2. Results and discussion
Here, we study the field enhancement in the gap between two silicon microdisks. The xz cross section of the structure is shown in Fig. 1 (a)
. The refractive index of silicon used in our calculations is 3.4. The Si disks used in our calculations are 2 μm in diameter and 200 nm thick. The disks are oriented with their axes in the y
direction and separated in the x
-direction. As we discuss below, the maximum enhancement is achieved for incident light propagating along the z
direction and polarized in the x
direction, and this is the incident plane wave direction and polarization used in all of the calculations reported here. We used the finite difference time domain method with typical grid sizes of 10nm, although we used adaptive meshing with grid sizes as small as 0.5 nm for those cases where the air gap was small. The incident plane wave used here has a Gaussian time profile, and, for the computation of the field enhancement, the fields are integrated over sufficient time at the point in the center of the gap, and then are Fourier transformed and normalized to the incident fields in order to obtain the frequency response of the fields.
(a) The xz cross section of the structure. (b). The magnitude of the x-component of the electric field in the center of the gap for the structure shown in Fig. 1a
. Two Si disks of 2 μm diameter and 200 nm thickness are separated by a 20nm air gap.
Figure 1 (b)
shows the field enhancement at the center of the gap. The disks are surrounded by vacuum (n=1) and the width of the gap between the two Si disks is 20 nm. Only the magnitude of the x component of the electric field is shown since the other components are much smaller. There are certain frequencies where the field has a maximum. These frequencies coincide with the resonances of the single Si disk. In particular the lowest frequency resonance at λ = 2.437 μm appears as a single broad peak within 2% of the single disk lowest resonance, which indicates that the two lowest disk modes are only weakly coupled. That lowest order resonance has a Q factor of 17, and the value of the resonant electric field in the gap region between the disks is enhanced 43 times relative to the incident field. In general two factors contribute to the field enhancement of a resonant field. The first one is precisely the Q factor, a temporal factor which measures of how long the field can build up coherently before being scattered and/or absorbed. The second factor has to do with how “concentrated” in space the field of one resonant photon is, and is measured in terms of an effective mode volume V expressed in units of (λ/2)3
S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture” Opt. Express
14, 1957 (2006). [CrossRef] [PubMed]
]. In the case of dielectric structures where absorption is negligible, the field enhancement is on the order of (σ.Q/V)1/2
where σ is the cross section of the resonant mode for the input wave measured in units of λ2
. In the present case, assuming a cross section σ of order 1, we find that the effective mode volume should be V ~ 0.01, which roughly corresponds to the physical volume of the gap region. This is consistent with the field distribution shown in logarithmic scale in figure 2 (a)
, where the field is mostly concentrated in the gap. Even though the lowest order resonance appears at the infrared region, the results can be scaled to any other region of the spectrum by appropriately scaling the dimensions of the structure provided that the refractive index of Si remains the same.
shows the magnitude of the electric field distribution at the longest wavelength resonance (2.437μm). Figure 2 (a)
shows the xz
cross-section of the magnitude of the electric field for the structure described in the previous paragraph. As shown in this figure, a bound state is formed in each disk corresponding to its lowest resonant mode, and the maximum magnitude of the field inside each disk is about 4 times higher than that of the incident field. The modes from each disk are coupled through the small gap, where the field reaches its maximum enhancement of 43 times the incident field amplitude. As a comparison, the longest wavelength resonance for a single Si disk is at 2.387 μm and the corresponding maximum magnitude electric field enhancement at the edges of the disk is 13. The xy
cross-section of the field is shown in Fig. 2 (b)
, illustrating that the field is almost uniformly enhanced along the y
-direction of the gap.
Fig. 2 (a). The xz cross-section of the magnitude of the electric field distribution at the middle of the disks for the longest wavelength resonance (2.437 μm). A logarithmic scale is used with red color corresponding to the maximum value. The structure consists of two Si disks of 2 μm diameter and 200nm thick separated by a 20nm gap. (b). The xy cross-section of the field distribution at the middle of the structure. A logarithmic scale is used with red color corresponding to the maximum value.
