Terahertz Polaritonics

Thomas Feurer, Institute for Applied Physics. 3012 Bern

 

In the recent past the terahertz gap between electronics (up to approximately 100 GHz) and optics (down to approximately 10 THz) has gained intense interest in connection with applications such as high-bandwidth signal processing, THz imaging, THz sensing, or THz spectroscopy. Most of these applications have become possible by the emergence of femtosecond laser-based methods for generation of THz pulses. So far none of these methods has provided an integrated platform for THz waveform generation, guidance, processing, and characterization. In the following we review our efforts to realize such a platform and which have led to THz polaritonics. The signal carriers are phonon-polaritons, coupled admixtures between electromagnetic waves and polar lattice vibrations of comparable frequency and wave vector [1]. In the polaritonics platform, they are generated by femtosecond optical pulses through a second-order nonlinear optical process and they propagate primarily in lateral directions, with the largest wave vector component perpendicular to the optical beam path. This key property, which enables interactions of the THz signals with further optical pulses or with integrated structures, arises because the THz dielectric constant is extremely high. There are three important ingredients of the polaritonics toolkit that distinguish it from more conventional THz methodology. Firstly, spatiotemporal imaging of polariton fields and their evolution, secondly, spatiotemporal coherent control over polariton generation and, thirdly, integrated polaritonic functional elements. Phonon-polaritons induce a change in the index of refraction at an optical probe wavelength, which is proportional to the lattice vibrational displacement and therefore to the THz electric field. The phase shift of a probe pulse, integrated through the crystal thickness, can be measured directly through interferometry. Moreover, with an expanded beam that is passed through the entire sample region of interest and transmitted to a CCD camera the THz field in the entire region can be visualized with unprecedented spatial and temporal precision. Recording such snapshots at progressively later times allows to assemble a movie of the THz field propagation through the crystal and also through functional elements embedded in the crystal. In the following three sections we discuss THz phonon-polariton propagation through advanced polaritonics structures, such as photonic bandgap materials and through so-called meta-materials.

I. 1D Polaritonic Bandgap Structure

We first illustrate the principle by analyzing an almost plane THz waveform propagating through a one-dimensional polaritonic bandgap structure. When an intense laser pulse passes through a LiNbO3 crystal it creates a nonlinear polarization which acts as a source for a THz field. In all experiments following, the excitation is a y-polarized laser pulse with a wave vector kpump || x which is focused to a line parallel to the y-axis as shown in Fig. 1(a). The optic axis of the LiNbO3 crystal is parallel to the y-axis and so is the THz electric field vector. If we assume that the excitation field is an infinitely tall line-source which moves parallel to the x-axis without changing its shape and, further, that the excitation field has a Gaussian spatial profile in the z-dimension and a Gaussian temporal profile, the THz waveform generated is an almost plane single-cycle wavelet. An example is shown in Fig. 1(b); the laser pulse propagates in the upward direction and has traveled about 400 µm into the crystal. When it enters the LiNbO3 crystal, the THz response generated near the front surface has traveled some measurable distance before the excitation pulse reaches the back of the sample and generates a THz field there. Because the radiation source moves faster than the phase velocity of the radiation generated, the THz field can be thought of Cherenkov-type radiation as indicated in Fig. 1(b); for LiNbO3 the Cherenkov angle is 64 degree. Figure 1(b) also shows that the temporal profile of both THz wavelets is that of a single-cycle pulse with a carrier frequency in the THz range and pulse durations on the order of a few picoseconds.

FIG. 1: (a) The laser pulse is focused to a line, propagates in the upward direction, and generates two nearly plane THz wavelets propagating almost perpendicular to the direction of the laser pulse.
(b) The THz response generated at the front surface has traveled some distance before the excitation pulse reaches the back of the sample. If the THz source moves faster than the phase velocity of the THz radiation generated, the THz field can be thought of Cherenkov-type radiation.

When the plane THz waveform impinges on the side surface of the LiNbO3 crystal partial reflection and transmission occur following the laws of Fresnel. By stacking different materials to the side of the LiNbO3 crystal a so-called one dimensional photonic crystal or, more conventional, a Bragg mirror is realized. Taking a single snapshot of the spatiotemporal field distribution at a time when most of the waveform has either returned from the 1D structure or has been transmitted through the structure allows to analyze its frequency response within the spectral range that is covered by the THz waveform spectrum. An example is shown in Fig. 2.

