Simulations of a liquid-crystal-based
electro-optical switch
M. Sluijter,1,*D. K. G. de Boer,1and H. P. Urbach2 1
Philips Research Europe, High Tech Campus 34, MS 31, 5656 AE Eindhoven, The Netherlands
2
Optics Research Group, Delft University of Technology, Department Imaging Science and Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
*Corresponding author: Maarten.Sluijter@philips.com
Received September 17, 2008; revised November 3, 2008; accepted November 20, 2008; posted December 8, 2008 (Doc. ID 101628); published December 29, 2008
We present simulations of a novel liquid-crystal-based electro-optical device that enables a switching effect owing to a backreflection phenomenon. In the simulations, we exploit the optical properties of a liquid-crystal layer with a Freédericksz alignment in an unconventional way. The resulting switching effect of the proposed optical design can be controlled by means of an external electric field. Possible applications of the liquid-crystal device can be found in, but are not restricted to, optical communication systems and lighting applications. © 2008 Optical Society of America
OCIS codes: 080.3095, 080.5692, 160.1190, 160.3710, 260.1440, 260.2710. In this Letter, we present simulations of a novel
liquid-crystal-based electro-optical device that en-ables a switching effect owing to a backreflection phenomenon.
A liquid crystal can be controlled by external elec-tric (or magnetic) fields. Interactions between bound-aries and liquid crystal also have a large controlling effect. Often, the influence of the boundaries opposes the response to an electric field. The result is a threshold phenomenon called the Freédericksz tran-sition (cf. [1], p. 202). Figure 1shows a Freédericksz alignment of a liquid-crystal layer in which the local optical axis (indicated by the director dˆ ) is rotated to-ward the direction of an external electric field. Note that a homeotropic alignment with opposite birefrgence can have the same effect. In this Letter, we in-vestigate Freédericksz alignments that have bend and splay deformations of liquid crystal but no twist deformations (cf. [1], p. 32).
In Fig. 1, the director dˆ lies in the x–z plane. It is convenient to write the director vector components as dˆx= cos关共x,z兲兴 and dˆz= sin关共x,z兲兴, where 共x,z兲 is
the angle between dˆ and the x axis. From the sym-metry of the system we can definemaxas the
maxi-mum value of at z = h / 2. Then the potential differ-ence V across the liquid-crystal layer that corresponds tomaxreads (cf. [1], p. 207)
V = 2K共m兲
冑
K110⌬
, 共1兲
where K共m兲 is the complete elliptic integral of the first kind with elliptic modulus m = sin2
max. Fur-thermore,⌬ is the anisotropy in the static dielectric permittivity of the liquid crystal. K11 is an elastic constant with which the associated splay deforma-tion energies scale. The threshold voltage below which the liquid crystal remains undistorted reads
Vth=
冑
K11/0⌬. Finally, it is important to realize that the liquid crystal profile depends only on the di-mensionless coordinate z / h (cf. [1], p. 208).For the simulation of inhomogeneous liquid-crystal configurations, we have developed a ray-tracing method in the geometrical-optics approach called the Hamiltonian method [2]. This method incorporates the fact that inside inhomogeneous anisotropic me-dia, ray paths of light rays are curved and is based upon one important assumption: the properties of an inhomogeneous anisotropic medium are assumed to change slowly over the distance of a wavelength. Therefore, the thickness h should be at least in the order of 20 wavelengths or more. Then the use of the Hamiltonian method is justified.
We consider the liquid-crystal layer of Fig.1. An in-cident light ray is linearly polarized in the x–z plane (TM mode) and propagates in the positive x direction. At 共x,z兲=共0,h/2兲 the light ray is injected in the liquid-crystal layer. For the calculation of the ray path inside the liquid crystal, we apply the Hamil-tonian method for extraordinary rays. With r the po-sition and pea vector with the direction of the local
phase velocity and length equal to the extraordinary refractive index, the Hamilton equations are given by (cf. [2], p. 1269)
dr共兲
d =ⵜpHe共dˆ兲, dpe共兲
d = −ⵜrHe共dˆ兲, 共2兲 where is a parameter that can be considered as time. In addition, the gradients ⵜrHe and ⵜpHe are
Fig. 1. Freédericksz alignment of a liquid-crystal layer ap-plied between two parallel glass plates, separated by a dis-tance h. The director is rotated by an angle in the direc-tion of the electric field E.
