Abstract

A boundary integral method [1] for calculating leaky and guided modes of microstructured optical fibers is presented. The method is rapidly converging and can handle a large number of inclusions (hundreds) of arbitrary geometries. Both, solid and hollow core photonic crystal fibers can be treated efficiently. We demonstrate that for large systems featuring closely spaced inclusions the computational intensity of the boundary integral method is significantly smaller than that of the multipole method. This is of particular importance in the case of hollow core band gap guiding fibers. We demonstrate versatility of the method by applying it to several challenging problems.

© 2007 Optical Society of America

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References

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  1. Matlab implementation of the code is available at http://www.photonics.phys.polymtl.ca/codes.html
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    [CrossRef]
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  16. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran "Multipole method for microstructured optical fibers. II. Implementation and results," J. Opt. Soc. Am. B 19, 2331-2340 (2002).
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    [CrossRef]
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    [CrossRef] [PubMed]

2006

2005

2004

2003

2002

K. Saitoh,M. Koshiba, "Full-Vectorial Imaginary-Distance Beam PropagationMethod Based on a Finite Element Scheme: Application to Photonic Crystal Fibers," IEEE J. Quantum Electron. 38, 297 (2002).
[CrossRef]

A. Cucinotta, S. Selleri, L. Vincent and M. Zoboli, "Holey fiber analysis through the finite element method," IEEE Photon. Technol. Lett. 14, 1530-1532 (2002).
[CrossRef]

S. V. Boriskina, T.M. Benson. P. Sewell and A. I. Nosich "Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides," IEEE J. Sel. Top. Quantum Electron. 8, 1225-1231 (2002).
[CrossRef]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke and L. C. Botten "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002).
[CrossRef]

B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran "Multipole method for microstructured optical fibers. II. Implementation and results," J. Opt. Soc. Am. B 19, 2331-2340 (2002).
[CrossRef]

2000

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).
[CrossRef]

1999

1997

IEEE J. Quantum Electron.

K. Saitoh,M. Koshiba, "Full-Vectorial Imaginary-Distance Beam PropagationMethod Based on a Finite Element Scheme: Application to Photonic Crystal Fibers," IEEE J. Quantum Electron. 38, 297 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

S. V. Boriskina, T.M. Benson. P. Sewell and A. I. Nosich "Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides," IEEE J. Sel. Top. Quantum Electron. 8, 1225-1231 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Cucinotta, S. Selleri, L. Vincent and M. Zoboli, "Holey fiber analysis through the finite element method," IEEE Photon. Technol. Lett. 14, 1530-1532 (2002).
[CrossRef]

Int. J. RF Microw. Comp. Eng.

R. Rodriguez-Berral, F. Mesa, and F. Medina, "Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides," Int. J. RF Microw. Comp. Eng. 14, 73-83 (2004).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Express

Opt. Fiber Technol.

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).
[CrossRef]

Opt. Lett.

Science

P. Russell "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

Other

A. Bjarklev, J. Broeng and A.S. Bjarklev "Photonic crystal fibers," Kluwer Academic Publishers, Boston, (2003).

Matlab implementation of the code is available at http://www.photonics.phys.polymtl.ca/codes.html

D. Colton and R. Kress "Integral equation methods in scattering theory," John Wiley & Sons, New York, (1983).

R. Kress "Linear integral equations," Springer-Verlag, New York, (1989).

M. Abramowitz and I. A. Stegun "Handbook of mathematical functions," Dover, New York, (1965).

M.C.J. Large, L. Poladian, G.W. Barton, M.A. van Eijkelenborg, "Microstructured Polymer Optical Fibres," Springer, Sydney, (2007)

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Figures (7)

Fig. 1.
Fig. 1.

Three structures studied in the paper. (a) Hollow coreMOF with five rings of circular holes; the pitch is Λ=2.74µm, the hole diameter is d=.95Λ and the core diameter is dc =2.5d. (b) Elliptic hollow core MOF with three layers of circular holes; the pitch is Λ=2µm, the hole diameter is d=0.9Λ and the core principal axis are a=2.3µm and b=4.6µm. (c) Solid core MOF with six silver coated elliptical holes; the outer hole principal axis are ao =0.84µm and bo =0.76mm, the inner hole principal axis are ai =0.74µm and bi =0.66µm, the pitch is Λ=1.5mm.

Fig. 2.
Fig. 2.

a) Schematic of aMOF cross section. b) Schematic of Re(G(s, s′)). Green’s function has a cusp when ss′ it also exhibits oscillations ( γ = n g 2 n e 2 ) . c ) Arbitrary shaped inclusion and a corresponding regularization circle.

Fig. 3.
Fig. 3.

