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Home , Crystal optics

FS 2020 - ETH Zurich with Intense Sources Prof. Manfred Fiebig

To be reviewed on 17.03.2020. Ques- Solution of Exercise sheet #3 tions and comments to be posted on https://crystaloptics.freeforums.net/

Exercise 3.1: Short questions (11 points) (a) Magnetic dipoles cannot follow the high frequencies of optical magnetic fields. Therefore the relative permeability is one at optical frequencies: µr = 1. Gyromagnetic resonances appear up to approximately 1 GHz. Optical frequencies are 1014 Hz. (b) Along the optical axis the light velocity/ does not depend on the light : The cut between the plane perpendicular to light propagation k and the indicatrix n(D) generally gives an oval, but for k parallel to the optical axis the cut is circular. This also means that the polarization of a beam travelling along an optical axis remains unchanged. For other directions differently polarized beams are refracted differently (and therefore travel in different directions depending on polarization). For biaxial the optical axes must not be in agreement with the crystallographic axes (and are in general wavelength dependent leading to a rotation of the indicatrix)! In uniaxial materials the optical axis lies along the high-symmetry axis of the crystal (restricted by symmetry). (c) The indicatrix is a general ellipsoid defined by

2 2 2 x1 x2 x3 2 + 2 + 2 = 1 . n1 n2 n3 It allows a description of the refractive index n as a function of the displacement vector D and is independent of the wave vector k. Originally the indicatrix describes the local shape at a given point of a curved surface in three dimensional space.

(d) For uniaxial crystals: The ordinary refractive index (no) is defined perpendicular to the optical axis (OA) no = n⊥, and therefore the polarization of ordinary rays (polarized perpendicular to the OA) remains unchanged (E||D). n|| = ne along the OA gives the extraordinary refractive index (ne). For arbitrary light polarizations (θ is the angle to the optical axis) one can use the indicatrix defined by 1 cos2 θ sin2 θ 2 = 2 + 2 . n (θ) no ne Biaxial crystals have two optical axes, leading to two possible ordinary refractive indices. A light beam can always be separated into an ordinary and an extraordinary ray, with the free choice of the ordinary direction.

(e) TiO2 has a higher refractive index compared to glass (nT iO2 = 2.62, nglass = 1.54), such that thinner glasses can be manufactured. However it becomes more expensive. (f) As the wavelength of light is usually much larger than unit cell parameters, optical effects are generally given by macroscopic and restricted only by the point symmetry of the crystal. Exceptions are optical activity (see formula in exercise before, the light rotation adds to a ma- croscopic value if the crystal has just one chirality), short wavelengths (x-ray), and maybe the predictions of the crystal field splitting (site symmetry more important than macroscopic crystal symmetry). (g) There are some points on the indicatrix with the same curvature known as isotropic points: for example, wavelength-dependent no and ne can obtain the same value for a certain wavelength. Other case one could think of would be a certain state of stress, such that no = ne. (h) The section between the indicatrix and the plane perpendicular to the light propagation direction k gives always an ellipse. The two main axes of this ellipse determine the two refractive indices, hence birefringent.

(i) The three different refractive indices define the ellipsoid of the indicatrix (na 6= nb 6= nc). Only under two special directions the section between the ellipsoid and the plane of polarization of the light beam describes a circle, i.e. the optical axis. (j) 6 constants are needed. 3 indices of diffraction and 3 directions of the ellipsoid with respect to the . √ (k) Because n = rµr but also c0 = nc with c0 the speed of light in vacuum, one can rewrite these two equations as: c = √ c0 to obtain the relationship. r µr

