SETIT 2009 th
5 International Conference: Sciences of Electronic, Technologies of Information and Telecommunications March 22-26, 2009 – TUNISIA
Ultra-Flattened Dispersion Honeycomb Lattice Photonic Crystal Fiber Nihal AREED Mansoura University, Mansoura, Egypt
[email protected] Abstract: A full vectorial Finite Difference Frequency Domain (FDFD) analysis is effectively applied to investigate the modal characteristics of photonic crystal fibers (PCFs). Comparison with a Full Vectorial Imaginary Distance Beam Propagation (FVIDBP) method based on finite element scheme is made and excellent agreement is achieved. Moreover, a detailed study of PCF with honeycomb lattice has been carried out with the purpose of optimizing both the confinement loss and the chromatic dispersion. Key words: photonic crystal fiber, holey fibers, finite difference frequency domain, confinement loss, dispersion.
1. Formulation Full vector finite difference mode solver that is based on discretization scheme first proposend by Yee[K. 66]. Yee's mesh is widely used in the FD analysis [K. 93]. Here Yee's 2-D mesh in frequency domain mode solver is used. The Yee's 2-D mesh is illustrated in Fig.1. The continuity conditions are automatically satisfied since all the transverse field components are tangential to the unit cell boundaries.
INTRODUCTION Over the past few years, photonic crystal fiber (PCF) technology has evolved from a strong research-oriented field to a commercial technology providing characteristics such as a wide single-mode wavelength range, a bend-loss edge at shorter wavelengths, a very large or small effective core area, anomalous group velocity dispersion at visible and near-infrared wavelengths [T. 03]. In PCFs, the guiding of light is obtained by an arrangement of air holes running along the length of the fiber. If a defect is introduced in the fiber, frequencies in the band gap can propagate only in the defect since they are forbidden from propagating in the lattice, and thus a waveguide is formed. Research over the last decade has generated a wide range of rigorous numerical algorithms for modeling PCF, such as plane-wave expansion (PWE) [A. 99][S. 01], Finite Difference Time Domain (FDTD) [M. 01][G. 01], Finite Element (FE) methods [F. 00][KUN 02]. These versatile algorithms have been applied with success to study issues such as dispersion and losses for PC wave-guide.
Fig. 1. (a) Yee’s 2-D mesh [Zha. 02]. Assume the fields have dependence of exp[i(βz-ωt)], and from Maxwell’s curl equations ( ∇×E=-∂B/∂t , ∇×H=∂D/∂t ), after scaling E by the free space impedance we have [Zha. 02]: µ zo = o εo
In this paper Finite Difference Frequency Domain (FDFD) Technique [ZHA 02] has been utilized to optimize the modal properties of honeycomb lattice PCFs. The paper is organized as follows. Following this introduction, a brief mathematical treatment of FDFD method is given in section 2. The results and their physical explanation are detailed in section 3. Finally, conclusions are drawn.
-1-
ik o H x = ∂E z / ∂y − i β E y ,
(1a)
ik o H y = i β E x − ∂E z / ∂x ,
(1b)
ik o H z = ∂E y / ∂x − ∂E x / ∂y ,
(1c)
SETIT2009
−ik o ε r E x = ∂H z / ∂y − i β H y ,
(2a)
−ik o ε r E y = i β H x − ∂H z / ∂x ,
(2b)
−ik o ε r E z = ∂H y / ∂x − ∂H x / ∂y .
