivity and accuracy and would not require routine and/or intensive calibration prior to the measurement.

In this review, we will introduce a selection of existing single-shot detection methodologies into three broad classes: (1) frequency-to-time mapping in Secs. II A and II B, (2) space-to-time mapping in Secs. II C and II D, and (3) angle-to-time mapping in Sec. II E. Figure 1 schematically compares conventional EO sampling [Fig. 1(a)] to a selection of existing THz single-shot techniques [Figs. 1(b)-1(f)]. For each method, we will discuss temporal resolution, feasibility of a 10 ps temporal window, the ability to accurately record the THz waveform, practicality to implement balancing for noise suppression, and the overall ease of implementation; the key characteristics are summarized in Table I in Sec. II F.

FIG. 1.

Comparison of single-shot THz methods to conventional EO sampling. (a) EO sampling with balanced detection using a quarter-wave plate (QWP), Wollaston prism (Woll.), two photodiodes (PD), and a scanning delay line to yield the pump-probe time delays. (b) Frequency-to-time mapping by spectrally encoding a THz pulse onto a linearly chirped readout pulse in a collinear geometry. The chirped readout is then measured on a spectrometer. (c) Amplitude encoding spectral interferometry time delays a temporally short readout pulse by δt relative to a chirped readout pulse and subsequently detects a spectral interference at the spectrometer. A suitably short readout pulse sets the time resolution and recovers the envelope of the THz-imprinted chirped optical readout pulse. (d) Non-collinear crossing of an 800 nm readout pulse with an expanded THz beam at the ZnTe detection crystal. The inset shows that crossing the readout beam of width w at an angle θ maps time to transverse position on the crystal, which is then imaged onto a camera. The projected readout beam onto the crystal has a width d, across which the THz field is ideally spatially and spectrally invariant. (e) Space-to-time mapping in time domain non-collinear SHG cross-correlation, in which a THz pulse is encoded onto a chirped readout pulse as in (b). The THz-imprinted chirped readout beam is then vertically focused to a line with a cylindrical lens (CL) and crossed with a short readout pulse in a SHG crystal at an angle of 2ϕ. The inset shows a zoomed in view of the intersection of the two beams in the SHG crystal, where different frequencies of the chirped readout pulse map to different transverse positions. The spatial profile of the SH signal, S(x), recovers the temporal intensity profile of the chirped readout pulse, which is detected by a camera. (f) Angle-to-time encoding using two crossed glass echelon optics, each with 20 steps, to create a pulse train of 400 beamlets that are incrementally time delayed relative to one another. The beamlets and THz beam are collinearly focused to the same spot at the ZnTe crystal, but the beamlets are imaged to different locations on the camera. The detected grid pattern is made up of individual square regions that sequentially read out the THz waveform.

FIG. 1.

Comparison of single-shot THz methods to conventional EO sampling. (a) EO sampling with balanced detection using a quarter-wave plate (QWP), Wollaston prism (Woll.), two photodiodes (PD), and a scanning delay line to yield the pump-probe time delays. (b) Frequency-to-time mapping by spectrally encoding a THz pulse onto a linearly chirped readout pulse in a collinear geometry. The chirped readout is then measured on a spectrometer. (c) Amplitude encoding spectral interferometry time delays a temporally short readout pulse by δt relative to a chirped readout pulse and subsequently detects a spectral interference at the spectrometer. A suitably short readout pulse sets the time resolution and recovers the envelope of the THz-imprinted chirped optical readout pulse. (d) Non-collinear crossing of an 800 nm readout pulse with an expanded THz beam at the ZnTe detection crystal. The inset shows that crossing the readout beam of width w at an angle θ maps time to transverse position on the crystal, which is then imaged onto a camera. The projected readout beam onto the crystal has a width d, across which the THz field is ideally spatially and spectrally invariant. (e) Space-to-time mapping in time domain non-collinear SHG cross-correlation, in which a THz pulse is encoded onto a chirped readout pulse as in (b). The THz-imprinted chirped readout beam is then vertically focused to a line with a cylindrical lens (CL) and crossed with a short readout pulse in a SHG crystal at an angle of 2ϕ. The inset shows a zoomed in view of the intersection of the two beams in the SHG crystal, where different frequencies of the chirped readout pulse map to different transverse positions. The spatial profile of the SH signal, S(x), recovers the temporal intensity profile of the chirped readout pulse, which is detected by a camera. (f) Angle-to-time encoding using two crossed glass echelon optics, each with 20 steps, to create a pulse train of 400 beamlets that are incrementally time delayed relative to one another. The beamlets and THz beam are collinearly focused to the same spot at the ZnTe crystal, but the beamlets are imaged to different locations on the camera. The detected grid pattern is made up of individual square regions that sequentially read out the THz waveform.

Close modal
TABLE I.

Summary of several THz single-shot detection methods in terms of the method’s highest time resolution, the theoretically attainable time window, the capability of incorporating balanced detection, technical implementation, and ease of alignment compared to the traditional scanning EO sampling apparatus, and experimental implications.

Method Time resolution Time window Balanced detection Technical implementation and ease of alignment Experimental implications
Frequency-to-time mapping 
Spectral encoding, Fig. 1(b) 25   τ 0 τ ch > τ 0   τch  Yes  • Requires a grating compressor or stretcher and spectrometer.  • Time resolution degrades with lengthening of time window. • Various distortions observed when time resolution is insufficient. 
Amplitude encoding spectral interferometry, Fig. 1(c) 34   τ0  τch  Yes  • Requires a grating compressor or stretcher and high-resolution spectrometer. • Requires an additional readout beam and delay line. • Requires interferometrically stable conditions and very precise alignment.  • Time window is limited by spectral resolution of the spectrometer. 
Space-to-time mapping 
Non-collinear crossing, Fig. 1(d) 39,40  τ0  w tan θ c 0   Difficult  • Minimal modification to standard EO sampling experimental design. • Intensive calibration needed to remove distortions due to spatial beam inhomogeneity and nonlinear response.  • Readout and THz beams are not focused at EO crystal. • Expansion of THz spot size needed for longer time windows reduces S/N significantly and hence difficult to detect in the linear regime. 
SHG cross correlation, Fig. 1(e) 49   τ0  2 w sin ϕ c 0   Difficult  • Requires a grating compressor or stretcher. • Requires an additional readout beam and delay line. • Requires sensitive alignment for proper phase matching in SHG crystal.  • Detection of nonlinear process is inherently noisier and weaker. • Expanded readout beams needed for larger time windows lowers SHG efficiency • Spatial chirp in laser distorts recovery of temporal waveform. 
Angle-to-time mapping 
Dual transmissive echelons, Fig. 1(f) 52   h Δ n c 0   m H Δ n c 0   Yes  • Requires custom-fabricated glass echelons, but with only minimal modification of EO sampling readout arm. • Imaging-based detection requires high optical quality components and imposes restrictions on experimental design.  • Pulse broadening in echelons for 10 ps time window is minimal for transform-limited pulse durations ≥50 fs. • Reflection-mode echelons are an option for very large time windows or broad bandwidth lasers. 
Method Time resolution Time window Balanced detection Technical implementation and ease of alignment Experimental implications
Frequency-to-time mapping 
Spectral encoding, Fig. 1(b) 25   τ 0 τ ch > τ 0   τch  Yes  • Requires a grating compressor or stretcher and spectrometer.  • Time resolution degrades with lengthening of time window. • Various distortions observed when time resolution is insufficient. 
Amplitude encoding spectral interferometry, Fig. 1(c) 34   τ0  τch  Yes  • Requires a grating compressor or stretcher and high-resolution spectrometer. • Requires an additional readout beam and delay line. • Requires interferometrically stable conditions and very precise alignment.  • Time window is limited by spectral resolution of the spectrometer. 
Space-to-time mapping 
Non-collinear crossing, Fig. 1(d) 39,40  τ0  w tan θ c 0   Difficult  • Minimal modification to standard EO sampling experimental design. • Intensive calibration needed to remove distortions due to spatial beam inhomogeneity and nonlinear response.  • Readout and THz beams are not focused at EO crystal. • Expansion of THz spot size needed for longer time windows reduces S/N significantly and hence difficult to detect in the linear regime. 
SHG cross correlation, Fig. 1(e) 49   τ0  2 w sin ϕ c 0   Difficult  • Requires a grating compressor or stretcher. • Requires an additional readout beam and delay line. • Requires sensitive alignment for proper phase matching in SHG crystal.  • Detection of nonlinear process is inherently noisier and weaker. • Expanded readout beams needed for larger time windows lowers SHG efficiency • Spatial chirp in laser distorts recovery of temporal waveform. 
Angle-to-time mapping 
Dual transmissive echelons, Fig. 1(f) 52   h Δ n c 0   m H Δ n c 0   Yes  • Requires custom-fabricated glass echelons, but with only minimal modification of EO sampling readout arm. • Imaging-based detection requires high optical quality components and imposes restrictions on experimental design.  • Pulse broadening in echelons for 10 ps time window is minimal for transform-limited pulse durations ≥50 fs. • Reflection-mode echelons are an option for very large time windows or broad bandwidth lasers. 

