Tunable Polarization-Induced Fano Resonances in Stacked Wire-Grid Metasurfaces
Xavier Romain, Riccardo Degl'Innocenti, Fadi I. Baida, Philippe Boyer
TTunable Polarization-Induced Fano Resonances in Stacked Wire-Grid Metasurfaces
Xavier Romain,
1, 2, ∗ Riccardo Degl’Innocenti, Fadi I. Baida, and Philippe Boyer D´epartement Optique - Institut FEMTO-ST UMR 6174,Universit´e Bourgogne Franche-Comt´e - CNRS, 25030 Besan¸con, France Department of Engineering, Lancaster University, Bailrigg, Lancaster LA1 4YW, UK (Dated: April 23, 2020)Stacked metasurfaces are being investigated in light of exploring exotic optical effects that can-not be achieved with single-layered metasurfaces. In this Letter, we theoretically demonstrate thatstacks of metallic wire-grid metasurfaces possessing specific polarization properties have the abilityto induce tunable Fano resonances. The developed original model - combining a circulating field ap-proach together with an extended Jones formalism - reveals the underlying principle that gives riseto the polarization-induced Fano resonances. The theoretical frame is validated in an experimentalproof of concept using commercially available wire-grids and a terahertz time domain spectrome-ter. This unexplored possibility opens an alternative path to the realization and control of Fanoresonances by using stacked metallic metasurfaces. Furthermore, these findings suggest that thepolarization can be used as an additional degree of freedom for the design of optical resonators withenhanced and tunable properties.
Fano resonances [1, 2] currently draw much attentionbecause of their remarkable and unique potential for ap-plications such as sensing with high quality factor [3].Over the past decades, Fano resonances have been re-ported in a large variety of experimental configurationsranging from electromagnetic structures [4–7] to elasticstructures [8]. Similar observations of Fano signaturein electromagnetic metamaterials [9] were made by de-liberately breaking the symmetry of the metamaterialunit-cell. These results have been rapidly followed byextensive studies on the coupling of trapped modes inmetamaterials in order to obtain Fano resonances [10].More recently, a vast literature focused on the excitationof tunable and/or multiple Fano lineshapes for increasedfunctionalities [11–13]. At the same time, metallic meta-materials have rised in popularity because of the widediversity of physical effects which they can exhibit suchas extraordinary optical transmission [14], negative re-fraction [15] or perfect absorption [16]. However, theperformances associated to these physical effects may belimited by the intrinsic physical properties of the unit-cell, or might require complex designs and fabrications.Stacked metasurfaces are currently proposed as an alter-native way to achieve complex functionalities [17]. Be-sides, the interaction between metasurfaces brings fur-ther degree of freedom and enables additional effects [18].For example, stacked structures are currently proposedto efficiently manipulate the polarization of light [19–22].In this Letter, we demonstrate that stacked metal-lic metasurfaces offer the possibility to realize tunableFano resonances that are induced by the specific polar-ization properties of the constitutive metasurfaces. Thepolarization-induced Fano resonance mechanism is theo-retically analyzed with an original cavity model whichcombines a circulating field approach [23] and an ex-tended Jones formalism [24]. The theoretical studyis experimentally supported by demonstrating tunable polarization-induced Fano resonances in the THz regimeby using a THz time domain spectrometer and imple-menting Metallic Wire-Grid Metasurfaces (MWGMs).The importance of the MWGMs linear polarizationproperties for exciting Fano resonances is first examined.An electromagnetic plane wave is considered to be propa-gating along the z-axis and falls in normal incidence on astack of two parallel and perfectly conducting MWGMs,as shown in Fig. 1 (a) , acting as linear polarizers. Thetransmission and reflection of the first MWGM, identi-fying the x and y axes respectively, serve as a reference.The period, the thickness and the aperture width of bothMWGMs are denoted by p , h and a respectively. The dis-tance between the two MWGMs is d and θ is the