4616:) provides a spatio-temporal picture of the involved phenomena, but from the spectral viewpoint its solutions correspond to KFC. The link between the topic of optical KFC and the LLE was theoretically developed in. These authors showed that the LLE (or generalizations including higher order dispersion terms) is the model which describes the generation of KFC and is capable of predicting their properties when the system parameters are varied. The spontaneous formation of spatial patterns and solitons travelling along the cavity described by the LLE is the spatiotemporal equivalent of the frequency combs and governs their features. The rather idealized conditions assumed in the formulation of the LLE, especially the high-Q condition, have been perfectly materialized by the spectacular technological progress that has occurred in the meantime in the field of photonics and has led, in particular, to the discovery of KFC.
4603:, has led to myriad applications. The work demonstrated the realization of broadband optical frequency combs exploiting the whispering gallery modes activated by a CW laser field injected into a high-Q microresonator filled with a Kerr medium, that gives rise to FWM. Since that time Kerr frequency combs (KFC), whose bandwidth can exceed an octave with repetition rates in the microwave to THz frequencies, have been generated in a wide variety of microresonators; for reviews on this subject see e.g. They offer substantial potential for miniaturization and chip-scale photonic integration, as well as for power reduction. Today KFC generation is a mature field, and this technology has been applied to several areas, including coherent telecommunications, spectroscopy, atomic clocks as well as laser ranging and astrophysical spectrometer calibration.
4902:: they are precisely correlated, for example in energy and momentum. This fact is fundamental for the quantum aspects of optical patterns. For instance, the difference between the intensities of the two symmetrical beams is squeezed, i.e. exhibits fluctuations below the shot noise level; the longitudinal analogue of this phenomenon has been observed experimentally in KFC. In turn, such quantum aspects are basic for the field of
4504:
1204:
731:
4557:
1503:
3057:
22:
4485:
4170:
4572:
4867:
338:
494:
4653:
122:
4567:
Figure 7 shows an example of the patterns that are generated, and travel along the cavity and out of the cavity. Like in the transverse case, also in the longitudinal configuration single or multiple Kerr cavity solitons can be generated; Figure 8 illustrates the case of a single cavity soliton that
945:
2601:
1876:
83:
for electromagnetic field and semiclassical equations of the two-level (or multilevel) atoms. For this reason the simplified theoretical approaches were developed for numerical simulation of laser beams formation and their propagation since the early years of laser era. The
726:{\displaystyle {\frac {\partial E}{\partial z}}+{\frac {\;\omega _{0}\ }{k_{0}c^{2}}}{\frac {\partial E}{\partial t}}-{\tfrac {1}{2k_{0}}}\ i\ \nabla _{\perp }^{2}E={\frac {1}{\varepsilon _{0}k_{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}}~}
2300:
is the normalized amplitude of the input field that is injected into the cavity, the second is the decay term, the third is the detuning term, the fourth is the cubic nonlinear term that takes into account the Kerr medium, the last term with the transverse
Laplacian
4582:
It is important to note that the instability which originates longitudinal patterns and cavity solitons in the LLE is a special case of the multimode instability of optical bistability, predicted by
Bonifacio and Lugiato in and first observed experimentally in.
1733:
1434:
4552:
are absorbed and, simultaneously, the system emits photon pairs corresponding to cavity modes symmetrically adjacent to the quasi-resonant mode, in such a way that the total photon energy, as well as the total longitudinal photon momentum, are conserved.
484:
4862:{\displaystyle {\frac {\;\omega _{0}\ }{k_{0}c^{2}}}{\frac {\partial E}{\partial t}}-{\tfrac {1}{2k_{0}}}\ i\ \nabla _{\perp }^{2}E={\frac {1}{\varepsilon _{0}k_{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}}~}
817:
333:{\displaystyle \nabla ^{2}{\cal {E({\vec {r}},t)}}-{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\cal {E({\vec {r}},t)}}={\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}},}
4499:
s (see Fig. 6). Since cavity solitons can be “written”and “erased” one by one in the transverse plane like in a blackboard, they are of great interest for applications to optical information processing and telecommunications.
828:
3379:
3575:
where one has three stationary states. However, the states that lie in the segment with negative slope are unstable, so that in the interval there are two coexisting stable stationary states: this phenomenon is called
2451:
2988:
1754:
4606:
A key impetus to these developments has been the realization of Kerr cavity solitons in microresonators, opening the possibility of utilizing Kerr cavity solitons in photonic integrated microresonators.
2211:
2342:) as the transverse LLE. Some years later than, there was the formulation of the longitudinal LLE, in which diffraction is replaced by dispersion. In this case one assumes that the envelope
4525:) the patterns arise from the interplay between FWM and dispersion. The FWM can give rise, for example, to processes in which pairs of photons of the longitudinal mode quasi-resonant with
4359:
4288:
1608:
1277:
2733:
115:
1014:” in microresonators, discovered in 2007 by Tobias Kippenberg and collaborators, that has raised a very lively interest, especially because of the applicative avenue it has opened.
3506:
2104:
2054:
4131:
4046:
3756:
2331:
3622:
3573:
3476:
3212:
3142:
2004:
1131:
343:
2823:
2660:
5474:
Leo, F.; Coen, S.; Kockaert, P.; Gorza, S.P.; Emplit, P.; Haelterman, M. (2010). "Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer".
4220:
3997:
3883:
3802:
2298:
2872:
1493:
4550:
3685:
3658:
2265:
2238:
1464:
3032:
1966:
1937:
1908:
1598:
1198:
3168:
2151:
2441:
4164:
4079:
3432:
3098:
2124:
4563:
Example of longitudinal pattern that travels along the cavity with the velocity of light in the medium and gives rise to a periodic sequence of pulses in the output.
4479:
4440:
4413:
4386:
3856:
3829:
739:
2765:
4642:
3922:
3529:
3236:
2928:
2908:
2843:
2785:
2680:
2400:
2380:
2360:
1551:
1531:
1268:
1248:
1228:
1151:
1080:
1060:
1040:
996:
The same equation governs two types of patterns: stationary patterns that arise in the planes orthogonal with respect to the direction of propagation of light (
968:
4591:
Optical frequency combs constitute an equidistant set of laser frequencies that can be employed to count the cycles of light. This technique, introduced by
5785:
Herr, T.; Brasch, V.; Jost, J.D.; Wang, C.Y.; Kondratiev, N.M.; Gorodetsky, M.L.; Kippenberg, T.J. (2014). "Temporal solitons in optical microresonators".
1153:
denotes time, is linearly polarized and therefore can be treated as a scalar, we can express it in terms of the slowly varying normalized complex envelope
940:{\displaystyle \displaystyle \left|\ {\frac {\partial ^{2}E}{\partial t^{2}}}\ \right|\ll \left|\ \omega _{0}\,{\frac {\partial E}{\partial t}}\ \right|\ }
5213:; Schliesser, A.; Arcizet, O.; Wilken, T.; Holzwarth, R.; Kippenberg, T.J. (2007). "Optical frequency comb generation from a monolithic microresonator".
4568:
circulates in the cavity and produces a sequence of narrow pulses in the output. Such solitons have been observed for the first time in a fiber cavity.
3262:
989:
in nonlinear optical systems. The patterns originate from the interaction of a coherent field, that is injected into a resonant optical cavity, with a
5695:
Herr, T.; Gorodetsky, M.L.; Kippenberg, T.J. (2015). "Chap. 6 : Dissipative Kerr
Solitons in Optical Microresonators". In Grelu, Philippe (ed.).
2682:
depends, in particular on the dispersion parameter at second order. Conditions of anomalous dispersion are assumed. An important point is that, once
40:
5309:
Brambilla, M.; Castelli, F.; Gatti, A.; Lugiato, L.A.; Prati, F. (1993). "Instabilities and quantum-noise reduction in nonlinear-optical mixing".
3953:) the pattern arises from the interplay of FWM and diffraction. The FWM can give rise, for example, to processes in which pairs of photons with
2596:{\displaystyle {\frac {\partial E}{\partial {\bar {t}}}}=E_{\text{in}}-E-i\theta E+i|E|^{2}E+i{\frac {\partial ^{2}E}{\partial {\bar {z}}^{2}}}}
5903:
Chembo, Y.K.; Menyuk, C.R. (2013). "Spatiotemporal
Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators".
5429:
Scroggie, A.J.; Firth, W.J.; McDonald s, G.S.; Tlidi, M.; Lugiato, L.A.; Lefever, R. (1994). "Pattern formation in a passive Kerr cavity".
1871:{\displaystyle \nabla _{\perp }^{2}E={\frac {\partial ^{2}E}{\partial {\bar {x}}^{2}}}+{\frac {\partial ^{2}E}{\partial {\bar {y}}^{2}}}}
2933:
1007:), travel along the cavity with the velocity of light in the medium and give rise to a sequence of pulses in the output of the cavity.
