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Letter
Nonreciprocal emission in magnetized epsilon-near-zero metamaterials Artur Davoyan, and Nader Engheta ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b01391 • Publication Date (Web): 11 Feb 2019 Downloaded from http://pubs.acs.org on February 17, 2019
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Nonreciprocal emission in magnetized epsilon-near-zero metamaterials Artur R. Davoyan1,2,* and Nader Engheta1,* 1Department
2Present
of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
address: Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, California 90024, USA
Abstract: We study light radiation in magneto-active media in the context of epsilon-near-zero (ENZ) metamaterials. We demonstrate theoretically that at the ENZ regime even small magnetization implies a significant transformation of the emission pattern by point sources. We reveal that the interplay of magnetization and near-zero permittivity leads to a nonreciprocal spirallike omnidirectional radiation. We show that such a dynamics is due to a magnetically induced rotating dipole. Finally, we suggest a scenario where the predicted effects may be observed with realistic materials. Key words: nonreciprocal, metamaterials, magneto-optics, time reversal symmetry breaking, quantum emission. The Lorentz reciprocity plays an important role in the physics of electromagnetic wave propagation and dictates basic operation principles of photonic and microwave devices1,2. Indeed, generic scattering, radiation and dispersion properties may well be explained by the underlying laws of associated time reversal symmetry1,2. By breaking reciprocity, novel physical principles may be uncovered to enable functional photonic systems of a new kind2. Despite the fact that there is a number of ways to break such a symmetry2, including magnetism, nonlinearity, and temporal variation, it appears that only magnetism, being of a microscopic origin, enables harnessing nonreciprocal light-materials interaction at the fundamental atomistic scale. Recent discovery of topological materials3 and growing interest in solid state defect physics4, where magneto-optical interactions are pronounced, stimulate further study of electromagnetic phenomena driven by magnetism. Notably, a broad spectrum of photonic system make use of magnetically induced nonreciprocity, including optical isolation, memory, Kerr spectroscopy, and magnetometry to name few. It is therefore important to explore the manipulation and control of nonreciprocal lightmaterials interaction in magnetized nanoscale photonic structures. In this regard, taming light emission and scattering by localized sources (e.g., nanoantennas, quantum dots, and defects in solids) is of a particular interest, as radiative phenomena are encountered in a broad range of disciplines from sensing and spectroscopy to quantum communications. Ability to stimulate nonreciprocal emission at the deep subwavelength level may further our understanding of lightmaterials interaction and pave the way to novel subwavelength devices. However, magnetooptical interaction occurring in naturally available materials is typically weak, resulting in rather weak nonreciprocal responses. This is where, metamaterials – judiciously engineered electromagnetic structures with properties not readily available in nature5-9 – may provide unique ways for tailoring lightwave dispersion9-14 and taming magneto-optical nonreciprocal phenomena.15,16 Hence, extreme parameter metamaterials with nontrivial light dispersion properties (e.g., hyperbolic metamaterials10,13,14, epsilon-near-zero9,17 and zero index metamaterials18,19) are of a particular interest, as with these materials limits of electrodynamics
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may be approached. For instance, materials with hyperbolic wave dispersion are utilized for a dramatic enhancement of spontaneous emission rate20,21 (i.e., Purcell enhancement), superresolution imaging beyond the diffraction limit,22 and for a long-range interaction.23 Materials with near-zero effective permittivity (and/or permeability), the so called epsilon-nearzero metamaterials (ENZ),9 in turn, possess conceptually new regimes of electromagnetic wave propagation, including “static optics",11 energy tunneling, 17 and long-range quantum supercoupling.25-27 Furthermore, ENZ metamaterials even in the presence of small magnetization reveal a significant transformation of the optical band structure leading to a substantial enhancement of some of the magneto-optical effects.15,16 However, detailed theory of light emission from a small source in ENZ media in the presence of magnetization is yet to be developed. In this Letter we study radiation of a point source immersed into a gyrotropic magneto-optically active media28,29 with a particular focus on the epsilon-near-zero regime of operation. We show theoretically that interplay of magneto-optical activity and near-zero optical wave dispersion significantly alters the radiation. We find that when both of the phenomena are of the same order of magnitude a significant transformation of the radiation pattern is possible. Specifically, we predict a spiral-like nonreciprocal radiation field, which is associated with a magnetically induced rotating dipole combined with the ENZ mediated wavelength increase. To illustrate these effects, for the sake of simplicity, we begin our discussion with a twodimensional (2D) scenario. Figure 1 illustrates the schematic of the studied geometry. In particular, we consider that a y-oriented 2D line dipole source is immersed into a magneto-optically active gyrotropic media with a magnetization perpendicular to the dipole moment, i.e., along the z-axis. All quantities are independent of the z coordinate, i.e., we assume that there is no field variation along the direction of magnetization. In this case the relative permittivity of the media is described by a tensor:
|| i 0
i
|| 0
0 0
(1)
where 𝜀 ∥ and 𝜀 ⊥ are in plane and out of plane components of the permittivity tensor, respectively, and the off-diagonal components 𝛼 correspond to the strength of the magneto-optical activity. In general the components of the permittivity tensor are dispersive and complex (i.e., lossy), however for the sake of concept demonstration in the following discussion we omit these effects if not stated otherwise. We note also that the effects predicted here by duality of Maxwell equations may be extended to the gyromagnetic systems (e.g., ferromagnetics at microwave frequencies) as well. In such a 2D geometry the TE 𝑯 = (𝐻𝑥,𝐻𝑦,0) and TM 𝑯 = (0,0,𝐻𝑧) waves are uncoupled and may be studied independently. The orientation of the dipole dictates that only TM waves are excited in such a system. We note that the case of z-oriented 2D electric current source is trivial and corresponds to the excitation of TE polarized waves with a well-familiar regular dipolar radiation which is not influenced by magnetism (i.e., determined only by 𝜀 ⊥ component of the permittivity tensor). In contrast, the radiation of the TM waves from the 2D source considered here is affected by magnetization of the system and this, as we show below, implies a significant transformation of the TM wave radiation pattern in the ENZ regime.
