Two-dimensional photonic crystal infiltrated by liquid ...

Two-dimensional photonic crystal infiltrated by liquid crystal: response to applied electric field P. Halevi, J. H. Arroyo-Nunez, Instituto Nacional de Astrofisica Optica y Electronica (Mexico); J.A. Reyes-Cervantes, Instituto de Física, Universidad Nacional Autonoma de Mexico.

ABSTRACT We have calculated the photonic band structure of a 2D photonic crystal whose empty cylinders are infiltrated by a liquid crystal; a D.C. electric field is applied in the direction parallel to the cylinders. The local dielectric constant within the cylinders is obtained by minimizing the free energy, which has elastic and electrostatic contributions. We have assumed strong anchoring of the molecules of the nematic liquid crystal at the cylinder boundaries and have averaged over the cross-sectional area of the cylinder. The resulting dielectric tensor is diagonal and depends on the applied field. Moreover, it has the same symmetry as a uniaxial material, so that the optical response of the H-modes and E-modes is given by different dielectric constants (“ordinary” and “extraordinary”). The photonic band structures exhibit a notable dependence on the applied field with shifts up to 6% of the bands. For the E-modes, with a careful choice of the filling fraction it is possible to design a complete photonic gap for a certain range of electric fields, and close the gap for other values of the field. Such behaviour could be applied to optical tuning, switching, and polarizing of light.

1. INTRODUCTION Photonic Crystals (PCs) have by now reached a mature state of development, with their optical properties well understood and with many realized and potential applications [1]. Most of this research deals with PCs whose characteristics are fixed, that is, once they have been fabricated there is no possibility to alter their optical response. A recent trend, however, concerns tunable or active PCs; by this we imply that, by means of some external agent, it becomes feasible to change the optical properties of the PC continuously and reversibly. Clearly this feature could lead to applications such as continuous control of Photonic Band (PB) gaps, optical switches and limiters, polarizers, etc. We can classify tunable PCs according to two broad categories. For one of these, the tuning causes structural changes with no alteration of the dielectric constants of the constituent materials. In the other category the configuration of the PC remains the same, and it is some material property of the PC that is affected by an external agent. As is evident from our list of references, some 35 papers have been published on tuning of PCs, which attests to the scientific and technological interest in this subject. Structural tuning has been proposed or accomplished by means of a variety of methods. Yoshino et al [2] applied mechanical stress to a polymer opal of nanoscale spheres, as a result of which reflectance peaks in the visible exhibited red shift; this was interpreted to be the result of changes in the lattice constant in the direction of the beam. Calculations by Kim and Gopalan [3] show that a 2D PC attached to a piezoelectric substrate can be strained by applying an electric field to the substrate; the hexagonal lattice becomes “pseudo-hexagonal” and substantial shifts in the PB gap edges and widths arise. Further, Lourtioz et al [4] did simulations on a 2D structure of wires (period on the order of mms), each of which has periodic insertions of diodes; depending on the polarity of a DC electric field applied to the wires they can be continuously connected or disconnected. This produces strong alterations in cm- and mm- wave transmission spectra. Golosovsky et al [5] reported that disk shaped magnetic particles can self-assembly into various 3D periodic structures; in continuation Saado et al [6] demonstrated that such magnetic PCs possess PB gaps that can be tuned in the microwave regime by means of an applied magnetic field. Next we consider tuning by means of the alteration of some material property of the PC. The following classification is convenient: (a) ferroelectric or ferromagnetic PCs that are tuned, respectively by an external electric or magnetic field; (b) PCs that incorporate a semiconductor with a substantial density of free electrons and/or holes, tuned by changing the

