Quantum Electronics, IEEE Journal of

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 8, AUGUST 1997

Self-Consistent Modeling of Beam Instabilities in 980-nm Fiber Pump Lasers Gen-Lin Tan, R. S. Mand, and Jimmy M. Xu, Senior Member, IEEE Abstract— Emission nonlinearities such as kinks in the L–I characteristic and beam steering have often been observed in semiconductor power lasers that were designed for single-mode operation. A physical model of these phenomena is presented in which they are attributed to effects of the lasing and coherent coupling of multiple lateral modes. This model has been implemented self-consistently in a finite-element scheme. Simulation results for a typical 980-nm fiber pump laser are described and are found to be in good qualitative agreement with experimental observations. This agreement includes bilateral steering of the peak field, nonsymmetric emission field intensity in a symmetric device, nonlinearity and kinks in the L–I characteristics, and beating patterns in the back plane image. Index Terms— Finite-element method, laser beam stability, laser beam steering, power laser, simulation.

I. INTRODUCTION

I

N THE operation of a typical ridge waveguide fiber pump laser, one often observes the so-called beam steering phenomena [1]–[3], in which the peak of the lateral far-field pattern shifts from side to side, and kinks appear in the – characteristics under high injection. This beam steering effect compromises the differential efficiency with which the laser emission can be coupled to an optical fiber in optical pumping applications. Even when the total power emitted by the laser is free of nonlinearities with respect to injected current, “kinks” in the coupled power characteristic can arise from the steering [4]. What might be the causes for these undesirable phenomena? How and why do they depend on the injection level? Can they be prevented or reduced, and how? These are the questions that motivated this paper. In a typical quantum-well (QW) power laser, the active ˚ usually supporting only layer is very thin (e.g., 50–200 A), a single mode in the vertical dimension, but the ridge width is intentionally kept wide (e.g., 2–4 m) in order to reduce heat and peak optical intensity at the facet to avoid COD. Therefore, the effective index step is carefully chosen to avoid multiple modes in the lateral direction [2]. While well-designed power lasers may support only a single lateral mode in the uninjected “cold” state, the effective waveguide may undergo significant changes under high injection, allowing higher order lateral modes to emerge. These changes can be brought about by electrooptical effects such as the Manuscript received October 7, 1996; revised April 18, 1997. This work was supported in part by the Ontario Laser and Lightwave Research Center, Bell Northern Research, Ltd., and Furukawa Electric, Ltd. G.-L. Tan and J. M. Xu are with the Department of Electrical Engineering, University of Toronto, Toronto, Ont., Canada M5S 1A4. R. S. Mand is with Furukawa Electric Technology, San Jose, CA USA. Publisher Item Identifier S 0018-9197(97)05443-2.

bandfilling effect and interactions between the high-intensity fundamental lateral mode and the carrier distribution in the active region. The lateral spatial hole burning (SHB) of the local gain profile by the fundamental mode is particularly significant in that the resulting local gain distribution, with twin peaks off axis, favors the pumping of higher order lateral modes. Observations have frequently shown that higher order lateral modes do indeed emerge, even in lasers scrupulously designed to operate only in a single lateral mode [5]–[7]. Beam instability and the emission nonlinearities have in the past been simulated [8]–[10] by one-dimensional (1-D) models which considered the effects of multiple modes. But in these studies, arbitrary structural asymmetries along the lateral direction had been introduced to produce the steering. The steering angles, which resulted from distortions to the fundamental lateral mode in these models, increased monotonically with current and the direction of steering was fixed by the assumed built-in asymmetry. The predictions of these earlier numerical models are, however, inconsistent with the bilateral swings about the axis with increasing current which have been observed in most fiber pump lasers that are designed to support a single lateral mode. In this paper, a two-mode laser model is presented that incorporates the dynamical evolution of the waveguide, the lasing of two lateral modes, and the coherent interaction between them. The new model has been implemented selfconsistently in a two-dimensional (2-D) finite element scheme in a code named finite-element light emitter simulator-2 modes (FELES-2). The results show that the near-field (and far-field) steers and becomes nonsymmetric after the first-order mode lases, even for an ideal symmetric laser structure. Simulations have also produced bilateral steering of the field peak and nonlinearities in the – characteristics. In addition, the new model predicts the occurrence of beating between two modes along the longitudinal direction, which is consistent with the observed back-plane luminescence image [6]. This paper is organized as follows. In Sections II and III, we describe the self-consistent two-mode model and its implementation in a 2-D finite-element scheme. In Section IV, we report the results obtained from extensive simulations of a representative structure and compare them to experimental observations. II. PHYSICAL MODEL OF TWO-LATERAL-MODE LASING In the two-mode model, we have made the following assumptions and approximations. 1) Although more than two modes could emerge in the evolving effective waveguide and contribute to beam

