1. Introduction

Random lasers are unique optical sources that generate narrowband radiation via the interplay of amplification and multiple scattering within a disordered environment [1–7]. Such systems exhibit various fascinating optical phenomena [1]. Furthermore, the fact that these sources can be controlled using electric fields [8], optical fields [9], or even temperature [10,11] assert their potential as externally controllable optical sources. Amongst the various features of a random laser, the most interesting behavior is exhibited by the lasing wavelength. Diffusive random lasers emit stable radiation at a fixed wavelength predicted by the gain maximum [3, 12]. In coherent random lasers based on Anderson localization, the eigenfrequency of the localized mode determines the lasing wavelengths [13, 14]. In a weakly disordered medium with coherent feedback, the cavity length associated with a two-scatterer cavity determines the resonant modes [15]. In a coherent random laser based on nonresonant feedback, however, the spontaneous emission events that excite amplified extended modes decide the lasing wavelength [16]. This implies, and has been demonstrated, that the peak wavelength fluctuates chaotically with every pulse [17]. Obviously, such fluctuations in wavelength present a major impediment in consideration of random lasers in potential applications. It is desirable that there is some control on the emission wavelength. Control is not fathomable in localization-based lasers because the frequency is decided by a mesoscopic resonance dictated by the random placement of scatterers. Even in two-scatterer cavities in weak systems, the cavity length is determined by the extent of the gain region in the random laser, which is not of well-defined length. In the case of nonresonant random lasers, however, the peak wavelength depends on two parameters, the spontaneous emission event and the lifetime of the corresponding extended mode excited by this photon. While the former is hard to control in a multiply scattering environment, the latter parameter offers a possibility of choosing the wavelength of lasing if the lifetime of the mode could somehow be controlled.

That diffusion of light can be influenced by resonant scatterers has been demonstrated experimentally in passive systems [18, 19]. Even in active systems, it was recently experimentally demonstrated that the lasing wavelength of a diffusive random laser can be tuned by employing resonant monodisperse scatterers [20]. It is of interest to examine a parallel situation in the regime of coherent random lasing. In this paper, we numerically investigate the emission properties of a coherent random laser when the scatterers are assumed spherical and monodisperse. We observe significant spectral changes in the emission. We find that such a system shows reduced wavelength fluctuations, in comparison to a conventional system. The random laser persistently emits coherent modes in a wavelength interval determined by the Mie resonance of the single scatterer wavelengths. The width of this range is determined by the quality factor of the underlying resonance. Although multimode emission continues to occur, the modes in the resonant interval overwhelm those in the vicinity of the maximum gain coefficient at all excitation energies. By changing the size of the scatterers, we demonstrate a tunability of the lasing interval across the flat emission band. Thus, a coherent random laser based on nonresonant feedback with deterministic lasing wavelength is demonstrated. Thus, we propose the exploitation of resonant scatterers as a possible handle on controlling the wavelength fluctuations.

2. System

Theoretically, nonresonant random lasers have been treated using light diffusion, which yields valuable information on the threshold behavior of these systems [21, 22], or their temporal evolution [23]. To reproduce actual spectral features including the occurrence of ultranarrow lasing modes, random walks of photons through the medium have to be studied. An access to individual modes is provided by a Monte Carlo simulation that computes photon propagation through the medium via a random walk, with deterministic and accurate scattering parameters [24]. This method is, thus, sensitive to the properties of the single scatterer, and hence ideally placed to study the effect of resonant scatterers. We study the random walk of numerous such photons in the medium, and reconstruct the spectral behavior of such a system.

