Optical Chirality Flux as a Useful Far-Field Probe of Chiral Near Fields

Article pubs.acs.org/journal/apchd5

Optical Chirality Flux as a Useful Far-Field Probe of Chiral Near Fields Lisa V. Poulikakos,† Philipp Gutsche,‡ Kevin M. McPeak,†,∥ Sven Burger,‡,§ Jens Niegemann,⊥ Christian Hafner,⊥ and David J. Norris*,† †

Optical Materials Engineering Laboratory and ⊥Institute of Electromagnetic Fields, ETH Zurich, 8092 Zurich, Switzerland ‡ Zuse Institut Berlin, 14195 Berlin, Germany § JCMwave GmbH, 14050 Berlin, Germany S Supporting Information *

ABSTRACT: To optimize the interaction between chiral matter and highly twisted light, quantities that can help characterize chiral electromagnetic fields near nanostructures are needed. Here, by analogy with Poynting’s theorem, we formulate the time-averaged conservation law of optical chirality in lossy dispersive media and identify the optical chirality flux as an ideal far-field observable for characterizing chiral optical near fields. Bounded by the conservation law, we show that it provides precise information, unavailable from circular dichroism spectroscopy, on the magnitude and handedness of highly twisted fields near nanostructures. KEYWORDS: optical chirality, plasmonics, chirality flux, circular dichroism, chirality conservation

C

plasmonic nanostructures. Rather, we wish to characterize the chirality of their intense near fields. This can be problematic with CD spectroscopy for two reasons: (i) excitation with circularly polarized light introduces local chirality in the system that is unrelated to the structure, and (ii) the differential extinction of left- and right-handed circularly polarized light can contain unwanted cancellation effects. Consequently, the amplitude and sign of the peaks in CD spectra do not directly relate to the chirality of the near field.20 Thus, a need exists for a more useful physical quantity for characterizing chiral near fields. Herein, we identify such a quantity, the optical chirality flux. By analogy with Poynting’s theorem, we formulate a timeaveraged conservation law for chirality in lossy dispersive media. We show how a chiral nanostructure can selectively dissipate linearly polarized light and generate an optical chirality flux in the far field. Further, we verify both analytically and numerically that this quantity can then provide useful information, not obtainable from CD spectroscopy, about the amplitude and handedness of the near-field chirality of the nanostructure. Therefore, the optical chirality flux provides a useful far-field observable for studying chiral near fields.

hiral shapes are those not superimposable upon their mirror image. Chirality is common in nature, e.g., in biomolecules that exist in either left- or right-handed forms (known as enantiomers). It is also seen in knotted and twisted fields, such as fluid vortices,1,2 magnetic flux tubes,3 and circularly polarized light.4 Indeed, chiral electromagnetic fields have long been exploited to characterize chiral matter. More recently, their use has also been proposed for enantioselective biosensing,5,6 enantioselective separation,7,8 asymmetric catalysis,9 and nonlinear spectroscopic imaging.10,11 In many of these applications, metallic nanostructures are utilized because their local plasmonic resonances can create concentrated electromagnetic fields.12 When the nanostructures have a chiral shape, they can exhibit a high degree of optical activity13−17 and induce intense highly twisted chiral near fields.18−20 However, it is not always clear how to utilize these near fields most effectively. For example, in enantioselective biosensing, a molecule should interact with the near field of the chiral plasmonic nanostructure.6 To maximize the effect, the near field should first be characterized through an appropriate farfield quantity. Typically, this has been done with circular dichroism (CD) spectroscopy, which records the difference in the optical extinction from a chiral object when it is illuminated with left- and right-handed circularly polarized light. (Alternative methods also exist.21,22) For most biomolecules, the differential absorption dominates the CD signal and provides information about the chiral arrangement of the molecular components. For larger objects, differential scattering becomes increasingly important, yielding data about longer range (≳20 nm) structure.23 While obtaining such knowledge is useful in many contexts, in applications such as enantioselective biosensing, we already know the nominal chiral shape of our © 2016 American Chemical Society

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RESULTS AND DISCUSSION Chirality Conservation. Chiral electromagnetic fields exhibit a twist around an axis, which in general can be referred to as optical chirality. More tightly wrapped field lines arise when a higher degree of local chirality exists.24 This can be quantified via the optical chirality density,25 Received: March 20, 2016 Published: July 11, 2016 1619

