IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 47, NO. 10, OCTOBER 2011 1283

Coupling Characteristics of Dual Liquid CrystalCore Soft Glass Photonic Crystal Fiber

Mohamed Farhat O. Hameed and Salah S. A. Obayya, Senior Member, IEEE

Abstract— In this paper, the coupling characteristics of a noveldesign of soft glass photonic crystal fiber with dual nematicliquid crystal core (SGLC-PCF) are introduced and analyzed.The index contrast between the core and cladding region ensuresthe index guiding through the reported coupler. The analysis iscarried out using the full vectorial finite difference method alongwith the full vectorial finite difference beam propagation method.The numerical results reveal that the SGLC-PCF coupler canbe used as a polarization splitter of length 6232 µm with lowcrosstalk better than −20 dB with great bandwidths of 250 nmand 60 nm for the quasi transverse electric and quasi transversemagnetic, modes, respectively. In addition, the proposed splitterhas a tolerance of ±3% in its length which makes the design lesssensitive to the perturbation introduced during the fabricationprocess.

Index Terms— Beam propagation method, couplers and split-ters, finite difference method, nematic liquid crystal, photoniccrystal fibers, soft glass.

I. INTRODUCTION

DUE TO their different uses in communication systems,the fiber couplers have attracted the interest of many

researchers in recent years. The fiber couplers can be usedto transfer, divide or combine the optical power in the com-munication systems. It has been shown by Mangan et al. [1]that it is possible to use the photonic crystal fiber (PCF) as anoptical fiber coupler. The PCF couplers [2]–[3] can be easilyrealized by introducing two adjacent defects. In addition, theyhave short coupling length and more flexibility design. Thedesign of PCF couplers as a polarization splitter [3]–[4],broadband directional coupler [5], wavelength division mul-tiplex components [2], [6] and filters [7] have been proposed.Saitoh et al. [2] evaluated the coupling characteristics of twodifferent dual core PCF couplers showing that it is possible torealize significantly shorter multiplexer-demultiplexer PCFs,compared to conventional optical fiber coupler. In addition,Florous et al. [3] numerically investigated the operation ofa polarization-independent splitters based on the PCF withelliptical air holes. However, small bandwidth of 5.1 nm and2.7 nm are achieved around wavelengths of 1.3 μm, and1.55 μm, respectively at a relatively low-level cross-talk of

Manuscript received May 5, 2011; revised July 29, 2011; accepted June 23,2011. Date of current version August 23, 2011.

M. F. O. Hameed is with the Faculty of Engineering, Mathematics andEngineering Physics Department, Mansoura University, Mansoura 35516,Egypt (e-mail: engmfarhat@mans.edu.eg).

S. S. A. Obayya is with the Faculty of Engineering, Electronics andCommunications Engineering Department, Mansoura University, Mansoura35516, Egypt (e-mail: sobayya@yahoo.co.uk).

Digital Object Identifier 10.1109/JQE.2011.2163702

−20 dB. In addition, polarization-independent splitter basedon all-solid silica PCF has been proposed by Chen et al.[4] with a device length of 10.69 mm and bandwidths of25.4 nm and 42.2 nm around wavelengths of 1.55 μm and1.31 μm, respectively. Moreover, polarization splitter of length20 mm based on a square-lattice PCF has been presented[8], showing crosstalk as low as −23 dB with bandwidths asgreat as 90 nm. Recently, the authors reported the couplingcharacteristics of a dual core soft glass PCF coupler infil-trated with a nematic liquid crystal (NLC-PCF) [9]. In addi-tion, polarization splitter [10] and multiplexer-demultiplexer[11] based on the soft glass NLC-PCF coupler have beenintroduced. The NLC-PCF splitter [10] of coupling length8.227 mm achieved low crosstalk better than −20 dB withgreat bandwidths of 30 nm and 75 nm for the quasi transverseelectric (TE) and quasi transverse magnetic (TM), modes,respectively. The NLC-PCF multiplexer-demultiplexer [11] oflength 3.265 mm also provided large bandwidths of 40 nmand 24 nm around the wavelengths of 1.3 μm and 1.55 μm,respectively.

In this paper, a novel design of high tunable coupler basedon soft glass PCF with air holes and dual nematic liquidcrystal core (SGLC-PCF) is presented and analyzed. Thesuggested design depends on using soft glass and nematicliquid crystal (NLC) of types SF57 (lead silica) and E7,

respectively. The refractive index of the SF57 material isgreater than the ordinary no and extraordinary ne refrac-tive indices of the E7 material. In addition, the propaga-tion through the SGLC-PCF coupler has taken place bythe modified total internal reflection mechanism due to theindex contrast between the core and cladding region. TheSGLC-PCF coupler has also strong polarization dependencedue to the infiltration of the NLC. Moreover, the SGLC-PCF coupler with the NLC has high tunability with thetemperature or external electric field. The high tunabilityfeature of the suggested coupler renders this design for theapplications ranging from optical communications to biosens-ing and bioimaging. Arc-fusion techniques [12] have beensuccessfully implemented for the infiltration of central defectcores therefore, the suggested design is easier for fabricationthan the NLCPCF [9] with filling all the cladding holes. Theeffect of the structure geometrical parameters, temperatureon the coupling characteristics of the reported coupler isinvestigated thoroughly. The numerical results reveal that theSGLC-PCF coupler can be used as a polarization splitterof length 6232 μm with low crosstalk better than −20 dBwith great bandwidths of 250 nm and 60 nm around the

