Plasmonics offers unique opportunities for the manipulation of light at the nanoscale [1]. The interaction of light with electrons in a metal provides tightly confined surface plasmon polarition (SPP) modes. SPPs are already widely used in chemical sensors [2]. Denser integration of SPP sensors is possible with the advent of nanostructuring. In addition, SPPs provide a natural interface between electronics and photonics, allowing for dense integration of photonic waveguides using metal stripes [3]. Several schemes have been developed for SPP waveguiding, such as SPP band gaps [4], Bragg mirrors [5], nanowires [6], grooves [7], and metal stripes [8]. Metal stripe waveguides are more compact than dielectric waveguides because of their natural ability to confine the light evanescently at the surface [8,9].

With many applications in sight, it is necessary to develop new methods for the analysis of photonic integrated circuits based on SPPs. A geometric optics dielectric waveguide model was used to obtain a physically intuitive description of SPP modes supported by a stripe [10]; that is, a surface plasmon stripe waveguide (SPSW). In that work, an approximation was made to base the phase of reflection upon the dielectric Fresnel relations, so that the SPP field distribution and polarization properties were neglected.

In this work, an analytic expression for the phase of reflection above the critical angle is derived that includes the polarization and mode-shape properties of SPPs. This improves the accuracy of the geometric optics method, while retaining the analyticity of the method. The modified geometrics approach is applied to analyze the propagation properties of a SPSW. A comprehensive vectorial finite-difference numerical computation is used to show the accuracy of the analytic method. The influence of material loss is included in the geometric optics method and this also agrees well with the numerical computations.

Previously, the reflection coefficient of a SPP on a semi-infinite metal boundary was derived for different angles of incidence [11]. By matching the electric and magnetic field distributions of the SPP to those of the free-space, including non-propagating evanescent modes, that work incorporated both the SPP field distribution and the SPP polarization properties. The reflection coefficient was expressed as:

where ε_d is the dielectric constant of the dielectric above the slab, ε_m is the dielectric constant of the metal, ε_p is the effective dielectric constant of the surface plasmon, ε_p=ε_mε_d/(ε_m+ε_d) and θ is the angle of incidence. This expression requires evaluation of the integral:

From Eqs. (1) and (2), it may be shown that above the critical angle the phase of reflection, ϕ, is given by the analytic expression:

where $p=\sqrt{-{\epsilon }_d⁄{\epsilon }_m},s={\epsilon }_d+{\epsilon }_m,q=\sqrt{1-p^4+p^2{\cos }^2\left(\theta \right)}$.

We applied this analytic form of the phase of reflection to the analysis of a SPSW. Figure 1 shows schematic of geometric optics approach to surface plasmon modes in a stripe waveguide. The metal stripe region (including material dispersion) is surrounded by a uniform dielectric.

 

Fig. 1. Schematic of surface plasmon stripe waveguide (SPSW) propagation

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The phase of reflection above cut-off was used in the self-consistency condition to calculate the effective index of the SPSW modes:

where $n_p=\sqrt{{\epsilon }_p}$.

Figure 2 shows, with solid lines, the effective index of the SPSW mode as a function of stripe width for ε_m=-24 at 800 nm free-space wavelength [13] and ε_d=1. The imaginary part of the metal’s dielectric response was neglected in this analysis, which is a common approximation for the visible region of the spectrum. The effects of losses are incorporated in an adhoc manner later in this work. For comparison, the equivalent dielectric waveguide (dashed line in Fig. 2) was calculated using an effective dielectric constant equal to that of the surface plasmon, ε_p, and the usual Fresnel relation for the phase of reflection. It is clear that the incorporation of the phase of reflection, given by Eq. 3, in the geometric method calculation modifies the effective index calculations, as compared with the dielectric model [10,11]. Of particular note is the deviation between the SPSW and dielectric calculations close to cut-off: The SPSW has a nearly flat effective index that is significantly smaller than the dielectric model. The flat region arises due to the rapid variation in the phase of reflection of the SPSW near the critical angle [11]. It is interesting to note that experimental works have strong losses for the modes in this region [12].

Figure 2 shows, with markers, the numerically calculated the propagation constants of the SPSW waveguide using a vectorial finite-difference mode solver. The slab was semi-infinite in the vertical direction. Since the finite difference method is sensitive to the grid size, convergence tests were applied to verify the accuracy of the results. The dependence on the domain size was also verified, ensuring that the fields vanished when approaching the boundaries and so the results were quantitatively the same for perfectly matched layer and perfect electric conductor boundary conditions. It is clear from Fig. 2 that good agreement was found between the comprehensive numerical calculations and the fully-analytic geometric optics method.

