Photomixing of two near-infrared lasers is well established for continuous-wave terahertz spectroscopy. Photomixing of three lasers allows us to measure at three terahertz frequencies simultaneously. Similar to Fourier spectroscopy, the spectral information is contained in an interferogram, which is equivalent to the waveform in time-domain spectroscopy. We use one fixed terahertz frequency ν ref to monitor temporal drifts of the setup, i.e., of the optical path-length difference. The other two frequencies are scanned for broadband high-resolution spectroscopy. The frequency dependence of the phase is obtained with high accuracy by normalizing it to the data obtained at ν ref , which eliminates drifts of the optical path-length difference. We achieve an accuracy of about 1 − 2 µm or 10 −8 of the optical path length. This method is particularly suitable for applications in nonideal environmental conditions outside of an air-conditioned laboratory.PACS numbers: 32.30.Bv Terahertz spectroscopy has been revolutionized by laser-based techniques and bears an enormous potential both for fundamental science and for a wide range of applications.[1, 2] One intriguing aspect of the terahertz range is that it allows for the determination of both amplitude and phase ϕ of an electromagnetic wave. The phase delay induced by a sample can be employed for measuring both the refractive index and the thickness [3][4][5][6][7] or for tomography, e.g. for the inspection of space shuttle foam insulation.[8-10] Reliable measurements of the phase require a high stability of the relevant experimental lengths, therefore (thermal) fluctuations or alignment drifts may not exceed a small fraction of the wavelength. Large wavelengths such as about 1 mm at 300 GHz thus facilitate the determination of the phase. However, thermal fluctuations cannot be fully suppressed even in an air-conditioned laboratory, and phase measurements in real-world applications in a less than ideal environment are challenging, in particular if they rely on a robust fiber-based system.[11] These difficulties are successfully surpassed in ellipsometry, which measures the phase difference between different polarization states. Here, we choose another route based on continuous-wave (cw) spectroscopy in the frequency domain and consider the phase difference of waves with different frequencies. Via photomixing of three lasers, we generate waves at three terahertz frequencies. The waves travel along the same path at the same time, their phases are measured simultaneously. We employ the phase at the fixed frequency ν ref to monitor length changes during the measurement. The normalized phases of the two other, scanning frequencies are nearly insensitive to thermal drifts. Without temperature stabilization of the laboratory, we achieve an accuracy which is equivalent to length changes of about 1-2 µm or 3-6 fs·c, where c denotes the speed of light.Continuous-wave terahertz radiation can be generated and coherently detected by illuminating two photomixers, transmitter and re...

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We report on the group delay observed in continuous-wave terahertz spectroscopy based on photomixing with phase-sensitive homodyne detection. We discuss the different contributions of the experimental setup to the phase difference ϕ(ν) between transmitter arm and receiver arm. A simple model based on three contributions yields a quantitative description of the overall behavior of ϕ(ν). Firstly, the optical path-length difference gives rise to a term linear in frequency ν. Secondly, the ultra-wideband log-spiral antennae effectively radiate and receive in a frequencydependent active region, which in the most simple model is an annular area with a circumference equal to the wavelength. The corresponding term changes by roughly 6π between 100 GHz and 1 THz. The third contribution stems from the photomixer impedance. In contrast, the derivative ∂ ϕ/∂ν is dominated by the contribution of periodic modulations of ϕ(ν) caused by standing waves, e.g., in the photomixers' Si lenses. Furthermore, we discuss the Fourier-transformed spectra, which are equivalent to the waveform in a time-domain experiment. In the time domain, the group delay introduced by the log-spiral antennae gives rise to strongly chirped signals, in which low frequencies are delayed. Correcting for the contributions of antennae and photomixers yields sharp peaks or "pulses" and thus facilitates a time-domain-like analysis of our continuous-wave data.

We present a study of the dielectric, structural, and magnetic properties of the multiferroic or linear magnetoelectric substitution series [(NH4)1−xKx]2[FeCl5(H2O)]. Pyroelectric currents, magnetic susceptibilities, and thermodynamic properties were examined on large single crystals of the erythrosiderite compounds and detailed magnetic-field versus temperature phase diagrams are derived for three different substitution levels. With increasing potassium concentration the material is tuned from a multiferroic (x ≤ 0.06) to a linear magnetoelectric (x ≥ 0.15) ground state. In contrast to the respective pure parent compounds with x = 0 or 1, however, the ferroelectric or linear magnetoelectric polarization in none of the substituted samples is switchable by external electric fields, because these samples exhibit a significant electric polarization already above the magnetic ordering transition. The polarization arises at a higher-lying structural phase transition that is examined by THz spectroscopy and, on a deuterated pure single crystal, by comprehensive neutron-diffraction experiments. The structural phase transition is attributed to an ordering of NH + 4 tetrahedra but does not break inversion symmetry in the pure material, while a finite K content causes pyroelectricity.

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In a continuous-wave terahertz system based on photomixing, the measured amplitude of the terahertz signal shows an uncertainty due to drifts of the responsivities of the photomixers and of the optical power illuminating the photomixers. We report on a simple method to substantially reduce this uncertainty. By normalizing the amplitude to the DC photocurrents in both the transmitter and receiver photomixers, we achieve a significant increase of the stability. If, e.g., the optical power of one laser is reduced by 10%, the normalized signal is expected to change by only 0.3%, i.e., less than the typical uncertainty due to short-term fluctuations. This stabilization can be particularly valuable for terahertz applications in non-ideal environmental conditions outside of a temperature-stabilized laboratory.

In a continuous-wave terahertz spectrometer, we employ photomixing of three lasers, allowing us to measure at three terahertz frequencies simultaneously. Whilst two of the terahertz frequencies can be tuned to obtain terahertz spectra, one terahertz frequency remains fixed. The data recorded at the fixed frequency is used to normalize the spectra of the two other frequencies. As a result, the accuracy of the terahertz phase measurement can be greatly enhanced.

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