Lecture Outline. Dispersion Relation Electromagnetic Wave Polarization 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3c
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1 Course Instructor Dr. Rymond C. Rumpf Office: A 337 Phone: (915) E Mil: rcrumpf@utep.edu EE 4347 Applied Electromgnetics Topic 3c Wve Dispersion & Polriztion Wve Dispersion These notes & Polriztion my contin copyrighted mteril obtined under fir use rules. Distribution of these mterils is strictly prohibited Slide 1 Lecture Outline Dispersion Reltion Electromgnetic Wve Polriztion Wve Dispersion & Polriztion Slide 2 1
2 Dispersion Reltion Wve Dispersion & Polriztion Slide 3 Derivtion in LHI Medi We strted with the wve eqution. 2 2 Ek E 0 We found the solution to be plne wves. E r Pe jk r If we substitute our solution bck into the wve eqution, we get n eqution clled the dispersion reltion. 2 2 n x y z c0 k k n k k k The dispersion reltion reltes frequency to wve vector. For LHI medi, it fixes the mgnitude of the wve vector to be constnt. Wve Dispersion & Polriztion Slide 4 2
3 Index Ellipsoids From the previous slide, the dispersion reltion for LHI mteril ws: kx ky kz k0 n This defines sphere clled n index ellipsoid. The vector connecting the origin to point on the surfce of the sphere is the k vector for tht direction. Refrctive index is clculted from this. k k n 0 ˆz index ellipsoid For LHI mterils, the refrctive index is the sme in ll directions. Think of this s mp of the refrctive index s function of the wve s direction through the medium. ˆx ˆy Wve Dispersion & Polriztion Slide 5 Wht About Anisotropic Mterils? Isotropic Mterils k k k k n b c 0 Unixil Mterils k k k k k k b c 2 b c 2 k 2 0 k no ne no Bixil Mterils k k k 1 k k n k k n k k n b c b 0 c Wve Dispersion & Polriztion Slide 6 3
4 Electromgnetic Wve Polriztion Wht is Polriztion? Polriztion is tht property of rdited electromgnetic wve which describes the time vrying direction nd reltive mgnitude of the electric field vector. Liner Polriztion (LP) Circulr Polriztion (CP) Left Hnd Circulr Polriztion (LCP) To determine the hndedness of CP, imgine wtching the electric field in plne while the wve is coming t you. Which wy does it rotte? Wve Dispersion & Polriztion Slide 8 4
5 Orthogonlity nd Hndedness We get from the curl equtions tht E H From the divergence equtions, we see tht E k nd H k E H k E We conclude tht,, nd form n orthogonl triplet. In fct, they follow the right hnd rule. H k Wve Dispersion & Polriztion Slide 9 Possibilities for Wve Polriztion Recll tht E k so the polriztion vector P must fll within the plne perpendiculr to k. We cn decompose the polriztion into two orthogonl directions, â nd ˆb. â P pˆ p bˆ b ˆb k Wve Dispersion & Polriztion Slide 10 5
6 Explicit Form to Convey Polriztion Our electromgnetic wve cn be now be written s jk r ˆ ˆ jk r E r Pe p p b e b p nd p b re in generl complex numbers in order to convey the reltive phse of ech of these components. j jb p E e p E e b b Substituting p nd p b into our wve expression gives j jb b ˆ ˆ ˆ jkr E r E ˆ E Ee b b e Ebe be e e The finl expression is: We interpret b s the phse difference between p nd p b. b E r E E e b e e ˆ ˆ b j j jkr We interpret s the phse common to both p nd p b. j j jkr Wve Dispersion & Polriztion Slide 11 Determining Polriztion of Wve To determine polriztion, it is most convenient to write the expression for the wve tht mkes polriztion explicity. E r E E e b e e ˆ ˆ b j j jkr We cn now identify the polriztion of the wve E mplite long ˆ E mplite long ˆ b b phse difference common phse Polriztion Designtion Mthemticl Definition Liner Polriztion (LP) = 0 Circulr Polriztion (CP) = ± 90, E = E b Right Hnd CP (RCP) Left Hnd CP (LCP) Ellipticl Polriztion = + 90, E = E b = - 90, E = E b Everything else Wve Dispersion & Polriztion Slide 12 6
7 Liner Polriztion A wve trvelling in the +z direction is sid to be linerly polrized if: z E x, y, z Pe jk z P sin xˆ cos yˆ For n rbitrry wve, jk r Er Pe Psinˆ cos b ˆ ˆ bˆ k All components of P hve equl phse. ˆb k â k Wve Dispersion & Polriztion Slide 13 Liner Polriztion Wve Dispersion & Polriztion Slide 14 7
8 Circulr Polriztion A wve trvelling in the +z direction is sid to be circulrly polrized if: z E x, y, z Pe jk z P x jy ˆ ˆ For n rbitrry wve, jk r Er Pe Pˆ jbˆ ˆ bˆ k The two components of P hve equl mplitude nd re 90 out of phse. RCP j j LCP k k Wve Dispersion & Polriztion Slide 15 LP x + LP y = LP 45 A linerly polrized wve cn lwys be decomposed s the sum of two linerly polrized wves tht re in phse. Wve Dispersion & Polriztion Slide 16 8