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1284 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 47, NO. 10, OCTOBER 2011

Liquid crystal

V

Vdo

x

y

z

n

Softglassd

Air

ϕ

Fig. 1. Cross section of the dual core SGLC-PCF coupler sandwichedbetween two electrodes and surrounded by silicone oil [25].

operating wavelength of 1.55 μm for the quasi TE andquasi TM, modes, respectively. The analysis is carried outusing the full vectorial finite difference method [13] alongwith the full vectorial finite difference beam propagationmethod [14].

In the next section, the numerical methods are describedbriefly. Following that, in the results section, the design andsimulated results obtained from a novel design of PCF couplerare presented. Then conclusions are drawn.

II. NUMERICAL APPROACHES

In order to study the coupling characteristics of the sug-gested coupler and calculate accurately the modal solutionof the input waveguide, the use of full vectorial numericalapproaches is mandatory. Over the last few years, a number ofdifferent numerical methods have been suggested to study theguiding characteristics of PCFs. These methods included thefinite difference method (FDM) [13], finite element method(FEM) [15], finite element based imaginary distance beampropagation method (IDBPM) [16] and multipole method [17].These methods can deal efficiently with arbitrarily shapedholes and on arbitrary numbers and arrangements of holes. Inthis study, the magnetic field formulation based full vectorialfinite difference method (FVFDM) [13] with perfect matchedlayer (PML) boundary conditions [18] is used to calculate thefull vectorial quasi TE and quasi TM modes for the couplerwaveguide. Simply, the fundamental quasi TE mode refers tothe fundamental Hy

11 or Ex11 modes, while the fundamental

quasi TM mode refers to the fundamental Hx11 or Ey

11 modesaccording to the Cartesian coordinate shown in Fig. 1. Throughall simulations, the grid sizes �x and �y in x and y directions,respectively are taken as 0.05 μm. It should be noted that aset of modes is calculated by the FVFDM, and the dominantmode is defined as the mode with the highest real effectiveindex value. Since, ∇.H = 0 and interface boundary conditionsare automatically satisfied in the formulation, then there is nochance for spurious (non physical) modes to appear in thespectrum of the solution.

To analyze the propagation through the coupler waveguidesection, a more suitable approach such as beam propagationmethod (BPM) is required. Due to its numerical efficiencyand versatility, some full vectorial BPM algorithms havebeen formulated based on the FEM [19], [20]. Moreover,many full vectorial BPM approaches based on the popularFDM have been reported [14], [21]. In this paper, the beam

propagation analysis of the reported coupler is evaluated bythe full vectorial finite difference beam propagation method(FVFD-BPM) [14]. Through all simulations, the transversestep sizes are also fixed to �x = �y = 0.05 μm while thelongitudinal step size �z is taken as 1 μm. In addition, thereference index no which is used to satisfy the slowly varyingenvelope approximation of the FVFD-BPM [14] is taken asthe effective index of the fundamental mode launched at theinput waveguide. Moreover, α is chosen within the range,0.5 ≤ α ≤ 1, at which the FVFD-BPM [14] is unconditionallystable. The α parameter is responsible for controlling thescheme which is used to solve the finite difference equations.

III. NUMERICAL RESULTS

Figure 1 shows cross section of the suggested triangularlattice dual core SGLC-PCF coupler. The two identical coresof diameter do have been infiltrated with a NLC of type E7.All the cladding air holes have the same diameter d andare arranged with a hole pitch �. The separation betweenthe two identical cores is equal to

√3 �. The NLCs are

anisotropic materials consisting of rod-like molecules whichare characterized by ordinary index no and extraordinary indexne. The following Cauchy models are used to calculate the noand ne of the E7 material [22]

ne,o = Ae,o + Be,o

λ2 + Ce,o

λ4 (1)

where Ae, Be, Ce, Ao, Bo and Co are the coefficients of theCauchy model. The Cauchy coefficients at T = 25 °C areAe = 1.6933, Be = 0.0078 μm2, Ce = 0.0028 μm4, Ao =1.4994, Bo = 0.0070 μm2, and Co = 0.0004 μm4. The relativepermittivity tensor εr of the E7 material is taken as [23]

