DOI: 10.22184/1993-7296.FRos.2022.16.4.306.317

The resonant scattering properties of a plane electromagnetic wave on a spherical dielectric particle with a dimensional parameter q of the order of 10 are briefly considered. Despite the long historical background, new solutions lead to the unexpected and sometimes unusual results demonstating a number of practically important properties, including high-order Fano resonance with generating of extremely high electromagnetic wave.

The resonant scattering properties of a plane electromagnetic wave on a spherical dielectric particle with a dimensional parameter q of the order of 10 are briefly considered. Despite the long historical background, new solutions lead to the unexpected and sometimes unusual results demonstating a number of practically important properties, including high-order Fano resonance with generating of extremely high electromagnetic wave.

Теги: fano resonance mesatronics metaphotonics surface nanostructuring мезатроника метафотоника наноструктурирование поверхностей резонанс фано

Optical Super-Resonance in the Dielectric Mesoscale Spheres

I. V. Minin, O. V. Minin

Tomsk Polytechnic University, Tomsk, Russia

The resonant scattering properties of a plane electromagnetic wave on a spherical dielectric particle with a dimensional parameter q of the order of 10 are briefly considered. Despite the long historical background, new solutions lead to the unexpected and sometimes unusual results demonstating a number of practically important properties, including high-order Fano resonance with generating of extremely high electromagnetic wave.

Keywords: Fano resonance, surface nanostructuring, metaphotonics, mesatronics

Received: 12.05.2022

Accepted: 01.06.2022

A rigorous solution to the problem of a plane electromagnetic wave diffraction on a homogeneous spherical particle with arbitrary size and an arbitrary dielectric capacitance value, was obtained by Gustav Mie back in 1908 [1] (and independently by Lava, Debye and Lorenz [2, 3, 4], it is also often called the Lorentz-Mie theory [5]). Nevertheless, the properties of this solution are still being studied [6–9].

In accordance with the Lorentz-Mie theory [1, 5], the scattering of a plane and linearly polarized electromagnetic wave on a spherical particle is represented as a slowly converging infinite series of partial components. In this case, each partial wave is shown as a sum of two modes, namely magnetic and electric modes [5, 10, 11]. The relevant efficiencies of external and internal scattering are mathematically related by fairly simple expressions to the pairs of complex scattering factors (an, bn) and (cn, dn), respectively [5, 12]. However, “Le bon Dieu est dans le detail” [13], these complex scattering factors are expressed using the combinations of Bessel and Neumann spherical functions and their derivatives and are rather complicated [5, 12, 14]. Even now, the clarification of these features and the relevant scattering properties on a spherical particle often leads to the unexpected and sometimes non-intuitive results.

At the early research stages, the scientists were focused mainly on the study of optical properties of the color of noble metal sols, atmospheric aerosols, and gloria [1, 6–11, 15]. In particular, in the late 1960s, the group of researchers under the guidance of professor V. F. Minin conducted a series of studies relating to the anomalous backscattering of dielectric mesoscale spheres and determined the optimal ranges for changing their dimensional parameter q (q = 2πa / λ, where a is the particle radius, λ is the radiation wavelength) and refractive index [16].

A new stage in the study of the dielectric sphere optical properties dates back to the early 2000s. In 2000, during the experiments on laser cleaning of flat surfaces, a group of scientists led by professor B. S. Lukianchuk [17] found that the nanoholes were formed on the plate surface directly under the spherical particles. Later, these effects became known as the photonic jet [18]. Note that the formation of “magnetic” photonic jets is possible for particles of arbitrary shape, in particular, cubic, and not only spherical [16]. These studies have inspired the scientists to further study the possibilities of mesoscale spheres in relation to the surface nanostructuring and development of high-resolution optical images [18–20].

Another modern area of research into the resonant scattering effects on the dielectric nanoparticles is related to the Mie-resonance metaphotonics as an alternative for the nanooptics, where the main light control mechanism is the use of Mie-resonances in such nanoparticles (q < 1) with a high refractive index [21]. This area was called Mie-tronics [22, 23].

Since then, such studies have made it possible to find a number of practical applications in the fields of microscopy, surface nanostructuring, nanolithography, Raman spectroscopy, nanoparticle manipulation, development of the structured near-field, etc. that have already been described in detail in the references [18–20, 24–26].

