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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 24163–24177
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A photonic-crystal optical antenna for extremely large local-field enhancement

Hyun-Joo Chang, Se-Heon Kim, Yong-Hee Lee, Emil P. Kartalov, and Axel Scherer  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 24163-24177 (2010)
http://dx.doi.org/10.1364/OE.18.024163


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Abstract

We propose a novel design of an all-dielectric optical antenna based on photonic-band-gap confinement. Specifically, we have engineered the photonic-crystal dipole mode to have broad spectral response (Q ~70) and well-directed vertical-radiation by introducing a plane mirror below the cavity. Considerably large local electric-field intensity enhancement ~4,500 is expected from the proposed design for a normally incident planewave. Furthermore, an analytic model developed based on coupled-mode theory predicts that the electric-field intensity enhancement can easily be over 100,000 by employing reasonably high-Q (~10,000) resonators.

© 2010 OSA

1. Introduction

In analogy to the radio-frequency antenna, an optical antenna can function either as a transmitter (emitter) or a receiver in the optical frequency domain. For use as a receiver, incident electromagnetic wave’s energy shall be efficiently captured by the antenna at a certain resonance condition, resulting in strongly-enhanced local-electromagnetic-fields. Recently, various metallic nano-antennas have been proposed and demonstrated [1

1. K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, “Optical antennas: Resonators for local field enhancement,” J. Appl. Phys. 94, 4632–4642 (2003). [CrossRef]

6

6. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005). [CrossRef] [PubMed]

] to obtain such strong local-fields (or hot-spots) for bio-chemical sensing applications [6

6. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005). [CrossRef] [PubMed]

,7

7. K. Kneipp., “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997). [CrossRef]

] and for boosting a wide range of nonlinear optics phenomena [8

8. A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003). [CrossRef] [PubMed]

,9

9. S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008). [CrossRef] [PubMed]

]. However, the performance of these antennas is often limited by their low Q < ~20, which is due to either huge radiation losses or absorption losses in the metallic structure itself [4

4. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002). [CrossRef] [PubMed]

]. Moreover, it has been known that resonance frequencies of plasmonic nanostructures do not have a strong dependence on scale [10

10. E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. 120(11), 5444–5454 (2004). [CrossRef] [PubMed]

], which makes it challenging to develop an array of diverse plasmonic resonances.

In this paper, we propose to use a 2-D photonic-crystal (PhC) slab structure [11

11. O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). [CrossRef]

] as a novel platform for the efficient optical antenna with strong local field enhancement. Until now, various 2-D PhC cavities have been extensively researched for achieving ultrasmall mode-volume (V) and high-Q enabled by the photonic-band-gap (PBG) in the in-plane directions [11

11. O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). [CrossRef]

15

15. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1, 49–52 (2007). [CrossRef]

]. Evidently, such non-metallic PhC cavities can provide much higher-Q (easily over 10,000) in comparison with their metallic counterparts. Furthermore, the resonances of PhC cavities are scalable with the lattice constant, allowing lithographic means to tune the resonances and to integrate them in a dense array within a small footprint.

2. Coupled mode theory for photonic-crystal optical antennas

Our scheme for a PhC optical antenna is shown in Fig. 1(a)
Fig. 1 All-dielectric photonic-crystal (PhC) antenna. (a) The 2-D PhC cavity is placed at a distance of d from the reflector. The linearly-polarized planewave is being illuminated from the top. (b) A coupled-mode theory diagram describes interaction between the incident planewave and the PhC cavity. The port k is defined for an imaginary two-way waveguide in the direction ofk=(θ,ϕ) with an angular extent defined by a differential solid angle, dΩ(θ,ϕ). α represents the energy amplitude of the cavity. S+k and S-k are the power amplitudes propagating in the port k. The PhC antenna is excited only through the 'port 0' with carried power of | S+0|2.
. A high-permittivity semiconductor PhC slab cavity is positioned nearby a plane mirror. By varying the gap distance, d, one can control the directionality of the far-field emission, which in turn will be used to optimize the antenna coupling efficiency [16

16. S.-H. Kim, S.-K. Kim, and Y.-H. Lee, “Vertical beaming of wavelength-small photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006). [CrossRef]

]. An electromagnetic planewave with intensity I0 is being illuminated at a normal direction (θ=0), where the incident electric-field is polarized along the x-axis. To understand the underlying physics, we first consider a simplified model in the spirit of coupled-mode theory (CMT) [17

17. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]

