Metal gratings for in-coupling a Gaussian beam incident from broadside to the long-range surface plasmon polariton (LRSPP) propagating in one direction along a membrane-supported Au slab bounded by air or water are proposed and modeled by the finite-difference time-domain method. Grating couplers for out-coupling the propagating LRSPP into free radiation directed along broadside are also investigated. Short grating designs consisting of a small number of Au bumps yield 15% to 20% in-coupling efficiencies, and about 60% out-coupling efficiencies. LRSPP back-reflections along the membrane waveguide caused by the out-coupling grating are also calculated and discussed.
©2010 Optical Society of America
Surface plasmon polaritons (SPPs) have stimulated much research interest in recent years, due in part to their interesting and unusual properties [1,2] which include sub-wavelength confinement  and large surface and bulk sensitivities , and due in part to their successful application to (bio)chemical sensors .
Single-interface SPPs are guided at optical wavelengths by a metal-dielectric interface . They have reasonably confined fields, a short propagation length, and they are heavily used in (bio)chemical sensors, conventionally in prism-coupled geometries . Alternatively, long-range surface plasmon polaritons (LRSPP) are guided by a thin metal slab or stripe bounded by media of similar refractive index forming a symmetric structure . They have comparatively less confined fields, a lower surface sensitivity, but much longer propagation lengths (1 to 2 orders of magnitude longer), and so are competitive for (bio)chemical sensing using, e.g., long interaction length structures such as the Mach-Zehnder interferometer .
For (bio)chemical sensing, the sensing solution is usually aqueous and in contact with one side of the metal. The symmetry required for propagation of the LRSPP can then satisfied by using a substrate which has an index close to that of water, such as Teflon or Cytop , to support the metal slab. Supporting the metal slab with a 1D photonic crystal, as demonstrated for air as the sensing medium , is an alternative. Another approach involves using an optically not-too-invasive dielectric membrane as the support, and allowing the sensing medium to bound the structure on both sides automatically ensuring symmetry, as demonstrated using an ultrathin free-standing Si3N4 membrane in air and water [9,10].
Many techniques can be used to couple the LRSPP to free-space beams or fibre modes, such as prism coupling [7–11], end-fire coupling , and coupling at broadside via an angle-cleaved  or tapered  optical fibre. End-fire coupling can be ~100% efficient , but requires access to a waveguide end facet which is not always possible or convenient, as in the case of the membrane-supported metal stripe [9,10]. Prism coupling is a broadside technique, and is efficient and reasonably convenient when used with a metal slab in an attenuated total reflection experiment [5,7,8], but it is difficult to implement with metal stripes and integrated structures [9–11] due to difficulties in controlling the spacing and parallelism between the base of the prism and the stripe, and the angle of incidence and divergence of the input beam. Broadside coupling with an angle-cleaved fibre is easier but requires that the fibre tip be cut accurately at a prescribed angle , and broadside coupling with a tapered fibre is easy but not very efficient . Metal gratings represent a potentially practical alternative.
Metal gratings to excite SPPs along the single metal-dielectric interface have been investigated extensively (e.g., [15–21]). Comparatively less work has been done on gratings to excite the coupled modes supported by the symmetric metal slab (ab and sb or LRSPP) (e.g., [22–26]). In  we proposed and modeled a metal grating to in-couple a 1D Gaussian beam incident from broadside (along the grating normal) to the LRSPP propagating in one direction. In the present paper we explore the design space of this in-coupling grating for a membrane-supported metal slab bounded by air or water. We also propose and model gratings for out-coupling the LRSPP into free radiation directed substantially at broadside. We quantify the performance of the gratings by computing their in- and out-coupling efficiencies, and we model the LRSPP back-reflection along the membrane waveguide caused by the out-coupling grating in order to compute its reflection coefficient. The gratings are envisaged to be monolithically integrated with the membrane waveguide, and we restrict ourselves to input beams that are incident at broadside and output beams that are substantially directed at broadside to facilitate alignment with external optical components.
2. Grating coupling scheme
The grating coupler scheme and the membrane waveguide are sketched in Fig. 1 (the structure is invariant along the z axis). The construction described in  is adopted, which consists of a t = 25 nm thick slab of Au (ε r,3 = −86.08 - j8.322)  on a d = 20 nm thick free-standing Si3N4 membrane (ε r,2 = 22), bounded on both sides by air ε r,1 = 1 or water ε r,1 = (1.3159 - j1.639 × 10−5)2  (ε r,i denotes the relative permittivity). The operating free-space wavelength is set to λ0 = 1310 nm. The LRSPP propagates along this structure, retaining its essential characteristics (i.e., long-range, symmetric E y field), as long as the membrane remains optically not too invasive, as is the case here.
