We investigate the effects of the space-time coupling (STC) on the nonlinear formation and propagation of Light Bullets, spatiotemporal solitons in which dispersion and diffraction along all dimensions are balanced by nonlinearity, through periodic media with a weak transverse modulation of the refractive index, i.e. waveguide arrays. The STC arises from wavelength dependence of the strength of inter-waveguide coupling and can be tuned by variation of the array geometry. We show experimentally and numerically that the STC breaks the spectral symmetry of Light Bullets to a considerable degree and modifies their group velocity, leading to superluminal propagation when the Light Bullets decay.
© 2011 OSA
Research in the last three decades has demonstrated that the concept of group velocity is a subtle one, which by no means is bounded to the limit of the speed of light in a medium. In fact, superluminal as well as subluminal propagation of light pulses were reported in various media [1–5] even raising doubts that signals could indeed propagate faster than the limits imposed by relativity . However, rigorous theoretical analysis has shown that group velocity and signal velocity are two different measurables, confirming the validity of the limits imposed by relativity . Nonetheless, in dispersive media pulsed beams may propagate at group velocities which are vastly different from those of a plane wave due to the direct coupling of spatial and temporal dispersion. Such pulses are also termed sub- or superluminal and are in the focus of this work.
From a practical viewpoint, the possibility to tune the group velocity of light in dispersive media reverts a huge potential for applications, e.g. for the retiming of Terabit/s OTDM data-packets [7–9] or for frequency conversion of ultrashort laser pulses in dispersive media, where long interaction lengths are desirable for high energy conversion [10–12].
A key ingredient to control the group velocity of pulses is the direct coupling of the spatial and temporal frequency components in the wavepackets’ spectrum resulting in non-factorizable spatiotemporal structure of the pulses. Some examples are tilted pulses , their axially symmetric generalization (X-like waves ), as well as helical beams in Raman media . Such pulses can be generated by use of passive optical elements or by means of the interplay of linear and nonlinear optical effects of ultrashort pulse propagation. Examples are quadratic  and Kerr nonlinear media with normal dispersion, which sustain the spontaneous generation of X-like waves [17–19] propagating sub- or superluminally. Furthermore in one dimensional arrays of nanowires this coupling of spatial and temporal frequency components leads to supermodal variation of the zero dispersion wavelength with heavy impact on quasisoliton-induced Supercontinuum-generation [20–22].
In this paper, we analyze for the first time space-time effects in the propagation of Light Bullets (LBs)  launched in periodic arrays of coupled waveguides . These discrete LBs are solitary wavepackets in which the nonlinearity balances dispersion and discrete diffraction simultaneously [25–32]. They exhibit a balance between spatial, temporal, and nonlinear effects and thus it is exciting to probe their complex interaction. In particular they are ideally suited to probe space time coupling (STC) effects because their formation and evolution is determined by the fibre arrays dispersion properties, which display a strong dependence of the discrete diffraction on the wavelength. We find that this discrete analogous of STC  combined with the self-focusing nonlinearity of the host material gives rise to a symmetry breaking of the LB spectrum and a consequent superluminal modification of group velocity during LB decay.
This feature and the possibility to enhance and tailor STC by a suitable waveguide design and array geometry proves that micro-structured media are excellent candidates to explore new effects of sub- and superluminal pulse propagation. If gain and/or loss are considered even more flexibility in the design of the dispersion velocities can be achieved .The paper is set up as follows: Section 2 investigates the nature of STC and its connection to the dispersion relation of a linear system, introduces a simplified evolution equation and LB solutions thereof and presents the samples and their properties. Section 3 discusses the breakdown of spectral symmetry associated with the solutions found in the prior section. Experimentally this asymmetry is measured using an extension to the established imaging cross correlator setup similar to the CROAK time-sepctral imaging method  and the results are shown to correspond with theoretical predictions. Section 4 discusses how STC influences the temporal evolution of LBs, giving rise to superluminal group velocity during decay. A simple explanation in terms of the dispersion relation of the array’s supermodes is given and shown to predict the behavior well. All predictions are verified in experiment and simulation. Section 5 draws conclusions on the results.
