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Physics & Astronomy International Journal

Review Article Volume 2 Issue 2

Analytical comparison of photonic crystal fibers for dispersion compensation with different structures using FDTD method

Faramarz E Seraji, Vajieh Arsang

Department of Communication Technology, Iran Telecom Research Center, Iran

Correspondence: Faramarz E Seraji, Faculty member Optical Communication Group, Department of Communication Technology, Iran Telecom Research Center, Tehran, Iran, Tel 98?21?84977723

Received: March 26, 2017 | Published: April 25, 2018

Citation: Seraji FE Arsang V. Analytical comparison of photonic crystal fibers for dispersion compensation with different structures using FDTD method. Phys Astron Int J. 2018;2(2):155-158. DOI: 10.15406/paij.2018.02.00078

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Abstract

In new generation of optical communication networks, simultaneous transmission of several information channels has become possible over a single optical fiber. But dispersion and dispersion slope of optical fibers are important factors, which limit high capacity optical fiber communication systems to perform high speed transmission.

In this paper, silica core photonic crystal fibers (PCFs) with three and six air–hole rings in the cladding are investigated for effective refractive index, propagation constant and dispersion characteristics, using FDTD method. It is shown that by increasing the number of the air–hole rings in the cladding around the core, the negative dispersion of the PCFs will increase, too. The obtained negative dispersion of –300 ps/nm.km is comparatively high with respect to the published results in the related literature.

Keywords: dispersion, modeling, FDTD method, photonic crystal fibers

Introduction

Photonic crystal fibers (PCFs) have attained popularity in designing of compact optical devices utilized in optical networks1,2 and optical sensing systems.3,4 The widely used applications are in designing of dispersion compensators commonly used in WDM optical networks reducing the number of required repeaters.5,6 One of the comparative characteristic features of the PCFs is their flexibilities in response to dispersion effects on propagating pulses present in a long–haul transmission optical fiber line.7,8

Till date, the photonic crystal fibers (PCFs) are known to be new types of optical fibers with different structures and profiles as compared to conventional optical fibers that are used in optical communication systems.9,10 Having different optical characteristics, the PCFs as optical devices have been used in various applications in optical systems, such as super continuum spectrum generation,11 high optical power transfer,12 dispersion compensations,13 tunable optical filters,14 Erbium–doped optical amplifiers,15 Bragg grating–based devices,16 optical nonlinear applications.17,18

Different methods are used to evaluate dispersion properties of the PCFs with various structures.19 In this paper, by using FDTD analysis, two different PCF structures are investigated for utilization in compensations of positive dispersion in optical fiber as a transmission link.

For determination of optimal parameters values of the PCF, analytical and numerical methods, such as the scalar effective index method,20 the vectorial effective index method (VEIM),21 the improved vectorial effective index method (IVEIM),22 and the finite difference frequency domain (FDFD) method,23,24 are usually utilized. Couple of reports on designs of dispersion compensating fibers (DCFs) are published on optimization of compensation of the dispersions using PCFs in optical transmission bands ranging from 1460 nm to 1675 nm.25,26

A survey shows a design of a pentagonal PCF with large flattened negative dispersion by using the full vector finite element method (FEM) where the average negative dispersions for two optimized designs were −611.9 ps/nm/km over 1,460–1,625 nm and −474 ps/nm.km over 1425–1675 nm wavelength bands, respectively.27 In another report a porous–core circular PCF with circular arrangement of air holes, both in the periodic cladding and the porous core is simulated by using an efficient FEM. It has shown a flattened dispersion of ± MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHXcqSaaa@3882@ 0.09 ps/THz/cm of 0.9–1.3 THz.28

An ultra–flattened dispersion over the range of –210 – 15 ps/nm.km is reported in29 by doping the core of a PCF with Germanium. In another report, a design of a hybrid PCF with elliptical and circular air holes of hexagonal layout with triangular lattice of five rings around the solid core is presented in,30 where they obtained large flattened dispersion is of the order of 4.88 ps/nm.km over a wavelength range of 1200–1800 nm.

