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Random lasing with spatially nonuniform gain
Optics Communications 371 (2016) 213–220
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Random lasing with spatially nonuniform gain Ting Fan a, Jiantao Lü b,n a b
School of Electronics and Information Engineering, Foshan University, Foshan 528000, China School of Science, Foshan University, Foshan 528000, China
art ic l e i nf o
a b s t r a c t
Article history: Received 12 January 2016 Received in revised form 4 March 2016 Accepted 28 March 2016
Spatial and spectral properties of random lasing with spatially nonuniform gain were investigated in two-dimensional (2D) disordered medium. The pumping light was described by an individual electric field and coupled into the rate equations by using the polarization equation. The spatially nonuniform gain comes from the multiple scattering of this pumping light. Numerical simulation of the random system with uniform and nonuniform gain were performed both in weak and strong scattering regime. In weak scattering sample, all the lasing modes correspond to those of the passive system whether the nonuniform gain is considered. However, in strong scattering regime, new lasing modes appear with nonuniform gain as the localization area changes. Our results show that it is more accurate to describe the random lasing behavior with introducing the nonuniform gain origins from the multiple light scattering. & 2016 Elsevier B.V. All rights reserved.
Keywords: Laser physics Random laser Nonuniform gain
1. Introduction Random lasing phenomenon is an attractive topic in the past decades after originally predicted by Letokhov [1]. The most difference between random laser and conventional one is that modes of random lasers are determined by multiple scattering and not by a laser cavity. To call a material a random laser, the multiple scattering process must play a determining role and the interference process determines the mode structure of it [2]. It is difficult to develop a theory that can describe all aspects of a random laser. A complete model has to include the dynamics of the gain mechanism and the interference effects. Researchers have developed different kinds of models for the random lasers, including the diffusive model [3], the time dependent theory [4], self-consistent multimode random laser theory [5], and so on. By combining Maxwell's equations with rate equations of electronic population, Jiang has developed the time dependent theory for one-dimensional (1D) case, and Sebbah extended it to two-dimensional (2D) case [6]. This model has successfully explained some experimental phenomenon [7–11]. In the previous work, gain is introduced in the system by pumping the four level atoms uniformly over the whole system. However, recent investigations have demonstrated that nonuniform gain can lead to the creation of new lasing mode [12,13], and local pumping allows the selection of spatially nonoverlapping modes [14,15]. In Jiang's n
Corresponding author. E-mail address: [email protected] (J. Lü).
http://dx.doi.org/10.1016/j.optcom.2016.03.078 0030-4018/& 2016 Elsevier B.V. All rights reserved.
model [4], optical gain is expressed by the pumping rate of the rate equations for electronic population, which can be written as the function of the position in the random media. One can change the pumping rate of different position and some new characteristics of random lasers were investigated, including the localized gain [7] and adaptive pumping [15]. But this is the artificial method to control the gain behavior for some special purpose, such as single mode selection. In the common random laser system, a lasing pulse is usually incident on the sample with a specific angle and serve as the pumping light. As the pumping light is irradiated onto the sample, it will be scattered in the random media and form some spatial distribution modes, which lead to the spatially nonuniform gain. For this reason, it is more accurate than using the uniform gain in the whole random sample. The limitation of Jiang's model is that he only used the phenomenological method to describe the gain behavior. Although it allows us to change the gain value for different positions, this method cannot truly represent the distribution of the gain value. In order to obtain the real spatial distribution of the pumping light, we must use an individual electrical field to describe the pumping pulse. Based on the fundamental laser theory, we suppose the gain coefficient is proportional to the strength of electric field. Once the spatial map of the pumping light is formed, it can be coupled into the rate equations by using the polarization equation to describe the stimulated absorption process. Our method not only provide a more accurate way to study the characteristic of the random laser, but also open a new path to investigate the relationship of the spatial modes between the pumping light and the emission one.
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Fig. 1. Spatial distribution of the electromagnetic-field intensity (left) and steady-state emission spectra (right) for uniform pumping with different pumping rates. (a) and (d) Epeak = 1 × 1010 V/m , (b) and (e) Epeak = 5 × 1010 V/m , and (c) and (f) Epeak = 1 × 1011 V/m .
In this paper, we report the computational results of the mode structure of the random laser with spatially nonuniform gain, which is caused by the multiple scattering of the pumping light. The rest of the article is organized as follows. In the next section we present the random media and the theoretical model. Section 3 shows the details of the numerical simulation results and the
interpretations. 2. Theoretical model We consider a 2D square random system with size L2 in the x–y plane. It consists of circular particles with a radius r and refractive
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transition rate is given by the term (E2/=ω2 )⋅(dP2/dt ) where ω2 is the transition frequency between level 2 and 1 and is chosen as 2π × 5.13 × 104 Hz (λ2 = 585 nm). The most difference between the previous work and ours, we use the term (E1/=ω1)⋅(dP1/dt ) in the rate equations to describe the excitation process instead of the pumping rate Wp , where ω1 is the center frequency of the excitation light and chosen as 2π × 5.64 × 104 Hz (λ2 = 532 nm). By coupling the polarization equation and rate equations, the excitation pulse can be described with an individual electric-field intensity. Therefore, the electrical field of the pumping pulse E1 and that of the emission one E2 were represented separately in the system. We considered a 2D transverse magnetic (TM) field in the x–y plane, thus the Maxwell's equations read ( i = 1, 2)
μ0
∂Hix ∂E = − iz ∂t ∂y
(5)
μ0
∂Hiy ∂E = iz ∂t ∂x
(6)
εj ε0
Fig. 2. Spatial distribution of the atomic population difference density ΔN between the upper and lower lasing levels for weak scattering regime.
