1. Introduction

Lasing phenomenon in disordered media is one of the most significant and interesting topic in the field of laser physics and technique, and has been widely studied both experimentally and theoretically [1–14]. For more details, one can see a review article [15] and the references therein. It is worth mentioning that, by combining Maxwell’s equations with four level rate equations of electronic population, a useful theoretical model was presented for one-dimensional (1D) case [5], and was further extended to two-dimensional (2D) case [6]. Many properties of random lasing have been explored via this model [5–14].

All the previous works have focused on the random laser in the optical band. Can we construct a model to reveal random lasing effects in terahertz (THz) domain? The search for efficient, high-power, inexpensive, and compact methods of generation of coherent THz radiation is one of the main topics in modern optoelectronics and photonics [16]. In this work, by combining Maxwell’s equations with three-level rate equations of electronic population in ruby for 1D case, we build a model to reveal the THz random lasing in disordered medium made of ruby grains. The refractive index of ruby for THz radiation is twice larger than the one for optics [17], thus allowing the disordered ruby grains to provide stronger confinement for THz wave, being beneficial to the formation of the THz random lasing.

The conventional ruby lasing takes place between the ground state of the Cr³⁺ ion (${}^{4}A{}_2$) and its first excited state (${}^{2}E$). Due to the trigonal crystal field and spin-orbit coupling, the upper level ${}^{2}E$ is split to a pair of sublevels $2\overline{A}$ and $\overline{E}$ (29.14 cm⁻¹ or 0.87 THz apart), forming two primary lines often denoted by R1 (694.3 nm) and R2 (693.9 nm), respectively. We hence can suggest that 0.87 THz random lasing could be operated in ruby grains using the levels $2\overline{A}$ and $\overline{E}$ by pumping the grains from the ${}^{4}A{}_2$ level to the $2\overline{A}$ level via a ruby laser operating on its R2 line. In fact, the feasibility of 29 cm⁻¹ far-infrared lasing based on the pumping of the $2\overline{A}$ level of ruby via the R2 line was discussed [17]. The generation of coherent 0.87 THz pulses in bulk ruby based on a V-type energy scheme pumped by both R1 and R2 lines was also proposed [18]. It should be noted that, with the generation of 29 cm⁻¹ photons, this scheme also supports the interaction of the electronics states $2\overline{A}$ (${}^{2}E$) and $\overline{E}$ (${}^{2}E$) of the excited Cr³⁺ with the resonant 29 cm⁻¹ phonons, such as the 29 cm⁻¹ phonon “bottlenecked” effect at low temperatures [19] and the reabsorption procession of 29 cm⁻¹ phonons [20]. Besides, the generation of THz radiation by difference-frequency mixing of the R1 and R2 lines in ZnTe was reported [21]. Random laser is also called mirrorless laser. Recently, a kind of THz mirrorless laser was made by a quasi-crystal semiconductor [22].

In view of achieving THz random lasing, we calculate the wave propagation in 1D disordered ruby medium and obtain the time-domain dependence of the electromagnetic fields firstly, and then through Fourier transformation to get the emission spectra in which exist many peaks that correspond to the electromagnetic wave modes supported by the system. The computed results reveal a threshold gain behavior for the modes in the 1D system in THz domain, which indicates that THz random lasing could occur in the 1D disordered ruby medium. Our results provide some theoretical predictions for future possible experimental observation, and propose a method to search for efficient, inexpensive, and compact THz source.

2. Theoretical model

The binary layers of the system are consisted of two dielectric materials, as shown in Fig. 1(a) . The white layer with a random variable thickness$a_n$and a dielectric constant ε₁ = ε₀ simulates the air, while the black layer with a fixed thickness b=90 μm and a dielectric constant ε₂ = 4ε₀ simulates the scatters that are also the gain media with a three-level atomic system, and the scheme of the energy levels is shown in Fig. 1(b). The random variable $a_n$ is described as $a_n=a(1+w\gamma )$, where a =100 μm, wis the strength of randomness, and γ is a random value in the range [-0.5, 0.5].

 

Fig. 1 (a) Schematic illustration of 1D random medium made of ruby grains, and (b) the scheme of the energy levels for Cr³⁺.

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For the 1D time-dependent Maxwell equations and for an active and non-magnetic medium, we have

For the three-level atomic system shown in Fig. 1(b), the rate equations read

For such a system, the polarization$P_z$obeys the quantum population equation of motion

This equation links Maxwell’s equations with rate equations. $\Delta N=N_2-N_3$ is the population difference density between the levels $2\overline{A}$ and $\overline{E}$. Amplification takes place when the external pumping mechanism produces population inversion$\Delta N<0$. The line width of the atomic transition is $\Delta {\omega }_l=1/{\tau }_{32}+2/T_2$ where the collision time $T_2$ is usually much smaller than the lifetime${\tau }_{32}$. The constant κis given by$\kappa =6\pi {\epsilon }_0{c}^3/{\omega }_{32}^2{\tau }_{32}$.

The values of those parameters in the above equations that will be used in simulating the active part in the following numerical calculations are taken as: $T_2$ = 2 × 10⁻¹⁴ s, ${\tau }_{32}$ = 1.1 × 10⁻⁹ s, ${\tau }_{21}$ = 3 × 10⁻³ s, ${\tau }_{31}$ = 3 × 10⁻⁶ s, and $N_T$ = ${\displaystyle \sum _{i=1}^3{N}_i}$ = 1.6 × 10²⁵ /m³.

When pumping is provided over the whole system, the electromagnetic fields can be calculated. In order to model such an open system, a Liao absorbing layer [23] is used to absorb the outward wave. The space and time increments have been chosen to be$\Delta x$=2×10⁻⁶ m and $\Delta t$≤$\Delta x$/(2c), where $\Delta t$ is taken to be 2×10⁻¹⁵ s, respectively. The pulse response is recorded during a time window of length $T_w$=5.2×10⁻¹⁰ s at all nodes in the system and Fourier transformed in order to obtain the intensity spectrum.

