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Polarization-Tailored Fano Interference in Plasmonic Crystals: A Mueller Matrix Model of Anisotropic Fano Resonance Subir K. Ray,† Shubham Chandel,† Ankit K. Singh, Abhishek Kumar, Arpita Mandal, Subhradeep Misra, Partha Mitra,* and Nirmalya Ghosh* Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Kolkata, Mohanpur 741246, India S Supporting Information *

ABSTRACT: Fano resonance is observed in a wide range of micro- and nano-optical systems and has been a subject of intensive investigations due to its numerous potential applications. Methods that can control or modulate Fano resonance by tuning some experimentally accessible parameters are highly desirable for realistic applications. Here we present a simple yet elegant approach using the Mueller matrix formalism for controlling the Fano interference effect and engineering the resulting asymmetric spectral line shape in an anisotropic optical system. The approach is founded on a generalized model of anisotropic Fano resonance, which relates the spectral asymmetry to physically meaningful and experimentally accessible parameters of interference, namely, the Fano phase shift and the relative amplitudes of the interfering modes. The differences in these parameters between orthogonal linear polarizations in an anisotropic system are exploited to desirably tune the Fano spectral asymmetry using pre- and postselection of optimized polarization states. The concept is demonstrated on waveguided plasmonic crystals using Mueller matrix-based polarization analysis. The approach enabled tailoring of several exotic regimes of Fano resonance in a single device, including the complete reversal of the spectral asymmetry, and shows potential for applications involving control and manipulation of electromagnetic waves at the nanoscale. KEYWORDS: Fano resonance, polarization, plasmonics, Mueller matrix, scattering dielectric objects, etc.3−14 Fano resonances in such a wide variety of micro- and nano-optical systems have been the subject of intensive investigations due to their numerous potential applications such as in sensing, switching, lasing, filters, robust color display, nonlinear and slow-light devices, and invisibility cloaking.2,4,5,19−22 Most of the aforementioned applications are known to critically depend upon the ability to control or modulate the asymmetry of the line shape by external means.4 Thus, tuning the Fano resonance via some experimentally accessible parameters is highly desirable for realistic applications4 as well as for fundamental studies.18 Here, we present a simple yet elegant approach for controlling the Fano interference effect and tuning the resulting asymmetric line shape in an anisotropic optical system by Mueller matrix23,24-based polarization analysis. The proposed method is founded on a generalized model of anisotropic Fano resonance and exploits the different polarization response

F

ano resonance is an universal phenomenon that exhibits a characteristic asymmetric spectral line shape, observed in a wide range of atomic, molecular,1,2 optical,3−15 nuclear,16 and solid-state systems.17 Originally described in the context of interacting quantum systems,1 the asymmetric spectral line shape in a Fano resonance emerges from the socalled configuration interaction due to the interference of a discrete excited state with a continuum of states.1,3,4 The coupling of the discrete and the continuum states in the configuration interaction is modeled using the so-called Fano asymmetry parameter, q.1 Despite voluminous literature available on this effect,1−17 there still remains considerable interest toward a fundamental understanding of this intriguing phenomenon using physically meaningful parameters.18 Although Fano resonance is regarded as a characteristic feature of interacting quantum systems, this can also be observed in classical optical phenomena because interference is ubiquitous in classical optics. Indeed, Fano-type spectral asymmetry has been observed in the scattered intensity from various optical systems, in plasmonic nanostructures, in electromagnetic metamaterials, in photonic crystals, in Mie scattering from © 2017 American Chemical Society

Received: November 3, 2016 Accepted: January 10, 2017 Published: January 10, 2017 1641

DOI: 10.1021/acsnano.6b07406 ACS Nano 2017, 11, 1641−1648

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⎡ (qeff + ϵ)2 ⎤ (B − 1)2 ⎥+ 2 Is(ω) = |Es(ω)|2 ≈ B2 × ⎢ 2 ⎢⎣ (ϵ + 1) ⎥⎦ (ϵ + 1)

(anisotropy) of the two interfering scattering modes to achieve control over the Fano resonance. The model suggests that the spectral asymmetry can be desirably tuned by modulating two experimentally accessible parameters of interference, namely, the Fano phase shift and the relative amplitudes of the interfering modes. Experimental control and tuning of the asymmetric spectral line shape is demonstrated on waveguided plasmonic crystals by pre- and postselection of optimized polarization states of light. The demonstrated control over Fano resonance offered by this remarkably simple approach may lead to promising applications of anisotropic Fano resonance.

