- Email: [email protected]
A numerical study of ZnO random lasers using FDTD method
Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
A numerical study of ZnO random lasers using FDTD method ⁎
Çağlar Duman , Fatih Kaburcuk
T
Department of Electric Electronics Engineering, Erzurum Technical University, Erzurum, Turkey
A R T IC LE I N F O
ABS TRA CT
Keywords: FDTD Method Rate equations ZnO Random laser
In this paper, a 2D strongly disordered a zinc oxide (ZnO) thin film is modelled and simulated to investigate behaviors of ZnO random lasers. In order to observe the scattering and interferes of the electromagnetic waves (EMW), Maxwell’s equations are solved for a transverse magnetic (TM) wave by using finite-difference time-domain (FDTD) method. Perfect electric conductor (PEC) boundary condition is used to truncate the FDTD problem space. The 2D random laser is formed with vertical aligned ZnO nanorods. The size and distribution of ZnO nanorods in the medium are randomly distributed in the air background medium. The intensity of modes and spectrum of ZnO random laser are observed for different EMW excitation values, and different numbers and radii of ZnO nanorods in the medium.
1. Introduction Random lasers were proposed in late 1960s [1]. Self-generation of electromagnetic wave (EMW) in an active medium filled with scatterers was proposed in [2] for the first time. In [2], EMW generation by an aggregate of scattering particles with negative absorption was considered and forming of stable frequency EMW oscillations in the medium was discussed theoretically [2,3]. In 1986, stimulated emission with influence of strong laser pulses in sodium lanthanum molybdate powders was observed in [4]. In the literature, many researchers [5–11] have been obtained sharp laser peaks from random lasers. In chronical order, important classes of random lasers can be sorted as neodymium random lasers [4], liquid dye random lasers [5], polymer random lasers [6], zinc oxide (ZnO) random lasers [7–9], electron-beam-pumped random lasers [10], liquid crystal random lasers [11], etc. Optical feedback occurs due to powerful and frequent scatterings for all type of random lasers. Multiple powerful scatterings can be obtained by random variation of the dielectric constant of the nanorods relative to background medium. This random medium is considered as a disordered medium. If the medium has sufficient gain, EMW can be amplified [12]. Low production costs, sample specific wavelength of operation, small flexible shape, and substrate compatibility of the random lasers increase the number of potential applications, but practical usage of random lasers is still very limited because of their low efficiencies. The random lasers do not need mirrors to form a feedback mechanism. Thus, they can be used for lasers which there is no available effective refractive elements like ultraviolet (UV) and X-Ray lasers [1,13]. There are several studies about random UV lasing from ZnO powder [7], ZnO thin film [8], and ZnO cluster [9]. ZnO is preferable for UV-blue region light emitters because of its direct wide band gap at 3.37 eV and large exciton binding energy (60 eV) [14,15]. Especially, ZnO thin films are the best option for the random lasers because of their high disorder. Thanks to the high disorder, optic waveguides can be obtained by minimizing nonaxial scattering of the EMW. Thus, the necessity of cleaving of semiconductor material can be avoided. Thanks to these features, random lasers formed by ZnO thin films can be driven by electrical current [16]. The high quality ZnO thin films can be fabricated by simple and cheap aqueous chemical methods at low temperatures. Thus, cheap plastic
⁎
Corresponding author. E-mail addresses: [email protected] (Ç. Duman), [email protected] (F. Kaburcuk).