The frequency responses for reflection and transmission show complementary band gaps, i.e. regions with high reflectivity or high transmission. Both measurements agree reasonably well with finite difference time domain simulations. Although Fig. 2 only shows the spectral amplitudes the experiments also give access to the spectral phases as the measurement itself yields the temporal amplitude of the electric field and not its intensity. That is, in principle one may also analyze the dispersive properties of such a 1D photonic structure.

FIG. 2: (a) Reflected and (b) transmitted spectrum. The blue lines correspond to the experimental results and the dashed green lines to the corresponding simulations. The 1D structure itself consisted of alternating layers of air and silicon with a thickness of 505 µm.

II. 2D Polaritonic Bandgap Structure

Next, we present results for two-dimensional photonic bandgap materials, i.e. LiNbO3 crystals with an array of air holes that form a two-dimensional lattice of regions of alternating dielectric properties. Much in the same way as a periodic potential in a crystal affects the electron motion, the periodic dielectric properties influence the propagation of electromagnetic waves [2]. Those frequencies or wavelengths which are allowed to propagate are called modes and groups of allowed modes are called bands; disallowed bands are called photonic band gaps. Photonic band gaps, in turn, give rise to fascinating optical phenomena, such as defect-assisted wave-guiding, inhibition of spontaneous emission, omnidirectional reflection, or apparent negative refraction to name but a few. Because the basic physical principle is diffraction the periodicity of any photonic structure has to be on the order of the wavelength of the electromagnetic wave. While in the visible region this poses quite a technological challenge, in the THz region, where free space wavelengths are on the order of tens to hundreds of microns, the fabrication of such structures is much more relaxed. Moreover, our imaging technique allows for measuring THz electric fields within and outside of the structure with sub-wavelength resolution.

An example is shown in Fig. 3; a plane single-cycle waveform impinges onto the twodimensional periodic structure from the left. From a sequence of electric field measurements the band diagram can be extracted. Figure 3 shows the band diagram of the transmitted part of the THz pulse with a few prominent modes in between the two light lines and a very pronounced band gap. The experimental results agree well with the corresponding simulations and show the strength of our imaging method.

FIG. 3: A plane single-cycle waveform travels from left to right through a hexagonal photonic crystal structure consisting of an array of air hole within a LiNbO3 crystal. Integrating the signal along the y dimension and stacking the resulting one-dimensional plots on top of each other yields an x-t plot with the color being proportional to the electric field strength. A two-dimensional Fourier transform results in a kx-ω plot, better known as a band diagram.

III. Meta-Materials

We close this contribution with an illustration of somewhat more complex structures, namely so-called meta-materials [3]. Such materials are composed of resonating subwavelength structures and exhibit remarkable electromagnetic properties. As a result, metamaterials facilitate various exciting new applications such as perfect lenses, perfect absorbers, or invisibility cloaking. Since the first experiments conducted at microwave frequencies rapid progress in fabrication technologies has allowed to build meta-materials for even higher frequencies, ranging from the THz to the mid-infrared and the near-infrared regime. The material properties of such materials result from the resonant electric or magnetic response of their constituting structures when interacting with an incident light wave and the associated formation of localized near-fields. Thus, a key to understanding is an experimental tool to measure such localized near-fields and our method of visualizing electromagnetic fields with λ/100 resolution may proof to be such a tool.

FIG. 4: The green arrows and the colored regions indicate the strength of the electric field at a frequency that is resonant with the outer ring and the solid lines are magnetic field lines created by the currents in the split ring. The resonant behavior is obvious when analyzing the current density in the two rings as a function of frequency.

So far, near-fields of individual resonators have mostly been analyzed through numerical simulations. In Fig. 4 we show the result of a double split-ring resonator; such a device exhibits magnetic resonances and may serve as an individual sub-wavelength building block for negative µ meta-materials. When performing such three-dimensional simulations for various frequencies the resonant behavior of the structure becomes obvious as shown in the lower part of Fig. 4. While the outer ring has a strong resonance at around 0.075 THz, the inner ring has its strongest resonance around 0.16 THz. In conclusion, we demonstrated the possibilities of our THz polaritonics platform to visualize electromagnetic fields within complex structures with sub-wavelength resolution. The method will be useful in analyzing photonic bandgap structures as well as novel meta-materials.

 

References

[1] Born M. and Huang K. 1954. Dynamical Theory of Crystal Lattices. Clarendon Press.
[2] Yablonovitch E., ”Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Physical Review Letters 58(20), 2059 (1987).
[3] Veselago V.G., ”The electrodynamics of substances with simultaneously negative values of ε and μ” Sov. Phys. Usp. 10(4), 509 (1968).

 

[Released: March 2009]