Temporal Fresnel diffraction induced by phase jumps in linear and nonlinear optical fibres

We analytically and numerically study the temporal intensity pattern emerging from the linear or nonlinear evolutions of a single or double phase jump in an optical fiber. The results are interpreted in terms of interferences of the well-known diffractive patterns of a straight edge, strip and slit and a complete analytical framework is provided in terms of Fresnel integrals for the case of purely dispersive evolution. When Kerr nonlinearity affects the propagation, various coherent nonlinear structures emerge according to the regime of dispersion.


Introduction
Diffraction is among the key effects of wave physics with applications in a broad range of technological domains spanning from imaging to spectroscopy, material sciences, mobile communications, test measurements or sensors. Propagation of a plane wave through a sharp edge is the unavoidable example taught in any physical optics courses to introduce the concepts and the theoretical tools available to handle Fresnel's diffraction [1,2]. But diffraction is not restricted to opaque screens that partially obstruct light, it is also involved when light is transmitted through a phase plate, i.e. a transparent medium imprinting a localized phase jump, leading to strong oscillations of the diffracted field [3,4].
The well-known free-space evolution of a beam can find analogs in the temporal domain. Indeed, the parabolic spectral phase induced by the dispersion of an ultrashort optical pulse is equivalent to the paraxial diffraction affecting the spatial propagation of a light beam [5][6][7]. This space/time duality has already been extremely fruitful and has stimulated numerous new concepts or interpretation in ultrafast optics such as temporal or spectral lenses [8,9], Fresnel lens [10], super resolution imaging [11], dispersion gratings [12] or two-wave temporal interferometers [13], to cite a few. Other studies have established links between the near-field propagation observed in diffraction and advanced applications for high repetition-rate sources when initial periodic phase modulation is converted into intensity pattern [14][15][16][17][18] in a process that can be linked to the Talbot array illuminators in the spatial domain [19]. However, despite the fact that coherent communications now heavily rely on the use of phase modulation, no explicit study of the space / time duality for a single phase step has been clearly reported so far. This is the scope of the present paper to fill this gap by providing a series of analytical and numerical results for a single phase jump evolving in a single mode optical fiber. We then extend the discussion to the case of a double phase jump and highlight some significant differences with respect to the pattern resulting from the dispersion of a temporal hole of light. In both cases, we also investigate the consequences of optical Kerr nonlinearity according to the regime of dispersion and demonstrate the emergence of coherent structures. Finally, the impact of the finite bandwidth of temporal modulation is discussed.

Situation under study and analytical treatment of the linear propagation
Before discussing our experiments, let us first recall the basis of the analogy between the spatial evolution of light affected by diffraction and the temporal changes experienced by light when dispersion is involved. We consider the simple case where a monochromatic plane wave with wavelength  and an amplitude ailluminates a phase pattern. It can be a light beam transmitted through a transparent plate with an abrupt change in thickness or refractive index. For a 1D transverse problem, the longitudinal evolution of light a(x,z) in the scalar approximation is ruled by the following differential equation : with t being the temporal coordinate. This shaped temporal waveform then propagates in a dispersive single mode waveguide, typically an optical fiber, that ensures that its spatial transverse profile is unaffected upon propagation. The temporal profile of the light in the approximation of the slowly varying envelope evolves according to: with 2 the second order dispersion coefficient. We have here considered a second order anomalous dispersion 2 = -20 ps 2 /km typical of the SMF-28 fiber used for optical telecommunications [20]. As this second-order dispersion coefficient is much higher than the third-order dispersion coefficient, it is possible, as a first approximation, to fully neglect the impact of higher order dispersive terms. The space-time duality readily appears in the mathematical structure of equations (1) and (3) where the transverse space coordinate and time are exchanged, the waveforms fulfilling the same normalized differential equation [5][6][7]. Consequently, both diffraction and dispersion imply the development of a quadratic spectral phase and lead formally to similar consequences. In order to better understand the evolution of temporally phase-sculpted waveform subject to dispersion, given the superposition property, it may be useful to rewrite the initial condition of our problem given by Eq.
(2) as a linear combination of two patterns illustrated in Fig. 1(a) : Expressed in the context of temporal evolution, this leads to: The temporal intensity profile of one semi-infinite edge is therefore characterized by strong oscillations of the plateau as can be seen in Fig. 1 The most pronounced ripple is obtained at tmax = text,0 and has a maximum equal to 1.37 which is independent of the propagation distance. It is followed by a local minimum at tmin = text,1.
