1. Introduction

A random laser (RL) usually consists of an ensemble of scattering centers randomly distributed inside an active material, this provides light confinement for lasing through multiple scattering and results in omnidirectional lasing emission [1,2]. From a practical point of view, RLs are attractive devices for their ease of fabrication and the wide range of materials that can be used, e.g. colloidal suspensions [3], semiconductor powders [4], optical fibers [5], biological tissues [6], liquid crystals [7] and polymers [8]. Due to their peculiar properties, RLs have found different fields of application, e.g. laser based speckle free imaging systems [9], optical tagging [10], lasing action at nanoscale [11] and optical information processing [12].

Two different spectral signatures from RLs have been reported: the “non-resonant” emission, associated to amplitude-only feedback [3], consisting of a smooth single peaked broad emission and the “resonant” emission, where optical phase contribute to lasing feedback, resulting in multiple narrow peaks, randomly positioned in frequency [4]. Different models have been proposed for understanding the origin of such narrow modes, see [13] and references therein. Frequency modes are usually associated to spatial modes which depend to the peculiar random distribution of scattering particles in a given device. Spatial gain distribution and spatial hole burning have been identified as control elements of the emission spectrum [14] and spatial modulation of the pump beam has been used as a tool for tailoring RL emission in frequency [15, 16] and space [17]. By changing the transverse profile of the pump beam it has been possible to observe the transition between “resonant” and “non-resonant” emission from the same laser and the broad spectrum in “non-resonant” emission regime has been attributed to the mutual coupling of many lasing modes rather than amplitude-only feedback [18, 19].

All the previously mentioned results with RLs are based on devices in which gain and scattering centers are embedded in the same element and the optical feedback is spatially distributed through the particle distribution inside the active material. Recently, we proposed a device architecture in which gain and scatterers are spatially separated, by placing the scattering centers on the edges of the active material, and typical spectral signature of a “resonant” RL has been obtained [20, 21]. We refer to this structure as a RL with spatially localized feedback, in order to differentiate it from the commonly diffused RLs, in which the feedback is distributed through thee active volume, as scattering centers and gain are spatially embedded. The evidence that lasing is occurring in the entire system is given by the observation of the same random spectrum from both scattering regions, which act as feedback elements and output couplers, as mirrors in a Fabry-Perot cavity. Considering the scattering elements as equivalent reflectors with arbitrary complex frequency response, the lasing frequency, threshold and slopes have been reproduced in simulations [20] from the well-known round trip condition for lasers [22].

In this manuscript, we extend our experimental and theoretical analysis of RL with spatially localized feedback, as we report both “resonant” and “non-resonant” emission spectra, obtained by varying the device geometry, with given scattering surfaces. Through numerical simulations, lasing frequencies and pump dependent mode intensities are understood as due to the complex frequency responses of the scattering elements which depend on the backscattering area. Strong coupling between adjacent modes is introduced in simulation, reproducing the single peaked spectrum of “non-resonant” emission, when a large number of modes are excited.

The manuscript is organized as follows. In Section 2, we present the experimental results and in Section 3 we describe the theoretical modelling approach. In Section 4, conclusions are drawn.

2. Experimental results

Experimental set-up and sample fabrication are described in detail in [20]. The active material in our devices is a liquid solution of Rhodamine B (0.5% vol. in equal parts of ethanol and ethylene glycol) and the scattering centers are titanium dioxide (TiO₂) polydisperse nanoparticles (Sigma Aldrich 224227) with average size of about 350 nm. Samples are fabricated by dropping a saturated water solution of TiO₂ particles onto a glass substrate, water is then evaporated by heating the sample, leaving a large circular spot (diameter of about 1 cm) of TiO₂ fixed to the glass substrate. A central part of the spot is mechanically removed obtaining an area of the sample free of TiO₂ and delimited by two rough “walls” of TiO₂. The liquid dye solution is dropped (10 µL) onto the sample, filling the region free of TiO₂ and infiltrating the areas with deposited TiO₂. The thickness of the TiO₂ walls and the gain region is 50 µm. An image of the sample is given in Fig. 1(a).