3170:. The stationary states in the segment with negative slope are unstable. The arrows show the hysteresis cycle which is covered when
6317:
Lugiato, L.A.; Prati, F.; Gorodetsky, M.L.; Kippenberg, T.J. "From the LLE to microresonator based soliton Kerr frequency combs".
4495:
A pattern displays an ordered array of intensity peaks. It is possible to generate also isolated intensity peaks, that are called
3238:
is constant, i.e. on the stationary solutions that are independent of all spatial variables. By dropping all derivatives in Eqs.(
2156:
5838:
Matsko, A.B.; Savchenkov, A.A.; Liang, W.; Ilchenko, V.S.; Seidel, D.; Maleki, L. (2011). "Mode-locked Kerr frequency combs".
4896:
The two photons that, as shown in Fig.4, are emitted in symmetrically tilted directions in the FWM process, are in a state of
5191:
4990:
3037:
85:
5274:
Haelterman, M.; Trillo, S.; Wabnitz, S. (1992). "Dissipative modulation instability in a nonlinear dispersive ring cavity".
5649:
4510:
A typical Kerr cavity soliton in the transverse plane showing a bright peak in the dark background with diffraction rings.
5337:
5166:
6241:
6173:
5722:
58:
5881:
Herr, T.; Brasch, V.; Gorodetsky, M.L.; Kippenberg, T.J. (2012). "Soliton mode-locking in optical microresonators".
1728:{\displaystyle {\frac {\partial E}{\partial {\bar {t}}}}=E_{\text{in}}-E-i\theta E+i|E|^{2}E+i\nabla _{\perp }^{2}E}
4293:
1429:{\displaystyle {\cal {E}}(x,y,z,t)\propto E(x,y,z,t)e^{i(\omega _{0}/{\tilde {c}})(z-{\tilde {c}}t)}+{\text{c.c.}}}
4225:
5960:(2013). "Modeling of octave spanning Kerr frequency combs using a generalized mean-field Lugiato Lefever model".
4938:
2878:
2999:
In some papers dealing with the longitudinal case one considers dispersion beyond the second order, so that Eq.(
3627:
If we refer to the modes of the empty cavity, in the case of the uniform stationary solutions described by Eq.(
1499:
that fills the cavity. For definiteness, consider a ring cavity (Fig. 2) of very high quality (High-Q cavity).
2685:
91:
76:
3481:
2059:
2009:
4084:
479:{\displaystyle {\cal {E}}({\vec {r}},t)\propto E({\vec {r}}_{\perp },z,t)e^{i(nk_{0})(z-ct)}+{\text{c.c.}}}
4491:
A typical pattern configuration that arises in the transverse planes in the output is a hexagonal pattern.
4002:
3699:
2304:
36:
5650:"Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis"
3583:
3534:
3437:
3173:
3103:
1971:
1085:
72:
5544:
Segard, B.; Macke, B. (1988). "Self-pulsing in intrinsic optical bistability with two-level molecules".
5509:
Bonifacio, R.; Lugiato, L.A. (1978). "Instabilities for a coherently driven absorber in a ring cavity".
5354:
5648:
Jones, D.J.; Diddams, S.A.; Ranka, J.K.; Stentz, A.; Windeler, R.S.; Hall, J.L.; Curdiff, S.T. (2000).
5184:
Self-organization in nonequilibrium systems. From dissipative structures to order through fluctuations
5159:
Self-organization in nonequilibrium systems. From dissipative structures to order through fluctuations
6343:
2790:
5451:
2622:
4179:
3956:
3861:
3761:
2276:
6052:
Lugiato, L.A.; Castelli, F. (1992). "Quantum noise-reduction in a spatial dissipative structure".
4166:
in such a way that the total energy of photons, and their total momentum, are conserved (Fig.4).
2848:
1469:
4528:
3663:
3636:
2243:
2216:
1442:
6258:
5073:
Lugiato, L.A.; Lefever, R. (1987). "Spatial
Dissipative Structures in Passive Optical Systems".
3008:
2845:-dependent stationary solution (stationary pattern) becomes a travelling pattern (with velocity
1942:
1913:
1884:
1556:
1156:
5446:
4876:
is applicable to modeling of the nonstationary evolution of the transverse mode pattern in the
3147:
2129:
2405:
2333:
describes diffraction in the paraxial approximation. Conditions of self-focusing are assumed.
4136:
4051:
3400:
3066:
2109:
812:{\displaystyle \displaystyle \left|\ \nabla ^{2}E\ \right|\ll \left|\ k_{0}\nabla E\ \right|}
4449:
6273:
6198:
6122:
6061:
6034:
5979:
5922:
5847:
5804:
5754:
5661:
5596:
5553:
5483:
5438:
5379:
5283:
5232:
5129:
5082:
5050:
4898:
4418:
4391:
4364:
3834:
3807:
5617:
8:
6224:
Lugiato, L.A.; Prati, F.; Brambilla, M. (2015). "Chapter 28: The
Lugiato Lefever Model".
5038:
4943:
4933:
3040:
2744:
1011:
6277:
6202:
6126:
6065:
6038:
6022:
5983:
5926:
5851:
5808:
5758:
5665:
5600:
5557:
5487:
5442:
5383:
5287:
5236:
5133:
5086:
5054:
6348:
6299:
6138:
6112:
6003:
5969:
5938:
5912:
5882:
5820:
5794:
5700:
5630:
5526:
5411:
5369:
5256:
5222:
5210:
4627:
3907:
3514:
3221:
2913:
2893:
2828:
2770:
2665:
2385:
2365:
2345:
1536:
1516:
1253:
1233:
1213:
1136:
1065:
1045:
1025:
953:
6165:
3696:), in the case of these stationary solutions E corresponds to a singlemode plane wave
6338:
6303:
6237:
6169:
6077:
5995:
5863:
5824:
5718:
5677:
5622:
5569:
5530:
5460:
5403:
5333:
5295:
5248:
5187:
5162:
5098:
4986:
4948:
3049:
986:
88:
of MBE follows from the standard nonlinear wave equation with nonlinear polarization
80:
6142:
6134:
5942:
5415:
5023:
4982:
6289:
6281:
6229:
6206:
6161:
6130:
6069:
6007:
5987:
5930:
5855:
5812:
5762:
5710:
5669:
5634:
5612:
5604:
5561:
5518:
5491:
5456:
5395:
5387:
5291:
5260:
5240:
5137:
5090:
5019:
4923:
3901:
3045:
3005:) includes also terms with derivatives of order higher than second with respect to
71:
The numerical models of lasers and the most of nonlinear optical systems stem from
6285:
6233:
6096:
5673:
4903:
4592:
3374:{\displaystyle |E_{\text{in}}|^{2}=|E|^{2}\left\{1+(\theta -|E|^{2})^{2}\right\}}
3251:
6073:
5122:
Philosophical
Transactions of the Royal Society of London B: Biological Sciences
5094:
1010:
The case of longitudinal patterns is intrinsically linked to the phenomenon of “
6210:
5957:
5934:
5587:
Udem, T.; Holzwarth, R.; Hänsch t, T.W. (2002). "Optical frequency metrology".
5565:
5391:
4928:
249:
237:
228:
162:
150:
141:
5714:
6332:
6100:
5816:
5767:
5742:
5495:
4975:
3858:
are the transverse components of the wave vector, exactly as the input field
982:
5008:"Propagation of a Light pulse in a Resonantly amplifying (absorbing) medium"
2877:
From a mathematical viewpoint, the LLE amounts to a driven, damped, detuned
6081:
5999:
5867:
5738:
5681:
5626:
5573:
5407:
5252:
5142:
5117:
5102:
4600:
4596:
3633:) the electric field is singlemode, corresponding to the mode of frequency
4586:
6257:
Castelli, F.; Brambilla, M.; Gatti, M.; Prati, F.; Lugiato, L.A. (2017).
5991:
5859:
5355:"Phase-space structure of two-dimensional excitable localized structures"
5039:"Effect of inhomogeneities on the operation regime of solid-state lasers"
3999:
are absorbed and, simultaneously, the system emits pairs of photons with
1496:
990:
5244:
1466:
is the frequency of the light beam that is injected into the cavity and
6294:
5697:
Nonlinear
Optical Cavity Dynamics: From Microresonators to Fiber Lasers
5522:
5007:
4953:
4877:
3924:
displays a spatial pattern: in the transverse plane in the case of Eq.(
1082:
are the transverse directions. If we assume that the electric field as
974:
describing the nonstationary evolution of the transverse mode pattern.
5399:
5743:"Kerr optical frequency combs: theory, applications and perspectives"
5374:
6189:
Kolobov, M.I. (1999). "The spatial behavior of nonclassical light".
