Table of Contents

Chapter 1: Introduction

Nearly all observational astronomy relies on electromagnetic waves. The radio and the submillimeter regions of the electromagnetic spectrum have become very important, beside the optical region. They allow the observation of the cold matter in the interstellar medium (ISM). The ISM has a great concentration of dust and gas. Such dust and gas form stars and planets. The emission of the elements which compose the ISM occurs in the submillimeter range (or THz range if we assume the 0.3THz-3THz band as the terahertz band). The observation of such emissions relies on bolometers or heterodyne receivers. Bolometers measure the power of the received light, and are wide-band. Instead, heterodyne receivers can easily achieve high spectral resolution. They are essential to detect the spectral line emissions and also absorption of the elements that compose the ISM. Water vapor is abundant in the earth atmosphere. Since H 2 O absorbs a large part of the radiation in the sub-millimeter and infrared range, see fig. 1.1.1, the THz region of the electromagnetic spectrum has been little explored for years. Recently few prototypes of heterodyne receivers have been developed to observe ISM emissions above 1 THz. They operate at very high altitudes, from air planes or space, where there is less or no water.

Several heterodyne receivers operating above 1 THz have been built for space and ground based telescopes, for example the heterodyne receiver of the Kuiper Airborne Observatory (KAO), of the Receiver Lab Telescope (RLT), the 1.4 THz CO N+ Deuterium Observations Receiver (CONDOR) for APEX, the Heterodyne Instrument for the Far-Infrared (HIFI) on Herschel, the German Receiver for Astronomy at Terahertz frequencies (GREAT/upGREAT) on SOFIA, and the THz receiver of the Stratospheric THz Observatory (STO).

The state of the art of the heterodyne detection consists largely of high spectral resolution for one or a few spatial pixels. There are very few heterodyne array receivers with many pixels up to now because they are very complex to build, as they essentially consist of many single pixel receivers.

The objective of my thesis is to allow the realization of heterodyne receivers with a great number (hundreds) of pixels. Such big arrays are intended to furnish 3D scans of the universe in the high frequency range. In order to do that it is necessary to radically rethink the structure of the heterodyne receiver. Bulk components such as double slot antennas on a lens, or horn antennas, that focus the astronomical signal onto the receiver cannot be used anymore since they require individually machining which make their realization very time consuming and expensive for big arrays. Instead we propose to use flat lenses for the receiver structure, which are much more interesting due to the possibility to build them for an entire array in a single process. The objective of this work is to search for ideas for flat elements that are easy to fabricate, to design them for the THz region and to investigate their properties by simulation and also by tests where possible.

This thesis is structured in 6 chapters. The first chapter explains the interest of heterodyne receivers for astronomy and briefly describes the heterodyne structure. The second chapter describes the various parts of heterodyne receivers, and introduces the state-of-the-art of current development.

The third chapter introduces the software used for the electromagnetic simulation of the tools developed in the thesis. Chapter 4 introduces the microstrip array of patch antennas, which is the first solution we found to substitute the horn or double slot antennas with flat lenses. It consists of an array of metallic rectangular antennas placed on a dielectric substrate which separates them from the ground plane. A microstip line network is used to feed each antenna. However the solution that has been found is quite challenging in terms of realization process, therefore, for the moment, we have opted for another option: the transmit-array. The transmit-array has been introduced in chapter 5, and represents another way to replace bulk lenses with flat components. It consists of an array of cells which provide a certain phase shift to an incoming wave to transform its planar front into a spherical one, with a certain focal point which depends on the design. My work focused in looking for mathematical models, in order to be able to design very big arrays without the necessity of EM optimizations, which are extremely time consuming. In chapter 6 introduces the zone plate. It is a particular kind of transmit-array where only two phase shifts of 0° and 180° are provided to an incoming wave, in order to create a constructive interference in the focal point. I show how I designed a zone-plate and simulate its focussing efficiency. In this case results show a good focusing effect but a high reflection coefficient of the incoming wave. Chapter 7 introduces the last tool developed in this work: the phase grating. A heterodyne receiver requires a reference signal, the Local Oscillator signal, to be send to each mixing element of the array. The phase grating is a beam divider which serves to split the incoming LO beam feed multiple elements of the array receiver.

For this application the most used phase gratings are the Dammann and the Fourier Gratings. The innovation of our approach consists in considering arbitrary electric field distributions as target field. In this way it is possible to design phase grating which split the LO beam into very close reflected beams, without loosing efficiency as it happens for Fourier and Dammann gratings. Chapter 2: Background

The heterodyne receiver

In the beginning of this chapter the various components of an heterodyne receiver are described in more details. The second half of this chapter gives an overview of heterodyne receiver arrays, i.e. special heterodyne receivers with several pixels. From the state of the art of arrays, I motivate the radically new approaches for heterodyne array receivers studied in my thesis. Heterodyne receivers in astronomy are tools which down-convert a high frequency input signal to a low frequency output signal, in our case from hundreds of GHz to a few GHz, see fig. 2.1.1. The signal from the sky is superimposed to an artificial, well-known, monochromatic signal generated by the local oscillator (LO), often with a beam splitter. In (sub)mm and THz receivers, the frequency of the signal from the LO is chosen to be close to the frequency of the sky signal. The mixer produces the beat frequency signal. It has a frequency equivalent to the difference between the LO and sky signal frequency. Thus the sky signal is translated to a lower frequency, which is easier to amplify and detect.

The diplexer

The diplexer (fig. 2.1.1.1) is necessary to superimpose the sky signal to the LO signal. Two kind of diplexers are used: the beam splitter and the Martin Puplett Interferometer (MPI). The beam splitter, which is generally composed of Mylar (polyethylene terephthalate), is the most common diplexer. It reflects part of the LO signal (~10%) and transmit most of the sky signal (~90%) to the mixer. The most of the LO signal is lost. This is a problem, especially at very high frequencies where the LO signal becomes too weak. The MPI can be used to overcome this problem, but it is more difficult to build and align, and has other constraints [START_REF] Defrance Fabien | Instrumentation of a terahertz heterodyne receiver[END_REF].

The focusing optics of the mixer

Usually, a horn antenna, or a lens and a double slot antenna, direct the radiation to the mixer. In the first case the signal is captured by the horn and sent to the mixer via a waveguide. In the second case the mixer is integrated inside the double slot antenna. The substrate is glued onto an elliptical lens(Feed Antenna of fig. 2.1.2.1(b)). A high dielectric medium is used for the lens, so that the field radiates upward the dielectric, and not downward.

The high frequency radiation from the sky and the LO arrives from the left of fig 2.1.2.1(b), and is collected towards the center of the double slot (in the center of fig. 2.1.2.1(a)), where there is the mixer. Such double slot antenna can reach a bandwidth of 16% of the central frequency.

The mixer

The mixer beats the sky and the LO signal and creates an intermediate frequency (IF) signal at | ν LO -ν sky | . The first mixer which has been used in heterodyne detection is the Schottky diode mixer, since it has the big advantage of being operating at room temperature. However it reaches the maximum sensitivity around 20K. It is operating from GHz frequencies up to several THz. The main disadvantage is that they need to be pumped with high LO power, from 100μW up to few mW Another big disadvantage is their high noise temperature, which can reach 4000K at 1THz. Even if it has a very large bandwidth, these limitations make the Schottky diode mixers inadequate to be used to general purpose applications, where a high sensitivity is needed. The state of the art of the mixers in the 0.3-1 THz range is the near quantum limited superconductor-insulator-superconductor (SIS) device, which is the most sensitive mixer. It has a very low noise temperature (of the order of 30K to 85K), but must be cooled down to 4K by liquid helium, which is really expensive. Besides the higher sensitivity, another advantage of the SIS mixer with respect to the Schottky diode is the lower LO power required, which is now of the order of 40μW to 100μW. Since the most of the SIS mixers are made of Niobium, they are frequency limited to 1.3 THz (twice the voltage gap of Niobium). Beyond the 1.3 THz SIS frequency limit, HEB mixers are used. Indeed, the Hot Electron Bolometer (HEB) is the most sensitive mixer operating above 1.3 THz. It is a cryogenic tool and needs to be cooled down to ≈ 4 K . It has a bandwidth of 3-4 GHz [START_REF] Defrance Fabien | Instrumentation of a terahertz heterodyne receiver[END_REF] with a noise temperature about 1200K between 1.4THz-1.9THz. A big advantage of the HEB is the need of a very low power from the LO to be pumped, which is about 1-2 μW.

As I designed HEBs into my patch antenna array, I will give a brief description: the HEB is a particular kind of bolometer. It consists of a 200 nm long NbN bridge located between two gold electrodes. It works at cryogenic temperature when the electrons and the cooper pairs of the bridge are thermally decoupled from the phonons. Therefore the HEB can be seen as a resistance between the two electrodes. When the radiation is absorbed by the electrons and the Cooper pairs, they become hot (hot electrons), and the resistance of the bolometer increase. Such resistance acts like a thermometer. Hot electrons can diffuse out of the bolometer into the electrodes when they are close enough. This happens on the time scale of the diffusion time τ diff . The electrons can also scatter with the lattice of the bolometer and cool down by emission of phonons into the substrate which happens on the scale of the escape time τ esc . Detectors can thus be separated into two categories, diffusion cooled (τ diff .<τ esc ) and phonon cooled (τ diff .>τ esc ) bolometers. When both the RF and the LO signal are received, the resistance oscillates with a frequency equal to the difference of the two incoming frequencies. Thus the beating of the HEB resistance generates the Intermediate Frequency (IF) signal, which is then amplified and detected via a spectrometer [START_REF] Cherednichenko | Terahertz superconducting hot-electron bolometer mixers[END_REF][START_REF] Wilms | Hotspot mixing: A framework for heterodyne mixing in superconducting hot-electron bolometers[END_REF][START_REF] Gao | Terahertz Superconducting Hot Electron Bolometer Heterodyne Receivers[END_REF][START_REF] Kollberg Erik | Impedance of Hot-Electron Bolometer Mixers at Terahertz Frequencies[END_REF][START_REF] Lobanov Yury | Heterodyne detection at near-infrared wavelengths with a superconducting NbN hotelectron bolometer mixer[END_REF][START_REF] Semenov | Hot-electron effect in superconductors and its applications for radiation sensors[END_REF][START_REF] Stockhausen | Optimization of Hot-Electron Bolometers for THz Radiation[END_REF].

The Local Oscillator

The LO (fig 2.1.4.1) must produce a high frequency well known monochromatic signal which has to be superimposed to the RF signal from the sky. The frequency multiplier chain is the most widely used for frequencies below 3 THz. A multiplier chain consists of several multipliers (and usually at least one amplifier) that frequency multiply an input wave at few GHz to the required THz frequency. It can operate at room temperature and is very stable. Horns at the output generate a linearly polarized Gaussian beam. It is possible to tune the frequency by 10 to 30% so that the heterodyne receiver can be used at different frequencies.

However every multiplier step losses some power, so that at the end a very low power is provided.

It is therefore very difficult to build multiplier chains that emit enough power above 3 THz and an alternative must be found. The Quantum Cascade Laser is generally used at these frequencies. The QCL is a semiconductor laser which emits radiation in the mid and far infrared range.

Conventionally, superconductor lasers emit radiation by a single photon passing from the conduction band to the valence band, where it is recombined with a hole. Instead the QCL does not contain bulk semiconductor, but is composed of thin varying-material layers which form a superlattice. The superlattice produces a certain potential along its length, which is traduced in a varying probability for the electron to occupy a certain position in the material. In this way each electron performs a series of quantum leaps of less energy, instead of a single jump, allowing in this way a reduction of the frequency at which the radiation is emitted. The variation of the thickness of the single layers allows the control of the frequency. In general, QCLs need to be cooled down to 10-70K, are very unstable, need to be phase locked and produce a beam which is not very Gaussian.

Due to these limitations, a great interest is currently addressed to increase the QCL performances [START_REF] Defrance Fabien | Instrumentation of a terahertz heterodyne receiver[END_REF]

The beam divider

In array receivers the reference signal from the LO needs to be send to each mixer. One could simply have one LO per pixel, but as this is very expensive, one usually uses a single LO and divides the beam into many beams, one for each mixer. Two kinds of beam dividers are currently in use on heterodyne receivers to split an incoming beam into different outgoing beams, which are the splitting waveguide, and the phase grating. In particular we are focusing on the last one (fig 2.1.5.1) which is the most promising one since it presents characteristics which can be easily improved. It consists of a grooved surface which modifies the phase profile of an incoming wave to generate a constructive interference, and give rise to a desired field pattern. Generally Dammann[10] and Fourier [START_REF] Graf Urs | STAR: SOFIA Terahertz Array Receiver[END_REF] gratings are used as beam dividers. In chapter 7, we will describe a novel design aiming to have small opening angles in order to feed very close spaced receivers.

State of the art of the heterodyne array receivers

With the turn of the century the first heterodyne receiver arrays have been developed, based on SIS mixers, with a growing attention to increasing frequency. A very nice description is given by Graf and Goldsmith [START_REF] Graf Urs | Terahertz Heterodyne Array Receivers for Astronomy[END_REF]. Here we shortly summarize a few of them • CHAMP/CHAMP+ CHAMP was an array of 4X4 pixel operating in the 480GHz atmospheric window. It has been installed at the Caltech Submillimiter Observatory on Mauna Kea, Hawaii in 1999. The array was composed of two subarrays of 8 pixels each with two different polarizations. The whole optic setup of CHAMP was cooled down to 15K to reduce the thermal noise. CHAMP+ is an upgrade of CHAMP operating in the 660GHZ and 850 GHz atmospheric windows, composed of two subarrays of 7 pixels each, and maintaining the same optomechanical design [START_REF] Graf Urs | Terahertz Heterodyne Array Receivers for Astronomy[END_REF].

• HERA HERA is a 3X3 pixel array operating at the frequency range between 215 and 270GHz. It has been installed in 2001 at the IRAM 30m telescope on Pico Veleta/Spain. Each pixel is covered by two mixers, one for each polarization. The detectors are tunable single SIS mixers. The signal from the LO is distributed to the pixels after splitting via a waveguide splitter. It has been the first detector to use a K-mirror image rotator to correct the optical field rotation [START_REF] Graf Urs | Terahertz Heterodyne Array Receivers for Astronomy[END_REF].

• SMART SMART presents a 2X4 pixel receiver operating at the two frequency bands of 460-490GHz and 800-880GHz in two polarizations. It has been first installed on the KOSMA 3 m telescope on Gornergrat/Switzerland and then moved to the NANTEN2 telescope on Pampa la Bola/Chile. It uses a single ended SIS mixer. The LO signal is coupled to the mixers via a diplexer. It has been the first pioneering instrument to use a Fourier phase grating to split the LO signal towards the mixers. A K-mirror type image rotator performs the image derotation [START_REF] Graf Urs | Terahertz Heterodyne Array Receivers for Astronomy[END_REF].

• HARP HARP consists of a 4X4 pixel array and works in the 325-375GHz band. It was installed at the James Clerk Maxwell Telescope on Mauna Kea/Hawaii in 2005. An interferometer is used for single sideband filtering. 16 beam splitters are used to distribute the LO signal to the mixers. They work by extracting directly the signal from the LO and sending that to the mixers [START_REF] Graf Urs | Terahertz Heterodyne Array Receivers for Astronomy[END_REF].

• SuperCam SuperCam is the first big array with more than 50 pixels. It consists of an array of 8X8 pixels. It is located at the Heinrich Hertz Telescope on Mount Graham/Arizona. Instead of using single individual mixer block, the focal plane consists of 8 monolithic rows containing 8 mixers each. The LO signal is coupled to the mixers by a Mylar beamsplitter. It is split towards the 8X8 pixel array by waveguide splitters [START_REF] Graf Urs | Terahertz Heterodyne Array Receivers for Astronomy[END_REF].

• upGREAT upGREAT is one of the first array receivers above 1THz. It has been flying on SOFIA for 1 year. It presents two arrays which operate at two different frequency ranges.

The low frequency range covering the 1.9-2.5 THz band with a 14 pixels array made of 2X7 hexagonal subarrays with two orthogonal polarizations, and the high frequency range of 4.745 THz with an array of 7 pixels.

All the arrays described here have only a few pixels to a few tens of pixels. Basically these arrays are mostly single pixel receivers packed close together into one array. For large arrays of hundreds of pixels this approach will become too labour intensive and expensive. My thesis investigates methods to radically simplify the design and lay the ground for very large (100-1000 pixel) arrays.

Simulation tools

The simulation softwares

All the mathematical models of this work are implemented in matlab. The designed projects are then exported in FEKO to be electromagnetically simulated. HFSS software is sometimes used as an aid.