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Photonic Crystal Materials and Devices, Ali Adibi, Axel Scherer, Shawn Yu Lin, Editors, Proceedings of SPIE Vol. 5000 (2003) © 2003 SPIE · 0277-786X/03/$15.00

temperature or the impurity density; (c) the presence of optical nonlinearities, with the tuning accomplished by laser illumination; and (d) infiltration of the PC with a liquid crystal - remarkably sensitive to temperature changes, to applied electric and magnetic fields, and to pressure. Below we briefly review each of these four mechanisms of tuning. The use of ferroelectric or ferromagnetic materials for tuning PCs was proposed in general terms by Figotin et al [7], who pointed out that the nature of wave propagation changes dramatically as the dielectric constant or magnetic permeability vary under the influence of an applied field. Kee et al [8] studied the changes in PB structure, induced by an external magnetic field, for 1D and 2D PCs that incorporate a ferrite; the magnetic permeability depends on the frequency, as well as on the field. As for ferroelectric substances, in the vicinity of a phase transition, they are also sensitive to temperature changes. This property was used by Zhou et al [9], who infilled a SiO2 colloid crystal with ferroelectric BaTiO3 and tuned the PB gap and the transmittance by varying the temperature. Recently Halevi and Ramos-Mendieta [10] proposed that, if a semiconductor is made to be constituent of a PC, then it is possible to tune the optical response of the PC by varying the density of the free electrons and/or holes in the semiconductor. This has been concluded on the basis of PB structure calculations for 2D PCs, by varying either the temperature or the density of impurities. Simulations for 1D PCs [11] show that even with realistic allowance for absorption, large changes in the reflectance are obtained with modest variations of the temperature or impurity density. Subsequently, Kee and Lim [12] extended this idea to a complete PB gap-for both TE and TM modes. Even in semiconductors, such as Si and GaAs, with a large band gap, electron-hole pairs can be generated by means of a sufficiently intensive pump beam or pulse. However, the densities of the free charge carriers are usually small, producing modest changes in the dielectric constant - that, of course, depend on the intensity of the pump beam. Such non-linear perturbations in the dielectric constant occur also for dielectrics, glasses, and liquids through the electro-optic effect and the photorefractive effect, for example. These ideas were applied to optical limiting and switching in both onedimensional [13] and two-dimensional [14] PCs. Switching by means of intensive laser light has been also studied for defect structures in 1D PCs [15] and a 3D PC in the mm-wave region [16]. The fourth and last category of tuning material properties is based on infiltration of a PC by a liquid crystal (LC). For a 1D PC with alternating layers of Si and a nematic LC, Chigrin et al [17] showed that an applied voltage can produce switching of the transmitted light; this is realized due to the Freedericksz phase transition which consists in the alignment of the nematic LC molecules parallel to a (sufficiently large) electric field. For the same type or structure, Ha et al [18] demonstrated that it is possible to tune a complete PB gap-for both TE and TM polarizations. Bush and John [19] considered a synthetic inverse opal whose voids are infiltrated with a nematic LC. Their 3D PB structure calculations show how the PB gap and density of states change substantially upon rotation of the LC molecules. Yoshino et al, Kang et al, and Meng et al [20] have confirmed experimentally that, indeed, synthetic opals infiltrated by liquid crystals, can be tuned by means of applied electric fields and also by variations of the temperature. Turning to 2D PCs, Leonard et al [21] infilled the pores of a macroporous Si PC with a nematic LC. With increased temperature the LC underwent a phase transition from the nematic to the isotropic phase. Thus, while below the transition the LC is described by an ordinary (no) and an extraordinary (ne) index of refraction, above the critical temperature it is characterized by a single index. The edges of the PB gap are shifted in the process, and this shift was measured for the TE polarized mode. Comparison with a PB structure calculation suggest that the anchoring of the LC molecules at the Si host walls is of the “escaped radial” type. Ki et al [22] did PB structure and transmittance calculations for Si and metallic rods in a nematic LC host. Assuming “axial” alignment of the LC molecules in the nematic phase, they studied the changes in the optical response as a result to the phase transition to the isotropic phase. Further, Susa [23] found large changes in the transmittance when increasing the refractive index of the LC from 1.4 to 1.6. A similar approach was taken by Pustai et al. [24] to simulate tuning of microcavities in a 2D PC. This paper also deals with 2D PCs infiltrated by a LC. Our calculations, for the first time, explicitly incorporate an electric field (applied parallel to the cylinders) as the agent of tuning. The most difficult theoretical aspect is the determination of the orientation of the rodlike nematic LC molecules at a given point within an LC infilled cylinder. The orientations of the “directors” depend on the LC itself, on the host material, on the geometry (here circular cylinders), and on special treatments that are frequently given to the contact (anchoring) surface [25]. Acording to Burylov [26], the

