June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS
3543
Chaotic oscillations of coupled nanobeam cavities with tailored optomechanical potentials Yue Sun1,2,* and Andrey A. Sukhorukov1 1
Nonlinear Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia 2 Laser Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia *Corresponding author: [email protected] Received February 7, 2014; accepted April 22, 2014; posted May 7, 2014 (Doc. ID 205769); published June 9, 2014 We reveal novel features of nonlinear optomechanical interactions in coupled suspended nanocavities that are driven by two detuned laser frequencies. Such driving enables simultaneous excitation of odd and even optical supermodes, which induce gradient forces of opposite signs, and the competition between these forces enables the realization of optomechanical potentials with large barriers and narrow wells. These types of potentials were suggested for precise displacement control, or “spectral bonding,” in the static regime. However we find that self-induced oscillations appear even at the deep global potential minima when the mechanical damping rate is below a certain threshold, including a new regime of chaotic switching between mechanical deformations of opposite signs. © 2014 Optical Society of America OCIS codes: (350.4238) Nanophotonics and photonic crystals; (200.4880) Optomechanics; (230.7370) Waveguides. http://dx.doi.org/10.1364/OL.39.003543
Optical force has been under intensive investigations for more than three decades since lasers were invented in 1970s [1]. The optically induced force is nowadays widely used for various practical applications in biomedicine and other disciplines. In the last decade, advances in fabrication technologies enabled creation of nanoscale structures, which can be deformed by optical forces, facilitating strong coupling of the mechanical and optical eigenmodes [2,3]. It was demonstrated that the mechanical structure deformities can be precisely controlled by tailoring optomechanical potentials [4]. Furthermore, a concept of “spectral bonding” was suggested, where an optical cavity resonance can be aligned to a specific wavelength through simultaneous excitation of different optical modes [5]. This approach, utilizing gradient optical forces between micro and nanoscale optical waveguides, opens new possibilities for reconfigurable photonic circuits, and further, optical signal shaping and routing based on all-optical tuning of the structure geometry [3]. Importantly, as Braginsky originally predicted [6], optomechanical interactions can lead to self-induced oscillations. Oscillations can arise when the mechanical damping is below a certain threshold due to the coupling between the optical and mechanical modes giving rise to complex nonlinear dynamics [7]. The type of oscillations can vary from periodic to chaotic [8–10]. In this Letter, we perform an analysis of optically induced static mechanical deformities and self-oscillations of coupled nanobeam cavities, considering a regime, where the distance between two cavities is large enough allowing the gradient force to dominate and two optical super-modes are excited simultaneously. We show that by selecting the two-mode excitation parameters, it is possible to realize a static optomechanical potential with a well- defined minimum corresponding to a “spectral bonding” regime [5]. Whereas it was previously shown that self-induced oscillations can appear in optomechanical structures [8–10], here we reveal that static solutions can become strongly unstable under finite mechanical damping rates even at the deep global minima of a static 0146-9592/14/123543-04$15.00/0
potential. Furthermore, we identify a new regime of chaotic switching between mechanical deformities of opposite signs when potential has only a single narrow well. We consider an optomechanical cavity that supports two optical modes, where each mode can exert force exciting the same mechanical mode. An example of such structure is a pair of closely spaced suspended nanobeam cavities [11] as schematically illustrated in Fig. 1(a). These two optical eigenmodes can interact with each other through one mechanical degree of freedom, the transverse separation of two beam cavities. We denote the amplitudes of cavity modes as A for even and A− for odd optical modes, and consider the mechanical mode whose deformity changes the separation between the cavities d by a distance ξ. The cavity modes A can be excited by the input s and their resonant frequencies are linearly tuned by the deformity ξ. Following the established procedures [9,11] we formulate the nonlinear coupled-mode equations in dimensionless form,
Fig. 1. (a) Schematic of an optomechanical cavity supporting even and odd optical modes that induce a mechanical deformity ξ to the transverse separation. (b) Dependence of two optical mode resonant angular frequencies on the mechanical deformity, ωp mark the angular frequencies of optical even and odd modes’ excitations. © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014
ξ̈ −Γm ξ_ − Ω2m ξ F − jA− j2 − F jA j2 ;
(1a)
A_ −ia bξ − Γo A s ;
(1b)
A_ − −ic dξ − Γo A− s− :
(1c)
