## Abstract

The ability of gratings made of dielectric ridges placed on top of flat metal layers to open gaps in the dispersion relation of surface plasmon polaritons (SPPs) is studied, both experimentally and theoretically. The gap position can be approximately predicted by the same relation as for standard optical Bragg stacks. The properties of the gap as a function of the grating parameters is numerically analyzed by using the Fourier modal method, and the presence of the gap is experimentally confirmed by leakage radiation microscopy. We also explore the performance of these dielectric gratings as SPP Bragg mirrors. The results show very good reflecting properties of these mirrors for a propagating SPP whose wavelength is inside the gap.

© 2010 OSA

## 1. Introduction

Plasmonics constitutes a new branch of nanophotonics dedicated to develop integrated optical devices based on surface plasmon polaritons (SPPs) [1,2]. SPPs are evanescent waves bounded to a metal-dielectric interface whose propagation is therefore confined in plane [3]. This confinement makes SPPs attractive for the miniaturization of photonic components and to overcome the diffraction limit. An important effort towards the development of plasmonic devices is necessarily dedicated to the design and understanding of systems able to control the SPP propagation. Although most attempts focus on the development of highly confined SPP waveguides [4–11], a deep comprehension on the strategies and mechanisms controlling the propagation of SPP on extended metallic surfaces is crucial to provide new approaches in the development and improvement of SPP waveguides and other plasmonic elements.

Different methods for controlling the propagation of SPP have been analyzed so far. One of these strategies relies in the interaction of SPPs with scattering defects and the design of periodic corrugated structures. Based on this principle, one- and two-dimensional gaps in the SPP dispersion relation can be engineered [12–15], and SPP optical elements such as Bragg mirrors [16–18], demultiplexors [19] and plasmonic crystals with guiding channels [20–22] can be obtained. The design of prisms and lenses based on the refraction induced by the surface corrugation is also possible [23]. Another approach is based on the use of dielectric loads on top of the metallic layer, to alter the effective refraction index of the propagating SPPs [24]. In this way, and applying the same refraction equations as in conventional optics [25], different elements such as lenses [24], prisms [26], in-plane Fresnel zone plates [27,28] or waveguides [7,29,30] can be designed.

In this paper, we analyze the possibility of using dielectric gratings on top of a continuous Au-film as Bragg mirrors for SPPs, in analogy to the conventional dielectric Bragg mirrors used in classical optics. The paper is divided in two main sections: In the first part we investigate the propagation of surface plasmons on interfaces consisting on flat metal layers covered by periodically structured dielectric loads. The alternation of effective refractive index opens a gap in the SPP dispersion relation, analogously to the propagation of light in periodically layered dielectrics. We study in detail the properties of the gaps opened in these systems as a function of different dielectric load parameters, such as height, relative thickness or the angle of SPP propagation. We also compare the gaps opened by the periodic dielectric loads with those opened by metallic gratings. The significant differences in the properties of these two kinds of gratings have been addressed and the underlying mechanisms for opening the gap in each case are discussed. In the second section, we demonstrate theoretical and experimentally the performance of this kind of systems as dielectric Bragg mirrors for SPPs.

## 2. SPP propagation along a periodically structured dielectric load on a metal surface

As mentioned above, SPPs are evanescent waves bounded to the interface of a metal and a dielectric, whose dispersion relation is determined by the dielectric permittivities of the metal and the dielectric composing that interface and its particular geometry [3]. In the case of a semi-infinite homogeneous dielectric on top of a semi-infinite homogeneous metal, the SPP dispersion relation is given by the analytical expression:

k_{SPP}being the SPP in-plane wavevector, k

_{0}the wavevector of light in vacuum, and ε

_{m}and ε

_{d}the permittivities of the metal and the dielectric, respectively. For more complex geometries (multilayers, corrugations, etc.), an analytical expression for the dispersion relation is difficult to obtain and this is usually calculated numerically. In this work, we have employed a Fourier Modal Method (FMM) [31,32] to compute the dispersion relations of SPPs propagating on Au surfaces covered by homogeneous and patterned dielectrics. Specifically, we calculated the reflectivity of the dielectric/metal film system when illuminated with p-polarized light from a glass substrate as a function of both the angle of incidence and wavelength. The minima in this reflectivity are associated with the transfer of energy to modes such as SPPs and therefore represent the SPP dispersion relation. This method has been demonstrated to provide good results in Au surfaces decorated with Au ridges gratings or Au plots arrays [18,21].

