Chapter 1 Introduction
1.3 Aims of the research
LCOS devices have great potential for microdisplay applications ranging from data projectors, large-screen rear-projection televisions (RPTVs) to near-to-eye (NTE) virtual projections devices. The major advantage of the single-crystalline silicon is the high electron mobility, which allows very small pixel structures to be fabricated on the silicon substrate. However, as we have mentioned, the fringing-field effect and the diffraction
Figure 1.10 Sketche of the fringing fields at the pixel edges when adjacent pixels are operated at different voltages.
Figure 1.11 Sketche of the diffraction effect of LCOS devices.
effect are critical as the pixel pitch becomes too small. Efficiency can be decreased drastically due to diffraction loss. Fringing-field effects often cause poor image sharpness and degrade the dynamic response [21,22].
In this dissertation, we investigate the influences of fringing fields in different boundary conditions and distinct LC cell structures. Varying these parameters can give rise to different optical effects when light passes through the LC layer. Starting from understanding the LC director distributions under the influence of fringing fields, we are able to solve this long-standing problem. On the aspect of diffraction effect in high resolution LCOS devices, we developed an extended beam propagation method to simulate the optical performance of the devices when the pixel pitches become comparable to the visible wavelengths. By using this powerful tool, it is capable of designing a LCOS panel with least diffraction loss. In the followings, the basic theories and algorithm of simulations for LC director orientation and optical performances of LCOS devices are described in Chap. 2. Fringing-field effects in some commonly employed LC modes are investigated in Chap. 3. The influences of changing LC cell structure on fringing-field effect are also addressed in this chapter. In Chap. 4, we present a novel design called the circularly polarized light illuminated vertically aligned (CPVA) LCOS device which firstly eliminates the annoying fringing-field effect in vertically aligned (VA) LC mode. In Chap. 5, the analyses on diffraction effect of LCOS devices are studied by using BPM. A phase-compensated finger-on-plane mode is presented to save the light which is lost due to the strong diffraction in a conventional finger-on-plane LCOS device [22]. Finally, we summarize our work and make a brief conclusion. Suggestions are also given for future study.
References
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“Ultra-high-efficiency LC projector using a polarized light illuminating system.” Soc.
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[12] J. A. Shimizu, “Scrolling color LCOS for HDTV rear projection.” Soc. Information
Display Tech. Digest, pp. 1072 (2001).
[13] T. Yamazaki, M. Tokumi, T. Suzuki, S. Nakagaki and S. Shimizu, “The Single-Panel D-ILA Hologram Device for ILATM Projection TV.” Intl. Display Workshop, pp. 1077 (2000).
[14] M. L. Jepsen, M. J. Ammer, M. Bolotski, J. J. Drolet, A. Gupta, Y. Lai, D. Huffman, H.
Shi and C. Vieri, “High resolution LCOS microdisplay for single-, double- or triple-panel projection systems.” Displays 23, pp. 109 (2002).
[15] G. R. Fowles, Introduction to Modern Optics 2nd Edition (Holt, Rinehart, and Winston, Inc., 1975).
[16] S. M. MacNeille, “Beam Splitter.” U.S. patent 2,403,731 (1946).
[17] E. H. Stupp and M. S. Brennesholtz, Projection Displays (John Wiley & Sons, 1999).
[18] S. Arnold, E. Gardner, D. Hansen and R. Perkins, “An Improving Polarizing Beamsplitter LCOS Projection Display Based on Wire-Grid Polarizers.” Soc.
Information Display Tech. Digest, pp. 1282 (2001).
[19] M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant Birefringence Optics in Multilayer Polymer Mirrors.” Science 287, pp.2451 (2000).
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[21] S. Zhang, M. Lu, and K. H. Yang, Soc. Information Display Tech. Digest, pp.898 (2000).
[22] W. Y. Chou, C. H. Hsu, S. W. Chang, H. C. Chiang and T. Y. Ho, Jpn. J. Appl. Phys.
41, pp.7386 (2002).
