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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 20 — Oct. 2, 2006
  • pp: 9467–9476
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Effective-index modeling of channel plasmon polaritons

Sergey I. Bozhevolnyi  »View Author Affiliations


Optics Express, Vol. 14, Issue 20, pp. 9467-9476 (2006)
http://dx.doi.org/10.1364/OE.14.009467


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Abstract

Effective-index approach is applied for modeling of channel plasmon polaritons (CPPs) propagating in rectangular grooves (trenches) and triangular (V-shaped) grooves in gold, accounting for the main features of CPP guiding and elucidating its underlying physics. The effective indexes of CPP modes along with the corresponding propagation lengths are calculated for different configurations and wavelengths while varying the groove depth. The results obtained allow one to identify the parameter range for realizing the single-mode CPP guiding featuring subwavelength confinement and moderate propagation loss at telecom wavelengths.

© 2006 Optical Society of America

1. Introduction

Surface plasmon polaritons (SPPs) having by nature subwavelength spatial periods and transverse dimensions perpendicular to the metal surface [1

1. H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

] offer the possibility of realizing subwavelength waveguiding and ultra-compact photonic components [2

2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]

]. The main issue in this context is to strongly confine the SPP field in the surface plane (perpendicular to the SPP propagation direction). Many different approaches have been investigated, including photonic band-gap structures [3

3. S. I. Bozhevolnyi, V. S. Volkov, K. Leosson, and A. Boltasseva, “Bend loss in surface plasmon polariton band-gap structures,” Appl. Phys. Lett. 79, 1076–1078 (2001). [CrossRef]

], metal stripes in a dielectric environment [4

4. B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidji, A. Leitner, F. R. Aussenegg, and J. C. Weeber, “Surface plasmon propagation in microscale metal stripes,” Appl. Phys. Lett. 79, 51 (2001). [CrossRef]

], metal particle waveguides [5

5. S. A. Maier, M. L. Brongersma, P. G. Kirk, S. Meltzer, A. A. G. Reguicha, and H. A. Atwater, “Plasmons — a route to nanoscale optical devices,” Adv. Mater. 13, 1501–1505 (2001). [CrossRef]

], and the use of polymer stripes placed on top of a metal surface [6

6. C. Reinhardt, S. Passinger, B. N. Chichkov, C. Marquart, I. P. Radko, and S. I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons, ”Opt. Lett. 31, 1307–1309 (2006). [CrossRef] [PubMed]

]. However, simultaneous realization of strong confinement and relatively low propagation loss (acceptable for practical purposes) has long been inaccessible. Channel SPP modes, or channel plasmon polaritons (CPPs) [7

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

], where the electromagnetic radiation is concentrated at the bottom of V-shaped grooves milled in a metal film, have been first predicted [8

8. D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85, 6323–6325 (2004). [CrossRef]

] and then experimentally shown [9

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]

] to exhibit useful subwavelength confinement and moderate propagation loss. Recently, we have realized various CPP-based subwavelength waveguide components, including Mach-Zehnder interferometers and waveguide-ring resonators [10

10. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonantors,” Nature 440, 508–511 (2006). [CrossRef] [PubMed]

], and demonstrated that the CPP guides can be used for large-angle bending and splitting of radiation [10

10. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonantors,” Nature 440, 508–511 (2006). [CrossRef] [PubMed]

, 11

11. V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Compact gradual bends for channel plasmon polaritons,” Opt. Express 14, 4494–4503 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-10-4494 [CrossRef] [PubMed]

], thereby enabling the realization of ultracompact plasmonic components and paving the way for a new class of integrated optical circuits. We have also developed a simple and efficient modeling approach [9

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]

] based on the effective-index method (EIM) [12

12. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977). http://www.opticsinfobase.org/abstract.cfm?URI=ao-16-1-113 [CrossRef] [PubMed]

] that has been used for designing the investigated CPP waveguide components [9

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]

11

11. V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Compact gradual bends for channel plasmon polaritons,” Opt. Express 14, 4494–4503 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-10-4494 [CrossRef] [PubMed]

].

