Nonlinear frequency response of the multi-resonant ring cavities

New theoretical approach describing a nonlinear frequency response of the multi-resonant nonlinear ring cavities (RC) to an intense monochromatic wave is developed. The approach closely relates the many-valuednesses of the RC frequency response and the dispersion relation of a waveguide, from which the cavity is produced. Arising of the multistability regime in the nonlinear RC is treated. The threshold and dynamic range of the bistability regime for an optical ring cavity with the Kerr nonlinearity are analytically derived and discussed.


Introduction
Resonant ring cavities (RC) belong to the core constructions of modern technology. They are utilized to produce various passive and active devices of optics, microwave electronics, spinwave electronics, plasmas, and other. As is well known, there are two conditions that are necessary to observe a resonant behavior. The first one is a phase condition. It consists in an in-phase addition of all the waves circulating in the ring. The second one is a dissipative condition. It claims in a small attenuation, which is necessary for multiple addition of the circulating waves. It is the multiple in-phase addition that leads to the resonant enhancement of the signal at a certain frequency, in case the both conditions are satisfied. The resonant enhancement provides a decrease in the nonlinear processes threshold, which makes the RC exceptionally useful constructions to study a variety of nonlinear phenomena.
The Lugiato-Lefever equation (LLE) is widely used to describe the nonlinear phenomena of the ring cavities. This equation was derived using both the mean-field approach and infinite-dimensional map (Ikeda map) [1][2][3][4]. In other words, in a representative LLE approach one plugs nonlinearity and integrates dispersion as a function of the modal index, so that despite dispersion is linear, one obtains bistability as a result of nonlinear shift of the resonance. Hence, the LLE is instrumental in modeling of the different nonlinear waveforms [5][6][7][8][9][10][11]. Recently the LLE was extended to consider the multiple nonlinear resonances appearing in the optical RC as well as the multi-valued stationary states [12][13]. In parallel, such states were independently investigated with a general Ikeda map [14]. The investigations done beyond the conventional LLE open new possibilities to describe the multistability in the solitary and coupled micro-rings [15][16][17]. As an example, we mention investigations of a super-cavity soliton formation [14], a soliton formation in the coupled micro-rings [18], and a cavity soliton formation in the micro-rings with active elements [19].
Two types of the nonlinear instabilities are distinguished in the RC studiesthe dispersive and absorptive ones [20]. Note that in existing RC literature they are indicated but not entirely viewed. Specifically speaking, the dispersive instability is described using RC frequency response, which does not ponder the wave-number many-valuednesses of the dispersion characteristic. Due to this, the important phase-dependent features of the nonlinear processes are lost. For example, when calculating the RC transmission spectra, its real part is only obtained.
Our theoretical approach, being the field of the extended LLE, does not challenge the existing literature, yet, at the same time, it introduces a new essential feature. This feature consists in a self-consistent unification of the observed frequency response and changing in the dispersion law. The aim of this paper is to propose a new theoretical approach, which bonds the effects of the nonlinear dispersion and frequency response of the multi-resonant nonlinear ring cavities. The approach enables one to relate the many-valuednesses of the RC frequency response and the dispersion characteristic of a waveguide the RC is made of.

Theoretical approach to describe nonlinear multi-resonant RC
In order to introduce the approach, let us consider a multi-resonant RC of the length l , which includes of a unidirectional coupler between a ring resonator and a waveguide shown in  (2)] were used in a number of investigations to describe the frequency response of the optical micro-resonators [21,22], the multiferroic active ring resonators [23], the optoelectronic rings [24], and the spin-wave optoelectronic rings [25][26][27]. A critical feature of the frequency response [Eq. (2)] is an enhancement of the intracavity field intensity close to the resonant frequencies, being the effect to take into account for the multi-resonant rings. To thoroughly study this effect, one should consider the dependence of an RC performance on the wave sensitivity of the waveguide parameters of which it is made.

