1. Introduction

Over the past decades, devices based on optical microcavities have already been indispensable for a wide range of applications and studies including cavity quantum electrodynamics [1,2], cavity optomechanics [3], photonics [4], biosensing [5] and nonlinear optics [6]. In these applications, typical transmission spectra and cavity ring-down measurement [7,8] are deployed in determining the Q-factor and the mode lifetime. Despite their high sensitivity, these techniques cannot provide spatiotemporal information when the electromagnetic field interacts with the cavity, therefore give no intuitive picture of the time-dependent process. In dealing with the evolution of wave in the cavity, real-space and real-time coherent imaging is preferred, which has rarely been investigated to date.

To fill this gap, we implemented a Fabry-Perot resonator structure for Terahertz (THz) wave and realized the coherent imaging of the real-time process in the resonator. The THz spectral regime, ranging from 0.1 to 10 THz, is one of the least explored yet most promising spectral regions [9,10]. At present, the numerous applications of THz wave and its significance to fundamental science makes finding ways to generate, control and detect THz radiation one of the key areas of modern research. Among the diverse sectors of THz research, ferroelectric crystal platform [11,12] is especially attractive due to its excellent performance in integrated operation of THz pulses, which is highly desirable here.

In our experiment, we generated THz waves in a LiNbO₃ (LN) slab via impulsive stimulated Raman scattering (ISRS) using ultrashort optical pulses and then recorded the electric field of THz waves by pump-probe imaging method [13]. With THz generation, guiding, control and detection totally integrated on a single LN micro-chip, the platform works as a full-featured THz system which could be applied in both THz signal processing and spectroscopy. We note that THz band gap crystals has been investigated in time domain [14]. Although it’s for acoustic waves, it still sheds the light on this work. It’s a new direction to investigate microcavity from the perspective of space-time evolution, which helps to understand transient optical phenomena. What’s more, our time-resolved experiments may provide some new ideas to other types of physical cavity systems, such as quantum cavity electrodynamics and cavity optomechanics in future.

2. Structure design and fabrication

The Fabry-Perot resonator is schematically shown in Fig. 1(a). The structure is fabricated in a polished x-cut LN slab (11mm × 10mm × 0.05mm), with its fabrication procedure described below.

 

Fig. 1 (a) Experimental scheme: The Fabry-Perot resonator is composed of a LN cavity at center and two Bragg grating structures as its reflective mirrors. Femtosecond pump laser was line-focused to the center of the structure to generate THz wave in the crystal slab. Coordinate system and C-axis of the crystal are also indicated. (b) Optical microscope image of the resonator after fabrication process and HF treatment. The LN crystal is transparent under normal illumination by blue light. The outlines of the carved structures are clear. Note that each Bragg grating section has ten units, of which only five are captured in the image. Scale bar: 500 μm. (c) Schematic illustration of the experimental setup.

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Firstly, protective coatings (SiO₂) were deposited on both surfaces of the LN slab by magnetron sputtering method at room temperature. Then femtosecond laser machining was introduced to fabricate the structure. We used amplified Ti:sapphire laser pulses (800 nm central wavelength, 120 fs duration, 1 kHz repetition rate) with about 4 μJ energy, focused by a lens with focal length 60 mm to a spot size of about 15 μm. A three-dimensional translation stage that held the 50μm-thick LN slab was programmed to move laterally such that the outline of the slot was irradiated and drilled through the depth of the slab, which is facilitated by the long Rayleigh range of the beam. Repeated operations of this sort were used to carve a series of periodic structures. After the machining, the SiO₂ layers were removed by a HF solution to get rid of the LN debris ejected in the machining process. With the aid of these steps, the optical-quality surface is guaranteed.

After the fabrication procedure, a row of cuboids (0.1mm × 1mm × 0.05mm) were cut off from the LN slab to form a periodic array with period 0.2 mm along X-axis. Figure 1(b) shows the micrograph of the fabricated structure recorded by an optical microscope. The outlines of the carved structures are clear under the illumination by blue light. Note that the Fabry-Perot resonator is composed of two parts: (a) LN cavity located at the center and (b) two reflective “mirrors” consisting of alternating cuboids of materials with different dielectric constants (air and LN). Due to the large size of the cuboid along Y-axis, the structure can be regarded as a one-dimensional (1-D) photonic crystal (PC) slab. This type of PC can act as a mirror for light with a frequency within a specified range, namely a uniform Bragg grating [15]. Ten units were fabricated in each Bragg grating to achieve good response. Sandwiched between the two gratings, the LN cavity at the center acts as a defect which breaks the translational symmetry of a periodic system, and therefore localized modes are permitted. The cavity length is designed to be 0.3 mm, which is comparable to the wavelength of THz regime that we are interested in.

