Instructor Dr. Raymond Rumpf (915) 747 6958 rcrumpf@utep.edu EE 5337 Computational Electromagnetics Lecture #3 Preliminary Topics Lecture 3These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Slide 1 Outline Review of Linear Algebra Wave Vectors Wave Polarization Index Ellipsoids Electromagnetic Behavior at an Interface Phase matching at an interface The Fresnel equations Visualization Image Theory Lecture 3 Slide 1
Review of Linear Algebra Lecture 3 Slide 3 Matrices Represent Sets of Equations A set of linear algebraic equations can be written in matrix form. a wa xa ya z b 11 1 13 14 1 a wa xa ya z b 1 3 4 a wa xa ya z b 31 3 33 34 3 a wa xa ya z b 41 4 43 44 4 a11 a1 a13 a14 w b1 a1 a a3 a 4 x b a31 a3 a33 a 34 y b 3 a a a a z b 41 4 43 44 4 Lecture 3 Slide 4
Interpretation of Matrices a11xa1 ya13 z b1 a1xa ya3z b a31xa3 ya33z b3 EQUATION FOR a11 a1 a13 x b1 a a a y b 1 3 a31 a3 a 33 z b 3 RELATION TO Equation for x Equation for y Equation for z a a a a a a a a a 11 1 13 1 3 31 3 33 a a a a a a a a a 11 1 13 1 3 31 3 33 From a purely mathematical perspective, this interpretation does not make sense. This interpretation will be highly useful and insightful because of how we derive the equations. Lecture 3 Slide 5 Compact Matrix Notation Matrices and vectors can be represented and treated as single variables. a11 a1 a13 a14 w b1 a1 a a3 a 4 x b a31 a3 a33 a 34 y b 3 a a a a z b 41 4 43 44 4 Ax b or Ax b a a a a a a a a A a a a a a a a a 11 1 13 14 1 3 4 31 3 33 34 41 4 43 44 square matrix w x x y z column vector b1 b b b3 b4 column vector Lecture 3 Slide 6 3
Matrices Require Special Algebra AB BA Commutative Laws AB BA ABBA Associative Laws ABC ABC ABCABC Distributive Laws A BC ABAC AB CACBC Multiplication with a Scalar Addition with a Scalar AB AB AB AB A B Matrix Inverses and Transposes 1 1 1 1 1 A A AB B A 1 T T 1 A A T T T T T T T T A A AB A B AB B A A? a11 a1 n IA am 1 a mn Lecture 3 Slide 7 Manipulating Matrix Equations Example #1: Dividing both sides by a matrix on the right ABCDE A B CC D E C 1 1 A B D E C 1 1 ABC DE 1 1 C C A B C D E Starting equation Post multiply both sides by C 1 Recall that CC 1 =I Example #: Dividing both sides by a matrix on the left C A B D E Example #3: Simplify an expression 1 1 C A DBCD 1 C A 1 1 A C A CC BC BC BCC 1 1 1 A B Starting equation Pre multiply both sides by C 1 Recall that C 1 C=I Starting equation Subtract D from both sides Recall inverse of a product rule Post multiply both sides by C 1 1 Recall that C 1 C=I Lecture 3 Slide 8 4
The Zero and the Identity Matrices Zero Matrix 0 0 0 0 0 0A A00 0A A0 A AA 0 Identity Matrix 1 I 0 0 1 IA AI A 1 1 A A AA I Lecture 3 Slide 9 Matrix Division It is very rare to calculate the inverse of matrix because it is very computationally intensive to do this. At first glance, matrix division appears to require calculating the inverse of a matrix, but highly efficient algorithms exist that do not need to do this. Pre Division: A = inv(b)*c; 1 A B C A = B\C; Post Division: A = C*inv(B); 1 A CB A = C/B; Despite that both of these equations are dividing by B, pre and postdivision do NOT lead to the same result. B C CB 1 1 Lecture 3 Slide 10 5
Matrix Division With More than Two Terms Suppose we must evaluate the following matrix expression. A 1 BC D Would the following MATLAB code work? A = B*C\D; So what is the correct code? A = B/C*D; A = B*(C\D); No! Remember the order of operations. MATLAB will first multiply B and C. A BC It will then backward divide on D. A BC D 1 It might be a good idea to time these two calculations to see which is faster. Lecture 3 Slide 11 Proper Notation for Matrix Division It is almost never correct to write matrix division as a fraction. A B Why? Does this represent predivision or postdivision? B A or AB??? 1 1 It is impossible to say, thus fraction notation is incorrect for matrices. One exception When both A and B are diagonal matrices, both pre and post division will give the same answer. 1 1 11 11 11 11 11 11 b a a b a b b a a b a b b a a b a b MM MM MM MM MM MM Lecture 3 Slide 1 6
Ax = b Vs. Ax = x Standard Linear Problem Ax b Eigen Value Problem Ax x Has only one answer Requires a source Has an infinite number of answers Modes No source Lecture 3 Slide 13 Wave Vectors Lecture 3 Slide 14 7