0018–9197/97$10.00  1997 IEEE

TAN et al.: SELF-CONSISTENT MODELING OF BEAM INSTABILITIES IN 980-NM FIBER PUMP LASERS

instability, the inclusion of only the two lowest lateral modes is sufficient to give a qualitatively correct picture of the underlying physics. It is also believed that the higher modes have much smaller effects. 2) Once the two lateral modes begin to lase, they either lock onto the same frequency due to a nonlinear interaction [11] or are spaced so closely in frequency that they are nearly the same. Albeit that other mechanisms of coupling may exist, this assumption is supported by the observed stable beating pattern in the back-plane luminescence image described in our earlier experimental report [6]. This point is also supported and elaborated in a more recent publication [12], where it was observed that the output light intensity of the laser shows almost periodic oscillations with changes in cavity length. These show that coherent coupling or interference between two modes exists. If the two modes considered were not in phase-locked operation, the interference would have been washed out over observation times longer than the inverse of the frequency difference. The actual causes of the phase lock remains an open issue and is to be investigated in the future. 3) The mirror reflectivities of the two modes are different and are structure dependent [13]. The calculation or measurement of the actual mirror reflectivity of the first-order lateral mode is nontrivial. Since the accurate determination of its value is not a main objective of the present paper, an estimated constant is used. The conclusions will not be affected, however, by the accuracy of this estimate. 4) As a key point where the new model differs from previous models, we would like to point out that the stimulated emission rate is proportional to the total optical field intensity, rather than the sum of intensities of the two modes. This can in fact be derived from Fermi’s golden rule. In an earlier publication [10], a 1-D twomode laser model was employed. However, it ignored intermodal interference and consequently produced symmetric carrier densities and optical intensity distributions without steering for the symmetric laser structure, which is not consistent with most experimental results. The numerical implementation of our two-mode model is based on our previous single-mode simulator, FELES [14], [15], but is more challenging in terms of numerical computation because it must treat instabilities of the laser operation. The two-mode code, FELES-2, self-consistently solves the 2-D Poisson’s equation, electron and hole continuity equations, the thermal conduction equation, the wave equation (with two modes), and two nonlinear rate equations of the fundamental and first-order-mode photon densities. Specifically, the following governing equations are used in the model (see [14], [15] and many references in them): (1) (2) (3)

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(4) (5)

(6) The first three equations are standard electronic device equations. The fourth is the self-heating and thermal interaction equation. The fifth is the wave equation with a local index dependent on carrier densities and gain. The last equation is in fact two rate equations for two lateral modes coupled to each other via the carrier density dependence of gain and loss. The following subsections offer more details. A. Device Equations In the device equations (1)–(3), , , , and follow the conventional definitions. In the thermal conduction equation, is the thermal conductivity, , is the material density, is the specific heat, is the electrical field, is the total current density, and is the nonradiative recombination rate. The electric model [(1)–(3)] includes Fermi–Dirac statistics, position-dependent band structure, and incomplete ionization. Moreover, it takes into account doping, temperature, composition, electrical-field-dependent mobilities, and recombinations from bulk, surface, Auger, spontaneous emission, and stimulated emission mechanisms. To avoid the otherwise exceedingly long computing time, we deactivate the thermal conduction equation in the simulations presented in this paper in order to concentrate on the interactions of the two modes which are already computationally difficult and demanding. The recombination model used is as follows: (7) which includes five important recombination mechanisms: spontaneous emission recombination ( ) and stimulated emission recombination ( ), which couple the electrical and optical equations, the conventional bulk recombination ( ), Auger recombination ( ), and impact ionization generation ( ). In the presence of interference between two modes, it can be shown that the recombination rate due to stimulated emission is given by (see the Appendix)

(8) is speed of light in vacuum, is the local where peak gain, is the effective index, is the normalized optical field, and is the photon density for the th optical mode, with corresponding to the fundamental and the first-order modes, respectively.

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where

Expanding the above expression, we obtain

is the optical mode density given by (12)

of which the first and second terms are the contributions from the fundamental and first-order modes, respectively, which are identical to [15, eq. (14)]. The third term is the contribution from the interference. It is this term that can lead to asymmetrical SHB, bilateral beam steering, and the beating pattern in back-plane image, as will be shown later. The spontaneous emission recombination rate is modeled as

From the peak gain as a function of carrier density calculated using (10), a first- or second-order polynomial [ ] for a bulk active layer or a logarithmic formula [ ] for a QW active layer is extracted. From the spontaneous emission rate as a function of carrier density calculated using (11), a third-order polynomial formula [ ] is extracted. These fitting formulas are used to calculate local emission in FELES-2 in order to save CPU time. B. Wave Equation In the wave equation (5), is the optical field, is the modal propagation constant, and is the complex refractive index given by [16]–[18]

(9) where and , the electron density in the valley in general and at equilibrium, respectively, are determined by (13)

is the rate constant, which depends on carrier density and temperature. In this model, the local gain and the spontaneous emission are calculated by considering -selection rules, reduced density of states including discrete and continuous energy spectra, and broadening effects due to intraband relaxation (for details, see [14]). This yields the local (material) gain

(10a) and the local maximum (peak) gain MAX

for all

(10b)

where the index refers to either hh (heavy hole) or lh (light hole). , where is the photon energy, and is the effective refractive index of the MQW structure. Although we solve the wave equation for a fixed wavelength, we use the local maximum gain instead of the local gain at the fixed wavelength in order to be consistent with the physical behavior of real laser devices. This treatment is generally adopted in other 2-D laser simulators (e.g., [16]–[18]). The spontaneous emission recombination rate is

