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Abstract
A dielectric (nano)particle can influence the local electromagnetic field and thereby alter its interaction with that field through the process of self-induced back-action. While this phenomenon is usually considered theoretically as a change in a cavity resonance frequency, such theoretical approaches are not as appropriate when considering systems away from resonance, such as with a subwavelength aperture in a metal film. Here we consider the interaction between an aperture, modelled with Bethe theory as a magnetic dipole, and a Rayleigh particle, modelled as an electric dipole. Using this magnetic dipole – electric dipole interaction, we quantify the self-induced back-action of the particle on the aperture transmission and the optical trapping potential. The model shows quantitative agreement with finite-difference time-domain simulations. This shows that the physics of self-induced back-action for an aperture and a nanoparticle can be understood in terms of dipole-dipole coupling.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Numerical simulations were first used to understand the trapping potential of an aperture in a metal film on a subwavelength particle [1]. That work was followed with experimental demonstrations of trapping near an aperture [2]. The self-induced back-action (SIBA) effect was demonstrated to allow for more efficient trapping of subwavelength particles by comparing numerical calculations including the particle and comparing with only the force from the field without the particle’s influence [3]. Since that time, shaped apertures have been used to trap smaller objects including inorganic nanoparticles (like colloidal quantum dots) and biomolecules (like proteins only a nanometer across) [3–20]. Despite these advances in the application of SIBA, the theoretical understanding of the SIBA effect for apertures and nanoparticles has been obscured by the complexity of the numerical simulations, and a simple theory which shows the underlying physics is desired.
Simple theories have been used to describe SIBA in other systems that invoke a cavity resonance, where the impact of the particle is to shift the cavity resonance [21]. That theoretical description is particularly suitable for photonic crystal cavities with a strong resonance [22–24]. Similar resonances can exist in aperture systems [25]; however, away from these resonances a simple theory is needed to understand the physics of SIBA. The current work is an attempt to bridge this gap by considering one such simple theory based on Bethe-Rayleigh aperture-particle theory. A finite-difference-time domain (FDTD) simulation is used to to show that the dipole model captures the basic physics of the system.
2. Bethe-Rayleigh SIBA theory
Here we consider the theoretical description to quantify SIBA when a Bethe [26] aperture interacts with a Rayleigh particle. The theory shows that interactions between the electric and magnetic dipoles of the particle and aperture can give rise to the SIBA effect. Figure 1 shows a schematic of the geometry under consideration with all the relevant geometric parameters such as the radius of the Bethe aperture $R$, the radius of the Rayleigh particle $a$, the distance $r$ of the Rayleigh particle above the Bethe aperture, and the direction of propagation $z$ of the exciting beam of light and of its polarization clearly labelled. Both the sphere and the aperture are subwavelength to allow for application of Bethe theory and Rayleigh theory. Using Bethe theory, the electric field produced by a subwavelength aperture in a perfect electric conductor can be modelled as coming from an effective magnetic dipole with magnetic polarizability:
assuming that the plane wave from below is normally incident.
Fig. 1. Geometry of an aperture in a metal film interacting with sphere with light incident from the opposite side.
The effective magnetic dipole at the aperture produces an electric field $\vec{E}_m$:
This electric field polarizes the dielectric sphere, producing an electric polarization
where the polarizability $\alpha$ is given by with $\epsilon _r$ being the relative permittivity of the dielectric sphere embedded in a medium of relative permittivity $\epsilon _b$, $\epsilon _0$ the permittivity of free space. The polarized dielectric creates a magnetic field,The electric field created by the aperture is modified from what it should have been in the absence of the sphere, or if the sphere did not interact with the aperture, by the change in the magnetic field at the aperture from $\vec{H}_0$ to $\vec{H}_a$ given by Eq. (6). In particular, the effective magnetic dipole moment at the aperture becomes
Inserting (8) and (1) into (9), we obtain
If we assume for simplicity that $\vec{n}$ is in the $z$-direction and $\vec{m}$ is in the $x$-direction, then Eq. (10) simplifies to
The transmission intensity $I$ through an aperture into the half-space, in terms of the effective magnetic dipole moment of the aperture, is given by
In the absence of back-action we use $\vec{m}_{eff}=\vec{m} = \frac {8R^3}{3}\vec{H_0}$ (14) to obtain
By inserting (13) into (14), simplifying and then dividing the result by (15), we obtain the transmission $T$ through the aperture, normalized to the transmission without back-action given by (15), as
A plot of $T$ as a function of $r$, calculated from the Bethe-Rayleigh theory for $a = R = 50~\mathrm {nm}$, and $\lambda = 500~\mathrm {nm}$, $\epsilon _r = 2.343$ and $\epsilon _b = 1$, is shown as a solid dark curve in Fig. 2 below. The theory is plotted only to 50 nm, so that the aperture and the particle are not overlapping and the results are physically meaningful.