In our previous report, by using two–dimensional finite difference time domain (2D–FDTD), three PCF structures were proposed to optimize the dispersions by considering the effects of the geometrical parameters, such as air–hole diameters and the center–to–center spacing between the holes (pitch). The air–holes were arranged in the cladding in the form of triangular lattice by constituting 11 rings around the core.31,32

FDTD numerical method

For a linear, isotropic, non–dispersive material with no source, the time–variant Maxwell’s equations are expressed as follows:9,33,34

Ht=1μ(r)×E MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabaqaamaabaabaaGcbaWaaSaaaeaajugibiabgkGi2QWaa8HaaeaajugibiaadIeaaOGaay51GaaabaqcLbsacqGHciITcaWG0baaaiabg2da9iabgkHiTOWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaeqiVd0MaaiikaiaadkhacaGGPaaaaiabgEGirlabgEna0QWaa8HaaeaajugibiaadweaaOGaay51Gaaaaa@4C4E@                 (1)

Et=1ε(r)×Hσ(r)ε(r)E MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabaqaamaabaabaaGcbaWaaSaaaeaajugibiabgkGi2QWaa8HaaeaajugibiaadweaaOGaay51GaaabaqcLbsacqGHciITcaWG0baaaiabg2da9OWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaeqyTduMaaiikaiaadkhacaGGPaaaaiabgEGirlabgEna0QWaa8HaaeaajugibiaadIeaaOGaay51GaqcLbsacqGHsislkmaalaaabaqcLbsacqaHdpWCcaGGOaGaamOCaiaacMcaaOqaaKqzGeGaeqyTduMaaiikaiaadkhacaGGPaaaaOWaa8HaaeaajugibiaadweaaOGaay51Gaaaaa@593B@   (2)

Whereε(r),μ(r),σ(r) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabaqaamaabaabaaGcbaqcLbsacqaH1oqzcaGGOaGaamOCaiaacMcacaGGSaGaaGPaVlaaykW7cqaH8oqBcaGGOaGaamOCaiaacMcacaGGSaGaaGPaVlaaykW7cqaHdpWCcaGGOaGaamOCaiaacMcaaaa@49AE@ are susceptibility, permeability, and conductivity of the dielectric material, respectively, andE MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabaqaamaabaabaaGcbaqcLbsaceWGfbGbaSaaaaa@36EE@ andH MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabaqaamaabaabaaGcbaqcLbsaceWGibGbaSaaaaa@3744@ are respective electric and magnetic fields vectors. Using Yee cell approach, the Maxwell’s equations can be written in the form of partial differential equations with respect to time and space.35

If we indicateβ MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGIYoaaaa@37A7@ as the propagation constant alongz MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG6baaaa@3765@ direction in a PCF, then the expression φ(x,y,z)=φ(x,y)ejβz MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgpGAcaGGOaGaamiEaiaacYcacaWG5bGaaiilaiaadQhacaGGPaGaeyypa0JaeqOXdOMaaiikaiaadIhacaGGSaGaamyEaiaacMcacaWGLbWcdaahaaqcbasabeaajugWaiaadQgacqaHYoGycaWG6baaaaaa@4A96@  will denote component of each field wherej2=1 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGQbqcfa4aaWbaaKqaafqabaqcLbmacaaIYaaaaKqzGeGaeyypa0JaeyOeI0IaaGymaaaa@3FA6@ . Therefore, the differentiation of Maxwell’s equations with respect toz MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG6baaaa@3785@ may be replaced byjβ MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGQbGaeqOSdigaaa@38F5@ and the fields would be expressed only in terms of transverse components.

A 2D Yee unit cell across the fiber cross section is shown in Figure 1.35 The x MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEaiabgkHiTaaa@37BF@ component of first curl of the Maxwell’s equation for magnetic field is derived as:33

Hxt=1μ(EzyEyz) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaWaaSaaaeaajugibiabgkGi2kaadIealmaaBaaajeaibaqcLbmacaWG4baajeaibeaaaOqaaKqzGeGaeyOaIyRaamiDaaaacqGH9aqpcqGHsislkmaalaaabaqcLbsacaaIXaaakeaajugibiabeY7aTbaakmaabmaabaWaaSaaaeaajugibiabgkGi2kaadweakmaaBaaajeaibaqcLbmacaWG6baaleqaaaGcbaqcLbsacqGHciITcaWG5baaaiabgkHiTOWaaSaaaeaajugibiabgkGi2kaadwealmaaBaaajeaibaqcLbmacaWG5baajeaibeaaaOqaaKqzGeGaeyOaIyRaamOEaaaaaOGaayjkaiaawMcaaaaa@5758@      (3)

Figure 1 2D yee unit cell across the fiber cross section.35

By discretization of Equation (3) in terms of time and space and by using Yee cell technique, we obtain:

