One-dimensional reflection by a semi-infinite periodic row of scatterers

Three methods are described in order to solve the canonical problem of the one-dimensional reflection by a semi-infinite periodic row of identical scatterers. The exact reflection coefficient R is determined. The first method is associated with shifting the domain byasingleperiodandsubsequentlyconsideringtwoscatterers,onebeingasinglescatterer and the second being the entire semi-infinite array. The second method determines the reflection coefficient R N associated with a finite array of N scatterers. The limit as N → ∞ is then taken. In general R N does not converge to R in this limit, although we summarize various arguments that can be made to ensure the correct limit is achieved. The third method considers direct approaches. In particular, for point masses, the governing inhomogeneous ordinary differential equation is solved using the discrete Wiener–Hopf technique. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY


Introduction
The wave bearing properties of periodic inhomogeneous media are important in many application areas in physics, engineering and applied mathematics and have been studied extensively [1].The standard procedure is to consider unbounded inhomogeneous media, for which Floquet-Bloch conditions are imposed in order to determine their band-gap properties.Ranges of frequencies where the predicted Bloch wavenumber is complex are associated with stopbands where no propagation is permitted in the structure.Of course, in practise, no periodic medium is of infinite extent, and so inspection of the propagation characteristics of such periodic materials involves inclusion of boundary or interface conditions.These are less-well studied models; for example, the wave propagation from a semi-infinite homogeneous material into a semi-infinite or finite-width heterogeneous periodic medium, where the aim is to predict the reflection and transmission coefficients, R and T , respectively.The link between this and the unbounded problem is the expectation that |R| = 1 (T = 0) for incident Fig. 1.Depiction of a time-harmonic wave incident on, and reflected from, a semi-infinite array of identical scatterers located at x = nd, n = 0, 1, 2, . . . .The scatterers are sketched as beads, although other kinds of inhomogeneities are permitted in the analysis.
frequencies that reside in band-gaps, i.e. all the incident energy is reflected.However, the reflection and/or transmission coefficients are measured by experimentalists to infer all the effective properties of the heterogeneous material, not just the band-gaps, and hence it is important to understand more fully the semi-infinite scattering problem.This article offers exact procedures for a simple canonical problem in order to illustrate possible approaches that may be applicable for tackling more complex heterogeneous periodic materials.
Here we consider the problem of steady one-dimensional reflection of waves by a semi-infinite periodic row of identical scatterers.As illustrated in Fig. 1 the scatterers are located at x = nd, where n = 0, 1, 2, . . ., and d > 0 is the spacing.
Between the scatterers, the motion is governed by a wave equation for the displacement U(x, t) = Re {u(x)e −iωt }, and so u(x) satisfies the reduced equation u ′′ (x) + k 2 u(x) = 0, where k is real and positive.As the field is steady, the time-dependence e −iωt is henceforth suppressed.The forcing for the problem is a unit-amplitude incident wave, u inc = e ikx , propagating towards the right (x increasing), the main aim is to calculate the reflected wave, Re −ikx , where R is the (complex) reflection coefficient.
The literature on this one-dimensional ''reflection problem'' is substantial; some of it will be discussed below.The problem itself is interesting for a number of reasons, not least that it can be solved explicitly, and so it is one of our purposes to give this solution.The goal is to express R in terms of the scattering properties (reflection and transmission coefficients) of the constituent scatterers and kd.Inevitably, properties of the periodic structure (passbands and stopbands) will play a role.Once R is known, it can be used for other related problems, such as scattering by additional scatterers to the left of the semi-infinite row, or for studying the effects of defects between two semi-infinite rows (one extending to x = +∞ and one extending to x = −∞).
We describe three ways to solve the reflection problem.The first way is very simple (Section 4).We exploit the consequences of shifting the semi-infinite row by one period (to the right or left).In effect, we regard the semi-infinite row as two scatterers, one of which is another semi-infinite row.This idea goes back to a series of papers by Millar in the 1960s, starting with [2].He used it for several two-dimensional grating problems.A similar approach was used for layered media by Shenderov [3].In our one-dimensional context, we obtain a quadratic equation for R; we show how to select the correct solution.We remark that there has been much recent interest in related two-dimensional waveguide problems; see, for example, [4][5][6], where the shifting-by-one-period idea is again employed, leading to a quadratic equation for a certain operator.
Another feature of the ''solution-by-shifting'' is that the Floquet-Bloch relation for periodic media, u(x + d) = e iqd u(x) (see Section 3), is not used.Nevertheless, we show subsequently that, in fact, it is satisfied (Section 4.2).Then, as an example, we show that our formula for R agrees with one obtained by Levine [7] for point scatterers (Section 4.3).
The second way to find R is to start with reflection (and transmission) by a finite periodic row of N scatterers, followed by taking the limit as N → ∞ (Section 5).Explicit formulas for the reflection and transmission coefficients, R N and T N , are known.It turns out that R N → R as N → ∞, where R comes from the shifting method, but only when we are in a stopband for the periodic structure.In a passband, R N does not have a well-defined limit as N → ∞.In the literature, methods have been devised for finding a sensible limiting solution, by introducing some kind of loss into the system: we show that doing this leads to R, as found by the shifting method of Section 4. The third way is to solve the reflection problem (with a semi-infinite row) directly.Some possible approaches are described in Section 6.After a description of Levine's work [8] (Section 6.1), we solve the problem using a discrete form of the Wiener-Hopf technique (Section 6.2).
Perhaps inevitably, there are other ways to solve the problem, which we do not discuss.For example, one such approach is developed in [9].

