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New Horizons in Cosmology with Spectral Distortions of the Cosmic Microwave Background

Following the pioneering observations with COBE in the early 1990s, studies of the cosmic mi- crowave background (CMB) have primarily focused on temperature and polarization anisotropies. CMB spectral distortions – tiny departures of the CMB energy spectrum from that of a perfect blackbody – provide a second, independent probe of fundamental physics, with a reach deep into the primordial Universe. The theoretical foundation of spectral distortions has seen major advances in recent years, highlighting the immense potential of this emerging field. Spectral distortions probe a fundamental property of the Universe – its thermal history – thereby providing additional insight into processes within the cosmological standard model(I) (CSM) as well as new physics beyond. Spectral distortions are an important tool for understanding inflation and the nature of dark matter. They shed new light on the physics of recombination and reionization, both prominent stages in the evolution of our Universe, and furnish critical information on baryonic feedback processes, in addition to probing primordial correlation functions at scales inaccessible to other tracers. In principle the range of signals is vast: many orders of magnitude of discovery space can be explored by detailed observations of the CMB energy spectrum. Several CSM signals are predicted and provide clear experimental targets that are observable with present-day technology. Confirmation of these signals would extend the reach of the CSM by orders of magnitude in physical scale as the Universe evolves from the initial stages to its present form. Their absence would pose a huge theoretical challenge, immediately pointing to new physics. Here, we advocate for a dedicated effort to measure CMB spectral distortions at the largest angular scales (greater than approximately 1°) within the ESA Voyage 2050 Program. We argue that an L-class mission with a pathfinder would allow a precise measurement of all the expected CSM distortions. With an M-class mission, the primordial distortions (created at z >~ 10(exp 3)) would still be detected at modest significance, while the late-time distortions will continue to be measured to high accuracy. Building on the heritage of COBE/FIRAS, a spectrometer that consists of multiple, cooled (approximately equal to 0.1 K), absolutely-calibrated Fourier Transform Spectrometers (FTS) with wide frequency coverage (ν approximately equal to 10 GHz to a few x THz) and all-sky spectral sensitivity at the level of 0.1 0.5 Jy/sr would be the starting point for the M-class option. A scaled and further optimized version of this concept is being envisioned as the L-class option. Such measurements can only be done from space and would deliver hundreds of absolutely-calibrated maps of the Universe at large scales, opening numerous science opportunities for cosmology and astrophysics. This will provide independent probes of inflation, dark matter and particle physics, recombination and the energy output of our Universe from at late times, turning the long-standing spectral distortion limits of COBE/FIRAS into clear detections.

Chluba, J.↗

The Primordial Inflation Explorer (PIXIE)

The Primordial Inflation Explorer is an Explorer-class mission to measure the gravity-wave signature of primordial inflation through its distinctive imprint on the linear polarization of the cosmic microwave background. PIXIE uses an innovative optical design to achieve background-limited sensitivity in 400 spectral channels spanning 2.5 decades in frequency from 30 GHz to 6 THz (1 cm to 50 micron wavelength). Multi-moded non-imaging optics feed a polarizing Fourier Transform Spectrometer to produce a set of interference fringes, proportional to the difference spectrum between orthogonal linear polarizations from the two input beams. The differential design and multiple signal modulations spanning 11 orders of magnitude in time combine to reduce the instrumental signature and confusion from unpolarized sources to negligible levels. PIXIE will map the full sky in Stokes I, Q, and U parameters with angular resolution 2.6 deg and sensitivity 0.2 uK per 1 deg square pixel. The principal science goal is the detection and characterization of linear polarization from an inflationary epoch in the early universe, with tensor-to-scalar ratio r <10(exp -3) at 5 standard deviations. In addition, the rich PIXIE data will constrain physical processes ranging from Big Bang cosmology to the nature of the first stars to the physical conditions within the interstellar medium of the Galaxy. We describe the PIXIE instrument and mission architecture needed to detect the signature of an inflationary epoch in the early universe using only 4 semiconductor bolometers.