2.1 In-coupling grating
The in-coupling grating is defined by a number of rectangular metal grating bumps deposited onto the waveguide, each of width W and height H, defining a unit cell of period Λ. A Transverse-Magnetic (TM- or x-) polarised 1D Gaussian line source having a waist width of 2w 0 = 8 µm (1/e 2 intensity) is assumed launched a distance s = 2 µm above the structure, from the broadside direction, and is incident onto the grating at an offset p from its centre (Fig. 1 (a)). We are interested in designs and beam offsets p that maximise coupling into the LRSPP propagating in the -x direction.Eq. (1) simplifies to the following approximate relation:
2.2 Out-coupling grating
The out-coupling grating scheme is shown in Fig. 1(b). The membrane waveguide dimensions are the same as in Fig. 1(a). The LRSPP is launched at a distance x = -L from the grating and it propagates in the + x direction. We are interested in maximising the coupling from the LRSPP to free radiation directed substantially at broadside (θ ~0°). Because the in- and out-coupling gratings are both radiation gratings, they should have the same dimensions (H, W, Λ) as long as they remain weakly perturbing  (again, this is not necessarily the case here).
3. Simulation results and discussion
The finite-difference time-domain (FDTD) method  was used to model the grating couplers and simulate their performance as a function of geometrical parameters. This method was selected because of the ease with which it handles complex electromagnetic structures, and because the gratings are not weakly perturbing. The mesh was set to 1 nm steps along the y direction in the region near the metal (−250 nm ≤ y ≤ 250 nm) in the case of the final simulations. The input beam consisted of a time-domain pulse lasting 8.65285 fs in duration (default for λ0 = 1310 nm). Perfectly matched layers were used to terminate all boundaries.
The accuracy of the method and the convergence of the results (relative to the mesh density) were verified by modelling the propagation of the LRSPP along a uniform section of membrane waveguide and comparing its field distribution and propagation characteristics (neff and mode power attenuation - MPA) with those computed using the transfer matrix method (TMM) . Errors of 0.01% and 6.0% relative to the TMM were noted for the neff and MPA, confirming that the LRSPP can be modelled with good accuracy.
3.1 In-coupling gratings
Simulations were performed initially using a small computational window (29 μm wide × 10 μm high) in order to rapidly generate good designs and find a good beam offset p. It was quickly found that about 11 metal bumps of a good height (H ~ 100 to 200 nm) were needed to achieve good coupling, and that the beam should be incident near the left edge of the grating to favour coupling to the LRSPP in the -x direction.
The effect of varying the grating period Λ was investigated by monitoring the normalised total power flowing in the x direction at x = −4 and −14 µm, Px(−4) and Px(−14). Px is computed by integrating the x-directed component of the real part of the Poynting vector over the vertical line at x spanning the height of the computational window, and then dividing by the power of the incident Gaussian beam. At x = −4 and −14 µm, the x-component of the Poynting vector points along -x. Figures 2(a) and (b) show Px(−4) and Px(−14) as a function of the grating period for air and water bounding the structure, respectively. The width and height of the bumps were held constant at W = 630 nm and H = 150 nm for air and at W = 490 nm and H = 110 nm for water, while the duty cycle was allowed to change to accommodate the changing period. From Fig. 2(a) it is observed that Px(−14) is maximum for Λ = 1200 nm, and that it does not vary strongly with Λ over the range Λ = 1180 to 1220 nm. From Fig. 2(b) it is observed that Px(−14) is maximum for Λ = 930 nm and that it varies more strongly with Λ because the fields are more confined in this case.
Figures 3(a) and 3(b) show Px(−14) as a function of grating dimensions W and H for air and water, respectively. In the case of air (Fig. 3(a)), the period of the grating was held constant to Λ = 1200 nm, W was held constant to W = 630 nm while H was varied, and H was held constant to H = 150 nm while W was varied. In the case of water (Fig. 3(b)), the period of the grating was held constant to Λ = 930 nm, W was held constant to W = 490 nm while H was varied, and H was held constant to H = 110 nm while W was varied. From Fig. 3 we observe that Px(−14) is maximised at W = 630 nm and H = 160 nm for air, and at W = 470 nm and H = 120 nm for water. Both of these grating designs have a duty cycle that is larger than 50%.