Hx|i,kn+1/2=Hx|i,kn1/2Δtμi,k(Ez|i,k+1nEz|i,knΔyjβEy|i,kn) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9D83@ (4)

Wheren MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUbaaaa@3779@ is the discrete time step,i MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGPbaaaa@3764@ andk MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRbaaaa@3766@ denote grid points of discretization inxy MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4bGaamyEaiabgkHiTaaa@396E@ plane, and Δt MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqqHuoarcaWG0baaaa@38D5@ , Δx MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqqHuoarcaWG4baaaa@38D9@ , and Δy MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqqHuoarcaWG5baaaa@38DA@  denote the time increment, distances between two adjacent points alongx MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4baaaa@3763@ andy MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5baaaa@3764@ directions, respectively. Similarly, the components of the other fields can also be derived.

To reduce the computational time, one prefers to consider the real numbers in above equation. For this particular reason, let us consider the real parts of Ez,Hx,andHy MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGfbWcdaWgaaqcbasaaKqzadGaamOEaaqcbasabaqcLbsacaGGSaGaaGPaVlaaykW7caWGibWcdaWgaaqcbasaaKqzadGaamiEaaqcbasabaqcLbsacaGGSaGaaGPaVlaaykW7caqGHbGaaeOBaiaabsgacaaMc8UaaGPaVlaadIealmaaBaaajeaibaqcLbmacaWG5baajeaibeaaaaa@4F4A@  to becos(βz+φ) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGJbGaai4BaiaacohacaGGOaGaaOOSdiaadQhacqGHRaWkcaGIgpGaaiykaaaa@3F08@ and that ofHz,Ex,andEy MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGibWcdaWgaaqcbawaaKqzadGaamOEaaqcbasabaqcLbsacaGGSaGaaGPaVlaaykW7caWGfbGcdaWgaaqcbawaaKqzadGaamiEaaWcbeaajugibiaacYcacaaMc8UaaGPaVlaabggacaqGUbGaaeizaiaaykW7caaMc8UaamyraSWaaSbaaKqaGfaajugWaiaadMhaaKqaGeqaaaaa@4FE7@ to besin(βz+φ) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGZbGaaiyAaiaac6gacaGGOaGaaOOSdiaadQhacqGHRaWkcaGIgpGaaiykaaaa@3F0D@ . Therefore, Equation (4) reduces to:35

Hx|i,kn+1/2=Hx|i,kn1/2Δtμi,k(Ez|i,k+1nEz|i,knΔyβEy|i,kn) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8DC4@     (5)

They MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5baaaa@3784@ andz MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG6baaaa@3785@ components of magnetic fields are obtained as follows:

Hy|i,kn+1/2=Hy|i,kn1/2+Δtμi,k(Ez|i+1,knEz|i,knΔxβEx|i,kn) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8F05@    (6)

Hz|i,kn+1/2=Hz|i,kn1/2+Δtμi,k(Ex|i,k+1nEx|i,knΔyEy|i+1,knEy|i,knΔx) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A36B@    (7)

Similarly, one can obtain the components of electrical fields. The stability of the FDTD numerical method is determined by the following expression:35

Δt1cΔx2+Δy2+(β/2)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabaqaamaabaabaaGcbaqcLbsacqqHuoarcaWG0bGaeyizImQcdaWcaaqaaKqzGeGaaGymaaGcbaqcLbsacaWGJbGcdaGcaaqaaKqzGeGaeuiLdqKaamiEaSWaaWbaaKqaGeqabaqcLbmacqGHsislcaaIYaaaaKqzGeGaey4kaSIaeuiLdqKaamyEaSWaaWbaaKqaGeqabaqcLbmacqGHsislcaaIYaaaaKqzGeGaey4kaSIaaiikaiabek7aIjaac+cacaaIYaGaaiykaSWaaWbaaKqaGeqabaqcLbmacqGHsislcaaIYaaaaaWcbeaaaaaaaa@535B@             (8)

For our boundary computational treatment, we use perfectly matched layers method. One of the key characteristic parameters of PCF is chromatic dispersion, which is summation of waveguide and material dispersions. The expression for total dispersion, consisting of material and waveguide dispersion is given as:

Dtot=λ2πc(2dβdλ+λd2βdλ2) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGebWcdaWgaaqcbasaaKqzadGaamiDaiaad+gacaWG0baajeaibeaajugibiabg2da9iabgkHiTOWaaSaaaeaajugibiabeU7aSbGcbaqcLbsacaaIYaGaeqiWdaNaam4yaaaakmaabmaabaqcLbsacaaIYaGcdaWcaaqaaKqzGeGaamizaiabek7aIbGcbaqcLbsacaWGKbGaeq4UdWgaaiabgUcaRiabeU7aSPWaaSaaaeaajugibiaadsgalmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabek7aIbGcbaqcLbsacaWGKbGaeq4UdWMcdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaaaakiaawIcacaGLPaaaaaa@5B89@                (9)

To simplify the above equation, we define normalized propagation constantβN(=β/k0=2πβ/λ) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacqaHYoGylmaaBaaajeaibaqcLbmacaGIobaajeaibeaajugibiaakIcacaGI9aGaeqOSdiMaaO4laiaakUgalmaaBaaajeaibaqcLbmacaGIWaaajeaibeaajugibiaak2dacaGIYaGaeqiWdaNaeqOSdiMaaO4laiabeU7aSjaakMcaaaa@4A72@ and substitute it in (9) to obtain:9,18,10

Dtot=λc(d2βNdλ2)π MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGebWcdaWgaaqcbasaaKqzadGaamiDaiaad+gacaWG0baajeaibeaajugibiabg2da9iabgkHiTOWaaSaaaeaajugibiabeU7aSbGcbaqcLbsacaWGJbaaaOWaaeWaaeaadaWcaaqaaKqzGeGaamizaOWaaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqaHYoGykmaaBaaaleaajugibiaak6eaaSqabaaakeaajugibiaadsgacqaH7oaBlmaaCaaajeaibeqaaKqzadGaaGOmaaaaaaaakiaawIcacaGLPaaajugibiabec8aWbaa@52B2@             (10)

Whereλ MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7oaBaaa@382A@ andc MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJbaaaa@375E@ are wavelength and velocity of light in a vacuum, respectively. The effective refractive index of fundamental mode is defined as neff=β(λ/2π) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabaqaamaabaabaaGcbaqcLbsacaWGUbGcdaWgaaqcbasaaKqzadGaamyzaiaadAgacaWGMbaaleqaaKqzGeGaeyypa0JaeqOSdiMaaOikaiabeU7aSjaac+cacaaIYaGaeqiWdaNaaOykaaaa@4524@ .36

To evaluate Equation (10) in terms ofβN MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacqaHYoGylmaaBaaajeaibaqcLbmacaWGobaajeaibeaaaaa@3A0A@ ,dβN/dλ MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGKbGaeqOSdiMcdaWgaaqcbasaaKqzadGaaOOtaaWcbeaajugibiaac+cacaWGKbGaeq4UdWgaaa@3EB9@ ,d2βN/dλ2 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGKbGcdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabek7aITWaaSbaaKqaGeaajugWaiaak6eaaKqaGeqaaKqzGeGaai4laiaadsgacqaH7oaBlmaaCaaajeaibeqaaKqzadGaaGOmaaaaaaa@43F4@ , we use FDTD method. The refractive index of the core material is determined by using Sellmeier formula as:18,37

nsilica2(λ)=1+k=13akλ2λ2bk2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabaqaamaabaabaaGcbaqcLbsacaWGUbWcdaqhaaqcbasaaKqzadGaam4CaiaadMgacaWGSbGaamyAaiaadogacaWGHbaajeaibaqcLbmacaaIYaaaaKqzGeGaaiikaiabeU7aSjaacMcacqGH9aqpcaaIXaGaey4kaSIcdaaeWbqaamaalaaabaqcLbsacaWGHbGcdaWgaaqcbasaaKqzadGaam4AaaWcbeaajugibiabeU7aSPWaaWbaaSqabKqaGeaajugWaiaaikdaaaaakeaajugibiabeU7aSTWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaeyOeI0IaamOyaSWaa0baaKqaGeaajugWaiaadUgaaKqaGeaajugWaiaaikdaaaaaaaqcbasaaKqzadGaam4Aaiabg2da9iaaigdaaKqaGeaajugWaiaaiodaaKqzGeGaeyyeIuoaaaa@63AE@     (11)