Fig. 2. Transmission through aperture calculated with Bethe-Rayleigh theory, normalized to transmission without SIBA. Theoretical result and FDTD simulation shown.
The trapping potential is given by
where all the terms are as previously defined and $\vec{E}_m$ is given by Eq. (2) in the absence SIBA or, in the presence of SIBA, by (8). The calculated difference between the trapping potentials with and without SIBA is shown as a solid dark curve in Fig. 3 below, for an incident intensity of $11.25\,{\mathrm{mW}}/\mu \mathrm{m}^2$.Fig. 3. Difference in trapping potential of aperture with and without SIBA calculated with Bethe-Rayleigh theory (black), and for the FDTD simulations with and without SIBA (green).
3. Numerical simulations
FDTD simulations were performed using Ansys Lumerical FDTD to show that the dipole model, presented in the preceding section, captures the basic physics of the system. The structure of this FDTD simulation was a 1 nm thick perfect electric conductor (PEC) with a circular aperture of 50 nm radius through its center. The PEC had an area of 1 $\mu \mathrm {m}^2$ in the $x-y$ plane. Because of the symmetry of the structure, a symmetric boundary condition was imposed at the minimum of the $x$-axis while anti-symmetric boundary condition was similarly imposed at the minimum of the $y$-axis. Perfectly Matched Layer (PML) boundary condition was used everywhere else. The simulation domain was made to go through the PEC to simulate an infinitely wide PEC plane. The simulation domain was 900 nm $\times$ 900 nm $\times$ 1000 nm. A total field scattered field (TFSF) source was used with 201 wavelengths in the range from 500 nm to 1500 nm. The source was $y$-polarized, with an amplitude of $2.911 \times 10^6~\mathrm {V/m}$ and propagated in the forward $z$-direction. The TFSF source was 640 nm $\times$ 640 nm $\times$ 682 nm. A dielectric sphere of 50 nm radius having a refractive index of 1.56 was moved along the direction of the beam, directly above the circular aperture of the PEC. A coarse uniform mesh with $dx = dy = 25~\mathrm {nm}$ and $dz = 1~\mathrm {nm}$ was initially set in the simulation region. To improve accuracy, however, a mesh override was used. The mesh override region was designed to match the dimensions of the TFSF source. For this mesh override, $dx = dy = 5~\mathrm {nm}$ while $dz = 0.25~\mathrm {nm}$. The mesh refinement setting in the simulation region was set to the conformal variant 1 type. The dt stability factor was set to 0.95, the time step to 0.0031638 fs and the minimum step size to 0.025 nm. Two 2-D monitors of the frequency domain field and power type were used. One $z$-normal monitor was fixed at $z = 0$ and measured the transmission while another $x$-normal monitor was fixed at $x = 0$ and measured the E-field. The $z$-position of the dielectric sphere was varied in steps of 10 nm from 51 nm to 261 nm, and for each position of the dielectric sphere, the transmission, $T$, and the $E$-field were recorded with and without the particle. The intensity at the desired wavelength (500 nm) was extracted from the record of intensity versus wavelength and the trapping potential, $U$, was computed in accordance with 17. Figure 4 below shows a $yz$-section of the simulation setup. Figures 5(a)-d show the distribution of the magnitudes of the electric and magnetic fields with and without the nanoparticle.
