High birefringence low-dispersion of nonlinear photonic crystal fiber

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High birefringence low-dispersion of nonlinear photonic crystal fiber夽 Qiang Xu a,b,∗ , Runcai Miao a , Yani Zhang b a b

College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China Department of Physics and Information Technology, Baoji University of Arts and Sciences, Baoji 721016, China

a r t i c l e

i n f o

Article history: Received 16 February 2012 Accepted 25 June 2012 Keywords: Fiber and waveguide optics Finite element method High birefringence Nonlinearity

a b s t r a c t The paper presents a theoretical design of high nonlinear photonic crystal fiber (PCF) with high birefringence and low dispersion. Its birefringence, dispersion and nonlinear coefficient are investigated simultaneously by using the full vectorial finite element method with anisotropic perfectly matched layers. Numerical results show that the proposed highly nonlinear low-dispersion fiber has a total dispersion as low as ±12.6 ps km−1 nm−1 over ultrabroad wavelength range from 1.21 to 1.61 ␮m, and the corresponding birefringence and nonlinear coefficient are about 2.1 × 10−3 and 28 W−1 km−1 at 1.55 ␮m, respectively. The proposed PCF with high birefringence, low dispersion and high nonlinearity can have important application in supercontinuum (SC) generation, dispersion compensation, wavelength (frequency) converter, polarization maintaining transmission and Raman amplifier. © 2012 Elsevier GmbH. All rights reserved.

1. Introduction In recent years, photonic crystal fibers (PCFs) have attracted much attention because of their unique properties [1–5], such as endless single mode, ultra flattened dispersion, polarization properties and new nonlinear effects, which cannot be realized in conventional fibers. PCFs are divided into two groups, named index guiding PCFs or holey fibers and photonic crystal band gap fibers. The index guiding PCFs have a silica core at the center of a structure consisting of multiple rows of air holes in silica with a triangular lattice, as cladding. The propagation of the lightwave in holey fibers, the same as conventional optical fibers, is by total internal reflection (TIR). In the other type of PCF a periodic structure consisting of holes in silica confine the lightwave with frequency in the structure photonic band gap (PBG) in a low-index core region [6]. High level of birefringence in PCFs is required to maintain the linear polarization state by reducing polarization coupling. Due to the large index contrast of PCF compared to the conventional fiber, highly birefringent (HB) PCFs have been reported by breaking the circular symmetry implementing asymmetric defect structures such as dissimilar air hole diameters along the two orthogonal axes [5,7], asymmetric core design [8] and designing an air hole lattice or a microstructure lattice with inherent anisotropic properties such

夽 Project supported by the Baoji University of Arts and Science Key Research Foundation (Grant No. ZK11142) and the Science and Technology Project of Shaanxi Province, China (Grant Nos. 2010K01-078 and 2011K02-08). ∗ Corresponding author at: College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China. E-mail address: [email protected] (Q. Xu).

as the elliptical-hole PCF [9], and squeezed hexagonal-lattice PCFs [10,11]. Modal birefringence in these HB PCFs has been predicted to have values an order magnitude of 10−3 higher than that of the conventional HB fibers (10−4 ) [12]. Control of chromatic dispersion in the PCFs is a very important issue for practical applications in dispersion compensation of optical communication systems and nonlinear optics. The dispersion properties of square lattice PCFs have been reported by Bouk et al. [13]. In the study, it has been demonstrated that the square lattice PCFs with the smallest pitch, that is 1 ␮m, have negative dispersion in the wavelength range around 1.55 ␮m. The square lattice PCFs with small pitch and large air hole diameter, whose dispersion slope is also negative, can be used to compensate the positive dispersion and dispersion slope of the traditional single-mode fibers in the C band. Nonlinear properties of PCFs are playing important role in the field of nonlinear optics. Fibers with high values of effective nonlinearity can reduce device length and the associated optical power requirements for fiberbased nonlinear optical devices. High nonlinearity can be efficiently used to generate supercontinuum pumped by ultra fast laser pulses and longer laser pulses [14]. The large variety of microstructured holes and its arrangements demand the use of numerical methods that can handle arbitrary cross-sectional shapes to analyze this kind of structures. Numerical studies of photonic crystal fiber employ a wide variety of techniques [15] including the finite difference time domain method (FDTDM), the Fourier transform method (FTM), the plane wave expansion method (PWEM), the scalar effective index method (SEIM), the beam propagation method (BPM), the finite element method (FEM), the multiple method (MPM), etc. In this paper, we propose a modified rectangular lattice PCF which exhibits low dispersion high birefringence (LDHB) and high

