Crystal Optics with Intense Light Sources Solution of Exercise Sheet #4

FS 2017 - ETH Zurich Crystal Optics with Intense Light Sources Prof. Manfred Fiebig

Solution of Exercise sheet #4 Discussed on 21.03.2018

Exercise 4.1: Short questions (15 points) ~ ~ (a) As di erence in the imaginary part of the refractive index ij(!; k) = ij(!) + i ijk(!)kl. ijk is sensitive to translational symmetry of the crystal, so it is a small e ect. Chirality can be induced in non-chiral crystals by the application of a magnetic eld (Faraday e ect or magneto-optical Kerr e ect, both originating in the Zeeman e ect). cos (b) The plane of polarization of linear polarized light will be rotated by an angle : D~ . sin This angle is independent of the angle of the incoming polarization. It just depends on the material and its thickness. (c) The dispersion relation generally is the relationship between energy and momentum of a partic- le/wave: !(~k). For an electromagnetic wave in vacuum it is linear ! = ck, while for free electrons ~k2 it is a parabola ! = 2m . For the light travelling inside a material it gets corrected by the refractive ck index ! = n(!) . (d) For an optical rotator the rotation angle is only given by the thickness of the material and it does not e ect the circularly polarized light. On the other hand a /2 will rotate the incoming light polarization depending on the angle that the optical axis is making vs the light polarization plane. It does also a ect circularly polarized light by changing its handedness + , . When light enters at 45, a Soleil-Babinet compensator can only perform 90 rotations. For a generic incoming polarization, all possible rotations and transformations between elliptical and linearly polarized light are possible. (e) In a zero order waveplate, the phase shift between the light polarization component is exact, ¡ = =2 for the case of /2. An higher order waveplate dephases the light more than one time so that ¡ = n + =2 with n being an integer. This kind of waveplate is more sensitive to wavelength shifts as a function of temperature but it is cheaper compared to the zero order. (f) Both describe the absorption dependence with respect to light polarizations. In uniaxial crystals we have dichorism while in biaxial crystals we have pleochroism. (g) Achromatic waveplates consist of two di erent materials that practically eliminate chromatic dispersion (e.g. MnF2 and SiO2). (h) A few percent Cr3+ (electronic con guration [Ar]3d3) replaces Al3+ (and also vanadium in beryl). The octahedral surrounding leads to a crystal eld splitting with three lower-lying levels t2g and two levels with higher energy (eg) { see also Tanabe-Sugano diagrams. The optical transition oc- curs between the three lled lower levels into one of the higher levels. In principle, this transition is forbidden in centrosymmetric crystals (Laporte rule) but can be allowed by coupling to vibronic oscillations. In corundum, violett and yellow-green light is absorbed, then lowered in energy by thermal excitations, and the crystal orescences in the complementary colour which is ruby red. Beryl has a weaker crystal eld than corundum, resulting in a smaller splitting and absorption of orange-red and blue-violet light. Green is transmitted, therefore the green emerald colour. Doping corundum with iron and titanium leads to the blue colour of a sapphire. It is important to note that the crystals appear colourful mainly because of their transparency and more or less seem colourless in powder form or streak colour. (i) In general a polariton is described as quasiparticles that results from the strong coupling of the electromagetic waves with an electric or magnetic dipole-like excitations, that appear as common quantum mechanical phenomenon like level repulsion or avoided crossing (anti-crossings). Depen- ding on the type of electromagnetic radiation and the interaction, polaritons can be of several types: a) Phonon polaritons: coupling of infrared photon with an optic phonon. b) Exciton polaritons: coupling of visible light with exciton (electron-hole pair). a) Intersubband polaritons: coupling of infrared or terahertz photon with intersubband transi- tions. a) Surface plasmon polaritons: coupling of light eith surface plasmons. Let us consider the case of a phonon polariton. The typical dispersion diagram is sketched below, where the three branches are shown and are di erently polarized (two transversal and one longi- tudinal).

Ne2 1 (j) In the relation: (!) = b + 2 2 , b describes the background dielectric constant m0 !0 ! +i ! caused by charge oscillations with resonance frequencies far above ! (their dispersion is at at lower frequencies). (k) See also M. Fox, Optical properties of solids\: " (l) The 4f levels involved in the laser transition are fairly independent on the host material because of the highly localized structure of the rare-earth 4f orbitals, which are shielded by outer 6s; 5s; 5p; 5d orbitals.