rotationangle of the second MWGM with respect to the x-axis.The geometrical notations of the structure are summa-rized in Fig. 1 (a) .The stack of two MWGMs is assumed to form a Fabry-Perot-like (FP-like) cavity. To accurately describe theresonance and polarization properties of the FP-like cav-ity, a circulating field approach [23] is associated to aJones formalism [24], as illustrated in Fig. 1 (b) . Ascalar model (excluding polarization properties), report-ing more details on the circulating field was thoroughlyinvestigated in [23]. The steady state forward circulatingfield, (cid:126)E c , is given by (cid:126)E c = J c U (cid:126)E launch (1)where (cid:126)E c corresponds to the infinite sum of waves incom-ing on the second MWGM, as highlighted by the dashedgrey ellipses in Fig. 1 (b) . J c is the Jones matrix thataccounts for the infinite round-trips in the cavity and itsexpression is detailed later in Eq. (4), Eq. (5) and Eq.(6). The propagation operator U = uI links the elec-tric fields from the first to the second MWGM inside theFP-like cavity. The term I is a (2 ×
2) identity matrix, u = e id π/λ represents the phase shift accumulated by a r X i v : . [ phy s i c s . op ti c s ] A p r the electric field in half a round-trip and λ denotes thewavelength. The electric field (cid:126)E launch , is the initial fieldlaunched in the cavity and its expression is (cid:126)E launch = J T (cid:126)E inc (2)where J T = (cid:18) t x t y (cid:19) and J R = (cid:18) r x r y (cid:19) (3)are the transmission and reflection Jones matrices of thefirst MWGM where J R is mentioned for completeness.The terms t x , t y and r x , r y are respectively the trans-mission and reflection coefficients along the x and y axesfor one MWGM. The polarization-induced effect can beinferred from the expression of J c which is written as J c = (cid:2) I − U J R J Rθ (cid:3) − (4)where J Rθ = R ( θ ) J R R ( − θ ) is the reflection Jones matrixof the second MWGM and R ( θ ) is the rotation matrix.The matrix J c can be reformulated as J c = (cid:18) J xxc J xyc J yxc J yyc (cid:19) (5)which can be expanded into J c = 1 D (cid:18) − u r y ( s r x + c r y ) u cs ( r x − r y ) r x u cs ( r x − r y ) r y − u r x ( c r x + s r y ) (cid:19) (6)where D = det ( I − U J R J Rθ ), c = cosθ and s = sinθ .For this complex general case given by Eq. (6), it isfundamental to remark that the coupling terms J yxc and J xyc are vanishing if r x = r y - as it would be the casewith polarization insensitive mirrors. In other words,the polarization properties of the MWGMs brings ad-ditional polarization coupling effects - inducing Fano res-onances - that are not achievable in classical FP cavi-ties. Other works using matrix formalism have analyzedanisotropic FP resonators [25, 26] or chiral FP interfer-ometers [27, 28]. None of these studies, however, consid-ered mirrors with polarization dependency, they ratheremployed classical FP cavities filled with anisotropic oroptically active media. The linear polarization depen-dency of the MWGMs ( r x (cid:54) = r y ) is the key differencethat permits the excitation of Fano resonances.The transmission of the optical arrangement of Fig.1 (a) is numerically simulated with a/p = 0 . h/p = 0 . d/p = 2 . θ = 10 ◦ . The transmission properties ofone MWGM are first investigated. The spectral range ofinterest is λ/p ≥ t x and r x are numerically calculated by Airy-like expressions. invertedlorentzianlineshape , Fanolineshape zad (a) x y ph ... J T � J R � J T J R ... ,, TransmittedElectric Fields E inc E c E launch ... z xy + + + (b) (d) m=1m=2m=3 ' d (c) FIG. 1. (a)
3D illustration of two MWGMs stacked alongthe z-axis where the second MWGM is rotated by an angle θ . (b) Principle of the FP-like cavity formed by two MWGMscharacterized by their Jones matrices J T , J R and J Tθ , J Rθ re-spectively. (cid:126)E launch is the initial electric field entering the cav-ity, and (cid:126)E c is the steady state forward circulating field in theFP-like cavity. The arrows are deliberately tilted to clearlyrepresent the round trips in the FP-like cavity. (c) Simulatedspectra of t x , the transmission coefficient along the x-axis forone MWGM. The real and imaginary parts of t x are depictedin solid blue line and dashed orange line respectively, andcomputed with a/p = 0 . h/p = 0 .
