Lecture 3 :
Electrooptic effect, optical activity
and basics of interference colors
with wave plates
NW optique physique II
• Electrooptic effect
– Electrooptic effect: example of a KDP Pockels cell
– Liquid crystals
• Optical activity
• Interference with polarized light: understanding the
interference colors of birefringent plates
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1 – Electrooptic effect
Principle
Anisotropy can be induced by external fields:
We will consider here only the effect of an electric field
Electrooptic effect
In general induced index changes are small and they require
high fields or large path lengths
However technological advances allow strong effects using low
fields (liquid crystals, electrooptic waveguides for telecoms,…)
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• All necessary tools have been seen in the case of natural
anisotropy, we only need now to connect the characteristic
eigen indices of the material to the applied external field
• This lecture will provide a few examples of existing effects and
their applications, other courses such as « Guided and Coupled
Waves » by Jean-Michel Jonathan will go over this subject in
more details
• Numerous technological applications to these induced optical
effects
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I. Electro-optical Effects
Modification of the index ellipsoïd
We can characterize the effect of the E field by the modification
of the index ellipsoid (n in the direction of D):
2
2
2
x
2
n xx
+
y
2
n yy
z
2yz 2xz 2xy
+ 2 + 2 + 2 + 2 =1
n zz n yz n xz n xy
Each 1/nij2 term may in general include
• terms proportional to E : Pockels effect
€
• and terms in E2 : Kerr effect
No linear terms in E if the medium is initially isotropic
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Pockels effect
The transformation of the index ellipsoïd can be calculated
from the electrooptic 3*6 matrix of the medium, according
to the following relationship:
⎛ 1
1 ⎞
−
⎜ 2
2 ⎟
n
n
x ⎟
⎜ xx
⎜ 1 − 1 ⎟
2
⎜ n yy
n y2 ⎟ ⎛ r11
⎜
⎜ 1
1 ⎟ ⎜ r21
⎜ 2 − 2 ⎟
n z ⎟ ⎜ r31
⎜ n zz
⎜ 1
⎟ = ⎜ r
⎜ 2 − 0 ⎟ ⎜ 41
⎜ n yz
⎟ ⎜ r51
⎜ 1 − 0 ⎟ ⎜⎝ r
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⎜ n 2
⎟
⎜ xz
⎟
1
⎜
⎟
⎜ n 2 − 0 ⎟
⎝ xy
⎠
r12
r22
r32
r42
r52
r62
r13 ⎞
⎟
r23 ⎟
⎛ Ex ⎞
r33 ⎟ ⎜ ⎟
⎟ ⎜ Ey ⎟
r43 ⎟ ⎜ ⎟
⎝ Ez ⎠
r53 ⎟
⎟
r63 ⎠
Matrix which
characterizes the
electrooptic
response of the
medium
NW optique physique II
Example: Pockels effect in KDP (see optics labs)
Electro-optic tensor
Section of the index ellipsoid
Initially uniaxial with optic axis z
y!
Y
X
Applied field
X
E // z
2
2
⎛
⎞
⎛
⎞
z2
1 x+y
1 x−y
+ 2 + 2 + 2xyr63 E = 1 = 2 ⎜
⎟ + 2 ⎜
⎟ + 2
2
no no ne
n X ⎝ 2 ⎠ nY ⎝ 2 ⎠ n e
x2
€
y2
z2
⎡ 1 2
⎤
n X = n o⎢1 − n o r63 E⎥
⎣ 2
⎦
⎡ 1 2
⎤
nY = n o⎢1 + n o r63 E⎥
⎣ 2
⎦
For a propagation along z
ϕ=
2π
2π 3
(nY − n X )e =
n o r63V
λ
λ
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NW optique physique II
The direction of the new axes can also depend on
the electric field
Same example of KDP but different direction of applied E
Applied field
Electro-optic tensor
E // y
x2
n o2
+
y2
n o2
+
z2
n e2
+ 2xzr41E = 1
Section of the index ellipsoid
z
€
β
x
β and Δn ∝ E
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NW optique physique II
Electrooptical effect: the case of liquid crystals
α
E
V=0
V
Birefringence
Birefringence ↓ when α ↑ (ie V ↑)
Optical axis //
molecules
α = 90° → no more birefringence
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NW optique physique II
Applications of the electrooptical effect
• Modulators: polarization states or intensity
• Deflectors
• Phase shifters
• Displays
• Optical switches
Modulation frequencies can be very high, up to a few GHz (not
for liquid crystals, which are slower)
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2 – Optical activity
A linear polarization is rotated by an angle α:
• proportional to the path length through the medium
• proportional to the concentration (for a solution)
• dependent on wavelength as 1/λ2
Certain substances cause a left handed rotation (levorotatory), other a right
handed rotation (dextrorotatory) with respect to the observer. A mixture with
equal concentrations (racemic) does not produce any rotation.
Examples of optically active media
 Cristalline quartz, used with light propagating in the direction of its optical axis
 Nicotine, turpentine, camphor, sugar in solution, etc.
