1. Introduction

Random lasers are among the most attractive and fascinating applications in disordered nanophotonics [1]. A random laser is basically a finite and open system constructed by a random assembly of scatterers dispersed into an optical gain medium [2]. Light is multiply scattered and amplified in the disordered medium. As the multiple scattering is completely random [3], the output usually exhibits an angular distribution over the full solid angle of 4π [4]. This special property makes random lasers in principle ideal for display and lighting applications [4]. However, the high pump threshold limits their practical applications [5]. Because the three primary colors (blue, green and red) are usually used in displaying color images, wavelength designability is necessary for practical display application of random lasers, too. Since the role of the cavity of a regular laser to trap light is taken over by the multiple scattering processes [6], it is impossible to tune the random lasing wavelength by adjusting the resonance frequency of a cavity as in regular lasers [2]. However, multiple scattering is the conditio sine qua non for random lasers [1], thus it is possible to control random lasing properties, in particular the lasing wavelength and the pump threshold by designing the scattering properties of every single scatterer.

Superscatterers [7–11 ] are artificial structures having very large total scattering cross-section (SCS), far exceeding their single resonance limit (2λ/π for 2D scatterers and 3λ ²/2π for 3D scatterers [7,8]). As the scattering properties of a superscatterer could be designed according to the actual need, it is a perfect candidate for controlling multiple scattering in random systems as well as the random lasing properties.

In this work, we adopt superscatterers proposed by Zhichao Ruan and Shanhui Fan [7] to construct random systems with optical gain media, and show how this kind of superscatterers affect multiple light scattering and random lasing properties including the lasing wavelength and the pump threshold. We first design three kinds of superscatterers with their superscattering wavelengths lying on red, green and blue, respectively. Then we use these superscatterers to compose random systems separately and analyze their multiple light scattering and random lasing properties. We find that the random lasing threshold could be lowered by two orders of magnitude (in our random configuration) at all trichromatic superscattering wavelengths in respect to other wavelengths or compared to random systems with conventional scatterers. The influence of metallic losses on scattering and lasing threshold is also discussed.

2. Design of superscatterers with specific wavelengths

The superscatterer used is a 2D metal-dielectric-metal heterostructure [7]. Superscattering of this structure only happens for TM waves. We first assume that the metal is lossless silver and the permittivity is described by the Drude model with $\epsilon =1-{\omega }_p^2/\left({\omega }^2+i{\gamma }_d\omega \right)$, where ω_p is the plasma frequency, ω is the frequency of the light wave, and γ_d is the damping constant. Because the superscattering wavelength of this kind of superscatterer is around the surface plasmon frequency, ω_sp ( ${\omega }_{\mathit{sp}}={\omega }_p/\sqrt{1+{\epsilon }_d}$, ε_d being the permittivity of the dielectric layer), of the surface plasmon polaritons at the interface of dielectric and metallic layers, we could achieve an intended superscattering wavelength by choosing the appropriate permittivity of the dielectric layer. Three different superscatterers at trichromatic wavelengths are designed, whose parameters are represented in Table 1. A comparison of the light scattering spectra, i.e., the total SCSs versus wavelength, of the three types of superscatterers with lossless metallic rods and dielectric rods is shown in Fig. 1(a). A sharp superscattering peak (maximum total SCSs) centered at the superscattering wavelength is superposed on a broad superscattering band for each specific designed superscatterer. Figure 1(b) presents the zoom-in view of the scattering peak of the three superscatterers. The sharp peaks have a full width at half maximum (FWHM) smaller than 1 nm, and the FWHMs of the broad bands are about 30 nm. The superscattering peaks and bands could be attributed to the surface plasmon resonances in higher and lower angular momentum channels respectively [7]. The superscattering peaks are at 700.2 nm, 546.1 nm, and 435.8 nm, respectively, which are the wavelengths of the monochromatic primary colors in the CIE 1931 color space. The total SCSs of the superscatterers are much larger than those of metallic rods and dielectric rods at the corresponding wavelengths. Some valleys appear on the superscattering bands, which result from the Fano resonance between different angular momentum channels.