Actually FEKO and HFSS are two electromagnetic simulation sofwares. FEKO is based on the Method of Moments (MoM) which is very good in simulating planar surfaces on a dielectric where the surface-over-volume ratio is small, indeed the MoM method is based on boundary conditions so that for example a volume is only defined by its surface contour, and the memory required by the meshing is reduced. Instead HFSS is based on a Finite Elements analysis (FE), which is better if we deal with arbitrary shaped objects. Since my work principally focuses on the necessity to substitute lens and horn antennas with planar receivers, FEKO is the fastest and most suited software to use. A brief description of the MoM method will follow in the next paragraph.

Optimization algorithms

The optimization process is a procedure to find a solution to a complex problem, which depends on many variables. A given solution is a set of variables that minimize a certain cost function. There are many algorithms which can be used to find a solution to a given problem. In this work we extensively make use of two main algorithms: the Genetic Algorithm (GA), and the Particle Swarm Optimization (PSO). The GA is very good in finding the optimal solution of the problem, but is very slow, while the PSO is faster but less accurate. Such algorithms are included in the FEKO software, which mixes them to achieve a better performance in terms of time required to get the solution, and quality of the result. Let's introduce first the Genetic Algorithm.

The Genetic Algorithm makes use of the Darwin's evolution theory to find the solution of a given problem. Each solution is considered as an individual whose genotype is given by its variables. A fitness function is assigned to any solution, and represents the probability for a given individual to reproduce. The higher values of the fitness function are assigned to the individuals who best match the requirements of the problem. Let's summarize the various steps of the GA [START_REF] Mitchell | An Introduction to Genetic Algorithms[END_REF]:

• A population of thousands of solutions/individuals is generated by assigning random values to the variables of each individual. The aim is to obtain a certain variety of the genotype with different values of the fitness function.

• After the fitness function has been calculated, a certain number of individuals are selected in pairs as parents to mate and to generate the offspring. The selection process is stochastic, where higher probabilities of reproduction are assigned to the individual with the highest fitness function.

• The genotype of the two parents is mixed by means of the crossover between the variables, as it happens in nature.

• Genetic mutation is necessary at this point to assure enough genomic variety in the population, and to avoid premature convergence to a non-optimal solution.

• Termination: the procedure is repeated until the difference between the precedent population and the current population is not significant. Other termination criteria can also be used such as a certain limit in the number of the generations, or until an acceptable value of the fitness function is reached.

Another algorithm which has been extensively used in this work is the Particle Swarm Optimization (PSO). It is an algorithm based on the observation of the movements of flock of birds. Let's consider a population of random solution, like for the GA. Each solution can be considered as a bird looking for food. Let's suppose that no bird knows where the food is. At every step each bird know what has been the best position it has ever assumed (i.e. the closest to the food), and also know the position of the bird which is closest to the food. The best chance to find some food is therefore to follow the movements of the bird which is closest to the food. Let's now explain the method more in detail: at every instant each solution has a set of variables which define its position, x i , and its velocity, v i . A fitness function is assigned to every solution, and depends on the distance of the solution from the objective. At each successive step the speed is modified according to the following formula [START_REF] Kennedy | Particle Swarm Optimization[END_REF]:

⃗ v i t + 1 =w ⃗ v i t + φ 1 ⃗ U 1 t ( ⃗ b i t -⃗ x i t )+ φ 2 ⃗ U 2 t ( ⃗ l i t -⃗ x i t ) ⃗ x i t + 1 =⃗ x i t + ⃗ v i t + 1 2.3.2.1
where ⃗ v i t + 1 is the speed of the particle i at the step t + 1 , w is defined as the inertia weight, φ 1 and φ 2 are two parameters called acceleration coefficients, ⃗ U 1 t and ⃗ U 2 t are two diagonal matrices whose element are random values between 0 and 1, ⃗ b i t is the best position (i.e. the position with the highest fitness function) that the particle i has ever reached in its movements, ⃗ l i t is the best position ever found by any particle in the solution domain, and ⃗ x i t is the actual position of the solution. The first term represents the inertia or momentum of the particle i, and is needed to avoid the solution to drastically change its direction. The second term is the cognitive component of the particle, and represent the tendency of the individual to return to its best solution. The third term represents the social component, i.e. the group norm or standard that should be attained. The calculated speed is used to update the position of the particle, as shown by the second line of eq.

3.2.1.

The best solution must be found inside the solution domain Θ . There are various methods to avoid the solution to go outside Θ . The least disruptive mechanism is to avoid the b i t term of eq. 3.2.1 to be updated outside the solution domain.

Chapter 3: The array of patch antennas

The receiver of the mixer

The chapter 2 introduced the various parts of a heterodyne receiver. In this chapter we are going to see in detail the antenna receiver. It usually consists of a horn antenna or a double slot antenna with a lens. Since we are aiming to an ease of fabrication for integration into arrays, we want to replace horns and lens with planar antennas. Since the antennas must be small (of the order of the micron to resonante at THz frequency), while the directivity increases with the size, it is necessary to use an array of antennas. The idea is to obtain a planar antenna array for each pixel of the heterodyne. In the following paragraphs I will introduce my work in the field of planar antenna arrays to be used for the heterodyne receiver at 600GHz. Some arrays exist which have been realized to be tested at lower frequencies (e.g. the 8X4 series fed array at 60GHz by M.S. Rabbani [42]). Rabbani's paper also provides some results of the 8X4 array rescaled at 100GHz, but no test has been carried out.

The main issue in realizing big arrays at high frequencies is due to the conductor losses. The highest frequency which has been simulated is 600GHz for a 2X2 array described in a paper from Ali Azarbar. In our case the conductor losses are not a big limitation, since we can dispose of cryogenic superconductor, such as the NbN, which can be cooled down without problems. To the author's knowledge, the design proposed in the following paragraphs represents the first attempt of a big array (16X16 elements) to be used for THz frequencies.

The array of planar antennas

The first and simplest receiving array which will be illustrated in this work is the planar antenna array. Such array is composed of many microstrip antennas connected together via a feed-line network (cf.

Rectangular patch

Every antenna can be seen as an equivalent circuit. The power which is radiated by an antenna is equivalent to the power dissipated by its equivalent impedance: Z L . Given Z s as the impedance of the feed port of the antenna, the maximum power is radiated when Z s =Z L * . The equivalence theorem is used, so that we consider only the radiating antenna case. For a preliminary analysis it is possible to assume Z s =50 Ω . Since the input impedance is assumed to be real, and since Z s =Z L * , the imaginary part of Z L must be zero. This condition also correspond to the resonant case, if the derivative of the imaginary part of the impedance, ∂ Z L ∂ f , is positive, with respect to the frequency.

The rectangular patch microstrip antenna is the simplest to realize and analyze. It consist of a rectangular conducting patch on a dielectric substrate. For a complete design, it is necessary to determine the width W of the patch, and its length L. An experimentally valid approximation for an efficient radiator is:

W = 1 2 f r √ μ 0 ϵ 0 √ 2 ϵ r + 1 (3.2.1.1)[17]
Due to the fringing effect, we introduce an equivalent dielectric constant for the microstrip line and the patch on the substrate.

It is given by:

W / h>1 ϵ reff = ϵ r + 1 2 + ϵ r -1 2 [ 1+ 12 h W ] -1 /2 (3.2.1.2)[17]
The effective length of the patch is extended by a value Δ L . Δ L is therefore a function of

ϵ reff : Δ L h =0,412 (ϵ reff + 0,3) ( W h + 0,264 ) (ϵ reff -0,258) ( W h + 0,8 ) (3.2.1.3)[17]
The effective length is the sum of the real length of the patch, and two times Δ L on both the sides of the patch: [START_REF] Balanis | Antenna Theory, Analysis and design[END_REF] The imaginary part of the impedance of the antenna must be zero, so the structure must be resonant:

L eff =L+ 2 Δ L (3.2.1.4)
λ 2 =L eff (3.2.1.5)
The impedance of the patch antenna, obtained by imposing the condition 3.2.1.5, is real because of the resonance, but it is not necessary equal to the characteristic impedance of the feeding network:

Z i . Since it is desirable to have a good matching, we want to modify the antenna impedance. To do so, it is necessary to choose an appropriate feeding point. Before that it is necessary to determine the impedance Z L of the antenna. It is given by:

Z i n = 1 Y i n = 1 2( G 1 +G 12 )
(3.2.1.6) [START_REF] Balanis | Antenna Theory, Analysis and design[END_REF] With:

G 1 = I 1 120 π 2 (3.2.1.7)[17] I 1 = ∫ 0 π [ sin ( k 0 W 2 cos θ ) cos(θ) ] 2 sin 3 (θ) d θ (3.2.1.8)[17] G 12 = 1 120π 2 ∫ 0 π [ sin ( k 0 W 2 cos(θ) ) cos(θ) ] 2 J 0 ( k 0 L sin θ)sin 3 θ d θ (3.2.1.9)[17]
where J 0 is the Bessel function of the first kind of order zero.

The input impedance decreases when the patch is fed closer to the center. It has been shown that the resonant input resistance can be changed by using an inset feed, recessed a distance y 0 from its edge. This technique can be used effectively to match the patch antenna using a microstrip-line feed. The input resistance for the inset feed is given approximately by:

R i n ( y= y 0 )= 1 2(G 1 +G 12 ) cos 2 ( π L y 0 ) =R i n ( y =0)cos 2 ( π L y 0 ) (3.2.1.10)[17]

Arrays of antennas

Usually the radiation pattern of the single patch element is relatively wide, and each element provides low directivity. For our applications it is necessary to design antennas with very directive characteristics. This can only be accomplished by increasing the electrical size of the antenna.

Enlarging the dimensions of single elements often leads to more directive characteristics. Another way to enlarge the dimensions of the antenna, without necessarily increasing the size of the individual elements, is to form an assembly of radiating elements in an electrical and geometrical configuration. This new antenna, formed by multielements, is referred to as an array. The array factor AF is given by [START_REF] Balanis | Antenna Theory, Analysis and design[END_REF]: 1) with:

AF = ∑ i 1 =1 N e j(i 1 -1) ψ x ⋅ ∑ i 2 =1 M e j (i 2 -1) ψ y =AF x ⋅AF y (3.2.2.
ψ x =k d x sin(θ)cos (ϕ)+ β x ψ y =k d y sin (θ) sin (ϕ)+ β y (3.2.2.2)
where d x and d y are the distances between two elements along the x and y direction respectively. β x and β y are the phase delay along x and y axis.

Feeding line

Let's considering an antenna in the transmitting mode. The radiation pattern, and the S parameter plot with respect to the frequency, describe its behavior. However the signal generated by the source does not reach directly the antenna. It must pass through a line, which matches the source to the Let's considering the impedance Z(x) which is seen by the source. It is given by eq. 3.2.3.1:

Z ( x)=Z 0 Z L cos ( 2π λ x ) + j Z 0 sin ( 2 π λ x ) Z 0 cos ( 2 π λ x ) + j Z L sin ( 2 π λ x ) 3.2.3.1
where λ is the guided wavelength within the transmission line. If Z ( x)≠ Z s , part of the signal generated by the source is reflected back. In this case, the reflecting coefficient Γ coincides with the S parameter. Γ Is given by eq 3.2.3.2:

Γ= Z ( x )-Z s Z ( x )+ Z s 3.2.3.2
It is clear that Z s must be equal to Z ( x) to have zero reflection. This condition is called impedance matching. The characteristic impedance Z 0 depends on the transmission line which is utilized. shown here [START_REF] Bahl | A Designer's Guide to Microstrip Line[END_REF]. Fig. 3.2.4.1.1 [19]: Microstrip line on a dielectric substrate with its design parameters.

Microstrip line

Introduction to microstrip lines

• Case W / h≤ 1 [START_REF] Bahl | A Designer's Guide to Microstrip Line[END_REF]

ϵ reff = ϵ r + 1 2 + ϵ r -1 2 [ 1 √ (1+ 12 h W ) + 0.04 ( 1-W h ) 2 ] 3.2.4.1.1 Z 0 = 60 √ (ϵ reff ) ⋅ln ( 8 h W + W 4 h ) 3.2.4.1.2 • Case W /h> 1 [17] ϵ reff = (ϵ r + 1) 2 + (ϵ r -1) 2 √ ( 1+ 12 h W ) 3.2.4.1.3 Z 0 = 120 π √ ϵ reff [ W h + 1.393+ 0.667 ln ( W h + 1.444 ) ] 3.2.4.1.4
Generally preliminary calculations utilize the standard value of 50Ω for both the source and the load impedance ( Z i and Z L in figure 4.2.3.1). Therefore the characteristic impedance Z 0 of the transmission line must be of 50Ω too. In this case we obtain by eq. 3.2.3.1: The depth y 0 of the feeding cavity is chosen to have Z L =50Ω (eq. 3.2.1.10). If y 0 =0 , the impedance of the patch will be greater and a microstrip line with a narrower width must be used.

Z ( x)≡ Z i =50Ω .
However it is often convenient to match the feed source to a 50Ω microstrip line. In this case it is necessary to use an adapter between the microstrip and the patch as shown in figure 3. The W 50 width of the strip line corresponds to Z 50 =50 Ω characteristic impedance. Let's suppose Z L > 50Ω as the impedance of the radiating patch. The impedance which is seen by Z 50 at the distance l qw by the patch is given by eq 3.2.3.1, with x=l qw :

Z (l qw )=Z qw Z L cos ( 2π λ l qw ) + j Z qw sin ( 2 π λ l qw ) Z qw cos ( 2 π λ l qw ) + j Z L sin ( 2 π λ l qx )
where Z qw is the impedance of the adapter. If l qw = λ 4 , then:

Z (λ / 4)= Z qw 2 Z L 3.2.4.1.5
Since the adapter must match the impedance of the Z 50 line, the following equation must be satisfied:

Z (λ /4)=Z 50 =50= Z qw 2 Z L
Therefore, the impedance of the l qw =λ / 4 adapter must be:

Z qw = √ 50⋅Z L 3.2.

Losses in the microstrip lines

Often the losses of the transmission line are neglected in the design, as first approximation.

However this is not the case for circuits operating at very high frequencies. Actually the losses become very high at terahertz frequencies, and the design of the circuit without taking them into account leads to very large errors. There are two kind of losses that occur in the transmission line circuits: the conductor loss and the dielectric loss. Let's consider first the conductor loss. The effective width of the feeding line must take into account the thickness of the conductor. It is given by, for W / h> π 2 [START_REF] Bahl | A Designer's Guide to Microstrip Line[END_REF]:

W e =W + t π ( 1+ ln 2 h t ) 3.2.4.2.1 (a)
for W / h⩽ π 2 [START_REF] Bahl | A Designer's Guide to Microstrip Line[END_REF]:

W e =W + t π ( 1+ ln 4 πW t ) 3.2.4.2.1 (b)
where W is the width of the line, h is the thickness of the dielectric, t is the thickness of the conductor and W e is the effective width. Let's define P , Q and R s as [START_REF] Bahl | A Designer's Guide to Microstrip Line[END_REF]:

P=1-( W e 4 h ) 2
3.2.4.2.2(a)

Q=1+ h W e + h π W e ( ln 2 h t - t h ) 3.2.4.2.2(b) R s = √ π f μ 0 σ c 3.2.4.2.2(c)
The attenuation constant per conductor loss is given by, for W / h⩽ π 2 [START_REF] Balanis | Antenna Theory, Analysis and design[END_REF]:

α c =8.68 Rs 2 π Z 0 h P [ 1+ h W e + h π W e ( ln 4 π w t + t W ) ] 3.2.4.2.3(a)
for π/ 2< W / h⩽2 [START_REF] Bahl | A Designer's Guide to Microstrip Line[END_REF]:

α c =8.68 R s 2 π Z 0 h P Q 3.2.4.2.3(b)
for W /h> 2 [START_REF] Bahl | A Designer's Guide to Microstrip Line[END_REF]: on the variable we consider (the tangent loss or the conductivity). Given tan δ≠ 0 we have [START_REF] Bahl | A Designer's Guide to Microstrip Line[END_REF]:

α c =8.68 Rs Z 0 h Q { W e h + 2 π ln [
α d =27,3 ϵ r √ ϵ eff ϵ eff -1 ϵ r -1 tan δ λ 0 3.2.4.2.4(a)
Equally, for σ≠ 0 , we have [START_REF] Bahl | A Designer's Guide to Microstrip Line[END_REF]: (which refers to the gold at 4K) with a dielectric loss tangent of 5e-4 (which is the HDPE at 600GHz). The theoretical resulting attenuation is therefore α c =1,14 dB/ cm for the conductor, and α d =0,004 dB/cm for the dielectric. Such results are in very good agreement with the numerical simulations. Even if the dielectric loss is negligible, this is not the case for the conductor.

α d =4.34 ϵ eff -1 √ ϵ eff (ϵ r -1) √ μ 0 ϵ 0 σ 3.
For this reason a superconductor is needed and will be considered in the following paragraphs. For this purpose the NbN is chosen which has ever shown (for example for the double slot antenna) a completely negligible resistivity at THz frequencies for a temperature of 16K.

