1. Introduction

Disordered photonic media [1] are featured with random scattering of light, leading to complex light propagation, diffusion, and localization phenomena, as well as the intriguing applications for solar energy harvesting [2], non-invasive imaging [3], and random lasing [4], etc. Random lasing relies on the multiple scattering of light for feedback, rather than the traditional well-defined optical cavity. The resulting output phenomenon exhibits multimode emission over a broad spectral range, which can be used for “fingerprinting” applications in document encoding, material labeling [5] and sensing [6]. The random laser can also be implemented as a powerful light source for display lighting and speckle-free imaging [7] due to its low-spatial coherence of ensemble phase-uncorrelated emissions from different locations [8]. Therefore, random laser has attracted intensive research interests. Various forms of disordered media, like fluorescent dye solutions [4], rare-earth ion doped fibers [9], ZnO powders [10,11], organic composites [12], and quantum-dot membranes [13], etc., were developed with gain material embedded in the scatterers or background. Depending on the selected gain materials, the operation frequency of random lasers spans from the UV to the infrared (IR) spectrum. In recent several years, mid-IR and far-IR random lasers based on III-V quantum cascade semiconductor materials experienced rapid development with diverse designs [14–18]. These are electrical-driven devices with compact chip footprints, making them highly promising for high-capacity integrated optoelectronic applications. Despite these progresses, the control of random laser output, especially the emission directionality is a long-standing challenge. The random emission is intrinsically non-directional due to the disordered resonances, which leads to low collection efficiency. Consequently, additional optical components, for example, bulky optical lenses and sophisticated optical systems, are required for beam alignment and power collection.

There have been some theoretical and experimental attempts to tame the angular properties of random optical modes. It was found that lasing output is preferentially obtained along the direction in which the gain medium is most extended [19]. Long optical waveguides made of ZnO [10] powders or fibers with transverse disorder [20] are reported to have directional edge emission due to the mode selectivity of the longitudinal Fabry-Perot cavity. By interfacing a random system of an organic-inorganic sol-gel film with dielectric Bragg reflectors [21] or gratings [22], the optical trapping by Bragg structures leads to directional emission. Despite the effectiveness, these additional photonic structures significantly alter the optical characteristics of disordered medium, which leads to much fewer emission modes, thereby losing advantageous features of the random laser. Using the wavefront shaping technique to iteratively optimize the pump profile can effectively select a localized mode with a given emission direction and directionality [23]. However, this method is unable to further reduce the divergence angle, which was intrinsically determined by the random optical mode. This technique is also difficult to be implemented for electrically pumped on-chip random lasers. In addition, the directional emission from the surface of a random cavity-based terahertz quantum cascade laser was recently observed without altering the random nature of optical modes, which is due to the intensity superposition of multiple modes in the far field [24]. However, the emission direction and divergence angle vary from sample to sample. And most of all, there is still a lack of control for in-plane random mode output.

Zero-index materials (ZIMs) with effective zero permittivity and/or permeability have remarkable characteristics in terms of their response to electromagnetic waves, namely, the electromagnetic waves experience little phase change as they propagate through the ZIMs [25–28]. However, the natural materials, for examples metals, doped semiconductors, polar materials, or ferrites, are highly lossy at their epsilon near zero (ENZ) or mu near zero (MNZ) frequencies, i.e., plasmon, phonon, polariton, or magnon frequencies [29]. While offering simultaneously effective zero permittivity and permeability at the same frequency, metallic metamaterials rely on the spoof electric and magnetic plasmon are also highly lossy due to the same reason as natural materials [30,31]. As research progresses, all-dielectric photonic crystals (PhCs) have emerged as low-loss and impedance-matched candidate for realizing ZIMs [32]. These all-dielectric PhC-based ZIMs are usually excited at the Dirac cone-like linear dispersion, where the photonic mode at the Γ point undergoes an underivable bend that gives rise to effective zero index, and the linear band structures make sure that the effective refractive index to be less dispersive ($d\omega /dk = const$) [33,34]. Based on the unprecedented flexibility for wave manipulation, all-dielectric photonic ZIMs can be applied for wavefront shaping [33,35], electromagnetic cloaks [36], large-area single-mode devices [37], etc. It is also investigated theoretically and experimentally to implement ZIMs for directional emission when emitters are embedded in the ZIMs [38–41]. To our knowledge, no one has yet used ZIMs for output control of random optical modes. Differing from previous studies with in-phase wave excitations [34,42], the optical modes in disordered medium radiate in all directions with irregular wavefronts. Especially, the uncontrolled wavevectors may couple with each other, leading to undesired emission angular lobes [43]. Therefore, whether the ZIMs can be used to effectively manipulate the emitting wavefronts of random optical modes is an open question. Namely, how ZIMs shape the output characteristics of random lasers requires a detailed study.

