Review of Basic Polarization Optics for LCDs Module

Review of Basic Polarization Optics for LCDs Module 4

Module 4 Goals • Polarization • Jones Vectors • Stokes Vectors • Poincare Sphere • Adiabadic Waveguiding

Polarization of Optical Waves Objective: Model the polarization of light through an LCD. Assumptions: • • Linearity – this allows us to treat the transmission of light independent of wavelength (or color). We can treat each angle of incidence independently. Transmission is reduced to a linear superposition of the transmission of monochromatic (single wavelength) plane waves through LCD assembly.

Monochromatic Plane Wave (I) A monochromatic plane wave propagating in isotropic and homogenous medium: A = constant amplitude vector = angular frequency k = wave vector = index of refraction = speed of light = wavelength in vacuum is related to frequency For transparent materials Dispersion relation

Monochromatic Plane Wave (II) • The E-field direction is always to the direction of propagation • Complex notation for plane wave: (Real part represents actual E-field) • Consider propagation along Z-axis, E-field vector is in X-Y plane: Y-axis EY X-axis independent amplitudes two independent phases Ex

Monochromatic Plane Wave (III) • There is no loss of generality in this case. • Finally, we define the relative phase as • Now in a position to look at three specific cases. 1. Linear Polarization 2. Circular Polarization 3. Elliptical Polarization

Linear Polarization • In this case, the E-field vector follows a linear pattern in the X-Y plane as either time or position vary. • Occurs when or • Y-axis AY Ax X-axis Important parameters: 1. Orientation 2. Handedness 3. Extent Linear polarized or plane polarized are used interchangeably

Circular Polarization • In this case, the E-field vector follows a circular rotation in the X-Y plane as either time or position vary. • Occurs when AY X-axis and • Y-axis Ax Important parameters: 1. Orientation 2. Handedness 3. Extent (-) CCW rotation = RH, (+) CW rotation = LH

Circular Polarization Equation of a circle

Elliptic Polarization States • This is the most general representation of polarization. The E-field vector follows an elliptical rotation in the X-Y plane as either time or position vary. • Occurs for all values of • Important parameters: 1. Orientation 2. Handedness 3. Extent of Ellipticity Y-axis AY a b Ax X-axis

Elliptic Polarization States eliminate t Transformation: x’ y’ b a Ax X-axis

d=3 p/4 d=p/2 d=p/4 d=0 d=p/4 d=p/2 d=3 p/4 d=p

Review Complex Numbers Im c = 3 – 4 i -2+2 i Re c = ei = cos + i. sin c = e-i = cos (- ) + i. sin (- ) = cos - i. sin Remember the identities: ex ey = ex+y ex / ey = ex-y d/dz ez = ez 3 -4 i

Complex Number Representation Polarization can be described by an amplitude and phase angles of the X-Y components of the electric field vector. This lends itself to representation with complex numbers: Im Re on x axis on y (imaginary axis)

Jones Vector Representation Convenient way to uniquely describe polarization state of a plane wave, using complex amplitudes as a column vector. is not a vector in real space, it is a mathematical abstraction in complex space. Jones Vector amplitude phases electric field Polarization is uniquely specified

Jones Vector Representation (II) If you are only interested in polarization state, it is most convenient to normalize it. A linear polarized beam with electric field vector oscillating along a given direction can be represented as: For orthogonal state,

Jones Vector Representation (III) Normalize Jones Vector Take

Jones Vector Representation (IV) The Jones matrix of rank 2, any pair of orthogonal Jones vectors can be used as a basis for the mathematical space spanned by all the Jones vectors. When y=0 for linear polarized light, the electric field oscillates along coordinate system, the Jones Vectors are given by: For circular polarized light: Mutually orthogonal condition

Polarization Representation Polarization Ellipse Jones Vector (d, y) (f, ) Stokes

Polarization Representation Polarization Ellipse Jones Vector (d, y) (f, ) Stokes

Jones Matrix Limitations Jones is powerful for studying the propagation of plane waves with arbitrary states of polarization through an arbitrary sequence of birefringent elements and polarizers. Limitations: • Applies to normal incidence or paraxial rays only • Neglects Fresnel refraction and surface reflections • Deficient polarizer modeling • Only models polarized light Other Methods: • 4 x 4 Method – exact solutions (models refraction and multiple reflections) • 2 x 2 Extended Jones Matrix Method (relaxes multiple reflections for greater simplicity)

Partially Polarized & Unpolarized Light We discussed monochromatic/polarization thus far. If light is not absolutely monochromatic, the amplitude and relative phase d between x and y components can vary with time, and the electric field vector will first vibrate in one ellipse and then in another. The polarization state of a polychromatic wave is constantly changing. If polarization state changes faster than speed of observation, the light is partially polarized or unpolarized. Optics – light of oscillation frequencies 1014 s-1 Whereas polarization may change 10 -8 s (depending on source)

Partially Polarized & Unpolarized Light Consider quasi monochromatic waves (D << ) Light can still be described as: Provided the constancy condition of A is relaxed. denotes center frequency A denotes complex amplitude Because (D << ), changes in A(t) are small in a time interval 1/D (slowly varying). If the time constant of the detector td>1/D , A(t) can change originally in a time interval td.

Partially Polarized & Unpolarized Light To describe this type of polarization state, must consider time averaged quantities. S 0 = <<Ax 2+Ay 2>> S 1 = <<Ax 2 -Ay 2>> S 2 = 2<<Ax. Ay cosd>> S 3 = 2<<Ax. Ay sind>> Ax, Ay, and d are time dependent << >> denotes averages over time interval td that is the characteristic time constant of the detection process. These are STOKES parameters.

