Numerical Simulation of Nonlinear Electromagnetic Wave Propagation in Nematic Liquid Crystal Cells

Transcription

1 Numerical Simulation of Nonlinear Electromagnetic Wave Propagation in Nematic Liquid Crystal Cells N.C. Papanicolaou 1 M.A. Christou 1 A.C. Polycarpou 2 1 Department of Mathematics, University of Nicosia 2 Department of Electrical and Computer Engineering, University of Nicosia 4th EAC for AMiTaNS, Varna, Bulgaria, June 11-16, 2012 In memory of Prof. C. I. Christov Ack.: The work of N.C. Papanicolaou and M.A. Christou was partially supported by grant DDVU02/71, NSF, Bulgarian Ministry of Education, Youth and Science Radio- Telecommunica.ons Lab RTeLab

2 Outline 1 Introduction 2 Formulation and Analysis Description and Geometry Maxwell Equations Mode-Matching Technique Non-Linear Equation for the Director Field Finite Difference Schemes and Boundary Conditions 3 Results Rigid Anchoring - Normal Incidence Soft Anchoring - Normal Incidence Rigid Anchoring - Oblique Incidence 4 Summary

3 Outline 1 Introduction 2 Formulation and Analysis Description and Geometry Maxwell Equations Mode-Matching Technique Non-Linear Equation for the Director Field Finite Difference Schemes and Boundary Conditions 3 Results Rigid Anchoring - Normal Incidence Soft Anchoring - Normal Incidence Rigid Anchoring - Oblique Incidence 4 Summary

4 Outline 1 Introduction 2 Formulation and Analysis Description and Geometry Maxwell Equations Mode-Matching Technique Non-Linear Equation for the Director Field Finite Difference Schemes and Boundary Conditions 3 Results Rigid Anchoring - Normal Incidence Soft Anchoring - Normal Incidence Rigid Anchoring - Oblique Incidence 4 Summary

5 Outline 1 Introduction 2 Formulation and Analysis Description and Geometry Maxwell Equations Mode-Matching Technique Non-Linear Equation for the Director Field Finite Difference Schemes and Boundary Conditions 3 Results Rigid Anchoring - Normal Incidence Soft Anchoring - Normal Incidence Rigid Anchoring - Oblique Incidence 4 Summary

6 Liquid Crystals are of particular interest because of their use in various devices such as LCDs, optical filters and switches, beam-steering devices, LC-thermometers. This is due to their electro-optical properties (anisotropic, birefringent and tunable). In this presentation, we develop an efficient and robust numerical method for the simulation of Electromagnetic Wave Propagation in Liquid Crystal Cells. We concentrate on Nematic Liquid crystals. The cases of both normal and oblique incidence are examined and interesting phenomena such as hysteresis and bistability are presented.

7 Liquid Crystals are of particular interest because of their use in various devices such as LCDs, optical filters and switches, beam-steering devices, LC-thermometers. This is due to their electro-optical properties (anisotropic, birefringent and tunable). In this presentation, we develop an efficient and robust numerical method for the simulation of Electromagnetic Wave Propagation in Liquid Crystal Cells. We concentrate on Nematic Liquid crystals. The cases of both normal and oblique incidence are examined and interesting phenomena such as hysteresis and bistability are presented.

8 Liquid Crystals are of particular interest because of their use in various devices such as LCDs, optical filters and switches, beam-steering devices, LC-thermometers. This is due to their electro-optical properties (anisotropic, birefringent and tunable). In this presentation, we develop an efficient and robust numerical method for the simulation of Electromagnetic Wave Propagation in Liquid Crystal Cells. We concentrate on Nematic Liquid crystals. The cases of both normal and oblique incidence are examined and interesting phenomena such as hysteresis and bistability are presented.

