Fano-Type Resonance of Waves in Periodic Slabs
FFano-Type Resonance of Waves in Periodic Slabs
Natalia PtitsynaLouisiana State University Stephen P. ShipmanLouisiana State University Stephanos VenakidesDuke UniversityJanuary 8, 2019
Abstract
We investigate Fano-type anomalous transmission of energy of plane waves across lossless slab scat-terers with periodic structure in the presence of non-robust guided modes. Our approach is based onrigorous analytic perturbation of the scattering problem near a guided mode and applies to very generalstructures, continuous and discrete.
Resonances with sharp peaks and dips have been observed in the graphs of the frequency dependence ofthe transmission of energy of plane waves through periodic slab structures. [3, 8, 10, 13]. These anomalieshave been connected to isolated real points on the complex dispersion relation D ( κ, ω ) = 0 that relates thefrequency ω of a sourceless field to its Bloch wave vector κ along the slab [8, 10]. The real point of thedispersion relation corresponds to a true guided mode in the slab, that is, one that decays exponentially awayfrom the slab. Because the point is isolated, the mode is non-robust with respect to perturbations of κ , ω , orthe physical or geometric parameters of the structure. This feature gives rise to sharp transmission anomaliesnear the frequency of the guided mode, as shown in Figs. 1 and 2.The anomalies observed resemble those investigated by U. Fano [4] in the context of quantum mechanics,in which an embedded eigenvalue is “dissolved” into the continuous spectrum upon perturbation of thesystem, resulting in resonant dynamical behavior. The shape that he derived for the resonance involves twoparameters and turns out to be a special case of a more general shape derived rigorously for the graph ofenergy transmission through photonic crystal slabs with a symmetry assumption on the slab geometry [11].Typically, symmetry gives rise to standing guided waves ( κ = 0 ), as discussed in [2, 12], and only standingwaves were analyzed in [11]. In the present work, we compute anomalies near traveling guided modes( κ (cid:54) = 0 ). The frequency of a non-robust guided mode is an embedded eigenvalue for a fixed value of κ .The dissolution of the eigenvalue into the continuous spectrum corresponds to the destruction of the guidedmode when κ is perturbed. We demonstrate the results through two examples, the scattering of polarizedEM fields by a lossless dielectric structure and the scattering of waves in a two-dimensional lattice by aone-dimensional periodic lattice attached along a line.1 a r X i v : . [ m a t h - ph ] A p r Analytic connection of scattering states and guided modes
In [11], we reformulate the Maxwell equations in a photonic crystal slab (Fig. 2) at fixed frequency ω andBloch wavenumber κ (the latter necessarily parallel to the slab) as a boundary-integral equation (see [6, 7],for example). The equation A ( κ, ω ) ψ = φ (1)relates two fields φ and ψ defined on the interface between the scatterer material (taken to be homogeneousand isotropic) and the ambient medium. The field ψ on the interface contains information that is necessaryand sufficient for the calculation of the EM field in the whole space; the calculation consists of the evaluationof an integral. Similarly, the field φ contains information for the calculation of the incident field. Solving theabove equation for ψ is equivalent to calculating the field in the photonic crystal and in the ambient mediumestablished as a result of the source field φ . We call ψ the total field to distinguish it from the scattered field,which is the difference between the total and incident fields in the ambient