- Email: [email protected]
Circuit-level modeling of quantum cascade lasers: Influence of Kerr effect on static and dynamic responses
Optik 127 (2016) 10303–10310
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Circuit-level modeling of quantum cascade lasers: Influence of Kerr effect on static and dynamic responses Mohsen Darman, Kiazand Fasihi ∗ Department of Electrical Engineering, Golestan University, Gorgan, Iran
a r t i c l e
i n f o
Article history: Received 17 June 2016 Accepted 23 August 2016 Keywords: Equivalent circuit model Kerr effect Quantum cascade lasers SPICE Static and dynamic characteristics Three-level rate-equations
a b s t r a c t The three-level rate-equations-based QCL models, considering the Kerr nonlinearity term, possess multiple DC solution regimes for nonnegative values of injection currents. We show that by giving the proper initial conditions, the DC bias point simulation results of the proposed QCL circuit-level model will converge to the physical positive output powers. We also show that the proposed model can accurately predict the influence of the Kerr effect on static and dynamic behaviors of the QCLs. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction The quantum cascade lasers (QCLs) have gained widespread attention due to their small sizes, large intensity modulation bandwidth, narrow linewidth, high operating temperatures and robust fabrication [1–3]. These advantages make them attractive for different applications, such as chemical sensors [4], anesthetic gas detecting, pollution monitoring, free-space optical communication systems, infrared spectroscopy [5], and so on. The operation of QCLs can be modeled and analyzed numerically using the multi-level rate-equations-based theoretical models. The used numerical approaches are accurate, but very computationally intensive and aren’t suitable for system-level designs and optimizations. Instead, it is possible to form a rate-equations-based equivalent circuit model and reduce drastically the complexity of the analysis [6–11]. We have shown that in the equivalent circuit model of QCLs, which are based on three-level rate-equations, considering the Kerr nonlinearity term, multiple solution regimes are obtained. It has been shown that the optical Kerr nonlinearity in QCLs can lead to some significant issues, such as mode locking and photon number dependency of the losses, hence, studies of QCL with optical Kerr nonlinearity are very important [12]. In this investigation, we propose a new equivalent circuit model for QCLs based on three-level rate-equations, which can predict the influence of the Kerr nonlinearity on the QCLs operations. We show that by giving the initial conditions, which can be obtained from a DC sweep simulation or an analytical expression, the proposed model can be used for both steady-state and dynamic responses. The proposed model is verified using the analytical and also numerical results from hamadou et al. [12] for the dynamics of electrons, in different energy levels, and photons in the cavity. The simulation results show an excellent agreement in all the comparisons. The paper is organized as follows: In Section II, the solution regimes of a three-level rate-equations-based model of the QCLs, considering the influence
∗ Corresponding author. E-mail addresses: [email protected] (M. Darman), [email protected] (K. Fasihi). http://dx.doi.org/10.1016/j.ijleo.2016.08.054 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
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Fig. 1. Schematic of a three-level QCL. Table 1 The used parametres for evaluating the solution regimes of the three-level rate-equations of the QCLs. The model parameters are from Hamadou et al. [12]. Symbol
Description
Value
32 31 21 sp out p 3 0 N L R1 R2 neff G ˇ Ith ˛m ˛w
Lifetimes representing the transitions from level 3 to 2 Lifetimes representing the transitions from level 3 to 1 Lifetimes representing the transitions from level 2 to 1 Spontaneous lifetime between level 3 and 2 Electron escape time Photon lifetime Electron lifetime in the upper level Power output coupling efficiency Free space wavelength Number of gain stages Lateral length of the cavity Reflecting power of facets 1 Reflecting power of facets 2 Effective refractive index of the cavity Gain coefficient per stage Fraction of spontaneous emission entering into lasing mode Threshold current Mirrors loss Waveguide loss of the cavity
2.1 × 10−12 (s) 4.2 × 10−12 (s) 3 × 10−13 (s) 3.55 × 10−9 (s) 10−12 (s) 3.36 × 10−12 (s) −12 1.4 × 10 (s) 0.19136 9 × 10−6 (m) 48 10−3 (m) 0.29 0.29 3.27 744 (1/s) 2 × 10−3 1.1108 (A) 1240 (1/m) (1/m) 2000
of Kerr nonlinearity, is presented. In Section III, the derivation and verification of the proposed equivalent circuit model of QCLs, that is based on the three-level rate-equations, with adding Kerr nonlinearity term, is demonstrated. 2. Solution regimes of the three-level rate-equations-based model of the QCLs, considering the influence of Kerr effect The rate equations of a three-level QCL, corresponding to Fig. 1, can be described in terms of the following three first-order differential equations [12–14] dN3 I N3 = − − (N3 − N2 )GNph , e 3 dt