9 Nematic Liquid Crystals Nematic liquid crystals owe their properties to their molecular structure. Their molecules are rod-like (calametic) and in this mesophase they macroscopically point in a preferred direction called the director. The orientation of the directors determines the electrical properties of the liquid crystal. Thus, the relative dielectric tensor of the LC is a function of the director angle. In the presence of an applied electric (or magnetic) field above a certain intensity the directors reorient, changing the optical properties of the liquid crystal. This is called the Fréedericksz Transition.

10 Nematic Liquid Crystals Nematic liquid crystals owe their properties to their molecular structure. Their molecules are rod-like (calametic) and in this mesophase they macroscopically point in a preferred direction called the director. The orientation of the directors determines the electrical properties of the liquid crystal. Thus, the relative dielectric tensor of the LC is a function of the director angle. In the presence of an applied electric (or magnetic) field above a certain intensity the directors reorient, changing the optical properties of the liquid crystal. This is called the Fréedericksz Transition.

11 Nematic Liquid Crystals Nematic liquid crystals owe their properties to their molecular structure. Their molecules are rod-like (calametic) and in this mesophase they macroscopically point in a preferred direction called the director. The orientation of the directors determines the electrical properties of the liquid crystal. Thus, the relative dielectric tensor of the LC is a function of the director angle. In the presence of an applied electric (or magnetic) field above a certain intensity the directors reorient, changing the optical properties of the liquid crystal. This is called the Fréedericksz Transition.

12 A Nematic Liquid Crystal Cell Exterior Region Exterior Region Figure: A nematic liquid crystal cell. At the boundaries rigid anchoring with homeotropic alignment is assumed.

13 Problem Dynamics Here, a uniform plane wave polarized along the plane of incidence (xz-plane) (pump beam, usually a laser) excites the LC, changing the orientation angle of the directors. This alters the dielectric tensor and hence affects the intensity of the electric and magnetic fields inside the cell. In turn, the new light intensity leads to corresponding new values for the director field. This a coupled problem. The set of Maxwell equations modeling the propagation of the electromagnetic wave is intrinsically linked to the Nonlinear Differential Equation for the director field.

14 Problem Dynamics Here, a uniform plane wave polarized along the plane of incidence (xz-plane) (pump beam, usually a laser) excites the LC, changing the orientation angle of the directors. This alters the dielectric tensor and hence affects the intensity of the electric and magnetic fields inside the cell. In turn, the new light intensity leads to corresponding new values for the director field. This a coupled problem. The set of Maxwell equations modeling the propagation of the electromagnetic wave is intrinsically linked to the Nonlinear Differential Equation for the director field.

15 Problem Dynamics Here, a uniform plane wave polarized along the plane of incidence (xz-plane) (pump beam, usually a laser) excites the LC, changing the orientation angle of the directors. This alters the dielectric tensor and hence affects the intensity of the electric and magnetic fields inside the cell. In turn, the new light intensity leads to corresponding new values for the director field. This a coupled problem. The set of Maxwell equations modeling the propagation of the electromagnetic wave is intrinsically linked to the Nonlinear Differential Equation for the director field.

16 Geometry z θ t H t E t i N d L y θ i θ r H r i 2 i 1 x H i E i E r Figure: Propagation of an EM-wave that is obliquely incident to a nematic LC-cell. Regions 1, 2 and 3: Before, inside and after the LC-cell respectively.

17 EM-wave propagation inside the cell is modeled by the time-harmonic Maxwell Equations E = jωµ 0 H, H = jωɛ 0ˆɛE, (1a) (1b) E : electric field intensity H : magnetic field intensity ɛ 0 : electric permittivity of free space µ 0 : magnetic permeability of free space η 0 : intrinsic impedance of vacuum (η 0 = µ 0 /ɛ 0 ) ˆɛ is the LC relative permittivity tensor, with components

