Review of Basic Polarization Optics for LCDs Module 4

1. Review of Basic PolarizationOptics for LCDsModule 4

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

3. 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.

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

5. 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 Ex independent amplitudes two independent phases

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

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

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

9. Circular Polarization Equation of a circle

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

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

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

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

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

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

16. 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,

17. Jones Vector Representation (III) Normalize Jones Vector Take

18. 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

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

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

21. 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: • 4x4 Method – exact solutions (models refraction and multiple reflections) • 2x2 Extended Jones Matrix Method (relaxes multiple reflections for greater simplicity)

22. Partially Polarized & Unpolarized Light Optics – light of oscillation frequencies 1014s-1 Whereas polarization may change 10-8s (depending on source) 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.

23. 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/Dw (slowly varying). If the time constant of the detector td>1/Dw, A(t) can change originally in a time interval td.

24. Partially Polarized & Unpolarized Light To describe this type of polarization state, must consider time averaged quantities. S0 = <<Ax2+Ay2>> S1 = <<Ax2-Ay2>> S2 = 2<<AxAy cosd>> S3 = 2<<AxAy 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.

25. 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.

26. Stokes Parameters Example: Unpolarized light No preference between Ax and Ay (Ax=Ay), d random S0 = <<Ax2+Ay2>>=2<<Ax2>> S1= <<Ax2-Ay2>>=0 since d is a random function of time S2,3=2<<AxAy cosd>>=2<<AxAy sind>>=0 if S0 is normalized to 1, the Stokes vector parameter is for unpolarized light. Example: Horizontal Polarized Light Ay=0, Ax=1 S0=<<Ax2>>=1 S1=<<Ax2>>=1 S2,3=2<<AxAy cosd>>=2<<AxAy sind>>=0

27. Stokes Parameters S0 = <<Ax2+Ay2>> = 2<<Ax2>> S1 = <<Ax2-Ay2>> = 0 S2 = 2<<AxAy cos(-1/2p)>> = 0 S3 = 2<<AxAy sin(-1/2p)>> = -1 Example: Vertically polarized light Ay=1, Ax=0 S0 = <<Ax2+Ay2>>=<<Ay2>>=1 S1 = <<Ax2-Ay2>>=<<-Ay2>>=-1 S2,3 = 2<<AxAy cosd>>=2<<AxAy sind>>=0 Example: Right handed circular polarized light (d=-1/2p) Ax=Ay

28. Stokes Parameters S0 = <<Ax2+Ay2>> = 2<<Ax2>> S1 = <<Ax2-Ay2>> = 0 S2 = 2<<AxAy cos(1/2p)>> = 0 S3 = 2<<AxAy sin(1/2p)>> = 1 Example: Left handed circular polarized light (d=1/2p) Ax=Ay Degree of polarization: Unpolarized S12 = S22 = S32 = 0 Polarized S12+S22+S32 = 1 useful for describing partially polarized light

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

30. 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:

31. 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.

32. Jones Matrix Method (IV) • By combining equations, the transformation due to the retarder • plate is: where W0 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 G and its azimuth angle y, and is represented by:

33. Examples f f f Polarizer with transmission axis oriented  to X-axis Polarization State Jones Vector E Y-axis 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: a b a b

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

35. Wave Plates Jones Matrices y x c-axis Remember: c-axis c-axis 450 Ingeneral: y c-axis

36. Polarizers Jones Matrices y transmissionaxis x transmissionaxis Remember: transmissionaxis 450 transmissionaxis y Ingeneral:

37. Birefringent Plates 45 45 Parallel polarizers Cross polarizers

38. Poincare’s Representatives Method

39. Poincare’ Sphere: Linear Polarization States

40. Poincare’ Sphere: Elliptic Polarization States

41. Polarization Conversion:

43. Polarization Conversion: Y-axis f s y X-axis Z-axis 

44. TN LCD Formulations Some Examples

45. General Matrix For LCD e – component || director o – component director • Twist angle  Phase retardation

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

47. Adiabatic Waveguiding • Notice that for TN displays since f<<G (twist angle muchsmaller than retardation G): • Then the outputpolarization reduces to: which means that the electric field vector “follows” the nematicdirector as beam propagates through medium – it rotates –

48. 90ºTwisted 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:

49. Transmission of Normal Black first minimum second minimum third minimum

50. 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: