Spatial solitons to mold random lasers in nematic liquid crystals [Invited]

Dye-doped nematic liquid crystals support random lasing under optical pumping, as well as reorientational optical spatial solitons acting as all-optical waveguides. By synergistically combining these two responses in a collinear pump-soliton geometry, the resulting soliton-enhanced random laser exhibits higher conversion efficiency and better directional properties. After a short account on random lasers and solitons in nematic liquid crystals – nematicons – we describe our experimental results on nematicon-molded random lasers. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
The last decades have witnessed substantial experimental and theoretical progress on random lasing in disordered systems, i. e., cavityless lasing via recurrent scattering [1]. The idea that a highly coherent process -such as laser action -can originate in disordered and diffusive systems has triggered several discussions about the fundamental mechanisms and the coherence of lasers. Nowadays, an overall consensus of possible random-lasing mechanisms has come forward and new challenges and opportunities in the area have been identified. At the same time, spatial optical solitons in nematic liquid crystals -nematicons-have reached a mature level of physical and technical understanding [2]. Hereby, after recalling the main features of reorientational nematicons and summarizing the state-of-the-art in the emerging field of random lasers, we describe how to mold and control the flow of random laser light by employing spatial solitons at a nonresonant wavelength in stimuli responsive complex fluids such as dye-doped nematic liquid crystals. We report our recent findings on demonstrating directional features and modulability of efficient random lasers which exhibit good beam quality and can be angularly steered via externally applied fields. / / 0.2 n n n ⊥ Δ = − ≥ . When extraordinarily-polarized light waves propagate in such dielectric, electric dipoles are induced in the elongated molecules and tend to react (through an orientation-dependent torque) to the electric field E, increasing both the orientation angle θ with respect to the wave-vector k and the corresponding refractive index in positive uniaxials, the change of n e versus θ provides a self-focusing Kerr-like response [4], which is saturable and nonlocal as it extends beyond the disturbance size [5].
If a finite light beam, in the extraordinary polarization, propagates in NLC, light induced reorientation yields a graded-index transverse profile which can confine the beam itself into a non-diffracting optical spatial soliton, so called nematicon [6,7]. Nematicons are stable nonlinear 2D + 1 wavepackets thanks to the NLC nonlocal and saturable responses [4]; they oscillate in amplitude and width as they propagate, and are walking solitons with Poynting with respect to the wave-vector k [8].

Random lasing
Since firstly proposed by Letokhov in 1968 [37], random lasers have been intensely studiedboth theoretically and experimentally -in a plethora of disordered systems (nanopowders, ceramics, liquid crystals, polymers and biological tissues, among others). Letokhov predicted that the combination of recurrent scattering and light amplification would lead to a new form of laser, spurring the creativity of scientists all over the world. The arbitrary walk of light waves inside random media, in fact, results in the confinement and localization of light within the material over distances long enough for optical amplification to overcome losses. While wave propagation in disordered media and light localization were described by P. Anderson with reference to vanishing propagation of electrons due to interference effects under very strong multiple scattering in disordered electronic lattices [38], this concept served as a model for metal-insulator transitions and has been widely adopted in optics to account for wave propagation in random media. In particular, when diffusive photon transport in completely disordered systems satisfies the condition k t ≤ 1 (where k is the magnitude of the local wavevector and  t is the transport mean free path of photons), almost complete localization of light waves occurs, termed strong localization. In this limit, light is brought to a stand still and engenders an inherent and cavity-free feedback mechanism. Conversely, weak optical localization is considered to be a particular case of interference, predicted and observed in random media and in partially ordered systems when k t >1 (Mott-Ioffe-Regel criterion [39,40]). Because of the bosonic nature of the light quanta, the possibility of coherent amplification or coherent absorption (attenuation) of light arises, as these are absent in the electronic (fermionic) case. Indeed, coherent amplification is a non-conservative scattering process where the temporal phase coherence of a wave is preserved despite gain. Active random media have repeatedly proven to be suitable candidates for diffusive laser action, mainly because the resonant feedback in conjunction with multiple scattering eliminates the need for an external cavity as in regular lasers. Light localization and interference effects, which survive multiple scattering, have been invoked to