In practice, metal stripes with a finite width are more useful than infinitely wide metal films because their modes are fully confined in the plane transverse to the
3.2 SPSW Modes: Propagation and Cut-off
I investigate the accuracy of this model, by applying the obtained analytic expres-sion of the phase of reflection to characterize a SPSW, and compare its results with
previously developed dielectric model [12] and finite-difference time-domain (FDTD)
simulation. Figure 3.2 shows a schematic of geometric optics approach to surfaceplasmon modes in a stripe waveguide. The metal stripe region (including material
dispersion) is surrounded by a uniform dielectric.The phase of reflection above cut-off is used in the self-consistency condition to
Ë
9
d m d
W
Figure 3.2: Schematic of SPSW propagation,
calculate the effective index of the SPSW modes:
. n„W ? TT N ??\ /r> r\
tan(p—^- coso — m— ) = tan(— )
? ZZ(3.5)
^eff-spsw = npsin0TO (3.6)
where np = ^Je~p.
Fig. 3.3 shows, with solid lines, the effective index of the SPSW mode as a function
of stripe width for em = —24 at 800 nm free-space wavelength [74] and e¿ = 1. The
imaginary part of the metal's dielectric response is 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 below. For comparison, the equivalent dielectric waveguide (dashed line in Fig. 3.3) is calculated using an effective dielectricconstant equal to that of the surface plasmon, e?, and the usual Fresnel relation for
the phase of reflection. It is clear that the incorporation of the phase of reflection,
given by Eq. (3.4), 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 SPSW1.025
Figure 3.3: 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).
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 [73]. It is interesting to note that experimental works have strong losses for the modes in this region [75].
3.2.1 Numerical Confirmation
Fig. 3.3 shows, with markers, the numerically calculated the propagation constants of the SPSW waveguide using a vectorial finite-difference mode solver. The slab is
semi-infinite in the vertical direction. Since the finite difference method is sensitive
to the grid size, convergence tests are applied to verify the accuracy of the results.
1.25
1.2 (D
?
SPSW w = ?.dµt?
SPSW w =1. Olmi
S ?(D 1 SPSW w = 2.0um
LlJ
Dielectric w=0.5 um
1.05
1
500 550 600 650 700 750 800
Wavelength (nm)
Figure 3.4: Dispersion of effective index for SPSW fundamental mode for various waveguide widths.
The dependence on the domain size is also verified, ensuring that the fields vanished when approaching the boundaries, and so the results are quantitatively the same for perfectly matched layer and perfect electric conductor boundary conditions. It is clear from Fig. 3.3 that good agreement is found between the comprehensive numerical calculations and the fully-analytic geometric optics method.
3.2.2 Dispersion
Fig. 3.4 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 the SPSW waveguide fundamental mode is calculated including the material dispersion
1
Figure 3.5: Dependence of the group velocity of SPSW on wavelength for various waveguide widths, including material dispersion.
of the metal [74]. 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. 3.4, as was done in Fig. 3.3, but including material dispersion in np. For most of the wavelength range shown, the equivalent dielectric calculation for the 0.5 //m width shows better agreement with the 1 /an results than the 0.5 //m results, as calculated with the geometric optics method including the SPP phase of reflection.
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. Fig. 3.5 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 µt? to 2.0 µ??, the group velocity drops significantly in the visible region. The 0.5 µ?? 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. 3.3.
Therefore, the dielectric model is quite accurate overall for wider waveguides.
3.3 SPSW Mode Loss
Next, I include the influence of losses in the geometric optics method, where the SPP loss is calculated along the path of propagation. Accounting for the Goos-Hänchen shift, s, as in [76], the loss can be found as:
7 = -^-
Zl + S(3-7)
where 7 is the loss of the SPSW, ? is loss of the SPP [15], I is the path length in
the lossy medium (i.e., over the metal stripe) between successive reflections, z¡ is the
projection of / onto the waveguide axis. An expression for s can be derived from the usual relation [77] as:S 2nnpcos0d0 { ' ;
1
Figure 3.6: Dependence of the loss of SPSW on the stripe width at the wavelength
of 800 nm.
Fig. 3.6 shows the calculated loss, using Eqs. (3.4), (3.7) and (3.8), with respect to
the stripe width at the free space wavelength of 800 nm. The loss of the SPP iscalcu-lated including the imaginary part of the dielectric constant of gold [29] . Numerical
calculations with the vectorial finite-difference mode solver are shown for compari-son. Good agreement is 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.
Loss is inevitable in a tightly confined SP waveguide. Basically, there are two channels for propagation losses: the radiation loss from the coupling into propagating modes in air and absorption by the metal. Since the light field widely extends into
the surrounding dielectric media, the modes in SP waveguides may be evanescent or leaky even when there are no material losses. Fig. 3.6 shows that the transmission loss has a maximum at 2.2 µta stripe width. From this figure one sees that the loss of the fundamental mode is smaller for a wider stripe. This loss calculation may be used to optimize SPSW design.
3.4 Conclusion
This section outlines, proposes and validates an improved geometric optics method to calculating surface plasmon propagation in patterned structures. An analytic ex-pression, Eq. 3.4, for the phase of reflection of surface plasmons at the boundary of a semi-infinite metal slab is found to improve the accuracy with respect to past approaches. I also incorporated the influence of material loss in the geometric optics method. All of the analytic calculations using the geometric optics method shows good agreement with comprehensive vectorial numerical calculations.