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Abstract
Surface plasmons usually take two forms: surface plasmon polaritons (SPP) and localized surface plasmons (LSP). Recent experiments demonstrate an interesting plasmon mode within plasmonic gaps, showing distinct characters from the two usual forms. In this investigation, by introducing a fundamental concept of SPP standing wave and an analytical model, we reveal the nature of the recently reported plasmon modes. The analytical model includes SPP propagating and SPP reflection within a metal-insulator-metal (MIM) cavity, which is rechecked and supplemented by numerical simulations. We systematically analyze SPP standing waves within various nanocavities. During the discussion, some unusual phenomena have been explained. For example, the hot spot of a nanodimer could be off-tip, depending on the order of standing wave mode; and that a nanocube on metal film can be viewed as a nanocube dimer with the same separation. And many other interesting phenomena have been discussed, such as dark mode of SPP standing wave and extraordinary optical transmission. The study gives a comprehensive understanding of SPP standing waves, and may promote the applications of cavity plasmons in ultrasensitive bio-sensings.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Surface plasmons are collective oscillations of free charges at the interface of a metal and an insulator. These oscillations are accompanied by electromagnetic waves, giving rise to two usual forms of excitations: surface plasmon polaritons (SPP) at unbounded interfaces and localized surface plasmons (LSP) in bounded geometries [1–6]. These two concepts have been widely used for more than one hundred years. Nowadays, surface plasmons remain a hot topic. This should be attributed to the continuous development of the nano-fabrication and synthesis methods, the measurement and characterization techniques, the powerful supercomputers, and to the fantastic potential applications of surface plasmons [1], such as, plasmonic nanoantennas [7,8], biosensing [9–12], photothermal therapy [13–15], photocatalysis [16,17], photoelectric devices [18–20], hot-carrier generation [21–23], surface-enhanced Raman scattering (SERS) [24–28], surface plasmon amplification by stimulated emission of radiation (SPASER) [29,30], nonlinear optics [31,32], subwavelength imaging [33], and so on. More than one hundred years have passed, and let’s discuss a less famous topic: SPP standing waves. SPP standing waves exist at specially designed nanostructures, such as nanocavities within a dimer or MI interface with periodic texturing [34,35]. Theoretically, SPP standing wave can form through SPP reflection at the end of an interface. But we cannot observe SPP standing wave at a flat and finite-length MI interface because the energy of SPP leaks out at the ends of the MI interface rather than bounces back. Also, some high order LSP resonance modes had been explained as standing wave formations [36–38]. Hence, we need a convincing model to show some solid evidences of SPP standing waves.
As another research background, recent experiments showed a peculiar surface plasmon mode sustained by configurations of nanoparticle-on-mirror and nanoparticle dimer, and this mode shows extremely high gap-width-dependent spectral sensitivity that is several orders of magnitude than the coupling of LSP [39–48]. At the same time, theoretical investigations confirmed the experimental results and proposed different interpretations for this surface plasmon mode [49–54]. For example, this new mode has been explained as coupled LSP mode [52,55], anapole states [39,56], patch antenna states [49,50], plasmonic coupling at the facets and the corners of the adjacent monomers [51], and so on. More essentially, this mode has been modeled as the semi-integer order reflection of SPPs in a Fabry-Perot cavity [40,41,43,45,46,53,57,58] or waveguide modes [42]. However, a rigorous model and a systematic analysis are still needed to better understand this mode.
