1. Introduction

Light transport in random systems has been extensively studied for a few decades, concerning rich fundamental physics and various potential applications [1]. Light scattering is a basic process that light encounters during its transport in a random system and causes light to deviate from its original transport route. The residence time of a photon is elongated by multiple scattering events; as a result, light signal can be amplified during the transport when an optical gain is introduced into the random system. The magnitude of amplification described by a diffusive equation with a gain relies on the scattering strength; and thus lasing oscillation occurs when the light scattering reaches a threshold at which the gain exceeds losses. Such a lasing oscillation is entitled a random laser since it operates based on multiple scattering that occurs in random systems [2]. The scattering strength is evaluated by the scattering mean free path (l _s) defined as an average distance between two consecutive scattering events and simply given by l _s = (ρσ _s)⁻¹, where ρ and σ _s represent the number density and the scattering cross section of scatterers, respectively. In accordance with the relation of l _s to the sample size L and the wavevector k = 2π/λ, where λ is the wavelength of the light, a random system operates in three regimes: (І) localized one known as the regime for Anderson localization of light when kl _s≤1; (ІІ) diffusive one when λ<l _s<L; and (ІІІ) ballistic one when L<l _s. Based on the relationship l _s = (ρσ _s)⁻¹, the l _s is reduced with increasing either ρ or σ _s. Since a large index contrast between scatterers and environment gives rise to a large value of σ _s according to the Mie theory [3], in the last few decades researchers have concentrated on use of high-refractive-index materials as scatterers in order to achieve random lasers. Random lasers have been observed in various systems such as suspensions of nanoparticles [4–7], semiconductor powders or films [8,9], nanoparticle-embedded polymers [10–12], rare-earth ion containing powders [13], liquid crystals [14], conjugated polymer films [15], photonic glasses [16], and so on. With respect to the type of feedback responsible for lasing oscillation, random lasers are classified into coherent (field, or resonant) or incoherent (intensity, or nonresonant) feedback ones. If coherent feedback is predominantly involved, random lasers demonstrate sharp spectral spikes with linewidth less than 1.0 nm superimposed on a broad amplified spontaneous emission (ASE) background. In contrast, the system merely shows narrowing of the emission spectrum to several nanometers above a pump threshold, when lasing is triggered by an incoherent feedback.

The origin of lasing spikes has been discussed extensively since the first observation of coherent random lasers in semiconductor ZnO powders by Cao and associates [8]. They tentatively proposed the optical confinement based on the principle of Anderson localization, which is understandable since kl _s is close to unity for ZnO powders (e.g. l _s≈0.8 λ in Ref. 8 and 0.5 λ in Ref. 9). Nevertheless, lasing spikes have been recently demonstrated in rather weakly scattering systems such as suspensions of nanoparticles and polymers embedded with nanoparticles [17,18], obviously unexplainable in terms of the localization model. Numerical simulations suggested that prelocalized modes [19], absorption induced localization [20], amplified extended modes [21], or intrinsic quasimodes [22], may contribute to the occurrence of the coherent feedback in such weakly scattering systems. Note two points in the above-mentioned weakly scattering systems showing coherent feedback lasing actions; (I) most of them operate in diffusive regimes (e.g. l _s≈12 µm in Ref. 17, l _s: 35 µm~1.07 cm in Ref. 18); (II) the scattering strength was exclusively controlled with the variation of ρ while σ _s is fixed. With regard to the spatial configuration of scatterers relative to the host, random lasing systems are described as either static or dynamic ones, where scatterers are spatially fixed in the former (e.g. nanoparticle-embedded solid systems) while they move ceaselessly in the later (e.g. suspensions of nanoparticles). The ever-examined random systems include both static and dynamic ones, simply by varying the value of ρ. For instance, Wu et al. examined TiO₂ particles-suspended dye solutions by varying ρ of TiO₂ particles, and revealed an optimal ρ in the diffusive regime showing the lowest pump threshold [18]. However, random lasing in static systems covering a wide range of scattering strength has not been examined yet, partially because of toughness in fabrication of such systems.

In this work we have adapted a different route to control the scattering strength of the system from ballistic to diffusive by varying σ _s instead of ρ and examined corresponding lasing oscillation behaviors. This system features static spatial configuration of scatterers extended in three dimensions. It has been revealed experimentally that an intermediate state exists where the coherent feedback for lasing oscillation is inhibited; as a consequence, sharp lasing spikes were not observed. When the scattering strength deviates from this regime, sharp peaks occur at a pump threshold. We believe that the present work is fruitful for the understanding of the fundamental physics in random lasers and will open up a route to design random lasers with a high optical gain.

