The paper discusses properties of flexural waves in elastic plates constrained periodically by rigid pins. A structured interface consists of rigid pin platonic gratings parallel to each other. Although the gratings have the same periodicity, relative shifts in horizontal and vertical directions are allowed. We develop a recurrence algorithm for constructing reflection and transmission matrices required to characterize the filtering of plane waves by the structured interface with shifted gratings. The representations of scattered fields contain both propagating and evanescent terms. Special attention is given to the analysis of trapped modes which may exist within the system of rigid pin gratings. Analytical findings are accompanied by numerical examples for systems of two and three gratings. We show geometries containing three gratings in which transmission resonances have very high quality factors (around 35 000). We also show that controlled lateral shifts of three gratings can give rise to a transmission peak with a sharp central suppression region, akin to the phenomenon of electromagnetic-induced transparency.
The use of resonant systems in filtering is an old topic in physics, with many applications. An emerging area in this field concerns the design of systems to achieve narrow band resonances for mechanical (Mohammadi et al. 2009) and optomechanical applications (Eichenfield et al. 2009). The ability of a system to give resonant wave effects is often quantified by its quality factor (Q), which has various definitions, one of which is the resonant frequency divided by the bandwidth. We describe here the properties and the underlying theory of simple geometric structures which give high Q-factor resonances in transmission and reflection for elastic flexural waves. We build upon previous results (Evans & Porter 2007; Movchan et al. 2009), in which systems composed of two gratings of rigid pins in a Kirchhoff plate were shown to be able to provide Q-factors of around 6000, comparable with that resulting from some of the best previous designs (Mohammadi et al. 2009). We extend this previous work by considering more complicated grating geometries, and, in particular, sets of three gratings. We show that the lateral shift parameter in grating stacks provides a powerful control over the transmission resonances, and can give rise to a variety of interesting physical effects. One of these arises when the careful control of grating shift can turn a high-quality transmission peak into a double peak separated by a zero transmission point. This phenomenon bears an interesting resemblance to the much studied electromagnetically induced transmission (Fleischhauer et al. 2005), a quantum mechanical effect, which has been shown to give rise to powerful control over electromagnetic waves, including the slowing of light to bicycle speed (Hau et al. 1999).
The system we study is not only of practical interest, but is also theoretically elegant. The system of rigid pins in a Kirchhoff plate can be modelled semi-analytically, and gives rise to simple but accurate expressions which enable the design of high Q resonant systems. Achieving optimal performance depends on an understanding of the consequences of symmetry of the systems on their properties (McPhedran & Maystre 1977; Botten et al. 1985; Popov et al. 1986; Mahmoodian et al. 2010; Ha et al. 2011) in a way easily understood within the semi-analytical framework.
The structure of the paper is as follows. In §2, we discuss the geometry of the system of gratings and the governing equations. Section 3 presents the recurrence algorithm for the evaluation of reflection and transmission matrices for a stack of shifted gratings. In §4, we discuss the trapped modes and resonances given by a pair of shifted gratings. In §5, we consider three gratings and the associated, more complicated resonant spectra, including the analogies between the transmittance properties of such systems and those in systems exhibiting electromagnetically induced transparency.
2. Geometry and governing equations
We consider an infinite thin elastic plate containing a finite number of parallel gratings. Each grating contains an infinite number of rigid pins arranged periodically with a separation d. An individual rigid pin is treated as a limiting case of a small circular hole, of radius a, with clamped edge, as the radius a tends to zero (figure 1). A finite number of these gratings can then be arranged parallel to one another with a separation distance η. In this paper, we study vertical (i.e. varying η) and horizontal shifts of these gratings and compare results with those derived by Movchan et al. (2009) for stacks of gratings in which all circular inclusions are directly below one another.
Although much of the theoretical material, such as the recurrence algorithm and energy conservation considerations, is valid for the general circular inclusion, our explicit representations and numerical results will be for the limiting case of . This special case of rigid pins will be shown to provide a simple analytical framework, yet offers a rich diversity of interesting physical effects.