Here, Γm is the mechanical damping, Ωm is the mechanical oscillation angular frequency, and F defines the strength of optical forces induced by two optical modes. The signs in front of F are chosen to represent an attractive force for the even mode and a repulsive force for the odd mode. Parameters a and c define the normalized angular frequency detunings of even and odd modes relative to two laser inputs that are tuned close to the corresponding mode resonances; see Fig. 1(b). Parameters b and d define the change rate of optical angular frequency shift due to the mechanical deformity. The values of s give the excitation amplitudes of two optical modes, and we assume that both modes exhibit the same decay rate Γo . We first analyze the static solutions for a continuous laser illumination with fixed amplitude and angular frequency. The mode amplitudes are found as ¯ Γo −1 ; A¯ s ia bξ ¯ Γo −1 ; A¯ − s− ic dξ
(2)
where ξ¯ is a static mechanical deformation corresponding to a local minimum or maximum of the effective optomechanical potential V eff , read as Eq. (3). ¯ − V eff ξ
Z
−Ω2m ξ¯ F − jA¯ − j2 − F jA¯ j2 dξ¯
F js j bξ¯ a ξ arctan 2 bΓo Γo 2 F js j dξ¯ c − − − arctan : dΓo Γo Ω2m ¯ 2
2
(3)
The optomechanical potential has a shape similar to a coupled microring structure, for which the “spectral bonding” concept was suggested [5]. Indeed, through the simultaneous excitation of two optical modes, this potential can be tailored to have a deep global minimum, which could then facilitate precise alignment of resonances [5]. We illustrate such a situation in Fig. 2(a), where the potential is shown with color shading. Static solutions corresponding to the potential maxima are indicated by black and minima by white lines. We note that the optomechanical potential is determined by three terms in Eq. (3), and we show their contributions in Figs. 2(b)–2(e)) for characteristic potential shapes at different laser detunings. The total potential is shown with solid black lines; minima are marked with open red circles and maxima with blue circles. The first term corresponds to the mechanical restoring force and has a parabolic shape centered at ξ 0, shown as magenta dashed lines. The second and third terms, shown as green and cyan dashed lines, indicate the contribution from optical even and odd mode induced force,
Fig. 2. (a) Optomechanical potential V eff versus displacement ξ and optical frequency detuning a shown in color map with local minima (white lines) and maxima (black lines). (b)–(e) Optomechanical potential profiles (solid black lines) and separate contributions corresponding to different terms in Eq. (3) shown with the magenta, green, and cyan dashed lines. The frequency detunings a are (b) −0.1, (c) 0.5, (d) 0.58, and (e) 1, corresponding to the white dashed lines in (a). Parameters are Ωm 1, Γo 0.01, c a − 1, b −0.8, d 0.7, and F 1.
respectively. In the vicinity of the crossing where optical modes strongly interact with each other, local minima are determined by these three terms together; see Figs. 2(c) and 2(d), while the fixed minima at ξ 0 from the parabolic term or the detuning frequency dependent minima close to ξ −a∕b and ξ −c∕d dominate away from the crossing; see Figs. 2(b) and 2(e). We now perform a stability analysis of static solutions. Whereas potential maxima naturally correspond to unstable configurations, we find that strong instabilities also occur at the minima of the static potential. We consider a small perturbation (δξ, δA , δA− ) applied to the static solutions of Eqs. (1). We substitute the static solutions with perturbations in Eqs. (1), and perform linearization with respect to the perturbation amplitudes. The resulting coupled linear differential equations have general solutions of the form δξ qeiγt q e−iγ t , δA α eiγt α− e−iγ t , and δA− β eiγt β− e−iγ t . Here
June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS
(q, α , α− , β , β− ) is the perturbation eigenvector, γ is the corresponding eigenvalue, and these values can be readily determined numerically. The eigenmode with the largest value of −Imγ determines the overall stability. If −Imγ ≤ 0, the perturbations decay or at least do not grow. However, if −Imγ > 0, the corresponding perturbation eigenmode grows exponentially as exp−Imγt. We numerically solve the eigenproblem with the generalized Schur decomposition method, find the maximum unstable perturbation growth rate, −Imγ, as a function of mechanical damping rate Γm and frequency detuning a along G3 , G5 , and G7 local minima branches, and summarize these results in Figs. 3(a)–3(c). In these figures, the blueish region corresponds to stable solutions as all perturbations decay. Interestingly, as frequency detuning approaches a very narrow single potential well (the minimum of G7 at smaller values of detuning a), much higher mechanical damping is required for stability as the well width is reduced, see Fig. 3(c). Therefore, whereas narrow wells are attractive for precise deformity control, higher mechanical damping needs to be provided in this regime. On the other hand, static deformity control can be achieved with much smaller damping for some of the wider wells, using stability islands visible in Figs. 3(b) and 3(c). The stability islands appear when two cavity modes strongly interact through the mechanical mode in the region where the mechanical and optical damping rates are comparable.
