## Numerical analysis of a SNOM tip based on a partially cladded optical fiber |

Optics Express, Vol. 19, Issue 23, pp. 23140-23152 (2011)

http://dx.doi.org/10.1364/OE.19.023140

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### Abstract

A Scanning Nearfield Optical Microscope (SNOM) tip with partial metallic cladding is presented. For its design, a very demanding 2D eigenvalue analysis of an optical waveguide with material and radiation losses is carried out by the Multiple Multipole Program (MMP) and by the Finite Element Method (FEM). These simulations require some special tricks that are outlined. The computed 2D MMP and FEM results are compared and discussed. This 2D analysis is followed by a full 3D FEM analysis of the SNOM tip. The obtained 3D results confirm the corresponding 2D predictions. Important conclusions regarding the guiding capabilities of the chosen structure and the efficiency of the applied numerical methods are presented.

© 2011 OSA

## 1. Introduction

3. P. Berini and R. J. Buckley, “On the convergence and accuracy of numerical mode computations of surface plasmon waveguides,” J. Comput. Theor. Nanosci. **6**(9), 2040–2053 (2009). [CrossRef]

*M(e)X(e)*= 0, where

*e*denotes the chosen eigenvalue that may be either the frequency or propagation constant of a guided mode. To find the eigenvalues and eigenvectors of this system the transcendental equation

7. S. Kagami and I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microw. Theory Tech. **32**(4), 455–461 (1984). [CrossRef]

10. Ch. Hafner, “OpenMaX: Graphic Platform for Computational Electromagnetics and Computational Optics”, http://openmax.ethz.ch/, ETH Zurich, 2010.

## 2. Finite element method (FEM) and multiple multipole program (MMP)

12. J. Smajic, Ch. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of numerical methods for the analysis of plasmonic structures,” J. Comput. Theor. Nanosci. **6**(3), 763–774 (2009). [CrossRef]

10. Ch. Hafner, “OpenMaX: Graphic Platform for Computational Electromagnetics and Computational Optics”, http://openmax.ethz.ch/, ETH Zurich, 2010.

10. Ch. Hafner, “OpenMaX: Graphic Platform for Computational Electromagnetics and Computational Optics”, http://openmax.ethz.ch/, ETH Zurich, 2010.

14. J. Smajic and C. Hafner, “Complex Eigenvalue Analysis of Plasmonic Waveguides,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuD2. http://www.opticsinfobase.org/abstract.cfm?URI=IPRSN-2010-ITuD2.

## 3. Considered structures and numerical models

16. V. Lotito, U. Sennhauser, and Ch. Hafner, “Effects of asymmetric surface corrugations on fully metal-coated scanning near field optical microscopy tips,” Opt. Express **18**(8), 8722–8734 (2010). [CrossRef] [PubMed]

18. COMSOL. Multiphysics, 4.1, Commercial FEM Solver, www.comsol.com.

## 4. Numerical results

_{0}= 1, which indicates that the field is rather strong in the surrounding air domain. Mode 4 has low loss for wavelength longer than 500 nm, which is most promising, but its field most strongly confined near the triple point, which is rather problematic. Furthermore, the direction of the electric field outside the cladding is opposite to the direction the electric field inside the cladding, which is different from the field of a HE11 mode of fiber without cladding. Thus, it is expected that the HE11 mode couples much better to Mode 2 than to Mode 4.

_{0}) of this mode is sometimes below 1, which indicates that some radiation loss is present. When tracing this mode with MMP, the Sommerfeld radiation condition becomes invalid at certain points and then the MMP solution becomes invalid. Since the mode curls around in the complex plane, one would need to run different MMP models (with and without Sommerfeld radiation condition) and patch them together.

_{n(1)}describing the guided and H

_{n(2)}describing the radiated modes. Hence in MMP, to switch between the guided and radiated modes, it is necessary to change only the kind of Hankel function. However, in these settings radiating modes have increasing field function with respect to the distance and in infinity the field becomes also infinite. This is of course physically not correct.

_{max}. The value R

_{max}depends on the propagation constant and on the distance to the beginning of the waveguide. For sufficiently large distances R

_{max}is relatively large and the MMP model is accurate enough for describing the field in the vicinity of the waveguide.

_{0}>1 and when the mode dispersion is not very strong. A similar disagreement is also found for Mode 3 when Real(γ)/k

_{0}is close to 1. Here, the MMP search shows some discontinuous behavior. In fact, the mode is lost (because it disappears completely when the Sommerfeld radiation condition is assumed to be valid) and the smart eigenvalue search routine of OpenMaX the switches to another “evanescent” mode. In this context, it should be mentioned that a proper distinction of guided and evanescent modes is only possible for loss-fee waveguides, where the propagation constant is real for guided modes and imaginary for evanescent modes. When material andradiation losses are present, all modes are characterized by a complex propagation constant. Modes that follow mostly the imaginary axis may then be called “evanescent”, although this is not a strict definition.

