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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 19 — Sep. 18, 2006
  • pp: 8779–8784
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Nano plasmon polariton modes of a wedge cross section metal waveguide

Eyal Feigenbaum and Meir Orenstein  »View Author Affiliations


Optics Express, Vol. 14, Issue 19, pp. 8779-8784 (2006)
http://dx.doi.org/10.1364/OE.14.008779


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Abstract

Optical plasmon-polariton modes confined in both transverse dimensions to significantly less than a wavelength are exhibited in open waveguides structured as sharp metal wedges. The analysis reveals two distinctive modes corresponding to a localized mode on the wedge point and surface mode propagation on the abruptly bent interface. These predictions are accompanied by unique field distributions and dispersion characteristics.

© 2006 Optical Society of America

1. Introduction

Surface Plasmon-Polariton (SPP) is a guided wave confined in 1D to a sub-wavelength cross-section, though infinite in the other transverse dimension [1

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003). [CrossRef] [PubMed]

]. Confinement in both the transverse dimensions is achievable using the plasmonic slow waves and specific topologies, analogous to “standard” dielectric waveguides – namely metallic nano-wires, which were discussed elsewhere [2

2. P. Berini, “Plasmon-polariton modes guided by a metal film of a finite width bounded by different dielectrics,” Opt. Express 7, 329 (2000). [CrossRef] [PubMed]

, 3

3. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” J. Nonlinear Physics and Materials 11, 65 (2002). [CrossRef]

]. However the surface guiding of SPP allows also for open contour topologies, relaxing the waveguide transversal dimension constraint.

Full fledged analytic solutions for optical modes of metallic wedge waveguides were never reported, and the most detailed analysis employed a scalar wave-equation assuming electrostatic regime for sharp [4

4. L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a Dielectric Wedge,” Phys. Rev. B 6, 3810 (1972). [CrossRef]

] and smooth [5

5. A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B 14, 5526 (1976). [CrossRef]

,6

6. A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: Analytical solution,” Phys. Rev. B 32, 6045 (1985). [CrossRef]

] topologies. Although simplifying the analysis, this eliminates the mere issue of the full vectorial nature of the hybrid modes and their structural dispersion while discarding the retardation effects. The electrostatic solution obtained in [4

4. L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a Dielectric Wedge,” Phys. Rev. B 6, 3810 (1972). [CrossRef]

] is an asymptotical approximation to the one obtained here.

Recently, localized plasmons propagating on metal wedge were demonstrated using FDTD simulation and in experiments [7

7. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005). [CrossRef]

]. The analysis presented here provides understanding of the structure of the modes and their parametric dependence, among which proposing the existence of a critical wedge angle similar to the one reported for modes of different symmetry, and should assist in enhanced design of such novel nano-waveguides.

2. Analysis of wedge plasmonic modes

The analysis of the metal wedge waveguide surrounded by a dielectric is performed in cylindrical coordinates system {r, θ, z} assuming indefinitely long wedge, both in the radial and axial directions, as shown in Fig. 1(a). Though actual wedges are finite, the plasmonic fields are located around the wedge tip, which validate this assumption.

Fig. 1. Metal wedge waveguide surrounded by air. (a) schematics (b) EeHo mode dispersion relations for 360 gold wedge.

Since plasmonic modes are slow waves, they exponentially decay into both air and metal, in the radial direction as well as azimuthally. In cylindrical coordinate system the radial solutions of the wave equation are the modified Bessel functions of the second kind, denoted by K. Satisfying the boundary conditions at each interface point is impossible for a single Modified Bessel function at each medium, thus the solution is a series of K-functions.

The order of the K-functions may be chosen as real or imaginary, i.e. set of basis function of the form ~Kv(qr)exp(ivθ) or ~Kiv(qr)exp(-vθ). q is the radial momentum, and v denoted the K-function order. Taking the order to be imaginary is more suitable since it implies azimuthal hyperbolic dependence, rather than harmonic, which better describes the azimuthal decay of the plasmonic solutions away from the interface. Moreover, the {Kiv(qr)}v is a complete set (in contrary to {Kv(qr)}v) with basis functions being square integrable and having vanishing power at infinity (decaying faster than r-1).

Using this set for the solution at each media: F(r,θ)=∫dv fv Kiv(qr)exp(-vθ), the expansion coefficients, as a function of the order, fv, define the v-spectrum of the solution. The general v-spectrum is continuous on the indefinite interval, [0,∞). However the compactness of this expansion allows for a sufficient representation on a limited interval, [0,vmax]. The criteria for sufficiently high vmax were the fulfillment of the boundary conditions to 1% accuracy. As the order of a basis function is increased, the azimuthal decay is enhanced while the radial one is decreased. The confinement of the plasmonic fields around the wedge tip is translated to effectively bounding the v-spectrum. This could be inferred also from the Kontorovich-Lebedev transform [11

11. G. Z. Forristall and J. D. Ingram, “Evaluation of distributions useful in Kontorovich-Lebedev transform theory,” SIAM J. Math. Anal. 3, 561 (1972). [CrossRef]

], having K-functions with imaginary order as basis set.