2. Space-time focusing, Light Bullets, and fibre arrays
Wave propagation in a linear medium can be described with the functional form of its dispersion relation and the shape of the corresponding modes. If the medium is invariant along the z-axis and periodic in x and y one can explicitly write the dispersion relation in the form , where is the band index, which will be set to and omitted from here on. is the propagation constant of the Bloch mode, which is periodic in and , the transverse wave vector components. These can be assumed to be continuous variables because of the large size of the array (see Fig. 1(a) ).The shape of the dispersion relation determines the evolution of any wavepacket which travels through this system. One speaks of direct space-time coupling (direct STC) if the dispersion relation contains terms that depend on both the transverse wave vector components and the frequency. Of special interest in this respect is the dispersion/diffraction, which is characterized by the local curvature of the dispersion relation, given by its Hessian matrix
One particular form of direct STC is the space-time focusing (STF) term, which derives from the chromatic dependence of the diffraction strength [33,36–38]. In a homogeneous dispersive medium, the STF term represents the first higher order correction to the paraxial wave equation (Schrödinger equation). The form of the STF term can be derived starting from the Helmoltz equation for the description of the propagation of the field in a homogeneous medium33,38], but for pulses shorter than a few tens to a hundred femtoseconds.
On the contrary, media featuring a weak periodic transverse modulation of the refractive index, such that the geometry can be understood as a (hexagonal) lattice of weakly coupled waveguides [39,40], have a dispersion relation of the form21,22]. This system, although paraxial, displays STF as can be seen fromFig. 1(a) with one finds that STC is influential unless41] and much more easily accessible than the limit for homogeneous materials imposed by Eq. (5). Furthermore one can tune the degree of STC easily by varying the wavelength dependence of the waveguide coupling, i.e. by variation of the array geometry.
The nonlinear evolution of light inside such an array under the action of the Kerr-nonlinearity can be described in a simplified manner in terms of the amplitudes of the light in each waveguide marked with the index . One therefore expands and of Eq. (6) into a Taylor series , and assumes that the modal structure is unaffected by the Kerr nonlinearity and thus, after back-transformation from frequency into temporal domain , gets the evolution equation42]. This equation is normalized such that all lengths are measured in units of , all times are measured in units of and amplitudes in units of , where is the instantaneous power in waveguide in watts. Thus energy units are , where is the dispersion coefficient of the waveguide mode and is its nonlinearity coefficient . The normalized STC constant is related to as .
Equation (9) is used to find families of LB solutions with the ansatz and consecutive numerical solution of the system of coupled, complex differential equations with a Newton-Raphson scheme. Results for various values of are displayed in Fig. 2 . LB families with a larger degree of STC require a slightly higher energy and peak power, are less localized and have a shorter upper limit of their temporal width for stable propagation. All LB families are tested to be stable if using linear stability analysis. They therefore obey the Vakhitov-Kolokolov theorem . The most striking difference, however, is displayed in Figs. 2(e) and 2(f). LBs with a higher level of STC are shifted in frequency, at a level which is different for the central waveguide and its nearest neighbours. This difference is due to spectral symmetry breaking, discussed in Section 3.
Among the various experimental implementations of 2D waveguide arrays, such as photorefractive lattices [45–47] or femtosecond-written waveguide arrays  we have chosen to use fibre arrays [48,49] because they combine high regularity, well-defined femtosecond-nonlinear response, high damage threshold, and well-known broadband dispersion properties. The fibre arrays are produced by a stack and draw technique, similar to the production of PCFs [50,51], from pure silica rods embedded in a background of fluoride doped silica with reduced refractive index. Special care [48,49] has been taken to suppress disorder-induced effects , such as Anderson localization . The level of STF can be tuned by variation of the array geometry, as can be seen from Fig. 1. A picture of the front facet of a typical fibre array is shown in Fig. 1(a). Figure 1(b) displays the coupling properties of the investigated arrays. The blue line displays the diffraction length , which depends heavily on the wavelength, because the modal overlap grows with increasing wavelength. The blue line displays the normalized STF level as displayed above, which is, in the wavelength range of , fairly constant with a value of for the samples with and a value of for the samples with .
