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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 6 — Mar. 20, 2006
  • pp: 2365–2371
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Curved long-range surface plasmon-polariton waveguides

Pierre Berini and Junjie Lu  »View Author Affiliations


Optics Express, Vol. 14, Issue 6, pp. 2365-2371 (2006)
http://dx.doi.org/10.1364/OE.14.002365


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Abstract

The propagation of the long-range surface plasmon-polariton mode in waveguides comprised of a curved thin metal film of finite width embedded in a homogeneous background dielectric is described. The curve and its mode are modelled in cylindrical coordinates using a rigorous vectorial numerical method and an absorbing boundary condition is applied on the radiating side of the bend. From the results obtained, it is confirmed that long-range structures are not incompatible with bending and that reasonably small radii of curvature can be used.

© 2006 Optical Society of America

1. Introduction

The design of curved waveguides is of central importance in integrated optics [1

1. S. E. Miller, “Integrated Optics: An Introduction,” Bell Syst. Tech. J. 48, 2059–2069 (1969).

] since achieving efficient directional change is necessary in numerous structures such as offsets (e.g.: S-bend), splitters and combiners (e.g.: Y-junction), Mach-Zehnder interferometers (e.g.: back-to-back Y-junctions) and couplers (e.g.: coupled S-bends). Achieving an efficient directional change implies minimising for instance the area or path length required for the curve while minimising its insertion loss. The circular curved waveguide is a suitable structure for achieving directional change and its analysis as a dielectric structure has been the subject of intense research for at least a few decades [2

2. E.A.J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).

].

2. Method

The curved waveguide of interest is shown schematically in cross-sectional view in Fig. 1(a) and in top view in Fig. 1(b), along with the cylindrical coordinate system (ρ, ϕ, z) used for its analysis. The metal film of thickness t and width w has an equivalent complex permittivity ε 2 and is embedded in an infinite background dielectric of permittivity ε 1. The circular curve has a radius of curvature r0 that is invariant with ϕ so the problem is two-dimensional and the structure is completely characterised via the modes that it supports along the propagation direction which is ϕ. The straight waveguide is obtained in the limit r0 → ∞.

Fig. 1. Curved waveguide in (a) cross-sectional and (b) top views.

The modes supported by the curved waveguide must satisfy the time-harmonic (e +jωt) Maxwell’s equations for source-free isotropic media [13

13. R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).

]:

×H=ε0εrE
(1)
×E=μ0H
(2)
H=0
(3)
(εyE)=0
(4)

with εr being a function of ρ and z. For convenience of analysis, an electric Hertzian vector potential e is introduced and its curl defined on the basis of the null divergence of H and the vector identity ∇ ∙ ∇ × A = 0 (where A is an arbitrary vector) as:

H=ε0×Πe
(5)

Using this definition and manipulating Maxwell’s equations (1)-(4) in the usual way [13

13. R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).

] yields the vector wave equation governing e :

2Πeεr1(εr)·Πe+εrβ02Πe=0
(6)

and the following relationship between E and e :

E=εr1××Πe
(7)

where β 0 = √μ 0 ε 0 is the phase constant of free-space. The modes supported by the curved waveguide propagate along the ϕ axis, so a mode exhibits an e -ϕϕ = e -jneffβ0r0ϕ dependency for propagation along + ϕ, where neff is the complex effective refractive index of the mode. The loss of the mode a is given in dB by:

α=20(log10e)θIm(βϕ)=20(log10e)θr0β0Im(neff)
(8)

for a curve subtending an angle of θ in radians, with r0 in m and β0 in m-1.

Equations (5)-(7) are solved numerically for any mode of interest using the Method of Lines (MoL) formulated in cylindrical coordinates [14

14. R. Pregla, “The Method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol. 14, 634–639 (1996). [CrossRef]

,15

15. J. Lu, “Modelling optical waveguide bends and application to plasmon-polariton waveguides,” M.A.Sc. Thesis, University of Ottawa, Ottawa, Canada, 2003.

]. An absorbing boundary condition positioned on the radiating side of the curve (right side in Fig. 1(a)) is required and used [16

16. T. G. Moore, J. G. Blaschak, A. Taflove, and G. A. Kriegsmann, “Theory and application of radiation boundary operators,” IEEE Trans. Antennas Propag. 36, 1797–1812 (1988). [CrossRef]

,17

17. R. Pregla, “MOL-BPM Method of lines Based Beam Propagation Method,” in Progress in Electromagnetics Research Vol. 11, J. A. Kong, Ed. (EMW Publishing, Cambridge, Mass., 1995).

