Abstract

A meshed index profile method, which is based on the localized function method, is demonstrated for analyzing modal characteristics of photonic crystal fibers with arbitrary air-hole structures. The index profile of PCF, which is expressed as a sum of meshed unit matrix, is substituted to full wave equation. By solving this full wave equation, we obtain the modal characteristics of the PCF such as the mode field distribution, the birefringence and the waveguide dispersion. The accuracy of the proposed meshed index profile method (MIPM) is demonstrated by examining the effective index and the birefringence of the two degenerate fundamental modes in the PCF with a triangular air-hole lattice. The MIPM is not restricted to the PCF structure and will be useful in designing various PCF devices.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  24. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).
  25. I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quant. Electron. 29, 2562-2567 (1993).
    [CrossRef]
  26. T.-L. Wu and C.-H. Chao, "Photonic Crystal Fiber Analysis Through the Vector Boundary-Element Method : Effect of Elliptical Air Hole," IEEE Photon. Tech. Lett. 16, 126-128 (2004).
    [CrossRef]
  27. A. Ferrando, E. Silvestre, P. Andres, and J. J. Miret, "Designing the properties of dispersion-�?attened photonic crystal �?bers," Opt. Express 9, 687-697 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-687.
    [CrossRef] [PubMed]
  28. M. J. Steel, P. T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Bottn, "Symmerty and degeneracy in microstructured optical fibers," Opt. Lett. 26, 488-490 (2001).
    [CrossRef]

2008 (1)

2007 (1)

2006 (2)

2005 (1)

2004 (2)

T.-L. Wu and C.-H. Chao, "Photonic Crystal Fiber Analysis Through the Vector Boundary-Element Method : Effect of Elliptical Air Hole," IEEE Photon. Tech. Lett. 16, 126-128 (2004).
[CrossRef]

W. Zhi, R, Guobin, and L. Shuqin, "A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers," J. Lightwave Technol. 22, 903-916 (2004).
[CrossRef]

2003 (3)

2002 (3)

2001 (4)

2000 (2)

1999 (3)

1998 (1)

1997 (1)

1993 (1)

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quant. Electron. 29, 2562-2567 (1993).
[CrossRef]

1990 (1)

Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in structures: Bloch wave solution of Maxwell's equation," Phys. Rev. Lett. 65, 2650-2653 (1990).
[CrossRef] [PubMed]

Albin, S.

Andres, P.

Bennett, P. J.

Birks, T. A.

Bottn, L. C.

Brechet, F.

F. Brechet et al, "Complete analysis of the propagation characteristics into photonic crystal fibers, be the finite element method," J. Optical Fiber Technol. 6, 181-201 (2001).
[CrossRef]

Broderick, N. G. R.

Brown, T. G.

Chai, L.

Chao, C.-H.

T.-L. Wu and C.-H. Chao, "Photonic Crystal Fiber Analysis Through the Vector Boundary-Element Method : Effect of Elliptical Air Hole," IEEE Photon. Tech. Lett. 16, 126-128 (2004).
[CrossRef]

Cucinotta, A.

Eggleton, B. J.

Elias, L. R.

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quant. Electron. 29, 2562-2567 (1993).
[CrossRef]

Ferrando, A.

Guo, S.

Guobin, R.

Hu, M.

Kimel, I.

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quant. Electron. 29, 2562-2567 (1993).
[CrossRef]

Knight, J. C.

Koshiba, M.

K. Saitoh and M. Koshiba, "Full-Vectorial Imaginary-Distance Beam Propagation Method Based on a Finite Element Scheme: Application to Photonic Crystal Fibers," J. Quantum Electron. 38, 927-933 (2002).
[CrossRef]

Kuhlmey, B. T.

Lee, K. S.

Li, H.

Li, L.

Li, Y.

Mafi, A.

McPhedran, R. C.

Midrio, M.

Miret, J. J.

Mogilevtsev, D.

Moloney, J. V.

Monro, T. M.

Nguyen, H. C.

Park, K. N.

