P. Piot, PHYS 630 – Fall 2008Index ellipsoid I• The energy density associated to an electromagnetic wave is• Remember that• So the energy density is• Which can be rewritten This defines an ellipsoidP. Piot, PHYS 630 – Fall 2008Index ellipsoid II• The ellipsoid in (α,β,γ) intersect the axis atis called– Index ellispoid– Optical indicatrix– Ellipsoid of wave normals• Correspondence between (α,β,γ) and (x,y,z)?A wave propagating along the x-axis ⇒ D vector is in (y,z) plane⇒ Dx=0 ⇒ α=0similarly a wave propagating along y (resp. z) ⇒ β (resp. γ ) =0P. Piot, PHYS 630 – Fall 2008Propagation along a principal axis• Nothing “exciting”: a linearly-polarized plane wave with E-fieldaligned with one of the principal axis i propagates with phasevelocity c/ni.P. Piot, PHYS 630 – Fall 2008Propagation along an arbitrary direction• If the wave travels in the crystal with an arbitrary direction, thenormal modes associated to the wave are linearly polarized.• These modes can be found from the index ellipsoid• Start with• Combining gives• Finally with Can be viewed as the projection of ηD on a planeorthogonal to u-P. Piot, PHYS 630 – Fall 2008Index ellipsoid• How to find the– 1: given the propagation direction of the ray draw thecorresponding ray in the (α,β,γ) plane.– 2: draw a plane normal to the ray andcontaining the origin– 3: the intersection of the plane withthe index ellipsoid gives an ellipse– 4: the principal axis of this ellipsegive the direction of the D vectorof the two lineareigenvectors. Thelength of the semi-minor and semi-majoraxis gives the index ofrefraction along the eigenpolarizations.P. Piot, PHYS 630 – Fall 2008Uniaxial crystals• A wave traveling with angle θ wrtthe ordinary axis give an index ellipseof the form• Normal modes have refractive indexesno and ne(θ)• Ordinary wave has index no regardlessof θ and E||D• Extraordinary wave mode has refractiveindex ne(θ) and E and D are not generallyparallel θnenon(θ)kHE,DθkDHθEo-wavee-waveP. Piot, PHYS 630 – Fall 2008Vector directions• Optical wave characterized by k, E, D, H, and B with power flowgiven by S.• …S ⊥ E and H• …D ⊥ k and H• …H ⊥ k and E• D, E, k, S are in the same plane ⊥ to (B and H)kDH,BESP. Piot, PHYS 630 – Fall 2008Dispersion relation• From Maxwell’s equations• Where the matrix M is• Non trivial solution is determinant of M vanishes i.e.• ω(k) is the dispersion relationP. Piot, PHYS 630 – Fall 2008k-surface• ω(k) generally describes a centro-symmetric surface comprising twosheets• The group velocity is• The Poynting vector is || to the group velocitykSwavefrontraykSwavefrontrayordinary extraordinaryP. Piot, PHYS 630 – Fall 2008Special case of uniaxial crystal• Take n1=n2=no and n3=ne the equation for the k-surface becomesnoin y-z planen(θ)noneθk3/kok2/kokSwavefrontrayordinarykSwavefrontrayextraordinaryHE,DDEHP. Piot, PHYS 630 – Fall 2008Double refraction• Explore refraction at the interface of a iso-tropic and anisotropic media• Snell’s law applieswhere θa+θ is the angle of the refracted wave with respect to thecrystal optical axis.• Case of a uni-axial crystal two refracted wavesθ1θaOpticalaxisinterfaceP. Piot, PHYS 630 – Fall 2008Polarizer• Linearly polarizedlight beam can beproduced via multiplereflection• Or using anisotropiccrystal arranged as aWollaston prism• Glan-Foucault prismP. Piot, PHYS 630 – Fall 2008Waveplate• Waveplate can be used to manipulate the polarization of an incomingwave• The phase shift between thetwo optical directions is• The correspondingJones matrix