On cogrowth function of algebras and its logarithmical gap

Let $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I${\it-reducible} if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An {\it obstruction} in a subword-minimal $I$-reducible word. A {\em cogrowth} function is number of obstructions of length $\le n$. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth.Comment: 5 page

Near-field characterization of conductive micro-resonators for terahertz sensing

Near-field (NF) terahertz (THz) time-domain spectroscopy (TDS) is an excellent tool for direct studies of THz electromagnetic resonances occurring on a micrometre scale. Micro-resonators are at the heart of numerous promising THz sensing and detecting solutions. Experimental studies of individual micrometrescale THz resonances are essential, yet extremely challenging for the common far-field spectroscopic methods due to extreme sensitivity requirements. NF THz spectroscopy and microscopy are non-contact techniques for spectroscopic studies of individual micro-resonators and mapping the field patterns of THz resonant modes excited in individual conductive or insulating micro-objects. They give access to essential parameters of micro-resonators, including their resonance frequency, local field enhancement and quality factors. It allows for material and structural characterisation of micro-objects. Using the example of carbon microfibres, we show the advantages of NF THz TDS for non-contact THz conductivity probing and direct observation of the fundamental and the third-order surfaceplasmon resonance modes in conductive THz micro-resonators

Terahertz near-field microscopy using the self-mixing effect in a quantum cascade laser

We demonstrate terahertz (THz) apertureless near-field microscopy exploiting the self-mixing effect in a quantum cascade laser (QCL). A THz wave is scattered by a sharp needle positioned above an object and coupled back into the QCL cavity resulting in detection of the THz near-field signal through the self-mixing effect. Using this technique we demonstrate two-dimensional imaging at 2.53 THz with a spatial resolution of 1 μm - the highest image resolution achieved with a THz frequency QCL to date. This method offers an experimentally simple approach to coherent, high-resolution THz imaging

Near-field spectroscopy and tuning of subsurface modes in plasmonic terahertz resonators

Highly confined modes in THz plasmonic resonators comprising two metallic elements can enhance light-matter interaction for efficient THz optoelectronic devices. We demonstrate that sub-surface modes in such double-metal resonators can be revealed with an aperture-type near-field probe and THz time-domain spectroscopy despite strong mode confinement in the dielectric spacer. The sub-surface modes couple a fraction of their energy to the resonator surface via surface waves, which we detected with the near-field probe. We investigated two resonator geometries: a λ/2 double-metal patch antenna with a 2 μm thick dielectric spacer, and a three-dimensional meta-atom resonator. THz time-domain spectroscopy analysis of the fields at the resonator surface displays spectral signatures of sub-surface modes. Investigations of strong light-matter coupling in resonators with sub-surface modes therefore can be assisted by the aperture-type THz near-field probes. Furthermore, near-field interaction of the probe with the resonator enables tuning of the resonance frequency for the spacer mode in the antenna geometry from 1.6 to 1.9 THz (~15%)

Designing an efficient hybrid optical cavity

We present an efficient terahertz (THz) detector based on an optically thin hybrid cavity. We use experimental and numerical methods to design efficient detectors, finding a hybrid cavity structure with a photoconductive (PC) layer as thin as 50 nm which absorbs almost 80% of light at the operation wavelength. These optically thin detectors are well suited to near-field microscopy and terahertz component integration