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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
1. Introduction
In 1958, Roger Penrose considered a two-dimensional cavity having mirrored walls as a mathematical model for illumination problems [1,2]. When a single point source of light is placed in the cavity, there will always remain unilluminated regions of the room. Such a cavity is therefore called a Penrose unilluminable room. In previous work, ray dynamical simulations have confirmed that the light is confined to specific regions depending on the initial conditions [3]. However the ray approximation assumes short wavelengths. At arbitrary wavelengths, it is necessary to account for diffraction and tunneling.
In the present work, numerical simulations are performed for wave propagation in a Penrose cavity using the finite-difference time-domain (FDTD) method [4]. It is found that some light propagates into the unilluminated regions due to diffraction. The relationship between the diffraction effect and the cavity size is studied. It is conjectured that the effect becomes more prominent as the cavity size decreases.
2. Model
Figure 1 shows the geometry of the Penrose cavity studied in this work [2,3]. The property of the unilluminable room does not depend on the cavity parameters. Accordingly, we used the same cavity shape as that presented in [3]. The cavity size is determined so that we can simulate the light propagation in a practical calculation time and with reliable accuracy. The left and right curved mirrors are half-ellipses whose semi-major and semi-minor axes are 4 μm and 2.5 μm, respectively. Points F1 and F2 are the foci of the left half-ellipse, while points F3 and F4 are the foci of the right half-ellipse. The cavity has four arms A, A', B, and B' that are connected just outside each focus. The curved mirrors connecting points F1 and F3 and connecting points F2 and F4 are half-ellipses whose semi-major and semi-minor axes are 2.5 μm and 1.6225 μm, respectively. The refractive index inside the cavity is 3.3. To obtain total internal reflection at the edges, the region outside the cavity is a perfect electric conductor.
The cavity has three kinds of chaotic ray trajectories. One set of rays starts from region A or A', and is confined to regions A, P, and A' as shown in Fig. 2(a). Another set starts from region B or B', and is confined to regions B, Q, and B' as shown in Fig. 2(b). The third kind consists of the trajectories starting from region M, and is confined to regions P, M, and Q as shown in Fig. 2(c). The cavity also has three kinds of stable ray trajectories, which propagate near periodic orbits. One is the set of axial trajectories shown in Fig. 2(d), another is the diamond-shaped trajectories in Fig. 2(e), and the third kind is the two V-shaped trajectories shown in Fig. 2(f). The yellow lines at the center of the green rays are the stable periodic orbits. These simulations show that rays launched from a point source in region A are confined to regions A, P, and A' and can never reach regions M, Q, B, and B'.
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.
To calculate the wave propagation corresponding to these rays, a point source is placed in the center of region A. It is a wire that contains magnetic current in the y direction so that the light is p polarized. Then a single optical pulse is generated at the point source so that the magnetic field in the y direction is proportional to a temporal function expressed by
where τ is the parameter to determine the width of the Gaussian envelope function, td the delay time, ω the angular frequency, which is expressed by 2πc/λ using the vacuum light velocity c and the vacuum wavelength λ, ϕ0 the initial phase at t = 0. The parameters are setas τ = 86 fs, td = 4τ, λ = 860 nm. The grid size is 2 nm to obtain convergence of the calculations. The calculated quantities are the temporal variation of the magnetic field distribution Hy and the electric field energy evaluated at monitoring points 1, 2, and 3 in Fig. 1. The monitor size is 1 μm1 μm. The commercially available software FullWAVE [5] developed by Rsoft design group is used for the FDTD calculations.Courvoisier et al. investigated output coupling mechanisms in a two-dimensional spiral microcavity, which has a notch along the rim, using femtosecond light pulses [6]. In their work, FDTD simulations explained the observed interaction of a light pulse with the notch very well. Therefore, we consider that the numerical approach using FDTD method is effective for investigating the propagation and diffraction of a light pulse in the Penrose cavity.
3. Results
Figure 3 is a short animation of the temporal variation of the magnetic field distribution Hy. The upper and lower limits of the color scale span the weak magnetic field distribution that escapes into regions M, Q, B, and B'. As a result, saturation occurs in the dark red and dark blue regions. After an optical pulse is generated at the point source, the magnetic field spreads strongly into regions A, P, and A'. However, a weak field also escapes into the unilluminated regions M, Q, B, and B'.
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.
Figure 4 plots the temporal variation of the electric field energy at each monitor. The energy at monitor 1 rapidly increases after a pulse is emitted by the source. Subsequently, the energies at monitors 2 and 3 also increases a little. These results prove that a small amount of light escapes into the unilluminable regions which the optical rays cannot reach.
Figure 5 shows the two-dimensional magnetic field distribution Hy at different times t. The color scale is the same as that in Fig. 3. First, the magnetic field is generated at the source in region A, as seen in Fig. 5(a). It then spreads toward region P, as shown in Fig. 5(b). One can already see that a small field is diffracted into region M at focus F1. Then the diffracted light spreads into the unilluminable regions M, Q, B, B', as seen in Figs. 5(c) and 5(d).
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.
4. Discussion
In this section, the relationship between the diffraction effect and the cavity size is discussed, by analogy with Fresnel diffraction [7] at a semi-infinite plane bounded by the straight edge shown in Fig. 6(a). The boundary is at z = 0 and is illuminated by a plane wave propagating along the z axis.
Fig. 6 Fresnel diffraction for the semi-infinite plane boundary sketched in panel (a). The resulting intensity distribution is plotted in panel (b).
The intensity distribution in the Fresnel diffraction pattern at z = L is plotted in Fig. 6(b). It is normalized to the intensity at . In the illuminated region x > 0, the intensity oscillates with diminishing amplitude as x increases. At the edge of the shadow region at x = 0, the normalized intensity is 0.25. In the shadow region x < 0, the intensity decreases monotonically to zero as x decreases. Defining the diffraction width Δx as the distance at which the normalized intensity decreases to any given value, then
where λ is the wavelength of the plane wave.Considering the diffraction at focus F1 in the Penrose cavity, it is then conjectured that the diffraction width at the horizontal centerline of the cavity is likewise proportional to the square root of the cavity size and thus to the width of region P along the horizontal centerline (i.e., to the width of the optical confinement region). Accordingly, the diffraction width relative to that confinement width decreases as the cavity size increases. The diffraction effect should thus be negligible for a Penrose cavity whose size is much larger than the wavelength of the light.
Although the three kinds of chaotic ray trajectories shown in Figs. 2(a)–2(c) are independent of each other, the simulations reveal that a small amount of coupling occurs between them due to diffraction. It is very important to describe the coupling effect theoretically. Studying the modal properties of the cavity also would be interesting from the viewpoint of the ray-wave correspondence of chaotic microcavities [8–11].
5. Conclusions
Optical wave propagation has been analyzed in a Penrose unilluminable room using the FDTD method. A small amount of light can propagate into the unilluminated regions (which optical rays never reach) by diffraction. It is conjectured that this diffraction effect will be more prominent for a smaller cavity. Future research should investigate the light propagation and diffraction theoretically. It is also important to investigate the modal properties of the Penrose unilluminable room microcavities.
Acknowledgments
T. F. first learned about the Penrose unilluminable room from Prof. A. E. Siegman in 2007. He also thanks Dr. S. Shinohara, Prof. S. Sunada, and Prof. T. Harayama for useful discussions.
References and links
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