94 OPTICS LETTERS / Vol. 34, No. 1 / January 1, 2009
functions of the director dˆ , the momentum pe, and
the ordinary and extraordinary index of refraction of the liquid crystal, noand ne, respectively. These
equa-tions are a set of six coupled first-order differential equations for the vector components of the position r共兲 and momentum pe共兲. Let r共0兲 and pe共0兲 be the boundary values for the position and momentum at 共x,z兲=共0,h/2兲. Then, by taking steps ⌬, the ray path r共0+ N⌬兲 and the corresponding momentum pe共0 + N⌬兲, with N苸N, are calculated by using the first-order Runge–Kutta method.
We simulate a nematic liquid crystal with no = 1.5266 and ne= 1.8181 (Merck BL009 mixture). We estimate K11= 10 pN and ⌬=10, yielding Vth = 1.0558 V. The ray paths are calculated with step size ⌬= 0.001 and h = 5. Figure 2 shows the results for four extraordinary ray paths. Each individual ray path is calculated for different values of Vr= V / Vth = 2 /K共m兲.
From Fig.2we can conclude that the ray paths are oscillatory and the angle at which the light is re-fracted at共x,z兲=共0,h/2兲 increases with Vrsincemax increases with Vr. As a result, we can expect that the period of the ray path decreases with increasing Vr. Figure2shows that this expectation is confirmed by the simulations: the optical system shows a light-guiding behavior. The minimum oscillatory period oc-curs at Vr= 1.1803, corresponding to a potential
dif-ference of V = 1.2461 V andmax=/ 4 rad. Apparently, the light rays are modulated in the vertical z direc-tion, but not in the horizontal x direction. This is due to the absence of a gradient in the x direction in the director profile. From these considerations, we can expect that light rays are modulated both in the ver-tical and horizontal direction if we induce an addi-tional horizontal gradient. In what follows, we will show that these expectations are confirmed by our simulations.
Now let us consider a liquid-crystal layer applied between two parallel ideal mirrors (100% reflec-tance). Similar to the configuration in Fig.1, the cor-responding director profile dˆ 共x,z兲 is a Freédericksz alignment. This time, however, we apply a horizontal gradient in the potential difference V = V共x兲 across
the liquid-crystal layer. This can be achieved with, e.g., the application of a resistive electrode structure. As a result, the director profile has an additional gra-dient in the horizontal direction. In Fig. 3, the pro-posed optical design is depicted in two different situ-ations. Figure 3(a) shows the situation where V共x兲 ⱕVth and dˆ 共x,z兲=共1,0兲. Hence, the properties of the liquid-crystal layer are homogeneous. On the other hand, in Fig. 3(b) we show the director profile for which V共x兲=Vth+ x / L共Vb− Vth兲, where L=150 is the total length of the optical system, Va= Vth= 1.0558 V, and Vb= 1.2461 V, corresponding tomax=/ 4 rad. In this case, the director profile has a gradient in both the vertical and the horizontal direction.
First, we will investigate the situation in Fig.3(a), where, at x = 0, an incident extraordinary ray enters the liquid crystal at approximately 12° with the ver-tical z direction. This extraordinary ray is repeatedly reflected by the two ideal parallel mirrors and propa-gates in the positive x direction. Since the director profile dˆ 共x,z兲=共1,0兲, the ray path of the light ray con-sists of straight lines. Figure4(a)shows the ray path of the extraordinary light ray inside the liquid crys-tal. Figure4(b)shows the angle of reflectioninside the optical system as a function of the number of re-flections. As expected, is constant throughout the system. After approximately 145 reflections, the light ray leaves the system at x = 150.
