1. Introduction

Harnessing of light, which means controlling photons in frequency-time and reciprocal-real space domains, is central to modern Photonics or Micro-Nano-Photonics. In practice, it is aimed at confining photons within tiny space (as compared to the wavelength) and for long time (as compared to the period), while achieving efficient light addressing (to), or collection (from) the wave-length scale photonic structures where they are meant to be confined.

The general approach to achieve strong confinement of photons consists in structuring high index materials at the wavelength scale, which is in the sub-micron range for the optical domain. Materials commonly used for that purpose are structured metals or high optical index dielectrics, immersed in low index dielectric material such as e.g. silica or air. Metals have recently attracted considerable attention in the framework of plasmonics (see for example [1,2]): very strong spatial confinement of photons can be attained in plasmonic nano-structures, such as in metallic nano-particules or nano-antennas (NA) [3–5], at the expanse however of a limited resonance strength (or confinement time) as a result of optical losses induced by metal absorption and of radiation to free space continuum. High index dielectric materials, such as semiconductors, have been widely used for photon confinement, along the so called refraction and Photonic Crystal (PC) based diffraction schemes: for the refraction scheme, use is made of total or partial internal reflection of photons at the semiconductor-low index material interface [6–8]; for the diffraction scheme, diffractive phenomena occurring in periodically structured materials are exploited to control the spatial-temporal trajectory of photons [9,10]. Two-dimensional (2D) planar photonic integration, based on both refraction and PC systems has been the scene of numerous outstanding achievements during the last 10 years: very strong temporal and spatial light localization has been achieved in confining micro-cavities [11–13], together with very efficient control over the collection and addressing of these resonant structures via tiny micro-guides [12]. 2D periodic metallic structures as well as 2D dielectric PC structures have also been proposed for efficient optical beam shaping in free space [14–16]. Coupling engineering is in the heart of all recent achievements of Micro-nano-Photonics: for example, the efficient optical beaming in free space reported in the latter references was obtained by appropriate coupling schemes with evanescent waves.

In the present contribution we address the issue of optical coupling to metallic micro-nano-structures or NA in free space. The usual way consists in focusing a Gaussian optical beam onto the micro-nano-structure [3,17,18]. However, this procedure is not very efficient to address metallic micro-nano-structures or NAs. The reason lies in the difficulty to achieve a sufficient coupling rate between the incoming optical beam and the NA in order to compensate for the rather large optical losses (radiation to free space and metallic absorption). A new approach is proposed in the present work for the optimum addressing of NA with a free space optical beam via the use of an intermediate coupling resonator structure, which is aimed at providing the appropriate modal conversion of the incoming beam, in the time domain. The basic idea consists in relaxing the stringency of the coupling conditions by increasing the optical energy density achieved in the vicinity of the NA, owing to the photon storage capabilities of the intermediate resonator. More specifically, the intermediate resonator is a PC membrane resonant structure based on surface addressable slow Bloch modes. Those structures are currently recognized as very versatile photonic structures in a sense that there are not only restricted to operate in the in plane wave-guiding regime, but can be opened to the third space dimension by controlling the coupling between wave-guided and radiation modes (see the review paper [19] and references there in). In brief, surface addressable PC membrane structures can be considered as ideal light harnessers. They have the capability to provide the optimum conversion of the incoming free space optical beam into a wave-guided slow Bloch mode with spatial and temporal confinement characteristics accurately designed to achieve the finest coupling conditions to metallic NA.

This paper reports on the theoretical demonstration of efficient addressing of high Purcell factor metallic NA, using a large quality factor PC membrane resonator (e.g. in silicon). In the next section a phenomenological description of the coupling scheme is given using the coupled mode formalism [20,21], offering a simple physical baseline and understanding of the coupling processes, although it is exclusively restricted to the temporal - frequency domain analysis. Section 3 is devoted to the simulation of the devices using FDTD analysis, which provides, in addition to the temporal-spectral characteristics, an accurate description of the spatial properties of the electromagnetic field (spatial confinement, radiation diagram).

