Breather solitons in highly nonlocal media
Abstract
We investigate the breathing of optical spatial solitons in highly nonlocal media. Generalizing the Ehrenfest theorem, we demonstrate that oscillations in beam width obey a fourth-order ordinary differential equation. Moreover, in actual highly nonlocal materials, the original accessible soliton model by Snyder and Mitchell [Science 276, 1538 (1997)] cannot accurately describe the dynamics of self-confined beams as the transverse size oscillations have a period which not only depends on power but also on the initial width. Modeling the nonlinear response by a Poisson equation driven by the beam intensity we verify the theoretical results against numerical simulations.
pacs:
42.65.Tg, 42.65.Jx, 05.45.YvI Introduction
Since the invention of the laser, optics has played an important role in nonlinear physics. One of the most known phenomena in nonlinear optics is the all-optical Kerr effect or an intensity-dependent refractive index Boyd et al. (2009). While in the simplest limit the change in refractive index depends on the local intensity value, in nonlocal media the nonlinear perturbation depends also on the intensity in neighboring points. Nonlocality strongly affects light propagation, leading to e.g. the stabilization of fundamental bright (2+1)D spatial solitons Dabby and Whinnery (1968); Suter and Blasberg (1993) as well as higher-order and vector solitons Hutsebaut et al. (2004); Alberucci et al. (2006); Skupin et al. (2006); Fratalocchi et al. (2007); Buccoliero et al. (2007); Assanto et al. (2008a); Buccoliero et al. (2008); Buccoliero and Desyatnikov (2009); Izdebskaya et al. (2015), complex dynamics and long-range interactions of solitons Peccianti et al. (2002); Rothschild et al. (2006); Conti et al. (2006) and between solitons and boundaries Alfassi et al. (2007); Alberucci et al. (2007); Peccianti et al. (2008). Optical nonlocality also entails the observation of fundamental phenomena, from soliton bistability Kravets et al. (2014) to spontaneous symmetry breaking Alberucci et al. (2015a), from turbulence to condensation Picozzi and Garnier (2011), from irreversibility and shock waves Barsi et al. (2007); Assanto et al. (2008b); Conti et al. (2009); Gentilini et al. (2015a) to gravity-like effects Bekenstein et al. (2015).
In general, even in the absence of losses, self-trapped beams in nonlocal media undergoes variations in transverse size owing to a dynamic balance between self-focusing and diffractive spreading Mitchell and Snyder (1999); Conti et al. (2004). Such behavior resembles the collective excitation phenomena in condensed matter, e.g. the collective modes in Bose-Einstein condensates where the center of mass or the condensate size in a harmonic trap undergo oscillations Pitaevskii and Stringari (2003). In optics, if the index well associated with the nonlinear response depends only on input power, nonlinear beam propagation can be described by a linear quantum harmonic oscillator and the breathing is purely periodic Snyder and Mitchell (1997). In actual media, however, self-focusing also depends on the transverse profile of the beam Conti et al. (2004). It was shown numerically that soliton breathing remains periodic in a (1+1)D simplified model, connecting this result with the existence of a (quasi) parabolic potential well Kaminer et al. (2007).
In this Paper we generalize the Ehrenfest theorem in order to find a set of ordinary differential equations ruling the evolution of a wave satisfying the Schrödinger equation. We demonstrate that, if the wave is subject to a parabolic potential, a single equation for the beam width can be derived. We then apply our new equation to the investigation of spatial optical solitons in highly nonlocal media. In such a limit, as first demonstrated by Snyder and Mitchell Snyder and Mitchell (1997) and later confirmed experimentally in nematic liquid crystals Conti et al. (2004); Alberucci et al. (2006) and thermo-optic media Rothschild et al. (2006), the light-induced waveguide can be satisfactorily approximated with a parabola, allowing the usage of all the mathematical tools developed for the quantum harmonic oscillator Sakurai (1994). This important result led to the coinage of the term accessible solitons Snyder and Mitchell (1997). The original model for accessible solitons predicts a breathing period depending only on the input power. Here we demonstrate that -in real media showing a non-differentiable response function- both extrema and period of the oscillations strongly depend on the input beam width. Numerical simulations with reference to a nonlinear response modeled by a Poisson equation, the latter modelling both nematic liquid crystals and thermo-optic materials Alberucci and Assanto (2007), support our findings.
