Extremely anisotropic boundary conditions and their optical applications

Authors


Abstract

[1] Following the idea of realizing an extremely anisotropic scatterer by combining positive and negative permittivity materials in a composite sphere, we discuss here some of the features and potential applications of extreme boundary conditions in optics. We analyze the scattering features and surface boundaries of an isolated sphere composed of two hemispheres with oppositely signed real parts of permittivity, whose local surface impedance can vary homogeneously from zero to infinity upon a simple 90 degree rotation of the impinging electric polarization, and we theoretically explore how combinations of these particles and other related geometries may tailor and boost the optical radiation of localized sources, e.g., quantum dots and molecular emitters.

1. Introduction

[2] Interaction of light with plasmonic materials has fascinated mankind since centuries [Bohren and Huffman, 1983], and today is at the basis of one of the most active research areas in optics, dealing with the nanoscale interaction of light with nanoparticles and nanostructured materials that are able to focus and enhance the optical fields well beyond the diffraction limit [Maier, 2007]. The anomalous interaction of light with plasmonic particles arises at the interface between materials with positive and negative real parts of permittivity, which is well known to support surface plasmon polaritons (SPP), i.e., confined resonant oscillations of electrons. In planar configurations such interfaces support SPP waves with longitudinal wave numbers smaller than that of free-space, whereas in 2-D or 3-D geometries they can sustain localized SPP resonances almost independent of the overall size of the object, allowing sharp scattering response from nanoparticles with physical cross sections much smaller than the wavelength of operation. The exotic features of plasmonic interfaces have also been recently explored for imaging purposes: a planar slab with permittivity opposite to the one of the background can realize a ‘poor-man’ version of the ideal perfect lens [Pendry, 2000] with sub-diffraction imaging properties beyond Abbe's classical limit [Fang et al., 2005; Taubner et al., 2006] and materials with complementary properties may support similarly anomalous transparency and resonant properties [Alù and Engheta, 2003]. The anomalous scattering features of combinations of plasmonic and non-plasmonic nanoparticles [Alù and Engheta, 2005] have also been used in complex arrangements to realize metamaterials with exotic bulk response [see, e.g., Engheta and Ziolkowski, 2006].

[3] As another exciting opportunity offered by combining positive- and negative-permittivity materials, we have recently discussed the anomalous scattering properties of a composite sphere formed by conjoining two hemispheres with oppositely signed real parts of permittivity [Alù and Engheta, 2009]. We have introduced this geometry to practically envision a simple series or parallel connection between a nanocapacitor and a nanoinductor within the optical nanocircuit paradigm [Engheta et al., 2005; Engheta, 2007]. In this framework, we have shown that such resonant nanoparticle indeed realizes the most basic ‘stereo-nanocircuit’, which can act as a short or an open circuit (respectively the resonant series and parallel combination of an inductor and a capacitor), depending on the polarization of the impinging electric field. From a scattering point of view, we have shown that a homogeneous surface impedance may be defined for this resonant composite sphere, which is identically zero when the electric field is polarized perpendicular to the interface between the two hemispheres and is infinite for tangential polarization, exactly as predicted by the nanocircuit interpretation of this geometry [Alù and Engheta, 2011]. The fascinating possibility of having a particle that may provide extreme and largely tunable local boundary conditions by a simple mechanical rotation or change of excitation may open several exciting possibilities for optical applications. Collections of such particles may realize electric or magnetic reflectors at frequencies for which conventional conductors and magnetic effects are not available, and different geometries may provide even wider degrees of flexibility for practical purposes. In the following, we first discuss the features of the optical nanoswitch introduced by Alù and Engheta [2009] from a scattering perspective, and then we envision alternative setups for which extreme and largely tunable boundary conditions based on similar resonant effects may tailor and boost the optical emission of optical dipole sources, such as molecules and quantum dots.

