## 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

It has long been known that a metal plane can support a surface plasmon-polariton (SPP) mode at optical wavelengths [1]. This waveguide consists of optically semi-infinite metallic (y<0) and dielectric (y>0) half-spaces, as shown in cross-sectional view in Fig. 1(a), and the SPP mode is guided along the plane as an evanescent wave with fields that peak at the interface. Barnes has recently produced a very lucid review of the single-interface SPP, emphasizing the characteristic lengths associated with the mode [2].

Over time, various architectures have been proposed for SPP waveguides. Popular 1-D architectures include the metal slab bounded by dielectric sketched in Fig. 1(b) and the dielectric slab bounded by metal sketched in Fig. 1(c) [3, 4, 5]. Representative 2-D architectures include the metal nanowire [6], the metal stripe [7–11] and the dielectric channel or gap within a metal film [12–14].

Despite decades of research on SPP waveguides, this theme remains an important and active one within the field of plasmonics as evidenced by current trends in the literature and as is apparent from recent review papers [2, 15, 16]. This is due to a number of factors including improvements in lithography and fabrication techniques making the nanoscale more accessible, the existence of many unresolved problems, and the interesting and sometimes unique peculiarities of SPPs (resonance, surface sensitivity, sub-wavelength confinement…) which create opportunities for new applications.

So given various SPP waveguide architectures, how does one determine what constitutes a “good” waveguide and mode of operation? Clearly, factors related to the intended application or experiment, or to the chosen input/output coupling means, can be determining. However, the question remains despite (or in concert with) external constraints.

Guiding SPPs means confining them along the plane (or an axis) transverse to the direction of propagation, preferably with low attenuation. The problem with SPP waveguides is that confinement and attenuation are in general correlated, meaning that increased confinement is accompanied by increased attenuation. This correlation was discussed in some detail for the case of the metal stripe [7] and for 1-D metal/dielectric structures [17]. It is also apparent from the expressions for the penetration depth and propagation length of the single-interface SPP [2]. This correlation necessarily leads to a confinement-attenuation trade-off in the design of a waveguide. Hence a “good” waveguide is one that optimises this trade-off for an intended application.

The desire to optimise the confinement-attenuation trade-off drives the requirement for a set of SPP waveguide quality measures or “figures of merit”. Global or local optimality could then be defined as maximizing (globally or locally) a figure of merit. Figures of merit would also be useful for comparing different architectures and modes of operation, or different designs, material choices and operating wavelengths. Steps in this direction were taken by Zia et. al. in [17] by proposing the use of “descriptors” like mode size, confinement factor [7] and propagation length for exploring the trade-off. Some preliminary work on figures of merit for SPP waveguides was also recently reported in [18]. Within the context of SPP nano-resonators, a figure of merit for single nanoparticle chemical sensors was defined by Sherry et. al. [19], and Maier defined a figure of merit for SPP cavities as the ratio *Q*/*V*_{eff}
, where *Q* is the quality factor and *V*_{eff}
is an effective mode volume [20, 21]. Other figures of merit for cavities in general are *Q*/*V*, *Q*^{2}
/*V* and *Q*/*V*^{1/2}
where *V* is the mode volume [22].

In this paper, three figures of merit are proposed for SPP waveguides and then used to compare and assess three popular 1-D structures. Section 2 defines the figures of merit and provides a rationale for each definition. Section 3 gives closed form expressions for the case of the single-interface SPP. Section 4 gives the computed wavelength response of 1-D structures and discusses the responses within the context of the figures of merit in order to illustrate their utility. A summary and concluding remarks are given in Section 5.

## 2. Definition of the figures of merit

It was deemed important for the figures of merit to have simple, intuitive and uncontentious definitions, and that they should be easy to compute from modal quantities. The definitions are inspired from economics, particularly from the “benefit-to-cost” ratio formed by consumers when comparing similar products from different manufacturers. In the case of SPP waveguides the benefit is confinement and the cost is attenuation, so the figures of merit are defined as the ratio of a confinement measure to attenuation for a particular mode. Clearly then, the figures of merit diverge (→∞) as the attenuation vanishes so lossless waveguides are “infinitely meritorious” according to this definition.

The confinement of a purely bound (non-radiative, non-leaky) mode can be measured in a number of ways, leading to different definitions for the figure of merit. Three ways of measuring confinement and three figures of merit are discussed in the following sub-sections. The intended waveguide application then determines which figure of merit is most relevant.

Henceforth, an *e*
^{+jωt} time dependence is assumed and modes propagate in the +*z* direction according to
${e}^{-{\gamma}_{z}z}$
. The complex propagation constant *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}
=${n}_{\mathit{1}}^{\mathit{2}}$
is the relative permittivity of the dielectric with *n*_{1}
its refractive index.

#### 2.1 Definition of the ${\mathrm{M}}_{1}^{1\mathrm{D}}$ figure of merit

A direct measure of confinement is the inverse mode size. But the measure used for the mode size depends on the dimensionality of the structure: for a 1-D waveguide the mode size is described by a linear measure such as width, for a 2-D waveguide it's an area, and for a 3-D structure such as a cavity it's a volume. Hence, a figure of merit defined on the basis of mode size must take into account the dimensionality of the structure. The definition of such a figure of merit for the 1-D case ${M}_{\mathit{1}}^{\mathit{1}D}$ is treated in detail here while the 2-D case ${M}_{\mathit{1}}^{\mathit{2}D}$ is relegated to future work.

In a 1-D structure, the mode width *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 ${M}_{\mathit{1}}^{\mathit{1}D}$
for 1-D waveguides, defined as a “benefit-to-cost” ratio, is:

where ${M}_{\mathit{1}}^{\mathit{1}D}$
is dimensionless. ${M}_{\mathit{1}}^{\mathit{1}D}$
is a field-based figure of merit and hence is useful for comparing waveguides in applications where achieving a prescribed or small mode width is important, such as end-fire coupling to a source or dense optical interconnects. The mode width *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.

The metal slab bounded by dielectric (Fig. 1(b)) and the dielectric slab bounded by metal (Fig. 1(c)) shall henceforth be referred to as the IMI (insulator-metal-insulator) and MIM (metal-insulator-metal) waveguides, respectively [17]. These three structures were chosen because the single-interface (Fig. 1(a)) is the simplest and most popular surface plasmon waveguide, while the IMI and MIM are modifications where the symmetric mode guided therein has lower loss or greater confinement, respectively, than the single-interface.

#### 2.2 Mode size and ${\mathrm{M}}_{1}^{1\mathrm{D}}$ of the single-interface SPP

Consider first the single-interface structure of Fig. 1(a). A sketch of Re{*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}
.

The mode width *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:

Eq. (2) is satisfied near the SPP resonance, where it is known that the field magnitude in the metal can indeed be large. When Eq. (2) is satisfied, the location of the 1/*e* point in the metal can easily be determined by assuming the usual exponential form for the field magnitude:

where *α*_{y,m}
=1/*δ*_{m}
, and from which the mode width *δ*_{w}
is derived:

Expressions for the penetration depths *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 ${\gamma}_{y,1}^{2}$ + ${\gamma}_{z}^{2}$=${\gamma}_{1}^{2}$ where ${\gamma}_{\mathit{1}}^{2}$=-${\beta}_{\mathit{0}}^{2}$${n}_{\mathit{1}}^{\mathit{2}}$
=-${\beta}_{\mathit{1}}^{2}$, yielding
${\gamma}_{y,1}=\sqrt{-{\beta}_{1}^{2}-{\gamma}_{z}^{2}}$
. The field magnitude follows an
${e}^{-{\alpha}_{y,1}y}$
dependence in the +*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:

Likewise, the constraint equation in the metal is ${\gamma}_{y\mathit{,}m}^{2}$+${\gamma}_{z}^{2}$=${\gamma}_{m}^{2}$ where ${\gamma}_{m}^{2}$=-${\beta}_{\mathit{0}}^{2}$
*ε*
_{r,m}, yielding
${\gamma}_{m}=\sqrt{-{\beta}_{0}^{2}{\epsilon}_{r,m}-{\gamma}_{z}^{2}}$
. The field magnitude follows an
${e}^{{\alpha}_{y,m}y}$
dependence along -*y* where *α*_{y,m}
=Re{*γ*_{y,m}
} so the field penetration depth *δ*_{m}
=1/*α*_{y,m}
is:

The figure of merit ${M}_{\mathit{1}}^{\mathit{1}D}$
is thus readily computed analytically for the single-interface waveguide (Fig. 1(a)) using Eq. (1) with (4)–(6), requiring only the propagation constant *γ*_{z}
of the SPP [7]:

In the low-loss case (*α*_{z}
≪*β*_{z}
, *ε*_{I}
≪*ε*_{R}
) and away from resonance where |*ε*_{r,m}
|≥*eε*_{r,1}
holds, Eqs. (5) and (6) can be simplified, and ${M}_{1}^{1\mathrm{D}}$ works out to:

#### 2.3 Mode size and ${\mathrm{M}}_{1}^{1\mathrm{D}}$ of the metal slab bounded by dielectric (IMI)

Consider next the symmetric IMI of thickness *t* shown in Fig. 1(b). Sketches of Re{*E*_{y}
(y)} associated with the *a*_{b}
and *s*_{b}
modes [4] are superposed onto the structure and *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, 7].

In the low-loss case (*α*_{z}
≪*β*_{z}
, *ε*_{I}
≪*ε*_{R}
) and away from resonance ${M}_{\mathit{1}}^{\mathit{1}D}$
works out to:

For *δ*_{D}
≫*t* the above further simplifies to:

For an asymmetric structure (different top and bottom dielectric claddings), *δ*_{w}
is determined directly from the computed field distribution.

#### 2.4 Mode size and ${\mathrm{M}}_{1}^{1\mathrm{D}}$ of the dielectric slab bounded by metal (MIM)

Consider as the last example the symmetric MIM of thickness *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:

which does not depend on the magnitude of the mode fields in the dielectric nor on whether 1/*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, 7].

In the low-loss case (*α*_{z}
≪*β*_{z}
, *ε*_{I}
≪*ε*_{R}
) and away from resonance where |*ε*_{r,m}
|≥*eε*_{r,1}
holds, ${M}_{\mathit{1}}^{\mathit{1}D}$
works out to:

For an asymmetric structure (different top and bottom metal claddings), *δ*_{w}
is determined directly from the computed field distribution.

#### 2.5 Role of the confinement factor

The confinement factor [7, 17] measures the fraction of mode complex power flowing through a prescribed area in the cross-section of the waveguide. Traditionally, this prescribed area is the core of a conventional dielectric waveguide, and the confinement factor varies from 0 to 1 with 1 indicating fields completely confinement to the core and 0 indicating an unconfined mode with fields extending infinitely into a cladding(s). In this structure the confinement factor is a clear measure of confinement.

However, SPPs are surface waves and many SPP waveguides do not have a core within which most of the mode is confined. For instance, no core is identifiable in the singleinterface waveguide of Fig. 1(a), making it awkward to define a confinement factor for this structure. In the case of the IMI of Fig. 1(b), the metal region might be considered the core, and hence the confinement factor clearly defined, but the mode fields are mostly in the dielectric claddings so the confinement factor remains close to 0 even for tightly bound (highly confined) modes. In the case of the MIM of Fig. 1(c), the dielectric region might be considered the core, and hence the confinement factor clearly defined, but the mode fields are mostly in the dielectric so the confinement factor remains close to 1 even for weakly bound (low confinement) modes.

Given these difficulties the inverse of the mode size was adopted in *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], or within another clearly defined region of the structure such as a layer with gain [24].

#### 2.6 Definition of the M2 figure of merit

Another direct measure of confinement is the distance between the mode's phase constant and the light line. For the structures of Fig. 1, the light line is defined by *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:

where *M*_{2}
is dimensionless. Advantageously, and contrary to ${M}_{\mathit{1}}^{D\mathit{1}}$
, *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].

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

When the confinement increases, the guided wavelength *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:

where *M*_{3}
is dimensionless and *n*_{eff}
>*n*_{1}
. Advantageously, and contrary to ${M}_{\mathit{1}}^{\mathit{1}D}$
, *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.

An intimate connection exists between the unloaded quality factor *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:

where *τ*_{spp}
is the SPP lifetime [26]:

and *v*_{g}
is the group velocity (velocity of energy transport) [23]:

For negligible dispersion in the metal and dielectric, *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

Useful approximate expressions for the figures of merit can be derived for the single-interface SPP of Fig. 1(a) using the following approximation to Eq. (7) for *n*_{eff}
and *k*_{eff}
[1, 2]:

Substituting the above into Eq. (8), which holds in the low-loss case (*α*_{z}
≪*β*_{z}
, *ε*_{I}
≪*ε*_{R}
) and away from resonance where |*ε*_{r,m}
|≥*eε*_{r,1}
holds, yields for ${M}_{\mathit{1}}^{\mathit{1}D}$
:

Assuming the Drude model for the relative permittivity of the metal [7]:

where *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}
≪${\omega}_{p}^{\mathit{2}}$
and *ω*^{2}
≫1/${\tau}_{D}^{\mathit{2}}$
:

From Eq. (22), it is noted that ${M}_{\mathit{1}}^{\mathit{1}D}$
does not depend on the wavelength of operation in the Drude region (neglecting dispersion in *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 ${M}_{\mathit{1}}^{\mathit{1}D}$
increases as *ε*_{r,1}
decreases, implying that the confinement (1/*δ*_{w}
) decreases less rapidly than the attenuation. Finally, it is observed that maximizing ${M}_{\mathit{1}}^{\mathit{1}D}$
means choosing a metal that maximizes the product *ω*_{p}*τ*_{D}
.

Following the same development for *M*_{2}
(Eq. (19) into Eq. (14)) yields:

Substituting the Drude model (Eq. (21)) into Eq. (23) for *ε*_{R}
≫*ε*_{r,1}
yields the following relationship under the conditions *ω*^{2}
≪${\omega}_{p}^{\mathit{2}}$
and *ω*^{2}
≫1/${\tau}_{D}^{\mathit{2}}$
:

From Eq. (24), it is observed that *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 ${M}_{\mathit{1}}^{\mathit{1}D}$
.

Finally, following the same development for *M*_{3}
(Eq. (19) into Eq. (15)) yields:

Substituting the Drude model (Eq. (21)) into Eq. (25) for *ε*_{R}
≫*ε*_{r,1}
, yields the following relationship under the conditions *ω*^{2}
≪${\omega}_{p}^{\mathit{2}}$
and *ω*^{2}
≫1/${\tau}_{D}^{\mathit{2}}$
:

From Eq. (26), it is observed that *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 ${\omega}_{p}^{\mathit{2}}$*τ*_{D}
and with decreasing *ε*_{r,1}
which is consistent with ${M}_{\mathit{1}}^{\mathit{1}D}$
. Approximating *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

The wavelength response and figures of merit were computed directly via the analytical solution for the single-interface SPP (Eq. (7)), while those of modes in the IMI and MIM were computed using the method of lines [7].

Ag and SiO_{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.

From the inset of Fig. 2(a), it is noted that *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] is observed for wavelengths shorter than the asymptote (*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.

Figure 2(c) and (d) give the computed values of *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.

Figure 3(a) shows the wavelength response of the mode width *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).

Figure 3(b) shows the wavelength response of the figure of merit ${M}_{\mathit{1}}^{\mathit{1}D}$
computed via Eq. (1). The approximation given by Eq. (20) (*ε*_{R}
≫*ε*_{r,1}
) for ${M}_{\mathit{1}}^{\mathit{1}D}$
of the single-interface SPP is also plotted, along with Eq. (22) for *λ*_{0}
>725 nm. For all structures, it is noted that ${M}_{\mathit{1}}^{\mathit{1}D}$
remains approximately flat for *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. ${M}_{\mathit{1}}^{\mathit{1}D}$
decreases rapidly for *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, ${M}_{\mathit{1}}^{\mathit{1}D}$
of the *s*_{b}
mode in the IMI is larger than that of the single-interface SPP, while ${M}_{\mathit{1}}^{\mathit{1}D}$
of the *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 ${M}_{\mathit{1}}^{\mathit{1}D}$
than the single-interface SPP indicating that the attenuation increased less rapidly than the confinement (1/*δ*_{w}
) as the thickness was reduced to *t*=50 nm.

Figure 3(c) shows the wavelength response of the figure of merit *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 ${M}_{\mathit{1}}^{\mathit{1}D}$
also hold in this case.

Figure 3(d) shows the wavelength response of the figure of merit *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 ${M}_{\mathit{1}}^{\mathit{1}D}$
also hold in this case, but not for the thickness of the MIM. It is noted that at a given wavelength the *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.

As already noted, ${M}_{\mathit{1}}^{\mathit{1}D}$
, *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.

The simple functions of *ε*_{R}
and *ε*_{I}
given by Eqs. (20), (23) and (25) for *ε*_{R}
≫*ε*_{r,1}
are good approximations to ${M}_{\mathit{1}}^{\mathit{1}D}$
, *M*_{2}
, *M*_{3}
and *Q* of the single-interface SPP and reasonable predictors of features, but not trends, in the wavelength response of other modes.

## 5. Summary and conclusions

Three methods of measuring confinement were discussed and three figures of merit (${M}_{\mathit{1}}^{\mathit{1}D}$
, *M*_{2}
, *M*_{3}
) were defined as benefit-to-cost ratios with the benefit being confinement and the cost being attenuation. ${M}_{\mathit{1}}^{\mathit{1}D}$
is defined using a field-based measure of confinement, the inverse mode width (1/*δ*_{w}
), and is limited to 1-D waveguides. ${M}_{\mathit{1}}^{\mathit{1}D}$
is useful for assessing and optimizing waveguides intended for application in, for example, end-fire coupling and optical interconnects. *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.

Approximations to the figures of merit were derived for the single-interface SPP resulting in simple expressions involving only the permittivities of the materials. These expressions were further simplified assuming the Drude model for the metal, leading to clear trends in the Drude region: ${M}_{\mathit{1}}^{\mathit{1}D}$
exhibits a flat wavelength response, *M*_{2}
increases with decreasing wavelength, and *M*_{3}
and *Q* decrease with decreasing wavelength.

The wavelength response of modes, and of their ${M}_{\mathit{1}}^{\mathit{1}D}$
, *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)${M}_{\mathit{1}}^{\mathit{1}D}$
exhibits a flat wavelength response in the Drude region.

## References and links

**1. **H. Raether, *Surface Plasmons on Smooth and Rough Surfaces and on Gratings* (Springer, Berlin, 1988).

**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]

**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. **25**, 844–846 (2000). [CrossRef]

**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. , **79**51–53 (2001). [CrossRef]

**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 **64**, 045411 (2001). [CrossRef]

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

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

**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. **440**, 508–511 (2006). [CrossRef]

**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. , **87**261114 (2005). [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]

**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]

**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]

**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]

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

**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]

**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]

**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]

**27. **E.D. Palik (Editor), *Handbook of Optical Constants of Solids*, (Academic Press, Orlando, Florida, 1985).

**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. **6**, 1583–1606 (1976). [CrossRef]

**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 **27**, 4654–4660 (1983). [CrossRef]

**30. **B. Brixner, “Refractive-index interpolation for fused silica,” J. Opt. Soc. Am. **57**, 674–676 (1967). [CrossRef]

**31. **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).

**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]