(11)

where is the bulk refractive index, and is the linewidth enhancement factor. The first term on the righthand side of (13) is the bulk refractive index contribution, the second term is the effect of the index reduction due to the carrier density, and the third and fourth terms are the contributions from local optical gain and free carrier absorption, respectively. As bias increases, both the real and imaginary parts of the complex refractive index change and the increasing fundamental mode intensity eventually causes SHB. For the two-mode model, we must solve the wave equation twice in order to obtain the propagation constants and , and for the zerothand the corresponding eigen vectors and first-order modes, respectively, if we use the Rayleigh quotient-inverse iteration algorithm [15]. One can prove that the two eigenvectors of the wave equation are not orthogonal in the device if the complex refractive index has a nonzero imaginary part. In laser devices, the refractive index will have a nonzero imaginary part above threshold, so we cannot ignore intermodal interference effects. C. The Rate Equations In the rate equations (6), is the vacuum light velocity, the photon density, the effective modal index, the the photon lifetime, the spontaneous modal gain, emission rate, and is the spontaneous emission factor for the th-order mode. Here 0 or 1, since we have two photon rate equations. Because of the lack of data for the spontaneous emission factor, and because we are primarily interested in the lasing characteristics above threshold, we neglect the contribution of


spontaneous emission to lasing power in (6) by assigning a zero value to this factor. In steady state, the photon rate equation, after neglecting the spontaneous emission, is

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We start from the surface integral of the complex Poynting vector and obtain

(14) is the th modal gain, and is the th modal where loss. The modal gain is calculated by averaging over the local gain weighted by the optical intensity

(15) (20) The modal gain can also be obtained from the imaginary part of the eigen value of the wave equation, but the present averaging method is consistent with the following calculation of the loss. The total loss includes the bulk loss , the free carrier absorption loss and the mirror loss :

where is the field outside the cavity at the mirror facet, is the field inside the cavity, and is the front mirror facet loss for the th mode, given by

(16) where and are the averages of the local bulk and free carrier absorption losses ( ) in the device and are calculated from

(21)

From the following relation between the field energy flow density and the photon density

(17)

we have (18) (22) The mirror loss,

is (19)

is the cavity length, and and are the where front and rear facet power reflectivities for the th mode, respectively. It should be noted that the facet reflectivities of the zerothorder and the first-order modes are not generally equal [13]. The exact values determine quantitatively the threshold of each mode but will not change the qualitative findings that will be presented here. While an accurate determination of the facet modal reflectivities is nontrivial and beyond the scope of the present work, it is recognized that including an experimentally or theoretically determined facet reflectivity for the first-order mode would be highly desirable for future work.

is the photon density and is the normalized where optical field. If we expand (20), we obtain

(23) The first two terms are the same as [15, eq. (28)], which represent the output powers of the zeroth- and first-order mode exclusive of interference. is the contribution due to the interference. III. NUMERICAL ALGORITHM

D. Output Power Due to the nonorthogonality of the two modes above threshold, we need to modify the conventional power formula (see [15, eq. (28)] to account for the interference between the two modes.

Having the physical model in place, one has to deal with the next and perhaps more difficult task of implementing it in a numerical scheme. In the following, we will briefly describe the main algorithms involved in FELES-2 [15], [19]–[21], then discuss the two coupled rate equations for the fundamental and

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first-order-mode photon densities, which are the critical aspect of the new model. With conventional computation methods, the simulation of a laser device would require an excessively fine mesh, an exceedingly large memory, and very long CPU times and would still often result in nonconvergence. To overcome these difficulties, more efficient computational methods are needed. In FELES-2, we used three efficient numerical techniques: 1) A modified finite element discretization method called FEM-QSG [i.e., the S-G scheme [19] embedded in the quadrature of a finite element method (FEM)]. It has a higher accuracy and a better rate of convergence than the conventional FEM. Thus, for a given accuracy, the new discretization method can use a coarser mesh than that required by the conventional FEM or the box integral and SG formula (BIM-SG) method [22] in a finite difference scheme. 2) A self-adaptive mesh refinement technique [20], that automatically creates denser grid points at the interfaces and in the transition regions where carrier concentrations and/or field change drastically, thus improving convergence considerably [20]. 3) An efficient preconditioning iteration method named ILUV-CGS [21], which solves the linear and nonlinear equations more efficiently, by employing an incomplete LU decomposition by value (ILUV) in conjunction with the conjugate gradient square (CGS) iteration scheme. In solving the wave equation, the FEM-QSG discretization method has also been used so that higher accuracy can be obtained on a coarser mesh. Moreover, we developed a generalized Rayleigh quotient-inverse iteration algorithm (RQIIM) [15] to solve the generalized eigenvalue equation obtained from finite element discretization of the wave equation, so that the solution of the wave equation requires only tens of seconds on an average workstation (less than the CPU time needed for solving the Poisson equation once) for each eigenvalue. It is well known that the eigenvalue problem is a difficult one for numerical solution, generally consuming a large amount of CPU time, and is the reason why some laser simulators have had to use the effective index approximation in solving the wave equation. In RQIIM, initial guesses of the eigenvalues for the fundamental and first-order modes are needed only at zero bias, following which the last eigenvalue and eigenvector are used as initial guess values at the start of the computation for each new bias point. At zero bias, the initial guesses of the eigenvalue are obtained by a preprocessor which uses the transfer matrix method and the effective index method. As a critically differing from the standard laser models, the present two-mode model has two coupled rate equations, one for each mode. The method of solving these rate equations is crucial to the success of simulation. Numerical convergence is found to be very sensitive to the iterative algorithm used to solve the coupled rate equations. The successful algorithm which has been adopted handles three distinct operation regimes differently to save CPU time. In the first regime, in which neither mode has reached threshold, the photon densities are set equal to zero. Thus, the rate equations are not solved