εr =⎛⎝

n2o sin2 ϕ + n2

e cos2 ϕ (n2e − n2

o) cos ϕ sin ϕ 0(n2

e − n2o) cos ϕ sin ϕ n2

o cos2 ϕ + n2e sin2 ϕ 0

0 0 n2o

⎞⎠

(2)

where ϕ is the rotation angle of the director n of the NLCas shown in Fig. 1. The proposed in-plane alignment of theNLC can be exhibited under the influence of an appropriatehomeotropic anchoring conditions [23], [24]. In addition,Haakestad et al. [25] demonstrated experimentally that in thestrong field limit, the NLC of type E7 is aligned in-plane incapillaries of diameter 5 μm with good accuracy. Moreover,T. T. Alkeskjold and A. Bjarklev [26] presented experimentallyin-plane alignment of the E7 material in PCF capillaries ofdiameter 3 μm with three different rotation angles, 0°, 45°and 90° using two sets of electrodes.

The background material of the reported SGLC-PCF cou-pler is a soft glass of type SF57 (lead silica). The Sellmeierequation of the soft glass of type SF57 [27] is given by

n2S F57 = Ao + A1λ

2 + A2

λ2 + A3

λ4 + A4

λ6 + A5

λ8 (3)

where nS F57 is the refractive index of the SF57 material, Ao =3.24748, A1 = −0.00954782 μm−2, A2 = 0.0493626 μm2,A3 = 0.00294294 μm4, A4 = −1.48144 × 10−4 μm6, andA5 = 2.78427 × 10−5 μm8 [27].

HAMEED AND OBAYYA: COUPLING CHARACTERISTICS OF DUAL LIQUID CRYSTAL CORE SOFT GLASS PCF 1285

7000

SGLC-PCF Coupler

Silica PCF Coupler6000

5000

4000

TM mode

Cou

plin

g le

ngth

LC

(μ

m)

TE mode

3000

2000

1000

00.7 0.75 0.8 0.85

d /�

Fig. 2. Variation of the coupling lengths of the two polarized modes withthe d/� ratio at constant � of 2.0 μm.

The fiber is placed between two pairs of electrodes allowingfor the arbitrary control of the alignment of the NLC directorvia an external voltage, as schematically shown in Fig. 1. Inaddition, two silica rods with appropriate diameter are usedto control the spacing between the electrodes and the fiber issurrounded by silicone oil, which has higher dielectric strengththan air [25]. Therefore, the external electric field will beuniform across the fiber cross section which results in goodalignment of the director of the NLC with constant rotationangle φ. Moreover the nonuniform electric field region willbe only at the edges far away from the core region where thelight will be propagating. As a result, the proposed coupleroverall performance will not be affected by the nonuniformfield distribution at the edges. Other layouts, such as thosedescribed in [28]–[29] can also be used to ensure better fielddistribution uniformity over the fiber cross section. In addition,L. Wei et al. [30] proved that by using sets of electrodes andcontrolling them independently, the direction of the electricalfield is rotatable under effective driving voltage of 50 Vrms.

There are many practical techniques that have been usedin manufacturing the nonsilica PCF such as capillary stack-ing [31], and extrusion [27], [32]. However, the extrusionapproach has been recently extended to the soft glasses such aslead silicate (SF57 glass) [27], [32]. The SF57 glass has lowerprocessing temperature of ∼520 °C [33] than 1500–1600 °Cof the silica glass. Therefore, it is possible to extrude the PCFpreform directly from the bulk glass. In addition, lead silicateglasses offer the highest thermal and crystallization stabilitymaking them particularly attractive for PCF fabrication.

Recently, some attention has been devoted to the possibilityof infiltration of the air holes of the PCF with differentmaterials such as liquid or liquid crystalline [25], [29],[34], [35]. However, PCF structure infiltrated with a LC hasunique and uncommon propagation and polarization prop-erties. Arc-fusion techniques have been successfully imple-mented for the infiltration of central defect cores [12], whileextensive control in the infiltration process of either core orcladding capillaries can be achieved by using UV curable poly-mers [35]. In addition. Haakestad et al. [25] experimentallyfilled all the cladding holes of the PCF with NLC by using

capillary forces and electrically tunable photonic bandgapguidance is reported. Moreover, tunable light switch using PCFwhose central defect and cladding holes are filled with NLCis studied by Fang et al. [29].