In this case, we will be interested in the spherical dielectric particles with a dimensional parameter q of the order of 10. Such particles occupy an underexplored niche between the nanoparticles (q < 1) and particles to which the geometric optics is applied (q ≈ 100) [18]. Recently, a number of practically important properties have been discovered for such particles [16, 18, 19, 27], and we will briefly consider one of the newest properties below. It should be noted that so far these studies are of a fundamental nature and their applied significance has yet to be assessed.

For the dielectric spherical particles, there will always be such refractive index at which the radiation wavelength inside the particle will be comparable to its diameter, i. e. q ≈ π / n. In this case, the particle becomes a resonator, since the process quasi-stationary condition is violated.

The optical properties of such a particle are described by the resonant frequencies and the relevant resonant modes considered in 1941 by Stratton [28]. Such a resonator has a high-quality factor subject to the low dissipation losses in the particle material. Inside the dielectric spheres, under the resonant conditions and parameters of size and material, the effects of a strong electromagnetic field enhancement can be observed. Thus, for a particle with the dimensional parameter q = 0.53 (that corresponds to the dipole resonance peak) and a high complex refractive index m = 7.07 + 0.07i, the magnetic field intensity excited in the sphere is about 400 times greater than the intensity in the incident wave [29, 30] in the field of spatial maxima. In this case, the resonant Mie scattering on dielectric spheres with a high refractive index and with a dimensional parameter of about 1 leads to a series of Fano resonances [19, 29, 31]. Other features of scattering on the small dielectric particles includes the availability of anapole effects, anomalous scattering, and occurrence of optical vortices due to the complex energy circulation both inside and near the particle surface [19, 32, 33]. The influence of environment surrounding the sphere was not investigated in these works.

A much less explored area of the electromagnetic wave scattering on a dielectric sphere relates to a dimensional parameter of the order of 10 [16, 19, 27]. For example, it was recently demonstrated in [34] that the weakly scattering mesoscale (the particle diameter is larger than the wavelength) dielectric spheres located in the vacuum can support the high-order Fano resonances related to the internal Mie modes. The relevant internal scattering efficiencies are expressed by the complex scattering Mie coefficients (cn, dn).

In the case of a dielectric mesoscale sphere, the high-order internal resonance mode interferes with a wide range of all other modes [34, 35]. This effect is shown on Fig.1 for a non-absorbing spherical particle with a refractive index n = 1.9 located in water [36] (in contrast to the results in [34, 37, 38] that were obtained for the particles located in vacuum). In this case, the optical contrast of the sphere with the dimensional parameter q = 32.27657 (the incident radiation wavelength of λ = 632.8nm) is 1.43. These parameters correspond to the resonant mode l = 55.

Figure 1a demonstrates the distribution of electric and magnetic fields in two planes, when all modes 1 ≤ l ≤ 77 were considered during the simulation. Figure 1b shows the same pattern, where all members are considered, except for the only resonant member l = 55. In this case, the radiation localization area has a view typical for a photon jet [18, 25]. Thus, in this case the only member l = 55 leads to an increase in the scattered radiation intensity by a factor of 1000.

The interference of wide and narrow spectral lines leads to the typical form of the high-order Fano resonance, as well as for the stress spectra of the electric and magnetic fields on the particle surface located in water, i. e. at the points (x = 0; y = 0; z = R), in the range of a dimensional parameter of the order q ≈ 30 [36], shown in Figure 2. The typical range of dimensional parameters required to obtain such resonances depends on the refractive index of the particle and the environment. For such Fano resonances, the field intensity enhancement factors in the case under consideration can reach extreme values of the order of 106. In this case, the typical form of Fano resonances is also observed in the scattering amplitude moduli [29, 34]. It should be noted that with an increase in the sphere material refractive index, the resonant mode number is decreased and the maximum achievable field intensity in the resonance is increased. However, for a refractive index greater than two, the field confinement shifts from the outer sphere boundary to its center that can be qualitatively explained on the basis of formula for the spherical lens focus. In turn, an increase in the sphere dimensions leads to the excitation of more pronounced super-resonance modes with the stronger field confinement.

It is also clearly seen in Fig. 2 that the asymmetry of the resonance intensities is mirrored when the magnetic or electric field prevails, and the resonance line width is greater for the field to which the intensity maximum corresponds. Moreover, the cascades of Fano resonances observed in the case of scattering on the weakly dissipative dielectric mesoscale spheres as a part of the Lorentz-Mie theory are accompanied by the singular interface effects with the large values of local wave-number vector and generation of the optical vortices with a typical core size that is much smaller than the diffraction limit [16, 18, 19, 27, 34, 35, 37, 38]. Such effects are found in the mesoscale dielectric particles when the particle dimensional parameter q exceeds a certain value that depends on both its optical contrast and environment optical parameters and the absolute value of the refractive index.