at all until one or both modes reach threshold. In the second regime, in which one mode has reached threshold, or its mode gain has exceeded its losses for a given iteration, the fundamental mode photon density is adjusted until before evaluating the relation of to . This approach is valid for any normal structure, for which the fundamentalmode lasing occurs before the first-order-mode lasing. In the third regime, both the fundamental and first-order modes have arrived at threshold, i.e., , and . It may appear that adjusting the first-order-mode photon density can balance , but this leads to a breakdown of equality for the fundamental mode. Therefore, the rate equations cannot be solved independently. In this regime, a Newton iteration procedure is used to solve the two coupled rate equations (24) where , , and . The Jacobian matrix is dynamically updated according to the POSM method [23] using the information for and obtained in previous iterations. IV. THE RESULTS

OF

SIMULATION

Having constructed the two-mode model and implemented it in the finite element scheme, we then proceed to the third task: simulate an example structure which is known to exhibit beam instability and emission nonlinearities. Through this example, we evaluate the validity of the multimode interaction hypothesis and gain insight into the dynamics of the field, carrier, and gain evolutions and thereby the underlying physics. For this purpose, we present simulation results of an aluminum-free InGaAs–GaAs–InGaP GRINSCH SL-SQW power laser at 0.98 m. This kind of laser uses InGaP instead of AlGaAs for cladding layers to avoid degradation due to aluminum oxidation during fabrication and laser operation. The structure of the simulated laser is very similar to those described in [24] and has a ridge with width 2.5 m and height 2.0 m. The GRINSCH layers are depicted in Fig. 1(a) and consist of 1) a 2.0- m n-InGaP cladding layer; 2) three 1.76, 1.65, and 1.58 150- InGaAsP grading layers ( ˚ In Ga As undoped SL-SQW eV, respectively); 3) a 90-A ˚ GaAs layer; 4) three 150-A ˚ player centered in a 390-A type grading layers; and 5) a p-type 2.2- m-thick InGaP cladding layer. The simulated device lateral span is 11 m in total. Measured fundamental lateral mode reflectivities of 0.03 (front mirror) and 0.95 (rear mirror), and estimated firstorder-mode reflectivities of 0.021 (front mirror) and 0.665 (rear mirror) [13], were used. Calculated material peak gain for the InGaAs QW versus carrier density are shown in Fig. 2(a). In the figure, we have marked three regions a, b, and c corresponding to computed emission wavelengths around 0.9864, 0.9826, and 0.9787 m, respectively. The wavelength change is not continuous. The wavelength jumps arise from changes in the dominant emission transition between the quantum energy levels of the conduction band and the heavyhole/light-hole valence bands. From the figure, we can see that the curve can be fitted with the logarithmic function when the carrier density is greater than


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(a)

(a)

(b)

(b)

(c) Fig. 1. (a) GRINSCH active region of the simulated laser, (b) fundamental-mode distribution, and (c) the first-order-mode distribution.

the transparency carrier density ( ). The function fits very well for medium and high injection levels, with a slight increase in error near threshold. When the carrier density is less than , we should use the fixed-wavelength gain value (in fact, loss), which cannot be fitted using a logarithmic function. However, this internal loss is negligible and set to zero in these simulations, because it is far less than the other losses near threshold. This approximation only influences the – curve shape near threshold and not in the high-power operation region. This does not influence the estimation of the threshold current, because the threshold current is obtained from extrapolation of the – curve at higher injection. In more general cases, FELES-2 can use a below-threshold fitting

(c) Fig. 2. (a) The computed and fitting curves of the maximum gain versus carrier density. (b) The computed and fitting curves of the spontaneous emission recombination rate versus carrier density. (c) The computed gain spectra.

formula ( , where , , and are fitting parameters), which fits well for a wider range of carrier densities (from 10 –10 cm ).

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TABLE I SOME GAIN AND ABSORPTION PARAMETERS USED FOR THE SIMULATION

TABLE II BAND PARAMETERS FOR In10x Gax Asy Py USED FOR THE SIMULATION

TABLE III THE EFFECTIVE MASSES, LATTICE CONSTANTS, AND REFRACTIVE INDEXES FOR THE ABOVE MATERIALS

Fig. 2(b) shows the computed and fitted spontaneous emission rate versus the carrier density, where the fitting formula , for the InGaAs QW structure. is Fig. 2(c) shows the gain spectra for different carrier densities around the lasing wavelength. The fitting parameters and the free carrier absorption coefficients used for the simulation are Ga As P listed in Table I. The band parameters of In matched to GaAs and strained In Ga As are obtained via interpolation of the theory-based results of Van de Walle’s “model-solid” theory [25], [26]. Some band parameter values used in the simulations are shown in Table II, where (conduction band), (light hole band), and (heavy hole band) are relative to the conduction band of GaAs. is the spin-orbit split-off, is the effective mass of is the biaxial strain parallel to the split-off band, and the heterointerface. The effective mass, lattice constant, and refractive index for these materials are shown in Table III. The refractive indices of these materials are obtained by [27, in QW eqs. (A1)–(A4)]. The linewidth enhancement factor lasers is a nonlinear function of carrier density and wavelength, for which experimental or calculated data remain scarce in , based on spontaneous literature. In [28], the measured emission spectra [28], ranges from 1.5 to 4.0 for the related versus carrier density based wavelengths. Our calculated on the Kramers–Kronig transformation from the gain spectrum gives a range of 1.5 to 3.9. In this simulation, a constant value has been chosen in order to simplify calculations of 3 for [16]–[18]. The other parameters used in the simulation are the usual ones listed in well-known literature. 1) Emission Characteristics: Fig. 1(b) and (c) shows the , ) of the fundamental mode distributions (i.e., and first-order modes, respectively, for the laser structure at a 1.58 V. Fig. 3 shows the light outputs of the fundabias

