## Figures of merit for surface plasmon waveguides

Optics Express, Vol. 14, Issue 26, pp. 13030-13042 (2006)

http://dx.doi.org/10.1364/OE.14.013030

Acrobat PDF (249 KB)

### Abstract

Three figures of merit are proposed as quality measures for surface plasmon waveguides. They are defined as benefit-to-cost ratios where the benefit is confinement and the cost is attenuation. Three different ways of measuring confinement are considered, leading to three figures of merit. One of the figures of merit is connected to the quality factor. The figures of merit were then used to assess and compare the wavelength response of three popular 1-D surface plasmon waveguides: the single metal-dielectric interface, the metal slab bounded by dielectric and the dielectric slab bounded by metal. Closed form expressions are given for the figures of merit of the single metal-dielectric interface.

© 2006 Optical Society of America

## 1. Introduction

2. W.L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. **8**, S87–S93 (2006). [CrossRef]

3. E. N. Economou, “Surface Plasmons in thin Films,” Phys. Rev. **182**, 539–554 (1969). [CrossRef]

4. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B , **33**, 5186–5201 (1986). [CrossRef]

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

6. J.-C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J.-P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B **60**, 9061–9068 (1999). [CrossRef]

7. 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]

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

2. W.L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. **8**, S87–S93 (2006). [CrossRef]

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

16. S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. **98**, 011101 (2005). [CrossRef]

17. R. Zia, M.D. Selker, P.B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A **21**, 2442–2446 (2006). [CrossRef]

7. 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]

18. L. Thylén and E. Berglind, “Nanophotonics and negative ε materials”, J. Zheijiang University: Science A **7**,
41–44 (2006). [CrossRef]

19. L. J. Sherry, S.-H. Chang, G. C. Schatz, and R. P. Van Duyne, “Localized Surface Plasmon Resonance Spectroscopy of Single Silver Nanocubes,” Nanoletters **5**, 2034–2038 (2005). [CrossRef]

*Q*/

*V*

_{eff}, where

*Q*is the quality factor and

*V*

_{eff}is an effective mode volume [20

20. S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quant. Elec. **38**, 257–267 (2006). [CrossRef]

21. S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express **14**, 1957–1964 (2006).
http://www.opticsexpress.org/abstract.cfm?URI=oe-14-5-1957. [CrossRef] [PubMed]

*Q*/

*V*,

*Q*

^{2}/

*V*and

*Q*/

*V*

^{1/2}where

*V*is the mode volume [22

22. D. Englund, I. Fushman, and J Vučković, “General Recipe for designing photonic crystal cavities,” Opt. Express **13**, 5961–5975 (2005).
http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-5961. [CrossRef] [PubMed]

## 2. Definition of the figures of merit

*e*

^{+jωt}time dependence is assumed and modes propagate in the +

*z*direction according to

*γ*

_{z}expands as

*γ*

_{z}=

*α*

_{z}+j

*β*

_{z}where

*α*

_{z}and

*β*

_{z}are the attenuation and phase constants, respectively. The complex effective index of the mode

*N*

_{eff}is given by

*N*

_{eff}=

*γ*

_{z}/

*β*

_{0}=

*α*

_{z}/

*β*

_{0}+j

*β*

_{z}/

*β*

_{0}=

*k*

_{eff}+j

*n*

_{eff}where

*β*

_{0}=

*2π*/

*λ*

_{0}is the phase constant of plane waves in free space and

*λ*

_{0}is the free-space wavelength. In Fig. 1,

*ε*

_{r,m}=-

*ε*

_{R}-j

*ε*

_{I}is the relative permittivity of the metal and

*ε*

_{r,1}=

*n*

_{1}its refractive index.