Figure 1(b) illustrates the spectral encoding method, in which a linearly chirped optical pulse is used to map different portions of a THz temporal profile onto different optical frequencies in a single laser shot.25 As in EO sampling, the THz field modulates the chirped readout pulse’s polarization, which is converted into an intensity modulation when it passes through an analyzing polarizer: ΔI/I0ETHz. The chirped readout is directed into a spectrometer, where an intensity-modulated spectrum is detected on a camera. The method is capable of implementing balancing by introducing a bias with a quarter-wave plate and a polarizing beam splitter. Consequently, the response is linear and quantitative compared to the initial demonstration in Ref. 25, which used the inherent static birefringence in the EO crystal26 for detection.

A caveat exists, however, since allocating a single frequency component to a single time delay violates the uncertainty principle. While increasing the chirp elongates the time window, imprinting the THz field onto a narrower range of frequency components degrades time resolution.27,28 The time resolution is given by τ 0 τ ch where τ0 is the transform-limited pulse duration and τch is the chirped pulse duration; for example, a 100-fs transform-limited input pulse that is chirped to 10 ps results in a time resolution of 1 ps. Moreover, in instances where the time resolution was insufficient for monitoring fast processes, the signal was shown to possess distortions that depended on whether the THz fields were strong or weak compared to the background.29,30 Only under strict conditions were these distortions eradicated, particularly by using a chirped readout pulse duration optimized for a fixed THz pulse duration,31 which only emphasizes the intricate balance between time resolution and the chirp rate. Finally, the mapping assumes a linear relationship between temporal chirp rate and frequency, which if not satisfied requires more careful characterization to avoid inaccuracies in recovering the THz waveform.

Using a laser with larger bandwidth coupled with a higher chirp rate was proposed as a way to lead to better time resolution,25 although high-bandwidth lasers are less efficient for THz generation.8 Alternatively, a distinct white-continuum readout pulse, independent of the pump arm, could be used, but this is not trivial to implement and yet still yields far from transform-limited pulse durations. It was later proposed that the direct time-frequency mapping could be deconvolved with in-line spectral interferometry,32,33 where the leak-through readout pulse, resulting from static birefringence in the EO crystal or slight detuning of the crossed polarizers, interferes in the frequency domain with the amplitude-modulated signal. With no direct association of frequency to time, the analysis should recover a time-resolution inversely proportional to the full spectral bandwidth of the readout pulse. However, this is a numerical reconstructive approach that relies on accurate characterization of the chirped readout pulse up to second-order dispersion (group delay dispersion). This revised method was experimentally demonstrated as a better diagnostic tool in terms of time resolution compared to spectral encoding, but ultimately, the imperfect deconvolution algorithm introduces numerical artifacts that are not conducive to reliable spectroscopic applications. In addition, this scheme does not allow for balancing since the small bias necessary for large fringe contrast (i.e., for high sensitivity) can only be introduced and subsequently detected in one polarization state. Despite these efforts to overcome the serious drawbacks described, we find that the method and its derivatives are unable to produce reliable THz waveforms to a degree suitable for quantitative spectroscopic applications.

Matlis and coworkers demonstrated a THz cross-correlation technique, shown in Fig. 1(c), which uses both a chirped readout pulse and a short readout pulse for linear spectral interferometry.34 First, the input optical beam is split and one of the two optical arms is chirped to ∼10 ps duration. Then, the chirped and transform-limited optical arms are recombined collinearly, separated by a relative time delay set to δt. The THz beam and recombined optical beams are focused collinearly onto the EO detection crystal with the appropriate time delay such that the THz signal is encoded onto the chirped readout pulse as either a polarization rotation or phase modulation (or a combination of both). Therefore, a THz signal can be encoded in either the amplitude or phase of the chirped readout pulse and subsequent extraction of a THz temporal waveform is possible given a reference interferogram in the absence of a THz pulse. In Fig. 1(c), after converting the polarization modulation into amplitude modulation with an analyzing polarizer, both the chirped and transform-limited optical pulses are passed into the spectrometer and the spectral interference is detected with a camera. The time resolution of the method is set by the pulse width of the readout pulse E-field, τ0, and the time window corresponds to the pulse width of the chirped readout pulse E-field, τch.

As for any highly sensitive interferometric measurement, there are stringent experimental conditions required for accurate implementation. The fidelity of the fringes is only maintained for collinear beams with high optical phase stability and good spatial beam mode matching. For the long temporal windows required, more precise alignment is needed since non-common-path optics must be used. In principle, the time window is freely adjustable by changing the chirp rate, but in practice is limited by the resolution of the spectrometer, which must be capable of resolving the finest fringe widths. Since the E-fields of the readout and chirped pulses do not overlap temporally, increasing τch for larger time windows requires a corresponding increase in time delay, δt. As a result, finer fringes are introduced in the interferogram since the fringe widths are inversely proportional to δt.

The retrieval of the full E-field of an optical readout pulse in this way has been demonstrated earlier in frequency domain holography35,36 and single-shot supercontinuum spectral interferometry.37 Alternate phase-encoding spectral interferometry with a single chirped pulse34 or with two chirped pulses in single-shot supercontinuum spectral interferometry37 is otherwise similar except the lack of a polarization-sensitive scheme prevents the use of balanced detection; also an additional measurement of the reference phase through cross-phase modulation is needed.37 The phase-encoding scheme becomes more pertinent to measurements of intense THz E-fields since very large phase differences can be detected38 without causing over-rotation (i.e., for induced phase differences greater than 90°) in the EO detection crystal. Therefore, we find that the technique described in Ref. 34 for amplitude encoding spectral interferometry is the most promising for high sensitivity measurements since detection occurs in the linear regime and allows for straightforward noise suppression through balanced detection.

For a time-space encoding method of non-collinear crossing39 or tilting of the optical readout intensity front with a prism or grating,40,41 the transverse profile of the optical readout samples the THz field at different times when they spatially overlap in the detection crystal as depicted in Fig. 1(d). The mapping between space-coordinate, x, to time-coordinate, t, is calculated geometrically, depending only on the angle θ between the THz beam and the optical readout beam: (t = xtanθ/c0), where c0 is the speed of light in vacuum. Consequently, the time window, Δt, is determined by the readout beam width, w, and the crossing angle, θ: (Δt = wtanθ/c0). For both of the described space-to-time encoding methods, a time window of 10 ps and 45° relative angle between the THz beam and optical beam requires a 3 mm readout beam width. In either the non-collinear or tilted pulse front geometry, there is minimal degradation of time resolution from the phase mismatch between the group index of the optical pulse and the phase index of the THz field in the EO crystal.42,43 The time resolution is thus dominated by dispersive propagation and absorption of the THz pulse in the crystal,43,44 just as in conventional EO sampling.