medium; clearly, a scattered fieldis required to satisfy a radiation condition. The integral operator A is of second-kind Fredholm type withindex zero, thus, it either has a bounded inverse or it has a nullspace of finite dimension and a range of thesame codimension.The integral operator A is parametrized by ω and κ . When either of these takes a nonreal value, a field ψ ,bounded on the interface, may produce an unbounded and hence unphysical field in space due to growth ofthe exponential e iκx or e iωt . Such fields are not only mathematically but also physically useful and form thefoundation of the so-called leaky modes. A (generalized) guided mode is a nonzero solution of the sourcelessequation A ( κ, ω ) ψ = . (2)The pairs ( κ, ω ) for which such a solution exists satisfy a dispersion relation D ( κ, ω ) = 0 . For pairs ( κ, ω ) such that κ is real and D ( κ, ω ) = 0 , we must have Im ω ≤ [8, 10]. In the case that Im ω = 0 , a solution to(2) represents an EM field that decays exponentially with distance from the scatterer and therefore representsa true guided mode. We shall work with a simple eigenvalue branch (cid:96) ( κ, ω ) of A ( κ, ω ) , for which wenecessarily have that (cid:96) ( κ, ω ) = 0 implies D ( κ, ω ) = 0 .The framework we have described explicitly for the Maxwell equations arises for very general dielectricand metal structures (not necessarily with homogeneous components) and for other continuous and discreteproblems of scattering by a slab; the methods of arriving at the formulation vary from problem to problem.For waves in an n -dimensional uniform lattice scattered by a ( n − -dimensional periodic lattice (Fig. 1),the problem is reduced to a finite-dimensional one, and A is a matrix.For the purpose of investigating anomalous scattering behavior near a guided mode, we let ( κ , ω ) be areal pair that satisfies (cid:96) ( κ , ω ) = 0 and use as an incident source field the plane wave φ inc ( x, z ) = (cid:96) ( κ, ω ) e iκx e iηz (incident from the left), (3)where x is the multi-variable in the directions parallel to the slab and z is the single variable perpendicularto the slab ( x and z may be discrete or continuous). The number η depends on κ , ω , and the structure itself.We shall work in the regime of only one propagating diffractive order, that is, one Fourier harmonic in theperiodic variable x that carries energy in the z -direction. In this case, the reflected and transmitted fields (atfar field) are simple: φ refl ( x, z ) = a ( κ, ω ) e iκx e − iηz (reflected on the left), (4) φ trans ( x, z ) = b ( κ, ω ) e iκx e iηz (transmitted on the right). (5)2t can be shown that the coefficients a ( κ, ω ) and b ( κ, ω ) are analytic functions of ( κ, ω ) near ( κ , ω ) [8, 11].At pairs ( κ, ω ) that satisfy (cid:96) ( κ, ω ) = 0 , the the right-hand side of (1) is zero, and the solution is a guidedmode. Thus we obtain an analytic connection between scattering states and generalized guided modes nearthe true guided mode at ( κ , ω ) . The analysis of the resonant transmission shape near a guided mode is based on the analysis of the threeanalytic functions (cid:96) ( κ, ω ) , a ( κ, ω ) , and b ( κ, ω ) near the real parameters ( κ , ω ) of a guided mode. Themode condition (cid:96) ( κ , ω ) = 0 , together with that of conservation of energy, namely | (cid:96) | = | a | + | b | forreal values of κ and ω , implies that ( κ , ω ) is a root of the three analytic functions (cid:96) , a , and b simultaneously: (cid:96) ( κ , ω ) = 0 , a ( κ , ω ) = 0 , b ( κ , ω ) = 0 , at ( κ , ω ) ∈ R . (6)The perturbation analysis of (cid:96) , a , and b near this common real root relies principally on the following twoconditions, which we