The interaction of Kerr nonlinearity with even-orders of dispersion: an infinite hierarchy of solitons
Antoine F. J. Runge, Y. Long Qiang, Tristram J. Alexander, Darren D. Hudson, Andrea Blanco-Redondo, C. Martijn de Sterke
aa r X i v : . [ phy s i c s . op ti c s ] S e p The interaction of Kerr nonlinearity with even-orders of dispersion: an infinitehierarchy of solitons
Antoine F. J. Runge , ∗ , Y. Long Qiang , Tristram J. Alexander , DarrenD. Hudson , Andrea Blanco-Redondo , and C. Martijn de Sterke , Institute of Photonics and Optical Science (IPOS),School of Physics, The University of Sydney, NSW 2006, Australia CACI-Photonics Solutions, 15 Vreeland Road, Florham Park, NJ 07932, USA Nokia Bell Labs, 791 Holmdel Road, Holmdel, NJ 07733, USA The University of Sydney Nano Institute (Sydney Nano),The University of Sydney, NSW 2006, Australia ∗ Corresponding author: [email protected]
Temporal solitons are optical pulses that arise from the balance of negative group-velocity disper-sion and self-phase modulation. For decades only quadratic dispersion was considered, with higherorder dispersion thought of as a nuisance. Following the recent reporting of pure-quartic solitons, wehere provide experimental and numerical evidence for an infinite hierarchy of solitons that balanceself-phase modulation and arbitrary negative pure, even-order dispersion. Specifically, we experi-mentally demonstrate the existence of solitons with pure-sextic ( β ), -octic ( β ) and -decic ( β )dispersion, limited only by the performance of our components, and show numerical evidence forthe existence of solitons involving pure 16 th order dispersion. Phase-resolved temporal and spectralcharacterization reveals that these pulses, exhibit increasing spectral flatness with dispersion order.The measured energy-width scaling laws suggest dramatic advantages for ultrashort pulses. Theseresults broaden the fundamental understanding of solitons and present new avenues to engineerultrafast pulses in nonlinear optics and its applications. INTRODUCTION
Solitons are among the most striking phenomena innonlinear physics and have been observed in a wide rangeof systems [1, 2]. In optics, these transform-limited,shape-maintaining pulses have been crucial in the devel-opment of numerous applications ranging from telecom-munications [3, 4], to frequency comb generation [5, 6]and mode-locked lasers [7, 8]. Traditionally, the forma-tion of these wavepackets relies on the balance betweenself-phase modulation (SPM) and negative quadratic dis-persion ( β < β k of order k atfrequency ω , the inverse group velocity v g for frequenciesclose to ω can be written as1 v g = 1 v g − k − | β k | ( ω − ω ) k − . (1)Here v g is the group velocity at ω , and k = 2 forquadratic dispersion, etc. For negative quadratic and infact for all high, even-order types of dispersion, (1) showsthat the group velocity monotonically increases with fre-quency. Consequently, both the SPM-generated low fre-quencies on the leading edge and the SPM-generated high frequencies on the trailing edge move towards the pulsecenter, leading to the formation of a soliton. This ar-gument suggests that temporal solitons should exist inthe presence of any negative even-order dispersion. In-deed, recent studies showed that optical solitons couldarise rom the balance between SPM and negative quar-tic dispersion ( β <
0) [17–19]. However, the existenceof solitons for higher, even dispersion order ( k >
4) andtheir properties are yet to be reported.Here we report experimental evidence for an infinite hi-erarchy of solitons, to which we refer as pure high, even-order dispersion (PHEOD) solitons, arising from the bal-ance between SPM and any negative even-order of dis-persion, of which conventional solitons and pure-quarticsolitons are its two lowest-order members. We experi-mentally demonstrate three new members of this hierar-chy, namely pure-sextic, -octic and -decic solitons (aris-ing, respectively, from the balance of SPM and negativedipsersion of order k = 6, 8, and 10). The experimentalresults agree well with numerical results found by solvingthe nonlinear Schr¨odinger equation (NLSE), modified tohigher orders of dispersion. In fact, in this way we pro-vide evidence PHEOD solitons of order 16.While previous theoretical works have studied the com-bined effects of high-order dispersion of order up to k = 8and nonlinear effects (see, e.g., [20–23]), they all considerequations with a large number of terms, many of whichrepresenting combined nonlinear and dispersive effects.Our aim differs in that we consider the Kerr nonlineareffect and pure, even, high-order dispersion (see (2) be-low).The PHEOD solitons are generated in a passivelymode-locked laser with tunable net-cavity dispersion [19].The solitons have spectral sidebands, associated with res-onant dispersive waves typical of fiber lasers, the spacingof which is directly, and quantitatively, related to thedispersion. By measuring the PHEOD solitons’ energywe find that they are related to the pulse duration, τ ,as E ∝ τ − ( k − . The strongly increasing pulse energywith decreasing pulse length demonstrates the potentialof high-order dispersion for unlocking innovations in non-linear optics. Our results establish a new degree of free-dom for the generation and study of ultrashort opticalpulses with potential applications in lasers [19] and fre-quency comb generation [24, 25]. NUMERICAL RESULTS