https://doi.org/10.1016/j.ijleo.2018.12.136 Received 7 December 2018; Accepted 26 December 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
substrates can be used in the production of UV laser diodes [17]. The ZnO random laser diodes fabricated in form of p-n homojunctions, heterojunctions, metal-insulator semiconductor, and Schottky junction were reported in [18–21]. There are two types of feedback mechanism in random lasers: nonresonant (incoherent) and resonant (coherent). In a random laser with the nonresonant feedback, lasing occurs at weak scattering regime. In a disordered medium, EMW scatters and walks randomly before leaving the media. If there is gain in the media, a photon can start stimulated emission and a secondary photon is created. When gain length is close to average light path, probability of creating a second photon becomes one and photon density increases. In a random laser with the resonant feedback, if the optical scattering is strong enough, the EMW can go back to the previous scatterer and a closed loop forms. While the amplification is bigger than lost at the closed loop, lasing starts. In the closed loop, phase shift should be integer multiples of 2π to create the stimulated emission. The EMW can return its starting position with several ways. Thus, all backscattered EMW interferes. In both feedback mechanism, if constructive interference occurs, the EMW amplifies and lasing frequency becomes equal to the frequency of the EMW. When the EMW scattering is weak, propagation of the EMW can be described by normal diffusion. With increasing of the scattering, recurrent scattering phenomenon increases. The interference between counter propagating EMW increases the backscattering and weak localization forms. If the scattering exceeds above a certain value, a transition to localized state occurs because the EMW propagation is prevented by multiple scatterings [22]. In random lasers, lasing occurs due to multiple scattering of EMW. Thus, morphology of the medium and precise tuning of the particle sizes are not needed. However, mode control is difficult for random lasers because of randomness of the scatterings. Therefore, several unexpected lasing modes are observed. Optical confinement of random lasers is insufficient and unstable. Thus, strong background luminescence and lasing emission intensity of random lasers show large fluctuations. In usual random lasers, specific excitation conditions can cause decreasing in mode number [13,15]. In this study, two dimensional (2D) strongly disordered ZnO thin film is modelled and simulated to investigate lasing action of ZnO random lasers using finite-different time-domain (FDTD) method. The 2D random laser is formed with vertical aligned ZnO nanorods. The intensity of modes and spectrum of the ZnO random laser are observed for different EMW excitation values, and different numbers and radii of ZnO nanorods in the medium. 2. Simulation method and problem 2.1. FDTD method The FDTD method [23] has one of the most popular numerical methods to solve Maxwell’s curl equations. This method has been used to solve many applications such as scattering, inverse scattering, antennas, plasma, waveguides, and coupling, etc. These Maxwell’s equations can be discretized both in time and space applying the second-order central difference formula. The electric (E) and magnetic (H) fields are sampled at discrete position both in time and space. A 2D problem space is considered in this work and terminated by perfect electric conductor (PEC) boundaries [24]. The E and H fields excited by the EMW will propagate in the problem space and they will hit the PEC boundaries. Then, they will propagate back to the problem space. Therefore, the field components on the outer boundaries are not calculated and are fixed to zero value during the FDTD calculations. The well-known two Maxwell’s curl equations are solved for a transverse magnetic (TM) wave [25].
μ0
μ0
∂Ez ∂Hx =− ∂t ∂y ∂Hy
εr ε0
∂t
=
(1)
∂Ez ∂x
(2)
∂Hy ∂Ez ∂Pz ∂Hx − + = ∂x ∂y ∂t ∂t
(3)
where P is the electric polarization density (from which amplification or gain can be obtained), μo, εo and εr are the permeability, permittivity of the free space, and relative permittivity of the materials, respectively. In this study, a four-level atomic system is considered for electron transition. The rate equations in (4) are used to take into consideration the gain of the medium. ∂N1 ∂t
=
N2 τ21
− Wp N1
∂N2 ∂t
=
N3 τ32
−
N2 τ21
−
Ez ∂Pz ℏωl ∂t
∂N3 ∂t
=
N4 τ43
−
N3 τ32
+
Ez ∂Pz ℏωl ∂t
∂N4 ∂t
= − τ 4 + Wp N1
N
(4)
43
where Ni, τi(i-1), WP, ℏ and, ωl are electron densities in the corresponding energy level, life times in the corresponding energy level, pumping rate, Planck constant, and transition frequency between energy levels 2 and 3, respectively. The (Ez ℏωl )(∂Pz ∂t ) term in (4) shows stimulated transition rate. The electric polarization density (Pz) is calculated by 994
Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
Fig. 1. Problem space.