,18

18. R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007). [CrossRef]

] as described in Fig. 1(b). The energy amplitude in the cavity denoted by α is being built up by the incident planewave while decaying into (infinitely many) multiple ‘ports’, which are imaginary two-way waveguides connecting the near-zone to the far-zone and vice versa. A symbol ‘k’ is assigned to the port lying in the direction of k=(θ,ϕ). The port k is extended within a differential solid angle, dΩ(θ,ϕ), at around k=(θ,ϕ). We have especially assigned a symbol ‘0’ to the port at a normal direction (θ=0). The CMT master equation describing how α is evolved in time is given by [17

17. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]

]
dαdt=iωcα(1τtot)α+κ0S+0.
(1)
Here ωc and 1/τtot (i.e., Qtot=ωcτtot/2) are the resonant frequency and the total decay rate of α, respectively. The total electromagnetic energy in the cavity is given by |α|2. The coupling from the planewave is described by the last two terms, κ0and S+0. For the planewave with an area of A, incident energy per unit time (power) to the ‘port 0’ is |S+0|2T=I0A=(1/2)cε0nbg|E0|2A, where ε0 and nbgare the permittivity of vacuum and the refractive index of the background medium, respectively and T denotes a time average over one optical cycle. Only a small fraction of the power, |S+0|2, will be coupled to the resonant mode, which is described by using the coupling constant, κ0. According to the CMT, κ0 can be expressed in terms of 1/τ0, the partial decay rate of α only through the port 0. Note that although κ0 is an infinitesimally small value, S+0 and A are infinitely large for an ideal planewave. Hence their product form of |κ0|2A becomes a certain finite value (see Appendix A).

Under continuous excitation, |α|2 rapidly grows within the time proportional to ~Qtot/ωc, then reaches a certain steady-state value. This saturation energy is obtained from Eq. (1),
|α|2T|t=τtot2|κ0|2|S+0|2T=12cε0nbgτtot2|E0|2|κ0|2A.
(2)
Note that the above equation contains the electric-field intensity of the incident wave, |E0|2. Another equivalent expression to the above equation can be obtained from the cavity quantum electrodynamics (cQED) definition of the mode volume (see Appendix B),
|α|2T|t=12Vmε0nc2|Eloc|2,
(3)
which now includes the maximum electric-field intensity in the cavity. To facilitate an explicit expression for |κ0|2A, we introduce a new term, namely ‘far-field coupling factor’ [19

19. C. A. Balanis, Antenna Theory: Analyses and Design (John Wiley and Sons, Inc., Hoboken, New Jersey, 2005).

],
η(θ,ϕ)dPdΩ(θ,ϕ)Ptot,
(4)
which is the ratio between the power radiated per unit solid angle in the direction of k=(θ,ϕ) and the total radiated power, Ptot. One can easily show that this dimensionless quantity varies from 0 to 1 and has the property such that η(θ,ϕ)dΩ=1. Furthermore, it can also be shown that, for a normally incident planewave, the product term, |κ0|2A, is directly proportional to η(θ=0) (see Appendix A) such that

|κ0|2A=2τtot(λnbg)2η(θ=0)cos2χ.
(5)

Combining Eqs. (2), (3), and (5), finally, we obtain the electric-field intensity enhancement factor,
ΦI|Eloc|2|E0|2=2πλ3nbgnc2QtotVmη(θ=0)cos2χ.
(6)
This result shows that ΦI is proportional to both Qtot/Vm and η. Therefore, a PhC cavity with the larger Purcell factor [20

20. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

] is more advantageous for the higher ΦI. However, for a resonant mode that has null vertical emission, η(θ=0)=0, or an incident planewave with χ=π/2, there is no enhancement (ΦI=0) regardless of how large Purcell factor is used. It may be noted that Qtot and η(θ=0)are not independent each other because η(θ=0)is an implicit function of Qtot (since Qtot=ω|α|2/Ptot). We also note that most high-Q PhC cavities (Qtot > 100,000) have either null vertical emission or very poor directionality [13

13. H.-Y. Ryu, M. Notomi, and Y.-H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83, 4294–4296 (2003). [CrossRef]

]. Therefore, we expect a certain trade-off relationship between Qtot and η(θ=0). In practical applications, the maximum Qtot may be limited by the bandwidth (Δω) requirement. For example, for nonlinear optics applications, where efficient light generation at frequencies different from that of a pumping beam is required, the bandwidth may be chosen to cover the wide enough frequency range of interests [6

6. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005). [CrossRef] [PubMed]

9

9. S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008). [CrossRef] [PubMed]

].