In the suggested design, the cladding air holes have thesame diameter d and are arranged with a hole pitch � =2.0 μm while the radius of the infiltrated NLC cores ro istaken as 0.5 μm. In addition, the rotation angle of the directorof the NLC and the temperature are fixed to 90° and 25 °C,respectively. Moreover, no, ne, nS F57 are taken as 1.5024,1.6970, and 1.802, respectively at the operating wavelengthλ = 1.55 μm. The refractive index of the SF57 material isgreater than no and neof the E7 material which guaranteesthe index guiding of the light through the high index coreSGLC-PCF coupler. In this study, the power is launched at theleft core and the coupling length is defined as the minimumlongitudinal distance at which maximum power is transferredfrom the left core to the right core. The coupling length LC canbe obtained using the operating wavelength λ, and effectiveindices of the even neff_e and odd modes neff_o as follows

LC = λ

2(neff_e − neff_o). (4)

In this evaluation, the effective indices of the even and oddmodes of the SGLC-PCF coupler are evaluated by the FVFDM[13] with PML boundary conditions [18]. The effect of thecoupler geometrical parameters, rotation angle of the directorof the NLC, and temperature on the coupling length of theproposed coupler is studied thoroughly. The influence of thed/� ratio is the first parameter to be considered. Figure 2shows the variation of the coupling lengths of the SGLC-PCFcoupler for the two polarized modes with the d/� ratio at theoperating wavelength of 1.55 μm. It is observed from thisfigure that the coupling lengths for the two polarized modesincrease with increasing the d/� ratio at constant hole pitch� of 2.0 μm. As the d/� ratio increases at constant �, thesoft glass bridge between the two cores decreases. Therefore,the distance taken by the modes to transfer from the left coreto the right core, and hence the coupling lengths of the twopolarized modes increase. As the d/� ratio increases from 0.7to 0.85, the coupling lengths of the quasi TE and quasi TMmodes of the SGLC-PCF coupler increase from 358 μm and1044 μm, to 1370 μm and 6373 μm, respectively.

The variation of the coupling length for the two polarizedmodes of the conventional silica PCF coupler with air holes isalso shown in Fig. 2. In this case, the infiltrated NLC holes ofdiameter do are removed from the core regions. The refractiveindex of the silica material is taken as 1.45 at λ = 1.55 μm andthe hole pitch is fixed to 2 μm. As the d/� ratio increases from0.7 to 0.85, the coupling lengths of the quasi TE and quasi TMmodes of the silica PCF coupler increase from 548 μm and754 μm to 1491 μm and 2294 μm, respectively. It should benoted that the birefringence, defined as the difference betweenthe effective indices of the quasi TE and quasi TM modes,is small for the conventional silica air holes PCF couplerswhile the SGLC-PCF coupler has high birefringence withoutusing elliptical holes or two bigger holes in the first ring [36].Therefore, the difference between the coupling lengths for the

1286 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 47, NO. 10, OCTOBER 2011

SGLC-PCF Coupler

Silica PCF Coupler

γ =

(L

CT

M −

LC

TE)/

LC

TM

0.70.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.75d /�

0.8 0.85

Fig. 3. Variation of the form birefringence of the SGLC-PCF coupler andconventional silica PCF coupler with the d/� ratio at constant � of 2.0 μm.

quasi TE and quasi TM modes of the SGLC-PCF coupler ata given d/� ratio is greater than that of the conventional PCFcoupler, as revealed in Fig. 2. The form birefringence [37] isdefined as the ratio of (LCTM–LCTE) to LCTM where LCTE,and LCTM, are the coupling lengths of the quasi TE and quasiTM modes, respectively. Figure 3 shows the variation of theform birefringence of the SGLC-PCF coupler and conventionalsilica air holes PCF coupler with the d/� ratio while the holepitch, rotation angle of the director of the NLC, temperature,and wavelength are fixed to 2.0 μm, 90°, 25 °C, and 1.55 μm,respectively. In addition, the radius of the NLC infiltrated dualcores ro is taken as 0.5 μm. It is revealed from Fig. 3 thatthe form birefringences of the conventional PCF coupler andSGLC-PCF coupler increase with increasing the d/� ratio. Asthe d/� ratio increases from 0.7 to 0.85, the form birefringenceincreases from 0.657 and 0.273 to 0.796 and 0.331 for theSGLC-PCF coupler and the conventional PCF coupler, respec-tively. It is also evident from Fig. 3 that the form birefringenceof the SGLCPCF coupler is approximately 2.2 times that ofthe conventional PCF coupler. In addition, the form birefrin-gence values indicate that the SGLC-PCF coupler has strongpolarization dependence therefore, it can be used as a polariza-tion splitter and its polarization dependence is stronger thanthose splitters presented in [37]. However, the conventionalsilica PCF coupler [2] has low birefringence and the highbirefringence can be realized by adjusting the size of the airholes around the two cores regions [37]–[39], which enlargesthe difference between the coupling lengths for the twopolarized modes. In [38], the two identical cores are formedby combination of large and small air holes which makesthe two cores birefringent. Zhang and Yang [39] reported apolarization splitter based on two non-identical cores withalso a combination of large and small air holes. However, in[37], two elliptic cores are used to improve the polarizationdependence of the conventional silica PCF coupler.