Beyond the technical simplicity of the technique, there are some considerations for this geometry in terms of applying it toward sensitive THz spectroscopy. Attaining larger time windows requires that the THz spot size at the detection crystal increase accordingly to d = w/cosθ (∼4 mm for the parameters given above), diminishing the THz field by a factor proportional to 1/d. This metric of d, however, only serves as a lower bound since in reality an ultra-broad-bandwidth Gaussian THz pulse has both a spatially dependent E-field amplitude and spectral content due to diffraction.45 In typical EO sampling, the spectrum of the THz field is essentially spatially invariant because the focused optical readout spot size is much smaller than that of the THz spot size. These concerns pose obstacles with regard to the complete mapping of nonlinear responses in multidimensional studies, where the preferred broadband THz sources46,47 are generally weaker compared to tilted pulse front generation in lithium niobate that yields a smaller THz bandwidth.48 

In this single-shot scheme, a near uniform spectral distribution across the readout pulse at the detection crystal is desired for optimal dynamic range. By approximation, for a variation in the THz field peak value by less than 1%, the THz beam FWHM for the highest frequency component must be >30 mm, assuming the THz beam is at normal incidence to the crystal and d = 4 mm. Compared to EO sampling that uses a focused THz beam, a beam diameter this large would lead to a decrease in THz field strength by more than an order of magnitude. Previous demonstrations of both the non-collinear geometry39 and collinear tilted readout pulse40 methods used a crossed-polarizer configuration that exploited the small static birefringence in the nonlinear crystal to maximize the detected modulation depth.26 The distortions introduced by detecting in the quadratic regime are small for weak signals, but typically an involved initial calibration that measures the nonlinearity of every pixel in the detector array is required.40 While the calibration also reports on the THz beam’s spatial profile, this would only normalize for amplitude variation across the THz focal spot, but neglects any spectral variation. These nonlinear detection schemes often have non-trivial balancing implementation or completely lack the ability as a trade-off for larger signals; consequently, shot-to-shot changes in the spatial beam profiles cannot be suppressed. Overall, this single-shot method has demonstrated merits for linear spectroscopy in systems where the absorptions are strong, but the barriers discussed hinder its practical implementation in nonlinear THz spectroscopy studies.

In this technique, the THz signal is first imprinted on a chirped optical readout pulse like in Sec. II A. However, in this case the temporal profile of the readout pulse is measured by crossing it at angle 2ϕ with a short readout pulse in a β-barium borate (BBO) crystal to produce a type I second-harmonic (SH) signal as shown in Fig. 1(e).49 The crossing of the two spatially and temporally coincident beams at an angle maps time delay onto transverse position, where different frequencies of the chirped pulse are projected onto different locations at the BBO crystal [see inset of Fig. 1(e)], similar to that in a single-shot optical autocorrelator.50 The spatial profile of the SH beam, S(x), is recorded on a CCD array, which is represented as an intensity cross-correlation between the two pulses: S ( x ) = I ch ( τ ) I r ( t + τ ) d τ , where Ich and Ir are the intensities of the chirped and short readout pulses, respectively. For a readout pulse with suitably short pulse duration, S(x) approaches the temporal intensity profile of the chirped readout pulse, Ich(t). The time window is given by Δt = 2wsinϕ/c0 and directly depends on the spatiotemporal overlap of the two beams across a width, w, in the BBO crystal, where ϕ is the angle in free space. For a 10 ps window at an angle of 15°, the overlap width needs to be ∼6 mm.

This method is similar to the space-to-time mapping described in Sec. II C, but it does not require any prior calibration of the THz spatial profile, consideration of spectral variation, or an expanded THz spot size, since both the chirped readout and THz beams are collinearly focused at the EO crystal. Also, no assumption is made about the frequency dependence of the chirp compared to spectral encoding,25 although spatial chirp could introduce inaccuracies in the recovery of the temporal waveform. In terms of degradation of time resolution, for a very thin SHG crystal, the dominant source arises from THz dispersion and absorption in the EO detection crystal, prior to the SH readout process. However, the method suffers from the complexity of relying on another nonlinear process, SHG, in addition to the EO effect, which makes the signal inherently noisier and weaker. Longer time windows call for larger beam widths, but this consequently reduces SHG efficiency and the sensitivity of the measurement. The most detrimental aspect to the method that makes it incompatible for low-noise measurements is that balanced detection cannot be implemented in an optimal or convenient way due to the strict conditions on proper phase matching and input polarizations for SHG.

This single-shot technique is built around a complementary pair of custom-made transmission-mode echelons,51,52 which are optics that look like glass stairways [see inset of Fig. 1(f)]. When the readout beam passes through a pair of a thin and a thick echelon, each with m steps, it is split into an array of tiny “beamlets” that are delayed incrementally in time relative to one another. The more glass a beamlet passes through, the longer the delay. By design, an integer number of small steps of height, h, in the thin echelon introduces the same time delay as a single large step of height, H, in the thick echelon or simply H = mh, where m is an integer. In the Fourier plane of the 2-lens 4f imaging system, the beamlets are all focused to the same spot at the detection crystal (as a pulse train), but they separate after the crystal and are spatially resolved at distinct regions of a 2D detector array at the image plane. Modulation of the pulse train in time is mapped onto modulation of intensity in the image, making it possible to record the entire THz trace with a single laser pulse.

The variation in signal intensity arising from the beamlets propagating at different angles through the detection crystal is negligible since the Rayleigh range of each optical readout beamlet is much larger than any crystal thickness used in practice. The time resolution resulting from the transmissive echelons is the larger of the readout pulse duration, τ0, or the time delay introduced by the smallest step increment, hΔn/c0, where Δn = nglassnair ≈ 0.5 is the refractive index difference. The time window is given by the total time delay resulting from all m steps of the large step increment, mHΔn/c0. For a 10 ps window with 20-step glass echelons, the small steps are 15 μm (or 25 fs) and the large steps are 300 μm (or 500 fs). For a pulse with ∼20 nm of bandwidth centered around 800 nm (a 50 fs transform-limited pulse) and a maximal path-length difference in fused silica of 6.3 mm, the broadening of the pulse duration is ∼1.5 fs,53 which is inconsequential when considering the time scale of THz dynamics. The degradation of time resolution only becomes noticeable for ultra-broad bandwidth lasers (<20 fs transform-limited pulses).

However, the method is inconvenienced by relying on custom-fabricated echelon optics with high optical quality, which have yet to become standard optical components. While the method retains the ability for balanced detection, it has the added requirement of in-focus imaging of both polarization states on the camera. As a result, the method is more susceptible to factors that degrade image resolution, such as aberrations in the echelons or other optical components, which affect the ability to accurately recover a THz waveform. Also, this imaging system requires an optimization of two opposing parameters: depth of focus and image resolution, since the echelons have a significant thickness and the recovery algorithm relies on being able to easily discern the steps from one another. Even with a sufficiently long focal length lens with the requisite numerical aperture, the focused beamlet spot size should be small relative to the THz beam at the EO crystal, which could be problematic for broad bandwidth THz sources.

The implementation of this method requires fairly minimal modification to the readout arm in a conventional free space THz detection setup and avoids the need for a high-resolution spectrometer, noncollinear geometries, stretchers or compressors, or more complex nonlinear optical measurements. With its relative ease of achieving high time resolution, a sufficiently long time window, and the possibility for detection in the linear regime and balance detection, the dual echelons present a convincing scheme for low-noise THz measurements.

Table I summarizes the methods presented above, highlighting the desired criteria for a fast-acquisition method with high S/N. With respect to these criteria, we have chosen to explore the viability of two techniques: amplitude encoding spectral interferometry described in Sec. II B and EO imaging with transmissive dual echelons in Sec. II E. These techniques possess high time resolution and reasonable time window limits, allow the THz and optical readout beams to be focused at the EO crystal, and allow for straightforward noise suppression with balanced detection. Sections III and IV provide analyses on the quantitative ability, noise floor, and technical ease of implementation of each method.

As stated previously, spectral interferometry is capable of attaining the transform-limited temporal resolution with a variable chirped readout pulse to control the time window. Here, we directly compare this method to the scanning delay line method to demonstrate the extent of its capabilities.