2. Extreme Boundary Conditions Around a Small Composite Nanosphere

[4] Consider the geometry introduced by Alù and Engheta [2009] and depicted in Figure 1, consisting of two conjoined hemispheres with radius a and permittivity ɛup, ɛdown, and same free-space permeability μ0, surrounded by free-space, with permittivity ɛ0. In the small-radius “quasi-static” limit of interest here, for which the impinging electric field may be considered uniformly distributed across the sphere with aλ0 (free-space wavelength of operation), we have shown that this boundary value problem has a closed-form solution for the quasi-static electric potential distribution in the special resonant condition for which ɛup = −ɛdown for the lossless scenarios (i.e., when the permittivities are all real quantities). For excitation as in Figure 1a, i.e., with electric field perpendicular to the common interface, we can write

equation image

where (r, θ, ϕ) are the spherical coordinates in the reference system centered with the nanosphere, ϕ0 is the potential distribution for r > a, ϕup and ϕdown are the potential distribution in the two hemispheres. Similarly, for the perpendicular polarization:

equation image

An example of the near-field potential distribution for ɛup = −ɛdown = 5ɛ0 is shown in Figure 2, consistent with the results of Alù and Engheta [2009]. Under the excitation in Figure 1a, the potential distribution outside and around the sphere (see Figure 2a) is identical to the one produced by a homogeneous perfectly electric conducting sphere, as the potential lines are all tangential to the surface of the sphere. Vice versa, for the perpendicular excitation the potential distribution in the surrounding outside region is the one produced by a perfectly magnetic conducting sphere. In terms of local boundary conditions on r = a, the two excitations produce a uniform, homogeneous local surface impedance with zero and infinite values, respectively, corresponding to perfectly normal and perfectly tangential electric field all over the spherical surface.

Figure 1.

A composite nanosphere obtained joining two hemispheres with different permittivity, as envisioned by Alù and Engheta [2009]. The impinging electric field is assumed uniformly distributed over the sphere volume, and polarized in the (a) perpendicular or (b) tangential direction with respect to the common interface.

Figure 2.

Potential distribution for the geometry of Figure 1 with ɛup = −ɛdown = 5ɛ0 under the two perpendicular excitations, consistent with that of Alù and Engheta [2009]. Corresponding to the excitation in (a) Figure 1a and (b) in Figure 1b.

[5] As discussed by Alù and Engheta [2009], this scattering response is consistent with the nanocircuit description of this composite particle, which is respectively a series and a parallel resonant LC tank. We have discussed [Alù and Engheta, 2011] how even other combinations of complementary nanoparticles may produce similar boundary effects. It should be noticed that, even though the external response of the composite sphere is identical to a homogeneous electric or magnetic conductor for r > a, the internal potential distribution is very peculiar, revealing the resonant features of the composite particle. In particular, the potential distribution sustains a pair of singularities at the origin, which represent the perfect virtual images of the charges at infinity sustaining the excitation fields. Relevant to the following discussion, these singularities do not affect the external potential for r > a, which is independent of the specific choice of ɛup and ɛdown, as long as the resonant condition ɛup = −ɛdown is ideally met. In practice, when losses and practical implementation limitations are considered, this condition cannot be exactly satisfied, the singularities disappear and the homogeneous response and extreme values of the surface impedance are expectedly limited. In the most general case, far from the sphere resonance, it makes little sense to even define a local surface impedance for such geometry, due to the inhomogeneous field variation around its surface.

[6] In order to verify how extreme boundary conditions may be sustained and realized in practice with this design, we analyze the special case of a hemisphere with permittivity ɛup = ɛrɛ0, i.e., we assume ɛdown = ɛ0. Equations (1)(2) still apply when ɛr = −1, but in Figure 3 we calculate the polarizability per unit volume α as a function of ɛr, using the mode-matching technique introduced by Kettunen et al. [2007]. By definition, α is a dimensionless quantity relating the induced electric dipole moment on the particle to the impinging electric field as p = ɛ0αVE0. We show the numerically calculated value of α using two different truncation orders N in our mode matching expansion as indicated in the legend, compared with the analytical value of polarizability calculated for a homogeneous sphere of same relative permittivity. It is verified that the two different truncation orders for the hemisphere calculations yield the same numerical result, confirming that the mode-matching expansion fully converges for any ɛr in the plot.

Figure 3.

Normalized polarizability per unit volume of a hemisphere with relative permittivity ɛr compared to that of a homogeneous sphere. Corresponding to the excitation (a) in Figure 1a and (b) in Figure 1b.

