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Abstract
We demonstrate electromagnetic field localization and enhancement effects on the non-structured planar surface of a two-dimensional gradient permittivity material. Surface plasmons are excited by a normally-incident Gaussian illumination beam and are confined to subwavelength rings on the surface of the gradient permittivity material. The performance of the surface is programmable by adjusting the permittivity distribution of the material and polarization of incident light. We show that field localization and enhancement effects can be realized at mid-infrared frequencies by conventional semiconductor materials with designed doping distributions. This demonstration suggests a compact and readily accessible platform for materials characterizations with spatially controlled illumination, providing a convenient approach to explore nanospectroscopy and light-matter interactions of nanomaterials, such as quantum dots, nanowires, and organic molecules.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Significant electromagnetic (EM) field localization and enhancement (LE) at the nanoscale has attracted tremendous attention for applications in sensing [1–5], surface enhanced spectroscopy [4–9], molecular electronics [10–12], super-resolved imaging [12–15], high harmonic generation [16–19], and superfocusing [20–22]. In practice, field LE involves the excitation of surface plasmons (SPs), electromagnetic waves formed due to collective oscillations of electrons at the interface between a dielectric and metal, to overcome the diffraction limit of light. Previous investigations have shown that through the excitation of SPs light energy can be bound to subwavelength or even deep subwavelength regions [23–25]. In addition, SP modes coupled between particle dimers, trimers and other cavity structures can further enhance the intensity of the EM field. Various metallic structures, such as pillars [26,27], bowties [28–31], particles [32], gratings [33–35], and even random structures [36], have been reported to obtain field LE effects with field enhancement of several orders of magnitude. However, because the above-mentioned nanoscale structures inevitably increase the complexity of plasmonic systems, practical challenges still remain to efficiently induce, detect, and harness the enhanced fields. In this regard, nano-gap-confined fields are not readily accessible [37,38]. Tip Enhanced Raman Scattering requires a bulk system and accurate alignments between the tip and nano-samples [21,22,31,39,40]. Surface roughness introduced from either the material itself or nanofabrication induces non-negligible loss that directly impacts plasmonic systems [41]. Thus, achieving field LE effects on a planar surface is required to explore molecular adsorption, orientation, and chemical reaction processes.
In our previous study [42], we analyzed the excitation and propagation of surface plasmon polaritons (SPPs) on a planar interface between a homogeneous dielectric and a gradient permittivity material (GPM). We also predicted one-dimensional (1D) field LE effects by propagating SPPs on a GPM. Here, we extend our analysis to two-dimensional (2D) systems and discuss the effects of permittivity distributions on field LE. We compare ideal materials and conventional semiconductor materials and propose a practical design for field LE at mid-infrared (MIR) frequencies. By using a non-structured planar surface with a permittivity gradient, we eliminate the loss arising from complex nanostructures. In addition, we demonstrate that GPMs could serve as a materials characterization platform with spatially controlled illumination to enhance both signal strength and spatial resolution.
2. Programmable field LE effects on a non-structured planar surface with a permittivity gradient
In our previous work, we showed that SPPs can be excited at the interface between a dielectric and a GPM, with constant permittivity ɛd and gradient permittivity ɛm, respectively [42]. When a TM polarized plane wave (the electric field is along the direction of the permittivity gradient, x) illuminates the surface of a GPM, it induces a single layer of oscillating electric dipoles at the interface. We can first write the radiating magnetic field of a single dipole located at a specific location on the surface of a GPM. The spatial spectra of the magnetic field radiated by dipoles across the whole interface can then be obtained by performing a Fourier transformation. Calculation of the integral as a function of kx gives the relative coefficient of the spatial frequency spectra, showing nonzero excited magnetic field components when kx > k0, indicating free space to SPP coupling. Excited SPPs propagate in the region where ɛm < -ɛd with local wave vectors described by ${k_{spp}}\textrm{ } = \textrm{ }{k_0}\sqrt {{\varepsilon _d}{\varepsilon _m}/({\varepsilon _d} + {\varepsilon _m})} $. As SPPs propagate toward a region where ɛm = -ɛd, their wave vectors approach infinity, corresponding to a very small group velocity and phase velocity. As a result, SPPs are confined to the planar surface of a GPM as illustrated in Fig. 1.
Fig. 1. Schematic illustrating field LE effects on a GPM. Free-space illumination excites SPPs, which propagate in both directions along the permittivity gradient. SPPs propagating towards higher permittivity are localized and enhanced at the location where ɛm = -ɛd. The green arrows indicate the propagation directions of excited SPPs. The green wave illustrates the localization and enhancement of SPPs.