∂2Pz ∂Pz + Δωl + ωl2 Pz = κΔNEz ∂t 2 ∂t
(5)
where Δωl = 1/τ32+2/T2 is linewidth of the atomic transition, where T2 is the mean time of between dephasing events, ΔN = N2-N3 is population density difference between second and third energy levels, and κ is constant and calculated by 6πε0c3/(ωl2τ32). In ZnO medium, excitonic recombination is dominant while optical gain forming, thus excitonic population can be considered as equal to carrier population in the medium [17]. 2.2. Problem space The size of the FDTD problem space is 2 μm × 2 μm. In order to satisfy the numerical stability in the FDTD method, the cell size is set to 10 nm in the x and y directions. The time step is 16.667 atto seconds. The distribution of ZnO nanorods is randomly distributed in the air background medium. The ZnO nanorods with different radii are distributed randomly in the FDTD problem space. The radii of ZnO nanorods are considered as vary between 80 nm and 120 nm. The number of the nanorods in the random media is set to around 200. We considered that the gaps between ZnO nanorods are air. The Fig. 1 shows the structure of problem space with randomly distributed ZnO nanorods. The Gaussian EM pulse [24] is used to excite the problem space. 2.3. Numerical results The spatial distribution of the optical electric field and emission pattern in the random media, shown in Fig. 2, are obtained when pumping rate is less than a threshold pumping rate which is about 4 × 1010m−3s−1. The stimulated emission starts when the pumping rate is greater than the threshold intensity. Therefore, higher light intensity in distinct regions of the random media starts forming. In Fig. 3, the spatial distribution of optical electric field and the optical emission are shown in the random media when pumping rate is just above the threshold pumping rate. If pumping rate is much higher than the threshold pumping rate, the localization of the optical emission in distinct regions seems more like apparent and also the emission intensity increases as seen from Fig. 4. The spectrum of output intensity in the point where a peak of optical emission is observed in Fig.4 is shown in Fig. 5-a. For the sake of the comparison, the spectrum of output intensity at the same point is shown in Fig. 5-b when the pumping rate is less than the threshold. As seen from Fig. 5-a, the optical wavelength of output intensity has a clear peak at 372 nm whereas there are several peaks in Fig. 5-b. The reason of this difference is nature of the emission. In Fig. 5-a, a narrow spectrum is observed because of stimulated emission while a broad spectrum is observed in Fig. 5-b because of spontaneous emission. In order to show the effect of the radii variation in the nanorods, the radii of the nanorods are decreased to half (40–60 nm). The decreasing in the radii causes increasing in the threshold pumping rate. Therefore, spontaneous emission is dominated in the random media as seen in Fig. 6. Fig. 7 shows the emission pattern when the pumping rate is above this threshold pumping rate. The stimulated emission and mode localization can also be obtained for the smaller radii. The reduction in the size of the nanorods in which the amplification of EMW is occurred causes increasing in the threshold. In order to show the effect of the number of the nanorods on the emission pattern, it has been double. It is realized from Fig. 8 that increasing the number of the nanorods changes the threshold pumping rate, the positions of the optical modes, and their intensities when the threshold is same as in Fig. 4. The variation of threshold pumping rate are not directly determined by the number of the nanorods. The changing number of nanorods changes the location of the nanorods, thus random nature of the medium also varies. 3. Discussions In this study, a 2D disordered medium with ZnO nanorods was modelled and simulated using the FDTD method to analyze lasing 995
Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
Fig. 2. (a) The spatial distribution of optical electric field and (b) emission pattern while pumping rate is smaller than the threshold value.
Fig. 3. (a) The spatial distribution of optical electric field and (b) emission pattern when pumping rate is just above the threshold value.
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Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
Fig. 4. (a) The spatial distribution of optical electric field and (b) emission pattern when pumping rate is much higher than threshold value.
Fig. 5. Spectrums of output intensity when (a) pumping rate is much higher than threshold value and (b) pumping rate is smaller than the threshold value.
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Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
Fig. 6. (a) The spatial distribution of optical electric field and (b) emission pattern when pumping rate is just above the threshold value.
Fig. 7. (a) The spatial distribution of optical electric field and (b) emission pattern when pumping rate is much higher than threshold value.
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Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
Fig. 8. Increasing the number of nanorods for same pumping rate in Fig. 4.
