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Electromagnetic field enhancement in a subwavelength rectangular open cavity

Consider the transverse magnetic polarization of the electromagnetic scattering of a plane wave by a perfectly conducting plane surface, which contains a two-dimensional subwavelength rectangular cavity. The enhancement is investigated fully for the electric and magnetic fields arising in such an interaction. The cavity wall is assumed to be a perfect electric conductor, while the cavity bottom is allowed to be either a perfect electric conductor or a perfect magnetic conductor. We show that the significant field enhancement may be achieved in both nonresonant and resonant regimes. The proofs are based on variational approaches, layer potential techniques, boundary integral equations, and asymptotic analysis. Numerical experiments are also presented to confirm the theoretical findings.


Introduction
The electromagnetic scattering by open cavities has significant applications in many scientific areas. For instance, the radar cross section (RCS) is a quantity which measures the detectability of a target by a radar system. Clearly, it is of high importance in military and civil use for a deliberate control in the form of enhancement or reduction of the RCS of a target. Since the cavity RCS can dominate the total RCS, it is critical to have a thorough understanding of the electromagnetic scattering by a target, particularly a cavity, for successful implementation of any desired control of its RCS. The underlying scattering problems have received much attention by many researchers in both the engineering and the mathematical communities. There has been a rapid development of the mathematical theories and computational methods in this area [3,8,9]. We refer to [26] for a survey of open cavity scattering problems.
In optics, rough optical device surfaces containing subwavelength cavities have created a lot of interest in electromagnetic field enhancement and extraordinary optical transmission effects in recent years, due to their important potential applications, such as in biological and chemical sensing, spectroscopy, Terahertz semiconductor devices [6,10,12,17,18,24,[31][32][33]35]. The amplification of the confined fields in the cavity can be strikingly high. These amazing features of light localization and enhancement can be quite useful in imaging, microscopy, spectroscopy, and communication [27,30]. However, there are still controversies on the mechanisms which contribute to anomalous field enhancement [18]. The complication arises from the multiscale nature of the structures and various enhancement behaviors that it induces. For instance, the enhancement can be attributed to surface plasmonic resonance [17,18], non-plasmonic resonances [33,35], or even without the resonant effect [28].
It is very helpful to provide a rigorous mathematical analysis for these phenomena, which turns out to be extremely challenging. The particular light patterns observed in the far-field or in the near-field may be the results of complex multi-interactions of surface waves, cavity resonances, resonant tunneling of plasmon waves, skin depth effects, etc. It requires the resolution of the full three-dimensional Maxwell equations in non-smooth geometries even for a qualitative description of the diffractive properties of electromagnetic waves in subwavelength structures. For most resonance phenomena, the localization or enhancement of light is very sensitive to geometrical parameters and the frequency. This motivates us to begin with simple geometries, where the analytical approaches are made possible to obtain an accurate description of the electromagnetic field. In [7,11], the field enhancement was considered for a single rectangle cavity and double rectangle cavities. Using asymptotic expansions of Green's function, they showed that the limiting Green function is a perfectly conducting plane with a dipole in place of the cavity when the width of the cavity shrinks to zero.
In [13,21,22], an asymptotic expansion method was proposed to study the solution of the Helmholtz equation in a domain including a single thin slot. A quantitative analysis was made in [28,29] for the field enhancement of the Helmholtz equation for a single narrow open slit. We refer [4,5] for the study of closely related Helmholtz resonators and resonances in bubbly media.
In this paper, we consider the electromagnetic scattering of a plane wave by a perfectly electrically conducting plane surface, which contains a subwavelength rectangular cavity. More specifically, the cavity is assumed to be invariant in the x 3 -axis and only the transverse magnetic polarization (TM) is studied. Hence the three-dimensional Maxwell equations can be reduced into the two-dimensional Helmholtz equation. The cavity wall is assumed to be a perfect electric conductor (PEC), and the cavity bottom is assumed to be either a perfect magnetic conductor (PMC) or a perfect electric conductor (PEC). A cavity is called the PEC-PMC type when it has a PEC wall and a PMC bottom, while a cavity is called the PEC-PEC type when it has a PEC wall and a PEC bottom. We investigate the electromagnetic field enhancement for both types of cavities. Using a combination of variational approaches, layer potential techniques, boundary integral equations, and asymptotic analysis, we demonstrate the field enhancement mechanisms for the underlying scattering problems different boundary conditions. Denote by λ, , d the wavelength of the incident field, the width of the cavity, and the depth of the cavity, respectively. For the PEC-PMC cavity, we prove in the nonresonant regime that the electric field enhancement has an order O(λ/d) and the magnetic field has no enhancement in the cavity provided that the length scales satisfy d λ. Moreover, if λ, we show that the Fabry-Perot type resonance occurs. In the resonant regime, we derive an asymptotic expansion for the resonant frequencies and deduce an order O(1/ ) enhancement for both the electric and magnetic fields. For the PEC-PEC cavity, we carry the same analysis in both the nonresonant and resonant regimes. The electric field is shown to have a weak enhancement of order O(λ /d) in the nonresonant regime with length scales satisfying d λ. In the resonant regime, a similar quantity analysis is done for the resonant frequencies and the same order O(1/ ) is achieved for the electric and magnetic field enhancement. These field enhancement results are summarized in Table 1. Numerical experiments are also done and used to confirm our theoretical findings.
The rest of the paper is organized as follows. The PEC-PMC cavity is examined in sections 2-3 and the PEC-PEC cavity is studied in sections 4-5. Sections 2 and 4 focus on the field enhancement for the non-resonant case. In each section, the model problems are introduced, the approximate problems are proposed, the error estimates are derived between the solutions of the exact and approximate problems, and the electric and magnetic field enhancement are discussed. Sections 3 and 5 address the field enhancement for the resonant case. In each section, the boundary integral equations are given, the asymptotic analysis is carried for the Fabry-Perot resonances, and the electric and magnetic field enhancement are presented under the resonant frequencies. The paper is concluded with some general remarks and directions for future work in section 6.

