The Study of the Bloch Transformed Fields Scattered by Locally Perturbed Periodic Surfaces
Ruming Zhang








The Study of the Bloch Transformed Fields Scattered by Locally Perturbed Periodic Surfaces
    Ruming ZhangCenter for Industrial Mathematics, University of Bremen
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Scattering problems from locally perturbed periodic surfaces have been studied both theoretically and numerically in recent years. In this paper, we will discuss more details of the structure of the Bloch transformed total fields. The idea is inspired by Theorem a in <cit.>, which considered how the total field depends on the incident plane waves, from a smooth enough periodic surface. We will show that when the incident field satisfies some certain conditions, the Bloch transform of the total field depends analytically on the quasi-periodicities in one periodic cell except for a countable number of points, while in the neighborhood of each point, a square-root like singularity exists. We also give some examples to show that the conditions are satisfied by a large number of commonly used incident fields. This result also provides a probability to improve the numerical solution of this kind of problems, which is expected to be discussed in our future papers.




§ INTRODUCTION


The problems that quasi-periodic incident fields scattered by periodic surfaces have been well studied in the past few years. A classical way to handle this kind of problems, which are defined in unbounded domains, is to reduce the problems into one periodic cell,  see <cit.>. However, when the incident field is no longer (quasi-) periodic, or the surface is no longer periodic, the scattered field is no longer (quasi-)periodic,  the classical way fails to work, so new  methods are required for these more difficult problems. Another way is to treat this kind of problems as incident fields scattered by rough surfaces, see <cit.>. In recent years, the Floquet-Bloch based method was applied to analyze the well-posedness of this kind of problems,  see <cit.>.  The Bloch transform build up a "bridge" between the non-periodic scattering problem and a family of quasi-periodic scattering problems, which also provided a computational method for numerical solutions. In these papers, the properties of the Bloch transformed total fields are important both theoretically and numerically, thus we will discuss some more properties in this paper.


This Floquet-Bloch based method was studied theoretically in <cit.> and <cit.>, for non-periodic incident fields, e.g. Herglotz wave functions or Green's functions, scattered by (locally perturbed) periodic surfaces. Based on the theoretical results, a numerical method was established to solve the scattering problems with a non-periodic incident field with a periodic surface, for cases in 2D space, see <cit.> and for  3D space, see <cit.>. Later on, the numerical method was extended for scattering problems with a locally perturbed surface in 2D space in <cit.>. The method was also applied to locally perturbed periodic waveguide problems in <cit.>. 

In the numerical solution of the scattering problems with (locally perturbed) periodic surfaces, for a chosen discrete space, the convergence rate depends on the regularity of the Bloch transformed total field. To improve the numerical method, a more detailed study of the regularity is required. In the paper <cit.>, Kirsch has studied how the total fields depend on the incident angle and wave number of incident plane waves, from smooth enough periodic surfaces. The method adopted in this paper is the perturbation theory. Inspired by this idea, we may extend the result to the dependence of the quasi-periodicity of the Bloch transformed total field, when the incident field and its Bloch transform satisfy some certain conditions, and the surface is locally perturbed. In future, it is hopefully that the result could be helpful to improve the numerical method to solve such kind of scattering problems, or might be extended to solve problems in higher dimensional spaces.





The rest of the paper is organized as follows. In Section 2,  the known results of the locally perturbed periodic problems will be shown. In Section 3, we will study extend the result in <cit.> to the non-perturbed cases, and finally, the result will be extended to the locally perturbed case in Section 4. For a more general case, we will show some results in Section 5. The definition and some properties of the Bloch transform will be shown in the Appendix. 











§ LOCALLY PERTURBED PERIODIC PROBLEMS






 §.§ Notations and domains


Let ζ be a Lipschitz continuous periodic function with period Λ, and the periodic surface is defined by

    Γ:={(x_1,ζ(x_1)): x_1∈}.
 
Let Λ^*=2π/Λ, define the periodic cells by

    =(-Λ/2,Λ/2],  =(-Λ^*/2,Λ^*/2]=(-π/Λ,π/Λ].

Let ζ_p be a Lipschitz local perturbation of ζ such that supp(ζ_p-ζ) is a compact domain in . For simplicity, assume that supp(ζ_p-ζ)⊂, then define the locally perturbed surface by

    Γ_p:={(x_1,ζ_p(x_1)): x_1∈}.

We can define the space above the surfaces by

    Ω:={(x_1,x_2): x_1∈, x_2>ζ(x_1)}
          Ω_p:={(x_1,x_2): x_1∈, x_2>ζ_p(x_1)}.