Exercise 3.2: Birefringent and gyrotropic materials (8 points) (a) This depends on the crystal system: Cubic structures are isotropic/not birefringent; orthorhom- bic/monoclinic/triclinic crystals are biaxial birefringent, all other systems are uniaxial. diamond (F d3m, cubic): isotropic; ZnS (P 63mc, hexagonal wurtzite): uniaxial; ZnS (43m, cubic zinc-blende): isotropic; BBO (R3c, rhomboidal): uniaxial; KDP (I42d above TC , tetragonal): uniaxial; Topaz (P bmn, orthorhombic): biaxial. (b) Optical active materials must be chiral (handedness, i.e., lost mirror operation). The direction of polarization rotation depends on the chirality of the passed specimen. Fused silica is a glass and is therefore achiral, calcite is achiral (R32/c), as is BBO (R3c). α-quartz comes in enantiomorphs with sym- metries P3121 or P3221 (chirality vs. screw axes!). Hexagonal β-quartz is also chiral. Both quartz types are optically active. Sodium bro- mate (with chiral group P213) is optically ac- tive. Glucose/dextrose molecules show circu- lar : the name dexter“(right) co- ” mes from the observed positive light rotation (also used to measure sugar concentration). Negative light rotation is observed in fructo- se/laevulose ( left“). ” Also optical active: certain phases of liquid crystals, that can be aligned by applied electric fields (used for calculator displays). (c) The wavelength 1064 nm is specific for a Nd:YAG-laser, 532 nm and 355 nm correspond to the second and third harmonic, respectively. Following the graph (just an exponential fit to the data!) one gets the following results (brackets give thickness of commercially available polarization rotators): 1064 nm: ≈ 6.1◦/mm ⇒ d =14.8 mm (14.2 mm). 532 nm: ≈ 27.5◦/mm ⇒ d =3.3 mm (3.4 mm). 355 nm: ≈ 60.5◦/mm ⇒ d =1.5 mm (0.7 mm). One sees that at lower wavelength the deviations are quite large – so just approximate data at high energies (lower wavelengths)!

3 0

1 3 0 1 2 0 1 1 0 Q u a r t z 1 0 0 f i t 9 0 m m

/ 8 0 °

n 7 0 o i t a ] t 6 0 o r

m c

i 5 0 f i c m

e 4 0 / p

2 0 s °

[ 3 0

n 2 0 o

i 1 0 t

a 0

t 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 1 2 0 0 o

r W a v e l e n g t h [ n m ]

c i f i c e p s

1 0 d a t a f r o m : h t t p : / / w w w . k a y e l a b y . n p l . c o . u k / g e n e r a l _ p h y s i c s / 2 _ 5 / 2 _ 5 _ 1 0 . h t m l p o i n t a t 3 5 5 n m f r o m : h t t p : / / w w w . u - o p t i c . c o m / g i d _ 5 2 _ p r d i d _ 6 6 _ p i d _ 2 9 9 _ l _ 1

5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 W a v e l e n g t h [ n m ] Exercise 3.3: Rotating light polarizations (3 points) (a) A or retarder is an optical device that alters the polarization state of a light wave travelling through it. A half-wave plate shifts the polarization direction of linearly polarized light. For example a half-wave plate at 45◦ will change an x-polarized light to y-polarized light and vice versa.

ω 2π (b) The accumulated phase can be written as φ = k(n) · d, where k(n) = n = nk0 = n is the c0 λ light wavevector in the material and d its thickness. The phases can be written as: 2π φ = k d + k d0 = (n d + n d0) , || e o λ e o 2π φ = k d + k d0 = (n d + n d0) , ⊥ o e λ o e 2π ∆φ = φ − φ = (n − n )(d0 − d) . || ⊥ λ o e

0 If the thicknesses are the same (d = d ) both polarization components E||, E⊥ experience the same retardation and thus the polarization remains unchanged. For a 90◦-rotation the accumulated phase must be ∆φ = π. The equation then reduces to (d0 − d) = λ/2∆n. For λ ≈ 600 nm, the difference in the refractive index is ∆n ≈ 0.01 which leads to (d0 − d) ≈ 30 µm, which can be well adjusted by sliding the quartz wedges using a micrometer screw.

E|| E E . d d’ .