DOI: 10.1021/acsphotonics.6b00201 ACS Photonics 2016, 3, 1619−1625

ACS Photonics χ=

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1 [D·(∇ × E) + B·(∇ × H)] 2

1

respectively. Because Im(we − wm) = − 4 (ϵ″|,|2 + μ″|/|2 ), the dissipation term is nonzero only for lossy media, i.e., those with imaginary components in the permittivity or permeability. Analogously (see Supporting Information), we can write the time-averaged, time-harmonic conservation law for optical chirality in lossy dispersive media as

(1)

where E and B are the time-dependent electric and magnetic fields. We assume linear dispersive media such that D = ϵ(ω)E and B = μ(ω)H with ϵ = ϵ′ + iϵ″ as the complex electric permittivity, μ = μ′ + iμ″ the complex magnetic permeability, and ω the angular frequency. In its time-averaged form, eq 1 yields ω χ ̅ = − Im(+*·)) (2) 2

−2ω

1 [E × (∇ × H) − H × (∇ × E)] 2

1

(7)

χm =

1 [/*·(∇ × )) + )·(∇ × /*)] 8

(8)

1 [, × (∇ × /*) − /* × (∇ × ,)] 4

(9)

Im(χe − χm )= 1 [−∇ϵ′·Im(, × ,*) − ∇μ′·Im(/ × /*)] 8 1 + ω(ϵ′μ″ + ϵ″μ′)Im(,*·/) 4

(3)

(10)

Equation 10 is then integrated over the entire structure. [In previous work, eq 2 was evaluated in the space surrounding the structure.] The first term on the right-hand side of eq 10 describes optical chirality dissipation via a gradient in ϵ′ and μ′. The examples presented here study piecewise homogeneous, isotropic media where ∇ϵ(ω) and ∇μ(ω) can only be nonzero between material domains. Thus, an interface is required for this dissipation mechanism.39 Note that the conservation law in eq 6 is not restricted to piecewise homogeneous, isotropic media, but is generally applicable to anisotropic systems. Additionally, the second term on the right-hand side of eq 10 is nonzero only for lossy media (ϵ″ ≠ 0 or μ″ ≠ 0) in analogy to the dissipation term in Poynting’s theorem [eq 5]. Figure 1 summarizes these results schematically. A chiral structure (a hand) made of a homogeneous isotropic lossy dispersive medium is excited with vertically polarized light. As linear polarization is an equal superposition of left- and righthanded circular polarization, the incoming optical chirality flux is zero. If the structure then selectively dissipates optical chirality of one handedness, a nonzero outgoing optical chirality flux must be generated according to eq 6. In this case, the only chiral light contained in the scattered field is due to the structure. Because the scattered optical chirality flux can be directly related to the polarization of the scattered light,34 it is an easily measurable far-field quantity.

(4)

∫V Im(we − wm)d3x + ∫V Re(∇·:) d3x = 0

1 [+*·(∇ × ,) + , ·(∇ × +*)] 8

Equations 6 to 9 form the central result of this work. This extension of the conservation law to lossy dispersive media identifies the mechanism of optical chirality dissipation upon interaction with matter [first term in eq 6]. Furthermore, by studying its time-averaged, time-harmonic form, we establish the optical chirality flux [second term in eq 6] as a physically measurable quantity. In general, the dissipation term in eq 6 can be written as

Previously, such a law was defined for lossless dielectrics25,36 and point sources.37 We extend this treatment and consider lossy dispersive media by analogy with Poynting’s theorem for time-harmonic fields. As shown below, this yields a physical interpretation of the dissipation of optical chirality and the optical chirality flux. In contrast to prior wavelength-dependent studies, the quantities obtained in this approach are uniquely defined and bounded by the conservation law. For time-averaged, time-harmonic fields, in source-free systems, Poynting’s theorem for conservation of energy in lossy dispersive media is38 −2ω

χe =

-=

a conservation law for χ analogous to Poynting’s theorem can be obtained:24,26,34,35 d3x + ∫ Σ·n da = 0 ∫V δχ δt S