18 and ɛ xx (z) 0 ɛ xz (z) ˆɛ(z) = 0 ɛ yy (z) 0, ɛ zx (z) 0 ɛ zz (z) ɛ xx = n 2 e sin 2 θ(z) + n 2 o cos 2 θ(z), ɛ xz = ɛ zx = (n 2 e n 2 o) sin θ(z) cos θ(z), ɛ yy = n 2 o, ɛ zz = n 2 e cos 2 θ(z) + n 2 o sin 2 θ(z). (2a) (2b) n o : the ordinary refractive index (wave polarized perpendicular to the directors) n e : the extraordinary refractive index (wave polarized parallel to the directors)

19 Because the incident wave is propagating in the xz-plane, the field components are independent of the y coordinate, i.e., F y = 0 where F = E x, E y, E z, H x, H y or H z. Thus, Eqs. (1,2) in component form read: E y z = jωµ 0H x, E x z = E z x jωµ 0H y, E y x = jωµ 0H z, H y z = jωɛ 0[ɛ xx E x + ɛ xz E z ], (3a) H x z = H z x + jωɛ 0ɛ yy E y, (3b) H y x = jωɛ 0[ɛ zx E x + ɛ zz E z ]. (3c)

20 Region 1 Incident electric and magnetic fields: E i = (a x cos θ i a z sin θ i ) E 0 e j(k i r), H i = n s η 0 (a ki E i ), (4a) (4b) k i = a ki k 0 n s : wave vector k 0 = ω µ 0 ɛ 0 : wavenumber in vacuum a ki = a x sin θ i + a z cos θ i : unit vector in direction of propagation n s : refractive index of exterior region r = a x x + a z z : position vector of observation point in xz-plane Therefore, E i = (a x cos θ i a z sin θ i ) E 0 e jk 0n s(x sin θ i +z cos θ i ), (5a) H i = a y n s E 0 η 0 e jk 0n s(x sin θ i +z cos θ i ). (5b)

21 Region 1 (continued) Snell s law of reflection implies θ r = θ i. Hence, the reflected wave propagates in the direction of the unit vector a kr = a x sin θ i a z cos θ i. Therefore, E r = ( a x cos θ i a z sin θ i ) ΓE 0 e jk 0n s(x sin θ i z cos θ i ), (6a) H r = n s η 0 a kr E r = a y n s ΓE 0 η 0 e jk 0n s(x sin θ i z cos θ i ), (6b) where Γ is the reflection coefficient in the plane of incidence. Consequently, in the lower exterior region, the total field is simply the superposition of the incident and reflected fields E = E i + E r = (a x cos θ i a z sin θ i ) E 0 e jk 0n s(x sin θ i +z cos θ i ) (a x cos θ i + a z sin θ i ) ΓE 0 e jk 0n s(x sin θ i z cos θ i ), H = H i + H r = a y n s E 0 η 0 e jk 0n s(x sin θ i +z cos θ i ) + a y Γn s E 0 η 0 e jk 0n s(x sin θ i z cos θ i ). (7a) (7b)

22 Region 3 Similarly, the transmitted fields in the upper region are given by E t = (a x cos θ t a z sin θ t ) TE 0 e jk 0n s(x sin θ t +z cos θ t ), H t = a y n s TE 0 η 0 e jk 0n s(x sin θ t +z cos θ t ) (8a) (8b) where θ t = θ i and T is the copolarized transmission coefficient at the upper interface, which separates the crystal and the exterior region.

23 Region 2 To satisfy continuity of the tangential fields for all x at a fixed z-plane, the fields inside the LC-cell are written: E(z, x) = E(z) e jk 0n sx sin θ i and H(z, x) = H(z) e jk 0n sx sin θ i. (9) So, the field components are of the form where F = E x, E y, E z, H x, H y or H z. F(x, z) = F(z) e j k 0n s x sin θ i (10) Clearly, F(x, z) = jk 0 SF (x, z), (11) x where S = n s sin θ i. Substituting (11) into (3), employing (10) and manipulating yields:

24 u 1 z = j k 0 [c 11 u 1 + c 14 u 4 ], u 3 z = j k 0 c 32 u 2, u 2 z = j k 0 u 3, u 4 z = j k 0 [c 41 u 1 + c 44 u 4 ]. (12a) (12b) where u 1 = E x, u 2 = E y, u 3 = η 0 H x, u 4 = η 0 H y, c 11 = (ɛ zx /ɛ zz )S, c 14 = (S 2 /ɛ zz ) 1, c 32 = ɛ yy S 2, c 41 = (ɛ xz ɛ zx /ɛ zz ) ɛ xx, c 44 = S(ɛ xz /ɛ zz ). The LC cell is then subdivided into N layers of thickness d. Each of these layers is assumed to be homogeneous and anisotropic. The dielectric properties of the layer are characterized by ˆɛ in Eq. (2a) and thus depend on the director field θ evaluated at the midpoint of the layer. The solutions of Eqs. (12) have the following generic form: u m (z) e jk 0nz, (13) where m = 1, 2, 3, 4 and n is the unknown refractive index inside the homogeneous LC layer.

25 Substituting Eq. (13) into Eqs. (12) results in the following system of algebraic equations: (n + c 11 )u 1 + c 14 u 4 = 0, (14a) nu 2 + u 3 = 0, c 32 u 2 + nu 3 = 0, c 41 u 1 + (n + c 44 )u 4 = 0. (14b) (14c) (14d) For the homogeneous system (14) to have a nontrivial solution, det (A) = 0, which leads to the following algebraic equation for n which admits solutions (n 2 c 32 ) [(n + c 11 )(n + c 44 ) c 41 c 14 ] = 0, (15) n 1,2 = ± c 32, n 3,4 = c 11 ± c 14 c 41. (16) Eqs. (14a, 14d) and solutions n 3,4 correspond to an incident plane wave polarized in a direction parallel to the plane of incidence. Eqs. (14b, 14c) and solutions n 1,2 correspond to an incident plane wave polarized in a direction perpendicular to the plane of incidence.

26 Region 2 (continued) Because ɛ xy = ɛ yx = ɛ yz = ɛ zy = 0, the two polarizations are fully decoupled (see Eqs. 14). This means that an incident plane wave polarized in a direction perpendicular to the plane of incidence will not generate field components inside the LC that are polarized in the parallel direction and vice-versa. To trigger the formation of an extraordinary wave inside the LC cell, the incident plane wave must be polarized in a direction parallel to the plane of incidence. Therefore: u 1 (z) = A e j k 0 n 3 z + B e j k 0 n 4 z, ( ) c41 u 4 (z) = A e j k 0 n 3 z n 3 + c 44 ( c41 n 4 + c 44 ) B e j k 0 n 4 z, (17a) (17b) where n 3 and n 4 are given by Eqs. (16). Note also that u 1 (x, z) = u 1 (z) e j k 0 x sin θ i, u 4 (x, z) = u 4 (z) e j k 0 x sin θ i. (18a) (18b)

27 Mode-Matching The unknown coefficients are obtained by enforcing the continuity of the tangential fields at the interfaces. As mentioned earlier, the LC is subdivided into N equally thin homogeneous layers. The first interface is between the lower exterior region (Region 1) and the first layer of the LC and the last interface is between the last LC-layer and Region 3. The total number of interfaces is equal to N + 1. This leads to 2N + 2 linear equations with 2N + 2 unknowns. Two of these correspond to the reflection and transmission coefficients in the plane of incidence. The remaining unknowns correspond to the modal expansion coefficients representing the fields inside the LC.