Kogut, Alan J.↗

The Primordial Inflation Explorer (PIXIE)

The Primordial Inflation Explorer is an Explorer-class mission to measure the gravity-wave signature of primordial inflation through its distinctive imprint on the linear polarization of the cosmic microwave background. PIXIE uses an innovative optical design to achieve background-limited sensitivity in 400 spectral channels spanning 2.5 decades in frequency from 30 GHz to 6 THz (1 cm to 50 micron wavelength). Multi-moded non-imaging optics feed a polarizing Fourier Transform Spectrometer to produce a set of interference fringes, proportional to the difference spectrum between orthogonal linear polarizations from the two input beams. Multiple levels of symmetry and signal modulation combine to reduce the instrumental signature and confusion from unpolarized sources to negligible levels. PIXIE will map the full sky in Stokes I, Q, and U parameters with angular resolution 2.6 deg and sensitivity 0.2 μK per 1 deg square pixel. The principal science goal is the detection and characterization of linear polarization from an inflationary epoch in the early universe, with tensor-to-scalar ratio r less than 10(exp −3) at 5 standard deviations. In addition, PIXIE will measure the absolute frequency spectrum to constrain physical processes ranging from inflation to the nature of the first stars to the physical conditions within the interstellar medium of the Galaxy. We describe the PIXIE instrument and mission architecture with an emphasis on the expected level of systematic error suppression.

Primordial Inflation Explorer is an Explorer-class↗

The Primordial Inflation Explorer (PIXIE)

The Primordial Inflation Explorer is an Explorer-class mission to open new windows on the early universe through measurements of the polarization and absolute frequency spectrum of the cosmic microwave background. PIXIE will measure the gravitational-wave signature of primordial inflation through its distinctive imprint in linear polarization, and characterize the thermal history of the universe through precision measurements of distortions in the blackbody spectrum. PIXIE uses an innovative optical design to achieve background-limited sensitivity in 400 spectral channels spanning over 7 octaves in frequency from 30 GHz to 6 THz (1 cm to 50 micron wavelength). Multi-moded non-imaging optics feed a polarizing Fourier Transform Spectrometer to produce a set of interference fringes, proportional to the difference spectrum between orthogonal linear polarizations from the two input beams. Multiple levels of symmetry and signal modulation combine to reduce systematic errors to negligible levels. PIXIE will map the full sky in Stokes I, Q, and U parameters with angular resolution 2.6 degrees and sensitivity 70 nK per 1degree square pixel. The principal science goal is the detection and characterization of linear polarization from an inflationary epoch in the early universe, with tensor-to-scalar ratio r < 10(exp. -3) at 5 standard deviations. The PIXIE mission complements anticipated ground-based polarization measurements such as CMBS4, providing a cosmic-variance-limited determination of the large-scale E-mode signal to measure the optical depth, constrain models of reionization, and provide a firm detection of the neutrino mass (the last unknown parameter in the Standard Model of particle physics). In addition, PIXIE will measure the absolute frequency spectrum to characterize deviations from a blackbody with sensitivity 3 orders of magnitude beyond the seminal COBE/FIRAS limits. The sky cannot be black at this level; the expected results will constrain physical processes ranging from inflation to the nature of the first stars and the physical conditions within the interstellar medium of the Galaxy. We describe the PIXIE instrument and mission architecture required to measure the CMB to the limits imposed by astrophysical foregrounds.

Kogut, Alan↗

Single-shot local measurement of terahertz correlated second harmonic generation in laser air plasma filaments

We present a single-shot detection method of terahertz correlated second harmonic generation in plasma-based sources by directly mixing an optical probe into femtosecond laser-induced plasma filaments in air. Furthermore, the single-shot second harmonic trace is obtained by measuring a second harmonic generation on a conventional CCD with a spatiotemporally distorted probe beam. The system shows a spectrometer resolution of 22 fs/pixel on the CCD and a true resolution on the order of the probe pulse duration. With considerable THz peak electric field strength, this formalism can open the door to single-shot THz detection without bandwidth limitations.

47 OTHER INSTRUMENTATION↗

Air- and oxygen-broadening coefficients for the O2 rotational line at 60.46/cm

Using an NBS laser-based tunable far-infrared spectrometer, the air- and oxygen-broadening coefficients were measured for the J = 10 - 10, N = 11 - 9 O2 rotational transition at 60.46/cm (1.812 THz). The air-broadening coefficient is 5.04 + or - 0.38 x 10 to the -7th/cm/Pa (0.0511 + or - 0.0039/cm/atm) (HWHM) at 245 + or - 6 K; the oxygen-broadening coefficient is 4.92 + or - 0.47 x 10 to the -7th/cm/Pa (0.0499 + or - 0.0048/cm/atm) (HWHM) at 259 + or -2 K. These direct experimental measurements of the air-broadening coefficient should improve the accuracy of retrieval calculations for far-infrared stratospheric balloon experiments which use O2 rotational lines to calibrate the viewing geometry.

