Noise temperature measurement of the receiver with a phase grating 

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Description of the different elements of a THz heterodyne receiver

The mixer is a non-linear device which mixes the LO and RF signals to down-convert the observed RF signal to a lower frequency, called intermediate frequency (IF). An antenna, connected to the mixer, is generally used to receive the RF and LO signals. This antenna can have a large frequency bandwidth (like log-spiral antennas), or be frequency selective (like twin-slot antennas or horns). Several non-linear devices are used as mixers. The three most common mixers used in THz heterodyne receivers are Schottky diodes, SIS (Superconductor-Insulator-Superconductor) junctions, and HEBs (Hot Electron Bolometer). These different kinds of mixers are presented below with their characteristics, frequency ranges, and appli-cations.

Down-conversion of the two RF side-bands

When the unbalanced standard mixer down-converts the RF spectrum, two side-bands, around the LO frequency, are down-converted. The lower side band (LSB), which is be-low the LO frequency, and the upper side band (USB), which is above the LO frequency, are both down-converted to the same IF band (figure 2.5).
When the two side bands are down-converted, they are superimposed in the IF band. Re-ceivers where both side bands are down-converted are called double-sideband (DSB) re-ceivers.

The pumping of mixers

Mixers are non-linear devices, which means that the relation between their current and voltage is non-linear. However, the relation between their input and output power is usually linear (when the input power level is not too high). They must receive enough LO power to be in a very sensitive state and efficiently mix the LO and RF signals to generate the IF signal. When they are in this state, we say that they are pumped. A mixer is correctly pumped when the conversion of the RF electromagnetic signal into the IF electrical signal is the most efficient. The pumping level has an influence on the sensitivity of the mixer. Different mixers, such as Schottky diodes, SIS junctions or HEBs do not require the same amount of LO power to be pumped. Sometimes, it can be problematic to find a LO which generates enough power to correctly pump a mixer, especially at frequencies above 1 THz.

The Schottky diode mixer

Schottky diode mixers can be used at frequencies from several GHz up to several THz, and have the huge advantage of being operational at room temperature. However, their sensi-tivity can be improved by cooling them down, where they reach their optimal performance around 20 K (cf. Chattopadhyay et al. [18]). However, they need to be pumped with a high LO power, in the range of hundreds of µW to a few mW, which is their main limita-tion. Schottky mixers can also be used as harmonic mixers, which means that they can mix the RF signal with harmonic multiples of the LO signal. The frequency of the IF signal can correspond to fI F = |k. f LO − l . fRF |, where k, l ∈ N. Schottky diode mixers have a large bandwidth of several GHz. The latest results obtained with Schottky diode mixers designed and manufactured at LERMA showed a DSB noise temperature of 870 K at 557 GHz at am-bient temperature. By cooling the mixer down to 134 K, this noise temperature was reduced by approximately 200 K (cf. Maestrini et al. [19] and Treuttel et al. [20]). At frequencies around 1 THz, harmonic Schottky diode mixers currently have a DSB noise temperature of 4000 K, at room temperature, as shown by the article from Thomas et al. [21]. A Schottky mixer has been used in the first heterodyne receiver above 1 THz (cf. Röser [3]). However, the next heterodyne receivers used cryogenic mixers, which have a lower noise temperature and require less LO power than Schottky mixers. Today, Schottky diode mixers are mainly used to analyze planets’ atmospheres, where we do not need the high sensitivity of SIS or HEB mixers. For such missions, their higher operating temperature is a big advantage because they do not require to be cooled down by cryogenic liquids, which evaporate with time. Pictures of a Schottky mixer circuit, and of a pair of Schottky diodes are shown in figure 2.6.

The SIS mixer

Superconductor-Insulator-Superconductor (SIS) junctions are very sensitive mixers at sub-millimeter wavelengths. However, as most SIS mixers use niobium or niobium nitride as superconducting material, they only work up to approximately 1.3 THz, twice the voltage gap of niobium. Practically, they are used as mixers for frequencies below 1 THz, where they are the most sensitive mixers. They have an excellent noise temperature (ie. 30 K at 100 GHz and 85 K at 500 GHz, cf. Carter et al. [22] and Chattopadhyay et al. [18]), and must be cooled down to approximately 4 K with liquid helium. Their bandwidth can be greater than 4 GHz and they need to be used with an LO which emits around 40 µW to 100 µW. As they offer the best sensitivity below 1 THz, they are used in nearly all sub-millimeter telescopes, such as ALMA [23] and NOEMA [24].