For simplicity, in this paper, we assume that Γ and Γ_p are all above the straight line {x_2=0}, and H be a positive number such that H>max{ζ_∞,ζ_p_∞}. Define the line Γ_H=×{H} and the spaces with between the surfaces Γ or Γ_p and Γ_H

    Ω_H:={(x_1,x_2): x_1∈, ζ(x_1)<x_2<H}
          Ω^p_H:={(x_1,x_2): x_1∈, ζ_p(x_1)<x_2<H}.

We can also define the curves and domains in one periodic cell: 

    Γ^Λ=Γ∩,   Γ^Λ_H=Γ_H∩;
          Ω^Λ=Ω∩×,  Ω^Λ_H=Ω_H∩×.





 §.§ Introduction to scattering problems


Given an incident field u^i that satisfies 

    Δ u^i+k^2 u^i=0  inΩ_p,

then it is scattered by the surface Γ_p, so there is a scattered field u^s such that it satisfies the Helmholtz equation 

    Δ u^s+ k^2 u^s=0   in Ω_p

and the Dirichlet boundary condition

    u^s=-u^i   on Γ_p.


In this paper, we only take the sound-soft surface as an example. In fact, the results could be extended to problems with some other different settings, for example, the impedance boundary conditions, or inhomogeneous mediums.

To guarantee that u^s is an out-going wave, i.e., u^s is propagating upwards, it is required that u^s satisfies the Upward Propagating Radiation Condition (UPRC) in the domain above Γ_H

    u^s(x)=1/2π∫_ e^ı x_1 ξ+ı√(k^2-|ξ|^2)(x_2-H)û^s(ξ,H)ξ̣   x_2≥ H,

where û^s(ξ,H) is the Fourier transform of u^s(·,H). The UPRC is equivalent to 

    ∂ u^s/∂ x_2=T^+ u^s   on Γ_H,

where T^+ is the Dirichlet-to-Neumann map defined by

    (T^+ϕ)(x_1)=ı/2π∫_√(k^2-|ξ|^2)e^ı x_1ξϕ̂(ξ)ξ̣,

where ϕ̂ is the Fourier transform of the function ϕ. It has been prove in <cit.> that the operator T^+ is bounded from H^1/2_r(Γ_H) to H^-1/2_r(Γ_H) for any |r|<1. Define the total field u=u^i+u^s, then it satisfies the following equations

    Δ u+k^2 u   =   0   in Ω_p
    
    u   =   0   on Γ_p
    ∂ u/∂ x_2   =   T^+u+[∂ u^i/∂ x_2-T^+u^i]   on Γ_H.

We can formulate the variational problem, i.e., for any u^i∈ H_r^1(Ω^p_H), seek for the weak solution u∈H^1_r(Ω_H^p) such that for any test function ϕ∈ H^1(Ω_H^p) with a compact support, satisfies the variational equation

    ∫_Ω_H^p[∇ u·∇ϕ-k^2uϕ]x̣-∫_Γ_HT^+(u|_Γ_H) ϕṣ=∫_Γ_H[∂ u^i/∂ x_2-T^+(u^i|_Γ_H)]ϕṣ.
 


The space H^1_r(Ω_H^p) with a tilde is defined as a subspace of H^1_r(Ω_H^p), i.e., H^1_r(Ω_H^p):={ϕ∈ H^1_r(Ω_H^p): ϕ|_Γ_p=0}. In the rest of this paper, we will always use the tilde to indicate the space of functions that satisfies homogeneous Dirichlet boundary conditions. 



In <cit.>, the uniquely solubility of the variational form (<ref>) was proved.

For any |r|<1, given any u^i∈ H^1_r(Ω_H^p), there is a unique u∈H^1_r(Ω_H^p) of the variational problem (<ref>).





 §.§ Bloch transformed fields


 As the Bloch transform only works on periodic domains, we have to transform the variational problem (<ref>) defined in a locally perturbed periodic domain Ω_H^p into the one defined in a periodic domain. Following <cit.>, define a  diffeomorphism Φ_p such that maps the periodic domain Ω_H_0 to Ω_H_0^p, where max{ζ_∞,ζ_p_∞}<H_0<H, by

    Φ_p(x_1,x_2)=(x_1,x_2+(x_2-H)^3/(ζ(x_1)-H)^3(ζ_p(x_1)-ζ_(x_1)),   (x_1,x_2)^⊤∈Ω_H_0.