PEC-PMC cavity without resonance
In this section, we discuss the electromagnetic field enhancement for the PEC-PMC cavity problem when the wavenumber is sufficiently small, i.e., no resonance occurs.
2.1. Problem formulation. As is shown in Figure 1, the problem geometry is assumed to be invariant in the d} the open aperture and the bottom side of the cavity, respectively. Let R 2 We assume that the cavity and the upper half-space are filled with the same homogeneous medium. The electromagnetic wave propagation is governed by the time-harmonic Maxwell equations (time dependence e −iωt ): where ω > 0 is the angular frequency, ε > 0 is the electric permittivity, µ > 0 is the magnetic permeability, E and H are the electric field and the magnetic field, respectively. Furthermore, we assume that the boundary ∂Ω \ Γ − is a perfect electrical conductor (PEC) while the boundary Γ − is a perfect magnetic conductor (PMC). Hence (E, H) satisfy the following boundary conditions: where ν = (ν, 0) is the unit outward normal vector on ∂Ω × R, i.e., ν is the unit outward normal vector on ∂Ω.
We consider a polarized time-harmonic electromagnetic wave (E inc , H inc ) that impinges on the cavity from above. In the transverse magnetic polarization (TM), the magnetic field is parallel to the x 3 -axis, which implies that the incident magnetic wave H inc = (0, 0, u inc ). Here u inc (x) = e iκx·d is a plane wave propagating in the direction d = (sin θ, − cos θ), where − π 2 < θ < π 2 is the incident angle and κ = ω √ εµ is the wavenumber. The corresponding incident electric field E inc is determined by Ampere's law E inc = i(ωε) −1 ∇ × H inc = µ/ε(cos θ, sin θ, 0)e iκx·d . Denote by λ = 2π/κ the wavelength of the incident wave. In this section, we assume that the length scale of the underlying geometry satisfies d λ. Let H = (0, 0, u ) be the total magnetic field. It can be verified from (2.1)-(2.2) that u satisfies Denote by u r the reflected field due to the interaction between the incident field u inc and the PEC plane Γ 0 . It can be shown that u ref (x) = e iκx·d , where d = (sin θ, cos θ). In R + 2 , the total field u consists of the incident wave u inc , the reflected wave u ref , and the scattered field u sc . The scattered fields u sc is required to satisfy the Sommerfeld radiation condition: 2. An approximate model. We introduce an approximate model for the problem (2.3)-(2.4) in order to estimate the field enhancement inside the cavity. Let be an orthonormal basis on the interval (0, ). It follows from the Neumann boundary condition in (2.3) that u can be expanded as the series of waveguide modes: where the coefficients β n are defined as If is small enough (in fact it only needs < λ/2 here), it follows that β n is are pure imaginary numbers for all n ≥ 1. It is easy to note that for each n, if nπ/ ≤ κ, the series consists of two propagating wave modes traveling upward and downward respectively; if nπ/ > κ the series consists of two evanescent wave modes decaying exponentially away from the bottom side and the open aperture of the cavity, respectively. Using the series (2.5), we may reformulate (2.3)-(2.4) into the following coupled problem: The continuity conditions are imposed along the open aperture Γ + , where 0+ and 0− indicate the limits taken from above and below Γ + , respectively.
For simplicity of notation, we define u + (x 1 ) = u (x 1 , 0) and u − (x 1 ) = u (x 1 , −d). Let u ± ,n = u ± , φ n Γ ± be the Fourier coefficients for u + and u − , respectively, where the inner product ·, · Γ ± is defined by For any s ∈ R, denote by H s (Γ + ) the trace functional space on Γ + with the norm given by Matching the series (2.5) on Γ + , Γ − and using the continuity conditions in (2.7), we get α + n + α − n e iβnd = u + ,n , α + n e iβnd + α − n = u − ,n . Solving the above equations by Cramer's rule yields that Therefore, in the cavity D , the total field Since u satisfies the homogeneous Dirichlet boundary condition on Γ − , we have It follows from (2.12) that Given a function u ∈ H 1 (R + 2 ) and let u + = u(x 1 , 0), we define a Dirichlet-to-Neumann (DtN) operator: where the Fourier coefficients u + n = u + , φ n Γ + .
Proof. For any u, w ∈ H 1/2 (Γ + ), they have the Fourier series expansions where u + n = u, φ n Γ + and w + n = w, φ n Γ + . It follows from (2.15) that Note that the coefficients β n are given by (2.6). For n = 0, a straightforward calculation yields that where we use the estimates Combining (2.16)-(2.17) and using (2.8), we obtain from the Cauchy-Schwarz inequality that which completes the proof.
Using (2.14)-(2.15) and the continuity of ∂ x 2 u on Γ + , we obtain the transparent boundary condition (TBC): Hence the problem (2.3)-(2.4) can be reduced to the following boundary value problem: (2.18) By calculating the Fourier coefficients u + ,n , which give the coefficients u − ,n by (2.13), we may obtain the solution in the cavity D from the formula (2.10).
To find an approximate model to (2.18), we examine the series (2.5) more closely. Note that if λ, the wave modes e −iβnx 2 φ n (x 1 ) and e iβn(x 2 +d) φ n (x 1 ) decay exponentially in the cavity, with a decaying rate of O(e −n/ ) for all n ≥ 1. Only the leading wave modes e −iκx 2 φ 0 (x 1 ) and e iκ(x 2 +d) φ 0 (x 1 ) propagate in the cavity. The observation motivates us to approximate the DtN map (2.15) by dropping the high order modes and define an approximate DtN map by Now we arrive at an approximate model problem: where v = 0 on Γ − and v sc = v − (u inc + u ref ) in R 2 + . Accordingly, we may approximate u by one single mode: where the coefficientsα + 0 andα − 0 are given bỹ Let v be the solution of the approximate model problem (2.20) and be given by (2.21) in the cavity, then there exist positive constants C 1 , C 2 independent of λ, , d such that Proof. In R 2 + , since the incident field u inc and the reflected field u ref satisfy the Helmholtz equation in (2.20), the scattered field v sc also satisfies Let G be the half-space Green function of the Helmholtz equation with Neumann boundary condition, i.e., it satisfies It is easy to note that where y = (y 1 , y 2 ), H 0 is the Hankel function of the first kind with order 0, and x is the reflection of the point x with respect to Γ 0 , i.e., x = (x 1 , −x 2 ).