Exercise 4.2: Absorption and colour (4 points) (a) See Optical properties of solids\, by M. Fox; with k =n!=c ~ : " 0

i(kz !t) E(k; z) = E0e

i(~n!z=c0 !t) = E0e

!z=c0 i(n!z=c0 !t) = E0e e

) = 2!=c0 = 4=

: Intensity (e), material-speci c : imaginary part of refractive index n = n0 + i = 0 + i00: dielectric constant (macroscopic electric eld) O.D. - optical density (log10), considers re ection and transmission, sample-speci c (i.e. spectrometer, laser safety goggles) optical conductivity = 0 + i00 (b) BG39 - tuquoise-blue transparent, OG515 - yellow transparent, RG715 - black, NG5 - gray transparent. (c) The color of a metal is determined by the frequency dependence of its re ectivity (that is, its conductivity). In the case of copper, excitations of d-electrons into the unoccupied conduction band happens around 2 eV that corresponds to 620nm.

Exercise 4.3: Second Harmonic Generation (SHG) (8 points)

Consider the following experimental setup to investigate the SHG response of BaTiO3: Laser light with wavelength 1064 nm travels along the x direction of the BaTiO3 sample. Using a half-wave plate the light polarization of the incoming light can be rotated (polarization rotator). Here, the vertical (horizontal) light polarization is chosen to be parallel to the crystallographic y (z) axis of the BaTiO3 crystal. The material produces SHG light with frequency 2! which is measured using a sensitive light detector. In addition, the polarization of the non-linear response can be analyzed using an optical polarization lter (polarization analyzer).

Polarization rotator Filter Detector

 

Polarization analyzer BaTiO3 sample

(a) BG39 should be placed behind the sample. BG39 lters the 1064nm eciently while letting the pass the second harmonic light (532nm). (b) In addition, nonlinear optical processes can occur in the polarization optics before and after the light passes the sample, and thus alter the measurement. To prevent this, additional lters can be used. One can place the RG715 to block frequency-doubled light from before the sample, and an OG515 lter after the sample blocks possible THG.

At room temperature BaTiO3 has the space group symmetry 4mm, which allows the following non-zero components of the SHG susceptibilities: zzz; xxz; zxx; yyz; and zyy. (c) The indices of the non-linear susceptibilities are related to the light polarization Pi(2!) ijkEj(!)Ek(!). Assume that the polarization analyzer is xed along the y-direction (0 ). If we now work out the allowed components for ijk with the fact that light is propagating along x-direction (see diagram), we see that only yyz and yzy are allowed with yyz = yzy. Thus we have:

Py (2!) = yyzEy (!) Ez (!) + yzyEz (!) Ey (!) = 2 yyzEy (!) Ez (!) Let us consider for the electric eld vector: 0 1 0 @ A E = E0 cos sin

We then obtain: 2 2 Py (2!) = 2 yyzE0 cos sin = yyzE0 sin (2)

Thus the measured intensity I2!() would depend on the rotation angle = [0 ; 360 ] of the polarization rotator in the following manner:

2 2 4 2 I2! () = Py (2!) = yyzE0 sin (2)

The corresponding polar plot of the measured intensity would look like:

(d) We have the SHG source term for ~kjjx: 0 1 0 ~ S(2!) = @ 2 yyzEy(!)Ez(!) A 2 2 zyyEy (!) + zzzEz (!)

If polarization analyzer is oriented along the z-axis (90), then the polarization is given by:

2 2 Pz (2!) = zyyEy (!) + zzzEz (!)

and with the electric eld vector: 0 1 0 @ A E = E0 cos sin we get: 2 2 2 2 Pz (2!) = zyyE0 cos + zzzE0 sin

Thus the angular dependence of I2!() is given by:

2 2 4 4 2 4 4 4 2 2 I2! () = Pz (2!) = zyyE0 cos + zzzE0 sin + 2 zyy zzzE0 sin cos On adding a phase di erence, we obtain:

2 2 4 4 2 4 4 4 2 2 I2! () = Pz (2!) = zyyE0 cos + zzzE0 sin + 2 zyy zzzE0 sin cos cos

With regards to the phase and assuming that zyy = zzz (not realistic!!), the corresponding polar plot of the measured intensity are given by:

(e) BaTiO3 is a ferroelectric material. Ferroelectricity is the phenomenon where certain materials have a spontaneous electric polarization that can be reversed by the application of an external electric eld. Such materials that exhibit ferroelectricity are called ferroelectric materials. Ferro- electricity breaks inversion symmetry so that SHG is allowed. Other techniques can be pyrocurrent measurements, PFM, Dielectric Measurements, XRD, etc.