action. The obtained results show that the spontaneous emission is dominant if pumping rate is under threshold, and if pumping rate is increased above the threshold, the stimulated emission occurs due to the optical gain being greater than the losses and forming of the optical closed loops. When pumping rate is under threshold, no closed loops are formed. If the radii of ZnO nanorods are decreased, the threshold pumping rate increases. Moreover, the number of ZnO nanorods slightly affects the threshold pumping rate value and changes the spatial distribution of electrical filed. References [1] M.A. Noginov, J. Novak, D. Grigsby, L. Deych, Applicability of the diffusion model to random lasers with non-resonant feedback, J. Opt. A: Pure Appl. Opt 8 (2006) 285–295. [2] V.S. Letokhov, P.N. Lebedev, Stimulated emission of an ensemble of scattering particles with absorption, JETP Lett. 5 (8) (1967) 262–265. [3] V.S. Letokhov, Generation of light by a scattering medium with negative resonance absorption, Sov. Phys. JETP. 26 (1968) 835–840. [4] V.M. Markushev, V.F. Zolin, C.M. Briskina, Luminescence and stimulated emission of neodymium in sodium lanthanum molybdate powders, Sov. J. Quantum Electron. 16 (2) (1986) 281–283. [5] B.R. Anderson, R. Gunawidjaja, H. Eilers, Low-threshold and narrow line width diffusive random lasing in rhodamine 6G dye-doped polyurethane with dispersed ZrO2 nanoparticles, Josai Shika Daigaku Kiyo 31 (10) (2014) 2363–2370. [6] S. Lia, L. Wang, T. Zhai, J. Tong, L. Niu, F. Tong, F. Cao, H. Liu, X. Zhang, A dual-wavelength polymer random laser with the step-type cavity, Org. Electron. 57 (2018) 323–326. [7] H. Cao, Y.G. Zhao, S.T.E. Ho, W. Seelig, Q.H. Wang, R.P.H. Chang, Random laser action in semiconductor powder, Phys. Rev. Lett. 82 (1999) 2278–2281. [8] S.F. Yu, C. Yuen, S.P. Lau, H.W. Lee, Zinc oxide thin-film random lasers on silicon substrate, Appl. Phys. Lett. 84 (2004) 3244–3246. [9] H. Cao, J.Y. Xu, D.Z. Zhang, S.H. Chang, S.T. Ho, E.W. Seeling, X. Liu, R.P.H. Chang, Spatial confinement of laser light in active random media, Phys. Rev. Lett. 84 (2000) 5584–5587. [10] G.R. Williams, S.B. Bayram, S.C. Rand, Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors, Phys. Rev. A (Coll Park) 65 (2001) 013807/1-6. [11] D.S. Wiersma, S. Cavalieri, Temperature-controlled random laser action in liquid crystal infiltrated systems, Phys. Rev. E 66 (2002) 56612/1-5. [12] P. Rafiee, V. Ahmadi, M.H. Yavari, Two-dimensional spectral-spatial analysis of ZnO nanoparticles random lasers, ICTON'09, Azores, Portugal (June 28 - July 2) (2009). [13] H. Cao, Lasing in random media, Waves Random Media 13 (2013) R1–R39. [14] Ü. Özgür, Ya.I. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Doğan, V. Avrutin, S.-J. Cho, H. Morkoç, A comprehensive review of ZnO materials and devices, J. Appl. Phys. 98 (4) (2005) 041301/1-103. [15] T. Nakamura, T. Yamamoto, S. Adachi, Temperature dependence of lasing characteristics of irregular-shaped-microparticle ZnO laser, Opt. Express 23 (2015) 28905–28913. [16] S.F. Yu, E.S.P. Leong, High-power single-mode ZnO thin-film random lasers, IEEE J. Quantum Electron. 40 (2004) 1186–1194. [17] E.S.P. Leong, S.F. Yu, S.P. Lau, A.P. Abiyasa, Edge-emitting vertically aligned ZnO nanorods random laser on plastic substrate, IEEE Photonics Technol. Lett. 19 (2007) 1792–1794. [18] J. Huang, S. Chu, J.Y. Kong, L. Zhang, C.M. Schwarz, G.P. Wang, L. Chernyak, Z.H. Chen, J.L. Liu, ZnO p–n homojunction random laser diode based on nitrogendoped p‐type nanowires, Adv. Opt. Mater. 1 (2013) 179–185. [19] E.S.P. Leong, S.F. Yu, UV random lasing action in p-SiC (4H)/i-ZnO-SiO2 Nanocomposite/n-ZnO:Al heterojunction diodes, Adv. Mater. 18 (2006) 1685–1688. [20] X. Ma, J. Pan, P. Chen, D. Li, H. Zhang, Y. Yang, D. Yang, Room temperature electrically pumped ultraviolet random lasing from ZnO nanorod arrays on Si, Opt. Express 17 (2009) 14426–14433. [21] S.B. Bashar, M. Suja, W. Shi, J. Liu, Enhanced random lasing from Au-ZnO nanowire schottky diode by using distributed Bragg reflector, 2016 IEEE Photonics Conference (IPC) (2016) 2–6 Oct. 2016. [22] H. Cao, Random lasers with coherent feedback, IEEE J. Sel. Top. Quantum Electron. 9 (2003) 111–119. [23] K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag. 14 (1966) 302–307. [24] A.Z. Elsherbeni, V. Demir, ACES series on computational electromagnetics and engineering, The Finite-difference Time-domain Method for Electromagnetics With MATLAB Simulations, second edition, SciTech Publishing, an Imprint of IET, Edison, NJ, 2016. [25] C. Wang, J. Liu, Polarization dependence of lasing modes in two-dimensional random lasers, Phys. Lett. A 353 (2006) 269–272.