It follows from Green's identity that By the fact that ∂ x 2 u inc + ∂ x 2 u ref = 0 on Γ 0 , especially on Γ + , we have It follows from the continuity of single layer potentials [15,16] and the above equality that Using (2.21)-(2.23) yields that Substituting the above equality into (2.25) and using the fact that φ 0 ( Therefore, the Fourier coefficients v + ,0 may be expressed as It follows from the fact u inc (x 1 , 0) = u ref (x 1 , 0) and (2.21) that Byα − 0 = −α + 0 e iκd and the above equation, we obtaiñ where c 0 = κ 2 √ h 1 , φ 0 Γ + . Note that x 1 , y 1 ∈ Γ + for small |x 1 − y 1 |, asymptotically, it holds that (cf. [34]) where γ 0 is the Euler constant defined by γ 0 = lim It follows from the assumption d λ that Using the definition of the incident wave gives where the positive constant C is independent of λ, , d. It follows from (2.21) that Noting that is small enough and κ(x 2 + d) 1, we have from (2.29)-(2.31) and (2.33) that there exists a positive constant C 1 such that which yields that which completes the proof.
Proof. It follows from (2.32) that Noting that d λ, we have It follows from (2.29)-(2.31) that for some positive constants c 1 and c 2 .
Combining the above estimates, we obtain that there exist constants C 3 and C 4 independent on λ, , d such that which completes the proof. Figure 2. The problem geometry with a transparent boundary.
The problem geometry is shown in Figure 2.
In the exterior domain R 2 + \B + R , by noting the PEC condition on Γ 0 \ Γ + R and the radiation condition (2.4), the scattered field can be expressed as the Fourier series: where u sc ,n = 2 π π 0 u sc (R, η) cos(nη)dη, H n is the Hankel function of the first kind with order n, and the polar coordinate r = ( we define the DtN operator on ∂B + R : It follows from (2.36) and (2.37) that the normal derivative of total field on ∂B + R can be written as . By the relation of the Hankel function and the modified Bessel function (cf. [1]), it yields that where K n is the modified Bessel function of the second kind. Using the formula zK n (z) = −nK n (z)− zK n−1 (z), we obtain where c is a positive constant. For any v, w ∈ H 1/2 (∂B + R ), we have from the definition of (2.37) that which completes the proof.
Proof. It follows from the definition (2.37) that Note that H (1) n = J n + iY n , where J n and Y n are the Bessel functions of the first and second kind, respectively, and the modulus of each Hankel function is decreasing function [1], then It follows from the Wronskian formula [1], where W (·, ·) denotes the Wronskian determinant of J n and Y n . The proof is completed by combining (2.39)-(2.41).
By the TBC operator (2.38), the model problem (2.18) can be reformulated as follows: (2.42) The variational formulation of (2.42) is to find where the sesquilinear form Theorem 2.6. The variational problem (2.43) has a unique solution u ∈ H 1 (B + R ). Proof. By the definition (2.15), we have for sufficiently small that It follows from Lemma 2.5 and the above inequality that , which shows that the sesquilinear form a(·, ·) satisfies a Gårding type inequality.
Next is show the uniqueness. It suffices to show that u = 0 if g = 0. A simple calculation yields ImA n |u ,n | 2 = 0.
It follows from Lemma 2.5 that u ,n = 0 for all n. Therefore u = 0 on ∂B + R . By the definition (2.37), we have T u = 0, which implies ∂ r u = T u = 0 on ∂B + R by (2.38). Hence we have u = 0 in R 2 + \B + R . It follows from the unique continuation [20] that u = 0 in B + R . The proof is completed by applying the Fredholm alternative theorem.
Correspondingly, the approximate model problem (2.20) can be written as follows: (2.45) The variational formulation of (2.45) is to where the sesquilinear form The well-posedness of the variational problem (2.46) can be shown similarly to Theorem 2.6. The proof is omitted for brevity. In the cavity D , the wave fields u and v are given by ( Lemma 2.7. Let u be the expansion of (2.10) inside the cavity with the Fourier coefficients given by the model (2.42), and let v be the expansion of (2.21)-(2.22) with the Fourier coefficients given by the approximate model (2.45), then the following estimates hold: Proof. The proof consists of five steps in order to show the estimates between u and v .
Step 1: we first give the estimate of u − u 0 where u 0 is the solution of (2.42) when there is no cavity, i.e., u 0 satisfies the variational problem It follows from (2.43) and (2.48) that (2.51) Denote by L be the induced operator for the sesquilinear form (2.44) such that where (·, ·) is the inner product on L 2 (B + R ). It follows from [28,Lemma A.3] that there exists a positive constant 0 such that where C(λ, d, R) is a constant independent of . Combing (2.49)-(2.52) yields Since v satisfies the variational problem (2.46), following the same arguments as above, we may show for sufficiently small that (2.53) Step 2: It follows from the standard regularity estimate for the Neumann problem that where the last inequality is obtained by using (2.53).
Step 3: This step is to estimate Using the estimates (2.50)-(2.51) gives For the second term in (2.56), we have from the estimate of For the variational problem (2.55), a combination of the above estimates and the boundedness of inverse of the linear operator L in (2.52) lead to the following estimate (2.57) Step 4: we estimate ∇u − ∇v L 2 (D ) . We have from the expansions of u in (2.7) and v in (2.21) that where the Parseval identity is used in the last inequality. For the first term (n = 0) in the above inequality, following from (2.9) and (2.22) and noting that d λ, we get For the second term (n ≥ 1), by (2.9) and iβ n d Consequently, we obtain It follows from (2.54) and (2.57) that On the other hand, we have from (2.57) that Combining the above estimates yields Step 5: the last step is to estimate u − v L 2 (D ) . Using the similar method as that in Step 4, we can obtain from the Parseval identity and (2.58)-(2.59) that where the estimates (2.60)-(2.61) are used.

