Mind the gap between theory and experiment Andrei Kiselev Jeonghyeon Kim and Olivier J. F. Martin Abstract After briefly introducing the surface integral equation method for the nu-

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Mind the gap between theory and experiment

Andrei Kiselev, Jeonghyeon Kim, and Olivier J. F. Martin

Abstract After briefly introducing the surface integral equation method for the nu-

merical solution of Maxwell’s equations, we discuss some examples where numerical

simulations based on effectively fabricated nanostructures can provide additional in-

sights into an experiment. Focusing on plasmonics, we study Fano resonant systems

for optical trapping, realistic dipole antennas for near-field enhancement, and hy-

brid nanostructures that combine plasmonic metals with dielectrics refractive index

sensing. For those systems, the same experimental detail can play a very different

role, depending on the type of physical observable. For example, roughness can

significantly influence the near-field, but be totally unnoticed in the far-field. It can

affect molecules adsorbed on the surface, while refractive index sensing can be fully

immune to such roughness. Approaching the experimental situation as closely as

possible is certainly a challenging task and we demonstrate a simple approach based

on SEM images for that. Altogether, bridging the gap between theory and experiment

is not such a trivial task. However, some of the simple steps illustrated in this chapter

can help build numerical models that match the experiment better.

Introduction

We did not have the pleasure to meet Werner S. Weiglhofer and only know of him

through his scientific publications. In spite of his too short career, they are extremely

numerous, diverse and impactful. Following the Web of Science categories, one

notices that these contributions do not only cover optics and electromagnetics, but

also reach out to applied physics, materials sciences and – of course – mathematics.

They are very well cited; his works on demystified negative index of refraction [1],

and that on light-propagation in helicoidal bianisotropic media [2], at the top of

his list of citations. Working at the Department of Mathematics of the University

Nanophotonics and Metrology Laboratory, Swiss Federal Institute of Technology Lausanne (EPFL),

EPFL-STI-NAM, Station 11, CH-1015 Lausanne, e-mail: olivier.martin@epfl.ch

2 Andrei Kiselev, Jeonghyeon Kim, and Olivier J. F. Martin

of Glasgow, it is not surprising that Werner’s publications have a strong theoretical

flavour and have inspired many theoretical works. Yet, analysing their citations

further indicates that these theoretical developments inspired numerous experimental

projects. As an example, among the citations of his work on light-propagation in

helicoidal bianisotropic media [2], a third of the citing articles report experiments.

This illustrates how well Werner succeeded in bridging the gap between theory and

experiments! Obviously, theory is very important and progress within the realm of

theoretical physics is often fascinating in itself. Evidently, new theories are often the

driver behind new experimental work. This chapter, however, focuses on the inverse

process, where experimental work requires numerical support as close as possible to

the experimental situation. After briefly presenting the numerical technique we have

developed for over a decade to solve Maxwell’s equations, we discuss three different

experimental situations where we attempted to model the real experiment as closely

as possible.

Computational electromagnetics

At the onset of studying a given experimental situation lies the fundamental question

of the choice of the most appropriate numerical method. It is fair to say that there is

not one single numerical technique that is fit for all situations and even for the narrow

field of plasmonics, which is the focus of our work, numerous approaches exist as

illustrated in a recent review article [3]. Furthermore, each numerical method can

be put to good use as long as it is utilized wisely and carefully. Especially, sufficient

efforts must be undertaken to characterize the algorithm beforehand, to make sure

that it will converge well for the problem at hand and is free from spurious behaviours.

This task is especially thankless, but of paramount importance if the numerical results

are to be trusted. Note that it does not only apply to home-developed numerical codes,

but should be equally undertaken with commercial packages that should never be

trusted blindly, even if they produce beautiful and colourful images!

To assess the accuracy of a numerical technique and obtain a metric to quantify it,

one usually resorts to canonical problems. Unfortunately, there are essentially only

two such problems for which a reference solution exists (the quasi analytical Mie

solution): the scattering by a sphere for three-dimensional (3D) problems or by a

cylinder for two-dimensional (2D) geometries [5]. Figure 1 illustrates this approach

for a 2D solution obtained with a volumetric Green’s tensor approach [4]. In this

case, two different incident polarizations must be considered, with the electric field

either perpendicular to the cylinder axis (transverse electric or TE field) or parallel

to the cylinder axis (transverse magnetic or TM field). The differential cross section

can be computed as a function of the scattering angle and compared with the quasi

analytical Mie solution, Fig. 1(a). This panel indicates that many features exist in

that response, which need to be reproduced accurately with the numerical method. A

more quantitative metric is obtained by integrating the difference between this cross

section and the Mie solution over all scattering angles and repeating the calculations

Mind the gap between theory and experiment 3

Mie (TM)

Numerical result (TM)

Mie (TE)

Numerical result (TE)

Error

with an increasing number of discretized elements, Fig. 1(b). In principle, the error

should decrease as the number of elements increases. However, this behaviour is

far from monotonous since it includes different facets of the numerical problem:

on the one side, a finer mesh approximates the scatterer better and should provide

a more accurate solution; on the other hand, it requires a larger numerical matrix

to be solved, which is more difficult, especially when the matrix condition number

increases, as is the case here [6,7]. Consequently, plateaus appear in the convergence

curve, Fig. 1(b). We also notice that the polarization influences the solution accuracy,

reminiscent that in electromagnetics all field components do not behave in the same

way: some are continuous across materials’ boundaries, others are not [8].