Ii The Schrödinger equation in the Heisenberg picture
In the scalar approximation, in the harmonic regime and for small nonlinear perturbations, the paraxial propagation of an optical wavepacket along is governed by
(1) |
where is the vacuum wave-number and the refractive index of the unperturbed medium. Due to its formal equivalence to the Schrödinger equation Longhi (2009), (1) states the equivalence between light propagation in space and temporal evolution of a quantum particle in a two-dimensional potential with and an effective mass : equation (1) can thus be analyzed with the tools of quantum mechanics. In the Heisenberg picture an operator evolves in space (or time in quantum mechanics) according to Sakurai (1994)
(2) |
where is the effective Hamiltonian operator and the square brackets indicate the commutator . In the definition of the effective Hamiltonian , the quadratic term in the operator and the term correspond to the effective kinetic energy and the photonic potential, respectively. Applying equation (2) to spatial operator and momentum operator leads to the Ehrenfest’s theorem Sakurai (1994)
(3) |
with and . The Ehrenfest theorem has been successfully applied in optics to derive the trajectory of finite-size beams Jisha et al. (2011) and the interaction of multiple filaments Conti et al. (2006). Here we aim to extend it and derive an ODE governing the beam width, the latter related with the operator . Equation (2) yields
(4) |
the latter providing
(5) |
after a derivative with respect to . Similarly, taking we find
(6) |
and
(7) |
The advantage of the Heisenberg picture is that the generic bra and ket are stationary Sakurai (1994) (invariant with in optics), thus all the equations dealing with operators hold valid for the average values , as well. For conciseness, hereafter we will omit the subscript when referring to average quantities related with the wave .
Iii Waves in a parabolic potential
The beam trajectory obeys the Ehrenfest theorem, whereas the beam width can be obtained from two (generally vectorial) second-order ODEs in two unknowns: the width of its transverse profile and the width of its Fourier transform . The solution is not straightforward, as a complete knowledge of the profile is needed to calculate the average refractive index well and its derivative. Stated otherwise, all momenta of - i.e., with - are required to get the second momentum evolution with . A substantial simplification applies when the index well is parabolic, that is, . In this case the beam width is governed by the fourth-order ODE
(8) |
Equation (8) must be solved with initial conditions on the beam width , its initial variation (vanishing in the presence of a flat phase profile), its convexity
Iv Self-trapped nonlinear waves in highly nonlocal media
Equation (8) is valid whenever the refractive index well is parabolic, in both linear ( independent of excitation) and nonlinear ( depending on wavepacket profile and amplitude) regimes Snyder et al. (1995). In the highly nonlocal limit the light-induced index well is much wider than the beam Snyder and Mitchell (1997); the photonic potential can be Taylor-expanded to the second-order and equation (8) accurately models light propagation. We solve it in Kerr media (refractive index dependent on intensity with the vacuum impedance) with reference to two common responses: Gaussian Snyder and Mitchell (1997); Krolikowski and Bang (2000) and diffusive-like Bekenstein et al. (2015). The two responses differ for the Green function linking the beam intensity to the nonlinear perturbation , with . Hereafter, for the sake of simplicity we refer to either (2+1)D structures with cylindrical symmetry or (1+1)D geometries.
iv.1 Ideal limit: differentiable Green function
When the Green function is twice differentiable in the origin, it is easy to obtain , with coefficients is computed in the origin. This exactly matches the Snyder-Mitchell model, with a nonlinear response exclusively dependent on input power Snyder and Mitchell (1997). The term is -independent, but it varies along through the local beam width , yielding an overall power-dependent phase shift of the beam Guo et al. (2004). The nonlinear lens, modeled by the transverse term proportional to , is constant with because the strength of the quantum harmonic oscillator is invariant across the sample. For planar phase fronts at the input, the beam breathing along follows Snyder and Mitchell (1997) where
(9) |
where is the radial waist of the soliton at power . According to (9) self-confined beams oscillate around and are affeced by both input power and width . The breathing period is proportional to , whereas the breathing amplitude increases with the difference . Figure 1 compares the predictions of (9) with BPM (Beam Propagation Method) simulations in (1+1)D (with power replaced by a power density in Wm), confirming that the Snyder-Mitchell model is correct in the Gaussian limit Guo et al. (2004).