Figure 1(a)
depicts one of these reflectivity maps for a 50-nm-thin Au layer on a glass substrate loaded by a homogeneous dielectric thin film (100 nm SiO_{2}, with refractive index n = 1.49) covered by air as sketched in the inset of the figure. The reflectivity minima curve, in blue, provides the dispersion curve of the SPP mode, which within the considered wavelength range is located well to the right of the light line due to the presence of the thin dielectric film. The SPP dispersion can be expressed in terms of effective index of the mode, *n _{eff}*, defined as

*n*k

_{eff}=_{SPP}/k

_{0}. The effective index plays for SPP the role of the refractive index for conventional radiative optics [24]. Figure 1(b) shows the evolution of

*n*with the thickness of the dielectric layer

_{eff}*t*for a fixed wavelength (λ

_{0}= 800 nm, with λ

_{0}being the wavelength in vacuum). As expected,

*n*augments as the thickness

_{eff}*t*increases and it reaches a saturation value for a given thickness (around 350 nm here), which is related to the exponential decay of SPP electric field within the dielectric medium.

If SPPs travel through a boundary between two differently thick dielectrics, *n _{eff}* will be discontinuous and the SPPs will experience Fresnel refraction and reflection [25]. In the same way, when SPPs propagate through a periodic stack of interfaces with different

*n*, the reflection from the subsequent boundaries will interfere and may give rise to constructive interference of the reflected waves which opens a gap in the SPP dispersion relation of the stack. Such a stack of interfaces with different

_{eff}*n*can be achieved by alternating dielectric ridges of different thicknesses on top of the metal surface, or in particular by alternating ridges of height

_{eff}*h*spaced by air (

*h*= 0 nm) as depicted in Fig. 1(d). As an example Fig. 1(c) shows the SPP dispersion curve for ridges of SiO

_{2}with

*h*= 100 nm on top of 50 nm Au. The SPP travels in the direction perpendicular to the grating (

*ϕ*= 0°). The condition for constructive interference, or equivalently for opening the gap is:

_{inc}*d*and

_{1}*d*correspond to the width of the different dielectric ridges [see Fig. 1(d)]. The widths

_{2}*d*and

_{1}*d*of the stack shown in Fig. 1(c) are calculated after Eq. (3) for providing a gap in the dispersion relation at λ

_{2}_{0}= 800 nm:

*d*= 150 nm,

_{1}*d*= 205 nm (${n}_{\mathit{eff}}^{1}$ = 1.27, ${n}_{\mathit{eff}}^{2}$ = 1.02). As can be seen in the figure, the dispersion relation presents a gap centered at λ

_{2}_{0}= 785 nm, in good agreement with the expected value. This result implies that Eq. (3), although not exact, provides a good approximation for the design of dielectric SPP Bragg mirrors, at least at normal incidence.

For the design of dielectric SPP Bragg mirrors for directions other than normal incidence, only the in-plane wavevector component perpendicular to the grating has to be taken into account in Eq. (2). This leads to the following general expression for any incident angle *ϕ _{inc}*:

*ϕ*,

_{inc}*ϕ*and

_{1}*ϕ*are related by Snell's law ${n}_{\mathit{eff}}^{inc}\mathrm{sin}{\varphi}_{inc}={n}_{\mathit{eff}}^{1}\mathrm{sin}{\varphi}_{1}={n}_{\mathit{eff}}^{2}\mathrm{sin}{\varphi}_{2}$. Throughout this paper we take ${n}_{\mathit{eff}}^{inc}={n}_{\mathit{eff}}^{2}$ and ${\varphi}_{inc}={\varphi}_{2}$.

