Antenna Beam Steering and Beam Forming at Mm-Wave and Thz Frequencies

Antenna beam steering and beam forming at mm-Wave and THz frequencies

Master Thesis submitted to the Faculty of the Escola Tècnicad’Enginyeria de Telecomunicacióde Barcelona Universitat Politècnicade Catalunya by Sara Vega Piña

In partial fulﬁllment of the requirements for the master in TELECOMMUNICATIONS ENGINEERING

Advisor: Mar´ıaConcepci´onSantos Co-Advisor: Daniel Nu˜no Barcelona, January 20th, 2021 Contents

List of Figures4

List of Tables6

1 Introduction8

2 Photonic Antenna Control 10 2.1 Phased Array Antennas Beam forming...... 10 2.2 Photoconductive Antennas for THz Radiation...... 11

3 Optical TTD Network 14 3.1 Multiwavelength Beam Steering Network...... 14 3.1.1 Effect of compensating delays...... 15 3.1.2 Geometrical Delays...... 16 3.1.3 Dispersive Delays...... 18 3.1.4 Maximum beam steered angle...... 19 3.2 CD-fading control...... 20 3.3 OTTDN Simulations...... 23

4 TeraHertz Systems 26 4.1 Definition of THz systems simulation methodology...... 26 4.1.1 Time-Domain simulation of PCAs (Primary fields)...... 26 4.1.2 Time-Domain simulation of Lens-coupled PCAs (Secondary fields).. 32 4.2 Validation of THz Simulation Methodology...... 35 4.2.1 PCAs modeling...... 35 4.2.2 Lens approach...... 36 4.2.3 Primary fields...... 38 4.2.4 Secondary fields...... 38 4.3 Simulation and measurement of THz systems...... 40 4.3.1 Fractal structures characteristics...... 40 4.3.2 Modeling of Sierpinski Fractal structures for THz radiation...... 40 4.3.3 Simulation of THz Far Fields with Sierpinski structures...... 44 4.4 Terahertz Beam Steering...... 49

5 Conclusions and future development 52

References 54

Appendices 59

A Matlab Codes 59 A.1 VPI Radiation Pattern .mat ﬁle...... 59 A.2 VPI Radiation Pattern Polar Plot...... 59 A.3 Radiation Pattern Projection...... 60 A.4 Radiation Pattern Cuts...... 61

2 B PCAs dimensions 63 B.1 Dipole dimensions list...... 63 B.2 Bow-Tie dimensions list...... 64

C Sierpinski PCAs Secondary Fields 65 C.1 Bow-Tie Secondary Fields...... 65 C.2 Sierpinski 1st-Order Secondary Fields...... 66 C.3 Sierpinski 2nd-Order Secondary Fields...... 67 C.4 Sierpinski 3rd-Order Secondary Fields...... 68 C.5 Sierpinski 4th-Order Secondary Fields...... 69

3 List of Figures

1 Vision of a future communication network architecture integrating THz wireless links into optical-fiber infrastructures...... 8 2 Currents distribution for an N elements PAA...... 10 3 Block scheme of PAA feeding network for beam steering...... 11 4 PCA parts. Biasing Voltage (1), Electrodes or antenna metalization (2), Sub- strate of photoconductive material, optical pulse (3) and the correspondent input Optical Pump (4) and output THz radiation (5)[10]...... 12 5 Radiated rays refraction with a single dielectric substrate (a) and a dielectric lens coupled (b)...... 13 6 PCA coupled to an Hyperhemispherical lens...... 13 7 OTTDN setup with DE-MZM...... 14 8 Group delay distribution against wavelength at the PAAEs: red line, right after the AWG (see Figure7) and green line, right before photodetection (see Figure 7)...... 15 9 Geometrical delays for each PAAE and a beam direction θ...... 17 10 Relative delays and beam direction for MW tuning...... 18 11 Maximum wavelength shift for δλ0 < 0 (a) and δλ0 > 0 (b)...... 19 12 Sketch of the typical options for external RF modulation at optical frequencies: DE-MZM configured for SSB modulation (a), Conventional PP-MZM (b), and DE-MZM for CD RF-amplitude free band control using the bias voltage (c)... 20 13 Comparison of fading function between a PP-MZM at Quadrature Point (blue) and a DE-MZM biased by condition (22) (red)...... 21 14 VPI setup of the MW-OTTDN...... 23 15 Array Factor diagram for a relative progressive wavelength shift δλ0 > 0 (a) and δλ0 < 0 (b)...... 25 16 Array Factor diagrams for θB = π/2 (a) and θB = 1.35π (b)...... 25 17 Primary fields (a) and Secondary fields (b) radiation scenarios...... 26 18 Back (a) and perspective (b) view of a PCA structure in CST...... 27 19 CST background configuration section...... 27 20 CST boundary conditions section...... 28 21 Discrete port connected to the electrodes (a) and excitation signal (b)...... 29 22 CST decoupling plane settings in the far field plot properties menu...... 29 23 Example of far field 2D plots without decoupling plane (a)-(c) and with a decoupling plane at Z = −20 (b)-(d). The Copolar components of the field are represented in (a)-(b) while the Crosspolar ones are plotted in (c)-(d)...... 30 24 Example of far field 3D plots without decoupling plane (a)-(c) and with a decoupling plane at Z = −20 (b)-(d). The Copolar components of the field are represented in (a)-(b) while the Crosspolar ones are plotted in (c)-(d)...... 31 25 Back (a), perspective (b) and left (c) view of a lens-coupled PCA structure in CST...... 32 26 Excitation signal for a frequency range 0.45 − 0.55 THz...... 32

4 27 Co-polar and Cross-polar far fields cuts for a mesh of 4 lines per wavelength (a), 8 lines per wavelength (b), 12 lines per wavelength (c) and 14 lines per wavelength (d) at f = 0.5 THz...... 33 28 Reference PCAs models for simulation methodology validation: H-shape Dipole (a) and Bow-Tie (b)...... 35 29 Dipole PCA schematic...... 35 30 Bow-Tie PCA schematic...... 36 31 Parameters setup of the Lens figure...... 36 32 Maximum lens angle...... 37 33 Co-polar and Cross-polar Primary fields projections comparison between the previous work [5] (left) and simulations (right) for a frequency f = 0.5 THz.. 38 34 Co-polar and Cross-polar Secondary fields cuts comparison between the previous work [5] (left) and simulations (right) for a frequency f = 0.5 THz..... 39 35 Schematic of bow-tie (a), sierpinski’s first-order (b), sierpinski’s second-order (c), and sierpinski’s third-order (d) structure...... 40 36 Example of sierpinski PCA schematic with La = 1200 µm and it = 4...... 41 37 Reflection coefficient of Bow-tie and sierpinski antennas of dimensions La = 400 µm (a) and La = 1200 µm (b)...... 44 38 Co-polar and Cross-polar Primary fields projections comparison between the Bow-Tie structure and different sierpinski orders at f = 0.5 THz...... 45 39 Secondary far field cuts of sierpinski PCAs: Co-polar component cut at φ = 90º (a) and Cross-polar component cut at φ = 45º (b) at f = 0.5 THz...... 46 40 Secondary far field cuts of a 3rd-Order sierpinski PCA of dimensions La = 400 µm: laboratory measurements (a) versus simulations (b) at f = 0.5 THz.. 47 41 Design of the optically steerable array with the dielectric lens [6]: Perspective view with dielectric lens and beam switching (a) and Back view with optical switch and fiber connections (b)...... 49 42 Rays direction representation when the PCA is positioned at x = 0 (a) and at x = ∆x (b)...... 49 43 CST setup of the lens-coupled PCA with an off-axis of ∆x = 1000µm (a), ∆x = 0µm (b) and ∆x = −1000µm (c)...... 50 44 Simulated radiation patterns for different off-axis displacements at f = 0.5 THz. 51 45 Co-polar and Cross-polar Secondary Fields projections comparison between Bow-Tie of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz. 65 46 Co-polar and Cross-polar Secondary Fields projections comparison between 1st-order Sierpinski of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz...... 66 47 Co-polar and Cross-polar Secondary Fields projections comparison between 2nd-order Sierpinski of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz...... 67 48 Co-polar and Cross-polar Secondary Fields projections comparison between 3rd-order Sierpinski of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz...... 68

5 49 Co-polar and Cross-polar Secondary Fields projections comparison between 4th-order Sierpinski of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz...... 69

Listings 1 Sierpinski Macro - Main Script...... 42 2 Sierpinski Macro - Iteration Script...... 42 3 Matlab Function to generate a .mat file from VPI output...... 59 4 Matlab Code to plot the polar radiation pattern from the .mat file...... 59 5 Matlab Code to plot radiation pattern circular projections from ASCII files exported from CST...... 60 6 Matlab Script to plot radiation pattern cuts from ASCII files exported from CST. 61

List of Tables

1 Summary of delays and wavelengths expressions for an array of N = 4 PAAE.. 16 2 Summary of simulation parameters...... 24 3 Summary of delays and wavelengths values for an array of N = 4 PAAE and the setup of Table2...... 24 4 Mesh cell size and computation time for different values of lines per wavelength and a bandwidth of 100 GHz at a central frequency f = 0.5 THz...... 34 5 List of lens dimensions values...... 37 6 List of ﬁxed sierpinski dimensions values...... 41 7 Total Radiated Power for the different sierpinski order PCAs simulated at f = 0.5 THz...... 46 8 Scanning angle for different off-axis values and an extension length Le = 1466 µm. 50 9 List of simulated dipole dimensions values...... 63 10 List of simulated bow-tie dimensions values...... 64

6 Abstract

The evolution of wireless communications and the emergence of novel applications will spur the demand of higher data rates, requiring, on one hand, unallocated spectrum in the millimeter Wave (mm-Wave) and Terahertz (THz) frequency bands to be exploited. On the other hand, the high propagation loss in the higher spectral bands will entail reduction of coverage areas, so that taking the signal in native format up to the radio interfaces leveraging existing optical fiber infrastructure will be advantageous. Along the same lines, beam steering techniques are expected to play an important role in the envisioned integrated wireless-photonic network. Photonic techniques for next generation fiber-wireless networks will be explored in this work. On one hand, a multiwavelength (MW) optical true time delay Network (OTTDN) to feed a phased array antenna (PAA) is presented. Beam steering capabilities can be demonstrated when applying MW tuning. A Dual Electrode Mach Zehnder Modulator (DE-MZM) as radio frequency (RF) external modulating stage allows to tune the operative band avoiding severe Chro- matic Dispersion (CD) fading to obtain a flattened response. An experimental setup for testing the technique in the UPC labs is described and studied in detail, showing the RF multiband spectral flat response potential of the technique, as well as the network conditions needed for beam steering with free-lobe operation. On the other hand, a simulation methodology to obtain the Co-Polar and Cross-Polar components of the far fields radiated by THz lens-coupled photoconductive antennas (PCAs) is presented and validated by applying and contrasting the solution with previous publications models, with excellent agreement, as well as by comparing simulated lens-coupled PCAs with measures made in our UPC labs with a Menlo time-domain spectrometer, showing the same pattern trends. Sierpinski structures are simulated at f = 0.5 THz using this method, observing that the power improvement with the fractal order, as well as the Airy pattern width depend on both the sampling aperture and the receiving antenna pattern. Finally, a simulation of an off-axis feeding example scenario shows the steering properties of the hyperhemispherical lens geometry.