One scatterer
Suppose that we have just one scatterer, located at the origin.Assume it is inside a ''cell'', |x| < a, where 0 < a < 1 2 d.The properties of the scatterer are encoded in the cell's reflection and transmission coefficients, r ± and t ± , defined as follows.For a wave incident from the left, we have ( Similarly, for a wave incident from the right, (2) We assume that the scatterers are lossless.Then it is well known that where the asterisk denotes complex conjugation.If t ̸ = 0 and r ̸ = 0, and we write r ± = |r|e iρ ± and t = |t|e iτ , the third of Eq. ( 3) reduces to a constraint on the phases, e 2iτ + e i(ρ + +ρ − ) = 0. ( If the cell is moved from x = 0 to x = b, it is easy to show that the new reflection coefficients are r + e 2ikb and r − e −2ikb , whereas the transmission coefficient remains unchanged.Henceforth, we assume that r ± and t are known or, at least, they can be computed.Then our problem reduces to calculating R in terms of r ± and t.
For a specific example, we can take a point scatterer at x = 0. Then a = 0, where M is a real constant.We find r This example was considered by Levine [7]; his α is our M.For a physical interpretation, the scatterer could be a bead of mass m 0 on a string of linear density ρ 0 , and then M = −m 0 k 2 /ρ 0 [8, Eq. (17.4)].

Periodic media: brief review
Wave propagation in periodic media is well studied.We review some aspects, because of their relevance to our problem.Classically, Floquet theory covers waves in periodic media, where the governing ordinary differential equation is v ′′ (x) + k 2 n(x)v(x) = 0 and n(x) is a periodic function with period d; v(x) and n(x) are defined for all x.This may not valid as we do not necessarily assume that the behaviour everywhere inside the cell is governed by a differential equation, e.g.we may allow points of discontinuity.
The main result of Floquet theory is that all solutions have the form v(x) = e iqx w(x) where q is a constant and w(x) is Extensions to other situations, including partial differential equations, are usually associated with the name of Felix Bloch: we shall refer to Eq. ( 7) as the Floquet-Bloch relation. 1 Returning to our problem, there are several ways to proceed.We can connect the fields in adjacent regions (where u ′′ + k 2 u = 0; see Section 3.1) across the cells, or we can connect the fields in neighbouring periods of the periodic structure (Section 3.2).Conceptually, these are different approaches but, of course, they lead to the same results, including a dispersion relation connecting qd to kd and properties of the constituent scatterers.

