1. INTRODUCTION

In our daily lives, multiple light scattering from disordered nanoparticles or structures is omnipresent, for instance, in milk, fog, or the insect-like white beetle [1]. When multiple light scattering becomes intense enough, light can be efficiently trapped within disordered media to form a virtual cavity. Then, a mirrorless laser, termed as a random laser (RL), was first theoretically proposed by Letokhov [2] to integrate recurrent light scattering events in a system with an active medium such as laser dyes [3,4], zinc oxide [5,6], and organic–inorganic halide perovskite [7]. As the scattered light propagates in a path to form a closed loop, a high degree of light confinement will lead to an interference effect and cause standing wave patterns. Thus, various narrow spikes would emerge above the broadband spontaneous emission spectrum to illustrate spatial resonance when the gain exceeds the loss. This is one of the characteristics of RLs resulting from coherent feedback. Unlike conventional lasers, RLs possess peculiar characteristics such as a low lasing threshold, multiple emission lasing wavelengths, a broad solid angle emission, and low spatial coherence [8] that can be applied to biomedical diagnosis [9] and low speckle noise light sources [10].

Many disordered materials or structures, including suspended nanoparticles [11,12], powder [13], polymer [14], or even biological tissues [9,15], have been adopted as an active medium or host to induce multiple light scattering and successfully produce RLs. However, the modulation of fundamental characteristics for these kinds of RLs, such as the output polarization and lasing wavelengths, is not easily achieved. Because of intrinsic birefringence and modulation possibilities, liquid crystals (LCs) have been widely used in optoelectronic devices including displays, phase or amplitude modulators, and sensors [16]. For instance, through polymer stabilized LCs inside wedge cells [17], speckle noise from laser projected images can be greatly reduced to produce a low speckle contrast of approximately 0.025. In addition, the wavelength tunability of band-edge lasing from dye-doped cholesteric LCs (DD-CLCs), with one-dimensional photonic crystal structures, has been demonstrated through external signals such as electric fields [18], magnetic fields, mechanical stress [19–21], and temperature [22,23].

Other researchers have applied nematic liquid crystals (NLCs) and polymer dispersed LCs as scattering materials, which were filled inside the capillary tube [4], hollow core fibers [4], and a glass cell [24,25] for RL generation. Through thermal control [3,4,25], the manipulation of emission wavelengths and the output intensity of dye-doped LC (DD-LC) RLs have been demonstrated. Because of the complicated mechanism regarding the unpredictable multiple light scattering and the interaction between light and nanoparticles in DD-LCs, these RLs possess abundant dynamics and are sensitive to the geometric shape of the filling containers. For example, linearly polarized light from a DD-LC laser in a wedge cell has been produced, and the polarization direction can be manipulated by the rubbing direction of coated polyimide (PI) on the glass plates [25,26]. Through the frequency alternation of the electric field, switchable characteristics between RLs and band-edge lasing have been reported in DD-CLCs with a negative dielectric anisotropy [24].

Unlike conventional laser systems, it is difficult to define the threshold from the regular to the characteristics of RLs. Recently, Lévy statistics defined by the power law has attracted great attention. It has been verified that a random lasing system with nonresonant feedback should exhibit transitions of Lévy and Gaussian intensity statistics through the variation of physical parameters such as particle size, mean free path [27], and excitation energy [28]. Generally, the output characteristics of RLs were related to the extent of light scattering within the disordering medium that can be quantitatively defined by the transport mean free path and measured by the coherent backscattering (CBS) technique [29,30].

In this paper, we investigate the intriguing characteristics of DD-LC lasers that can be related to the configuration of glass cells and rubbed PI on glass plates. To distinguish the operational state and underlying statistical behavior of an RL in a DD-LC laser, the $\alpha$-stable distribution was adopted to fit the entire intensity distribution. We also demonstrated the light scattering in a disordered LC mixture through the transport mean free path by the measurement of CBS. In addition, reduction of speckle noise is also an important issue in laser projection that has been investigated from different output plates of DD-LC lasers.