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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
A numerical study of ZnO random lasers using FDTD method ⁎
Çağlar Duman , Fatih Kaburcuk
T
Department of Electric Electronics Engineering, Erzurum Technical University, Erzurum, Turkey
A R T IC LE I N F O
ABS TRA CT
Keywords: FDTD Method Rate equations ZnO Random laser
In this paper, a 2D strongly disordered a zinc oxide (ZnO) thin film is modelled and simulated to investigate behaviors of ZnO random lasers. In order to observe the scattering and interferes of the electromagnetic waves (EMW), Maxwell’s equations are solved for a transverse magnetic (TM) wave by using finite-difference time-domain (FDTD) method. Perfect electric conductor (PEC) boundary condition is used to truncate the FDTD problem space. The 2D random laser is formed with vertical aligned ZnO nanorods. The size and distribution of ZnO nanorods in the medium are randomly distributed in the air background medium. The intensity of modes and spectrum of ZnO random laser are observed for different EMW excitation values, and different numbers and radii of ZnO nanorods in the medium.
1. Introduction Random lasers were proposed in late 1960s [1]. Self-generation of electromagnetic wave (EMW) in an active medium filled with scatterers was proposed in [2] for the first time. In [2], EMW generation by an aggregate of scattering particles with negative absorption was considered and forming of stable frequency EMW oscillations in the medium was discussed theoretically [2,3]. In 1986, stimulated emission with influence of strong laser pulses in sodium lanthanum molybdate powders was observed in [4]. In the literature, many researchers [5–11] have been obtained sharp laser peaks from random lasers. In chronical order, important classes of random lasers can be sorted as neodymium random lasers [4], liquid dye random lasers [5], polymer random lasers [6], zinc oxide (ZnO) random lasers [7–9], electron-beam-pumped random lasers [10], liquid crystal random lasers [11], etc. Optical feedback occurs due to powerful and frequent scatterings for all type of random lasers. Multiple powerful scatterings can be obtained by random variation of the dielectric constant of the nanorods relative to background medium. This random medium is considered as a disordered medium. If the medium has sufficient gain, EMW can be amplified [12]. Low production costs, sample specific wavelength of operation, small flexible shape, and substrate compatibility of the random lasers increase the number of potential applications, but practical usage of random lasers is still very limited because of their low efficiencies. The random lasers do not need mirrors to form a feedback mechanism. Thus, they can be used for lasers which there is no available effective refractive elements like ultraviolet (UV) and X-Ray lasers [1,13]. There are several studies about random UV lasing from ZnO powder [7], ZnO thin film [8], and ZnO cluster [9]. ZnO is preferable for UV-blue region light emitters because of its direct wide band gap at 3.37 eV and large exciton binding energy (60 eV) [14,15]. Especially, ZnO thin films are the best option for the random lasers because of their high disorder. Thanks to the high disorder, optic waveguides can be obtained by minimizing nonaxial scattering of the EMW. Thus, the necessity of cleaving of semiconductor material can be avoided. Thanks to these features, random lasers formed by ZnO thin films can be driven by electrical current [16]. The high quality ZnO thin films can be fabricated by simple and cheap aqueous chemical methods at low temperatures. Thus, cheap plastic
⁎
Corresponding author. E-mail addresses: [email protected] (Ç. Duman), [email protected] (F. Kaburcuk).
