EQUIVALENT CIRCUIT MODEL OF SEMICONDUCTOR LASERS TAKING ACCOUNT OF GAIN SUPPRESSION

Transcription

1 EQUIVALENT CIRCUIT MODEL OF SEMICONDUCTOR LASERS TAKING ACCOUNT OF GAIN SUPPRESSION Kaiz Aedi and Mohsen Khanzadeh Departent of Electrical Engineering, Faculty of Electrical and Coputer Engineering, Shahid Beheshti University, G. C., Evin , Tehran, Iran ABSTRACT In this paper, the rate equation-ased equivalent luped eleent circuit odel of seiconductor lasers is used to study the effect of gain suppression on characteristics of intensity odulation of sall signal of Fary perotseiconductor lasers (FP-SLs). Modeling is firstly perfored with the siple solution of rate equations of seiconductor lasers (SL), which can e used to odel asic laser ehavior under oth direct current (DC) and alternating current (AC) conditions. Then the odel is ipleented in conventional siulation progra with integrated circuit ephasis (SPICE) circuit siulators, such as advanced design syste (ADS), and it is used to siulate the sall signal intensity odulation features of FP-SLs. The results of circuit siulations are copared with those of nuerical siulations. This siple theoretical odel is especially suitale for coputer aided design (CAD), and greatly siplifies the design of optical counication systes. KEYWORDS: Equivalent Circuit Model, Seiconductor Lasers, Gain Suppression. I. INTRODUCTION There has een growing interest in developing direct odulation of seiconductor diode in high speed and long haul optical counication systes over the past decade. Moreover a ajor advantage of seiconductor lasers is that they can e directly odulated. In contrast, any other lasers are continuous wave sources and cannot e odulated directly at all. The other advantages of seiconductor lasers are including their low cost, copact size, low power consuption and high optical output power [1, 2]. On the other hand, when the laser is iased aove threshold its operation and dynaics are influenced y property of gain suppression, which originates fro intra-and relaxation processes of injected carriers [3]. For instance the daping rate of laser relaxation oscillations and odulation andwidth are deterined y gain suppression [2]. Building an accurate laser diode odel ecoes ore iportant in odern high-speed optoelectronic integrated circuit (OEIC) design. Various efforts have een ade to get a well-estalished laser odels such as a rate equation-ased odel and a finite-difference tie-doain (FDTD)-ased odel for OEIC design, however, the difficulty of extracting accurate odel paraeters in the rate equationased odel case and the long siulation tie in the FDTD-ased odel reain ajor ostacles to applying the to actual OEIC design [4]. Accurate extraction of the sall signal equivalent circuit for laser diodes (LDs) is extreely iportant for optiizing the device perforance. The rate-equation odel paraeters can e otained fro odulation response y using nuerical optiization techniques. However, the accuracy of the nuerical optiization ethods - that iniize the difference etween easured and odeled data can vary depending upon the optiization ethod and starting values, while the analytical ethods allow us to extract the equivalent circuit odel paraeters in a straightforward anner [5]. The theoretical work presented here is ased on the siulation odel proposed y Ahed et al. [6]. Then ased on the result of theoretical analysis of rate equations, sall signal equivalent circuit odel of InGaAsP seiconductor lasers is proposed. Because the ain goal of this work is estalishing a 460 Vol. 6, Issue 1, pp