In this study, by introducing a fundamental concept of SPP standing wave within plasmonic nanocavities, we reveal the nature of the new plasmon mode sustained by the very narrow plasmonic gaps. The concept of SPP standing wave actually means that SPP propagates along the cavity and bounces back at the end of the cavities, suffering a reflection phase shift. SPP propagation and reflection are rigorously modeled. The propagation of SPP within the cavity is modeled as the low-energy branch of the coupled SPP mode sustained by an metal-insulator-metal (MIM) cavity; and SPP reflection at the end of the cavity can be solved using Fourier-transformed electric and magnetic field [59–65]. Our model indicates that SPP won’t be reflected at the end of an MI interface to form a standing wave but it leaks out into the free space, so it is impossible to create a standing wave. However, SPP reflection coefficient at the ends of an MIM cavity increases dramatically as the cavity width becomes several nanometers [59,62], which enables SPP standing waves. SPP standing wave modes are observed (but not rigorously confirmed) only in recent years because of the advanced techniques for nano-synthesis and nano-characterization. Finally, we prove that SPP standing waves are supported in various cavities with different geometries, ranging from flat cavities to varying width cavities, from straightly-ended cavities to circularly-ended cavities, from plasmonic dimers to particle-on-films, as long as the cavities are narrow enough. For flat cavities, the model is supported by an analytical analysis of SPP propagation and reflection and by numerical simulations, while other cavities are revealed by numerical simulations. During the discussion, some unusual phenomena about the surface plasmons within nanocavities have been explained. 1. For a nanodimer, considered from the point of view of SPP standing wave, the locations of hot spots could be off-tip, depending on the orders of the standing waves. 2. The configuration of a nanocube on a metal surface with a separation of $a$ can be approximately treated as a nanocube dimer with the same separation $a$. This phenomenon can be viewed as: nanocube on the metal surface won’t couple with the image in the “mirror”, but it couples with the image directly at the surface. 3. We proved in theory that SPP standing waves could exist within various cavities. In this work, a systematic investigation of SPP standing waves is made. And many other interesting phenomena have been discussed, such as dark mode of SPP standing wave and extraordinary optical transmission. The study gives a comprehensive understanding of SPP standing waves, and may promote the applications of cavity plasmons in ultrasensitive bio-sensings.
2. Analytical model and numerical simulations
Analytical model of SPP standing waves. For a 1D optical cavity along $x$-axis, light propagates within the cavity and bounces back at the reflecting surfaces. Light after multiple reflections produces standing waves for certain resonance wavelengths. The simplest cavity is a Fabry–Perot interferometer whose plane mirrors are aligned parallelly. As mechanically shown in Fig. 1(a), the total electric field at $x$ to the left boundary of the cavity is
where the incident electric field intensity is normalized to 1, $r$ is the reflection coefficient at the inner boundary of the mirror, $k = 2\pi /\lambda$ is the propagating constant of the light with wavelength $\lambda$ in the cavity, $L$ is the length of the cavity. For a resonant cavity, the denominator is a minimum. So, the resonance condition is $k L = n \pi$ or $L = n \lambda /2$, where $n$ is an integer. This is the famous half-wavelength resonance. And due to the half-wave loss upon light reflection for normal incidence, there is a half-wave loss and $r \approx -1$. So, it must be a node of the standing wave at the end of the cavity. Finally, the electric field within the cavity is as shown in Fig. 1(b) for a given resonance condition: $n = 3$.
Fig. 1. Comparison of optical and SPP cavities. (a) Electric field at $x$ to the left boundary of a Fabry–Perot interferometer. (b) Optical standing wave within the Fabry–Perot interferometer. (c) SPP propagation at the MI interface, as well as reflection and leaks at the end of the interface. (d) SPP standing wave within an MIM cavity.
Now three things should be different for SPP standing waves. First, the “mirrors” are different. For light, the mirror is an interface of two different materials. However, the mirror for SPP reflection is simply the end of the MI interface, because SPP can only exist at the MI interface. Second, at the ends of the cavity are no longer the wave nodes, because half-wave loss doesn’t exist for reflection from high refractive index medium to low refractive index medium. In return, at each end of the cavity is an antinode. Third, the propagating wavelength $\lambda _{\rm SPP}$ of SPP is smaller than the light wavelength $\lambda _0$ of the same frequency, and they are connected by the dispersion relation, which have the form: $\lambda _{\rm SPP}=\lambda _0 \sqrt {\frac {\varepsilon _{1} + \varepsilon _{2}} {\varepsilon _1 \varepsilon _1}} = \frac {1}{k}$ [1]. $\varepsilon _1$ and $\varepsilon _2$ are the dielectric functions of the insulator and metal, respectively; and $k$ is the propagation constant. If the reflection phase is neglected, when cavity length $L$ is semi-integer times the SPP wavelength $\lambda _{\rm SPP}$, SPP standing wave forms. And incident light that can excite such an SPP standing wave is called the resonance light, and the wavelength of this light is called the resonance wavelength.
Although the above descriptions of SPP standing waves seem reasonable, there is a severer problem: why we can’t observe this mode in finite-length MI interfaces. This is because the reflection coefficient $r$ is too low and most of the energy leaks out into the free space, as schematically shown in Fig. 1(c). That is to say, most energy of SPP is dispersed before forming a standing wave.
