Limits to surface-enhanced Raman scattering near arbitrary-shape scatterers: erratum

Jérôme Michon; Jérôme Michon; Mohammed Benzaouia; Mohammed Benzaouia; Wenjie Yao; Owen D. Miller; Steven G. Johnson

doi:10.1364/OE.401577

Abstract

In this erratum, we correct two minor algebraic errors from our previous published manuscript [Opt. Express 27, 35189 (2019) [CrossRef] ], which do not affect the main results or conclusions, and make a corresponding small change to one figure.

In our previously published manuscript [1], the final framed equation of Appendix C as well in Eq. (9) $\epsilon _0$ should be replaced with $\epsilon$, as should be clear from the preceding derivation. As a consequence, the right-hand terms of equations (10, 11, 15, 16) should be multiplied by $n_b^{4}$. Note that all of the paper’s computations were done in vacuum ($n_b=1$), so they are not affected by this modification.

The constant term in Eq. (11) should be replaced with the background periodic LDOS. The equation becomes:

(11)$$ q_\mathrm{loc} \leq \frac{2Sk^{2}\cos\theta}{3\pi} \left[\frac{\rho^{\textrm{per}}_{b,\mathbf{p}}}{\rho_b}+\frac{3\pi n_b^{2}}{2k^{3}} \frac{|\chi|^{2}}{\textrm{Im} \chi}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert {\mathbf{G^{per}}\mathbf{U^{*}_\mathrm{R}}} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2} \right] ,$$

where $\rho ^{\textrm {per}}_{b,\mathbf {p}}$ is the polarized periodic LDOS related to the power radiated by a dipole $\mathbf {p}$. If only one diffraction channel is supported (for example, when the period $a$ is smaller than $\lambda /2$), we have:

(E1)$$ \rho^{\textrm{per}}_{b,\mathbf{p}} = \frac{\rho^{\textrm{per}}_{b}}{2}\left( 1-\frac{|\mathbf{\hat{p}}^{{\dagger}} \mathbf{\hat{k}_+}|^{2}+|\mathbf{\hat{p}}^{{\dagger}} \mathbf{\hat{k}_-}|^{2}}{2} \right) \leq \frac{\rho^{\textrm{per}}_{b}}{2} = \frac{n_b}{2\pi c S \cos\theta}$$

where $\rho ^{\textrm {per}}_{b}$ is the total periodic LDOS (sum of $\rho ^{\textrm {per}}_{b,\mathbf {p}}$ over three orthogonal directions) and $\mathbf {k_\pm }=-\mathbf {k_{0\parallel }}\pm k_{0z}\mathbf {\hat {z}}$ (directions of far-field emission). In this case, Eq. (11) becomes:

(E2)$$ q_\mathrm{loc} \leq 2+\frac{Sn_b^{2}\cos \theta}{k} \frac{|\chi|^{2}}{\textrm{Im} \chi}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert {\mathbf{G^{per}}\mathbf{U^{*}_\mathrm{R}}} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^{2} \,.$$

The constant factor of $2$ simply reflects the enhancement in the case of a perfect back-reflector.

It is important to note that it is only the constant term (independent of the scatterer) that was changed in the periodic limit. This leads to small quantitative changes in Fig. 3, but does not affect the conclusions of the paper.

Fig. 3. Updated near-field enhancement bounds for an isolated Ag sphere and a square array of Ag spheres with varying period $a$. The spheres have a radius $R = 13$ nm, and the emitter is located $d = 20$ nm away from their surface along the lattice axis. The incident field’s polarization is aligned with the sphere-emitter direction and $\lambda = 350$ nm. Inset: period at the points P, Q, and Q’ as a function of sphere radius and lattice period, for $d = 20$ nm.

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Disclosures

The authors declare no conflicts of interest.

References

1. J. Michon, M. Benzaouia, W. Yao, O. D. Miller, and S. G. Johnson, “Limits to surface-enhanced Raman scattering near arbitrary-shape scatterers,” Opt. Express 27(24), 35189–35202 (2019). [CrossRef]

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Figures (1)

Fig. 3.

Fig. 3. Updated near-field enhancement bounds for an isolated Ag sphere and a square array of Ag spheres with varying period

$a$

. The spheres have a radius

$R = 13$

nm, and the emitter is located

$d = 20$

nm away from their surface along the lattice axis. The incident field’s polarization is aligned with the sphere-emitter direction and

$\lambda = 350$

nm. Inset: period at the points P, Q, and Q’ as a function of sphere radius and lattice period, for

$d = 20$

nm.

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Equations (3)

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q_{l o c} \leq \frac{2 S k^{2} \cos θ}{3 π} [\frac{ρ_{b, p}^{per}}{ρ_{b}} + \frac{3 π n_{b}^{2}}{2 k^{3}} \frac{| χ |^{2}}{Im χ} {| | | G^{p e r} U_{R}^{*} | | |}^{2}],

ρ_{b, p}^{per} = \frac{ρ_{b}^{per}}{2} (1 - \frac{| {\hat{p}}^{†} {\hat{k}}_{+} |^{2} + | {\hat{p}}^{†} {\hat{k}}_{-} |^{2}}{2}) \leq \frac{ρ_{b}^{per}}{2} = \frac{n_{b}}{2 π c S \cos θ}

q_{l o c} \leq 2 + \frac{S n_{b}^{2} \cos θ}{k} \frac{| χ |^{2}}{Im χ} {| | | G^{p e r} U_{R}^{*} | | |}^{2} .