Effective-index approach is applied for modeling of channel plasmon polaritons (CPPs) propagating in rectangular grooves (trenches) and triangular (V-shaped) grooves in gold, accounting for the main features of CPP guiding and elucidating its underlying physics. The effective indexes of CPP modes along with the corresponding propagation lengths are calculated for different configurations and wavelengths while varying the groove depth. The results obtained allow one to identify the parameter range for realizing the single-mode CPP guiding featuring subwavelength confinement and moderate propagation loss at telecom wavelengths.
©2006 Optical Society of America
Modern communication systems dealing with huge amounts of data at ever increasing speed try to utilize the best sides of electronic and optical circuits. Electronic circuits are tiny but their operation speed is limited, whereas optical circuits are much better in transmitting data at high speed but their sizes are limited by diffraction (i.e., propagating light cannot be squeezed in the lateral cross section below its wavelength λ in the medium used). Note that the lateral light confinement, i.e. the waveguide mode width w, determines not only the minimum separation between neighbor waveguide components but also the largest bend angle θ~λ/w (allowed by the acceptable level of the bend loss) that sets in turn the limit on the length of an elementary component L~w/θ. The above constraints make the maximum density of components being strongly dependent on the mode lateral confinement: n~(Lw)-1~λ/w 3, impelling thereby further the search for waveguide materials and configurations that would minimize this confinement.
Surface plasmon polaritons (SPPs) having by nature subwavelength spatial periods and transverse dimensions perpendicular to the metal surface  offer the possibility of realizing subwavelength waveguiding and ultra-compact photonic components . The main issue in this context is to strongly confine the SPP field in the surface plane (perpendicular to the SPP propagation direction). Many different approaches have been investigated, including photonic band-gap structures , metal stripes in a dielectric environment , metal particle waveguides , and the use of polymer stripes placed on top of a metal surface . However, simultaneous realization of strong confinement and relatively low propagation loss (acceptable for practical purposes) has long been inaccessible. Channel SPP modes, or channel plasmon polaritons (CPPs) , where the electromagnetic radiation is concentrated at the bottom of V-shaped grooves milled in a metal film, have been first predicted  and then experimentally shown  to exhibit useful subwavelength confinement and moderate propagation loss. Recently, we have realized various CPP-based subwavelength waveguide components, including Mach-Zehnder interferometers and waveguide-ring resonators , and demonstrated that the CPP guides can be used for large-angle bending and splitting of radiation [10, 11], thereby enabling the realization of ultracompact plasmonic components and paving the way for a new class of integrated optical circuits. We have also developed a simple and efficient modeling approach  based on the effective-index method (EIM)  that has been used for designing the investigated CPP waveguide components [9–11].
In this work, the EIM is applied for modeling of CPPs propagating in rectangular grooves (trenches) and triangular (V-shaped) grooves in gold. First, the validity of the EIM for modeling of SPP modes supported by two-dimensional (2D) metal waveguide structures is investigated by comparing with the finite-difference (FD) simulations conducted for trenches in metal films embedded in dielectric media . The EIM is then used for calculating the effective indexes of CPP modes at several wavelengths along with the corresponding propagation lengths for trenches and V-grooves, while varying the groove depth. The results obtained allow one to identify the parameter range for realizing the single-mode CPP guiding featuring subwavelength confinement and moderate propagation loss (i.e., exhibiting the propagation length of the order of 100 µm) at telecom wavelengths.
2. Comparison of EIM and FD simulations
The main attractive feature of the EIM is that it allows one to combine the results of modeling conducted for one-dimensional (1D) waveguiding configurations so that the characteristics of 2D (channel) waveguides can be described . For a rectangular-core waveguide, one should first analyze a planar (slab) waveguide obtained by letting one dimension of the original 2D waveguide approach infinity. Thus obtained mode propagation constant(s) is then used to define the corresponding effective dielectric index(s) assigned to the core index(s) of another 1D waveguide considered in the perpendicular direction. The propagation constant(s) of this second waveguide are taken to represent those of the original rectangular waveguide . The EIM is one of the most extensively used approaches for modeling of dielectric channel waveguides in integrated optics, where its validity has been scrutinized and various corrections have been introduced . The EIM has also been successfully applied to modeling of weakly guided SPP modes supported by thin metal stripes of finite width [15, 16]. Here, three 2D metal waveguide configurations that are able of supporting SPP modes with subwavelength confinement (Fig.1) are analyzed by making use of the EIM.