Convergence analysis and comparison with the multipole method for the three simple test structures: (a) six circular holes; diameter d=5µm, pitch L=6.75µm (b) six elliptic holes; axis a=2.5µm b=1.5µm, pitch L=6.75µm (c) six metal coated cylinders; outer diameter do =0.8µm, inner one di =0.7µm, pitch Λ = 1.5µm.

Fig. 4.
Fig. 4.

Hollow core MOF with 5 rings of holes in the reflector. (a) Dispersion curve of the fundamental mode. (b) Loss as a function of the number of reflector layers.

Fig. 5.
Fig. 5.

Birefringence of the fundamental mode of a PCF with elliptic hollow core. (b) Outset: Sz fluxes for the x and y polarizations of the fundamental mode at λ=1.42µm.

Fig. 6.
Fig. 6.

Loss dispersion curves for the two polarizations of the fundamental mode of a MOF with one ring of metallized elliptic holes. Outset: Sz fluxes for the x and y polarizations of the fundamental mode at the wavelengths of the two plasmon excitation peaks.

Fig. 7.
Fig. 7.

Schematic of a coated inclusion.

Tables (4)

Tables Icon

Table 1. Performance comparison of the multipole and boundary integral methods.

Tables Icon

Table 2. Effective refractive index of a mode (of a symmetry class p=1 as defined in [15]) of a solid core MOF featuring one ring of six holes (see Fig. 3(a)).

Tables Icon

Table 3. Effective refractive index of a mode of a solid core MOF featuring one ring of six elliptic inclusions (see Fig. 3(b)). The results are for the fundamental mode where the nodal line of the Ez field is horizontal. For the other polarization the value 1.446429072+ 2.9898E-6i is obtained by us and 1.446427235+2.9601E-6i by [17].

Tables Icon

Table 4. Effective refractive index of a mode of a solid core MOF with one ring of six coated holes (see Fig. 3(c)). Results are for the fundamental core guided mode.

Equations (35)