In this work, the EIM is applied for modeling of CPPs propagating in rectangular grooves (trenches) and triangular (V-shaped) grooves in gold. First, the validity of the EIM for modeling of SPP modes supported by two-dimensional (2D) metal waveguide structures is investigated by comparing with the finite-difference (FD) simulations conducted for trenches in metal films embedded in dielectric media [13

13. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13, 6645–6650 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6645 [CrossRef] [PubMed]

]. The EIM is then used for calculating the effective indexes of CPP modes at several wavelengths along with the corresponding propagation lengths for trenches and V-grooves, while varying the groove depth. The results obtained allow one to identify the parameter range for realizing the single-mode CPP guiding featuring subwavelength confinement and moderate propagation loss (i.e., exhibiting the propagation length of the order of 100 µm) at telecom wavelengths.

2. Comparison of EIM and FD simulations

The main attractive feature of the EIM is that it allows one to combine the results of modeling conducted for one-dimensional (1D) waveguiding configurations so that the characteristics of 2D (channel) waveguides can be described [12

12. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977). http://www.opticsinfobase.org/abstract.cfm?URI=ao-16-1-113 [CrossRef] [PubMed]

]. For a rectangular-core waveguide, one should first analyze a planar (slab) waveguide obtained by letting one dimension of the original 2D waveguide approach infinity. Thus obtained mode propagation constant(s) is then used to define the corresponding effective dielectric index(s) assigned to the core index(s) of another 1D waveguide considered in the perpendicular direction. The propagation constant(s) of this second waveguide are taken to represent those of the original rectangular waveguide [12

12. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977). http://www.opticsinfobase.org/abstract.cfm?URI=ao-16-1-113 [CrossRef] [PubMed]

]. The EIM is one of the most extensively used approaches for modeling of dielectric channel waveguides in integrated optics, where its validity has been scrutinized and various corrections have been introduced [14

14. K. S. Chiang, C. H. Kwan, and K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes of rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999). http://www.opticsinfobase.org/abstract.cfm?URI=JLT-17-4-716 [CrossRef]

]. The EIM has also been successfully applied to modeling of weakly guided SPP modes supported by thin metal stripes of finite width [15

15. 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). http://www.opticsinfobase.org/abstract.cfm?URI=JLT-23-1-413 [CrossRef]

, 16

16. R. Zia, A. Chandran, and M. L. Brongersma, “Dielectric waveguide model for guided surface polaritons,” Opt. Lett. 30, 1473–1475 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-12-1473 [CrossRef]

]. Here, three 2D metal waveguide configurations that are able of supporting SPP modes with subwavelength confinement (Fig.1) are analyzed by making use of the EIM.

Fig. 1. Schematic of the surface plasmon waveguides under consideration.

Fig. 2. The effective indexes of SPP modes and their propagation lengths for the configuration shown in Fig. 1(a) as a function of the metal film thickness. The results of simulations with the FD method [13] are also shown for comparison.

It is seen that the agreement between the EIM and FM calculations is very good even when the SPP modes are close to cutoff where the EIM approximation usually becomes rather poor [12

12. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977). http://www.opticsinfobase.org/abstract.cfm?URI=ao-16-1-113 [CrossRef] [PubMed]

, 14

14. K. S. Chiang, C. H. Kwan, and K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes of rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999). http://www.opticsinfobase.org/abstract.cfm?URI=JLT-17-4-716 [CrossRef]

]. A possible reason for this encouraging observation might be the fact that the considered SPP modes are tightly confined to the gap in the metal film, decreasing thereby the influence of corner regions (i.e. the regions situated above and below the metal film but away from the gap) contributing to the EIM errors [12

12. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977). http://www.opticsinfobase.org/abstract.cfm?URI=ao-16-1-113 [CrossRef] [PubMed]

, 14

14. K. S. Chiang, C. H. Kwan, and K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes of rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999). http://www.opticsinfobase.org/abstract.cfm?URI=JLT-17-4-716 [CrossRef]

]. It is therefore expected that the SPP modes supported by trenches cut into a metal, i.e. in the configuration shown in Fig. 1(b), can be adequately described using the EIM.

3. Trench CPP modes

The first step in the EIM applied to the trench [Fig. 1(b)] or V-groove [Fig. 1(c)] configuration is the same as before, i.e., one should analyze the SPP guiding in the air gap between two metal surfaces. The following simulations are concerned with grooves in gold and several wavelengths chosen in the interval between visible and telecom wavelengths. The following dielectric constants of gold were used in the simulations: n=0.166+3.15i (λ=0.653 µm), 0.174+4.86i (0.775 µm), 0.272+7.07i (1.033 µm) and 0.55+11.5i (1.55 µm) [18

18. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

]. The corresponding characteristics of the fundamental gap SPP mode are shown in Fig. 3.