Whereλ MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabaqaamaabaabaaGcbaqcLbsacqaH7oaBaaa@3818@ is the wavelength of light, andak MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGHbGcdaWgaaqcbasaaKqzadGaam4AaaWcbeaaaaa@39CB@ andbk MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGIbGcdaWgaaqcbasaaKqzadGaam4AaaWcbeaaaaa@39CC@ are the Sellmeier coefficients. The practical values of ak MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabaqaamaabaabaaGcbaqcLbsacaWGHbWcdaWgaaqcbasaaKqzadGaam4Aaaqcbasabaaaaa@3A09@  and bk MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabaqaamaabaabaaGcbaqcLbsacaWGIbWcdaWgaaqcbasaaKqzadGaam4Aaaqcbasabaaaaa@3A0A@  for k=1,2,3 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabaqaamaabaabaaGcbaqcLbsacaWGRbGaeyypa0JaaGymaiaacYcacaaMc8UaaGOmaiaacYcacaaMc8UaaG4maaaa@3F05@  are given as:38

a1=0.6961663,a2=0.4079426,a3=0.8974794, MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabaqaamaabaabaaGcbaqcLbsacaWGHbWcdaWgaaqcbasaaKqzadGaaGymaaqcbasabaqcLbsacqGH9aqpcaaIWaGaaiOlaiaaiAdacaaI5aGaaGOnaiaaigdacaaI2aGaaGOnaiaaiodacaGGSaGaaGPaVlaaykW7caaMc8UaamyyaOWaaSbaaKqaGeaajugWaiaaikdaaSqabaqcLbsacqGH9aqpcaaIWaGaaiOlaiaaisdacaaIWaGaaG4naiaaiMdacaaI0aGaaGOmaiaaiAdacaGGSaGaaGPaVlaaykW7caWGHbWcdaWgaaqcbasaaKqzadGaaG4maaqcbasabaqcLbsacqGH9aqpcaaIWaGaaiOlaiaaiIdacaaI5aGaaG4naiaaisdacaaI3aGaaGyoaiaaisdacaGGSaGaaGPaVdaa@643E@

b1=0.0684043,b2=0.1162414,b3=9.896161. MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabaqaamaabaabaaGcbaqcLbsacaWGIbWcdaWgaaqcbasaaKqzadGaaGymaaqcbasabaqcLbsacqGH9aqpcaaIWaGaaiOlaiaaicdacaaI2aGaaGioaiaaisdacaaIWaGaaGinaiaaiodacaGGSaGaaGPaVlaaykW7caWGIbWcdaWgaaqcbasaaKqzadGaaGOmaaqcbasabaqcLbsacqGH9aqpcaaIWaGaaiOlaiaaigdacaaIXaGaaGOnaiaaikdacaaI0aGaaGymaiaaisdacaGGSaGaaGPaVlaaykW7caaMc8UaamOyaSWaaSbaaKqaGeaajugWaiaaiodaaKqaGeqaaKqzGeGaeyypa0JaaGyoaiaac6cacaaI4aGaaGyoaiaaiAdacaaIXaGaaGOnaiaaigdacaGGUaaaaa@61FD@

In this paper, by using the above equations and other related expressions in FDTD method, two PCF samples with different numbers of air–hole rings are selected for design in order to study their dispersion behaviors.

PCF with three rings of air–holes

The selected PCF structure with air–hole of equal diametersd=0.75μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGKbGaeyypa0JaaGimaiaac6cacaaI3aGaaGynaiaaykW7caaMc8UaeqiVd0MaaOyBaaaa@4107@ and air–hole spacing ofΛ=2μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacqqHBoatcqGH9aqpcaaIYaGaaGPaVlaaykW7cqaH8oqBcaGITbaaaa@3F63@ arranged in three rings in the form of triangular lattice is illustrated in Figure 2. The core diameter is taken as 1.45μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaaIXaGaaiOlaiaaisdacaaI1aGaaGPaVlaaykW7cqaH8oqBcaGITbaaaa@3F36@ .

The spectrum of chromatic dispersion of fundamental mode in the wavelength range of 1100–2000 nm is illustrated in Figure 3 (Red curve). The dispersion at 1510 nm is obtained as –140 ps/nm.km. The dispersion of the proposed PCF in this case is nearly twice the dispersion of conventional dispersion compensating fiber (i.e.,–80 ps/nm.km).

Figure 2 The designed PCF with three rings of air–holes.

Figure 3 Dispersion spectra of the PCFs with three and six circular air–hole rings in the cladding region.