While our previous demonstration focused on a 1D permittivity gradient, we predict that by constructing a GPM with a 2D permittivity distribution, it is possible to achieve field LE with programmable locations and enhancement. To understand how a 2D GPM affects field LE, we performed numerical simulations using the finite-element based commercial software COMSOL Multiphysics. The GPM is defined as a cylinder with fixed radius (R = 30 µm) and height (H = 10 µm) in a Cartesian coordinate system with its center located at (x = 0 µm, y = 0 µm) and axis of symmetry along the z-axis. For illustration purposes, we first set the imaginary part of the GPM permittivity to a small constant value of ɛi = 0.01. The dielectric permittivity above the GPM and between the GPM surface and illumination source is set to that of air (ɛd = 1).
We first employ a GPM with monotonically increasing permittivity along the + x direction. We use a tanh function ${\varepsilon _m}\textrm{ } = \textrm{ }\delta \tanh (\alpha x)\textrm{ } + \textrm{ }\beta $, where δ = 5, α = 1×105 /m, and β = -1 to describe its permittivity distribution. A Gaussian beam centered at x = y = 0 with a waist of 30 µm and electric field magnitude of Ex = 1 V/m is normally incident on the GPM. The incident frequency is 18 THz and the electric field is linearly polarized along the x direction. Figure 2(a) shows the permittivity distribution of the material and the electric field amplitude distribution (Ex0) of the incident beam at y = 0. For any particular x position, the permittivity distribution is homogeneous along the y direction. At x = 0, the permittivity ɛm = -1 = -ɛd. Figure 2(b) shows the distribution of the electric field magnitude in a plane 1 nm above the surface of the GPM on the side of illumination (hereinafter all the electric magnitude distributions are extracted from the same plane). The radius of the region shown is r = 1 µm. We note a bright blue band along the line of x = 0 perpendicular to the polarization of the incident beam. This bright blue band corresponds to regions of enhanced electric field, in this case a line-shape field LE effect. Figure 2(c) shows the electric field magnitude distribution along the x-direction at y = 0. The peak electric field magnitude of Ex is 250 V/m, much greater than the incident electric field magnitude of 1 V/m.
Fig. 2. (a) The permittivity distribution of a GPM and the electric field magnitude distribution of the incident linearly polarized Gaussian beam along the x-direction; Electric field distributions (Ex) (b) in a plane 1 nm above the surface of the GPM, and (c) along the diameter in the x-direction; (d), (e) and (f) are the counterparts of (a), (b) and (c), respectively, for a 2D permittivity distribution and a linearly polarized Gaussian beam; (g), (h) and (i) are the counterparts of (a), (b) and (c), respectively, for a 2D permittivity distribution and a radially polarized Laguerre-Gaussian beam. The radii of the regions shown in (b), (c) and (d) are 1 µm.
To demonstrate the shape-programmability of field LE effects, we next employ a GPM with a 2D radially-symmetric permittivity distribution given by ${\varepsilon _m} = \textrm{ }\delta \tanh (\alpha (r - {r_0}))\textrm{ } + \textrm{ }\beta $, where δ = -5, α = 1×105/m, β = -ɛd = -1, r0 = 0.5 µm, and $r\textrm{ } = \textrm{ }\sqrt {{x^2} + {y^2}} $ is the radial distance from the center of the GPM. The incident beam remains unchanged. Figure 2(d) shows the permittivity distribution and the incident electric field along the x-direction at y = 0. Note that, here, the permittivity distribution is radially symmetric whereas the permittivity varied only along the x-direction in the previous example. Along a ring of r = 0.5µm, ɛm = -1 = -ɛd and when r > 0.5 µm, ɛm is less than -1. Figure 2(e) shows the electric field magnitude distribution on a plane. The radius of the region shown is r = 1 µm. Figure 2(f) gives the electric field magnitude distribution along the line of y = 0. As expected, the EM field at the position where ɛm = -1 = -ɛd is localized and its shape is controlled by the 2D GPM. Because the incident beam is polarized along the x direction and the permittivity gradient is along the radial direction, at x = 0 and y ≠ 0, there is no TM component to excite SPPs. Thus, no field LE effect occurs at these positions and the result is an imperfect ring of field LE. This example demonstrates that the shape of field LE can be controlled by programming the distribution of the permittivity and the polarization of the incident light.
For the same permittivity distribution shown in Fig. 2(d), axisymmetric field LE can also be obtained by using a radially polarized illumination beam. The electric field magnitude profile for the radially polarized beam is shown in Fig. 2(g) along with the permittivity distribution of the GPM. This method of illumination also leads to a relative increase in the field enhancement factor. Inspired by the “bull’s eye” structure which concentrates excited SPPs towards its center slit [43], we simulate illumination of the 2D GPM using a radially polarized Laguerre-Gaussian beam. Thus, SPPs are excited equally along all radial directions and propagate towards or away from the center of the GPM. Figures 2(h) and 2(i) show the electric field magnitude distribution on a plane and along the line of y = 0, respectively. Unlike previous examples where a linearly polarized illumination beam was considered, the radially polarized beam induces an unbroken and localized ring of electric field enhancement at r = 0.5 µm. More importantly, there is an order of magnitude increase in the EM field intensity enhancement.
