## Spectral behavior of partially pumped weakly scattering random lasers |

Optics Express, Vol. 19, Issue 4, pp. 3418-3433 (2011)

http://dx.doi.org/10.1364/OE.19.003418

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### Abstract

Stochastic noise is incorporated in the numerical simulation of weakly scattering random lasers, which qualitatively captures lasing phenomena that have been observed experimentally. We examine the behavior of the emission spectrum while pumping only part of the entire one-dimensional random system. A decrease in the density of lasing states is the dominant mechanism for observing discrete lasing peaks when absorption exists in the unpumped region. Without such absorption, the density of lasing states does not reduce as dramatically but the statistical distribution of (linear) lasing thresholds is broadened. This may facilitate incremental observation of lasing in smaller-threshold modes in the emission spectrum with fine adjustments of the pumping rate.

© 2011 Optical Society of America

## 1. Introduction

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19. X. Wu, J. Andreasen, H. Cao, and A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B **24**, A26–A33 (2007). [CrossRef]

20. J. Andreasen, C. Vanneste, L. Ge, and H. Cao, “Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems,” Phys. Rev. A **81**, 043818 (2010). [CrossRef]

13. C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. **87**, 183903 (2001). [CrossRef]

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10. S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B **59**, R5284–R5287 (1999). [CrossRef]

12. X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A **74**, 053812 (2006). [CrossRef]

26. L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. **95**, 043902 (2005). [CrossRef] [PubMed]

27. J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. **3**, 88–127 (2011). [CrossRef]

11. Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A **64**, 063808 (2001). [CrossRef]

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30. E. V. Chelnokov, N. Bityurin, I. Ozerov, and W. Marine, “Two-photon pumped random laser in nanocrystalline ZnO,” Appl. Phys. Lett. **89**, 171119 (2006). [CrossRef]

20. J. Andreasen, C. Vanneste, L. Ge, and H. Cao, “Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems,” Phys. Rev. A **81**, 043818 (2010). [CrossRef]

31. H. Cao, X. Jiang, Y. Ling, J. Y. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B **67**, 161101 (2003). [CrossRef]

32. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science **320**, 643–646 (2008). [CrossRef] [PubMed]

13. C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. **87**, 183903 (2001). [CrossRef]

33. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. **85**, 70–73 (2000). [CrossRef] [PubMed]

34. J. Andreasen and H. Cao, “Finite-different time-domain formulation of stochastic noise in macroscopic atomic systems,” J. Lightwave Technol. **27**, 4530–4535 (2009). [CrossRef]

*et al.*compared the statistical distribution of resonance decay rates and lasing thresholds under local pumping without absorption [35

35. X. Wu and H. Cao, “Statistical studies of random-lasing modes and amplified-spontaneous-emission spikes in weakly scattering systems,” Phys. Rev. A **77**, 013832 (2008). [CrossRef]

*et al.*also observed the absolute degree of lasing threshold fluctuations increases with a decrease of the pump size [36

36. X. Wu, A. Yamilov, H. Noh, H. Cao, E. W. Seelig, and R. P. H. Chang, “Random lasing in closely packed resonant scatterers,” J. Opt. Soc. Am. B **21**, 159–167 (2004). [CrossRef]

## 2. Numerical Methods

### 2.1. Generation of One-Dimensional Random Structures

*N*= 41 layers. Dielectric material with index of refraction

*n*

_{1}= 1.05 separated by air gaps (

*n*

_{2}= 1) results in a spatially modulated index of refraction

*n*(

*x*). Outside the random media

*n*

_{0}= 1. The system is randomized by specifying different thicknesses for each of the layers as

*d*

_{1,2}= 〈

*d*

_{1,2}〉 (1 +

*ηζ*) where 〈

*d*

_{1}〉 and 〈

*d*

_{2}〉 are the average thicknesses of the layers, 0 <

*η*< 1 represents the degree of randomness, and

*ζ*is a random number uniformly distributed in (−1,1). The average thicknesses are 〈

*d*

_{1}〉 = 100 nm and 〈

*d*

_{2}〉 = 200 nm giving a total average length of 〈

*L*〉 = 6100 nm. The grid origin

*x*= 0 is at the left boundary of the structure and the length of the random structure

*L*is normalized to 〈

*L*〉. The degree of randomness is set to

*η*= 0.9. The localization length 〈

*ξ*〉 = 220

*μ*m was calculated from the dependence of ensemble-averaged transmittance

*T*on the system lengths

*L*as

*ξ*

^{−1}= −

*d*〈ln

*T*〉/

*dL*and averaged over the wavelength range of interest (500 nm to 750 nm). Different realizations of random structures are generated using different random seeds for

*ζ*.

