Abstract

Metamaterials have already been used to model various exotic optical spaces. Here we demonstrate that mapping of monochromatic extraordinary light distribution in a hyperbolic metamaterial along some spatial direction may model the flow of time. This idea is illustrated in experiments performed with plasmonic hyperbolic metamaterials. The appearance of the statistical arrow of time is examined in an experimental scenario that emulates a big-bang-like event.

© 2011 Optical Society of America

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef] [PubMed]
  3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
    [CrossRef]
  4. I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402(2010).
    [CrossRef] [PubMed]
  5. Z. Jakob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
    [CrossRef]
  6. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699–1701 (2007).
    [CrossRef] [PubMed]
  7. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
    [CrossRef] [PubMed]
  8. Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008).
    [CrossRef]
  9. A. S. Davydov, Quantum Mechanics (Pergamon, 1976).
  10. L. Landau and E. Lifshitz, Statistical Physics (Elsevier, 2004).

2010 (1)

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402(2010).
[CrossRef] [PubMed]

2008 (1)

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008).
[CrossRef]

2007 (2)

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699–1701 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

2006 (4)

Z. Jakob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[CrossRef]

Alekseyev, L. V.

Davis, C. C.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699–1701 (2007).
[CrossRef] [PubMed]

Davydov, A. S.

A. S. Davydov, Quantum Mechanics (Pergamon, 1976).

Hung, Y. J.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699–1701 (2007).
[CrossRef] [PubMed]

Jacob, Z.

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008).
[CrossRef]

Jakob, Z.

Landau, L.

L. Landau and E. Lifshitz, Statistical Physics (Elsevier, 2004).

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[CrossRef]

Lifshitz, E.

L. Landau and E. Lifshitz, Statistical Physics (Elsevier, 2004).

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Narimanov, E.

Narimanov, E. E.

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402(2010).
[CrossRef] [PubMed]

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Smolyaninov, I. I.

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402(2010).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699–1701 (2007).
[CrossRef] [PubMed]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008).
[CrossRef]

New J. Phys. (1)

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[CrossRef]

Opt. Express (1)

Phys. Rev. Lett. (1)

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402(2010).
[CrossRef] [PubMed]

Science (4)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699–1701 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Other (2)

A. S. Davydov, Quantum Mechanics (Pergamon, 1976).

L. Landau and E. Lifshitz, Statistical Physics (Elsevier, 2004).

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Figures (4)

Fig. 1
Fig. 1

Schematic views of (a) wired and (b) layered hyperbolic metamaterials. (c) Hyperbolic dispersion relation is illustrated as a surface of constant frequency in k space. When the z coordinate is timelike, k z behaves as effective energy.

Fig. 2
Fig. 2

Experimental demonstration of straight world lines in a hyperbolic metamaterial: (a) Image of the plasmonic hyperbolic metamaterial obtained using an optical microscope under white-light illumination. The defect used as a plasmon source is shown by an arrow. (b) Atomic force microscope (AFM) image of the metamaterial shows stripes of PMMA formed on the gold film surface using electron-beam lithography. (c) Straight plasmonic ray, or world line, is emitted from the defect under illumination with 532 nm laser light. The ray direction is indicated by the arrow. For the sake of clarity, light scattering by the edges of the PMMA pattern is partially blocked by semitransparent rectangles. (d) Schematic view of a particle world line in a ( 2 + 1 )-dimensional Minkowski spacetime.

Fig. 3
Fig. 3

Experimental demonstration of world line behavior in an “expanding universe” using a plasmonic hyperbolic metamaterial. (a) Optical and (b) AFM images of the plasmonic hyperbolic metamaterial based on PMMA stripes on gold. The defect used as a plasmon source is shown by an arrow. (c) Plasmonic rays or world lines increase their spatial separation as a function of a timelike radial coordinate. The point (or moment) r = τ = 0 corresponds to a toy big bang. For the sake of clarity, light scattering by the edges of the PMMA pattern is partially blocked by semitransparent triangles. (d) Schematic view of world lines behavior near the big bang.

Fig. 4
Fig. 4

(a) (b), (c) Effect of various degrees of disorder on field distribution inside the metamaterial structure shown in Fig. 3. The cross section of image (c) near the launch point (d) and near the outer rim of the structure (e) demonstrates the increase of entropy. (f) Entropy [as defined by Eqs. (8, 9)] plotted as a function of a timelike radial coordinate for the experiment presented in (c) demonstrates that statistical and cosmological arrows of time do coincide in our experiments.

Equations (11)

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ω 2 c 2 D ω = × × E ω , D ω = ε ω E ω .
ω 2 c 2 φ ω = 2 φ ω ε 1 z 2 + 1 ε 2 ( 2 φ ω x 2 + 2 φ ω y 2 ) .
2 φ ω ε 1 z 2 + 1 | ε 2 | ( 2 φ ω x 2 + 2 φ ω y 2 ) = ω 0 2 c 2 φ ω = m * 2 c 2 2 φ ω ,
1 g x i ( g i k g φ x k ) = m 2 c 2 2 φ .
2 E r ε θ r 2 + 1 | ε r | ( 2 E r z 2 + 2 E r r 2 θ 2 ) E r ε θ r r + E r ε θ r 2 + E θ ε r r 2 θ = 2 E r t 2 .
2 φ ω ε θ r 2 + 1 | ε r | ( 2 φ ω z 2 + 2 φ ω r 2 θ 2 ) = ω 0 2 c 2 φ ω = m * 2 c 2 2 φ ω ,
2 φ ω ε θ r 2 + 1 | ε r | 2 φ ω r 2 θ 2 = m * 2 c 2 2 φ ω .
S = k ln G = k ln N ! N 1 ! N 2 ! N m ! ,
S I ln I I m ln I m ,
2 E θ ε r r 2 θ 2 + 1 | ε θ | ( 2 E θ z 2 + 2 E θ r 2 ) 2 E r ε θ r 2 θ = 2 E θ t 2 .
2 φ ω ε r r 2 θ 2 + 1 | ε θ | ( 2 φ ω z 2 + 2 φ ω r 2 ) = ω 0 2 c 2 φ ω = m * 2 c 2 2 φ ω ,

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