Connect across one cell
Consider the cell at x = md.To its left, we can write and to its right, 1 The terminology varies.Floquet [10] studied solutions of systems of first-order ordinary differential equations with periodic coefficients, but he did not consider wave motion.That was done later by Bloch [11] in the context of the three-dimensional Schrödinger equation with a periodic potential.For further discussion, see [1].
Using the cell's transmission and reflection coefficients, t and r (m) ± = r ± e ±2ikmd , we obtain Rearranging, using r Now, in a periodic medium, the Floquet-Bloch relation, Eq. ( 7), says that for some q.This gives A m+1 e ikd = A m e iqd and B m+1 e −ikd = B m e iqd .Substituting in Eq. ( 8) gives an eigenvalue problem for q.Thus, if we put λ = e iqd , we find that where ξ is a real parameter defined by The quadratic equation ( 10) can be written as cos qd = ξ , giving the familiar interpretation in terms of passbands (|ξ | < 1) and stopbands (|ξ | > 1).In particular, for point scatterers (see Eq. ( 6)), we obtain

Connect across one period
Consider neighbouring cells, at x = md and x = (m + 1)d.In the interval where A m and B m are coefficients, and u ± are fields defined by Eqs. ( 1) and ( 2) when the cell is moved from x = 0 to x = md; this movement changes r ± to r (m) ± = r ± e ±2imkd .Similarly, in the next interval, from X to X + d (which contains the cell at x = (m + 1)d), we can write The fields in Eqs. ( 13) and ( 14) should match at x = X (as should their derivatives): As X can vary, we obtain which we write as These determine A m+1 and B m+1 from A m and B m .In our scattering problem we know A 0 (from the specified incident wave) but not B 0 (in fact, R = r + + B 0 t).For a finite row of scatterers, a second piece of information comes from the radiation condition: there are only right-going waves to the right of the row.However, for a semi-infinite row, we do not have an obvious radiation condition.We could impose the Floquet-Bloch condition, Eq. ( 9), as appropriate for periodic media, but it is not clear that this is correct.Doing this is equivalent to requiring that A m+1 e ikd = λA m and B m+1 e −ikd = λB m for some λ.Imposing this condition gives an eigenvalue problem for λ, Setting the determinant of this system to zero gives precisely Eq. ( 10).

First method: move the semi-infinite row
If we move the semi-infinite row by one period to the right, the new reflection coefficient is Re 2ikd = R 1 , say.But we can also view the original row as comprising a single cell located at x = 0 to the left of a semi-infinite row with cells located at x = nd, n = 1, 2, . . . .Between the cells at x = 0 and x = d (in fact, for a < x < d − a), we can write u(x) = A 1 e ikx + B 1 e −ikx .The wave incident on the cell at x = d, A 1 e ikx , is reflected as A 1 R 1 e −ikx .Thus The wave incident on the cell at x = 0 consists of e ikx from the left and B 1 e −ikx from the right.Thus
When ξ 2 > 1, we are in a stopband for the periodic structure and we have perfect reflection: |R| = 1.When ξ 2 < 1, we are in a passband, and then |R| < 1; we examine this case in more detail in Section 4.3.

Field between the scatterers
We can calculate the field between the cells: does it satisfy the Floquet-Bloch relation, Eq. ( 9)? Let us start by calculating the field between the cells at x = 0 and x = d in terms of R. We have Next, consider the field between the cells at x = d and x = 2d, and write it as u(x) = A 2 e ikx + B 2 e −ikx .The wave incident on the cell at x = 2d, A 2 e ikx , is reflected as The wave incident on the cell at x = d consists of A 1 e ikx from the left and B 2 e −ikx from the right.Thus These can be rearranged to give that is, A 2 and B 2 are written proportional to A 1 and B 1 , respectively.Now, if the Floquet-Bloch relation, Eq. (9), is satisfied, we should have A 2 e ikd = λA 1 and B 2 e −ikd = λB 1 , for some λ (see below Eq. ( 9)).Using these relations, and then eliminating λ, we obtain e ikd t −1 {e 2iτ + r − R} = e −ikd t −1 {1 − r + R −1 }, which reduces precisely to Eq. ( 18): the fields inside the semi-infinite row do indeed satisfy the Floquet-Bloch relation.

