## Abstract

The photonic crystal with a square lattice of Si cylinders surrounded by nematic liquid crystals is investigated. The directors of the liquid crystal are parallel to the two dimensional PC plane. The transversed electric (TE) modes are studied. The plane wave expansion method is used to calculate the equi-frequency surface (EFS). The negative refraction phenomenon is found and the refractive angle can be tuned by changing the directors of the nematic liquid crystals.

©2006 Optical Society of America

## 1. Introduction

The negative refraction and the negative phase velocity of electromagnetic waves in left-handed materials have attracted a lot of attention, recently. In 1968, Veselago [1] first analyzed a material that had a negative index. In the pass few years, new developments in structured composite materials have given rise to negative refractive index materials which have both negative dielectric permittivity and negative magnetic permeability in some frequency ranges. In the microwave regime, negative refractive index materials can be fabricated from metallic wires and split-rings [2] assembled in a periodic cell structure.

In contrast to the split ring resonators (SRR), the photonic crystals (PhCs) consist of periodically modulated dielectric constants. It was shown that diffraction effects of PhCs can produce effective negative refraction or even negative index [3–6], hence the applications of PhCs are not limited in the band gap region. Generally, there are two kinds of negative refraction in PhCs [7]. One is the left-handed behavior as being described by Veselago. The other negative refraction is realized without negative index or a left-handed behavior, but by the high order Bragg scattering [5] or anisotropy [6]. In the first case, the wave vector **k** and the Poynting vector S are antiparallel (i.e., **S** · **k** < 0). In the latter case, the PhCs behave much like a right-handed medium (i.e., **S** · **k** > 0). The losses can be much smaller comparing to meta-materials due to the use of non-conducting dielectric materials. Both negative refraction and left-handedness in PhCs were experimentally demonstrated in the microwave region [8–10].

The idea of negative refraction opens up new frontiers in optics. Pendry mentioned the concept of a perfect lens [11] that enables imaging with sub-wavelength image resolutions. The focusing properties of PhCs have become a hot topic of scientific research over the past few years. Its resulted focusing effects, caused by anisotropy in square-lattice PhCs, have been investigated by Lou *et al* [6] and Chien *et al* [12]. Moussa *et al*. [13] have showed both theoretically and experimentally that a triangular lattice of rectangular dielectric rods can have behavior such as negative refractions and superlensing.

The anomalous refractive properties of photonic crystal are usually analyzed by equifrequency surface (EFS). From the fact that the EFS varied when its refractive index changed, we speculate that the propagation direction in photonic crystal structures could change. Therefore, we could use the index-tunable materials to control the refractive direction. The tunable materials such as ferroelectrics [14–15] and liquid crystals [16–19] etc. have been proposed. Except for changing refractive indices, the refraction angle could varied mechanically [20], but the structure is too complicated and indirectly.

The properties of LCs can be changed easily by applying an external electric fields. Hence, they are adaptable for tuning. Bush and John [16] predicted the tunability of band structure in photonic crystals utilizing liquid crystals. Takeda and Yoshino [18] showed the tunable refraction effects in the LC infiltrated PC structures. Liu and Chen [19] demonstrated that the photonic band gap of a 2D PC can be modulated by the nematic liquid crystal.

In this communication, we consider a square lattice of Si cylinders surrounded by the nematic liquid crystal. The cases of the directors of LCs parallel to the two dimensional planes and transversal electric (TE) mode are studied. The electric field of the optical wave oscillates only in the 2D plane and can be affected by rotating directors of LCs under the applied electric field. Therefore, we could control the refraction of PhCs.

In the next section, the plane wave expansion method is used to calculate the EFS for predicting the refractive angle. The appropriate frequencies which have negative refraction are discussed in the third section. The tunable negative refraction is demonstrated and the results are compared with those obtained by the finite difference time domain method in the fourth section.

## 2. Numerical method

The simplified Maxwell’s equation can be written as:

Since light waves are transmitted in periodic structures, we can use Bloch’s theorem to expand the H field in plane waves:

where **k** is the wave vector in the Brillouin zone, *e*
_{G} is the direction perpendicular to the 2D plane, and **G** is the reciprocal lattice vector. The dielectric constant of photonic structure is periodic with respect to the lattice vector. Similarly, we can expand it in a Fourier series on **G**:

Liquid crystals generally possess two dielectric constants. One is the ordinary dielectric constant ε^{o} and the other is the extraordinary dielectric constant ε^{e}. The light waves with electric fields perpendicular and parallel to the director of the LC have ordinary and extraordinary dielectric constants, respectively. When the director rotates on the 2D plane, the components of the dielectric tensor can be represented as:

where Φ is the rotation angle of the director. The director is presented by n = (cosΦ, sinΦ). By inserting Eq. (3) and (2) into Eq. (1) and multiplying by e_{G}, we can get the following infinite eigenvalue matrix problem:

The main numerical problem in obtaining eigenvalue is the evaluation of the Fourier coefficients of the inverse dielectric tensors. The best method shown by Ho *et al*. (HCS) is to calculate the matrix of Fourier coefficients of real-space tensors and take its inverse to obtain the required Fourier coefficients. The error is less than 1% with HCS methods for 441 plane waves [21].

