Electromagnetic field expansion in a Wilson basis
Citation for published version (APA):Floris, S. J., & Hon, de, B. P. (2012). Electromagnetic field expansion in a Wilson basis. In Proceedings of the 42nd European Microwave Conference (EuMC), October 29-November 1 2012, Amsterdam (NL) (pp. 88-91). Institute of Electrical and Electronics Engineers.
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Electromagnetic field expansion in a Wilson basis
Abstract—A Wilson basis is an ingenious modification of a Gabor frame, with basis functions that have a notionally compact support in phase space. We shall give a short description of an algorithm for the construction of a Wilson basis. Through spatial scaling of the Wilson basis relative to a higher-order LP-mode, the basis may appear to vary from effectively local to effectively global. For mode-matching purposes local is advantageous. How-ever, the field expansion requires fewer coefficients in the more global basis.
Index Terms—computational electromagnetics, Wilson bases.
I. INTRODUCTION
I
N computational physics, many solution strategies for the calculation of a physical field involve the expansion of that field in a basis that spans the appropriate linear space.Further, a priori truncation of the number of basis functions is usually inevitable, and hence it is desirable that the basis functions resemble possible field solutions. For example, if the physical problem is a small perturbation of a field problem for which the basis functions are solutions, then the resulting truncated system of equations for the actual field will be small, even if high accuracy is required. However, such global (entire-domain) bases are inherently specific, and the construction may be difficult and computationally expensive in itself.
A local basis is often relatively simple to construct, and much more flexible, especially if it is a conformal mesh that closely follows the contours or surfaces in the actual configuration. Depending on the problem, it may be prudent to employ higher-order basis functions to improve convergence upon refining the mesh. However, the resulting system of equations is often very large, albeit sparse.
The eigencurrent approach to the linear embedding via Green’s operators technique [1] comprises a hybrid global-local basis set-up, in which a large electromagnetic computa-tional domain is divided into smaller domains called bricks, which interact through equivalent sources on their bound-aries. Initially, these equivalent sources are expanded using local Rao-Wilton-Glisson (RWG) functions. Subsequently, the eigenfunctions for an isolated brick are constructed as linear combinations of these RWG functions. The rationale is that many of the equivalent-source eigenfunctions do not radiate, rendering the overall linear system of equations quite sparse. Beam-based phase-space source representations [2] also combine advantages of the local and global formulations, and have additional favourable traits, especially in that they offer the flexibility and physical transparency of geometrical optics, while retaining the uniform properties germane to spec-tral techniques. Such phase-space representations have been rigorously formalized in the theory of frames, in particular Gabor frames [3]. Furthermore, the use of iso-diffracting
Gaussian windows [4] turns out to produce the snuggest frame representations for all frequencies of interest [3]. Gabor frames have also been successfully employed as expansion and test functions in the method of moments [5].
Unless Gabor frames are tight, they comprise an overcom-plete family of expansion functions. A tight Gabor frame is not redundant, and hence it constitutes a basis. However, the Balian-Low theorem [8] states that tight Gabor frames either have infinitely long tails in the spatial or in the spectral do-main. Further, tight Gabor frames lead to numerically unstable expansions [8], which is unacceptable.
A Wilson basis is an ingenious modification of a Gabor frame such that it loses its redundancy and hence becomes a basis, without sacrificing stability. The spectral localization becomes duo-localization in that positive and negative spatial frequencies are combined. The Wilson basis functions are composed of the same analytical constituents as the Gabor frames. Hence, a method of moments implementation is still feasible. So far, the use of Wilson bases in electromagnetics has only been advocated by Arnold [6]. We shall provide an accessible introduction into Wilson bases, and shall explore their use in modal field expansions for optical fibers.
