1. Introduction and motivation

Understanding the glass transition is an enduring challenge, attracting continued interest [1, 2, 3]. A supercooled glass will manifest anomalous relaxation, a non-equilibrium condition that can persist in time beyond the duration of any reasonable experiment. The persistent non-equilibrium state has unique properties that can be used to achieve innovative functional materials. For example, glassy systems can have anomalously enhanced responses, such as huge viscosity, a strong history-dependence, and aging, i.e., a continuous evolution even with fixed external conditions. An expanding and promising field is that of using material complexity and glassy response to affect optical linear and nonlinear response [4, 5]. Ferroelectrics, such as LiNbO₃, form one important class of photonic linear and nonlinear materials. Interestingly, dielectric spectroscopy indicates that a wide class of ferroelectric crystals, the so-called relaxors, manifest something that can best be understood as a glassy phase [6, 7]. In a glass, the non-equilibrium occurs between the liquid and crystalline phases, leading to the so-called glass-forming-liquid or glass-former behavior. In a glassy ferroelectric, the non-equilibrium is between the high-symmetry paraelectric and the low-symmetry polar phase, and the resulting system behaves like a dipolar glass. Glass-forming-liquid physics in relaxors is most evident in samples that have relevant compositional disorder, that is, where ions are randomly substituted in the lattice without altering the lattice geometry, for example replacing a Nb with a Ta atom in a single lattice cell of a KNbO₃ crystal. Microscopically, polar-nanoregions (PNRs) form around the compositionally disordered sites and give rise to a complex dipole-dipole interaction that ultimately causes the slowdown in relaxation and glassy behavior.

Non-stoichiometric composition has allowed the growth of variety of new optically transparent relaxors with a room-temperature glassy phase [8]. PNRs introduce a strong local anisotropy in optical response that can be oriented electrically, producing a strong electro-optic effect that affects light propagation[9, 10, 11]. When the samples are further doped with photosensitive impurities, such as Cu atoms in K_1−xLi_xTa_1−yNb_yO₃ (KLTN) [12], these can allow the formation of light-induced electric fields and an ensuing strong photorefractive (PR) nonlinearity. Transparent relaxors have recently opened the way to new optical phenomena, such as scale-free propagation [13, 14] and reprogrammability of optical response [15, 16, 17, 18].

Ferroelectrics with photosensitive impurities at equilibrium support optical PR spatial solitons [19, 20, 21], a specific example of the general class of nonlinear waves that propagate without distorting and changing [22, 23, 24]. Solitons form when material nonlinearity exactly balances linear wave-spreading processes, a fact that fixes their shape, size and intensity to the so-called soliton solution and to the “existence curve” in the wave parameter space.

To date, no study has been reported describing the behavior of solitons in a glassy system undergoing aging. In this paper we detect and study optical solitons in out-of-equilibrium super-cooled compositionally disordered Cu-doped photorefractive KNTN (K_1−xNa_xTa_1−yNb_yO₃). Analyzing the conditions leading to solitons in the glassy system, we find that these change in time without relaxing to a stable time-independent state typical of equilibrium soliton formation. Whereas, in general, this leads to soliton break-up, we also observe that appropriately changing the wave parameters, i.e., normalized width and intensity, allows to compensate crystal relaxation and leads to a stable non-spreading optical waveform. Reconstructing, from these parameters, the equivalent soliton solution in the normalized width-intensity parameter plane indicates that the solution is exploring, in time, the parameter space beyond the constraints of the soliton existence curve, undergoing what can best be termed “soliton aging”.