7 1 Introduction

Data traffic in wireless networks is experiencing an explosive growth which implies future link data rates of hundreds of Gbps. Achieving these levels of demand will require to employ unallocated spectrum at the millimiter Wave (mm-Wave) and Terahertz (THz) frequency bands, allowing transmission through wider channels. An objective in the evolution of wireless systems has been to improve radio links to achieve increasingly higher speeds and transmission capac- ities. Nevertheless, it is of great interest to explore the possibility of integrating these wireless systems with existing optical-fiber infrastructures, taking advantage of the broadband and high capacity of optical systems and the huge market of fiber optic communications. 5G has driven the emergence of new applications, from Internet of Things (IoT) networks, which will be the mainstay of smart cities, to autonomous driving. Radar imagers at mm-Waves have acquired a high relevance for applications as unmanned vehicles, traffic control or security scanning, for which the design of phased array antennas (PAA) for beam steering has become a very active research topic. The use of optical True Time Delay Networks (OTTDN) to feed the PAAs is advantageous because it presents low-losses, has a small footprint and it is immune to interferences, besides it allows direct interfacing with high-capacity fiber networks through Radio-over-Fiber (RoF) fronthauling in communication networks [1]. The evolution towards 6G makes necessary to explore the THz frequency band, for which new transceivers models are needed. Moreover, integration of THz-over-Fiber (ToF) architectures is of high interest, leading the concept of a future network that combines geographically distributed THz transceivers (TRx) connected through widely deployed optical-fiber infrastructures, relying on direct Optical-to-THz conversion [2].

Figure 1: Vision of a future communication network architecture integrating THz wireless links into optical-ﬁber infrastructures.

The main goal of this thesis is to explore different photonic methods to achieve beam forming and beam steering at mm-Wave and THz frequencies. It will be focused on simulated demon- strations of optically feed radiation elements, with the ultimate goal of putting those ideas into practice in realistic contexts and lead to compact chips with mm-Wave and THz antennas con- trolled by optical ﬁber and which connect to the network. The objectives of this thesis are summarized below. • Analyse a new OTTDN architecture to feed PAAs with beam steering capabilities, con-

8 sidering an alternative configuration to the one presented in our previous work [3]. • Explore photonic techniques to overcome the severe CD RoF amplitude fading. • Characterize THz transceivers (Lens-coupled photoconductive antennas). • Define a simulation methodology to obtain the Co-polar and Cross-polar components of THz radiation. • Analyse beam forming and beam steering capabilities of THz systems. In Chapter2 different radiation elements and structures are introduced and characterized, focusing on mm-Wave PAAs, on the one hand, and on the other, on lens-coupled Photoconductive Antennas (PCAs) An OTTDN is presented in Chapter3[4], where multiwavelength (MW) tuning is theoretically developed to prove its beam steering principle. A Dual-Electrodue Mach Zehnder Modulator (DE-MZM) is proposed as an alternative to conventional Push-Pull Mach Zehnder Modulator (PP-MZM) to achieve a multiband operation without RF amplitude fading effects. An example of MW beam steering network is defined, characterized and simulated, demonstrating the predicted capabilities. Focusing on the THz band, a lens-coupled PCA scenario is studied in Chapter4. An Elec- tromagnetic (EM) simulation solver setup is characterized in order to define a THz analysis methodology, which is, in the first place, validated with the analysis of previous publications models [5]. Fractal Antennas geometries are characterized and simulated with the described methodology to compare their properties. Finally, an array optical switch will be considered to change the origin of the lens feeding in order to study the beam steering capabilities [6]. This thesis concludes in Chapter 5, with a summary of the obtained analysis results, demonstrating the potential of the systems presented, and a proposal of future work lines.

9 2 Photonic Antenna Control

In this chapter, optical techniques to control wireless antenna transmissions are introduced. For the mm-Wave band, the focus will be the optical feeding of PAAs to achieve beam forming and beam steering. In the case of THz band, Lens-coupled PCAs will be analysed.

2.1 Phased Array Antennas Beam forming Elementary antennas are quite limited in terms of directivity due to their electrical dimensions, which are usually of the wavelength order at most. Moreover, in practice the radiation pattern cannot be modeled as desired by applying any current distribution, which is the objective of beam forming and beam steering techniques. PAAs make it possible by aligning and feeding N elementary antennas with speciﬁc amplitudes and phases, as shown in Figure2, so that the interference is constructive in a desired direction θ [7].

Figure 2: Currents distribution for an N elements PAA.

The PAA Elements (PAAEs) signal overlapping is known as the Array Factor (AF) which is expressed as

N−1 jiψ FA(ψ) = ∑ aie (1) i=0 where an is the current amplitude feeding of each PAAE, and ψ is the phase difference between two consecutive elements. For constructive interference along direction theta the condition is

ψ = kd cosθ + α (2)

2π fRF with d the inter-element spacing, α the progressive phase and k = c the wavenumber, where fRF is the signal frequency and c the wave velocity in vacuum. The maximum radiation amplitude points towards the angle at which all the radiated waves are combined constructively, that is to say, when all the elements are in phase, which happens

10 when ψ = 0. This means that by controlling the progressive phase applied to the PAAE α, the radiation pattern can be pointed to a desired angle θ. In the following chapters, an optical system approach to control the phase feeding of a PAA (see Figure3) by taking advantage of the photonic capabilities will be presented, considering True Time Delay (TTD) techniques [8].

Figure 3: Block scheme of PAA feeding network for beam steering.

2.2 Photoconductive Antennas for THz Radiation The THz frequencies lie between the electronics region (radio, microwaves and millimeter waves) and the photonics region (infrared, visible, UV and x-ray). Generation of THz signals has become a challenging objective in the wireless applications, due to the obstacles emerged from trying to adapt the existing technologies to the THz frequency band. On one hand, due to the carrier mobility of oscillating conductors, radio frequency (RF) conventional sources are not capable of generating carriers at THz velocities. On the other hand, the THz photons energy c generated by electron transitions in semiconductors, which follows the equation E = h λ , is low as compared to the environment temperature (hv < kT), so generating photons as is done in optics, jumping electrons from the conduction band to the valence band is not effective because it is competing with the noise that makes them go up and down randomly. Currently, sources based on PCAs, as that represented in Figure4, are commonly used due to the emergence of picosecond and femtosecond laser pulses, able to generate pulses of length duration below 100 f s [9]. These ultrashort pulses may be used to illuminate a semiconductor in which a high voltage bias is maintained in between two electrode contacts. For photogenerated carriers lifetimes below the picosecond, a THz pulse may be radiated by an antenna imprinted over the semiconductor. The usual semiconductor substrate is Low Temperature Grown Gallium Arsenide [LT-GaAs] with an energy gap in between the valence and conduction band of around 1.42 eV and a refractive index n = 3.42. In order to generate a THz signal, the optical pulse is introduced in the antenna through the electrodes gap. The signal is absorbed, generating photocarriers inside the photoconductor and thereby by changing the material conductivity. These carriers are accelerated in the substrate by the Direct Current (DC) bias, generating a transient photocurrent, which induces an electromagnetic ﬁeld and radiation in the THz band.

11 Figure 4: PCA parts. Biasing Voltage (1), Electrodes or antenna metalization (2), Substrate of photoconductive material, optical pulse (3) and the correspondent input Optical Pump (4) and output THz radiation (5)[10].

On the other hand, to detect THz radiation, a portion of the optical pumped pulse is drifted towards the receiver and focused to the receiving PCA gap by the electrodes side, generating photocarriers in the substrate. When the THz beam is pointed to the PCA gap from the substrate side, those carriers are accelerated inducing a signal current between the PCA electrodes. The energy band gap restricts the range of wavelengths that may be used to those yield photon energies higher than the energy bandgap. That unfortunately excludes, the typical wavelengths in fiber optic communications λ = 1550 nm. The literature shows examples of alternative PCA substrates that allow exploitation of fiber optic devices and equipment, which usually involve a trade-off between the carrier lifetime and the band gap [11]. A main drawback may be observed when coupling antennas with dielectric substrates with high refractive index values. Radiated rays are deflected due to the change of medium (see Figure5), following the Snell’s law

nsub sinθr = sinθi (3) nair where θi and θr are the plane incident and refracted angles from the normal, and nsub, nair the refractive index of the substrate and air medium respectively. As the substrate refractive index will be always higher than the vacuum one, the rays will be strongly deﬂected, which means that for a certain incident angle, known as critical angle, the rays will be completely reﬂected as shown in Figure 5a, letting escape only the rays transmitted inside the solid angle of width

α Ω = 4π sin2 (4) 2 with α = arcsin(n−1) the boundary angle for total reﬂection.

12 (a) (b)

Figure 5: Radiated rays refraction with a single dielectric substrate (a) and a dielectric lens coupled (b).

