## Abstract

We propose a novel SPP coupling scheme capable of high SPP throughput and high SPP coupling efficiency based on a slit of width greater than the wavelength, immersed in a uniform dielectric. The dispersive properties of the slit are engineered such that the slit sustains a low-loss higher-order waveguide mode just above cutoff, which is shown to be amenable to wavevector matching to the SPP mode at the slit exit. The SPP throughput and SPP coupling efficiency are quantified by numerical simulations of visible light propagation through the slit for varying width and dielectric refractive index. An optimal SPP coupling configuration satisfying wavevector matching is shown to yield an order-of-magnitude greater SPP throughput than a comparable slit of sub-wavelength width and a peak SPP coupling efficiency ≃ 68%. To our knowledge, this is the first investigation of coupling between higher-order waveguide modes in slits of super-wavelength width and SPP modes.

© 2010 Optical Society of America

## 1. Introduction

Surface-plasmon-polariton (SPP) modes, transverse-magnetic (TM) electromagnetic waves that exist at a metal-dielectric interface, hold promise for the miniaturization of optical devices [1, 2]. Due to the lack of readily-available sources directly emitting SPP modes, designing methods to couple plane-wave modes to SPP modes with high efficiency and high throughput remains an important objective. Plane-wave modes directly incident onto a metal-dielectric interface cannot efficiently couple into SPP modes due to a mismatch between the SPP wavevector and the component of the plane-wave wavevector along the interface. Scatterers have been used to bridge the wavevector mismatch between plane-wave and SPP modes. When a scatterer is illuminated, enhancement of the incident plane-wave wavevector along the metal-dielectric interface by the Fourier spatial frequency components of the scatterer geometry in the plane of the interface enables wavevector matching between the incident light and the SPP mode.

A widely-used scatterer-based SPP coupling technique is to illuminate a slit in a metal film. When a slit is illuminated by a TM-polarized plane wave, a small portion of the incident wave excites a guided mode in the slit. The guided mode propagates through the slit and subsequently diffracts at the slit exit. The total light intensity leaving the slit exit defines the total throughput, the SPP intensity leaving the slit exit defines the SPP throughput, and the ratio of the SPP throughput to the total throughput defines the SPP coupling efficiency. The throughput and efficiency of a slit are highly dependent on the width of the slit relative to the wavelength of the incident plane wave. A slit of width less than the wavelength has inherently low total throughput and low SPP throughput, but is capable of high SPP coupling efficiency. We showed in our previous work [3] that the SPP coupling efficiency of a sub-wavelength slit is increased up to ≃ 80% by coating the slit with a nanoscale dielectric layer; the dielectric layer, however, does not significantly affect the SPP throughput. The objective of this work is to design an “ideal” SPP coupling scheme capable of both high throughput and high efficiency.

Increasing the total throughput of a slit can be achieved by simply increasing the slit width. Increasing the SPP throughput, on the other hand, is more challenging because the SPP throughput is mediated by coupling between the guided mode in the slit and the SPP mode, which is not fully understood and remains a topic of current research. Recently, several theoretical [4, 5, 6] and semi-analytical [3, 7, 8, 9] models have been developed to describe coupling from a guided mode in a single slit to a SPP mode. The SPP mode is defined by a single, unique solution to Maxwell’s Equations after imposed the magnetic field boundary condition at the metal surface. The guided mode in the slit, on the other hand, generally consists of a superposition of infinite TM-polarized waveguide eigenmodes, which each correspond to solutions to Maxwell’s Equations after imposing the magnetic field boundary condition at the slit edges. The width of the slit dictates the eigenmode composition of the guided mode in the slit. All previous models [3, 4, 5, 6, 7, 8, 9] describing SPP coupling by a slit have assumed a slit width less than the wavelength. When the slit width is less than the wavelength, all except the zeroth-order TM_{0} waveguide eigenmode are attenuated and the guided mode is accurately and simply approximated as the TM_{0} eigenmode. The TM_{0}-eigenmode approximation becomes increasingly inaccurate [5, 7] as the slit width is increased to values comparable to and/or larger than the wavelength and higher-order eigenmodes become predominant. To date, accurate models of SPP coupling from super-wavelength slits sustaining higher-order eigenmodes have yet to be realized.