]. The formulation of the MoL for curves is vectorial and well suited to this waveguide problem since it makes use of a 1-D discretisation along the ρ axis and analytical forms along z. The analytical forms make it easy to handle either a very large or an infinitesimal metal thickness t, while the width of the metal w and the radius of curvature r0 set the discretisation. Using the MoL, the complex effective index neff of a mode, along with the spatial distribution of its six field components, can be determined.

3. Results

The formulation and software developed [15

15. J. Lu, “Modelling optical waveguide bends and application to plasmon-polariton waveguides,” M.A.Sc. Thesis, University of Ottawa, Ottawa, Canada, 2003.

] were validated by: (i) reproducing the 1-D field distributions and radiation losses computed for dielectric curves using the MoL [14

14. R. Pregla, “The Method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol. 14, 634–639 (1996). [CrossRef]

] (the radiation losses reported therein were also validated experimentally [18

18. R. J. Deri and R. J. Hawkins, “Polarization, scattering, and coherent effects in semiconductor rib waveguide bends,” Opt. Lett. 13, 922–924 (1988). [CrossRef] [PubMed]

]); (ii) comparing favourably with the radiation losses computed for dielectric curves using the finite element [20

20. T. Yamamoto and M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by the finite-element method,” J. Lightwave Technol. 11, 1579–1583 (1993). [CrossRef]

] and finite difference [21

21. S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996). [CrossRef]

] methods (the radiation losses computed therein were also validated experimentally [19

19. M. W. Austin, “GaAs/GaAlAs Curved rib waveguides,” IEEE J Quant. Elect. 18, 795–800 (1982). [CrossRef]

]); (iii) reproducing the transition losses computed using the finite-difference method [21

21. S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996). [CrossRef]

] for a straight dielectric waveguide end-coupled to a curved one as a function of r0 ; (iv) reproducing the phase and attenuation constant versus thickness t (and the field distribution at t = 100 nm) of the ssb0 , asb0 , aab0 , sab0 modes for straight 1 μm wide metal stripe waveguides [3

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

] in the limit r0 → ∞; and (v) making comparisons with experiment for S-bends fabricated using thin narrow Au stripes in SiO2 and propagating the long-range ssb0 mode at λ0 = 1550 nm [11

11. R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, and P. Berini, “Passive integrated optics elements based on long-range surface plasmon-polaritons,” J Lightwave Technol. 24, 477–494 (2006). [CrossRef]

].

In order to remain consistent with the results reported in Ref. [3

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

] for the straight metal stripe, a set of curves were analysed at the same wavelength of λ0 = 633 nm and assuming the same materials: an Ag film having εr,2 = - 19 -j0.53 surrounded by a dielectric of relative permittivity εr,1 = 4. The radius of curvature r0 is variable and the cross-sectional dimensions (w, t) of the film considered are: w = 0.5, 1 μm with t = 10, 11, 12, 13, 14, 15 nm. The highest loss, highest confinement waveguide among this set is the w = 1 μm, t = 15 nm structure, for which the attenuation of the long-range ssb0 mode is 26 times lower than that of the surface plasmon supported by the corresponding single interface (compare the ssb0 mode for w = 1 μm, t = 15 nm with the ssb0 mode for t → ∞ in Fig. 2(b) of Ref. [3

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

]). Thus, the ssb0 mode remains long ranging for all (w, t) combinations considered.

Fig. 2. (a) Insertion loss, (b) radiation loss and (c) Re{neff } of the ssb0 mode versus radius of curvature for w = 1 μm and t = 15 nm. (d) Summary of r0,opt and ILmin for all (w, t) cases considered.