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directors of the molecules usually have one of three types of configurations: planar polar; planar radial (the directors lying, in both cases, in the plane perpendicular to the cylinders); and escaped radial. Other, as yet unrealised configurations, have been also studied for cylindrical cavities [21, 26]. The escaped radial geometry, which we will assume to hold, is realized when the nematic LC molecules, while anchored to the walls of the cylinders, “escape” toward the axial orientation at the cylinder axis. A further difficulty in the modeling is the question of the anchoring strength; we choose to assume the strong anchoring limit. Unfortunately, the nature and strength of anchoring for different cases of interest are mostly unknown, even in the absence of an applied field. We made numerous assumptions, discussed in sec. 2. Based on these, in sec. 3 we calculate the dielectric tensor, in the presence of an axial electric field, at a given point inside a circular cylinder. This results in a local and anisotropic (non-diagonal) dielectric tensor. As an additional simplifying step, we average this tensor over the cross-sectional area of the cylinder; as a consequence, a diagonal dielectric tensor -with uni-axial symmetry- is obtained. Thus, even with a DC electric field present, the basic symmetry of the 2D PC is preserved and it still supports independent TM and TE modes. In sec. 4 we present the band structures for each of these modes, with the applied field as a parameter. Finally, aspects of tuning and possible applications are discussed in sec. 4.

2. ASSUMPTIONS AND APPROXIMATIONS This being the first simulation of tuning of a PC by means of an external field we will make no attempt at generality and do not claim that our results hold for some particular set of LC and host materials. Rather, our aim is to highlight the interesting theoretical issues involved and motivate further research on the subject. It is important to realize that the dielectric tensor of a nematic LC bounded by a cylindrical medium is much different from the bulk dielectric tensor (with all the molecules aligned in one direction). So, in this section, rather than considering the PC as a whole, we concentrate on a single, circular and infinitely long, cylinder of LC that is contained within an isotropic dielectric material. Here’s the list of the numerous assumptions and approximations involved in our calculation: a) Homeotropic anchoring of the nematic LC molecules at the cylinder walls. This means that the easy direction for the molecular orientation is the direction perpendicular to the wall at every point. According to Burylov [26], “in essentially every experiment to date the anchoring is of homeotropic type”. This, however, does not mean that the directors are perpendicular to the walls; the actual angle also depends on elastic forces due to neighboring molecules, on external fields, and on the strength of anchoring. b) This leaves us with three possible structures [26], namely the “planar polar”, the “planar radial”, and the “escaped radial” (ER). In the first two structures all the directors of the LC lie in the plane perpendicular to the cylinder. In case of the ER structure the directors form a certain angle with the wall; the angle gradually increases as the cylinder axis is approached and, on the axis itself, the directors are parallel to it. Rather arbitrarily, we assume the ER structure, which was also found to be plausible by Leonard et al [21]. It has been shown [26] (in the absence of an applied field) that the ER structure is favored energetically for cylinders that have a sufficiently large radius. c) The molecules can “escape” in either one of the two orientations parallel to the cylinders, and this is associated with the formation of point defects. We assume that there are no point defects and, in fact, for infinite cylinders this structure is energetically more stable [26]. d) We assume that the anchoring of the LC molecules at the cylinder walls is extremely strong, which is to say that the force on a molecule (that is adjacent to the wall) due to molecules in the wall is far greater than the forces due to neighboring molecules and the applied electric field. In this strong anchoring limit the actual direction of the director at the wall coincides with the easy direction. Having assumed homeotropic anchoring, this means that all the directors at the cylinder walls are perpendicular to it. Of course, strong anchoring is not the rule, for example Leonard et al [21] assumed weak anchoring for their particular choice of LC and host materials. e) The macroscopic description of the Van der Waals forces between the LC molecules is given in terms of the following formula [26] for the elastic contribution to the free-energy density: f el (r) = (1 / 2) K11 (∇ ⋅ n) 2 + (1 / 2) K 22 (n ⋅ ∇ × n) 2 + (1 / 2) K 33 (n × ∇ × n) 2 − K 24 ∇ ⋅ (n∇ ⋅ n + n × ∇ × n).

