Supplementary Information Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons Jae K. Jang, Miro Erkintalo, Stéphane Coen, and Stuart G. Murdoch The Dodd-Walls Centre for Photonic and Quantum Technologies, and Physics Department, The University of Auckland, Private Bag 99, Auckland, New Zealand This article contains supplementary theoretical and experimental information to the manuscript entitled Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons. Specifically, we derive the rate of phase-modulation induced drift using the mean-field Lugiato- Lefever equation, and show how tweezing can be reproduced in full numerical simulations. We also derive the equations describing the impact of asynchronous phase modulation. Finally, we experimentally and theoretically demonstrate that temporal tweezing is distortionless. SUPPLEMENTARY FIGURE mode-locked laser,.8 ps GHz, 53 nm Signal/pattern generator & RF electronics ALNAIR PRITEL Pattern generator GHz MZM cw laser, 55 nm 5 ns EOM MZM Single pulse Addressing beam WDM EDFA isolator m SMF controller Fibre cavity 9/ coupler 99/ coupler Driving beam EDFA BPF polarization controller BPF GSa s scope Supplementary Figure Experimental setup. EOM: Electro-optic phase modulator; EDFA: Erbium-doped fibre amplifier; BPF: Bandpass filter; SMF: Single-mode fibre; MZM: Mach-Zehnder intensity modulator; WDM: Wavelength division multiplexer.
SUPPLEMENTARY FIGURE a 5.5 b 5.5 5 Electronic 5 Optical Laboratory time (ms).5 3.5 3.5.5.8.6.. Electronic signal (a.u.) Laboratory time (ms).5 3.5 3.5.5.8.6.. Optical intensity (a.u.).5.5 Fast time (ps) Fast time (ps) Supplementary Figure Illustrative numerical simulations of temporal tweezing. a, Phase profile φ(τ, t) used in the simulations to illustrate the temporal tweezing of CSs. The second phase peak shifts back and forth in sequence at a rate of ± 5 fs/roundtrip with intermediate stationary periods. b, The corresponding dynamical evolution of two.6 ps-long temporal CSs. The trailing CS always drifts towards the shifting phase maximum. These simulations use experimental parameters, with t R =.8 µs, α =.6, δ =. rad, L = m, β =. ps km, γ =. W km, θ =., P in = 96 mw. The phase peaks are Gaussian with 9 ps FWHM and. rad amplitude.
SUPPLEMENTARY FIGURE 3 a.3 Measured FROG trace, without PM b.3 Measured FROG trace, with PM Frequency (THz).5.5 Frequency (THz).5.5.3.3 c Spectrum (db) 6 Measured spectra With PM Without PM 56 58 55 55 55 Wavelength (nm) d Autocorrelation (a.u.)..9.8.7 Measured normalised autocorrelations With PM Without PM.6 e Power (W) 8 6 Simulated CS profiles With PM Without PM Maximum phase gradient 5 5 5 5 Supplementary Figure 3 Demonstration of the distortionless nature of temporal tweezing. a, Measured FROG trace of our temporal CSs without phase modulation (PM). b, The same measurement but with phase modulation. c, Measured average spectra with and without phase modulation. d, Corresponding intensity autocorrelation functions extracted from the FROG measurement. e, Simulation results, comparing the temporal intensity profiles of a CS when trapped to a phase peak (black line), in the absence of phase modulation (red line), and when it sits at the point of maximum phase gradient (blue line). Simulation parameters are identical to those of Supplementary Fig.. 3
SUPPLEMENTARY NOTE : THEORY OF TEMPORAL TWEEZING Temporal tweezing relies on picosecond CSs exhibiting an attractive time-domain drift towards the maxima of the intracavity phase profile. In the main manuscript, we explain the physics underlying the attraction in terms of the CSs shifting their instantaneous frequencies in reaction to a phase modulation. Here we elaborate on this aspect by presenting a more formal mathematical treatment. The dynamics of light in a high-finesse passive fibre cavity is governed by the mean-field Lugiato-Lefever equation (LLE),, t R E(t,τ) t = [ α iδ i L β ] + iγ L E E + θ E τ in. () Here t is the slow time describing the evolution of the intracavity field envelope E(t,τ) over subsequent cavity roundtrips, while τ is a fast time describing the temporal profile of the field envelope in a reference frame travelling at the group velocity of the holding beam in the cavity. E in is the field of the holding beam with power P in = E in. The parameter α = π/f accounts for all the cavity losses, with F the cavity finesse. Denoting φ as the linear phase-shift acquired by the intracavity field over one roundtrip with respect to the holding beam, δ = πl φ measures the phase detuning of the intracavity field to the closest cavity resonance (with order l). Finally, L is the cavity length, β and γ are, respectively, the dispersion and nonlinear coefficient of the fibre, and θ is the input coupler power transmission coefficient. To analyse the