Opt Quant Electron (2011) 42:487–497DOI 10.1007/s11082-010-9430-y

Optimal tunability of waveguides based on siliconphotonic crystals infiltrated with liquid crystals

J. Cos · J. Ferré-Borrull · J. Pallarès · L. F. Marsal

Received: 19 July 2010 / Accepted: 17 December 2010 / Published online: 6 January 2011© Springer Science+Business Media, LLC. 2011

Abstract In this work we study the optimization of the tunability range in waveguidesbased on two-dimensional silicon photonic crystal infiltrated with liquid crystal. The ana-lyzed structure consists of a two-dimensional silicon photonic crystal with a triangular latticeof circular holes where a line of scatterers in the direction �–K has been replaced by a lineof circular holes with different radius infiltrated by E7 liquid crystal. To this end, we use theplane-wave expansion method considering anisotropy and modelling supercells to accountfor the lattice defects that define the waveguide. Finally we study the field distributions ofthe guided modes in order to analyze their symmetries and confinement.

Keywords Liquid crystal · Photonic crystal · Waveguides ·Plane-wave expansion method

1 Introduction

Tunable photonic crystal (PC) technology is a promising technology for the development ofphotonic circuits. In this way, several works propose the combination of photonic crystalswith nonlinear optical (NLO) materials or liquid crystal (LC) materials (Busch and John1999; Takeda and Yoshino 2002; Maksymov et al. 2004). In the former case, a high-inten-sity control signal with frequency outside the bandgap changes the properties of the crystal(Cuesta-Soto et al. 2004). Among the possibilities for tuning the optical properties of silicon-based photonic crystals, liquid crystals represent a very suitable material for achieving thisproperty for different reasons. In one hand, an easy tuning of the optical properties can beachieved by means of the control of the anisotropy with external applied electric fields (Weisset al. 2005). On the other hand, there is a good material availability because its applicationin the display industry.

J. Cos · J. Ferré-Borrull (B) · J. Pallarès · L. F. MarsalNanoelectronic and Photonic Systems, Department of Electronic,Electric and Automatic Control Engineering, Universitat Rovira i Virgili,Avda. Països Catalans 26, 43007 Tarragona, Spaine-mail: josep.ferre@urv.cat

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Besides the tuning properties of the liquid crystals, the optical anisotropy is another funda-mental property than can be taken as a degree of freedom in the design of applications. In thelast years there are several works that exploit this anisotropy in the theoretical (Kopperschmidt2001; Arriaga et al. 2008) or the experimental (Schuller et al. 2005; Sun et al. 2007) analysisof photonic crystals. However, only a few studies on LC-infiltrated two-dimensional (2D)photonic structures are devoted to study the gap maps and tunable capabilities as a functionof the geometrical parameters of the photonic crystal for direct and inverse structures.

Despite the advantages of using liquid crystals for the tuning of the optical properties,several weak points have to be considered. First, liquid crystals usually used in the displayindustry have a larger refractive index than air and consequently the infiltration reduces thecontrast index with the consequence of a decrease in the performance of the devices. Second,these liquid crystals have a small birefringence in its anisotropic state (E7 liquid crystal:no = 1.52, ne = 1.70,�n = 0.18), which results in an even smaller tunability range. Con-sequently, in order to obtain actual working devices, such as couplers (Liu and Chen 2004),Y-shaped waveguides (Takeda and Yoshino 2003) or optical circuits (Maksymov et al. 2004),it is determinant to know how to obtain the optimal design from the available materials.

In this work we study the optimization of the tunability range in waveguides as a functionof the radius of the line of scatterers that defines the waveguide. To this end, we use theplane-wave expansion method and a supercell formulation of a 2D photonic crystal to takeaccount of the lattice defects that define the waveguide.