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(1)

Here n = n(r ) is the director, which –having unit magnitude– depends on the position vector r only through its direction. The elastic moduli K11, K22, and K33 describe, respectively, transverse bending (splay), torsion (twist), and longitudinal bending (bend) deformations. Frequently they are of the same order of magnitude for nematic LCs, and sometimes the calculations are simplified by the equal elastic constants approximation K11 = K22 = K33 ≡ K.

(2)

The last term in eq. (1) contributes to the free energy only at the cylinder walls and is negligible for strong anchoring. Hence we assume that K24 = 0. f)

(3)

We assume that the wave field E is much weaker than the applied field E0: E 1 the field Eo essentially overcomes the Van der Waals forces between the molecules. Note that the effectiveness of the field E o is greatly augmented for large radii R of the tubes. Eq.(11) can be seen as a special case of a formula for the free energy that has been derived by Reyes and Rodriguez [27], who studied guided waves in a LC fiber. In their eq.(4) we made the replacements E z Eo, Er 0, and H φ 0 for TM wave fields. Also, the elastic constant K there is replaced here by K/2 to conform with the conventional formula eq. (1) for f el (r ) .

The stationary orientational configuration θ (x) is determined by minimizing the free energy, eq. (11). This minimization leads to the Euler-Lagrange equation [27]

228


d 2θ

dθ − sin2θ(x ) − qx 2 sin2θ(x ) = 0 . (13) dx dx On the axis of the cylinder (x = 0) this equation reduces to sin2θ(0) = 0, which has the solutions θ = π/2 or θ = 0. The first possibility corresponds to the planar polar or the planar radial structures. Our choice of the escaped radial configuration (assumption (b)) dictates the boundary condition θ(0) = 0. because we also assumed homeotropic hard anchoring (assumptions (a) and (d)), the second boundary condition is θ(1) = π/2, namely the directors are perpendicular to the cylinder walls at every point. Eq.(13) is then solved numerically by the shooting method [28] for a series of values of the dimensionless parameter q, defined by eq.(12). The results are given in Fig. 1. 2x 2

2

+ 2x

Fig. 1 The angle between the nematic liquid crystal director and the cylinder axis as function of the normalized radial distance (x = r/R) from the axis for seven values of the field parameter q: 0, 3, 6, 10, 20, 40, 80.

Fig. 1 clearly shows that, for any value of the field parameter q, the directors are constrained to the axial direction on the cylinder axis and to the perpendicular direction at its walls. Not surprisingly, the larger E 0 the more effective is the axial alignment. In the absence of the applied field (q = 0) the last term in eq. (13) vanishes; the resulting equation was solved analytically by Cladis and Kleman [29]: θq (x) = 2 tan -1x, q = 0

(14)

Lim θq(x) = 0,

(15)

Also, from eq. (13), q →∞

x ≠ 1.

With θq(r) thus calculated numerically, the eqs. (9) completely determine the dielectric tensor ε ij (r ) at every point of the LC and for all values of the field. In principle, now one could procede to the calculation of the PB structure, with allowance for the inhomogeniety and anisotropy of the LC cylinders, as done by Hornreich et al [30]. However, the treatment greatly simplifies by averaging the dielectric tensor over the cross-sectional area of the cylinder. That is, we replace ε ij (r ) by


229

ε ij = Upon substituting eqs. (9) it is readily seen that

ε ij

1

πR 2

R

2π

rdr dϕε ij (r ) .

∫ ∫ 0

(16)

0

= 0 for i ≠ j. The remaining elements of

ε ij

are

ε xx = ε yy = ε o + (1/2) ε a Φ(q) ε zz = ε o + ε a [1 − Φ(q)]

(17) (18)

1

Φ(q ) = 2∫ dx ⋅ xsin 2 θ q (x ) ,

(19)

0

with θq(x) given by Fig.1. The integral eq.(19) is solved numerically and the function Φ (q) is plotted in Fig.2.

Fig.2 The function Φ (q), defined by eq. (19), determines the dielectric response of a nematic liquid crystal tube as function of the applied electric field E0, parallel to the tube. The parameter q is given by eq. (12).

