Boundary integral equation methods for simulation and design of photonic devices

Garza participated in the conception of the project, the development of the integration strategy, coded the numerical method and performed the presented simulations. Garza contributed to the conception of the project, provided input on the implementation of the WGF method and edited the figures.

LIST OF TABLES

Background

INTRODUCTION

• Maxwell’s equations
• Time-harmonic solutions: The frequency domain
• Units and electromagnetic parameters
• Boundary integral equation methods
• Dielectric waveguides
• Windowing of improper integrals
• Thesis outline

One of the central themes in this thesis concerns the propagation of fields in the presence of dielectric waveguide structures. An important observation from this numerical experiment is that the convergence of the window integrals generally depends on the parameter A/eλ, where.

Figure 1.1: Convergence of the integrals from equations (1.12), (1.13) and (1.15).

Two-dimensional problems

WINDOWED GREEN FUNCTION METHOD FOR 2D WAVEGUIDES

Mathematical Framework for 2D Open Waveguides

Here, the SIW is one of the two parts formed when a perfectly uniform waveguide is cut by a straight line (plane) perpendicular to the axis of the waveguide. With linearity, simultaneous illumination with multiple SIWs can be achieved directly by adding appropriate solutions for single SIW illumination.

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

• Error estimates for a simplified windowed integral
• Windowed integral equations

A strategy was devised to obtain these integrals that does not rely on the oscillatory nature of the integrands. If we rely on the exponential decay of the RHS integrands of equation (2.27), on the other hand, we obtain the integrals over Γ∞j by truncating the integrals over Γ⊥j.

Table 2.1: Convergence of the windowed integrals I tr (A) and I w (A) with κ = 2 π and window parameter α = 0

Numerical examples

The core and cladding wavenumbers are kco = 2π and kcl = 4π/3, respectively, and the barrier circular wavenumber is kob = 4π. The wavenumbers of the core and cladding are respectively given by kco = 2π and kcl = 4π/3, while the waveguide half width ish=0.5.

Figure 2.2: Super-algebraic convergence of the WGF method for various test con- con-figurations

MODE FINDER ALGORITHM FOR ELECTROMAGNETIC WAVEGUIDES

Mode equations

The bound modes enjoy the unique property of concentrating all the energy to a vicinity of the waveguide. Once the longitudinal components of the field have been found, the transverse components can be calculated from the relationship.

Figure 3.1: Illustration of a uniform dielectric waveguide with arbitrary cross sec- sec-tion.

Bound modes of a circular waveguide

Then, to find the modes of the circular waveguide, one must find the values ​​of m andkz for which Amin equation (3.10) has non-trivial zero space. This can be done by performing a root search on the minimum singular value of Am as a function of kz - the corresponding right-singular vectors can be used to find the values ​​of A1, A2, B1 and B2 that can be used to calculate the longitudinal fields of the state at using equation (3.8). The specific implementation of the mode-finding algorithm used for the circular case is shown in Algorithm 3.1, and in fact it works as a simplified case of the mode-finding algorithm shown in Section 3.4.

Given that Ez and Hz satisfy the Helmholtz equation (3.3), these fields can be expressed in terms of the single and double layer potential [32]:. is chosen so that the system of integral equations involves only weakly singular kernels. A first approach to finding the modes of a waveguide based on the integral representation of the previous section can be employed by an algorithm similar to the one we used for the circular case (Algorithm 3.1), but instead instead of taking the minimum singular value of equation (3.10), one would take the minimum singular value of AB. In [10] a method to avoid spurious solutions of a similar kind was introduced for the method of definite solutions, and in [3] the same idea was successfully implemented to regulate the σmin curve in the context of the integral equation formulation of the Zaremba eigenvalue problem.

Figure 3.2: Minimum singular value as a function of the propagation constant obtained using Algorithm 3.1

Numerical examples

• Circular waveguide
• Flower waveguide
• Multiple core waveguide

We see that the σmin curve behaves similarly to that of the circular waveguide, while the field values ​​associated with the different modes reflect the geometric symmetry. The values ​​of σminas a function of the propagation constant kzare shown in (a), EzandHz for the second mode are shown in (b) and (d), while (c) and (e) also show the Ezand Hz fields for the fifth mode found tone . For the purpose of demonstrating the versatility of the method we have presented in this chapter, the last example is that of a waveguide consisting of two separate cores.