The enhancement factor of the field LE effect can be tuned by adjusting the steepness, or “slope”, of the permittivity gradient, which directly impacts the coupling efficiency of the incident beam to the SPPs [42]. We investigate this relationship using the above-demonstrated 2D permittivity distribution with a radially polarized Laguerre-Gaussian beam. Figure 3(a) shows the permittivity distribution with various values of α, i.e. the “slope” of the permittivity gradient, while keeping the other parameters in Fig. 2(g) unchanged. By definition, α also defines the length of the area over which a permittivity gradient extends, corresponding to the SPPs excitation area. The larger α, the shorter the SPPs excitation area. Furthermore, the slope of the gradient affects the size of localized EM fields. The larger α is, the faster the velocity of the SPPs decreases and, as a result, the narrower the width of localized EM fields. Figure 3(b) shows the electric field magnitude distribution for different values of α. From the above analysis, a larger “slope” induces higher SPP coupling efficiency and narrower EM localization rings, increasing field LE effects. The localized electric field magnitudes increase with increasing α in the first three field distributions in Fig. 3(b). However, when α is very large, the length of the region over which the permittivity gradient extends becomes very short. At this stage, the short SPPs excitation area plays a more important role than high SPP coupling efficiency, resulting in weaker field LE effects as shown in the fourth field distribution in Fig. 3(b) for α = 10×105/m. Therefore, we should find a tradeoff between these roles to optimize field LE effects in practical applications.
Fig. 3. (a) Permittivity distributions as a function of radius r with different values of α for a 2D GPM; (b) Electric field magnitude distributions for 2D GPMs with different values of α. The radii of the regions shown are r = 1 µm.
3. Practical design with a conventional semiconductor
In practical applications, losses are unavoidable in all materials and structures. One advantage of our system is that scattering loss is negligible because of atomically flat surfaces. The main loss mechanism of our system influencing field LE effects stems from material absorption. Here, we again use the 2D permittivity distribution and radially polarized Laguerre-Gaussian beam illumination in Fig. 2(g) to investigate the effects of material absorption on field LE. While the real parts of the permittivity are unchanged, we adjust the imaginary parts of the permittivity and plot electric field magnitude distributions along a radial segment of the GPM in Fig. 4. We find that as the imaginary part of the permittivity increases, the electric field magnitude of the peak EM field decreases. At low absorption, e.g. ɛi = 0.01, the peak electric field magnitude is nearly 103 V/m, corresponding to an intensity enhancement factor of 106. Even at high absorption, e.g. ɛi = 0.5, the peak electric field magnitude is 3 times larger than the incident electric field magnitude of 1 V/m, corresponding to a nine-fold intensity enhancement. Thus, the field LE effect has great tolerance to absorption loss and is suitable for practical applications using a conventional material with permittivity inhomogeneity, for example, a doped semiconductor.
Fig. 4. Electric field magnitude distributions as a function of radius for different imaginary parts of permittivity.
With the development of advanced fabrication techniques, materials with gradient permittivity can be readily obtained. Either gas phase thermal annealing or monolayer doping can be employed to achieve 2D GPMs. For example, in a monolayer doping approach, an indium arsenide (InAs) substrate is covered by a monolayer of ammonium sulfide, capped by a layer of silicon dioxide, and then annealed to drive in the sulfur atoms [44]. By depositing a layer of dielectric mask with suitable thickness, the lateral distribution of sulfur doping is fully controllable. To model such a system, we assign a carrier concentration distribution of $n(r)\textrm{ } = \textrm{ }({n_{\max }} + \textrm{ }{n_{\min }})/2\textrm{ } + \textrm{ }({n_{\max }} - \textrm{ }{n_{\min }})\tanh (\alpha r)/2$ to a cylindrically doped InAs substrate, where nmax and nmin are the maximum and minimum carrier concentration, respectively. We set α = 2×105 /m, nmin = 3.6 × 1024 /m3, and nmax = 9.4 × 1024 /m3 as described by the dashed curve in Fig. 5(a). The permittivity of the doped InAs can be described by the Drude model as follows [45]:
Fig. 5. (a) Real and (b) imaginary parts of the permittivity of doped InAs at different frequencies. Dashed curves are doping distribution. (c) Electric field magnitude distributions along the diameter 1 nm above the surface of the GPM for different frequencies with small absorption (ɛi = 0.03); (d) Electric field magnitude distributions along the diameter 1 nm above the surface of the GPM with physical absorptions for different frequencies.