The limiting behavior of Φ (q) may be found with the help of eqs.(14) and (15). It is

Φ(0) = 4ln2 − 2 ≅ 0.772 Lim Φ(q ) = 0 , q →∞

(20) (21)

well corresponding to Fig. 2. This also sets the limiting values of the averaged dielectric tensor. In the absence of the external field, by eqs. (17), (18), and (20), we have

ε xx (0) = ε yy (0) = ε o + (2ln2 − 1) ε a 230


(22a)

ε zz (0) = ε o + (3 − 4ln2) ε a

(22b)

while, in the strong field limit, with the help of eqs.(17), (18), and (21), the averaged dielectric tensor reduces to eq.(6). Because, in this limit, all the nematic LC molecules (ignoring the singularity at r = R) are parallel to the cylinder this could have been expected. It is interesting to consider how the dielectric tensor elements ε ij (q) evolve with increasing electric field. To be specific, for a positive nematic LC ( ε a > 0), from eqs. (22) we find that ε zz (0 ) − ε xx (0) ≅ −0.158ε a < 0 .

(23)

This means that the anchoring of the molecules at the cylinder walls converts the intrinsically (bulk) positive LC to a negative LC in the average. Now, as the field increases, Φ (q) and ε xx (q) decrease monotonously, while ε zz (q) increases monotonously. Clearly, for sufficiently large fields, the LC will become positive again – as, of course, it is in the limit q ∞. Then there must be a specific value of q for which the LC is neither negative, nor positive, namely when

(24) ε xx (q∗) = ε zz (q∗) . From egs. (17) and (18) we easily find that this occurs for Φ (q*) = 2/3, which has the solution q* ≈ 5 independently of the values of ε o and ε a . For the corresponding value of E0 the LC becomes isotropic! Just the opposite happens with an intrinsically negative LC ( ε a < 0). Without an electric field it is, in the average, a positive LC. With increasing field it becomes isotropic for q ∗ ≈ 5 and, for q > 5 turns to be a negative LC. The averaged dielectric tensor of the nematic LC cylinder is completely specified by eqs. (17) – (19) for any value of the field Eo. The LC tube is uni-axially anisotropic for all values of Eo except one – given by Φ(q ) = 2/3 – for which it is isotropic. Table 1 gives ε xx (q) and ε zz (q) for seven values of q. We have chosen ε o = 2.2201 and ε a = 0.6360 corresponding to the LC “E7” [21].

q 0 2 ≈5 10 50 100 1000 ∞

Φ(q ) 0.7726 0.7280 0.6667 0.5588 0.2891 0.1993 0.0619 0

ε xx (q )

ε zz (q )

Nature of LC

2.4627 2.4475 2.4267 2.3899 2.2983 2.2677 2.2210 2.2201

2.3546 2.3850 2.4267 2.5000 2.6834 2.7445 2.8379 2.8561

negative nematic negative nematic Isotropic positive nematic positive nematic positive nematic positive nematic positive nematic

Table 1. Illustrating the transition between three phases of the liquid crysta l with increasing electric field.

4. BAND STRUCTURE. Now let us consider the two-dimensional PC as a whole. It is characterized by a position-dependent dielectric tensor whose non-vanishing elements are


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ε + (1/2)ε a Φ(q), x and y in LC cylinder ε xx (x, y ) = ε yy (x, y ) =  o  ε h , x and y in host material  ε + ε a [1 −Φ(q)], x and y in LC cylinder ε zz (x , y ) =  o ε h , x and y in host material

(25)

(26)

where the function Φ (q) is defined by eq. (19). Here we have used the eqs. (17) and (18) and have assumed that the host material that surrounds the LC cylinders is isotropic, having a dielectric constant ε h . This dielectric tensor has two important properties: (a) it has uni-axial symmetry and (b) it is translationally invariant in the direction z parallel to the cylinders. As a consequence, the determination of the normal modes of the system is greatly simplified. For a nonmagnetic system and harmonic time dependence the general Maxwell wave equation is ∇ × ∇ × E = (ω/c) 2 D .

(27)

First we take the z – component of this equation. Because the fields must be independent of z this reduces to − ∇ 2 E z = (ω/ c )2 D z = (ω/ c )2 ε zz (x, y ) E z .

