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552 INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL TECHNOLOGY, VOL.13, NO.6, NOVEMBER 2018

Highly Birefringent Photonic Crystal Fibers with Flattened Dispersion and Ultralow Confinement loss Pranaw Kumar* and Jibendu Sekhar Roy School of Electronics Engineering KIIT- Deemed to be University, Bhubaneswar, India E-mail: [email protected] Abstract-In this paper photonic crystal fibers (PCFs) with different refractive index profiles have been investigated. A PCF structure with triangular refractive index profile of seven hexagonal rings show better performance compared to other simulated structures having same geometry but with different refractive index profiles. Dispersion characteristics of the triangular refractive index PCFs have nearly zero flattened dispersion with 2

birefringence of the order of 10 . Full vector finite difference time domain (FDTD) method has been applied for simulation using OptiFDTD software. These fibers may be used in optical communication system and optical sensors. Index Terms-Photonic crystal fibers, dispersion, birefringence, effective mode area, non linear coefficient.

I.

INTRODUCTION

Characteristics of propagating light through PCFs are determined by its refractive index profile. Refractive index of fibers usually changes during fabrication. Accuracy in measurement of refractive index during designing of PCFs is essential. Propagation of electromagnetic waves in PCFs is controlled by the refractive indices of materials used. As known PCFs is a multilayer structure with periodic arrangement of dielectric materials [1-2]. Model index of PCF structure without defect decides its optical performance. Such modal index is termed as effective cladding index and can be expressed as function of operating wavelength. Refractive index profile varies in special range compared to the wavelength of light propagated. It characterizes PCFs and has great impact in the field of communication. Special class of fiber of silica material with holes in their cladding region

which lies along the length of the fiber is termed as Holey Fibers. Due to its structures, it is also called as Honey Comb structures. Lower refractive index of cladding due to presence of holes allows modified total internal reflection to take place. Guiding mechanism categorizes PCFs in index guided fibers and photonic band gap fibers. Index guided fibers have solid core surrounded by holes present in the cladding region. Photonic band gap fibers have hollow cores. Lack of physical boundary between core and cladding of PCFs causes the light to get trapped in space between holes present in the cladding. This creates a loss termed as leakage loss or confinement loss [3-4]. In order to reduce space between two consecutive holes, its dimensions are being increased. Effective indices contrast has a great impact on the unique feature of PCFs [5]. Features like dispersion, large effective mode area, birefringence mode and confinement loss makes PCFs superior to conventional optical fiber. In order to reduce such losses, different methods have been proposed by researchers. Initially they put their efforts in fabrication techniques of fibers [6-7], as changes in refractive index have been observed during fabrication. Later on PCFs with different refractive index profile became a point of attraction [8-10]. PCFs with refractive index profile like step raised core [11] and depressed clad single mode have been reported [12]. Data carrying capacity of optical fiber gets limited due to chromatic dispersion. Inter symbol interference caused due to spreading of light pulses have been an issue, when rate of transmission exceed 10Gbps [13-14]. Large air filling fraction in cladding region helps in achieving tight confinement into small cores. It results waveguide dispersions of higher

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magnitude which even can cancel large negative dispersion of silica glass [15-16]. Zero dispersion at desired wavelength can be obtained by adjusting pitch factor (hole to hole spacing) and by varying air filled fraction into cladding region. It results non-linear applications like super continuum generation, SPM and CPM [17-18]. Any distortion in the structure of core to create two folds symmetry results high birefringence. Large difference between glass and air index results even higher degree of birefringence. Introducing asymmetricity [19] in the structures of core and utilizing capillaries of different thickness near core, result more than ten times birefringence than that of conventional optical fiber [20-21]. Fibers are highly birefringent due to its less sensitivity towards temperature variations as compared to the conventional optical fiber, makes them useful in many applications [22]. Flexibility in designing the PCFs structure results many high birefrigent fibers [23]. High refractive index [24] difference in between core and cladding region for the PCFs allow strong mode confinement. Thus it attracts high nonlinearity. Moreover flexibility in tailoring of dispersion helps in application using nonlinear effects of fibers. Compound glasses reports material nonlinearity of high order of longer magnitude than that of silica fiber. Combination of high nonlinear compound having high nonlinearity, together cause increase in nonlinearity of fiber. Lead silicate (SF57) with high nonlinearity and low loss has been reported recently [25].