(6)

where χe and χm are defined as the complex harmonic electric and magnetic optical chirality densities and - is the corresponding optical chirality flux:

where + and ) represent complex time-harmonic fields. χ̅ is a time-even and parity-odd pseudoscalar.26 Recently, χ has been interpreted as the degree of asymmetry in the excitation rate between a molecule and its enantiomer in a chiral local field.24 Further, a significant increase in enantioselectivity was found when chiral molecules interact locally with electromagnetic fields exhibiting a high ratio between χ and the electric energy density.5,27 Consequently, χ has since been used to study chiral plasmonic nanostructures.6,18,19,28 It has identified near fields of high optical chirality enhancement, defined as the time-averaged optical chirality density [eq 2] normalized by the corresponding value for circularly polarized light.18 It has also been used to calculate wavelength-dependent quantities by spatially averaging near a structure of interest19,29,30 or by evaluating at a single spatial coordinate.31 Although this can provide an indication of the spectral dependence of χ (e.g., for comparison with CD spectra29), the results vary with choice of integration volume or spatial coordinate. Note that a significant enhancement in the CD response of chiral molecules was found when they are placed in the hot spot of an achiral plasmonic antenna.32,33 Instead of χ, we focus on the conservation of optical chirality. If χ is combined with the optical chirality flux, Σ=

∫V Im(χe − χm )d3x + ∫V Re(∇·-) d3x = 0

(5)

1

Here we = 4 (, ·+*) and wm = 4 ()·/*) are the harmonic electric- and magnetic-energy densities; , , + , ) , and / again represent the complex time-harmonic fields; and : = 1 (, × /*) is the complex Poynting vector. The two terms 2 in eq 5 represent energy dissipation and energy flux, 1620


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wavelength, the conservation law [eq 6] states that a maximum optical chirality flux is generated at the same wavelength. This flux can then be measured experimentally by means of the third Stokes parameter [eq 13]. For most applications, it is desirable to obtain a high optical chirality density [eq 2] in the near fields surrounding a chiral structure, which are easily accessible by an analyte. We now consider whether the physical quantities resulting from the conservation law in eq 6 can provide information about the magnitude and handedness of these chiral near fields. For this, we draw a parallel between electromagnetic energy and optical chirality. We consider a general system where a structure (ϵ1, μ1) is surrounded by a medium (ϵ2, μ2 with ϵ″2 and μ″2 = 0, as in most practical cases). Table 1 shows the dissipation (computed

Figure 1. Conservation of optical chirality for a chiral structure (hand) made of a lossy dispersive medium. The structure is excited (left) with achiral linearly polarized light. Due to the conservation law [eq 6], dissipation of optical chirality in the structure causes an outgoing optical chirality flux (right).

Table 1. Comparison of Dissipation (Left Column) and Density (Right Column) Terms for Electromagnetic Energy (Top Row) and Optical Chirality (Bottom Row)a

Indeed, for an electromagnetic plane wave, - is proportional to the third Stokes parameter, an experimentally observable quantity describing the degree of circular polarization.38 The amplitude of an electric field can be written as A = lALCPL + rARCPL, where l and r are the weighting factors of left- and righthanded circular polarization. Thus, - is a linear combination of components of each handedness: - = |l|2 -LCPL + |r |2 -RCPL (analogous to the power flux : ). By applying eq 9 to left- and right-handed circularly polarized light (with μ = μ0), - can be directly related to : : ω -LCPL = :LCPL (11) c ω -RCPL = − :RCPL (12) c ω - = (|l|2 :LCPL − |r |2 :RCPL) (13) c

a The dissipation terms are evaluated within the studied structure (ϵ1, μ1), while the density terms are evaluated in the surrounding medium (ϵ2, μ2, with ϵ2″ and μ2″ = 0). ,1, /1 and , 2 , /2 are the complex timeharmonic fields in the structure and surrounding medium, respectively. For simplicity, the expressions in homogeneous isotropic media are shown. See Supporting Information for treatment of anisotropic media.

