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Electromagnetic Waves & Polarization 9/13/2017 Course Instructor Dr. Raymond C. Rumpf Office: A‐337 Phone: (915) 747‐6958 E‐Mail: [email protected] EE 4347 Applied Electromagnetics Topic 3a Electromagnetic Waves & Polarization Electromagnetic Waves & Polarization These notes may contain 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 9/13/2017 Maxwell’s Equations Electromagnetic Waves & Polarization Slide 3 Recall Maxwell’s Equations in Source Free Media In source‐free media, we have and .J 0 v 0 Maxwell’s equations in the frequency‐domain become Curl Equations Divergence Equations Ej B D 0 Hj D B 0 Constitutive Relations DE BH Electromagnetic Waves & Polarization Slide 4 2 9/13/2017 The Curl Equations Predict Waves After substituting the constitutive relations into the curl equations, we get Ej H Hj E A time‐harmonic magnetic field will A time‐harmonic electric field will induce a time‐harmonic electric induce a time‐harmonic magnetic field circulating about the magnetic field circulating about the electric field. field. A time‐harmonic circulating electric A time‐harmonic circulating field will induce a time‐harmonic magnetic field will induce a time‐ magnetic field along the axis of harmonic electric field along the circulation. 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 Magnetic Field Electric Field Electromagnetic Waves & Polarization Slide 6 3 9/13/2017 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. Ej H Hj E Solve for H 1 1 H E j 1 1 Ej E j Electromagnetic Waves & Polarization Slide 8 4 9/13/2017 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 E 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 EE 2 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. 12 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. E 2 E Electromagnetic Waves & Polarization Slide 10 5 9/13/2017 Wave Equation in LHI Media (2 of 2) We now apply the vector identity A AA 2 E 2 E E 22EE In LHI media, the divergence 22E EE equation can be written in terms of E. 22EE 0 D 0 E 0 E 0 E 0 Electromagnetic Waves & Polarization Slide 11 Wave Number k and Propagation Constant We can define the term as either k 22 or 22 k 2 This let’s us write the wave equation more simply as 22EkE 0 or 22EE 0 Electromagnetic Waves & Polarization Slide 12 6 9/13/2017 Solution to the Wave Equation Electromagnetic Waves & Polarization Slide 13 Components Decouple in LHI Media We can expand our wave equation in Cartesian coordinates. 22EkE 0 22 2222 Eaxxˆˆˆˆˆˆ Ea yy Ea zz k Ea xx k Ea yy k Ea zz 0 22 22 22 EkEaxxxyyyzzzˆˆˆ EkEa EkEa 0 We see that the different field components have 22 decoupled from each other. EkExx 0 All three equations have the same numerical form so 22EkE 0 they all have the same solution. yy 22 Therefore, we only need the solution to one of them. EkEzz 0 22EkE 0 Electromagnetic Waves & Polarization Slide 14 7 9/13/2017 General Solution to Scalar Wave Equation Our final wave equation for LHI media is 22EkE 0 This could be handed off to a mathematician to obtain the following general solution. Er Eejkr Ee jkr 00 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 Erxx Ee Ee x jkr jkr Eryy Ee Ee y jkr jkr Erzz Ee Ee z We can assemble these three equations into a single vector equation. E r Ex raˆˆˆxy E ra yz E ra z Eejk r Ee jk r 00 forward wave backward wave Electromagnetic Waves & Polarization Slide 16 8 9/13/2017 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. Er Pejk r Frequency‐domain E rt,cos P t k r Time‐domain We define the various parameters as rxayazaˆˆˆposition xyz k wave vector E total electric field intensity 2f 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 Er Pejkr The magnetic field component is derived by substituting this solution into Faraday’s law. E jH 1 PejHjk r HkPe jkr Electromagnetic Waves & Polarization Slide 18 9 9/13/2017 Solution in Terms of the Propagation Constant The wave equation and it solution in terms of is 22EkE 0 Er Eerr Ee 00 forward wave backward wave The general expressions for a plane wave are Er Pe r Frequency‐domain E rt,cos P t r Time‐domain The magnetic field component is The wave vector and propagation 1 r constant are related through HPe j 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 20 10 9/13/2017 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 9/13/2017 Fundamental Vs. Intuitive Parameters Fundamental Parameters Intuitive Parameters These parameters are fundamental to These parameters collect specific solving Maxwell’s equations, but it is information about a wave from the difficult to specify how they affect a fundamental parameters. wave. This is because all of they all affect all properties of a wave. Refractive index, n Magnetic Permeability, Impedance, Electric Permittivity, Wavelength, v Electrical Conductivity, Velocity, 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 1700’s to be 2 wave disturbance 2 0 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. 22EE 0 1 2 v 2 v 0 v Electromagnetic Waves & Polarization Slide 24 12 9/13/2017 Speed of Light in Vacuum, c0 In a vacuum, = 0 and = 0 and the velocity becomes the speed of light in a vacuum. 11 vc 0 299,792,458 m s 00 When not in a vacuum, = 0r and = 0r and the velocity is reduced by a factor n called the refractive index. 11 11c v 0 00 rr 00 rr n n rr Electromagnetic Waves & Polarization Slide 25 Frequency is Constant, Wavelength Changes Frequency is the most When a wave enters a different fundamental constant about a material, its speed and thus its wave. It never changes in linear wavelength change. materials. c v 0 0 n n Electromagnetic Waves & Polarization Slide 26 13 9/13/2017 Speed, Frequency & Wavelength The speed of a wave, its frequency, and its wavelength are related through vf We are now in a position to derive an expression for wavelength. 112 f f Electromagnetic Waves & Polarization Slide 27 Wavelength & Wave Number k Recall that we defined the wave number as 1 kn0r0r 00 rr c0 The angular frequency is related to wavelength through the ordinary frequency f. cc 22f 00 2 0 n Substituting this into the first equation gives 112c0 2 kn 2 n k cnc00 Electromagnetic Waves & Polarization Slide 28 14 9/13/2017 Wave Vector, k The wave vector conveys two pk ieces 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. kkakakax ˆˆˆxyyzz 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 2 2 k1 k2 1 2 Electromagnetic Waves & Polarization Slide 30 15 9/13/2017 Magnitude May Convey Refractive Index When the frequency of a wave is known, the magnitude of the wave vector conveys refractive index. 1 2 2 n 2 n 2 kkn1 kkn2 k 101 2020 0 0 0 Electromagnetic Waves & Polarization Slide 31 Material Impedance, (1 of 3) Impedance is defined as the relationship between the amplitudes of E and H. EH00 Recall the relationship between E and H.
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