In this work, we propose and characterize a new SPP coupling scheme consisting of a slit of super-wavelength width immersed in a uniform dielectric. The width of the super-wavelength slit is selected to sustain a first-order TM_{1} eigenmode just above cutoff, which then couples to the SPP mode at the slit exit. This contrasts to previously-explored SPP coupling configurations using sub-wavelength slits that sustain only the lowest-order TM_{0} eigenmode [3, 4, 5, 6, 7, 8, 9]. We show that the TM_{1} mode just above cutoff is advantageous for SPP coupling because it possesses a transverse wavevector component (lying in the plane of the metal surface) that is larger than that achievable with a TM_{0} in a slit of sub-wavelength width. It is proposed that if the transverse wavevector component of the TM_{1} mode, added with the peak Fourier spatial frequency component (due to diffraction at the slit exit), equals to the wavevector of the SPP mode on the metal surface, high SPP coupling efficiency is achievable. The hypothesis is tested by numerical simulation of visible light propagation through a slit as a function of the slit width and refractive index. An optimized geometry is discovered that satisfies the predicted wavevector matching condition, yielding a peak SPP coupling efficiency of ≃ 68% and an SPP throughput that is over an order of magnitude greater that achieved with a sub-wavelength slit. Compared to a sub-wavelength slit, the optimized super-wavelength slit geometry is easier to fabricate, has comparable SPP coupling efficiency and an over order-of-magnitude greater SPP throughput.

## 2. Hypothesis: SPP Coupling Using a Super-Wavelength Slit Aperture Immersed in a Dielectric

Consider a semi-infinite layer of metal (silver) with relative permittivity *ɛ** _{m}* extends infinitely in the

*x*- and

*y*-directions and occupies the region −

*t*<

*z*< 0. A slit of width

*w*oriented parallel to the

*z*-axis and centred at

*x*= 0 is cut into the metal film. The metal film is immersed in a homogeneous dielectric medium with relative permittivity

*ɛ*and refractive index $n\hspace{0.17em}=\hspace{0.17em}\sqrt{{\varepsilon}_{d}}$. The slit is illuminated from the region below it with a TM-polarized electromagnetic plane wave of wavelength

_{d}*λ*=

*λ*

_{0}/

*n*and wavevector

*k⃗*=

_{p}*kẑ*, where

*k*= 2

*π*/

*λ*. The +

*z*-axis defines the longitudinal direction, and the

*x*-axis defines the transverse axis. The electromagnetic wave couples into a guided mode in the slit having complex wavevector

*k⃗*=

*k*

*+*

_{z}ẑ*k*

*, where*

_{x}x̂*k*

*and*

_{z}*k*

*are the longitudinal and transverse components of the complex wavevector, respectively. The attenuation of the guided mode in the slit can be characterized by a figure of merit (FOM) defined as*

_{x}*x*-propagating SPP modes. The SPP modes have complex wavevector ±

*k*

*, where Re[*

_{spp}x̂*k*

*] and Im[*

_{spp}*k*

*] describe the spatial periodicity and attenuation, respectively, of the SPP field along the transverse direction.*

_{spp}A SPP coupling scheme based on a slit structure is designed by first mapping *k** _{z}* and

*k*

*of the TM*

_{x}_{0}and TM modes sustained in the slit for varying slit width. The longitudinal wavevector components

*k*

*of the TM*

_{z}_{0}and TM

_{1}modes in the slit are calculated by solving the exponential and oscillatory forms of the complex eigenvalue equation [10], respectively, for an infinite metal-dielectric-metal waveguide using the Davidenko method with an iterative solving scheme [11].

*ɛ*

*is modeled by fitting to experimental data of the real and imaginary parts of the permittivity of silver [12], and*

_{m}*ɛ*is assumed to be real and dispersion-less. Figure 1(a) shows FOM curves for TM

_{d}_{0}and TM

_{1}modes in slits of varying width for the representative case where the slit is immersed in a dielectric with a refractive index

*n*= 1.75. The FOM values for the TM

_{0}modes are largely insensitive to variations in the slit width and gradually decrease as a function of increasing frequency. FOM curves for the TM

_{1}modes are characterized by a lower-frequency region of low figure of merit and a higher-frequency region of high figure of merit, separated by a kneel located at a cutoff frequency. The cutoff slit width

*w*for the TM

_{c}_{1}mode at a given frequency

*ω*is the threshold slit width value below which the TM

_{1}mode is attenuating. At a fixed visible frequency

*ω*= 6.0 × 10

^{14}Hz (

*λ*= 285nm),

*w*∼ 300nm. The dominant mode in the slit can be identified at a particular frequency and slit width by the mode with the largest FOM. The TM

_{c}_{0}mode is dominant for

*w*<

*w*, and the TM

_{c}_{1}mode is dominant for

*w*>

*w*.