(2c)
where P is a square matrix defined as [Zha. 02]:
⎛ Pxx Pxy ⎞ P =⎜ ⎟ ⎝ Pyx Pyy ⎠
Solving the eigenvalue equation (7) for the eign value β; the effective modal index is computed from neff =β/ko. The eign vectors are the modal fields of the guided modes. In this paper the refractive indices and field distributions of PCFs are calculated using a designed 2D FDFD package for mode solver. The confinement losses and the dispersion curves are then calculated. An especially MATLAB codes are designed to evaluate these parameters through the simulation of Eq. (9) and Eq. (10) [S. 05][F. 04][KRI 05] [J. 03]:
where, k o = ω µo ε o , ω is the angular frequency, µo, and εo are the permeability and the permittivity of the free space, respectively, εr is the relative permittivity of the medium considered, and β is the modal propagation constant. Discretizing Eqs. (1) and (2) yields
iko Hx ( j , l ) = ⎡⎣Ez ( j , l +1) − Ez ( j , l )⎤⎦ / ∆y −i βEy ( j , l ) , (3a) ikoHy ( j ,l ) = i βEx ( j ,l ) −⎡⎣Ez ( j +1, l ) −Ez ( j ,l ) ⎤⎦ / ∆x ,
(8)
loss ( dB / m ) = 8 .868 × 10 6 × k o × Im( n eff )(9)
(3b)
Dispersion ( Ps / km .nm ) =
ikoHz ( j,l ) =⎣⎡Ey ( j +1,l) −Ey ( j,l )⎤⎦/∆x −⎡⎣Ex ( j,l +1) −Ex ( j,l )⎤⎦/∆y, (3c)
−
−ikoεrx Ex ( j ,l ) = ⎡⎣Hz ( j ,l ) −Hz ( j ,l −1) ⎤⎦ / ∆y −i βHy ( j ,l ) , (4a)
λ c
(10)
∂ Re( n eff ) 2
×
∂λ 2
−ikoεry Ey ( j ,l ) =i βHx ( j ,l ) −⎡⎣Hz ( j ,l ) −Hz ( j −1,l )⎤⎦ / ∆x , (4b)
where λ is the wavelength in units of µm and neff is the effective mode index.
−ikoεrzEz ( j,l) =⎣⎡Hy ( j,l) −Hy ( j −1,l)⎦⎤/∆x−⎡⎣Hx ( j,l) −Hx ( j,l −1)⎤⎦/∆y. (4c)
2. Numerical results First, the numerical method FDFD outlined above has been used to calculate the modal characteristics of a HF with a two hexagonal rings of eighteen holes symmetrically arranged around the core as shown in Fig. 2.a In this figure d is the hole diameter, the a is the lattice constant and the back ground index n=1.45. Because of the symmetrical nature of the system only one-quarter of the fiber cross section is analyzed. The computational window size (x×y) is 9µm×9µm. the field profiles of the y dominant Hy component, of the fundamental HE11 mode for a hole pitch value of 1.8 µm and the d/a ratio of 0.6 is plotted in Fig. 2.b. Figures (3.a) and (3.b) show the wavelength dependence of the real part of the complex effective index and the wavelength dependence of the confinement loss, respectively. The results of the imaginary distance full vectorial finite element based beam propagation method (IDVFEBPM) [SAL 05] also shown on the same figures. The comparison between the two methods illustrates that, the two results are close to each other. However, the FDFD method is simpler and needs less computational resources.
where,
ε rx ( j , l ) = ⎡⎣ε r ( j , l ) + ε r ( j , l − 1) ⎤⎦ / 2, (5a) ε ry ( j , l ) = ⎡⎣ε r ( j , l ) + ε r ( j − 1, l ) ⎤⎦ / 2, (5b) (5c)
+ ε r ( j − 1, l − 1) + ε r ( j , l − 1)] / 4 In writing Eqs. (5a)~(5c), the refractive indices have been approximated by averaging the refractive indices of adjacent cells. A perfect matched layer with the following impedance matching condition [Chin. 04] is applied to bound the computational window.
σe σ = m 2 εo n µo
(6)
where σe and σm are the electric and magnetic conductivities of the PML, respectively, and n is the refractive index of the adjacent computing domain. Which means that the wave impedance of a PML medium exactly equals to that of the adjacent medium in the computing window regardless of the angle of propagation.
×10-3
a d
Equations (3) and (4) can be written in matrix form and after some algebra, the problem is reduced to an eign value equation in terms of the transverse electric fields [Zha. 02]:
⎡E x ⎤ ⎡E x ⎤ P ⎢ ⎥ = β2 ⎢ ⎥ ⎣E y ⎦ ⎣E y ⎦
n
(7)
(a)
-2-
y(µm)
ε rz ( j , l ) = [ε r ( j , l ) + ε r ( j − 1, l )
x(µm) λ=1.55 µm
(b)
Fig.2. PCF of two rings of 18 air holes, (a) Details: d/a=0.6, n=1.45, (b) The dominant Hy field distribution at 1.8µm [SAL 05].
