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  • Teemu Niemi

    Polarization transformations inbianisotropic arrays

    School of Electrical Engineering

    Thesis submitted for examination for the degree of Master ofScience in Technology.

    Espoo 21.3.2012

    Thesis supervisor:

    Prof. Sergei Tretyakov

    Thesis instructor:

    M.Sc. (Tech.) Antti Karilainen

    A? Aalto UniversitySchool of ElectricalEngineering

  • aalto universityschool of electrical engineering

    abstract of themasters thesis

    Author: Teemu Niemi

    Title: Polarization transformations in bianisotropic arrays

    Date: 21.3.2012 Language: English Number of pages:8+62

    Department of Radio Science and Technology

    Professorship: Radio Science Code: S-26

    Supervisor: Prof. Sergei Tretyakov

    Instructor: M.Sc. (Tech.) Antti Karilainen

    This thesis studies regular arrays of the most general bianisotropic particles. Thegoal is to develop a systematic way to find required polarizabilities for these scat-ters so that, when ordered into periodical lattices, they exhibit any wanted polar-ization transformation for a plane wave. The method is verified by synthesizingtwo different devices: a circular polarization selective surface and a twist polarizer.A circular polarization selective surface is a device that transmits one handednessof circular polarization and reflects the other. A twist polarizer rotates the planeof polarization of a linearly polarized plane wave by 90. The operation of thesynthesized devices is verified by analytical models, numerical simulations, andexperimental measurements. The results indicate that the developed method canbe efficiently used for developing novel polarization transformers.

    Keywords: Polarization transformation, arrays of bianisotropic particles, inter-action field, circular polarisation selectivity, twist polarizer

  • aalto-yliopistosahkotekniikan korkeakoulu

    diplomityontiivistelma

    Tekija: Teemu Niemi

    Tyon nimi: Polarisaatiomuunnokset bianisotrooppisissa hiloissa

    Paivamaara: 21.3.2012 Kieli: Englanti Sivumaara:8+62

    Radiotieteen ja -tekniikan laitos

    Professuuri: Radiotiede Koodi: S-26

    Valvoja: Prof. Sergei Tretyakov

    Ohjaaja: DI Antti Karilainen

    Taman diplomityon tavoitteena on tutkia mahdollisimman yleisista bianisotroop-pisista sirottajista muodostettuja saannollisia hiloja. Tyossa on kehitettymenetelma, jota kayttamalla voidaan ratkaista sellaiset polarisoituvuudet naillesirottajille, jotta niista muodostetut saannolliset hilat totetuttavat toivotunpolarisaatiomuunnoksen tasoaallolle. Menetelman toiminta varmistetaan suun-nittelemalla kaksi erilaista laitetta ympyrapolarisaatiolle selektiivinen pintaseka kiertopolarisaattori. Ympyrapolarisaatiolle selektiivinen pinta on laite, jokapaastaa toisen ympyrapolarisaation katisyyden lapi ja heijastaa toista. Kierto-polarisaattori kiertaa lineaarisesti polarisoidun tasoaallon sahkokentan suuntaa 90astetta. Suunniteltujen laitteiden toiminta varmistetaan analyyttisilla malleilla,numeerisilla simulaatioilla seka kokeellisilla mittauksilla. Tulokset osoittavat, ettakehitetty menetelma voi olla tehokas tyokalu erilaisten polarisaatiomuuntimiensuunnittelussa.

    Avainsanat: Polarisaatiomuunnos, bianisotrooppinen hila, vuorovaikutuskentta,ympyrapolarisaatioselektiivisyys, kiertopolarisaattori

  • iv

    Preface

    This thesis was made in the Aalto University School of Electrical Engineering at theDepartment of Radio Science and Engineering in the research group of AdvancedArtificial Materials and Smart Structures in 2011 2012. I would like to thank mysupervisor Prof. Sergei Tretyakov for the continuing guidance and for the possibilityto work in this research group for past three years on many interesting topics. I alsothank my instructor M.Sc. Antti Karilainen and others who have guided me duringthe past few years, especially Prof. Constantin Simovski and Dr. Pekka Alitalo.

    Finally, I would like to thank my family and my friends for supporting me in thecourse of my studies.

    Otaniemi, 21.3.2012

    Teemu Niemi

  • v

    Contents

    Abstract ii

    Abstract (in Finnish) iii

    Preface iv

    Contents v

    Symbols and abbreviations vii

    1 Introduction 1

    2 Polarization transformations, literature overview 32.1 Circular polarization selective surfaces . . . . . . . . . . . . . . . . . 42.2 LP to CP polarizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Polarization rotators . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    3 Reflection and transmission from a bianisotropic array 93.1 Dyadic reflection and transmission coefficients . . . . . . . . . . . . . 103.2 Effective polarizabilities . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 The fields radiated by the induced currents . . . . . . . . . . . . . . . 12