2 Fary perot-seiconductor lasers (FP-SLs) odel that can e used in coputer aided design (CAD) of optoelectronic systes, we try to ipleent this odel in siulation progra with integrated circuit ephasis (SPICE)-like siulators. Then we can siulate the FP-SLs coined with other electrical devices such as laser drivers. The realization of SPICE siulation depends on the transforation fro the odel equations into the equivalent circuit representation. The solution of this equivalent circuit odel is copared with the nuerical siulations done y MATLAB software and the results presented y Ahed et al. [6]. To the author s knowledge, reports on the evaluation of gain suppression in circuit odel of seiconductor lasers have not een yet reported. This paper is structured as follows. Theoretical solutions of rate equations are introduced in Section 2. In Section 3, Rate equation-ased odel is investigated. The results and discussion of this work appear in Section 4. Finally, Section 5 contains the conclusion. II. THEORETICAL SOLUTIONS OF RATE EQUATIONS The equivalent circuit odel is ased on the theoretical solution of rate equations. Derivation of these equations originates fro Maxwell equations with a quantu echanical approach for the induced polarization [4]. However, the rate equations could also e derived y considering physical phenoena. Through this approach, the rate equations with the nuer of injected electrons into the active layer N(t), through the current I(t) and photon nuer S(t) are given y [2]: dn dt 1 t N I AS (1) e e ds C G Gth S N (2) dt r Equation (1) relates the rate of change in the electron nuer to the drive current I(t), the stiulated photon nuer S and carrier recoination rate. Equation (2) associates the rate of change in photon nuer to photon loss and the rate of coupled recoination into the lasing ode. G in the Eq. (2) is the optical gain (s -1 ), and is defined in the nonlinear for as [2]: G A BS (3) With the coefficients of linear gain A and nonlinear gain (gain suppression) B defined as [7]: B a a A N N g (4) V πc n V 0 in 2 a R 2 cv N N where a is the tangential gain, is the confineent factor of the optical field in the active layer with volue of V and refractive index of n a, Ng is the electron nuer at transparency, N s is the electron nuer characterizing B, in is the intra-and relaxation tie, c is the Speed of light in free space, R cv is the dipole oent, is the reduced Planck s constant and 0 is the dielectric constant in free space. λ 0 is the lasing wavelength and for this FP-SL is assued 1.55 µ. The nonlinear gain coefficient (B 0) is approxiately calculated and set to a value of 683 s -1 [6]. Influence of gain suppression on odulation characteristics is exained y varying the coefficient k in Eq. (6) to adjust value of B relative to the fixed value B 0. kb 0 B (6) Exact analytical solution of the full rate equations cannot e otained. Therefore soe approxiations are needed to find analytical solutions. It is possile to assue that the dynaic changes in the electron and photon nuer away fro their steady-state values are sall. Under this assuption, the s (5) 461 Vol. 6, Issue 1, pp

3 sall-signal responses of one variale in ters of a perturation to another can e expressed y taking the differential rate Eqs [4]. I S N t I I cosω t (7) t S S cosω t (8) t N N cosω t (9) where I, N, and S are the ias coponents, and I, N, and S are the agnitudes of the corresponding sall-signal perturations. By sustituting Eqs.(7) (9) into rate Eqs.(1) and (2), separating equations of oth ias and odulation ters, applying several nuerical operations, derivations of coefficients, and neglecting the ters of higher haronics, following pair of equations for the ias coponents are otained [6]: A BS G S N 0 th I e A S 0 e And another pair of linear equations for the odulation coponents as [6]: N C Γ jω S S N 0 S a V where A the ias coponent of linear gain A is [6]: Γ S and III. Γ N r CV a r (10) (11) (12) I A j S ΓN Ω N 0 (13) e A a N N g (14) V are the daping rates of S(t) and N(t), respectively, and are given y [6]: Γ CN S BS (15) rs a RATE EQUATION-BASED MODEL 3.1 Modeling ias coponents 1 Γ N S (16) V e In this paper, equivalent circuit odel is derived fro the solution of rate equation for FP-SL, which has the advantage of siple ipleentation of odel and short siulation tie. FP-InGaAsP Seiconductor Laser eitting at wavelength λ = 1.55 µ are considered in the calculations. Typical values of the paraeters used in odeling and siulations are listed in Tale 1. At the first step, it is necessary to deterine the value of threshold current, which specify the lower liit of injected current. When the SL is iased aove the threshold, the electron nuer N(t) is claped just aove the threshold electron nuer Nth hence the threshold nuer of carriers siply deterined y [2]: 462 Vol. 6, Issue 1, pp