2. Phenomenological description based on the coupled mode theory

The formalism of coupled mode theory can provide a faithful description of the global temporal-spectral characteristics of the optical power transfers and storages within a photonic system, on the basis of a few key kinetics parameters governing the rates of optical power exchanges.

2.1 General basic equations

Let us first consider the simple case of a NA directly addressed with an incoming optical beam, as shown schematically in Fig. 1 . $S_{+1},S_{-1},S_{-2}$ are proportional to the amplitudes of, respectively, the incoming, reflected and transmitted fields. a is proportional to the amplitude of the field in the NA. ${\left|S_{+1}\right|}^2,{\left|S_{-1}\right|}^2,{\left|S_{-2}\right|}^2$ stand for the incoming, reflected and transmitted optical powers respectively and ${\left|a\right|}^2$ represents the optical energy stored in the NA. The kinetics of the system is described by the coupled mode equations:

 

Fig. 1 Schematic representation of direct coupling to the NA (symbols are defined in the text).

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From above equations it is possible to derive, for a given input optical power ${\left|S_{+1}\right|}^2,$ the spectral characteristics of the transmitted / reflected optical powers and of the optical energy ${\left|a\right|}^2$ stored in the NA.

We consider now the case where the proposed coupling scheme, via the use of an intermediate resonator, e.g. a PC membrane, is applied, as illustrated in Fig. 2 .

 

Fig. 2 Schematic representation of coupling to NA through an intermediate resonator (symbols are defined in the text).

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The kinetics of the system is now described by the following coupled mode equations:

b is proportional to the amplitude of the field in the intermediate resonator, ${\left|b\right|}^2$ being the stored optical energy. ${\omega }_{0a}$ and ${\omega }_{0b}$ are respectively the resonant pulsations of the NA and of the intermediate resonator. A variety of situations can be accounted for by the equations above, depending upon the relative magnitudes of the kinetic parameters associated to the various coupling rates, and upon the values of the resonant pulsations of the two resonators. In the following, for sake of simplifying the presentation, the analysis will be restricted to the case of equal resonant pulsations, ${\omega }_{0a}={\omega }_{0b}={\omega }_0.$ Also, the photonic system will be analysed under steady state harmonic conditions.

2.2 Efficient addressing of a metallic NA with free space optical beam

Localized surface plasmon modes, associated with metallic nano-particles, have the capability to provide very strong spatial photon confinement at the expanse, however, of moderate temporal confinement strength, as a result of radiation and metallic absorption induced optical losses. Addressing metallic nano-structures, which means feeding electromagnetic energy into them, is therefore a challenging issue: one has to find ways to funnel photons into the nano-structure at high rate to compensate for the rather large dissipation rate $\frac{1}{{\tau }_0}$ of the structures into the radiation continuum and into the metal through absorption phenomena. This is made further difficult by the small nano-metric size of the structure. The usual way of addressing metallic nano-structures consists in focusing a light beam onto its close environment. Coupled mode theory can be applied for a first order description of this simple coupling process, on the basis of the set of Eqs. (1). If $S_{+1}$ is the incoming light field (assumed to be carried by a single optical mode) meant to be funneled into the nano-structure, the energy stored can easily be expressed as:

From coupled mode theory (set of Eqs. (2)), the energy stored can easily be expressed as:

If ${\tau }_c\ge \ge {\tau }_0,$ which is easily achievable with classical dielectric resonators, given the rather small value of ${\tau }_0,$ as discussed above, the relation $\tau =\sqrt{{\tau }_0{\tau }_c}$ to be met for optimum energy insertion in the nano-structure is therefore not as stringent as in the case where light is directly funnelled from free space. Temporal confinement of photons provided by the intermediate resonator is used to increase the optical energy density in the vicinity of the NA and, consequently, to ease the energy transfer from the incoming beam into the latter. For example, the typical spectral width of the resonance of a metallic NA is about 100nm (for a resonant wave-length in the µm range): this corresponds to a quality factor (proportional to ${\tau }_0$) of around 10; if the quality factor of the intermediate resonator (proportional to ${\tau }_c$) is in the 10³-10⁴ range, which is readily achievable with a variety of dielectric resonators, the requested coupling rate $\frac{1}{\tau }$ to achieve the maximum storable energy ${\left|a\right|}_{\max }^2$ in the NA is reduced by more than one order of magnitude.