Our results arise from modelling the propagation of flexural waves through the structured interface. We consider plane waves that propagate freely through the homogeneous material until they reach the stack of gratings whereupon they are reflected or transmitted (figure 1). We derive expressions for the reflected and transmitted fields, taking into account both evanescent and propagating waves. We use p, an integer, to represent the diffraction order of the scattered field, and we define ΩH to represent the set of propagating orders of Helmholtz type. Special attention is given to resonance modes, which enhance transmittance across the stack of gratings.
(a) Governing equations
Let the displacement W be a solution of the scattering problem for the biharmonic operator (see Movchan et al. 2009): 2.1where . Within this expression, D=Eh3/(12(1−ν2)) is the flexural rigidity of the plate, h denotes the plate thickness, E is the Young modulus and ν is the Poisson ratio. In addition, ρ is the mass density and ω is the angular frequency.
The boundary conditions on the circular boundaries of the inclusions are Dirichlet clamping conditions: 2.2We consider the limit as , corresponding to an array of rigid pins constraining the plate.
We use the quasi-periodic Green's function for the biharmonic operator. As in Movchan et al. (2009), we employ the Green's function quoted in McPhedran et al. (2009): 2.3where ρ=r−r′ and ρ=|ρ|. This is the solution of the general equation 2.4Here is the Hankel function of the first kind, of order 0 and K0 is a related modified Bessel function. This Green's function, unlike that for the Laplacian, is bounded and not logarithmic near r=r′.
For the grating stack problem, the displacement field W is the sum of the incident wave and the scattered field (to be determined): 2.5The field W satisfies the Bloch quasi-periodicity condition along the horizontal x-axis: 2.6where , d is the period, and α0 is the Bloch parameter , where θi is the angle of incidence (figure 1).
The solution of equation (2.1) is conveniently expressed in the form of 2.7where WH and WM are solutions of the Helmholtz and modified Helmholtz equations, respectively, 2.8
The incident field is represented by plane waves of two types:
Propagating (or evanescent) solution of the Helmholtz equation 2.9where . Here χ0 is real and positive for a propagating solution. For the evanescent solution, χ0 is pure imaginary, with positive imaginary part.
Evanescent solution of the modified Helmholtz equation 2.10where , , τ0>0.
As in Movchan et al. (2009), we also use similar plane wave series expansions to describe the reflected and transmitted waves.
3. Recurrence algorithm for shifted gratings
We follow a procedure similar to that outlined by Movchan et al. (2009). For photonic crystal structures, the scattering matrices were constructed and analysed by Botten et al. (2001). For a single grating, we define the matrices R± and T± to characterize the reflection and transmission of an incident wave meeting the grating from above (+) or below (−): 3.1
Here H and M refer to Helmholtz and modified Helmholtz type waves, respectively. For the R+ matrix, which consists of four block matrices of order (2N+1)×(2N+1), where N denotes the order of truncation of the multipole expansions, the (p,q)th element of the block R+HH gives the reflection coefficient Rp when the incident wave comes from above the grating, and its only non-zero amplitude is for channel q. The (p,q)th element of R+MH contains the reflection coefficient for the same incident wave. The (p,q)th entries of R+HM and R+MM are the reflection coefficients Rp and for the incident wave from above the grating with its only non-zero amplitude being for channel q. The matrices R− and T± are filled in the same way.
We then build a stack of gratings by placing an additional grating on top of the preceding layers. Compared with Movchan et al. (2009), we introduce an additional propagation matrix to account for the horizontal shifts of gratings relative to each other. For electromagnetic waves in photonic crystal structures, a similar procedure was employed by Botten et al. (2001).
(a) Propagation matrices for shifted gratings
When we change from a single grating to a pair of gratings within the stack, it is convenient for the system of coordinates to have its origin on the symmetry line between the gratings. In this case, we use propagation matrices to shift the phase origin of the reflection and transmission matrices by ±η/2 in the vertical direction and ξ/2 in the horizontal direction, where η is the vertical separation and ξ is the relative horizontal shift of the neighbouring gratings.
We define two propagation matrices, one for the vertical direction and the other to account for the lateral shifts: 3.2with if p corresponds to a Helmholtz type wave and if p corresponds to a plane wave of modified Helmholtz type, and 3.3where 3.4varies with the diffraction order p of the wave. As the matrices are diagonal, their multiplication is commutative and this allows convenient notation for the reflection and transmission matrices on the central line between successive gratings.