have already mentioned: | (cid:96) ( κ, ω ) | = | a ( κ, ω ) | + | b ( κ, ω ) | for real κ and ω , (7)if (cid:96) ( κ, ω ) = 0 for κ ∈ R , then Im ω ≤ . (8)The following conditions are assumed because they hold generically: ∂(cid:96)∂ω ( κ , ω ) (cid:54) = 0 , ∂a∂ω ( κ , ω ) (cid:54) = 0 , ∂b∂ω ( κ , ω ) (cid:54) = 0 . (9)The Weierstraß preparation theorem for analytic functions of several variables [5] then provides the followingforms (with (cid:36) = ω − ω and ˜ κ = κ − κ ): (cid:96) ( κ, ω ) = e iθ (cid:2) (cid:36) + (cid:96) ˜ κ + (cid:96) ˜ κ + O (˜ κ ) (cid:3) [1 + O ( | ˜ κ | + | (cid:36) | )] , (10) a ( κ, ω ) = e iθ (cid:2) (cid:36) + r ˜ κ + r ˜ κ + O (˜ κ ) (cid:3) [ r + O ( | ˜ κ | + | (cid:36) | )] , (11) b ( κ, ω ) = e iθ (cid:2) (cid:36) + t ˜ κ + t ˜ κ + O (˜ κ ) (cid:3) [ t + O ( | ˜ κ | + | (cid:36) | )] . (12)The function (cid:96) has been normalized so that the constant in the second factor is equal to 1. All factors in theseforms are analytic, and the numbers r and t are real and positive.Consequences of (8) are that (cid:96) is real and Im (cid:96) ≥ : r > , t > , (cid:96) ∈ R , Im (cid:96) ≥ . (13)Using the forms (10,11,12), we can examine various terms in the expansion of the equation of conservationof energy (7) for real values of κ and ω . Computation of | (cid:96) | yields (cid:96) ¯ (cid:96) = (cid:2) (cid:36) + (cid:96) ˜ κ + 2 (cid:96) (cid:36) ˜ κ + 2 Re (cid:96) (cid:36) ˜ κ + 2 (cid:96) Re (cid:96) ˜ κ ++ (2 (cid:96) Re (cid:96) + | (cid:96) | )˜ κ + ... (cid:3) [1 + O ( | ˜ κ | + | (cid:36) | )] , (14)and computation of | a | and | b | yield analogous expressions. In the case that (cid:96) (cid:54) = 0 , three terms of theexpansion of | (cid:96) | = | a | + | b | are determined in terms of the coefficients expressed explicitly in the forms(10,11,12). In the case that (cid:96) = 0 , some of these terms become trivial and others become relevant. Weanalyze these two cases separately. 3 .1 Case 1: (cid:96) (cid:54) = 0 The three terms in the expansion of (7) that give information about the coefficients in the forms (10,11,12)are (cid:36) term: r + t , ˜ κ term: (cid:96) = r | r | + t | t | ,(cid:36) ˜ κ term: (cid:96) = r Re r + t Re t . (15)Because of the convexity of the function x (cid:55)→ x , the (cid:36) -term and the (cid:36) ˜ κ -term imply that (cid:96) ≤ r ( Re r ) + t ( Re t ) , (16)with equality if and only if Re r = Re t , and the ˜ κ -term is expanded to (cid:96) = r ( Re r ) + t ( Re t ) + r ( Im r ) + t ( Im t ) . (17)From conditions (16) and (17) together, we infer thatIm r = Im t = 0 and r = t = (cid:96) . (18)We thus obtain useful expressions for the zero-sets of (cid:96) , a , and b near the guided mode pair ( κ , ω ) : (cid:96) ( κ, ω ) = 0 ⇐⇒ ω = ω − (cid:96) ( κ − κ ) − (cid:96) ( κ − κ ) − . . . , (19) a ( κ, ω ) = 0 ⇐⇒ ω = ω − (cid:96) ( κ − κ ) − r ( κ − κ ) − . . . , ( (cid:96) ∈ R ) (20) b ( κ, ω ) = 0 ⇐⇒ ω = ω − (cid:96) ( κ − κ ) − t ( κ − κ ) − . . . . (21)We observe that, if all coefficients r n and t n vanish, then the curve a ( κ, ω ) = 0 (resp. b ( κ, ω ) = 0 ), forreal values of κ , describes real frequencies ω for which transmission T is equal to (resp. ). Wedo know that (cid:96) is real, so that these curves lie at most a distance O (( κ − κ ) ) from the real ω -axis, andone can deduce that the real parts of the frequencies on these curves correspond to peaks and dips of thetransmission, which do not necessarily reach exactly or .These results allow us to make two important observations about the shape of the transmission resonanceas a function of frequency. Both of these are illustrated on the graphs in Fig. 1.1. For κ (cid:54) = κ , either both the peak and dip of T as a