Emission characteristics of random lasers Ravitej Uppu Under the guidance of Dr. Sushil Mujumdar Abstract We studied emission characteristics of a coherent random laser based on incoherent feedback via nonresonant multiple scattering. The observed spectral line shape fluctuations were quantified in terms of correlations of an individual spectrum with the ensemble-averaged spectrum, which infers the signature of the gain profile of the medium. These correlations are studied in relation to the intensity of the highest coherent modes. A scatter plot of a statistical distribution of intensities is constructed, the centroid of which is used to quantify the coherent emission efficiency of a random laser. This quantification scheme is used to study the behavioral dependence of the coherent emission on the gain parameter.
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Emission characteristics of random lasers
Coherent emission is a generic feature of a resonator. A resonator with an amplifying medium embedded between the mirrors gives us a laser. Random lasing is an interesting phenomenon that can generate coherent light from a system with innate structural disorder through non-resonant multiple scattering. This is in stark contrast to the operational principles of a laser. The genesis of the concept of random lasing is credited to Letokhov [1] who studied the phenomenon of non-resonant feedback of light in a gain medium. He showed that in an open disordered amplifying medium, the mode that lases is characterized by the gain profile of the medium and not the configuration of the disorder. This claim was experimentally verified in a dye-scatterer system under conditions of varying disorder strengths [2]. In the current scenario, random lasing has an ubiquitous occurrence in a large range of materials and sizes [3–6] starting from nano-crystalline powders, to micron-sized nematic crystals, and all the way up to kilometers of fiber. The following study deals with random lasers that are realized in systems with spatially random nanoscopic scatterers embedded in an amplifying medium. The scatterers are dielectric or metallic nano-particulate powders that are chemically inert to the surrounding medium, and participate only in scattering and not in the emission. The amplifying medium is a laser dye, such as Rhodamine. In such a system, when the disorder strength is low enough to forbid localization effects, coherent emission is realized via amplification of extended modes through non-resonant feedback of light [7]. The multiple scattering events enhance the path lengths of the photons and thereby provide an avalanche gain for the spontaneously emitted fluorescence photons. Alternative explanations, based on the formation of random resonances between the scatterers also elucidate this phenomenon [8]. Due to the disorder in the system, strong interaction exists between the modes [9]. Consequently, the system (even when the disorder is static) exhibits several kinds of fluctuations, such as frequency fluctuations [10] and intrinsic intensity fluctuations [11]. In random lasing experiments involving ultra-narrow modes via non-resonant feedback, the coherent emission is necessarily accompanied by incoherent radiation from the random system. The two components are coupled through the gain, and any fluctuations in one are expected to occur in the other as well. Photon statistics experiments on random lasers have indeed demonstrated a superposition of coherent and incoherent components [12]. These fluctuations are directly reflected in the spectral line shape, which show a typical profile consisting of ultra-narrow modes, comprising the coherent energy, residing on an incoherent pedestal arising from the fluorescence of the amplifying medium. We investigate the line shape fluctuations in terms of the correlation of an individual spectrum with the ensemble-averaged spectrum that carries the signature of the gain profile. We relate these fluctuations to the maximum coherent intensity emitted from the system, after separating the incoherent part. We portray the coherent emission efficiency in the radiation through a distribution of ratio of coherent to incoherent intensities. Based on these, we propose a graphical depiction of the intensity fluctuations through a scatter plot [13]. The centroid of this scatter plot serves as a characterization parameter for the laser. The dependence of the spectral line shape on the gain parameter is reported. The coherent emission intensity fluctuations are studied using the aforementioned characterization parameter to quantify the “goodness” of the random laser.
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Emission characteristics of random lasers
Figure 1: Experimental Setup - Lens L1 focuses the pump beam onto the cuvette holding the sample. Lens L2 focuses the collected emission into the spectrometer. (a) shows a schematic of the scatterers with an illustrative extended mode that supports the avalanche gain of a photon generated in the system. The resulting ultra-sharp spectral line is shown in inset (b).