(1)
dN2 = dt
(2)
1 1 + 32 sp
N3 −
N2 + (N3 − N2 )GNph , 21
N3 N2 N1 dN1 = + − , 31 21 out dt dNph dt
= N (N3 − N2 )GNph + Nˇ
(3) Nph N3 − (1 − 0 Nph ), p sp
(4)
where N1 , N2 and N3 are the instantaneous numbers of electrons in energy level 3, i.e. the upper lasing state, energy level 2, i.e. the lower lasing state, and energy levels 1, i.e. the ground state, respectively. Nph denotes the photon numbers in the cavity, N is the number of cascade stages, G is the gain coefficient, and 0 is the dimensionless coefficient, which describes the magnitude of nonlinear effects. ˇ denotes the spontaneous emission coupling coefficient, e is the magnitude of electronic charge and I is the injection current. 32 , 31 and 21 are the lifetimes representing the transitions from level 3 to 1, 3 to 2 and 2 to 1, respectively, and out represents the electron escape time between two adjacent stages. sp denotes the radiative spontaneous relaxation time between levels 3 and 2, and 3 is the electron lifetime in the upper level and is given by 1/3 = 1/32 + 1/31 + 1/sp . Furthermore, p is photon lifetime that can be expressed as p = (c (˛w + ˛m ))−1 , where ˛w and ˛m are the losses of waveguide cavity and mirrors, respectively, and c = c/neff is the average velocity of light in the
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Fig. 2. (a) Graphs of (8) and (9) versus the ϑNph , when I/Ith = 0.3 . (b) The corresponding graphs of (N2 )1 and (N2 )2 versus ϑNph .(c) The enlarged graph of (b) aroundNph = 0..
system, in which neff and c are the effective refractive index of the cavity and the speed of light in vacuum, respectively. The mirrors loss can be given by ˛m = − ln(R1 R2 )/(2L), where R1 and R2 are the reflectivity of the facets 1 and 2, respectively, and L is the lateral length of the cavity. The output power and the threshold current can be calculated analytically from the QCL rate-equations. The output power obeys the Pout /Nph = 0 h/p = ϑ relation, where 0 is the power output coupling efficiency and assuming that R1 = R2 ,is given by [15] 0 =
1 ˛m . 2 ˛m + ˛w
(5)
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Fig. 3. (a) Graphs of (8) and (9) versus the ϑNph , when I/Ith = 1.45 . (b) The corresponding graphs of (N2 )1 and (N2 )2 versus ϑNph .(c) The enlarged graph of (b) aroundNph = 0..
Also, the threshold current can be written as [7] Ith =
e NGp 21
+ + sp 1 − − sp
,
(6)
where = 21 /31 , = 21 /32 and sp = 21 /sp . Now, the solution regimes of the QCLs are evaluated, under the steady-state conditions. From (1), it can be shown that N2 =
N3 3
+ GN3 Nph − GNph
I e
.
(7)
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Fig. 4. Circuit-level implementation of a three-level QCL model, considering the Kerr nonlinearity.
Fig. 5. The graphs of different solution regimes for 0 = 1.25 × 10−10 ..
By substituting (7) in (2) and (4), one can find (N3 )1 =
I e 1 3
1 + GNph 21
+ GNph + GNph 21
,
(8)
31
and
(N3 )2 =
I e
−
(1−0 Nph )Nph Np 1 3
−
ˇ sp
,
(9)
respectively. Using the QCL parameters found in Table 1, the threshold injection current can be calculated as Ith = 1.1108 A. The graphs of (8) and (9) versus the output power, Pout = ϑNph , for the injection current of 0.3 Ith is shown in Fig. 2(a). Furthermore, by substituting (8) and (9) in (7), respectively, (N2 )1 and (N2 )2 can be obtained and graphed in Fig. 2 (b). Fig. 2 (c) shows the enlarged graph of Fig. 2(b) aroundNph = 0. As the same way, Fig. 3 shows the corresponding graphs, when I = 1.45 Ith . From Figs. 2 and 3, one can see that, in both cases, when I < Ith or I > Ith , the QCL rate-equations have three different solution regimes: (a) a negative solution regime, (b) a positive physical solution regime, which is desired, and (c) a positive nonphysical solution regime. In the general case, by considering these solution regimes, it can be deduced that using of a SPICE based circuit simulator, which is based on the Newton-Raphson iterative technique, depending the value of the initial condition, the circuit-level simulation result can be converged to any one of the solution regimes. This problem can be resolved by using the proper initial conditions which can be obtained using a DC sweep simulation or from a DC bias point analytical expression [12]. In fact, it can be shown that in a DC sweep simulation (with a sufficiently small sweeping step), which always starts with zero initial condition, the static simulation results of the three-level rate-equations-based equivalent circuits (considering the Kerr nonlinearity term), are always converged to the physical solution regime. Furthermore, Hamadou et al. [12] showed that when the normalized injection current verifies the well-known optical stability domain condition: 1<
I Ith
<
0 Nph.sat 1 1 + + 2 4 40 Nph.sat
,
(10)
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Fig. 6. (a) Time evolution of the electron numbers in each of the three energy levels, for different values of 0 , when J/Jth = 1.45. (b) The corresponding time evolution plot of the photon numbers. (c) The corresponding time evolution plot of the population inversion between energy level 3 and 2.