effect of a phase-modulation of the holding field, we follow the approach of Ref. 3. Specifically, assuming a phase-modulation temporal profile φ(τ) that repeats periodically with a period equal to the cavity roundtrip time t R (or an integer fraction of it), we can write E in (τ) = F in exp[iφ(τ)], where F in is a constant scalar. Substituting this expression into Supplementary Eq. (), together with the ansatz E(t, τ) = F(t, τ) exp[iφ(τ)], yields F(t,τ) t R β Lφ F [ t τ = α F iδ F i L β ] + iγ L F F + θ F τ in. () where α F = α β Lφ / (3) δ F = δ β L(φ ) / () with φ = dφ/dτ and φ = d φ/dτ. For typical parameters, corrections to α and δ are small (see Supplementary Note ), and therefore we can first observe that the background intracavity field on which the CSs are superimposed has the same phase modulation as that imposed on the external holding beam (to within a constant phase shift). Next, we consider the second term on the left-hand-side of Supplementary Eq. () which gives rise to a local change of the group-velocity of the field. Indeed, with the change of variable τ τ F = τ + β Lφ t/t R (and assuming that the phase modulation is slow in comparison to the duration of CSs), the second term on the left-hand-side cancels out and Supplementary Eq. () can be recast into a form identical to the LLE [Supplementary Eq. ()], but with a timeindependent holding field F in. Such an equation, in the τ F frame, admits stationary CS solutions. 3 The CS solutions
of Supplementary Eq. () thus drift in the fast time domain τ, and move with respect to the phase modulation, with a rate V drift = dτ dt = β Lφ t R. (5) V drift is generally small enough for the total change of the temporal position of a CS over one roundtrip to be approximated as τ drift V drift t R = β Lφ. In our experiments β < (anomalous dispersion), τ drift = β Lφ, and a CS overlapping with an increasing part of the phase modulation profile (φ > ) will be temporally delayed (τ drift > ) whilst another overlapping with a decreasing part (φ < ) will be advanced (τ drift < ). Both scenarios result in CSs approaching the phase maxima, where the drift ceases (φ = τ drift = ). It is worth noting that τ drift coincides exactly with the temporal delay accrued by a signal with an instantaneous angular frequency shift δω = φ, propagating through a length L of fibre whose group-velocity dispersion is β. Therefore the above LLE-analysis is fully consistent with the simple physical description presented in our main manuscript. If the phase modulation φ(τ) is dynamically varied [such that φ(τ, t) also depends on the slow-time variable t] then the intracavity phase profile will adjust accordingly, with a response time governed by the cavity photon lifetime t ph (with our parameters t ph 3. t R ). This is the principle of our temporal tweezer that allows trapped CSs to be temporally shifted, as is demonstrated experimentally in the main manuscript. Temporal tweezing of CSs can also be readily studied using direct numerical simulations of Supplementary Eq. (). In Supplementary Fig., we show an example of numerical results where we start with two CSs at the top of two 9 ps wide (FWHM) Gaussian-shaped phase pulses initially separated by ps. After 5 roundtrips, we start shifting the second phase pulse at a rate of 5 fs per roundtrip, first bringing it ps closer, then moving it away and back again until we eventually stop. We can see how the corresponding CS follows these manipulations. At the end of the sequence, it gets trapped again stably at the phase peak. Note that all the parameters used in this simulation (listed in the caption of Supplementary Fig. ) match values used in our experiments. The only difference is the much faster manipulation of the phase profile, performed over a much smaller number of roundtrips, to expedite the computations and to facilitate visualization of the transient dynamics. In particular, we remark that the simulation here captures the actual two orders of magnitude difference between the durations of the CSs (.6 ps) and the RF pulses ( 9 ps), whilst the experiments reported in our main manuscript are subject to limited bandwidth of the detection electronics. SUPPLEMENTARY NOTE : ASYNCHRONOUS PHASE-MODULATION Here we show theoretically that CS trapping can be achieved even if the phase modulation frequency is not an exact multiple of the cavity FSR. In particular, we derive the equation that allows the CS trapping position to be predicted. Consider phase modulation with a frequency f PM = N FSR + f slightly mismatched with respect to the Nth harmonic of the FSR. If f >, the intracavity field takes effectively too long to complete one cavity roundtrip and be in synchronism with the phase modulation. The temporal delay that the intracavity field accumulates with respect 5