Similar computations for an incident plane wave propagating along the axis of the disks (i.e., in the y
-direction) and polarized in the x
direction show that the lowest resonance appears at the same wavelength (2.437μm), but that the field enhancement between the disks is 18 times the amplitude of incident field, or almost 2.5 times smaller than the field enhancement in the case studied in Fig. 1 (b)
, where the incident k
-vector is parallel to the z
axis. This is due to the increased coupling of the incident plane wave with the bounded modes in each disk. However, it should be noted that this difference is due to the symmetry of the disk structure, and other geometries such as spheres or cylinders will provide the same enhancement for both incident propagation directions. Also, calculations for plane waves incident along the z
-direction and with electric fields polarized in the y
-direction generate an enhancement factor of only 2.7 relative to the incident field. This is an indication that, in addition to the dielectric confined states that one needs to form (in this case the individual Si disks resonances), the fields have to be normal to the vacuum-high index interface inside the gap. Therefore the discontinuity of the normal component of the electric field seems to be one mechanism [6
J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall Mode Volumes in Dielectric Optical Microcavities,” Phys. Rev. Lett.
95, 143901 (2005). [CrossRef] [PubMed]
] contributing to the concentration of the field in the region between the disks.
We confirmed the importance of the individual bound disk modes in achieving the highest enhancement by introducing small artificial absorbing regions within the disks but in the other side of the gap. In this case, the enhancement inside the gap is reduced significantly even though the absorbing region is about 1500 nm away from the gap. Therefore, it is apparent that for the generation of very high electric fields in nanometer size areas, we must have structures with bound states separated by a small gap to allow the wave from one bound state to interact with the other through the lower refractive index gap.
(a) The xz cross section of the perturbed structure. (b). The magnitude of the x-component of the electric field (normalized to the input field magnitude) in the center of the gap for the structure shown in Fig. 3a
. Two Si disks of 2 μm diameter and 200 nm thickness are separated by a 30 nm and 60 nm (green and red lines, respectively) gap and two smaller Si disks of 10nm and 40nm diameter are separated by 20nm air gap. The blue line correspond to the same structure as in Fig. 1b
The field enhancement inside the gap can increase even more by moving the disks further away and modifying them close to the air gap in order to get a coupling between the modes in each disk. Figure 3 (a)
shows the xz cross section of the structure. The 2μm diameter Si disks are separated by a gap of width d1
. Smaller disks are placed with their centers at the edges of the bigger disks. The width of the gap between the smaller disks is d2
. The green line in Fig. 3 (b)
shows the normalized magnitude of the x component of the electric field at the middle of the gap for the case where the disks are separated by d1
=30 nm and smaller Si disks with diameter of 10nm are placed with their centers at the edges of the bigger disks and separated by d2
=20nm. The longest wavelength resonance appears at almost the same wavelength as in Fig. 1b
(2.431μm) and the field enhancement at the middle of the gap is 49 times relative to the input field. Moving the 2μm disk further away creating a gap with d1
=60nm, placing smaller Si disks of 40nm diameter at the edges of the bigger Si disk and keeping the gap between the smaller Si disks at d2
=20nm [red line in Fig. 3 (b)
], the longest wavelength resonance appears at 2.427μm and gives 57 times field enhancement at the middle of the gap.
The maximum of the x
-component of the electric field normalized to the amplitude of the incoming field, measured at the center of the gap for different values of the gap width. The blue circles correspond to the case of two Si disks, each with a diameter of 2 μm and a thickness of 200 nm (see Fig. 1a
). The red crosses correspond to the case of two Si disks separated by d1
=30nm and two smaller Si disks separated by a gap of width, d2
(see Fig. 3a
A previous study has shown that the higher the refractive index of the materials, the higher the enhancement at the center of the gap [9
M. M. Sigalas, R. S. Williams, D. A. Fattal, S.Y. Wang, and R. G. Beausoleil, “Comparison of field enhancement of scattered waves from dielectric and metallic nanoparticles,” to be submitted.
