1. Introduction

A random laser is a peculiar type of laser that utilizes multiple scattering to trap light [1–4]. The narrow-spectrum emission from a disordered gain medium, particularly when related to mesoscopic interference effects in complex scattering media [5], has intrigued researchers. Furthermore, the practical advantages of random lasers have been explored. Random lasers do not require transparent gain media with a carefully aligned laser cavity; diverse gain media, such as ceramics [6], powders [7–9], and films [10], have been introduced, which variegate available lasing frequencies, including ultraviolet [9] and terahertz [11,12].

Despite the advantages of laser gain media, random lasers have seldom been utilized in practical applications. The first reason is its inherently low power efficiency. In most random lasers, multiple scattering not only traps the emission light but also rejects pumping light effectively [13]. Although one- [14] and two-dimensional [12,15,16] random lasers can avoid this issue by separating the pumping and lasing dimensions, confining the direction of multiple scattering to lower dimensions is not always possible. Second, random lasers have low directionality (or low spatial coherence). Although temporally coherent but spatially incoherent light is advantageous for specific applications, such as speckle-free coherent imaging [17], it is typically challenging for random lasers to replace conventional lasers in most applications. Random lasers with controllable directionality have been studied to address this issue [12,15,18].

Recently, a scattering cavity was introduced within the scattering gain medium to improve the power efficiency and directionality of random lasers [19]. Inspired by a fish trap, the cavity is composed of a spacious internal volume and small entrance [ Fig. 1(a)]. Owing to the easy-to-get-in but hard-to-get-out structure, the pumping light is eventually absorbed into the gain medium. Contrastingly, the emitted light is successively amplified through reflections from the cavity wall, which is made of a gain medium. When the gain exceeds the scattering loss, it becomes a laser. Unlike conventional lasers, this laser cannot retain a stable resonant mode in the cavity owing to random scattering and is called a non-resonant laser (NRL) to be distinguished from random lasers without an explicit cavity [20,21]. Notably, most random lasers are NRLs, except random lasers with coherent feedback [22].

 

Fig. 1. Two cavity designs for non-resonant lasers. Light trapping is enhanced by diffuse reflection on the cavity wall in both cavities: (a) a spherical vacancy with a small outlet, (b) a vertical hole with a sufficient depth. The green and red lines indicate the pump and emission light, respectively.

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The incorporation of a scattering cavity successfully enhances the power efficiency and directionality of the laser. However, carving a spherical cavity within a scattering medium requires a complicated fabrication process. Because a simple configuration is one of the major advantages of random lasers, such practical difficulties severely undermine their merits and hinder their popularization. Furthermore, the lasing properties related to the cavity scale, such as the lasing threshold and spatial coherency, are practically limited as the fabrication difficulty increases for smaller cavities.

To mitigate these practical difficulties, we introduced a deep hole on the surface of the scattering medium as a laser cavity [Fig. 1(b)]. Even with an open-top cavity structure, efficient light trapping can be realized based on diffuse reflection that randomly redirects light. Deep-hole cavities were fabricated by drilling a micro-sized hole on the surface of a porous Nd:Y₃Al₅O₁₂ (Nd:YAG) ceramic. The successful operation of NRLs in deep-hole cavities was enabled by the introduction of pumping light within the cavities. Significant improvements in directionality, threshold, and power efficiency were observed compared to those in previous work [19]. The lasing characteristics at various cavity diameters and depths were explored and discussed in comparison with numerical results.

2. Principle

Unlike a spherical cavity, a deep-hole cavity does not have a narrower entrance than its interior space. The straight and open-top structure appears ineffective for trapping light, and this intuition is true if the cavity wall is a mirror surface. However, we found that the exit probability diminishes dramatically when the cavity is made of a scattering medium with a diffusely reflecting surface. Assuming that the cavity wall is a perfect diffuser, the direction of the reflected light obeys Lambert’s cosine law independent of its incident direction [23] [ Fig. 2(a)]. Because every wall reflection randomly redirects the light trajectory, the internal-cavity photons can be regarded as one-dimensional random walkers along the depth direction, which has a rapidly decreasing probability distribution with increasing displacement. The random walk scheme suggests a rapidly increasing trapping ability with an increase in the aspect ratio (AR) of the cavity.