 

Fig. 1 Sample pumped with W = 780 µm (a) with outlined in white the geometries with minimum and maximum widths. Experimental set-up (b), details are given in the text.

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The experimental set-up is shown in Fig. 1(b). The optical pump is a frequency doubled (532 nm) Nd:YAG pulsed pump with 10 ns Full Width Half Maximum (FWHM) pulse duration at 10 Hz (Litron NanoT250). A beam expander (BE) is used in order to shine with uniform intensity a Spatial Light Modulator (SLM, Holoeye LCR-1080). Amplitude modulation of the pump beam profile is obtained by selection of an active mask of pixels of the SLM screen. By placing before and after the SLM two polarization dichroic mirrors (M₁, M₂) with orthogonal axis and imposing variable gray level to the SLM, beam shape and intensity can be arbitrarily tuned. White pixels rotate 90 degrees the incident light polarization and black pixels let the polarization unchanged [23]. After the SLM, the beam size is reduced (x0.15) with a pair of convex lenses (f₁, f₂) and directed to the sample.

By using an imaging lens (IL) and a beam splitter (BS), the sample is imaged onto a CCD camera and a movable optical fiber termination (diameter 105 µm) connected to our spectrometer (SPEC, Andor Shamrock 303). A longpass edge filter (F) is used to discard residual pump light. The optical fiber termination operates as a spatial filter that can be moved on the sample along the vertical and horizontal directions, by two computer controlled translation stages.

Different lasing geometries are obtained by varying the width W of the pumped area while keeping the length L constant, with L = 2.4 mm. The length of the pumped region is kept lower than the distance d between the two TiO₂ walls, with d = 2.6 mm, so that the residual dye infiltrated into the TiO₂ region is not pumped. The width is varied between 85 µm and 780 µm and the emission spectra are measured as a function of the pump energy. Results are shown in Figs. 2(a)–2(h). We observe “resonant” and “non-resonant” emission regimes, depending on the width W. For widths below 195 µm, narrow modes with sub-nanometer FWHM are randomly placed in the entire gain spectral region. For widths greater than 195 µm, modes are placed around the gain peak, the wings of the spectrum become smooth and as width is further increased, the emission finally consists of a single peaked spectrum with its maximum at the gain peak.

 

Fig. 2 Spectra obtained as a function of the pump flux with L = 3.5 mm and different values of W: W = 85 µm (a), W = 98 µm (b), W = 110 µm (c), W = 150 µm (d), W = 195 µm (e), W = 250 µm (f), W = 520 µm (g), W = 780 µm (h). The full set of measurements is shown as a sequence in Visualization 1.

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For W < 195 µm, all obtained spectra are characterized by distant modes in the entire gain window. In this regime, as W is increased from 72 µm to 195 µm, more modes appear in the spectrum and modes lasing for small widths are still present at larger widths, however with different thresholds and slopes. As an example, consider the peak observed at 608 nm and 610 nm for W = 110 µm and W = 150 µm, shown in Figs. 2(c) and 2(d), respectively. For W = 110 µm, emission at 608 nm is weaker than at 610 nm, for W = 150 µm, this situation is reversed: mode at 608 nm has lower threshold and higher slope than mode at 610 nm. This general behavior is observed at other wavelengths with a random redistribution of balance between gain and loss at each width.

For W > 195 µm, the spectral range of emission is reduced. For W = 98 µm, see Fig. 2(b), lasing modes are distributed in a wider window of about 10 nm. For W = 250 µm, see Fig. 2(f), modes are densely distributed in a 5 nm window around 607 nm. At large values of W, narrow and well separated frequency peaks disappear and a single smooth profile is measured.