However, proper numerical treatment of light propagation inside such arrays has to include effects beyond those which are treated in Eq. (9), which serves as a simplified model to help understand the underlying physics. Numerical simulations are treated within the framework of the generalized unidirectional Maxwell Equations [54–56], containing additional terms, including higher order dispersion, higher order nonlinear effects, and the explicit treatment of the carrier wave dynamics. Details are discussed in .
3. Spectral Symmetry Breaking of LBs
Nonlinear evolution of wavepackets is always influenced by the properties of the dispersion relation of the medium in which the wavepackets travel. Among the countless examples are the generation of supercontinua [57–59] in photonic crystal fibres, and the generation of nonlinear X-waves in continuous and discrete media [16,60–62]. In homogeneous media, the STC and self-focusing nonlinearity leads to formation of sub-luminal pulses  related to the breakdown of spectral symmetry, whereas in arrays of strongly coupled nanowires STC induces nonlinear spectral symmetry breaking .
The asymmetric reshaping of the spatiotemporal spectra of LBs is shown in Fig. 3(a) , where the square modula of the spectra are plotted for LB solutions with various levels of DSC, governed by the parameter .
As a measure of the asymmetry of the spatiotemporal spectrum of a particular wavepacket we define the angular mean frequency and the difference of this quantity from the centre of the first Brillouin Zone to its edge as a measure of the spectral asymmetry as:
These quantities can now be calculated for the LB solutions of Eq. (9) and are plotted in Figs. 3(a) and 3(b) for various values of as a function of the nonlinear phase shift . It can be seen that is almost independent of the nonlinear phase shift and thus of the peak power of the LB (compare Fig. 2) but depends on the level of STC given by . This dependence is displayed in Fig. 3(c), which confirms that the spectral asymmetry is a good measure of the level of STC as it depends linearly on such that
3.1 Experimental Observation
A simplified sketch of the experimental setup is depicted in Fig. 4 . It is similar to a CROAK, setup , an extension to the imaging cross-correlation technique discussed in [63,64]. Pulses emitted from a Ti:Sa amplifier with a width of 60 fs at 800 nm are split into two parts. The first part is used to pump an OPA which in turn emits pulses at , with a pulse length of 170 fs, which are focused into the central waveguide of a fibre array. The output of which is imaged onto an infrared camera and also onto a thin BBO crystal. The pulses propagate through this crystal collinearly with the second part of the 800 nm pulse, with a relative delay that can be tuned by a motion stage. The crystal is cut and oriented for broadband sum-frequency (SF) generation at . This SF field has a spatial distribution proportional to the time-slice of the wavepacket of the light leaving the array at the time determined by . This delay is tuned to select the time slice of the wavepacket with the peak intensity, thereby acting as a temporal gate, which discriminates all radiation from the SF process, except for a 60 fs window around the centre of the LB, which is much shorter [24,41].
The spatial spectrum of the SF wavepacket is now generated by means of a lens in a 2f setup. The part of the spectrum corresponding to is selected with a slit. The light which passes the slit is then imaged onto an imaging spectrometer, which in turn records the spatiotemporal spectrum , from which the angular mean wavelength and the spectral asymmetry can be calculated and compared to the values calculated above.Results for a fibre array with and a length of for various input power levels are displayed in Fig. 5 . Note that from previous experiments  we know that for this geometry an input energy of is sufficient for the generation of a single LB, whereas is needed to generate two LBs which are not yet fully developed and not temporally separated at this length. The results displayed here are in remarkable agreement to the predictions from above. For low input energies, depicted in Fig. 5(a), there is no measurable spectral symmetry breaking, which is clear because the exciting pulse is spectrally symmetric, and there is not enough input power for the spectrally asymmetric LB to act as a nonlinear attractor. If, however, the input power is increased sufficiently to generate a LB, as can be seen in Figs. 5(b) and 5(c), spatiotemporal asymmetry is observed, as expected. The observed value of coincides well with the expected value of , derived from data taken from Fig. 1(b) and Eq. (11). The weaker asymmetry at might be either due to the stronger redshift at higher input powers, with or the existence of more temporally non-separated dispersive waves, which are spectrally symmetric. If the input energy is increased to be sufficient for the excitation of a second LB, as has been done in Fig. 5(d), no spectral asymmetry is observed, which can be explained by the fact that the second LB is not yet fully developed and not temporally separated.