However, the leaks are not inevitable. If two $z$-axis stacked MI interfaces get close (equivalent to an MIM structure), as shown in Fig. 1(d), SPP at two interfaces will couple through the near field, giving rise to two new SPP modes: one at the higher energy state and the other at the lower energy state. The lower energy SPP state dominates the optical spectrum if the width of the cavity is small enough. In the following context, SPP of MIM configurations will refer in particular to this lower energy SPP mode for simplification.
The dependency of the reflection phase and amplitude on the gap width and photon energy is shown in Fig. 2(a) and (b), respectively. The method is briefly described below. The total electric field within the cavity excited with a pane wave of unit amplitude is
where, $t$ is the coefficient of electric field transmitted to the cavity, $r$ is the reflection coefficient of SPP at the end of the cavity, $E_{z}^{\rm sp}(x)$ is the electric field of SPP wave of an infinite-length MIM cavity at position $x$, $k$ is the propagating constant of SPP with the dispersion relation $e^{-k_{1 z} d}=\frac {k_{1 z} \varepsilon _{2}+k_{2 z} \varepsilon _{1}}{k_{1 z} \varepsilon _{2}-k_{2 z} \varepsilon _{1}}$ and $k_{i z}^{2}=k^{2}-\varepsilon _{i} k_0^2$ [1]. $t$ and $r$ can be calculated within an MIM cavity using continuity boundary conditions of the the field and Poynting flux at two ends of the cavity and using Fourier-transformed electric and magnetic field [59–61]. The corresponding solution for $t$ and $r$ are:SPP gain phase through propagating and reflection, considering $r$ is a complex number. The resonance condition is that $1-r^2 e^{2 i k L}$ has a minimum. At small-gap and long-wave limit, when the reflection phase shift $\varphi$ can be neglected and the dielectric function of noble metal is approximately $\varepsilon _{2} \approx -\frac {\omega _{\rm p}^{2}}{\omega ^{2}}$, the dispersion relation of SPP is approximate:
Using Eq. (6) and proper boundary conditions, we can find the explicit expression of the approximate resonance wavelength of the cavities.Numerical simulations for other cavities. In case of irregular cavities, numerical simulation method can be a good supplement. The numerical simulations can also be used to recheck the analytical model.The simulations are performed using software Comsol Multiphysics. The simulations are based on the classical electromagnetic theory, and we have neglected quantum tunnelling and nonlocal effects, which will cause spectral red-shift and weakening [59]. The scattering and absorption spectra are obtained by integrating the scattered energy and resistive losses, respectively. The transmission spectra of periodic structures are collected using a collecting port on the opposite side of the exciting port. Perfectly matched layers (PML) are used at the outermost non-periodic boundaries. In this article, the noble metal used is silver with Drude parameters: $\epsilon _\infty = 4.039$, $\hbar \omega _{\rm p} = 9.172$ eV, and $\hbar \gamma = 0.0207$ eV.
3. Results and discussion
3.1 Four types of SPP standing waves within plasmonic dimers
SPP standing waves are supported within various plasmonic cavities. Within the cavities, the SPP propagating wavelength can be a constant or a variable, depending on whether the cavity width is a constant or not. Also, at the ends of the cavity, SPP can be reflected perpendicularly or obliquely, depending on whether the directions of SPP propagating and the cavity ends are perpendicular or not. According to the different ways of SPP propagation and reflection, four types of usual plasmonic cavities are shown in Fig. 3, a constant propagation and perpendicular reflection, b constant propagation and oblique reflection, c variable propagation and perpendicular reflection, d variable propagation and oblique reflection. We will not make a further discussion about the fourth type due to its complexity. In the following, we will discuss detailedly the SPP standing waves supported within the first three cavities.
Fig. 3. SPP standing waves within different cavities. (a) SPP standing waves by constant propagation and perpendicular reflection. (b) SPP standing waves by constant propagation and oblique reflection. (c) SPP standing waves by variable propagation and perpendicular reflection. (d) SPP standing waves by variable propagation and oblique reflection. The red arrows indicate the directions of SPP propagation, and the yellow structures indicate the plasmonic nanostructures.