The measurements were conducted with a 1-kHz amplified titanium-sapphire laser with a center wavelength of 800 nm, a pulse energy of 1.5 mJ, and a 50 fs transform-limited pulse duration. The experimental setup in Fig. 2 was designed for convenient and accurate comparison between the single-shot method and the conventional delay line method. The optical pump beam was directed into a 2 mm thick ZnTe crystal (Ingcrys) to generate THz pulses via optical rectification47 and the THz output was imaged onto a 1 mm thick ZnTe crystal with a pair of parabolic mirrors. The identical THz generation setup was also used in the dual echelons technique that will be described later in Sec. IV. The first beam splitter (90%R) separates the pump and readout beams while the second beam splitter (70%R) creates the chirped and short readout arms. Prior to entering the second beam splitter, the beam was spatially filtered with a pinhole to decrease spatial interference upon recombination of both beams downstream. The chirped and short readout pulses must recombine at the third beam splitter (50%R) with equal intensities in order to ensure the strongest modulation depths of the resulting interferogram; the second beam splitter was set to account for the loss upon emerging from the dual-grating compressor. This compressor consisted of two 830.8 grooves/mm pulse compression gratings that introduced linear spectral chirp while carefully minimizing spatial chirp. The chirped optical readout pulse had a temporal duration of τch ≈ 7 ps.

FIG. 2.

THz spectral interferometry experimental setup using an amplitude encoding scheme. The input laser is initially split into an optical pump for THz generation and the optical readout pulse. The optical readout arm is further split to give a chirped readout pulse (via a dual-grating compressor) and a short readout pulse, which are time delayed relative to one another by a stage. The THz pulse is imprinted onto the amplitude of the chirped readout pulse at the ZnTe detection crystal, where both beams are focused. Both the chirped and short readout pulses are directed into a spectrometer and the spectral interference is imaged onto a camera. In the single-shot method, both stages are fixed during the measurement. For the conventional delay line method, the chirped readout arm is blocked, and the short readout is used for EO sampling.

FIG. 2.

THz spectral interferometry experimental setup using an amplitude encoding scheme. The input laser is initially split into an optical pump for THz generation and the optical readout pulse. The optical readout arm is further split to give a chirped readout pulse (via a dual-grating compressor) and a short readout pulse, which are time delayed relative to one another by a stage. The THz pulse is imprinted onto the amplitude of the chirped readout pulse at the ZnTe detection crystal, where both beams are focused. Both the chirped and short readout pulses are directed into a spectrometer and the spectral interference is imaged onto a camera. In the single-shot method, both stages are fixed during the measurement. For the conventional delay line method, the chirped readout arm is blocked, and the short readout is used for EO sampling.

Close modal

Spectral interferometry requires excellent collinearity between the interfering beams in order to eliminate spatial interference artifacts from inhomogeneous beam profiles or beams crossing at an angle. Interferometers with common path optics would circumvent the collinearity difficulties, but phase masks or two-dimensional pulse shapers cannot achieve the long temporal chirp widths desired here. Following the recombination beam splitter, we aligned each arm through a 25-μm pinhole and a 10-μm spectrometer entrance slit. In addition, a motorized stage was incorporated into the readout arm to set the time delay, δt, between the two pulses. Once set, this stage is immobilized for the entirety of the measurements. Similarly, the stage in the pump arm for THz generation is set to temporally overlap with the chirped readout pulse and then also fixed for the single-shot measurement.

Balanced detection was implemented in both the single-shot technique and scanning EO sampling with a quarter-wave plate and a Wollaston prism. In spectral interferometry, a small splitting-angle Wollaston prism was used to ensure that both spectrograms were imaged onto the camera. The emerging beams of orthogonal polarization were focused onto the slit of the spectrometer. A half-wave plate was used to compensate for the polarization-dependent efficiency of the grating in the spectrometer by rotating the polarization of the beams to 45°, yielding equal intensities at the detector. The single-shot detector was a homebuilt spectrometer coupled to a high-speed 10-bit CMOS camera (Photron, Fastcam 1024 PCI, 1024 × 1024-pixel chip). The spectrometer used a 1200 grooves/mm grating to yield a wavelength range of Δλ = 30 nm and a spectral resolution of 0.03 nm/pixel. The camera is capable of recording images at 1000 frames per second while synchronized to the 1 kHz laser (its performance is detailed in Ref. 54). For the conventional scanning delay line method, the signal was recorded using a data acquisition card and a pair of low-noise photodiodes (Hamamatsu, S2281) by a method described in Ref. 55. For both methods, the optical pump beam was chopped at the maximum modulation frequency of 500 Hz such that alternate readout pulses recorded the “signal” (THz pulse present) and “reference” (THz pulse absent). Detection at this high frequency allows for acquisition of an interferogram on a single-shot basis, which effectively reduces 1/f noise associated with pointing instability and laser drift. Most importantly, these factors degrade the optical phase stability of the system and consequently wash out the fringe pattern. The high frequency detection scheme is diagrammed in Fig. 3, where a 1 kHz sample clock is set by a digital delay generator that synchronizes the acquisition with the arriving laser pulse; the acquisition is triggered by the rising edge of the 500 Hz transistor-transistor logic (TTL) output from the chopper controller.

FIG. 3.

The single-shot detection hardware includes a 1 kHz sample clock and 500 Hz trigger source. The data are then streamlined off the camera to either computer RAM or hard drive for processing.

FIG. 3.

The single-shot detection hardware includes a 1 kHz sample clock and 500 Hz trigger source. The data are then streamlined off the camera to either computer RAM or hard drive for processing.

Close modal

In this section, we will briefly discuss the key concepts of EO detection, how it applies in THz spectral interferometry, and the experimental treatment of the data required to recover the THz signal.

1. Electro-optic detection

In the linear EO effect in zincblende crystals like ZnTe, the THz E-field induces a time-dependent EO phase retardation, Δφ, in the optical readout pulse that passes through the crystal,

Δ φ ( t ) = 2 π λ opt Δ n ( t ) ,
(1)

where Δ n ( t ) = r 41 n 0 3 E THz ( t ) / 2 is the THz-induced birefringence, ℓ is the crystal thickness, n0 is the index of refraction in the absence of a THz field at λopt (the optical readout wavelength), and r41 is the appropriate coefficient of the EO tensor.56 When the polarization of the readout pulse is oriented 45° relative to the principal axes of the THz-induced index ellipsoid, the measured signal intensity, I, in the limit of weak THz modulation strengths (i.e., small Δφ) is

I ± ( t ) = I 0 ( t ) [ 1 ± sin Δ φ ( t ) ] I 0 ( t ) [ 1 ± Δ φ ( t ) ] ,
(2)

where I0 is the intensity in the absence of a THz E-field and “+” and “−” signify horizontal and vertical polarizations of the optical readout, respectively. This expression signifies a conversion of the induced change in the readout pulse’s polarization to an amplitude modulation following transmission through both a quarter-wave plate and Wollaston prism. Furthermore, the implementation of balanced detection doubles the detected signal modulation while reducing common mode noise,

Δ I + ( t ) I 0 ( t ) Δ I ( t ) I 0 ( t ) = 2 sin Δ φ ( t ) 2 Δ φ ( t ) ,
(3)

where ΔI±(t) = I±(t) − I0(t).

2. THz spectral interferometry

In THz spectral interferometry, a THz E-field may be encoded in either the amplitude or the phase of the chirped readout pulse, depending on the polarization of the chirped pulse relative to the principal axes of the THz-induced birefringence.34,44 Time delaying a short readout pulse by δt from the THz-imprinted chirped pulse introduces a frequency-dependent phase difference in the readout field equivalent to ωδt and gives rise to a spectral interference pattern. The measured interferogram at the spectrometer may be written as

T ( ω ) = E ̃ ch ( ω ) + E ̃ r ( ω ) e i ω δ t 2 = E ̃ ch ( ω ) 2 + E ̃ r ( ω ) 2 + E ̃ ch ( ω ) E ̃ r ( ω ) e i ω δ t + c . c . ,
(4)

where E ̃ ch ( ω ) = E ̃ ch ( ω ) e i ϕ ch ( ω ) and E ̃ r ( ω ) = E ̃ r ( ω ) e i ϕ r ( ω ) are the amplitude and phase of the E-fields of the chirped and short readout pulses, respectively, and c.c. is the complex conjugate of the preceding cross-term. This expression may be rewritten to include the additional relative phase difference between the pulses in the spectral domain, Δϕ(ω) = ϕr(ω) − ϕch(ω),

T ( ω ) = I ch ( ω ) + I r ( ω ) + I ch ( ω ) I r ( ω ) cos Δ ϕ ( ω ) + ω δ t ,
(5)

where Ich(ω) and Ir(ω) are the spectral intensities of the chirped and short readout pulses, respectively.

