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Using the effective-index approximation along with the explicit expression for the propagation constant of gap surface plasmons established recently [Opt. Express 16, 2676-2684 (2008)], we develop an analytic description of channel plasmon polariton (CPP) modes guided along V-grooves and gaps between two metal cylindrical surfaces. The results obtained for V-grooves are compared with those reported previously. Dispersion of the main CPP characteristics is calculated for air-gold configurations, allowing one to design graded-gap waveguides for the single-mode operation supporting well-confined fundamental CPP modes in a broad wavelength range.
©2009 Optical Society of America
1. Introduction
Photonic components utilizing surface plasmon polaritons (SPP) modes whose lateral confinement is dictated by the extent of metal nanostructure (and can be, in principle, decreased without cutoff) open the enticing perspective of developing nanometer-sized optoelectronic circuits functioning well below the diffraction limit [1]. Among various SPP guiding configurations, waveguides utilizing SPP modes supported by a dielectric gap between two metal surfaces [2] promise the possibility of achieving a better trade-off between the lateral confinement and the propagation loss [3]. Gap-SPP (G-SPP) based configurations include waveguides having the gap width varying in the lateral direction [4–6], trench [3,7] and V-groove [8–10] waveguides. V-groove waveguides (fabricated with focused ion-beam milling [10–13] and nanoimprinting [14] techniques) have recently been experimentally investigated and exploited to realize various subwavelength waveguide components, including Mach-Zehnder interferometers and waveguide-ring resonators [11], add-drop multiplexers and grating filters [12] and efficient nanofocusing tapers [13].
Modeling of SPP waveguides allowing for the two-dimensional (2D) mode field confinement (in the cross section perpendicular to the propagation direction) is, in general, a rather complicated problem requiring the usage of sophisticated computational techniques [4–9,15–17], which is often time-consuming due to the very detailed and fine mesh required near metal edges. Even though very careful and detailed simulations are crucial for understanding intricate physical phenomena involved (e.g., hybridization of channel and wedge SPP modes [15]), the effective-index method (EIM), in which the results of modeling conducted for one-dimensional (1D) waveguide configurations are combined to obtain the characteristics of 2D (channel) waveguides [18], was found being quite helpful in judging upon the existence of guided (bound) SPP modes [3,10,19,20]. Quite recently, the EIM has been employed to introduce a normalized waveguide parameter (normalized frequency [21]) characterizing the mode field confinement and to obtain the corresponding formulae for various G-SPP based waveguides by making use of the analytic expression for the G-SPP propagation constant found for moderate gap widths [22]. The obtained relations were tested with a finite-element method (FEM), and it was shown that waveguides with different dimensions and operating at different wavelengths, but having the same normalized parameter, feature indeed very similar field (normalized) field distributions. This important circumstance was advantageously used for designing efficient CPP tapers for nanofocusing of radiation [13].
In this work, using the EIM along with the explicit expression for the G-SPP propagation constant [22], we develop an analytic description of channel plasmon polaritons (CPPs) supported by graded-gap waveguides: V-grooves and gaps formed by two metal cylindrical surfaces, for which we obtain the closed-form solutions. The results obtained for V-grooves are compared with the previous simulations [3,9,15,16] including those utilizing the finite-element method [22]. We calculate the main CPP characteristics for air-gold configurations in a broad wavelength range, allowing one to design CPP graded-gap waveguides for single-mode operation supporting a well-confined fundamental mode.
2. CPP modes in V-grooves
We begin our consideration with the V-groove geometry [Fig. 1(a) ] assuming that the groove angle is sufficiently small: θ << 1. In this case, the configuration can essentially be treated as that of 2D layered geometry, allowing thereby the usage of TE- and TM-wave representations. SPP modes can be found only for TM-modes [2], whose main (in dielectric) electric field component is parallel to the z-axis, i.e. E z. Within the EIM framework [18], the spatial distribution of the latter for a mode propagating along the x-axis is sought in the following form:
Fig. 1 (a). Geometry of the considered V-groove configuration. (b) Fundamental and (c) second CPP mode E z-field distributions (the size of panels: 6 × 10 µm2) calculated for an infinitely deep air-filled groove in gold with the angle θ = 25°at the wavelength λ = 1.55 µm.