SPP standing waves by constant propagation and perpendicular reflection. For a flat cavity with planar ends, as shown in Fig. 3(a), the SPP vector is a constant for a given wavelength. Considering the phase shift $\varphi$ by SPP reflection at the sharp ends, the accurate resonance condition is
Here, the phase shift $\varphi$ is calculated by the complex reflection coefficient $r$. The accuracy of the analytical model is limited by the difference between the ideally defined field distributions in the analytical model and the geometry-dependent field distributions in reality. Also, when Fourier transforms are performed, the singularity within the integration area and the integral limit would also influence the accuracy. For the considered cavity, the analytical model shows nice agreement with the numerical simulations within in a broad spectral range for cavities of different gap width, as shown in Fig. S1 in Supplement 1. At small-gap and long-wave limit, the explicit resonance wavelength is approximate: Also, the corresponding calculations can be done numerically. The corresponding configurations can be a 1D cavity within a plasmonic nanoslit shown in Fig. 4(a), or a 2D cavity within a plasmonic nanocube dimer shown in Fig. 4(d). The SPP standing waves within the nanoslit are investigated using the analytical model and numerical simulations. When excited with the resonance wavelength, SPP standing wave forms and gives rise to a maximum in the transmission spectrum. In Fig. 4(b), the optical spectra by the analytical method and by numerical simulation are shown by the solid black line and dotted red line, respectively. Four peaks appear in the spectral range of 500-1700 nm. The peak positions match perfectly for these two methods. The relative intensities of four peaks are different for the two methods due to the different natures of transmission and scattering. Figure 4(c) shows the $z$-components of the electric field $E_z$ at the four peaks marked by the corresponding symbols in Fig. 4(b). Clearly, there are 1, 2, 3 and 4 wave nodes respectively at the four resonance wavelengths, corresponding to the $n$-th longitudinal mode of SPP standing waves.Fig. 4. SPP standing waves by constant propagation and perpendicular reflection. A 1D cavity within a plasmonic nanoslit (a) and a 2D cavity within a plasmonic nanocube dimer (d). The red and green arrows show the propagation and polarization directions of the incident wave, respectively. In (a), the slit width 2$a$ is 2 nm, and the slit thickness $L$ is 200 nm. In (d), the side length $L$ of the two identical silver nanocubes is 100 nm, and dimer gap width $2a$ is 2 nm. The dimer is emerged in the water. The sharp edges of the nanocubes are replaced with arcs of radius of 5 nm. (b) Optical spectra of the nanoslit using the analytical model (black line) and from numerical simulations (red line). (e) Simulated optical spectra of the nanocube dimer. The inset shows the surface charge densities at the dimer at peak $\alpha$, and for a better view one nanocube is rotated by 90$^ \circ$. The spectrum is shown in the double logarithmic coordinates. (c) Normal components of the electric field at the corresponding peaks in (b). (f) Surface charge densities at the facing surface of a nanocube at the corresponding peaks in (e).
The standing waves supported by a nanocube dimer induce eight absorption peaks in the shown wavelength range, as shown in Fig. 4(e). The inset shows the surface charge densities of the rightmost peak, and for a better view one particle is rotated by 90$^\circ$ along one edge. There is one node in the middle of the cavity, which is the fundamental longitudinal mode of the SPP standing wave modes. It is worth noting that the event-order longitudinal standing waves are accompanied by transverse modes, as shown by the surface charge distributions in Fig. 4(f). For odd-order standing waves, the wave nodes only appear along the propagation direction. For even-order standing wave modes, the wave nodes appear both along the longitudinal and transverse directions. The transverse modes are resulted from SPP reflections at the round corners. The transverse modes should have a symmetric feature, so only even-order SPP standing wave modes have transverse excitations.
SPP standing waves by constant propagation and oblique reflection.
If the ends of a fixed-width cavity are not straight, the resonance condition will depend on the shape of the ends. Within a cylinder dimer stacked head-to-head is a fixed-width and circular-ended cavity, as shown in Fig. 3(b). For this cavity, the wave equation within the cavity ($r\le R$) is
where $\epsilon _{\rm eff}$ is the effective dielectric function defined by the dispersion relation, and $R$ is the radius of the cylinder. Neglecting the little variation of the electric field in the $z$-axis, we will have a 2D system in the $x$-$y$ plane. Solving Eq. (9) in the polar coordinates, we have the canonical solutions: Bessel functions of the first kind $J$ of integer order $m$Fig. 5. SPP standing waves within a cylindrical dimer stacked head-to-head. (a) Configuration of the cylindrical cavities. (b) Simulated optical spectra of the cylindrical cavities. (c) 0 and 1 order Bessel Functions of the first kind as a function of the wave vector of SPP. The pentacle symbols indicate the locations of the resonances. (d) Normal components of the electric field at the central plane of the dimer at the corresponding peaks.