2.5.
Field enhancement without resonance. The electromagnetic field enhancement factor is defined as the ratio of the energy between the total field and the incident field in the cavity. Hence the electric and magnetic field enhancement factors are given by . Theorem 2.8. There exists a constant 0 depending on λ and d such that for all 0 < < 0 the following estimates hold: where C 1 and C 2 are some positive constants independent of λ, 0 , , d.
Remark 2.9. The electric field enhancement has an order O (λ/d), which is enormous due to the length scale d λ, while the magnetic field has no significant enhancement in this situation.
Proof. Recalling the scale assumption d λ, we have from Theorem 2.2 and Lemma 2.7 that where we use the fact that C(λ, d, R) √ d ≤ C 2 2 for sufficiently small . On the other hand, A straightforward calculation yields Combining the above estimates gives Noting that |∇ × H | = |∇ × (0, 0, u )| = |∇u | and ∇ × H = −iωεE , we obtain For the magnetic field, it follows from Theorem 2.3 and Lemma 2.7 that where we use C(λ, d, R) | ln |/d ≤ min {C 1 /2, C 2 /2} for sufficiently small . Clearly, we have Therefore, we obtain which completes the proof.

PEC-PMC cavity with resonances
This section is to discuss the Fabry-Perot type resonance and derive the asymptotic expansions for those resonance; analyze quantitatively the field enhancement at the resonance frequencies in a narrow cavity, i.e., λ. Recall the model problem (2.3)-(2.4): It is known that the scattering problem (3.1) has a unique solution for a complex wavenumber k with Imk ≥ 0 (cf., [3]). By analytic continuation, the solution has a meromorphic extension to the whole complex plane, except for a countable number of points, which are poles of the resolvent associated with the scattering problem (3.1). These poles are called the resonances of the scattering problem, and the associated nontrivial solutions are called resonance modes. If the frequency of the incident wave is taken to be close to the real part of the resonance, an enhancement of the field is expected if the imaginary part of the resonance is small. Since the width of the cavity tends to zero, we may assume that κ 2 is not the eigenvalue of the Laplacian with the above boundary conditions in the cavity. Thus the Green function of the Helmholtz operator in D exists and can be expressed as [14] g where c m,n = 1 Lemma 3.1. The scattering problem (3.1) is equivalent to the following boundary integral equation: It follows from Green's identity in R + 2 that where the half-space Green function G is given in (2.24). Using Green's identity inside the cavity yields By the continuity of the single layer potential, we have from (3.4) and (3.5) that The proof is finished by imposing the continuity of the solution on the open aperture Γ + .
It is clear to note that Introduce new variables by rescaling with respect to : X = x 1 / , Y = y 1 / . Define three functions: c m,n α m,n d cos(mπX) cos(mπY ) (3.6) and the boundary integral operators: Hence the boundary integral equation (3.3) is equivalent to the following operator equation:

Asymptotics of the integral operators.
In this subsection, we study the asymptotic properties of the integral operators in (3.7). For clarity, we first introduce several notation below.
, and γ 0 is the Euler constant. For the fixed depth of the cavity d and the wave number κ, if small enough, we have the following asymptotic expansions for the kernels G e and G i . where Here C j , j = 1, 2, 3 are positive constants independent of .
Proof. The proof of (i) follows directly from the definitions of the operators T e , T i in (3.7) and the asymptotic expansions of their kernels given by Lemma 3.2. The proof of (ii) follows directly from [11, Theorem 4.1, Lemma 4.2].

3.3.
Asymptotics of the resonances. The scattering resonance of (3.1) is defined as a complex wavenumber κ with negative imaginary part such that there is a nontrivial solution to (3.1) when the incident field is zero. This is the characteristic value of the operator ΓP + K + K 1 ∞ + K 2 ∞ with respect to the variable κ. For simplicity, we write By Lemma 3.3, it is easy to see that L is invertible for sufficiently small . Now assume that there exists ϕ 0 such that where I is the identity operator. It is straightforward to check that the eigenvalues of operator L −1 P + I are λ(κ, ) = 1 + Γ(κ, ) L −1 1 (0,1) , 1 (0,1) L 2 (0,1) . Therefore, the characteristic values of the operator P + L are the roots of the analytic functions λ(κ, ), and the associated characteristic function is given by Proof. For given roots of λ(κ, ), it is easy to check that they are the characteristic values of the operator P + L with corresponding characteristic function defined above. It follows from the asymptotic expansions in Lemma 3.3 and the Neumann series that which gives (3.11).
Next, we present the asymptotic expansion of resonances for the scattering problem (3.1).
Using Lemma 3.4, we obtain where ρ(κ) = 1 π (2 ln 2 + ln κ + γ 1 ). Note that the axis {z ∈ C : argz = ± π 2d } is the branch cut for p(κ, ), hence we choose a small number θ 0 > 0 and consider the domain {z ∈ C : − π 2d + θ 0 < argz < π 2d − θ 0 or π 2d + θ 0 < argz < 3π 2d − θ 0 }. On the other hand, we are only interested in those resonances which are not in the high frequency regime. Therefore, we only need to find all the roots of p(κ, ) in the domain for some fixed number M > 0. Since p(κ, ) blows up as κ → 0 or κ → (j + 1/2)π/d, j ∈ Z. As a result, there exists δ 0 > 0 such that all the roots of p in Ω θ 0 ,M actually lies in the smaller domain It is clear that − tan κd κ is analytic in Ω δ 0 ,θ 0 ,M and its roots are given by k n,0 = nπ/d, n = 1, 2, · · · . Note that each root is simple. Denote by k n the roots of λ(κ, ). From Rouche's theorem, we deduce that k n are also simple and are close to k n,0 if is sufficiently small. We now derive the leading order asymptotic terms for these roots. Define (3.14) Expanding p 1 (κ, ) at k n,0 yields We can conclude that p 1 (κ, ) has simple roots in Ω δ 0 ,θ 0 ,M which are close to k n0 's. Moreover, these roots are analytic with respect to the variables and ln . Denoting the roots of p 1 by k n1 and expanding them in term of and ln , we obtain Using Rouche's theorem, we claim that k n1 gives the leading order for the asymptotic expansion of the roots k n . More precisely, we have (3. 16) In the following, we give a proof of the claim. Note that and Hence there exists a constant C n such that |p(κ, ) − p 1 (κ, )| < |p 1 (κ, )|, ∀κ ∈ |κ − k n1 | = C n 2 ln .
3.4. The field enhancement with resonance. First we give an asymptotic expansion of p(κ, ) at the resonant frequencies.
Lemma 3.6. At the resonant frequencies κ = Rek n where k n is given in (3.12), we have Proof. Assume that |κ − Rek n | ≤ ln . It follows from the definition of p 1 in (3.14) that Recalling γ 1 = γ 0 − ln 2 − πi 2 , we have from the expansion of k n1 in (3.15) that When κ = Rek n , we have which completes the proof.
Combining Lemmas 3.6 and Lemma 3.7, we obtain at the resonant frequencies κ = Rek n that 3.4.1. Enhancement in the far field. We first investigate the scattered field in the domain R + 2 \B + R . Recall that Here we use the fact ∂ ν u inc + ∂ ν u ref = 0 on Γ 0 , especially on Γ + . Using Hence In the resonance case when κ = Rek n , the enhancement comes from the term 1 p . It follows from Lemma 3.6 that Correspondingly, we have which shows that the enhancement of the scattered magnetic field in the far field region has an order O(1/ ) compared to the non-resonance case when κ = Rek n . It follows from the Ampere law ∇ × H = −iωεE and H = (0, 0, u ) that the enhancement of the scattered electric field also has an order O(1/ ) in the far field region when κ = Rek n .