28 Lower Exterior Region: u 1 (z) = E 0 cos θ i e jnsk 0z cos θ i ΓE 0 cos θ i e jnsk 0z cos θ i, u 4 (z) = n s E 0 e jnsk 0z cos θ i + n s ΓE 0 e jnsk 0z cos θ i. (19a) (19b) LC Region: u 1,m (z) = A m e j k 0 n 3 (z d m) + B m e j k 0 n 4 (z d m), u 4,m (z) = C A A j e j k 0 n 3 (z d m) C B B m e j k 0 n 4 (z d m), (20a) (20b) where C A = c 41 / (n 3 + c 44 ), C B = c 41 / (n 4 + c 44 ), and d m = (m 1)d ; m = 1, 2,... N, where the index m indicates the layer number. Upper Exterior Region: u 1 (z) = TE 0 cos θ i e jnsk 0(z d N+1 ) cos θ i, u 4 (z) = n s TE 0 e jnsk 0(z d N+1 ) cos θ i. (21a) (21b) Note: The z coordinate was transformed to z z 0, where z 0 corresponds to the z coordinate of the lower layer interface.

29 Enforcing the continuity of u 1 and u 4 at each of the N + 1 interfaces yields z = 0 : ΓE 0 cos θ i + A 1 + B 1 = E 0 cos θ i, : n s ΓE 0 C A A 1 C B B 1 = n s E 0, z = d : A 1 e j k 0 n 3 d + B 1 e j k 0 n 4 d A 2 B 2 = 0, A 1 C A e j k 0 n 3 d + B 1 C B e j k 0 n 4 d A 2 C A B 2 C B = 0,.. z = (m 1)d : A m 1 e j k 0 n 3 d + B m 1 e j k 0 n 4 d A m B m = 0, A m 1 C A e j k 0 n 3 d + B m 1 C B e j k 0 n 4 d A m C A B m C B = 0,.. z = Nd : A N e j k 0 n 3 d + B N e j k 0 n 4 d E 0 cos θ i T = 0, : C A A N e j k 0 n 3 d + C B B N e j k 0 n 4 d n s E 0 cos θ i T = 0. The above system is solved using LU decomposition to obtain the expansion coefficients and the reflection and transmission coefficients at the lower and upper interfaces, respectively.

30 Non-Linear ODE for the Director Field The orientation of the directors inside a LC in the presence of an electromagnetic field is governed by the following functional which represents the total free energy per unit volume: [ 1 F = 2 k 11( ˆn) k 22[ˆn ( ˆn)] k 33 ˆn ( ˆn) 2 I ] c ñ(θ) d 3 r, where n 0 n e ñ = n0 2 sin2 θ + ne 2 cos 2 θ (22) and ˆn = (sin θ, 0, cos θ), (23) c the speed of light in a vacuum, I = 1 2 R[(E H ) a z ] = 1 2 R[E xh y ] the local light intensity, and k 11, k 22 and k 33 are the splay, twist and bend elastic constants. Functional F attains its minimal value when the Euler-Lagrange equation holds, where f is the integrand in Eqn. (22) f θ d f = 0 (24) dz θ z

31 Substituting Eqn. (23) into the integrand of (22) yields f = 1 2 k 11θz 2 sin 2 θ k 33θz 2 cos 2 θ I n 0 n e. (25) c n0 2 sin2 θ + ne 2 cos 2 θ Replacing (25) into the Euler-Lagrange equation and rearranging leads to the nonlinear ODE for the directors θ zz k sin 2θ 2(1 k sin 2 θ) θ2 z + α(z) sin 2θ (1 k sin 2 θ)(1 β sin 2 = 0, (26) θ) 3/2 where k = (k 33 k 11 )/k 33, β = 1 (n 0 /n e ) 2 and α(z) = βn oi(z) 2ck 33. Utilizing the definition of the Fréedericksz threshold intensity, I Fr, (26) can be expressed as I Fr = ck 33π 2 n o βl 2 (27) θ zz 1 k sin 2θ 2 (1 k sin 2 θ) θ2 z + 1 ( π ) 2 I sin 2θ 2 L I Fr (1 k sin 2 θ)(1 β sin 2 = 0. (28) θ) 3/2

32 Boundary Conditions The orientation of the directors versus the z coordinate is posed as a two-point boundary value problem, governed by (28), and a set of boundary conditions at the walls of the cell. For strong anchoring, the two boundary conditions are of Dirichlet type: θ(z = 0) = 0 and θ(z = L) = 0. (29) For soft anchoring, the boundary conditions are of Robin type expressed as first-order differential equations: ( ) dθ k 33 (1 k sin 2 θ) + 1 ( ) df = 0 at z = 0, L, (30) dz 2 dz where F(θ) = C cos 2 θ + C 4 cos 4 θ is known as the interfacial potential.