explain the random laser action observed in quite a few exotic and complex active systems [41][42][43][44][45][46][47][48]. Initial experimental investigations on the Letokhov prediction reported strong amplification at the transition frequency of the gain medium, albeit no discrete line narrowing was observed [41,42,49]. Later measurements, performed by tightly focusing the pump beam on smaller disordered systems, allowed observing discrete and narrow-band lasing lines [43,44,[50][51][52][53]. In these experiments, typical gain saturation was observed by analyzing the photon statistics, whereas the emission distribution was found to be random in time and in space. In the search for a mechanism that could explain the origin of these discrete laser peaks [54], various scenarios have been proposed. An early picture suggested that feedback arises from multiple random scattering events leading to the spontaneous formation of closed optical paths within the disordered medium [54,55]. As an alternative to this early picture, loop resonators with large refractive index contrast were proposed and it was found that finite size scatterers could substantially increase the refractive index in these resonators because of disorder correlation [56,57]. Lately, single shot experiments in weakly scattering systems provided another interesting point of view: in fact, the appearance of random spikes suggested that spontaneously emitted photons accumulate gain along extended paths [58,59]. These "providential photons" -generated at a given point in space-were thought to acquire enough gain by returning to the point where they were originated, activating a new lasing mode at each pump shot. Using time resolved spectroscopy, aiming to evaluate the decay rates of the modes, anomalous diffusion was found and confirmed the existence of long-lived modes [60]. More recently, Fallert et al also reported experimental evidence of coexisting extended and localized modes in diffusive systems characterized by L/ t >>1, being L the size of the specimen [61]. Although extended and long lived modes appear to be responsible for random laser action, they can only be obtained in specific configurations and cannot explain the experimental observations in their entirety. The next natural step in order to evaluate the role of extended and localized modes was to study their structure and distribution in disordered systems without gain; when gain is then introduced, the modes with the longest lifetime and lasing threshold become discrete and intense laser lines [62].
Nonetheless, the most debated arguments about random lasing remain the role of interference effects, whether the distinction between diffusive and random lasers is needed and whether coherent feedback is required to produce random lasing. The distinction between diffusive and coherent random lasers is hampered by the fact that multiple scattering can be regarded as an elastic process, thus interference effects are inherent to it. A simplified diffusive model enters the problem when interference effects are averaged out during long excitation pulses compared to the characteristic time of the scattering dynamics. However, narrow-band spectra in random laser materials need be modeled accounting for interference effects. On a different note, coherent feedback is not required for random lasing as optical cavities are not essential for obtaining coherent random laser emission. The characteristic 'Poissonian' photon statistics that characterizes the coherent emission of a laser source is observed when light is first-order (field) and second-order (intensity) coherent. Any mechanism that selects a specific narrow banded mode (for example a band-pass filter) gives rise to first-order coherence, whereas second-order coherence is obtained by gain saturation. A random laser can exhibit coherent emission irrespective of the localization of the modes and the amount of 'coherent' feedback because of the amplification of spontaneously emitted photons by stimulated emission. When the gain is large, the intensity will grow until the gain medium is saturated. In general, this tends to suppress the intensity fluctuations and give rise to second-order coherence. To these extents, after the first studies on micro and nano-powders dissolved in gain media [43,44], a plethora of other experiments followed on ceramic random laser [63], biological tissues and bovine semen [47,64], nano-imprinted DNA [65], conjugated polymers [66], semiconductor polycrystalline films [67], disordered photonic crystals [68], perovskites [69], and thermotropic liquid crystals [70][71][72][73][74][75], among many others. Liquid crystals, in particular, are stimuli responsive complex fluids, turbid in appearance, characterized by a scattering cross section up to six orders of magnitude larger than conventional isotropic fluids [76]. The external stimuli responsiveness of this class of materials makes them very promising for controlling the flow of random laser light. The spontaneous fluctuations of the molecular director in nematic liquid crystals lead to fluctuations in the local dielectric tensor, which is the main responsible for the (recurrent multiple) scattering in these systems, which can also be doped with active molecules to provide amplification.