3.4.2.
Enhancement in the cavity. Now we study the field enhancement in the cavity. It follows from (2.5) that the total field u may be expanded as the sum of waveguide modes: Lemma 3.8. The coefficients in (3.18) have the following expansions: 2iκ cos(κd)p + O( 2 ln ), (3.19) and where the positive constant C independents of , κ and m.
Proof. Recall that u = 0 on Γ − then equation (3.18) becomes Taking the derivative of (3.20) with respect to x 2 yields which gives Multiplying (3.21) with φ 0 (x 1 ) and integrating it over Γ + , we get which gives the formulas for α + 0 and α − 0 in (3.19). For m ≥ 1, it follows from multiplying(3.21) with cos mπx 1 and integrating over Γ + that Note that iβ m = O m for m ≥ 1, and by (3.17) which completes the proof.
In the following, we give the field enhancement inside the cavity.
Noting when κ = Rek n , we have which yields that Therefore, the magnetic field enhancement due to the resonance is of an order O(1/ ) inD . We conclude from Ampere's law that the electric field enhancement is also of an order O(1/ ).
It follows from the asymptotic expansion of G e in Lemma 3.2 and Lemma 3.7 that where h 2 (X) : Note that at the resonant frequencies κ = Rek n , 1 p = 2i q 0 (1 + O( ln 2 )).
Thus we have It is clear to note that the leading order of the resonant mode is a constant on the open aperture with an order of O(ln ) and the magnetic field enhancement is of an order O(1/ ). On the other hand, since which implies that at the resonant frequencies κ = Rek n , on the top aperture Γ + , the scatter field satisfies By Ampere's law, we know the electric field enhancement is also of an order O(1/ ) on the top aperture Γ + when κ = Rek n .

Numerical experiments.
In this section, we present some numerical experiments to verify the theoretical studies of the field enhancement for the PEC-PMC cavity. As a representative example, we take the cavity width = 0.005, the cavity depth d = 1, and the angle of incidence θ = π/3 in the numerics. Figure 3 (left) and (right) show the plot of the electric field enhancement Q E the the magnetic field enhancement factor Q H against the wavenumber κ, respectively. We observe that both the electric and magnetic enhancement factors do reach peaks at the resonant frequencies  Figure 3. The field enhancement factor is plotted against the wavenumber κ for the PEC-PMC cavity: (left) the electric field enhancement factor Q E ; (righ) the magnetic field enhancement factor Q H .
(3.12), which are close to κ = nπ, n = 1, 2, . . . . Besides the peaks at the resonant frequencies, the electric field enhancement factor Q E also displays a peak when the wavenumber κ is sufficiently small, i.e., when the wavelength of the incident field λ is sufficiently large. In this situation, the scales satisfy d λ. The electric field should have an enormous enhancement according to our theoretical result in Theorem 2.8. In contrast, the magnetic field enhancement factor Q H does not show a peak when the wavenumber approaches zero, which is also confirmed in Theorem 2.8 that the magnetic field has no enhancement when d λ.