### 2.2. Stochastic Maxwell-Bloch equations: FDTD parameters

34. J. Andreasen and H. Cao, “Finite-different time-domain formulation of stochastic noise in macroscopic atomic systems,” J. Lightwave Technol. **27**, 4530–4535 (2009). [CrossRef]

38. J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C. qi Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A **77**, 023810 (2008). [CrossRef]

39. D. M. Sullivan, *Electromagnetic Simulation Using the FDTD Method* (IEEE Press, 2000). [CrossRef]

40. R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A **52**, 3082–3094 (1995). [CrossRef] [PubMed]

*c*-number equations that are derived from the quantum Langevin equations in the many-atom and many-photon limit [41

41. P. D. Drummond and M. G. Raymer, “Quantum theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A **44**, 2072–2085 (1991). [CrossRef] [PubMed]

*k*= 10.5

_{a}*μ*m

^{−1}, the corresponding wavelength

*λ*= 600 nm. The lifetime of atoms in the excited state

_{a}*T*

_{1}and the dephasing time

*T*

_{2}are included in the Bloch equations. The width of the gain spectrum is given by Δ

*k*= (1/

_{a}*T*

_{1}+ 2/

*T*

_{2})/

*c*[42]. We set

*T*

_{1}= 1.0 ps. The value of

*T*

_{2}is chosen such that the gain spectrum spans ten resonances of the passive system. With an average frequency spacing Δ

*k*= 0.5

*μ*m

^{−1}, Δ

*k*= 5.0

_{a}*μ*m

^{−1}, and

*T*

_{2}= 1.3 fs. We also include incoherent pumping of atoms from level 1 to level 2. The rate of atoms being pumped is proportional to the population of atoms in level 1 (

*ρ*

_{11}), and the proportionality coefficient

*P*is called the pumping rate. The stochastic simulations solve for the population of excited atomic states

_{r}*ρ*

_{22}and atomic polarization

*ρ*

_{1}=

*ρ*

_{12}+

*ρ*

_{21}and

*ρ*

_{2}=

*i*(

*ρ*

_{12}–

*ρ*

_{21}). With

*T*

_{2}≪

*T*

_{1}, we neglect pump fluctuations on the polarization because they are orders of magnitude smaller than noise due to dephasing. The stochastic Maxwell-Bloch (SMB) equations are solved through a parallel FDTD implementation.

### 2.3. Implementation of Three Pumping Cases

*x*≤

*ℓ*(still in both the higher-index dielectric layers and the air gaps). We choose

_{G}*ℓ*/

_{G}*L*= 1/3. In the unpumped region (

*x*>

*ℓ*), there are no atoms nor is there absorption of any kind. There is only scattering due to the passive random structure. In both of these cases, the output field is sampled at the grid point

_{G}*x*=

*L*at the right boundary of the random system. We Fourier-transform the output field to obtain the emission spectra.

*x*>

*ℓ*) is achieved by placing two-level atoms there in the ground state. Light emitted from the pumped region is reabsorbed in the unpumped region. The average decay length of the intensity in this region yields an absorption length

_{G}*ℓ*≈ 170 nm. This is much smaller than the length of the unpumped region (

_{a}*L*–

*ℓ*= 4067 nm). With strong absorption and no pumping, the noise terms are small in the unpumped region. We neglect them when the excited state population is less than a threshold value

_{G}*α*. Incrementally decreasing

*α*and monitoring the change of physical quantities, we found results to converge when

*α*= 10

^{−12}. When absorption is included at

*x*>

*ℓ*, fields emitted from the pumped region are significantly absorbed so that no signal reaches

_{G}*x*=

*L*. Thus, in this case, the field at

*x*= 0 is used to obtain the emission spectra. Without absorption, results from sampling at

*x*=

*L*are identical in character to those from sampling at

*x*= 0.

*t*= 267 ps, we find a steady state is reached by 16.6 ps for all pumping rates considered here.