2. Annealed versus quenched experiments

Our experiments are carried out in a transparent zero-cut 1.17^(x) × 1.90^(y) × 2.43^(z) mm sample with K_0.89Na_0.11Ta_0.63Nb_0.37. In Fig. 1(a), we report the low-frequency dielectric constant ε_r versus temperature T. The Curie-Weiss mean-field law breaks below T_B = 305K (i.e., the Burns temperature which we can identify with the dynamical glass transition temperature) and pronounced hysteresis in the linear response is detected in proximity of T_max = 285K where ε_r has its peak for the cooling trajectory. The Curie temperature is T_C = 275K (fitting data for T > T_B with the Curie-Weiss law ε_r = C/(T − T_C)), and we carry out our experiments in the glass-forming-liquid region (T_max < T < T_B) at the equilibration temperature T_e = 301K.

 

Fig. 1 Conventional PR solitons fail to form. (a) Low frequency normalized dielectric constant ε_r versus T manifesting thermal hysteresis for the KNTN sample. Dark-green regions indicate the paraelectric (PE) and ferroelectric (FE) ergodic (i.e., where no glassy phenomena emerge) phases, the light-green region indicates the crossing region where glassy physics emerges. (b) Steady-state screening solitons in annealed KNTN and insets reporting transverse intensity distributions of the beam at input (z = 0) and at the output (z = L_z) of the sample at different times. (c) Beam evolution in the same conditions of (b) in quenched KNTN, leading to modulation instability (M.I.) and a large variability in the trajectories (shaded region indicates the values observed in 10 successive experiments). (d) Quasi-steady-state solitons in the annealed case compared to the phenomena observed in the (e) quenched case. The slight systematic asymmetry along x in the annealed and quenched cases is caused by nonlocal terms in the response [29].

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The KNTN is also Cu-doped and manifest strong PR response for visible wavelengths. It is biased on the x facets with L_x = 1.17 mm, and its temperature T is controlled through a Peltier-junction. The crystal is initially slowly cooled to the operating temperature T_e with a cooling rate of α_a = 2.5 · 10⁻³K/s (annealed dipolar glass) from the initial T_i = 310K. It is inserted in a standard setup for the observation of one-dimensional (1D) PR screening solitons. Specifically, a Gaussian beam from a doubled-Nd-YAG laser (wavelength λ =532 nm, power before the crystal in the soliton beam 50 μW) is focused down onto the input facet of the sample and diffracts along the z axis, as shown in Fig. 1(b), from the initial minimum beam waist of 8 μm to 27 μm, along the L_z =2.43 mm propagation in the sample (expected Gaussian linear diffraction is 33 μm for n₀ = 2.35). In Figs. 1(b)–1(e) we report one example of observed beam phenomenology. As the voltage is applied, for V = 550V the x–polarized (i.e., parallel to E₀ = V/L_x) beam converges to a soliton after 45 min when a background illumination (a co-propagating plane-wave polarized in the y-direction, orthogonal to external bias field) such that the intensity ratio $u_0^2=I(0)/I_b={\left(5.2\right)}^2$, where I(x) is the beam intensity distribution and I_b the background intensity (Fig. 1(b)). The steady-state (SS) soliton appears in all identical to solitons generally observed in conventional ergodic ferroelectrics [25]. If the background illumination is absent (i.e., $u_0^2\gg 1$), the beam leads to a quasi-steady-state (QSS) soliton, as shown in Fig. 1(d) [26, 27]. In turn, the glassy nature is activated through super-cooling to T_e from T_i. Specifically, in Fig. 1(c) we show the exact same experiment reported in Fig. 1(b) in conditions of cooling rate α_q = 0.07K/s (quenched dipolar glass), and in Fig. 1(e) we report the result for the $u_0^2\gg 1$ case (to be compared to the Fig. 1(d) case). Evidently, the picture is fundamentally altered. In the two conditions in which the annealed case manifests SS and QSS solitons, the quenched beams appear to behave in the same manner. As expected from the general picture of an enhancement of nonlinearity due to aging, both situations lead to a runaway process that culminates and ends with beam break-up typical of modulation instability, a phenomenon normally observed either for extremely long propagation or elevated nonlinearity. Congruently, beam breakup appears on time scales that are longer than the SS soliton formation time and QSS life cycle, so the mechanism at work is intrinsically different to the conventional transient involved in QSS [28].

The impossibility of forming conventional PR solitons in the quenched dipolar glass is a flag to the fact that the nonlinear response of the material is in fact aging and changing the balance between the linear diffraction and nonlinear self-focusing, hampering the formation of a stable soliton. We repeated the experiment modifying the value of static applied voltage V but the picture is not altered in any significant way: no stable SS soliton is observed.

3. Soliton aging

Assuming therefore an aging nonlinear response for the supercooled crystal equilibrated at a fixed T_e, no truly stable soliton can form. However, as the material response changes, we can ideally change adiabatically the soliton waveform to once again satisfy the perfect balance between diffraction and self-focusing. This would form our definition of an aging soliton. In the PR nonlinearity, this soliton aging is even more effective since the spatial scale involved in the soliton formation process is not directly the actual size of the input launched laser beam Δx, but the rescaled size Δξ = Δx/d, where d depends on the applied electric field and low frequency dielectric constant ε_r [25]. This means that solitons have self-similar properties that allow the compensation of the aging nonlinearity without actually altering the optical waveform. In Fig. 2(a) we report the observed output intensity Δx_out compared to the input stabilized in time by applying the V(t) and consequent bias E₀(t) of Fig. 2(b). Assuming that the soliton existence parameters remain, during the process, fixed, we can deduce the equivalent hump in the ε_r caused by the supercooling, as shown in the lower blue curve of Fig. 2(b). The soliton solution is preserved through the transient response of the aging KNTN.

 

Fig. 2 Solitons that age. (a) Output FWHM indicates a stable soliton following supercooling. (b) E₀(t) required to follow the soliton as the susceptibility ages and equivalent ε_r(t). The shaded region indicates the transient hump in response. (c) Self-similar waveforms during the aging. (Center) Annealed existence curve (blue squares) and equivalent existence conditions for the aging KNTN in the quenched case (green asterix points) at different instants of time, in the conditions of stabilized soliton. Insets, top-left, input (red) and output (blue) beam profiles in the linear case (at t = 0); top-right, output profile for a SS soliton in the annealed case in A; right, profiles of the output at t₁, t₂, t₃.

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4. Discussion

We consider a material whose nonlinearity ages because of built-in disorder. Paraxial propagation is described by the generalized nonlinear Schrödinger equation [22]. The slowly varying envelope of the optical beam A(x, z), localized in the x direction and diffracting along the z direction, obeys the nonlinear wave equation

To test this picture, in Fig. 2(c) we report the annealed screening soliton existence curve (blue squares), where each point corresponds to the observation of a stable SS soliton in the annealed KNTN. As the compensated soliton evolves through the aging, the soliton itself wanders in the parameter plane, for example passing through the points indicated with the symbol “*”. These are plotted onto the plane considering the value of V applied at the given t and using the nominal ε_r at the equilibration T_e (as per Fig. 1(a)). The basic existence curve is violated. In turn, the beam profiles are self-similar, indicating that all these points actually refer to one single point in the soliton parameter plane, as the beam profiles of three selected snap-shots after the initial transient (45 min) indicate (t₁ = 55 min, t₂= 85 min, t₃=115 min). Hence, the nonlinearity in action is not that associated with the equilibrium susceptibility, but to an aging ε_r, as discussed in Fig. 2(c), validating the general soliton aging picture.

5. Conclusions

We have experimentally analyzed what happens to solitons when the nonlinearity is relaxing. In supercooled PR transparent relaxors we have identified soliton aging, that is, the formation of a stable optical soliton waveform even as the material ages, supported by a time-dependent rescaling of soliton parameters to allow for a self-similar waveform. In terms of the general properties of soliton-supporting equations, our study sheds light on the interaction between nonlinear waves and complex states of matter.