It is also interesting to note that the cascade of super-resonance modes (Fig. 2) demonstrates the pronounced typical alternating cascades with the periods of about ∆q ≈ 0.35 and ∆q ≈ 0.20. However, the absolute intensity values of the magnetic and electric fields differ slightly depending on the mode (magnetic or electric) and a dimensional parameter. In this case, the spectral position of the resonance peaks can be changed in a controllable way by changing the dimensional parameter and the relative refractive index of the sphere material, as well as the environmental conditions. This quasi-periodicity can be used in a number of interesting applications, including the sphere selection to achieve the best field confinement, improve the subwavelength focusing resolution, and possibly to develop a quasi-periodic frequency comb.

The decisive role in the occurrence of Fano resonances is played by the magnetic dipole resonances of isolated dielectric particles. The magnetic dipole mode of a dielectric particle excited at the magnetic resonance wavelength can be stronger than the electric dipole response, and thus make the main contribution to the scattering efficiency (see Fig. 2). Such spherical particles have a unique location of hot points at the sphere poles and are due to the specific behavior of internal Mie modes. It follows from the rigorous analytical calculations [34] that for the high-order Fano resonances at certain dimensional parameter values greater than 10 and the refractive index greater than 2, the increase order in the field strength inside a particle located in the vacuum can reach about 104–108 that is related, as indicated, with a sharp narrowing of the resonance line width [16, 19, 27, 34, 36–39].

An important feature of the high-order Fano resonances in such particles is the high confinement degree of magnetic and electric fields that exceeds the diffraction limit, both inside the particle and on its surface [34, 37–39]. The latter is related to the formation of regions with the enormous values of local wave-number vectors [16, 27, 34], similar to the super-oscillation effects [34, 40]. In connection with these “super resonances”, the occurrence of magnetic photonic jets [16–19] and giant magnetic fields [19, 27, 34, 36–39] has been shown that can be attractive for many photonic applications. Moreover, for spherical particles with the dimensional parameters of the order of 20 and higher and a refractive index of less than 2, in the super-resonance mode, the focusing over-enhancement effect in the particles provides for about 4000 times stronger field strength (i. e., greater by more than an order than for the small particles with q ≈ 0.5 [29, 30]) than for the incident radiation. It confirms the possibility of overcoming the diffraction limit [27, 34, 38], despite the high sensitivity to the dissipation loss value in the particle material.

A naive answer to the question “How to maximize the radiation confinement in a dielectric particle?” is to “make it so as to minimize the dissipative constant of the particle material”. However, surprisingly enough, it turns out that the correct answer is exactly the opposite: a small energy dissipation in the sphere material can also contribute (rather than worsen) the sub-diffraction field confinement [38], i. e. its dissipative constant must be low, but not equal to zero.

The new physical effect mentioned above, namely the optical super-resonance in the messcale dielectric spheres due to the high-order Fano resonance, can become a new way to obtain the ultrahigh magnetic fields. This possibility can be explained as follows. As noted previously, the optical vortices occur [34] in the mesoscale dielectric particles, when the particle dimensional parameter q exceeds a certain value depending on its refractive index. Therefore, according to the Biot-Savart law, the relevant ring currents develop the magnetic fields. The simulation based on the Lorentz-Mie theory showed that inside a mesoscale weakly dissipative dielectric particle placed in the vacuum, the magnetic field could be potentially enhanced by more than 4 orders that could provide the magnetic induction values of the order of 105 T [34]. This value is close to the interatomic magnetic fields [41, 42] and is comparable with the capabilities of magnetic-cumulative generators [43, 44]. In this case, such fields can demonstrate the effects of magnetic nonlinear optics [34], when any changes in the refractive index are caused by magnetic effects. However, the question of dispersion in the spherical particle material under such conditions still remains open.

