Electromagnetic Formalisms for Optical Propagation in Three-Dimensional Periodic Liquid-Crystal Microstructures

Following the targeted word direction of Opto- and Nanoelectronics, the field of science and technology related to the development of new display technology and organic materials based on liquid crystals ones is meeting the task of replacing volume inorganic electro-optical matrices and devices. An important way in this direction is the study of promising photorefractive materials, conducting coatings, alignment layers, as well as electric schemes that allow the control of liquid crystal mesophase with good advantage. This book includes advanced and revised contributions and covers theoretical modeling for optoelectronics and nonlinear optics, as well as includes experimental methods, new schemes, new approach and explanation which extends the display technology for laser, semiconductor device technology, medicine, biotechnology, etc. The advanced idea, approach, and information described here will be fruitful for the readers to find a sustainable solution in a fundamental study and in the industry.


Introduction
Nanoscale structures have achieved novel functions in liquid crystal devices such as liquid crystal displays, optical filters, optical modulators, phase conjugated systems, optical attenuators, beam amplifiers, tunable lasers, holographic data storage and even as parts for optical logic systems over the last decades (Blinov et al. (2006;2007); Sutkowski et al. (2006)). Many theoretical works also have been reported on liquid crystal (LC) optics. Jones method (Jones (1941)) is first proposed for an easy calculation, which stratifies the media along the cell normal while remains the transverse LC orientation uniform, and hence supplies a straightforward way to analyze the forward propagation at normal incidence. This was later followed by the extended Jones method (Lien (1997)), which allows to trace the forward waves at an oblique incidence. The Berreman method (Berreman (1972)) then provides an alternative process to include forward and backward waves. A further step in LC optics is to consider rigorously the LC variation both along the cell normal and along a single transverse direction, leading to a two-dimensional treatment of light propagation. This step is fulfilled by implementing the finite-difference time-domain method (Kriezis et al. (2000a); Witzigmann et al. (1998)), the vector beam propagation method (Kriezisa & Elston (1999 ); Kriezis & Elston (2000b)), coupled-wave theory (Galatola et al. (1994); Rokushima & Yamakita (1983)), and an extension of the Berreman approach (Zhang & Sheng (2003)), and has proven to be successful in demonstrating the strong scattering and diffractive effects on the structures with transverse LC variation lasting over the optical-wavelength scale. For three-dimensional LC medium with arbitrary normal and transverse LC variations, Kriezis et al. (2002) proposed a composite scheme based on the finite-difference time-domain method and the plane-wave expansion method to evaluate the light propagation in periodic liquid-crystal microstructures. Olivero & Oldano (2003) applied numerical calculations by a standard spectral method and the finite-difference frequency-domain method for electromagnetic propagation in LC cells. Glytsis & Gaylord (1987) gave three-dimensional coupled-wave diffraction algorithms via the field decomposition into ordinary and extraordinary waves, although the transverse variation of the ordinary/extraordinary axis raises the complexity. Alternatively, this work neglects the multiple reflections and gives a coupling-matrix algorithm that is much easier to manipulate algebraically for three-dimensional LC media, yet accounts for the effects of the Fresnel refraction and 4 www.intechopen.com the single reflection at the surfaces of the media. The detailed derivations are described in appendix A. Furthermore, analogous with the Berreman approach (Berreman (1972)) to consider the multiple reflections for one-dimensional layered media (i.e. stratifying the media along the cell normal while remaining the transverse LC orientation uniform), another supplementary formulae including the influences of multiple reflections for three-dimensional media (i.e. stratifying the media along the cell normal and simultaneously including the varying LC orientation along the transverse) are also addressed in the appendix A. The program code of wolfram mathematica for coupling-matrix method is appended in appendix B for references. In this section, extended Jones matrix method is revisited first due to its similar underlying concepts can supply an accessibility to understand the coupling-matrix method. In the extended Jones matrix method, the liquid crystal cell (Figure 1(a)) is decomposed into multiple one-dimensional (z) independent stripes (Figure 1(c)), treating the transverse LC orientation uniform within each stripes and being irrelevant each other. Each stripe is further divided into N layers along the z direction, including two separate polarizer and analyzer layers. In the layer, there are four eigen-mode waves: two transmitted and two reflected waves; while at the interface of the layer, the boundary condition is that the tangential components of the electric field are continuous. Without loss of generality, considering the propagation of waves in the xz plane at angle angle θ related to z axis, it specifies k =( k 0 sinθ,0,k 0 cosθ),e x t e n d e d Jones Matrix can relate the electric fields at the bottom of the ℓ th layer to the fields at the top of the ℓ th layer of each strip by:

Extended Jones matrix method revisited
Here, k 0 = ω/c = 2π/λ with λ the wavelength of the incident light in free space. dz ℓ is the thickness of the the ℓ th layer. θ o and φ o are the orientation angles of the LC director defined in the spherical coordinate. ε i,j∈{x,y,z} is the dielectric tensors defined in appendix A. Equation (1) can be understood as follow. A −1 ℓ transforms the electric fields at the bottom of the ℓ th layer into the eigen-mode fields. Ξ ℓ then propagates the eigen-mode fields from the bottom of the ℓ th layer to the top of the ℓ th layer through the distance dz ℓ . Finally, A ℓ transform the eigen-mode fields at the top of the the ℓ th layer back into the electric fields at the top of the ℓ th layer, which is equal to the electric fields at the bottom of the (ℓ + 1) th layer by boundary condition. Grouping all layers, the extended Jones matrix formula that relates the incident electric fields (ℓ = 0) and the emitted electric fields (ℓ = N + 1) is given by Electromagnetic Formalisms for Optical Propagation in Three-Dimensional Periodic Liquid-Crystal Microstructures www.intechopen.com with θ p = sin −1 (sin θ/ℜ n p ) in which ℜ n p stands for the average of the real parts of the two indices of refraction (n e and n o ) of the polarizer. The total transmission for the stripe is calculated by The total transmission of the three-dimensional LC media then can be evaluated by summing up the contributions from the individual stripe.