_{2}We have then computed the SPP dispersion relation for gratings with *h* = 100 nm for different angles *ϕ _{inc}* and parameters

*d*and

_{1}*d*calculated from Eq. (4) to provide a gap centered at λ

_{2}_{0}= 800 nm. The results are shown in Table 1 , where we compare the intended gap center position with that obtained from the dispersion relations. According to the results, Eq. (4) predicts quite accurately the gap position for small angles of incidence, but the deviation becomes more significant as

*ϕ*increases, being as strong as 100 nm for

_{inc}*ϕ*= 60°. This deviation of the predicted results from the ones obtained by calculating the dispersion relation can be attributed to our approximated estimation of the effective index of the dielectric ridges. We use an isotropic

_{inc}*n*value obtained from continuous dielectric layers of the same height than the ridges (

_{eff}*n*= 1.27 in this case). However, we can expect that

_{eff}*n*is anisotropic as is the structure itself. The

_{eff}*n*= 1.27 value makes sense only when SPPs propagate perpendicularly to the ridge. SPPs travelling along the ridge, on the other hand, become guided inside the ridge and

_{eff}*n*of this guided mode is ~1.06 for the particular ridge dimensions we deal with here [30]. Therefore, the

_{eff}*n*for SPPs going through the ridges at angles other than 0° or 90° would be in between these two extreme values, moving towards smaller values as the angle of propagation increases. The actual gap position should then move towards smaller wavelengths as

_{eff}*ϕ*increases, as can be seen in Table 1. A more accurate determination of the ridges

_{inc}*n*for the different angles of propagation would improve the validity of Eq. (4).

_{eff}#### 2.1 Gap properties for different grating parameters

After gaining some insight into how the dielectric gratings can be designed, we now analyze the dependence of the gap position and width on the grating parameters: filling factor, height of the dielectric ridges and angle of incidence.

The grating filling factor, FF, is defined as the ratio of the surface occupied by the dielectric ridge to the length of the unit cell or periodicity *d* = *d _{1}* +

*d*: FF =

_{2}*d*/

_{1}*d*. Figure 2(a) shows how the gap width evolves as the filling factor ranges from 0.1 to 0.9 for a grating of 50-nm-high SiO2 ridges with

*d*= 355 nm (

*d*then varies from 35 to 320 nm). The SPP propagates perpendicularly to the grating

_{1}*ϕ*= 0°. These results show that the maximum gap width is obtained when the filling factor is around 0.50. In fact, for both very small and very high FF the perturbation of the grating is so small that the open gap is very narrow. The grating effect is stronger for FF around 0.50, therefore providing the bigger gap width around these values. Moreover, for traditional Bragg mirrors it is known that the optimum ratio between thicknesses of the composing layers corresponds to the “lambda/4” condition (where lambda refers to λ in the media). In our case of dielectric SPP Bragg mirrors, the obtained most favorable FF is 0.42. Taking into account that ${n}_{\mathit{eff}}^{1}$ is higher than ${n}_{\mathit{eff}}^{2}$, this indicates that the “lambda/4” condition, although not exact, can be established as a good condition for the design of these mirrors. Interestingly, the gap width is higher for FF = 0.1 than for FF = 0.9, where the gap disappears. This reveals that very narrow dielectric stripes on top of the Au surface are more efficient as Bragg mirrors than a dielectric layer engraved with very narrow air grooves.

_{inc}In Fig. 2(b), the evolution of the gap position as a function of FF is shown. The graph shows that the gap center is red shifted almost linearly as the filling factor increases. This result is in good agreement with Eq. (4). It can be explained taken into account that, for a fixed period, an increase in FF implies an increase in *d _{1}* and a decrease in

*d*. This represents that the SPP travels longer through the medium with bigger

_{2}*n*. Therefore, the dispersion relation of the SPP propagating through the grating shifts towards higher in-plane wavevectors as FF increases. As the first Brillouin zone boundary is fixed for a given period, the gap position moves towards lower energies. Regarding the gap edges, they are accordingly red shifted as FF increases. Figure 2(c) accounts for the evolution of gap width as the height of the grating,

