1. INTRODUCTION

Many applications of terahertz (THz)-waves [1,2], such as non-destructive evaluation [3,4], biomedical diagnostics [5,6], security monitoring [7,8], ultrafast nonlinear spectroscopy [9,10], and communication technologies [11,12], are the subject of intense research. For photonics-based THz technologies, time-domain sampling acquisition systems still present significant challenges for applications requiring real-time data [13]. Fortunately, recent advances in hardware and computational algorithms have significantly improved the performance of THz systems, targeting higher acquisition speeds [14,15], greater sensitivity [16,17], wider dynamic range [18,19], and optimal adaptation to the needs of specific applications [20]. THz time-domain spectroscopy (THz-TDS) has played a key role in these advances [21]. Yet it remains difficult to achieve real-time performance and high-frequency resolution simultaneously, due to the inherent trade-offs between measurement time and sample number [13]. Conversely, THz spectroscopy in the frequency domain offers high-frequency resolution and a wide dynamic range [7,18,22–25], but at the cost of the lack of real-time broadband acquisition capability. Essentially, there is still to this day a clear trade-off in desired performance between THz approaches in the time domain and in the frequency domain [13].

Unlike the Fourier transform of a THz waveform, an ideal system for THz spectroscopy must be able to detect broadband THz-waves with high-frequency resolution while working in real time without the need for digital processing or sampling. In optics, processing mathematical functions in an analog way is well-known to accelerate the processing speed of signals or information [26,27]. For THz-waves, there are at least two conceivable approaches to achieve this: one involves the spatial separation of broadband THz-waves by frequency using diffraction prisms or gratings [28]; the other employs up-conversion of broadband THz-waves to near-infrared range (NIR) light through difference-frequency optical mixing [16,17]. Developing a THz spectrometer based on diffraction gratings entails several drawbacks, such as reflection losses at the grating surface and the need to use power meters in the THz range, which are generally inefficient, have limited working bandwidth, and always require a compromise between frequency selectivity and sensitivity [28]. On the other hand, difference-frequency generation through up-conversion of THz photons to the NIR shows promising results [16,17], with the potential to even open the road towards quantum optics in this frequency range [16,17]. Nonetheless, detecting broadband THz-waves generated by a broadband fs laser necessitates a similarly broadband fs probe light. Given that both the THz-waves and the probe light are broadband, complex difference-frequency optical mixing occurs [29], complicating the retrieval of high-resolution THz-wave spectral information. Moreover, no single-shot broadband parametric spectrometer has been demonstrated to date. In short, neither method discussed above currently facilitates the real-time and high-resolution spectroscopy of broadband THz-waves, thus necessitating the development of a novel spectroscopic technique.

Here, we propose a method that involves real-time measurement of the wavenumber of broadband THz-waves, thereby bridging the gap and leveraging the capabilities of both time-domain and frequency-domain approaches. By focusing on wavenumber—a vector quantity—instead of frequency—a scalar quantity—we have developed a THz spectroscopy method that achieves high-frequency resolution while preserving real-time measurement capabilities through one-shot acquisition in an analog manner.

2. THEORETICAL BACKGROUND

A. THz Parametric Detection

In this investigation, we employ THz parametric detection [22], where broadband THz-waves are up-converted to near-infrared light, to ascertain the wavenumber of THz-waves. This process of THz parametric wavelength conversion adheres to the principles of energy and momentum conservation, involving the pump beam (?pump), Stokes beam (?Stokes), and THz-wave (?THz). Firstly, for a collinear phase matching condition, where the three waves, i.e., the pump beam (?pump), Stokes beam (?Stokes), and THz-wave (?THz), are aligned with no angular separation, a phase mismatch (Δ?) inevitably appears due to the broadband nature of the pump beam. Since the THz wavenumber only affects the intensity of the Stokes beam, and the Stokes beam is generated coaxially with the pump beam, it becomes difficult to distinguish the different wavenumber signals from the high background noise of the pump beam.