II. WILSONBASES
High-frequency electromagnetic fields may often be char-acterized by rapidly varying phases and slowly varying am-plitudes. In [6], it is demonstrated that such fields experience strong localization in phase-space. Phase-space in this regard is the space spanned by position and spatial frequency. A popular choice for describing wavefields in phase-space is the windowed spatial Fourier transform, with windowing function g(x) as well as its Fourier transform ˆg(ξ) both having localized support. Now, suppose that both g and ˆg are centered about zero. Then, by discretizing the windowed Fourier transform in the following form [7]
gmn(x) = e2πiαmxg(x − βn), m, n ∈ Z, α, β ∈ R, (1)
the window can be moved in phase-space to an arbitrary coordinate on a grid spanned by (αm, βn) [8]. We would like to expand any function f ∈ L2
(R) in coefficients cmn, such
that
f =X
m,n
cmn(f )gmn, (2)
which is only possible if the functions gmn span the space,
or equivalently, αβ ≤ 1. This implies that the set of functions gmn constitute a frame [7]. A frame spans a vector-space V ,
but unlike a basis for V , the frame vectors may be dependent. A frame can be thought of as a basis possibly supplied with additional elements [9]. By definition, a set of functions gmn
S.J. Floris
TE Connectivity, ’s-Hertogenbosch, The Netherlands Sander.Floris@TE.com
B.P. de Hon
Eindhoven University of Technology, The Netherlands B.P.d.Hon@TUe.nl
constitutes a frame if there are nonvanishing finite frame bounds A and B, such that for all functions f ∈ L2
(R), Akf k2≤X m,n |hgmn, f i| 2 ≤ Bkf k2, (3)
where the inner-products are linear in the second argument. If αβ = 1, the frame would comprise an orthonormal basis [7]. To eliminate redundancy, we would like to select αβ = 1. However, this has adverse consequences for the localization, since the Balian-Low theorem states that in that case either the function g(x) or the function ˆg(ξ) has an infinitely long tail, i.e., eitherR x2|g(x)|2dx = ∞ orR ξ2|ˆg(ξ)|2dξ = ∞.
As an example of windowing functions that are used in practice, we mention the Gabor expansions. In a Gabor ex-pansion, the windowing function g(x) in Eq. (1) is chosen to be Gaussian, which has good phase-space localization if αβ is well below one. However, the functions gmn are
highly redundant when αβ is small. To avoid redundancy, one may choose to use an orthonormal basis by choosing αβ = 1. Unfortunately, in addition to the loss of phase-space localization, that scheme is numerically unstable [8].
To achieve an orthonormal basis with strong localization in both domains, somehow the Balian-Low theorem has to be circumvented. In [10], Wilson proposes to construct a basis with exponential decay that is similar to Eq. (1), i.e., a basis with basis functions of the form
ψmn(x) = fm(x − n), m ∈ N\{0}, n ∈ Z. (4)
However, he introduces two peaks in ˆfm(ξ). This amounts to
an effective duo-localization, for which the Balian-Low theo-rem does not apply. In [8], the Fourier transformed expression for fm is chosen to be dependent on only one real-valued
function φ, and is of the form ˆ f1(ξ) = φ(ξ) ˆ f2l+κ(ξ) = 1 √ 2φ(ξ − l) + (−1) l+κφ(ξ + l) eiπκξ, (5)
where l ∈ N\{0}, κ ∈ {0, 1}. The derivation of the function φ
is given in the next section.
The given set of functions is a frame, but not yet an orthonormal basis. By cleverly combining pairs of functions, the redundancy is removed and the set of orthonormal basis functions remains, which reads [8],
wln(x) = ˆ φ(x − n), l = 0, √ 2 ˆφ(x − n/2) cos(2πlx), l + n even, √ 2 ˆφ(x − n/2) sin(2πlx), l + n odd. (6)
Indeed, the inner-product of any two basis functions yields
hwln(x), wl0n0(x)i =
∞
Z
−∞
wln(x)wl0n0(x)dx = δll0δnn0. (7)
The appeal of working with the Wilson basis is that the pertaining functions are exponentially decaying functions with strong localized support in both domains, and that the basis functions in Eq. (6) constitute an orthonormal basis which allows for an optimum sampling of phase-space.
In [6], it is demonstrated that an expansion in a Wilson basis is particularly useful for wavefunctions of geometrical optics type, that have strong localization in phase-space. The propagation of such a field is accomplished by the forward propagation of the Gaussian window constituents in Eq. (1). The forward propagated constituents remain orthogonal, be-cause [6] wln(x, 0) → wln(x, z), ˆ wln(ξ, 0) → ˆwln(ξ, z) = ˆwln(ξ, 0)ei2πz √ k2−ξ2 , (8)
is again a Wilson basis. We will not use this aspect of the Wilson basis in this paper. Instead, we exploit the strong localization and orthonormality of the basis functions, to expand a high-frequency field in terms of coefficients.
