Some dynamical aspects of fluorescence and lasing have been studied in a dye-doped cholesteric liquid crystal by measuring the response of the material to nanosecond optical pumping. It has been found that as the pumping energy is increased the fluorescence pulse duration decreases, reaching a minimum at the lasing threshold. Above the threshold the temporal profiles are irregular and consist of a set of narrow pulses whose measured duration is limited by the detector risetime (1 ns). The results are interpreted in terms of a recently proposed model [JETP, 118, 822 (2014)] that makes use of rate equations to account for the laser generation in cholesteric liquid crystals. The prediction of such equations for an experimental configuration appropriate for fluorescence lifetime measurements is analyzed.
© 2015 Optical Society of America
Cholesteric liquid crystals (CLC's) are one-dimensional photonic bandgap crystals . Due to this characteristic, CLC's can be used to construct lasers with distributed feedback, just by mixing a small proportion of an organic dye into a cholesteric matrix [2–7]. In fact, CLC lasers have attracted great attention due to their simple fabrication, low threshold and possibility of wavelength tuning [8–17]. Current theoretical understanding of CLC lasers is based on the concept of optical density of states (DOS), which regulates the modification of fluorescence emission in the presence of a photonic bandgap and determines the laser characteristics. Most of the physics of CLC lasers is well established both theoretically and experimentally, but there are still some open questions. For example, it has been shown that the emission intensities of a dye are substantially modified if the dye is dissolved in a CLC compared to the case of an isotropic solvent [5,18,19]. However, fluorescence lifetimes do not suffer appreciable alteration . This lack of modification occurs regardless of the light polarization, sample thickness and wavelength, all of which affect the DOS and, in principle, should have a counterpart in the lifetimes.
Up to now, rather few experimental results have been published on luminescence lifetimes and, in general, about the kinetic aspects of light emission in photonic materials [20–23]. In this paper we present some studies on this respect by examining the response of a CLC laser to nanosecond optical pumping. The temporal profiles of the output pulses have been found to depend greatly on the pumping power, thus giving information about the dynamics of the fluorescence and laser emission.
2. Experimental procedure and results
The CLC was a mixture of the nematic material E7 and a chiral twisting agent (compound 2 in ) in proportions 94.9:5.1 wt %. The dye was dye 4-(dicyanomethylene)-2-methyl-6-(p-dimethylaminostyryl)-4H-pyran (DCM), with a concentration of 1%. The sample cell was made of two glass plates separated a distance L = 10 μm. The material was in the Cano geometry with the helix axis perpendicular to the substrates in the CLC phase. A Nd:YAG laser operating at the second-harmonic frequency (wavelength 532 nm) was used as a pump source. The pulses had a width of 14 ns and a repetition rate of 5 Hz. A lens of 20 cm of focal length focused the laser on the sample.
For the spectroscopic measurements the emission at the sample was collected along the substrate normal by using an optical-fiber spectrometer with a resolution of 0.5 nm. At room temperature (22 °C) the reflection band of the mixture was between 559 nm and 634 nm and the helical pitch was p = 367 nm. When the CLC sample was excited above a threshold of about 2 μJ/pulse, laser emission was produced. The lasing wavelength corresponded to the long-wavelength edge of the stop band at 634 nm [see Fig. 1(a)]. The light was circularly polarized and had the same handedness as the CLC helix. Below this pump energy a fluorescence spectrum was observed, with fringes at the edges outside the gap [Fig. 1(b)]. The highest peak occurs at the lasing wavelength.
For the dynamical measurements the emitted light around the cell normal was collected using a lens of 50 mm of diameter that focused the light on a fast photodiode (rise and fall times 1ns). A filter centered at 634 nm with a narrow bandpass (FWHM 4 nm) was used to avoid the averaging effect of the lifetime in a wider wavelength range. Circularly polarized light with the same handedness as the helix was selected with a circular polarizer. To eliminate artificial pulse broadening during pulse averaging (due to jitter effects) single shot data were taken.
Figure 2 shows the pulse duration as a function of the pumping pulse energy (black dots). For low pumping levels the pulses have similar widths to the excitation pulses (14 ns). As the pumping energy increases, the duration decreases, attaining a minimum of about 2 ns, which corresponds to the laser onset. For energies higher than the laser threshold the width of the emitted pulses shows again a slight increasing trend.
Figure 3 shows some representative pulses for different pumping powers. Well below the threshold intensity, the pulse shape is similar to that of the pumping laser [Fig. 3(a)]. At the threshold, the narrowing of the pulse is limited by the detector time resolution (2 ns). When lasing starts, the pulse shape becomes more irregular, and the output consists of individual bursts of about 2 ns with a fluctuating pattern formation [Fig. 3(b)].
We now turn to interpret these results. For this aim, we will use a recently proposed model for laser generation in CLC's that makes use of kinetic equations for the excited states populations and generated light . The approach is standard in ordinary laser physics and only incorporates the peculiarity of the CLC's through a single parameter τc, the radiation dwelling time in the CLC layer. This parameter is given by
Typically ρ undergoes rapid oscillations as a function of wavelength, with maxima at the edge modes where approximately ρ = (αL/sp)2/2 . Here, α is given by α = Δn/n where Δn is the CLC birefringence and s = 1,2,3... accounts for the successive edge modes. It turns out that the laser threshold is very sensitive to τc, in such a way that, typically, lasing is achieved only for the wavelength corresponding to the absolute maximum of τc . This is the first edge mode of the long wavelength band edge (s = 1).
