Abstract

In this paper a new technique for numerical analysis of microstructured optical fibers is proposed. The technique uses a combination of model order reduction method and discrete function expansion technique. A significant reduction of the problem size is achieved (by about 85%), which results in much faster simulations (up to 16 times) without affecting the accuracy.

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References

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  1. P. Kowalczyk, M. Wiktor, and M. Mrozowski, “Efficient finite difference analysis of microstructured optical fibers,” Opt. Express13, 10349–10359 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10349 .
  2. P. Kowalczyk and M. Mrozowski, “A new conformal radiation boundary condition for high accuracy finite difference analysis of open waveguides,” Opt. Express15, 12605–12618 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-20-12605 .
    [CrossRef] [PubMed]
  3. B. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Trans. Automat. Contr.,  AC-26, 1732 (1981).
  4. P. Feldmann and R. W. Freund, “Efficient Linear Circuit Analysis by Pade Approximation via the Lanczos Process,” IEEE Transactions on Computer-Aided Design,  14, 639–649 (1995).
    [CrossRef]
  5. A. Odabasioglu, M. Celk, and L. T. Pileggi, “PRIMA: Passive Reduced-order Interconnect Macromodeling Algorithm,” 34th DAC, 58–65 (1997).
  6. B. N. Sheehan, “ENOR: Model Order Reduction of RLC Circuits Using Nodal Equations for Efficient Factorization,” in Proc. IEEE 36th Design Automat. Conf., 17–21 (1999).
  7. L. Kulas and M. Mrozowski, “Macromodels in the frequency domain analysis of microwave resonators,”Microwave and Wireless Components Letters, IEEE, 14, 94–96 (2004), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1278378&isnumber=28582
    [CrossRef]
  8. L. Kulas and M. Mrozowski, “Multilevel model order reduction,” Microwave and Wireless Components Letters, IEEE, 14, 165–167 (2004), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1291452&isnumber=28762
    [CrossRef]
  9. L. Kulas and M. Mrozowski, “A fast high-resolution 3-D finite-difference time-domain scheme with macromodels,” IEEE Trans. Microwave Theory Tech. 52, 2330– 2335 (2004).
    [CrossRef]
  10. J. Podwalski, L. Kulas, P. Sypek, and M. Mrozowski, “Analysis of a High-Quality Photonic Crystal Resonator,” Microwaves, Radar & Wireless Communications. MIKON 2006. International Conference on, 793–796 (2006) http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4345301&isnumber=4345079 .
  11. A. C. Cangellaris, M. Celik, S. Pasha, and Z. Li,“Electromagnetic model order reduction for system-level modeling,” Microwave Theory Techniques, IEEE Transactions on, 47, 840–850 (1999), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=769317&isnumber=16668 .
    [CrossRef]
  12. Y. Zhu and A. C. Canellaris, Multigrid Finite Element Methods for Electromagnetic Field Modeling (John Wiley & Sons, Inc., 2006).
    [CrossRef]
  13. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express10, 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 .
    [PubMed]
  14. S. Guo, F. Wu, S. Albin, H. Tai, and R. S. Rogowski, “Loss and dispersion analysis of microstructured fibers by finite-difference method,” Opt. Express12, 3341–3352 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3341 .
  15. L. Kulas, P. Kowalczyk, and M. Mrozowski, “A Novel Modal Technique for Time and Frequency Domain Analysis of Waveguide Components,” Microwave and Wireless Components Letters, IEEE, 21, 7–9 (2011), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5659497&isnumber=5680668 .
    [CrossRef]

2004

L. Kulas and M. Mrozowski, “A fast high-resolution 3-D finite-difference time-domain scheme with macromodels,” IEEE Trans. Microwave Theory Tech. 52, 2330– 2335 (2004).
[CrossRef]

1995

P. Feldmann and R. W. Freund, “Efficient Linear Circuit Analysis by Pade Approximation via the Lanczos Process,” IEEE Transactions on Computer-Aided Design,  14, 639–649 (1995).
[CrossRef]

1981

B. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Trans. Automat. Contr.,  AC-26, 1732 (1981).

Canellaris, A. C.

Y. Zhu and A. C. Canellaris, Multigrid Finite Element Methods for Electromagnetic Field Modeling (John Wiley & Sons, Inc., 2006).
[CrossRef]

Celk, M.

A. Odabasioglu, M. Celk, and L. T. Pileggi, “PRIMA: Passive Reduced-order Interconnect Macromodeling Algorithm,” 34th DAC, 58–65 (1997).

Feldmann, P.

P. Feldmann and R. W. Freund, “Efficient Linear Circuit Analysis by Pade Approximation via the Lanczos Process,” IEEE Transactions on Computer-Aided Design,  14, 639–649 (1995).
[CrossRef]

Freund, R. W.

P. Feldmann and R. W. Freund, “Efficient Linear Circuit Analysis by Pade Approximation via the Lanczos Process,” IEEE Transactions on Computer-Aided Design,  14, 639–649 (1995).
[CrossRef]

Kulas, L.

