1. 1 Optics of LC displays

2. 2 Chap.2 Polarization of optical waves

3. 3 A monochromatic plane wave is For mathematical simplicity Only the real part represents the actual electric field. The polarization state of a light beam is specified by its electric field. –Consider a light advances along z axis, then its electric field vectors must lie in xy plane.

4. 4 Linear polarization state Consider the time evolution of the electric filed vector at the origin z=0. Define relative phase as Linear polarized, or plane polarized if The vibration of the electric field vector is confined in a specific plane.

5. 5 Now examine the space evolution of the electric field vector at a fixed point in time (say t=0) Linear polarized, or plane polarized if Linear polarization is most widely used in optics.

6. 6 Circular polarization state A beam of light is said to be circularly polarized if the electric field vector undergoes uniform rotation in the xy plane.

7. 7 The notation is different from Hecht.

8. 8 2.3 Jones vector representation The plane wave is expressed in terms of its complex amplitude as a column vector [ignore time domain] To obtain the real x component of the electric field, we must perform the operation.

9. 9 If we are interested only in the polarization state of wave, we will use the normalized Jones vector that satisfy the condition. Thus, a linearly polarized light with electric field oscillating along a given direction is x y

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11. 11 右 左 Hecht representation Yeh representation

12. 12 A powerful technique. In LCDs, the orientation of the director depends on the applied voltage and the boundary condition. The polarization state of the input light beam can be converted to any other polarization state by means of a suitable retardation plate, or LC plate. Assume there is no reflection of light from either surface of the plate 4. Jones Matrix Method

13. 13 Two coherent P-states in the principal coordinate system, some how caused to have a phase lag. 光程差 相位差

14. 14 Incident linear polarized light described by Jones vector V x, V y are complex numbers, x and y axes are fixed lab. axes. Ψis the angle between the s-f coordinate and the x-y coordinate, with the z axis as the axis of the coordinate rotation. Slow axis (LC director, C axis, n e axis) Fast axis Lab axis

15. 15 Decompose the light into the linear combination of the fast and slow normal modes in the LC cell. This is done by coordinate transformation. After entering the medium, due to the difference in phase velocity, one component is retarded relative to the other. The retardation changes the polarization state of the emerging beam.

16. 16 The polarization state of a light can be converted to any other polarization state by suitable LC plate.

17. 17 The polarization state of the emerging beam in the medium s-f coordinate system is

18. 18 The mean absolute phase change is defined as Define the phase retardation which is the relative phase change due to propagation, not the absolute change.

19. 19 The birefringence of a typical retardation is small, that is the absolute change >> the relative phase retardation For LCs, n s ~1.74, n f ~1.52 n s -n f ~ 0.22

20. 20 The polarization state of the emerging beam can be rewritten as The polarization state of the emerging beam in the x-y coordinate is by transforming back from the s-f coordinate system

21. 21 Finally, the polarization state of the emerging light in the lab axes is R(  ) is the coordinate rotation matrix and W 0 is the Jones matrix for the retardation plate (LC plate) in the s-f coordinate The phase factor e -i  can be neglected if interference effects due to multiple reflection are not important, or not observable.

22. 22 The Jones matrix of a retardation plate is characterized by its phase retardation  and its azimuth angle , and is represented by the product of three matrices

23. 23 Coordinate transformation is included.

24. 24 Neglecting the absolute phase  ‘, the matrix representations of the polarizer’s transmitting axis parallel to the x or y axes, is The Jones matrix of a linear polarizer orientated with transmission axis parallel to the lab x axis  ‘ is the absolute phase

25. 25 The transmission axis of a polarizer has an angle  with respect to lab x axis is

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27. 27 Example: a half wave retardation plate Incident light, vertically polarized, Jones matrix for the half wave plate The result is horizontal polarized light Azimuthal angle  =45°

28. 28 Regardless of the azimuth angle

29. 29 Example: a quarter wave retardation plate Incident light, vertically polarized Jones matrix for the quarter wave plate The result is Left handed circularly polarized light Azimuthal angle  =45°

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31. 31 Intensity transmission spectrum Incident light Emerging light

32. 32 A birefringent plate between parallel polarizers LC plate VV’ y-axis x-axis

33. 33 The corresponding Jones matrix is The incident light be unpolarized, thus when it passes through the first polarizer Assume the intensity of the incident light is unity and only half of the intensity passes through the polarizer

34. 34 The emerging beam The transmitted light is y polarized, because the light goes through a polarizer. analyzer Birefringent plate After pass the 1st polarizer

35. 35 A birefringent plate between crossed polarizers The transmitted light is x polarized

36. 36 Parallel Aligned Cells y x

37. 37 A birefringent plate between a pair of polarizers The analyzer forms an angle  with x axis Homework

38. 38 Optical properties of a twisted nematic liquid crystal (TN-LC) The orientation of the director is a function of position. Thus, the director is twisted. Subdivide the medium into a large number of thin plates. Each of the thin plate is approximated by a homogeneous medium. Assume the twisting is linear and the azimuth angle is  is a constant

39. 39 Twisted Nematic Liquid Crystal Displays: Normally White (e-mode) Twisted Nematic Liquid Crystal Displays: Normally White (e-mode)

