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We report fabrication of a THz phonon-polariton resonator in a single crystal of LiNbO3 using femtosecond laser machining with high energy pulses. Fundamental and overtone resonator modes are excited selectively and monitored through spatiotemporal imaging. The resonator is integrated into a single solid-state platform that can include THz generation, manipulation, readout and other functionalities.
©2004 Optical Society of America
1. Introduction
In recent years considerable progress has been made in the development of pulsed sources of THz radiation, methods for their characterization, and their applications in THz spectroscopy. 1–6 Biased semiconductor surfaces or nonlinear crystals may be irradiated with femtosecond optical pulses to generate broadband (roughly single-cycle) THz wavepackets, and a variety of methods including the use of femtosecond pulse sequences or periodically poled nonlinear crystals have been introduced to produce multiple-cycle THz waves with narrower bandwidths. These methods result in THz radiation that is projected into the free space adjacent to the THz source. The radiation then may be directed toward a sample of interest, and again through free space toward a detector at which the field is analyzed.
For many applications, a fully integrated platform that supports THz generation, manipulation, propagation into a sample of interest, and spectroscopic analysis would be more versatile and convenient. Generation of THz phonon-polariton waves (admixtures of lattice vibrational and electromagnetic modes that propagate coherently at light-like speeds) in nonlinear crystals including LiTaO3 and LiNbO3 through impulsive stimulated Raman scattering (ISRS)7, 8 offers strong prospects for such a system, since the THz responses remain within the solid-state hosts where their spatial and temporal evolution can be fully characterized through spatiotemporal imaging.9 Generation of single-cycle, broadband THz wavepackets or multiple-cycle, frequency-tunable narrowband waves, or in fact nearly arbitrary THz waveforms, can be conducted with such a platform through the use of spatial, temporal, and spatiotemporal femtosecond pulse shaping methods.10 In addition, THz functional elements including waveguides, interferometers, and diffraction gratings have recently been fabricated in these materials through femtosecond laser machining.11, 12 Thus THz generation, delivery to and from a sample, detection, and spectroscopic analysis may all be integrated into a single solid-state platform. Information processing in which coherent THz signals are used to modulate photonic signals or as the information carriers themselves, or in which incoming THz radiation is detected and analyzed, may also be facilitated through the use of an integrated platform.
Among the elements needed for a fully functional integrated THz signal processing or spectroscopy system, structures that are resonant at selected frequencies will play important roles. Frequency-selective filters and resonators will be used for spectral analysis, photonic or optical signal modulation, and a wide range of other purposes, just as resonant structures are widely used in integrated MHz (ultrasonic) and GHz (microwave) signal processing, photonics, and some forms of optical spectroscopy. In this paper we demonstrate femtosecond laser fabrication of an integrated THz resonator, and we illustrate its operation through spatiotemporal imaging which permits direct visualization of the THz waveforms within it.
2. Experimental setup
2.1. Laser machining
In many applications, femtosecond laser machining has been conducted with submicrojoule pulse energies and high (~1 MHz) repetition rates, with the emphasis on small regions of damage whose optical properties are altered.13, 14 For applications with THz phonon-polaritons that have wavelengths in the 20–200 µm range, in which we wish to completely remove substantial regions of LiNbO3 single crystalline material, we need substantially larger machined features. We have used amplified Ti:sapphire laser pulses (790 nm central wavelength, 50 fs duration, 1 kHz repetition rate) with about 200 µJ energy, focused by a microscope objective with numerical aperture 0.1 to a spot size of about 40 microns. The pulse-to-pulse fluctuations in energy were on the order of 2%. In some cases we have carved waveguides and similar structures by translating the sample laterally such that each region is irradiated by just one or a small number of pulses, each of which ablates material to a depth of about 10 microns. Repeated ‘milling’ operations of this sort have been used to carve slots that may extend through several hundred microns of material.11
Fig. 1. (a) Experimental setup used to generate and monitor narrow-band phonon polaritons inside the resonator. A phase mask (LHS illustration) or a third cylindrical lens (RHS) is used to project several optical interference fringes or a single line of excitation light onto the resonator, permitting generation of a selected harmonic or the fundamental frequency respectively. (b) Machined structure. The resonator is formed by the crystalline material between the two carved out rectangles. The distances indicated are approximate values. As evident, the resonator walls are not perfect and it turns out that the side surfaces are tilted by about 6 to 8 degrees due to the laser machining process. The air gaps also form resonators which may be coupled to the central crystalline resonator.
