Electromagnetic Waves & Polarization

Transcription

1 Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 3a Electromagnetic Waves & Polarization Electromagnetic These notes Waves may & contain Polarization copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Slide 1 Lecture Outline Maxwell s Equations Derivation of the Wave Equation Solution to the Wave Equation Intuitive Wave Parameters Dispersion Relation Electromagnetic Wave Polarization Visualization of EM Waves Electromagnetic Waves & Polarization Slide 2 1

2 Maxwell s Equations Electromagnetic Waves & Polarization Slide 3 Recall Maxwell s Equations in Source Free Media In source free media, we have J and. Maxwell s equations in the frequency domain become v Curl Equations E jb H jd Divergence Equations D B Constitutive Relations D E B H Electromagnetic Waves & Polarization Slide 4 2

3 The Curl Equations Predict Waves After substituting the constitutive relations into the curl equations, we get E jh H j E A time harmonic magnetic field will induce a time harmonic electric field circulating about the magnetic field. A time harmonic circulating electric field will induce a time harmonic magnetic field along the axis of circulation. A time harmonic electric field will induce a time harmonic magnetic field circulating about the electric field. A time harmonic circulating magnetic field will induce a timeharmonic electric field along the axis of circulation. An H induces an E. That E induces another H. That new H induces another E. That E induces yet another H. Ando so on. Electromagnetic Waves & Polarization Slide 5 Visualization of Curl & Waves Electric Field Magnetic Field Electromagnetic Waves & Polarization Slide 6 3

4 Derivation of the Wave Equation Electromagnetic Waves & Polarization Slide 7 Wave Equation in Linear Media (1 of 2) Since the curl equations predict propagation, it makes sense that we derive the wave equation by combining the curl equations. E j H H 1 j Solve for H 1 E H j E 1 1 E j E j Electromagnetic Waves & Polarization Slide 8 4

5 Wave Equation in Linear Media (2 of 2) The last equation is simplified to arrive at our final equation for waves in linear media. 1 E 2 This equation is not very useful for performing derivations. It is typically used in numerical computations. Note: We cannot simplify this further because, in general, the permeability is a function of position and cannot be brought outside of the curl operation. 1 2 E E E Electromagnetic Waves & Polarization Slide 9 Wave Equation in LHI Media (1 of 2) In linear, homogeneous, and isotropic media two important simplifications can be made. First, in isotropic media the permeability and permittivity reduce to scalar quantities. 1 2 E E Second, in homogeneous media is a constant and can be brought to the outside of the curl operation and then brought to the right hand side of the equation. 2 E E Electromagnetic Waves & Polarization Slide 1 5

6 Wave Equation in LHI Media (2 of 2) We now apply the vector identity 2 E E 2 2 EE E 2 2 E E E 2 2 EE A A A 2 In LHI media, the divergence equation can be written in terms of E. D E E E Electromagnetic Waves & Polarization Slide 11 Wave Number k and Propagation Constant We can define the term k 2 2 as either This let s us write the wave equation more simply as 2 2 Ek E or or k E E Electromagnetic Waves & Polarization Slide 12 6

7 Solution to the Wave Equation Electromagnetic Waves & Polarization Slide 13 Components Decouple in LHI Media We can expand our wave equation in Cartesian coordinates. 2 2 Ek E Eaˆ Eaˆ Eaˆ k Eaˆ k Eaˆ k Eaˆ x x y y z z x x y y z z E k E aˆ E k E aˆ E k E aˆ x x x y y y z z z We see that the different field components have decoupled from each other. All three equations have the same numerical form so they all have the same solution. Therefore, we only need the solution to one of them. 2 2 Ek E E k Ex k E 2 2 x 2 2 Ey E k E 2 2 z y z Electromagnetic Waves & Polarization Slide 14 7

