Controlled transportation of light by light at the microscale

We show how light can be controllably transported by light at microscale dimensions. We design a miniature device which consists of a short segment of an optical fiber coupled to transversely-oriented input-output microfibers. A whispering gallery soliton is launched from the first microfiber into the fiber segment and slowly propagates along its mm-scale length. The soliton loads and unloads optical pulses at designated input-output microfibers. The speed of the soliton and its propagation direction is controlled by the dramatically small, yet feasible to introduce permanently or all-optically, nanoscale variation of the effective fiber radius.

Transportation of objects by other objects, both at the macroscale and microscale, is an evident constituent of the evolution of nature in general and living beings in particular. At the macroscale, we travel and carry things from one place to another and use machines to make it easier and faster [1]. At the microscale, human-guided transportation and manipulation of objects is of great multidisciplinary importance with applications ranging from medical and life science to nanomaterial science, bionanotechnology, and nanoelectronics [2][3][4].
In microphotonics, addressed in this paper, we can separate the micro-objects under study into those constituted of matter (e.g., waveguides, microresonators, and micro/nanoparticles) and those constituted of light (e.g., optical waves, pulses, and localized states). Consequently, at the microscale we distinguish the transportation and manipulation of (a) matter by matter, (b) light by matter, (c) matter by light, and (d) light by light.
There are numerous examples when matter micro/nano-objects are controllably transported and manipulated by other matter micro/nano-objects. The developed approaches often resemble the manipulation of macroscopic objects with mechanical, electro-mechanical, and magnetic tools [2][3][4]. In particular, at the atomic-scale dimensions, the transportation and manipulation of atoms and nano-objects are possible with an atomic force microscope (AFM) [5][6][7] (Fig. 1(a)).
Transportation of light by material micro-objects is possible as well. For example, optical microresonators are used to confine and manipulate light at the microscale [8,9] (Fig. 1(b)).
They are commonly considered at rest with respect to the laboratory system of reference [8][9][10]. Generally, the translation of a microresonator with constant speed does not affect the behavior of localized states residing in it [11]. However, accelerated translation, vibration and rotation can significantly perturb the resonant states [12][13][14][15][16]. In the simplest case, light confined in a microresonator can be transported mechanically using a "truck" in the form of a translation stage. In another example, controlling the perturbation of an eigenstate in a rotating microresonator allows one to use it as a miniature gyroscope [14,15].
In turn, light in the form of optical tweezers can confine and manipulate matter microobjects [17,18]. For example, light waves can localize microparticles close to their antinodes by the gradient and scattering forces ( Fig. 1(c)). In addition, propagation of light through nonlinear media allows the manipulation of light itself, such as modification of its spectrum and self-localization at the microscale. Examples of current significant interest include fre- quency comb generation [19,20], optomechanical processes [21] and formation of solitons [20,[22][23][24]. Conventionally, for telecommunication applications, broadband picosecond and sub-picosecond temporal solitons are used [22,23]. Broadband solitons have been studied in application to the frequency comb generation [19,20]. Slow broadband solitons with picosecond duration have been demonstrated in photonic crystals [25,26]. Slow narrowband solitons, which may have much smaller propagation speed, can be realized in periodic microstructures provided that the soliton central frequency is close to the band gap edge [27][28][29].
However, can we use light as optical tweezers for light? Is it possible to confine and translate light controllably and all-optically at the microscale? The natural approach to answer this question is to consider a soliton as a moving microresonator which can confine and transport weaker light and, thus, be used as a micro-truck for light ( Fig. 1(d)). This may be possible since the electric field E of a soliton propagating along an optical fiber induces a change in the refractive index ∆n ∼ |E| 2 due to the non-linear Kerr effect [18,19]. Therefore, the soliton field (as well as the field of any other sufficiently strong optical pulse) can act as a moving effective potential well which traps and transports an optical signal. Three decades ago, transportation of a localized optical state by an optical soliton was proposed [30,31]. This beautiful idea did not attract much attention because a realistic device, which enables the all-optical transportation of a relatively weak state of light including its loading and unloading, has not been suggested to date.
In this paper, we describe a microdevice where relatively weak optical pulses and eigenstates are transported between input and output ports by a soliton (Fig. 2 A pulse with central angular frequency ω s is coupled into the FS from MF 0 forming a whispering gallery mode, which is enhanced due to the constructive self-interference. As a result, a WGS with central frequency ω s is formed. It is assumed that ω s is close to the cutoff frequency ω s,e (z) − ω s,e assumed here, the expression for slowly propagating whispering gallery modes can be factorized as R ms,eps,e (r)e ims,eϕ ψ s,e (z, t)e −iωs,et where m s,e and p s,e are azimuthal and radial quantum numbers. Consequently, the propagation of a narrow bandwidth WGS with central frequency ω s and a WGE with frequency ω e = ω s along the fiber axis coordinate z is determined by their amplitudes ψ s (z) and ψ e (z). These functions are defined by a system of coupled nonlinear Schrödinger equations, which are similar to those commonly used in nonlinear fiber optics [22,30,31] where the temporal and spatial coordinates are interchanged [27,28,36,37].
