Abstract
Optical beams are generally unbound in bulk media and propagate with a velocity approximately amounting to the speed of light in freespace. Guidance and full spatial confinement of light are usually achieved by means of waveguides, mirrors, resonators and photonic crystals. Here we theoretically demonstrate that nonlinear selforganization can be exploited to freeze optical beams in bulk nearzeroindex media, thus enabling threedimensional selftrapping of still light without the need of optical resonators. Light is stopped to a standstill owing to the divergent wavelength and the vanishing group velocity, effectively rendering, through nonlinearity, a positiveepsilon trapping cavity carved in an otherwise slightlynegativeepsilon medium. By numerically solving Maxwell’s equations, we find a solitonlike family of still azimuthal doughnuts, which we further study through an adiabatic perturbative theory that describes soliton evaporation in lossy media or condensation in actively pumped materials. Our results suggest applications in optical data processing and storage, quantum optical memories and solitonbased lasers without cavities. Additionally, nearzeroindex conditions can also be found in the interplanetary medium and in the atmosphere, where we provide a complementary explanation to the rare phenomenon of balllightning.
Introduction
Spatial and temporal selftrapping^{1,2,3,4} occur in several optical systems, including photorefractive media^{5,6}, liquid crystals^{7,8} and metamaterials^{9,10}. Remarkably, nonlinearity can act simultaneously on temporal and spatial domains to compensate for both diffraction and dispersion, thus enabling the formation of light bullets, spatiotemporal doughnuts and Xshaped waves^{11,12,13,14,15}.
Physical systems enabling either slow or fast light^{16,17,18} naturally enhance radiationmatter interaction, thus boosting nonlinear processes that can be efficiently used for active light control^{19}, alloptical switching and modulation^{20,21}. In particular, nearzeroindex (NZI) media can slow down light propagation^{22,23,24} and enable extreme nonlinear dynamics^{25}, enhanced second and third harmonic generation^{26}, active control of tunneling^{27}, optical switching and bistable response^{28}. These materials naturally exist in nature, for example plasmas, transparent conductors and metals near their bulk plasma frequency ^{29}. Besides, they can be artificially realized as waveguides close to modal cutoff^{30}, using surface phonon polaritons in GaAs quantum wells^{31}, or by engineering subwavelength metallic nanowires, nanospheres, or nanocircuits embedded in dielectric matrices. The latter strategy has enabled the development of epsilonnearzero (ENZ) metamaterials, which have been investigated for applications such as enhanced transmission^{32}, cloaking^{33}, energy squeezing in narrow channels^{34} and subwavelength imaging^{35,36}. The ENZ regime is inevitably associated with high dispersion and is therefore accompanied by absorption, which can be suppressed by embedding externally pumped active inclusions in the NZI medium^{37}.
Here we theoretically investigate selforganization of light in NZI media with Kerrlike instantaneous nonlinearity. In particular, we reveal the existence of fully confined doughnutshaped solitons with vanishing Poynting vector and angular momentum. In practice nonlinearity enables digging a threedimensional cavity for light, which in turn remains frozen and selftrapped. We study the effect of loss on stationary light doughnuts by developing a fully numerical soliton perturbative theory, finding that they evaporate over time due to absorption: their amplitude decreases, their frequency blueshifts slightly and their radius increases. Conversely, if externally pumped active inclusions with inversion of population are embedded within the NZI medium, the opposite scenario takes place and azimuthal doughnuts condensate over time. These findings demonstrate the possibility to freeze light beams in ENZ media, with potential applications in optical data processing and storage, quantum optical memories and NZI lasers operating without cavities. Interestingly, ENZ conditions are found also in the interplanetary medium and in the atmosphere and we argue that our theoretical results may provide insight into balllightning (BL) formation^{38,39,40}.
Results and Discussion
Still light
We consider a generic NZI medium with Drude temporal response and instantaneous Kerrlike nonlinearity (see Methods). Both of these ingredients ensue from freeparticle temporal dynamics, which is characteristic of plasmas, metals, transparent conductors and ENZ metamaterials, all examples of NZI media. In particular, Kerrlike nonlinearity naturally arises from the ponderomotive force in plasmas and metals^{41}.