Emission detected for W = 780 µm is compared with the ASE emission with the same pumped area of the active device. We fabricated a sample consisting of a large drop of the dye solution used as active material and free of scatterers. We pumped a rectangular region of the sample with dimension L = 2.4 mm and W = 780 µm near the drop edge so that deflected light was collected by the spectrometer. In Fig. 3, we show the measured ASE and the lasing emission of the device with W = 780 µm. In both case, the pump flux is 50 pJ/µm². We observe that the two obtained spectral profiles differ in shape and the FWHM of the lasing emission for W = 780 µm, equal to 3.3 nm, is four times smaller than the FWHM of the ASE emission, equal to 14 nm.

 

Fig. 3 Spectra obtained for lasing device with L = 3.5 mm and W = 780 µm (solid line) and ASE experiment with same geometry (dashed line). Pump flux is 50 pJ/µm².

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3. Theoretical model and numerical simulations

The single device is understood as working on the principle described in [20], where the scattering elements (i.e. TiO₂ walls) are modeled with arbitrarily shaped amplitude and phase spectral response. From the resulting round trip condition [22], lasing modes are randomly distributed in frequency and losses are frequency dependent.

Here we expand this theoretical approach, introducing (i) the variation of the spectral responses of the scattering elements with device geometry and (ii) including the non-linear interaction between modes, as modelled in the theoretical framework of coupled mode theory [24] for RLs [18,19].

In our structures, the ASE flux exits the pumped region, hits the scattering agglomerations of TiO₂ and is backscattered in all directions. Some frequencies inside the gain spectral window close a phase matched loop, after traveling back and forward the active area, and result in lasing modes. If the active area is increased in width, a larger region of the backscattering elements (TiO₂ walls) is illuminated and more backscattering paths (i.e. more phase delays) are available, thus increasing the probability that more frequencies can close the loop and lase.

This vision based on straight trajectories among scattering centers is useful for an intuitive understanding of phase condition in which an integer number of wavelength periods fits the trajectory length. However, both amplitude and phase responses are spatially distributed and spatially dependent properties which depend on the porosity and roughness of the TiO₂ walls. In our model, we translate the spatial complexity of the problem to the frequency domain: each frequency will see a different reflection coefficient as the illuminated area is changed and will be able to travel on a larger number of backscattering paths (more phase values available at each frequency) as this area is increased. In this way, the effect of the roughness and porosity of the TiO₂ walls is expressed by the numerically constructed amplitude and phase profiles of their frequency responses.

In simulations, we introduce the integer number n, with 1≤n≤N, for representing the step increment of width W in experiments. In devices which share a portion of the backscattering reflectors, allowed frequencies for W_n-1 will be available for W_n = W_n-1 + ΔW, with ΔW an increment in width, but with a different balance of gain and losses.

For each simulated device, the complex frequency responses of left and right backscattering reflectors are expressed in the form R(ν)·exp(iϕ(ν)), where R(ν) and ϕ(ν) are the amplitude and phase spectral profiles, ν the frequency variable and i the imaginary unit.

We construct the amplitude responses of left and right reflectors of the device with width W_n as R_n(ν) = 0.5·(R_n-1(ν) + R_RAND(ν)), where R_n-1(ν) is the amplitude response of the device with W = W_n-1 and R_RAND(ν) is an arbitrarily shaped contribution associated to the width increment ΔW. In this way, the reflectivities of consecutive widths are correlated through the spectral contribution R_RAND(ν), which represents the spectral properties of ΔW. R_RAND(ν) is numerically constructed from the sum of 40 Gaussian curves with 0.6 nm FWHM which are randomly distributed and free to overlap in a spectral range of 10 nm, as in [20]. Maximum and minimum reflection values are imposed by rescaling the obtained profile. By varying the number of curves and the FWHM, smooth or abrupt profiles can be obtained and qualitatively the same results are observed in both cases. The amplitude profile of the smallest width considered, W₁, is constructed following this procedure.