2. Experimental

The random system used in this work was comprised of a macroporous silica disk immersed in a dye solution. The porous silica monolith was prepared using the sol–gel method accompanied by phase separation as reported previously [23]. In a typical synthetic procedure, a prescribed amount of tetramethoxysilane was added to a 0.01M acetic acid aqueous solution in the presence of poly (ethylene glycol) (molecular weight ~10,000). The mixture was stirred vigorously in an ice bath for 30 min, and the resultant transparent solution was sealed and kept at 40 °C for gelation. The wet gel was aged for 24 h after gelation, and transferred to a stainless-steel autoclave to be kept at 200 °C for 24 h. The gel was then dried and heat treated at 1000°C for 5 h. The resultant porous structure exhibits a sharp pore size distribution because the phase separation and sol–gel transition take place homogeneously under isothermal conditions. The SEM image in Fig. 1(a) shows that the sample is composed of interconnected pores and silica skeletons, with the central pore size of ~3 µm and the porosity of ~0.5. The 1.0 mm-thick silica disk was attached to the interior wall of a polystyrene cuvette (10 × 10 × 40 mm) using polystyrene resin, and then infiltrated with a 5.0 mM solution of rhodamine 6G (R6G). The solvent was composed of methanol and benzyl alcohol mixed in various ratios, allowing for a well control over the index contrast to silica since the refractive index of silica (n _s = 1.459 at λ = 570 nm [24]) is between those of methanol (n _m = 1.334 [25]) and benzyl alcohol (n _b = 1.540 [26]). Taking the volume ratio of methanol as x, the refractive index of the mixture solution (n _bs) was approximately given by n _bs = xn _m + (1-x)n _b, and the index contrast between silica and solvent is given by ∆n = n _s-n _bs. In this way, random lasing systems with various scattering strength were obtained so that the appearance of the immersed porous silica disk ranged from rather opaque to very transparent. Using measurements of coherent backscattering (CBS) the l _s of porous silica was evaluated to be ~8.0 µm. Although we could not quantitatively estimate the value of l _s via CBS when porous silica was immersed in the solution because of rather weak scattering, we can infer that l _s should be much longer than 8.0 µm due to decrement in refractive index mismatch by the solvent infiltration. As ∆n approaches zero, the transparency of the porous silica increases gradually, implying a scattering regime extended from diffusive to the ballistic regime. For instance, the sample with x = 0.398 (Δn = 0.001) looks rather transparent, while x = 0.500 (Δn = 0.022) appears turbid. Therefore, the random systems herein span diffusive and ballistic regimes.

 

Fig. 1 (a) SEM image of the macroporous silica. The central pore size is about 3 µm. (b) Experimental pump-probe scheme for random lasing. The excitation source is a picoseconds Nd³⁺: YAG laser (wavelength: 532 nm, pulse duration: 25 ps, repetition rate: 10 Hz). The backward emission was collected with the fiber bundle into a monochromator and analyzed with a CCD system.

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Figure 1(b) depicts the experimental scheme for emission measurement, where the second-harmonic output of a mode-locked picosecond Nd³⁺: YAG laser (wavelength: 532 nm, pulse duration: 25 ps, repetition rate: 10 Hz) was focused at the front surface of the porous silica disk by a lens of 5.0 cm focal length to excite R6G molecules, and the backward emission was collected with a fiber bundle into a monochromator (SPEX 270M, Jobin Yvon) coupled to a liquid nitrogen cooled charge-couple-device (CCD 3000, Hamamatsu photonics, the spectral resolution was 0.07 nm).

3. Results and discussion

Figure 2(a) depicts the evolution of emission spectra against pump energy for the sample with Δn = 0.001. At low pump energy the sample exhibits a broad spontaneous emission band centered around λ = 570 nm with full width at half maximum (FWHM) of 25 nm, while the FWHM significantly reduces to ~9 nm when the pump energy increases up to 0.7 µJ. As the pump energy increases further to 0.86 µJ, multiple sharp peaks are observed at around λ = 566 nm with FWHM much less than 1.0 nm. The emergence of such sharp peaks indicates the onset of lasing oscillation with coherent feedback. The emission behavior is very different from that for a neat dye solution, wherein only a broad ASE appears even at higher pump energies. The number of sharp peaks increases with the increment of the pump energy, which is a typical characteristics involved in random lasing with coherent feedback. Figure 2(b) plots the integrated emission intensity versus pump energy, clearly showing a threshold behavior around 0.86 µJ where an abrupt increase of emission intensity occurs. If the pump position is changed, the lasing frequencies vary from position to position, as depicted in Fig. 2(c), which is caused by the spatially-random distribution of pores.