Conversely, THz parametric detection using a non-collinear phase matching condition is a background-free technique [22], in which each Stokes beam is emitted at a distinct angle relative to the pump beam to comply with the law of momentum conservation. This configuration enables the measurement of the THz-wave’s wavenumber by monitoring the angle of Stokes beam generation using a two-dimensional sensing device, such as a charge-coupled device (CCD) or a complementary metal-oxide semiconductor (CMOS) camera. As an example, this technique has recently demonstrated ultra-high-detection sensitivity at the zeptojoule (zJ) level at room temperature for detecting monochromatic THz-waves, utilizing sub-nanosecond pump pulses with a multi-stage amplification configuration [18]. Indeed, ns-duration optical pumping has so far made it possible to effectively exploit lattice vibrations (Stokes) in lithium niobate for this purpose [18,22,23,25], thanks to a process onset that takes just over a hundred femtoseconds [30].

To our knowledge, non-collinear parametric detection geometry remains the only method capable of accurately measuring THz wavenumbers, and this fundamental measurement should be extended to broadband THz pulse detection.

B. Frequency Resolution by Wavenumber Measurement

In the collinear up-conversion technique using ultrafast laser pulses, which measures THz information by differential frequency generation (DFG) or sum frequency generation (SFG), the bandwidth of the pump beam must be limited using filters to improve frequency resolution [17], as the large bandwidth inherent in fs laser pulses has a negative impact on frequency resolution. Even with the deployment of several narrow-band filters, the achievable frequency resolution remains limited [17]. In addition, the reduction in bandwidth significantly reduces the amount of pump energy, potentially degrading the signal-to-noise ratio (SNR). In contrast, the parametric detection method proposed here eliminates the need for filters, even when using a broadband femtosecond (fs) pump laser. The method also significantly improves frequency resolution by directly measuring wavenumbers. The underlying principle is elucidated qualitatively as follows.

Initially, the influence of the broadband nature of the pump beam is analyzed in Fig. 1(a). For a given THz frequency, the angular broadening Δ? of the Stokes beam can be approximated by the angle formed by the two Stokes beams (?Stokes1) and (?Stokes2) at each end of the pump’s spectral width. This is illustrated by the two green lines in Fig. 1(a) when the longest (?pump1) and shortest (?pump2) wavelengths of the pump beam interact with the same THz-wave (?THz), indicated by the yellow lines. Naturally, the angle of generation of each Stokes beam is governed by the law of conservation of momentum, as illustrated by the equation

pumpTHzStokes?pump−?THz=?Stokes.

Fig. 1. Concept figure. (a) Non-collinear phase matching condition using broadband pump beam. (b) Interaction between pump beam and THz-waves in a non-collinear geometry. (c) Frequency resolution attainable through wavenumber measurement using a 280 fs optical pulse duration. (d) Up-converted THz-waves observed with a commercial IR sensor card

The angleformed between the pump beam and the Stokes beam is determined by the following equation:

?=?Stokes??Stokes?=|?pump?−?THz?||?pump?−?THz?|,

Given that the wavenumber of the THz-wave is significantly smaller (approximately 1/300th) than that of near-infrared light, ?Stokes? is markedly less than ?Stokes?. As a result, the phase matching angle Δ? between the Stokes beams generated by the pump’s broad bandwidth for the same THz frequency becomes nearly negligible.

In a second step, we examine the effect of the interaction length (?) between the pump size and the THz-waves, as illustrated in Fig. 1(b). In practical scenarios, both the pump and THz-waves possess a finite beam diameter, and the resultant wavenumber vector is dictated by the uncertainty principle as formulated in Eq. (3) as follows:

This condition acts as a limit to the system’s frequency resolution, attributable to the diffraction limit of the generated Stokes beam. This condition is illustrated by ? in Fig. 1(b). Basically, the more the pump beam overlaps the THz-wave, the better the frequency resolution of the different wavenumber components in the output Stokes signal. This geometry creates an effect similar to that of sampling a wider time vector when using a THz-TDS system, but in analog form and with a single pump pulse.

