1. Introduction

Surface plasmon polaritons (SPPs) are transverse magnetic (TM) polarized optical surface waves that propagate, typically, along a metal–dielectric interface [1, 2, 3], henceforth termed the single interface. One purely bound (nonradiative) SPP wave (mode) is supported by this structure, with fields that peak at the interface and decay exponentially away into both media. Its fields are associated with a charge density wave in the metal; so the SPP is a coupled excitation (plasmon polariton).

The single-interface SPP exhibits interesting and useful properties such as an energy asymptote in its dispersion curve, very high surface and bulk sensitivities, and subwavelength confinement near its energy asymptote. But it is also characterized by a high attenuation, especially near the energy asymptote, limiting the scope for applications. For a metal bounded by an ideal dielectric, the attenuation is caused primarily by free-electron scattering in the metal and, at short enough operating wavelengths, by interband transitions. Roughness along the interface causes additional attenuation.

A practical way of reducing the SPP attenuation is to use a thin metal film or stripe bounded on all sides by the same dielectric and operating the structure in the so-called long-range SPP (LRSPP) mode. “Long-range” is taken to mean that the LRSPP attenuation is at least a factor of 2 to 3 lower than that of the single-interface SPP, resulting in propagation over a longer distance. Indeed, attenuation reduction factors, or equivalently, range extension factors, greater than 100 have been demonstrated experimentally with the LRSPP. The range extension mitigates an important limitation of the single-interface SPP, but this comes at the expense of reduced confinement. Nevertheless, the extended range may outweigh the reduced confinement potentially enabling better and more competitive applications.

Our purposes in this paper are to highlight the attributes of the LRSPP, review its origins and the body of work that has been conducted on this wave, and to discuss potential applications based on its use. The LRSPP has appeared in a large number of studies, either as the focal point or peripherally within another context. Literary contributions that include consideration of the LRSPP are cited, and, generally, only the salient points involving the LRSPP are described. In many cases, there is more to a cited contribution than this narrow perspective, so the reference list should not be regarded as listing exclusively LRSPP studies. The discussion is at times broadened to include the single-interface SPP, or the short-range (high-attenuation) modes supported by the structures considered, in order to provide context and comparison or to underline important points of differentiation. The references are organized by subsection, then chronologically.

The literature on SPPs in general is of course much broader. The texts edited by Boardman [1] and by Agranovich and Mills [2], and the monograph by Raether [3], review early work and remain important literary landmarks in the field. Papers by Sambles et al. [4], Welford [5], and Barnes [6], serve as good introductions to SPPs, as does a recent textbook by Maier [7]. Good recent reviews of the field include those of Barnes et al. [8], Zayats et al. [9], Maier and Atwater[10], Ozbay [11], and Atwater [12]. Ebbesen et al. [13] and Degiron et al. [14] have recently discussed integrated optical circuitry based on SPPs.

Work conducted on or with the LRSPP can be organized in many ways. The main division adopted here follows the dimensionality of the guiding structures: Section 2 reviews the LRSPP in the metal slab, which provides confinement in one dimension, and Section 3 reviews the LRSPP in the metal stripe, which provides confinement in two dimensions. Subsections 2.1, 3.1 briefly and didactically summarize the properties of the bound modes in these structures in order to provide context and introduce relevant symbols and terminology. Section 4 discusses the prospects for applications, and Section 5 gives concluding remarks.

1.1. Notation

Throughout this paper, an ${\mathrm{e}}^{+j\omega t}$ time dependence is assumed with modes propagating in the $+z$ direction according to ${\mathrm{e}}^{-\gamma z}$. The complex propagation constant γ in inverse meters expands as $\gamma =\alpha +\mathrm{j}\beta$, where α and β are the attenuation and phase constants, respectively. The normalized propagation constant is ${\gamma }_{\mathrm{eff}}=\gamma ∕{\beta }_0=\alpha ∕{\beta }_0+\mathrm{j}\beta ∕{\beta }_0=k_{\mathrm{eff}}+\mathrm{j}{n}_{\mathrm{eff}}$, where ${\beta }_0=2\pi ∕{\lambda }_0=\omega ∕c_0$ is the phase constant of plane waves in free space, ${\lambda }_0$ the wavelength in free space, and $c_0$ the speed of light in free space. The complex effective index of a mode $N_{\mathrm{eff}}$ is then given by $N_{\mathrm{eff}}=-\mathrm{j}{\gamma }_{\mathrm{eff}}=n_{\mathrm{eff}}-\mathrm{j}{k}_{\mathrm{eff}}$, where $n_{\mathrm{eff}}$ is the effective index and $k_{\mathrm{eff}}$ the normalized attenuation. The mode power attenuation (MPA) in decibels per meter is given by $\mathrm{MPA}=\alpha 20{\log }_{10}\mathrm{e}$.

The propagation length $L_e$ is the distance from the launch point where the mode power decays by a factor of $1∕e$ and is given by $L_e=1∕\left(2\alpha \right)$. (A less used definition of the propagation length, the $1∕e^2$ power decay length, is also found in the literature and yields a larger length for the same attenuation.)

The group velocity of a mode $v_g$ is given by $\partial \omega ∕\partial \beta$, and its lifetime is given by $\tau =L_e∕v_g$.

The relative permittivity ${\epsilon }_r$ is related to the permittivity ε in the usual way, ${\epsilon }_r=\epsilon ∕{\epsilon }_0$, where ${\epsilon }_0$ is the permittivity of free space. The relative permittivity is related to the optical parameters n, k via ${\epsilon }_r={(n-\mathrm{j}k)}^2$. A numerical subscript i identifies these quantities in a specific region. The relative permittivity of a metal at optical wavelengths is decomposed into real and imaginary parts as ${\epsilon }_r=-{\epsilon }_R-\mathrm{j}{\epsilon }_I$. For good metals ${\epsilon }_R\gg {\epsilon }_I$ in the infrared and over at least part of the visible (e.g., Au, Ag, Cu, and Al).

2. Metal Slab $(w=\infty )$

2.1. Modes of the Metal Slab

The single interface is sketched in Fig. 1(a), and the structures sketched in Figs. 1(b), 1(c) are important variations. The variation shown as Fig. 1(b) consists of a thin metal film of thickness t and relative permittivity ${\epsilon }_{r,2}$ bounded by optically semi-infinite dielectrics (claddings) of relative permittivity ${\epsilon }_{r,1}$ and ${\epsilon }_{r,3}$, henceforth referred to as the metal slab. Another variation consists of a thin dielectric film of thickness t and relative permittivity ${\epsilon }_{r,1}$ bounded by optically semi-infinite metals of relative permittivity ${\epsilon }_{r,2}$, as sketched in Fig. 1(c), and henceforth referred to as the metal clads. The metal slab is said to be symmetric when ${\epsilon }_{r,1}={\epsilon }_{r,3}$ and asymmetric otherwise (${\epsilon }_{r,1}>{\epsilon }_{r,3}$ or ${\epsilon }_{r,1}<{\epsilon }_{r,3}$). The distribution of the main transverse electric field component of the modes $\left(E_y\right)$ is sketched as the red curves over the cross section of each structure. Mode propagation occurs along the $+z$ axis (upward perpendicular to the page).

In the symmetric metal slab, the bound single-interface SPPs supported by the individual metal–dielectric interfaces at large t, couple as t is reduced, forming two TM-polarized $(E_x=H_y=H_z=0)$ bound supermodes, sometimes termed coupled modes, that exhibit distinct dispersion characteristics and a distinct evolution with structure parameters (ε and t). These supermodes are denoted herein as $a_b$ and $s_b$ for asymmetric bound and symmetric bound, respectively, since their main transverse electric field component $\left(E_y\right)$ varies either asymmetrically or symmetrically across the structure (along the y axis), as sketched in Fig. 1(b). The $H_x$ field component of the modes has the same symmetry as the corresponding $E_y$ field, but the longitudinal electric field $E_z$ has the opposite symmetry. The charge density in the metal linked to the $a_b$ mode has a symmetric distribution over t, as indicated by the pluses in Fig. 1(b), whereas the charge density associated with the $s_b$ mode is asymmetric over t, as indicated by the plus and minus signs.

In a symmetric structure with lossless claddings, the attenuation and effective index of the $s_b$ mode decrease smoothly as t is reduced, with its mode fields increasingly expelled from the metal film and penetrating more deeply into the claddings. Indeed, as $t\to 0$, the confinement and attenuation of this mode vanish as it evolves smoothly into the vertically polarized TEM (transverse electromagnetic) wave of the background. The LRSPP is the ${\mathit{s}}_{\mathit{b}}$ mode of the thin metal slab. The $a_b$ mode exhibits increasing confinement and penetration into the metal with decreasing t and, consequently, increasing attenuation. For large t the $a_b$ and $s_b$ modes are degenerate with the single-interface SPPs supported by the uncoupled top and bottom metal–dielectric interfaces.

The trends are similar in an asymmetric slab except that (i) the $s_b$ mode cuts off below a certain thickness t that depends on the permittivities and operating wavelength, and (ii) with increasing t the $a_b$ mode evolves into the SPP supported by the metal interface with the high-index cladding, while the $s_b$ mode evolves into the SPP at the interface with the low-index cladding. These trends with t are apparent from Fig. 2, which plots the effective index and normalized attenuation of the $a_b$ and $s_b$ modes in an asymmetric slab at ${\lambda }_0=632.8\mathrm{nm}$ assuming Ag for the metal ${\epsilon }_{r,2}={(0.0657-\mathrm{j}4)}^2$ and claddings of relative permittivity ${\epsilon }_{r,1}={1.5}^2$ and ${\epsilon }_{r,3}={1.55}^2$. The symmetric structure may be more convenient for working with the LRSPP because the mode remains nonradiative (purely bound) for $t\to 0$, whereas in an asymmetric structure the LRSPP ($s_b$ mode) remains nonradiative only for t greater than a cutoff thickness.

The $a_b$ and $s_b$ modes exhibit distinct dispersion characteristics if t is small enough. For example, Fig. 3 shows normalized dispersion curves for the $a_b$ and $s_b$ modes propagating along a lossless metal film modeled as a Drude metal and bounded symmetrically by semi-infinite vacuum for three normalized thicknesses. For very thin metal films $(U=0.1)$ the dispersion of the $a_b$ and $s_b$ modes are distinct, but for thick films $(U=2)$ they merge and coincide with the corresponding single-interface SPP. The dispersion curves are asymptotic with the light line as $\beta \to 0$, and they are asymptotic with $\Omega =1∕\surd 2\Rightarrow \omega ={\omega }_p∕\surd 2$ as $\beta \to \infty$, where ${\omega }_p$ is the plasma frequency of the Drude model. The $\beta \to \infty$ limit is termed the SPP energy asymptote. The dispersion curve of the $s_b$ mode is above that of the $a_b$ mode, so a given β occurs at a higher ω for this mode. The confinement increases, $v_g$ tends to zero, and the optical density of states diverges near the energy asymptote. A region of negative $v_g$ occurs for the $s_b$ mode before the energy asymptote. The trends are the same for structures having symmetric dielectric claddings, with the asymptotic limits taking on appropriate values in the dielectric [light line, $\omega =\beta c_0∕n_1$; energy, $\omega ={\omega }_p{(1+{\epsilon }_{r,1})}^{-1∕2}$]. The trends are similar for real metals, except that the attenuation of the modes increases dramatically toward the energy asymptote, and the depth of the asymptote is limited by bendback. The $s_b$ mode is generally long range away from the energy asymptote (and for small t).

The nomenclature used for identifying the $a_b$ and $s_b$ modes varies greatly throughout the literature, and occasionally erroneous assignments are made. The nomenclature that we have adopted follows the integrated optics convention of identifying a mode by features in the spatial distribution of its main transverse electric field ($E_y$ in the structures of Fig. 1). Other mode nomenclatures are based on identifying features in the longitudinal electric field, in the charge distribution across the metal, or on the mode’s location on a dispersion diagram. Thus, in the literature, one finds the $s_b$ mode sometimes termed the asymmetric mode based on $E_z$ or on charge distribution (Fig. 1(b)), or the ${\omega }_{+}$ mode because it corresponds to the highest curve on a dispersion diagram (Fig. 3). Likewise, the $a_b$ mode has also been termed the symmetric mode or the ${\omega }_{-}$ mode.

The LRSPP can be excited by p-polarized light by using a high-index prism as sketched in longitudinal cross-sectional view in Fig. 4 (case 1, ${\epsilon }_{r,3}={\epsilon }_{r,1}$). In typical experiments, the angle of incidence θ is varied beyond the critical angle of the ${\epsilon }_{r,4}|{\epsilon }_{r,3}$ interface, and the reflected power is monitored by using a detector. The excitation of a mode occurs when the in-plane wavenumber of the incident light ${\beta }_0{n}_4\sin \theta$ equals the mode’s wavenumber β, resulting in a drop of the reflected power. If the reflectance is plotted versus θ, then dips appear for each excited mode as shown in Fig. 5. The $s_b$ mode is excited at a smaller θ than the $a_b$ mode, since the former has a smaller β at the operating ω (Figs. 2, 3). Lower attenuation results in a narrower dip, a signature of the LRSPP. As t increases, the $a_b$ and $s_b$ dips merge into one, because the modes become degenerate and identical to the single-interface SPP. The plot of Fig. 5 is termed an attenuated total reflection (ATR) angular spectrum, since the excitation of modes results in attenuation of the reflected light as measured by the detector.

The LRSPP can also be excited by a TM-polarized optical beam via end-fire coupling as sketched in longitudinal cross-sectional view in Fig. 6. In this arrangement the beam is focused onto the end facet of the structure such that it overlaps well with the fields of the LRSPP. End-fire coupling can be efficient, since the transverse mode fields of the LRSPP are symmetrically distributed over the structure cross section, as are the exciting fields in such arrangements. The technique can be easier to implement than prism coupling, and it eliminates problems associated with prism loading (mode perturbation and unwanted outcoupling); however, it requires access to high-quality end facets, which can be difficult or inconvenient to create in some structures, and it is not β selective; thus all modes that overlap the input beam will be excited to some extent (including radiative modes); so outputs must be interpreted carefully.

The metal clads depicted in Fig. 1(c) also support coupled modes, such as the symmetric mode for which the main transverse electric field distribution is also sketched. Table 1 summarizes modal information for the three structures of Fig. 1, assuming ${\lambda }_0=1550\mathrm{nm}$, Ag for the metals, $\mathrm{Si}{\mathrm{O}}_2$ for the dielectrics, $t=20\mathrm{nm}$ for the metal slab, and $t=50\mathrm{nm}$ for the metal clads; ${\delta }_w$ is the $1∕e$ mode field width. The trade-off between confinement and attenuation across these structures is evident from these data. The LRSPP in the metal slab is at one end of the trade-off, having a low attenuation but also low confinement (smaller $n_{\mathrm{eff}}-n_1$ and larger mode size ${\delta }_w$), whereas the symmetric mode of the metal clads is at the other end of the trade-off (higher attenuation and confinement). The single-interface SPP is between them. Neither the metal clads nor the single interface support LRSPPs.

2.2. Origins of the LRSPP [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]

The LRSPP builds on a number of previous studies involving the single interface and the metal slab, including, for example, the work of Ritchie [15], Kliewer and Fuchs [17], Otto [18, 19], Economou [20], Kretschmann [23], and Abelès and Lopez-Rios [25].

The coupled modes in the symmetric metal slab were studied long ago through dispersion computations, for example, such as those produced by Kliewer and Fuchs [17] for an ideal thin metal film bounded by vacuum (Fig. 3) and by Economou [20] for a similar structure and a number of variants thereof. These early studies did not include the effects of damping; so the range of mode propagation could not be assessed.

Otto described and demonstrated prism coupling as a means of exciting the single-interface SPP [18] (the Otto configuration), by introducing a low-index gap between the prism and the metal film (Fig. 4, case 2). Otto also considered, theoretically, prism coupling to the $a_b$ and $s_b$ modes of the metal slab, including damping in the metal, bounded symmetrically by low-index claddings and high-index prisms [19], and computed the reflection and transmission responses of the system. Interestingly, in Fig. 3 of [19], he showed a narrow linewidth in the reflection and transmission resonances of the system mediated by the $s_b$ mode (${\omega }_{+}$ in the notation of [19]) at ${\lambda }_0=546.1\mathrm{nm}$ for a Ag slab $30\mathrm{nm}$ thick and for a prism-metal spacing of $400\mathrm{nm}$ (the largest considered). He also showed this resonance narrowing as the prism-metal spacing increases, a behavior later understood to be characteristic of the LRSPP (e.g., [48]). He showed theoretically that the transmittance of the system could be very high, 70% in the case of a $70\mathrm{nm}$ thick Ag film, and that the transmission was mediated primarily by the $s_b$ mode. (Transmission through a Ag film of this thickness bounded by air is essentially zero.) He suggested using the system in transmission as a polarizer. The arrangement was studied experimentally shortly thereafter [22], confirming the main findings of Otto [19], and exploring its possible application as a polarizing spectral filter. (Dragila et al. studied a similar system [43]).

During the same time, Tien et al. [21] proposed and demonstrated the prism coupling approach to excite dielectric waveguide modes, and Kretschmann [23] demonstrated a variant of the Otto configuration [18] (the Kretschmann–Raether configuration) for exciting the single-interface SPP where the metal film is deposited directly onto the base of the prism (Fig. 4, case 3). (This configuration was also considered by Turbadar years earlier [16], as pointed out by Welford [5].) The Otto [18] and Kretschmann–Raether [23] configurations are not suitable for exciting LRSPPs, but Otto’s other configuration [19] is suitable.

Abelès and Lopez-Rios [25] combined the Otto and Kretschmann–Raether configurations (Fig. 4, case 4) to excite the (essentially) uncoupled single-interface SPPs supported at the opposite interfaces of a highly asymmetric thin metal slab.

Although much had been learned regarding the coupled modes of the metal slab, their attenuation (and range) had remained unexplored until the work of Kovacs [26, 28]. Kovacs computed the propagation constant of the $a_b$ and $s_b$ modes at ${\lambda }_0=450\mathrm{nm}$ in In slabs $({\epsilon }_{r,2}=-20.358-\mathrm{j}6.019)$ of thickness $t=50,30,10\text{\hspace{0.17em}nm}$, bounded symmetrically by semi-infinite $\mathrm{Mg}{\mathrm{F}}_2$ $(n_1=n_3=1.382)$ (pp. 97–99 of [26] or Table 1 of [28]). His computations show that $n_{\mathrm{eff}}$ and α of the $a_b$ mode are larger than those of the $s_b$ mode and that they move further apart with decreasing t. Specifically, his results show $n_{\mathrm{eff}}$ and α of the $a_b$ mode both increasing with decreasing t, and $n_{\mathrm{eff}}$ and α of the $s_b$ mode both decreasing with decreasing t. From his computations, it is also noted that α of the $s_b$ mode for $t=10\mathrm{nm}$ is $29\times$ lower than for $t=50\mathrm{nm}$. Prism coupling to the coupled modes was also investigated theoretically and experimentally for different In thicknesses $(t=19,27,42\mathrm{nm})$.

Kovacs also explored theoretically and experimentally prism coupling to the $a_b$ and $s_b$ modes of thin Ag slabs bounded by identical dielectrics (Fig. 4, case 1; Fig. 5) [26, 27]. The experimental structures comprised an Ag slab $40–55\mathrm{nm}$ thick bounded by cryolite and were excited at ${\lambda }_0=632.8\mathrm{nm}$ by angle scanned $\left(\theta \right)$ prism coupling. These structures allowed the experimenters to unambiguously differentiate and identify the $s_b$ and $a_b$ modes, observing that their angular separation (in θ) decreased with increasing t as expected. Informative plots of the Poynting vector through the structure, and of the current density in the metal, were also given, from which one notes, for instance, that the power flow in the metal has a large component in the direction antiparallel $(-z)$ to the direction of modal propagation $(+z)$. Effects caused by the finite thickness of the claddings were also investigated, revealing the importance of the gap s (Fig. 4, case 1) on the coupling efficiency of the modes. The $s_b$ mode was not really long range in these structures because of the large thickness of the metal film, the roughness of the interfaces, and inhomogeneities in the bounding cryolite films.

Thus, many essential features of the LRSPP were uncovered by Kovacs [26, 28]. And by making slight modifications to then-known prism coupling techniques [18, 19, 21, 23, 25], particularly by removing the second prism in Otto’s other geometry [19], he demonstrated experimentally via optical means the existence of the coupled modes in symmetric metal slabs [26, 27]. (Many of his results [26] were subsequently summarized in Chap. 4 of [1].)

At about the same time, Fukui et al. [29] (in collaboration with G. I. Stegeman) included damping in the metal and computed the lifetime τ of the $a_b$ and $s_b$ modes in thin unsupported Ag films bounded symmetrically by vacuum $({\epsilon }_1={\epsilon }_3=1)$ as a function of the Ag film thickness t. They predicted a propagation length of $L_e=3\mathrm{cm}$ for the $s_b$ mode on a $16\mathrm{nm}$ thick Ag film in vacuum at ${\lambda }_0=1.06\mu \mathrm{m}$. The propagation length of the corresponding single-interface SPP is $0.06\mathrm{cm}$; so the predicted range extension was a factor of 50. They also showed the $s_b$ mode guided as $t\to 0$, with no cutoff thickness being apparent in their results.

Sarid [30, 31] then studied theoretically the $a_b$ and $s_b$ modes in the metal slab bounded by asymmetric dielectrics (${\epsilon }_{r,1}={1.5}^2$, ${\epsilon }_{r,3}={1.55}^2$). From his results, reproduced here as Fig. 2, it is apparent that the LRSPP does not exist below a cutoff thickness, $t\sim 20\mathrm{nm}$ in this case. He modeled the prism coupling arrangement (Fig. 4, case 1), noting that the half-width angle corresponding to the excitation of the LRSPP was very narrow, about 0.004°. He also commented on the criticality of the gap s, pointing out that if it is too large the coupling efficiency suffers, but if it is too small then the prism loads and wipes out the LRSPP. The terminology “long-range” for identifying the low-loss version of the $s_b$ mode originated here.

Experiments on improved structures [32, 33, 34, 35] were reported shortly after [27], leading to long (measured) $s_b$ propagation lengths, and thus to the first experimental observations of the LRSPP.

The structure explored experimentally by Kuwamura et al. [32] consisted of a $14.6\mathrm{nm}$ thick Ag film deposited on $\mathrm{Ca}{\mathrm{F}}_2$ with index-matching oil used as the other cladding. The structure was excited at ${\lambda }_0=632.8\mathrm{nm}$ in a prism-coupled configuration. The measured propagation length of the $s_b$ mode in this structure $(L_e\sim 100\mu \mathrm{m})$ is approximately $10\times$ longer than that of the SPP on the corresponding single interface.

Craig et al. [33] reported a propagation length of $L_e=250\mu \mathrm{m}$ for the $s_b$ mode supported by a $10\mathrm{nm}$ thick Ag film, bounded by glass on one side and index-matching oil on the other, and excited at ${\lambda }_0=632.8\mathrm{nm}$ by using a retroreflecting prism [36]. This measured propagation length is $63\times$ longer than that of the corresponding single-interface SPP.

Quail et al. [34] characterized a $17\mathrm{nm}$ thick Ag film and a $14.5\mathrm{nm}$ thick Al film, both on glass and covered by index-matching oil, at ${\lambda }_0=632.8\mathrm{nm}$ in a prism-coupling configuration. They measured propagation lengths for the $s_b$ mode that are about $10\times$ longer than the corresponding single-interface SPPs.

Dohi et al. [35] measured a propagation length of $L_e\sim 265\mu \mathrm{m}$ for the $s_b$ mode supported by an Ag film about $15\mathrm{nm}$ thick on Pyrex and covered with oil having a slightly different index than that of the Pyrex (i.e., in a slightly asymmetric structure ${\epsilon }_{r,1}\ne {\epsilon }_{r,3}$). Their structures were excited at ${\lambda }_0=632.8\mathrm{nm}$ in a prism coupling arrangement. Their range extension is comparable with that achieved by Craig et al. [33].

2.3. Prism Coupling and Field Enhancement [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53]

Difficulties with prism coupling to the LRSPP (Fig. 4) were identified early on [27, 30, 31, 32, 33, 34, 35, 36]. Experimental difficulties include controlling the gap spacing s and parallelism between the base of the prism and the metal slab, as well as the angle of incidence of the input light θ, while ensuring that the permittivities of both claddings remain closely matched $({\epsilon }_1={\epsilon }_3)$. These requirements suggest the need for high-performance optomechanical components and working with index-matching oils or custom-fabricated structures.

Another difficulty is that the mode fields of the LRSPP extend deeply into the claddings; so the loading effect of the higher-index prism on the mode is quickly apparent, perturbing its fields and propagation characteristics and rendering it radiative into the prism, as is readily apparent from the plane wave computations of Wendler and Haupt [48]. If the prism is too far from the metal slab, the coupling efficiency is poor, but if it is too close, loading destroys the LRSPP (cf. Figs. 1 and 7 of [48]). Under conditions of perfect coupling to the LRSPP, radiative damping equals intrinsic damping (damping without the prism), so its range is halved (i.e., its attenuation is doubled) [40, 48], as is the case for the single-interface SPP in the Otto [18, 37] and Kretschmann–Raether [3, 23] configurations.

Thus, the reflected field in a prism coupling experiment includes a specularly reflected contribution as well as an outcoupled (reradiated) contribution due to the propagating LRSPP. Since the LRSPP has a long propagation length, the outcoupled contribution extends over a large spatial cross section and can interfere with the specular reflection. The contributions can, however, be distinguished if a finite-size input beam is used [41, 42, 45] (as for the single-interface SPP [38] in the Otto [18] and Kretschmann–Raether [23] configurations). For example, Fig. 7(a) shows a finite-width input beam prism coupled to the LRSPP on a Ag film supported by a Pyrex substrate and covered by index-matching oil, and Fig. 7(b) shows the measured intensity profile at the observation plane (${\lambda }_0=632.8\mathrm{nm}$, $w=1.27\mathrm{mm}$, $s=1.35\mu \mathrm{m}$, $t=16\mathrm{nm}$, $L_e=274\mu \mathrm{m}$, $L_e∕w=0.2157$) [42]. From Fig. 7(b) two bumps separated by a null are noted, the leftmost one due to specular reflection of the input beam and the rightmost one due to the outcoupled LRSPP. In such a situation, the measured profile of the outcoupled contribution follows the decay of the LRSPP [42, 45]. The importance of considering the finite size of the input beam and of its angular spread when interpreting prism coupling experiments with LRSPPs has also been highlighted in other papers [46, 49, 50, 51, 53].

Barnes and Sambles [47] excited the LRSPP via prism coupling in a $40\mathrm{nm}$ thick Ag slab bounded symmetrically by Langmuir–Blodgett layers (22-tricosenoic acid) about $600\mathrm{nm}$ thick at ${\lambda }_0=632.8\mathrm{nm}$. They achieved a modest increase in propagation length over the corresponding single-interface SPP, due in part to the large thickness of the Ag film, to loss in the Langmuir–Blodgett layers, and to suspected damage caused to one of the Langmuir–Blodgett layers during Ag evaporation. However, it is interesting to note that a multilayer structure incorporating organic claddings could be fabricated with some success.

The LRSPP was also seemingly observed in a thin Ag film bounded on both sides by thin layers of Teflon and excited by using prism coupling, presumably at ${\lambda }_0=632.8\mathrm{nm}$ [52].

Sarid et al. [40] computed the field enhancement in a prism coupling arrangement at ${\lambda }_0=632.8\mathrm{nm}$ assuming Ag films 70 and $10\mathrm{nm}$ thick bounded by dielectrics of index 1.5. The field enhancement was defined as the ratio of the squared magnitude of the magnetic field in the vicinity of the metal surface to the squared magnitude of the magnetic field in the prism. They reported field enhancements of 40 and 600 at the excitation angle of the $s_b$ mode in the 70 and $10\mathrm{nm}$ thick films, respectively, for near perfect coupling into the modes. The $s_b$ mode in the $70\mathrm{nm}$ thick case is not long-range and corresponds almost to the single-interface SPP in the Otto geometry [18] (distinct $s_b$ and $a_b$ modes still occur at this thickness); the $s_b$ mode in the $10\mathrm{nm}$ thick case corresponds to the LRSPP. They mentioned computing the field enhancement in the Kretschmann–Raether geometry and observing that they are similar to the Otto case. Thus, they concluded that the field enhancement associated with the excitation of the LRSPP via prism coupling $(\sim 600)$ can be 1 order of magnitude larger than that in the Otto and Kretschmann–Raether geometries $(\sim 40)$. Indeed, assuming that Eq. (10) of [39] holds, then the field enhancement of the LRSPP relative to the single-interface SPP should follow the ratio of their propagation lengths ($L_e$’s) or the inverse ratio of their attenuations (α’s). LRSPP field enhancements were later shown to exist on corrugated gratings as well (e.g., [56]).

Lévy et al. [44] compared theoretically the prism-coupled LRSPP field enhancement to that of the prism-coupled ${\mathrm{TE}}_0$ and ${\mathrm{TM}}_0$ modes supported by a dielectric slab waveguide (equivalently, a 1D cavity). Results for the structures investigated show that the field enhancement of the ${\mathrm{TE}}_0$ and ${\mathrm{TM}}_0$ modes along the core center was about $6\times$ greater than the enhancement of the LRSPP fields.

This enhancement [39, 40, 44] might loosely be termed an intensity enhancement, since another definition for the field enhancement found in the literature forms the ratio of the field magnitudes, yielding the square root of the former.

2.4. Corrugated Gratings [54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70]

Symmetric periodically corrugated metal slabs supporting LRSPPs have also been explored as an alternative to prism coupling and as structures having rich and interesting properties in their own right. Generally (TM) p-polarized light is used to illuminate a corrugated grating, with the plane of incidence oriented perpendicular to the grooves, and the response in reflection and/or transmission recorded as a function of the angle of incidence (angle scan) or the wavelength of operation (wavelength scan).

Different theoretical approaches have been used to model grating responses (cf. Barnes et al. [63] for a good discussion), many rooted in Chandezon’s coordinate transformation formalism [54], whereby corrugated boundaries are mapped to flat surfaces allowing the straightforward application of boundary conditions and transfer matrices for handling many layers. Cotter et al. [62] improved on the method by using scattering matrices instead of transfer matrices to eliminate numerical instability when modeling thick structures.

Inagaki et al. [55] reported the excitation of the $s_b$ and $a_b$ modes at ${\lambda }_0=632.8\mathrm{nm}$ in a thin free-standing $44\mathrm{nm}$ thick corrugated Ag slab in air, with the measured resonance width of the $s_b$ mode being considerably narrower than that of the $a_b$ mode. Further experiments were conducted by the same group [57] on free-standing corrugated Ag slabs having thicknesses in the range of $26–212\mathrm{nm}$, showing splitting of the $s_b$ and $a_b$ modes for a slab thickness of about $91\mathrm{nm}$. The measured phase and attenuation constants of these modes compared qualitatively well with theoretical expectations for a flat slab over the thickness range investigated, but larger phase and attenuation constants were measured throughout. The coupling efficiency was also found to be strongly dependant on the thickness of the slab.

This symmetric free-standing corrugated slab was subsequently studied theoretically by Dutta Gupta et al. [58], supporting the main conclusions of Inagaki et al. [55, 57]. They also study the effects of damping by outcoupling to free radiation for various corrugation amplitudes.

Cavalcante et al. [59] studied theoretically the metal slab bounded symmetrically by dielectrics where one of the metal–dielectric interfaces is smooth and the other interface has a sinusoidal profile. They computed the reflectivity and the intensity of the field near the metal slab, both as a function of the angle of incidence for various grating amplitudes. They observed, among other points, an increase in the linewidth of the reflectivity dip for both the $a_b$ and $s_b$ modes, indicating increased attenuation, as the corrugation amplitude increases.

The study of Chen and Simon [60] also pointed out that an additional loss contribution to the LRSPP in corrugated gratings may be expected that is due to scattering from the grating grooves.

Bryan-Brown et al. [61] measured the reflection response of symmetrically cladded thin corrugated Ag slabs through the LRSPP excitation angle and deduced the grating groove depth as well as the thickness and optical parameters of the embedded Ag films via comparisons with a theoretical model of the experiment.

Salakhutdinov et al. [64] investigated sinusoidally corrugated metal slabs where the upper and lower surfaces of a slab are corrugated in phase or phase shifted by π. They computed the perturbed propagation constant of the LRSPP for corrugations of both types in a Cu film, as a function of groove depth and t. They found that the corrugations increase the attenuation of the mode, but much more so for π-shifted corrugations than for in-phase corrugations. They found that $n_{\mathrm{eff}}$ increases for the π-shifted corrugations but that it either increases or decreases for in-phase corrugations depending on t. They modeled and demonstrated experimentally anomalous reflection (interference between reflected and outcoupled guided fields) from in-phase corrugations.

Hooper and Sambles [65] reported a theoretical study pertaining to the excitation and nature of the modes in unsupported (air on both sides) corrugated thin Ag slabs where both surfaces of the slab were corrugated either conformally (identical profiles, in phase) or nonconformally (identical profiles, phase shifted). Sinusoidal profiles with and without the first harmonic in grating wavenumber were explored. The reflection, transmission, and absorption of incident radiation were determined for a variety of structure parameters. Four coupled modes (two $a_b$-like and two $s_b$-like) were found in the nonconformal case that incorporates the first harmonic in grating wavenumber, which are due to new symmetries for the charge distribution that are allowed by the grating features. The second harmonic also leads to anticrossings and bandgaps in the dispersion diagram. This grating architecture for small t $\left(30\mathrm{nm}\right)$ is capable of an unusually large transmittance over a wide frequency band, almost independently of the angle of incidence, with the transmittance mediated by one of the $a_b$ modes.

Chen et al. [67] further explored this grating concept experimentally and theoretically by way of a $27\mathrm{nm}$ thick Ag slab deposited conformally onto a corrugated $\mathrm{Si}{\mathrm{O}}_2$ substrate, covered with index-matching fluid, and clamped with a pair of $\mathrm{Si}{\mathrm{O}}_2$ prisms (one on each side of the slab). A sinusoidal grating profile with the first harmonic in grating wavenumber was implemented. The origin of anticrossings and bandgaps was discussed in terms of structure parameters and the role of the $a_b$-like and two $s_b$-like modes.

Lévêque and Martin [66] investigated theoretically the excitation of the LRSPP by a Gaussian beam normally incident onto gratings consisting of either periodic rectangular grooves etched into the top surface of an Au slab, or periodic rectangular Au protrusions deposited onto the top surface of an Au slab. The Au slab was assumed to be free standing (bounded by vacuum), $70\mathrm{nm}$ thick, and the groove depth or protrusion height in the range of about $40–60\mathrm{nm}$. The length of the gratings was set to 5 periods, the width of the incident Gaussian beam matched the grating length, and the other grating parameters were varied (period, duty cycle, and groove depth or protrusion height). Lévêque and Martin predicted an optimal coupling efficiency of 33% from the Gaussian beam into LRSPPs propagating in both directions along the slab (i.e., along $\pm z$) at ${\lambda }_0=633\mathrm{nm}$, using a grooved grating. (They also investigated coupling into the single-interface SPP, using similar grating structures.)

Sellai and Elzain [70] computed the total p-polarized reflectivity from gratings formed by modulating the thickness of the metal slab, predicting sharp reflection dips at specific operating wavelengths where the incident light is efficiently coupled into the $a_b$ and $s_b$ modes of the structure.

Agarwal [56] studied a different kind of structure, consisting of a smooth metal slab bounded by dielectrics of the same index but with the top dielectric being of finite thickness and having a sinusoidal corrugation applied to its top surface. He showed that the excitation of the LRSPP under plane wave illumination leads to a field enhancement near the grating surface.

A less studied geometry for exciting corrugated gratings is where the plane of incidence is rotated 90° in the plane of the grating (azimuthally) such that it is parallel to the grating grooves. This case was investigated experimentally and theoretically by Chen et al. [69] for a $23\mathrm{nm}$ thick Ag slab deposited conformally onto a sinusoidally corrugated $\mathrm{Si}{\mathrm{O}}_2$ substrate, covered with index-matching fluid, and clamped with $\mathrm{Si}{\mathrm{O}}_2$ prisms (a structure similar to those investigated in [67] but without the harmonic in the grating vector). They measured a coupling efficiency of about 24% into the $a_b$ mode and predicted strong coupling into the LRSPP but under different design and excitation conditions.

Korovin [68] developed a formulation for modeling multilayer corrugated gratings based on a curvilinear coordinate transformation, but, in contrast to Chandezon [54], he used an established solution to Maxwell’s equations in Cartesian coordinates having eigenvalues that are determined analytically, thus improving on the efficiency and accuracy of the method. He then applied this formulation to model the reflectance from a $50\mathrm{nm}$ thick Au film bounded by glass ($400\mathrm{nm}$ thick on one side, semi-infinite on the other) with all interfaces conformally corrugated. Reflectance dips due to coupling to the $s_b$ and $a_b$ modes are apparent from his computations.

2.5. Modal Studies [71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104]

A good number of studies have been reported concerning the modes supported by metal slabs [Fig. 1(b)] and variations thereof. The modes are typically obtained by solving a suitably defined 1D boundary value problem based on the wave equations written in the frequency domain for lossy isotropic media. Necessary boundary conditions are applied between layers, and modes are found as solutions to a characteristic (transcendental) equation much in the same way as for 1D dielectric waveguides. The permittivity of the metal layer is usually obtained from the Drude model of the metal or from measurements. Additional 1D layers (metallic or dielectric) can be included in the analysis, and transfer matrices can be developed for the stack (e.g., [91]). The solution approach typically does not involve the discretization of spatial variables, and although the transcendental equation is solved numerically, the approach is essentially analytical.

Tomaš and Lenac [71] derived expressions to estimate the damping properties (lifetime and propagation length) of the LRSPP in thin unsupported metal slabs at long wavelengths.

The study by Stegeman et al. on asymmetric metal slabs $({\epsilon }_1\ne {\epsilon }_3)$ [72] included consideration of leaky modes in addition to the bound modes supported by the structure. The mode nomenclature, $s_b$ or $a_b$, originated here. A cutoff thickness exists in their asymmetric structures below which the LRSPP no longer propagates as a purely bound mode, corroborating the results of Sarid [30].

The follow-up study by Burke et al. [80] reports a comprehensive treatment of the structure and its modes. The bound modes in symmetric structures, and the bound and leaky modes in asymmetric structures, are discussed in detail, and their evolution over good ranges of structure parameters $(t,{\epsilon }_1,{\epsilon }_3)$ and operating wavelength are given. A physical interpretation is given to the leaky modes.

Burton and Cassidy [84] presented modal results for the metal slab at ${\lambda }_0=1.3\mu \mathrm{m}$ for various metals and cladding indices near silica, as a function of slab thickness and asymmetry.

Smith et al. [102] also investigated the metal slab, emphasizing the power dissipation spectra of a dipole into the modes supported, and modal dispersion.

The excitation of the modes was also considered by Burke et al. [80], and a suggestion was made to use end-fire coupling, as sketched in Fig. 6, as the means to excite the modes (following [73], where Stegeman et al. computed high excitation efficiencies for the single-interface SPP). This coupling technique was used by Vaicikauskas [93] to obtain Fourier-transform infrared spectra of the LRSPP in a $\mathrm{Ga}\mathrm{As}∕\mathrm{Au}∕\text{Air}∕\mathrm{Ge}∕\mathrm{Ga}\mathrm{As}$ multilayer 1D structure.

Stegeman and Burke [74] explored the bound modes of two coupled thin metal slabs as a function of their separation for a few refractive index values of the intervening dielectric. They found supermodes of the system consisting of symmetrical and asymmetrical couplings of the $a_b$ mode, and of symmetrical and asymmetrical couplings of the $s_b$ mode. The supermodes formed from coupled $s_b$ modes can be long-range. Yoon et al. [100] computed the dispersion of the symmetrically coupled $s_b$ modes supported by the same structure as a function of the separation between the thin metal slabs for an intervening dielectric that has a higher or lower index compared with the claddings. They found for a high-index intervening dielectric that the dispersion curves insect a common point for all separations between the metal slabs. At the intersection frequency, the effective index of the mode is independent of separation and equal to the index of the intervening dielectric, the propagation length varies linearly with separation, the transverse mode fields have a constant magnitude between the metal slabs (i.e., they are uniform), and the mode fields normalized to the separation are invariant.

Economou [20] also studied the modes of two and three coupled thin metal slabs, but without loss in the metals. Avrutsky et al. [98] investigated the supermodes supported by a multilayer stack constructed from (five) alternating thin metal and (six) dielectric slabs and found (among other modes) one long-range supermode consisting of symmetrical couplings of $s_b$ modes. Davis [103] investigated a similar structure and found two long-range supermodes for their stack (three metal and four dielectric layers), symmetrically and asymmetrically distributed.

Stegeman and Burke [75] investigated the effects of one air gap on the propagation of the LRSPP. In this study, the gap was positioned along the top surface of the metal film and various parameters of the system were altered. In general, the effects of the air gap were found to be deleterious, causing the LRSPP to become cut off for very thin gaps $(\sim 5\mathrm{nm})$. Methods for mitigating air gaps include using index-matched fluids to fill them [75] or carefully controlling the thickness of the gap while compensating for its effects by using a high-index layer deposited on the opposite surface to form an index-matched effective medium [93].

Lenac and Tomaš [77, 78, 79] studied theoretically the metal–dielectric slab system and found that the modes of the system were hybridized or coupled modes of the individual constitutive structures. They found that the LRSPP was very sensitive to the presence of a thin dielectric layer, leading to increased damping. They also studied the frequency dispersion of the system applying Drude [77, 78, 79] and dielectric [79] dispersion models to the metal and dielectric layers, respectively. Yang et al. [82] studied a similar system.

Kou and Tamir [83] proposed placing a high-index dielectric layer within one of the claddings near the metal film, essentially forming a dielectric waveguide therein that couples with the metal slab. When the structural parameters are properly selected, one of the coupled modes of the system retains the character of the LRSPP but has a longer propagation length. Prism coupling to this structure (among others) was discussed in [49, 51].

Guo and Adato [95, 97] proposed to extend the range of the LRSPP in the metal slab by introducing thin low-index layers on both sides of the metal film with high-index outer claddings bounding the system and operating the LRSPP near cutoff, where it has lower loss. In [95], they also investigated the effects of placing thin high-index layers alongside the film, finding that they increase both the confinement and attenuation of the LRSPP. Durfee et al. [101] investigated similar structures and, additionally, the case where only one low-index layer is placed alongside the metal film. Adding low-index layers in this manner [95, 97, 101] allows a thicker metal slab to be used with lower attenuation compared with a corresponding conventional symmetric slab, but care must be taken to operate sufficiently far from cutoff to avoid radiation loss ($n_{\mathrm{eff}}>\text{index}$ of outer claddings). As with the LRSPP in the conventional symmetric structure, consideration to mode size and coupling efficiency must also be given.

Yun et al. [104] investigated theoretically the modes supported by a buried rectangular dielectric waveguide with a thin metal slab bisecting the structure along its horizontal plane of symmetry. The modes investigated are similar in character to the $s_b$ and $a_b$ modes of the metal slab but perturbed by the high-index dielectric core which provides strong horizontal and vertical confinement. Propagation lengths and mode areas are reported revealing that the $s_b$-like mode is capable of long-range propagation when the metal slab is thin enough.

Stegeman [76] studied the metal slab bounded symmetrically by birefringent media $(n_e-n_o=0.05)$, finding that the effective index of the LRSPP was determined primarily by the perpendicular (to the metal slab) refractive index and that birefringence was essentially negligible. Mihalache et al. [89] investigated the modes of thin metal films on a uniaxial substrate having its optic axis in the plane of the film, covered with an isotropic cladding. They computed mode cutoff thicknesses assuming quartz as one cladding ($n_e=1.553$, $n_o=1.544$), glass as the other, various metals, and various orientations of the optic axis.

Wendler and Haupt [81] studied the evolution of the LRSPP as a function of structure asymmetry $({\epsilon }_1\ne {\epsilon }_3)$, showing that the LRSPP cuts off in an asymmetric structure, and that $L_e\to \infty$ as the LRSPP nears cutoff, in agreement with other studies (e.g., [30, 35, 80]). They proposed operating the LRSPP near cutoff, suggesting that an increase in $L_e$ of up to 3 orders of magnitude over the LRSPP in a symmetric structure is achievable. However, in subsequent work [48], they considered the excitation of the LRSPP near cutoff by plane waves assuming prism coupling, and they showed that prism loading limited the increase in $L_e$ to a factor of about 3 over the LRSPP in the corresponding prism-loaded symmetric case, and to a factor of about 2 over the LRSPP in the corresponding unloaded symmetric case (Fig. 11 of [48]). Interestingly, the measurements of Dohi et al. [35], obtained at the same operating wavelength in a system similar to that modeled [48], are in good agreement with these prism-loaded computations: an increase in $L_e$ by a factor of about 3 is apparent for the LRSPP near cutoff compared with the LRSPP of their symmetric case (Fig. 1 of [35]).

Breukelaar and Berini [94] modeled a section of asymmetric metal slab $({\epsilon }_1\ne {\epsilon }_3)$ placed between two butt-coupled symmetric ones $({\epsilon }_1={\epsilon }_3)$, and they computed the insertion loss through the system for end-fire excitation while varying the asymmetry from none $({\epsilon }_1={\epsilon }_3)$ through well beyond cutoff of the LRSPP. Radiation spreading through the asymmetric portion was modeled via normal mode decomposition by discretizing the radiation continuum into an appropriate orthonormal basis. The lowest insertion losses were always obtained for no asymmetry $({\epsilon }_1={\epsilon }_3)$. This is due to the fact that the end-fire coupling efficiency into the LRSPP decreases as the mode nears cutoff because its fields expand more deeply into the higher-index cladding. Indeed, at cutoff the mode fields extend infinitely into the higher-index cladding, and the end-fire coupling efficiency is zero. Thus, while it is true that $L_e\to \infty$ near cutoff [81], this range extension is not readily accessible owing to difficulties in coupling the mode to sources [48, 94]. At present, the literature suggests no extension [94], or extension by a factor of about 3 [35, 48], for the LRSPP in asymmetric structures compared with in symmetric ones.

Zervas [85] presented computations suggesting that a bound LRSPP could be supported by a thin metal slab $\left(16\mathrm{nm}\right)$ in a highly asymmetric structure, well beyond the $s_b$ cutoff point (see the curve labeled $s_b$ in region I of Figs. 2 and 3 of [85]). However, one notes from his computations that the effective index of the mode in the highly asymmetric region is far below the index of one of the claddings $(\beta ∕{\beta }_0<1.2<n_1=1.462)$; so the mode is radiative. A subsequent study by Liu et al. [99] ignored the effective index and this consideration, focusing only on the attenuation constant of the $s_b$ mode.

Tournois and Laude [90] found negative group velocities for the $s_b$ mode in the lossless metal slab $({\epsilon }_I=0)$ when ${\epsilon }_R$ becomes similar to ${\epsilon }_{r,1}$. For metals this condition occurs near the energy asymptote, as is apparent from the computations of Kliewer and Fuchs (Fig. 3 [17]).

As mentioned in Subsection 2.1, the confinement and attenuation of the LRSPP vanish together as the thickness of the metal film is reduced, leading to a trade-off between these two fundamental mode properties (this is apparent from many of the modal studies conducted for the slab, e.g., [80]). Zia et al. [92] reported computations illustrating this trade-off for the metal slab and the metal clads. Berini [96] proposed three figures of merit (FoMs) for 1D waveguides, defined as benefit-to-cost ratios, where three different confinement measures were used as the benefit and attenuation was used as the cost. Closed-form expressions of the FoMs were derived for the single-interface SPP. The FoMs and the quality factor (Q) were used to assess and compare the single interface, the metal slab, and the metal clads, implemented by using Ag and $\mathrm{Si}{\mathrm{O}}_2$, over ${\lambda }_0$’s ranging from the infrared to the SPP energy asymptote. Preferred ${\lambda }_0$’s emerged depending on which FoM was used and thus on how confinement was measured. The largest FoMs were obtained for the LRSPP, but they can also be large for the symmetric mode in the metal clads. Q’s of about 10,000 were found for the LRSPP.

Al-Bader and Imtaar studied the bound [86], leaky [87], and bound hybrid [88] modes of cylindrical structures, comprising a metal film wrapped around a core and surrounded by a cladding. In the case where the cladding is index matched to the core, a long-range mode similar to the $s_b$ mode in the slab is supported [86]. They showed that some of the modes in this structure evolve into those of the corresponding metal slab at large radii.

2.6. Rough Metal Films [105, 106]

There are few studies of the effects of roughness on the propagation of the LRSPP, due largely to the arduous task of producing an accurate theoretical treatment. The effects of roughness are that it modifies the LRSPP’s propagation characteristics, outcoupling (scattering) it into free radiation. These effects are of considerable importance to the LRSPP, especially as attempts are made to produce lower-loss waveguides.

Farias and Maradudin [105] computed the effect of roughness on the propagation length of the $s_b$ mode supported by a rough Ag film of variable thickness and bounded by vacuum at the optical frequencies of $\omega =0.94\times {\omega }_{p,\mathrm{Ag}}$ $({\lambda }_0\sim 330\mathrm{nm})$ and $\omega =0.031\times {\omega }_{p,\mathrm{Ag}}$ $({\lambda }_0\sim 10\mu \mathrm{m})$, where ${\omega }_{p,\mathrm{Ag}}$ is the plasma frequency of Ag. The Ag roughness profiles were taken as identical along both surfaces and characterized by an RMS (root mean squared) deviation of $2\mathrm{nm}$ and correlation lengths of 50 and $100\mathrm{nm}$. They found, as expected, that the propagation length is reduced by roughness, but not very significantly for their chosen (typical) roughness parameters. Extrapolating their results down to a $20\mathrm{nm}$ Ag thickness, one finds that roughness reduces the propagation length by about 10% and 4% at ${\lambda }_0\sim 330\mathrm{nm}$ and ${\lambda }_0\sim 10\mu \mathrm{m}$, respectively, compared with the corresponding smooth films.

Paulick [106] constructed a smooth surface model to account for roughness, by introducing an equivalent surface current incorporating one empirical roughness parameter. Results generated by the model were then compared with the experiments of Inagaki et al. [55, 57], yielding encouraging agreement for the dispersion characteristics of the $a_b$ and $s_b$ modes. He also considered enhanced transmission through an otherwise opaque metal film, in the spirit of [19, 22, 43], but using corrugations instead of prisms to couple the $a_b$ and $s_b$ modes with free radiation. He found strong transmission at normal incidence through a corrugated free-standing $60\mathrm{nm}$ thick Ag film and demonstrates that the $s_b$ mode mediates the transmission, more so than the $a_b$ mode.

Modeling roughness as sinusoidal corrugation [56, 58, 59, 60] simplifies the theoretical treatment and seems reasonable for generating approximate estimates (or better) of the effects of roughness on the propagation of the LRSPP and other modes. However, much work remains to be done to determine the range of validity, and indeed the appropriateness, of such models for the LRSPP subject to real roughness profiles.

2.7. Islandized Metal Films [107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122]

It is natural to try very thin metal slabs in order to increase the propagation length. However, as the thickness of a metal film decreases below a threshold value, its (as-deposited, unannealed) optical parameters begin to differ [108, 110, 112, 113, 114, 115] from the bulk values [109, 111]. The main cause is that the volume mass density of metal films typically decreases from the bulk as the thickness is reduced beyond a threshold value because of the formation of voids. Further reduction in thickness eventually leads to a discontinuous (islandized) film. The volume mass density can start decreasing at a thickness of $t\sim 30\mathrm{nm}$, and as it does, ${\epsilon }_R$ decreases while ${\epsilon }_I$ increases. This trend for the permittivity persists as the film turns into a collection of islands if the film is treated as an equivalent continuous plane parallel film and its thickness and permittivity are interpreted as effective optical values. For an islandized metal film in air (or vacuum), as $t\to 0$, ${\epsilon }_R$ goes through 0, changes sign, and hits a peak value before approaching $-1$, whereas ${\epsilon }_I$ hits a peak value then approaches 0; i.e., ${\epsilon }_r\to 1$ as $t\to 0$ (e.g., [114]). If unmitigated, the main implications of this for the LRSPP are that its attenuation does not drop as rapidly with thickness as suggested by using the bulk permittivity of the metal, and a thickness is eventually reached where the metal permittivity is such that the LRSPP is no longer supported. Annealing treatments postdeposition have a strong impact on the microstructure and thus can significantly improve the quality of a thin metal film [107], so the density problem can be mitigated to some extent. Figure 8 shows trends for the resistivity $\left(\rho \right)$ and the relative permittivity $({\epsilon }_r=-{\epsilon }_R-\mathrm{j}{\epsilon }_I)$ of Au at ${\lambda }_0=632.8\mathrm{nm}$, measured in situ as a function of Au thickness during vacuum deposition (by thermal evaporation at a rate of $0.02\mathrm{nm}∕\mathrm{s}$ onto borosilicate glass substrates held at room temperature); here the bulk values (annealed) are achieved for $t\sim 15\mathrm{nm}$ [110].

Interestingly, long-range surface modes propagating along islandized metal films bounded symmetrically by dielectrics have been demonstrated [116, 117, 118, 119, 120, 121, 122].

Experiments were reported by Yang et al. [116], where a propagation length of $L_e=66\mu \mathrm{m}$ was measured for the long-range mode propagating along an islandized Ag film on quartz covered with index-matching fluid and excited at ${\lambda }_0=632.8\mathrm{nm}$ via prism coupling (the experimental arrangement is sketched in the inset to Fig. 9(a)). Figure 9(a) compares the measured ATR spectrum (crosses) to the fitted theoretical response (solid curve) from which the effective relative permittivity and thickness of the islandized Ag film were extracted [${\epsilon }_{r,2}=-0.55-\mathrm{j}17.19$ and $t=9.2\mathrm{nm}$—the relative permittivity follows the trends shown in Figs. 8(b), 8(c)]. Figure 9(b) shows the computed distribution of the z-directed component of the Poynting vector at $\theta =55.78°$, from which it is noted that essentially no energy propagates within the islandized film, explaining the long range of the mode (the field distribution is otherwise similar to that of the $s_b$ mode in a continuous film).

Takabayashi et al. [117] studied theoretically long-range modes supported by symmetric ultrathin $(<10\mathrm{nm})$ Ag films at ${\lambda }_0=632.8\mathrm{nm}$, using the thickness-dependant permittivity measurements for Ag reported in [114] combined with an effective medium theory. They reported that under certain conditions, s-polarized light can excite a surface wave in this structure. In [118], Takabayashi et al. investigated the impact of absorbing claddings in such structures.

Long-range surface mode propagation lengths in the range of $35–70\mu \mathrm{m}$ were measured by Wu et al. [119] for Au, Cu, Al, and Fe islandized films on a glass substrate, covered with index-matching fluid, and excited at ${\lambda }_0=632.8\mathrm{nm}$ via prism coupling.

Takabayashi et al. [121] measured long-range surface mode propagation lengths of 300 and $480\mu \mathrm{m}$ for islandized Ag films 4 and $2.7\mathrm{nm}$ thick, respectively, on BK7 glass covered with index-matching fluid and excited at ${\lambda }_0=632.8\mathrm{nm}$ by using prism coupling; these propagation lengths are longer by factors of 75 and 120, respectively, than that of the corresponding single-interface SPP.

Kume et al. [122] observed both (TM) p- and (TE) s-polarized long-range modes along a composite layer comprising dispersed and isolated Ag spherical nanoparticles about $4\mathrm{nm}$ in diameter embedded in an $\mathrm{Si}{\mathrm{O}}_2$ film, as a function of the composite layer thickness, which was varied from 13 to $47\mathrm{nm}$. The volume ratio of Ag to the total volume of the $\mathrm{Si}{\mathrm{O}}_2$ film was about 0.05, and the composite film was bounded slightly asymmetrically by $\mathrm{Si}{\mathrm{O}}_2$ claddings. Modes were excited by using prism coupling at ${\lambda }_0=632.8\mathrm{nm}$. The longest propagation length measured was $77\mu \mathrm{m}$ for the p-polarized long-range mode in the $13\mathrm{nm}$ thick composite. They also investigated the effect of prism loading on their responses by varying the thickness of the intervening layer. Theoretical responses to the measurements using a relative permittivity for the composite layer estimated based on Maxwell–Garnett theory $(2.624-\mathrm{j}0.096)$ agreed reasonably well.

Wood et al. [120] studied the long-range mode propagated by islandized $2\mathrm{nm}$ thick Ni films on $\mathrm{Si}{\mathrm{O}}_2$ covered with index-matching fluid and excited via prism coupling at ${\lambda }_0=632.8\mathrm{nm}$. They then attempted to fit a theoretical response to the measurements in order to uniquely determine the optical effective permittivity and thickness of their Ni layer only to find that degenerate fits (i.e., different combinations of ${\epsilon }_R$, ${\epsilon }_I$, t produce equally good fits). They concluded generally that the thickness and permittivity of films cannot both be determined from theoretical fits to prism coupling experiments conducted with any long-range mode and that the thickness is required in order to determine the permittivity.

Though long propagation lengths are evidently achievable on islandized metal films, it is noted that fabricating such structures reproducibly is challenging and that an islandized film eliminates applications where a continuous metal is essential. However, the ability to support s-polarized waves in such structures is an interesting and potentially useful attribute.

2.8. Long-Range Surface Exciton Polariton[116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134]

Kovacs [26, 123] investigated an unusual system, consisting of a thin continuous Fe film bounded symmetrically by cryolite claddings. What renders the system unusual is that the relative permittivity of Fe at his chosen operating wavelength $({\lambda }_0=540\mathrm{nm})$ is ${\epsilon }_R\sim 0$ and ${\epsilon }_I\sim 17$. Modal computations for semi-infinite claddings revealed that the $s_b$ mode had $\beta \gg \alpha$ and $n_{\mathrm{eff}}>n_1$ only for the case of thin films ($t\sim 20\mathrm{nm}$ or less). These inequalities no longer held as t was increased and did not hold for the $a_b$ mode at any t. (The corresponding single-interface does not support a purely bound SPP.) Additionally, it was noted for the $s_b$ mode that no energy propagates in the Fe film in directions parallel or antiparallel to that of modal propagation (i.e., $\pm z$), in contrast to the LRSPP where energy propagates in the metal film along the antiparallel direction. The existence of this mode was confirmed experimentally via prism coupling at ${\lambda }_0=540\mathrm{nm}$ in a structure consisting of an Fe film $18\mathrm{nm}$ thick bounded by about $550\mathrm{nm}$ of cryolite.

Yang et al. [124] also showed that a thin slab of material satisfying ${\epsilon }_R\ll {\epsilon }_I$ can support a long-range surface mode when cladded symmetrically, but rather surprisingly, that its range increases as ${\epsilon }_I$ increases. Since the condition ${\epsilon }_R\ll {\epsilon }_I$ is readily met in continuous materials near an excitonic resonance, they termed the mode long-range surface exciton polariton (LRSEP). Experimental excitation of the mode was conducted at ${\lambda }_0=3.391\mu \mathrm{m}$ via prism coupling to a $\sim 45\mathrm{nm}$ thick V film $({\epsilon }_{r,2}\sim 9-\mathrm{j}48)$ deposited onto a quartz substrate and covered with an index-matched fluid. (The effective optical permittivity of islandized metal films can also satisfy ${\epsilon }_R\ll {\epsilon }_I$; see Subsection 2.7, Fig. 8 and [108, 110, 112, 113, 114]; so the long-range surface wave propagating along them is often termed an LRSEP, although no excitonic resonances are involved.)

In a subsequent paper, Yang et al. [125] reported useful small-t asymptotic expressions for the long-range surface mode and explored them for various limiting cases of ${\epsilon }_{r,2}$. Slight asymmetry $({\epsilon }_{r,1}\cong {\epsilon }_{r,3})$ and absorption in the claddings were also considered. An expression for the propagation constant of the LRSPP in the symmetric structure $({\epsilon }_{r,1}={\epsilon }_{r,3})$ is given, from which it is observed that $\alpha \propto {(t∕{\lambda }_0)}^2$. They found that long-range surface modes exist in a symmetric structure for almost any value of ${\epsilon }_{r,2}$, including the case ${\epsilon }_I=0$ and ${\epsilon }_R<{\epsilon }_{r,1}$, which does not support a single-interface SPP. Expressions for the cutoff thickness and cutoff asymmetry ($\max [{\epsilon }_{r,1}-{\epsilon }_{r,3}]$ and $\max [{\epsilon }_{r,3}-{\epsilon }_{r,1}]$) are derived for the LRSPP in slightly asymmetric structures; the expressions exhibit ${\lambda }_0$ and ${\lambda }_0^{-2}$ dependencies, respectively, indicating that a larger t is required at longer wavelengths to maintain guidance given an asymmetry, and that the cladding permittivities must be more closely matched at longer wavelengths for a given t. They also discussed the role of the light line in the case of absorbing structures, indicating, as had been previously noted [72, 80], that it does not clearly separate the radiative and nonradiative regions of the dispersion curve; this was also observed in [94]. Prism coupling experiments were conducted at ${\lambda }_0=3.391\mu \mathrm{m}$ on $43.5\mathrm{nm}$ thick Pd and $50.4\mathrm{nm}$ thick V films deposited onto a quartz substrate with index-matched fluid forming the other cladding. The Pd (${\epsilon }_{r,2}=-110-\mathrm{j}142)$ and V $({\epsilon }_{r,2}=10-\mathrm{j}49.3)$ films were observed to support LRSPP and LRSEP waves, respectively.

At about the same time, Prade et al. [128] produced a study that included consideration of lossless metal slabs bounded by dielectrics where the permittivities satisfy ${\epsilon }_R<{\epsilon }_{r,1},{\epsilon }_{r,3}$ or ${\epsilon }_{r,3}<{\epsilon }_R<{\epsilon }_{r,1}$ with ${\epsilon }_R$, ${\epsilon }_{r,1}$, and ${\epsilon }_{r,3}$ similar in magnitude. They showed $a_b$ and $s_b$ modes existing for ${\epsilon }_R<{\epsilon }_{r,1},{\epsilon }_{r,3}$ as long as t remains below a cutoff thickness (recall that the purely bound single-interface SPP is not supported for ${\epsilon }_R<{\epsilon }_{r,1}$). In the asymmetric case $({\epsilon }_{r,1}\ne {\epsilon }_{r,3})$ the $s_b$ mode has an additional cutoff thickness at smaller t. The $a_b$ mode has a negative group velocity. For the case ${\epsilon }_{r,3}<{\epsilon }_R<{\epsilon }_{r,1}$, only the $s_b$ mode exists, exhibiting a cutoff thickness at small t and evolving into the single-interface SPP at the ${\epsilon }_R-{\epsilon }_{r,3}$ interface with increasing t.

Yang et al. [126] applied the virtual mode treatment [48] to study theoretically prism coupling to the LRSPP and LRSEP. They demonstrated that the perturbation caused by the prism could increase the propagation length of the LRSEP, as opposed to only reducing it as is generally observed for the LRSPP.

Bryan-Brown et al. [127] excited the LRSEP at ${\lambda }_0=1.52\mu \mathrm{m}$ along Cr films 33.6, 20.9, 17.8 and $11.2\mathrm{nm}$ thick deposited onto quartz substrates and covered by an index-matched fluid. Flat films excited by using prism coupling and corrugated films excited by grating coupling were investigated.

Bryan-Brown et al. [129] investigated at ${\lambda }_0=632.8\mathrm{nm}$ the LRSPP in thick corrugated Pd films and the LRSEP in thin corrugated Pd films both excited by grating coupling. The thickness dependence of the Pd permittivity allowed exploration of both types of wave by using the same material.

Giannini et al. [134] investigated the LRSEP along a thin $(t<50\mathrm{nm})$ amorphous Si layer cladded symmetrically by $\mathrm{Si}{\mathrm{O}}_2$. They investigated a few wavelengths of operation but emphasize the ultraviolet (${\lambda }_0=318$ and $375\mathrm{nm}$) where the relative permittivity of Si satisfies ${\epsilon }_R<{\epsilon }_I$. They found that the LRSEP in this structure slightly outperforms the LRSPP in the corresponding Au structure at these wavelengths, by comparing the propagation length, the field extension, and the associated FoM [96] of each wave. Prism coupling experiments were conducted with broadband light and with a laser source at ${\lambda }_0=375\mathrm{nm}$, on structures comprising a $t=13$ or $20\mathrm{nm}$ thick amorphous Si layer cladded by $\mathrm{Si}{\mathrm{O}}_2$, confirming the excitation of the LRSEP.

Other related studies include those of Crook et al. [130, 131] on an organic thin film, of Takabayashi et al. [132] on a thin Si film, and of Yang and Sambles [133] on indium tin oxide films.

2.9. Nonlinear Interactions [135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168]

The large field enhancement [40] and the long propagation length [28, 29, 30] are the main attributes of the LRSPP motivating interest in nonlinear interactions.

Sarid et al. [135] computed the intensity-dependant propagation constant of the $a_b$ and $s_b$ modes in a symmetric structure consisting of a Cu film bounded by InSb at ${\lambda }_0\sim 5\mu \mathrm{m}$ as a function of the Cu thickness. They found that the power required for a phase shift of $\pi ∕2$ over a distance of $L_e$ is about 1 order of magnitude lower for the LRSPP at $t=15\mathrm{nm}$ compared with the corresponding single-interface SPP.

Stegeman et al. [136] computed the cross sections for copropagating and counterpropagating second-harmonic generation, degenerate four-wave mixing, and intensity-dependant phase shifts, driven by LRSPPs. Ag films cladded symmetrically by CdSe or InSb were considered. Ag thicknesses in the range of $15–35\mathrm{nm}$ were found to be optimal for the nonlinear processes considered. Except for copropagating second-harmonic generation, which is limited by the inability to achieve phase matching, they obtain cross sections that are significantly larger than those related to focused Gaussian beams, stating as a reason that a better trade-off between confinement and $L_e$ can be achieved for LRSPPs.

Deck and Sarid [137] investigated theoretically second-harmonic generation by LRSPPs in a prism coupling arrangement excited at a fundamental ${\lambda }_0$ of $1.06\mu \mathrm{m}$, assuming a nonlinear medium (quartz) as one cladding and an index-matched linear medium as the other. They predicted that the intensity of the outcoupled second-harmonic wave would be 2–4 orders of magnitude greater than in the corresponding Kretschmann–Raether and Otto configurations for the same incident field, and they attributed this improvement to the strongly enhanced fields associated with the LRSPP in this arrangement [40].

Quail et al. [138] verified this prediction [137], measuring an outcoupled second-harmonic signal having an intensity that is 2 orders of magnitude larger than in the Kretschmann–Raether and Otto geometries. The LRSPP was excited at a fundamental ${\lambda }_0$ of $1.06\mu \mathrm{m}$ by using prism coupling in a structure consisting of a $13.5\mathrm{nm}$ thick Ag film on an X-cut quartz substrate covered with index-matching fluid, as shown in Fig. 10(a). The outcoupled second-harmonic signal was observed at a 5° offset compared with the angle of the fundamental because of dispersion in the prism. Figure 10(b) compares the measured second-harmonic reflection coefficient R (defined as the ratio of the second-harmonic irradiance to the square of the fundamental irradiance) to that expected for the single-interface SPP. In a subsequent study, Quail and Simon [146] monitored the transmitted second-harmonic signal generated by the LRSPP in a similar experimental arrangement, noting greater generation efficiency.

Stegeman et al. [139] computed the efficiency of second-harmonic generation by two counterpropagating LRSPPs excited at a fundamental ${\lambda }_0$ of $1.06\mu \mathrm{m}$ by prism coupling in a structure consisting of a $15\mathrm{nm}$ thick Ag film on MNA (2-methyl-4-nitroaniline, an organic crystal; see Ref. 15 of [139]) used as the nonlinear cladding and assuming a linear index-matched material as the other cladding. Second-harmonic generation efficiencies of about ${10}^{-4}$ were predicted.

Moshrefzadeh et al. [140] computed the second-harmonic signal generated by an LRSPP propagating along thin ($t=15\text{\hspace{0.17em}nm}$ and $t=25\mathrm{nm}$) Ag films bearing a $0.5\mathrm{nm}$ thick nonlinear adlayer and bounded symmetrically by benzene. The LRSPP was excited at a fundamental ${\lambda }_0$ of $1.06\mu \mathrm{m}$ by prism coupling. An increase in second-harmonic intensity of ${10}^5$, compared with simple reflection off the corresponding metal surface, was predicted for the $15\mathrm{nm}$ thick Ag case.

Stegeman and Karaguleff [141] investigated theoretically degenerate four-wave mixing via LRSPPs excited by prism coupling at ${\lambda }_0=1.06\mu \mathrm{m}$, predicting conversion efficiencies into the degenerate fourth wave of about 10%–50% for an Ag film $15\mathrm{nm}$ thick on a nonlinear substrate (PTS, bis-(p-toluene sulphonate) of 2,4-hexadiyne-1,6-diol; see Ref. 15 of [141]) and covered with a linear index-matched material.

Liao et al. [142] computed the cross sections for second-harmonic generation by counterpropagating waves in dielectric waveguides and Ag slabs supporting LRSPPs. Fundamental ${\lambda }_0$’s of 1.06 and $10.6\mu \mathrm{m}$ were considered, assuming MNA and CdSe as the nonlinear materials, respectively. The Ag slab was bounded symmetrically by the nonlinear material and the dielectric waveguides incorporated the nonlinear material as either the core or the lower cladding. They find the highest cross sections when the nonlinear medium is used as the core of a dielectric waveguide.

Karaguleff and Stegeman [143] conducted a similar study for degenerate four-wave mixing, at ${\lambda }_0=1\mu \mathrm{m}$ assuming PTS, and at ${\lambda }_0=5.5\mu \mathrm{m}$ assuming InSb, as the nonlinear materials. They reached essentially the same conclusion as for second-harmonic generation [142], which is that the highest cross sections are obtained when the nonlinear medium is used as the core of a dielectric waveguide. They attributed the difference to the better confinement–attenuation trade-off available in the dielectric waveguides. They also noted that the LRSPP cross section in the corresponding prism-coupled geometry [141] is greater by a factor of about two compared with the freely guided (no prism) LRSPP.

Stegeman and Seaton [144] investigated theoretically the modes supported by the metal slab bounded on one or both sides by media with intensity-dependant refractive indices $(n=n_L+n_I{\left|H\right|}^2)$. They assumed lossless Cu films of thicknesses 50 and $20\mathrm{nm}$, and InSb ($n_I<0$, self-defocusing) as the nonlinear medium and give results at ${\lambda }_0=5.3\mu \mathrm{m}$. They found in this case $(n_I<0)$ that the modes approach cutoff with increasing mode power. The details of the approach are different depending one whether one or both claddings are nonlinear, but the $s_b$ mode in the thinnest film (i.e., the LRSPP) bounded on both sides by the nonlinear material approaches cutoff most rapidly. Assuming $n_I>0$ (self-focusing) generates a more complex power dependency, including the observation of a maximum in mode power transmission, mode fields developing maxima away from the film, and new modes existing above minimum power thresholds. Stegeman and Seaton searched for TE-polarized SPP’s in these structures $(n_I>0)$ but did not find them. However, subsequent work [145] revealed their existence for very thin metal films $(t\sim 10\mathrm{nm})$ having a small ${\epsilon }_R$. Ariyasu et al. [150] expanded on this work [144, 145], considering different combinations for the nonlinear claddings, including attenuation.

Mihalache et al. [153, 157] also found TE-polarized SPPs in the self-focusing case, for slightly asymmetric claddings. Boardman and Twardowski [156, 162] studied the interaction between TE- and TM-polarized modes in similar intensity-dependant nonlinear waveguides and considered to some extent the thin metal slab.

Hickernell and Sarid [151] developed a theory for prism coupling to the LRSPP in structures having a metal slab on a substrate exhibiting an intensity-dependant refractive index and covered by an index-matched medium. They found that an intensity 2 orders of magnitude lower than that required for the single-interface SPP is needed to observe bistability via the LRSPP.

Agarwal and Dutta Gupta [152] developed a theory for a multilayer structure on a similar nonlinear substrate and studied bistability with prism-coupled single-interface SPPs (Kretschmann–Raether), and the $a_b$ and LRSPP modes in a thin metal slab. They found that LRSPP bistability occurs at much lower intensity thresholds (by at least 1 order of magnitude).

Nunzi and Ricard [148] observed optical phase conjugation using single-interface SPPs and attributed the main nonlinear contribution in their experiments to heating from the metal film (i.e., thermal effects) rather than to field-induced nonlinearities in the media. They made the points that all of the energy coupled into the SPP is converted into heat in the metal film and that the adjacent dielectric region is where nonlinear interactions are sought. They commented that thermal aspects would remain important in comparable experiments conducted with the LRSPP even though the latter exhibits less attenuation. Sambles and Innes [155] raised similar points, emphasizing the need for consideration of thermal issues even when working with the LRSPP.

Yang and Sambles [164] measured, via prism coupling at ${\lambda }_0=514.5\mathrm{nm}$, optical power-dependant changes in the excitation curve and angle of LRSEPs in a Pd film $4\mathrm{nm}$ thick deposited onto BK7 and bounded on the other side by index-matched fluid. High- and low-power angle scans were conducted, along with power scans at fixed angles. The optical nonlinearities were thermally induced by heating of the Pd film and were deemed to be changes in the index of the matching fluid (thermo-optic effect) and changes in the thickness of the fluid layer (coupling gap).

Quail and Simon [147] observed from modal computations that the $s_b$ mode at the second harmonic had the same phase constant as the $a_b$ mode at the fundamental for a prescribed Ag thickness in their experimental arrangement. Based on this observation, they demonstrated phase-matched copropagating second-harmonic generation by exciting the $a_b$ mode at the fundamental and measuring the prism outcoupled $s_b$ mode (LRSPP) at the second harmonic.

Fukui et al. [149] considered the effects of a finite-width beam in prism-coupled second-harmonic generation experiments, computing the spatial profile of the reflected second-harmonic signal for an incident $2\mathrm{mm}$ wide square profile (stepped intensity) beam. The profiles show a global maximum for a prescribed metal film thickness.

Building on this theory, Schmidlin and Simon [161] measured the profile of the reflected second-harmonic signal in the same experimental situation as that reported in [138], but using a $0.2\mathrm{mm}$ wide output slit (moved via a translation stage) and a $0.5\mathrm{mm}$ wide input beam. They deduced the propagation length of the LRSPP (excited at the fundamental) from the measured reflected second-harmonic spatial profile in the region outside the incident beam width.

Chen and Simon [154, 158] studied experimentally and theoretically second-harmonic generation from LRSPPs excited at a fundamental ${\lambda }_0$ of $\sim 1\mu \mathrm{m}$ in a corrugated Ag film bounded on both sides by quartz, although only one of the quartz claddings was responsible for the generation of the second harmonic owing to the presence of index-matching fluid at the other Ag/quartz interface. They observed enhanced second-harmonic generation due to the LRSPP but conclude that scattering from the grating grooves significantly limits the possible enhancement compared with flat films in a prism-coupled geometry. Phase matching between the fundamental $a_b$ and the second-harmonic LRSPP was also observed in one of the corrugated structures having the appropriate Ag film thickness [158].

Tzeng and Lue [159, 160] studied theoretically second-harmonic generation via prism-coupled excitation of the LRSPP in Ag [159], and Ag, Au, Cu, Al [160] films, bounded by linear dielectrics, where the second harmonic is generated by nonlinearities in the metal (bulk and selvedge regions) and outcoupled by the prism. Electron gas hydrodynamic theory [167] was used to describe the nonlinear response of electrons in the metal. They found an increase in the second-harmonic generation that is 2 orders of magnitude larger for the LRSPP compared with the corresponding single-interface SPP. They also noted effects due to the thickness of the coupling gap. Lue and Dai [166] extended the study to include coupled LRSPPs in structures comprising a pair of Ag slabs bounded on all sides by $\mathrm{Si}{\mathrm{O}}_2$.

Simon et al. [163] and Wang and Simon [165] reported the backscattering of the second-harmonic wave from excitation of the LRSPP via prism coupling at a fundamental ${\lambda }_0$ of $1.06\mu \mathrm{m}$ in thin Ag films on quartz, bounded on the other side by index-matched fluid. The measured backscattered second-harmonic signal exhibited a peak in the direction of the incident beam (antispecular) with a slight angular offset $(\sim 30\mathrm{mrad})$ due to dispersion in the coupling prism.

Li and Zhang [168] compared theoretically the nonlinear coefficients of the single interface, the metal slab, and the metal clads.

2.10. Biosensors [169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195]

The high surface sensitivity of the single-interface SPP has been heavily exploited in (bio)chemical sensors ever since the initial demonstrations [169]. The conventional and mainstream approach to SPP sensing [169, 170] rests on the Kretschmann–Raether configuration (Fig. 4, case 3), where efficient coupling to the single-interface SPP depends strongly on the thickness and index of an adlayer located at the metal/fluid interface (surface sensing) and on the index of the sensing fluid itself (bulk or refractometric sensing). Various modifications and alternatives to this approach have been proposed over time in the quest to achieve greater sensitivity and lower detection limits. Reviews discussing different SPP sensor architectures and interrogation schemes were published by Homola and co-workers [171, 173] and by Chien and Chen [172].

Some of the alternative approaches proposed involve using the LRSPP instead of the single-interface SPP to perform the sensing function along metal slabs. The typical approach uses a prism-coupled arrangement such as that depicted in Fig. 4, case 1, with the lower cladding $\left({\epsilon }_{r,1}\right)$ being the sensing fluid (usually an aqueous buffer) contained within a flow cell, and the upper cladding $\left({\epsilon }_{r,3}\right)$ being a dielectric that has a refractive index closely matched to that of the sensing fluid. Popular materials for this cladding layer are Cytop or Teflon, which have a refractive index slightly above and slightly below (respectively) that of de-ionized water, so the sensing structures are typically slightly asymmetric. Au is normally used as the sensing surface given its chemical stability and the availability of good surface chemistries for this metal.

The use of the LRSPP in a prism-coupled arrangement was proposed as a sensor by Matsubara et al. [174]. They computed the angular response at ${\lambda }_0=632.8\mathrm{nm}$ for structures comprising Ag as the metal film, bounded by ethanol on one side, and by $\mathrm{Mg}{\mathrm{F}}_2$ on the other side also in contact with the prism. $\mathrm{Mg}{\mathrm{F}}_2$ was selected because its index approximately matches that of ethanol. Corresponding Kretschmann–Raether configurations were also modeled. Matsubara et al. measured the angular response of the arrangements in ethanol, demonstrating a narrower response in the case of the LRSPP.

Similar LRSPP structures excited via prism coupling were reported in [178], where Teflon and $\mathrm{Mg}{\mathrm{F}}_2$ were compared as cladding materials and Au was used for the metal film. The approach was demonstrated in bulk sensing experiments.

A similar structure was reported in [180], where Cytop was selected for the cladding. The approach was demonstrated in bioaffinity sensing experiments on Au spots using an imaging prism-coupled arrangement. An improvement of 20% over the conventional Kretschmann–Raether configuration was claimed.

Slavík and Homola [185] demonstrated bulk sensing using a Teflon-Au-water structure. The optimal thickness of the Au film $\left(24\mathrm{nm}\right)$ and of the Teflon layer $\left(1200\mathrm{nm}\right)$ were determined from modeling at ${\lambda }_0=900\mathrm{nm}$ and selected such that the highest coupling and bulk sensitivity would be achieved. Interrogation was conducted by using an LED centered at ${\lambda }_0=830\mathrm{nm}$ and a spectrometer. They reported a bulk sensing detection limit of $2.5\times {10}^{-8}$ and stated that this is the best value reported to date for any bulk sensor based on SPPs. The measured spectral position of the LRSPP dip is shown in Fig. 11 as aqueous sensing fluids of different refractive indices are injected into the sensor. The baseline noise signal is also shown on an expanded scale, from which the noise was deduced to be $1.4\mathrm{pm}$.

Vala et al. [195] compared a similar Teflon-Au-water structure with a conventional Kretschmann–Raether structure for detecting large analytes such as biotinylated latex beads (through a streptavidin bridge) and bacteria (heat-killed Escherichia coli through antibodies). Cutoff of the LRSPP was noted in the case of latex beads for the test structure having the thinnest Au film. The sensitivity of the LRSPP sensor was found to be about $5.5\times$ greater than the conventional Kretschmann–Raether configurations for the detection of E.coli.

Slavík and Homola [183] considered the Teflon-Au-water and $\mathrm{Mg}{\mathrm{F}}_2–\mathrm{Au}$-water structures, optimizing the layer thicknesses ($\mathrm{Mg}{\mathrm{F}}_2$, Teflon, Au) such that efficient prism coupling could be obtained into both the $a_b$ and $s_b$ modes of the same structure under either angular or wavelength interrogation. They found for a properly optimized structure that near perfect coupling can be achieved into both modes and verify this experimentally for a Teflon-Au-water structure under wavelength interrogation. They demonstrated bulk sensing and argue that the availability of two sensing modes having different bulk and surface sensitivities is advantageous for removing unwanted perturbations in biosensor applications. Further work on this approach was reported in [184], including noise and cross-sensitivity analyses and biosensing experimentation with IgE. The approach was compared with a two-channel compensated Kretschmann–Raether configuration.

Hastings et al. [189] also considered dual-mode ($a_b$ and $s_b$) operation in the Teflon-Au-water structure under wavelength interrogation. They showed that monitoring both resonances allows changes in adlayer thickness and changes in the bulk index of the sensing solution to be monitored separately. Through modeling they found optimal designs that minimize the detection limit for surface and bulk sensing by using a Cramer–Rao lower bound to estimate the smallest detectable shift in the coupling dips. They conducted experiments demonstrating the ability to monitor changes in bulk index and changes in surface coverage, the former by alternating between two buffer solutions and the latter via the biotin-streptavidin system (with biotin immobilized onto the sensor surface). The detection limits for bulk and surface sensing were estimated to be $2.3\times {10}^{-5}$ RIU (refractive index units) and $11\mathrm{pg}∕{\mathrm{mm}}^2$, respectively. In [190] they investigated the same concept but under angular interrogation and found a reduction in cross sensitivity compared with wavelength interrogation.

Enhanced fluorescence from tagged antibodies pumped with LRSPPs was demonstrated by Kasry and Knoll [181]. The LRSPPs were excited by prism coupling in a Teflon-Au-water structure and the tagged antibodies were adsorbed onto a $30\mathrm{nm}$ thick polysterene spacer layer deposited on the Au surface. Fluorescence intensities about 22 times larger were observed for the LRSPP pump compared with pumping with the single-interface SPP in corresponding experiments. The enhanced fluorescence intensity was attributed to the enhanced field of the LRSPP [40]. The authors also pointed out that the larger penetration depth into the sensing medium of the LRSPP could lead to further enhancement if a thick matrix were used to bind tagged analyte to the sensor along the third dimension (into the sensing medium). In a subsequent paper, Knoll et al. [192] reported the binding of a hydrogel matrix to a Cytop-Au structure and then demonstrated sensing using the LRSPP of free prostate specific antigen to its antibody immobilized in the matrix.

Dostálek et al. [188] compared prism-coupled Cytop-Au-water and Teflon-Au-water structures at ${\lambda }_0=632.8\mathrm{nm}$. They explored the sensitivity of both structures for bulk and surface sensing and deduced the detection limits from their measured responses and from the width of their reflection dips. Both structures improve on the detection limits of the conventional Kretschmann–Raether configuration, with the Teflon structure being about $14\times$ better for bulk sensing if operated near cutoff, and the Cytop structure being about $2.4\times$ better for surface sensing. They measured the fluorescence intensity of dye pumped by the LRSPP (following [181]) as a function of dye distance from the metal film set by a spacer layer consisting of a protein stack or a thin layer of Cytop. Quenching was noted for a distance of $4\mathrm{nm}$, maximum intensity was measured for distances of $20–40\mathrm{nm}$, and significantly more fluorescence was measured compared with the conventional Kretschmann–Raether configuration. Their setup for conducting fluorescence studies in this manner is sketched in Fig. 12. They also deposited a series of Au films of thickness ranging from $t=15\mathrm{nm}$ to $t=45\mathrm{nm}$ on Cytop and on Teflon and extracted the optical parameters of their films by fitting measured ATR spectra to theoretical responses. They observed some substrate dependence in the extracted optical parameters, probably due to differences in the roughness of the starting surfaces, but otherwise observe the same trends in the relative permittivity with decreasing t as discussed in Subsection 2.7. The bulk optical parameters for Au were achieved in their cases at $t\sim 25\mathrm{nm}$.

Wang et al. [193] demonstrated a biosensor capable of detecting aflatoxin ${\mathrm{M}}_1$ $\left({\mathrm{AFM}}_1\right)$ in milk using a Cytop-Au structure. The sensor detects LRSPP pumped fluorescence emitted from labeled antibodies (Cy5-GAR) bound to the sensor surface following the inhibition immunoassay format (Au/thiol/BSA-${\mathrm{AFM}}_1$ (where BSA is bovine serum albumin) immobilized onto the sensor followed by detection of ${\mathrm{aAFM}}_1$/Cy5-GaR).

Dostálek et al. [194] proposed a biosensor using a pair of broadside coupled thin Au slabs separated from each other and from the prism (on one side) by thin layers of Cytop. As discussed in Subsection 2.5 (with regards to [74]), coupled thin slabs support supermodes consisting of symmetrical and asymmetrical couplings of $a_b$ and $s_b$ modes, and when the metals are thin, the two supermodes involving the $s_b$ mode are long range. In [194] the symmetric and asymmetric long-range supermodes are excited at different angles of incidence at the same operating wavelength. The modes have different probing depths into the sensing medium; so they exhibit different bulk and surface sensitivities. Thus, by fitting measured ATR spectra involving both supermodes to theoretical responses (computed using the transfer matrix method), the thickness and refractive index of an adlayer can be extracted. Dostálek et al. demonstrated the approach by characterizing hydrogel layers (thickness and index) deposited onto their sensors and by observing the diffusion of BSA into the hydrogels.

Rajan et al. [186] modeled a sensor consisting of a multimode step-index fiber with the cladding removed over a length, whereupon an LRSPP supporting structure is deposited directly onto the core, the structure consisting of a Teflon layer followed by a thin Au film covered with the sensing fluid. The bulk sensitivity of the sensor was determined under wavelength interrogation as a function of structure parameters. In a subsequent paper, Jha et al. [191] studied a similar structure with tapered fiber sections added on either side of the sensing region.

Chen et al. [187] investigated theoretically the temperature stability of prism-coupled LRSPP sensors in a Cytop-Ag-water configuration by assuming a temperature-dependant model for the permittivity of the Ag film. They found that the configuration is thermally stable over a large temperature range $\left(250\mathrm{K}\right)$, but they ignored the temperature dependence of the prism (SF10), and of Cytop and water (which have large thermo-optic effects).

In the above-described studies, the index symmetry required to ensure propagation of the LRSPP was provided by selecting a cladding material that is closely index-matched to the index of the sensing fluid. An alternative approach involves using a thin high-index layer, which when taken in combination with the low index of the sensing fluid, creates an effective index that closely matches the index of the material on the other side [175]. Such a structure is physically asymmetric but appears symmetric from an effective medium point of view. Sensing using the LRSPP in such a structure has been considered, with coupling provided by a corrugated grating [176, 179], and where one of the claddings comprises two layers, a thin high-index dielectric $\left({\mathrm{Si}}_3{\mathrm{N}}_4\right)$ layer followed by the sensing medium (air or water), such that the effective index of the combination matches the index of the silica cladding on the other side of the metal film.

A similar idea was applied in [182] to create an LRSPP sensor based on prism coupling, but where a Teflon/${\mathrm{Ta}}_2{\mathrm{O}}_5$ bilayer system was used to match (effectively) the index of the aqueous medium on the other side of an Au film. This approach was demonstrated in bulk sensing experiments.

The LRSPP in Kou and Tamir’s structure [83] was considered theoretically in [64] for sensing by using a corrugated grating as the coupling means. Liao et al. [177] explored theoretically prism-coupled LRSPPs in a multilayer system and commented on the suitability for chemical sensing. (Supporting the metal slab on a 1D finite photonic crystal as discussed in Subsection 2.L [237] or suspending the metal slab as discussed in Subsection 2.4 [55] and Subsection 3.9 [342] represents a further alternative to satisfying the index symmetry requirement for sensing with the LRSPP.)

2.11. Emission and Molecular Scattering [196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217]

The decay of excited molecular emitters directly into SPPs via the near field [196, 197] has been a subject of intense study [198]. Reports involving the LRSPP in this process (and the like) are reviewed here.

Gruhlke et al. [199] observed optical emission mediated by the LRSPP and $a_b$ modes in symmetric corrugated Ag films of various thicknesses $(18–61\mathrm{nm})$, whereby molecular emitters adjacent to the Ag film excited directly the modes, which were then outcoupled into free radiation by the corrugation. The dispersion of the modes over a broad spectral range $(580–880\mathrm{nm})$ was also measured via this emission process. The LRSPP in these experiments was responsible for most of the observed emission.

In a subsequent publication [200], Gruhlke and Hall indicated that in their experiments [199, 200] the excited molecules are on one side of the metal film and the emission is observed on the other side. They emphasized that the process provides a dispersive channel through their otherwise (nearly) opaque metal films (as in studies involving prism coupling [19, 22, 43]). Figure 13 shows the measured radiation spectrum emitted from the symmetric structure sketched at the origin of the polar plot. Fluorescent molecules are located in the lower photoresist layer and are pumped at ${\lambda }_0=514.5\mathrm{nm}$ through the backside at normal incidence. The spectrum is normalized to the intensity measured from a sample that contains only the lower (fluorescent) photoresist layer. The emitted intensity is $\sim 30\times$ stronger at ${\lambda }_0=780\mathrm{nm}$ owing to mediation by the LRSPP. Indeed, the fluorescence reradiated by the LRSPP into this peak is $\sim 30\%$ of that radiated isotropically (at the same wavelength) in the absence of the metal layer. They stated that field enhanced fluorescence likely had a role to play.

Leung et al. [201] computed the decay rate of a dipole placed $20\mathrm{nm}$ above asymmetrically and symmetrically cladded (free-standing) corrugated gratings. Among all the cases considered, the strongest enhancement of the rate was obtained for coupling into the LRSPP of a $10\mathrm{nm}$ thick free-standing corrugated Ag grating, where the decay rate was about $100\times$ that of a free dipole.

Lenac and Tomaš [202, 203] derived the cross sections for absorption of the $s_b$ and $a_b$ modes by molecules placed above the metal in a symmetric configuration as a function of t. They found decreasing and increasing cross sections with decreasing t for the $s_b$ and $a_b$ modes, respectively. They also found larger cross sections at shorter ${\lambda }_0$’s. The differences were attributed to differences in modal confinement.

Tomaš and Lenac [204] also derived the cross sections for scattering of the $s_b$ mode $(\beta ,\omega )$ into the $s_b$ mode $({\beta }^{\prime },{\omega }^{\prime })$ (as well as other modal combinations) by molecules placed above the metal in a symmetric metal slab as a function of t. They found that the cross section decreases as t decreases and estimated that the LRSPP scattering cross section along an Ag film $10\mathrm{nm}$ thick is 1–2 orders of magnitude smaller than that for scattering between single-interface SPPs. They also commented that the LRSPP cross section would compete with the LRSPP field enhancement in prism-coupled arrangements [40], such that surface-enhanced Raman scattering (SERS) mediated by prism-coupled LRSPPs is not likely to be significantly more enhanced than SERS mediated by prism-coupled single-interface SPPs.

In a subsequent paper [205], Lenac and Tomaš derive the power lost by a molecular dipole to the $s_b$ and $a_b$ modes of a symmetric metal film as a function of t, with and without prism outcoupling. The dipole was located near and below the metal film within the bottom cladding. Their computations without prism outcoupling (freely guided) showed decreasing and increasing power coupled into the $s_b$ and $a_b$ modes, respectively, with decreasing t. For an Ag film $15\mathrm{nm}$ thick, about $30\times$ more power is coupled into the $a_b$ mode than into the LRSPP at ${\lambda }_0=514.5\mathrm{nm}$; about $5\times$ more power would be coupled into the corresponding single-interface SPP. However, for a dipole located farther from the metal, the LRSPP is preferentially excited, since its fields penetrate more deeply into the cladding than those of the $a_b$ mode. The angular distribution of power $(\mathrm{d}P∕\mathrm{d}\Omega )$ emitted into the prism by outcoupling from the $s_b$ and $a_b$ modes, themselves excited by the dipole, was computed, showing that the emission is dominated by the $s_b$ mode rather than the $a_b$ mode. The computations for the $s_b$ mode with decreasing t showed a decrease in the angular width of emission and an increase in the peak of the angular distribution of emitted power (i.e., in $\max \{\mathrm{d}P∕\mathrm{d}\Omega \}$). The peak is about $1000\times$ greater for mediation via the LRSPP in an Ag film $15\mathrm{nm}$ thick than for a free dipole in the same medium as the claddings. Computations for mediation via the single-interface SPP in the corresponding (optimized) Kretschmann–Raether configuration yield a peak about $200\times$ greater, so mediation via the LRSPP produces an improvement of about $5\times$. They also estimate a SERS intensity enhancement of about $30\times$ for mediation via the LRSPP versus the single-interface SPP in prism-coupled geometries. Thus, although the coupling of a dipole to the freely guided LRSPP (no prism) and the molecular scattering cross section of freely guided LRSPP’s [204] are lower than for the corresponding single-interface SPP, the field enhancement associated with prism coupling [40] compensates sufficiently to produce greater and sharper emission peaks for the LRSPP. Subsequent calculations comparing SERS mediated by single-interface SPPs in the Otto geometry with mediation by the prism-coupled LRSPP also yield an enhancement of about $30\times$ in the peak for the LRSPP [206].

Barnes and co-workers have addressed emission extraction through metal films in light emitting diodes (e.g., [198, 207, 208, 209, 210, 211]) for different device architectures and from different perspectives, such as the direct coupling of molecular emitters to SPP modes and the conversion of SPPs into free radiation. The most promising approaches to date involve mediation via the coupled modes of an effectively symmetrical thin metal contact [208, 209, 210, 211], where symmetry is achieved by coating the top of the metal contact with a thin dielectric overlayer of a prescribed thickness, which taken in combination with air yields an effective medium that is index matched to the underlying active medium. Outcoupling of the $a_b$ and $s_b$ modes to free radiation is achieved via corrugations in the full structure or in the thin dielectric overlayer. The metal contact in these studies is $43–55\mathrm{nm}$ thick (Ag).

Chiu et al. [216] compared the measured emission from a two-layer structure consisting of $\mathrm{Au}∕\mathrm{Al}{\mathrm{q}}_3$ on photoresist on Si with the emission from a four-layer structure consisting of $\mathrm{Au}∕\mathrm{Al}{\mathrm{q}}_3∕\mathrm{Au}∕\mathrm{Al}{\mathrm{q}}_3$ on photoresist on Si. The $\mathrm{Al}{\mathrm{q}}_3$ layers were $50\mathrm{nm}$ thick, and the Au layers were $20\mathrm{nm}$ thick. Each structure was formed into a 1D rectangular lamellar grating having the purpose of outcoupling the surface plasmons propagating along the structure into free radiation. The structures were pumped by $405\mathrm{nm}$ light, and the $\mathrm{Al}{\mathrm{q}}_3$ molecules decay into the modes of the structure including the surface plasmon modes supported therein. Enhanced emission in the range of ${\lambda }_0=500\mathrm{nm}$ to ${\lambda }_0=650\mathrm{nm}$was measured from the four-layer structure compared with the two-layer structure and a flat (planar) structure. The enhanced emission was attributed to mediation by LRSPPs. The modes supported by these multilayer structures were not investigated at the operating wavelengths.

Andrew and Barnes [212] demonstrated the transfer of energy between donor and acceptor molecules (dipoles) through thin Ag films, with the transfer being mediated by the $a_b$ and $s_b$ modes. The Ag film was cladded below by a $60\mathrm{nm}$ thick layer of $\mathrm{Al}{\mathrm{q}}_3$-doped polymethyl methacrylate (PMMA, donor) on $\mathrm{Si}{\mathrm{O}}_2$, and above by a $60\mathrm{nm}$ thick layer of R6G-doped PMMA (acceptor, where R6G is Rhodamine 6G) , as sketched in Fig. 14(a). The structure is essentially symmetric at the wavelengths of interest. Figure 14(b) shows the measured emission spectrum of an $\mathrm{Al}{\mathrm{q}}_3$-doped layer only, and the emission and absorption spectra of an R6G-doped layer only. The emission spectrum of the $\mathrm{Al}{\mathrm{q}}_3$-doped layer overlaps strongly with the absorption spectrum of the R6G-doped layer (in the range ${\lambda }_0\sim 525\mathrm{nm}$ to ${\lambda }_0\sim 550\mathrm{nm}$); so, when pumped from the back side (at ${\lambda }_0=408\mathrm{nm}$), the excited $\mathrm{Al}{\mathrm{q}}_3$ molecules can donate their energy to the R6G molecules through interaction with the $a_b$ and $s_b$ modes of the structure. The R6G molecules then decay spontaneously at ${\lambda }_0\sim 575\mathrm{nm}$, and part of this emission is captured by the optical detection setup aligned with the top of the structure. Figure 14(c) shows the measured spectrum obtained from the structure for the case of a $t=60\mathrm{nm}$ thick Ag film (black curve), along with the spectra of two control samples, each having only one type of dipole—donor only (no R6G, blue curve) and acceptor only (no $\mathrm{Al}{\mathrm{q}}_3$, red curve). The emission was significantly larger than in the case of the control samples, indicating efficient energy transfer through the Ag film via mediation by the $a_b$ and $s_b$ modes. The mediation accounted for 70% of the total emission.

Okamato et al. [213] suggested that lasing in the LRSPP in a corrugated grating bounded symmetrically by gain media [4-dicyanomethylene-2-methyl-6- (p-dimethylaminostyryl)-4H-pyran (DCM)-doped $\mathrm{Al}{\mathrm{q}}_3$, peak ${\lambda }_0=620\mathrm{nm}$] could be achieved, by pointing out that the loss of the LRSPP in flat Ag films thinner than $60\mathrm{nm}$ is lower than the available gain in the medium considered. In their concept, standing LRSPP waves would be amplified by stimulated emission and partially outcoupled by the grating forming an output laser beam.

Winter et al. [214] investigated further the concept proposed by Okamoto et al. [213], considering the existence of the $a_b$ mode as well as the LRSPP. Photoluminescence measurements on (effectively) symmetric corrugated structures similar to those reported in [210], where the Ag film thickness was varied from 20 to $90\mathrm{nm}$, show that a significant amount of power is indeed coupled into both the $a_b$ and $s_b$ modes. They also compute the fraction of total power coupled by a dipole emitting at ${\lambda }_0=620\mathrm{nm}$, and positioned 20 and $90\mathrm{nm}$ away from a flat symmetrically cladded Ag film, into the $a_b$ and $s_b$ modes of the structure, as a function of t. For a dipole separation of $20\mathrm{nm}$ and for $t=20\mathrm{nm}$, about 80% of the power is coupled into the $a_b$ mode while about 7% couples into the LRSPP. However, for a dipole separation of $90\mathrm{nm}$ and for $t=20\mathrm{nm}$, about 15% of the total power is coupled into the $a_b$ mode, and about 15% is coupled into the LRSPP. For larger dipole separations, more power would (proportionally) be coupled into the LRSPP than into the $a_b$ mode, a trend consistent with earlier computations [205], owing to the greater extension of the LRSPP mode fields into the cladding. They then estimate the gain available to each mode by ascribing to each a fraction of the total gain taken to be the same as the fraction of total power coupled in from a dipole at a particular location. Taking into account in this manner the gain lost to the $a_b$ mode, their results suggest that lasing in the LRSPP would be possible, but an Ag film thinner than about $15\mathrm{nm}$ would be needed.

Wang and Zhou [215] considered the prospects for amplification of LRSPPs in a structure consisting of an Au film on Si and covered with a multilayer system consisting of alternating Si and Er:Si nanolayers. Ignoring the $a_b$ mode, they predicted net gain into the LRSPP at ${\lambda }_0=1500\mathrm{nm}$ for Au films thinner than about $20\mathrm{nm}$.

De Leon and Berini [217] proposed a model for SPP amplification that accounts for the nonuniform gain distribution of a dipolar gain medium pumped at broadside and placed along a symmetric metal slab. The model takes into account four channels for excited state decay (coupling to $a_b$, $s_b$ and radiative modes, and coupling to electron–hole pairs in the metal), leading to a position-dependant dipole lifetime. Additionally, the model uses a realistic pump irradiance distribution within the gain medium as computed by using a transfer matrix method. The rate equations for the standard four-level pumping model are then applied locally, with the lifetime and irradiance taking on their position-dependant values, leading to a nonuniform gain distribution. The distribution is then incorporated into a multilayer waveguide mode solver from which mode power gains are computed for the modes of the system. Using this approach, they predict that net amplification of the LRSPP is possible in the visible $({\lambda }_0=560\mathrm{nm})$ by using a reasonable concentration of R6G molecules $\left(3\mathrm{mM}\right)$ and a reasonable pump irradiance ($210\mathrm{kW}∕{\mathrm{cm}}^2$ at ${\lambda }_0=532\mathrm{nm}$, pulsed) assuming a $20\mathrm{nm}$ thick Ag slab bounded by Cytop on one side and the index-matched gain medium on the other.

2.12. Other Studies [218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245]

Other studies of the LRSPP in the metal slab include investigations involving magnetic materials [218, 219, 220, 233, 236], electro-optic materials [221, 226, 229, 230, 235] and photonic crystals [237, 245], studies of polarizing devices [222, 223, 224, 225, 228, 231, 234] and pulse reshaping devices [227], a study of propagation through gaps in the metal slab [238], and a study of bending out of plane [241]. Electro-optic and filtering devices involving the LRSPP propagating across an array of metal nanowires have been explored [242, 244]. The LRSPP supported by semiconductor heterostructures has been investigated [240]. Approaches to extend the range of the single-interface SPP by involving a large multimode waveguide [232, 243] or by using an anisotropic photonic crystal [239] have been reported.

Sarid [218, 219] modeled the modes supported by a metal film bounded by identical magnetic semiconductor claddings subject to a magnetization field in the plane of the metal film and perpendicular to the direction of propagation of the modes (transverse magnetization). He found that the range of the LRSPP decreases with increasing magnetization owing to the destruction of symmetry in the dielectric claddings. Cutoff thicknesses were noted for the LRSPP.

Hickernell and Sarid [220] investigated LRSPPs in a thin magnetic metal film (Ni) bounded symmetrically by dielectrics under transverse magnetization. They computed the change in the propagation constant of the LRSPP as a function of t due to the magnetization, showing that the change decreases as t decreases. The differential reflectance of a thin Ni film due to the application of a magnetic field was measured by using prism coupling through the excitation angle of the LRSPP. A small magnetically induced modulation was detected.

Sepúlveda et al. [233] studied the metal slab bounded on both sides by magneto-optic dielectric claddings (yttrium iron garnet, YIG), or bounded on one side by YIG and on the other by an index-matched nonmagnetic dielectric. Both configurations were considered under different magnetization directions (transverse, longitudinal). Nonreciprocity was predicted for both the $s_b$ and the $a_b$ mode in certain cases and was quantified with decreasing t. Their results indicate that nonreciprocal LRSPPs can be obtained, in particular, for transverse magnetizations of opposite direction in the lower and upper claddings for the YIG cladded film, and for both transverse magnetization directions in the case of the YIG/dielectric cladded structure. They also discussed the LRSPP in a ferromagnetic metal film (Co) bounded by dielectrics.

Khurgin [236] studied a similar system, consisting of a thin metal film bounded on one side by a magneto-optic material (bismuth-doped gadolinium iron garnet, Bi:GdIG) and on the other by an index-matched dielectric. He also concluded that nonreciprocal LRSPPs are possible in this structure under transverse magnetization.

Device applications using electro-optic materials include modulators [221, 226, 230] and a tunable filter [229], where the LRSPP is excited via prism coupling in a thin metal film bounded on one side by an electro-optic dielectric, and on the prism side by an index-matched dielectric. The electric field is applied to the electro-optic medium via the metal film and an additional electrode deposited onto its other surface. The narrow width of the LRSPP reflection dip improves the performance of the devices compared with the single-interface SPP.

Liu and Xiao [235] proposed and modeled an electro-optic switch consisting of a lossless metal slab bounded on both sides by an electro-optic material (${\mathrm{Ba}}_{0.5}{\mathrm{Sr}}_{0.5}\mathrm{Ti}{\mathrm{O}}_3$, BST). Switching occurs by varying electro-optically the phase (beating) between the $a_b$ and the $s_b$ mode of the slab and taking as the outputs of the switch the top cladding in isolation and the bottom in isolation.

Konopsky and Alieva [237] reported the excitation of LRSPPs on a thin $\left(5\mathrm{nm}\right)$ Au film bounded by air on one side and on the other by a finite 1D photonic crystal implemented as a multilayer ${\mathrm{Ta}}_2{\mathrm{O}}_5∕\mathrm{Si}{\mathrm{O}}_2$ stack on a BK7 prism. Semi-infinite 1D photonic crystals support Bloch surface waves in the bandgap, having fields that oscillate and decay exponentially into the crystal. Thus the full structure (finite 1D photonic crystal, intermediate dielectric layer, metal slab) was designed such that essentially the same LRSPP field distribution was achieved within the metal and air regions as in the corresponding hypothetical air–metal–air structure. An LRSPP propagation length estimated to be about 2 orders of magnitude longer than that of the single-interface SPP in the corresponding Kretschmann–Raether configuration was measured. In subsequent work [245], the authors described a similar structure, shown in Fig. 15, but using an $8\mathrm{nm}$ thick layer of Pd as the slab. The computed ATR angular reflection spectrum is sketched as the red curve in Fig. 15, showing the intensity dip due to coupling to the LRSPP and fringes due to interference between the outcoupled field and the reflected beam (see also Subsection 2.3). They demonstrated sensing of 3% ${\mathrm{H}}_2$ in a ${\mathrm{N}}_2$ atmosphere by using this structure. A finite 1D photonic crystal provides a practical means to support the metal slab for an arbitrary dielectric bounding its other side (air, water, etc.) [237, 245], but the design is specific to the operating wavelength, metal slab (thickness and index), and index of the bounding medium.

The LRSPP has a role to play in polarizers and in polarizing couplers and splitters [222, 223, 224, 225, 231, 234], principally because of the strong coupling that can be achieved with fibers and dielectric waveguides in a broadside arrangement. In [228], specific modes of a dual-mode fiber were selectively excited by coupling to the LRSPP in a structure similar to that of Kou and Tamir [83], and where the LRSPP was excited using prism coupling.

Andaloro et al. [227] studied theoretically the reshaping of pulses on reflection from prism-coupled metal films where either LRSPPs or single-interface SPPs are excited.

Sidorenko and Martin [238] investigated the tunneling of SPPs across a perpendicular gap in the metal slab filled with the background dielectric, for symmetrically and asymmetrically cladded structures, as a function of the gap length and metal slab thickness. Results at ${\lambda }_0=785\mathrm{nm}$ show that the LRSPP on a $40\mathrm{nm}$ thick Au film can tunnel through a $5\mu \mathrm{m}$ long gap while retaining 80% of its field amplitude. Standing wave patterns in front of the gap are caused by reflection of the LRSPP by the gap. Transmission of the $a_b$ mode through the gap is much less efficient and excites the LRSPP on the other side.

Sun [241] investigated bending along metal slabs that are curved out of plane [radius of curvature in the $y\text{–}z$ plane, Fig. 1(b)]. The metal slab was cladded symmetrically by thin dielectrics, and the structure was bounded by air. The dielectric/air interfaces provide additional confinement to the LRSPP along this direction, allowing low-radiation-loss bends. High transmission levels were computed by using the finite difference time domain method for the LRSPP around 90° bends having radii on the order of the free-space wavelength of operation.

Wu et al. [242] considered a structure where the metal slab was replaced with an array of ($10\mathrm{nm}$ thick) metal nanowires, cladded on one side by an electro-optic polymer and on the other by an index-matched glass substrate. They computed ATR angular spectra for the structure under prism coupling, showing a narrower dip compared with a continuous metal slab of the same thickness, due to the excitation of the LRSPP across the wires. Low-loss surface waves can be supported along such a structure, as in discontinuous and islandized films (Subsection 2.7). The narrower dip leads to improved modulator performance compared with, e.g., [221]. In a subsequent study [244] they investigated a similar structure, with the electro-optic polymer replaced by glass, used as a notch filter under prism-coupled wavelength interrogation. They reported filter designs having a $3\mathrm{nm}$ bandwidth and tunable over about $45\mathrm{nm}$ by varying the angle of incidence of the input light.

Plumridge and Phillips [240] considered theoretically the SPPs and the dielectric modes guided by a multiple quantum well heterostructure. The quantum wells are modeled as a quasi-2D electron gas, where the electrons are free to move in the plane of the well as a gas of almost free electrons, but are confined and restricted to intersubband transitions in the perpendicular plane. The main electric field component of the LRSPP is perpendicular to the wells (i.e., not in the plane of the electron gas); so damping is due principally to intersubband transitions (i.e., not to free-electron scattering), which by design can have energies far from that of the LRSPP, leading to lower propagation loss. Structures were modeled by using an anisotropic permittivity for the quantum wells, based on the Drude model for the permittivity components in the plane, and a Lorentz oscillator model for the permittivity along the perpendicular. The model was applied to a $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ multilayer structure, and the LRSPP (among other modes) was explored in the infrared (${\lambda }_0=6\mu \mathrm{m}$ to ${\lambda }_0=20\mu \mathrm{m}$). Propagation lengths of many centimeters were predicted.

Lee and Gray [232] proposed a Kretschmann–Raether type of structure for re-exciting single-interface SPPs, whereby the thin metal film is placed directly onto the core of a large multimode dielectric waveguide. The multimode waveguide effectively traps the reradiated light outcoupled by the prism (and normally lost) and redirects it toward the metal film at the proper angle for re-excitation of the single-interface SPP. Simulations confirm the viability of range extension via this approach. Montgomery and Gray [243] further explored the concept and its design space through finite difference time domain simulations and modal analyses.

Krokhin et al. [239] derived the dispersion relation for the SPPs propagated along a thin metal film bounded by vacuum on one side and a strongly anisotropic photonic crystal on the other. They showed for the SPP localized at the metal–crystal interface that its range can be increased by 50% in the infrared, compared with the corresponding isotropic case, by orienting the optical axis of the substrate along the perpendicular to the metal film. The penetration depth of the SPP into the substrate increases with its range.

3. Metal Stripe $(w<\infty )$

3.1. Modes of the Metal Stripe

The thin metal film of thickness t, finite width w, and relative permittivity ${\epsilon }_{r,2}$, bounded by optically semi-infinite dielectrics of relative permittivity ${\epsilon }_{r,1}$ and ${\epsilon }_{r,3}$, is sketched in front cross-sectional view in Fig. 16 and is henceforth referred to as the metal stripe. The metal stripe is obtained from the metal slab (Fig. 1(b)) by limiting its width. This increase in dimensionality leads to major changes, principally, the creation of lateral confinement and thus of an enriched mode spectrum, and an LRSPP having a lower attenuation than its counterpart in the corresponding metal slab. Another important change is that modal solutions to Maxwell’s equations must be obtained numerically, increasing considerably the analysis effort, whereas the modes of the slab can be derived analytically rather straightforwardly. Despite this difficulty, the metal stripe can be handled by well-established numerical techniques and by some commercial modeling tools, if appropriate care is taken. Theoretical studies using vectorial formulations of the method of lines (MoL), the finite element method (FEM), and the finite difference method (FDM) have been reported. The effective index method (EIM) has also been shown to approximate reasonably well some of the modes, including the LRSPP.

There are four fundamental modes supported by the metal stripe, labeled $aa_b{}^0$, $as_b{}^0$, $sa_b{}^0$, and $ss_b{}^0$. Given the finite width of the structure, higher-order modes having extrema along x in their field distribution can also be supported. The $E_y$ field component dominates for all modes when $w∕t\gg 1$; so the modes are TM in character, but not purely TM because all field components including $H_z$ are always nonzero.

The nomenclature adopted extends that used for the slab and describes the $E_y$ field component of the mode: a and s refer to asymmetric and symmetric, respectively, the first position being associated with the horizontal dimension $\left(x\right)$ and the second with the vertical one $\left(y\right)$; b signifies purely bound (nonradiative), and the superscript counts the number of extrema in the horizontal distribution of $E_y$, not counting the corner peaks.

The evolution of the modes with dimensions $(w,t)$, materials $({\epsilon }_{r,1},{\epsilon }_{r,2},{\epsilon }_{r,3})$ and operating wavelength $\left({\lambda }_0\right)$ is complex, especially for asymmetric structures $({\epsilon }_{r,1}\ne {\epsilon }_{r,3})$. The difficulty arises because all modes are supermodes (coupled modes) created from the coupling of elemental corner and/or edge (or finite-width interface) modes, with the selection of specific ones depending on the similarity of their phase constants. Since the elemental modes also change with structure parameters and operating wavelength, the supermodes can at times evolve unpredictably. However, trends have been noted over a range of studies pertaining to symmetric structures $({\epsilon }_{r,1}={\epsilon }_{r,3})$, as discussed in the following paragraphs.

The evolution of the modes as $t\to$ 0 resembles the evolution of the $s_b$ and $a_b$ modes of the symmetric metal slab in that all modes eventually become partitioned into either lower-attenuation ($ss_b{}^m$ and $as_b{}^m$ are $s_b$-like) or higher-attenuation ($aa_b{}^m$, $sa_b{}^m$ are $a_b$-like) modes, as determined by the distribution (symmetric or asymmetric) of $E_y$ along y. This partitioning is readily observed from Fig. 17 in the case of a $w=1\mu \mathrm{m}$ wide stripe (compare Fig. 2). Unlike in the metal slab, the modes in the metal stripe are not asymptotic with the single-interface SPP as w and t increase. Instead, the four fundamental modes are asymptotic and degenerate with an elemental corner mode, and the higher-order modes are asymptotic with elemental edge modes, as is also readily observed from Fig. 17.

One of the fundamental modes, the $ss_b{}^0$ mode, evolves smoothly and predictably as w, $t\to 0$ into the vertically polarized TEM wave of the background. Its $E_y$ and $H_x$ fields evolve from being highly localized near the metal corners, as shown in Figs. 18(a), 18(c), to being spread out over the waveguide cross section, as shown in Figs. 18(b), 18(d). (Figure 18 plots the normalized $\mathrm{Re}\left\{S_z\right\}$, where $S_z=(E_{x}H_y{}^{*}-E_{y}H_x{}^{*})∕2$ is the complex power density carried by the mode.) The resulting field distribution [Figs. 18(b), 18(d)] can be well matched to Gaussian-like fields, such as those emerging from dielectric waveguides (e.g., single-mode fiber, SMF), leading to efficient end-fire excitation. This modal transformation as $w,t\to 0$ is accompanied by a reduction in confinement and attenuation, as noted from Fig. 17 [red curve in Fig. 17(b)], due to reduced field penetration into the metal. The $ss_b{}^0$ mode in the thin symmetric metal stripe is the fundamental long-range mode and, following convention, is termed a LRSPP.

The LRSPP in wide metal stripes has an attenuation similar to that of the LRSPP in the corresponding metal slab, but narrowing the width can reduce the attenuation further by orders of magnitude. However, as in the metal slab, confinement must also be traded off against attenuation. Furthermore, it is important for the structure to be symmetric in order for the $ss_b{}^0$ mode to remain purely bound, long range, and well behaved with varying structure parameters and operating wavelengths. The degree of asymmetry that can be tolerated depends principally on the confinement provided by the stripe.

Efficient end-fire excitation of the LRSPP in the metal stripe can be achieved by using a free-space beam in the manner depicted in Fig. 6 or by the polarization-aligned fundamental mode of a SMF, butt coupled directly to the input of the structure as depicted in longitudinal cross-sectional view in Fig. 19(a). Butt coupling can also be achieved with a polarization-maintaining SMF (PM-SMF) or tapered SMF.

One of the other fundamental modes, the $as_b{}^0$ mode, evolves in a similar manner as $t\to 0$, except that its $E_y$ field develops two extrema along x, and the mode becomes unguided below cutoff dimensions $(w,t)$ that depend on the materials and operating wavelength. This mode, which is fundamental for large w and t, evolves into the first long-range higher-order mode as $t\to 0$. Long-range modes of orders higher than the $as_b{}^0$ mode may also exist, originating from the mode families $ss_b{}^m$ ($m>0$, odd) and $as_b{}^m$ ($m>0$, even). As observed from Fig. 17, they have cutoff dimensions that increase with mode order m and are long-range only near cutoff.

All modes having the $aa_b{}^m$ and $sa_b{}^m$ symmetries $(m\ge 0)$ increase in attenuation as $t\to 0$ and do not couple efficiently with Gaussian-like fields in an end-fire arrangement because $E_y$ is asymmetric along y. The $aa_b{}^0$ and $sa_b{}^0$ modes are guided for all film dimensions and remain localized near the corners.

In asymmetric structures $({\epsilon }_{r,1}\ne {\epsilon }_{r,3})$, true field symmetries exist only with respect to the y axis. Mode fields exhibit symmetriclike or asymmetriclike distributions with respect to the x axis, and localization to either the top or bottom metal–dielectric interfaces. As in the slab, the symmetriclike modes are localized along the interface with the lowest dielectric, while the asymmetriclike modes are localized along the interface with the highest dielectric. The evolution of modes with structure parameters is not obvious, since different elemental modes can merge in and out of a supermode as parameters change. Also, different numbers of extrema may occur along the top and bottom interfaces (they are counted along the interface where the field is localized, and this number is used in the nomenclature). Long-range modes can exist in asymmetric structures but only near cutoff and having perturbed field distributions that compromise excitation in an end-fire arrangement. The long-range modes are localized on the low-index side with fields that extend deeply into the high-index cladding.

3.2. Straight Waveguides [246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282]

The symmetric metal stripe provides confinement in the plane transverse to its longitudinal axis. The stripe can be dimensioned such that the $ss_b{}^0$ mode propagates with low loss (as the LRSPP) and couples efficiently in an end-fire arrangement to TM-polarized Gaussian-like beams (in fiber or free space) while all long-range high-order modes are cut off and any remaining high-loss modes are excited with very low efficiency.

The majority of the experimental work conducted to date has used butt-coupling to SMF to excite the waveguides; this is in contrast to the metal slab (Section 2), where prism and grating coupling are normally used. As mentioned in Subsection 2.1, end-fire coupling is easier to implement than prism coupling, but all modes (including radiative ones) that overlap to some extent with the input fields will be excited; outputs must therefore be interpreted carefully, especially in cases where the input coupling is inefficient. Fortunately, LRSPP butt-coupling excitation efficiencies greater than 90% are readily achievable, rendering the excitation of any other modes (usually) immaterial.

Work conducted on straight waveguides propagating the LRSPP [246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282] is reviewed in this subsection. Passive integrated structures [283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302] are reviewed in Subsection 3.3, but some of the work also reports results on straight waveguides that are discussed here.

Berini [246] studied theoretically the four fundamental bound modes of the symmetric metal stripe $({\epsilon }_{r,1}={\epsilon }_{r,3})$ using the MoL and extended the mode nomenclature of [72] to identify them. The study was conducted at ${\lambda }_0=633\mathrm{nm}$ on a $w=1\mu \mathrm{m}$ wide Ag stripe for a range of t, with the stripe embedded in an optically infinite dielectric background having ${\epsilon }_{r,1}={\epsilon }_{r,3}=4$. The constitution of modes from elemental corner and edge modes was discussed. The partitioning of modes with decreasing t into $a_b$-like and $s_b$-like modes was observed. The evolution of the $ss_b{}^0$ mode into the LRSPP, including the extension of its fields from the corners into a Gaussian-like distribution, was described. The need to trade off attenuation versus confinement was noted, end-fire coupling was suggested for exciting the mode, and its use for optical signal transmission was proposed. (In [246], parts (a) and (b) of Fig. 2 were inverted with respect to its caption; this is corrected here as Figs. 18(c), 18(d).)

A subsequent study on symmetric structures [248] investigated via the MoL the fundamental and higher-order modes as a function of stripe dimensions, background permittivity, and operating wavelength. Figure 17 gives the effective index and normalized attenuation of the first eight modes of the stripe $(ss_b{}^0,as_b{}^0,sa_b{}^0,aa_b{}^0,ss_b{}^1,as_b{}^2,aa_b{}^2,sa_b{}^1)$ and of the $a_b$ and $s_b$ modes of the corresponding slab, and Fig. 18 shows some mode intensity distributions. Cutoff dimensions were noted for the higher-order long-range modes. It was found that decreasing the stripe dimensions (w and t) and the background permittivity, and operating at longer wavelengths, decreased the attenuation and confinement of the LRSPP. MPAs of $0.1–0.01\mathrm{dB}∕\mathrm{mm}$ were predicted for the LRSPP in the infrared $({\lambda }_0\sim 1550\mathrm{nm})$. It was suggested that the stripe operating in the LRSPP could be used to implement integrated structures such as interconnects, splitters, couplers, and interferometers.

Charbonneau et al. [249] demonstrated propagation of the LRSPP in the metal stripe by measuring output intensity distributions from $8\mu \mathrm{m}$ wide, $20\mathrm{nm}$ thick, $\sim 3.5\mathrm{mm}$ long Au stripes embedded in $\mathrm{Si}{\mathrm{O}}_2$. The structures were excited at ${\lambda }_0\sim 1550\mathrm{nm}$ via end-fire coupling to a standard PM-SMF as sketched in Fig. 19(a). Outputs were captured for various polarization angles of the input light, demonstrating efficient coupling for vertically (TM) polarized light and substantially no coupling for horizontally (TE) polarized light, as shown in Fig. 19(b). The propagation constants of the $ss_b{}^0$ and $sa_b{}^0$ modes were also computed via the MoL as a function of t, along with those of the $a_b$ and $s_b$ modes of the corresponding slab, yielding an MPA of $0.9\mathrm{dB}∕\mathrm{mm}$ for an $8\mu \mathrm{m}$ wide $20\mathrm{nm}$ thick stripe.

Theoretical studies using the MoL on the bound modes of asymmetric stripes $({\epsilon }_{r,1}\ne {\epsilon }_{r,3})$ were reported by Berini [250, 251]. The study in [250] was conducted at ${\lambda }_0=633\mathrm{nm}$ on a $1\mu \mathrm{m}$ wide Ag stripe for a range of t, with the stripe bounded on both sides by optically semi-infinite dielectric claddings having ${\epsilon }_{r,1}=2^2$ and ${\epsilon }_{r,3}={1.9}^2$. None of the fundamental modes, including the $ss_b{}^0$ and $as_b{}^0$ modes, were found to be long-range in this structure. One of the higher-order modes was long range but only near cutoff, and it had a perturbed field distribution. The subsequent study [251] considered higher-order modes, other widths, and more cases of cladding asymmetry. Modes were found to change character as t was reduced, because elemental modes evolve and merge in or out of supermodes. Long-range modes were again found but only near cutoff and having perturbed field distributions. Other studies involving the (short-range) modes of the asymmetric metal stripe include, for example, [247, 252, 253, 255, 257, 260, 265].

Charbonneau [283] measured the MPA of the LRSPP at ${\lambda }_0=1550\mathrm{nm}$ via the cutback method, as well as its output intensity distribution. The waveguides consisted of $20\mathrm{nm}$ thick Au stripes, of widths 8, 6 and $4\mu \mathrm{m}$, embedded in $\mathrm{Si}{\mathrm{O}}_2$, and the measured MPAs were 2.4, 1.6, and $1.4\mathrm{dB}∕\mathrm{mm}$, respectively. The Au stripes in these structures (and in those of [249]) were deposited on a $5\mu \mathrm{m}$ thick $\mathrm{Si}{\mathrm{O}}_2$ lower cladding thermally oxidized from the underlying Si substrate. The upper cladding consisted of a $2\mu \mathrm{m}$ thick layer of plasma-enhanced chemical vapor deposition $\mathrm{Si}{\mathrm{O}}_2$ covered with index-matching fluid. The claddings in these structures were likely slightly index mismatched and insufficiently thick to allow the modes their full expansion, resulting in higher than expected attenuation. However, the trend of decreasing attenuation with decreasing width was observed, as well as long-range single-mode guidance.

It was surmised in [248] that narrowing the width of the stripe would lead to reduced polarization sensitivity. This was confirmed by Berini [254], who modeled via the MoL square cross-section stripes $(w=t)$ in a homogeneous dielectric background, and showed that the $ss_b{}^0$ mode had two degenerate orthogonally polarized counterparts in this structure, with their main transverse electric field component polarized along x (TE polarized) and y (TM polarized). The modes were named $ss_{b,x}{}^0$ and $ss_{b,y}{}^0$, respectively, and modeling revealed that they exhibit the same qualitative evolution with decreasing dimensions as the $ss_b{}^0$ mode of the rectangular stripe. MPAs in the range of $3–0.01\mathrm{dB}∕\mathrm{mm}$ were computed at ${\lambda }_0=1550\mathrm{nm}$ for stripe dimensions ranging from $w=t=250\mathrm{nm}$ to $w=t=150\mathrm{nm}$ with Au used for the stripe and $\mathrm{Si}{\mathrm{O}}_2$ as the background dielectric. Coupling losses to SMF in the range of $0.7–6\mathrm{dB}$ per facet were computed, with the smallest coupling loss occurring for $w=t=250\mathrm{nm}$ and largest one occurring for $w=t=150\mathrm{nm}$. A trade-off between MPA and coupling loss is necessary, and it was suggested that the case $w=t=180\mathrm{nm}$ ($\mathrm{MPA}\sim 0.14\mathrm{dB}∕\mathrm{mm}$, coupling loss $\sim 3\mathrm{dB}$) would provide a good compromise.

Nikolajsen et al. [256] reported insertion loss measurements at ${\lambda }_0=1550\mathrm{nm}$ for the LRSPP in Au stripes of various lengths, $3–12\mu \mathrm{m}$ wide and $10\mathrm{nm}$ thick, deposited on $15\mu \mathrm{m}$ of benzocyclobutene (BCB) spin coated and cured on Si and covered by $15\mu \mathrm{m}$ of BCB deposited by the same process. They reported MPA values in the range of about $0.6–0.8\mathrm{dB}∕\mathrm{mm}$ and coupling losses to PM-SMF of about $0.5\mathrm{dB}$. The vertical and horizontal profiles of the output mode were also measured. The MPA of $0.6\mathrm{dB}∕\mathrm{mm}$ amounts to a $78\times$ increase in propagation length over the corresponding single-interface SPP.

Al-Bader [258] modeled symmetric Ag stripes in Si at ${\lambda }_0=1550\mathrm{nm}$, using the FDM. The width of the stripe was fixed to $w=1\mu \mathrm{m}$, and t was varied. The propagation constants of the fundamental modes $ss_b{}^0$, $as_b{}^0$, $aa_b{}^0$, $sa_b{}^0$, and of the first higher-order mode $sa_b{}^1$, were computed and compared with the $a_b$ and $s_b$ modes of the corresponding slab. He computes an MPA of about $3–7\mathrm{dB}∕\mathrm{mm}$ for the LRSPP in stripes $15–20\mathrm{nm}$ thick.

Boltasseva and co-workers [285, 286] report MPA measurements at ${\lambda }_0=1550\mathrm{nm}$ for $8\mu \mathrm{m}$ wide Au stripes having $t\sim 8.5\mathrm{nm}$ to $t\sim 35\mathrm{nm}$ and cladded on both sides by $13–15\mu \mathrm{m}$ of BCB. The lowest MPA measured among their waveguides was $\sim 0.6\mathrm{dB}∕\mathrm{mm}$ for $t=15\mathrm{nm}$. The coupling loss to SMF was measured for $t=15\mathrm{nm}$ and found to vary from 0.5 to $1.5\mathrm{dB}$ per facet for $w=10\mu \mathrm{m}$ to $w=4\mu \mathrm{m}$, respectively. Mode outputs and profiles were measured for various stripe dimensions. The EIM was proposed for modeling the metal stripe, and propagation constants for the fundamental $\left(ss_b{}^0\right)$ and first three higher-order long-range modes were computed as a function of w for $t=10\mathrm{nm}$.

Charbonneau et al. [287] measured LRSPP mode outputs at ${\lambda }_0=1550\mathrm{nm}$ for $7\mathrm{mm}$ long, $25\mathrm{nm}$ thick Au stripes of various widths ($w=2,4,6,8\mu \mathrm{m}$) on $20\mu \mathrm{m}$ of $\mathrm{Si}{\mathrm{O}}_2$ on Si and covered with index-matched polymer. The mode outputs captured under identical measurement conditions show reduced mode confinement and attenuation with decreasing w.

Zia et al. [259] applied the EIM to the metal stripe and compared their computed propagation constants for the $ss_b{}^0$, $as_b{}^0$, and $sa_b{}^1$ modes with those of Al-Bader [258] over his range of t. Good agreement was achieved for the $ss_b{}^0$ and $as_b{}^0$ modes, but not for the $sa_b{}^1$ mode (the authors reported achieving a better agreement with [248] for this mode). They also applied the FDM to extend the comparison to other stripe dimensions ($w=0.5\mu \mathrm{m}$ to $w=4\mu \mathrm{m}$ and $t=24,50,100\mathrm{nm}$), again achieving good agreement with the EIM for the $ss_b{}^0$ and $as_b{}^0$ modes.