The effect of the radius ro of the two identical NLC coresis also investigated. In this study, the hole pitch, d/� ratio,rotation angle of the director of the NLC, temperature, andwavelength are fixed to 2.0 μm, 0.8, 90°, 25 °C, and 1.55 μm,respectively. Figure 4 shows the variation of the couplinglength of the two polarized modes with the wavelength at

TM mode

ro � 0.4 μm

ro � 0.45 μm

ro � 0.5 μm

TE mode

Wavelength λ (μm)

Cou

plin

g le

ngth

LC (μ

m)

10 000

8000

6000

4000

2000

01.3 1.35 1.4 1.45 1.5 1.55 1.6

Fig. 4. Variation of the coupling length of the two polarized modes of theSGLC-PCF coupler with the wavelength at different ro values.

different ro, 0.4 μm, 0.45 μm, and 0.5 μm. As the wavelengthincreases, the confinement of the two polarized modes throughthe core regions decreases. Consequently, the distance takenby the two polarized modes and hence the coupling lengthdecrease by increasing the wavelength. In addition, the indexcontrasts seen by the two polarized modes decreases andhence the confinement through the core regions decreases byincreasing ro. As a result, the distance taken by the modes andthen the coupling length decrease by increasing ro.

It is also observed from Fig. 4 that the coupling length ofthe quasi TE mode at ϕ = 90° is shorter than that of the quasiTM mode. At ϕ = 90°, the relative permittivity εr of the E7material has the diagonal form [εxx, εyy, εzz] where εxx = n2

o,εyy = n2

e and εzz = n2o. In this case, εyy is greater than εxx

therefore, the index contrast seen by the quasi TM modes isgreater than that for the quasi TE modes. Consequently, thequasi TM modes are more confined in the core regions than thequasi TE modes. As a result, the quasi TM modes take longerdistance than the quasi TE modes to transfer from the leftcore to the right core. Consequently, the coupling length of thequasi TM mode is longer than that of the quasi TE mode. Thesituation is reversed at ϕ = 0° at which the relative permittivityεr of the E7 material has the diagonal form [n2

e, n2o, n2

o]. In thiscase, the coupling length of the quasi TE mode is longer thanthat of the quasi TM mode. Figure 5 shows the variation of theform birefringence with the wavelength at different ro, 0.4 μm,0.45 μm and 0.5 μm. It is evident from this figure that the formbirefringence increases with increasing ro. At λ = 1.55 μmthe form birefringence are equal to 0.7267, 0.7491, and 0.7699at ro = 0.4 μm, 0.45 μm and 0.5 μm, respectively.

The effect of the deformation of the two identical NLCinfiltrated holes into elliptical cores on the performance of thesuggested coupler is further studied. Here, ao and bo are theradii of the elliptical holes in x and y directions, respectively,as shown in the inset of Fig. 6. Figure 6 shows the variationof the coupling length of the two polarized modes with thewavelength at different bo, 0.4 μm, 0.45 μm and 0.5 μm. Inthis study, the hole pitch, d/� ratio, rotation angle, temperatureare fixed 2.0 μm, 0.8, 90°, and 25 °C, respectively. In addition,the radius in x direction ao is fixed to 0.5 μm. It is found thatcoupling lengths of the two polarized modes decrease with

HAMEED AND OBAYYA: COUPLING CHARACTERISTICS OF DUAL LIQUID CRYSTAL CORE SOFT GLASS PCF 1287

0.4 μm

0.45 μm

ro � 0.5 μm

ϕ � 90°

Wavelength λ (μm)

1.30.7

0.72

0.72

0.76

0.78

0.8

1.35 1.4

d /� � 0.8 μm�� 2.0 μm

1.45 1.5 1.55 1.6

γ =

(L

CT

M −

LC

TE)/

LC

TM

Fig. 5. Variation of the form birefringence of the SGLC-PCF coupler withthe wavelength at different ro values.

10 000

8000

6000

Cou

plin

g le

ngth

LC (μ

m)

4000

TM mode

bo = 0.4 μm

bo = 0.45 μm

bo = 0.5 μm

ao = 0.5 μm

2ao

2bo

TE mode2000

01.3 1.35 1.4 1.45

Wavelength λ (μm)

1.5 1.55 1.6

Fig. 6. Variation of the coupling length of the two polarized modes of theSGLC-PCF coupler with the wavelength at different bo while ao is taken as0.5 μm where ao and bo are the radii of the elliptical NLC holes in x and ydirections, respectively.

increasing bo value. However, the numerical results reveal thatthe form birefringence increases with increasing the radiusin y direction bo. At the operating wavelength of 1.55 μm,the form birefringence increases from 0.7239 to 0.7699 as boincreases from 0.4 μm to 0.5 μm. The effect of the radiusao in x direction on the performance of the reported coupleris also investigated. In this case, the hole pitch, d/� ratio,rotation angle, temperature are fixed 2.0 μm, 0.8, 90°, and25 °C, respectively. In addition, the radius in y direction bo isfixed to 0.5 μm. It is found that the effect of the ao variationis the same effect of the bo variation on the performance ofthe SGLC-PCF coupler.

