Formation of Bright Solitons from Wave Packets with Repulsive Nonlinearity

Formation of bright envelope solitons from wave packets with a repulsive nonlinearity was observed for the first time. The experiments used surface spin-wave packets in magnetic yttrium iron garnet (YIG) thin film strips. When the wave packets are narrow and have low power, they undergo self-broadening during the propagation. When the wave packets are relatively wide or their power is relatively high, they can experience self-narrowing or even evolve into bright solitons. The experimental results were reproduced by numerical simulations based on a modified nonlinear Schr\"odinger equation model.

observes the broadening of the spin wave packet during its propagation. At certain large input pulse widths and high power levels, however, the spin wave packet undergoes self-narrowing and evolves into a bright envelope soliton.
The formation of this soliton is contradictory to the prediction of the standard NLSE model, but was reproduced by numerical simulations with a modified NLSE model that took into account damping and saturable nonlinearity. Figure 1 shows representative data on the formation of bright solitons from surface spin-wave packets. Graph (a) shows the experimental configuration. The YIG film strip was cut from a 5.6-mm-thick (111) YIG wafer grown on a gadolinium gallium garnet substrate. The strip was 30 mm long and 2 mm wide. The magnetic field was set to 910 Oe. The input and output transducers were 50-mm-wide striplines and were 6.3 mm apart. The input microwave pulses had a carrier frequency of 4.51 GHz. Note that, in Fig. 1 and other figures as well as the discussions below, Pin denotes the nominal microwave pulse power applied to the input transducer, tin denotes the half-power width of the input microwave pulse, Pout is the power of the output signal, and tout represents the half-power width of the output pulse. In Fig. 1, graphs ( Pin andtin values, as indicated. The circles in (d) shows a fit to the hyperbolic secant squared function. 1,3 Graph (e) shows the corresponding phase (q) profile of the signal shown in (d). Here, the profile shows the phase relative to a reference continuous wave whose frequency equals to the carrier frequency of the input microwave. 6 Graph (h) shows the change of tout with Pin for a fixed tin, as indicated, while graph (i) shows the change of tout with tin for a fixed Pin, as indicated.
The data in Fig. 1  These results indicate that the spin-wave packet experiences strong self-narrowing when it is relatively broad. (3) The pulses shown in (d) and (g) are indeed bright solitons. As shown representatively in (d) and (e), they have a hyperbolic secant shape and a constant phase profile at their centers, which are the two key signatures of bright solitons. 1,6 The data from Fig. 1 clearly demonstrate the formation of bright solitons from surface spin-wave packets when the energy of the initial signals (the product of Pin and tin) is beyond a certain level. This result is contradictory to the predictions of the NLSE model. One possible argument is that the width of the YIG strip might play a role in the observed formation of bright solitons. To rule out this possibility, similar measurements were carried out with an YIG strip that is an order of magnitude narrower. The main data are as follows.  Fig. 1, the data here were measured by a 50-W inductive probe, 12 rather than a secondary microstrip transducer. The distance between the input transducer and the inductive probe was about 2.6 mm. The magnetic field was set to 1120 Oe. The input microwave pulse had a carrier frequency of 5.07 GHz.
The data in Fig. 2 show results very similar to those shown in Fig. 1. Specifically, the low-power, narrow spinwave packets undergo self-broadening, as shown in (b), (c), (f), and (h); as the power and width are increased to certain levels, the spin-wave packets experience self-narrowing, as shown in (h) and (i), and can also evolve into solitons, as shown in (d), (e), and (g). Therefore, the data in Fig. 2 clearly confirm the results from Fig. 1. This indicates that the formation of solitons reported here is not due to any effects associated with the YIG strip width. Note that the solitons shown in Fig. 2 are narrower than those shown in Fig. 1. This difference results mainly from the fact that the spinwave amplitudes and dispersion properties were different in the two experiments. The spin-wave dispersion differed in the two experiments because the magnetic fields were different and the wave numbers of the excited spin-wave modes were also not the same.
Turn now to the spatial formation of solitons from surface spin-wave packets. Figure 3 shows representative data.
Graph (a) gives the profile of an input signal. The power and carrier frequency of the input signal were 700 mW and 5.07 GHz, respectively. Graphs (b)-(f) give the corresponding output signals measured with the same experimental configuration as depicted in Fig. 2(a). The signals were measured by placing the inductive probe at different distances (x) from the input transducer, as indicated. The red curves in (b)-(f) are the corresponding phase profiles.
The data in Fig. 3 show the spatial evolution of a spin-wave packet. At x=1.1 mm, the packet has a width similar to that of the input pulse. As the packet propagates to x=2.1 mm, it develops into a soliton, which is not only much narrower than both the initial pulse and the packet at x=1.1 mm but also has a constant phase at its center portion, as shown in (c). At x=2.6 mm, the packet has a lower amplitude due to the magnetic damping but still maintains its solitonic nature, as shown in (d). As the packet continues to propagate further, it loses its solitonic properties and undergoes self-broadening, as shown in (e) and (f), due to significant reduction in amplitude. Note that the phase profiles for all the signals in (b), (e), and (f) are not constant. These results support the above-drawn conclusion, namely, that it is possible to produce a bright soliton from a surface spin-wave packet.
The data in Fig. 3 also indicate the other two important results. (1) The development of a soliton takes a certain distance, about 2 mm for the above-cited conditions, due to the fact that the nonlinearity effect needs a certain propagation distance to develop. (2) The soliton exists only in a relatively short range, about 1-2 mm for the abovecited conditions, due to the damping of carrier spin waves. To increase the "life" distance or lifetime of a spin-wave soliton, one can take advantage of parametric pumping 13 or active feedback 9 techniques.
As mentioned above, the soliton formation presented here is contradictory to the standard NLSE model. However, it can be reproduced by numerical simulations based on the equation where u is the amplitude of a spin-wave packet, x and t are spatial and temporal coordinates, respectively, vg is the group velocity, h is the damping coefficient, D is the dispersion coefficient, and N and S are the cubic and quintic nonlinearity coefficients, respectively. The quantic nonlinearity term is included because the cubic nonlinearity is insufficient to capture the experimental observations presented above. This additional term is an expansion to the lowest order of saturable nonlinearity. The simulations used the split-step method to solve the derivative terms with respect to x and used the Runge-Kutta method to solve the equation with the rest of the terms. 14,15 A high-order Gaussian profile was taken in simulations for the input pulse because it is much closer to the experimental situation than a squared pulse. The use of a square pulse as in the input pulse gave rise to numerical noise due to the discontinuity at the pulse's edges. The use of a fundamental Gaussian function did not onsiderably change the simulation results. It should be noted that both the standard and modified NLSE models are for nonlinear waves in one-dimensional (1D) systems, and previous work had demonstrated the feasibility of using the 1D NLSE models to describe nonlinear spin waves in quasi-1D YIG film strips. 16 The profiles in Fig. 4 indicate that, at low initial power, the pulse is broader than the initial pulse and has a phase profile which is not constant at the pulse center, as shown in (a) and (b); at relatively high power, however, the pulse is not only significantly narrower than the initial pulse but also has a constant phase across its center portion, as shown in (c). These results agree with the experimental results presented above.