1. The greyed areaspecifies the spectral region ( λ/p ≤ .
8) for which the modelhas lower accuracy. (d)
Simulated transmission spectra ofthe FP-like cavity with θ = 10 ◦ and d/p = 2 .
0. The integer m denotes the FP harmonic order where the Fano resonancesoccurs. The black curve shows the spectrum of the transmis-sion T computed with the numerical values extracted from themonomode modal method. The dashed pink curve gives thespectrum of the transmission T (cid:48) computed with the specialcase where t x = 1 and r x = 0. The perfectly conducting metal hypothesis imposes thatthe two other coefficients t y and r y are 0 and − (cid:126)E launch is polarized along the x-axis. Figure 1 (c) represents thespectra of the real and imaginary parts of t x in solid blueline and dashed orange line respectively. The transmis-sion T of the FP-like cavity is defined as T = | J T,xxF P | + | J T,yxF P | (7)with J TF P = J Tθ J c U J T (8)where J TF P is the transmission Jones matrix of the FP-like cavity and J Tθ = R ( θ ) J T R ( − θ ) is the transmissionJones matrix of the second MWGM. Note that the trans-mission results achieved by this formalism are strictlyidentical to the transmission values computed by the S-matrix propagation algorithm [29]. Figure 1 (d) yieldsthe transmission spectrum of the cavity in solid black linecomputed for d/p = 2 . θ = 10 ◦ . Clear Fano-like res-onances appear at the FP resonance condition expressedas λ = 2 dm (9)where m denotes the FP harmonic order with m ∈ N .It is worth emphasizing that the Fano dip asymmetryincreases as λ decreases. However, the greyed area forwhich λ/p ≤ . t x .To further explain the origin of these Fano resonances,the lower FP harmonic m = 1 is analyzed in more details.At λ/p = 4 .
0, the coefficients are t x = 0 .
985 + 0 . i and r x = 0 . × − + 0 . i . For explanatory purposes, itis now assumed that t x = 1 and r x = 0. In this ideal andalmost realistic scenario the matrix J c reduces to J c = − u cs − u c − u c . (10)This specific peculiar case is interesting from a theo-retical perspective as it helps elucidating the physicalmechanism which gives rise to Fano resonances. To high-light the core principle responsible for the polarization-induced Fano resonance , it is now essential to distinguishtwo cases:1) When the MWGMs are aligned ( θ = 0 ◦ ), the cou-pling terms are J yxc = J xyc = 0. This configurationyields J yyc = 1 / (1 − u ) corresponding to the elec-tric field Airy distribution of a perfect y-polarizedFP resonance referred to as FP y . At the same time,Eq. (8) becomes J TF P = J T J c U J T = (cid:18) u
00 0 (cid:19) (11) which represents an x-polarized single pass propa-gation operator. The x-polarized field (cid:126)E launch doesnot couple to the FP y resonance and simply propa-gates through the structure. Therefore, the FP y resonance is not excited or, in other word, it is”trapped” in the FP-like cavity.2) When the MWGMs are not aligned ( θ (cid:54) = 0 ◦ ), thecoupling of (cid:126)E launch to the FP y resonance occurs viathe term J yxc . As it is numerically demonstrated inFig. 1 (d) , such coupling induces a Fano resonanceaccording to the FP resonance condition given byEq. (9). The single pass x-polarized electric fieldand the ”trapped” FP y resonance might be re-garded as an analogue of bright and dark modesrespectively, that are exploited to excite Fano res-onances [10].Finally, for this special case where t x = 1 and r x = 0,the transmission T (cid:48) through the FP-like cavity is T (cid:48) = (cid:12)(cid:12)(cid:12)(cid:12) u (1 − u )1 − u cos θ (cid:12)(cid:12)(cid:12)(cid:12) cos θ (12)which is consistent with the transmission expression re-ported in ref. [29]. The FP resonance condition, as