Wave Vector k The wave vector k conveys two pieces of information: (1) Magnitude conveys the wavelength inside the medium, and () 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 Lecture 3 Slide 15 Magnitude Conveys Wavelength Most fundamentally, the magnitude of the wave vector conveys the wavelength of the wave inside of the medium. 1 k 1 1 k Electromagnetic Waves & Polarization Slide 16 8
Magnitude May Convey Refractive Index When the frequency of a wave is known, the magnitude of the wave vector conveys refractive index. 1 k n k n 1 1 0 1 0 k n k n 0 0 k0 0 Electromagnetic Waves & Polarization Slide 17 The Complex Wave Number A wave travelling the +z direction can be written in terms of the wave number k as E z Pe jkz k k jk Substituting this back into the wave solution yields jk jk z kz jkz E z E0e E0e e attenuation oscillation Electromagnetic Waves & Polarization Slide 18 9
and A wave travelling the +z direction can also be written in terms of a phase constant and an attenuation coefficient as kz jkz E z E0 e e z j z Ez E0 e e attenuation oscillation We now have physical meaning to the real and imaginary parts of the wave vector. k j k = Re[k] phase term n k = Im[k] attenuation term 0 Electromagnetic Waves & Polarization Slide 19 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) This implies that these are the only.5 configurations that electromagnetic fields can take on. Electromagnetic Waves & Polarization Slide 0 10
D Waves with Doubly Complex k Real k x Imaginary k x Complex k x Real k y x y x y x y Imaginary ky x y x y x y Complex ky x y x Lecture 3 Slide 1 y x y Wave Polarization Lecture 3 Slide 11
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? Lecture 3 Slide 3 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 a p bˆ b ˆb k Electromagnetic Waves & Polarization Slide 4 1
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 5 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) = 0 Circular Polarization (CP) = ± 90, E a = E b Right Hand CP (RCP) Left Hand CP (RCP) Elliptical Polarization = + 90, E a = E b = - 90, E a = E b Everything else Electromagnetic Waves & Polarization Slide 6 13
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 3 http://en.wikipedia.org/wiki/circular_polarization Slide 7 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 Lecture 3 Slide 8 14
Linear Polarization Electromagnetic Waves & Polarization Slide 9 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 90 out of phase. RCP j j LCP k k Lecture 3 Slide 30 15
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. Lecture 3 Slide 31 LP x + jlp y = CP A circularly polarized wave is the sum of two orthogonal linearly polarized waves that are 90 out of phase. Lecture 3 Slide 3 16
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 wave. Lecture 3 Slide 33 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 structures that are 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 3 Slide 34 17
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. 4. 45 LP RCP 90 LP 0 LP +45 LP Lecture 3 Slide 35 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 3 Slide 36 18
Calculating the Polarization Vectors Incident Wave Vector Surface Normal Unit Vectors of Polarization Directions sin cos 0 aˆ y 0 kinc k0n inc sin sin ˆ n 0 ˆ nˆ k ate inc 0 cos 1 nˆ k inc aˆ TE kinc Right handed aˆ TM coordinate system aˆ TE kinc x y z Composite Polarization Vector P p aˆ p aˆ TE TE TM TM In CEM, we usually make P 1 Lecture 3 Slide 37 Index Ellipsoids Lecture 3 Slide 38 19
Dispersion Relations The dispersion relation for a material relates the wave vector to frequency. Essentially, it tells us the refractive index as a function of direction through a material. It is derived by substituting a plane wave solution into the wave equation. For a linear, homogeneous, and isotropic (LHI) material, the dispersion relation is: k k k k n a b c 0 This can also be written as: k k k n a b c k 0 0 Lecture 3 Slide 39 Index Ellipsoids From the previous slide, the dispersion relation for a LHI material was: k k k k n a b c 0 This defines a sphere called an index ellipsoid. The vector connecting the origin to a point on the sphere is the k vector for that direction. Refractive index n is calculated from its magnitude. k k n 0 For LHI materials, the refractive index is the same in all directions. ĉ index ellipsoid Think of this as a map of the refractive index as a function of the wave s direction through the medium. â ˆb Lecture 3 Slide 40 0