Moreover, the super-resonance properties are determined not only by the sphere dimensional parameter and its complex refractive index, but also by the environment. Therefore, if the environment is changed (for example, there will be air or another gas or liquid instead of vacuum), the super-resonance properties will be also changed. On the basis of rigorous analytical calculations according to the Lorentz-Mie theory, the high sensitivity of the super-resonance effect based on the mesoscale spherical particles to the efficient refractive index value of the environment was demonstrated for the first time [36, 39] (see Fig. 3). Thus, in the case of air, in contrast to the vacuum [34, 37, 38], a twofold drop in intensity of the magnetic and electric fields at hot points is observed with a change in the medium refractive index of the order of 10–7. In this case, the resonant values of the particle dimensional parameter do not coincide with those for vacuum and are shifted to the blue region [36, 39].

Thus, for a sphere located in water [36], a change in the medium efficient refractive index by 2 · 10–6 (that is equivalent, for example, to a change in water temperature by approximately ΔT = 0,0106 °C) also leads to a twofold drop in the field intensity for the same dimensional parameter. In this case, the resonant values of the dimensional parameter are also changed in comparison with the vacuum [36]. It should be noted that in the case of a relevant change in the sphere dimension resonant parameter located in a medium (relative to that for vacuum), the field intensity values at the hot points will be almost the same as for a sphere in vacuum [36, 39].

In our opinion, further prospects for studying the super-resonance effect are related to the fact that for a multilayer spherical particle (with a concentric layer arrangement), the scattered field outside the particle is also represented as a multipole dissociation, when the expressions connecting the various partial cross sections with the scattering factors formally coincide with those for a single-layer particle [45–47]. However, the scattering factors are determined on the basis of recurrent relations, the number of which depends on the number of sphere layers. Moreover, in such a configuration, an additional degree of freedom is developed: any changes in the spatial arrangement of layers and their complex refractive indices can control the position of various resonances, including the resonance degeneration. Apparently, the super-resonance effects in this case can obtain new properties, the consideration of which is the subject of future research.

Conclusions

Despite the long-term historical background, the clarification of solutions of the Lorentz-Mie equations and the relevant scattering properties on a spherical particle is still far from being considered completed, and sometimes it leads to the unexpected and even unusual results, considered for the small particles in [15, 19, 30, 48, 49], etc. Moreover, new effects are being discovered for the larger mesoscale particles [16–20, 25–27, 34, 36–39, 50].

All the results briefly described above were obtained exclusively as a part of the classical Lorentz-Mie theory without any modifications or generalizations. The discovery of new effects became possible when studying the problem parameters and such aspects of the problem that had not previously been given due attention, i. e. with a change in the “angle of view” to the well-known Lorentz-Mie theory formulas or, as defined by L. I. Mandelstam, with the “second degree of understanding” [19]. This made it possible to reveal the unusual physics of the new phenomenon and its application prospects.

To sum it up, we can conclude that the weakly dissipative mesoscale dielectric spherical particles make it possible to efficiently control the magnetic and electric components in a simultaneous way, support both the non-resonant field confinement mode (photon jet formation) and the super-resonant effect different from the whispering gallery mode [51]. The latter is valid only for the spheres and is not applicable to the cylinders. In the super-resonance mode, a record-high stress level of electric and magnetic fields can be generated that is not inferior to that for the plasmonic structures [52, 53], but without involving any plasmonic effects and materials. In view of medicine and biophotonics, such an effect can be useful for fighting viruses [54].

However, the super-resonance effect is extremely sensitive to the changes in the particle dimensional parameter and the environmental refractive index [36, 39]. Therefore, the results of studies of the super-resonance effect for the spheres in vacuum are of certain academic interest, but cannot be directly applied in the real conditions.

From a practical perspective, the super-resonance effect has great potential for the extreme photonics, research of the material properties under the extreme conditions, Raman scattering enhancement, sensors and nonlinear optics, development of a quasi-periodic frequency comb (metrology), etc. being the current trends in the field of modern mesoscale dielectric photonics [27]. It is noteworthy that this effect was demonstrated not only in the optical, infrared, and terahertz ranges, but it was also transferred to acoustics [55] based on the solution of the Helmholtz equations [56]. However, these and other practical applications are yet to be explored. Nevertheless, with due regard to other new optical effects recently discovered in the field of mesoscale particles both with medium and high refractive index [16–19, 25–27, 36–39], and with low [57], it can be stated that a new scientific field has been formed called “mesotronics” [16, 58].

Acknowledgement

The work was performed within the framework of the Tomsk Polytechnic University development program and partially as a part of the RFBR grant (project No.21-57-10001). The authors also express their gratitude to Song Zhou from the Jiangsu Key Laboratory of Advanced Manufacturing Technology, Huaiyin Institute of Technology, China for assistance in calculations [36, 39].

