Abstract. A three-dimensional numerical model of a vertical- cavity surface-emitting laser (VCSEL) containing a reso- nance grating of quantum wells (QWs) is developed. The Helmholtz equation for a éeld and the diffusion equation for a medium, in which an electron beam is the source of charge carriers, are solved self-consistently, which allowed us to énd the longitudinal and radial proéles of the generated éeld, its frequency, and the threshold pump current. The character- istics of the higher-order modes are calculated against the background of the frozen medium formed by the generated mode. The stability limit of the single-mode regime and the type of a mode at which lasing begins to develop with increasing pump power are found from calculations of the gain balance and losses for higher-order modes. An iteration algorithm is developed for calculating the parameters of a VCSEL with many QWs, the calculation time increasing linearly with the number of QWs. The proéles of the resonator modes and their frequency spectrum are calculated for a cylindrically symmetric VCSEL. The stability limits of single-mode lasing are determined. The results are compared qualitatively with experiments. Keywords: resonance heterostructure, method of counterpropa- gating beams, eigenvalues, nonlinear operator. 1. Introduction Heterostructures containing many quantum wells (QWs) are of practical interest for application in vertical-cavity surface-emitting lasers (VCSELs). They can be pumped either by an electron beam or laser diodes [1]. Longitudi- nally electron-beam-pumped semiconductor lasers can be used as quasi-continuous monochromatic radiation sources in display tech-nologies. A distinctive feature of such lasers is the absence of optical conénement in the direction perpendicular to the resonator axis. The laser éeld distribution in the transverse direction is determined by a change in the complex per-mittivity, which in turn is controlled by the current density distribution in an electron beam spot, the scattering of electrons in a semiconductor and the diffusion of charge carriers. The érst theoretical studies of the mode composition and radiation directivity taking into account the spatial transverse inhomogeneity of excitation were performed for a transversely pumped semiconductor laser [2, 3]. Analytic solutions were constructed for the éeld in the resonator for some characteristic pump rate distributions and the param- eters of conéguration losses was introduced. These solutions described qualitatively the radiation pattern for the basic types of oscillations upon pumping slightly exceeding the lasing threshold. This approach applied to longitudinally pumped lasers showed [4] that the characteristic transverse size of the éeld intensity distribution for the fundamental mode at the lasing threshold is smaller than the diameter of an electron beam with the Gaussian current density function. The laser radiation divergence in experiments noticeably exceeded the diffraction limit, especially when the lasing threshold was considerably exceeded. It was assumed that this was caused by excitation of higher-order modes. The authors of paper [5] calculated the éeld intensity proéles for transverse modes in the WKB approximation, found their excitation thresholds, and analysed the inhomogeneities of the gain distribution and the role of a thermal lens. The divergence angle of a laser beam for the given current was equated to the divergence angle of the highest excited type of oscillations. The theory explained qualitatively the increase in the divergence angle from  5  at the lasing threshold up to  15  in the case of a considerable excess over the threshold in laser electron-beam tubes with a single-crystal active region and the electron-beam spot diameter 35 mm. However, these papers neglected the inêuence of the gener- ated éeld on the distribution of the complex permittivity, which is considerable in semiconductor lasers [6]. In the last years the attention of researchers has shifted from single-crystal structures to nanoheterostructures in which the lasing threshold at room temperature can be considerably reduced by using resonantly periodic amplié- cation [7]. The simulation of a QW grating VCSEL is a challenging computing problem because of a great number of layers with boundaries partially reêecting light and due to the nonlinear type of eigenvalue equations. In addition, a mathematical model of the laser should adequately take into account diffraction intracavity phenomena and match the solutions for the electromagnetic éeld in QWs with the solu- tions of nonlinear equations of the diffusion type for current carriers. D.V. Vysotskii, N.N. Elkin, A.P. Napartovich State Research Center of the Russian Federation, Troitsk Institute for Innovation and Fusion Research, ul. Pushkovykh 12, 142190 Troitsk, Moscow region, Russia; e-mail: dima@triniti.ru; V.I. Kozlovsky, B.M. Lavrushin P.N. Lebedev Physics Institute, Russian Academy of Sciences, Leninsky prosp. 53, 119991 Moscow, Russia Received 19 May 2009; revision received 8 July 2009 Kvantovaya Elektronika 39 (11) 1028 ë 1032 (2009) Translated by M.N. Sapozhnikov PACS numbers: 42.55.Px; 42.60.Da; 41.75.Fr DOI:10.1070/QE2009v039n11ABEH014159 Simulation of a longitudinally electron-beam-pumped nanoheterostructure semiconductor laser D.V. Vysotsky, N.N. Elkin, A.P. Napartovich, V.I. Kozlovsky, B.M. Lavrushin 961/810 ë Ososkov ë 15/i-10 ë SVERKA ë 5 ÒÑÎÑÔ ÍÑÏÒ. å 1 Quantum Electronics 39 (11) 1028 ë 1032 (2009) ß2009 Kvantovaya Elektronika and Turpion Ltd