Quantification of lineshape fluctuations in coherent random lasers

Coherent emission is generally associated with resonators, in which light travels back and forth because of reflections between two mirrors. When an amplifying medium is placed between the mirrors, the light that remains inside for the longest duration interacts most with the medium and acquires maximum intensity. The distance between the mirrors determines the resonance wavelength of the laser light. A resonance is ‘hit’ when an integral number of half-wavelengths exactly matches this distance, and the emitted light is coherent. In contrast, random lasing is an exciting optical phenomenon that can generate coherent light in an environment that is structurally disordered.

The amplifying medium routinely used in random lasers is a laser dye dissolved in alcohol. When a homogeneous dye solution is excited, its molecules re-emit the excitation energy through spontaneous fluorescent emission. When scatterers are added to the homogeneous solution, the fluorescent light is scattered within the sample and its exit is delayed. This delay enhances the interaction of the light with other excited dye molecules and realizes stimulated emission and, thus, amplification. Experiments on random lasers have yielded both broad- and narrow-band coherent spectra.

Numerical simulations have uncovered contributions of delayed and, hence, amplified photons in the spectral profile of a random laser. These photons inside the medium grow in intensity for the wavelength with the highest gain coefficient. This happens at the cost of gain at other wavelengths, thus narrowing down the spectrum and yielding a stable emission profile. When the same system is reduced in size, the smooth, stable spectrum transforms into a multipeaked, fluctuating spectrum.^1,2 In this case, only a few photons acquire high intensities and create ultra-narrow-band peaks in the spectrum. Since temporal coherence of a light wave is inversely proportional to its bandwidth, these modes possess a high degree of coherence. The rest of the excited molecules supplement the emission with lower-intensity fluorescence. While one may occasionally observe no fluorescence, that is more of an exception than a rule. To achieve coherent random lasing, relevant physical parameters (such as the gain level, the strength of disorder, or the system's size) must be optimized.

Figure 1 shows outputs from the same system calculated using time-independent Monte Carlo calculations. These reproduce a large set of experimentally observed features. The difference in the calculations was that the sample that generated smooth, broad-band spectra in Figure 1(A) had a larger excited extent, approximately 1cm³. Upon reducing the size to a few thousand cubic micrometers, the excitation is concentrated in this smaller volume. Thus, the gain level is raised and the spectrum yields sharp lines, indicating coherent emission. The small size also prevents averaging of the high-intensity modes. Similar averaging is expected to happen when the excitation energy is provided at a slower rate, which can allow simultaneous fluorescence, leaking the gain out of the system.

Figure 1. Calculated random lasing spectra. The smooth spectrum in (A) transforms into a multipeaked spectrum with coherent modes in (B) upon system-size reduction and increase in gain. Δλ: Line width.

One feature that becomes immediately apparent is that the system is entirely statistical. There is no control over the amount of amplification that a photon experiences. This is entirely decided by the photon's path length, its location in the system, and the gain at that position. These statistics manifest themselves as fluctuations in the spectra. The amount of coherent energy emitted fluctuates in every emission pulse, as does the wavelength of the coherent light and, consequently, the lineshape of the spectrum.

We have now quantified the lineshape fluctuations by correlating the individual with the ensemble-averaged lineshapes.³ We found that smooth, incoherent spectra have a correlation coefficient of nearly unity for all spectra. In comparison, for coherent spectra, the coefficient exhibits a wide distribution (see Figure 2) with a mean at 0.92, with some spectra having a coefficient of less than 0.8. We also found that the larger contribution to the coefficient comes from the relatively broad base seen in the spectra from coherent random lasers, which originates from depleted gain in the system. These fluctuations are the cost of the system's high gain. As a consequence of the latter, there is no a priori information nor control over the frequency of the intense peaks. Thus, the high gain, while responsible for the coherent modes, is also the source of frequency instability. This mars the potential applicability of this unique source of coherent light.

Figure 2. Distribution of the coefficient of correlation between an individual multimode spectrum and the averaged spectrum.

The strong fluctuations in random lasers make it hard to characterize the source or even predict its behavior. For instance, what should be the optimal combination of size, excitation energy, and randomness in the source to achieve a certain amount of coherent emission? Such a question is impossible to answer until the universal, statistical behavior of random lasers has been quantified. This, too, is in contrast to conventional lasers, which are sufficiently well understood theoretically to exactly predict features like lineshape and width, as well as emission intensity. Our continuing experiments are directed to catalog these statistics and provide inputs toward a complete statistical theory of this system. The other problem of interest is the fluctuating frequency of random lasers. We are addressing wavelength fluctuations in random lasers to test whether modifications can be implemented to minimize these variations.

An alternative explanation for the coherent modes is based on random resonators formed in the system.⁴ Regardless of the origin of these modes, the fluctuations born from randomness make the random laser an intriguing subject for further research activity, from the viewpoint of both fundamental and applied physics.