The operator Φ_p-I_2 (where I_2 is the identity operator in ^2) has a support which is a subset of Ω^Λ_H_0⊂Ω^Λ_H. Then let u_T=u∘Φ_p, then it satisfies the following variational problem

    ∫_Ω_H[A_p∇ u_T·∇ϕ_T-k^2 c_p u_Tϕ_T]x̣-∫_Γ_HT^+(u_T|_Γ_H)ϕ_Tṣ
            =∫_Γ_H[∂ u^i/∂ x_2-T^+(u^i|_Γ_H)]ϕ_Tṣ

for all ϕ_T∈H^1(Ω_H) with compact support, where

    A_p(x)=|∇Φ_p(x)|[(∇Φ_p(x))^-1((∇Φ_p(x))^-1)^T]∈ L^∞(Ω_H,^2× 2),
          c_p(x)=|∇Φ_p(x)|∈ L^∞(Ω_H).

From the definition of Φ_p, the supports of A_p-I and c_p-1 are both subsets of Ω^Λ_H.

Suppose r≥ 0, let w=_Ω u_T, from the mapping property of the Bloch transform and u_T∈ H_r^1(Ω_H)⊂ H^1(Ω_H), w∈ L^2(;H^1_α(Ω^Λ_H)), and satisfies the following variational problem for any ϕ∈ L^2(;H^1_α(Ω^Λ_H))

    ∫_ a_α(w(α,·),ϕ(α,·))α̣+b(^-1_Ω w,^-1_Ωϕ)=∫_∫_Γ^Λ_Hf(α,·)ϕ(α,·)ṣα̣,

where 

    a_α(w(α,·),ϕ(α,·))   =∫_Ω^Λ_H[∇ w(α,·)·∇ϕ(α,·)-k^2w(α,·)ϕ(α,·)]x̣
       -∫_Γ_H^ΛT^+_α[w(α,·)|_Γ_H^Λ]ϕ(α,·)ṣ
          b(ξ,ψ)=[Λ/2π]^1/2∫_Ω^Λ_H(A_p-I)∇ξ·∇ψṣ-k^2[Λ/2π]^1/2∫_Ω^Λ_H(c_p-1)ξψx̣,
          f(α,·)=∂_Ω u^i(α,·)/∂ x_2-T^+_α[(_Ω u^i)(α,·)|_Γ^Λ_H],

where the α-quasi-periodic DtN map T^+_α is defined by

    T^+_α(ψ)=ı∑_j∈√(k^2-|Λ^* j-α|^2)ψ̂(j)e^ı(Λ^*j-α)x_1,  ψ=∑_j∈ψ̂(j)e^ı(Λ^*j-α)x_1.


At the end of this subsection, we will list some of  the useful results in <cit.>. The first one is the equivalence between the original problem (<ref>) and the one with Bloch transform (<ref>).


Assume u^i∈ H^1_r(Ω_H^p) for some r∈[0,1), then u_T∈H^1_r(Ω_H) satisfies (<ref>) if and only if w=_Ω u_T∈ H_0^r(;H^1_α(Ω^Λ_H)) satisfies (<ref>).


Then the variational problem is uniquely solvable for a Liptschiz interface Γ_p. 

If ζ_p is Lipschitz continuous, then (<ref>) is uniquely solvable in H^r_0(;H^1_α(Ω^Λ_H)) for all u^i∈ H^1_r(Ω_H^p), r∈[0,1).







There is an equivalent form of (<ref>) if the u^i∈ H_r^1(Ω_H^p) for r∈(1/2,1).


If ζ_p is Lipschitz continuous and u^i∈ H^1_r(Ω_H^p) for r∈(1/2,1), then w∈ H_0^r(;H^1_α(Ω^Λ_H)) equivalently satisfies for all α∈ and ϕ_α∈ H^1_α(Ω^Λ_H) that 

    a_α(w(α,·),ϕ_α)+b(^-1_Ω w,ϕ_α)=∫_Γ^Λ_Hf(α,·)ϕ_αṣ,









§ SCATTERING FROM NON-PERTURBED PERIODIC SURFACES



In Section <ref>, we have shown some known results for the solvability  of the scattering problems from locally perturbed periodic surfaces. The regularity of the Bloch transform of the total field w(α,·)  depends on the decaying rate of the incident field. In this section, we will turn to a further study of the structure of the Bloch transformed field, for non-perturbed periodic surfaces. Based on the results in this section, the discussion for perturbed surfaces will be carried out in the next section.