Here, ?THz and ?pump represent the refractive indices of THz and near-infrared light in LiNbO3, respectively. Δ?? is the frequency bandwidth of the pump, ? is the angle between the pump and Stokes beam, ? is the angle between the pump and THz-waves, and ? is the speed of light.

In Fig. 1(c), the blue line represents the frequency resolution determined by the bandwidth of the pump beam [Fig. 1(a) and the left-hand term in Eq. (4)], while the orange line represents the frequency resolution determined by the interaction length between the pump and the THz-wave [Fig. 1(b) and the right-hand term in Eq. (4)]. Using Eq. (4), we estimated the frequency resolution with a pump beam having a center wavelength of 1024 nm, a bandwidth of 6.1 nm, and an interaction length of 480 µm, corresponding to the THz beam’s full width at half maximum (FWHM). The frequency and spatial intensity distributions of the pump beam were assumed to follow a Gaussian profile. With our experimental conditions, the overall frequency resolution achievable through wavenumber measurement is approximately 40 GHz with a single laser pulse of 280 fs in pulse duration, as indicated by the green line in Fig. 1(c).

Finally, the use of an intense THz source [31] to study the detection of broadband parametric up-conversion creates a unique situation where the Stokes signal generated is intense enough to be visualized on a commercial NIR viewing card. This is demonstrated in Fig. 1(d), where the separation between the pumping beam on the left (?pump) and the Stokes signal on the right (?Stokes) can be observed. In Supplement 1, we present the complete Visualization 1 of this Stokes signal capture while varying the delay between the pump beam and the THz-wave.

3. RESULTS

A. Non-collinear Up-Conversion Detection

Figure 2 shows the intensity of the Stokes beam under different conditions, with the delay line positioned at the peak position of the THz-wave. When the THz pulse is blocked, only the scattered components of the pump beam appear on the left-hand side of the near-infrared camera, as shown in Fig. 2(a). In contrast, when a single THz pulse overlaps a pump pulse during detection, the Stokes beam with angular dispersion corresponding to the different THz frequencies is captured by the camera with an integration time of 100 µs, i.e., for a real single-shot measurement, as shown in Fig. 2(b). Note that the signal intensity is so high that an OD 2 neutral density filter is required to avoid saturation, underlining the high sensitivity and merit of this detection scheme. Regarding the acquisition speed capability in the current demonstration, the camera operates in reduced mode in the vertical axis, with a resolution of 128 pixels in height and 2160 pixels in width. This mode enables acquisition at a repetition rate of kHz. Ideally, to improve detection of each 25 kHz laser shot from our laser, the use of a line-scan camera should be considered. These sensors can achieve measurement rates of the order of hundreds of kHz, or even MHz, while maintaining a very high digital dynamic range. For example, the Teledyne DALSA 16 k TDI line-scan camera offers such capabilities.

Fig. 2. Up-converted signal. Image of the Stokes beam: (a) without THz-waves, (b) with THz-waves, and (c)–(e) with THz-waves passing through a bandpass filter.

Fig. 3. Time-frequency mapping. (a) Temporal waveforms of THz-waves at each camera pixel (b) Temporal waveforms of THz-waves at camera pixels 45, 196, and 464. (c) Fourier transforms of the THz temporal waveforms at each camera pixel. (d) Fourier transforms of the THz temporal waveforms at camera pixels 45, 196, and 464.

Figures 2(c)–2(e) show the effect of inserting a THz bandpass (BP) filter into the beam. By observing the Stokes beam intensity with a camera, we can acquire information about the THz-waves, including its wavenumber (which is equivalent to frequency) based on the generation angle. However, even though we know the exact bandpass of these THz filters, we need to accurately calibrate the camera pixels to the expected THz frequency in order to obtain absolute frequency measurements, as detailed below.

