The electromagnetic field of elementary time-dependent toroidal sources

Afanasiev, G. N.; Stepanovsky, Yu. P.

doi:10.1088/0305-4470/28/16/014

Cited by 115 publications

(162 citation statements)

References 25 publications

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“…Despite being physically distinct from the electric dipole, the toroidal dipole emits radiation with the same angular momentum and parity properties as the former 5,13 , and therefore the two multipoles cannot be distinguished by a distant observer. Nevertheless, the toroidal dipole emission presents a distinct frequency dependence, and in the harmonic case where T(t)∝e iωt , its intensity has an additional scaling factor k 2 = ω 2 /c 2 as compared to the electric dipole 15 :…”

Section: Tutorial Insets Inset I: Toroidal Multipolesmentioning

confidence: 99%

“…Indeed, electromagnetic fields scattered by the toroidal dipole moment of a molecule with a characteristic size R scale with the free-space wavelength λ as ~(R/λ) 3 , while electric and magnetic dipole scattering scales as ~(R/λ) and ~(R/λ) 2 , respectively 13,15,51 . Hence, the toroidal response of a system will become increasingly important in comparison to the standard dipoles, as the size of the system or molecule increases.…”

Section: Dynamic Toroidal Multipolesmentioning

confidence: 99%

“…Moreover, the vector potential, A T (r,t), of the toroidal emission differs from the vector potential, A p (r,t), emitted by an electric dipole, and, in fact, their difference ∆A = A T -A p cannot be eliminated via a gauge transformation 13 . Indeed, the electromagnetic fields emitted by a point-like toroidal dipole, superimposed with a point-like electric dipole p = ikT vanish at any distance away from the source (but not at the source), whereas for the corresponding vector potential remains nonzero 51…”

Section: Tutorial Insets Inset I: Toroidal Multipolesmentioning

confidence: 99%

“…1) and studied in the context of nuclear 6 , atomic 7 , and molecular physics 8 , classical electrodynamics 9,10 , and solid state physics 3 . In the field of electromagnetism, in particular, a number of works have led to the development of a complete theoretical framework for toroidal electrodynamics [11][12][13][14][15] and the prediction of exotic effects, including dynamic non-radiating charge current configurations 10 and non-reciprocal interactions 9 . Following recent experimental observations of toroidal contributions in the response of materials across the electromagnetic spectrum, dynamic 2 toroidal multipoles are now the focus of substantial research efforts.…”

Section: Introductionmentioning

confidence: 99%

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Electromagnetic toroidal excitations in matter and free space

et al. 2016

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The toroidal dipole is a localized electromagnetic excitation, distinct from the magnetic and electric dipoles. While the electric dipole can be understood as a pair of opposite charges and the magnetic dipole as a current loop, the toroidal dipole corresponds to currents flowing on the surface of a torus. Toroidal dipoles provide physically significant contributions to the basic characteristics of matter including absorption, dispersion, and optical activity. Toroidal excitations also exist in free-space as spatially and temporally localized electromagnetic pulses propagating at the speed of light and interacting with matter. We review recent experimental observations of resonant toroidal dipole excitations in metamaterials and the discovery of anapoles, non-radiating current configurations involving toroidal dipoles. While certain fundamental and practical aspects of toroidal electrodynamics remain open for the moment, we envision that exploitation of toroidal response can have important implications for the fields of photonics, sensing, energy and information. IntroductionThe interactions of electromagnetic radiation with matter underpin some of the most important technologies today -from telecommunications to information processing and data storage; from spectroscopy and imaging to light-assisted manufacturing. Our understanding and description of the electromagnetic properties of matter traditionally involves the concept of electric and magnetic dipoles, as well as their more complex combinations, known as multipoles. Introduced by Maxwell and Lorentz and later refined by Jackson and Landau, this framework, termed the multipole expansion, is central in physics and is being routinely applied in the study of optical, condensed matter, atomic, nuclear phenomena and beyond 1 . Within this framework, electromagnetic media can be represented by a set of point-like multipole sources [2][3][4][5] . The commonly used set of multipoles comprises the electric and magnetic families, which can be represented by oscillating charges and loop currents respectively. Dynamic toroidal multipoles constitute a third independent family of elementary electromagnetic sources, rather than an alternative multipole expansion or higher-order corrections to the conventional electric and magnetic multipoles (see tutorial inset I).Introduced in 1958 by Y. B. Zeldovich, toroidal moments have been considered in systems of toroidal topology (see Fig. 1) and studied in the context of nuclear 6 , atomic 7 , and molecular physics 8 , classical electrodynamics 9,10 , and solid state physics 3 . In the field of electromagnetism, in particular, a number of works have led to the development of a complete theoretical framework for toroidal electrodynamics [11][12][13][14][15] and the prediction of exotic effects, including dynamic non-radiating charge current configurations 10 and non-reciprocal interactions 9 . Following recent experimental observations of toroidal contributions in the response of materials across the electromagnetic spectrum, dyn...