SETIT2009
×10-2
neff=2.512113
×10-2
y(µm)
y(µm)
neff=2.630982
λ=0.5 µm
λ=1.55 µm
x(µm)
x(µm)
(a) (b) Fig.5. Longitudinal components of the poynting vector x of the fundamental HE11 mode at different wavelengths for a honeycomb Lattice PCF of one cladding layer of 12 air holes. (a)
Figure 6 shows the modal characteristics of the honeycomb-lattice PCF with d/a=0.3, d/a=0.375 and d/a=0.45; respectively, for the wavelengths between 1µm to 2µm. The figure demonstrates that, the minimum dispersion and maximum confinement losses are obtained with the PCF characterized by minimum d/a ratio. Moreover, Fig. 7 indicates how the modal characteristics of the honeycomb-lattice PCF shown in Fig.4 change as a function of the substrate refractive index. Notice that, the increase in the value of the substrate refractive index causes a significant decrease in the dispersion, and increase in the confinement loss curves.
(b) Fig.3. Variation of (a) Effective index and (b) Confinement loss of a two-ring PCF with the hole pitch, a, with d/a =0.6. Next, the structure to be studied is a PCF having a honeycomb lattice of 12 air holes surrounding a silica defect as shown in Fig.4. Because of the symmetrical nature of the system only one-right half of the fiber cross section is analyzed. Thus, the computational window size (x×y) is set to 1.95µm×2.04µm. Figures 5.a and 5.b show the longitudinal components of the poynting vectors of the fundamentalx HE11 mode at the wavelengths 0.5µm and 1.55µm, respectively. The figures indicate that, at shorter wavelengths, the electric field is well confined into the core region; while at longer wavelengths, the electric field penetrates into the air holes and extends to the cladding region. 3.9µm d
(a)
a
2.04µm
n
Fig.4. Schematic diagram of a honeycomb lattice PCF of one ring of 12 air holes: n=2.65, d=0.1125µm, - 3 a=0.75µm.
(b) Fig. 6 Variation of the model characteristics of the considered PCF with the hole pitch a, with d/a as parameter (a)Waveguide dispersion, (c)Confinemnet loss.
SETIT2009 The confinement loss and the waveguide dispersion x of the fundamental mode HE11 for the considered fibers are shown in figures (9.a) and (9.b), respectively. It is seen that, the loss characteristics are significantly improved with increasing the number of cladding layers. Another interesting observation from Fig.9.b is that deviations of the dispersion curves with respect to the total number of layers, when the operating wavelength is shorter around 1µm, are smaller than these of the dispersion values, when the operating wavelength is longer around 2µm. The reason is that when the operating wavelength is shorter, electromagnetic fields are more confined to the core region and only the core region has major impact on the propagation properties. On the other hand, when the operating wavelength is longer, fields spread more to the cladding region and the index profile of the cladding region has more influence on the normalized propagation constant.
(a)
(b)
Fig.7. Variation of the model characteristics of considered PCF with the substrate refractive index n: (a)Waveguide dispersion, (c)Confinemnet loss. Next, HFs having different number of cladding layers of 24, 36, and 48 air holes surrounding a silica defect shown in Fig. 8 will be analyzed. (a)
Two Cladding Layers
Three Cladding Layers
(b) Fig. 9. Variation of the model characteristics of considered PCF with the substrate refractive index n: (a)Waveguide dispersion, (c)Confinemnet loss.
d
a
n
The main concept of this paper is optimizing the values of the modal characteristics through of modifying the geometrical parameters. Material dispersion has been taken into account in these results through the Sellmeier equation expressed as in
Four Cladding Layers "Original PCF"
Fig. 8. Schematic diagrams of honeycomb lattice PCF with different cladding layers. -4-
SETIT2009 The optimized PCF is HF with a four cladding layers of a=0.75µm, d=0.3375µm and is characterized by, the diameter of the holes on the inner cladding layer has a value of 0.1231875µm. The optimized results shown in Fig.11 indicate that zero flattened dispersion at 1.55µm is obtained as well as the accompanying loss improvement reaches about 98% compared with the original PCF shown in Fig.8.
reference [R. 92]: 0 . 6961663 λ + 2 − 0 . 0684043 0 . 4079426 λ2 . + 2 2 λ − 0 . 1162414 0 . 8974794 λ2 2 λ 2 − 9 . 896161
1 + n (λ ) =
2
λ
2
(11)
The chromatic dispersion is calculated by adding the material dispersion and the waveguide (geometrical) dispersion as expressed in reference [Ferr. 01]: D(λ)≈Dm(λ)+Dwg(λ)
(a)
(12)
Where Dwg(λ)=GVD│
[ n(λ)=const.=1.45]
The proper choice of the radius of the holes on the inner cladding layer would be possible to optimize the modal characteristics of honeycomb lattice PCF with four cladding layers. To this end we run a series of simulations to determine the optimum diameter of the holes and the whole simulation period took around 2 hours. Figure 10 shows the dispersion as a function of holes diameter, and demonstrates that the diameter of the holes would have to be modulated from the original 0.3375µm to 0.1231875µm to efficiently obtain zero chromatic dispersion at the center of telecommunication c band 1.55µm.