2 Numerical method

In this work we analyze the structure of the Fig. 1a consisting of a two dimensional siliconphotonic crystal with a triangular lattice of circular holes with radius R and lattice constanta, where a line of holes in the direction �–K has a different radius Rdef and is infiltratedby E7 liquid crystal. The triangular structure has a larger photonic band gap (PBG) for TEmodes than other geometries such as square, honeycomb, kagome,… (Susa 2002), for thesame refractive index contrast.

We consider the x–y plane as the periodicity plane of the PCs. In the case of isotropic2D photonic crystal, the Maxwell equations are decoupled in TE and TM polarizations. TEcomponents are Ex , Ey and Hz while TM components are Hx , Hy and Ez (Joannopoulos etal. 2008). However the inclusion of anisotropic elements causes Maxwell equations to becoupled except for certain liquid crystal optical axis orientations.

Following (Alagappan et al. 2006), Maxwell’s equations can only be decoupled in twocases: (1) when the optical axis is oriented along the rods of the 2D photonic crystal, and (2)

Fig. 1 Schematic view of the 2Dphotonic crystal waveguide.Dotted line and numbers indicatesthe supercell and the index in theEq. (8). The inset shows thedefinition of the angle α thatcharacterizes the LC optical axisorientation with respect x axis

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when the optical axis is perpendicular to the rods. Furthermore, only in the second case thedirection of the optical axis influences the photonic band structure. In these conditions wecan write the following eigenequations for the TM and TE polarizations, respectively:

− ε−1z (r)

(∂2

∂x2 + ∂2

∂y2

)Ez(r) = ω2μ0ε0Ez(r). (1)

∇xy × ε−1r (r) × ∇xy × Hz(r)ez = ω2μ0ε0Hz(r)ez . (2)

where ∇xy = ex · ∂/∂x + ey · ∂/∂y, and εz(r) and εr (r) are the position-dependent compo-nents of the dielectric tensor. For a position r corresponding with the liquid crystal these aredefined as

εz(r) = εo

εLC = εr (r) =(

εosin2α + εecos2α (εo − εe) · cosα · sin α

(εo − εe) · cosα · sin α εocos2α + εesin2α

)(3)

where α is the angle between the optical axis and the x-axis as it is stated in Fig. 1. For aposition r corresponding with an isotropic materal (silicon or air in the studied structure)εz(r) and reduce εr (r) to the corresponding dielectric constant.

In both Eqs. (1) and (2) is assumed the solution Az = A(r)e−iωt where ω is the angularfrequency. Applying plane-wave expansion method on Eqs. (1) and (2) we can obtain a pairof eigenequations for TE and TM respectively (Alagappan et al. 2006)∑

G′ε−1

zGG′(G − G′) · |k + G| · |k + G′| Fz,k(G′) = (ω/c)2 Fz,k(G), (4)

∑G′

(k + G) · ε−1rGG′(G − G′) · (k + G′)Hz,k(G′) = (ω/c)2 Hz,k(G), (5)

where ε−1zGG′ and ε−1

rGG′ are, respectively, the Fourier Transform of the inverse of the dielec-

tric constants εz(r) and εr (r). It is important to note that ε−1rGG′ is a 2x2 matrix, details about

the obtaining of ε−1rGG′ can be found in Cos et al. (2010). In order to produce a symmetric

eigenvalue problem it is defined Fz,k(G) as

Fz,k(G) = |k + G| · Ez,k(G). (6)

The plane wave expansion method is based on the expansion of the fields within the photoniccrystal as a superposition of plane waves propagating along the directions defined by thereciprocal lattice vectors. However, the inclusion of defects breaks the periodicity in one ormore directions, thus in order to apply periodic boundary conditions is necessary to use asupercell formulation. In the supercell formulation, the unit cell corresponds to several pho-tonic crystal unit cells, where some are replaced in order to introduce the defects. We havechosen a 7 × 1 supercell (dotted line in Fig. 1a) in order to model our structure. We havetested the accuracy of this supercell size by carrying out simulations also for bigger sizes(9 × 1 and 11 × 1 supercells) and confirmed that they lead to the same conclusions.