In the time domain, the response is simply the inverse Fourier transform of (5),

F 1 [ T ( ω ) ] ( t ) = I ch ( t ) + I r ( t ) + f ( t δ t ) + f ( t δ t ) ,
(6)

where f ( t δ t ) = E ch ( τ ) E r ( t δ t + τ ) d τ is the cross-correlation between the chirped and short readout pulses. The first two terms are the non-interfering autocorrelation components that are centered at t = 0, and make up the dc response. In contrast, the cross-correlation terms are separated from the dc response and are centered at ±δt and are derived from the spectral interference. For a short readout pulse with negligible spectral phase variation, the value of the cross-term approaches the chirped E-field in amplitude and phase: f(tδt) ≈ Ech(t).

Here, we only discuss the amplitude modulation in the chirped readout pulse because phase encoding does not enable balanced detection. In the time domain, the chirped readout pulse’s E-field in the absence of a THz pulse can be written as: Ech,0(t) = ϵch,0(t) eiΦch,0(t), where ϵch,0(t) and Φch,0(t) are its time-dependent amplitude and phase, respectively. As described previously, when the chirped readout pulse’s polarization lies between the principal axes of the crystal at 45° and following phase-to-amplitude conversion with polarization-sensitive optics, the THz E-field is imprinted on the amplitude of the transmitted chirped pulse: E ch,THz ± ( t ) = ϵ ch,THz ± ( t ) e i Φ ch , 0 ( t ) , where ϵ ch,THz ± ( t ) = 1 ± Δ φ ( t ) ϵ ch , 0 ( t ) . Consequently, the signal modulation is conveniently extracted from the envelopes of both Ech,0 and Ech,THz,

Δ I ± ( t ) I 0 ( t ) = ϵ ch,THz ± ( t ) ϵ ch , 0 ( t ) 2 1 .
(7)

Following the implementation of balancing according to (3), the THz E-field may directly be calculated since ΔI/I0 ∝ ΔφETHz as shown by (1)-(3).

An example of a pair of raw interferograms with balancing implemented and THz signal present is shown in Fig. 4(a). We integrated over the (+) and (−) polarization interferograms to recover a spectrum from each; the spectrum for the (+) polarization is shown in Fig. 4(b). The fringe spacing changes because the time delay between the readout pulse and the blue edge of the chirped pulse is much shorter than the delay between the readout pulse and the red edge of the chirped pulse. Similarly, a reference interferogram and its respective spectra for both polarizations are acquired in a successive laser shot in the absence of a THz pulse. In accordance with (6), the spectra in Fig. 4(b) were inverse Fourier transformed to the time domain; the modulus is shown for the (+) polarization in Fig. 4(c). Since the cross-correlation peak at either ±δt ≈ 5 ps represents the envelope of the chirped pulse, the THz signal modulation can be calculated using (7) by simply squaring and dividing the signal and reference time domain traces in Fig. 4(c) (taken with two successive laser pulses) and subtracting unity. A zoomed-in view around +δt is shown for both polarizations in Fig. 4(d), however, in these traces the THz signal is barely distinguishable from the noise. Because the noise is largely correlated, it is effectively suppressed by balancing as seen in Fig. 4(e), where the noise is reduced and the THz signal is doubled as predicted by (3).

FIG. 4.

THz spectral interferometry data processing. (a) Raw interferogram derived from a single laser pulse, showing (+) and (−) detected polarizations. (b) The spectrum is calculated by integrating over the vertical dimension in the (+) interferogram in (a). (c) Modulus of the Fourier transform of the spectral interference in (b), which corresponds to the amplitude of the cross-correlation between the chirped and short readout pulses’ E-fields (i.e., approximately the envelope of the chirped readout pulse) in the time domain. (d) Unbalanced time traces collected in two laser pulses for both polarization states by dividing the signal and reference traces in (c) and subtracting unity. With no averaging, the THz signal is indistinguishable from the noise. (e) Balanced time trace collected in two laser pulses following subtraction of (+) and (−) traces in (d), which reveals the THz signal.

FIG. 4.

THz spectral interferometry data processing. (a) Raw interferogram derived from a single laser pulse, showing (+) and (−) detected polarizations. (b) The spectrum is calculated by integrating over the vertical dimension in the (+) interferogram in (a). (c) Modulus of the Fourier transform of the spectral interference in (b), which corresponds to the amplitude of the cross-correlation between the chirped and short readout pulses’ E-fields (i.e., approximately the envelope of the chirped readout pulse) in the time domain. (d) Unbalanced time traces collected in two laser pulses for both polarization states by dividing the signal and reference traces in (c) and subtracting unity. With no averaging, the THz signal is indistinguishable from the noise. (e) Balanced time trace collected in two laser pulses following subtraction of (+) and (−) traces in (d), which reveals the THz signal.

Close modal

The spectral interference, as given by (4), shows fast modulations introduced by the linear phase factor, ωδt, on top of a large, slowly varying background corresponding to the sum of the spectral intensities of the chirped and short readout pulses. Consequently, the signal interferogram does not change significantly compared to the reference interferogram due to a relatively small signal modulation caused by the THz E-field. Experimentally, it was non-trivial to achieve full modulation depths across the spectrum, but the optical alignment was fine-tuned until we observed reliable measurement of the THz E-field. We expect that further improvements should yield larger modulation depths and an increase in the S/N.

Our most important evaluation was to determine whether amplitude encoding spectral interferometry reproduced the THz signal quantitatively with a lower noise floor for a fixed acquisition time compared to conventional EO sampling. In Fig. 5(a), a direct comparison of the THz traces recorded using the two methods is shown, where we used the same total number of laser pulses to record both traces. Spectral interferometry maintains the high temporal resolution of the scanning method by incorporating a short readout pulse, differing from the result obtained by solely using a chirped readout pulse.25 Although the temporal dynamics were closely matched, the absolute THz signal measured differed from the delay line method and depended strongly on the alignment of the interferometer. The THz spectrum in Fig. 5(b), given by the Fourier transform of Fig. 5(a), has approximately the same bandwidth as that measured using the delay line, although it is difficult to resolve spectral features because of noise and a short experimental time window that yielded poor spectral resolution.

FIG. 5.

Direct comparison between THz spectral interferometry and conventional EO sampling. (a) The averaged and balanced THz time traces for THz spectral interferometry and scanning delay line method show relatively good agreement in signal strength and temporal dynamics. (b) The Fourier transforms of the time traces given in (a) show similar spectral bandwidth. The poor spectral resolution is a result of the short temporal window used in the experiment.

FIG. 5.

Direct comparison between THz spectral interferometry and conventional EO sampling. (a) The averaged and balanced THz time traces for THz spectral interferometry and scanning delay line method show relatively good agreement in signal strength and temporal dynamics. (b) The Fourier transforms of the time traces given in (a) show similar spectral bandwidth. The poor spectral resolution is a result of the short temporal window used in the experiment.

Close modal

It is evident in Fig. 5(a) that THz spectral interferometry yields a noisier time trace than the delay line method for the same acquisition time. Since single-shot acquisition records all n time points simultaneously, for the same acquisition time it should be able to average a factor of n more traces than the delay line method (disregarding data transfer and computational processing time) and reduce random noise by factor of n . Ideally, the noise of one single-shot trace would be the same as a single trace acquired by the delay line method, but expectedly there is a tradeoff between acquisition speed and sensitivity. We found that the measured noise of the single-shot trace (two laser pulses) compared to that of a delay line trace (a factor of n more laser pulses) was too large, such that noise suppression even by extensive averaging of traces would never yield the requisite noise floor. The respective noise of each method, defined here as the uncertainty in ΔI/I0 across a trace acquired in the absence of THz pulses (not shown), was measured to be 0.13% using spectral interferometry and 0.06% using the delay line method for the same number of laser pulses (a total of 6400 laser pulses).

In this system, laser noise presents itself in similar ways to conventional EO sampling with a photodiode (e.g., intensity fluctuations, electronic detector noise), but also in new respects. More stringent conditions are required for low-noise interferograms since laser pointing fluctuations compounded with spatial chirp dynamically change the spectral interference compared to just measuring intensity changes with a large-area photodiode. Additionally, spectral phase and amplitude profiles of the pulses change on a shot-to-shot basis. The last, but most important, concern is with regard to optical phase stability in the system that also drifts from shot to shot, between successive signal and reference acquisitions. Relative phase shifts between the signal and reference interference patterns manifest themselves strongly when the two time domain traces in Fig. 4(c) are divided to yield Fig. 4(d). In spite of single-shot spectral recording, ultra-high phase stability is still desired to achieve good S/N.