index n2, which are randomly distributed in a background medium with a refractive index n1. The scattering strength is varied by adjusting the refractive index contrast Δn = n2 − n1 and the surface-filling fraction Φ , while Φ = Nπr 2/L2 can be modified by changing the particle radius r and amount N . This system can be considered as the multiple scattering light is confined in the x–y plane among a 3D sample, which resulting in a quasi-2D type of light transport. In order to control the randomness and the gain independently, the background medium is chosen as the active part and modeled as a simplified four-level atomic system. The electrons in the ground level 0 are transferred to the upper level 3 by an external laser pulse. Electrons in level 3 flow downward to level 2 by means of nonradiative decay process with a characteristic time τ 32 = 10−13 s. The depopulation of highest excited state to ground state through the spontaneous emission is not considered. The non-radiative transition from level 3 to 0 and level 2 to 1 are also neglected. The intermediate level 2 and 1 are the upper and lower levels of the laser transition, respectively. The lifetime of these two levels are τ 21 = 10−10 s and τ10 = 5 × 10−12 s , respectively. We describe the dynamics of atomic populations by rate equations as follow:
dN3 N E dP = − 3 + 1 ⋅ 1 dt τ 32 =ω1 dt
(1)
dN2 N E dP N = 3 − 2 ⋅ 2 − 2 dt τ 32 =ω2 dt τ 21
(2)
dN1 N E dP N = 2 + 2 ⋅ 2 − 1 τ 21 =ω2 dt τ10 dt
(3)
dN0 N E dP = 1 − 1 ⋅ 1 dt τ10 =ω1 dt
(4)
∂Hiy ∂Eiz ∂P ∂Hix + iz = − ∂t ∂t ∂x ∂y
The polarization density equations
(7) Pi (i = 1, 2) obeys the following
d2P1 dP + Δω1 1 + ω12P1 = κ1 (N0 − N3 ) E1 dt 2 dt
(8)
d2P2 dP + Δω2 2 + ω22P2 = κ 2 (N2 − N1) E2 dt 2 dt
(9)
where Δω1 = 6.67 × 1012 Hz is the linewidth of the excitation light, Δω2 = 1/τ 21 + 2/T2 is the linewidth of the atomic transition, T2 = 2 × 10−14 s is the collision time. The constant κi (i = 1, 2) is given by κ 2 = 6πϵ0 c 3/ω12 τ 32 and κ 2 = 6πε0 c 3/ω22 τ 21, respectively. By coupling the electric-field of the excitation pulse and population density in atomic levels through the polarization density equation, we can describe the pumping process by the external excitation pulse, once the excitation ended, and the pumping process stopped. Amplification occurs when the external pumping mechanism produces population inversion, i.e. N2 − N1 > 0. The excitation pulse is introduced as a Gaussian in temporal regime
⎛ 4π (t − t0 )2 ⎞ E1 (t ) = Epeak cos (ω1t ) exp ⎜ − ⎟ ⎝ ⎠ τ2
(10)
where τ is the width of the Gaussian pulse, Epeak is the peak value of the pumping, and t0 is the time corresponding to the peak value. The electromagnetic fields in the 2D active random medium can be obtained by FDTD methods. Perfectly matched layer (PML) is used to model an open system as absorbing conditions.
3. Results and discussion
where Ni is the population density in level i, i¼ 0–3. The stimulated
In this section, we will study the mode structure of a random laser when considering the spatial nonuniform of the excitation light, which comes from the multiple scattering in the disordered media. We select a 2D sample with S = 4 × 4 μm2 and r = 50 nm . A laser pulse with the TM polarization was introduced onto the 2D sample as the excitation source. The parameters of the excitation pulse is chosen as follow: τ = 10 ns, t0 = 5 ns. The gain of the whole system can be controlled by adjusting the peak value of the pumping pulse Epeak . The input light was scattered by the particles and form a spatially nonuniform distribution. In some area, the
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Fig. 3. Spatial distribution of the electromagnetic-field intensity (left) and steady-state emission spectra (right) for nonuniform gain with different pumping rates. (a) and (d) Epeak = 1 × 1010 V/m , (b) and (e) Epeak = 5 × 1010 V/m , (c) and (f) Epeak = 1 × 1011 V/m .
local gain value is high where the pumping light is strong. For each set of FDTD simulations, the spatial distribution of the field intensity of the pumping light and stimulated emission were obtain. The intensity spectrum of the laser field can be obtained by integrating the electrical-field intensity of the whole system and
carrying out the Fourier transformation. In a 2D random system, the scattering strength is controlled by the surface-filling fraction Φ and the refractive index contrast Δn = n2 − n1. Once these two parameters exceed the respective critical value, the transition from diffusion state to localization
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Wavelength (nm) Fig. 4. Spatial distribution of the electromagnetic-field intensity (left) and steady-state emission spectra (right) for uniform gain with different pumping rates. (a) and (d) Epeak = 1 × 1010 V/m , (b) and (e) Epeak = 5 × 1010 V/m , (c) and (f) Epeak = 1 × 1011 V/m .
state of the system occurs. In what follows, we will present the simulation results of both the weak and strong scattering regime. In order to compare the difference between the uniform and nonuniform gain situation, the mode structure with the uniform gain was also shown.
