# lecture15 - Lecture 15 Polarization States of Plane Waves...

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1 ECE 303 – Fall 2005 – Farhan Rana – Cornell University Lecture 15 Polarization States of Plane Waves In this lecture you will learn: • More complex mathematics for plane waves • Polarization states of plane waves (linear, circular, elliptical) ECE 303 – Fall 2005 – Farhan Rana – Cornell University Review: Maxwell’s Equations for Phasors Time-harmonic E and H-fields are given as: () ( ) [ ] t j e r E t r E ω r r r r Re , = ) [ ] t j e r H t r H r r r r Re , = ) [ ] t j e r t r ρ r r Re , = ) [ ] t j e r J t r J r r r r Re , = Maxwell’s equations for the vector phasors of time-harmonic fields are then: () ( ) r r E o r r r ε = . () 0 . = r H o r r µ r H j r E o r r r r = × () ( ) ( ) r E j r J r H o r r v r r r + = × Gauss’ Law: Gauss’ Law for the Magnetic Field: Faraday’s Law: Ampere’s Law:

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2 ECE 303 – Fall 2005 – Farhan Rana – Cornell University Review: Plane Wave Phasors and Complex Poynting Vector For a plane wave we know the E-field and H-field phasors to be: () r k j o e E n r E r r r r . ˆ = r k j o o e E n k r H r r r r . ˆ ˆ × = η The complex Poynting vector was defined as: () () ( ) r H r E r S r r r r r r * × = The time-average power per unit area is one-half of the real part of the complex Poynting vector For a plane wave: [] () () o o o o E k E n k n r H r E r S t r S 2 ˆ ˆ ˆ ˆ Re 2 1 Re 2 1 Re 2 1 , 2 2 * = × × = × = = r r r r r r r r E H k r = 377 o o o ε µ ECE 303 – Fall 2005 – Farhan Rana – Cornell University Review: More Calculations in the Complex Notation Example: Consider a plane wave with E-field of amplitude E o and pointing in a direction 45-degrees w.r.t. the x-axis (as shown) and traveling in the + z -direction z x y E H Write expression for the E-field phasor: z k j o e E y x r E + = 2 ˆ ˆ r r Write expression for the H-field phasor: z k j o o z k j o o z k j o o z k j o o o e E x y r H e E x y k r H e E y x z k j j r H e E y x j r H r E j r H = = + × = + × = × = ω 2 ˆ ˆ 2 ˆ ˆ 2 ˆ ˆ ˆ 2 ˆ ˆ r r r r r r r r r r r r z k k ˆ = r k r
3 ECE 303 – Fall 2005 – Farhan Rana – Cornell University Review: More Calculations in the Complex Notation - II The E-field phasor is: () z k j o e E y x r E + = 2 ˆ ˆ r r z k j o o e E x y r H = η 2 ˆ ˆ r r The H-field phasor is: z x y E H Find the complex Poynting vector: () () () o o o o E z E x y y x r H r E r S 2 2 * ˆ 2 ˆ ˆ 2 ˆ ˆ = × + = × = r r r r r r Find the time-average power per unit area: [] o o E z r S t r S 2 ˆ Re

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## This note was uploaded on 02/02/2008 for the course ECE 3030 taught by Professor Rana during the Fall '06 term at Cornell University (Engineering School).

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lecture15 - Lecture 15 Polarization States of Plane Waves...

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