Linearly polarized emission from random lasers with anisotropically amplifying media Sebastian Knitter,1 Michael Kues,1 Michael Haidl, and Carsten Fallnich∗ Institute of Applied Physics, Westf¨alische Wilhelms-Universit¨at M¨unster, Corrensstraße 2, 48149 M¨unster, Germany 1 Authors share credit in equal parts ∗ [email protected]
Abstract: Simulations on three-dimensional random lasers were performed by finite-difference time-domain integration of Maxwell’s equations combined with rate-equations providing gain. We investigated the frequency-dependent emission polarization of random lasers in the far-field of the sample and characterized the influence of anisotropic pumping in orthogonal polarizations. Under weak scattering, the polarization states of random lasing modes were random for isotropic pumping and linear under anisotropic pumping. These findings are in accordance with recent experimental observations. A crossover was observed towards very strong scattering, in which the scattering destroys the pump-induced polarizationanisotropy of the random lasing modes and randomizes (scrambles) the mode-polarization. © 2013 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (290.5855) Scattering, polarization.
References and links 1. H. Cao, Y. Zhao, S. Ho, E. Seelig, Q. Wang, and R. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999). 2. H. Cao, J. Y. Xu, D. Z. Zhang, S. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000). 3. K. van der Molen, R. Tjerkstra, A. Mosk, and A. Lagendijk, “Spatial extent of random laser modes,” Phys. Rev. Lett. 98, 143901 (2007). 4. X. Wu, W. Fang, A. Yamilov, A. A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006). 5. H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. A Math. Gen. 38, 10497–10535 (2005). 6. D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008). 7. M. Noginov, Solid-State Random Lasers, Springer Series in Optical Sciences (Springer, 2005). 8. B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6, 355–359 (2012). 9. S. Knitter, M. Kues, and C. Fallnich, “Spectro-polarimetric signature of a random laser,” Phys. Rev. A 88, 013839 (2013). 10. N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012). 11. M. Leonetti, C. Conti, and C. Lopez, “The mode-locking transition of random lasers,” Nat. Photonics 5, 615–617 (2011). 12. S. Knitter, M. Kues, and C. Fallnich, “Emission polarization of random lasers in organic dye solutions.” Opt. Lett. 37, 3621–3623 (2012).
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13. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966). 14. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000). 15. C. Wang and J. Liu, “Polarization dependence of lasing modes in two-dimensional random lasers,” Phys. Lett. A 353, 269–272 (2006). 16. H. Liu, J. Liu, J. L¨u, Q. Zhang, and K. Wang, “The research on polarization states with local pumping in twodimensional active random medium,” Opt. Commun. 282, 1004–1008 (2009). 17. Y. Liu, J.-S. Liu, and K.-J. Wang, “A method to control the polarization of random terahertz lasing in twodimensional disordered ruby medium,” Chin. Phys. B 20, 094205 (2011). 18. D. Bicout, C. Brosseau, A. Martinez, and J. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994). 19. A. Kim and M. Moscoso, “Influence of the relative refractive index on the depolarization of multiply scattered waves,” Phys. Rev. E 64, 026612 (2001). 20. N. Ghosh, A. Pradhan, P. Gupta, S. Gupta, V. Jaiswal, and R. Singh, “Depolarization of light in a multiply scattering medium: Effect of the refractive index of a scatterer,” Phys. Rev. E 70, 066607 (2004). 21. C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. 4, 794–798 (2008). 22. M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197–204 (2013). 23. A. G. Ardakani, A. R. Bahrampour, S. M. Mahdavi, and M. G. G. Ali, “Numerical study of random lasing in three dimensional amplifying disordered media,” Opt. Commun. 285, 1314–1322 (2012). 24. W. S. Jodrey and E. M. Tory, “Computer simulation of close random packing of equal spheres,” Phys. Rev. A 32, 2347–2351 (1985). 25. U. M. Scheven, “Intrinsic dispersivity of randomly packed monodisperse spheres,” Phys. Rev. Lett. 99, 054502 (2007). 26. R. El-Dardiry, S. Faez, and A. Lagendijk, “Classification of light sources and their interaction with active and passive environments,” Phys. Rev. A. 83, 031801(R) (2011). 27. J. Andreasen, “Finite-difference time-domain formulation of stochastic noise in macroscopic atomic systems,” J. Lightwave Technol. 27, 4530–4535 (2009). 28. R. G. S. El-Dardiry and A. Lagendijk, “Tuning random lasers by engineered absorption,” Appl. Phys. Lett. 98, 161106 (2011). 29. V. Markushev, M. Ryzhkov, and C. Briskina, “Characteristic properties of ZnO random laser pumped by nanosecond pulses,” Appl. Phys. B 84, 333–337 (2006). 30. H. van de Hulst, Light Scattering by Small Particles, Structure of Matter Series (Dover, 1957). 31. A. Penzkofer and W. Falkenstein, “Theoretical investigation of amplified spontaneous emission with picosecond light pulses in dye solutions,” Opt. Quantum Electron. 10, 399–423 (1978). 32. F. Sch¨afer and K. Drexhage, Dye Lasers, Topics in Applied Physics (Springer, 1973). 33. T. Vincenty, “Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations,” Surv. Rev. 23, 88–93 (1975). 34. R. El-Dardiry, R. Mooiweer, and A. Lagendijk, “Experimental phase diagram for random laser spectra,” New J. Phys. 14, 113031 (2012). 35. M. Xu and R. Alfano, “Random walk of polarized light in turbid media,” Phys. Rev. Lett. 95, 213901 (2005). 36. P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
1.
Introduction
The interplay of optical gain and multiple scattering in randomly arranged dielectric structures allows for the amplification of characteristic spatial field distributions with distinct optical frequencies visible as peaks in the optical spectrum and referred to as random lasing modes. Since the first report of these narrow peaks in emission spectra emerging from semiconductor powder [1], numerous experimental studies followed to investigate the phenomenon called “random lasing”, e.g., concerning the threshold behavior [2] or the spatial/spectral mode distribution [3]. Random lasing was demonstrated in a plethora of different materials, ranging from scatterers immersed in laser-dye [4] to intrinsically amplifying semiconductor nano-crystallites [5]. Reviews on the properties and applications of random lasers can be found in [5–7]. Common to all realizations is the presence of optical amplification and multiple-scattering, with the latter enabling the feedback needed to build up laser operation. The properties of random lasers allow for versatile applications: above a certain pump-threshold energy, random lasers convert light
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31592
more efficiently than plain fluorescence [6], rendering them interesting for certain illumination applications [8]. In addition, the usually omnidirectional emission and the small size of random lasers [2] makes them attractive for display applications. As a result of the random assembly of (static) scatterers, random lasers can emit recognizable emission spectra and polarization, characteristic to any given sample [9]. These properties could be exploited to create markers for anti-counterfeiting applications. Currently the random media community is concerned with the question, how random lasers, as diverse as their emission characteristics are, can be controlled, e.g., in terms of the emission spectrum [10], the spatial coherence [8] and the relative phases [11] of random lasing modes. Only recently the manipulation of the emission polarization has received attention in the form of experimental studies [9, 12]. The polarization-dependent interference processes generating random lasing modes can in principle be manipulated by the use of anisotropic amplification, i.e., field amplification in xor y-direction only. The potential of the pump-anisotropy as a control parameter for a random laser stimulated the series of simulations presented in this article. The finite-difference timedomain method (FDTD) [13] solving the Maxwell equations coupled to the rate equations of a four-level laser system [14], enabled studies of the field evolution depending on anisotropic amplification. Our simulations allowed to distinctly change arbitrary sample parameters beyond the constraints of actually available materials, e.g., a free choice of refractive index, optical gain and gain-bandwidth. Numerical studies on the polarization of two-dimensional random lasers were performed before [15–17], and were focused on polarization-dependent threshold energy, mode profile and emission spectrum. These investigations were limited to polarization parallel or perpendicular to the slab, due to the two-dimensional slab geometry. However, most experimental random lasers are three-dimensional and the diffusive properties in such systems are considerably different from systems of lower dimensionality, e.g., in terms of depolarization [18–20] or wave-localization [21, 22]. It is therefore necessary to employ a three-dimensional model. In a recent study on three-dimensional random lasers such a model was demonstrated [23], but the investigations were mainly concerned with the spatial distribution of modes and their emission spectra, without giving particular attention to the mode-polarization. In this paper we present three-dimensional numerical simulations on the polarization characteristics of random lasing modes in the far-field of the emission. We consider the difference of isotropic and anisotropic amplification on the polarization states of the random lasing modes. Experimentally, an anisotropic amplification can be achieved for example by orienting the dipole moment of the molecules used for amplification or by selectively pumping only molecules in a particular orientation as shown in Ref. [12]. Furthermore, the dependence of the polarization characteristics on the scattering strength was characterized. The remaining paper is structured as follows: First the considered medium and the numerical model is presented (Sec. 2) followed by a validation regarding the random lasing threshold and the stability of spectral modes (Sec. 3). Then in Sec. 4, the polarization characteristics of the random lasing modes are investigated considering isotropic and anisotropic amplification. The effect of the scattering strength on the polarization characteristics, leading to a polarization scrambling for anisotropically amplified random laser modes is investigated in Sec. 5. 2.