 

Fig. 2 (a) Normalized emission spectra obtained at an identical pump position while at various pump energies: E _p = 0.57 (black), 0.70 (red), 1.06 (blue), 1.30 (dark cyan), and 1.97 µJ (magenta). (b) Integrated emission intensity versus E _p. The lasing threshold (E _th) occurs at E _p = 0.86 µJ, as seen in the onset of an abrupt increase. The sample has Δn = 0.001. (c) Normalized emission spectra obtained at various pump positions, while keeping pump energy E _p = 1.30 µJ.

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To examine the effect of Δn on spectral features, we prepared samples having a higher Δn and performed random lasing experiments. When Δn = 0.008 (x = 0.433), sharp peaks are not observed until the pump energy reaches 1.06 µJ. The number of laser modes is obviously less than that of the sample with Δn = 0.001. As Δn increases up to 0.012 (x = 0.450), sharp lasing peaks are not observed even if the pump energy is very high. It is surprising to see that the spectral feature of sharp peaks revivals when Δn is further increased to 0.022 (x = 0.500), while the threshold is ~1.30 µJ, higher than those for the samples with Δn = 0.001 and 0.012. Note that the number of laser modes reaches a maximum for the sample with Δn = 0.001. For clear demonstration, emission spectra recorded at 1.30 µJ are shown in Fig. 3(a) for samples with Δn = 0.008 (black), 0.012 (red), and 0.022 (blue), respectively. The pump energy-dependent emission intensity is plotted in Fig. 3(b), where a threshold transition is not clearly observed for the samples with 0.008, 0.012, and 0.022.

 

Fig. 3 (a) Normalized emission spectra of the samples with Δn = 0.008 (black), 0.012 (red), and 0.022 (blue), respectively. E _p = 1.30 µJ. (b) Integrated emission intensity versus E _p.

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Considerable attention has been paid to the origin of random lasing with coherent feedback in weakly scattering systems both experimentally and theoretically. For instance, Polson et al. revealed that lasing oscillation cavities in conjugated polymer films are formed by closed loop paths due to multiple scattering [17], which is supported by the fact that the cavity length is integral times the scattering mean free path. The cavity length is determined from a Fourier transform of the emission spectrum, which will be discussed below. Yamilov et al. suggested that optical absorption significantly contributes to localization of light via numerical simulation because light absorption limits the number of modes that can lase; as a result, random lasing with coherent feedback can be observed [20]. In that report, a tight focus of the pump beam on the sample surface is essential to the observation of sharp peaks. In our experiment, optical absorption could contribute to the coherent feedback because of the high concentration of R6G molecules, and the sample is pumped by a tightly focused laser beam in order to obtain high pump density. When the excitation spot becomes large, the intensity of featureless background emission increases, while the number of discrete peaks does not drastically vary. Recently, Mujumdar et al. discussed the role of amplified extended modes which experience abnormally long transport distance before escaping the random system [21]. This model involves the amplification of loosely-correlated individual scattering events so that the pulse-to-pulse single-short emission spectra should demonstrate totally chaotic behaviors with each other. We have experimentally measured such single-short spectra, as shown in Fig. 4 . The lasing frequencies of dominant laser modes remain almost the same from spectrum to spectrum for Δn = 0.001, while the mode intensity exhibits fluctuation. This fluctuation probably arises from mode competition since a large number of laser modes could be excited simultaneously at a high pump rate, as reported previously [27]. Meanwhile, the lasing frequencies exhibit strongly chaotic behaviors for Δn = 0.008 and 0.022, respectively. The results indicate that the lasing for Δn = 0.008 and 0.022 probably arises from amplified extended modes, while other mechanisms have to be considered for Δn = 0.001.

 

Fig. 4 Single-shot emission spectra recorded at 1.97 µJ for the samples with Δn = 0.001 (a), 0.008 (b), and 0.022 (c). The arrows marked in (a) denote lasing modes reproduced from pulse to pulse. All the spectra are normalized.