iv.2 Real case: singular Green function
Since actual highly nonlocal materials obey a diffusion-like equation with a Green function non-differentiable in the origin, this leads to discrepancies and quantitative inaccuracies when they are described by the original Snyder-Mitchell model Guo et al. (2004); Alberucci et al. (2014). In thermo-optic materials or nematic liquid crystals in the perturbative regime Peccianti and Assanto (2012), for instance, the nonlinear index well in the perturbation regime stems from a Poisson equation (we neglect the derivative along for simplicity)
(10) |
with the factor ( for self-focusing) an equivalent nonlocal Kerr coefficient accounting for the ratio between the beam amplitude and the corresponding index perturbation. From (10), the nonlinear index well for an arbitrary beam profile can be Taylor expanded as , with the intensity in the origin Conti et al. (2004). For a Gaussian beam it is , thus
(11) |
Substitution of (11) in equation (8) provides
(12) |
Equation (12) shows that the fourth-order ODE equation (8) turns into a second-order ODE when the medium nonlinearity is governed by equation (11). Applying the proper boundary conditions we find
(13) |
with .
As expected, in the linear regime it is , i.e., a shape-preserving soliton is excited.
Equation (13) corresponds to the motion of a classical particle subject to a conservative force ; when
(14) |
with the latter depending on both the normalized excitation and the initial width . The force vanishes when . Then equation (13) can be recast in the form
(15) |
with an effective power-dependent potential acting on the beam width
(16) |
Figure 2(a) illustrates the potential , asymmetric with respect to the local minimum and therefore sustaining non-sinusoidal oscillations of the momentum . Such dynamics is confirmed by direct numerical integration of equation (13), as plotted in figure 2(b). Integrating the energy conservation law over one half-period yields the breathing period
(17) |
where is the extremum opposite to the initial value during one single oscillation. Results from (17) are graphed in figure 2(c) together with the direct numerical integration of equation (13) (corresponding to the beam width graphed in figure 2(b)): the match is nearly perfect. At variance with equation (9), the oscillation period depends not only on input power but also on input width . In particular, has a local minimum
iv.3 Full numerical simulations in a Poisson material
To check our predictions we integrated equations (1) and (10) in a radially symmetric geometry, using a standard BPM in log-polar coordinates Alberucci et al. (2014, 2015b). The results for a given input power corresponding to a soliton of width m are summarized in figure 3. Noteworthy, now it is , i.e., the normalized input power had to be doubled with respect to the theoretical value (11) because the intensity profile overlaps with higher polynomial terms of the self-induced potential Alberucci et al. (2014); Ouyang et al. (2006).
The intensity evolution shows a periodic to aperiodic transition for varying input widths. The case m does not excite a shape-preserving soliton because in real Poisson media the exact soliton profile slightly differs from a Gaussian profile Alberucci et al. (2014); Ouyang et al. (2006).
We start analyzing the wavepacket behavior when close to the input, i.e., for short propagation length. In the interval 2.5 mm the excitation is close enough to the soliton state (i.e., m for the chosen power) and the self-trapped beam oscillates quasi-periodically (see figure 3 and figure 4(a)). Figure 4(b) shows the first oscillation period, computed doubling the position of the first local extremum in width versus . In agreement with theory the oscillation period depends on , with shorter when . The numerical results resemble quite closely the predictions from equation (17), with quantitative discrepancies arising when the input beam is much wider than the soliton (see figure 4(b)). As visible in figure 4(a), on longer propagation distances the wavepacket evolution departs from theory: when the
difference is small, the oscillation period slightly varies along ; conversely, both for very narrow () and very wide () inputs, the oscillations become markedly aperiodic. The discrepancy can be ascribed to two main causes: i) the effective shape of the self-induced index well is not perfectly parabolic, as discussed above; ii) the beam shape strongly departs from Gaussian due to the nonlinear interaction between a large number of modes, in turn breaking the validity of equation (11), the relationship between and now requiring a more involved approach.