PCF structures with seven ring of hexagonal lattice have been investigated. Different types of glasses have been used. All the four structures studied, differ in their refractive index profile as shown in Table 1. All the structures have dense flint glass with refractive index 1.7847 as wafer. Table 1: Refractive index of all flint glass used. Flint Glass n1(SF5) n2(SF15) n3 (SF18) n4(SF10) n5(SF14) n6(SF11)

Refractive index 1.672 1.6990 1.722 1.7283 1.762 1.7847

Fig. 1(a) Proposed PCF structure

In this paper, PCF structures with seven ring of hexagonal lattice have been investigated. Different types of glasses have been used and propagation characteristics of light through them have been studied. Structure B has shown better result than other structures simulated. However all structures investigated reports nearly zero flattened dispersion with high birefringence. Thus each structure simulated in this paper has wide applications. II. MODELLING PCF STRUCTURE

Fig. 1 (b) Refractive index distribution of designed fibres

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Structure A has holes filled with different dense flint glass (SF15) with refractive index 1.699. Structure B has a triangular refractive index distribution in the cladding region. The holes of central line and holes below the central line are filled with dense flint glass SF5. The two line, the first and second innermost layers facing core have holes filled with glass SF15. Similarly the third and fourth line has holes filled with glass SF18. Again the fifth and sixth layer has holes filled with glass SF10. The most exterior layer have glass SF14 in their holes. Similarly structure C has rectangular shape refractive index profile. It has two exterior vertical layers filled with glass SF10. Moreover the central rectangular profile also have four exterior vertical layers (aligned perpendicular to the core) have holes filled with glass SF14. Besides these all the holes are filled glass SF 15. In a similar fashion, structure D has the same refractive index distribution as in structure C. But in addition, it has two more vertical layers, each on either side of the central vertical layer (aligned perpendicular to the core) which has holes filled with SF10. Investigated structure is shown in Fig. 1(a). Different refractive index profile distribution of all four structures has been explained with the help of schematic diagram in Fig.1(b). III. THEORETICAL ANALYSIS PCFs, being a complex structure due to the large refractive index difference between glass and air have made its analysis a challenging task. Maxwell's equations are solved numerically using developed techniques [26-27]. A set of modes with different frequency, sharing similar axial wave vector component  is less utilized than a set of equal frequency mode sharing the same. Hence it would be convenient to arrange Maxwell's equation with  2 as Eigen values [28]. (2  k 2 ( T)  [ ln  ( T)] )   2 (1) Here, we have all the field vectors as of the form, Q  QT (rT ) e j z (rT )  (x, y) position in the , transverse plane, vacuum wave vector considered is k   / c and dielectric constant is T (rT ) . It

allows easy inclusion of material dispersion. Two equations relating hx and hy can be obtained from equation1, in Cartesian coordinates [28-29]. 2hx  2hx  ln   hx hy  (2a)      k 2   2  h  0   x y 2  y x   2 hy  2 hy  ln   hx hy  2 2   2      k    hy  0 x2 y 2 x  y x 

y 2

x2

(2b)

In paraxial scalar approximation middle terms of equation (2a) and 2(b) are determined from the second term of equation (1). By neglecting the coupling between vector components of the field obtained from equation (1), equation (2a) and 2(b) in paraxial scalar approximation we can obtain a scalar wave equation [29].  2 H r  (k 2 (rT )   2 ) HT  0 (3) In the photonic crystal cladding maximum axial refractive index (nmax ) is strongly frequency dependent. For a scalar approximation by averaging the square of reflective index in photonic crystal cladding it can be estimated that [30].

nmax  (1  F )ng2  Fna2

(4)

Equation (4) where, F is the air filling fraction and na is refractive index in holes. For a long wavelength limitation Maximum axial refractive index varies with frequency and can be calculated by broadening the fields in terms of Bessel's function and applicable symmetry centered on the air holes. As per Helmholtz equation [31-32] for a source free, transparent, time invariant and non permeable space can be expressed as equation (5) and (6).