In order to obtain a better coupling of the THz radiation to free space, dielectric lenses of high resistivity and a refraction index similar to the photoconductive substrate are used, reducing the back reflections due to critical angles incidence, as represented in Figure 5b. In this work, hyperhemispherical lenses, as the one shown in Figure6, will be modeled to prevent back reflections and to efficiently couple the radiated power into air.

Figure 6: PCA coupled to an Hyperhemispherical lens.

Currently, the THz studies and measurements are mainly focused on emission and detection by time-domain spectroscopy (TDS) systems, but frequency domain systems could also be found of interest [12]. THz generation will be analysed in the following chapters, by deﬁning a methodology to model and simulate the scenario presented in Figure6.

13 3 Optical TTD Network

In this ﬁrst part of the thesis, a MW OTTDN to feed a PAA will be presented in order to demonstrate beam steering capabilities when applying MW tuning. A DE-MZM as the RF external modulating stage will be proposed, allowing to tune the operative band avoiding severe CD fading and obtaining a ﬂattened response.

3.1 Multiwavelength Beam Steering Network Figure7 shows a sketch of the targeted OTTDN, which resembles the distribution of wavelength channels to users in access Wavelength Division Multiplexing-Passive Optical Network (WDM- PONs) [13].

An array of tunable input lasers, provides a MW signal comprising wavelengths λi, i = 1,2,...,N, numbered in descending order in Figure7, with N the number of PAA elements. For clarity, we will consider in this analysis an even number of array elements, but extension to an odd number is straightforward. After RF modulation with an external DE-MZM, which will be analyzed in more detail in the following sections, the signal propagates through a dispersive medium for which Dispersion Compensating Fiber (DCF) with negative dispersion coefficient D is conveniently chosen to provide large values of CD in a low volume and over a wide optical band. For the PAA steering, each laser wavelength will be conveniently tuned inside a specific channel of an Arrayed Waveguide Grating (AWG) whose outputs are connected to the PAAEs through an array of N photodetectors. In order to exploit WDM-PON equipment, the C-band of optical communications around λ0 = 1.55 µm is considered. Figure7 shows the group delay acquired by the RF signal over the different wavelength optical carriers after travelling through the fiber. This delay is proportional to the optical carrier wavelength difference, with the dispersion coefficient D, the proportionality constant. A typical value D = −85 ps/(nm · Km) is used for the dispersion compensating fiber.

Figure 7: OTTDN setup with DE-MZM.

14 3.1.1 Effect of compensating delays Owing to CD, the signal fed to every PAAE will have suffered a delay that will depend on the wavelength value inside its channel. Let the center wavelength of each OTTDN channel be

λ0i = λ01 + (i − 1)∆λAWG (5) with ∆λAWG the AWG channel spacing in wavelength units and λ01 the nominal center wavelength of the ﬁrst element of the array (placed at the top in Figure7), which is considered here the one with the lowest wavelength value, as shown in Figure8.

Figure 8: Group delay distribution against wavelength at the PAAEs: red line, right after the AWG (see Figure7) and green line, right before photodetection (see Figure7).

To properly exhaust the channel bandwidth provided by the AWG for a PAA symmetrically steered along both positive and negative angles, when all the input lasers are located at the center of their respective AWG channels, the PAA should radiate towards the broadside direction. This is accomplished by matching the arrival time of each center channel wavelength through the ﬁxed delays Ti placed before each PAAE, either before or after the photodetection stage. For lasers wavelengths in their AWG channel center, the value of the proportional delay in between PAAEs will be

TAWG = |D|L∆λAWG (6) with D the CD parameter and L the ﬁber length, so that the compensating delays for each PAAE need to be

Ti = (i − 1)TAWG (7)

In Figure8, we provide a graphical explanation to illustrate the effect of the compensating delays over the delay dependence on wavelength for this OTTDN structure proposal. The red

15 line values represent the delays suffered from the multiwavelength signal modulation with the RF envelope until it is ﬁltered in the AWG, τ0i, while the green line represents the delays after the compensating delay stage (check Figure7), corresponding to

τi = τ0i + Ti (8)

In the same way that the wavelengths have been deﬁned in (5), and following the representation of Figure8, τ01 may be taken as the absolute timing reference for RF signal arrival at the PAAEs input.

The speciﬁc Ti expressions to obtain the required delay values for a PAA of N = 4 elements are summarized in Table1.

Compensating Geometrical Wavelength i Delay (Ti) Delay (∆τGi) shift (δλi) 3 d 3 d 1 0 − 2 c sin(θ) 2 DLc sin(θ) 1 d 1 d 2 TAWG − 2 c sin(θ) 2 DLc sin(θ) 1 d 1 d 3 2TAWG 2 c sin(θ) − 2 DLc sin(θ) 3 d 3 d 4 3TAWG 2 c sin(θ) − 2 DLc sin(θ)

Table 1: Summary of delays and wavelengths expressions for an array of N = 4 PAAE.

3.1.2 Geometrical Delays The far ﬁeld radiated by the PAA will result from the interference of the PAAE individual far ﬁeld patterns. Considering the relative delay in between PAA elements, the radiation from each one will contribute to the array factor with a term

Ei = ai exp(− j2π fRF ∆τGi) (9) where ai is the feeding current amplitude, fRF is the frequency of the RF signal, and ∆τGi the geometrical delay due to propagation from each position in the array (see Figure9), which should be compensated with the delay of the current feeding each element. Those that are going to be more delayed when spreading, should be ahead in the OTTDN feeding. The delay ∆τGi will be different as seen from each direction theta due to the different elements position in the array, which have an inter-element spacing d. For an even number of channels N, it may be written in compact form as

N + 1 d ∆τ = i − sin(θ) (10) Gi 2 c with c the free-space wave velocity, d the a PAA inter-element spacing, and θ the ﬁeld point angle referred to the PAA broadside direction (see Figure7). Lobe-free operation among the full

16 λRF 180º beam steering of the PAA requires d < 2 [20]. Uniform distributions will be considered and therefore ai = a. The summation over all the array elements currents distributions provides the AF (1), that when convolved with the individual beam pattern of the antennas in the array Ei provides the PAA beam pattern

EPAA(θ) = AF(θ) ∗ Ei(θ) (11)

Isotropic radiators will be considered in the analysis, to focus on the PAA response, and therefore Ei(θ) = Ei.

Figure 9: Geometrical delays for each PAAE and a beam direction θ.

The mechanism by which the OTTDN achieves the beam steering towards angle θ consists in providing each PAAE with a RF signal replica which has the same amplitude and a progressive delay that compensates the geometrical delay due to the PAAE position in the array, see Figure 9. We may then write the complex signal of each PAAE as

Ii = Aexp(− j2π fRF ∆τNi) (12) with A the common RF amplitude and ∆τNi its corresponding network delay. The condition for beam steering is that network delays compensate the geometrical delays due to the position in space of each PAAE, pointing at the intended direction of maximum directivity.

∆τGi + ∆τNi = 0 (13)

Note that while for the network delays we measure the time starting from the first element in the array, here it is preferable to take the center of the array as the time reference due to the geometrical symmetry. As an example, the geometrical delays expressions for a PAA with N = 4 elements are summarized in Table1. As long as all the channels are synchronized to the same absolute time reference for the broadside condition, the relative delays between the PAAEs will define the beam direction by fulfil- ment of the condition (13).

17 3.1.3 Dispersive Delays Referring to Figure9, for radiation into angle θ, the required relative network delays for each PAAE need to be ∆τNi = −∆τGi, with ∆τGi the geometrical delay given by (10). The strategy for steering the direction of the PAA beam is to exploit dispersion to achieve the required delays into each PAAE. Therefore, the condition to ﬁnd the position of the wavelengths into each OTTDN channel for the targeted beam steering angle is

∆τNi = DLδλi (14)

With our numbering, and the references in Figure7, the relative delays need to decrease along the array for theta positive and to increase for theta negative. From (10), (13) and (14) we obtain the required phase shift with respect to the channel center for a beam steering angle theta as

N + 1 d δλ = i − sin(θ) (15) i 2 DLc

The PAAEs corresponding wavelengths are expressed in terms of δλi as

λi = λ0i + δλi (16)

A useful deﬁnition is the relative progressive wavelength shift in between contiguous PAAE for a speciﬁc angle of steering (See Figure 10 for a graphical explanation), which will of course depend on the sign of D, negative for hihgly dispersive DCF.

d δλ = sin(θ) (17) 0 DLc ´

Figure 10: Relative delays and beam direction for MW tuning.

18 3.1.4 Maximum beam steered angle The limit for the maximum steered angle, considering that the wavelengths are always equidis- tant to obtain the progressive delays, will come from the required wavelength shift of any of the PAAEs reaching the limit of the AWG channel half-bandwidth.

(a) (b)

Figure 11: Maximum wavelength shift for δλ0 < 0 (a) and δλ0 > 0 (b).

As shown in Figure 11, the channels that experience the greater wavelength shift are the ones further apart from the PAA center, which suffer a maximum displacement δλi = ±∆λAWG/2. Therefore, letting i = N in (15) the maximum steered angle with respect to the broadside direction is

c|D|∆λ θ = arcsin AWG (18) max d (N − 1)

It could be checked that, the higher the number of elements in the PAA or the narrower the AWG channel bandwidth, the larger the CD value needed to fulfill the requirement of a specific maximum steering angle. On the other hand, the higher the CD, the shorter the Distributed Feedback Bragg (DFB) tuning ranges to achieve the same angle shift, which means better accuracy and stability required for the tuning. Depending of specifications, the proper trade-off for the chosen value of D should be found.

19 3.2 CD-fading control A significant downside of a high value of CD is the RF amplitude fading effect which reduces the RF operative bandwidth [19]. In this section we will show how the RF amplitude fading typical of dispersive propagation of Double Side-Band (DSB) signals may be avoided at a specific operative frequency through appropriate biasing of a DE-MZM with no specific requirement for RF signal electrical shift, allowing the OTTDN to be tuned over a wide frequency band. Figure 12 illustrates different choices for the electrical connections in the RF modulation stage of the OTTDN.

(a) (b)

(c)

Figure 12: Sketch of the typical options for external RF modulation at optical frequencies: DE-MZM conﬁgured for SSB modulation (a), Conventional PP-MZM (b), and DE-MZM for CD RF-amplitude free band control using the bias voltage (c).

