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

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References

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  1. 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).
  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. 6(9), 2040–2053 (2009).
    [CrossRef]
  4. 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).
  5. A. Taflove, Advances in Computational Electrodynamics, The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 1998).
  6. J. Jin, The Finite Element Method in Electromagnetics (Wiley: Chichester, UK 1993).
  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]
  8. Ch. Hafner, Post-Modern Electromagnetics Using Intelligent MaXwell Solvers (Wiley: Chichester, UK 1999).
  9. Ch. Hafner, MaX-1: A Visual Electromagnetics Platform (Wiley: Chichester, UK 1998).
  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 Nanoclusters and Nanostructured Surfaces; A. K. Ray, Ed., (American Scientific Publishers: Valencia, CA, 2010).
  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]
  13. G. H. Golub and Ch. F. Van Loan, Matrix Computations, 3rd ed.; (Johns Hopkins University Press: Baltimore, MD, 1996).
  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. 4(3), 692–703 (2007).
  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]
  17. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
    [CrossRef]
  18. COMSOL. Multiphysics, 4.1, Commercial FEM Solver, www.comsol.com .

2010

2009

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]

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]

2007

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).

1984

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]

1972

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Berini, P.

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]

Buckley, R. J.

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]

Christy, R.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Fukai, I.

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]

Hafner, Ch.

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]

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]

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).

Herzig, H. P.

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).

Johnson, P.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Kagami, S.

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]

Lotito, V.

Mishrikey, M.

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]

Nakagawa, W.

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).

Raguin, L.

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]

Sennhauser, U.

Smajic, J.

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]

Tavzarashvili, K.

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]

Vaccaro, L.

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).

IEEE Trans. Microw. Theory Tech.

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]

J. Comput. Theor. Nanosci.

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]

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]

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).

Opt. Express

Phys. Rev. B

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Other

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

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 .

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.

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).

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).

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Figures (10)

Fig. 1
Fig. 1

Geometry of the considered optical fiber with metallic cladding (a) is presented. The core of the fiber is made of silica (εr = 2.25) and for the symmetric cladding two silver (εr = εr(λ)) strips are used. The measured permittivity data for silver modeling were used (b) [17].

Fig. 2
Fig. 2

3D geometry of the considered SNOM tip based on the cladded optical waveguide presented in Fig. 1 is depicted. At the end of the fiber one silver strip is 40nm and the other one 90nm elongated beyond the core end in order to form a narrow field enhancement region.

Fig. 3
Fig. 3

MMP (a) and FEM (b) modeling of the waveguide shown in Fig. 1. The MMP field sources placed along different interfaces are marked by crosses. The FEM model also requires a termination of the surrounding air and the corresponding ABC. Both models use the same PEC and PMC boundary conditions along the symmetry planes.

Fig. 4
Fig. 4

3D FEM modeling of the SNOM tip waveguide termination presented in Fig. 2. The straight section of the waveguide (500nm) was meshed by using a special meshing technique (sweep method) based on a triangular surface mesh and prisms generated in the perpendicular direction. By following the modeling from Fig. 3b the PMC boundary condition was used over the symmetry plane and the first order ABC was used over the cylindrical (the straight waveguide section) and spherical boundaries (the tips) air truncating boundaries.

Fig. 5
Fig. 5

2D FEM eigenvalue analysis results in the complex propagation constant plane (left) and the corresponding eigenfields (right) of the four guiding modes are depicted. Red circles in marked with M1 – M4 in the complex plane show the locations of the eigenmodes visualized. Since different modes cover different wavelength ranges and since the wavelength scale along the mode-traces is highly nonlinear, the wavelengths at the beginning and the end of each mode trace are given.

Fig. 6
Fig. 6

Comparison of the 2D eigenvalue results between MMP and FEM. Evidently a very good agreement is obtained – except at areas where radiation loss may cause strong differences.

Fig. 7
Fig. 7

The behavior of a MMP search function (the residual norm of the MMP over-determined linear system of equations divided by the guided electromagnetic power as presented in [14]) in the complex propagation plane is shown. Sharp dips (dark areas) of this function correspond to the eigenvalues

Fig. 8
Fig. 8

The behavior of Mode 2 in the γ-plane (left) and the associated normalized radiation and material losses (right).

Fig. 9
Fig. 9

Three different wavelengths of Mode 2 were chosen (a), the corresponding radiation and material losses computed (b), and the corresponding eigenfields depicted (c). The eigensolutions are used for feeding the 3D structure (d).

Fig. 10
Fig. 10

The results of the 3D analysis are presented. The absolute value of the electric field is depicted (a) and the electric field values 5nm away from the silver tips are compared (b). The field values are normalized according to the highest field value obtained.

Equations (1)

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f( e )= Residual( e ) / Amplitude( e )

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