### 2.4. Transfer Matrix Method

20. J. Andreasen, C. Vanneste, L. Ge, and H. Cao, “Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems,” Phys. Rev. A **81**, 043818 (2010). [CrossRef]

*M*. Linear gain, independent of frequency, is simulated by appending an imaginary part to the dielectric function

*ɛ*(

*x*) =

*ɛ*(

_{r}*x*) +

*iɛ*(

_{i}*x*), where

*ɛ*(

_{r}*x*) =

*n*

^{2}(

*x*). The complex index of refraction where

*n*(

_{y}*x*) < 0 for gain and in turn modifies the real part [20

**81**, 043818 (2010). [CrossRef]

*n*(

_{y}*x*) is considered to be spatially constant within the random system. This is similar to the SMB simulations in the previous section and yields a gain length

*ℓ*= 1/|

_{g}*n*|

_{y}*k*(

*k*= 2

*π*/

*λ*is the vacuum frequency of a TLM) which is the same in the dielectric layers and the air gaps. Partial pumping is implemented with a step function

*n*(

_{y}*x*) =

*n*(−

_{i}H*x*+

*ℓ*), where

_{G}*x*= 0 is the left edge of the structure and

*x*=

*ℓ*specifies the right edge of the pumping region.

_{G}*n*(

_{y}*x*) = (2

*kℓ*)

_{a}^{−1}, where

*n*(

_{y}*x*) > 0 for absorption and

*ℓ*is the absorption length. For partial pumping with absorption

_{a}*ñ*(

*x*) =

*n*(

*x*)+

*i*(1/2

*kℓ*) in the unpumped region. Such absorption is not included in the pumped region.

_{a}*M*

_{22}] = 0 and Im[

*M*

_{22}] = 0 [20

**81**, 043818 (2010). [CrossRef]

*k*,

*n*). The crossing of a real and imaginary zero line in the (

_{i}*k*,

*n*) plane results in

_{i}*M*

_{22}= 0 at that location. The values of

*k*and

*n*at these locations correspond to the frequency and threshold gain of a lasing mode, respectively. The benefit of this method is that the lasing thresholds may be estimated quickly and easily relative to one another. Moreover, without gain saturation and noise included the effects of partial pumping are isolated.

_{i}## 3. Stochastic Maxwell-Bloch Simulations of Random Lasers

### 3.1. Uniform Pumping

*E*(

*k*)|

^{2}for uniform pumping with increasing pumping rates. At a pumping rate of

*P*= 1.00 [Fig. 1(a)], there is no net gain. The number of ground-state atoms is

_{r}*ρ*

_{11}and the number of excited-state atoms is

*ρ*

_{22}. Without stimulated emission and noise, the number of atoms pumped from the ground to excited state is

*P*

_{r}ρ_{11}/

*T*

_{1}. Meanwhile, the decay rate of atoms is

*ρ*

_{22}/

*T*

_{1}. Thus, when

*P*= 1.00,

_{r}*ρ*

_{11}=

*ρ*

_{22}and the atomic system is at the transparency point (

*ρ*

_{22}–

*ρ*

_{11}= 0). Noise reduces the excited state population, and it is just below the transparency point for

*P*= 1.00. The steady-state emission spectra in this case has a broad peak and is centered at the atomic transition frequency

_{r}*k*= 10.5

_{a}*μ*m

^{−1}, resembling the spontaneous emission spectrum. On top of it there are many fine spikes whose frequencies change chaotically from one time window of Fourier transform to the next. They result from the stochastic emission process with their spectral width determined by the temporal length of the Fourier transform. We have found [44

44. J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A **82**, 063835 (2010). [CrossRef]

*P*= 1.10 [Fig. 1(b)], there is net gain. The broad emission peak grows and narrows spectrally. Since optical gain is frequency dependent, the emission intensity closer to

_{r}*k*is amplified more than that away from

_{a}*k*, leading to a spectral narrowing. This behavior is typical of ASE.

_{a}*P*increases [Figs. 1(c) – 1(g)], discrete peaks begin to form amidst the broad emission peak. They correspond to resonances of the passive system. We mark eight visible peaks in Fig. 1(d). The frequency of these peaks is stable with respect to the pumping rate. They also become narrower and more distinct at higher pumping rates. All of these modes are constantly excited by noise and subsequently amplified in the presence of population inversion.