3. Design of a photonic-crystal optical antenna

3.1 Effects of a nano-slot

As a practical example, we have designed a compact, stand-alone, label-free Raman sensor [7

7. K. Kneipp., “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997). [CrossRef]

,21

21. A. Campion and P. Kambhampati, “Surface-enhanced Raman scattering,” Chem. Soc. Rev. 27, 241–250 (1998). [CrossRef]

] operating at a wavelength of ~1 μm. For this purpose, however, Qtot should usually be limited by ~100 in consideration of typical magnitudes of Stokes energy shifts. Designing such a low-Q PhC cavity could be challenging as it should have both moderately large η(θ=0)and small Vm.

Figure 2(a)
Fig. 2 Effects of the nano-slot on local-field enhancement. The photonic-crystal (PhC) cavity is completely immersed in water (nbg = 1.33), where the bottom reflector is excluded. (a) The left panel shows a schematic illustration of the nano-slot positioned at the center of the PhC cavity. The nano-slot has a rectangular geometry whose width is fixed to 0.05 a while its length, L, is to be varied. The right panel shows electric-field intensity (|E|2) distribution around the nano-slot. (b) Quality factor, Qtot, mode volume, Vm, and the far-field coupling factor, η(θ=0), calculated as a function of L. (c) The electric-field intensity enhancement factors, ΦI, obtained from the coupled-mode theory (CMT) are compared with those obtained from the rigorous finite-difference time-domain (FDTD) simulation. The maximum ΦI of ~750 (~783) is obtained from CMT (FDTD) at L = 0.8 a.
shows our design, where the conventional single-defect PhC cavity (also known as H1) has been modified (see Appendix C). The nano-slot structure introduced in the centre of the cavity creates highly-concentrated electromagnetic-fields inside the gap so that it can be used to optimize the overlap with an analyte molecule [22

22. J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95(14), 143901 (2005). [CrossRef] [PubMed]

,23

23. M. Loncar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82, 4648–4650 (2003). [CrossRef]

]. Hereafter, the entire PhC cavity is assumed to be immersed in a medium having a refractive index of 1.33 (that of water) for real-time sensing applications. The three key parameters, Qtot, Vm, and η, (see Fig. 2(b)) can be obtained by using the finite-difference time-domain method (FDTD) [24

24. A. Taflove, and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (2nd ed) (Artech House, 2000).

]. Note that far-field emission from our PhC cavity mode is x-polarized at θ=0 hence χ=0 in Eq. (6).

First, we consider a PhC cavity without having a bottom reflector. The length of the nano-slot, L, is varied while the width of it has been fixed to 0.05 a, which corresponds to ~20 nm for the operational wavelength (see Appendix D). When L is chosen to be 0.8 a, Vm can be minimized to be ~0.004 λ3, which shows about 2.2 fold reduction from that of the cavity without the nano-slot (L = 0). We note that, as the nano-slot width decreases further, the reduction factor for Vm can be brought up to ~7 times [22

22. J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95(14), 143901 (2005). [CrossRef] [PubMed]

] (see Appendix E). Interestingly, the condition minimizing Vm maximizes Qtot so that Qtot/Vm can be maximized at the same time. Furthermore, η(θ=0) in the range of 0.15 ~0.18 is already larger than that from the conventional dipole-antenna radiation, 3/8π0.119 (see Appendix F), indicating that the efficient inhibition of radiation in the directions parallel to the PhC slab indeed improves the free-space coupling. Inserting all the above values, Qtot, Vm, and η, into Eq. (6), we get considerably large ΦI of ~783, which is already comparable to those obtained from the metallic SERS active substrates [25

25. J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90(5), 057401 (2003). [CrossRef] [PubMed]

,26

26. H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006). [CrossRef] [PubMed]

].

To validate our analytical expression of Eq. (6), we compare our CMT predictions (Fig. 2(c)) to the rigorous FDTD simulation results that incorporate realistic incoming planewaves and subsequent time-evolution of the stored field energy (see Appendix G). As shown in Fig. 2(c), both results agree well with each other indicating that our CMT formalism is valid even for a resonance with Q < 100. It should be noted that, however, it may not be applicable to the conventional metallic antennas, because they usually support too low Q (typically, <10) [1

1. K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, “Optical antennas: Resonators for local field enhancement,” J. Appl. Phys. 94, 4632–4642 (2003). [CrossRef]

4

4. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002). [CrossRef] [PubMed]

,25

25. J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90(5), 057401 (2003). [CrossRef] [PubMed]

,26

26. H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006). [CrossRef] [PubMed]

] to justify the weak-coupling assumption of CMT [18

18. R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007). [CrossRef]

].