Chapter 2
Theory and Numerical Simulation
2.1 Theories of deformations and optical properties of liquid crystals
To investigate the distributions of LC molecules under external fields, the continuum theory is often employed. It provides an insight of LC physic and was initiated by Oseen [1]
and Zocher [2] back in 1933. Afterward, Frank had reexamined the theory critically and presented the theory of curvature elasticity [3]. With regard to the optical properties of nematic LCs, the optical rotation and birefringence effects are often induced. These phenomena had been studied by many authors, the most clear, rigorous and accessible reference being that of de Vries [4].
2.1.1 Continuum theory of liquid crystals
Many important physical properties of LC can be explained by continuum theory disregarding the details of the structure on the molecular scale. When external fields are applied, the LC molecules are rotated according to the field strength; when the fields are withdrew, the LC molecules return back to their original alignments. This effect is similar to the elastic deformations in solids. Therefore, it can be predicted that the energy generated by the deformations of LC molecules has the same form as that described by Hooke’s law.
LCs are either rod-like or disk-like molecules. In macroscopic point of view, LC molecules tend to align in a common direction, which is called “director” n. The continuum theory describes the distortions of n under some external forces. A good starting point is to explore the minimum energy condition. At first, some assumptions have to be addressed according to the characteristics of nematic:
1. n is defined as a unit vector, i.e. n⋅n=1.
2. n and –n are indistinguishable, i.e. there is no permanent dipole in nematic.
3. All properties are identical when the coordinate is rotated around n, which is usually called C¶ symmetry.
4. The reflective symmetry of the plane including n. The properties of the system at two sides of the plane including n are the same.
Despite the influence of the surface energies, the free energy density glc due to the distortion of n can be expressed as:
l k j i ijkl j i
ij a
a , , ,
lc n n n
g = + , (2.1)
where aij and aijkl are the constant coefficients depend on the material, and i, j, k, l =x, y ,z which are the axes of a right-handed Cartesian coordinate system. This equation uses second-order approximation and omits the higher order terms. Based on the assumptions listed above, one can obtain a finally form of free energy density of a nematic as:
[
33 2]
2 22
2 11
lc ( ) ( ) ( )
2
g =1 k ∇⋅n +k n⋅∇×n +k n×∇×n , (2.2)
where kii, i=1, 2, 3 are the elastic constants associated with three basic types of LC deformations, i.e. splay, twist and bend. It is shown in this equation that g=0 when there is no deformation of n. there
When nematic is doped with some chiral material, the LC directors present a helical structure as shown in Fig. 2.1. P0 denotes the natural pitch of the structure which represents the distance along the helical axis for the directors to rotate 360°. A right-handed twist structure gives a positive P0 and the left-handed twist structure gives a negative P0. In this case, the forth assumption is not satisfied. The orientations of LC directors are varied along the z axis in the natural balanced state. Therefore, an additional term must be added in Eq.
(2.2) to interpret this phenomenon. The modified free energy density is given as:
⎥⎦
⎢ ⎤
⎣
⎡ ∇⋅ + ⋅∇× + + ×∇×
= 2 33 2
0 22
2 11
lc ( ) ( ) ( )
2
g 1 2 n n
n n
n k
k p
k π
. (2.3)
n
The effect of static electric field
The influence of static electric field on nematic LC can be explored by understanding the anisotropy of its dielectric constant. Usually, the dielectric constant ε can be expressed as a tensor: dielectric constant measured along n and ε⊥ is the dielectric constant measured perpendicular to n. When external electric fields are applied on liquid crystal, the electric displacement D can is given as:
) ( ε
ε E n E
D= ⊥ + a ⋅ . (2.5)
Hence, the electric free energy ge can be expressed as:
( )
Since the first term in Eq (2.6) is not related to the director distribution and introduce a constant value for the energy of the system, it can be ignored when deriving the minimum
Figure 2.1 Structure of chiral nematic liquid crystals. P0 is the natural pitch.