Coupling matrix method
Parallel to the equation (9) by one-dimensional treatments for strips, an analogous coupling-matrix formulae for the propagations of waves through the three-dimensional periodic microstructures can be given as: Here, E + q,ℓ and M + q,ℓ ( E − q,ℓ and M − q,ℓ ) represent the physical forward (backward) TE and TM fields, i.e. transverse electric and transverse magnetic fields corresponding to the planes of the diffraction waves in the incident (ℓ = 0) and emitted (ℓ = N + 1) regions. In which the components of the vectors E + q,ℓ , M + q,ℓ , E − q,ℓ ,o r M − q,ℓ define the diffraction waves along the direction n gh = n xgî + n yhĵ + ξ ghk : with ε I = n 2 I (ε E = n 2 E ) being the dielectric coefficient in the incident (emitted) region. Note that the components with imaginary ξ gh values are ignored for studied cases due to the decaying natures along the electromagnetic propagations parallel to the z direction. Λ x (Λ y ) is the periodicity of the LC structure along the x (y)d i r e c t i o n . S ℓ∈{1∼N} is the matrix representing the propagations of waves through the ℓ th structured layer. It consists of the matrix T (a) ℓ , which is the (column) eigen-vector matrix of the characteristic matrix G ℓ for the ℓ th layer , and the diagonal matrix exp iκ (a) ℓ dz ℓ relates to the eigen-value κ (a) ℓ of G ℓ with dimensionless dz ℓ = dz ℓ k 0 : (18) In this context, the notation E (or M) denotes the N g N h × 1 vector with components E gh (or M gh ) describing the wave along n gh .ñ x (ñ y )a r eN g N h × N g N h diagonal matrices with N g N h diagonal elements n xg (n yh ) being the same (g, h) sequence as that of E and M,a n d are calculated by Equations (14-15).ε ij∈{x,y,z} are N g N h × N g N h matrices with elements ε ij,αβ being the Fourier transform of the spatial dielectric coefficients ε ij (x, y; z),i nw h i c ht h e indexes α, β are arranged by the relation Above N g(h) define the number of considered total Fourier orders g (h)i nt h ex (y) direction. 1 represents the N g N h × N g N h identity matrix. One may understand the Equation (17) for the ℓ th layer by the similar way as described in extended Jones method: the (T (a) ℓ ) −1 term represents the coordinate transformation from the spatial tangential components of fields fˆt ,ℓ =[ e x,ℓ h y,ℓ e y,ℓ h x,ℓ ] t denoted by Equations (46)-(47) at ℓ th interface into the orthogonal components of the eigen-modes in the ℓ th layer; the exp iκ a ℓ dz ℓ term describes eigen-mode propagation over the distance dz ℓ (thickness of the ℓ th layer); the T (a) ℓ term then is the inversely coordinate transformation from the eigen-mode components back to the spatial tangential components of fields at the next interface. Considering the continuum of tangential fields on interfaces, these fields emitted from the ℓ th layer hence can be straightforwardly treated as the incident fields fˆt ,ℓ+1 for the (ℓ + 1) th layer, and allow to follow the next transfer matrix S ℓ+1 to describe the sequential propagations of fields through the (ℓ + 1) th layer as in Equation (13). For the matrices S ent and S ext defined for the (isotropic) uniform incident (ℓ = 0) and emitted (ℓ = N + 1) regions, respectively, the eigen-modes are specially chosen (and symbolized) as E + q and M + q ( E − q and M − q ) (Ho et al. (2011); Rokushima & Yamakita (1983)), representing the physical forward (backward) TE and TM waves as the above-mentioned. In which the transform matrix T (i) ε I between the eigen-mode components and the tangential components fˆt ,0 =[ e x,0 h y,0 e y,0 h x,0 ] t for the isotropic incident region (ℓ = 0) is given as: Here,ṅ y andṅ x are N g N h × N g N h diagonal matrices with normalized elements n yh m gh and n xg m gh respectively. ξ −1 is the diagonal matrix with elements 1/ξ gh (not the inverse of the matrix ξ), in which m gh =(n yh n yh + n xg n xg ) 1/2 , ξ gh =(ε I − n yh n yh − n xg n xg ) 1/2 ,andε I = n 2 I have been applied for the incident region. A similar transform for fˆt ,N+1 in the emitted region can be derived straightforwardly by replacing all the ε I in Equation (19) with ε E and can be denoted w it hξ gh =( ε E − n yh n yh − n xg n xg ) 1/2 ,a n d ε E = n 2 E . S ent is the matrix representing the light propagation from the incident region into the medium, and indicates the essential refraction and the reflection at the first interface of the medium. To consider these effects in a simple way, a virtual (isotropic) uniform layer, which has zero thickness and effective dielectric coefficient ε a = n 2 avg ,e . g . n avg =( n e + n o )/2 for the liquid-crystal grating, is assumed to exist between the incident region and the 1 st layer. S ent thereby can be approximately evaluated as: Here, T (i) ε a is formulated as equation (19) with the replacements of ε I by ε a , ξ gh =( ε a − n yh n yh − n xg n xg ) 1/2 ,andε a = n 2 avg . Similar to the argument of S ent ,anothervirtual(isotropic) uniform layer is included between the emitted region and the N th layer to consider the effects of refraction and the reflection at the last interface. Here, S ext is approximated as: Put everything together, and the propagation of fields through three-dimensional periodic microstructures hence can be evaluated as in Equation (13).