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2 E z ( c , g ) + k 0 2 γ c , g 2 E z ( c , g ) = 0
2 H z ( c , g ) + k 0 2 γ c , g 2 H z ( c , g ) = 0 ,
E z ( c ) = E z ( g )
H z ( c ) = H z ( g )
E t ( c ) = E t ( g ) i k 0 γ c 2 ( n e E z ( c ) τ H z ( c ) n ) = i k 0 γ g 2 ( n e E z ( g ) τ H z ( g ) n ) ,
H t ( c ) = H t ( g ) i k 0 γ c 2 ( n e H z ( c ) τ + ε c E z ( c ) n ) = i k 0 γ g 2 ( n e H z ( g ) τ + ε g E z ( g ) n )
E z ( r ) = L e ( r s ) G ( r , r s ) d l s
H z ( r ) = L h ( r s ) G ( r , r s ) d l s .
G ( r , r s ) = i 4 H 0 ( 1 ) ( k 0 γ r r s ) ,
1 : 0 2 π e c ( j ) ( s ) G c ( s , s ) J ( j ) ( s ) d s = k = 1 N c 0 2 π e g ( k ) ( s ) G g ( s , s ) J ( k ) ( s ) d s
2 : 0 2 π h c ( j ) ( s ) G c ( s , s ) J ( j ) ( s ) d s = k = 1 N c 0 2 π h g ( k ) ( s ) G g ( s , s ) J ( k ) ( s ) d s
3 : 1 γ c 2 ( n e 0 2 π e c ( j ) ( s ) G c ( s , s ) τ J ( j ) ( s ) d s 0 2 π h c ( j ) ( s ) G c ( s , s ) τ J ( j ) ( s ) ds h c ( j ) ( s ) 2 ) =
1 γ c 2 ( n e k = 1 N c 0 2 π e g ( k ) ( s ) G g ( s , s ) τ J ( k ) ( s ) d s k = 1 N c 0 2 π h g ( k ) ( s ) G g ( s , s ) τ J ( k ) ( s ) d s + h c ( j ) ( s ) 2 ) ,
4 : 1 γ c 2 ( n e 0 2 π h c ( j ) ( s ) G c ( s , s ) τ J ( j ) ( s ) d s + ε c 0 2 π e c ( j ) ( s ) G c ( s , s ) n J ( j ) ( s ) ds + ε c e c ( j ) ( s ) 2 ) =
1 γ g 2 ( n e k = 1 N c 0 2 π h g ( k ) ( s ) G g ( s , s ) τ J ( k ) ( s ) d s + ε g k = 1 N c 0 2 π e g ( k ) ( s ) G c ( s , s ) n J ( k ) ( s ) d s ε g e c ( j ) ( s ) 2 )
ψ ( k ) ( s ) = t = 0 2 n ( k ) 1 ( 1 2 n ( k ) m = n ( k ) n ( k ) 1 e i m ( s s t ) ) ψ ( k ) ( s t ) .
I ( j ) = 0 2 π ψ ( j ) ( s ) Φ ( s , s ) J ( j ) ( s ) d s t = 0 2 n ( j ) 1 ( 1 2 π m = n ( j ) n ( j ) 1 e i m s t 0 2 π e i m s Φ ( s , s ) a j d s ) ψ ( j ) ( s t ) ,
0 2 π e i m s G ( s , s ) d s ' = i π 2 J m ( k 0 γ a j ) H m ( 1 ) ( k 0 γ a j ) e i m s
0 2 π e i m s G ( s , s ' ) n d s ' = [ 1 2 a j + i k 0 γ π 2 J m ( k 0 γ a j ) H m ( 1 ) ( k 0 γ a j ) e i m s ] e i m s ,
0 2 π e i m s G ( s , s ' ) n d s ' = m π 2 a j J m ( k 0 γ a j ) H m ( 1 ) ( k 0 γ a j ) e i m s
I ( k ) = 0 2 π ψ ( k ) ( s ) Φ ( s , s ) J ( k ) ( s ) d s t = 0 2 n ( k ) 1 ( a k 2 π 2 n ( k ) Φ ( s , s t ) ) ψ ( k ) ( s t ) .
A ( n e ) · X = 0 .
Ψ ( k ) ( s ) = t = 0 2 n ( k ) 1 ( 1 2 n ( k ) m = n ( k ) n ( k ) 1 e i m ( s s t ) ) Ψ ( k ) ( s t ) .
I ( j ) = 0 2 π Ψ ( j ) ( s ) Φ ( s , s ) d s t = 0 2 n ( j ) 1 ( 1 2 n ( j ) m = n ( j ) n ( j ) 1 e i m s t 0 2 π e i m s Φ a ( s , s ) d s ) Ψ ( j ) ( s t ) ,
0 2 π e i m s G a ( s , s ) d s = 0 2 π e i m s [ G a ( s , s ' ) G c ( s , s ) ] d s + 0 2 π e i m s G c ( s , s ) d s ,
0 2 π e i m s Φ a ( s , s ) d s = 0 2 π e i m s [ Φ a ( s , s ' ) a j J ( j ) ( s ) Φ c ( s , s ) ] d s + a j J ( j ) ( s ) 0 2 π e i m s Φ c ( s , s ) d s ,
H 0 ( 1 ) ( k 0 γ R ) n = k 0 γ x s ( y s y s ) y s ( x s x s ) J ( s ) R H 1 ( 1 ) ( k 0 γ R ) .
H 0 ( 1 ) ( k 0 γ R ) τ = k 0 γ x s ( x s x s ) + y s ( y s y s ) J ( s ) R H 1 ( 1 ) ( k 0 γ R ) .
H 0 ( 1 ) ( 2 a k 0 γ sin s s 2 ) n = k 0 γ sin s s 2 H 1 ( 1 ) ( 2 a k 0 γ sin s s 2 ) ,
H 0 ( 1 ) ( 2 a k 0 γ sin s s 2 ) τ = k 0 γ sin ( s s ) 2 sin s s 2 H 1 ( 1 ) ( 2 a k 0 γ sin s s 2 ) .
lim s s [ H 0 ( 1 ) ( k 0 γ R ) H 0 ( 1 ) ( 2 a k 0 γ sin s s 2 ) ] = 2 i π ln J ( s ) a
lim s s [ H 0 ( 1 ) ( k 0 γ R ) n a J ( s ) H 0 ( 1 ) ( 2 a k 0 γ sin s s 2 ) n ] = 1 κ ( s ) J ( s ) i π J ( s ) ,
lim s s [ H 0 ( 1 ) ( k 0 γ R ) τ a J ( s ) H 0 ( 1 ) ( 2 a k 0 γ sin s s 2 ) τ ] = 0
L o ( j ) e o ( j ) ( s ) G m ( s , s ) J ( s ) d s + L i ( j ) e i ( j ) ( s ) G m ( s , s ) J ( s ) d s = k = 1 N c L 0 ( k ) e g ( k ) ( s ) G g ( s , s ) J ( s ) d s ,
L i ( j ) e c ( i ) ( s ) G c ( s , s ) J ( s ) d s = L i ( j ) e i ( j ) ( s ) G m ( s , s ) J ( s ) d s + L o ( j ) e o ( j ) ( s ) G m ( s , s ) J ( s ) d s .

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