_{r}### 3.2. Partial Pumping

*E*(

*k*)|

^{2}for partial pumping with increasing pumping rates. At

*P*= 1.00 [Fig. 2(a)], the system is near the transparency point (

_{r}*ρ*

_{22}–

*ρ*

_{11}≲ 0) in the pumped region. The steady-state emission spectra again has a broad featureless peak and is centered at the atomic transition frequency

*k*= 10.5

_{a}*μ*m

^{−1}. On top of it there are many fine spikes resulting from the stochastic emission process.

*P*= 1.10 [Fig. 2(b)]. These peaks grow as

_{r}*P*increases further [Figs. 2(c) – 2(g)]. We mark seven visible peaks in Fig. 2(d). The frequency of these peaks is stable with respect to the pumping rate. They also become narrower and well separated due to amplification. There is one less peak here compared to the uniform pumping case, but there appears to be some correspondence between the peak frequencies for partial and uniform pumping. The relation of these peaks to peaks in the uniform pumping case will be examined in detail in Sec. 4.

_{r}44. J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A **82**, 063835 (2010). [CrossRef]

### 3.3. Absorption in the Unpumped Region

*E*(

*k*)|

^{2}for partial pumping with absorption in the unpumped region as the pumping rate increases. At

*P*= 1.00 [Fig. 3(a)], near the transparency point in the pumped region, the broad ASE peak is seen at

_{r}*k*= 10.5

_{a}*μ*m

^{−1}. Like the previous partial pumping case, resonance peaks emerge clearly in the emission spectrum for

*P*= 1.10 [Fig. 3(b)] and they grow as

_{r}*P*increases further [Figs. 3(c) – 3(g)]. In contrast to the previous partial pumping case, there are far fewer peaks in the emission spectra with absorption included. We mark four visible peaks in Fig. 3(d). The frequency of these peaks is stable with respect to the variation of the pumping rate. However, the peak frequencies in this case are notably different from those in the uniform and partial pumping cases without absorption. The relation between these peaks will be examined in detail in Sec. 4.

_{r}45. P. J. Bardroff and S. Stenholm, “Quantum theory of excess noise,” Phys. Rev. A **60**, 2529–2533 (1999). [CrossRef]

## 4. Threshold Lasing Modes With Linear Gain

*k*and thresholds

*n*with uniform and partial pumping implemented via Eq. (1). With uniform pumping (

_{i}*ℓ*/

_{G}*L*= 1), the separation of thresholds between neighboring modes marked by diamonds is quite small. This results in all the modes having very similar behavior as the pumping rate is increased in the SMB simulations in Fig. 1. The eight peaks marked by arrows in Fig. 1(d) are associated with the threshold lasing modes (TLMs) marked by filled diamonds in Fig. 4(a). There are nine filled diamonds because the two TLMs closest to

*k*in Fig. 4(a) appear only as a “composite” peak in Fig. 1(d). The finite linewidths of the two modes exceed their frequency spacing, which is reduced by the frequency pulling effect. Consequently, the two modes are indistinguishable and appear to be merged. Nevertheless, there is a clear correspondence between the TLMs and the peaks seen in the SMB simulations. The frequency pulling effect merely shifts the frequencies toward the center of the gain curve in the SMB simulations by 10–20% (compared to the TLM frequencies).

_{a}*ℓ*/

_{G}*L*= 1/3), the lasing thresholds |

*n*| increase significantly as illustrated by the circles in Fig. 4(a). The seven peaks marked by arrows in Fig. 2(d) are associated with the TLMs marked by filled circles in Fig. 4(a). Aside from slight frequency pulling (∼ 16%), there is a clear correspondence between the TLMs and the peaks seen in the SMB simulations. The increased frequency separation between the TLMs (e.g., at

_{i}*k*= 9.26

*μ*m

^{−1}and

*k*= 10.4

*μ*m

^{−1}) allows them to be visible in the SMB simulations at

*P*= 1.10 [Fig. 2(b)].