 E(r ) 

2 c2

 ( r ) E (r )

(5)

1 2 (6)  H (r )]  2 H (r )  (r ) c Where  (r ) is waveguide dielectric function,  [

 is angular frequency and E and H are electric and magnetic field respectively.

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Time step allowed for reliable computation, assuming speed of light as radiation propagation speed can be calculated as:

1 1 1 1  t      2 2 c  (x) (y) (z )2 

1 2

(7)

Where ∆x, ∆y and ∆z are the dimension of mesh cell. Large air filled fraction in the cladding region determines waveguide dispersion of fibers. It cancels out the effects of material dispersion. Hence dispersion profiles of PCFs are better than conventional optical fibers. Appropriate dispersion profile encourages non linear application of fiber. It has huge impact on walk off parameter, broadening of optical pulses, bandwidth of the fiber [33]. The model index number (neff ) is drawn from Maxwell’s equation by solving Eigen value problem using FDTD method. Total dispersion of any fiber is the result of addition of waveguide dispersion and material dispersion [34]. It can be obtained by using [34]

D  

Dg         Dm   

(8)

Where Γ is the confinement factor of silica, which is to unity for PCF. Material dispersion (Dm) can also calculated from three term Sellemier formula. However waveguide dispersion can be obtained by [34]. 2   Re[n eff ] D     c  2

x y   Re(n eff  n eff )

Where Re n

x eff

(10) and Re n are the real part of y eff

effective refractive index of x and y polarized mode respectively and B is the birefringence. PCFs having pitch factor larger then 1μm and with large air filled fraction results high nonlinearity and low confinement loss in fibers [38-39]. Non linearity can be enchased by reducing the effective modes through a core of small diameter. By choosing appropriate size and pattern of PCF structure, effective mode area can be reduced by which the non linearity of the fiber can be increased [40]. Effective non linear mode area is given by [40]  

eff  ( 



  2

E dxdy)2 /



4

E dxdy

    (11) Where E in equation is the electric field derived by solving Maxwell equation. Similarly the effective nonlinear coefficient can be calculated by formula [40]



n2 2 n2  cAeff  Aeff

(12) Where n2 is the non-linear refractive index 20

2

coefficient is 3.01X 10 m / W (for Silica), c is the velocity of light, and  is the operating wavelength.

(9)

Re[ neff ] is the real part of neff , λ is the operating wavelength and c is the velocity of light in vacuum. Large refractive index difference and small core in PCFs show high birefringence even at little asymmetric core. Degree of birefringence of any fiber can be easily determined by the assymetricity of the core. Moreover it can also be determined by the size of hole in the cladding region. Assuming cross section profile to be perfect, any fiber having structure greater than two fold symmetric is not birefringent [35-36]. However any imperfections in structure of fiber along with features possible in designing PCFs can lead to show birefriengent behavior [37]

In PCFs losses gets affected by fraction of light in the glass and roughness of glass air interfaced. Bloch waves in cladding region are evanescent for a guided mode. In fibers where cladding is thinner, these evanescent fields at the cladding boundaries are substantial and cause attenuation. In index guiding PCFs, smaller value of d/∧ result larger loss [40]. Confinement loss can be obtained by using the formula [41].

Lc  8.686[k0 I m neff ]

(13)

Step index approximation complicates the definition values of refractive indices in case of PCFs due to non circularity of its core region. Moreover, its effective indices have strong dependency on wavelength caused due to air-

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glass matrix. Pitch factor (  ) , that is hole to hole spacing is used instead of core radius Veff of PCFs can be obtained as [42].  2 2 Veff  2 ncore  neff

has been shown in Fig. 3. Structure B reports highest birefringence as compared to the other structures. Besides the other three structures also 3 report high birefringence is of order 10 .



