Abstract

A supercell lattice method, believed to be novel, deduced from the plane-wave expansion method and the localized basis function method, is presented for analyzing photonic crystal fibers (PCFs). The electric field is decomposed by use of Hermite—Gaussian functions, and the dielectric constant of PCFs missing a central air hole is considered as the sum of two virtual different periodic dielectric structures of perfect photonic crystals (PCs). The structures of both virtual PCs are expanded in cosine functions. From the wave equation and the orthonormality of the Hermite—Gaussian functions, the propagation characteristics of the PCFs, such as the mode field distribution, the effective area, and the dispersion property, are obtained. The accuracy of the novel method is demonstrated as we obtain the same results when the dielectric constant is split into two virtual ideal PCs in different ways.

© 2003 Optical Society of America

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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 J. Selected Topics in Quantum. Elec (1)

J.M. Dudley, and S. Coen, �??Numerical simulations and coherence properties of supercontinuum Generation in Photonic Crystal and Tapered Optical Fibers,�?? IEEE J. Selected Topics in Quantum. Electron. 8, 651-659 (2002).
[CrossRef]

IEEE Photon. Tech. Lett (1)

L. P. Shen, C. L. Xu, and W. P. Huang are preparing a manuscript to be called �??Modal characteristics of index-guiding photonic crystal fibers: a comparison between scalar and vector analysis.�??

IEEE Photon. Tech. Lett. (2)

N.A. Mortensen, J.R. Folken, P.M.W. Skovgaard, and J. Broeng, �??Numerical aperture of single-mode photonic crystal fibers,�?? IEEE Photon. Tech. Lett., 14, 1094-1096 (2002).
[CrossRef]

P. Shen, W. P. Huang, and S. S. Jian, �??Design of photonic crystal fibers for dispersion-related applications�?? IEEE Photon. Tech. Lett. (to be published).

J. Lightwave Technol (1)

D. Mogilevtsev, T. A. Birks, P. St. Russell, �??Localized function method for modeling defect mode in 2-d photonic crystal,�?? J. Lightwave Technol. 17, 2078-2081 (1999).
[CrossRef]

J. Lightwave Technol. (3)

Opt. Express (4)

Opt. Lett. (3)

Optical Fiber Technol. (1)

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, �??Photonic crystal fibers: a new class of optical waveguides,�?? Optical Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Other (13)

S. E. Barkou, J. Broeng, and A. Bjarklev, �??Dispersion properties of photonic bandgap guiding fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG5, pp. 117-119.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).

T. A. Birk, D. Mogilevtsev, J. C. Knight, and P. St. J. Russell, �??Single material fibers for dispersion compensation,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), FG2, pp. 108-110.

A. Bjarklev, J. Broeng, K. Dridi, and S. E. Barkou, �??Dispersion properties of photonic crystal fibres,�?? in European Conference on Optical Communication, (Madrid, Spain, 1998), pp. 135-136.

R. Guobin, L. Shuqin, W. Zhi, and J. Shuisheng are preparing a manuscript to be called �??Study on dispersion properties of photonic crystal fiber by effective-index model�?? (in Chinese).

A. W. Snyder, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

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

W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and J. Shuisheng, �??the mode characteristics of the photonic crystal fibers,�?? to be published by ACTA OPTICA SINICA (in Chinese).

M. Koshiba, �??Full vector analysis of photonic crystal fibers using the finite element method,�?? IEICE Electron, E85-C, 4, 881-888 (2002). (C) 2003 OSA 5 May 2003 / Vol. 11, No. 9 / OPTICS EXPRESS 980 #2290 - $15.00 US Received March 31, 2003; Revised April 15, 2003

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, �??The analogy between photonic crystal fibres and step index fibres,�?? in Optical Fiber Communication Conference, (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114-116.

R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian are preparing a manuscript to be called �??Study on dispersion properties of photonic crystal fiber by effective-index model.�??

J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, �??Polarization-preserving holey fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2001), MA1.3, pp. 6-7.

T.M. Monro, D.J. Richardson, N.G.R. Broderick, �??Efficient modeling of holey fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), FG3, 111-113.