(b) Recurrence relations for a stack of gratings
For our recurrence procedure for an incident wave from above the grating stack, we use relations from Movchan et al. (2009) for the reflection and transmission coefficients for a stack of s+1 gratings. The sole difference is that the introduction of and its conjugate affects the forms of the reflection and transmission matrices, which we use in the recurrence procedure and subsequent numerical calculations: 3.5
Here the subscript s identifies reflection and transmission matrices for a stack consisting of s gratings. For an incident wave from below the stack, two additional matrices are required: We consider the addition of a single grating to a finite stack of s gratings. This stack is characterized by four scattering matrices, R+s, on the upper side of the stack, and , below the stack. The additional grating is characterized by R+1, on its upper side and , on its lower side. An incident wave with amplitude δi arrives at the stack of s+1 gratings. The products of these matrices and the propagation matrices, described by equations (3.5), are used to characterize the repeated reflection and transmission of a plane wave within the structured interface in the form of sums involving geometric series, one for reflection from the upper side of the stack of s+1 gratings, and the other for transmission through the stack. These expressions are simplified to derive the following recurrence relations: 3.6and 3.7where , , , , P and Qξ are defined as in equations (3.2)–(3.5).
We convert the matrices to the forms incorporating the phase shifts as in equations (3.5) and then we start the recurrence. There is explanation for the recurrence procedure without the horizontal shifts in the study of Movchan et al. (2009). To evaluate the matrices , , we re-phase with accordingly.
The building block of our recurrence procedure is a single grating. For completeness, we give the conservation of energy relation which governs the scattering of flexural waves by a single grating, derived by Movchan et al. (2009): 3.8As mentioned earlier, we use an integer p to represent the diffraction order of the scattered field, and we define ΩH to represent the set of propagating orders of Helmholtz type. The sets and contain evanescent waves, of Helmholtz and modified Helmholtz type, respectively. The set comprises all integers p since all modified Helmholtz waves are evanescent. We use δp to denote the amplitude of the Helmholtz type incident wave, and to represent the amplitude of the incident wave of modified Helmholtz type. The direction is specified using equation (3.4) for both.
4. Trapped modes for a pair of shifted gratings
In the study of Movchan et al. (2009), attention was drawn to the existence of trapped modes between a pair of gratings, characterized by very sharp transmission resonances at specific frequencies. This analysis was carried out for a symmetrical pair of gratings where the rigid pins were aligned directly above one another, and the angle of incidence of 30° was highlighted for its particularly striking example of enhanced transmission. We also consider this θi, and investigate how shifting the gratings relative to one another affects the resonance frequency and hence the transmittance across the stack.
We consider identical gratings of rigid pins with periodicity d, but the upper grating is placed so that the pins are not aligned directly above one another (figure 2). The relative grating separation is determined by η/d, where η is the vertical distance between the gratings. We denote the horizontal shift by ξ. We proceed with the recurrence procedure outlined in the previous section and derive expressions for the reflection and transmission coefficients characterizing this pair of gratings: 4.1and 4.2where , , , , P and Qξ are defined as in equations (3.2)–(3.5), and U and L represent the upper and lower grating, respectively.
Re-phasing is carried out to obtain and using the appropriate inverse propagation matrices. It is important to remember that as a shifted layer is added onto the existing stack, alternative propagation matrices are required and therefore re-phasing with the corresponding inverse propagation matrices must be carried out at each stage of the recurrence algorithm. This is obviously more relevant as the size of the stack increases from a single pair.
(a) Numerical results for shifted pairs
We use numerical methods to identify the existence of localized modes for various shifted pairs. In figure 3, we show the normalized transmitted energy as a function of β for a pair of gratings of rigid pins with unit periodicity (d=1), for various relative horizontal shifts ξ of the gratings. The subscript p is an integer representing the diffraction order of the scattered field, ΩH represents the set of propagating orders and therefore all p∈ΩH contribute to . The subscript H refers to Helmholtz-type propagating waves. For convenience, we denote the normalized transmitted energy by 4.3
We note in figure 3 that, irrespective of the shift of the gratings, a sharp spike in transmittance occurs near β=4.2, where the reflectance of the single grating is zero. This is a Rayleigh anomaly, where the order −1 passes off (i.e. where α−1=−β) and occurs at β=4π/3.
