Chapter 1 Introduction
2.4 Convergence Issues of the PWE Method
In the opening section of this chapter, we alluded to some known convergence issues with the PWE method. From working out the Helmholtz equation in the plane wave basis, the connection is clear between the plane wave method and Fourier analysis so it is not surprising that any difficulties in Fourier analysis (see appendix C) will lead to difficulties here. However, as it applies to solving the photonic bands problem, these issues were studied as early as 1992 by S¨oz¨uer, Haus and Inguva [13]. This was in an era when the field of photonic crystals was still in its infancy, and tremendous amount of effort went into accurate calculations of the band structure in search of true band gaps. In an effort to correct a common misconception, they wrote the following:
It is clear that, just because increasing N does not produce visible differ- ences in the resulting band structure, one has not necessarily converged to the ‘true’ values. In this case, it is merely an indication of the slow
convergence of the Fourier series.
Here, N refers to the number of terms in the summation. Unfortunately, most con- vergence analysis on the PWE method fails to address the problem, and even research published over a decade later [14, 15, 16] still fails to grasp the difference between convergence and accuracy. After all, what is the significance of ‘rapid convergence’
of the calculation if it does so to a wrong value? The problem is, of course, the slow convergence of the underlying dielectric function that the finite Fourier series is supposed to model. Additional terms in the series do not change the model enough so it only appears as though the calculation has converged.
In the literature, there was also some discrepancy as to how to treat the η =²−1 term in the Helmholtz equation (eqn. (2.15)). While some have treated the Helmholtz operator as we have, others [11] expand²r(r) in the plane wave basis, and then invert the matrix instead. For an infinite Fourier series, the matrices ηk,k0 and ²k,k0 would be each other’s inverse, but that is no longer true once we truncate. It appeared that the convergence was more rapid (i.e. used a fewer number of plane waves) using the matrix inversion treatment [17], thus justifying the computationally intensive matrix inversion, but bear in mind the caveat about convergence from above. Of course, in the early 1990’s, there were more constraints on CPU and memory resources than there are today. This issue is addressed nicely by Li in 1996 [18], and we will review his work in section C.4. In the end, the point is moot, since Steven Johnson et al. found a way [4] to implement the plane wave method without explicitly forming either matrix, and have since made their software, the MIT Photonic Bands (MPB) package, freely available.
There are three key ideas to their approach. First, rather than solving the eigen- value problem explicitly by diagonalization (computational work required O(N3)), they use an iterative eigensolver. Secondly, they noticed that in the Helmholtz equa- tion, the curl operator is diagonal in Fourier space, while the division by² is diagonal in real space, so they make use of the Fast Fourier Transform (FFT) algorithm to transform to the appropriate basis so they do not actually store the entireN×N op- erator. They were thus able to reduce their storage requirement fromO(N2) toO(N).
This allowed them to use a much higher number of plane waves than otherwise prac- tically achievable. However, representation of discontinuities in a Fourier basis still poses a problem. The final ingredient is an averaging technique that smoothes out the discretized elements that encloses a discontinuity. It makes use of effective medium theory and the grid elements containing the discontinuities are assigned a dielectric tensor instead of a scalar. Therefore, they have replaced the sharp scalar dielectric discontinuity with a smoothed dielectric tensor, making the Fourier representation less objectionable.
We have chosen to highlight some of the key contributions of that work here for two reasons. The first reason is that we were unable to take advantage of their ideas in the inverse problem, so we still face the same convergence issues described in [13].
It should become clear when we formulate the inverse Helmholtz problem in chapter 6 why we could not capitalize on their wisdom. The other reason is that though their work is often cited, there still appears to be confusion about the validity of the PWE method, especially the key components to making the method work. Other more recent articles on PWE [14, 16, 15, 19] either omit the reference entirely, or if it does cite it, the work demonstrates a complete lack of appreciation for the results by Johnson et al.. A recent article (2006) on these ‘fast Fourier factorization methods’
[20] still have not caught on to the fact that their method is at best equivalent if not inferior to the tensorial averaging in the Johnson reference, except that they do not even realize the NlogN scaling, requiring O(N2) storage. For the reader interested in performing these computations, it is important to understand the significance of Johnson’s work and to evaluate other PWE method research in light of their results.
Chapter 3
Inverse Problems
Mathematicians often cannot help but cringe when physicists are doing math, and even more so whenever physicists claim to be ‘rigorous’ with their math. This chapter is not written to satisfy the mathematicians for two reasons. First, I am not a mathematician, so I am quite certain that they will cringe despite my best efforts.
More importantly though, this is meant to be accessible for engineers and physicists, and often what mathematicians consider ‘special cases’ are the only ones we happen to care about. So with apologies to any mathematicians reading this, the goals of this chapter are threefold: First, we want to help the reader develop an appreciation for what inverse problems are and what makes them difficult. Second, we want to introduce the specialized tools that are used to solve these inverse problems. Finally, we bring the focus back to our particular application, and fine tune the ideas developed for the purpose of photonic device design.
There are many excellent references on inverse problems. A standard reference is the textbook by Engl [21] which gives a thorough overview of the subject, but the mathematics is quite formal. A very nice introduction to the subject for physicists can be found in a series of lecture notes by Sze Tan and Colin Fox at the University of Auckland [22]. The work by Per Christian Hansen is more focused on discrete and finite dimensional problems, and hence particularly suitable to our application.
He has also written a package of matlab functions for inverse problems available for download as well [23]. The ideas presented in this chapter are mostly taken from his work, although the discussion of the role of noise in distinguishing between a forward
and inverse problem has to our knowledge not been articulated elsewhere. Arnold Neumaier also provides a concise treatment similar to Hansen’s, but bridges the gap to the infinite dimensional treatment of inverse problems with more mathematical rigor [24].
We begin the chapter by attempting to define what an inverse problem is through some examples of simple physical problems. We introduce the concept of anill-posed problem to distinguish between theforward ordirect problem vs. the inverse problem.
In section 3.2, we restrict our discussion to finite dimensional linear operators, allow- ing us to illustrate the pathologies of inverse problems in the linear algebra formalism.
A numerical example is provided to help illustrate the effects of ill-conditioning. We make use of the singular value decomposition (SVD) to explain why standard tech- niques will fail to solve inverse problems. The SVD also allows us to utilize the condition number as a quantifying metric for how ill-posed a particular problem is.
In section 3.3 we introduce regularization as a tool for solving inverse problems. We conclude with a glimpse of the difficulties we expect to encounter for the purpose of PBG device design.