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Dispersion Relation & Index Ellipsoids 8/27/2020 Advanced Electromagnetics: 21st Century Electromagnetics Dispersion Relation & Index Ellipsoids Lecture Outline • Dispersion relation • Dispersion surfaces • Index ellipsoids Slide 2 1 8/27/2020 Dispersion Relation Slide 3 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 conveys the spatial period (i.e. wavelength) of the wave inside the material. When the frequency is known, the magnitude conveys the material’s refractive index n (more to be said later). 22 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). ˆ kkakbkcabcˆˆ Slide 4 2 8/27/2020 The Dispersion Relation The dispersion relation for a material relates the wave vector to frequency . Essentially, it sets a rule for the values of as a function of direction and frequency. For an ordinary linear, homogeneous and isotropic (LHI) material, the dispersion relation is: 2 222n kkkabc c0 222 kkkabc 2 This can also be written as: 2 kk000 nc0 Slide 5 How to Derive the Dispersion Relation (1 of 2) The wave equation in a linear homogeneous anisotropic material is: 2 Assume no magnetic response i. e. 1 . Ek 00 r E 0 The solution to this equation is still a plane wave, but the allowed values for (modes) are more complicated. jk r ˆ E Ee00 E Eaabcˆˆ Eb Ec Substituting this solution into the wave equation leads to the following relation: 2 kk E000r0 kE k E 0 This equation has the form: abcˆˆ ˆ 0 Each (•••) term has the form: EEEabc 0 Each vector component must be set to zero independently. aEEEˆ component: abc 0 Matrix form… ˆ bEEE component: abc 0 Ea E 0 cEEEˆ component: abc 0 b Ec Slide 6 3 8/27/2020 How to Derive the Dispersion Relation (2 of 2) Solutions for 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: 222 kkkabc 2221 22 22 22 kknkknkkn000abc It can also be shown that given the wave vector , the polarization of the electric field is: kkkabcˆ E0 222abcˆˆ 22 22 22 kkn kkn kkn 000abc Slide 7 Dispersion Relation for Anisotropic Media Given the dielectric tensor… 2 aa00n 00 0000n2 r bb 2 00 cc 00n The general form of the dispersion relation is: 222 kkkabc 2221 22 22 22 kknkknkkn000abc This can be written in a more useful form as: 2 2 2 22 22 22 2 ka k b k c24 kk bcacab kk kk kk22 22 22 00 2 2 2 k1 nnbc nn ac nn ab n a n b n c Slide 8 4 8/27/2020 Dispersion Relation for Uniaxial Crystals Uniaxial crystals have nnnaboo n ordinary refractive index nnc ee nextraordinary refractive index The general dispersion relation reduces to: 222 22 2 kkkabc22 kk ab k c This has two solutions corresponding kk000 nnn222to the two polarizations (TE and TM). eoe Sphere Ellipse Ordinary Wave Extraordinary Wave This has two solutions corresponding to the two polarizations (TE and TM). The first solution is the same solution for an isotropic material. The wave behaves like it is propagating through a isotropic material with index 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 and . Slide 9 Dispersion Surfaces Slide 10 5 8/27/2020 Dispersion Surface Observe that the dispersion relation for a LHI material is the equation for a sphere: 222 22 kkkknabc0 kkn 0 This sphere has many names: dispersion surface, k‐surface, and momentum surface. For LHI materials, the index ellipsoid is a sphere dispersion surface indicating that the magnitude of the wave vector is constant in all directions. This implies the refractive index is constant is all directions for LHI media. Slide 11 Dispersion Surfaces for Uniaxial Crystals (1 of 2) 222 22 2 Ordinary Wavekkkabc22 kk ab k c Extraordinary Wave 222kk000 nnneoe 22 2 222 kkab k c 2 kkk 2 k 0 abck 0 220 2 0 nnoe ne nneo nneo Slide 12 6 8/27/2020 Dispersion Surfaces for Uniaxial Crystals (2 of 2) Positive Uniaxial Observations • Both solutions share a common axis. • This common axis looks isotropic with refractive index regardless of direction. • Since both solutions share only a single common axis, these crystals are called uniaxial. • The common axis is called: o Optic axis o Ordinary axis o C axis o Uniaxial axis Negative Uniaxial • Deviation from the optic axis will result in two separate possible modes. Slide 13 Dispersion Surface for Biaxial Crystals (1 of 2) Biaxial crystals have three unique refractive indices. Most texts adopt the convention where nnnabc The general dispersion