mental mode, the first-order mode, and the total, as a function of injection current. Before the lasing of the first-order mode, the laser emission is purely from the fundamental mode with a threshold current of 13.38 mA and a quantum efficiency of 0.93 mW/mA, in close agreement with experimental data. The first-order-mode threshold is about 126 mA with a quantum efficiency of about 0.15 mW/mA. Simulations show that the first-order mode has a higher loss (see Fig. 4) and lower gain than the zeroth-order mode. Physically, this is because the relatively loose confinement of the first-order mode results in higher internal losses, with greater mirror loss due to its lower reflectivity. The loss increases and then saturates with forward bias. The situation for gain is different. Carrier distribution does not favor the first-order mode at low bias. However, due to SHB above the fundamental mode threshold, the firstorder mode is increasingly pumped due to an improving match between the mode intensity profile and the distorted gain profile. As a result, we observe clear appearances of kinks and knees in the total power in Fig. 2 in the range of current from 200 to 300 mA, which is consistent with experimental observations [7]. At still higher biases, the experimentally measured power output decreases rapidly as bias increases, even more than indicated in Fig. 2. This difference between simulation and experiment is due more to heating, as noted in [7], than to the SHB and multimode excitations modeled here. From these results, one can clearly see that the kinks or knees of light–current characteristics can indeed arise from the coexistence and coupling of two lateral modes. Because the design of the simulated laser diode is not optimized to suppress the beam-steering, it has a strong lateral confinement even at an unpumped state and a relatively large reflectivity of the first-order mode that result in early lasing of the first-order mode and considerably kinkier – characteristics than usual. On the other hand, it is more illustrative of the effect and easier to handle computationally, and these observations are consistent with experimental results. Next, we will show that the occurrence of nonsymmetric far- and near-field distributions and bilateral beam instability/steering, which have all been observed experimentally, can also be the results of interference of two lateral modes in a symmetric structure. 2) Near-Field Distribution: Fig. 5 shows the superimposed near-field optical intensities at various biases along the center of the active layer. One can see that the near field is symmetric before the lasing of the first-order mode [Fig. 5(a)]. Beyond the lasing threshold of the first-order mode, two phenomena occur: the peaks of the optical intensities shift away from the center of the device and the field distribution becomes asymmetric; and the direction of the shift varies with bias (e.g., see the two curves for 149 and 171 mA). The bilateral steering reflects an inherent instability which has been observed in experiments with 980-nm pump lasers and which cannot be accounted for by assuming a built-in structural asymmetry as in [10]. The fact that these phenomena have been revealed in our numerical simulation is of considerable significance in both device physics and numerical computation, as it is well known that the intrinsic instability is a challenging task for numerical analysis. Physically, this instability is due to


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Fig. 3. Simulated light–current characteristics. (a)

Fig. 4. The comparison of losses of two modes. The solid line is for the zeroth-order mode, the dashed line is for the first-order mode.

a change in relative phase between the two modes with current, as can be seen in Fig. 6(a) and (b) when the nearfield distributions of the zeroth-order and the first-order lateral mode are decomposed along the center of the active layer at 149 and 171 mA, respectively. From this figure, it is clearly seen that when the phase of the first-order mode shifts by , the near-field peak shifts to the opposite side. 3) Far-Field Intensity: Fig. 7 shows the lateral far-field intensities at several biases, obtained through a Fourier transform of the total near field. Alternate shifts of the far-field intensities, in agreement with experimental results, are displayed. The results strongly suggest that the occurrence of higher order modes can be a dominant factor restricting the usable fibercoupled power, and call for the development of more advanced laser waveguide designs that take into account the dynamic evolutions of local index, gain, and field under high injection. 4) SHB: For a better understanding, we now look into the dynamic evolution of SHB as bias is increased. In the current range 44–105 mA, the first-order mode has not yet lased, and the laser operates in a single mode as designed. The local gain distribution along the center of active layer is symmetric, and the hole in the middle of the gain profile deepens as the

(b) Fig. 5. Total intensities of the lateral near field (a) from 44 to 194 mA and (b) from 216 to 297 mA.

bias increases [Fig. 8(a)].1 At 126 mA, however, the first-order mode starts lasing, and the local gain distribution becomes nonsymmetric [Fig. 8(b)]. This is due to the nonsymmetric near-field intensity. For higher biases [Fig. 8(c)], alternate shifts of the local gain peaks occur due to the alternating phase of the near field. 5) Longitudinal Optical Beat Phenomenon: When backplane electroluminescent images are taken [6], it is observed that the bright spots of the light intensity are displayed periodically along the cavity and occur not at the center axis but instead swing about the axis. This bilaterally symmetric beating phenomenon can easily be explained by the time average of the superposition of two lateral modes. Let us assume the fundamental and first-order modes have near-field distributions and in the – plane, 1 The rapid oscillations in Fig. 8 are caused from numerical noise and interpolation from 2-D data computed on a coarser mesh than desirable in the absence of concerns of computation times.

