Solions In Optomechanical Arrays

Nonlinear science has been a fast-developing topic in the nearly half of a cen-tury.It impresses deeply not only the understanding of the nature on us,but also promote the development of the modern technology.For the last decades,the op-tomechanical system which concerns the interaction between photon and phonon via radiation pressure has attracted extensive attention.Its unique optomechanical non-linearity has shown luxuriant contents and wide applicability.Here we study the propagating problem of the optical field in the optomechanical array,which include the self-focusing phenomenon and modulation instability and so on.We discuss not only the connection between optomechanical nonlinearity and Kerr nonlinearity,but also the unique behaviours of the optical fields induced by optomechanical nonlin-earity.The main contents are:1,We deduce the evolution equation of the optical and mechanical field in op-tomechanical array.And we apply two methods to show the existence of optical soli tons in optomechanical arrays.First we utilize the traveling method to get the accurate soliton solution.Secondly,we take advantage of an analytically approxi-mate method to get the connection between optomechanical nonlinearity and Kerr nonlinearity.It shows that when the intrinsic change of the system is much smaller than the characteristic frequency of the mechanical mode,the evolution of the optical field can be well approximated by the nonlinear Schrodinger equation.We also ver-ify our analyses with numerical simulations.It must been pointed out that here the optical soltions stem from a new mechanism,photon-phonon interaction,and they can exhibit a very low group velocity which depends on the photon hopping rate.2,We study the modulation instability phenomenon in optomechanical arrays.We find that under the optomechanical nonlinearity,the modulation instability ex-hibits a very unique behaviour.First,the modulation instability happens even in nor-mal dispersion region,and this modulation instability is oscillatory.Secondly,con-tinuous version and discrete version with optomchanical nonlinearity have essential difference.The modulation instability happens anyway in the continuous situation but need a minimum background in the discrete form.Thirdly,We discuss the effect of the phonon dispersion.We observe that in a manner the phonon dispersion can be regarded as a correction of the optical dispersion.When they have the same sign,the optical dispersion is strengthened or conversely weakened.This property expound a very interesting phenomenon,when the optical dispersion is weaker than the phonon dispersion and have the opposite sign,even in the continuous form,the modulation instability will occur only when the background is stronger than a specific value in normal dispersion region.3,We finally study the discrete solitons of the optomechanical array.In the s-ingle pulse situation,the optomechanical nonlinearity will sustain the same discrete breather as the kerr nonlinearity,but they have quite different linearly stability proper-ty.Analytically we find when the mechanical frequency is large enough,the stability judgement equation of the optomechanical nonlinearity have the same eigenvalues as the kerr nonlinearity when the eigenvalues is much smaller than the mechanical frequency.And when the mechanical frequency is small enough,the eigenvector with eigenvalue much larger then the mechanical frequency is always stable.This property can be clearly verified by the numerical simulation with the inter-site mode.We also find a stable on-site mode in kerr nonlinearity becomes unstable under some circumstance.We finally briefly discuss the simplest example of the optomechanical array:a dual optomechanical system.We find when the mechanical frequency is the same order of the photon hopping magnitude,then we will observe a inter-conversion phenomenon of self-localized state and transition state,and this conversion is quite sensitive to the initial value.In general,here we show that the optical solitons can be obtained in the optomechanical array.And they stem from a new mechanism——photon phonon interaction or optomechanical nonlinearity.We not only expound the connection of optomechanical nonlinearity and Kerr nonlinearity,but also reveal a series of unique phenomena induced by optomechanical nonlinearity.It includes the oscillatory modulation instability in normal dispersion region,the essential difference between continuous and discrete form,the modulation action of the phonon dispersion and the linearly instability of discrete soliton,dynamical behaviour of the dual optome-chanical system and so on.This work may extend the range of optomechanics and nonlinear optics,and provide a new field to study soliton theory and develop the corresponding applications in all-optical switch and on-chip optical architecture.