Fig. 3. Results of stability analysis. (a)–(c) The growth rate of the most stable perturbations −Imγ versus mechanical damping rate Γm for static solutions along the G3 , G5. and G7 branches of local minima, horizontal white line marks Γm 3.5622 corresponding to (d), (e). (d) Self-sustained oscillation regions, colored in blue, red, and green. Lines show static states; local minima (solid gray lines) and local maxima (dashed gray lines), and the boundaries of static optomechanical potential barriers are indicated with thin dotted gray lines. (e) Frequency of the oscillatory states with the line colors corresponding to shadings in (d).
3545
Next, we examine the dynamics of growing perturbations on top of unstable static solutions. The linear stability analysis predicts exponential growth of small perturbations, however as perturbations grow larger complex nonlinear dynamics can emerge. We explore the dynamics of growing perturbations by numerically solving Eqs. (1) using the Dormand-Prince method and identify the appearance of self-sustained periodic oscillations with deformations shifted away from the static minima G3, G5, G7 and frequencies ranging from 0.063 to 0.121 times of the natural frequency of the structure. We show the different ranges of mechanical oscillations with blue, red, and green shadings in Fig. 3(d), and the corresponding oscillation frequencies are presented in Fig. 3(e) for the mechanical damping rate at Γm 3.5622, which is indicated by the white lines in Figs. 3(a)–3(c). These deformity oscillating regions demonstrate nonlinear dynamics that cannot be described by the static optomechanical potential. In particular, small perturbations applied to global minima G7 in frequency detuning range a 0.58–0.76 grow into two oscillatory stable states oscillating at different frequencies (0.11353 and 0.12012 times natural frequency of the structure) and outside the static optomechanical potential well. While approaching the static potential crossing, only the upper (blue-shaded) state shows discontinuity in time-averaged deformity and oscillation frequency at a 0.56, which is close but doesn’t coincide with the crossing detuning a 0.55. For the lower (red) oscillatory state, a transition region appears in the vicinity of detuning a 0.5, where both the amplitude and frequency of the self-sustained oscillation drop dramatically to zero while the time-averaged deformity converges to the static stable states on the branch G5 . For the range of detunings a 0.76–1.14 there are no periodic oscillations, and we leave this region blank in Figs. 3(d) and 3(e). An example of perturbation evolution at a 0.9 in this region is shown in Figs. 4(a)–4(c), which demonstrate complex aperiodic dynamics of the mechanical displacement and cavity mode amplitudes. The deformities mostly stay positive or negative, with sharp switchings between them. Positive deformities correspond to the excitation of both optical modes, whereas for negative deformities primarily even mode is excited. The aperiodic nature of oscillations is confirmed by the frequency spectrum of the optical odd mode amplitude, shown in Fig. 4(d). The broad triangle-shape spectrum suggests that the temporal signal is chaotic. To further characterize the dynamics, we determine the local maxima of deformities, marked by circles in Fig. 4(a). Then, we construct a Lorenz map [12] showing a mapping between the successive maxima in Fig. 4(e). We see the presence of regular regions with almost deterministic mapping for negative and positive deformity amplitudes, and an irregular region in the middle that gives rise to chaotic mixing. We find that the regular mapping regions can be approximated as ξmax n 1 ≃ 0.05 0.99034ξmax n (blue dashed line) for negative and ξmax n 1 ≃ 0.02 0.94233ξmax n (red dashed line) for positive deformities. This mapping is above the diagonal for negative deformations, and below the diagonal for positive deformities. It means that oscillations will feature accordingly growing or decaying peak amplitudes
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mixture of positive and negative deformity oscillations, which is different from the previously considered scenario of period doubling [8,10]. In summary, we demonstrate that optomechanical potentials with deep and narrow wells can be achieved in coupled suspended nanobeam cavities through the simultaneous excitation of odd and even optical supermodes. We predict the appearance of strong optomechanical instabilities even at a global minimum of static optomechanical potentials when the mechanical damping rate is below a certain threshold, and identify a broad parameter region where chaotic oscillations emerge with aperiodic switching between positive and negative deformities. These results suggest new possibilities for flexible static and dynamic control. Fig. 4. Aperiodic chaotic oscillations. (a)–(c) Time evolution of a small perturbation excited at a 0.9 in local minima branch G7 . (a) mechanical deformity, (b) intensity of optical even and (c) odd modes, (d) spectrum of optical odd mode shown in (c), and (e) Lorenz map between the successive maxima of mechanical deformity, marked with circles in (a). Blue and red dashed lines show approximate mapping in the regular regions of negative and positive deformities, respectively. (f) Histogram of mechanical deformity with unit total counts at each optical frequency detuning a.
