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Virtual Journal for Biomedical Optics

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  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 11 — Nov. 26, 2007
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Electric field enhancement between two Si microdisks

M. M. Sigalas, D. A. Fattal, R. S. Williams, S.Y. Wang, and R. G. Beausoleil  »View Author Affiliations


Optics Express, Vol. 15, Issue 22, pp. 14711-14716 (2007)
http://dx.doi.org/10.1364/OE.15.014711


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Abstract

The field enhancement in the gap between two Si microdisks is theoretically investigated using the finite difference time domain method. We show that the electric field within this gap increases as the distance between the two disks decreases, and it can be enhanced by as much as two orders of magnitude. By perturbing the Si microdisks to force the field leakage into an ever smaller volume, the field enhancement can reach a value as high as 238 with a deep sub-wavelength mode volume. This behavior is comparable to what can be observed in gap plasmons between metal nanoparticles, but is produced here in purely dielectric structures.

© 2007 Optical Society of America

1. Introduction

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.

Fig. 1. (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. [4

4. 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.

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

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.

Fig. 3. (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.

Fig. 4. 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).

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.

3. Conclusions

In conclusion, the field enhancement in the gap between two Si microdisks has been studied using the finite difference time domain method. The local field inside the gap generally increases with decreasing gap dimension, and it can be enhanced up to 92 times for 1 nm gaps. By creating a small perturbation in the shape of the Si disks in order to allow penetration of the evanescent field from one disk into the other over a smaller volume, the field enhancement can increase up to 238 times. These values are comparable to the local electric field enhancements in the gaps between metal particles under similar illumination conditions [5

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 Q factors and mode volumes can be also very similar to what is observed for gap plasmons, however the physics behind the enhancement in qualitatively different in the two cases. For gap plasmons, the electron resonance at the metal surface constitute the enabling factor for strong field confinement, and the Q factor is determined by absorption in the metal mostly. In the present case of a gap between dielectric structures, the Q is purely radiative, and the field concentration seems to necessitate the presence of confined dielectric modes coupled to the gap region. The high field enhancement found in these dielectric structures suggests that they could be used for a range of applications traditionally addressed by plasmonic structures, such as surface-enhanced Raman scattering, nanometer-scale lasers and electro-optic modulators.

References and links

1.

H. Xu, E. J. Bjerneld, M. Kall, and L. Borjesson, “Spectroscopy of single hemoglobin molecules by surface enhanced Raman scattering” Phys. Rev. Lett. 83, 4357 (1999). [CrossRef]

2.

J. Jiang, K. Bosnick, M. Maillard, and L. Brus, “Single molecule Raman spectroscopy at the junctions of large Ag nanocrystals,” J. Phys. Chem. B 107, 9964 (2003). [CrossRef]

3.

Hartschuh, E. J. Sanchez, X. S. Xie, and L. Novotny, “High resolution near field Raman microscopy of single-walled carbon nanotubes,” Phys. Rev. Lett. 90, 095503 (2003). [CrossRef] [PubMed]

4.

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture” Opt. Express 14, 1957 (2006). [CrossRef] [PubMed]

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]

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]

7.

J. Vahala, “Oprical microcavities,” Nature 424, 839 (2003). [CrossRef] [PubMed]

8.

V. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081 (2004). [CrossRef] [PubMed]

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.

OCIS Codes
(170.5660) Medical optics and biotechnology : Raman spectroscopy
(230.5750) Optical devices : Resonators

ToC Category:
Physical Optics

History
Original Manuscript: May 18, 2007
Revised Manuscript: August 16, 2007
Manuscript Accepted: August 17, 2007
Published: October 24, 2007

Virtual Issues
Vol. 2, Iss. 11 Virtual Journal for Biomedical Optics

Citation
M. M. Sigalas, D. A. Fattal, R. S. Williams, S. Y. Wang, and R. G. Beausoleil, "Electric field enhancement between two Si microdisks," Opt. Express 15, 14711-14716 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-22-14711


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References

  1. H. Xu, E. J. Bjerneld, M. Kall, and L. Borjesson, "Spectroscopy of single hemoglobin molecules by surface enhanced Raman scattering" Phys. Rev. Lett. 83, 4357 (1999). [CrossRef]
  2. J. Jiang, K. Bosnick, M. Maillard, and L. Brus, "Single molecule Raman spectroscopy at the junctions of large Ag nanocrystals," J. Phys. Chem. B 107, 9964 (2003). [CrossRef]
  3. Hartschuh, E. J. Sanchez, X. S. Xie, and L. Novotny, "High resolution near field Raman microscopy of single-walled carbon nanotubes," Phys. Rev. Lett. 90, 095503 (2003). [CrossRef] [PubMed]
  4. S. A. Maier, ``Plasmonic field enhancement and SERS in the effective mode volume picture’’ Opt. Express 14, 1957 (2006). [CrossRef] [PubMed]
  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]
  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]
  7. J. Vahala, ``Oprical microcavities,’’ Nature 424,839 (2003). [CrossRef] [PubMed]
  8. V. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, ``All-optical control of light on a silicon chip,’’ Nature 431, 1081 (2004). [CrossRef] [PubMed]
  9. M. M. Sigalas, R. S. Williams, D. A. Fattal, S.Y. Wang, R. G. Beausoleil, ``Comparison of field enhancement of scattered waves from dielectric and metallic nanoparticles,’’ to be submitted.

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