_{eff}*h*, is varied from 10 nm to 250 nm, assuming grating period and width of 355 and 150 nm, respectively. From the results plotted, we observe that the gap width increases monotonically with

*h*. Besides, the gap width reaches values as large as 125 nm for

*h*= 250 nm. Since the increase in

*h*means that ${n}_{\mathit{eff}}^{1}$ augments, the index contrast of the media composing the Bragg grating raises, and as a consequence the gap width is enlarged. In Fig. 2(d) the dependence of the gap center and edges position on the grating height is plotted. The center wavelength increases linearly with

*h*until the height is about 120 nm. Thereafter the increase is slower towards a saturation point. The upper and lower edges follow the same evolution. The red shift of the gap center can be understood from Eq. (4): as

*h*increases, ${n}_{\mathit{eff}}^{1}$ also does and therefore λ

_{0}/2 increases. Since

*n*reaches a saturation value as

_{eff}*h*increases [see Fig. 1(b)], related with the exponential decay of the SPP electric in the dielectric, the same behavior takes place for the gap center wavelength.

Figures 2(e) and 2(f) show the evolution of the gap width with the angle of incidence for gratings of height *h* = 100 nm and FF = 0.42. In Fig. 2(e) we used the periodicity *d* calculated from Eq. (4) to obtain a gap centered at λ_{0} = 800 nm. As we know, the actual gap position deviates strongly from this wavelength for large angles of incidence. So, in order to prevent conclusions associated with the variation of the gap center position, for Fig. 2(f) we corrected accordingly the periods so that the centre wavelength was the same for all angles of incidence. We observe that these variations do not considerably affect the gap width which remains around 65 nm. The main parameters governing the gap width are FF and the height of the ridges, which determine the optical contrast along the grating. There is however a weak tendency of increasing the gap width with ϕ_{inc}, as it also happens with classical optics dielectric Bragg mirrors.

#### 2.2 Comparison with gaps opened in metallic corrugated layers

As has been mentioned above, the properties of SPP Bragg mirrors based on metallic gratings have already been analyzed in the literature, both theoretically and experimentally [16–18]. In this subsection, we compare the behavior of the dielectric and metallic gratings in order to gain insight in their differences and similarities. In particular, we present the dependence of the gap properties on the grating filling factor and on the angle of incidence.

Figures 3(a)
and 3(b) show respectively the gap width and gap position evolution with the mirror filling factor for a grating consisting on 50-nm-high Au stripes on a 50-nm-thick Au film. The grating periodicity is *d* = *d _{1}* +

*d*= 355 nm. The design of metallic based SPP Bragg mirrors is based on the purely geometrical Bragg relationship [17]:

_{2}*d*= 355 nm is designed for showing a gap centered at λ

_{0}= 724 nm for normal incidence. The results plotted in Figs. 3(a) and 3(b) point out several differences between metallic and dielectric based SPP Bragg mirrors. First, the gap width is much larger for metallic mirrors: it can achieve 130 nm for a 50-nm-high Au grating, while the maximum value obtained for a 50-nm-high SiO

_{2}grating is only 26 nm. This implies that the scattering power of the Au grating is much stronger than that of the SiO

_{2}one. We associate this higher scattering power of the Au grating to the fact that the SPP has to “jump” from the Au ridge surface to the Au film surface while propagating through the grating, since a SPP is always bounded to the interface between a metal and a dielectric. On the other hand, in a dielectric Bragg mirror the SPP changes smoothly from one media to another, and it is the contrast in effective index between the two media which determines the gap width. Nevertheless, a gap width of 125 nm can also be attained with dielectric mirrors by using 250-nm-high dielectric stripes. Moreover, being the propagation of the SPP through the dielectric ridges much smoother since it does not involve as much scattering as in the case of metallic ones, we expect less losses associated with the dielectric gratings. Regarding the evolution of the gap width with the grating filling factor, the behavior is similar for both kinds of gratings, showing the maximum gap width for filling factors around 0.50. Indeed, the maximum gap width occurs exactly at FF = 0.5 for the metallic grating, in perfect agreement with the “lambda/4”condition (since here ${n}_{\mathit{eff}}^{1}$ = ${n}_{\mathit{eff}}^{2}$).

