Preliminary Topics in EM

Transcription

1 ECE 53 1 st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) rcrumpf@utep.edu Lecture #1 Preliminary Topics in EM Lecture 1 1 Lecture Outline Maxwell s equations Wave vectors and polarization Scattering at an interface Scattering from multiple interfaces Phase, group, and energy velocity Bonus Topics Lenses Gaussian beams Lecture 1 1

2 Maxwell s Equations Maxwell s Equations

3 All Together Now Divergence Equations B 0 D v Curl Equations D H J t B E t What produces fields Constitutive Relations Dt t Et B t t H t means convolution means tensor How fields interact with materials Lecture 1 Slide 5 Physical Boundary Conditions and 1 1 and Tangential components of E and H are continuous across aninterface. E1,T E,T H1,T H,T Fields normal to the interface are discontinuous across an interface. Note: Normal components of D and B are continuous across the interface. Tangential component of the wave vector is continuous across an interface. 1E1,N 1H1,N E,N H,N k1,t k,t These are more complicated boundary conditions, physically and analytically. Lecture 1 Slide 6 3

4 Wave Vectors and Polarization Wave Vector k A wave vector conveys two pieces of information at the same time. First, its orientation describes the direction of the wave. It is perpendicular to the wave front. Second, its magnitude conveys the spatial period of the wave. It is divided by the spatial period of the wave (wavelength). E r E jk r exp 0 r xxˆ yyˆzzˆ position vector n k 0 k k xˆk yˆk zˆ If the frequency of the wave is known and constant, and x y z it usually is, the magnitude of the k vector conveys the Lecture 1 refractive index of the material the wave is in. Slide 8 k 4

5 The Complex Wave Vector A wave travelling the +z direction can be written in terms of the wave number k as E z E e 0 jkz k k jk Substituting this back into the wave solution yields jk jk z jkz kz E z E0e E0e e oscillation growth/decay Lecture 1 Slide 9 and A wave travelling the +z direction can also be written in terms of a propagation constant and an attenuation coefficient as z j z E z E0 e e growth/decay oscillation We now have physical meaning to the real and imaginary parts of the wave vector. k j k = Re[k] phase term k = Im[k] attenuation term n 0 Lecture 1 Slide 10 5

6 1D Waves with Complex k Purely Real k Purely Imaginary k Complex k Uniform amplitude Oscillations move power Considered to be a propagating wave Decaying amplitude No oscillations, no flow of power Considered to be evanescent Decaying amplitude Oscillations move power Considered to be a propagating wave (not evanescent) Lecture 1 Slide 11 What is Polarization? Polarization refers to the orientation of the electric field relative to the direction of the wave. 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? Lecture 1 Slide 1 6

7 Handedness Convention As Viewed From Source Polarization is taken as the time varying electric field view with the wave moving away from you. Primarily used in engineering and quantum physics. As Viewed From Receiver Polarization is taken as the time varying electric field view with the wave coming toward you. Primarily used in optics and physics. source RCP source RCP receiver receiver source LCP source LCP receiver receiver Lecture 1 Slide 13 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, â E r P a b aˆ bˆ k jk r Pe sin ˆ cos ˆ All components of P have equal phase. ˆb k k Lecture 1 Slide 14 7

8 Circular Polarization A wave travelling in the +z direction is said to be linearly polarized if: z E x, y, z Pe jk z P x jy ˆ ˆ 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 90 out of phase. LCP j j RCP k k Lecture 1 Slide 15 Summary In general, a wave travelling in the +z direction can be written as j j x y j z Ex, y, ze ˆ ˆ xe x Eye ye 1 j j x y j z Hx, y, z E ˆ ˆ xe x Eye ye LP x LP y RCP LCP Elliptical E x and E y E y = 0 E x = 0 E x =E y =E R E x =E y =E L E y = E R E L RH: E R >E L E x = E R + E L LH: E R <E L x y 0 0 -/ +/ Lecture 1 Slide 16 8

9 LP x + LP y = LP 45 A linearly polarized wave can always be decomposed as the sum of two orthogonal linearly polarized waves that are in phase. Lecture 1 Slide 17 LP x + jlp y = CP A circularly polarized wave is the sum of two linearly polarized waves that are 90 out of phase. Lecture 1 Slide 18 9

10 RCP + LCP = LP A linearly polarized 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. Lecture 1 Slide 19 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 are different Reflection/transmission can be different Frequency of resonators Cutoff conditions for filters, waveguides, etc. Lecture 1 Slide 0 10

11 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 90 LP 0 LP +45 LP Lecture 1 Slide 1 LCP TE and TM We use the labels TE and TM when we are describing the orientation of a linearly polarized wave relative to a device. TE/perpendicular/s the electric field is polarized perpendicular to the plane of incidence. TM/parallel/p the electric field is polarized parallel to the plane of incidence. Lecture 1 Slide 11

