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Optical singularities associated with the energy flow of two closely spaced dielectric-core gold-shell nanocylinders are studied by two-dimensional finite difference time domain method. The simulation results show that optical vortices as well as saddle points can be observed in the energy flow pattern of light interacting with the core-shell nanocylinder pair in its in-phase symmetric dipolar plasmon mode. The rotating direction of the optical vortices can be tuned by varying the width of the gap between the nanocylinder pair and the value of the permittivity of the dielectric core.
©2009 Optical Society of America
1. Introduction
The energy flow of an electromagnetic (EM) wave can exhibit unusual and complex pattern in the neighborhood of points where the amplitude of the field is zero. The phase singularities in the energy flow that was addressed in the seminal paper by Nye and Berry [1], has developed into an exciting field of optics, called singular optics [2]. For a metallic nanoparticle, it is shown that light scattering by such a particle exhibits features such as optical whirlpools [3], and optical vortices near the nanoparticle [4]. Optical vortices have also been observed in optical fibers, in nonlinear Kerr media [5], in super lens [6], and in the near field diffracted by subwavelength apertures [7–9]. The application of the singular optics includes nanoscale resolution in the focal plane of an optical microscope and optical trapping of viruses [10] and small particles [11].
A seemingly unrelated field in the study of the core-shell nanostructures such as nanorices [12], nanorings [13], and nanoshells [14] is of special interest due to the highly tunable localized surface plasmon modes. Such structures give rise to very large enhancement in the junction formed by the pair of the particles, which make them attractive as SERS substrate [15]. The dependence of the optical properties of the core-shell nanostructures on the size, dielectric core, shape, and surrounding medium has been an active field of research and can be found in numerous studies in the literature. However, the optical singularities associated with the energy flow in these systems have not been investigated in detail yet. In this work we report our studies on the optical singularities associated with the energy flux of two closely spaced dielectric-core gold-shell nanocylinders.
Two-dimensional finite difference time domain method (FDTD) was employed for this study. For light polarized parallel to the axis connecting the pair, there are four zero energy points in the energy flow diagram, two saddle points and two optical vortices. Among these four points, the saddle points are outside the cylinders and the optical vortices are inside the cylinders. Both the width of the gap between the nanocylinder pair and the value of the permittivity of the dielectric core were found to have strong influence on the rotating direction of the optical vortices and their effects on the properties of the optical vortices were studied in details.
2. The phase of the Poynting vector
The phase of the Poynting vector, , is defined as
where is the modulus of . At the point where the amplitude of the energy flux is zero, the phase at that point is undefined according to (1) and thus it is a phase singular point. Singular points can occur at the following positions in an energy flow diagram: (a) the center of a rotating energy flow (optical vortex), (b) a saddle point, and (c) a sink or a source for the EM wave. In this work, the magnitude of the rotation and direction of the rotation of the optical vortex is called the circulation of the optical vortex and is defined as,where the surface integral of the magnitude of is summed over the surface inside the dielectric core of the core-shell nanocylinder. When the circulation is larger than zero, it means that the rotating direction of the optical vortex is counter clockwise and the optical vortex is a right-handed optical vortex (RV). On the other hand, if the circulation is smaller than zero, it means that the rotating direction of the optical vortex is clockwise and the optical vortex is called a left-handed optical vortex (LV).3. Numerical method
FDTD [16] was used to investigate the near field optical properties of the dielectric-core gold-shell nanocylinder pair. The optical response of gold is modeled using the three critical point pole pairs (CP3) model [17,18]. The CP3 model can be expressed as
And the codes have been checked with the analytical theory [19]. Our FDTD simulation domain is separated into three regions from outside to inside: absorbing boundary, scattered field region, and the total field region. Perfectly Matched Layers (PMLs) are used as absorption boundary to prevent reflections of scattered waves back into the simulation domain. The FDTD calculations were done using a mesh size of 0.5 nm, a time step of 8.3333e-19s, and a Courant number of 0.5. The geometrical arrangement of the incident EM wave and the dielectric-core gold-shell nanocylinders are shown schematically in Fig. 1 . In this configuration, the incident wave is TM polarized, i.e., the only non-zero component of the magnetic field is in the y-direction. The inner radius and outer radius of the core-shell nanocylinder used in this work are 112.5nm and 135nm, respectively.
Fig. 1 Schematic diagram of the geometric arrangement of the incident EM wave and the dielectric-core gold-shell nanocylinder pair. The propagating direction of the incident wave is in the z-direction and is perpendicular to the axis connecting the nanocylinder pair (x-direction). The electric field of the incident wave is in the x-direction.