As for the gap center position, the behavior of the two kinds of gratings is completely different. For dielectric gratings [Fig. 2 (b)], the evolution of the gap center is monotonic with FF and the gap edges movement compensate for the gap width evolution. This is not the case for metallic gratings. From Eq. (5), the gap center should not change with FF. However, it decreases slightly as FF increases. On top, the gap edges evolution is quite asymmetric. The lower gap edge moves with the filling factor as expected, favoring a larger gap width at FF around 0.5. The upper gap edge, on the other hand, moves very little. In fact, this can be explained in terms of the SPP dispersion relation [12]. The SPP Au/air dispersion relation lies very close to the light line and the movement of the upper edge is highly limited, lest the mode should turn into a radiative one [see Fig. 3(c)]. The lower edge, in contrast, is free to move as much as the grating parameters determine. Since this effect is more significant in the conditions providing a bigger gap width, such as FF < 0.6, the gap center position is shifted towards longer wavelengths, as can be seen in Fig. 3(b). As a consequence, the gap center position is higher than the design value λ_{0} = 724 nm: the limited opening of the upper gap edge artificially shifts the gap center to lower energy. In the case of dielectric mirrors, the limitation of movement of the gap upper edge is less strict. We associate this to the fact that the SPP dispersion relation is shifted towards larger in-plane wavevector values (away from the light line) due to the presence of the dielectric load [see Fig. 1(c)].

Figures 3(d) and 3(e) show the evolution of the gap width and position with the angle of incidence. The grating height was *h* = 50 nm for all the cases and the grating period was calculated using Eq. (5) for getting a gap centered at λ_{0} = 724 nm in each case. We observe a strong reduction of gap widths with *ϕ _{inc}* in this case [see Fig. 2(e)]. This behavior had already been predicted in metallic gratings [33,34]. We associate it with a reduction of the scattering power of the metallic defects (ridges) with the angle of incidence [34]. The dielectric gratings, on the other hand, show a much smaller dependence of the gap width with

*ϕ*. Therefore, when considering gratings for the design of SPP Bragg mirrors, the dielectric based design is more consistent for different angles of incidence while the properties of the metallic ones are strongly affected by the angular change, making the first system more flexible.

_{inc}Back again with the metallic gratings, the gap center position [Fig. 3(e)] becomes closer to the nominal value for bigger angles of incidence, as a consequence of the asymmetry of the gap edges movement. The upper gap edge remains almost insensitive to the angle of incidence, all the influence of the reduction in scattering power being taken by the lower gap edge.

#### 2.3 Experimental verification of the presence of gaps in dielectric gratings

After presenting the theoretical basics for the design of dielectric SPP mirrors, we experimentally prove the creation of a gap. For that, we have employed leakage radiation microscopy (LRM). LRM is a far-field technique based on recovering the radiation leaking towards the substrate while the SPP propagates on a thin metal film deposited on glass [24,35–37]. This technique gives images whose intensity is proportional to that of the propagating SPP, and has already been demonstrated to provide quantitative information on SPP devices, such as mirrors [18,38] and waveguides [30]. With an appropriate lens configuration, the leakage radiation microscope can image not only the real plane but also the conjugated (Fourier) plane [36,37], which allows a direct observation of the k-space. For a fixed wavelength, this provides images of the two-dimensional iso-frequency SPP dispersion relation and hence it discloses the gap [39,18].

In the experiment we imaged the Fourier plane for surface plasmons propagating on Au surfaces decorated with several dielectric gratings of different periods and widths. The gratings on the measured sample are 95-nm-high PMMA ridges deposited on top of a 60-nm-thick Au film. The gratings have been fabricated by means of electron beam lithography on top of a Au film deposited by thermal evaporation. The gratings parameters have been designed to open gaps for SPPs of λ_{0} around 800 nm at 0° and 45° with respect to the grating normal (*d* = 360, 466 nm respectively). The chosen filling factor is FF = 0.42, the most optimized value in order to obtain the widest gap width as has been shown in the section 2.1.

