Experimental demonstration of metamaterial multiverse in a ferrofluid

Transcription

1 Experimental emonstration of metamaterial multiverse in a ferroflui Igor I. Smolyaninov, 1,* Braley Yost, Evan Bates, an Vera N. Smolyaninova 1 Department of Electrical an Computer Engineering, University of Marylan, College Park, MD 074, USA Department of Physics Astronomy an Geosciences, Towson University, 8000 York R., Towson, MD 15, USA * smoly@um.eu Abstract: Extraorinary light rays propagating insie a hyperbolic metamaterial look similar to particle worl lines in a +1 imensional Minkowski spacetime. Magnetic nanoparticles in a ferroflui are known to form nanocolumns aligne along the magnetic fiel, so that a hyperbolic metamaterial may be forme at large enough nanoparticle concentration n H. Here we investigate optical properties of such a metamaterial just below n H. While on average such a metamaterial is elliptical, thermal fluctuations of nanoparticle concentration lea to transient formation of hyperbolic regions (3D Minkowski spacetimes) insie this metamaterial. Thus, thermal fluctuations in a ferroflui look similar to creation an isappearance of iniviual Minkowski spacetimes (universes) in the cosmological multiverse. This theoretical picture is supporte by experimental measurements of polarization-epenent optical transmission of a cobalt base ferroflui at 1500 nm. OCIS coes: ( ) Metamaterials; ( ) Nanomaterials. References an links 1. I. I. Smolyaninov an Y. J. Hung, Moeling of time with metamaterials, JOSA B 8, (011).. I. I. Smolyaninov, Critical opalescence in hyperbolic metamaterials, Journal of Optics 13, (011). 3. D. A. Genov, S. Zhang, an X. Zhang, Mimicking celestial mechanics in metamaterials, Nature Physics 5, (009). 4. E. E. Narimanov an A. V. Kilishev, Optical black hole: Broaban omniirectional light absorber, Appl. Phys. Lett. 95, (009). 5. A. Greenleaf, Y. Kurylev, M. Lassas, an G. Uhlmann, Electromagnetic wormholes an virtual magnetic monopoles from metamaterials, Phys. Rev. Lett. 99, (007). 6. I. I. Smolyaninov, Metamaterial-base moel of the Alcubierre warp rive, Phys. Rev. B 84, (011). 7. T. G. Mackay an A. Lakhtakia, Towars a metamaterial simulation of a spinning cosmic string, Phys. Lett. A 374, (010). 8. I. I. Smolyaninov an Y. J. Hung, Minkowski omain walls in hyperbolic metamaterials, Phys. Letters A 377, (013). 9. I. I. Smolyaninov, Metamaterial Multiverse, Journal of Optics 13, (010). 10. I. I. Smolyaninov an E. E. Narimanov, Metric signature transitions in optical metamaterials, Phys. Rev. Letters 105, (010). 11. I. I. Smolyaninov, E. Hwang, an E. E. Narimanov, Hyperbolic metamaterial interfaces: Hawking raiation from Rinler horizons an spacetime signature transitions, Phys. Rev. B 85, 351 (01). 1. I. I. Smolyaninov, Vacuum in strong magnetic fiel as a hyperbolic metamaterial, Phys. Rev. Letters 107, (011). 13. I. I. Smolyaninov, Planck-scale physics of vacuum in a strong magnetic fiel, Phys. Rev. D 85, (01). 14. I. I. Smolyaninov, Quantum electromagnetic black holes in a strong magnetic fiel, J. Phys. G: Nucl. Part. Phys. 40, (013). 15. Y. Gao, J. P. Huang, Y. M. Liu, L. Gao, K. W. Yu, an X. Zhang, Optical negative refraction in ferrofluis with magnetocontrollability, Phys. Rev. Letters 104, (010).