In the linear limit, homogeneous transverse electromagnetic (TEM) waves are solutions of Maxwell’s equations with a complex electric field given by , where . The angular frequency ω and the wavevector k satisfy the dispersion relation , where c is the speed of light in free space and is the frequencydependent dielectric constant, which is given by the Fourier transform of the Drude temporal response function (see Methods). The material dispersion basically depends on two constants: the plasma frequency and the damping rate γ. The linear dispersion relation of TEM waves is depicted in Fig. 1 in the lossless limit , together with the phase and group velocities . Note the cutoff of TEM waves at the plasma frequency , where the medium enters the ENZ regime, the phase velocity diverges and the group velocity vanishes^{22,23,24}.
Homogeneous nonlinear modes
Owing to the vanishing group velocity, nonlinear effects are dramatically enhanced in the ENZ regime^{25,26}. For , homogeneous modes with vanishing wavenumber, infinite phase velocity and zero group velocity can be found by neglecting damping and setting , with and being the material’s Kerr coefficient (see Methods). The resulting dispersion relation is plotted in Fig. 2(a). We find zeroindex homogeneous modes to have a cutoff at the plasma frequency , where the electric field amplitude drops to zero. In order to evaluate the stability of homogeneous modes, we perturb them with smallamplitude waves: , where are the perturbation amplitudes with wavevector q and temporal growth eigenvalue α. Inserting this expression in the Maxwell’s equations and retaining only the lowestorder terms in and , we find a homogeneous system of linear equations, whose nontrivial solutions are signaled by the vanishing of the secular determinant [see supplementary information (SI) for more details on the technical aspects of the theory]. This condition determines the complex temporal eigenvalues α. Instabilities are then associated with positive real parts of the eigenvalue α, indicating unbound amplification of the perturbation. We plot results of the stability analysis in Fig. 2(b–d) and in particular, we depict the maximum of the real part of the eigenvalue, . In analogy to standard modulation instability in 1D paraxial systems^{3}, the gain spectrum of the perturbations is nonvanishing within a finite wavevector window and is peaked at a characteristic wavevector modulus. However, in contrast to 1D paraxial systems, the gain spectrum is 3D and has a nontrivial dependence on polar and azimuthal angles (θ,ϕ) of the perturbation wavevector q.
Still azimuthal doughnuts
The modulation instability scenario strongly suggests the presence of still 3D solitons in NZI media. In order to verify this hypothesis, we transform Maxwell’s equations into spherical coordinates and search for azimuthallypolarized solutions: . As Maxwell’s equations are invariant under a constant phase shift (see Methods), without any loss of generality we can assume that the electric field envelope is real , meaning that we are seeking nonpropagating solutions which are not accompanied by a phase flow. Indeed, assuming that such solutions exist, we show that the Poynting vector vanishes thoroughly (see SI). Besides, we seek localized solitonlike solutions vanishing at and at , owing to the azimuthal polarization. Upon examination of the asymptotical expansion of Maxwell’s equations for , we find that 3D solitonlike azimuthal solutions can actually exist only in the ENZ regime (see SI). Thus, we discretize derivatives with respect to the radius and the polar angle and then transform the differential wave equation for the electric field into a nonlinear algebraic system for the electric field amplitudes in the twodimensional grid (see SI). We solve this nonlinear algebraic system by means of an iterative NewtonRaphson algorithm and find a family of still azimuthal doughnuts [see Fig. 3(a)] for , which presents a cutoff at , where the soliton loses localization and its amplitude vanishes. The frequencydependent maximum amplitude and the corresponding radius of the still doughnut family are plotted in Fig. 3(b), while a rθ contourplot of the squared electric field profile (normalized to the scaling field of the still doughnut at is depicted in Fig. 3(c). The total dielectric permittivity profile is shown in Fig. 3(d). Importantly, in the soliton existence domain , the linear dielectric constant is negative and thus, at long radius where the electric field amplitude is small, the NZI medium is metallike. Conversely, in the volume around the radius for which the electric field is maximum, nonlinearity is nonnegligible and the total dielectric permittivity is positive (dielectriclike). From here we see that the existence of still azimuthal doughnuts originates in the extraordinary ability of nonlinearity to dig a dielectriclike 3D cavity within a metallike environment. This scenario is unique of NZI media, which prevent propagation of the fields outside the induceddielectric trapping cavity. We emphasize that modulation instability enables the excitation of nonpropagating solitons starting directly from unstable homogeneous waves with frequency falling in the ENZ regime.