[7] For ɛr > 1, the hemisphere and sphere curves are practically coincident, ensuring that the induced dipole moment per unit volume is weakly affected by the shape of the object, but it is simply given by the integral of the induced polarization vector in the material, as expected for small dielectric objects. For negative polarization (i.e., ɛr < 1), however, the shape significantly affects the scattering properties of these small objects, and the plasmonic effects arise. In this case, the hemispherical shape leads to a different response compared to the sphere, as particularly evident for the negative permittivity. As is well known, the sphere polarizability has a vertical asymptote for ɛr = −2, strictly associated with its shape, and it has a finite response for any other value of ɛr.

[8] The hemisphere polarizability has a more complex resonant response [Kettunen et al., 2008]: its resonant asymptotes vary with the field polarization, as expected due to the shape anisotropy, and the mode-matching expansion does not converge at all, for both polarizations, in the range −3 < ɛr < −1/3. The reason for this lack of convergence resides in the distributed plasmonic resonance at the interface between the two hemispheres, which can produce singular field concentration, in this lossless scenario for all this range of permittivities. Indeed, even the closed-form solution (1)(2), obtained for ɛr = −1 may be derived only after introducing a singularity at the origin. It is interesting that this exact non-convergence range is obtained in the homogenization of a square lattice of touching square cylinders [Perrins and McPhedran, 2010], possibly associated with similar extreme boundary conditions at the corners of the squares. An analogous range of non-convergence may be derived in the general case of conjoined hemispheres, as a function of the combination of ɛup and ɛdown.

[9] This is shown in Figures 4 and 5, in which we have calculated the near-field potential distribution around a composite sphere for different values of ɛup and ɛdown, as indicated in the different panels, using the same mode-matching expansion used in Figure 3 with N = 50. Figure 4 corresponds to the excitation of Figure 1a, and Figure 5 corresponds to that in Figure 1b. It is seen that for a simple dielectric hemisphere with ɛr = 5 the potential distribution is only mildly distorted compared to the one in free-space (top left panels). For a negative permittivity ɛr = −5 (top right panels) the distribution slightly changes in Figure 4, due to oppositely signed charges induced on the hemisphere cap, whereas charge concentration at the hemisphere edges produce some resonant singularities in Figure 5. The potential distributions are fully converged in both scenarios, as predicted by Figure 3. If we now add a second hemisphere with positive permittivity ɛdown = 3ɛ0 (Figures 4 and 5, bottom left), we notice that the convergence is more challenging in both scenarios, and numerical artifacts arise in the calculated response, in particular around the spherical surface and at the interface edges. This is because we are in the range of permittivities that produce inherent instabilities and plasmonic resonances near the interface between plasmonic and dielectric materials. When the second permittivity ɛdown = 4.9ɛ0 gets closer to the resonant condition ɛdown = −ɛup (bottom right panels), the potential distributions tend to converge to the ones predicted by equations (1)(2) and depicted in Figure 2. The field singularity, correctly predicted by the mode-matching expansion of the internal fields, starts moving toward the origin in this limit and the convergence of the solution improves. It is relevant to notice that, while the potential distributions around the composite sphere are very similar to those in Figure 2 for this scenario, the singular field distribution predicted at the ideal condition ɛdown = −ɛup is only weakly approached when ɛdown = 4.9ɛ0. This is because the singular extreme behavior of the internal fields at the resonance condition is highly sensitive to the matched condition at the joint interface. For the outside potential, of most interest for the applications described here, it is recognized how in this last scenario, and partially also for ɛdown = 3ɛ0, the boundary conditions on the sphere surface are indeed extreme and largely tunable by the external excitation: the equi-potential lines vary from tangential to the surface (Figure 4, near-zero surface impedance) to normal to the surface (Figure 5, near-infinite surface impedance) upon rotation by 90°. Conversely, an external observer can hardly capture any indication of inhomogeneity around the composite sphere (except around the common equatorial interface). This confirms that extreme boundary conditions and largely tailorable optical response from this composite inclusion may support a fairly robust response to frequency dispersion and design modifications. We have discussed the influence of shape and possible presence of losses in these setups of by Alù and Engheta [2009, 2011]. Finally, before concluding this section, it should be stressed that the analysis presented here is based on a quasi-static solution applicable only in the limit of small spheres. In this limit, retardation effects and non-uniformity of the exciting field may be neglected, and the resonance condition ɛdown = −ɛup appears to be independent of the frequency of operation. It is obvious, however, that passivity and causality conditions on the involved materials require a significant frequency dispersion of the permittivity when its real part is negative [Skaar and Seip, 2006; Lind-Johansen et al., 2009; Gustafsson and Sjöberg, 2010], which implies an unavoidable bandwidth restriction on these effects.