4 GthV N th Ng (17) a And consequently the threshold current is calculated fro the following relation [2]: I th en e th (18) Tale 1 Typical value of the paraeters of a 1.55 µ InGaAsP laser Syol Meaning Circuit Value Paraeter e Electron charge e1 1.6x10 19 e Electron lifetie t_e 2.83x10 9 G th Threshold gain G th 7.81x10 10 a Tangential gain coefficient a Field confineent factor in zeta 0.2 the active layer V Volue of the active region V 60x10 18 N g Electron nuer at Ng 5.31x10 7 transparency B 0 Nonlinear gain B 683 C Spontaneous eission factor C 2.5x10 5 r Radiative recoination lifetie t_r 7.772x10 9 Sustituting G th fro Tale 1 into Eq. (17) and then using the result of N th in the Eq. (18) leads to I th = 3.33 A. By solving the Eqs.(10) and (11), the ias coponent of the photon nuer (S ) can e evaluated as: S I I th (19) egth To otain a circuit odel for ias coponents of SL, one straight approach is to define a circuit with two nodes, one represents electron nuer N and the other is for odeling photon nuer S, so the nodes are laeled with N and S, respectively. The solution of this circuit and finding the voltages of two aove entioned nodes is equal to calculate the values of ias coponents S and N. It s ovious that two KCL equations are required to copletely descrie the circuit. Equation (19) is one of the requisite equations, and y oving toward the KCL equation, it can e rearranged as: S I I R th (20) 1 which R 1 is defined as 1/(e*G th), and due to characteristic of the KCL equation, all coponents of this expression are current. So three ranches are intersected at node S and each one ust through aount of current corresponding to each ter of the Eq. (20), as illustrated in Figure 1. The other necessary equation for siulating N is acquired fro Eq. (11), y noralizing with a coefficient of e and sustituting Eq. (14) it provides: N N GS I 0 (21) R2 R3 where R 3 is defined as 1 / (azv*e*s) and R 2 is equal to τ e/e while azv is defined as a constant factor a /V and G is equal to e*azv*ng. 463 Vol. 6, Issue 1, pp

5 In this research, odels of ias and sall signal equations are ipleented using the Hewlett Packard Advanced Design Syste (HP-ADS) syolically defined devices (SDDs) odel to InGaAsP Fary- Perot lasers. The enefit of the SDD is that once the odel is defined, any circuit siulator in Advanced Design Syste can use the odel, and derivatives are also calculated autoatically in the process of siulation. These kinds of odels need the physical paraeters of the laser diodes. Thus, the paraeters for the equivalent circuit in Figure 1 should e known efore aking a odel. These paraeters are shown in Figure 2. Figure 1. Equivalent circuit odel for ias coponents Figure 2. Paraeters used in the circuit odel Rate equations have various levels of coplexity to express ore accurate laser operations. The presented odel is one of the siplest equivalent circuit odels. In other words, ore coplex equations and any paraeters are required to iprove odel s perforance. The ain prole with the rate equation-ased odel is that circuit designers need to know the physical farication paraeters of the laser, which includes the volue of the active region. Although laser anufacturers provide such data, it is typically liited and insufficient for the circuit design. In addition, the reaining paraeters still need a lot of easureent facilities and a long easureent tie [4]. Finally, rate equation-ased odels could have an advantage in expressing nonlinear characteristics like near threshold operation; which are critical for OEIC design due to the slow speed and signal distortion fro operation in the nonlinear region [4]. 3.2 Circuit odel of sall signal odulation Sall signal odulation coponents are S and N, which can e derived fro Eqs. (12) and (13) and ultiplying oth equations y e: eγ jω ea a CV - e S S V a r e N Γ N jω S 0 I (22) 464 Vol. 6, Issue 1, pp