Here, we explore the possibility of using a ZIM to manipulate in-plane emission of random optical modes in the context of terahertz (THz) quantum cascade random lasers (QCRLs). The THz QCRLs operate in transverse-magnetic (TM) polarization due to the double-metal waveguide configuration. The modes are strongly confined in a planar disordered medium with a subwavelength thickness, sandwiched by two mental layers in the vertical direction. Therefore, two-dimensional (2D) structural model is a good approximation to investigate the in-plane angular property of random optical modes. In this work, we devised a ZIM based on all-dielectric PhC with zero effective zero index at ∼100 µm, which is integration-compatible with the THz QCRLs. Using a passive ZIM to enclose a 2D planar disordered dielectric pillar system, the emission directionality of random optical modes is improved significantly. At the same time, the field distributions inside the random cavity region are merely perturbed. Meanwhile, the Q factors of random cavity modes are even slightly improved, which is beneficial for random lasing with lower threshold. Spectral scanning indicates that this improvement of radiation directionality applies to a broad wavelength range with a maximum enhancement factor of 30, a 6-fold increase compared to that of original random optical modes. Close to the zero-index wavelength, the divergence angle is reduced to ∼6°. When multiple random optical modes superimpose in the far field over a frequency range, the divergence angle can be further compressed to ∼3.5°. Statistical study proves that ZIM generally works well, regardless of different disordered patterns and the filling fractions of random scatterers. Further simulation results show that the directionality improvement by using zero-index PhC also applies to random medium designs, featured with strongly localized or weakly extended random modes. By tailoring the boundary of zero-index PhC, the emission direction of random modes can also be manipulated, providing flexibility for on-chip functional integrations.

2. Design of on-chip THz zero-index material

It was reported that a PhC made of high-index dielectric rods can exhibit zero effective permittivity and permeability at the frequency corresponding to the apex of a Dirac cone, which is accidentally formed at the $\mathrm{\Gamma }$ point in its band diagram [33]. Considering the compatibility with THz QCRLs, the devised zero-index PhC in this work is a square lattice made of quantum cascade material pillars with an effective refractive index of 3.60 at the interested frequency range around 100 µm. The lattice has a constant of 53.78 µm and the pillar has a diameter of 21.28 µm. At the Γ point, the PhC typically supports three degenerate TM-polarized eigenmodes, i.e., electric monopole (ED), transverse magnetic dipole (TMD), and longitudinal magnetic dipole (LMD), as shown in Fig. 1(a), respectively. The band structure is depicted in Fig. 1(b). The TM2 and TM4 bands with a linear dispersion cross at the Brillouin zone centre and intersect a flat TM3 band, thereby forming a triple degeneracy point at 100 µm with a Dirac cone-like dispersion. Off the Γ point, the TM3 corresponds to LMD, while both TM2 and TM4 bands show a combination of ED and TMD. The isofrequency contours in Fig. 1(c) over the wavelength range from 100 µm to 95 µm show nearly circular profiles, suggesting that the homogenization criterion is satisfied [38,41], so that the PhC can be treated as a homogeneous medium. The homogeneous dielectric property of PhC is crucial for the directionality control of random optical modes, which will be shown in the following sections. To further analyze the optical properties of the ZIM, finite-difference time-domain (FDTD) simulations were performed. We calculated the reflection and transmission coefficients of a finite PhC with a lateral thickness of 8 units. The S-parameter retrieval method [44] was used to calculate the effective refractive index:

 

Fig. 1. (a) Schematic of a zero-index material (ZIM) unit cell based on quantum cascade material pillar and its three degenerate modes close to Dirac cone point with transverse-magnetic polarization. (b) Photonic band structure of the ZIM. One can see Dirac-cone dispersion at the Γ point at λ=100.34 µm. The shaded area denotes the region below the light line. (c) Isofrequency contours with increasing frequency from the Dirac point of the ZIM. (d) The calculated effective refractive index of the designed ZIM.