The first term represents the Fano-type asymmetric spectral line shape with an effective asymmetry parameter qeff = q/B. The second term corresponds to a Lorentzian background, previously reported in the context of Fano resonance on diverse optical systems.6,25 The introduction of the parameters (q − i) and B in eq 1 is therefore crucial, as it leads to the useful expression (eq 3) for Fano resonance in the scattered intensity and enables intuitive interpretation of the essential spectral features by two physically meaningful parameters of interference, the Fano phase shift (φF) and the relative amplitude (B) of the interfering scattered fields. The parameter B controls the contrast of the Fano interference, and for the ideal case of B = 1, perfect destructive interference (Is(ωF) = 0) occurs at the qγ Fano frequency ωF = ω0 − 2 (corresponding to ϵ = −q,

THEORY Fano Resonance in Scattering from Anisotropic Systems: A Generalized Formalism. We begin with a phenomenological model where a Fano-type spectral asymmetry in the scattered intensity profile naturally arises due to the interference of the scattered fields of a narrow resonance with a broad spectrum (continuum). A recent temporal phase formalism introduced to describe Fano resonance in an absorbing system18 showed that the spectral asymmetry of Fano resonance (described by the q-parameter) arises due to an additional phase shift (φF) of the temporal dipole response of the discrete state with respect to the continuum. On conceptual grounds, a similar type of phase factor can also be introduced to describe the Fano resonance in scattering of electromagnetic waves. We thus model the interfering scattered field of the narrow resonance by a complex Lorentzian (jR(ω)), which is shifted by an additional phase factor with respect to the field of the broad spectrum with relative field amplitude (B), assumed to be independent of frequency ω (ideal continuum). The resultant electric field is given by

(

Here, ϵ = ϵ(ω) =

ω − ω0 , (γ / 2)

(

intensity is maximum for the frequency ωm = ω0 +

γ 2q

)

corresponding to ϵ = 1/q and ωm > ω0 (ω0) for positive (negative) values of q, and as q → ∞ (φF → 0), one obtains the symmetric Lorentzian line shape. It also follows from eq 3 that the

⎡ (q − i ) ⎤ + B⎥ Es(ω) ≈ [j R (ω) + B] = ⎢ ⎣ (ϵ+i) ⎦ ⎡ 2 ⎤ q + 1 iψ (ω) ⎢ = + B )⎥ e ⎢ ϵ2 +1 ⎥ ⎣ ⎦

(3)

(2)

The resulting expression for the scattered intensity can be obtained as 1642

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ACS Nano vector S (a 4 × 1 real vector describing the state of polarization of light through measurable intensities) into another one and fully characterizes the polarization properties of any optical system. The Mueller matrix corresponding to the Jones matrix of eq 4 is a matrix of a diattenuating retarder exhibiting linear diattenuation and retardance effects23,26,27 (the form of the Mueller matrix is provided in the Supporting Information, eq S7). Here, the linear diattenuation is defined as the differential attenuation of orthogonal linear polarizations either by scattering or by absorption, and linear retardance is defined as the phase difference between orthogonal linear polarizations. The corresponding magnitudes of linear diattenuation (d) and linear retardance (δ) are given by d=

|J11|2 − |J22 |2 2

2

|J11| + |J22 |

=

qeff =

M34 M44

(5)

The information on the amplitude and the phase anisotropy of Fano resonance is encoded in the parameters d and δ, respectively, which can be determined from experimental Mueller matrices, as we demonstrate subsequently. Equations 1, 3, and 4 reveal that the polarization state offers a convenient handle to directly control the Fano phase shift (φF) and the relative amplitude (B) parameters and the resulting spectral asymmetry (qeff) in anisotropic system. This can be achieved by pre- and postselection of polarization states, and the corresponding expressions for the scattered intensity can be written as Is(ω) = |E+β J Eα|2 =

1 T Sβ M Sα 2

q , B

⎛ sin α sin β ⎞ Beff = ⎜1 + ⎟B cos α cos β ⎠ ⎝

(8)

RESULTS AND DISCUSSION Anisotropic Fano Resonance in a Waveguided Plasmonic Crystal: Mueller Matrix Analysis. The waveguided plasmonic crystal samples consisted of a two-dimensional periodic array of gold (Au) nanodisks (or nanoellipses) on top of an indium tin oxide (ITO) waveguiding layer coated on a quartz substrate (see Materials and Methods section). The elastic scattering spectral Mueller matrices M from the waveguided plasmonic crystal samples were recorded using a home-built spectroscopic Mueller matrix system integrated with a dark-field microscope. Focused annular-shaped white light was used as an excitation source in this arrangement (see Materials and Methods section). Typical SEM images of a Au circular disk array and an elliptical disk array patterned on an ITO-coated quartz substrate are shown in Figure 1a and b, respectively. It has been shown that for such waveguided plasmonic crystals the localized surface plasmons (of metallic nanostructures) can couple to the guided modes of the underlying waveguide (the ITO dielectric layer).8−12,29,30 Thus, in the presence of the periodic metal nanostructures, the bound guided modes couple to the photon continua and become leaky, forming quasiguided hybrid modes.29,30 The interference of the scattered fields of these quasiguided modes (acting as the narrow resonance peaking at E0 = ℏω ≈ 1.777 eV, λ0 ≈ 698 nm) and the dipolar plasmon resonance of the Au disk array (acting as the broad continuum) leads to Fano resonance in the scattering spectra (peak at Em ≈ 1.896 eV, λm ≈ 654 nm), as evident from the observed spectral asymmetry in the polarization-blind scattering spectra (M11 element) from a Au circular disk array (Figure 1c). Fitting the data with eq 3 yields the effective asymmetry parameter qeff ≈ +0.90 and the Fano phase factor φF ≈ −63.2° (determined using eq 2). The corresponding parameters in a similar Au ellipse array (Figure 1c) were qeff ≈ +1.562 → φF ≈ −47.5°. The origin of anisotropic Fano resonance in the waveguided plasmonic crystals is schematically illustrated in Figure 1d. The anisotropy