The temporal phase difference T between 1 and 2 can be expressed as T(t,z) = arg(2(t,z)) -arg(1(t,z)) = arg((t,z)) -arg((-t,z)) +  =  +  with  = arg((t,z)) -arg((-t,z)) plotted with black line in panel (b2) and that can be analytically derived as : with the normalized coordinate u being  is null at t = 0 and is close to an even multiple of  around tmax. This formula can be closely adjusted around t = 0 by the following linear fit (black dashed line) : The intensity profile obtained after propagation is given by the following expression that can be interpreted exploiting Cornu's spiral [4,22] : One particularly interesting case is when  equals =  (see Fig. 2, panel (a) and red curve in panel (b)). In this case, T(t) equals  around the central position so that the interference between 2(t) and 1(t) is destructive around t=0. As 2(0) and 1(0) have identical intensities, the destructive interference is complete and the intensity drops to zero. Therefore, the intensity in the central part is lowered compared to what would have been expected from the incoherent sum of the intensities of the two diffraction pattern |1| 2 + |2| 2 ( Fig. (2), black line). On the contrary, around tmax,  becomes an odd multiple of , so that the interference becomes constructive and the peak intensity is enhanced by 31%. As a consequence, the prediction of Eq. (6) initially derived for an abrupt intensity edge also applies for a step phase edge as can be checked in panel 2(a) (see dashed line). Note that for  = , the expression of the resulting temporal intensity pattern becomes analytically extremely simple [23]: binary intensity modulation of the initial condition as discussed in [24]. Indeed, in this last configuration, a central spot was progressively growing at the centre of the light hole due to constructive interference.
One advantage of temporal optics compared to spatial optics (that requires fine tuning of precision optics [25]) is that it is quite straightforward to adjust optoelectronic devices to modify . As an example, we have considered   /2 in panel (b) of Indeed, when  = /2, the maxima are obtained for  -/2 or 3/2, i.e., for times that are -18.6 or 41.2 ps after 10 km of propagation (see Fig. 1(b2)). The amplitude of the ripple is also affected and the temporal fringes that develop on each side of the dip are not identical: for , the peak at t = -35.4 ps is lowered as the interference process involves a tail of |1| 2 with a reduced intensity. On the contrary, the bump at t = 24.1 ps, is increased as the tail linked to |2| 2 is more powerful and therefore stimulates a constructive interference with a higher efficiency. We also note that the intensity does not drop to zero in the central part. The destructive interference condition is obtained for i.e. at t = T/2) so that 2(t) and 1(t) have significantly different values that lowers the efficiency of the destructive process. Those trends are fully confirmed by the evolution of the pulse pattern recorded after 10 km of propagation according to  and summarized in panel (c) of Fig. 2. We observed the change in the visibility of the central fringe [25] as well as the continuous shift of the maxima. In order to qualitatively predict the temporal location tmin of this dip, we can take advantage of the linear approximation of [Eq. (8)] to propose the following empirical prediction: Panel (c) of Fig. 2 confirm the reasonable agreement of Eq. (12) which is close to linear shift of tmin according to the initial amplitude of the phase offset. Note that if the regime of dispersion is normal instead of anomalous, the pattern will be flipped in the temporal domain (see Fig. 2(b), dashed blue curve). We can note that the amplitude of only the first central bumps is affected: for larger time, the intensity of the tails of 1 or 2 becomes negligible so that the influence of the interference process on the ripple becomes much more negligible.

Nonlinear propagation
We are now interested in the propagation occurring when nonlinear effects become significant. Indeed, contrary to the usual diffraction in free space, propagation in a waveguide can also involve nonlinear effects. In this context, optical fibers represent an ideal testbed: thanks to a very low level of losses, the Kerr nonlinearity of silica may be accumulated over several kilometers. The temporal evolution of a waveform in an optical fiber resulting from the interaction between nonlinearity and dispersion can be taken into account through an additional term accounting for self-phase modulation in Eq.(3), leading to the well-known nonlinear Schrödinger equation (NLSE) [26]: with  is the Kerr coefficient of the fiber, typically  = 1.1 /W/km for the SMF-28 fiber.