Overall, amplitude encoding spectral interferometry is able to detect THz E-fields with high temporal resolution and close to quantitative field strengths, but its poor noise performance and oversensitivity to alignment, spectral fluctuations, and the requirement of high optical phase stability cannot be ignored in the case of measuring small nonlinear signals. While spectral interferometry is also routinely used in 2D electronic spectroscopy,57 the methodology is significantly different. In that case, only a readout pulse is always present and the signal is indicated by the mere existence of a cross-correlation peak (i.e., spectral interference). This is a background-free measurement, which is not the case here where we measure small changes in an already present cross-correlation peak formed by interference between the short readout pulse and the chirped pulse. In the current geometry, we conclude that spectral interferometry is not a promising candidate for multidimensional THz measurements.

Previous studies using dual echelons have focused on optical pump-probe measurements for irreversible processes58,59 and also more relevantly on diagnostics for real-time characterization of THz pulses.52 In the optical pumping scenario,60 the optical pump and probe spot sizes are on the same order of magnitude, which means that pointing fluctuations largely impact the S/N or there must be a tradeoff between using larger pump spot sizes (i.e., lower fluences) and the signal strength. In contrast, the THz pumping scenario allows for both the THz pump and optical readout beams to be focused since the THz spot size is approximately three orders of magnitude larger. As a result, the THz detection scheme is less susceptible to laser pointing fluctuations and long term drift, and enables the possibility for signal averaging over an extended period of time. Here, we intend to transform the method into a robust technique for THz spectroscopy by implementing detection at a higher frequency with balanced detection and multi-shot averaging.

We used two transmissive echelons, each of which is a right-angled prism with a staircase structure along its hypotenuse surface: one with large steps (step height H = 300 μm, time step = HΔn/c0 = 500 fs) and one with small steps (step height h = 15 μm, time step = hΔn/c0 = 25 fs) each having transverse dimensions of 15 mm × 15 mm (see inset of Fig. 6). Each echelon has 20 steps to yield a total of 400 pulses incrementally delayed in ∼25 fs time steps with an overall time window of 10 ps. The echelons were fabricated by depositing index-matched epoxy onto the hypotenuse of a right-angled fused silica prism and molding the epoxy with steps of the appropriate size. A competing fabrication technology for higher optical quality transmissive echelons stacks and chemically bonds thin layers of fused silica, which can be obtained from Okamoto Optics (Japan). Alternatively, reflection-mode echelons, which are more favorable when much larger temporal windows are desired (for higher spectral resolution) or for setups with very broad bandwidth lasers, can be obtained from Sodick F.T. (Japan); these echelons have recently been demonstrated to measure THz E-fields.61 

FIG. 6.

The experimental setup showing convenient switching between dual echelons and delay line methods, each with balanced EO detection. The input laser is split into the optical pump for THz generation in ZnTe and the optical readout arm. An expanded and spatially filtered optical readout beam is directed through a pair of transmissive echelons. The inset shows a schematic illustration of dual 20-step echelons used to generate 400 pulses incrementally delayed in time (not drawn to scale). The large echelon (right) has a step height of H = 300 μm and the small echelon (left) has a step height of h = 15 μm. The emerging beamlets and THz beam are both focused to the same spot at the ZnTe detection crystal. Then, the beamlets are passed through phase sensitive optics (i.e., a quarter-wave plate and polarizing beam splitter), before being imaged onto a camera. Alternatively, in conventional EO sampling, the optical readout beam is re-directed to bypass the echelons and detected with a pair of photodiodes.

FIG. 6.

The experimental setup showing convenient switching between dual echelons and delay line methods, each with balanced EO detection. The input laser is split into the optical pump for THz generation in ZnTe and the optical readout arm. An expanded and spatially filtered optical readout beam is directed through a pair of transmissive echelons. The inset shows a schematic illustration of dual 20-step echelons used to generate 400 pulses incrementally delayed in time (not drawn to scale). The large echelon (right) has a step height of H = 300 μm and the small echelon (left) has a step height of h = 15 μm. The emerging beamlets and THz beam are both focused to the same spot at the ZnTe detection crystal. Then, the beamlets are passed through phase sensitive optics (i.e., a quarter-wave plate and polarizing beam splitter), before being imaged onto a camera. Alternatively, in conventional EO sampling, the optical readout beam is re-directed to bypass the echelons and detected with a pair of photodiodes.

Close modal

The optical layout for the echelon experiments is shown in Fig. 6. In order to quantitatively compare single-shot echelon measurements and conventional point-source measurements with a scanning delay line, the setup could be rapidly switched between the detection modalities using magnetic base plates. A spatially filtered and expanded optical readout beam passed through the echelons and was imaged onto a camera for single-shot detection using a standard 2-lens 4f imaging system, with the ZnTe detection crystal in the Fourier plane and the camera in the image plane. We used a relatively long focal length first lens (f1 = 50 cm) and demagnified by 2 × after the second lens (f2 = 25 cm) at the image plane. The long focal length was used for an increased depth of focus, while allowing for sufficient image resolution and a relatively tightly focused optical readout spot size (∼50 μm diameter) compared to the THz spot size (∼1 mm diameter) at the ZnTe detector crystal. In the delay line method, a mirror on a magnetic base plate redirected the expanded readout beam through a 1 mm diameter pinhole, replicating the beam diameter of a beamlet from the dual echelons. Then, the single beam was focused into the ZnTe detection crystal and detected with photodiodes for point-source measurements.

The THz beam and either the multiple optical beamlets from the echelons or a single optical beam for standard detection were combined with a pellicle beam splitter such that they propagated collinearly through the detection crystal. THz-induced modulation in the readout beam was detected using the standard EO sampling configuration with a quarter-wave plate and polarizing beam splitter. The high optical quality ZnTe crystal produced images with relatively sharp features and tolerable degradation in image quality.

For balanced detection in the single-shot echelon setup, both polarization states emerging from the polarizing beam splitter are imaged onto the camera. A raw image showing the pair of orthogonal polarization states acquired in a single laser shot is shown in Fig. 7(a). For both images to be in focus, the path length from the detection crystal to the camera must be the same for the transmitted and reflected beams emerging from the polarizing beam splitter. The beam transmitted through the polarizing beam splitter was reflected directly to the detector while the reflected beam was routed to the detector slightly off right-angle using a pair of mirrors (see Fig. 6). A polarizing beam splitter was used due to the high optical quality required for near aberration-free imaging; alternatively, a small-splitting-angle, high optical quality Wollaston prism could be used.54 As for the spectral-interferometry method described above, data were recorded on a 10-bit CMOS camera which was synchronized to the laser and recorded images at 1 kHz. The THz pulses were modulated at 500 Hz, so alternating pulses were used to record signal and reference images.

FIG. 7.

Data processing in dual echelon single-shot THz method. (a) An example of a raw pair of signal images (S±) with the THz pulse present for (+) and (−) polarization states. A pair of reference images (R±) with the THz pulse blocked looks very similar. (b) An image pair (taken with two laser pulses) computed by dividing the signal image by the reference image (S±/R±). The THz modulation is visible in several columns near the center of the grid, where time is increasing from top to bottom (25 fs steps) and left to right (500 fs steps) for each square of the grid. In practice, data processing follows a different algorithm. (c) Using a custom extraction algorithm, THz time traces were retrieved for left (+) and right (−) echelon images shown in (a). The unbalanced traces have opposite signs as expected. (d) THz time trace with balancing implemented by subtraction of traces in (c). This doubles the magnitude of signal and suppresses common mode noise.

FIG. 7.