(6)

Here, Eα = [cos α sin α eiΦα], TEβ = [cos β sin β eiΦβ]T are Jones vectors for general elliptical polarizations, and Sα and Sβ are their corresponding Stokes vectors.23,24 Although both the narrow resonance and the continuum modes may exhibit the anisotropy effect, since the spectral asymmetry is exhibited around the spectral window of the narrow resonance, the anisotropic polarization response of the narrow resonance mode is more pertinent for tuning the Fano spectral asymmetry using polarization states of light. In this regard, two types of anisotropic systems are of potential importance: type (a), where the narrow resonance exhibits both phase and amplitude anisotropy (qx ≠ qy, ϵx ≠ ϵy),10−12 and type (b), where the narrow resonance is perfectly diattenuating: jRx ≠ 0 and jRy = 0.8,9 Equation 6 can be approximated in the form of eq 3 with effective asymmetry parameter qeff and effective relative amplitude parameter Beff. For type (a) anisotropy, qeff can be controlled by directly modifying φF (or q = −cot φF) using preand postselections in linear polarization basis (Φα = Φβ = 0). In the limit (ω0,x − ω0,y) ≪ γ, the corresponding parameters can be approximated as qeff =

B

, eff

For simplicity, we have assumed the continuum mode to be isotropic (Bx = By) although Bx ≠ By can also be incorporated. Note that the above formalism is based on Jones algebra (eq 4), which cannot encompass partial polarization states and depolarizing interactions (loss of degree of polarization). The corresponding Mueller matrix M (eq 6) is a nondepolarizing Mueller−Jones matrix. However, in practical situations, depolarization may arise due to several causes such as spatial averaging (incoherent addition) of polarized intensities and averaging over many scattering angles.23,26 This is the case for our microscopic experimental geometry, as described subsequently (see also Supporting Information). In such a situation, the above analysis can be implemented by filtering out the depolarized component of the scattered light using decomposition of the experimental Mueller matrix into basis matrices of a depolarizer (MDepol) and a nondepolarizing diattenuating retarder (Mueller−Jones matrix MPol),23,26−28 and the latter can be subjected to analysis using eq 6 (see Supporting Information). In what follows, we experimentally demonstrate the validity of the model of anisotropic Fano resonance in waveguided plasmonic crystals by Mueller matrix analysis. We then illustrate tuning of Fano spectral asymmetry by Mueller matrix-based polarization state control.

M12 M11

δ = arg(J11) − arg(J22 ) = tan−1

q

⎡ (q × cos α cos β + q × sin α sin β) ⎤ x y ⎥ q≈⎢ ⎢⎣ ⎥⎦ cos(α − β) (7) eff

For type (b) anisotropy, q can be controlled by changing Beff with 1643

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Figure 1. Fano resonance in the scattering spectra of waveguided plasmonic crystals. Typical SEM images of (a) a Au circular disk array and (b) a Au elliptical disk array. (c) The scattering spectra (E = ℏω = 1.710−3.1 eV, corresponding to λ = 400−725 nm) from the Au disk array (blue solid line) and ellipse array (red solid line) exhibit Fano spectral asymmetry. Dimensions of the disk and ellipse array are (D = 160 nm, L = 480 nm) and (Dx = 134 nm, Dy = 95 nm, L = 480 nm), respectively. Theoretical fits of the spectra (for E = 1.710−2.066 eV) with eq 3 are shown by dotted lines, and the values of the fitted parameters (qeff, B) are noted. The peaks of the narrow Lorentzian resonance and the resulting Fano resonance were at (E0 = 1.777 eV, λ0 = 698 nm; Em = 1.896 eV, λm = 654 nm) and (E0 = 1.784 eV, λ0 = 696 nm; Em = 1.850 eV, λm = 671 nm) for the circular disk and the ellipse array, respectively. (d) Illustration of the origin of anisotropic Fano resonance in the waveguided plasmonic crystals (see text for details).

