## Analysis of microstructured optical fibers using compact macromodels |

Optics Express, Vol. 19, Issue 20, pp. 19354-19364 (2011)

http://dx.doi.org/10.1364/OE.19.019354

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### 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.

© 2011 OSA

## 1. Introduction

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]

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]

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]

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]

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]

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.

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. Formulation

*z*direction, so we can assume that

**E**(

*ρ*,

*ϕ*,

*z*) =

**E**(

*ρ*,

*ϕ*)

*e*

^{−γz}and

**H**(

*ρ*,

*ϕ*,

*z*) =

**H**(

*ρ*,

*ϕ*)

*e*

^{−γz}, where

*γ*is the propagation coefficient. In the analysis a standard Yee’s mesh in the cylindrical coordinate system is applied (Fig. 2). We assume that the computational domain is discretised into

*M*and

*N*points in the radial and angular direction, respectively. Consequently, Yee’s mesh measures

*K*=

*M*×

*N*points.

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.

*ρ*, Δ

*ϕ*are the space discretization in the

*ρ*and

*ϕ*direction, respectively.

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.

*γ*one gets a system of state-space equations which have the form that makes the application of the MOR technique possible.

**e**

*,*

_{M}**h**

*) from the ones located outside (*

_{M}**e**

*,*

_{U}**h**

*). In such a case equations (8) can be rewritten in the following form where*

_{U}**S**

^{(}

^{e}^{)},

**S**

^{(}

^{h}^{)}are coupling matrices [9

**52**, 2330– 2335 (2004). [CrossRef]

**h**

*represents the field samples placed at the boundary of the model and matrix*

_{B}**L**chooses them from whole set of

**h**

*(*

_{M}**h**

*=*

_{B}**L**

^{T}**h**

*). It is important to note, that matrix*

_{M}**S**

^{(e)}chooses the field samples placed at the boundary of the outside region. In such a case, relation (12) represents electromagnetic properties of the selected region in a compact form. Note that matrix

**S**

^{(}

^{e}^{)}may be regarded as an equivalent of generalized admittance matrix of a region, except that

*γ*rather than

*ω*is a sweep parameter.

**C**,

**G**and

**T**are symmetric semi-positive matrices describing circuit reactive or resistive components,

**X**is a vector of nodal voltages and

**J**is a vector of current sources at ports whose location is described by the incidence matrix

**B**and

*s*is the Laplace variable

*s*=

*jω*(a complex angular frequency). It has been show that ENOR algorithm produces an orthogonal projection basis

**V**, which spans the solution subspace for the assumed frequency band (

*s*). The projection where

**X̂**(

*s*) =

**V**

^{T}**X**(

*s*), significantly reduces a dimension of the problem (dim

**X̂**(

*s*) << dim

**X**(

*s*)). As a result, one looses the information about the internal state variables, however the relation between output and input ports is preserved. The dimension of

**X̂**(

*s*) depends on the frequency bandwidth and on the assumed accuracy. If

*q*is an order of the reduction and

*p*is a number of ports, then relation (13) is approximated with respect to frequency up to its

*q*moments (with a center point

*s*

_{0}) [6]. As a result of the projection with basis

**V**, dim

**X̂**(

*s*) is equal

*p*×

*q*. Since dim

**X̂**(

*s*) << dim

**X**, the inversion in (12) is performed for a significantly smaller matrix.

*s*with a propagation coefficient

*γ*. Let

**V**be a projection matrix obtained from ENOR algorithm for relation (11), then it can be reduced to the following form where

**ĥ**

*=*

_{M}**V**

^{T}**h**

*.*

_{M}**52**, 2330– 2335 (2004). [CrossRef]

*γ*. To improve the accuracy or widen the bandwidth of

*γ*a higher reduction order

*q*is required.

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]

**Q**

*and*

_{e}**Q**

*are orthonormal matrices, whose columns corresponding to the boundary contain the discretized modal functions (in Yee’s mesh). As a result, Maxwell’s grid equations are projected two times - the first time using orthogonal basis (17) and the second time using basis obtained via ENOR algorithm. Due to DFE, the number of ports is considerably reduced: from the number of field samples at the boundary of the macromodel*

_{h}*p*to the number of expansion functions

*N*.