5887:
5608:
2890:) is in 2D from the spatial viewpoint. In a waveguide configuration
5705:
1513:
In the original LLE, one assumes conditions such that the envelope
6117:
5974:
5917:
5799:
5227:
3056:
4484:
4169:
2983:{\displaystyle {\frac {\partial ^{2}E}{\partial {\bar {x}}^{2}}}}
6156:
Gatti, A.; Brambilla, E.; Lugiato, L.A. (2008). Wolf, E. (ed.).
4503:
3624:
is increased and then decreased, one covers a hysteresis cycle.
1203:
5209:
4481:
assumes the configuration of a hexagonal pattern (see Fig.5).
5880:
4556:
1502:
246:
243:
159:
156:
5837:
5308:
6316:
5955:
5428:
4514:
3904:(FWM), which can generate other modes, so that the envelope
2990:
and one has the transverse LLE in 1D. The longitudinal LLE (
6319:
6256:
4571:
3939:
4880:(1966) . Under condition of stationary Kerr nonlinearity
2206:{\displaystyle \theta =(\omega _{c}-\omega _{0})/\kappa }
6259:"The LLE, pattern formation and a novel coherent source"
6094:
5694:
5353:
Gomila, D.; Jacobo, A.; Matias, M.A.; Colet, P. (2007).
4872:
Rigorous procedure demonstrated that this precursor of
4446:
Actually further FWM processes enter into play, so that
1000:) and patterns that form in the longitudinal direction (
5647:
5352:
5273:
4587:
Microresonator Kerr frequency combs and cavity solitons
6160:. Progress in Optics. Vol. LI. pp. 251–348.
4723:
587:
488:
resulting in the standard "parabolic" wave equation:
6223:
6155:
6023:"Generation Dynamics of a Giant Coherent Light Pulse"
5586:
4656:
4630:
4531:
4452:
4421:
4394:
4367:
4296:
4228:
4182:
4176:
A four-wave mixing process in which two photons with
4139:
4087:
4054:
4005:
3959:
3910:
3864:
3837:
3810:
3764:
3702:
3666:
3639:
3586:
3537:
3517:
3484:
3440:
3403:
3265:
3224:
3176:
3150:
3106:
3069:
3011:
2936:
2916:
2896:
2851:
2831:
2793:
2773:
2747:
2688:
2668:
2625:
2454:
2408:
2388:
2368:
2348:
2307:
2279:
2246:
2219:
2159:
2132:
2112:
2062:
2012:
1974:
1945:
1916:
1887:
1757:
1611:
1559:
1539:
1519:
1472:
1445:
1280:
1256:
1236:
1216:
1159:
1139:
1088:
1068:
1048:
1028:
956:
832:
831:
743:
742:
497:
346:
125:
94:
5473:
3063:
Stationary curve of the normalized output intensity
1968:
are normalized temporal and spatial variables, i.e.
6014:
1022:Figure 1 shows a light beam that propagates in the
31:
may be too technical for most readers to understand
5784:
4974:
4861:
4648:equation where longitudinal derivative is absent:
4636:
4544:
4473:
4434:
4407:
4380:
4353:
4282:
4214:
4158:
4125:
4073:
4040:
3991:
3916:
3877:
3850:
3823:
3796:
3750:
3679:
3652:
3616:
3567:
3523:
3500:
3470:
3426:
3373:
3230:
3206:
3162:
3136:
3092:
3026:
2982:
2922:
2902:
2866:
2837:
2817:
2779:
2759:
2727:
2674:
2654:
2595:
2435:
2394:
2374:
2354:
2325:
2292:
2259:
2232:
2205:
2145:
2118:
2098:
2048:
1998:
1960:
1931:
1902:
1870:
1727:
1592:
1545:
1525:
1487:
1458:
1428:
1262:
1242:
1222:
1192:
1145:
1125:
1074:
1054:
1034:
962:
939:
811:
725:
478:
332:
109:
4914:For reviews on the subject of the LLE, see also.
2126:being the cavity decay rate or cavity linewidth,
985:and René Lefever as a paradigm for spontaneous
6330:
5508:
5156:
4619:
3100:as a function of the normalized input intensity
2741:), one must come back to the original variables
6051:
6020:
5330:Optical bistability: controlling light by light
5005:
3531:-shaped and there is an interval of values of
3218:Let us focus on the case in which the envelope
5072:
3038:Uniform stationary solutions. Connection with
2930:, and the transverse Laplacian is replaced by
4354:{\displaystyle k_{x}=-{\bar {k}}_{x},k_{y}=0}
2996:) is equivalent to the transverse LLE in 1D.
6250:
5902:
4283:{\displaystyle k_{x}={\bar {k}}_{x},k_{y}=0}
1533:is independent of the longitudinal variable
6310:
6217:
5780:
5778:
5699:. Wiley-VCH Verlag GmbH. pp. 129–162.
5543:
5502:
5302:
5267:
5068:
5066:
5064:
4999:
4624:The averaging over longitudinal coordinate
2362:is independent of the transverse variables
950:The averaging over longitudinal coordinate
5896:
5831:
5731:
5422:
5346:
5332:. Academic Press, Inc., Orlando, Florida.
5205:
5203:
5150:
5030:
4660:
3508:we obtain a curve as that shown in Fig.3.
2910:depends only on one spatial variable, say
524:
6293:
6182:
6149:
6116:
6045:
5973:
5916:
5886:
5798:
5766:
5704:
5641:
5616:
5450:
5373:
5226:
5141:
4515:Longitudinal patterns and cavity solitons
1553:(i.e. uniform along the cavity), so that
904:
59:Learn how and when to remove this message
43:, without removing the technical details.
6088:
5949:
5874:
5775:
5688:
5580:
5537:
5061:
4570:
4555:
4502:
4483:
4168:
3660:quasi-resonant with the input frequency
3055:
2728:{\displaystyle E({\bar {z}},{\bar {t}})}
1501:
1202:
110:{\displaystyle \mathbf {P} ^{\text{NL}}}
6188:
5956:Coen, S.; Randle, H.G.; Sylvestre, T.;
5467:
5321:
5200:
5109:
5036:
4972:
4442:are the components of the wave-vectors.
3930:), along the cavity in the case of Eq.(
3690:In the transverse configuration of Eq.(
2213:is the cavity detuning parameter, with
6331:
5737:
5175:
5115:
3501:{\displaystyle \theta >{\sqrt {3}}}
3254:, one obtains the stationary equation
2240:being the cavity frequency nearest to
2153:the diffraction length in the cavity.
2099:{\displaystyle {\bar {y}}=y/\ell _{d}}
2049:{\displaystyle {\bar {x}}=x/\ell _{d}}
6103:(2015). "On-chip optical squeezing".
6095:Dutt, A.; Luke, K.; Manipatruni, S.;
5327:
5181:
5118:"The chemical basis of morphogenesis"
4126:{\displaystyle k_{x}=-{\bar {k}}_{x}}
86:Slowly varying envelope approximation
41:make it understandable to non-experts
6021:Letokhov, V S; Suchkov, A F (1966).
5006:Kryukov, P G; Letokhov, V S (1970).
4981:. University Science Books. p.
4041:{\displaystyle k_{x}={\bar {k}}_{x}}
3888:The cubic Kerr nonlinearity of Eqs.(
3751:{\displaystyle e^{i(k_{x}x+k_{y}y)}}
3256:
2445:
2326:{\displaystyle \nabla _{\perp }^{2}}
1602:
15:
5157:Nicolis, G.; Prigogine, I. (1977).
3617:{\displaystyle |E_{\text{in}}|^{2}}
3568:{\displaystyle |E_{\text{in}}|^{2}}
3471:{\displaystyle |E_{\text{in}}|^{2}}
3397:If we plot the stationary curve of
3207:{\displaystyle |E_{\text{in}}|^{2}}
3137:{\displaystyle |E_{\text{in}}|^{2}}
1999:{\displaystyle {\bar {t}}=\kappa t}
1126:{\displaystyle {\cal {E}}(x,y,z,t)}
1017:
13:
4909:
4891:
4828:
4818:
4756:
4710:
4702:
4578:Longitudinal Kerr cavity solitons.
4222:are absorbed and two photons with
2955:
2941:
2568:
2554:
2466:
2458:
2309:
1843:
1829:
1797:
1783:
1759:
1708:
1623:
1615:
1283:
1210:A light beam propagates along the
1091:
916:
908:
859:
845:
794:
753:
692:
682:
620:
574:
566:
509:
501:
349:
299:
289:
234:
225:
207:
197:
147:
138:
127:
14:
6360:
4846:
4519:In the longitudinal case of Eq.(
3214:is increased and then decreased.
977:The model usually designated as
710:
317:
97:
20:
6266:The European Physical Journal D
6135:10.1103/PhysRevApplied.3.044005
5024:10.1070/PU1970v012n05ABEH003957
4939:Nonlinear Schroedinger equation
2879:nonlinear Schroedinger equation
2818:{\displaystyle z-{\tilde {c}}t}
2267:. In the righthand side of Eq.(
972:Suchkov-Letokhov equation (SLE)
6228:. Cambridge University Press.