 

Fig. 2. Light trapping capability of the proposed cavity. (a) Illustration of Lambert’s cosine law. The distribution of radiant intensity after reflection (I_Ω) is proportional to the angle of reflection (θ), independent of the incident angle. (b) Overlapping trajectories of photons in the deep-hole cavities with different aspect ratios (ARs). Light trajectories at each AR are accumulated for a hundred photons departing from the cavity bottom, and diffusely reflecting until they exit the cavity. Higher AR yields denser and longer trajectories. (c) The probability that a photon leaves a deep-hole cavity after each reflection without reflection loss (red line) and including reflection loss from diffuse reflectance (R) (gray line). The probability drops drastically as AR increases. Spherical cavities and corresponding exit probabilities (black dots) are plotted together. β indicates the diameter ratio of the cavity to the entrance. The y-axis is in log scale.

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To visualize the trapping ability of the deep-hole cavities, we numerically calculated the photon trajectories inside the cavities [Fig. 2(b)]. The gain or loss from the wall reflections has been excluded here for simplicity, without losing generality. One hundred photons were introduced at the bottom of cavities with various ARs (1, 3, and 5), and the better trapping ability for larger AR was well presented by the denser photon trajectories. A higher photon density for a deeper cavity depth was observed, which was also anticipated from the random walker analogy.

To quantify the trapping ability of the deep-hole cavity, we calculated the expected exit probability for each reflection $\left\langle {{P_{\textrm{loss}}}} \right\rangle $, including photon leak into the bulk medium quantified by diffuse reflectance (R) [Fig. 2(c)]. We found that $\left\langle {{P_{\textrm{loss}}}} \right\rangle $ diminished dramatically as AR increased. This is a consequence of the increasing photon density and decreasing exit probability with an increase in the cavity depth. The cavity with AR = 2 already provides $\left\langle {{P_{\textrm{loss}}}} \right\rangle $ < 0.1 if R is greater than 95%, implying that a photon experiences >10 wall reflections on average before it leaves the cavity. We also found that photon loss from the open top is negligible for AR greater than 8 because $\left\langle {{P_{\textrm{loss}}}} \right\rangle $ is almost equal to the diffuse reflectance of the cavity wall. To compare the trapping abilities directly, spherical cavities that provide the same $\left\langle {{P_{\textrm{loss}}}} \right\rangle $ without reflection loss have been depicted over the graph with the corresponding cavity-to-entrance diameter ratio (β). It is worthy noting that successful NRLs have been demonstrated with spherical cavities of β = 2–4 [19]. Figure 2(c) provides an approximate reference for required gain levels to support laser action depending on ARs of deep-hole cavities. However, the above numerical results do not consider the effect of gain on the internal distribution of photons, which are important for characterizing the lasing threshold and efficiency. To properly estimate the lasing performance, a more complex numerical model that considers uneven distributions of pumping light, emission light, and gain within the cavity is given in Supplement 1.

3. Materials and methods

In the experiments, we fabricated deep-hole cavities by micro-drilling porous Nd:YAG bulk ceramics. To explore lasing characteristics, deep-hole cavities with various diameters and ARs were engraved on a single ceramic. A coupling lens was used to deliver pumped light to the bottom of the cavities and collimate laser emission.

3.1 Synthesis and characterization of Nd:YAG powders

To obtain high-purity Nd:YAG particles, a co-precipitation method was used. The starting precursors were yttrium nitrate tetrahydrate (Y(NO₃)₃·4H₂O, ≥ 99.99%, Sigma-Aldrich Inc., USA), neodymium nitrate hexahydrate (Nd(NO₃)₃·6H₂O, $ \ge $ 99.9%, Sigma-Aldrich Inc., USA), and aluminum ammonium sulfate dodecahydrate (NH₄Al(SO₄)₂·12H₂O, reagent grade, Alfa Aesar Inc., USA). The raw materials were homogeneously mixed and dissolved in deionized (DI) water at 40°C. In this study, the concentration of Nd³⁺ ions incorporated into the YAG phase was fixed at 1 at.%, according to the efficient doping concentration in conventional laser experiments [24,25]. Owing to the high light intensity inside the cavity, a higher doping concentration may be more advantageous in our laser design to relieve saturation of absorption and gain. For precipitation, a solution was prepared by dissolving ammonium bicarbonate (NH₄HCO₃, ≥ 99%, Sigma-Aldrich Inc., USA) in a mixed solvent of ethanol and DI water. The volume ratio of ethanol to DI water was 0.6. The precursor solution was added dropwise to the precipitant solution at a dripping rate of 3 ml/min at room temperature. The suspension was aged for 24 h, centrifuged, and washed repeatedly with water and ethyl alcohol in sequence to obtain the precipitate. It was then dried at 85°C for 24 h in an oven. To achieve crystallization and eliminate residual organics, the particles were calcined in air at 1,250°C for 4 h. The calcined powders were sieved through a 200-mesh screen.