Application to point scatterers
Let us consider point scatterers in a passband: we have Eqs.( 6) and (12), with qd real.Suppose that S and sin qd are positive.(Care with signs is needed because of the square-roots.)Then Eqs. ( 21) and (19) give where µ = 1 2 M/k = ir/t.This elegant formula was found by Levine [7, Eq. (38)], using a different method (see Section 6).
Let us return to the matter of care with signs.We find that Re ikd = cos qd − cos kd 1 − cos qd cos kd + sin kd | sin qd | sgnS , with S = |t|{sin kd − µ cos kd}.Thus, we obtain Eq. ( 26) when S sin qd > 0 but the fraction on the right-hand side must be inverted when S sin qd < 0.

Second method: start with a finite row
Consider a periodic row of N identical cells, centred at x = nd, n = 0, 1, 2, . . ., N − 1.We define reflection and transmission coefficients, R N and T N , by x > (N − 1)d + a.We investigate letting N → ∞ so as to solve the semi-infinite-row problem.
The quantities R N and T N are known exactly, where U n is a Chebyshev polynomial of the second kind, defined by and ξ is the real parameter defined by Eq. ( 11).

Comments on the formulas for R N and T N
The formulas in Eqs.(27) see [14].Note that the eigenvalues of P are given by Eq. ( 10).
Direct calculation shows that |R N | 2 + |T N | 2 = 1, as required (for any ξ ).To see this, use the recurrence relation
Let us compare R ∞ with R, defined by Eq. (24).Using e −η = ξ −  ξ 2 − 1, we have which agrees precisely with the expression for R obtained in Section 4.1.

Letting N → ∞ when |ξ | < 1
When |ξ | < 1, we are in a passband.Put ξ = cos qd, whence Eq. ( 27) gives These formulas do not have well-defined limits as N → ∞, and so we must seek ways to interpret them.
''For very large N, and real φ, sin 2 Nφ will oscillate rapidly around the value 1 2 '' [15, p. 1121], giving a large-N average estimate for |T N | 2 (replacing sin 2 (Nqd) in Eq. ( 32) by 1 2 ), which in turn gives the estimate This is incorrect: it does not agree with Eq. ( 21), which gives This appears rather concerning: the obvious ''averaging'' of the transmitted wave as N → ∞ does not lead to the ''correct'' answer obtained in the previous section.Indeed, Markoš and Soukoulis offered the pessimistic remark [16, p. 88] concerning point scatterers: ''In no case can we claim that an infinite crystal can be obtained in the limit N → ∞ of δ-function barriers''.
However, we now show a way to overcome this difficulty.
In the context of acoustic waves in a layered half-space, Gilbert [17] and Schoenberg and Sen [18] start by introducing a small amount of loss and then reason as follows.The matrix P, Eq. ( 29), can be diagonalized using its eigenvalues, λ 1 and λ 2 , so that P N can be written in terms of λ N 1 and λ N 2 .When |ξ | < 1, λ 1 and λ 2 are on the unit circle in the complex plane.
Introducing some loss moves one eigenvalue (λ 1 , say) inside the circle and one outside.Eliminating the effect of the growing term, λ N 2 , leads to a formula for R. Thus, the eigenvector (x 1 , x 2 ) T corresponding to the eigenvalue λ 1 = ξ +i Then R is given by which agrees precisely with Eq. ( 20).Similar approaches have been used for more complicated problems [19][20][21][22][23].In fact, the idea of adding some damping so as to extract a physically meaningful solution goes back to Rayleigh; see, for example, [24, Section 1.5] or [25, p. 259 and p. 478].

Third method: direct treatments
Another possibility is to tackle the reflection problem directly, with a semi-infinite row of scatterers.The first difficulty is specifying and then enforcing some kind of radiation condition as x → ∞.Explicit recognition of this difficulty is fairly recent.In a stopband, u(x) → 0 exponentially as x → ∞.In a passband, it is not so clear what to do: waves propagate back and forth between each cell, as within a finite row of scatterers.Potel et al. [26] require that the power flux be in the +x direction; Levine [8, Section 53] has calculated this quantity for point scatterers.Again, another option is to introduce some loss, leading to a form of ''limiting absorption principle''.