Popov et al. (1986) have discussed the consequences of the symmetry of structures incorporating lossless diffraction gratings, in the range of frequency for which only zeroth orders in reflection and transmission are propagating. They concluded that if the grating structure has a centre of symmetry, then a 100 per cent transmission is achievable. If it has symmetry about a vertical axis, then a 100 per cent reflection is achievable. If both these properties exist, or alternatively, if it has symmetry with respect to a horizontal axis, then a resonant region combining a 100 per cent reflection together with a 100 per cent transmission is achievable.
In figure 3, we note the extremely sharp transmittance resonance near β=3.6. This full transmittance, irrespective of the shift value, is in keeping with the discussion of Popov et al. (1986), since the double grating structure has a centre of symmetry irrespective of the shift value.
The extremely high transmittance for a pair of gratings corresponds to nearly 100 per cent reflectance for a single grating (dotted-dashed curve), indicating the possibility of a trapped mode. The value of β for which the trapped mode occurs, β* say, increases with increasing shift up to half of the period, where it attains its maximum. Its minimum corresponds to the case of zero shift. In §5b, where we compare pairs and triplets, we include figure 9, which shows the frequency of enhanced transmission β* versus shift ξ for shifted pairs. The curve is symmetric about ξ=0.5d, as expected for a pair of identical gratings incorporating a relative lateral shift.
(b) Quality factors of transmittance resonance
We focus most of our attention on a shift of ξ=0.5d for unit periodicity (d=1) and unit grating separation (η=d), since this returns the highest value for β* and the highest quality factor for the transmittance resonance. The quality factor or Q factor is a dimensionless parameter which characterizes a resonance's bandwidth relative to the frequency of the maximum energy. We use the standard definition 4.4where β* is the resonance frequency and Δβ is the frequency difference between the half-power points. For a given β*, the narrower or sharper a peak is, the higher the Q. Typically, an equivalent definition is 4.5where β*+iγ* represents a complex resonant frequency, and this definition allows us to determine how far we have to go into the complex plane to find a resonance. A third definition of Q expresses it as the ratio of the energy density inside the resonant structure to that outside it.
The resonance frequencies for the pairs of unshifted and shifted pairs are 3.58221 and 3.65970, respectively. The quality factor for the unshifted pair is around 5400 (Movchan et al. 2009) and we also find an extraordinarily large quality factor of around 9400 for the grating pair with a shift of half the period. Even sharper transmittance spikes are observed when we double the grating separation to η=2d. We observe a quality factor of around 35 000 for the unshifted pair, and 19 000 for a shifted pair with ξ=0.5d=η/4. This increase in Q is consistent with the reduction in the effect of evanescent modes owing to the increased separation of the gratings, relative to their periodicity.
In figure 4, we show the resonance transmittance spikes for the unshifted and shifted pairs for η=d. Both exhibit the extreme sharpness of the resonance, which arises at these specific frequencies because of the reflectance characteristics of the single grating.
We also note the separation shape of the resonant peaks, which is characteristic of Fano resonances (Fano 1961). These arise when there is a slow background variation of transmittance, in the presence of a rapid foreground variation. In our case, the slow variation is that of the transmittance of a single grating, whereas the rapid variation is caused by the change of phase between the gratings as β varies (see the terms P and Qξ in equations (3.2) and (3.3)).
In this analysis, we have focused on a specific angle of incidence θi=30°, and discussed the resulting enhanced transmission effects, and the quality factors which characterize the resonances. The angle at which the plane waves are incident on the system is an important parameter of the model, and some comparison of the trapped modes arising for 30° and 27° was made by Movchan et al. (2009).
5. Analysis of trapped modes for stacks of three gratings
We extend the stack to triplets of gratings, one consisting of three parallel, aligned gratings and one with a shifted middle grating. We analyse the effect of relative shifts on the frequency and the quality factor of the resonance. We compare the nature of the trapped modes arising from both doublets and triplets.