8 General Solution to Scalar Wave Equation Our final wave equation for LHI media is 2 2 Ek E This could be handed off to a mathematician to obtain the following general solution. jk r jk r Er E e E e forward wave backward wave Electromagnetic Waves & Polarization Slide 15 General Solution to Vector Wave Equation Given the solution to scalar wave equation, we can write solutions for all three field components. jkr jkr Exr Exe Exe jkr jkr Eyr Eye Eye jkr jkr E r E e E e z z z We can assemble these three equations into a single vector equation. E r E ˆ ˆ ˆ x r ax Ey r ay Ez r az jk r jk r Ee Ee forward wave backward wave Electromagnetic Waves & Polarization Slide 16 8

9 General Expression for a Plane Wave The solution to the wave equation gave us two plane waves. From the forward wave, we can extract a general expression for plane waves. jk r Er Pe Frequency domain E rt, Pcos tkr Time domain We define the various parameters as r xaˆ ˆ ˆ x yay zaz position k wave vector E total electric field intensity 2 f angular frequency P polarization vector t time Electromagnetic Waves & Polarization Slide 17 Magnetic Field Component Given that the electric field component of a plane wave is written as jk r E r Pe The magnetic field component is derived by substituting this solution into Faraday s law. E j H jk r Pe j H H kpe 1 jk r Electromagnetic Waves & Polarization Slide 18 9

10 Solution in Terms of the Propagation Constant The wave equation and it solution in terms of is 2 2 r r Ek E Er Ee Ee r Er Pe, cos forward wave The general expressions for a plane wave are Frequency domain E rt P t r Time domain backward wave The magnetic field component is 1 H Pe r j The wave vector and propagation constant are related through jk Electromagnetic Waves & Polarization Slide 19 Visualization of an EM Wave (1 of 2) We tend to draw and think of electromagnetic waves this way Electromagnetic Waves & Polarization Slide 2 1

11 Visualization of an EM Wave (2 of 2) However, this is a more realistic visualization. It is important to remember that plane waves are also of infinite extent in all directions. Electromagnetic Waves & Polarization Slide 21 Intuitive Wave Parameters Electromagnetic Waves & Polarization Slide 22 11

12 Fundamental Vs. Intuitive Parameters Fundamental Parameters These parameters are fundamental to solving Maxwell s equations, but it is difficult to specify how they affect a wave. This is because all of they all affect all properties of a wave. Magnetic Permeability, Electric Permittivity, Electrical Conductivity, Intuitive Parameters These parameters collect specific information about a wave from the fundamental parameters. Refractive index, n Impedance, Wavelength, Velocity, v Wave Number, k Propagation Constant, Attenuation Coefficient, Phase Constant, Electromagnetic Waves & Polarization Slide 23 Wave Velocity, v The scalar wave equation has been known since the 17 s to be 2 wave disturbance 2 angular frequency v v wave velocity If we compare our electromagnetic wave equation to the historical wave equation, we can derive an expression for wave velocity. 2 2 EE v v v Electromagnetic Waves & Polarization Slide 24 12

13 Speed of Light in Vacuum, c In a vacuum, = and = and the velocity becomes the speed of light in a vacuum. 1 1 v c 299,792, 458 m s When not in a vacuum, = r and = r and the velocity is reduced by a factor n called the refractive index c v n r r r r n r r Electromagnetic Waves & Polarization Slide 25 Frequency is Constant, Wavelength Changes Frequency is the most fundamental constant about a wave. It never changes in linear materials. When a wave enters a different material, its speed and thus its wavelength change. v c n n Electromagnetic Waves & Polarization Slide 26 13

14 Speed, Frequency & Wavelength The speed of a wave, its frequency, and its wavelength are related through v f We are now in a position to derive an expression for wavelength f f Electromagnetic Waves & Polarization Slide 27 Wavelength & Wave Number k Recall that we defined the wave number as 1 k r r r r n c The angular frequency is related to wavelength through the ordinary frequency f. c c 2 f 2 2 n Substituting this into the first equation gives c 2 k k n n c n c Electromagnetic Waves & Polarization Slide 28 14

