1. Introduction

Light trapping near optical resonators have been shown to enhance nonlinear processes [1] and hyperspectral light localization [2]. Pioneering efforts in light-trapping include using cold atomic gases [3], photonic crystal nanocavities [4], gradient index materials [5,6], and geometric tapering of the structure [7]. Rainbow-trapping has been proposed as a means of localized storage of broadband electromagnetic radiation in metamaterials and plasmonic heterostructures [8–10].

The methods and structures needed for rainbow-trapping, however, remain a great challenge and are often quite elaborate and difficult to fabricate. A gradient in groove depth (L in Fig. 1), for example, as reported by [8,11], requires nanofabrication methods with nanometer-precision control over each groove depth.

 

Fig. 1 p-polarized radiation (E-field in the z-direction) can launch SPPs traveling in the x-direction into the grooves, as well as SPPs traveling in the z-direction on the top surface between grooves. In narrow grooves when w ≲ 150 nm, SPP fields within the grooves overlap, resulting in coupling of SPPs on the sidewalls (shown as a squiggly line between two down-traveling SPPs facing each other on the opposite walls of the groove). Likewise, SPPs can become coupled through the metal when d is comparable in size, to the skin depth of SPPs in the metal. However, coupling through the metal requires still smaller d values ∼ 25 nm, approximately the skin depth of SPPs in the metals in the visible and near infrared range.

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2. Introducing a width-based gradient

This article uncovers a new parameter for rainbow-trapping gratings, enabling a simple standard planar fabrication process over a large area. Unlike groove depth, which is directly correlated with the resonator length, this new parameter, i.e. the groove width, is fundamentally linked to the surface plasmon polariton (SPP) interactions on the sidewalls of the groove (shown with downward arrows in Fig. 1). As the name suggests, the energy of surface plasmon modes is concentrated very near the metal-dielectric interface and exponentially decays away from the boundary. Thus, the proximity of surfaces supporting SPPs can significantly affect the optical properties of the waveguide such as a nanogroove. Figure 1 depicts p-polarized light impinging on a metallic grating comprised of nanogrooves with width w and depth L and assumed infinitely long in the y direction. Unless the groove widths are comparable to the evanescent tail of the SPPs in the dielectric or smaller (∼150 nm) the SPPs on each groove wall are essentially decoupled, and the waveguide takes on a conventional dielectric slab nature, with a waveguide dispersion independent of the groove width. However, as grooves become quite narrow and comparable in size to the exponential tail of SPPs in the dielectric, w is conceived as an additional parameter modifying the dispersion relation of the groove as opposed to directly changing the cavity resonance. This is unintuitive at first, because a typical cavity resonator is sensitive to L, not w. Therefore, at very small cavity widths, another mechanism is at play. Although this effect has been known to exist in abnormal optical absorption [12,13], we show here that extraordinary properties such as rainbow-trapping ensue when w is used as a gradient parameter of a grating. Conversely, at groove-widths much larger than the exponentially decaying SPP tail in the dielectric, the effect of the width on the response of the structure becomes negligible, and grooves assume a conventional waveguide nature. This evanescent reach of the field into the dielectric is typically ∼ 50 nm for a single metal-dielectric interface, or ∼100 nm in a sandwich structure of the grooves, and unlike the relatively fixed skin-depth of about ∼25 nm in the metal, it is quite dependent on the wavelength. This defines the interval wherein w becomes unfrozen and takes on an active role in the cavity response. Beyond this interval, the contribution of w to the groove dispersion relation becomes negligible as the SPPs effectively decouple. Since most graded gratings reported thus far fall in the latter regime where w is frozen, the use of w as a design parameter of gratings has remained unintuitive and unexplored.

Working against this understanding, we propose including the use of w as a powerful design parameter for light-trapping applications, rendering a two-dimensional landscape of design parameters for light-trapping and waveguiding. Further, this approach naturally accommodates unintended variations in groove dimensions (both L and w) that are sometimes unavoidable in practice. This powerful analytic and predictive tool opens up the possibility of easily producible and large area surfaces for rainbow-trapping, while simultaneously offering a quick mnemonic design tool that can often circumvent the need for full simulations for designing hyperspectral-gratings.

3. Theory and discussion

It is sometimes useful to consider subwavelength volumes such as nanogrooves as the building blocks of subwavelength gratings. When all grooves are identical, the emergent optical response of the grating can often be analyzed based on a single groove’s response. We extend this groove-wise analysis to include graded gratings with a gradually changing groove geometry along the direction of light propagation. This enables the study of the rainbow-trapping behavior of the grating by analytically and numerically considering the effects of SPP coupling within grooves of various widths. The approach introduced here, not only explains the behavior of an individual groove, but also extends to the behavior of the grating as a whole. Note that our groove-wise method of analysis differs from approaches that treat the entire grating as a homogeneous effective medium approximation and field-localization as a spoof-SPP [14]. Also, throughout this work, we will refer to the trapping of spatially resolved wavelengths of broadband light over the grating as a “trapped-rainbow effect”, even when this spectral range lies outside the visible range, in keeping with the usage of this jargon in the literature [15].