https://doi.org/10.1016/j.ijleo.2018.12.136 Received 7 December 2018; Accepted 26 December 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
substrates can be used in the production of UV laser diodes [17]. The ZnO random laser diodes fabricated in form of p-n homojunctions, heterojunctions, metal-insulator semiconductor, and Schottky junction were reported in [18–21]. There are two types of feedback mechanism in random lasers: nonresonant (incoherent) and resonant (coherent). In a random laser with the nonresonant feedback, lasing occurs at weak scattering regime. In a disordered medium, EMW scatters and walks randomly before leaving the media. If there is gain in the media, a photon can start stimulated emission and a secondary photon is created. When gain length is close to average light path, probability of creating a second photon becomes one and photon density increases. In a random laser with the resonant feedback, if the optical scattering is strong enough, the EMW can go back to the previous scatterer and a closed loop forms. While the amplification is bigger than lost at the closed loop, lasing starts. In the closed loop, phase shift should be integer multiples of 2π to create the stimulated emission. The EMW can return its starting position with several ways. Thus, all backscattered EMW interferes. In both feedback mechanism, if constructive interference occurs, the EMW amplifies and lasing frequency becomes equal to the frequency of the EMW. When the EMW scattering is weak, propagation of the EMW can be described by normal diffusion. With increasing of the scattering, recurrent scattering phenomenon increases. The interference between counter propagating EMW increases the backscattering and weak localization forms. If the scattering exceeds above a certain value, a transition to localized state occurs because the EMW propagation is prevented by multiple scatterings [22]. In random lasers, lasing occurs due to multiple scattering of EMW. Thus, morphology of the medium and precise tuning of the particle sizes are not needed. However, mode control is difficult for random lasers because of randomness of the scatterings. Therefore, several unexpected lasing modes are observed. Optical confinement of random lasers is insufficient and unstable. Thus, strong background luminescence and lasing emission intensity of random lasers show large fluctuations. In usual random lasers, specific excitation conditions can cause decreasing in mode number [13,15]. In this study, two dimensional (2D) strongly disordered ZnO thin film is modelled and simulated to investigate lasing action of ZnO random lasers using finite-different time-domain (FDTD) method. The 2D random laser is formed with vertical aligned ZnO nanorods. The intensity of modes and spectrum of the ZnO random laser are observed for different EMW excitation values, and different numbers and radii of ZnO nanorods in the medium. 2. Simulation method and problem 2.1. FDTD method The FDTD method [23] has one of the most popular numerical methods to solve Maxwell’s curl equations. This method has been used to solve many applications such as scattering, inverse scattering, antennas, plasma, waveguides, and coupling, etc. These Maxwell’s equations can be discretized both in time and space applying the second-order central difference formula. The electric (E) and magnetic (H) fields are sampled at discrete position both in time and space. A 2D problem space is considered in this work and terminated by perfect electric conductor (PEC) boundaries [24]. The E and H fields excited by the EMW will propagate in the problem space and they will hit the PEC boundaries. Then, they will propagate back to the problem space. Therefore, the field components on the outer boundaries are not calculated and are fixed to zero value during the FDTD calculations. The well-known two Maxwell’s curl equations are solved for a transverse magnetic (TM) wave [25].
μ0
μ0
∂Ez ∂Hx =− ∂t ∂y ∂Hy
εr ε0
∂t
=
(1)
∂Ez ∂x
(2)
∂Hy ∂Ez ∂Pz ∂Hx − + = ∂x ∂y ∂t ∂t
(3)
where P is the electric polarization density (from which amplification or gain can be obtained), μo, εo and εr are the permeability, permittivity of the free space, and relative permittivity of the materials, respectively. In this study, a four-level atomic system is considered for electron transition. The rate equations in (4) are used to take into consideration the gain of the medium. ∂N1 ∂t
=
N2 τ21
− Wp N1
∂N2 ∂t
=
N3 τ32
−
N2 τ21
−
Ez ∂Pz ℏωl ∂t
∂N3 ∂t
=
N4 τ43
−
N3 τ32
+
Ez ∂Pz ℏωl ∂t
∂N4 ∂t
= − τ 4 + Wp N1
N
(4)
43
where Ni, τi(i-1), WP, ℏ and, ωl are electron densities in the corresponding energy level, life times in the corresponding energy level, pumping rate, Planck constant, and transition frequency between energy levels 2 and 3, respectively. The (Ez ℏωl )(∂Pz ∂t ) term in (4) shows stimulated transition rate. The electric polarization density (Pz) is calculated by 994
Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
Fig. 1. Problem space.