Figure 4.

Near-field electric potential distribution for different values of ɛup and ɛdown, as indicated, for excitation as in Figure 1a.

Figure 5.

Near-field electric potential distribution for different values of ɛup and ɛdown, as indicated, for excitation as in Figure 1b.

3. Optical Applications and Extreme Planar Boundary Conditions

[10] The concepts outlined in section 2 suggest that it may be possible to realize extreme boundary conditions over a composite spherical surface by joining two complementary hemispheres with oppositely signed permittivity. In the absence of loss, ideally zero or infinite values of surface impedance may be of great interest in scattering and radiation applications, as they provide the possibility to strongly interact with the impinging radiation. A plane wave hitting such nanosphere would ‘feel’ a perfectly electric conducting or perfectly magnetic conducting sphere simply depending on the orientation of the common equatorial plane with respect to the polarization of the impinging electric field, and without the use of conducting electric or magnetic materials. Furthermore, this effect may be particularly interesting in the case of localized optical emitters, such as quantum dots or molecules, placed near by the surface of the nanosphere, which may provide significant gain and directivity enhancement as electric or magnetic conductors do in radio-frequency antenna applications. In addition, the large fields induced inside the nanosphere suggest that this composite particle may be used to boost optical radiation or optical nonlinearities, by properly placing optical sources or nonlinear materials around or inside the composite sphere. It should be however stressed that a nonlinearity or radiating element extracting some of the energy focused by the particle would necessarily affect its overall resonance, partially reducing the expected field enhancement. We are currently investigating these issues in more detail.

[11] These concepts may be extended to more practical setups and applications if we consider arrays and combinations of these particles. For instance, a planar array composed of spheres as in Figure 1 may be able to provide extremely tunable optical response upon mechanical rotation of its constituents, or for different light polarizations, which may be of great interest for optical radiation or filtering applications. We are currently investigating some of these concepts in more detail, considering the coupling among such resonant elements and their collective response. An analogous, but arguably simpler, geometry may be obtained by alternating thin layers of plasmonic and dielectric materials, which may be regarded as the one-dimensional equivalent of the composite sphere analyzed in section 2. In the following, we discuss the scattering and boundary condition properties of semi-infinite substrates obtained by alternating thin layers of materials with oppositely signed permittivity, as depicted in Figure 6.

Figure 6.

A semi-infinite layered substrate formed by alternating metallic and dielectric thin planar slabs with permittivity ɛm and ɛd, respectively, and same thickness, excited by a plane wave.

[12] We analyze two possible topologies for a semi-infinite metallo-dielectric substrate, in which the interface between neighboring layers is oriented parallel (Figure 6a) or orthogonally (Figure 6b) to the interface with free-space. We use a simple homogenization approach to model the layered geometry, valid for thin layers and for sake of simplicity we assume equal thicknesses for the alternating layers. In this scenario, it is well known that such metallo-dielectric geometry may be modeled as an anisotropic material with permittivity tensor [Ramakrishna et al., 2003]

equation image

where the first two elements correspond to the electric polarizations transverse to the layers and the last element to the perpendicular one. Applications of this layered geometry have been numerous in the metamaterial community [see, e.g., Pendry and Ramakrishna, 2003; Alù and Engheta, 2009b; Mattiucci et al., 2010; Salandrino and Engheta, 2006]. Here we analyze the surface boundary conditions at the interface with free-space, extending the 3-D concepts explored in section 2 to a planar scenario. Following the results of section 2 and inspecting equation (3), it is obvious that of particular interest is the limit for which ɛd = −ɛm, which provides extreme values of effective permittivity, maximum anisotropy and expectedly anomalous optical response.

[13] It is simple to prove that the magnetic field reflection coefficient for a plane wave with arbitrary transverse-magnetic (TM) polarization with respect to the planar interface between free space and an anisotropic medium described by the permittivity tensor (3), impinging with arbitrary transverse wave number kequation image, may be written as

equation image

where ɛt and ɛn are the tangential and normal components, respectively, of the tensor equation image with respect to the interface.