The first considered structure [Fig. 1(a)], in which both a substrate and a cladding are dielectric, corresponds to the configuration analyzed previously using the FD mode solver . The first step in the EIM is to analyze the SPP guiding in the gap between two metal surfaces filled with the dielectric cladding. For an individual metal surface, the SPP propagation constant (wave number) β is given by β=(2π/λ) [ε m ε d/(ε m+ε d)]0.5, where λ is the wavelength in air, εm and εd are the dielectric constants of metal and dielectric, respectively . The propagation constant β can be related to the SPP effective refractive index, i.e., N eff =βλ/(2π), whose real and imaginary parts determine the SPP wavelength and propagation length. For two close metal surfaces, the SPPs associated with individual metal surfaces become coupled, and the dispersion relation for β becomes implicit though its solution can be readily obtained . The effective index of this gap SPP is then used to represent a (lossy) dielectric core sandwiched between the substrate and the cladding. Such a three-layer structure is analyzed searching for TE-modes having the electric field perpendicular to the metal surface in the gap (as the gap SPP does) and thereby parallel to the interfaces in the considered structure. Detailed simulations were carried out for the working wavelength of 632.8 nm, the cladding being a polymer with the refractive index of 1.49, the metal — silver with the index of 0.119+3.964i and the substrate — silica glass with the index of 1.47 . The results obtained are shown in Fig. 2 along with the mode effective indexes obtained with the FD method (as retrieved from Fig. 2 in Ref. ).
It is seen that the agreement between the EIM and FM calculations is very good even when the SPP modes are close to cutoff where the EIM approximation usually becomes rather poor [12, 14]. A possible reason for this encouraging observation might be the fact that the considered SPP modes are tightly confined to the gap in the metal film, decreasing thereby the influence of corner regions (i.e. the regions situated above and below the metal film but away from the gap) contributing to the EIM errors [12, 14]. It is therefore expected that the SPP modes supported by trenches cut into a metal, i.e. in the configuration shown in Fig. 1(b), can be adequately described using the EIM.
3. Trench CPP modes
The first step in the EIM applied to the trench [Fig. 1(b)] or V-groove [Fig. 1(c)] configuration is the same as before, i.e., one should analyze the SPP guiding in the air gap between two metal surfaces. The following simulations are concerned with grooves in gold and several wavelengths chosen in the interval between visible and telecom wavelengths. The following dielectric constants of gold were used in the simulations: n=0.166+3.15i (λ=0.653 µm), 0.174+4.86i (0.775 µm), 0.272+7.07i (1.033 µm) and 0.55+11.5i (1.55 µm) . The corresponding characteristics of the fundamental gap SPP mode are shown in Fig. 3.