5618:11858/00-001M-0000-000F-C239-D
4966:
4468:
4456:
4320:
4249:
4111:
4026:
3947:In the transverse case of Eq.(
3743:
3711:
3604:
3588:
3555:
3539:
3458:
3442:
3414:
3405:
3357:
3346:
3337:
3327:
3306:
3297:
3283:
3267:
3194:
3178:
3124:
3108:
3080:
3071:
3018:
2965:
2858:
2806:
2722:
2716:
2701:
2692:
2655:{\displaystyle {\bar {z}}=z/a}
2632:
2578:
2531:
2522:
2475:
2430:
2418:
2192:
2166:
2069:
2019:
1981:
1952:
1923:
1894:
1853:
1807:
1688:
1679:
1632:
1587:
1569:
1479:
1413:
1404:
1389:
1386:
1380:
1356:
1345:
1321:
1312:
1288:
1187:
1163:
1120:
1096:
993:medium that fills the cavity.
979:Lugiato–Lefever equation (LLE)
463:
448:
445:
429:
418:
394:
384:
375:
363:
354:
77:Partial Differential Equations
1:
6166:10.1016/S0079-6638(07)51005-X
4959:
4644:results in the "mean-field"
4620:The Suchkov-Letokhov equation
4215:{\displaystyle k_{x}=k_{y}=0}
3992:{\displaystyle k_{x}=k_{y}=0}
3878:{\displaystyle E_{\text{in}}}
3797:{\displaystyle k_{x}=k_{y}=0}
2443:. The longitudinal LLE reads
2293:{\displaystyle E_{\text{in}}}
1495:of the light velocity in the
1270:are the transverse directions
6234:10.1017/CBO9781107477254.032
5674:10.1126/science.288.5466.635
5461:10.1016/0960-0779(94)90084-1
5431:Chaos, Solitons and Fractals
5296:10.1016/0030-4018(92)90367-Z
4973:Siegman, Anthony E. (1986).
2867:{\displaystyle {\tilde {c}}}
1488:{\displaystyle {\tilde {c}}}
7:
6074:10.1103/PhysRevLett.68.3284
5186:. Berlin: Springer Verlag.
5095:10.1103/PhysRevLett.58.2209
4917:
4612:
4545:{\displaystyle \omega _{0}}
4521:
3949:
3932:
3926:
3896:
3890:
3692:
3680:{\displaystyle \omega _{0}}
3653:{\displaystyle \omega _{c}}
3629:
3387:
3246:
3240:
3001:
2992:
2886:
2737:
2735:is obtained by solving Eq.(
2609:
2338:
2269:
2260:{\displaystyle \omega _{0}}
2233:{\displaystyle \omega _{c}}
1741:
1459:{\displaystyle \omega _{0}}
10:
6365:
6286:10.1140/epjd/e2017-70754-1
6211:10.1103/RevModPhys.71.1539
5935:10.1103/PhysRevA.87.053852
5566:10.1103/PhysRevLett.60.412
5392:10.1103/PhysRevE.75.026217
3027:{\displaystyle {\bar {z}}}
1961:{\displaystyle {\bar {y}}}
1932:{\displaystyle {\bar {x}}}
1903:{\displaystyle {\bar {t}}}
1593:{\displaystyle E=E(x,y,t)}
1193:{\displaystyle E(x,y,z,t)}
981:was formulated in 1987 by
6226:Nonlinear Optical Systems
6191:Reviews of Modern Physics
5893:(arXiv version of Ref.18)
5715:10.1002/9783527686476.ch6
3580:. If the input intensity
3163:{\displaystyle \theta =4}
2146:{\displaystyle \ell _{d}}
970:results in "mean-field"
733:, under conditions :
5817:10.1038/nphoton.2013.343
5768:10.1515/nanoph-2016-0013
5511:Lettere al Nuovo Cimento
5496:10.1038/nphoton.2010.120
3940:Transverse patterns and
2436:{\displaystyle E=E(z,t)}
75:(MBE). This full set of
6105:Physical Review Applied
6054:Physical Review Letters
5546:Physical Review Letters
5075:Physical Review Letters
4159:{\displaystyle k_{y}=0}
4074:{\displaystyle k_{y}=0}
3427:{\displaystyle |E|^{2}}
3093:{\displaystyle |E|^{2}}
2119:{\displaystyle \kappa }
1509:Top view of ring cavity
73:Maxwell–Bloch equations
5143:10.1098/rstb.1952.0012
5116:Turing, A.M.. (1952).
4863:
4638:
4610:The longitudinal LLE (
4579:
4564:
4546:
4511:
4492:
4475:
4474:{\displaystyle E(x,y)}
4443:
4436:
4409:
4382:
4355:
4284:
4216:
4160:
4127:
4075:
4042:
3993:
3918:
3879:
3852:
3825:
3798:
3752:
3681:
3654:
3618:
3569:
3525:
3502:
3472:
3428:
3375:
3232:
3215:
3208:
3164:
3138:
3094:
3028:
2984:
2924:
2904:
2868:
2839:
2819:
2781:
2761:
2729:
2676:
2656:
2597:
2437:
2396:
2376:
2356:
2327:
2294:
2261:
2234:
2207:
2147:
2120:
2100:
2050:
2000:
1962:
1933:
1904:
1872:
1729:
1594:
1547:
1527:
1510:
1489:
1460:
1430:
1271:
1264:
1244:
1224:
1194:
1147:
1127:
1076:
1056:
1036:
964:
941:
813:
727:
480:
334:
111:
5276:Optics Communications
5037:Suchkov, A F (1966).
4864:
4639:
4574:
4559:
4547:
4506:
4487:
4476:
4437:
4435:{\displaystyle k_{z}}
4410:
4408:{\displaystyle k_{y}}
4383:
4381:{\displaystyle k_{x}}
4356:
4285:
4217:
4172:
4161:
4128:
4076:
4043:
3994:
3919:
3880:
3853:
3851:{\displaystyle k_{y}}
3826:
3824:{\displaystyle k_{x}}
3799:
3753:
3682:
3655:
3619:
3570:
3526:
3503:
3473:
3429:
3376:
3233:
3209:
3165:
3139:
3095:
3059:
3029:
2985:
2925:
2905:
2869:
2840:
2820:
2782:
2762:
2730:
2677:
2657:
2598:
2438:
2397:
2377:
2357:
2328:
2295:
2262:
2235:
2208:
2148:
2121:
2101:
2051:
2001:
1963:
1934:
1905:
1873:
1730:
1600:. The equation reads
1595:
1548:
1528:
1505:
1490:
1461:
1431:
1265:
1245:
1225:
1206:
1195:
1148:
1128:
1077:
1057:
1037:
965:
942:
814:
728:
481:
335:
112:
5992:10.1364/OL.38.000037
5860:10.1364/OL.36.002845
5328:Gibbs, H.M. (1985).
4899:quantum entanglement
4654:
4628:
4529:
4450:
4419:
4392:
4365:
4294:
4226:
4180:
4137:
4085:
4052:
4003:
3957:
3908:
3862:
3835:
3808:
3762:
3700:
3664:
3637:
3584:
3535:
3515:
3482:
3438:
3401:
3263:
3222:
3174:
3148:
3104:
3067:
3009:
2934:
2914:
2894:
2884:The transverse LLE (
2849:
2829:
2791:
2771:
2745:
2686:
2666:
2623:
2452:
2406:
2386:
2366:
2346:
2305:
2277:
2244:
2217:
2157:
2130:
2110:
2060:
2010:
1972:
1943:
1914:
1885:
1755:
1609:
1557:
1537:
1517:
1470:
1443:
1278:
1254:
1234:
1214:
1157:
1137:
1086:
1066:
1046:
1026:
1012:Kerr frequency combs
954:
829:
740:
495:
344:
123:
92:
6278:2017EPJD...71...84C
6203:1999RvMP...71.1539K
6127:2015PhRvP...3d4005D
6099:; Nussenzveig, P.;
6066:1992PhRvL..68.3284L
6039:1966JETP...23..764L
5984:2013OptL...38...37C
5927:2013PhRvA..87e3852C
5852:2011OptL...36.2845M
5809:2014NaPho...8..145H
5759:2016Nanop...5...13C
5666:2000Sci...288..635J
5601:2002Natur.416..233U
5558:1988PhRvL..60..412S
5488:2010NaPho...4..471L
5443:1994CSF.....4.1323S
5384:2007PhRvE..75b6217G
5288:1992OptCo..91..401H
5245:10.1038/nature06401
5237:2007Natur.450.1214D
5221:(7173): 1214–1217.
5161:. Wiley, New York.