_{r}*M*

_{22}. The zero line crossings are marked by circles which correspond to TLM solutions [the same solutions marked by the circles in Fig. 4(a)]. From small to large thresholds (top to bottom), real and imaginary zero lines may cross once but may also cross again at larger thresholds. Thus, two classes of TLMs can be seen: those associated with first-crossings of the zero lines and those associated with additional crossings. Only the first class of TLMs have any correspondence to TLMs with uniform pumping [20

**81**, 043818 (2010). [CrossRef]

46. J. Andreasen and H. Cao, “Creation of new lasing modes with spatially nonuniform gain,” Opt. Lett. **34**, 3586–3588 (2009). [CrossRef] [PubMed]

46. J. Andreasen and H. Cao, “Creation of new lasing modes with spatially nonuniform gain,” Opt. Lett. **34**, 3586–3588 (2009). [CrossRef] [PubMed]

*k*(for uniform pumping) and because of an extra higher-frequency mode that shifted from

_{a}*k*= 13.2

*μ*m

^{−1}to

*k*= 13

*μ*m

^{−1}. Note that the shift is not caused by frequency pulling because

*n*is frequency-independent. This is a shift caused by partial pumping because

_{i}*n*modifies the real part of the refractive index

_{i}*n*[some modes shift away from

_{r}*k*as seen in Fig. 4(a)]. Frequency pulling due to the finite-width gain spectrum in the SMB simulations shifts this mode further to

_{a}*k*= 12.7

*μ*m

^{−1}in Fig. 2.

*ℓ*= 170 nm, which was the approximate absorption length in the SMB simulations in Sec. 3.3. The lasing thresholds

_{a}*n*shown in Fig. 4(a) increase compared to partial pumping without absorption, as expected. The four peaks marked by arrows in Fig. 3(d) are associated with the TLMs marked by filled squares in Fig. 4(a). More frequency pulling occurs in this case with a shift of 30–40% compared to the TLM frequencies. Nevertheless, there is a clear correspondence between the TLMs and the peaks seen in the SMB simulations.

_{i}*P*= 1.10 [Fig. 2(b)]. The reason for the increased frequency separation in this case is entirely different from the partial pumping case without absorption. With absorption, the peaks are associated with modes confined to the pumped region [18

_{r}18. A. Yamilov, X. Wu, H. Cao, and A. L. Burin, “Absorption-induced confinement of lasing modes in diffusive random media,” Opt. Lett. **30**, 2430–2432 (2005). [CrossRef] [PubMed]

*ℓ*+

_{G}*ℓ*. Feedback from the random structure beyond

_{a}*ℓ*+

_{G}*ℓ*is suppressed due to absorption. Without absorption, additional feedback in the unpumped region plays a role in determining the modes.

_{a}*P*and the experimentally limited pump step (e.g., by power fluctuations) is Δ

_{t}*P*, then the relative pump step is

*δP*= Δ

*P*/

*P*. Larger

_{t}*P*means a smaller allowable adjustments of

_{t}*δP*, thereby allowing a finer tuning of the pumping rate with respect to the threshold value.

## 5. Effects of Inhomogeneous Pumping and Absorption on Threshold Statistics

*ℓ*[implemented via Eq. (1)] is always chosen to coincide with an interface between the higher-index dielectric material and air. This results in a partial pumping length of

_{G}*ℓ*/

_{G}*L*= 0.33 ±0.011 over the 10000 realizations. With the number of modes reduced by 3 on average in the partial pumping case with absorption, we consider 30000 structure realizations in order to maintain roughly the same number of modes. Different realizations of random structures are generated using different random seeds for

*ζ*. The frequency range for these calculations is limited to

*k*± 2

_{a}*μ*m

^{−1}, the same range as the SMB simulations in Sec. 3. The solutions are pinpointed precisely by using the Secant method. Locations of minima of |

*M*

_{22}|

^{2}and a random value located closely to these minima locations are used as the first two inputs to the Secant method. Once a solution converges or |

*M*

_{22}| < 10

^{−12}, a solution is considered found. Verification of these solutions is provided by the phase of

*M*

_{22}, calculated as

*θ*= atan2(Im

*M*

_{22}, Re

*M*

_{22}). Locations of vanishing

*M*

_{22}give rise to phase singularities since both the real and imaginary parts of

*M*

_{22}vanish. The phase change around a path surrounding a singularity is ±2

*π*. Thus, if the phase change around a proposed solution is not ±2

*π*, that solution is discarded.

*M*

_{22}. Convergence for these modes is limited numerically by machine precision.

*n*< 0, we hereafter refer |

_{i}*n*| to

_{i}*n*for brevity. The optimal bin size Δ

_{i}*n*for lasing thresholds was found using the Scott formula [47

_{i}47. D. W. Scott, “On optimal and data-based histograms,” Biometrika **66**, 605–610 (1979). [CrossRef]

*n*= 0.001 and for partial pumping Δ

_{i}*n*= 0.002. The histograms are normalized yielding the probability distribution

_{i}*P*(

*n*) so that ∫

_{i}*P*(

*n*)

_{i}*dn*= 1.

_{i}*P*(

*n*) for uniform and partial pumping. No absorption is included for partial pumping. Figure 5(a) shows a large-threshold tail for

_{i}*n*> 0.175 in the partial pumping case. The reason for the sharp kink between small and large-threshold modes is seen clearly in Fig. 4(b). The large-threshold modes are formed predominantly by secondary crossings of the real and imaginary zero lines of

_{i}*M*

_{22}. These secondary crossings have

*n*well above the first crossings. This tail highly distorts the threshold statistics which is evident in the skewness

_{i}*S*that characterizes the degree of asymmetry around the mean value. The skewness increases from

*S*= 1.4 for uniform pumping to

*S*= 2.2 for partial pumping.

*n*≤ 0.175 with partial pumping and re-normalize the histogram to obtain a new probability distribution. The inset in Fig. 5(a) shows this re-normalized threshold distribution (the distribution for uniform pumping is left unchanged). Due to asymmetry (even uniform pumping has

_{i}*S*> 1), we characterize the first moment of the distribution using the most probable threshold

*n*rather than the mean threshold 〈

_{m}*n*〉.

_{i}*n*shifts from 0.047 for uniform pumping to 0.112 for partial pumping, a factor of 2 increase.

_{m}*n*increases from

_{m}*σ*= 0.012 for uniform pumping to

*σ*= 0.023 for partial pumping, nearly twice as large. This indicates the fluctuation of thresholds increases for smaller pumping sizes. Furthermore, the inset of Fig. 5(a) shows the slope of the rising part of the re-normalized threshold distribution with partial pumping. The number of lasing modes

*dN*within a threshold range

_{l}*dn*is proportional to the slope

_{i}*m*(

*dN*=

_{l}*mdn*). With partial pumping, the slope is 4.5 times smaller than with uniform pumping (including the large-threshold tail for partial pumping gives a slope 6 times smaller). If the pumping rate is gradually increased from zero, the number of available lasing modes can be less with partial pumping. In Sec. 4 we discussed how the relative pump step

_{i}*δP*= Δ

*P*/

*P*is smaller for larger

_{t}*P*in experiments. The most probable threshold

_{t}*n*gives a good representation of

_{m}*P*by showing the increased lasing thresholds are a general occurrence and not limited to one random realization. The relative pump step

_{t}*δn*=

_{i}*dn*/

_{i}*n*may allow a finer tuning of the pumping rate, thereby making it easier to see modes begin lasing incrementally.

_{m}44. J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A **82**, 063835 (2010). [CrossRef]

*n*. The inset of Fig. 5(b) reveals that the two distributions are nearly identical at smaller thresholds. Because there is no absorption, feedback from scattering in the unpumped region of the random structures still occurs. Thus, it is not surprising that the results are quite similar.

_{m}*available*small-threshold lasing modes for partial pumping is roughly 35% less than for uniform pumping. With the number of available lasing modes reduced, the frequency spacing between them increases. The enlarged filled diamonds in Fig. 4(a) shows modes which exist for uniform pumping but not partial pumping. These modes disappear, as described in [20

**81**, 043818 (2010). [CrossRef]

*k*.

*ℓ*/

_{G}*L*= 1/3) with and without absorption in the unpumped region. The absorption length is

*ℓ*= 170 nm. The large-threshold modes have completely disappeared by adding absorption in Fig. 6(a). The large-threshold tail without absorption has been excluded in order to compare the distributions directly. We have found [46

_{a}46. J. Andreasen and H. Cao, “Creation of new lasing modes with spatially nonuniform gain,” Opt. Lett. **34**, 3586–3588 (2009). [CrossRef] [PubMed]

*ℓ*+

_{G}*ℓ*which is roughly three times less than the uniform pumping case. We consider three times as many random structures when absorption is included to obtain comparable sampling. For the 30000 realizations, a total of 76673 modes were found for partial pumping with absorption (comparable to the 81396 modes found for uniform pumping with only 10 000 realizations).

_{a}*λ*≤ 750 nm) and the use of frequency-independent gain. The two peaks correspond to different mode numbers. The smaller-threshold peak is composed mostly of higher-frequency modes while the larger-threshold peak is composed of lower-frequency modes. There is not enough fluctuation in their thresholds to completely wash out the bi-modal distribution. Thus we take an average of the two most probable thresholds to find

*n*. This results in

_{m}*n*= 0.129 with absorption, which is nearly identical to the mean threshold 〈

_{m}*n*〉 = 0.130.