]. Decreasing the volume of the gap can also increase the local enhancement of the electric field. Figure 4
shows the enhancement inside the gap as a function of the gap width. For the first structure [shown in Fig. 1 (a)
] of the 2μm diameter disks (circles in Fig. 4
), the field increases as the disks draw closer, reaching a maximum of 92 at a separation of 1 nm, and then decaying rapidly as the disk surfaces touch. The field enhancements are even higher when the overlap of the evanescent modes emerging from each disk occurs in an even smaller volume. As a test case [see Fig. 3 (a)
], we set the distance of closest approach of the two 2 μm diameter disks to be d1
=30 nm, but then centered two much smaller Si disks at the closest edges of the large disks. We varied the diameter of the smaller disks as the width of the gap, d2
, changes, and recomputed the field enhancement at the exact center of the gap. As shown by the red crosses in Fig. 4
, in this case the enhancement obtained at each value of the width is greater than that of the two large disks themselves, reaching a maximum of 238 for a gap width d2
=1 nm air gap, and decaying to 106 as the two smaller disks touch (d2
=0). These electric field enhancements inside the gap are comparable to the ones obtained in the gap between two metal nanorods where the enhancements of several hundred times have been reported [5
J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. Garcia de Abajo, “Optical properties of coupled metallic nanorods for field enhanced spectroscopy,” Phys. Rev. B
71, 235420 (2005). [CrossRef]
The field enhancement can increase further by placing small metal nanoparticles in the region between the two large disks. Replacing the small Si disks described in the previous paragraph [see Fig. 3 (a)
] with Ag disks of the same size, at a gap width d2
=5nm the enhancement at the center is 160, compared to 113 for Si nanodisks. Our computations show that there is a negligible change in either the resonance frequency or the Q factor when the Si nanoparticles are replaced by Ag. This suggests that using dielectric structures with very high Q
(due to the very low intrinsic absorption in these materials) and metal nanoparticles in the region of maximum field amplitude appears to enhance the field further without reducing the Q
-factors significantly due to absorption in the metal.
So far we have studied the lowest frequency resonance where the Q
factors are small, and the times needed for the convergence of the field computations are relatively small. Higher resonance frequencies corresponding e.g. to coupled whispering gallery modes of individual disks show much higher Q
factors, and we could expect equally higher field enhancement factors for these modes. However, this seems not to be the case. For example, in the case of the 2 μm diameter disks separated by d1
=60 nm with two smaller Si disks of 40nm diameter and d2
=20nm on the edge [shown by the red line in Fig. 3 (b)
], the resonance at 1.251μm has a Q
factor (505) that is about 30 times higher than that of the lowest frequency resonance, but shows a field enhancement (78) which is only double that of the lowest frequency resonance. This suggests that either the effective mode volume of the high Q resonance is much larger than that of the low Q resonance, or that the cross section of the high Q mode for the incident plane wave is much smaller than that of the low Q mode - or a combination of both. This issue will be clarified in a subsequent publication. Whatever the reason, this fact suggest that the low Q resonance formation with its small mode volume cannot be explained by the usual coupled mode theory even though the disks are weakly coupled.
Placing the Si disks on a SiO2 substrate reduces the field enhancement inside the air gap because the fields are not as localized within the disks when attached to a material with a refractive index higher than that of vacuum. In particular, for the 2 μm Si disks separated by a 20 nm gap and placed on a SiO2 (rather than vacuum) substrate, the longest wavelength resonance increases to 2.538 μm, and the electric field enhancement in the center of the gap is reduced by 45% to 23.4.