The feeding networks

The feeding network is extremely important in an array of microstrip antennas. A bad network can compromise the behavior of the entire array. There are essentially two main criteria that feeding networks must satisfy: first the antennas must be feed with the same phase for a broadside radiation;

second the impedance of the network connected to the single elements must match that of the source. Let's analyze the two cases separately.

Phase of the feeding networks

Let's consider a transmitting array (reciprocity theorem). The feeding signal must reach each element of the array with the same phase. This can be accomplished by imposing that the length of every transmission line, which connects the source to the single antenna element, must be a multiple of the wavelength. If the line which links the port 1 with the source is N λ , and the line from the source to the port 2 is M λ , the phase is the same for both the ports. Nevertheless, this criteria makes the array narrow-band, since it is designed only for the frequency c 0 / λ . It is often more convenient, even if more complicated, to design the network such that the distance from the source to every port is the same. The procedure to accomplish such condition will be illustrated.

Let's consider a 2X2 array. An example of feeding network that achieves the same distance from the feeding to the elements is the following (fig 3 The same procedure can be repeated infinitely to get a generic 2 n X2 n array.

Impedance of the feeding network

The feeding network is a structure which consists of a series of microstrip lines connected together to bring the signal from the antennas to the hot electron bolometer, or equivalently, in the transmitting mode, from the HEB to the antennas. An example is given in fig. 3 The impedance after the split is equal to a parallel between the two impedances of 50Ω , which results in an impedance of 25Ω An adapter is necessary to match the 50Ω impedance, before the split, to the 25Ω after the split. Normally the length of the adapter is λ / 4 , therefore, by using eq. 3.2.3.1, the impedance which is seen by the port A A in fig. 3.2.5.2.2 is:

Z A = Z adp 2 25Ω
Since the matching condition requires Z A =50Ω , Z adp is therefore:

Z adp =√ 25 ⋅50 Ω 3.2.5.2.1
The feeding networks are designed by using these simple rules. There exist also other shapes which can be used to match a narrow line with a wider one. However such shapes are determined experimentally or numerically. A numerical solver software must be used, like Altair FEKO or HFSS.

Design and simulation of the patch antenna array

The first design parameter we need to define is the operating frequency of the patch antenna. Let's assume f =600GHz , which is a typical value for astronomical observation with heterodyne receivers. The maximum allowed thickness of the antenna substrate to be operative is h=λ r /8

where λ r is the wavelength inside the substrate. Thicker substrates present too many losses in term of surface wave, while too thin substrates present too narrow band. Let' assume as first approximation exactly h= λ r /8 . Since λ r depends on the dielectric constant of the substrate, we need to define ϵ r . High values of ϵ r mean small wavelength inside the dielectric and, therefore, thinner substrate. Since it is easier to fabricate thick substrate than thinner substrate (mechanical etching), we opt for a low ϵ r value. The best match is given by the "high-density polyethylene" (HDPE) with a dielectric constant of ϵ r =2,26 . Thus we have:

h=λ r /8= c 0 8 f √ ϵ r ≃41μ m 3.2.6.1
Given f , ϵ r and h we can calculate the width W of the patch antenna by eq 3.2.1.1, and its length L by eq. 3.2.1.2, 3.2.1.3 and 3.2.1.4.

The second parameter to be designed is the impedance of the receiver (or the source in the transmitting mode by using the equivalence theorem). In our case, the receiver is the Hot Electron Bolometer (HEB) mixer, with an impedance of R s =50 Ω . It is necessary to match the impedance of the HEB with the impedance R i n of the patch antenna. Eq 3.2.1.6, 3.2.1.7, 3.2.1.8 and 3.2.1.9

give an input impedance of: R i n =219Ω . Since R s ≠ R i n , the feeding line of the patch antenna must be shifted towards the center. The position of the feeding line can be determined by eq.

3.

2.1.10; we have: y 0 =13μ m . Thus the shape of the patch antenna has been determined. Now it is necessary to set up the width of the feeding line. To avoid an impedance mismatching (and thus the reflection of the signal back to the source), the characteristic impedance of the feeding line must be equal to the impedance of the patch. The width of the feeding line must be adjusted accordingly.

By imposing Z 0 =Z i n =50Ω , the width can be calculated by eq 3. The final goal of the project is to assemble the patch antennas together into a 16X16 array configuration. In order to avoid unwanted sidelobes, the intra-element distance must be lower than the wavelength: d < λ .

A source impedance of 200Ω has been used for the simulation. In reality, it is necessary to add a λ / 4 length microstrip adapter, with a characteristic impedance of Z adp =√ 50⋅200Ω , between the 200Ω feeding line and the 50Ω source. Now the single element of the array of patch antennas has been defined. The design parameters which have been obtained can be used as a first guess for the Genetic Algorithm (GA) FEKO optimization, which will run in the contour of the guess point. The obtained patch will be integrated in the 2X2 patch array configuration, shown in figure 3.2.5.1.1 If the widths of the feeding lines are chosen correctly, the entire 2X2 configuration will present a minimum of the reflection coefficient around 600GHz . Another GA optimization is necessary to adjust the design parameters. After that, the 2X2 array can be considered as a single element itself, and can be integrated in another An impedance matching bandwidth of 0.8 % is obtained which is quite narrow for our application. Another drawback of this design is the high side-lobe level in the XZ plane. Actually figure 3.2.6.6 shows a sidelobe of -7.2 dB . It is important to note that such sidelobe is present in the E plane. Since the asymmetry of the patch array with respect to the H plane is only given by the feeding network, it is easy to conclude that such feeding network is the cause of the sidelobe.

The array which has been illustrated in this chapter is a result of a very long and time consuming optimization. The main part of the cost function of the optimization consisted in the impedance matching to minimize the reflection coefficient. Then the obtained result has been adjusted to give a better shape to far field by decreasing for example the sidelobe level. The two requirements have shown to be quite contrasting.

In absence of losses, the impedance matching condition corresponds to the equality between the gain, realised gain and directivity, since the whole input power is accepted by the array. This condition can always be accoplished by placing an adapter between the source and the load. For this reason the most important parameter becomes the directivity only, and not the S 11 parameter.

Therefore in some cases it could be more convenient to optimize only the directivity. A -3dB bandwidth of 4 % is obtained which is much larger than in the preceding case. Such larger band is due to the fact that the source is assumed to be always matched to the load. In reality this condition cannot be achieved since the adapter which will be put between the source and the load is working only at a certain frequency.

The most important advantage of this procedure is the lower values of the sidelobes. The far field at 592GHz is shown in fig 3.2.6.9 and 3.2.6.10 for the E and H plane respectively: The highest sidelobe is at -11 dB which is below the -10dB limit.

Fig. 3.2.6.9 and 3.2.6.10 show the directivity as a function of the declination θ . Another important parameter is the magnitude of the electric field. Actually a good far field plot is expected to be as Gaussian as possible with respect to θ . Let's consider for example the bivariate normal distribution:

G= 1 (2 π σ x σ y √ 1-ρ 2 ) e - 1 2 (1-ρ 2 ) [ (X -μ x ) 2 σ x 2 + (Y -μ y ) 2 σ y. 2 - 2ρ( X-μ x )(Y -μ y ) σ x σ y ] 3.2.6.2
where σ x , y is the standard deviation, μ x , y is the mean of the distribution, and ρ is the correlation between X and Y. Let's assume: μ x =μ y =ρ=0 . Let's assume the following Cartesian coordinate system, where the angle θ becomes the radius, and φ the azimuth:

X =θ cos(φ ) Y =θsin (φ ) 3.2.6.3
We can now calculate the Gaussian distribution by substituting eq.3.2.6.3 in eq.3.2.6.2 Figure

Conclusions

A 16X16 array of patch antennas has been designed on a HDPE substrate with a ground plane to match the beam width requirement of the heterodyne receiver ( < 5°). Superconducting NbN has been assumed and has been represented by PEC material in the numerical simulation, in order to minimize the losses. A λ r / 8 thick substrate is considered at 600GHz operating frequency as a compromise between the field pattern and the bandwidth. Two different design procedures have been used consisting in an impedance optimization and directivity optimization respectively. The first procedure is very long and time consuming. Since the RF source impedance is fixed at 50 Ω , the resulting bandwidth is very small (0.8%). Moreover such procedure comports a high sidelobe level, due to the fact that the directivity is not optimized. The second procedure leads to a better directivity, a lower sidelobe level (<10%), and a larger -3dB gain bandwidth of 4%. Changes in the radiating element shape don't modify the band significantly. Such band is quite low for generalpurpose receivers, where it should be around 30%. However, for astronomical receivers targeting only one or a few specific lines, a narrow RF bandwidth is sufficient, and patch -antenna arrays are a promising option.

In the clean room currently available, the substrate we require is too thick to be realized by chemical deposition (possible to about 1-3 microns), and too thin for mechanical etching (possible to about 100-300μm). Every attempt has been made to modify the thickness (for example by modifying the dielectric constant), each one leading to unacceptable results in terms of far field and reflection coefficient. At the moment these factors don't allow the patch array to be fabricated, but further studies could be carried out by other researchers to find a way to fabricate such a thin substrate.

Chapter 4: The transmit-array

Introduction

The arrays of patch antennas can be used to reduce the beamwidth of the receiver. Another possibility is to use only a single antenna, such as a double slot antenna. In this case the beam will be wider, and a lens becomes necessary to reduce its width (fig. shifted by a certain quantity in comparison to the phase in H. Given φ F the phase in F and λ the wavelength of the free space wave, we have from the definition of the wave vector:

φ H =φ F + 2 π λ FH 4.1.1
since Fp=FH we have, for the point P:

φ P =φ F + 2 π λ FP =φ F + 2 π λ FH + 2 π λ Pp =φ H + 2 π λ Pp 4.1.2
Given φ 0 the phase of the flat wave before the transmit-array, eq. 4.1.2 can be written as:

φ P -φ 0 =φ H -φ 0 + 2 π λ Pp
that is, given Δ φ P =φ P -φ 0 and Δ φ H =φ H -φ 0 :

Δ φ P =Δ φ H + 2 π λ Pp 4.1.3
It is clear, from fig. 4.1.2, that:

FP=√FH 2 + HP 2 since:
Pp=FP-Fp=FP -FH we have:

Δ φ P =Δ φ H + 2 π λ (√FH 2 + HP 2 -FH ) 4.
HP=√x 2 + y 2 .
The phase-shift that a certain cell located at ( x , y) must provide is therefore

given by [START_REF] Yu Nanfang | Flat optics with designer metasurfaces[END_REF]:

Δ φ P =Δ φ H + 2 π λ (√f 2 + x 2 + y 2 -f ) 4.1.5
The design of the single cell of the transmit-array is an argument which has been widely discussed in literature. Generally the design parameters of each cell are their reflection coefficient (or equivalently the transmission coefficient) and phase shift. Lower reflection coefficients are desired, which mean that the most part of the radiation passes through the cell without being reflected at the lens input. Therefore the bandwidth is defined as the frequency interval where the reflection coefficient is lower than a certain fixed value, and the phase shift is constant. The phase shift must be optimized to fit eq. 4.1.5. We will see in detail these aspects in the next paragraphs. Several solutions exist for the design of the generic cell. Let's consider some practical cases [START_REF] Abdelrahman Ahmed | Transmitarray Antenna Design using Slot-TypeElement[END_REF][START_REF] Laurent | Circularly-Polarized Discrete Lens Antennas in the 60-GHz Band[END_REF]24,[START_REF] Malhat Hend | Nano-Dielectric Resonator Antenna Reflectarray/Transmittarray for Terahertz Applications[END_REF][START_REF] Ryan Colan | A Wideband Transmitarray Using Dual-Resonant Double Square Rings[END_REF][START_REF] Wu Rui Yuan | High-Gain Dual-Band Transmitarray[END_REF].

• "L. Dussopt [START_REF] Laurent | Circularly-Polarized Discrete Lens Antennas in the 60-GHz Band[END_REF] proposed a circularly polarized transmit-array design operating at 60 GHz, generating a broadside beam and a 30° tilted beam respectively. The transmit-array includes four geometries of the single cell to provide a phase shift of 0°, 90°, 180° and 270°. An than -10dB at 59.5-61.9 GHz, an insertion loss of 0.35 dB at 60.4 GHz, and a 1-dB transmission bandwidth of 3.9 GHz (6.5%). Their theoretical radiation patterns at 60 GHz show a maximum gain of 4.85 dBi and a 3-dB beamwidth of 88° and 131° in E-and Hplanes, respectively." [START_REF] Laurent | Circularly-Polarized Discrete Lens Antennas in the 60-GHz Band[END_REF] Note that this configuration has been designed for 60GHz. It is not applicable to our case since it comports the presence of vertical via holes connecting the two layers, which cannot be realized with the available fabrication process. accomplished by varying the radius of the dielectric cylinders." [START_REF] Malhat Hend | Nano-Dielectric Resonator Antenna Reflectarray/Transmittarray for Terahertz Applications[END_REF] Since the antenna scaling is not valid anymore (due to the plasmonic effect), the proposed design is not very suited to be rescaled at other frequencies. Moreover, the difficulty to realize a such thin metal layer (h=50nm), and to couple it to such a fine dielectric shape make this solution impracticable to our purposes.

• "Colan G. M. Ryan proposed a transmit-array, operating at 30 GHz, whose elements consist of square rings, since they have already been demonstrated to have wide bandwidth performance in reflect-array antennas. In the transmit-array configuration, the single cells of "In the "fixed gap" approach, the gap size between the inner and the outer rings was set to a constant value and the width of the elements was varied in order to control the phase variation. In an effort to improve the bandwidth, the second strategy employed a "variablegap", in which the gap size is a function of the element width. A "double sided" design was also investigated to determine if an increased coupling between the rings could either widen the element bandwidth or lead to a larger per-layer phase change. Finally, an element with a fixed outer ring width was simulated; in this case the phase variation is controlled by specifying the width of the inner ring. This last design is unique since it combines two different geometries which together are meant to cover the full 360° phase range. These two geometries differ in the width of their outer rings, but both implement the phase compensation by varying the width of their respective inner rings. Covering different, nonoverlapping phase ranges, both geometries are to be used on each transmit-array layer depending upon what phase compensation value is required.

Among all of the four unit cell designs proposed here, the fixed-gap and the variable-gap show the largest phase variation (approximately 80°) as a function of the ring width.

However, the transmission magnitude of the fixed gap option drops rapidly as its width is varied around its resonant dimension; the same is true for the double sided design. Such behavior implies a small element transmission bandwidth which will consequently limit the bandwidth of a multilayer antenna based on these two structures. The variable-gap approach is an appealing option due to its large phase variation and relatively slowly varying magnitude response. However, precise fabrication control is necessary to implement this choice since the gap dimension is calculated as a percentage of the overall element width.

Therefore, it was deemed to be infeasible by the author. Over its two geometries, the fixedouter-ring unit cell produces approximately 70° of phase variation with a transmission magnitude that is either equal to or better than that of the competing designs. Furthermore, applying its two different geometries results in a larger number of phase points that can be applied to compensate the incoming wave, thus decreasing the quantization error of the phase response. As was shown in earlier work, smaller phase errors of the fixed-outer-ring unit cell lead to higher antenna gain. Therefore, the fixed-outer-ring element was chosen by the author for the transmit-array unit cell." [START_REF] Ryan Colan | A Wideband Transmitarray Using Dual-Resonant Double Square Rings[END_REF] Despite the promising characteristics of the proposed unit cells, the transmit-array design involves the use of free space between the layers, realized with foam. Since the foam has thermal insulation characteristics, it cannot be used in cryogenic applications like for our case. Moreover the characterization of the cells in terms of phase shift and transmission magnitude has been performed via numerical simulation, while the method proposed in this thesis involves the use of simple analytical formulas.

• "Different slot length L s , with periodicity P=0.62 λ=16.46mm and slot width W =2mm are considered. Four layers of this cross-slot element are used for this design with separation between layers equal to H =λ/ 4=6.64mm ." [START_REF] Abdelrahman Ahmed | Transmitarray Antenna Design using Slot-TypeElement[END_REF] Also in this case no dielectric is considered between layer, resulting in the same thermal insulation problem of the preceding case. Moreover no analytical model is provided for its design, which is based only on numerical simulations.