(a) Symmetric triplets of gratings
A triplet is constructed wherein the separation between each grating is unity and an overall symmetric geometry is obtained. Two geometries are analysed here. The first contains three parallel, aligned gratings (figure 5a), and therefore may be considered to consist of two pairs of aligned gratings. As the separation of the outer gratings is 2d, comparison is also made with the pair with η=2d. The other triplet contains three identical gratings, but with the middle layer incorporating a shift of ξ=0.5d (figure 5b). Therefore, this configuration comprises two shifted pairs but with the upper grating shifted in the direction opposite to the central one. Once again, the outer gratings are effectively aligned gratings with η=2d.
Referring to figure 5, we observe that the horizontal axis y=0 acts as a line of symmetry for the grating stack. In treating diffracting systems with up-down symmetry, it is of value to break up the diffraction problem into symmetric and anti-symmetric parts (figure 6), where two types of trapped modes are characterized by their symmetry and anti-symmetry about y=0 (McPhedran & Maystre 1977; Botten et al. 1985; Popov et al. 1986). The general incidence, illustrated in figure 6a, is the superposition of the anti-symmetric and symmetric incidences shown in figure 6b,c, subject to a constant multiple, i.e. (a)=1/2[(b)+(c)]. For the frequency region where only zeroth orders in reflection and transmission propagate and where complex amplitudes are evaluated with respect to an origin at the centre of symmetry, the consequences of symmetry were discussed by McPhedran & Maystre (1977). We note that in our problem, the reflected and transmitted zero-order amplitudes differ in phase by ±π/2, as in McPhedran & Maystre (1977).
With our generalized treatment, we use the recurrence procedure described in §3. For the geometry of figure 5a, this is straightforward since the propagation matrices for adding the third grating are identical to the ones used to add the second grating. More care is required for the geometry of figure 5b. The first addition involves adding a grating with a shift to the right but after re-phasing with appropriate propagation matrices, the second shift is in the opposite direction so that the pins of the two outer gratings are aligned. Therefore, different propagation matrices are required for this stage of the procedure and the corresponding re-phasing is carried out to obtain expressions for reflection and transmission coefficients for a stack of three gratings.
(b) Numerical results for symmetric triplets
We focus our numerical analysis on the unshifted triplet, and on a shift of half the period ξ=0.5d for unit periodicity and unit grating separation η (figure 7), since these geometries have the full symmetry (Popov et al. 1986) which permits resonances in which the transmittance varies from zero to unity. We consider more general shifts later. In figure 7a, we plot Ttot versus β. Each triplet generates two localized resonances for β in the range 1.2–4.2, with one fixed resonance always at β*=3.61747, and the other varying in frequency with the shift.
We also observe transmittance resonance for the frequency β*=3.61747 for the pair of unshifted gratings with η=2d. This is to be expected since the triplet is made up of an outer pair of parallel gratings and a shifted middle grating whose effect is cancelled out by the anti-symmetry, as in figure 6b. In effect, the middle grating is not ‘seen’. All three enhanced transmission resonances are shown in detail in figure 7b.
The other two resonant frequencies for the triplets occur at β*=3.56573 and 3.68599, the first for the unshifted triplet and the latter for the triplet with the shifted middle grating. These frequencies are similar to the resonance frequencies for the corresponding pairs of unshifted and shifted gratings (3.58221 and 3.65970, respectively).
This correspondence is explained by the symmetric geometry of the triplet. It is effectively made up of two pairs of shifted pairs. Both these pairs allow for ‘trapped’ modes at the same resonance frequency β* and thereby support the enhanced transmission across the three-grating system. The numerical difference (less than 1%) arises in the transmission problem owing to the inclusion of evanescent modes in our model, which allows for some field coupling between the gratings.
In figure 8, we show these transmission resonances for a shift of half the period. The proximity of the peaks is evident, as is the Fano nature of the resonances, which is more pronounced for the resonance arising from the pair of shifted gratings (solid line). This spike displays the higher Q (over 9000).
For the general shift, we observe both the fixed resonance at β*=3.61747, and the same correlation for the other transmission peak and the corresponding shifted pair. This is shown in figure 9, where we plot β* versus ξ for both shifted pairs, and the symmetric trapped modes for the triplets with a shifted middle grating.
For a triplet with outer gratings separated by η=2d, but with the middle grating placed away from y=0, we break the symmetry that was discussed in §§4a and 5a. (McPhedran & Maystre 1977; Popov et al. 1986). This removal of the centre of symmetry results in the reduction of the system's ability to facilitate enhanced transmission. Indeed, our numerical computations stop producing enhanced transmission spikes with high Q for small deviations from the symmetrical geometries we have discussed.