### 2.1 Definition of the M 1 1 D figure of merit

*δ*

_{w}decreases with confinement so the inverse of the mode width (1/

*δ*

_{w}) is adopted as the confinement measure. In keeping with convention,

*δ*

_{w}is determined by the 1/

*e*mode field magnitude decay points and the main transverse electric field component of the mode is used. Hence the figure of merit

*δ*

_{w}can be identified directly from the spatial distribution of |

*E*

_{y}(y)| associated with the SPP mode of interest. However, useful closed-form expressions for

*δ*

_{w}can be obtained for 1-D structures and so they are derived in the following subsections for the three example waveguides of Fig. 1.

17. R. Zia, M.D. Selker, P.B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A **21**, 2442–2446 (2006). [CrossRef]

### 2.2 Mode size and M 1 1 D of the single-interface SPP

*E*

_{y}(y)} associated with the SPP mode is superposed onto the structure, and two field points are identified:

*E*

_{y,1}(0

^{+}) and

*E*

_{y,m}(0

^{-}) which are the values of Re{

*E*

_{y}(y)} in the dielectric and metal regions, respectively, on either side of the interface. In the lossless case,

*δ*

_{D}and

*δ*

_{m}are the 1/

*e*field penetration depths into the dielectric and metal regions relative to

*E*

_{y,1}(0

^{+}) and

*E*

_{y,m}(0

^{-}), respectively. When losses are taken into account, |

*E*

_{y,1}(0

^{+})| and |

*E*

_{y,m}(0

^{-})| are used to define

*δ*

_{D}and

*δ*

_{m}.

*δ*

_{w}is the distance between the 1/

*e*field magnitude points relative to the

*global*maximum which is |

*E*

_{y,1}(0

^{+})|. A 1/

*e*point relative to |

*E*

_{y,1}(0

^{+})| may or may not exist in the metal depending on how far the operating wavelength is from the SPP resonance. The condition that determines whether a 1/

*e*point exists in the metal is obtained by inspecting the boundary condition applicable to the normal fields on either side of the discontinuity [23]:

*D*

_{y,m}(0

^{-})=

*D*

_{y,1}(0

^{+}) or

*E*

_{y,m}(0

^{-})=

*E*

_{y,1}(0

^{+})

*ε*

_{r,1}/

*ε*

_{r,m}. Now a 1/

*e*field magnitude point occurs in the metal if |

*E*

_{y,m}(0

^{-})|>|

*E*

_{y,1}(0

^{+})|/

*e*, which yields the following condition on the permittivities upon application of the boundary condition:

*e*point in the metal can easily be determined by assuming the usual exponential form for the field magnitude:

*α*

_{y,m}=1/

*δ*

_{m}, and from which the mode width

*δ*

_{w}is derived:

*δ*

_{D}and

*δ*

_{m}are easily derived from the constraint equations [23], which must hold independently within each region of the waveguide. The constraint equation in the dielectric is

*y*direction where

*α*

_{y,1}=Re{

*γ*

_{y,1}} so the field penetration depth into the dielectric

*δ*

_{D}=1/

*α*

_{y,1}is:

*ε*

_{r,m}, yielding

*y*where

*α*

_{y,m}=Re{

*γ*

_{y,m}} so the field penetration depth

*δ*

_{m}=1/

*α*

_{y,m}is:

*γ*

_{z}of the SPP [7

7. 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]

### 2.3 Mode size and M 1 1 D of the metal slab bounded by dielectric (IMI)

*t*shown in Fig. 1(b). Sketches of Re{

*E*

_{y}(y)} associated with the

*a*

_{b}and

*s*

_{b}modes [4

4. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B , **33**, 5186–5201 (1986). [CrossRef]

*δ*

_{D}is the 1/

*e*field magnitude penetration depth into the dielectric claddings given by Eq. (5). For the symmetric structure the field magnitude is largest and equal along both metal dielectric interfaces so the width of the

*a*

_{b}and

*s*

_{b}modes

*δ*

_{w}is taken straightforwardly as:

*δ*

_{w}does not depend on the magnitude of the mode fields in the metal nor on whether 1/

*e*points exist therein. The figure of merit is then readily computed using Eqs. (1), (5) and (9) with the propagation constant

*γ*

_{z}determined from an appropriately defined boundary-value problem representing the waveguide [4

4. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B , **33**, 5186–5201 (1986). [CrossRef]

**61**, 10484–10503 (2000). [CrossRef]

*α*

_{z}≪

*β*

_{z},

*ε*

_{I}≪

*ε*

_{R}) and away from resonance

*δ*

_{D}≫

*t*the above further simplifies to:

*δ*

_{w}is determined directly from the computed field distribution.

