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Metamaterial model of fractal time Igor I. Smolyaninov Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA phone: 301-405-3255; fax: 301-314-9281; e-mail:
[email protected]
While numerous examples of fractal spaces may be found in various fields of science, the flow of time is typically assumed to be one-dimensional and smooth. Here we present a metamaterial-based physical system, which can be described by effective three-dimensional (2+1) Minkowski spacetime. The peculiar feature of this system is that its time-like variable has fractal character.
The fractal
dimension of the time-like variable appears to be D=2.
Recent advances in metamaterial optics [1-3] enable researchers to design novel physical systems, which can be described by effective space-times having very unusual metric and topological properties. Examples include constructing analogues of black holes [4-7], wormholes [8,9], spinning cosmic strings [10], and even the metric of Big Bang itself [11]. More unusual examples include building a physical system, which can be described by (-,-,+,+) metric signature, which differs from the usual Minkowski signature (-,+,+,+) of physical spacetime [12]. The latter advance is based on the unusual optics of hyperbolic metamaterials, which have different signs of dielectric
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permittivity ε along different spatial directions. Mapping of monochromatic extraordinary light distribution in a hyperbolic metamaterial along some spatial direction may model the “flow of time” in a three dimensional (2+1) effective Minkowski spacetime [11]. To better understand this effect, let us start with a nonmagnetic uniaxial anisotropic material with dielectric permittivities εx= εy= ε1 and εz = ε2, and assume that this behaviour holds in some frequency range around ω=ω0. Any electromagnetic field propagating in this material can be expressed as a sum of the “ordinary” and “extraordinary” contributions, each of these being a sum of an arbitrary
r
number of plane waves polarized in the “ordinary” ( E perpendicular to the optical axis)
r
and “extraordinary” ( E parallel to the plane defined by the k–vector of the wave and the optical axis) directions. Let us define our “scalar” extraordinary wave function as
ϕ=Ez so that the ordinary portion of the electromagnetic field does not contribute to ϕ. Since metamaterials exhibit high dispersion, let us work in the frequency domain and write the macroscopic Maxwell equations as [13]
ω2 r c2
r r r r t r Dω = ∇ × ∇ × Eω and Dω = ε ω Eω ,
(1)
which results in the following wave equation for ϕω if ε1 and ε2 are kept constant inside the metamaterial:
−
ω2 c2
ϕω =
∂ 2ϕω 1 ⎛ ∂ 2ϕω ∂ 2ϕω + ⎜ + ∂y 2 ε 1∂z 2 ε 2 ⎜⎝ ∂x 2
⎞ ⎟⎟ ⎠
(2)
In hyperbolic metamaterials [14] ε1 and ε2 have opposite signs. Let us consider the case of constant ε1 >0 and ε2