Random medium and numerical model
The three-dimensional random medium considered in this paper was composed of 13652 dielectric monodisperse spherical scatterers with radius r p = 60 nm arranged by an adapted sphere dropping algorithm [24] within a cubic volume of size L3 = 3.2 × 3.2 × 3.2 µm3 . This distribution of scatterers results in a filling fraction of φ = 0.37, being the maximal possible filling
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31593
fraction for randomized structures of monodisperse spherical particles [25]. The background medium had a refractive index of nbg = 1, whereas the refractive index of the particles np can take different values in order to alter the scattering strength of the medium, i.e., the scattering mean free path. The particles were considered to supply optical gain, modeled by a four-level laser system. The electric field evolution was described by the three-dimensional Maxwell equations, numerically solved by exploiting a FDTD approach [13] including perfectly matched layer absorbing boundary conditions to consider an open system. For detailed informations on the model see Appendix A. The Maxwell equations were coupled to the rate equations of the four-level laser system via the equation of motion of the occurring material polarization density, allowing for optical gain. The corresponding gain curve had a center frequency at ωa = 2π ·600 THz and a width of ∆ωa = 2π ·14 THz. The strength of the amplification was controlled by the pump rate p. This approach models an external pumping process, e.g., optical or electrical pumping, that is generating population inversion for gain in a four-level laser system. An anisotropic amplification (used in Sec. 4) was accomplished by considering three sets of rate equations coupling separately to each spatial direction of the material polarization density. This approach leads to separately addressable amplification in each spatial direction, controllable by according pump rates px , py , and pz . The model was implemented for parallel computation on multiple general purpose graphics processing units (gpGPU: NVIDIA, M2090, 6 GByte). 3.
Random lasing in three-dimensional, isotropically pumped media
The first part of the study was focused on media with isotropic gain, in order to reproduce the most prominent signs of random lasing: the pronounced pump-threshold energy and the occurrence of narrow lines in the emission spectrum. Random lasing was stimulated in a medium with np = 2.0. The simulation was performed for a light propagation time of 10 ps. We found this time to be a reasonable tradeoff between simulation time and the validity of the random lasing spectra (later discussed). To start the field dynamics, the system was excited with a broadband pulse of 2 fs duration. The start pulse was intended to model spontaneous emission at an amplitude of 100 V/m, small compared to the amplitude of random lasing modes of approx. 108 V/m. In order to determine the pump threshold intensity the pump rate p was increased from 0 to 0.7 ps−1 . For each simulation a mean intensity could be derived from the R time integration of the squared electric field (|Ex |2 + |Ey |2 + |Ez |2 ) dt at 27 evenly distributed points in the medium, representing an approximate value proportional to the overall intensity within the medium. Figure 1(a) shows the mean intensity as a function of the pump rate. For pump rates below 0.08 ps−1 , only negligible laser emission intensity could be observed, thus the diffusive losses in the system surmount the gain, preventing the system from laser emission. However, for pump rates above 0.08 ps−1 the mean intensity increased almost linearly with the pump rate, evidencing a threshold behavior, analogous to experimental realizations of random lasers [6]. Considering the spectrum of the field evolution shown in Fig. 1(b) for a pump rate of 1 ps−1 , distinct spectral modes occurred, which are characteristic for random lasing and comparable to spectra from three-dimensional experimental random lasers [1]. The observed threshold behaviour would be less prominent if spontaneous emission would be part of the simulation [26]. In order to investigate whether the random lasing emission depends on spontaneous emission and to simultaneously check the algorithm, simulations with different initial conditions were performed. One set of simulations was started with different spatial position of the broadband starting pulse, while another set was simulated with a noise term added to the rate equations (analogous to Ref. [27]). For each approach ten realizations were simulated. As an example, Fig. 2 shows a selection of three spectra for a pump rate of p = 10 ps−1 , whereas (a) and (b)
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31594
1 (b)
18 (a)
← 600.8 THz (498.9 nm)
12 10
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Spektral intensity (a.u.)
Intensity (a. u.)
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0.8 0.6 ← 603.1 Thz (497.1 nm)
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← 598.3 THz → (501.0 nm)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 Pump rate (ps −1) 1
590
601.6 THz (498.3 nm)
600 595 605 Frequency (THz)
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Fig. 1. (a) Mean (a)integrated spectral intensity as a function of the pump rate. (b) Spectral intensity: distinct modes occurred at different frequencies for a pump rate of 1 ps−1 . 0 590
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result from simulations with different initial positions of the start(THz) pulse with a separation of Frequency 1 µm and (c) results from a simulation with noise-perturbed rate equations. As can be seen, the peak intensities in the spectra were slightly different, whereas the spectral positions of the modes remained fixed. This behavior suggests that the spectral modes are not defined by the field evolution generated by spontaneous emission in the beginning, but rather by the dielectric arrangement of the scattering structure within the medium.
Spectral intensity (a.u.)
1
0 1
0 1
a)
b)
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0 590
595 600 605 Frequency (THz)
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Fig. 2. Spectra for three different initial conditions for a pump rate of p = 10 ps−1 : (a) and (b) for two different spatial positions of the exciting pulse (1 µm separation) and (c) for noise-perturbed rate equations.
These results show that the considered model and its numerical implementation is an appropriate tool to describe the generation of lasing modes in a three-dimensional structure. While for the simulations described above, the amplification was isotropic (acting equally on fields in the x-, y- and z-direction) it will be shown in the next section how random lasing modes can be manipulated by means of anisotropic pumping. #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31595
4.
Dependencies of random lasing modes on anisotropic pumping
The absorption and amplification properties of the gain medium of a random laser are very important for the emission properties of the entire system. These properties limit the emission bandwidth [28], have a major influence on the mode stability [29] and may determine the mode polarization [12] of random laser emission. In a conventional laser system the emission polarization is usually determined in the resonator by polarizing elements, e.g., Brewster-plates or a Lyot-filter. Random lasers are missing such elements, but display polarization-dependent multiple scattering [30], which would usually randomize the emission polarization [18–20]. In the following section, simulations will be presented that show how random lasers can emit linearly polarized light in all modes, as long as the gain medium amplifies anisotropically and the scattering strength is moderate (np /nbg = np = 1.5). Without loss of generality, the x-y-plane (positive z) of the cubic simulation domain was chosen to emit light, that was then spectro-polarimetrically characterized. In order to mimic an instrument to measure the frequency-dependent Stokes vector ~S(ω) = (S0 (ω), S1 (ω), S2 (ω), S3 (ω)), the fields Ei (x, y,t) in the i ∈ {x, y}-direction in the output plane (x, y) were recorded over time t during the simulation (e.g., Ex (x0 , y0 ,t 0 ) stands for the electric field-component in the x-direction at position x0 , y0 and time t 0 ). A two-dimensional Fourier transform yielded the angular spectrum over time, of which only the zero-order peak Ei0 (t) = F2d (Ei (x, y,t))|kx =ky =0 was used, with the wave-vector components kx and ky . The fields Ei0 (t) represent light that was emitted from the output plane as a plane wave, which then traveled in positive z-direction. Additionally, a one-dimensional Fourier transform was applied to this time signal and supplied the emission spectrum for each polarization direction E˜i (ω) = F [Ei0 (t)], including the relative phases for each frequency component. Knowledge of E˜i allowed to calculate the complete frequency-dependent Stokes vector ~S(ω) with: S0 S1 S2 S3
= |E˜x |2 + |E˜y |2 , = |E˜x |2 − |E˜y |2 , = 2 · Re(E˜x∗ E˜y ), = 2 · Im(E˜x∗ E˜y ).
(1) (2) (3) (4)
The simulation was set up with three independent sets of laser rate-equations and allowed to calculate how random lasing modes developed if the pumping occurred anisotropically in the x-, y- or z-direction or in any superposition. This strongly anisotropic amplification resembles a simplified model of real-world systems, in which linearly polarized pump light is absorbed by molecules with asymmetric absorption momenta [31] only in a particular orientation [32]. Diagonally oriented molecules, absorbing partial amounts of energy from horizontally or vertically polarized pump light, are not considered here. Even though this aspect of the simulation might underestimate competition between differently polarized modes, it allows an unobstructed view on the scattering-induced polarization mixing, as the only process allowing energy flow from fields polarized in x-direction to fields polarized in the y-direction and vice versa. As the sample emission was only recorded in the z-direction, anisotropic pumping was simulated in the xand y-direction with the according pump rates px = p · cos2 (α)
(5)
and py = p · sin2 (α) ,
(6)
conserving the total pump rate p = px + py (see Appendix A for details). For the anisotropy parameter α equal to 0° or 90° the pumping was anistotropic purely in x- or y-direction, respectively. In the isotropic case (α = 45°) both axes were pumped at the same rate of 0.5p. #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31596
This case can be compared to optical pumping of anisotropically oriented molecules with a circular pump polarization, which does not introduce any predominant direction for amplification. The simulation was performed for different degrees of pump anisotropy (α = 0° . . . 90°). The spectra E˜i (ω) = F (Ei0 (t = 5 ps . . . 10 ps)) were calculated from a 5 ps long time trace after transient oscillations have decayed, which resulted in a spectral resolution of 0.2 THz (≈ 0.2 nm).