Levine's approach
Levine has given direct treatments of the problem, first for point scatterers [7] (see Section 4.3 and [8, Section 52]) and then [8, Section 62] for a more general situation governed by with u(x) = e ikx + Re −ikx for x ≤ 0; here, η is a given d-periodic function.A special role is played by the function ϕ [8, Eq. (62.8)], defined by Eliminating η(x)u(x) in Eq. ( 34) using the differential equation, Eq. ( 33), followed by two integrations by parts gives [8, Eq. (62.9)], Assuming that u(x) and u ′ (x) are continuous across x = 0, u(0 Now, to find u, Levine [8, Eq. (62.5)] starts by writing down a Lippmann-Schwinger equation, but this is suspect within a passband when k is real.Instead, we can use Laplace transforms.Thus, let U(s) = L{u} =  ∞ 0 u(x)e −sx dx be the Laplace transform of u(x).Then Eq. ( 33) gives This is a Volterra integral equation of the second kind for u.For similar treatments, see [27,28].
At this stage, we have not used the d-periodicity of η or the Floquet-Bloch relation, Eq. ( 9), u(x + d) = e iqd u(x).Levine assumes that this relation holds for x ≥ 0, whence u(d) = e iqd u(0) and u ′ (d) = e iqd u ′ (0).Then Eq. (35) gives which can be used to obtain R in terms of qd, kd and ϕ(k).To make further progress, we need u(x) for 0 < x < d. Levine uses various Fourier series but does not obtain explicit formulas (except for point scatterers).Another option would be to solve the integral equation, Eq. (36); recall that such equations can always be solved by iteration.
To conclude, we might ask: how did Levine obtain the correct result for point scatterers, as discussed in Section 4. 1 − e i(q+k)d , which is valid provided Im (q + k) > 0, i.e. some loss was introduced without comment!