Figure 2 shows in Part (a) the total insertion loss (IL) of the ssb0 mode for θ= 90°, w = 1 μm and t = 15 nm as a function of r0 . The total insertion loss is comprised of the propagation and radiation losses. One immediately notes the existence of an optimum radius of curvature r0,opt where the insertion loss is minimised (ILmin ), occurring in this case at r0,opt ~ 130 μm. An optimum radius exists because the waveguide includes an absorbing medium (the metal). For r0 > r0,opt the insertion loss is dominated by the propagation loss, which increases with r0 due to the increasing arc length of a fixed angle curve (say 90°), while for r0 < r0,opt radiation loss dominates and increases with decreasing r0 . Part (b) gives the radiation loss component, which is obtained by repeating the computations with Im{εr,2 } = 0. It is noted that some radiation loss indeed occurs at the optimum radius of r0,opt ~ 130 μm. Part (c) plots the Re{neff } of the ssb0 mode of the curve as a function of r0 , showing that Re{neff } converges to the value of the corresponding straight case as r0 → ∞, and diverges as r0 → 0. This divergence stems from the definition of neff for the curve. Consider that neffβ0 gives the phase constant in rad/m of the mode propagating along the arc r0ϕ. Since the mode peak shifts outward as r0 → 0, then neffβ0 must increase if the phase along r0ϕ is to be the same as the phase of the mode peak, leading to the observed divergence as r0 → 0. The difference between the effective indices of the curved and straight metal stripes seems to follow an r02 dependence as in dielectric waveguides [22

22. E. F. Kuester and D. C. Chang, “Surface-wave radiation loss form curved dielectric slabs and fibers,” IEEE J. Quant. Elect. 11, 903–907 (1975). [CrossRef]

]. Part (d) summarizes r0,opt and ILmin for all combinations (w, t) considered and shows a clear trend: the higher confinement, higher attenuation structures (e.g.: w = 1 μm, t = 15 nm) provide a lower ILmin and a smaller r0,opt .

Fig. 3. Normalised contours of Re{Ez } of the ssb0 mode for w = 1 μm, t = 15 nm, (a) r0 = 1 m and (b) r0 = r0,opt . (c) Normalized distributions of Re{Ez } along a horizontal cut immediately above the metal for w = 1 μm, t = 15 nm and r0 = 1 m, 100 μm and 50 μm. (d) Summary of the transition loss at r0 = r0,opt for all (w, t) cases considered.

Parts (a) and (b) of Fig. 3 show normalised contours of the Re{Ez } of the ssb0 mode (w = 1 μm, t = 15 nm) for r0 = 1 m and r0 = r0,opt respectively, the 1 m radius being used to model the waveguide as straight (r0 → ∞). Part (c) shows normalized distributions of the Re{Ez } along a horizontal cut immediately above the metal film (w = 1 μm, t = 15 nm) for three radii of curvature: r0 = 1 m, 100 μm and 50 μm. Two principal effects are noted from these plots as r0 decreases: (i) the mode peak shifts from the center towards the outside of the bend, and (ii) the field becomes oscillatory along the outside of the bend. The first effect causes a transition (coupling) loss to occur at the junction between waveguides of differing radii of curvature, say curved and straight, as can be appreciated by comparing the contours in Part (a) with those of Part (b). This transition loss can be computed via an overlap integral on the participating normalised mode fields, bearing in mind that the curved mode is a radiation mode [11

11. R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, and P. Berini, “Passive integrated optics elements based on long-range surface plasmon-polaritons,” J Lightwave Technol. 24, 477–494 (2006). [CrossRef]

]. Part (d) summarizes the transition loss for curves designed at r0,opt and joined to their corresponding straight waveguide (r0 → ∞) for all combinations (w, t) considered. A clear trend is apparent from Part (d): the lower confinement, lower attenuation structures (e.g.: w = 0.5 μm, t = 10 nm) provide a lower transition loss. This trend leads away from the trend observed Fig. 2(d), however, transition losses can be substantially eliminated by simply laterally offsetting the curved and straight waveguide sections.

4. Conclusions

The main conclusions drawn from this work regarding the long-range ssb0 mode propagating along curves in metal stripe waveguides are that: (i) long-range structures are not incompatible with bending, (ii) the higher attenuation, higher confinement structures considered have a lower minimum insertion loss at a smaller optimal radius of curvature, (iii) reasonably small radii of curvature can be used, and (iv) the effects caused by bending are the same as those encountered in conventional dielectric waveguides. The analysis technique discussed in this paper could also be used to study bends in other surface plasmon waveguide structures [23

23. B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides”, Appl. Phys. Lett. 88, 094104 (2006). [CrossRef]

,24

24. J.-C Weeber, M. U. Gonzälez, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips”, Appl. Phys. Lett. 87, 221101 (2005). [CrossRef]

].