2 1 0 0
d) localized distribution
10
localized distribution
isotropic distribution
20
30 40 50 Pump anisotropy
60 in degree
70
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Fig. 3. Polarization states of single random lasing modes projected onto the Poincar´e sphere with (a) amplification in x-, (b) equal amplification in x- and y- and (c) amplification in the y-direction. States on the positive-S1 hemisphere of the Poincar´e sphere are marked with blue dots and states on the negative-S1 hemisphere are marked with red dots to avoid ambiguity in the two-dimensional projection. The polarization states cluster around the pump polarization induced by anisotropic amplification and are randomly distributed over the sphere under isotropic amplification. (d) The coverage of modes was almost isotropic over a large range of α = 25 . . . 65° and strictly localized for α = 0 . . . 15° and α = 75 . . . 90°.
Performing the operations above, gives direct access to the polarization of individual modes, which can be plotted on the Poincar´e sphere (see Fig. 3). An automated peak-finding algorithm was employed to the spectral traces, with a positive peak criterion at ω j if |E˜x (ω j )|2 + |E˜y (ω j )|2 = I(ω j ) > I(ω j±1 ). If the intensity I(ω j ) of the so found peaks was smaller than 10% of the maximal peak intensity in the spectrum, the considered peak was discarded. While the polarizations of modes clustered around the linear pump polarizations (α = 0° and 90°, Figs. 3(a) and 3(c)), for isotropic pumping (α = 45°, Fig. 3(b)) the polarization states of the modes were distributed all over the surface of the Poincar´e sphere without an apparent pattern. It is worth mentioning, that for anisotropic amplification the individual spectra for the x- and y-component show peaks in the same spectral positions. This is only possible in three-dimensional systems, where scattering couples light from modes in one polarization to the other and therefore seeds modes of the same frequency in the other polarization. The spread of the modes was additionally characterized by first calculating the mean greatcircle distance [33] between the polarization of random lasing modes and the pump polarization ! p 2 + (cos φ sin φ − sin φ cos φ cos Λ)2 (cos φ sin Λ) 1 2 1 2 1 (7) ∆s(~S1 , ~S2 ) = arctan sin φ2 sin φ1 + cos φ2 cos φ1 cos Λ #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31597
with φ1 = 2Ψ1 and φ2 = 2Ψ2 being the azimuthal coordinates and Λ = 2(χ2 − χ1 ) being the polar separation of two polarization states ~S1 and ~S2 on the Poincar´e sphere. Secondly, these distances were averaged over all permutations of random lasing peaks in one emission spectrum, constituting the mean mode separation ∆s in rad. The parameter ∆s is an indicator for the spread of modes on the Poincar´e sphere: ∆s is zero when all modes are identical and approaches π/2 for a homogeneous coverage of the Poincar´e sphere. As shown in Fig. 3(d), the spread of modes ∆s is small for strongly anisotropic pumping (α = 0 . . . 15° and α = 75 . . . 90°). On the other hand, the mode-distribution on the Poincar´e sphere was almost isotropic over a wide range of α = 25 . . . 65°. The results of the simulation match recent experimental observations well [12]. In these experiments anisotropic pumping was accomplished by exciting dye-molecules in a ethanoldye-solution with linearly polarized laser light. The highly anisotropic absorption momenta of these molecules led to an anisotropy in which molecules in an orientation parallel to the pump polarization were pumped very efficiently, while molecules perpendicular to the pump polarization were only meagerly pumped. When such molecules emit or amplify radiation, they will mainly contribute to fields parallel to their own orientation. In this way it could be shown experimentally, that lasers with a spatially randomized feedback mechanism can be forced to emit in a linear polarization for all modes (Fig. 4(a)). The so prepared random dye-laser could also be pumped isotropically, by using circularly polarized pump light, essentially exciting molecules disregarding their current orientation. Similarly to the behaviour of the simulated medium in this study, the polarization of random lasing modes covered the surface of the Poincar´e sphere isotropically (Fig. 4(b)).
Fig. 4. Polarization of random lasing modes in a glycerol solvent projected on the Poincar´e sphere: (a) under linear and (b) under circular pump polarization (view in S1-direction). Red squares indicate the pump polarization. For details see Ref. [12]
The main difference between the simulation and existing experimental realizations is the scattering strength of the investigated material. While the scattering mean free path-length ls in the simulation was on the order of a few µm, the scattering mean free path in the experiments was about two orders of magnitude larger (i.e. weaker scattering). The choice of these parameters has two main reasons: First, the currently available computer memory is insufficient to simulate a three-dimensional system as large as employed in the experiment. Second, the experimental system made of particles immersed in an organic dye-solution would not show distinct random lasing modes if the scattering strength were to be as high as in the simulation. Though the origins of this behaviour are not yet completely understood, which can be extracted from Ref. [34], presenting a detailed experimental study on the matter. Besides of these differences, the qualitative agreement between simulation and experiment is striking. Considering again the presented simulations, in order to estimate the intensity ratio of
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31598
0°
c)
45°
d)
90°
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in degree
Fig. 5. (a) Integrated spectral intensity in x- and y-polarization, normalized to intensity in both polarization channels for each α. Red-full and blue-dashed lines guide the eye. For anisotropic pumping (α = 0° and 90°), the random lasing emission is polarized only in the direction of the pump. For isotropic pumping in the x- and y- direction simultaneously (α = 45°), modes develop into both polarization directions. The emission spectra are different for pumping in (b) x- and (d) y-direction. The emission spectrum under (c) isotropic pumping is linearly independent of the two anisotropic cases.
emitted radiation in both polarizations, the intensities emitted in the x- and y-polarization R IRi = 1/Itot |E˜i (ω)|2 dω were determined for each α, as depicted in Fig. 5(a), with Itot = (|E˜x |2 + |E˜y |2 ) dω being the total emitted intensity in both polarization channels. For the anisotropic cases α = 0° and 90° the emission polarization was concentrated in the plane of pumping. The two curves intersect at α ≈ 45° and Ix ≈ Iy ≈ 0.5, which means, that light was evenly distributed in x- and y-direction. The slight asymmetry of both curves with respect to α = 45° may be attributed to the remaining inhomogeneities of the computer-generated medium. It must be noted, that at α = 0° and 90° a non-zero component of about one percent remained in the respective orthogonal polarization. This can be attributed to polarization mixing due to scattering. This effect gets stronger if the scattering strength increases. As will be shown in Sec. 5, scattering-induced polarization crosstalk can lead to a complete polarimetric randomization of the emitted random lasing modes. Emission spectra I(ω) = |E˜x (ω)|2 + |E˜y (ω)|2 for different anisotropies α are shown in Figs. 5(b)–5(d). The location of the main peaks in the emission spectrum changed significantly with α, without an apparent pattern. The emission spectrum at α = 45° is linearly independent of the two anisotropic cases, indicating the nonlinear nature of the system. Additionally, the different formed spectra for the different values of the #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31599
anisotropy parameter α, leading to different occurring spectral modes having different thresholds, could explain the fluctuations of the integrated spectral intensity in x- and y-direction with varying α (see Fig. 5(a) in the interval [20°, 70°]). Although scattering facilitated the mixing of polarization components, the polarization states of random lasing modes were found to be almost parallel to the direction of the anisotropic pump (see Figs. 3(a) and 3(c)). It appeared unlikely to see modes mainly in the same linear polarization state across a scattering structure. This at a first glance unobvious fact can be explained with the net losses that modes might experience: modes with a polarization in the direction of anisotropic amplification experience gain, that compensates for diffusive losses in the scattering medium and leads to laser emission. Modes having field components only orthogonal to the pumping direction leave the sample through the boundaries, without amplification and without any chance of contributing to laser emission. In other words, the anisotropic amplification allows to specifically excite modes in the same polarization as the polarization of the pump. On the other hand, in the case of isotropic pumping modes of both polarization directions (x and y) can be stimulated simultaneously. Given a random phase between these polarization components, any polarization state could be occupied, as shown in Fig. 3(b). While the polarization state is homogeneously linear throughout the sample under anisotropic pumping, under isotropic pumping the polarization in different locations of the sample might differ strongly. Since however these differently polarized field components have the same optical frequency, they can interfere and a random, but well-defined polarization-state can be detected in the far-field emission. This was observed in the experiments on the spectro-polarimetric emission of microscopic random lasers with isotropic amplification [9], where modes in random polarization states but with a high degree of polarization between 0.8 and 0.9 were detected. 5.