The paper <cit.> provided a very good inspiration for the problem in this section. Firstly, we will define some sets and functions spaces. Define the set of all singularities with a fixed wave number k and a period Λ by

    :={α∈: ∃  n∈, s.t., |Λ^*n-α|=k},

which is, definitely, a countable set located periodically on the real line. Let ⊂ be an interval (the case that = is included), W⊂^2 be a bounded periodic cell of a periodic domain, and define the space of functions defined in the domain ×Ω^Λ_H that depends analytically on the first variable

    C^ω(;S(W)):={f∈𝒟'(× W): ∀ α_0∈, ∃ δ>0, s.t., ∀α ∈ (α_0-δ,α_0+δ)∩,.
    .
    ∃  C>0,  f_n∈ S(W) s.t., f(α,x)=∑_n=0^∞ (α-α_0)^n f_n(x), f_n_ S(W)≤ C^n},

where S(W) is a Sobolev space defined on W. In this paper, S(W) is either H^n(Ω^Λ_H) or H^n(Ω^Λ_H), n=0,1,2. 
Define the set of functions defined in the domain × W that depends C^n-continuously on the first variable

    C^n(;S(W)):={f∈𝒟'(× W): , ∀α∈, j=0,…,n, ∂^j f(α,·)/∂α^j∈ S(W),.
    .∂^j f(α,·)/∂α^j_S(W) is uniformly bounded for α∈},

thus we can define the space C^∞(;S(W)) in the same way

    C^∞(;S(W)):={f∈𝒟'(× W): , ∀α∈, j=0,1,…,∞, ∂^j f(α,·)/∂α^j∈ S(W),.
    .∂^j f(α,·)/∂α^j_S(W) is uniformly bounded for α∈},




With the definitions of the new spaces and sets, Theorem a in <cit.> could be rewritten in the following form with a fixed wave number k.  

Suppose the incident field u^i(α,x) is the plain wave, i.e., u^i(α,x):=e^ıα x_1-ı√(k^2-α^2)x_2, and it is scattered by a smooth enough sound soft surface.
 The total field, denoted by u(α,x), belongs to the space C^0((-k,k);H^1(Ω^Λ_H)).  For any interval ⊂(-k,k)∖, the total field u belongs to C^ω(;H^1(Ω^Λ_H)). Moreover, for any α_0∈∩(-k,k), i.e., there is an n_0∈ such that |Λ^*n_0-α|=k, there is a neighborhood U of α_0 and quasi-periodic functions v,w∈ C^ω(U∩(-k,k); H^1(Ω^Λ_H)) such that u=v+β_n_0(α)w, where β_n_0=√(k^2-|Λ^* n_0-α|^2) with non negative real and imaginary parts.


We will just show the main idea of the proof in <cit.>.  
The proof is based on the variational form of the function v(α,x):=e^-ıα x_1u(α,x), i.e., for any ϕ∈H_0^1(Ω^Λ_H),

    a_α(v(α,·),ϕ)=-2ı√(k^2-α^2)∫_Γ^Λ_He^-ı√(k^2-α^2)x_2ϕṣ,

where 

    a_α(ψ,ϕ):=∫_Ω^Λ_H[_αψ·_αϕ-k^2 v(α,·)ϕ]x̣-∫_Γ_H^ΛT^+_α[ψ|_Γ^Λ_H]ϕṣ,

_α=+ıα(1,0)^⊤, and the operator T^+_α is the DtN map defined on periodic functions in Γ^Λ_H by

    T^+_α(ψ)=ı∑_j∈√(k^2-|Λ^*j-α|^2)ψ̂(j)e^ıΛ^* j x_1,  ψ=∑_j∈ψ̂(j)e^ıΛ^* j x_1.
 
Each term in the variational form depends analytically on α∈(-k,k) except for   the term ∫_Γ_H^ΛT^+_α[v(α,·)|_Γ^Λ_H]ϕṣ. The operator T^+_α depends analytically on α∈(-k,k)∖, thus the solution v(α,·) depends analytically in α∈(-k,k)∖. For each α_0∈, there is a neighbourhood of U such that T^+_α could be split into one operator depends analytically on α∈ U and the one with a √(k^2-|Λ^*n_0-α|^2)-singularity. Thus the singularity of v(α,·) with respect to α∈ U could be deduced from the Neumann series.



For any α_0∈, there is either 1) an n_0∈ such that |Λ^*n_0-α_0|=k or 2) two n_1, n_2∈ such that |Λ^*n_1-α_0|=|Λ^*n_2-α_0|=k.

 Take Case 1) for example, i.e., |Λ^*n_0-α_0|=k, then either Λ^*n_0-k=α_0 or Λ^*n_0+k=α_0. If Λ^*n_0-k=α_0, β_n_0=√(α-α_0)·√(2k+α_0-α) where the second term, which is independent of n_0, is analytic for α in a small enough neighborhood of α_0. If k+Λ^*n_0=α_0, 
β_n_0=ı√(2k-α_0+α)·√(α-α_0), where the first term, which is independent in n_0, is analytic for α in a small enough neighborhood of α_0 as well. Then for α in a small neighborhood U of α_0, the total field u in Theorem <ref> has the form of 

    u=v+√(α-α_0) w

where v, w are both  functions in C^ω(U∩(-k,k);H^1(Ω^Λ_H)).

