## Efficient simulation of subwavelength plasmonic waveguides using implicitly restarted Arnoldi

Optics Express, Vol. 14, Issue 16, pp. 7291-7298 (2006)

http://dx.doi.org/10.1364/OE.14.007291

Acrobat PDF (137 KB)

### Abstract

In this paper, we present a full-vector finite difference method to solve for optical modes in one and two dimensional subwavelength plasmonic waveguides. We have used the Implicitly Restarted Arnoldi method to directly calculate the propagation constants of the dominant modes. The method has low computational complexity and can be applied to accurately model complex geometries and structures with fast-varying field profiles. When applied to solve for purely bounded modes, our method automatically separates evanescent and low-loss guided modes.

© 2006 Optical Society of America

## 1. Introduction

1. M. Hochberg, T. Baehr-Jones, C. Walker, and A. Scherer, “Integrated plasmon and dielectric waveguides,” Opt. Express **12**, 5481–5486, (2004). [CrossRef] [PubMed]

2. K. Tanaka, M. Tanaka, and T. Sugiyama, “Simulation of practical nanometric circuits based on surface plasmon polariton gap waveguide,” Opt. Express **13**, 256–266, (2005). [CrossRef] [PubMed]

3. G. I. Stegeman, R. F. Wallias, and A. Maradudin, “Excitation of surface polaritons by end-fire coupling,” Opt. Lett. **8**, 386-(1983). [CrossRef] [PubMed]

4. R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width,” Opt. Lett. **25**, 844–846, (2000). [CrossRef]

6. S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron **40**, 325–329, (2004). [CrossRef]

7. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric strucutres,” J. Phys. Rev. B **61**, 10484–10503, (2000). [CrossRef]

8. P. Berini, A. Stohr, K. Wu, and D. Jager, “Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects,” J. Lightwave Technol. **14**, 2422–2435, (1996). [CrossRef]

9. R. Zia, M. D. Selker, and M. L. Brongersma, “Leaky and Bound Modes of Surface Plasmon Waveguide,” Phys. Rev. B **71**, 165431, (2005). [CrossRef]

10. C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express **7**, 260–272, (2000). [CrossRef] [PubMed]

5. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A **21**, 2442–2446, (2004). [CrossRef]

11. J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” J. Phys. Rev. B **72**, 075405, (2005). [CrossRef]

12. I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelength,” J. Phys. Rev. B **62**, 15299–15302, (2000). [CrossRef]

## 2. Numerical solution approach

### 2.1. Theory

13. P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, “ Analysis of vectorial mode fields in optical waveguides by new finite difference method,” J. Lightwave Technol. **12**, 487–494, (1994). [CrossRef]

6. S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron **40**, 325–329, (2004). [CrossRef]

*β*is the propagation constant assuming a functional form of

*exp*(

*i*(

*βz*-

*ωt*)) for the steady state modes,

*k*

_{0}is the wave number in the free space,

*ε*is the dielectric constant, and

*n*

_{eff}=

*β*/

*k*

_{0}is the effective index. If the derivatives in these equations are approximated with their equivalent finite differences, all the discrete equations can be arranged into a linear system

*H̅*is a vector containing field values at all the points on the mesh [13

13. P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, “ Analysis of vectorial mode fields in optical waveguides by new finite difference method,” J. Lightwave Technol. **12**, 487–494, (1994). [CrossRef]

13. P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, “ Analysis of vectorial mode fields in optical waveguides by new finite difference method,” J. Lightwave Technol. **12**, 487–494, (1994). [CrossRef]

*L*motivate the use of the Arnoldi method to iteratively calculate the eigenvalues. Alternative iterative eigenvalue algorithms such as [14

14. K. Ramm, P. Lusse, and H.-G. Unger, “Multigrid Eigenvalue Solver for Mode Calculation of Planar Optical Waveguides,” IEEE Photonics Technol. Lett. **9**, 967–969, (1997). [CrossRef]

### 2.2. Eigenvalue computation

*L*in Eq. (5) can exceed 10000. Direct methods for solving the eigenvalue decomposition problem typically require

*O*(

*n*

^{2}) storage and

*O*(

*n*

^{3}) operations. Therefore, solving eigenvalue problems for matrices with order

*n*larger than several thousand elements is intractable. Since we are interested in the dominant resonant modes of the simulated structure, we only need to calculate a few eigenvalues (

*β*

^{2}) that meet the following criteria: (a)

*Re*(

*β*

^{2}) > 0, (b)

*Im*(

*β*

^{2}) > 0, and (c)

*Re*(

*β*

^{2}) >>

*Im*(

*β*

^{2}).