 

Fig. 1 (a) Light scattering spectra of the three designed superscatterers. (Inset) Scheme of the superscatterer, dark and light grey regions correspond to metallic and dielectric materials, respectively. The magenta and black dashed lines are the total SCSs of the lossless silver rod and ZnO rod with a diameter of 80 nm, respectively. (b) The zoom-in views of the three scattering peaks of the scattering spectra. The total SCSs are calculated by the finite element method software package(Comsol Multiphysics)

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Table 1. Parameters for the three superscatterers.(λ_p = 2πc/ω_p)

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3. Lasing thresholds in random systems based on quasi-stationary leaky modes

Random systems can be constructed with these three types of superscatterers and gain media. In a finite open random system, light is multiply scattered and can be described by a set of quasi-stationary leaky modes [5,12] or Anderson localized modes [13]. When sufficient optical gain is introduced into such systems, the leakage of one or more modes will be completely compensated, and then lasing action takes place [5, 12]. Therefore it is the quasi-stationary leaky mode that determines the radiation direction, frequency and pump threshold of a random laser. The mode with the smallest energy loss or highest quality factor is most prone to lase [5]. As the quality factor Q is inversely proportional to the energy loss rate of a mode, the inverse quality factor (1/Q) could be referred to as the lasing threshold [5, 12].

The quasi-stationary leaky mode could be characterized by the projected local density of state (PLDOS) spectrum given by [13–16 ]

In order to investigate the quasi-stationary leaky modes and random lasing threshold numerically, we consider systems formed by random assembly of 2D scatterers (three types of super-scatterers, metallic rods and dielectric rods, respectively) dispersed into some kind of dielectric media. The configuration of a random system is composed of 50 scatterers (superscatterers shown in Table 1, lossless silver rods or ZnO rods) in a circle region with a radius of 5 μm. The permittivity of the material around the scatterers is 1. Figure 2(a) illustrates such a 2D system with superscatterers. Rods are infinitely long in z direction, and light waves are assumed to be TM waves propagating perpendicular to z direction. Figure 2(b) shows a top view of the random system. Random systems with metallic rods or dielectric rods are the same as those in Fig. 2. The PLDOS spectra are calculated by the finite element method software package(Comsol Multiphysics). A dipole current source, J(r, ω, p̂) = δ(r − r ₀)e^iωt, at r ₀ is introduced into the random system (the red dot in Fig. 2(b)). A perfect matched layer (PML) is used to absorb the light going out, which means that the light is transporting and scattered in an open region. We compute the power radiated by the dipole current source, P(r ₀, ω, p̂), then the PLDOS is given by [15]

 

Fig. 2 (a) Stereogram sketch and (b) the top view of the random system with superscatterers. The dark and light grey regions correspond to metallic and dielectric materials, respectively.

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Figure 3(a) shows the PLDOS spectra of random systems with trichromatic superscatterers, metallic rods and dielectric rods respectively, that are calculated in the center of the circle area with the direction p̂ of the dipole current along the +y direction. One can see that all PLDOS spectra are composed of a set of peaks, each of them being a signature of the quasi-stationary mode. The section around the wavelength where there exists the maximum total SCS for green superscatterers is magnified and displayed in Fig. 3(b). The peaks in Fig. 3(b) could be perfectly fitted by a sum of the Lorentzian function series, as revealed by Eq. (1). Thus the quasi-stationary leaky modes can be characterized by the fitted central frequency ω_m, the effective line width Γ_m, the Purcell factor $F_p^m$, and the Q factor Q_m = ω_m/Γ_m. Figure 3(d) represents the Q factors extracted from the Lorentzian fitting of the PLDOS spectra. In the PLDOS spectrum related to the green superscatterers, there are many narrow peaks around 546.03 nm, shown in Fig. 3(c), whose widths are so small that their Q factors far exceeds all the others. But the energy of these modes mainly exists inside the superscatterers, which means that they are the resonance of surface plasmon polaritons inside the superscatterers. Thus these modes will not make considerable contribution to the random lasing processes. Therefore the Q factors of these modes are not taken into account in Fig. 3(d). Figure 4(a)–4(c) show the electric amplitude distributions (EADs) of the quasi-stationary leaky modes with the largest Q factors of the random systems with superscatterers of Type Blue, Type Green and Type Red, respectively. Figure 4(d) and 4(e) show EADs of the quasi-stationary leaky modes of the random systems with metallic rods at 620.8 nm and with dielectric rods at 514.6 nm respectively. All these five random systems have same distributions of scatterers, but their EADs are different due to the different scattering properties of corresponding scatterers. The central dark spot in each picture is the dipole current source.