within the structure) and density (computed in the surrounding medium) terms for electromagnetic energy (top row) and optical chirality (bottom row). ,1, /1 and , 2 , /2 are the complex time-harmonic fields in the structure and surrounding medium, respectively. For simplicity, the bottom row of Table 1 applies to homogeneous, isotropic media (see Supporting Information for anisotropic media). While the dissipation terms (left column Table 1) are bounded by the conservation law, the density terms (right column Table 1) vary with integration volume. For electromagnetic energy, Table 1 shows that the dissipation (left) and density (right) terms both depend on the sum of |,i|2 and |/i|2 with i = {1, 2} and the corresponding material functions. This connection between dissipation and density has significant physical implications. For example, at resonance, metallic nanoparticles exhibit maxima in dissipation of electromagnetic energy. Further, the plasmon-induced surface charges occurring at resonance lead to enhancement of the electromagnetic near fields.42 Due to this effect, it has been observed for electromagnetic energy43−45 that the nearand far-field resonances match precisely in the small-particle limit. As the particle size increases, damping effects lead to a red-shift between near and far field.43,45 The results in Table 1 show that for optical chirality an analogous relationship exists between near- and far-field resonances. Similar to electromagnetic energy, the optical chirality dissipation (left) and density (right) are both proportional to Im(,*i ·/i) with i = {1, 2} and the

Equation 13 is proportional to the third Stokes parameter. In contrast to CD spectroscopy, where the total extinction spectrum is recorded upon excitation with left- and righthanded circularly polarized light, the measurement of requires the selective analysis of the degree of circular polarization of the scattered fields. Before proceeding, we briefly compare - to another quantity that is potentially useful for characterizing chiral fields, the optical helicity.35,40,41 Because the helicity depends on the electric vector potential, it is only uniquely defined for divergence-free D fields. Further, although the optical helicity can be gauge invariant if integrated over all space,35 it cannot be determined using only quantities obtained directly from Maxwell’s equations. In contrast, the optical chirality flux can be calculated directly from the vector fields. So far, we have identified the optical chirality flux as a uniquely defined, far-field observable. The conservation law of optical chirality in lossy, dispersive media, introduced in eqs 6−9, reveals the equality of optical chirality dissipation and optical chirality flux. Thus, the solutions to Maxwell’s equations across material boundaries can now be implemented to determine the optical chirality of the entire system studied. The relationship between optical chirality dissipation and optical chirality flux has important implications in the design and optimization of chiral nanostructures for enantioselective applications. For instance, if a structure is engineered to yield a maximum optical chirality dissipation [eq 10] at a given 1621


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where E0 is the amplitude of the incoming electric field, k the wavenumber, r the radial coordinate, and θ the polar angle. Equations 15 and 16 describe the spatial dependence of the optical chirality density and optical chirality flux in this system. Equation 15 contains two terms, corresponding to the optical chirality density of the incoming and scattered fields, respectively. The same two terms are in the optical chirality flux [eq 16]. For the scattered field, this means that both the optical chirality density and flux exhibit the same resonance condition, defined by the real part of αe. This corresponds to the Fröhlich condition where Im(ϵ1) is small or slowly varying at resonance49 (see Supporting Information). Figure 2a verifies this analytical treatment via numerical simulations (see Supporting Information) of a 10-nm-diameter

corresponding material functions. Again for metallic nanoparticles, the plasmon-induced surface charges connect the spectral properties of the electromagnetic fields inside the structure (,1, /1) and the surrounding near fields (, 2 , /2 ). Thus, in the small-particle limit, wavelengths of maximum optical chirality dissipation [∝Im(,1*·/1)] match the wavelengths of maximum optical chirality density [∝Im(,*2 ·/2)]. As for electromagnetic energy, spectral shifts between near- and far-field resonances will occur with increasing particle size due to damping effects. The respective conservation laws of electromagnetic energy and optical chirality [eqs 5, 6] enable the detection of maxima in dissipation by means of the flux in the far field. We demonstrate this with specific examples below. It should be noted that the relationship between optical chirality dissipation and density enabled by our formalism applies to systems beyond homogeneous metallic nanoparticles (see Supporting Information for expressions in anisotropic media). Examples include systems supporting Fano resonances, which arise due to constructive and destructive interference of narrow discrete resonances with broad spectral lines or continua.46 Due to interference effects, these systems can exhibit polarization-invariant far-field responses despite their polarization-tunable near fields.47 Specifically, for planar chiral oligomers, it has been shown that a nonreciprocal differential dissipation of left- and right-handed circularly polarized light can be obtained by modal interference and Fano resonances.48 Quasistatic Electric-Dipole Limit. We now demonstrate the connection between optical chirality density in the near field and optical chirality flux in the far field of a structure by examining metallic nanoparticles. As a proof of concept, we focus on the small-particle regime, where damping, phase retardation, and multipolar effects are negligible.43,45 For this, we first treat a small sphere in the quasistatic electric-dipole limit analytically. In this system, both the local electric-field intensity and the scattered power exhibit a resonance due to the denominator in the electric-dipole polarizability, αe = 4π ϵ0a3{ϵ1(ω) − ϵ2}/{ϵ1(ω) + 2ϵ2}