2. SAMPLE PREPARATION AND EXPERIMENTAL SETUP

The DD-LC mixtures were prepared by doping 0.5 wt.% pyrromethene 597 (PM597, Exciton Inc.) as a gain medium into the NLCs (MDA-981602, $n_e=1.7779$, $n_o=1.5113$, clearing point at 109°C, Merck Inc.) as the main scattering material. The DD-LCs were continuously stirred in a small vessel to produce uniformly distributed mixtures and heated to ensure that LCs were in the isotropic phase. Then the produced LC mixtures were filled into two cells that comprised two separated indium tin oxide (ITO) glass plates with parallel (Sample-I) and wedge (Sample-II) configurations, respectively, as shown in Figs. 1(a) and 1(b). For the parallel cell [Fig. 1(a)], the separation between two ITO glasses, with antiparallel rubbing PI, was about 110 μm. For the wedge cell [Fig. 1(b)], the two ITO glass plates were separated by plastic spacers with two different sizes of about 60 and 170 μm at both ends. Only one side of the ITO glass (R-plate) was coated with PI for Sample-II as shown in Fig. 1(b).

 

Fig. 1. Side view of alignment of LCs in (a) normal cell which comprised two parallel glass plates with coating of antiparallel rubbing PI on both of them, and (b) wedge cell which comprised two nonparallel glass plates with only one side coating of rubbing PI on the R-plate. The wedge angle is 0.31°. The experimental setup for the generation of RL in (c) Sample-I, and (d) Sample-II. The pictures shown in the inset of (c) and (d) reveal the projected pattern from the output beam of FPL and RL.

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The schematic setups to generate random lasing from two DD-LC cells are shown in Figs. 1(c) and 1(d). The DD-LC cells were optically pumped by a linearly polarized and frequency-doubling $Q$-switched Nd:YAG laser with a central wavelength of 532 nm. The output pulses from the pump source revealed a 10 Hz repetition rate and a 2.9 ns pulse duration. After beam expansion, the pump beam was focused by a lens with a focal length of 15 cm. The estimated spot size on the LC cell was around 10 μm. The emission spectrum of the DD-LC laser was collected by a fiber tip and then measured by a spectrometer (HR-4000, Ocean Optics Inc.) with a resolution of around 0.3 nm.

3. RESULT AND DISCUSSION

In order to confirm the alignment of the LCs inside the cell, polarization optical microscopy (POM) with a cross polarizer was used to measure the produced samples as shown in Figs. 2(a)–2(c). In this measurement, the samples were rotated to obtain the POM image at different angles ($\theta =0°$, 15°, 30°, and 45°), which are defined as the included angles between the rubbing direction on one substrate of produced sample and the orientation of the input polarizer. It is obvious that the contrast of the POM image between $\theta =0°$ and 15° is relatively large for Sample-I [Fig. 2(a)] which reveals the ordering alignment of LCs inside the cell. Owing to the geometric shape of the wedge cell from R-plate of Sample-II, the image contrast between $\theta =0°$ and 15° decreases in Fig. 2(b). Because of no rubbing PI on the un-rubbed plate (UR-plate) on Sample-II, the alignment of the LCs is relatively random, and the image contrast between all the measured angles is unapparent in Fig. 2(c).

 

Fig. 2. POM images of DD-LCs from (a) parallel cell and (b) R-plate as well as (c) UR-plate of the wedge cell. The emission spectrum of the DD-LC laser reveals (d) FP characteristic from Sample-I and random lasing characteristic from (e) R-plate, and (f) UR-plate of Sample-II, respectively. The inset shows the polar plots of the output spectrum component from each DD-LC cell.

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Through the pump of a $Q$-switched laser [Figs. 1(c) and 1(d)], the output spectra of the DD-LC laser from two produced cells of the same thickness (110 μm) are revealed in Figs. 2(d)–2(f). Due to the Fabry–Perot (FP) effect resulting from two parallel ITO glass plates in Sample-I, the regular interference fringe with a relatively large contrast is shown in Fig. 2(d). The picture of the projected mode pattern from Sample-I [upper side of Fig. 1(c)] reveals an obvious interference fringe due to the FP configuration. Nevertheless, the disordering alignment of the LCs inside the cell is increased in the wedge cell to induce recurrent light scattering and coherent feedback. As the pump pulse incidence from the UR-plate of Sample-II, the output from two plates (R-plate and UR-plate) reveals the characteristic of the RL, showing irregular emission spikes on the top of the pedestal from the measured spectrum as shown in Figs. 2(e) and 2(f). Besides, the interference fringes of the projected mode patterns from two plates of Sample-II [upper and lower sides of Fig. 1(d)] became unapparent. In comparison with the R-plate, the amplitude fluctuation and aperiodicity of the emission spikes from the UR-plate of Sample-II became obvious, as shown in Fig. 2(f). This is attributed to the increase in the disordering alignment of the birefringent LCs near the UR-plate without the restriction of rubbing PI. Therefore, multiple light scattering would efficiently enhance and cause the projected mode pattern [lower side of Fig. 1(d)] to become more distorted.