Random lasing polarization in different scattering regimes
In this section it will be shown, how the scattering strength of an anisotropically pumped medium can be used to switch between two distinct regimes: one in which the random lasing modes occupy polarization states close to the linear pump polarization, as it was shown in the previous section, and another one, where strong scattering randomizes the mode polarization completely. The pump was chosen to be anisotropic in the x-direction (α = 0°) to investigate the impact of the scattering-induced polarization scrambling. In order to increase the scattering strength of the medium, the refractive index of the particles np was increased and the polarimetric distance ∆s between the polarization of the anisotropic pump ~Spump = (1, 1, 0, 0) and all peaks within one emission spectrum was calculated and averaged. To give additional information about the scattering properties of the medium, the scattering mean free path (without gain, i.e. in a passive medium) ls (np ) = (ρσ )−1 (8) as a function of the refractive index was plotted in the same diagram (y-axis on the right side). Here, ρ = 416 µm−3 resembles the number of particles per unit volume and σ (n) =
128 π 5 r6p (n2p − 1)2 3 λ 4 (n2p + 2)2
(9)
the scattering cross-section of an individual spherical particle (particle radius r p = 60 nm, wavelength λ = 2πc/ω, speed of light in vacuum c = 3 × 108 m/s, optical frequency ω = 2π · 600 THz, refractive index of particles np ) calculated from Mie theory [30]. The results are depicted in Fig. 6: for low refractive indices of the scattering particles (np < 1.8) the spread of modes was small (∆s < 0.2) and the polarization states were clustered around the linear pump #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31600
Fig. 6. Average great circle distance of random lasing modes and the linear pump polarization on the Poincar´e sphere for different refractive indices. The dashed-blue line guides the eye. Strong scattering (on the right side of the vertical dashed line) facilitated a wide spread of modes on the Poincar´e sphere. The mean free path was calculated from Eqs. (8) and (9).
polarization, as reported in the previous section. For np > 1.8 the mode spread rose rapidly until np ≈ 2.7 and reached a value of about ∆s ≈ 1.6 ∼ = π/2 which is indicative of a homogeneous coverage of the Poincar´e sphere. Further increasing the refractive index beyond np > 2.7 would be unphysical with existing transparent materials, but would also not increase ∆s further, as the mode distribution is already homogeneous on the Poincar´e sphere. We attribute the strong fluctuations of ∆s in this region to the small number of lasing modes (< 10) contributing to ∆s. This behavior can be understood by comparing the relevant length scales of this system: the size of the medium L = 3.2 µm, the transport mean-free path length lt (≈ 2ls for Mie-scattering [5]) and the isotropization length l p . While lt is the propagation length after which the direction of an incident light field is completely randomized, l p determines the randomization length with respect to the polarization. The isotropization length is proportional to the transport mean-free path l p = glt . However, the parameter g is strongly dependent on the size-parameter q = 2πr p /λ and the refractive index n p of the particles [35]. For the physical system here (q ≈ 0.8 and n p = 1.5 . . . 4) g is on the order of two. When emission with purely linear polarization was detected, the refractive index was low and covered a range of n p = 1.5 . . . 1.8, corresponding to a transport mean-free path range of 6.0 to 2.8 µm and an isotropization length of 12 to 5.6 µm. Whereas lt was still on the order of the medium size, l p was larger than the medium. That means that scattering in the medium was apparently strong enough to allow random lasing, but the medium was too small to significantly disturb the linearly polarized emission by scattering. In the region of strong scattering (n p = 1.8 . . . 4.0) lt was less than 1.5 µm and l p covered a range of 3.0 to 1.6 µm. Of course lt was still small enough to allow for lasing, but l p was now on the order of the medium size. That means that, even disregarding the optical amplification taking place inside of the medium, a photon would display a completely randomized state of polarization after traveling through the structure in this regime. From the derivation in [35] one can deduce, that for a reasonable n p -range from 1.5 to 2.5, l p is always larger than lt (the actual value of g is still depending on q). Therefore, it should
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31601
always be possible to construct a random laser (anisotropic amplification) with lt < L < l p , so that the emission of such a laser may be of arbitrary linear polarization, which is equal to the polarization of the pump. If a random laser with random polarization should be desired, one might simply prepare a larger system lt < l p < L or take measures to increase the refractive index contrast between the background medium and the scattering structure. 6.
Conclusion
We presented a numerical model capable of simulating random lasing with anisotropic gain in three dimensions. Random lasing spectra and a distinct threshold were observed, similar to experimental observations. Our calculations revealed, that the emission spectra are only slightly dependent on initial conditions emerging from noise. While the spectral locations of random lasing peaks were fixed, the peak magnitudes fluctuated for different initial conditions, which would be measured as shot-to-shot power fluctuation in a laboratory experiment. Random lasing modes have been found to have distinct polarization states in the far-field emission of the sample in accordance with experimental studies on the matter [9,12]. Simulations with different degrees of amplification anisotropy were performed, mimicking the pumping of an anisotropically absorbing medium (e.g., an organic laser-dye) in linear (α = 0° or 90°) and circular pump polarization (α = 45°). Our simulations showed a strong dependence of the emission spectra and the random lasing mode polarization in the far-field on the pump anisotropy and on the orientation of the anisotropic pump. The transport properties, i.e., the scattering mean free path of the sample, appear to have a strong influence on the polarization of individual random lasing modes: for weak scattering (n p < 1.8) and anisotropic amplification (linear pump polarization) modes clustered in close proximity to the pump polarization, as it was recently reported in [12]. For very strong scattering we observed a crossover towards a complete randomization of mode polarization, independent of the pump polarization. This crossover enables the construction of random lasers that are switchable between linear and random emission polarization by means of changing the refractive index contrast between the scattering structure and the background medium. Such a light source might be useful in lab-on-a-chip type experiments with chiral or birefringent specimens, where the polarization dependent response of a specimen may be of interest. Appendix A. Three-dimensional model The model for a three-dimensional random laser, used in this paper, is similar to the one considered for two dimensions in Ref. [14]. The temporal field evolution was described by the Maxwell equations for a medium without free charges:
∂ ∂t
∇~E (~r,t) = 0, ~ (~r,t) = 0, ∇H ~D (~r,t) = ∇ × H ~ (~r,t) ,
(10) (11) (12)
∂ ~ ~ √1 ∂t H (~r,t) = − ε0 µ0 ∇ × E (~r,t) ,
(13)
~E (~r,t) = [~D (~r,t) − ~P (~r,t)]/ε0 εr (~r) ,
(14)
~ the magnetic field induction, ~D the electric displacement field, where ~E is the electric field, H ~P the polarization density, ε0 the vacuum permittivity, µ0 the permeability of free space, and εr (~r) the relative permittivity representing the dielectric structure of the medium.
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31602
The four-level system was modeled by the rate equations: ∂ N3i (~r,t) , N3i (~r,t) = pi N0i (~r,t) − ∂t τ32 ∂ N3i (~r,t) Ei (~r,t) ∂ N2i (~r,t) + , N2i (~r,t) = Pi (~r,t) − h¯ ωa ∂t ∂t τ32 τ21 ∂ N2i (~r,t) Ei (~r,t) ∂ N1i (~r,t) − , N1i (~r,t) = Pi (~r,t) − h¯ ωa ∂t ∂t τ21 τ10 N1i (~r,t) ∂ N0i (~r,t) = − ∑ pi N0i (~r,t) + ∑ ∂t τ10 i i
(15) (16) (17) (18)
with N1i , N2i , and N3i being the atomic states of the four-level system for the respective spatial direction i ∈ [x, y, z], N0 being the ground-state population, and N0 (~r,t = 0) = Ntot = 3.311 · 1024 being the amount of the participating atoms. The level lifetimes were τ32 = 10−13 s, τ21 = 10−10 s and τ10 = 10−12 s analog to Ref. [36]. The gain was considered by the equation of motion for the material polarization density: ∂ 6πε0 c3 ∂2 2 P (~ r,t) + ∆ω ∆Ni (~r,t)Ei (~r,t) , P (~ r,t) + ω P (~ r,t) = i a i i a ∂t 2 ∂t ωa2
(19)
with center frequency ωa , amplification bandwidth ∆ωa , and ∆Ni = N1i (~r,t) − N2i (~r,t) representing the inversion of levels that are responsible for the laser transition. Acknowledgments This work was supported by the German Federal Ministry of Education and Research (BMBF) within the compound project PEARLS under grant number 13N10154. We acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publication Fond of University of M¨unster.