In a typical configuration, the external RF modulation stage is implemented using a conventional PP-MZM characterized by a 180º phase difference between RF signals applied to each interferometer arm. The usual outcome is a RF DSB amplitude modulation, for which frequency dependent CD-induced RF amplitude fading may cause significant amplitude distortion and RF amplitude notches at specific frequencies [19][21]. A DE-MZM as the RF modulating element of a dispersive OTTDN for PAA beam steering has been proposed in [17] to achieve a Single Side-Band (SSB) modulation of the data by driving each MZ arm with the same RF signal delayed by 90º. However the requirement of a specific electrical phase shift makes the technique intrinsically narrow band [22] and no frequency re- configuration is possible. We are going to make a comparison of the different modulations in order to demonstrate how the biasing technique can be used with a DE-MZM to switch the fading notch frequency. Taking as delay reference the time of arrival when all wavelengths are centered into their AWG,

20 τ01, the normalized amplitude ﬁeld in the small signal approximation of the signal at each AWG port when a DE-MZM with independent feed of each branch is used is given by

θ θ j φ− B E ≈ cos B + jme 2 cos(2π f (t − δλ DL)) (19) i 2 RF i

πVB πVRF where θB = , and m = with VB, VRF respectively the biasing voltage and RF signal peak Vπ Vπ amplitude applied to different electrodes of the DE-MZM, (see Figure 12c), with Vπ its half- wave voltage, and φ the phase due to CD expressed in (20), with λ0 the reference wavelength which is conveniently set to the center of the C-band [19].

πDLλ 2 f 2 φ = 0 RF (20) c The small-signal approximation in (19) implies that small enough RF amplitudes are applied to the modulators, that is VRF Vπ . From there, the normalized photodetected current at frequency fR at each PAAE will be

θ θ I ≈ cos B sin B − φ cos(2π f (t − δλ D )) (21) PDi 2 2 RF i L

The biasing voltage VB may be adjusted to compensate the CD-induced amplitude fading given by φ, using the condition

θB = (2n − 1)π + 2φ (22)

Figure 13: Comparison of fading function between a PP-MZM at Quadrature Point (blue) and a DE-MZM biased by condition (22) (red).

21 In Figure 13 an example comparison for the operation frequency 8GHz, between the fading function when biasing different modulators is represented. For a physical system with a dispersion coefﬁcient DL, the notch of the PP-MZM fading function (blue) will remain in the same position despite the biasing applied, while the notch of the DE-MZM fading function (red) can be displaced to ﬂatten the photodetected current amplitude around the desired frequency.

22 3.3 OTTDN Simulations A photonic simulation software (VPI TransmissionMaker) has been used to assess the potential of the technique. Figure 14 shows the system schematic developed to simulate the radiation pattern of the PAA.

Figure 14: VPI setup of the MW-OTTDN.

The system has been constructed by different stages. The ﬁrst one starting by the left side of the schematic consists on the Lasers feeding, whose wavelengths have been set as

N + 1 λ = λ + (i − 1)∆λ + i − δλ (23) i 01 AWG 2 0 where the central wavelength of the ﬁrst channel λ01 is deﬁned as

∆λ λ = λ − (N − 1) AWG (24) 01 0 2 An optical coupler is added to introduce the N carriers to the modulation stage, for which a MZM element has been used. A sinusoidal signal source is used to inject the RF signal at fRF through the upper branch, while the bias is introduced by a DC source through the lower one. The remaining DC sources are configured with a null voltage. It is worth noting that setting these inputs to ground will result in errors which are neither directly nor easily linked to this fact. An optical Fiber of L fib length and with a dispersion coefficient D is used for the dispersive stage. The modulated carrier wavelengths need to be filtered before detection in order to separate them and send each one towards a different PAAE, for which an AWG element with N outputs is used. A delay element with the values defined in (7) is introduced before each photodetector to accomplish the delay (8).

23 Once the modulated signals are photodetected, the amplitude Ai and phase Phi need to be obtained to represent the radiation pattern, for which Two Port Analyzers are used. A last Simula- tion Interface element is used to run a Matlab function (see Appendix A.1 and A.2), introducing Ai and Phi values as inputs. This function plots and saves a 2D radiation pattern over θ angle. λ Assuming a PAAE spacing of d = 4 complying with the secondary lobe-free radiation condition, a full 180º beam steering requires a minimum dispersion |D|minL = 58 ps/nm, according to the maximum beam condition (18). Considering standard DCF with distributed dispersion coefﬁcient D = −85 ps/(nm · Km), it corresponds to roughly 0.68 Km.

Parameter Deﬁnition Value

λ0 Reference wavelength 1550 nm N Number of PAAEs 4

∆λAWG AWG channel spacing 1.6 nm ps D Dispersion −85 nm·Km L Fiber Length 4 Km

Table 2: Summary of simulation parameters.

In order to focus the study towards an experimental test, the values have been adapted to the available resources, yielding the parameters listed in Table2 with an inter-element spacing d = 4λ/5 at 8 GHz and a dispersion of DL = −340 ps/nm. Table3 lists the required delays by setting these simulation parameters.

Compensating Geometrical Wavelength i Delay (Ti) Delay (∆τGi) shift (δλi) 1 0 ns −0.15sin(θ) ns 0.45sin(θ) nm 2 0.54 ns −0.05sin(θ) ns 0.15sin(θ) nm 3 1.08 ns 0.05sin(θ) ns −0.15sin(θ) nm 4 1.62 ns 0.15sin(θ) ns −0.45sin(θ) nm

Table 3: Summary of delays and wavelengths values for an array of N = 4 PAAE and the setup of Table2.

A ﬁrst set of simulations have been run to demonstrate the steering capabilities of the system. Figure 15 shows the Array Factors resulting from tuning the relative progressive wavelength shift δλ0. The lobe-free operation is limited to ±14º, which implies a theoretical relative progressive wavelength shift of δλ0 = ±73.5 pm. It can be observed that performing a simulation with a relative progressive wavelength shift δλ0 = 80 pm, a second lobe with the same amplitude of the primary one shows up, proving the restriction.

24 (a) (b)

Figure 15: Array Factor diagram for a relative progressive wavelength shift δλ0 > 0 (a) and δλ0 < 0 (b).

The biasing technique is demonstrated by a second set of simulations. In this case, the relative progressive wavelength shift δλ0 has been fixed, while the RF frequency is shifted around 8 GHz. In order to obtain a flat response along the frequency band, the bias is optimized using condition (22). It is observed in Figure 16a how the Quadrature Point (QP) biasing θB = π/2 results in a significant CD fading amplitude penalty of around 5.1 dB when the RF frequency is shifted from 7 GHz to 9 GHz. In order to shift the CD-fading free band the MZM is biased at θB = 1.35π, resulting in the flattened amplitude response shown on Figure 16b.

(a) (b)

Figure 16: Array Factor diagrams for θB = π/2 (a) and θB = 1.35π (b).

25 4 TeraHertz Systems

The goal of this second part of the thesis is to deﬁne a methodology to study and analyse THz radiation using lens-coupled PCAs and use it to analyse different antenna models. In the following sections, different kinds of PCAs geometries are going to be modeled and characterized with and without the lens coupling for, on the one hand, validate our methodology with the existing literature and, on the other one, analyse the properties of some geometries of interest in order to prove the beam forming and beam steering properties of the lens.

4.1 Definition of THz systems simulation methodology The THz systems under analysis, as explained in Section 2.2, consist of different lens-coupled photoconductive antennas, from which the Co-polar and Cross-polar components of the radiated fields want to be obtained. CST microwave studio [23] will be used as the EM simulation solver in charge of computing the Maxwell Equations to obtain the THz radiated signals of the modeled antennas. In order to validate our methodology, some comparisons with previous works results will be carried out [5], for which two simulation scenarios are defined to obtain both the Primary and Secondary fields. The Primary fields are those ones radiated by the PCA inside the dielectric lens, while the Secondary ones are the ones that come out of the lens, as represented in Figure 17.

(a) (b)

Figure 17: Primary ﬁelds (a) and Secondary ﬁelds (b) radiation scenarios.

4.1.1 Time-Domain simulation of PCAs (Primary fields) The first step taken to analyse the radiation of the THz systems is to simulate the Primary fields radiated by the PCAs into the dielectric lens medium. PCA structures composed by a Perfect Electric Conductor (PEC) mask over a Low-Temperature Grown Gallium Arsenide (LT- GaAs) substrate, like the one shown in Figure 18, will be modeled for it. It is assumed that this

26 dielectric medium occupies a semi-inﬁnite half space, which may be approached by applying the following settings.

(a) (b)

Figure 18: Back (a) and perspective (b) view of a PCA structure in CST.

First of all, the background material should be set. This is the material that is assumed to lie at the boundaries of the simulation box. This is appropriate for the substrate side, as emulating radiation into a semi-infinite substrate. The problem is that with CST, only one background material may be defined, which is assumed the same at all boundaries. The lens material is considered the same as the substrate, so the GaAs dielectric constant ε = 12.94 is set in the background configuration section shown in Figure 19.

Figure 19: CST background conﬁguration section.

27 The substrate boundary is not appropriate for the antenna metallization side because this is in contact with air, so an air box of thickness Lair = 20 µm is placed at the bottom of the antenna structure for the simulation. This will be complemented later by considering a decoupling plane to eliminate the effect of back reflections produced at the boundary in between air and the background material. In order to simplify the computation memory and runtime needed for the simulation, symmetry planes are defined at the Boundary Conditions configuration window (see Figure 20a). A magnetic plane is configured in the YZ plane due to the even symmetry of the antenna, while an electric plane is assigned to the XZ plane due to the odd symmetry produced by the bias port. No symmetry configuration is valid for the XY plane, due to the difference of the substrate and air materials.

(a) (b)

Figure 20: CST boundary conditions section.

In the practice, the PCA electrodes gap is fed by an optical pumped pulse coming from a laser source combined with a bias voltage. Different electrodes shapes, like interdigitated and plasmonic contacts, have been analysed to improve the antenna performance [24, 25]. To simulate the laser optical pump and bias applied to the electrodes, a S-parameter discrete port is used, which will inject current into the mesh central cells of the antenna, where the gap is located. Figure 21 shows the discrete port included to the PCA schematic and the Gaussian pulse used as the excitation signal. As the optical illumination cannot be added in the simulation, basic electrodes lines will be modeled.

