9/10/2020 Electromagnetics: Electromagnetic Field Theory Electromagnetic Wave Polarization Lecture Outline • Definition of Electromagnetic Wave Polarization • Orthogonality & Handedness • Polarization Classification: Linear, Circular & Elliptical • Poincaré Sphere • Polarization Explicit Form • Dissection of a Plane Wave Slide 2 1 9/10/2020 Definition of Electromagnetic Wave Polarization Slide 3 What is Polarization? Polarization is that property of a electromagnetic wave which describes the time‐varying direction and relative magnitude of the electric field vector. Er Pejkr Polarization Vector Slide 4 2 9/10/2020 Why is Polarization Important? • Different polarizations can behave differently in a device • Orthogonal polarizations will not interfere with each other • Polarization becomes critical when analyzing devices on the scale of a wavelength • Focusing properties of lenses can depend on polarization • Reflection/transmission can depend on polarization • Frequency of resonators can depend on polarization • Cutoff conditions for filters, waveguides, etc., can depend on polarization Slide 5 Orthogonality & Handedness Slide 6 3 9/10/2020 Plane Waves in LHI Media are TEM In linear, homogeneous and isotropic (LHI) media, the electric field , magnetic field , and direction of the wave are all perpendicular to each other. These are called transverse electromagnetic (TEM) waves. E kH Further , , and follow a right‐hand rule. Slide 7 Proof That ⊥ Start with Gauss’ law. D 0 Put this in terms of the electric field intensity E. E 0 E 0 E 0 Substitute in expression for a plane wave. Pejk r 0 jk Pejk r 0 kP0 kP0 kP According to the dot‐product test, if then . Ek Slide 8 4 9/10/2020 Proof That ⊥ Start with Gauss’ law for magnetic fields. B 0 Put this in terms of the magnetic field intensity H. H 0 H 0 H 0 Substitute in expression for a plane wave. jk r jk H ejk r 0 kH 0 He0 0 0 0 According to the dot‐product test, if kH0 then kH . 0 0 Hk Slide 9 Proof That ⊥ Start with Faraday’s law. E jH Substitute in expression for a plane wave. jk r jk r Ee00 j He jkr jkr jk E00 e j H e kE H 00 kE According to the properties of cross products, H 0 0 will be perpendicular to both and . Slide 10 5 9/10/2020 Polarization Classification: Linear, Circular & Elliptical Slide 11 Linear Polarization (LP) An electromagnetic wave has linear polarization if the electric field oscillation is confined to a single plane. jkz E Ea0 ˆx e Slide 12 6 9/10/2020 Linear Polarization (LP) An electromagnetic wave has linear polarization if the electric field oscillation is confined to a single plane. jkz E Ea0 ˆ y e Slide 13 Linear Polarization (LP) An electromagnetic wave has linear polarization if the electric field oscillation is confined to a single plane. jkz E Ea0 cosˆˆxy sin ae Slide 14 7 9/10/2020 Circular Polarization (CP) Suppose two LP waves are brought together… jkz E10 Eaˆx e jkz EEae20 ˆ y Slide 15 Circular Polarization (CP) …and then one LP wave is phase shifted by 90. ˆ jkz ˆ jkz Ea0 x e E0aey Before phase shift (LP) E1 E2 Eaˆ e jkz jE aˆ e jkz After phase shift (RCP) 0 x 0 y Slide 16 8 9/10/2020 Circular Polarization (CP) The overall electric field will appear to rotate with respect to both time and position. jkz E12EEajae 0 ˆˆxy Slide 17 Circular Polarization (CP) An electromagnetic wave has circular polarization if the direction of the electric field rotates with time and has a constant magnitude. jkz E Ea0 ˆˆxy jae Slide 18 9 9/10/2020 Right Circular Polarization (RCP) An electromagnetic wave has right circular polarization if the electric field rotates clockwise when viewed from behind. jkz E Ea0 ˆˆxy jae Slide 19 Left Circular Polarization (LCP) An electromagnetic wave has left