statedby Eq. (9), corresponds to the case where u = 1 andtherefore leads to T (cid:48) = 0 , ∀ θ (cid:54) = 0 (mod π ). The expres-sion of T (cid:48) at the FP resonance condition produces in-verted Lorentzian line shapes, as depicted by the dashedpink curve in Fig. 1 (d) and as predicted by Eq. (17)of ref. [6]. The results presented in Fig. 1 (d) are alsocompatible with ref. [6]. Indeed, T → T (cid:48) when the order m decreases since t x → λ → ∞ , as shown in Fig.1 (c) . It is important to note that, in Eq. (10), the cou-pling term J yxc can be controlled by θ . It further suggeststhat the spectral width of the polarization-induced Fanoresonances can be tuned by acting on θ , as previouslyobserved in ref. [29].The polarization-induced Fano resonance effect andits spectral tunability are experimentally demonstratedin the THz region. The THz range is particularly rele-vant and suited to verify the developed theoretical model (a) (b) ������� ° ������� ° ������� ° ������� ° ������� ° ������� ° FIG. 2. (a)
Measured and (b) computed transmission spectrafor p = 35 µm, a = 25 µm, h = 10 µm and d = 1 , cm . as metals are close to be considered as perfect electricconductors at these frequencies. A pair of commer-cially available MWGMs from PureWavePolarizer [30]is placed in the same configuration as previously de-picted in Fig. 1 (a) . The geometrical parameters are p = 35 µm, a = 25 µm, h = 10 µm, and d = 1 , cm .The two MWGMs are inserted into a Menlo Systems TeraK15 THz time domain spectrometer based on photocon-ductive antennas [31]. The maximal spectral resolution ofthe spectrometer is 1.2 GHz in the range of 0.1 THz to 3.5THz. The first MWGM’s transmission axis is aligned tothe polarization direction of the photoconductive emit-ter. A set of transmission measurements, denoted by T ( θ ), are acquired at different relative angles of the sec-ond MWGM, i.e. for θ = 20 ◦ , ◦ and 45 ◦ . The trans-mission T , measured at θ = 0 ◦ , serves as a reference forthe normalized transmission T exp which is defined as T exp = T ( θ ) T . (13)Figure 2 (a) represents the normalized transmission spec-tra for the different angles θ and Fig. 2 (b) depicts thecomputed counterpart using the FP-like cavity model. Inorder to be commensurate with the detector’s polariza-tion sensitivity, the numerical transmission is given by | J T,xxF P | . The measured and simulated results reportedin Fig. 2 (a) and (b) respectively, exhibit a remarkableagreement only limited by the available spectrometer res-olution and by the signal to noise ratio.In conclusion, stacked MWGMs with linear polariza-tion dependency have been theoretically investigated andexperimentally confirmed to be an efficient approach toinduce and control Fano resonances. The experimen-tal measurements were performed in the THz spectralrange showing very good agreement with the predictedtrends. It is worth stressing the versatility and general-ization of this alternative way to realize Fano resonancesby using FP-like cavities. This different approach doesnot require metasurfaces that are specially designed withcomplex unit-cells or custom-made materials. Rather,the polarization-induced Fano resonance addresses fun-damental concepts of optics as it essentially relies onsimple polarization properties and basic resonance ef-fects. This principle could therefore be extended to othermetasurfaces made of unit-cells featuring the same linearpolarization properties [24]. Likewise, the polarization-induced Fano resonances could be scaled to other fre-quency ranges and they are currently being investigatedin the visible region where the metal absorption has tobe considered. 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