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31603
Abstract: Simulations on three-dimensional random lasers were performed by finite-difference time-domain integration of Maxwell’s equations combined with rate-equations providing gain. We investigated the frequency-dependent emission polarization of random lasers in the far-field of the sample and characterized the influence of anisotropic pumping in orthogonal polarizations. Under weak scattering, the polarization states of random lasing modes were random for isotropic pumping and linear under anisotropic pumping. These findings are in accordance with recent experimental observations. A crossover was observed towards very strong scattering, in which the scattering destroys the pump-induced polarizationanisotropy of the random lasing modes and randomizes (scrambles) the mode-polarization. © 2013 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (290.5855) Scattering, polarization.
References and links 1. H. Cao, Y. Zhao, S. Ho, E. Seelig, Q. Wang, and R. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999). 2. H. Cao, J. Y. Xu, D. Z. Zhang, S. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000). 3. K. van der Molen, R. Tjerkstra, A. Mosk, and A. Lagendijk, “Spatial extent of random laser modes,” Phys. Rev. Lett. 98, 143901 (2007). 4. X. Wu, W. Fang, A. Yamilov, A. A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006). 5. H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. A Math. Gen. 38, 10497–10535 (2005). 6. D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008). 7. M. Noginov, Solid-State Random Lasers, Springer Series in Optical Sciences (Springer, 2005). 8. B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6, 355–359 (2012). 9. S. Knitter, M. Kues, and C. Fallnich, “Spectro-polarimetric signature of a random laser,” Phys. Rev. A 88, 013839 (2013). 10. N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012). 11. M. Leonetti, C. Conti, and C. Lopez, “The mode-locking transition of random lasers,” Nat. Photonics 5, 615–617 (2011). 12. S. Knitter, M. Kues, and C. Fallnich, “Emission polarization of random lasers in organic dye solutions.” Opt. Lett. 37, 3621–3623 (2012).
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31591
13. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966). 14. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000). 15. C. Wang and J. Liu, “Polarization dependence of lasing modes in two-dimensional random lasers,” Phys. Lett. A 353, 269–272 (2006). 16. H. Liu, J. Liu, J. L¨u, Q. Zhang, and K. Wang, “The research on polarization states with local pumping in twodimensional active random medium,” Opt. Commun. 282, 1004–1008 (2009). 17. Y. Liu, J.-S. Liu, and K.-J. Wang, “A method to control the polarization of random terahertz lasing in twodimensional disordered ruby medium,” Chin. Phys. B 20, 094205 (2011). 18. D. Bicout, C. Brosseau, A. Martinez, and J. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994). 19. A. Kim and M. Moscoso, “Influence of the relative refractive index on the depolarization of multiply scattered waves,” Phys. Rev. E 64, 026612 (2001). 20. N. Ghosh, A. Pradhan, P. Gupta, S. Gupta, V. Jaiswal, and R. Singh, “Depolarization of light in a multiply scattering medium: Effect of the refractive index of a scatterer,” Phys. Rev. E 70, 066607 (2004). 21. C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. 4, 794–798 (2008). 22. M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197–204 (2013). 23. A. G. Ardakani, A. R. Bahrampour, S. M. Mahdavi, and M. G. G. Ali, “Numerical study of random lasing in three dimensional amplifying disordered media,” Opt. Commun. 285, 1314–1322 (2012). 24. W. S. Jodrey and E. M. Tory, “Computer simulation of close random packing of equal spheres,” Phys. Rev. A 32, 2347–2351 (1985). 25. U. M. Scheven, “Intrinsic dispersivity of randomly packed monodisperse spheres,” Phys. Rev. Lett. 99, 054502 (2007). 26. R. El-Dardiry, S. Faez, and A. Lagendijk, “Classification of light sources and their interaction with active and passive environments,” Phys. Rev. A. 83, 031801(R) (2011). 27. J. Andreasen, “Finite-difference time-domain formulation of stochastic noise in macroscopic atomic systems,” J. Lightwave Technol. 27, 4530–4535 (2009). 28. R. G. S. El-Dardiry and A. Lagendijk, “Tuning random lasers by engineered absorption,” Appl. Phys. Lett. 98, 161106 (2011). 29. V. Markushev, M. Ryzhkov, and C. Briskina, “Characteristic properties of ZnO random laser pumped by nanosecond pulses,” Appl. Phys. B 84, 333–337 (2006). 30. H. van de Hulst, Light Scattering by Small Particles, Structure of Matter Series (Dover, 1957). 31. A. Penzkofer and W. Falkenstein, “Theoretical investigation of amplified spontaneous emission with picosecond light pulses in dye solutions,” Opt. Quantum Electron. 10, 399–423 (1978). 32. F. Sch¨afer and K. Drexhage, Dye Lasers, Topics in Applied Physics (Springer, 1973). 33. T. Vincenty, “Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations,” Surv. Rev. 23, 88–93 (1975). 34. R. El-Dardiry, R. Mooiweer, and A. Lagendijk, “Experimental phase diagram for random laser spectra,” New J. Phys. 14, 113031 (2012). 35. M. Xu and R. Alfano, “Random walk of polarized light in turbid media,” Phys. Rev. Lett. 95, 213901 (2005). 36. P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
1.
Introduction
The interplay of optical gain and multiple scattering in randomly arranged dielectric structures allows for the amplification of characteristic spatial field distributions with distinct optical frequencies visible as peaks in the optical spectrum and referred to as random lasing modes. Since the first report of these narrow peaks in emission spectra emerging from semiconductor powder [1], numerous experimental studies followed to investigate the phenomenon called “random lasing”, e.g., concerning the threshold behavior [2] or the spatial/spectral mode distribution [3]. Random lasing was demonstrated in a plethora of different materials, ranging from scatterers immersed in laser-dye [4] to intrinsically amplifying semiconductor nano-crystallites [5]. Reviews on the properties and applications of random lasers can be found in [5–7]. Common to all realizations is the presence of optical amplification and multiple-scattering, with the latter enabling the feedback needed to build up laser operation. The properties of random lasers allow for versatile applications: above a certain pump-threshold energy, random lasers convert light
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31592
more efficiently than plain fluorescence [6], rendering them interesting for certain illumination applications [8]. In addition, the usually omnidirectional emission and the small size of random lasers [2] makes them attractive for display applications. As a result of the random assembly of (static) scatterers, random lasers can emit recognizable emission spectra and polarization, characteristic to any given sample [9]. These properties could be exploited to create markers for anti-counterfeiting applications. Currently the random media community is concerned with the question, how random lasers, as diverse as their emission characteristics are, can be controlled, e.g., in terms of the emission spectrum [10], the spatial coherence [8] and the relative phases [11] of random lasing modes. Only recently the manipulation of the emission polarization has received attention in the form of experimental studies [9, 12]. The polarization-dependent interference processes generating random lasing modes can in principle be manipulated by the use of anisotropic amplification, i.e., field amplification in xor y-direction only. The potential of the pump-anisotropy as a control parameter for a random laser stimulated the series of simulations presented in this article. The finite-difference timedomain method (FDTD) [13] solving the Maxwell equations coupled to the rate equations of a four-level laser system [14], enabled studies of the field evolution depending on anisotropic amplification. Our simulations allowed to distinctly change arbitrary sample parameters beyond the constraints of actually available materials, e.g., a free choice of refractive index, optical gain and gain-bandwidth. Numerical studies on the polarization of two-dimensional random lasers were performed before [15–17], and were focused on polarization-dependent threshold energy, mode profile and emission spectrum. These investigations were limited to polarization parallel or perpendicular to the slab, due to the two-dimensional slab geometry. However, most experimental random lasers are three-dimensional and the diffusive properties in such systems are considerably different from systems of lower dimensionality, e.g., in terms of depolarization [18–20] or wave-localization [21, 22]. It is therefore necessary to employ a three-dimensional model. In a recent study on three-dimensional random lasers such a model was demonstrated [23], but the investigations were mainly concerned with the spatial distribution of modes and their emission spectra, without giving particular attention to the mode-polarization. In this paper we present three-dimensional numerical simulations on the polarization characteristics of random lasing modes in the far-field of the emission. We consider the difference of isotropic and anisotropic amplification on the polarization states of the random lasing modes. Experimentally, an anisotropic amplification can be achieved for example by orienting the dipole moment of the molecules used for amplification or by selectively pumping only molecules in a particular orientation as shown in Ref. [12]. Furthermore, the dependence of the polarization characteristics on the scattering strength was characterized. The remaining paper is structured as follows: First the considered medium and the numerical model is presented (Sec. 2) followed by a validation regarding the random lasing threshold and the stability of spectral modes (Sec. 3). Then in Sec. 4, the polarization characteristics of the random lasing modes are investigated considering isotropic and anisotropic amplification. The effect of the scattering strength on the polarization characteristics, leading to a polarization scrambling for anisotropically amplified random laser modes is investigated in Sec. 5. 2.