_{c}The real part of the transverse wavevector component, Re[*k** _{x}*], of the guided mode in the slit describes the component of electromagnetic momentum in the transverse plane parallel to the plane of the metal surface. Values of

*k*

*are obtained from the relation*

_{x}*k*=

*nk*

_{0}is the magnitude of the wavevector in the dielectric core of the slit. Figure 1(b) shows Re[

*k*

*] values over the visible-frequency range for the TM*

_{x}_{0}mode in a slit of width

*w*= 200nm and for the TM

_{1}mode in slits of widths

*w*= 350nm and

*w*= 500nm. At the frequency

*ω*= 6.0 × 10

^{14}Hz, Re[

*k*

*] for the TM*

_{x}_{0}mode in the

*w*= 200nm slit is nearly two orders of magnitude smaller than Re[

*k*

*] for the TM*

_{x}_{1}mode in the

*w*= 350nm and

*w*= 500nm slits. Values of Re[

*k*

*] for the TM*

_{x}_{1}mode generally increase for decreasing slit width. Given the parameters in Fig. 1(b) and for a fixed

*ω*= 6.0 × 10

^{14}Hz, Re[

*k*

*] for the TM*

_{x}_{1}mode increases from 8.5 × 10

^{6}m

^{−1}to 1.3 × 10

^{7}m

^{−1}as the slit width decreases from 500nm to 350nm.

Diffraction at the slit exit generates transverse spatial frequency components, *κ*. The diffraction spectrum is a distribution of transverse spatial frequencies generated by diffraction at the slit exit. We calculate the diffraction spectrum by Fourier transformation of the transverse field profiles of the guided mode [13]. Figure 1(c) shows the normalized diffraction spectrum for slit widths *w* = 200nm, *w* = 350nm, and *w* = 500nm at a fixed frequency *ω* = 6.0 × 10^{14} Hz. The peak transverse spatial frequency component, *κ _{p}*, is the spatial frequency at which the diffraction spectrum peaks. For the parameters in Fig. 1(c),

*κ*shifts from 1.6 × 10

_{p}^{7}m

^{−1}to

*κ*= 8.3 × 10

_{p}^{6}m

^{−1}as the slit width increases from

*w*= 200nm to

*w*= 500nm. It is noteworthy that

*κ*< Re[

_{p}*k*

*] for all slit width values.*

_{spp}A simple picture of diffraction-assisted SPP coupling based on the data in Figs. 1(a)–(c) for *w* = 200nm, *w* = 350nm and *w* = 500nm at a fixed *ω* = 6.0 × 10^{14} Hz is presented in Fig. 1(d). SPP coupling at the slit exit is mediated by diffraction of the guided mode, yielding a net real transverse wavevector component Re[*k** _{x}*] +

*κ*. Coupling from the diffracted mode at the slit exit to the SPP mode adjacent to the slit exit is optimized when the wavevector-matched condition Re[

_{p}*k*

*] +*

_{x}*κ*= Re[

_{p}*k*

*] is satisfied. Because Re[*

_{spp}*k*

*] is generally larger than both Re[*

_{spp}*k*

*] and*

_{x}*κ*, large and commensurate contributions from both Re[

_{p}*k*

*] and*

_{x}*κ*are required to fulfill wavevector matching. In a sub-wavelength slit, the TM

_{p}_{0}mode has Re[

*k*

*] <<*

_{x}*κ*and SPP coupling at the slit exit requires a sufficiently small slit width to generate large diffracted spatial frequency components to match with Re[

_{p}*k*

*]. On the other hand, a super-wavelength slit sustains a TM*

_{spp}_{1}mode with Re[

*k*

*] ≃*

_{x}*κ*. The large contributions of Re[

_{p}*k*

*] to the net real transverse wavevector component reduces the required contributions from*

_{x}*κ*needed for wavevector matching. As a result, wavevector matching with the SPP mode adjacent to the slit exit can be achieved with a relatively large slit aperture.

_{p}## 3. Methodology

SPP coupling efficiency of a slit immersed in a dielectric is modeled using finite-difference-time-domain (FDTD) simulations of Maxwell’s Equations. The simulation grid has dimensions of 4000 × 1400 pixels with a resolution of 1nm/pixel and is surrounded by a perfectly-matched layer to eliminate reflections from the edges of the simulation space. The incident beam is centered in the simulation space at *x* = 0 and propagates in the +*z*-direction, with a full-width-at-half-maximum of 1200nm and a waist located at *z* = 0. The incident electromagnetic wave has a free-space wavelength *λ*_{0} = 500nm and is TM-polarized such that the magnetic field, *H _{y}*, is aligned along the

*y*-direction.