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Please cite this article in press as: Q. Xu, et al., High birefringence low-dispersion of nonlinear photonic crystal fiber, Optik - Int. J. Light Electron Opt. (2012), http://dx.doi.org/10.1016/j.ijleo.2012.06.088

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nonlinear single mode operation in the C band simultaneously. The optical properties of proposed fiber are numerically studied by adopting full vectorial finite element method (FV-FEM) [16] with anisotropic perfectly matched layers (PMLs) [17]. This fiber will have important applications in the fields of polarization maintaining transmission system and dispersion compensation, and also in the design of widely tunable wavelength converter based on four-wave mixing.

To accurately simulate this PCF architecture we have adopted an efficient full-vector FEM with anisotropic perfectly matched layers (PMLs) to predict all the propagation characteristics of the waveguide with high accuracy. From Maxwell’s equations the basic equation for the FEM can be expressed as (1)

where E denotes the electric field, and [εr ] and [r ] are the relative dielectric permittivity and magnetic permeability tensors, respectively. When applying a FV-FEM to PCFs, a curvilinear hybrid edge element is very useful for avoiding spurious solutions and for accurately modeling curved boundaries of air holes. Dividing the fiber cross section into a number of curvilinear hybrid elements, from (1) we can obtain the following eigenvalue equation [K]{E} =

k02 n2eff [M]{E},

(2)

where [K] and [M] are the finite element matrices, {E} is the discretized electric field vector consisting of the edge and nodal variable, neff is the effective index, and anisotropic PLMs as absorbing boundary condition is used to evaluate the confinement losses. Utilizing sparse nature of [K] and [M], (2) is solved with the multifrontal method, and neff is obtained. 2.1. Phase and group modal birefringence To analyze the polarization properties of PCF, normally two kinds of modal birefringence are defined. That is, the phase modal birefringence is expressed as y

B() = |Re(neff ()) − Re(nxeff ())|,

(3)

which is associated with the polarization beat length LB () = /B(), and the group modal birefringence G() =

dˇx dˇy dˇ() − = B() −  , dk dk dk

2.3. Nonlinearity coefficient The nonlinearity coefficient () is another important parameter for fiber design. The nonlinearity coefficient () of PCFs can be defined as [19]

2. Theoretical method

 − k2 [εr ]E = 0, ∇ × ([r ]−1 ∇ × E) 0

where  is the confinement factor in silica. To most index-guided PCFs, the modal power is almost confined in the silica core and  is close to unity [18]. The material dispersion can be obtained directly from the three-term Sellmeier formula, while the waveguide dispersion can be calculated the same as in Eq. (5).

() =

2n2 Aeff

(7)

where n2 = 3.0 × 10−20 m2 W−1 is the nonlinear refractive index of silica material, and Aeff is the effective mode area, which is calculated by Aeff =