3. Experimental setup

Figure 1(c) shows a sketch of the pump-probe experimental setup. We used pump-probe technique to record the THz transient in the LN slab. Laser pulse from a Ti:sapphire regenerative amplifier (800 nm central wavelength, 120 fs duration, 1 kHz repetition rate) was divided into a pump beam (450 mW) and probe beam (50 mW). The pump pulse was directed to a mechanical time delay line and then focused to a “line” on the cavity region by a cylindrical lens whose focal length was 300 mm, subsequently generated THz waves by optical nonlinear effect in the LN slab.

In bulk LN, THz waves propagate in Cherenkov-like pattern since the phase velocities of the waves are far too slow to match optical group velocities [16]. In our experiment, however, the LN slab is thin enough to act as a subwavelength waveguide, and the generated THz wave travels and forms waveguide mode rapidly, which is propagating laterally and orthogonal to the direction of pump propagation [17]. An 800 nm-stop filter was exploited to block the transmitted pump beam immediately after the sample. In the probe branch, the laser was frequency-doubled to 400 nm via a BBO crystal, spatially filtered and expanded to illuminate the whole sample at normal incidence. As the THz wave travels in the slab, it changes the crystal's refractive index by electro-optic effect. As a result, the expanded probe beam develops a spatially dependent phase shift proportional to the refractive index change immediately after the sample. Because the collecting camera is sensitive only to intensity, it is essential to perform phase-to-amplitude conversion beforehand. We accomplish this by imaging the sample with phase contrast imaging technique. The imaging resolution as small as 3 μm is achieved, which is about one-several tenths of the THz wave wavelength in LN crystal, providing an extra fine spatial resolution throughout.

By changing the time delay between the arrival of the pump and probe pulses on the sample, a full spatiotemporal evolution of THz wave was extracted from the image sequence. (Visualization 1)

4. Results and discussion

4.1 Temporal evolution

Figure 2 shows some images from Visualization 1 captured by the imaging system at different time, which gives an intuitive demonstration of THz field evolution. The central sections enclosed by the dashed white boxes are enlarged for readability. In addition, line graphs indicating THz pulse electric field amplitude are plotted on each enlarged image. Figure 2(a) shows the moment when THz pulse is just launched (0.2 ps). It's clear that the position of line-focusing is located in the cavity, close to the center. After the THz pulse is generated in the slab, it develops into two wave packages with opposite propagation direction along X-axis, as is shown in Fig. 2(b). The left one is slightly brighter than the right one, which means that THz amplitudes of both sides are not equivalent. The difference of field strengths arises from the oblique incidence configuration of the pump laser on the sample (clear in Fig. 1(c)). In our setup, the incident pump laser was tilted by θ = 20° relative to the probe beam to avoid imaging disturbance of the 400-nm light generated by second harmonic generation (SHG) with the 800-nm pump focusing light on the LN slab. Figures 2(c)-2(e) describe the interaction process of two THz pulses travelling in opposite directions covering a time range of 1.6 ps. In Fig. 2(c), the two waves are reflected back by the Bragg grating structures and propagate towards each other. As the waves travel and overlap, they superpose to form a resultant wave and the phenomenon of constructive interference is observed. From Fig. 2(d), we see that the crests of the two pulses meet at the center of the cavity, making the resultant amplitude the sum of individual ones. Soon afterwards, the two waves separate from each other and regain their wave profiles, as shown in Fig. 2(e). The interference between waves that are reflected back and forth by the Bragg gratings leads to standing waves in the cavity. Figure 2(f) demonstrates the long-lifetime of the formed standing wave. We note that in the cavity region the confined light field is still clearly discernible at 35.8 ps, whereas in the unstructured region, the propagating waves are free to spread along X-axis and arrive at the edge of the Bragg gratings (x = ± 2mm) at the same time. With the spatiotemporal data collected by the time-resolved imaging system, the frequency information of the standing waves in the cavity would be straightforward to acquire and analyze.