We now vary the beam offset p and monitor Px 15 µm to the left of the first grating bump, Px(−15-0.5W), and 15 µm to the right of the eleventh (last) grating bump, Px(15+10Λ+0.5W). The x-component of the Poynting vector points along -x and +x at these locations, respectively. We also monitor the normalised total power flowing in the y direction Py (defined similarly to Px) at y = ± 5 µm. We are interested in maximising Px(−15-0.5W) and in understanding the power flow at the other monitoring positions. The best grating dimensions found from the previous simulations were used (Λ = 1200 nm, W = 630 nm, H = 160 nm for air; Λ = 930 nm, W = 470 nm, H = 120 nm for water). Figure 4 plots these monitored powers as a function of beam offset, from p = 0 (beam aligned with the centre of the grating, middle of the sixth bump) to p = 5.5Λ (beam aligned with the left edge of the grating) in steps of 0.5Λ. The plots are symmetric as p ranges from the centre of the grating to its right edge.
We note from Fig. 4 that Px(−15-0.5W) = Px(15+10Λ+0.5W) for p = 0, indicating that the grating sends power into equal parts in the -x and + x directions when the beam is aligned with its centre. As the offset increases, Px(−15-0.5W) increases to a maximum value (at p = 4Λ or x = 1.2 µm for air, and p = 3.5Λ or x = 1.395 µm for water), then decreases as the beam impinges beyond the left edge of the grating. Px(15+10Λ+0.5W) decreases monotonically as p increases, and is significantly smaller than Px(−15-0.5W) at the best offsets where essentially unidirectional coupling occurs.
The power transmitted through the grating, Py(−5), remains essentially constant at ~2.5% for air and ~2.9% for water. It is dependent on the metal thickness so a thicker metal could fully block the transmission. Py(5) is located above the source, so it corresponds to the reflected power. As the offset increases, Py(5) decreases to a minimum value near the offset that maximises Px(−15-0.5W) (p = 3.5Λ for air and p = 3Λ for water), then increases sharply as the beam impinges beyond the left edge of the grating on what is a essentially a good mirror.
Using the best grating parameters obtained from the previous simulations (Λ = 1200 nm, W = 630 nm, H = 160 nm, p = 4Λ for air, and Λ = 930 nm, W = 470 nm, H = 120 nm, p = 3.5Λ for water), a larger simulation window was used in order to monitor the LRSPP over a long propagation length and observe its fields away from the grating region where spatial transients can dominate the response. This is necessary in order to remove the contribution of radiative modes and of the ab mode from the total fields and thus from the calculations of the coupling efficiency (the ab mode is weakly excited at best and it attenuates very rapidly). We emphasise here that the values of Px plotted in Figs. 2-4 were computed in regions of spatial transients (x = −4, −14, −15-0.5W and 15+10Λ+0.5W), and so they do not correspond to the coupling efficiency into the LRSPP.
Figure 5 shows the distribution of Ey along y, normalised to the field amplitude of the incident beam, at various locations (x) from the left edge of the grating. From these plots it is noted that Ey evolves into a stable and unvarying distribution far from the grating, where it is slightly localised to the metal surface opposite the membrane as expected for this structure [9, 10]. In the region close to the grating (x > −25 μm, Figs. 5(a) and (c)), the field distribution is distorted because it is comprised of the LRSPP and spatial transients. In the case of air, the LRSPP needs to travel almost 200 μm before clearing this region (Fig. 5(b)), whereas for water it needs to travel about 100 μm (Fig. 5(d)).
