# Lecture 10

## Full text

(1)

### Light Valves

Monochromatic Plane Waves

Electromagnetic Propagation in Anisotropic Media

Spatial Light Modulators

(2)

### Electromagnetic Propagation in Anisotropic Media

M H

H B

P E E

D B D

t J H D

t E B

o o

++

µµ

==

µµ

==

++

εε

==

εε

==

==

••

ρρ

==

••

∂∂ ==

−− ∂∂

××

∂∂ ==

++ ∂∂

××

Equations ve

Constituit 0

0 E=Electric field vector

H=Magnetic field vector D=Electric displacement B=Magnetic induction ρ=Electric charge density J= Current density

P=electric polarization M=Magnetic polarization ε=permitivity tensor

εo=permitivity of vacuum µ=permeability tensor

µo=permeability of vacuum

















εε εε εε

 ==















εε

==

εε εε

==

z y x

z y x o

j ij i

n n n

E D

0 0

0 0

0 0

0 0

0 0

0 0

2 2 2

(3)

### Plane Wave in Homogeneous Media

Electric Field Vector Magnetic Field Vector

s c n

k == ωω

### s == unit vecto r in propagatio n direction

Plugging these in Maxwell’s equation reduces to

2

### k

This leads to a relation between ω and k

0

2 2

2 2

2 2

2 2

2

==

−−

−−

µε µε ω ω

−−

−−

µε µε ω ω

−−

−−

µε µε ω ω

y x

z y

z x

z

z y z

x y

x y

z x y

x z

y x

k k

k k k

k

k k k

k k

k

k k k

k k

k det

(4)

### Normal Surface

Solution for nx(εx) at a given

frequency represents a 3D surface in k space known as normal surface

consists of two shells having four points in common

optic axis = two lines that go through the origin and these four points

Given a direction of propagation, there are two k values that are intersections of propagation direction and normal surface

k values different phase velocities (ω/k) of waves

propagating along the direction

Along an arbitrary direction of propagation, s, there can exist two independent plane waves linearly polarized propagating with phase

velocities (±c/n1 and (±c/n2 ) Yeh & Gu

(5)

### Classification of Media

• Normal surface is determined by the principal indices of refraction, nx, ny, nz

• nx≠ny≠nz ⇒ biaxial material

– Two optical axes

• no2xoyo and ne2= εzo

uniaxial material (z-axis) – Normal surface consists of a

sphere and ellipsoid of revolution

– no is ordinary index and ne extraordinary index

– ne – no either +ve or -ve

• nx=ny=nz ⇒ isotropic material

– Normal surface degenerate in a single sphere

Yeh & Gu Other Method

(6)

Yeh & Gu

(7)

2 2

e o z

o o y

x

Normal surface

2

2 2

2 2

2 2

2 2

2 2

### kk

o o

z e

y x

• Sphere gives the relationship

between ω and k for the ordinary (O) wave

• Ellipsoid of revolution gives the relationship between ω and k for the extraordinary (E) wave

• The two surfaces touch at two points on z-axis

Eigen-refractive indices are

2 2 2

2 2

o

n wave 1

E

n n

wave -

O

e

o n

sin n

cos θθ ++ θθ

==

−−

==

θ is the angle between propagation direction and optic axis

(8)

### Phase Retardation

• Inside a uniaxial medium, a phase retardation develops between O-wave and E-wave

– Due to diff. in phase velocity

• Phase retardation leads to a new polarization state

⇒Birefringent plates can be used to alter polarization state of light

o o

o

e e

e

Hecht

(9)

## [[ikr]]Czexp[[ikr]]

o

o

e

e

### ••

Wave propagating in uniaxial medium perpendicular to z(c-) axis

Assuming Co = Ce =1

e e

o e

λλ λλ λλ

λλ λλ

2 4 2

4 4

### At

Linearly polarized

Circularly polarized

Linearly polarized but ± to original

• Birefringent plate with thickness dλ/4 is known as quarter-wave plate and it is used to convert a linear polarization to circular polarization

• Birefringent plate with dλ/4 is known as half-wave plate and it is used to change direction of linear polarization

(10)

### Polarization by Selective Reflection

Reflection of light from the

boundary between two dielectric materials is polarization dependent

At the Brewsters angle of incidence

Light of TM polarization is totally refracted

Only TE component is reflected

t i B

B t

B i

B t

t t

B i

n n tan

cos n sin

n

sin n sin

n

==

θθ

θθ

==

θθ

θθ

−−

==

θθ θθ

==

θθ 90o

Saleh & Teich

(11)

