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, Q² /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 +jβ_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 /β₀ =α_z /β₀ +jβ_z /β₀ =k_eff +jn_eff where β₀ =2π/λ₀ is the phase constant of plane waves in free space and λ₀ is the free-space wavelength. In Fig. 1, ε_r,m =-ε_R -jε_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 n₁ 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.

 

Fig. 1. 1-D SPP waveguides considered: (a) single-interface, (b) symmetric dielectric-cladded metal slab, and (c) symmetric metal-cladded dielectric slab.

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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 M₁ 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 β₁ =ωn₁ /c₀ where c₀ is the velocity of light in free-space. The distance then is given by β_z -β₁ , or in terms of the effective index, by n_eff -n₁ . Hence the figure of merit M₂ , defined as a “benefit-to-cost” ratio, is:

where M₂ is dimensionless. Advantageously, and contrary to $M_{\mathit{1}}^{D\mathit{1}}$ , M₂ holds for 1- and 2-D waveguide structures. M₂ 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 M₃ , defined as a “benefit-to-cost” ratio, using the inverse guided wavelength 1/λ_g as the measure of confinement, is:

where M₃ is dimensionless and n_eff >n₁ . Advantageously, and contrary to $M_{\mathit{1}}^{\mathit{1}D}$ , M₃ holds for 1- and 2-D waveguide structures. M₃ 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 M₃ . 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_gω/(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 M₃ given by Eq. (15), reveals that Q=πM₃ . Hence, although M₃ 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 M₃ 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 ω² ≪${\omega }_p^{\mathit{2}}$ and ω² ≫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 λ₀ 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_pτ_D .

Following the same development for M₂ (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 ω² ≪${\omega }_p^{\mathit{2}}$ and ω² ≫1/${\tau }_D^{\mathit{2}}$ :

From Eq. (24), it is observed that M₂ 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 -β₁ ), increases more rapidly than the attenuation with decreasing λ₀ . It is also noted that M₂ increases with τ_D , which is consistent with $M_{\mathit{1}}^{\mathit{1}D}$ .

Finally, following the same development for M₃ (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 ω² ≪${\omega }_p^{\mathit{2}}$ and ω² ≫1/${\tau }_D^{\mathit{2}}$ :

From Eq. (26), it is observed that M₃ 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 M₂. This implies that the confinement, measured as the inverse guided wavelength (1/λ_g ), increases more slowly than the attenuation with decreasing λ₀ . It is also noted that M₃ 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=πM₃ with M₃ 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 SiO₂ 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 =1.26×10¹⁶ rad/s and τ_D =8.40×10^-15 s. The Drude region was taken as λ₀ ≥725 nm and λ₀ =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 λ₀ =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 SiO₂. The SPP bend-back [32] is observed for wavelengths shorter than the asymptote (λ₀ <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 <λ₀ /(2n₁ )) 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 ≅n₁ , so the energy asymptote and the non-radiative portion of the bend-back curve are included in the analysis.

 

Fig. 2. (a) Relative permittivity of Ag (ε_r,m =-ε_R -jε_I ) and SiO₂ (ε_r,1 =$n_{\mathit{1}}^{\mathit{2}}$ ) over the range 230≤λ₀ ≥2000 nm; the inset zooms-in on the range 230≤λ₀ ≤500 nm. (b) Dispersion curve of the single-interface SPP. (c) and (d) n_eff and k_eff of modes supported by the structures of Fig. 1, respectively.

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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 λ₀ <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 λ₀ 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 λ₀ >725 nm. For all structures, it is noted that $M_{\mathit{1}}^{\mathit{1}D}$ remains approximately flat for λ₀ >725 nm, implying that any change in confinement (1/δ_w ) is balanced by a change in attenuation as λ₀ is varied in this region. $M_{\mathit{1}}^{\mathit{1}D}$ decreases rapidly for λ₀ <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 M₂ computed via Eq. (14). The approximation given by Eq. (23) (ε_R ≫ε_r,1 ) for M₂ of the single-interface SPP is also plotted, along with Eq. (24) for λ₀ >725 nm. For all structures, it is noted that M₂ increases with decreasing λ₀ up to a maximum value from which M₂ then decreases rapidly as the modes tend toward their asymptote. Hence an optimal wavelength of operation exists that maximizes M₂ . Clearly, the confinement (β_z -β₁ ) increases more rapidly than the attenuation with decreasing λ₀ 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 M₂ 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 M₃ computed via Eq. (15). The approximation given by Eq. (25) (ε_R ≫ε_r,1 ) for M₃ of the single-interface SPP is also plotted, along with Eq. (26) for λ₀ > 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 M₃ and Q converge to essentially the same measure (Q=πM₃ ). 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 M₃ and Q near 840 nm. The other modes follow the same trend in that M₃ and Q increase linearly with λ₀ . 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 M₃ 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, $M_{\mathit{1}}^{\mathit{1}D}$ , M₂ , M₃ and Q eventually tend to zero with decreasing λ₀ , 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 $M_{\mathit{1}}^{\mathit{1}D}$ , M₂ , M₃ and Q of the single-interface SPP and reasonable predictors of features, but not trends, in the wavelength response of other modes.

 

Fig. 3. (a) δ_w , (b) $M_{\mathit{1}}^{\mathit{1}D}$ , (c) M₂ , (d) M₃ , (e) v_g , and (f) Q of modes supported by the structures of Fig. 1.

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5. Summary and conclusions

Three methods of measuring confinement were discussed and three figures of merit ($M_{\mathit{1}}^{\mathit{1}D}$ , M₂ , M₃ ) 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. M₂ and M₃ are defined using the distance of the mode from the light line (β_z -β₁ ) and the inverse guided wavelength (1/λ_g ) as their measures of confinement, respectively, and they are applicable to 1- and 2-D waveguides. M₂ and M₃ are useful for assessing and optimizing waveguides intended for application in, for example, Bragg gratings and resonators, respectively. The intimate connection between M₃ 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, M₂ increases with decreasing wavelength, and M₃ and Q decrease with decreasing wavelength.

The wavelength response of modes, and of their $M_{\mathit{1}}^{\mathit{1}D}$ , M₂ , M₃ 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) M₂ and M₃ can exhibit a peak versus wavelength indicating that a preferred wavelength of operation exists (with respect to M₂ and M₃ ); (vii)$M_{\mathit{1}}^{\mathit{1}D}$ exhibits a flat wavelength response in the Drude region.