The HEB mixer

Hot Electron Bolometer (HEB) mixers are currently the most sensitive mixers for frequen-cies above 1.3 THz. They need to be cooled down to approximately 4 K and can reach a bandwidth of 3 or 4 GHz. They have a noise temperature better than Schottky mixers (ap-proximately 1200 K between 1.4 THz and 1.9 THz for HIFI [8]). They only require 1 or 2 µW of LO power to be pumped, a lot less than Schottky and SIS mixers. It enables them to be used with high frequency LO which only emit a few µW. They are a good alternative to Schottky diode mixers for high THz frequencies, when a high sensitivity is needed, or when there is not a lot of LO power available. All actual heterodyne receivers for astronomy above 1 THz use HEB mixers. HEB mixers have been used in HIFI [8] on the Herschel satellite, and are used on GREAT [10] and upGREAT [11], which operate from SOFIA airplane.

The Martin Puplett Interferometer

The Martin Puplett interferometer (MPI) is a diplexer which can superimpose the LO and RF signals with very little losses, for both signals. The functioning of the MPI is extensively described in chapter 4, because it is an important part of our 2.6 THz heterodyne receiver. The MPI is composed of two wire grids, G1 and G2, and two roof-top mirrors T1 and T2, as shown in figure 2.12. An ellipsoidal mirror (MLO) is added to focus the LO signal.
The MPI has already been used as diplexer in several major heterodyne receivers, such as GREAT [10] and CONDOR [5]. However, as it is a lot more difficult to align than a simple beam splitter, it is only used in THz heterodyne receivers for high frequencies, where there is little LO power available.

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The IF chain and the spectrometer

At the output of the mixer, the intermediate frequency (IF) signal needs to be amplified and filtered before being processed by a spectrometer. The first amplifier, just after the mixer, is usually a low noise cryogenic amplifier because it is important to add as little noise to the IF signal as possible. Then, ambient temperature low noise amplifiers (LNA) and a bandpass filter are often used to amplify further the IF signal and filter it. Finally, a spectrometer is used to analyze the down-converted spectrum of the IF signal. Because of recent technological developments, digital Fourier transform spectrometers (DFTS) have become the standard spectrometers for heterodyne receivers. A DFTS uses an analog to digital converter (ADC) card to digitize the input signal, and a FPGA to perform a fast Fourier transform (FFT) of the data, in real time. The spectral data can be directly transmitted to a computer. With the increasing speed of the FPGAs and ADC cards, DFTS are improving fast and some 5 GHz bandwidth DFTS are currently available (ie. the second generation of DFTS from Omnisys company).

Description of our 2.6 THz heterodyne receiver

During this PhD, I built, tested and improved a 2.6 THz heterodyne prototype receiver (fig-ure 2.14), whose elements are described below.
• The LO: I use a 2.6 THz frequency multiplier chain from VDI (Virginia diodes Inc.) which emits a maximum of 2 µW.
• The mixer: For our tests, we used a HEB using a log spiral antenna which was designed and produced at LERMA and LPN laboratories and works well for frequencies up to several THz (cf. Delorme et al. [28] and Lefèvre et al. [29]). It uses a NbN (niobium nitride) bridge on a silicon substrate and is phonon cooled. However, the final mixer will be a HEB with a twin-slot antenna optimized for 2.5 to 2.7 THz. In both cases, we add a silicon lens in front of the HEB to focus the signal.
• The mixer bias supply: The bias supply for the HEB has been manufactured at LERMA, according to the plans elaborated at SRON to build the bias supply for the HIFI instru-ment of the Herschel satellite.
• The diplexer: A Martin Puplett interferometer (MPI) is used as diplexer. I have specif-ically designed it for our 2.6 THz receiver and it is extensively described in chapter 4.