The supercell dielectric constant can be described as

εr (�r) =6∑

m=1

[εr,air (�r − �rm)

] + εr,LC(�rm), (7)

Where εr,air (�r) corresponds to the dielectric constant function of a unit cell of the periodicphotonic crystal, εr,LC (�r) corresponds to the dielectric constant function of the defect unit

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cell, �rm is the position of the photonic crystal scatterer, and m = 1, . . .6 corresponds to theindex of the unit cells as it is stated in Fig. 1a.

Thus, the Fourier Transform of the inverse of the dielectric constant of the supercell canbe expanded as

ε−1rGG =

6∑m=1

[ε−1

air (G − G ′) · e−i((G−G′)·�rm )]

+ ε−1LC (G − G ′), (8)

where ε−1air (G) and ε−1

LC (G) are defined respectively by

ε−1air (G) =

{( f · ε−1

air + (1 − f ) · ε−1b ) f orG = 0

(ε−1air − ε−1

b ) · S(G) f orG �= 0

}(9a)

ε−1LC (G) =

{( f · ε−1

LC + (1 − f ) · ε−1b ) f orG �= 0

(ε−1LC − ε−1

b ) · S(G) f orG = 0

}(9b)

with S(G) = 2 f J1(G R)(G R)

and J1 is the first-order Bessel function.εb and εair are two 2x2 identity matrixes multiplied by the dielectric constant of the

isotropic background and air respectively and εLC is defined in Eq. (3).

3 Results and discussion

As previously stated, the liquid crystal director is always in the x-y plane, thus guided modesare tuned by changing the angle between the direction �–K and the LC director, α. Accord-ing to Johnson et al., calculations for the supercell formulation of the waveguide have tobe carried out between � and K’ points, where K’ corresponds to the projection of the Mpoint on the �–K direction (Johnson et al. 2000). The tunability range can be maximized byfinding the optimal radius for the infiltrated scatterers. We use 1225 plane waves in order toguarantee convergence for the two guided modes considered within an error smaller than 1%for radii of the defects bigger than 0.1a.

The photonic band structure for TE modes of the 2D photonic crystal without defects isshown in Fig. 2, the radius of the scatterers has been chosen so that the bandgap is maxi-mum (R/a = 0.440). There is a bandgap between 1st and 2nd modes in the �–K direction

Fig. 2 Photonic band structureof a 2D silicon photonic crystalwith a triangular lattice of holes(R/a = 0.440)

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Optimal tunability of waveguides based on silicon photonic crystals 491

Fig. 3 Dispersion relation forthe two existing guided modes ofa photonic crystal waveguidewith defect size Rdef /a = 0.440and LC optical axis orientationα = 0◦. Shadow regions indicatethe projected band structure onthe �-K’ direction of thephotonic crystal

Fig. 4 Frequencies of the guidedmodes GM1 (solid lines) andGM2 (dotted lines) of a photoniccrystal waveguide with defectsizes Rdef /a = 0.440 versus theLC optical axis orientation ata �–point and b K’-point

with a width of �ωa/2πc = 0.218 between normalized frequencies ωa/2πc = 0.511 andωa/2πc = 0.293.

The dispersion relations for the two existing guided modes, for a waveguide whereRdef /a = 0.440 and for an LC optical axis orientation α = 0◦ are shown in Fig. 3. We

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Fig. 5 Maximum tunabilityrange of the guided modes versusthe defect radius. a GM1. b GM2

label the two guided modes (and the corresponded field distribution) as following: at the�-point G M1 is the lower frequency mode and G M2 is the higher frequency mode whereasat the K’-point G M1 is the higher frequency mode and G M2 is the lower frequency mode.Additionally, the shadow regions indicate the projected band structure of the periodic struc-ture, it is, the projection of the bands at all the directions of the first Brillouin zone in the�-K direction.