Data processing in dual echelon single-shot THz method. (a) An example of a raw pair of signal images (S±) with the THz pulse present for (+) and (−) polarization states. A pair of reference images (R±) with the THz pulse blocked looks very similar. (b) An image pair (taken with two laser pulses) computed by dividing the signal image by the reference image (S±/R±). The THz modulation is visible in several columns near the center of the grid, where time is increasing from top to bottom (25 fs steps) and left to right (500 fs steps) for each square of the grid. In practice, data processing follows a different algorithm. (c) Using a custom extraction algorithm, THz time traces were retrieved for left (+) and right (−) echelon images shown in (a). The unbalanced traces have opposite signs as expected. (d) THz time trace with balancing implemented by subtraction of traces in (c). This doubles the magnitude of signal and suppresses common mode noise.

Close modal

In the dual echelon method, two successive laser pulses yield two pairs of images, where each pair contains both polarization states (+) and (−): one pair with the THz pulse present (the signal, S±) and the other with the THz pulse blocked (the reference, R±) (see Fig. 7(a) for an example image pair). For the reference, the “+” and “−” only denote which polarization state it belongs to since no sign dependence exists unlike for the signal; the images corresponding to each polarization are similar but not identical, and therefore, separate references are required to accurately record the signal and suppress noise effectively. For demonstrative purposes, Fig. 7(b) shows an image resulting from dividing a signal image by a reference image, which was collected with two successive laser pulses. The readout pulse’s intensity is visibly modulated by the THz E-field near the center of the grid, and the sign of modulation is opposite in the left and right images. Here, the directionality of time is increasing from up to down in 25 fs steps and left to right in 500 fs steps.

In practice, a time trace is first extracted from both signal and reference images like those in Fig. 7(a) before dividing signal by reference. The first step involves averaging of the signal and reference images, which are collected in alternating succession (i.e., odd-numbered images correspond to the signal and even-numbered images correspond to the reference since the pump is blocked for every even-numbered laser pulse). For typical measurements of systems with reversible dynamics, numerous signal and reference images can be collected in succession and averaged to suppress noise. After recording N images, we summed the N/2 odd-numbered signal images and the N/2 even-numbered reference images pixel-by-pixel. This order of operations is the least computationally intensive and minimizes the total acquisition time.

Prior to extraction, a “projective” transformation was applied in Matlab to each of the post-cropped (+) and (−) grids; this corrects for any tilt or skew in the images and allows for methodical extraction of the time-dependent signal associated with each square region of the echelon grid. The same transformation matrix for the (+) grid was applied to both signal and reference grids and similarly for the (−) grid, in order to yield the corrected grids, referred to as S avg ± and R avg ± .

Next, we determined the intensity value at each of the 400 squares in the 2D grid by integrating the intensity across all of the pixels contained within a specified region for each square. The integrated intensities were recorded at every ith-row and jth-column coordinate of the S avg ± grid, given by I S ± ( i , j ) , and in the R avg ± grid, given by I R ± ( i , j ) . The selected dimensions of the square were optimized to discard pixels near the edges that were more susceptible to fluctuations while enough pixels were integrated over to suppress the electrical noise of the detector; approximately 60% of the central region of each square was used. In particular, the minimization of noise in the traces required optimization of the location and size of squares in two steps: first, individually for each polarization state, and then, together with balancing implemented. The optimization was conducted by calculating the noise floor from extracted intensity traces in the absence of any THz signal and the image processing parameters determined for each polarization state.

Subsequently, the 20 × 20 grid can be unwrapped into a 400 × 1 vector of intensity values, which corresponds to sequential time delays with 25 fs steps, i.e., I±(i, j) → I±(t). In this experiment, because the last row of the echelons we used were damaged, we discarded every other row to instead yield a 200 × 1 time sequence with 50 fs steps. The time-dependent signal modulation that corresponds to a THz time trace was calculated for each polarization state according to

Δ I ± ( t ) I 0 ( t ) = I S ± ( t ) I R ± ( t ) I R ± ( t ) ,
(8)

where I S ± ( t ) and I R ± ( t ) are the recovered signal and reference intensity time traces, respectively. A single-shot THz time trace for each polarization state is shown in Fig. 7(c). Balancing was implemented according to (3) by subtracting these two time traces of orthogonal polarizations. A marked improvement in S/N is demonstrated in the resulting balanced single-shot THz trace shown in Fig. 7(d), where the signal modulus is doubled and common mode noise is reduced.

Following the successful demonstration of the dual echelon method in extracting a THz waveform, further analysis is required to conclude on its potential as a viable tool for multidimensional THz spectroscopy. To reiterate, we require that the single-shot method: (1) must be capable of quantitatively recording the THz time trace and (2) for a given acquisition time, the S/N must be better than the conventional scanning delay line method. Below, we examine these criteria.

1. Accurate THz waveform extraction

The first criterion was evaluated by acquiring THz time traces with the dual echelon method and the conventional scanning delay line method. Here, the number of traces averaged, N, the number of laser pulses per time delay, κ, and the number of time points per trace, n, were kept constant between both methods. In the dual echelon method, after every κ laser pulses, the images were averaged to yield a single trace and repeated for a total of N traces. If both methods were limited by identical noise sources, their S/N would be matched when they have the same number of laser pulses per time delay: Nκn total pulses for the delay line method and total pulses for the dual echelon method. We will show that for a 1D trace, the acceleration of acquisition time to achieve a matched S/N is less than the optimal reduction of 1/nth of the time, but still provides substantial timesaving.

For each method, Fig. 8 shows the average of many traces and resulting spectra with the same number of averages per time delay (N = 55, κ = 250), however, the dual echelon method uses 1/nth of the number of laser pulses compared to with the delay line. As seen in the time traces in Fig. 8(a) and their Fourier transforms in Fig. 8(b), the dual echelon method (solid pink line) was able to replicate a THz time trace collected by the delay line method (dashed green line). In addition to the closely matched time dynamics, the signal modulation amplitudes also agreed quantitatively (traces in Figs. 8(a) and 8(b) have not been normalized to each other). This demonstrates the dual echelon method’s ability to accurately record THz spectra, which enables quantitative E-field measurement with reliable spectral features. Small deviations in the time traces and Fourier domain spectra in Fig. 8 may be a result of physical imperfections in the echelons themselves that may be seen in the images in Fig. 7(a), the unique sources of noise in each system discussed below, or laser drift or a change in lab humidity between the two measurements.

FIG. 8.

Direct comparison of dual echelon method and conventional EO sampling. (a) THz time traces for dual echelon method and scanning delay line method for the same number of averages per time delay (i.e., the dual echelons use a factor of n fewer total number of laser pulses than the delay line method). This comparison demonstrates a high degree of agreement both in temporal dynamics and THz signal modulation strengths. (b) Fourier transforms of the time traces given in (a) show that the dual echelon method is capable of recording a reliable THz spectrum.

FIG. 8.

Direct comparison of dual echelon method and conventional EO sampling. (a) THz time traces for dual echelon method and scanning delay line method for the same number of averages per time delay (i.e., the dual echelons use a factor of n fewer total number of laser pulses than the delay line method). This comparison demonstrates a high degree of agreement both in temporal dynamics and THz signal modulation strengths. (b) Fourier transforms of the time traces given in (a) show that the dual echelon method is capable of recording a reliable THz spectrum.

Close modal

2. Noise sources and suppression

The remaining, but most important test, was to quantify the potential time savings from using the echelons instead of a scanning delay line. In a dual echelon THz trace, like that shown in Fig. 7(c), the most noticeable feature was the unique periodic structure of the noise. This observation led to the final evaluation on expediting THz acquisition using the dual echelons, which was conducted with the THz pulse blocked and only the readout pulse measured. Additionally, the readout-noise-limited baseline was normalized for the total number of pulses (i.e., the same total acquisition time) between both methods and calibrated with no detection crystal in order to simulate the best-case scenario which uses very high optical quality crystals (aberrations degrade image quality and consequently interfere with the signal extraction process).

Figure 9(a) shows the unbalanced averaged readout-only traces for each polarization state, where the strongly periodic noise components are visible. The structural noise occurs at the period of a large echelon step (i.e., 500 fs or 2 THz), with its largest magnitudes, specified by the spikes, coinciding with the intensities derived from the perimeters of the 2D grid. The imaging-based detection is more susceptible to shot-to-shot beam pointing fluctuations and slow laser drift that result in jitter and long-term drift of the spatial 2D image and consequently negatively impact the extraction process. The phenomenon may be explained for a beam with a Gaussian intensity profile that varies across the transverse dimensions of the echelons; the spatial variation will be most noticeable in moving from an echelon step located at the last row to one at the top row that corresponds to sequential time delays. The increased sensitivity to laser fluctuations is not surprising since compared to large area photodiodes used in scanning EO sampling, the echelons have a reduced active detection area per extracted time delay on the camera chip.

