Electromagnetic Properties of Materials Part 2

Transcription

1 ECE st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) rcrumpf@utep.edu Lecture #3 Electromagnetic Properties of Materials Part 2 Nonlinear and Anisotropic Materials Lecture 3 1 Lecture Outline Nonlinear materials Anisotropic materials Tensor rotation and unrotation Dispersion relation and index ellipsoids Lecture 3 Slide 2 1

2 Nonlinear Materials Nonlinear Materials All materials are nonlinear; some just have stronger nonlinear behavior than others. For radio frequencies, materials tend to breakdown before they exhibit nonlinear properties. Nonlinear properties are commonly exploited in optics. In general, the polarization of a material is a nonlinear function of the electric field and can be expressed as P E E E e 0 e 0 e Linear response This is what we have done so far. Nonlinear response Lecture 3 Slide 4 2

3 Potential Well Description V is the push required to displace the charge by a distance r. 2 Linear materials have a parabolic potential well. V r This leads to a resonator that oscillates as a perfect sinusoid. r Lecture 3 Slide 5 Potential Well for Nonlinear Materials Nonlinear for large r. This type of resonance does not oscillate as a sinusoid. It can be seen as a series of sinusoids. Linear parabolic region for small r. As bound charges are pushed very hard, the restoring force is no longer linear. This results in anharmonic oscillation. The oscillator is effectively resonating at multiple frequencies at the same time. Lecture 3 Slide 6 3

4 Nonsymmetric Potentials This is a non symmetrical potential and therefore exhibits a preferred direction for polarization. The result is a rectified signal along with potentially other frequency content. DC component Lecture 3 Slide 7 Applications of Nonlinear Materials (1) Materials Ordinary linear materials (2) Materials Rectification can create a DC potential from an optical wave Frequency doubling (second harmonic generation) can make lasers at otherwise impossible wavelengths. Parametric mixing can provide sum and difference frequencies Pockel s effect can introduce birefringence from an applied electric field. Lacks inversion symmetry (3) Materials Kerr effect field dependent dielectric constant Third harmonic generation can generate very short wavelength waves. Raman scattering the mechanical vibration of a molecule can shift the frequency of the wave Brillouin scattering dielectric response changes with applied pressure Two photon absorption two photons are absorbed simultaneously where as a single photon would not be absorbed. Has inversion symmetry Lecture 3 Slide 8 4

5 Anisotropic Materials Anisotropic Materials In some materials, charges are more easily displaced along certain directions. For this reason, the material can become polarized in a direction slightly different than the applied field. In this case, the susceptibility is a tensor quantity. Imagine pushing at an angle against a sliding glass door. The door has a preferred direction for displacement. Real materials are not this dramatic! P E 0 e Isotropic materials are good for simple transmission of electromagnetic waves. Anisotropic materials are good for controlling and manipulating the waves. Lecture 3 Slide 10 5

6 Atomic Scale Picture Charges will oscillate differently in these two directions. Therefore, the susceptibility changes as a function of direction in the lattice. large small Lecture 3 Slide 11 Tensor Relations for Anisotropic Materials The material polarization is now expressed as: Px xx xy xzex Py 0 yx yy yz Ey Pz zx zy zz Ez This is most often treated through a dielectric tensor. r I e The constitutive relation between E and D is then Dx xx xy xz Ex D E y 0 yx yy yz y Dz zx zy zz Ez The dielectric tensor has Hermitian symmetry for the general lossy case. ij * ji This means the field components are no longer independent. But it presents many new possibilities!! Lecture 3 Slide 12 Materials can become polarized in directions that are slightly different than the electric field! 6

7 Principle Axes It is always possible to choose a different coordinate system such that the dielectric tensor becomes diagonal. xˆ yˆ zˆ aˆ bˆ cˆ a, b, and c are called the Principal Axes of the Crystal. a, b, and c are not necessarily at 90 to each other. Da a 0 0 Ea Db 0 0 b 0 Eb Dc 0 0 c Ec Alternative Description: There are only three degrees of freedom for 3D tensors. Numbers can only occur in the off diagonal elements when the tensor is rotated. Lecture 3 Slide 13 Maxwell s Equations with Anisotropic Materials Maxwell s equations remain unchanged. E jb H jd D 0 B 0 The constitutive relations now include tensors D E B H D D x xx xy xz Ex E y yx yy yz y D z zx zy zz E z Lecture 3 Slide 14 7

8 Symmetry and Anisotropy Anisotropy Crystal Symmetry Dielectric Properties Isotropic Cubic Uniaxial Biaxial Hexagonal Tetragonal Monoclinic Triclinic Orthorhombic O O E positive birefringence E E O O negative birefringence a b c a b c Lecture 3 Slide 15 Tensor Rotation and Unrotation 8