12 CAUTION: Another Place You Will See TE and TM Labels The labels TE and TM are also used to describe modes in homogeneous metallic waveguides. There really is no comparing TE and TM waveguide modes to TE and TM plane waves incident on a interface. It is a completely different concept. TE 10 TE 01 TE 0 TE 30 TE 11 TM 11 TE 1 TM 1 Lecture 1 Slide 3 Scattering at an Interface 1

13 Reflection, Transmission and Refraction at an Interface Angles inc ref 1 n sin n sin 1 1 Snell s Law Lecture 1 n, 1 1 n, TE Polarization r t 1r TE TE TE cos11cos cos cos 1 1 cos1 cos cos t 1 1 TE TM Polarization r t 1r TM TM TM cos 1cos1 cos cos 1 1 cos1 cos cos 1 1 cos t cos TM 1 Slide 5 Simplifications at Normal Incidence For normal incidence, we have inc ref 1 0 In isotropic materials, the different polarizations reflect and transmit the same. 1 r rt E rtm 1 t tte ttm 1 R r n T n 1 t It really does not make sense to talk about TE and TM for normal incidence because there is no plane of incidence from which to define it. All polarizations scatter the same. 1r t This is NOT conservation of energy because these are field amplitude quantities, not power quantities. Lecture 1 Slide 6 13

14 The Critical Angle (Total Reflection) Above the critical angle c, reflection is 100% r r TE TM cosc 1cos 1 cos cos c 1 cos 1cosc cos cos 1 c 1 This will happen when cos( ) is imaginary. These conditions are derived from Snell s Law. n1 cos 1sin 1 sin c n n sin 1 1 c n1 n cos sin n n 1 1 c n sin c 1 sin n n sin n sin 1 c 1 c n1 Lecture 1 Slide 7 Brewster s Angle (Total Transmission) TE Polarization cosb 1cos rte 0 cos cos B 1 sin 1 1 B We see that as long as 1 = then there is no Brewster s angle. Generally, most materials have a very week magnetic response and there is no Brewster s angle for TE polarized waves. TM Polarization cos 1cos1 rtm 0 cos cos 1 1 sin 1 1 B We see that if 1 = then there is no Brewster s angle. For materials with no magnetic response, the Brewster s angle equation reduces to tan Lecture 1 n This is the most well known equation. B 1 1 n1 Slide 8 14

15 Notes on a Single Interface It is a change in impedance that causes reflections Snell s Law quantifies the angle of transmission Angle of transmission and reflection does not depend on polarization. The Fresnell equations quantify the amount of reflection and transmission Amount of reflection and transmission depends on the polarization For incident angles greater than the critical angle, a wave will be completely reflected regardless of its polarization. When a wave is incident at the Brewster s angle, a particular polarization will be completely transmitted. Lecture 1 Slide 9 Validity of Law of Reflection and Snell s Law There are no plane wave sources. Beam sources are more realistic. A beam can be decomposed into a plane wave spectrum. The Fresnel equations predict that each of the component plane wave will reflect and transmit with a different amplitude depending on its angle and polarization. This means that the amplitude profile of a beam will be modified after reflection and transmission. ref??? inc If the amplitude profile is modified, then the beam will propagate and diffract differently.??? n sin n sin 1 1 This ultimately means the reflected and transmitted beams will propagate at different angles than the law of reflection and Snell s law predict. Lecture

16 More Accurate Picture of Reflection and Transmission temporarily a surface wave Lecture 1 31 Simulation Example A Gaussian beam 5 0 wide is incident from air onto glass with n = 1.5. theoretical modeled inc ref trn inc ref trn inc ref trn 60?? Lecture

17 Scattering from Multiple Interfaces Reflection and Transmission from a Slab incident n ref, ref n, n trn, trn These add together to be the total reflection. These add together to be the total transmission. r re r 1 jk0nl 1 jk0nl rre 1 trn r r ref 1 ref Lecture 1 trn L For small reflections, r r re 1 jk0nl Slide 34 17

18 The Fabry-Perot Cavity Linear Response Low Finesse R R 4% 1 L 5 T R R T FP FP R1R 1 A RR 1 1 A cos 1RR RR cos TT 1 1 RR RR cos Linear Response High Finesse R R 67% 1 L 5 R T R reflectance at first interface 1 R reflectance at second interface T 1R transmittance at first interface 1 1 T 1R transmittance at second interface A power loss through cavity round trip phase shift in the cavity Lecture 1 For small reflections, R T FP FP R 1 cos 1R 1cos Slide 35 Anti-Reflection Coatings n1, 1 n, n, ar ar n1, 1 n, L Lecture 1 General Case ar 1 0 L 4n ar n nn ar 1 0 L 4n ar No magnetic response Slide 36 18