Peyghambarian, N.

Richardson, D. J.

Russel, P. St. J.

Russell, P. St. J.

Saitoh, K.

K. Saitoh and M. Koshiba, "Full-Vectorial Imaginary-Distance Beam Propagation Method Based on a Finite Element Scheme: Application to Photonic Crystal Fibers," J. Quantum Electron. 38, 927-933 (2002).
[CrossRef]

Satpathy, S.

Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in structures: Bloch wave solution of Maxwell's equation," Phys. Rev. Lett. 65, 2650-2653 (1990).
[CrossRef] [PubMed]

Schülzgen, A.

Selleri, S.

Shuisheng, J.

Shuqin, L.

Silvestre, E.

Singh, M. P.

S. K. Varshney, M. P. Singh, and R. K. Sinha, "Propagation Characteristics of Photonic Crystal Fibers," J. Opt. Commun. 24, 192-198 (2003).

M. Midrio, M. P. Singh, and C. G. Someda, "The Space Filling Mode of Holey Fibers : An Analytical Vectorial Soultion," J. Lightwave Technol. 18, 1031-1037 (2000).
[CrossRef]

Sinha, R. K.

S. K. Varshney, M. P. Singh, and R. K. Sinha, "Propagation Characteristics of Photonic Crystal Fibers," J. Opt. Commun. 24, 192-198 (2003).

Someda, C. G.

Steel, M. J.

Sterke, C. M.

Temyanko, V. L.

Varshney, S. K.

S. K. Varshney, M. P. Singh, and R. K. Sinha, "Propagation Characteristics of Photonic Crystal Fibers," J. Opt. Commun. 24, 192-198 (2003).

Vincetti, L.

Wang, C.

White, P. T. P.

White, T. P.

T. P. White et al, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 11, 721-732 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-721.
[CrossRef]

Wu, T.-L.

T.-L. Wu and C.-H. Chao, "Photonic Crystal Fiber Analysis Through the Vector Boundary-Element Method : Effect of Elliptical Air Hole," IEEE Photon. Tech. Lett. 16, 126-128 (2004).
[CrossRef]

Xing, Q.

Yao, Y.

Zhang, N.

Zhang, Z.

Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in structures: Bloch wave solution of Maxwell's equation," Phys. Rev. Lett. 65, 2650-2653 (1990).
[CrossRef] [PubMed]

Zhi, W.

Zhu, Z.

Zoboli, M.

Appl. Opt. (2)

IEEE J. Quant. Electron. (1)

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quant. Electron. 29, 2562-2567 (1993).
[CrossRef]

IEEE Photon. Tech. Lett. (1)

T.-L. Wu and C.-H. Chao, "Photonic Crystal Fiber Analysis Through the Vector Boundary-Element Method : Effect of Elliptical Air Hole," IEEE Photon. Tech. Lett. 16, 126-128 (2004).
[CrossRef]

J. Lightwave Technol. (8)

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Holey Optical Fibers: An Efficient Modal Model," J. Lightwave Technol. 17, 1093-1102 (1999).
[CrossRef]

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Modeling large air fraction holey optical fibers," J. Lightwave Technol. 18, 50-56 (2000).
[CrossRef]

A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, "Perturbation Analysis of Dispersion Properties in Photonic Crystal Fibers Through the Finite Element Method," J. Lightwave Technol. 20, 1433-1442 (2002).
[CrossRef]

H. Li, A. Mafi, A. Schülzgen, L. Li, V. L. Temyanko, N. Peyghambarian, and J. V. Moloney, "Analysis and Design of Photonic Crystal Fibers Based on an Improved Effective-Index Method," J. Lightwave Technol. 25, 1224-1230 (2007).
[CrossRef]

M. Midrio, M. P. Singh, and C. G. Someda, "The Space Filling Mode of Holey Fibers : An Analytical Vectorial Soultion," J. Lightwave Technol. 18, 1031-1037 (2000).
[CrossRef]

W. Zhi, R, Guobin, and L. Shuqin, "A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers," J. Lightwave Technol. 22, 903-916 (2004).
[CrossRef]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, "Localized Function Method for Modeling Defect Modes in 2-D Photonic Crystals," J. Lightwave Technol. 17, 2078-2081 (1999).
[CrossRef]

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Holey optical fibers : An efficient modal model," J. Lightwave Technol. 17, 1093-1101 (1999).
[CrossRef]

J. Opt. Commun. (1)

S. K. Varshney, M. P. Singh, and R. K. Sinha, "Propagation Characteristics of Photonic Crystal Fibers," J. Opt. Commun. 24, 192-198 (2003).

