Lecture 10

Lecture 10

6.976 Flat Panel Display Devices

Light Valves

• Monochromatic Plane Waves

• Electromagnetic Propagation in Anisotropic Media

• Spatial Light Modulators

Outline

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

ε_o=permitivity of vacuum µ=permeability tensor

µ_o=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

Plane Wave in Homogeneous Media

Electric Field Vector Magnetic Field Vector

(( ))

[[ ^{i t k r} ]]

exp

E ω ω −− •• ^{H exp} [[ ⁱ (( ^{ω ω t −− k •• r} )) ]]

s c n

k == ωω

s == unit vecto r in propagatio n direction

Plugging these in Maxwell’s equation reduces to

(( ^×× )) ^{++ ω ω}

^{µε µε == 0}

×× k E E

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

Normal Surface

• Solution for n_x(ε_x) 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

Classification of Media

• Normal surface is determined by the principal indices of refraction, n_x, n_y, n_z

• n_x≠n_y≠n_z ⇒ biaxial material

– Two optical axes

• n_o²=ε_x/ε_o=ε_y/ε_o and n_e²= ε_z/ε_o

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

sphere and ellipsoid of revolution

– n_o is ordinary index and n_e extraordinary index

– n_e – n_o either +ve or -ve

• n_x=n_y=n_z ⇒ isotropic material

– Normal surface degenerate in a single sphere

Yeh & Gu Other Method

Yeh & Gu

Light Propagation in Uniaxial Media

2 2

e o z

o o y

x

== εε == εε n ; εε == εε n εε

Normal surface

2

0

2 2

2 2

2 2

2 2

2 2

 ==



 



 −− ω ω

 





 



 ++ ++ −− ω ω

c n

k c

n k n

k k

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

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

[[ ^{i k r} ]] ^{C D exp} [[ ^{i k r} ]]

exp D

C

D ==

−−

•• ++

−−

••

Hecht

Phase Retardation

[[ ^{i k r} ]] ^{C z exp} [[ ^{i k r} ]]

exp x

C

E ==

−−

•• ++

−−

••

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

Assuming C_o = C_e =1

(( ))

(( )) ^{[[ ]]}

(( ^ik )) ^{[[ x z ]]}

exp E

z ix ik

exp E

n n

c z

x E

e e

o e

++

−−

==

==

==

++

==

−−

ω ω

== ππ

==

++

==

==

λλ λλ λλ

λλ λλ

2 4 2

4 4

d d 2 d

y At

d d 2

y At

0 y

At

Linearly polarized

Circularly polarized

Linearly polarized but ± to original

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

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

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 90^o

Saleh & Teich

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

Wave Retarders

(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

(( ⁿ

−− ⁿ

)) ^d

λλ

== ππ ΓΓ 2

Yeh & Gu

Wave Retarders

(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 n_f, n_s

• Intensity modulated by relative phase retardation

(( ⁿ

−− ⁿ

)) ^d == ^k

(( ⁿ

−− ⁿ

)) ^d

λλ

== ππ ΓΓ 2

Saleh & Teich

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. κ_e=0

Transmittance of unpolarized light through polarizer

Transmittance of unpolarized light through pair of // polarizers

Transmittance of unpolarized light through pair of ± polarizers

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

Spatial Light Modulators

• Spatial light modulators (or light valves) are the buiding blocks of optical information processors and display

systems.

• Consider a plane monochromatic wave of the form:

• A spatial light modulator (SLM) is a device that can

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

(optically addressed SLM)

(( ))

[[ ]]

{{ ^{ω ω −− •• ++ φφ} }}

== eˆ Re E exp j t k r )

t , r (

E

Examples of Light Modulation Schemes

• 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

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

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

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

Membrane Mirror Light Modulator

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

• Readout light diffracts from membrane mirror

Warde

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

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