Scientists are so far not only curious to demonstrate the occurrence of RLs in various kinds of nanostructures but are also interested in controlling their output characteristics such as polarization. In a previous report [25], cross linearly polarized light from dye-doped twist nematic LCs has been demonstrated, which depends on the rubbing PI on the output plate. It is recognized that the orientation of the transition dipole moment of the dye molecules would follow the parallel alignment of the local nematic director inside the wedge cell. To confirm the output polarization from the two sides of the DD-LC lasers, a linear polarizer (LP) was placed between the DD-LC cell and the fiber tip as shown in the experimental setup in Figs. 1(c) and 1(d). Through a rotation of the LP behind the LC cell, the spectrum component at a specific angle $\theta$ relative to the $y$ axis was successfully measured.

The polar plots of integrated intensity from the emission spectrum of the DD-LC laser as a function of the angle $\theta$ are also revealed in the inset of Figs. 2(d)–2(f). For Sample-I and the R-plate of Sample-II, the spectrum components parallel to the rubbing direction of PI onto the glass plate ($\theta =90°$ and $\theta =270°$) reveal the largest intensity as shown in the insets of Figs. 2(d) and 2(e). As the included angle between the direction of the LP and the rubbing direction increased, the measured intensity gradually decreased, as shown in the polar plots. The orthogonal spectrum components relative to the PI rubbing direction ($\theta =0°$ and $\theta =180°$) exhibited a minimum value.

The degree of polarization ($P_{\text{deg}}$) can be estimated by the formula mentioned previously [25,26], in which $P_{\text{deg}}=1$ represents the linear polarization and $P_{\text{deg}}=0$ represents unpolarization or circular polarization. From the estimation, the degree of polarization from Sample-I was about 0.92, which demonstrated nearly linearly polarized light from the rubbing plate of the DD-LCs. Owing to the slight disordering of the alignment of the LCs from the R-plate of Sample-II, with only a single-sided rubbing plate, the estimated degree of polarization at approximately 0.86 is slightly lower compared to that of Sample-I. In contrast, the unpolarized emission light from the UR-plate of Sample-II has also been demonstrated because of the relatively disordered alignment of the NLCs; the emission spectrum component from the polar plot represents almost the same integrated intensity at all angles, as shown in the inset of Fig. 2(f).

In order to confirm the lasing action from the DD-LC laser, the in–out relation (Sample I: black squares; R-plate of Sample-II: red circles; UR-plate of Sample-II: blue triangles) from three plates of the DD-LC lasers are shown in Fig. 3(a). When the pump energy increased above a certain threshold, the integrated intensity increased obviously to demonstrate the lasing action of the DD-LC laser. The output intensity from the two plates of Sample-II reveals almost the same value at the same pump energy. In comparison to the FP laser with lasing threshold about 3.2 μJ, the lasing threshold of RL from Sample-II declined to 2.6 μJ owing to the enhancement of multiple-light scattering.

 

Fig. 3. (a) Output intensity as a function of pump energy from three output plates of the DD-LC laser, and the histograms of intensity distribution of the DD-LC RL from the UR-plate of Sample-II with pump energy (b) below the lasing threshold ($E=1.4\mathrm{\mu J}$), (c) around the threshold ($E=2.6\mathrm{\mu J}$), and (d) above the threshold ($E=3.5\mathrm{\mu J}$), in which the red curves indicate the $\alpha$-stable fit.

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To quantitatively define the threshold of an RL, the econophysical function, termed as the $\alpha$-stable distribution, was applied by Uppu and Mujumdar [27]. By a statistical measurement of intensity fluctuation from the dynamic system, Lévy fluctuations (power-law tailed) have been produced to represent the Gaussian or Lévy statistics. Generally, the $\alpha$-stable distribution [27,28,31] utilized four parameters to fit the statistical data, which was constructed by the intensity distribution of the scattering system and can completely illustrate the heavy tail distribution. Here, the parameter $\alpha$ describes the exponent tails of the distribution, in which $\alpha =2$ reveals Gaussian behavior and $\alpha <2$ shows Lévy behavior. The other parameters $\mu$, $\beta$, and $\sigma$ describe the location, skewness, and width of the distribution, respectively.