3.2 Fabrication of Nd:YAG porous bulk ceramic

The synthesized powders were pelletized by uniaxial pressing at 30 MPa into a 6-mm-diameter stainless steel mold with a sample thickness of 2 mm. The green bodies were then cold isostatic pressed (CIP) at 200 MPa for 5 min. To prepare porous Nd:YAG pellets, they were sintered at 1,350°C for 10 h at a heating/cooling rate of 5°C/min in air. The consolidated Nd:YAG ceramics were characterized using conventional X-ray diffraction (XRD) with Cu Kα radiation at a scan rate of 5°/min between 10° and 80° (SmartLab, Rigaku Inc., Japan). The observed X-ray diffraction patterns were consistent with the standard diffraction peaks for the cubic YAG phase (JCPDS #33-0040) [ Fig. 3(a)]. Microstructure images of the densified Nd:YAG ceramics were obtained by scanning electron microscopy (SEM, Model Philips XL 30 FEG, Koninklijke Philips N.V., Netherlands) after diamond polishing [Fig. 3(a), inset]. The average domain size of the consolidated sample was determined by multiplying the linear-intercept length of 200 grains by 1.56 [26]. Bulk density was measured using Archimedes’ method. Samples with a relative density of 70% and an average domain size of 400 nm were obtained because the sintering temperature was much lower than that of conventional Nd:YAG laser ceramics with full density.

 

Fig. 3. The proposed laser system (a) X-ray diffraction (XRD) result of the sintered Nd:YAG ceramics (black) and standard diffraction peaks of the cubic YAG phase (red), showing fair agreement between measured results and expected peaks. The inset is a scanning electron microscopy (SEM) image of the microstructure of the ceramic. (b) The arrangement of cavities fabricated on the Nd:YAG ceramic pellet. ARs increase from left to right (3, 4, 5, 6, 7, and 8) while diameters increase from top to bottom every two rows (30, 50, and 70 µm). (c) SEM image of the inner surface of the vertically drilled cavity. (d) The optical system. A coupling lens focuses pump light into the bottom of the cavity fabricated on the ceramic pellet and collimates the emission light. A dichroic mirror is placed to separate the pumping and emission lights.

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3.3 Fabrication of deep-hole cavities

The surfaces of the fabricated Nd:YAG ceramic pellets were polished and attached to a Cu holder for machining. Deep-hole cavities were fabricated using a commercial microdrilling machine (G. I. Tech Co., Ltd, Cheonan, Republic of Korea). Two copies of cavities of three diameters (30, 50, and 70 µm) with six different ARs (3, 4, 5, 6, 7, and 8) –total 36– were engraved on the same surface of the ceramic pellet [Fig. 3(b)]. The drilled cavities show a rough and porous surface, which is essential for optical feedback from diffuse reflection [Fig. 3(c)]. The bottom faces of the drilled cavities were indented at 118°, owing to the point angle of the drill bits. After machining, the ceramic holder assembly was immersed in acetone and sonicated for 30 min to detach the ceramic from the holder and remove debris. Subsequently, the ceramic was baked again at 800 °C for 2 h to remove potential organic contamination during the cavity fabrication process.

3.4 Optical setup

The experimental setup comprises a coupling lens, dichroic mirror, and pump source, which are only necessary for the laser action of the proposed cavity [Fig. 3(d)]. The coupling lens (f = 20.0 mm, NA = 0.52, AL2520H-B, Thorlabs, Inc.) was used to focus the whole pumping power inside the cavity and to collimate the emission light. A dichroic mirror (FF875-Di01-25 × 36, Semrock, Inc.) was used to separate the laser emission from pumping light. A femtosecond Ti: sapphire laser (> 2 W, FWHM = 5 nm at 808 nm, beam diameter = 1.2 mm, Chameleon Vision-S, Coherent Inc.) was used as the pumping source. The delivered pump power was calibrated by measuring the optical power transmitted through a dichroic mirror.

We measured the power and spectrum of laser emission as a function of the delivered pump power. A silicon photodiode (S121C, Thorlabs, Inc.) with a bandpass filter (λ_c = 1064 nm, Δλ_FWHM = 10 nm, FL1064-10, Thorlabs, Inc.) was used for power measurements. An optical spectrum analyzer (δν = 7.5 GHz, δλ = 28.4 pm, at 1064 nm, OSA201C, Thorlabs, Inc.) was used for the spectrum measurements.