4. Phase singularities inside a core-shell nanocylinder pair
We first study the spectral characteristics of the core-shell nanocylinder pair in the wavelength range between 700nm and 1500nm. In this study the permittivity of the dielectric core is taken as 4.1, and the distance between the two cylinders is set at 27nm. The extinction, scattering, and absorption cross section spectra for this system are shown in Fig. 2(a) . We can find in these spectra that there are three major plasmon modes for the nanocylinder pair. The plasmon mode near 1161 nm is the in-phase symmetric dipolar (SD) mode. This plasmon mode is originated from the electrons at the inner surface of the core-shell nanocylinder aligned and oscillates symmetrically with the electrons at the outer surface [20]. The other two peaks in the figure are the in-phase antisymmetric dipolar plasmon mode near 935 nm, and the in-phase symmetric quadrupolar plasmon mode near 800nm. The charge density distributions associated with these plasmon modes are also shown in Fig. 2(a). In this work, we’ll focus on the energy flow distribution in its in-phase SD interaction mode.
Fig. 2 (a) The FDTD calculated extinction, scattering, and absorption spectra of a dielectric-core gold-shell nanocylinder pair with separation distance of 27nm. The permittivity of the dielectric core is 4.1. The charge distributions of these plasmon modes are shown in the inset. (b) The energy flow pattern near the nanocylinder pair in its in-phase SD interaction mode.
The energy flow pattern of light scattering from such a core-shell nanocylinder pair of a monochromatic plane wave with incident wavelength of 1161nm is shown in Fig. 2(b). We can see in this figure that there are four zero energy points in the energy flux diagram and these phase singularities are marked as S1, S2, RV, and LV, respectively. Among these four points, S1 and S2 are outside the cylinders and RV and LV are inside the cylinders. We can find from the energy flow pattern in the figure that energy flows into S1 in z-direction and flows out of S1 in x-direction, flows into S2 in x-direction and flows out of S2 in z-direction, but for RV and LV, the energy flow circulates around these two points and deceases in magnitude as it get closer toward these two points. From the characteristics of the energy flow pattern around these four points we can identify S1 and S2 as saddle points, RV and LV as the centers of optical vortices. It is evident from the figure that optical singularities exist in the energy flux associated with two closely spaced core-shell nanocylinder pair.
5. The dependence of the circulating direction of the optical vortex on the gap width
In order to have a better understanding on the properties of the optical vortex associated with the nanocylinder pair, the effect of the width of the gap between the nanocylinder pair on the rotating direction of the optical vortex is studied. Figure 3(a)-(d) are the energy flow patterns of the core-shell nanocylinder pair that have gap width of 20 nm, 35 nm, 70 nm, and 135 nm, respectively. As the geometrical arrangement is symmetrical with respect to the z-axis that passes through the center of the nanocylinder pair, it is not surprising to see in these figures that the rotating directions of the two optical vortices in the same figure are also symmetrical with respect to the z-axis that passes through the center of the nanocylinder pair.
Fig. 3 The energy flow patterns of the core-shell nanocylinder pair. The gap width between the nanocylinder pair is (a) 20nm, (b) 35nm, (c) 70nm, (d) 135nm. The optical vortices disappear when the separation distance is 70nm. (e) The variation of the circulation inside each core-shell nanocylinder with the gap width. The circle-dotted red line and the square-dotted blue line represent the circulation inside the upper and the lower core-shell nanocylinders, respectively.
For the convenience of discussion we’ll hence call the configuration with upper optical vortex rotating in a left-handed direction (LV) and lower optical vortex rotating in a right-handed direction (RV) as inward-type rotation, and the configuration with upper right-handed optical vortex (RV) and lower left-handed optical vortex (LV) as outward-type rotation. From the energy flow patterns shown in these figures we can find that when the gap width between the nanocylinder pair are 20 nm and 35 nm the rotational type of the optical vortices are inward-type rotation, for gap width equals to 70 nm, no optical vortex could be found, but when the gap width is 135nm, the rotational type of the vortices change to the outward-type rotation.
Examine these four figures carefully we can find that the relative intensity between the EM waves that flow through the gap and the outer edge (the edge that is opposite to the edge near the gap) of the cylinders play a major role in determining the circulating direction of the vortices. When the intensity of the EM wave is larger at the outer edge of the cylinder than that is in the gap, the circulating direction of the vortex follows the energy flow direction of the EM wave at the outer edge and thus the circulating direction for the upper vortex is left-handed and for the lower vortex it is right-handed (Fig. 3(a)-(b)). When the intensity of the EM wave is larger in the gap than that is at the outer edge of the cylinder, the circulating direction of the vortex follow the energy flow direction of the EM wave in the gap and thus the circulating direction of the upper vortex is right-handed and the lower vortex is left-handed (Fig. 3(d)). But when the strengths of the two energy flows are comparable, no vortex can be found in the energy flow pattern shown in Fig. 3(c). Therefore, we can conclude that the gap width between the nanocylinder pair influences the strength of the energy flows in different regions and that in turn determines the rotational type of the optical vortices. A detail study of the rotating type of the optical vortices and their circulation defined by Eq. (2) as a function of the gap width was presented in Fig. 3(e).
Figure 3(e) shows that when the gap width between the nanocylinder pair is larger than 70nm, the vortices are outward-type rotation. This is because the direction of the energy flow tends to flow toward the gap due to mutual coupling of the two plasmonic nanocylinders. Therefore, except for the gap width between the nanocylinder pair much smaller than the incident wavelength, the intensity of the EM wave is larger in the gap than that is at the outer edge of the cylinder, and thus the vortices are outward-type rotation. When the gap width between the nanocylinder pair is larger than 270nm, the circulation of the outward optical vortices reduces due to the reduction of the near-field coupling.
6. The dependence of the optical vortices on the permittivity of the dielectric core
In addition to the gap width between the cylinder pair, we find that the value of the permittivity of the dielectric core can also be used to tune the rotation type of the optical vortex. Figure 4(a) shows the extinction cross section spectra of the core-shell nanocylinder pair with gap width of 27 nm and with dielectric cores having permittivity of 2.1, 3.1, and 4.1, respectively, in the wavelength range between 700nm and 1600nm. The three peaks marked by the arrows at 852nm, 996nm, and 1161nm are the in-phase symmetric dipolar plasmon modes for these nanocylinder pairs. Because of the dielectric screening effect [21], the resonance wavelengths of these plasmon modes are red-shifted when the permittivity of the dielectric core increases.
Fig. 4 (a) The calculated extinction cross section spectra for the core-shell nanocylinder pair with dielectric permittivity of 4.1, 3.1, and 2.1, respectively. The energy flow patterns associated with the core-shell nanocylinder pair with dielectric permittivity of (b) 4.1, (c) 3.1, and (d) 2.1. Note that the rotating direction of the energy flow in (b) is opposite to the rotating direction of the energy flow in (c) and (d).
The energy flow patterns of these core-shell nanocylinder pairs, calculated at their respective resonance wavelengths, are shown in Fig. 4(b)-(d). We can see in these figures that the rotation type of the vortices changes from the inward-type to the outward-type when the value of the permittivity of the dielectric core changes from 4.1 to 2.1. To explain the change in the rotating direction of the optical vortices, we first note that in Fig. 4(a) the intensity of the EM wave at S1, a saddle point, is zero and this is because the gap width is much smaller than the wavelength of the incident light and thus EM wave can’t transmit through the gap. In this circumstance, the rotating direction of the optical vortex is dictated by the direction of the energy flow at the outer edge of the nanocylinders as was discussed in the previous section, and thus the rotational type is inward-type when the value of the permittivity of the dielectric core is 4.1. However, as the permittivity of the dielectric core decrease, the EM wave is more likely to be trapped inside the nanocylinders than just penetrating through the nanocylinders, and the intensity of the EM wave inside the nanocylinders increases quite dramatically as the permittivity of the dielectric core decreases. This is because the cavity resonance inside the nanocylinders is similar to one-dimensional Fabry-Perot resonance. As the permittivity of the dielectric core increases, the reflectance at the metal/dielectric boundary decreases due to the closer match in the magnitude of the real part of the refractive index, and this result in the enhancement of EM field inside the nanocylinders [22].
This EM field confinement effect can be seen quite clearly in our simulation. The intensity of the EM field inside the nanocylinders increases quite substantially from Fig. 4(b) to (d). As the EM wave that penetrates into and stays in the nanocylinders increases, the energy flow direction inside the nanocylinder will eventually be determined by the direction of EM wave that penetrates into the nanocylinders. We can see quite clearly in Fig. 4(c) and (d) that when the intensity of the EM wave inside the nanocylinder is larger than the intensity of the EM wave at the outer edge of the cylinders, the energy flow direction inside the nanocylinders is determined by the propagating direction of the EM wave that flows into the nanocylinder and the rotation type of the optical vortices in these cases became outward-type.
7. Summary
In conclusion, the energy flow associated with a pair of dielectric-core gold-shell nanocylinders were studied by two-dimensional finite difference time domain method with light polarized to the axis connecting the nanocylinder pair. There are four zero energy (optical singular) points in the energy flow diagram, two saddle points and two optical vortices. Among these four points, the saddle points are outside the cylinders and the optical vortices are inside the cylinders. Both the width of the gap between the nanocylinder pair and the value of the permittivity of the dielectric core are found to have strong influence on the rotating direction of the optical vortices. The gap width between the nanocylinder pair affects the energy distribution near the nanocylinders and that in turn determines the rotational type of the optical vortices. The permittivity of the dielectric core determines the magnitude of the EM field intensity inside the nanocylinders and thus can be used to control the energy flow direction inside the nanocylinders. Our studies show for the first time that optical vortices can be found in a core-shell nanocylinder pair and the rotating direction of the optical vortices associated with the nanocylinder pair can be controlled by changing either the spacing between the nanocylinder pair or by changing the permittivity of the dielectric core.
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