 

Fig. 2 (Visualization 1) Images of the THz electric field distribution at different time delays. For readability, the central sections are enlarged and red graphs indicating THz pulse electric field amplitude are attached. Note that yellow arrows are plotted to denote the propagation directions of THz waves. (a) The THz pulse is just launched at the center of the cavity after the initial time point (0.2 ps). (b) The generated THz waves breaks into two branches and propagates laterally along X-axis at 1.4 ps. (c)-(e) The two THz waves in opposite directions interact with each other, revealing the process of interference in time domain. (f) Remaining waves confined in the cavity are still discernible at 35.8 ps. In vivid contrast, THz wave in unstructured LN slab has propagated far away.

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4.2 Frequency response

Since the pump beam is line-focused in Y-axis and the resonator is uniform along the same direction, it is reasonable to average the THz signal over Y direction at each X position. With the simplification, each 2D matrix containing XY coordinate information is compressed to a 1D data set relating to X-axis position only. Repeat the compression process at different time and we obtain a series of 1D images which record propagation profiles. All the 1D data sets are then joined together in time sequence, forming a full temporal and spatial map E(x, t) in an individual 2D graph, as shown in Fig. 3(a) where the horizontal axis indicates propagation distance and the vertical axis indicates time evolution. Based on the space-time plot, we also conducted the frequency analysis to build a comprehensive understanding of the mode behavior. Portion data of Fig. 3(b) in a time range from 40 ps to 90 ps is handled to get rid of the effect of the unstable process at the beginning. Figure 3(b) gives the FT result of such a portion along the t axis with a spectral resolution of 0.02 THz. Figures 3(a) and 3(b) reveal the nature of the resonant cavity in a complementary way. Figure 3(a) demonstrates real-space information that THz waves are confined in the resonator and reflect multiple times from the boundaries, and in Fig. 3(b), signal in frequency domain is extracted. Due to interference, certain patterns and frequencies of radiation are sustained by the resonator, with the others being suppressed by destructive interference.

 

Fig. 3 (a) Space-time plot generated by averaging over the vertical dimension of frames in Fig. 2. The center of LN cavity is set as the origin. (b) Fourier transform result along the t axis of the portion of (a) enclosed by the dashed white lines (in time range from 40 ps to 90 ps). Three resonance modes are clearly shown.

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4.3 THz confinement

The resonance frequencies of the three modes represented in Fig. 3(b) are 0.36 THz, 0.44 THz and 0.54 THz, respectively. It is known that in an ideal resonance system such as a string which is held at both ends, only waves whose frequencies are multiples of the fundamental are permitted to exist stably. Obviously, the cavity demonstrated here has significant difference from a traditional ideal system. To explain the difference, we suggest two major factors leading to the difference.

One factor is related to the confinement by the slab waveguide. As the THz pulse propagates in the LN slab, strong index guiding comes into play. Considering the propagating modes, the subwavelength slab waveguide affects the dispersion curve and thus the effective refractive indices in a fundamental and significant way. In the previous work, we have introduced a general solution to a uniaxial LN slab waveguide with air cladding systematically [18]. Figure 4(a) shows the dispersion curves of the first three TE propagating modes (TE₀ TE₁ and TE₂) for a 50 μm-thick LN slab calculated by this solution, which corresponds to our experimental condition. It's noticeable that the dispersion curve of the localized modes in the slab is different from light line in air or bulk LN. We also note that the TE₀ mode has no cut-off frequency while the high-order ones have. For the TE₁ mode and TE₂ mode, the cut-off frequencies are 0.6 THz and 1.2 THz, respectively, which are higher than all the resonance frequency we discuss here (0.36 THz, 0.44 THz and 0.54 THz). Therefore, both the TE₁ mode and the TE₂ mode have no effects on the resonance frequencies and thus only the fundamental TE₀ mode needs to be considered. Phase effective refractive index (ERI), defined by $n_p=ck/2\pi$, can be extracted directly from the dispersion curve shown in Fig. 4(a). The calculated phase ERI of the propagating TE₀ mode in the slab waveguide are demonstrated in Fig. 4(b). The three marked points on the curve indicate resonance frequencies of the cavity mode.