 

Fig. 3 (a) PLDOS spectra. (b) A magnification of the green line in the dashed box in (a). (c) A magnification of the PLDOS in the dashed box in (b). (d) Q factors of the random systems. Blue, green, red, magenta and black colors correspond to random systems with trichromatic superscatterers, metallic rods and dielectric rods, respectively.

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Fig. 4 (a)–(c) The EADs of the modes with the largest Q factors of the random systems with superscatterers of Type Blue, Type Green and Type Red, respectively. (d) and (e) EADs of the quasi-stationary leaky modes of the random systems with metallic rods at 620.8 nm and with dielectric rods at 514.6 nm, respectively.

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The Q factors for a random system with superscatterers are much larger than those with conventional scatterers. This is caused by the larger total SCSs of superscatterers compared to conventional scatterers. The Q factors of modes inside the superscattering band for a system with superscatterers could be one order of magnitude larger than those of modes with their central wavelengths outside the superscattering band in the same system or of modes belonging to random systems with conventional superscatterers. Especially, the modes inside the superscattering peak have much larger Q factors than modes outside the superscattering band and the largest Q factor could be two orders of magnitude larger. This could be attributed to the resonance between the quasi-stationary modes and the surface plasmon inside the superscatterers.

As mentioned above, the inverse quality factor (1/Q_m) could be referred to as the lasing threshold. The lasing threshold of the random system with superscatterers and gain media is two orders of magnitude lower at the trichromatic superscattering wavelengths with respect to other wavelengths or comparing to those with conventional scatterers.

4. Laser actions of the random systems

In order to investigate the lasing characteristics, we introduce a gain into the random systems, assuming the medium around the scatterers to be an optically active material. The population inversion density of the optically active material can be modeled by the negative imaginary part of the relative permittivity −γ (γ > 0) [18]. Thus we assume that the relative permittivity of the active material is 1 + i(−γ). We set an electric point dipole oscillating in y direction as the spontaneous light source at the center of the 2D random system (red dot in Fig. 2(b)). Using the finite element method software package (Comsol Multiphysics), the radiation spectrum of the random system is calculated, which is defined by the fluxes of the Poynting vectors of light directed outward of the 2D random system: $F={\int }_0^{2\pi }\Re \left(\frac{\mathit{E}\times {\mathit{H}}^{*}}{2}\right)\cdot {\mathit{n}}_{\mathit{out}}d\theta$, where n_out is the unit outward normal vector of the circle region, as shown in Fig. 2(b). Figure 5 displays the radiation spectra of the active random systems with γ = 0.0027. There are obvious lasing peaks (435 nm, 546 nm, 700 nm) in the spectra radiated by the modes with the largest Q factors in the systems with superscatterers, while no lasing action occurs in the random systems with metallic rods or ZnO rods.

 

Fig. 5 Radiation spectra of the random systems with gain media. Blue, green, red, magenta and black curves correspond to random systems with trichromatic superscatterers, metallic rods and dielectric rods, respectively.

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As the largest Q factor for a random system with superscatterers lies in the superscattering peak, one could design superscatterers with a specific superscattering wavelength and utilize them to construct random lasers with their lasing wavelength at that specific wavelength. Since three primary colors are the basis for color displaying, we have designed random lasers with trichromatic lasing wavelengths.

