A photonic-crystal optical antenna for extremely large local-field enhancement.

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.

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 is being illuminated at a normal direction ( ), 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 . The port k is extended within a differential solid angle, , at around . We have especially assigned a symbol ‘0’ to the port at a normal direction ( ). The CMT master equation describing how α is evolved in time is given by [17] Here and (i.e., ) 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, and . For the planewave with an area of A, incident energy per unit time (power) to the ‘port 0’ is , where and are the permittivity of vacuum and the refractive index of the background medium, respectively and denotes a time average over one optical cycle. Only a small fraction of the power, , will be coupled to the resonant mode, which is described by using the coupling constant, . According to the CMT, can be expressed in terms of , the partial decay rate of α only through the port 0. Note that although is an infinitesimally small value, and A are infinitely large for an ideal planewave. Hence their product form of becomes a certain finite value (see Appendix 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 of with an angular extent defined by a differential solid angle, . α represents the energy amplitude of the cavity. S and S are the power amplitudes propagating in the port k. The PhC antenna is excited only through the 'port 0' with carried power of | S|2.

Under continuous excitation, rapidly grows within the time proportional to ~ , then reaches a certain steady-state value. This saturation energy is obtained from Eq. (1), Note that the above equation contains the electric-field intensity of the incident wave, . Another equivalent expression to the above equation can be obtained from the cavity quantum electrodynamics (cQED) definition of the mode volume (see Appendix B), which now includes the maximum electric-field intensity in the cavity. To facilitate an explicit expression for , we introduce a new term, namely ‘far-field coupling factor’ [19], which is the ratio between the power radiated per unit solid angle in the direction of and the total radiated power, . One can easily show that this dimensionless quantity varies from 0 to 1 and has the property such that . Furthermore, it can also be shown that, for a normally incident planewave, the product term, , is directly proportional to (see Appendix A) such that

Combining Eqs. (2), (3), and (5), finally, we obtain the electric-field intensity enhancement factor, This result shows that is proportional to both and η. Therefore, a PhC cavity with the larger Purcell factor [20] is more advantageous for the higher . However, for a resonant mode that has null vertical emission, , or an incident planewave with , there is no enhancement ( ) regardless of how large Purcell factor is used. It may be noted that and are not independent each other because is an implicit function of (since ). We also note that most high-Q PhC cavities ( > 100,000) have either null vertical emission or very poor directionality [13]. Therefore, we expect a certain trade-off relationship between and . In practical applications, the maximum may be limited by the bandwidth ( ) 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, 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 and small .

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, , , 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 hence in Eq. (6).

Fig. 2

Effects of the nano-slot on local-field enhancement. The photonic-crystal (PhC) cavity is completely immersed in water ( = 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 ( ) distribution around the nano-slot. (b) Quality factor, Q, mode volume, V, and the far-field coupling factor, , calculated as a function of L. (c) The electric-field intensity enhancement factors, , obtained from the coupled-mode theory (CMT) are compared with those obtained from the rigorous finite-difference time-domain (FDTD) simulation. The maximum of ~750 (~783) is obtained from CMT (FDTD) at L = 0.8 a.

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, can be minimized to be ~0.004 , 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 can be brought up to ~7 times [22] (see Appendix E). Interestingly, the condition minimizing maximizes so that can be maximized at the same time. Furthermore, in the range of 0.15 ~0.18 is already larger than that from the conventional dipole-antenna radiation, (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, , , and η, into Eq. (6), we get considerably large 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 as well as the final intensity enhancement, (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 by the reflector (see Fig. 4a ) [16]. As a result, both far-field radiation (or ) and can be modified. For example, as shown in Fig. 3(d), if d equals to the effective 1.0 λ distance ( ), 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 ~0.67 at (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 , , and η as a function of d with a periodicity of ~ are clearly visible, indicating that the far-field interference is associated with the gap size [16].

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 ( ), where the mapping between (x,y) and (θ,ϕ) is given by and . 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 .

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 which 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, Q, mode volume, V, and the far-field coupling factor, , calculated as a function of d. (c) The electric-field intensity enhancement factors, , obtained from the coupled-mode theory (CMT) are compared with those obtained from the rigorous finite-difference time-domain (FDTD) simulation. of ~3,437 (~4,500) is obtained from CMT (FDTD) at ( ).

Similar comparison between our CMT results and rigorous FDTD results has been performed for the PhC antenna with the bottom reflector. For , 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 will occur at with ~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 . 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 .

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 is achievable. By employing high-Q ( > 10,000) modified dipole modes, > 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]. 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.

Table 1

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

	Without a mirror	Effective gap 0.95 λ	Effective gap 1.00 λ	Effective gap 1.05 λ
Qtot	5812	5464	5261	5256
Vm (λ3)	0.00315	0.00316	0.00315	0.00315
η(θ = 0)	0.1956	0.2267	0.2989	0.2850
| E loc|2/| E 0|2	97,682	106,246	134,948	128,632