## 5. Conclusions

## References and links

1. | H. A. Atwater, J. A. Dionne, and L. A. Sweatlock, “Subwavelength-scale Plasmon Waveguides,” in |

2. | Ch. Hafner, C. Xudong, A. Bertolace, and R. Vahldieck, “Multiple multipole program analysis of metallic optical waveguides,” Proc. of SPIE Vol. 6617, pp. 66170C–1, SPIE Europe: Cardiff, UK, 2007. |

3. | P. Berini and R. J. Buckley, “On the convergence and accuracy of numerical mode computations of surface plasmon waveguides,” J. Comput. Theor. Nanosci. |

4. | W. H. Press, S. A. Teukolski, W. T. Vetterling, and B. P. Flannery, |

5. | A. Taflove, |

6. | J. Jin, |

7. | S. Kagami and I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microw. Theory Tech. |

8. | Ch. Hafner, |

9. | Ch. Hafner, |

10. | Ch. Hafner, “OpenMaX: Graphic Platform for Computational Electromagnetics and Computational Optics”, http://openmax.ethz.ch/, ETH Zurich, 2010. |

11. | Ch. Hafner, J. Smajic, and M. Agio, “Numerical Methods for the Electrodynamic Analysis of Nanostructures”,in |

12. | J. Smajic, Ch. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of numerical methods for the analysis of plasmonic structures,” J. Comput. Theor. Nanosci. |

13. | G. H. Golub and Ch. F. Van Loan, |

14. | J. Smajic and C. Hafner, “Complex Eigenvalue Analysis of Plasmonic Waveguides,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuD2. http://www.opticsinfobase.org/abstract.cfm?URI=IPRSN-2010-ITuD2. |

15. | W. Nakagawa, L. Vaccaro, H. P. Herzig, and Ch. Hafner, “Polarization mode coupling due to metal-layer modifications in apertureless near-field scanning optical microscope probes,” J. Comput. Theor. Nanosci. |

16. | V. Lotito, U. Sennhauser, and Ch. Hafner, “Effects of asymmetric surface corrugations on fully metal-coated scanning near field optical microscopy tips,” Opt. Express |

17. | P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

18. | COMSOL. Multiphysics, 4.1, Commercial FEM Solver, www.comsol.com. |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(180.4243) Microscopy : Near-field microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: August 31, 2011

Revised Manuscript: October 8, 2011

Manuscript Accepted: October 17, 2011

Published: October 31, 2011

**Virtual Issues**

Vol. 7, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Jasmin Smajic and Christian Hafner, "Numerical analysis of a SNOM tip based on a partially cladded optical fiber," Opt. Express **19**, 23140-23152 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23140

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### References

- H. A. Atwater, J. A. Dionne, and L. A. Sweatlock, “Subwavelength-scale Plasmon Waveguides,” in Surface Plasmon Nanophotonics, M.L. Brongersma, P.G. Kik, eds. (Springer: Dordrecht, The Nederlands, 2007).
- Ch. Hafner, C. Xudong, A. Bertolace, and R. Vahldieck, “Multiple multipole program analysis of metallic optical waveguides,” Proc. of SPIE Vol. 6617, pp. 66170C–1, SPIE Europe: Cardiff, UK, 2007.
- P. Berini and R. J. Buckley, “On the convergence and accuracy of numerical mode computations of surface plasmon waveguides,” J. Comput. Theor. Nanosci. 6(9), 2040–2053 (2009). [CrossRef]
- W. H. Press, S. A. Teukolski, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, Second Edition (Cambridge University Press, Port Chester, NY, USA, 1997).
- A. Taflove, Advances in Computational Electrodynamics, The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 1998).
- J. Jin, The Finite Element Method in Electromagnetics (Wiley: Chichester, UK 1993).
- S. Kagami and I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microw. Theory Tech. 32(4), 455–461 (1984). [CrossRef]
- Ch. Hafner, Post-Modern Electromagnetics Using Intelligent MaXwell Solvers (Wiley: Chichester, UK 1999).
- Ch. Hafner, MaX-1: A Visual Electromagnetics Platform (Wiley: Chichester, UK 1998).
- Ch. Hafner, “OpenMaX: Graphic Platform for Computational Electromagnetics and Computational Optics”, http://openmax.ethz.ch/ , ETH Zurich, 2010.
- Ch. Hafner, J. Smajic, and M. Agio, “Numerical Methods for the Electrodynamic Analysis of Nanostructures”,in Nanoclusters and Nanostructured Surfaces; A. K. Ray, Ed., (American Scientific Publishers: Valencia, CA, 2010).
- J. Smajic, Ch. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of numerical methods for the analysis of plasmonic structures,” J. Comput. Theor. Nanosci. 6(3), 763–774 (2009). [CrossRef]
- G. H. Golub and Ch. F. Van Loan, Matrix Computations, 3rd ed.; (Johns Hopkins University Press: Baltimore, MD, 1996).
- J. Smajic and C. Hafner, “Complex Eigenvalue Analysis of Plasmonic Waveguides,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuD2. http://www.opticsinfobase.org/abstract.cfm?URI=IPRSN-2010-ITuD2 .
- W. Nakagawa, L. Vaccaro, H. P. Herzig, and Ch. Hafner, “Polarization mode coupling due to metal-layer modifications in apertureless near-field scanning optical microscope probes,” J. Comput. Theor. Nanosci. 4(3), 692–703 (2007).
- V. Lotito, U. Sennhauser, and Ch. Hafner, “Effects of asymmetric surface corrugations on fully metal-coated scanning near field optical microscopy tips,” Opt. Express 18(8), 8722–8734 (2010). [CrossRef] [PubMed]
- P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
- COMSOL. Multiphysics, 4.1, Commercial FEM Solver, www.comsol.com .

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