Table of contents :

Contents
Abbreviations
1 Introduction 
2 Terahertz heterodyne receivers 
2.1 Motivation
2.2 THz heterodyne receivers in astronomy
2.2.1 Main characteristics of heterodyne receivers
2.2.2 Overview of existing THz heterodyne receivers
2.3 General principle of heterodyne receivers
2.3.1 The heterodyne principle
2.3.2 Sensitivity of heterodyne receivers
2.4 Description of the different elements of a THz heterodyne receiver
2.4.1 The mixer
2.4.2 The local oscillator
2.4.3 The diplexer
2.4.4 The IF chain and the spectrometer
2.5 Our 2.6 THz heterodyne receiver
2.5.1 Description of our 2.6 THz heterodyne receiver
2.5.2 Main aspects of this PhD
3 Stability of the heterodyne receiver 
3.1 Introduction
3.1.1 Motivation
3.1.2 Influence of the noise on the optimal integration time
3.2 The Allan variance
3.2.1 Background and theory
3.2.2 Allan variance theory
3.2.3 Total power and spectral Allan variance
3.2.4 The calculation algorithm
3.3 Stability of our heterodyne receiver
3.3.1 Warm intermediate frequency chain and DFTS
3.3.2 Stability of the bias circuit and the cryogenic amplifier
3.3.3 Stability of the local oscillator and the HEB mixer
3.4 Conclusion
4 The Martin Puplett Interferometer (MPI) 
4.1 Motivation
4.2 Gaussian beam optics
4.2.1 Context and motivation
4.2.2 Electric field distribution of a Gaussian beam
4.2.3 Gaussian beam characteristics
4.2.4 Conclusion
4.3 Description of the MPI
4.3.1 Input of the MPI
4.3.2 Detailed description of the elements of the MPI
4.3.3 The rotation of the polarization in the MPI
4.3.4 The bandwidth of the MPI
4.4 Design of our MPI
4.4.1 Calculation of the ellipsoidal mirror (MLO)
4.4.2 Calculation of the grids’ required characteristics
4.5 Test and evaluation of each individual component of the MPI
4.5.1 The ellipsoidal mirror (MLO)
4.5.2 The polarizing grids
4.5.3 Efficiency of the roof-top mirrors
4.5.4 Air absorbance
4.6 Efficiency of the whole MPI
4.6.1 Presentation of the experiment
4.6.2 Different steps of the experiment
4.6.3 Conclusion
4.7 Conclusion
5 Phase gratings 
5.1 Background and theory
5.1.1 Motivation
5.1.2 Presentation of the phase gratings
5.2 The stepped phase gratings
5.2.1 Overview of the stepped phase gratings
5.2.2 Theory of Dammann gratings
5.2.3 Test of a transmissive Dammann grating
5.3 The Fourier grating
5.4 The Global phase grating
5.4.1 General presentation
5.4.2 Numerical calculation
5.4.3 Conversion of a phase profile into a grating’s surface
5.4.4 Electromagnetic simulations
5.5 Reflective and transmissive phase grating prototypes
5.5.1 Design considerations for the two prototypes
5.5.2 Numerical calculation
5.5.3 Design of the transmissive and reflective phase gratings
5.5.4 Electromagnetic simulations
5.5.5 Mechanical design
5.5.6 Geometrical measurements of the 2 prototypes
5.5.7 Electromagnetic simulation of the manufactured reflective grating
5.5.8 Test of the 2 prototypes
5.5.9 Noise temperature measurement of the receiver with a phase grating
5.6 Conclusion
6 Conclusion 
A Gaussian beam optics 
A.1 Theory of the Gaussian beam optics
A.1.1 The wave equation
A.1.2 The Helmholtz equation
A.1.3 The paraxial wave equation
A.1.4 The fundamental Gaussian mode equation
A.1.5 Expression of the beam parameter
A.1.6 The radius of curvature and the beam radius
A.1.7 The phase shift factor
A.1.8 Final expression of the fundamental Gaussian mode
A.1.9 Electric field distribution of a Gaussian beam
A.2 Gaussian beam characteristics
B Functioning of the Martin Puplett interferometer 
B.1 Rotation of the polarization of the RF signal in the MPI
B.2 Bandwidth of the MPI
B.3 Water vapor absorption
C Theory of the Dammann grating 
C.1 Approximation of the Maxwell’s equations
C.2 Detail of the phase modulation generated by the Dammann grating
C.3 Example of a 1×2 Dammann grating
Bibliography 

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