Abstract

Using the finite-difference time-domain method, propagation of light waves is studied in a Penrose unilluminable room. Such a room always has dark (unilluminated) regions, regardless of the position of a point source in it. However, in contrast to the predictions of ray dynamical simulations, a small amount of light propagates into the unilluminated regions via diffraction. We conjecture that this diffraction effect becomes more prominent as the size of the room decreases.

© 2015 Optical Society of America

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References

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  1. H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry (Springer-Verlag, 1991).
  2. E. W. Weisstein, “Illumination problem,” MathWorld-A Wolfram Web Resource ( http://mathworld.wolfram.com/IlluminationProblem.html ).
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  4. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).
  5. Synopsys, Inc., “FullWAVE Product Overview” ( http://optics.synopsys.com/rsoft/rsoft-passive-device-fullwave.html ).
  6. F. Courvoisier, V. Boutou, J. P. Wolf, R. K. Chang, and J. Zyss, “Deciphering output coupling mechanisms in spiral microcavities with femtosecond light bullets,” Opt. Lett. 30(7), 738–740 (2005).
    [Crossref] [PubMed]
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  8. T. Fukushima, S. Shinohara, S. Sunada, T. Harayama, K. Sakaguchi, and Y. Tokuda, “Lasing of TM modes in a two-dimensional GaAs microlaser,” Opt. Express 22(10), 11912–11917 (2014).
    [Crossref] [PubMed]
  9. T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photonics Rev. 5(2), 247–271 (2011).
    [Crossref]
  10. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
    [Crossref] [PubMed]
  11. Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80(4), 041807 (2009).
    [Crossref]

2014 (1)

2011 (1)

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photonics Rev. 5(2), 247–271 (2011).
[Crossref]

2009 (1)

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80(4), 041807 (2009).
[Crossref]

2005 (1)

1998 (1)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Boutou, V.

Cao, H.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80(4), 041807 (2009).
[Crossref]

Capasso, F.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Chang, R. K.

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Courvoisier, F.

Faist, J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Fang, W.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80(4), 041807 (2009).
[Crossref]

Fukushima, T.

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Harayama, T.

Ho, S.-T.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80(4), 041807 (2009).
[Crossref]

Liu, B.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80(4), 041807 (2009).
[Crossref]

Narimanov, E. E.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Nöckel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Sakaguchi, K.

Shinohara, S.

Sivco, D. L.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Solomon, G. S.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80(4), 041807 (2009).
[Crossref]

Song, Q.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80(4), 041807 (2009).
[Crossref]

Stone, A. D.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Sunada, S.

Tokuda, Y.

Wolf, J. P.

Zyss, J.

Laser Photonics Rev. (1)

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photonics Rev. 5(2), 247–271 (2011).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80(4), 041807 (2009).
[Crossref]

Science (1)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[Crossref] [PubMed]

Other (6)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed.(Pergamon Press, 1980).

H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry (Springer-Verlag, 1991).

E. W. Weisstein, “Illumination problem,” MathWorld-A Wolfram Web Resource ( http://mathworld.wolfram.com/IlluminationProblem.html ).

T. Fukushima, S. Shinohara, S. Sunada, T. Harayama, K. Sakaguchi, and Y. Tokuda, “Ray dynamical simulation of Penrose unilluminable room cavity,” in Frontiers in Optics (FiO) and Laser Sicence (LS) XXIX Meetings (Optical Society of America, Washington, DC, 2013), JW3A.19.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

Synopsys, Inc., “FullWAVE Product Overview” ( http://optics.synopsys.com/rsoft/rsoft-passive-device-fullwave.html ).

Supplementary Material (1)

NameDescription
» Visualization 1: MOV (9731 KB)      Short animation

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Figures (6)

Fig. 1
Fig. 1

Model of the Penrose cavity.

Fig. 2
Fig. 2

Ray trajectories confined in the Penrose cavity. Chaotic trajectories in (a) regions A, P, and A', (b) regions B, Q, and B', and (c) regions P, M, and Q. Stable trajectories that are (d) axial, (e) diamond-shaped, and (f) V-shaped. The central yellow lines are stable periodic orbits.

Fig. 3
Fig. 3

Click on the link (Visualization 1) to start an animation of the temporal variation of the magnetic field distribution in the Penrose cavity.

Fig. 4
Fig. 4

Temporal variation of the electric field energy at each monitor.

Fig. 5
Fig. 5

Magnetic field distribution in the Penrose cavity at (a) t = 0.229 ps, (b) t = 0.315 ps, (c) t = 0.573 ps, and (d) t = 14.333 ps.

Fig. 6
Fig. 6

Fresnel diffraction for the semi-infinite plane boundary sketched in panel (a). The resulting intensity distribution is plotted in panel (b).

Equations (2)

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g ( t ) = exp [ ( t t d τ ) 2 ] sin ( ω t + ϕ 0 ) ,
Δ x λ L

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