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Figure 1. Schematic illustration of the two-dimensional geometry studied in this Letter. A point y-oriented 2D line dipole is immersed into a magnetically active medium magnetized along the z axis with emphasis on the ENZ regime. To study the radiation and its transformation with magnetization we consider wave propagation equation for TM polarization:
(
∂2
∂2
+ ∂𝑦2 + ∂𝑥2
𝜔2 𝑐
2
)
𝑖𝛼 ∂𝑗𝑦
𝜀𝑒𝑓𝑓 𝐻𝑧 = 𝜀 ∥ ∂𝑦 ―
∂𝑗𝑦
∂𝑥 ,
(2)
here 𝑗𝑦(𝜌) = ―𝑖𝜔𝑝𝛿(𝜌) is the y-oriented 2D line source current, p is the dipole moment, 𝛿(𝜌) = 𝛿 (𝑥)𝛿(𝑦) is the 2D Dirac delta function, ω is the frequency of operation, H is the z component of z the TM wave’s magnetic field, and 𝜀𝑒𝑓𝑓 = (𝜀2∥ ― 𝛼2)/𝜀 ∥ denotes the effective permittivity inside the magneto-active medium, and 𝜌 is the radial coordinate. The wave propagation in a source free medium (i.e., when 𝑗y = 0) is determined by the value of the 𝜀𝑒𝑓𝑓.16 In this case the wave dispersion in the (x,y) plane is given as 𝑘2𝑥 + 𝑘2𝑦 = 𝑘2𝑒𝑓𝑓 =
𝜔2
𝜀 , 𝑐2 𝑒𝑓𝑓 and as 𝛼→𝜀 ∥ the effective permittivity approaches the ENZ regime, 𝜀𝑒𝑓𝑓→0, such that 𝑘2𝑒𝑓𝑓 = 𝑘2𝑥 + 𝑘2𝑦→0.16 Note that the dispersion in the (x,y) plane stays isotropic despite the presence of magnetization, which is due to the rotational symmetry of the geometry with respect to the magnetization axis. We search for the solution of the Eq.(2) utilizing the Green’s function theory. With this
[
𝛼 ∂𝑗𝑦
]
∂𝑗𝑦
1
approach 𝐻𝑧(𝒓) = ∫Ω𝐺(𝒓,𝒓′) ―𝑖𝜀 ∥ ∂𝑦′ + ∂𝑥′ 𝑑𝑥′𝑑𝑦′, where 𝐺(𝒓,𝒓′) = ― 4𝑖ℋ(1) 0 (𝑘𝑒𝑓𝑓|𝒓 ― 𝒓′|) is the Green’s function of the 2D Helmholtz equation and ℋ(1) 𝑂 is the Hankel function of the first kind, 𝒓 is the radial vector to the observation point, 𝒓′ is the radial vector determining the current source, and integration is over the space occupied by the source. Performing integration by parts and making use of the properties of the Dirac delta and Hankel functions we finally obtain the following expressions for the magnetic field: 𝐻𝑧(𝜌,𝜙) =
𝜔𝑝 𝑘𝑒𝑓𝑓 4
[
𝛼
ℋ(1) 1 (𝑘𝑒𝑓𝑓𝜌) cos (𝜙) ― 𝑖𝜀 ∥ sin (𝜙)
]
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𝐸𝜃 =
[ℋ
𝜔𝑝2𝜋 𝜇0 4𝑖 𝜆 𝜀0
(1)( 0 𝑘𝑒𝑓𝑓𝜌)
(cos (𝜙) ― 𝑖
)―
𝛼 𝜀 ∥ sin (𝜙)
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ℋ(1) 1 (𝑘𝑒𝑓𝑓𝜌)𝜀𝑒𝑓𝑓 𝑘𝑒𝑓𝑓𝜌
𝜀∥
]
cos (𝜙)
(3)
According to Eq. (3) one can clearly see that in the presence of magnetization an additional effective dipole moment with strength 𝛼/𝜀 ∥ orthogonal and out-of phase with that of the source is induced. Such an interplay of the intrinsic source dipole and that of magnetically induced one leads to an interesting change in the radiated field configuration. In Figures 2a-d we plot the intermediate radiation zone profile of the 𝐻𝑧 for different values of the ratio 𝛼/𝜀 ∥ assuming, 𝛼
without loss of generality, 𝜀 ∥ = 1. As expected for 𝜀 ∥ →0 magnetization can be neglected and the radiation is just that of a point y-oriented 2D dipole line source in conventional media, Figure 2a. With the increase of the magnetization strength, i.e., increase of the ratio 𝛼/𝜀 ∥ , two effects act simultaneously. First, the interference between the two orthogonal and out-of phase currents leads to the formation of “spiral-like" radiation (e.g., Figures 2b,c show radiations for 𝛼/𝜀 ∥ = 0.5 and 𝛼/𝜀 ∥ = 0.9, respectively). Such spiral-like patterns may be understood as a superposition of a field