(

 2  dx dy)2 |E|   4 dx dy |E|

(8)

where E is the electric field of the medium. A low effective area provides a high density of power needed for nonlinear effects to be significant. 3. Design model and simulation results Fig. 1 shows the proposed PCF design. The whole fiber is based on pure silica and the all air-holes are arranged by using square lattice structure along the fiber length, and a central air-hole is eliminated to form a light propagation region. It is characterized by the lattice constants , an air-filling fraction f = d/ , the small air-holes diameter D and the big air-holes diameter d, which represent the x-directional and y-directional the air hole diameters, respectively. In general, the modal birefringence is an important parameter to define the polarization property of birefringent fibers. Fig. 2 shows the calculated phase birefringence by changing the air-filling fraction f from 0.6 to 0.9 in step size 0.1 with fixing = 2.0 ␮m. From Fig. 2 we can see that phase birefringence value increases with the wavelength and up-shift with the increase of the air-filling fraction f. when D = 0.92 ␮m, = 2.0 ␮m, f = 0.9, the maximum of the phase birefringence can be up to 2.1 × 10−3 for the operating wavelength 1.55 ␮m.

(4)

where  is the wavelength of the light, ˇx () and ˇy () are the propagation constants of two orthogonal polarization modes, with the x,y x,y relationship of neff () = ˇx,y ()/k0 to the refractive indices neff () and the free-space wave number k0 = 2/.

2.2. Chromatic dispersion Once the modal effective indexes neff are solved, the waveguide dispersion parameter Dw () can be obtained: Dw () = −

 ∂2 |Re(neff )| , c ∂2

(5)

where c is the velocity of the light in a vacuum. The total dispersion is calculated as the sum of waveguide dispersion (or geometrical dispersion) and the material dispersion in the first-order approximation: D() ≈ Dw () +  ()Dm (),

(6)

Fig. 1. Schematic cross-section of the proposed PCF structure.

Please cite this article in press as: Q. Xu, et al., High birefringence low-dispersion of nonlinear photonic crystal fiber, Optik - Int. J. Light Electron Opt. (2012), http://dx.doi.org/10.1016/j.ijleo.2012.06.088

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f=0.6 f=0.7 f=0.8 f=0.9

-3

2.5x10

Birefringence of phase

Dispersion (ps·km-1·nm-1)

3.0x10

-3

2.0x10

-3

1.5x10

-3

1.0x10

-4

5.0x10

0.0 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

3

40 30

10

Dm( ) +Dw( )

0 -10

Dw( )

-20 -30 -40 1.2

Wavelength( μm)

Dm( )

20

1.3

1.4

1.5

1.6

1.7

Wavelength (μm)

Fig. 2. Birefringence of phase as a function of wavelength . Fig. 5. Total dispersion of the PCF when D = 0.92 ␮m, = 2.0 ␮m and f = 0.9. -3

2.0x10

-3

-3

1.0x10

f=0.6 f=0.7 f=0.8 f=0.9

-4

5.0x10

0.0 -4

-5.0x10

-3

-1.0x10

-3

-1.5x10

-3

-2.0x10

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Wavelength( μm) Fig. 3. Birefringence of group as a function of wavelength .

7.0

0

70

6.5 6.0

60

-4 5.5

50

2

Aeff( μm )

Waveguide dispersion( ps·km-1·nm-1)

Fig. 3 shows the calculated group birefringence curves by fixing D = 0.92 ␮m and = 2.0 ␮m, while changing the air-filling fraction f from 0.6 to 0.9 in step size 0.1. It can be seen from Fig. 3 that the proposed PCF has a negative group birefringence parameter and a negative group birefringence slope in the wavelength range around 1.55 ␮m. The dependences of waveguide dispersion Dw () on various incremental values of the design parameter of the proposed PCF are plotted in Fig. 4. Here the silica index is assumed to be 1.45 for calculating dispersion. Fig. 4 shows the waveguide dispersion curves by fixing D = 0.92 ␮m and = 2.0 ␮m, while changing the air-filling fraction f from 0.6 to 0.9 in step size 0.1. It can be seen from Fig. 4 that the proposed PCF has a negative dispersion parameter and a negative dispersion slope in the wavelength range around 1.55 ␮m, which demonstrates an excellent dispersion