40. 40 Γis the phase retardation of the plate when it is untwisted. The director is parallel to the surface plate d is cell thickness The total twist angle is Divide the cell into N equally thin plate. Each plate has a phase retardation of  =  /N. The plate are orientated at a azimuth angle , 2 , 3 , ……(N-1) , N , with  =  /N. The overall Jones matrix for these N plates is given by R is the coordinate rotation matrix and W m is the Jones matrix for the mth plate, W 0 is

41. 41 Using  =  /N and R(  1)R(  2)=R(  1+  2), the overall Jones matrix can be written as Finally, the overall Jones matrix becomes

42. 42 Further simplified and in the limit when N  ∞ M is the exact expression for the Jones matrix of a linearly twisted nematic LC plate. V: initial polarization, V’: polarization state after exiting the plate  V’=MV, and T=(V’/V) 2 =M 2,

43. 43 It is often useful to examine the polarization states in the local principal coordinate system. In the local principal system, the e component is along the direction of the director, and the o component is perpendicular to the director. The results can be written as

44. 44 Adiabatic following (waveguiding in TN_LCD)

45. 45 The polarization of the input light is parallel to c axis (the director) of LCs in the entrance plane  E-mode operation In the principal coordinate system, the Jones vector of the input beam is

46. 46 In the principle coordinate system, the Jones vector of the output beam is (4-3-15)

47. 47 In TN_LC, if φ << Γ – Ex: LC cell of E7 with 20  m, twist angle  /2.  n=0.23. –  / Γ =1/37 at =500nm. The 2nd term (o-component) in eq. (4.3-15) is near zero, thus, the output Jones vectors is approximated as

48. 48 The electric field vector of the beam remain parallel to the local director as the beam propagates in the twisted nematic LC medium. –The same in the O-mode operation. –Adiabatic following or waveguiding phenomenon. Adiabatic following occurs if the incident beam is polarized either parallel or perpendicular to the director at the entrance plane, and the twist rate is small. Strictly, the polarization state in the medium is elliptical as shown in the figure.

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50. 50 90º twisted nematic liquid crystal-1 90º twisted nematic liquid crystal sandwiched between a pair of parallel polarizers. –So called normally black(NB) configuration. We consider the e-mode operation, thus the input Jones vector is

51. 51 The transmission axis of the second polarizer is perpendicular to the local director at the exit plane. According to eq(4-3-15), the output intensity is Where ψ is the twist angle of the cell, u is the so-called Mauguin parameter The transmission is near 0 for LCs with ψ<< 

52. 52 90º twisted nematic liquid crystal-2 90º twisted nematic liquid crystal sandwiched between a pair of parallel polarizers. –So called normally black(NB) configuration. We consider the o-mode operation, thus the input Jones vector is

53. 53 The output polarization state in terms of the Jones vectors in the local principal coordinate is Again, the transmission is near 0 for LCs with ψ<<  0

54. 54 Transmission properties of a general TN-LCD  

55. 55 The input and the output polarization, as determines by the orientation angles of the polarizer transmission axes, are The transmission of the system, according to Jones matrix is V and V’ are the input and output Jones vectors

56. 56 After some steps In various forms, define

57. 57 The transmission become Also, it can be written as This expression is particular useful in the design of TN- LCDs or STN-LCDs

58. 58 4.4 phase retardation at oblique incidence In general, two independent modes of propagation given a direction of propagation in an anisotropic medium. –Ordinary and extraordinary (mutual orthogonal). –O-mode is independent of the propagation direction, but e- mode depends on the incident direction. –Thus, phase retardation depends on the incident direction. Consider an a plate, c-axis (director) parallel to the surface plate.  homogeneous LC cell. –For normally incident, the phase retardation is: –A general expression is –Such general expression can be derived as follows.

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60. 60 1. Wave approach Given an arbitrary incident plane wave, both ordinary and extraordinary waves are generated in the medium. The electric field amplitude can be written as –Tangential components ( ,  ) are the same due to continuity condition at the boundary. –Thus, the phase difference between the two modes is

61. 61 2. Ray approach Referring to figure 4.10, the phase retardation is Using the continuity condition (snell’s law) and simple trigonometry. We obtain Which is exactly

62. 62 4.5 conoscopy A conoscopy is a polarizing microscope designed to examine the transmission property of a sample plate at various directions of incidence simultaneously. For uniaxial media, a conoscope can be used to determine the orientation of the c- axis (director).

63. 63 Employing a convergent or divergent light as the source, which consists of many plane waves over a cone of solid angle. A lens converts each component of the wave into a point at the focal plane. The transmission sandwiched between polarizers depends on the sample orientations and the angle of incidence.

64. 64 4.5.1 a plate of uniaxial crystals For small incident angle  <<1 For small birefringence, Γ 0 : phase retardation at normal incidence.

65. 65 If the phase retardation is 2  m (full wave), the emerge beam at the exit face have the same polarization as that at the entrance face. This beam will be blocked by the second polarizer and leading to a dark fringe at the focal plane.

66. 66 4.5.2 c plate of uniaxial crystals At normal incidence, the wave propagating along the c-axis. Both components of the wave are propagating at exactly the same phase velocity, the polarization state remain unchanged, the transmission is zero. At a general incident angle, the phase retardation is introduced between the two modes. Since the symmetry of the structure, the phase retardation is a function of , forming a series of dark and bright rings. Provided  <<1

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