SPP standing waves by variable propagation and perpendicular reflection.
SPP standing waves are also supported within cavities of varying gap width as shown in Fig. 3(c). Two configurations as designed: a cylindrical dimer stacked side-by-side shown in Fig. 6(a) and (a) spherical dimer shown in Fig. 6(d). It is not difficult to find from the dispersion relation that the square of SPP wave vector $k^2$ is approximately inversely proportional to the gap width $2a$ of a cavity. Furthermore, the reflection coefficient $r$ is also very sensitive to the gap width. When the gap width is varying as shown in Fig. 3(c), it is hard to obtain the resonance wavelength using the aforementioned analytical model. The alternative is to use the transformation optics approach [66–69]. In this work, the problems are solved using the numerical method.
Fig. 6. SPP standing waves by variable propagation and perpendicular reflection. Configurations of a cylindrical dimer stacked side-by-side (a) and a spherical dimer (d). In (a), the radius $R_2$ of the cylinder is 200 nm, and their separation is 1 nm. In (d), the radius of the nanospheres is 100 nm, and their separation is 1 nm. The incident wave propagates along their central plain and polarizes perpendicularly to the dimer. Simulated optical spectrum of the cylindrical dimer (b) and the spherical dimer (e). (c) Normal components of the electric field at the central plane of the cylinder dimer at the corresponding peaks in (b). (f) Surface charge densities at the facing surface of one nanosphere at the corresponding peaks in (e).
When the incident wave propagates along the central plain of the cylinder dimer and polarizes perpendicularly to the dimer, the induced SPP will propagate within the gap and bounce back at the openings. The propagating wavelength varies depending on the gap width and SPP reflection phase is determined by the radii of the nanorods. Here, the problem is tackled numerically. The simulated optical spectrum is shown in Fig. 6(b). We can find rich optical resonances in the visible optical range. Five peaks are picked out. The normal components of the electric field corresponding to these five peaks are shown in Fig. 6(c). The $n$-th peak has a node number of $2n+1$, which is the $n+1$-th SPP standing wave mode. Another feature is that the wave nodes are denser at the narrow part. This is because the SPP waves are more compressed at smaller gaps according to the dispersion relation. It should be noted that, in the simulation, the length of the nanorods is set to be infinite, which suppresses the excitation of SPP standing wave modes along the direction of the cylinder axis. If the rod dimer has a finite length, depending on the length of the nanorods, at certain wavelengths there will also be SPP standing wave modes along the direction of the cylinder axis, which will not be detailed.
Figure 6(e) shows the optical spectrum of a spherical dimer illuminated by an electromagnetic wave polarizing perpendicularly to the dimer. The optical spectrum is obtained by numerical method. Five peaks appear in the visible light range, which is much complex than the usual case observed in the experiments. For example, in the experiment, the most common excitation of the dimer is the longitudinal coherent mode, and the two nanoparticles oscillate in phase inducing a spectral red-shift. To check the plasmon modes, Fig. 6(f) shows the surface charge distributions on one of the nanospheres. For mode $o$, there is only one type of charge, so it is the common coherent mode of the dimer. For mode $\alpha$, the charges at the center and at the boundary become different, which is the fundamental mode of SPP standing waves. And for $\beta$, $\chi$, and $\phi$, wave nodes also appear along the azimuthal direction. They are all high-order SPP standing wave modes. On the one hand, it is interesting to see such a rich optical spectrum of a simple sphere dimer just by narrowing the separation. So, SPP standing waves are not only supported by a long and flat cavity but also by a sharper connection, as long as the separation is narrow enough. On the other hand, it is not necessarily a ‘hot’ spot at the center of a dimer. Possibly, there may be a wave node for high-order SPP standing waves at the center of two nanotips. That is to say, the hot spots could be off-tip. The results remind the experimentalists to find a proper position and excitation wavelength for the ‘hot’ spot between a close dimer.
3.2 SPP standing waves within the cavity between a plasmonic nanoparticle and a metal film
The most common nanocavity fabricated in experiments is the gap between a nanoparticle and a film separated by an insulator interlayer. Depending on the geometry of the nanoparticle, three cavities are shown: Fig. 7(a) a squared-nanowire on a film, Fig. 7(d) a circular-nanowire on a film, Fig. 7(g) a nanosphere on a film.