### 2.4 Mode size and M 1 1 D of the dielectric slab bounded by metal (MIM)

*t*shown in Fig. 1(c). A sketch of Re{

*E*

_{y}(y)} associated with the symmetric mode is superposed onto the structure and

*δ*

_{m}is the 1/

*e*field penetration depth into the metal claddings given by Eq. (6). The width of the mode

*δ*

_{w}is taken as:

*e*points exist therein. However, it is possible for 1/

*e*points to extend into the metal claddings, as in the single-interface case. The figure of merit is readily computed using Eqs. (1), (6) and (12) with the propagation constant

*γ*

_{z}determined from an appropriately defined boundary-value problem representing the waveguide [5

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

**61**, 10484–10503 (2000). [CrossRef]

*α*

_{z}≪

*β*

_{z},

*ε*

_{I}≪

*ε*

_{R}) and away from resonance where |

*ε*

_{r,m}|≥

*eε*

_{r,1}holds,

*δ*

_{w}is determined directly from the computed field distribution.

### 2.5 Role of the confinement factor

**61**, 10484–10503 (2000). [CrossRef]

17. R. Zia, M.D. Selker, P.B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A **21**, 2442–2446 (2006). [CrossRef]

*M*

_{1}as the field-based measure of confinement. The confinement factor, however, remains useful if one is interested in assessing the fraction of mode power carried within the metal region [7

**61**, 10484–10503 (2000). [CrossRef]

24. C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Longwavelength (λ≈8-11.5 µm) semiconductor lasers with waveguides based on surface plasmons,” Opt. Lett. **23**, 1366–1368 (1998). [CrossRef]

### 2.6 Definition of the M2 figure of merit

*β*

_{1}=

*ωn*

_{1}/

*c*

_{0}where

*c*

_{0}is the velocity of light in free-space. The distance then is given by

*β*

_{z}-

*β*

_{1}, or in terms of the effective index, by

*n*

_{eff}-

*n*

_{1}. Hence the figure of merit

*M*

_{2}, defined as a “benefit-to-cost” ratio, is:

*M*

_{2}is dimensionless. Advantageously, and contrary to

*M*

_{2}holds for 1- and 2-D waveguide structures.

*M*

_{2}is useful for comparing waveguides in applications where achieving a prescribed or large distance from the light line is important, such as in the design of Bragg gratings [25

25. S. Jetté-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides,” Opt. Express **13**, 4674–4682 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4674. [CrossRef] [PubMed]

### 2.7 Definition of the M3 figure of merit and connection to the quality factor

*λ*

_{g}=

*2π*/

*β*

_{z}generally decreases. Hence the figure of merit

*M*

_{3}, defined as a “benefit-to-cost” ratio, using the inverse guided wavelength 1/

*λ*

_{g}as the measure of confinement, is:

*M*

_{3}is dimensionless and

*n*

_{eff}>

*n*

_{1}. Advantageously, and contrary to

*M*

_{3}holds for 1- and 2-D waveguide structures.

*M*

_{3}is useful for comparing waveguides in applications where achieving a small guided wavelength is important, such as in the design of nano-resonators or nanoscale grating couplers.

