NW optique physique II Electrooptic effect, optical ...

NW optique physique II 1 Lecture 3 : Electrooptic effect, optical activity and basics of interference colors

with wave plates

• 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

NW optique physique II 2

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,…)

1 – Electrooptic effect


• 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


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):

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

€

x 2

nxx2 +

y 2

nyy2 +

z 2

nzz2 +

2yznyz2 +

2xznxz2 +

2xynxy2 = 1


The transformation of the index ellipsoïd can be calculated

from the electrooptic 3*6 matrix of the medium, according

to the following relationship:

€

1nxx

2 − 1nx

2

1nyy

2 − 1ny

2

1nzz

2 − 1nz

2

1nyz

2 − 0

1nxz

2 − 0

1nxy

2 − 0

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

=

r11 r12 r13

r21 r22 r23

r31 r32 r33

r41 r42 r43

r51 r52 r53

r61 r62 r63

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

ExEyEz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Pockels effect

Matrix which characterizes the

electrooptic response of the

medium


Example: Pockels effect in KDP (see optics labs) Electro-optic tensor Section of the index ellipsoid

Initially uniaxial with optic axis z y!

X �Applied field

E // z

€

x 2

no2 +

y 2

no2 +

z 2

ne2 + 2xyr63E = 1=

1nX2

x + y2

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+1nY2

x − y2

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+z 2

ne2

X Y

€

nX = no 1−12no2r63E

⎡

⎣ ⎢ ⎤

⎦ ⎥

nY = no 1+12no2r63E

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

ϕ =2πλ(nY − nX )e =

2πλno3r63V

For a propagation along z


The direction of the new axes can also depend on

the electric field Same example of KDP but different direction of applied E

Electro-optic tensor

€

x 2

no2 +

y 2

no2 +

z 2

ne2 + 2xzr41E = 1

Applied field

E // y

z

x β and Δn ∝ E β

Section of the index ellipsoid


Electrooptical effect: the case of liquid crystals

V=0 V

E

α

Birefringence

Optical axis // molecules

Birefringence ↓ when α ↑ (ie V ↑)

α = 90° → no more birefringence


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)


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)

2 – Optical activity


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

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


• 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

Interference with polarized waves

With polarized light E1 et E2 are complex amplitudes:


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


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)


YES, so that the constructive interference has an

intensity that is maximum

We already have (max contrast)

Maximum intensity:

Imax is maximum for α=45°

IS THERE A BETTER CHOICE FOR THE INCIDENT POLARIZATION?

α : angle between the incident polarization and the neutral axes (ordinary and

extraordinary) of the birefringent medium:


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

BEST POLARIZATION CONDITION FOR INTERFERENCE


• 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


Observation of fringes with a small angle Wollaston prism

Small angle θ ⇒we neglect the ray deviation

δ=2(ne-no)xθ

Straight fringes // y axis

Dark fringe (if P ⊥ A) for x=0

P A

x

y

θ

Observation on the slide projector

with green light + parallel or crossed polarizers : 10 fringes over 30mm*30mm of quartz

⇒ Calculate the angle θ


P A

We add the unknown phase plate with its axes parallel to

the Wollaston’s axes

Unknown plate

Application to the measurement of the phase shift of a wave plate: Babinet compensator

The plate induces a translation of the fringes by

δplate/λ*period of the fringes


Babinet translated by d to bring back the dark fringe

at the center (P ⊥ A):

δplate=(ne-no)dθ

d

P A

Unknown plate

Babinet compensator

We can translate one prism of the Wollaston to bring back the central fringe at x=0


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


δ in nm


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

Interference with white light :

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

NW optique physique II Electrooptic effect, optical ...

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