circular polarization if the electric field rotates counterclockwise when viewed from behind. jkz E Ea0 ˆˆxy jae Slide 20 10 9/10/2020 Elliptical Polarization (EP) An electromagnetic wave has elliptical polarization if the electric field rotates with time to form an ellipse. Slide 21 Continuum of LP+LP Polarizations As the phase difference between the two LP waves changes, the resulting polarization… jkz j jkz LPxxLPyyEae0 ˆ Ee0 aeˆ Slide 22 11 9/10/2020 Continuum of LP+LP Polarizations As the phase difference between the two LP waves changes, the resulting polarization oscillates between RCP, LCP, LP45, LP‐45, and EP. jzk j jkz LPx LPy Eae0 ˆx Ee0 aˆye jjkz Eaeae0 1ˆxy ˆ Slide 23 Continuum of CP+CP Polarizations As the phase difference between the two CP waves changes, the resulting polarization is... jkz j jkz RPC LCP Ea0 1ˆˆxy jae Ea0 1ˆˆxy jaee Slide 24 12 9/10/2020 Continuum of CP+CP Polarizations As the phase difference between the two CP waves changes, the resulting polarization is LP where the phase translates into a tilt angle. jkz jjkz RCPLCP Ea0 1ˆˆxy jae Ea0 1ˆˆx jaey e ˆˆ jkz Ea0 cosxy sin ae Slide 25 Poincaré Sphere Slide 26 13 9/10/2020 Poincaré Sphere The polarization of a wave can be mapped to a unique point on the Poincaré sphere. RCP Points on opposite sides of the sphere are ‐45° LP 90° LP orthogonal. See Balanis, Chap. 4. 0° LP +45° LP LCP Slide 27 Continuum of All Polarizations Slide 28 14 9/10/2020 Polarization Explicit Form Slide 29 Possibilities for Wave Polarization Recall that ⊥so the polarization vector must fall within the plane perpendicular to . The polarization can be decomposed into two orthogonal directions, and . aˆ ˆ Rules for Choosing aˆ and b 1. Orthogonality akbˆ ˆ 2. Handedness abˆ ˆˆ k bˆ k ˆ P paabˆ p b Slide 30 15 9/10/2020 Explicit Form to Convey Polarization The expression for the electromagnetic wave can now be written as jk rˆ ˆ jk r Er Pe paab pbe pa and pb are in general complex numbers in order to convey the relative phase of each of these components. jjab pEepEeaa bb Substituting pa and pb into the wave expression gives jjjkr j j jkr Er Eeab aˆ Eebeˆ Eaˆ Eeba bˆ ea e ab ab Interpret b – a as the phase Interpret a as the phase difference between pa and pb. common to both pa and pb. ba a The final expression is: ˆ jˆ jjkr Er Eaab Eebee Slide 31 Determining Polarization of a Wave To determine polarization, it is most convenient to write the expression for the wave that makes polarization explicit. Ea amplite along aˆ E amplite along bˆ Er Eaˆ Eebeejˆ jjkr b ab phase difference common phase Given , , and the following table can be used to identify the polarization of the wave… Polarization Designation Mathematical Definition Linear Polarization (LP) = 0° Circular Polarization (CP) = ± 90° and Ea = Eb Right‐Hand CP (RCP) = + 90° and Ea = Eb Left‐Hand CP (LCP) = - 90° and Ea = Eb Elliptical Polarization Everything else Slide 32 16 9/10/2020 Dissection of a Plane Wave Slide 33 Dissection of a Plane Wave Complex Amplitudes Impedance HkPH ˆ ˆ 0 0 E0 ˆ H0 Constitutive Parameters EP0 E0 n n 0 rr r Hrt,cos H t k r 0 0 r n 0 r r 0 E rt,cos E0 t k r Direction Speed Ordinary Frequency k kˆ v f 2 f k Free Space Wavelength Wavelength in Medium Refractive Index cf0 0 2 c k n 00 v Slide 34 17 9/10/2020 Reading Off the Parameters Ert,cosPk t r VmStandard Form 9 Ert, 3 aˆˆxy 7 a cos 2.4 10 t 5.1 x 3.5 z V m Polarization Vector Angular Frequency V rad Pa37ˆˆ a 2.4 109 xym sec Wave Vector Step 1 –Factor out negative sign. 99 cos 2.4 10txz 5.1 3.5 cos 2.4 10 t 5.1 xz 3.5 Step 2 –Set expression in parentheses equal to kr kxkykzxyz kxxyz ky kz 5.1 x 3.5 z Step 3 –Read off components of k kkk5.1, 0, 3.5 kaa 5.1ˆˆ 3.5 m1 xyz xz Slide 35 18.