Figures 4(a)
and 4(b) show the obtained images corresponding to a SPP of energy 1.6 eV (λ_{0} = 775 nm) propagating on the gratings with periodicities 360 and 466 nm respectively. The SPPs are launched by focusing a circularly polarized Ti:sapphire laser on top of the dielectric grating. The presence of defects on the Au surface provides the needed momentum to excite the SPP [18]. For a Au thin film having small defects, the Fourier image would show a circle whose radius is the SPP in-plane wavevector, k_{SPP}. The presence of the dielectric grating opens a gap in the dispersion relation, as can be seen in the images. This gap appears at around 0° for Fig. 4(a) and 45° for Fig. 4(b), as expected. By using the FMM method, the iso-frequency SPP dispersion relations of dielectric gratings have also been calculated for comparison with the experimental images. Figures 4(c) and 4(d) show the calculated maps for the same gratings as the ones measured in Figs. 4(a) and 4(b), and they agree very well. In order to get the best match in the angular position of the gap for a given SPP energy, we set n_{PMMA} = 1.47, which is in good conformity with the optical properties of this resist. Figures 4(a) and 4(c) show gaps centered at 0° and Figs. 4(b) and 4(d) at 41.5°. Regarding the gap angular width, in the first case we have an experimental width of Δα = 32° and a predicted value Δα = 42°, and in the second case Δα = 6° for both situations. This angular gap provides us information on the angular acceptance of these systems when used a SPP Bragg mirrors [18]. The angular acceptance decreases as the angle of incidence increases. The good agreement between the experimental Fourier images and the simulated ones indicates the excellent suitability of FMM to describe these gratings.

## 3. Design and characterization of dielectric based SPP Bragg mirrors

In the previous section we have studied and demonstrated the gap properties of dielectric ridges on a metal. In this section we extend the idea by assuming a finite grating of dielectric ridges on a metallic surface to create an effective mirror for propagating SPPs. These gratings of dielectric ridges can act as SPP Bragg mirrors due to the presence of a gap in the dispersion relation, which means that the propagation of the SPP through the structured dielectric media is forbidden for the energy range inside this gap. To study the efficiency of this system we have calculated, by means of FMM, the reflectivity of different Bragg mirrors composed of SiO_{2} ridges and designed for *ϕ _{inc}* = 45° (

*h*= 100 nm,

*d*= 200 nm,

_{1}*d*= 466 nm). The calculated gap of this dielectric grating is centered at λ

_{0}= 750 nm, and the gap width is 65 nm. The suitability of the FFM method to compute the SPP Bragg mirror's reflectivity has been demonstrated in Ref. [17].

Figure 5(a)
depicts the schematic of the calculated configuration and Fig. 5(b) presents the electric field map distribution of the SPP impinging on a Bragg mirror made of 10 ridges. The SPP energy corresponds to λ_{0} = 765 nm, well inside the gap of this grating. As can be seen in the image, after reaching the mirror the SPP is reflected at 2 × 45°. The mirror reflectivity, as calculated from the field map following the procedure described in [17], is R = 72% ± 2%. We should note that the reflectance calculation takes into account the intrinsic SPP propagation losses, since it is based on the ratio of the SPP intensity after a certain propagation distance with and without mirror. The obtained result, R = 72%, implies that dielectric based Bragg mirrors can act as efficient mirrors for SPP. The transmittance of the mirror is zero, as can be seen in Fig. 5(b) from the absence of intensity at the other side of the mirror. The mirror introduces then a 26% of losses, which could be due to the absorption when traveling through the mirror, or due to the scattering from SPPs to radiative light propagating in air or in the glass substrate. Figure 5(c) shows the evolution of the calculated reflectance, transmittance and losses with the number of dielectric ridges. All the curves show saturation when the number of dielectric ridges approaches 20, from which can be interpreted that a SPP Bragg mirror composed of 20 dielectric ridges is optimum to maximize the reflection of a propagating SPP. This also enables us to get transmittance values very close to zero.

