Mode illumination

WINDOWED GREEN FUNCTION METHOD FOR 3D WAVEGUIDES

6.4 Mode illumination

present chapter, the dielectric formulation derived in Chapter 5 can be used for this problem. When combined with the windowing function described in Section 6.2 the following three-dimensional windowed integral equations result:

m(r)+R^∆ε[WAm](r)+S^∆α[WAj](r)+T^∆α[WAj](r)= 2εe

εe+εi

E^inc×n, (6.3a) j(r)+R^∆µ[WAj](r) −S^∆α[WAm](r) −T^∆α[WAm](r)= 2µe

µe+µi

H^inc×n, (6.3b) for r ∈Γ.

In order for the system (6.3) to provide a good approximation in the vicinity of the windowed region, it is necessary for the integrands in these operators to have non-vanishing oscillations as d(r) → ∞. Remark 6.3.1 addresses this important point.

Remark 6.3.1 In view of the definitions of the densities j and m in terms of the interior scattered fields, together with the radiation conditions, the densities behave asymptotically as the tangential components of a superposition of outgoing bound modes. Them-th mode contribution contains a factor ofe^+ikzm|z|(where k^m_z denotes the propagation constant of them-th mode). Since the kernels oscillate with a factor ofe^+ik|r−r0|, and since bothk^m_z andk are positive, the product of the kernels and the densities result in non-vanishing oscillations asd(r) → ∞.

6.4 Mode illumination

In many instances, illuminating a waveguide with an incoming bound mode is desirable—this can model incoming fields from other structures that are connected through a waveguide. Historically, the sourcing of bound modes has been a non- straightforward matter, and alternative approximations are usually used—such as mode bootstrapping, illumination by Gaussian beams that approximate the mode, or by exciting the modes with point sources [76, 81]. These techniques usually require either additional simulations or large propagation distances for the incoming waves to shed away the undesired radiative or modal components, or the simulation is restricted to single-mode waveguides to avoid spurious modes being excited.

However, it is highly advantageous to be able to directly source anymode at will, incurring into as little extra computation work and error as possible. With this goal in mind, this section proposes an integral equation methodology to accurately simulate the scattering of incident bound modes. In particular, we can use as incident field any given bound mode of the relevant waveguides—on the basis of an auxiliary

representation for the incident fields which, at minimal expense, incur errors that are exponentially smaller with regards to a certain approximation parameter.

To describe the proposed incident-mode approach, let us consider a three- dimensional nonuniform waveguide structure composed of one or more SIWs. For clarity, we consider the case in which there is only one exterior and one interior domain, denoted as usual by Ωe and Ωi, respectively. Let one of the SIW carry an incoming bound mode as an incident excitation. The region Ω^inc wherein the incoming mode is defined coincides with the SIW itself—including both the core and cladding region. Thus, using the associated indicator function

χ^inc(r)=











1 for r ∈Ω^inc,

0 for r <Ω^inc, (6.4)

the total fields are given by E = E^incχ^inc+E^scat andH = H^incχ^inc +H^scat. Next, define the interior incident and scattered densities

m^inc≡ m^inc_i = χ^incE^inc_i ×n, (6.5a) j^inc ≡ j^inc_i = χ^incH^inc_i ×n, (6.5b) m^scat ≡ m^scat_i =E^scat_i ×n, (6.5c) j^scat ≡ j^scat_i =H^scat_i ×n, (6.5d) and the exterior incident and scattered densities

m^inc_e = −χ^incE^inc_e × n, (6.6a) j^inc_e = −χ^incH^inc_e ×n, (6.6b)

m^scat_e =−E^scat_e ×n, (6.6c)

j^scat_e =−H^scat_e × n, (6.6d)

respectively. Additionally, given the necessary continuity of the tangential compo- nents of the field induced by the boundary conditions, we obtain that the relations

m^inc_e =−m^inc_i , (6.7a)

j^inc_e =−j^inc_i , (6.7b)

m^scat_e =−m^scat_i , (6.7c)

j^scat_e =−j^scat_i , (6.7d) hold.