FIG. 9.

Noise traces where the THz pump is blocked and the total number of pulses acquired are maintained between both dual echelon and delay line methods. (a) The averaged unbalanced traces from the dual echelons, where periodic noise is strong. (b) The balanced traces from both methods, where the dual echelon method achieves a lower noise floor than the delay line method. With balancing implemented, the periodic noise arising from the dual echelons, shown in (a), is greatly suppressed. The dual echelons average a factor of 200 times more traces (equivalent to the number of time points in a trace, n = 200) in the same total acquisition time. (c) The rms noise of the dual echelons system in response to averaging a number of traces given by Nech, which when plotted on a logarithmic scale yields close to 1 / N ech scaling. The dual echelons attain the same noise floor of the delay line method at Nech = 145 traces (290 pulses), which represents approximately 4.5% of the total laser pulses required by the delay line method (6400 pulses).

FIG. 9.

Noise traces where the THz pump is blocked and the total number of pulses acquired are maintained between both dual echelon and delay line methods. (a) The averaged unbalanced traces from the dual echelons, where periodic noise is strong. (b) The balanced traces from both methods, where the dual echelon method achieves a lower noise floor than the delay line method. With balancing implemented, the periodic noise arising from the dual echelons, shown in (a), is greatly suppressed. The dual echelons average a factor of 200 times more traces (equivalent to the number of time points in a trace, n = 200) in the same total acquisition time. (c) The rms noise of the dual echelons system in response to averaging a number of traces given by Nech, which when plotted on a logarithmic scale yields close to 1 / N ech scaling. The dual echelons attain the same noise floor of the delay line method at Nech = 145 traces (290 pulses), which represents approximately 4.5% of the total laser pulses required by the delay line method (6400 pulses).

Close modal

However, the periodic noise itself does not necessarily stifle the single-shot method since averaging effectively suppresses random noise from pointing fluctuations. Additionally, high-frequency chopping that reduces 1/f noise is important for suppressing noise arising from a spatiotemporally fluctuating intensity of the readout pulse. Finally, structural noise is a prime candidate for the implementation of balancing since the noise profiles in the (+) and (−) polarization traces are highly correlated, as seen in Fig. 9(a), while any THz signal would appear with opposite sign. Through balancing, the rms noise was effectively suppressed by more than a factor of two [compare traces in Fig. 9(a) with the balanced trace, solid pink, in Fig. 9(b)]. These noise suppression techniques were equally effective in the delay line method [see dashed green line in Fig. 9(b)].

In our conclusion on noise performance, in particular to evaluate the effects of noise sources unique to the dual echelons, we made a direct comparison of the noise floors while conserving the total number of laser pulses between the two methods. Figure 9(b) clearly shows that the dual echelons achieve a lower noise floor than that of the delay line in the same total acquisition time. Although the dual echelons did not yield a reduction in noise by the optimal factor of 200 estimated from averaging a factor of 200 more traces (equivalent to the number of time points in a trace, n = 200), the lower noise floor indicates that there still may be a potential for significantly reducing experimental acquisition time. We quantified the absolute time saved by recording the rms noise as a function of the number of traces averaged, Nech, shown in Fig. 9(c), where two laser pulses yield one trace. We found that the dual echelon method was able to recover an equivalent level of S/N after averaging 145 traces (290 laser pulses), which represents only 4.5% of the averaging time required by the delay line method (6400 laser pulses) or approximately 22 times faster. Therefore, in addition to being quantitatively reliable, we find that the dual echelon method has a potential for expediting acquisition in time-demanding multidimensional THz experiments.

Our last observation of noise suppression in the dual echelon system concerns the trending behavior of the noise floor in response to variable periods of averaging time. In Fig. 9(c), the data are plotted on a logarithmic scale and compared to the 1 / N ech scaling (dotted-dashed line) that is expected from averaging. While noise suppression due to averaging works effectively at short time scales, where averaging many laser pulses at a high acquisition rate effectively suppresses beam-pointing instability, we observed a deviation after Nech ≈ 220 (or ≈440 ms acquisition time). The deviation results from uncompensated slow laser drift in the system occurring on the several hundreds of milliseconds time scale. An iterative optimization algorithm that retrieves the extraction parameters on this time scale is likely to benefit noise performance for scans acquired over an extended period of time.

We reviewed the existing single-shot THz techniques and identified two that were the most promising for fast and accurate acquisition of THz traces: spectral interferometry and the dual echelon approach. For these two single-shot methods, we carefully evaluated the following central criteria that would enable their use as a viable tool for multidimensional THz spectroscopy: signal reproducibility and a low noise floor. Unsurprisingly, there was a tradeoff between fast acquisition and measurement noise, but effective noise suppression techniques enabled the single-shot dual echelon method to meet both criteria.

THz spectral interferometry was able to accurately reproduce THz time traces as recorded by a point-source readout with a scanning delay line, but presented a multitude of technical difficulties in terms of accurate alignment and interferometric phase stability. This led to signal strengths that were overly sensitive to optical alignment and a higher noise floor than standard scanning EO sampling, which ultimately precludes spectral interferometry as a competitive method for multidimensional THz spectroscopy. Common path optics, for example, using phase masks and spatial light modulators, would improve phase stability and could greatly aid the method, but it is not clear that these strategies can produce a sufficiently chirped optical pulse.

The dual echelon method was also able to accurately record a THz signal with a much better S/N than the classic scanning delay line approach. The THz field strength, spatio-temporal waveform, and spectral content were all well-matched, and could attain the same noise floor as the delay line method in 4.5% of the number of laser pulses otherwise required (≈22 times faster). The amount of time saved will be even greater for experiments that require additional time points for larger time windows (e.g. for temporally long resonant features with narrow spectral linewidths) or finer temporal sampling (e.g. for high-frequency resonances) since acquisition time increases for the delay line method, but ideally not for the single-shot method. The dual echelons have straightforward technical experimental implementation, although it requires custom echelon optics and a camera with particular specifications (e.g., with regard to data transfer time, electronic noise performance, and synchronization capabilities). The largest difference compared to traditional EO sampling arises in the data processing, where more diligent care is needed to effectively suppress additional sources of noise in a reliable and expedient manner.

Because this fast-acquisition method gives quantitative traces with a low noise floor, the dual echelons are a convincing alternative to scanning delay lines when transitioning to higher-dimensional THz experiments. Specifically, there are advantages to recording a 2D trace quickly since pump-induced fluctuations (typically the dominant source of noise in strong pumping regimes) are largely minimized. Additionally, the method easily lends itself to measurements of small, nonlinear signals, where higher sensitivity is needed, or of small changes in large THz signals, where both high sensitivity and a high dynamic range are needed. For experiments that require finer spectral resolution, we are transitioning towards a reflective large echelon for larger time windows since it allows for in-focus imaging of the beamlets, where in the optimal configuration, all steps lie in the same image plane. Furthermore, we believe that possible improvement may arise from implementing sampling with a high-repetition-rate oscillator that is electronically gated to the THz pulse.62 In this scheme, noise resulting from laser amplification and 1/f noise may be significantly reduced; oscillators also typically produce a more homogeneous spatial mode profile compared to that of an amplified pulse, which is key to low-noise EO imaging with the dual echelons. More recently, a distortion-free THz signal enhancement technique that allows for balanced detection63,64 has been developed, and this may be incorporated to yield higher S/N. In light of these recent measurements, we find that expediting THz detection using dual echelons is a powerful technique that may be implemented in a straightforward manner for a variety of new experiments.

The authors thank Dr. Sharly Fleischer and Dr. Dylan Arias for useful discussions in the presented work. K.A.N. acknowledges encouraging discussions with Dr. C.-W. Baik and Dr. O. Gurel of Samsung Advanced Institute of Technology. The work was supported by NSF Grant No. CHE1111557 and the Samsung GRO Program. B.K.O. was supported in part by NSF GRFP.

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