The numerically computed transmission as a function of $r$ is shown as the solid green curve in Fig. 2 above, while the difference between the numerically computed trapping potentials with and without SIBA is shown as a solid green curve in Fig. 3 above.
4. Discussion
Quantitative agreement between theory and simulation is found in this study, thus providing support that the dipole theory captures the relevant SIBA physics for this geometry. While off-axis excitation is possible, we will need to include electric dipole at the aperture that is normal to the aperture, so we have left it out in this simple theory to keep things as clear as possible. We moved the particle off-axis in the simulation and found a slight increase in the transmission of less than 1% and an increase in the difference in trapping potential relative to the no-SIBA case of about 2% above the on-axis value for every $5^\circ$ tilt at a wavelength of 500 nm. For normal incidence, as in the current study, the first order effect is the magnetic-electric dipole interaction considered.
As can be seen from Figures 2 and 3, SIBA enhances both the transmission and the trapping potential. In particular, about 17% increase in the transmission and about 28% increase in the difference in trapping potential relative to the no-SIBA case can be observed in Figures 2 and 3. The enhancement shown explains the already known and exploited ability of aperture tweezers to work at lower optical intensities, a quality which makes them useful for trapping photosensitive samples.
The observation of SIBA with a PEC is interesting because it indicates that surface plasmons are not necessary, while it does not deny the existence of plasmons in real systems. Surface plasmons are particularly interesting close to the plasmonic resonance, which is not of primary interest for this off-resonance work. Shaping the aperture can also be captured within the present theory by using the effective magnetic polarizability of a shaped aperture.
The Bethe polarizability in the present model does not account for plasmonic resonance effects. Plasmonic resonances can shift the peak in the transmission, and the corresponding polarizability for larger apertures (for example, [27], [28]). It is possible then to consider that this theory can be modified to include these resonance effects; however, the simple Bethe theory does not present any resonance peak (and actually over-estimates the transmission for larger apertures due to the strong power scaling). The polarizability of the nanoparticle does contain implicitly the Frölich resonance condition that accounts for the plasmonic resonance of the particle, but that is not explored in this work. The plasmon resonance of apertures in real metals are around 300 nm are in the visible regime for vacuum and Ag [29] and in the infrared for supported apertures in Au and in water [3]. At much longer wavelengths, the transmission through the aperture is better represented by a Bethe theory than the tail end of a Lorentzian, so we believe that the present theory would be better suited for that regime.
While equal nanoaperture and particle size was chosen as a starting point in this study to ensure that we were in the dipole limit for both the particle and aperture, we attempted different geometries as well. For example, doubling the aperture radius increased the change in transmission (normalized to the no-particle case) by over 5 times, and the potential increased by 64 times.
In this work, we have shown that there is a SIBA contribution within Bethe-Rayleigh theory for a particle-aperture system. By monitoring the transmission changes, it is possible to get information related to the particle size and position, which has already been done in experiments to size proteins [30]. It is also possible that this can be applied to tracking [31], and possibly to near-field polarization mapping [32]. To achieve polarization and full position tracking, the present theory should be extended to include off-axis positions (including the electric dipole of the aperture as well).
While the agreement between analytical and numerical solutions found in this study is close, it is not exactly perfect. Since we performed a convergence study and found that the solution was fully converged (showing negligible variation with smaller mesh sizes) for the simulation parameters used, we believe that the differences remaining between theory and experiment are the result of the dipole approximation made in the theory.
5. Conclusions
Here we have shown the most basic description of SIBA in an aperture system that uses an effective magnetic dipole of the aperture and the electric dipole of a particle within the interaction range of the aperture-particle system. The magnetic dipole of the Bethe aperture creates an electric field, which polarizes the electric dipole of the Rayleigh scatterer, which in turn creates a magnetic field back at the aperture. This can enhance the transmission and trapping potential significantly for the non-resonant case and benefit trapping. This simple theory may be extended to different aperture shapes, or different locations of the aperture with respect to the particle (and perhaps even multiple apertures or multiple particles).
Funding
Natural Sciences and Engineering Research Council of Canada (RGPIN-2023-04108).
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.
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