The general trends with can be confirmed by computing light propagation over longer distances. The results in figure 4(c) show soliton breathing over a propagation length of 5 mm. First, the evolution smoothly changes with , ruling out the presence of chaotic dynamics Alberucci et al. (2015b); Aschiéri and Doya (2013). Second, the yellow portions in figure 4(c) (bottom and top) correspond to strongly aperiodic dynamics. Between them, in the center of the panel, the dynamics is quasi-periodic with a comb-like structure: each tooth is tilted towards the left (smaller ), showing that the oscillation period changes and tends to a minimum when . Consistently with theory, the oscillation amplitude is proportional to . In addition, the oscillation amplitude unexpectedly drops with due to an effective dissipation (in the framework of the effective potential defined via equation (16)) arising from the nonlinear interaction between the modes of the structure, as modeled by the higher-order polynomial terms in the light induced index well.
Next we study beam breathing in the frequency domain. To carry out this analysis we use a wavelet transform, as the evolution is not periodic and extends over a finite domain. Wavelets allow to address the temporal fluctuations in the spectrum of a signal. Such goal is achieved by using a basis composed by functions localized both in time and frequency. The family of wavelets is found by shifting and stretching a given function, named the mother wavelet. Local components of the spectrum are found compressing/dilating the mother wavelet, the compression factor used to determine at which scale we are analyzing the signal. Here we choose the wavelet transform Daubechies db6 Daubechies (1992). To avoid artifacts due to the boundaries, we limit our analysis in the interval 1 mm mm. For both m (figure 4(d)) and m (figure 4(e)) the peak of the wavelet transform does not move on the frequency axis with . The wavelet transform is also strongly localized on the scale axis, demonstrating that no diffusion effects occur in the frequency domain. The absolute value of the transform decreases with , in line with the emerging dissipative mechanism described above. Comparing the two cases, it is evident that the spectral components are higher when m due to larger oscillation amplitude. Noteworthy, for m the higher frequency components are much more relevant than for m. In fact, in the former case the oscillation around the average value is heavily affected by the non-Gaussian profile of the soliton Ouyang et al. (2006), a contribution neglected in deriving equations (13) and (16). This is evident in figure 4(f) showing the average period in the range 1 mm mm, computed from the wavelet transform. The shape is very close to figure 4(b), except near where a spurious peak appears. Physically, close to the input the action of the higher-order modes can be neglected; for long distances their effect accumulates and cannot be neglected anymore.
V Conclusions
In conclusion, using tools from quantum mechanics we derived a general equation ruling the nonlinear evolution of the beam width in a parabolic index well. Applying this model to light propagation in highly nonlocal media, we investigated how soliton breathing departs (both qualitatively and quantitatively) from the ideal Snyder-Mitchell law in real materials. In particular, we showed that the beam width dynamics can be modeled as a classic particle subject to a potential which depends on the width of the input beam. Thus, although the beam itself introduces a longitudinal change in the index well Aschiéri and Doya (2013), remarkably no aperiodic or chaotic evolution Alberucci et al. (2015b) takes place within the validity of our model. Numerical simulations verify that the breathing period depends on the width of the input beam and confirm the absence of chaos. Moreover, the simulations indicate the emergence of novel and intriguing effects due to the nonlinear interaction of several modes, assessing the suitability of nonlocal nonlinear optics for the investigation of many-body physics Gentilini et al. (2015b); Vocke et al. (2015).
Acknowledgments
AA and GA thank the Academy of Finland for financial support through the FiDiPro grant no. 282858. JCP gratefully acknowledges Fundação para a Ciência e a Tecnologia, POPH-QREN and FSE (FCT, Portugal) for the fellowship SFRH/BPD/77524/2011.