28 (a) (b)

Figure 21: Discrete port connected to the electrodes (a) and excitation signal (b).

Finally, the frequency range under analysis should be limited by defining the minimum and maximum frequency in the simulation settings. Field monitors can be inserted to characterize the antenna at certain frequencies. For Primary fields simulations, the frequency range is set from 0 to 2 THz in order to obtain the characteristic reflection and energy curves, and far field monitors at specific frequencies are added to obtain the radiation patterns. CST Time-Domain (TD) solver is used to compute the simulations. Once the TD simulation has successfully finished, the output fields are available in the Farfields directory. We are interested in differentiating the Co-polar and Cross-polar components of the radiated THz signal, which can be obtained by choosing the Linear Directional polarization when displaying the far field plot.

Figure 22: CST decoupling plane settings in the far ﬁeld plot properties menu.

29 As mentioned previously, the decoupling plane option prevents reflection at the air-substrate boundary to arrive at the substrate. An interesting discussion about the use of decoupling planes for PCA simulation may be found in [26]. Figure 22, shows hot to select the decoupling plane option and position inside the Properties section of the far field Plot. For the Primary Field simulations, a decoupling plane at the limit of the air box (Z = −20) is added. In Figures 23 and 24 the comparisons of an example far field’s 2D and 3D plots when considering a decoupling plane and when not are presented. As said, the decoupling plane prevents interactions in between fields at each side of the plane, and in particular, since the antenna radiation is mainly towards the substrate side, when se- lecting the decoupling plane option, no fields appear at the air side. The relevant fields for our investigation are those radiated towards the substrate and this is where we have to assess the effect of the decoupling plane. In the Figures 23 and 24, the substrate radiation features a more significant effect of the decoupling plane in (a), (b), than in (c), (d).

(a) (b)

Figure 23: Example of far ﬁeld 2D plots without decoupling plane (a)-(c) and with a decoupling plane at Z = −20 (b)-(d). The Copolar components of the ﬁeld are represented in (a)-(b) while the Crosspolar ones are plotted in (c)-(d).

30 (a) (b)

Figure 24: Example of far ﬁeld 3D plots without decoupling plane (a)-(c) and with a decoupling plane at Z = −20 (b)-(d). The Copolar components of the ﬁeld are represented in (a)-(b) while the Crosspolar ones are plotted in (c)-(d).

31 4.1.2 Time-Domain simulation of Lens-coupled PCAs (Secondary fields) The fields of interest of the THz systems are those that come out of the lens (Secondary fields), which are the ones that we are actually able to measure with a physical setup. To simulate this scenario, a lens is added to the PCA structures of Figure 18 to obtain a lens-coupled structure like the one shown in Figure 25.

(a) (b) (c)

Figure 25: Back (a), perspective (b) and left (c) view of a lens-coupled PCA structure in CST.

In this case, the background medium is considered a semi-infinite space of vacuum in all directions, with a refraction index n = 1 and a dielectric constant ε = 1, which is configured in the Background menu of Figure 19. The symmetry planes and the ports are added in the same way as for the Primary fields simulations. When the lens is added to the antenna structure, the electrical size of the model increases, and so do the number of mesh cells to obtain an accurate result. This may lead to a quite high memory usage due to the significant computer RAM required, as well as a long computation time. This limitations may be faced by adjusting the simulation bandwidth and the number of mesh cells used in order to chose a proper input signal duration and time step width respectively [27]. To perform our simulations, a bandwidth of 100 GHz will be chosen for low frequencies ( f < 1 THz), where multiple resonance is expected, generating the excitation pulse observed in Figure 26.

Figure 26: Excitation signal for a frequency range 0.45 − 0.55 THz.

32 For the mesh optimization, a trade off between the runtime and the accuracy of the results is needed. A lens-coupled dipole PCA has been chosen for this analysis. It is expected that the greater the number of mesh cells, the more accurate the results. The number of mesh cells may be shifted by modifying the number of lines per wavelength. Nevertheless, when increasing the frequency, the number of mesh cells for a chosen value of lines per wavelength will grow signiﬁcantly. For this reason, and taking into account that the lens that we will measure have their optimal performance at low THz frequencies, the analysis will be focused on f < 1 THz. At the central frequency of interest f ∼ 0.5 THz, the number of lines per wavelength has been increased from 4 to 14, showing the values of mesh cells and approximated computation time summarized in Table4. The resulting far ﬁeld cuts from the simulations with different number of mesh cells are presented in Figure 27.

(a) (b)

Figure 27: Co-polar and Cross-polar far ﬁelds cuts for a mesh of 4 lines per wavelength (a), 8 lines per wavelength (b), 12 lines per wavelength (c) and 14 lines per wavelength (d) at f = 0.5 THz.

The simulations carried out have made it possible to verify the improvement in the accuracy of the obtained far fields when increasing the number of mesh cells. However, the improvement curve flattens when reaching 12 lines per wavelength in the mesh settings. If we observe Figure 27c and Figure 27d, corresponding to a mesh of 12 lines per wavelength and 14 lines per wavelength respectively, there are no significant differences. The power of the side lobes increases a bit with the mesh, but the radiation pattern shape is the same. Nevertheless, the runtime suffers

33 an increase from 24 up to 72 hours, which means that this last value of mesh cells does not add value to the simulations. For this reason, a value of 12 lines per wavelength is selected to set the number of mesh cells.

Lines per Number of mesh Computation time wavelength cells (millions) (hours) 4 3.9 1 8 28.0 3 12 91.3 24 14 142.2 72

Table 4: Mesh cell size and computation time for different values of lines per wavelength and a bandwidth of 100 GHz at a central frequency f = 0.5 THz.

In the same way as with the Primary fields Simulation, CST TD solver is used to compute the fields which will be available in the Farfields directory once the simulation has successfully finished.

34 4.2 Validation of THz Simulation Methodology In order to validate our simulation methodology to obtain THz radiation results, previous publications have been taken as reference [5]. Two PCAs models have been chosen for the analysis: an H-Shape Dipole (Figure 28a) and a Bow-Tie (Figure 28b). These models are presented in the following section, which will consist of a PEC plane structure modeled over a LT-GaAs substrate brick of sz = 0.1 µm thickness.

(a) (b)

Figure 28: Reference PCAs models for simulation methodology validation: H-shape Dipole (a) and Bow-Tie (b).

4.2.1 PCAs modeling Despite different geometries of the PCA can be considered, there are two main parameters that should be taken into account when modeling the antennas. In one hand, the antenna length La, which would determine the resonance condition, and in the other hand, it is also important to size the electrode gap g smaller than the laser pointer to have an uniform illumination. In the ﬁrst place, an H-shape Dipole has been modeled using basic brick shapes with the dimensions shown in Figure 29 with a gap size gx = gy = 10 µm. A dipole length Ld = 30 µm and a bias line width wby = 10 µm have been used (see Appendix B.1 for the remaining dimension values).

Figure 29: Dipole PCA schematic.

35 The second model contrasted with previous work follows a Bow-Tie geometry, with structure and dimensions shown in Figure 30. Brick shapes have been used to model a gap size of gx = gy = 10 µm and a ﬂare angle a = 90º, as well as the remaining structure (see Appendix B.2 for the remaining dimension values).

Figure 30: Bow-Tie PCA schematic.

The structures have been designed following a parametric approach, based on the deﬁnition of variables allowed by CST so that resizing is automatic and straightforward. The list of parameters can be found in AppendixB.

4.2.2 Lens approach The lens deployed for the simulations follows a substrate hyperhemispherical model, which is composed by a half sphere of radius R plus an extended sphere cut of length Le, as is represented in Figure 31.

Figure 31: Parameters setup of the Lens ﬁgure.

36 Figure 32: Maximum lens angle.

The lens is required to minimize total internal reﬂection at the substrate-air interface, and therefore the condition is that all incident angles at the lens-air interface must be below the critical angle. As can be observed in Figure 32, the maximum angle of incidence is related to the di- L mensions of the lens by sin(θmax) = R . Therefore

L 1 e = sinθ < sinθ = (25) R max c n where n is the substrate refractive index. Taking the extreme case of equation (25), the extension length Le must be

R L = (26) e n A GaAs substrate hyperhemispherical lens is modeled for the simulations with a refractive index n = 3.42 and a radius R = 5 mm.

Name Value (µm) Description R 5000 lens radius n 3.42 refractive index

Le 1466.28 extension length

Table 5: List of lens dimensions values.

37 4.2.3 Primary fields The Primary fields of the dipole and the bow-tie have been simulated, obtaining the far fields at f = 0.5 THz. The exported ASCII files of the results have been used to plot the circular projections of Figure 33 (right ones) using Matlab (see Appendix A.3), which are compared with those obtained from the same PCAs shapes in previous analysis (left ones) [5].

H Dipole

(a) (b)

Bow-Tie

Figure 33: Co-polar and Cross-polar Primary ﬁelds projections comparison between the previous work [5] (left) and simulations (right) for a frequency f = 0.5 THz.

It can be seen that following our methodology, the projections have similar patterns to those of the previous publications. On the one hand, the dipole maintains a slotted pattern for the Co-polar component, while four maximums appear around φ = ±15º and φ = 180 ± 15º for the Cross-polar component. On the other hand, the Bow-tie shows a uniform pattern for the Co-polar component while the four maximums, in that case of lower amplitude, are observed at φ = ±45º and φ = ±135º. It is worth mentioning that the results used for comparison [5], have also been obtained with simulations carried out in CST and therefore the small differences observed have to be attributed to slightly different choices of the simulation parameters.

4.2.4 Secondary fields The Secondary fields of the dipole and the bow-tie have been obtained, the far field cuts of which are represented in Figure 34 (right ones) by using Matlab (see Appendix A.4), which are also compared with those obtained from the same PCAs shapes in previous analysis (left

38 ones) [5]. In this case, the simulation results in [5] were obtained by including the effect of the lens through an approximate method based on a combination of Physical Optics (PO) and Geometrical Optics (GO) to the Primary ﬁelds obtained with CST, whereas we simulated in CST the complete structure including the lens. H Dipole

(a)

Bow-Tie

(b)

Figure 34: Co-polar and Cross-polar Secondary ﬁelds cuts comparison between the previous work [5] (left) and simulations (right) for a frequency f = 0.5 THz.