To calibrate the camera pixels with THz frequencies, the pump beam was scanned with time delays, condition illustrated in Fig. 2(b). This process is similar to time-domain spectroscopy for each camera pixel, but corresponding to individual wavenumbers. During this process, the camera outputs were vertically integrated to convert the Stokes beam intensity into a one-dimensional format. Then, as shown in Fig. 3(a), the data for each delay time were displayed vertically, allowing the temporal waveforms for each pixel to be visualized. It is important to note that the temporal waveform obtained in Fig. 3(a) does not represent a THz pulse in space, but a time convolved in an 8.0 ps time window. The reason why the DC component increases with the high-frequency component is that the parametric gain of LiNbO3 is higher at higher frequencies [32].

To evaluate the frequency resolution, the sweep step of the delay stage was set to a value lower than the frequency resolution estimated from Eq. (4) (step: 66.7 fs, sampling points: 1024). Figure 3(b) shows the Stokes beam corresponding to low-frequency THz-waves at the bottom and high-frequency THz-waves at the top. Specifically, Fig. 3(b) shows temporal waveforms at pixels 45, 196, and 464, corresponding to THz frequencies of 250 GHz, 475 GHz, and 900 GHz, respectively. These waveforms indicate modulations of Stokes beam intensity at different frequencies across the camera pixels.

The results of the Fourier transform of the temporal waveforms obtained for each pixel are shown in Fig. 3(c). In this figure, the vertical axis represents the THz-wave frequency information for each camera pixel, while the horizontal axis shows the corresponding frequency obtained from the FFT operation. The diagonal yellow line stretching from the bottom-left to top-right indicates the information of the Stokes beam. For example, the temporal waveforms at camera pixels 45, 196, and 464, shown in Fig. 3(b), correspond to the positions marked by the dotted lines in Figs. 3(a) and 3(c), with their respective spectral profiles displayed in Fig. 3(d). From Fig. 3(d), a frequency resolution of approximately 120 GHz was obtained within the 200–900 GHz range. This resolution is over 16 times higher compared to the 1.6 THz frequency resolution achievable with direct observation using a single 280 fs pump beam pulse. Furthermore, when detection was performed with a single pump pulse, the system dynamic range was equivalent to that of normal EO sampling (see Supplement 1). Note that EOS detection using a regenerative laser to acquire a THz trace with a frequency resolution of around 10 GHz takes several minutes.

B. Single-Shot Spectroscopic Detection

Finally, one-shot spectroscopic detection was conducted using broadband THz parametric detection. With the delay stage fixed at the peak position of the THz pulse, bandpass filters transmitting 480 GHz (BP-F1), 660 GHz (BP-F2), and 800 GHz (BP-F3) were inserted at the focal point of the THz-wave after the off-axis elliptical mirror (OAEM), as illustrated in Fig. 5(a) in Section 5. The appearance of the Stokes beam with BP filters is demonstrated in Figs. 2(c)–2(e). Compared to the reference shown in Fig. 2(b), the intensity of the Stokes beam is attenuated overall in each figure, except for light in the specific range of the filter passband. These results suggest that the THz absorption spectrum can be obtained solely from the positional data of the camera, without the need for special data processing such as FFT.

Fig. 4. Single-shot spectrometer. (a) Spectroscopic results from broadband THz parametric detection using three types of bandpass filters. (b) Spectroscopic results from electro-optic sampling (EOS).

Fig. 5. Experimental setup comprises (a) an intense THz source coupled with broadband THz parametric detection, (b) measurement of the THz-wave beam diameter at the detector position, and (c) measurement of the pump beam diameter passing through the detector.

To confirm this, we used the relation between the camera pixels and THz frequencies obtained in Fig. 3(c). Using this calibration, a THz spectrograph with the previously defined spectral resolution is retrieved for each image captured by the camera. The Stokes intensity profiles for the reference and the three THz filters, each obtained with and without the THz pump in two laser pulses, are shown in Fig. 4(a). It should be noted that the spectrum of the three bandpass THz filters obtained using broadband, single-shot THz parametric detection [see Fig. 4(a)] closely matches with the ones measured by conventional electro-optic (EO) sampling [see Fig. 4(b)]. It is important to note that the spectrum presented in Fig. 4(b) required more than 10 million laser pulses, whereas the result in Fig. 4(a) was obtained with a single unmodified optical pulse of 280 fs duration. This contrasts also with other single-pulse approaches that rely on techniques such as chirping the probe beam spectrum [14] or creating a tilted pulse front [33].