(14) Where  is wavelength in vacuum, the ncore is the refractive index of core, and neff is the effective refractive index. IV. SIMULATION AND RESULTS Structures with different refractive index profile have been simulated. A full vector method with transparent boundary condition has been used to study the propagation characteristics of designed fibers. An efficient finite difference time domain (FDTD) method has been applied. All the holes are of equal dimensions. Holes have a diameter of 1.2  m . Pitch factor taken for all structure

Fig. 2 Dispersion behaviour

is 2  m . Difficulties have been found for maintaining phase matching in long fibers due to dispersion. In order to overcome this problem, nearly zero flattened dispersion fibers are desired. Dispersion properties of PCFs are considered to be the most important parameter. In order to generate supercontinnum light, high nonlinearity with nearly zero flattened dispersion behavior is highly desirable. Dispersion as a function of wavelength for all the structure simulated in this paper is shown in Fig. 2. Structure A reported negative dispersion. Structure B shows almost zero dispersion. However structure B has shown flattened zero dispersion over a wide range of wavelength. Structure C and D have also reported nearly zero dispersion beyond wavelength 0.85 micrometer. Simulation results show an increase in birefringence with increase in wavelength. PCFs with high birefringence and nearly zero flattened dispersion have been widely accepted for nonlinear applications. Moreover highly birefringent fibers are used in optical sensors. Birefringence property of investigated structure

Fig. 3 Birefringence behaviour

Fig. 4 Effective mode area

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Area occupied by the fundamental mode of the fibre is termed as effective mode area. Dimension of fundamental mode increases with the increase in wavelength. Nonlinear co-efficient and effective mode area is inversely related. Variation of effective mode area with the increase in wavelength can be seen in Fig. 4. Moreover non linear co-efficient as a function of wavelength has been shown in Fig. 5.

Fig. 5 Nonlinear coefficient as a function of wavelength

investigated has reported ultra low loss. However structure B reports very low loss compared to other structure. Structure C, D and A also reports loss of the order of 106 . Effective refractive index decreases with the increase in the wavelength as shown in Fig. 7. It shows that investigated structure has a good agreement from theory [41].

Fig. 7 Effective index number

Investigated PCF structures allow single mode propagation over telecommunication wavelength due to fine confinement of mode. However, the value of Veff  4.1 assures the fibers to support single mode propagation as shown in Fig. 8.

Fig. 6 Confinement loss behaviour

Guided modes of PCFs are inherently leaky. It is due to low index contrast between core and cladding without holes filled with air. These leaky modes are the major contributors to losses observed in PCFs than other factors like physical imperfection loss, absorption loss and scattering loss. Confinement loss of structures studied in this paper is shown in Fig. 6. All the structure

Fig. 8 Normalized frequency

The simulated results for different PCFs

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structure are compared with other published results for other PCF structures in Table 2.

2.

Table 2: Comparison with previously reported PCF structures:

3.

PCF

Ref [14]

Metho d and Theore tical/ Experi mental FDTD

Ref [16]

FDTD

Ref [17] Ref [22] Ref [35]

FEM FEM FEM (Experi mental) FDTD (Experi mental) FDTD

2.10μm 1.6μm 1.2μm

65 55 60

10-4 10-3 10-3

10-1 ---

2μm

-700

--

10-1

1.25μm

-750

--

--

FDTD

1.2μm

Nearl y Zero

10-2

10-6

Ref [43]

Ref [44] Proposed structure B

Dimensi on of holes (Diamet er)

Dispe rsion in ps/n m.km

Birefri ngence

Confine ment loss in dB/Km

1.60μm to 2.40μm 1.2μm

55

10-3

10-1

-120

10-3

10-4

4.

5.

6.

7.

V. CONCLUSION Structure B with triangular refractive index profile reports good results in comparison to other structures. It shows nearly flattened zero dispersion with very high birefringence of the 2 order of 10 . Moreover, it has low effective mode area and exhibits very low loss than other structures. However, other structures investigated in this paper also report almost zero dispersion at first and second optical window. Also these structures reports high birefringence and very low loss. All structures simulated allow single mode propagation. Thus it can be said that though all the structures investigated in this paper shows good propagation characteristics of fibre. However, structure B has shown excellent and desirable results. Hence it can be used in optical communications high data rate transfer, optical sensor and for medical applications.

8.

9.

10.

11.

12. 13.

14.

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