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Abstract
We present a fully analytic theory to study the power and field enhancement inside a real metal slit. A generalized formula for the reflection coefficient at the interface of the slit is derived. The resulting expression is purely analytic and the reflection coefficient can be simply evaluated to provide physical insight, while not requiring complicated numerical simulations. The calculated values of reflection phase and amplitude are then used in the Fabry-Pérot formalism to compute the electric field and the power inside the slit. It is shown that the power attains its maximum value when the scattering and the absorption cross-sections of the slit are equal, a confirmation of the maximum power transfer theorem for this case. The analytic results agree well with numerical simulations, which is promising for optimizing performance in applications ranging from modulators to optical tweezers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Slits in a metal are desired for extreme subwavelength confinement of light [1–7]. This has many applications including spectroscopy [8–10], nonlinear optics [11–13], optical tweezers [14–16], and sensors [17–21]. In addition, the metal layers can also serve as contacts that can be used in optoelectronic applications like modulators [22–25]. When light is confined to the subwavelength (nanometer) scale in metals, there is an obvious trade-off between field enhancement and losses since both increase when reducing the slit width. In the recent years, others have explored the maximum power transfer theorem for plasmonic nanostructures, but the analysis was focused on nanoantennas [26]. Given the growing applications of slits, there is a need for determining the optimal slit configurations that balance the field intensity and the absorption in the metal-insulator-metal (MIM) geometry.
A common approach in plasmonic design is to use numerical simulations, which is akin to doing experiments on a computer. However, simulations are resource-intensive, while 2D simulations typically take less time than 3D simulations, narrow slits can be challenging for time-domain approaches because the grid size needs to be small to accurately capture the rapid field decay as well as the time step to satisfy Courant stability with small grids. Furthermore, the simulation should run for a long time for high quality resonances. Typical time-domain runs last several hours in our experience. In addition, a new simulation is required for any change in parameters, such as geometry, materials, or wavelength (depending upon the simulation method). Therefore, they lack the desired insights for design, and what researchers need is a simple theory that can rapidly calculate the optimal slit configuration for a given desired property, such as the maximum power transfer.
In this work, we develop a theoretical framework to assess the variation of power and field enhancement inside a 2D slit of vanishing width. One might presume that there is a direct relationship between the two; i.e. they keep increasing until quantum effects come into play. But here we show that the power inside the slit saturates at a specific gap width, much larger than the subnanometer scale where tunneling sets in. We explain this behaviour on the basis of effective coupling of light with the nanoaperture, characterized by the scattering ($\sigma _s$) and the absorption ($\sigma _a$) cross-sections. They are defined as the ratio of scattered power and absorbed power to the incident intensity [27].
An analytic expression for reflection at the interface of a metallic slit, accounting for losses, has been derived in the past but still requires numerical integration [28]. Here, we introduce a fully analytic approach (without numerical integration) using the reflection coefficient of a slit in perfect electric conductor (PEC) [29] but loading with an effective dielectric to incorporate the finite conductivities of metals. After the expression is found, we use it to calculate the transmission Fabry-Pérot model, which has been previously used to explain extraordinary transmission and absorption in slits [30,31]. The surface plasmon resonances (SPR) are found as zero-order Fabry-Pérot modes to derive expressions for transmission, field enhancement, power, and optical cross-sections. Using these equations, slit configurations of varying gap widths are analyzed and the results of the theory are verified by comprehensive numerical simulations.
2. Description of the metallic slit geometry
Figure 1(a) shows the schematic of the metallic slit extending infinitely in $x$ and $y$ directions. The length ($L$) of the slit is 100 nm and the width ($a$) is varied between 1 nm and 20 nm. The surrounding medium is air with a permitivitty ($\epsilon _d$) of 1 and the permittivity of metal ($\epsilon _m$) is a function of the wavelength [32]. We assume that only the fundamental TM mode propagates in the slit since it does not have a cut-off wavelength for any gap dimensions. As shown in the figure, this mode propagates in the $z$-direction and the transverse magnetic field is along the $y$-direction.
Fig. 1. (a) Schematic of the Au slit of permittivity $\epsilon _m$ surrounded by air ($\epsilon _d = 1$). The width of the slit is $a$ and length is $L$, the metal extends infinitely in $x$ and $y$ directions (b) $x-z$ plane of the slit, $t_{12}$ is the transmission coefficient from air into the slit, $t_{21}$ is the transmission coefficient from slit to air, and $r_{21}$ is the reflection coefficient from the slit back into the slit.
3. Reflection and transmission coefficients
Figure 1(b) shows the $x-z$ view of the slit along with the transmission and reflection coefficients of the slit as a Fabry-Pérot resonator. $t_{12}$ and $t_{21}$ are the transmission coefficients for transmission from surrounding to slit and slit to surrounding respectively. $r_{21}$ is the reflection coefficient for reflection from the slit back into the slit. The expressions for the coefficients presented below were derived in an earlier work using single mode matching to continuum [29,33].
where $I$ is given as:In Eq. (3), $a$ is the width of the slit normalized to the free space wavelength ($\lambda$).