In the present case, we used a different procedure. The same region of the 250 µm thick LiNbO3 sample was left in the beam path for 3 seconds, during which time a hole was drilled through the depth of the crystal, which is facilitated by the long Rayleigh range of the beam. The sample was then translated laterally by a distance of 10 µm, irradiated for 3 seconds, and the process was repeated until a trench of specified length had been carved. The resonator structure was machined in this manner by carving out two 0.5 mm×0.6 mm rectangular pieces separated by 300 µm. After machining all four sides, each rectangle was removed by applying minimal pressure. A microscope image of the structure is shown in Fig. 1(b). The front side of the LiNbO3 resonator turns out to be (290±5) µm wide and the width of the back side is (220±5) µm, that is, the side surfaces are tilted by about 8 degrees.
2.2. Polariton generation and detection
Phonon-polaritons were generated with single or crossed excitation pulses and monitored with an expanded probe pulse that illuminated the entire crystal and was projected onto a CCD camera as shown in Fig. 1(a). Binary phase masks with various fringe spacings were inserted into the excitation beam path (along with a 1:1 cylindrical 2-lens telescope) to project 2–5 interference fringes within the resonator. Alternatively, a third cylindrical lens was used to project a single line of excitation light, roughly 40 µm wide, onto the structure. The pump energy was typically on the order of 20 µJ. Even if we assume that all absorbed pump energy, which is on the order of 10-6 to 10-4 of the incident energy, is eventually converted to heat and radiation is the only cooling mechanism, the crystal’s temperature will change at most by a few tens of degrees (K). The probe pulse (2 µJ) was frequency-doubled with a 100 µm thick β- barium borate crystal, illuminating the whole crystal including the resonator structure with an angle of incidence of 10-degrees to partially compensate for group velocity mismatch with the 800 nm excitation pulse(s), and projected onto a CCD camera with a 0.55 numerical aperture microscope objective. The CCD camera had 640×480 pixels, a frame rate of 30 Hz, and the integration time was typically 200 ms. The polariton-induced refractive index changes in the sample result in spatially varying modulations in the phase of the probe light, and these can be detected as amplitude modulations by placement of the camera slightly out of the focal plane (the Talbot effect).15, 16 The resulting images, recorded with variably delayed probe pulses, may be viewed in rapid succession to present a ‘movie’ of phonon-polariton propagation. An undesired feature of Talbot imaging is some loss of resolution, leading to blurring of the resonator structure. In order to illustrate the signal recurrences within the resonator with optimal clarity, all images were rescaled as they were recorded so that weaker signals recorded after several recurrences could still be observed clearly. This procedure sacrifices quantitative information about the resonator loss from polariton damping and partial transmission.