Fig. 2. (Color online) Ray paths of extraordinary rays for four different values of Vr. In (a), Vr= 1. Here, the liquid crystal is homogeneous and the ray path is a straight line. In (b)–(d), Vr increases to 1.0062, 1.0252, and 1.1803, re-spectively. The ray paths are oscillatory, and the period de-creases with increasing Vr.
Fig. 3. Liquid-crystal layer applied between two ideal par-allel mirrors, separated by a distance h. In (a), V共x兲ⱕVth
and dˆ =共1,0兲. In (b), V共x兲=Vth+ x / L共Vb− Vth兲, where L
= 150 is the total length of the optical system and Vb⬎Va.
Fig. 4. (Color online) In (a), the ray path of the extraordi-nary light ray (TM mode) is depicted inside the liquid-crystal layer as defined in Fig.3(a). In (b), the angle of re-flection of the ray path is constant, and eventually, the ray leaves the system at x = 150.
Second, we examine the configuration of Fig. 3(b), where the light ray enters the liquid crystal again at
x = 0 at 12°. The light ray is reflected by the two ideal
parallel mirrors and initially propagates in the posi-tive x direction. Since in this case the director profile dˆ 共x,z兲 is inhomogeneous, the ray path of the light ray is curved and affected in the horizontal direction. Owing to the presence of a lateral gradient in the liquid-crystal profile and the well-known fact that light bends toward regions with high refractive in-dex, decreases after each reflection; see Fig. 5(b). After 120 reflections共x75兲,is reduced to negative values. These negative values for imply that the ray is now propagating in the negative x direction. After 240 reflections, the light ray leaves the system at x = 0, as can be seen in Fig.5(a). From these obser-vations, we can conclude that the horizontal direction of propagation of the ray is backreflected owing to the lateral gradient in the director profile.
To realize an acceptable efficiency in a real appli-cation, we suggest the following measures. For a high reflectivity of the mirrors, we suggest the use of di-electric stacks with a reflectivity of 99.9%, similar to mirrors used in laser cavities. Then, after 240
reflec-tions, the total reflectivity is reduced to 78% versus a reduction to below 1% for mirrors with 90% reflec-tance. To minimize absorption and scattering, elec-trode structures could be fabricated outside the liquid-crystal layer on top of the dielectric stack. To optimize the optical response and minimize the num-ber of reflections, the birefringence⌬n=兩ne− no兩 of the
liquid crystal should be approximately 0.3 or higher. Moreover, the gradients in the liquid crystal can be optimized with the thickness h and the voltage V共x兲. With these suggestions, we believe that a total effi-ciency of 10% or higher is feasible.
In a three-dimensional design, the director is par-allel to the x–z plane. Since the director has no gra-dient in the y direction, light is not affected by the liquid crystal in the y direction. Therefore, the idea of multiple switches in parallel is possible and mainly depends on the collimation of a light source and any residual scattering effects. In view of this discussion, we point out the self-confinement of spatial optical solitons in nematic liquid-crystal cells: in [3] simula-tions reveal an oscillatory self-confined light beam in a 75m-thick liquid crystal cell, similar to the opti-cal behavior presented in Fig.2.
We have shown that a two-dimensional gradient in a liquid-crystal layer with a Freédericksz alignment applied between two ideal parallel mirrors enables a backreflection phenomenon in which the direction of propagation of extraordinary light rays (TM mode) is reversed. As a result, the propagation direction of light in the proposed optical system can be switched by means of an external electric field. Hence, the op-tical system behaves like an electro-opop-tical switch. References
1. P. J. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics (Taylor & Francis, 1997).
2. M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, J. Opt. Soc. Am. A 25, 1260 (2008).
3. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, Opt. Quantum Electron. 37, 95 (2005).
Fig. 5. (Color online) In (a), the ray path of the extraordi-nary light ray (TM mode) is depicted inside the liquid-crystal layer as defined in Fig.3(b). From (b), we conclude that the angle of reflection decreases along the ray path: the direction of propagation is reversed, and the light ray leaves the system at x = 0.