the nonzero solutions for the photon number can be expressed as
Nph,± =
−0 Nph,sat − 1 ±
2
0 Nph,sat + 1 20
− 40 Nph,sat I I
th
,
(11)
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Fig. 7. Normalized modulation response (in decibel) versus modulation frequency for different values of normalized injection current, when 0 = 1.25 × 10−10 ..
where, Nph,+ is a nonphysical (unstable) solution, Nph,− is a physical (stable) solution and Nph,sat is the photon saturation number that is given by Nph,sat =
1
.
(12)
3 1 + G
During the circuit simulation, one can see that for a given value of I/Ith , using the initial condition obtained from (11) for Nph,− , always, the DC bias simulation result of the proposed circuit-level model is converged to the physical bias point and as a result, the other static and dynamic simulation results will be converged to desired results. 3. Derivation and verification of the equivalent circuit model for the QCLs, based on the three-level rate-equations with considering the Kerr effect As mention previously, the three-level rate-equations-based model of QCLs, considering the Kerr nonlinearity term, have a multiple DC solution regime, and the convergence to the desired solution requires the initial conditions which can be obtained using a DC sweep simulation or an analytical expression. Fig. 4 shows the circuit-level implementation of the proposed QCL model. This equivalent circuit model is obtained through manipulation of the three-level rate-equations (1)–(4). For facilitating the interpretation of the rate-equations as the electric circuit equations, we take Ni = vni , i = 1, 2 , 3, in which vni can be taken as the voltage of the node ni . After substituting the above transformations, the modified rate-equations are C3
dvn3 vn + 3 + Istim3 = I, R3 dt
(14)
C2
dvn2 vn + 2 = I2 + Istim2 , R2 dt
(15)
C1
dvn1 vn + 1 = I1 + I 1 , R1 dt
(16)
d vm vm + = Ispon + Istim + Ik , Rm dt
(17)
Cm
Where C1 = C2 = Cm = 1, C3 = e, R1 = out , R2 = 21 ,Rm = p ,R3 = 3 /(e), I1 = vn3
Ispon = Nˇ sp , Ik =
2 −1 , I 0 Nph stim p
vn2
21
, I1 =
vn3
31
, I2 = (32 −1 + sp −1 ) vn3 ,
= NG(vn3 − vn2 )Nph , Istim2 = G(vn3 − vn2 )Nph , and Istim3 = eG(vn3 − vn2 )Nph .