Figure 2. Manifestation of the anisotropic nature of Fano resonance in the Au circular disk array (corresponding to Figure 1) in the spectral scattering Mueller matrix. (a) Experimental scattering Mueller matrix M (E = 1.710−2.254 eV shown here) of the Au circular disk array (corresponding to Figure 1). The elements are normalized by the M11 element. (b) Spectral variation of the Mueller matrix-derived linear diattenuation d (blue solid line, left axis) and linear retardance δ (red dashed line, right axis). The δ and d parameters exhibit rapid variations across the narrow resonance peak of the quasiguided mode (E ≈ 1.8 eV or λ ≈ 690 nm) as a characteristic signature of anisotropic Fano resonance. The magnitudes of the rapidly varying components dFano and δFano across the peak of the narrow resonance, which are pertinent to anisotropic Fano resonance, are noted.

may arise from two distinct scenarios: (i) the differential polarization response of the transverse magnetic (TM) and the transverse electric (TE) quasiguided resonance modes excited by x/y linear polarizations (respectively) in the dark-field illumination geometry. The scattered fields corresponding to the TM/TE quasiguided narrow resonance modes may accordingly exhibit both phase and amplitude differences between orthogonal linear polarizations. (ii) The anisotropic shape of the Au-disk array may lead to additional anisotropy of the dipolar plasmon resonance mode (broad continuum). For the Au ellipse array, the differential polarization response of both the narrow quasiguided mode and the broad dipolar plasmon mode may thus contribute to the resulting anisotropy. In the case of the Au circular disk array, on the other hand, the differential polarization response of the quasiguided mode is expected to be the primary source of the Fano resonance anisotropy. However, the anisotropy of the narrow resonance mode is more pertinent for tuning the Fano spectral asymmetry (using eq 6), and this also enables a relatively simple recipe for tuning the spectral asymmetry by pre- and postselection of polarization states (as shown in eq 7 and eq 8). For the experimental demonstration of the proposed approach (based on eqs 6−8), the Au circular disk array instead of the ellipse array is therefore chosen for subsequent Mueller matrix analysis. The anisotropy of Fano resonance is manifested as nonzero off-diagonal elements in the Mueller matrix M of the Au circular disk array (Figure 2a). Additionally, M contains depolarization contributions, which are reflected in the diagonal elements (whose magnitudes are less than unity).23,26 Since in our high numerical aperture (NA) microscopic geometry excitation and collection of scattered light is performed over a range of angles, incoherent addition of polarized intensities

corresponding to each scattering angle leads to the depolarization effect (see Supporting Information). These depolarizing contributions of the scattered light are efficiently filtered out (via MDepol) using polar decomposition of the Mueller matrix28 (see Supplementary Figure S2). The resulting nondepolarizing MPol matrix (shown in Supplementary Figure S2) is used to glean the phase and the amplitude anisotropy effects, signatures of which are characteristically encoded in the (M34/M43, M24/ M42) elements and (M12/M21, M13/M31) elements, respectively. The phase and the amplitude anisotropy effects are subsequently quantified through linear retardance δ and diattenuation d parameters (Figure 2b), respectively (determined using eq S4 in the Supporting Information). As previously noted, the scattered fields corresponding to the TM(x)/TE(y) quasiguided modes may exhibit both phase difference (φF,x ≠ φF,y or qx ≠ qy) and amplitude difference (due to both qx ≠ qy and E0,x ≠ E0,y or ω0,x ≠ ω0,y) between orthogonal linear polarizations. This differential polarization response of the TM/TE quasiguided modes appears to be the primary source of the observed Fano resonance anisotropy in the Au circular disk array. This is evident from the corresponding rapid variation of δ and d anisotropy parameters across the narrow resonance peak of the quasiguided mode (E ≈ 1.8 eV or λ ≈ 690 nm). Moreover, since the spectral asymmetry is exhibited around the spectral window of the narrow resonance of the quasiguided mode, the rapidly varying components (δFano and dFano in Figure 2b) around this spectral range are pertinent to the anisotropy in the Fano spectral 1644

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ACS Nano asymmetry. Note that the δ parameter additionally possesses a frequency-independent background ∼0.8 radian, which is not pertinent to the anisotropic Fano resonance. It is also observed (from MPol in Supplementary Figure S2) that M34/M43 ≫ M24/ M42 (yields orientation angle of anisotropy axis ∼5 ± 0.6°), confirming that the TM(x) and TE(y) linear polarizations are the system eigenstates. We now turn to inspect whether the anisotropy in the Fano eff spectral asymmetry (qeff x ≠ qy , differences in the effective asymmetry parameters between orthogonal linear polarizations) are indeed linked to the Mueller matrix-derived anisotropy parameters δFano and dFano, as predicted by eqs 4 and 5. The MPol matrix is thus utilized further to determine the q-parameters for the two orthogonal linear polarizations (TM, x; TE, y polarization) (Figure 3a and b). For this purpose, the