Experimental procedure The experimental setup (shown above) consists of a suspension of nano-particles (ZnO) in a 3.6 mM solution of Rhodamine 6G dissolved in methanol. The concentration of the ZnO particles (d 10 nm) was 3 mg/ml. The mean free path l was calculated to be ∼300 µm, which is very large in comparison to λ. This implies that the system is weakly scattering. This suspension was excited by a focused Nd:YAG laser beam (λ =532.8 nm) with a pulsewidth of ∼30 ps. The diameter of the focal spot was measured using a razor-edge setup to be ∼40 µm. Due to the strong absorption of Rhodamine 6G, the pump beam experiences attenuation over a length of about 500 µm in the system. This gives an effective system size of the order of 10−4 mm3 . The emission was analyzed using a 50 cm focal length spectrograph. 2500 emission pulses were grabbed for analysis. The emission was captured for various pump energies so as to study the excitation dependence of the system.
Spectral line shape fluctuations Figure 2(a) and (b) show two typical spectra captured from the system when excited with a pump energy (Ep ) ∼2.1 µJ. They illustrate the typical line shape of emission with distinct, well-separated ultra-narrow peaks riding on an incoherent pedestal. The emission when resolved using a 1200 lines/mm grating gave a resolution limited coherent peak bandwidth of 0.2 nm. In order to fathom the actual bandwidth of the peaks, the emission was resolved using a 2400 lines/mm grating. Even then, the peak width was resolution limited to ∼0.12 nm. Thus, we could only set a lower limit on the bandwidth of the coherent peaks. The coherence length lc is related to the bandwidth as lc =
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Figure 2: Typical emission spectra from the system showing ultra-narrow peaks. The reconstructed incoherent pedestals are depicted by the red lines in (a) and (b). (c) and (d) are the corresponding coherent components.
From, this the coherence length was calculated to be ∼3 mm, which is only the lower limit. This kind of coherent emission from a disordered system which is a few orders of magnitude smaller than a mm3 is very intriguing, and encouraging for practical applications. To separate out the coherent emission, we used a spline interpolation algorithm involving the valleys inbetween the peaks to reconstruct the incoherent pedestals (shown as red curved in Figure 2(a) and (b)). The pedestals thus constructed were correlated to the gain curve of the amplifying medium, which is constructed by taking the ensemble-average of the captured spectra. These correlations were iteratively maximized by varying the parameters of the peak-finding algorithm that was used to locate the valleys. Clearly, the pedestals are a good fit for the incoherent part as seen from the figures. The coherent emission (shown in Figure 2(c) and (d)) was the result of the separation of the pedestals from the spectra. The line shape fluctuations are quantified as the departure from the ensemble-averaged line shape, which carries the characteristics of gain profile. The quantification is done through the calculation of the correlation coefficient η of the individual spectrum S with the ensemble-averaged spectrum hSi, using the relation P ¯ ¯ (Sm − S)(hSi m − hSi) η=qP m P ¯ 2 )( ¯ 2 ( m (Sm − S) n (hSin − hSi) )
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where m and n index the elements of S and hSi and the overbars denote their respective means. The distribution of the correlation coefficients is shown in Figure 3(a). The mean correlation coefficient is be 0.92, indicating a good correlation. This shows that the signature of the gain profile persists even in coherent random lasing. An independent calculation of correlation between the pedestals and hSi, yielded a mean value of 0.96. Evidently, a substantial contribution to η comes from incoherent pedestal comprising of the fluorescence from the depleted-gain system. In the model of the random laser based on extended modes, the occurrence of intense ultra-sharp peaks depletes the inversion in the system. Stronger the peak, larger is the depletion and hence stronger the suppression of subsequent peaks. This suggests that the spectral profile is sensitive to the largest coherent peak in the spectrum. This is indeed found to be true as show in Figure 3(b), which shows a 4
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Figure 3: (a) Histogram of correlation coefficient (η) of the spectra with the ensemble-averaged spectrum. (b) Scatter plot of η versus the peak intensity suggesting a weaker correlation for spectra with large coherent peaks.
