Windowed integral equations

WINDOWED GREEN FUNCTION METHOD FOR 2D WAVEGUIDES

2.2 Windowed Green Function Method (WGF) .1 Integral Equation Formulation.1Integral Equation Formulation

2.2.4 Windowed integral equations

Per Remark 2.2.1, for a fixed r ∈Γ, the integrands associated with the operators on the left-hand side of equation (2.18) are oscillatory and slowly decaying at

Table 2.1: Convergence of the windowed integralsI_tr(A)andI_w(A)withκ= 2πand window parameterα=0.5

A |I−I_tr(A)| |I−I_w(A)|

10 5.0×10⁻² 3.6×10⁻³ 20 3.6×10⁻² 5.8×10⁻⁵ 25 3.2×10⁻² 9.3×10⁻⁶ 50 2.2×10⁻² 3.1×10⁻⁹ 75 1.8×10⁻² 3.1×10⁻¹² 100 1.6×10⁻² 1.0×10⁻¹⁴

infinity—just like the integrand in the simplified example presented in section 2.2.3.

Thus, it is expected that use of windowing in the integrands of these operators should result in convergence properties analogous to the ones described in that section. Since the integrands on the right-hand side (RHS) of equation (2.18) are known functions, and since, as shown below, the full RHS can be evaluated efficiently (by relying on equation (2.27)), the use of windowing may be restricted to the left-hand side operators.

These considerations lead to the following system of “windowed” integral equa- tions on theboundeddomaineΓ= Γ∩ {wA(r),0}:

E(r)Φ^scat_w (r)+T[wAΦ^scat_w ](r)= −E(r)Φ^inc(r) −T[Φ^inc](r), (2.25) for r ∈ eΓ. A discrete version of equations (2.25) can be obtained by substituting all left-hand side integrals by adequate quadrature rules (as mentioned above, the Nyström method [32, Sec. 3.5] is used for this purpose). The windowing function wA(r)used here is selected as follows: (i)wA(r)=1 on any portion ofΓthat is not part of a SIW; (ii)wA(r)=weA(dq(r))along any portion ofΓthat is contained in the q-th SIW, and where dq(r)is the distance to the edge of the aforementioned SIW (see Figure 2.1b). As demonstrated in section 2.3 through a variety of numerical results, the solutionΦ^scat_w of equation (2.25) provides a super-algebraically accurate approximation toΦ^scatthroughout the region {wA(r)=1}.

Note that the net wavenumber of the integrands in the RHS of equation (2.25) can be arbitrarily small, since the phase of the exponentials in (2.5) can be arbitrarily close to the negative of the phase of the kernels (see Remark 2.2.1). Thus a direct windowed computation ofT[Φ^inc](r)generally presents a considerable challenge.

In order to obtain these integrals, a strategy that doesn’t rely on the oscillatory nature of the integrands was devised. To introduce this alternative strategy, consider the

2.2. Windowed Green Function Method (WGF) 25 discussion leading to equation (2.18). Clearly, the first (resp. second) component in the quantity[ξ, η]^T=T[Φ^inc](r)for r ∈eΓis given by the limit as r tends toeΓof the fieldu(resp. the normal derivative of the fieldu) that results as the pair[φ, ψ]^T in (2.10) is replaced by the incident densities [φ^inc, ψ^inc]^T(= Φ^inc). Note that, per equation (2.14), Φ^inc vanishes identically outside Γ^inc_j = Γj ∩Ω^inc. Now, given that the illuminating structure is a single SIW whose optical axis coincides with the z−axis (as indicated in section 2.2.1), the corresponding integration domainΓ^inc_j can be decomposed as the unionΓ^inc_j = eΓ^inc_j ∪Γ^∞_j of the two disjoint segments eΓ^inc_j = Γ^inc_j ∩ {wA(r) , 0} andΓ^∞_j = Γ^inc_j ∩ {wA(r) = 0}. (With reference to Figure 2.1b note thatΓ^inc_j =

Ð^j−1

`=1Γ<sub>`</sub>^inc_j Ð

ÐN

`=j+1Γ<sup>inc</sup><sub>j`</sub>

whereΓ^inc_{j`</sub> = Γ<sub>j`}∩Ω^inc; similarly,eΓ^inc_j andΓ^∞_j are decomposed as the unions of the curveseΓ^inc_{j`</sub> = Γ<sup>inc</sup><sub>j`} ∩ {wA(r), 0}and Γ^∞_{j`</sub> = Γ<sup>inc</sup><sub>j`} ∩ {wA(r) = 0}, respectively. Figure 2.1b only displays the independent componentseΓ^inc_{j`</sub> and Γ<sup>∞</sup><sub>j`}.) Using the fact that the incident densities ϕ^inc andψ^inc vanish outsideΓ^inc_j , forr ∈Ω_j the relation