## 3. PhCs surrounded by LC

We consider that a 2D PC is composed of square-lattice Si cylinders surrounded by the
nematic liquid crystal. The liquid crystal is chosen to be 5CB and the ordinary and extraordinary refraction index of 5CB are *n ^{o}* = 1.522 and

*n*=1.706, respectively. The liquid crystals become isotropic when the external electric field is not applied and the average refractive index is

^{e}*n*= (2

^{av}*n*+

^{o}*n*)/3 = 1.583. Because the refractive index of Si cylinders compared with that of LC is high, we can not get high anisotropies caused by LC. In order to have high anisotropies, a smaller radius r = 0.25a is adopted to improve the anisotropies. The TE mode of electromagnetic wave of the Si photonic crystal with the nematic liquid crystal as a background material is considered. The directors of the LC are parallel to the 2D plane, as shown in Fig. 1. The electric fields of electromagnetic wave exist only in the 2D plane and are affected by the direction of the directors of LC. When the directors are rotated by applying external electric fields in 2D planes, the anisotropies of the PhC will be changed. Then, the constant frequency contours are changed due to the variation of the anisotropies and the refractive angles will be different.

^{e}The band structure of the PhC surrounded by the isotropic LC is shown in Fig. 2. At low frequency region, the wavelength is much larger than the lattice constant, therefore the PhC behaves as a homogenous medium. The anomalous refraction does not appear at frequencies below the normalized frequency ω=0.25(ωa/2πc). It is because that the anisotropies caused by the LC are not remarkable and the constant frequency contours do not vary obviously as the directors rotate. Fig. 3 shows the EFS of the first band in different director angles of Φ=0°, 45°, and 90°. We can see that the shapes of EFS do not change obviously with different director angle Φ. On the contrary, the wavelengths are more close to the lattice constant at high frequencies, so that the anisotropies are apparent and the constant frequency contours are more complicated. At the high lattice constant to wavelength ratio a/λ, the effect of evanescent wave [22] would become so strong that the EFS can not predict the refractive direction accurately. In order to have more significant effect, the 2^{nd} band is chosen to study the negative refraction phenomenon.

## 4. Tunable Refraction

The incident lights of two normalized frequencies ω=0.43 and ω=0.44 are adopted to investigate the negative refraction phenomenon. Fig. 4 shows the constant-frequency contours of the second band of Φ=0°, 30°, 45°, and 90°. The contours of EFS are seen to change more significantly at the higher frequency when the directors rotate. In Fig. 4(a), one can see that the contour of ω=0.44 is an ellipse and the main axis of the ellipse is parallel to the x-axis. As the director rotates, the contour is deformed continuously and finally the main axis is perpendicular to x-axis in Fig. 4(d).

The light-wave propagation direction in 2D PhCs is oriented to the group velocity vector:

and is always perpendicular to the EFS in the direction along which ω is increasing. The tangential components of wave vectors of incident waves and refractive waves are always conserved at an interface between two materials. By this momentum conservation, we can set up a construction line which is normal to the interface and intersect the constant frequency contour. Therefore, the normal direction at the intersecting point on the EFS is the group velocity direction. By this way, the refractive angle can be determined. The construction line for the tangential component **k**
_{y}=0.1 and the corresponding refractive direction is shown in Fig. 4(b).

We consider that the light propagates from air to PhC and smaller incident angles θ_{i} = 6.7° for ω=0.43 and 6.5° for ω=0.44, i.e. the tangential component **k**
_{y}=0.1 normalized by π/a, are chosen to prevent the construction line intersecting the EFS of the 3^{rd} band which causes other refractions. The relation between the refractive angle and the rotation angle Φ is shown in Fig. 5. The rotation angles Φ of the director of the LC are changed from 0°to 90°. The refractive angles are more sensitive to the rotation angle of LC at the higher frequency. At the normalized frequency ω=0.43, the refractive angles are changed in the range from -10° to 6°. At the other normalized frequency ω=0.44, the range of the refraction is larger and between -10° and -70°. The refractive angles are always negative.

When the rotation angle is small, we find that the construction line does not intersect the constant frequency contour, as shown in Fig 4(a), i.e. there exists partial band gap and the light can not transmit. An electrically controlled optical switch can be designed by this concept.

The finite difference time domain (FDTD) method is also used to simulate the light propagation in the PhC and examine the refractive direction predicted by EFS. The distribution of the magnetic field of the light at ω=0.43 is shown in Fig. 6(a) and (b). The refractive angle is from negative to positive when LCs rotate from Φ= 0° to 50°. The simulating results of FDTD are quite identical to the prediction by EFS. The magnetic field distributions of light at ω=0.44 are presented in Fig. 6(c) and (d). The propagation direction predicted by FDTD at Φ=90° is fit to the result of EFS, but the light beam disperse seriously at Φ=20° which is near the partial band gap of EFS. The dispersion of the light beam may be caused by evanescent effect [22].

## 5. Conclusion

In summary, we have demonstrated the tunable effects of a square lattice photonic crystal structure surrounded by LCs. Because of the anisotropic properties of LCs, the symmetric EFS of PhCs disappears. The constant frequency contour is affected by the direction of the director of LCs. Therefore, the refractive angles can be controlled by rotating the director of liquid crystals. The refraction angle can be modulated from negative to positive at ω=0.43. The more significant tunable negative refraction is achieved at ω=0.44. This behavior which is presented may bring new applications for designing the tunable devices like the optical switches in photonic integrated circuits. The refraction predicted by EFS is almost identical to the simulation results of FDTD. Further theoretical investigations and many experimental efforts are needed to bring the tunable refraction into reality.

## Acknowledgments

The authors sincerely acknowledge the financial assistance of the national science council of Taiwan under the Grant No. NSC 94-2212-E006-025.

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