The power of the present approach is that a simple model for a single groove can explain the behavior of an entire functionally graded grating, with a non-homogeneous effective index. More specifically, this is accomplished by ensuring that the groove geometry changes across the grating (along the z-direction as shown in Fig. 1), in such a manner that the light-trapping response of the extended structure follows the light-trapping behavior of the groove. Finally, the approach allows us to provide a simple and powerful pictorial representation of predicting the response of graded gratings without having to perform simulations.

In general, when there is a refractive index gradient within an optical structure, light is naturally guided along the direction of the increasing index. At the same time, an optical resonator near the resonance significantly decreases the group velocity of light propagating through the resonant medium. (This condition is realized, for instance, near the band edges of periodic structures such as photonic crystals [16]).

We model a groove as a metal-insulator-metal (MIM) waveguide. The non-oscillatory p-polarized bound modes of this waveguide result in the coupling of the localized modes in the core ( $-\frac{w}{2}<z<\frac{w}{2}$ in the inset of Fig. 2) (for details see [17]). The electric and magnetic field-components of the coupled-SPPs are given by solving Maxwell’s equations:

 

Fig. 2 This plot shows λ_sp as a function of dielectric core thickness w with permittivity ∊₁, and a metal cladding with permittivity of ∊₂. The MIM waveguide is unbounded in both x and y directions defined in Fig. 1. At point (c) corresponding to inset (c) shown under the curve, the waveguide is almost outside the regime of influence of plasmonic coupling (w > 200 nm), where it can be treated as a decoupled MIM waveguide with a wavelength of λ_c ≃ λ_decoupled = 9 μm. At point (b), λ_sp is compressed to about 75% of λ_decoupled due to the intragroove coupling effect, and at point (a) this compression is a further 50% of λ_decoupled. Inset shows the fundamental and higher order modes of a plasmonic cavity given by Eqs. (4) and (5) together. The MIM waveguide is now bounded in both x and z directions and only assumed to be infinite along y axis. The effect shown in the main plot equally applies to the bounded case depicted in the inset.

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Figure 2 shows the MIM SPP wavelength ${\lambda }_{\mathit{sp}}=\frac{2\pi }{\text{Re}(\beta )}$, calculated from Eq. (4) as a function of the core thickness w [13]. Whereas the main plot shows longitudinal cross sections of unbounded MIM waveguides, bounded grooves forming a Fabry-Perot type resonator cavity are shown in the inset of Fig. 2. Applying the continuity conditions: E_z1|_x=0 + E_z2|_x=0 = 0 due to the perfect electric conductor at the bottom of the resonator, and ${\frac{\partial (H_{y1}+H_{y2})}{\partial x}|}_{x=L}=0$ due to the near unity reflection from the top of the resonator where $\beta \gg \sqrt{∊}{k}_0$, yields the relation between the plasmonic wavelength λ_sp, cavity length L, and cavity modes n :

As the groove width decreases to the point where SPPs on the opposite walls of the groove become coupled, the propagation length first increases, since the electric field within the plasmonic MIM waveguide approaches the electrostatic regime where the field is maximally stored in the dielectric gap. As w is further decreased, the confined field is driven deeper into the metal, thus increasing the losses. The long propagation lengths due to the coupling of SPPs, and the high effective index courtesy of the metal cladding, strike a balance at an optimum groove width. Of course, this optimal width depends on the wavelength, and losses generally improve in the mid-IR range. Details of this optimal balance are discussed elsewhere [21].

At the other extreme, w → ∞ (separating the groove walls from each other and resulting in a single vertical wall) leads to $n_{\mathit{eff}}=\sqrt{\frac{{∊}_1{∊}_2}{{∊}_1+{∊}_2}}$, a single SPP interface. Engineering a grating with a graded n_eff through slow (adiabatic) tapering of w, therefore, results in a metasurface with a spatially varying effective index. This gradient index guides light in the direction of the increasing index until it comes to a stop at when the Fabry-Perot resonant condition is satisfied at the groove of appropriate depth (L). Subsequent grooves are below cutoff and backscatter the wave in the direction of the on-resonance groove.