It should be noted that the relevant factor of merit to assess the capability of the intermediate resonator to provide high optical energy density is its Purcell factor: this factor gives a quantitative evaluation of the combined temporal and spatial confinement properties of the resonator. In the present work, the spatial confinement strength of the intermediate resonator should be kept within reasonable limits in order to remain easily addressable with an incoming beam of a reasonable lateral size, eg the width (around 10µm) of a Gaussian beam provided by a standard optical fiber.

PC membrane resonators can ideally combine the requested temporal-spatial confinement characteristics discussed above for the optimal addressing of metallic NA. This is confirmed by the results of FDTD simulations presented in the next section.

3. Addressing of a metallic NA with an intermediate PC membrane resonator: FDTD analysis

A PC membrane can be used as a wavelength selective transmitter / reflector: when light is shined on this photonic structure, in an out-of-plane (normal or oblique) direction, resonances in the reflectivity spectrum can be observed. These resonances, so called Fano resonances, arise from the resonant coupling of external radiation to the guided modes in the structures, whenever there is wavelength and in-plane k vector component matching. Accurate tailoring of the spectral characteristics of the Fano resonances (shape, spectral width) is made possible by the design of the PC membrane (PC type, strength of the periodic corrugation, symmetry of the wave-guided mode, membrane thickness… [22,23]). If the lateral size of the illuminated membrane is infinite, the spectral width of the resonance is like the inverse of its lifetime ${\tau }_c,$ that is the lifetime of the wave-guided mode, where ${\tau }_c$ is simply the coupling time constant between wave-guided and radiated plane-wave modes.

In real devices, the lateral size of the illuminated area is limited, and the resonant coupling efficiency of incoming photons to the guided modes is also controlled by the lateral escape rate of the wave-guided mode out of this area. The lateral kinetics of photon within the membrane and below the beam must be slowed down to fully preserve constructive interferences of the incoming light with the wave-guided photons. The ability of high index contrast PC to slow down photons via the excitation of slow Bloch modes, and to confine them laterally, especially at the high symmetry points (or extremes) of the dispersion characteristics (see [22] for example), allows for a very good control over the lateral escape rate and lends itself to the production of devices with compact lateral size. Detailed accounts on PC membrane reflectors/resonators can be found in [22,23].

We consider here a 1D PC membrane which consists in a periodic array of low index material slits (silica) formed in a high index silicon membrane. The structure, which is designed in order to accommodate a slow Bloch mode Fano resonance [24], with a large quality factor, addressable in the vertical direction, is shown in Fig. 3 , where its topological parameters (membrane thickness, period and silica filling factor of the slit array) are given. The transmission spectrum obtained by 2D FDTD simulation is also reported and reveals a resonance near 1.506 µm. The lateral size of the incoming excitation beam is 12µm and its polarization is TM (electric field perpendicular to the slits: blue arrow in Fig. 3).

 

Fig. 3 Schematic view of the 1D PC membrane reflector used as an intermediate coupling resonator. (The topological parameters of the structure are given in the figure). The field intensity distribution at resonance and the transmission spectrum are also shown.

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The strength of the resonance is given by the spectrum bandwidth $\delta \lambda$ which is related to its quality factor $Q_{PC}=\frac{{\lambda }_0}{\delta \lambda }={\omega }_0{\tau }_c,$ where ${\lambda }_0$ is the Fano resonance wavelength. $Q_{PC}$ is around 4800, which corresponds to a rather strong resonance. The reflectivity of the structure is close to 100% at the resonance wavelength: this indicates that the above discussed optimum coupling, or impedance matching, conditions between the incoming optical beam and the wave-guided slow Bloch mode are achieved (in-plane k-vector and lateral size matching). The field intensity distribution of the Fano resonance is shown in Fig. 3. This paper is meant to provide a proof of concept of the general proposed coupling principles. Consequently, a fully illustrative simulation of the optical properties can be obtained assuming that the size of the structure along the slit direction is infinite, that is to say by performing a simplified, in terms of time consuming, 2D FDTD simulation of the structure. We will, therefore, restrict the rest of this paper to 2D FDTD simulations, while retaining the generality of the derived conclusions.