scatter of highest coherent intensity against η of a spectrum. The scatter suggests an anti-correlation between the parameters, asserting the trend that larger the maximal coherent intensity is, larger is the deviation from hSi and hence smaller η. After removal of the incoherent pedestal, the distribution of the intensities of the coherent peaks was investigated (shown in Figure 4). The intensity distribution follows an exponential decay. This can be related to the exponential distribution of the path lengths of photons in an open, non-resonant disordered system with gain [7]. The actual gain coefficient experienced by a certain path depends on the local inversion, which has strong temporal as well as spatial fluctuations due to the dynamics of the photons. This causes the departure from exponential decay, which is also observed in the tail end of the distribution.
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Figure 4: Distribution of the coherent peak intensities after removal of the incoherent component from the spectra.
When the inversion level in the system depletes to an extent that the system cannot support avalanche amplification that results in coherent peaks, the remaining emission assumes the form of fluorescence [7]. There is no control on these dynamics and hence the fraction of the coherent emission. The emission from random laser is intrinsically fluctuating [11]. We observed that even for very small pump energy Ep fluctuations of <6%,
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Figure 5: Intrinsic fluctuations - Three emission spectra captured at the same pump energy of the laser. The inset shows the fluctuations in pump energy. The random laser emission shows strong intensity fluctuations in emission (18%) even for relatively small pump fluctuations (< 6%).
the emission showed surprisingly strong fluctuations of nearly 18%. The spectra from Figure 5 illustrate these fluctuations. Note that the spectra were captured at low resolution using a 600 lines/mm grating to include the green pump beam in the wavelength range, and hence the coherent peaks are concealed. We observe that the spectrum represented by the black curve that has the highest emission has only intermediate pump energy among the three spectra. A distribution of ratio of coherent to incoherent emission intensities was constructed to measure the fraction of coherent energy emitted in each excitation of the system, as shown in Figure 6. The distribution is asymmetric with a mean ratio of 0.12, indicating that, on an average, only about 11% of the emission is coherent. Even though the distribution is asymmetric, the number of spectra on either side of the mean are comparable, but the probability of finding a spectrum with coherent intensity above 11% decays exponentially (as shown by the red curve). The incoherent energy is primarily due to the fluorescence of the gain medium. A choice of amplifying medium with a large gain cross-section and a small fluorescence cross-section might reduce the incoherent part, thus coupling more energy into coherent modes of the random laser. 0.12
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Figure 6: Distribution of the ratio of the coherent(Icoh ) to incoherent intensities(Iincoh ).
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Figure 7: Emission from the system at different pump energies. At very low pump energies (Ep ), 1 µJ, there are no coherent peaks. As Ep is increased, coherent peaks appear. Evidently, the coherence in the emission gets averaged out as Ep is increased further.