In the same way as with the Primary ﬁelds, similarities can also be observed in the Secondary ﬁelds. On the one hand, in the case of the dipole, when comparing the CST results with the PO-GO it can be observed that the position of sidelobes for the cut at φ = 0º of the Co-polar component is the same in both methods, θ = ±35º approximately, with higher sidelobe levels found in the PO-GO case. For the Cross-polar component, it can also be observed how the lobes appear at φ = ±45º and φ = ±135º. On the other hand, a single lobe appears in the Co-polar component of the bow-tie, pointing towards the broadside direction, while the four lobes remain at φ = ±45º and φ = ±135º with an amplitude 15dB below the maximum, following similar trends for both methods. The use of different approximation and simulation tools may explain the differences observed in the resulting patterns, although both methods seem to provide the same basic features.

39 4.3 Simulation and measurement of THz systems THz radiated far ﬁelds’ characteristics, such as the bandwidth and the power amplitude of the generated signal, is inﬂuenced by the PCA shape. Focusing on the Bow-tie shape, recent studies have demonstrated an enhancement of the radiated power by modeling a fractal structure of the antenna. This structure changes the surface current distribution to form individual sub- wavelength radiators which are coherently coupled [28]. In this section, an analysis of different order Fractal PCAs following the methodology described above will be carried out.

4.3.1 Fractal structures characteristics Mathematically, the term fractal, coming from the Latin word fractus ’broken, irregular’, is used to deﬁne a self-similar subset of a base pattern. A self-similar object consists on the replication of its structure at different scales, normally made of smaller copies of itself, which is also known as unfolding symmetry [29]. One fractal structure that has been demonstrated to possess multi-band frequency characteristics is the sierpinski triangle, composed by a set of equilateral smaller triangles. It is built from a base equilateral triangle, from which a central smaller triangle, whose vertices are located at the midpoints of the base triangle, is removed to achieve the ﬁrst order, and so on with the remaining triangles to reach the ith order, as represented in Figure 35.

(a) (b) (c) (d)

Figure 35: Schematic of bow-tie (a), sierpinski’s ﬁrst-order (b), sierpinski’s second-order (c), and sierpinski’s third-order (d) structure.

4.3.2 Modeling of Sierpinski Fractal structures for THz radiation The sierpinski PCA structure that will be modeled and analysed is shown in Figure 36. The PEC mask consists of four pads, of which the upper ones are connected by a bias line, in the same way as the lower ones. Two symmetric sierpinski triangles are placed on top of each bias line, which are separated by a vacuum gap.

40 Figure 36: Example of sierpinski PCA schematic with La = 1200 µm and it = 4.

In order to build the structure, basic shapes are used to model the pads, bias lines and alignment marks in the ﬁrst place, setting the dimension values listed in Table6. The antenna length La will be switched to analyse different sierpinski dimensions.

Value (µm or Name Description deg) gx 10 x dimension of the gap

gy 10 y dimension of the gap a 60 Equilateral triangle angle

Lchip 2000 chip size

xpad 400 x dimension of the pad

ypad 200 y dimension of the pad

wb 2 y width of the bias line

wa 20 width of the alignment mark

xa 150 x length of the alignment mark

ya 150 y length of the alignment mark

xa sp 200 x void space for alignment mark

ya sp 200 y void space for alignment mark

Table 6: List of ﬁxed sierpinski dimensions values.

A CST Macro is developed to construct the sierpinski triangle structure based on the antenna

41 length La and iteration order it. The Main Script presented in Listing1 is executed when calling the Macro. It calculates the base triangle side length as a function of the antenna length La and computes the position of the triangle vertices, which will be passed as parameters to the sub function sierpinskia, next to the triangle angle a, the iteration order it and an initialized counter ii.

1 Sub Main () 2 Dim wwa As Double 3 Dim P1x,P1y,P2x,P2y,P3x,P3y As Double 4 Dim dg As Integer 5 Dim ii As Integer 6 7 ii=1 8 9 wwa=La*Tan(aa/2*pi/180) 10 11 P1x=0 12 P1y=0 13 P2x=wwa/2 14 P2y=−La/2 15 P3x=−wwa/2 16 P3y=−La/2 17 18 sierpinskia(P1x,P1y,P2x,P2y,P3x,P3y,a,it,ii) 19 20 End Sub Listing 1: Sierpinski Macro - Main Script.

The sub function sierpinskia presented in Listing2, will iterate as many times as the sierpinski order it set, computing in each iteration the ith-order subtriangles vertices from the base ones. A tolerance parameter tole is used to overlap the triangles vertices, which is needed to allow the current ﬂows. When the counter reaches the total of iterations conﬁgured, each single subtriangle is built.

1 Sub sierpinskia (P1x,P1y,P2x,P2y,P3x,P3y,aa As Double,dg As ... Integer,ByRef ii As Integer) 2 3 4 Dim b, b1, b2, b3, h, xo, yo, P1tx, P1ty, P2tx, P2ty, P3tx, P3ty ... As Double 5 6 If dg<1 Then 7 8 ' drawa triangle 9 10 ' find equal length side 11 12 b1=Sqr((P1x−P2x)ˆ2+(P1y−P2y)ˆ2) 13 b2=Sqr((P2x−P3x)ˆ2+(P2y−P3y)ˆ2) 14 b3=Sqr((P1x−P3x)ˆ2+(P1y−P3y)ˆ2)

42 15 16 b=(max(max(b1,b2),b3)+min(min(b1,b2),b3))/(1+2*Sin(aa/2*pi/180)) 17 18 ' find height 19 20 h=tole*b*Cos(aa/2*pi/180) 21 22 ' find center 23 24 xo=(P1x+P2x+P3x)/3 25 yo=(P1y+P2y+P3y)/3 26 27 '' triangle vertices coordinates 28 29 P1tx=xo 30 P1ty=yo+2/3*h 31 32 P2tx=xo−tole*b*Sin(aa/2*pi/180) 33 P2ty=yo−h/3 34 35 P3tx=xo+tole*b*Sin(aa/2*pi/180) 36 P3ty=yo−h/3 37 38 With Extrude 39 .Reset 40 .Name"t"& cstr(ii) 41 .Component"component1" 42 .Material" \ac{pec}" 43 .Mode"Pointlist" 44 .Height 0 45 .Point P1tx,P1ty 46 .LineTo P2tx,P2ty 47 .LineTo P3tx,P3ty 48 .LineTo P1tx,P1ty 49 .Create 50 End With 51 52 End If 53 54 ii=ii+1 55 56 If dg>0 Then 57 58 sierpinskia(P1x, P1y, (P1x+P2x)/2, (P1y+P2y)/2, (P1x+P3x)/2, ... (P1y+P3y)/2, aa, dg−1, ii) 59 sierpinskia(P2x, P2y, (P1x+P2x)/2, (P1y+P2y)/2, (P2x+P3x)/2, ... (P2y+P3y)/2, aa, dg−1, ii) 60 sierpinskia(P3x, P3y, (P3x+P2x)/2, (P3y+P2y)/2, (P1x+P3x)/2, ... (P1y+P3y)/2, aa, dg−1, ii) 61 62 End If 63 64 End Sub Listing 2: Sierpinski Macro - Iteration Script.

43 Once the sierpinski triangle is obtained, it is manually replicated using the mirror tool to form the bow tie antenna, a gap of dimensions gx = gy = 10 µm is subtracted and a substrate box of the chip dimensions with a width sz = 0.1 µm and of GaAs dielectric material is added.

4.3.3 Simulation of THz Far Fields with Sierpinski structures Both simulation of the Primary and Secondary fields of different order sierpinski antennas (Bow-Tie, sierpinski 1st-Order, sierpinski 2nd-Order, sierpinski 3rd-Order and sierpinski 4th- Order) have been carried out, for which antennas of dimensions La = 400 µm and La = 1200 µm have been modeled using the Macro explained above. Before showing the Fields obtained from the simulations, an analysis of the antennas resonance has been carried out. The sierpinski structures are characterised for its multi-band performance, which can be observed in the reflection coefficient curves (see Figure 37), presenting n + 1 resonant bands for the nth fractal order. It can be also observed that the resonance frequencies are log-periodically spaced with a factor δ = 2. Moreover, the longer the antenna, the lower in frequency are the resonance peaks.

(a) (b)

Figure 37: Reﬂection coefﬁcient of Bow-tie and sierpinski antennas of dimensions La = 400 µm (a) and La = 1200 µm (b).

After taking a look to the Primary ﬁelds components of the analysed PCAs, some trends may be observed (see Figure 38). On the one hand, the Co-polar component concentrates its power along the y axis (antenna length direction) when the sierpinski Order is increased, tending to form lobes towards θ = 45º. For sierpinski antennas of higher dimensions, these lobes are even sharper, showing a valley around the broadside direction. On the other hand, when increasing the sierpinski Order, the Cross-polar component lobes are better deﬁned. For larger scale antennas, these ones are more spaced, moving away from the x axis (antenna gap axis).

44 Bow-Tie

(a) La = 400µm (b) La = 1200µm

Sierpinski 2nd-Order

Sierpinski 3rd-Order

(e) La = 400µm (f) La = 1200µm

Sierpinski 4th-Order

(g) La = 400µm (h) La = 1200µm

Figure 38: Co-polar and Cross-polar Primary ﬁelds projections comparison between the Bow-Tie structure and different sierpinski orders at f = 0.5 THz.

45 A GaAs hyperhemispherical lens of radius dimension R = 5mm has been added to the sierpinski structure to simulate the Secondary ﬁelds. Some interest cuts of the radiation patterns obtained from the lens-coupled structures (see AppendixC) are plotted in Figure 39.

(a) (b)

Figure 39: Secondary far ﬁeld cuts of sierpinski PCAs: Co-polar component cut at φ = 90º (a) and Cross-polar component cut at φ = 45º (b) at f = 0.5 THz.