Random medium and numerical model
The three-dimensional random medium considered in this paper was composed of 13652 dielectric monodisperse spherical scatterers with radius r p = 60 nm arranged by an adapted sphere dropping algorithm [24] within a cubic volume of size L3 = 3.2 × 3.2 × 3.2 µm3 . This distribution of scatterers results in a filling fraction of φ = 0.37, being the maximal possible filling
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31593
fraction for randomized structures of monodisperse spherical particles [25]. The background medium had a refractive index of nbg = 1, whereas the refractive index of the particles np can take different values in order to alter the scattering strength of the medium, i.e., the scattering mean free path. The particles were considered to supply optical gain, modeled by a four-level laser system. The electric field evolution was described by the three-dimensional Maxwell equations, numerically solved by exploiting a FDTD approach [13] including perfectly matched layer absorbing boundary conditions to consider an open system. For detailed informations on the model see Appendix A. The Maxwell equations were coupled to the rate equations of the four-level laser system via the equation of motion of the occurring material polarization density, allowing for optical gain. The corresponding gain curve had a center frequency at ωa = 2π ·600 THz and a width of ∆ωa = 2π ·14 THz. The strength of the amplification was controlled by the pump rate p. This approach models an external pumping process, e.g., optical or electrical pumping, that is generating population inversion for gain in a four-level laser system. An anisotropic amplification (used in Sec. 4) was accomplished by considering three sets of rate equations coupling separately to each spatial direction of the material polarization density. This approach leads to separately addressable amplification in each spatial direction, controllable by according pump rates px , py , and pz . The model was implemented for parallel computation on multiple general purpose graphics processing units (gpGPU: NVIDIA, M2090, 6 GByte). 3.
Random lasing in three-dimensional, isotropically pumped media
The first part of the study was focused on media with isotropic gain, in order to reproduce the most prominent signs of random lasing: the pronounced pump-threshold energy and the occurrence of narrow lines in the emission spectrum. Random lasing was stimulated in a medium with np = 2.0. The simulation was performed for a light propagation time of 10 ps. We found this time to be a reasonable tradeoff between simulation time and the validity of the random lasing spectra (later discussed). To start the field dynamics, the system was excited with a broadband pulse of 2 fs duration. The start pulse was intended to model spontaneous emission at an amplitude of 100 V/m, small compared to the amplitude of random lasing modes of approx. 108 V/m. In order to determine the pump threshold intensity the pump rate p was increased from 0 to 0.7 ps−1 . For each simulation a mean intensity could be derived from the R time integration of the squared electric field (|Ex |2 + |Ey |2 + |Ez |2 ) dt at 27 evenly distributed points in the medium, representing an approximate value proportional to the overall intensity within the medium. Figure 1(a) shows the mean intensity as a function of the pump rate. For pump rates below 0.08 ps−1 , only negligible laser emission intensity could be observed, thus the diffusive losses in the system surmount the gain, preventing the system from laser emission. However, for pump rates above 0.08 ps−1 the mean intensity increased almost linearly with the pump rate, evidencing a threshold behavior, analogous to experimental realizations of random lasers [6]. Considering the spectrum of the field evolution shown in Fig. 1(b) for a pump rate of 1 ps−1 , distinct spectral modes occurred, which are characteristic for random lasing and comparable to spectra from three-dimensional experimental random lasers [1]. The observed threshold behaviour would be less prominent if spontaneous emission would be part of the simulation [26]. In order to investigate whether the random lasing emission depends on spontaneous emission and to simultaneously check the algorithm, simulations with different initial conditions were performed. One set of simulations was started with different spatial position of the broadband starting pulse, while another set was simulated with a noise term added to the rate equations (analogous to Ref. [27]). For each approach ten realizations were simulated. As an example, Fig. 2 shows a selection of three spectra for a pump rate of p = 10 ps−1 , whereas (a) and (b)
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31594
1 (b)
18 (a)
← 600.8 THz (498.9 nm)
12 10
1
8 6
Spektral intensity (a.u.)
Intensity (a. u.)
14
4 2 0
0 1
Spectral Intensity (a. u.)
16
(b)
0.8 0.6 ← 603.1 Thz (497.1 nm)
0.4 0.2
← 598.3 THz → (501.0 nm)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 Pump rate (ps −1) 1
590
601.6 THz (498.3 nm)
600 595 605 Frequency (THz)
610
Fig. 1. (a) Mean (a)integrated spectral intensity as a function of the pump rate. (b) Spectral intensity: distinct modes occurred at different frequencies for a pump rate of 1 ps−1 . 0 590
595
600
605
610
result from simulations with different initial positions of the start(THz) pulse with a separation of Frequency 1 µm and (c) results from a simulation with noise-perturbed rate equations. As can be seen, the peak intensities in the spectra were slightly different, whereas the spectral positions of the modes remained fixed. This behavior suggests that the spectral modes are not defined by the field evolution generated by spontaneous emission in the beginning, but rather by the dielectric arrangement of the scattering structure within the medium.
Spectral intensity (a.u.)
1
0 1
0 1
a)
b)
c)
0 590
595 600 605 Frequency (THz)
610
Fig. 2. Spectra for three different initial conditions for a pump rate of p = 10 ps−1 : (a) and (b) for two different spatial positions of the exciting pulse (1 µm separation) and (c) for noise-perturbed rate equations.
These results show that the considered model and its numerical implementation is an appropriate tool to describe the generation of lasing modes in a three-dimensional structure. While for the simulations described above, the amplification was isotropic (acting equally on fields in the x-, y- and z-direction) it will be shown in the next section how random lasing modes can be manipulated by means of anisotropic pumping. #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31595
4.
Dependencies of random lasing modes on anisotropic pumping
The absorption and amplification properties of the gain medium of a random laser are very important for the emission properties of the entire system. These properties limit the emission bandwidth [28], have a major influence on the mode stability [29] and may determine the mode polarization [12] of random laser emission. In a conventional laser system the emission polarization is usually determined in the resonator by polarizing elements, e.g., Brewster-plates or a Lyot-filter. Random lasers are missing such elements, but display polarization-dependent multiple scattering [30], which would usually randomize the emission polarization [18–20]. In the following section, simulations will be presented that show how random lasers can emit linearly polarized light in all modes, as long as the gain medium amplifies anisotropically and the scattering strength is moderate (np /nbg = np = 1.5). Without loss of generality, the x-y-plane (positive z) of the cubic simulation domain was chosen to emit light, that was then spectro-polarimetrically characterized. In order to mimic an instrument to measure the frequency-dependent Stokes vector ~S(ω) = (S0 (ω), S1 (ω), S2 (ω), S3 (ω)), the fields Ei (x, y,t) in the i ∈ {x, y}-direction in the output plane (x, y) were recorded over time t during the simulation (e.g., Ex (x0 , y0 ,t 0 ) stands for the electric field-component in the x-direction at position x0 , y0 and time t 0 ). A two-dimensional Fourier transform yielded the angular spectrum over time, of which only the zero-order peak Ei0 (t) = F2d (Ei (x, y,t))|kx =ky =0 was used, with the wave-vector components kx and ky . The fields Ei0 (t) represent light that was emitted from the output plane as a plane wave, which then traveled in positive z-direction. Additionally, a one-dimensional Fourier transform was applied to this time signal and supplied the emission spectrum for each polarization direction E˜i (ω) = F [Ei0 (t)], including the relative phases for each frequency component. Knowledge of E˜i allowed to calculate the complete frequency-dependent Stokes vector ~S(ω) with: S0 S1 S2 S3
= |E˜x |2 + |E˜y |2 , = |E˜x |2 − |E˜y |2 , = 2 · Re(E˜x∗ E˜y ), = 2 · Im(E˜x∗ E˜y ).
(1) (2) (3) (4)
The simulation was set up with three independent sets of laser rate-equations and allowed to calculate how random lasing modes developed if the pumping occurred anisotropically in the x-, y- or z-direction or in any superposition. This strongly anisotropic amplification resembles a simplified model of real-world systems, in which linearly polarized pump light is absorbed by molecules with asymmetric absorption momenta [31] only in a particular orientation [32]. Diagonally oriented molecules, absorbing partial amounts of energy from horizontally or vertically polarized pump light, are not considered here. Even though this aspect of the simulation might underestimate competition between differently polarized modes, it allows an unobstructed view on the scattering-induced polarization mixing, as the only process allowing energy flow from fields polarized in x-direction to fields polarized in the y-direction and vice versa. As the sample emission was only recorded in the z-direction, anisotropic pumping was simulated in the xand y-direction with the according pump rates px = p · cos2 (α)
(5)
and py = p · sin2 (α) ,
(6)
conserving the total pump rate p = px + py (see Appendix A for details). For the anisotropy parameter α equal to 0° or 90° the pumping was anistotropic purely in x- or y-direction, respectively. In the isotropic case (α = 45°) both axes were pumped at the same rate of 0.5p. #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31596
This case can be compared to optical pumping of anisotropically oriented molecules with a circular pump polarization, which does not introduce any predominant direction for amplification. The simulation was performed for different degrees of pump anisotropy (α = 0° . . . 90°). The spectra E˜i (ω) = F (Ei0 (t = 5 ps . . . 10 ps)) were calculated from a 5 ps long time trace after transient oscillations have decayed, which resulted in a spectral resolution of 0.2 THz (≈ 0.2 nm).
2 1 0 0
d) localized distribution
10
localized distribution
isotropic distribution
20
30 40 50 Pump anisotropy
60 in degree
70
80
90
Fig. 3. Polarization states of single random lasing modes projected onto the Poincar´e sphere with (a) amplification in x-, (b) equal amplification in x- and y- and (c) amplification in the y-direction. States on the positive-S1 hemisphere of the Poincar´e sphere are marked with blue dots and states on the negative-S1 hemisphere are marked with red dots to avoid ambiguity in the two-dimensional projection. The polarization states cluster around the pump polarization induced by anisotropic amplification and are randomly distributed over the sphere under isotropic amplification. (d) The coverage of modes was almost isotropic over a large range of α = 25 . . . 65° and strictly localized for α = 0 . . . 15° and α = 75 . . . 90°.
Performing the operations above, gives direct access to the polarization of individual modes, which can be plotted on the Poincar´e sphere (see Fig. 3). An automated peak-finding algorithm was employed to the spectral traces, with a positive peak criterion at ω j if |E˜x (ω j )|2 + |E˜y (ω j )|2 = I(ω j ) > I(ω j±1 ). If the intensity I(ω j ) of the so found peaks was smaller than 10% of the maximal peak intensity in the spectrum, the considered peak was discarded. While the polarizations of modes clustered around the linear pump polarizations (α = 0° and 90°, Figs. 3(a) and 3(c)), for isotropic pumping (α = 45°, Fig. 3(b)) the polarization states of the modes were distributed all over the surface of the Poincar´e sphere without an apparent pattern. It is worth mentioning, that for anisotropic amplification the individual spectra for the x- and y-component show peaks in the same spectral positions. This is only possible in three-dimensional systems, where scattering couples light from modes in one polarization to the other and therefore seeds modes of the same frequency in the other polarization. The spread of the modes was additionally characterized by first calculating the mean greatcircle distance [33] between the polarization of random lasing modes and the pump polarization ! p 2 + (cos φ sin φ − sin φ cos φ cos Λ)2 (cos φ sin Λ) 1 2 1 2 1 (7) ∆s(~S1 , ~S2 ) = arctan sin φ2 sin φ1 + cos φ2 cos φ1 cos Λ #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31597
with φ1 = 2Ψ1 and φ2 = 2Ψ2 being the azimuthal coordinates and Λ = 2(χ2 − χ1 ) being the polar separation of two polarization states ~S1 and ~S2 on the Poincar´e sphere. Secondly, these distances were averaged over all permutations of random lasing peaks in one emission spectrum, constituting the mean mode separation ∆s in rad. The parameter ∆s is an indicator for the spread of modes on the Poincar´e sphere: ∆s is zero when all modes are identical and approaches π/2 for a homogeneous coverage of the Poincar´e sphere. As shown in Fig. 3(d), the spread of modes ∆s is small for strongly anisotropic pumping (α = 0 . . . 15° and α = 75 . . . 90°). On the other hand, the mode-distribution on the Poincar´e sphere was almost isotropic over a wide range of α = 25 . . . 65°. The results of the simulation match recent experimental observations well [12]. In these experiments anisotropic pumping was accomplished by exciting dye-molecules in a ethanoldye-solution with linearly polarized laser light. The highly anisotropic absorption momenta of these molecules led to an anisotropy in which molecules in an orientation parallel to the pump polarization were pumped very efficiently, while molecules perpendicular to the pump polarization were only meagerly pumped. When such molecules emit or amplify radiation, they will mainly contribute to fields parallel to their own orientation. In this way it could be shown experimentally, that lasers with a spatially randomized feedback mechanism can be forced to emit in a linear polarization for all modes (Fig. 4(a)). The so prepared random dye-laser could also be pumped isotropically, by using circularly polarized pump light, essentially exciting molecules disregarding their current orientation. Similarly to the behaviour of the simulated medium in this study, the polarization of random lasing modes covered the surface of the Poincar´e sphere isotropically (Fig. 4(b)).
Fig. 4. Polarization of random lasing modes in a glycerol solvent projected on the Poincar´e sphere: (a) under linear and (b) under circular pump polarization (view in S1-direction). Red squares indicate the pump polarization. For details see Ref. [12]
The main difference between the simulation and existing experimental realizations is the scattering strength of the investigated material. While the scattering mean free path-length ls in the simulation was on the order of a few µm, the scattering mean free path in the experiments was about two orders of magnitude larger (i.e. weaker scattering). The choice of these parameters has two main reasons: First, the currently available computer memory is insufficient to simulate a three-dimensional system as large as employed in the experiment. Second, the experimental system made of particles immersed in an organic dye-solution would not show distinct random lasing modes if the scattering strength were to be as high as in the simulation. Though the origins of this behaviour are not yet completely understood, which can be extracted from Ref. [34], presenting a detailed experimental study on the matter. Besides of these differences, the qualitative agreement between simulation and experiment is striking. Considering again the presented simulations, in order to estimate the intensity ratio of
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31598
0°
c)
45°
d)
90°
595
600
605
in degree
Fig. 5. (a) Integrated spectral intensity in x- and y-polarization, normalized to intensity in both polarization channels for each α. Red-full and blue-dashed lines guide the eye. For anisotropic pumping (α = 0° and 90°), the random lasing emission is polarized only in the direction of the pump. For isotropic pumping in the x- and y- direction simultaneously (α = 45°), modes develop into both polarization directions. The emission spectra are different for pumping in (b) x- and (d) y-direction. The emission spectrum under (c) isotropic pumping is linearly independent of the two anisotropic cases.
emitted radiation in both polarizations, the intensities emitted in the x- and y-polarization R IRi = 1/Itot |E˜i (ω)|2 dω were determined for each α, as depicted in Fig. 5(a), with Itot = (|E˜x |2 + |E˜y |2 ) dω being the total emitted intensity in both polarization channels. For the anisotropic cases α = 0° and 90° the emission polarization was concentrated in the plane of pumping. The two curves intersect at α ≈ 45° and Ix ≈ Iy ≈ 0.5, which means, that light was evenly distributed in x- and y-direction. The slight asymmetry of both curves with respect to α = 45° may be attributed to the remaining inhomogeneities of the computer-generated medium. It must be noted, that at α = 0° and 90° a non-zero component of about one percent remained in the respective orthogonal polarization. This can be attributed to polarization mixing due to scattering. This effect gets stronger if the scattering strength increases. As will be shown in Sec. 5, scattering-induced polarization crosstalk can lead to a complete polarimetric randomization of the emitted random lasing modes. Emission spectra I(ω) = |E˜x (ω)|2 + |E˜y (ω)|2 for different anisotropies α are shown in Figs. 5(b)–5(d). The location of the main peaks in the emission spectrum changed significantly with α, without an apparent pattern. The emission spectrum at α = 45° is linearly independent of the two anisotropic cases, indicating the nonlinear nature of the system. Additionally, the different formed spectra for the different values of the #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31599
anisotropy parameter α, leading to different occurring spectral modes having different thresholds, could explain the fluctuations of the integrated spectral intensity in x- and y-direction with varying α (see Fig. 5(a) in the interval [20°, 70°]). Although scattering facilitated the mixing of polarization components, the polarization states of random lasing modes were found to be almost parallel to the direction of the anisotropic pump (see Figs. 3(a) and 3(c)). It appeared unlikely to see modes mainly in the same linear polarization state across a scattering structure. This at a first glance unobvious fact can be explained with the net losses that modes might experience: modes with a polarization in the direction of anisotropic amplification experience gain, that compensates for diffusive losses in the scattering medium and leads to laser emission. Modes having field components only orthogonal to the pumping direction leave the sample through the boundaries, without amplification and without any chance of contributing to laser emission. In other words, the anisotropic amplification allows to specifically excite modes in the same polarization as the polarization of the pump. On the other hand, in the case of isotropic pumping modes of both polarization directions (x and y) can be stimulated simultaneously. Given a random phase between these polarization components, any polarization state could be occupied, as shown in Fig. 3(b). While the polarization state is homogeneously linear throughout the sample under anisotropic pumping, under isotropic pumping the polarization in different locations of the sample might differ strongly. Since however these differently polarized field components have the same optical frequency, they can interfere and a random, but well-defined polarization-state can be detected in the far-field emission. This was observed in the experiments on the spectro-polarimetric emission of microscopic random lasers with isotropic amplification [9], where modes in random polarization states but with a high degree of polarization between 0.8 and 0.9 were detected. 5.