Index Ellipsoids for Uniaxial Materials ne â n O ĉ n O ne n O ˆb Observations Both solutions share a common axis. This common axis looks isotropic with refractive index n 0 regardless of polarization. Since both solutions share a single axis, these crystals are called uniaxial. The common axis is called: o Optic axis o Ordinary axis o C axis o Uniaxial axis Deviation from the optic axis will result in two separate possible modes. k k k k k a b c a b k 0 k 0 0 no ne Lecture 3 Slide 41 Index Ellipsoids for Biaxial Materials Biaxial materials have all unique refractive indices. Most texts adopt the convention where n n n a b c The general dispersion relation cannot be reduced. optic axes ĉ Notes and Observations The convention n a <n b <n c causes the optic axes to lie in the a c plane. The two solutions can be envisioned as one balloon inside another, pinched together so they touch at only four points. â ˆb Propagation along either of the optic axes looks isotropic, thus the name biaxial. Lecture 3 Slide 4 1
Direction of Power Flow Isotropic Materials y P k x Anisotropic Materials y k P x Phase propagates in the direction of k. Therefore, the refractive index derived from k is best described as the phase refractive index. Velocity here is the phase velocity. Power propagates in the direction of P which is always normal to the surface of the index ellipsoid. From this, we can define a group velocity and a group refractive index. Lecture 3 Slide 43 Illustration of k versus P k P P k Negative refraction into an electromagnetic band gap material. We don t need a negative refractive index to have negative refraction. Lecture 3 Slide 44
Phase Matching at an Interface Lecture 3 Slide 45 Illustration of the Dispersion Relation k k k k n x y 0 Index ellipsoid y k k y k x x The dispersion relation for isotropic materials is essentially just the Pythagorean theorem. It says a wave sees the same refractive index no matter what direction the wave is travelling. Lecture 3 Slide 46 3
Index Ellipsoid in Two Different Materials Material 1 (Low n) Material (High n) k k k k n x,1 y,1 1 0 1 k k k k n x, y, 0 n n 1 Lecture 3 Slide 47 Phase Matching at the Interface Between Two Materials Where n 1 < n Material 1 k k k k n x,1 y,1 1 0 1 n n 1 Material k k k k n x, y, 0 Lecture 3 Slide 48 4
Summary of the Phase Matching Trend for n 1 < n n n 1 Material 1 Material k k k k n x,1 y,1 1 0 1 k k k k n x, y, 0 1 3 4 5 6 Properly phased matched at the interface. Lecture 3 Slide 49 Phase Matching at the Interface Between Two Materials Where n 1 > n n n 1 inc Material 1 c k inc x,1 ky,1 k1 k0n1 c Material k k k k n x, y, 0 Lecture 3 Slide 50 5
Summary of the Phase Matching Trend for n 1 > n n n 1 Material 1 Material k k k k n x,1 y,1 1 0 1 k k k k n x, y, 0 1 inc c inc c 3 4 inc Properly phased matched at the interface. c Lecture 3 Slide 51 inc c Reflection and Transmission: The Fresnel Equations Lecture 3 Slide 5 6
Reflection, Transmission, and Refraction at an Interface n, 1 1 n, Angles TE Polarization cos11cos rte cos cos 1r 1 1 TM TM cos Lecture 3 1 Slide 53 t TE TE cos1 cos cos t 1 1 TE TM Polarization cos 1cos1 rtm cos cos t 1r inc ref 1 n sin n sin 1 1 TM 1 1 cos1 cos cos 1 1 cos t Snell s Law Reflectance and Transmittance Reflectance The fraction of power R reflected from an interface is called reflectance. It is related to the reflection coefficient r through RTE rte RTM r R r TE TE TM RTM rtm Transmittance The fraction of power T transmitted through an interface is called transmittance. It is related to the transmission coefficient t through T TE t cos 1 TE cos1 T TM t cos 1 TM cos1 T T TE TM t t TE TM Lecture 3 Slide 54 7
Amplitude Vs. Power Terms Wave Amplitudes The reflection and transmission coefficients, r and t, relate the amplitudes of the reflected and transmitted waves relative to the applied wave. They are complex numbers because both the magnitude and phase of the wave can change at an interface. Eref reinc Etrn teinc Wave Power The reflectance and transmittance, R and T, relate the power of the reflected and transmitted waves relative to the applied wave. They are real numbers bound between zero and one. Eref R Einc Etrn T Einc Often, these quantities are expressed on the decibel scale R 10log R T 10log T db 10 db 10 Lecture 3 Slide 55 Conservation of Power When electromagnetic wave is applied to a device, it can be absorbed (i.e. converted to another form of energy), reflected and/or transmitted. Without a nuclear reaction, nothing else can happen. A RT 1 Reflectance, R Fraction of power from the applied wave that is reflected from the device. Transmittance, T Fraction of power from the applied wave that is transmitted through the device. Absorptance, A Fraction of power from the applied wave that is absorbed by the device. Applied Wave Reflected Wave Absorbed Wave Transmitted Wave Lecture 3 Slide 56 8