*Q*of a resonant mode and

*M*

_{3}. The

*Q*of a lossy dispersive waveguide mode, resonating due to perfect reflectors placed at the input and output of the waveguide, is [23]:

*Q*=

*ω*(

*Energy stored in the mode*)/(

*Mode power dissipated*). This works out to:

*τ*

_{spp}is the SPP lifetime [26

26. M. Fukui, V. C. Y. So, and R. Normandin, “Lifetimes of Surface Plasmons in thin Silver Films”, Phys. Stat. Sol. (b) **91**, K61–K64 (1979). [CrossRef]

*v*

_{g}≅

*v*

_{p}=

*λ*

_{g}

*ω*/(

*2π*) where

*v*

_{p}is the phase velocity. Using

*v*

_{p}for

*v*

_{g}in Eq. (17), and comparing the resulting expression for

*Q*obtained via Eq. (16) with the definition for

*M*

_{3}given by Eq. (15), reveals that

*Q*=

*πM*

_{3}. Hence, although

*M*

_{3}and

*Q*are independent quality measures, they become the same to within a factor of

*π*when dispersion is negligible. Of course

*Q*itself can be used as a waveguide figure of merit but

*M*

_{3}is easier to compute.

## 3. Expressions for the single-interface SPP

*n*

_{eff}and

*k*

_{eff}[1, 2

2. W.L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. **8**, S87–S93 (2006). [CrossRef]

*α*

_{z}≪

*β*

_{z},

*ε*

_{I}≪

*ε*

_{R}) and away from resonance where |

*ε*

_{r,m}|≥

*eε*

_{r,1}holds, yields for

**61**, 10484–10503 (2000). [CrossRef]

*ω*

_{p}is the plasma frequency and

*τ*

_{D}the relaxation time, and substituting this into Eq. (20) for

*ε*

_{R}≫

*ε*

_{r,1}, yields the following relationship under the conditions

*ω*

^{2}≪

*ω*

^{2}≫1/

*ε*

_{r,1}). This implies that any increase in confinement, measured as the inverse mode size (1/

*δ*

_{w}), is perfectly balanced by an increase in attenuation as

*λ*

_{0}decreases. It is also noted that

*ε*

_{r,1}decreases, implying that the confinement (1/

*δ*

_{w}) decreases less rapidly than the attenuation. Finally, it is observed that maximizing

*ω*

_{p}

*τ*

_{D}.

*ε*

_{R}≫

*ε*

_{r,1}yields the following relationship under the conditions

*ω*

^{2}≪

*ω*

^{2}≫1/

*M*

_{2}is inversely proportional to the wavelength of operation in the Drude region. Hence the confinement, measured as the distance from the light line (

*β*

_{z}-

*β*

_{1}), increases more rapidly than the attenuation with decreasing

*λ*

_{0}. It is also noted that

*M*

_{2}increases with

*τ*

_{D}, which is consistent with

*ε*

_{R}≫

*ε*

_{r,1}, yields the following relationship under the conditions

*ω*

^{2}≪

*ω*

^{2}≫1/

*M*

_{3}is proportional to the wavelength of operation in the Drude region (neglecting dispersion in

*ε*

_{r,1}), a trend that is opposite to that observed for M

_{2}. This implies that the confinement, measured as the inverse guided wavelength (1/

*λ*

_{g}), increases more slowly than the attenuation with decreasing

*λ*

_{0}. It is also noted that

*M*

_{3}increases with

*τ*

_{D}and with decreasing

*ε*

_{r,1}which is consistent with

*Q*from Eq. (16) in like manner reveals that

*Q*=

*πM*

_{3}with

*M*

_{3}given by Eq. (26); hence these trends also hold for Q.

## 4. Wavelength response of 1-D structures

**61**, 10484–10503 (2000). [CrossRef]

_{2}were adopted as the materials and measured optical parameters (

*n*,

*k*) were used [27–30]. The

*n*and

*k*values were splined and interpolated at the desired wavelengths, and then used to compute the relative permittivities, which are plotted in Fig. 2 (a) for both materials. The plasma frequency and relaxation time of Ag were obtained by fitting Eq. (21) to the relative permittivity in the Drude region (following [31]), yielding

*ω*

_{p}=1.26×10

^{16}rad/s and

*τ*

_{D}=8.40×10

^{-15}s. The Drude region was taken as

*λ*

_{0}≥725 nm and

*λ*

_{0}=2000 nm was taken the upper limit of the wavelength range considered.