Numerical analyses
In this section, a simple case is applied to demonstrate the algorithms and is verified by finite-difference time-domain (FDTD) method. Consider a one-layer film (N = 1) with liquid-crystal orientation . By the Fourier transform defined in equations (42-45), the non-zero Fourier components for the dielectric elements ε ij,gh are: . For simplicity, we only consider three Fourier components of fields, i.e. (g, h) = (±1, 0) and (0, 0), for this case. The corresponding transfer-matrix formula in equation (13) are then given as:  (13), showing comparable results.
which relates to the eigen-vector matrices T 0 /T 2 for the isotropic incident/emitted layer in the equation (19), and the eigen-values κ (a) 1 and eigen-vector T (a) 1 matrices of G in equation (18) for the liquid-crystal film. Here, z lc = z lc /k 0 is the thickness of the liquid-crystal film. In this case, we simply choose the unit-amplitude normal TE incidence with respect For the isotropic incident/emitted air layer (ε = 1), the associatedṅ x ,ṅ y , ε,andξ in T 0 /T 2 are referred to equations (14,15,16), and are given as: Note we have used a small incident angle (θ = 10 −5 , φ = 0) to avoid the numerical instability at θ = 0. For the layer of liquid-crystal film, the associatedñ x ,ñ y and ε ij∈{x,y,z} in G in equation (18) are written out as below: Consequently, the eigen-values κ (a) 1 and eigen-vector T (a) 1 matrices of G can be numerically evaluated and a similar process for S ent and S ext can be followed straightforwardly. Together with all these definitions of matrixes in equation (24), the transmittance fields E + q,2 and M + q,2 then can be decided. Here, we set λ = 0.55um, Λ x = 2.0um, n o = 1.5, and n e = 1.6. The numerical results of far-field diffractions for this case by RCWA ignoring the influences of the multiple reflections are shown in figure 2(b), and are in agreement with these obtained by FDTD. Besides, an alternative consideration described by the equations (89-92) in appendix A, which includes the multiple reflections, is shown in figure 2(a), and clarifies the effectiveness of the easy-manipulated algorithm in equation (13) for the three-dimensional periodic LC media.

A. Derivation of the coupling matrix
In this appendix, detailed derivations of the coupling matrix method are demonstrated for references.

A.1 Maxwell's equations in spatial-space descriptions
Without charges and currents, Maxwell's equations can be read as: and the equations can be derived as: Here, all the field components are assumed to have time dependence of exp (iωt) and are omitted everywhere. The relative permeability of the medium is assumed to be μ = 1. Note that ε ij∈{x,y,z} are defined as functions of position (x, y, z)andε ij are of (x, y, z). λ is the vacuum wavelength of the incident wave. Variables x, y, z generally represent spatial positions while these appeared in suffix, e.g ε ij∈{x,y,z} , denote the orientations along the directionsx,ŷ,ẑ. Moreover, the variable i is the imaginary constant number i = √ −1 and that appeared in suffix, e.g. dz i , is an integer indexing number. For liquid crystals, the dielectric matrix ε is associated with the orientation of director (θ o , φ o ): where n e and n o are extraordinary and ordinary indices of refraction of the birefringent liquid crystal, respectively, θ o is the angle between the director and the z axis, and φ o is the angle between the projection of the director on the xy plane and x axis.