It should be noted that the ordinary no and extraordinarynerefractive indices of the E7 material are temperaturedependent [9], [22]. Therefore, the effect of the temperaturevariation on the coupling length is the next parameter tobe considered. Figure 7 shows the variation of the couplinglength for the two polarized modes with the temperature at arotation angle ϕ, of 90°, while the other parameters are fixedto � = 2.0 μm, d/� = 0.8, ro = 0.5 μm and λ = 1.55 μm.It can be seen from this figure that the coupling length of thequasi TM mode decreases with increasing the temperaturewhile the coupling length of the quasi TE mode is nearly

5000

4000

3000

Cou

plin

g le

ngth

LC (μ

m)

2000

TM modeϕ = 90°

TE mode1000

015 20 25 30

Temperature T (°C)

35 40 45

Fig. 7. Variation of the coupling length for the two polarized modes of thedual core SGLC-PCF coupler with the temperature.

constant. As T increases from 15 °C to 45 °C, the couplinglength of the quasi TM mode at ϕ = 90° decreases from4587 μm to 3489 μm. The dependence of the coupling lengthon the temperature can be explained by analyzing the dominantfield components of the quasi TE and quasi TM modes and thedirection of the director of the NLC. At ϕ = 90°, the relativepermittivity tensor εr of the E7 material has the diagonal form[εxx, εyy,εzz] where εxx = n2

o, εyy = n2e and εzz = n2

o. Asthe temperature increases from 15 °C to 45 °C, εyy decreasesfrom 2.9227 to 2.7569 while εxx slightly changes from 2.2602to 2.2641. Therefore, the index contrast seen by the quasi TMmodes decreases with increasing the temperature while theindex contrast seen by the quasi TE modes is nearly constant.Consequently, the confinement of the quasi TM modesinside the core regions and hence the distance needed by thequasi TM modes to transfer from the left core to the rightcore decrease with increasing the temperature. Therefore,the coupling length of the quasi TM mode decreases withincreasing the temperature while the coupling length of thequasi TE mode is nearly invariant, as shown in Fig. 7.

The situation is reversed at ϕ = 0° at which εr of the E7material has the diagonal form [n2

e, n2o, n2

o]. As the temperatureincreases εxx decreases while εyy is nearly invariant. Therefore,the index contrast seen by the quasi TE modes decreases withincreasing the temperature while the index contrast seen by thequasi TM modes is nearly constant. Consequently, the confine-ment of the quasi TE modes inside the core regions and hencethe distance needed by the quasi TE modes to transfer fromthe left core to the right core decrease with increasing the tem-perature. Therefore, the coupling length of the quasi TE modedecreases with increasing the temperature while the couplinglength of the quasi TM mode is nearly invariant. The controland tunability of the rotation angle of the NLC can be achievedwith a good accuracy in a strong field limit [25] with stets ofelectrodes as successfully experimentally shown in [25], [26].

Figure 2 and Fig. 3 reveal that the SGLC-PCF coupler hasstrong polarization dependence due to the infiltration of theNLC which increases the birefringence between the two fun-damental polarized modes. Therefore, the SGLC-PCF couplercan be easily designed as a polarization splitter and its polar-

1288 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 47, NO. 10, OCTOBER 2011

3

2.8

2.6

1.4 1.45 1.5Cladding hole radius r (μm)

1.55 1.6 1.65 1.7

2.4

2.2

� = 4.3 μmR

= (

LC

TM/L

CT

E)

2

1.8

� = 4.4 μm

� = 4.5 μm

Fig. 8. Variation of the coupling length ratio R for the quasi TE and quasiTM modes with the cladding hole radius r at different hole pitches.

ization dependence is stronger than those splitters presentedin [37]–[39]. The fiber coupler can separate the two polarizedstates, quasi TE and quasi TM modes at a given wavelengthif the coupling lengths LCTE and LCTM of the quasi TE andquasi TM modes, respectively satisfy the coupling ratio [4]

R = LCTM : LCTE = i : j (5)

where i and j are two integers of different parities. In thiscase, the length of the coupler is equal to Lf = LCTE × i/j.Therefore, to achieve the shortest splitter, the optimal valueof R should be 2. Figure 8 shows the coupling length ratiobetween the coupling lengths of the quasi TE and quasi TMmodes as a function of the cladding hole radius r at differenthole pitch values, 4.3 μm, 4.4 μm and 4.5 μm. In this study,the central hole radius ro, operating wavelength, rotationangle of the director of the NLC and temperature are takenas 1.0 μm, 1.55 μm, 90° and 25 °C, respectively. It is foundthat the coupling length ratio R increases with increasing thecladding hole radius at a given hole pitch. As can be seenfrom Fig. 8, the coupling length ratio equals to 1.9850 atcladding hole radius of 1.5 μm and hole pitch of 4.5 μm. Thecoupling lengths calculated by the FVFDM are 3127 μm and6207 μm for the quasi TE and quasi TM modes, respectivelyat the operating wavelength λ = 1.55 μm.