Figure 6 shows movies of the time evolution of the fields over a portion of the computational domain for the same gratings. The movies of Figs. 6(a) and 6(c) show the Ex field component of the (x-polarised) source reflecting strongly from the gratings, and transmitted weakly through, for air and water as the bounding media, respectively. Two reflected waves are observed because the incident beam overlaps partly with the left edge of the grating (so 2 reflecting surfaces are involved). The movies of Figs. 6(b) and 6(d) show the Ey field component of the LRSPP. Essentially unidirectional coupling is achieved, with the Ey field of the LRSPP propagating in the -x direction being much larger than that of the LRSPP propagating in the + x direction, as observed. Field transients are also noted from these movies to be strong near the gratings.Eq. (3) becomes increasingly dominated by its first term:
Figure 7 plots Px(x) in dB as a function of inverse distance from the grating (−1/x) over the ranges of −425 ≤ x ≤ −25 µm for air and −225 ≤ x ≤ −25 µm for water. A larger simulation window was needed for air because the Ey field converges more slowly to the field of the LRSPP (eg, Fig. 5). Also plotted in Fig. 7 are the mode power attenuation in dB/mm (MPA = 0.02αlog10e) estimated from the decay of Px(x) (using a spline interpolation of Px(x) evaluated Δx = 0.1 µm about a position x), and Px,0 computed from Eq. (4) using the corresponding α (and plotted in dB). Recall that Px is normalised to the power of the incident beam, so Px,0 is also normalised and corresponds directly to the coupling loss of the beam to the LRSPP at the left edge of the grating. The MPA and Px,0 are erroneous in the region of spatial transients because Px,r(x) ≠ 0, but they become increasingly correct as x → -∞, as does Eq. (4). The MPA decreases in magnitude and Px,0 converges monotonically as x → -∞ because Px,r(x) → 0. Richardson’s extrapolation was used to estimate converged values of Px,0 from their values at x = −25, −50, −100 and −200 µm, yielding −7.76 dB for air and −7.12 dB for water; these values are plotted at −1/x = 0 in Fig. 7. The associated in-coupling efficiencies are 16.7% for air and 19.5% for water, respectively. These efficiencies are lower than the ~100% coupling efficiency that is achievable via end-fire excitation , but higher than the efficiencies reported to date for other out-of-plane in-coupling techniques on this structure [11, 13, 14]. The MPA computed from the FDTD results at x = −400 µm for air and at x = −200 µm for water are well converged to the MPA computed using a finite-difference mode solver and the same mesh ; the errors are 3.1% for air and 1.7% for water, respectively.
3.2 Out-coupling gratings
The LRSPP propagating along the structure can be out-coupled into (TM-) x-polarised light propagating substantially at broadside (in the + y direction) by a grating of similar design. Out-coupling gratings can be analysed by the volume current method  if the grating is weakly perturbing, meaning that the bump perturbation should be less than about 2% . This is not the case here so we use the FDTD method to simulate their performance.
As indicated in Fig. 1(b), the LRSPP propagating in the +x direction is launched at a distance L to the left of the grating (L = 5 μm initially), and the out-coupled fields propagating in the +y direction are monitored. The LRSPP mode fields were computed using a finite-difference mode solver on the same mesh , then used as the source in the FDTD simulations. To reduce eventual fabrication complexity we constrain the height of the out-coupling grating bumps to be identical to that of the best corresponding in-coupling gratings (H = 160 or 120 nm for air or water), and we assume initially 11 grating bumps.
The effects of varying the grating period Λ and the bump width W were investigated first by monitoring the normalised total power flowing in the +y direction at y = 4 µm, Py(4). Figure 8(a) reveals that the maximum Py is achieved at Λ = 1200 nm and W = 580 nm for the case of air. As discussed earlier, the in-coupling and out-coupling gratings should be very similar. Indeed, the best period for both couplers is the same, but the width of the bumps and the duty cycle of the gratings are different. The duty cycle of the best out-coupling grating is lower than 50% (W = 580 nm) whereas that of the best in-coupling grating is larger than 50% (W = 630 nm). One reason for the different duty cycles is that the height of the out-coupling grating is not optimised. Figure 8(b) reveals that the maximum Py is achieved at Λ = 920 nm and W = 460 nm for the case of water. The duty cycle in this case is 50% and the grating period and bump width are both less than those of the best grating in-coupler (which are Λ = 930 nm and W = 470 nm). It is also noted that the out-coupling grating bounded by water (Fig. 8(b)) produces a larger Py compared to that bounded by air (Fig. 8(a)), because the fields of the LRSPP are more confined in the case of the former.
Figure 9 plots Py as the number of grating bumps increases from 11 to 31 (in steps of 5 bumps) for the best grating dimensions of the previous simulations (Λ = 1200 nm, W = 580 nm, H = 160 nm for air; Λ = 920 nm, W = 460 nm, H = 120 nm for water). Py increases with the number of grating bumps and rapidly saturates to a maximum value due to the large radiation loss induced by propagation along the grating. 25 to 30 bumps are sufficient to reach saturation in both cases.
Figure 10 shows the distribution of Ex along x, normalised to the field amplitude of the incident LRSPP, as observed at different locations y above the structure. The same grating designs as in the previous simulations (Fig. 9) were used along with 31 grating bumps in both cases. From Figs. 10(a) and 10(b) we note that the main radiation lobe is located near the first grating bump. (This differs from the prediction of the volume current method stating that the main lobe should be located near the centre of the grating , verifying that this method can’t be used for strongly perturbing gratings). As the observation distance y increases, the main lobe broadens but the full width of the radiation narrows as the fields emitted further along the grating begin to contribute.