polarization (qeff y = 0.861, qy = 0.471 → φF,y ≈ −64.8°) provide conclusive evidence of the anisotropic nature of the Fano spectral asymmetry, albeit with relatively weaker magnitude (small differences in the parameters). Note that in addition to the differences in (qx − qy) or Fano phase (φF,x − φF,y), the resonance frequencies of the quasiguided mode are also slightly different for x/y polarizations (E0,x ≠ E0,y or λ0,x ≠ λ0,y noted in the caption of Figure 3). The estimated Fano phase and amplitude parameters (qx/qy, Bx/By) were subsequently used to predict (using eqs 4 and 5) the spectral variations of the retardance (δFano) and diattenuation (dFano) parameters (shown in Figure 3c and d). The computed δFano and dFano parameters exhibit rapid spectral variation across the narrow resonance peak of the quasiguided mode and are in good agreement with the corresponding variations obtained independently from the experimental Mueller matrix (δFano and dFano shown in Figure 2b). This establishes self-consistency of our analysis and demonstrates that the Mueller matrix approach enables one to check the accuracy of the Fano asymmetry parameters and their eff anisotropy (qx/qy, qeff x /qy ), which are determined by fitting the spectral variation of the intensity to eq 3. The physical connection between the retardance and the diattenuation parameters with the Fano phase (φF) and the relative amplitude (B) of the interfering modes in the waveguided plasmonic crystal samples is worth a brief mention here. It can be interpreted from eqs 4 and 5 that the observed rapid spectral variation of the retardance (phase anisotropy) parameter δFano is primarily related to the difference in the Fano phase between orthogonal linear polarizations (φF,x − φF,y or qx − qy), which is the phase anisotropy of the quasiguided narrow resonance mode. The diattenuation (amplitude anisotropy) parameter dFano, on the other hand, is influenced by both qx ≠ qy associated with the quasiguided narrow resonance mode and Bx ≠ By of the broad dipolar plasmon mode. However, the amplitude anisotropy of the broad dipolar plasmon mode (Bx ≠ By) yields a smoothly varying background only, and thus the amplitude anisotropy of the narrow resonance mode (qx ≠ qy) is primarily responsible for the observed rapid spectral variation of dFano. The above results demonstrate that the experimentally observed effective spectral asymmetry can be mapped to two physical parameters of interference, the Fano phase and the relative amplitude parameter (qeff → φF, B). The Fano resonance anisotropy originating from the differences in these parameters for orthogonal polarizations leaves its characteristic signature as rapidly varying spectral retardance (δ) and diattenuation (d) effects in the Mueller matrix. The δ and d parameters thus hold promise as useful experimental metrics for probing and analyzing the phase and the amplitude anisotropy effects (respectively) of the Fano resonance. Polarization-Controlled Tuning of the Fano Resonance. We now demonstrate that the Fano spectral asymmetry can be tuned by Mueller matrix-based polarization state control. The results presented in Figures 2 and 3 revealed that the waveguided plasmonic crystal sample exhibits anisotropy in the Fano spectral asymmetry, and the corresponding Mueller matrix analysis demonstrated the validity of our model of anisotropic Fano resonance. The general recipe provided by the model (eq 6−8) may now be explored for desirable tuning of Fano resonances in such anisotropic systems. Tuning of the Fano asymmetry parameter (qeff) by pre- and postselection of optimized polarization states (using eq 6) is illustrated in Figure 4a. Here, the preselected state is optimized to be elliptical (α =

Figure 3. Quantification of the anisotropy in the Fano asymmetry (q) parameters. The spectral variation of scattered intensities for (a) TM (x polarization) and (b) TE (y polarization). These are obtained by pre- and postselection (using eq 6) of corresponding polarization states (Stokes vector: [1 1 0 0]T for x and [1 −1 0 0]T for y) on the MPol matrix (corresponding to Figure 2). Theoretical fits of the spectra (for E = 1.710−2.066 eV) with eq 3 are shown by eff dotted lines, and the estimated values for (qeff x , qx, Bx)/(qy , qy, By) parameters are noted. The parameters of the resonance were estimated to be (E0,x = 1.773 eV, λ0,x = 699 nm; Em,x = 1.890 eV, λm,x = 656 nm) for x and (E0,y = 1.779 eV, λ0,y = 697 nm; Em,y = 1.904 eV, λm,y = 651 nm) for y polarization. Differences in the qparameters between x polarization and y polarizations provide conclusive evidence of anisotropic Fano resonance. The theoretical predictions (using the estimated (qx/qy, Bx/By) parameters in eqs 4 and 5) of spectral variation of (c) linear diattenuation dFano and (d) linear retardance δFano. The rapid spectral variations of these parameters across the narrow resonance peak of the quasiguided mode and good agreement with the corresponding variations obtained independently from the experimental Mueller matrix (δFano and dFano in Figure 2b) establish self-consistency of the analysis.

spectral variations of the scattered intensities Is(ω) for the two orthogonal linear polarizations (x and y) are obtained by preand postselection (using eq 6) of corresponding polarization states (Stokes vector: [1 1 0 0]T for x and [1 −1 0 0]T for y) on the MPol matrix (corresponding to Figure 2). The resulting spectra are fitted with eq 3 to obtain the q-parameters. Differences in the q-parameters and the Fano phases between x polarization (qeff x = 0.931, qx = 0.542 → φF,x ≈ −61.5°) and y 1645