Tailoring gratings using this SPP coupling through a w-variation, as shown above, enables the rainbow-trapping structures illustrated in the inset of Fig. 3. The main plot of Fig. 3 shows the resonant condition of the grooves while accounting for the wavelength compression due to the coupling of SPPs on the groove walls. A single point anywhere on this plot defines a groove with a width w_g and a length Lg, and accordingly a sequence of points on this plot represent a series of grooves, or a grating. Two such gratings are shown in insets (a) and (b), where the groove depths and groove widths are gradually changing along the z-direction. The corresponding schematics of such gratings are shown next to each dotted-line. Since the solid-plot of this curve represents the groove resonant condition, any point that lies on this curve denotes a locus of light-trapping. And any dotted line that intersects this plot is a light trapping graded grating. Note that both axes are plotted as functions of z, or the direction of light propagation to simply indicate that the groove dimensions (w, and L) within each grating are varied in the direction of light propagation. We shall henceforth refer to these dotted lines as cut-lines, as they indicate graded gratings that intersect the light-trapping curve. The cut-lines shown in insets (a), where only depth is varied, and (b), where only width is changed represent only two simple examples of light-trapping graded gratings. Slanted cut-lines of various angles, as well as curved cut-lines corresponding to nonlinear gradient profiles can also be used, representing gratings with concomitant variation of depth and width.

 

Fig. 3 Plot of the fundamental cavity mode (n=0) as a function L and w for a single frequency. Dotted cut-lines (a) and (b) outline the discrete nature of the grating possessing linear gradients in L and w, respectively, with each point corresponding to a nano-groove of particular dimensions (w_ng, L_ng). Inset (a): a grating with a linear gradient in depth variation corresponding to the vertical dotted line (a). Inset (b): a grating corresponding to the horizontal dotted line (b), where the gradient is strictly based on groove width. Inset (c) plots the resonant dispersion curve for several frequencies in the visible range. Each spectral component intersects at a different location with the horizontal dotted line pictorially representing the grating. The result is the formation of a rainbow trapping effect over the grating. In order to compare the E-field profiles of light trapping based on depth variation shown in inset (A), to that based on width variation of inset (B), COMSOL simulations for two structures of comparable dimensions are shown. Structures (A) and (B) correspond to the COMSOL simulation of the E-field and correspond to cut-lines (a) and (b), respectively. The simulation frequency for both structures is at 30 THz. It can be seen that the profile of the light localization is more symmetric in the width-based structures compared to a depth-based grating.

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The schematic shown under the horizontal cut-line (b) in Fig. 3 represents a grating comprised of grooves numbered with Latin numerals (i through v). Each point of the dotted line corresponds to a single groove of the grating, with dimensions immediately known from its coordinates. The position of each point with respect to the solid plot, predicts the groove response to the excitation. Grooves i and ii are below resonance, groove iii is at resonance (positioned exactly on the solid curve), and grooves iv and v lie above resonance. Consequently, grating (b) is a uniform-depth grating comprised of grooves of constant depth L but gradually changing width w. Light is guided from groove v towards groove i by grooves that are above the cutoff until it reaches the location of a resonant groove iii where it is trapped—beyond which the grooves are below the cutoff and scatter the light back towards the location of the trap, which is groove iii.

As we are considering gratings with spatial gradients along the z-direction, it is useful to reintroduce L and w as functions of z, as in Fig. 3. This is easily accomplished by scaling L and w axes by the same function that defines the spatial changes of L and w. As a result, the coordinates of a point such as point iii on the cut-line (b) of Fig. 3, simultaneously convey the dimensions of a single groove as well as the relative location of the groove within the grating. That is, cut-line (b) outlines a light-trapping graded grating while preserving complete information about the dimensions of individual constituent grooves, and their changes along z. As previously mentioned, this ensures that the light-trapping behavior of the grating follows the light-trapping behavior of the groove, when L and w are both functions of z.

Therefore, individual points of the cut-lines shown in Fig. 3, correspond to the dimensions of a single groove while the entire cut-line outlines the contour of a rainbow-trapping grating, such as the two gratings depicted right next to the dotted lines (a) and (b). By maintaining the one-to-one correspondence between the groove geometry and its placement within the grating, the intersection of the resonant curve with any such cut-line, reveals the locus of light-trapping over that grating. For clarity, the resonant curve in Fig. 3 is only plotted for a single frequency of light, while inset (c) shows multiple frequencies in the visible part of the spectrum. For this reason Fig. 3 serves as a powerful mnemonic key for designing light-trapping gratings with groove width w, groove depth L, or an arbitrary combination thereof. The following procedure can be used to design light trapping structures in a particular region of interest, without having to run simulations. First, draw a cut-line with practically acceptable groove dimensions that intersects the resonant plots such as those shown in the inset (c) of Fig. 3. Next, position this cut-line line with respect to the resonant dispersion plot so that their intersection, which represents light-trapping, occurs in the region of interest (such as the center) of the grating. A cut-line can involve a combination of L and w and should certainly take into account the feasibility constraints and fabrication methods of choice, when choosing groove dimensions. The resulting cut-line provides the detailed design parameters of the light-trapping grating of interest. It is worth mentioning that the cut-lines outlining the grating profiles given here are simple examples and arbitrary grating profiles (such as nonlinear profiles) can be just as easily analyzed based on this mnemonic design key.