A.2 Maxwell's equations in k-space descriptions
Consider the general geometry illustrated in Figure 3 of stacked multi-layer two-dimensional periodic microstructures. To apply the rigorous coupled-wave theory to the stack, all of the layers have to define the same periodicity: Λ x along the x direction and Λ y along the y direction. The thickness for the ℓ th layer is dz ℓ , and these layers contribute to a total thickness of the stack Z N = ∑ N ℓ=1 dz ℓ . The periodic permittivity of an individual layer in the stack can be expanded in Fourier series of the spatial harmonics as: A similar transform for the fields in the stack can be expressed in terms of Rayleigh expansions: where n I (n E ) is the refraction index for the isotropic incident (emitted) region. θ, φ are the incident angles defined in sphere coordinates, and z is the normal direction for the xy plane of periodic structures. Here, the electric field of an incident unit-amplitude wave has been introduced by E inc = u × exp (−ik · r) as illustrated in figure 3, in which the wave vector k as well as the unit polarization vector u are given by: k = k 0 n I (sin θ cos φx + sin θ sin φŷ + cos θẑ) (50) u = u xx + u yŷ + u zẑ = (cos Ψ cos θ cos φ − sin Ψ sin φ)x (51) + (cos Ψ cos θ sin φ + sin Ψ cos φ)ŷ − (cos Ψ sin θ)ẑ with the Ψ angle between the electric field vector and the incident plane. Now we can express Maxwell's equations by the (g, h) Fourier components in k-space descriptions. For simplicity, we introduce the definitions of the tangential and normal fields at the interfaces as Here, e i∈{x,y,z} = e i (z ℓ ) and h i∈{x,y,z} = h i (z ℓ ) are column matrices with Fourier components e i,gh (z ℓ ) and h i,gh (z ℓ ), respectively. In the following context, a straightforward calculation to obtain the infinite set of coupled-wave equations corresponding to the infinite Fourier components is fulfilled. First, we express Maxwell's curl equations (40)-(41) in terms of the spatial x, y, z components: Next, we introduce the Fourier representations of E, H,a n dε (r) as defined in Equations (44)-(47). Maxwell's curl equations (53)-(54) can thereby be regrouped by the components of fˆt and fn. For the component h z,gh (z ℓ ), the equation can be derived as: It is simplified to be: h z,gh (z ℓ ) = n xg e y,gh (z ℓ ) − n yh e x,gh (z ℓ ) For the component , the equation can be derived as: and is simplified to be: For the component , the equation can be derived as: and is simplified to be: For the component e z,gh , the equation can be derived as: and is simplified to be: For the component , the equation can be derived as: and is simplified to be: For the component , the equation can be derived as: and is simplified to be: To solve the fields systematically, these equations are reformulated in terms of the full fields fˆt and fn in the following context, and show an eigen-system problem for studied periodic structures.
In the following contexts, we continue to describe (a) the solutions of the transition fields within stack layers via equation (70), especially for these uniform layers with isotropic materials which bring in the degenerate eigen-states, and (b) the continuum of fields conditioned at interfaces between stack layers. Consequently, a complete analysis for fields through all stacks can be fulfilled, and the associated near/far field optics can be evaluated.

A.5 Eigen-system solutions
As indicated in equation (70), the tangential fields fˆt within the layers proceed an eigen-system process, in which the eigen-states are independent to each other and allow individual/straightforward analyses to evaluate the transition behaviors through the layers. At the interfaces among the layers, the tangential fields fˆt associated with the composite phases/amplitudes of the eigen-states follow the physical continuous conditions in the laboratory framework. These characteristics lead to the necessary transform between the laboratory and eigen-system frameworks as described below. Besides, for these uniform layers with isotropic materials, especially for the incident and emitted regions, the eigen-system shows the degenerate status, and a reasonable choice of the eigen-states corresponding to the physical conditions is emphasized below. Implemented with all these, the behaviors of the tangential fields fˆt through all stacks layers including the in-between interfaces can be decided. The normal fields fn are then obtained by equation (69), and thereby the complete light waves are understood.