4. Results and discussion

4.1 Directionality (spatial coherency)

Because NRLs do not have resonant modes that constrain the spatial lasing profile, the laser emission is as spatially incoherent as possible within the constraints of a given beam diameter and divergence. Thus, the directionality of NRLs can be simply quantified by the number of available (incoherent) spatial modes, ${N_x} = {\left( {{\textstyle{\pi \over 2}}{\textstyle{{\textrm{NA}} \over \lambda }}D} \right)^2}$, where D is the outlet diameter, and NA is the numerical aperture of the coupling lens [27]. A smaller N_x signifies a more directional laser emission after the coupling lens. For a given coupling lens (NA = 0.52), N_x = 530, 1473, and 2888 for D = 30 µ, 50 µ, and 70 µm, respectively. These values are much smaller than the N_x values of the spherical cavities (21800 in Ref. [19]) and random lasers (∼10⁷ in Ref. [6]). Notably, the directionality of the NRL is related to the laser output power owing to the NA dependence of N_x. For example, if we decrease the NA of the coupling lens, the directionality of the laser emission increases (as N_x decreases), whereas the laser output power decreases.

4.2 Spectral properties

The laser action was confirmed by narrowing the emission spectrum [ Fig. 4(a)]. The uneven spectrum of the stimulated emission cross-section caused the narrowing of the linewidth during successive amplifications in the cavity, leaving only a single peak in the emission spectrum [28]. Accordingly, we observed the narrowing of the strongest peak of the Nd:YAG photoluminescence spectrum (1064 nm) as the pump power increased, while the other peaks (e.g., 1061.5 nm) were suppressed. For instance, the linewidth narrowed to 53 pm for a 50-µm-diameter cavity with AR = 8 at 1.0 W pump power, which is more than 10 times smaller than the fluorescence linewidth. This implies that spontaneous emission is overwhelmed by stimulated emission. Thus, we conclude that the proposed deep holes can operate as a laser.

 

Fig. 4. Spectral properties for cavities with different aspect ratios (ARs). (a) Normalized spectra for a 50-µm-diameter cavity with AR = 6. The corresponding pump powers are denoted on the left. (b) Linewidths and (d) peak wavelengths of 50-µm-diameter cavities with different ARs. (c) Linewidths and (e) peak wavelengths of AR = 6 cavities with different diameters. Each error bar in (c, e) represents the minimum and maximum measured values obtained from two identical cavities.

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Narrower linewidths were observed for cavities with larger ARs [Fig. 4(b)] and diameters [Fig. 4(c)]. However, when the pump power increased further, the linewidth was again broadened. Such linewidth rebroadening was observed regardless of cavity geometry, but a lower pump power was required to reach the pivot point for a cavity with a smaller AR or diameter.

Peak wavelengths increased consistently as the pump power increased [Figs. 4(d) and 4(e)]. We expect that the thermal effect is responsible for such a redshift in the spectrum [29]. A more significant redshift was observed for cavities with smaller ARs or diameters, which is reasonable as the pump power is localized to smaller volumes. For instance, the estimated temperature of the cavity from the redshifted peak was 305°C for the 30-µm-diameter cavity with AR = 3 at 1.6 W pump power. We suspect that this temperature rise may also be responsible for the linewidth rebroadening. As the temperature goes up, a decrease in the stimulated emission cross-section alongside redshift [29] leads to a reduction in light amplification, which effectively broadens the linewidth by adding more spontaneous emission.

4.3 Power properties

The laser output powers for various deep-hole cavities are shown in Fig. 5. The output power curves exhibit a smooth transition near the lasing threshold, as in random lasers [30,31]. This is due to the spatial and spectral overlap between spontaneous and stimulated emissions [32]. After the threshold, the output power increased linearly, as in conventional lasers. However, as the pump power increased further, we observed that the slope efficiently decreased and eventually became negative. Such an output power drop generally occurs at lower pump powers, as the cavity diameter or ARs decrease. This is because light amplification in a deep-hole cavity is confined to a much smaller volume than conventional lasers. When a high pump power is applied, the localization of high energy into a small volume causes an unwanted temperature rise as well as significant optical nonlinear effects. While temperature rise deteriorates light amplification by stimulated emission [29,33], the nonlinear effect may alter scattering properties in such a way that gain decreases or loss increases unexpectedly. The decreasing gain drops the slope efficiency and eventually prohibits lasing when gain cannot compete against loss, resulting in negative efficiency.