Doughnut evaporation/condensation
In standard transparent media, the main quantity accounting for optical propagation is the Poynting vector, representing the temporal rate of energy transfer per unit area. For our trapped solitons, the Poynting vector is thoroughly vanishing (see SI), so we describe doughnut selftrapping through the opticalcycleaveraged density of electromagnetic energy. Now, if absorption is taken into account, the energy density is expected to be damped and vanish exponentially over time. A numerical verification of this hypothesis could consist in temporally evolving Maxwell’s equation with the doughnut initial condition. However, temporal evolution requires nonlinear 3D finitedifferencetimedomain (FDTD) numerical simulations, which are computationally demanding. Besides, traditional approaches used in dielectric and plasmonic waveguides^{42,43,44} relying on the slowlyvaryingenvelope approximation (SVEA) can not be used, as the SVEA does not hold in the ENZ regime^{25}. Instead, we have developed a soliton perturbation theory (see SI) capable of accounting for both damping and amplification (e.g., in systems containing externally pumped active inclusions within the NZI medium) under the assumption that (i) damping or (ii) gain are much smaller than the soliton angular frequency ω. We further assume that the temporal evolution of the still doughnut adiabatically follows the soliton family, finding that the soliton amplitude (i) decays or (ii) increases over time following the exponential law , where is the initial field amplitude and γ is a phenomenological absorption/pumping rate. Accordingly, the doughnut (i) expands and blueshifts or (ii) shrinks and redshifts in either case (see SI). The timedependent field amplitude (blue left yaxis) and doughnut radius (red right yaxis) are plotted in Fig. 4(a) for a representative example, along with three snapshots of the isosurface at different times in Fig. 4(b–d), where we have assumed as initial condition the doughnut of Fig. 3(a,i) damping (For the full temporal evolution see movie in the SI). The doughnut evaporates over time, as its amplitude decreases and its radius increases. The gain scenario (ii) can be interpreted by inverting the temporal direction, so that the doughnut condensates over time, as its amplitude increases and its radius decreases.
Balllightnings?
BLs are rare lightning events with hitherto unknown theoretical explanation^{38,39,40}. BLs emit broadband radiation and can either propagate or stand still. Initially considered as myth, BLs have puzzled scientists for centuries and their existence has been questioned until the first recent experiment able to measure their spectrum^{40}. Understanding of the nature of BLs is still unsatisfactory as they can not be easily reproduced in laboratory. Among the several theories trying to explain their nature, the socalled masercaviton theory^{38} suggests that BLs are localized highfield solitons forming a cavity surrounded by plasma. Indeed, during thunderstorms, atmosphere can get ionized and become a NZI medium with a plasma frequency falling in the terahertzmicrowave spectral region, where rotational levels of water can be excited. The ensuing emitted radiation is thought to remain selftrapped and heat up the air, thus emitting broadband blackbody radiation^{38}. This theory explains several aspects of BLs, e.g., their typical size and their motion due to plasma density perturbations, but it does not provide any quantitative description of the selfinduced soliton cavity. Following our rigorous calculations, as suggested by the masercaviton theory, we speculate that BLs may actually ensue from a selforganization process in the ENZ regime, where we theoretically demonstrate the existence of still doughnut solitons, as discussed above. The actual spherical shape of BLs observed in experiments^{40} may be due to mixed polarization, heating and higher order nonlinear effects, or the intrinsically incoherent nature of radiation emitted in the atmosphere. The ENZ condition would explain the infrequency of the phenomenon and provides an insightful signature for experimental investigations.