6.4. Mode illumination 93 Using these definitions and the representation formulas1presented in Chapter 5, we obtain the representation for thetotalfields:

iωµiSi[j_i^inc+ j^scat_i ](r)+ i ωεi

Ki[j^inc_i + j^scat_i ](r)+Di[m^inc_i +m^scat_i ](r)=











Ei(r), r ∈Ωi,

0, r <Ω_i, (6.8a)

−iωεiSi[m^inc_i + m^scat_i ](r) − i

ωµiKi[m^inc_i + m^scat_i ](r)+Di[j_i^inc+ j^scat_i ](r)=











Hi(r), r ∈Ω_i, 0, r <Ω_i.

(6.8b)

iωµeSe[j^inc_e + j^scat_e ](r)+ i

ωεeKe[j^inc_e + j^scat_e ](r)+De[m^inc_e +m^scat_e ](r)=











Ee(r), r ∈Ω_e,

0, r <Ω_e, (6.9a)

−iωεeSe[m^inc_e +m^scat_e ](r) − i

ωµeKe[m^inc_e +m^scat_e ](r)+De[j^inc_e + j^scat_e ](r)=











He(r), r ∈Ω_e, 0, r <Ωe.

(6.9b) Following the same steps as in the bounded-obstacle case discussed in section 5.3, we obtain the following system of integral equations

m(r)+R^∆ε[m](r)+S^∆ε[j](r)+T^∆ε[j](r)=

−

m^inc(r)+R^∆ε[m^inc](r)+S^∆ε[j^inc](r)+T^∆ε[j^inc](r)

, (6.10a)

j(r)+R^∆µ[j](r) −S^∆µ[m](r) −T^∆µ[m](r)=

−

j^inc(r)+R^∆µ[j^inc](r) −S^∆µ[m^inc](r) −T^∆µ[m^inc](r)

, (6.10b)

1Although the representation theorems presented are for bounded obstacles, a limiting form akin to that in [67] can be used to justify that an integral representation exists for waveguide structures with infinite boundaries.

where all the operators involved are given by integrals along the completeinfinite boundary Γof the waveguide. The operators on the left-hand side act only on the scattered densities; in virtue of section 6.1, the product of the densities times the relevant kernels have non-vanishing oscillations along the surface of the waveguide, hence the windowing technique provides an effective way to numerically truncate and evaluate these operators. On the contrary, the operators on the right-hand side act on the incident densities, for which the waves travel in the direction opposite to those inherent in the kernels, so that the net oscillations of the integrands may vanish. A strategy designed to evaluate these right-hand operators in spite of the potential loss of integrand oscillatory character is presented in the following section.

6.4.1 Evaluation of incident contributions

In order to overcome the difficulty associated with the evaluation of the right- hand side expressions in equation (6.10), we make use of an auxiliary representation for the incident mode. To do this, we denote byΓ^wthe portion of the waveguide for which the window function is greater than zero,

Γ^w = Γ∩ {r :wA(r)> 0}, (6.11) and we let Γ^∞ denote the portion of the SIW that carries the incident mode that is not contained inΓ^w:

Γ^∞ =Ω^inc∩ (Γ\Γ^w). (6.12)

Further, we define the auxiliary (infinite) boundaryΓ^⊥to be the plane perpendicular to the SIW with the incident mode, and that crosses the waveguide exactly at the junction betweenΓ^wandΓ^∞. Additionally, we considerΩ^∞to be the portion ofΩ^inc that goes fromΓ^⊥ to infinity in the direction opposite to the incoming mode—this notation is analogous to that used for the one presented in Chapter 2 and depicted in Figure 2.1 for the two-dimensional case. Using these definitions together with the representation formulas from section 5.2, we obtain

iωµi(S_i^∞+S_i^⊥)[j^inc_i ](r)+ i ωεi

(K_i^∞+K_i^⊥)[j^inc_i ](r)+(D_i^∞+D_i^⊥)[m^inc_i +](r)=











Ei(r), r ∈ (Ω_i∩Ω^∞), 0, r <(Ω_i∩Ω^∞), (6.13a)