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Figure 4. (a) Polarization controlled tuning of Fano spectral asymmetry in a moderately anisotropic experimental waveguided plasmonic crystal sample (corresponding to Figures 2 and 3) and (b) theoretical predictions for a strongly anisotropic system. The preselected state is elliptical (α = +45°, Φα − Φβ ≈ 0.8 radian, Sα = [1 0 0.697−0.717]T) in (a) and linear (α = +45°, Φα − Φβ = 0, Sα = [1010]T) in (b). Postselections are at different linear polarization angles β (Sβ = [1 cos 2β sin 2β 0]T) (marked by arrows). The spectral variations of the scattered intensities for different β (obtained using respective Sα and Sβ in eq 6) were fitted to eq 3, and the resulting parameters (qeff, Fano phase φF, or q) are noted. In (a), the pre- and postselections are done on the experimental MPol matrix. In (b), the scattered intensities are simulated using eqs 4 and 6 with input parameters (qx = 1.5, qy = 0.5, Bx = By = B = 0.6) and (E0,x = 1.894 eV, λ0,x = 654 nm; E0,y = 1.882 eV, λ0,y = 659 nm). In (a), the reversal of the Fano spectral asymmetry (qeff: positive → negative) for the postselected linear polarization angle β = 130° is shown by reversed direction of the arrow, and the corresponding changes in the spectral line shapes are highlighted in the inset. The inset of (b) highlights the spectral shapes corresponding to the four interesting regimes of Fano resonance tailored in the strongly anisotropic system: qeff > 0, qeff → 0 exhibiting a symmetric Lorentzian dip, qeff → ∞ exhibiting a symmetric Lorentzian peak, and qeff < 0 exhibiting a reversal of the Fano asymmetry.

positive → negative) by making postselections in the range of linear polarization angles 90° < β < 135°. The reversal of the effective spectral asymmetry is expected to arise due to the reversal of the sign of the resulting q-parameter or the Fano phase (φF) in eq 7. As evident from Figure 4a, this reversal of the Fano spectral asymmetry is indeed observed even for our moderately anisotropic waveguided plasmonic crystal sample, where the Fano phase is reversed: φF: −61° for β = 0° → +88° for β = 130°. This is of particular practical interest because potentially a Fano spectral dip (energy/wavelength EF/λF corresponding to the intensity minima) can be turned to a spectral peak (Em/λm) or vice versa, enabling a large tunability of EF/λF. In this example ∼60 nm tunability in λF is achieved (λF ≈ 720 nm for β = 0° to λF ≈ 660 nm for β = 130°). These features can be seen more prominently in the Fano resonant part of the fitted spectral intensity profile (fitted first term in eq 3 is shown separately in Supplementary Figure S4). The ability to desirably tailor the Fano spectral asymmetry from the same system using pre- and postselection of optimized polarization states, also for a moderately anisotropic system such as the

+45°, Φα − Φβ ≈ 0.8 radian). The choice of this elliptical state enables us to compensate for the additional wavelengthindependent background anisotropy of the sample (retardance δ ≈ 0.8 radian; see Figure 2b), which is not pertinent to the anisotropy in the Fano spectral asymmetry. The postselections are done at varying linear polarization angles β. This is equivalent to the type (a) anisotropic system with pre- and postselection in a linear polarization basis (eq 7). In conformity with the corresponding predictions, the results demonstrate tuning of the effective asymmetry parameter (qeff) by directly modifying the Fano phase φF (or q) (the corresponding values for φF and the resulting qeff parameters are noted in the figure), albeit for a limited range permitted by the moderate level of the anisotropy of the sample (due to relatively small difference in φF,x − φF,y). Note that unlike an ideal situation, the Beff parameter also slightly varies here due to Bx ≠ By as noted in Figure 3a and b (see Figure S3 in the Supporting Information for the details of the parameters). It can be envisaged by using the estimated values for the (qx/qy, Bx/By) parameters of Figure 3 in eq 7 that for the preselected polarization angle α = +45° it is possible to reverse the spectral asymmetry parameter (qeff: 1646

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much potential, but its practical benefits remain to be rigorously evaluated.