3. Theoretical background
3.1. Phonon-polaritons
In crystalline solids, electromagnetic radiation may be coupled to polar lattice vibrations of similar frequency and wave vector. The coupled modes, called phonon polaritons, travel at light-like speeds with partly mechanical and partly electromagnetic energy. In the absence of phonon-polariton coupling, at the low wave vectors and (THz) frequencies of interest here, the phonon mode is essentially dispersionless while the electromagnetic mode exhibits essentially linear dispersion described by ω=[c/n(ω)]k, where the refractive index n varies only weakly with frequency ω or wave vector k. Strong mixing between the two modes occurs when their frequencies and wave vectors are similar. Phenomenologically, the coupling between vibrational displacement Q̄ and polarization P̄ can be described by the following pair of equations:17
where j labels the mode, ε f is the permittivity of vacuum, Γ is the phenomenological damping, ωT is the natural oscillator frequency, and S is the coupling strength. LiNbO3 is a uniaxial crystal and the three axes can be divided into two ordinary and one extraordinary. In this coordinate system the tensors ω̿ T , S̿, ε̿∞, and G̿ are diagonal. At room temperature the crystals are below the Curie point and have lattice symmetry C3v. There are four A1 symmetry IR and Raman active modes with the electric field polarized along the optic axis. In principle, those are the modes that we can excite when the excitation laser light is parallel to the optic axis. The lowest frequency mode has by far the dominant oscillator strength. Furthermore, the highest frequency modes have frequencies too high to be driven impulsively with the available optical pulse widths. Therefore the equation system (1) can be simplified to just two coupled equations involving only the lowest frequency optic phonon mode in each case. Combining (1) with Maxwell’s equations then fully describes the propagating phonon-polariton response.
In the present experiments we excite phonon-polaritons through impulsive stimulated Raman scattering (ISRS) which has been demonstrated and exploited for many years in great detail.7, 8 The polarization of the excitation light is parallel to the optic axis, and so is the polarization of the phonon-polariton response that is generated. The ISRS force is given by
where N is the oscillator density and ELaser the slowly varying envelope of the laser electric field. The reduced mass µ and differential polarizability (∂α/∂x) correspond to the lowest frequency A1 mode.
3.2. Forward wave vector component
The phonon-polaritons generated by ISRS have a non-negligible forward wave vector component. Thus although their propagation direction is primarily in the transverse dimension of the waveguide (to the right and left in the plane of Fig. 1(b), there is also a forward component (downward into the page of Fig. 1(b), with an angle of approximately 24 degrees. The polariton wavefronts therefore approach the back of the sample at a 66-degrees angle, which exceeds the critical angle for total internal reflection. Thus the polaritons reflect without loss off the back and front crystal faces as they move laterally between the side faces. Despite their 6–8 degrees bevel, these faces are approached at an angle which is larger than the critical angle (14 degrees) and total internal reflection should occur even at the sides. Note, however, that the LiNbO3 crystal thickness is only 250 microns. The crystalline element acts as a planar waveguide, and depending on the polariton wavelength relative to the crystal thickness, the forward wave vector component may be reduced after propagation within it. This may establish a coupling between crystalline and air gap resonators, since the polaritons reach the sides of the crystalline resonator with essentially no forward propagation component, in which case total internal reflection at the sides will be averted.
The phonon-polariton forward wave vector component satisfies the phase matching condition for the probe pulse which is propagating through the crystal in the forward direction, but the backward wave vector component of the polaritons after reflection off the back surface of the sample does not. Thus the signal level is reduced significantly when the polaritons are moving in the opposite direction of the probe pulse.
In order to determine whether the wavefront is indeed totally internally reflected at the crystal-air interface, we have performed 3D finite-difference time-domain (FDTD) simulations18–20 of a phonon-polariton wave generated in a 250 µm thick waveguide slab. The simulations solve Maxwell’s equations coupled to a damped harmonic oscillator that represents the corresponding phonon mode, as described by the system of coupled equations (2). The ISRS excitation process is explicitly simulated to ensure the correct spatiotemporal excitation profile. The values of the parameters used in the simulations are listed in table 1. The simulations show that for wavelengths larger that the waveguide thickness, the angle between the wave vector (averaged over a cross section) and the crystal surface rapidly decreases as as the polariton propagates away from the excitation region. The angle drops below the critical angle for total internal reflection within 100 to 200 µm and then approaches zero. That is, after a relatively short propagation distance the system must be treated as three coupled resonators.
Table 1. Values of parameters used in FDTD simulations.
3.3. System resonances
The resonator of interest is a LiNbO3 bridge in between two air gaps with tilted side surfaces, as shown in Fig. 2(a).