energy condition. Therefore, the free energy density with external electric field can be expressed as:
( )
Note that this equation is under the assumption of strong boundary condition. For weak boundary condition, a surface energy term must be included. There are several various approaches [5-9], the details are beyond the scope of this thesis and will not be addressed.Euler-Lagrange equation
To determine the director distribution, we have to minimize the energy generated by the LC deformations. For simplicity, we suppose that all parameters in Eq. (2.7) depend on z-direction only. Therefore, the total free energy is given as:
( )
where i=x, y ,z and d denotes the thickness of the LC layer. This is a function related to the integral path. Our purpose is to find a path, ni(z), from a to b which gives the minimum integral value Gm. In this case, an arbitrary path, ni(z)+αδni(z), from a to b gives the value0
The second term in the integrand can be integrated by parts:
dz
The integrated term vanishes because δni(a)= δni(b)=0. Therefore, Eq. (2.11) becomes:
Because δni is an arbitrary function, the integrand in Eq. (2.13) must vanish itself:
0
This result is known as Euler-Lagrange equation which is commonly used for solving the static problems of LC director distribution.
2.1.2 Optical properties of an ideal helix: de Vries theory
As we mentioned previously, the LC directors present a helical structure when molecules possessing chirality are doped. The same structure is also found in pure cholesteric esters.
Therefore, this helical structure is termed cholesteric. The general optical properties of an ideal cholesteric LCs (CLCs) contains many effects occurred in common LC devices [10,11]. The optical behaviors of light propagating in some commonly used LC modes, e.g.
twisted nematic (TN), super-twisted nematic (STN) and homogenous, etc., are only the special cases of that in CLCs.
Let us recall the structure shown in Fig. 2.1. In this specific case, n is rotated with a helix axis, i.e. the z-axis. We can describe n in this ideal state as:
constant z
q 0 n
sin n
cos n
0 z y x
+
=
=
=
=
φ
φ φ
(2.15)
where q0=2π/P0. In this configuration, n is horizontal everywhere and it is usually called the
“planar texture”. To analyze the optical properties of CLCs, the local dielectric constant at any point r must be given at first:
), ( )n ( )n ε ε ( δ ε ) (
εij r = ⊥ ij + // + ⊥ i r j r (2.16)
where i, j=x, y, z. It is noteworthy that although this equation looks the same as Eq. (2.5), the physical meanings are different. Eq. (2.16) accounts for the optical behaviors of light propagating inside a uniaxial material and Eq. (2.5) accounts for the anisotropy of electric properties with external fields. In Eq. (2.16), the dielectric constants can be related associated with the refraction indices as:
, n ε
ε⊥ = 0 2o ε// =ε0n2e (2.17)
where ε0 is the permittivity of vacuum, no is the ordinary refractive index of LC and ne is the extraordinary refractive index of LC. When an electromagnetic wave of frequency (ω)
propagates along the helix axis (i.e. the z-axis), the field components are:
where t represents time and Re stands for the real part. The electric field vectors must satisfy the well-known wave equation:
⎟⎟⎠
The wave equation can be simplified by using circularly polarized light:
y
Hence, Eq. (2.19) becomes:
⎟⎟⎠
Solving Eq. (2.22) give the following forms of propagating waves:
)z
where a and b are constants. Substitute Eq. (2.24) into Eq. (2.22) gives the relations between the a and b:
( )
The non-trivial solution of this equation is obtained only when the determinant vanishes:
0
This equation gives the dispersion relation of CLCs, i.e. the relation between frequency ω and wavenumber l. Figure 2.2 (a) shows a typical dispersion relation of CLCs with ne=1.5578, no=1.4748 and P0=0.25 µm. It is shown that the curve is separated into two branches by a band gap. The range of frequency of this band gap can be determined by setting l=0 in Eq. (2.26):
Using Eq. (2.23), the boundaries of the band gap can be given as:
e
where ω+(0) corresponds to the upper branch and ω-(0) corresponds to the lower branch.
For a incident wave with fixed ω, we can have four eigenmodes. Two propagate along +z direction (transmissive modes) and two propagate along –z direction (reflective modes).
Here we focus on the transmissive modes, i. e. ∂ω/∂l>0.