• "Rui Yuan Wu proposed a dual band transmit-array in the two frequency bands of 11.8-12.2 and 17.5-18.1 GHz. As shown in fig. 4.1.8, the unit cell consists of four identical metallic layers without any dielectric substrates. In each layer, there are three slots which can be grouped into two kinds. The middle one is longer, and two identical shorter slots are on the two sides. The size of the square cell is P=15mm, and the gap between two layers is H=5mm. The width of slots is marked by W as 1.5mm, and there is a 1-mm-wide space between two slots. Since there is no substrate to support the structure, the mechanical strength of the metallic layers needs to be taken into consideration, in which the thickness is chosen as 0.3mm. The frequency-domain solver in commercial software, CST Microwave There is no interaction when the values of L1 and L2 change uniquely. Thus, it is worthy to present the behaviors of the unit cell at 18 GHz when L1 changes and the behavior at 12

GHz when L2 changes, respectively." [START_REF] Wu Rui Yuan | High-Gain Dual-Band Transmitarray[END_REF] No dielectric is assumed between the layers: if a dielectric was used, the phase shift provided by the cell would decrease, and the full range of 360° would not be covered. Thus we have again the drawback of the thermal insulation and the lacking of a mathematical model for the phase shift.

• The ideal transmit-array cell would minimize the incoming beam reflection so that most of the power would be transmitted with a phase shift given by eq. 4.1.5.

Z 03 = Z 3 Z 0 Z 3 + Z 0 4.2.2
Afterward, the impedance Z 03 must be translated towards the second layer, through the transmission line Z l . Given l 2 the thickness which separates the bottom layer from the second layer, the translated impedance is given by (we are considering lossless dielectrics):

Z ' 2 =Z l Z 03 cos (β l 2 )+ 1i⋅Z l sin (βl 2 ) Z l cos (β l 2 )+ 1i⋅Z 03 sin (βl 2 ) 4.2.3
where β=2 π/ λ .We can now calculate the parallel impedance between Z 2 and Z ' 2 :

Z ' 22 = Z ' 2 Z 2 Z ' 2 + Z 2 4.2.4
Given l 1 the thickness of the dielectric which separates the second layer from the first layer, the impedance of eq. 4.2.4 can be translated to the first layer:

Z ' 1 =Z l Z ' 22 cos(βl 1 )+ 1i⋅Z l sin (βl 1 ) Z l cos(βl 1 )+ 1i⋅Z ' 22 sin (βl 1 ) 4.2.5

The impedance calculated by eq.4.2.5 is in parallel with the impedance of the metal pattern on the first layer. The total equivalent impedance is therefore given by:

Z eq = Z 1 Z ' 1 Z 1 + Z ' 1 4.2.6
The reflection coefficient can now be calculated by eq 4.2.1. In particular we are interested in the absolute value of the reflection coefficient expressed in dB:

| Γ | dB =20log 10 | Z eq -Z 0 Z eq + Z 0 | 4.2.7
Now let's calculate the phase shift from the input to the output of the cell. We introduce the input signal V i on the left of the circuit illustrated in fig. 4.2.2. Therefore the input current can be calculated as

I i =V i /Z eq 4.2.8
where Z eq is given by eq. 4.2.6. Part of the input signal is reflected back V i -, and part is transmitted through the cell: V i + . We can calculate V i + as:

V i + = V i + Z 0 I i 2 4.2.9
Part of the input current passes though the impedance of the metal pattern of the first layer:

I 1 =V i / Z 1 4.2.10
The remaining part is transmitted towards the second layer through the dielectric. It is given by

I 12 =I i -I 1 4.2.11
Now we want to translate the input signal toward the second layer. To do so we need to split it in its backwards and forwards components. Note that these two components are different from V i -and V i + , since V i -and V i + refer to the signal before the layer 1. Therefore we define a signal downstream of the layer 1, as: V 12 =V i . We can now calculate:

V 12 -= V 12 + Z l I 12 2 4.2.12 V 12 + = V 12 + Z l I 12 2 4.2.13
Now we can translate V 12 towards the second layer:

V 2 =V 12 + e -iβ l 1 + V 12 -e i βl 1 4.2.14
The total current produced by the voltage V 2 is given by V 2 /Z ' 22 where the term Z ' 22 is the equivalent impedance seen by the second layer, and given by eq. 4.2.4. Part of the current passes through the second layer. The remaining part goes towards the third layer, and is given by the total current minus the current that passes in the second layer:

I 23 =V 2 / Z ' 22 -V 2 /Z 2 4.2.15
Thus, given V 23 =V 2 as in the precedent case, we can calculate the backwards and the forwards components after layer 2 as:

V 23 -= V 23 -Z l I 23 2 4.2.16 V 23 + = V 23 + Z l I 23 2 4.2.17
We can now calculate the voltage on layer 3:

V 3 =V 23 + e -iβ l 2 + V 23 -e i βl 2 4.2.18
Again we have V o =V 3 . We can therefore calculate the current which passes through the free space as:

I o =V o /Z 0 4.2.19
We can now calculate the forward component of the output signal as:

V o + = V o + Z 0 I o 2 4.2.20
The phase-shift that the cell must provide is equal to the difference between the phase of the forwards component of the output signal, minus the forwards component of the input signal:

circuit model is not Z 0 anymore, instead it is Z ' 0 =Z 0 k k  .
Therefore the new impedance is

given by[29]:

Z ' 0 =Z 0 /cos (θ) 4.2.22
If the electric field is polarized along the X axis, we are in the Transverse Magnetic (TM) mode. In this case[29]:

Z ' 0 =Z 0 cos(θ) 4.2.23
In our case we consider only the TE polarization because the receiver (or the source if we consider the transmitting mode) in the focal plane is linearly polarized. Consequently we can only use eq.

4.2.23

The transmit-array concept

In the previous paragraphs, the impedances of the three layers were assumed in order to give a The metal is shown in gray. The wave-vector is assumed to be normal to the pattern. Let's assume the electric field polarized along the y direction (or x direction, by considering the symmetry of the structure). M.I. Kontorovich (1950s), derived the boundary condition for such meshes to be:

Ê y tot =i η eff 2
α Ĵ y where Ĵ y is the averaged surface current density along the y-axis and α is the grid parameter. The averaging is made over the period D, and the current density is given by the jump of the magnetic field across the plane of the pattern. The surface impedance is therefore given by[29,30]:

Z=i η eff 2 α 4.3.1
where[29,30]

α= k eff F π ln ( 1 sin π w 2 D ) 4.3.2
is the grid parameter, and

k eff = 2π f c √ ϵ eff 4.3.3
is the wave factor. In general the metal pattern can be inside a dielectric, or on the interface between two dielectrics. In this last case, we have[29]:

ϵ eff = ϵ 1 + ϵ 2
Z Z c = η eff 2 4
from which, we have[29,30]:

Z c =-i η eff 2 α 4.3.6
We have now all the equations necessary to design the transmit-array. With eq. 4.2.

Transmit-array design and simulation

The algorithm to design the transmit-array will be now described. The dielectric substrate which separate the first layer from the second layer is assumed to be the same of that which separates the second layer from the third layer. The input data are the following:

• dielectric constant of the medium in which the transmit-array is situated (which must necessarily be the free space, as will be discussed later): ϵ 1 , and the dielectric constant of the substrate: ϵ 2 .

• The thickness of the substrate which separates the first layer from the second: l 1 , and that which separates the second from the third layer: l 2 .

• The phase variation between input and output of the cell in the center of the transmit-array: Δ φ H

• The reflection coefficient Γ

• The operating frequency from which we can calculate the wave vector β inside the dielectric as: β= 2 π(freq) c √ ϵ 2

• The focal distance f

• The dimensions of the generic cell in the X and Y directions: D

• The size of the whole transmit-array (which is limited by the maximum phase shift that can be achieved, as will be shown later)

The steps for the design are the following (the electric field is assumed to be polarized in the y dimension):

• Compute the phase shift required to be added by each cell with eq. 4.1.5.

• Compute the impedance of the inclined beam at the output of the generic cell (located at the coordinates x,y from the central point), by eq. 4.2.22 and 4.2.23, where

θ TE =tan -1 ( x / f )
for the TE mode and θ TM =tan -1 ( y / f ) for the TM mode.

• Compute the three impedances of the three layers for every cell, by eq. 4.2.7 and 4.2.21. The Particle Swarm Optimization is used. The existence of a solution is not guaranteed for a given Γ and Δ φ .

• Compute the cell parameters for every given value of the impedance by inverting eq 4.3.1, if img(Z)>0, or eq. 4.3.6, if img(Z)<0.

• Draw the cell

The transmit-array is an array of cells which must provide a certain phase shift. Given a certain value for the phase shift of the central cell, the phase shift will increase as we move away from the center. However the phase variation range is not 360 o , since the solution does not exist for every value of Γ and Δ φ . A trade off is necessary. In general the phase range increases as the reflection coefficient increases. The design parameters which are reported here are the best compromise for Γ and Δ φ :

• ϵ 1 =1 (vacuum) and ϵ 2 =11.58 (sapphire) below the transmit-array, with respect to the power incident on the same area but before the transmit-array. Since the magnitude of the electric field is E=1V / m for the incident planar wave, the incident power is given by:

• l 1 =l 2 =100 μ m (dielectric substrate thickness) • Δ φ H =100 o (
P i = 1 2 ( Siz e) 2 E 2 Z 0 4.4.1
where Siz e is the surface of the transmit-array. Let's now calculate the transmitted power. FEKO provides the electric and magnetic field phasor in the focal plane shown in fig. 4.4.3. We can calculate the time averaged Poynting vector as

S =ℜ ( 1 2 E∧H  ) 4.4.2
The transmitted power is given by: Finally we have:

P t = ∫ - siz e 2
η t =P t /P i ≈ 63 % 4.4.4
The rest of the power is in part reflected away and in part falls outside the area of the plane shown in fig. 4.4.3. Since we are interested in the power focused in the beam waist, it is more convenient to consider the efficiency as the ratio between the power which goes through the waist area, with respect to the incident power. It is given by:

η f = ∬
waist S (x , y)dx dy/ P i ≈ 27 % 4.4.5

Concerning the focusing efficiency, we should consider that the converging of the phase fronts towards the focal point is reduced by their interaction with the external field. In the reality we don't have this problem, since the incoming beam is Gaussian. Unfortunately in FEKO we have a planar beam everywhere, so we are underestimating the actual efficiency. In order to obtain more realistic values, a frame can be used in the simulation to reduce such interaction and to increase the focusing of the beam. The disadvantage in the use of the frame is a reduction of the transmitted power. Fig. The transmitted power is about η t ≈ 44 % , while, for a 500μ m beam waist radius, the focused power is η f ≈ 33 % . Therefore, the use of the frame increases the percentage of radiation that is focused.

Eq. 4.1.5 defines the phase shift distribution over the transmit-array area. Since the phase shift varies more rapidly at the edges of the transmit-array, our array results to be down-sampled. Since the cell dimensions must be constant over the surface area (ideally the metasurface is composed by an infinite array of equal elements), the solution is to reduce the dimensions of every cell. However, if the cells are too small (< 0.3λ), eq. 4.3.1 (or 4.3.6) provide too small values of W , which can not be simulated (a too fine mesh is required), and cannot be fabricated with adequate accuracy. A better sampling leads to a better focused beam. To investigate how well a beam can be focused with our fabrication technology we consider a transmit-array which focuses the light only in one dimension (1D transmit-array), orthogonal to the polarization. Since the dimension of the cell which is orthogonal to the electric field is not important, it can be reduced as shown in fig. consider the main effect on the X axis (normal to the electric field). In this case the beam waist (along the X axis) is larger than the transmit-array length. The reason for a so large beam waist is due to the interaction of the field downstream the transmit-array with the external field. The power which is transmitted by the array, calculated with eq. 4.4.4, results to be 108% of the incident power. This effect is due to the curving of the field around the edges of the transmit-array, so that part of the power which comes from the external of the transmit-array is forced to go to the focal point (fig. 4.4.11). The approach to calculate the efficiency is therefore jeopardized and a metallic frame will be used as in the previous case to obtain more realistic results. The fact that the transmit-array focuses on a line is very clear. A beam waist of 350 μ m can be assumed which corresponds to the 45% of the incident power (it is the overall efficiency as that given in eq.4.4.5). The transmitted power is 47% of the incident one (eq. 4.4.3). The reduction of the transmitted power is entirely due to the presence of the frame.

It is now clear that the near field result is a combination of the effects of the transmit-array and the frame. In order to acquire a wider understanding of the focusing effect, it is therefore necessary to calculate the field of the frame only on the dielectric surface. Then it must be compared with the field given by the transmit-array only and the transmit-array with frame. Therefore it has 25% less of the transmitted power with respect to the case of the frame with the transmit-array. Also in this case it is clear that the frame helps the focusing effect at the expense of the transmitted power. Since the focal length is much longer than the 1,5mm case, the curvature of the phase front is less accentuated. The focusing effect is weakened and the field becomes more sensitive to the frame geometry. We will see later that this fact makes the experimental set up very tricky to align. We define therefore the following efficiencies:

• η trans-frame = P frame / P i =5 %±2 % efficiency of the frame in terms of transmitted power It is evident from the results that the power transmitted by the transmit-array and the frame is very low with respect to the incident power. This is due to the big absorbing frame which has been put around the transmit-array and the metallic frame, and which is necessary to obtain a good resolution of the field measured by the QCM. Such structure is too big and complex to be simulated with FEKO or HFSS. However more useful results are obtained if we consider the efficiency relative to the transmitted power. A reduction of the beam waist area is observed for the transmit-array with respect to the frame only. However the transmissivity of the transmit-array is not very good and a lot of radiation doesn't pass through it. This reduces the signal to noise ratio, and consequently the signal results to be less concentrated in the focal point and more dispersed around it. As a result the relative efficiency of the transmit-array results to be lower than that of the frame only both if we consider the beam waist of the frame, or the same beam waist of the transmit-array. At this stage results don't provide a clear evaluation of the transmit-array functioning. More precise instruments, such as a near field probe, are necessary.

• η trans-TA =P TA / P i =3 %±2 % of

Summary and conclusions for the transmit-array

The transmit-array concept has been introduced in this chapter. It consists of a tool which is able to convert an incoming planar wave into a spherical wave. In order to do that, the transmit-array is divided into several cells. Each cell must provide a certain phase shift which depends on its position in the array, according to eq. 4.1.5.We first simulated a square transmit-array with equal spaced cells. It is already possible to observe the curving of the waves downstream of the transmit-array meaning that the mathematical model we used to design the array is correct. A better focusing can be achieved by placing a frame around the transmit-array to block the interaction of the field downstream of the array with the field outside. Efficiencies of 27% and 33% are achieved without the frame, and with the frame respectively. Such efficiencies are too low for application in heterodyne receivers, since the received signal is already very low. The main disadvantage of this kind of transmit-array is the too large spacing between the cells. Smaller cells involve a better sampling of the phase profile, however the cells cannot be too small because otherwise their mathematical model provides geometries which contain too small features, such as gaps or patches, and are very difficult to realize with our instrumentation. In any case the dimension that cannot be reduced is only that which is parallel to the electric field, the other one can be reduced arbitrarily.

This fact allows us to design an array which is very fine sampled in the direction orthogonal to the electric field. In this case a better curvature can be achieved in the plane orthogonal to the electric field. Thus the focal point becomes a focal line. Transmit-arrays of this kind have been designed, realized and tested with a QCM. Simulations are made for transmit-array with a 1.5mm focal distance and with 20mm focal distance, with and without frame. In the case of 1.5mm focal distance frame, the totality of the incident field is transmitted by the array, but the beam waist of the focused beam is larger than the transmit-array itself; for this reason a frame has been added to increase the focusing effect, at the cost of the transmitted power which is now only the 47% of the incident field. In this case a small beam waist is obtained but the overall efficiency is only 45%, which is still not so good for our application. However this study has demonstrated that the concept works, and possible improvements could be made by improving the study of new metamaterial concepts. Since the 1.5mm focal length array is too small to be measured, simulations and measurements of a 20mm focal length prototype have been performed. In this case the focusing effect is very reduced due to the larger radius of curvature of the beam downstream of the array.

Experimental results for this prototype show a weak focusing effect, which is also limited by the instrumentation tolerance. We suggest further experimentations with a near field probe to calculate the field in proximity of the array and to analyze its curvature also, if possible, for the 1.5mm focal length transmitarray.

Due to all these drawbacks, another solution will be investigated in the next chapter, which also overcomes the problem of the focusing on a line: the zone plate (it is a particular kind of transmitarray).

Chapter 5: The zone plate

Introduction

The After that a great number of works occurred in this field, leading to a widespread of literature and contributions from all over the world. In this thesis we propose a zone plane constituted by the same The goal is to locate source points that constructively interfere at the focal point. In order to do that, the difference between the optical path length and the focal length shouldn't be greater than half a wavelength [START_REF] Center For | Fresnel zone plate theory, generation, tolerancing, fabrication, and applications[END_REF]:

l-l 0 < λ 2 6.2.3
The 1 st zone is defined by the source points which satisfy this criterion, it is shown in red below: In general, the n th zone defined as the locus of the points satisfying the following condition [START_REF] Center For | Fresnel zone plate theory, generation, tolerancing, fabrication, and applications[END_REF]:

(n-1) λ 2 <l-l 0 < n λ 2 6.2.5
n indicates the number of the zone considered.