J. Opt. Soc. Am. B (1)

J. Optical Fiber Technol. (1)

F. Brechet et al, "Complete analysis of the propagation characteristics into photonic crystal fibers, be the finite element method," J. Optical Fiber Technol. 6, 181-201 (2001).
[CrossRef]

J. Quantum Electron. (1)

K. Saitoh and M. Koshiba, "Full-Vectorial Imaginary-Distance Beam Propagation Method Based on a Finite Element Scheme: Application to Photonic Crystal Fibers," J. Quantum Electron. 38, 927-933 (2002).
[CrossRef]

Opt. Express (5)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in structures: Bloch wave solution of Maxwell's equation," Phys. Rev. Lett. 65, 2650-2653 (1990).
[CrossRef] [PubMed]

Other (2)

M. A. R. Franco, H. T. Hattori, F. Sircilli, A. Passaro, and N. M. Abe, "Finite Element Analysis of Photonic Crystal Fiber," in PROC IEEE MTT-S IMOC, 5-7 (2001).

I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).

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Figures (10)

Fig. 1.
Fig. 1.

(a) Meshed transverse dielectric structure of a PCF and (b) the detailed definition of index profile function.

Fig. 2
Fig. 2

x-polarized fundamental mode field intensities (|Ex |2) of PCFs with different air-hole periods Λ and d/Λ=0.2.

Fig. 3
Fig. 3

x-polarized contour plots of the PCFs with the same parameters shown in Fig. 2. The contour plots are spaced by 2dB(a,b) and 1dB(c,d) from -30dB.

Fig. 4
Fig. 4

x-polarized mode intensity distribution and contour plot of the PCF with an elliptical air-hole lattice. The contour plots are spaced by 2dB from -40dB.

Fig. 5.
Fig. 5.

Transverse structure of the PM-PCF with Λ=4.4µm, D1=4.5µm and D2=2.2µm.

Fig. 6.
Fig. 6.

x-polarized mode intensity distribution and contour plot of the PM-PCF shown in Fig. 5. The contour plots are spaced by 2dB from -30dB

Fig. 7.
Fig. 7.

Birefringence of a PM-PCF as a function of wavelength.

Fig. 8.
Fig. 8.

Waveguide dispersions of PCFs with different parameters.

Fig. 9.
Fig. 9.

Rms error of the effective index as a function of wavelength in the range between 0.5µm and 1.8µm.

Fig. 10.
Fig. 10.