Next we make an approximation that is the basis of this work: for a real metallic slit, we obtain the coefficients by dielectrically loading the slit with a material of refractive index $n_{\mathrm{eff}}$, which is the effective index of a MIM structure given by ${k_z}/{k_0}$ and has both real and imaginary parts. Here, $k_z$ is the propagation constant of the TM mode in the slit obtained by solving the dispersion relation:
Therefore, as we transition from a slit in PEC to a real metallic slit, the wavelength in the slit changes from $\lambda$ to $\lambda /n_{\mathrm{eff}}$. So, $I$ is now formulated as:
This equation is the main result of this work, and its utility in showing a physical phenomenon will be shown in the following sections.
The reflection phase is validated against our existing work that calculates the coefficients by taking the EM fields inside and outside a real metallic slit into consideration [34]. Compared to that past work, the need for numerical calculation of the index has been removed by invoking the effective index approximation. This approximation is most accurate when field is almost constant and confined to the slit region, which is applicable for many practical cases (particularly in the infrared). Figure 2(a) shows the reflection amplitude for slits of widths 5 nm and 20 nm. The amplitude increases with increasing wavelength and narrowing gaps. Figure 2(b) presents the phase of reflection. We compare this phase with past numerical simulations [34] and find good agreement for wavelengths between 700 nm to 1300 nm, to within 0.06 rad for 5 nm and 0.15 rad for 20 nm. For shorter wavelengths, however, the theoretical values are not as well matched with the numerical simulation results due to the penetration of the electric field into the metal when the frequency of electromagnetic radiation approaches the plasma frequency. In this regime, the metal no longer acts as a perfect electric conductor, which undermines our assumption of a constant field in the slit region.
Fig. 2. (a) Reflection amplitude and (b) Reflection phase at the interface of metallic slit for widths of 5 nm and 20 nm calculated from Eqs. (1) and (5).
4. Scattering and absorption cross-sections
The transmission through a Fabry-Pérot resonator is calculated by summing the multiply reflected TM waves inside the slit. The final expression for transmittance is given as:
The same approach is followed to derive the expression for the electric field inside the slit. The constant field approximation allows us to calculate the power in the slit directly from the field intensity, and the variation of both with the gap width is shown in the subsequent section. For a PEC slit where there is no absorption, the scattering cross-section reaches the single channel limit at resonance. The single channel limit defines the maximum absolute scattering cross-section of a subwavelength scatterer. It depends upon the wavelength of the incident radiation and is given by $2\lambda /\pi$ for a 2D scatterer and $3\lambda ^2/2\pi$ for a 3D scatterer. This fundamental quantity has also been extended to nanophotonic systems and recent efforts have focused on overcoming the limit by introducing multiple channels [37–40].
Figure 3(a) shows the scattering cross-section of a PEC slit plotted against its length for three different slit widths, at a fixed wavelength of 1570 nm. This is obtained by setting the imaginary part of the propagation constant to zero ($k_z'' = 0$). The values of reflection coefficient used in this analysis are calculated from Eq. (3). At resonance, it approaches the single channel limit, which is calculated to be 999.5 nm ($\approx$1000 nm). Although all three configurations scatter light at the single channel limit, the narrower slits are seen exhibiting sharper resonance peaks. The transmission through the slit at resonance is $\lambda /(\pi a)$, which implies that in the absence of absorption, the narrowest slit before the onset of tunneling allows the maximum amount of transmission. But this changes significantly when losses are considered.
Fig. 3. (a) Scattering cross-section ($\sigma _s$) of PEC slits of widths 5 nm, 10 nm, and 20 nm for wavelength of 1570 nm, (b) $\sigma _s$ for widths of 4 nm, 10 nm, and 20 nm for wavelength of 890 nm calculated from Eq. (6) after setting the absorption to zero ($k_z'' = 0$). $\sigma _s$ approaches the single channel limit ($2\lambda /\pi$) at the resonance condition for each configuration.
Figure 3(b) shows the result for a wavelength of 890 nm, with 4 nm, 10 nm, 20 nm slit widths, to be comparable with later optimal resonance conditions. We observe multiple resonances for the three configurations, each approaching the single channel limit (567 nm).
The subsequent analysis focuses on the real metal slit. Here, we use a 100 nm long slit with widths ranging 1 nm to 20 nm and the wavelength of the incident radiation varies between 500 nm and 1650 nm. The values of reflection coefficients are obtained from Eq. (5). The losses are a result of absorption of electromagnetic waves into the metal as they propagate back and forth inside the slit. They are taken into account by considering the non-zero imaginary part of the propagation constant ($k_z''$) in the Fabry-Pérot formalism. The propagation lengths range from 2.6 nm for a 1 nm wide slit at 500 nm to 7465 nm for a 20 nm wide slit at 1650 nm and the corresponding absorption cross-sections are calculated from Eq. (7).