 Similar conclusion could be obtained for the second case. Thus we get a simpler form (<ref>) of the representation of the regularity form near the singularity α_0∈. 



The result in Theorem <ref> can be extended to a wider family of incident fields with the form ϕ(α,x)=e^ıα x_1-ı√(k^2-α^2)x_2 for α∈, where both the plane waves and the evanescent waves are included. 




If the incident field ϕ(α,x)=e^ıα x_1-ı√(k^2-α^2)x_2, where α∈, then the total field u(α,·)  belongs to the space C^0(;H^1(Ω^Λ_H)) and for any interval ⊂∖, u∈ C^ω(;H^1_α(Ω^Λ_H)). Moreover, for any α_0∈, there is a small neighbourhood U of α_0 and a pair of functions v(α,·), w(α,·)∈ C^ω(U∩; H^1(Ω^Λ_H)) such that 

    u(α,·)=v(α,·)+√(α-α_0) w(α,·).





It is easy to extend the result of Theorem <ref> to the case that α∈ (-∞,-k)∪(k,+∞). The only case that may make problem is when α in the neighbourhood of ± k, for the plain wave is no longer analytic in (-k-δ,-k+δ) or (k-δ,k+δ), where δ>0 is a small enough positive number. 

Let α∈ (k-δ,k+δ), and δ is small enough such that ∩(k-δ,k+δ)={k} then the incident wave

    ϕ(α,x)=exp(ıα x_1-ı√(k^2-α^2)x_2).


As e^-ıα x_1ϕ(α,x)=e^-ı√(k^2-α^2)x_2 has the form

    e^-ıα x_1ϕ(α,x)=cosh(-ı√(k^2-α^2)x_2)+sinh(-ı√(k^2-α^2)x_2).

Define the functions

    ϕ_1(α, x)=e^ıα x_1cosh(-ı√(k^2-α^2)x_2),
          ϕ_2(α, x)=- e^ıα x_2sinc(-ı√(k^2-α^2)x_2)√(α+k)  x_2,

where sinc is a function defined in  by

    sinc(z)=sinh(z)/z      z≠ 0
    
    1      z=0
    .

As cosh and sinc have Taylor's series at z=0

    cosh(z)=∑_n=0^∞z^2n/(2n)!,  sinc(z)=∑_n=0^∞z^2n/(2n+1)!,

the functions cosh(-ı√(k^2-α^2)x_2)  and sinc(-ı√(k^2-α^2)x_2) has the Taylor's expansion

    cosh(-ı√(k^2-α^2)x_2)=∑_n=0^∞(α^2-k^2)^n x_2^2n/(2n)!
          sinc(-ı√(k^2-α^2)x_2)=∑_n=0^∞(α^2-k^2)^n x_2^2n/(2n+1)!,

thus they are both analytic functions in α in a small neighbourhood of k. Then the incident field is written into the form of 

    ϕ(α,x)=ϕ_1(α,x)+√(α-k) ϕ_2(α,x).

From the proof of Theorem <ref> in <cit.>, the total field with the incident field ϕ_j∈ C^ω((k-δ,k+δ);H^1(Ω^Λ_H)), j=1,2, u_j has the decomposition that

    u_j=v_j+√(α-k)  w_j

where v_j, w_j∈ C^ω((k-δ,k+δ);H^1_α(Ω^Λ_H)) are all analytic functions in α. Then the total field u satisfies

    u   =u_1+u_2
       =v_1+√(α-k)  w_1+√(α-k) [v_2+√(α-k)  w_2]
       =[v_1+(α-k)w_2]+√(α-k) [w_1+v_2].

The terms v_1+(α-k)w_2 and w_1+v_2 are both in the space C^ω((k-δ,k+δ);H^1(Ω^Λ_H)), let v=v_1+(α-k)w_2 and w=w_1+v_2, then

    u=v+√(α-k)  w,

the case that α_0=k is proved. The case that α_0=-k could be proved similarly. The proof is finished. 


In the rest of this section, we will consider the scattering problems with non-perturbed periodic surfaces. Let u∈H_r^1(Ω^H) be the total field with the incident u^i∈ H_r^1(Ω^H) and w=_Ω u∈ H_0^r(;H^1_α(Ω^Λ_H)). As w is Λ^*-periodic in α, we only consider the regularity in one periodic cell . For any α_0∈, α_0+Λ^*∈, thus  distributes periodically in . Let α_1∈ be a fixed number, there are finite number of elements of  that lies in [α_1,α_1+Λ^*]. Let S be the number, then α_1<α_2<⋯<α_S=α_1+Λ^* where [α_1,α_S]∩={α_1,…,α_S}. 