Jennings, D. A.↗

Submillimeter Laboratory Investigations: Spectroscopy and Collisions

Currently, millimeter-wave and submillimeter-wave spectroscopy is conducted in our laboratory on several different types of spectrometers. Our standard spectrometer utilizes the output of a phase-locked klystron operating in the 40-60 GHz region, which is sent into a crossed-waveguide harmonic generator, or "multiplier". The high frequency millimeter-and submillimeter-wave radiation is transmitted via quasi-optical techniques through an absorption cell and then onto a detector, which is either an InSb hot electron bolometer cooled to 1.4 K or a Si bolometer cooled to 0.3 K. The detector response is sent to a computer for measurement and analysis. The frequency range produced and detected in this manner goes from 80 GHz to upwards of 1 THz. Spectra are normally taken with source modulation, with line frequencies typically measured to an accuracy of 50-100 kHz. Higher accuracy is available when needed. Recently, we developed a new, broad-band spectrometer in our laboratory based on a free-running backward wave oscillator (BWO) of Russian manufacture as the primary source of radiation. The so-called FASSST (fast-scan submillimeter spectroscopic technique) system uses fast-scan and optical calibration methods rather than the traditional locking techniques. The output power from the BWO is split such that 90% goes into the absorption cell while 10% is coupled to a 40-meter Fabry-Perot cavity, which yields fringe? for frequency measurement. Results from this spectrometer on the spectrum of nitric acid (HNO3) show that 100 GHz of spectral data can be obtained in 5 seconds with a measurement accuracy of 50 kHz. Currently, the frequency range of the FASSST system in our laboratory is roughly 100-700 GHz.

Herbst, Eric↗

Pulse Length Monitor for Breakdown Diagnostics in THz and Mm-Wave Accelerators

The development of novel high-gradient accelerating structures operating at THz frequencies is critical for future free-electron lasers and TeV scale linear colliders. To reach high energies with reasonable length requires high accelerating gradients of ~100 MV/m. The main limitation to reaching these high-energy gradients is the vacuum RF breakdown phenomenon, which disrupts normal accelerator operations. For stable operations and to understand the breakdown microscopic dynamics, a new device capable of detecting the breakdown occurrences is required. In this paper, we provide the design of a pulse length monitor based on an analog to digital converter for fast signal digitization without the need to use high-speed digitizers to be used in a commercial mm-wave heterodyne spectrometer.

43 PARTICLE ACCELERATORS↗

A 2 THz Heterodyne Array Receiver for SOFIA: Summary of Research

We proposed to perform a comprehensive design study of a 16-element heterodyne array receiver for SOFIA. The array was designed to utilize hot-electron bolometers in an efficient, low-cost waveguide mount to achieve low noise performance between approximately 1500 and 2400 GHz. Due to the prevailing physical conditions in the interstellar medium, this frequency range is one of the richest in the Far-Infra Red (FIR) portion of the spectrum. An array designed for this wavelength range will make excellent use of the telescope and the available atmospheric transmission, and will provide a new perspective on stellar, chemical, and galactic evolution in the present as well as past epochs. A few of the most important molecular and atomic species which the instrument will sample are CII, OI, CO, OH, NII, and CH. The system used the most sensitive detectors available in an efficient optical system. The local oscillator was a compact CO2 pumped far-infrared laser currently under development for SOFIA. The backend spectrometer was an array acousto-optic spectrometer (aAOS). The spectrometer utilizes proven hardware and technologies to provide broadband performance (> 1 GHz per AOS channel) and high spectral resolution (1 MHz) with the maximum sensitivity and minimum complexity and cost. The proposed instrument would be the fastest and most sensitive heterodyne receiver ever to operate in the 1.5 - 2.4 THz band. One of the key technologies developed for the proposed instrument is the laser micromachining of waveguide structures. These structures provide both the optical link between the instrument and the telescope (via an array of efficient feedhorns) and the impedance transformation between the detectors and free space. With the assistance of funds provided from this grant, we were able to fabricate and test the world's first laser micromachined feedhorns. The quality of the waveguide structure is far better than that obtainable using any other fabrication technique. The beam parameters are an excellent match to what is expected from theory. The success of this experiment demonstrates the viability of using laser micromachined components in the development of high performance, large format array receivers. We have continued our development efforts and will propose to build a 2nd generation instrument based on the same design concepts.