experimental waveguided plasmonic crystal sample, underscores the robustness of the proposed approach. Note that the anisotropy effects of the plasmonic crystals can be significantly enhanced by changing the periodicity of the disk array or by other means.29 As demonstrated in Figure 4b, a much more dramatic control on the Fano phase (φF) and the resulting effective spectral asymmetry (qeff) parameter can be obtained in such a strongly anisotropic system having larger differences in φF,x − φF,y and E0,x − E0,y. Theoretical predictions using α = +45°, varying β in eqs 4 and 6, reveal that several exotic regimes of Fano resonance can be tailored: (1) beginning with a moderate level of spectral asymmetry, a high degree of asymmetry can be obtained (qeff ≈ +0.83, φF = −63° for β = 90°), (2) the symmetric Lorentzian dip in the scattering spectra can be tailored (qeff → 0, φF → 90° for β = 110°), (3) the symmetric dip can be reversed to a symmetric Lorentzian peak (qeff → ∞, φF ≈ 0° for β = 135°), and importantly, (4) the spectral asymmetry can be completely reversed (qeff ≈ −1.21, φF = +26° for β = 120°). This enables a large (∼80 nm) tunability in the Fano spectral dip (λF ≈ 690 nm for β = 90° to λF ≈ 610 nm for β = 120°). A similar type of control on Fano resonance can also be obtained in a type (b) (perfectly diattenuating) anisotropic system, where qeff can be controlled by changing the relative amplitudes of the interfering modes Beff (as predicted by eq 8). An example of tuning the effective Fano spectral asymmetry (qeff) by modifying the Beff parameter is illustrated in Figure S5 of the Supporting Information. These are illustrative examples, and many other interesting possibilities emerge, wherein the anisotropy and the spectral parameters of both the narrow resonance and the broad continuum modes can be appropriately designed to enable much larger tunability of the Fano spectral dip. For this purpose, the relevant Fano parameters (qx/qy, Bx/By) can be extracted from the Mueller matrix and subsequently used in combination with eqs 6−8 to optimize the polarization states for tailoring the desirable effective spectral asymmetry (qeff).

MATERIALS AND METHODS Fabrication of the Waveguided Plasmonic Crystal Samples. The waveguided plasmonic crystal samples consisted of a twodimensional periodic array of gold nanodisks (or nanoellipses) on top of a ∼190 nm thick ITO waveguiding layer coated on a quartz substrate. We used electron beam lithography and metal deposition by the thermal evaporation technique to fabricate these nanostructures (see Supporting Information). The dimensions of the fabricated Au circular disk array were (diameter D = 160 nm, height = 30 nm, center to center distance L = 480 nm) and the corresponding dimensions of the Au ellipse array were (Dx = 134 nm, Dy = 95 nm, height = 30 nm, L = 480 nm). Dark-Field Mueller Matrix Spectroscopy System. The elastic scattering Mueller matrices M from the samples were recorded using a home-built spectroscopic Mueller matrix system integrated with an inverted microscope operating in the dark-field mode (shown in Supplementary Figure S1). The system essentially comprises a (i) conventional inverted microscope (IX71, Olympus) operating in the dark-field imaging mode, (ii) polarization state generator (PSG) and polarization state analyzer (PSA) units, and (iii) a spectrally resolved signal detection (spectroscopy) unit. Collimated white light from a halogen lamp (JC12 V100WHAL-L, Olympus) is used as an excitation source and is passed through the PSG unit for generating the input polarization states. The PSG unit consists of a horizontally oriented fixed linear polarizer P1 and a rotatable achromatic quarter-wave plate (Q1, AQWP05M-600, Thorlabs, USA) mounted on a computercontrolled rotational mount (PRM1/M-Z7E, Thorlabs, USA). The light is then focused to an annular shape at the sample site using a dark-field condenser (Olympus U-DCD, NA = 0.8−0.92). The sample-scattered light is collected by the microscope objective (MPlanFL N, NA = 0.8), passed through the PSA unit, and then relayed to spectrometer (HR 4000, Ocean Optics, USA). The darkfield arrangement facilitates detection of exclusively the samplescattered light (scattering spectra). The PSA unit essentially comprises the same components with a fixed linear polarizer (P2, oriented in a vertical position) and a computer-controlled rotating achromatic quarter-wave plate (Q2), but positioned in a reverse order. The specifics of the Mueller matrix measurement strategy can be found elsewhere.26 Briefly, the 4 × 4 spectral Mueller matrices are constructed by combining 16 sequential spectrally resolved intensity measurements (spectra) for four different combinations of the optimized elliptical polarization state generator (using the PSG unit) and analyzer (using the PSA unit) basis states. The four elliptical polarization states are generated by sequentially changing the fast axis of Q1 to four angles (ϑ = 35°, 70°, 105°, and 140°) with respect to the axis of P1. Similarly, the four elliptical analyzer basis states are obtained by changing the fast axis of Q2 to the corresponding four angles (35°, 70°, 105°, and 140°). These 16 polarization-resolved scattering spectra are combined to yield the scattering Mueller matrix of the sample (see the Supporting Information).26

CONCLUSIONS In conclusion, we have presented a Mueller matrix-based method for controlling the Fano interference effect and engineering the resulting asymmetric spectral line shape in an anisotropic optical system. The method is founded on a generalized model of anisotropic Fano resonance. The model provides a general recipe for desirably tuning the spectral asymmetry by modulating two parameters of interference (Fano phase and relative amplitude parameter) using pre- and postselection of optimized polarization states of light. The principle is demonstrated on waveguided plasmonic crystals exhibiting a moderate level of anisotropy, and a much more dramatic control is envisaged in strongly anisotropic systems. The large tunability in the Fano spectral dip in an appropriately designed strongly anisotropic system may enhance active Fano resonance-based applications,4,20,31−33 e.g., by enabling a polarization-optimized multisensing platform,31,32 developing polarization-controlled Fano switching devices,4,20 and xtending applications in filtering and robust color displays.33 This promising approach should therefore stimulate further studies enabling applications of polarization-optimized anisotropic Fano resonant systems and may provide useful insights in the analysis/interpretation and control of Fano resonance in diverse systems. Finally, the proposed Mueller matrix-based method represents a fundamentally interesting approach with