Fig. 3. The gap SPP mode effective index and its propagation length as a function of the width w of air gap in gold for several light wavelengths. The insert shows the gap configuration and the SPP magnetic field orientation along with the example of field distribution across a 1-µm-wide gap (λ=1.55 µm).

R(u)exp(u)(2u+exp(u)exp(u))(1+exp(u))2.
(1)

This function reaches its maximum value, which is by 20% larger than that at w→∞, for u≈2.4, resulting in the following expression for the optimum width: w opt≈0.4λ|Re(ε m)+1|0.5, a value which is quite close to that found above from numerical simulations.

The CPP mode supported by the trench [Fig. 1(b)] can now be described by considering TE-modes in a three-layer structure, in which a dielectric core having the effective index of the corresponding gap SPP is sandwiched between the air cladding and the gold substrate. This is a relatively straightforward procedure, and the results for two gap widths and the telecom wavelength of 1.55 µm are shown in Fig. 4. It is readily seen that the subwavelength lateral confinement of the trench CPPs can be achieved simultaneously with relatively long propagation distances, a feature which is similar to that found for V-groove CPPs [8

8. D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85, 6323–6325 (2004). [CrossRef]

10

10. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonantors,” Nature 440, 508–511 (2006). [CrossRef] [PubMed]

].

Fig. 4. The trench CPP mode effective index and its propagation length as a function of the trench depth d in gold for two trench widths w (λ=1.55 µm) in the parameter ranges corresponding to the single-mode waveguiding regime. The insert shows the trench configuration and the electric field orientation.

Fig. 5. The trench CPP field magnitude distributions for 500-nm-wide trenches of different depths, including the close to cutoff depth of 1.5 µm (Fig. 4).

The circumstance that the trench CPP modes are obtained from the consideration of a strongly asymmetric waveguide structure (because of the metal substrate) can be used to deduce a simple relation for the mode cutoff condition. Utilizing the normalized waveguide parameters and setting the asymmetry parameter to infinity, one obtains the following cutoff condition for the mth-mode [21

21. H. Kogelnik and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Appl. Opt. 13, 1857- (1974). http://www.opticsinfobase.org/abstract.cfm?URI=ao-13-8-1857 [CrossRef] [PubMed]

]:

V(w,d)=2πλdNeff2(w)1=π2+mπ,m=0,1,2,
(2)

Fig. 6. The trench parameter range ensuring the single-mode CPP guiding at the wavelength range of 0.775-1.55 µm evaluated using Eq. (2) and Fig. 3.

Finally, it should be borne in mind that, as the CPP mode approaches the cutoff and the CPP field becomes progressively larger at the trench (or groove) edges, one should expect the occurrence of CPP coupling to other SPP modes: plane (conventional) SPPs, propagating away from the groove, edge [22

22. A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B 24, 5703–5712 (1981). [CrossRef]

] and coupled wedge [23

23. D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, “Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap,” J. Appl. Phys. 100, 013101 (2006). [CrossRef]

] SPP modes. Such a coupling might become efficient, especially when the propagation constants (all being larger than that of in air) would match, resulting in additional loss. On the other hand, these modes have rather different polarization properties (e.g., the main electric field components of CPP and SPP modes are orthogonal), a circumstance that would decrease the coupling strength.

4. V-groove CPP modes

Let us now consider a V-groove whose width is monotonously decreasing with the increase of the depth [Fig. 1(c)]. Since light tends to be confined in regions with higher refractive indexes, sufficiently deep grooves support CPP modes (formed by gap SPPs) that are confined to the groove bottom, where the gap SPP index is at maximum (Fig. 3). Using the EIM, one can find the V-groove CPP modes by analyzing a one-dimensional layered (in depth) guiding structure, in which the top layer of air and the bottom layer of metal abut a stack of layers having refractive indexes determined by the layer depth: an index is equal to the gap SPP effective index N eff for a gap width corresponding to the groove width at this depth [9

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]

]. The corresponding results obtained for the telecom wavelength of 1.55 µm are shown in Fig. 7.

Fig. 7. The effective indexes of fundamental CPP modes at 1.55 µm and their propagation lengths as a function of the groove depth d for different groove angles θ. The insert shows the groove configuration and dominant orientation of the CPP electric field. The plane SPP effective index is also indicated.