The numerical simulation was modeled to emulate a system comprising spherical scatterers suspended randomly in a box (10 mm×10 mm×1 mm) containing a 5 mM solution of Rhodamine-6G in methanol. The simulation consists of two parts: excitation and the subsequent re-emission. During the excitation phase, a bunch of pump photons (wavelength 532 nm) undergoes a random walk through the sample after entering the front face of the box in a 60 μm diameter spot, simulating a laser beam. Essentially, each random walk comprises a set of rectilinear paths, the lengths of which are determined by the number density and scattering cross-section of the scatterers. At the end of each path, a scattering event is assumed that propels the photons into another direction. During the propagation process, the photon bunch experiences attenuation due to the absorption cross-section of the ground state molecules. The process continues till the bunch either exits the box, or is absorbed in the sample. The excitation energy determines the number of such photon bunches to be simulated. This excitation leads to local population inversion which is recorded as a function of depth. Azimuthal symmetry around the sample normal is assumed to simulate experiments. The second stage considered the process of fluorescence followed by diffusion-mediated amplification. The spatial inversion profile, along with the normalized fluorescence profile of the dye, was used to select the position and wavelength of a spontaneous emission photon. This photon was then allowed to perform a similar three dimensional random walk through the active disordered medium. During the walk, the photon incurred either stimulated emission and or possible reabsorption depending on the local inversion. After every scattering event, the local concentration N₀(r⃗) and N₁(r⃗) of the dye molecules in the ground and excited state, respectively, were recorded. This took into account the local population dynamics and hence the gain competition between different modes. After complete de-excitation of the sample, spectra were constructed from the recorded values of the number of emitted photons at various wavelengths, maintaining a spectral resolution of 0.1 nm.

In these calculations, the random laser is characterized by the transport mean free path ℓ and the gain length ℓ_g. These parameters are macroscopic in nature and parameterize the overall scattering and amplifying properties of the sample. In conventional samples, the individual scatterers (dielectric powders) are arbitrarily shaped and sized, and all single particle effects are averaged out. The average description does away with any wavelength dependence in the multiple scattering process. The only wavelength dependence then originates from the gain length, which is determined by the gain profile. For amplifying systems such as laser dyes, the gain profile is usually 40–50 nm wide. In the random laser, coherent modes appear over a wide range, determined by a combination of the wavelengths with high gain, and the spectral width of the detector. Figure 1 shows this behavior from an experimental sample, comprising randomly shaped, polydisperse ZnO scatterers in a solution of Rhodamine 6G in methanol (concentration 5 mM). Panel [A] exemplifies two characteristic experimental spectra, which exhibit coherent modes that differ in wavelength. In Panel [B], we show the histogram of wavelengths of these coherent peaks. A continuous distribution is seen with a width of about 10 nm, corresponding to the central flat region of the gain profile where the gain cross-section of the dye is maximum. The right panels indicate the corresponding data for theoretically computed spectra, using the photon random walks described above. In these simulations, the system was assumed to have arbitrarily shaped nonresonant scatterers, which meant that a uniform transport mean free path was assumed for all wavelengths. Clearly, the spectral features and the distribution of the lasing wavelengths is reproduced by the calculations. The larger width in the simulations originate from the fact that the simulations study the entire emission band, while in the experimental setup, the observable band is limited by the spectral range of the spectrometer.

 

Fig. 1 (Color online.) [A]: Two exemplary spectra showing coherent modes from a conventional random laser. Inset: Schematic of the random lasing sample signifying arbitrarily shaped and sized scatterers. [B]: Continuous distribution of the wavelength of the lasing modes observed over 100 spectra. [C] Two spectra calculated from the Monte Carlo simulation, showing coherent modes at random positions. [D] The histogram of the wavelengths of calculated peaks, over 40 spectra.

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We invoked resonant effects via the transport mean free path ℓ, which is calculated as

 

Fig. 2 (Color online.) Q_sca(λ) for TiO₂ microsphere of diameter 1.14 μm immersed in Methanol is shown as the black curves. The broken blue line shows the corresponding transport mean free path ℓ of a suspension of these spheres. The normalized gain cross-section (σ_emm(λ)) of the dye is shown superimposed in red.