9 Definition of a Rotation Matrix b a Rotation matrix [R] is defined as b R a The rotation matrix should not change the amplitude. This implies that [R] is unitary. b a H 1 R R H H RR R R I Lecture 3 17 Derivation of a 2D Rotation Matrix b bx cos sinax b y sin cos a y R a Start with vector a at angle. a a ˆ ˆ xxayy a cos xˆsin yˆ Add angle to rotate the vector b acos xˆsin yˆ Apply trig identities cos cos sinsin xˆ b a sincos cossinyˆ a cos sin ˆ x ay x a cos sin ˆ y ax y Lecture

10 3D Rotation Matrices Rotation matrices for 3D coordinates can be written directly from the previous result. a b b a a b cos 0 sin cos sin 0 Rx 0 cos sin R y R z sin cos 0 0 sin cos sin 0 cos Lecture 3 19 Rotation Matrix that Rotates a Onto b Normalize Vectors a ˆ ˆ b a b a b Algorithm v aˆ bˆ s v i.e. sin ab caˆ bˆ i.e. cos ab 0 v3 v2 v v3 0 v 1 v2 v1 0 1 c 2 s RIv v 2 [R] is such that Ra ˆ bˆ Lecture

11 Tensor Rotation Tensors are rotated in a slightly different manner than vectors. R 1 Rz R z Lecture 3 21 Combinations of Rotations Suppose we wish to first rotate about the x axis by some angle, second rotate about the y axis by some angle, and third rotate about the z axis by some angle. For vectors b R R R z y x a For tensors R 1 R R R R R R 1 1 z y x x y z Lecture

12 Composite Rotation Matrix We can combine multiple rotation matrices into a single composite rotation matrix [R]. R R R R b Ra Tensor rotation using the composite rotation matrix is R 1 y z x R R This equation implies we will rotate first about the x axis, second about the z axis, and third about the y axis. Vector rotation using the composite rotation matrix is Lecture 3 23 Animation of Tensor Rotation Lecture

13 Order of Rotations Matter The order that we multiply the rotation matrices controls the order that the rotations are performed. b Rz45 Ry220 Rx10 a 1. First rotates about the x axis by Second rotates about the y axis by Third rotates about the z axis by 45. In general, we do not get the same result when the order of rotation is changed. R R R R y x x y Lecture 3 25 Numerical Examples (1 of 2) Given: r Rotate about x axis by 20 R x Rotate about y axis by 45 R y Rotate about z axis by 60 R z r r r Lecture

14 Numerical Examples (2 of 2) Given: r Rotate first about x axis by 20 and second about the y axis by 45 R y x R Rotate first about z axis by 60 and second about the y axis by Ry45Rz r Rotate first about x axis by 20, second about the y axis by 45, and third about the z axis by Rz60Ry45Rx Lecture 3 27 r r Tensor Unrotation (1 of 2) A tensor can always be diagonalized along its principle axes. a b c Principle Axes: is along aˆ a is along bˆ b is along cˆ c Convention: But suppose we are given a general nine element tensor. a b c xx xy xz rot yx yy yz zx zy zz How do we determine a, b, c and the principle axes a, b, and c? Lecture

15 Tensor Unrotation (2 of 2) We can relate the original tensor and the rotated tensor through the composite rotation matrix [R]. 1 rot R R We calculate the eigen vectors and eigen values of [ rot ]. rot R is the eigen-vector matrix of rot is the eigen-value matrix of rot Lecture 3 29 Determining Principle Axes (1 of 2) What are the principle axes of [ rot ]? Let s put the original principal axes a, b, and c into a matrix. a x x x P ay by c y az bz c z aˆ b bˆ c cˆ Now let s rotate them according to [R] so they correspond to that of [ rot ]. P R P rot We are not rotating a tensor here so we don t use [R][P][R] -1. Lecture

16 Determining Principle Axes (2 of 2) For the special case of orthorhombic symmetry, the principle axes start off as P aˆ bˆ cˆ [P] is just the identity matrix here. The principle axes of the rotated tensor are then P rot I R R We now have a second interpretation of the eigenvector matrix [R]. These are the principle axes of the rotated tensor (for orthorhombic symmetry). Lecture 3 31 Directly Setting the Principle Axes We may know the orientation that we would like to place our tensor directly in terms of the desired crystal axes a, b, and c. We place these into a matrix and interpret it as the composite rotation matrix. ax bx cx R ay by c y a z bz c z aˆ bˆ cˆ The rotated tensor can be directly calculated from [R]. 1 rot R R Lecture

17 Dispersion Relation & Index Ellipsoids The Wave Vector The wave vector (wave momentum) is a vector quantity that conveys two pieces of information: 1. Wavelength and Refractive Index The magnitude of the wave vector tells us the spatial period (wavelength) of the wave inside the material. When the free space wavelength is known, k conveys the material s refractive index n (more to be said later). 2 2 n k 0 free space wavelength 0 2. Direction The direction of the wave is perpendicular to the wave fronts (more to be said later). k k aˆk bˆ k cˆ a b c Lecture 3 Slide 34 17