Truncating the v-spectrum components above vmax may cause errors mostly at the interfaces. Thus, constructing and testing the solution, according to the boundary conditions, on the interface assures of the solution accuracy on the entire domain.

3. Modal characteristics

{(i)E˜M=E˜D(iii)θE˜M=f1θE˜Df2rrH˜D(ii)H˜M=H˜D(iv)θE˜M=f3rrE˜Df4θH˜D
(1)

The general solution is:

Eze=exp(iβz){dv{avKiv(qMr)cosh(v(θα))}θ[0,2α]dv{bvKiv(qDr)cosh(v(θαπ))}θ[2α,2π]
(2)

Where the radial momentum: qD,M=(β2-k02εD,M)0.5, k0 is the free-space wave number, and ε is the relative dielectric constant. For Hzo a,b,cosh are replaced by c,d,sinh in Eq. (2), v which denotes the K-function imaginary order, is real and continuous values. Since each of the basic field functions in both sides of the interface has a different r-dependence, the boundary conditions for all r values dictate a superposition of wave solutions. This is a major complication absent in the electrostatic analysis, for which one assumes substantial β-values resulting with similar basis functions on both sides of interface (i.e. qM=qD).

Approximating the v-spectrum integral as N-sized discrete series, projecting the four boundary conditions onto the base function of the dielectric {Kiv(qDr)}, using orthogonality of Kiv(qr)/r, and characteristics of the K-function integral, yields 4N algebraic equations:

vavcosh(vα)G(v,s;qMqD)=bscosh(s(α+π))π22ssinh(πs)
(3-a)
vcvsinh(vα)G(v,s;qMqD)=bssinh(s(α+π))π22ssinh(πs)
(3-b)
vavvsinh(vα)G(v,s;qMqD)=f1bssinh(s(α+π))π22sinh(πs)f2vdvsinh(v(α+π)H(v,s)
(3-c)
vcvvcosh(vα)G(v,s;qMqD)=f3vbvcosh(v(α+π))H(v,s)+f4dscosh(s(α+π))π22sinh(πs)
(3-d)

Expressions for G and H functions are given at the Appendix. The solution of this algebraic equation set is self-consistent for a vanishing determinant – resulting in the dispersion relation of the guided modes and the field v-spectrum, from which the field distribution is extracted.

Typically the v-spectrum peak value is observed around v=0 and v=3/α for the fields in the dielectric and metal respectively and the spectrum is rapidly decaying with v. Broadest span of v-spectrum is required for the solution on interfaces.

To elucidate the basic plasmonic mode characteristics, the metal losses are not taken into account in the followings. It can be done due to the relative smallness of loss in respect to metal dispersion and for short propagation distance, as is commonly done. We use the lossless Drude model for the metal permittivity, with plasma wavelength of λP=137nm. The same calculation scheme may support the case of complex permittivity, namely complex εM and β.

The distinct characteristics of the two dispersion branches are associated with the ‘dual role’ that the wedge plays. The one dimensional discontinuity formed at the wedge tip serves as indefinitely thin plasmonic waveguide “core”. On the other hand, a metal wedge is comprised of two metal surfaces, each serves as a single surface SPP mode waveguide, which are coupled at their merging point. These two distinct modes of propagation supported by the wedge configuration are manifested by the two dispersion curves. The propagation associated with wedge tip 1D discontinuity is relegated to the upper branch, while the lower curve is more of the two coupled surface modes.

Fig. 2. Electric field components (absolute value, A.U.) at pt. I (β=6k0;ω=0.74ωp).
Fig. 3. Electric field components (absolute value, A.U.) at pt. II (β=6k0;ω=0.69ωp).
Fig. 4. Tangential Pointing vector (Sz) (absolute value, A.U.) at: (a) pt. II, (b) pt. I
Fig. 5. Electric field components (absolute value, A.U.) at pt. III (β=8k0;ω=0.7ωp).

Equipped with the dispersion characteristics, sensitivity to structural parameters can be conjectured. As the surrounding dielectric constant increases, the dispersion curves are pulled down, as ωspp decreases. The cutoff frequency should increase with the wedge angle, as the lower dispersion branch may diverge less from the single surface SPP curve, due to reduced coupling between the modes near the wedge point. A cutoff angle is thus expected at each given wavelength, above which a solution does not exist. This resembles the cutoff angle that was observed in some specific FDTD simulations for different symmetry modes (EeHe) [7

7. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005). [CrossRef]

] and also for the complementary structure of V-grooved channel polaritons [10

10. D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29, 1069 (2004). [CrossRef] [PubMed]

]. It is noteworthy that the modes of the symmetry reported here are have higher effective index (i.e., enhanced plasmonic effect) and tighter confinement, than those reported in [7

7. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005). [CrossRef]

].