It is interesting that, even though the gap SPP modes have been considered (in different contexts) during the last three decades [17, 19, 20], it has apparently been unnoticed that the propagation length first increases when the gap width decreases from large values corresponding uncoupled individual SPPs (Fig. 3). This surprising fact indicates that gap SPP modes with better confinement can exhibit longer propagation lengths. The longest propagation length seems to be ~10% larger than the SPP propagation length (for w→∞) independently on the wavelength, though it is achieved for different (gap) width-to-wavelength ratios: w opt/λ≈1.4 (λ=0.653 µm), 2.6 (λ=0.775 µm), 3.3 (λ=1.033 µm) and 5.7 (λ=1.55 µm). It is seen that, with the accuracy of 10% for the above wavelengths, the optimum width w opt can be evaluated simply as w opt≈0.5λ|Re(ε m)+1|0.5. The explanation of this remarkable effect is related to the fact that, with the decrease of the gap width, the electric field of the gap SPP (composed of two exponents [17, 20]) approaches quickly to the electrostatic (capacitor) mode, which is constant across the gap (see the insert in Fig. 3). Consequently, the fractional electric field energy concentrated in the gap first increases (and the damping in metal decreases) when the gap width decreases, reaches its maximum and then starts to decrease with the field being squeezed from the gap into the metal. The exact analysis of the process of energy redistribution is quite complicated because the corresponding dispersion relation is implicit . If one considers only the main electric field components and approximates the gap SPP propagation constant by that of the SPP (for large gap widths), one finds that the fraction of the electric field energy in the gap can be expressed as a function of only one parameter u=w(2π/λ)|Re(ε m)+1|-0.5:
This function reaches its maximum value, which is by 20% larger than that at w→∞, for u≈2.4, resulting in the following expression for the optimum width: w opt≈0.4λ|Re(ε m)+1|0.5, a value which is quite close to that found above from numerical simulations.
The CPP mode supported by the trench [Fig. 1(b)] can now be described by considering TE-modes in a three-layer structure, in which a dielectric core having the effective index of the corresponding gap SPP is sandwiched between the air cladding and the gold substrate. This is a relatively straightforward procedure, and the results for two gap widths and the telecom wavelength of 1.55 µm are shown in Fig. 4. It is readily seen that the subwavelength lateral confinement of the trench CPPs can be achieved simultaneously with relatively long propagation distances, a feature which is similar to that found for V-groove CPPs [8–10].
It should be emphasized that the ability of subwavelength (trench and V-groove) CPPs to propagate with moderate propagation loss is a direct consequence of the aforementioned property of gap SPPs to most efficiently fill the available dielectric space (gap) between the metal walls. Note that the aspect ratio (the depth-to-width ratio) required to support the fundamental trench CPP mode is quite large (typically ≥ 3), whereas the V-grooves used in the experiments (also at telecom wavelengths) were relatively shallow [9–11]. The difference in CPP guiding abilities of trench and V-grooves is related to the fundamental difference in the CPP field distributions in depth for these groove configurations. The field of trench CPP decreases to (nearly) zero at the trench bottom (Fig. 5) because of a large magnitude of the dielectric constant of the metal substrate (and the boundary condition for the electric field), whereas the (fundamental) CPP field is predicted to reach its maximum at the groove bottom [7, 8]. The V-groove CPPs are thereby better confined in the depth direction than the trench CPPs, utilizing more efficiently the groove space (in the depth direction).
The circumstance that the trench CPP modes are obtained from the consideration of a strongly asymmetric waveguide structure (because of the metal substrate) can be used to deduce a simple relation for the mode cutoff condition. Utilizing the normalized waveguide parameters and setting the asymmetry parameter to infinity, one obtains the following cutoff condition for the mth-mode :
Here V(w, d) is the normalized waveguide frequency (or normalized thickness), and Neff (w) is the effective index of the corresponding gap SPP mode determined for the gap width w in the first step of the EIM (Fig. 3). Using Eq. (2) the condition of the single-mode (trench) CPP guiding can be simply written down as follows: 0.5π<V(w, d)<1.5π. Using the calculated gap SPP characteristics shown in Fig. 3, one can easily identify the parameter range for the well-confined single-mode guiding of trench CPPs (V(w, d)~π) and even estimate the CPP propagation length, which is close to but longer than the propagation length of the gap SPP. The parameter ranges for the single-mode (trench) CPP guiding are shown in Fig. 6 for the wavelengths of 0.775 and 1.55 µm, where the area allowing for the single-mode guiding in the whole wavelength range between 0.775 and 1.55 µm is highlighted. Note that this is not a trivial fact that such an area can be found (see the next section), because the normalized frequency V scales with the ratio d/λ [Eq. (2)] and the gap SPP effective index Neff decreases with the increase of the wavelength (Fig. 3).