Fig. 7. SPP standing waves within the cavities between single plasmonic nanoparticles and a metal film. Configurations of a squared-nanowire on a film (a), a circular-nanowire on a film (d), and a nanosphere on a film (g). In (a), the side length of the cross-section of the nanowire is 200 nm, and the separation is 2 nm. In (d), the radius of the silver nanowire is 200 nm, and the separation is 1 nm. In (g), the radius of the nanosphere is 100 nm, and the separation is 0.5 nm. The light propagates perpendicular to the film and polarizes along the film. (b) Optical spectrum of the squared-nanowire on the film excited by oblique (black solid line) and normal (black dash line) incidence angles, compared with the spectrum of a nanowire dimer with the same side-length and the same separation (gray solid line). The insets bordered with the corresponding colors show the three configurations. The nanocube on the film and the nanocube dimer have the same inter-particle separations. (e) Optical spectra of the circular-nanowire on the film. (h) Optical spectral of the nanosphere on the film. (c),(f),(i) Normal components of the electric field ($E_z$) at the cutting plane of the cavities at the corresponding peaks respectively in (b),(e),(h).
Figure 7(b) shows the optical spectrum of the squared-nanowire on the film excited by oblique (black solid line) and normal (black dash line) incidence angles, compared with the spectrum of a nanowire dimer with the same side-length and the same separation (gray solid line). The insets show the corresponding configurations. The detailed comparison of their spectra shows that the resonance wavelengths of the corresponding SPP standing wave modes are close for the nanowire-on-film and nanodimer, while some standing wave modes are missing for the nanowire-on-film. Figure 7(c) shows the normal components of the electric field by oblique incidences to show the SPP standing wave modes. The wave nodes get denser from the longer wavelength to the shorter; while the even-order SPP standing wave modes are missing for normal incidence. This is because the even-order SPP standing wave modes are dark modes for normal incident waves.
The results indicates that, the configuration of a nanocube on a metal surface with a separation of $a$ can be approximately treated as a nanocube dimer with the same separation $a$. This phenomenon can be viewed as: nanocube on the metal surface won’t couple with the image in the “mirror”, but it couples with the image directly at the surface. The underlying physics is that SPP propagations within the two cavities are the same and that SPP reflection phases at the ends of the two narrow cavities approach zero. So, the conclusion, that the configuration of a nanocube on a metal surface can be treated as a nanocube dimer with the same separation, is inapplicable to other configurations, such as a nanorod on a film. Besides, our results indicate that a nanocube on a metal film can be approximately solved using the same analytical model of a finite-length MIM cavity. And the resonance condition is the sum of the propagation phase and the reflection phase equals an integer multiple of $\pi$.
The optical spectrum of the circular-nanowire on the film excited by normal incidence is shown in Fig. 7(e). The spectrum shows multi-peaks. The spectrum is similar to Fig. 7(b), except that the peaks are broadened. In Fig. 7(f), the modes at the corresponding peaks of the normal incidence are recognized by the normal components of the electric field. The number of the nodes from the longer wavelength to the shorter wavelength is 2$n$+1, and $n$ is the $n$-th peak in the spectrum from the longer wavelength to the shorter. Here the even-order dark modes are also missing. The even-order modes cannot be excited by normal incident waves, but they can be excited by obliquely incident waves, as discussed above. The wave nodes get denser at the narrower part of the gap.
The optical spectrum of the nanosphere on the film in Fig. 7(h) shows a series of peaks. The spacing between the nanosphere and the metal film can be viewed as a 3D cavity. Also, the SPP standing wave modes are confirmed by the normal component of the electric field in Fig. 7(i). The cavity keeps the three common features of SPP standing waves. First, the number of the nodes is odd. Second, the modes have denser wave nodes at the narrower parts of the cavity. Third, when the wavelength becomes shorter, the nodes become condenser.
