Abstract

A new technique of numerical analysis of microstructured optical fibers is presented. The technique combines a standard 2D finite difference equations with the discrete function expansion. By doing this one gets a matrix eigenvalue problem of a smaller size and a simple formulation of radiation boundary condition. The new algorithm was tested for the microstructures of different types and excellent agreement of the obtained results with other methods was achieved.

© 2005 Optical Society of America

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References

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AEU (1)

T.Weiland, "VerlustbehafteteWellenleiter mit beliebiger. Randkontur und Materialverteilung," AEU 33, 170-174 (1979).

Applied Computational Electromagnetics (1)

M.Wiktor, M. Mrozowski,"Discrete Projection for Finite Difference Methods," 20th Annual Review or Progress in Applied Computational Electromagnetics, Syracuse NY 2004, Conf. proceedings, S04P08.

Electronics Letters (1)

M. Mrozowski, M. Okoniewski, M.A. Stuchly, "A hybrid PEE- FDTD method for efficient field modeling in cyllindrical coordinates," Electronics Letters 32, 194-195 (1996).
[CrossRef]

IEEE Microwave and Guided Wave Letters (1)

M. Mrozowski, "A Hybrid PEE-FDTD Algorithm for Accelerated Time Domain Analysis of Electromagnetic Waves," IEEE Microwave and Guided Wave Letters 4, 323-325 (1994).
[CrossRef]

IEEE Microwave and Wireless Components L (1)

M. Wiktor, M. Mrozowski,"Efficient Analysis of Waveguide Components Using a Hybrid PEE-FDFD Algorithm," IEEE Microwave and Wireless Components Letters 13, 396-398 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

N. Kaneda, B. Houshmand, T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

J.P. Berenger, "Perfectly matched layer for the FDTD solution of wave-structure. Interaction problems," IEEE Trans. Microwave Theory Tech. 44, 110-117 (1996).

H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Microstrip Substrates Terminated by a Perfectly Matched Layer," IEEE Trans. Microwave Theory Tech. 49, 712-715 (2001).
[CrossRef]

J. Comput. Phys. (1)

J.P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

J. Lightwave Technol. (1)

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

Opt. Express (5)

D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, "Leakage properties of photonic crystal fibers," Opt. Express 10, 1314-1319 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314</a>.
[PubMed]

H.P. Uranus, H.J.W.M. Hoekstra, "Modeling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express 12, 2795-2809 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2795">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2795</a>.
[CrossRef] [PubMed]

Z.Zhu, T.G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10, 853-864 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853</a>.
[PubMed]

Shangping Guo, Feng Wu, Sacharia Albin, Hsiang Tai, Robert S. Rogowski,"Loss and dispersion analysis of microstructured fibers by finite-difference method," Opt. Express 12, 3341-3352 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3341">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3341</a>.
[CrossRef] [PubMed]

C. Kerbage, B. J. Eggleton, "Numerical analysis and experimental design of tunable birefringence in microstructured optical fiber," Opt. Express 10, 246-255 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-5-246">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-5-246</a>.
[PubMed]

Opt. Lett. (1)

Science (1)

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

SIAM (1)

C.D. Meyer, "Matrix analysis and applied linear algebra", SIAM, Philadelphia (2000).

SIAM J. Appl. Math. (1)

A. Bayliss, M. Gunzburger, E. Turkel, "Boundary conditions for the numerical solution of elliptic equations in exterior regions," SIAM J. Appl. Math. 42, 430-451 (1982).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

The discrete field components in the n-th cell (n = 1..K).

Fig. 2.
Fig. 2.

The media interface

Fig. 3.
Fig. 3.

Example of the structure used to illustrate the scheme of the DFE method implementation (different colors represents different media).

Fig. 4.
Fig. 4.

Three different types of the microstructures used to testing presented technique.

Fig. 5.
Fig. 5.

Illustration of the computation domain - DFE method for some outer and inner rings is used.

Fig. 6.
Fig. 6.

The relative error of the calculated effective index: a)real part b)imaginary part. Current work - solid line, FD with rectangular mesh and PML [8] - dashed line.

Tables (3)

Tables Icon

Table 1. The effective indices of the structure with 6 circular holes obtained by different methods

Tables Icon

Table 2. The effective indices of the structure with 3 angular-shaped holes obtained by different methods.

Tables Icon

Table 3. The effective indices of the cored structure from Fig. 4(c).

Equations (32)