3.2 Effects of a bottom reflector

We now consider the effect of a bottom reflector on far-field radiation patterns. In the absence of the bottom reflector, radiation losses occur symmetrically with respect to the PhC slab as shown in Fig. 3(c)
Fig. 3 Far-field radiation patterns. Far-field patterns are calculated by using the finite-difference time-domain method (FDTD) and the near- to far-field transformation formulae presented in ref. 11. The two far-field patterns (x,y) in the upper row are for the upper hemispherical surface (0θ90), where the mapping between (x,y) and (θ,ϕ) is given byx=θcosϕ and x=θsinϕ. In the two polar plots in the lower row, the dotted line (solid line) shows the radiation pattern in the x = 0 (y = 0) plane. (a, c) The photonic-crystal cavity without the bottom reflector, where the length of the nano-slot is 0.8 a. (b, d) The same photonic-cavity cavity as before but with the bottom reflector, where the gap size is d=λ/nbg.
. If the downward-emitting components could be suppressed, one would double η(θ=0) as well as the final intensity enhancement, ΦI (see Eq. (4)). In fact, the gap distance, d, is a key parameter to control such enhancement through interferences between the originally upward- and downward-emitting waves, both of which can be met at θ=0 by the reflector (see Fig. 4a
Fig. 4 Effects of the bottom reflector on local-field enhancement. The length of the nano-slot is fixed to be 0.8 a, at whichΦI has been optimized in the absence of the bottom reflector. (a) Electric-field intensity distribution in the x-z plane. The downward-propagating waves can be redirected to the top by the reflector and they can interfere with the originally upward-propagating waves. The interference condition can be determined by the gap size, d. (b) Quality factor, Qtot, mode volume, Vm, and the far-field coupling factor, η(θ=0), calculated as a function of d. (c) The electric-field intensity enhancement factors, ΦI, obtained from the coupled-mode theory (CMT) are compared with those obtained from the rigorous finite-difference time-domain (FDTD) simulation. ΦIof ~3,437 (~4,500) is obtained from CMT (FDTD) at d=0.9λ/nbg (d=1.0λ/nbg).
) [16

16. S.-H. Kim, S.-K. Kim, and Y.-H. Lee, “Vertical beaming of wavelength-small photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006). [CrossRef]

]. As a result, both far-field radiation (or η(θ=0)) and Qtot can be modified. For example, as shown in Fig. 3(d), if d equals to the effective 1.0 λ distance (d=λ/nbg=λ/1.33), fairly-good unidirectional beaming can be obtained [16

16. S.-H. Kim, S.-K. Kim, and Y.-H. Lee, “Vertical beaming of wavelength-small photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006). [CrossRef]

]. In this regard, our design is reminiscent of the Yagi-Uda antenna [5

5. T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi-Uda antenna,” Nat. Photonics 4, 312–315 (2010). [CrossRef]

] in the radio-frequency domain. What is interesting here is that one could achieve very high η(θ=0) ~0.67 at d=0.95λ/nbg (see Fig. 4(b)), which is about 4 fold enhancement in comparison with that of the PhC cavity without a bottom reflector. Strong modulations in Qtot, Vm, and η as a function of d with a periodicity of ~0.5×(λ/nbg) are clearly visible, indicating that the far-field interference is associated with the gap size [16

16. S.-H. Kim, S.-K. Kim, and Y.-H. Lee, “Vertical beaming of wavelength-small photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006). [CrossRef]

].

Similar comparison between our CMT results and rigorous FDTD results has been performed for the PhC antenna with the bottom reflector. For d<0.9λ/nbg, the both results seem to agree well with each other. However, as d increases further, the CMT prediction begins to fall below the exact FDTD result. The CMT model predicts that the maximum of ΦI will occur at d=0.9λ/nbgwith ΦI ~3,437. Clearly, the PhC optical antenna outperforms or at least shows similar performances compared with metallic nano-structures [1

1. K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, “Optical antennas: Resonators for local field enhancement,” J. Appl. Phys. 94, 4632–4642 (2003). [CrossRef]

6

6. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005). [CrossRef] [PubMed]

,8

8. A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003). [CrossRef] [PubMed]

,9

9. S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008). [CrossRef] [PubMed]

,25

25. J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90(5), 057401 (2003). [CrossRef] [PubMed]

,26

26. H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006). [CrossRef] [PubMed]

]. However, the exact FDTD simulation predicts higher enhancement of ~4,500 at a slightly different gap size of d=1.0λ/nbg. Such discrepancy between the two predictions might be due to the fact that the gap region may act as a Fabry-Perot like cavity enclosed by the top PhC slab mirror [16

16. S.-H. Kim, S.-K. Kim, and Y.-H. Lee, “Vertical beaming of wavelength-small photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006). [CrossRef]

]. This broad resonance may interfere with the narrow resonance of the PhC cavity to produce a Fano-like spectral feature [17

17. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]

]. The result shown in Fig. 4(c) suggests that the more rigorous CMT model incorporating all those interactions should be developed for d>~λ/nbg.

4. Concluding remarks

The proposed design still leaves room for further improvement. For example, by narrowing the width of the nano-slot, additional factor of 2~3 times ΦI is achievable. By employing high-Q (Qtot> 10,000) modified dipole modes, ΦI > 105 is expected from our CMT model (see Appendix H). For enhancing nonlinear signal generation based on such high-Q antennas, one may utilize the multitude of resonances; one for the pumping wavelength and others for the nonlinear-converted wavelengths [27

27. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15(12), 7303–7318 (2007). [CrossRef] [PubMed]

]. Note that such dramatic results do not rely on technologically challenging geometries, such as extremely sharp tips and/or small gaps, which are common features in many metallic antenna designs [1

1. K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, “Optical antennas: Resonators for local field enhancement,” J. Appl. Phys. 94, 4632–4642 (2003). [CrossRef]

3

3. E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett. 89, 093120 (2006). [CrossRef]

,6

6. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005). [CrossRef] [PubMed]

,19

19. C. A. Balanis, Antenna Theory: Analyses and Design (John Wiley and Sons, Inc., Hoboken, New Jersey, 2005).

]. Furthermore, one may develop a miniaturized antenna for free-space communication. The similar design can also be applied to high-efficiency photovoltaic devices for driving nano-electronics [28

28. B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449(7164), 885–889 (2007). [CrossRef] [PubMed]

]. The proposed scheme is fully compatible with the matured silicon nanofabrication technique and could be used in a wide range of applications where strong light-and-matter interactions are essential.

Appendix A - Calculation of |κ0|2A.

According to the known result from Fraunhofer diffraction theory, the finite extent (area A) of an incoming planewave inevitably involves the angular spread, ΔΩ, such that
ΔΩ=(λ/nbg)2A
(A-1)
where λ is the wavelength in vacuum. From the well-known result of CMT [17

17. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]

], we obtain the relation between the coupling constant κ0 and the partial decay rate 1/τ0 such that |κ0|2=(2/τ0)cos2χ, where χ is the angle between the incident wave’s polarization and the outgoing radiation’s polarization. The ratio between the power radiated into the solid angle, ΔΩ, and the total radiated power is given by

ΔP(θ0)Ptot=1/τ01/τtot.
(A-2)

Combining all the above information,
|κ0|2A=2τ0(λnbg)21ΔΩcos2χ=2τtot(λnbg)2ΔP(θ0)ΔΩ1Ptotcos2χ
(A-3)
In the limit of infinitely large area, as both ΔP and ΔΩ become infinitesimally small, the above formula equals to Eq. (5).

Appendix B - Definition of mode volume Vm.

The cQED mode volume is defined as the ratio of the total electromagnetic energy stored in the cavity and the peak electromagnetic energy density, such that
Vm=12ε0ε(r)|E(r,t)|2+12μ0|H(r,t)|2d3rmax{12ε0ε(r)|E(r,t)|2+12μ0|H(r,t)|2},
(B-1)
where we have assumed that the cavity is consisting of non-magnetic materials (μ(r)=1). Taking time averages on both the denominator and the numerator and using the fact, for harmonically oscillating fields, (1/2)ε0ε|E|2T=(1/2)μ0|H|2T, one can rewrite the above expression as
Vm=|α|2Tmax{ε0ε(r)|E(r,t)|2}T=|α|2T12ε0nc2|Eloc|2
(B-2)
where nc2and Elocare the refractive index and the electric-field at the location where the term braced is maximized.

Appendix C - Design of the dipole-mode cavity with Q < 100

A single-defect PhC cavity is formed in a 2-D triangular-lattice PhC slab. The slab thickness, T, and the refractive index of the slab, nslab, are chosen to be 0.6 a and 3.5 (that of Si at near-infrared wavelengths), respectively, where ‘a’ denotes the lattice constant of the PhC. For potential real-time bio-sensing applications, the refractive index of a background material is assumed as water, so nbg = 1.33. For a triangular air-hole type PhC slab, the radius of air-holes is usually chosen to be R ~ 0.35 a, which allows mechanical robustness as well as wide-enough PBG in the in-plane directions. Throughout the paper, we fix R to be 0.36 a, which gives the in-plane PBG from ωn = 0.27 ~ 0.35, where ωn is the normalized frequency, a/λ.