The main challenges for the development of random lasing applications are: i) the electrical excitation; ii) the directional control of a collimated beam, since the light is usually emitted in an unpredictable way and over a broad range of angles. Meeting the first challenge would allow applications in display and lighting technology, the main issue being the electrical conductance of the used materials because of their intrinsic disorder and oftenporous character; Williams et al performed initial studies of rare-earth-doped oxide powders that can be excited electrically [77]. The directional control is crucial for a wide range of random laser applications, from imaging to security scanning. Various solutions have been tried over the years, including fibers [78][79][80][81], microchannels [82], tailored pump [83], and nanostructures [84].

Random lasing with collinear near-infrared soliton and visible pump
In order to face the limitations about directionality and transverse profile of random laser emission in liquid crystals, we resorted to a configuration encompassing both a reorientational nonlinear response and random laser action under optical pumping; such combination, in which reorientation supports spatial solitons, could provide suitable solutions to the lack of beam-like features by the presence of an optical spatial soliton. At variance with the configuration demonstrated in Ref. [85] with orthogonal wave-vectors for pump and nematicon beams, we used a collinear geometry: a pulsed pump laser at wavelength within the absorption region of the guest-host and a continuous-wave (cw) near-infrared (NIR) source were both injected with wave-vectors along the z-axis of a 100μm thick glass cell containing the soft matter.
In such planar configuration, illustrated in Fig. 1, light propagated in the bulk of a thick sample of dye-doped NLC with optic axis n pre-oriented by mechanical rubbing at π/4 with respect to the input wave-vectors k g // k NIR // z in the principal plane yz [86]. The guest-host was the commercial E7 (Merck) doped with 0.3 wt % Pyrromethene 597 dye (Exciton), with refractive indices || n = 1.71 and n ⊥ = 1.52 for electric field parallel and perpendicular to n, respectively. Fluorescen means of outwhere undesi typical examp ( Fig. 2(a)), a ( Fig. 2(c)); th and wavelength ollinearly with m), both gently beam was abl ption and re-e p-generated fl mission were e ed within the s along the 2 m guided emissio xtraordinary-wa ematicon could at the various w d be filtered propagating in ave ( Fig. 2(b)) ° is apparent.
) an ordinary-wa y-wave beam diffr mW extraordinary off.