Random lasing polarization in different scattering regimes
In this section it will be shown, how the scattering strength of an anisotropically pumped medium can be used to switch between two distinct regimes: one in which the random lasing modes occupy polarization states close to the linear pump polarization, as it was shown in the previous section, and another one, where strong scattering randomizes the mode polarization completely. The pump was chosen to be anisotropic in the x-direction (α = 0°) to investigate the impact of the scattering-induced polarization scrambling. In order to increase the scattering strength of the medium, the refractive index of the particles np was increased and the polarimetric distance ∆s between the polarization of the anisotropic pump ~Spump = (1, 1, 0, 0) and all peaks within one emission spectrum was calculated and averaged. To give additional information about the scattering properties of the medium, the scattering mean free path (without gain, i.e. in a passive medium) ls (np ) = (ρσ )−1 (8) as a function of the refractive index was plotted in the same diagram (y-axis on the right side). Here, ρ = 416 µm−3 resembles the number of particles per unit volume and σ (n) =
128 π 5 r6p (n2p − 1)2 3 λ 4 (n2p + 2)2
(9)
the scattering cross-section of an individual spherical particle (particle radius r p = 60 nm, wavelength λ = 2πc/ω, speed of light in vacuum c = 3 × 108 m/s, optical frequency ω = 2π · 600 THz, refractive index of particles np ) calculated from Mie theory [30]. The results are depicted in Fig. 6: for low refractive indices of the scattering particles (np < 1.8) the spread of modes was small (∆s < 0.2) and the polarization states were clustered around the linear pump #196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31600
Fig. 6. Average great circle distance of random lasing modes and the linear pump polarization on the Poincar´e sphere for different refractive indices. The dashed-blue line guides the eye. Strong scattering (on the right side of the vertical dashed line) facilitated a wide spread of modes on the Poincar´e sphere. The mean free path was calculated from Eqs. (8) and (9).
polarization, as reported in the previous section. For np > 1.8 the mode spread rose rapidly until np ≈ 2.7 and reached a value of about ∆s ≈ 1.6 ∼ = π/2 which is indicative of a homogeneous coverage of the Poincar´e sphere. Further increasing the refractive index beyond np > 2.7 would be unphysical with existing transparent materials, but would also not increase ∆s further, as the mode distribution is already homogeneous on the Poincar´e sphere. We attribute the strong fluctuations of ∆s in this region to the small number of lasing modes (< 10) contributing to ∆s. This behavior can be understood by comparing the relevant length scales of this system: the size of the medium L = 3.2 µm, the transport mean-free path length lt (≈ 2ls for Mie-scattering [5]) and the isotropization length l p . While lt is the propagation length after which the direction of an incident light field is completely randomized, l p determines the randomization length with respect to the polarization. The isotropization length is proportional to the transport mean-free path l p = glt . However, the parameter g is strongly dependent on the size-parameter q = 2πr p /λ and the refractive index n p of the particles [35]. For the physical system here (q ≈ 0.8 and n p = 1.5 . . . 4) g is on the order of two. When emission with purely linear polarization was detected, the refractive index was low and covered a range of n p = 1.5 . . . 1.8, corresponding to a transport mean-free path range of 6.0 to 2.8 µm and an isotropization length of 12 to 5.6 µm. Whereas lt was still on the order of the medium size, l p was larger than the medium. That means that scattering in the medium was apparently strong enough to allow random lasing, but the medium was too small to significantly disturb the linearly polarized emission by scattering. In the region of strong scattering (n p = 1.8 . . . 4.0) lt was less than 1.5 µm and l p covered a range of 3.0 to 1.6 µm. Of course lt was still small enough to allow for lasing, but l p was now on the order of the medium size. That means that, even disregarding the optical amplification taking place inside of the medium, a photon would display a completely randomized state of polarization after traveling through the structure in this regime. From the derivation in [35] one can deduce, that for a reasonable n p -range from 1.5 to 2.5, l p is always larger than lt (the actual value of g is still depending on q). Therefore, it should
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31601
always be possible to construct a random laser (anisotropic amplification) with lt < L < l p , so that the emission of such a laser may be of arbitrary linear polarization, which is equal to the polarization of the pump. If a random laser with random polarization should be desired, one might simply prepare a larger system lt < l p < L or take measures to increase the refractive index contrast between the background medium and the scattering structure. 6.
Conclusion
We presented a numerical model capable of simulating random lasing with anisotropic gain in three dimensions. Random lasing spectra and a distinct threshold were observed, similar to experimental observations. Our calculations revealed, that the emission spectra are only slightly dependent on initial conditions emerging from noise. While the spectral locations of random lasing peaks were fixed, the peak magnitudes fluctuated for different initial conditions, which would be measured as shot-to-shot power fluctuation in a laboratory experiment. Random lasing modes have been found to have distinct polarization states in the far-field emission of the sample in accordance with experimental studies on the matter [9,12]. Simulations with different degrees of amplification anisotropy were performed, mimicking the pumping of an anisotropically absorbing medium (e.g., an organic laser-dye) in linear (α = 0° or 90°) and circular pump polarization (α = 45°). Our simulations showed a strong dependence of the emission spectra and the random lasing mode polarization in the far-field on the pump anisotropy and on the orientation of the anisotropic pump. The transport properties, i.e., the scattering mean free path of the sample, appear to have a strong influence on the polarization of individual random lasing modes: for weak scattering (n p < 1.8) and anisotropic amplification (linear pump polarization) modes clustered in close proximity to the pump polarization, as it was recently reported in [12]. For very strong scattering we observed a crossover towards a complete randomization of mode polarization, independent of the pump polarization. This crossover enables the construction of random lasers that are switchable between linear and random emission polarization by means of changing the refractive index contrast between the scattering structure and the background medium. Such a light source might be useful in lab-on-a-chip type experiments with chiral or birefringent specimens, where the polarization dependent response of a specimen may be of interest. Appendix A. Three-dimensional model The model for a three-dimensional random laser, used in this paper, is similar to the one considered for two dimensions in Ref. [14]. The temporal field evolution was described by the Maxwell equations for a medium without free charges:
∂ ∂t
∇~E (~r,t) = 0, ~ (~r,t) = 0, ∇H ~D (~r,t) = ∇ × H ~ (~r,t) ,
(10) (11) (12)
∂ ~ ~ √1 ∂t H (~r,t) = − ε0 µ0 ∇ × E (~r,t) ,
(13)
~E (~r,t) = [~D (~r,t) − ~P (~r,t)]/ε0 εr (~r) ,
(14)
~ the magnetic field induction, ~D the electric displacement field, where ~E is the electric field, H ~P the polarization density, ε0 the vacuum permittivity, µ0 the permeability of free space, and εr (~r) the relative permittivity representing the dielectric structure of the medium.
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31602
The four-level system was modeled by the rate equations: ∂ N3i (~r,t) , N3i (~r,t) = pi N0i (~r,t) − ∂t τ32 ∂ N3i (~r,t) Ei (~r,t) ∂ N2i (~r,t) + , N2i (~r,t) = Pi (~r,t) − h¯ ωa ∂t ∂t τ32 τ21 ∂ N2i (~r,t) Ei (~r,t) ∂ N1i (~r,t) − , N1i (~r,t) = Pi (~r,t) − h¯ ωa ∂t ∂t τ21 τ10 N1i (~r,t) ∂ N0i (~r,t) = − ∑ pi N0i (~r,t) + ∑ ∂t τ10 i i
(15) (16) (17) (18)
with N1i , N2i , and N3i being the atomic states of the four-level system for the respective spatial direction i ∈ [x, y, z], N0 being the ground-state population, and N0 (~r,t = 0) = Ntot = 3.311 · 1024 being the amount of the participating atoms. The level lifetimes were τ32 = 10−13 s, τ21 = 10−10 s and τ10 = 10−12 s analog to Ref. [36]. The gain was considered by the equation of motion for the material polarization density: ∂ 6πε0 c3 ∂2 2 P (~ r,t) + ∆ω ∆Ni (~r,t)Ei (~r,t) , P (~ r,t) + ω P (~ r,t) = i a i i a ∂t 2 ∂t ωa2
(19)
with center frequency ωa , amplification bandwidth ∆ωa , and ∆Ni = N1i (~r,t) − N2i (~r,t) representing the inversion of levels that are responsible for the laser transition. Acknowledgments This work was supported by the German Federal Ministry of Education and Research (BMBF) within the compound project PEARLS under grant number 13N10154. We acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publication Fond of University of M¨unster.
#196328 - $15.00 USD Received 23 Aug 2013; revised 18 Oct 2013; accepted 29 Oct 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031591 | OPTICS EXPRESS 31603