(c) Quality factors for the trapped modes
We have noted the correspondence between trapped modes arising at the same frequency for pairs and triplets. It is interesting to determine which geometry produces the sharper transmittance resonance. In §4b, we noted the extraordinarily large quality factors arising from pairs of gratings. For the triplets, we still observe sharp resonances but they are not as pronounced as for the equivalent geometry with two gratings.
For the parallel triplet (figure 5a), we record a quality factor of around 3800 for the symmetric trapped mode, which is below that of the equivalent localized mode for a pair of aligned gratings (around 5400). For the shifted triplet (figure 5b), the symmetric mode's resonance quality factor is also down on its counterpart with two gratings, being around 1600 compared with 9400 for the shifted pair. This difference arises because of the increased coupling effect of evanescent modes since the triplets consist of two of the pairs with which we are comparing them. However, the anti-symmetric modes arising for both triplets which are a more precise fit with the equivalent localization for two aligned gratings as η=2d for the outer pair in both cases, have extremely large and almost identical quality factors to the pair, being around 35 000. The aligned triplet results in the marginally higher Q, being closer to 36 000.
(d) Double resonances and the analogy with electromagnetic-induced transparency
As illustrated in figure 7, enhanced transmission occurs at frequencies corresponding to resonances of symmetric and anti-symmetric modes within the triplet of gratings. The frequency of the anti-symmetric mode is independent of the shift of the middle grating, whereas that of the symmetric mode changes with the shift ξ, as shown in figure 9. For an angle of incidence θi=30°, the frequencies of symmetric and anti-symmetric modes coincide for the specific value of ξ=0.25200d, and have the common value β*=3.61747. We will now discuss the double resonance effect associated with this particular geometry.
In figure 10, we show the transmittance for the three grating stack with the optimized shift ξ, and that for the pair of gratings constituting one half of the stack (i.e. with shift ξ and spacing η=d). It will be noted that the pair of gratings has a transmission resonance reaching unity, while for the triplet the peak is suppressed, and replaced by a sharp minimum reaching down to zero. The presence of this minimum shows that the phases of the symmetric and anti-symmetric parts of the solution referred to in §5a differ by either 0 or π. In the first case, the symmetric and anti-symmetric parts will add in the upper half of the triplet, but cancel in the lower half, preventing energy flow into the transmitted field region. In the second case, they will cancel in the upper half of the structure, leading to total reflectance off its upper surface.
The Q factor for the transmission peak for the shifted pair (ξ=0.25200d) is 1130, and is considerably lower for the triplet with the middle grating shifted by ξ=0.25200d because of the broader peak arising from the double resonance. We also note that the normalized transmission energy is less than one (around 0.92) for the two identical peaks shown in figure 10. This is because of the extreme dip to zero transmission observed at β*=3.61747, and this notch has an extraordinarily high Q factor of over 36 000. The dashed lines, indicating the half-power and full-power points, clearly illustrate the extreme sharpness of this dip.
The phenomenon of electromagnetic-induced transparency (EIT) is a quantum-mechanical effect which arises in three-level atomic systems. It involves a peak in absorption of a probe beam, which is suppressed by the addition of a coupling beam which creates a ‘window of transparency’ for the probe beam. The curve of absorption versus frequency for EIT bears a striking resemblance to the transmittance curve of figure 10. Corresponding factors between EIT and our double resonance phenomenon are the three levels (read three gratings), the creation of destructive interference (in EIT owing to the choice of atomic system and the coupling beam, in our double resonance, the accurate control of the shift parameter) and the very high Q factors attainable. Differing factors are that our double resonant system does not absorb, and that it does not need separate probe and control beams. Nevertheless, this analogy between EIT and the double resonant grating stack illustrates the intrinsic wave nature of quantum mechanics.
S.G.H. gratefully acknowledges the financial support of the Duncan Norman Charitable Trust through the Duncan Norman Research Scholarship. The participation of R.C.M. in this project was facilitated by a grant from the Research Centre in Mathematics and Modelling of the University of Liverpool.
- Received May 19, 2010.
- Accepted July 21, 2010.
- This journal is © 2011 The Royal Society