The Co-polar cuts present a pattern composed by a primary lobe pointing towards the broadside direction, accompanied by two side lobes of lower amplitude, which form an Airy pattern. For small sierpinski PCAs (La = 400 µm), almost nothing changes in the pattern shape and amplitude when increasing the fractal order. In the case of bigger structures (La = 1200 µm) the side lobes amplitude does show a slight increment for higher sierpinski order antennas. In addition, the Airy pattern width seems to depend on the antenna dimensions, showing that for smaller sierpinski dimensions, the pattern is more directive. In the case of the Cross-polar far ﬁeld there are no big differences. A quadrupole behaviour with lobes around −15dB below the Co-polar lobe amplitude is observed, presenting a null along the axis. The lobes direction is maintained for the different order antennas, pointing towards θ = 5º. Only a slight amplitude decay may be observed when increasing the sierpinski iterations.

La = 400 µm La = 1200 µm Bow-Tie 0.476 W 0.455 W Sierpinski 1st-Order 0.472 W 0.479 W Sierpinski 2nd-Order 0.463 W 0.475 W Sierpinski 3rd-Order 0.450 W 0.472 W Sierpinski 4th-Order 0.440 W 0.434 W

Table 7: Total Radiated Power for the different sierpinski order PCAs simulated at f = 0.5 THz.

46 As it has been mentioned above, fractal structures have been modeled to increase the radiation of THz emitters in terms of power, therefore the total radiated power of the simulated antennas has been summarized in Table7 to analyse the improvement at a central frequency f = 0.5 THz. An upturn may be observed for sierpinski structures of dimensions La = 1200 µm, reaching its better power performance for the ﬁrst fractal order. However, in the case of the smaller sierpinski (La = 400 µm), the radiated power decays when increasing the fractal order. It may be substantiated for the perforations dimensions, which for this antenna length may be smaller than the wavelength reassembling a solid surface current distribution [30].

(a)

(b)

Figure 40: Secondary far ﬁeld cuts of a 3rd-Order sierpinski PCA of dimensions La = 400 µm: laboratory measurements (a) versus simulations (b) at f = 0.5 THz.

The measured results in Figure 40 feature asymmetries which are specially apparent in the Cross-polar components. These asymmetries are an indication of the need for further adjustment of the laser feed and have been proposed as a measure of the laser alignment quality in practical setups. Considering the details of the measurement system [31] we ﬁnd a reasonable agreement with simulated patterns, with similar Co-polar 3 dB beam widths of respectively 4º and 3º in the 0º and 90º cuts, and side-lobes located around 5º in the 90º cut. As for the Cross-polar component, apart from the mentioned laser misalignment asymmetry the position of side-lobes

47 in the Cross-polar cuts at 0º / 90º is seen to coincide around 3º / 5º in both simulation and measures. Further adjustment of the physical setup is presently underway.

48 4.4 Terahertz Beam Steering To end this chapter, it is of high interest for the THz frequency band to study the steering potential of PCAs. Optically switched lens integrated photodiodes-antennas arrays have been proposed as a beam steering technique in the mm-Wave band [6]. The idea of this technique is to connect an array of antennas to an optical switch, through which a single array element will be fed at each time, focusing the beam towards different steering angles depending on the antenna selected (see Figure 41). The objective is to analyse the off-axis performance of our lens-coupled photoconductive antenna structure to replicate this idea in the THz frequency band.

(a) (b)

Figure 41: Design of the optically steerable array with the dielectric lens [6]: Perspective view with dielectric lens and beam switching (a) and Back view with optical switch and ﬁber connections (b).

It is expected that when switching the PCA position with respect to the lens center, and therefore, the origin of the fields introduced to the substrate medium, the rays direction will be changed. This means that at the different points of the substrate-air surface the incident angle of the fields will be shifted, leading to deflection of the transmitted rays, as shown in Figure 42, which will be constructively overlapping towards a different direction than the one obtained with the centered PCA.

(a) (b)

Figure 42: Rays direction representation when the PCA is positioned at x = 0 (a) and at x = ∆x (b).

49 In order to analyse the steering properties of the substrate lens used for the simulations, the lens feeding point has been displaced along the x axis. An H-shape Dipole PCA of dimensions Lchip = 2 mm has been chosen to simplify the computation resources needed. The position of the chip has been deﬁned in equation (27), with i = 1,2,...,N, considering that the chip is displaced from left to right as shown in Figure 43.

N + 1 Lchip ∆x = i − (27) i 2 2

(a) (b) (c)

Figure 43: CST setup of the lens-coupled PCA with an off-axis of ∆x = 1000µm (a), ∆x = 0µm (b) and ∆x = −1000µm (c).

It should be observed how the primary lobe of the radiation pattern is shifted from the broadside direction when feeding the different chips at the positions ∆xi. Equation (28) gives the relationship that follows the scanning angle θi with respect to the chip offset, which is inversely proportional to the Extension Length Le of the Lens.

∆xi tan(θi) = − (28) Le Setting the lens dimensions of Table5, the expected scanning angles for each chip position are the ones summarized in Table8. The resulting radiation patterns for the different off-axis displacements simulated are represented in Figure 44, showing the expected scanning angles at ±34º.

∆x(µm) θ(deg) −1000 34.29 0 0 1000 −34.29

Table 8: Scanning angle for different off-axis values and an extension length Le = 1466 µm.

50 As seen in previous simulations, the H-Dipole pattern presents high side-lobes at the direction ±35º, which coincides with the steering angle for the off-axis chosen and may cause confusion. For the analysis carried out, the lobes at which the attention should be payed are the higher ones, which are switched from the broadside direction for the central chip selection, to ±34.29º when the chips at ∆x = ±1000 µm are fed.

Figure 44: Simulated radiation patterns for different off-axis displacements at f = 0.5 THz.

51 5 Conclusions and future development

To sum up, this thesis has been divided in two main parts to analyse different models of optically feed antenna elements to achieve beam forming and beam steering at mm-Wave and THz frequencies. In the first part, we have presented a proposal for an optical network to control the direction of maximum directivity of a PAA, for which the beam steering is achieved through the wavelength tuning of an array of input lasers and dispersive propagation. The free-lobe beam steering operation is mainly restricted by the PAA inter-elements spacing, which will also determine the maximum pointing direction along with the AWG channel bandwidth. Moreover, a double-branch DE-MZM configuration for optical modulation has been proposed. The simulation results show the ability of the biasing technique to avoid the CD-fading penalty, of around 5dB in the example system biased at QP, and enable large bandwidth and high count PAAs by switching the notch position of the photodetected signal. The method is tunable, allowing to reconfigure the target frequency band. In the second part, a simulation solver setup has been described to define a methodology to obtain the far fields of THz radiation elements, including the solver configuration to reproduce the radiated fields inside the lens (Primary Fields) and the transmitted fields from the lens surface towards the air (Secondary Fields). Despite a trade off between the runtime of the solver and the accuracy of the results has been applied, the solution has shown excellent results, which have been validated by comparing the far fields obtained from simulating replicas of previous publications models with remarkable similarity, even when with the present resources it is limited to the simulation of frequencies below 1 THz and some simulations may take up to more than 24 hours of computer time. This methodology has been used to analyse Fractal Antennas. After simulating different order sierpinski PCAs, the resonance curves have been obtained, proving the multi-band characteristics of the shapes. An analysis around the frequency f = 0.5 THz of different sierpinski order and dimensions has been carried out. Smaller structures (La = 400 µm) seem to be more directive, but little improvement may be seen when increasing the iteration order, while larger antennas (La = 1200 µm) show a wider airy pattern and slight power improvement when shift- ing the fractal order, with an optimal power transmission for the first fractal order. This analysis has helped us to see the scope of the methodology presented, even when we are limited to frequencies below 1 THz by the computer power required to simulate with CST large structures. Finally, the lens-coupled H Dipole off-axis performance has been analysed by carrying out simulations where the PCA is displaced along the x axis, showing the steering capability of the hyperhemispherical lens substrate and its dependence with the extension length, which will define the limits of the steering range. Our simulations have yielded a ±34º steering with a typical H dipole PCA with a spatial shift of ∆x = 1000 µm. Besides the results summarized above, the results of this work may constitute the foundations for future developments, in the path towards portraying this principles into real communication systems. Possible future work lines are summarized below. A first step to evolve towards mm-Wave optical feed arrays is to build the setup of the MW

52 OTTDN system simulated in the laboratory in order to measure the steering and reconfigu- ration performance and determine the physical limitations. Microwave operating frequencies have been considered for the analysis in order to replicate the available antenna elements performance, so a next step could be to design photonic integrated antenna arrays to work at the mm-Wave band. On the other hand, and thinking of a functional model of the presented solution, an automatic bias control method may be the implementation key. This would add the flexibility needed to work with adaptive resources by adjusting the bias as a function of the operating frequency. It can be observed that the proposed solution presents unused spectrum in the central channels due to the wavelength spacing conditions. Another work line could be to find a way of reducing the spectrum occupied by looking for an alternative to the AWG filter with non-uniform channels. The THz band is still a novel study field and great advances are yet to come. The methodology presented could be a tool to study in depth different antennas and lens shapes characteristics, as it has been done with the sierpinski antennas. In particular, further work should consider the development of simulation strategies that may allow to increase the simulation frequencies by reducing the computer resources and time required. Approximate methods to include the effect of the lens and obtain the Secondary fields as a function of the Primary fields are worth exploring. The steering capabilities have been analysed with a dipole PCA, which presents high secondary lobes. A more optimal PCA shape could be analysed to fit in the free-lobe condition. Laboratory measurements of the lens-coupled antenna with the PCA feeding displaced could be also carried out to observe the physical performance of the lens steering capability. This could be the basis to implement the optically switched PCAs array proposed in previous publications to feed the lens and switch the beam towards specific directions when feeding each one of the array PCAs.