Walker, Christopher K.↗

A 2 THz Heterodyne Array Receiver for SOFIA

We proposed to perform a comprehensive design study of a 16-element heterodyne array receiver for SOFIA. The array was designed to utilize hot-electron bolometers in an efficient, low-cost waveguide mount to achieve low noise performance between approx. 1500 and 2400 GHz. Due to the prevailing physical conditions in the interstellar medium, this frequency range is one of the richest in the FIR portion of the spectrum. An array designed for this wavelength range will make excellent use of the telescope and the available atmospheric transmission, and will provide a new perspective on stellar, chemical, and galaxy evolution in the present as well as past epochs. A few of the most important molecular and atomic species which the instrument will sample are CII, OI, CO, OH, NII, and CH. The system used the most sensitive detectors available in an efficient optical system. The local oscillator was a compact CO2 pumped far-infrared laser currently under development for SOFIA. The backend spectrometer was an array acousto-optic spectrometer (aAOS). The spectrometer utilizes proven hardware and technologies to provide broadband performance (greater than or equal to 1 GHz per AOS channel) and high spectral resolution (1 MHz) with the maximum sensitivity and minimum complexity and cost. The proposed instrument would be the fastest and most sensitive heterodyne receiver ever to operate in the 1.5 - 2.4 THz band. One of the key technologies developed for the proposed instrument is the laser micromachining of waveguide structures. These structures provide both the optical link between the instrument and the telescope (via an array of efficient feedhorns) and the impedance transformation between the detectors and free space. With the assistance of funds provided from this grant, we were able to fabricate and test the world's first laser micromachined feedhorns. Figure 1 is a photograph of the 2 THz double feedhorn structure designed and constructed under the auspices of this grant. The quality of the waveguide structure is far better than that obtainable using any other fabrication technique. Figure 2 is the beam pattern obtained from it. The beam parameters are an excellent match to what is expected from theory. The success of this experiment demonstrates the viability of using laser micromachined components in the development of high performance, large format array receivers. Figure 3 is an illustration of the focal plane array concept we developed lot- SOFIA base upon this work. Un- fortunately, our instrument proposal was not selected as a first generation SOFIA instrument. However, we have continued our development efforts and will propose to build a 2nd generation instrument based on the same design concepts. Our work under NAG 2-1057 was very rewarding and we appreciate the opportunity provided by NASA to pursue it. The research led directly to publishing 2 papers (listed below) and the award of an NSF grant to our group to construct a laser micromachining system on the University of Arizona campus.

Walker, Christopher K.↗

A 2 THz Heterodyne Array Receiver for SOFIA

We proposed to perform a comprehensive design study of a 16-element heterodyne array receiver for SOFIA. The array was designed to utilize hot-electron bolometers in an efficient, low-cost waveguide mount to achieve low noise performance between approx. 1500 and 2400 GHz. Due to the prevailing physical conditions in the interstellar medium, this frequency range is one of the richest in the FIR portion of the spectrum. An array designed for this wavelength range will make excellent use of the telescope and the available atmospheric transmission, and will provide a new perspective on stellar, chemical, and galaxy evolution in the present as well as past epochs. A few of the most important molecular and atomic species which the instrument will sample are CII, OI, CO, OH, NII, and CH. The system used the most sensitive detectors available in an efficient optical system. The local oscillator was a compact CO2 pumped far-infrared laser currently under development for SOFIA. The backend spectrometer was an array acousto-optic spectrometer (aAOS). The spectrometer utilizes proven hardware and technologies to provide broadband performance (greater than or equal to 1 GHz per AOS channel) and high spectral resolution (1 MHz) with the maximum sensitivity and minimum complexity and cost. The proposed instrument would be the fastest and most sensitive heterodyne receiver ever to operate in the 1.5 - 2.4 THz band. One of the key technologies developed for the proposed instrument is the laser micromachining of waveguide structures. These structures provide both the optical link between the instrument and the telescope (via an array of efficient feedhorns) and the impedance transformation between the detectors and free space. With the assistance of funds provided from this grant, we were able to fabricate and test the world's first laser micromachined feedhorns. The quality of the waveguide structure is far better than that obtainable using any other fabrication technique. The beam parameters are an excellent match to what is expected from theory. The success of this experiment demonstrates the viability of using laser micromachined components in the development of high performance, large format array receivers. Unfortunately, our instrument proposal was not selected as a first generation SOFIA instrument. However, we have continued our development efforts and will propose to build a 2nd generation instrument based on the same design concepts.