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Section: Tutorial Insets Inset I: Toroidal Multipolesmentioning

confidence: 99%

Section: Dynamic Toroidal Multipolesmentioning

confidence: 99%

Section: Tutorial Insets Inset I: Toroidal Multipolesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Electromagnetic toroidal excitations in matter and free space

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“…FD pulses can be seen as propagating counterparts of the localized toroidal dipole excitations in matter [11]. The toroidal dipole is distinct from the conventional electric and magnetic dipoles [12,13] and have attracted significant interest in recent years as important contributors to the electromagnetic properties of media with non-local response or elements of toroidal symmetry [11,[14][15][16][17][18][19][20][21]. Superposition of dynamic electric and toroidal dipoles leads to anapoles which, through destructive interference, exhibit vanishing radiated fields outside the source [20,22].…”

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confidence: 99%

Exciting dynamic anapoles with electromagnetic doughnut pulses

Raybould

Fedotov

Papasimakis

et al. 2017

Applied Physics Letters

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As was predicted in 1995 by Afanasiev and Stepanofsky, a superposition of electric and toroidal dipoles can lead to a non-trivial non-radiating charge current-configuration, the dynamic anapole. The anapoles were recently observed first in microwave metamaterials and then in dielectric nanodisks. However, spectroscopic studies of toroidal dipole and anapole excitations are challenging owing to their diminishing coupling to transverse electromagnetic waves. Here, we show that anapoles can be excited by electromagnetic Flying Doughnut (FD) pulses. First described by Helwarth and Nouchi in 1996, FD pulses are space-time inseparable exact solutions to Maxwells equations that have toroidal topology and propagate in free-space at the speed of light. We argue that FD pulses can be used as a diagnostic and spectroscopic tool for the anapole excitations in matter.Flying Doughnut (FD) pulses were introduced by Hellwarth and Nouchi in 1996 [1] as exact solutions to Maxwells equations in free-space [2][3][4][5][6][7][8][9]. They are necessarily few-cycle, wideband electromagnetic perturbations with toroidal field topology that can exist in transverse electric (TE) and transverse magnetic (TM) configurations [1,10]. FD pulses can be seen as propagating counterparts of the localized toroidal dipole excitations in matter [11]. The toroidal dipole is distinct from the conventional electric and magnetic dipoles [12,13] and have attracted significant interest in recent years as important contributors to the electromagnetic properties of media with non-local response or elements of toroidal symmetry [11,[14][15][16][17][18][19][20][21]. Superposition of dynamic electric and toroidal dipoles leads to anapoles which, through destructive interference, exhibit vanishing radiated fields outside the source [20,22]. Anapoles have been observed in microwave metamaterials [16] and dielectric nanoparticles [18], and their excitation has also been predicted in core-shell nanoparticles [23] and nanowires [24]. Moreover, anapoles have been employed to enhance nonlinear effects [25] and realize high-Q microwave metamaterials [26]. However, anapole modes are weakly coupled to freespace radiation, which renders experimental observations particularly challenging. In this paper, we demonstrate numerically the excitation of anapoles in a spherical dielectric particle driven by an FD pulse.We consider a dispersionless spherical dielectric particle interacting with a TM FD pulse, as depicted in Fig. 1. The FD pulse is brought to focus on the dielectric sphere located at the origin of the coordinate system. The interactions between FD pulses and dielectric spherical particles are investigated by a finite element solver of Maxwell's equations in three dimensions. The simulations are conducted in the transient domain. We define the incident FD pulse as per the field prescriptions in ref. (1)where σ = z + ct and τ = z − ct, and f 0 is an arbitrary normalisation constant. The parameters q 1 and q 2 have dimensions of length and represent respectively the ...

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Sharp Toroidal Resonances in Planar Terahertz Metasurfaces

et al. 2016

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A toroidal dipole in metasurfaces provides an alternate approach for the excitation of high-Q resonances. In contrast to conventional multipoles, the toroidal dipole interaction strength depends on the time derivative of the surrounding electric field. A characteristic feature of toroidal dipoles is tightly confined loops of oscillating magnetic field that curl around the fictitious arrow of the toroidal dipole vector.

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The electromagnetic field of elementary time-dependent toroidal sources

Cited by 115 publications

References 25 publications

Electromagnetic toroidal excitations in matter and free space

Electromagnetic toroidal excitations in matter and free space

Exciting dynamic anapoles with electromagnetic doughnut pulses

Sharp Toroidal Resonances in Planar Terahertz Metasurfaces

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