Assuming |ψ s (z)| |ψ e (z)| we have: Here κ s,e = c 2 /(n 2 s,e ω s,e ), c is the speed of light, n s and n e are the refractive indices of the FS at frequencies ω s and ω e , n 2 is its nonlinear refractive index, δ(z) is the delta-function, and A ss , A se are the effective mode areas defined in [22] and Appendix A1. The terms J s (t)δ(z − z 0 ) and J e (t)δ(z − z 1 ) in Eqs. (1a) and (1b) determine the soliton and weak pulse sources at microfibers, which are specified below. Parameters D sj and D ej are the coupling of the FS to microfibers at frequencies ω s and ω e determined in Ref. [34]. For a single input-output microfiber with a source, Eq. (1a) coincides with that obtained previously in [37].
To estimate the characteristic parameters of our device, we assume that the FS is uniform.
Then Eqs. (1a) and (1b) can be solved analytically [22] yielding for WGS: where P s is the soliton peak power, L s is the soliton characteristic width and v s is the soliton velocity. After the substitution of |ψ Eq. (3) determines the maximum variation of the cutoff frequency caused by the WGS.
Assuming that the WGE have the same speed as the WGS, we look for the solution of Eq. (1b) in the form ψ (0) e (z, t) = Φ(x)e iαx+iβt , which depends on the dimensionless relative coordinate x = z−vst Ls . Then Φ(x) satisfies the equation    [34,41]. We found that the real parts of these coupling parameters with the same order of magnitude do not noticeably modify the WGE (blue curve in Fig. 3(e1), (e2)). This is the reason why they are set to zero.  Fig. 3(b2) shows the propagation of the WGS when its speed is reduced to 1.3 · 10 5 m/s = 0.00044c in between microfibers. In Fig. 3(b3), the WGS stops in between MF 1 and MF 2 and returns back to MF 1 to unload the WGE (see [32] for the analogue dispersionless propagation of a linear pulse  Fig. 3(e1)-(e3)). Taking this dissipation into account, we find that, for the D e2 = i0.05 µm −1 chosen, more than 70% of the WGE power is unloaded into MF 2 .
Obviously, the amount of unloaded power can be improved by increasing D e2 . Generally, the evolution of a WGS-trapped WGE pulse with frequencies distributed within the quantum well bandwidth (rather than coinciding with its eigenvalue) can be quite complex [42][43][44].
Thus, as in the case of slow linear propagation of whispering gallery modes [32][33][34][35], the and Refs. [46,47]). Therefore, further optimization of the carrier pulse parameters and its speed may be required. However, according to our estimates (see Appendix A3), these pulses do not introduce a significant temperature variation. For chalcogenide and hydrogenated amorphous silicon (a-Si:H) fibers, which have larger nonlinearity, the peak power of the WGS can be two orders of magnitude smaller [48,49]. As it is well-known from quantum mechanics [50], the one-dimensional potential well induced by a WGS can always hold at least one optical eigenstate despite of its shallowness. One of the intriguing conclusion of our findings is that the WGS speed can be fully controlled by unexpectedly small variation of the cutoff wavelength ∆ω (cut) s ∼ 1 GHz, which, for the fiber radius r 0 ∼ 20 µm, corresponds to an effective radius variation of r 0 ∆ω (cut) s /ω s ∼ 0.1 nm. The fabrication precision achievable in SNAP technology [32,35] makes the introduction of such dramatically small variations feasible. Furthermore, these variations can be induced all-optically. In fact, the amplitude of mechanical vibrations of an optical microresonator, which are induced by whispering gallery modes, can be tuned up to 10 nm [51]. For the microresonator with radius r 0 ∼ 20 µm considered in [51], this corresponds to a cutoff frequency variation exceeding 10 GHz.
In the equations, ∆ω s,e (z) = ω where F s,e (r, φ) are the transversal modal distribution of WGS and WGE [1]. In order to calculate the effective mode areas, we need to evaluate these integrals. The transversal modal distribution are approximated by the Airy functions: Here azimuthal quantum numbers m s,e are related to the frequencies and the fiber radius We have chosen a Gaussian pulse as the source J s (t). In particular, for our simulations, where t 0 = 1 ns and τ 0 = 0.3 ns. The weak signal source J e (t) corresponds to a CW signal with frequency ω e + δω e , where δω e is the detuning chosen so that ω e + δω e is equal to the eigenfrequency of the WGS potential.