4. Conclusion

Appendix

The boundary conditions coefficients:

f1=εDεMqM2qD2;f2=ωμβ(εM+εD)qD2εM;f3=ωε0β(εM+εD)qD2;f4=qM2qD2;
(A-1)

Expressions for G and H functions:

G(ξ1,ξ2;τ)=π2cos(ξ1ln(τ))2ξ1sinh(πξ1)δ(ξ1ξ2)+π216[cosh(πξ1)cosh(πξ2)]1g(ξ1,ξ2;τ)
g(ξ1,ξ2;τ)=[τ21]τiξ12𝔽1(1iξ1+ξ22,1iξ1ξ22;2;1τ2)
H(ξ1,ξ2)=π22[cosh(πξ1)cosh(πξ2)]1;G(ξ,ξ;τ)=0;H(ξ,ξ)=0;
(A-2)

2F1 denotes the Hypergeometric function.

Acknowledgment

We acknowledge the Israeli Ministry of Science for support.

References and Links

1.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003). [CrossRef] [PubMed]

2.

P. Berini, “Plasmon-polariton modes guided by a metal film of a finite width bounded by different dielectrics,” Opt. Express 7, 329 (2000). [CrossRef] [PubMed]

3.

V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” J. Nonlinear Physics and Materials 11, 65 (2002). [CrossRef]

4.

L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a Dielectric Wedge,” Phys. Rev. B 6, 3810 (1972). [CrossRef]

5.

A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B 14, 5526 (1976). [CrossRef]

6.

A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: Analytical solution,” Phys. Rev. B 32, 6045 (1985). [CrossRef]

7.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005). [CrossRef]

8.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002). [CrossRef]

9.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005). [CrossRef] [PubMed]

10.

D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29, 1069 (2004). [CrossRef] [PubMed]

11.

G. Z. Forristall and J. D. Ingram, “Evaluation of distributions useful in Kontorovich-Lebedev transform theory,” SIAM J. Math. Anal. 3, 561 (1972). [CrossRef]

12.

B. Prade and J. Y. Vinet, “Guided optical waves in fibers with negative dielectric constant,” J. Light. Tech. 12, 6 (1994). [CrossRef]

13.

H. Reather, Surface plasmon (Springer, Berlin, 1988).

14.

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polariton on planar metallic waveguides,” Opt. Express 12, 4072 (2004). [CrossRef] [PubMed]

OCIS Codes
(200.4650) Optics in computing : Optical interconnects
(230.7370) Optical devices : Waveguides
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: June 20, 2006
Revised Manuscript: August 21, 2006
Manuscript Accepted: August 28, 2006
Published: September 18, 2006

Citation
Eyal Feigenbaum and Meir Orenstein, "Nano plasmon polariton modes of a wedge cross section metal waveguide," Opt. Express 14, 8779-8784 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8779


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References

  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824 (2003). [CrossRef] [PubMed]
  2. P. Berini, "Plasmon-polariton modes guided by a metal film of a finite width bounded by different dielectrics," Opt. Express 7, 329 (2000). [CrossRef] [PubMed]
  3. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes in metal nanowires and left-handed materials," J. Nonlinear Phys. Mater. 11, 65 (2002). [CrossRef]
  4. L. Dobrzynski, and A. A. Maradudin, "Electrostatic edge modes in a Dielectric Wedge," Phys. Rev. B 6, 3810 (1972). [CrossRef]
  5. A. Eguiluz, and A. A. Maradudin, "Electrostatic edge modes along a parabolic wedge," Phys. Rev. B 14, 5526 (1976). [CrossRef]
  6. A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, "Electrostatic edge modes of a hyperbolic dielectric wedge: Analytical solution," Phys. Rev. B 32, 6045 (1985). [CrossRef]
  7. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, "Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding," Appl. Phys. Lett. 87,061106 (2005). [CrossRef]
  8. I. V. Novikov, and A. A. Maradudin, "Channel polaritons," Phys. Rev. B 66, 035403 (2002). [CrossRef]
  9. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005). [CrossRef] [PubMed]
  10. D. F. P. Pile, and D. K. Gramotnev, "Channel plasmon-polariton in a triangular groove on a metal surface," Opt. Lett. 29, 1069 (2004). [CrossRef] [PubMed]
  11. G. Z. Forristall and J. D. Ingram, "Evaluation of distributions useful in Kontorovich-Lebedev transform theory," SIAM J. Math. Anal. 3, 561 (1972). [CrossRef]
  12. B. Prade and J. Y. Vinet, "Guided optical waves in fibers with negative dielectric constant," J. Lightwave Technol. 12, 6 (1994). [CrossRef]
  13. H. Reather, Surface plasmon (Springer, Berlin, 1988).
  14. M. P. Nezhad, K. Tetz, and Y. Fainman, "Gain assisted propagation of surface plasmon polariton on planar metallic waveguides," Opt. Express 12, 4072 (2004). [CrossRef] [PubMed]

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