It should be also mentioned that the difference observed between the reference signals (without sample) comes from the different crystals used for the measurements. EO sampling data were obtained using a thin ?-cut LN crystal [31], while the spectrum from parametric detection was acquired using a bulk LN crystal. When a bulk LN crystal is used for electro-optical detection, the detectable bandwidth is limited to 1.5 THz due to high absorption and poor coherence length [34]. Still, the improvement in sensitivity at high frequencies, up to 2 THz, also underlines the merits of parametric amplification.

4. DISCUSSION

One-shot spectroscopic measurements were conducted using broadband THz parametric detection. Indeed, more than 10 years ago, Kawada et al. demonstrated single-shot EO detection in a non-collinear geometry [33]. Unfortunately, this type of EO detection requires a cross-polarization analyzer to operate close to the optical zero polarization point [33]. This requirement distorts the linearity of the EO’s response to the electric field and can make this method less attractive, if not impractical. In our work, the direct observation of frequency spectra using up-conversion detection in a non-collinear geometry with signal amplification by optical parametric amplification is, to our knowledge, the first demonstration. This feature is ideally suited for THz systems operating with low-repetition lasers, as is often the case with intense THz sources [35] or in other applications where a lock-in amplifier cannot be used. As mentioned earlier, a fast line camera could also enable real-time THz spectroscopy for high-repetition-rate lasers exceeding 100 kHz on a shot-by-shot basis, which would be essential for studying dynamic samples such as biological targets.

We must emphasize that our method is different from parametric detection demonstrations carried out in the past, as the pump beam and the THz-wave are broadband in relation to each other. For example, in collinear broadband THz-wave detection in the frequency domain, the pump beam must be monochromatic [17], which implies a short temporal overlap between the quasi-monochromatic pump beam and the broadband THz-waves. Under such conditions, noise increases due to spontaneous parametric downward conversion, as reported by Tobias Pfeiffer [36]. Therefore, non-collinear broadband THz parametric detection is expected to be less noisy and more suitable for sensitive detection, such as photon counting of broadband THz pulses.

In our scheme, frequency resolution can be further improved by increasing the nonlinear interaction length, simply by increasing the diameters of the THz-wave beams. Interactions between the pump beam and THz-waves take place inside a lithium niobate crystal, which has a high refractive index (around 5). For example, by setting the diameter of the pump beam and that of the THz-wave beam at 4.5 mm, we can create an interaction window of 75 ps. This extended interaction window improves frequency resolution and enables the exploration of pulse delays and single-shot detection of multiple pulses, as indicated in recent research [12].

In conclusion, this study proposes a new method for single-shot spectroscopy of broadband THz-waves. This breakthrough opens up new possibilities for broadband THz-wave detection, offering a technique that simultaneously achieves high-frequency resolution, real-time capability, and the ability to include several successive parametric amplification stages [18]. The potential applications of this method are numerous, particularly in the fields of security, materials science, and communications technology, promising significant advances in all these areas.