The wavelength dispersion of CPP guiding in V-grooves along with the occurrence of higher-order CPP modes is illustrated with the results of simulations conducted for the groove angle of 250 at several wavelengths (Fig. 8). It is readily seen that, whereas the single-mode CPP guiding is possible in the wavelength range of 1.033–1.55 µm (for the groove depths d in the range of ~0.9–1.9 µm), one cannot expect to achieve the single-mode V-groove guiding in the whole range of 0.775–1.55 µm unlike the CPP guiding in trenches (Fig. 6). Another interesting feature, which is seen also in Fig. 7 and in Fig. 2 of Ref. 9 (but not commented upon), is that the CPP propagation length is first decreases with the decrease of the groove depth, starting to rapidly increase only when the depth becomes close to the cutoff value. This feature is also counterintuitive, because shorter propagation length is associated with better mode confinement that one would expect to be better for deeper grooves. Note that this feature is not found for trench CPP modes whose propagation length increases monotonously with the decrease of the trench depth (Fig. 4).

Fig. 8. (a) The effective indexes of V-groove CPP modes and (b) the corresponding CPP propagation lengths as a function of the 250-angle groove depth d for several light wavelengths.

One can venture the following physical explanation of this effect. As the groove depth becomes smaller, the groove width at the groove top becomes smaller as well, increasing thereby the index contrast between the upper (air) half-space and the upper part of the groove. This increase in the index contrast forces the mode field to be closer to the groove bottom, where the damping is larger. Note that, at the same time, there is a tendency of increasing the part of the mode field propagating above the groove (in air) and thereby decreasing the damping. It is only due to a drastic increase in the damping (of gap SPPs) for very small gap widths (Fig. 3) that the first tendency appears stronger than the second one. Such an explanation is consistent with the fact that trench CPP propagation length increases monotonously with the decrease of the trench depth, since a trench has a constant width and hence introduces a constant damping. The evolution in the CPP field distributions with the groove depth is illustrated in Fig. 9 for the fundamental and second modes at 1.033 µm.

Fig. 9. The CPP amplitude distributions for 250-angle V-grooves of different depths, including the close to cutoff (for TE1 mode) depth of 2.14 µm (Fig. 8).

5. Conclusion

In summary, the EIM has been applied for modeling of CPPs propagating in rectangular grooves (trenches) and triangular (V-shaped) grooves in gold. The validity of the EIM for modeling of SPP modes supported by 2D metal waveguide structures was investigated by comparing with the finite-difference simulations conducted for trenches in metal films embedded in dielectric media [13

13. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13, 6645–6650 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6645 [CrossRef] [PubMed]

]. The EIM was then used to determine the trench and V-groove CPP characteristics at several wavelengths for trenches and V-grooves, accounting for the main features of CPP guiding and elucidating its underlying physics. When considering the SPP guiding in the dielectric gap between two metal surfaces (this being the first step in the EIM applied to grooves), it has been found that that gap SPP modes with better confinement can exhibit longer propagation lengths. This remarkable feature seems to be contradicting the well-known tradeoff between the SPP mode confinement and its propagation length [2

2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]

, 4

4. B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidji, A. Leitner, F. R. Aussenegg, and J. C. Weeber, “Surface plasmon propagation in microscale metal stripes,” Appl. Phys. Lett. 79, 51 (2001). [CrossRef]

, 5

5. S. A. Maier, M. L. Brongersma, P. G. Kirk, S. Meltzer, A. A. G. Reguicha, and H. A. Atwater, “Plasmons — a route to nanoscale optical devices,” Adv. Mater. 13, 1501–1505 (2001). [CrossRef]

]. It has been explained by the fact that, with the decrease of the gap width, the electric field of the gap SPP approaches quickly to the electrostatic (capacitor) mode, which is constant across the gap, increasing thereby the fractional electric field energy concentrated in the gap (before it starts to decrease with the field being squeezed from the gap into the metal). It has been further conjectured that the ability of subwavelength (trench and V-groove) CPPs to propagate with moderate propagation loss is a direct consequence of this property of gap SPPs to most efficiently fill the available dielectric space (gap) between the metal walls.