3.1. Weak scattering regime We consider the weak scattering situation with the sample parameters chosen as follow: Φ = 40% (N = 815), n1 = 1.2 and n2 = 1.25. According to the previous work [16,17], the refractive index difference between the background media and scatterer is low and this system can be considered as in weak scattering
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Fig. 5. Spatial distribution of the atomic population difference density ΔN between the upper and lower lasing levels for strong scattering regime.
regime. The uniform gain situation was first simulated by use of the model in Ref. [6]. Fig. 1 shows the spatial distribution of the electromagnetic-field intensity and steady-state emission spectra for uniform pumping with increasing pumping rates. At a pumping rate of Epeak = 1 × 1010 V/m [Fig. 1(d)], the net gain is below the threshold. The emission spectra in this case has a broad peak and is centered at the atomic transition frequency ω2 = 2π × 5.13 × 104 Hz (λ2 = 585 nm), resembling the spontaneous emission spectrum. On top of the spectrum there are many fine spikes whose frequencies change chaotically from one time window to another. A more detailed study of the threshold condition was performed. Based on the input–output curve and the steadystate emission spectra, we conclude these phenomenon comes from the stochastic emission process within the disordered medium and is similar to the statistical characteristics in the previous works [13,18]. As the pumping rate increases to Epeak = 5 × 1010 V/m [Fig. 1(e)], the broad emission peak grows and becomes spectrally narrower. As the optical gain is frequency dependent, the emission intensity closer to ω2 is amplified more than that away from it, resulting in a spectral narrowing. As the pumping rate increases further to Epeak = 1 × 1011 V/m [Fig. 1(f)], the gain is well above the threshold and discrete peaks emerge among the broad emission band. These peaks correspond to resonances of the passive system and the frequency of them is stable with respect to the pumping rate. We then study the spatial distribution of the electromagneticfield intensity with different pumping rates. As shown in Fig. 1(a)– (c), the lasing mode is distributed over the whole system and has a large traveling component. This is in accordance with the simulation result in the previous work [16]. In order to ensure these results are not limited to the particular sample configuration, the simulations are performed with different realization of the random structure. The results are qualitatively similar and with slight differences arise from stochasticity. Next, we will investigate the mode structure for nonuniform gain situation with the new model. In order to eliminate the influence of the different realization of the random structure, we use the same sample configuration as the above. The excitation pulse
was incident onto the sample and scattered over the plane. Since the non-uniformity of the gain is a crucial aspect discussed in this paper, we firstly study the maps of the gain distribution. As the gain coefficient is proportional to the atomic population difference density ΔN = N2 − N1 between the upper and lower lasing levels, one can extract the ΔN and represent the gain coefficient, which is shown in Fig. 2. From the map of the gain distribution, we can see clearly that the multiple scattering will lead to the spatial nonuniform gain among the sample. All the gain maps for different pumping rates are nearly identical. Fig. 3 shows the spatial distribution of the electromagnetic-field intensity and steady-state emission spectra for nonuniform gain situation with increasing pumping rates. The evolution of the emission spectra is quite similar to that of the uniform gain situation. When the pumping rate is low, the system is also in the ASE state. As the pumping rate increases, some discrete peaks appear in the broad emission band, this indicate that the stimulated emission occurs. Although the spatial distribution of the electromagnetic-field intensity for the nonuniform gain case is similar to that of the uniform gain [Fig. 3 (a)–(c)], the mode structure is altered. The difference between the uniform and nonuniform gain situations is that under a pumping rate well above the threshold, the quantity of the lasing modes decrease with the nonuniform gain. It is apparent that the frequency separation between lasing peaks is increased for the nonuniform gain case. The spectral overlap has been reduced and the resonance peaks are more distinguishable. Some of the lasing mode vanished due to the decrease of the spatial mode overlapping. We mark five visible peaks in Fig. 1(f). The frequency of these peaks is stable with the nonuniform gain situation (see Fig. 3 (f)). This indicates that when consider the nonuniform gain, there is no new lasing modes emerge in the weak scattering regime. All the lasing modes correspond to resonances of the passive system and amplified in the presence of population inversion. 3.2. Strong scattering regime We choose the sample parameters for strong scattering regime as follow: Φ = 40% (N = 815), n1 = 1.2 and n2 = 2.0. The refractive index of the scatterer chosen here is based on the results of the relational work [6,10,11], this system can be considered as in Anderson localization regime. In order to eliminate the influence of the particular configuration on the lasing mode, a same realization of the random structure was used as the weak scattering regime, i.e., the position of each particles keep unchanging. Fig. 4 shows the spatial distribution of the electromagnetic-field intensity and steady-state emission spectra for uniform pumping with increasing pumping rates. At a pumping rate of Epeak = 1 × 1010 V/m [Fig. 4(d)], the system is near the threshold point. The steady-state emission spectra has a broad featureless peak and is centered at the atomic transition frequency ( λ2 = 585 nm ). The spikes on top of the broad band result from the stochastic emission process. Similar to the weak scattering case, as the pumping rate increase to Epeak = 5 × 1010 V/m , the broad emission peak grows and becomes spectrally narrower [Fig. 4(e)]. As the pumping rate increase further to Epeak = 1 × 1011 V/m , well discrete peaks begin to form and only eight visible modes is left [marked in Fig. 4(f)]. The frequency of these peaks is stable with respect to the pumping rate and well separated due to amplification. The spatial distribution of the electromagnetic-field intensity with different pumping rates is shown in Fig. 4(a)–(c). The lasing modes are localized in certain area due to the interference effect. The propagation of waves comes to a halt in this case arise from the multiple scattering process. As the pumping rate increase, the lasing mode was confined in a smaller area. Both the spectral and spatial behavior are in agree with the previous theoretical and experimental works [7,19].
