In a scattering problem, an incident wave impinges on an obstacle resulting in a scattered field. Optimal design in this context describes the search for a scatterer that is best suited to produce a desired effect when interacting with the incident wave. Both the material and the shape of the scatterer may contribute.
Usually, an object is called chiral if it cannot be superimposed onto its mirror image. In the context of a scattering problem involving an electromagnetic wave, the term must be viewed more generally due to the complex interaction of wave field and scatterer. Helicity is a property of electromagnetic fields that extends the concept of circular polarization handedness from individual plane waves to general solutions of the homogeneous Maxwell system. The concept of electromagnetic chirality (em-chirality) has recently been introduced to characterize interactions of scattering objects or media with electromagnetic waves of positive and negative helicity. In contrast to the traditional binary notion of geometric chirality, em-chirality can be quantified directly in terms of the object’s interaction with electromagnetic waves using em-chirality measures.
Helices are typical examples of geometrically chiral objects:
They cannot be superimposed on their mirror images through
rotations and translations.Both incident and scattered field can be decomposed into
components of one helicity.
Consequently, such measures may be used to rank different designs with respect to their em-chirality. Those objects that are extremal in this ordering, are called maximally em-chiral and have the desirable property of being invisible to fields of one helicity. This gives rise to a range of applications in optical sciences and, potentially, chemistry. The goal of CRC Project C5 is to design such maximally em-chiral objects, and to further understand and develop the concept of em-chirality.
Mathematical background
In this section we will provide some background on the mathematical formulation of scattering problems. In particular, we will describe how the em-chirality of a scatterer can be defined and how measures of em-chirality can be constructed. Continue reading.Collapse content.
Our standard setting is the propagation of time-harmonic electromagnetic waves at a single frequency $\omega$ in a homogeneous isotropic medium described by the electric permeability $\varepsilon_0$ and the magnetic permeability $\mu_0$. Thus, the incident field, consisting of the electric field $\boldsymbol{E}$ and the magnetic field $\boldsymbol{H}$, is a solution of the Maxwell system \begin{equation*} \text{curl} \boldsymbol{E} - \text{i} \omega \mu_0 \boldsymbol{H} = 0 \, , \quad \text{curl} \boldsymbol{H} + \text{i} \omega \varepsilon_0 \, \boldsymbol{E} = 0 \, . \end{equation*} A particularly simple solution is a plane wave. More complex fields that are formed by linear superpositions of such plane waves, are called Herglotz wave pairs. We typically consider these as incident fields.
We are considering scattering problems: The incident Herglotz wave pair impinges on some obstacle and is scattered. The exact physical nature of the scatterer is not important here: it may be penetrable, or impenetrable with some boundary condition satisfied by the total field. The interaction of scatterer and obstacle gives rise to the scattered field which also is a solution to the Maxwell system outside of the scatterer. Moreover, it satisfies the Silver-Müller radiation condition. This condition implies an asymptotic behaviour like an outgoing spherical wave at large distances from the scatterer with an amplitude dependent on the direction of observation. This amplitude is called the far field pattern.
The map of the amplitude density of the incident Herglotz wave pair to the far field pattern of the scattered field can be viewed as the full mathematical description of the scattering process. It is called the far field map. An alternative description works with expansions in vector spherical harmonics: By expanding both incident and scattered fields in bases of such functions, one may consider the $\boldsymbol{T}$-matrix that maps the coefficients of the incident field onto those of the scattered field. Both far field operator and $\boldsymbol{T}$-matrix may be viewed as realizations of the scattering operator that abstractly maps incident to scattered field.
Whatever the formulation, the incident and the scattered field each may be decomposed into fields of just one helicity. Thus, we obtain a decomposition of the far field operator into four components that each map incident fields of just one helicity to scattered fields of just one helicity. A scatterer is em-achiral if the components of the operator are related to the ones obtained by inverting the helicities of both incident and scattered field by certain unitary relations, and em-chiral if this is not the case. In particular, zero em-chirality implies that the singular values of these operators coincide. A scalar measure may be constructed by forming some norm of the difference of these sequences of singular values.