where β denotes the CPP mode propagation constant. Substituting this field form in the Helmholtz equation and neglecting the partial derivatives ∂Z/∂y and ∂2 Z/∂y 2 that can be made arbitrarily small by decreasing the groove angle θ, one obtains the following pair of coupled equations describing two 1D waveguide configurations:
where ε d and ε m are the susceptibilities of dielectric and metal [Fig. 1(a)], respectively, k 0 = 2π/λ is the wave number and λ is the radiation wavelength in vacuum. The propagation constant q(y) appearing in Eq. (2)a), which fundamentally describes the gap-SPP waveguide configuration [Fig. 1(a)], defines the effective refractive index profile of the second waveguide configuration described by Eq. (2)b) – hence the name of method [18]. It should be noted that the y-coordinate appearing in Eq. (2)a) acts as a parameter that determines the gap width: w(y) = 2y tan(θ/2) ≈θ y.
Hereafter we make use of the (approximate) analytic expression for the G-SPP propagation constant established previously [22] and found to be valid for not too small gap widths, i.e. when :
providing thereby one with the solution of Eq. (2)a) along with the well-known G-SPP field distribution Z[w(y), z] (see, e.g [2].) that can readily be written down:
where A is the normalization constant and . Consequently, the next step in dealing with the CPP waveguide problem becomes solving Eq. (2)b) with Eq. (3) taken into account:
In the case of linear variation of the V-groove width, as is our case: w(y) = 2y tan(θ/2) ≈θ y, this waveguide problem [Eq. (5)] is analogous to the well-known problem of Coulomb potential in quantum mechanics (see, e.g [23].), whose solution can be represented as follows:
with the mode field distributions found recursively:
where α n are the normalization constants. The complete 2D field distributions for different CPP modes can readily be found as products of the gap-SPP field distribution [Eq. (4)] and the corresponding depth profiles [Eq. (7)]. As an example, the patterns of first two CPP mode supported at λ = 1.55 µm by a gold V-groove with the angle θ = 25° were calculated (Fig. 1) using the following dielectric constant of gold: n = 0.55 + 11.5i [24]. Note a very close resemblance of the present mode patterns and those reported previously [15].
It is seen that the second CPP mode field [Fig. 1(c)] is concentrated substantially farther away from the groove bottom than the fundamental CPP mode [Fig. 1(b)], implying that the single-mode CPP guiding can be achieved by properly truncating (cutting off) a groove [9]. It is expected that the CPP mode characteristics will be essentially the same for finite-depth V-grooves provided that the truncation occurs well away from the mode maximum. We have calculated the fundamental CPP mode characteristics using the above relation [Eq. (6)] for different air-filled V-grooves in gold and found a good quantitative correspondence with those calculated previously [3] for relatively large groove depths (a few pertinent results are shown in Fig. 2 and discussed below). Conversely, the cutoff depth d can be evaluated as the depth corresponding to the position of mode maximum, leading to the following cutoff condition for the fundamental CPP mode [Eq. (7): ξ c = k|r|d/θ > 1. Since this analysis is conducted within the same framework as that used to introduce the normalized waveguide parameter [22], one should expect to find that the same parameter plays equally important role in the current consideration. Indeed, the above cutoff condition can be most conveniently presented by using the waveguide parameter V cpp [22]:
Fig. 2 The fundamental CPP mode characteristics, i.e. (a) the effective index and propagation length along with (b) cutoff groove depth and normalized mode width, as a function of light wavelength for the groove angles θ = 20° and 25°. Filled circles represent the results of EIM simulations obtained previously [3] for the groove angle θ = 25°.
In fact, the present analysis provides a formal justification of the waveguide parameter [22] by introducing the normalized depth coordinate ξ [Eq. (7)] that can be cast in the form ξ = (V cpp 2/8)⋅(y/d). This form in turn implies that, for well-guided modes away from cutoff, the CPP waveguides with the same V cpp should exhibit not only the same field confinement [22] but also the same mode field distributions along the normalized (to the groove depth) depth coordinate, as indeed was verified by comparing the CPP depth profiles calculated with the FEM method for two different V-groove waveguides with V cpp = 1.34π [22] and the depth profile given by Eq. (7): Y 0(y) ~(y/d) exp(- 2.2y/d), since V cpp 2/8 ≈2.2. The correspondence of the field profiles was found quite good within the groove (y/d ≤ 1), confirming that the obtained formulae can be applied to the CPP modes located well inside the groove so as to avoid their hybridization with the wedge modes [15] whose existence is disregarded within the EIM framework [3,22].