D[βjϕ^inc] −S h

βjν⁻_j¹ψ^inci

=

∫

eΓ^inc_j ∪Γ^∞

j

βj

×

∂Gj(r,r⁰)

∂n(r⁰) ϕ^inc(r⁰) −Gj(r,r⁰)ν⁻_j¹ψ^inc(r⁰)

dsr⁰, (2.26) results. The evaluation of the right-hand integral is discussed in what follows.

The integral over the bounded curveeΓ^inc_j in (2.26) can be computed using any of the standard singular-integration techniques mentioned in section 2.2.2. The curve Γ^∞_j extends to infinity, on the other hand, with an integrand that decays slowly (see Remark 2.2.1). Fortunately, however, theΓ^∞_j integration problem can be significantly simplified by relying on the fact that[ϕ^inc, ψ^inc]^T(= Φ^inc) actually coincides with the boundary values of the incident field and its normal derivative, as indicated in equation (2.14). To evaluate theΓ^∞_j integral, consider the identity

∫

Γ^∞_j

βj

∂Gj(r,r⁰)

∂n(r⁰) ϕ^inc(r⁰) −Gj(r,r⁰)ν⁻_j¹ψ^inc(r⁰)

ds_r⁰

=−

∫

Γ^⊥_j

βj

∂Gj(r,r⁰)

∂n(r⁰) u^inc(r⁰) −Gj(r,r⁰)∂u^inc

∂n (r⁰)

dsr⁰, (2.27) for r ∈ {z > −A} that results by using Green’s theorem in a manner akin to [37]

(the corresponding bounded-domain result can be found e.g. in [31, theorem 3.1]).

(Here Γ^⊥_j = Ω_j ∩ {z = −A} is such that Γ^⊥_j ∪ Γ^∞_j is the boundary of the region Ω_j∩ {z < −A}, see Figure 2.1b.) Equation (2.27) provides an alternative approach

for the evaluation of the integrals overΓ^∞_j in equation (2.26). While some of theΓ^⊥_j curves are unbounded, along such unbounded curves the fields decayexponentially fast(see equation (2.6)), and, thus, arbitrary accuracy can efficiently be achieved in the corresponding integrals by truncation of the integration domain to a relatively small bounded portion ofΓ^⊥_j. It is worthwhile to emphasize that the substitution of the right-hand side of (2.25) by quantities evaluated in terms of equations (2.27) and (2.26), as explained above in this section, amounts to a non-standard procedure which significantly facilitates the implementation of the proposed methodology in the case of bound-mode excitations.

The overall WGF method for open waveguides is summarized in points (1) to (3) below. As mentioned above, the bounded-interval numerical integrations men- tioned in the points (1) to (3) can be effected by means of any numerical integration method applicable to the kinds of singular integrals (with singular kernels and, at corners, unknown singular densities) associated with the problems under considera- tion. The implementation used to produce the numerical results in section 2.3 relies on numerical integration methods derived from [32, Sec. 3.5].

The WGF algorithm proceeds as follows:

(1) Evaluate the RHS of equation (2.25) by decomposing the integrals in the operators (2.20) into integrals over eΓ^inc_j and Γ^∞_j . The integrals involving the bounded integration domainseΓ^inc_j are computed by direct numerical in- tegration. Relying on the exponential decay of the integrands of the RHS of equation (2.27), on the other hand, the integrals overΓ^∞_j are obtained by truncation of the integrals overΓ^⊥_j .

(2) Solve forΦ^scat_w in equation (2.25) by either inverting directly the corresponding discretized system (as is done here), or by means of a suitable iterative linear- algebra solver, if preferred.

(3) Evaluate the approximate fieldsuw using the representation formula (2.10) in conjunction with equation (2.27):

u_w(r)=D[βjwAϕ^scat_w ](r) −S h

βjν⁻_j¹wAψ_w^scati (r) +D[βjϕ^inc](r) −S h

βjν⁻_j¹ψ^inci

(r), (2.28)

forr ∈Ω_j, and where the layer potentials involvingϕ^incandψ^incare computed as in point 1, that is, by direct evaluation of the integrals over eΓ^inc_j and via equation (2.27) for the integrals overΓ^∞_j .