To analyze the lasing characteristics of the two produced samples, 1000 spectrum slots from each output plate of samples were grabbed. First, we obtained the histogram of intensity distribution with $\lambda =572\mathrm{nm}$ from the UR-plate of Sample-II at different pump energies [$E=1.4\mathrm{\mu J}$ in Fig. 3(b), $E=2.6\mathrm{\mu J}$ in Fig. 3(c), and $E=3.5\mathrm{\mu J}$ in Fig. 3(d)]. It is noted that the intensity fluctuation below the lasing threshold reveals near Gaussian distribution with $\alpha =1.9$, which is close to 2. This demonstrates that the intensity distribution from the rubbing plate of the wedge cell is similar to the Gaussian but still within the Lévy regime. The obtained value declined to $\alpha =1.26$ around the lasing threshold but increased slightly to $\alpha =1.34$ at a higher pump energy.

In addition, we also used the $\alpha$-stable distribution to distinguish the operation state from three output plates around the lasing threshold $E=2.8\mathrm{\mu J}$. The time varied intensity distribution with wavelength $\lambda =572\mathrm{nm}$ is shown in Fig. 4(a) (black line: Sample-I; green line: R-plate of Sample-II; blue line: UR-plate of Sample-II). The fluctuations in Fig. 4(a) show nonmonotonic behavior and reveal the largest variation from the UR-plate of Sample-II. The intensity probability distributions from Sample-I as well as the R-plate and UR-plate of Sample-II are shown by the histograms in Figs. 4(b)–4(d). For the Fabry–Perot laser (FPL) from Sample-I, the Gaussian distribution was demonstrated by the $\alpha$-stable distribution fit that yields $\alpha =2.0$. However, the emission characteristic from the R-plate of Sample-II in Fig. 4(c) shows a value of $\alpha =1.77$. Finally, a strong Lévy characteristic is illustrated by the UR-plate of Sample-II as shown in Fig. 4(d). It shows a quick initial fall and manifests a fat-tailed distribution that can fit well to obtain the value $\alpha =1.11$.

 

Fig. 4. (a) Intensity fluctuation from three output plates of DD-LC lasers with a central wavelength $\lambda =572\mathrm{nm}$, and the corresponding statistics of intensity distribution from (b) Sample-I ($\alpha =2.00$; Gaussian distribution), (c) R-plate ($\alpha =1.77$; Lévy distribution), and (d) UR-plate of Sample-II ($\alpha =1.11$, strong Lévy distribution), in which the red curves indicate the $\alpha$-stable fit.

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The obviously different dynamics of the alternated configurations of the DD-LC lasers might be related to the light scattering within the LC mixtures near the output plates. In order to quantify the extent of the light scattering, the transport mean free path $l_t$ can be estimated [32] with the relation $l_t=l_s/(1-\cos \theta )$, where $l_s$ is the scattering mean free path, and cos $\theta$ is the average cosine of the scattering angle. Experimentally, the transport mean free path $l_t$ has been measured by the CBS technique [30] as illustrated in Fig. 5(a). A collimated green laser beam with a central wavelength $\lambda$ at 532 nm was divided by a beamsplitter (BS) into two beams, namely, the signal ($D_{\text{Sig}}$) and the reference beam ($D_{\text{Ref}}$). The signal beam was reflected by the DD-LC sample and recorded by a detector ($D_{\text{Sig}}$), which was moved along the direction perpendicular to the beam propagation through a motorized translation stage. In addition, a mask with a small pinhole (0.5 mm in diameter) was placed in front of the detector. The transmitted beam through the BS was detected by the other detector ($D_{\text{Ref}}$) as a reference beam. The distance between $D_{\text{Sig}}$ and the DD-LC sample was around 20 cm. To increase the signal-to-noise ratio, a mechanical chopper in cooperation with a lock-in amplifier was used. The scattering signal of the DD-LC mixtures from $D_{\text{Sig}}/D_{\text{Ref}}$ as a function of the divergent angle $\theta$ is shown in Figs. 5(b)–5(d).

 

Fig. 5. (a) Schematic setup for CBS measurements using a continuous-wave green laser with a central wavelength of 532 nm. The measured CBS cone from three output plates of DD-LC lasers including (b) Sample-I, (c) R-plate, and (d) UR-plate of Sample-II, respectively.