As a first approximation, we have neglected the impact of the optical losses that are reduced in the telecommunication spectral window of telecommunication fiber (around 0.2 dB/km) and can be ideally compensated used Raman distributed amplification [27].
We solve the scalar NLSE using numerical simulations based on the well-established split-step Fourier method [26].
The longitudinal evolution of the intensity profile for an initial power of 290 mW is plotted in Fig. 3 propagation length of 10 km. We can note that the width of the coherent structures that surround the central gap is also affected by the input power with the degree of compression increasing with the initial power. The pattern seems to asymptotically tend to a modulated soliton train with unequal temporal spacing and with a peak power close to four times the initial average power, as predicted for other initial conditions in [32,35]. The temporal location of those soliton-like peaks is also affected by the peak power. Whereas the temporal shift induced by nonlinearity is quite limited on the most central solitons (see also dotted white line in panel (a1)), we note that the temporal location of other bright structures emerging from secondary ripples is more powerdependent.
The picture gets very different when normal dispersion is involved (panel (a2), 2 = 20 ps 2 /km). Instead of a central gap broadening with propagation distance according to Eq.
(11), we observe a dip that evolves with its temporal duration unaffected, while its minimum goes down to a null intensity. The details of the intensity profiles plotted in Fig. 3(b2) for a propagation distance of 5 and 10 km stress that the width of the central part does not evolve and is good agreement with a black soliton BS(t), which is characterized by a full hole of light and a phase offset of  at its center [38], with an analytical expression provided by : with TBS the temporal duration of the black soliton and PBS the power of the continuous background linked by Due to the imperfect initial profiles, this dark soliton is surrounded by radiative waves that progressively move away from the central part. Consequently, it confirms that imprinting an initial temporal phase singularity of  is a possible approach to generate black solitons [39], which contrasts with the others technics that have focused on the advanced shaping of the temporal intensity and phase profile [40,41] or on the nonlinear interaction of a pair of delayed pulses [42,43]. In the normal regime of dispersion, the initial non vanishing dip that appears upon linear propagation is converted into a grey soliton-like structure that has a reduced contrast. Both the reduced initial phase offset as well as the reduced depth of the main dip induced by dispersion contribute to generate a grey soliton with a reduced greyness [26]. Note that the grey soliton has a non-zero velocity, as also stressed in spatial optics [44]. Regarding the evolution recorded in the anomalous dispersion regime for  = /2, we note that the fluctuations of peak power and widths of the localized ultrashort structures according to the input power becomes pronounced. This breathing behavior is dominated by solitons over finite background such as Akhmediev breathers, Kutnetsov-Ma solutions or superregular structures [45] that may exist both in temporal optics [46] but also in spatial optics [47]. It is worthy to note that the highest peak power is not always achieved for negative times as we could have expected from the asymmetry existing in the linear propagation.

Situation under study and dispersive propagation
The diffraction of 2D transparent phase objects such as square or circular samples have been the subjects of past investigations in spatial wave optics, with applications to metrology [4]. We now consider the case where the temporal phase shift imprinted on the continuous wave  is limited to a duration T0. The spatial analog of this initial condition is a 1D is a transparent stripe of width x0 with a height leading to a phase offset . This stripe has two abrupt edges and is illuminated in normal incidence. Our ideal temporal object can be analytically described as: Once again, we can take advantage of the superposition principle to facilitate the discussion of the dispersion phenomena and the analytical calculations [25,48]. Indeed, there are various ways to rewrite Eq. (16). One can see this problem as the temporal coherent addition of two abrupt phase jumps of similar amplitude but with opposite temporal orientations overlapping by the duration T0. Another way to interpret the initial condition is illustrated in Fig. 4(a). It is convenient to rewrite this initial condition as the sum of three elements: H(t-T0/2) and 3(t) = rect(t/T0) exp(i ), rect being the rectangular function of width 1 (purple curve). The pattern made by A(t) = '1(t) + '2(t) (green curve) corresponds to an opaque light hole of width T0. This temporal analogue of an opaque stripe of constant width [49][50][51] has been the subject of our recent paper dealing with the observation of the temporal Arago spot in optical fibers [24]. Therefore, our problem resumes to the coherent superposition of the temporal Arago pattern and the temporal pattern induced by an aperture of width T0 with a phase offset of . The intensity profiles linked to both waves can be analytically derived [48] and are plotted on Fig. 4(b1) for a propagation distance of 10 km. We observe the central intensity bump typical of the Arago spot that will interfere with the temporally broadened pattern induced by the rectangular phase offset. The problem can be solved analytically and the temporal profile can be once again predicted using a combination of Fresnel integrals: 1 cos sin The phase difference ' = arg((t)) -arg((t)) - can be analytically predicted as (see Fig. 4(b2), black curve): The linear evolution for an initial phase offset of  and a duration T0 of 40 ps is provided in Fig. 5(a1) and is compared with the temporal Arago spot (panel 5(a2)). We can note several features that are directly linked to the interference process that may exist between  and . First, due to constructive interference (' being close to ), which should be compared with the central intensities of the considered aperture and the 1D Arago spot (purple and green lines respectively) that are given by [12] : All these quantities can be easily graphically predicted using the Cornu's spiral as directly linked to the distance between the point of curvilinear coordinate Z and the . The intensity at the center can be up to 2.8 times the average power of the illumining light, which is 58 % higher than the maximum of |(0)| 2 taken alone.