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b07406. Schematic diagram of the experimental system, Mueller matrix construction and decomposition scheme, decomposed nondepolarizing and depolarizing Mueller matrices, dependence of the derived effective Fano asymmetry parameter on the postselection of linear polarization angle, tuning of Fano spectral asymmetry for perfectly diattenuating case (PDF) 1647

DOI: 10.1021/acsnano.6b07406 ACS Nano 2017, 11, 1641−1648

Article

ACS Nano

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AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Shubham Chandel: 0000-0002-0727-8879 Author Contributions †

S. K. Ray and S. Chandel contributed equally to this work.

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by IISER-Kolkata, an autonomous institute funded by MHRD, Govt. of India. S.K.R. acknowledge UGC, Govt. of India, S.C acknowledge IISER Kolkata, A.K.S and A.M acknowledge CSIR, Govt. of India for research fellowships. REFERENCES (1) Fano, U. Effects of Configuration Interaction on Intensities and Phase Shifts. Phys. Rev. 1961, 124, 1866−1878. (2) Shafiei, F.; Wu, C.; Wu, Y.; Khanikaev, A. B.; Putzke, P.; Singh, A.; Li, X.; Shvets, G. Plasmonic Nano-Protractor Based on Polarization Spectro-Tomography. Nat. Photonics 2013, 7, 367−372. (3) Miroshnichenko, A. E.; Flach, S.; Kivshar, Y. S. Fano Resonances in Nanoscale Structures. Rev. Mod. Phys. 2010, 82, 2257−2298. (4) Luk’yanchuk, B.; Zheludev, N. I.; Maier, S. A.; Halas, N. J.; Nordlander, P.; Giessen, H.; Chong, C. T. The Fano Resonance in Plasmonic Nanostructures and Metamaterials. Nat. Mater. 2010, 9, 707−715. (5) Wu, C.; Khanikaev, A. B.; Shvets, G. Broadband Slow Light Metamaterial Based on a Double-Continuum Fano Resonance. Phys. Rev. Lett. 2011, 106, 107403. (6) Gallinet, B.; Martin, O. J. F. Ab initio Theory of Fano Resonances in Plasmonic Nanostructures and Metamaterials. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 235427. (7) Fan, J. A.; Wu, C.; Bao, K.; Bao, J.; Bardhan, R.; Halas, N. J.; Manoharan, V. N.; Nordlander, P.; Shvets, G.; Capasso, F. SelfAssembled Plasmonic Nanoparticle Clusters. Science 2010, 328, 1135. (8) Shcherbakov, M. R.; Vabishchevich, P. P.; Komarova, V. V.; Dolgova, T. V.; Panov, V. I.; Moshchalkov, V. V.; Fedyanin, A. A. Ultrafast Polarization Shaping with Fano Plasmonic Crystals. Phys. Rev. Lett. 2012, 108, 253903. (9) Christ, A.; Tikhodeev, S. G.; Gippius, N. A.; Kuhl, J.; Giessen, H. Waveguide-Plasmon Polaritons: Strong Coupling of Photonic and Electronic Resonances in a Metallic Photonic Crystal Slab. Phys. Rev. Lett. 2003, 91, 183901. (10) Zhu, Y.; Hu, X.; Huang, Y.; Yang, H.; Gong, Q. Fast and LowPower All-Optical Tunable Fano Resonance in Plasmonic Microstructures. Adv. Opt. Mater. 2013, 1, 61−67. (11) Gantzounis, G.; Stefanou, N.; Papanikolaou, N. Optical Properties of Periodic Structures of Metallic Nanodisks. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77, 035101. (12) Milana, L.; Justin, N.; Philip, B.; Gregory, T. F.; Drew, F. D.; Roper, D. K. Modulation of Plasmonic Fano Resonance by the Shape of the Nanoparticles in Ordered Arrays. J. Phys. D: Appl. Phys. 2013, 46, 485103. (13) Sonnefraud, Y.; Verellen, N.; Sobhani, H.; Vandenbosch, G. A. E.; Moshchalkov, V. V.; Van Dorpe, P.; Nordlander, P.; Maier, S. A. Experimental Realization of Subradiant, Superradiant, and Fano Resonances in Ring/Disk Plasmonic Nanocavities. ACS Nano 2010, 4, 1664−1670. (14) Fan, X.; Zheng, W.; Singh, D. J. Light Scattering and Surface Plasmons on Small Spherical Particles. Light: Sci. Appl. 2014, 3, e179. 1648

DOI: 10.1021/acsnano.6b07406 ACS Nano 2017, 11, 1641−1648