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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

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Definition of Electromagnetic Wave Polarization

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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 Pejkr Polarization Vector

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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

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Orthogonality & Handedness

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Plane Waves in LHI Media are TEM

In linear, homogeneous and isotropic (LHI) media, the electric field 𝐸, magnetic field 𝐻, and direction of the wave 𝑘 are all perpendicular to each other. These are called transverse electromagnetic (TEM) waves.  E kH

Further 𝐸, 𝐻, and 𝑘 follow a right‐hand rule.

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Proof That 𝐸 ⊥𝑘 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.          Pejk r 0  jk  Pejk r 0 kP 0     kP0 kP According  to the dot‐product test, if then . Ek

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Proof That 𝐻 ⊥𝑘 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 ejk r 0 kH  0 He0 0 0 0     According to the dot‐product test, if kH0 then kH .   0 0 Hk

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Proof That 𝐸 ⊥𝐻 Start with Faraday’s law. E jH 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 H  0 0  will be perpendicular to both 𝑘 and 𝐸. Slide 10

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Polarization Classification: Linear, Circular & Elliptical

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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

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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

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Linear Polarization (LP) An electromagnetic wave has linear polarization if the electric field oscillation is confined to a single plane.   jkz E Ea0 cosˆˆxy sin ae

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Circular Polarization (CP) Suppose two LP waves are brought together…

  jkz E10 Eaˆx  e

  jkz EEae20  ˆ y 

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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

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Circular Polarization (CP)

The overall electric field will appear to rotate with respect to both time and position.

  jkz E12EEajae 0 ˆˆxy  

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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

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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

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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

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Elliptical Polarization (EP) An electromagnetic wave has elliptical polarization if the electric field rotates with time to form an ellipse.

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Continuum of LP+LP Polarizations As the phase difference  between the two LP waves changes, the resulting polarization…

 jkz j  jkz LPxxLPyyEae0 ˆ Ee0 aeˆ

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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

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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 jae Ea0 1ˆˆxy jaee

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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  RCPLCP Ea0  1ˆˆxy jae Ea0 1ˆˆx  jaey  e ˆˆ jkz  Ea0 cosxy sin ae

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Poincaré Sphere

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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

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Polarization Explicit Form

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Possibilities for Wave Polarization

Recall that 𝐸 ⊥𝑘so the polarization vector 𝑃 must fall within the plane perpendicular to 𝑘. The polarization can be decomposed into two orthogonal directions, 𝑎 and 𝑏.

aˆ ˆ Rules for Choosing aˆ and b 1. Orthogonality  akbˆ ˆ 2. Handedness  abˆ ˆˆ k

 bˆ  k ˆ P paabˆ p b

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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.

jjab pEepEeaa bb

Substituting pa and pb into the wave expression gives       jjjkr 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 jˆ 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 Eebeejˆ jjkr b   ab   phase difference   common phase

Given 𝐸, 𝐸, and 𝛿 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

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Dissection of a Plane Wave

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Dissection of a Plane Wave

Complex Amplitudes Impedance  HkPH 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

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Reading Off the Parameters    Ert,cosPk 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 Pa37ˆˆ 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  kkk5.1,  0,  3.5  kaa  5.1ˆˆ  3.5 m1 xyz xz Slide 35

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