1. Introduction

Terahertz (THz) waves present attractive applications, such as sensing, biomedical analysis, and illicit drugs inspection [1–3], while the development of quasi-monochromatic, intense and tunable coherent THz sources is beneficial to promote their precision, sensitivity and flexibility [4]. Hence, efficient narrowband, high amplitude and compact THz sources are extremely urgent. In the last decades, numerous different schemes have been proposed [5–14], and it is obvious that an unprecedented advancement has occurred in this field. For example, Kung-Hsuan Lin and associates employed the unique dispersion properties of THz waveguide modes in LN crystal and then generated intense, frequency-tunable multiple-cycle THz waves by tilting optical pulse fronts [5]. A train of optical laser pulses produced by phase masks and interferometers effectively create narrowband, tunable sources by adjusting the number and timing [6,7]. Metamaterial structures also make it possible to realize narrowband thermal THz emitters [8]. THz quantum cascade lasers (QCL) [9] are compact, while photoconductive switches (PCSS) can be efficient [10] for the generation of narrowband THz waves. In the recent years, considerable progress has also occurred in generating narrowband THz pulses with high frequencies and low frequencies by using organic crystals [11,12]. It is especially worth to note that employing specially designed periodically poled lithium niobate (PPLN) waveguides generates THz waves with linewidth of 20 GHz [13], and the conversion efficiency of optical to THz is expected up to 10% [14]. However, there still have several drawbacks among existing methods. For instance, the relatively large area of the pump for tilting optical pulse fronts depresses the THz pulse intensity density; limited energy efficiency, stability and complexity are negative for a train of optical laser pulses. Metamaterial structures can also realize narrowband THz emitters but suffer from the complex experimental setup. Specially designed PPLN waveguides and QCL are required at very low temperatures, while PCSS is relatively challenging to scale to high pulses energies. So far, at room temperature, the minimum linewidth of terahertz waves is 50 GHz, while narrower THz radiation is desirable in the basic sciences and applications, such as high-precision molecular spectroscopy, finger-printing gas molecules, coherent control of vibrational modes [15], drivers of compact particle acceleration [16], and coherent X-ray generation [17].

In this letter, we propose a new configuration for efficient generation and frequency modulation of quasi-monochromatic THz waves using LE setup in a subwavelength LN slab waveguide. On the basis of phase matching between the pump laser and the THz waves, the field enhancement of narrowband THz waves, with frequency linewidth narrower than 38 GHz, is realized at a characteristic frequency. By changing the dispersion of THz waves, which has been realized by adjusting the thickness of the host LN slab, the central frequency and bandwidth are modulated within certain range. To achieve narrower linewidth and dynamical modulation of the central frequency, we combine the individual LN slab with double gold mirrors on both sides symmetrically to form an Au-Air-LN-Air-Au multilayer structure. As a result of the dramatic change of the dispersion in the multilayer LE structure, THz waves with linewidth narrower than 10 GHz are obtained. Additionally, the frequency and linewidth of quasi-monochromatic THz waves can be tuned dynamically by varying the distance between LN slab and Au mirrors flexibly. This work should serve as a reference for generating and amplifying tunable narrowband THz sources.

2. LE Model design

In the general experimental setup, the THz waves generated in the ferroelectric crystal do not propagate collinearly with the pump femtosecond laser beam, because of the large difference of refractive index between the pump laser and THz waves. Instead the THz waves propagate primarily in the lateral direction, shown as a Cherenkov radiation pattern. Furthermore, in the subwavelength waveguide, the THz waves propagate perpendicular to the optical pump beam [18], due to the strong confinement effect of the waveguide. What’s more, taking advantage of the intense dispersion of subwavelength waveguide, it is possible to fulfill phase matching between THz wave and femtosecond laser by adopting lateral excitation model.

As schematically illustrated in Fig. 1(a), pump beam is line-focused on the lateral surface of LN, and under the interaction between femtosecond laser pulses and LN, THz radiation is excited via impulsive stimulated Raman scattering [19, 20]. Both the polarizations of the pump pulses (red) and the generated THz waves (blue) are along the crystal optical axis (z axis of coordinate system). The size of LN used in the model is 1cm × d × 1.1cm (y-cut) and d is several tens of microns. In this structure, the dispersion of femtosecond laser pulse is tolerable because the sample thickness is far outweigh than the wavelength of pump, then the velocity is considered to be 1.36 × 10⁸ m/s in LN waveguide, while the effective refractive index of THz waves is decreased rapidly in subwavelength waveguide compared to bulk LN [21]. When velocity matching is fulfilled between femtosecond laser pulses and certain frequency THz waves, they will propagate synchronously, which make it possible to realize the quasi-monochromatic coherent THz waves in this LE configuration.