2 16. M. Tegmark, "Parallel Universes". In "Science an Ultimate Reality: from Quantum to Cosmos," honoring John Wheeler's 90th birthay. J. D. Barrow, P.C.W. Davies, & C.L. Harper es. Cambrige University Press (003). 17. R. Wangberg, J. Elser, E. E. Narimanov, an V. A. Poolskiy, Nonmagnetic nanocomposites for optical an infrare negative-refractive-inex meia, J. Opt. Soc. Am. B 3, (006). 18. L. D. Lanau, E. M. Lifshitz, an L. P. Pitaevskii, Course of Theoretical Physics (Ree, 1984). Vol CRC Hanbook of Chemistry an Physics, D. R. Lie es. (CRC Press, 005). 0. T. Tumkur, G. Zhu, P. Black, Yu. A. Barnakov, C. E. Bonner, an M. A. Noginov, Control of spontaneous emission in a volume of functionalize hyperbolic metamaterial, Appl. Phys. Letters 99, (011). 1. I. I. Smolyaninov an A. V. Kilishev, Light propagation through ranom hyperbolic meia, Optics Letters 38, (013). 1. Introuction Recent avances in electromagnetic metamaterials an transformation optics gave rise to consierable progress in moeling unusual spacetime geometries, such as spacetime geometry near the big bang [1], black holes [-4], wormholes [5], Alcubierre warp rive [6], spinning cosmic strings [7], Minkowski omain wall [8], an even metamaterial multiverse [9]. Hyperbolic metamaterials are especially interesting in this respect since extraorinary rays in a hyperbolic metamaterial behave as particle worl lines in a three imensional (+1) Minkowski spacetime [1,10]. When this spacetime is curve, metamaterial analogs of black holes [11] an the big bang [1] may be create. It appears that in a very strong magnetic fiel physical vacuum itself behaves as a hyperbolic metamaterial [1-14], so the metamaterial spacetime analogs appear to be quite meaningful. While theoretical progress of this fiel is impressive, experimental evelopments appear to be consierably slower. Fabrication of 3D metamaterial structures require to accurately represent exotic spacetime moels is extremely ifficult, so experimental emonstrations are mostly limite to simplifie D geometries [1,8,11]. Here we investigate a promising way to bypass these experimental ifficulties by using ferrofluis. Magnetic nanoparticles in a ferroflui are known to form nanocolumns aligne along the magnetic fiel [15], so that a wire array hyperbolic metamaterial may be forme at large enough magnetic nanoparticle concentration n H. We investigate optical properties of such a metamaterial just below n H. While on average such a metamaterial is elliptical, thermal fluctuations of the nanoparticle concentration lea to transient formation of hyperbolic regions insie this metamaterial. Extraorinary light rays insie these regions look similar to particle worl lines in a +1 imensional Minkowski spacetime [1]. Thus, thermal fluctuations in a ferroflui give rise to transient Minkowski spacetimes which are somewhat analogous to iniviual Minkowski universes which appear an isappear as part of the larger cosmological multiverse [16]. This theoretical picture is supporte by experimental measurements of polarization-epenent optical transmission of a cobalt base ferroflui at 1500 nm.. Hyperbolic metamaterials as Minkowski spacetime analogs As a first step, let us recall the analogy between extraorinary light propagation in hyperbolic metamaterials an worl lines in 3D Minkowski spacetime, which is escribe in etail in refs. [1,10]. To better unerstan this analogy, let us start with a non-magnetic uniaxial anisotropic material with ielectric permittivities x = y = 1 an z =. Any electromagnetic fiel propagating in this material can be expresse as a sum of orinary an extraorinary contributions, each of these being a sum of an arbitrary number of plane waves polarize in the orinary ( E 0 ) an extraorinary ( z E z 0) irections. Let us efine our scalar extraorinary wave function as ϕ=e z so that the orinary portion of the electromagnetic fiel oes not contribute to ϕ. Maxwell equations in the frequency omain results in the following wave equation for ϕ ω if 1 an are kept constant insie the metamaterial [1,10]:

3 ω ϕ c ϕω z 1 ϕ ω ϕω + x y ω = 1 While in orinary elliptic anisotropic meia both 1 an are positive (corresponing to effective spacetime being Eucliean space), in hyperbolic metamaterials 1 an have opposite signs. These metamaterials are typically compose of multilayer metal-ielectric or metal wire array structures, as shown in Fig. 1. Let us consier the case of constant 1 >0 an <0, an assume that this behavior hols in some frequency range aroun ω=ω 0. Let us assume that the metamaterial is illuminate by coherent CW laser fiel at frequency ω 0, an we stuy spatial istribution of the extraorinary fiel ϕ ω at this frequency. Uner these assumptions Eq. (1) coincies with the 3D Klein-Goron equation escribing a massive scalar fiel ϕ ω in a +1 imensional Minkowski spacetime. Note that the spatial coorinate z=τ behaves as a timelike variable in Eq. (1). When a metamaterial is built an illuminate with a coherent extraorinary CW laser beam at frequency ω=ω 0, the stationary pattern of light propagation insie the metamaterial represents a complete history of a toy (+1) imensional Minkowski spacetime. This history is written as a collection of particle worl lines along the timelike z coorinate. If aiabatic variations of 1 an are allowe insie the metamaterial, worl lines of massive particles in some well known curvilinear spacetimes can be emulate, incluing the worl line behavior near the beginning of time at the moment of big bang [1]. Thus, mapping of monochromatic extraorinary light istribution in a hyperbolic metamaterial along some spatial irection may moel the flow of time in an effective three imensional (+1) spacetime. (1) Fig. 1. Typical geometries of hyperbolic metamaterials: (a) multilayer metalielectric structure (b) metal wire array structure (c) effective meium parameters of the ferroflui metamaterial. 3. Ferrofluis in external magnetic fiel as hyperbolic metamaterials Ferrofluis are colloial suspensions of nanoscale (<10 nm) ferromagnetic particles in a carrier flui. Each ferromagnetic nanoparticle is coate with a surfactant to inhibit clumping. Due to this coating, magnetic attraction of nanoparticles becomes weak enough so that no particle agglomeration occurs in the absence of external magnetic fiel. On the other han, when magnetic fiel is applie, ferromagnetic particles form nanocolumns, which are aligne along the magnetic fiel irection (see [15] an references therein). As a result, ferroflui geometry becomes similar to metal wire array hyperbolic metamaterial structure shown in Fig. 1(b) with the column iameter approximately the same (10 nm) as nanoparticle size. Let us apply the stanar metamaterial escription to this meium. Diagonal components of the ferroflui ielectric tensor may be obtaine using Maxwell-Garnett approximation [17]: = z = n m + ( 1 n) () n m + (1 n) ( + m) 1 = x, y = (3) 1 n ( + ) + n ( ) m

4 where n is the average volume fraction of the ferromagnetic nanoparticle phase, an m an are the ielectric permittivities of the ferromagnetic an liqui phase, respectively. We are intereste in experimental situations which arise when m <0, which correspon to ferromagnetic particles being metallic or being coate with metal. In such situations the ferroflui becomes a hyperbolic metamaterial if n > n H = (4) m At this nanoparticle concentration changes sign from positive to negative, while 1 remains positive if m >>. We are intereste in optical properties of ferrofluis just below n H. While on average such a ferroflui remains usual elliptical material, thermal fluctuations of the nanoparticle concentration n lea to transient formation of hyperbolic regions insie the ferroflui. As escribe above, extraorinary light rays insie these regions look similar to particle worl lines in a +1 imensional Minkowski spacetime [1]. Thus, thermal fluctuations in a ferroflui give rise to transient Minkowski spacetimes which are somewhat analogous to iniviual Minkowski universes which appear an isappear as part of the larger cosmological multiverse [16]. Accoring to the concept of chaotic inflation, an infinite multiverse must contain Hubble volumes realizing all kins of physical laws an initial conitions. In particular, an infinite multiverse will contain an infinite number of 3+1 imensional Minkowski spacetimes (Hubble volumes), some of them being virtually ientical to ours. It appears that fluctuating ferrofluis exhibit some similarity to this theoretical picture, albeit on a smaller scale an in smaller number of spacetime imensions: insie the ferroflui fluctuating +1D Minkowski spacetimes appear an isappear insie a larger 3D Eucliean space. 4. Thermal fluctuations of the nanoparticle volume fraction in ferrofluis Let us evaluate thermal fluctuations of the ielectric tensor in ferrofluis. In principle, both m an experience thermal fluctuations ue to thermal fluctuations of metal an liqui ensities. However, ensity fluctuations in liquis are proportional to ( V / P) T [18], which is very small in a typical (incompressible) liqui far from its critical temperature. Thus, fluctuations of 1 an must be ominate by thermal fluctuations of the nanoparticle volume fraction n (see Eqs. () an (3)). If N is the number of nanoparticles in a given volume V of the ferroflui, its stanar eviation ue to thermal fluctuations is [18] ( ΔN ) = N (5) Thus, stanar eviation of the nanoparticle volume fraction ue to thermal fluctuations is 1/ 1/ ( ) 1/ n v Δ n = (6) 1/ V where v is the volume of iniviual nanoparticle. In orer for the macroscopically average metamaterial escription to be vali, we nee V>>v. Assuming V>10v limitation, the range 1/ of acceptable volume fraction fluctuations is ( Δ n) n 1/ / 3. At this fluctuation level the metamaterial escription remains applicable. This consieration emonstrates that it is possible to choose a ferroflui having average n<n H, so that on average this ferroflui will be a usual elliptical material, while consierable fraction of its volume will behave as a hyperbolic metamaterial ue to thermal fluctuations of n. The local value of n in the hyperbolic areas may temporarily excee n H ue to thermal fluctuations. We shoul also note that characteristic time scale of these fluctuations is much larger than the inverse light frequency at 1500 nm. Therefore, macroscopic electroynamics escription of these areas as hyperbolic metamaterials remains vali.