Control variables of this study include the type of metal (chosen as silver), the thickness of the metal layer (set at *t* = 300nm), the polarization of the incident electromagnetic wave (TM), the angle of incidence of the incident electromagnetic wave (normal), and the wavelength of the incident electromagnetic wave (*λ*_{0} = 500nm). The independent variables include the width of the slit, *w*, which varies from 100nm to 800nm, and the refractive index of the surrounding dielectric *n*, which varies from 1.0 to 2.5. The dependent variables are the time-averaged intensity of the SPP modes coupled to the metal surface at the slit exit, *I _{spp}*, the time-averaged intensity of the radiated modes leaving the slit region,

*I*, and the SPP coupling efficiency,

_{r}*η*. The dependent variables are quantified by placing line detectors in the simulation space to capture different components of the intensity pattern radiated from the exit of the slit, similar to the method employed in Ref. [3]. The

*I*detectors straddle the metal/dielectric interface, extending 50nm into the metal and

_{spp}*λ*

_{0}/4nm into the dielectric region above the metal, and are situated adjacent to the slit exit a length

*λ*

_{0}away from the edges of the slit. The

*I*detector captures the intensity radiated away from the slit that is not coupled to the surface of the metal. The coupling efficiency is then calculated by the equation

_{r}## 4. Results and discussion

The numerical simulations provide evidence of high-throughput and high-efficiency SPP coupling from a slit of super-wavelength width. Figure 2 displays representative snap-shots of the instantaneous |*H _{y}*|

^{2}intensity and time-averaged |

*H*|

_{y}^{2}angular distribution calculated from FDTD simulations for plane-wave, TM-polarized, normal-incidence illumination of a slit immersed in a dielectric (

*n*= 1.75) for slit width values

*w*= 200nm,

*w*= 350nm, and

*w*= 500nm. Radiative components of the field in the dielectric region above the slit propagate away from the metal-dielectric interface, and plasmonic components propagate along the metal-dielectric interface. For

*w*= 200nm [Fig. 2(a)], the incident plane wave couples into a propagative TM

_{0}mode in the slit, which is characterized by intensity maxima at the dielectric-metal sidewalls. Diffraction of the TM

_{0}mode at the exit of the slit yields a relatively strong radiative component with an angular intensity distribution composed of a primary lobe centred about the longitudinal axis and a relatively weak plasmonic component. For

*w*= 350nm [Fig. 2(b)] and

*w*= 500nm[Fig. 2(c)], the incident plane wave couples primarily into the TM

_{1}mode in the slit, which is characterized by an intensity maximum in the dielectric core of the slit. The high-throughput SPP coupling is evident by the large SPP intensities observed for

*w*= 350nm. Diffraction of the TM

_{1}mode at the

*w*= 350nm slit exit yields a relatively weak radiative component with an angular intensity distribution skewed at highly oblique angles and a relatively strong plasmonic component. Further increasing the slit width to

*w*= 500nm increases the total throughput through the slit, but reduces the efficiency of SPP coupling. Diffraction of the TM

_{1}mode at the

*w*= 500nm slit exit yields a strong radiative component with an angular intensity distribution composed of two distinct side lobes and a relatively weak plasmonic component.

Trends in the SPP coupling efficiencies calculated from the FDTD simulations are compared to qualitative predictions from the model of diffraction-assisted SPP coupling described in Fig. 1. Figure 3(a) plots the FDTD-calculated SPP coupling efficiencies as a function of the optical slit width *nw* for dielectric refractive index values ranging from *n* = 1.0 to *n* = 2.5. For sub-wavelength slit width values *nw* < *λ*_{0}, highest SPP coupling efficiency is observed for the smallest optical slit width. This trend is consistent with diffraction-dominated SPP coupling predicted to occur for sub-wavelength slit widths, in which small slit width is required to yield large diffracted spatial frequencies to achieve wavevector matching. For super-wavelength slit width values *nw* > *λ*_{0}, the SPP coupling efficiencies exhibit periodic modulations as a function of optical slit width, qualitatively agreeing with the general trends observed in experimental data measured for a slit in air [7] and theoretical predictions based on an approximate model for SPP coupling from a slit [5]. The data in Fig. 3 reveals that the magnitude of the fluctuations in the SPP coupling efficiencies are highly sensitive to the dielectric refractive index. For refractive index values *n* = 1.0, 1.5, 1.75, and 2.0, the SPP coupling efficiency rises as *nw* increases above *λ*_{0} and reaches local maxima of *η* = 14%, 44%, 68%, and 48% at a super-wavelength optical slit width *nw* ≃ 600nm, respectively. The rapid increase *η* as the slit width increases from sub-wavelength slit width values to super-wavelength slit width values is attributed to the disappearance of the TM_{0} mode in the slit and the emergence of the TM_{1} mode in the slit, which boosts the net real transverse wavevector component at the slit exit to enable wavevector matching. It is interesting to note that the SPP coupling efficiency peak at *nw* = 600nm observed for lower refractive index values is absent for *n* = 2.5.