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Figures (8)

Fig. 1.
Fig. 1.

Schematic of the way in which the transverse dielectric structure is constructed. (a) supercell lattice PCF, (b) PC1, (c) PC2.

Fig. 2.
Fig. 2.

Cross sections along the y=0 axis of the dielectric structure of PC1, PC2, and PCF.

Fig. 3.
Fig. 3.

x-polarized mode intensity (|Ex |2) distribution of the triangular lattice circular-hole PCF, with structural parameters D=2.3 µm; d=0.69 µm; and P1 =50, N=10, P2 =500.

Fig. 4.
Fig. 4.

Contour lines of the x-polarized mode intensity (|Ex |2) of the PCF; the structural parameters are the same as in Fig. 3. The dielectric constant profile is superimposed. The intensity contours are spaced by 2 dB from -30 dB.

Fig. 5.
Fig. 5.

Effective modal area for triangular-lattice circular-hole PCFs with different D and d/D.

Fig. 6.
Fig. 6.

Dispersion profile of x-polarized mode of the triangular-lattice circular-hole PCFs. All the solid lines in different colors are the waveguide dispersions; the dotted curves are the material dispersion.

Fig. 7.
Fig. 7.

Modal birefringence Δn=n x -n y .

Fig. 8.
Fig. 8.

Simulated dielectric structure of an elliptical-hole PCF.

Tables (1)

Tables Icon

Table 1. Structure Parameters of Two Perfect Photonic Crystals (n si and n air are the refractive index of pure silica and air at the operation wavelength)

Equations (40)

