Download Crystal Optics with Intense Light Sources Exercise sheet #4

Crystal Optics with Intense Light Sources
Exercise sheet #4
FS 2017 - ETH Zurich
Prof. Manfred Fiebig
To be discusses on 05.04.2017
Exercise 4.1: Short questions
(15 points)
(a) How does circular birefringence manifests itself mathematically in the refractive index? (1)
(b) Briefly describe how a linearly polarized light beam is affected by an optical rotator. (1)
(c) What is described by the dispersion relation? (1)
(d) Compare optical rotator, half-wave plate and Soleil-Babinet compensator: What determines the
rotation angle of the polarization? How do they affect circularly polarized light? (2)
(e) Get information on zero and higher order waveplates. What is their difference? (1)
(f) What is the difference between dichroism and pleochroism? (1)
(g) Find out the design of achromatic wave plates and sketch it? (1)
(h) Corundum (Al2 O3 ) and beryl (Be3 Al2 Si6 O18 , also known as helidor) are colorless in their pure
state. However, if some of the aluminum ions are replaced by Cr3+ one gets red ruby and green
emerald, respectively. In both cases the chromium ions are surrounded by an oxygen octahedron.
What creates the color impression and why is it different despite featuring the same doping ions?
(2)
(i) What is a polariton? Explain its three branches. (1)
(j) What is described by b in the following solution of the simple oscillator model? (1)
(ω) = b +
N e2
1
m0 ω02 − ω 2 + iΓω
(k) Draw the frequency dependence of the dielectric constant (ω) (real and imaginary part) for a
material with two resonances. (2)
(l) Consider the following materials used for lasers: Nd:YAG (cubic, 1064 nm), Nd:glass (1062 nm),
Nd:YCOB (monoclinic, 1060 nm), Nd:YVO4 (tetragonal, 1064 nm). Why are the lasing wavelengths so very similar despite their different structure? (1)
Exercise 4.2: Absorption and colour
(4 points)
(a) Relate the imaginary part κ of the complex refractive index ñ = n+iκ to the absorption coefficient
α as known from the Lambert-Beer law: I(z) = I0 e−αz . What other quantities can be used to
describe optical absorption? How are these related to the absorption coefficient? (2)
(b) On the next page transmittance curves of four different optical filters are shown. What are the
colors of these filters? (1)
(c) Why does copper have its characteristic red color? (1)
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 2ω which is measured using a sensitive light detector.
In addition, the polarization of the non-linear response can be analyzed using an optical polarization
filter (polarization analyzer).
Polarization rotator
Filter
Detector


Polarization analyzer
BaTiO3
sample
(a) Think about the light you want and the light you do not want to measure. Which filter would
you place behind the sample? Use the filter curves given on the next page. (1)
(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 filters can
be used. Propose filter type(s) and its/their placement(s) to improve the setup! Use again the
curves provided below. (1)
At room temperature BaTiO3 has the space group symmetry 4mm, which allows the following non-zero
components of the SHG susceptibilities: χzzz , χxxz , χzxx , χyyz , and χzyy .
(c) Remember that the indices of the non-linear susceptibilities are related to the light polarization
Pi (2ω) ∼ χijk Ej (ω)Ek (ω). Assume that the polarization analyzer is fixed along the y-direction
(0◦ ). How does the measured intensity I2ω (φ) depend on the rotation angle φ = [0◦ , 360◦ ] of the
polarization rotator? Draw a polar plot of the measured intensity. (2)
(d) How is the angular dependence of I2ω (φ) if polarization analyzer is oriented along the z-axis
(90◦ )? Draw again a polar plot of the measured intensity. (2)
SHG source term for ~k||x:


0
~

2χyyz Ey (ω)Ez (ω)
S(2ω)
=
χzyy Ey2 (ω) + χzzz Ez2 (ω)
(e) BaTiO3 is a ferroelectric material. Describe the term ferroelectric and think about different ways
how to measure the transition temperature from the paraelectric to the ferroelectric phase. (2)
NG5
RG715
OG515
BG39
source: SCHOTT
To exercise 4.2 (b) and 4.3 (a,b):