 

Fig. 5. Output powers for (a) 30-µm-, (b) 50-µm-, and (c) 70-µm-diameter cavities with different aspect ratios.

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The measured output powers for a given pump power were highly dependent on the cavity diameters and ARs (Fig. 5). For instance, we observed monotonically increasing slopes for increasing ARs in 30-µm-diameter cavities, while the slopes reached a maximum around AR = 5–6 in 50- and 70-µm-diameter cavities.

To compare the lasing power efficiencies, the slope efficiencies were quantified in the linearly increasing region [ Fig. 6(a)]. Moreover, we estimated lasing thresholds by extrapolating the linear region, which is marked by triangles on the bottom line of Fig. 6(a). We achieved slope efficiencies of 3.40%, 3.95%, and 3.76% and lasing thresholds of 143 mW, 159 mW, and 174 mW for the 30-, 50-, and 70-µm-diameter cavities with AR = 6, respectively.

 

Fig. 6. (a) Output powers for cavities of aspect ratio (AR) = 6 with different diameters, with the best result of spherical cavities in Ref. [19] (gray line). The linear regions of the power curves are extrapolated (dashed line) with their x-intercepts marked as triangles. The slope efficiencies estimated from the slope of each extrapolated line are shown near the line. The lasing thresholds estimated from the x-intercepts are 143 (cyan, 30 µm diameter), 159 (blue, 50 µm diameter), 174 (magenta, 70 µm diameter), and 790 mW (black, Ref. [19]), respectively. Each error bar represents the minimum and maximum measured values obtained from two identical cavities. (b) Slope efficiencies obtained from the numerical model (black line) and the experiment (dots). The left y-axis is for the slope efficiencies predicted by the numerical model, while the right y-axis is for those measured in the experiment.

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From the observations in Fig. 6(a), the deep-hole cavity design shows much lower threshold powers and even higher slope efficiencies than the previous spherical cavity design [19]. The reduction of laser threshold can be explained by the concentration of optical power in a smaller cavity. We were able to make holes down to 30 µm in diameter, which is much smaller than the previous design (300–1500 µm in diameter). In the case of power efficiency, the open-top structure of the proposed cavity enabled direct free-space coupling, which helped minimize the undesired coupling loss. In contrast, the spherical cavity design requires an optical fiber to transmit optical power between the inside and outside of the cavity engraved in the middle of a gain medium, causing unavoidable coupling loss.

The quantified slope efficiencies for the various deep-hole cavities are shown in Fig. 6(b). Furthermore, we provide an expected slope efficiency curve calculated from the numerical model described in Supplement 1. The slope efficiencies predicted by the numerical model were independent of the diameter within the 30–70 µm range. The maximum slope efficiency around AR = 5–6 is expected from the numerical model, which is consistent with the experimental results for the 50- and 70-µm-diameter cavities. We suspect that the inconsistency between the results from the 30-µm-diameter cavities and the measurement values lower than expected could be derived from various sources, such as temperature-dependent material properties and potential contamination during the fabrication steps.

5. Conclusion

In this study, we propose and demonstrate an NRL using a deep-hole cavity. We show that the deep-hole geometry outperforms the spherical geometry because it is not only considerably easier to fabricate but also provides a lower lasing threshold and better slope efficiency. This scalable cavity design allows controlling the lasing threshold, laser linewidth, and number of spatial modes (i.e., directionality) by adjusting the cavity diameter and depth. In this sense, our laser is promising for speckle-free imaging [17,34], holographic display [35,36], and quantitative phase imaging [37,38], in which the spatiotemporal coherence of light sources is carefully controlled to balance resolution and speckle reduction [39]. Nonetheless, an adequate temperature control scheme is required to provide better and more stable lasing. Recently, a wide variety of new materials, including nanoparticles [40], graphene [41], perovskites [42–45], and dye materials [46] have been explored for the use in random lasers. We expect the deep-hole geometry to be easily applicable to these materials by either fabricating a hole on the surface or depositing the material on a hole structure, which significantly enhances lasing characteristics.

The fundamental limitation on the power efficiency comes from the optical pumping process in the used laser system. A large portion of pump light exits the cavity via the outlet before it is absorbed. In addition, the gain material may show saturable absorption owing to the high intensity of pumping light inside the cavity. We believe the above problems can be mitigated if our laser system is pumped electrically, like a laser diode.