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z + ( n si 2 x ) = n air 2 , z + ( ln n si 2 y ) = 0 .
ε ( x , y ) = ε PC 1 + ε PC 2 , ln ε = ( ln ε ) PC 1 + ( ln ε ) PC 2 ,
ε PC 1 = a , b = 0 P 1 P 1 ab cos 2 π a x D cos 2 π b y 3 D ,
ε PC 2 = a , b = 0 P 2 P 2 ab cos 2 π a x N D cos 2 π b y 3 N D ,
( ln ε ) PC 1 = a , b = 0 P 1 P 1 ab ln cos 2 π a x D cos 2 π b y 3 D ,
( ln ε ) PC 2 = a , b = 0 P 2 P 2 ab ln cos 2 π a x N D cos 2 π b y 3 N D ,
ε ( x , y ) PC 1 = m , n = P 1 P 1 F ( K mn ) cos ( k x x ) cos ( k y y ) ,
F ( K mn ) = n air 2 δ ( K mn ) + 2 ( n air 2 n si 2 ) f J 1 ( K mn R ) K mn R ,
F ( 0 ) = n air 2 + f ( n air 2 n si 2 )
K mn = ( m + n ) k x i ( m n ) k y ( i is the imaginary unit ) , k x = 2 π D , k y = 2 π ( 3 D ) .
f = 2 π R 2 3 D 2 , ( R = d 2 , the hole radius ) .
ε ( x , y ) PC 1 = m , n = 0 2 P 1 F ( K mn ) cos 2 π ( m + n 2 P 1 ) x D cos 2 π ( m n ) y 3 D .
P 1 ab = F ( K a + b + 2 k 2 , a b + 2 k 2 ) + F ( K a b + 2 k 2 , a + b + 2 k 2 )
+ F ( K a + b + 2 k 2 , a b + 2 k 2 ) + F ( K a b + 2 k 2 , a + b + 2 k 2 ) ,
P 1 ab = F ( K a + b + 2 k 2 , a b + 2 k 2 ) + F ( K a b + 2 k 2 , a + b + 2 k 2 ) , when a = 0 or b = 0 ,
P 100 = F ( K k , k ) when a = 0 and b = 0 .
e t ( x , y ) = e x ( x , y ) x ̂ + e y ( x , y ) y ̂ .
e x ( x , y ) = a , b = 0 F ε ab x ψ a ( x ) ψ b ( y ) , e y ( x , y ) = a , b = 0 F ε ab y ψ a ( x ) ψ b ( y ) ,
ψ i ( s ) = 2 i π 1 4 ( 2 i ) ! ω exp ( s 2 2 ω 2 ) H 2 i ( s ω ) ,
M s ε s = β s 2 ε s , ( s = x , y corresponding to e x , e y ) ,
I abcd ( 1 ) = + ψ a ( x ) ψ b ( y ) t 2 [ ψ c ( x ) ψ d ( y ) ] d x d y ,
I abcd ( 2 ) = + ε 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 ε x ] d x d y ,
I abcd ( 3 ) y = + ψ a ( x ) ψ b ( y ) y [ ψ c ( x ) ψ d ( y ) ln ε y ] d x d y .
I abcd ( 1 ) = 2 a + 2 b + 1 ϖ 2 δ ac δ bd + 2 b ( 2 b + 1 ) 2 ϖ 2 δ ac δ b 1 , d + 2 a ( 2 a + 1 ) 2 ϖ 2 δ a 1 , c δ bd
+ 2 ( b + 1 ) ( 2 b + 1 ) 2 ϖ 2 δ ac δ b , d 1 + 2 ( a + 1 ) ( 2 a + 1 ) 2 ϖ 2 δ a , c 1 δ bd ,
I abcd ( 2 ) = f , g = 0 P 1 P 1 fg I fac ( 21 ) x I gbd ( 21 ) y + f , g = 0 P 2 P 2 fg I N fac ( 21 ) x I N gbd ( 21 ) y ,
I abcd ( 3 ) x = f , g = 0 P 1 P 1 fg ln I fac ( 32 ) x I gbd ( 21 ) y f , g = 0 P 2 P 2 fg ln I N fac ( 32 ) x I N gbd ( 21 ) y ,
I abcd ( 3 ) y = f , g = 0 P 1 P 1 fg ln I fac ( 21 ) x I gbd ( 32 ) y f , g = 0 P 2 P 2 fg ln I N fac ( 21 ) x I N gbd ( 32 ) y ,
I i 1 i 2 i 3 ( 21 ) s = + cos ( 2 π i 1 s l s ) ψ i 2 ( s ) ψ i 3 ( s ) d s ,
I N i 1 i 2 i 3 ( 21 ) s = + cos ( 2 π i 1 s N l s ) ψ i 2 ( s ) ψ i 3 ( s ) d s ,
I i 1 i 2 i 3 ( 32 ) s = + cos ( 2 π i 1 s l s ) s ψ i 2 ( s ) s ψ i 3 ( s ) d s ,
I N i 1 i 2 i 3 ( 32 ) s = + cos ( 2 π i 1 s N l s ) s ψ i 2 ( s ) s ψ i 3 ( s ) d s , ( s = x , y )
M abcd s M s [ ( a 1 ) F + b , ( c 1 ) F + d ] .
A eff = [ + + E ( x , y ) 2 d x d y ] 2 + + E ( x , y ) 4 d x d y .
A eff = 2 [ π ω a 1 , a 2 = 0 F 1 ε a 1 , a 2 2 ] 2 a 1 , a 2 , a 3 , a 4 = 0 b 1 , b 2 , b 3 , b 4 = 0 F 1 ε a 1 , b 1 ε a 2 , b 2 ε a 3 , b 3 ε a 4 , b 4 ξ · ζ a · ζ b ,
ξ = ( 2 a 1 ) ! ( 2 a 2 ) ! ( 2 a 3 ) ! ( 2 a 4 ) ! ( 2 b 1 ) ! ( 2 b 2 ) ! ( 2 b 3 ) ! ( 2 b 4 ) ! ,
ζ s = t 1 = 0 min ( 2 s 1 , 2 s 2 ) t 2 = 0 min ( 2 s 3 , 2 s 4 ) ( 1 ) κ s Γ ( κ s ) t 1 ! ( 2 s 1 t 1 ) ! ( 2 s 2 t 1 ) ! t 2 ! ( 2 s 3 t 2 ) ! ( 2 s 4 t 2 ) ! ,
κ s = s 1 + s 2 + s 3 + s 4 t 1 t 2 , ( s = a , b ) .
D w ( λ ; M D , f ) = 1 M D w ( λ M ; D , f ) .

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