3.2 Numerical Verification
Verification of this very indirect measurement technique for STC is achieved by means of numerical solution of the set of coupled, unidirectional Maxwell-equations [21,24,41,54–56], including dispersion to all orders and the time-delayed Raman response for fused-silica glass . The wavelength dependence of the waveguide coupling and thus STC can be turned on and off, to allow for direct observation of related effects.
Evaluation of the numerical data is kept as close as possible to the experiment described above. A frequency filter, corresponding to the wavelength response of the sum-frequency crystal is applied, the position with peak power is determined and then filtered in time by means of a Gaussian, with 60 fs temporal with, centered onto the peak. Fourier-transformation in space and time generates the spatiotemporal spectra, of which the spatiotemporal centres of gravity are determined, which are in turn used to calculate the level of spatiotemporal symmetry breaking .Results for a fibre array with are shown Fig. 6 as a function of the propagation length z and the input energy . Subfigure (a) depicts results from the simulation with a realistic level of STC, whereas those in subfigure (b) where determined with STC switched off, by setting . Whereas almost no symmetry breaking of the spatiotemporal spectra is visible for the simulations without STC, a clear symmetry breaking can be observed, if STC is switched on, such that takes values between , which agrees well with the expected value of , predicted by Eq. (11) and a level of taken from Fig. 1(b) for a fibre array with . From this data it also becomes clear that the spectral asymmetry develops together with the LBs, because it is initially zero and starts to form at , which is the approximate propagation length after the initial wavepacket has shaped itself into a LB [24,41]. Note that these values are lower than the values reported in the experiments in section 3 due to the longer diffraction lengths of these samples, but in excellent agreement with the expected value of .
Thus STC is shown to be directly responsible for the breaking of the spectral symmetry of ultrashort wavepackets propagating through waveguide arrays. The symmetry is broken because STC imprints asymmetry onto the spectra of LBs, to which all excitations of sufficient power are attracted. In the next section we will show how this spectral asymmetry influences the LB’s transition into linear waves, which occurs during their decay.
4. Superluminal Decay of LBs
One intriguing property of LBs, which we initially reported together with their first observation , is their unique decay mechanism, which is related to the redshift driven by the stimulated Raman effect, which is induced by the non-instantaneous fraction of the nonlinear response. LBs decay because the strength of the inter-waveguide coupling and the level of dispersion grow rapidly for increasing wavelength, up to a point that the energy of the LB can no further sustain solitonic propagation. Additional details on the decay mechanism and the scalability of LB propagation are discussed in . However, in section 2 we have shown that this growth is the very effect, which is responsible for STC. Thus STC and LB decay are intricately linked and the LB decay will be heavily influenced by STC.To verify the impact of the STC on the LB propagation we investigate the motion of the pulse’s centre of gravity, termed luminality , which is negative for superluminal propagation and positive for subluminal propagation. For a wavepacket evolving according to Eq. (9) it represents the difference between the inverse group velocity of the isolated waveguide and the actual group velocity and is given by24,41]. Indeed, after some straightforward, albeit lengthy, calculation it can be shown that the acceleration due to the first term , so that any acceleration of the pulse centre occurs with a luminality proportional to , thus the acceleration depends entirely on the strength of STC. It can further be shown that and thus the pulse acceleration has only nonlinear contributions, which thus vanishes after the LB has decayed into a low-power linear wave. This behaviour is therefore rooted in the interplay of STC with nonlinearity. A closer analysis of term immediately reveals limiting values, being , whereas the lower limit applies for a mutually out-of-phase field on the edge of the first Brillouin zone and the upper limit applies for a mutually in-phase field as acquired in the centre of the first Brillouin zone.These considerations already give a pretty good idea on the group-velocity evolution of a LB. The LB initially propagates with , with both summands in Eq. (12) being in equilibrium, as long as the wavepacket is a LB. During decay, however, this equilibrium slowly breaks and acceleration, proportional to is possible for a limited time, until power levels have dropped sufficiently to bring the acceleration to a halt, with the wavepacket now propagating at a new luminality . As the LB bifurcates from the top of the band-structure, and thus from the mode in the centre of the first Brillouin zone, we can expect the LB to preferably decay into this mode. Therefore will actually take a value close to, but below .
All these predictions are verified by a simulation displayed in Fig. 7 . Here we propagate a LB solution of Eq. (9) with and an initial energy of , through a waveguide array described by Eq. (9) with an additional linear loss term leading to a reduction of the energy of . Decay of the LB is expected at , because LBs with have an energy threshold of . Figure 7(a) displays the pulse shape in the central waveguide as a function of the propagation distance together with the position of the temporal centre of gravity . The graph can be split into three distinct regions. For the pulse propagates virtually unchanged, as expected for a LB. Close to the point of decay one can observe rapid pulse decay connected with an acceleration of the wavepacket. After the pulse has decayed at the wavepacket again propagates at a constant, albeit increased velocity, undergoing strong diffraction and dispersion. More details of the acceleration process are shown in Fig. 7(b), which displays the luminality , as calculated from the derivative of the centre of gravity curve of Fig. 7(a) as well as by using Eq. (12). Both methods give equal results and display the three regions defined above. Near-constant for , acceleration for and again near-constant luminality for . Also shown is the upper limit of the luminality , which is about two times bigger than the actual one reached by the wavepacket but still a good estimate. Figure 7(c) displays a reduced phase-space trajectory of the wavepacket together with those points in phase-space which correspond to a LB solution, taken from Fig. 2(a). The reduced phase-space is spanned by the pulse energy, and the nonlinear phase shift , which can both be easily measured from the simulation data. During propagation there is a constant reduction of energy, induced by the additional loss term. For , the LB can adapt to this loss by adiabatically reducing its nonlinear phase shift and adapting its field to that of a LB with lower energy, thus the trajectory in phase space remains pinned to the curves of LB solutions. If, however, the energy drops below the threshold energy at contact to the LB curve is lost and immediate decay into linear waves is observed.
Although Eq. (12) and the following discussion can be used to predict the superluminality, it does not give insight into the underlying physical reason of the superluminal velocity of the linear waves. As mentioned before, the key to understand the physics of STC induced superluminal velocities in waveguide arrays is the fact that the LBs bifurcate from the supermode at the centre of the first Brillouin zone, whose propagation properties are defined by Eq. (6), with . There is, however, one important difference between this supermode and any stable LB, which is their degree of localization. The linear supermode is, of course, non-localized, whereas all stable LBs have more than 80% of their energy localized in the central waveguide (see Fig. 2(d)) and thus behave somewhat similar to a pulse propagating through a (nonlinearly) isolated waveguide. Both the Bloch-mode’s group velocity and the waveguide mode’s group velocity are associated with derivatives of their respective propagation constants:22].
If the luminality is rewritten to get the difference of velocities one gets the result
4.1 Experimental Observation
Experimental observation of superluminal propagation is not straightforward. The complete discussion from above only applies for inside of the array and direct measurements of a local propagation velocity with the required accuracy of is unpractical because it would require very fine cut-backs of the array under extremely reproducible excitation conditions.Time-of-flight measurements are, however, feasible. Under the conservative assumption of a relative speedup of during decay and a propagation length of 20 mm after the point of decay one can assume a difference in the time of arrival in the order of 20 fs. Such a quantity is well above the level of accuracy and repeatability of our cross correlation scheme, which is in the order of 1 fs.
However, there is a lack of a reference for such a time-of-flight measurement, because STC cannot be switched off in a sample. We therefore resort to use the dispersive waves propagating through the same sample as a reference. Their spatiotemporal spectra do not undergo nonlinear symmetry breaking and therefore they do not experience speedup. As the LB is concentrated mostly in the central waveguide, whereas the dispersive waves are spread throughout the waveguide array, we measure the delayFig. 4, with an imaging lens and a CCD camera in place of the parts behind the BBO crystal. Note that this quantity is also independent of the pulse length of the reference beam.