Figure 2.2 (a)The dispersion relation of a CLC with ne=1.5578, no=1.4748 and P0=0.25 µm; the insert denotes the characteristics of eigenmodes at some specific frequencies; (b) Calculated ellipticity with respect to l2/q0 when –1<l2/q0<0; (c) Calculated ellipticity with respect to l2/q0 when 0< l2/q0<103 ; (d) Calculated ellipticity with respect to l1/q0 when 0< l1/q0<600.
The reflective modes have similar behaviors. The wavenumber of the modes can be derived
The polarization states of the eigenmodes can be deduced using the definition of ellipticity:
.
according to Eq. (2.25), we have
2
Therefore, Eq. (2.30) becomes
2
Using Eq. (2.33), we calculate the frequency-dependent ellipticities of the eigenmodes as shown in Fig. 2.2 (b)-(d). They can be roughly divided into three regimes:
1. Elliptical Regime: ω->ω≥0 and 2q0c/∆n>>ω>ω+. There are two elliptical regimes separated by the band gap of Bragg reflection. The eigenmodes are elliptically polarized with opposite sense of handness. For ω=0, the ellipticities become ±1, which indicates the eigenmodes become circularly polarized in this particular case , i. e. denoted by A1
and A2 in Fig. 2.2 (a).
2. Bragg Regime: ω+>ω>ω-. In this regime, l2 is imaginary which indicates that the corresponding eigenmode becomes an evanescent wave. On the other hand, the eigenmode correspond to l1 has its ellipticity close to -1. Thus, the transmitted wave is almost (but not really) left-handed circularly polarized. Recall that P0=0.25 µm>0 corresponds to a right-handed helix. Hence, it can be concluded that the transmitted wave is almost circularly polarized with opposite rotational sense to that of the helix structure, and the reflective wave will have its rotational sense identical to that of the helix structure. For an extreme case when ne=no, the band gap shrinks to one frequency ω=cq0/no. The corresponding wavelength λ= noP0 is in agreement with the usual Bragg reflection condition. For the evanescent wave, the wavenumber can be expressed as:
( )
From Eq. (2.24), we have the field vector written as:
.
where a and b can be imaginary which are then expressed as real numbers, A and B, multiplied by phasors, exp(iφa) and exp(iφb). Based on Eq. (2.25), we have
Therefore, we can derive the relationship between A and B, that is
b
When |A|=|B|, the light is linearly polarized. Otherwise, the light is in general
elliptically polarized. The angle between the long axis of polarization ellipse and the local director is (φa-φb)/2. From Eq. (2.38), the angle can be derived as:
]. κ k q [
κq ) 2
tan( 2 2
0 2 0
0
−
= −
− a
b φ
φ (2.39)
Figure 2.3 shows the calculate B/A and φb-φa with respect to frequency. As shown in the Fig., the evanescent wave is linearly polarized. The vibration direction of electric field is varied from parallel to perpendicular to the local director n when changing the frequency from ω- to ω+.
3. Maugauin Regime: ω>>2q0c/∆n. In this high frequency regime, the eigenmodes have their ellipticities either near-zero or near-infinity, which means they are almost linear polarized parallel or perpendicular to the local LC director. As shown in the insert diagram of Fig. 2.2 (a), E1 and E2 indicate the case when ω=500q0c, the ellipticities become -0.024 and 42.693.
Table 2.1 summarizes the properties of eigenmodes in each regimes.
Fig. 2.3 The calculate (a) B/A and (b) φb-φa with respect to frequency.
Table 2.1 The properties of eigenmodes in each regimes.
2.2 Numerical Simulations
Numerical simulation is often utilized to analyze the performance of liquid crystal displays. To calculate the distribution of LC director, one can use Eq. (2.14) for searching the minimum energy condition once the initial and boundary conditions are given. In this case, the free energy density is expressed in vector-form as Eq. (2.7). Some literatures expressed the free energy density in tensor-form to maintain the equivalence of n and -n [12-14]. However, simulations using tensor-form was reported to cause transformations between topologically inequivalent states [15]. In view of this, the vector form is adopted in our research.