The trench CPP guiding has been discussed (for the first time to our knowledge) in detail, and a simple relation for the cutoff condition of CPP modes has been established. It has been found that the trench CPP guiding in the single-mode regime can be achieved in the whole range of 0.775–1.55 µm, a circumstance that can be advantageously exploited for realization of nonlinear electromagnetic interactions. The V-groove CPP guiding has been considered for different groove angles and light wavelengths, highlighting the main features and discussing the physical phenomena involved. The results obtained allow one to identify the parameter range for realizing the single-mode CPP guiding featuring subwavelength confinement and moderate propagation loss (i.e., exhibiting the propagation length of the order of 100 µm) at telecom wavelengths. Finally, the EIM validity when describing V-groove CPPs was discussed emphasizing the need for further detailed investigations.

Acknowledgments

The author is grateful to A. Bouhelier, A. Dereux, J.-C. Weeber and R. Zia for fruitful discussions during his leave of absence at the University of Burgundy (Dijon, France) where this work was largely prepared, and acknowledges the support by the European Network of Excellence, PLASMO-NANO-DEVICES (FP6-2002-IST-1-507879) and by the Danish Research Agency (contract No. 272-05-0450).

References and links

1.

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

2.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]

3.

S. I. Bozhevolnyi, V. S. Volkov, K. Leosson, and A. Boltasseva, “Bend loss in surface plasmon polariton band-gap structures,” Appl. Phys. Lett. 79, 1076–1078 (2001). [CrossRef]

4.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidji, A. Leitner, F. R. Aussenegg, and J. C. Weeber, “Surface plasmon propagation in microscale metal stripes,” Appl. Phys. Lett. 79, 51 (2001). [CrossRef]

5.

S. A. Maier, M. L. Brongersma, P. G. Kirk, S. Meltzer, A. A. G. Reguicha, and H. A. Atwater, “Plasmons — a route to nanoscale optical devices,” Adv. Mater. 13, 1501–1505 (2001). [CrossRef]

6.

C. Reinhardt, S. Passinger, B. N. Chichkov, C. Marquart, I. P. Radko, and S. I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons, ”Opt. Lett. 31, 1307–1309 (2006). [CrossRef] [PubMed]

7.

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

8.

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85, 6323–6325 (2004). [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.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonantors,” Nature 440, 508–511 (2006). [CrossRef] [PubMed]

11.

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Compact gradual bends for channel plasmon polaritons,” Opt. Express 14, 4494–4503 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-10-4494 [CrossRef] [PubMed]

12.

G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977). http://www.opticsinfobase.org/abstract.cfm?URI=ao-16-1-113 [CrossRef] [PubMed]

13.

L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13, 6645–6650 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6645 [CrossRef] [PubMed]

14.

K. S. Chiang, C. H. Kwan, and K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes of rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999). http://www.opticsinfobase.org/abstract.cfm?URI=JLT-17-4-716 [CrossRef]

15.

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). http://www.opticsinfobase.org/abstract.cfm?URI=JLT-23-1-413 [CrossRef]

16.

R. Zia, A. Chandran, and M. L. Brongersma, “Dielectric waveguide model for guided surface polaritons,” Opt. Lett. 30, 1473–1475 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-12-1473 [CrossRef]

17.

I. P. Kaminow, W. L. Mammel, and H. P. Weber, “Metal-clad optical waveguides: analytical and experimental study,” Appl. Opt. 13, 396–405 (1974). http://www.opticsinfobase.org/abstract.cfm?URI=ao-13-2-396 [CrossRef] [PubMed]

18.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

19.

K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82, 1158–1160 (2003). [CrossRef]

20.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]

21.

H. Kogelnik and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Appl. Opt. 13, 1857- (1974). http://www.opticsinfobase.org/abstract.cfm?URI=ao-13-8-1857 [CrossRef] [PubMed]

22.

A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B 24, 5703–5712 (1981). [CrossRef]

23.

D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, “Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap,” J. Appl. Phys. 100, 013101 (2006). [CrossRef]

OCIS Codes
(230.7380) Optical devices : Waveguides, channeled
(240.6680) Optics at surfaces : Surface plasmons
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Optics at Surfaces

History
Original Manuscript: July 21, 2006
Manuscript Accepted: September 7, 2006
Published: October 2, 2006

Citation
Sergey I. Bozhevolnyi, "Effective-index modeling of channel plasmon polaritons," Opt. Express 14, 9467-9476 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9467


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References

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