Lecture 4: Anisotropic Media. Dichroism. Optical Activity. Faraday Effect in Transparent Media. Stress Birefringence. Form Birefringence

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1 Lecture 4: Anisotropic Media Outline Dichroism Optical Activity 3 Faraday Effect in Transparent Media 4 Stress Birefringence 5 Form Birefringence 6 Electro-Optics Dichroism some materials exhibit different absorption properties for orthogonal polarization states (e.g. tourmaline, sheet polarizers might assume complex indices of refraction ñ x,y,z to describe anisotropic absorption (dichroism (in general not correct since scalar complex index comes from combining (real dielectric constant ɛ with electrical conductivity σ anisotropic materials may have anisotropic conductivity tensor σ principal axes of dielectric and conductivity tensors may not coincide for uniaxial crystals both principal axes systems coincide choose prinicpal coordinates where both tensors are diagonal repeat derivations as before Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media Dichroism in Uniaxial Crystals Fresnel equation ratio of electric field components k,l E k = s (ñ k µ ɛ l E l s (ñ l µ ɛ k same as for non-absorbing materials, but different interpretion ñ, ɛ l,k complex ratios are (in general complex ordinary and extraordinary waves are elliptically polarized finite conductivity non-vanishing current density j displacement vectors not orthogonal to wave vector can be neglected if real part of ɛ k much larger than imaginary part Ordinary and Extraordinary Waves in Uniaxial Dichroic Materials for uniaxial media with small conductivities, two solutions to Fresnel equation have analogous form ñ = ñ o, ñ = cos θ ñ o + sin θ ñ e polarization states of waves almost linear because ratios of electric field components dominated by real parts attenuation of ordinary wave independent of wave direction attenuation of extraordinary wave depends on angle between wave vector and optic axis θ separate equation into real and imaginary parts using n k = cos θ o + sin θ ne, k n 3 = k o cos θ n 3 o + k e si θ n 3 e Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 3 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 4

2 Optical Activity Extraordinary Wave in Uniaxial Dichroic Materials separate equations for real (n and imaginary (k parts k n 3 = cos θ o = k o cos θ n 3 o + sin θ e + k e si θ n 3 e real part of index obeys same law as for non-absorbing material extinction coefficient of extraordinary wave shows similar dependence on angle between wave vector and optic axis will understand behavior of sheet polarizers at non-normal incidence with these equations Introduction plane of polarization of linearly polarized light is rotated when passing an optically active medium rotation angle is proportional to path length sense of rotation does not change for reverse light path quartz is best-known optically active crystal; optical activity comes from helical arrangement of layers in quartz crystals helical arrangement can have two distinctly different forms: right-handed and left-handed isotropic substances can also be optically active (natural sugar desolved in water; optical activity comes from handedness of molecules treating optical activity consistently is not trivial here only phenomenology Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 5 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 6 Optical Activity in Jones Formalism linearly polarized wave (coherently decomposed into left-, right-circularly polarized waves: ( 0 = ( i + ( i in optically active medium, left- and right-circularly polarized waves experience different indices of refraction after distance d in optically active medium ( Ex = e iωn r d c E y ( i + e iωn l d c ratio of electric field components given by ( i E y E x = tan γ, γ = ω c (n l n r d = πd λ (n l n r Optical Activity in Mueller Formalism ratio of electric field components is real wave still linearly polarized, but plane of polarization is rotated by angle γ optically active medium acts as circular retarder or rotator Mueller matrix for optical activity: M OA = cos β siβ 0 0 siβ cos β , β = d λ π (n r n l d: geometrical path length; λ wavelength β d : specific rotary power (.7 per mm at 589 nm for quartz Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 7 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 8

3 Faraday Effect in Transparent Media Mueller Matrix for Faraday Effect Mueller matrix for Faraday effect M F = cos α sin α 0 0 sin α cos α , α = VHd magnetic field applied to transparent, isotropic medium (solids, liquids, neutral gases, plasmas rotation of linearly polarized light parallel to magnetic field back-forth travel through Faraday cell doubles angle of rotation (optical isolators V : material constant called Verdet constant d: geometrical path length in medium H: signed magnitude of magnetic field parallel to line of sight light along (against magnetic field vector: H > 0 (H < 0 V typically degrees per mm and Gauss Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 9 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 0 Stress Birefringence mechanical stress induces anisotropy in isotropic material stress (force per area can be external (mechanical mounting, inhomogeneous temperature or intrinsic (due to manufacturing glass has no fixed melting temperature stress can be frozen in 00 to 000 hours of annealing reduces stress refractive indices for polarization parallel and perpendicular to stress n = n 0 + K σ, n = n 0 + K σ n 0 : index of refraction of unstressed matieral σ: stress K, : stress optical coefficients for polarimetry: interested in birefringence n n = (K K σ = K σ Stress Optical Coefficients typical values for K for different materials material stress optical coefficien thermal expansion coefficient N-BK mm /N /K fused silica mm /N /K SF mm /N /K N-SF mm /N /K K only slightly dependent on wavelength in visible intrinsic birefringence < nm/cm for the very best glass < nm/cm for fused silica 5 nm/cm is typical for precision optics Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media