∂2Pz ∂Pz + Δωl + ωl2 Pz = κΔNEz ∂t 2 ∂t
(5)
where Δωl = 1/τ32+2/T2 is linewidth of the atomic transition, where T2 is the mean time of between dephasing events, ΔN = N2-N3 is population density difference between second and third energy levels, and κ is constant and calculated by 6πε0c3/(ωl2τ32). In ZnO medium, excitonic recombination is dominant while optical gain forming, thus excitonic population can be considered as equal to carrier population in the medium [17]. 2.2. Problem space The size of the FDTD problem space is 2 μm × 2 μm. In order to satisfy the numerical stability in the FDTD method, the cell size is set to 10 nm in the x and y directions. The time step is 16.667 atto seconds. The distribution of ZnO nanorods is randomly distributed in the air background medium. The ZnO nanorods with different radii are distributed randomly in the FDTD problem space. The radii of ZnO nanorods are considered as vary between 80 nm and 120 nm. The number of the nanorods in the random media is set to around 200. We considered that the gaps between ZnO nanorods are air. The Fig. 1 shows the structure of problem space with randomly distributed ZnO nanorods. The Gaussian EM pulse [24] is used to excite the problem space. 2.3. Numerical results The spatial distribution of the optical electric field and emission pattern in the random media, shown in Fig. 2, are obtained when pumping rate is less than a threshold pumping rate which is about 4 × 1010m−3s−1. The stimulated emission starts when the pumping rate is greater than the threshold intensity. Therefore, higher light intensity in distinct regions of the random media starts forming. In Fig. 3, the spatial distribution of optical electric field and the optical emission are shown in the random media when pumping rate is just above the threshold pumping rate. If pumping rate is much higher than the threshold pumping rate, the localization of the optical emission in distinct regions seems more like apparent and also the emission intensity increases as seen from Fig. 4. The spectrum of output intensity in the point where a peak of optical emission is observed in Fig.4 is shown in Fig. 5-a. For the sake of the comparison, the spectrum of output intensity at the same point is shown in Fig. 5-b when the pumping rate is less than the threshold. As seen from Fig. 5-a, the optical wavelength of output intensity has a clear peak at 372 nm whereas there are several peaks in Fig. 5-b. The reason of this difference is nature of the emission. In Fig. 5-a, a narrow spectrum is observed because of stimulated emission while a broad spectrum is observed in Fig. 5-b because of spontaneous emission. In order to show the effect of the radii variation in the nanorods, the radii of the nanorods are decreased to half (40–60 nm). The decreasing in the radii causes increasing in the threshold pumping rate. Therefore, spontaneous emission is dominated in the random media as seen in Fig. 6. Fig. 7 shows the emission pattern when the pumping rate is above this threshold pumping rate. The stimulated emission and mode localization can also be obtained for the smaller radii. The reduction in the size of the nanorods in which the amplification of EMW is occurred causes increasing in the threshold. In order to show the effect of the number of the nanorods on the emission pattern, it has been double. It is realized from Fig. 8 that increasing the number of the nanorods changes the threshold pumping rate, the positions of the optical modes, and their intensities when the threshold is same as in Fig. 4. The variation of threshold pumping rate are not directly determined by the number of the nanorods. The changing number of nanorods changes the location of the nanorods, thus random nature of the medium also varies. 3. Discussions In this study, a 2D disordered medium with ZnO nanorods was modelled and simulated using the FDTD method to analyze lasing 995
Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
Fig. 2. (a) The spatial distribution of optical electric field and (b) emission pattern while pumping rate is smaller than the threshold value.
Fig. 3. (a) The spatial distribution of optical electric field and (b) emission pattern when pumping rate is just above the threshold value.
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Fig. 4. (a) The spatial distribution of optical electric field and (b) emission pattern when pumping rate is much higher than threshold value.
Fig. 5. Spectrums of output intensity when (a) pumping rate is much higher than threshold value and (b) pumping rate is smaller than the threshold value.
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Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
Ç. Duman, F. Kaburcuk
Fig. 6. (a) The spatial distribution of optical electric field and (b) emission pattern when pumping rate is just above the threshold value.
Fig. 7. (a) The spatial distribution of optical electric field and (b) emission pattern when pumping rate is much higher than threshold value.
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Optik - International Journal for Light and Electron Optics 181 (2019) 993–999
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Fig. 8. Increasing the number of nanorods for same pumping rate in Fig. 4.
