## A new conformal radiation boundary condition for high accuracy finite difference analysis of open waveguides

Optics Express, Vol. 15, Issue 20, pp. 12605-12618 (2007)

http://dx.doi.org/10.1364/OE.15.012605

Acrobat PDF (296 KB)

### Abstract

A highly accurate radiation boundary condition for finite difference analysis of open waveguides is introduced. The boundary condition is applicable to the structures embedded in a homogeneous medium and fitted to the cross section of the structure. The numerical tests carried out for a few types of waveguides including microstructured fibers showed that the proposed approach improves the accuracy by about an order of magnitude in comparison with the PML technique and eliminates all its disadvantages.

© 2007 Optical Society of America

## 1. Introduction

1. 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]

2. Shangping Guo, Feng Wu, Sacharia Albin, Hsiang Tai, and Robert 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. [CrossRef] [PubMed]

3. 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. [CrossRef] [PubMed]

5. J.P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

*dB*for TBC and about -120

*dB*for PML. However, TBC are still applied in other techniques (beam propagation method [6

6. G.R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron., **28**, 363–370 (1992). [CrossRef]

7. H.P. Uranus and 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2795. [CrossRef] [PubMed]

8. H. Rogier and 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]

9. H. Rogier and D. De Zutter, “Berenger and Leaky Modes in Optical Fibers Terminated with a Perfectly Matched Layer,” J. Lightw. Technol., **20**, 1141 – 1148 (2002). [CrossRef]

3. 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. [CrossRef] [PubMed]

## 2. Formulation

### 2.1. One dimensional propagation problems

*yz*plane, characterized by permittivity

*ε*(

*x*). Assuming that the fields variation along the

*z*direction is represented by a term e

^{-γz}and there is no variation in the

*y*direction, Maxwell’s equations can be separated into two independent systems for

*TE*and

^{x}*TM*modes.

^{x}*TE*modes

^{x}*e*=

_{m}*E*(

_{y}*m*Δ

*x*),

*ε*=

_{m}*ε*(

*m*Δ

*x*) and Δ

*x*denotes the discretization step (see fig. 1).

*R*

_{zy}^{(e)},

*R*

_{yz}^{(H)}and

*ε*are square

*M*×

*M*matrices

*e*= [

*e*

_{1},

*e*

_{2},⋯,

*e*]

_{M}^{T},

*h*= [

*h*

_{1},

*h*

_{2},⋯,

*h*]

_{M}^{T}.

### 2.2. Boundary conditions for ID analysis

*ε*

_{2}=

*ε*

_{2}(

*x*)).

*κ*= [-

_{i}*γ*

^{2}- k

^{2}

_{0}

*ε*]

_{i}^{1/2},

*k*

_{0}=

*ω*(

*μ*

_{0}

*ε*

_{0})

^{1/2}and

*A*

_{-},

*A*

_{+}are unknown coefficients. For evanescent modes

*κ*, is real and grater than zero, however for leaky modes

_{i}*κ*, is complex and the proper sheet of the Riemann surface must be chosen to satisfy the Sommerfeld radiation condition [10].

_{i}*a*

_{-}= e

^{-κ1Δx}and

*a*

_{+}= e

^{-κ3Δx}.

*A*=

*R*

_{yz}^{(h)}

*R*

_{zy}^{(e)}-

*ωμ*

_{0}

*ε*, then

*a*

_{-}and

*a*

_{+}are the functions of

*γ*above problem becomes nonlinear

*γ*

_{0}and denote the solution as

*γ*

_{1}:

*γ*

_{2}, substituting

*γ*

_{1}into operator

*Ã*. The procedure should be repeated until the difference between two sequential values of

*γ*is less than the assumed error.

*F*(

*γ*) =

*γ*-

*, where*γ ^

*is an eigenvalue of the problem*γ ^

*Ã*(

*γ*)

*ẽ*=

γ ^

^{2}

*ẽ*. A zero of this function is simultaneously the solution of eigenproblem (12). Since complex plane of

*γ*can be replaced by two dimensional real space and function

*F*(

*γ*) by log|

*F*(

*γ*)| solving of (12) transforms to a problem of finding the minimum of the function.

### 2.3. Boundary conditions for 2D analysis

1. 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]

2. Shangping Guo, Feng Wu, Sacharia Albin, Hsiang Tai, and Robert 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. [CrossRef] [PubMed]

*z*direction in a form e

^{-γz}the electric and magnetic field can be expressed by only two components

*E*(

_{z}*ρ*,

*φ*,

*z*) and

*H*(

_{z}*ρ*,

*φ*,

*z*). The other components are unambiguously defined by the relations:

*H*

_{m}^{(2)}(∙) is a Hankel function of the second kind,

*κ*= [

*γ*

^{2}+

*k*

^{2}

_{0}

*με*]

^{1/2},

*η*= [

*μ*/

*ε*)

^{1/2}and

*A*,

_{m}*B*,

_{m}*C*,

_{m}*D*are arbitrary coefficients. In practice we have to reduce the series to a finite number of terms, so let us assume that

_{m}*m*= 0,1, ...,

*Q*.