*ε*

_{R}=

*ε*

_{r,1}near

*λ*

_{0}=360 nm thus placing the single-interface SPP energy asymptote (resonance) near 3.4 eV. The asymptote is indeed observed in Fig. 2 (b), which plots its dispersion curve and the light line in SiO

_{2}. The SPP bend-back [32

32. E. T. Arakawa, M. W. Williams, R. N. Hamm, and R. H. Ritchie, “Effect of Damping on Surface Plasmon Dispersion,” Phys. Rev. Lett. **31**, 1127–1129 (1973
). [CrossRef]

*λ*

_{0}<360 nm, E>3.4eV) and links the non-radiative SPP to the radiative one on the left side of the light line.

*n*

_{eff}and

*k*

_{eff}, respectively, for the single-interface SPP, the

*a*

_{b}and

*s*

_{b}modes of the IMI, and the symmetric (henceforth denoted

*s*

_{b}) mode of the MIM.

*t*of the IMI structure was taken as 20 nm, since for this thickness, the

*s*

_{b}and

*a*

_{b}modes remain distinct from each other and from the single-interface SPP. From Fig. 2(d) it is noted that these modes are longer- and shorter-range, respectively, than the single-interface SPP.

*t*of the MIM structure was taken as 50 nm, ensuring that the mode maintains sub-wavelength confinement (

*δ*

_{w}<

*λ*

_{0}/(2

*n*

_{1})) over the wavelength range of analysis. From Fig. 2(d) it is noted that this mode is shorter-range than the single-interface SPP. The short wavelength limit to the range of analysis of each mode was taken as the wavelength where

*n*

_{eff}≅

*n*

_{1}, so the energy asymptote and the non-radiative portion of the bend-back curve are included in the analysis.

*δ*

_{w}computed via Eqs. (4), (9) and (12). At any given wavelength,

*δ*

_{w}of the

*s*

_{b}mode in the IMI is much greater than that of the others, while

*δ*

_{w}of the

*s*

_{b}mode in the MIM structure is much smaller.

*δ*

_{w}of the

*a*

_{b}mode in the IMI follows very closely

*δ*

_{w}of the single-interface SPP, despite the attenuation being greater by about one order of magnitude. For

*λ*

_{0}<445 nm, the inequality |

*ε*

_{r,m}|<

*eε*

_{r,1}is satisfied and

*δ*

_{w}of the

*s*

_{b}mode in the MIM begins to increase as shown and prescribed by Eq. (12). Also,

*δ*

_{w}of the single-interface SPP decreases less rapidly with

*λ*

_{0}below 445 nm as observed and prescribed by Eq. (4).

*ε*

_{R}≫

*ε*

_{r,1}) for

*λ*

_{0}>725 nm. For all structures, it is noted that

*λ*

_{0}>725 nm, implying that any change in confinement (1/

*δ*

_{w}) is balanced by a change in attenuation as

*λ*

_{0}is varied in this region.

*λ*

_{0}<600 nm as the modes head towards their energy asymptote, indicating that the attenuation is increasing more rapidly than the confinement (1/

*δ*

_{w}). At ant given wavelength,

*s*

_{b}mode in the IMI is larger than that of the single-interface SPP, while

*a*

_{b}mode in the same structure is smaller. Hence, the attenuation decreased more rapidly than the confinement (1/

*δ*

_{w}) in the case of the

*s*

_{b}mode as the thickness was reduced to

*t*=20 nm, while the attenuation increased more rapidly than the confinement (1/

*δ*

_{w}) in the case of the

*a*

_{b}mode. It is also noted that at a given wavelength the

*s*

_{b}mode in the MIM has a slightly better

*δ*

_{w}) as the thickness was reduced to

*t*=50 nm.