(14)

Here a is the sphere radius and ϵ0, ϵ1, and ϵ2 are the permittivity in vacuum, the structure, and the surrounding medium, respectively.12 Because the sphere is achiral, we introduce chirality to this system by exciting it with circularly polarized light. Specifically, we study the electric field Etot = Einc + Escat and the magnetic field Htot = Hinc + Hscat ≈ Hinc, where the incident field corresponds to that of left-handed circularly polarized light propagating in +z. Furthermore, in the quasistatic limit, where the near zone of an electric dipole extends to infinity, the scattered fields are defined as the static dipole fields with harmonic time dependence.38 The optical chirality density χ̅ [eq 2] and optical chirality flux - [eq 9] are then χd̅ =

E02k 16πr 3

{16πr 3ϵ0 − Re(αe)[1 + 3 cos 2θ ]}

⎛ cos θ ⎞ ω ⎜ ⎟ χ ̅ −sin θ ⎟ -d = ⎟ 2k d ⎜⎜ ⎝ 0 ⎠

Figure 2. (a) The scattered power, :scat , and scattered optical chirality flux, -scat , for a 10-nm-diameter Ag sphere excited with left-handed circularly polarized light (inset). Both quantities are integrated over a slightly larger sphere volume and normalized. Their spectra match exactly according to the resonance in the electric-dipole polarizability αe (vertical arrow). The peak in -scat corresponds to the near-field resonance of the optical chirality density calculated analytically in the electric-dipole limit. (b) Optical chirality flux, -scat , for a 10-nmdiameter Ag nanosphere, upon excitation with left- (-scat,LCPL ) and right-handed (-scat,RCPL ) circularly polarized light. The sign of -scat changes with the handedness of the excitation source.

(15)

Ag nanosphere. The scattered power and scattered optical chirality flux are integrated over a volume slightly larger than the nanosphere to yield :scat and -scat , respectively. We note that for optical chirality, magnetic dipolar contributions must also be considered, as discussed in the Supporting Information. As expected, a peak appears in the spectrum of each quantity at

(16) 1622


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of-principle calculations, as their small size allows a physical interpretation of resonances within the dipole limit, where damping, phase retardation, and multipolar effects are negligible.43,45 With finite-element simulations (see Supporting Information), we excite the structure with x-polarized light propagating along +z. Figure 3a maps the optical chirality enhancement of the dimer at a transverse (359 nm) and longitudinal (500 nm) resonance for a slice in the x−y plane at z = 0 (the center of the rods). (Further transverse resonances at 347 and 369 nm are shown in the Supporting Information.) The three panels of Figure 3b plot the optical chirality density [eq 2] integrated over the space near the dimer, the scattered component of the dimer’s CD spectrum (again excited along +z), and the optical chirality flux scattered in the forward direction (i.e., forward half-space). We chose this last parameter because the polarization of forward scattered light can be detected in an experimental transmission measurement. Due to the conservation law [eq 6], the optical chirality flux can be determined with a surface integral of eq 9 over the boundaries of the computational domain, following Gauss’s law. Thus, unlike the volume integral of the optical chirality density [top panel in Figure 3b], the optical chirality flux remains constant for any computational domain containing the entire structure (see Supporting Information). According to the plots in Figure 3a, the transverse resonance exhibits primarily positive (left-handed) optical chirality density, while the longitudinal resonance is primarily negative (right-handed). This change of sign is also seen in the integrated optical chirality density and flux [top and bottom panels of Figure 3b]. However, the integrated optical chirality density (top panel) does not match the polarity of the sign in Figure 3a, i.e., positive at 359 and negative at 500 nm. Indeed, as expected, we found that the shape and sign of the χ̅ spectrum depended on the integration volume chosen. Meanwhile, the CD spectrum (center panel), computed upon excitation in the +z direction, does not even exhibit a resonance at 500 nm. The signals at this longitudinal resonance for left- and right-handed circularly polarized light are equal and cancel. This effect is confirmed in Figure 4a and b, where the

