## 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 Optical Society of America

## 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 [1–6] to obtain such strong local-fields (or hot-spots) for bio-chemical sensing applications [6,7] and for boosting a wide range of nonlinear optics phenomena [8,9]. 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]. Moreover, it has been known that resonance frequencies of plasmonic nanostructures do not have a strong dependence on scale [10], 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] 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 [11–15]. 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)
. 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]. An electromagnetic planewave with intensity ${I}_{0}$ is being illuminated at a normal direction ($\theta =0$), 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) [17,18] 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 $k=(\theta ,\varphi )$. The port *k* is extended within a differential solid angle, $d\Omega (\theta ,\varphi )$, at around $k=(\theta ,\varphi )$. We have especially assigned a symbol ‘0’ to the port at a normal direction ($\theta =0$). The CMT master equation describing how *α* is evolved in time is given by [17]

*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, ${\kappa}_{0}$and ${S}_{+0}$. For the planewave with an area of

*A*, incident energy per unit time (power) to the ‘port 0’ is ${\u3008{\left|{S}_{+0}\right|}^{2}\u3009}_{T}={I}_{0}A=(1/2)c{\epsilon}_{0}{n}_{bg}{\left|{\overrightarrow{E}}_{0}\right|}^{2}A$, where ${\epsilon}_{0}$ and ${n}_{bg}$are the permittivity of vacuum and the refractive index of the background medium, respectively and ${\u3008\u3009}_{T}$ denotes a time average over one optical cycle. Only a small fraction of the power, ${\left|{S}_{+0}\right|}^{2}$, will be coupled to the resonant mode, which is described by using the coupling constant, ${\kappa}_{0}$. According to the CMT, ${\kappa}_{0}$ can be expressed in terms of $1/{\tau}_{0}$, the partial decay rate of

*α*only through the port 0. Note that although ${\kappa}_{0}$ is an infinitesimally small value, ${S}_{+0}$ and

*A*are infinitely large for an ideal planewave. Hence their product form of ${\left|{\kappa}_{0}\right|}^{2}A$ becomes a certain finite value (see Appendix A).

Under continuous excitation, ${\left|\alpha \right|}^{2}$ rapidly grows within the time proportional to ~${Q}_{tot}/{\omega}_{c}$, then reaches a certain steady-state value. This saturation energy is obtained from Eq. (1),

Combining Eqs. (2), (3), and (5), finally, we obtain the electric-field intensity enhancement factor,

*η*. Therefore, a PhC cavity with the larger Purcell factor [20] is more advantageous for the higher ${\Phi}_{I}$. However, for a resonant mode that has null vertical emission, $\eta (\theta =0)\text{\hspace{0.17em}}=0$, or an incident planewave with $\chi =\pi /2$, there is no enhancement (${\Phi}_{I}=0$) regardless of how large Purcell factor is used. It may be noted that ${Q}_{tot}$ and $\eta (\theta =0)\text{\hspace{0.17em}}$are not independent each other because $\eta (\theta =0)\text{\hspace{0.17em}}$is an implicit function of ${Q}_{tot}$ (since ${Q}_{tot}=\omega {\left|\alpha \right|}^{2}/{P}_{tot}$). We also note that most high-

*Q*PhC cavities (${Q}_{tot}$ > 100,000) have either null vertical emission or very poor directionality [13]. Therefore, we expect a certain trade-off relationship between ${Q}_{tot}$ and $\eta (\theta =0)\text{\hspace{0.17em}}$. In practical applications, the maximum ${Q}_{tot}$ may be limited by the bandwidth ($\Delta \omega $) 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 [6–9].

## 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 [7,21] operating at a wavelength of ~1 μm. For this purpose, however, ${Q}_{tot}$ 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 $\eta (\theta =0)\text{\hspace{0.17em}}$and small ${V}_{m}$.