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(a)

Fig. 7. The far-field evolution from 149 to 239 mA.

(b) Fig. 6. The lateral near field distributions of the 0-order made and the first order mode and their superposition at two bias currents at (a) 149 mA and (b) 171 mA, respectively.

respectively, with normalized propagation constants and . Thus, the total optical intensity may be written as follows: (25) where

indicates complex conjugation. When or ( ), the optical intensity is (26)

This corresponds to the case where the light spot is on one side. When or [ ], the optical intensity is (27) This corresponds to the case where the light spot is on the other side. So the beat period is (28)

From solving the 2-D wave equation, is about 0.0109 immediately following the first-order-mode lasing, yielding 45 m, as compared to the measured value 52 m. Considering that no fitting was done for better numerical agreement, the result is rather encouraging. Simulation shows that modal propagation constants of both modes decrease as the bias increases (Fig. 9). More important, however, is how the difference between the propagation constants of the two modes changes as the bias increases. Fig. 9 also shows a plot of the difference between the propagation constants versus bias. We observe that below threshold, the difference decreases as the bias increases (due to the carrierinduced index decrease). This shows that before lasing the effective confinement weakens as the bias increases. But after the fundamental-order-mode lases, the difference monotonically increases (due to SHB), even beyond the zero-bias value. This shows that a well-designed laser diode with a single mode at zero bias can still have higher order modes at higher bias, leading to beam steering. After the first-order-mode lases, the increase slows down and almost saturates, while exhibiting a more complicated pattern. These features are qualitatively consistent with the experimental observation that the longitudinal beat period first decreases and then slightly increases with bias. In [29] and [30], it was pointed out that from the beam propagation method (BPM) several solutions including a symmetric and two asymmetric ones (which are each other’s image) can be found in a stripe laser. We can now see that these asymmetric solutions are results of the superpositions of the fundamental and first-order mode fields that are not resolved in the BPM approach. In addition, the BPM results have the drawbacks of increased error in the presence of loss/gain and of using mode-independent facet reflectivity. V. SUMMARY In this paper, we present a new model for power laser device and have implemented it in a 2-D finite element scheme.


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Fig. 9. The modal propagation constants of two modes and their difference versus bias. (a)

(b)

(c) Fig. 8. The local gain distribution along the center of the active layer (a) before the first-order-mode lasing, (b) just above the threshold of the first-order-mode lasing (medium bias), and (c) well above the threshold of the first-order-mode lasing (high bias).

The simulation results are consistent with the experimental observations of beam steering and emission nonlinearities in 980-nm power lasers. In implementing the model, an effective

computational procedure was developed and its key steps are summarized as follows. 1) Solve the wave equation under injection to obtain the first two modes: the fundamental and first-order modes. 2) Solve two coupled photon rate equations after reaching the threshold of the first-order mode. 3) In solving the current continuity equation, use total optical intensity for calculating the stimulated emission rate. It is the interference of two modes that leads to the beam-steering phenomenon. 4) Compute the optical output power from the total field, instead of the sum of the optical powers of the two modes. Finally, attention must be given to the fact that the two modes have different internal and mirror losses and modal gains. The new model has been implemented in a code named finite element light emitter simulator-2 (FELES-2), which has been used extensively in our lab at the University of Toronto and in some industry research institutes, and has proven to be a robust, versatile, and self-consistent tool for modeling and design. We have used the model and FELES-2 to study a typical power laser structure. It is found that a power laser designed to operate in single lateral mode could in fact evolve under the stress of current and optical power to allow the emergence and lasing of higher order modes, and consequently exhibit intrinsic beam instability and emission nonlinearity. A main cause of these undesirable and often unexpected phenomena is the dynamic and substantial changes in the complex refractive index and the coupling between multilateral modes under high injection. Because the present model is 2-D, it does not model the effects arising from the strong differences in intensity and magnitude of SHB between the front and back facets of an AR/HR coated laser diode. Thus the effect of the kink levels’ dependence on the ratio of the facet reflectivities would not appear in the present model. Moreover, the linewidth enhancement factor , which depends strongly on the carrier density and wavelength and strongly influences mode shifts due to its effect on index change in the waveguide, has been neglected. The constant approximation of used here

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influences the details of beam steering in the far field. A more refined model would be needed in the future with more detailed studies that are sensitive to this effect. APPENDIX STIMULATED EMISSION RATE FOR MULTIPLE MODES According to Fermi’s golden rule [31], the transition probability can expressed as (A1) and the stimulated emission rate as (A2) Assume the optical field is and from , we obtain the modulus of the vector potential : (A3) where is amplitude of electrical field and is the optical frequency. The modulus of the Poynting vector is Re (A4) In order to connect the amplitude of field, density , we use according to [30]

, with photon (A5)

where

is the group velocity at material. So we obtained

of which is the normalized optical field. Considering two modes, vector potential should be

(A6) So we obtain the transition probability for two modes:

(A7) is the momentum matrix elewhere ment. While the stimulated emission rate becomes

(A8)

where is the gain of the active material. Because classical textbooks usually only consider the cases of single mode in bulk material, they use in (A5). It is, however, more reasonable to use instead of for multimode waveguides, which will give rise to a small change of the above formula. ACKNOWLEDGMENT The authors thank J. Guthrie for his contributions in the early stage of this work, and Dr. Suda and M. Fay for their assistance in preparing this manuscript. They also thank the reviewer for his comments for improving the quality of the manuscript. REFERENCES [1] T. L. Paoli, “Nonlinearities in the emission characteristics of stripegeometry (AlGa)As double-heterostructure junction lasers,” IEEE J. Quantum Electron., vol QE-12, pp. 770–776, 1976. [2] H. Asonen, J. Nappi, A. Ovtchinnikov, P. Pavolainen, G. Zhang, R. Ries, and M. Pessa, “High-power operation of aluminum-free ( 0.98 m) pump laser for erbium-doped fiber amplifier,” IEEE Photon. Technol. Lett., vol. 5, pp. 589–590, 1993. [3] O. Imafuji, T. Takayama, H. Sugiuya, M. Yuir, H. Naito, M. Kume, and K. Itoh, “600 mW CW single-mode GaAlAs triple-quantum-well laser with a new index guided structure,” IEEE J. Quantum Electron., vol. 29, pp. 1889–1894, 1993. [4] S. Ishikawa, K. Fukagai, H. Chida, T. Miyazaki, H. Fujii, and K. Endo, “0.98–1.02 m strained InGaAs/AlGaAs double quantum-well high-power lasers with GaInP buried waveguides,” IEEE J. Quantum Electron., vol. 29, pp. 1936–1942, 1993. [5] J. Guthre, G. L. Tan, M. Ohkubo, T. Fukushima, Y. Ikegami, T. Ijichi, M. Irikawa, R. S. Mand, and J. M. Xu, “Beam steering in 980 nm high power laser diode,” in LEOS’94 Conf. Proc., Oct. 31–Nov. 3, 1994, vol. 2, pp. 43–44. [6] , “Beam instability in 980 nm power laser: Experiment and analysis,” IEEE Photon. Technol. Lett., pp. 1409–1411, Dec. 1994. [7] H. Jaeckel, G. L. Bona, P. Buchmann, H. P. Meier, D. Bettiger, W. J. Kozlovsky, and W. Lenth, “Very high-power (425 mW) AlGaAs SQWGRINSCH ridge laser with frequency-doubled output (41 mW at 428 nm),” IEEE J. Quantum Electron., vol. 27, pp. 1560–1567, 1991. [8] J. Buus, “Multimode field theory explanation of kinks in the characteristics of DH lasers,” Electron. Lett., vol. 14, pp. 127–128, 1978. [9] R. Lang, “Lateral transverse mode instability and its stabilization in stripe geometry lasers,” IEEE J. Quantum Electron., vol. QE-15, pp. 718–726, 1979. [10] J. Buus, “Models of the static and dynamic behavior of stripe geometry lasers,” IEEE J. Quantum Electron., vol. QE-19, pp. 953–959, 1983. [11] S. H. Strogatz and I. Stewart, “Coupled oscillators and biological synchronization,” Scientific American, vol. 269, no. 12, pp. 102–109, 1993. [12] M. F. C. Schemmann, C. J. van der Poel, B. A. H. van Bakel, H. P. M. M. Ambrosius, and A. Valster, “Kink power in weakly index guided semiconductor lasers,” Appl. Phys. Lett., vol. 66, no. 8, pp. 920–922, 1995. [13] A. Hardy, “Formulation of two-dimensional reflectivity calculations based on the effective-index method,” J. Opt. Soc. Amer. A, vol. 1, no. 5, May 1984. [14] G. L. Tan, K. Lee, and J. M. Xu, “Finite element light emitter simulator (FELES): A new 2D software design tool for laser devices,” Jpn. J. Appl. Phys., vol. 32, no. 1B, pt. 1, pp. 583–589, Jan. 1993. [15] G. L. Tan, N. Bewtra, K. Lee, and J. M. Xu, “A two dimensional nonisothermal finite element simulation of laser diodes,” IEEE J. Quantum Electron., vol. 29, pp. 822–835, 1993. [16] T. Ohtoshi, K. Yamaguchi, C. Nagaoka, T. Uda, Y. Murayama, and N. Chinone, “A two-dimensional device simulator of semiconductor lasers,” Solid-State Electronics, vol. 30, pp. 627–638, 1987. [17] K. B. Kahen, “Two-dimensional simulation of laser diodes in the steady state,” IEEE J. Quantum Electron., vol. 24, pp. 641–651, 1988. [18] M. Ueno, S. Asada, and S. K. Shiro, “Two-dimensional numerical analysis of lasing characteristics for self-aligned structure semiconductor lasers,” IEEE J. Quantum Electron., vol. 26, pp. 972–981, 1990. [19] G. L. Tan, X. L. Yuan, Q. M. Zhang, W. Ku, and A. Shey, “Two dimensional semiconductor device analysis based on a new finite element

=


[20] [21]