(a)
d
d'
(a)
zero chromatic dispersion at d'=0.365d
(b) Fig.11. Comparison between the original PCF and optimized PCF: (a)Chromatic Dispersion, (b) Confinemnt Loss,.
3. Conclusion
××d (b) Fig.10. Simulation for obtaining zero chromatic dispersion at 1.55µm: (a) Geometry, d=0.3375µm, a=0.75µm, n=1.45, (b) Dispersion vs. holes diameters of the inner cladding layer. -5-
Eignvalue equations for solving full-vector modes of optical waveguides have been formulated using Yee-mesh based finite difference algorithm incorporated with perfectly matched layer absorbing boundary conditions. A comparison with published data has been performed, and excellent agreement is obtained. From the numerical results, it has been shown that, the confinement loss increases rapidly with increasing wavelength. It has been confirmed that increasing the number of cladding layers of PCF
SETIT2009 Department of Electromagnetic Institute of Technology, Sweden.
will decrease confinement loss more than the geometrical dispersion. The confinement loss is drastically reduced by increasing hole radius with respect to lattice constant. Moreover, ultra zero flattened dispersion is obtained by optimizing holes on the inner cladding layer surrounding the silica defect. Further, the accompanying loss improvement reaches about 98%.
Theory,
Lund
[STE 01] Steven. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190, 2001. [SAL 05] Salah S. A. Obayya, B.M.A. Rahman and K.T.V. Grattan “Accurate finite element modal solution of photonic crystal fibers,” IEE Proc.-Optoelectron., Vol. 152, No.5, Oct. 2005.
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[T. 03] T. Fujisawa and M. Koshiba, “Finite element characterization of waveguide dispersion in nonlinear holey fibers,” Opt. Express, Vol. 11, No. 13, 30 June 2003.
[Chin. 04] Chin-Ping Yu and Hung-Chun Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express, Vol. 12, No. 25, 13 December 2004.
[ZHA. 02] Zhaoming Zhu and Thomas G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 853, Vol.10, No.17, 2002.
[F. 00] F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite element method,” Opt. Fiber Technol. 6, 181-191, 2000. [Ferr. 01]Ferrando, E. Silvestre, and P. Andres, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express, vol. 9, pp. 687, 2001. [F. 04] F. Poli, A. Cucinotta, S. Selleri and A. H. Bouk, “Tailoring of flattened dispersion in highly nonlinear photonic crystal fibers,” IEEE Photon. Technol. Lett., to be published (2004). [G. 01] G. E. Town and J. T. Lizer, “Tapered holey fibers for spot size and numerical-aperture conversion,” Opt. Lett. 26, 1042-1044, 2001. [J. 03] J. I. Kim, “Analysis and Applications of Microstructure and Holey Optical Fibers”, PhD Thesis, Faculty of the Virginia, polytechnic Institute and State Univ. , Sept. 2003. [KEE. 66] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302307, 1966. [K. 93] K. S. Kunz and R. J. Luebbers, the Finite Difference Time Domain Method for Electromagnetics, (CRC, Boca Raton, 1993.. [KUN. 02] Kunimasa Saitoh and Masanori Koshiba, “Full- Vectorial Imaginary-Distance Beam Propagation Method Based on a Finite Element Scheme: Application to Photonic Crystal Fibers,” IEEE Journal of Quantum Electronics, Vol.38, No.7, July 2002. [KRI 05] Kristen Lantz Reichenbach and Chris Xu, “The effects of randomly occurring nonuniformities on propagation in photonic crystal fibers,” Opt. Express, Vol. 13, No. 8, 18 April 2005. [M. 01] M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference timedomain method,” Microwave Opt. Technol. Lett. 30, 327330 (2001). [R. 92] R. Lundin, “Minimization of the chromatic dispersion over a broad wavelength range in a single-mode optical fiber”, Report, Code:LUTEDX/(TEAT-7018)/1-15/(1992) Sweden
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