Experimental situations are usually much more complicated than a sphere or a

cylinder and we will show in Sec. 3.2 that it is possible to use reciprocity to assess

the accuracy of numerical results produced for complex geometries.

In this chapter, we focus on the surface integral equation (SIE) method for the

numerical solution of Maxwell’s equations. An interesting feature of such a formu-

lation is that the boundary conditions at the edge of the computation window are

included in the equations and need not be taken care of by using ad hoc prescriptions,

such as absorbing boundary conditions or perfectly matched layers [9]. Indeed, these

boundary conditions are already included in the kernel of the integral equation, and

can take different forms, like infinite homogeneous space [10], surfaces or stratified

media [11, 12], or waveguide cavities [13]. There is of course a price to pay for this:

except for infinite homogeneous space where the kernel is known analytically [10],

(a) (b)

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

−6

0.6

0.4

0.2

0.0

120

180

Scattering angle [degree]

100

1000

Number of elements

Fig. 1 Measuring the numerical solution accuracy for light scattering by an infinite cylinder in

vacuum illuminated by a plane wave normal to the cylinder. Two polarizations are considered:

transverse electric (TE) with the electric field normal to the cylinder axis and transverse magnetic

(TM) with the electric field along the cylinder axis. (a) Comparison between the numerical solution

and the reference Mie solution for a dielectric cylinder with relative permittivity  = 4 and a size

parameter  = √   = 10.43, where  is the cylinder diameter,  its relative permittivity and

 the wavelength in vacuum. (b) Relative error between the Mie and numerical solutions (defined

as the square of the difference between the numerical and Mie far-field amplitudes, normalized to

the square of the analytical amplitude) as a function of the number of discretized elements for two

different materials , for TE (solid lines) and TM (dashed lines) polarizations. Adapted from Ref.

4 with permission, copyright IEEE 2000.

=λ6 64+0 23

=4 0

Differential scattering cross section

−

4 Andrei Kiselev, Jeonghyeon Kim, and Olivier J. F. Martin

it must be evaluated numerically, usually by resorting to plane waves or eigenmodes

expansions [11, 14].

The SIE is constructed from the combination of an equation for the electric

field and one for the magnetic field; different weighted combinations can be used

here [15]. The volume integral form of Maxwell’s equations is transformed into a

surface equation using Gauss’ theorem and the solution is computed from unknown

electric and magnetic currents defined only on the surface of the scatterer [16]. This

is very advantageous since only that surface needs to be discretized; on the other

hand, a limitation of this approach is that the resulting matrix is dense since each

mesh is connected to all the other meshes in the system, different, e.g., from the finite

difference time domain method, where only nearest neighbours are connected [17].

The resulting system of linear equations is constructed through a Galerkin procedure,

where Rao-Wilton-Glisson functions are used both as basis and test functions [18].

The accuracy of the method strongly depends on the order used for the quadrature in

the Galerkin scheme [19]. Once the surface currents are known, different observables

can be computed, from the near-field to the different cross-sections [20], or even the

force and torque produced by the incident light on the nanostructure [21, 22].

Approaching experimental situations

Having settled for the numerical technique, we wish to address the question of

modelling the geometry of a real experiment as accurately as possible and will

do that in the context of three different plasmonic systems. This field of research

studies the interaction of light with coinage metals, like gold, silver, aluminium or

heavily doped semiconductors [23]. When light impinges on a nanostructure made

from such a metal, it resonantly excites the free electrons in the metal, producing

a very strong near-field at the vicinity of the nanostructure [24, 25]. It is quite

remarkable that nanostructures much smaller than the wavelength can exhibit such

strong resonances; the reason being the localisation of the free charges in a specific

pattern associated with each optical resonance [26, 27].

3.1

Fano-resonant systems

In principle, any resonant system has an optical response with a Lorentzian shape [5].

This is also true for a plasmonic nanostructure, as long as only one single resonance

is excited like in a small particle or a dipole antenna [28]. On the other hand, as soon

as more than one resonances are present, the lineshape can become very complicated

with several different peaks. A prominent family of such irregular responses is the so-

called Fano lineshape, following the name of Ugo Fano who discovered them while

interpreting atomic spectroscopy experiments [29]. In the context of plasmonics,

Fano resonances occur when two modes are present in the system, often a bright

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MindthegapbetweentheoryandexperimentAndreiKiselev,JeonghyeonKim,andOlivierJ.F.MartinAbstractAfterbrieflyintroducingthesurfaceintegralequationmethodforthenu-mericalsolutionofMaxwell’sequations,wediscusssomeexampleswherenumericalsimulationsbasedoneffectivelyfabricatednanostructurescanprovideadditional...

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