Birefringence of a PCF in the wavelength range between 0.5µm and 1.8µm

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

E t = ( x , y ) = E x ( x , y ) x ̂ + E y ( x , y ) y ̂ ,
E x ( x , y ) = m = 0 S n = 0 S e mn x ψ m ( x ) ψ n ( y ) ,
E y ( x , y ) = m = 0 S n = 0 S e mn y ψ m ( x ) ψ n ( y ) ,
ψ i ( s ) = 2 i 2 π 1 4 i ! w exp ( s 2 2 w 2 ) H i ( s w ) ,
i = m , n s = x , y
[ 2 k 2 β x 2 k 2 + n 2 ( x , y ) ] E x = 1 k 2 x [ E x ln n 2 ( x , y ) x + E y ln n 2 ( x , y ) y ] ,
[ 2 k 2 β y 2 k 2 + n 2 ( x , y ) ] E y = 1 k 2 y [ E x ln n 2 ( x , y ) x + E y ln n 2 ( x , y ) y ] ,
n 2 ( x , y ) = n 2 ( x , y N ) + n 2 ( x , y ( N 1 ) ) + + n 0 2 ( x , y 0 ) + + n 2 ( x , y N 1 ) + n 2 ( x , y N )
= n N 2 ( x ) + n ( N 1 ) 2 ( x ) + + n 0 2 ( x ) + + n N 1 2 ( x ) + n N 2 ( x ) = i = N N n i 2 ( x ) , or
n 2 ( x , y ) = n 2 ( x N , y ) + n 2 ( x ( N 1 ) , y ) + + n 0 2 ( x 0 , y ) + + n 2 ( x N 1 , y ) + n 2 ( x N , y )
= n N 2 ( y ) + n ( N 1 ) 2 ( y ) + + n 0 2 ( y ) + + n N 1 2 ( y ) + n N 2 ( y ) = i = N N n i 2 ( y ) .
M abcd x e x = ( I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) x ) e x = β x 2 e x ,
M abcd y e y = ( I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) y ) e y = β y 2 e y ,
I abcd ( 1 ) = ψ a ( x ) ψ b ( y ) 2 [ ψ c ( x ) ψ d ( y ) ] d x d y ,
I abcd ( 2 ) = n 2 ( x , y ) ψ a ( x ) ψ b ( y ) ψ c ( x ) ψ d ( y ) d x d y ,
I abcd ( 3 ) x = ψ a ( x ) ψ b ( y ) x [ ψ c ( x ) ψ d ( y ) ln n 2 ( x , y ) x ] d x d y ,
I abcd ( 3 ) y = ψ a ( x ) ψ b ( y ) y [ ψ c ( x ) ψ d ( y ) ln n 2 ( x , y ) y ] d x d y ,
I abcd ( 1 ) = 2 a + 2 b + 1 w 2 δ ac δ bd + 2 b ( 2 b + 1 ) 2 w 2 δ ac δ b 1 , d + 2 a ( 2 a + 1 ) 2 w 2 δ a 1 , c δ bd
+ 2 ( b + 1 ) ( 2 b + 1 ) 2 w 2 δ ac δ b , d 1 + 2 ( a + 1 ) ( 2 a + 1 ) 2 w 2 δ a , c 1 δ bd ,
I abcd ( 2 ) = i = N N Δ y n i 2 ( x ) ψ a ( x ) ψ b ( y ) ψ c ( x ) ψ d ( y ) d x d y ,
= i = N N [ Δ y ψ b ( y i ) ψ d ( y i ) d y n i 2 ( x ) ψ a ( x ) ψ c ( x ) d x ]
= i = N N [ ψ b ( y i ) ψ d ( y i ) Δ y n i 2 ( x ) ψ a ( x ) ψ c ( x ) d x ] ,
I abcd ( 3 ) x = ψ a ( x ) ψ b ( y ) x [ ψ c ( x ) ψ d ( y ) i = N N ln n i 2 ( x ) x ] d x d y
= i = N N [ Δ y ψ b ( y ) ψ d ( y ) d y ln n i 2 ( x ) x ψ a ( x ) x [ ψ c ( x ) ] d x ]
= i = N N [ ψ b ( y i ) ψ d ( y i ) Δ y ln n i 2 ( x ) x ψ a ( x ) x [ ψ c ( x ) ] d x ] ,
I abcd ( 3 ) y = ψ a ( x ) ψ b ( y ) y [ ψ c ( x ) ψ d ( y ) i = N N ln n i 2 ( y ) y ] d x d y
= i = N N [ Δ x ψ a ( x ) ψ c ( x ) dx ln n i 2 ( y ) y ψ c ( y ) y [ ψ d ( y ) ] dy ]
= i = N N [ ψ a ( x i ) ψ c ( x i ) Δ x ln n i 2 ( y ) y ψ c ( y ) y [ ψ d ( y ) ] dy ] .
D w ( λ , M Λ , f ) = 1 M D w ( λ M , Λ , f ) ,

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