1. Introduction

There has been considerable interest towards experimental studies on the correlation properties of light produced by exotic sources, and possible methods to control their coherence. Coherence is one of the fundamental properties of light, and it was thought for a long time that laser light always exhibits high spatial and temporal coherence compared to thermal light. Novel sources, such as random lasers [1–4], have shown that laser radiation can have varying degrees of coherence. Particularly, lasers with a low degree of spatial coherence have been of interest for their potential applications in imaging, due to high brightness and the absence of speckle noise [5].

The coherence properties of light produced within disordered media are determined by the internal scattering processes. Experimental determinations of the spatial coherence of random laser emission have been demonstrated by measuring the emission from active material powders [6], and liquid active media solutions with suspended scatterers [7]. Typically, the spatial coherence has been estimated by making use of either the single coordinate pair Young’s double-slit experiment [7], or speckle analysis of the output beam [8]. However, complete characterization requires the measurement of the two coordinate spatial coherence function, which can be found in Young’s experiment only by scanning the slits over all possible spatial coordinate combinations on the wavefront.

The novelty of our research is that we experimentally determine the two-coordinate coherence function for a quasi-random laser operating in pulsed mode with the use of a novel configuration of double-grating interferometer with a stationary camera and z-scanning gratings [9].

The quasi-random laser we utilize, is based on transparent wood (TW) as a semi-ordered host matrix, which was doped with Rhodamine 6G (Rh6G) dye as the gain medium. This type of TW laser was introduced recently by Vasileva et al. [10]. The TW is a composite material with excellent mechanical properties [11], structural anisotropy [11, 12], as well as high transmittance (about 90 %) [12], with a simultaneously high haze value (up to 95 %) [13, 14], which indicates strong scattering within the substance. The scattering is caused by the fact that natural wood has a complex hierarchical structure scaling from nanometers to tens of micrometers (vessels, fibers, rays, etc.) [15] and constituting of several components (cellulose, hemicellulose, lignin, etc.) of different refractive indices varying from 1.53–1.61 [16]. Chemical modification of wood (delignification) followed by infiltration with a polymer of matching refractive index is required to make natural wood transparent. After delignification, the refractive index difference of the inner components is reduced and the average refractive index of the template is around 1.53 [17]. To convert the TW host material into an optically active medium, a laser dye (Rh6G) is introduced in the material together with the infiltrating polymer. More details about the material used in this work including preparation and characterization can be found elsewhere [10].

The lasing action from the TW-Rh6G has been attributed to the collective effect of cellulose fibers working as an assembly of small Fabry-Perot type resonators [10] that are partially ordered due natural growth of internal wood components [11, 18]. On the other hand, since optical feedback is realized via scattering on the wood fiber boundaries and is not based on a classical external resonator, the emission has features of random lasing. Thus, we find it appropriate to define this type of source as a quasi-random laser.

2. Experimental setup

We consider the emission from three different samples, in order to make sure that the measured coherence properties are not caused by the measurement setup itself. For this, we need one non-lasing sample (low coherence), one lasing sample (higher coherence) and the TW-Rh6G sample (unknown). We prepared reference samples based on Rh6G in poly-methyl-metacrylate (PMMA) [19] and compare these results with the TW-Rh6G. The first reference sample was a simple PMMA slab with Rh6G and the second one was otherwise similar, but it had a gold mirror at one end. The two comparison samples are functionally the same Fabry-Perot type cavities, but the sample with a gold mirror has a higher quality factor, and thus lasing is easier to achieve with it.

All the samples were of a rectangular shape, as is shown in Fig. 1, with facets polished to diminish output scattering. We always used the same side-pumping scheme for the samples, which is depicted in Fig. 1 (d). The pumping intensity in transversal direction decreased with the penetration depth due to the absorption of light by Rh6G molecules (cross-section inset in Fig. 1 (d)). Further on, we will refer to this feature to understand some of the experimental findings.