A Wiener-Hopf approach
For a semi-infinite periodic row of point scatterers, an exact solution can be obtained using the Wiener-Hopf technique [24].The governing equation is We write the solution u in terms of a new function v, defined by u(x) = e ikx − e −ikx + v(x) for x < 0 and u(x) = v(x) for x > 0. The continuity conditions on u at each scatterer give continuity of v at x = nd for n = 0, 1, 2, . . .; the term −e −ikx is included in the definition of v(x) for x < 0 so as to ensure continuity of v(x) at x = 0.At the first scatterer, the condition u ′ (0 + ) − u ′ (0 − ) = Mu(0) (see Eq. ( 5)) gives (nd) for n = 1, 2, . . . .Using Green's function associated with the homogeneous string, e ik|x−y| /(2ik), we find where v n = v(nd).This equation cannot be solved directly using Floquet-Bloch theory as the geometry is not periodic in x for all x.The semi-infinite structure suggests using the Wiener-Hopf technique, which we do by employing the z-transform.For other applications of the discrete Wiener-Hopf technique see, for example, [29][30][31][32][33][34][35].
First set x = md in Eq. (37) to yield the infinite algebraic system Note that for m ≥ 0, Eq. ( 38) gives a closed system to solve for v m .For m < 0, Eq. ( 38) defines v m in terms of v n with n ≥ 0. Now, apply the z-transform, i.e. multiply Eq. ( 38) by z m and sum over all m ∈ Z, to give The order of summation can be reversed in the final term on the right-hand side, and hence shifting the counter m to where With k real, we see that the series defining F + converges for |z| < 1 whereas the series defining F − converges for |z| > 1.In order to obtain a common region (annulus) of the complex z-plane for convergence of these sums we add a small amount of damping, replacing kd by kd + iε = κd, say, with 0 < ε ≪ 1.Then the series defining F + and F − converge for |z|e −ε < 1 and |z|e ε > 1, respectively, and as ε ≪ 1, the overlap region is approximately 1 − ε < |z| < 1 + ε.Summing the geometric series yields where we reiterate that the subscript + indicates that the function is analytic in the domain |z| < e ε .Similarly for F − we find that Using these results for F ± , we write the system, Eq. (39), as where we have introduced the notation As for F + , function V + is analytic in a disc; but here we assume this region is |z| < e δ + ε for some δ + > 0, which contains the unit circle.Similarly, V − is taken to be analytic outside the disc |z| > e −δ − ε for some δ − > 0. These assumptions, which will be verified a posteriori, mean that Eq. ( 40) is valid in the annular region D: exp(− min{δ − , 1}ε) < |z| < exp(min{δ + , 1}ε).
Eq. ( 40) can be rearranged into the Wiener-Hopf equation where and we have introduced the complexified version of the dispersion relation for the periodic medium, Eq. ( 12), cos Qd = cos κd + M 2κ sin κd, which defines Qd in terms of κd = kd + iε.In a passband, qd is real, and as 0 < ε ≪ 1 we can show that Qd = qd + iδε, for some δ > 0; hence e iQd lies inside the unit circle in the z-plane.In a stopband, qd is purely imaginary and a consistent choice of the branch of Eq. ( 12) yields Im (qd) > 0, so that e iQd will also lie inside the unit circle.So, we can write K (z) = K + (z)K − (z) where which have simple zeros and poles outside/inside the annular domain D, respectively.Now, divide both sides of Eq. ( 42) by which can be rearranged into the form The left-hand and right-hand sides analytically continue the function E(z) into the whole of the z-plane, and hence E must be entire.This function is easily evaluated by examining the asymptotic behaviour of the left-hand side.From Eqs. ( 41) and (44), V − (z) = O(z −1 ) and K − (z) → 1 as |z| → ∞.Thus, E(z) tends to a constant at infinity, and by the extended form of Liouville's theorem we obtain Hence, Eq. ( 46) yields the explicit solution We note that V + (z) has a single simple pole at z = e −iQd , whereas V − (z) has a single simple pole at z = e iκd .Thus, we can confirm that δ + = δ and δ − = 1 and hence the annulus D has finite width and contains the unit circle as required.
Finally, to determine the solution in the physical domain we apply the inverse z-transform where C is the unit circle in D which is traversed in an anticlockwise direction.
For m ≥ 0 the contour C can be taken off to infinity, picking up the residue contribution from the pole of V + (z) (zero of K + (z)) at z = e −iQd .This yields, after a little algebra and after letting ε → 0, the solution v m = 4k M e imqd e i(q−k)d/2 sin([k − q]d/2) = e imqd e i(q−k)d/2 sin kd sin( where the relation between the first and second forms are established using Eq. (12).Note that v m+1 = e iqd v m .Similarly, for m < 0, the residue contribution from the pole of V − (z) in agreement with Levine's formula, Eq. (26).

Conclusions
Three methods have been described in order to determine the reflection coefficient R associated with the canonical problem of the one-dimensional reflection by a semi-infinite periodic row of identical scatterers located at x = nd, for n = 0, 1, 2, . . . .The first approach successfully determined R by shifting the array by one period.The third approach used a direct attack on the governing ordinary differential equation for point scatterers, and also yielded the correct R.Both methods treated directly the problem of a semi-infinite array.On the other hand the second approach considered a finite number (N) of scatterers, determined the associated reflection coefficient R N , and then took the limit as N → ∞.This limit gives R correctly when the frequency of the incident wave resides in a stopband.In a passband, methods were described that yield the correct exact result for R; they require the artificial introduction of a small amount of loss into the problem.
(We remark that an approach based on the discrete Wiener-Hopf technique can be used to solve the problem with N point scatterers.This yields the solution in a slightly different form to that given in Section 5, but it also requires a small amount of dissipation to recover the correct limit as N → ∞.) Of interest in future work is the consideration of how the time-harmonic problem can be used in the time domain in order to yield the reflected and transmitted fields due to transient incidence such as an incoming pulse.Indeed, very few transient problems associated with heterogeneous media have been considered.