_{i}*n*is 15% larger for partial pumping when absorption is included in the unpumped region. This increase of

_{m}*n*shows the lasing threshold increase is a general occurrence and not limited to one random realization.

_{m}*σ*= 0.023 for partial pumping without absorption to

*σ*= 0.015 when absorption is included. This indicates the fluctuation of thresholds decreases when absorption is included. Furthermore, the inset of Fig. 6(a) shows the slope of the rising part of the re-normalized distribution for partial pumping. With absorption, the slope is 1.3 times greater (including the large-threshold tail without absorption means the slope with absorption is 1.8 times greater). Compared to the uniform pumping case where

*σ*= 0.012, the absolute fluctuation of thresholds is still larger with partial pumping even when absorption is included. The rising slope of the distribution with absorption is roughly 3.4 times smaller than for uniform pumping.

*n*. In this case, when absorption is included, the distribution is narrower. Measuring the half-width

_{m}*σ*for each case yields

_{n}*σ*= 0.232 for uniform pumping,

_{n}*σ*= 0.200 for partial pumping, and

_{n}*σ*= 0.145 for partial pumping with absorption. Without absorption, the distribution narrows as well which may not be surprising if spatially inhomogeneous gain is considered to enhance scattering feedback from within the pumped region [19

_{n}19. X. Wu, J. Andreasen, H. Cao, and A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B **24**, A26–A33 (2007). [CrossRef]

## 6. Discussion and Conclusion

*ℓ*. If the size of the unpumped region

_{G}*L*–

*ℓ*is less than the absorption length

_{G}*ℓ*, the total number of modes with and without absorption may be similar.

_{a}*dn*is thus smaller with partial pumping. The threshold fluctuations for partial pumping without absorption in the unpumped region were nearly twice as large as those for uniform pumping. With larger lasing thresholds, stronger pumping is required to reach lasing in partially pumped systems, making the amplified spontaneous emission (ASE) stronger. The SMB simulations show that noise tends to smear out the differences in thresholds as it constantly excites all modes within the gain spectrum. With partial pumping, all TLMs in the small-threshold regime resulted in well-defined peaks in the SMB emission spectra. This result, however, clearly depends on the absolute strength of the threshold fluctuations and the tunability of the pumping rate. Larger threshold fluctuations would make the selection of fewer small-threshold modes for lasing possible, even in the presence of noise. The tunability of the pumping rate, experimentally, depends on the lasing threshold

_{i}*P*. Given a fixed pump step

_{t}*δP*, the relative pump step

*δP*= Δ

*P*/

*P*is smaller for larger

_{t}*P*. The most probable threshold

_{t}*n*gives a good representation of

_{m}*P*.

_{t}*n*was found to increase for partial pumping. Thus, a finer tuning of the pumping rate (smaller steps

_{m}*δn*=

_{i}*dn*/

_{i}*n*) is possible. This facilitates the observation of an incremental increase of lasing modes with the pumping rate.

_{m}10. S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B **59**, R5284–R5287 (1999). [CrossRef]

12. X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A **74**, 053812 (2006). [CrossRef]

48. O. Frazão, C. Correia, J. L. Santos, and J. M. Baptista, “Raman fibre Bragg-grating laser sensor with cooperative Rayleigh scattering for strain-temperature measurement,” Meas. Sci. Technol. **20**, 045203 (2009). [CrossRef]

49. S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castañón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” *Nat. Photonics*4, 231–235 (2010). [CrossRef]

## Acknowledgments

## References and links

1. | V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP |

2. | V. M. Markushev, V. F. Zolin, and C. M. Briskina, “Powder laser,” Zh. Prikl. Spektrosk. |

3. | C. Gouedard, D. Husson, and C. Sauteret, “Generation of spatially incoherent short pulses in laser-pumped neodymium stoichiometric crystals and powders,” J. Opt. Soc. Am. B |

4. | N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature |

5. | W. L. Sha, C.-H. Liu, and R. R. Alfano, “Spectral and temporal measurements of laser action of rhodamine 640 dye in strongly scattering media,” Opt. Lett. |

6. | M. A. Noginov, H. J. Caulfield, N. E. Noginova, and P. Venkateswarlu, “Line narrowing in the dye solution with scattering centers,” Opt. Commun. |