function of ω lie to the left of the frequency ω orboth lie to the right of ω . On which side they lie depends on the sign of (cid:96) and κ − κ .2. The order in which the peak and dip in T occur on the real ω -axis is the same for κ < κ as it is for κ > κ (assuming r (cid:54) = t ). This is because the coefficients of the linear terms in (20) and (21) areequal. If r < t , then the peak comes to the right of the dip, and if t < r , then the peak comes tothe left of the dip.The transmission coefficient (square root of transmitted energy) as a function of real κ and ω is T ( κ, ω ) = (cid:12)(cid:12)(cid:12)(cid:12) b ( κ, ω ) (cid:96) ( κ, ω ) (cid:12)(cid:12)(cid:12)(cid:12) = t | (cid:36) + (cid:96) ˜ κ + t ˜ κ + · · · || (cid:36) + (cid:96) ˜ κ + (cid:96) ˜ κ + · · · | | η (cid:36) + η ˜ κ + · · · |≈ t (cid:12)(cid:12)(cid:12)(cid:12) (cid:36) + (cid:96) ˜ κ + t ˜ κ (cid:36) + (cid:96) ˜ κ + (cid:96) ˜ κ (cid:12)(cid:12)(cid:12)(cid:12) (1 + Re η (cid:36) + Re η ˜ κ ) near ( κ , ω ) . (22)The coefficients have the following significance: 4. (cid:96) is the rate at which the anomaly in ω moves past the guided-mode frequency ω as a function of κ .2. lim ˜ κ,(cid:36) → T ( κ, ω ) = t . In particular, T ( κ, ω ) is continuous at the guided mode pair ( κ , ω ) .3. ∂∂ω T ( κ , ω ) | ω = t Re ( η ) . This relates Re η to the slope of the non-resonant transmission at κ = κ .4. ∂∂κ T ( κ, ω ) | κ = t Re (cid:16) t − (cid:96) (cid:96) + η (cid:17) .5. The coefficients t and (cid:96) control the spreading of the peak and dip of the anomaly as κ is perturbedfrom κ . Fig. 1. Plane waves in the uniform 2D lattice are incident upon the coupled periodic 1D lattice from the left and aretransmitted to the right. The square root of the percentage of transmitted energy as a function of frequency ω is shownfor several values of the Bloch wave vector κ near the parameters ( κ , ω ) = (0 . , . of a guided mode.The exact calculation and the theoretical formula are practically identical. (cid:96) = 0 This case occurs when the dispersion relation (cid:96) ( κ, ω ) = 0 , solved for ω , is symmetric in the variable κ about κ . In particular, this occurs at resonant frequencies for κ = 0 if the structure has symmetry about a line orplane perpendicular to the plane of the scatterer [2, 11, 12, 13], as is the case in Fig. 2.From the second relation in (15), we observe that (cid:96) = 0 implies r = 0 and t = 0 also, and we find thatthe following three terms in the expansion give information about the coefficients: (cid:36) term: r + t ,(cid:36) ˜ κ term: Re (cid:96) = r Re r + t Re t , ˜ κ term: | (cid:96) | = r | r | + t | t | . (23)Because (cid:96) = 0 , the expansions (19,20,21) reduce to (cid:96) ( κ, ω ) = 0 ⇐⇒ ω = ω − (cid:96) ( κ − κ ) − . . . , (24) a ( κ, ω ) = 0 ⇐⇒ ω = ω − r ( κ − κ ) − . . . , (25) b ( κ, ω ) = 0 ⇐⇒ ω = ω − t ( κ − κ ) − . . . . (26)5he second of the relations (23) tells us that the curve (cid:96) ( κ, ω ) lies between the curves a ( κ, ω ) = 0 and b ( κ, ω ) = 0 . In particular, if r and t are real and (cid:96) is imaginary, then the transmission peak and dip movein opposite directions away from the guided mode frequency ω as κ is perturbed from κ .In [11], we demonstrate that the following approximate expression for the transmission anomaly near ( κ , ω ) matches numerical simulations for the scattering of E -polarized fields by a periodic lossless dielec-tric slab: T ( κ, ω ) ≈ t | (cid:36) + t ˜ κ | (1 + η (cid:36) ) r | (cid:36) + r ˜ κ | + t | (cid:36) + t ˜ κ | (1 + η (cid:36) ) near ( κ , ω ) . (27)Here, the coefficients have the following significance:1. lim ω → ω T ( κ , ω ) = t and lim κ → κ T ( κ, ω ) = t (cid:12)(cid:12)(cid:12)(cid:12) t (cid:96) (cid:12)(cid:12)(cid:12)(cid:12) .2. η = 1 t r ∂T∂ω ( κ , ω ) | ω .3. The coefficients t and r control the spreading of the peak and dip of the transmission anomaly as κ is perturbed from κ . xy z ! =0.12 ! =0.09 ! =0.06 ! =0.03 ! =0.01 ! =0.00 T r a n s m i ss i on ( T ) Real frequency ( " ) Fig. 2. Transmission of E-polarized EM fields through a periodic array of infinitely tall lossless rods ( (cid:15) = 12 , µ = 1 )as a function of the reduced frequency ω , for various of values of the wave vector κ near the parameters (0 , . ofa guided mode. The exact calculation and the theoretical formula are practically identical. One of the classic examples of anomalous scattering behavior is observed in the excitation of the noble gasesnear characteristic energies of the atom (the Auger states [9]). The anomalies exhibit a peak and a dip andare called “Fano resonances” after the work of U. Fano [4], in which he derived a formula for this shape: σ = const. ( q + f ) f , f = ω − ω res Γ / , (28)where the parameters q and Γ control the locations of the peak and dip and f is the deviation of the frequencyfrom resonance normalized to a characteristic width Γ .6here is a connection between formula (27) and the Fano shape (28) that can be expressed in concreteterms. Namely, the six real parameters in (27) ( t , η , and the real and imaginary parts of t and r ) can bereduced to two if certain conditions are satisfied, resulting in the Fano shape. The conditions are (in additionto (cid:96) = 0 )1. r and t are real (meaning that the extremal values of T are and ),2. η = 0 (the background transmission is flat),3. r r + t t = 0 (for small real ˜ κ , the dispersion relation given by (24) is purely imaginary).The resonance (27), as a function of frequency ω , now reduces to the Fano shape (28) with Γ = 2¯ κ (cid:112) ( rr ) + ( tt ) and q = t/ (cid:112) ( rr ) + ( tt ) . (29)The first condition is satisfied by the example in Fig. 2, while the second and third are not. The expansions (19,20,21) can be utilized also to yield a formula for the shape of the argument of the complextransmission coefficient near the guided mode parameters ( κ , ω ) ,phase of transmitted field = arg b ( κ, ω ) − arg (cid:96) ( κ, ω ) , (30)which exhibits sharp spike as a function of ω for values of κ close to κ . This quantity is closely related tothe effective density of states associated with transmission of incident waves through a periodic structure [1].A phenomenon that is directly associated to the interaction of plane waves with guided modes is the res-onant amplitude enhancement of the scattered field inside the structure. The scattering problem is nearlysingular for small perturbations, so the solution in the scatterer is large compared to the incident field andresembles a guided mode. We have established in [11] a leading-order result for the behavior of this en-hancement as a function of κ : amplitude enhancement ∼ const . | κ | ( κ → . (31)One of the essential ingredients in the analysis is the fact that there is no resonant enhancement at κ = κ .This fact is tantamount to the fact that the scattering problem has a solution at the parameters of the guidedmode, although it is not unique [2]; in other words, the source field has no “resonant component” at normalincidence. Acknowledgment
This work was supported by NSF grant DMS-0505833 (S. Shipman and N. Ptitsyna) and NSF grantsDMS-0207262 and DMS-0707488 (S. Venakides). 7 eferences [1] Jon M. Bendickson, Jonathan P. Dowling, and Michael Scalora. Analytic expressions for the electro-magnetic mode density in finite, one-dimensional, photonic band-gap structures.
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