## 3. Numerical results

*r*= 2.5

*μm*, the pitch length is Λ = 6.75

*μm*and the refractive index of the background medium is

*n*= 1.45. In the analysis the vacuum wavelength

_{bg}*λ*= 1.45

*μm*is assumed. Due to the symmetry of the structure only a quarter of the cross section (circle radius

*R*= 10

*μm*) is analysed. Hence, the boundary consist of a PEC (Perfect Electric Conductor) and/or PMC (Perfect Magnetic Conductor) at the symmetry planes and the radiation condition at the arc boundary [1

*ρ*direction and 160 along

*ϕ*. For homogeneous regions (rings/layers 1 – 80 and 190 – 200) the DFE method with 20 basis functions is used. The macromodel is defined in the inhomogeneous part of the domain (rings 78 – 192). An application of the DFE technique at the boundary of the macromodel reduces the number of ports

*p*from

*p*= 960 (6 rings of 160 samples) to

*p*= 120 (6 rings of 20 amplitudes). The propagation coefficient corresponding to an effective refractive index

*n*

_{0}= 1.44 is assumed as a center of the “

*γ*band”. The results obtained for different orders

*q*are collected in Table 1 and compared to the ones obtained from standard FD scheme and FD scheme with DFE. The combination of the MOR and DFE techniques can significantly reduce the problem size which results in shorter computation time (about 14 times).

*q*and the number of basis functions

*N*on the results is shown in Fig. 5. As expected, the accuracy of the calculations depends on the field distribution - for higher order modes the higher number

*N*is required. However, the reduction order

*q*must be grater than 5 in both cases.

*q*= 6 and

*q*= 7 are compared with the ones from the standard FD method and a very good agreement is achieved (error does not exceed 0.5% for

*q*= 7).

*q*= 6 and

*N*= 20 are collected in Table 2. All the parameters of the simulation (size of the reduced problem, eigenproblem solution time) are the same as for undistorted structure.

*r*

_{1}= 1

*μm*, the outer radius

*r*

_{2}= 2

*μm*and the angular width of 108°. The refractive index of the background medium is

*n*= 1.44402362. In this case, the analysis is carried out for vacuum wavelength

_{bg}*λ*= 1.55

*μm*. Also a shape of the numerical domain is different for this example - only a half of the cross section can be analyzed (circle radius

*R*= 2.25

*μm*) - see Fig. 8b. The numerical domain is divided into 140 cells along

*ρ*direction and 210 along

*ϕ*. Also in this case the DFE method with 20 basis functions is used for the homogeneous regions (rings 1 – 45 and 120 – 140). The macromodel is defined in the inhomogeneous part of the domain (rings 42 – 123). A propagation coefficient corresponding to the effective refractive index

*n*

_{0}= 1.28 is assumed as a center of the “

*γ*band”. The results obtained for different orders

*q*are collected in Table 3. The projections reduced the number of state variables from almost 60000 to 3000 which significantly shortens the time of eigenvalue problem solution (from 8s to 0.5s).

*r*= 2

*μm*and the pitch length is Λ = 5

*μm*. The refractive index of the background medium is

*n*= 1.42, core

_{bg}*n*= 1.45. The vacuum wavelength

_{c}*λ*= 1.5

*μm*is assumed. The discretization along

*ρ*is

*M*= 200 and along

*ϕ*is

*N*= 160. Similarly to the first fiber, the analysis is limited to a quarter of a circle of radius

*R*= 8

*μm*- see Fig. 9b. The DFE method with 20 basis functions is used for homogeneous regions (rings 1 – 80 and 190 – 200) and the macromodel is defined in inhomogeneous part of the domain (rings 78 – 192). A propagation coefficient corresponding to an effective refractive index

*n*

_{0}= 1.41 is assumed as a center of the “

*γ*band”. The results obtained for different orders

*q*are collected in table 4. Also for this PCF the results well agree with the ones obtained from the standard FD method. The problem was reduced from over 60000 to just 3900 variables and the analysis was 16 times shorter.

*q*, which has to be high enough (

*q*> 5). Small discrepancy in the results can be observed for propagation coefficients which differs from the center of the “

*γ*band”. However, it is a consequence of the assumed approximation and to improve the accuracy a higher reduction order must be used.

## 4. Conclusion

## Acknowledgments

## References and links

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. , |

4. | P. Feldmann and R. W. Freund, “Efficient Linear Circuit Analysis by Pade Approximation via the Lanczos Process,” IEEE Transactions on Computer-Aided Design , |

5. | A. Odabasioglu, M. Celk, and L. T. Pileggi, “PRIMA: Passive Reduced-order Interconnect Macromodeling Algorithm,” |

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. |

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, |

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] |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 14, 2011

Revised Manuscript: June 23, 2011

Manuscript Accepted: June 28, 2011

Published: September 22, 2011

**Citation**

P. Kowalczyk, L. Kulas, and M. Mrozowski, "Analysis of microstructured optical fibers using compact macromodels," Opt. Express **19**, 19354-19364 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19354

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### References

- P. Kowalczyk, M. Wiktor, and M. Mrozowski, “Efficient finite difference analysis of microstructured optical fibers,” Opt. Express 13, 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. Express 15, 12605–12618 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-20-12605 . [CrossRef] [PubMed]
- B. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Trans. Automat. Contr. , AC-26, 1732 (1981).
- 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]
- 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]
- 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]
- 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. Express 10, 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. Express 12, 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]

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