At this point, we would like to point out that, even if the gap width opened by the metallic gratings is much higher than the one for dielectric gratings of comparable height, as has been shown in the previous section, the efficiency of the system as SPP Bragg mirrors is not that better. From the results shown in Ref [17], the calculated reflectivity for a mirror composed of 20 Au ridges of t = 50 nm is R = 70%. The reflectance obtained by 20 SiO_{2} ridges is found to be higher [R = 76%, see Fig. 5(c)]. We associate this reduced efficiency of the metallic gratings to the higher losses that they introduce, again linked to the scattering mechanism responsible of the gap opening in this system. For dielectric gratings, less scattering is expected and we attribute the losses mainly to the smaller propagation distance of the SPP through a dielectric load than through air, implying then that losses are higher where transmission is higher. In this way, the losses decrease with the number of lines for dielectric mirrors, contrary to the behavior on metallic ones where the losses increase as the number of scattering center increases.

Finally, to confirm that the effect of the Bragg mirror is due to the gap opening, Fig. 5(d) shows the electric field map for a SPP of λ_{0} = 800 nm hitting a mirror designed for a different wavelength (*h* = 50 nm, *d* = 416 nm; *d _{1}* = 150 nm, calculated gap centered on 586 nm). Being the SPP completely out of the gap in this case, it travels through the mirror with a very small portion being reflected (R = 1%). The transmittance through the mirror is 71%, and we attribute the losses mainly to the reduced propagation distance on the SPP on a dielectric load.

#### 3.1 Experimental characterization of dielectric based SPP Bragg mirrors

The LRM technique has been used to characterize experimentally the performance of the dielectric based SPP Bragg mirrors. For that, we have fabricated by means of electron beam lithography mirrors composed by different numbers of dielectric ridges on top of a 60-nm-thick Au substrate. To reduce the fabrication time, we have used a negative resist (SU-8, n = 1.57) instead of PMMA. The SU-8 ridges height was fixed to *h* = 100 nm. The period of the grating was chosen to be *d* = 436 nm and the width *d _{1}* = 184 nm (FF = 0.42). The nominal gap center associated to these parameters, according to Eq. (4), is 775 nm so the actual gap is expected around λ

_{0}= 725 nm following Table 1. The same geometry as in Ref [18]. has been employed to image the SPPs interacting with the dielectric mirrors. The SPPs on the Au surface are excited by focusing a linearly polarized laser on top of a 200 nm wide line [40], and the SPPs travel 25 μm until it reaches the mirror [see Fig. 6(a) ].

Figure 6(b) shows the direct space leakage radiation image of a SPP impinging on a mirror made of 20 SU-8 ridges. The laser excitation wavelength is λ_{0} = 750 nm. The image shows how most of the SPP is nicely reflected when hitting the mirror and no transmission through it is observed. In the case of Fig. 6(b), the objective used to focus the light on the launching line has a large numerical aperture, N.A. = 0.65, resulting in a broad angular distribution of the excited SPPs. This becomes even more obvious when looking at the conjugated (Fourier-) plane as shown in Fig. 6(c). The SPPs excited at the launching line travel in opposite directions, giving rise to the two opposite bright circular segments [36]. The significant spreading of the circular segments reflects the broad angular distribution of the excited SPPs. The bright circle in the center is just the directly transmitted light from the excitation, and the grey band is due to light diffracted in all angles by the excitation line. Once excited, the downwards-going SPP travels towards the mirror, where it becomes reflected. This generates another circular segment in the Fourier image located at 90° with respect to the incident one. The reduced angular width of the reflected beam indicates that the angular acceptance of the Bragg mirror was smaller than the angular distribution of the incident SPP beam. This large angular spreading of the incident and reflected beams prevents any quantitative analysis of image 6(b). Therefore, we reduced the numerical aperture of the focusing objective and the obtained direct and conjugated space images are shown in Figs. 6(d) and 6(e). Here, the incident and reflected SPPs beams appear much more collimated. The direct transmission of the laser located in the center of the Fourier plane has been blocked by a mask to image only the leakage radiation coming from the SPPs. Still some diffraction coming from the launching line and the grating lines is visible, as shows the grey stripe on the Fourier image [Fig. 6(e)].