However, without introducing any structural modification, the cavity supports doubly-degenerate dipole-modes (the two dipole-modes are at the exactly same frequency) due to the perfect six-fold symmetry. In order to break the inherent degeneracy and to lower the Q a bit more, we have changed positions of the four-holes as a function of their radii (Rm) such that (see Fig. 5(a)
Fig. 5 (a) Structure of the low-Q dipole mode cavity. (b) Normalized frequency and Quality factor as a function of the modified hole radius, Rm. (c) Vector plot of the electric-fields. (d) Far-field radiation pattern from the mode shown in Fig. 5(c).
)

Modified hole1: (12,32)a(1+RRm2,0.932)a
Modified hole2: (12,32)a(1+RRm2,0.932)a
Modified hole3: (12,32)a(1+RRm2,0.932)a
Modified hole4: (12,32)a(1+RRm2,0.932)a

We have investigated how Q and ωn are evolved as we vary Rm (Fig. 5(b)). Maximizing the strength of in-plane confinement is desirable for higher η(θ=0). Therefore, we choose Rm to be 0.36 a, which results in Q of ~45. For the deformed dipole mode (Rm = 0.36 a), electric-field intensity distribution (|E|2) as well as its electric-field lines are shown in Fig. 5(c). We can see that it has the highest |E|2around the centre of the cavity, where electric-fields are mostly parallel along the x direction. This is the main reason why the dipole-mode has strong vertical radiation with the well-defined linear polarization (x-pol) (see Fig. 5(d)). This fact also provides us an additional hint for further optimization of the mode volume, which will be explained in Appendix E. It is a known fact that the other degenerate-split dipole pair is orthogonally polarized (y-pol) to this dipole-mode [11

11. O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). [CrossRef]

].

Appendix D - Tuning of the PhC cavity by varying the length of a nano-slot

Figure 6
Fig. 6 Resonant wavelength (nm) and normalized frequency (a/λ) as a function of the slot length (a), where a = 350 nm was assumed.
shows evolution of the normalized frequency (ωn) and the resonant wavelength (in case of a = 350 nm) as a function of the slot length (L). In this paper, the width of the nano-slot is fixed to 0.05 a, which corresponds to 17.5 nm. When L is varied between 0.6 a and 1.0 a, the resonant wavelength is quite stable at ~ 1,095 nm. Slight tuning the lattice constant by about 2% will bring down the resonant wavelength to 1,064 nm, which is suited for Nd:YAG laser pumping.

Appendix E - Introduction of a nano-slot into a PhC cavity

As shown in Fig. 2(a), we introduce the nano-slot in the region where electric-fields are aligned along the x direction. If the width of it is sufficiently small not to perturb the original field distribution too much, then we can safely assume that the electric-fields are still aligned in the same direction. However, the amplitude of the electric-field within the slot can be changed noticeably due to the boundary condition for the normal components of the electric-fields. The similar idea was proposed by J. T. Robinson, et al. and applied to 1-D nanobeam cavities [22

22. J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95(14), 143901 (2005). [CrossRef] [PubMed]

]. Here, we briefly discuss how the nano-slot can reduce the mode volume.

First let us recall the boundary condition for normal components of the electric-field on either side of the boundary surface (at x=±Δ/2),
nslab2Ex(x=Δ/2)nbg2Ex'(x=Δ/2).
(E-1)
Here Ex and Ex' are the electric-fields before and after introducing the nano-slot at the centre of the cavity, respectively. We also assume that the width of the slot is so small (Δ0) that it could not change the total electromagnetic energy contained in the cavity. From the definition of the cQED mode volume, before introducing the nano-slot, we have

Vm=|α|2T12ε0nslab2|Ex(x=0)|2.
(E-2)

After introducing the nano-slot, however,
Vm'=|α|2T12ε0nbg2|Ex'(x=0)|2(nbgnslab)2Vm.
(E-3)
Therefore, for an ideal nano-slot (Δ0), the maximum reduction in the mode volume could be up to (nslab/nbg)2=(3.5/1.33)26.9.

Appendix F - Examples of the far-field coupling factor, η

For isotropic radiation pattern over the entire 4π steradian (ideal spherical source)

Because dP/dΩ is a constant, Ptot=dPdΩdΩ=dPdΩdΩ=4πdPdΩ. Therefore, we get η(θ,ϕ)=η(θ=0)=1/4π0.0796

For classical dipole antenna radiation

dP/dΩ is given by dP/dΩ=(3Ptot/8π)sin2(π/2θ), where the direction of the dipole oscillation is assumed to be in the x-y plane. Therefore, η(θ=0)=3/8π0.119.

Appendix G - Calculation of ΦI using FDTD

In order to estimate the electric-field intensity enhancement (ΦI) through Eq. (6), the three important parameters, Q, Vm, and η(θ=0), should be obtained by using FDTD.

Cavity Q

Ex-field component at the center of the cavity has been detected as a function of time. Then the harmonic inversion algorithm (harminv) [29] is used to obtain Q.

Mode volume Vm

For sufficiently high-Q modes, both of the following definitions will produce the almost same values.
Vm(1)=ε0ε(r)|E(r,tm)|2d3rmax{ε0ε(r)|E(r,tm)|2},
(G-1)
Vm(2)=ε0ε(r)|E(r,t)|2d3rTmax{ε0ε(r)|E(r,t)|2}T
(G-2)
Here, in the first definition, tm is the time when harmonically varying electric-fields are maximized. In the second definition, time averages are performed over one optical period. However, for low-Q modes, differences between Vm(1) and Vm(2) become more pronounced because of the fast decay of the fields within one optical cycle. We have used the second expression, Vm(2), for our definition of the mode volume (see also Appendix B).

Far-field coupling factor, η(θ=0)

As depicted in Fig. 7(a)
Fig. 7 (a) FDTD simulations used to obtain the far-field coupling factor, η. (b) The rigorous FDTD simulation used to directly obtain ΦI.
, we set up a small area (S0 = (a/40)×(a/40)) which locates sufficiently far away from the cavity (Lz > 3λ). Then, the power radiated through the whole surface (Stot) and the power radiated through the small area (S0) are obtained during FDTD time stepping. Using the definition of η(θ=0) (see Eq. (4)),

η(θ=0)=dPdΩ|θ=0Ptot=(Power radiated through S0S0/Lz2)×1Power radiated through Stot
(G-3)

Rigorous FDTD simulation for ΦI

As depicted in Fig. 7(b), a continuous-wave (CW) planewave source (|E0|2) which is polarized along the x direction is generated and propagated onto the PhC cavity. We detect |Eloc(t)|2as a function of time at the centre of the slot until it reaches a certain steady-state value. Then, we compute the ratio between |Eloc(t)|2T and |E0|2T to get the enhancement factor, ΦI.

Appendix H - Extremely large local-field enhancement > 105 from the modified dipole mode

We have designed a high-Q dipole mode by modifying the six nearest neighbor holes as depicted in Fig. 8(a)
Fig. 8 (a) Structure of the high-Q PhC dipole mode cavity. (b,c) Electric-field intensity (|E|2) distributions (b) in the x-y plane and (c) in the x-z plane.
. The slab thickness, T, and the refractive index of the slab, nslab, are the same as before. The entire device is assumed to be completely immersed in water (nbg = 1.33). The radius of background air-holes is chosen to be r = 0.35 a and the six nearest holes are slightly pushed away from the cavity center after reducing their radii to rm = 0.28 a. To estimate resultant local electric-field intensity enhancement, we have used Eq. (6). Three important parameters, Q, Vm, and η(θ=0), along with the resultant field enhancement values are presented in Table 1

Table 1. Electric-field intensity enhancement from the high-Q dipole mode.

table-icon
View This Table
for several representative cases with and without a bottom mirror.

It is noteworthy that η(θ=0) has been compromised with the high-Q, to a relatively low value of ~0.3. However, owing to their high Q/V values, one can achieve extremely high electric-field intensity enhancements over 100,000. Highly localized electric-field intensity distributions within the nano-slot are shown in Fig. 8(b) and Fig. 8(c).

Acknowledgments

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-357-C00042), the Defense Advanced Research Projects Agency through W911NF-07-1-0277, the Boeing through CT-BA-GTA-1, and the National Institutes of Health through 4R00EB007151-03.

References and links

1.

K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, “Optical antennas: Resonators for local field enhancement,” J. Appl. Phys. 94, 4632–4642 (2003). [CrossRef]

2.

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308(5728), 1607–1609 (2005). [CrossRef] [PubMed]

3.

E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett. 89, 093120 (2006). [CrossRef]

4.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002). [CrossRef] [PubMed]

5.

T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi-Uda antenna,” Nat. Photonics 4, 312–315 (2010). [CrossRef]

6.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005). [CrossRef] [PubMed]

7.

K. Kneipp., “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997). [CrossRef]

8.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003). [CrossRef] [PubMed]

9.