6 These nodal voltage equations identically descrie the relations of the circuits shown in Fig. 3(a) and. 3(), therefore analysis of the circuits with software such as ADS should lead to analogous results otained fro the nuerical siulation of rate equations. According to the nodal voltage equations, diagonal eleents of nodal adittance atrix in Eq. (22) specify the adittance etween corresponding node and ground, while ecause of non-equal expression in the off-diagonal entries of the atrix, there is no coponent placed etween nodes S and N. These expressions can e odeled as voltage-controlled-current-sources (VCCS). Identical to ias coponents there is two nodes (S and N ) corresponding to sall-signal paraeters. Thus each eleent of the atrix is an adittance attached to nodes. At this oent it is expected to introduce the. The first expression is es which can e otained fro Eqs. (10) and (15). Equation (10) can e rewritten as: CN BS A Gth (23) rs Now it can e sustituted in Eq. (15) then ultiplied y e as well, produce expanded for of eг S in the first entry of the atrix as: eγs e2bs A Gth (24) To derive a luped-eleent sall signal equivalent circuit odel, several theoretical coponents should e defined. As the atrix is naturally adittance, all entries of the atrix are also adittance. So Eq. (24) is reverse value of R4. The expression appeared in the 1st and 4th entries of the atrix can e ejω odeled as the capacitor C1 with the capacitance value of e (1.6x10 19 ). Moreover the ter -ea / V( S CV /( a r )) which is ultiplied y N, can e odeled as voltage-controlled current source SRC14. The equivalent circuit odel of sall-signal seiconductor laser is illustrated in Figure 3 [8-11]. (a) 465 Vol. 6, Issue 1, pp

7 () Figure 3. Actual scheatic ipleentation for sall signal paraeters; (a) S and () N According to the second row of the atrix, there are two paraeters left to e identified. One is A which can e otained fro Eq. (14) and is odeled as the voltage-controlled current source SRC15; the other paraeter is eγ N which is siply odeled as resistor R 5 using the Eq. (16). The ter I in the right side of Eq. (22) is the agnitude of the corresponding sall-signal perturation and is odeled as an AC current source SRC16 with aplitude 1 and phase 0. As entioned efore capacitor C2 is represented for the ter ejω. At a given ias current I, the odulation response H (Ω ) at a specified odulation frequency Ω is defined as the ratio of the odulated photon nuer S (Ω ) to the corresponding un-odulated value S (0) [6, 9-11]. S H Ω Ω (25) S 0 To achieve the transfer function of Eq. (25), the circuit has een siulated with two conditions. These conditions are syolized y two AC siulation coponents in the ADS software which is shown in Figure 3(a). The first one (AC1) siulates the nuerator of the transfer function with frequency sweep fro 100 MHz to 10GHz, and the other (AC2) corresponds to the denoinator of transfer function which sets in the frequency near zero. After siulating the circuit to plot the function, the expression in Eq. (26) is used. (db(ac1.ac.s)-db(ac2.ac.s))/2 (26) At which the 1/2 is used for transforing default db function (20 log x) to (10 log x). The result of this siulation is shown in Figure 4 which is confored to the result of nuerical siulation done with MATLAB software illustrated in Figure Vol. 6, Issue 1, pp

8 Figure 4. Modulation response otained fro the siulation of the circuit odel with ADS software Figure 5. Result of nuerical siulation with MATLAB software These figures plot frequency spectru of the odulation response H (f ) when I = 3 * I th. The figures show that H (f ) exhiits a pronounced peak at the frequency f (peak) = 4.9 GHz. In this case, the odulation andwidth is f 3dB = 8.69 GHz. for ore details on this ehavior refer to section 3.1 of the reference [6]. IV. RESULTS AND DISCUSSION Variation of the response H (f ) with the ias current I is shown in Figure 6. I changes fro I = 1.1 * I th to I = 27.1 * I th which corresponds to f 3dB(ax). The spectra exhiit the coon feature that the low-frequency coponents are flat with H (f ) = 1. Figure 6 shows that the peak value H (peak) decreases with the increase of I, and the spectru ecoes flat when I = 27.1I th. 467 Vol. 6, Issue 1, pp

9 Figure 6. Variation of H (f ) with ias current I acquired fro siulation with MATLAB Sweeping the coefficient k in relation I = k * I th is accoplished with the tuning capaility of ADS software and the result is shown in Fig. 7. Figure 7. 4 responses of H (f ) acquired fro circuit siulation with ADS Figure 8 is generated y defining Eqn. Max1 for etter vision on relation etween increase of I = k * I th and reduction of the peak value of the odulation response. Figure 8. Variation of H (f ) with ias current I The gain suppression ter BS increases with the photon nuer S, and consequently with I. Here, we illustrate the influence of gain suppression on the odulation characteristics y varying the nonlinear gain coefficient B relative to its value B 0. Figure 9 plots the odulation response H (f ) as 468 Vol. 6, Issue 1, pp