 

Fig. 4 (a) Blue curves: calculated dispersion curves of the THz wave propagating in the LN slab waveguide. The wave is TE-polarized in our experimental condition. Green/purple lines: light lines in air/bulk LN crystal, respectively. (b) The frequency-dependent phase ERI for the fundamental TE mode. In this figure, positions for three cavity resonance modes are marked. (c) The simulated electric field strength vs. time at the cavity boundary (x = 150 μm) with and without the Bragg mirror. Signal in cavity without Bragg gratings (blue curve) decays faster than in cavity containing Bragg grating structures (red curve). (d) The simulated dispersion curve of the transmitted THz wave. It's obvious that some frequency components cannot propagate through the PC structure. The white dashed box is drawn to guide the eyes. (e) The calculated band diagram of the Bragg grating. Two bands are shown and the band gaps are drawn to guide the eyes.

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From Fig. 4(b), we find distinct differences of the phase ERI between the three resonance frequencies. Information on the resonance frequency, phase ERI and the corresponding effective wavelength is summarized in Table 1. Note that the standing wavelength and symmetry in Table 1 will be discussed later in the next subsection.

Table 1. Experimental information for the three cavity modes

View Table

The other factor is the influence of the boundary. We suggest that the Bragg grating structure plays a vital role in THz confinement in the cavity. To prove it, we implemented an additional experiment using another cavity where the Bragg grating sections are replaced by air [19,20]. Figure 5 shows the experimental results of the altered cavity where the Bragg grating sections are replaced by air. Now the LN cavity has two material void as its boundary, the microphotograph of which is shown in Fig. 5(a). Follow the steps of data processing described before to display the experimental results of space-time and space-frequency plots, shown in Figs. 5(b) and 5(c), respectively. From Fig. 5(c), it's clear that no distinguishing cavity mode exists in the resonator in the same time range (40ps to 90ps). According to the experimental results, the setup without PC structure performs poorly in confining light compared with the cavity consisting of Bragg gratings.

 

Fig. 5 Experimental results of individual cavity without Bragg gratings as its boundary. (a) Optical microscope image of the altered cavity. Scale bar: 0.2 mm. (b) The space-time plot in the structure shown in (a). Region enclosed by the white lines indicates the LN cavity. The THz signal attenuates faster than that shown in Fig. 3(a). (c) The result of a Fourier transform along the time axis of (b) in the time range from 40 ps to 90 ps. Compared with Fig. 3(b), no cavity mode is observed.

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In dealing with the comparison between the two configurations, three-dimensional FDTD software (Lumerical Solutions) is used to ensure all conditions imposed on both samples are unified. All parameters are set exactly the same with experimental configuration. Figure 4(c) shows the simulated time-varying THz electric fields at the cavity boundary (x = 150 μm). During the first 10 ps, THz waves oscillate with nearly identical amplitudes at the boundary of both cavities. After several oscillation periods, the THz wave in cavity without Bragg grating structures attenuates dramatically (blue curve). However, in the cavity containing the Bragg grating structure, the attenuation is much smaller (red curve). Thus THz signal in the time range from 40 ps to 90 ps is still strong enough to be detected and analyzed, which is consistent with the experimental results shown in Fig. 5.

It should be pointed out that even without a Bragg grating the THz appears to be confined in the LN cavity for a long time (~20 ps in Figs. 4(c) and 5(b)). We attributed the case to the large refraction index difference between air and LN. According to the Fresnel reflection equation, large index difference leads to high reflectivity at the interface, which helps to confine light in the LN resonator.