33 Finite Difference Schemes 3-point explicit scheme: [ θ k+1 i = 1 θi+1 k 3 + θk i + θi 1 k k sin 2θk i (θi+1 k ] θk i 1 )2 8(1 k sin 2 θi k ) + 1 ( π ) 2 I h 2 sin 2θi k 6 L I Fr (1 k sin 2 θi k )(1 β sin 2. (31) θi k ) 3/2 5-point explcit scheme: Derivation: θ k+1 i = 1 [ θi+2 k 8 + θk i+1 + 4θk i + θ k i 1 + θk i 2 5k sin 2θi k 64(1 k sin 2 θi k ) (θk i+1 θk i 1 )2 + 5h2 ( π ) 2 I sin 2θi k 16 L I Fr (1 k sin 2 θi k )(1 β sin 2. (32) θi k ) 3/2 θ = θ i+2 + θ i+1 4θ i + θ i 1 + θ i 2 5h 2 + O(h 2 ). (33) ]

34 3-point semi-implicit scheme: ( ) (1 ω 1 ) θ m+1 i+1 2θm+1 i + θ m+1 i 1 + ω 1 (θi+1 m 2θm i + θi 1 m ) k sin 2θ m [ i 8(1 k sin 2 (1 ω θi m 2 )(θi+1 m ) θm i 1 )(θm+1 i+1 θm+1 i 1 ) + ω 2(θi+1 m θm i 1 )2] + 1 ( π ) 2 I h 2 sin 2θi m 2 L I Fr (1 k sin 2 θi m )(1 β sin 2 = 0, (34) θi m ) 3/2 where ω 1, ω 2 are the relaxation parameters for the second and first derivatives, respectively. Note: The 5-point scheme reverts to a 3-point scheme near the boundaries. The Robin type boundary conditions for soft anchoring, (30) are discretized as follows: θ k+1 i = θi+1 k h sin θ ( i C cos θi + 2C 4 cos 3 ) θ i k 33 (1 k sin 2, at z = 0, (35a) θ i ) θ k+1 i = θi 1 k h sin θ ( i C cos θi + 2C 4 cos 3 ) θ i k 33 (1 k sin 2, at z = L. (35b) θ i )

35 Convergence Rate Figure: Comparison of the convergence rates of the proposed finite difference schemes for Methoxybenzylidene butylaniline (MBBA) and homeotropic alignment.

36 θ = θ(z) for θ i = 0 and homeotropic b.c.s (rad) I Inc /I Fr = I Inc /I Fr = I Inc /I Fr = I Inc /I Fr = z/l Figure: The directors tilt angle as a function of space (0 z L/2) for four different biased intensities.

37 m MBBA (our approach) PAA Increasing (our approach) PAA Decreasing (our approach) MBBA (ref [22]) PAA (ref [22]) I Inc /I Fr Figure: θ max vs scaled incident intensity for MBBA (λ = nm, n o = 1.544, n e = 1.758, k 11 = N, k 33 = N) and PAA (λ = 480 nm, n o = 1.595, n e = 1.995, k 11 = N, k 33 = N). For both cases, the refractive index of the exterior region n s = n o. For MBBA B = 0.25(1 k 2.25β) > 0, whereas for PAA B < 0.

