Lecture 10 6.976 Flat Panel Display Devices
Light Valves Outline • • •
Monochromatic Plane Waves Electromagnetic Propagation in Anisotropic Media Spatial Light Modulators
6.976 Flat Panel Display Devices - Spring 2001
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References • Jin Au Kong, Electromagnetic Wave Theory, EMW Publishing, Cambridge, MA, USA • B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, John Wiley & Sons, New York • E. Hecht, Optics, Addison-Wesley Publishing • Cardinal Warde, Spatial Light Modulators for Optically Controlled Phased-Array Radar Signal Processors, Chapter 6 of Photonic Aspects of Modern Radar, Ed. H. Zmuda and E.N. Toughlian, p.163. • Cardinal Warde, Spatial Light Modulators for Optically Processing and Displays, SPIE International Symposium on Optical Science, Engineering and Instrumentation, 13 July 1995. • P. Yeh and C. Gu, Optics of Liquid Crystal Displays, John Wiley & Sons, New York, 1999.
6.976 Flat Panel Display Devices - Spring 2001
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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
Warde
6.976 Flat Panel Display Devices - Spring 2001
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Monochromatic Plane Wave E = E (r,t) = A cos (ωt − k • r ) = Re{A exp[i (ωt − k • r )]} k •E = 0
Transverse wave
ϕ = ωt − k • r
phase
ω= angular frequency • For transparent material n=n(λ) and is k = wave vector real Ã= constant vector representing amplitude • Dependence of n on wavelength is optical dispersion or chromatic ω 2π dispersion k =n =n c λ • For materials with absortion, n is complex n=index of refraction • Transverse wave c=speed of light in vacuum λ=wavelength of light in vacuum 6.976 Flat Panel Display Devices - Spring 2001
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Monochromatic Plane Wave
Hecht
E = E (r,t) = A cos(ω t − k • r ) = Re{A exp[i (ωt − k • r )]} 6.976 Flat Panel Display Devices - Spring 2001
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Polarization State • Polarization state specified by the electric field vector E(r,t) • Assuming propagation in the z-direction – Transverse wave lies in xy-plane – Two mutually independent components are E x = A x cos(ωt − kz + δ x )
E y = A y cos(ωt − kz + δ y ) – Ax, Ay are independent positive amplitudes – δx, δy are independent phases
• These corresponds to elliptic polarization with relative phase δ=δy-δx. 6.976 Flat Panel Display Devices - Spring 2001
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Linear Polarization • Beam of light is linearly polarized if the electric field vector vibrates in a constant direction in the xy-plane – δ=δy-δx=0 or π Ey Ex
=
Ay Ax
or
Ey Ex
=−
Ay Ax
• Since Ax, A are independent, the electric field vector of linearly polarized light can vibrate along any direction in the xy-plane • Linearly polarized light also known as plane polarized light. Hecht 6.976 Flat Panel Display Devices - Spring 2001
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Circular Polarization • Beam of light is circularly polarized if the electric field vector undergoes uniform rotation in the xy-plane – δ=δy-δx=π/2 – Ay = Ax
Hecht
6.976 Flat Panel Display Devices - Spring 2001
• Beam of light is right-hand circularly polarized when δ=π/2 which corresponds to counter-clockwise rotation of the E field vector in xy-plane • Beam of light is left-hand circularly polarized when δ=+π/2 which corresponds to clockwise rotation of the E field vector in xy-plane Lecture 10
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Circular Polarization States
Yeh & Gu 6.976 Flat Panel Display Devices - Spring 2001
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Elliptic Polarization • Beam of light is elliptically polarized if the curve traces by the tip of the electric field vector is an ellipse in the xyplane – Linear and circular polarization are special cases of the elliptic polarization At z = 0 2
E x E y cos δ + −2 Ex E y = sin 2 δ Ax A y Ax A y Can do transform ation such that 2
Yeh & Gu
+ve e corresponds to right handed E field -ve e corresponds to left-handed E field 6.976 Flat Panel Display Devices - Spring 2001
2
E x′′ Ey ′′ =1 + a b 2A A tan 2φ = 2 x y2 cos δ A x − Ay 2
ellipticit y e = ± Lecture 10
b a 10
Elliptic Polarization
Yeh & Gu 6.976 Flat Panel Display Devices - Spring 2001
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Jones Vector Representation • The plane wave is expressed in terms of its complex amplitudes as column vectors A e iδ x
J = A ye iδ x
y
• Thus linearly polarized light is given by
cos ψ J = ψ sin
– where ψ=tan-1 (Ay/Ax) is the azimuth angle of the oscillation direction with respect to the x-axis
• Special case of ψ=0 (linearly polarized light) 1 0 x = , y = 0 1
• Special case for right and left circularly polarized waves R=
1 1 1 1 , L = 2 − i 2 i
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In general cosψ J (ψ , δ) = iδ e sin ψ
Yeh & Gu 6.976 Flat Panel Display Devices - Spring 2001
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Electromagnetic Propagation in Anisotropic Media ∂B ∇×E + =0 ∂t ∂D ∇×H − =J ∂t ∇•D = ρ
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
∇•B= 0 Constituit ve Equations D = ε E = ε oE + P B = µH = µ o H + M
D i = εijE j n 2x ε = εo 0 0
6.976 Flat Panel Display Devices - Spring 2001
0 n 2y 0
0 εx 0 = 0 n 2z 0 Lecture 10
0 εy 0
0 0 ε z 14
Plane Wave in Homogeneous Media Magnetic Field Vector
Electric Field Vector
[
H exp[i (ωt − k • r )]
]
E exp i (ωt − k • r ) k=
ω ns c
s = unit vector in propagation direction
Plugging these in Maxwell’s equation reduces to
k × (k × E ) + ω 2µε E = 0
This leads to a relation between ω and k ω 2µε x − k 2y − k 2z det k yk x k zk x 6.976 Flat Panel Display Devices - Spring 2001
k xk y ω2µε y − k 2x − k 2z
k xk z k yk z
k zk y
