## Abstract

The problem of the absorption of light by a nanoscale dielectric cone is discussed. A simplified solution based on the analytical Mie theory of scattering and absorption by cylindrical objects is proposed and supported by the experimental observation of sharply localized holes in conical silicon tips after high-fluence irradiation. This study reveals that light couples with tapered objects dominantly at specific locations, where the local radius corresponds to one of the resonant radii of a cylindrical object, as predicted by Mie theory.

©2013 Optical Society of America

The absorption and scattering of light by objects with subwavelength dimensions is one of the most fundamental problems in electrodynamics. The strong sensitivity of the optical properties of such objects to their shape and size has also taken this problem to the forefront of many technological advances and challenges (Refs. 1,2 and references therein). More particularly, owing to their combined focusing and guiding properties, particles with a conical shape have led to many successful applications such as nanoscale optical tweezers or quasi-non-dispersive antireflection coatings [3,4], but have surprisingly received little theoretical attention comparatively to their spherical and cylindrical counterparts. This is most likely due to the problematic presence of a singularity at the cone apex [5], combined with their large aspect ratio [6]. Nevertheless, the relevance of this problem has recently arisen in the fields of metrology and microscopy [7], and, more particularly, in the context of the laser-assisted Atom Probe Tomography (APT) technique [8,9]. In this paper, we first uncover the interaction between light and infinite axisymmetric particles by extending Mie theory of light absorption by cylindrical particles [10,11], assuming that an axisymmetric particle is a mere stack of infinitesimal cylinders with a varying radius. Focusing our study on the case of a conical particle, we then deduce theoretically and show experimentally that the absorption of light occurs preferentially at specific locations on the particle, where the local radius corresponds to one of the resonance absorption/scattering radii of an illuminated cylinder at the considered wavelength. The impact of the finiteness of the particle, i.e. of the light diffracted at the apex and fed into the particle, is also discussed.

As we shall show, the light absorption by conical nanoparticles can be understood based on the analytical Mie theory of absorption by infinitely long cylindrical particles [10,11]. Since the absorbed power is proportional to the (2n)^{th} power [n = number of photon(s) involved in the transition] of the electric field ${\overrightarrow{E}}_{\mathrm{int}}$ induced inside the particle by the incident electromagnetic wave [12], this absorption problem can be summarized by the expression of ${\overrightarrow{E}}_{\mathrm{int}}(r,\vartheta )$, where $r$and $\vartheta $ are respectively the radial and azimuthal coordinates in the local axis of the cylinder cross-section. For a transverse magnetic (TM) plane wave incident normally to the axis z of symmetry of a circular cylinder of radius $R$, ${\overrightarrow{E}}_{\mathrm{int}}(r,\vartheta )$ is oriented along $z$ and its amplitude reads [10,11]

^{th}-order Bessel function of first kind [13]. The ${d}_{k}(R/{\lambda}_{0})$ coefficients are determined by the boundary conditions along the surface $r=R$ [10,11]. A factor $\mathrm{exp}(-i\omega t)$ expressing the time dependence of the wave is assumed, $\omega $ being the angular frequency of the incident wave. To illustrate Eq. (1), Fig. 1 plots the behavior of $\left|{E}_{z,\mathrm{int}}\right|$ in a cylindrical silicon particle as a function of the cylinder radius when illuminated with green (${\lambda}_{0}=515$ nm) and infrared (IR, ${\lambda}_{0}=1030$ nm) light. The complex refractive index follows the experimental data of [14]. Figure 1 shows that, as a result of the resonant behavior of the ${d}_{k}(R/{\lambda}_{0})$ coefficients, $\left|{E}_{z,\mathrm{int}}\right|$ also presents resonant values – or maxima - in cylinders with radii of specific values. The position of these maxima as well as their spacings depend upon the incident wavelength. This resonance behavior can be understood as the result of the constructive interference between the transmitted and (multiply) internally reflected waves inside the cylinder, as also occurs in thin films of specific thicknesses [15]. Interestingly, however, the resonant thicknesses of a slab differ somewhat from the resonant diameters of a cylinder, the latter proving to be roughly $0.257\times {\lambda}_{0}$ smaller than the former [16].