18 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 an ordinary 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 2 k Lecture 3 Slide 35 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 surface of the sphere is the k vector for that direction. Refractive index is calculated from this. k k n 0 ĉ 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. â ˆb Lecture 3 Slide 36 18

19 How to Derive the Dispersion Relation (1 of 2) The wave equation in a linear homogeneous anisotropic material is: 2 Ek00rE 0 We are ignoring the magnetic response here. The solution to this equation is still a plane wave, but our allowed values for k (modes) are more complicated. jk r E Ee ˆ ˆ ˆ 0 E0 Ea a Eb b Ec c Substituting this solution into the wave equation leads to the following relation: k k E k E k E 2 r This equation has the form: ˆ aˆ b cˆ 0 Each ( ) term has the form: E E E 0 a b c Each vector component must be set to zero independently. aˆ component: Ea Eb Ec 0 Matrix form bˆ component: Ea Eb Ec 0 cˆ component: Ea Eb Ec 0 Ea E b 0 E c Lecture 3 Slide 37 How to Derive the Dispersion Relation (2 of 2) Solutions for k are the eigen values of the big matrix and derived by setting the determinant to zero. det 0 This leads to the following general equation: k k k 1 k k n k k n k k n a b c a 0 b 0 c It can also be shown that given the wave vector, the polarization of the electric field is: ka kb ˆ kc E ˆ ˆ 0 a b c k k0n a k k0n b k k0n c Lecture 3 Slide 38 19

20 Generalized Dispersion Relation Given the dielectric tensor, 2 a 0 0 n 0 0 a 2 r 0 0 b 0 nb c 0 0 n C The general form of the dispersion relation is: k k k 1 k k n k k n k k n a b c a 0 b 0 c This can be written in a more useful form as: 2 k k k k k k k k k k k k nn nn nn n n n a b c 2 b c a c a b b c a c a b a b c Lecture 3 Slide 39 Dispersion Relation for Uniaxial Crystals Uniaxial crystals have na nb no no ordinary refractive index n n n extraordinary refractive index c E E The general dispersion relation reduces to: ka kb k 2 c ka kb kc 2 k 2 0 k no ne no Sphere Ellipse Ordinary Wave Extraordinary Wave This has two solutions corresponding to the two polarizations (TE and TM). This has two solutions corresponding to the two polarizations (TE and TM). This first solution is the same as for an isotropic material. It acts like it is propagating through a isotropic material with index n O so it is called the ordinary wave. The second solution is an ellipsoid. Depending on its direction, the effective refractive index will be somewhere between n O and n E. Lecture 3 Slide 40 20

21 Index Ellipsoids for Uniaxial Crystals (1 of 2) Ordinary Wave k k k k k k a b c 2 a b c 2 k 2 0 k no ne no Extraordinary Wave k k k a b c 2 k no k k k a b c 2 k ne no n O ne no ne Lecture 3 Slide 41 Index Ellipsoids for Uniaxial Crystals (2 of 2) 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. Lecture 3 Slide 42 21

22 Dispersion Relation for Biaxial Crystals (1 of 2) Biaxial crystals have all unique refractive indices. Most texts adopt the convention where n n n a b c The general dispersion relation cannot be reduced. ĉ Notes and Observations The convention n a <n b <n c causes the optic axes to lie in the a c plane. optic axes 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 43 Dispersion Relation for Biaxial Crystals (2 of 2) There are three special cases when it can be simplified. We can cause these conditions by launching electromagnetic waves at the proper orientation kb k 2 c ka 0: kb kc k0na k nc nb ka kc 2 kb 0: ka kc k0nb k nc na ka kb 2 kc 0: ka kb k0nc k nb na We see that each case has two separate solutions corresponding to the two polarizations (TE and TM). Lecture 3 Slide 44 22

23 Dispersion Surfaces of Magnetoelectric Materials Magnetoelectric materials can exhibit up to 16 singularities. D E H and B H E Alberto Favaro and Friedrich W. Hehl, Light propagation in local and linear media: Fresnel Kummer wave surfaces with 16 singular points, arxiv preprint arxiv: (2015). Lecture 3 Slide 45 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. Energy 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 46 23

24 Illustration of k versus Raymond C. Rumpf "Engineering the Dispersion and Anisotropy of Periodic Electromagnetic Structures," Solid State Physics, Vol. 66, pp , Negative refraction into a photonic crystal. Lecture 3 Slide 47 Double Refraction in Anisotropic Materials Isotropic Materials y Anisotropic Materials y k 1 x k 1 x k e k 2 k 0 Anisotropic materials have two index ellipsoids one for each polarization. Wave energy can split between the two produce an ordinary and an extraordinary wave. Lecture 3 Slide 48 24