*b*by powers of a given matrix

*L*in order to find the largest eigenvalue of

*L*. The power method explicitly forms the Krylov Subspace (

*K*

_{n}), which is defined as

16. W. J. Stewart and A. Jennings, “Algorithm 570: LOPSI: A Simultaneous Iteration Method for Real Matrices [F2],” ACM Trans. Math. Softw. **7**, 230–232, (1981). [CrossRef]

*L*are explicitly generated, the resulting basis vectors that span

*K*

_{n}become numerically ill-conditioned, which can ultimately lead to inaccurate results. In contrast to the power method, the Arnoldi iteration projects- the matrix

*L*onto the Krylov Subspace (

*K*

_{n}) iteratively forming a set of orthogonal basis vectors (

*V*

_{m})

*V*

_{m}=

*I*and

*f*

_{m}

*H*

_{m}) of order

*m*where

*m*<<

*n*,

18. D. C. Sorensen, “Implicit application of polynomial filters in a K-step Arnoldi method,” SIAM Journal on Matrix Analysis and Applications **13**, 357–385, (1992). [CrossRef]

17. R. B. Lehoucq and D. C. Sorensen, “Deflation techniques within an implicitly restarted iteration,” SIAM Journal on Matrix Analysis and Applications **17**, 789–821, (1996). [CrossRef]

*V*

_{m}and

*H*

_{m}grows as the number of iterations increases, using Eq. (7) directly to compute the eigenvalues can become expensive. Therefore, restarting the Arnoldi iteration based on the information obtained in the previous iterations can lead to a substantial performance improvement since the size of

*H*

_{m}does not exceed

*k*where

*k*=

*m*+

*p*,

*m*is the number of desired eigenvalues, and

*p*is the number of extra iterations performed. The algorithm is implicitly restarted by using the

*p*extra eigenvalue estimates as shifts in a manner similar to the implicitly shifted QR algorithm. The shifted

*QR*factorization effectively reduces the subdiagonal elements of

*H*

_{m}to 0 to form the matrix

*Ĥ*. Therefore, an approximate partial Schur decomposition is obtained, Q

*Ĥ*

*Q*

^{*}, where

*Q*is an orthogonal vector and

*Ĥ*is an upper triangular matrix. Based on the properties of Schur decomposition, the diagonal entries of

*Ĥ*are the eigenvalue estimates of

*L*at the end of each iteration of the algorithm. More information on the Implicitly Restarted Arnoldi algorithm and critical implementation considerations can be found in [17

17. R. B. Lehoucq and D. C. Sorensen, “Deflation techniques within an implicitly restarted iteration,” SIAM Journal on Matrix Analysis and Applications **17**, 789–821, (1996). [CrossRef]

*L*matrix and the 13 eigenvalues of largest real part calculated using Implicitly Restarted Arnoldi. It is apparent that only a few of the calculated eigenvalues correspond to bounded propagation modes. Therefore, we signifcantly reduce the computational complexity by calculating only the eigenvalues that satisfy the criteria. For the 1680×1680 matrix, the solution to the eigenvalue problem using direct methods requires approximately 86.8 seconds of CPU time while the iterative eigenvalue computation only needs 1.1 seconds. As the order of

*L*is increased, the computational savings provided by the iterative method becomes even more significant. For a 2100×2100

*L*matrix, the direct eigenvalue solution requires 677.8 seconds while the iterative method only needs 2.25 seconds to calculate 20 eigenvalues. Given the efficiency of iterative eigenvalue computation using Implicitly Restarted Arnoldi, we are able to calculate the desired eigenvalues and their associated eigenvectors for matrices with

*n*greater than 100,000, which are required for complex structures.

## 3. Results and discussion

### 3.1. Planar multilayered waveguide

10. C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express **7**, 260–272, (2000). [CrossRef] [PubMed]

*λ*= 1.55

*μm*. The metal is silver with a dielectric constant of

*ε*

_{m}= -125.72 +

*i*3.23. A dielectric constant of

*ε*

_{d}= 12.25 is assumed for the dielectric material. In order to solve for the bounded electromagnetic modes, zero boundary conditions are applied to the points lying on the boundary of the computational window. Also, as described in the left inset of Fig. 2, if periodic boundary conditions are imposed, a 1D waveguide with infinite width can be modeled. The dispersion curves for the slab fundamental (symmetric bound

*S*

_{b}) mode of a 1D metal-insulator-metal (MIM) structure with a dielectric region thickness of 100

*nm*is calculated using our method and the method proposed in [10

10. C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express **7**, 260–272, (2000). [CrossRef] [PubMed]

**7**, 260–272, (2000). [CrossRef] [PubMed]

*n*

_{eff}values using our modeling technique with a moderate resolution of 20 is less than 0.3% with respect to the values obtained using [10

**7**, 260–272, (2000). [CrossRef] [PubMed]

*L*, the required CPU time, the

*n*

_{eff}values obtained from our simulation method for an MIM waveguide of 100

*nm*thickness with different resolutions. With a resolution of 10, the error is less than 0.5%, and the waveguide is simulated in 26 seconds.