In general, all the odd zones interfere constructively with other odd zones, as even zones interfere constructively with even zones, while odd and even zones interfere destructively between them. For this reason the zone plate is constructed by alternating absorbing and transparent zones. Note that a zone plate where the odd zones are absorbers is completely equivalent to a zone plate where the even zones are absorbers [START_REF] Center For | Fresnel zone plate theory, generation, tolerancing, fabrication, and applications[END_REF].

Fig 5 .2.5 [START_REF] Center For | Fresnel zone plate theory, generation, tolerancing, fabrication, and applications[END_REF] Inverting the metal zones with the free space zones doesn't change the focusing effect

In both the cases, the edges between opaque and transparent zones are given by the following equation [START_REF] Center For | Fresnel zone plate theory, generation, tolerancing, fabrication, and applications[END_REF]:

l-l 0 = n λ 2 5.2.6
Equation 5.2.6 can be rewritten in terms of source point radii as [START_REF] Center For | Fresnel zone plate theory, generation, tolerancing, fabrication, and applications[END_REF]:

√ f 2 + r n 2 -f = n λ 2 5.2.7
where r n represents the transition to the n th zone. Solving for r n gives [START_REF] Center For | Fresnel zone plate theory, generation, tolerancing, fabrication, and applications[END_REF] r n

2 =n λ ( f + n λ 4 ) 5.2.8

Design and simulation of the zone plate

The zone plate geometry is constructed by alternating transparent and opaque zones, where the radius of the n th zone is given by eq. 5.2.8. All zone plates that are defined by eq. 5.2.8 and where either all odd or even rings are blocked will bring plane wave radiation with wavelength λ to focus at a distance f from the lens. However this kind of zone plate presents a big disadvantage, which is due to the fact that half of the field which reaches the zone plate is blocked, thus reducing its efficiency. It is more convenient not to block the radiation from the even (or odd in the inverse configuration) zones, but it is better to shift such radiation by half of the wavelength. Therefore a phase shift of π is introduced between odd and even zones. Such phase shift can be realized by using the same cell structure than the transmit-array. The only difference is that square patches are used instead of the strips, to make the structure independent of polarization. It has been shown for the transmit-array that it is not easy to obtain a wide phase range from the square cell structure.

Therefore finding two cell structures with a phase shift of π is very tricky. PSO optimization shows no solution for a Γ=-25 dB reflection coefficient. After increasing Γ to -15dB a solution has been found in the limit of the feasibility. Referring to the D and W parameters of fig. 4.3.4, the resulting cell structures present the following characteristics Please note that the has the shape of a square with rounded corners. Now let's define an area coincident with such a square in the focal point downstream of the zone plate. This area will be used to calculate the field transmitted by the zone plate. Let's define:

Δ φ Z 1 Z 2 Z 3 W 1 D 1 W 2 D 2 W 3 D 3 cell 1 -160°-297 -470 -282 34

•

A=17.64 mm 2 as the area of the defined square

• P id = A 2 ⋅ E 0 2
Z 0 the ideal power of a planar wave that passes through the defined area

• P t the power transmitted by the zone plate

The simulation with FEKO leads to the following results:

• η t =P t / P id =43 % percentage of the power transmitted by the zone plate define the waist area as: A w =π r 2 , we can define P f as the power which is focused on the waist area (it is the integral of the power flux on the waist area). Therefore we can calculate:

• η ft =P f / P id =14 % percentage of the ideal incident power which is focused on the waist area • η fr =P f / P t =33 % percentage of the transmitted power that is focused inside the waist area

The low value of the focused power is due to the fact that the radiation passing through one zone is constructive but not perfectly in phase. It can be imagined as a vector sum with the vectors pointing in different directions within [0 π ], so the length of the summed up vector is smaller than the sum of the amplitude of the vectors. The beam waist is assumed to be the same of the preceding case: r =450μ m . In this case we have:

• η t =P t / P id =39 % percentage of the power transmitted by the zone plate • 1-η r =61 % percentage of the power reflected by the zone plate

• η ft =P f / P id =15 % percentage of the incident power which is focused on the waist area • η fr =P f / P t =40 % percentage of the transmitted power that is focused inside the waist area Results show that the transmitted power decreases due to the blocking effect of the frame. However the percentage of the transmitted power inside the waist area increases. This is an effect due to the fact that the frame blocks part of the radiation that would interfere destructively with the field in the focal spot (the same focusing effect as for the transmit-array). However, these two contrasting effects lead to an overall increase of η ft . In order to evaluate the performance of the transmissive zone plate, it is now necessary to compare it with the reflective zone plate, where the odd zones are replaced by reflecting circles. where the black circles represent the beam waist of 450μ m For the reflective zone plate without the frame, we have: The behavior of the reflective zone plane with respect to the frame is the same as in the preceding case. However it is clear that in absolute term the focusing efficiency of the transmissive zone plate is much higher. This means that our model leads to an improvement of the classical concept of zone by a factor around 2.

• η t =P t / P id =44
The reflected power in tab.5.3.2 is calculated by considering the integral of the transmitted power in a plane parallel to the zone plate downstream of it. Another way to calculate the reflected power consists in the inversion of the relation which links the VSWR with Γ . It is given by:

reflected power ≈ | Γ | 2 ≈ ( VSWR-1 VSWR+ 1 ) 2 5.3.1
The VSWR can be calculated by considering the field on a line perpendicular to the zone plate placed at x=0 and y=0 (i.e. passing through the center of the zone plate). By averaging the VSWR for z>0 (upstream of the zone plate) we obtain the following results:

VSWR Reflected power ZP 3±2 30 %±20 % ZP with frame 3±2 30 %±10 % RZP 4±2 30 %±20 %
RZP with frame 3±1 20 %±6 % Tab 5.3.3: Estimation of the power reflected by the zone plates. The high standard deviation doesn't allow a comparison between them The reflected power calculated by eq. 5.3.1 is more reliable than that calculated previously, so that the values of the reflected power calculated in tab 5.3.2 are probably overestimated. However the results in tab. 5.3.3 do not allow a comparison between the various kinds of zone plates due to the high standard deviation.

Conclusions

In this chapter the zone plate has been introduced which has the property to focus the radiation in a point. The proposed zone plate has been designed with the same metamaterial than the transmitarray of the previous chapter. It is a transmitting-zone plate since it is designed to transmit the radiation for all the adjacent Fresnel zones, while the original design was intended to block the radiation at an interval of one zone. In the transmitting zone plate a ̝ π phase shift between adjacent zones is added and is well accomplished. Moreover simulations also show that the overall efficiency of the transmitting zone plate (which is the ratio of the power inside the beam waist with respect to the incident power) is about twice as much as the one for the reflecting one. This result was expected since in the transmitting zone plate none of the radiation is blocked. Even if the result is still far from application for the heterodyne receivers, it is a good progress in the design of transmitting zone plate.

Simulations show that the main drawback of the proposed transmitting zone plate is the high value of the reflected power. A further development of the metamaterials technology could overcome this problem, therefore a deeper study of the metamaterial physics is highly suggested. Another step that should be done is the realization and test of the proposed zone plate: indeed the fact that the radiation is more concentrated in the focal point with respect to the transmit-array allows an easier experimental analysis. In common receivers there will be Gaussian beam illumination. This needs to be simulated and is expected to improve the coupling efficiency.

Chapter 6: The phase grating

Introduction

Since our objective consists in the realization of multipixel heterodyne receivers, it is necessary to feed each mixer with a LO signal. There are 3 possibilities for N pixels. The first possibility is to have N LO sources, one for each pixel, but this solution is very expensive. The second possibility is to split the signal in the waveguide. Finally the third option is to split the radiated field of the LO.

We are dealing with this last approach. In this case a phase grating is used to split the LO beam towards multiple mixers (fig. 6.1.1 shows the whole heterodyne scheme; the phase grating is shown split the signal from the LO into different beams). Fig. 6.1.1: the signal from the LO is split by the phase grating, so that multiple pixels can be fed at the same time In our application, there is no constraint on the radiated phase of each beam. However the field intensity in each beam must be identical.

Phase grating are commonly used [START_REF] Heyminck | Structured surface reflector design for oblique incidence beam splitter at 610 Ghz[END_REF]34,[START_REF] Graf | A Novel Type of Phase Grating for Thz Beam Multiplexing[END_REF][START_REF] Groppi | SuperCam: A 64 pixel heterodyne array receiver for the 350 Ghz Atmospheric Window[END_REF] as beam dividers; they consists of a surface with several grooves as sown in fig. 6.1.2

The Dammann's grating

The Dammann's grating is a periodical structure where the single cell consists of a transmission function which only assumes the values -1 and +1, corresponding to a phase shift of the incident radiation of 0 and π. As it happens for all the periodical structures, a single beam incident on it is split into different orders. The angles between such orders with the normal to the grating can be calculated by inverting the grating equation [10,[START_REF] Defrance Fabien | Instrumentation of a terahertz heterodyne receiver[END_REF]:

n λ=Δ u sin(θ n ) (6.2.1)
where n is the considered diffractive order, λ is the wavelength, Δ u the cell size, and θ n the angle. Since no order can be seen for θ> 90 °, the number of the diffractive orders is limited. As the cell size determines the number of the order and their direction, the intensity of the light radiated in each order is given by the shape of the cell. Let's consider for example only the one dimensional case. The transmission function is uniquely determined by its transition points where the function passes from -1 to 1 and vice versa. We indicate with ξ 1,... , ξ N z such transition points normalized to the single cell dimension, so that ξ 1 =-1/2 and ξ N z =1/ 2 (fig 6 .2.1). The transmission function can be calculated as[10,1]:

T (ξ)= ∑ k=1 N Z (-1) k+ 1 rect ( ξ-( ξ k+ 1 + ξ k )/ 2 z k+ 1 + z k ) (6.2.2)
The intensity and the phase of the light radiated in the order n are given by the Fourier transform of the transmission function. We obtain[10,1]: to typical efficiencies of 50-60%, where the efficiencies are defined as the ratio between the power radiated in the diffraction order, with respect to the incident power.

t( n=0)= ∑ k=1 N z -1 (-1) k+ 1 (ξ k + 1 -ξ k ) t( n)= 1 2 π n ∑ k=1 N z -1 (-1) k+ 1 {sin (2π n ξ k+ 1 )-sin (2 π n ξ k )+ i[cos(2 π n ξ k )-cos(2 π n ξ k+ 1 )]} (

The Fourier grating

The Fourier grating is an extension of the Dammann's grating. While eq. 6.2.1 is still valid for the angles of the orders, a better intensity distribution of the radiation can be achieved by optimizing the shape of the cell. Each cell can be modeled as a combination of its Fourier components as [START_REF] Heyminck | Structured surface reflector design for oblique incidence beam splitter at 610 Ghz[END_REF]:

φ (ξ)= ∑ n=0 ∞ a n cos ( n 2π
Δ ξ ξ ) (6.3.1) such that the transmission function becomes [START_REF] Heyminck | Structured surface reflector design for oblique incidence beam splitter at 610 Ghz[END_REF]:

T (ξ)=e i∑ n=0 ∞ a n cos ( n 2 π Δ ξ ξ ) (6.3.2)
whose Fourier transform can be expressed by as [START_REF] Heyminck | Structured surface reflector design for oblique incidence beam splitter at 610 Ghz[END_REF]:

t( n)=⊗ k =0 ∞ [ ∑ q=-∞ ∞ J q (a k )⋅δ ( k -q 2 π Δ ξ ) ] (6.3.3)
Even if eq.(6.3.3) represents a nice mathematical expression, it is easier and less time consuming to perform a direct FFT transform. Given |t (n)| as the required intensity for the generic n-th order, it is therefore possible to calculate the a n Fourier components of the single cell. Generally the following algorithm is performed [START_REF] Heyminck | Structured surface reflector design for oblique incidence beam splitter at 610 Ghz[END_REF]:

• 4 N sets of N Fourier components are defined as starting points. The FFT is then performed in order to evaluate which are the sets which best fit the required intensity distribution among the various orders [START_REF] Heyminck | Structured surface reflector design for oblique incidence beam splitter at 610 Ghz[END_REF].

• The best (e.g. 20) sets of the Fourier components are then optimized to fit the intensity distribution. Normally all the optimizations converge to the same cell structure, which is chosen as the best one [START_REF] Heyminck | Structured surface reflector design for oblique incidence beam splitter at 610 Ghz[END_REF] This algorithm works well for 1D structures with N coefficients to be calculated. It becomes prohibitive for 2D structures where the number of coefficients is N 2 . In this case the speed of the random search is not acceptable anymore. It is necessary to start from the desired beams and to invert the light path of the system, which now represents a beam combiner. The phase variation of the combined field provides a good phase grating shape for the desired diffraction pattern. In this way it is possible to obtain efficiencies of the order of 80% which are well above the Dammann's gratings. The bandwidth is approximately ±5 % of the frequency design. The dominating bandwidth limiting effect is the angular dispersion. [START_REF] Heyminck | Structured surface reflector design for oblique incidence beam splitter at 610 Ghz[END_REF]

Our phase grating design concept

During my thesis I developed a matlab program to generate a phase grating for heterodyne applications. My work is the development of an idea of prof. Massimiliano CASALETTI which essentially consists of a series of Fourier and inverse-Fourier transformation of a target field.

In general the input of our phase grating design process is the target near field that we want on the receivers, while the output is an array which contains the depth of each groove of the grating. The concept is shown in fig. 6.4.1:

The relationship between the aperture field and the target field

Let's assume that the aperture field on the grating surface is X polarized. In this case the wave equation is given by:

∇ 2 E x ( x , y , z )+ k 2 E x ( x , y , z )=0 (6.5.1)
where k is the wave number. By Fourier transforming in the xy plane, keeping the z axis the same as before, we get:

( -k x 2 -k y 2 + ∂ 2 ∂ z 2 ) Ẽ x ( k x , k y , z)+ k 2 Ẽx (k x , k y , z )=0 (6.5.2)
where k x and k y are the spatial angular frequencies that can be interpreted as the x and y components of the wave vector. The general solution of eq. 6.5.2 is given by:

Ẽx (k x , k y , z )=c 1 ( k x , k y ) e -k z z i + c 2 ( k x , k y ) e k z z i (6.5.3)
where c 1 and c 2 are found by imposing the appropriate boundary conditions. By imposing the radiation at infinity condition: Ẽx (k x , k y ,∞)=0 , we must have c 2 ≡ 0 , so that we have:

Ẽx (k x , k y , z )=c 1 ( k x , k y ) e -k z z i (6.5.4)
which, for z=0, becomes: Ẽx (k x , k y ,0)=c 1 (k x , k y ) (6.5.5) finally we have:

Ẽx (k x , k y , z )= Ẽx ( k x , k y ,0) e -k z z i (6.5.6)
that furnishes the relation between the aperture field (z=0) and the target field (z=h).