AUTHORS

I. V. Minin, Doctor of Engineering Sciences, Prof.minin@gmail.com, Tomsk Polytechnic University, Tomsk, Russia.

ORCID 0000-0002-6108-8419

O. V. Minin, Doctor of Engineering Sciences, Tomsk Polytechnic University, Tomsk, Russia.

ORCID 0000-0002-9749-2106

I. V. Minin, O. V. Minin

Tomsk Polytechnic University, Tomsk, Russia

The resonant scattering properties of a plane electromagnetic wave on a spherical dielectric particle with a dimensional parameter q of the order of 10 are briefly considered. Despite the long historical background, new solutions lead to the unexpected and sometimes unusual results demonstating a number of practically important properties, including high-order Fano resonance with generating of extremely high electromagnetic wave.

Keywords: Fano resonance, surface nanostructuring, metaphotonics, mesatronics

Received: 12.05.2022

Accepted: 01.06.2022

A rigorous solution to the problem of a plane electromagnetic wave diffraction on a homogeneous spherical particle with arbitrary size and an arbitrary dielectric capacitance value, was obtained by Gustav Mie back in 1908 [1] (and independently by Lava, Debye and Lorenz [2, 3, 4], it is also often called the Lorentz-Mie theory [5]). Nevertheless, the properties of this solution are still being studied [6–9].

In accordance with the Lorentz-Mie theory [1, 5], the scattering of a plane and linearly polarized electromagnetic wave on a spherical particle is represented as a slowly converging infinite series of partial components. In this case, each partial wave is shown as a sum of two modes, namely magnetic and electric modes [5, 10, 11]. The relevant efficiencies of external and internal scattering are mathematically related by fairly simple expressions to the pairs of complex scattering factors (an, bn) and (cn, dn), respectively [5, 12]. However, “Le bon Dieu est dans le detail” [13], these complex scattering factors are expressed using the combinations of Bessel and Neumann spherical functions and their derivatives and are rather complicated [5, 12, 14]. Even now, the clarification of these features and the relevant scattering properties on a spherical particle often leads to the unexpected and sometimes non-intuitive results.

At the early research stages, the scientists were focused mainly on the study of optical properties of the color of noble metal sols, atmospheric aerosols, and gloria [1, 6–11, 15]. In particular, in the late 1960s, the group of researchers under the guidance of professor V. F. Minin conducted a series of studies relating to the anomalous backscattering of dielectric mesoscale spheres and determined the optimal ranges for changing their dimensional parameter q (q = 2πa / λ, where a is the particle radius, λ is the radiation wavelength) and refractive index [16].

A new stage in the study of the dielectric sphere optical properties dates back to the early 2000s. In 2000, during the experiments on laser cleaning of flat surfaces, a group of scientists led by professor B. S. Lukianchuk [17] found that the nanoholes were formed on the plate surface directly under the spherical particles. Later, these effects became known as the photonic jet [18]. Note that the formation of “magnetic” photonic jets is possible for particles of arbitrary shape, in particular, cubic, and not only spherical [16]. These studies have inspired the scientists to further study the possibilities of mesoscale spheres in relation to the surface nanostructuring and development of high-resolution optical images [18–20].

Another modern area of research into the resonant scattering effects on the dielectric nanoparticles is related to the Mie-resonance metaphotonics as an alternative for the nanooptics, where the main light control mechanism is the use of Mie-resonances in such nanoparticles (q < 1) with a high refractive index [21]. This area was called Mie-tronics [22, 23].

Since then, such studies have made it possible to find a number of practical applications in the fields of microscopy, surface nanostructuring, nanolithography, Raman spectroscopy, nanoparticle manipulation, development of the structured near-field, etc. that have already been described in detail in the references [18–20, 24–26].

In this case, we will be interested in the spherical dielectric particles with a dimensional parameter q of the order of 10. Such particles occupy an underexplored niche between the nanoparticles (q < 1) and particles to which the geometric optics is applied (q ≈ 100) [18]. Recently, a number of practically important properties have been discovered for such particles [16, 18, 19, 27], and we will briefly consider one of the newest properties below. It should be noted that so far these studies are of a fundamental nature and their applied significance has yet to be assessed.

For the dielectric spherical particles, there will always be such refractive index at which the radiation wavelength inside the particle will be comparable to its diameter, i. e. q ≈ π / n. In this case, the particle becomes a resonator, since the process quasi-stationary condition is violated.