We investigate the static and dynamic behaviors of the proposed QCL equivalent circuit model using of the Winspice simulator. In the circuit simulation, we use the parameters provided in Table 1. Fig. 5 shows the plot of the light-current characteristics of the proposed QCL model when 1.1 < I/Ith < 4, for 0 = 1.25 × 10−10 . The used initial condition values,Nph , for obtaining the physical and nonphysical regimes are 1.0238 × 108 and 7.0016 × 109 , respectively, which can be obtained from (11) for I/Ith = 1.1. It can be shown that the results of the bias point simulation results (Fig. 5) are in excellent agreement with the analytical results from hamadou et al. [12]. It must be known that a gradual increase of 0 , due to the increase of frequency detuning away from resonance, leads to an increment of saturable absorber mechanism, so the optical stability domain decreases gradually and the area of the z-shaped region becomes narrower. In following, we investigate the dynamic behaviors of the proposed model. Fig. 6(a), (b) and (c) shows the time evolution of the electron numbers in each of the three energy levels, the photon numbers and the population inversion between energy level 3 and 2, for different values of 0 and I/Ith = 1.45, respectively. In this case, we set the initial condition Nph = 4.875 × 108 . The simulation results of the proposed
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circuit-level model are in good agreement with numerical results, which obtained by solving the rate-equations using the 4th order Runge-Kutta method, from hamadou et al. [12]. As can be seen (from Fig. 6(a)) in contrast of level 2 and 3, the steady state population in level 1 is independent of 0 . Furthermore, it can be seen that (from Fig. 6(b) and (c)) the time needed to establish the stable stationary regime increases with increasing 0 .Finally, Fig. 7 shows the normalized modulation response of the proposed QCL circuit (in decibel) versus the modulation frequency, for the different values of normalized injection current. As an important note, it can be seen that the variation of modulation bandwidth respect to the high values of the normalized injection current (I/Ith = 2.25, 2.5 and 2.7) is descending. 4. Conclusion In this paper, a new circuit-level model of QCLs, considering the Kerr nonlinearity term, was proposed. It was demonstrated that a three-level rate-equations-based model, taking the Kerr effect account, leads to multiple solution regimes: a negative solution regime, a positive physical solution regime and a positive nonphysical solution regime. It was shown that the problem of convergence to the nonphysical solutions can be resolved using the initial conditions which can be obtained from a DC sweep simulation or an analytical expression. The static and dynamic behaviors of the QCLs, considering the Kerr nonlinearity term, have been investigated. It was shown that the proposed circuit-level model can accurately predict the Kerr effect characteristics of QCLs, and so can useful for studying of the optical bistability related applications in QCLs. References [1] G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A.L. Hutchinson, D.L. Sivco, A.Y. Cho, High-power infrared (8-micrometer wavelength) superlattice lasers, Science 276 (1997) 773–776. [2] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, A.Y. Cho, Quantum cascade laser, Science 264 (1994) 553–556. [3] C. Gmachl, F. Capasso, D.L. Sivco, A.Y. Cho, Recent progress in quantum cascade lasers and Applications, Rep. Prog. Phys. 64 (2001) 1533–1601. [4] E.L. Holthoff, D.A. Heaps, P.M. Pellegrino, Development of a MEMS-scale photoacoustic chemical sensor using a quantum cascade laser, IEEE. Sens. J. 10 (2010) 572–577. [5] B.G. Lee, M.A. Belkin, R. Audet, J. MacArthur, L. Diehl, C. Pflugl, F. Capasso, Widely tunable single-mode quantum cascade laser source for mid-infrared spectroscopy, Appl. Phys. Lett. 91 (2007) 231101–231103. [6] P.C. Zhang, C. Qi, X.Y. Jiang, X.Z. Shi, G. Wang, A new circuit model of quantum cascade lasers, J. Optoelectron. Laser 22 (2011), p. 1313. [7] K.S.C. Yong, M.K. Haldar, J.F. Webb, An equivalent circuit for quantum cascade lasers, J. Infrared. Millim. TE. 34 (2013) 586–597. [8] C. Qi, X. Shi, S. Ye, G. Wang, Equivalent circuit-level model of quantum cascade lasers: influence of doping density on steady state and dynamic responses, IEEE J. Opt. Quantum Electron. 49 (2013). [9] G.C. Chen, G.H. Fan, S.T. Li, Spice simulation of a large-signal model for quantum cascade laser, IEEE J. Opt. Quantum Electron. 40 (2008) 645–653. [10] A. Biswas, P.K. Basu, Equivalent circuit models of quantum cascade lasers for SPICE simulation of steady state and dynamic response, J. Opt. A Pure App. Opt. 9 (2007), p. 26. [11] J.F. Webb, M.K. Haldar, Improved two level model of mid-infrared quantum cascade lasers for analysis of direct intensity modulation response, J. Appl. Phys. 111 (2012) 043110–043115. [12] A. Hamadou, J.L. Thobel, Modelling of optical Kerr effects on the static and dynamic behaviors of quantum cascade laser, Opt. Commun. 284 (2011) 2972–2979. [13] A. Hamadou, Analytical investigation of the dynamics behaviors of quantum cascade laser, Opt. Commun. 335 (2015) 271–278. [14] A. Hamadou, S. Lamari, J.L. Thobel, Dynamic modeling of a midinfrared quantum cascade laser, J. Appl. Phys. 105 (2009) 093116-1–093116-6. [15] A. Hamadou, J.L. Thobel, S. Lamari, Modelling of temperature effects on the characteristics of mid-infrared quantum cascade lasers, Opt. Commun. 281 (2008) 5385–5388.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Circuit-level modeling of quantum cascade lasers: Influence of Kerr effect on static and dynamic responses Mohsen Darman, Kiazand Fasihi ∗ Department of Electrical Engineering, Golestan University, Gorgan, Iran