The relationship between the Fourier transform and the matlab FFT

The algorithm for the optimization of the aperture field (i.e. the tangential electric field on the phase grating surface) requires the calculation of some Fourier (and inverse Fourier) transforms. Since the FFT defined by matlab is different from the definition of the Fourier transform, it is necessary to find the relation between them, so that it becomes possible to express the Fourier transform in terms of the FFT. The Fourier transform of the electric field is defined by: where k x and k y are the x and y components of the wave vector. Matlab defines the FFT of a generic function F as:

E (k x , k y , z)= ∬ ∞ E ( x ,
F (k 1 , k 2 )= ∑ n=1 N x ∑ m=1 N y e - 2 π N x i(n-1)(k 1 -1) e - 2 π N x i (m-1)(k 2 -1)
F (n , m) (6.6.2) where F is an array of N x X N y components, n and m are the indexes of the function F in the space domain, and k 1 and k 2 are the indexes of F in the spectral domain. In order to approximate the integral 7.6.1 with the expression 7.6.2, it is necessary to find the relationship between n , m and x , y , and between k 1 , k 2 and k x , k y We have, for the spatial domain:

x=(n-1)Δ x + x 0 (6.6.3) y=(m-1) Δ y + y 0 (6.6.4)

where x 0 and y 0 are the starting point of the spatial coordinates, and Δ x and Δ y are the spatial steps. For the spectral domain, we have: (7.6.5) k y =(k 2 -1)Δ k y + k y,0 (7.6.6) where k x,0 and k y,0 represent the coordinate of the reference spectral point, and Δ k x and Δ k y are the spectral step. By substituting eq. 7.6.3 and 7.6.4 in eq. 7.6.1 we get:

k x =(k 1 -1) Δ k x + k x,0
E (k x , k y )= ∑ n ∑ m E(n , m)e -i {k x [ (n-1)Δ x + x 0 ] + k y [(m-1)Δ y + y 0 ]} Δ x Δ y (7.6.7)
by collecting the term e -i (k x x 0 + k y y 0 ) and replacing eq. 7.6.5 and 7.6.6 in the remaining terms, we get:

Δ x Δ y E(k x , k y )=e -i(k x x 0 + k y y 0 ) ∑ n ∑ m E (n , m)e -i [kx, 0 (n-1)Δ x + k y,0 (m-1)Δ y] e -i [ (k 1 -1)(n-1)Δ x Δ k x ] e -i [ (k 2 -1)(m-1)Δ y Δ k y ] 7.6.8
at this point we can observe that eq. 7.6.8 includes somehow eq. 7.6.2. In order to express eq. 7.6.8

in terms of eq. 7.6.2, we have to impose the last two exponentials of eq. 7.6.8 to be equal to the two exponentials of eq. 7.6.2. Such condition is:

e -i [(k1-1)(n-1)Δ x Δ kx ] =e - 2 π N x i (n-1)(k 1 -1) e -i [(k2-1)(m-1)Δ y Δ ky] =e - 2 π N y i (n-1)(k 2 -1)
which is equivalent to:

Δ x Δ k x = 2 π N x (7.6.9)
Δ y Δ k y = 2 π N y (7.6.10) under this condition, and assuming F ( n , m)=E (n , m)e i [k x,0 (n-1)Δ x + k y ,0 (m-1)Δ y] , we get:

E (k 1 , k 2 )≈ Δ x Δ y e
i(k x x 0 + k y y 0 ) FFT {E (n , m)e -i [k x,0 (n-1)Δ x + k y ,0 (m-1)Δ y ] } (7.6.11) Eq.7.6.11 represents the link between the Fourier transform and the FFT. Similarly we can derive the link between the inverse Fourier transform and the IFFT. The inverse Fourier transform is given by:

E( x , y)= 1 4 π 2 ∬
∞ E(k x , k y ) e i(k x x+ k y y) dk x dk y (7.6.12) while for the IFFT we have:

F (n , m)= 1 N x N y ∑ n ∑ m e 2 π N x i(n-1)(k 1 -1) e 2π N y i(m-1)(k 2 -1)
F ( k 1 , k 2 ) (7.6.13) by substituting eq. 7.6.3 and 7.6.4 in eq. 7.6.12 we get:

E(n , m)= 1 4 π 2 ∬ ∞ E(k x , k y )e i { k x [(n-1)Δx+ x 0]
+k y[ (m-1)Δ y + y 0] } dk x dk y (6.6.14) by collecting the term e i(k x x 0 + k y y 0 ) , and replacing eq. 6.6.5 and 6.6.6 in the remaining terms, we get:

E(n , m)= 1 4 π 2 ∬ ∞ E(k x , k y ) e i (k x x 0 +k y y 0 ) e i {[ (k 1 -1)Δ k x + k x,0 ] (n-1)Δ x + [ (k 2 -1)
Δ k y + k y ,0] (m-1)Δ y } dk x dk y 6.6.15 which can be rewritten as:

E (n , m)= 1 4 π 2 e i [ (n-1)k x ,0 Δ x + (m-1) K y,0 Δ y ] ∑ n ∑ m E (k x , k y )e i(k x x 0 + k y y 0 ) e i [(k 1 -1)Δ k x (n-1)Δ x + (k 2 -1)Δ k y (m-1)Δ y ] Δ x Δ k y 7.6.1 6
By imposing eq 6.6.9 and 6.6.10, we obtain the relation between the inverse FT and the IFFT.:

E (n , m )= 1 Δ x Δ y e i [(n-1)k x ,0 Δ x + (m-1)k y,0 Δ y ] IFFT [E (k 1 , k
2 )e i (k x x 0 + k y y 0 ) ] 6.6.17

Now we know how to calculate the Fourier transform and the inverse Fourier transform with matlab. In the next paragraph, the algorithm for the optimization of the aperture field will be illustrated.

The otpimization algorithm

The optimization algorithm allows to calculate the electric field on the phase grating surface which is necessary to obtain a desired target field, at the distance h from it. Therefore as a first step, we have to define the target field. We can assume for example a target field which consists of four spots At this point we define the "focal window" as a function of the spatial coordinates, which is equal to 1 when the following condition is satisfied:

√ ( x-x p ) 2 + ( y-y p ) 2 < w p 6.7.1 and 0 in the other points. In eq. 6.7.1, x p and y p represent the coordinates of the center of one generic spot in the target field, while w p represents the radius of the spot. Eq. 6.7.1 represents a filter which must be multiplied with the field on the target plane to allow only the radiation on the four spot to pass, as we will see later in the optimization process We also define the "aperture window" as a function which is equal to 1 when x and y belong to the phase grating area. Both the "focal window", and the "aperture window", are shown in fig. 6.3.1, where the yellow part represents the points where the relative function is equal to 1, and 0 for the blue part. It should be noted that the "focal window" can coincide, as in this case, with the target field. The "focal window", and the "aperture window" are related to the electric field in the spatial domain. Now we define another function in the spectral domain: the "spectral window". It is equal to 1 when the following condition is valid:

√ k x 2 + k y 2 < k 0 6.7.2
and 0 elsewhere. Such condition represent the part of the spectral domain which is not attenuated if we look at it from an infinite distance. At this point we can start the optimization. The algorithm is the following:

1. The focal window is taken as the initial point for the algorithm optimization 2. focal window is Fourier transformed to obtain it in the Fourier domain 3. The Fourier transformed focal window is then translated down towards the phase grating by multiplying it times e k z h i , accordingly to eq. 6.5.6, where we assume x=-h . Since the target field is very distant from the phase grating, we assume that the transformed focal window doesn't contain any dissipative component. For this reason we also filter out the higher order modes by multiplying the transformed focal window times the spectral window described above. Therefore we obtain the spectral aperture field as:

Ẽaperture =transformed focal window⋅spectral window⋅e k z h i 4. We inverse transform the spectral aperture field to obtain the aperture field in the spatial domain E aperture .

5. The phase grating reflects the incident beam by transforming it into E aperture . It must be observed that the magnitude of E aperture must be the same of the incident beam, since we only have reflection. The only quantity that is changed is the phase. For this reason we have to take the phase only of E aperture and to multiply it times the magnitude distribution of the incident wave (which in general will be a Gaussian distribution). We obtain E ' aperture =∢E aperture ⋅|Gauss|

6.

E ' aperture represents the field reflected by an ideal infinite phase grating. Since the phase grating has a finite extension, we have to multiply E ' aperture times the aperture window:

E ' ' aperture =E ' aperture ⋅aperture window .

7. E ' ' aperture represents the starting value of the aperture field, from which we can start the optimization process. We Fourier transform it to obtain the spectral aperture field Ẽ ' ' aperture .

8. We translate Ẽ ' ' aperture towards the target plane, by multiplying it times e k z h i (eq. 6.5.6).

Thus we obtain the Fourier transformed focal field on the target plane: 13. aperture field is back transformed in the spatial domain: E ' ' ' aperture .

Ẽ ' ' focal = Ẽ ' ' aperture ⋅e -k z hi .
14. Again the obtained aperture field must be modified in order to take into account the Gaussian distribution of the incident field magnitude. Therefore we take the phase of the obtained aperture field and multiply it times the Gaussian distribution: The algorithm illustrated above allows to calculate a target field composed by a certain number of spots within which the intensity is kept constant. A more realistic approach can be done by assuming a Gaussian distribution inside the single spot. Moreover, we empirically observed that it is often convenient to assume an initial guess of the target field where each spot has a different phase in order to improve the convergence. Now we know how to calculate the aperture field. The next step consists in the design of the phase grating and will be illustrated in the next paragraph.

Phase grating design

An optimization algorithm has been introduced to calculate the magnitude and phase of the aperture field which generates a desired target field at the distance h from the aperture field plane. It is now necessary to calculate the shape of the grating necessary to generate such aperture field. In general we have an incoming beam with a certain phase distribution, which is then reflected by the phase grating, and assumes another phase distribution which coincides with the aperture field distribution.

It is therefore necessary to know both the incoming beam phase distribution and the aperture field The phase of the Gaussian beam incoming field, in local coordinates, is given by:

φ i =-( k 0 z '+ k 0 x ' 2 + y ' 2 2R -ψ ) 6.8.1
where

Z r =π w 0 2 / λ , R=z ' [1+ ( Z r. / z ' ) 2 ] , w z =w 0 √ 1+ ( Z r / z ' ) 2 , ψ=tan -1 (z ' /Z r )
and w 0 is the minimum beam waist. In order to express the incident phase distribution in the grating plane, we have to substitute the global coordinates in eq.6.8.1. The coordinate change is the following:

x ' =-x cos (θ)+ z sin(θ) y ' = y z ' =-z cos(θ)-x sin(θ)+ d 6.8.2
At this point we know the incident phase distribution, φ i , and the aperture field phase distribution, φ a =angle ( E ' ' aperture ) . The phase grating cell consists of a groove etched on a generated the optimized design respectively. In this last case, the efficiency of the total power calculated inside the radius of the four spots is 52%. Such low value is due to the fact that the size of the four spots is greater than the optimized one, and it is also due to the fact that the four spots are very close to each other. By modifying the position of the spots to P( x , y )=(±4 λ ;±4 λ) , and by increasing the radius of the measured spot to 1.3λ , efficiencies of 80 % are reached, which are very good compared to the Fourier grating. The overall result is good, even for a very inclined beam. However the dimensions of the four spots and the distance of the focal plane from the phase grating are too small for a practical case of interest. Typical dimensions of these parameters (given by the receiver configuration) are:

• F=200λ (distance between the focal plane and the grating)

• P( x , y )=(±18 λ ;±18λ) (position of the four spots)

• r =8 λ (radius of the spots)

Due to the greater distance between the focal plane and the phase grating, the dimension of the grating must increase, otherwise a too high truncation error will affect the aperture field (point 6 of paragraph 6.7). The increased dimension of the grating will comport a very high increase of the computational cost of the physical optics simulation. Such effort cannot be achieved with our simulation tools. For this reason the cell width has been increased in the simulation from c=0,25λ to c=1,5λ . Simulations of the field generated by such grating show good results for a normal incident Gaussian beam (fig 6.9.5, x and y dimensions are in m), with an efficiency of 63%. In this case the efficiency is limited due to the too small spacing between the spots (like in the preceding case), and due to the too large cells. However such efficiency is not very far compared to requested efficiency of 70%. The large cell approximation is not good anymore for larger incidence angles: in such cases the surface of the cell is too large and will work as a mirror, reflecting all the radiation in the opposite direction. The resulting target field will be distorted with too high sidelobe levels.

Conclusions

A new method has been introduced for the design of a phase gratings to split the LO signal towards different mixers at the same time. Conventional phase gratings (such as the Dammann and the Fourier gratings) are optimized only in the far field. This means that the radiation which is reflected by the grating is divergent. Since the focusing of the split beams towards the mixers is highly desired, we propose a phase grating whose output consists in a near field radiation pattern. In this way it is possible to define the size of the spots in the target plane to match the beam waist of the receivers. Another advantage of the proposed design relies to the possibility to create arbitrary near field shapes, so that it is possible to obtain beams which are very close to each other, while the beam distance in classical Dammann and Fourier grating is limited by the cell size which define the diffractive orders.

Simulations show good results (efficiencies up to 80%) for small geometries which require a small number of cell elements, while bigger geometries require too many cells to be simulated with our instruments. However a rough estimation of their functioning can be done by increasing the cell dimension in the simulation tool. In this case the resulting simulated efficiency will be down estimated. Since the under estimated efficiency is 63% for the required heterodyne configuration, there are good probabilities that a device realized with smaller cells will allow efficiencies up to 70%, which fall within the acceptable limit. Therefore we suggest as successive steps to simulate big geometries with smaller cells, if the instrumentation is adequate, otherwise to realize directly the small cell configuration to be tested in an anechoic chamber. The realization of the phase grating can be performed by drilling a flat metal surface with a drill. The smallest the drill, the highest the precision. We plan to send the design of the phase grating to the french company Vuichard, which reach a resolution of 1-3μm.

Chapter 7: Conclusions

This thesis has the objective to prepare for the next generation of very large heterodyne receiver arrays with a great number of pixels (n>100) for THz frequencies. In order to make the fabrication of large arrays possible with reasonable resources, it is necessary to rethink the design of the components of heterodyne receivers. In particular, I focused on the design of the antenna structure that is to be connected to the mixer, and the beam divider for the local oscillator. For the antenna structure, the idea is to design planar structures which can be easily fabricated for a whole array in a single process, instead of having bulky elements such as double slot antennas on a lens, or horn antennas, which have to be machined individually one by one. The first solution which has been investigated is an array of planar antennas. This consists of an array of patch antennas connected via a micro-strip feeding network and separated from the ground plane by a dielectric substrate. In this way the lens is not necessary anymore since the focusing effect is determined by the array factor (more elements create a narrower beam). Two methods have been employed to design an array for 600 GHz, the first consisting of an impedance optimization with a Genetic Algorithm followed by some adjustments to increase the directivity, and the second consisting of a direct optimization of the directivity. The first method is very long and time consuming, since the solution must be computed for every frequency of the domain, and leads to a solution which is extremely narrowband ( ≈ 1 % ). The second method is much faster and leads to a wider band design ( ≈ 4 % ), and has lower side-lobe levels. Such a bandwidth is quite low for general-purpose receivers, where it should be around 30%. However, for astronomical receivers targeting only one or a few specific lines, a narrow RF bandwidth is sufficient, and patch -antenna arrays are a promising option. The main drawback of this design is the required HDPE substrate thickness ( 42μ m ), which is too thick for chemical deposition and too thin for mechanical etching. It is not possible to increase the thickness because surface waves would arise, nor is it possible to decrease it because that would lead to an even narrower band. A great number of attempts have been also made by modifying the dielectric constant of the substrate, but results are unsatisfying in terms of the far field beam pattern and RF bandwidth. Another challenge of this configuration is the high sensitivity of the array behavior to the design parameters (due to the narrow band), which also makes fabrication difficult. Technical advancements in clean room technology might well overcome the fabrication challenges, in particular that of the ground plane, and make patch arrays an attractive option. In the meantime, we have analyzed other solutions, such as the transmit-array.

The transmit-array represents another way to replace the lens of the receiver with a flat element, also called "flat lens". The transmit-array is composed of an array of cells which provide a certain phase shift to an incoming wave, in order to transform a flat front into a spherical one. Therefore each cell is uniquely characterized by its reflection coefficient (or transmission coefficient), which must be small (or high) in order to transmit most of the signal which is incident on the array, and by the phase shift provided. A great number of cell structures have been proposed in literature for much lower frequencies, but all of them are impracticable at submillimeter/THz frequencies. Either they contain parts which cannot be rescaled to very small dimensions, or parts that cannot be easily fabricated. Some include free space between the layers. None of them provide a mathematical model to predict their behavior. Indeed, the big innovation of our approach is to provide simple analytical equations which can be used in place of numerical simulations (which are time consuming) to design the whole structure. First results for an array focusing on a point show an efficiency of 27 % . Such a low value is due to the interaction between the field that passes through the transmit array and the field that passes around it. A frame that blocks the beam outside the transmit array can be used to reduce such interaction, with the drawback of a reduced transmitted power. However such reduction is compensated by an overall increase of the efficiency to 33%.

A better focusing effect can be achieved by reducing the cell dimensions, to allow a finer phase sampling. However such dimensions cannot be reduced arbitrarily, due to the limitations in the fabrication tolerances. Moreover only the dimension transverse to the electric field polarization vector can be reduced. For this reason the focusing effect can be increased only in one dimension. A design with a focal distance of 1.5mm has been simulated with FEKO, leading to efficiencies of about 45% (with a frame). Even if the 1.5mm focal distance design is interesting from an applicative point of view, it cannot easily be verified in the laboratory with our test equipment. For this reason a 20mm focal distance prototype has been designed, simulated, fabricated and tested. In order to evaluate the functioning of such array, the experimental results are compared with those obtained for a blank, empty frame filled only with dielectric. Unfortunately the measured efficiencies are lower than expected, due to the blocking effect of the absorber which has been put around the device, and the instrument limitations. It would be interesting see whether the transmit array with the 1.5mm focal length, would yield better result, but this requires the design of a complex measurement set-up, possibly integrating the transmit array with a cryogenic mixer. An alternative would be to design, simulate and test low frequency prototypes. The effect of illuminating the transmit array with a Gaussian beam, as is usual in heterodyne receivers, should also be studied. It is likely that this would reduce much of the interference, allow us to remove the frame and increase the efficiency.