Application of the theoretical approach to a nonlinear optical ring
In this work, we use an optical wave intensity dependent refractive index as the nonlinear parameter of the waveguide substance. In the case of the Kerr nonlinearity, we write   02 n is a second-order refractive index, and I is an optical wave intensity. For investigation of the nonlinear RC, we substitute the  . The latter formula shows that the refractive index is enhanced at all the resonant frequencies. The next step in introducing of our approach should be done utilizing some dispersion law for a regular waveguide fabricating the resonant RC. For demonstration of a generality and advantages of our approach, we will use below the simplest approximation for the linear dispersion law in a form cn   (3) where c is the speed of light. Note that this step is valid for the frequency band, in which the dispersion impact on the wave process is relatively weak. Usually it works well for analysis of several neighboring resonant frequencies. Provided it is necessary to study a nonlinear RC in a wide frequency range, one should consider a higher-order dispersion.
To study the nonlinear effects, we substitute the intensity dependent refractive index is a linear frequency response at the resonant frequency, which defines the maximum of the transfer function. Analyzing the obtained data, it is possible to do valuable conclusions on the nonlinear resonant RC characteristics. Below we discuss some of them only.
One can see a fascinating behavior of the nonlinear RC characteristics that manifest themselves with increasing in the input optical intensity Expression for the threshold intensity will be obtained and discussed below. Note that an increase in the input intensity 0 I higher than th I provides appearance of the region with two stable and one unstable state of the intracavity intencity, which corresponds to the bistability phenomenon. The magenta lines in the both figures show the dispersion characteristics and the transmission coefficient frequency response for the intensities higher than the threshold value ( 0 15 th II  ). Further increase in the input intensity (up to 0 100 th II  ) extends the frequency band of the instable behavior. Provided this band covers the frequency range located between two neighboring resonant frequencies, then additional stable and unstable states appear.
Our calculations show that the new unstable states can develop progressively with increasing of the circulating power. Such a behavior corresponds to the multistability regime (see blue lines in Fig 2.a and 2.b). So, for the nonlinear frequency shift ( 0 100 th II  ), which is more than two free spectral ranges ( 2   ), four stable and three unstable values of the intracavity intensity and corresponded wave-number multivaluedness appear. Furthermore, more and more unstable states may advance with increasing of the circulating power.
It is clear physically that in order to have multistability, it is necessary to observe the conditions, in which the nonlinear frequency shift is more than the distance between two adjacent frequencies. In case of a single ring, this requires a sufficiently high input power, which in the real-life resonant ring cavities cannot be achieved due to a nonlinear damping. However, the threshold for the multistability observation can be significantly reduced in some cases like coupled resonator systems [28]. But this effect is out of the paper scope.

Optical ring bistability threshold
As an example, we consider the bistability phenomenon in an optical ring cavity. As was already mentioned, one of the distinctive features of the bistability behavior is an appearance of the two-valuedness in the relationship between the intracavity and input intensities. Note that these extremums correspond to the group velocity zeros for the wave under consideration. As is seen from Fig. 3.a, the nonlinear dispersion characteristics coincide with the linear one in the frequency range situated far enough from the given resonance.   (7)] and the nonlinear frequency shift coincide with the results obtained in a series of investigations devoted to the Kerr nonlinearity [11,30,31].

Dynamic range of the bistability phenomenon
For analysis of the bistability dynamic range, we use the diagram of the nonlinear behavior [29]. As in the catastrophe theory, such diagram is a curvilinear surface, which demonstrates all real roots of [Eq. (6) Fig. 4 and it corresponds to the square mark in Fig. 3 Further increasing of the input intensity leads to extending of the multi-valued range. One can see the positions of the kinks on the characteristics in Fig. 4, marked with the blue and orange circles, that are also shown in Fig. 3. For the illustrative purposes, the extension of the bistability frequency band with increasing of the input intensity is shown by the projection of the nonlinear diagram onto a plane (  Fig. 3). Note that diagrams akin to Fig. 4 are widely used for the description of bifurcation maps [31,32].

Summary
A general theoretical approach uniting the dispersion and the transmission response of the multi-resonant nonlinear ring cavities is proposed for the first time. The approach provides an opportunity to study the nonlinear wave processes in wide frequency range, significantly exceeding the distance between the adjacent resonant RC frequencies. To utilize this, it is enough to know the linear dispersion law and the dependence of material parameters of the ring on the wave amplitude. As an illustration, the main characteristics of the bistability phenomenon in an optical ring with the Kerr nonlinearity are considered. The proposed approach demonstrates the existence of not only two stable and one unstable value of intracavity intensity, but also many-valuedness of stable and unstable wave-numbers.