a r t i c l e
i n f o
Article history: Received 17 June 2016 Accepted 23 August 2016 Keywords: Equivalent circuit model Kerr effect Quantum cascade lasers SPICE Static and dynamic characteristics Three-level rate-equations
a b s t r a c t The three-level rate-equations-based QCL models, considering the Kerr nonlinearity term, possess multiple DC solution regimes for nonnegative values of injection currents. We show that by giving the proper initial conditions, the DC bias point simulation results of the proposed QCL circuit-level model will converge to the physical positive output powers. We also show that the proposed model can accurately predict the influence of the Kerr effect on static and dynamic behaviors of the QCLs. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction The quantum cascade lasers (QCLs) have gained widespread attention due to their small sizes, large intensity modulation bandwidth, narrow linewidth, high operating temperatures and robust fabrication [1–3]. These advantages make them attractive for different applications, such as chemical sensors [4], anesthetic gas detecting, pollution monitoring, free-space optical communication systems, infrared spectroscopy [5], and so on. The operation of QCLs can be modeled and analyzed numerically using the multi-level rate-equations-based theoretical models. The used numerical approaches are accurate, but very computationally intensive and aren’t suitable for system-level designs and optimizations. Instead, it is possible to form a rate-equations-based equivalent circuit model and reduce drastically the complexity of the analysis [6–11]. We have shown that in the equivalent circuit model of QCLs, which are based on three-level rate-equations, considering the Kerr nonlinearity term, multiple solution regimes are obtained. It has been shown that the optical Kerr nonlinearity in QCLs can lead to some significant issues, such as mode locking and photon number dependency of the losses, hence, studies of QCL with optical Kerr nonlinearity are very important [12]. In this investigation, we propose a new equivalent circuit model for QCLs based on three-level rate-equations, which can predict the influence of the Kerr nonlinearity on the QCLs operations. We show that by giving the initial conditions, which can be obtained from a DC sweep simulation or an analytical expression, the proposed model can be used for both steady-state and dynamic responses. The proposed model is verified using the analytical and also numerical results from hamadou et al. [12] for the dynamics of electrons, in different energy levels, and photons in the cavity. The simulation results show an excellent agreement in all the comparisons. The paper is organized as follows: In Section II, the solution regimes of a three-level rate-equations-based model of the QCLs, considering the influence
∗ Corresponding author. E-mail addresses: [email protected] (M. Darman), [email protected] (K. Fasihi). http://dx.doi.org/10.1016/j.ijleo.2016.08.054 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
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Fig. 1. Schematic of a three-level QCL. Table 1 The used parametres for evaluating the solution regimes of the three-level rate-equations of the QCLs. The model parameters are from Hamadou et al. [12]. Symbol
Description
Value
32 31 21 sp out p 3 0 N L R1 R2 neff G ˇ Ith ˛m ˛w
Lifetimes representing the transitions from level 3 to 2 Lifetimes representing the transitions from level 3 to 1 Lifetimes representing the transitions from level 2 to 1 Spontaneous lifetime between level 3 and 2 Electron escape time Photon lifetime Electron lifetime in the upper level Power output coupling efficiency Free space wavelength Number of gain stages Lateral length of the cavity Reflecting power of facets 1 Reflecting power of facets 2 Effective refractive index of the cavity Gain coefficient per stage Fraction of spontaneous emission entering into lasing mode Threshold current Mirrors loss Waveguide loss of the cavity
2.1 × 10−12 (s) 4.2 × 10−12 (s) 3 × 10−13 (s) 3.55 × 10−9 (s) 10−12 (s) 3.36 × 10−12 (s) −12 1.4 × 10 (s) 0.19136 9 × 10−6 (m) 48 10−3 (m) 0.29 0.29 3.27 744 (1/s) 2 × 10−3 1.1108 (A) 1240 (1/m) (1/m) 2000
of Kerr nonlinearity, is presented. In Section III, the derivation and verification of the proposed equivalent circuit model of QCLs, that is based on the three-level rate-equations, with adding Kerr nonlinearity term, is demonstrated. 2. Solution regimes of the three-level rate-equations-based model of the QCLs, considering the influence of Kerr effect The rate equations of a three-level QCL, corresponding to Fig. 1, can be described in terms of the following three first-order differential equations [12–14] dN3 I N3 = − − (N3 − N2 )GNph , e 3 dt