The results of applying Algorithm 3.2 to this problem are shown in Figure 3.8, which clearly shows the aforementioned coupling effects. For this case, we used a set of interior points - for the modified algorithm - along the spiral turns in the interior of both cores. The Ez and Hz fields for the fourth mode found are shown in (b) and (d), respectively, while (c) and (e) depict the Ez and Hz fields (respectively) of the seventh mode.

Figure 3.4: Convergence in the 13 different propagation constants for the case of a circular waveguide

Three-dimensional problems

RECTANGULAR-POLAR METHOD FOR SINGULAR INTEGRAL OPERATORS

Closed surfaces

This formulation is guaranteed to provide a unique density solution to the scattering problem discussed here [32], and due to the nature of this second kind of integral equation, the number of iterations for GMRES remains essentially bounded as it is increased.

Open surfaces

Surface representation

4.12) It is clear that any r-dependent combination I = I(r) of integrals over Γ, of the types considered in sections 4.1.1 and 4.1.2, can be decomposed as a sum of integralsIq(r) about the different spots. In particular, the integral representations and boundary operators considered in those sections can be expressed in the form 4.15). In the next section, we present a methodology for accurate numerical evaluation of the integral Iq(r) for a given discrete approximation of the density eϕ(r0).

The solution to the integral equation problem then follows via an application of the iterative linear-algebra solver GMRES.

Integration strategy

• Density singularities along edges
• Singular “rectangular-polar” integration algorithm and a new edge- resolved integral unknownresolved integral unknown
• Computational cost
• Patch splitting for large problems

Integration strategy 55 The proposed strategy for evaluating the integral in equation (4.16) depends on the proximity of the point rs to the q-th patch. In view of the smoothness of the integrands for the non-adjacent cases currently considered (r is far from the integration patch), the integral in (4.23) can be evaluated exactly on the basis of a given high-order quadrature rule. Like section 4.3.2, this section deals with the problem of evaluating the quantity Iq(r) on the basis of the reformulation (4.23).

The evaluation problem of Iq for r ∈Ωcq poses a significant challenge given the singularity of the kernel H(r,r0) at r = r0. We now turn our attention to the precise evaluation of the integrals in equation (4.37). Relying on the coordinates (4.39) of the projection point in the almost singular case, and using the same notation.

Figure 4.1: Changes of variables (equations (4.21) and (4.22)) used to resolve edge-singularities in the density.

Numerical results

• Forward map convergence
• Edge geometries
• Open surfaces
• CAD geometries

The accuracy of the overall solver critically depends on the accuracy of the forward map calculation. It is worth noting that for the double-layer operator the evaluation of the quantity n(r0) · (r− r0)/|r −r0|2 is particularly sensitive to cancellation errors, and to reach small errors (10− 6 or smaller), special handling is required. The maximum far field value for the reference solution is equal to 2.144. b) Maximum (absolute) far-field error for the problem of scattering by a disk of radius 1 with k = 1.

The intensity profile |U|2 is shown in (a) and (b), while (c) shows the real part of the total field. Figure (a) shows the discretization of the patch, Figure (b) shows the real part of the field and Figures (c)-(e) show three different intensity views. In particular, these figures show the appearance of the well-known Poisson spot (also known as the Arago spot and the bright Fresnel spot) clearly visible in the center of the figure (c).

Table 4.1: Errors in the forward map (relative to the maximum forward map value) of the combined field operator for various patch splitting configurations and a spherical harmonic density (5,2)

ELECTROMAGNETIC BOUNDARY INTEGRAL EQUATIONS

• Background on three-dimensional EM-BIE
• Some concepts from differential geometry
• Electromagnetic potentials and integral operators
• Dielectric integral equations
• Incident electromagnetic fields
• Multipole solutions
• Electromagnetic beams
• Rectangular-polar electromagnetic solver
• Implementation validation: Scattering by a sphere

For a wave number k and a density d in the interface of the surface Γ we define the potential vector operators [63]. As mentioned above, the treatment of the three-dimensional time harmonic Maxwell equations by means of integral equations is similar to the treatment for the case of acoustic scattering. In this section we present a derivation of the integral equations for a transmission problem defined in what follows.