We will define the space of the functions defined in × W that satisfy the property of u(α,·) in  Corollary <ref> as follows. 

    ^ω(;S(W);):={u∈ C^0(;S(W)):  for any  subinterval _0⊂∖, .
       u∈ C^ω(_0;S(W)); ∀α_j∈∩, there is a neighbourhood U of α_j
       .and a pair v,w∈ C^ω(U∩,S(W)) such that u=v+√(α-α_j)  w}.





We assume that the incident field u^i∈ H_r^1(Ω_H) satisfies that the Bloch transform  (_Ω u^i)(α,·)∈^ω(;H^1(Ω^Λ_H);).



 Assumption <ref> is satisfied by a large number of incident waves.  In fact, the Green's functions in a half-space satisfies Assumption <ref>. We will take this for example. In the following examples, we set Λ=2π thus Λ^*=1.

 The half-space Green's function has the form of

    u^i(x,y)=ı/4[H_0^(1)(k|x-y|)-H_0^(1)(k|x-y'|)]

where y=(y_1,y_2)^T lies above Γ_H. Bloch transform of the Green's function has the representation of

    ( u^i)(α,x)=1/2π∑_j∈e^ıα_j (x_2-y_2)+ıβ_j y_2sinc(β_j x_2)x_2.

As the number of elements in S_N is finite, suppose the number to be M, then for each j=1,…,M, define

    u_j(α,x)=1/2π∑_j∈e^ıα_j (x_2-y_2)e^ıβ_j y_2sinc(β_j x_2)x_2,
          u^1_j(α,x)=ı/2π∑_j∈e^ıα_j (x_2-y_2)sinc(β_j y_2)y_2sinc(β_j x_2)x_2,
          u^2_j(α,x)=1/2π∑_j∈e^ıα_j (x_2-y_2)cos(β_j y_2)sinc(β_j x_2)x_2.

then u_j=β_j u_j^1+u_j^2. As sinc and cos has the Taylor's expansions that

    sinc(x)=∑_n=0^∞(-1)^n x^2n/(2n+1)!,  cos(x)=∑_n=0^∞(-1)^n x^2n/(2n)!,

u_j^1, u_j^2 are analytic functions in both α and x. 
Note that (_Ω u^i)(α,x)-∑_j=1^M u_j(α,x) is the convergent sum of analytic functions, _Ω u^i∈^ω(;H^1(Ω^Λ_H);).





With Assumption <ref>, the regularity result for the Bloch transform of the total field comes directly from Corollary <ref>. 


Let r>1/2. Suppose the incident filed u^i∈ H_r^1(Ω_H) satisfies Assumption <ref>, with a total field u∈H_r^1(Ω_H). Then the Bloch transform of the total field w(α,x)=(_Ω u)(α,x)∈^ω(;H^1(Ω^Λ_H);). 



When r>1/2, from Theorem <ref>, the variational form is equivalent to the decoupled system 

    a_α(w(α,·),ϕ_α)=∫_Γ^Λ_Hf(α,·)ϕ_αṣ.

Then the periodic function v(α,x):=e^ıα x_1w(α,x) satisfies the variational form for any periodic function ϕ∈H^1_0(Ω^Λ_H)

    a_α(v(α,·),ϕ)=∫_Γ^Λ_He^ıα x_1f(α,·)ϕṣ.

Thus the conclusion could be obtained from the same argument of the proof of Corollary <ref>.































With the regularity results for non-perturbed periodic surfaces, we will discuss the locally perturbed cases in the next section.




§ SCATTERING FROM LOCALLY PERTURBED PERIODIC SURFACES


In this section, we still hold the assumption that r>1/2, then from Theorem <ref>, the variational form is equivalent to 

    a_α(w(α,·),ϕ_α)+b(u_T,ϕ_α)=∫_Γ^Λ_Hf(α,·)ϕ_αṣ.

Let v(α,x)=e^ıα x_1w(α,·), ϕ=e^ıα x_1ϕ_α(x), then v is the solution of the following system

    a_α(v(α,·),ϕ)=∫_Γ^Λ_He^ıα x_1f(α,·)ϕṣ-b_α(u_T,ϕ),

where the sesquilinear form b_α is defined by

    b_α(ξ,ψ)=[Λ/2π]^1/2∫_Ω^Λ_He^ıα x_1[(A_p-I)ξ·_αψ-k^2(c_p-1)ξψ]x̣,

which depends analytically on α, thus the system (<ref>) has a right hand side that depends analytically on α. So from similar arguments in Theorem <ref>.


Suppose u^i∈ H_r^1(Ω_H^p) for some r>1/2 satisfies Assumption <ref>, then the Bloch transform  w=_Ω u_T∈ H_0^r(;H^1(Ω^Λ_H)) belongs to the space ^ω(;H^1(Ω^Λ_H);).