(1)
dN2 = dt
(2)
1 1 + 32 sp
N3 −
N2 + (N3 − N2 )GNph , 21
N3 N2 N1 dN1 = + − , 31 21 out dt dNph dt
= N (N3 − N2 )GNph + Nˇ
(3) Nph N3 − (1 − 0 Nph ), p sp
(4)
where N1 , N2 and N3 are the instantaneous numbers of electrons in energy level 3, i.e. the upper lasing state, energy level 2, i.e. the lower lasing state, and energy levels 1, i.e. the ground state, respectively. Nph denotes the photon numbers in the cavity, N is the number of cascade stages, G is the gain coefficient, and 0 is the dimensionless coefficient, which describes the magnitude of nonlinear effects. ˇ denotes the spontaneous emission coupling coefficient, e is the magnitude of electronic charge and I is the injection current. 32 , 31 and 21 are the lifetimes representing the transitions from level 3 to 1, 3 to 2 and 2 to 1, respectively, and out represents the electron escape time between two adjacent stages. sp denotes the radiative spontaneous relaxation time between levels 3 and 2, and 3 is the electron lifetime in the upper level and is given by 1/3 = 1/32 + 1/31 + 1/sp . Furthermore, p is photon lifetime that can be expressed as p = (c (˛w + ˛m ))−1 , where ˛w and ˛m are the losses of waveguide cavity and mirrors, respectively, and c = c/neff is the average velocity of light in the
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Fig. 2. (a) Graphs of (8) and (9) versus the ϑNph , when I/Ith = 0.3 . (b) The corresponding graphs of (N2 )1 and (N2 )2 versus ϑNph .(c) The enlarged graph of (b) aroundNph = 0..
system, in which neff and c are the effective refractive index of the cavity and the speed of light in vacuum, respectively. The mirrors loss can be given by ˛m = − ln(R1 R2 )/(2L), where R1 and R2 are the reflectivity of the facets 1 and 2, respectively, and L is the lateral length of the cavity. The output power and the threshold current can be calculated analytically from the QCL rate-equations. The output power obeys the Pout /Nph = 0 h/p = ϑ relation, where 0 is the power output coupling efficiency and assuming that R1 = R2 ,is given by [15] 0 =
1 ˛m . 2 ˛m + ˛w
(5)
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Fig. 3. (a) Graphs of (8) and (9) versus the ϑNph , when I/Ith = 1.45 . (b) The corresponding graphs of (N2 )1 and (N2 )2 versus ϑNph .(c) The enlarged graph of (b) aroundNph = 0..
Also, the threshold current can be written as [7] Ith =
e NGp 21
+ + sp 1 − − sp
,
(6)
where = 21 /31 , = 21 /32 and sp = 21 /sp . Now, the solution regimes of the QCLs are evaluated, under the steady-state conditions. From (1), it can be shown that N2 =
N3 3
+ GN3 Nph − GNph
I e
.
(7)
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Fig. 4. Circuit-level implementation of a three-level QCL model, considering the Kerr nonlinearity.
Fig. 5. The graphs of different solution regimes for 0 = 1.25 × 10−10 ..
By substituting (7) in (2) and (4), one can find (N3 )1 =
I e 1 3
1 + GNph 21
+ GNph + GNph 21
,
(8)
31
and
(N3 )2 =
I e
−
(1−0 Nph )Nph Np 1 3
−
ˇ sp
,
(9)
respectively. Using the QCL parameters found in Table 1, the threshold injection current can be calculated as Ith = 1.1108 A. The graphs of (8) and (9) versus the output power, Pout = ϑNph , for the injection current of 0.3 Ith is shown in Fig. 2(a). Furthermore, by substituting (8) and (9) in (7), respectively, (N2 )1 and (N2 )2 can be obtained and graphed in Fig. 2 (b). Fig. 2 (c) shows the enlarged graph of Fig. 2(b) aroundNph = 0. As the same way, Fig. 3 shows the corresponding graphs, when I = 1.45 Ith . From Figs. 2 and 3, one can see that, in both cases, when I < Ith or I > Ith , the QCL rate-equations have three different solution regimes: (a) a negative solution regime, (b) a positive physical solution regime, which is desired, and (c) a positive nonphysical solution regime. In the general case, by considering these solution regimes, it can be deduced that using of a SPICE based circuit simulator, which is based on the Newton-Raphson iterative technique, depending the value of the initial condition, the circuit-level simulation result can be converged to any one of the solution regimes. This problem can be resolved by using the proper initial conditions which can be obtained using a DC sweep simulation or from a DC bias point analytical expression [12]. In fact, it can be shown that in a DC sweep simulation (with a sufficiently small sweeping step), which always starts with zero initial condition, the static simulation results of the three-level rate-equations-based equivalent circuits (considering the Kerr nonlinearity term), are always converged to the physical solution regime. Furthermore, Hamadou et al. [12] showed that when the normalized injection current verifies the well-known optical stability domain condition: 1<
I Ith
<
0 Nph.sat 1 1 + + 2 4 40 Nph.sat
,
(10)
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Fig. 6. (a) Time evolution of the electron numbers in each of the three energy levels, for different values of 0 , when J/Jth = 1.45. (b) The corresponding time evolution plot of the photon numbers. (c) The corresponding time evolution plot of the population inversion between energy level 3 and 2.