4. Comparison with simulations

In order to compare the theory presented here with simulations, a horizontal cut-line with w spanning from 3 nm to 40 nm, is moved up to vertically sweep over the resonant plot shown in the top plot of Fig. 4. As this cut-line moves up, it intersects the resonant plot at different locations corresponding to the loci of light trapping over the grating. The location of the predicted trap is then compared with simulations. The simulated trap locations are shown as points on the plot, with error bars corresponding to the full-width-at-80%-maximum of the trapped field distribution. (See insets (a) and (b)). Ten different simulations performed in COMSOL Multiphysics are overlaid to form the composite image shown in the top plot of Fig. 4. Full-blown simulation results for two such points (a) and (b) are shown below the main plot. Permittivity values were taken from Johnson & Christy [22] and Rakic̀ [23]. These simulations assume silver as the grating material and air as the dielectric, at an excitation wavelength of 1 μm. The position of the trap over that grating corresponds to the location of the maximum field intensity within the resonant grooves, as shown in insets (a) and (b) of Fig. 4. The horizontal error bars in Fig. 4 are based on the field distribution at the location of resonance (trap), which spatially extends over a few adjacent grooves on either side of the principal resonant groove. This distribution is asymmetric, since grooves below cutoff (past the location of the trap) do not admit the fields within, while grooves before the trap do. Due to this asymmetric nature of the trap, full-width-at-80%-maximum of the trapped field distribution is taken as the error bar values in determining the trap location. The values for two such points are shown above the gratings in insets (a) and (b).

 

Fig. 4 Ten simulation frames are overlaid to produce this compound image. The width-gradient profile is constant across all frames (w changing from 3 nm to 35 nm). That is, the structures in each frame utilize only groove width as the gradient parameter. In each successive frame the structure gets deeper; that is, the length (L) of the structure increases from left to right. Points (a) and (b) show the loci of light-trapping for two such frames. For example, point (a) in the figure shows a structure with a depth (L) of 1.1 μm, and the location of the trap at a groove with a width of 11 nm, corresponding to the grating shown in inset (a); the same logic applies for inset (b). The solid curve on the main plot shows the exact analytical solution of λ_sp as a function of L and w (groove dimensions). Insets (a) and (b) show the field distribution of the trap over the structures that corresponds to points (a) and (b) on the main graph. The locations of the trap determined from the simulation, closely agree with the analytical solution.

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The darker regions indicate high field localization. Each simulation is performed with a structure having a constant profile of w variation. L is successively increased in each subsequent simulation from 0.5 μm to 1.5 μm, corresponding to the sweeping of a long horizontal cut-line representing a width-based grating (similar to cut-line (b) in Fig. 3), over the resonant dispersion plot, revealing the loci of light-trapping. When the resulting points are compared to the analytical model (solid curve), the agreement is remarkable. The simulated values are very slightly higher than the theory, as expected, since the theory only includes the coupling of surface plasmons within the groove and ignores any coupling between the adjacent grooves. In reality, there is a weak but nontrivial coupling between neighboring grooves.

We note that the structures shown in Fig. 4 are for the purposes of verifying and comparing our approach with simulated results along a large portion of the resonant curve. From a practical standpoint of designing rainbow-trapping structures, these dimensions can easily be chosen to meet the fabrication criteria, an example of which is shown in Fig. 5. Structures of similar dimensions have been recently produced reliably [24].

 

Fig. 5 Simulation of light trapping by a graded grating where the groove depths are fixed at 60 nm with groove widths varying between 20 nm to 50 nm. The direction of energy flow is overlaid on the plot of the E-field. The excitation is through a current-source at 500 THz (600 nm), as seen to the left of grating. The dielectric function of gold is taken from Johnson & Christy [22]. This illustration depicts realistic dimensions for a rainbow-trapping grating operating in the visible range such as the one shown in the inset of Fig. 3. (A series of simulations for a range of frequencies is shown in Visualization 1.)

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5. Conclusion

Uncovering width as a grating design parameter reveals the full two dimensional landscape of rainbow-trapping gratings amenable to many applications in bio-imaging, surface enhanced Raman spectroscopy, optoelectronics, and infrared sensors. Using width variation instead of depth can lead to single-step fabrication of gratings whose groove depths are simply defined by the thickness of the resist used for developing the gratings. The theory developed here accounts for a generalized combination of groove width and depth, as a combination of both is often inescapably present in practice.