We consider now the coupling scheme to the NA based on the use of the intermediate PC membrane resonator, as shown in Fig. 4 . The NA is a gold dipole which has been designed to match the wavelength of the PC resonator with the following geometrical parameters: 50nm thickness, 200nm arm length and 40nm feed gap width. The gold dielectric function was obtained by fitting on the optical bulk experimental data reported in [24], for the frequency of interest. Other metals (eg silver) could have been chosen as well to illustrate the proposed coupling scheme. The PC membrane is addressed with a Gaussian optical beam (diameter 12µm) from the bottom and the NA is located above the membrane at a variable distance, which results in a variable coupling time constant τ between the NA and the PC membrane resonator: the topological parameters of the structure are given in the figure. An example of transmission spectrum of the system is also shown (NA put directly on top of the PC membrane): the narrow Fano resonance response shows up within the much wider background of the NA spectral response, which cannot be made visible in the narrow wavelength range of the spectrum. This is a clear indication that indeed, as announced in the previous section, the PC membrane resonance is much stronger than the one of the NA (${\tau }_c\ge \ge {\tau }_0$). The transmission ratio is also significantly increased (as compared to the plain PC membrane) as expected and can be derived from the coupled mode theory (see annex, section 1).

 

Fig. 4 Schematic view of the 1D PC membrane reflector used as an intermediate coupling resonator. (Topological parameters of the structure are given in the figure). The transmission spectrum is also shown.

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A plot of the maximum electric field intensity in the NA as a function of the distance between the latter and the PC membrane is shown in Fig. 5 , at the resonance wavelength (1.51µm). As expected from coupled mode theory, the electric field reaches a maximum for an optimum coupling constant ($\tau =\sqrt{{\tau }_0{\tau }_c}:$ see section 2.2), achieved for an optimum distance between the NA and the PC membrane resonator. The field intensity distributions of the structure are also shown in Fig. 5. The gloss of the NA reaches its maximum under the optimum coupling conditions. For decreasing coupling rates, that is for increasing coupling distances, the NA darkens to the PC membrane advantage, which brightness increases gradually, before leveling off at a saturation value, as predicted by the coupled mode theory.

 

Fig. 5 Plot of the maximum electric field in the NA as a function of the distance between the latter and the PC membrane. The field intensity distributions in the structure at resonance are also shown

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FDTD simulation shows, in addition, that the maximum achievable electric field in the NA can be further increased by the use of PC membranes with larger Q_PC factors: for increasing Q_PC factors, the optimum coupling distance is observed to increase, which means that the optimum coupling time constant τ is also increasing precisely as anticipated from the coupled mode theory (according to Eq. (2), with ${\tau }_c\approx Q_{PC}$). However, according again to the coupled mode theory, the maximum achievable electric field in the NA under optimum coupling conditions, should be independent of Q_PC (see Eq. (3)). The reason for this discrepancy lies in that the coupled mode theory does not account for the additional radiation losses induced by the coupling. It turns that these coupling induced losses tend to vanish for decreasing coupling rates $\frac{1}{\tau },$ which is confirmed by the fact that maximum achievable electric field tends to saturate for increasing Q_PC (around 450 a.u.).

For sake of comparison, we have carried out FDTD simulations of the direct addressing of the NA, positioned at the waist of a focused optical beam under the diffraction limit conditions (waist size down to 1.5µm): the optimum coupling conditions could not be attained; the maximum field intensity (normalized to the total input optical power) achievable in the NA was a factor 3 below the one achieved with the help of the PC membrane, which does not require light focusing. This comparison is illustrated in Fig. 6 , where the electric field intensity distributions are shown for direct addressing and PC membrane mediated addressing of the NA, respectively: for the case of direct addressing, the field colour scale intensity has been artificially multiplied by a factor of 3, in order to make clearly visible (i) the focused incoming free space beam and (ii) the factor 3 enhancement of the maximum achievable electric field intensity in the NA, with the help of the PC membrane.