The same numerical approach is used for building the arbitrarily shaped phase profiles, for each device width. Additionally, the possibility that phase delays from small devices comprised in larger ones are still present in the latter is granted by the following provision of the model. First, N different phase profiles are constructed as ϕ₁(ν), ϕ₂(ν) .. ϕ_N(ν), corresponding to N nested devices of increasing width; then, for the n-th device, the available phase values at a given frequency ν₀ are given by the ensemble of all previously constructed phases evaluated at that frequency, i.e. Φ_n = {ϕ₁(ν₀), ϕ₂(ν₀) ... ϕ_n(ν₀)}.In this way, the fact that Φ_n ⊂ Φ_{n + 1} ensures that all phase delays allowed for a given device are also allowed for a device that contains it.

Introducing the previously defined amplitude and phase spectral profiles into the round trip equation [22], we obtain the conditions for lasing in amplitude:

and phase:

In simulations, we construct a frequency vector spanning 18 nm centered at 600 nm with a resolution of 1.5 pm. Amplitude values range between 0.1 and 0.8 and phase values are set between –π and π with a resolution of 1 mrad.

Simulations are performed for N = 20 and results are shown for n = 1, 5, 10 and 20 in Figs. 4(a)–4(d). These values of n are arbitrarily chosen and correspond to incremental variation of the width W in experiments. As n is increased, an increasing number of allowed modes is obtained, varying from 6 modes for n = 1 to 176 modes for n = 20. Modes lasing for n-1 are still present for n, as expected from Eq. (2). Total losses are calculate as the right side of Eq. (1) and are plotted in Figs. 4(a)–4(d). As a consequence of the arbitrarily built amplitude profiles of R₁ and R₂ for each n, losses change with n and their variation around the mean term decrease as n increase due to the average operation at each increment of n. From the results in Fig. 4, the following behavior is expected when gain is added to simulations: modes that are observed at small values of n (small widths) are also available at greater values of n (greater widths) although with different thresholds and slopes due to the varying losses profile. If N tends to infinity, one would expect that all the available frequencies will be allowed to lase with the resulting emission consisting of a single peaked spectrum with its maximum at the gain peak (as frequency dependent losses average with n) and with the same FWHM of the gain window. In experiments, we observed a single peaked spectrum centered at the gain maximum but with a FWHM about 4 times lower than the gain spectral region with the same pumping level and geometry, obtained with ASE measurements.

 

Fig. 4 Total losses (black lines, left axis) and allowed modes (red lines, right axis) calculated for n = 1 (a), 5 (b), 10 (c) and 20 (d).

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In RL with distributed feedback, this spectral signature has been understood as due to the mutual coupling of a large number of modes that lock in phase, resulting in a collapsed spectrum [18,19]. In our structure, modes overlap spatially into the gain area and into the porous TiO₂ walls: mode interaction and competition are treated theoretically within the coupled mode theory approach [24], in which the temporal evolution of frequency modes is described through a set of coupled differential equations.

In our simulations, for each device with width W_n, we consider:

where a_k(t) is the k^th mode, placed at frequency (2π)⁻¹ω_k, with losses α_k, c_j,k are the coupling coefficients with the other j modes, g(t,ω_k) is the time dependent gain (i.e. the pump in experiments) at frequency (2π)⁻¹ ω_k, γ_k are the gain saturation coefficients and t is the time variable. Gain saturation in Eq. (3) is considered as not affected by cross saturation of overlapping spatial modes, as we limit our analysis to the time-frequency domain. The key element in Eq. (3) is the term including the coupling coefficients, in which the real and imaginary parts of all a_j(t), i. e. their intensity and frequency contents, mutually interact, changing the emission spectrum with respect to the uncoupled case.

Allowed frequencies for different device widths are chosen so that modes active for small widths are still present at greater widths, as expressed in Eq. (2). In numerically implementing Eq. (3), we limit the maximum number of simulated modes to 40 in order to lower the simulation speed required.