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3. In-plane directionality improvement for random optical modes

2D QCRL, which has randomly dispersed dielectric pillars made of quantum cascade material in a low-index benzocyclobutene (BCB) background, was demonstrated to have fairly high scattering efficiency and good performances [15]. We start from this structure to study the angular features of random optical modes with and without ZIM in its periphery. The dielectric pillars in circular shape are distributed over an area of 830 µm × 830 µm without long-range correlations. The pillar radius is 6.4 µm and its areal filling fraction (FF) is 20%, leading to multiple localized mode emission. The ZIM with eight periods of all-dielectric PhC mentioned above is placed adjacent to the disordered structure without a special spacing requirement between the two structures. The 2D Finite-Element-Method (FEM) simulations by COMSOL Multiphysics were conducted to study the modal properties. The disordered system is set to be lossless, simulating the random laser with balanced gain and loss. The ZIM is used as a passive wavefront shaper in this study, therefore the dielectric pillars of ZIM are slightly lossy with its imaginary part of refractive index to be 0.01. This induces little influence on the random optical modes which will be shown below.

For a disordered structure, the typical optical modes have random electric field profiles localized to different spatial regions. In the first row of Fig. 2, the field profiles at ∼99 µm show that the random modes radiate power to all the directions, as shown by the far-field calculation results at the lowest row of Fig. 2, which is the typical characteristic of random lasing. By contrast, the angular emission properties of random modes are completely different once the zero-index PhC is introduced. From the second row of Fig. 2, we can see that the outgoing waves from the PhC show nearly-parallel wavefronts with wavevectors normal to the interface boundary. According to the law of momentum conservation, the optical waves, generated from the disordered structure, are mostly “reflected” and can only be transmitted to the zero-index PhC if the incident angle is close to zero. As a result, the output of random optical modes with a zero-index PhC is highly directional. This phenomenon is still pronounced when the random resonating wavelength deviates from the Dirac wavelength where the effective index is small. The directionality enhancement by ZIM based on all-dielectric PhC can also be seen in the far field regime. Once the ZIM is added, the emission lobes in the longitudinal and/or transverse directions dominate the output, which is determined by the geometry of ZIM domain. The relative intensities of the four lobes are determined by the random mode profile and the “filtering effect” of ZIM. The output divergence angle is around 6°for each emission lobe, significantly narrower than that of the original random optical modes, and the emission power is mainly concentrated on the longitudinal and/or transverse directions without discernible side lobes.

 

Fig. 2. (a)-(c) Near-field and the corresponding far-field properties of three pair of random optical modes in a disordered structure with and without ZIM in its periphery. The squares inside the field distribution figures outline the region of disordered structure and ZIM based on all-dielectric PhC, respectively. In the polar figures, the blue and orange patterns show angular far-field intensities of random optical modes without and with ZIM.

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However, it is worth mentioning that the corresponding random optical modes with and without ZIM for emission wavefront shaping show negligible wavelength shift and very similar spatial profiles inside the disordered structure region. This indicates that the zero-index PhC slightly perturbs the random mode resonating. The influence of zero-index PhC on random optical modal properties has also been investigated in terms of the modal loss. We have calculated the Q factors of the random optical modes over a broad wavelength range. As shown in Fig. 3, the modes have random oscillating wavelengths with Q factors in a range of 200∼10000. The optical properties seem not purely random, since the Q factor tends to increase with longer wavelengths. This is mainly due to the Mie scattering of pillars leading to higher scattering efficiency in the wavelength range from 90 µm to 100 µm. These are also the typical features of a strongly scattering medium. By enclosing the disordered medium with a zero-index PhC, these features are not changed. However, through inspecting the results carefully, it can be found that the Q factors of most random optical modes with zero-index PhC in the surrounding are improved, despite the fact that zero-index PhC is lossy. The inset of Fig. 3 shows the Q factor comparison of three pairs of random modes in Fig. 2. The wavelength shift is ignorable. The Q factor enhancement can be attributed to the reflection of the off-normal incident wave from disordered structure and thus reduces the overall modal loss.

 

Fig. 3. Calculated Q factors for random optical modes with and without a ZIM in the periphery of a disordered structure. The inset shows the five pairs of modes highlighted by the black dashed rectangle. The three pairs of random optical modes in Fig. 2 are marked with black dashed arrows. One can see apparent Q factor enhancement. The correspondence for random optical modes with and without a ZIM is identified by calculating the correlation coefficients of field profiles in the disordered structure.

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The above calculation results and analysis reveal that the ZIM based on all-dielectric PhC can significantly improve the in-plane emission directionality of the random optical modes, meanwhile maintaining the intrinsic multimode and broadband nature of a disordered system. In addition, the Q factors can be improved with the aid of off-normal reflection, which is beneficial for low-threshold random lasing.