References
- Boyd et al. (2009) R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds., Self-focusing: Past and Present (Springer, New York, 2009).
- Dabby and Whinnery (1968) F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
- Suter and Blasberg (1993) Dieter Suter and Tilo Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993).
- Hutsebaut et al. (2004) X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Optics Communications 233, 211–217 (2004).
- Alberucci et al. (2006) A. Alberucci, M. Peccianti, G. Assanto, A. Dyadyusha, and M. Kaczmarek, “Two-color vector solitons in nonlocal media,” Phys. Rev. Lett. 97, 153903 (2006).
- Skupin et al. (2006) S. Skupin, O. Bang, E. Edmundson, and W. Królikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 066603 (2006).
- Fratalocchi et al. (2007) A. Fratalocchi, A. Piccardi, M. Peccianti, and G. Assanto, “Nonlinearly controlled angular momentum of soliton clusters,” Opt. Lett. 32, 1447 (2007).
- Buccoliero et al. (2007) Daniel Buccoliero, Anton S. Desyatnikov, Wieslaw Krolikowski, and Yuri S. Kivshar, “Laguerre and hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
- Assanto et al. (2008a) Gaetano Assanto, Noel F. Smyth, and Annette L. Worthy, “Two-color, nonlocal vector solitary waves with angular momentum in nematic liquid crystals,” Phys. Rev. A 78, 013832 (2008a).
- Buccoliero et al. (2008) Daniel Buccoliero, Anton S. Desyatnikov, Wieslaw Krolikowski, and Yuri S. Kivshar, “Spiraling multivortex solitons in nonlocal nonlinear media,” Opt. Lett. 33, 198–200 (2008).
- Buccoliero and Desyatnikov (2009) Daniel Buccoliero and Anton S. Desyatnikov, “Quasi-periodic transformations of nonlocal spatial solitons,” Opt. Express 17, 9608–9613 (2009).
- Izdebskaya et al. (2015) Yana V. Izdebskaya, Gaetano Assanto, and Wieslaw Krolikowski, “Observation of stable vector vortex solitons,” Opt. Lett. 40, 4182–4285 (2015).
- Peccianti et al. (2002) M. Peccianti, K. Brzadkiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460 (2002).
- Rothschild et al. (2006) C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769 (2006).
- Conti et al. (2006) Claudio Conti, Marco Peccianti, and Gaetano Assanto, “Complex dynamics and configurational entropy of spatial optical solitons in nonlocal media,” Opt. Lett. 31, 2030–2032 (2006).
- Alfassi et al. (2007) B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Boundary force effects exerted on solitons in highly nonlocal nonlinear media,” Opt. Lett. 32, 154 (2007).
- Alberucci et al. (2007) Alessandro Alberucci, Marco Peccianti, and Gaetano Assanto, “Nonlinear bouncing of nonlocal spatial solitons at the boundaries,” Opt. Lett. 32, 2795–2797 (2007).
- Peccianti et al. (2008) Marco Peccianti, Andriy Dyadyusha, Malgosia Kaczmarek, and Gaetano Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
- Kravets et al. (2014) Nina Kravets, Armando Piccardi, Alessandro Alberucci, Oleksandr Buchnev, Malgosia Kaczmarek, and Gaetano Assanto, “Bistability with optical beams propagating in a reorientational medium,” Phys. Rev. Lett. 113, 023901 (2014).
- Alberucci et al. (2015a) Alessandro Alberucci, Armando Piccardi, Nina Kravets, Oleksandr Buchnev, and Gaetano Assanto, “Soliton enhancement of spontaneous symmetry breaking,” Optica 2, 783–789 (2015a).
- Picozzi and Garnier (2011) Antonio Picozzi and Josselin Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107, 233901 (2011).
- Barsi et al. (2007) Christopher Barsi, Wenjie Wan, Can Sun, and Jason W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930–2932 (2007).
- Assanto et al. (2008b) Gaetano Assanto, T. R. Marchant, and Noel F. Smyth, “Collisionless shock resolution in nematic liquid crystals,” Phys. Rev. A 78, 063808 (2008b).