4 Form Birefringence birefringence in media that are isotropic at molecular level but anisotropic on scales smaller than wavelength (form birefringence periodic arrangement of two different, isotropic, dielectric materials in plane-parallel layers layers extend in x, y much further than wavelength layer interfaces z-axis indices of refraction, thicknesses and layers λ, composite medium acts as uniaxial medium with optic axis z-axis z Effective Indices of Refraction effective indices for plane wave, wave vector in xy-plane case electric field perpendicular to layers: E = (0, 0, E z T displacement vector component normal to interface must be continous across interface D = D = D z D E = n, E D = n Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 3 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 4 Effective Extraordinary Index of Refraction electric fields in media and D E = n, E D = n Effective Ordinary Index of Refraction z mean electric field E = E + E and mean displacement field D related by n D = ne E = E f n + ( f n with f = d d +d, filling factor of material (effective extraordinary index of refraction because linear polarization of wave is parallel to optic axis case electric field perpendicular to z-axis: E = (E x, E y, 0 T tangential components of E continuous across interfaces E = E = E mean electric field E and mean displacement field D related by D = o E = f + ( f n E Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 5 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 6

5 Electro-Optics Form Birefringent Negative Uniaxial Media mean electric field E and mean displacement field D related by negative uniaxial medium o e = D = o E = f + ( f n E ( f f ( n f + ( f 0 maximum birefringence for equal amounts of materials, f = birefringence independent of layer thickness waveplate from surface grating in dielectric material depth of grooves have to be much deeper than their width Introduction external fields isotropic media can become anisotropic electro-optics: effects of external electric field applied to isotropic media and crystals for isotropic crystals (n the same for all 3 principal axes r: Pockels coefficient R: Kerr coefficient δn = n e n o n3 (re + RE quadratic (Kerr effect occurs in all crystals, isotropic media linear (Pockels effect only in some crystals Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 7 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 8 Modification of Dielectric Tensor electric field change in elements of the inverse dielectric tensor ( ɛ ij linear and quadratic electro-optical effects described by tensors ( ( δ = δ ɛ = r ijk E k + R ijkl E k E l. ij ij many elements of r ijk,r ijkl are zero in crystallographic coordinate system because of symmetries of crystal, non-zero elements are tabulated determine new principal axis system in which dielectric tensor is diagonal again in piezoelectric crystals (quartz, KDP and ferroelectric crystals (LiNbO 3, linear effect much larger than quadratic effect electro-optical coefficients depend strongly on frequency because of mechanical resonances Kerr effect discovered 875 by John Kerr in a piece of glass also seen in liquids, gases isotropic medium becomes uniaxially anisotropic with optic axis parallel to electric field field perpendicular to line of sight. Kerr cells not used much anymore quadratic dependence on electric field strength very high voltages some of media are toxic or explosive birefringence n = BλE with wavelength λ, B Kerr constant Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 9 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 0

6 Pockels effect Kerr Constant Kerr constant, index of refraction for liquids at 550 nm and C liquid n B i0 6 V/m nitrobenzene (C 6 H 5 NO acetone (CH 3 COCH carbon disulfide (CS chloroform (CHCl fast light switches, of the order of a few ps effect about one order of magnitude smaller in glasses, often also induced elasto-optical effect due to field-induced deformation. effect about 3 orders of magnitude smaller in gasses in KDP crystallographic and principal coordinate systems coincide electric field along beam, waves polarized at ±45 propagate with different speeds phase retardation φ (units of waves and voltage V φ = n3 or 63 V λ retardation is independent of distance between electrodes Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media Pockels Cell Properties to minimize field-of-view limitations, distance should be as small as possible to minimize sparking, distance should be as large as possible depending on needs of application, optimize distance Crystal Aberrations Crystal Astigmatism: A converging beam passing a uniaxial, plane-parallel plate is subject to crystal astigmatism. The two focal points along the center ray are separated by Dn o si Ω ( ne o x = n e (n o si Ω + ne cos Ω 3 x Longitudinal astigmatism in units of the plate thickness D Plate thickness Ω Angle between optic axis and interface normal n o n e Ordinary index of refraction Extraordinary index of refraction. Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 3 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 4: Anisotropic Media 4