7. | D. S. Wiersma, M. P. van Albada, and A. Lagendijk, “Random laser?” Nature |

8. | H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. |

9. | H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. |

10. | S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B |

11. | Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A |

12. | X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A |

13. | C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. |

14. | X. Jiang and C. M. Soukoulis, “Localized random lasing modes and a path for observing localization,” Phys. Rev. E |

15. | C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. |

16. | M. Patra, “Decay rate distributions of disordered slabs and application to random lasers,” Phys. Rev. E |

17. | V. M. Apalkov and M. E. Raikh, “Universal fluctuations of the random lasing threshold in a sample of a finite area,” Phys. Rev. B |

18. | A. Yamilov, X. Wu, H. Cao, and A. L. Burin, “Absorption-induced confinement of lasing modes in diffusive random media,” Opt. Lett. |

19. | X. Wu, J. Andreasen, H. Cao, and A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B |

20. | J. Andreasen, C. Vanneste, L. Ge, and H. Cao, “Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems,” Phys. Rev. A |

21. | P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B |

22. | M. Terraneo and I. Guarneri, “Distribution of resonance widths in localized tight-binding models,” Eur. Phys. J. B |

23. | F. A. Pinheiro, M. Rusek, A. Orlowski, and B. A. van Tiggelen, “Probing anderson localization of light via decay rate statistics,” Phys. Rev. E |

24. | A. D. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Phys. Rep. |

25. | A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, “Breakdown of diffusion in dynamics of extended waves in mesoscopic media,” Phys. Rev. Lett. |

26. | L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. |

27. | J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. |

28. | G. van Soest, M. Tomita, and A. Lagendijk, “Amplifying volume in scattering media,” Opt. Lett. |

29. | M. Bahoura, K. J. Morris, G. Zhu, and M. A. Noginov, “Dependence of the neodymium random laser threshold on the diameter of the pumped spot,” IEEE J. Quantum Electron. |

30. | E. V. Chelnokov, N. Bityurin, I. Ozerov, and W. Marine, “Two-photon pumped random laser in nanocrystalline ZnO,” Appl. Phys. Lett. |

31. | H. Cao, X. Jiang, Y. Ling, J. Y. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B |

32. | H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science |

33. | X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. |

34. | J. Andreasen and H. Cao, “Finite-different time-domain formulation of stochastic noise in macroscopic atomic systems,” J. Lightwave Technol. |

35. | X. Wu and H. Cao, “Statistical studies of random-lasing modes and amplified-spontaneous-emission spikes in weakly scattering systems,” Phys. Rev. A |

36. | X. Wu, A. Yamilov, H. Noh, H. Cao, E. W. Seelig, and R. P. H. Chang, “Random lasing in closely packed resonant scatterers,” J. Opt. Soc. Am. B |

37. | A. Taflove and S. Hagness, |

38. | J. Andreasen, H. Cao, A. Taflove, P. Kumar, and C. qi Cao, “Finite-difference time-domain simulation of thermal noise in open cavities,” Phys. Rev. A |

39. | D. M. Sullivan, |

40. | R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A |

41. | P. D. Drummond and M. G. Raymer, “Quantum theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A |

42. | A. E. Siegman, |

43. | G. J. de Valcárcel, E. Roldán, and F. Prati, “Semiclassical theory of amplification and lasing,” Rev. Mex. Fis. |

44. | J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A |

45. | P. J. Bardroff and S. Stenholm, “Quantum theory of excess noise,” Phys. Rev. A |

46. | J. Andreasen and H. Cao, “Creation of new lasing modes with spatially nonuniform gain,” Opt. Lett. |

47. | D. W. Scott, “On optimal and data-based histograms,” Biometrika |

48. | O. Frazão, C. Correia, J. L. Santos, and J. M. Baptista, “Raman fibre Bragg-grating laser sensor with cooperative Rayleigh scattering for strain-temperature measurement,” Meas. Sci. Technol. |

49. | S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castañón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(140.3460) Lasers and laser optics : Lasers

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 20, 2010

Revised Manuscript: February 3, 2011

Manuscript Accepted: February 3, 2011

Published: February 7, 2011

**Citation**

Jonathan Andreasen and Hui Cao, "Spectral behavior of partially pumped weakly scattering random lasers," Opt. Express **19**, 3418-3433 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3418

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### References

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