We consider the propagation of optical pulses in amedium with Kerr nonlinearity and k th order dispersion,where k is an even integer. This evolution can be de-scribed by the modified NLSE i ∂ψ∂z = − ( i ) k | β k | k ! ∂ k ψ∂T k − γ | ψ | ψ, (2)where ψ ( z, T ) is the pulse envelope, z is the propaga-tion coordinate, T is the local time, β k is the dispersioncoefficient, which is taken to be negative, and γ is thenonlinear parameter. For k >
2, (2) is non-integrableand has no known analytic pulse-like solutions. How-ever, we can look for stationary solutions of (2) of theform ψ ( z, T ) = A ( T ; µ ) e iµz , which satisfy µA − ( i ) k | β k | k ! ∂ k A∂T k − γA = 0 , (3)so the shape is preserved during propagation, and A canbe taken to be real. We solve (3) using the Newton-conjugate-gradient method [18, 26].Solving (3) provides a single solution for each disper-sion order, but using a scaling argument we can obtainan entire family of solutions. Since (2) is invariant underthe transformation A → αA τ → α − /k τ µ → α µ, (4)there exists a continuous family of PHEOD solitons foreach dispersion order k , each with the same pulse shape,but with different amplitudes and widths, that can beparameterized by their values of µ [18]. From (4) it can beshown straightforwardly that the energy E k of a solitonwith pure even dispersion order k scales as E k ∝ τ − ( k − .The numerically calculated temporal and spectral in-tensity profiles of the resulting stationary solutions for k = 6 , ,
10 and 16, with the same temporal pulse dura-tion at full width half maximum (FWHM) τ = 1 ps, are -2 -1 0 1 2Time (ps) N o r m a li z ed po w e r ( a . u ) (a) -4 -2 0 2 4Time (ps) P o w e r ( d B / d i v ) (b)-1 -0.5 0 0.5 1Frequency (THz) S pe c t r a l i n t en s i t y ( a . u ) (c) -1 -0.5 0 0.5 1Frequency (THz) S pe c t r a l i n t en s i t y ( d B / d i v ) (d) FIG. 1. Numerically calculated temporal stationary solutionsfor k = 6 (blue) , ,
10 (green) and 16 (black) with asame pulse width (FWHM) of τ = 1 ps. Temporal profilesin linear (a) and logarithmic (b) scales. Corresponding linear(c) and logarithmic (d) spectra. The different solutions havebeen shifted vertically for clarity. shown in Fig. 1. We note that similar to PQSs, the pulsetemporal shapes of the PHEOD solitons exhibit oscilla-tions in the tails [18, 27], which become more prominentwith increasing dispersion order, as seen in Fig. 1(b). Si-multaneously the central part of the associated spectrumbecomes increasingly flat (see Fig. 1(c) and particularlyin Fig. 1(d)). This increased spectral flatness can qual-itatively be understood from (1) for the group velocitynear frequency ω . It shows that for high dispersion or-ders k the group velocity remains approximately constantaround ω before changing rapidly. There is thus a fre-quency interval around ω for which the dispersion isessentially irrelevant, and in this frequency interval thespectral intensity does not need to vary significantly tobalance the dispersion. To understand the effect of thespectral flatness in the time domain, recall that the sec-ond derivative of a function corresponds to the secondmoment of its Fourier transform. Thus the Fourier trans-form of a flat function, i.e., a function with a small secondderivative, must have a small second moment, which canonly be achieved by sign changes. For the pulses we areconsidering this corresponds to oscillations in time [18]. -30-20-100 S pe c t r u m ( d B ) (a)-30-20-100 S pe c t r u m ( d B ) (b)1552 1556 1560 1564 1568Wavelength (nm)-30-20-100 S pe c t r u m ( d B ) (c) (d)-10010 D e l a y ( p s ) (e)-10010 D e l a y ( p s ) (f)1556 1558 1560 1562 1564Wavelength (nm)-10010 D e l a y ( p s ) I n t en s i t y ( a . u ) (g) -2-1012 P ha s e () I n t en s i t y ( a . u ) (h) -2-1012 P ha s e () -4 -2 0 2 4 Time (ps) I n t en s i t y ( a . u ) (i) -2-1012 P ha s e () FIG. 2. Spectral and temporal measurements of pure high-order dispersion solitons sextic (top row), octic (middle row) and decic(bottom row) dispersion. The applied dispersion is β = −
500 ps / km, β = − × ps / km and β = − × ps / km,respectively. (a)-(c) Measured (blue), and calculated (red-dashed) spectra. (d)-(f) Measured spectrograms. (g)-(i) Retrievedtemporal intensity (blue), phase (orange) and corresponding calculated temporal shapes (red-dashed). EXPERIMENTAL SETUP AND RESULTS
The intrinsic dispersion of conventional optical waveg-uides is dominated by quadratic contribution ( β ) whilethe effects of higher-order dispersion are usually weak.In fact a complex structure was required just to achievedominant negative quartic dispersion [17, 28]. Thisstrongly limits the possibility of observing higher-orderdispersion soliton propagation in waveguides. To over-come this limitation, and achieve the dominant negativehigh-order dispersion required for the generation of thesenovel solitons, we used a passively mode-locked fiber lasersimilar to the one reported by Runge et al. [19]. Thelaser incorporates an intracavity programmable spectral-shaper, which is used to adjust the net-cavity dispersion[19, 29, 30]. The applied phase mask compensates forthe intrinsic second, third and fourth order dispersion ofthe fiber components, and applies a large negative, higheven-order dispersion. The applied phase profile can bewritten as φ ( ω ) = L X n =2 β n ( ω − ω ) n n ! + β k ( ω − ω ) k k ! ! , (5)where L = 18 .
17 m is the cavity length, β n is the n th dispersion order for n = 2 , β = +21 . / km, β = − .
12 ps / km, and β = +2 . × − ps / km. These values are chosen tocompensate for the dispersion of the SMF used in oursetup and are based on values reported in [31, 32]. Thesecond term on the right-hand side of (5) correspondsto the negative high, even-order dispersion required forthe generation of sextic ( k = 6), octic ( k = 8) or de-cic ( k = 10) PHEOD solitons. To obtain the completespectral and temporal characterization of the pulses, weused a frequency resolved electrical gating (FREG) setupwhich allows for the measurement of the pulse spectro-gram [17, 33]. The temporal intensity and phase of thepulses are then retrieved using a conventional blind de-convolution numerical algorithm [34]. Spectral and temporal characterization
The results of the spectral, temporal and phase-resolved measurements of the output pulses for the laseroperating with pure-sextic, octic and decic dispersion areshown in the first, second and third rows of Fig. 2, respec-tively. Figure 2(a)-(c) (left column) show the measuredoutput spectra (blue curves) and the corresponding nu-merically calculated pulse shapes (red-dashed curves), forthe three different dispersion orders. The measured spec-tral − λ = 3 . , . . ω . Since the spectral fluctuationsthus appear far from the central frequency and at least10 dB below the peak (see Fig. 2(c)), we are confidentthat they do not affect the pulse dynamics significantly.This assertion is confirmed by the corresponding mea-sured spectrograms in Fig. 2(d)-(f), which show clearunchirped pulses for all three cases. The vertical streaksat short and long wavelengths correspond to the first side-bands [35]. Finally, the temporal intensity and phaseprofiles of the sextic, octic and decic PHEOD solitonsare shown in Fig. 2(g)-(i), respectively. The retrievedFWHM pulse durations of the pure-sextic, -octic and -decic solitons are τ = 1 . , .
69 and 1 .
77 ps, respec-tively. For all cases, the measured temporal intensities(blue curves) are in good agreement with the correspond-ing numerical solutions (red-dashed curves) for similarFWHM. The retrieved temporal phase (orange curve) in-dicates that the emitted pulses are slightly chirped. Thisis because our experimental setup is a lumped systemin which the required dispersion is applied at a singlepoint in the cavity, just before the output coupler [19].Note that the numerically predicted oscillations in thetails of the temporal profiles (see Fig. 1(b)) are not ob-served since these are expected to appear approximately20 dB below the pulse’s maximum which is below thebackground in our experiments.As discussed in Section 3, the central part of thePHEOD soliton spectrum becomes increasingly flat withincreasing order of dispersion. The flatness or peakednessof a function is often expressed in terms of the kurtosis,but this measure has recently been discredited [36]. In-stead we introduce the flatness F , which we define tobe the fraction of the pulse energy that is within itsspectral FWHM. Since it is a fraction, F is intrinsicallynormalized: it has maximum F = 1 for a rectangularfunction and 0 < F < F has the additional advantage that it is an in-trinsic property of a function and does not depend onits parameters. For example, for all Gaussian functions F = erf( √ ln 2) ≈ . x ) is the error func-tion, and F = 1 / √ ≈ .