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3. Results and analysis

As a consequence of shorter mean free paths, the resonant photons diffuse slowly through the random medium. This wavelength sensitivity is reflected in the pathlength distribution P(l) of photons. In the following analysis, we study the random laser with resonant scatterers in comparison to that with polydisperse nonresonant scatterers, the latter assumed to have no wavelength sensitivity. The behavior at two wavelengths, namely, the wavelength of maximum gain cross-section (λ_max), and the resonant wavelength (λ_res) for the resonant scatterers, is discussed. Figure 3(a) summarizes P(l) in the nonresonant-scatterers-random-laser (henceforth described as NSRL). No discernible variation is observed in P(l) for the photons at λ_max (black curve) and λ_res (red curve) in the plot. Figure 3(b) depicts the same for the resonant-scatterers-random-laser (RSRL). In this plot, the smaller ℓ for λ_max is seen to manifest as a slower decay, consistent with slower diffusion in the stronger disorder. Furthermore, an increased probability of longer paths for λ_max is noticeable in the tail of the distribution. Figures 3(c) and 3(d) capture the behavior of emitted intensity at λ_max and λ_res in the above scenarios. In the NSRL (Fig. 3(c)), the intensity contribution at λ_max is clearly larger. This can be traced to the obvious difference in the number of fluorescent photons and the gain cross-section. Figure 3(d), on the other hand, shows that the red curve overwhelms the black at all path lengths. The tail is seen to have a stronger component of the resonant intensity, while the smaller pathlengths also contribute a larger intensity at λ_res. Thus, the wavelength-sensitivity of resonant scatterers overcomes the dominance of photons at the maximum emission cross-section, despite their higher probability of occurrence and larger gain cross-section.

 

Fig. 3 (Color online.) The pathlength distribution P(l) of photons in the random medium for (a) the NSRL (nonresonant scatterers random laser) and (b) the RSRL (resonant scatterers random laser) system. The black line represents the photons generated at maximum of the fluorescence cross-section (λ_max), and the red line indicates the photons at the resonant wavelength λ_res. (b) shows a mildly weaker decay for the resonant photons, indicating a slower diffusion. (c) and (d) depict the intensity contributed by the photons to the emission spectrum as a function of their pathlengths, for (c) the NSRL system and (d) the RSRL system, with the color-code as earlier. For the NSRL system, the intensity contribution from the photons at λ_max dominates, whereas, for the RSRL system, the resonant photons’ contribution is larger over the entire range of pathlengths.

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In a coherent random laser system with nonresonant feedback, the photons in the tail region seen in Fig. 3(d) create the rare events that realize intense coherent peaks, while the photons with smaller paths contribute to the incoherent pedestal. The consequence of the wavelength-sensitivity on the emission spectrum is shown in Fig. 4, which depicts comparative spectra at an excitation energy of 7.2 μJ. The top row illustrates the variation in ℓ as a function of wavelength. The left column of plots describes the NSRL, which show a multitude of coherent modes distributed over the emission band. The spectra from the RSRL (right column) show the existence of at least one coherent peak within the interval of λ_res, demarcated with vertical dotted lines. The interval width is about 4 nm. Within this interval, the coherent peaks continue to show the characteristic frequency and intensity fluctuations that are associated with coherent random lasing.

 

Fig. 4 (a) and (b) denote the transport mean free path ℓ for the NSRL and the RSRL system, respectively. (c) and (d) depict corresponding emission spectra at four different shots at an excitation energy E_p= 7.2 μJ. The black vertical lines highlight the resonance that persistently yields a coherent mode.

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An important observation from the spectra is that the number of coherent peaks in the RSRL is smaller. We carried out quantitative analysis of the modal density of the final spectral output in the two cases. The total emission band is approximately 25 nm in width. The modal density in this band does not appreciably differ in the two systems upto an excitation energy of 7 μJ. Thereafter, the modal density of the NSRL increases faster than the resonant system. In comparison, a contradictory behavior was observed within the persistence interval. In this case, the modal density in the persistence interval is larger in the case of resonant scatterers. These observations indicate the influence of gain competition in the resonant system. Evidently, a larger number of modes is emitted in the persistence interval, which suppresses the modes outside of the interval, due to which the overall density is lowered. Effectively, the wavelength fluctuations in the random laser are also diminished when resonant scatterers are used.

