1. Introduction

The development of intense terahertz (THz) pulse sources has advanced in recent years [1]. Optical rectification with tilted-pulse-front pumping [2] has enabled the development of table-top sources of single-cycle, phase-stable THz pulses with field amplitude > 1 MV/cm [3]. These developments have stimulated studies on coherent control over matter and light in the THz frequency range [4]; for example, there have been studies on Rydberg states in silicon [5], spin waves [6], molecular orientation and alignment [7], the soft mode of lattice vibrations in SrTiO₃ [8], and two-quantum rotational coherences [9]. However, very few studies have been conducted on coherent phenomena under narrowband (quasi-monochromatic) THz fields [10–12], because the only available sources for intense, tunable, narrowband THz pulses are free-electron lasers, which require large facilities and have poor temporal resolution owing to large timing jitter. Therefore, it is desirable to develop table-top sources of intense, tunable, narrowband, chirp-free THz pulses.

The chirped laser pulse beating method [13, 14] has been widely used for generating tunable, narrowband THz pulses. Unlike the more sophisticated method of generating THz pulses with synthesized waveforms using liquid crystal spatial light modulators [15, 16], the setup for the chirped pulse beating method is simple and cost-efficient. Recently, we formulated an expression for the direct relationship between the dispersion of the laser pulse stretcher and the chirp of the THz pulse [17]. In this formulation, the third-order spectral phase of the laser pulse stretcher produces the chirp of the THz pulse. Based on this theory, we have demonstrated the generation of negatively chirped THz pulses using the chirped laser pulse beating method with a grating-pair laser pulse stretcher [18]; however, there is a lower limit on the chirp of the THz pulse generated using this technique [19]. As such, it is impossible to generate a transform-limited, chirp-free THz pulse with this method.

For a low-power pump laser, a fiber-based pulse stretcher can be used to generate a chirp-free THz pulse [19]. However, this method is not applicable to high-power pump lasers, because of unwanted nonlinear effects within the fiber.

In the study presented here, we show that it is possible to cancel the third-order spectral phase of the laser pulse stretcher by combining two grating pairs. First, we show that positively chirped THz pulses can be generated using a grating pair with internal lenses. We then combine this with a grating pair that does not contain internal lenses. Since these two grating pairs have opposite signs for their spectral phases and different ratios for their second- to third-order spectral phases, the total third-order spectral phase can be canceled out with leaving the non-zero total second order spectral phase, thereby enabling the generation of chirp-free THz pulses. The applicability of this method is not limited to the generation of weak THz pulses with a low-power pump laser and photoconductive antennas as demonstrated in this study, but extends to various schemes of intense THz pulse generation: e.g., the scheme with an amplified Ti:sapphire laser and optical rectification with tilted-pulse-front pumping. Therefore, this method will enable us to investigate the coherent transient dynamics of interaction between matter and intense narrow-band THz pulses.

2. Theory

We first refer to the relationship between the spectral phase of the pulse stretcher and the chirp of THz pulses, which was investigated in our previous study [17]. Figure 1(a) shows the schematic of the setup for generating tunable narrowband THz pulses with the chirped pulse beating method. In this study, we assume that the input laser pulse has a Gaussian shape with an electric field given by

 

Fig. 1 (a) Schematic of the setup for generating tunable narrowband THz pulses with the chirped pulse beating method. (b) Double-pass grating pair without internal lenses. (c) Single pass grating pair with internal lenses; the schematic is drawn so that the effective perpendicular separation l < 0. The inset shows the case for l > 0. In (b) and (c), the red and blue arrows schematically show the traces of the lower and higher frequency components of the input laser pulse, respectively.

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Next we consider the spectral phase of the grating pairs. Figure 1(b) shows the schematic of a grating pair without internal lenses in a double-pass configuration, the quadratic and cubic spectral phases of which are given by [20, 21]

Figure 1(c) shows the schematic of a grating pair with an internal lens in a single-pass configuration, the quadratic and cubic spectral phases of which are given by [22]

3. Experiment

Based on the above theory, we perform an experiment to generate chirp-free THz pulses with a pulse stretcher that consists of two kinds of grating pairs. The experimental setup is the same as that used in our previous work [17], except for the modifications in the pulse stretcher. Specifically, we use a pulsed fiber laser with a Gaussian intensity spectrum with a center wavelength λ₀ = 805 nm, a FWHM of 9.2 nm, a repetition rate of 50 MHz, and a average power of 10 mW. We measure the pulse duration with an interferometric autocorrelation technique based on two-photon photoemission in a photomultiplier tube [23] and obtain the e⁻¹ half-width σ = 110 fs. Dipole-type photoconductive antennas on low temperature-grown GaAs substrates are used as the THz emitter and the detector in a standard THz time-domain spectroscopy setup. A probe pulse is focused on the detector antenna to excite photocarriers in the substrate. A current proportional to the instantaneous THz electric field is thus generated, and electrically detected. By scanning the delay between the THz pulse and the probe pulse, the temporal profile of the THz electric field is measured.