19 Bragg Gratings LH n H L L nl LH n H LL nl LH n H A Bragg grating is typically composed of alternating layers of high and low refractive index. Each layer is /4 thick. Higher index contrast provides wider stop band. More layers improves suppression in the stop band. LL nl L H n H LL nl B LL 4nL B LH 4n Bragg center wavelength B H stop band B Lecture 1 Slide 37 Multilayer Filters in d 1 d d3 d d5 4 d d For small reflection coefficients, the overall reflection coefficient can be written as e e e j j4 jm in 0 1 M where i is the reflection coefficient at the i th interface. Z i Z Z Z i1 i i1 i i1 i i1 i d d We can design any filter response we want by appropriate selection of s and incorporating enough segments. The design process is essentially the same as for designing digital filters. 1 1 M M Lecture 1 Slide 38 19

20 Phase, Group, and Energy Velocity Phase Velocity The phase velocity of a wave is the speed at which the phase of a single frequency wave propagates through space. It is defined in terms of the angular frequency (number of oscillations per unit time) and the wave number k (number of oscillations per unit distance). v p k Lecture

21 Derivation of Phase Velocity We start with the expression for a wave travelling in the x direction. ; sin E zt kx t kx t 0 kx t x t k x vp t k Wave moves at a speed that keeps the argument of the sine function constant. We set the argument equal to zero and rearrange the terms. This has units of distance/time, which is velocity. We derived this from the phase of the wave sin(). Lecture 1 41 Phase Refractive Index We can characterize a medium by it phase refractive index n p. This is the factor describing how much slower than the speed of light that the phase is propagating. v p c n 0 p Lecture 1 4 1

22 Phase Velocity Can Approach Infinity in a Waveguide (1 of ) A low order mode can be thought of as the sum of two plane waves at small angles relative to the longitudinal direction of the waveguide. Phase propagates slowly. Lecture 1 43 Phase Velocity Can Approach Infinity in a Waveguide ( of ) A high order mode can be thought of as the sum of two plane waves at large angles relative to the longitudinal direction of the waveguide. Phase propagates quickly. Lecture 1 44

23 Group Velocity The group velocity is the speed and direction in which the envelope of the wave s amplitude propagates. It is defined as Here, the wave appears to be very fast, but the overall package of energy propagate slowly. v g k Lecture 1 45 Group Refractive Index We can characterize a medium by it group refractive index n g. This is the factor describing how much slower than the speed of light that the envelop of the wave is propagating. v g c n 0 g Lecture

24 Phase Vs. Group Velocity By their definitions, the phase velocity applies only to a wave at a single frequency. The group velocity applies to a packet of waves covering some spectrum. The phase and group velocities are often the same, but they can be different. Lecture 1 47 Derivation of Group Velocity (1 of 7) By definition, group velocity applies to a wave composed of more than one frequency. A wave composed of two frequencies can be written as ; sin sin E zt kx 1 1t kx t Wave 1 Wave Lecture

25 Derivation of Group Velocity ( of 7) Using the following identity from trigonometry, A B AB sin Asin Bsin cos we get ; sin 1 1 sin kx tkxt kx tkxt E z t k x t k x t sin cos k k1 1 k k1 1 sin x t cos x t Lecture 1 49 Derivation of Group Velocity (3 of 7) The last equation can be written in terms of the center frequency c and the bandwidth. ; sin cos E z t k x t k x t c c k k1 k k1 kc k 1 1 c Lecture

26 Derivation of Group Velocity (4 of 7) We interpret this as a sine wave at frequency c and wave number k c that is modulated by a cosine function. ; sin cos E zt kx t k x t c c Lecture 1 51 Derivation of Group Velocity (5 of 7) We have identified the cos(kx - t) term as the envelope. How quickly does this move? It moves at a speed that keeps the argument of the cosine function constant. kx t constant Lecture 1 5 6

27 Derivation of Group Velocity (6 of 7) To derive a velocity term (dx/dt), we differentiate this equation. kx t 0 kdx dt 0 The equation can now be manipulated to derive a quantity with units of velocity. kdx dt 0 kdx dt dx dt k v dx dt k Lecture 1 53 Derivation of Group Velocity (7 of 7) We now take the limit as the deltas become very small. v g lim 0 k0 k d dk The equivalent equation in more than one dimension is v g k k k dispersion relation Lecture