Conclusions
Our investigation of selforganization phenomena in NZI media with cubic nonlinearity has resulted in the demonstration that zeroindex nonlinear waves are unstable in all spatial directions and that still azimuthally polarized selftrapped doughnuts can be excited. We have discussed the existence domain of this 3D soliton family with thoroughly vanishing Poynting vector and provided details on its characteristics. Besides, we have studied the effect of loss/amplification, finding that still light doughnuts evaporate/condensate over time, respectively. Our model applies to any NZI medium with cubic nonlinearity and our results are universal as they are rescaled to the relevant physical quantities (plasma frequency , plasma wavevector , Kerr coefficient of any specific medium in this regime (e.g., metals, transparent conductors, plasmas and metaldielectric ENZ metamaterials). Our findings pave the way for the development of novel applications in optical data processing and storage, the realization of quantum optical memories and the design of solitonbased lasers without cavities. Incidentally, NZI conditions can be found also in the interplanetary medium and in the atmosphere and we have discussed possible relationships between our results and balllightning formation.
Methods
Model
In our investigations we have considered a generic NZI medium with Drude temporal response and instantaneous Kerrlike nonlinearity. Both of these ingredients ensue from freeparticle temporal dynamics, which is characteristic of plasmas, metals, transparent conductors and ENZ metamaterials, all examples of NZI media. In particular, Kerrlike nonlinearity naturally arises from the ponderomotive force in plasmas and metals^{41} and is well represented by the constitutive relation between the displacement vector and the electric field :
where is the vacuum permittivity, is the nonlinear susceptibility of the medium, is the Drude temporal response function, is the Dirac deltafunction, is the plasma frequency and γ is the temporal damping rate due to inelastic collisions. Optical propagation is governed by the wave equation
where is the vacuum permeability.
Still homogeneous waves
Homogeneous nonlinear modes with vanishing group velocity and diverging phase velocity have been calculated by inserting the Ansatz in Eq. (2), which in turn enables the calculation of the nonlinear dispersion.
In order to evaluate the stability of homogeneous modes, we have perturbed them with smallamplitude waves
where and are the perturbation amplitudes with wavevector q and temporal growth eigenvalue α. We have thus numerically calculated the complex eigenvalues α (which real parts represent the instability growth rates) of the ensuing homogeneous system of algebraic equations (see SI).
Still azimuthal doughnuts
Given the isotropic nature of the system, the most natural coordinates to calculate 3D solitons are spherical , where r is the modulus of the position vector and are its polar and azimuthal angles (see SI). In our calculations we have assumed that the electric field does not depend on the azimuthal angle and thus it is polarized along the azimuthal direction . Still nonparaxial solitonlike solutions of Maxwell’s equations have been calculated numerically by transforming the continuous variables into a discrete twodimensional grid with steps and the azimuthal electric field into an ordered vector . Approximating derivatives by finite differences, Eq. (2) becomes a nonlinear system of algebraic equations, which we have numerically solved through the NewtonRaphson method.
Doughnut evaporation/condensation
We have accounted for the effect of damping/amplification in the temporal domain through an electric field amplitude oscillating with angular frequency ω and exponentially decaying/increasing over time at a rate . The soliton perturbative theory is then developed by assuming that, under the assumption of small damping/amplification , at every time t, the field pattern adiabatically follows the unperturbed soliton family with timedependent maximum amplitude , radius and angular frequency . Inserting the expression for the electric field into Eqs. (1) and (2) and making use of the adiabatic approximation, one obtains the timedependent soliton parameters , and
where , and are the field amplitude, radius and angular frequency of the soliton at the initial time , respectively.
Additional Information
How to cite this article: Marini, A. and Abajo, F. J.G. Selforganization of frozen light in nearzeroindex media with cubic nonlinearity. Sci. Rep. 6, 20088; doi: 10.1038/srep20088 (2016).
References
Trillo, S. & Torruellas, W. E. (eds) Spatial Solitons. (Springer, Berlin, 2001).
Agrawal, G. P. Nonlinear Fiber Optics. (Academic, New York, 2001).