(28)

Now taking the transverse (in-plane) component of eq. (27) we get ∇ t (∇ t ⋅ E t ) − ∇ 2t E t = (ω/c) 2 D t = (ω/c) 2 ε xx (x, y) E t

(29)

where E t = E - E z zˆ and ∇ t = ∇ − zˆ ∂/∂z is the transverse gradient operator. We note that eq. (28) is a wave equation for the longitudinal wave field E z ; it does not involve the transverse field E t . One the other hand, E z is absent from eq. (29); this wave equation governs the behavior of the in-plane field E t . Thus we reach the important conclusion that there are two independent modes: the “E-mode” with E// zˆ and H⊥ zˆ and the “Hmode” with H // zˆ and E⊥ zˆ . This is a generalization of the well known situation for 2D PCs constituted with isotropic materials; now the LC cylinders are uni-axially anisotropic. As seen from eqs. (28) and (29), the eiqenvalue problem for the E-mode depends only on the dielectric tensor element ε zz (x, y ) ; on the other hand, for the H-mode it depends only on the element ε xx (x, y ) . By eqs. (26) and (25) then, the dielectric constant that characterizes the LC is

()

[

( )]

εE q ≡ εo + εa 1− Φ q

()

()

for the E - modes

ε H q ≡ ε o + (1/2 ) ε a Φ q for the H - modes .

(30)

(31)

The final conclusion is that the PB structure problems can be solved as if the LC were isotropic, however with the above, different, dielectric constants for the two modes. Now we can procede to apply the usual theory for the E – and H – mode [1]. The host material is assumed to be silicon ( ε = 11.7 ). A square array of circular cylinders is infilled with the nematic LC “E7”; its ordinary and extraordinary dielectric constants are ε o = 2.2201 and ε e = 2.8561. We are assuming that the wave propagation is limited to the plane of periodicity. First we present the band structure for the H-mode, using eq. (31), in Fig.3. The cylinder filing fraction is f = 0.5 in (a), while in (b) the cylinders are almost touching. The P B structures are plotted for no applied electric field present (Eo= 0 or q = 0, thin line), for a field of “medium magnitude” (q = 10, 232


medium thickness), and for a very strong field (q = 1000, thick line). We have not found a complete PB gap for the Hpolarization for any value of the filling fraction and any value of the field. For f = 0.5 there are partial gaps for propagation in the ΓX and ΓM directions. However, the latter gap is completely closed for almost touching cylinders (b). We note that, for q =1000, some bands shift by as much as 6%. 0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0

r = 0.4 a ( a) Γ

X

M

Γ

0.0

r = 0.49 ( b) Γ

a X

M

Γ

Fig.3 Band structures for the H-mode of a nematic liquid crystal–infilled two–dimensional photonic crystal. The field parameter q takes the values 0 (thin line), 10 (medium thickness), and 1000 (thick line). (a) R = 0.4a, (b) R = 0.49a, where a is the lattice period.

The PB structures for the E-mode have been calculated using eq. (30), Fig. 4. The same three values of q, as in Fig. 3, have been selected, and the cylinder radii are R = 0.3 a (a), R = 0.425 a (b), and R = 0.49 a (c). We see that, for a relatively small R(a) the situation is similar to that in Fig. 3 (a), namely, there is no complete PB gap for any value of q. On the other hand, for almost touching cylinders (c) there is a complete PB gap – for all values of q. The most interesting situation in exhibited in (b), where the existence or non-existence of a complete gap depends on the value of q. This can be better appreciated in part (d) that renders the maximum of the first PB and the minimum of the second PB as function of the field parameter q. We note that for a q–value a little above 20 the band gap just vanishes. For smaller q´s the gap is open and for lager q´s the bands overlap. Clearly then, for cylinder radii R = 0.425a one can open and close the PB gap by changing the magnitude of the electric field applied parallel to the cylinders. Moreover, the width of the gap (for q < 20) can be regulated by the field Eo; the smaller Eo the larger is the PB gap, up to a maximum value of about 0.004 (2 π c/a) with the field shut off. We see then that the PB gap for the E-modes can be continuously tuned by relatively small fields Eo. Further, this system can also function as an optical switch: for instance, light of frequency 0.226 (2 π c/a) polarized parallel to the cylinders will be completely reflected if the field is off, but will be (partially) transmitted if a sufficiently large field (say, for q = 100) is turned on. In addition, this tunable PC can also act as a field-sensitive polarizer: for example, unpolarized light of frequency ω = 0.225 (2 π c/a) will be transmitted if q < 10; however, only the H–polarized component will be transmitted if q > 10 (as long as q < 25), as can be seen from Fig. 4 (c). This is because there is no complete PB gap for the H–modes for any value of q.