Microscopic origin: the atomic arrangement in the molecule is asymmetric, the
molecule is not identical to its image in a mirror (for quartz it is an asymmetry in the
crystal structure)
NW optique physique II
Interpretation in terms of circular birefringence
In terms of the modification induced on a polarization state, wwe can
interpret optical activity as a phase shift induced between the left handed
and right handed circular eigen polarizations :
α = (ϕR-ϕL)/2=π/λ(nR-nL)e
Note that for quartz the circular birefringence is 128 times smaller than the
linear birefringence (ne-no)
Applications
• Sugar concentration measurement (saccharimetry): 100°Z=34,626° (for
λ=589,44nm) for 26g of sucrose in 100ml for a thickness of 200mm
• Dose or control of purity for different substances in the food industry,
pharmaceutical, cosmetic and chemical industries
• Penumbra analyzer: Soleil biquartz
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NW wave optics (polarization)
3 – Interference with polarized light
Main properties of two wave interference
• Two wave interference: superposition of two fields with a
phase shift
• Shape of the fringes: dark fringes for
• Best contrast for equal intensities of the two beams
NW wave optics (polarization)
Interference with polarized waves
With polarized light E1 et E2 are complex amplitudes:
• The interference term 2Re(E1*.E2.eiϕ) is zero if the two
polarizations are orthogonal
• In the case of linear polarizations (E1 and E2 real vectors),
the interference term becomes 2 E1.E2cosϕ
• If E1//E2 we get back to the unpolarized case: e.g. a Michelson
interferometer with incident light TE or TM, the only dependance with
polarization comes from the Fresnel coefficients R and T of the beamsplitter
NW wave optics (polarization)
Interference using birefringent media
The amplitude of an incident wave is split between two
waves: the ordinary and extraordinary waves
CAN THEY INTERFERE?
• In principle NO because the fields are orthogonal
• BUT using a linear polarizer (called the « analyzer ») at
the exit of the birefringent medium, the two fields are
projected onto the same axis, that of the analyzer,
and
YES they can thus interfere
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WHAT IS THE BEST POSITION FOR THE ANALYZER?
Maximize the contrast
The two interfering waves should have the same intensity
so that the destructive interference is zero: Imin=0
Two possibilities
• E1.E2>0
• E1.E2<0
The 2 fields after the analyzer must have the
same amplitude (to within a sign)
NW wave optics (polarization)
IS THERE A BETTER CHOICE FOR THE INCIDENT
POLARIZATION?
YES, so that the constructive interference has an
intensity that is maximum
α : angle between the incident polarization and the neutral axes (ordinary and
extraordinary) of the birefringent medium:
We already have
(max contrast)
Maximum intensity:
Imax is maximum for α=45°
NW wave optics (polarization)
BEST POLARIZATION CONDITION FOR
INTERFERENCE
In conclusion two situations maximize both contrast and intensity:
the birefringent medium should be placed
between a polarizer P and an analyzer A such that :
1. P and A parallel and at 45° of the neutral axes
2. P and A orthogonal and at 45° of the neutral axes
NW wave optics (polarization)
• In both cases, Iin is the intensity arriving on the birefringent medium, i.e.
after the first polarizer.
• P and A orthogonal is often a better choice because the contrast is always
maximum even if the neutral axes are not at 45°
• Interference does not always imply « fringes » :
if the birefringent medium is a simple plate with plane parallel sides,
is uniform over the whole observation field;
the interference results is a uniform intensity (no fringes) as for a Michelson with
an incident plane wave
NW wave optics (polarization)
Observation of fringes with a small angle
Wollaston prism
Small angle θ ⇒we neglect
the ray deviation
x
θ
δ=2(ne-no)xθ
Straight fringes // y axis
y
Dark fringe (if P ⊥ A) for x=0
Observation on the slide projector
P
A
with green light + parallel or crossed
polarizers : 10 fringes over 30mm*30mm of
quartz
⇒ Calculate the angle θ
NW wave optics (polarization)
Application to the measurement of the phase
shift of a wave plate: Babinet compensator
We add the unknown phase plate with its axes parallel to
the Wollaston’s axes
The plate induces a
translation of the fringes by
δplate/λ*period of the
fringes
Unknown plate
P
A
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Babinet compensator
We can translate one prism of the Wollaston to bring back the
central fringe at x=0
d
Babinet translated by d to
bring back the dark fringe
at the center (P ⊥ A):
δplate=(ne-no)dθ
Unknown plate
P
A
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Interference with white light :
Newton’s color scale and channeled spectrum
The phase ϕ varies because of the different λ. We neglect the dependence of
ne-no with λ.
Observation with the naked eye
→ superposition of different I(λ)
→ Resultant color if δ is not too large
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δ in nm
NW wave optics (polarization)
NW wave optics (polarization)
Interference with white light :
Newton’s color scale
White fringe at the center for P//A, dark fringe for P⊥A
Advantage of the sensitive color (purple): sensitivity <100nm
Esay to observe in polarized light because δ=(ne-no)e is small
→ observation on the slide projector with layers of scotch tape
Channeled spectrum
Observation with spectrometer → dispersion of the different λ
P//A: dark fringes if δ=λ/2+kλ
P⊥A: dark fringes si δ=kλ