In their comment [1] on our recent article [2], Meunier and colleagues reject our conclusion that the Poynting vector magnitude is a direct estimate for the laser ablation threshold reduction when using plasmonic nanoparticles. They argue that there is no theoretical basis for using the Poynting vector magnitude for calculating the deposition of energy and that they believe the electric field magnitude (|E|²) should be used instead. We reject their assertions which we believe are a result of their invalid assumptions. Here we present a detailed theoretical basis for our conclusions and address their experimental data, which we believe is inconclusive.

The authors of [1] start their theoretical justification with an equation that describes the work (W) done by an electromagnetic field on a system of charged particles [3]:

The importance of this new formulation is clearly stated in [3], “This necessary addition to Ampere’s law is of crucial importance for rapidly fluctuating fields. Without it, there would be no electromagnetic radiation.” From Eqs. (1) and (2), the Poynting theorem can be derived for describing the conservation of energy by relating the work, the change in internal energy (dissipation), and the flow of energy into the system through electromagnetic fields in terms of the divergence of the Poynting vector, ∇∙S = ∇∙(E × H) [3].

Meunier and colleagues [1] recognize the Poynting theorem and write; “|S| is not in general a measure of [the work] W in the case of an harmonic field. However, ∇∙S would provide the correct result.” While they agree that ∇∙S is the measure of energy transfer, it is unclear why they still disagree with our conclusion that the Poynting vector magnitude |S|, specifically S_z for surface ablation, is the predictor of energy deposition for ablation. It is possible that Meunier and colleagues [1] may have overlooked the fact that, by the Divergence Theorem, ∇∙S for an arbitrary volume is exactly mathematically equivalent to S∙n over the surface of that volume. Given a control volume that consists of an optically thick section of substrate with an expanse large in comparison with the laser spot size, then the only electromagnetic energy transfer into the control volume will occur through the top surface. If the normal from the top surface of that substrate is considered in the z direction, then ∇∙S = S_z, which is precisely what we showed in our work [2] for the ablation of a silicon substrate. This value S_z represents the exact amount of electromagnetic energy locally crossing the surface at any given time.

In a contradictory statement, Meunier and colleagues [1] also argue that “|S| is not a good metric to evaluate energy absorption” because “the resulting field is by far not a plane wave” as support for their argument that |E|² should be used as a measure of electromagnetic field intensity. While they recognize that the plane wave approximation is not valid in the near-field of plasmonic nanostructures, they fail to recognize that the |E|² magnitude represents the intensity of electromagnetic fields only for cases where the plane wave approximation is valid.

To demonstrate that |E|² is not proportional to |S| when the plane wave approximation is invalid, as in the near-field of a plasmonic particle, we present Mie theory calculations for two different cases in Fig. 1 . For a large ‘lens-like’ dielectric sphere, the ratio of the electric to magnetic field enhancements – |E|²/|B|² is nearly constant (Fig. 1d) and thus the |E|² and |S| distributions have nearly identical enhancements and distributions (Figs. 1a and 1c). This result indicates that the |E|² enhancement can be used instead of |S| enhancement values to evaluate the ablation threshold reduction when the plane wave approximation is reasonable. On the other hand, for a small plasmonic particle, there are significant differences in the magnitude and distribution of |E|² and |S| (Figs. 1e and 1g). Due to the contribution of evanescent fields, the ratio of |E|²/|B|² varies over two orders of magnitude in the near-field region (Fig. 1h), implying that the Poynting vector enhancement should be used as the value to evaluate the ablation threshold reduction when using plasmonic nanoparticles. The Poynting vector is the general term for evaluating the electromagnetic energy flow in either case.

 

Fig. 1 The enhanced values of electromagnetic fields calculated for a 2µm SiO₂ sphere (a – d) and a 150 nm gold particle (e – h) in a water environment at a wavelength of 780 nm. The polarization is in the x direction and the light is propagating in the positive z direction. Note that the x and y axes have been scaled by the radius a for each sphere.

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As for the removal of individual gold nanoparticles, we argue that ablation occurs at the surface of a nanoparticle at the threshold fluence and can initiate complete nanoparticle ablation at fluences slightly above the threshold value. In single particle studies, Plech and colleagues experimentally showed that ultrafast laser ablation of gold nanoparticles begins at the surfaces with the highest field enhancements [4]. In our study [2], we show that the |S|, and not |E|², surface enhancement values match well with the ablation threshold reduction of gold. This result suggests that the Poynting vector magnitude at the surface is a reasonable estimate of the local electromagnetic field intensity for particle ablation. For an exact description of nanoparticle ablation, however, we need to develop a time-resolved analysis of the processes involved in plasmonic laser ablation of nanoparticles, which is currently under investigation.

The experimental results shown in [1] are interesting, but not sufficient to disprove our earlier conclusions. First, since no before ablation image is shown, the authors cannot verify that ablation occurred with a single rod and not with multiple rods. Second, the ~450 nm wide ablation crater, which is 5 × larger than the long axis of the nanorod, cannot reflect the true nanoscale signature of the near-field confinement of enhanced electromagnetic fields around a plasmonic particle. The near-field enhancement around a plasmonic nanoparticle typically decays rapidly within a radius away from the particle surface. For the accurate study of the mechanism of plasmon-enhanced ablation, one needs to examine ablation sites that are closer to the scale of the nanoparticle using laser fluences closer to the threshold, as we have done in our studies [2,5]. Third, they present neither calculations nor experimental results to show that their ablation results actually fit better to |E|² and not to |S|. Finally, and most importantly, the observed enhancement values for surface ablation in our experiments fit the |S| magnitude and do not fit the |E|² magnitude [2,5]. Other groups have also found that the |E|² enhancement on the surface does not effectively match experimental comparisons for laser threshold reduction [6,7]. Specifically, Luk’yanchuk et al. [6] found ablation morphologies to be consistent with the S_z enhancement for ultrafast laser near-field ablation, in agreement with our results.

In conclusion, and in opposition to [1], we believe that the use of the Poynting vector magnitude |S|, specifically S_z for surface ablation, is theoretically justified as a measure of energy deposition for plasmonic laser nanoablation. We reiterate our claim presented in [2] that the Poynting vector magnitude is a better predictor of the enhancement and shape for ultrafast laser plasmonic surface ablation than |E|².