We propose to generalize Eq. (1) to the case of an infinite axisymmetric particle with an arbitrary surface $R(z)$ illuminated normally to its axis (referred to as normal incidence in the rest of this paper). This is physically acceptable provided the resulting z-dependence of the ${d}_{k}[R(z)/{\lambda}_{0}]$ coefficients is such that $\left|{\partial}^{2}{E}_{z,\mathrm{int}}/\partial {z}^{2}\right|<<{\left|2\pi \tilde{n}/{\lambda}_{0}\right|}^{2}\left|{E}_{z,\mathrm{int}}\right|$. When this condition is verified, the wave equation determining ${E}_{z,\mathrm{int}}$ is indeed hardly perturbed by the z-dependence of the ${d}_{k}$ coefficients. In other words, if the particle surface is smooth enough along its symmetry axis, $\left|{\partial}^{2}{E}_{z,\mathrm{int}}/\partial {z}^{2}\right|$ is small and the particle can be considered as locally cylindrical, i.e. the coupling between cross-sections with different radii can be neglected. An extra benefit of this condition for the normally illuminated cones considered in this paper is that each local cylinder is also illuminated normally to its axis. Consequently, the contributions to the internal field oriented along the $\overrightarrow{r}$ and $\overrightarrow{\vartheta}$ directions can be expected to be small relative to${E}_{z,\mathrm{int}}$and thus be ignored. Considering a conical particle with cone angle $\alpha $ and a surface defined by

the above condition reduces toTo experimentally validate our theory summarized by Eqs. (1) and (2), conical silicon tips have been irradiated with high-fluence (~100 mJ/cm^{2}) TM IR light in a Laser-Assisted Wide-Angle Tomographic Atom Probe (LAWATAP) tool of Cameca [17]. The Scanning Electron Microscope (SEM) image of Fig. 3
shows that, after irradiation, the tips present sharp holes, indicating strongly localized absorption. The calculated value of ${\left|{E}_{z,\mathrm{int}}[r=R(z),\vartheta =\pi ,z]\right|}^{4}$ obtained with Eqs. (1) and (2) for silicon irradiated with IR light is reproduced on the left of Fig. 3, making it obvious that these holes are precisely located at the theoretically predicted maxima of ${\left|{E}_{z,\mathrm{int}}\right|}^{4}$. Similar holes are also observed in the case of green irradiation but are packed much more tightly together (Fig. 1), making their discussion more difficult (not shown). The sharp localization of the holes at the resonance radii predicted by Mie theory for the cylindrical particle proves that the light is coupled into the cone dominantly at these locations, hence confirming the validity of the present theory. The relatively large cone angle ($\alpha ~10\xb0$) of the tip in Fig. 3, also suggests that the theory remains valid, at least in first order, even when perturbations of the wave equation induced by a non-negligible $\left|{\partial}^{2}{E}_{z,\mathrm{int}}/\partial {z}^{2}\right|$ are present.

Although the agreement between theory and experiment in Fig. 3 is clear, three important remarks need to be made for its complete and unambiguous understanding. Firstly, it has to be stressed that the holes are sharply located inside each resonant cross-section of the cone. Their apparent elliptical shape in Fig. 3 is merely due to the 45° tilt angle of the primary beam of the SEM with respect to the tip axis. This tilt is also responsible for the small (downward) apparent misalignment of the holes with respect to the corresponding resonance peaks. Actually, only the leftmost point of each hole is reliable for measurement of the local radius $R(z)$ of the holes (locations of the 3 measured values of 113, 178 and 241 nm). Secondly, the use of the fourth power of $\left|{E}_{z,\mathrm{int}}\right|$ is justified by the dominant two-photon absorption process of high-fluence IR radiation by silicon [18]. For the sake of completeness, note also that the inclusion of this nonlinear effect in the value of the complex refractive index in Eq. (1) proves to slightly reduce the heights of the resonance $\left|{E}_{z,\mathrm{int}}\right|$ peaks while keeping them at the same position (not shown). Thirdly and finally, the reason for the one-sided shape of the observed holes is still under discussion, with two competing possible explanations. This shape may indeed be due to a transient nonlinear absorption phenomenon common to pulsed-laser ablation of Si, wherein free-carrier absorption becomes very intense and dominant in the course of the laser pulse, hence strongly decreasing the light penetration depth [19]. It is, however, also suspected that the hole shape may result from an optical interference between ${E}_{z,\mathrm{int}}$ and the perturbation field induced by the local $\left|{\partial}^{2}{E}_{z,\mathrm{int}}/\partial {z}^{2}\right|$ highlighted in Fig. 2. This question is still under investigation.

The impact of the finiteness of a particle such as shown in Fig. 3 remains to be determined. The agreement between the developed theory for infinitely long particles and the experimental observation of holes summarized in Fig. 3 seems to infer that the apex does not have much impact upon its absorption characteristics in the case of incidence normal to the cone axis. The same conclusion is reached in recent theoretical studies of the electromagnetic field inside an illuminated truncated cylinder with planar endings [20]. Let us, however, investigate the amplitude ${E}_{z,apex}$ of the light diffracted at a rounded apex, considered here as an additional independent contribution by invoking the linearity of Maxwell’s wave equations. To understand why ${E}_{z,apex}$ is likely to be negligible, two determining factors must be investigated, namely the coupling efficiency of the apex and the guiding ability of the cone. First, to determine how efficiently light can be coupled into the apex, let us consider the hemispherical apex as an isolated silicon sphere. It is then easy to show, based on Mie theory for spherical particles [10], that only a few percent of the incident IR light is coupled into the apex when its radius ${R}_{\text{apex}}<<145$ nm, i.e. far from the first resonance peak of the spherical particle. Note, furthermore, that similar conclusions would be reached using Bethe-Bouwkamp diffraction theory by assimilating the apex as a hole in a conductive screen [21]. In other words, it can be understood that the approximate 35-nm apex radius of the silicon nanoparticle shown in Fig. 3 is too small to significantly couple to IR light. Second, let us study how much electromagnetic power a conical particle, acting as a dielectric fiber [22], can guide away from the apex. In such a waveguiding structure, indeed, only bound modes propagate with a significant electric field inside the guide and therefore have a chance of being partially absorbed. The relevant parameter to be tested is the normalized frequency $V=2\pi {R}_{\text{apex}}/{\lambda}_{0}\sqrt{{\left(\Re (\tilde{n})\right)}^{2}-1}$ [22]. Noting that the condition for single-mode operation $V<2.4$ is reached for ${R}_{\text{apex}}<120$ nm when the guide is excited by IR light, it is clear that only the fundamental HE11 mode can be excited at the apex and be guided through the nanoparticle of Fig. 3 [22,23]. This obviously further reduces the power actually fed into the particle via the apex, in particular far from the incidence parallel to the axis, measured to be optimal for coupling to HE11 [24]. Similar conclusions can be reached for green light although the 35-nm apex radius is closer to the first resonance peak of the spherical silicon particle (${R}_{\text{apex}}=$59 nm) and to the single-mode condition (${R}_{\text{apex}}<50$ nm). It can thus be expected that more electromagnetic power can be fed into the cone through its apex in case of green illumination. In summary, the light diffracted at the apex can be ignored because the coupling efficiency and guiding ability of a normally illuminated cone with small apex radius are low. It is important to note that the situation would be different for light incident (quasi-) parallelly to the axis, where the coupling through the ending/apex is indeed significant [20,24]. This coupling is e.g. used in recent successful applications where arrays of cylindrical and conical Mie resonators serve as antireflection coating [25,26].

To conclude, we have developed a simplified light-absorption theory for conical nanoparticles based on the assumption that an axisymmetric particle is a stack of cylinders with varying radius. This theory predicts that light couples and is therefore absorbed in these objects at specific locations, where the local radius corresponds to a resonance of the Mie theory developed for cylindrical particles. We have shown theoretically that, for this approximation to be good, the particle surface should be sufficiently smooth and the wavelength sufficiently short. Further, we have shown that the finiteness of the particle, i.e. the presence of an apex, has little impact on its absorption efficiency provided the radius-to-wavelength ratio is small and the incidence is close to normal to the cone axis. An experimental confirmation of the proposed theory has been found in the observation of sharply localized holes in nanoscale conical silicon particles illuminated with high-fluence IR light. The location of these holes is in excellent agreement with the developed theory.

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