 

Fig. 1 (a) Schematic illustration of the LE model. The pump is line-focused by a cylindrical lens onto the lateral surface of LN waveguide, which is propagating along x direction, parallel to its surface, polarized along optical axis (z axis). The THz wave is also polarized along z, propagating with pump collinearly. (b) The corresponding simulation structure. d is the sample thickness, and dipole source excited by femtosecond laser pulse to produce THz wave.

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Along the z axis, the distribution of pump energy has little effect on frequency selection, so we choose x-y plane to implement 2-D simulations employing a commercial finite difference time domain (FDTD) software (Lumerical Solutions), and the corresponding model is shown in Fig. 1(b). Along the propagation direction, the number of dipole sources is 1000 with a step of 10 μm, in which the dipole with linewidth of 370 GHz denotes phonon polariton excited by femtosecond laser pulse to produce THz wave [22], and the time delay of adjacent dipole is 73.3 fs given by above velocity of femtosecond laser pulse. In addition, the theoretical dispersion and absorption also have been taken into the material setting, thus it is reasonable to propose this model, which is pretty close to the physical structure and experimental setup.

3. Result and discussion

For a LN waveguide with thickness of 30 μm, it has been obtained how electric field evolution is related to a temporal and spatial context, just as shown in Fig. 2(a). An interesting phenomenon is that the oscillations of THz waves increase from single-cycle to multi-cycle as THz waves propagate forward. And little reflection can also be observed at the interface between LN and air. For a more intuitive understanding of THz propagation characteristics, such as bandwidth and amplitude, numerical simulation in time and frequency domain have been done for different positions along x coordinate, as shown in Figs. 2(b) and 2(c). As the propagation distance increases, the oscillation time of THz waves becomes longer and the corresponding spectral widths become narrower. At the same time, the field of THz waves, with a certain frequency, gains enhancement continuously. The efficient generation of narrowband THz waves is attributed to phase matching, which can be characterized by dispersion curves, as shown in Fig. 2(d). The intersection point of cyan and green curves is the phase matching point of pump and THz waves. In this structure, the refractive index of femtosecond laser pulse is nearly a constant, so the dispersion of dipole sources (DS) acts as a straight line approximately (cyan curve). But the effective refractive index of THz waves decreases in subwavelength waveguide compared with bulk LN, and corresponding dispersion can be calculated by planar waveguide theory (green curves) [21,23,24]. The white and red curves indicate the dispersion relationship of THz waves in air and bulk LN respectively. It can be clearly seen that the brightest intersection is precisely the velocity matching point of femtosecond laser pulse and THz waves, and the frequency-selective point of numerical simulation is a high coincidence with theoretical derivation. Thus, based on this mechanism of long-distance phase matching, only specific frequency of THz waves synchronously propagate with pump, which make broadband THz waves that radiated by dipole sources, selected to a narrower one and gained higher electric field.

 

Fig. 2 (a) E-field evolution of THz waves as a function of space and time. (b) Normalized field distribution in time domain along x coordinate. (c) Corresponding Fourier spectra of (b). (d) The numerical simulated dispersion curves for an individual waveguide, with thickness of 30 μm. The colorbar gives the spectral intensity. Overlaid on the simulated data are the theoretical dispersion curves for THz waves in air (white line), bulk LN (red line), subwavelength waveguide (green line), and dipole sources (DS) excited by pump (cyan line). η₀ is the angle of dispersion curves between femtosecond laser pulses and THz waves.

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Since maximum efficiency frequency-selective excitation has been achieved on the basis of long-distance phase matching, accordingly, the center frequency and linewidth can be modulated efficiently by controlling the dispersion of THz waves. It can be operated by changing the thickness of subwavelength waveguide. Figure 3(a) shows the frequency-dependent field profiles of THz waves with different thickness range from 10 μm to 100 μm with a step of 10 μm. The central frequency can be tuned from 0.09 to 0.82 THz, and bandwidth is decreased from 36.8 to 32.3 GHz, as Fig. 3(b) shows. Obviously, the center frequency and bandwidth have very big concern with dispersion of THz waves, which can be adjusted by changing the waveguide thickness. Figure 3(c) gives the theoretical dispersion curves of LN with thickness of 30 μm, 50 μm, 70 μm, and 90 μm, respectively. The center frequency is gradually reduced as the thickness increases, while the bigger angle η₀, the angle of dispersion curves between femtosecond laser pulses and THz waves, leads to the linewidth of THz waves narrower than 38 GHz. From another perspective, so narrow linewidth is related closely to the coherence length, which can be shown as:

 

Fig. 3 (a) The frequency domain spectra of different thickness samples at the positon x = 7 mm. (b) The thickness-dependent variation of frequency and bandwidth. (c) The theoretical dispersion curves of LN waveguide with the thickness of 30 μm,50 μm,70 μm,and 90 μm, respectively. (d) The calculation of the effective THz refractive index for different thickness from 10 μm to 90 μm, step for 20 μm, and the black line represents the group index of femtosecond laser pulses.

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4. Optimization and analysis of the model

For individual LE model, quasi-monochromatic THz waves can be generated efficiently when phase matching is fulfilled. And the frequency of narrowband THz waves can be broadly tuned through adjusting the thickness of host LN slab. Nevertheless, for many applications, such as sensing, biomedical analysis, and illicit drugs inspection, it is expected to generate a narrower one which can be modulated dynamically. Consequently several efforts have been made as following.

As Fig. 4(a) shows, two gold mirrors are put symmetrically along y axis of the sample, which forms an Au-Air-LN-Air-Au multilayer LE structure. The air gap between Au and LN is denoted by h, which can be adjusted from 0 μm to 100 μm. In this configuration, we use the same excitation way to line focus the pump onto the lateral surface of LN waveguide as before, which subsequently results in guide mode of THz waves propagating in the waveguide. Because of the change of boundary condition, the stricter phase matching makes the bandwidth narrower compared with individual waveguide. In addition, by adjusting the gap between Au mirrors and LN waveguide symmetrically, the frequency of THz waves can be tuned dynamically and easily.

 

Fig. 4 (a) The multilayer LE configuration of Au-Air-LN-Air-Au for narrow bandwidth. (b) The frequency domain THz spectra for different air gap. (c) The selected frequency and bandwidth vary with the gap size. (d) The dispersion curves for multilayer waveguide with the gap of 30 μm. The solid lines are the same as Fig. 2(d) explains. η₀ and η₁ are the angle of dispersion curves between DS and THz waves, corresponding to individual waveguide, multilayer waveguide with the gap of 30 μm, respectively.

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For the LN with thickness of 30 μm, the frequency domain profiles for different air gap at position of x = 7 mm are shown in Fig. 4(b), in which h = 0 μm and no gold denote two limiting cases: (1) the gold mirrors close to LN infinitely; (2) individual LN slab waveguide. In the optimized model, adjusting the gap between Au and LN from 0 to 100 μm, the frequency and linewidth can be modulated efficiently by the changes of mode dispersion, which can be seen clearly in Fig. 4(c). As the gap increases, the localized energy in LN waveguide is weakened, and the lower oscillation of THz waves makes the selected frequency decreased from 0.9 to 0.27 THz. Simultaneously, the linewidth is tuned narrower than 10 GHz when gap is smaller than 30 μm, which hasn’t been reported up to now. The realization of THz waves with such narrow linewidth is caused by the dramatic variation of mode dispersion after adding double gold mirrors, as shown in Fig. 4(d). In the optimized structure of h = 30 μm, not only the dispersion curve of THz waves is thinner than that of individual waveguide, but the angle of the dispersion curves between DS and THz waves η₁ is more than η₀. Thus, the overlapping area will be narrower than before, which makes quasi-monochromatic THz waves generated. Adjusting the gap between the gold mirrors and LN slab waveguide is very efficient for dispersion regulation, so that the quasi-monochromatic THz waves can be modulated dynamically in this optimized structure. Thus, in multilayer LE structure, it has been demonstrated that the frequency with narrower linewidth of THz waves can be tuned dynamically by varying the gap between LN slab and Au mirrors flexibly, which is beneficial to the development of high-resolution and real-time measurement.

5. Conclusion

In this work, a kind of LE configuration has been proposed for quasi-monochromatic terahertz generation via impulsive stimulated Raman scattering in a LN subwavelength slab waveguide. Using finite different time domain method and planar waveguide theory, the spectrum information of THz waves, such as central frequency and linewidth, is analyzed. Simultaneously, the frequency and linewidth modulations in different structure are thoroughly discussed. We have demonstrated that the electric field enhancement of narrowband THz waves can be realized when long-distance phase matching is fulfilled between femtosecond laser pulse and THz waves. For individual waveguide with thickness of 30 μm, the frequency linewidth of THz sources is as narrow as 36 GHz, which gains enhancement continuously because of enough constructive interference. With reducing thickness of LN sample, the changes of mode dispersion make frequency and linewidth modulated effectively. Nonetheless, the linewidth is still not narrow enough and in time frequency modulation is difficult. Thus, we design an Au-Air-LN-Air-Au multilayer LE set up, in which boundary condition has changed the mode dispersion of THz waves dramatically, so that the frequency linewidth narrower than 10 GHz is realized. Additionally, the central frequency of quasi-monochromatic THz waves can also be tuned dynamically by varying the distance between LN slab and Au mirrors flexibly, which can be manipulated and implemented in actual experiment. In the view of this work, the optimized LE structure will be great significance for the development of narrowband THz sources.