PEC-PEC cavity without resonance
In this section, assuming d λ, we show that there is a weak enhancement of the electric field and there is no enhancement of the magnetic field in the PEC-PEC cavity.
Here the incident field u inc = e iκx·d is a plane wave propagating in the direction d = (sin θ, − cos θ) and the reflected field u r = e iκx·d with the propagating direction d = (cos θ, sin θ).
Let φ 0 (x 1 ) = 1 √ and φ n (x 1 ) = 2 cos nπx 1 (n ≥ 1) be an orthonormal basis on the interval (0, ). Since the total field satisfies the Neumann boundary condition on the bottom and lateral sides of the cavity, we have from (4.1) that u can be expressed as the sum of waveguide modes: where the coefficients β n are defined as β n = κ, n = 0, i (nπ/ ) 2 − κ 2 , n ≥ 1.
If < λ/2, then β n are pure imaginary numbers for all n ≥ 1. For each n, it can be seen if nπ/ ≤ κ, the expansion consists of two propagating wave modes traveling upward and downward respectively; if nπ/ > κ, the expansion consists of two evanescent wave modes decaying exponentially away from the bottom side and open aperture of the cavity, respectively. Using the expansion (2.5), we may reformulate the model (4.1) into the following coupled problem: By matching the expansion series (4.2) and using the continuity conditions in (4.3), we may obtain Therefore, the wave field A straightforward calculation yields It follows from (4.6) that Substituting (4.7) into (4.5) yields iβ n e iβnd − e −iβnd e iβnd + e −iβnd u + ,n φ n (x 1 ) on Γ + .
Given a function u ∈ H 1/2 (Γ + ), we let u + = u(x 1 , 0) and define a DtN map on Γ + : iβ n e iβnd − e −iβnd e iβnd + e −iβnd u + n φ n (x 1 ), (4.8) where the Fourier coefficients u + n = u + , φ n Γ + . Following a similar proof to Lemma 2.1, we may show that the DtN map B PEC is also bounded. The proof is omitted for brevity.
where the constant C = max κ sin(κd) By (4.8), we derive the TBC: which helps to reduce (4.1) into to a boundary value problem in R + 2 : (4.9) The solution u in the cavity D may be obtained from (4.4) once the Fourier coefficients u + ,n are available (note that u − ,n can be computed from from (4.7)). To find an approximate the model problem to (4.9), we similarly drop all the nonzero wave modes and define Hence we may consider the following problem: where Inside the cavity, we may approximate u with one single mode: v (x) = α + 0 e −iκx 2 +α − 0 e iκ(x 2 +d) φ 0 (x 1 ), (4.12) where the coefficientsα + 0 andα − 0 are given bỹ (4.14) 4.2. Enhancement of the approximated field. This section presents the estimates of the solution for the approximate model problem (4.11).
Let v be the solution of (4.11) and be given by (4.12) inside the cavity, then there exist positive constants C 1 , C 2 independent on λ, , d such that Proof. For the approximate model (4.11), we obtain from Green's formula that where G is the half-space Green function satisfying the Neumann boundary condition and is given in (2.24). It follows from the continuity of the single layer potential that In light of (4.12)-(4.14), it yields that Substituting the above equality into (4.15) and using the fact that φ 0 ( where h 1 (x 1 ) is given in (2.26). Therefore, the Fourier coefficients v + ,0 may be expressed as It follows from the fact u inc (x 1 , 0) = u ref (x 1 , 0) and the inner product of (4.12) with φ 0 (x 1 ) that Usingα − 0 =α + 0 e iκd and the above equation gives where c 0 = κ 2 √ h 1 , φ 0 Γ + . Using the assumption d λ and expression of c 0 in (2.28), it gives that Recalling the definition of the incident wave, we get A simple calculation yields Noting that d λ, we also have (2.34)-(2.35) holds. It follows from (4.17)-(4.18) and the estimates in (2.34)-(2.35) that there exist constants C 1 and C 2 independent on , λ, d such that which completes the proof.
Let v be the solution of (4.11) and be given by (4.12) inside the cavity, then there exist positive constants C 3 , C 4 independent on λ, , d such that It follows from d λ that In light of (4.17)-(4.18), there exist constants C 3 , C 4 such that which completes the proof.