to the phase profile of the holding beam over one roundtrip is given by τ = t R NT PM, with T PM = /f PM the period of the phase modulation. Given that t R = FSR, and assuming f FSR, we have τ f/[fsr f PM ]. A CS can remain effectively trapped provided that, during each roundtrip, the phase-modulation induced drift exactly cancels the relative delay accrued from the synchronization mismatch: τ drift + τ =. Recalling that τ drift = β Lφ, we can see that in the presence of a non-zero frequency mismatch f, the CSs will not be trapped to the peak of the phase profile (where the gradient φ vanishes). Rather, they will be trapped at a temporal position τ CS along the phase profile that satisfies the equation: φ (τ CS ) = f β L FSR f PM. (6) This also allows us to obtain the limiting value for the frequency mismatch that the trapping process can tolerate: f < β L FSR f PM φ (τ) max f max, (7) where φ (τ) max is the maximum of the gradient of the phase profile, and for brevity we have assumed φ(τ) to be symmetric. When f = f max, the CSs will be trapped to the steepest point along the phase profile, but when f > f max the effective temporal drift arising from the frequency mismatch is too large to be overcome by the phase-modulation attraction. SUPPLEMENTARY NOTE 3: ASYNCHRONOUS PHASE-MODULATION IN THE LLE MODEL Asynchronous phase-modulation can also be analysed in the mean-field LLE by considering a driving field modulated with a travelling phase profile. Transferring the intracavity field into a reference frame where the phase pattern is stationary, ξ = τ + ( τ/t R )t, and injecting E in = F in exp[iφ(ξ)] and E = F exp[iφ(ξ)] into Supplementary Eq. () we obtain: t R F(t,τ) t ( τ + β Lφ ) F [ ξ = α F iδ FF i L β ] + iγ L F F + θ F ξ in, (8) where δ FF = δ β L(φ ) / + τφ and φ = dφ/dξ. It can again be seen that the CS solutions will be stationary in the reference-frame moving with the intracavity phase profile, provided that the second term on the left-hand side vanishes. This occurs when the solitons are located at a temporal coordinate ξ that solves the equation: φ (ξ) = τ β L = f β L FSR f PM. (9) This result coincides with that in Supplementary Eq. (6), derived using simple physical arguments. SUPPLEMENTARY NOTE : TEMPORAL TWEEZING IS DISTORTIONLESS As mentioned above, phase modulation of the holding beam modifies the effective cavity loss [α F = α β Lφ /, Supplementary Eq. (3)] and detuning [δ F = δ β L(φ ) /, Supplementary Eq. ()], which could be expected to 6
distort the temporal CS characteristics. However, for typical parameters, the corrections to α and δ are negligible, so that both trapping and temporal tweezing are distortionless in practice. As an example, for the parameters listed in the caption of Supplementary Fig., we find that max{ β Lφ / }.5 α =.6, and max{ β L(φ ) / }. δ =.. We have tested this proposition further, experimentally and numerically. Specifically, we have measured the characteristics of the temporal CSs supported by our cavity both with and without phase modulation. Results are summarized in Supplementary Fig. 3. Here we compare their spectrograms [Supplementary Figs 3a,b] measured using a second-harmonic generation (SHG) frequency-resolved optical gating (FROG) apparatus, average spectra [Supplementary Fig. 3c] measured with an optical spectrum analyser, and finally normalized intensity autocorrelation functions [Supplementary Fig. 3d] extracted from the FROG traces, as indicated. The phase modulation, when applied, consists of a GHz sine wave of.5 rad amplitude similar to what we have used in our other experiments. In each case, the measurements obtained with and without phase modulation match completely, indicating that phase modulation does not meaningfully influence the temporal CS temporal intensity profile or peak power. Similar conclusions can be drawn from numerical simulations, and in Supplementary Fig. 3e we show simulated temporal intensity profiles of CSs when (i) the soliton is trapped to a phase peak (black line); (ii) no phase modulation is applied (red line) and (iii) when the soliton is drifting towards the peak at the point of maximum phase gradient. The curves are largely undistinguishable, confirming the distortionless nature of our scheme. REFERENCES. Lugiato, L. A. & Lefever, R. Spatial dissipative structures in passive optical systems. Phys. Rev. Lett. 58, 9 (987).. Haelterman, M., Trillo, S. & Wabnitz, S. Dissipative modulation instability in a nonlinear dispersive ring cavity. Opt. Commun. 9, 7 (99). 3. Firth, W. J. & Scroggie, A. J. Optical bullet holes: Robust controllable localized states of a nonlinear cavity. Phys. Rev. Lett. 76, 63 66 (996).. Agrawal, G. P. Nonlinear Fiber Optics, th ed. (Academic Press, 6). 7