In order to confirm the polarization splitter based on theSGLC-PCF coupler, the FVFD-BPM [14] is used to studythe propagation along its axial direction. Initially, at z = 0,the fundamental components Hy and Hx of the quasi TE andquasi TM modes, respectively of single core soft glass PCFwith air holes obtained by the FVFDM [13] at λ = 1.55 μmare launched into the left core of the SGLC-PCF coupler.These input fields, in turn, start to transfer to the right coreof the coupler and at the corresponding coupling lengths,the fields are completely transferred to the right core. Thecoupling lengths calculated by the FVFD-BPM are 3128 μmand 6208 μm for the quasi TE and TM modes respectivelywhich are in excellent agreement with those obtained by theFVFDM. The ratio between the coupling lengths LCTM toLCTE is slightly less than 2.0. Therefore, the length of theproposed splitter is Lf = (6208 + 2∗3128)/2.0 = 6232 μmat which the two polarized states are separated well. Figure 9

1

0.8

0.6

Nor

mal

ized

pow

er

0.4

0.2

00 1000 2000 3000

TM mode

TE mode

Lf

Lf = 6232 μm

4000

Propagation distance z (μm)

5000 6000 7000

Fig. 9. Evolution of the normalized powers at the left core for the quasiTE and quasi TM modes at the operating wavelength of 1.55 μm along thepropagation direction.

20(a)

Hy

(b)

Hx

15

Hei

ght (

μm

)

10

10 15 20 25

0.8

0.6

0.4

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20

15

Hei

ght (

μm

)

10

10 15 20 25

0.8

0.6

0.4

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Width (μm)20

(c)

Hy

(d)

Hx

15

Hei

ght (

μm

)10

10 15 20 25

0.8

0.6

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20

15

Hei

ght (

μm

)

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10 15 20 25

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(e)

Hy

(f)

Hx

15

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ght (

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)

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10 15 20 25

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20

15

Hei

ght (

μm

)

10

10 15 20 25

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Fig. 10. Field contour patterns for Hy and Hx of the quasi TE and quasiTM modes, respectively, at (a, b) z = 0, (c, d) z = 3128 μm, and (e, f)z = 6232 μm at λ = 1.55 μm.

shows the power transfer normalized to the input power forthe quasi TE and quasi TM modes at the operating wavelengthof 1.55 μm in the left core of the SGLC-PCF coupler. It isevident from Fig. 9 that the two polarized modes are separatedwell after a propagation distance equals to Lf = 6232 μm. Thenormalized powers of the quasi TE mode in the right and leftcores of the coupler are 0.0005 and 0.9995, respectively atz = 6232 μm. However the normalized powers of the quasiTM mode in the right and left cores of the coupler are 0.9990and 0.0010, respectively.

The field distributions of the dominant field component Hyand Hx of the quasi TE and quasi TM modes, respectivelyat λ = 1.55 μm are shown in Fig. 10 at different waveguidesections z, 0, 3128 μm and 6208 μm. It is evident from thisfigure that, at z = 0, the input fields are launched into the leftcore and as the propagation distance increases, the normalizedpower in the right core increases and that in the left coredecreases. At z = 3128 μm, which is equal to the couplinglength of the quasi TE mode, the normalized power of quasiTE mode is approximately completely transferred to the rightcore. The normalized powers of the quasi TE mode in theleft and right cores of the coupler are 0.0038 and 0.9962,respectively. However, the normalized powers of the quasi TM

HAMEED AND OBAYYA: COUPLING CHARACTERISTICS OF DUAL LIQUID CRYSTAL CORE SOFT GLASS PCF 1289

−5

−10

−15

−20

−25

Cro

ssta

lk C

T (

dB)

−30

−35

1.4 1.45 1.5

Wavelength λ (μm)

BW (TM) = 60 nm

BW (TE) = 250 nm

1.55 1.6 1.65 1.7−40

Fig. 11. Wavelength dependence crosstalks of the SGLC-PCF coupler aroundthe operating wavelength λ = 1.55 μm for the quasi TE and quasi TM modes.

mode in the left and right cores of the coupler are 0.4940and 0.5056, respectively at z = 3128 μm. Finally, the twopolarized modes are separated after a propagation distanceequals to Lf = 6232 μm.