The optical properties of such a particle are described by the resonant frequencies and the relevant resonant modes considered in 1941 by Stratton [28]. Such a resonator has a high-quality factor subject to the low dissipation losses in the particle material. Inside the dielectric spheres, under the resonant conditions and parameters of size and material, the effects of a strong electromagnetic field enhancement can be observed. Thus, for a particle with the dimensional parameter q = 0.53 (that corresponds to the dipole resonance peak) and a high complex refractive index m = 7.07 + 0.07i, the magnetic field intensity excited in the sphere is about 400 times greater than the intensity in the incident wave [29, 30] in the field of spatial maxima. In this case, the resonant Mie scattering on dielectric spheres with a high refractive index and with a dimensional parameter of about 1 leads to a series of Fano resonances [19, 29, 31]. Other features of scattering on the small dielectric particles includes the availability of anapole effects, anomalous scattering, and occurrence of optical vortices due to the complex energy circulation both inside and near the particle surface [19, 32, 33]. The influence of environment surrounding the sphere was not investigated in these works.

A much less explored area of the electromagnetic wave scattering on a dielectric sphere relates to a dimensional parameter of the order of 10 [16, 19, 27]. For example, it was recently demonstrated in [34] that the weakly scattering mesoscale (the particle diameter is larger than the wavelength) dielectric spheres located in the vacuum can support the high-order Fano resonances related to the internal Mie modes. The relevant internal scattering efficiencies are expressed by the complex scattering Mie coefficients (cn, dn).

In the case of a dielectric mesoscale sphere, the high-order internal resonance mode interferes with a wide range of all other modes [34, 35]. This effect is shown on Fig.1 for a non-absorbing spherical particle with a refractive index n = 1.9 located in water [36] (in contrast to the results in [34, 37, 38] that were obtained for the particles located in vacuum). In this case, the optical contrast of the sphere with the dimensional parameter q = 32.27657 (the incident radiation wavelength of λ = 632.8nm) is 1.43. These parameters correspond to the resonant mode l = 55.

Figure 1a demonstrates the distribution of electric and magnetic fields in two planes, when all modes 1 ≤ l ≤ 77 were considered during the simulation. Figure 1b shows the same pattern, where all members are considered, except for the only resonant member l = 55. In this case, the radiation localization area has a view typical for a photon jet [18, 25]. Thus, in this case the only member l = 55 leads to an increase in the scattered radiation intensity by a factor of 1000.

The interference of wide and narrow spectral lines leads to the typical form of the high-order Fano resonance, as well as for the stress spectra of the electric and magnetic fields on the particle surface located in water, i. e. at the points (x = 0; y = 0; z = R), in the range of a dimensional parameter of the order q ≈ 30 [36], shown in Figure 2. The typical range of dimensional parameters required to obtain such resonances depends on the refractive index of the particle and the environment. For such Fano resonances, the field intensity enhancement factors in the case under consideration can reach extreme values of the order of 106. In this case, the typical form of Fano resonances is also observed in the scattering amplitude moduli [29, 34]. It should be noted that with an increase in the sphere material refractive index, the resonant mode number is decreased and the maximum achievable field intensity in the resonance is increased. However, for a refractive index greater than two, the field confinement shifts from the outer sphere boundary to its center that can be qualitatively explained on the basis of formula for the spherical lens focus. In turn, an increase in the sphere dimensions leads to the excitation of more pronounced super-resonance modes with the stronger field confinement.

It is also clearly seen in Fig. 2 that the asymmetry of the resonance intensities is mirrored when the magnetic or electric field prevails, and the resonance line width is greater for the field to which the intensity maximum corresponds. Moreover, the cascades of Fano resonances observed in the case of scattering on the weakly dissipative dielectric mesoscale spheres as a part of the Lorentz-Mie theory are accompanied by the singular interface effects with the large values of local wave-number vector and generation of the optical vortices with a typical core size that is much smaller than the diffraction limit [16, 18, 19, 27, 34, 35, 37, 38]. Such effects are found in the mesoscale dielectric particles when the particle dimensional parameter q exceeds a certain value that depends on both its optical contrast and environment optical parameters and the absolute value of the refractive index.