 

Fig. 1 Schematic representation of the experimental samples, (a) reference sample; (b) reference sample with a gold mirror (Au M) attached to a facet; (c) TW-dye sample; (d) schematics of the pumped area in the sample. The intensity distribution of the pump beam is nonuniform due to absorption and depth of focus.

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We used the setup depicted in Fig. 2 for our experiments. A side-pumping scheme (Fig. 2 (a)) with the second harmonic of Nd:YAG (532 nm) at 0.70 mJ pulse energy was realized, where the active material was pumped using a 2 mm long and ∼300 µm wide line, and the pumping conditions were kept constant over all measurements. The lens L1 expanded the pump beam, and the square aperture A was used to sample the central part, where the spatial intensity distribution was nearly uniform. The cylindrical lens CL focused the pump light to a line, and the lens L2 collimated the output emission. To remove scattered pump light, a long-pass filter F was implemented.

 

Fig. 2 (a) Pumping scheme, lens L1 expands the beam towards a square aperture A, after which the cylindrical lens CL focuses a line on the studied sample. The emission is collected by lens L2 and filter F removes any remaining pump light. (b) Interferometer, grating G₁ disperses the beam and G₂ recombines ±1 orders at the image plane, W being the zeroth order block. Electronic shutters ES₁ and ES₂ are used to find the intensity from individual arms, and the microscope objective MO images the interference pattern onto the detector C.

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Luminescence emitted by the sample was then directed to a modified double grating interferometer [9] (Fig. 2 (b)). It consisted of binary phase gratings G₁ and G₂, electronic shutters ES₁ and ES₂, a microscope objective MO, and a camera C. The camera was a typical square-law CCD-array detector, with a broad spectral response. The light first passed through the binary grating G₁, where it was split to several diffraction orders and the orders ±1 were maximized by the grating design. All other orders were either blocked (W being the zeroth order block) or propagated outside the measurement system. The two beams were then recombined by the second grating G₂ at the image plane, where they interfered and the interference pattern was imaged with a microscope objective onto the camera. By scanning the gratings in the z-direction, it was possible to shear the replicas of the emission with respect to each other, while keeping the overall propagation distance (from the source to the camera) approximately constant. Spatial coherence information was obtained from the visibility of the interference pattern.

Since the sources we consider are quasi-monochromatic, we can ignore the spectral response of our system and the visibility of the interference fringes, $\mathcal{V}$, in the grating interferometer is then related to the space–time domain degree of coherence as [9]

To conveniently quantify the results with a single numerical value, we define the overall degree of spatial coherence as [20]

The displacement Δx between the two interfering fields is related to the scanning of the two gratings in the z-direction as

To avoid excessive diffractive spreading, the length of the device was minimized, in accordance with [9]. By employing trigonometry and the grating equation, the parameters of the interferometer were chosen as D₁ = 50 mm and D₂ ≈ 158 mm. For the other parameters, we set W = 4 mm, so that the period of the first and second gratings become d₁ ≈ 7.94 µm and d₂ ≈ 6.02 µm, respectively. The interference was imaged with a microscope objective MO – featuring 5× magnification – onto a Thorlabs DCC 1545M monochromatic camera. The fringe period was set to d = 12.5 µm, corresponding to ∼12 pixel widths on the camera and thus avoiding undersampling.

An important feature of the double grating interferometer is the absence of time delay between the two interfering beams, in which case temporal coherence of the emission does not affect spatial coherence measurements [9]. The two gratings should be parallel in the ideal case, but because the propagation direction of the diffracted orders is dependent on the angle of incidence, the system is also self correcting to a certain degree. The main advantages of using the grating interferometer instead of a Young’s interferometer with moving slits [21], are the speed and efficiency of the device, to avoid dye photobleaching during measurement. Significant photobleaching occurs after more than 50 000 shots, which is ∼1.4 hours at 10 Hz repetition rate. Young’s interferometer has to scan over all possible combinations of spatial positions with its pinholes, meaning that the measurement is time consuming and has a low light efficiency. On the other hand, the employed grating interferometer features a light efficiency of ~32 %, and it measures all sets of points along the x direction for a given Δx value. This allows for a measurement time of ~45 minutes in the present case.