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The CBS trace can be theoretically fitted by the albedo $\alpha (\theta )$ with the following formula [29]:

Here, $q_{\perp }=2\pi \theta /\lambda$ is the component of $q$ normal to the $z$ axis, and $z_0\cong 0.7l_t$ (for pointlike scatter). With the good fit of Eq. (1) [red solid curves in Figs. 5(b)–5(d)], the obvious differences in $l_t$ between the three cases of about 19.9, 9.9, and 5.5 μm, respectively, were obtained. Approximately, the full width at half-maximum (FWHM) ($W$) of the CBS cones is related to $l_t$ by the formula $l_t\approx 0.7\frac{\lambda }{2\pi W}$ as shown in the inset of Figs. 5(b)–5(d). The smallest $l_t$ from the UR-plate of Sample-II (about half of the R-plate of Sample-II and one-quarter of Sample-I) demonstrated that the LC alignment inside the cell was related to the configuration, parallel or wedge cell, and the rubbing of PI to alter the light scattering within the LCs.

Based on Young’s double slit experiment, the spatial coherence of the emitted light from DD-LC lasers with different configurations was also investigated [8]. In this work, a double slit with 70 μm width and 250 μm separation was adopted to generate the interference patterns. After passing through a double slit and focusing by a cylindrical lens, the far-field interference pattern of the DD-LC laser with different configurations was imaged onto a screen and recorded by a charge-coupled device. The grabbed images of interference fringes in the central part of the images from Sample-I as well as the R-plate and UR-plate of Sample-II are shown in Figs. 6(a)–6(c). In addition, the one-dimensional intensity distributions of the interference fringes are shown in the second row [Figs. 6(d)–6(f)]. Theoretically, the projected interference fringes with high contrast imply that the laser has a relatively high spatial coherence. Thus, we computed the mutual coherence function $\gamma$ from the grabbed data of interference fringes by the following formula [8]:

 

Fig. 6. Projected interference fringe images from (a) Sample-I, (b) R-plate, and (c) UR-plate of Sample-II, respectively. One dimensional intensity distributions of interference fringes from (d) Sample-I, (e) R-plate, and (f) UR-plate of Sample-II, respectively. The images of the monochromatic speckle pattern from (g) Sample-I, (h) R-plate, and (i) UR-plate of Sample-II.

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In previous reports [17], a low speckle noise and a high quality image can be produced from an RL with the characteristic of low spatial coherence. In this work, the monochromatic speckle patterns from the DD-LC lasers were obtained with the schematic setup of the subjective speckle measurement (according to the IEC 62906-1-2:2015 [33]). The lasing beam of the DD-LC cell was focused by the lens with focal length $f_2=25\mathrm{mm}$ onto the diffuser to produce a monochromatic speckle pattern and was then projected onto a screen. The illuminated monochromatic speckle pattern was then recorded by a digital camera (pixel size: $4496\times 3000$). Due to the slightly high spatial coherence from the output of the Sample-I with FP configuration to produce the interference from beamlets, many dazzling spots can clearly be seen in Fig. 6(g), with an estimated speckle contrast $C$ of about 0.093. In contrast, the granular patterns on the projected image in Figs. 6(h) and 6(i) are greatly reduced from the R-plate and UR-plate of Sample-II with a speckle contrast $C$ of about 0.053 and 0.039, respectively. In addition to the low spatial coherence resulting from a disordered alignment of LCs, the great reduction of speckle contrast below 0.04 in Fig. 6(i) is also attributed to the unpolarized light from the UR-plate of Sample-II.

4. CONCLUSION

In this work, we have demonstrated the obviously different output characteristic of a DD-LC laser, including Fabry–Perot or random lasing, which can be manipulated by the configuration of the glass cell and rubbed PI on an ITO glass plate. Besides, the output polarization of the DD-LC laser can be controlled by the alignment of LCs at the output plate to produce linearly polarized or unpolarized output emissions from the cell with or without rubbing PI on the glass plate. Moreover, the threshold of the DD-LC RL from the wedge cell can be obtained by $\alpha$-stable distribution with different pump energies to show a near Gaussian distribution and Lévy distribution around the lasing threshold. In addition, we also distinguished the operation state of the DD-LC laser from an FPL with a parallel cell having $\alpha =2$ but a Lévy distribution from the RL having $\alpha =1.77$ and $\alpha =1.11$, respectively, with and without rubbed PI on the glass plate of a wedge cell. The obvious difference is attributed to light scattering within the LC mixtures that can be quantitatively measured by the coherent backscattering technique to show the smallest transport mean free path of approximately 5.5 μm from the wedge cell without rubbing PI. Furthermore, the low spatial coherence of a DD-LC RL was estimated from Young’s double slit experiment to produce the low contrast monochromatic speckle patterns of approximately 0.039 that was far smaller than the FPL with a value of approximately 0.093. The results indicate not only the superior physical properties of a DD-LC laser, but also a practical device that can be used for laser projection and imaging.