Results of such measurements for increasing input energy in a sample with of a length of and are depicted in Fig. 8 . Whereas there is a clear trend for an increasing delay with increasing input energy, which is due to the effect the of the Raman-redshift, there is one obvious difference between the short sample, where the LBs have been barely generated and the long sample where they have already decayed for 10 to 20 mm before the end of the sample. For the short sample the increase of the delay is more or less monotonic. However, it is interrupted by periodic decreases for the longer sample. These ditches have a periodicity of roughly 80 nJ and begin at and . These energy levels have been identified with the excitation of the first and second LB in a previous work . The depths of the ditches have a value of roughly 20 to 30 fs and coincide well with the expectation of the reduction of the delay of roughly 20 fs, mentioned above.
We therefore argue that the observed ditches are an experimental proof for superluminal group velocities acquired during the decay of the LBs and thus a direct consequence of STC. They occur only for energies which are sufficient for the creation of LBs, only for lengths where LB have already decayed, and with a magnitude very close to the expected value.
4.2 Numerical Verification
Numerical verification of superluminal LB decay is based on the simulations discussed in Sec. 3.2. We simulate the propagation of light through a waveguide array with and various input energies, modelling a realistic level of STC behaviour as experienced by a real waveguide array (see Fig. 1(b)) and for a model without STC, thus .
Results are depicted in Fig. 9 . Subfigure (a) displays the delay, as defined in Eq. (16) for sample lengths and according to the ones used in the experiment described above. The curves for the short samples display almost no difference between the two models, which is consistent with the expectation that STC does not lead to superluminal propagation unless the LBs have decayed. However, as expected, there is a strong difference for the longer samples, where the LBs have already decayed. The model with STC is characterised by a reduced average delay, thus increased average speed and displays the periodic ditches observed in the experiment, whereas the curve which corresponds to the model without STC, does not display such ditches. The position of the ditches is, however shifted to and , which is possibly related to an overestimation of the coupling efficiency into the fibre array for the experimental data. The depth of the ditches is roughly 50 fs and therefore somewhat higher than observed in the experiment, but still very close to the expected value.
A more systematic picture of the differences between the delays measured for the two models is given in Fig. 9(b), which displays this difference as a function of the input power and the sample length. It can be clearly seen that both models produce very similar results for lengths up to , which is the point at which LB decay starts to set in . Longer lengths are characterized by an earlier arrival of the wavepackets in the model with STC, thus they have undergone speedup. This speedup is slightly undulating with input power, corresponding to energy dependent differences in the excitation position and number of LBs. If the sample length is chosen to be very long, such as very similar results are again observed. This can be explained by noting that the LBs have then decayed for a very long time and their energy has spread over the complete array. Thus the argumentation which has lead us to the definition of Eq. (16), is no longer valid and is no longer a meaningful measure for superluminal propagation.
We have investigated STC in periodic media. We have found that in periodic media, in contrast to STF in homogeneous ones, STC can be engineered to a desirable strength for considerably longer pulses propagating under small angles. If the modulation contrast of the periodic medium is low STC can be understood as being related to the wavelength dependence of the coupling between the fundamental modes of adjacent unit cells in an array of waveguides. This dependence is tuneable by changing the array geometry.
STC leads to the direct coupling of spatial and temporal effects. LBs are an ideal model system to study this interplay, because they naturally evolve into a state, in which spatial and temporal effects are balanced.
We have shown for the first time that the nonlinear dynamics of LBs leads to STC-induced symmetry breaking of the spatiotemporal spectra of LBs, the strength of which is found to be proportional to the level of STC. We have devised an experimental scheme to measure the level of spatiotemporal asymmetry and shown that the results agree well with analytic predictions, as well as with numerical simulations. We have further shown that numerical simulations without STC do not display spatiotemporal symmetry breaking.
We have also shown for the first time that STC and the spectral asymmetry imprinted by it leads to speedup to superluminal velocities during LB decay. We have derived a simple model for this effect, which is related to the difference in group velocities of the waveguide array’s Bloch-modes, the fundamental waveguide mode, and a nonlinearly induced transition between the two, triggered by the decay of the LBs. We have further derived a simple analytic expression of the maximum observable speedup. Experiments which determine the delay using a cross-correlation setup clearly show signs of superluminal decay for samples which are longer than the decay length of LBs but not for shorter samples. This is in agreement with our expectations and is supported by numerical simulations, which unequivocally link the measurements to STC.
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