6.4. Mode illumination 95

iωµe(Se^∞+Se^⊥)[j^inc_e ](r)+ i ωεe

(Ke^∞+Ke^⊥)[j^inc_e ](r)+(De^∞+De^⊥)[m^inc_e +](r)=











Ee(r), r ∈ (Ω_e∩Ω^∞), 0, r <(Ω_e∩Ω^∞), (6.13b) where the superscripts∞ and ⊥in the potential operators denote integration over Γ^∞and Γ^⊥ (normal pointing away fromΩ^∞), respectively. Similar expressions for HiandHehold by exchangingE→H, j → m, m→ jand µ→ −ε.

The idea of this auxiliary formulation for the field was introduced for the two-dimensional case in Chapter 2 (see also [22]), and in its three-dimensional variant—equation (6.13))—can be used to evaluate the challenging right-hand side in equation (6.10). In detail, consider a point r ∈ Γ^w, then, in view of r <((Ω_e∩Ω^∞) ∪ (Ω_i∩Ω^∞)), together with equation (6.13), we have

R^∆ε[m^inc](r)+S^∆ε[j^inc](r)+T^∆ε[j^inc](r)= (R^∆,wε −R^∆,⊥ε )[m^inc](r)+

(S^∆,wε −S^∆,⊥ε )[j^inc](r)+(T^∆,wε −T^∆,⊥ε )[j^inc](r), (6.14a)

R^∆µ[j^inc](r) −S^∆µ[m^inc](r) −T^∆µ[m^inc](r)=(R^∆,wµ −R^∆,⊥µ )[j^inc](r)−

(S^∆,wµ −S^∆,⊥µ )[m^inc](r) − (T^∆,wµ −T^∆,⊥µ )[m^inc](r), (6.14b) where, again, the superscripts∞ and⊥in the integral operators denote integration overΓ^∞andΓ^⊥, respectively. The identity in equation (6.14) is an important result from this thesis in view of Remark 6.4.1.

Remark 6.4.1 The operators in the right-hand side of equation(6.14)involve inte- grals that can be accurately computed. The integrals over the bounded surface Γ^w can be treated as in the bounded obstacle case. On the other hand, the integrals over Γ^⊥, in spite of involving an infinite surface, the integrands are exponentially de- caying towards infinity—alongΓ^⊥—due to the nature of the incident bound modes.

This exponential decay allows us to truncateΓ^⊥ to perform the integrals, incurring in an exponentially small truncation error.

Putting it all together, we obtain the following WGF system of integral equations for a waveguide structure illuminated by a bound mode:

m(r)+R^∆ε[WAm](r)+S^∆ε[WAj](r)+T^∆ε[WAj](r)=b^⊥_m(r), r ∈ Γ^w, (6.15a) j(r)+R^∆µ[WAj](r) −S^∆µ[WAm](r) −T^∆µ[WAm](r)=b^⊥_j(r), r ∈Γ^w, (6.15b) where

b^⊥_m(r)= −

m^inc+(R^∆,wε −R^∆,⊥ε )[m^inc](r)+

(S^∆,wε −S^∆,⊥ε )[j^inc](r)+(T^∆,wε −T^∆,⊥ε )[j^inc](r)

, (6.16a)

b^⊥_j(r)=

j^inc+(R^∆,wµ −R^∆,⊥µ )[j^inc](r)−

(S^∆,wµ −S^∆,⊥µ )[m^inc](r) − (T^∆,wµ −T^∆,⊥µ )[m^inc](r)

. (6.16b)