L. Kulas and M. Mrozowski, “A fast high-resolution 3-D finite-difference time-domain scheme with macromodels,” IEEE Trans. Microwave Theory Tech. 52, 2330– 2335 (2004).
[CrossRef]

Moore, B.

B. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Trans. Automat. Contr.,  AC-26, 1732 (1981).

Mrozowski, M.

L. Kulas and M. Mrozowski, “A fast high-resolution 3-D finite-difference time-domain scheme with macromodels,” IEEE Trans. Microwave Theory Tech. 52, 2330– 2335 (2004).
[CrossRef]

Odabasioglu, A.

A. Odabasioglu, M. Celk, and L. T. Pileggi, “PRIMA: Passive Reduced-order Interconnect Macromodeling Algorithm,” 34th DAC, 58–65 (1997).

Pileggi, L. T.

A. Odabasioglu, M. Celk, and L. T. Pileggi, “PRIMA: Passive Reduced-order Interconnect Macromodeling Algorithm,” 34th DAC, 58–65 (1997).

Sheehan, B. N.

B. N. Sheehan, “ENOR: Model Order Reduction of RLC Circuits Using Nodal Equations for Efficient Factorization,” in Proc. IEEE 36th Design Automat. Conf., 17–21 (1999).

Zhu, Y.

Y. Zhu and A. C. Canellaris, Multigrid Finite Element Methods for Electromagnetic Field Modeling (John Wiley & Sons, Inc., 2006).
[CrossRef]

IEEE Trans. Automat. Contr.

B. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Trans. Automat. Contr.,  AC-26, 1732 (1981).

IEEE Trans. Microwave Theory Tech.

L. Kulas and M. Mrozowski, “A fast high-resolution 3-D finite-difference time-domain scheme with macromodels,” IEEE Trans. Microwave Theory Tech. 52, 2330– 2335 (2004).
[CrossRef]

IEEE Transactions on Computer-Aided Design

P. Feldmann and R. W. Freund, “Efficient Linear Circuit Analysis by Pade Approximation via the Lanczos Process,” IEEE Transactions on Computer-Aided Design,  14, 639–649 (1995).
[CrossRef]

Other

A. Odabasioglu, M. Celk, and L. T. Pileggi, “PRIMA: Passive Reduced-order Interconnect Macromodeling Algorithm,” 34th DAC, 58–65 (1997).

B. N. Sheehan, “ENOR: Model Order Reduction of RLC Circuits Using Nodal Equations for Efficient Factorization,” in Proc. IEEE 36th Design Automat. Conf., 17–21 (1999).

L. Kulas and M. Mrozowski, “Macromodels in the frequency domain analysis of microwave resonators,”Microwave and Wireless Components Letters, IEEE, 14, 94–96 (2004), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1278378&isnumber=28582
[CrossRef]

L. Kulas and M. Mrozowski, “Multilevel model order reduction,” Microwave and Wireless Components Letters, IEEE, 14, 165–167 (2004), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1291452&isnumber=28762
[CrossRef]

J. Podwalski, L. Kulas, P. Sypek, and M. Mrozowski, “Analysis of a High-Quality Photonic Crystal Resonator,” Microwaves, Radar & Wireless Communications. MIKON 2006. International Conference on, 793–796 (2006) http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4345301&isnumber=4345079 .

A. C. Cangellaris, M. Celik, S. Pasha, and Z. Li,“Electromagnetic model order reduction for system-level modeling,” Microwave Theory Techniques, IEEE Transactions on, 47, 840–850 (1999), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=769317&isnumber=16668 .
[CrossRef]

Y. Zhu and A. C. Canellaris, Multigrid Finite Element Methods for Electromagnetic Field Modeling (John Wiley & Sons, Inc., 2006).
[CrossRef]

Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express10, 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 .
[PubMed]

S. Guo, F. Wu, S. Albin, H. Tai, and R. S. Rogowski, “Loss and dispersion analysis of microstructured fibers by finite-difference method,” Opt. Express12, 3341–3352 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3341 .

L. Kulas, P. Kowalczyk, and M. Mrozowski, “A Novel Modal Technique for Time and Frequency Domain Analysis of Waveguide Components,” Microwave and Wireless Components Letters, IEEE, 21, 7–9 (2011), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5659497&isnumber=5680668 .
[CrossRef]

P. Kowalczyk, M. Wiktor, and M. Mrozowski, “Efficient finite difference analysis of microstructured optical fibers,” Opt. Express13, 10349–10359 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10349 .

P. Kowalczyk and M. Mrozowski, “A new conformal radiation boundary condition for high accuracy finite difference analysis of open waveguides,” Opt. Express15, 12605–12618 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-20-12605 .
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

The idea of model order reduction algorithm.

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Boundary between the model (yellow background) and a regular grid (white background) with coupling ports.

Fig. 4
Fig. 4

A holey PCF: a) cross section, b) numerical domain.