We first confirm that the sign of the chirp of the THz pulses can be controlled by varying the value of l using a grating pair with an internal lens as the pulse stretcher. The pulse stretcher has a fixed grating constant $d_{\text{L}}^{-1}=1200$ lines/mm and a fixed incident angle γ_L = 3°, and six perpendicular separations l = 4.1, 2.4, −0.8, −1.6, −2.4, and −3.3 cm. Figure 2(a) represents the measured relationship between the perpendicular separation l and the quadratic spectral phase shift ${\mathrm{\Phi }}_{\text{L}}^{(2)}$. The obtained results correspond well to the theoretical values. Figures 2(b) and (c) represent the time traces of the generated THz field for the l = −2.4 cm and 2.4 cm cases, respectively. Figures 2(d) and (e) represent their short-time Fourier transforms, respectively. The black dashed lines show the theoretical values of the instantaneous frequency, ω_T − 3Φ⁽³⁾ω_Tt/(2Φ⁽²⁾²). It can be clearly seen that positively chirped THz pulses are generated for l < 0.

 

Fig. 2 (a) The experimental dependence of the value of ${\mathrm{\Phi }}_{\text{L}}^{(2)}$ on the perpendicular separation l. The black solid line represents the theoretical value of ${\mathrm{\Phi }}_{\text{L}}^{(2)}$ as a function of l, and the blue dots show the experimental results. (b), (d) THz field waveform and (c), (e) Short-time Fourier transform, for the cases of the effective perpendicular separations l = −2.4 and 2.4 cm, respectively, with a center frequency of 0.5 THz. The black dashed lines show the theoretical values.

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Next we combine the grating pair with an internal lens in series with another grating pair without internal lenses. For the grating pairs we use the fixed parameter values l = −1.6 cm, γ_L = 3°, d_L = 1200 lines/mm, γ_NL = 0°, and d_NL = 830 lines/mm, and four values of b = 20, 35, 45, and 53 cm. Under our experimental condition, the diffraction efficiencies of the gratings are between 75% and 80%. A total of six diffractions from the gratings, combined with the losses in the Michelson interferometer, results in an overall efficiency of 10%. The pulse energy of the pump laser pulse applied to the emitter antenna is 12 pJ.

Since the values of the frequency sweep rates of the THz pulses are too small to be estimated with the short-time Fourier transforms, we evaluate the chirp of the THz pulses by measuring the dependence of the linewidth on the center frequency. Figure 3 shows the measured linewidths Δ/(2π) as functions of the center frequencies f_T = ω_T/(2π). Note that we are not able to obtain the linewidths at some of the center frequencies due to the distortion of the THz power spectrum caused by water vapor absorption [24]. The solid lines show the theoretical curves of Eq. (4). Note that there are no fitting parameters, and all experimental measurements agree well with these theoretical curves. It can be seen that the linewidths have no observable dependence on the center frequency for the cases of b = 35 and 45 cm, and therefore chirp-free THz pulse are generated, in contrast to the cases of b = 20 and 53 cm. Note that the perfect elimination of ${\mathrm{\Phi }}_{\text{total}}^{(3)}$ theoretically occurs for b = 36.6 cm.

 

Fig. 3 The dependence of the linewidth Δ/(2π) on the center frequency f_T = ω_T/(2π). Solid lines represent the theoretical value, and the dots show the experimental results. The black, blue, red, and green dots show the results for a perpendicular separation b = 20, 35, 45, and 53 cm, respectively.

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4. Conclusion

In conclusion, we have experimentally demonstrated the chirp-free THz pulse generation using a pulse stretcher comprised of two grating pairs that cancel out the third-order spectral phase when used in tandem. This method is also applicable to various schemes of THz pulse generation. Specifically, it is straightforward to apply this method to the scheme of intense THz pulse generation using the optical rectification of amplified Ti:sapphire laser pulses in LiNbO₃ [3], thereby enabling us to investigate the coherent transient dynamics of the interaction between matter and intense narrowband THz pulses.