### 3.2. Subwavelength metallic strip in a dielectric medium

6. S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron **40**, 325–329, (2004). [CrossRef]

*λ*= 1.55

*μm*,

*ε*

_{m}= -125.72+

*i*3.23,

*ε*

_{d}= 12.25, we calculate the

*n*

_{eff}values of the first three guided modes,

*M*

_{00}(Fig. 4),

*M*

_{01}(Fig. 5), and

*M*

_{10}(Fig. 6), for a single metallic strip of fixed width

*w*= 1

*μm*and varying thickness. Two almost degenerate corner modes are represented by curve

*M*

_{C}. This degeneracy originates from the fact that the absolute value of the field profiles of these two modes are almost the same. These modes are concentrated at the four corners and are highly lossy [6

**40**, 325–329, (2004). [CrossRef]

*S*

_{b}) and antisymmetric (

*A*

_{b}) slab modes shown in Fig. 3 correspond to the guided modes of an insulator-metal-insulator (IMI) planar waveguide.

*M*

_{00}and

*M*

_{10}dispersion diagrams are the same as those obtained in [6

**40**, 325–329, (2004). [CrossRef]

*M*

_{01}mode and the corner modes curves are slightly different. However,

*M*

_{01}shown in this figure is well predicted by the dielectric model for guided surface polaritons that use the effective index of a planar IMI waveguide to predict the effective indices of the guided modes for a 2D metallic strip waveguide of the same thickness [20

20. R. Zia, A. Chandran, and M. L. Brongersma, “ Dielectric waveguide model for guided surface polaritons,” Opt. Lett. **30**, 1473–1475, (2005). [CrossRef] [PubMed]

20. R. Zia, A. Chandran, and M. L. Brongersma, “ Dielectric waveguide model for guided surface polaritons,” Opt. Lett. **30**, 1473–1475, (2005). [CrossRef] [PubMed]

*t*= 100

*nm*and

*w*= 1

*μm*with a separation of 100

*nm*, we found that the fundamental mode has an effective index of

*n*

_{eff}= 4.24+

*i*0.012. The field profile associated with this mode is shown in Fig. 7. Interestingly, the field profile and effective index of this mode are similar to those of the fundamental mode of a single dielectric strip in a metallic medium [5

5. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A **21**, 2442–2446, (2004). [CrossRef]

## 4. Conclusion

## References and links

1. | M. Hochberg, T. Baehr-Jones, C. Walker, and A. Scherer, “Integrated plasmon and dielectric waveguides,” Opt. Express |

2. | K. Tanaka, M. Tanaka, and T. Sugiyama, “Simulation of practical nanometric circuits based on surface plasmon polariton gap waveguide,” Opt. Express |

3. | G. I. Stegeman, R. F. Wallias, and A. Maradudin, “Excitation of surface polaritons by end-fire coupling,” Opt. Lett. |

4. | R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width,” Opt. Lett. |

5. | R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A |

6. | S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron |

7. | P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric strucutres,” J. Phys. Rev. B |

8. | P. Berini, A. Stohr, K. Wu, and D. Jager, “Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects,” J. Lightwave Technol. |

9. | R. Zia, M. D. Selker, and M. L. Brongersma, “Leaky and Bound Modes of Surface Plasmon Waveguide,” Phys. Rev. B |

10. | C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express |

11. | J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” J. Phys. Rev. B |

12. | I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelength,” J. Phys. Rev. B |

13. | P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, “ Analysis of vectorial mode fields in optical waveguides by new finite difference method,” J. Lightwave Technol. |

14. | K. Ramm, P. Lusse, and H.-G. Unger, “Multigrid Eigenvalue Solver for Mode Calculation of Planar Optical Waveguides,” IEEE Photonics Technol. Lett. |

15. | V. Hernandez, J. E. Roman, A. Tomas, and V. Vidal, “A Survey of Software for Sparse Eigenvalue Problems,” Technical report, Universidad Politecnica de Valencia, (2005). |

16. | W. J. Stewart and A. Jennings, “Algorithm 570: LOPSI: A Simultaneous Iteration Method for Real Matrices [F2],” ACM Trans. Math. Softw. |