    4 Synthesizing polarization transformers 164.1 Polarizabilities for a RHCPSS . . . . . . . . . . . . . . . . . . . . . . 16

    4.1.1 CPSS without magnetic polarizabilities . . . . . . . . . . . . . 184.1.2 A reciprocal canonical helix as a RHCPSS . . . . . . . . . . . 19

    4.2 Polarizabilities for a twist polarizer . . . . . . . . . . . . . . . . . . . 22

    5 Synthesizing arrays of uniaxial particles 245.1 Uniaxial twist polarizer . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    6 A CPSS using an array of chiral particles 266.1 Analytical model for chiral particles . . . . . . . . . . . . . . . . . . . 266.2 Numerical simulations for canonical helices . . . . . . . . . . . . . . . 276.3 Comparison of the results for arrays of canonical helices . . . . . . . . 28

    6.3.1 Linearly polarized normal incidence . . . . . . . . . . . . . . . 286.3.2 Circularly polarized incident field . . . . . . . . . . . . . . . . 29

    6.4 Possible modifications for canonical helix . . . . . . . . . . . . . . . . 336.4.1 The effect of the unit cell size . . . . . . . . . . . . . . . . . . 336.4.2 Effect of the particles orientation . . . . . . . . . . . . . . . . 336.4.3 Array of horizontal 0/2 true helices . . . . . . . . . . . . . . 36

    6.5 Numerical simulations for practical PCB realization . . . . . . . . . . 38

  • vi

    7 An array of chiral elements as a twist polarizer 417.1 Numerical study of an idealized twist polarizer . . . . . . . . . . . . . 417.2 Numerical study of practical PCB realization . . . . . . . . . . . . . . 437.3 Experimental verification of the twist polarizer . . . . . . . . . . . . . 447.4 Comparison of the numerical and experimental results . . . . . . . . . 47

    8 Discussion and conclusions 52

    References 54

    Appendix A Measuring transmission through a slab 58

  • vii

    Symbols and abbreviations

    Symbols

    Einc Incident electric fieldHinc Incident magnetic fieldEloc Local electric fieldHloc Local magnetic fieldEt Transmitted electric fieldEr Reflected electric field0 Vacuum permittivity, 0 8.854 1012 As/Vm0 Vacuum permittivity, 0 = 4 107 Vs/Amc0 Speed of light in free space, c0 = 1/

    00

    0 Free space wavelengthk0 Wave vector in free space, k0 =

    00 = 2/0.

    p Electric dipole moment induced in the unit cellm Magnetic dipole moment induced in the unit cellAR Axial ratio in linear scale, AR = 0 . . . 1x, y, z Unit vectors of the cartesian coordinate system

    I Unit dyadic, I = xx + yy + zz

    Jt Transversal rotation dyadic, Jt = n Itme Polarizability dyadic, i.e, the linear mapping from Eloc to m

    em Effective polarizability dyadic, i.e, the linear mapping from Hinc to pyxme The yx component of me

    e Interaction dyadicn Normal vector of an arraya Lattice constant of a square arrayS Area of a square unit cell, S = a2

    l Length of a dipole arm (dipole length is 2l)rl Loop radiusL Total lenght of the wirer0 Wire radius

    Operators

    z Complex conjugate of z, (a+ jb) = a jbab Dyadic product of vectors a and b(

    A)1

    Inverse of dyadic A, A (

    A)1

    = I(A)T

    Transpose of dyadic A

    ab cd = (b c)ad Dot-product of two dyadsab c = a(b c) Cross-product of a dyad and a vector

  • viii

    Abbreviations

    CP Circular polarizationCPSS Circular polarization selective surfaceDGR Dual gridded reflectorFEM Finite element methodLH Left handedLHCP Left-hand circular polarizationLHCPSS Left-hand circular polarization selective surfaceLP Linear polarizationPCB Printed circuit boardPEC Perfect electrical conductorRH Right handedRHCP Right-hand circular polarizationRHCPSS Right-hand circular polarization selective surfaceVNA Vector network analyzer

  • 1 Introduction

    In the literature, one can find a wide variety of definitions for metamaterials. Usu-ally, metamaterials are defined as materials whose electromagnetic properties aredetermined by their artificial structure instead of the properties of the materialsthey are made of [1, 2, 3]. One of the interesting applications of the antenna andmetamaterial research is the possibility to manipulate the polarization state of aplane wave [4]. These polarizers have numerous applications in practical antennaengineering.

    One example of an interesting polarization transformer is a circular polarizationselective surface (CPSS). An ideal CPSS is a device that reflects the wave of onehandedness of circular polarization (CP) and is invisible to the other. An idealCPSS would have numerous practical applications. The initial motive for this studywas in satellite communications: a surface that would reflect only wave with onesense of polarization would be a circular polarization equivalent for dual griddedreflector (DGR). DGR reflector antennas can steer the radiation beam in differentdirections, depending on the polarization state of the linearly polarized (LP) wave.This reduces the weight of the antenna as only one reflector is