Modal losses α_k are assigned to each mode following the spectral behavior depicted in Figs. 4(a)–4(d): as the device width is increased, loss spectral profile varies randomly and tends to smear out.

The coupling coefficient matrix is constructed by assigning decreasing values with mode distance, so that adjacent and distant modes are strongly and poorly coupled, respectively. Coupling coefficient for strongly coupled adjacent modes is set to 1.77·10⁻³ then it linearly decreases towards the most distant mode with constant slope for all modes. Gain has gaussian spectral and temporal intensity profiles: pulse duration and gain bandwidth are set in accordance with experimental conditions.

We consider four simulated devices in which 5, 10, 20 and 40 modes are allowed to lase. We solve Eq. (3) for each device and for increasing values of the gain. The spectrum of each mode is obtained after Fourier transforming. In Fig. 5, we show the total intensity spectrum of emission obtained with the four devices considered.

 

Fig. 5 Simulation results obtained after solving Eq. (3), for different devices with 5 (a), 10 (b), 20 (c) and 40 (d) modes.

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In Fig. 5(a), the five active modes lase with different threshold and slopes given by their particular losses. In Fig. 5(b), the same modes are observed together with the new lasing modes available for this configuration, the loss-gain equilibrium is varied and accordingly threshold and slope of each mode. For 20 lasing modes, in Fig. 5(c), we observe in the emission spectrum only five intense peak, around the central frequency (i.e. the gain maximum). Each allowed frequency strongly interacts with its neighbors and adjacent modes collapse at frequency where the gain/losses balance is favorable. For higher n (i.e. greater W in experiments), this occurs around the gain peak, due to the averaging of losses. This strong coupling is not verified when lasing frequencies are well separated, lowering the strength of modal interaction as it happens when few modes are activated as in Figs. 5(a) and 5(b).

In Fig. 5(d), the density of excited modes is further increased resulting in emission centered at the gain peak frequency and a spectral bandwidth which is about 7 times smaller than the FWHM of gain in simulations. This is qualitatively in agreement with our experimental results for the largest width considered, where the recorded lasing linewidth is of 3.3 nm and the ASE FWHM is of 14 nm. The difference ratio of band narrowing is attributed to the greater number of modes involved in experiments, with respect to the 40 modes in simulation.

We also compared the results shown in Figs. 5(a)–5(d) with the results obtained when c_j,k = 0. We observed no relevant differences for the case of 5 and 10 modes, this confirms that distant modes barely interact together. For 20 and 40 modes, all allowed modes lase at their frequencies, filling the gain window and resulting in a FWHM of lasing emission equal to the FWHM of the gain curve.

4. Conclusions

We observed the emission regimes of RLs with localized feedback as a function of the illuminated surface of the scattering elements. Spectra consisting of narrow modes randomly distributed in frequency are obtained for small illuminated areas. A single peaked smooth profile is measured for device with large back-scattering areas. The emission spectra have been measured as a function of the pumping energy, for each device geometry considered. The number of lasing modes increases as the illuminated back-scattering area is increased and their threshold and slopes vary with the illuminated area. This is understood as due to the increasing number of back-scattering paths available and to the frequency dependent losses which vary with the back-scattering region considered. When many modes are allowed to lase, strong coupling between adjacent modes in frequency takes place, resulting in a single peaked emission centered at the gain peak.

We propose a theoretical interpretation of the observed behavior by modelling the back-scattering reflectors with arbitrarily shaped spectral profiles in amplitude and phase. Coupling mode theory is employed for accounting mutual interactions between spectrally adjacent modes. A good agreement between experiments and numerical simulations is obtained.

This is, as far as we know, the first theoretical treatment of the emission regimes of RLs with spatially localized feedback, confirming a clear similarity with commonly employed RLs with distributed feedback. In our vision, the reported results are useful both from a practical and a theoretical point of view, as we demonstrate experimentally that the emission regime can be tuned by varying the device geometry and the proposed model is a useful tool for the understanding of random lasing devices.