Below, we list the results that have been accomplished in this project.
Numerical design and experimental characterization of metallic helices optimized for large em-chirality
Helices made of noble metals are prime examples of scatterers with high em-chirality at the microwave frequency range. By carrying out extensive simulations of electromagnetic scattering problems, parameters such as wire thickness, coil diameter, pitch and number of revolutions, may be chosen optimally with respect to em-chirality up to the optical range. Continue reading.Collapse content.
A shape optimization has been carried out to design silver helicies with large em-chirality at many discrete frequencies ranging from the optical band to the far infrared. In simulations, the designs found achieve more than 90 percent of the upper bound of em-chirality when the wavelength is larger than or equal to $3\,\mu$m, while performance of the helices decreases toward the optical band.
Together with the group of P. Fischer (U. Stuttgart) we fabricated and characterized helices for operation at 800nm and identified some of the imperfections that affect the performance [Gar22]. In particular, and due to the small features of the target helix, the shape of the fabricated helices was rather different from the theoretical target. Our work clearly indicates that both theoretical designs and, potentially more critically, fabrication techniques, must be improved for fabricating maximally em-chiral objects for operation in the optical band.
Figure reproduced from [Gar22]. (a) Fabrication of optimized helices for operation at 800nm. (b) Theoretical optimized helix at 800nm.
(c) Model of actual helices generated through fabrication.
Maximizing the em-chirality of thin free-form nanowires using an asymptotic wire model
To overcome the decrease of performance of silver helices towards the optical frequency band, we went beyond helical shapes for the scatterers and developed a shape optimization scheme to design highly em-chiral thin dielectric and metallic free-form nanowires at optical frequencies. Here, each evaluation of the smooth objective functional and of its shape derivative in a gradient based shape optimization requires the evaluation of the far field operator and of its shape derivative at each iteration step. This is very expensive computationally.
Optimization of a twisted gold nanowire with elliptical cross section with respect to the value of a
measure of em-chirality $J_{\text{HS}}$. The value of a different measure $J_2$ is also given.
The maximum value of both measures is $1$.
To reduce the computational complexity, we used in [FGKR23, AGK21] an asymptotic wire model [CGK21] for thin dielectric wires with arbitrary cross-sections that may also twist along the wire. The shape, the relative electric permittivity, and the relative magnetic permeability of the wire enter this formula by means of the spine curve of the wire and two electric and magnetic polarization tensors.
Based on this asymptotic model, we have developed a quasi-Newton optimization scheme to determine the shape of thin dielectric wires with circular cross-sections [AGK21] with large measures of em-chirality. Not a single Maxwell system has to be solved during the whole optimization process. Our numerical results suggest that thin helical structures are at least locally optimal among this class of scattering objects. However, the optimized dielectric nanowires do not possess extremely high values of em-chirality at optical frequencies.
To improve the designs, we considered nanowires with arbitrary cross-sections that may also twist along the nanowire in [CGK21]. The asymptotic wire model was extended from dielectric materials to noble metals, such as gold or silver. The distinguishing feature of these materials is the negative real part of their electric permittivity at optical frequencies combined with a small positive imaginary part. This permits the excitation of plasmonic resonances, which are not observed for dielectric nanowires, and that turn out to be an important ingredient in the design of thin highly chiral nanowires.
Numerical experiments for silver and gold nanowires with elliptical cross-sections gave very large chirality values at frequencies close to the plasmonic resonance frequency of the nanowire. We constructed highly em-chiral nanowires for discrete frequencies across the whole optical band. In addition to the shape of the spine curve of the nanowire, the twist rate of the elliptical cross-section of the nanowire along the spine curve turned out to have a considerable effect, and thus has to be optimized.
