1. Review of Basic Polarization Optics for LCDs Module 4

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

3. 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. Polarization of Optical Waves

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

5. 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: independent amplitudes two independent phases Monochromatic Plane Wave (II) Y-axis X-axis ExEx EYEY

6. 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 Monochromatic Plane Wave (III)

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

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

9. Circular Polarization Equation of a circle

10. 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. Elliptic Polarization States Y-axis X-axis AxAx AYAY Important parameters: 1.Orientation 2.Handedness 3.Extent of Ellipticity Occurs for all values of a b

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

12.  =3  /4  =  /2  =  /4  =0  =  /4  =  /2  =3  /4 ==  =  /2  =  /4  =0  =  /4  =  /2  =3  /4 ==

13. Review Complex Numbers -2+2i 3-4i Im Re  = 3 – 4i  = e i  = cos  + i.sin   = e -i  = cos (-  ) + i.sin (-  ) = cos  - i.sin  Remember the identities: e x e y = e x+y e x / e y = e x-y d/dz e z = e z

14. 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: Complex Number Representation Im Re on x axis on y (imaginary axis)

15. 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

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. When  =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 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.

19. Polarization EllipseJones Vector (  )(  ) Stokes Polarization Representation

20. Polarization EllipseJones Vector (  )(  ) Stokes Polarization Representation

21. 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) Jones Matrix Limitations

22. We discussed monochromatic/polarization thus far. If light is not absolutely monochromatic, the amplitude and relative phase  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 10 14 s -1 Whereas polarization may change 10 -  s (depending on source) Partially Polarized & Unpolarized Light

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

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

25. 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

26. Example: Unpolarized light No preference between Ax and Ay (Ax=Ay),  random S 0 = >=2 > S 1 = >=  S 2,3 =2 >=2 >=  since  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 = >=1 S 1 = >=1 S 2,3 =2 >=2 >=  Stokes Parameters

27. Example: Vertically polarized light Ay=1, Ax=0 S 0 = >= >=1 S 1 = >= >=-1 S 2,3 = 2 >=2 >=  Example: Right handed circular polarized light (  =-1/2  ) Ax=Ay S 2 = 2 > =  S 0 = > = 2 > S 1 = > =  S 3 = 2 > = -1 Stokes Parameters

28. Example: Left handed circular polarized light (  =1/2  ) Ax=Ay S 2 = 2 > =  S 0 = > = 2 > S 1 = > =  S 3 = 2 > = 1 Degree of polarization: Unpolarized S 1 2 = S 2 2 = S 3 2 =  Polarized S 1 2 +S 2 2 +S 3 2 = 1 useful for describing partially polarized light Stokes Parameters

29. Jones Matrix Method (I) Z-axis Y-axis s X-axis f   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 n s and n f 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  is the wavelength

30. For a “simple” retardation film, the following phase changes occur: (relative phase retardation) (mean absolute phase change) Jones Matrix Method (II) 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. By combining equations, the transformation due to the retarder plate is: where W 0 is the Jones matrix for the retarder plate and R (  ) 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 , and is represented by: Jones Matrix Method (IV)

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

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

35. y x c-axis 45 0  Jones MatricesWave Plates Remember: In general:

36. y x 45 0  transmission axis Jones MatricesPolarizers transmission axis In general: Remember:

37. Birefringent Plates 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. Z-axis Y-axis s X-axis f  

44. Some Examples TN LCD Formulations

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

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

47. Notice that for TN displays since  <<  (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 – Adiabatic Waveguiding

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

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

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

51. Normal White Mode (II)

52. E n (n) n n nn n n n n n E E E E E E E E E E Y-axis X-axis

53. z d A F B D C  ee oo Phase Retardation at Oblique Incidence: Complicating Matters

54. 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 4x4 Methods Summary of Optics