Finally, it should be borne in mind that, as the CPP mode approaches the cutoff and the CPP field becomes progressively larger at the trench (or groove) edges, one should expect the occurrence of CPP coupling to other SPP modes: plane (conventional) SPPs, propagating away from the groove, edge  and coupled wedge  SPP modes. Such a coupling might become efficient, especially when the propagation constants (all being larger than that of in air) would match, resulting in additional loss. On the other hand, these modes have rather different polarization properties (e.g., the main electric field components of CPP and SPP modes are orthogonal), a circumstance that would decrease the coupling strength.
4. V-groove CPP modes
Let us now consider a V-groove whose width is monotonously decreasing with the increase of the depth [Fig. 1(c)]. Since light tends to be confined in regions with higher refractive indexes, sufficiently deep grooves support CPP modes (formed by gap SPPs) that are confined to the groove bottom, where the gap SPP index is at maximum (Fig. 3). Using the EIM, one can find the V-groove CPP modes by analyzing a one-dimensional layered (in depth) guiding structure, in which the top layer of air and the bottom layer of metal abut a stack of layers having refractive indexes determined by the layer depth: an index is equal to the gap SPP effective index N eff for a gap width corresponding to the groove width at this depth . The corresponding results obtained for the telecom wavelength of 1.55 µm are shown in Fig. 7.
It is seen that the CPP effective index increases while the propagation length decreases with the decrease of the groove angle θ . Note that the CPP mode index Neff determines the mode confinement, since the mode penetration depth in air (above the sample surface) is given by dpen =(λ/2π)·(-1)-0.5, so that a larger effective index corresponds to a smaller penetration depth and thereby better field confinement in depth (in width, the CPP mode is confined within the groove walls). In this respect, the CPP guiding in V-grooves is counterintuitive: the groove guiding abilities (such as the CPP mode confinement) are better for narrower grooves, which have smaller (!) cross sections . Here, better confinement is achieved, as usual, at the expense of larger damping (shorter propagation lebgth). It is also seen that the CPP propagation length starts to rapidly increase when the groove depth decreasing approaches a certain (cut-off) value, below which no CPP mode could be found. Since, at the same time, the CPP effective index approaches that of air, the increase of propagation length signifies the tendency of the CPP mode field being extended progressively outside the groove. As discussed in the end of the previous section, one might expect an increase in the CPP propagation loss associated with the CPP coupling to other SPP modes (the plane SPP index is indicated in Fig. 7). Finally, it should also be mentioned that, in Fig. 7, the CPP characteristics are shown in the range of groove depths corresponding to the single-mode CPP guiding in the groove with the angle θ=25° .
The wavelength dispersion of CPP guiding in V-grooves along with the occurrence of higher-order CPP modes is illustrated with the results of simulations conducted for the groove angle of 250 at several wavelengths (Fig. 8). It is readily seen that, whereas the single-mode CPP guiding is possible in the wavelength range of 1.033–1.55 µm (for the groove depths d in the range of ~0.9–1.9 µm), one cannot expect to achieve the single-mode V-groove guiding in the whole range of 0.775–1.55 µm unlike the CPP guiding in trenches (Fig. 6). Another interesting feature, which is seen also in Fig. 7 and in Fig. 2 of Ref. 9 (but not commented upon), is that the CPP propagation length is first decreases with the decrease of the groove depth, starting to rapidly increase only when the depth becomes close to the cutoff value. This feature is also counterintuitive, because shorter propagation length is associated with better mode confinement that one would expect to be better for deeper grooves. Note that this feature is not found for trench CPP modes whose propagation length increases monotonously with the decrease of the trench depth (Fig. 4).