5134:1952RSPTB.237...37T
5087:1987PhRvL..58.2209L
5055:1966JETP...22.1026S
4944:Optical bistability
4934:Kerr frequency comb
4769:
3578:optical bistability
3041:optical bistability
2760:{\displaystyle z,t}
2322:
1772:
1721:
998:transverse patterns
633:
5523:10.1007/bf02763162
5437:(8–9): 1323–1354.
5182:Haken, H. (1983).
4859:
4755:
4744:
4634:
4601:mode-locked lasers
4580:
4565:
4542:
4512:
4493:
4471:
4444:
4432:
4405:
4378:
4351:
4280:
4212:
4156:
4123:
4071:
4038:
3989:
3914:
3875:
3848:
3821:
3794:
3748:
3677:
3650:
3614:
3565:
3521:
3498:
3468:
3424:
3371:
3250:), and taking the
3228:
3216:
3204:
3160:
3134:
3090:
3024:
2980:
2920:
2900:
2864:
2835:
2815:
2777:
2757:
2725:
2672:
2652:
2593:
2433:
2392:
2372:
2352:
2323:
2308:
2290:
2257:
2230:
2203:
2143:
2116:
2096:
2046:
1996:
1958:
1929:
1900:
1868:
1758:
1725:
1707:
1590:
1543:
1523:
1511:
1485:
1456:
1426:
1272:
1260:
1240:
1220:
1190:
1143:
1123:
1072:
1052:
1032:
960:
937:
936:
809:
808:
723:
619:
608:
476:
330:
107:
6060:(22): 3284–3286.
5905:Physical Review A
5660:(5466): 635–639.
5595:(6877): 233–237.
5362:Physical Review E
5311:SUSSP Proceedings
5193:978-3-642-88338-5
5081:(21): 2209–2211.
4992:978-0-935702-11-8
4949:Pattern formation
4858:
4853:
4842:
4813:
4754:
4748:
4743:
4717:
4697:
4673:
4637:{\displaystyle z}
4323:
4252:
4114:
4029:
3917:{\displaystyle E}
3872:
3599:
3550:
3524:{\displaystyle S}
3496:
3453:
3434:as a function of
3395:
3394:
3278:
3231:{\displaystyle E}
3189:
3119:
3050:pattern formation
3021:
2978:
2968:
2923:{\displaystyle x}
2903:{\displaystyle E}
2861:
2838:{\displaystyle z}
2809:
2780:{\displaystyle z}
2719:
2704:
2675:{\displaystyle a}
2635:
2617:
2616:
2591:
2581:
2494:
2482:
2478:
2395:{\displaystyle y}
2375:{\displaystyle x}
2355:{\displaystyle E}
2287:
2072:
2022:
1984:
1955:
1926:
1897:
1866:
1856:
1820:
1810:
1749:
1748:
1651:
1639:
1635:
1546:{\displaystyle z}
1526:{\displaystyle E}
1482:
1424:
1407:
1383:
1263:{\displaystyle y}
1243:{\displaystyle x}
1223:{\displaystyle z}
1146:{\displaystyle t}
1075:{\displaystyle y}
1055:{\displaystyle x}
1042:direction, while
1035:{\displaystyle z}
987:pattern formation
963:{\displaystyle z}
935:
927:
923:
893:
877:
873:
840:
802:
783:
767:
751:
722:
717:
706:
677:
618:
612:
607:
581:
561:
537:
516:
474:
397:
366:
324:
313:
284:
240:
221:
192:
153:
104:
81:Maxwell equations
69:
68:
61:
6356:
6344:Nonlinear optics
6323:
6322:
6314:
6308:
6307:
6297:
6263:
6254:
6248:
6247:
6221:
6215:
6214:
6197:(5): 1539–1589.
6186:
6180:
6179:
6153:
6147:
6146:
6120:
6092:
6086:
6085:
6049:
6043:
6042:
6018:
6012:
6011:
5977:
5953:
5947:
5946:
5920:
5900:
5894:
5892:
5890:
5878:
5872:
5871:
5835:
5829:
5828:
5802:
5787:Nature Photonics
5782:
5773:
5772:
5770:
5735:
5729:
5728:
5708:
5692:
5686:
5685:
5645:
5639:
5638:
5620:
5584:
5578:
5577:
5541:
5535:
5534:
5506:
5500:
5499:
5476:Nature Photonics
5471:
5465:
5464:
5454:
5426:
5420:
5419:
5377:
5359:
5350:
5344:
5343:
5325:
5319:
5318:
5306:
5300:
5299:
5282:(5–6): 401–407.
5271:
5265:
5264:
5230:
5207:
5198:
5197:
5179:
5173:
5172:
5154:
5148:
5147:
5145:
5113:
5107:
5106:
5070:
5059:
5058:
5034:
5028:
5027:
5003:
4997:
4996:
4980:
4970:
4924:Four-wave mixing
4868:
4866:
4865:
4860:
4856:
4855:
4854:
4851:
4849:
4843:
4841:
4840:
4839:
4826:
4825:
4816:
4814:
4812:
4811:
4810:
4801:
4800:
4791:
4790:
4777:
4768:
4763:
4752:
4746:
4745:
4742:
4741:
4740:
4724:
4718:
4716:
4708:
4700:
4698:
4696:
4695:
4694:
4685:
4684:
4674:
4671:
4670:
4669:
4658:
4643:
4641:
4640:
4635:
4551:
4549:
4548:
4543:
4541:
4540:
4480:
4478:
4477:
4472:
4441:
4439:
4438:
4433:
4431:
4430:
4414:
4412:
4411:
4406:
4404:
4403:
4387:
4385:
4384:
4379:
4377:
4376:
4360:
4358:
4357:
4352:
4344:
4343:
4331:
4330:
4325:
4324:
4316:
4306:
4305:
4289:
4287:
4286:
4281:
4273:
4272:
4260:
4259:
4254:
4253:
4245:
4238:
4237:
4221:
4219:
4218:
4213:
4205:
4204:
4192:
4191:
4165:
4163:
4162:
4157:
4149:
4148:
4132:
4130:
4129:
4124:
4122:
4121:
4116:
4115:
4107:
4097:
4096:
4080:
4078:
4077:
4072:
4064:
4063:
4047:
4045:
4044:
4039:
4037:
4036:
4031:
4030:
4022:
4015:
4014:
3998:
3996:
3995:
3990:
3982:
3981:
3969:
3968:
3923:
3921:
3920:
3915:
3902:four-wave mixing
3900:) gives rise to
3884:
3882:
3881:
3876:
3874:
3873:
3870:
3857:
3855:
3854:
3849:
3847:
3846:
3830:
3828:
3827:
3822:
3820:
3819:
3803:
3801:
3800:
3795:
3787:
3786:
3774:
3773:
3757:
3755:
3754:
3749:
3747:
3746:
3739:
3738:
3723:
3722:
3686:
3684:
3683:
3678:
3676:
3675:
3659:
3657:
3656:
3651:
3649:
3648:
3623:
3621:
3620:
3615:
3613:
3612:
3607:
3601:
3600:
3597:
3591:
3574:
3572:
3571:
3566:
3564:
3563:
3558:
3552:
3551:
3548:
3542:
3530:
3528:
3527:
3522:
3507:
3505:
3504:
3499:
3497:
3492:
3477:
3475:
3474:
3469:
3467:
3466:
3461:
3455:
3454:
3451:
3445:
3433:
3431:
3430:
3425:
3423:
3422:
3417:
3408:
3389:
3380:
3378:
3377:
3372:
3370:
3366:
3365:
3364:
3355:
3354:
3349:
3340:
3315:
3314:
3309:
3300:
3292:
3291:
3286:
3280:
3279:
3276:
3270:
3257:
3237:
3235:
3234:
3229:
3213:
3211:
3210:
3205:
3203:
3202:
3197:
3191:
3190:
3187:
3181:
3169:
3167:
3166:
3161:
3143:
3141:
3140:
3135:
3133:
3132:
3127:
3121:
3120:
3117:
3111:
3099:
3097:
3096:
3091:
3089:
3088:
3083:
3074:
3046:Four-wave mixing
3033:
3031:
3030:
3025:
3023:
3022:
3014:
2989:
2987:
2986:
2981:
2979:
2977:
2976:
2975:
2970:
2969:
2961:
2953:
2949:
2948:
2938:
2929:
2927:
2926:
2921:
2909:
2907:
2906:
2901:
2873:
2871:
2870:
2865:
2863:
2862:
2854:
2844:
2842:
2841:
2836:
2824:
2822:
2821:
2816:
2811:
2810:
2802:
2786:
2784:
2783:
2778:
2766:
2764:
2763:
2758:
2734:
2732:
2731:
2726:
2721:
2720:
2712:
2706:
2705:
2697:
2681:
2679:
2678:
2673:
2661:
2659:
2658:
2653:
2648:
2637:
2636:
2628:
2611:
2602:
2600:
2599:
2594:
2592:
2590:
2589:
2588:
2583:
2582:
2574:
2566:
2562:
2561:
2551:
2540:
2539:
2534:
2525:
2496:
2495:
2492:
2483:
2481:
2480:
2479:
2471:
2464:
2456:
2446:
2442:
2440:
2439:
2434:
2401:
2399:
2398:
2393:
2381:
2379:
2378:
2373:
2361:
2359:
2358:
2353:
2336:We refer to Eq.(
2332:
2330:
2329:
2324:
2321:
2316:
2299:
2297:
2296:
2291:
2289:
2288:
2285:
2266:
2264:
2263:
2258:
2256:
2255:
2239:
2237:
2236:
2231:
2229:
2228:
2212:
2210:
2209:
2204:
2199:
2191:
2190:
2178:
2177:
2152:
2150:
2149:
2144:
2142:
2141:
2125:
2123:
2122:
2117:
2105:
2103:
2102:
2097:
2095:
2094:
2085:
2074:
2073:
2065:
2055:
2053:
2052:
2047:
2045:
2044:
2035:
2024:
2023:
2015:
2005:
2003:
2002:
1997:
1986:
1985:
1977:
1967:
1965:
1964:
1959:
1957:
1956:
1948:
1938:
1936:
1935:
1930:
1928:
1927:
1919:
1909:
1907:
1906:
1901:
1899:
1898:
1890:
1877:
1875:
1874:
1869:
1867:
1865:
1864:
1863:
1858:
1857:
1849:
1841:
1837:
1836:
1826:
1821:
1819:
1818:
1817:
1812:
1811:
1803:
1795:
1791:
1790:
1780:
1771:
1766:
1743:
1734:
1732:
1731:
1726:
1720:
1715:
1697:
1696:
1691:
1682:
1653:
1652:
1649:
1640:
1638:
1637:
1636:
1628:
1621:
1613:
1603:
1599:
1597:
1596:
1591:
1552:
1550:
1549:
1544:
1532:
1530:
1529:
1524:
1494:
1492:
1491:
1486:
1484:
1483:
1475:
1465:
1463:
1462:
1457:
1455:
1454:
1435:
1433:
1432:
1427:
1425:
1422:
1417:
1416:
1409:
1408:
1400:
1385:
1384:
1376:
1373:
1368:
1367:
1287:
1286:
1269:
1267:
1266:
1261:
1249:
1247:
1246:
1241:
1229:
1227:
1226:
1221:
1199:
1197:
1196:
1191:
1152:
1150:
1149:
1144:
1132:
1130:
1129:
1124:
1095:
1094:
1081:
1079:
1078:
1073:
1061:
1059:
1058:
1053:
1041:
1039:
1038:
1033:
1018:The LLE equation
969:
967:
966:
961:
946:
944:
943:
938:
933:
932:
928:
925:
924:
922:
914:
906:
903:
902:
891:
882:
878:
875:
874:
872:
871:
870:
857:
853:
852:
842:
838:
825:
821:
818:
816:
815:
810:
807:
803:
800:
793:
792:
781:
772:
768:
765:
761:
760:
749:
732:
730:
729:
724:
720:
719:
718:
715:
713:
707:
705:
704:
703:
690:
689:
680:
678:
676:
675:
674:
665:
664:
655:
654:
641:
632:
627:
616:
610:
609:
606:
605:
604:
588:
582:
580:
572:
564:
562:
560:
559:
558:
549:
548:
538:
535:
534:
533:
522:
517:
515:
507:
499:
485:
483:
482:
477:
475:
472:
467:
466:
444:
443:
405:
404:
399:
398:
390:
368:
367:
359:
353:
352:
339:
337:
336:
331:
326:
325:
322:
320:
314:
312:
311:
310:
297:
296:
287:
285:
283:
282:
281:
272:
271:
258:
253:
252:
242:
241:
233:
222:
220:
219:
218:
205:
204:
195:
193:
191:
190:
181:
180:
171:
166:
165:
155:
154:
146:
135:
134:
116:
114:
113:
108:
106:
105:
102:
100:
64:
57:
53:
50:
44:
24:
23:
16:
6364:
6363:
6359:
6358:
6357:
6355:
6354:
6353:
6329:
6328:
6327:
6326:
6315:
6311:
6261:
6255:
6251:
6244:
6222:
6218:
6187:
6183:
6176:
6158:Quantum Imaging
6154:
6150:
6093:
6089:
6050:
6046:
6027:Sov. Phys. JETP
6019:
6015:
5954:
5950:
5901:
5897:
5879:
5875:
5836:
5832:
5783:
5776:
5736:
5732:
5725:
5693:
5689:
5646:
5642:
5609:10.1038/416233a
5585:
5581:
5542:
5538:
5517:(15): 510–516.
5507:
5503:
5472:
5468:
5452:10.1.1.594.1475
5427:
5423:
5357:
5351:
5347:
5340:
5326:
5322:
5307:
5303:
5272:
5268:
5208:
5201:
5194:
5180:
5176:
5169:
5155:
5151:
5114:
5110:
5071:
5062:
5043:Sov. Phys. JETP
5035:
5031:
5004:
5000:
4993:
4971:
4967:
4962:
4920:
4912:
4910:Review articles
4904:quantum imaging
4894:
4892:Quantum aspects
4850:
4845:
4844:
4835:
4831:
4827:
4821:
4817:
4815:
4806:
4802:
4796:
4792:
4786:
4782:
4781:
4776:
4764:
4759:
4736:
4732:
4728:
4722:
4709:
4701:
4699:
4690:
4686:
4680:
4676:
4675:
4665:
4661:
4659:
4657:
4655:
4652:
4651:
4629:
4626:
4625:
4622:
4593:Theodor Haensch
4589:
4536:
4532:
4530:
4527:
4526:
4517:
4451:
4448:
4447:
4426:
4422:
4420:
4417:
4416:
4399:
4395:
4393:
4390:
4389:
4372:
4368:
4366:
4363:
4362:
4339:
4335:
4326:
4315:
4314:
4313:
4301:
4297:
4295:
4292:
4291:
4268:
4264:
4255:
4244:
4243:
4242:
4233:
4229:
4227:
4224:
4223:
4200:
4196:
4187:
4183:
4181:
4178:
4177:
4144:
4140:
4138:
4135:
4134:
4117:
4106:
4105:
4104:
4092:
4088:
4086:
4083:
4082:
4059:
4055:
4053:
4050:
4049:
4032:
4021:
4020:
4019:
4010:
4006:
4004:
4001:
4000:
3977:
3973:
3964:
3960:
3958:
3955:
3954:
3945:
3942:cavity solitons
3909:
3906:
3905:
3869:
3865:
3863:
3860:
3859:
3842:
3838:
3836:
3833:
3832:
3815:
3811:
3809:
3806:
3805:
3782:
3778:
3769:
3765:
3763:
3760:
3759:
3734:
3730:
3718:
3714:
3707:
3703:
3701:
3698:
3697:
3671:
3667:
3665:
3662:
3661:
3644:
3640:
3638:
3635:
3634:
3608:
3603:
3602:
3596:
3592:
3587:
3585:
3582:
3581:
3559:
3554:
3553:
3547:
3543:
3538:
3536:
3533:
3532:
3516:
3513:
3512:
3491:
3483:
3480:
3479:
3462:
3457:
3456:
3450:
3446:
3441:
3439:
3436:
3435:
3418:
3413:
3412:
3404:
3402:
3399:
3398:
3360:
3356:
3350:
3345:
3344:
3336:
3320:
3316:
3310:
3305:
3304:
3296:
3287:
3282:
3281:
3275:
3271:
3266:
3264:
3261:
3260:
3252:squared modulus
3223:
3220:
3219:
3198:
3193:
3192:
3186:
3182:
3177:
3175:
3172:
3171:
3149:
3146:
3145:
3128:
3123:
3122:
3116:
3112:
3107:
3105:
3102:
3101:
3084:
3079:
3078:
3070:
3068:
3065:
3064:
3054:
3013:
3012:
3010:
3007:
3006:
2971:
2960:
2959:
2958:
2954:
2944:
2940:
2939:
2937:
2935:
2932:
2931:
2915:
2912:
2911:
2895:
2892:
2891:
2853:
2852:
2850:
2847:
2846:
2830:
2827:
2826:
2801:
2800:
2792:
2789:
2788:
2772:
2769:
2768:
2746:
2743:
2742:
2711:
2710:
2696:
2695:
2687:
2684:
2683:
2667:
2664:
2663:
2644:
2627:
2626:
2624:
2621:
2620:
2584:
2573:
2572:
2571:
2567:
2557:
2553:
2552:
2550:
2535:
2530:
2529:
2521:
2491:
2487:
2470:
2469:
2465:
2457:
2455:
2453:
2450:
2449:
2407:
2404:
2403:
2387:
2384:
2383:
2367:
2364:
2363:
2347:
2344:
2343:
2317:
2312:
2306:
2303:
2302:
2284:
2280:
2278:
2275:
2274:
2251:
2247:
2245:
2242:
2241:
2224:
2220:
2218:
2215:
2214:
2195:
2186:
2182:
2173:
2169:
2158:
2155:
2154:
2137:
2133:
2131:
2128:
2127:
2111:
2108:
2107:
2090:
2086:
2081:
2064:
2063:
2061:
2058:
2057:
2040:
2036:
2031:
2014:
2013:
2011:
2008:
2007:
1976:
1975:
1973:
1970:
1969:
1947:
1946:
1944:
1941:
1940:
1918:
1917:
1915:
1912:
1911:
1889:
1888:
1886:
1883:
1882:
1859:
1848:
1847:
1846:
1842:
1832:
1828:
1827:
1825:
1813:
1802:
1801:
1800:
1796:
1786:
1782:
1781:
1779:
1767:
1762:
1756:
1753:
1752:
1716:
1711:
1692:
1687:
1686:
1678:
1648:
1644:
1627:
1626:
1622:
1614:
1612:
1610:
1607:
1606:
1558:
1555:
1554:
1538:
1535:
1534:
1518:
1515:
1514:
1474:
1473:
1471:
1468:
1467:
1450:
1446:
1444:
1441:
1440:
1421:
1399:
1398:
1375:
1374:
1369:
1363:
1359:
1352:
1348:
1282:
1281:
1279:
1276:
1275:
1255:
1252:
1251:
1235:
1232:
1231:
1215:
1212:
1211:
1158:
1155:
1154:
1138:
1135:
1134:
1090:
1089:
1087:
1084:
1083:
1067:
1064:
1063:
1047:
1044:
1043:
1027:
1024:
1023:
1020:
955:
952:
951:
915:
907:
905:
898:
894:
890:
886:
866:
862:
858:
848:
844:
843:
841:
837:
833:
830:
827:
826:
823:
819:
788:
784:
780:
776:
756:
752:
748:
744:
741:
738:
737:
714:
709:
708:
699:
695:
691:
685:
681:
679:
670:
666:
660:
656:
650:
646:
645:
640:
628:
623:
600:
596:
592:
586:
573:
565:
563:
554:
550:
544:
540:
539:
529:
525:
523:
521:
508:
500:
498:
496:
493:
492:
471:
439:
435:
425:
421:
400:
389:
388:
387:
358:
357:
348:
347:
345:
342:
341:
321:
316:
315:
306:
302:
298:
292:
288:
286:
277:
273:
267:
263:
262:
257:
232:
231:
224:
223:
214:
210:
206:
200:
196:
194:
186:
182:
176:
172:
170:
145:
144:
137:
136:
130:
126:
124:
121:
120:
101:
96:
95:
93:
90:
89:
65:
54:
48:
45:
37:help improve it
34:
25:
21:
12:
11:
5:
6362:
6352:
6351:
6346:
6341:
6325:
6324:
6309:
6249:
6242:
6216:
6181:
6174:
6148:
6087:
6044:
6013:
5962:Optics Letters
5948:
5895:
5873:
5846:(15): 2845–7.
5840:Optics Letters
5830:
5793:(2): 145–152.
5774:
5753:(2): 214–230.
5730:
5723:
5687:
5640:
5579:
5552:(5): 412–415.
5536:
5501:
5482:(7): 471–476.
5466:
5421:
5345:
5339:978-0122819407
5338:
5320:
5301:
5266:
5199:
5192:
5174:
5168:978-0471024019
5167:
5149:
5128:(641): 37–72.
5108:
5060:
5029:
5018:(5): 641–672.
5012:Sov. Phys. Usp
4998:
4991:
4964:
4963:
4961:
4958:
4957:
4956:
4951:
4946:
4941:
4936:
4931:
4929:Frequency comb
4926:
4919:
4916:
4911:
4908:
4893:
4890:
4848:
4838:
4834:
4830:
4824:
4820:
4809:
4805:
4799:
4795:
4789:
4785:
4780:
4775:
4772:
4767:
4762:
4758:
4751:
4739:
4735:
4731:
4727:
4721:
4715:
4712:
4707:
4704:
4693:
4689:
4683:
4679:
4668:
4664:
4633:
4621:
4618:
4588:
4585:
4539:
4535:
4516:
4513:
4497:cavity soliton
4470:
4467:
4464:
4461:
4458:
4455:
4429:
4425:
4402:
4398:
4375:
4371:
4350:
4347:
4342:
4338:
4334:
4329:
4322:
4319:
4312:
4309:
4304:
4300:
4279:
4276:
4271:
4267:
4263:
4258:
4251:
4248:
4241:
4236:
4232:
4211:
4208:
4203:
4199:
4195:
4190:
4186:
4155:
4152:
4147:
4143:
4120:
4113:
4110:
4103:
4100:
4095:
4091:
4070:
4067:
4062:
4058:
4035:
4028:
4025:
4018:
4013:
4009:
3988:
3985:
3980:
3976:
3972:
3967:
3963:
3944:
3938:
3913:
3868:
3845:
3841:
3818:
3814:
3793:
3790:
3785:
3781:
3777:
3772:
3768:
3745:
3742:
3737:
3733:
3729:
3726:
3721:
3717:
3713:
3710:
3706:
3674:
3670:
3647:
3643:
3611:
3606:
3595:
3590:
3562:
3557:
3546:
3541:
3520:
3495:
3490:
3487:
3465:
3460:
3449:
3444:
3421:
3416:
3411:
3407:
3393:
3392:
3383:
3381:
3369:
3363:
3359:
3353:
3348:
3343:
3339:
3335:
3332:
3329:
3326:
3323:
3319:
3313:
3308:
3303:
3299:
3295:
3290:
3285:
3274:
3269:
3227:
3201:
3196:
3185:
3180:
3159:
3156:
3153:
3131:
3126:
3115:
3110:
3087:
3082:
3077:
3073:
3053:
3036:
3020:
3017:
2974:
2967:
2964:
2957:
2952:
2947:
2943:
2919:
2899:
2860:
2857:
2834:
2814:
2808:
2805:
2799:
2796:
2776:
2756:
2753:
2750:
2724:
2718:
2715:
2709:
2703:
2700:
2694:
2691:
2671:
2651:
2647:
2643:
2640:
2634:
2631:
2615:
2614:
2605:
2603:
2587:
2580:
2577:
2570:
2565:
2560:
2556:
2549:
2546:
2543:
2538:
2533:
2528:
2524:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
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2254:
2250:
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2176:
2172:
2168:
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2140:
2136:
2115:
2093:
2089:
2084:
2080:
2077:
2071:
2068:
2043:
2039:
2034:
2030:
2027:
2021:
2018:
1995:
1992:
1989:
1983:
1980:
1954:
1951:
1925:
1922:
1896:
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1862:
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1665:
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1656:
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1625:
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1589:
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1580:
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1565:
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1481:
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1259:
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1183:
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1122:
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1107:
1104:
1101:
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1093:
1071:
1051:
1031:
1019:
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959:
931:
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913:
910:
901:
897:
889:
885:
881:
869:
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861:
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851:
847:
836:
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799:
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791:
787:
779:
775:
771:
764:
759:
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747:
735:
734:
712:
702:
698:
694:
688:
684:
673:
669:
663:
659:
653:
649:
644:
639:
636:
631:
626:
622:
615:
603:
599:
595:
591:
585:
579:
576:
571:
568:
557:
553:
547:
543:
532:
528:
520:
514:
511:
506:
503:
470:
465:
462:
459:
456:
453:
450:
447:
442:
438:
434:
431:
428:
424:
420:
417:
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411:
408:
403:
396:
393:
386:
383:
380:
377:
374:
371:
365:
362:
356:
351:
329:
319:
309:
305:
301:
295:
291:
280:
276:
270:
266:
261:
256:
251:
248:
245:
239:
236:
230:
227:
217:
213:
209:
203:
199:
189:
185:
179:
175:
169:
164:
161:
158:
152:
149:
143:
140:
133:
129:
99:
67:
66:
28:
26:
19:
9:
6:
4:
3:
2:
6361:
6350:
6347:
6345:
6342:
6340:
6337:
6336:
6334:
6320:
6313:
6305:
6301:
6296:
6291:
6287:
6283:
6279:
6275:
6271:
6267:
6260:
6253:
6245:
6243:9781107477254
6239:
6235:
6231:
6227:
6220:
6212:
6208:
6204:
6200:
6196:
6192:
6185:
6177:
6175:9780444532114
6171:
6167:
6163:
6159:
6152:
6144:
6140:
6136:
6132:
6128:
6124:
6119:
6114:
6111:(4): 044005.