(
𝛼
)
due to a y oriented dipole with strength 𝒑𝑑 = 𝑝 1 ― 𝜀 ∥ 𝒚 and of a field caused by a rotating dipole 𝛼
𝐩𝑟𝑜𝑡 = 𝑝𝜀 ∥ (𝒚 +𝑖𝒙). Note that such a radiation is nonreciprocal: the handedness of the spiral associated with the direction of the dipole rotation is reversed upon the time reversal operation. Furthermore, the direction of emission is flipped upon the magnetization reversal (i.e., 𝛼→ ― 𝛼 transformation). An ideal spiral radiation is the characteristic of a rotating point source, for which 𝜋 the x and y dipole moments are equal in amplitudes and have a 2 phase shift, i.e., 𝑝𝑥 =± 𝑖𝑝𝑦.30,31 This regime is in principle achieved for 𝛼 = 𝜀 ∥ . However, as 𝛼→𝜀 ∥ the effective epsilon is approaching the near-zero regime, such that the wavelength in the medium 𝜆 = 2𝜋/𝑘𝑒𝑓𝑓→∞. The effect of the wavelength increase with the increase of magnetization is clearly seen in Figures 2ad.
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Figure 2. (a-d) show magnetic field (H component) distribution maps for 2D line dipole z radiation in an unbounded medium for different strengths of magneto-optical activity: (a) α/ε ∥ = 0, (b) α/ε ∥ = 0.5, (c) α/ε ∥ = 0.9, and (d) α/ε ∥ = 0.99, respectively. Here we assume 𝜀 ∥ = 1, see text. Panel (d) shows the far-zone radiation patterns for different values of α/ε ∥ . Next we study the far-zone radiation patterns, see Figure 2e. For this purpose we calculate the angular distribution of the radial power flow caused by the interplay between the intrinsic dipole and the induced one: 1 2𝜋𝜔2𝑝2 𝜇0 𝜀0 𝜌
𝑆𝜌(𝜙,𝑘𝑒𝑓𝑓𝜌 ≫ 1) = 16𝜋 𝜆
[cos2 𝜙 +
α 2 ε∥
( ) sin2 𝜙]
(4)
As can be seen from this expression the far-field is just the superposition between two orthogonal dipoles, which is due to the exact π/2 phase shift between the sources. The variation of the far-field radiation pattern with α/ε ∥ is shown in Figure 2e. The transition from a dipolar radiation to an omnidirectional one is apparent as the ratio 𝛼/𝜀 ∥ increases. Worth noting that the 1 2𝜋
2
net radiated power is given as 𝑃 = 16 𝜆 𝜔 𝑝2
𝜇0 𝜀0
(1 + ( ) ), and at 𝛼 = 𝜀 𝛼 2
𝜀∥
∥
it is twice that of a
single emitter in nonmagnetized medium.
Figure 3. (a) Schematic illustration of a geometry: A point 3D electric dipole source is placed in the middle of a parallel plate waveguide filled with magneto-active Bismuth Iron Garnet (Bi3Fe5O12). (b-d) distributions of magnetic field (Hz component) for different values of the waveguide thickness h: (b) ℎ = 1.5ℎ𝐸𝑁𝑍, (c) ℎ = 1.2ℎ𝐸𝑁𝑍,, and (d) ℎ = ℎ𝐸𝑁𝑍,, respectively. (e) Magnetic field distribution for a silver – Bi3Fe5O12– silver waveguide at λ=1550nm for real material parameters for ℎ = 0.76ℎ𝐸𝑁𝑍.
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As we showed above the radiation of the point dipole in such a two-dimensional geometry may be strongly affected by the magnetization, especially when 𝛼→𝜀 ∥ , i.e., when approaching magnetically induced effective ENZ regime. However, for regular magneto-active materials at visible frequencies typically 𝛼 ≪ 𝜀 ∥ (e.g., for a Bismuth Iron Garnets (Bi3Fe5O12) α