Figure 7
shows the profile of the measured leakage intensity along the incident and reflected SPP beams for the mirror shown in Fig. 6(d) (for the location of the intensity profile, see inset in the figure). The SPP intensity decays exponentially, as expected [3]. By fitting the experimental data, the SPP propagation distance (L_{SP}) can be obtained. In this case, we get L_{SP} = (11.5 ± 0.6) μm, smaller than the expected value for a film of 60 nm Au at λ_{0} = 750 nm, L_{SP} (theoretical) = 24 μm. This can be attributed to resist residues remaining on the Au surface after the lithography process or to thinner than nominal Au layers. The mirror position can be identified by the presence of some small peaks in the intensity. Interestingly, the intensity profile evolution after the mirror shows an interference pattern, indicating that the reflected SPP interacts with the scattered light produced when the incident SPP hits the first line of the grating. We observe that this scattered light dominates the signal as the reflected SPP propagates and decays further away from the grating: unlike the SPPs, the light scattered by the dielectric ridges into the substrate does not decay exponentially and hence contributes stronger at larger distances. Taking into account this effect, we calculate the reflectivity of the mirror by dividing the exponential fit of the intensity profile after the mirror (violet line) by the fit before the mirror (green line): R = (85 ± 7) %. The large error is mainly caused by the arbitrariness on the chosen position of the fitted curves. We have also performed measurements at λ_{0} = 800 nm, which is expected to be already close to the band edge. In this case, L_{SP} increases to L_{SP} = 16 μm, as expected for longer wavelengths (L_{SP} (theoretical) = 32 μm). Again, the mirrors reflectivity for N = 5 lines is very high and close to 85%. A similar result can be obtained for mirrors designed for other angles of incidence. Thus, we conclude that the dielectric Bragg mirrors are indeed very efficient for reflecting SPPs propagating on planar surfaces, even more efficient than theoretically expected, a result that was also found with the metallic mirrors [17].

## 4. Conclusions

This paper shows how the concept of dielectric stacks coming from classical radiative optics also applies for the context of propagating surface plasmons. We have demonstrated that a grating composed of dielectric ridges on a flat Au layer can be used to control the propagation of SPPs by opening a gap in its dispersion relation. The position of this gap has been predicted by directly applying the analogy with standard optical Bragg stacks for different angles of incidence. Even though the prediction is quite accurate for normal incidence, there is a deviation of the calculated results from the theory for bigger angles. The properties of the gap have been analyzed in terms of varying grating parameters. The influence of the dielectric grating height and the filling factors conform well to the analogy with the standard optical system. From the comparison of these gratings with their metallic counterparts, we have observed notable differences which we attribute to the different mechanisms responsible for the opening of the gap in each case. The gap is opened in the case of dielectric mirrors by the contrast in effective index of the SPP, whereas the phenomena mostly responsible for gap opening in metallic mirrors is the strong scattering power of these gratings. The gaps have been experimentally confirmed by leakage radiation microscopy in Fourier plane, and a good agreement has been obtained between experiments and simulations.

We have gone a step further and extended the idea of using a grating of dielectric ridges as mirrors for propagating SPPs. The properties of these SPP Bragg mirrors have been analyzed both experimental and theoretically. The results show very good reflecting properties of these mirrors for SPPs with wavelengths inside the gap of the Bragg reflector.

## Acknowledgments

The authors acknowledge the support from the EU through the STREP project PLASMOCOM (FP6-IST-034754) and from La fundació CELLEX Barcelona. The authors would also like to thank Jean-Claude Weeber for his help with the numerical calculations.

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