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CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114207 Large Signal Circuit Model of Two-Section Gain Lever Quantum Dot Laser Ashkan Horri1** , Seyedeh Zahra Mirmoeini2 , Rahim Faez3 2 1 Young Researchers Club, Arak Branch, Islamic Azad University, Arak, Iran Department of Electrical Engineering, Arak Branch, Islamic Azad University, Arak, Iran 3 Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran (Received 20 July 2012) An equivalent circuit model for the design and analysis of two-section gain lever quantum dot (QD) laser is presented. This model is based on the three level rate equations with two independent carrier populations and a single longitudinal optical mode. By using the presented model, the effect of gain lever on QD laser performances is investigated. The results of simulation show that the main characteristics of laser such as threshold current, transient response, output power and modulation response are affected by differential gain ratios between the two-sections. PACS: 42.55.Px, 78.67.Hc DOI: 10.1088/0256-307X/29/11/114207 Quantum dot lasers have attracted much attention in recent years because they exhibit excellent properties such as low threshold current, high modulation bandwidth, and low frequency chirp.[1,2] In recent studies, the gain-lever effect[3,4] is a method used to enhance the efficiency of amplitude modulation (AM) and optical frequency modulation (FM) at microwave frequencies by taking advantage of the sub-linear nature of the gain versus carrier density. In this Letter, we describe a circuit level implementation of a two-section gain lever QD laser. This circuit model is based on the three level rate equations. The main advantages of the circuit modeling approach include that the circuit model gives an intuitive idea of the physics of the device, and helps us to understand the static and dynamic behavior of laser. Indeed, circuit modeling is known as a useful approach for design and analysis in optical systems.[5−10] The agreement of simulation results with experimental results verifies the accuracy of our model. The gain-lever effect can be realized in a twosection device as shown in Fig. 1.[3] Using asymmetric current injection, the short section, which is referred to as the modulation section, is dc-biased at a lower gain level than the long section, termed the gain section. This scheme provides a high differential gain under small signal rf modulation. The gain section is only dc-biased and supplies most of the amplification but at a relatively smaller differential gain. The fractional length of the long section is denoted by β. Due to the gain clamping at threshold and the dependence of gain with carrier density, small changes in carrier density in the short section produce a drastically larger variation in carrier density in the long section to maintain the threshold gain condition. The outcome is that the modulation efficiency and 3-dB bandwidth depend on the differential gain ratios between the two sections.[3] Next, we present a circuit model for the two-section QD laser in order to analyze the characteristics of the device in HSPICE. The rate equations for semiconductor lasers are used to investigate the static and dynamic characteristics of lasers. The rate equations for the two-section QD laser are[3−4] πΌ1 π1 ππ 1 = − − π£π 1 π, (1) ππ‘ π π1 πΌ2 π2 ππ 2 = − − π£π 2 π, (2) ππ‘ π π2 π ππ = Γ π£π 1 (1 − β)π + Γ π£π 2 βπ − ππ‘ ππ (οΈ π π2 )οΈ 1 +π½ + , (3) π1 π2 where for section π, ππ is the carrier number, πΌπ is the injection current, ππ is the carrier lifetime, ππ is the material optical gain, Γπ is the optical confinement factor, π is the photon numbers, π£ is the group velocity, ππ is the photon lifetime, π½ is the spontaneous emission factor, β is the fractional length of gain section, and π is the electron charge. We define an approximate formula for the optical gain as[5,6] ππ = π΄π (ππ − πtri ), (4) where π΄π is the differential optical gain and πtri is the carrier number at transparency for section π. The inverse photon lifetime is defined as ππ−1 = π£Γ [(1 − β)π 1 + βπ 2 ]. (5) We define the carrier population in the short and long cavity sections using[5,6] (οΈ ππ )οΈ 1 , (6) π1 = πe1 exp π1 ππ (οΈ ππ )οΈ 2 π2 = πe2 exp . (7) π2 ππ In this case, πe1 and πe2 are the equilibrium carrier numbers in the short and long sections, respectively, while π1 and π2 are the corresponding diode ideality factor, typically set equal to 2.[5,6] π1 and π2 are the voltages across the laser. In order to eliminate the incorrect solution regimes, we transform π into a new ** Corresponding author. Email: [email protected] © 2012 Chinese Physical Society and IOP Publishing Ltd 114207-1 CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114207 ππ e1 πΌC11 = πΌC12 = , 2π 1 (οΈ ππ )οΈ ππ e1 (οΈ 1 exp −1 πΌD12 = 2π 1 π1 ππ (οΈ ππ )οΈ ππ )οΈ 2ππ 1 1 1 + , exp π1 ππ π1 ππ ππ‘ )οΈ (οΈ ππ )οΈ ππ e2 (οΈ 2 πΌD21 = −1 , exp 2π 2 π2 ππ ππ e2 πΌC21 = πΌC22 = , 2π 2 (οΈ ππ )οΈ ππ e2 (οΈ 2 exp −1 πΌD22 = 2π 2 π2 ππ )οΈ (οΈ 2ππ 2 ππ 2 ππ 2 )οΈ + , exp π2 ππ π2 ππ ππ‘ 2 variable ππ via π = (π π + πΏ) , where πΏ is an arbitrary constant. We assume that the value of πΏ is about 10−60 . This transformation enables the simulation to converge to a correct numerical solution.[5,6] i1 I2 I1 ο±Φh Gain h A2 g1 g2 A1 N1 N2 Carrier Density (19) (20) (21) 2 (22) 2 π΅s2 = ππΌ2 (Θ 2 πΌT2 )(ππ + πΏ) , (23) ππ π ππ πΆπβ + = π΅π1 + π΅π2 + π΅π 21 + π΅π 22 , ππ‘ π πβ (24) ππ π½1 Θ1 πΌπ 1 , (25) π΅π1 = π1 (π π + πΏ) ππ π½2 Θ2 πΌπ 2 π΅π2 = , (26) π2 (π π + πΏ) πΏ π΅π 21 = Γ (1 − β)π π πΌ1 (Θ 1 πΌT1 )(ππ + πΏ) − , 2 (27) Substituting Eqs. (6) and (7) into Eqs. (1) and (2), and applying appropriate manipulations, we obtain (οΈ ππ )οΈ (οΈ ππ )οΈ ππ )οΈ ππ e1 (οΈ 2ππ 1 1 1 1 exp −1+ exp 2π 1 π1 ππ π1 ππ π1 ππ ππ‘ (οΈ ππ )οΈ )οΈ ππ ππ e1 (οΈ 1 e1 + exp −1 + 2π 1 π1 ππ π1 2 + ππΌ1 (π )(π π + πΏ) , (8) (οΈ ππ )οΈ ππ )οΈ (οΈ ππ )οΈ ππ e2 (οΈ 2ππ 2 2 2 2 exp πΌ2 = exp −1+ 2π 2 π2 ππ π2 ππ π2 ππ ππ‘ (οΈ ππ )οΈ )οΈ ππ e2 (οΈ 2 + exp −1 2π 2 π2 ππ ππ π2 2 + + ππΌ2 (π )(π π + πΏ) , (9) π2 πΏ π΅π 22 = Γ βπ π πΌ2 (Θ 2 πΌT2 )(ππ + πΏ) − , 2 πΆπβ = 2π π , π πβ = 1. + I1 IC11 D11 By using the new variable ππ in Eq. (3), one can write V1 ππ π + ππ ππ‘ = Γ (1 − β)π π πΌ1 (π1 )(ππ + πΏ) D12 ID11 IT1 2π π + Γ βπ π πΌ2 (π2 )(ππ + πΏ) π½ 1 π1 π½2 π2 + ππ + ππ − πΏ, π1 (ππ + πΏ) π2 (π π + πΏ) (18) π΅s1 = ππΌ1 (Θ 1 πΌT1 )(ππ + πΏ) , Fig. 1. Schematic view of the two-section gain-lever quantum dot laser. πΌ1 = (17) (28) (29) IC12 ID12 Bs1 VT1 - + I2 D21 V2 (10) D22 ID22 ID21 IT2 IC22 Bs2 VT2 - where Vm πΌ1 (π1 ) = π£π΄1 (π1 − ππ‘π1 ), πΌ2 (π2 ) = π£π΄2 (π2 − ππ‘π2 ). (11) After setting Θ1 = 2ππ 1 , Θ2 = 2ππ 2 and using the fact that π1 = Θ1 πΌT1 and π2 = Θ2 πΌT2 , we can define πΌ1 = πΌT1 + πΌD12 + πΌC12 + π΅s1 πΌT1 = πΌD11 + πΌC11 , πΌ2 = πΌT2 + πΌD22 + πΌC22 + π΅s2 , πΌT2 = πΌD21 + πΌC21 , (οΈ ππ )οΈ )οΈ ππ e1 (οΈ 1 πΌD11 = exp −1 , 2π 1 π1 ππ (12) (13) (14) (15) (16) Sout Br1 Rph Cph Br2 Bs21 Bs22 + - Sout Fig. 2. Large signal circuit model of the two-section gain lever quantum dot laser. The previous equations can be mapped directly into a circuit as shown in Fig. 2. Diodes π·11 and π·12 , and current