In order to overcome the problem of the focusing only on a line, another solution has been proposed which consists of the zone plate. Traditionally, a zone plane has rings which transmit and others that block the radiation, such that only the radiation that has constructive interference at the focal point passes. We designed one of the first of these zone plate for 600 GHz radiation. We found a large reflection of over 50%. Of the radiation that passes through the zone plate 25% is focused on the focal radius when a frame is used to block interfering radiation. We improved on this design by introducing a 180° phase shifts instead of blocking the destructively interfering rings. The 180°p hase shifts were obtained by using the same cell structure proposed for the transmit-array, on a sapphire dielectric substrate. A reduction of the designed-for reflection coefficient to -15dB was necessary to allow for such a large phase shift. This allowed us to increase the focusing efficiency of the transmitted radiation to 40%. Unfortunately, the overall efficiency of the zone plate with the frame is only 15%. In further studies simulations using a Gaussian illumination should be carried out, that will be more realistic and diminish any interference.

The final part of this thesis deals with the beam divider, which is used to split the LO signal to feed multiple pixels at the same time. Two kinds of LO dividers are currently in use in heterodyne receivers, which are waveguide splitters, and the phase grating. In particular, we are focusing on the last one which is the most promising one, since it can be more easily scaled to THz frequencies. It consists of a grooved surface which modifies the phase profile of an incoming wave to generate constructive interference, and give rise to the desired field pattern. Generally, Dammann and Fourier gratings are used as beam dividers, while we propose a more versatile design. Indeed in both the Dammann and Fourier gratings the angular position of the beams are only dependent on the grating periodicity, while in our design arbitrary distributions can be achieved. Moreover Dammann and Fourier gratings only work in the far field, while our grating allows the formation of arbitrary near field patterns with any possible shape. Such versatility is advantageous as it allows for the formation of closely packaged beams required by the heterodyne architecture, where the receivers are closely spaced (e.g. for our prototype 18mm at the focal distance of 100mm, which corresponds approximately to an angle of 10°). Our design is based on a series of FFT and iFFT transforms to fit the desired field distribution (which is our input), with a certain aperture field distribution on the phase grating plane. The difference between the phase of the obtained aperture field and the phase of the incoming field determines the phase grating structure. For the computation, the structure is described as an array of numbers which correspond to the distance of the generic cell basis from the reference plane. This array is then imported into a matlab program to 123 simulate the structure with a code based on the physical optics approximation, which is faster than a full MoM simulation with FEKO. Finally the resulting field is compared to the desired field. The main drawback of this concept is the reduced size required for the generic cell, which corresponds to λ / 4 . This increases a lot the simulation time, so that only gratings with a small number of cells can be simulated with reasonable computation time. Results show efficiencies of about 52%

for closely packaged beam configurations, while they increase up to 80% for more widely spaced beams. Such efficiencies are very good compared to the Fourier gratings. However, for the configurations required by typical heterodyne receivers, bigger gratings are necessary, which require a very high number of cells. A rough estimation of the efficiency of such gratings can be performed by increasing the cell size up to 3/ 2 λ In this case the resulting efficiency would be underestimated. Since values of about 63% are calculated, it is reasonable to assume that 70% efficiencies could be reached with a finer cell sampling, and this result is perfectly compatible with the heterodyne requirements. Therefore we suggest as subsequent steps to simulate big geometries with smaller cells, if the simulation tools are adequate and fast, otherwise to realize directly the small cell configuration and test it in the laboratory.

In summary, I explored three different approaches to focus the radiation onto the mixers. All three work and can be produced for an entire array at once. They are very interesting alternatives, once their efficiency can be increased. I also designed a new kind of beam splitter for the local oscillator, which will allow to pump a closely packed heterodyne array with a single LO.

This thesis is only a first step towards large heterodyne arrays. In this thesis, I carried out different designs for the focusing onto a single pixel of the heterodyne receiver. It would be desirable to advance the design even further, especially to increase the efficiency of the focusing. The next step consists in the test of an array of few pixels. The purpose is to verify that close packaging of the optics does not cause any fabrication challenges, mounting of the mixer, and optical interference.

Such an array is expected to focus the light into an array of spots, where the mixers will be placed.

If the focusing is good, then it is necessary to start thinking how to modify the fabrication process of the mixer, in order that an array of mixers could be easily fabricated in a single process. Once that the array of mixers has been obtained, a study of the disposition of the amplifiers with respect to the mixers must be carried on. Indeed the mixers must be cooled down to cryogenic temperatures and great care needs to be taken that the amplifiers do not heat the mixers. A configuration where the mixer array and the amplifier array are separated might be easiest. (The amplifier noise increases with temperature, but there is no cut-off and it is acceptable if they are slightly warmer than the mixers.) An alternative would be integrated mixer arrays, with very low power consuming amplifiers (e.g. SiGe technology) and good thermal connection to the cryocooler. At some point it is necessary to connect the mixers to the amplifiers, and the amplifiers to the intermediate frequency circuit and spectrometers. It is desirable to have single cables for several channels, to reduce the number of connections. An interference study must be carried on.

For multiple pixels it is necessary to divide the signal from the local oscillator into N beams to pump the N mixers. For this purpose Fourier gratings are commonly used, while my thesis introduced a new kind of grating, which allows the split of the LO beam towards very closely packaged mixers with a high efficiency. At this point the power provided by the LO could become too low to feed hundreds of mixers, and an improvement of the LO technology (both the amplifier multiplier chain or the QCL) becomes necessary.

Future sub-millimeter/ far-infrared telescopes all call for large heterodyne array receivers. At the conference [44], in preparation of the AtLAST telescope a large ground based sub-millimeter telescope to be built in the Atacama desert in Chile, a wish for array receivers with 1000 pixels was expressed. Such large arrays are currently not available, and increasing the number of pixels with designs would render the array prohibitively expensive.

For the airplane observatory SOFIA the SHASTA (Stratospheric Heterodyne Array System for Terahertz Astronomy) instrument has just been selected as a third generation instrument. It will have 64 pixels in a closely packed array to observe between 1.9 to 2.1 THz. Its design concept uses mixer blocks for 8X8 pixels, but with horns drilled in the blocks and traditional mounting of the mixers [START_REF]microruban et séparés du plan de masse par un substrat diélectrique. Cette configuration profite du facteur du réseau pour réduire la largeur de faisceau du signal collecté. Cependant, nos simulations nous montrent que la bande RF des réseaux d'antennes patch est étroite[END_REF]. Such an array could be substantially simplified with the technology proposed in this thesis.

Future proposals of satellite missions include the Origins Space Telescope [46], which calls for heterodyne arrays between 8 and 64 pixels. Though this size of arrays are still just feasible with the standard technology, an implementation of the approach proposed in this thesis would reduce weight, simplify fabrication and testing and would hence considerably reduce the cost or a space mission.

125

Chapter 1 :

 1 Introduction.........................................................................................................................5 Chapter 2: Background.........................................................................................................................9

Fig. 1 . 1 [ 1 ]

 111 Fig. 1.1[1] Spectral windows. The Terahertz gap is shown between 0.1 THz and 10 THz which has been unexplored for years due to the high H 2 O absorption.

Finally chapter 8

 8 furnishes a summary and describe the overall result of this work. This thesis has been co-financed by the "CNES" and the "ED 127". It is the result of the collaboration of researchers of the instrumentation group in LERMA (Laboratoire d'Etudes du Rayonnement et de la Matière en Astrophysique et Atmosphères) at the Observatory of Paris and the L2E (Laboratoire d'Electronique et Electromagnétisme) at the Sorbonne Université.

Fig. 2 .

 2 Fig. 2.1.1[1]: Heterodyne receiver. The sky signal is superimposed to the LO signal through the beam splitter, and both signals are sent to the mixer. The resulting IF signal is then amplified and analyzed.

Fig. 2 .

 2 Fig. 2.1.1.1: the diplexer

Fig. 2 .

 2 Fig. 2.1.2.1(a): Design of the double slot antenna seen from the top. The LO and sky signal are sent to the mixer in the center of the double slot. The mixer becomes an IF signal source. The IF signal is sent to the amplifier through the low pass filter on the right

Fig. 2 .

 2 Fig. 2.1.3.1: HEB scheme. It is composed of two gold plates separated by a superconducting NbN strip. The whole structure acts as a non linear resistance which is responsible of the signal modulation.

Fig. 2 .

 2 Fig. 2.1.4.1: the Local Oscillator (LO) Two different kinds of LO are commonly used depending on the frequency.

Fig. 2 .

 2 Fig. 2.1.5: the phase grating. The radiation incident on the grating is split into several reflected beams with different directions. The phase grating can also be transmissive. In this case, the incident beam is diffracted into different beams.

Figure 3 .

 3 Fig. 3.2.1[16]: Photo of a 4X4 planar antenna array. All the patches are connected via a metallic microstrip line network.

  Fig. 3.2.3.1 Scheme of the equivalent circuit of the system:"source+antenna".

  Microstrip lines (fig. 3.2.4.1.1) are a particular kind of transmission line. They are a common way to feed patch antennas. Their characteristic impedance only depends on the width of the line. Let's define W the width of the line, and h the thickness of the dielectric. The thickness of the metal strip is neglected. The equations to calculate the characteristic impedance of the transmission line are

Fig. 3 .

 3 Fig. 3.2.4.1.2: patch antenna adapted with a feeding cavity.

2 . 4 . 1 . 3 Fig. 3 .

 24133 Fig. 3.2.4.1.3[20] patch antenna adapted with a λ /4 long microstrip adapter placed between the 50Ω impedance microstrip line, and the radiating patch.

  Fig. 3.2.5.1.1 feeding network of a 2X2 array. The distance from the source to the every element must be the same.

Fig. 3 . 2 . 5 . 2 . 1 :

 32521 Fig. 3.2.5.2.1: Simple T junction. After the split the impedance of the microstrip line is doubled to avoid an impedance mismatching.

1 L

 1 Fig. 3.2.6.1: far field in log scale of a patch antenna adapted to 200 Ω , D≃10 dB .

2 Fig. 3 .

 23 Fig. 3.2.6.3: 16X16 array configuration: the signal is received by the 256 antennas and sent to the HEB which becomes an IF source. The IF signal is sent to the via out. A stepped impedance filter, and two gaps are used to decouple the RF and IF signal

Fig. 3 .Figures 3 . 2 . 6 . 5 , 3 . 2 . 6 . 6 ,Fig. 3 . 2 . 6 . 6 :

 3326532663266 Fig. 3.2.6.4 b: scheme of the signal route from the source to the via-out circuit.

  Fig. 3.2.6.8: -3dB directivity bandwidth of the 16X16 array optimized for the directivity.

Fig. 3 .

 3 Fig. 3.2.6.9 far field of the 16X16 array optimized for the directivity at the project frequency; E plane

3. 2 . 6 .

 26 Fig. 3.2.6.11:comparison between the electric field distribution of the 16X16 directivity optimized array, and the Gaussian curve (E plane).

Fig. 3 .

 3 Fig. 3.2.6.12: comparison between the electric field distribution of the 16X16 directivity optimized array, and the Gaussian curve (H plane).

  Fig. 4.1.1 (a): Focusing effect for a classic bulk dielectric lens.

1 . 4 Fig. 4 . 1 . 3

 14413 Fig. 4.1.3 Representation of the position of the center of the single cell with respect to the coordinate system.

  example of unit cell is shown in fig. 5.1.4. The size of the cell is 2.5X2.5mm 2 (λ / 2X λ / 2 at 60GHz ) and it consists of two identical square patch antennas (1.55X1.55 mm 2 ) connected by a vertical via. The bottom patches (focal source side) have the same orientation for every cells, while the top patches (free space side) are rotated by an angle α ( α=0 °,90°,180°and 270°) for the four different unit cells, generating a circularly polarized wave through sequential rotation. The simulated S-parameters of these unit cells under normal incidence show a reflection coefficient lower

Fig. 4 .

 4 Fig. 4.1.4[44]: Dussopt's transmit-array cell. The phase shift is obtained bu modifying the orientation of the two patches at the input and the output planes.

  Fig. 4.1.6[26] Ryan's transmit-array cell. The signal passes through the cell composed by two metal square rings placed on the same side, or the opposite side of a dielectric. The size of the rings is changed in order to obtain the required phase shift.

  Fig. 4.1.7 a[3]: Abdelrahman's transmit-array cell (top view) The shape of the cross etched on the metal determines the phase shift.

Fig. 4 .

 4 Fig. 4.1.7 b[3]: Abdelrahman's transmit-array cell (side view) The air gap between the layers causes a thermal insulation which is not optimal for cryogenic applications.

  Fig. 4.1.8[27] Rui Yuan Wu's transmit-array cell. Two different kinds of slots are considered to work in two bands. The phase shift is provided at a certain frequency by modifying the relative slot length.

4 . 2

 42 Fig. 4.1.9[24]:first Haipeng's transmit-array cell: the phase variation is obtained by varying r 1n

Figure 4 .

 4 2.1(b) shows in more details the single cell structure. Each cell is composed of a dielectric substrate and three metal patterns. The shape of each pattern is optimized for the reflection coefficient and the phase shift requested. In general the cell can be seen as an electric circuit, where each layer is represented by means of an equivalent impedance: Z 1 , Z 2 , and Z 3 .

Figure 4 . 2 .

 42 Fig. 4.2.1 (a): Reflection and transmission scheme of a single cell of the transmit-array. The two components of the input signal ( V i ± ) are transformed to the output components ( V o ± ).

Fig. 4 .

 4 Fig. 4.2.1 (b): Scheme of the single cell structure of the transmit-array. The phase shift is provided by the three metal motives on the top, in the middle, and on the bottom of the cell.

Fig. 4 . 2 . 2 :

 422 Fig. 4.2.2: Representation of the equivalent circuit of the cell: the three metal motives are modeled by the three impedance at the input, in the middle, and at the output of the equivalent circuit. The dielectric substrate is modeled by the transmission lines between such impedances.

Figures 4 .Fig. 4 . 2 . 3 :

 4423 Figures 4.1.1 (b) shows that the beam at the output of the transmit-array is not perpendicular to the dielectric, but it is inclined. In this case the polarization angle of the beam is crucial to calculate the characteristic impedance of the medium in which the wave propagates. Let's consider figure 4.2.3:

  Fig. 4.3.1[29]: Inductive pattern composed of metal strips (metal part in gray).

2 4 . 3 . 4 2 Fig. 4 .

 43424 Fig. 4.3.2[29]: capacitive metal pattern composed of rectangulare patches (metal part in gray).

Fig. 4 . 3 . 7

 437 Fig. 4.3.3: Empty grid representing the transmit-array cells. Each cell contain the relative metal motif.

  Fig. 4.4.1 (a,b,c) show the three layers of the designed transmit-array (both dielectric and metal are shown in black by the autocad picture; the area inside each rectangle is in metal):

Fig. 4 .

 4 Fig. 4.4.1 (b) : Second layer of the transmitarray with focal length of 1.5mm.

Fig. 4 .Fig. 4 .

 44 Fig. 4.4.1 (c) : Third layer of the transmit-array with focal length of 1.5mm.

Fig. 4 .

 4 Fig. 4.4.2 (b) E plane electric field of the 1.5mm focal distance transmit-array with no frame.

2 S

 2 ( x , y) dx dy4.4.3

4. 4

 4 Fig. 4.4.4 (a) H plane electric field of the 1.5mm focal distance transmit-array with frame

4 . 4 . 6 Fig. 4 . 4 . 6 :

 446446 Fig. 4.4.6: Scheme of a transmit-array focused in one dimension only: in this case the shape of the beam downstream the array is cylindrical

Fig. 4 .

 4 Fig. 4.4.7 (b) : Second layer of the 1D focusing transmit-array with focal length of 1.5mm.

Fig. 4 .

 4 Fig. 4.4.7 (c) Third layer of the 1D focusing transmit-array with focal length of 1.5mm.

Fig. 4 .

 4 Fig. 4.4.7 (a,b,c) shows the three layer generated automatically by the matlab code. The automation of the design algorithm requires that all the cells must be equal. Since the X dimension of the cells is very small, the designed structure requires a lot of simulation time. In general it is not necessary to keep the X dimension constant: indeed the phase variation given by eq. 4.1.5 is slower near the center of the transmit-array, and becomes faster near the edge. For this reason larger cells can be used in the center, and smaller cells near the edge. Fig.4.4.8 shows an example of a transmit-array designed with this criterion:

Fig 4 . 4 .

 44 Fig 4.4.10 (a) Linear scale (focusing at 1.5mm, without frame)

Fig. 4 . 4 . 11 :

 4411 Fig. 4.4.11: Curving effect at the edge of the transmit-array.

Fig. 4 .

 4 Fig. 4.4.13 (b) Electric field in linear scale for a transmit-array focusing in 1D with frame.

Fig. 4 .

 4 4.14 (a,b) shows the averaged and instantaneous electric field respectively, in the H plane, for the frame only: (a) Frame only averaged field. Fig. 4.4.14 (b) Frame only instantaneous field. The standing wave shown in figure 4.4.14 (a) for Z>0 is an index of high reflection coefficient.

Figure 4 . 4 .