A direct corollary of Theorem <ref> is described as follows.

Suppose α_1,…,α_s∈∩, there are S+1 functions defined in ×Ω^Λ_H, i.e.,  w_1(α,·),…,w_S(α,·),w_S+1(α,·) that belong to the space C^∞(;H^1(Ω^Λ_H)) such that

    w(α,·)=w_S+1(α,·)+∑_n=1^S√(α-α_n) w_n(α,·).










§ FURTHER REGULARITY RESULTS OF SCATTERING PROBLEMS


In this section, we will consider a more generalized class of incident fields. In the following part, we only want to show the regularity results without proving them, for the proofs are quite similar.


Although the Green's function u^i(x,y) satisfies Assumption <ref>, there is a large class of incident fields that does not satisfy this assumption. We will firstly study the regularity property of the Bloch transform of the Herglotz wave function. Some of the results are from <cit.>. Define the weighted Hilbert space L^2_cos(-π/2,π/2) as the closure of C_0^∞(-π/2,π/2) in the norm

    ϕ_L^2_cos(-π/2,π)=[∫_-π/2^π/2|ϕ(θ)|^2/cosθθ̣]^1/2.

 The downward propagating Herglotz wave function with the density in L^2_cos(-π/2,π/2) is defined by

    (Hg)(x)=∫__-e^ı k x· dg(d)ṣ(d)=∫_π/2^π/2e^ı k(sinθ x_1-cosθ x_2)ϕ(θ)θ̣,

where ϕ∈ L^2_cos(-π/2,π/2), _- is the lower half of one unit circle in ^2. From <cit.>, the Bloch transform of the Herglotz wave functions has the representation of 

    (_Ω Hg)(α,x)=√(Λ^*)∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2ϕ(arcsin(Λ^*j-α)/k)/√(k^2-|Λ^*j-α|^2)

For some choice of the function ϕ, i.e., if 
ϕ(t)=h[cos(t)]cos(t), where h is an analytic function defined in [0,1] with h(0)=0, then

    (_Ω Hg)(α,x)=√(Λ^*)/k∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2h[√(k^2-|Λ^*j-α|^2)/k].

By differentiating _Ω Hg with respect to α,

    ∂/∂α(_Ω Hg)(α,x)
    
    =   -ı√(Λ^*)/k x_1∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2 h[√(k^2-|Λ^*j-α|^2)/k]
       -ı√(Λ^*)/kx_2∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2Λ^*j-α/√(k^2-|Λ^*j-α|^2)h[√(k^2-|Λ^*j-α|^2)/k]
       +√(Λ^*)/k^2∑_|Λ^*j-α|<ke^ı(Λ^*j-α)x_1-ı√(k^2-|Λ^*j-α|^2)x_2Λ^*j-α/√(k^2-|Λ^*j-α|^2)h'[√(k^2-|Λ^*j-α|^2)/k],

then ∂/∂α(_Ω Hg)(α,x)∈ L^p(;H^1(Ω^Λ_H)) for any 1≤ p<2, thus the function _Ω Hg∈ W^1,p_0(;H^1(Ω^Λ_H)). From the Sobolev embedding theorem, _Ω Hg∈ H^r_0(;H^1(Ω^Λ_H)) for some r∈[1/2,1). Thus from the property of the inverse Bloch transform, Hg∈ H_r^1(Ω_H).

As in each open interval ⊂∖, the terms in the finite sum do not change, and each term is an analytic function in both α and x. Suppose α_0∈ is a singularity point, then when δ>0 is small enough such that (α_0-δ,α_0+δ)∩={α_0}. Consider the domain (α_0-δ,α_0], it is easy to prove that (_Ω Hg)(α,x)=v(α,x)+√(α-α_0) w(α,x) where v,w∈ C^ω([α_0-δ,α_0];H^1(Ω^Λ_H)). Similar result could be obtained in the interval [α_0,α_0+δ).


We will define the following space.

    ^ω_c(;S(W);):={u∈ C^0(;S(W)): for any subinterval _0⊂∖,.
       u∈ C^ω(_0;S(W)); ∀α_j∈∩, there is a small enough δ>0 and two pairs 
       v_1,w_1∈ C^ω([α_j-δ,α_j];S(W)) and v_2,w_2∈ C^ω([α_j,α_j+δ];S(W)) such that
       .u=v_1+√(α-α_j) w_1 for α∈(α_j-δ,α_j]; u=v_2+√(α-α_j) w_2 for α∈[α_j,α_j+δ).},

then the Herglotz wave function discussed above belongs to this space.