sources πΌC11 and πΌC12 model the 114207-2 CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114207 Output photon numbers (106) 1.5 A1/A2=1 A1/A2=2 A1/A2=3 A1/A2=4 1 0.5 0 0 0.01 0.02 0.03 0.04 4 A1/A2=1 A1/A2=2 3 A1/A2=3 2 1 0 0 1 2 3 4 5 Time (10-10 s) Fig. 5. Transient response of the two-section QD laser for π΄1 /π΄2 = 1, 2, 3. Based on the three-level rate equations, we have proposed an equivalent circuit model of a two-section QD laser for analysis of the steady state and dynamic responses. By using the presented model, the effects of differential gain ratio on laser performance, such as threshold current, transient response, output power and modulation response are investigated. Simulation results show that the differential gain ratio has considerable effects on these characteristics. The results indicate that the presented circuit model can be a very useful tool for simulation of a two-section QD laser in an optoelectronic device simulator. 2.5 2 creases for higher differential gain ratio π΄1 /π΄2 . Figure 4 shows the frequency response of the QD laser for different values of differential gain ratio π΄1 /π΄2 . It can be found that, by increasing the differential gain ratio, the modulation bandwidth is extended. Indeed, it has been observed theoretically that the modulation efficiency enhancement is proportional to the ratio of differential gains of the two sections.[8] The effect of differential gain ratio on the transient response of QD laser is shown in Fig. 5. It is observed that the increment in differential gain ratio leads to decrease of the switch on delay time. Results obtained by using the present circuit model are in good agreement with the experimental results reported by other researchers. Output photon numbers (106) charge storage and carrier capture for the short section. Diodes π·21 and π·22 , and current sources πΌC21 and πΌC22 model the charge storage and carrier capture for the long section. π΅s1 and π΅s2 model the effect of stimulated emission on the short and long sections, respectively. π ph and πΆph model the time variation of the photon number under the effect of spontaneous and stimulated emission, which are denoted by π΅π1 , π΅π2 , π΅s21 , and π΅s22 . πout produces the output photon number of the laser in the form of a voltage. The device investigated was grown by MBE on an n+ GaAs substrate. The active region consists of 10-stacked layers of InAs QDs covered by 5nm In0.15 Ga0.75 As QWs, separated by 33-nm GaAs spacers, of which 10-nm is carbon p-type doped. The cladding layers are step-doped 1.5-µm-thick Al0.35 Ga0.65 As. The laser structure is capped with a 400-nm-thick GaAs. Two-section lasers with 1.5mm cleaved cavity lengths πΏcav , and 3-µm-wide ridge waveguides were fabricated. The electrical isolation between the 0.5 and 1-mm segments was achieved using proton implantation.[3,4] The fraction length of gain section β is 0.8. The carrier recombination lifetimes π1 and π2 are 5 ns. The spontaneous emission factor π½ is 10−5 . The circuit model of Fig. 2 is used to analyze the characteristics of the two-section QD laser in HSPICE. 0.05 I1 (A) Modulaion response (dB) Fig. 3. Output photon number of the two-section QD laser versus differential gain ratios π΄1 /π΄2 = 1, 2, 3, 4. References 15 A1/A2=1 10 5 A1/A2=2 A1/A2=3 A1/A2=4 0 −5 −10 107 108 109 1010 Frequency (Hz) Fig. 4. Modulation response of the two-section QD laser for different differential gain ratios π΄1 /π΄2 = 1, 2, 3, 4. Figure 3 shows the light-current characteristics when the differential gain ratio changes from 1 to 4. 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