15 Wave Vector, k The wave vector k conveys two pieces of information: (1) Magnitude conveys the wavelength inside the medium, and (2) direction conveys the direction of the wave and is perpendicular to the wave fronts. k k aˆ k aˆ k aˆ x x y y z z Electromagnetic Waves & Polarization Slide 29 Magnitude Conveys Wavelength Most fundamentally, the magnitude of the wave vector conveys the wavelength of the wave inside of the medium. 1 2 k k Electromagnetic Waves & Polarization Slide 3 15

16 Magnitude May Convey Refractive Index When the frequency of a wave is known, the magnitude of the wave vector conveys refractive index. 1 2 k 2 n k n k 2 n k n k Electromagnetic Waves & Polarization Slide 31 Material Impedance, (1 of 3) Impedance is defined as the relationship between the amplitudes of E and H. E H Recall the relationship between E and H. jk r 1 jkr Er Pe and H kpe We can derive an expression for impedance by collecting all of the amplitude terms together in our expression for H. 1 ˆ ˆ jk r Ek ˆ ˆ jkr H kk E P e kpe This term is the amplitude of H Electromagnetic Waves & Polarization Slide 32 16

17 Material Impedance, (2 of 3) From the last slide, the amplitude of H is Ek ˆ ˆ jk r Ek H kp e H Dividing both sides of our expression by E gives an expression for impedance. E H k Since k, the final expression for impedance is k Electromagnetic Waves & Polarization Slide 33 Material Impedance, (3 of 3) We can now revise our expression for the electric and magnetic field components of a wave as ˆ jkr k P jkr E r Pe and H e where E Vacuum Impedance H Electromagnetic Waves & Polarization Slide 34 17

18 Dispersion Relation Electromagnetic Waves & Polarization Slide 35 Derivation in LHI Media We started with the wave equation. 2 2 Ek E We found the solution to be plane waves. E r Pe jk r If we substitute our solution back into the wave equation, we get an equation called the dispersion relation. 2 2 n x y z c k k n k k k The dispersion relation relates frequency to wave vector. For LHI media, it fixes the magnitude of the wave vector to be a constant. Electromagnetic Waves & Polarization Slide 36 18

19 Index Ellipsoids From the previous slide, the dispersion relation for a LHI material was: kx ky kz k n This defines a sphere called an index ellipsoid. The vector connecting the origin to a point on the surface of the sphere is the k vector for that direction. Refractive index is calculated from this. k k n aˆz index ellipsoid For LHI materials, the refractive index is the same in all directions. Think of this as a map of the refractive index as a function of the wave s direction through the medium. aˆx aˆy Electromagnetic Waves & Polarization Slide 37 What About Anisotropic Materials? Isotropic Materials k k k k n a b c Uniaxial Materials k k k k k k a b c 2 a b c 2 k 2 k 2 2 no ne no Biaxial Materials k k k 1 k k n k k n k k n a b c a b c Electromagnetic Waves & Polarization Slide 38 19

20 Electromagnetic Wave Polarization What is Polarization? Polarization is that property of a radiated electromagnetic wave which describes the time varying direction and relative magnitude of the electric field vector. Linear Polarization (LP) Circular Polarization (CP) Left Hand Circular Polarization (LCP) To determine the handedness of CP, imagine watching the electric field in a plane while the wave is coming at you. Which way does it rotate? Electromagnetic Waves & Polarization Slide 4 2

21 Orthogonality and Handedness We get from the curl equations that E H From the divergence equations, we see that E k and H k E H k E We conclude that,, and form an orthogonal triplet. In fact, they follow the right hand rule. H k Electromagnetic Waves & Polarization Slide 41 Possibilities for Wave Polarization Recall that E k so the polarization vector P must fall within the plane perpendicular to k. We can decompose the polarization into two orthogonal directions, â and ˆb. â P paˆ p bˆ a b ˆb k Electromagnetic Waves & Polarization Slide 42 21