[14] In the case of Figure 6a, in the limit ɛm → −ɛd we get

equation image

ensuring that the interface behaves equivalently as a perfect magnetic conductor (PMC) for any angle of incidence, analogous to the sphere in Figure 1b (it should be noted that RTMH here is the reflection coefficient for magnetic fields, so RTMH = −1 corresponds to the reflection from a PMC). A similar result is obtained for transverse-electric (TE) polarization, ensuring that the boundary of Figure 6a offers infinite surface optical impedance for any impinging electromagnetic wave and zero transmission. It is interesting that the exotic propagation properties across a finite slab of such metamaterial have been used for imaging [Salandrino and Engheta, 2006] and radiation [Alù et al., 2007] purposes. Focusing only on the extreme boundary condition of one interface between a semi-infinite substrate, we obtain an ideal, angle-independent perfect magnetic conductor. PMC boundaries are of great interest for antenna purposes at radio frequencies [Feresidis et al., 2005], and this geometry may offer the possibility to translate these concepts into optical frequencies, for which magnetic or electric conducting materials are not available.

[15] Conversely, the configuration of Figure 6b, in the limit ɛm → −ɛd yields

equation image

In this scenario, the interface behaves as a perfect-electric conductor (PEC) [RTMH = 1] with zero surface impedance (analogous to Figure 1a) for normal incidence k = 0, but provides reduced reflection for oblique incidence. In particular, for:

equation image

a Brewster-like condition is obtained and zero reflection may be expected from such interface. For TE excitation, the surface behaves as a PEC for all angles of incidence, as expected. Similar to the nanosphere scenario, different orientation of the layers provides extremely different optical boundary conditions, which may be of interest for optical radiation applications.

[16] When an optical emitter is placed in front of the substrates in Figure 6, interesting optical response may be expected. If the configuration of Figure 6a would behave as a regular PMC for any emitter, expectedly suppressing the radiation from emitters polarized in the direction normal to the interface and boosting the radiation of an emitter polarized along the interface, the geometry of Figure 6b has a more atypical response. Using the reciprocity theorem, it is possible to show that the radiation patterns from an optical emitter polarized along [FH(θ)] or normal [FV(θ)] to the substrate in free-space (see geometry in Figure 7) is given by

equation image

The patterns are proportional to each other, and identical when ɛd = ɛ0. Their radiation pattern is shown in Figure 7 for two values of ɛd0. For both polarizations, the radiation is identically zero at broadside and grazing angle, and the rather directive pattern has a maximum, independent on the polarization of the optical emitter, at the Brewster angle condition (7). This configuration may provide directive radiation from localized optical emitters at specific angles, independent on the polarization of the source.

Figure 7.

Horizontal and vertical dipoles placed on the surface of the layered metamaterial of Figure 6. Normalized polar radiation pattern FH(θ) for different values of ɛd. The pattern is identical for FV(θ), apart from a different normalization factor, consistent with equation (8).

4. Conclusions

[17] In this paper, we have discussed the realization and the potential optical applications of extreme boundary conditions obtained using the combination of plasmonic and dielectric materials with oppositely signed permittivity. We have discussed the extreme, highly tunable homogeneous boundary condition realized by a composite sphere, whose averaged surface impedance can, in the ideal lossless case, vary from zero to infinity upon 90 degree rotation of the impinging electric polarization. We have then extended this design to a planar interface between free-space and alternating metallo-dielectric layers. We have shown that in this configuration exotic boundary conditions may also stem from the localized resonances supported by this combination of plasmonic and dielectric materials, providing extreme boundary conditions of interest to boost and tailor the radiation of localized optical sources.

Acknowledgments

[18] This work was supported by the National Science Foundation (NSF) CAREER grant ECCS-0953311 (to A.A.), by the U.S. Air Force Office of Scientific Research (AFOSR) with grant FA9550-08-1-0220 (to N.E.) and with YIP award FA9550-11-1-0009 (to A.A.) and by the U.S. Office of Naval Research Multidisciplinary University Research Initiative (ONR-MURI) grant N00014-10-1-0942.

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