the resonance in Re(αe). At this frequency, a near-field resonance also occurs in the optical chirality density [eq 15]. Thus, a direct connection exists between the optical chirality density in the near field of a structure and the optical chirality flux in the far field. In Figure 2b we show that excitation with left- or righthanded circularly polarized light results in mirror symmetric -scat spectra for an achiral nanosphere. This is expected from the chirality conservation law because the sphere is excited with an incoming optical chirality flux of positive or negative sign, respectively. Chiral Plasmonic Nanorod Dimer. Due to the connection between optical chirality density in the near field and optical chirality flux in the far field of a structure, one might conclude that these quantities are equivalent for characterizing the chiral near field. However, - is clearly superior because it is bounded by the conservation law. We confirm this with a specific example. For simplicity, we treat a Ag nanorod dimer that exhibits only two-dimensional chirality (see Figure 3). Similar configurations of chiral nanorod dimers have been studied in prior work50−52 and have proven particularly suitable for proof-

Figure 3. (a) Near-field maps (z = 0 plane) of the optical chirality enhancement of a Ag nanorod dimer with hemispherically capped cylindrical nanorods (40 nm diameter, 80 nm length, 20 nm spacing). The right nanorod is shifted 40 nm in the −y direction to create a twodimensionally chiral structure. The dimer is excited with x-polarized light propagating along +z at a transverse (359 nm) and longitudinal (500 nm) resonance. (Further transverse resonances at 347 and 369 nm are shown in the Supporting Information.) The transverse (longitudinal) resonance shows primarily positive (negative) optical chirality density. (b) Top: integrated optical chirality density [eq 2] in newtons. Center: scattered component of the dimer’s CD spectrum in watts. Bottom: forward-scattered optical chirality flux in N/s. For comparison, − -scat,forward is plotted. The dominant sign of the field maps in (a) is also seen in the optical chirality flux. In the CD spectrum, the longitudinal resonance is absent.

Figure 4. (a) Absorption spectra upon excitation with left- (absLCPL) and right-handed circularly polarized light (absRCPL). (b) Scattering spectra upon excitation with left- (scatLCPL) and right-handed circularly polarized light (scatRCPL). (c) Scattered power spectrum (scatxPL) upon excitation with x-polarized light. 1623


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Notes

absorbed and scattered spectra are shown for left- and righthanded circularly polarized light separately. The CD spectrum results from the difference between these two components, which are identical at long wavelengths. Similar cancellation effects have been observed for chiral plasmonic nanorod dimers in prior work52 and can be explained within the coupled-dipole oscillator model.50,51 [For comparison, the scattered power spectrum upon x-polarized light excitation, scatxPL, is shown in Figure 4c.] Figure 4a and b also show that the largest differential absorption and scattering occurs at 369 nm, leading to a large CD amplitude. However, due to cancellation effects, this differential component is not necessarily related to the optical chirality density in the near field of the chiral nanorod dimer. Further details on the CD spectra are discussed in the Supporting Information. Finally, we note that a nonzero optical chirality flux represents a dominance of either left- or right-handed optical chirality that is scattered into the far field. High optical chirality flux can occur even for a resonance where the local near-field enhancement is small. The nanorod dimer displays such a resonance at 500 nm, as indicated by the high magnitude in -scat despite low values in the optical chirality enhancement [Figure 3a] and scattered power [Figure 4c]. Finding such resonances can be extremely useful (e.g., in enantioselective separation, where the handedness of chiral optical forces is determined by the gradient in optical chirality density and the optical chirality flux7) and is not possible via the local optical chirality density, which scales with the field enhancement.53

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank C. Genet, A. Canaguier-Durand, M. Schäferling, E. De Leo, F. Prins, B. le Feber, and F. Schmidt for helpful discussions. This work was supported by the Swiss National Science Foundation (Award No. 200021-146747) and the Einstein Foundation Berlin (ECMath, project SE6). Computations were performed at the ETH High-Performance Computing Cluster Brutus.