10 a function of B when I = 3I th. It shows that when I = 3I th, increase of gain suppression B fro 0 to 3B 0 gradually decrease the spectra around f (peak). V. CONCLUSIONS Figure 9. Decrease of H (f ) with increase of gain suppression B In this paper, the SL s equivalent circuit odel was developed ased on siple rate-equations which utilize an AC current source to account for the sall signal perturation. The coparison in Section 4 showed that our odel was useful for descriing the SL s sall signal characteristics taking account of gain suppression. Just as entioned efore, such a odel is very helpful in the siulation and design of optoelectronic systes with SLs. By ipleenting the SPICE-like equivalent circuit odels it can e conveniently coined with the electrical coponents in large-scale EDA designs. REFERENCES [1] R. Raasway, K. Sivarajan, G. Sasaki, (2008) Optical Networks: A Practical Perspective, Morgan Kaufann, Third Edition. [2] S. W. Z. Mahoud, (2007) Influence of gain suppression on static and dynaic characteristics of laser diodes under digital odulation, Egypt. J. Sol., Vol. 30, No. 2, pp [3] M. Ahed, M. Yaada, (2012) Modeling and siulation of dispersion-liited fier counication systes eploying directly odulated laser diodes, Indian J. Phys., Vol. 86, No. 11, pp [4]. J. H. Ki, (2005) Wide-Band and Scalale Equivalent Circuit Model for Multiple Quantu Well Laser Diodes:, Ph.D. dissertation, Georgia Institute of Technology. [5]. J. Gao, X. Li, J. Flucke, G. Boeck, Direct paraeter-extraction ethod for laser diode rate-equation odel, (2004) J. Lightwave Technol., Vol. 22, No. 6, pp [6]. M. Ahed, A. Ellafi, (2008) Analysis of sall-signal intensity odulation of seiconductor lasers taking account of gain suppression, Praana - Journal of Physics, Vol. 71, No. 1, pp [7] M. F Ahed, S. W. Z. Mahoud, M. Yaada, (2003). Influence of the spectral gain suppression on the intensities of longitudinal odes in 1.55 µ InGaAsP lasers, Egypt. J. Sol., Vol. 26, No. 2, pp [8] K. Aedi, and M. B. Nasrollahnejad, Analysis and Circuit Model of Optical Injection-Locked Seiconductor Lasers, IREMOS, Vol. 4, No. 4, pp , [9] M. Ahed, and M. Yaada, Mode oscillation and haronic distortions associated with sinusoidal odulation of seiconductor lasers, European Physical Journal D, Vol. 66, No. 9, pp ,246-9, [10] S. Oderatt, B. Eitzigann, and B. Schithuse, Haronic alance analysis for seiconductor lasers under large-signal odulation, Opt. Quant. Electron., Vol. 38, pp , [11] B. Schithüsen, S. Oderatt, and B. Witzigann, Large-signal siulation of seiconductor lasers on device level: nuerical aspects of the haronic alance ethod, Opt. Quant. Electron., Vol. 40, No. 5-6, pp , Vol. 6, Issue 1, pp

11 AUTHOR Kaiz Aedi was orn in Ahar, Iran, in He received his B.S. degree fro University of Tehran, Iran, in 1992, his M.S. degree fro Iran University of Science and Technology, Tehran, Iran in 1995, and his Ph.D. degree fro Tariat Modares University, Tehran, Iran, in 2008, all in electrical engineering. His research interests include design, circuit odeling and nuerical siulation of optoelectronic devices, seiconductor lasers, optical odulators, optical aplifiers and detectors. Dr. Aedi is currently an Assistant Professor at Shahid Beheshti University, Tehran, Iran. Mohsen Khanzadeh was orn in Bafgh, Iran, on August 12, He received the B.S. degree in electronics engineering fro Guilan University, Rasht, Iran, in 2003, and is currently working toward the M.S. degree in electrical engineering at the University of Shahid Beheshti, Tehran, Iran. He worked on equivalent circuit odel of LASER devices design and analysis. His research interests are optical fier counication and odeling of the SLs. 470 Vol. 6, Issue 1, pp