Abstract
Subwavelength structures demonstrate many unusual optical properties which can be employed for engineering of a new generation of functional metadevices, as well as controlled scattering of light and invisibility cloaking. Here we demonstrate that the suppression of light scattering for any direction of observation can be achieved for a uniform dielectric object with high refractive index, in a sharp contrast to the cloaking with multilayered plasmonic structures suggested previously. Our finding is based on the novel physics of cascades of Fano resonances observed in the Mie scattering from a homogeneous dielectric rod. We observe this effect experimentally at microwaves by employing high temperaturedependent dielectric permittivity of a glass cylinder with heated water. Our results open a new avenue in analyzing the optical response of highindex dielectric nanoparticles and the physics of cloaking.
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Introduction
In the past decade, the study of cloaking and invisibility has attracted a lot of attention^{1}. Several approaches for achieving the cloaking regime have been proposed on the basis of metamaterials^{1,2,3,4,5,6,7,8,9,10,11,12,13,14} and they employ transformation optics^{2,3,4,5}, cancellation of light scattering^{6,7,8}, nonlinear response of multishell structures^{9}, as well as the use of multilayered plasmonic particles^{10}, graphene^{11} and magnetooptical effect with an external magnetic field^{12}.
The fact that metamaterials can distort straight light rays is based on the transformation optics and Fermat's principle^{2,3}. Fermat's principle states that light rays follow not geometrical shortest paths but extremal optical paths these are the geometrical length multiplied by the refractive index of the material. If the refractive index varies in space for nonuniform media the optimal paths are no longer straight lines, but are curved, which may cause invisibility of an object. A simplest cylindrical invisibility device can be achieved by a linear transformation that transforms a cylindrical volume of radius R_{2} to cylindrical ring of thickness R_{2} − R_{1} by equation r = R_{1} + r′(R_{2} − R_{1})/R_{2} within or, equivalently, . Such transformation medium acts as an invisibility cloak, guiding light around the interior of the cloak and anything placed inside the inner radius R_{1} is hidden^{15}.
For hiding small subwavelength particles, Alù and Engheta suggested to use plasmonic coatings in order to reduce drastically the total scattering crosssection of an object^{16}. The effect is based on the resonant cancellation of the dipole moment of the particle if the polarization vector in a plasmonic shell is antiparallel with respect to that in a dielectric for a given wavelength. To increase the bandwidth of the particle invisibility, it was suggested to cover the particle with several shells of various materials^{17}.
The nonlinear response of multishell structures was used to study the scattering properties of plasmonic nanoparticles with a nonlinear layer^{9}. It was demonstrated that the cloak can be made nonlinear and the cloaking performance of multishell plasmonic structures can be controlled by changing the amplitude of the incident wave.
All those approaches require employing specially designed ‘covering shells’ with engineered parameters making difficult a practical realization of many invisibility concepts^{8}. Here we suggest a novel approach for the realization of tunable invisibility cloaking at all angles of observation that allows a direct switching from visibility to invisibility regimes and back. Our approach is based on the cancelation of scattering from a homogeneous highindex dielectric object without additional coating layers. The main idea of our approach is based of the properties of the characteristic lineshape of the Fano resonance^{18} and the cloaking effect is based on the novel physics of the resonant Mie scattering from a homogeneous dielectric rod that appears as cascades of Fano resonances with each individual resonance described by the conventional Fano formula^{19}. We demonstrate this novel cloaking effect and its tunability experimentally through a heatinginduced change of dielectric permittivity of a glass tube filled with water.
Results
General concept and formalism
The wellknown Mie scattering is described by analytical solutions of the Maxwell equations for elastic scattering of electromagnetic waves by a sphere^{20,21,22}. If the sphere diameter is comparable with the wavelength of incident light , the Mie scattering will be driven by resonances in the dielectric sphere. This leads to the emission of electromagnetic waves by the particle and interference between the nonresonant scattering from the particle and scattering by narrow Mie modes. If a spectrally narrow Mie band interacts constructively or destructively with a broad radiation spectrum, we can expect a Fanotype resonance^{18}, a resonant wave phenomenon wellknown across many different branches of physics^{23}. Some characteristic examples of Fano resonance are found in the studies of magnetization^{24} and electronic polarization properties^{25}, semiconductor optics^{26,27}, electronphonon coupling in superconductors^{28,29,30} and scattering in photonic crystals^{31,32} and plasmonic nanostructures^{33,34,35}.