Figure 11 shows the field distributions over a portion of the computational domain. Figures 11(a) and 11(b) show the Ey and Ex field distributions for the structure bounded by air, whereas Figs. 11(c) and 11(d) show the fields for water. The LRSPP was launched at x = −200 μm for air and at x = −150 μm for water. From Figs. 11(a) and 11(c), we note a standing wave pattern in Ey for x < 0, caused by back reflection from the grating. We also note a rapid decrease in |Ey| for x > 0 as the grating induces radiation. Figures 11(b) and 11(d) show that the radiation is directed along +y, and that Ex is largest near the first bump, as noted above.
From Figs. 10 and 11 we note that an optical fibre could be aligned along the broadside direction, near the left edge of the grating, to capture the main radiated lobe. In this case out-coupling efficiencies similar to the in-coupling efficiencies would be expected. An alternative arrangement would be to place a photodetector (or array) above the grating such that all of the radiation would be captured. In this case, Py plotted in Fig. 9 corresponds directly to the out-coupling efficiencies, which are at best 62% and 58% for air and water, respectively.
As noted from Figs. 11(a) and 11(c), the grating causes the incident LRSPP to be partially reflected in the -x direction. The reason for this is attenuation along the grating due to the strong perturbation of each bump. At the input of the grating, the Ey fields reflected from the 2k-1 (k = 1, 2,…) bumps differ in phase by π compared to the Ey fields reflected from the neighbouring 2k (k = 1, 2,…) bumps (by design). If these reflected fields had the same amplitude, then the total reflected field would sum to zero (or very close to zero for a weakly perturbing grating). But this is not the case in these gratings, where the field amplitude differs significantly between neighbouring bumps, leading to observed LRSPP reflection. This reflection can be reduced by varying the strength of the grating along its length such that the amplitude of neighbouring partial reflections is the same at the input of the grating.
The reflection coefficient of the LRSPP at the input of the grating can be determined from the computed field distributions. Figure 12 shows the distribution of Ey along x at y = 30 nm. The maximum and minimum values of Ey (Ey,max and Ey,min) can be deduced from these plots at a distance L from the grating away from the spatial transients, and so are associated with the LRSPP only. Following , the standing wave ratio s at L is then calculated via:Fig. 12), yielding 15.40% and 20.10% for air and water, respectively. Using Eq. (7) with the α’s computed using the finite-difference mode solver, we find |Γ0| = 15.74% and 20.94%, respectively. The corresponding mode power reflection coefficients are |Γ0|2 = 2.48% and 4.38%. Such levels of reflection may be deemed low enough depending on the application.
4. Concluding remarks
Au grating couplers for in-coupling a Gaussian beam incident from broadside to the LRSPP propagating in one direction along a (Si3N4) membrane-supported Au slab bounded by air or water on both sides, and for out-coupling the LRSPP into light directed along broadside, were proposed and modelled at λ0 = 1310 nm using the FDTD method. The gratings consist of a periodic array of rectangular Au bumps deposited onto the Au slab. The number of bumps, their dimensions, and the grating period were varied in order to find designs that provide good coupling efficiencies. The location of incidence of the beam relative to the in-coupling grating centre (offset) was also investigated. The grating couplers should be easy to fabricate and will facilitate the alignment of ancillary optical components.
A good in-coupling grating design for the structure bounded by air consists of W = 630 nm wide, H = 160 nm high bumps arranged in a period of Λ = 1200 nm, whereas a good design for the structure bounded by water consists of W = 470 nm wide, H = 120 nm high bumps arranged in a period of Λ = 930 nm. The computed in-coupling efficiencies are 16.7% and 19.5% for the structures bounded by air and water, respectively, into the LRSPP propagating along a single direction when the grating is excited near one of its edges. 11 bumps were found to be sufficient for both in-coupling gratings.
A good out-coupling grating design for the structure bounded by air consists of W = 630 nm wide, H = 160 nm high bumps arranged in a period of Λ = 1200 nm, whereas a good design for the structure bounded by water consists of W = 460 nm wide, H = 120 nm high bumps arranged in a period of Λ = 920 nm. The designs are similar but not identical to the in-coupling gratings. The computed out-coupling efficiencies are 62% and 58% for air and water respectively, from the LRSPP propagating along the waveguide to radiation directed along broadside, assuming that all radiated fields can be captured by a photodetector. The strong perturbation of the out-coupling grating causes the incident LRSPP to be back-reflected with mode power reflection coefficients of 2.48% and 4.38% for air and water, respectively; this reflection can likely be reduced by optimising the grating design.
Techniques to accurately extract the coupling efficiencies and reflection coefficient from the FDTD simulation data are described.
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