 44 Figure 4.4.14 (b) shows a significant focusing effect due to the frame only. However the curvature of the beam downstream the frame is less accentuated than that in fig.4.4.12. Fig 4.4.15 (a,b) show the linear and dB scaled fields in the focal plane parallel to the transmit-array. In this case, the power which passes through the aperture of the frame is only 22% of the incident power (eq.4.4.4).

Fig. 4 .

 4 Fig. 4.4.15(a): Frame only electric field in linear scale

Fig. 4 .

 4 Fig. 4.4.16(b) Electric field transmitted by the frame with the transmit-array.

Fig. 4 .

 4 Fig. 4.4.17 (b) Far field of a dipole source transmitted by the transmit-array with the frame.

Fig. 4 .

 4 Fig. 4.4.18 (c): Average electric field transmitted the 20mm focusing transmit-array with frame.

Fig. 4 .Fig. 4 .

 44 Fig. 4.4.19 (a): Instantaneous electric field transmitted by the 20mm focusing transmitarray without frame.

Fig. 4 .

 4 Fig. 4.4.1 (a) Detail of the transmit-array focusing at 20mm with its frame.

  the transmit-array in terms of transmitted power • η frame =P frame →W frame / P i =4 %±3 % efficiency of the frame inside the beam waist of the frame • η TA =P TA →W TA / P i =1 %±5 % total efficiency of the transmit-array inside the beam waist of the transmit-array • η relative : frame→ W frame =P frame →W frame / P frame =72 %±3 % efficiency of the frame relative to the transmitted power in the beam waist of the frame • η relative : frame →W TA =P frame →W TA / P frame =55 %±3 % efficiency of the frame relative to the power in the same area of the beam waist of the transmit-array • η relative : TA →W TA =P TA →W TA / P TA =41 %±5 % efficiency of the transmit-array relative to the transmitted power inside the beam waist of the transmit-array

  In 1936 a US patent from A. G. Clavier and R. H. Darbord introduced the zone plate at microwave frequencies. Such patent describes both the alternate zone blocking and the half wave shifting zone plates. Another zone plate has also been introduced the same patent, with a one quarter distant plane mirror behind the zone plate to produce the desired phase shift[31]. A 1939 patent from E. Bruce (referring to the Clavier/Darbord patent of 1936) deals with both the reflecting and the half wave shift zone plates, the last one realized with a ϵ=20 dielectric constant material[31]. In 1960 Sobel and Wiltse developed and tested many zone plates at millimeter wavelengths. Moreover they also introduced phase corrections smaller than half a wavelength for the phase shifting zone plate[31]. In 1961 Van Burskirk reproduced the prototype proposed in 1936 by Clavier and Darbord, with a mirror behind the zone plate to perform the desired phase shift, to operate in the X band[31]. In 1985 Wiltse published a paper proposing a flat lens consisting of a transmitting zone plate of alternate dielectric zones adjusted to have the same thickness and providing the required phase shift between them. This paper, together with a paper from D.N.Black and J.C.Wiltse in 1987, and other two papers from J.C.Wiltse and J.E.Garret in 1991 have constituted the state of the art at that time and have been referenced dozens of times[31].

Fig 5 . 2

 52 Fig 5.2.2[32]: The optical path length from a source in the zone plate center to the focal point equal to the focal distance

Fig 5 . 2 . 3 [ 32 ]

 52332 Fig 5.2.3[32] In red the locus of the sources creating a constructive interference in the focal point

1

 1 Parameters of the two cells performing a π phase shift. where Δ φ is the phase shift between the output and the input of the cell; Z 1 , Z 2 and Z 3 are the imaginary part of the impedances of the first, second and third layer; D 1 , D 2 and D 3 are the dimension (in μ m of the cells for the first, second and third layer; W 1 , W 2 and W 3 are the spacing (in μ m ) between the patches for the first, second and third layer. The table is similar to tab. 4.4.1. A zone plate with a 1.5mm focal distance has been designed and simulated. It consists of 5 zones with the following radii: r 1 =900μ m , r 2 =1.3mm , r 3 =1.7mm , r 4 =2.0mm and r 5 =2.3mm . Fig 6.3.1 a,b,c shows a picture of the first, second and third layer of the zone plate respectively.

Fig 5 .

 5 Fig 5.3.1 (a) First layer of the zone plate focusing at 1.5mm

• 1 -

 1 η r =57 % percentage of the power reflected by the zone plate The low value of the transmitted power is due to the fact that the equations that define the impedance of the metasurface (which composes three layers of the cells) are valid for infinite structures. Such structure are composed of an infinite array of patches or square rings as those shown in fig 4.3.1 or 4.3.2. In our case we are assuming that the dimensions of the patches (or rings) is varying with the position. This blocking effect is much more evident here, for the zone plate, that with respect to the transmit-array, because the zone plate is bigger and the curving of the light at the edges of the zone plate (so that it can pass on the other side) is only a marginal effect what happens with the transmit-array.

Figure 5 .

 5 Figure 5.3.2 (a,b) shows the power flux of the field on the area A in linear and dB scale respectively:

  Fig 5.3.3 (a) Time averaged electric field of the transmitting zone plate without frame (in linear scale)

  Fig 5.3.5 (a) Time averaged electric field of the reflecting zone plate without frame (in dB scale)

6 . 2 . 3 ) 1

 6231 fig 6.2.1 Scheme of the transmission function of the grating. Its Fourier transform determines the power radiated in each order

Fig 6 . 4 . 1 :

 641 Fig 6.4.1: The incident beam from the source is reflected into several beams focused on the target plane

  y , z )e j(k x x+ k y y) dx dy(6.6.1)

  Fig 6.7.1: The aperture window and the focal window, where the relative function is equal to 1, are shown in yellow

9 .

 9 The obtained transformed focal field is inverse Fourier transformed, to obtain it in the spatial domain: E ' ' focal 10. The focal field in the spatial domain E ' ' focal is filtered by multiplying it by the focal window. The filtered focal field is then: E filtered =E ' ' focal ⋅focal window . 11. The filtered field is transformed in the spectral domain Ẽ filtered 12. The spectral filtered field is translated towards the phase grating plane. The evanescent are filtered away. We obtain: Ẽ ' ' ' aperture = Ẽ filtered ⋅spectral window⋅e k z hi .

16 .

 16 reflected by an infinite ideal phase grating. The finite aperture must be taken into account by multiplying E aperture IV times the aperture window. We obtain We substitute E ' ' aperture =E aperture V and we go back to step 7 until a certain number of steps is reached (in our case 20, since after 20 step the plot of the aperture field doesn't change anymore) simplified and synthetic scheme of the algorithm is shown in fig. 6.7.2.

Fig 6 . 7 . 2 :

 672 Fig 6.7.2: Scheme of the optimization algorithm representing the steps necessary to pass from the aperture field to the target field, and vice versa, until convergence

Fig 6 . 8 . 1 :

 681 Fig 6.8.1:Detail of the Gaussian beam: beam waist of the incident beam ( w 0 )is shown in blue, while d represents the path length traveled by the wave to reach the phase grating (in orange).

Fig 6 . 9 . 1 :

 691 Fig 6.9.1: Configuration of the phase grating: d represents the optical distance traveled by the wave between its beam waist (in blue) and the phase grating (in orange); the signal is then reflected into different beams focused on the focal plane (in yellow) by the phase grating

6. 9

 9 Fig 6.9.3 (b) Simulated electric field reflected by the phase grating (colour scale in V/m)

Fig 6 . 9 . 5 :

 695 Fig 6.9.5: Simulated electric field reflected by the phase grating for a bigger distance between it and the focal plane, and for larger spots (scale in V/m)

  Position of the generic observation point in the far field in the Cartesian coordinate system.

	Fig. 3.2.2.1
	The coordinate system which is
	assumed is shown in fig 3.2.2.1:

  2 π e ( W e

	2 h	+ 0.94 ) ]} -2	{	W e h	+	[W e /(π h)] W e 2 h + 0.94 } 3.2.4.2.3(c)
	where Z 0 is given by eq.3.2.4.1.2 (or 3.2.4.1.4). Eq. 3.2.4.2.3 a,b,c are expressed in dB/cm.
	Let's now calculate the dielectric loss. Two different formulas can be used indifferently, depending

  =0.1μ m and ϵ r =2.26 . The conductivity is assumed to be 4.4843 GS / m

	2.4.2.4(b)
	Eq 3.2.4.2.4 a,b are expressed in dB/cm too. The conductor and dielectric losses have been
	calculated at 600GHz for a microstrip line with the following characteristics: W =10μ m ,
	h=13μ m , t

  zone plate is an idea of Augustin Fresnel, a French engineer, which originates from his work on and diffraction, over one hundred years ago. He divided the aperture plane in Fresnel zones that are in phase opposition. Later Lord Rayleigh demonstrated that it is possible to increase the focusing effect by blocking the radiation of alternate zones. This information has been recorded in his personal notebooks, but he did not publish it in open literature[31].The first zone plate paper dates back to 1875 by Jacques Louis Soret. He proposed a zone plate with 98 alternating black and white zones. Professor Robert W. Wood improved such design by suggesting a half period phase correction in place of blocking the radiation from alternate zones. This would lead to a four fold improvement of the radiation at the focal point. R.W. Wood constructed such zone plate in 1898. After that the zone plate has remained a laboratory curiosity for many years[31].

(6.2.4)

The positive values of the imaginary impedances are shown in red. The eight, tenth and twelfth columns are constant and represent the cell dimension in the Y direction (the direction of the polarization of the electric field). The seventh, ninth and eleventh columns provide the values of W given by eq.4.3.6 (in black) or 4.3.1 (in red). The thirteenth and fourteenth columns show the abscissas (in μ m ) of the edges of each cell. The last column shows the inclination of the output beam as shown in fig. 4.2.3. In this case theta coincides with θ TE and is given by eq. 4.2.22. The length of the transmit-array is equal to twice the maximum abscissa in the first column ( 2⋅0,73 mm=1,46mm ). Fig. 4.4.9 shows the instantaneous electric field in the H plane: which is greater than the length itself of the transmit-array. However it can be shown that the 90% of the power is concentrated inside a radius of 1,3mm (the transmit-array length being 5.2mm).

Experimental measurements

For the experimental set up, four kinds of transmit-arrays have been realized, with a focal distance of 1,5mm, 5mm 10mm and 20mm, as well as a blank and empty frame with sapphire only, which provides a comparison to see the improved focusing due to the transmit-array. Even if the 1,5mm focal distance transmit-array is more interesting from the applicative point of view, it is extremely difficult to test it close to the device. For this reason we have only tested the 20mm transmit-array, even if the focusing effect is much less strong. All the four transmit-arrays that were fabricated focus the light on a line parallel to the transmit-array, at the focal distance. Such a line is parallel to the electric field polarization, and is normal to the transmit-array strips. The physical realization of the transmit-array has been carried out by François Joint, in a clean room at Orsay. It consists in a photolithography process with gold on sapphire. As a first step a photoresist layer is placed on the sapphire. Successively a mask with the transmit-array pattern is placed on the photoresist and UV radiation at 580nm is projected on the mask. At this point the part of the photoresist that was covered by the mask is cleaned up with water and leaves a portion of sapphire corresponding to the transmit-array pattern uncovered. After that a layer of gold is deposited on the remaining photoresist and the uncovered sapphire. Finally the remaining photoresist is cleaned with acetone and only the gold pattern of the transmit-array remains on the sapphire.

For the experimental set-up we need a high frequency signal generator, which is the RPG chain, and a detector, which is the terahertz pyroelectric detector or QCM, from the QCM instruments Ltd. The first experimental set-up is shown in fig. 5.5.1: the signal generated by the RPG is collimated by a lens with a 50mm focal distance. The power collimation is necessary since the beam generated by the RPG is strongly diverging, and would lead to a too low signal to be detected by the QCM as well as a curved wave front at the transmit-array. The transmit-array is placed between the lens and the QCM. The distance between the array and the QCM is equal to the focal distance of the transmit-array, therefore it is 20mm. The resulting measured field is obtained by scanning the XY plane parallel to the transmit-array at the focal distance from it. A metallic disc with an aperture hole of 1,5mm in the center is applied to the QCM to increase its resolution. The smaller the hole, the higher the spatial resolution, however the hole cannot be too small because the signal measured by the QCM would be too small. metamaterial used for the transmit-array of the preceding chapter (see ref 29.) In this way a zone plate will be which is easy to manufacture at very high frequencies. To the best of the author's knowledge, this is the first attempt of creating a simple metamaterial, whose behaviour is entirely described by analytical equations (no full wave simulation is necessary), to be used as flat lens operating at THz frequencies (the design has been proposed at 600GHz but it can be easily rescaled to higher frequencies e.g. up to 1.5THz).

Physics of the zone plate

A zone plate is an optic device constituted of alternating circles of opaque and transparent materials.

In this way the radiation passing through the zone plate is partially blocked. The alternating between light and shadow is performed in such a way that a constructive interference is generated at the desired focal point. Let's now consider a source located at a distance r from the center of the zone plate, and in the same plane. The distance l between the focal point and the source is called "optical path length" and is given by [START_REF] Center For | Fresnel zone plate theory, generation, tolerancing, fabrication, and applications[END_REF]:

The optical path length between the center of the zone plate and the focal point is simply equal to the focal length [START_REF] Center For | Fresnel zone plate theory, generation, tolerancing, fabrication, and applications[END_REF]:

reflecting surface. The depth t of the groove produces a phase offset due to the increase of the path length (fig 6.8.2). Such path length is equal to δ 1 + δ 2 =2 t cos θ . Therefore the phase variation due to the cell is given by: φ a -φ i =2k 0 t cos θ+ π where the term π takes into account the phase shift due to the reflection. Finally we can calculate the depth t as: The output of the described procedure is an array which contains the thickness distribution of the single cells as a function of the spatial coordinates. Such array will be imported in a matlab file created by prof. Massimiliano Casaletti, UPMC, to simulate its electromagnetic behavior with the physical optics approximation. The obtained aperture field will be then compared with that obtained by the aperture field optimization.

Simulations of the phase gratings

consider the geometry as illustrated in fig. 6.9.1:

ABSTRACT

The observation of the interstellar medium is very important at mm/(sub)mm/THz frequencies to understand how stars and planets form. Generally such observations rely on heterodyne receivers. These are instruments that achieve very high spectral resolution by down converting a high frequency signal towards a lower frequency one. In a heterodyne receiver the incoming signal is superimposed onto an artificial, well-known, monochromatic signal generated by the local oscillator (LO), chosen to be close to the frequency of the sky signal. The mixer produces the beat frequency signal. It has a frequency equivalent to the difference between the LO and sky signal frequency. Thus the sky signal is translated to a lower frequency, and it is easier to amplify and detect. Usually heterodyne receivers have only one spatial pixel with many frequency channels. prototypes have been realized recently with few pixels. Our objective is to develop arrays of hundreds of pixels.

In order to do that, some components which compose the heterodyne receiver must be radically rethought, such as the receiving antenna and the LO beam divider. Indeed the receiving antenna generally consists of a double slot antenna on a lens, or a horn antenna. Such antennas are not the best choice for arrays of many pixels since they have to be machined and mounted individually. Instead it is convenient to develop planar structures which can be easily produced in bulk in a single process. In particular we designed and simulated arrays of patch antennas, transmit-arrays and zone plates. The array of patch antennas consists of an array of metallic patches connected via a microstrip line and separated from the ground plane by a dielectric substrate. This configuration takes advantage of the array factor to reduce the beamwidth ABSTRACT of the incoming signal in place of the lens. However our simulations showed the array of patch antennas to be quite narrowband for a general purpose application, and quite difficult to realize. For this reason we also analyzed the possibility to use another solution such as the transmit-array. It is an array of several cells which provide a certain phase shift to an incoming wave in order to transform its phase front from planar to spherical. The purpose of the transmit-array is to focus the incoming beam towards a double slot antenna and a mixer placed below it. The simulations showed that a good focusing effect can be reached on a line. We fabricated such a transmit-array and tested it in the laboratory. Because of the small dimensions of a few millimeters these tests are difficult to carry out. Within the measurement error design and simulations are consistent. A third option of a planar lens was studied in the thesis: the zone plate. This is a particular kind of transmit-array which presents only two phase shift of 0° and 180°. The zone plates focus well, but are unfortunately not very efficient. The final part of the thesis introduces a particular kind of beam divider which allows beam splitting of the LO signal towards an array of four very closely packed mixers. To split the beam with such small relative angles is very difficult with the classical Fourier and Dammann grating, for this reason the method we proposed to design such a beam divider is very innovative. Indeed it allows the forming of arbitrary shaped beam patterns, which are not limited by the diffraction orders. Simulations show efficiencies up to 80% are very good in comparison with classical gratings. In summary in this thesis I have tried several radically different approaches to simplify heterodyne receivers and made a first step towards for large heterodyne arrays with hundreds of pixel