It is also interesting to note that the cascade of super-resonance modes (Fig. 2) demonstrates the pronounced typical alternating cascades with the periods of about ∆q ≈ 0.35 and ∆q ≈ 0.20. However, the absolute intensity values of the magnetic and electric fields differ slightly depending on the mode (magnetic or electric) and a dimensional parameter. In this case, the spectral position of the resonance peaks can be changed in a controllable way by changing the dimensional parameter and the relative refractive index of the sphere material, as well as the environmental conditions. This quasi-periodicity can be used in a number of interesting applications, including the sphere selection to achieve the best field confinement, improve the subwavelength focusing resolution, and possibly to develop a quasi-periodic frequency comb.

The decisive role in the occurrence of Fano resonances is played by the magnetic dipole resonances of isolated dielectric particles. The magnetic dipole mode of a dielectric particle excited at the magnetic resonance wavelength can be stronger than the electric dipole response, and thus make the main contribution to the scattering efficiency (see Fig. 2). Such spherical particles have a unique location of hot points at the sphere poles and are due to the specific behavior of internal Mie modes. It follows from the rigorous analytical calculations [34] that for the high-order Fano resonances at certain dimensional parameter values greater than 10 and the refractive index greater than 2, the increase order in the field strength inside a particle located in the vacuum can reach about 104–108 that is related, as indicated, with a sharp narrowing of the resonance line width [16, 19, 27, 34, 36–39].

An important feature of the high-order Fano resonances in such particles is the high confinement degree of magnetic and electric fields that exceeds the diffraction limit, both inside the particle and on its surface [34, 37–39]. The latter is related to the formation of regions with the enormous values of local wave-number vectors [16, 27, 34], similar to the super-oscillation effects [34, 40]. In connection with these “super resonances”, the occurrence of magnetic photonic jets [16–19] and giant magnetic fields [19, 27, 34, 36–39] has been shown that can be attractive for many photonic applications. Moreover, for spherical particles with the dimensional parameters of the order of 20 and higher and a refractive index of less than 2, in the super-resonance mode, the focusing over-enhancement effect in the particles provides for about 4000 times stronger field strength (i. e., greater by more than an order than for the small particles with q ≈ 0.5 [29, 30]) than for the incident radiation. It confirms the possibility of overcoming the diffraction limit [27, 34, 38], despite the high sensitivity to the dissipation loss value in the particle material.

A naive answer to the question “How to maximize the radiation confinement in a dielectric particle?” is to “make it so as to minimize the dissipative constant of the particle material”. However, surprisingly enough, it turns out that the correct answer is exactly the opposite: a small energy dissipation in the sphere material can also contribute (rather than worsen) the sub-diffraction field confinement [38], i. e. its dissipative constant must be low, but not equal to zero.

The new physical effect mentioned above, namely the optical super-resonance in the messcale dielectric spheres due to the high-order Fano resonance, can become a new way to obtain the ultrahigh magnetic fields. This possibility can be explained as follows. As noted previously, the optical vortices occur [34] in the mesoscale dielectric particles, when the particle dimensional parameter q exceeds a certain value depending on its refractive index. Therefore, according to the Biot-Savart law, the relevant ring currents develop the magnetic fields. The simulation based on the Lorentz-Mie theory showed that inside a mesoscale weakly dissipative dielectric particle placed in the vacuum, the magnetic field could be potentially enhanced by more than 4 orders that could provide the magnetic induction values of the order of 105 T [34]. This value is close to the interatomic magnetic fields [41, 42] and is comparable with the capabilities of magnetic-cumulative generators [43, 44]. In this case, such fields can demonstrate the effects of magnetic nonlinear optics [34], when any changes in the refractive index are caused by magnetic effects. However, the question of dispersion in the spherical particle material under such conditions still remains open.

Moreover, the super-resonance properties are determined not only by the sphere dimensional parameter and its complex refractive index, but also by the environment. Therefore, if the environment is changed (for example, there will be air or another gas or liquid instead of vacuum), the super-resonance properties will be also changed. On the basis of rigorous analytical calculations according to the Lorentz-Mie theory, the high sensitivity of the super-resonance effect based on the mesoscale spherical particles to the efficient refractive index value of the environment was demonstrated for the first time [36, 39] (see Fig. 3). Thus, in the case of air, in contrast to the vacuum [34, 37, 38], a twofold drop in intensity of the magnetic and electric fields at hot points is observed with a change in the medium refractive index of the order of 10–7. In this case, the resonant values of the particle dimensional parameter do not coincide with those for vacuum and are shifted to the blue region [36, 39].