Figure 3(b) displays the time-averaged radiative intensity *I _{r}*, SPP intensity

*I*, and total intensity

_{spp}*I*=

_{t}*I*+

_{spp}*I*, as a function of the optical slit width for

_{r}*n*= 1.75. Although the smallest optical slit width generally yields high SPP coupling efficiency, the total throughput and the SPP throughput is low. As the optical slit width increases to

*w*≃

*λ*

_{0}from sub-wavelength values, an increase in

*I*and a decrease in

_{r}*I*yield low SPP coupling efficiency. In the super-wavelength range of optical slit width values, 520nm <

_{spp}*nw*< 700nm, concurrently high SPP throughput and high SPP coupling efficiency (

*η*> 50%) are observed. For the optical slit width value

*nw*≃ 600nm,

*I*is about an order of magnitude larger than

_{spp}*I*for the smallest slit width value

_{spp}*nw*= 175nm. As the optical slit width is further increased

*nw*> 700nm,

*I*is significantly greater than

_{r}*I*, resulting again in low SPP coupling efficiencies.

_{spp}Variations in the peak SPP coupling efficiency at a fixed optical slit width *nw* = 600nm for varying *n* can be qualitatively explained by the mismatch between the net real transverse wavevector component Re[*k** _{x}*] +

*κ*and the real SPP wavevector Re[

_{p}*k*

*]. Figure 4 plots the transverse wavevector mismatch Re[*

_{spp}*k*

*] – (Re[*

_{spp}*k*

*] +*

_{x}*κ*) as a function of the dielectric refractive index at a constant optical slit width value

_{p}*nw*= 600nm. The wavevector mismatch increases monotonically from −0.4 × 10

^{7}m

^{−1}to 2.4 × 10

^{7}m

^{−1}as the refractive index increases from

*n*= 1.0 to

*n*= 2.5, crossing zero at

*n*= 1.75. Coincidence between the

*n*value that yields peak SPP coupling efficiency at

*nw*= 600nm and that which yields zero wavevector mismatch supports the hypothesis that optimal SPP coupling efficiency occurs when Re[

*k*

*] = (Re[*

_{spp}*k*

*] +*

_{x}*κ*), and that this condition can be achieved using a super-wavelength slit aperture immersed in a dielectric. The relatively large wavevector mismatch for

_{p}*n*= 2.5 is also consistent with the noted absence of a SPP coupling efficiency peak at

*nw*= 600nm.

## 5. Conclusions

In conclusion, we have presented a theoretical proposal and a numerical study of a high-throughput and high-efficiency SPP coupling method. The crux of the method is a super-wavelength slit aperture immersed in a uniform dielectric sustaining a TM_{1} mode just above cutoff. High SPP coupling efficiency is achieved when the transverse wavevector component of the TM_{1} mode, added with the peak diffracted spatial frequency component, equals to the wavevector of the SPP mode on the metal surface. Based on numerical simulations of light propagation through a slit of varying slit width and varying surrounding dielectric refractive index, an optimal slit width and refractive index combination is found that satisfies wavevector matching. Under optimal conditions, it has been shown that a super-wavelength slit is capable of larger SPP intensity throughput than that achievable with a sub-wavelength slit and an SPP coupling efficiency of 68%. Our work is the first to explore SPP coupling from super-wavelength slits by explicitly treating the interaction between the higher-order eigenmodes (which become non-evanescent when the slit width is increased) and SPP modes. The conclusions will assist in the continued development of SPP devices by providing a new high-throughput and high-efficiency method for coupling to SPP modes.

## Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors acknowledge helpful discussions with L. Sweatlock, R. D. Kekatpure, and H. Lezec.

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