In our numerical simulations, we use the dimensionless version of Eqs. (A1) by introducing dimensionless variables which gives We choose time T 0 by setting κsT 0 In addition, we choose P 0 by setting ωsn 2 T 0 P 0 nsAss = 1 which gives Finally, we introduce the following dimensionless parameters and functions: ∆ω s,e = ∆ω s,e · T 0 ;γ s,e = γ s,e · T 0 ;D s = D s · L 0 ;D e = D e · r −1 ω L 0 ;Ĵ s,e = J s,e · T 0 L 0 √ P 0 (A9) As the result, Eqs. (A6) are presented in the dimensionless form: where r ω = ω e /ω s = 0.87 is the ratio of frequencies, η = 2n e ω 2 e A ss n s ω 2 is the dimensionless height of the soliton potential (see Eq. (4)) where r A = A ss /A se ∼ 0.95 is the ratio of effective mode areas and r n = ne ns = 1 is the ratio of refractive indices. For these parameters, the dimensional height of the WGS-induced potential is η = 1.435.
Under quite general conditions [2], we assume that the relative cutoff frequency variations of the FS at the WGS and WGE frequencies are equal: Taking this relation into account, we finally obtain: MHz (corresponding to Q = 2 × 10 6 at frequency ω/2π = 190 THz). The plot shows that the WGS is not fully formed and does not reach MF 1 . If we reduce the attenuation factor to γ s = 30 MHz (Q = 2 × 10 7 ), the WGS is formed as shown in Fig. A1(b). However, the soliton quickly decays, in particular, after passing MF 1 where it experiences additional losses of energy due to the coupling with MF 1 . In Fig. A1(c), γ s = 3 MHz (Q = 2 × 10 8 ).
In this case, which has been considered in the main text, a WGE can be formed and survive a few millimeters of transportation. In Fig. A1(d), the attenuation is reduced further to γ s = 0.3 MHz (Q = 2 × 10 9 ). This value for the attenuation factor is feasible [3] and allows our microdevice to transport the WGE from one microfiber to the other without significant losses. Finally, for comparison, Fig. A1(e) shows the linear and lossless propagation of a pulse launched from the vicinity of MF 1 , which has the same original shape as the WGS which has just passed MF 1 in Fig. A1(d). The propagation of this pulse was calculated by setting γ s = 0 and n 2 = 0 in Eq. (A1a). It can be seen that, due to dispersion, this pulse strongly decays before reaching MF 1 . In addition, as it follows from the inset in Fig. A1(e), the speed of the pulse spreading is comparable with its group velocity.

A3. The damage threshold and temperature effects
The damage on the optical fiber induced by the CW radiation is usually due to melting as a result of light energy absorption. However, pulses shorter than ∼ 1 ns can damage the optical fiber by other processes including dielectric breakdown in the material (electron avalanche) caused by the strong electric field (see [4][5][6] and references therein). The threshold for optical damage of short pulses depends on the pulse central frequency and its intensity distribution in space and time. The threshold values experimentally determined previously are characterized by the value of fluence F defined as the average energy of the pulse per its unit cross-sectional area. Roughly assuming that F ∼ ω 0.4 [4] and using the data of Ref.
[5] we find that the threshold fluence for a pulse with duration of 0.1-1 ns is 10-50 J/cm 2 .
These values might not be directly applicable to the WGS due to the specific geometry of our problem and they can significantly vary depending on experimental conditions. However, we believe that they can serve as reasonable estimates for the problem considered here.
We calculate the fluence F of the WGS using the soliton model described by Eq. (2) of the main text which yields Here L and v are the FWHM and the speed of the WGS. In Fig. A2   The propagation of WGS and WGE is controlled by dramatically small variations of the cutoff frequencies along the FS length. Therefore, we have to ensure that these variations are not affected by the temperature variation caused by the WGS propagation or, alternatively, take them into account in our modeling. Attenuation of the WGS power is primary due to absorption and scattering effects, while only the absorption of power contributes to fiber heating. Let us assume that the latter effect is determined by an attenuation coefficient γ abs = 1 MHz. Since the characteristic cutoff frequency variation and WGS bandwidth γ abs , we can estimate the WGS spatial attenuation constant in the linear approximation as (see e.g. [2]) α = 2 1/2 n s c −1 ω 1/2 s Im ∆ω + iγ abs 1/2 ∼ = 2 −1/2 n s c −1 ω 1/2 s ∆ω −1/2 γ abs = 4 m −1 (A15) From our numerical simulations shown in Fig. A1(c) and Fig. A1(d