The PCF with six rings of air–holes

As of previous analysis, a PCF structure is considered with the same air–hole diameter ofd=0.75μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGKbGaeyypa0JaaGimaiaac6cacaaI3aGaaGynaiaaykW7caaMc8UaeqiVd0MaaOyBaaaa@4107@ , core diameter of 1.45μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaaIXaGaaiOlaiaaisdacaaI1aGaaGPaVlabeY7aTjaak2gaaaa@3D9B@ and a different air–hole spacing ofΛ=1.1μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacqqHBoatcqGH9aqpcaaIXaGaaiOlaiaaigdacaaMc8UaaGPaVlabeY7aTjaak2gaaaa@40CF@ . In this case, six air–hole rings are considered in the cladding region (in this case, figure not shown).36

By using the home–made programming code, in this case as well, the chromatic dispersion spectrum has been depicted in Figure 3 (Blue curve). In this case, the dispersion value of –300 ps/nm.km is obtained, which twice the value is obtained in the previous design of the PCF with three circular air–hole rings. The results obtained here is comparable to the ones reported,18 in which the proposed structure with seven air–hole rings were used. The reported dispersion value was –150 ps/nm.km.

In the simulations performed by FDTD method, by varyingβ MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacqaHYoGyaaa@3789@ values in each case, the resonance fundamental frequencies, wavelengths, and effective refractive indices are determined and tabulated in Table 1.17,18

×106β MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaacbeqcLbsaqaaaaaaaaaWdbiaa=DnacaWFXaGaa8hmaSWdamaaCaaajeaibeqaaKqzadWdbiaa=zdaaaqcLbsapaGaa8NSdaaa@3D7A@

neff MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaaiaabaqaamaabaabaaGcbaacbmqcLbsacaWFUbWcdaWgaaqcbasaaKqzadGaa8xzaiaa=zgacaWFMbaajeaibeaaaaa@3B6F@

λ(μm) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaacbeqcLbsacaWF7oGaaGPaVlaa=HcacaWF8oGaa8xBaiaa=Lcaaaa@3CBB@

Fundamental Frequency (THz)

5

1.3010

1.6349

183.50

6

1.3343

1.3973

214.70

7

1.3575

1.2185

246.20

8

1.3745

1.0795

277.90

9

1.3884

0.9631

309.50

12

1.4116

0.7391

405.90

15

1.4239

0.59641

503.00

Table 1 Calculated effective refractive index used for determination of dispersion of PCF with six air-hole rings

Conclusion

In this paper, by using FDTD method, two proposed photonic crystal fibers have been analyzed with three and six air–hole rings in the cladding arranged in triangular lattices. In the case of three air–hole rings in the cladding of the PCF, for air–hole with equal diameters of d=0.75μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGKbGaeyypa0JaaGimaiaac6cacaaI3aGaaGynaiaaykW7caaMc8UaeqiVd0MaaOyBaaaa@4107@ and air–hole spacing ofΛ=2μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacqqHBoatcqGH9aqpcaaIYaGaaGPaVlaaykW7cqaH8oqBcaGITbaaaa@3F63@ , the dispersion value is determined to be –140ps/nm.km at wavelength 1510nm.

In case of the PCF with six air–hole rings, air–hole spacing ofΛ=1.1μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacqqHBoatcqGH9aqpcaaIXaGaaiOlaiaaigdacaaMc8UaaGPaVlabeY7aTjaak2gaaaa@40CF@ , and with the same air–hole of diameter ofd=0.75μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaWGKbGaeyypa0JaaGimaiaac6cacaaI3aGaaGynaiaaykW7caaMc8UaeqiVd0MaaOyBaaaa@4107@ , the dispersion value is found to be –300ps/nm.km. In both the cases, the diameters of the PCF core are taken as1.45μm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacaaIXaGaaiOlaiaaisdacaaI1aGaaGPaVlabeY7aTjaak2gaaaa@3D9B@ .

It is further shown that when the air–hole spacingΛ MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaceGacaGaaiaabiqaamaabaabaaGcbaqcLbsacqqHBoataaa@37DC@ is decreased by a ratio of 55% and by doubling the number of air–hole rings, the dispersion will increase more than twice the value. The PCF with high negative dispersion value is suitable for design of high dispersion compensating fiber used in long–haul optical transmission links.

Acknowledgements

The authors greatly acknowledge the allotment of the post–graduate student by GJK Institute of Higher Education, Qazvin, Iran, for collaboration in the running academic project of optical group at ITRC.

Conflict of interest

Authors declare there is no conflict of interest.

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