The first property that we analyze is the tuning of the guided modes with the LC optical axisorientation, α. Figure 4 shows the dependence of the two guided modes with this angle for aradius of the defects of Rdef /a = 0.440. As it can be seen, for the �-point (Fig. 4a) varyingα from 0◦ to 180◦ the frequency of G M1 can be tuned from ωa/2πc = 0.462 to ωa/2πc =0.443 while the frequency of G M2 can be tuned simultaneously from ωa/2πc = 0.507 toωa/2πc = 0.537. For the K’-point (Fig. 4b) G M1 can be tuned from ωa/2πc = 0.471to ωa/2πc = 0.454 while G M2 can be tuned simultaneously from ωa/2πc = 0.399 toωa/2πc = 0.422. It is important to note that there is a periodicity of 180◦, and the frequencyshows a maximum or a minimum (depending on the mode and the on the wavevector) atα = 0◦, 90◦ and 180◦.

The main goal of this work is to maximize the tunability range optimizing the radius ofthe scatterers that define the waveguide. This tunability range is defined for each guidedmode at the �-point or at the K’-point as the difference between the maximum and minimumnormalized frequencies. These values can be obtained analyzing the band structure for the

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Optimal tunability of waveguides based on silicon photonic crystals 493

Fig. 6 Field distribution for themodulus of the Z-component ofthe magnetic field (Hz) for GM1at the �-point for a photoniccrystal waveguide with defectssize Rdef = 0.420 and LCoptical axis orientations:a α = 0◦. b α = 90◦

angles α = 0◦, and α = 90◦ for each radius. We have limited the range of the radius analyzedbetween Rdef /a = 0.1 and Rdef /a = 0.5, since lower values of Rdef /a requires a highernumber of plane waves to reach the desired accuracy.

The tunability ranges of the two guided modes for defect radius between Rdef /a = 0.100and Rdef /a = 0.500 at �-point and K’-point are shown in Fig. 5. It is important to notethat G M1 is within the bandgap for all Rdef /a while G M2 is within the bandgap betweenRdef /a = 0.100 and Rdef /a = 0.360 for the �-point and between Rdef /a = 0.335 andRdef /a = 0.500 for the K’-point. For the remaining radius values, for some orientation ofthe liquid crystal optical axis, the mode frequency lies out of the bandgap and the modebecomes leaky. For G M1 (Fig. 5a) it can be seen that maximum tunability range appears atRdef /a = 0.420 for the �-point and Rdef /a = 0.360 for the K’-point. At these optimal val-ues a compromise is reached between the amount of the liquid crystal and the quality of thewaveguide. For G M2 (Fig. 5b) the tunability range is monotonically increasing with increas-ing defect radius. Consequently, the maximum tunability range corresponds to the highestdefect radius value for which the mode is not leaky for any α, this is: Rdef /a = 0.360 forthe �-point and Rdef /a = 0.500 for the K’-point.

Finally, the field distributions corresponding to these guided modes at the �-point and theK’-point for the defect radius that maximizes the tunability range are presented in Figs. 6, 7,

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Fig. 7 Field distribution for themodulus of the Z-component ofthe Magnetic Field (Hz) for GM2at the �-point for a photoniccrystal waveguide with defectssize Rdef = 0.360 and LCoptical axis orientations:a α = 0◦. b α = 90◦

8, and 9. Figure 6 shows the amplitudes of the Hz field for the first guided mode G M1 at the�-point for a photonic crystal with a radius of the defects Rdef = 0.420 and an orientationof the LC optical axis of α = 0◦ (Fig. 6a), and α = 90◦ (Fig. 6b). White circles represent aschematic view of the photonic crystal. In these figures it can be observed how the field isconfined to the waveguide region due to the photonic gap effect. This guided mode presentsone node mainly oriented in the �–K direction.