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(1) Moffatt, H. K. The degree of knottedness of tangled vortex lines. J. Fluid Mech. 1969, 35, 117−129. (2) Scheeler, M. W.; Kleckner, D.; Proment, D.; Kindlmann, G. L.; Irvine, W. T. M. Helicity conservation by flow across scales in reconnecting vortex links and knots. Proc. Natl. Acad. Sci. U. S. A. 2014, 111, 15350−15355. (3) Berger, M. A. Introduction to magnetic helicity. Plasma Phys. Controlled Fusion 1999, 41, B167. (4) Kong, J. A. Electromagnetic Wave Theory, 2nd ed.; John Wiley & Sons, 1942. (5) Tang, Y.; Cohen, A. E. Enhanced enantioselectivity in excitation of chiral molecules by superchiral light. Science 2011, 332, 333−336. (6) Hendry, E.; Carpy, T.; Johnston, J.; Popland, M.; Mikhaylovskiy, R. V.; Lapthorn, A. J.; Kelly, S. M.; Barron, L. D.; Gadegaard, N.; Kadodwala, M. Ultrasensitive detection and characterization of biomolecules using superchiral fields. Nat. Nanotechnol. 2010, 5, 783−787. (7) Canaguier-Durand, A.; Hutchison, J. A.; Genet, C.; Ebbesen, T. W. Mechanical separation of chiral dipoles by chiral light. New J. Phys. 2013, 15, 123037. (8) Tkachenko, G.; Brasselet, E. Optofluidic sorting of material chirality by chiral light. Nat. Commun. 2014, 5, 3577. (9) Takano, Y.; Takahashi, J.-i.; Kaneko, T.; Marumo, K.; Kobayashi, K. Asymmetric synthesis of amino acid precursors in interstellar complex organics by circularly polarized light. Earth Planet. Sci. Lett. 2007, 254, 106−114. (10) Rodrigues, S. P.; Lan, S.; Kang, L.; Cui, Y.; Cai, W. Nonlinear imaging and spectroscopy of chiral metamaterials. Adv. Mater. 2014, 26, 6157−6162. (11) Rodrigues, S. P.; Cui, Y.; Lan, S.; Kang, L.; Cai, W. Metamaterials enable chiral-selective enhancement of two-photon luminescence from quantum emitters. Adv. Mater. 2015, 27, 1124− 1130. (12) Novotny, L.; Hecht, B. Principles of Nano-Optics, 2nd ed.; Cambridge University Press, 2012. (13) Gansel, J. K.; Thiel, M.; Rill, M. S.; Decker, M.; Bade, K.; Saile, V.; von Freymann, G.; Linden, S.; Wegener, M. Gold helix photonic metamaterial as broadband circular polarizer. Science 2009, 325, 1513− 1515. (14) Hentschel, M.; Schäferling, M.; Weiss, T.; Liu, N.; Giessen, H. Three-dimensional chiral plasmonic oligomers. Nano Lett. 2012, 12, 2542−2547. (15) Fan, Z.; Govorov, A. O. Chiral nanocrystals: plasmonic spectra and circular dichroism. Nano Lett. 2012, 12, 3283−3289. (16) McPeak, K. M.; van Engers, C. D.; Blome, M.; Park, J. H.; Burger, S.; Gosálvez, M. A.; Faridi, A.; Ries, Y. R.; Sahu, A.; Norris, D. J. Complex chiral colloids and surfaces via high-index off-cut silicon. Nano Lett. 2014, 14, 2934−2940. (17) Hu, T.; Isaacoff, B. P.; Bahng, J. H.; Hao, C.; Zhou, Y.; Zhu, J.; Li, X.; Wang, Z.; Liu, S.; Xu, C.; Biteen, J. S.; Kotov, N. A. Selforganization of plasmonic and excitonic nanoparticles into resonant chiral supraparticle assemblies. Nano Lett. 2014, 14, 6799−6810. (18) Schäferling, M.; Dregely, D.; Hentschel, M.; Giessen, H. Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures. Phys. Rev. X 2012, 2, 031010.

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CONCLUSION In summary, we have formulated the conservation law of optical chirality in its time-averaged form for lossy dispersive media. We identify the optical chirality flux as a physically useful observable that provides information on the magnitude and dominant handedness of the optical chirality in the near field of a structure. The same information cannot be obtained with conventional CD spectroscopy. Thus, the optical chirality flux can be exploited as a far-field probe of local resonances at which one handedness dominates the chiral electromagnetic fields. This can allow more effective use of chiral plasmonic nanostructures in enantioselective applications.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.6b00201. Derivation of the time-averaged conservation law of optical chirality, discussion on cancellation effects in CD spectra, calculations for the quasistatic electric-dipole limit, information on the finite-element simulations (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address ∥

Louisiana State University, Baton Rouge, Louisiana 70803, United States. 1624


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