A.5.1 Uniform layers with isotropic materials
For the uniform layers with isotropic materials, i.e. ε(r) is a scalar constant, the coupled-wave equation of the tangential fields fˆt in equation (70) can be simplified as: Here, all the submatrices in C are diagonal and thereby the component states are independent. By straightforward calculation, its eigen-values as well as the corresponding eigen-vectors for (g, h)-order component can be obtained: with ξ gh = ε − n yh n yh − n xg n xg while the corresponding eigen-vector matrix are: −ξ gh n xg n yh n xg n xg −ε gh n xg n yh ξ gh n xg n yh n xg n xg −ε gh n xg n yh n yh n yh −ε gh n xg n yh −ε gh ξ gh n xg n yh n yh n yh −ε gh n xg n yh ε gh ξ gh n xg n yh Due to the degeneracy in (κ gh,1 , κ gh,2 ) and (κ gh,3 , κ gh,4 ), the eigenvector can be remixed by arbitrary linear combinations. Choosing m gh = n yh n yh + n xg n xg the equation (73) is then shown as: Hence, v gh1 and v gh2 correspond to the (g, h)-order forward TE and TM (transverse electric and transverse magnetic) representations (with respect to the plane of the diffraction wave), respectively. v gh3 and v gh4 then correspond to the backward TE and TM ones. For example, with equation (69) and (79), v gh1 denotes the component fields: along the direction n gh = n xgî + n yhĵ + ξ ghk . It can be seen that the characteristic fields in equations (80)-(81) associated with the eigen-solution ∝ exp(−iξ gh z) and constitutes the forwards TE wave. It is noted that the field amplitudes are normalized to |e gh | = 1, |h gh | = √ ε,a n de gh · n gh = h gh · n gh = e gh · h gh = 0 -that is, n gh , e gh ,a n dh gh are mutually perpendicular. Similarly, the remaining eigen-vectors can characterize the forwards and backwards TE/TM waves and are omitted here. In this way, a unit-amplitude incident wave then can be given as E + q =[0...1...0] t , M + q = 0forTEwave,and M + q =[0...1...0] t , E + q = 0 for TM wave as defined below. Considering the full components g min ≤ g ≤ g max and h min ≤ h ≤ h max ,t h ec o u p l e d -w a v e equation (71) can be straightforwardly written as: Note thatṅ y andṅ x are the N g N h × N g N h diagonal matrices with diagonal elements n yh m gh and n xg m gh respectively. ξ −1 is the matrix with elements 1/ξ gh , not the inverse of the matrix ξ. Moreover, E + q and M + q ( E − q and M − q ) correspond to the physical forward (backward) TE and TM waves, respectively. The transition of fields qˆt within the considered layer are now solved as:

A.5.2 Periodic-structured layers with isotropic/birefringent materials
For the in-between periodic layers, the transition equations of tangential fields fˆt in equation (70) can be generally expressed as: with the transition of qˆt Here, T (a) is the eigen-vector matrix of G of equation (70) with column eigen-vectors, and κ (a) is the corresponding diagonal eigen-value matrix.

A.6 Boundary conditions
Now for each layer, we have been able to independently solve the transition of electromagnetic fields in the individual layers, but the continuum of fields on interfaces has still not been included. Considering the tangential components in fˆt are continuous across i th interface at z i , the constriction equations can be shown as Grouping this condition into the fields qˆt in equation (86), and introducing two virtual layers to consider the Fresnel refraction and reflection at surfaces of the media as described in the texts, a general expression for N−multilayered periodic structures can be obtained as in equation (13). This argument ignores the effects of multiple reflections as applied by (extended) Jones method, and similarly supplies as a easy-manipulated method. Further, an alternative process to consider the multiple reflections is described as below for references. Similarly, group the equation (87) with (86), the consecutive matrix equation with undecided diffraction/reflection waves can be written as: where the first boundary is indexed as 0. Consequently, the relation between fields in the incident region 0 and in the emitted region N + 1 can be obtained as: ) or alternatively: Consider that the reflective field in the emitted region is zero, i.e.
The transmittance field in the emitted region can be obtained as: and the reflective field in the incident region is:

A.7 Diffraction efficiency
To evaluate the diffraction efficiency with the obtained q ± t ,thex, y,andz components of the transmittance/reflection fields of the diffraction order (g, h) can be calculated by equations 19 (or 82, 83) and 69 for emitted/incident regions, and thereby the standard definitions of diffraction efficiency can be followed. Note that the incident fields should be excluded when calculating the reflection fields in the incident region.

B. Program codes of Wolfram Mathematica for Coupling Matrix Method
In this appendix, the program codes of Wolfram Mathematica for the (numerical) study case in the previous section are added as follows. It could be able to do the simulations by copy and paste the codes, while few characters may need to be adjusted, e.g., the superscript of W ′ (W ′′ ) and the power symbol on no ∧ 2(ne ∧ 2).