22 Explicit Form to Convey Polarization Our electromagnetic wave can be now be written as jk r ˆ ˆ jk r E r Pe p a p b e a b p a and p b are in general complex numbers in order to convey the relative phase of each of these components. ja jb p E e p E e a a b b Substituting p a and p b into our wave expression gives ja jb ˆ ˆ jk r j ba ˆ ˆ ja E r Eae aebe be EaaEbe be e The final expression is: jkr We interpret b a as the phase difference between p a and p b. b a a E r E a E e b e e ˆ ˆ a b We interpret a as the phase common to both p a and p b. j j jkr Electromagnetic Waves & Polarization Slide 43 Determining Polarization of a Wave To determine polarization, it is most convenient to write the expression for the wave that makes polarization explicity. E r E a E e b e e ˆ ˆ a b j j jkr We can now identify the polarization of the wave E amplite along ˆ a a E amplite along ˆ b b phase difference common phase Polarization Designation Mathematical Definition Linear Polarization (LP) = Circular Polarization (CP) = ± 9, E a = E b Right Hand CP (RCP) Left Hand CP (LCP) Elliptical Polarization = + 9, E a = E b = - 9, E a = E b Everything else Electromagnetic Waves & Polarization Slide 44 22

23 Linear Polarization A wave travelling in the +z direction is said to be linearly polarized if: z E x, y, z Pe jk z P sin xˆ cos yˆ P is called the polarization vector. For an arbitrary wave, jk r Er Pe Psinaˆ cos b ˆ aˆ bˆ k All components of P have equal phase. ˆb k â k Electromagnetic Waves & Polarization Slide 45 Linear Polarization Electromagnetic Waves & Polarization Slide 46 23

24 Circular Polarization A wave travelling in the +z direction is said to be circularly polarized if: z E x, y, z Pe jk z P xˆ jyˆ P is called the polarization vector. For an arbitrary wave, jk r Er Pe Paˆ jbˆ aˆ bˆ k The two components of P have equal amplitude and are 9 out of phase. RCP j j LCP k k Electromagnetic Waves & Polarization Slide 47 LP x + LP y = LP 45 A linearly polarized wave can always be decomposed as the sum of two linearly polarized waves that are in phase. Electromagnetic Waves & Polarization Slide 48 24

25 LP x + jlp y = CP A circularly polarized wave is the sum of two orthogonal linearly polarized waves that are 9 out of phase. Electromagnetic Waves & Polarization Slide 49 RCP + LCP = LP A LP wave can be expressed as the sum of a LCP wave and a RCP wave. The phase between the two CP waves determines the tilt of the LP wave polarization. Electromagnetic Waves & Polarization Slide 5 25

26 Circular Polarization (1 of 2) Engineering Right Hand Circular Polarization (RCP) x y z Physics/Optics Left Hand Circular Polarization (LCP) Electromagnetic Waves & Polarization Slide 51 Circular Polarization (2 of 2) Engineering Left Hand Circular Polarization (LCP) x y z Physics/Optics Right Hand Circular Polarization (RCP) Electromagnetic Waves & Polarization Slide 52 26

27 Poincaré Sphere The polarization of a wave can be mapped to a unique point on the Poincaré sphere. Points on opposite sides of the sphere are orthogonal. See Balanis, Chap LP RCP 9 LP LP +45 LP Electromagnetic Waves & Polarization Slide 53 LCP 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 are different Reflection/transmission can be different Frequency of resonators Cutoff conditions for filters, waveguides, etc. Electromagnetic Waves & Polarization Slide 54 27

28 Example Dissect a Wave (1 of 9) The electric field component of a 5.6 GHz plane wave is given by: j x j y j z Er, taˆ x.4915 j.855e e e j x j y j z aˆ j.472 e e e y aˆ.7844 j e e e z j x j y j z 1. Determine the wave vector. 2. Determine the wavelength inside of the medium. 3. Determine the free space wavelength. 4. Determine refractive index of the medium. 5. Determine the dielectric constant of the medium. 6. Determine the polarization of the wave. 7. Determine the magnitude of the wave. Electromagnetic Waves & Polarization Slide 55 Example Dissect a Wave (2 of 9) Solution Part 1 Determine Wave Vector The standard form for a plane wave is jk r E r Pe Comparing this to the expression for the electric field shows that ˆ ˆ ˆ x y z jk r j x j y j z Pa j a j a j e e e e The polarization vector P will be use again later. The wave vector k is determined from the second expression above to be jk r jk y j x j y j z jkxx y jkz z e e e e e e e k aˆ aˆ aˆ m x y z 1 Electromagnetic Waves & Polarization Slide 56 28