To quantify the persistence of coherent modes in the wavelength interval, we calculate the ratio (r) of the maximum intensity in the resonant interval (I_res) to the spectral maximum (I_max) for various spectra. The behavior of the ensemble mean of this ratio (〈r〉) as a function of excitation energy is shown in Fig. 5(a). The mean ratio was calculated over 40 spectra. The squares indicate the magnitude of the ratio while the vertical error bars indicate the standard deviation in peak intensity. The NSRL system shows the ratio decreasing systematically, as the spectral maximum grows with excitation energy. The RSRL system, on the contrary, starts to show an enhancement of this ratio at an excitation energy of 3.7 μJ. Importantly, this energy also corresponds to the threshold excitation energy of the random laser, which implies that the influence of resonant scatterers on the spectra occurs immediately after the system crosses threshold. Subsequently, the enhancement of the ratio grows with excitation energy, as is seen in the tail of Fig. 5(a). Figures 5(b) and 5(c) illustrate this enhancement in the ensemble-averaged spectra. In conventional random lasers based on nonresonant feedback, ensemble-averaging leads to a smoothened spectrum correlated with the gain profile [25]. This behavior is observed in the NSRL (left spectra) at all excitation energies. The RSRL system (right spectra) shows the peak created by the resonant light in the ensemble-averaged spectrum. The peak grows with excitation energy, while the rest of the spectrum smoothens out. The survival of the peak after averaging is evidence to the fact that the modes seen within the persistence interval are not sheer happenstance, and are a consequence of the Mie resonance in each scatterer. Thus, we observe that a random laser can be forced to persistently emit coherent modes within a desired wavelength interval, by exploiting single particle resonances. Furthermore, the scheme of using resonant scatterers can lead to characterization of random lasers using their emission wavelength band, a feature not possible with conventional systems.

 

Fig. 5 (a) Variation of the ensemble-averaged ratio (〈r〉 = I_res/I_max) with excitation energy ranging from 1.5 μJ to 8.2 μJ for the NSRL (black squares) and the RSRL system (red circles). (b) and (c) depict the ensemble-averaged spectra from the two systems at the said excitation energies. The persistence interval for the RSRL system is prominent on averaging.

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It is desirable to consider a situation where the Mie resonance of a scatterer overlaps the maximal gain cross-section of the amplifying medium. This can be achieved by employing the right scatterer with the appropriate size parameter. In Fig. 6, we discuss tunability of the persistence interval by changing the scatterer size, which alters the wavelength and the quality factor of the resonance. Three ensemble-averaged spectra are depicted, for scatterer diameters 1.09 μm, 1.17 μm and 1.3 μm. Clearly, the persistence interval is seen to tune with varying scatterer diameters. Notably, the enhancement in the intensity for diameter 1.17 μm is not the maximum despite it being located closest to the gain maximum, precisely because the participating Mie resonance has a lower quality factor compared to the other diameters. Nonetheless, a tunability of about 10 nm in the persistence interval is observed.

 

Fig. 6 Black, red and blue lines represent the ensemble-averaged spectra from the RSRL system for scatterer diameters of 1.09, 1.17 and 1.30 μm respectively. The persistence interval could be tuned over a range of ∼10 nm within this small range of scatterer diameters.

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4. Conclusions

In summary, we have studied the influence of resonant scatterers on the emission from a coherent random laser based on nonresonant feedback. We find that the system preferentially emits lasing modes within a persistence interval of wavelengths corresponding to a Mie resonance. Since persistent emission occurs in this interval, the number of modes occurring over the entire emission band is reduced, thus reducing the wavelength fluctuations that have always been associated with random lasers. By choosing the scatterers of the appropriate size, the persistence interval can be tuned across the gain profile. Accordingly, a tunable system can be achieved that enables coherent random laser emission that is stabilized to a narrow band of wavelengths. Such random lasers can be characterized with their emission wavelength interval. Such an improvement can boost the application potential of the coherent random laser. The important parameter for consideration here is the Q-factor of the underlying Mie resonance. Accordingly, very narrow resonances can be exploited for experiments which can offset the broadening effects brought about by the polydispersity in experimental samples, which is reported to range from 2% (Polystyrene) [20] to ±3% (Titania) [26] in spherical scatterers. Scatterers with size parameters larger than those considered here can also offer high-Q resonances. On a different note, the recently demonstrated gain engineering technique [27] can be employed to match the gain profile of the medium to a desired, high-Q Mie resonance. Finally, it would also be of interest to investigate whether and how resonant scatterers influence the phenomenon of random lasing with coherent feedback, wherein, resonant cavities formed between two scatterers are exploited for lasing. In such a system, equi-spaced lasing modes are observed in the spectrum. An interesting question is whether, in this equi-spaced set of modes, a particular mode is preferred over others when it matches the Mie resonance of the scatterer. Such questions make the phenomenon of random lasing using resonant scatterers an interesting candidate for further research and applications.