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Wavelength (nm) Fig. 6. Spatial distribution of the electromagnetic-field intensity (left) and steady-state emission spectra (right) for nonuniform gain with different pumping rates. (a) and (d) Epeak = 1 × 1010 V/m , (b) and (e) Epeak = 5 × 1010 V/m , (c) and (f) Epeak = 1 × 1011 V/m .
We then investigate the mode structure for nonuniform gain situation. We use the same sample configuration and excitation fashion as the above. Firstly we study the gain map for this situation. As can be seen in Fig. 5, the gain distribution is nonuniform and show a localized property. This is caused by the strong multiple scattering of the excitation light. The spatial distribution of the emission intensity for nonuniform gain with
increasing pumping rates were shown in Fig. 6(a)–(c). It is clear that under a low pumping rate ( Epeak = 1 × 1010 V/m ) in Fig. 6(a), the spatial lasing mode is still in a diffusion state. This is different from the result in Fig. 4(a). As the pumping rate increase further (Fig. 6(b) and (c)), the lasing mode transform into the localized state. The spectral intensity corresponding to Fig. 6(a)–(c) is shown in Fig. 6(d)–(f). The principal variation tendency of the spectra
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with different pumping rate is similar to the result in Fig. 4. The broad emission peak grows and narrows spectrally. However, in contrast to the uniform gain case, the resonance peaks are more visible in the emission spectrum for higher pumping rate. It is obviously that the frequency separation between lasing peaks is increased from the uniform gain case. We mark some lasing peaks in Fig. 6(e) and (f) and find that the frequency of these peaks is also stable with the increasing of the pumping rate. This means the lasing modes come from the resonances of the passive system. But once we compare the lasing modes between the uniform and nonuniform gain case, we find that the frequency of them is quite different. The wavelengths of the visible lasing peaks in Fig. 6 (f) are λ1 = 580.90 nm , λ2 = 585.02 nm and λ3 = 589.29 nm . There is no one wavelength that match the visible lasing modes marked in Fig. 4(f). This indicates that although we use the same configuration of the sample in these two simulations, the mode structure is different. Due to the spatially nonuniform gain, it will create some new lasing modes. This result can be explained as follow. When the system is in the weak scattering regime, the spatially lasing mode keeps in the diffusive state with increasing the pumping rate. It means that whether consider the nonuniform gain caused by the multiple scattering or not, the emission light will spread all over the whole sample. Because there is no dominant area in the random media for the emission light, the frequency of the lasing peaks arise from the passive system modes. However, the situation is quite different for strong scattering regime. The spatially lasing mode transform into the localized state and some of the stimulated emission process was confined in a certain area. As discussed in the earlier work, random laser action in localized state was determined by quasi-state modes [8]. It means that different lasing mode was excited in different localized area. If we compare the spatially distribution of the emission light in Figs. 4 and 6, we found that the localized area is very different between uniform and nonuniform gain. As the particle position is stochastic in the sample, different area will support a unique lasing mode and a corresponding lasing peak in spectral regime. This can explain why the frequency of the visible peaks in Fig. 6(f) is entirely different from those in Fig. 4(f). It should be emphasized that the spatially nonuniform gain here is not caused by the artificial method, but comes from the inherent multiple light scattering of the pumping pulse. As discussed in Section 1, the purpose for introducing the nonuniform gain in the previous work is to control the lasing properties of the random system. If one want to reveal the emission characteristics of a random laser, the inherent nonuniform gain comes from multiple scattering must be considered. Because we cannot obtain a absolute uniform gain sample in real experiment, it is difficult to compare the lasing frequencies and test if our new model is correct. However, the nonuniform gain which origins from the multiple scattering do exist in all the random lasing systems. This means it is more accurate to investigate the lasing properties of random laser by using our improved model. The lasing frequency and spatially distribution can be quite different from those of the lasing modes calculated by the previous method.
4. Conclusion The spatial and spectral behavior with nonuniform gain was studied in both weak and strong scattering random lasers. The previous time dependent theory of random laser is improved to nonuniform gain regime by introducing an independent electric field to describe the excitation pulse. The nonuniform gain of the 2D sample comes from the inherent multiple light scattering of
the pumping pulse. By use of the FDTD method, a numerical study on lasing behavior in 2D active random media was performed. Simulation results of a random system with uniform and nonuniform gain were compared, both in weak and strong scattering regime. For weak scattering regime, all the lasing modes correspond to the resonances of the passive system whether the inherent nonuniform gain is considered. The amount of the lasing peaks decreases and the frequency separation between them increases for the nonuniform gain case. For strong scattering regime, the result is quite different. The system transform from diffusive to localization as the multiple scattering process increase. Because the random laser action in localized state was determined by quasi-state modes, different spatial distribution of the lightwave will lead to a different mode structure. Once the inherent nonuniform gain is introduced, the spatial distribution of the emission changes, which will result in the creation of new lasing modes in the random system. Because the nonuniform gain caused by the multiple scattering of the pumping light is universal in all the random laser systems, we think our modified model produce more accurate results than the previous time dependent random laser theory.
Acknowledgment This work was supported by the Guangdong Natural Science Foundation (No. 2014A030313618) and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. 2014KQNCX180).