of the mechanical deformities in the corresponding regions, until entering the central region of the map where chaotic change in oscillation occurs. We have calculated the Lyapunov exponents; the largest two eigenvalues of 0.0373 and −0.0002 indicate the chaotic behavior which is similar to that of a Lorenz system. In order to sketch a panoramic view of this chaotic region, we plot out the histogram of deformity perturbation time evolution with unit total counts at each frequency detuning a in Fig. 4(f). Throughout this region the distributions of the deformity are primarily localized around two discrete positions in the vicinity of negative (ξ −0.5) and positive (ξ 1.5) values. This further confirms that the origin of the whole chaotic region is the
This work was supported by the Australian Research Council programs (including Future Fellowship FT100100160 and Discovery Project DP130100086). References 1. S. Chu, Science 253, 861 (1991). 2. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005). 3. I. Favero and K. Karrai, Nat. Photonics 3, 201 (2009). 4. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, Nature 462, 633 (2009). 5. P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, Nat. Photonics 1, 658 (2007). 6. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Phys. Lett. A 287, 331 (2001). 7. F. Marquardt, J. G. E. Harris, and S. M. Girvin, Phys. Rev. Lett. 96, 103901 (2006). 8. T. Carmon, M. C. Cross, and K. J. Vahala, Phys. Rev. Lett. 98, 167203 (2007). 9. G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, 043603 (2011). 10. F. Marino and F. Marin, Phys. Rev. E 87, 052906 (2013). 11. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009). 12. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview, 1994).
3543
Chaotic oscillations of coupled nanobeam cavities with tailored optomechanical potentials Yue Sun1,2,* and Andrey A. Sukhorukov1 1
Nonlinear Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia 2 Laser Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia *Corresponding author: [email protected] Received February 7, 2014; accepted April 22, 2014; posted May 7, 2014 (Doc. ID 205769); published June 9, 2014 We reveal novel features of nonlinear optomechanical interactions in coupled suspended nanocavities that are driven by two detuned laser frequencies. Such driving enables simultaneous excitation of odd and even optical supermodes, which induce gradient forces of opposite signs, and the competition between these forces enables the realization of optomechanical potentials with large barriers and narrow wells. These types of potentials were suggested for precise displacement control, or “spectral bonding,” in the static regime. However we find that self-induced oscillations appear even at the deep global potential minima when the mechanical damping rate is below a certain threshold, including a new regime of chaotic switching between mechanical deformations of opposite signs. © 2014 Optical Society of America OCIS codes: (350.4238) Nanophotonics and photonic crystals; (200.4880) Optomechanics; (230.7370) Waveguides. http://dx.doi.org/10.1364/OL.39.003543
Optical force has been under intensive investigations for more than three decades since lasers were invented in 1970s [1]. The optically induced force is nowadays widely used for various practical applications in biomedicine and other disciplines. In the last decade, advances in fabrication technologies enabled creation of nanoscale structures, which can be deformed by optical forces, facilitating strong coupling of the mechanical and optical eigenmodes [2,3]. It was demonstrated that the mechanical structure deformities can be precisely controlled by tailoring optomechanical potentials [4]. Furthermore, a concept of “spectral bonding” was suggested, where an optical cavity resonance can be aligned to a specific wavelength through simultaneous excitation of different optical modes [5]. This approach, utilizing gradient optical forces between micro and nanoscale optical waveguides, opens new possibilities for reconfigurable photonic circuits, and further, optical signal shaping and routing based on all-optical tuning of the structure geometry [3]. Importantly, as Braginsky originally predicted [6], optomechanical interactions can lead to self-induced oscillations. Oscillations can arise when the mechanical damping is below a certain threshold due to the coupling between the optical and mechanical modes giving rise to complex nonlinear dynamics [7]. The type of oscillations can vary from periodic to chaotic [8–10]. In this Letter, we perform an analysis of optically induced static mechanical deformities and self-oscillations of coupled nanobeam cavities, considering a regime, where the distance between two cavities is large enough allowing the gradient force to dominate and two optical super-modes are excited simultaneously. We show that by selecting the two-mode excitation parameters, it is possible to realize a static optomechanical potential with a well- defined minimum corresponding to a “spectral bonding” regime [5]. Whereas it was previously shown that self-induced oscillations can appear in optomechanical structures [8–10], here we reveal that static solutions can become strongly unstable under finite mechanical damping rates even at the deep global minima of a static 0146-9592/14/123543-04$15.00/0
