Nonlinear dynamical tunneling of optical whispering gallery modes in the presence of a Kerr nonlinearity

Jeong-Bo Shim, Peter Schlagheck, Martina Hentschel, and Jan Wiersig

Phys. Rev. A 94, 053849 – Published 28 November 2016

Abstract

The effect of a Kerr nonlinearity on dynamical tunneling is studied, using coupled whispering gallery modes in an optical microcavity. The model system that we have chosen is the “add-drop filter,” which is comprised of an optical microcavity and two waveguides coupled to the cavity. Due to the evanescent fields scattering on the waveguide, the whispering gallery modes in the microcavity form doublets, which manifest themselves as splittings in the spectrum. As these doublets can be regarded as a spectral feature of dynamical tunneling between two different dynamical states with a spatial overlap, the effect of a Kerr nonlinearity on the doublets is numerically investigated in the more general context of the relationship between cubic nonlinearity and dynamical tunneling. Within the numerical realization of the model system, it is observed that the doublets show a bistable transition in their transmission curve as the Kerr nonlinearity in the cavity is increased. At the same time, one rotational mode becomes dominant over the other one in the transmission, since the two states in the doublet have uneven linewidths. By using coupled-mode theory, the underlying mode dynamics of the phenomena is theoretically modeled and clarified.

Received 16 December 2015

DOI:https://doi.org/10.1103/PhysRevA.94.053849

Physics Subject Headings (PhySH)

Kerr effect Nonlinear optics Whispering gallery mode resonators

Nonlinear DynamicsGeneral PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Jeong-Bo Shim^1,2, Peter Schlagheck¹, Martina Hentschel³, and Jan Wiersig²

¹Département de Physique, University of Liège, 4000 Liège, Belgium
²Institut für Theoretische Physik, Otto-von-Guericke-Universität Magdeburg, D-39016 Magdeburg, Germany
³Institut für Physik, Technische Universität Illmenau, D-98693 Illmenau, Germany

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Vol. 94, Iss. 5 — November 2016

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Images

Figure 1
Structure of the add-drop filter: a circular microcavity with radius $R$ is coupled to two waveguides. The four branches given by the waveguides are indexed by $i = 1, 2, 3$ , and 4. The coupling is through the scattering of evanescent fields on the waveguides. The width of the waveguides is $0.04 R$ and the closest gap between the cavity and the waveguides is $0.1 R$ . The transmission from channel 1 to channel 2 are numerically computed by the ratio of the outgoing wave $S_{2 -}$ to the incoming wave $S_{1 +}$ .
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Figure 2
(a) Transmission amplitude at channel 2 without Kerr nonlinearity. Due to the scattering on the waveguides and the high $Q$ factor of the resonance, the transmission dip is split. The excited waves in the cavity on the left-hand side and the right-hand side of the split dips (denoted by 1 and 3) exhibit clockwise rotation, whereas the wave between the split dips (denoted by 2) shows counterclockwise rotation. The result of the comparison with Eq. (9) is superimposed as a dotted curve. Each dip exhibits different parity [(b) odd, (c) even] with respect to the symmetry axes of the system.
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Figure 3
FDTD simulation of the transmission amplitude associated with the split whispering gallery modes. (a) $χ^{(3)} = 1.0 \times 10^{- 7}$ , (b) $2.0 \times 10^{- 7}$ , (c) $3.0 \times 10^{- 7}$ , (d) $4.0 \times 10^{- 7}$ , (e) $5.0 \times 10^{- 7}$ , and (f) $6.0 \times 10^{- 7}$ . The intensity of the whispering gallery modes lies in the range of ${| E |}^{2} = 2 \sim 6 \times 10^{4}$ .
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Figure 4
Results of the CMT analysis for the split transmission under $χ^{(3)}$ nonlinearity. With increasing nonlinearity, stationary states are identified and plotted. Stable and unstable branches of the profile are denoted by blue and red colors, respectively. (a) $χ^{(3)} = 1.0 \times 10^{- 7}$ , (b) $2.0 \times 10^{- 7}$ , (c) $3.0 \times 10^{- 7}$ , (d) $4.0 \times 10^{- 7}$ , (e) $5.0 \times 10^{- 7}$ , and (f) $6.0 \times 10^{- 7}$ . The superimposed black dotted lines are connecting the branches that are most likely to be observed in numerics and in an experiment. They show good agreement with Fig. 3.
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Figure 5
Relative amplitudes of clockwise (dotted line) and counterclockwise (solid line) rotating modes with respect to the driving amplitude $S_{+}$ . The stability of each branch is denoted by the same color code as in Fig. 4 (stable: blue; unstable: red). (a) $χ^{(3)} = 1.0 \times 10^{- 7}$ , (b) $2.0 \times 10^{- 7}$ , (c) $3.0 \times 10^{- 7}$ , (d) $4.0 \times 10^{- 7}$ , (e) $5.0 \times 10^{- 7}$ , and (f) $6.0 \times 10^{- 7}$ . The range where the counterclockwise rotating mode is dominant is shaded by gray color, and the corresponding amplitude that is likely dominant in this range is denoted by a thick line.
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Figure 6
The same computation as in Fig. 5 with practical parameters. For this computation, we set $ω = 48.901725$ and $g = 4.63 \times 10^{- 7}$ with $0.1 W$ for pumping. For each mode, dampings are set equal to $γ = 4.22 \times 10^{- 8}$ and $Γ = 3.38 \times 10^{- 8}$ . As the Kerr nonlinearity increases from (a) $χ^{(3)} = 1.0 \times 10^{- 7}$ to (b) $5.0 \times 10^{- 7}$ , the spectral range where the counterclockwise mode is dominant (shaded by gray color) reduces.
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