3. Numerical results

Firstly the wavelength dependence of the electromagnetic field intensity in the 1D system is calculated at different pump rates. As can be seen from Fig. 2(a) , when the pump rate is quite low ($W_p$=10 s⁻¹), there are many discrete peaks near 344 μm (or 0.87 THz), each having quite weak peak intensity in the spectrum. Each peak corresponds to a lasing mode supported by the disordered medium, for which four peaks are indicated by their central wavelengths λ₀, λ₁, λ₂ and λ₃, respectively. As the pump rate increases, these peak intensities are nearly unchanged until the pump rate increases to a value ($W_p$=30 s⁻¹) at which only those indicated modes are effectively amplified, as shown in Figs. 2(b) and 2(c). When the pump rate $W_p$ further increases, the mode λ₀ dominates the spectrum, as shown in Fig. 2(d). This means the mode λ₀ perhaps has the minimum lasing threshold among the modes. Meanwhile, let us pay attention to the spectral width of the modes. The width becomes quite broader at $W_p$=30 s⁻¹ than those at lower $W_p$. As the pump rate further increases, accompanied by the increase of the peak intensity, the spectral width becomes narrower and narrower until to a stable value.

 

Fig. 2 The spectral intensity in arbitrary units versus the wavelength for 1D disordered ruby medium shown in Fig. 1 at (a) $W_p$ = 10s⁻¹, (b) $W_p$ = 30 s⁻¹, (c) $W_p$ = 100 s⁻¹, and (d) $W_p$ = 1000 s⁻¹.

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In order to obtain the information about the threshold gain behavior for the modes in detail, numerical calculations are performed at different pump rates from which we can obtain the curves of the peak intensity and the spectral width vs the pump rate, as shown in Fig. 3 . According to the traditional method, the pump thresholds for the four modes can be measured from the intensity curves in Fig. 3(a) as $W_{I0}$=30 s⁻¹, $W_{I1}$=60 s⁻¹, $W_{I2}$=50 s⁻¹ and $W_{I3}$=45 s⁻¹, which shows that different modes have different pump thresholds. Note that the mode λ₀ has the minimum lasing threshold, which is easy to understand because the central wavelength of the mode λ₀ is very near the transition wavelength of the active medium. The above numerical results clearly indicate that this 1D ruby disordered system could support random lasing phenomenon in THz domain.

 

Fig. 3 The plot of the peak intensity and spectral width of the lasing modes vs the pump rate $W_p$. (a) The peak intensity for the four indicated modes, and the lasing threshold measured from the plots are $W_{I0}$=30 s⁻¹, $W_{I1}$=60 s⁻¹, $W_{I2}$=50 s⁻¹ and $W_{I3}$=45 s⁻¹; (b) the peak intensity and spectral width for the mode λ₀; and (c) the spectral width for the four indicated modes, and the lasing threshold measured from the plots are $W_{W0}$=43 s⁻¹, $W_{W1}$=82 s⁻¹, $W_{W2}$=78 s⁻¹ and $W_{W3}$=61 s⁻¹.

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As can be seen from Fig. 3(b), within a pump regime near the pump threshold, a jump occurs for the spectral width for the mode λ₀. The peak value of the jump appears at the point that is very near the threshold of the mode. A method was proposed to determine the lasing threshold for a 2D random laser based on the spectral width [11]. That is, the lasing threshold is defined as such a pumping energy at which the spectral width becomes half of its maximal value. This method was used to analyze the threshold gain behavior for random lasing in a 2D disordered medium [8]. Based on this method and the width curves in Fig. 3(c), the thresholds are measured as $W_{W0}$=43 s⁻¹, $W_{W1}$ = 82 s⁻¹, $W_{W2}$ = 78 s⁻¹ and $W_{W3}$ = 61 s⁻¹. Obviously, the results from the two methods are consistent. Note that for a given mode, the specific value for its pump threshold depends on the defining fashion.

In order to check whether the results above are universal or not, we take huge amount calculation for different combinations of system parameters. All the results demonstrate that the conclusion we have obtained is universal.

It is important to note that the proposed method can be applied to other solid materials with suitable transitions in the THz range, such as Cr³⁺:BeAl₂O₄ (alexandrite), having R line splittings of 41 cm⁻¹ (1.23 THz) [24]. The bulk ruby was proposed as a coherent THz source based on a V-type energy scheme, which needs two monochromatic optical beams (694.3 nm and 693.9 nm) as pump beams [18]. In contrast to this, the proposed method in this work only needs one monochromatic beam at 693.9 nm as the pump beam. The obvious advantages of the proposed THz emitter over other THz sources are compactness, cheap, diminutive, and ease of handling, thus opening a way for orders of magnitude increase in THz tomography, nondestructive quality control, medical diagnostics, biomaterial characterization, and so on. A rigorous challenge is the proposed THz emitter could work better at low temperature 4.2K because the absorption coefficient of ruby for the 344 μm wavelength is about 0.4 cm⁻¹ for the ordinary ray and 0.5 cm⁻¹ for the extraordinary ray at 300K, and less than 0.01 cm⁻¹ at 4.2K [17]. It is necessary to perform some experiments to search some ways to overcome this weak point.

Conclusions

In this work, we build a novel model for 1D active disordered medium made of ruby grains with a three-level atomic system to discuss random lasing phenomenon in THz domain. Our results suggest that THz random lasing could occur under suitable conditions. Our findings are significant for expanding random laser concept into THz domain and have potential application to construct new type of THz source.