Figure 2(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 [22,23]. 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, ${Q}_{tot}$, ${V}_{m}$, and *η*, (see Fig. 2(b)) can be obtained by using the finite-difference time-domain method (FDTD) [24]. Note that far-field emission from our PhC cavity mode is *x*-polarized at $\theta =0$ hence $\chi =0$ 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, ${V}_{m}$ can be minimized to be ~0.004 ${\lambda}^{3}$, 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 ${V}_{m}$ can be brought up to ~7 times [22] (see Appendix E). Interestingly, the condition minimizing ${V}_{m}$ maximizes ${Q}_{tot}$ so that ${Q}_{tot}/{V}_{m}$ can be maximized at the same time. Furthermore, $\eta (\theta =0)$ in the range of 0.15 ~0.18 is already larger than that from the conventional dipole-antenna radiation, $3/8\pi \cong 0.119$ (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, ${Q}_{tot}$, ${V}_{m}$, and *η*, into Eq. (6), we get considerably large ${\Phi}_{I}$ of ~783, which is already comparable to those obtained from the metallic SERS active substrates [25,26].

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) [1–4,25,26] to justify the weak-coupling assumption of CMT [18].

#### 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)
. If the downward-emitting components could be suppressed, one would double $\eta (\theta =0)$ as well as the final intensity enhancement, ${\Phi}_{I}$ (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 $\theta =0$ by the reflector (see Fig. 4a
) [16]. As a result, both far-field radiation (or $\eta (\theta =0)$) and ${Q}_{tot}$ can be modified. For example, as shown in Fig. 3(d), if *d* equals to the effective 1.0 λ distance ($d=\lambda /{n}_{bg}=\lambda /1.33$), fairly-good unidirectional beaming can be obtained [16]. In this regard, our design is reminiscent of the Yagi-Uda antenna [5] in the radio-frequency domain. What is interesting here is that one could achieve very high $\eta (\theta =0)$ ~0.67 at $d=0.95\lambda /{n}_{bg}$ (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 ${Q}_{tot}$, ${V}_{m}$, and *η* as a function of *d* with a periodicity of ~$0.5\times (\lambda /{n}_{bg})$ are clearly visible, indicating that the far-field interference is associated with the gap size [16].

Similar comparison between our CMT results and rigorous FDTD results has been performed for the PhC antenna with the bottom reflector. For $d<0.9\lambda /{n}_{bg}$, 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 ${\Phi}_{I}$ will occur at $d=0.9\lambda /{n}_{bg}$with ${\Phi}_{I}$ ~3,437. Clearly, the PhC optical antenna outperforms or at least shows similar performances compared with metallic nano-structures [1–6,8,9,25,26]. However, the exact FDTD simulation predicts higher enhancement of ~4,500 at a slightly different gap size of $d=1.0\lambda /{n}_{bg}$. 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]. This broad resonance may interfere with the narrow resonance of the PhC cavity to produce a Fano-like spectral feature [17]. The result shown in Fig. 4(c) suggests that the more rigorous CMT model incorporating all those interactions should be developed for $d>~\lambda /{n}_{bg}$.

## 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 ${\Phi}_{I}$ is achievable. By employing high-*Q* (${Q}_{tot}$> 10,000) modified dipole modes, ${\Phi}_{I}$ > 10^{5} 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]. 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 [1–3,6,19]. 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]. 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 ${\left|{\kappa}_{0}\right|}^{2}A$.

According to the known result from Fraunhofer diffraction theory, the finite extent (area *A*) of an incoming planewave inevitably involves the angular spread, $\Delta \Omega $, such that

*λ*is the wavelength in vacuum. From the well-known result of CMT [17], we obtain the relation between the coupling constant ${\kappa}_{0}$ and the partial decay rate $1/{\tau}_{0}$ such that ${\left|{\kappa}_{0}\right|}^{2}=\left(2/{\tau}_{0}\right){\mathrm{cos}}^{2}\chi $, 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, $\Delta \Omega $, and the total radiated power is given by

Combining all the above information,

## Appendix B - Definition of mode volume ${V}_{m}$.

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

## 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, *n _{slab}*, 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

*n*= 1.33. For a triangular air-hole type PhC slab, the radius of air-holes is usually chosen to be

_{bg}*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 ω

_{n}= 0.27 ~ 0.35, where ω

_{n}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 (*R _{m}*) such that (see Fig. 5(a)
)

We have investigated how *Q* and ω_{n} are evolved as we vary *R _{m}* (Fig. 5(b)). Maximizing the strength of in-plane confinement is desirable for higher $\eta (\theta =0)$. Therefore, we choose

*R*to be 0.36 a, which results in

_{m}*Q*of ~45. For the deformed dipole mode (

*R*= 0.36 a), electric-field intensity distribution (${\left|\overrightarrow{E}\right|}^{2}$) as well as its electric-field lines are shown in Fig. 5(c). We can see that it has the highest ${\left|\overrightarrow{E}\right|}^{2}$around the centre of the cavity, where electric-fields are mostly parallel along the

_{m}*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].