action. The obtained results show that the spontaneous emission is dominant if pumping rate is under threshold, and if pumping rate is increased above the threshold, the stimulated emission occurs due to the optical gain being greater than the losses and forming of the optical closed loops. When pumping rate is under threshold, no closed loops are formed. If the radii of ZnO nanorods are decreased, the threshold pumping rate increases. Moreover, the number of ZnO nanorods slightly affects the threshold pumping rate value and changes the spatial distribution of electrical filed. References [1] M.A. Noginov, J. Novak, D. Grigsby, L. Deych, Applicability of the diffusion model to random lasers with non-resonant feedback, J. Opt. A: Pure Appl. Opt 8 (2006) 285–295. [2] V.S. Letokhov, P.N. Lebedev, Stimulated emission of an ensemble of scattering particles with absorption, JETP Lett. 5 (8) (1967) 262–265. [3] V.S. Letokhov, Generation of light by a scattering medium with negative resonance absorption, Sov. Phys. JETP. 26 (1968) 835–840. [4] V.M. Markushev, V.F. Zolin, C.M. Briskina, Luminescence and stimulated emission of neodymium in sodium lanthanum molybdate powders, Sov. J. Quantum Electron. 16 (2) (1986) 281–283. [5] B.R. Anderson, R. Gunawidjaja, H. Eilers, Low-threshold and narrow line width diffusive random lasing in rhodamine 6G dye-doped polyurethane with dispersed ZrO2 nanoparticles, Josai Shika Daigaku Kiyo 31 (10) (2014) 2363–2370. [6] S. Lia, L. Wang, T. Zhai, J. Tong, L. Niu, F. Tong, F. Cao, H. Liu, X. Zhang, A dual-wavelength polymer random laser with the step-type cavity, Org. Electron. 57 (2018) 323–326. [7] H. Cao, Y.G. Zhao, S.T.E. Ho, W. Seelig, Q.H. Wang, R.P.H. Chang, Random laser action in semiconductor powder, Phys. Rev. Lett. 82 (1999) 2278–2281. [8] S.F. Yu, C. Yuen, S.P. Lau, H.W. Lee, Zinc oxide thin-film random lasers on silicon substrate, Appl. Phys. Lett. 84 (2004) 3244–3246. [9] H. Cao, J.Y. Xu, D.Z. Zhang, S.H. Chang, S.T. Ho, E.W. Seeling, X. Liu, R.P.H. Chang, Spatial confinement of laser light in active random media, Phys. Rev. Lett. 84 (2000) 5584–5587. [10] G.R. Williams, S.B. Bayram, S.C. Rand, Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors, Phys. Rev. A (Coll Park) 65 (2001) 013807/1-6. [11] D.S. Wiersma, S. Cavalieri, Temperature-controlled random laser action in liquid crystal infiltrated systems, Phys. Rev. E 66 (2002) 56612/1-5. [12] P. Rafiee, V. Ahmadi, M.H. Yavari, Two-dimensional spectral-spatial analysis of ZnO nanoparticles random lasers, ICTON'09, Azores, Portugal (June 28 - July 2) (2009). [13] H. Cao, Lasing in random media, Waves Random Media 13 (2013) R1–R39. [14] Ü. Özgür, Ya.I. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Doğan, V. Avrutin, S.-J. Cho, H. Morkoç, A comprehensive review of ZnO materials and devices, J. Appl. Phys. 98 (4) (2005) 041301/1-103. [15] T. Nakamura, T. Yamamoto, S. Adachi, Temperature dependence of lasing characteristics of irregular-shaped-microparticle ZnO laser, Opt. Express 23 (2015) 28905–28913. [16] S.F. Yu, E.S.P. Leong, High-power single-mode ZnO thin-film random lasers, IEEE J. Quantum Electron. 40 (2004) 1186–1194. [17] E.S.P. Leong, S.F. Yu, S.P. Lau, A.P. Abiyasa, Edge-emitting vertically aligned ZnO nanorods random laser on plastic substrate, IEEE Photonics Technol. Lett. 19 (2007) 1792–1794. [18] J. Huang, S. Chu, J.Y. Kong, L. Zhang, C.M. Schwarz, G.P. Wang, L. Chernyak, Z.H. Chen, J.L. Liu, ZnO p–n homojunction random laser diode based on nitrogendoped p‐type nanowires, Adv. Opt. Mater. 1 (2013) 179–185. [19] E.S.P. Leong, S.F. Yu, UV random lasing action in p-SiC (4H)/i-ZnO-SiO2 Nanocomposite/n-ZnO:Al heterojunction diodes, Adv. Mater. 18 (2006) 1685–1688. [20] X. Ma, J. Pan, P. Chen, D. Li, H. Zhang, Y. Yang, D. Yang, Room temperature electrically pumped ultraviolet random lasing from ZnO nanorod arrays on Si, Opt. Express 17 (2009) 14426–14433. [21] S.B. Bashar, M. Suja, W. Shi, J. Liu, Enhanced random lasing from Au-ZnO nanowire schottky diode by using distributed Bragg reflector, 2016 IEEE Photonics Conference (IPC) (2016) 2–6 Oct. 2016. [22] H. Cao, Random lasers with coherent feedback, IEEE J. Sel. Top. Quantum Electron. 9 (2003) 111–119. [23] K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag. 14 (1966) 302–307. [24] A.Z. Elsherbeni, V. Demir, ACES series on computational electromagnetics and engineering, The Finite-difference Time-domain Method for Electromagnetics With MATLAB Simulations, second edition, SciTech Publishing, an Imprint of IET, Edison, NJ, 2016. [25] C. Wang, J. Liu, Polarization dependence of lasing modes in two-dimensional random lasers, Phys. Lett. A 353 (2006) 269–272.
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