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Random lasing with spatially nonuniform gain Ting Fan a, Jiantao Lü b,n a b
School of Electronics and Information Engineering, Foshan University, Foshan 528000, China School of Science, Foshan University, Foshan 528000, China
art ic l e i nf o
a b s t r a c t
Article history: Received 12 January 2016 Received in revised form 4 March 2016 Accepted 28 March 2016
Spatial and spectral properties of random lasing with spatially nonuniform gain were investigated in two-dimensional (2D) disordered medium. The pumping light was described by an individual electric field and coupled into the rate equations by using the polarization equation. The spatially nonuniform gain comes from the multiple scattering of this pumping light. Numerical simulation of the random system with uniform and nonuniform gain were performed both in weak and strong scattering regime. In weak scattering sample, all the lasing modes correspond to those of the passive system whether the nonuniform gain is considered. However, in strong scattering regime, new lasing modes appear with nonuniform gain as the localization area changes. Our results show that it is more accurate to describe the random lasing behavior with introducing the nonuniform gain origins from the multiple light scattering. & 2016 Elsevier B.V. All rights reserved.
Keywords: Laser physics Random laser Nonuniform gain
1. Introduction Random lasing phenomenon is an attractive topic in the past decades after originally predicted by Letokhov [1]. The most difference between random laser and conventional one is that modes of random lasers are determined by multiple scattering and not by a laser cavity. To call a material a random laser, the multiple scattering process must play a determining role and the interference process determines the mode structure of it [2]. It is difficult to develop a theory that can describe all aspects of a random laser. A complete model has to include the dynamics of the gain mechanism and the interference effects. Researchers have developed different kinds of models for the random lasers, including the diffusive model [3], the time dependent theory [4], self-consistent multimode random laser theory [5], and so on. By combining Maxwell's equations with rate equations of electronic population, Jiang has developed the time dependent theory for one-dimensional (1D) case, and Sebbah extended it to two-dimensional (2D) case [6]. This model has successfully explained some experimental phenomenon [7–11]. In the previous work, gain is introduced in the system by pumping the four level atoms uniformly over the whole system. However, recent investigations have demonstrated that nonuniform gain can lead to the creation of new lasing mode [12,13], and local pumping allows the selection of spatially nonoverlapping modes [14,15]. In Jiang's n
Corresponding author. E-mail address: [email protected] (J. Lü).
http://dx.doi.org/10.1016/j.optcom.2016.03.078 0030-4018/& 2016 Elsevier B.V. All rights reserved.
model [4], optical gain is expressed by the pumping rate of the rate equations for electronic population, which can be written as the function of the position in the random media. One can change the pumping rate of different position and some new characteristics of random lasers were investigated, including the localized gain [7] and adaptive pumping [15]. But this is the artificial method to control the gain behavior for some special purpose, such as single mode selection. In the common random laser system, a lasing pulse is usually incident on the sample with a specific angle and serve as the pumping light. As the pumping light is irradiated onto the sample, it will be scattered in the random media and form some spatial distribution modes, which lead to the spatially nonuniform gain. For this reason, it is more accurate than using the uniform gain in the whole random sample. The limitation of Jiang's model is that he only used the phenomenological method to describe the gain behavior. Although it allows us to change the gain value for different positions, this method cannot truly represent the distribution of the gain value. In order to obtain the real spatial distribution of the pumping light, we must use an individual electrical field to describe the pumping pulse. Based on the fundamental laser theory, we suppose the gain coefficient is proportional to the strength of electric field. Once the spatial map of the pumping light is formed, it can be coupled into the rate equations by using the polarization equation to describe the stimulated absorption process. Our method not only provide a more accurate way to study the characteristic of the random laser, but also open a new path to investigate the relationship of the spatial modes between the pumping light and the emission one.
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Fig. 1. Spatial distribution of the electromagnetic-field intensity (left) and steady-state emission spectra (right) for uniform pumping with different pumping rates. (a) and (d) Epeak = 1 × 1010 V/m , (b) and (e) Epeak = 5 × 1010 V/m , and (c) and (f) Epeak = 1 × 1011 V/m .
In this paper, we report the computational results of the mode structure of the random laser with spatially nonuniform gain, which is caused by the multiple scattering of the pumping light. The rest of the article is organized as follows. In the next section we present the random media and the theoretical model. Section 3 shows the details of the numerical simulation results and the
interpretations. 2. Theoretical model We consider a 2D square random system with size L2 in the x–y plane. It consists of circular particles with a radius r and refractive
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transition rate is given by the term (E2/=ω2 )⋅(dP2/dt ) where ω2 is the transition frequency between level 2 and 1 and is chosen as 2π × 5.13 × 104 Hz (λ2 = 585 nm). The most difference between the previous work and ours, we use the term (E1/=ω1)⋅(dP1/dt ) in the rate equations to describe the excitation process instead of the pumping rate Wp , where ω1 is the center frequency of the excitation light and chosen as 2π × 5.64 × 104 Hz (λ2 = 532 nm). By coupling the polarization equation and rate equations, the excitation pulse can be described with an individual electric-field intensity. Therefore, the electrical field of the pumping pulse E1 and that of the emission one E2 were represented separately in the system. We considered a 2D transverse magnetic (TM) field in the x–y plane, thus the Maxwell's equations read ( i = 1, 2)
μ0
∂Hix ∂E = − iz ∂t ∂y
(5)
μ0
∂Hiy ∂E = iz ∂t ∂x
(6)
εj ε0
Fig. 2. Spatial distribution of the atomic population difference density ΔN between the upper and lower lasing levels for weak scattering regime.