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

Figure 6
shows evolution of the normalized frequency (ω_{n}) 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]. 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 $x=\pm \Delta /2$),

Here ${E}_{x}$ and ${E}_{x}^{\text{'}}$ 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 ($\Delta \to 0$) 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 haveAfter introducing the nano-slot, however,

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

#### For isotropic radiation pattern over the entire $4\pi $ steradian (ideal spherical source)

Because $dP/d\Omega $ is a constant, ${P}_{tot}={\displaystyle \int \frac{dP}{d\Omega}d\Omega =}\frac{dP}{d\Omega}{\displaystyle \int d\Omega}=4\pi \frac{dP}{d\Omega}$. Therefore, we get $\eta (\theta ,\varphi )=\eta (\theta =0)=1/4\pi \cong 0.0796$

#### For classical dipole antenna radiation

$dP/d\Omega $ is given by $dP/d\Omega =(3{P}_{tot}/8\pi ){\mathrm{sin}}^{2}(\pi /2-\theta )$, where the direction of the dipole oscillation is assumed to be in the *x-y* plane. Therefore, $\eta (\theta =0)=3/8\pi \cong 0.119$.

## Appendix G - Calculation of ${\Phi}_{I}$ using FDTD

In order to estimate the electric-field intensity enhancement (${\Phi}_{I}$) through Eq. (6), the three important parameters, *Q*, ${V}_{m}$, and $\eta (\theta =0)$, should be obtained by using FDTD.

#### Cavity Q

${E}_{x}$-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 V_{m}

For sufficiently high-*Q* modes, both of the following definitions will produce the almost same values.

*Q*modes, differences between ${V}_{m}^{(1)}$ and ${V}_{m}^{(2)}$ become more pronounced because of the fast decay of the fields within one optical cycle. We have used the second expression, ${V}_{m}^{(2)}$, for our definition of the mode volume (see also Appendix B).

#### Far-field coupling factor, $\eta (\theta =0)$

As depicted in Fig. 7(a)
, we set up a small area (*S _{0}* = $(a/40)\times (a/40)$) which locates sufficiently far away from the cavity (

*L*> 3λ). Then, the power radiated through the whole surface (

_{z}*S*) and the power radiated through the small area (

_{tot}*S*) are obtained during FDTD time stepping. Using the definition of $\eta (\theta =0)$ (see Eq. (4)),

_{0}#### Rigorous FDTD simulation for ${\Phi}_{I}$

As depicted in Fig. 7(b), a continuous-wave (CW) planewave source (${\left|{\overrightarrow{E}}_{0}\right|}^{2}$) which is polarized along the *x* direction is generated and propagated onto the PhC cavity. We detect ${\left|{\overrightarrow{E}}_{loc}(t)\right|}^{2}$as a function of time at the centre of the slot until it reaches a certain steady-state value. Then, we compute the ratio between ${\u3008{\left|{\overrightarrow{E}}_{loc}(t\to \infty )\right|}^{2}\u3009}_{T}$ and ${\u3008{\left|{\overrightarrow{E}}_{0}\right|}^{2}\u3009}_{T}$ to get the enhancement factor, ${\Phi}_{I}$.

## Appendix H - Extremely large local-field enhancement > 10^{5} 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)
. The slab thickness, *T*, and the refractive index of the slab, *n _{slab}*, are the same as before. The entire device is assumed to be completely immersed in water (

*n*= 1.33). The radius of background air-holes is chosen to be

_{bg}*r*= 0.35 a and the six nearest holes are slightly pushed away from the cavity center after reducing their radii to

*r*= 0.28 a. To estimate resultant local electric-field intensity enhancement, we have used Eq. (6). Three important parameters,

_{m}*Q*, ${V}_{m}$, and $\eta (\theta =0)$, along with the resultant field enhancement values are presented in Table 1 for several representative cases with and without a bottom mirror.

It is noteworthy that $\eta (\theta =0)$ 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.

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