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As shown in previous reports, the power Fourier transform (PFT) spectrum of single-shot emission spectra offer information on lasing oscillation cavity in weakly scattering systems [28]. The lasing oscillation cavity length is related to Fourier components in accordance with the formula L _c = d _mπ/mn, where L _c is the lasing oscillation cavity length, d _m is the Fourier component, m denotes the order of Fourier component, and n is the average refractive index of the system. Figure 5 represents individual single-shot emission spectra ((a), E _p = 1.97µJ) and corresponding PFT spectra (b) for Δn = 0.001, 0.008, and 0.022, respectively. For Δn = 0.001, neither emission spectrum nor PFT exhibits well-defined periodicity. The irregularity could be caused by the fact that there exist a multiple number of oscillation cavities and each discrete spectral peak comes from different cavities. Using the first PFT component d ₁~7.9 µm and n = 1.458, L _c = 17.02 µm is obtained. For Δn = 0.008, the PFT spectrum appears similar to that for Δn = 0.001 although the regularity increases a little. The corresponding L _c can be calculated to be 22.29 µm. As Δn increases further, although obvious periodicity cannot be seen in the emission spectra, the PFT spectra become more regular. This may suggest the reduction of the number of lasing oscillation cavities as Δn increases. The corresponding L _c can be calculated to be 63.86 µm, which is larger than those for Δn = 0.001 and 0.008.

 

Fig. 5 (a) Single-shot emission spectra and (b) corresponding power PFT spectra derived from (a).

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We intend to ascribe the coherent lasing oscillation observed when Δn = 0.001 to naturally occurring microcavities formed by chance. Although the cavity size derived from analysis of PFT spectra in Fig. 5 is much larger than the pore size, the interconnected geometry of the pores makes it possible to form microcavities for lasing oscillation. Possibility of finding a microcavity in weakly scattering media was recently discussed by Tulek et al. [29], who argued that there are two types of disorder in random lasers. The first type is a smooth and long-range disorder that promotes lasing by trapping the stimulated emission, while the second type is rough and short-range one that suppresses lasing by scattering the emitted photons outside the excited area. In the present case, the smooth and long-range disorder may be developed when Δn = 0.001 to support lasing. One possibility is that silica skeletons act as waveguides to trap the stimulated emission because the refractive index of skeletons is higher than that of the solvent. Strong absorption of dye dissolved in solvent could assist the formation of microcavities by limiting the effective sample volume. Such cavities embedded in the sample are responsible for the appearance of less chaotic peaks in emission spectra (Fig. 4(a)). As Δn is increased, light scattering at the interfaces of silica skeletons and pores becomes intense, which can act as second-type disorder that breaks down the microcavity. As the contribution of the second-type disorder becomes dominant, coherent lasing oscillation disappears, which accounts for the absence of sharp peaks at Δn = 0.012. This is also evidenced by the fact that the number of lasing modes for the system with Δn = 0.008 is less than that with Δn = 0.001. As Δn increases further, the revival of the coherent feedback can be assigned to a different mechanism related to light scattering in diffusive regime (e.g. amplified extended modes or short-lived quasimodes). The larger L _c for the system when Δn = 0.022 (68 µm) indicates the longer residue time of photons inside the system because of the multiple scattering.

Our lasing mechanism in a ballistic regime may be relevant to that examined by Kumar and associates [30]. They proposed that the rare sub-diffusive scattering events could significantly contribute to lasing despite of rather weakly scattering strength for an amplifying random medium. The probability of undergoing at least one scattering back in the finite amplifying random medium of size L is given by [1-exp(-L/l _t)], where l _t is a transport mean free path after which the propagation direction of light is randomized. The corresponding gain factor is expressed by exp(L/l _g), where l _g is a gain length defined as a path length after which the intensity of a photon is amplified by a factor of e⁺¹. Therefore, for the amplification to occur, the optical gain given by the product should exceed unity, i.e., [1-exp(-L/l _t)] × exp(L/l _g)>1. At rather large l _t, the optical gain can be much larger than unity and assist a coherent lasing oscillation [28]. Since the gain factor can be large in our system due to the high efficiency of R6G, a photon reflected by the end face of the microcavity can be amplified to a large extent, leading to the coherent feedback random lasing.

4. Conclusions

In summary, we have systematically investigated lasing oscillation properties in porous silica-based systems. Controlling the refractive index of the mixed solvent, we can vary the scattering strength over a wide range. When the refractive index contrast between the solvent and silica is very small, so that the system is in ballistic regime, random lasing with coherent feedback is observed. However, when the refractive index contrast is increased to a critical value, the coherent feedback emission signal vanishes. A further increase in the refractive index contrast results in the revival of the coherent feedback. We suggest that the existence of underlying microcavities plays an important role in the ballistic regime while other mechanism such as amplified extended modes may lead to the coherent feedback in lasing oscillation when the scattering strength increases.