The analytic expressions obtained [Eq. (6-8)] allows one to easily and reliably estimate the main CPP characteristics and to design V-grooves supporting well-confined CPP modes in a broad wavelength range. As an example, using the well-documented gold refractive index dispersion [25], we have calculated the fundamental CPP mode characteristics, i.e. the effective index, N 0 = Re{1 + (r/θ)2}0.5, propagation length, L = λ/[4π Im{1 + (r/θ)2}0.5], cutoff groove depth (d c = θ /k|r|) and the normalized mode width (w = d c θ /λ) for two air-filled V-grooves in gold having different angles θ = 20° and 25° (Fig. 2). It seen that the mode confinement deteriorates with the increase in wavelength and groove angle allowing, on the other hand, for longer propagation length, a trend which is consistent with the results of previously reported investigations [9,10,15,16].
The advantage of the obtained closed-form solutions [Eqs. (6-8)] is that one can quickly conduct similar evaluations of the CPP characteristics for other groove angles and different material (metal and dielectric) configurations and design V-grooves for the single-mode operation supporting well-confined fundamental CPP modes. Note, that their accuracy is commensurate with that of the EIM employed previously [3,10] and that, in general, the above results (Figs. 1 and 2) are consistent with the available accurate simulations [9,15,16].
3. CPP modes confined by cylindrical surfaces
The preceding analysis of V-grooves can be extended to other graded-gap waveguides provided that the variation of gap width is sufficiently slow. One immediate candidate is a waveguide formed by two cylindrical surfaces [Fig. 3(a) ], whose modes will hereafter be designated as CyPPs to distinguish them from CPPs supported by V-grooves. Considering the gap region in the vicinity of the smallest separation, i.e. for |y| << min(R 1,R 2) with R 1 and R 2 being the cylinder radii, the gap width variation can be approximated as follows:
Fig. 3 (a) Geometry of the graded-gap configuration formed by two cylindrical surfaces. (b) CyPP field profiles calculated for the first three modes [Eq. (12)].
and w 0 is the smallest gap width. Inserting the above dependence w(y) in Eq. (5) and making further approximations valid for |y| << (w 0 R)0.5, one obtains the following equation for the CyPP propagation constant β and the mode field distribution Y(y) in width [Fig. 3(a)]:
Close inspection reveals that this waveguide problem is analogous to another well-known problem in quantum mechanics, viz. the linear harmonic oscillator, with the solution being readily expressed in the following form (see, e.g [26].):
with the mode field distributions given by:
where α n are the normalization constants.
The CyPP field distributions in width can be described by using the normalized width coordinate ξ [Eq. (12)], and are shown for the first three modes in Fig. 3(b). The complete 2D field distributions for different CyPP modes can readily be found as products of the gap-SPP field distribution [Eq. (4)] and the corresponding width profiles [Eq. (12)]. As an example, the patterns of CyPP modes guided by different gaps between identical 2-µm-radius gold cylinders at the light wavelengths of 1.033 and 1.55 µm were calculated (Fig. 4 ) using the following dielectric constants of gold: n = 0.272 + 7.07i (λ = 1.033 µm), and 0.55 + 11.5i (1.55 µm) [24]. It seen that the mode confinement (to the gap center) deteriorates with the increase in gap width and wavelength, albeit the influence of the wavelength is considerably less pronounced. The latter feature can be explained by the fact that the normalized width coordinate ξ is weakly dependent on the wavelength [Eq. (12)].
Fig. 4 (a,b,c) Fundamental and (d) second CyPP mode E z-field magnitude distributions (the size of panels: 2 × 1 µm2) calculated for different gaps between 2-µm-radius gold cylinders with the smallest separations: w0 = (a,c,d) 100 and (b) 200 nm and wavelengths: λ = (a,b) 1.55 and (c,d) 1.033 µm.