6110:
6106:
6102:
6098:
6091:
6083:
6079:
6075:
6071:
6067:
6063:
6059:
6055:
6048:
6040:
6036:
6032:
6028:
6024:
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6009:
6005:
6001:
5997:
5993:
5989:
5985:
5981:
5976:
5971:
5967:
5963:
5959:
5958:Erkintalo, M.
5952:
5944:
5940:
5936:
5932:
5928:
5924:
5919:
5914:
5911:(5): 053852.
5910:
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5899:
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5869:
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5801:
5796:
5792:
5788:
5781:
5779:
5769:
5764:
5760:
5756:
5752:
5748:
5747:Nanophotonics
5744:
5740:
5734:
5726:
5724:9783527413324
5720:
5716:
5712:
5707:
5702:
5698:
5691:
5683:
5679:
5675:
5671:
5667:
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5659:
5655:
5651:
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5636:
5632:
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5401:
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5368:(2): 026217.
5367:
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4482:
4465:
4462:
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4453:
4427:
4423:
4400:
4396:
4373:
4369:
4361:are emitted.
4348:
4345:
4340:
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4332:
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4317:
4310:
4307:
4302:
4298:
4277:
4274:
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3645:
3641:
3632:
3631:
3625:
3609:
3593:
3579:
3560:
3544:
3518:
3511:The curve is
3509:
3493:
3488:
3485:
3463:
3447:
3419:
3409:
3391:
3384:
3382:
3367:
3361:
3351:
3341:
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3330:
3324:
3321:
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3293:
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3272:
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3258:
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2994:
2972:
2962:
2950:
2945:
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2889:
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2882:
2880:
2875:
2855:
2832:
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2803:
2797:
2794:
2774:
2754:
2751:
2748:
2740:
2739:
2713:
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2698:
2689:
2669:
2649:
2645:
2641:
2638:
2629:
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2606:
2604:
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2563:
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2536:
2526:
2518:
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2509:
2506:
2503:
2500:
2497:
2488:
2484:
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2447:
2444:
2427:
2424:
2421:
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2369:
2349:
2341:
2340:
2334:
2318:
2313:
2281:
2272:
2271:
2252:
2248:
2225:
2221:
2200:
2196:
2187:
2183:
2179:
2174:
2170:
2163:
2160:
2138:
2134:
2113:
2091:
2087:
2082:
2078:
2075:
2066:
2041:
2037:
2032:
2028:
2025:
2016:
1993:
1990:
1987:
1978:
1949:
1920:
1891:
1860:
1850:
1838:
1833:
1822:
1814:
1804:
1792:
1787:
1776:
1773:
1768:
1763:
1751:
1750:
1745:
1738:
1736:
1722:
1717:
1712:
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1693:
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1669:
1666:
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1660:
1657:
1654:
1645:
1641:
1629:
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1605:
1604:
1601:
1584:
1581:
1578:
1575:
1572:
1566:
1563:
1560:
1540:
1520:
1508:
1504:
1500:
1498:
1476:
1451:
1447:
1418:
1410:
1401:
1395:
1392:
1377:
1370:
1364:
1360:
1353:
1349:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1318:
1315:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1274:
1273:
1257:
1237:
1217:
1209:
1205:
1201:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1160:
1140:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1069:
1049:
1029:
1015:
1013:
1008:
1006:
1003:
999:
994:
992:
988:
984:
983:Luigi Lugiato
980:
975:
973:
957:
948:
929:
919:
911:
899:
895:
887:
883:
879:
867:
863:
854:
849:
834:
804:
797:
789:
785:
777:
773:
769:
762:
757:
745:
700:
696:
686:
671:
667:
661:
657:
651:
647:
642:
637:
634:
629:
624:
613:
601:
597:
593:
589:
583:
577:
569:
555:
551:
545:
541:
530:
526:
518:
512:
504:
491:
490:
489:
486:
468:
460:
457:
454:
451:
440:
436:
432:
426:
422:
415:
412:
409:
406:
401:
391:
381:
378:
372:
369:
360:
327:
307:
303:
293:
278:
274:
268:
264:
259:
254:
215:
211:
201:
187:
183:
177:
173:
167:
131:
118:
117:as a source:
87:
82:
78:
74:
63:
60:
52:
42:
38:
32:
29:This article
27:
18:
17:
6318:
6312:
6269:
6265:
6252:
6225:
6219:
6194:
6190:
6184:
6157:
6151:
6108:
6104:
6090:
6057:
6053:
6047:
6030:
6026:
6016:
5968:(1): 37–39.
5965:
5961:
5951:
5908:
5904:
5898:
5876:
5843:
5839:
5833:
5790:
5786:
5750:
5746:
5739:Chembo, Y.K.
5733:
5696:
5690:
5657:
5653:
5643:
5592:
5588:
5582:
5549:
5545:
5539:
5514:
5510:
5504:
5479:
5475:
5469:
5434:
5430:
5424:
5375:nlin/0703011
5365:
5361:
5348:
5329:
5323:
5314:
5310:
5304:
5279:
5275:
5269:
5218:
5214:
5211:Del’Haye, P.
5183:
5177:
5158:
5152:
5125:
5121:
5111:
5078:
5074:
5046:
5042:
5032:
5015:
5011:
5001:
4976:
4968:
4913:
4897:
4895:
4885:
4881:
4873:
4871:
4650:
4645:
4623:
4611:
4609:
4605:
4590:
4581:
4575:
4566:
4560:
4520:
4518:
4507:
4496:
4494:
4488:
4445:
4173:
3948:
3946:
3941:
3931:
3925:
3895:
3889:
3887:
3691:
3689:
3628:
3626:
3577:
3510:
3396:
3385:
3245:
3239:
3217:
3060:
3039:
3000:
2998:
2991:
2885:
2883:
2876:
2825:, so that a
2767:and replace
2736:
2618:
2607:
2337:
2335:
2268:
1880:
1739:
1512:
1506:
1438:
1207:
1200:in this way
1021:
1009:
1004:
1002:longitudinal
1001:
997:
995:
978:
976:
971:
949:
736:
487:
340:where :
119:
70:
55:
46:
30:
6295:2434/502714
6097:Gaeta, A.L.
5888:1211.0733v1
5049:(5): 1026.
4884:reduces to
1497:Kerr medium
1230:direction.
6333:Categories
6101:Lipson, M.
6033:(4): 764.
5706:1508.04989
5400:10261/6146
5317:: 115–136.
4960:References
4954:Disk laser
4878:Disk laser
2402:, so that
6349:Photonics
6304:126088543
6272:(4): 84.
6118:1309.6371
5975:1211.1697
5918:1210.8210
5825:118546909
5800:1211.0733
5531:120619908
5447:CiteSeerX
5228:0708.0611
4829:∂
4819:∂
4784:ε
4761:⊥
4757:∇
4720:−
4711:∂
4703:∂
4663:ω
4597:John Hall
4576:Figure 8.
4561:Figure 7.
4534:ω
4508:Figure 6.
4489:Figure 5.
4321:¯
4311:−
4250:¯
4174:Figure 4.
4112:¯
4102:−
4027:¯
3669:ω
3642:ω
3486:θ
3334:−
3331:θ
3152:θ
3061:Figure 3.
3019:¯
2966:¯
2956:∂
2942:∂
2859:~
2807:~
2798:−
2717:¯
2702:¯
2633:¯
2579:¯
2569:∂
2555:∂
2510:θ
2504:−
2498:−
2476:¯
2467:∂
2459:∂
2314:⊥
2310:∇
2249:ω
2222:ω
2201:κ
2184:ω
2180:−
2171:ω
2161:θ
2135:ℓ
2114:κ
2088:ℓ
2070:¯
2038:ℓ
2020:¯
1991:κ
1982:¯
1953:¯
1924:¯
1895:¯
1854:¯
1844:∂
1830:∂
1808:¯
1798:∂
1784:∂
1764:⊥
1760:∇
1713:⊥
1709:∇
1667:θ
1661:−
1655:−
1633:¯
1624:∂
1616:∂
1507:Figure 2.
1480:~
1448:ω
1405:~
1396:−
1381:~
1361:ω
1316:∝
1208:Figure 1.
917:∂
909:∂
896:ω
884:≪
860:∂
846:∂
795:∇
774:≪
754:∇
693:∂
683:∂
648:ε
625:⊥
621:∇
584:−
575:∂
567:∂
527:ω
510:∂
502:∂
455:−
402:⊥
395:→
379:∝
364:→
300:∂
290:∂
265:ε
238:→
208:∂
198:∂
168:−
151:→
128:∇
79:includes
49:July 2018
6339:Solitons
6143:16013174
6082:10045663
6000:23282830
5943:16050188
5868:21808332
5741:(2016).
5682:10784441
5627:11894107
5574:10038540
5416:38460064
5408:17358415
5253:18097405
5103:10034681
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