*M*

_{2}computed via Eq. (14). The approximation given by Eq. (23) (

*ε*

_{R}≫

*ε*

_{r,1}) for

*M*

_{2}of the single-interface SPP is also plotted, along with Eq. (24) for

*λ*

_{0}>725 nm. For all structures, it is noted that

*M*

_{2}increases with decreasing

*λ*

_{0}up to a maximum value from which

*M*

_{2}then decreases rapidly as the modes tend toward their asymptote. Hence an optimal wavelength of operation exists that maximizes

*M*

_{2}. Clearly, the confinement (

*β*

_{z}-

*β*

_{1}) increases more rapidly than the attenuation with decreasing

*λ*

_{0}on the long-wavelength side of the maximum, and vise versa on the short wavelength side. For the materials and thicknesses selected the wavelengths that maximize

*M*

_{2}are: 830 nm for the

*a*

_{b}mode in the IMI, 840 nm for the single-interface SPP and 850 nm for the

*s*

_{b}mode in the MIM; the

*s*

_{b}mode in the IMI has two peaks at 870 and 1120 nm. The observations made with respect to the thickness of the IMI and MIM in the case of

*M*

_{3}computed via Eq. (15). The approximation given by Eq. (25) (

*ε*

_{R}≫

*ε*

_{r,1}) for

*M*

_{3}of the single-interface SPP is also plotted, along with Eq. (26) for

*λ*

_{0}> 725 nm. Figure 3(e) shows the group velocity

*v*

_{g}of the modes computed via Eq. (18) and Fig. 3(f) plots the quality factor

*Q*computed via Eq. (16). As discussed in Section 2.7, when dispersion is low then

*M*

_{3}and

*Q*converge to essentially the same measure (

*Q*=

*πM*

_{3}). This is indeed observed in the Drude region for all cases, especially the less dispersive ones such as the

*s*

_{b}mode in the IMI and the singleinterface SPP. It is noteworthy that the

*s*

_{b}mode in the MIM does not follow the same trend with decreasing wavelength as the other modes, in that it exhibits a maximum in

*M*

_{3}and

*Q*near 840 nm. The other modes follow the same trend in that

*M*

_{3}and

*Q*increase linearly with

*λ*

_{0}. The observations made with respect to the thickness of the IMI in the case of

*s*

_{b}mode in the MIM has a significantly worse

*M*

_{3}and

*Q*than the single-interface SPP indicating that the attenuation increased more rapidly than the confinement (1/

*λ*

_{g}) as the thickness was reduced to

*t*=50 nm. Finally, it is noted that reasonably high Q values (10,000) are achievable with the

*s*

_{b}mode in the IMI.

*M*

_{2},

*M*

_{3}and

*Q*eventually tend to zero with decreasing

*λ*

_{0}, as the modes tend toward their energy asymptote. Clearly then, the attenuation increases more rapidly than the confinement in this region, regardless of how the confinement is measured. This is caused by the rapidly increasing fraction of mode power within the metal(s). Also, the modes are nearing the band edge of Ag (~310 nm), so the onset and increase of interband absorption prevents

*ε*

_{I}from becoming small while doing nothing to increase the confinement.

## 5. Summary and conclusions

*M*

_{2},

*M*

_{3}) were defined as benefit-to-cost ratios with the benefit being confinement and the cost being attenuation.

*δ*

_{w}), and is limited to 1-D waveguides.

*M*

_{2}and

*M*

_{3}are defined using the distance of the mode from the light line (

*β*

_{z}-

*β*

_{1}) and the inverse guided wavelength (1/

*λ*

_{g}) as their measures of confinement, respectively, and they are applicable to 1- and 2-D waveguides.

*M*

_{2}and

*M*

_{3}are useful for assessing and optimizing waveguides intended for application in, for example, Bragg gratings and resonators, respectively. The intimate connection between

*M*

_{3}and

*Q*(the unloaded quality factor) was also discussed.

*M*

_{2}increases with decreasing wavelength, and

*M*

_{3}and

*Q*decrease with decreasing wavelength.

*M*

_{2},

*M*

_{3}and

*Q*, were then computed for three popular 1-D SPP waveguides: the single-interface, the IMI and the MIM. The following conclusions are drawn from these responses: (i) the attenuation increases at a greater rate than the confinement (regardless of how it's measured) as a mode tends towards its energy asymptote; (ii) the attenuation of the

*s*

_{b}mode in the IMI drops more rapidly than its confinement (regardless of how it's measured) as the thickness of the metal film is reduced; (iii) the confinement measured as the inverse mode width and distance from the light line of the

*s*

_{b}mode in the MIM can increase more rapidly than the attenuation as the thickness of the dielectric film is reduced; (iv) the

*s*

_{b}and

*a*

_{b}modes in a thin IMI have large and small figures of merit and

*Q*factors, respectively; (v)

*Q*factors of about 10,000 are achievable for the

*s*

_{b}mode in a thin IMI in the Drude region; (vi)

*M*

_{2}and

*M*

_{3}can exhibit a peak versus wavelength indicating that a preferred wavelength of operation exists (with respect to

*M*

_{2}and

*M*

_{3}); (vii)

## References and links

1. | H. Raether, |

2. | W.L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. |

3. | E. N. Economou, “Surface Plasmons in thin Films,” Phys. Rev. |

4. | J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B , |

5. | J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization” Phys. Rev. B |

6. | J.-C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J.-P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B |

7. | P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B |

8. | R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmonpolariton waves supported by a thin metal film of finite width,” Opt. Lett. |

9. | B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, and F.R. Aussenegg, “Surface plasmon propagation in microscale metal stripes,” Appl. Phys. Lett. , |

10. | J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B |

11. | R. Nikolajsen, K. Leosson, I. Salakhutdinov, and S.I. Bozhevolnyi, “Polymer-based surface-plasmonpolariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. |

12. | I. V. Novikov and A.A. Maradudin, “Channel polaritons,” Phys. Rev. B |

13. | S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nat. |

14. | D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. , |

15. | W.L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. |

16. | S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

17. | R. Zia, M.D. Selker, P.B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A |

18. | L. Thylén and E. Berglind, “Nanophotonics and negative ε materials”, J. Zheijiang University: Science A |

19. | L. J. Sherry, S.-H. Chang, G. C. Schatz, and R. P. Van Duyne, “Localized Surface Plasmon Resonance Spectroscopy of Single Silver Nanocubes,” Nanoletters |

20. | S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quant. Elec. |

21. | S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express |

22. | D. Englund, I. Fushman, and J Vučković, “General Recipe for designing photonic crystal cavities,” Opt. Express |

23. | R.E. Collin, |

24. | C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Longwavelength (λ≈8-11.5 µm) semiconductor lasers with waveguides based on surface plasmons,” Opt. Lett. |

25. | S. Jetté-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides,” Opt. Express |

26. | M. Fukui, V. C. Y. So, and R. Normandin, “Lifetimes of Surface Plasmons in thin Silver Films”, Phys. Stat. Sol. (b) |

27. | E.D. Palik (Editor), |

28. | P. Winsemius, F. F. van Kampen, H. P. Lengkeek, and C. G. van Went, “Temperature dependence of the optical properties of Au, Ag and Cu,” J. Phys. F: Metal Phys. |

29. | G. Leveque, C. G. Olson, and D. W. Lynch, “Reflectance spectra and dielectric functions for Ag in the region of interband transitions,” Phys. Rev. B |

30. | B. Brixner, “Refractive-index interpolation for fused silica,” J. Opt. Soc. Am. |

31. | D. J. Nash and J. R. Sambles, “Surface plasmon-polariton study of the optical dielectric function of silver,” J. Mod. Opt. |

32. | E. T. Arakawa, M. W. Williams, R. N. Hamm, and R. H. Ritchie, “Effect of Damping on Surface Plasmon Dispersion,” Phys. Rev. Lett. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: October 2, 2006