5. MATERIALS AND METHODS

A. Experimental Setup

Figure 5(a) illustrates the experimental setup for broadband THz parametric detection. In our experiment, we used an amplified Ytterbium (Yb) solid-state laser (Pharos model: PH1-10W from Light Conversion) that produces optical pulses at a wavelength of 1.024 µm with a pulse duration of 280 fs. The maximum energy per pulse is 400 µJ with a repetition rate of 25 kHz at an average power of 10 W. A beam splitter (BS 7T93R in the figure) with a ratio of 93% for THz-wave generation and 7% for parametric detection was used for this experiment. The THz source produces THz pulses with a 400 kV/cm electric field at focus with an average power of 75 mW and a broadband characteristic up to 3 THz, as previously reported [31]. Figure 5(b) shows an image of the THz spot size at focus with a FWHM characteristic of 480 µm in beam diameter. This beam irradiates a MgO:LiNbO3 crystal (dimensions: 62.5mm×4mm×5mm) for detection, where the fs pump beam [with a FWHM characteristic of 1.6 mm in diameter, as shown in Fig. 5(c)] is incident under non-collinear phase matching conditions, converting the THz-waves into a Stokes beam (near-infrared) via THz parametric detection, subsequently observed with a camera (Excelitas Technologies’ PCO.Edge 5.5 scientific CMOS camera). Given the finite beam diameter of the Stokes beam, direct observation of the generation angle with a standard camera is challenging. Therefore, a short pass edge filter (BSP01-532R-25) from Semrock company has been inserted in the way to suppress the generation of the second harmonic of the pump created in the crystal and a convex lens with a focal length of 300 mm is positioned between the MgO:LiNbO3 and the camera [37] to focus the signal, facilitating spatial separation by generation angle.

B. Interference Effect

The absolute assessment of the generation angles is complex in THz parametric detection due to the indistinct generation position of a Stokes beam within the MgO:LiNbO3. To overcome this, a delay stage is introduced to the detection pump beam, allowing the generation angles (camera pixels) and THz frequencies to be calibrated using time-domain spectroscopy. With a 1.5 ps time window in time-domain spectroscopy, only THz waveforms below 335 GHz can be captured according to Nyquist’s theorem. Calibrating frequencies above 335 GHz remains challenging using only the dynamics of broadband THz parametric detection described so far. Consequently, we will discuss phase measurement of THz-waves using the interference effects produced in broadband THz parametric detection.

A Stokes beam is generated through the interaction between the pump beam and THz-waves, with its phase dependent on the phase of the THz-wave electric field. The generated Stokes beam interferes with the corresponding wavelength component of the pump beam, resulting in a composite wave represented by

(5)?=?1??(∅1(?)+?Stokes?)+?2??(∅2(?)+?Stokes?),

(6)?=?12+?22+2?1?2cos?(∅1−∅2).

Here, since ∅2(?) is constant, modulating the phase ∅1(?) of the pump beam through the delay stage reflects the phase of the THz-wave in the modulation of the Stokes beam. Although methods for obtaining the phase of THz-waves using interference in THz parametric detection have been reported [38], this is the first method to acquire the phase using interference between a pump beam and Stokes beam. Thus, time-domain spectroscopy in broadband THz parametric detection enables phase measurement through optical interference without the need for a specialized optical system.

C. Sample Properties

In recent years, THz filters have become easier to manufacture [39]. The THz bandpass filters used in this work are based on polarization-insensitive frequency-selective surfaces (FSS). The substrate is made of stainless steel with a thickness of 100 µm, and a matrix of periodic holes was etched by laser cutting using the LPKF ProtoLaser U3 system. The microscope image of the BP-F1 filter at 480 GHz is shown in Supplement 1 together with the filter characteristics.

Funding

Japan Society for the Promotion of Science (24K00943); Japan Science and Technology Agency (JPMJAP2340, JPMJFR212J); Canada Research Chairs (CRC-2019-127); Natural Sciences and Engineering Research Council of Canada (ALLRP 590951-23, 2023-03322, RGPIN-2023-03322).

Acknowledgment

The authors acknowledge Redwan Ahmad for lending us the THz filter samples. The authors also acknowledge useful discussions with professors Kodo Kawase and Chiko Otani. S. M. acknowledges assistance from the JSPS; K. M. acknowledges assistance from JST ASPIRE Program, JST FOREST Program, and JSPS KAKENHI. F. B. gratefully acknowledges financial support from NSERC, the CRC tier2 on Spatiotemporal encryption of THz light, and the Alliance International.