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008). [CrossRef] [PubMed]

10.

E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. 120(11), 5444–5454 (2004). [CrossRef] [PubMed]

11.

O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). [CrossRef]

12.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]

13.

H.-Y. Ryu, M. Notomi, and Y.-H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83, 4294–4296 (2003). [CrossRef]

14.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nature photon. 1, 449–458 (2007).

15.

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1, 49–52 (2007). [CrossRef]

16.

S.-H. Kim, S.-K. Kim, and Y.-H. Lee, “Vertical beaming of wavelength-small photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006). [CrossRef]

17.

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]

18.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007). [CrossRef]

19.

C. A. Balanis, Antenna Theory: Analyses and Design (John Wiley and Sons, Inc., Hoboken, New Jersey, 2005).

20.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

21.

A. Campion and P. Kambhampati, “Surface-enhanced Raman scattering,” Chem. Soc. Rev. 27, 241–250 (1998). [CrossRef]

22.

J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95(14), 143901 (2005). [CrossRef] [PubMed]

23.

M. Loncar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82, 4648–4650 (2003). [CrossRef]

24.

A. Taflove, and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (2nd ed) (Artech House, 2000).

25.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90(5), 057401 (2003). [CrossRef] [PubMed]

26.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006). [CrossRef] [PubMed]

27.

A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15(12), 7303–7318 (2007). [CrossRef] [PubMed]

28.

B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449(7164), 885–889 (2007). [CrossRef] [PubMed]

29.

S. G. Johnson, http://ab-initio.mit.edu/wiki/index.php/Harminv

OCIS Codes
(190.4360) Nonlinear optics : Nonlinear optics, devices
(230.5750) Optical devices : Resonators
(160.5293) Materials : Photonic bandgap materials
(230.5298) Optical devices : Photonic crystals
(290.5825) Scattering : Scattering theory

ToC Category:
Photonic Crystals

History
Original Manuscript: September 13, 2010
Revised Manuscript: October 16, 2010
Manuscript Accepted: October 21, 2010
Published: November 3, 2010

Citation
Hyun-Joo Chang, Se-Heon Kim, Yong-Hee Lee, Emil P. Kartalov, and Axel Scherer, "A photonic-crystal optical antenna for extremely large local-field enhancement," Opt. Express 18, 24163-24177 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24163


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References

  1. K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, “Optical antennas: Resonators for local field enhancement,” J. Appl. Phys. 94, 4632–4642 (2003). [CrossRef]
  2. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308(5728), 1607–1609 (2005). [CrossRef] [PubMed]
  3. E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett. 89, 093120 (2006). [CrossRef]
  4. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002). [CrossRef] [PubMed]
  5. T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi-Uda antenna,” Nat. Photonics 4, 312–315 (2010). [CrossRef]
  6. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005). [CrossRef] [PubMed]
  7. K. Kneipp and ., “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997). [CrossRef]
  8. A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003). [CrossRef] [PubMed]
  9. S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008). [CrossRef] [PubMed]
  10. E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. 120(11), 5444–5454 (2004). [CrossRef] [PubMed]
  11. O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). [CrossRef]
  12. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]
  13. H.-Y. Ryu, M. Notomi, and Y.-H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83, 4294–4296 (2003). [CrossRef]
  14. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nature photon. 1, 449–458 (2007).
  15. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1, 49–52 (2007). [CrossRef]
  16. S.-H. Kim, S.-K. Kim, and Y.-H. Lee, “Vertical beaming of wavelength-small photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006). [CrossRef]
  17. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]
  18. R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007). [CrossRef]
  19. C. A. Balanis, Antenna Theory: Analyses and Design (John Wiley and Sons, Inc., Hoboken, New Jersey, 2005).
  20. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  21. A. Campion and P. Kambhampati, “Surface-enhanced Raman scattering,” Chem. Soc. Rev. 27, 241–250 (1998). [CrossRef]
  22. J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95(14), 143901 (2005). [CrossRef] [PubMed]
  23. M. Loncar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82, 4648–4650 (2003). [CrossRef]
  24. A. Taflove, and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (2nd ed) (Artech House, 2000).
  25. J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90(5), 057401 (2003). [CrossRef] [PubMed]
  26. H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006). [CrossRef] [PubMed]
  27. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15(12), 7303–7318 (2007). [CrossRef] [PubMed]
  28. B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449(7164), 885–889 (2007). [CrossRef] [PubMed]
  29. S. G. Johnson, http://ab-initio.mit.edu/wiki/index.php/Harminv

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