potential. Furthermore, we identify a new regime of chaotic switching between mechanical deformities of opposite signs when potential has only a single narrow well. We consider an optomechanical cavity that supports two optical modes, where each mode can exert force exciting the same mechanical mode. An example of such structure is a pair of closely spaced suspended nanobeam cavities [11] as schematically illustrated in Fig. 1(a). These two optical eigenmodes can interact with each other through one mechanical degree of freedom, the transverse separation of two beam cavities. We denote the amplitudes of cavity modes as A for even and A− for odd optical modes, and consider the mechanical mode whose deformity changes the separation between the cavities d by a distance ξ. The cavity modes A can be excited by the input s and their resonant frequencies are linearly tuned by the deformity ξ. Following the established procedures [9,11] we formulate the nonlinear coupled-mode equations in dimensionless form,
Fig. 1. (a) Schematic of an optomechanical cavity supporting even and odd optical modes that induce a mechanical deformity ξ to the transverse separation. (b) Dependence of two optical mode resonant angular frequencies on the mechanical deformity, ωp mark the angular frequencies of optical even and odd modes’ excitations. © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014
ξ̈ −Γm ξ_ − Ω2m ξ F − jA− j2 − F jA j2 ;
(1a)
A_ −ia bξ − Γo A s ;
(1b)
A_ − −ic dξ − Γo A− s− :
(1c)
Here, Γm is the mechanical damping, Ωm is the mechanical oscillation angular frequency, and F defines the strength of optical forces induced by two optical modes. The signs in front of F are chosen to represent an attractive force for the even mode and a repulsive force for the odd mode. Parameters a and c define the normalized angular frequency detunings of even and odd modes relative to two laser inputs that are tuned close to the corresponding mode resonances; see Fig. 1(b). Parameters b and d define the change rate of optical angular frequency shift due to the mechanical deformity. The values of s give the excitation amplitudes of two optical modes, and we assume that both modes exhibit the same decay rate Γo . We first analyze the static solutions for a continuous laser illumination with fixed amplitude and angular frequency. The mode amplitudes are found as ¯ Γo −1 ; A¯ s ia bξ ¯ Γo −1 ; A¯ − s− ic dξ
(2)
where ξ¯ is a static mechanical deformation corresponding to a local minimum or maximum of the effective optomechanical potential V eff , read as Eq. (3). ¯ − V eff ξ
Z
−Ω2m ξ¯ F − jA¯ − j2 − F jA¯ j2 dξ¯
F js j bξ¯ a ξ arctan 2 bΓo Γo 2 F js j dξ¯ c − − − arctan : dΓo Γo Ω2m ¯ 2
2
(3)
The optomechanical potential has a shape similar to a coupled microring structure, for which the “spectral bonding” concept was suggested [5]. Indeed, through the simultaneous excitation of two optical modes, this potential can be tailored to have a deep global minimum, which could then facilitate precise alignment of resonances [5]. We illustrate such a situation in Fig. 2(a), where the potential is shown with color shading. Static solutions corresponding to the potential maxima are indicated by black and minima by white lines. We note that the optomechanical potential is determined by three terms in Eq. (3), and we show their contributions in Figs. 2(b)–2(e)) for characteristic potential shapes at different laser detunings. The total potential is shown with solid black lines; minima are marked with open red circles and maxima with blue circles. The first term corresponds to the mechanical restoring force and has a parabolic shape centered at ξ 0, shown as magenta dashed lines. The second and third terms, shown as green and cyan dashed lines, indicate the contribution from optical even and odd mode induced force,
Fig. 2. (a) Optomechanical potential V eff versus displacement ξ and optical frequency detuning a shown in color map with local minima (white lines) and maxima (black lines). (b)–(e) Optomechanical potential profiles (solid black lines) and separate contributions corresponding to different terms in Eq. (3) shown with the magenta, green, and cyan dashed lines. The frequency detunings a are (b) −0.1, (c) 0.5, (d) 0.58, and (e) 1, corresponding to the white dashed lines in (a). Parameters are Ωm 1, Γo 0.01, c a − 1, b −0.8, d 0.7, and F 1.