Fig. 5
Fig. 5

Error of the effective refractive index as a function of the reduction order and the number of basis function (it is evaluated in reference to the result obtained from standard FD method).

Fig. 6
Fig. 6

GVD of the fundamental mode evaluated from standard and reduced problem.

Fig. 7
Fig. 7

Distortion of the microstructure presented in Fig. 4a.

Fig. 8
Fig. 8

A holey PCF with 3 annular-shaped holes: a) cross section, b) numerical domain.

Fig. 9
Fig. 9

An air-hole assisted PCF: a) cross section, b) numerical domain.

Tables (4)

Tables Icon

Table 1 Effective refractive index neff obtained for PCF from Fig. 4a. Eigenproblem solution time - round brackets, problem size - square brackets.

Tables Icon

Table 2 Effective refractive index neff obtained for distorted PCF shown in Fig. 7.

Tables Icon

Table 3 Effective refractive index neff obtained for PCF from Fig. 8a. Eigenproblem solution time - round brackets, problem size - square brackets.

Tables Icon

Table 4 Effective refractive index neff obtained for PCF from Fig. 9a. Eigenproblem solution time - round brackets, problem size - square brackets.

Equations (18)

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[ T t ( e ) 0 0 T z ( e ) ] [ γ R t t ( e ) R t z ( e ) R z t ( e ) 0 ] [ E t E z ] = j ω μ 0 [ H t H z ] , [ T t ( h ) 0 0 T z ( h ) ] [ γ R t t ( h ) R t z ( h ) R z t ( h ) 0 ] [ H t H z ] = j ω ε 0 [ P t 0 0 P z ] [ E t E z ] .
T t ( e ) = d i a g ( ρ 1 ( h ) 1 , , ρ K ( h ) 1 , ρ 1 ( e ) , , ρ K ( e ) ) , T z ( e ) = d i a g ( ρ 1 ( e ) 1 , , ρ K ( e ) 1 )
T t ( h ) = d i a g ( ρ 1 ( e ) 1 , , ρ K ( e ) 1 , ρ 1 ( h ) , , ρ K ( h ) ) , T z ( h ) = d i a g ( ρ 1 ( h ) 1 , , ρ K ( h ) 1 )
P t = d i a g ( ε ρ 1 , , ε ρ K , ε ϕ 1 , , ε ϕ K ) , P z = d i a g ( ε z 1 , , ε z K )
R t t ( e ) = [ 0 I K × K I K × K 0 ] = R t t ( h ) , R t z ( e ) = [ R ϕ R ρ ] = R z t ( h ) T , R z t ( e ) = [ R ϕ R ρ ] = R t z ( h ) T
R ρ = Δ ρ 1 { 1 , m = n , 1 , m = n 1 , 0 , otherwise , R ϕ = Δ ϕ 1 { 1 , m = n , 1 , m = n N , 0 , otherwise ,
E t = [ E ρ 1 E ρ K ρ 1 ( e ) E ϕ 1 ρ K ( e ) E ϕ K ] , E z = [ E z 1 E z K ] , H t = [ H ρ 1 H ρ K ρ 1 ( h ) H ϕ 1 ρ K ( h ) H ϕ K ] , H z = [ H z 1 H z K ]
F ( h ) h t = γ e t , F ( e ) e t = γ h t ,
F ( h ) = R t t ( e ) 1 T t ( e ) 1 ( j ω μ 0 I + 1 j ω ε 0 T t ( e ) R t z ( e ) P z 1 T z ( h ) R z t ( h ) ) , F ( e ) = R t t ( h ) 1 T t ( h ) 1 ( j ω ε 0 P t + 1 j ω μ 0 T t ( h ) R t z ( h ) T z ( e ) R z t ( e ) ) .
[ F U ( h ) S ( h ) 0 F M ( h ) ] [ h U h M ] = γ [ e U e M ] , [ F U ( e ) 0 S ( e ) F M ( e ) ] [ e U e M ] = γ [ h U h M ] ,
( γ I + 1 γ F M ( e ) F M ( h ) ) h M = S ( e ) e U .
h B = L T ( γ I + 1 γ F M ( e ) F M ( h ) ) 1 S ( e ) e U ,
( s C + G + 1 s T ) X ( s ) = BJ ( s ) ,
( s V T CV + V T GV + 1 s V T TV ) X ^ ( s ) = V T BJ ( s ) ,
( γ V T IV + 1 γ V T F M ( e ) F M ( h ) V ) h ^ M = V T S ( e ) e U ,
[ F U ( h ) S ( h ) V 0 F M ( h ) V ] [ h U h ^ M ] = γ [ e U e M ] , [ F U ( e ) 0 V T S ( e ) V T F M ( e ) ] [ e U e M ] = γ [ h U h ^ M ] .
e ˜ U = Q e e U , h ˜ M = Q h h M ,
F ˜ ^ ( e ) F ˜ ^ ( h ) h ˜ ^ t = γ 2 h ˜ ^ t ,

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