5 5. Experimental sample an setup For our experiments we have chosen cobalt magnetic flui from Strem Chemicals compose of 10 nm cobalt nanoparticles in kerosene with AOT (soium ioctylsulfosuccinate) an LP4 (a fatty aci conensation polymer). The average volume fraction of cobalt nanoparticles in this ferroflui is 8.%, which is below n H at 1500 nm light wavelength: the value of n H= 17% can be calculate using Eq. (4) base on the ielectric constants of kerosene =1.93 an cobalt Re m =-9.0 at 1500 nm [19]. Effective meium parameters of the ferroflui in external magnetic fiel calculate using Eqs. (,3) are plotte in Fig. 1(c). Accoring to Eq. (6), for a (50 nm) 3 1/ volume of the ferroflui ( Δn ) = Thus, at any given time.3% of such volume elements shoul have the local value of n above n H (this correspons to ~σ eviation from the average nanoparticle concentration). A similar estimate for a (30 nm) 3 1/ volume of the ferroflui prouces ( Δn ) = so that 16% of such volume elements shoul have the local value of n above n H (this correspons to ~1σ eviation). These volume elements shoul exhibit transient hyperbolic behavior. The cutoff of the hyperbolic ispersion law has been analyze in ref. [10]. It is efine by the metamaterial lattice constant. In our case, 17% volume concentration of nanoparticles results in average inter-particle istance of 18 nm. Thus, escription of (50nm) 3 volumes as volumes occupie by a hyperbolic metamaterial oes make sense. Typical instantaneous istribution of fluctuating hyperbolic regions in our sample calculate using Eqs. (6) an () is presente in Fig.. The instantaneous xy epenence of the particle number was calculate using a ranom number generator base on the stanar eviation given by Eq. (5). If the local instantaneous value of n(x,y) exceee n H, the ispersion law is locally hyperbolic. These hyperbolic regions are shown in re. As escribe above, hyperbolic regions in a ferroflui shown in Fig. behave as transient +1 imensional Minkowski spacetimes, which temporarily appear an isappear insie a larger metamaterial multiverse. Fig.. Typical instantaneous istribution of fluctuating hyperbolic regions in the cobalt-base ferroflui sample calculate using Eq. (6). Hyperbolic regions are shown in re. These regions behave as transient +1 imensional Minkowski spacetimes which temporarily appear an iappear insie a larger metamaterial multiverse.

6 Note that ferrofluis base on gol an silver-coate ferromagnetic nanoparticles propose in [15] woul require consierably smaller values of n~%, an therefore woul exhibit much smaller losses an much more pronounce hyperbolic behavior at 1500 nm. Therefore, similar experiments with gol an silver-coate ferromagnetic nanoparticles may be conucte much closer to the critical value n H, leaing to consierably larger size of transient hyperbolic regions insie the ferroflui. We have examine polarization-epenent optical transmission of the cobalt base ferroflui at 1500 nm as a function of external magnetic fiel. Our experimental setup is shown schematically in Fig. 3(a). Linear polarize light from a 1500 nm laser has been sent onto the ferroflui sample via a λ/4 plate, which mae the illuminating light circular polarize. The cobalt base ferroflui was place in a 10 μm thick optical cuvette locate between the poles of a magnet. Polarization state of the transmitte light has been analyze using a polarizer as a function of external magnetic fiel. Temporal fluctuations of the transmitte signal have been also evaluate as a function of applie magnetic fiel. Fig. 3. (a) Schematic view of our experimental setup. (b) Photo of the ferroflui metamaterial sample next to a permanent magnet. The inset shows excessive ferroflui on the sie of the cuvette, which forms spikes along the applie magnetic fiel. Fig. 4. Experimentally measure transmission of the cobalt base ferroflui as a function of external magnetic fiel an polarization angle. Transmission signal was average over minutes.