index n2, which are randomly distributed in a background medium with a refractive index n1. The scattering strength is varied by adjusting the refractive index contrast Δn = n2 − n1 and the surface-filling fraction Φ , while Φ = Nπr 2/L2 can be modified by changing the particle radius r and amount N . This system can be considered as the multiple scattering light is confined in the x–y plane among a 3D sample, which resulting in a quasi-2D type of light transport. In order to control the randomness and the gain independently, the background medium is chosen as the active part and modeled as a simplified four-level atomic system. The electrons in the ground level 0 are transferred to the upper level 3 by an external laser pulse. Electrons in level 3 flow downward to level 2 by means of nonradiative decay process with a characteristic time τ 32 = 10−13 s. The depopulation of highest excited state to ground state through the spontaneous emission is not considered. The non-radiative transition from level 3 to 0 and level 2 to 1 are also neglected. The intermediate level 2 and 1 are the upper and lower levels of the laser transition, respectively. The lifetime of these two levels are τ 21 = 10−10 s and τ10 = 5 × 10−12 s , respectively. We describe the dynamics of atomic populations by rate equations as follow:
dN3 N E dP = − 3 + 1 ⋅ 1 dt τ 32 =ω1 dt
(1)
dN2 N E dP N = 3 − 2 ⋅ 2 − 2 dt τ 32 =ω2 dt τ 21
(2)
dN1 N E dP N = 2 + 2 ⋅ 2 − 1 τ 21 =ω2 dt τ10 dt
(3)
dN0 N E dP = 1 − 1 ⋅ 1 dt τ10 =ω1 dt
(4)
∂Hiy ∂Eiz ∂P ∂Hix + iz = − ∂t ∂t ∂x ∂y
The polarization density equations
(7) Pi (i = 1, 2) obeys the following
d2P1 dP + Δω1 1 + ω12P1 = κ1 (N0 − N3 ) E1 dt 2 dt
(8)
d2P2 dP + Δω2 2 + ω22P2 = κ 2 (N2 − N1) E2 dt 2 dt
(9)
where Δω1 = 6.67 × 1012 Hz is the linewidth of the excitation light, Δω2 = 1/τ 21 + 2/T2 is the linewidth of the atomic transition, T2 = 2 × 10−14 s is the collision time. The constant κi (i = 1, 2) is given by κ 2 = 6πϵ0 c 3/ω12 τ 32 and κ 2 = 6πε0 c 3/ω22 τ 21, respectively. By coupling the electric-field of the excitation pulse and population density in atomic levels through the polarization density equation, we can describe the pumping process by the external excitation pulse, once the excitation ended, and the pumping process stopped. Amplification occurs when the external pumping mechanism produces population inversion, i.e. N2 − N1 > 0. The excitation pulse is introduced as a Gaussian in temporal regime
⎛ 4π (t − t0 )2 ⎞ E1 (t ) = Epeak cos (ω1t ) exp ⎜ − ⎟ ⎝ ⎠ τ2
(10)
where τ is the width of the Gaussian pulse, Epeak is the peak value of the pumping, and t0 is the time corresponding to the peak value. The electromagnetic fields in the 2D active random medium can be obtained by FDTD methods. Perfectly matched layer (PML) is used to model an open system as absorbing conditions.
3. Results and discussion
where Ni is the population density in level i, i¼ 0–3. The stimulated
In this section, we will study the mode structure of a random laser when considering the spatial nonuniform of the excitation light, which comes from the multiple scattering in the disordered media. We select a 2D sample with S = 4 × 4 μm2 and r = 50 nm . A laser pulse with the TM polarization was introduced onto the 2D sample as the excitation source. The parameters of the excitation pulse is chosen as follow: τ = 10 ns, t0 = 5 ns. The gain of the whole system can be controlled by adjusting the peak value of the pumping pulse Epeak . The input light was scattered by the particles and form a spatially nonuniform distribution. In some area, the
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Fig. 3. Spatial distribution of the electromagnetic-field intensity (left) and steady-state emission spectra (right) for nonuniform gain with different pumping rates. (a) and (d) Epeak = 1 × 1010 V/m , (b) and (e) Epeak = 5 × 1010 V/m , (c) and (f) Epeak = 1 × 1011 V/m .
local gain value is high where the pumping light is strong. For each set of FDTD simulations, the spatial distribution of the field intensity of the pumping light and stimulated emission were obtain. The intensity spectrum of the laser field can be obtained by integrating the electrical-field intensity of the whole system and
carrying out the Fourier transformation. In a 2D random system, the scattering strength is controlled by the surface-filling fraction Φ and the refractive index contrast Δn = n2 − n1. Once these two parameters exceed the respective critical value, the transition from diffusion state to localization
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Wavelength (nm) Fig. 4. Spatial distribution of the electromagnetic-field intensity (left) and steady-state emission spectra (right) for uniform gain with different pumping rates. (a) and (d) Epeak = 1 × 1010 V/m , (b) and (e) Epeak = 5 × 1010 V/m , (c) and (f) Epeak = 1 × 1011 V/m .
state of the system occurs. In what follows, we will present the simulation results of both the weak and strong scattering regime. In order to compare the difference between the uniform and nonuniform gain situation, the mode structure with the uniform gain was also shown.