the nonzero solutions for the photon number can be expressed as
Nph,± =
−0 Nph,sat − 1 ±
2
0 Nph,sat + 1 20
− 40 Nph,sat I I
th
,
(11)
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Fig. 7. Normalized modulation response (in decibel) versus modulation frequency for different values of normalized injection current, when 0 = 1.25 × 10−10 ..
where, Nph,+ is a nonphysical (unstable) solution, Nph,− is a physical (stable) solution and Nph,sat is the photon saturation number that is given by Nph,sat =
1
.
(12)
3 1 + G
During the circuit simulation, one can see that for a given value of I/Ith , using the initial condition obtained from (11) for Nph,− , always, the DC bias simulation result of the proposed circuit-level model is converged to the physical bias point and as a result, the other static and dynamic simulation results will be converged to desired results. 3. Derivation and verification of the equivalent circuit model for the QCLs, based on the three-level rate-equations with considering the Kerr effect As mention previously, the three-level rate-equations-based model of QCLs, considering the Kerr nonlinearity term, have a multiple DC solution regime, and the convergence to the desired solution requires the initial conditions which can be obtained using a DC sweep simulation or an analytical expression. Fig. 4 shows the circuit-level implementation of the proposed QCL model. This equivalent circuit model is obtained through manipulation of the three-level rate-equations (1)–(4). For facilitating the interpretation of the rate-equations as the electric circuit equations, we take Ni = vni , i = 1, 2 , 3, in which vni can be taken as the voltage of the node ni . After substituting the above transformations, the modified rate-equations are C3
dvn3 vn + 3 + Istim3 = I, R3 dt
(14)
C2
dvn2 vn + 2 = I2 + Istim2 , R2 dt
(15)
C1
dvn1 vn + 1 = I1 + I 1 , R1 dt
(16)
d vm vm + = Ispon + Istim + Ik , Rm dt
(17)
Cm
Where C1 = C2 = Cm = 1, C3 = e, R1 = out , R2 = 21 ,Rm = p ,R3 = 3 /(e), I1 = vn3
Ispon = Nˇ sp , Ik =
2 −1 , I 0 Nph stim p
vn2
21
, I1 =
vn3
31
, I2 = (32 −1 + sp −1 ) vn3 ,
= NG(vn3 − vn2 )Nph , Istim2 = G(vn3 − vn2 )Nph , and Istim3 = eG(vn3 − vn2 )Nph .