7 We shoul emphasize that unlike typical 3D hyperbolic metamaterial fabrication techniques escribe in the literature (see for example [0]), preparation of sample shown in Fig. 3(b) requires much less effort. 6. Experimental results Experimentally measure transmission of the cobalt-base ferroflui for three ifferent values of the external magnetic fiel B 0 is plotte in Fig. 4 as a function of polarization angle (α=0 o correspons roughly to E fiel of the electromagnetic wave being perpenicular to B 0 ). While at zero magnetic fiel 1500 nm light transmission through the ferroflui is isotropic (weak polarization epenence in zero fiel observe in Fig. 4 may be attribute to the quartz cuvette), it becomes strongly anisotropic when magnetic fiel is applie to the ferroflui nm light transmission exhibits pronounce minima when E fiel of the wave is parallel to the irection of external magnetic fiel. This behavior is natural an expecte if cobalt nanocolumns are inee forme insie the ferroflui. On the other han, nonzero light transmission in these minima inicates that the ferroflui at n<n H remains an elliptic material. Accoring to our estimates, a 10 μm thick layer of hyperbolic metamaterial woul have zero transmission when E fiel is parallel to the metallic nanocolumns. Temporal fluctuations of the transmitte signal have been also evaluate as a function of applie magnetic fiel an light polarization as shown in Fig. 5. These measurements reveal consierable increase in temporal fluctuations of the transmitte signal in applie DC magnetic fiel. These results are consistent with the expecte strong fluctuations of the ielectric tensor of the ferroflui metamaterial, which is escribe in Section 4 above. Fig. 5. Measure temporal fluctuations of the sample transmission as a function of light polarization an applie magnetic fiel: (a) measurements in zero fiel, (b) measurements in B 0 =1400G. Angle between B 0 an E is inicate for each time epenency. It appears from Fig. 5 that in external magnetic fiel temporal fluctuations exhibit strong epenence on the polarization state of incient light. They are the strongest when E fiel of the incient linear polarize light is irecte along the external magnetic fiel. In aition to general increase in temporal fluctuations as a function of external magnetic fiel, short bursts of increase fluctuations are also observe. This behavior may be explaine by suen breakown of nanocolumn orering followe by nanocolumn rearrangement. It is interesting to note that the observe strong polarization epenence of transmission fluctuations isappears at much lower concentrations of cobalt nanoparticles in the ferroflui. Normalize fluctuations of optical transmission measure for ifferent polarization states of 1500 nm light at 1% an 8.% volume concentrations of cobalt nanoparticles are compare in

8 Fig. 6. This figure emonstrates that the observe effect inee isappears far from the hyperbolic ege. Fig. 6. Temporal fluctuations of normalize transmission in applie magnetic fiel at ifferent concentrations of cobalt nanoparticles in the ferroflui: (a) measure fluctuations as a function of light polarization at 1% volume concentration of cobalt nanoparticles, (b) measure fluctuations as a function of light polarization at 8.% volume concentration of cobalt nanoparticles. 7. Conclusion We have investigate optical properties of ferrofluis in the sub-hyperbolic n<n H range of volume fractions of metallic ferromagnetic nanoparticles. Polarization-epenent optical transmission of such a cobalt base ferroflui at 1500 nm has been stuie as a function of applie magnetic fiel. While on average such ferroflui metamaterials are elliptical, thermal fluctuations of the nanoparticle concentration lea to transient formation of hyperbolic regions insie this metamaterial. These regions behave as transient +1 imensional Minkowski spacetimes which temporarily appear an isappear insie a larger metamaterial multiverse. The escribe experimental system may also be stuie as yet another example of ranom hyperbolic meium [1]. As emonstrate recently in ref. [1], stuy of light propagation through such meia may give important insights into electromagnetic properties of our universe immeiately after the electro-weak phase transition leaing to new estimates on the magnitue of magnetic fiels which existe in the early universe. Acknowlegments This work is supporte by the NSF grant DMR We are grateful to J. Klupt for experimental help.