9 LP x + jlp y = CP A circulrly polrized wve is the sum of two orthogonl linerly polrized wves tht re 90 out of phse. Wve Dispersion & Polriztion Slide 17 RCP + LCP = LP A LP wve cn be expressed s the sum of LCP wve nd RCP wve. The phse between the two CP wves determines the tilt of the LP wve polriztion. Wve Dispersion & Polriztion Slide 18 9
10 Circulr Polriztion (1 of 2) Engineering Right Hnd Circulr Polriztion (RCP) x y z Physics/Optics Left Hnd Circulr Polriztion (LCP) Wve Dispersion & Polriztion Slide 19 Circulr Polriztion (2 of 2) Engineering Left Hnd Circulr Polriztion (LCP) x y z Physics/Optics Right Hnd Circulr Polriztion (RCP) Wve Dispersion & Polriztion Slide 20 10
11 Poincré Sphere The polriztion of wve cn be mpped to unique point on the Poincré sphere. Points on opposite sides of the sphere re orthogonl. See Blnis, Chp LP RCP 90 LP 0 LP +45 LP Wve Dispersion & Polriztion Slide 21 LCP Why is Polriztion Importnt? Different polriztions cn behve differently in device Orthogonl polriztions will not interfere with ech other Polriztion becomes criticl when nlyzing devices on the scle of wvelength Focusing properties of lenses re different Reflection/trnsmission cn be different Frequency of resontors Cutoff conditions for filters, wveguides, etc. Wve Dispersion & Polriztion Slide 22 11
12 Exmple Dissect Wve (1 of 9) The electric field component of 5.6 GHz plne wve is given by: j x j y j z Er, tˆ x j0.8550e e e j x j y j z ˆ j e e e y ˆ j e e e z j x j y j z 1. Determine the wve vector. 2. Determine the wvelength inside of the medium. 3. Determine the free spce wvelength. 4. Determine refrctive index of the medium. 5. Determine the dielectric constnt of the medium. 6. Determine the polriztion of the wve. 7. Determine the mgnitude of the wve. Wve Dispersion & Polriztion Slide 23 Exmple Dissect Wve (2 of 9) Solution Prt 1 Determine Wve Vector The stndrd form for plne wve is jk r E r Pe Compring this to the expression for the electric field shows tht ˆ ˆ ˆ x y z jk r j x j y j z P j j j e e e e The polriztion vector P will be use gin lter. The wve vector k is determined from the second expression bove to be jk r jk y j x j y j z jkxx y jkz z e e e e e e e k ˆ ˆ ˆ m x y z 1 Wve Dispersion & Polriztion Slide 24 12
13 Exmple Dissect Wve (3 of 9) Solution Prt 2 Wvelength inside the medium The wvelength inside the medium is relted to the mgnitude of the wve vector through 2 2 k k The mgnitude of the wve vector is k k k k x y z m m m m The wvelength is therefore m cm Wve Dispersion & Polriztion Slide 25 Exmple Dissect Wve (4 of 9) Solution Prt 3 Free spce wvelength The free spce wvelength is 8 c0 310 m s c0 f cm 9 1 f s Solution Prt 4 Refrctive index It follows tht the refrctive index of the medium is cm n 6.0 n cm Alterntively, we could determine the refrctive index through k k k c0 k k k n n m s m k0 c0 2 f s 6.0 Wve Dispersion & Polriztion Slide 26 13
14 Exmple Dissect Wve (5 of 9) Solution Prt 5 Dielectric constnt Assuming the medium hs no mgnetic response, 2 n 2 r r n Solution Prt 6 Wve Polriztion To determine the polriztion, the electric field is written in the form tht mkes polriztion explicit. j ˆ ˆ j jkr E r EEbe be e The choice for â nd ˆb is rbitrry, but they most both be perpendiculr to k â P pˆ p bˆ b ˆb k Wve Dispersion & Polriztion Slide 27 Exmple Dissect Wve (6 of 9) Solution Prt 6 Wve polriztion (cont d) We determine vlid choice for â by first picking ny vector tht is not in the sme direction s k v 1ˆ 2ˆ 3ˆ x y z The cross product will give us vector perpendiculr to k kv ˆ ˆ ˆ ˆ x y z kv We determine vlid choice for ˆb using the cross product so tht it is perpendiculr to both â nd k ˆ k ˆ b ˆ ˆ ˆ x y z k ˆ Wve Dispersion & Polriztion Slide 28 14
15 Exmple Dissect Wve (7 of 9) Solution Prt 6 Wve polriztion (cont d) To determine the component of the polriztion vector P in the â nd ˆb directions using the dot product. p ˆ P V m p Pbˆ j V m b We cn now write E nd E b from p nd p b by incorporting the phse difference into the prmeter. E V m Eb V m 90 The common phse between p nd p b is simply 0. 0 Wve Dispersion & Polriztion Slide 29 Exmple Dissect Wve (8 of 9) Solution Prt 6 Wve polriztion (cont d) Finlly, we hve E r E ˆ E e bˆ e e b j j jkr E V m Eb V m 90 0 k ˆ ˆ ˆ m x y z From this, we determine tht we hve circulr polriztion (CP) becuse E = E b nd = ±90. More specificlly, this is left hnd circulr polriztion (LCP) becuse = Wve Dispersion & Polriztion Slide 30 15
16 Exmple Dissect Wve (9 of 9) Solution Prt 7 Mgnitude of electric field The mgnitude of the wve is simply the mgnitude of the polriztion vector E r P E E 2 2 b V m V m 2.4 V m Wve Dispersion & Polriztion Slide 31 16
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