compensating property. The Dw () decreases gradually with the wavelength as well as present down-shift with the increase of f. Here, In order to obtain better dispersion compensating, we fixed f = 0.9. The total dispersion of PCF can be easily tailored based on the following design procedures. First, calculate the material dispersion Dm () according to Sellmeier formula; then obtain the waveguide dispersion of PCF and plot curves of Dm () and Dw (). Finally, achieve the total dispersion D() according to Eq. (6). Fig. 5 shows that the total dispersion curve, waveguide dispersion curve and material dispersion curve versus wavelength for the proposed PCF with the optimal design parameter D = 0.92 ␮m, = 2.0 ␮m and f = 0.9. Obviously, the material dispersion curve, which is monotonically increasing with the increase of wavelength, is anti-symmetric curves with the waveguide dispersion curve. As you can see, the total dispersion is an ultra-flattened within ±12.6 ps km−1 nm−1 for the wavelengths in the range from 1.21 to 1.61 ␮m. In Fig. 6 we plot the calculated effective mode area and the nonlinearity coefficient as a function of the wavelength , for the optimized design parameters D = 0.92 ␮m, = 2.0 ␮m and f = 0.9. From the results in Fig. 6, we can clearly observe the remarkably small effective mode area of the proposed PCF with only six airhole-rings exhibits 4.4 ␮m2 at the wavelength of  = 1.55 ␮m. The lower effective mode area will induce higher nonlinearity coefficient which is about 28 W−1 km−1 , that will have an important application in nonlinear four wave maxing effect. Finally, from the electric field distribution of the fundamental mode in Fig. 7 at the wavelength of  = 1.55 ␮m, we can see the strong confinement of light in the core of the PCF. The stronger mode confinement can give rise to lower confinement leakage loss. Hence, the proposed PCF has a high birefringence and negative

-8

-12

5.0 40

4.5 4.0

f=0.6 f=0.7 f=0.8 f=0.9

30

γ ( km-1·W-1)

Birefringence of group

1.5x10

3.5

20 3.0 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

-16 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Wavelength( μm) Fig. 4. Dispersion as a function of wavelength .

Wavelength (μm) Fig. 6. The effective mode area and the nonlinearity coefficient as a function of the wavelength  for the proposed PCF with optimized design parameters D = 0.92 ␮m, = 2.0 ␮m and f = 0.9.

Please cite this article in press as: Q. Xu, et al., High birefringence low-dispersion of nonlinear photonic crystal fiber, Optik - Int. J. Light Electron Opt. (2012), http://dx.doi.org/10.1016/j.ijleo.2012.06.088

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References

Fig. 7. Electric field distribution of the fundamental mode at a wavelength of 1.55 ␮m for the proposed PCF.

dispersion effect and low effective mode area with inducing high nonlinearity coefficient, which will has important applications in the fields of polarization maintaining transmission, chromatic dispersion management, dispersion compensation, SC generation, wavelength (frequency) converter and Raman amplifier. 4. Conclusions A type of high-birefringence low-dispersion of nonlinear PCF is proposed and optimized by using the FV-FEM. The wavelength of the proposed highly nonlinear low-dispersion PCF can be controlled artificially by adjusting the air-filling fraction f. When the parameters of the proposed fiber are optimized to be D = 0.92 ␮m, = 2.0 ␮m and f = 0.9, the optimized PCF has a low dispersion of about ±12.6 ps km−1 nm−1 from 1.21 to 1.61 ␮m and a high nonlinear coefficient of 28 W−1 km−1 at 1.55 ␮m. Also, the optimized PCF is single mode with a birefringence of 2.1 × 10−3 . There will be no doubt that the proposed PCF can obtain the admirable applications in the fields of the chromatic dispersion controlling, the dispersion compensation and the nonlinear optical transmission system.

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Please cite this article in press as: Q. Xu, et al., High birefringence low-dispersion of nonlinear photonic crystal fiber, Optik - Int. J. Light Electron Opt. (2012), http://dx.doi.org/10.1016/j.ijleo.2012.06.088