3.3 SPP standing waves within crooked slits
SPP standing waves within slits bent in the wave propagating direction. The behavior of SPP standing wave within a bent cavity whose gap width keeps a constant is investigated in Fig. 8. The schematic diagram is shown in Fig. 8(a). The SPP wave may experience a longer cavity length than the vertical spacing between two cavity ends when SPP propagates along a crooked cavity. If the cavity is not bent severely, the dispersion relation of SPP may not be influenced too much. Equivalently, the length of the cavity is stretched. If the standing wave condition is satisfied, SPP must have a longer wavelength. In return, the resonant light should have a longer wavelength. Then we design a simple crooked cavity by making a cosine curve along the cavity. The cavity has two tangential ends, so that the incident light couples to the SPP waves in the same way as it couples to SPP waves supported within a straight cavity. Figure 8(b) shows the simulated optical spectra of the crooked cavities with different curve nodes. As expected, when the curve nodes increase, the resonance wavelengths show obvious red-shifts. A more detailed analytical calculation can confirm that when the cavity is not crooked too much, the crooked cavity has the same resonance wavelength as the straight cavity with the same effective length. In Fig. 8(c), the electric field distributions for these four cavities at the corresponding resonance wavelengths are shown. All these four cavities exhibit a fundamental SPP standing wave mode. So, SPP waves can propagate along a rough cavity and bounce back at the ends to form a standing wave excitation.
Fig. 8. SPP standing waves within slits bent in the wave propagating direction (a)-(c) and bent in the normal direction of SPP propagating (d)-(i). (a) Schematic diagram of the SPP waves propagating along the bent slit. (b) Simulated optical spectra of the bent cavities with different curve nodes. The amplitude of the curve is 1 nm; the cavity length is 100 nm; and the gap width is 2 nm. (c) The electric field distributions at the corresponding peaks of the four cavities. (d),(h) Cavities with cross-sections patterned by a circle and a square. (e),(h) and (f),(i) are the corresponding optical spectra and electric field distributions, respectively. Insets in (e),(h) show their corresponding charge distributions on the surface of the cores. The thickness of the films (i.e., the length of the cavities) is 200 nm and the width of the slits is 10 nm. The radius of the circle is $\frac {1600}{2\pi }$ nm; and the side length of the square is 400 nm. The patterns in (d)-(i) are designed within the unit cells of the hexagonal gratings whose lattice period are 800 nm.
SPP standing waves within slits bent in the normal direction of SPP propagating. Now, we consider the cavities those have crooked cross-sections perpendicular to the direction of the SPP wave propagation. The crooked cross-sections will not influence the length of SPP propagation along the cavity. However, the crooked cross-sections will cause a transverse flow of SPP waves. Two configurations with crooked cross-sections are designed in Fig. 8(d)-(i). The slits are designed in the hollow out silver films, and they may have different patterns: a circular pattern in Fig. 8(d) and (a) squared pattern in Fig. 8(g). Their corresponding optical spectra are shown in Fig. 8(e) and (h). There is a spectral shift between two slits. The spectral shift may be attributed to their different cross-sections. The insets show their corresponding charge distributions on the surface of the cores. Both cavities show a fundamental SPP standing wave mode along the slits. Also, the fundamental standing wave modes are confirmed by the electric field distributions in Fig. 8(f) and (i), each of which shows a wave node in the middle of the cavity. It is worth noting that the transmission of the slit reaches a maximum of about 0.6, which is more than 5 times the area ratio of the slit to the whole. The reason is the same as observed in experiment 100 years ago [65]: at the surface of the slit forms surface plasmons which concentrate light within the cavity area and allow the transmission much larger than the area ratio of the cavity to the whole film.
4. Conclusions
In conclusion, we systematically analyze the SPP standing waves within various plasmonic cavities. A model of SPP standing wave is built by analyzing the SPP wave propagation along the MIM cavity and reflection at the cavity boundary. Compared with the optical cavity, the SPP cavity has three different features. First, SPP propagating wavelength is much smaller than light, which is its subwavelength nature. Second, SPP reflection doesn’t suffer half-wave loss but it suffers phase shift due to energy leak, so at the end of the cavity is a wave antinode rather than a wave node. Third, SPP reflection coefficient is not close to one naturally but it only approaches one when the cavity width is much smaller than the wavelength of light, which explains the nature of a new plasmon mode found recently within very close dimers. Various cavities are proved to support SPP standing waves when the cavity widths are narrow enough. During the discussion, many interesting phenomena are discussed, e.g., SPP dark mode excitation, off-tip hot spots, special ‘mirror’ of the metal film, and extraordinary optical transmission. This investigation helps to enrich the concept of surface plasmons by introducing a fundamental concept of SPP standing wave and may find potential applications of ultra-narrow plasmonic cavities in bio-sensing and characterizations.
Funding
National Key Research and Development Program of China (2017YFA0303402, 2017YFA0303404); National Natural Science Foundation of China (11934003, NSAF U1930402); Fundamental Research Funds for the Central Universities (2021MS046).
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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