The crosstalk is a measure of the unwanted power, remain-ing at the end of the SGLC-PCF coupler. The crosstalk CTaround the operating wavelength λ = 1.55 μm for the quasiTE and quasi TM modes are shown in Fig. 11. The crosstalkin decibel [3] for the desired quasi TE mode at the left coreis defined such that

CTTE = 10 log10

(PuTM

PdTE

)(6)

where PdT E and PuT M are the normalized power of the desiredquasi TE and undesired quasi TM modes, respectively at theleft core. However, the crosstalk [3] of the desired quasi TMmode at the right core is given by

CTTM = 10 log10

(PuTE

PdTM

)(7)

where PuTE and PdTM are the normalized power of the unde-sired quasi TE and desired quasi TM modes, respectively at theright core. It is revealed form Fig. 11 that the proposed splitterhas large bandwidths (BWs) of 250 nm and 60 nm for the quasiTE and quasi TM modes, respectively at which the crosstalksare better than −20 dB. Therefore, the proposed polarizationsplitter is less sensitive to the perturbation introduced duringthe fabrication process due to the low level crosstalks withwide wavelength ranges. The BWs of the SGLCPCF splitterare much larger than those reported in [3], [6]. The BW of thequasi TE mode in [3] is 2.7 nm around λ = 1.55 μm, whilethe BW in [6] is 2.0 nm. In addition, the proposed splitter isshorter than those reported in [3] and [6] of lengths 15.4 mmand 9.08 mm, respectively. Moreover, the SGLC-PCF splitterhas wide wavelength range larger than the reported splitterby Chen et al. [4] of BW = 25.4 nm around λ = 1.55 μmfor the quasi TE mode. Furthermore, the splitter in [4] haslonger length of 10.69 mm than that of the SGLC-PCF splitter.Additionally, the reported splitter is shorter than the NLC-PCF splitter [10] of coupling length 8.227 mm. Furthermore,the SGLC-PCF splitter has greater BWs than the NLC-PCF

splitter [10]. Therefore, the SGLC-PCF has advantages ofshorter coupling length, greater bandwidths and easier forfabrication than that presented in [10].

It is also shown from Fig. 11, that the bandwidth of the quasiTM mode is less than that of the quasi TE mode. At ϕ = 90°,the quasi TM mode is more confined through the core regionthan the quasi TE mode. Therefore, the quasi TE mode is moreaffected by the wavelength variation around λ = 1.55 μm thanthe quasi TM mode. As a result, the undesired normalizedpower of the quasi TE mode at the right core at the devicelength of 6232 μm increases with the wavelength variationaround λ = 1.55 μm more than the undesired normalizedpower of the quasi TM mode at the left core. Therefore, theBW of the quasi TM mode at the right core is less than thatof the quasi TE mode at the left core.

The tolerances of the fiber length, rotation angle ϕ of thedirector of the NLC and temperature are also investigated. Itis worth noting that the tolerance of a specific parameter iscalculated while the other parameters of the proposed designare kept constant. It is found that the fiber length and rotationangle ϕ allow a tolerance of ±3% and ±5°, respectively atwhich the crosstalks are still better than −20 dB. In addition,the cross talk for the quasi TE and quasi TM modes are betterthan −20 dB through the range of T from 15 °C to 40 °C. Thetemperature can be controlled by using thermo-electric moduleas experimentally described by T. R. Wolinski et al. [40], [41]allowing for temperature control in the 10–120 °C range with0.1 °C long-term stability and electric field regulation in the0–1000 V range with frequencies from 50 Hz to 2 KHz.

Finally, coupling the light in this new type of PCF polar-ization splitter is considered. It is found that the best way tocouple light in this PCF device is by splicing to a standardsingle mode fiber (SMF) and then lunching the light froma laser source direct to the SMF [42]. This approach is veryeffective for making low-loss interface between SMF and PCFas experimentally reported by Leon–Saval et al. [42].

IV. CONCLUSION

The coupling characteristics of a novel design of high tun-able SGLC-PCF coupler have been numerically investigated.The dual core SGLC-PCF coupler has stronger polarizationdependence than the conventional silica air holes PCF coupler.In addition, a novel type of polarization splitter based onSGLC-PCF coupler has been presented and analyzed. TheSGLC-PCF splitter has advantages in terms of its shortcoupling length as well as low crosstalks over large opticalbandwidths. The suggested splitter has a length of 6.232 mmwith a crosstalk better than −20 dB with bandwidths of250 nm and 60 nm for the quasi TE and quasi TM modes,respectively. Additionally, the splitter has a tolerance of ±3%in its length which makes the design more robust to theperturbation introduced during the fabrication. Moreover, therotation angle of the director of the NLC has a toleranceof ±5° at which the crosstalks are better than −20 dB.Furthermore, the crosstalk for the quasi TE and quasi TMmodes are better than −20 dB through the temperature rangefrom 15 °C to 40 °C.

1290 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 47, NO. 10, OCTOBER 2011

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Mohamed Farhat O. Hameed author photograph and biography not availableat the time of publication.

Salah S. A. Obayya author photograph and biography not available at thetime of publication.