Propagation distance at which this maximum appears can be well predicted by a crossing point of two extrema located at  T0/2  tmax which are given by Eq. (6). This observation provides scaling properties of the resulting pattern meaning that we can tailor the position of the maximum by carefully choosing the temporal extend of the initial phase jump. Note that the constructive interference process between  and  is not optimally efficient due to the very different intensity levels of the two waves.
Moreover, the phase difference '(0) that can be derived analytically as : is not an odd multiple of  (see Fig. 5(b2) as well as Fig. 4(b2)). For asymptotic propagation distance and similarly to the Arago spot, the central intensity tends to 1.
Regarding the first ripples that surround the central peaks (see Fig. 6(a), red line), we note that their intensity is enhanced compared to || 2 + || 2 (black line). Indeed, for the position of the lateral maxima at tmax + T0/2, the interference between  and  is constructive, ' being an odd multiple of  after 10 km of linear propagation (see Fig.   4(b2)).
The influence of  is illustrated in Fig. 6 for a dispersive propagation distance of 10 km and we compare the pattern achieved for  =  (red line) and = 1.37 (blue line). The initial phase offset significantly impacts the visibility of the fringes as well as the amplitude of the maximum of the oscillations. A more systematic study of the influence of  is provided in panel (b). The temporal intensity pattern is fully symmetric, whatever  is. We can make out that, after 10 km of propagation, the most pronounced dips surrounding the central peak are not achieved for  =  but for  = 1.37 . The intensity of the central part is also strongly influenced by  as can also be seen in panel (b2) where we can make out that the peak intensity at t = 0 follows a sinusoidal evolution typical of a two wave interference process. Once again, ' helps us to understand why  =  is not the optimum value to achieve the highest central  Fig. 7(b2)).
The influence of the initial average power on the resulting pattern is summarized in Fig. 8(a) for anomalous and normal dispersion. Compared to the results discussed in section 2.2 (see Fig. 3), we can note several differences regarding the evolution in the anomalous dispersion regime. First of all, we note that, contrary to the peak intensity of the lateral peaks that remains more or less constant with the input average power in the case of a single phase jump, we observe here some significant fluctuations in the peak power, denoting a breathing behavior. The temporal position of these structures is affected by the power, higher the power is, closer the structures are from the initial abrupt phase jumps. Regarding the pattern in the central part, contrary to [24], the evolution is not monotonic with power. We observe that a well-defined double peak structure appears for a well-chosen power (around 0.9 W), which is reminiscent from the recent investigation of the generation of a pair of pulses from an initial super-Gaussian pulse [55]. We can expect the initial temporal duration T0 to be an efficient mean to control the number of structures in this central part [56] and that more complex interactions could be observed for longer durations [57]. On the contrary, in the normal regime of propagation, two black solitons are visible with temporal location in the vicinity of the initial phase offsets and a normalized intensity equals to 1 between these two phase shifts.
The pattern achieved for  = /2 is also provided in panel (b). Quite remarkably, the differences that were very pronounced in the case of a single phase jump are here attenuated in the anomalous regime of dispersion so that we retrieve, at least qualitatively, the different features we previously discussed for panel (a). However, in the normal regime of dispersion, we can note, that the dark solitons that emerge are not black and move one away from each other with an intensity pattern that is symmetric.