38 m, o (rad) m (our approach) o (our approach) m (ref [26]) o (ref [26]) H/H th Figure: Maximum director angle (θ m) and interfacial director angle (θ o) versus the normalized applied magnetic field. MBBA liquid crystal cell at a temperature T = 39.5 having the following specifications: L = 2.9µm, λ = nm, n o = , n e = , k 11 = N, k 33 = N. The exterior region is glass. Soft anchoring is applied at the liquid crystal-to-wall interface with an interfacial potential F (θ) = 47.0cos 2 θ 18.0cos 4 θ µn/m.

39 Maximum director angle (rad) degrees 30 degrees 60 degrees Scaled incident intensity Figure: Maximum director angle θ m as a function of the scaled incident intensity I inc /I Fr for MBBA at different angles of incidence.

40 Scaled Inc. Intens. = 1.33 Scaled Inc. Intens. = 1.74 Scaled Inc. Intens. = 2.20 Maximum director angle (rad) Incident angle (degrees) Figure: Maximum director angle θ m as a function of the incident angle for MBBA at different values of the scaled incident intensity I inc /I Fr.

41 Summary A numerical method is presented for treating electromagnetic wave propagation in Nematic Liquid Crystal cells. The problem is governed by the Maxwell time-harmonic equations coupled with a nonlinear ODE for the tilt angle of the directors. These are solved iteratively; the Mode-Matching Technique is used for the Maxwell equations and a semi-implicit Finite Difference scheme is employed for the ODE governing the director field. The proposed method was validated by comparing the obtained results for some indicative parameter values with published data and were found to be in very good agreement. Finally, the method is used to treat a variety of cases revealing including finite anchoring and oblique incidence.

42 Summary A numerical method is presented for treating electromagnetic wave propagation in Nematic Liquid Crystal cells. The problem is governed by the Maxwell time-harmonic equations coupled with a nonlinear ODE for the tilt angle of the directors. These are solved iteratively; the Mode-Matching Technique is used for the Maxwell equations and a semi-implicit Finite Difference scheme is employed for the ODE governing the director field. The proposed method was validated by comparing the obtained results for some indicative parameter values with published data and were found to be in very good agreement. Finally, the method is used to treat a variety of cases revealing including finite anchoring and oblique incidence.

43 Summary A numerical method is presented for treating electromagnetic wave propagation in Nematic Liquid Crystal cells. The problem is governed by the Maxwell time-harmonic equations coupled with a nonlinear ODE for the tilt angle of the directors. These are solved iteratively; the Mode-Matching Technique is used for the Maxwell equations and a semi-implicit Finite Difference scheme is employed for the ODE governing the director field. The proposed method was validated by comparing the obtained results for some indicative parameter values with published data and were found to be in very good agreement. Finally, the method is used to treat a variety of cases revealing including finite anchoring and oblique incidence.

44 Summary A numerical method is presented for treating electromagnetic wave propagation in Nematic Liquid Crystal cells. The problem is governed by the Maxwell time-harmonic equations coupled with a nonlinear ODE for the tilt angle of the directors. These are solved iteratively; the Mode-Matching Technique is used for the Maxwell equations and a semi-implicit Finite Difference scheme is employed for the ODE governing the director field. The proposed method was validated by comparing the obtained results for some indicative parameter values with published data and were found to be in very good agreement. Finally, the method is used to treat a variety of cases revealing including finite anchoring and oblique incidence.

45 For Further Reading I P. de Gennes and J. Prost. The Physics of Liquid Crystals. 2nd ed, Oxford: Clarendon Press, H. L. Ong. Optically induced Fréedericksz transition and bistability in a nematic liquid crystal. Phys. Rev. A, 28(4): , V. Ilyina, S. J. Cox, and T. J. Sluckin. A computational approach to the optical Fréedericksz transition. Optics Communications, 260: , K. H. Yang and C. Rosenblatt. Determination of the anisotropic potential at the nematic liquid crystal-to-wall interface. Appl. Phys. Lett., 43(1):62 64, July 1983.