Kivshar, Y. S. & Agrawal, G. P. Optical Solitons: From Fibers to Photonic Crystals. (Academic, New York, 2003).
Chen, Z., Segev, M. & Christodoulides, D. N. Optical spatial solitons: historical overview and recent advances. Rep. Prog. Phys. 75, 086401 (2012).
DelRe, E. et al. Onedimensional steadystate photorefractive spatial solitons in centrosymmetric paraelectric potassium lithium tantalate niobate. Opt. Lett. 23, 421–423 (1998).
Ciattoni, A., Rizza, C., DelRe, E. & Marini, A. Lightinduced dielectric structures and enhanced selffocusing in critical photorefractive ferroelectrics. Opt. Lett. 34, 3295–3297 (2009).
Conti, C., Peccianti, M. & Assanto, G. Observation of Optical Spatial Solitons in a Highly Nonlocal Medium. Phys. Rev. Lett. 92, 113902 (2004).
Peccianti, M., Conti, C., Assanto, G., De Luca, A. & Umeton, C. Routing of anisotropic spatial solitons and modulational instability in liquid crystals. Nature 432, 733–737 (2004).
Shadrivov, I. V. & Kivshar, Y. S. Spatial solitons in nonlinear lefthanded metamaterials. J. Opt. A: Pure Appl. Opt. 7, S68 (2005).
Dong, H., Conti, C., Marini, A. & Biancalana, F. Terahertz relativistic spatial solitons in doped graphene metamaterials. J. Phys. B: At. Mol. Opt. Phys. 46, 155401 (2013).
Mihalache, D., Mazilu, D., Crasovan, L.C., Malomed, B. A. & Lederer, F. Threedimensional spinning solitons in the cubicquintic nonlinear medium. Phys. Rev. Lett. 61, 7142 (2000).
Conti, C. et al. Nonlinear Electromagnetic X Waves. Phys. Rev. Lett. 90, 170406 (2003).
Malomed, B. A., Mihalache, D., Wise, F. & Torner, L. Spatiotemporal optical solitons. J. Opt. B: Quantum Semiclass. Opt. 7, R53 (2005).
Torner, L. & Kartashov, Y. V. Light bullets in optical tandems. Opt. Lett. 34, 1129–1131 (2009).
Minardi, S. et al. Threedimensional light bullets in arrays of waveguides. Phys. Rev. Lett. 105, 263901 (2010).
Tsakmakidis, K. L., Boardman, A. D. & Hess, O. ‘Trapped rainbow storage of light in metamaterials. Nature 450, 397–401 (2007).
Boyd, R. W. Slow and fast light: fundamentals and applications. J. of Mod. Opt. 56, 1908–1915 (2009).
Kim, K.H., Husakou, A. & Herrmann, J. Slow light in dielectric composite materials of metal nanoparticles. Opt. Express 20, 25790–25797 (2012).
Vlasov, Y. A., O’Boyle, M., Hamann, H. F. & McNab, S. J. Active control of slow light on a chip with photonic crystal waveguides. Nature 438, 65–69 (2005).
Mingaleev, S. F., Miroshnichenko, A. E., Kivshar, Y. S. & Busch, K. Alloptical switching, bistability and slowlight transmission in photonic crystal waveguideresonator structures. Phys. Rev. E 74, 046603 (2006).
Bajcsy, M. et al. Efficient alloptical switching using slow light within a hollow fiber. Phys. Rev. Lett. 102, 203902 (2009).
Ciattoni, A., Marini, A., Rizza, C., Scalora, M. & Biancalana, F. Polariton excitation in epsilonnearzero slabs: Transient trapping of slow light. Phys. Rev. A 87, 053853 (2013).
D’Aguanno, G. et al. Frozen light in a nearzero index metasurface. Phys. Rev. B 90, 054202 (2014).
Newman, W. D. et al. Ferrellâ Berreman Modes in Plasmonic EpsilonnearZero Media. ACS Photon. 2, 2–7 (2015).
Ciattoni, A., Rizza, C. & Palange, E. Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity. Phys. Rev. A 81, 043839 (2010).