The closed-form expressions obtained for the CyPP propagation constants [Eq. (11)] and mode field profiles [Eq. (12)] allow one to quickly evaluate the CyPP characteristics. It is important to notice that the number of CyPP modes supported by this configuration is limited. The cutoff condition can readily be expressed with respect to the effective cylinder radius R, which is introduced in Eq. (9), as follows [see Eq. (11)]:
It is interesting that, within the considered framework, the cutoff condition is independent on the smallest gap width w 0. To get an idea on the dispersion of fundamental CyPP mode we have considered the air-gold configuration with w 0 = 100 nm and R = 2 µm and calculated the mode characteristics, i.e., its effective index N 0 = (λ/2π)⋅Re{β 0} and propagation length L = 1/(2⋅Im{β 0}), using Eq. (11). Another important parameter is the mode cross section, whose extension along the y-axis can be taken equal to the interval: -√2 < ξ < √2, when expressed via the normalized width coordinate ξ [see Eq. (12) and Fig. 3(b)], resulting in the following mode dimensions [see Eqs. (9) and (12)]:
The normalized (to the light wavelength) mode sizes, w y/λ and w z/λ, and the mode characteristics, N 0 and L, along with the cutoff radius [Eq. (13)] for the fundamental CyPP mode, R 0, are shown in Fig. 5 in a broad wavelength range. Note that even though the mode confinement along the gap is worse than that across the gap: w y >> w z, both mode sizes can be designed to be sufficiently small by a proper choice of the cylinder separation w 0 [Eq. (14)]. The difference in mode sizes reflects merely the difference in the physical mechanisms of the mode confinement: the confinement across the gap is dictated by material boundaries whereas that along the gap is determined by the magnitude of the effective mode index.
Fig. 5 The fundamental CyPP mode characteristics, i.e. (a) the effective index and propagation length along with (b) cutoff cylinder radius and normalized mode size, as a function of light wavelength for the cylinder gap w 0 = 100 nm and the effective cylinder radius R = 2µm.
The closed-form expressions obtained for the CyPP modes are useful for quick evaluation of their characteristics, and, for example, the properties of the fundamental mode for other parameters than used above (Fig. 5) can be obtained using the scaling provided by the above formulae [Eqs. (11-14)]. Overall scaling requires the knowledge of the normalized waveguide parameter, similar to the case of CPP modes in V-grooves [22]. The case of CyPPs is however more complicated because the approximation of linearly polarized modes can be justified only in the vicinity of the smallest separation, i.e. for |y| << min(R 1,R 2), as mentioned above. One can nevertheless extend formally the usage of the EIM up to the boundaries dictated by the smallest (of the two) cylinder radius, i.e. for |y| < R min = min(R 1,R 2), for the sake of obtaining a single parameter combination that can be used to judge upon the guiding potential of a given system. The corresponding expression can readily be obtained by using Eqs. (3) and (4) in the limit of small gaps (w 0 << R min):
We believe that the above relation for the normalized waveguide parameter can be found useful when comparing the CyPP guiding for different systems, but its accuracy should be carefully investigated by using accurate numerical modeling of this waveguide configuration for various system parameters, similarly to what has been done in the case of V-grooves [22].
4. Conclusion
In summary, using the EIM along with an explicit expression for the G-SPP propagation constant [22], we have derived closed-form solutions describing the CPP modes supported by graded-gap waveguides: V-grooves and gaps formed by two metal cylindrical surfaces. The analytic expressions were obtained both for the mode field distributions and propagation constants. Dispersion of the main CPP characteristics was calculated for air-gold configurations, allowing one to design graded-gap waveguides for the single-mode operation supporting well-confined fundamental CPP modes (i.e., being far away from the cutoff) in a broad wavelength range. The latter is very important not only for minimizing the lateral mode width but also for avoiding additional loss, since, in practice, any structural irregularities would result in coupling of waveguide modes (especially those close to the cutoff) to plane SPPs propagating away from the waveguide, causing thereby additional propagation loss.
Acknowledgments
The authors acknowledge the support by the Danish Agency for Science, Technology and Innovation (contract No. 274-07-0258).
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