4. Directionality analysis of the random optical modes

To have a quantitative understanding of the in-plane far-field improvement, the output directionality of an optical mode is defined by

From Fig. 4(a), we can observe that the ${D_{\textrm{max}}}$ of random optical modes in a disordered structure outlines a flat envelope over the interested wavelength range. Most of the directionality indexes are in the range of 3-5 with an average value of ∼4. For the same disordered structure with zero-index PhC in its periphery, the ${D_{\textrm{max}}}$ of random optical modes is increased almost over the whole spectral range. The maximum output directionality enhancement factor can reach 30, which is increased by ∼6 times compared to that of the original structure. The directionality enhancement gets more remarkable when the wavelength moves towards the Dirac wavelength of PhC and the corresponding effective refractive index approaches zero. The directionality index dramatically drops off at a critical point of ∼99 µm, which corresponds to the maximum frequency of the TM3 band. As the TM3 band is nearly flat, the random optical modes with large off-normal incident angle will couple to this band and then exit the photonic structure. This leads to deteriorated emission directionality of random modes. In contrast, the random optical modes only couple to TM4 band when the wavelength is shorter than 99 µm. As the TM4 band is linear, isotropic, and close to the Γ point, the random wave coupling occurs for a small incident angle with regard to the interface of PhC and disordered structure. As a result, the random optical modes mainly emit to the longitudinal and transverse directions with a narrow divergence angle.

 

Fig. 4. (a) Statistically calculated ${D_{\textrm{max}}}$ for random optical modes at the wavelength range from 78 µm to 103 µm. Obviously, ${D_{\textrm{max}}}$ of modes with zero-index PhC is much larger, especially near the Dirac point wavelength. (b) The far-field superposition intensity of random optical modes among 95 µm to 99 µm, where the directionality enhancement is the most pronounced. (c) The far-field superposition intensity of random optical modes among 80 µm to 99 µm.

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As the multimode and broadband operation is the key feature of random lasers, we calculated the electric field superposition of all the random modes in the far field within a spectral range from 95 µm to 99 µm. As shown in Fig. 4(b), disordered system without ZIM emits in almost all directions. By contrast, the emission directionality is sharply enhanced when the ZIM is included. The emission power is concentrated on the transverse and longitudinal lobes with significantly improved intensity. An even smaller divergence angle of ∼3.5° is observed. This effect will contribute to the high collection efficiency for broadband random emission, which is beneficial for on-chip integration applications. Although spatial hole burning, gain competition, and many other complex dynamics may exist in a random laser, the calculation without considering the relative emission intensity of practical random lasing modes still implies the effectiveness of a ZIM for directionality improvement due to its broadband property. The random laser may also have a broader emission spectral range if the gain material with a large gain bandwidth is used. So the superimposed far-field intensity of all the random optical modes from 80 µm to 99 µm is also calculated. As shown in Fig. 4(c), the result still shows improved output directionality with suppressed emission to the directions around 45°, 135°, 225°, 315°. This further validates the broadband directionality enhancement of ZIM based on all-dielectric PhC.

5. Statistical analysis for disordered structures with different FFs

To confirm the in-plane directionality improvement applies to the random optical modes of different disordered structures, statistical analysis has been conducted for FFs ranging from 20% to 35% with five randomly generated patterns for each FF. The FF is an important parameter of a disordered structure, which shifts the optimal operation wavelength range with a tighter mode profile and higher Q factor for random lasing [15]. By comparing the results in Fig. 4(a) and Fig. 5(a), one can first conclude that the different random patterns produce similar optical properties, as the wavelength distributions of optical modes and the trend of output directionality are essentially the same. Meanwhile, one can see that the insensitivity of random mode properties to the pattern generation is preserved when the ZIM based on all-dielectric PhC is introduced for directionality enhancement. Obviously, the wavefront tailoring effect of zero-index PhC also applies to other FFs. The most prominent directionality enhancement of random optical modes always appears at the same wavelength range. This indicates that only the zero-index PhC contributes to the directionality enhancement.

 

Fig. 5. (a)-(d) ${D_{\textrm{max}}}$ of FF = 20%, 25%, 30%, and 35%, respectively. The results of five different random patterns are plotted in each figure. The orange and blue dots represent the ${D_{\textrm{max}}}$ with and without ZIM, respectively.