### Polarization by Selective Refraction

• In an anisotropic crystal, two polarizations of light refract at different angles

– Spatially separation

• Devices are usually two cemented prisms of uniaxial crystals in different orientations

Saleh & Teich

(12)

### (Wave Plates)

• Retarders change the

polarization of an incident wave

• One of the two constituent polarization state is caused to lag behind the other

– Fast wave advanced – Slow wave retarded

• Relative phase of the two

components are different at exit

• Converts polarization state into another

– Linear to circular/elliptical

– Circular/elliptical to linear

s

f

Yeh & Gu

(13)

### (Wave Plates)

Wave retarders are often made of anisotropic materials

– uniaxial

• When light wave travels along a principal axis, the normal

modes are linearly polarized pointing along the other two principal axes (x, y)

– Travel with principal refractive indices nf, ns

Intensity modulated by relative phase retardation

s

f

o

s

f

Saleh & Teich

(14)

### Anisotropic Absorbtion and Polarizers

To take care of absorbtion, generalize the refractive index to complex number

t coefficien extinction

e are

o

e e

e

o o

o

,

i n nˆ

i n nˆ

κκ κκ

κκ

−−

==

κκ

−−

==

Define

T1=transmission with polarization // to the transmission axis T2=transmission with polarization ± to the transmission axis

2 2 T 2

T Ratio T

Extinction

2 1

2 2 2

1

2 1

o

1 2

T T T

T T T

T T

x p

==

== ++

== ++

==

O-type polarizer transmits ordinary waves and attenuates extraordinary wave i.e. κo=0

E-typepolarizer transmits extraordinary waves and attenuates ordinary wave i.e. κe=0

Transmittance of unpolarized light through polarizer

Transmittance of unpolarized light through pair of // polarizers

Transmittance of unpolarized light through pair of ± polarizers

(15)

### Optical Activity

• Optically active materials are substances that rotate a beam of light traversing through them in the direction of the optical axis.

– Usually given in º/mm

• Could be induced by external signal such as

– Electric field (electro-optic effect)

– Optical signal (photo-refractive effect)

Yeh & Gu

(16)

### modify the phase, polarization and/or amplitude of a 2D light beam as a function of either:

– A time varying electrical drive signal (electrically addressed SLM) – The intensity distribution of another time-varying optical signal

o

(17)

### • Electro-optic effect

– Pockels effect ∆n ∞ E (LiNbO3) – Kerr effect ∆n ∞ E2 (PLZT)

### • Photorefractive effect

– ∆n ∞ Exposure (LiNbO3,BaTiO3)

### • Molecular alignment by electric field

– Torque=PXE (Liquid Crystals)

### • Micromechanical

– Electrostatic deformation (membranes, gels, oil films)

### • Thermal

– Thermoplastics, smectic A & nematic liquid crystals

### • Electrophoresis

– motion of charged particles in an electric field

(18)

### Electro-optic Modulation of Light

• Electric field induced birefringence rotates plane of polarization

• Field applied transversely to direction of propagation of light

• Analyzer transmits an amplitude proportional to cosine of polarization angle wrt analyzer

Warde

(19)

### Electro-Optic SLMs

• Write light illuminates photodetector & accumulates charge

• Voltage built up across electro-optic crystal

• Electric field induced birefringence

• Phase retardation

• Reflects intensity modulated signal

Warde

(20)

### Liquid Crystal Devices

• Nematic liquid crystal

– The re-orientation of

molecules with the electric field alters the birefringence of the material.

• Electroclinic Smetic liquid crystal

– The tilt angle of molecules is linear with applie delectric field

• Surface stabilized Smetic Ferro- electric liquid crystal

– Molecules switch between two surface stabilized states

because of the torque resulting from coupling of ferro-electric polarization to the applied E

Warde

(21)

### Membrane Mirror Light Modulator

• Electrostatic forces deform mirror into wells etched in supporting surface.

• Readout light diffracts from membrane mirror

Warde

(22)

### Deformable Mirror Device

• Array of micro-mirrors integrated with SRAM array

• Signal stored in each SRAM cell applies voltage to each mirror

• Applied voltage deflects Mirror and hence direct light

Courtesy Texas Instruments

(23)

### Summary of Today’s Lecture

• Light valves modulate light coming from an independent light source of high intensity

• Modulation derived from phenomena causing

– Reflection – Diffraction – Scattering

– Polarization change

• Modulation of

– Amplitude – Phase

– Polarization – Intensity

Warde

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## References

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