5. The influence of metallic loss

In fact metals are lossy, which will greatly affect the superscattering properties. As the core and shell sizes of the superscatterer are smaller than the bulk electron mean free path in silver, the size-dependent electron scattering will contribute to the permittivity and the damping constant in the Drude model should be modified as γ_d = γ_bulk + A × V_F/l_r, where γ_bulk = 0.002ω_p, A ≈ 1, V_F = 7.37 × 10⁻⁴ λ_pω_p, l_r = ρ ₁ for the metallic core, and l_r = ρ ₃ − ρ ₂ for the metallic shell [7, 19, 20]. Here we consider only the green superscatterer. The damping constants γ_d are equal to 0.0036ω_p and 0.0121ω_p for the metallic core and shell respectively. The black solid line in Fig. 6(a) is the scattering spectrum of green superscatterers. There only exists a broad superscattering band, while the narrow superscattering peak in the lossless case disappears, since the loss rate dominates over the radiation leakage rate in the higher angular momentum channels contributing to the superscattering peak [7]. The colored squares in Fig. 6(a) represent the Q factors of four different random systems constructed by green superscatterers, which show that Q factors increase with the total SCS statistically. The largest Q factor lies inside the superscattering band. For example, the green squares with the largest Q factor appearing at 522.4 nm. In some configurations, the enhancement behavior may be relatively weak, as indicated by the magenta squares on the left side of the dashed line, which might arise from only 50 superscatterers in the random system. The stars in Fig. 6(a) represent the Q factors of the random system with the lossy silver rods, which shows that these Q factors are obviously smaller than those of the random systems with lossy trichromatic superscatterers. Figure 6(b) and 6(c) show EADs of the quasi-stationary leaky modes of the random systems with lossy supperscatters at 541.3 nm and with lossy silver rods at 575.0 nm, respectively. Two random systems have different distributions of scatterers.

 

Fig. 6 (a) Q factors of random systems with superscatterers of type Green (Colored squares) compared with those of the lossy silver rods (Stars). Black solid line is the scattering spectrum of the green superscatterers. (b) and (c) EADs of the modes of the random systems with lossy supperscatterers at 541.3 nm and with lossy metal at 575.0 nm, respectively.

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6. The influence of nonuniformity and misalignment of the superscatterers

The theoretical analysis and numerical simulations should help us to perform experiments, therefore it is definitely necessary to discuss the experimental difficulties. In a real random system, one must take into account the nonuniformity and misalignment of the superscatterers.

For nonuniform superscatterers, each superscatterer has its own SCS. From numerical analysis, we know that the total SCS varies much more greatly with the thickness of the dielectric layer than with the thickness of the outer metallic layer and the radius of the inner metallic sphere. Thus we consider only the influence of the dielectric layer thickness. The thickness of the dielectric layer is assumed to have a normal distribution with an expectation μ and a standard deviation σ. Then a polydispersity index can be defined as 2σ/μ. We construct different random systems with the same distributions but different polydispersities of the superscatterers. Figure 7(a) shows the change of the Q factor at 522.4 nm in Fig. 6(a) with the polydispersity. When the polydispersity is smaller than 3%, the Q factor keeps larger than 110 and the shift of the resonant wavelength is smaller than 1.5 nm. When the polydispersity is larger than 3%, the Q factor will decrease drastically. That is to say, if the polydispersity of the superscatterers can be controlled within 3%, the experimental results will be consistent with the numerical simulations.

 

Fig. 7 Q factor and resonant wavelength of random systems versus (a) the polydispersity and (b) location deviation of the superscatterers.

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For the misalignment of the superscatterers, we construct different random systems of disturbed distributions of superscatterers, whose random deviations are normally distributed with a standard deviation σ′. Figure 7(b) presents the variation of the Q factor at 522.4 nm with σ′. When the deviation is less than 10 nm, the Q factor is larger than 120 and the change of the resonant wavelength is smaller than 1.8 nm. When the deviation is larger than 10 nm, the Q factor will decrease rapidly. In addition, the largest Q factor might occur at another wavelength in the superscattering band. Therefore, if the location deviation of the superscatterers can be controlled within 10 nm, the experimental results will coincide with the simulations.

7. Conclusion

In summary, we have introduced superscatterers into active random systems and numerically demonstrated that the multiple scattering and lasing properties in such systems can be controlled by specially designed superscatterers. In the scattering spectrum of superscatterers, a narrow superscattering peak is superposed on a broad superscattering band. The random lasing threshold of the random system with lossless superscatterers has been lowered by two orders of magnitude at trichromatic wavelengths in respect to other wavelengths or comparing with a random laser with conventional scatterers. Though the narrow superscattering peak disappears for lossy superscatterers, enhancement of Q factors inside the superscattering band is observed. Based on this special dependence of the random lasing wavelength on the total SCS, we could design the random lasing wavelength according to the actual need by utilizing superscatterers with designed superscattering wavelength. The lasing wavelength designablity and low pump threshold are advantageous for the practical applications such as display and lighting.