707 for all squared hyperbolicsecants. The circles in Fig. 3 give the flatness F of the nu-merically calculated spectra from Fig. 1 for different evendispersion orders k . The measured values of F (blue dia-monds) from the spectra from Fig. 2 and Ref. [19], agreeto the numerically calculated within 2%. This confirmsthat the flatness of the spectrum increases monotonicallywith k . It has been shown that flatter spectra can lead toenhanced pump-comb conversion and smaller line-to-linepower variations in frequency combs [25]. F pa r a m e t e r Num.Exp.
FIG. 3. Numerical (red circles) and experimental (blue dia-monds) values of the flatness F of PHEOD soliton spectrumversus dispersion order. S pe c t r u m ( d B / d i v ) (a) 0510152025 ( p s - ) (d) S pe c t r u m ( d B / d i v ) (b) 0246 ( p s - ) (e)-5 0 5 ( 10 s -1 ) S pe c t r u m ( d B / d i v ) (c) 1 2 3 4 5Sideband order0246810 ( p s - ) (f) FIG. 4. Measured PHEOD soliton output spectra for differentdispersion orders. (a) Sextic PHEOD soliton spectra for β = −
50 (yellow), β = −
100 (orange), and β = −
500 ps / km(blue). (b) Octic PHEOD soliton spectra for β = − β = −
200 (green) and β = − / km (purple).(c) Decic PHEOD soliton spectra for β = − β = − × (cyan), and β = − × ps / km (red).Coloured circles show the k th power of the measured side-bands positions versus sideband order for the (d) sextic; (e)octic; and (f) decic PHEOD soliton spectra. The solid linescorrespond to linear fits. Sideband analysis
To confirm the nature of the cavity’s linear dispersion,we analyzed the position of the spectral sidebands of theemitted pulses. These dispersive waves arise from theconstructive interference between the solitons and thelinear waves emitted by the soliton while it propagatesinside the cavity [19, 35]. Constructive interference oc-curs when β sol − β lin = 2 mπ/L where m is a positiveinteger. For k th order dispersion the linear waves sat-isfy β lin = −| β k | ( ω − ω ) k /k !, while the PHEOD solitonshave a constant dispersion across its entire bandwidth of β sol = C k | β k | /τ k [18, 37], where C k are constants of orderunity that depend on the dispersion order. Thus, we findthat the spectral position of the m th spectral sideband ω m is given by ω m = ± τ (cid:20) k ! (cid:18) m π τ k | β k | L − C k (cid:19)(cid:21) /k . (6)Following this argument, it is straightforward to showthat for a pure k th order dispersion soliton, the k th powerof two consecutive sidebands is constant and given by2 πk ! / ( | β k | L ), irrespective of the value of C k . To checkthis prediction, for each dispersion order we measured theoutput spectrum for three different values of the disper-sion coefficient β k and we measured the spectral positionsof the low frequency sidebands.The results of these measurements are shown in Fig. 4.In Fig. 4(a)-(c), we show three measured spectra for eachdispersion order k . The sextic PHEOD soliton spectra forthree different values of β are shown in Fig. 4(a). Cor-responding results for octic and decic PHEOD solitonsare shown in Fig. 4(b) and (c), respectively. The circlesmark the spectral positions of the low frequency side-bands. The k th power of these measured positions as afunction of the sideband order for the nine PHEOD soli-ton spectra are shown in Fig. 4(d)-(f). In all cases thespacings follow a linear relationship as expected. Thepredicted and measured spectral spacing for all the spec-tra shown in Fig. 4(a)-(c) are summarized in Table I.Note that the experimental values agree within 4% to thecorresponding expected values calculated from (6) andbased on the net-cavity dispersion that was applied bythe pulse-shaper. Since taking a high power of a datasetamplifies the noise, the agreement between the measuredand expected results is remarkable, confirming the typeand magnitude of the cavity dispersion. Energy-width scaling
Finally, we study the energy-width scaling relationshipof the PHEOD solitons. Following the scaling argumentof (4) and by dimensional analysis, we find the energy-width scaling relation of pure high-order dispersion Kerrsolitons for k th order of dispersion E k = M k | β k | γτ k − , (7)where M k is a constant found numerically. For thesextic, octic and decic PHEOD solitons, we found that Dispersion Applied β k Predicted Measuredorder k (ps k / km) spacing (ps − k ) spacing (ps − k )-50 4.98 × × × × -500 4.98 × × -100 1.39 × × × × -1000 1.39 × × -5000 2.51 × ×
10 -10 × × × -50 × × × TABLE I. Predicted (from (6)) and measured sideband spac-ing values for different values of applied dispersion k . M = 0 . M = 0 .