4.3.
Field enhancement without resonance. Following from the same steps as those in subsection 2.4, we may also show that where C(R) is a positive constant depending on R but is independent of λ, , d. The details are omitted for brevity. The following estimates are the main results of the electromagnetic field enhancement for the PEC-PEC cavity when d λ.
Theorem 4.5. If is small enough such that 2C(R) √ d < min C 1 (κd) 2 , C 2 (κd) 2 , then the electric field has no enhancement. If d λ such that d λ ≤ min C(R) d , then the electric field enhancement factor satisfies and the magnetic field enhancement factor satisfies where the constant C(R) is given in (4.20), the constants C 1 , C 2 are given in Theorem 4.2, and the constants C 3 , C 4 are given in Theorem 4.3.
Proof. By Theorem 4.2 and (4.20), we can obtain We consider two cases: Since ∇u inc 2 L 2 (D ) = κ 2 d, we obtain (ii) If d λ such that d λ ≤ min C(R) then we get which gives Noting that |∇ × H | = |∇u | and using Ampere's law ∇ × H = −iωεE , we obtain , which shows that for case (i) the electric field E has no enhancement; for case (ii) the electric field enhancement factor has an order O(λ /d).

4.4.
Numerical experiments. We present a numerical example to illustrate and verify the results stated in Theorem 4.5. The width of the cavity is = 0.005, the depth of the cavity is d = 1, and the incident angle is θ = π/3. Figure 4 plots the electric field enhancement factor Q E against the wavenumber κ ∈ (0, 10 −8 ). It shows that when κ is sufficient small, i.e., when the wavelength λ is sufficiently large, the electric field does have an enhancement, which corresponds to case (ii) in Theorem 4.5; when the wavenumber κ increases (the wavelength λ decreases) but is still small, the electric field has no enhancement, which corresponds to case (i) in Theorem 4.5. The plot is not shown for the magnetic field since it has no enhancement in this region.

PEC-PEC cavity with resonance
In this section, we show the field enhancement at resonant frequencies for the PEC-PEC cavity. Consider the model problem: (5.1) Here we assume that λ. It can be verified that where c m,n = 1 Using Green's formula, we obtain It follows from the continuity of the single layer potential that we have By imposing the continuity of the solution on Γ + , we have the following lemma.
Lemma 5.1. The scattering problem (5.1) is equivalent to the boundary integral equation: Similarly, we introduce the rescaling variables X = x 1 / , Y = y 1 / and define the following boundary integral operators: where G e , G i , ϕ are give in (3.6) with g PMC being replaced by g PEC . Then the boundary integral equation (5.3) is equivalent to the operator equation: where f = (u inc + u ref ) Γ + = 2e iκ sin θ X .

Asymptotics of the integral operators.
In this subsection, we study the asymptotic properties of the integral operators in (5.4).
Recalling the definitions of c m,n and α m,n in (5.2), we let Using the fact (cf. [19]) that for m = 0, we get Following from the identity we have for m ≥ 1 that Here we use the asymptotic expression of (1 − x 2 ) −1/2 and coth(d (mπ/ ) 2 − κ 2 ) → 1 as → 0.
which completes the proof.
Using Rouche's theorem again, we obtain k n = k n1 + O( 2 ln ), which completes the proof.
5.4. The field enhancement with resonance. In the far field region, we can follow the same steps as those in Subsection 3.4.1 with the new p(κ, ). Then we have the following theorem with the detailed proof omitted.
Theorem 5.4. At the resonant frequencies κ = Rek n where k n is given in (5.6), the scattered electric and magnetic fields have enhancement of an order O(1/ ) in the far field region.
The following result gives the field enhancement inside the cavity.   Figure 5. The field enhancement factor is plotted against the wavenumber κ for the PEC-PEC cavity: (left) the electric field enhancement factor Q E ; (righ) the magnetic field enhancement factor Q H .
Therefore, the magnetic field enhancement has an order of O(1/ ). We conclude from Ampere's law that the electric field enhancement also has an order of O(1/ ).
Finally, on the open aperture Γ + , we may follow similar steps as those in subsection 3.4.3 and show the electromagnetic field enhancement. The proof is omitted again for brevity.
Theorem 5.7. At the resonant frequencies κ = Rek n where k n is given in (5.6), the enhancement of the scattered electric and magnetic fields has an order O(1/ ) on the open aperture Γ + .

Numerical experiments.
We show some numerical experiments to verify the theoretical findings of the field enhancement for the PEC-PEC cavity. We take the cavity width = 0.005, the cavity depth d = 1, and the angle of incidence θ = π/3. Figure 5 (left) and (right) show the plot of the electric field enhancement Q E the the magnetic field enhancement factor Q H against the wavenumber κ, respectively. We observe that both the electric and magnetic enhancement factors do obtain peaks at the resonant frequencies (5.6), which are close to κ = (n + 1 2 )π, n = 0, 1, 2, . . . .

Concluding remarks
We have studied the electromagnetic field enhancement for the scattering of a plane wave by a subwavelength rectangular open cavity. The TM polarized wave model problem is considered for both the PEC-PMC and the PEC-PEC cavities. The electric and magnetic field enhancements are examined in both the nonresonant and resonant regimes. These results provide an understanding on some of the mechanisms for the wave field amplification and localization in a subwavelength structure. This work focuses on the TM polarization of the electromagnetic wave in a single cavity.