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6. Directionality enhancement for other disordered structures

To further verify the generality of our proposal, we have also explored the angular property of random mode output from another two typical structure designs of THz QCRLs. Metallic pillar has high scattering efficiency with low ohmic losses at the THz frequency range even if it is embedded in the high-index background of semiconductor gain materials [16]. Therefore, the random optical modes in this system are featured with strongly localized modes over a broad spectral range. We replace the random dielectric pillars with metallic pillars while keeping structure size, FF and pillar radius the same as in section 3. The typical mode is less leaky and its emission is divergent at a certain range of polar angles, as shown in Fig. 6(a). Statistical simulation in Fig. 6(c) shows that the ${D_{\textrm{max}}}$ of random optical modes in this system has a large variation in the range of 2.5 to 7. However, when the zero-index PhC is introduced in the periphery, the random optical mode emits to 0° and 90° with a divergence angle smaller than 8°. Similarly, the field distribution inside the random medium remains almost intact. This far-field beam collimating effect also works most efficiently at ∼99 µm with a directionality enhancement factor up to 30.

 

Fig. 6. (a),(b) Near and far-field profiles of random optical modes in disordered structures with metallic pillar and air hole as scatterers, respectively. (c),(d) The output directionality enhancement factor of random optical modes for random metallic pillar and air hole structures with and without ZIM, respectively.

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Another classical design for the on-chip random laser is the disordered structure with air holes randomly distributed in the high-index background of semiconductor gain materials [24]. By further replacing the random metallic pillars with air holes and maintaining other geometry parameters unchanged, we have calculated random optical modes over a broad frequency range. As depicted in Fig. 6(b), most optical modes are extended across the whole structure due to their lower scattering strength. Therefore, the output covers almost 360°of polar angle with no dominant far-field lobe. Similarly, the zero-index PhC collimates the output to the longitudinal and transverse directions with a small divergence angle. The directionality enhancement is less efficient than the metallic pillar case, which is due to the fact that the more extended modes tend to leak through four output channels. Whereas, the broadband directionality enhancement still applies to this weakly scattering system.

7. Output direction manipulation of random optical modes

In addition, we also explore the possibility to manipulate the output direction of random optical modes by using the ZIM based on all-dielectric PhC. As the electromagnetic waves experience little phase change when propagating through the low-index medium, the emission wavefront of random optical modes can be altered by changing the geometry of ZIM in the periphery of a disordered structure. In Fig. 7, the zero-index PhC was arranged as a triangular prism adjacent to a random pillar structure. The random optical waves exit the structure with wavefronts nearly parallel to the boundary of PhC, no matter for strongly localized modes or extended modes. Therefore, the output direction is ∼45°, as indicated by the arrow. The plane wave incidence has been implemented for wave manipulation by ZIM in many other reports, whereas this is the first application for the emission directionality as well as direction control of random optical modes.

 

Fig. 7. Output direction control of random optical modes by using the zero-index PhC. The PhC shaped to be a triangular prism is in adjacent to one side of the disordered structure. The red arrow indicates the direction normal to the PhC boundary. (a) Localized mode. (b) Extended mode.

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8. Conclusion

In conclusion, we have demonstrated a simple, effective, and universal approach to tuning random optical modes’ in-plane emission in a disordered structure. By juxtaposing a 2D ZIM based on all-dielectric PhC next to a disordered dielectric structure, we achieved broadband emission directionality enhancement for random optical modes, while the random properties are preserved and the Q factors are even slightly improved. The ZIM works most efficiently when the wavelength approaches the top of the flat quasi-longitudinal TM3 band of PhC. The maximum output directionality enhancement factor can reach ∼30, increased by 6 folds over that of random modes without ZIM. For a single random optical mode, the divergence angle is ∼6°and can be further reduced to ∼3.5° for far-field superposition of multiple random optical modes over a frequency range. The approach can also be applied to disordered medium with different structural parameters and scatterer inclusions for strongly localized or weakly extended random optical modes. In addition, we demonstrated that the emission direction of random optical modes can be tuned by altering the boundary of ZIM. This directionality enhancement and direction control can be attributed to Snell’s law with wave propagation through a low-index medium. Although the research is conducted based on the structural configuration of THz QCRLs, the scheme can be in principle implemented to random lasers in other frequency ranges. The effectiveness of ZIM for broadband directionality enhancement of random optical modes will significantly boost the miniaturization of random lasers with higher power and spectral efficiency. By using a ZIM, more optical power from random emission modes with distinct wavelengths and uncorrelated phases will be directed to a limited angular range, which would reduce the spatial coherence of a random lasing system. The control of emission direction further provides flexibility for more complex integration with other functional devices, which can be potentially used for on-chip random sensing, spectroscopy, and multichannel signal processing applications.