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discretization employing the S–G scheme,” IEEE Trans. ComputerAided Design, vol. 8, pp. 468–478, 1989. G. L. Tan, Q. M. Zhang, and J. M. Xu, “Computation of field and charge transport in compound semiconductor devices—Some new features and methods,” IEEE Trans. Magn., vol. 27, pp. 4158–4161, 1991. Z. Y. Zhao, Q. M. Zhang, G. L. Tan, and J. M. Xu, “A new preconditioner for CGS iteration in solving large sparse nonsymmetric linear equation in semiconductor device simulation,” IEEE Trans. ComputerAided Design, vol. 11, pp. 1432–1440, 1991. D. L. Scharfetter and H. K. Gummel, “Large-signal analysis of a silicon read diode oscillation,” IEEE Trans. Electron Devices, vol. ED-16, pp. 64–77, 1969. G. L. Tan, S. W. Pan, W. H. Ku, and A. J. Shey, “ADIC-2.C: A generalpurpose optimization program suitable for integrated circuit design applications using the pseudo objective function substitution method (POSM),” IEEE Trans. Computer-Aided Design, vol. 7, pp. 1150–1163, 1988. M. Ohkubo, T. Ijichi, A. Iketani, and T. Kikuta, “Aluminum free InGaAs/GaAs/InGaAsP/InGaP at 0.98 m,” Electron. Lett., vol. 28, pp. 1149–1150, 1992. C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B, vol. 39, no. 3, pp. 1871–1883, 1989. M. P. C. M. Krijn, “Heterojunction band offsets and effective masses in III–V quaternary alloys,” Semicond. Sci. Technol., vol. 6, pp. 27–31, 1991. S. Adachi, “Material parameters of In10x Gax Asy P10y and related binaries,” J. Appl. Phys., vol. 53, no. 12, pp. 8775–8792, 1982. N. K. Dutta, J. Wynn, D. L. Sivco, and A. Y. Cho, “Linewidth enhancement factor in strained quantum well lasers,” Appl. Phys. Lett., vol. 56, no. 23, pp. 2293–2294, 1990. R. Baets and J. Buus, “Comparison of a beam propagation model for laser diodes with other modeling techniques,” Proc. Inst. Elect. Eng., vol. 132, no. 4, pt. J, pp. 205–206, 1985. R. Baets, J.-P. Van De Capelle, and P. E. Lagasse, “Longitudinal analysis of semiconductor lasers with low reflectivity facets,” IEEE J. Quantum Electron., vol. QE-21, pp. 693–699, 1985. H. C. Casey, Jr., and M. B. Panish, Heterostructure Lasers. New York: Academic, 1978.

Gen-Lin Tan graduated from the Department of Engineering Physics, Qinghua University, China, in 1961. After graduation, he was engaged in teaching and research with the Department of Electrical Engineering, Beijing Polytechnic University, Beijing, China, where he was a Professor. His research concerned microcomputer application, integrated circuit analysis and optimiation, semiconductor device simulation, and layout optimization of IC designs. He is the author or coauthor of more than 40 articles and three books. During 1987–1988, he was a Visiting Scholar with the Department of Electrical Engineering and Computer Science, University of California at San Diego, La Jolla. Since 1989, he has been a Senior Research Associate with the Department of Electrical Engineering, University of Toronto, Toronto, Ont., Canada. His current research interests include simulation and optimization of high-speed semiconductor devices, optoelectronic devices, and integrated circuits.

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R. S. Mand, photograph and biography not available at the time of publication.

Jimmy M. Xu (S’86–M’87–SM’91) received the Ph.D. degree in electrical engineering from the University of Minnesota, Minneapolis, in 1987. Currently, he is a Professor in the Department of Electrical and Computer Engineering, University of Toronto, Ont., Canada, and holds the endowed chair of Nortel Professor in Emerging Technology. He is a principal investigator of the Ontario Laser and Lightwave Research Centre and the Ontario Centre for Material Research and is a key associate of the Information Technology Research Centre. He has authored and co-authored more than 100 refereed papers in physics and engineering journals, and more than 60 refereed conference papers. He has been granted 10 patents on electronic and photonic devices. His current research interests include semiconductor physics, nanostructures, quantum electronics, and compound semiconductor device design, modeling, and measurements. Currently, he leads a group of 12 researchers and graduate students at the Optoelectronics Laboratory, University of Toronto, and conducts research primarily in the areas of optoelectronics, quantum electronics, nanostructure physics, heterostructure transistors, quantum-well electronics, and photonic devices, as well as largescale computer simulations. The Optoelectronics Laboratory is sponsored by Nortel Technology and conducts research projects funded by agencies and companies in Canada, the U.S., France, and Japan. He is an editor for the IEEE TRANSACTIONS ON ELECTRON DEVICES. Dr. Xu won the 1996 Award of Merits by the FCCP Education Foundation, the 1995 Steacie Prize for contributions to fundamental and applied quantum electronics, and a 1995 Conference Board Canada—NSERC Award for ’Best Practices in University—Industry R&D” (Honorable Mention). One of his students received the NSERC Doctoral Thesis Prize (Engineering), another was given “The Best Student Paper Award LEOS ’94,” and a third won the 1996 Centennial Thesis Award. He is a member of the IEEE Electron Device Society Meetings Committee and ex-officio of the IEEE Electron Device Society Administration Committee.

Quantum Electronics, IEEE Journal of - IEEE Xplore

Quantum Electronics, IEEE Journal of - IEEE Xplore

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