Note added in proof: A paper was recently published on the modelling of surface plasmon waveguides, including bend structures, using a 3-D electromagnetic field solver [25

25. A. Degiron and D. Smith, “Numerical simulations of long-range plasmons”, Opt. Express 141611–1625 (2006) http://www.opticsexpress.org/abstract.cfm?id=88076 [CrossRef] [PubMed]

].

References and links

1.

S. E. Miller, “Integrated Optics: An Introduction,” Bell Syst. Tech. J. 48, 2059–2069 (1969).

2.

E.A.J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).

3.

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

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]

5.

R. Nikolajsen, K. Leosson, I. Salakhutdinov, and S.I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82, 668–670 (2003). [CrossRef]

6.

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmon-polariton waveguides,” J. Appl. Phys. 98, 043109 (2005). [CrossRef]

7.

R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Long-range plasmon-polariton wave propagation in thin metal films of finite-width excited using an end-fire technique,” Proc. SPIE 4087, 534–540 (2000). [CrossRef]

8.

T. Nikolajsen, K. Leosson, and S.I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5836 (2004). [CrossRef]

9.

R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express 13, 977–984 (2005) http://www.opticsexpress.org/abstract.cfm?id=82563 [CrossRef] [PubMed]

10.

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated Optical Components Utilizing Long-Range Surface Plasmon Polaritons,” J. Lightwave Technol. 23, 413–422 (2005). [CrossRef]

11.

R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, and P. Berini, “Passive integrated optics elements based on long-range surface plasmon-polaritons,” J Lightwave Technol. 24, 477–494 (2006). [CrossRef]

12.

H. S. Won, K. C. Kim, S. H. Song, C.-H. Oh, P. S. Kim, S. Park, and S. I. Kim, “Vertical coupling of long-range surface plasmon polaritons,” Appl. Phys. Lett. 88, 011110 (2006). [CrossRef]

13.

R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).

14.

R. Pregla, “The Method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol. 14, 634–639 (1996). [CrossRef]

15.

J. Lu, “Modelling optical waveguide bends and application to plasmon-polariton waveguides,” M.A.Sc. Thesis, University of Ottawa, Ottawa, Canada, 2003.

16.

T. G. Moore, J. G. Blaschak, A. Taflove, and G. A. Kriegsmann, “Theory and application of radiation boundary operators,” IEEE Trans. Antennas Propag. 36, 1797–1812 (1988). [CrossRef]

17.

R. Pregla, “MOL-BPM Method of lines Based Beam Propagation Method,” in Progress in Electromagnetics Research Vol. 11, J. A. Kong, Ed. (EMW Publishing, Cambridge, Mass., 1995).

18.

R. J. Deri and R. J. Hawkins, “Polarization, scattering, and coherent effects in semiconductor rib waveguide bends,” Opt. Lett. 13, 922–924 (1988). [CrossRef] [PubMed]

19.

M. W. Austin, “GaAs/GaAlAs Curved rib waveguides,” IEEE J Quant. Elect. 18, 795–800 (1982). [CrossRef]

20.

T. Yamamoto and M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by the finite-element method,” J. Lightwave Technol. 11, 1579–1583 (1993). [CrossRef]

21.

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996). [CrossRef]

22.

E. F. Kuester and D. C. Chang, “Surface-wave radiation loss form curved dielectric slabs and fibers,” IEEE J. Quant. Elect. 11, 903–907 (1975). [CrossRef]

23.

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides”, Appl. Phys. Lett. 88, 094104 (2006). [CrossRef]

24.

J.-C Weeber, M. U. Gonzälez, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips”, Appl. Phys. Lett. 87, 221101 (2005). [CrossRef]

25.

A. Degiron and D. Smith, “Numerical simulations of long-range plasmons”, Opt. Express 141611–1625 (2006) http://www.opticsexpress.org/abstract.cfm?id=88076 [CrossRef] [PubMed]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: February 3, 2006
Revised Manuscript: March 7, 2006
Manuscript Accepted: March 14, 2006
Published: March 20, 2006

Citation
Pierre Berini and Junjie Lu, "Curved long-range surface plasmon-polariton waveguides," Opt. Express 14, 2365-2371 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2365