The total electromagnetic field - that is, the sum of the incident and the scattered field - satisfies the transmission boundary conditions. Define auxiliary fields to be such that they satisfy Maxwell's equations in Ωi with wavenumber ke (note that this is reversed), and in the interior they are the continuation of the incident fields. On the other hand, Algorithm 5.2 presents the steps needed to perform one forward map (the action) of the right-hand side in equation (5.33), which is necessary for the GMRES solver.

Figure 5.1: Scattering by a dielectric sphere of radius 2 λ 0 . The convergence in the far field is shown in (a)

WINDOWED GREEN FUNCTION METHOD FOR 3D WAVEGUIDES

• Radiation conditions for 3D EM waveguides
• Window function for 3D SIWs
• Beam illumination
• Mode illumination
• Evaluation of incident contributions
• Numerical examples

In order for the system (6.3) to provide a good approximation near the window region, the integrands in these operators must have non-vanishing fluctuations as d(r). A strategy designed to evaluate these right-hand operators despite the possible loss of the oscillatory character of the integrand is presented in the next section. To overcome the difficulty associated with evaluating the terms on the right-hand side of equation (6.10), we use an auxiliary representation for the incident mode.

To do this, we denote by Γw the part of the waveguide for which the window function is greater than zero. In fact, this example is representative of the "launch mode" problem, for which a waveguide is illuminated in order to produce such coupling to the modes propagating in the waveguide. In this mode-launching problem, several modes are excited and the simulation shows "jumping" of trapped fields within the waveguide.

Figure 6.1 (a) presents the error of the solution along the center of the waveguide core; for this particular case, where a discretization of 18 × 18 ( ∼ 9 points per mode’s wavelength) is used, accuracies of order 10 − 4 are achieved by the WGF method

Shape optimization of electromagnetic devices

BIE GRADIENT-BASED OPTIMIZATION OF ELECTROMAGNETIC DEVICES

• Device optimization in a boundary integral setting
• Adjoint computation of the gradient
• Numerical approximation of directional derivatives
• Adjoint operators
• Numerical examples: Metasurface design

Consider the inner product defined by Φ,Ψ. 7.7) Furthermore, we aim to maximize the objective functions of the form. With this notation, we can write the directional derivative of the reduced cost functional atα in the β direction. On the other hand, the second term can be approximated by certain estimates of the forward map—ie.

In the context of our integral equation problem, the directional derivatives in equation (7.24) depend nonlinearly on the explicit parametrization of the dielectric barriers, which is defined by the parameter vector α. Instead, to solve the adjoint problem, we follow the same approach as in the case of the direct problem. In order to do this, we then need to find a way to calculate the operation of the associative operator.

Figure 7.1: Metasurface consisting of an array of 10 × 10 × 10 nanoposts. The design is optimized to focus light at a point in a given focal plane.

Concluding remarks

CONCLUSIONS AND FUTURE WORK

Future work

The orthogonal-polar method for singular integrals presented in Chapter 4 is the basis for most of the work presented in this thesis. The numerical examples presented in the thesis show the enabling nature of the proposed approaches. A natural continuation of the work presented in this thesis concerns the application of the three-dimensional WGF method in Chapter 6 in conjunction with the BIE optimization framework developed in Chapter 7 for the design and optimization of fully three-dimensional waveguide structures.

Future work 121 For reference in this context we present Figures 8.1 to 8.3 from [74], which demonstrate the nature of applied problems that can be addressed with these methods. In this case, a toothed structure is illuminated by a beam, and the sizes of the "teeth" of the grating are designed to combine as much energy as possible in the output mode. Figures 8.2 and 8.3 present the results of the optimization runs obtained for the mode split and mesh coupler, respectively.

Figure 8.1: Illustrations of the two-dimensional mode splitter (a) and grating cou- cou-pler (b) problems [74].

Back matter

BIBLIOGRAPHY

A fast high-order algorithm for solving surface scattering problems: basic implementation, tests and applications. A high-order integral solver for scalar diffraction problems through screens and apertures in three-dimensional space. Windowed Green function method for the Helmholtz equation in the presence of multilayered media. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences June 2017.

Three-dimensionally shifted quasi-periodic Green's function across the spectrum, including wood anomalies. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences November 2017. Fast high-order numerical methods for distribution problems by surfaces and periodic groups of particles - including wood anomalies. A dynamical theory of the electromagnetic field. Philosophical Transactions of the Royal Society of London January 1865.