*E*(

_{x}*x*,

*y*,

*z*) and

*E*(

_{y}*x*,

*y*,

*z*) are needed, hence

*x*=

*ρ*cos(

*φ*) and

*y*=

*ρ*sin(

*φ*).

*z*= 0, we get

*S*be a set of points of Yee’s mesh (where tangential electric field is defined) inside the computational domain close to the boundary (white circles in fig. 2) and

^{C}*S*be a set of points on the boundary (crosses in fig. 2). Let us denote by

^{B}*E*and

^{C}_{t}*E*the field values from set

^{B}_{t}*S*and

^{C}*S*, respectively. From relations (23) and (24) we get

_{B}*C*= [

*A*

_{0},... ,

*A*,

_{Q}*B*

_{0},... ,

*B*,

_{Q}*C*

_{0},... ,

*C*,

_{Q}*D*

_{0},...,

*D*]

_{Q}^{T}. The relation between field values on the boundary and inside the computational domain is

*M*

_{C}^{(inv)}=

*M*

_{C}^{-1}if the number of the

*S*elements is equal to the size of

^{C}*C*vector, otherwise to invert

*M*the singular value decomposition (SVD) algorithm must be applied [11] (

_{C}*M*

_{C}^{(inv)}

*M*=

_{C}*I*).

*A*in eigenvalue problem (14). The new problem involves only fields located inside the structure - all points outside (and on) the boundary can be neglected.

*M*and

_{B}*M*

_{C}^{(inv)}are functions of

*γ*, we get a nonlinear eigenvalue problem similar to (12). Again simple iteration is in most cases enough to reach the solution, but if this fails the convergence is restored by applying minimalization procedure described in the previous paragraph.

## 3. Numerical results

*m*= 4,

*n*is a refractive background index and Δ is a step of the discretization [2

2. Shangping Guo, Feng Wu, Sacharia Albin, Hsiang Tai, and Robert 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. [CrossRef] [PubMed]

12. N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Trans. Microwave Theory Tech. **45**, 1645–1649 (1997). [CrossRef]

*ε*

_{r1}=

*ε*

_{r3}= 1.21,

*ε*

_{r2}= 1,

*b*= 1

*nm*,

*λ*

_{0}= 0.2

*nm*are presented in Table 1 (corresponding fields distributions are shown in fig. 3 – 5).

*ε*

_{r1}=

*ε*

_{r3}= 9,

*ε*

_{r2}= 1,

*b*= 1

*nm*,

*λ*

_{0}= 1.5

*nm*are collected in Table 2.

*n*exceeds 10% and is at least two orders of manitude greater than for the new technique.

_{eff}9. H. Rogier and D. De Zutter, “Berenger and Leaky Modes in Optical Fibers Terminated with a Perfectly Matched Layer,” J. Lightw. Technol., **20**, 1141 – 1148 (2002). [CrossRef]

*R*= 0.5

*μm*, the core index

*n*= 2.9, the background index

_{c}*n*= 1. 55 and

_{bg}*λ*

_{0}= 1

*μm*. The discretization assumed in PML simulation was 200 × 200 cells (80,000 of variables) and the domain size was 2

*μm*× 2

*μm*. The PML region was placed 40 cells from the core. This was necessary since a smaller distance to PML results in a significant increase of the simulation error. The alternative analysis was carried out with the same discretization and the proposed boundary condition (

*Q*= 20). The number of variables was only about 20,000, because the boundary condition was imposed very close to the structure

*R*= 0.55

_{B}*μm*(see fig. 6). All numerical results are collected and compared with the theoretical values in Table 3. A significant inaccuracy of the imaginary part of the effective index obtained from PML technique is shown, especially for guided modes.

*r*= 2.5

*μm*and the pitch length is Λ = 6.75

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

*n*= 1.45 and

_{bg}*n*= 1 for holes. The vacuum wavelength used in calculation is

_{a}*λ*= 1.45

*μm*. The radius of the computation domain is

*R*= 9.5

*μm*, but due to the symmetry of the structure only a quarter of the circle has to be analysed. The boundary conditions consist of a perfect electric and/or magnetic conductor at the structure symmetry planes and the radiation condition presented in section 2 at the curved boundary. The discretization assumed in simulation is 150 × 150 cells. The PML approach requires 45,000 variables (the domain is a square 11

*μm*× 11

*μm*). The new boundary condition allows one to fit a boundary to the contour, which reduces the number of variables by 33% to 30,000.

13. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Ranversez, C.M. de Sterke, L.C. Botten, and M.J. Steel, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

*r*

_{1}= 1

*μm*, the outer radius

*r*

_{2}= 2

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

*n*= 1.44402362. The vacuum wavelength is

_{bg}*λ*= 1.55

*μm*. This time, the computational domain is a half-circle with a radius

*R*= 2.25

*μm*. The discretization was 300×150 cells and the size of the operator was reduced to about 60,000 from 90,000 used in simulation with PML technique (with a rectangular domain 5

*μm*× 2.5

*μm*).

14. N.A. Issa and L. Poladian, “Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers,” J. Lightwave Technol. **21**, 1005–1012 (2003). [CrossRef]

*HE*

_{21}- like mode.

*s*for PML and 30

*s*for the proposed algorithm (on a standard 2.8GHz desktop). For the microstructured fibers (only partial use of boundary conditions) the difference was smaller (70

*s*versus 40

*s*and 140

*s*vs. 90

*s*).

14. N.A. Issa and L. Poladian, “Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers,” J. Lightwave Technol. **21**, 1005–1012 (2003). [CrossRef]

9. H. Rogier and D. De Zutter, “Berenger and Leaky Modes in Optical Fibers Terminated with a Perfectly Matched Layer,” J. Lightw. Technol., **20**, 1141 – 1148 (2002). [CrossRef]

## 4. Conclusions

## Acknowledgment

## References and links

1. | Z. Zhu and T.G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express |

2. | Shangping Guo, Feng Wu, Sacharia Albin, Hsiang Tai, and Robert S. Rogowski,“Loss and dispersion analysis of microstructured fibers by finite-difference method,” Opt. Express |

3. | P. Kowalczyk, M. Wiktor, and M. Mrozowski, “Efficient finite difference analysis of microstructured optical fibers,” Opt. Express |

4. | A. Taflove and S.C. Hagness, “Computational electrodynamics: the finite-difference time-domain method,” Artech House, Boston (2005), 3rd edn. |

5. | J.P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

6. | G.R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron., |

7. | H.P. Uranus and H.J.W.M. Hoekstra, “Modeling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions,” Opt. Express |

8. | H. Rogier and D. De Zutter, “Berenger and Leaky Modes in Microstrip Substrates Terminated by a Perfectly Matched Layer,” IEEE Trans. Microwave Theory Tech. |

9. | H. Rogier and D. De Zutter, “Berenger and Leaky Modes in Optical Fibers Terminated with a Perfectly Matched Layer,” J. Lightw. Technol., |

10. | E.M. Kartchevski, A.I. Nosich, and G.W. Hanson, “Mathematical Analysis of the Generalized Natural Modes of an Inhomogeneous Optical Fiber,” J. Appl. Math. |

11. | C.D. Meyer, “Matrix analysis and applied linear algebra”, SIAM, Philadelphia (2000). |

12. | N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Trans. Microwave Theory Tech. |

13. | T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Ranversez, C.M. de Sterke, L.C. Botten, and M.J. Steel, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B |

14. | N.A. Issa and L. Poladian, “Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers,” J. Lightwave Technol. |

**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 4, 2007

Revised Manuscript: April 27, 2007

Manuscript Accepted: April 27, 2007

Published: September 17, 2007

**Citation**

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/oe/abstract.cfm?URI=oe-15-20-12605

Sort: Year | Journal | Reset

### References

- Z. Zhu, 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, Feng Wu, S. Albin, H. Tai, 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. [CrossRef] [PubMed]
- P. Kowalczyk, M. Wiktor, 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. [CrossRef] [PubMed]
- A. Taflove, S.C. Hagness, "Computational electrodynamics: the finite-difference time-domain method," Artech House, Boston (2005), 3rd edn.
- J.P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
- G.R. Hadley, "Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron., 28, 363-370 (1992). [CrossRef]
- 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2795. [CrossRef] [PubMed]
- 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]
- H. Rogier, D. De Zutter, "Berenger and Leaky Modes in Optical Fibers Terminated with a Perfectly Matched Layer," J. Lightwave Technol., 20, 1141 - 1148 (2002). [CrossRef]
- E.M. Kartchevski, A.I. Nosich, G.W. Hanson, "Mathematical Analysis of the Generalized Natural Modes of an Inhomogeneous Optical Fiber," J. Appl. Math. 65, 2033 - 2048 (2005).
- C.D. Meyer, "Matrix analysis and applied linear algebra", SIAM, Philadelphia (2000).
- N. Kaneda, B. Houshmand, T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997). [CrossRef]
- T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, R. Ranversez, C.M. de Sterke, L.C. Botten, M.J. Steel, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002). [CrossRef]
- N.A. Issa, L. Poladian, "Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers," J. Lightwave Technol. 21, 1005-1012 (2003). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.