Thus, for a sphere located in water [36], a change in the medium efficient refractive index by 2 · 10–6 (that is equivalent, for example, to a change in water temperature by approximately ΔT = 0,0106 °C) also leads to a twofold drop in the field intensity for the same dimensional parameter. In this case, the resonant values of the dimensional parameter are also changed in comparison with the vacuum [36]. It should be noted that in the case of a relevant change in the sphere dimension resonant parameter located in a medium (relative to that for vacuum), the field intensity values at the hot points will be almost the same as for a sphere in vacuum [36, 39].

In our opinion, further prospects for studying the super-resonance effect are related to the fact that for a multilayer spherical particle (with a concentric layer arrangement), the scattered field outside the particle is also represented as a multipole dissociation, when the expressions connecting the various partial cross sections with the scattering factors formally coincide with those for a single-layer particle [45–47]. However, the scattering factors are determined on the basis of recurrent relations, the number of which depends on the number of sphere layers. Moreover, in such a configuration, an additional degree of freedom is developed: any changes in the spatial arrangement of layers and their complex refractive indices can control the position of various resonances, including the resonance degeneration. Apparently, the super-resonance effects in this case can obtain new properties, the consideration of which is the subject of future research.

Conclusions

Despite the long-term historical background, the clarification of solutions of the Lorentz-Mie equations and the relevant scattering properties on a spherical particle is still far from being considered completed, and sometimes it leads to the unexpected and even unusual results, considered for the small particles in [15, 19, 30, 48, 49], etc. Moreover, new effects are being discovered for the larger mesoscale particles [16–20, 25–27, 34, 36–39, 50].

All the results briefly described above were obtained exclusively as a part of the classical Lorentz-Mie theory without any modifications or generalizations. The discovery of new effects became possible when studying the problem parameters and such aspects of the problem that had not previously been given due attention, i. e. with a change in the “angle of view” to the well-known Lorentz-Mie theory formulas or, as defined by L. I. Mandelstam, with the “second degree of understanding” [19]. This made it possible to reveal the unusual physics of the new phenomenon and its application prospects.

To sum it up, we can conclude that the weakly dissipative mesoscale dielectric spherical particles make it possible to efficiently control the magnetic and electric components in a simultaneous way, support both the non-resonant field confinement mode (photon jet formation) and the super-resonant effect different from the whispering gallery mode [51]. The latter is valid only for the spheres and is not applicable to the cylinders. In the super-resonance mode, a record-high stress level of electric and magnetic fields can be generated that is not inferior to that for the plasmonic structures [52, 53], but without involving any plasmonic effects and materials. In view of medicine and biophotonics, such an effect can be useful for fighting viruses [54].

However, the super-resonance effect is extremely sensitive to the changes in the particle dimensional parameter and the environmental refractive index [36, 39]. Therefore, the results of studies of the super-resonance effect for the spheres in vacuum are of certain academic interest, but cannot be directly applied in the real conditions.

From a practical perspective, the super-resonance effect has great potential for the extreme photonics, research of the material properties under the extreme conditions, Raman scattering enhancement, sensors and nonlinear optics, development of a quasi-periodic frequency comb (metrology), etc. being the current trends in the field of modern mesoscale dielectric photonics [27]. It is noteworthy that this effect was demonstrated not only in the optical, infrared, and terahertz ranges, but it was also transferred to acoustics [55] based on the solution of the Helmholtz equations [56]. However, these and other practical applications are yet to be explored. Nevertheless, with due regard to other new optical effects recently discovered in the field of mesoscale particles both with medium and high refractive index [16–19, 25–27, 36–39], and with low [57], it can be stated that a new scientific field has been formed called “mesotronics” [16, 58].

Acknowledgement

The work was performed within the framework of the Tomsk Polytechnic University development program and partially as a part of the RFBR grant (project No.21-57-10001). The authors also express their gratitude to Song Zhou from the Jiangsu Key Laboratory of Advanced Manufacturing Technology, Huaiyin Institute of Technology, China for assistance in calculations [36, 39].

AUTHORS

I. V. Minin, Doctor of Engineering Sciences, Prof.minin@gmail.com, Tomsk Polytechnic University, Tomsk, Russia.

ORCID 0000-0002-6108-8419

O. V. Minin, Doctor of Engineering Sciences, Tomsk Polytechnic University, Tomsk, Russia.

ORCID 0000-0002-9749-2106

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