ω2µε z − k 2x − k 2y Lecture 10
=0
15
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 )
6.976 Flat Panel Display Devices - Spring 2001
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Classification of Media • Normal surface is determined by the principal indices of refraction, nx, ny, nz • nx≠ny≠nz ⇒ biaxial material – Two optical axes
• no2=εx/εo=εy/εo and ne2= εz/εo
– ⇒ 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 6.976 Flat Panel Display Devices - Spring 2001
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Yeh & Gu 6.976 Flat Panel Display Devices - Spring 2001
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Light Propagation in Uniaxial Media ε x = ε y = ε on o2 ; ε z = ε o ne2 Normal surface
k 2x + k 2y k 2z ω2 k 2 ω2 + 2 − 2 2 − 2 = 0 2 ne no c n c o • 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 6.976 Flat Panel Display Devices - Spring 2001
Eigen-refractive indices are
O - wave
n = no
E − wave
1 cos 2 θ sin 2 θ = + 2 2 n no n 2e
θ is the angle between propagation direction and optic axis
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Phase Retardation
[
]
[
D = Co Do exp − ik o • r + Ce De exp − ik e • r
]
• 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
Hecht 6.976 Flat Panel Display Devices - Spring 2001
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Phase Retardation Wave propagating in uniaxial medium perpendicular to z(c-) axis
[
]
[
E = Co x exp − ik o • r + Ce z exp − ik e • r
]
Assuming C o = Ce =1
At y = 0 E =x+z At y = d λ 4 =
Linearly polarized
π2 ω c(n e − n o )
E = exp(ik e d λ 4 )[ix + z ] Circularly polarized At y = d λ 2 = 2d λ 4
E = exp(ik e d λ 2 )[− x + z ] 6.976 Flat Panel Display Devices - Spring 2001
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
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Anisotropic Absorbtion and Polarizers To take care of absorbtion, generalize the refractive index to complex number nˆo = no − iκ o nˆe = ne − iκ e κ o , κ e are extinction coefficient
O-type polarizer transmits ordinary waves and attenuates extraordinary wave i.e. κo=0 E-type polarizer transmits extraordinary waves and attenuates ordinary wave i.e. κe=0
Define T1=transmission with polarization // to the transmission axis T2=transmission with polarization ± to the transmission axis
Extinction Ratio = T1 + T2 2 T12 + T22 Tp = 2 T1T2 Tx = 2 To =
T2 T1
Transmittance of unpolarized light through polarizer
Transmittance of unpolarized light through pair of // polarizers
Transmittance of unpolarized light through pair of ± polarizers
6.976 Flat Panel Display Devices - Spring 2001
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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
Yeh & Gu
6.976 Flat Panel Display Devices - Spring 2001
– Electric field (electro-optic effect) – Optical signal (photo-refractive effect)
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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:
E (r , t ) = eˆ Re{Eo exp[j(ωt − k • r + φ )]}
• 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) 6.976 Flat Panel Display Devices - Spring 2001
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Typical SLM Structures Signal multiplying and amplifying functional SLM structures A = optical wave amplitudes I=intensities or currents
Warde 6.976 Flat Panel Display Devices - Spring 2001
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Functional Classes of SLMs Amplitude-only Modulation A m ( x, y , t ) = A r ( x, y )Ta [I w ( x, y, t )] φm ( x, y ) = φ r ( x, y ) + φmo ( x, y ) eˆ m ( x, y ) = M ij eˆ r ( x, y )
[ ]
Phase-only Modulation A m ( x, y ) = Ar ( x, y ) [1 − α ( x, y )]
φm ( x, y , t ) = φ r ( x, y ) + φ (I w ( x, y )) eˆ m ( x, y ) = [M ij ]eˆ r ( x, y )
6.976 Flat Panel Display Devices - Spring 2001
Warde
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Functional Classes of SLMs Polarization Modulation A m ( x, y , t ) = A r ( x, y ) [1 − α ( x, y, t ) ] φm ( x, y ) = φ r ( x, y ) + φ mo ( x, y ) eˆ m ( x, y , t ) = [M ij ( I w ( x, y , t ) )]eˆ r ( x, y )
Intensity Modulation Yeh & Gu
Im ( x, y , t ) = Ir ( x, y ) + Ti (I w ( x, y ) ) eˆ m ( x, y ) = M ij eˆ r ( x, y )
[ ]
Warde
6.976 Flat Panel Display Devices - Spring 2001
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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 6.976 Flat Panel Display Devices - Spring 2001
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Electro-optic Modulation of Light
Warde
• 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
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Electro-Optic SLMs
Warde
• • • • •
Write light illuminates photodetector & accumulates charge Voltage built up across electro-optic crystal Electric field induced birefringence Phase retardation Reflects intensity modulated signal
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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
Warde
6.976 Flat Panel Display Devices - Spring 2001
• Surface stabilized Smetic Ferroelectric liquid crystal – Molecules switch between two surface stabilized states because of the torque resulting from coupling of ferro-electric polarization to the applied E Lecture 10
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Membrane Mirror Light Modulator
Warde
• Electrostatic forces deform mirror into wells etched in supporting surface. • Readout light diffracts from membrane mirror
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Deformable Mirror Device
Courtesy Texas Instruments
• 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 6.976 Flat Panel Display Devices - Spring 2001
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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
6.976 Flat Panel Display Devices - Spring 2001
Warde
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