III. DERIVATION OF THE FUNCTIONφ
In this section, we give a streamlined summary of the work done by Daubechies et al. [8]. One may choose as a starting point, a function g(x), such that gmn in Eq. (1) with α = 0.5
and β = 1 constitutes a frame. Then, with aid of the Zak transform, the function φ can be constructed by applying an operator P on g, such that
φ =√2P−1/2g, (9) where
P =X
m,n
Pmn, Pmng = hgmn, gi gmn. (10)
In order to obtain gmn such that it constitutes a frame, we
require that the bounds are such as in Eq. (3), i.e.,
AI ≤ P ≤ BI, 0 < A ≤ B < ∞, (11) where I is the identity operator. As a first estimated for P, we choose Pest= A+B2 I, and as such, we find that the norm
I − 2P A + B = B − A A + B < 1, (12) is bounded, so that we know that the operator P−1/2exists and is bounded as well [8]. The approach we adopt to determine P is to rewrite P−1/2 as P−1/2= r 2 A + B I − I − 2P A + B −1/2 . (13) In view of Eq. (12), the Taylor expansion of Eq. (13) converges and in combination with Eq. (9) leads to
φ =√2P−1/2g = √ 2 A + B ∞ X k=0 (2k)! 22k(k!)2 I − 2P A + B k g. (14)
In this expression, we already recognize a recurrence relation in the repeated application of the operators I and P. The first fraction in the sum can be calculated efficiently using the hypergeometric recurrence relation
f [k] = (2k)! 22k(k!)2 =
2k − 1
2k f [k − 1], f [0] = 1. (15)
Upon substituting Eq. (10) into Eq. (14), we obtain a numerical scheme to evaluate the operators for a windowing function g,
φ = √ 2 A + B ∞ X k=0 f [k]X m,n gmnbkmn, (16) where bk
mn satisfies the following recurrence relation
bkmn= b k−1 mn − 2 A + B X m0,n0 wmn;m0n0bk−1m0n0, (17)
where wmn;m0n0 = hgmn, gm0n0i. Note that Eq. (16) evaluated
for k = 0 requires that b0
mn= δm0δn0.
We choose α = 0.5 and β = 1 and choose a Gaussian window described by
g(x) = (2ν)1/4e−νπx2= e−π2x 2
, for ν = 1/2. (18) For ν = 1/2, the inner-product evaluated for this Gaussian function gives wmn;m0n0 = exp h i(m0− m)(n + n0)π 2 i × exph−π 4(n − n 0)2−π 4(m − m 0)2i. (19)
To determine the coefficients A and B, we make use of the Zak transform UZ, which is defined in [8] as
(UZg)(t, s) =
√ 2X
l∈Z
e2πitlg (2(s − l)) . (20)
The frame bounds A and B are then found by calculating A = inf s,t∈[0,1]|(UZ g)(t, s)|2+ |(UZg)(t, s + 1/2)|2 , B = sup s,t∈[0,1] |(UZg)(t, s)|2+ |(UZg)(t, s + 1/2)|2 . (21)
In view of the exponential decay of the window g, and the small domain of s and t, the infinite sum in Eq. (20) can be reduced to incorporate only a few terms around l = 0. For our choice of the windowing function, we obtain A = 1.6693 and B = 2.3607. To find the function φ, we bounded the sum over m and n to ±31. This is justified, because the function w has Gaussian decrease in m and n as seen in Eq. (19). To avoid having to evaluate the entire program for every value of x, we changed the order of summation as expressed in Eqs. (14) and (16), so that φ(x) = √ 2 A + B X m,n amngmn(x), (22) where amn=Pkf [k]b k
mn needs to be computed only once.
The sum over k is evaluated until then norm of a deviates less then 0.05% with respect to the norm in the previous iteration. The function φ(x) as well as its Fourier transform
ˆ
φ(ξ) are depicted in Fig. 1. The function ˆφ(ξ) can be found in a similar way as expressed in Eq. (22) upon replacing gmn(x)
by gmn(ξ) = (−1)mn21/2expi2πnξ − 2π(ξ + m/2)2.
These graphs for ν = 1/2 are different from those presented by Daubechies et al. in [8], although the algorithm and the coefficients for A and B are the same. We have strong reasons to suspect that the graphs presented in [8] are wrong [11].
−4 −3 −2 −1 0 1 2 3 4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x φ ( x ) −4 −3 −2 −1 0 1 2 3 4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ξ ˆ φ(ξ )
Fig. 1. The top figure shows the function φ(x). The lower figure shows ˆ
φ(ξ), the Fourier transform of φ(x).
