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Coherence Converting Plasmonic Hole Arrays

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Abstract

Simulations are presented that demonstrate that the global state of spatial coherence of an optical wavefield can be altered on transmission through an array of subwavelength-sized holes in a metal plate that supports surface plasmons. It is found that the state of coherence of the emergent field strongly depends on the separation between the holes and their scattering strength. Our findings suggest that subwavelength hole arrays on a metal film can be potentially employed as a plasmon-assisted coherence converting device, useful in modifying the directionality, spectrum, and polarization of the transmitted wave.

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Acknowledgments

Choon How Gan was supported by the Department of Energy under grant no. DE-FG02-06ER46329. Yalong Gu was supported by Air Force Office of Scientific Research under grant no. FA9550-08-1-0063. Taco Visser acknowledges support from The Netherlands Foundation for Fundamental Research of Matter (FOM).

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Correspondence to Greg Gbur.

Appendix: Derivation of the Estimate of |β|

Appendix: Derivation of the Estimate of |β|

Here we provide details of the assumptions and approximations used to arrive at the order of magnitude estimate of the scattering strength of |β| ∼ 5. The derivation adheres closely to the electrostatics approximation employed by Bohren and Huffman in the problem of scattering of light by a sphere that is small compared with the wavelength [30, Sec 5.2]. A spherical cavity in a metal background medium is used as an approximation for a hole in a metal plate.

We begin by considering the electric field radiated by a monochromatic electric dipole into the far field [32, Sec 9.2], with spatial dependence

$${\bf E}_s = \frac{k^2}{4\pi\epsilon_m } \left(\textrm{{\bf \^{r}}} \times {\bf p}\right) \times \textrm{{\bf \^{r}}}\,\, \frac{{\rm exp}\,({\rm i}kr)}{r} $$
(18)

with k is the wavenumber, ϵ m is the dielectric constant of the medium in which the wave propagates, p is the electric dipole moment, and is a unit vector in the direction of r, which is the position vector of the point of observation as measured from the dipole (r = |r|). We may treat the scattering of the sphere by replacing it with an electric dipole with dipole moment p = \(\epsilon_m \,\alpha \, E_0 \,{\rm exp}\,\left({\rm i}\omega t\right) \,\,\textrm{{\bf \^{p}}}\), where E 0 is the amplitude of the incident field illuminating the dipole, polarized along the unit vector , and the polarizability of the sphere α is given by [32, Sec 4.4]

$$ \alpha = 4 \pi a^3 \frac {\epsilon _s - \epsilon _m }{\epsilon _s + 2 \epsilon _m} $$
(19)

with ϵ s and a the dielectric constant and radius of the sphere, respectively. Substituting for p into Eq. 18 and rearranging, we have

$${\bf E}_s = E_0{\bf X}\frac{{\rm exp}\,({\rm i}kr)}{{\rm i}kr}, $$
(20)

where

$${\bf X} = \frac{{\rm i}k^3}{4\pi}\,\alpha\,\left(\textrm{{\bf \^{r}}} \times \textrm{{\bf \^{p}}}\right) \times \textrm{{\bf \^{r}}} $$
(21)

is the vector scattering amplitude, a dimensionless quantity. In our model, we treat the plasmon scattering parameter β as a scalar analog to the vector scattering amplitude X. To estimate |β| , we thus attempt to derive an estimate for |X| , i.e., |β| ∼ |X|. We note that α as given in Eq. 19 is the polarizability for a sphere of ε s embedded in a medium of ϵ m . we are interested in the case of an air-filled spherical cavity (ϵ s  = ϵ 0 = 1) for which the cavity polarizability takes on the form

$$ \alpha = 4 \pi a^3 \frac {1 - \epsilon _m/\epsilon_0 }{1 + 2 \epsilon _m/\epsilon_0} . $$
(22)

Substituting this expression for α in Eq. 21, it is found that

$$ \begin{array}{rll} \mid{\bf X}\mid & = & \frac{k^3}{4\pi}\,\mid\alpha \mid \\ & = & (2\pi)^3 \, \frac{a^3}{\lambda^3 } \, \bigg| \frac{1 - \epsilon _m/\epsilon_0 }{1 + 2 \epsilon _m/\epsilon_0} \bigg|. \end{array} $$
(23)

For metals that are conducive to the generation of surface plasmons, \(\big|{\rm Re}\,(\epsilon _m)\big| \gg \big|{\rm Im}\,(\epsilon _m)\big|\), and \(\big|{\rm Re}\,(\epsilon _m)\big| \gg 1\), typically. Taking this into account, then for a subwavelength sphere with radius a = λ/3, it is obtained that

$$ \mid \beta \mid \, \sim \, \mid{\bf X}\mid \, \sim 5. $$
(24)

It is to be noted that while the dipole approximation is shown to be valid by Bohren and Huffman for a small sphere in a three-dimensional homogeneous medium, we have applied the formalism to obtain an order of magnitude estimate for |β|, which describes the scattering by a cylindrical hole in a two-dimensional plate. Provided the depth of the hole is not significantly larger than its diameter (i.e., the hole is as wide as it is tall), we expect that the scattering results for a sphere will not differ significantly from that of a cylinder, especially considering the assumed subwavelength size of the scatterer. It should be noted that the depth of the hole is not specified in the model used in this paper, but for plasmonic transmission experiments is often taken to be comparable to the width (see, for instance, [9]). The result |β| ∼ 5 lies within the range of values found by others for similar plasmonic systems; for instance, [27] finds a value of β ≈ 3, while [33] finds values as large as β ≈ 60.

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Gan, C.H., Gu, Y., Visser, T.D. et al. Coherence Converting Plasmonic Hole Arrays. Plasmonics 7, 313–322 (2012). https://doi.org/10.1007/s11468-011-9309-1

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