We investigate the static and dynamic behaviors of the proposed QCL equivalent circuit model using of the Winspice simulator. In the circuit simulation, we use the parameters provided in Table 1. Fig. 5 shows the plot of the light-current characteristics of the proposed QCL model when 1.1 < I/Ith < 4, for 0 = 1.25 × 10−10 . The used initial condition values,Nph , for obtaining the physical and nonphysical regimes are 1.0238 × 108 and 7.0016 × 109 , respectively, which can be obtained from (11) for I/Ith = 1.1. It can be shown that the results of the bias point simulation results (Fig. 5) are in excellent agreement with the analytical results from hamadou et al. [12]. It must be known that a gradual increase of 0 , due to the increase of frequency detuning away from resonance, leads to an increment of saturable absorber mechanism, so the optical stability domain decreases gradually and the area of the z-shaped region becomes narrower. In following, we investigate the dynamic behaviors of the proposed model. Fig. 6(a), (b) and (c) shows the time evolution of the electron numbers in each of the three energy levels, the photon numbers and the population inversion between energy level 3 and 2, for different values of 0 and I/Ith = 1.45, respectively. In this case, we set the initial condition Nph = 4.875 × 108 . The simulation results of the proposed
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circuit-level model are in good agreement with numerical results, which obtained by solving the rate-equations using the 4th order Runge-Kutta method, from hamadou et al. [12]. As can be seen (from Fig. 6(a)) in contrast of level 2 and 3, the steady state population in level 1 is independent of 0 . Furthermore, it can be seen that (from Fig. 6(b) and (c)) the time needed to establish the stable stationary regime increases with increasing 0 .Finally, Fig. 7 shows the normalized modulation response of the proposed QCL circuit (in decibel) versus the modulation frequency, for the different values of normalized injection current. As an important note, it can be seen that the variation of modulation bandwidth respect to the high values of the normalized injection current (I/Ith = 2.25, 2.5 and 2.7) is descending. 4. Conclusion In this paper, a new circuit-level model of QCLs, considering the Kerr nonlinearity term, was proposed. It was demonstrated that a three-level rate-equations-based model, taking the Kerr effect account, leads to multiple solution regimes: a negative solution regime, a positive physical solution regime and a positive nonphysical solution regime. It was shown that the problem of convergence to the nonphysical solutions can be resolved using the initial conditions which can be obtained from a DC sweep simulation or an analytical expression. The static and dynamic behaviors of the QCLs, considering the Kerr nonlinearity term, have been investigated. It was shown that the proposed circuit-level model can accurately predict the Kerr effect characteristics of QCLs, and so can useful for studying of the optical bistability related applications in QCLs. References [1] G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A.L. Hutchinson, D.L. Sivco, A.Y. Cho, High-power infrared (8-micrometer wavelength) superlattice lasers, Science 276 (1997) 773–776. [2] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, A.Y. Cho, Quantum cascade laser, Science 264 (1994) 553–556. [3] C. Gmachl, F. Capasso, D.L. Sivco, A.Y. Cho, Recent progress in quantum cascade lasers and Applications, Rep. Prog. Phys. 64 (2001) 1533–1601. [4] E.L. Holthoff, D.A. Heaps, P.M. Pellegrino, Development of a MEMS-scale photoacoustic chemical sensor using a quantum cascade laser, IEEE. Sens. J. 10 (2010) 572–577. [5] B.G. Lee, M.A. Belkin, R. Audet, J. MacArthur, L. Diehl, C. Pflugl, F. Capasso, Widely tunable single-mode quantum cascade laser source for mid-infrared spectroscopy, Appl. Phys. Lett. 91 (2007) 231101–231103. [6] P.C. Zhang, C. Qi, X.Y. Jiang, X.Z. Shi, G. Wang, A new circuit model of quantum cascade lasers, J. Optoelectron. Laser 22 (2011), p. 1313. [7] K.S.C. Yong, M.K. Haldar, J.F. Webb, An equivalent circuit for quantum cascade lasers, J. Infrared. Millim. TE. 34 (2013) 586–597. [8] C. Qi, X. Shi, S. Ye, G. Wang, Equivalent circuit-level model of quantum cascade lasers: influence of doping density on steady state and dynamic responses, IEEE J. Opt. Quantum Electron. 49 (2013). [9] G.C. Chen, G.H. Fan, S.T. Li, Spice simulation of a large-signal model for quantum cascade laser, IEEE J. Opt. Quantum Electron. 40 (2008) 645–653. [10] A. Biswas, P.K. Basu, Equivalent circuit models of quantum cascade lasers for SPICE simulation of steady state and dynamic response, J. Opt. A Pure App. Opt. 9 (2007), p. 26. [11] J.F. Webb, M.K. Haldar, Improved two level model of mid-infrared quantum cascade lasers for analysis of direct intensity modulation response, J. Appl. Phys. 111 (2012) 043110–043115. [12] A. Hamadou, J.L. Thobel, Modelling of optical Kerr effects on the static and dynamic behaviors of quantum cascade laser, Opt. Commun. 284 (2011) 2972–2979. [13] A. Hamadou, Analytical investigation of the dynamics behaviors of quantum cascade laser, Opt. Commun. 335 (2015) 271–278. [14] A. Hamadou, S. Lamari, J.L. Thobel, Dynamic modeling of a midinfrared quantum cascade laser, J. Appl. Phys. 105 (2009) 093116-1–093116-6. [15] A. Hamadou, J.L. Thobel, S. Lamari, Modelling of temperature effects on the characteristics of mid-infrared quantum cascade lasers, Opt. Commun. 281 (2008) 5385–5388.