Manuscript Accepted: December 11, 2006

Published: December 22, 2006

**Citation**

Pierre Berini, "Figures of merit for surface plasmon waveguides," Opt. Express **14**, 13030-13042 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-13030

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### References

- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).
- W. L. Barnes, "Surface plasmon-polariton length scales: a route to sub-wavelength optics," J. Opt. A: Pure Appl. Opt. 8, S87-S93 (2006). [CrossRef]
- E. N. Economou, "Surface Plasmons in thin Films," Phys. Rev. 182, 539-554 (1969). [CrossRef]
- J. J. Burke, G. I. Stegeman and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986). [CrossRef]
- 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]
- J.-C. Weeber, A. Dereux, C. Girard, J. R. Krenn and J.-P. Goudonnet, "Plasmon polaritons of metallic nanowires for controlling submicron propagation of light," Phys. Rev. B 60, 9061-9068 (1999). [CrossRef]
- 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]
- 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]
- B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner and F. R. Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett., 7951-53 (2001). [CrossRef]
- J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, "Near-field observation of surface plasmon polariton propagation on thin metal stripes," Phys. Rev. B 64, 045411 (2001). [CrossRef]
- 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]
- I. V. Novikov and A. A. Maradudin, "Channel polaritons," Phys. Rev. B 66, 035403 (2002). [CrossRef]
- S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006). [CrossRef]
- D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett., 87261114 (2005). [CrossRef]
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- S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 011101 (2005). [CrossRef]
- R. Zia, M. D. Selker, P. B. Catrysse and M. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2006). [CrossRef]
- L. Thylén and E. Berglind, "Nanophotonics and negative ε materials," J. Zheijiang University: Science A 7, 41-44 (2006). [CrossRef]
- L. J. Sherry, S.-H. Chang, G. C. Schatz and R. P. Van Duyne, "Localized surface plasmon resonance Spectroscopy of single silver nanocubes," Nanoletters 5, 2034-2038 (2005). [CrossRef]
- S. A. Maier, "Effective mode volume of nanoscale plasmon cavities," Opt. Quantum Electron. 38, 257-267 (2006). [CrossRef]
- S. A. Maier, "Plasmonic field enhancement and SERS in the effective mode volume picture," Opt. Express 14, 1957-1964 (2006). [CrossRef] [PubMed]
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- C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson and A. Y. Cho, "Long-wavelength (λ ≈ 8 - 11.5 µm) semiconductor lasers with waveguides based on surface plasmons," Opt. Lett. 23, 1366-1368 (1998). [CrossRef]
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- M. Fukui, V. C. Y. So, and R. Normandin, "Lifetimes of Surface Plasmons in thin Silver Films," Phys. Stat. Sol.(b) 91,K61-K64 (1979). [CrossRef]
- E. D. Palik, ed., Handbook of Optical Constants of Solids, (Academic Press, Orlando, Florida, 1985).
- P. Winsemius, F. F. van Kampen, H. P. Lengkeek and C. G. van Went, "Temperature dependence of the optical properties of Au, Ag and Cu," J. Phys. F: Met. Phys. 6, 1583-1606 (1976). [CrossRef]
- G. Leveque, C. G. Olson and D. W. Lynch, "Reflectance spectra and dielectric functions for Ag in the region of interband transitions," Phys. Rev. B 27, 4654-4660 (1983). [CrossRef]
- B. Brixner, "Refractive-index interpolation for fused silica," J. Opt. Soc. Am. 57, 674-676 (1967). [CrossRef]
- D. J. Nash and J. R. Sambles, "Surface plasmon-polariton study of the optical dielectric function of silver," J. Mod. Opt. 43, 81-91 (1996).
- E. T. Arakawa, M. W. Williams, R. N. Hamm and R. H. Ritchie, "Effect of damping on Surface Plasmon Dispersion," Phys. Rev. Lett. 31, 1127-1129 (1973). [CrossRef]

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