3.1. Weak scattering regime We consider the weak scattering situation with the sample parameters chosen as follow: Φ = 40% (N = 815), n1 = 1.2 and n2 = 1.25. According to the previous work [16,17], the refractive index difference between the background media and scatterer is low and this system can be considered as in weak scattering
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Fig. 5. Spatial distribution of the atomic population difference density ΔN between the upper and lower lasing levels for strong scattering regime.
regime. The uniform gain situation was first simulated by use of the model in Ref. [6]. Fig. 1 shows the spatial distribution of the electromagnetic-field intensity and steady-state emission spectra for uniform pumping with increasing pumping rates. At a pumping rate of Epeak = 1 × 1010 V/m [Fig. 1(d)], the net gain is below the threshold. The emission spectra in this case has a broad peak and is centered at the atomic transition frequency ω2 = 2π × 5.13 × 104 Hz (λ2 = 585 nm), resembling the spontaneous emission spectrum. On top of the spectrum there are many fine spikes whose frequencies change chaotically from one time window to another. A more detailed study of the threshold condition was performed. Based on the input–output curve and the steadystate emission spectra, we conclude these phenomenon comes from the stochastic emission process within the disordered medium and is similar to the statistical characteristics in the previous works [13,18]. As the pumping rate increases to Epeak = 5 × 1010 V/m [Fig. 1(e)], the broad emission peak grows and becomes spectrally narrower. As the optical gain is frequency dependent, the emission intensity closer to ω2 is amplified more than that away from it, resulting in a spectral narrowing. As the pumping rate increases further to Epeak = 1 × 1011 V/m [Fig. 1(f)], the gain is well above the threshold and discrete peaks emerge among the broad emission band. These peaks correspond to resonances of the passive system and the frequency of them is stable with respect to the pumping rate. We then study the spatial distribution of the electromagneticfield intensity with different pumping rates. As shown in Fig. 1(a)– (c), the lasing mode is distributed over the whole system and has a large traveling component. This is in accordance with the simulation result in the previous work [16]. In order to ensure these results are not limited to the particular sample configuration, the simulations are performed with different realization of the random structure. The results are qualitatively similar and with slight differences arise from stochasticity. Next, we will investigate the mode structure for nonuniform gain situation with the new model. In order to eliminate the influence of the different realization of the random structure, we use the same sample configuration as the above. The excitation pulse
was incident onto the sample and scattered over the plane. Since the non-uniformity of the gain is a crucial aspect discussed in this paper, we firstly study the maps of the gain distribution. As the gain coefficient is proportional to the atomic population difference density ΔN = N2 − N1 between the upper and lower lasing levels, one can extract the ΔN and represent the gain coefficient, which is shown in Fig. 2. From the map of the gain distribution, we can see clearly that the multiple scattering will lead to the spatial nonuniform gain among the sample. All the gain maps for different pumping rates are nearly identical. Fig. 3 shows the spatial distribution of the electromagnetic-field intensity and steady-state emission spectra for nonuniform gain situation with increasing pumping rates. The evolution of the emission spectra is quite similar to that of the uniform gain situation. When the pumping rate is low, the system is also in the ASE state. As the pumping rate increases, some discrete peaks appear in the broad emission band, this indicate that the stimulated emission occurs. Although the spatial distribution of the electromagnetic-field intensity for the nonuniform gain case is similar to that of the uniform gain [Fig. 3 (a)–(c)], the mode structure is altered. The difference between the uniform and nonuniform gain situations is that under a pumping rate well above the threshold, the quantity of the lasing modes decrease with the nonuniform gain. It is apparent that the frequency separation between lasing peaks is increased for the nonuniform gain case. The spectral overlap has been reduced and the resonance peaks are more distinguishable. Some of the lasing mode vanished due to the decrease of the spatial mode overlapping. We mark five visible peaks in Fig. 1(f). The frequency of these peaks is stable with the nonuniform gain situation (see Fig. 3 (f)). This indicates that when consider the nonuniform gain, there is no new lasing modes emerge in the weak scattering regime. All the lasing modes correspond to resonances of the passive system and amplified in the presence of population inversion. 3.2. Strong scattering regime We choose the sample parameters for strong scattering regime as follow: Φ = 40% (N = 815), n1 = 1.2 and n2 = 2.0. The refractive index of the scatterer chosen here is based on the results of the relational work [6,10,11], this system can be considered as in Anderson localization regime. In order to eliminate the influence of the particular configuration on the lasing mode, a same realization of the random structure was used as the weak scattering regime, i.e., the position of each particles keep unchanging. Fig. 4 shows the spatial distribution of the electromagnetic-field intensity and steady-state emission spectra for uniform pumping with increasing pumping rates. At a pumping rate of Epeak = 1 × 1010 V/m [Fig. 4(d)], the system is near the threshold point. The steady-state emission spectra has a broad featureless peak and is centered at the atomic transition frequency ( λ2 = 585 nm ). The spikes on top of the broad band result from the stochastic emission process. Similar to the weak scattering case, as the pumping rate increase to Epeak = 5 × 1010 V/m , the broad emission peak grows and becomes spectrally narrower [Fig. 4(e)]. As the pumping rate increase further to Epeak = 1 × 1011 V/m , well discrete peaks begin to form and only eight visible modes is left [marked in Fig. 4(f)]. The frequency of these peaks is stable with respect to the pumping rate and well separated due to amplification. The spatial distribution of the electromagnetic-field intensity with different pumping rates is shown in Fig. 4(a)–(c). The lasing modes are localized in certain area due to the interference effect. The propagation of waves comes to a halt in this case arise from the multiple scattering process. As the pumping rate increase, the lasing mode was confined in a smaller area. Both the spectral and spatial behavior are in agree with the previous theoretical and experimental works [7,19].