17. | R. B. Lehoucq and D. C. Sorensen, “Deflation techniques within an implicitly restarted iteration,” SIAM Journal on Matrix Analysis and Applications |

18. | D. C. Sorensen, “Implicit application of polynomial filters in a K-step Arnoldi method,” SIAM Journal on Matrix Analysis and Applications |

19. | R. Radke, “A MATLAB implementation of the implicitly restarted Arnoldi method for solving large scale eigenvalue problems,” Technical report, Dept. of Applied and Computational Mathematics, Rice University, Houston, TX, (1996). |

20. | R. Zia, A. Chandran, and M. L. Brongersma, “ Dielectric waveguide model for guided surface polaritons,” Opt. Lett. |

**OCIS Codes**

(240.0310) Optics at surfaces : Thin films

(240.5420) Optics at surfaces : Polaritons

(240.6680) Optics at surfaces : Surface plasmons

(240.6690) Optics at surfaces : Surface waves

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: June 5, 2006

Revised Manuscript: July 20, 2006

Manuscript Accepted: July 20, 2006

Published: August 7, 2006

**Citation**

Amir Hosseini, Arthur Nieuwoudt, and Yehia Massoud, "Efficient simulation of subwavelength plasmonic waveguides using implicitly restarted Arnoldi," Opt. Express **14**, 7291-7298 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7291

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### References

- M. Hochberg, T. Baehr-Jones, C. Walker, and A. Scherer, "Integrated plasmon and dielectric waveguides," Opt. Express 12, 5481-5486 (2004). [CrossRef] [PubMed]
- K. Tanaka, M. Tanaka, and T. Sugiyama, "Simulation of practical nanometric circuits based on surface plasmon polariton gap waveguide," Opt. Express 13, 256-266 (2005). [CrossRef] [PubMed]
- G. I. Stegeman, R. F. Wallias, and A. Maradudin, "Excitation of surface polaritons by end-fire coupling," Opt. Lett. 8, 386 (1983). [CrossRef] [PubMed]
- R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, "Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000). [CrossRef]
- R. Zia,M. D. Selker, P. B. Catrysse, andM. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2004). [CrossRef]
- S. J. Al-Bader, "Optical transmission on metallic wires-fundamental modes," IEEE J. Quantum Electron 40, 325-329 (2004). [CrossRef]
- P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric strucutres," J. Phys. Rev. B 61, 10484-10503 (2000). [CrossRef]
- P. Berini, A. Stohr, K. Wu, D. Jager, "Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects," J. Lightwave Technol. 14, 2422-2435 (1996). [CrossRef]
- R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguide," Phys. Rev. B 71, 165431 (2005). [CrossRef]
- C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media," Opt. Express 7, 260-272 (2000). [CrossRef] [PubMed]
- J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," J. Phys. Rev. B 72, 075405 (2005). [CrossRef]
- I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Metallic photonic crystals at optical wavelength," J. Phys. Rev. B 62, 15299-15302 (2000). [CrossRef]
- P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by new finite difference method," J. Lightwave Technol. 12, 487-494 (1994). [CrossRef]
- K. Ramm, P. Lusse, and H.-G. Unger, "Multigrid eigenvalue solver for mode calculation of planar optical waveguides," IEEE Photonics Technol. Lett. 9, 967-969 (1997). [CrossRef]
- V. Hernandez, J. E. Roman, A. Tomas, and V. Vidal, "A Survey of Software for Sparse Eigenvalue Problems," Technical report, Universidad Politecnica de Valencia, (2005).
- W. J. Stewart and A. Jennings, "Algorithm 570: LOPSI: A Simultaneous Iteration Method for Real Matrices [F2]," ACM Trans. Math. Softw. 7, 230-232 (1981). [CrossRef]
- R. B. Lehoucq and D. C. Sorensen, "Deflation techniques within an implicitly restarted iteration," SIAM J. Matrix Anal. Appl. 17, 789-821 (1996). [CrossRef]
- D. C. Sorensen, "Implicit application of polynomial filters in a K-step Arnoldi method," SIAM J. Matrix Anal. Appl. 13, 357-385, (1992). [CrossRef]
- R. Radke, "A MATLAB implementation of the implicitly restarted Arnoldi method for solving large scale eigenvalue problems," Technical report, Dept. of Applied and Computational Mathematics, Rice University, Houston, TX, (1996).
- R. Zia, A. Chandran, and M. L. Brongersma, "Dielectric waveguide model for guided surface polaritons," Opt. Lett. 30, 1473-1475 (2005). [CrossRef] [PubMed]

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