Fano resonance is observed when the wave scattering process can reach the same final state via two different paths^{23}. The first scattering path corresponds to the formation of a narrow resonant band, where the wave phase changes by ≈ π. The second scattering path corresponds to a broad background, where the wave phase and amplitude are nearly constant in the spectrum range of interest. The resonant band can be described as a complex Lorentz function L(Ω) = (Ω + i)^{−1}, where Ω = (ω − ω_{0})/(Γ/2), while ω_{0} and Γ correspond to the position and the width of the band. The continuum is defined as B exp(iφ_{B}). The resulting wave takes a form A exp (iφ_{A})/(Ω + i) + B exp (iφ_{B}) where A(ω), B(ω), φ_{A}(ω) and φ_{B}(ω) are real functions, frequency dependence of which can be neglected compared to the Lorentz function. If there is no parts of background avoiding the above interaction, following Fano relation takes the form
where q = cot Δ is the Fano asymmetry parameter, sin^{2}[D(ω)] represents a background produced by a plane wave, Δ(ω) = φ_{A}(ω) − φ_{B}(ω) is the phase difference between the narrow resonant and continuum states, B(ω) = sin[Δ(ω)] At q = 0, instead of the conventional Mie peak, the resonant Mie dip with symmetrical Lorentzian shape I(ω)~Ω^{2}/(1 + Ω^{2}) is observed with the scattering intensity vanishing at the eigenfrequency ω_{0}. Destructive interference results in complete suppression of the scattering intensity at a given frequency^{18}. When the Fano parameter q deviates from zero, the Fano lineshape becomes asymmetric and the zerointensity frequency becomes shifted away from the resonance ω_{0}. It is important to emphasize that at any finite value of q there exists zerointensity frequency ω_{zero} = ω_{0} − qΓ/2, see Eq. (1). Note that ω_{zero} < ω_{0} at q > 0 and ω_{zero} > ω_{0} at q < 0 that is clearly seen in Fig. 1b. We employ this property of the Fano lineshape in order to realize the invisibility cloaking for a highindex dielectric rod in free space.
Fano resonances in the Mie scattering
Our study outlined below is limited to the case of an infinitely long dielectric circular rod. However, the suggested concept and subsequent analysis are rather general and they can be applied to other types of “bodies of revolution”. The case of an infinite rod, referred to as the LorenzMie theory, corresponds to a twodimensional problem of scattering in the plane normal to the symmetry axis z and it involves cylindrical rather than spherical functions in the infinite series of the analytical Mie solution^{20,21,22}.
We consider the Mie scattering by a single homogeneous infinite circular rod with the radius r and the purely real dielectric permittivity ε_{1} embedded in the transparent and homogeneous surrounding medium with the dielectric permittivity of ε_{2} = 1. The Mie scattering by a cylinder can be expanded into orthogonal electromagnetic dipolar and multipolar terms, with cylindrical LorenzMie coefficients a_{n} and b_{n}^{21}. For the TEpolarization considered here, the scattered fields are defined by coefficients of only one type a_{n}, while b_{n} are equal to zero. In addition, we introduce new coefficients d_{n} to characterize the field inside the cylinder; more details are given in Methods. The resonances are denoted as TE_{nk} where n is multipole order and k is resonance number (n is integer and k is positive integer).