 

Fig. 2. Effective index as a function of stripe width for SPSW modes (solid line), TE modes of a dielectric waveguide with the same refractive index as the SPP (dashed line), and numerical values (points).

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Figure 3 shows the influence of waveguide and material dispersion on the effective index as a function of wavelength for various waveguide widths. The effective index of SPSW waveguide fundamental mode was calculated including the material dispersion of the metal [13]. Material dispersion plays an important role in SPSWs due to the strong variation in the dielectric constant of the metal.

For comparison, the equivalent dielectric waveguide is also shown in Fig. 3, as was done in Fig. 2, but including material dispersion in n_p. For most of the wavelength range shown, the equivalent dielectric calculation for the 0.5 µm width shows better agreement with the 1 µm results than the 0.5 µm results, as calculated with the geometric optics method including the SPP phase of reflection.

 

Fig. 3. Dispersion of effective index for SPSW fundamental mode for various waveguide widths.

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Fig. 4. Dependence of the group velocity of SPSW on wavelength for various waveguide widths, including material dispersion

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The analytic geometric optics method proposed here allows for efficient calculation of other propagation properties of the SPSW, such as the group velocity and the group velocity dispersion. Figure 4 shows the dependence of the group velocity of SPSW as a function of wavelength for various stripe widths. As stripe width is increased from 0.5 µm to 2.0 µm the group velocity drops significantly in the visible region. The 0.5 µm wide equivalent dielectric waveguide, including material dispersion, is shown for comparison, and it is found to have a smaller group velocity than the SPSW model proposed in this work. Strong variation as a function of stripe width is observed just above optical wavelength. More detailed analysis (not shown in the figure) found that for wider waveguides, the group velocity of both SPSW proposed here and equivalent dielectric waveguide approached the same value, which is consistent with Fig 2. Therefore, the dielectric model is quite accurate overall for wider waveguides.

Next we included the influence of losses in the geometric optics method, where the SPP loss was calculated along the path of propagation. Accounting for the Goos-Hänchen shift, s, as in [14], the loss can be found as:

where γ is the loss of the SPSW, ξ is loss of the SPP [15], l is the path length in lossy medium (i.e., over the metal stripe) between successive reflections, z_l is the projection of l onto the waveguide axis. An expression for s can be derived from the usual relation [16] as:

Figure 5 shows the calculated loss, using Eqs. (3), (6) and (7), with respect to the stripe width at the free space wavelength of 800 nm. The loss of the SPP was calculated including the imaginary part of the dielectric constant of gold [15]. Numerical calculations with the vectorial finite-difference mode solver are shown for comparison. Good agreement was found between the comprehensive numerical calculations and the fully-analytic geometric optics method. Since the dielectric constant imaginary part is very small in comparison with its real part, almost exactly the same value is obtained for effective index.

 

Fig. 5. Dependence of the loss of SPSW on the stripe width at the wavelength of 800 nm.

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This loss calculation may be used to optimize SPSW design. Figure 5 shows that the transmission loss has a maximum at 2.2 µm stripe width.

Recently there have been works to consider edge states in SPP waveguides [17] and grooves [18]. Our comprehensive vectorial numerical calculations naturally include these edge states and for the conditions considered in this work, it also agrees well with the analytic model. For narrower stripe widths, the edge states will play an important role and these are strongly dependent upon the shape of the edges. As a result, the analytic model will be less accurate in this regime.

In conclusion, we have proposed and validated an improved geometric optics method to calculating surface plasmon propagation in patterned structures. An analytic expression for the phase of reflection of surface plasmons at the boundary of a semi-infinite metal slab was found to improve the accuracy with respect to past approaches. The geometric optics method was used to calculate the effective index of SPSWs as a function of waveguide width and wavelength for different modes and including the effects of material dispersion. The group velocity and group velocity dispersion may be calculated efficiently using this method, which is relevant to recent pulse-propagation methods in this regime [19]. We also incorporated the influence of material loss in the geometric optics method. All of the analytic calculations using the geometric optics method showed good agreement with comprehensive vectorial numerical calculations. This work may be extended to a number of other plasmonic structures, such as ring resonators, microdisks and stripe resonators, and it has utility in the design of surface plasmon integrated circuits and integrated biosensors.

The authors thank Rashid Zia for valuable discussions. The authors acknowledge funding from NSERC (Canada).