Influence of the finite modulation bandwidth
In order to end this discussion and to get perspectives on potential experimental demonstration, we take into account the finite bandwidth of the initial phase modulation. Indeed, whereas in spatial optics, use of very straight edges is technically feasible, temporal optics faces the practical limitations of the optoelectronics that will ultimately limit the steepness of the fronts applied on the initial continuous wave. In the article investigating the temporal Arago spot, we have shown using super-Gaussian intensity profiles instead of ideal rectangular one has only limited impact on the resulting temporal pattern. As a first approximation, we consider here that the bandwidth limitations affecting the generation of the phase profile can be modelled by Gaussian filter with a full width at half maximum of 80 GHz. The smoothened phase profile is shown in Fig. 9(a) for a modulation of  and -red solid and dashed lines respectivelyThe pattern observed after a propagation distance of 10 km is plotted in Fig. 9(b) where we note that major differences affect the linear propagation. Indeed, given the strong interference process that occurs near the phase jumps, any change in the phase profile severely impacts the emergence of the features we have previously identified. We also note that the pattern induced by a softened phase shift of strongly differs from the one emerging from a softened phase shift of - which can be explained by the impact of the temporal gradient present in the transition region. Once again, the exact identification of the nature of the coherent structures is challenging and requires dedicated analytical tools.
The changes are also significant when considering the propagation in presence of nonlinearity, as stressed by panel (c1) of Fig. 9 that should be compared to Fig. 8(a).
In the anomalous regime of propagation, we do not observe the central doublet that was generated in the ideal case for a power around 1W. The differences are even more striking when considering the nonlinear propagation in the normal regime of dispersion.
Whereas the ideal -phase step leads to the generation of two black solitons, taking the bandwidth limitations, the dark solitons are now grey. They have a non-null velocity and move away from each other. Note that such a pattern may, to some extent, qualitatively recall the trends observed in linearly frequency modulated signal as analyzed in nonlinear optics [36,58] based on Witham approaches initially used in fluids [59].
The nonlinear dynamics gets very different when considering a phase step of -.
Given the initial temporal chirp, the two grey solitons tend to move towards each other's and collide after 10 km for an initial power of 0.38W. Taking advantage of the finite bandwidth could therefore be a simple but efficient mean to generate experimentally a doublet of dark solitons with opposite velocities that could then collide [60]. The pattern observed in the central region for anomalous propagation is also drastically impacted by the sign of the phase offset. The initial perturbation turns into an expanding nonlinear oscillatory structure with a higher number of coherent structures being present in the central part and experiencing growth and decay cycles.

Conclusion
To conclude, we have studied the temporal intensity pattern emerging from the linear or nonlinear evolution of a single or double phase jump. We have provided an interpretation of the pattern in terms of interferences of the well-known diffractive patterns of a straight edge, strip and slit. A complete analytical framework has been provided in terms of Fresnel integrals for the case of purely dispersive evolution. This has enabled us to stress the similarity as well as the differences that exist in the pattern resulting from a phase shift and from an intensity modulation.
We The present work can be extended in many aspects. First, the initial wave is not restricted to a continuous fully coherent wave. Similarly to the spatial case, we can indeed consider the impact of a temporal phase shift imprinted on an initial pulse [23,61,62], on a wave that has an initial chirp [63,64] or a wave that is only partially coherent [3]. With the progress of coherent transmissions, it is also possible to combine intensity and phase modulation, therefore mimicking a partially transparent phase object [22,48,65]. Moreover, it is possible to benefit from the vectorial properties of light in fibers to explore new degrees of freedom and other nonlinear structures [66]. We also believe that the present discussion can be of help to better understand the evolution of the vectorial shock waves we recently described [67] and to catch the way cross-phase modulation may affect through dispersion the temporal pattern of a continuous wave [68].
As the NLSE is a universal mathematical model that also accurate to describe wave propagation in other fields of physics such as Bose-Einstein condensates [69] or hydrodynamics [70], our conclusions dealing with the impact of phase jumps can be extended to the nonlinear evolution of water waves. Indeed, in recent years, the link between temporal optics and hydrodynamics has been a fruitful driving force in stimulating the understanding of coherent structures in focusing and defocusing regimes of nonlinear propagation [45,53,71,72].