5. CONCLUSION. On the basis of PB structure calculations, we have demonstrated that a 2D PC, infilled with a nematic LC, can be (continuously) tuned and switched by means of an electric field, applied parallel to the cylinders. This is true for the E– mode (whose electric field is also parallel to the cylinders) in the sense that the PB gap gradually decreases as the Proc. of SPIE Vol. 5000

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0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

r = 0.3 a ( a) Γ

r = 0.425 ( b) X

Γ

M

a

Γ

X

Γ

M

10

100

1000

0.230

0.5

0.229

m a x i m u m of 1st band m i n i m u m of 2nd band

0.228 0.4

0.227 0.226 0.225

0.3 0.224 0.223 0.222

0.2

r = 0.49a ( c) Γ

0.221

X

M

Γ

( d) 10

q

100

1000

Fig.4 As in Fig.3 for the E-mode and for three values of the cylinder radius: (a) R = 0.3 a, (b) R = 0.425a, (c), R = 9.49a. In (d) we plot the maximum of the first band and the minimum of the second band as function of the field parameter q for R = 0.425a [as in (b)]. There is a pass band for q < 30 and a stop band for q > 30.

234


external field Eo is increased, until it closes completely for a certain value of Eo. A switching action is also feasible, with the light being completely reflected when the applied voltage is off, and being transmitted when the voltage is switched on to a sufficiently high field value. On the other hand, for the parameters considered we found no PB gap for the H– mode. This fact could be exploited for a field–sensitive polarizer, with unpolarized light being transmitted below a certain critical value of Eo, however with only the H–polarized component transmitted for higher field values. Microscopically, this behavior stems from the LC molecules becoming more and more aligned with the cylinder axis as the external field is increased, until, for extremely high fields (or, what amounts to the same, very wide cylinders) all the molecule directors are parallel to the cylinders. Our calculations have been based on numerous assumptions and approximations, and we would hesitate to claim that the results hold for this LC or that host material … In particular, much further study is needed of the anchoring of the nematic LC molecules at the cylindrical walls of the host material. It would be interesting and useful to perform PB structure calculations, similar to those presented here, for other types of configurations of the molecules and for the anchoring strength as an arbitrary parameter. Further, the approximation of equal elastic constants should be discarded, and our averaging of the dielectric tensor of the LC should be replaced by a PB structure calculation based on the complete, inhomogeneous and anisotropic, dielectric tensor for the LC. Hopefully, such improvements will be realized in the future and will yield results that can predict and explain experimental behavior of LC infilled PCs. In this paper the applied field E o was represented by the parameter q, defined in eq. (12), that is proportional to E o2 and to the radius of the cylinders squared, R 2 . As can be seen from Table 1, the difference between ε zz (100) and ε zz ( ∞ ) is

about 4%, and the difference between ε xx (100) and ε xx (∞ ) is less than 2%. Thus we can consider q = 100 as representing a very strong electric field. What would be the actual value of such a field? If we take ε a = 0.6 and K = 10 6 dyn, and express R in µm and E o in V/cm then eq. (12) gives E o(V/cm) ≅ 450/Ro (µm). Then, for a very small radius R = 0.1 µm, Eo ≅ 4,500 V/cm. Considering a capacitor (“cell”) of a width of 30 µm, this corresponds to an applied voltage Vo ≅ 13 Volts. This is well below the voltage that would produce an electric arc (~ 250 V). This calculation shows that, even for very narrow cylinders, it is experimentally possible to achieve the “high-field” value q = 100. Finally we note that the reorientation times of the molecules can be as short as miliseconds, see Leonard et al [14].

ACKNOWLEDGEMENTS We wish to thank Rosalio F.Rodriguez-Zepeda, Felipe Ramos-Mendieta and Adan S.Sanchez for very helpful advice. This project was supported by CONACyT grant #32191-E.

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