It is worth mentioning that propagation through free space or optical components can modify the spatial coherence of light [22]. For example, light passed through a small aperture features increased spatial coherence, even if the input light is completely incoherent. In our experiments, we avoided these effects by sampling the emitted light with a large aperture and keeping the propagation length constant (the camera and sources had constant distance to each other), so that we could measure only the inherent properties of the light.

3. Results and discussion

The results for the emission from the first reference sample are depicted in Fig. 3. Its emission spectrum is narrowed to 12–14 nm (Fig. 3 (a)) in comparison to the typical 50 nm-width of the photoluminescent spectrum of Rh6G [23–26]. The narrowing of the emission line is caused by the influence of the low quality Fabry-Perot resonator formed by the sample facets, with a reflection of about 4 % [10]. As the optical amplification in the system is not enough to compensate the losses, amplified spontaneous emission (ASE) is observed. In fact, the cavity had such low quality that the lasing threshold would not be reached without considerably damaging the sample, and therefore, its value was not estimated. It is however, considerably higher than our pump energy of 0.70 mJ.

 

Fig. 3 PMMA + Rh6G sample below lasing threshold, (a) typical spectrum, (b) the measured spatial coherence, $\overline{\gamma }=0.11\pm 0.01$.

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The measurement of the spatial coherence was performed over an ensemble of 10 pulses at each z-coordinate, and the single-shot interference pattern was observed to be stable. The measured spatial coherence function of ASE was nearly incoherent, with the coherent area located mainly at zero separation (Fig. 3 (b)). The coherent area found here was produced by the propagation through the optical system, which is in agreement with the Van Cittert–Zernike theorem [22]. The overall degree of spatial coherence for this sample was found to be $\overline{\gamma }=0.11\pm 0.01$.

To increase the Q-factor of the Fabry-Perot cavity, a gold mirror was placed at the other facet of the Rh6G-PMMA sample as was depicted in Fig. 1 (b). With such design, optical losses in the cavity were reduced and the Q-factor of the resonator was increased in comparison with the first case. This reduced the threshold energy to around 0.44 mJ, which we could easily exceed. Lasing peaks appeared at the background of an ASE spectrum, shown in Fig. 4 (a), although the pumping conditions were the same as for the previous sample.

 

Fig. 4 PMMA + Rh6G sample above lasing threshold, (a) typical spectrum, (b) the measured spatial coherence, $\overline{\gamma }=0.34\pm 0.02$.

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The measurement of the spatial coherence was performed in a similar manner as before, with averaging over 10 pulses and no large fluctuations in the single shot interference patterns were observed. The measured spatial coherence function (Fig. 4 (b)) is clearly more coherent than the one measured for the low quality cavity, as can be seen from the prominent side-lobes, and with the overall degree of coherence being $\overline{\gamma }=0.34\pm 0.02$. This coherence function has similarities to the broad area laser diode (BALD) [27, 28], which is a perfect example of multimode lasing.

As for the TW material, the internal structure is much more complicated. It is essentially a stack of cellulose fibers of various lengths (orders of mm) and diameters (in the range of 10–50 µm). In this medium, the cellulose fibers can act as small uncoupled Fabry-Perot resonators, which contribute to the output radiation. At the same time, the fibers are only semi-ordered, and strong scattering can occur at their boundaries, adding a certain degree of randomness to the emission. Therefore, for careful interpretation of the experimental results, we first imaged the emission surface of the TW sample (Fig. 5(a)) by substituting the collimating lens (L2 in Fig. 2 (a)) with a microscope objective.

 

Fig. 5 Images of the emission surface, (a) the intensity distribution where some of the individual cells are visible, (b) portion of the image interfered with itself at zero separation, and (c) at a separation of ∼10 µm.