respectively. In the vicinity of the crossing where optical modes strongly interact with each other, local minima are determined by these three terms together; see Figs. 2(c) and 2(d), while the fixed minima at ξ 0 from the parabolic term or the detuning frequency dependent minima close to ξ −a∕b and ξ −c∕d dominate away from the crossing; see Figs. 2(b) and 2(e). We now perform a stability analysis of static solutions. Whereas potential maxima naturally correspond to unstable configurations, we find that strong instabilities also occur at the minima of the static potential. We consider a small perturbation (δξ, δA , δA− ) applied to the static solutions of Eqs. (1). We substitute the static solutions with perturbations in Eqs. (1), and perform linearization with respect to the perturbation amplitudes. The resulting coupled linear differential equations have general solutions of the form δξ qeiγt q e−iγ t , δA α eiγt α− e−iγ t , and δA− β eiγt β− e−iγ t . Here
June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS
(q, α , α− , β , β− ) is the perturbation eigenvector, γ is the corresponding eigenvalue, and these values can be readily determined numerically. The eigenmode with the largest value of −Imγ determines the overall stability. If −Imγ ≤ 0, the perturbations decay or at least do not grow. However, if −Imγ > 0, the corresponding perturbation eigenmode grows exponentially as exp−Imγt. We numerically solve the eigenproblem with the generalized Schur decomposition method, find the maximum unstable perturbation growth rate, −Imγ, as a function of mechanical damping rate Γm and frequency detuning a along G3 , G5 , and G7 local minima branches, and summarize these results in Figs. 3(a)–3(c). In these figures, the blueish region corresponds to stable solutions as all perturbations decay. Interestingly, as frequency detuning approaches a very narrow single potential well (the minimum of G7 at smaller values of detuning a), much higher mechanical damping is required for stability as the well width is reduced, see Fig. 3(c). Therefore, whereas narrow wells are attractive for precise deformity control, higher mechanical damping needs to be provided in this regime. On the other hand, static deformity control can be achieved with much smaller damping for some of the wider wells, using stability islands visible in Figs. 3(b) and 3(c). The stability islands appear when two cavity modes strongly interact through the mechanical mode in the region where the mechanical and optical damping rates are comparable.
Fig. 3. Results of stability analysis. (a)–(c) The growth rate of the most stable perturbations −Imγ versus mechanical damping rate Γm for static solutions along the G3 , G5. and G7 branches of local minima, horizontal white line marks Γm 3.5622 corresponding to (d), (e). (d) Self-sustained oscillation regions, colored in blue, red, and green. Lines show static states; local minima (solid gray lines) and local maxima (dashed gray lines), and the boundaries of static optomechanical potential barriers are indicated with thin dotted gray lines. (e) Frequency of the oscillatory states with the line colors corresponding to shadings in (d).