29 Example Dissect a Wave (3 of 9) Solution Part 2 Wavelength inside the medium The wavelength inside the medium is related to the magnitude of the wave vector through 2 2 k k The magnitude of the wave vector is k k k k x y z m m m m The wavelength is therefore m cm Electromagnetic Waves & Polarization Slide 57 Example Dissect a Wave (4 of 9) Solution Part 3 Free space wavelength The free space wavelength is 8 c 31 m s c f cm 9 1 f 5.61 s Solution Part 4 Refractive index It follows that the refractive index of the medium is cm n 6. n cm Alternatively, we could determine the refractive index through k k k c k k k n n m s m 9 1 k c 2 f s 6. Electromagnetic Waves & Polarization Slide 58 29

30 Example Dissect a Wave (5 of 9) Solution Part 5 Dielectric constant Assuming the medium has no magnetic response, 2 n 2 r r n Solution Part 6 Wave Polarization To determine the polarization, the electric field is written in the form that makes polarization explicit. j ˆ ˆ j jkr E r EaaEbe be e The choice for â and ˆb is arbitrary, but they most both be perpendicular to k â P paˆ p bˆ a b ˆb k Electromagnetic Waves & Polarization Slide 59 Example Dissect a Wave (6 of 9) Solution Part 6 Wave polarization (cont d) We determine a valid choice for â by first picking any vector that is not in the same direction as k v 1aˆ 2aˆ 3aˆ x y z The cross product will give us a vector perpendicular to k kv aˆ.2896aˆ.8381ˆ.4622 ˆ x ay az kv We determine a valid choice for ˆb using the cross product so that it is perpendicular to both â and k ˆ k aˆ b.538aˆ.2771ˆ.8182 ˆ x ay az k aˆ Electromagnetic Waves & Polarization Slide 6 3

31 Example Dissect a Wave (7 of 9) Solution Part 6 Wave polarization (cont d) To determine the component of the polarization vector P in the â and ˆb directions using the dot product. p ˆ a Pa V m p Pbˆ j V m b We can now write E a and E b from p a and p b by incorporating the phase difference into the parameter. Ea V m Eb V m 9 The common phase between p a and p b is simply. Electromagnetic Waves & Polarization Slide 61 Example Dissect a Wave (8 of 9) Solution Part 6 Wave polarization (cont d) Finally, we have E r E aˆ E e bˆ e e a b j j jkr Ea V m Eb V m 9 k aˆ aˆ aˆ m x y z From this, we determine that we have circular polarization (CP) because E a = E b and = ±9. More specifically, this is left hand circular polarization (LCP) because = Electromagnetic Waves & Polarization Slide 62 31

32 Example Dissect a Wave (9 of 9) Solution Part 7 Magnitude of electric field The magnitude of the wave is simply the magnitude of the polarization vector E r P E E 2 2 a b V m V m 2.4 V m Electromagnetic Waves & Polarization Slide 63 Visualization of EM Waves 32

33 Waves in Materials (1 of 3) Waves in Vacuum H is 377 smaller than E. E H E and H are in phase Im E H H k P Amplitude does not decay Electromagnetic Waves & Polarization Slide 65 Waves in Materials (2 of 3) Waves in Dielectric H is larger now, but still smaller than E. 1 E and H are still in phase Im E H H k P Amplitude still does not decay Electromagnetic Waves & Polarization Slide 66 33

34 More Realistic Wave (E Only) It is important to remember that plane waves are of infinite extent in the x and y directions. Electromagnetic Waves & Polarization Slide 67 More Realistic Wave (E & H) It is important to remember that plane waves are of infinite extent in the x and y directions. Electromagnetic Waves & Polarization Slide 68 34