53 References

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56 Acronyms

AF Array Factor AWG Arrayed Waveguide Gratings CD Chromatic Dispersion DC Direct Current DCF Dispersion Compensating Fiber DE-MZM Dual Electrode Mach Zehnder Modulator DFB Distributed Feedback Bragg DSB Double Side-Band EM Electromagnetic GO Geometrical Optics IoT Internet of Things LT-GaAs Low-Temperature Grown Gallium Arsenide mm-Wave millimeter Wave MW multiwavelength OTTDN Optical True Time Delay Network PAA Phased Array Antenna PAAE Phased Array Antenna Element PCA Photoconductive Antenna PEC Perfect Electric Conductor PO Physical Optics PP-MZM Push-Pull Mach Zehnder Modulator QP Quadrature Point RF Radio Frequency RoF Radio-over-Fiber SSB Single Side-Band TD Time Domain TDS Time-Domain Spectroscopy THz Terahertz ToF THz-over-Fiber

57 TRx Transceiver TTD True Time Delay WDM-PON Wavelength Division Multiplexing-Passive Optical Network

58 Appendices

A Matlab Codes

A.1 VPI Radiation Pattern .mat ﬁle

1 function [Ph1,Ph2,Ph3,Ph4] = Diagrama Rad(A1,Ph1,A2,Ph2,A3,Ph3,A4,Ph4) 2 3 % A1..A4 photodetected signal amplitudes 4 % Ph1..Ph4 photodetected signal phases 5 6 save('\path\File Name','a','p'); 7 FA=FA/max(abs(FA)); 8 9 end Listing 3: Matlab Function to generate a .mat ﬁle from VPI output.

A.2 VPI Radiation Pattern Polar Plot

1 clear all 2 load('Beam 0.mat') 3 4 p1 = p; 5 a1 = a; 6 7 ph1 = p1*pi/180; 8 d = 4/5;% PAAE spacing asa function of wavelength 9 10 th=[−90:2:90]; 11 th = th*pi/180; 12 N=length(a1); 13 FA1=zeros(size(th)); 14 15 for l=1:N 16 n=l−(N+1)/2; 17 FA1=FA1+a1(l)*exp(i*(n*(2*pi*d*sin(th))+ph1(l))); 18 end 19 20 FA1 abs = abs(FA1)/max(abs(FA1)); 21 figure; 22 pax = polaraxes; 23 polarplot(th,FA1 abs); 24 pax.ThetaZeroLocation='top'; 25 thetalim([−90 90]); 26 title('Array Factor'); Listing 4: Matlab Code to plot the polar radiation pattern from the .mat ﬁle.

59 A.3 Radiation Pattern Projection

1 clear 2 close all 3 load('ASCII file.txt','−ascii') 4 a=ASCII file; 5 6 % radiation pattern choice 7 % rp=3 abs, rp=4 Xpol, rp=6 copol 8 9 b=a(:,6); 10 t=[0:0,1:180]; 11 p=[0:0,1:360]; 12 tr=t*pi/180; 13 pr=p*pi/180; 14 c=reshape(b,length(t),length(p)−1); 15 d=[c,c(:,1)]; 16 aux = [a(:,3) a(:,4) a(:,6)]; 17 18 %normalized to zero dB 19 20 dm=max(max(aux)); 21 e=d−dm; 22 23 % to plot only the positivez fields 24 25 tr2=tr(1:91); 26 x12=sin(tr2')*cos(pr); 27 y12=sin(tr2')*sin(pr); 28 29 figure 30 colormap('jet') 31 h=pcolor(x12,y12,e(1:91,:)) 32 caxis([−30 0]) 33 set(h,'edgecolor','none') 34 set(gca,'FontSize',18) 35 xlabel('sin(\theta)cos(\phi)','FontSize',18) 36 ylabel('sin(\theta)sin(\phi)','FontSize',18) 37 colorbar Listing 5: Matlab Code to plot radiation pattern circular projections from ASCII ﬁles exported from CST.

60 A.4 Radiation Pattern Cuts

1 clear 2 close all 3 load('ASCII file 0deg.txt','−ascii') 4 a1=ASCII file 0deg; 5 6 load('ASCII file 45deg.txt','−ascii') 7 a2=ASCII file 45deg; 8 9 load('ASCII file 90deg.txt','−ascii') 10 a3=ASCII file 90deg; 11 12 load('ASCII file 135deg.txt','−ascii') 13 a4=ASCII file 135deg; 14 15 % radiation pattern choice 16 % rp=3 abs, rp=4 Xpol, rp=6 copol 17 18 % Co−polar components 19 20 copol1=a1(:,6); 21 copol2=a2(:,6); 22 copol3=a3(:,6); 23 copol4=a4(:,6); 24 25 % Cross−polar components 26 27 crosspol1=a1(:,4); 28 crosspol2=a2(:,4); 29 crosspol3=a3(:,4); 30 crosspol4=a4(:,4); 31 32 % Extract the part of the radiation pattern pointing towards the top ... side of the PCA 33 34 copol1=copol1(1:90); 35 copol2=copol2(1:90); 36 copol3=copol3(1:90); 37 copol4=copol4(1:90); 38 39 crosspol1=crosspol1(1:90); 40 crosspol2=crosspol2(1:90); 41 crosspol3=crosspol3(1:90); 42 crosspol4=crosspol4(1:90); 43 44 %normalized to zero dB 45 46 aux = [copol1 copol2 copol3 copol4 crosspol1 crosspol2 crosspol3 ... crosspol4]; 47 dm=max(max(aux)) 48 49 copol1=copol1−dm; 50 copol2=copol2−dm;

61 51 copol3=copol3−dm; 52 copol4=copol4−dm; 53 54 crosspol1=crosspol1−dm; 55 crosspol2=crosspol2−dm; 56 crosspol3=crosspol3−dm; 57 crosspol4=crosspol4−dm; 58 59 % Construct the other half pattern by symmetry 60 61 N=length(copol1); 62 for n=1:N 63 copol1 neg(n)=copol1(N+1−n); 64 copol2 neg(n)=copol2(N+1−n); 65 copol3 neg(n)=copol3(N+1−n); 66 copol4 neg(n)=copol4(N+1−n); 67 68 crosspol1 neg(n)=crosspol1(N+1−n); 69 crosspol2 neg(n)=crosspol2(N+1−n); 70 crosspol3 neg(n)=crosspol3(N+1−n); 71 crosspol4 neg(n)=crosspol4(N+1−n); 72 end 73 74 copol1 total=[copol1 neg, copol1(1), copol1.']; 75 copol2 total=[copol2 neg, copol2(1), copol2.']; 76 copol3 total=[copol3 neg, copol3(1), copol3.']; 77 copol4 total=[copol4 neg, copol4(1), copol4.']; 78 79 crosspol1 total=[crosspol1 neg, crosspol1(1), crosspol1.']; 80 crosspol2 total=[crosspol2 neg, crosspol2(1), crosspol2.']; 81 crosspol3 total=[crosspol3 neg, crosspol3(1), crosspol3.']; 82 crosspol4 total=[crosspol4 neg, crosspol4(1), crosspol4.']; 83 84 % Set thex axis 85 86 T=length(copol1 total); 87 t=[0:0,1:T−1]−90; 88 89 figure; 90 plot(t, copol1 total,'b', t, copol2 total,'b−.', t, copol3 total, ... 'b:', t, copol4 total,'b−−', t, crospol1 total,'r', t, ... crosspol2 total,'r−.', t, crosspol3 total,'r:', t, ... crosspol4 total,'r−−','LineWidth', 2); 91 legend('Copolar \phi=0','Copolar \phi=45','Copolar \phi=90', ... 'Copolar \phi=135','Crosspolar \phi=0','Crosspolar \phi=45', ... 'Crosspolar \phi=90','Crosspolar \phi=135'); 92 set(gca,'FontSize',26) 93 title('Far−field','FontSize',34); 94 xlabel('\theta(degrees)','FontSize',30); 95 ylabel('Magnitude(dB)','FontSize',30); 96 ylim([−30 0]); 97 xlim([−90 90]); Listing 6: Matlab Script to plot radiation pattern cuts from ASCII ﬁles exported from CST.

62 B PCAs dimensions

B.1 Dipole dimensions list

Name Value (µm) Description

gx 10 x dimension of the gap

gy 10 y dimension of the gap

La 1000 y dimension of the antenna

Lchip 2000 chip size

Ld 30 dipole length

Lpad 500 y dimension of the pad

wbx 10 x width of the bias line

wby 10 y width of the bias line

wali 30 width of the alignment mark

xali 100 x length of the alignment mark

yali 100 y length of the alignment mark

xali space 200 x void space for alignment mark

yali space 350 y void space for alignment mark

Table 9: List of simulated dipole dimensions values.

63 B.2 Bow-Tie dimensions list

Value (deg or Name Description µm) a 90 x ﬂare angle

gx 10 x dimension of the gap

gy 10 y dimension of the gap

La 1200 y dimension of the antenna

Lchip 2000 chip size

wa 30 width of the alignment mark

xa 150 x length of the alignment mark

ya 150 y length of the alignment mark

xa sp 200 x void space for alignment mark

ya sp 250 y void space for alignment mark

Table 10: List of simulated bow-tie dimensions values.

64 C Sierpinski PCAs Secondary Fields

C.1 Bow-Tie Secondary Fields

(a) La = 400µm

(b) La = 1200µm

Figure 45: Co-polar and Cross-polar Secondary Fields projections comparison between Bow-Tie of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz.

65 C.2 Sierpinski 1st-Order Secondary Fields

(a) La = 400µm

(b) La = 1200µm

Figure 46: Co-polar and Cross-polar Secondary Fields projections comparison between 1st-order Sierpinski of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz.

66 C.3 Sierpinski 2nd-Order Secondary Fields

(a) La = 400µm

(b) La = 1200µm

Figure 47: Co-polar and Cross-polar Secondary Fields projections comparison between 2nd-order Sierpinski of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz.

67 C.4 Sierpinski 3rd-Order Secondary Fields

(a) La = 400µm

(b) La = 1200µm

Figure 48: Co-polar and Cross-polar Secondary Fields projections comparison between 3rd-order Sierpinski of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz.

68 C.5 Sierpinski 4th-Order Secondary Fields

(a) La = 400µm

(b) La = 1200µm

Figure 49: Co-polar and Cross-polar Secondary Fields projections comparison between 4th-order Sierpinski of dimensions La = 400 µm (a) and La = 1200 µm (b) at f = 0.5 THz.