The air gaps, as well as the crystalline region, can act as THz resonators, so the whole system consists of three coupled resonators. The degree of coupling depends on the reflectivity at the crystal-air interfaces between them. The resonance frequencies of the system of coupled resonators were obtained through FDTD simulations incorporating the excitation process as well as probing. The probe intensity at the center of the LiNbO3 resonator was extracted as a function of time, and the frequency response was obtained by Fourier transformation. In order to cover a broad frequency range with a single simulation, the full width at half maximum of the excitation pulse was chosen to be about 1/5 (50 µm) of the LiNbO3 resonator width. Therefore, what is shown in Fig. 2(b) is the frequency response convolved with the spectrum of polaritons generated. Additional simulations with just the LiNbO3 bridge show that only the resonances below approximately 0.3 THz, which corresponds to a wavelength of about 200 µm, are visibly affected by the air gap resonators. While the line widths of resonances for frequencies below 0.6 THz are dominated by reflection losses, for higher frequencies absorption starts to play a major role.
Fig. 2. (a) Schematic illustration of the coupled system of resonators with tilted side surfaces. The excitation pulse (in red) passes through the center of the LiNbO3 resonator and generates two polariton waveforms (in green) propagating in opposite lateral direction with a forward component. (b) Calculated frequency response. The two modes observed in the experiments are indicated by two gray bars, where the width corresponds to the uncertainty in the measured frequency.
4. Results
We first present results of almost single-mode excitation, i.e. the spatial excitation pattern almost perfectly matches that of a longitudinal resonator mode. We then show results of a single line excitation where the width of the line is about a fifth of the resonator width and the phonon-polariton response bounces back and forth between the two resonator walls producing periodic recurrences of the original spatial pattern.
By using a phase mask grating with 80 µm spacing, a narrowband polariton response is generated inside the resonator, and its evolution is monitored for over 25 ps. By careful positioning of the excitation grating, we excite a response that is very close to one of the eigenmodes of the resonator corresponding to a standing wave with a wavelength which is a third of the central resonator width (see Fig. 2). The intensity of the mode reaches a maximum at the two resonator walls. When electromagnetic radiation is reflected from an interface transitioning from a high to low dielectric medium, the reflected wave undergoes no phase shift, and the polariton waveform generated sets up a standing wave within the resonator.
The recurrent behavior of the standing wave is summarized in Fig. 3 and is shown in the corresponding movie. The two terminal quarter wavelengths at the walls of the resonator are not as clearly imaged due to scattering of the 400 nm probe pulse at the walls of the resonator and the out-of-focus Talbot imaging. Figure 4 is constructed by averaging the signal in the uniform direction along the length of the resonator for each image. Note that Fig. 4 makes it seem that the waveform persists uniformly for some time and then abruptly switches sign. This artifact is a result of the bitmap recalibration done at every image for maximum contrast.
Figure 4 reveals that between approximately 5 ps and 8 ps the signal intensity seems to pass through a local minimum. As mentioned above, phonon-polaritons propagate with a slight forward component and the length of phasematched overlap with the probe pulse is large when they propagate toward the back side but very small when travelling toward the front side. Therefore, the signal intensity will decrease or increase depending on whether the phonon-polariton is propagating in the forward or backward direction. Due to the bitmap recalibration this effect becomes visible only if the noise level exceeds the signal level in which case the oscillations seem to disappear for a number of delay steps.
Fig. 3. (a) A phonon-polariton response with approximately 80 µm wavelength has been generated inside the resonator. A well-defined standing wave and its confined oscillations at a frequency of (0.67±0.02) THz ((1.50±0.05) ps period) are clearly observed. (b) (1544 KB) Evolution of the polariton inside the resonator.
The standing-wave response has a period of (1.50±0.05) ps, which corresponds to a frequency of (0.67±0.05) THz. This is in good agreement with the system resonance at 0.67 THz, as seen in Fig. 2(b).