Figure 1c presents the total Mie scattering efficiency and the spectral dependence of the Mie scattering efficiency of individual modes in the lowfrequency part of the spectrum at ε_{1} = 60 where we observe the modes TE_{0k}, TE_{1k}, TE_{2k} and TE_{3k}. Intensity of the multipole modes TE_{4k}, TE_{5k}, (n > 3) differs from zero for higher frequencies for x > 1. Spectra of the Mie scattering presented in Fig. 1c demonstrate different possible lineshapes that can be described by the Fano formula (1). Namely, TE_{01} mode is quite symmetrical (q is big enough), while TE_{02} mode has typical Fano lineshape with negative q. Both TE_{1} modes have Fano lineshape with negative q. The higher order modes (TE_{2} and TE_{3}) are very narrow in the long wavelength spectral range, however it is straightforward to identify qparameter for these peaks.
We show in Fig. 2 and in Methods how the spectrum of the Mie scattering can be presented in the form of an infinite series of the Fano profiles. We consider the Maxwell boundary condition for the tangential components of E field at the cylinder surface for the TE polarization The interference of the background and resonant scattered fields creates a cascade of Fano profiles (see Fig. 1c or 2). In contrast to the spectra of d_{0}^{2} describing magnetic field inside the rod, the scattering spectra outside the rod a_{0}^{2} demonstrate asymmetric profiles with either sharp increase or drop of a_{0}^{2} values at the resonance frequencies of the cylinder eigenmodes. To demonstrate that we really have the Fano resonance, we calculated the spectral dependence of the Fano parameter q for the dipole mode TE_{0k} using the profiles of 2700 resonances (for and for with the step of Δε = 0.5) in the broad spectral range at ε_{2} = 1. Figure 2d shows an example of such a fitting for TE_{02} mode at ε_{1} = 60 with Fano parameter q = 3.55 at x = 0.71. Finally we obtained an important result: all 2700 values of the Fano parameter q forms a general dependence q(x) (Fig. 2e). As follows from the fitting results, the roots of the background function define the points of infinity q(x) → ±∞ while the maxima of the background define the roots of q(x) (see common vertical dashed lines in Figs. 2b and 2e). The Fano parameter demonstrates characteristic cotangenttype dependence that is the key feature of the Fano approach demonstrating that the Fano lineshape depends only on the position of the resonance on the frequency scale with respect to the maximum (or minimum) of the background. Each Mie resonance is phase shifted with respect to the maximum of the sin  type background by phase Δ(ω). In particular, Mie resonances positioned close to the background maxima (at x ≈ 2.2, 5.4 etc) exhibit symmetric Lorentziantype dips while Mie resonances positioned close to the background minima (at x = 0, ≈ 3.8 etc) exhibit symmetric Lorentziantype peaks. On the both sides of the maxima mirrorlike asymmetric resonance profiles are observed. This law is valid independently on the rod permittivity ε_{1}.
Numerically calculated structure of the magnetic field in the regimes of Fano cloaking and strong Mie scattering are shown in Fig. 3a–h. We observe a practically complete suppression of scattering at frequencies corresponding to the dips in the function Q_{sca}(x) (Fig. 3i. It means that the incident TEpolarized light passes the cylinder without scattering making the cylinder invisible from any angle of observation. To analyze the invisibility dynamics, we calculated the dip intensity as a function of dielectric constant ε_{1} for the lowest dip in the frequency scale. For a reference scattering intensity we choose standard scattering efficiency for cylinder made from perfect conducting metal (PEC). The calculations show that the scattering efficiencies of dielectric cylinder and PEC coincide when cylinder permittivity is about 10. At ε_{1} > 10 the invisibility dynamics appears for x around 0.505.
Experimental verification of tunable invisibility
In the case of highindex dielectric materials and weak losses, a typical asymmetric Fano profile has a local maximum and a local minimum located close to each other, as shown in Fig. 3i. The existence of the first strong dip at ω_{cloak} = 0.505c/r (ε_{1} = 60) is caused by several reasons: the perfect zerointensity condition at ω_{zero} > ω_{0} (q < 0) for nearby Fanotype TE_{11} mode, narrowness of the neighboring TE_{21} mode and longdistance location to two intense scattering bands TE_{01} and TE_{02} (Fig. 1c). This proximity can be employed for the demonstration of the switching between visability and invisibility, tuning the scattering from the uncloaked to cloaked regimes. This can be achieved by modulating the parameters of an object or by changing the wavelength of the incoming radiation. To demonstrate the concept of Fano cloaking in experiment, we employ the advantages of strong temperature dependence of the dielectric permittivity of water^{36}.