Dependence on pump energy We observed that the spectral line shape has a dependence on various system parameters like the pump energy Ep , strength of disorder ls , strength of the amplifying medium lg and the system size. Here, we report on the effect of pump energy on the random laser emission. Figure 7 illustrates an exemplary spectrum captured when the system is pumped at pump energies of 1 µJ, 2.1 µJ, 3.5 µJ and 5.1 µJ. We observe that at a very low value of Ep , the inversion in the system is not sufficient to support avalanche amplification and hence the emission profile follows the fluorescence profile of pure dye. As Ep is increased beyond a certain threshold energy, the spectra feature ultra-narrow peaks. On further increment of Ep , the emission profile tends to get smoothened, with the coherent peaks growing smaller in height and prominence. The width of the spectrum at 5.1 µJ is ∼10 nm, an evidence to the fact that the coherence is lost. The line shape of this spectrum also shows strong resemblance with the amplified fluorescence emission profile of Rhodamine 6G. As discussed earlier, the presence of coherent peaks in the spectrum leads to a departure from the amplified fluorescence line shape, which is reconstructed by averaging a large number of spectra. We measured η = 0.98 at Ep = 5.1 µJ, in comparison to η = 0.89 at Ep = 2.1 µJ. The smoothening is believed to be the consequence of self-averaging that builds up in the system due to the increase in the number of photons that can excite long path modes in the amplifying medium. An ideal scenario of the random laser emission is when the emission is maximally coherent and preferably in a single peak. But, as the results above depict, the coherent random laser shows strong fluctuations in emission intensity. To capture the essence of these fluctuations and to characterize their behavioral dependence on pump energy, we create a scatter plot of the intensity of the strongest coherent mode in the spectrum versus the total emission intensity from the random laser as shown in Figure 8. The black squares represent the centroids of the distributions. The distribution shown in Figure 8(a) corresponds to the regime where the
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Emission characteristics of random lasers
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Figure 8: Scatter plots of maximum coherent intensity versus total emitted intensity from the random laser. Centroids are marked as black squares. (a) Distribution at very low pump energies when the system does not exhibit random lasing. (b) Distribution in the random lasing regime is skewed with a large dispersion into the first quadrant of the axes setup about the centroid of the distribution. (c) and (d) Distribution as the Ep is increased further forces the system to self-average. The distribution tends down to compactify itself as the coherent peaks get washed out.
inversion is very weak to support avalanche amplification. The distribution is very compact and show very small variation about the centroid. The distribution in Figure 8(b) corresponds to the coherent random laser regime. Here, the distribution is skewed with a very large dispersion along both axes. The vertical and horizontal lines about the centroid demarcate four regions of the space, which indicate spectra with different characteristics. The first quadrant, in which the dispersion is stronger, implies large fluctuations of intensities due to the coherent peaks. In this regime, the system shows a “blossoming” in the scatter plot. This blossoming can be considered as a signature for the system tending towards a good random laser, with larger energy channeled into coherent modes. The distributions in Figure 8(c) and (d) correspond to higher pump energies. They get increasingly compact about the centroid as Ep is raised. This compactness is due to the near absence of distinct coherent peaks as a result of self-averaging. Another interesting feature about this compactness is the decrease in fluctuations of total emitted intensity in random lasers. This tells that the self-averaging not only subdues the coherent modes but also pushes the system towards a simple dye-like emission, except for an enhanced gain narrowing due to excess amplification arising from extended paths. This behaviour is well quantified by the shift of the centroid as seen from the coordinates of the centroids in Figure 8. Accordingly, the position of the centroid can be used to characterize the emission regime of a random laser as shown in Figure 9. The motion of the centroid shows a sharp dip when the energy crosses the threshold level. The valley point corresponds to the threshold regime where 8
Emission characteristics of random lasers
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the fluctuations are very strong. As Ep is increased, the system still exhibits strong fluctuations while emitting coherent radiation. The trough portion of the curve represents the “good” random lasing regime of the system. As the system is pumped further, the centroid travels back to higher values because of the subdual of the coherent modes due to self-averaging. In summary, we report statistical studies of the intensity fluctuations in a coherent random laser based on non-resonant feedback. The line shape fluctuations are studied and quantified in terms of their correlations with ensemble-averaged profile which shows that spectra with strong coherent emission have large deviations from ensemble-averaged line shape. A quantification scheme is constructed to gauge the goodness of the coherent emission from the system. This scheme is used to calibrate the coherent emission intensity which shows strong dependence on pump energy. Based on this, we demarcate the emission from the random laser into fluorescing, good random lasing and self-averaged regimes.
Acknowledgements I would like to acknowledge Dr. Sushil Mujumdar for his guidance and constructive criticism during the course of the project. I would also like to extend my gratitude to Mr. Anjani Kumar Tiwari for all the support in the lab and for the discussions which have helped in clarifying many issues.
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REFERENCES
Emission characteristics of random lasers
REFERENCES
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