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Wavelength (nm) Fig. 6. Spatial distribution of the electromagnetic-field intensity (left) and steady-state emission spectra (right) for nonuniform gain with different pumping rates. (a) and (d) Epeak = 1 × 1010 V/m , (b) and (e) Epeak = 5 × 1010 V/m , (c) and (f) Epeak = 1 × 1011 V/m .
We then investigate the mode structure for nonuniform gain situation. We use the same sample configuration and excitation fashion as the above. Firstly we study the gain map for this situation. As can be seen in Fig. 5, the gain distribution is nonuniform and show a localized property. This is caused by the strong multiple scattering of the excitation light. The spatial distribution of the emission intensity for nonuniform gain with
increasing pumping rates were shown in Fig. 6(a)–(c). It is clear that under a low pumping rate ( Epeak = 1 × 1010 V/m ) in Fig. 6(a), the spatial lasing mode is still in a diffusion state. This is different from the result in Fig. 4(a). As the pumping rate increase further (Fig. 6(b) and (c)), the lasing mode transform into the localized state. The spectral intensity corresponding to Fig. 6(a)–(c) is shown in Fig. 6(d)–(f). The principal variation tendency of the spectra
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with different pumping rate is similar to the result in Fig. 4. The broad emission peak grows and narrows spectrally. However, in contrast to the uniform gain case, the resonance peaks are more visible in the emission spectrum for higher pumping rate. It is obviously that the frequency separation between lasing peaks is increased from the uniform gain case. We mark some lasing peaks in Fig. 6(e) and (f) and find that the frequency of these peaks is also stable with the increasing of the pumping rate. This means the lasing modes come from the resonances of the passive system. But once we compare the lasing modes between the uniform and nonuniform gain case, we find that the frequency of them is quite different. The wavelengths of the visible lasing peaks in Fig. 6 (f) are λ1 = 580.90 nm , λ2 = 585.02 nm and λ3 = 589.29 nm . There is no one wavelength that match the visible lasing modes marked in Fig. 4(f). This indicates that although we use the same configuration of the sample in these two simulations, the mode structure is different. Due to the spatially nonuniform gain, it will create some new lasing modes. This result can be explained as follow. When the system is in the weak scattering regime, the spatially lasing mode keeps in the diffusive state with increasing the pumping rate. It means that whether consider the nonuniform gain caused by the multiple scattering or not, the emission light will spread all over the whole sample. Because there is no dominant area in the random media for the emission light, the frequency of the lasing peaks arise from the passive system modes. However, the situation is quite different for strong scattering regime. The spatially lasing mode transform into the localized state and some of the stimulated emission process was confined in a certain area. As discussed in the earlier work, random laser action in localized state was determined by quasi-state modes [8]. It means that different lasing mode was excited in different localized area. If we compare the spatially distribution of the emission light in Figs. 4 and 6, we found that the localized area is very different between uniform and nonuniform gain. As the particle position is stochastic in the sample, different area will support a unique lasing mode and a corresponding lasing peak in spectral regime. This can explain why the frequency of the visible peaks in Fig. 6(f) is entirely different from those in Fig. 4(f). It should be emphasized that the spatially nonuniform gain here is not caused by the artificial method, but comes from the inherent multiple light scattering of the pumping pulse. As discussed in Section 1, the purpose for introducing the nonuniform gain in the previous work is to control the lasing properties of the random system. If one want to reveal the emission characteristics of a random laser, the inherent nonuniform gain comes from multiple scattering must be considered. Because we cannot obtain a absolute uniform gain sample in real experiment, it is difficult to compare the lasing frequencies and test if our new model is correct. However, the nonuniform gain which origins from the multiple scattering do exist in all the random lasing systems. This means it is more accurate to investigate the lasing properties of random laser by using our improved model. The lasing frequency and spatially distribution can be quite different from those of the lasing modes calculated by the previous method.
4. Conclusion The spatial and spectral behavior with nonuniform gain was studied in both weak and strong scattering random lasers. The previous time dependent theory of random laser is improved to nonuniform gain regime by introducing an independent electric field to describe the excitation pulse. The nonuniform gain of the 2D sample comes from the inherent multiple light scattering of
the pumping pulse. By use of the FDTD method, a numerical study on lasing behavior in 2D active random media was performed. Simulation results of a random system with uniform and nonuniform gain were compared, both in weak and strong scattering regime. For weak scattering regime, all the lasing modes correspond to the resonances of the passive system whether the inherent nonuniform gain is considered. The amount of the lasing peaks decreases and the frequency separation between them increases for the nonuniform gain case. For strong scattering regime, the result is quite different. The system transform from diffusive to localization as the multiple scattering process increase. Because the random laser action in localized state was determined by quasi-state modes, different spatial distribution of the lightwave will lead to a different mode structure. Once the inherent nonuniform gain is introduced, the spatial distribution of the emission changes, which will result in the creation of new lasing modes in the random system. Because the nonuniform gain caused by the multiple scattering of the pumping light is universal in all the random laser systems, we think our modified model produce more accurate results than the previous time dependent random laser theory.
Acknowledgment This work was supported by the Guangdong Natural Science Foundation (No. 2014A030313618) and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. 2014KQNCX180).
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