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[ ρ 1 0 0 0 ρ 0 0 0 ρ 1 ] [ 0 γ ϕ γ 0 ρ ϕ ρ 0 ] [ E ρ ρ E ϕ E z ] = μ 0 [ H ρ ρ H ϕ H z ]
[ ρ 1 0 0 0 ρ 0 0 0 ρ 1 ] [ 0 γ ϕ γ 0 ρ ϕ ρ 0 ] [ H ρ ρ H ϕ H z ] = ε 0 [ ε ρ 0 0 0 ε ϕ 0 0 0 ε z ] [ E ρ ρ E ϕ E z ]
[ T t ( e ) 0 0 T z ( e ) ] [ γ R tt ( e ) R tz ( e ) R zt ( e ) 0 ] [ E t E z ] = μ 0 [ H t H z ]
[ T t ( h ) 0 0 T z ( h ) ] [ γ R tt ( h ) R tz ( h ) R zt ( h ) 0 ] [ H t H z ] = ε 0 [ P t 0 0 P z ] [ E t E z ]
T t ( e ) = diag ( ρ 1 ( h ) 1 , , ρ K ( h ) 1 , ρ 1 ( e ) , , ρ K ( e ) ) , T z ( e ) = diag ( ρ 1 ( e ) 1 , , ρ K ( e ) 1 )
T t ( h ) = diag ( ρ 1 ( e ) 1 , , ρ K ( e ) 1 , ρ 1 ( h ) , , ρ K ( h ) ) , T z ( h ) = diag ( ρ 1 ( h ) 1 , , ρ K ( h ) 1 )
P t = diag ( ε ρ 1 , , ε ρK , ε ϕ 1 , , ε ϕK ) , P z = diag ( ε z 1 , , ε zK )
R tt ( e ) = [ 0 I I 0 ] = R tt ( h ) , R tz ( e ) = [ F R ] = R zt ( h ) T , R zt ( e ) = [ F R ] = R tz ( h ) T
R = Δ ρ 1 [ 1 1 1 1 ] , F = Δ ϕ 1 [ 1 1 1 1 ]
E t = [ E ρ 1 E ρK ρ 1 ( e ) E ϕ 1 ρ K ( e ) E ϕK ] , E z = [ E z 1 E zK ] , H t = [ H ρ 1 H ρK ρ 1 ( h ) H ϕ 1 ρ K ( h ) H ϕK ] , H z = [ H z 1 H zK ]
γ 2 T t ( h ) R tt ( h ) T t ( e ) R tt ( e ) E t + γ T t ( h ) R tt ( h ) T t ( e ) R tz ( e ) E z + T t ( h ) R tz ( h ) T z ( e ) R zt ( e ) E t = k 0 2 P t E t
γ P z E z = D t E t
D t = T z ( e ) [ R T F T ] T t ( e ) 1
A E t = γ 2 E t
A = T t ( h ) R tt ( h ) T t ( e ) R tz ( e ) P z 1 D t P t + T t ( h ) R tz ( h ) T z ( e ) R zt ( e ) k 0 2 P t
ε ρ = Δ ϕ 1 ln ρ 1 ρ 2 [ ρ 1 ρ 2 ρ [ ( ϕ ( ρ ) ϕ 1 ) ε a + ( ϕ 2 ϕ ( ρ ) ) ε b ] ] 1
ε ϕ = Δϕ [ ln ρ 2 ρ 1 ϕ 1 ϕ 2 ε a ln ρ ( ϕ ) ρ 1 + ε b ln ρ 2 ρ ( ϕ ) ] 1
ε z = p ε b + ( 1 p ) ε a
{ f 1 = g 1 sin ( π N ) + g 2 sin ( 2 π N ) + + g q sin ( N ) f 2 = g 1 sin ( π 2 N ) + g 2 sin ( 2 π 2 N ) + + g q sin ( 2 N ) f N 1 = g 1 sin ( π ( N 1 ) N ) + g 2 sin ( 2 π ( N 1 ) N ) + + g q sin ( ( N 1 ) N )
E ˜ t = [ E ρ 1 E ρ ( l 1 ) N E ˜ ρ 1 E ˜ ρq E ρ ( l + 1 ) N E ρK
ρ 1 ( e ) E ϕ 1 ρ ( l 1 ) N ( e ) E ϕ ( l 1 ) N ρ ( l 1 ) N + 1 ( e ) E ˜ ϕ 1 ρ l N ( e ) E ˜ ϕq
ρ l N + 1 ( e ) E ϕ ( l + 1 ) N ρ K ( e ) E ϕK ] T
E t = Q ˜ E ˜ t
Q ˜ = [ Q ˜ ρ Q ˜ ϕ ]
Q ˜ ρ = [ 1 1 0 0 0 sin π 2 N 1 sin 3 π 2 N 1 sin ( 2 q 1 ) π 2 N 1 sin π ( N 2 ) 2 N 1 sin 3 π ( N 2 ) 2 N 1 sin ( 2 q 1 ) π ( N 2 ) 2 N 1 1 1 ]
Q ˜ ϕ = [ 1 1 cos π 2 ( 2 N 1 ) cos 3 π 2 ( 2 N 1 ) cos ( 2 q 1 ) π 2 ( 2 N 1 ) cos π ( 2 N 2 ) 2 ( 2 N 1 ) cos 3 π ( 2 N 2 ) 2 ( 2 N 1 ) cos ( 2 q 1 ) π ( 2 N 2 ) 2 ( 2 N 1 ) 0 0 0 1 1 ]
A ˜ E ˜ t = γ 2 E ˜ t
A ˜ = Q ˜ T A Q ˜
E ρ = m = 1 a m H m ( 2 ) ( κ ρ ) sin ( m ϕ ) e γ z
E ϕ = m = 0 b m H m ( 2 ) ( κ ρ ) cos ( m ϕ ) e γ z
E ˜ ρ m = H m ( 2 ) ( κ ρ n ( e ) ) H m ( 2 ) ( κ ρ n + N ( e ) ) E ˜ ρ m + q
E ˜ ϕ m = ρ n ( 2 ) H m ( 2 ) ( κ ρ n ( h ) ) ρ n + N ( h ) H m ( 2 ) ( κ ρ n + N ( h ) ) E ˜ ϕ m + q

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