3545
Next, we examine the dynamics of growing perturbations on top of unstable static solutions. The linear stability analysis predicts exponential growth of small perturbations, however as perturbations grow larger complex nonlinear dynamics can emerge. We explore the dynamics of growing perturbations by numerically solving Eqs. (1) using the Dormand-Prince method and identify the appearance of self-sustained periodic oscillations with deformations shifted away from the static minima G3, G5, G7 and frequencies ranging from 0.063 to 0.121 times of the natural frequency of the structure. We show the different ranges of mechanical oscillations with blue, red, and green shadings in Fig. 3(d), and the corresponding oscillation frequencies are presented in Fig. 3(e) for the mechanical damping rate at Γm 3.5622, which is indicated by the white lines in Figs. 3(a)–3(c). These deformity oscillating regions demonstrate nonlinear dynamics that cannot be described by the static optomechanical potential. In particular, small perturbations applied to global minima G7 in frequency detuning range a 0.58–0.76 grow into two oscillatory stable states oscillating at different frequencies (0.11353 and 0.12012 times natural frequency of the structure) and outside the static optomechanical potential well. While approaching the static potential crossing, only the upper (blue-shaded) state shows discontinuity in time-averaged deformity and oscillation frequency at a 0.56, which is close but doesn’t coincide with the crossing detuning a 0.55. For the lower (red) oscillatory state, a transition region appears in the vicinity of detuning a 0.5, where both the amplitude and frequency of the self-sustained oscillation drop dramatically to zero while the time-averaged deformity converges to the static stable states on the branch G5 . For the range of detunings a 0.76–1.14 there are no periodic oscillations, and we leave this region blank in Figs. 3(d) and 3(e). An example of perturbation evolution at a 0.9 in this region is shown in Figs. 4(a)–4(c), which demonstrate complex aperiodic dynamics of the mechanical displacement and cavity mode amplitudes. The deformities mostly stay positive or negative, with sharp switchings between them. Positive deformities correspond to the excitation of both optical modes, whereas for negative deformities primarily even mode is excited. The aperiodic nature of oscillations is confirmed by the frequency spectrum of the optical odd mode amplitude, shown in Fig. 4(d). The broad triangle-shape spectrum suggests that the temporal signal is chaotic. To further characterize the dynamics, we determine the local maxima of deformities, marked by circles in Fig. 4(a). Then, we construct a Lorenz map [12] showing a mapping between the successive maxima in Fig. 4(e). We see the presence of regular regions with almost deterministic mapping for negative and positive deformity amplitudes, and an irregular region in the middle that gives rise to chaotic mixing. We find that the regular mapping regions can be approximated as ξmax n 1 ≃ 0.05 0.99034ξmax n (blue dashed line) for negative and ξmax n 1 ≃ 0.02 0.94233ξmax n (red dashed line) for positive deformities. This mapping is above the diagonal for negative deformations, and below the diagonal for positive deformities. It means that oscillations will feature accordingly growing or decaying peak amplitudes
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OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014
mixture of positive and negative deformity oscillations, which is different from the previously considered scenario of period doubling [8,10]. In summary, we demonstrate that optomechanical potentials with deep and narrow wells can be achieved in coupled suspended nanobeam cavities through the simultaneous excitation of odd and even optical supermodes. We predict the appearance of strong optomechanical instabilities even at a global minimum of static optomechanical potentials when the mechanical damping rate is below a certain threshold, and identify a broad parameter region where chaotic oscillations emerge with aperiodic switching between positive and negative deformities. These results suggest new possibilities for flexible static and dynamic control. Fig. 4. Aperiodic chaotic oscillations. (a)–(c) Time evolution of a small perturbation excited at a 0.9 in local minima branch G7 . (a) mechanical deformity, (b) intensity of optical even and (c) odd modes, (d) spectrum of optical odd mode shown in (c), and (e) Lorenz map between the successive maxima of mechanical deformity, marked with circles in (a). Blue and red dashed lines show approximate mapping in the regular regions of negative and positive deformities, respectively. (f) Histogram of mechanical deformity with unit total counts at each optical frequency detuning a.
of the mechanical deformities in the corresponding regions, until entering the central region of the map where chaotic change in oscillation occurs. We have calculated the Lyapunov exponents; the largest two eigenvalues of 0.0373 and −0.0002 indicate the chaotic behavior which is similar to that of a Lorenz system. In order to sketch a panoramic view of this chaotic region, we plot out the histogram of deformity perturbation time evolution with unit total counts at each frequency detuning a in Fig. 4(f). Throughout this region the distributions of the deformity are primarily localized around two discrete positions in the vicinity of negative (ξ −0.5) and positive (ξ 1.5) values. This further confirms that the origin of the whole chaotic region is the
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