28 Ordinary Materials In an ordinary material, the dispersion relation is c0 n k The phase velocity is c0 vp n The group velocity is v g d c0 dk n or c k n 0 k We see that the phase velocity and group velocity are equal. When are they not equal? Lecture 1 55 Dispersive Materials (1 of ) In a dispersive material, the refractive index n can be different at different frequencies. We differentiate the dispersion relation as follows. n c0k dn nd c dk We rearrange the terms to arrive at c0 dn d n n dk dk 0 This is group velocity, v g This is phase velocity, v p Lecture

29 Dispersive Materials ( of ) We now have an expression relating group and phase velocity. v p dn v ndk g Solving this for group velocity yields dn k dn vg vp vp1 ndk ndk k We see it is the dn/dk term that is responsible for v g v p. Any time the refractive index n is not constant, the medium is said to have dispersion and the group velocity will deviate from the phase velocity. Lecture 1 57 v p Phase and Group Refractive Indices From the previous equations we can write the phase refractive index n p and group refractive index n g as n p kc0 n g dn p np d dnp np 0 d 0 Lecture

30 Summary of Phase, Group and Energy Velocity Phase Velocity Phase velocity describes the speed and direction of the phase of a wave. v ˆ p s c 0 np k v Group Velocity Group velocity describes the speed and direction of the envelope of a pulse. c0 vg k k ng vg v v no dispersion g p Energy Velocity Energy velocity describes the speed and direction of the energy. P c0 ve ne U ve v v p linear materials Lecture 1 e g Slide 59 k y ŝ k v g v p k x Lenses Lecture 1 Slide 60 30

31 Lenses Lenses are structures that focus electromagnetic waves. Lenses are also used to collimate a beam or diverge a beam. Optical Lens Microwave Lens W. Chalodhorn, D. R. Deboer, Use of Microwave Lenses in Phase Retrieval Microwave Holography of Reflector Antennas, IEEE Trans. Ant. Prop., vol. 50, no. 9, pp , 00. Lecture 1 Slide 61 Ray Tracing Ray tracing is a graphical technique to determine the direction of a beam that passes through the lens. Lecture 1 Slide 6 31

32 Ray Tracing Definitions Thin lens focal plane focal plane focal point optical axis focal point Lecture 1 Slide 63 Ray Tracing: Rule #1 The direction of a ray passing through the center of the lens remains unchanged. Lecture 1 Slide 64 3

33 Ray Tracing: Rule # A ray parallel to the optical axis will pass through the focal point. Lecture 1 Slide 65 Ray Tracing: Rule #3 An arbitrary ray will pass through the focal plane at the same point as a parallel ray passing through the center of the lens. Lecture 1 Slide 66 33

34 Lens s Makers Formula The focal length of a thin lens is approximately 1 nl ens n F n0 R1 R R R 1 F F Lecture 1 Slide 67 Death Rays From a Skyscraper The curved glass on a skyscraper in London acts like a lens. In late August / early September, the sun is at just the right angle to focus light down onto the street. Here, it melted part of an expensive Jaguar. 0 Fenchurch Street, London. Lecture 1 Slide 68 34

35 Gaussian Beams Lecture 1 Slide 69 Ray Matrix A ray matrix relates the height and slope of the input and output rays of an optical system. x, y h A Bh1 C D 1 Ray Matrix h 1 h 1 z Optical System Lecture

36 Common Ray Matrices Unperturbed Distance Thin Lens 1 d f 1 d z f f z 1 1 Lecture 1 71 Combining Ray Matrices Ray matrices are combined using standard matrix multiplication. A C B D A B C D = A C B D A3 B3 A BA1 B1 C3 D 3 C D C1 D 1 Note the order of these matrices. Distance Distance 1 d1 d1 1 d1 d d 1 d Lens Distance 1 d d f d f 1 1 f 1 d z f f z 1 1 Lecture

37 The Gaussian Beam Equation w 0 r 1 z kr Ex, y, z E0 exp exp j kz tan exp j wz w z z0 Rz amplitude amplitude profile longitudinal phase radial phase w z R z z 0 w0 1 z0 nw 0 0 z z0 z1 z Beam width (radius) Radius of curvature Division between near field and far field. Beam diverges linearly for z > z 0. w z z 0 0 w0 minimum spot size r distance from z-axis r x y Lecture 1 73 Geometry of the Gaussian Beam far field near field x, y x, y far field w z w 1 e 0 z 0 z 0 w 0 z planar wavefront Gaussian beam has circular wavefront planar wavefront Lecture

38 ABCD Law The ABCD law allows us to carry radius of curvature R and beam width w through a system s ray matrix. The ABCD law is q z 1 1 Aq1 z B 1 C D q1 z or Cq z D q z A B q z Here the beam parameters R(z) and w(z) are combined into the complex beam parameter q(z) j q z R z nw z w w1 Lecture 1 75 A C B D R R1 q1 q 38