IV. FIELD EXPANSION IN AWILSON BASIS
With the basis functions in Eq. (6) readily evaluated, any modal field distribution f (x, y) can be expanded according to
f (x, y) =X
l,n
X
l0,n0
Clnl0n0wlnl0n0(x, y), (23)
where wlnl0n0(x, y) = wln(x)wl0n0(y). By the orthonormality
of the basis functions, the corresponding Wilson basis coeffi-cients are determined uniquely by
Clnl0n0 = ∞ Z −∞ ∞ Z −∞ f (x, y)wlnl0n0(x, y) dxdy. (24)
We introduce a scaling parameter d, so that the width of the basis functions can be reduced to the same order of magnitude as the modal fields. To preserve the orthonomality, the amplitudes of the Wilson functions are scaled as well, i.e., wln(x) = d−1/2wln(x/d).
By the orthonormality of the basis functions, the inner-product of any two real-valued functions f1(x, y) and f2(x, y)
is then computed in the Wilson basis by hf1, f2i =
X
l,n,l0,n0
(C1)lnl0n0(C2)∗lnl0n0. (25)
We will make use of this property in the next section to demonstrate the density in Wilson coefficients (phase-space) to describe a higher-order electromagnetic field.
V. EXAMPLES
In Fig 2, we have depicted a Gauss-Laguerre modal LP6,7
field ψ that is associated with a optical waveguide with a parabolic refractive index profile. The amplitude is chosen such that the field carries unit power. We have expanded ψ in terms of coefficients C according to Eq. (24). Via Eq. (25), we have verified that the field indeed carries unit power, i.e. hψ, ψi = P
l,n,l0,n0Clnl2 0n0 = 1. The real coefficients Clnl2 0n0
describe a confined energy density distribution in phase-space, spanned by Wilson basis functions that are themselves strongly localized in phase-space.
Fig. 2. The Gauss-Laguerre modal LP6,7field ψ.
Fig. 3. Example of a basis function w1,0,2,0.
To give an example of a higher-order Wilson basis function, we have plotted the function w1,0,2,0 in Fig. 3. We have
eval-uated the expansion of ψ for three distinct scaling parameters d. To demonstrate the confinement of the energy density, we focus on the distribution in the x-direction by introducing a summation over all coefficients that are associated with the y-coordinate, i.e. Dln=Pl0n0Clnl2 0n0. In Fig. 4, it is clear that
the scaling parameter d affects the density distribution in the Wilson basis indexed by n and l. By increasing d, the Wilson
−20 −15 −10 −5 0 5 10 15 20 10−6 10−4 10−2 100 n d= 3 µm −20 −15 −10 −5 0 5 10 15 20 10−6 10−4 10−2 100 n d= 6 µm −20 −15 −10 −5 0 5 10 15 20 10−6 10−4 10−2 100 n d= 10 µm l= 0 l= 1 l= 2 l= 3 l= 4 l= 5
Fig. 4. The energy density Dlnfor d = 3 µm, d = 6 µm and d = 10 µm.
basis used for expanding the LP6,7mode changes in character
from local to global. As such, fewer spatial and more
higher-order basis functions are required to accurately describe the field ψ. To gain insight in the required number of coefficients, we have shown the cumulative distributions for the three energy density distributions in Fig. 5. For d = {3, 6, 10} µm, one requires 839, 354 and 216 coefficients, or 1338, 528 and 340 coefficients to capture 95% or 99% of the power. For
0 200 400 600 800 1000 1200 1400 70 75 80 85 90 95 100 Number of coefficients Normalized power [%] d=3 µm d=6 µm d=10 µm 95% and 99%
Fig. 5. The accumulated power versus the required number of coefficients.
mode-matching purposes, it is advantageous to use spatially local bases, whereas from Fig. 5 we conclude that using a more global basis reduces the required number of coefficients.
VI. CONCLUSIONS
We have provided a streamlined construction of Wilson bases. By means of an example, we have demonstrated that by spatial scaling of a Wilson basis relative to a higher-order LP-mode, that basis may appear to vary from effectively local to effectively global. One may choose to minimize the required number of coefficients by selecting the effectively global basis. On the other hand, one may benefit from a more local basis, e.g., for mode-matching purposes.
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