Animation of the optimization prozess to obtain a nanowire
optimized with respect to a measure of em-chirality.
Optimal design of maximally em-chiral objects without simplifying assumptions
Good candidates for general scatterers with high em-chirality are objects that in some sense generalize the nanowires that can be treated using the asymptotic representation formula. Hence, we focus on objects with a circular cross section following a spine curve, but with variable thickness. This can be done by carrying out shape optimization based on the full scattering model and using the boundary element method as the numerical solver. Continue reading.Collapse content.
Shape derivatives were implemented and used for solving an inverse scattering problem already in [HABH19], but only for star-shaped domains. For each evaluation of the domain derivative, the approach requires the solution of multiple exterior boundary value problems in every iteration step. These, we carry out using the boundary element method. The implementation of this approach for the general class of objects described above is currently in progress. One necessary tool are formulations of electromagnetic scattering problems and corresponding shape derivatives that lead to robust integral equation formulations for wide range of material parameters. [AHA19].
In further related work, characterizations of second order domain derivatives for electromagnetic scattering problems were derived [HH20]. These can be used for second degree iterative methods which show better convergence properties on initial iterations. Also, domain derivatives for scattering problems for objects with impedance boundary conditions were investigated [Hag19].
Extending the concept and theoretical foundation of em-chirality: Chirality signatures
It is a well known phenomenon that it is possible to continuously transform a geometrically chiral object into its mirror image without ever attaining an achiral configuration. As a consequence, many continuous measures for distinguishing between chiral objects and their mirror images suffer from chiral zeros, where the measure becomes zero for a chiral object, and/or from unhanded states, where two physically different objects produce the same value of the measure. Our research has provided possible solutions for this long standing problem. Continue reading.Collapse content.
A complex extension of em-chirality that solves the problem of false chiral zeros is introduced in [VFC22]. The em-chirality and its complex extension are also shown to have the largest possible invariance in electromagnetism: conformal invariance. Additionally, [VFC22] introduces a conformally invariant multidimensional pseudovector, called chirality signature, which for the first time allows to continuously distinguish between any mirror pair of objects.
Moreover, an explicit multi-frequency formulation of the far field operator was also introduced in [VFC22], which allows us to treat polychromatic illuminations and frequency-changing transformations such as Lorentz boosts and conformal transformations.
Figure from [VFC22]. The sequence of transformations change the initial configuration onto its mirror image without ever becoming achiral.
Such a phenomenon, called chiral connectedness, causes severe difficulties in chirality quantification.
The work in [VFC22] solves such difficulties in the context of light-matter interaction.
Helicity in matter
Chirality as a physical quantity can be seen as a property of a scattering problem for electro- magnetic waves. Hence, it is an important question, if this quantity remains invariant under transformations that leave the underlying Maxwell system invariant. While it was known for the Poincaré group that this is indeed the case, we addressed the conformal group within this project. The consideration of the conformal group unexpectedly led to establishing the theoretical grounds for studying the permanent transfer of helicity between dynamic electromagnetic fields and magnetic materials hosting chiral magnetization textures. Continue reading.Collapse content.
The conformal group is the largest group of space-time transformations that leaves the Maxwell system invariant. A conformally invariant derivation of the integrated optical helicity expression was reported in [Fer19]. The new expression has been one of the keys for obtaining a generalization of helicity that includes matter [Fer21], and establishes a new link between optics and magnetism.
Reproduced from [Fer22] with permission. Exchange of electromagnetic helicity and material chirality.
(a) The absorption of enough helicity coming from a beam of light can switch the handedness of
chiral magnetization textures. (b) The loss of chirality in the magnetization produces a beam of
light that contains non-zero helicity.
M. Vavilin. Polychromatic T-Matrix: Group-theoretical derivation and applications. PhD thesis, Karlsruhe Institute of Technology (KIT), February2024.[bibtex]