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The majority of luminescent dye molecules are located on the walls of the cellulose fibers as is seen in Fig. 5 (a), creating a honeycomb-like structure, also reported in [10]. To examine if there is any significant correlations between the individual emitters (cellulose fibers), two copies of the near zone images were overlapped at the image plane by using the grating interferometer. The produced interference patterns at zero (Fig. 5 (b)) and ~10 µm (Fig. 5 (c)) separations were recorded. The interference fringes rapidly disappear with increasing separation distance, and therefore show that the emitters are mutually uncorrelated. This type of behavior is also reminiscent of BALD, where several mutually uncorrelated cavity modes along the wide axis [28] cause a complicated spatial coherence function [9, 27].

To perform the spatial coherence measurements for the TW output emission, the collimating lens L2 was placed back into the setup instead of the microscope objective. Again, the measurements were averages over 10 pulses, and almost no fluctuations were found from single-shot interference patterns. The measured values for the emission from the TW sample are shown in Fig. 6. We estimate the threshold energy to be about 0.54 mJ, which is slightly higher than for the Rh6G sample with gold mirror, but is still easily exceeded in our experiment. The main difference between coherent and incoherent feedback in random lasers is the presence or absence of narrow spectral lines. In our case, the narrow peaks are absent from the output emission (Fig. 6 (a)) and the spectral line is broadened (FWHM ≈ 6 nm) due to collective lasing action by a set of uncorrelated wood fibers. Thus, the spectrum indicates quasi-random laser action with incoherent feedback.

 

Fig. 6 TW above lasing threshold, (a) typical spectrum, (b) the measured spatial coherence, $\overline{\gamma }=0.16\pm 0.01$, and (c) a comparison of the coherence from all three samples corresponding to the position x = −200 µm.

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The measured spatial coherence function is displayed in Fig. 6 (b). It is clear from these results that the spatial coherence function has some multimode structure, and that the overall degree of coherence is between the values measured for the two reference samples, which was found to be $\overline{\gamma }=0.16\pm 0.01$. This intermediate case is clearly seen in Fig. 6 (c), where the comparison of coherence functions of all three cases is presented.

Comparison with the reference samples shows that the measurement setup does not significantly modify the coherence of the radiation, since in the opposite case the spatial coherence would yield systematically similar shape and overall degree of coherence for all of the considered samples. Thus, it is safe to conclude that the measured spatial coherence functions characterized intrinsic properties of the sources themselves.

One immediately sees that for the second reference sample (Fig. 4 (b)) and TW (Fig. 6 (b)), the coherence function has lower values for x > 0, which is caused by the pump beam absorption in the samples. The pump beam intensity exponentially decreases while entering the active medium, as was shown in Fig. 1 and the varying distribution of the pump power has been shown to affect the properties of the output light [29]. The output consists of laser emission from areas with high pump intensities (the side of pump incidence) and ASE and photoluminescence from less pumped areas (away from the pump). In other words, the output beam features high (laser emission) and low (ASE and photoluminescence) coherence areas, with the degree of coherence decreasing across the beam along the direction of the pump.

4. Conclusions

To summarize, we have demonstrated experimental determination of the two coordinate spatial coherence function for a quasi-random laser based on a TW-Rh6G material which has neither perfectly ordered nor random structure. We have used double-grating interferometer of modified configuration with a stationary camera and z-scanning gratings in the short pulse regime. The presented technique applied for this type of source can open new opportunities in the characterization of random lasers. The nature of the emission produced by gain media with randomly distributed scatters has been under discussion, and measuring the two-coordinate spatial coherence function can be used as a fundamental parameter to distinguish between ASE and lasing, as we have shown here.

The emitted light from the TW sample was found to be a mixture of laser emission produced by a number of uncorrelated laser cavities on the strongly pumped side of the sample, and low coherence light in the areas of less intense pumping. These types of quasi-random lasers can be constructed from any semi-ordered material that features optical gain, with coherence properties determined by the ordering of the internal structure. It is conceivable that such sources can offer novel solutions in a wide range of imaging and lighting applications.