Transistor-laser operation
As noted above, when pumping the system with pulses above the lasing threshold, the presence of a collinear NIR-soliton modifies the output spectra through improved photon collection and interactions, as well as modified scattering. This is apparent not only in the characteristics graphed in Fig. 3(c), but also form single-pulse spectra collected at the output. Figure 4 displays several examples of individual RL spectral realizations versus near-infrared soliton power and for three values of input pump energy above threshold. It is clear that, at a given pumping level, a higher-power collinear soliton tends to enhance the random occurrence of sharp and intense lasing peaks over the background consisting of fluorescence and amplified spontaneous emission. Such trend can be interpreted in terms of better confinement of emitted light afforded by nematicon waveguides induced by higher NIR power. Such behavior can be exploited for all-optical modulation of the random laser, i. e., the control of the RL operation by means of low-power (mW) non-resonant input beams collinear with the pump but in the extraordinary-wave polarization. Unlike previous reports where random lasing was modulated via light-induced absorption or heating [91] or via trans-cis light-induced isomerization [92], the soliton control entails the possibility of a non-dissipative NIR-driven transistor random laser, a random trans-laser [93,94]. Figure 5 shows the operation of such random trans-laser when switching on or off a 6 mW NIR soliton. As the soliton is formed, the RL in-out characteristic shifts to a lower pump threshold, promoting either the turn-on of the random laser or its increased efficiency, depending on the bias point represented by the pumping level. This mechanism is illustrated by extracting the peak intensity counts after either averaging over a large number (N = 200) of single-shot spectra or taking the average of the whole spectra. In either case the trans-laser can be switched-on by the presence of a collinear soliton (cases "α" in Fig. 5) or its efficiency substantially enhanced (cases "β" in Fig. 5). Below RL threshold, conversely, the nematicon changes the input-output conversion of the system in a negligible manner. Remarkably, the NIR-driven tr increasing fro from 160 pJ to gating RL e guidedoverlap at e solitonded-wave (d), after Fig. 6(b). emission [93]. The our gueste position Fig. 6(d).  [12]. ong x, the abatically ng vector ence of a ickness is rom 7° to well above om panels in Fig. (7)). S well as pump- As the be displaced in th of measured a plane yz) of th nematicon to reorientation w nonlinear resp  iles (plane xy) of an applied voltag tput location corre c) Calculated (soli voltage (see also [9 he nematicon p ptic reorientatio d, therefore, th g k and half ap δ ≈ 0° but -in r the nematicon y reorient the N mm diameter) on a rotating m eorient at θ = θ ping θ m from nd the walk-o o a reduced eff applied bias. output spectra of R ering for (a) V = 0 NIR power for the y 7°, its outpu = 0 and V = 2V k-off (i. e., obs s worth stressin d guide the R ecause in that l RL emission abo ge of (a) V = 0 V a esponding to the in id blue line) and m 93]). path and steer on illustrated he Poynting v pex angle equa n between -it n-assisted rand NLC optic axi ) with field stre mount. Its rotat θ m in the yz pla −50° to 50°, t off δ from −7° ficiency as sca RL emission well V, (b) V = 0.5 V e nematicon was 5 ut profile gets V in Fig. 8 apply an s aim, we T about 2 th respect eeping the n θ could le the RL emission spot shifted with the yz plane of observation, as shown for the limiting cases in Fig.  9(a-b).
Typical averaged RL spectra, obtained at pumping levels above threshold and with a 7 mW nematicon, are displayed in Fig. 9(c) together with measured and calculated walk-off of the RL-beam versus magnetic field orientation [97]. Spectra acquired for opposite angles θ m are similar, demonstrating that the magnetic reorientation was quite efficient within the NLC volume of interest, despite the initial (boundary defined) anchoring of the optic axis n.  [97]).

Conclusions and future work
The synergy of two nonlinear optical responses, namely self-focusing through reorientation and light-matter interaction for optical amplification, in conjunction with randomness and scattering has been demonstrated in a soliton-aided and soliton-controlled random laser to improve the overall performance and add novel features to this cavityless light source. Such device is able to "beam" random laser emission, improving its conversion efficiency and adding modulation capabilities in transistor-like operation driven by a small non-resonant continuous-wave input. Moreover, the emitted RL beam can be steered at will within the active medium itself, using e. g., an external voltage or magnet.
Several questions, however, remain to be answered and will require extensive future work. A model accounting for anisotropic scattering and feedback in a spatially nonhomogeneous and birefringent gain medium is a formidable theoretical (and numerical) task to be undertaken. On the experimental side, the spectral features of each point within the emitted RL profile need be investigated, both in forward and in backward propagation, addressing the role of mode evolution in the graded-index light-induced channel. In terms of applications, a thorough study of time responses and their optimization when modulating the nematicon-assisted RL with electric or magnetic fields could open perspectives in sensing and speckle-free imaging.

Funding
Academy of Finland, Finland Distinguished Professor (grant no. 282858).