Walker, Christopher K.↗

Hot-Electron Bolometer Mixers on Silicon-on-Insulator Substrates for Terahertz Frequencies

A terahertz Hot-Electron Bolometer (HEB) mixer design using device substrates based on Silicon-On-Insulator (SOI) technology is described. This substrate technology allows very thin chips (6 pm) with almost arbitrary shape to be manufactured, so that they can be tightly fitted into a waveguide structure and operated at very high frequencies with only low risk for power leakages and resonance modes. The NbTiN-based bolometers are contacted by gold beam-leads, while other beamleads are used to hold the chip in place in the waveguide test fixture. The initial tests yielded an equivalent receiver noise temperature of 3460 K double-sideband at a local oscillator frequency of 1.462 THz and an intermediate frequency of 1.4 GHz.

bolometer↗

Terahertz EPR spectroscopy using a 36-tesla high-homogeneity series-connected hybrid magnet

Electron Paramagnetic Resonance (EPR) is a powerful technique to study materials and biological samples on an atomic scale. High-field EPR in particular enables extracting very small g-anisotropies in organic radicals and half-filled 3d and 4f metal ions such as Mn II (3d 5 ) or Gd III (4f 7 ), and resolving EPR signals from unpaired spins with very close g-values, both of which provide high-resolution details of the local atomic environment. Before the recent commissioning of the high-homogeneity Series Connected Hybrid magnet (SCH, superconducting + resistive) at the National High Magnetic Field Laboratory (NHMFL), the highest-field, high-resolution EPR spectrometer available was limited to 25 T using a purely resistive “Keck” magnet at the NHMFL. Herein, we report the first EPR experiments performed using the SCH magnet capable of reaching the field of 36 T, corresponding to an EPR frequency of 1 THz for g = 2. The magnet’s intrinsic homogeneity (25 ppm, that is 0.9 mT at 36 T over 1 cm diameter, 1 cm length cylinder) was previously established by NMR. We characterized the magnet’s temporal stability (5 ppm, which is 0.2 mT at 36 T over one-minute, the typical acquisition time) using 2,2-diphenyl-1-picrylhydrazyl (DPPH). This high resolution enables resolving the weak g-anisotropy of 1,3-bis(diphenylene)-2-phenylallyl (BDPA), Δg = 2.5 × 10 –4 obtained from measurements at 932 GHz and 33 T. Subsequently, we recorded EPR spectra at multiple frequencies for two Gd III complexes with potential applications as spin labels. In conclusion, we demonstrated a significant reduction in line broadening in Gd[DTPA], attributed to second order zero field splitting, and a resolution enhancement of g-tensor anisotropy for Gd[sTPATCN]-SL.

47 OTHER INSTRUMENTATION↗

Pressure broadening of the 2.5 THz H(Cl-35) rotational line by N2 and O2

The pressure broadening coefficients of the 2.5 THz rotation line of H(Cl-35) by N2 and O2, for application to the analysis of far infrared spectra obtained in the stratosphere, are reported. The broadening coefficients were measured in absorption at 296 and 201 L using a tunable far infrared spectrometer. Results show that at room temperature the N2 pressure broadening coefficient is 0.0669 +/-0.0021/cm/atm (2sigma), with a temperature coefficient B of 0.58 +/-0.02.