- Conti et al. (2009) Claudio Conti, Andrea Fratalocchi, Marco Peccianti, Giancarlo Ruocco, and Stefano Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
- Gentilini et al. (2015a) Silvia Gentilini, Maria Chiara Braidotti, Giulia Marcucci, Eugenio DelRe, and Claudio Conti, “Nonlinear gamow vectors, shock waves, and irreversibility in optically nonlocal media,” Phys. Rev. A 92, 023801 (2015a).
- Bekenstein et al. (2015) Rivka Bekenstein, Ran Schley, Maor Mutzafi, Carmel Rotschild, and Mordechai Segev, “Optical simulations of gravitational effects in the newton–schrödinger system,” Nat. Phys. 11, 872–878 (2015).
- Mitchell and Snyder (1999) D. J. Mitchell and A. W. Snyder, “Soliton dynamics in a nonlocal medium,” J. Opt. Soc. Am. B 16, 236 (1999).
- Conti et al. (2004) C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
- Pitaevskii and Stringari (2003) Lev. P. Pitaevskii and Sandro Stringari, Bose-Einstein Condensation (Oxford University Press, New York, 2003).
- Snyder and Mitchell (1997) A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538 (1997).
- Kaminer et al. (2007) Ido Kaminer, Carmel Rotschild, Ofer Manela, and Mordechai Segev, “Periodic solitons in nonlocal nonlinear media,” Opt. Lett. 32, 3209–3211 (2007).
- Sakurai (1994) J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, MA, 1994).
- Alberucci and Assanto (2007) Alessandro Alberucci and Gaetano Assanto, ‘‘Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions,” J. Opt. Soc. Am. B 24, 2314–2320 (2007).
- Longhi (2009) S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
- Jisha et al. (2011) Chandroth P. Jisha, Alessandro Alberucci, Ray-Kuang Lee, and Gaetano Assanto, “Optical solitons and wave-particle duality,” Opt. Lett. 36, 1848–1850 (2011).
- Snyder et al. (1995) A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, “Periodic solitons in optics,” Phys. Rev. E 51, 6297–6300 (1995).
- Krolikowski and Bang (2000) W. Krolikowski and O. Bang, “Solitons in nonlocal nonlinear media: exact solutions,” Phys. Rev. E 63, 016610 (2000).
- Guo et al. (2004) Qi Guo, Boren Luo, Fahuai Yi, Sien Chi, and Yiqun Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004).
- Alberucci et al. (2014) Alessandro Alberucci, Chandroth P. Jisha, and Gaetano Assanto, “Accessible solitons in diffusive media,” Opt. Lett. 39, 4317–4320 (2014).
- Peccianti and Assanto (2012) Marco Peccianti and Gaetano Assanto, “Nematicons,” Phys. Rep. 516, 147 – 208 (2012).
- Alberucci et al. (2015b) Alessandro Alberucci, Chandroth P. Jisha, Noel F. Smyth, and Gaetano Assanto, “Spatial optical solitons in highly nonlocal media,” Phys. Rev. A 91, 013841 (2015b).
- Ouyang et al. (2006) Shigen Ouyang, Qi Guo, and Wei Hu, ‘‘Perturbative analysis of generally nonlocal spatial optical solitons,” Phys. Rev. E 74, 036622 (2006).
- Aschiéri and Doya (2013) Pierre Aschiéri and Valérie Doya, “Snake-like light beam propagation in multimode periodic segmented waveguide,” J. Opt. Soc. Am. B 30, 3161–3167 (2013).
- Daubechies (1992) Ingrid Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992).
- Gentilini et al. (2015b) Silvia Gentilini, Maria Chiara Braidotti, Giulia Marcucci, Eugenio DelRe, and Claudio Conti, “Physical realization of the glauber quantum oscillator,” Sci. Rep. 5, 15816 (2015b).
- Vocke et al. (2015) David Vocke, Thomas Roger, Francesco Marino, Ewan M. Wright, Iacopo Carusotto, Matteo Clerici, and Daniele Faccio, “Experimental characterization of nonlocal photon fluids,” Optica 2, 484–490 (2015).