Under a laser scheme based on three electron bands: ground singlet level, ground triplet level and excited singlet level, with average populations per unit volume n1, n3 and n2, respectively, the rate equations result to be 
Numerical integration of Eqs. (2) permits to study the influence of the optical parameters of the CLC's and dyes on a great variety of laser characteristics. In particular we present here the results for the shape of the emission pulses for several pumping-pulse energies for a pumping source of duration 14 ns. Figure 4 shows the normalized temporal profiles for some representative pumping levels. The parameters used in the simulation are given in Table 1. At low pumping, luminescence duration is close to that of the excitation source [Fig. 4(a)]. As pumping increases the width of the fluorescence pulses decreases. At the threshold the emission is extremely narrow, clearly below 1 ns. For powers above the threshold relaxation oscillations (spiking) take place [Fig. 4(b)], with an increasing number of spikes as the pumping level is raised.
These predictions are in qualitative agreement with the experimental results (Fig. 2 and Fig. 3). At low pumping levels, the emission pulse is similar to that of the pumping laser. The narrowest pulse is obtained at the threshold, and its duration is experimentally limited by the photodiode response time. The observed increasing trend for the pulse widths above the threshold (Fig. 2) is also explained by the photodiode time resolution, since only the envelope of the spikes can be detected and the increasing number of spikes [Fig. 3(b)] produces an increasing apparent width. The irregularity of the profiles observed in this regime is also in accordance with the idea that lasing takes place through spiking. In many solid-state lasers, mechanical and thermal disturbances act to continually re-excite the spiking behavior, which often occurs in an unpredictable-fashion . Here we expect to have a similar phenomenon that must produce the observed output fluctuations.
Although it is not evident how to define a precise duration for pulses like those in Fig. 3(b), we have made a rough evaluation of the widths of the pulses obtained in the simulations for the whole range of pumping energies. Open circles in Fig. 2 are the resulting data. The agreement with the measurements (closed circles) can be considered as satisfactory, especially if one keeps in mind the roughness of the computed widths and the fact that some of the parameters used for the simulations (Table 1) are mere estimates.
Now it is interesting to examine the predictions of the model for some other cases of especial interest. In particular we want to analyze the issue of the fluorescence lifetimes of a dye dissolved in a CLC. As previously mentioned, the fluorescence spectrum of a dye inside a CLC is deeply modified respect to the case of an isotropic solvent. However, as has been mentioned above, it has been experimentally shown  that the kinetic of the fluorescence is essentially unaltered by the CLC photonic structure. The question that now arises is whether the theoretical frame of Eqs. (2) can properly account for these phenomena.
Let us assume that the sample is illuminated with short pulses of low energy. Under these conditions, a small population n20 is generated in the sample. The population then decays exponentially according toEqs. (2)]. This deexitation gives rise to light emission, whose power can be calculated by considering just the last two terms of the right-hand side in the last of Eqs. (2). If for the initial time we take Pe (0) = 0, that equation has an analytical solution
Besides, if τf >> τc, as is the case in practical situations, Eq. (4) simplifies to
Finally, the light power outside the cavity Pout is a fraction of the light power inside the cavity Pe. This fraction is equal to the ratio between the time that light takes to travel along the sample and the dwelling time of the cavity τc. Then we get
Thus, as expected, the fluorescence spectrum of the doped dye is significantly modified in the CLC sample because Pout is proportional to 1/τr, i.e., according to Fermi's golden rule, to ρ [see e.g. Equation (3) of ref .]. On the other hand, the time evolution of the detected light must be a simple exponential with characteristic time τf. Experimentally it is found that τf is not substantially influenced by the CLC . Therefore the sum of the radiative and nonradiative deexitation rates (equal to 1/τf) must remain unaltered. In other words, if an excited molecule in a photonic material has an enhanced radiative deexitation rate it must show a reduced nonradiative deexitation probability and vice versa.
It must be commented that although the photonic material can, in principle, influence the emission kinetics through τc, [see Eq. (4)], deviations from the simple exponential law [Eq. (5)] are unlikely. This would require cavity lifetimes τc comparable to fluorescence lifetimes τf. If we suppose τf = 1 ns, this condition implies a coefficient of distributed losses as small as β = 0.05 cm−1 even if the CLC samples are extremely thick [see Eq. (1)]. The light-scattering coefficient of a nematic is already orders of magnitude higher than this value [29–31].
In summary, we have studied some kinetic aspects of light emission in dye-doped CLC's by measuring the shape of the emitted pulses as a function of the pump energy. For nanosecond pumping the emitted pulse width is narrower than the pump laser, and has a minimum at the laser threshold. Above the threshold the temporal profiles suggest the appearance of relaxation oscillations. The results can be accounted for within a model of coupled rate equations, which incorporates the CLC peculiarities simply through the radiation dwelling time of the cavity τc. Within this model, an alteration of the fluorescence spectrum of the dye is predicted if it is dissolved in a CLC structure. On the other hand, the fluorescence lifetime is still τf because usually τf >> τc. Since according to the experiments of other authors τf is not substantially altered by the CLC cavity, this implies that the corresponding modifications of the radiative and nonradiative dexitation rates must be opposite to each other.
This work is supported by MINECO-FEDER (Project MAT2012-38538-CO3-02) of Spain-UE, and the Basque Government (Project GI/IT-449-10). G.S-E thanks the UPV/EHU for a grant.
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