The Critical Angle (Total Reflection) Above the critical angle c, reflection is 100% r r TE TM cosc 1cos cos cos c 1 cos 1cosc cos cos 1 c 1 1 This will happen when cos( ) is imaginary. The condition for the critical angle is derived from Snell s Law. n1 cos 1sin 1 sin c n n sin 1 1 c n1 Condition for Total Internal Reflection (TIR) 1 1 sin 1 c n n n 1 sin c 1 1 c sin n1 n sin n sin 1 c Lecture 3 Slide 57 n n Brewster s Angle (Total Transmission) TE Polarization cosb 1cos rte 0 cos cos B 1 sin 1 1 B 1 1 1 1 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 r TM cos 1cos1 0 cos cos 1 1 sin 1 1 B 1 1 1 1 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 n This is the most well known equation. B 1 1 n1 Lecture 3 Slide 58 9
Notes on a Single Interface It is a change in impedance that causes reflections Law of reflection says the angle of reflection is equal to the angle of incidence. Snell s Law quantifies the angle of transmission as a function of angle of incidence and the material properties. Angle of transmission and reflection do not depend on polarization. The Fresnel equations quantify the amount of reflection and transmission, but not the angles. Amount of reflection and transmission depends on the polarization and angle of incidence. 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 3 Slide 59 Visualization of Wave Scattering at an Interface Lecture 3 Slide 60 30
1/8/018 Longitudinal Component of the Wave Vector 1. Boundary conditions require that the tangential component of the wave vector is continuous across the interface. Assuming kx is purely real in material 1, kx will be purely real in material. We have oscillations and energy flow in the x direction.. Knowing that the dispersion relation must be satisfied, the longitudinal component of the wave vector in material is calculated from the dispersion relation in material. k x, k y, k0 n k y, k0 n k x, We see that k y will be purely real if k0 n k x,. We see that k y will be purely imaginary if k0 n k y, z. Lecture 3 Slide 61 Field at an Interface Above and Below the Critical Angle (Ignoring Reflections) 1.. 3. 4. n1 n n1 n n1 n No critical angle 1 C 1 C The field always penetrates material, but it may not propagate. Above the critical angle, penetration is greatest near the critical angle. Very high spatial frequencies are supported in material despite the dispersion relation. In material, energy always flows along x, but not necessarily along y. Lecture 3 Slide 6 31
Simulation of Reflection and Transmission at a Single Interface (n 1 <n ) n 1 =1.0, n =1.73 B =60 Lecture 3 Slide 63 Simulation of Reflection and Transmission at a Single Interface (n 1 >n ) n 1 =1.41, n =1.0 C =45 Lecture 3 Slide 64 3
Field Visualization for C =45 inc = 44 inc = 46 inc = 67 inc = 89 Lecture 3 Slide 65 Electromagnetic Tunneling If an evanescent field touches a medium with higher refractive index, the field may no longer be cutoff and become a propagating wave. This is a very unusual phenomenon because the evanescent field is contributing to power flow. This is called electromagnetic tunneling and is analogous to electron tunneling through thin insulators. Lecture 3 Slide 66 33
Image Theory Lecture 3 Slide 67 Image Theory Reduces Size of Models When fields are symmetric in some manner about a plane, it is only necessary to calculate one half of the field because the other half contains only redundant information. Sometimes more than one plane of symmetry can be identified. Image theory can dramatically reduce the numerical size of the model being solved. Reduced model Device is 75% smaller. Model is 94% smaller. G. Bellanca, S. Trillo, Full vectorial BPM modeling of Index Guiding Photonic Crystal Fibers and Couplers, Optics Express 10(1), 54 59 (00). Lecture 3 Slide 68 34
Summary of Image Theory Electric Fields Magnetic Fields Electric Fields Magnetic Fields perfect electric conductor (PEC) perfect magnetic conductor (PMC) image fields Duality Lecture 3 Slide 69 Image Theory Applied to an Airplane Lecture 3 Slide 70 35