relation cannot be reduced. Notes and Observations • The convention causes the optic axes to lie in the ‐̂ plane. • The two solutions can be envisioned as one balloon inside another, pinched together so they touch at only four points. • Propagation along either of the two optic axes looks isotropic, thus the name biaxial. optic axes Slide 14 7 8/27/2020 Dispersion Surface for Biaxial Crystals (2 of 2) There are three special cases when the biaxial case can be simplified. These conditions can be produced in practice by launching electromagnetic waves at the proper orientation. Each special case has two separate solutions corresponding to the two possible polarizations. 22 2222kkbc 2 kkkknkabca0: 0022 0 nncb 22 2222kkac 2 kkkknkbacb0: 0022 0 nnca 22 2222kkab 2 kkkknkcabc0: 0022 0 nnba Slide 15 Dispersion Surface for Biaxial Crystals (2 of 2) There are three special cases when the biaxial case can be simplified. These conditions can be produced in practice by launching electromagnetic waves at the proper orientation. Each special case has two separate solutions corresponding to the two possible polarizations. 22 2222kkbc 2 kkkknkabca0: 0022 0 nncb 22 2222kkac 2 kkkknkbacb0: 0022 0 nnca 22 2222kkab 2 kkkknkcabc0: 0022 0 nnba Slide 16 8 8/27/2020 Dispersion Surface for Biaxial Crystals (2 of 2) There are three special cases when the biaxial case can be simplified. These conditions can be produced in practice by launching electromagnetic waves at the proper orientation. Each special case has two separate solutions corresponding to the two possible polarizations. 22 2222kkbc 2 kkkknkabca0: 0022 0 nncb 22 2222kkac 2 kkkknkbacb0: 0022 0 nnca 22 2222kkab 2 kkkknkcabc0: 0022 0 nnba Slide 17 Dispersion Surface for Biaxial Crystals (2 of 2) There are three special cases when the biaxial case can be simplified. These conditions can be produced in practice by launching electromagnetic waves at the proper orientation. Each special case has two separate solutions corresponding to the two possible polarizations. 22 2222kkbc 2 kkkknkabca0: 0022 0 nncb 22 2222kkac 2 kkkknkbacb0: 0022 0 nnca 22 2222kkab 2 kkkknkcabc0: 0022 0 nnba Slide 18 9 8/27/2020 Dispersion Surfaces of Magnetoelectric Materials Magnetoelectric materials can exhibit up to 16 singularities. DEH and BHE Alberto Favaro and Friedrich W. Hehl, “Light propagation in local and linear media: Fresnel‐Kummer wave surfaces with 16 singular points,” arXiv preprint arXiv:1510.05566 (2015). Slide 19 Index Ellipsoid Slide 20 10 8/27/2020 Index Ellipsoid for LHI Media Dispersion surfaces and index ellipsoids are essentially the same thing. They are just scaled by a constant. 222 22 cˆ kkkknabc0 kkn 0 index ellipsoid x2222yzn The surface becomes a map of refractive index as a bˆ function of direction of the waves. aˆ Slide 21 Physical Interpretation of Index Ellipsoid in Isotropic Media Refractive index is the same in all directions. nnnabc Wavelength is the same in all directions. Slide 22 11 8/27/2020 Physical Interpretation of Index Ellipsoid in Positive Uniaxial Media Refractive index is higher in one direction than the other two directions. nnnabc Wavelength is smaller for waves propagating in the high‐index direction. Slide 23 Physical Interpretation of Index Ellipsoid in Negative Uniaxial Media Refractive index is lower in one direction than the other two directions. nnnabc Wavelength is larger for waves propagating in the low‐index direction. Slide 24 12 8/27/2020 Direction of Power Flow Isotropic Materials Anisotropic Materials y y k k x x Phase propagates in the direction of . Therefore, the refractive index derived from || is best described as the phase refractive index . Velocity here is the phase velocity ⃗. Power propagates in the direction of the Poynting vector ℘ which is always normal to the surface of the index ellipsoid. From this, group velocity ⃗ and group refractive index can be defined. Slide 25 Animation of versus℘ ℘ Medium 1 Medium 2 ℘ Slide 26 13 8/27/2020 Double Refraction in Anisotropic Materials Isotropic Materials Anisotropic Materials y y k1 k1 x x ke k2 ko Anisotropic materials have two index ellipsoids –one for each polarization. Wave power can split between the two to produce both an ordinary and an extraordinary wave. Slide 27 14.
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