Park, K.↗

All-Solid-State 2.45-to-2.78-THz Source

Sources in the THz range are required in order for NASA to implement heterodyne instruments in this frequency range. The source that has been demonstrated here will be used for an instrument on the SOFIA platform as well as for upcoming astrophysics missions. There are currently no electronic sources in the 2 3- THz frequency range. An electronically tunable compact source in this frequency range is needed for lab spectroscopy as well as for compact space-deployable heterodyne receivers. This solution for obtaining useful power levels in the 2 3- THz range is based on utilizing power-combined multiplier stages. Utilizing power combining, the input power can be distributed between different multiplier chips and then recombined after the frequency multiplication. A continuous wave (CW) coherent source covering 2.48 2.75 THz, with greater than 10 percent instantaneous and tuning bandwidth, and having l 14 W of output power at room temperature, has been demonstrated. This source is based on a 91.8 101.8-GHz synthesizer followed by a power amplifier and three cascaded frequency triplers. It demonstrates that purely electronic solid-state sources can generate a useful amount of power in a region of the electromagnetic spectrum where lasers (solid-state or gas) were previously the only available coherent sources. The bandwidth, agility, and operability of this THz source has enabled wideband, high-resolution spectroscopic measurements of water, methanol, and carbon monoxide with a resolution and signal-to-noise ratio unmatched by other existing systems, providing new insight in the physics of these molecules. Further - more, the power and optical beam quality are high enough to observe the Lamb-dip effect in water. The source frequency has an absolute accuracy better than 1 part in 1012, and the spectrometer achieves sub-Doppler frequency resolution better than 1 part in 108. The harmonic purity is better than 25 dB. This source can serve as a local oscillator for a variety of heterodyne systems, and can be used as a method for precision control of more powerful but much less frequency-agile quantum mechanical terahertz sources.

Mehdi, Imran↗

Far-Infrared Heterodyne Spectrometer for Sofia

The project goal was to evaluate the scientific capabilities and technical requirements for a far-infrared heterodyne spectrometer suitable for the SOFIA Airborne Observatory, which is now being developed by NASA under contract to the Universities Space Research Association (USRA). The conclusions detailed below include our specific recommendations for astronomical observations, as well as our intended technical approach for reaching these scientific goals. These conclusions were presented to USRA in the form of a proposal to build this instrument. USRA subsequently awarded the University of Colorado a 3-year grant to develop the proposed Hot-Electron micro-Bolometer (HEB) mixer concept for high frequencies above 3 THz, as well as other semiconductor mixer technologies suitable for high sensitivity receivers in the 2-6 THz frequency band.

Betz, A. L.↗

Very High Current Density Nb/AlN/Nb Tunnel Junctions for Low-Noise Submillimeter Mixers

We have fabricated and tested submillimeter-wave superconductor-insulator-superconductor (SIS) mixers using very high current density Nb/AlN/Nb tunnel junctions (J(sub c) approximately equal 30 kA/sq cm) . The junctions have low resistance-area products (R(sub N)A approximately 5.6 Omega.sq micron), good subgap to normal resistance ratios R(sub sg)/R(sub N) approximately equal 10, and good run-to-run reproducibility. From Fourier transform spectrometer measurements, we infer that omega.R(sub N)C = 1 at 270 GHz. This is a factor of 2.5 improvement over what is generally available with Nb/AlO(x)/Nb junctions suitable for low-noise mixers. The AlN-barrier junctions are indeed capable of low-noise operation: we measure an uncorrected receiver noise temperature of T(sub RX) = 110 K (DSB) at 533 GHz for an unoptimized device. In addition to providing wider bandwidth operation at lower frequencies, the AlN-barrier junctions will considerably improve the performance of THz SIS mixers by reducing RF loss in the tuning circuits.

Kawamura, Jonathan↗

Far-Infrared Heterodyne Spectrometer for SOFIA

This report summarizes work done under NASA Grant NAG2-1062 awarded to the University of Colorado. The project goal was to evaluate the scientific capabilities and technical requirements for a far-infrared heterodyne spectrometer suitable for the SOFIA Airborne Observatory, which is now being developed by NASA under contract to the Universities Space Research Association (USRA). The conclusions detailed below include our specific recommendations for astronomical observations, as well as our intended technical approach for reaching these scientific goals. These conclusions were presented to USRA in the form of a proposal to build this instrument. USRA subsequently awarded the University of Colorado a 3-year grant (USRA 8500-98-010) to develop the proposed Hot-Electron micro-Bolometer (HEB) mixer concept for high frequencies above 3 THz, as well as other semiconductor mixer technologies suitable for high sensitivity receivers in the 2-6 THz frequency band.

Betz, A. L.↗