Stokes Parameters Note: All four Stokes Parameters have the same dimension of intensity. They satisfy the relation: the equality sign holds only for polarized light.

Stokes Parameters Example: Unpolarized light No preference between Ax and Ay (Ax=Ay), d random S 0 = <<Ax 2+Ay 2>>=2<<Ax 2>> S 1= <<Ax 2 -Ay 2>>=0 S 2, 3=2<<Ax. Ay cosd>>=2<<Ax. Ay sind>>=0 since d is a random function of time if S 0 is normalized to 1, the Stokes vector parameter is for unpolarized light. Example: Horizontal Polarized Light Ay=0, Ax=1 S 0=<<Ax 2>>=1 S 1=<<Ax 2>>=1 S 2, 3=2<<Ax. Ay cosd>>=2<<Ax. Ay sind>>=0

Stokes Parameters Example: Vertically polarized light Ay=1, Ax=0 S 0 = <<Ax 2+Ay 2>>=<<Ay 2>>=1 S 1 = <<Ax 2 -Ay 2>>=<<-Ay 2>>=-1 S 2, 3 = 2<<Ax. Ay cosd>>=2<<Ax. Ay sind>>=0 Example: Right handed circular polarized light S 0 = <<Ax 2+Ay 2>> = 2<<Ax 2>> S 1 = <<Ax 2 -Ay 2>> = 0 S 2 = 2<<Ax. Ay cos(-1/2 p)>> = 0 S 3 = 2<<Ax. Ay sin(-1/2 p)>> = -1 (d=-1/2 p) Ax=Ay

Stokes Parameters Example: Left handed circular polarized light S 0 = <<Ax 2+Ay 2>> = 2<<Ax 2>> S 1 = <<Ax 2 -Ay 2>> = 0 S 2 = 2<<Ax. Ay cos(1/2 p)>> = 0 S 3 = 2<<Ax. Ay sin(1/2 p)>> = 1 Degree of polarization: Unpolarized S 12 = S 22 = S 32 = 0 Polarized S 12+S 22+S 32 = 1 useful for describing partially polarized light (d=1/2 p) Ax=Ay

Jones Matrix Method (I) f Y-axis y y s X-axis Z-axis • The polarization state in a fixed lab axis X and Y: • Decomposed into fast and slow coordinate transform: (notation: fast (f) and slow (s) component of the polarization state) rotation matrix • If ns and nf are the refractive indices associated with the propagation of slow and fast components, the emerging beam has the polarization state: Where d is the thickness and l is the wavelength

Jones Matrix Method (II) • For a “simple” retardation film, the following phase changes occur: (relative phase retardation) (mean absolute phase change) • Rewriting previous retardation equation:

Jones Matrix Method (III) • The Jones vector of the polarization state of the emerging beam in the X-Y coordinate system is given by transforming back to the S-F coordinate system.

Jones Matrix Method (IV) • By combining equations, the transformation due to the retarder plate is: where W 0 is the Jones matrix for the retarder plate and R(Y) is the coordinate rotation matrix. (The absolute phase can often be neglected if multiple reflections can be ignored) A retardation plate is characterized by its phase retardation and its azimuth angle y, and is represented by:

Examples Polarizer with transmission axis oriented to X-axis Polarization State Y-axis E f X-axis f’ is due to finite optical thickness of polarizer. If polarizer is rotated by y about Z ignoring f’ polarizers transmitting light with electric field vectors to x and y are: b b a f f Jones Vector

Examples ¼ Wave Plate and the thickness and incident beam is vertically polarized: Polarization State Y-axis E X-axis f Incident Jones Vector Emerging Jones Vector

Wave Plates y c-axis Jones Matrices x Remember: c-axis In general: c-axis 450 y

Polarizers Jones Matrices y transmission axis x transmission axis Remember: transmission axis 450 In general: transmission axis y

Birefringent Plates 45 Parallel polarizers 45 Cross polarizers

Poincare’s Representatives Method

Poincare’ Sphere: Linear Polarization States

Poincare’ Sphere: Elliptic Polarization States

Polarization Conversion:

Polarization Conversion: Y-axis f s y X-axis Z-axis

Some Examples • TN LCD Formulations

General Matrix For LCD e – component || director o – component director f Twist angle Phase retardation

Adiabatic Waveguiding • Consider light polarized parallel to the slow axis of a twisted LC twisted structure: • Then, the output polarization will be: 90° Twist

Adiabatic Waveguiding • Notice that for TN displays since f<< (twist angle much smaller than retardation ): • Then the output polarization reduces to: which means that the electric field vector “follows” the nematic director as beam propagates through medium – it rotates –

90º Twisted Nematic (Normal Black) • Consider twisted structure between a pair of parallel polarizers and consider e-mode operation. e-mode input • The transmission after the second polarizer:

Transmission of Normal Black first minimum second minimum third minimum

Normal White Mode (I) • Consider twisted structure between a pair of parallel polarizers and consider e-mode operation. e-mode input • The transmission after the second polarizer:

Normal White Mode (II)

Y-axis E n n X-axis E E n n E E n E (n) n E n n E

Phase Retardation at Oblique Incidence: Complicating Matters z B D F qo qe d A C

Summary of Optics Vital to understanding LCD’s and their viewing angle solutions: • Linear, circular, elliptical polarization • Jones Vector • Stokes Parameters • Jones Matrixes • Adiabatic Waveguiding • Extended Jones and 4 x 4 Methods