We now consider the nonnegligible imaginary part of InAs’s permittivity and plot corresponding electric field magnitude distributions in Fig. 5(d). Frequency-dependent field LE effects are still observable with a trend similar to that of Fig. 5(c), i.e. the smaller the radius of the ring, the stronger the EM field. The peak intensity is still 60 times larger than the incident intensity. The widths of peaks (rings) are about 1 µm, which are much smaller than 1/10 of incident wavelengths. These results show that field LE effects may be realized on the planar surface of a doped semiconductor. In the ideal case, one could achieve field localization and enhancement effects for arbitrarily high frequency by increasing the doping concentration. However, for most highly doped semiconductors, electron mobility decreases and the effective mass increases with increasing doping concentration, where the increase of effective mass defines the upper limit of the plasma frequency. Most importantly, a decrease in electron mobility increases the collision frequency, γ, contributing to the imaginary part of ɛm, which reduces field LE effects as shown in Fig. 4. There are various semiconductors whose permittivity can be tuned over a wide wavelength range in the mid-infrared. By analyzing published experimental data of those heavily doped semiconductors [49–53], we anticipate the lower limit of working wavelengths is ∼4 µm. The upper limit of working wavelengths can reach 300 µm (1 THz). However, the enhancement is weaker in THz range due to weakened light-matter interactions. In addition, because the electron mobility of doped semiconductors is higher at a lower temperature when T > ∼70 K, one direct way to reduce loss is to lower the working temperature. Furthermore, we anticipate observing stronger enhancement with the development and practical design of novel high mobility materials, such as 2D electron gases and graphene [54,55].
4. A materials characterization platform with spatially controlled illumination
From the above analysis, we demonstrate the EM field LE effects on the planar surface of a doped semiconductor, InAs. In this section, we show that this planar surface can serve as a materials characterization plat form to probe nanoparticles. In simulation, we use two water droplets located at x = 1 µm and x = -3 µm to represent sample particles on the top surface of the doped InAs. The radii of the water droplets are both set to 200 nm and the permittivity of water is taken to be 2.25. The parameters of the platform are the same as those in Fig. 5(d).
Electric field magnitude distributions for various incident frequencies are shown in Fig. 6(a). The radius of the region shown is r = 4 µm. In addition to the expected frequency-dependent ring-shape EM field LE effects, the fields surrounding and within water droplets are influenced by the distribution of EM fields. When one water droplet is located outside but close to the EM field localization rings, the electric field magnitude at the edge of the droplet is greatly enhanced. The closer the water droplet is to the rings, the stronger the internal EM field. For example, for frequencies ranging from 18.0 THz to 18.4 THz, both water droplets are located outside the enhanced field rings. The electric field magnitude of the water droplet located at x = -3 µm is nearly the same as that of the background, while the electric field magnitude of the water droplet located at x = 1 µm is greatly enhanced, almost to the same magnitude as the EM field localization ring. The maximum internal electric field magnitude as a function of frequency for each water droplet is shown in Fig. 6(b). The peak electric field magnitude is much larger than the incident electric field magnitude of 1 V/m. This EM field enhancement is very useful for weak signal detection, such as surface-enhanced Raman spectroscopy detection. The mechanism of the EM field enhancement at the edge of the water droplets is a complicated process that includes scattering of SPPs, scattering of incident light, and mutual interaction between the water droplet and the doped InAs. Among these processes, the scattering of SPPs should be the dominant mechanism because of the forbidden SPP band inside EM field localization rings. For example, if one water droplet is located inside of an EM field localization ring, no matter how close it is to the ring, there is no enhancement of the electric field inside the water droplet as shown for the water droplet located at x = 1 µm for frequencies ranging from 18.8 THz to 19.4 THz. Because ɛm is larger than -ɛd inside an EM field localization ring, no SPP can be excited or propagates in this region. As a result, no EM field enhancement can be observed.
Fig. 6. (a) Electric field magnitude distributions for different frequencies on a plane 1 nm above the surface of the InAs platform with two water droplets located on its surface at x = 1 µm and x = -3 µm, respectively. The radii of the regions are 4 µm. (b) Maximum electric field magnitudes inside each water droplet as a function of incident frequency.
5. Conclusions
In conclusion, we have demonstrated programmable EM field LE effects on a non-structured planar interface between a homogeneous dielectric and a GPM. We are able to modify the location and magnitude of enhancements by tuning the permittivity distribution of a GPM and by adjusting the polarization of incident light. We have demonstrated that GPMs could be realized using a conventional semiconductor, InAs, with a designed doping profile. This system has great potential to function as a wavelength-dependent illumination platform with spatially controlled illumination at MIR frequencies. We envision this platform may be used for materials characterization and near field signal detection.
Funding
National Natural Science Foundation of China (11504336); Fundamental Research Funds for the Central Universities (265201430).
Disclosures
The authors declare no conflicts of interest.
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