Rotating Optical Microcavities with Broken Chiral Symmetry

Raktim Sarma, Li Ge, Jan Wiersig, and Hui Cao

Phys. Rev. Lett. 114, 053903 – Published 5 February 2015

Abstract

We demonstrate in open microcavities with broken chiral symmetry that quasidegenerate pairs of copropagating-wave resonances are transformed by rotation to counterpropagating ones, leading to a striking change of emission directions. The rotation-induced relative change in output intensity increases exponentially with cavity size, in contrast to the linear scaling of the Sagnac effect. By tuning the degree of spatial chirality with cavity shape, we are able to maximize the emission sensitivity to rotation without spoiling the quality factor.

Received 15 April 2014

DOI:https://doi.org/10.1103/PhysRevLett.114.053903

Authors & Affiliations

Raktim Sarma¹, Li Ge^2,3, Jan Wiersig⁴, and Hui Cao^1,*

¹Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
²Department of Engineering Science and Physics, College of Staten Island, CUNY, Staten Island, New York 10314, USA
³The Graduate Center, CUNY, New York, New York 10016, USA
⁴Institut für Theoretische Physik, Universität Magdeburg, Magdeburg, Postfach 4120, Germany

^*hui.cao@yale.edu

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Vol. 114, Iss. 5 — 6 February 2015

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Images

Figure 1
Comparison of Sagnac effect in a rotating microcavity with chiral symmetry ( $η = 1$ , dashed line) and without chiral symmetry ( $η = 0.1$ , solid line). The value of $η | V |^{2}$ is kept the same. (a) (Dimensionless) frequency splitting for a pair of quasidegenerate modes as a function of rotation frequency $Ω$ . [(b) and (c)] Evolution of CW (thick line) and CCW (thin line) traveling-wave components in the quasidegenerate modes with rotation. In the symmetric cavity ( $η = 1$ ), at low rotation speed the eigenmodes remain standing-wave modes with equal weights of CCW and CW components, and their frequency difference is barely changed by rotation. When the rotation speed is sufficiently high, one mode evolves to a CCW traveling-wave mode and the other one to a CW traveling-wave mode, and their frequency difference starts to grow significantly with $Ω$ . In a chiral cavity ( $η = 0.1$ ), the evolution of frequency splitting with rotation is identical to that of the symmetric cavity. Without rotation, both modes are dominated by CCW traveling waves, but one of them (b) transforms into a CW traveling wave mode at high $Ω$ .
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Figure 2
Far-field emission intensity patterns of a pair of quasidegenerate modes ( $λ = 598 nm$ ) in a nonrotating dielectric disk ( $n = 3.0$ ) of asymmetric limaçon shape ( $R = 591 nm$ , $ε_{1} = 0.1$ , $ε_{2} = 0.075$ , $δ = 1.94$ ). (a) Angular distributions of emission intensities at a distance of $r = 50 R$ from the cavity center for both modes, which have similar output directions. (b) Far-field patterns of CW (red solid line) and CCW (blue dashed line) wave components in the resonances, exhibiting distinct output directions.
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Figure 3
Emission from the rotating asymmetric limaçon cavity with the same parameters as the stationary one in Fig. 2. [(a), (b)] Spatial distributions of field intensities for a pair of degenerate modes, which correspond to the stationary modes in Fig. 2, at the normalized rotation frequency $Ω R / c = 1 0^{- 3}$ . The intensities outside the cavity are enhanced to illustrate that the main output directions of the two modes are different, even though they have the same output directions without rotation [Fig. 2]. (c) Spatial distribution of field intensity for one of the quasidegenerate modes in the nonrotating cavity. It is dominated by CW wave. When the cavity rotates in the CCW direction, this mode switches from CW to CCW wave, and the main output direction is changed dramatically. (d) Angular distribution of far-field intensity for the pair of modes shown in (a) and (b) in the rotating cavity.
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Figure 4
Rotation-induced change in emission pattern of the same cavity as the one in Fig. 3, when both quasidegenerate modes are excited simultaneously. (a) Angular distribution of the emission intensity $I_{e}$ at a distance of $r = 3 R$ from the cavity center at three rotation speeds. To show the change in the emission profile, $I_{e} (θ)$ is normalized ( $\int_{0}^{2 π} I_{e} (θ) d θ = 1$ ). (b) Relative changes in the main emission peak intensity (at $θ = 0.73$ ) (solid squares and solid line) and in the ratio of main peak intensity over the secondary peak intensity (at $θ = 2.79$ ) (crosses and dashed line) vs the normalized rotation frequency $Ω R / c$ . Both peak intensities are integrated over a range of emission angles marked by the double-arrowed segments in (a). For comparison, relative changes of resonant frequencies $Δ ω / ω_{0}$ are plotted for circular cavities with the same area and refractive index (open circles and dotted line). The symbols represent the numerical data, and the straight lines are linear fit of the data in the log-log plot, which gives the slope. The values of the slope are (from top to bottom) $2.4 \times 1 0^{3}$ , $1.2 \times 1 0^{3}$ , and $5.7 \times 1 0^{- 1}$ , respectively. The rotation-induced changes of output intensity are much larger than that of the resonance frequency.
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Figure 5
Output sensitivity to rotation for the asymmetric limaçon cavity with varying degree of spatial chirality. The cavity parameters are the same as those in Fig. 2 except for the value of $δ$ . (a) Relative change of the emission intensity in the main output direction (solid squares and dashed line) as a function of spatial chirality $α$ for the quasidegenerate modes in Fig. 2. The rotation frequency is fixed at $Ω R / c ≃ 1.5 \times 1 0^{- 5}$ . The difference between the emission patterns for CW and CCW waves in the nonrotating cavity is quantified by $β$ (solid circles and solid line), which is also plotted against $α$ . With increasing spatial chirality $α$ , CW and CCW outputs become more distinct, enhancing the emission sensitivity to rotation. [(b), (c)] Far-field patterns for CW wave (red solid line) and CCW wave (blue dashed line) in two cavities with $δ = 0$ (b) and 2.75 (c). The dotted line marks the cavity boundary. At $δ = 0$ , both CW and CCW waves emit predominantly in the direction close to $θ = π / 2$ (b), and the slight difference of their emission directions is a result of wave effects in the wavelength-scale cavity [33]. As $δ$ increases from 0 to $π$ , the main emission direction of the CW wave moves towards $θ = 0$ and the CCW wave towards $θ = π$ ; meanwhile, the secondary emission peak, which is in the opposite direction of the main peak, grows monotonically.
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