16 and M = 0 . β k , for each order of dispersion con-sidered. Concretely, we adjusted the pump power in thelaser cavity and measured the output pulse energy afterdeducting the portion of energy in the spectral sidebandsby integrating the optical spectrum. The results of thesemeasurements for k = 6, 8 and 10 are shown in Fig. 5.The circles show the measured pulse energies versus thepulse duration ( τ ) for three different values of dispersion β k for each dispersion order k . All results are in goodagreement with (7) once we account for the output cou-pling and the insertion loss of the pulse-shaper. Thisshows that the pulse energy E ∝ τ − ( k − , consistentwith (7) and confirm that these pure high-order disper-sion solitons follow a different energy-width scaling rela-tion that could be used for the generation of ultrashortoptical pulses with high energy. CONCLUSION AND DISCUSSION
We report the experimental discovery of an entire fam-ily of optical solitons arising from the balance betweenSPM and higher-order dispersion. One can consider con-ventional optical solitons to be the lowest-order memberof this family of PHEOD solitons. All these pulses fun-damentally arise from similar physical effects: the non-linearity generates frequencies on the pulses’ leading andtrailing edges, and these shift towards the pulse centerunder the effect of dispersion. Our investigation com-bines numerical results following from solving (3), andexperimental results obtained using a fiber laser. Thefiber laser incorporates a spectral pulse-shaper which isused to apply a large negative pure high, even-order dis-persion [19]. We find that the numerical and experimen-tal results are in very good agreement.As the order of dispersion increases, the soliton spec- E ( p J ) (a)204060 E ( p J ) (b)1.4 1.5 1.6 1.7 1.8 1.9 2 2.1Pulse duration (ps)204060 E ( p J ) (c) FIG. 5. Measurement energy-width scaling properties of thepure high-order dispersion Kerr solitons. The circles markthe measured pulse energy E versus pulse duration. (a) Pure-sextic soliton energy for β = −
500 (red), β = − β = − / km (green). (b) Pure-octic soliton en-ergy for β = − × (red), β = − × (blue) and β = − × ps / km (green). (c) Pure-decic soliton en-ergy for β = − × (red), β = − × (blue) and β = − × ps / km (green). The solid curves are fits for(a) pure-sextic; (b) -octic; and (c) -decic solitons from (7). tra become increasingly flat. In addition to this, thepeak power of the pulses increases as τ − k , where k isthe dispersion order, so that the energy increases as τ − ( k − . These features could find applications in high-energy laser systems. Alternatively, the spectral flatnessof PHEOD solitons could be used to generate frequencycombs with small tooth power variations [25, 38].Apart from possible laser applications, we have shownthat our setup provides a powerful tool to open up newroutes for the generation and study of a wide range ofnovel optical pulses [39]. This approach allows for thesimple yet precise tailoring of the net-cavity dispersionand the enhancement of any high-order dispersion ef-fects, which is currently impossible through conventionalwaveguide dispersion engineering [17]. While we onlydemonstrate PHEOD solitons up to the 10 th order of dis-persion, we emphasize that our approach is only limitedby the specifications of the pulse-shaper. This limitationcould be overcome by using a device with higher spectralresolution and bandwidth so enabling the generation ofPHEOD solitons of order higher than 10 th .In addition to direct quantitative insights on novel soli-ton pulses provided by our experiments, the approach it-self is expected to become an established tool for the gen-eration and study of ultrafast pulses [19, 39]. We expectour results to stimulate future investigations and discov-eries in other areas of physics, engineering and applied mathematics. FUNDING INFORMATION
Australian Research Project (ARC) Discovery Project(DP180102234); Asian Office of Aerospace R&D(AOARD) grant (FA2386-19-1-4067).
DISCLOSURES
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