At microwave frequencies, we use a glass cylinder filled with water that is characterized by dielectric constant of ε = 80 at 20°C and ε = 50 at 90°C in the frequency range from 1 GHz to 6 GHz. A rectangular horn antenna (TRIM 0.75 GHz to 18 GHz; DR) connected to a transmitting port of the vector network analyzer Agilent E8362C is used to approximate a planewave excitation. The water cylinder with radius 12 mm and height 42 cm is placed into the farfield region of the antenna (on the distance approximately 2.5 m) and the similar horn antenna (TRIM 0.75 GHz to 18 GHz) is employed as a receiver. The scattering efficiency is yielded from the imaginary part of the forward scattering amplitude (due to the optical theorem). The latter is proportional to E/E_{0} − 1, where E_{0} is measured electric field in the free space and E is the electric field in the presence of the cylinder with water. Note that the dielectric permittivity of the glass tube is much smaller than that for water for microwaves and the tube has no effect on the measured scattering efficiency.
Figure 3i shows strong suppression of scattering (of the order of 20 dB) from a glass tube filled with water and measured in microwave experiment; the data agree well with the theoretical predictions. A strong temperature dependence of the dielectric permittivity of water ε_{1} leads to a profound shift of the scattering maxima (the uncloaked regime) and minima (the cloaked regime) located in the spectrum near the resonant frequencies of the Mie resonance modes TE_{01} (the spectral region ω~ 1.25 GHz) and TE_{11} (ω ~ 2 GHz), as shown in Fig. 4.
Discussion
Fano resonance is known to be extremely sensible to variation of parameters due to proximity of the maxima and minima in the frequency scale. We have changed the rod permittivity to shift TE_{11} resonance position. When sharp wing of Fano lineshape moves through a fixed frequency we observe a dramatic change of scattering intensity from a high to negligible value. Here the Fano resonance allows us to switch a glass tube with water from the visible regime of strong Mie scattering (T = 90°C) to the regime of strong invisibility (T = 50°C) at the same frequency of 1.9 GHz. Numerical calculations of the component H_{z} of the electromagnetic field confirm directly the switching effect and demonstrate both perfect invisibility in Fig. 4b,d and strongly distorted scattered fields in Fig. 4a,c.
So, we have presented the first experimental observation of tunable invisibility of a macroscopic object transformed from visible to invisible states and back, without any coating layers. We have shown that the total intensity of the Mie scattering for waves of a certain polarization vanishes under the condition of the Fano resonance at any angle of observation. Remarkable that highindex dielectric materials are available for different wavelength ranges or can be engineered at will^{37}. Our study reveals a novel physics behind the seemingly wellknown Mie scattering of light and it may open a novel route towards manipulation and control of electromagnetic waves in alldielectric nanophotonics.
Methods
Fano formula
First, we consider a classical problem of the Fano resonance that appears as a result of interference between a narrow (resonant) band and a continuum background known as the configuration interaction in the physics of quantum phenomena. The resonant band can be described by a complex Lorentzian function L(Ω) = (Ω + i)^{−1}, where Ω = (ω − ω_{0})/(Γ/2), while ω_{0} and Γ corresponds to the position and width of the narrow frequency band.