We assume that the incident field u^i∈ H_r^1(Ω^H_p) for r>1/2 satisfies that the Bloch transform _ω u^i∈^ω_c(;H^1(Ω^Λ_H);).



 With the incident field satisfying Assumption <ref>, we can obtain the following corollary of Theorem <ref>.


Suppose u^i satisfies Assumption <ref>, then the total field u with the Bloch transform w=_Ω u∈^ω_c(;H^1(Ω^Λ_H);).  


Recall the definition of the set  of singularities, and let α_j, j∈ be an ascending series of , then we can get the corollary of the regularity result in each interval (α_j,α_j+1).


Suppose u^i satisfies Assumption <ref>, then the Bloch transform w=_Ω u belongs to the space C^0(;H^1(Ω^Λ_H)). Moreover, for each j∈, for α in the interval [α_j,α_j+1], there are three functions  w^j_0, w^j_1, w^j_2∈ C^∞([α_j,α_j+1];H^1(Ω^Λ_H)) such that

    w=w^j_0+√(α-α_j) w^j_1+√(α-α_j+1)w^j_2.









§ APPENDIX: BLOCH TRANSFORM


A useful tool to handle the (locally perturbed) periodic scattering problems is the Bloch transform, see <cit.>, which build up a "bridge" between the infinitely defined problem and a family of  coupled quasi-periodic problems defined in one  single periodic cell. For any function ϕ∈ C_0^∞(Ω), the one dimensional Bloch transform is defined as

    (_Ωϕ)(α,x)=[Λ/2π]^1/2∑_j∈ϕ(x+(Λ j
    0
    ))e^-ıαΛ j,  α∈,  x_1∈,

where Ω is a Λ-periodic (where Λ>0) strip  or curve in x_1-direction in ^2, which satisfies that if (x_1,x_2)^T∈Ω, then (x_1±Λ,x_2)^T∈Ω. 

Though Ω does not always have the same definition throughout this paper, we all denote the one dimensional Bloch transforms by _Ω. 

Thus _Ωϕ(α,x) is α-quasiperiodic in x_1 for period Λ, for a fixed α, i.e.,

    (_Ωϕ)(α,x+(Λ
    0
    ))=e^ıαΛ(_Ωϕ)(α,x),

 and 2π/Λ-periodic in α∈ for a fixed x, i.e.

    (_Ωϕ)(α+2π/Λ,x)=(_Ωϕ)(α,x).
 
  Let Λ^*=2π/Λ and set

    =(-Λ/2,Λ/2],  =(-Λ^*/2,Λ^*/2]=(-π/Λ,π/Λ],
 
and let Ω^Λ=×∩Ω. Then _Ω maps a function in C_0^∞(Ω) into a function that belongs to the space C^∞(×Ω^Λ). 

To introduce more about the properties of the Bloch transform _Ω, we need some notations and spaces. In the rest of this section, Ω is assumed to be bounded in x_2-direction. Firstly, for any s,r∈, define the weighted space

    H_r^s(Ω)={ϕ∈𝒟'(Ω): x→(1+|x_1|^2)^r/2ϕ(x)∈ H^s(Ω)}

with the norm

    ϕ_H_r^s(Ω)=(1+|x_1|^2)^r/2ϕ(x)_H^s(Ω).

We can also define the function space in ×Ω^Λ. For some integers r∈ and some real number s∈, define the space

    H^r_0(;H^s_α(Ω^Λ))={ϕ∈𝒟'(×Ω^Λ): ∑_ℓ=0^r∫_∂^ℓ_αϕ(α,·)^2_H^s()α̣}

with the norm defined as

    ϕ_H_0^r(;H^s_α(Ω^Λ))=[∑_ℓ=0^r∫_∂^ℓ_αϕ(α,·)^2_H^s()α̣]^1/2.

With the interpolation in r, the definition could be extended to all r≥ 0. We can also define the spaces with a negative r by duality arguments, thus the space H^r_0(;H^s_α(Ω^Λ)) is well defined for any r, s∈.   With these definitions, the following properties for Bloch transform holds. For proofs see Theorem 8 in <cit.>, and we will omit them here.


The Bloch transform extends to an isomorphism between H^s_r(Ω) and H^r_p(;H^s_α(Ω^Λ)) for all s,r∈. When s=r=0, _Ω is an isometry. The inverse operator 

    (_Ω^-1w)(x+(Λ j
    0
    ))=[Λ/2π]^1/2∫_ w(α,x)e^ıΛ jαα̣,   x∈Ω^Λ, j∈.

Moreover, the L^2-adjoint operator _Ω^* equals to its inverse _Ω^-1.




§ ACKNOWLEDGEMENTS.
 The author was supported by the University of Bremen and the European Union FP7 COFUND under grant agreement n^∘600411.


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