The amplitudes of the Hz field for the second guided mode G M2 at the �-point for aphotonic crystal with a radius of the defects Rdef = 0.360 and an orientation of the LCoptical axis of α = 0◦ (Fig. 7a), and α = 90◦ (Fig. 7b) are shown in Fig. 7. Again, it canbe observed how the field is confined in the region of the waveguide due to the photonic gapeffect. This guided mode presents a region of high energy between the defect cylinders in thedirection �–K. It is important to observe that the field distributions for these two optical axisorientations are quite similar, although there are small differences that produce the frequencytuning.

The amplitudes of the Hz field for the first guided mode G M1 at the K’-point for a pho-tonic crystal with a radius of the defects Rdef = 0.360 and an orientation of the LC opticalaxis of α = 0◦ (Fig. 8a), and α = 90◦ (Fig. 8b) are shown in Fig. 8. Again, it can be observedhow the field is confined in the region of the waveguide due to the photonic gap effect. This

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Optimal tunability of waveguides based on silicon photonic crystals 495

Fig. 8 Field distribution for themodulus of the Z-component ofthe magnetic field (Hz) for GM1at the K’-point for a photoniccrystal waveguide with defectssize Rdef = 0.360 and LCoptical axis orientations:a α = 0◦. b α = 90◦

guided mode has the same distribution as in Fig. 6 because it corresponds to the same guidedmode but for a different defect size.

Finally, the amplitudes of the Hz field for the second guided mode G M2 at K’-point fora photonic crystal with a radius of the defects Rdef = 0.500 and an orientation of the LCoptical axis of α = 0◦ (Fig. 9a), and α = 90◦ (Fig. 9b) are shown in Fig. 9. Again, it canbe observed how the field is confined in the region of the waveguide due to the photonic gapeffect. This guided mode has the same distribution of the one of Fig. 7 because it correspondsto the same guided mode but for a different size of the defects. In this case, the distributionsfor these two LC optical axis orientations are more different between them than previouscases due to bigger proportion of liquid crystal.

4 Conclusions

In this work we have studied the tunability range of a Si-based 2D photonic crystal wave-guide. The waveguide consists of a triangular lattice of holes in silicon where a line of holesin the direction �–K has a different radius and is infiltrated with E7 liquid crystal. By means

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Fig. 9 Field distribution for themodulus of the Z-component ofthe Magnetic Field (Hz) for GM2at the K’-point for a photoniccrystal waveguide with defectssize Rde f = 0.500 and LCoptical axis orientations:a α = 0◦. b α = 90◦

of the plane wave expansion method and the definition of a 7x1 supercell we have studiedthe optimization of this structure varying the size of the defect scatterers and studying thefrequency of two existing guided modes. Also, the field distribution of these two guidedmodes has been studied.

The calculation reveals that maximum and minimum values of the frequency of the guidedmodes appear for liquid crystal optical axis orientations of α = 0◦ and α = 90◦. Therefore,studying the difference between the frequencies of each guided mode for these two orienta-tions, the tunability range can be estimated for each defect size. For the first guided mode thereare maxima in the tunability range for Rdef /a = 0.420 for the �-point and Rdef /a = 0.360for the K’-point. For the second guided mode the tunability range increase with the defectradius, thus the maximum tunability range corresponds to the maximum defect radius forwhich the mode is not leaky for any α: Rdef /a = 0.360 for the �-point and Rdef /a = 0.500for the K-point.

The study of the field distribution shows a good confinement of the modes in the waveguideand high field concentration in the waveguide regions of high refractive index, correspondingto the silicon. Additionally, the guided modes show the same field distribution of the highenergy regions at the �-point and at the K’-point for all the cases.

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Optimal tunability of waveguides based on silicon photonic crystals 497

Acknowledgments This work was supported by Spanish Ministry of Ciencia e Innovacion (MICINN) undergrant number TEC2009-09551, HOPE CSD2007-00007 (Consolider-Ingenio 2010) and AECID-A/024560/09.

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