Two numerical methods are often employed, i.e. finite difference method [16] and finite element method [17]. In two dimensional (2D) simulations, finite difference method splits the simulated area into grids, which can be mismatched with boundaries of the area. This problem can be solved by using finite element method, which uses polygons to fit in the shapes of the area. In our simulations, a commercial software, 2dimMOS (from autronic-MELCHERS GmbH) [18], which uses finite element method is employed to calculate the director profile at first. The optical performance is then calculated by Jones matrix method or beam propagation method as will be introduced in the followings. The latter one includes the diffraction effect which may cause serious problem in LCOS projection system.
2.2.1 Jones matrix method
The most commonly used method to simulate the optical performance is the Jones matrix method [11,19]. Consider a plane wave propagates in a homogenous medium, its electric field is a function of space r and time t, which can be written as:
( )
r =A ( −k⋅r+ϕ)E ,t eiωt (2.40)
where ω is the angular frequency, k is the wavevector, A is a constant vector and ϕ is the initial phase. Due to the nature of transverse waves, the electric field vector is always perpendicular to the direction of propagation. For a wave propagates along the z-direction, the field vectors can be written as:
( )
The phase difference between the x and y components are:
(
y k2z) (
x k1z)
'y 'xδ= ϕ − − ϕ − =ϕ −ϕ (2.42)
If we neglect the common parts related to time in Eq. (2.41), the field vectors can be expressed as a column vector named Jones vector:
⎟⎟⎠
Note that Jones vector is not a vector in real physical space; rather it is a vector in an abstract mathematical space which denotes the polarization state of light. Table 2.2 lists various polarization states represented by Jones vector.
Polarization State Jones Vector
φ x
Polarization State Jones Vector
φ x
Table. 2.2 Jones vectors corresponding to various polarization states.
When light propagates in a birefringence material, the field vector can be given as:
where k0 represents the wavenumber in vacuum and d is the thickness of the material. Eq.
(2.44) can be written in the matrix form as:
⎟⎟⎠
This is so-called the Jones matrix method. The Jones matrix, M, is a 2×2 transformation matrix, it characterizes the modulation of polarization state of a normally incident light.
When light propagates through a twisted nematic (TN) liquid crystal layer with twist angle φt and phase difference Γ=k0d(neff-no). In this condition, the local optical axis is a function of position. The represented Jones matrix can be derived by subdividing the LC layer into a large number, N, of thin layers as shown in Fig. 2.4. Each layer can be treated as a homogeneous wave plate. Assume the variations between the thin layers are linear, i.e.
each sub-layer has a phase retardation of Γ/N and is oriented at azimuthal angle of ρ, 2ρ, 3ρ, … (N-1)ρ, Nρ with ρ=φt/N. Under this assumption, the Jones matrix of the nth sub-layer is:
( )
R( )
nρwhere R represents the rotational matrix:
( )
⎟⎟By multiplying all the sub-layer matrices together, we obtain
( ) ( )
When N tends to infinite (N→∞), by using Chebyshev’s identity [20], the above equation can be further simplified to be
( )
When the LC layer is sandwiched by two crossed polarizers, the output electric field can be given as:
where β is the angle between the transmission axis of linear polarizer and the entrance LC director. Finally, the normalized transmittance can be calculated as T =|E |2.
Fig. 2.4The represented Jones matrix can be derived by subdividing the LC layer into a large number, N, of thin layers.
For reflective LC device, the incident light passes through the LC layer twice. The director distribution encountered by reflective waves is the mirror image of the original one.
For example, in Fig. 2.5, a reflective LC device with left-handed twist structure is equivalent to a tranmissive LC layer with right-handed twist structure plus a tranmissive LC layer with left-handed twist structure. The normalized reflectance can be expressed as [21]:
In numerical calculations, the LC layer is divided into slabs. Each slab can be treated as an uniform TN cell. For LCOS system, PBS is employed to meet the crossed-polarizer
In numerical calculations, the LC layer is divided into slabs. Each slab can be treated as an uniform TN cell. For LCOS system, PBS is employed to meet the crossed-polarizer