We can define the continuum spectrum as B exp(iφ_{B}), so that the resulting combined wave takes the form
Here A(ω), B(ω), φ_{A}(ω) and φ_{B}(ω) are real functions which relative changes in the frequency range of interest are negligible in comparison with the Lorentzian function. The intensity of the resulting wave is given by
where Δ(ω) = φ_{A}(ω) − φ_{B}(ω) is the phase difference between the resonant and continuum states. The resulting extended Fano formula can be written in the form
In Eq. (4), the parameter q is defined as the Fano asymmetry parameter that characterizes a relative transition strength for the discrete state vs. continuum set of states. The first term in Eq. (4) describes a narrow band and the additional background spectrum in the region of the narrow band is presented by the second term. The background component that does not interfere with the narrow band is accounted for by the introduction of an interaction coefficient η ∈ [0..1]. Such noninteracting background is observed experimentally in the light scattering from different physical systems^{28,29,30} and photonic crystals^{31,38}. Comparing Eq. (3) and Eq. (4), we obtain
where F = A/B is the relative intensity of the narrow band and background component.
For η = 1 and A = 1, we obtain the Fano lineshape profile,
where q = cotΔ and B = sinΔ. We notice that the absolute value of the Fano parameter q is a measure of the relative strength of the amplitude of the resonant scattering compared to its nonresonant value.
Mie scattering as a cascade of Fano resonances
To demonstrate that the spectrum of Mie scattering can be presented in the form of an infinite series of the Fano profiles, we consider elastic scattering of electromagnetic waves by a homogeneous infinite dielectric rod of the radius r and the purely real dielectric permittivity ε_{1}. The surrounding medium have the dielectric permittivity of ε_{2}. The farfield scattered by a cylindrical rod can be expanded into orthogonal electromagnetic dipolar and multipolar terms, with cylindrical LorenzMie coefficients a_{n} and b_{n}^{21}. For the TEpolarization the scattered fields are defined by coefficients of only one type a_{n}, while b_{n} are equal to zero. The Maxwell boundary conditions for the tangential components of H and E fields at the rod's surface for the TE polarization can be presented as
where x = rω/c = 2π r/λ, E_{n}, A_{n} and D_{n} are cylindrical harmonic amplitudes of the incident, scattered and internal magnetic fields, respectively, expressed in terms of the Bessel J_{n}(ζ) and Hankel functions. The LorenzMie coefficients a_{n} = A_{n}/E_{n} and coefficients d_{n} = D_{n}/E_{n} for the scattered and internal magnetic fields can be determined from the system (7) as:
Figure 2 demonstrates that the spectrum of Mie scattering can be presented in the form of an infinite series of the Fano profiles. We write simultaneously two relations (i) the expression for the LorenzMie coefficient a_{n} which defines the scattered field in accord to (7), (ii) analytical expression describing the Fano resonance at the interference of a narrow resonance (symmetric Lorentzian) with a slow varying background (plane wave).
We identify two terms of different linewidth with the first one as slowly changed background and second one, as symmetric Lorentzian, both oscillating with the resonance frequency ω. Therefore we have a Fano resonance between background scattering from the rod and resonant Mie mode within the same cylindrical harmonic (see Fig. 2). The interference of the incident and scattered fields creates a complicated nearfield pattern and it may give rise either to strong enhancement (constructive interference) or strong suppression (destructive interference) of the electromagnetic field around the rod.
The similar analysis can be performed in the case of the TM polarized waves and for any “body of revolution” including a sphere.
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Acknowledgements
We acknowledge fruitful discussions with A.E. Miroshnichenko, A.N. Poddubny, Yu.A. Baloshin and A.P. Slobozhanyuk. This work was supported by the Government of the Russian Federation (grant 074U01), the Ministry of Education and Science of the Russian Federation, Russian Foundation for Basic Research, Dynasty Foundation (Russia) and the Australian National University.
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M.R. developed a theoretical model and conducted simulations and data analysis. D.F. performed experimental measurement. M.L, P.B. and Y.K. provided a guidance on the theory, numerical analysis and experiment. All authors discussed the results and contributed to the writing of the manuscript.
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Rybin, M., Filonov, D., Belov, P. et al. Switching from Visibility to Invisibility via Fano Resonances: Theory and Experiment. Sci Rep 5, 8774 (2015). https://doi.org/10.1038/srep08774
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DOI: https://doi.org/10.1038/srep08774
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