Figure 4(a) shows variation of scattering and absorption cross-sections, calculated from Eqs. (6) and (7). The resonant peaks are red-shifted when the slit width is reduced due to an increase in the effective index of the TM mode inside the slit. The higher values of effective index are a result of larger penetration of the EM-fields into the metal, which also causes increased Ohmic losses [41]. The scattering cross section at long wavelengths rises slightly. This is because the phase of propagation is getting smaller, as is the phase of reflection, so it is closer to a zeroth order resonance condition. The absorption cross-section is maximized at the resonance of a 4 nm gap, where it becomes comparable to the scattering cross-section.
Fig. 4. (a) Scattering cross-section ($\sigma _s$) and absorption cross-section ($\sigma _a$) for gap widths ranging from 1 nm to 20 nm, calculated analytically using Eqs. (6) and (7). Red lines represent $\sigma _s$ and blue lines represent $\sigma _a$. (b) $\sigma _s$ and $\sigma _a$ obtained from Lumerical FDTD simulations.
4.1 Comparing with FDTD simulations
We perform the 2D numerical simulations in Lumerical FDTD within perfectly matched layer (PML) boundaries using a uniform mesh size of 0.2 nm in both the directions. A frequency-domain power monitor is used to visualize the fields in the region. Two cross-section analysis objects, one in the total field region and the other one in the scattered field are added to measure the absorption and scattering cross-sections.
Figure 4(b) shows the scattering and absorption cross-sections obtained from our Lumerical FDTD simulations. They match closely with the analytical results, however, the values in the simulations are slightly higher than the theoretical values which is explained in part by the presence of higher order modes in the simulations and by the finite penetration into the metal.
5. Maximum power transfer in the slit
Figure 5(a) shows the peak values of the optical cross-sections. The vertical line indicates that the scattering and the absorption cross-sections become approximately equal at a width of 4 nm. We extend this study to 80 nm and 120 nm long slits which give us the same results (see Supplement 1). Additionally, it is observed that longer slits produce equal scattering and absorption cross-sections for wider gaps.
Fig. 5. (a) The peak values of $\sigma _s$ (red squares) and $\sigma _a$ (blue circles) at the resonance condition, taken from Fig. 4(a). $\sigma _s$ and $\sigma _a$ are equal at a gap of around 4 nm. The vertical dashed line indicates the 4 nm mark. (b) Field enhancement (red circles) and power (blue squares) inside a metallic slit computed analytically using the Fabry-Pérot formalism. Field varies inversely with gap width, power saturates at 4 nm for equal values of $\sigma _s$ and $\sigma _a$ at resonance, marked by the vertical dashed line.
Figure 5(b) shows the variation of the field enhancement and power with the gap width. The electric field varies inversely with the width as narrow slits lead to tighter confinement and greater local density of optical states. The power on the other hand, attains its peak at 4 nm and starts declining if the gap widths are further reduced, which is a manifestation of the maximum power transfer theorem. For the 80 nm and 120 nm long slits, the power inside the slit peak at 3 nm and 5 nm respectively, for equal values of scattering and absorption cross-sections. Numerical simulations are performed for the 100 nm long slit and they match well with our analytical work (see Supplement 1).
6. Discussion
The optimal configuration of slits is of interest for modulation in ultrafast communication systems. While modulators based on MIM structures have achieved smaller size, lower power consumption, and lower driving voltages than their semiconductor counterparts [22–25,42], they still suffer from losses. So, achieving the optimum design of the slit is important for practical large-scale integration with CMOS-electronics.
A lot of research in plasmonics has been directed towards understanding the nonlocal and quantum tunneling effects observable at extremely small length scales. They are seen as the fundamental limits bounding the performance of plasmonic devices [43–46]. In metallic slits, the narrowest gap where tunnelling sets in has been experimentally shown to be under 1 nm [46,47]. However, the fact that metallic absorption also tends to increase with narrowing gaps is often overlooked. Losses are paramount for practical devices and here we have shown that the optimal gap width (4 nm for a 100 nm long slit) is much larger than the limit set by tunnelling, which has been noted previously [48].
7. Conclusion
In conclusion, we have introduced a fully analytic theory to study the dependence of power and field enhancement on the width of a real metal slit. The main component of our theory is the derivation of a purely analytic expression for reflection coefficient at the slit interface (based on the effective index approximation) that does not rely on any numerical techniques (such as finite-difference, or numerical integration) for calculating the values of reflection amplitude and phase. For slits of varying gap widths, the power obtained by the Fabry-Pérot model is maximized for the configuration that has equal scattering and absorption cross-sections at resonance, an expected outcome of the maximum power transfer theorem. Here, we have shown that the power in a 100 nm long slit peaks for a 4 nm wide gap at the resonant wavelength of 890 nm and this result is verified by FDTD simulations. Our theory allows the determination of optimum slit performance by balancing the field intensity and losses, both of which grow with narrowing gaps. We believe the knowledge of the power inside the slit will be key in designing MIM geometries and optimizing their performance for applications including modulators, optical tweezers and sensors.
Funding
Natural Sciences and Engineering Research Council of Canada (CREATE Quantum Computing, RGPIN-2017-03830).
Acknowledgments
The authors thank Zohreh Sharifi for useful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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