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References

  1. S. E. Miller, "Integrated Optics: An Introduction," Bell Syst. Tech. J. 48, 2059-2069 (1969).
  2. E.A.J. Marcatili, "Bends in Optical Dielectric Guides," Bell Syst. Tech. J. 48, 2103-2132 (1969).
  3. P. Berini, "Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000). [CrossRef]
  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]
  5. R. Nikolajsen, K. Leosson, I. Salakhutdinov and S.I. Bozhevolnyi, "Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths," Appl. Phys. Lett. 82, 668-670 (2003). [CrossRef]
  6. P. Berini, R. Charbonneau, N. Lahoud and G. Mattiussi, "Characterization of long-range surface-plasmon-polariton waveguides," J. Appl. Phys. 98, 043109 (2005). [CrossRef]
  7. R. Charbonneau, P. Berini, E. Berolo and E. Lisicka-Shrzek, "Long-range plasmon-polariton wave propagation in thin metal films of finite-width excited using an end-fire technique," Proc. SPIE 4087, 534-540 (2000). [CrossRef]
  8. T. Nikolajsen, K. Leosson and S.I. Bozhevolnyi, "Surface plasmon polariton based modulators and switches operating at telecom wavelengths," Appl. Phys. Lett. 85, 5833-5836 (2004). [CrossRef]
  9. R. Charbonneau, N. Lahoud, G. Mattiussi and P. Berini, "Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons," Opt. Express 13, 977-984 (2005) http://www.opticsexpress.org/abstract.cfm?id=82563 [CrossRef] [PubMed]
  10. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen and S. I. Bozhevolnyi, "Integrated Optical Components Utilizing Long-Range Surface Plasmon Polaritons," J. Lightwave Technol. 23, 413-422 (2005). [CrossRef]
  11. R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, and P. Berini, "Passive integrated optics elements based on long-range surface plasmon-polaritons," J Lightwave Technol. 24, 477-494 (2006). [CrossRef]
  12. H. S. Won, K. C. Kim, S. H. Song, C.-H. Oh, P. S. Kim, S. Park and S. I. Kim, "Vertical coupling of long-range surface plasmon polaritons," Appl. Phys. Lett. 88, 011110 (2006). [CrossRef]
  13. R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).
  14. R. Pregla, "The Method of lines for the analysis of dielectric waveguide bends," J. Lightwave Technol. 14, 634-639 (1996). [CrossRef]
  15. J. Lu, "Modelling optical waveguide bends and application to plasmon-polariton waveguides," M.A.Sc. Thesis, University of Ottawa, Ottawa, Canada, 2003.
  16. T. G. Moore, J. G. Blaschak, A. Taflove and G. A. Kriegsmann, "Theory and application of radiation boundary operators," IEEE Trans. Antennas Propag. 36, 1797-1812 (1988). [CrossRef]
  17. R. Pregla, "MOL-BPM Method of lines Based Beam Propagation Method," in Progress in Electromagnetics Research Vol. 11, J. A. Kong, Ed. (EMW Publishing, Cambridge, Mass., 1995).
  18. R. J. Deri and R. J. Hawkins, "Polarization, scattering, and coherent effects in semiconductor rib waveguide bends," Opt. Lett. 13,922-924 (1988). [CrossRef] [PubMed]
  19. M. W. Austin, "GaAs/GaAlAs Curved rib waveguides," IEEE J Quant.Elect. 18, 795-800 (1982). [CrossRef]
  20. T. Yamamoto and M. Koshiba, "Numerical analysis of curvature loss in optical waveguides by the finite-element method," J. Lightwave Technol. 11, 1579-1583 (1993). [CrossRef]
  21. S. Kim and A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightwave Technol. 14, 2085-2092 (1996). [CrossRef]
  22. E. F. Kuester and D. C. Chang, "Surface-wave radiation loss form curved dielectric slabs and fibers," IEEE J. Quant. Elect. 11, 903-907 (1975). [CrossRef]
  23. B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner and J. R. Krenn, "Dielectric stripes on gold as surface plasmon waveguides", Appl. Phys. Lett. 88, 094104 (2006). [CrossRef]
  24. J.-C. Weeber, M. U. González, A.-L. Baudrion and A. Dereux, "Surface plasmon routing along right angle bent metal strips," Appl. Phys. Lett. 87, 221101 (2005). [CrossRef]
  25. A. Degiron and D. Smith, "Numerical simulations of long-range plasmons", Opt. Express 141611-1625 (2006) http://www.opticsexpress.org/abstract.cfm?id=88076 [CrossRef] [PubMed]

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