Fig. 4. Each horizontal line is constructed from images at different time steps, such as those in Fig. 3, by averaging the signal along the resonator length where the signal extends.
The response to a single line of excitation light, incident at the middle of the resonator, is shown in Fig. 5 and in the accompanying movie. The selected images come from a series of snapshots recorded at different probe delays. They show the recurrences typical of resonator behavior. The excitation light extends through the 600 µm tall resonator and into the surrounding (bulk) material. The first frame (0 ps) shows the single-cycle ‘ripple’-like polariton response which similarly extends through and outside of the resonator structure. The break in the response is due to the damaged part of the structure, evident in Fig. 1(b). The movie shows that within the resonator, the response reflects back and forth between the side walls and yields recurrences which, from the selected images shown in Fig. 5, are seen to occur with a period of approximately (3.9±0.4) ps. The corresponding frequency is (0.26±0.03) THz, which, within the error bars, corresponds to the system resonance at 0.25 THz.
Fig. 5. (a) Line excitation inside the resonator and subsequent recurrences. As evident from the figure, recurrences occur on average every (3.9±0.4) ps. The dotted line between the last two images indicates that they are separated by two recurrence periods. (b) (1929 KB) Evolution of the polariton inside the resonator.
Figure 6 is constructed by selecting a region within the resonator from each of the images like those in Fig. 5. Each region is essentially a horizontal line segment, contained within the resonator region, defined by its vertical position between the top and bottom of the resonator. The signal along such a line is essentially uniform, so it is averaged and reduced to a point, which represents the signal level at that vertical position inside the resonator at a particular time. The process is repeated for the image recorded at the next time step, and so on, and all of the resulting points are placed in time order to produce a horizontal line in Fig. 6. This line displays the time-dependent signal at the selected vertical position in the resonator averaged across the horizontal extent of the resonator. This entire process is repeated for the different vertical positions in the resonator, and the resulting times are placed above each other to generate Fig. 6. Both figures and the movie show that up to 6–7 recurrences can be clearly observed. The slight differences in the details of the recurrence patterns result from the fact that the initial line excitation is not perfectly centered in the resonator.
The decay of the signal is mostly due to partial transmission through and scattering at the resonator boundaries which, due to limitations of the laser machining process, are not perfectly smooth, uniform, or perpendicular to the front and back crystal surfaces.
Meanwhile, the polariton responses outside the resonator structure propagate away from their regions of origin without constraints. In the low wave vector range included within the bandwidth of the single-cycle polariton response, the group velocity shows very little dispersion.21 The images clearly show that the single-cycle pulse does not spread significantly over millimeter distances.
Fig. 6. Each horizontal line is constructed from images at different time steps, such as those in Fig. 5, by averaging the signal along the resonator length where the signal extends.
5. Conclusions
We have used femtosecond laser machining of LiNbO3 to fabricate a functional THz resonator whose properties were demonstrated through direct spatiotemporal imaging of the phonon-polariton responses within it. A broadband response was generated with a single line of excitation light, and a higher-order resonator mode was excited, with narrower bandwidth, using a grating excitation pattern that was well matched to the resonator width. Phonon-polariton resonators and resonator arrays may be used for photonic signal modulation, for electrooptic generation of ultrahigh-frequency electrical signals, or for spectroscopic or signal processing applications in which the phonon-polaritons (or radiation produced as they exit through a crystal face) are used as the signal carriers. Recent further progress in femtosecond laser machining of these materials and substantially reduced phonon-polariton dephasing and damping rates at low temperatures (e.g. 77K)22 suggest that far more highly resonant structures may be constructed than those demonstrated here. Finally, recent results from thin (10-µm) LiNbO3 films23 indicate that these will be suitable for applications of the type demonstrated here, greatly facilitating the laser machining process and eliminating the issue of forward and backward wave vector components.
Acknowledgments
This work was supported in part by the MRSEC Program of the National Science Foundation under award number DMR 02-13282.
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