One can venture the following physical explanation of this effect. As the groove depth becomes smaller, the groove width at the groove top becomes smaller as well, increasing thereby the index contrast between the upper (air) half-space and the upper part of the groove. This increase in the index contrast forces the mode field to be closer to the groove bottom, where the damping is larger. Note that, at the same time, there is a tendency of increasing the part of the mode field propagating above the groove (in air) and thereby decreasing the damping. It is only due to a drastic increase in the damping (of gap SPPs) for very small gap widths (Fig. 3) that the first tendency appears stronger than the second one. Such an explanation is consistent with the fact that trench CPP propagation length increases monotonously with the decrease of the trench depth, since a trench has a constant width and hence introduces a constant damping. The evolution in the CPP field distributions with the groove depth is illustrated in Fig. 9 for the fundamental and second modes at 1.033 µm.
Both tendencies discussed above can be observed in the behavior of the fundamental CPP mode: the (normalized) field maximum increases with the decrease of the groove depth and so does the portion of the mode field outside the groove (for negative depth coordinates). It should be however borne in mind that insofar the validity of the EIM when describing V-groove CPPs has not been scrutinized. On one hand, good agreement was found when comparing the EIL simulations for V-grooves in silver for the wavelength of 0.633 µm  with the finite-difference time-domain calculations . The EIM simulations of V-grooves in gold for telecom wavelengths were also found in good agreement with the experimental results [9, 10]. On the other hand, the EIM usage forces the V-groove CPP fields to approach zero at the groove bottom (Fig. 9) similarly to the case of trench CPPs (Fig. 5) and for the same reason (see the previous section), whereas the CPP field is expected to reach its maximum at the V-groove bottom [7, 8]. It remains to be seen to what extent the latter difference influences the calculated CPP characteristics and what approach is better suited to model experiments, where one should not expect to find sharp-angled bottoms in fabricated V-grooves.
In summary, the EIM has been applied for modeling of CPPs propagating in rectangular grooves (trenches) and triangular (V-shaped) grooves in gold. The validity of the EIM for modeling of SPP modes supported by 2D metal waveguide structures was investigated by comparing with the finite-difference simulations conducted for trenches in metal films embedded in dielectric media . The EIM was then used to determine the trench and V-groove CPP characteristics at several wavelengths for trenches and V-grooves, accounting for the main features of CPP guiding and elucidating its underlying physics. When considering the SPP guiding in the dielectric gap between two metal surfaces (this being the first step in the EIM applied to grooves), it has been found that that gap SPP modes with better confinement can exhibit longer propagation lengths. This remarkable feature seems to be contradicting the well-known tradeoff between the SPP mode confinement and its propagation length [2, 4, 5]. It has been explained by the fact that, with the decrease of the gap width, the electric field of the gap SPP approaches quickly to the electrostatic (capacitor) mode, which is constant across the gap, increasing thereby the fractional electric field energy concentrated in the gap (before it starts to decrease with the field being squeezed from the gap into the metal). It has been further conjectured that the ability of subwavelength (trench and V-groove) CPPs to propagate with moderate propagation loss is a direct consequence of this property of gap SPPs to most efficiently fill the available dielectric space (gap) between the metal walls.
The trench CPP guiding has been discussed (for the first time to our knowledge) in detail, and a simple relation for the cutoff condition of CPP modes has been established. It has been found that the trench CPP guiding in the single-mode regime can be achieved in the whole range of 0.775–1.55 µm, a circumstance that can be advantageously exploited for realization of nonlinear electromagnetic interactions. The V-groove CPP guiding has been considered for different groove angles and light wavelengths, highlighting the main features and discussing the physical phenomena involved. The results obtained allow one to identify the parameter range for realizing the single-mode CPP guiding featuring subwavelength confinement and moderate propagation loss (i.e., exhibiting the propagation length of the order of 100 µm) at telecom wavelengths. Finally, the EIM validity when describing V-groove CPPs was discussed emphasizing the need for further detailed investigations.
The author is grateful to A. Bouhelier, A. Dereux, J.-C. Weeber and R. Zia for fruitful discussions during his leave of absence at the University of Burgundy (Dijon, France) where this work was largely prepared, and acknowledges the support by the European Network of Excellence, PLASMO-NANO-DEVICES (FP6-2002-IST-1-507879) and by the Danish Research Agency (contract No. 272-05-0450).
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