 

Fig. 6 Field intensity distributions at resonance for a NA directly addressed by a focused Gaussian beam (left) and PC membrane mediated addressing of the NA (right).

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Other characteristics of optical coupling between NA and PC membranes can be also usefully exploited. For example, coupling the NA to the PC membrane at the resonance wavelength of the latter, allows for transmission of the light which is, otherwise, fully reflected by the PC membrane alone. A significant amount of the incoming light can be transmitted, theoretically up to 100%, for increasingly large coupling rates $\frac{1}{\tau },$ according to the coupled mode theory (see annex, section 1). The transmitted light is a combination of two components, (i) the light radiated by the NA ($\approx \frac{{\left|a\right|}^2}{{\tau }_0}$) and (ii) the light transmitted directly by the PC membrane, as a result of the phase shift induced by the former (term $S_{-2}$ in the set of Eqs. (2), section 2). The coupled mode theory tells us in addition that the second component dominates for large coupling rates (see annex sections 1 and 2). This is confirmed by FDTD simulations, which can provide the far field transmission pattern of the structure, as shown in Fig. 7 , for the case where the NA is directly put on top of the PC membrane. The two transmission spectra with and without the NA respectively (already shown in Figs. 3 and 4) are given again on the same graph, for sake of easy comparison. The transmission ratio can reach amounts as high as about 60% and the radiation pattern is very directive: it mimics accurately the field pattern of the incoming optical beam. The radiation lobes of the NA hardly show up in the over-all radiation pattern. This behaviour constitutes a striking specific illustration of the more general physical phenomenon, so-called Electromagnetic Induced Transparency and could be usefully applied for the resonant detection / imaging of metallic nano-particules, with high sensitivity.

 

Fig. 7 Far field transmission pattern at resonance of the NA / PC structure. The transmission spectra with (blue line) and without (red line) NA is also reported.

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4. Conclusion

The flow of innovations whose threshold has been initiated in the late 1980 by the introduction of the concept of photonic crystals is still very close to its source and will inflate in the future to an extent which is certainly beyond our full consciousness. In the present contribution, we propose a new generic approach for 3D light harnessing based on coupling engineering between 1D-2D Photonic Crystal membranes and nano-photonic structures: in this new approach, 1D-2D Photonic Crystal membranes are used as a generic photonic platform for light addressing of nano-photonic structures from free space. No only does this approach allow for very efficient light insertion in tiny micro-nano-resonator structures, but also it opens the way to the production of a wide range of functionality. In the present contribution, this approach is illustrated with the demonstration of efficient light addressing of low quality factor and very high Purcell factor metallic NA; it can be fruitfully extended to the addressing of high quality factor micro-nano-resonators, thus widening the range of accessible coupling regimes, including the strong coupling regime. Attractive outcomes of the proposed approach can be contemplated for such applications where strong light concentration and/or accurate beam shaping is requested with sub-wavelength spatial resolution: they include low optical power consuming non-linear optics, sensing and bio-photonics, highly sensitive resonant near-field spectroscopy and imaging, light trapping with sub-wavelength spatial resolution, etc…

5. Annexe

5.1 Transmission ratio of the NA / PC membrane system at resonance

From the set of Eqs. (2) the transmission ratio $\frac{{\left|S_{-2}\right|}^2}{{\left|S_{+1}\right|}^2}$ can be derived easily at the resonance frequency$\omega ={\omega }_{0a}={\omega }_{0b}={\omega }_0:$

5.2 Light radiated by the NA at resonance

$\frac{{\left|a\right|}^2}{{\tau }_0}$ represents the total optical loss rate of the NA, including the radiation to free space and metallic absorption losses. From Eq. (1), it can be written: