Optical frequency combs generated mechanically

It is shown that a highly equidistant optical frequency comb can be generated by the parametric excitation of an optical bottle microresonator with nanoscale effective radius variation by its natural mechanical vibrations.

In addition to the previously developed approaches, here we show that highly equidistant and moderately broadband combs can be generated by parametric excitation of an optical mode of a bottle resonator by its natural mechanical vibrations. It is well known that mechanical vibrations of optical microresonators can be exceptionally monochromatic (like, e.g., those in quartz watches [13,14]) and possess a remarkably high Q-factor [15,16]. Usually, mechanical oscillations, being very small in magnitude, only slightly affect the spectrum of optical resonator by perturbing its physical dimensions and refractive index. In the first order of perturbation theory, mechanical oscillations with frequency f modify optical eigenfrequencies m can give rise to the parametric excitation of optical modes with eigenfrequencies { } m  and creation of an optical frequency comb. The effect is similar to that observed in electro-optical modulation of whispering gallery mode (WGM) microresonators with quadratic optical nonlinearity [2]. However, it is based on natural rather than externally modulated excitations of optical modes and is not associated with special electro-optical properties of the microresonator material.
In practice, the eigenfrequencies { } m  cannot be perfectly equidistant and it should be assumed that In this paper, we show that an elongated optical bottle microresonator with nanoscale variation of effective radius can have an equidistant spectrum which matches its natural mechanical vibrations. We calculate the inelastic resonant transmission amplitude through parametrically excited bottle microresonator and determine the structure of generated OFC with exceptionally accurate spacing f . The power of OFC teeth depends on the variations m  and can be large if the optical and mechanical Q-factors of the microresonator are high and deviations m  are small.

A bottle microresonator with equidistant spectrum
An optical bottle microresonator is defined as a segment of an optical fiber with a bottle-shaped effective radius variation (ERV) ( ) r z (Fig. 1) [17]. A strongly elongated bottle microresonator with ERV possess In Eqs.
center z the refr the Airy respecti depend radius 0~1 R m to intro parabol km).

Feas resonat
The idea equidist with na core of m f ( [19,22]. In spite of so small nanoscale height, this resonator supports a large number of axial modes. Here we will consider only a single series of modes (i.e., fixed m in Eq. (2)) with eigenfrequencies localized near frequency 0  . For characteristic 0~4 00  THz and fiber radius 0~2 0 r µm, the bandwidth of this series is It follows from Eq. (3) that the spacing between eigenfrequencies of this series is ~200 ax   MHz and, thus, it includes ~ 1000 elements. Here we consider transitions between modes of such series generated by mechanical vibrations of frequency f which is close to the eigenfrequency spacing ax   . More precisely, we consider an optical bottle microresonator, which has a series of eigenfrequencies q  with almost equidistant separation, i.e., The timedependent resonance amplitude of a WGM propagating along the optical fiber with nanoscale radius variation is described by the Schrödinger equation [23]: We assume that an eigenfrequency of mechanical oscillation of this microresonator, f , is close to the separation of optical eigenfrequencies by setting In the presence of this oscillation, the behavior of the resonator is described by Floquet quasi-states [24] ( , ) exp( 2 ) ( , ) where q  is a quasi-frequency and is a periodic function of time with period In this equation, we introduced Im f which characterized the attenuation and Q-factor Im / Re mech Q f f  of mechanical oscillations. Due to the proximity of frequency f and spacing between optical eigenfrequencies, 1 q q     , we will assume that the transition amplitudes between quasi-states ( , ) q t  r is much greater than transitions between these quasi-states and other quasi-states of the resonator. Then, the expression for the non-stationary Green's function of the resonator [25] will include only the series { ( , )} m t  r of our interest and can be written as The eigenfrequencies of a high Q-factor optical microresonator are usually detected by measuring the resonant transmission amplitude of light evanescently coupled to the resonator though a waveguide (Fig. 1). For a resonant WGM which circulates and slowly propagates along the bottle resonator axis z , the propagation constant ( ) z  is small , 0 ( ) z k   , and, therefore, the characteristic variation length  of WGM along axis z is large compared to the wavelength of radiation, Typically, the width of the coupling waveguide d is comparable with the radiation wavelength so that 0 d    . Under these conditions, the input light with frequency 1  can be introduced by adding the following source term to the right hand side of Eq. (5): where ( ) z  is the delta-function and 1   characterizes the switching time, which is usually much greater than the life time of eigenstates, i.e., | Im( ) | m    . Then, for weak coupling between the input-output waveguide and resonator (strongly under-coupling regime [26]), the inelastic output amplitude ( Selecting the sequence of resonance terms in Eq. (10) with varying 1 m and fixed q and 2 m and setting the input radiation frequency 1  equal to the minimum value of denominator in this equation, 1 2 Re( ) Re we determine the equidistant comb in the output spectrum with teeth at where the range of 1 m is limited by the bandwidth of the resonator spectrum (Fig. 2). Eqs. (11) and (12) are symmetric with respect to substitution 1 2  . However, due to the asymmetry of Eq. (10) with respect to this substitution, fixing the output frequency 2  and varying the input frequency 1  will not generate the comb of similar strength. Eqs. (10)- (12) show that there exist equally spaced combs of the transmission amplitude corresponding to {Re( ) Re } q m f   with fixed q and variable m . These combs are analyzed below for the special case of a bottle microresonator with time dependent quadratic radius variation. The power of the comb teeth defined by Eqs. (11) and (12) is strongest when the input frequency is equal to the resonance, (i.e., 2 0 m  ). In this case The power of teeth of this comb is determined from Eq. (10) as 2 * 0 2 ,0 2 , 2 , 2 2 (Im Im ) It follow the opti

Param
The ma with qu ., 0 0 z  . upling to (Fig. 3). acing of (16) e Eq. (5) with radius variation defined by Eq. (15) is a well-known Schrödinger equation with time-dependent quadratic potential that can be solved analytically [24]. The normalized solutions of this equation are: Here ( ) y t satisfies the Mathieu equation: The Floquet solution of this equation found in the first order in  and  is In derivation of Eq. (19), it was assumed that the Floquet solution is stable, which is valid for relatively small amplitude of vibrations,    . Using Eqs. (17) and (19) we expand the even solutions at 0 z  , 2 (0, ) q t  , into Fourier series: The spectrum of combs defined by Eqs. (13) and (14) depends only on the ratio of parameters  and  which formally can be arbitrarily small. Fig. 4 shows the power of the frequency comb   [7][8][9][10][11]. Not of Fig. 4  ). We assume that, due to the smoothness of the bottle resonator radius variation ( ) r z  and the semiclassical nature of the axial spectrum, the deviation from equidistance of the spectral series considered can be made very small, at least within a relatively small bandwidth. Tuning of eigenfrequencies by CO2 and femtosecond laser post-processing is possible as well [28,29]. We suggest that the maximum deviation from the equidistance of the spectral series considered, max | | m  has the order of 1 MHz, similar to [30]. In this case, the exactly solvable model of section 6 can be applied for ~1 In the absence of noise [31], the frequency of mechanical vibrations f slightly changes during the relaxation time mech  due to nonlinear processes. We estimate this change using the one-dimensional nonlinear wave equation which describes the radial deformation of microresonator ( ) u r [32]: Here 0  is the material density and  and  are the second and third order elastic constants.

Summary
It is shown that an OFC can be generated mechanically by excitation of an optical bottle microresonator with equidistant spectrum by its natural mechanical vibrations. In practice, small deviations from the spectral equidistance are introduced by fabrication errors. However, these deviations affect the OFC teeth power rather than their equidistance, which is defined by the natural frequency of vibrations f . As an example, we determine the OFC generated by a bottle microresonator with parabolic ERV excited by harmonic oscillations. It is found that the power of OFC teeth depends on the ratio of the perturbation of the ERV and deviation of the excitation frequency rather than their actual values. The bandwidth of the generated OFC is equal to 1.3q f  where q is the axial quantum number corresponding to the input optical frequency q  . Generally, the power of OFCs generated mechanically is inverse proportional to the squared product of their optical and mechanical Q-factors. Provided that these Q-factors are large enough, the power required for the generation of these combs can be remarkably small and only limited by the sensitivity of the optical detectors. A bottle resonator with parabolic ERV is not the only one which possesses equidistant spectral series. Other structures potentially enabling the generation of OFCs mechanically include specially designed bottle resonators with non-parabolic ERV [23] and series of coupled microresonators [35].