Vincenti, M. A., de Ceglia, D., Ciattoni, A. & Scalora, M. Singularitydriven secondand thirdharmonic generation at εnearzero crossing points. Phys. Rev. A 84, 063826 (2011).
Powell, D. A. et al. Nonlinear control of tunneling through an epsilonnearzero channel. Phys. Rev. B 79, 245135 (2009).
Argyropoulos, C., Chen, P.Y., D’Aguanno, G., Engheta, N. & Alù, A. Boosting optical nonlinearities in εnearzero plasmonic channels. Phys. Rev. B 85, 045129 (2012).
Raether, H. Excitation of Plasmons and Interband Transitions by Electrons (Springer, Berlin, 1980).
Vesseur, E. J. R., Coenen, T., Caglayan, H., Engheta, N. & Polman, A. Experimental verification of n = 0 structures for visible light. Phys. Rev. Lett. 110, 013902 (2013).
Vassant, S. et al. Epsilonnearzero mode for active optoelectronic devices. Phys. Rev. Lett. 109, 237401 (2012).
Alù, A., Bilotti, F., Engheta, N. & Vegni, L. Metamaterial covers over a small aperture. IEEE Trans. Ant. Propag. 54, 1632–1643 (2006).
Alù, A. & Engheta, N. Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623 (2005).
Silveirinha, M. G. & Engheta, N. Tunneling of electromagnetic energy through subwavelength channels and bends using εnearzero materials. Phys. Rev. Lett. 97, 157403 (2006).
Alù, A., Silveirinha, M. G., Salandrino, A. & Engheta, N. Epsilonnearzero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern. Phys. Rev. B 75, 155410 (2007).
Castaldi, G., Savoia, S., Galdi, V., Alù, A. & Engheta, N. Analytical study of subwavelength imaging by uniaxial epsilonnearzero metamaterial slabs. Phys. Rev. B 86, 115123 (2012).
Rizza, C., Di Falco, A. & Ciattoni, A. Gain assisted nanocomposite multilayers with near zero permittivity modulus at visible frequencies. Appl. Phys. Lett. 99, 221107 (2011).
Handel, P. H. & Leitner, J.F. Development of the masercaviton ball lightning theory. J. Geophys. Res.: Atmospheres 99, 10689–10691 (1994).
Abrahamson, J. & Dinniss, J. Ball lightning caused by oxidation of nanoparticle networks from normal lightning strikes on soil. Nature 403, 519–521 (2000).
Cen, J., Yuan, P. & Xue, S. Observation of the optical and spectral characteristics of ball lightning. Phys. Rev. Lett. 112, 035001 (2014).
Ginzburg, P., Hayat, A., Berkovitch, N. & Orenstein, M. Nonlocal ponderomotive nonlinearity in plasmonics. Opt. Lett. 35, 1551–1553 (2010).
Afshar, S. V. & Monro, T. M. A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity. Opt. Express 17, 2298–2318 (2009).
Marini, A., Hartley, R., Gorbach, A. V. & Skryabin, D. V. Surfaceinduced nonlinearity enhancement in subwavelength rod waveguides. Phys. Rev. A 84, 063839 (2011).
Skryabin, D. V., Gorbach, A. V. & Marini, A. Surfaceinduced nonlinearity enhancement of TM modes in planar subwavelength waveguides. J. of the Opt. Soc. of Am. B 28, 109–114 (2011).
Acknowledgements
A.M. is supported by an ICFOnest+ Postdoctoral Fellowship (Marie Curie COFUND program). A.M. acknowledges fruitful discussions about ENZ media with Alessandro Ciattoni and Carlo Rizza. A.M. acknowledges interesting discussions about globular lightnings with Mario Raparelli and Andrea Aiello.
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A.M. conceived the idea, worked out the theory and performed numerical calculations. F.J.G.d.A. supervised the research. A.M. and F.J.G.d.A. discussed the results and wrote the paper.
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Marini, A., García de Abajo, F. Selforganization of frozen light in nearzeroindex media with cubic nonlinearity. Sci Rep 6, 20088 (2016). https://doi.org/10.1038/srep20088
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