The Bragg grating's ability to control light derives from its unique band structure. Figure 4(d) shows the calculated dispersion diagram of the transmitted THz wave from Bragg grating structure by FDTD method. We see that two photonic band gaps lie in the dispersion curve. One gap is located at a large range from 0.27 THz to 0.46 THz, which includes the two low-order resonance mode (0.36THz and 0.44 THz). The other gap contains frequency range higher than 0.52 THz, where the high-order resonance mode (0.54 THz) is covered. The results are coincident with the band diagram calculated by FEM software (COMSOL), which is shown in Fig. 4(e). As we mainly focus on the frequency range from 0.2 THz to 0.6 THz, the calculated diagram is given to cover this range. The low-frequency and high-frequency bands shown in the diagram are band 1 and band 2, respectively. We can see from the diagram that there are two band gaps, one located at a range from 0.29 THz to 0.45 THz and the other located at a range above 0.52 THz, which is in accordance with Fig. 4(d). In this way, THz waves of these frequencies can’t propagate in the Bragg grating regions as Bloch waves and are largely reflected. Consequently, the Bragg grating works as a high reflectance mirror for resonance modes in the cavity. As the band structure is dependent on the geometry size of the grating, we are able to monitor the band gap flexibly by changing the parameters of the Bragg structure. If the band gap fails to overlap the resonance frequencies of the cavity, then the resonance modes cannot be sustained in the central cavity due to the low reflectivity.

4.4 Standing waves

Besides the THz temporal evolution and frequency response, the data provides rich information of field details of the standing wave modes in the resonator. Figure 6 shows the experimental results of spatial distribution of the three standing wave mode in the resonator, which is obtained from Fourier transform of the experimental results. The 300 μm-long cavity region is marked by dotted black lines. Caused by inversion symmetry, the structure supports either symmetric (even-mode) or an anti-symmetric (odd-mode) field distribution. Here the even and odd modes are defined with respect to the inversion symmetry center of the cavity. The fundamental resonance occurs in the even mode, and the first higher order resonance in an odd mode, and so forth. However, these two resonance modes are not observed in our experiment due to the weak confinement by Bragg gratings in low frequency range. The lowest order mode revealed in Fig. 6 is resonant at 0.36 THz and possesses two nodes and three antinodes, indicating it's the 3rd-order mode (even). Likewise, the resonance modes at 0.44 THz and 0.54 THz are 4th-order (odd) and 5th-order (even), respectively. As shown in Fig. 6, our cavity distributes a few energies in the Bragg gratings, which is different from an ideal resonance system in which all of the vibration energy is confined. We recorded the distance between two adjacent antinodes for each resonance mode and compared them with effective wavelengths in Table 1. The results show good agreements, which provide strong validation for our analysis on ERI in waveguide discussed above.

 

Fig. 6 Spatial distribution along X-axis of THz electric fields for cavity modes resonant at (a) 0.36 THz (b) 0.44 THz and (c) 0.54 THz, which is obtained from Fourier transform of the experimental results. The three standing wave are 3rd-order, 4th-order and 5th-order, respectively. The order can be identified from its waveform in the resonator. For example, in (a) the THz wave has three antinodes and two nodes, therefore it is 3rd-order.

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4.5 Potential applications

This design of Fabry-Perot resonator, fabricated in the platform for integrated THz experiments, may have some potential applications. As many materials have characteristic spectra in THz range which are distinguishable from other materials, the THz spectra can be used as fingerprint and allow the identification of various materials [21]. In contrast to traditional THz system like THz-TDS and FT-FIR method working in free space, the cavity demonstrated here has a size of hundred microns which is favorable in micro-scale detection, especial relating to bio-sensing and drug inspection. As shown in Fig. 3(a), the strength of the generated THz field reaches the order of 1kV/cm, strong enough for conventional sensing. The Q-factor of the cavity is observed to be about 20. We propose the Q-factor can be increased by both improving the fabrication quality and employing high-Q structures with novel design.

5. Conclusion

In conclusion, we demonstrate here a compact way to achieve real-time information in a cavity system using an individual micro-chip. By combining a LN cavity with two Bragg grating mirrors, standing wave modes and confinement of THz wave in the cavity are realized. Using coherent detection and phase contrast imaging method, the spatiotemporal information of THz fields is recorded and analyzed. After that, the temporal evolution, frequency information and field distribution of different cavity modes are thoroughly discussed. This method introduces an effective paradigm which can be easily extended to some more complex cavities such as hexagonal photonic crystal cavity and whisper gallery mode cavity. In this way, it provides a powerful toolkit to investigate the physics in cavity in time regime. In addition, the results presented in this paper may facilitate capabilities of functional devices such as THz sensor.