IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 40, NO. 9, SEPTEMBER 2004 1163 Spatially Dependent Noise Model for Vertical-Cavity Surface-Emitting Lasers Johan S. Gustavsson, Jörgen Bengtsson, and Anders Larsson Abstract—We present a comprehensive noise model for vertical- cavity surface-emitting lasers (VCSELs). The time-domain model accounts for the stochastic fluctuations in the local carrier density in the separate confinement heterostructure and quantum wells, and in the modal intensity and phase of both the internal and the out-coupled optical field. In this work, we consider these fluctu- ations to be caused by the temporal uncertainty of the processes that supply or consume carriers and photons, such as carrier diffu- sion and photons escaping the cavity, and the processes that create or annihilate carriers and photons, such as stimulated emission and absorption. The noise model is based on a deterministic quasi- three-dimensional dynamic model that treats the carrier transport, heat generation and dissipation, and optical fields in the device. Langevin noise terms are derived and added to the rate equations in the numerical solution. The noise model is applied to simulate the noise characteristics of fundamental-mode stabilized VCSELs with a shallow, mode discriminating, surface relief. The relative intensity noise and frequency noise of the output are calculated. From the latter, the linewidth of the VCSEL can be estimated. The results are compared with those of conventional multimode VCSELs. Index Terms—Dynamics, modeling, noise, transverse modes, vertical-cavity surface-emitting laser (VCSEL). I. INTRODUCTION T HE vertical-cavity surface-emitting laser (VCSEL) has become an important component for communication and sensing. This is due to fabrication and testing advantages related to the geometry of the VCSEL. Also related to the geometry is the fact that VCSELs are characterized by a strong spatial variation of injection current, carrier density, photon density, and temperature. Therefore, spatial effects have a large impact on the behavior of VCSELs, and models that account for these spatial dependencies are needed to accurately predict the performance of VCSELs. Previous work on VCSEL modeling has accounted for the transverse variation of injection current, carrier density, and photon density (including single and mul- tiple transverse modes). Here we report on a comprehensive time-domain VCSEL model that also accounts for the spatial dependence of the noise. The noise characteristics of semiconductor lasers are of great importance for optical links in that noise often limits link perfor- mance. In digital links, laser noise degrades the signal-to-noise ratio and introduces extra timing jitter, which ultimately limits Manuscript received November 6, 2003; revised May 28, 2004. The authors are with the Photonics Laboratory, Department of Microtech- nology and Nanoscience, Chalmers University of Technology, SE-412 96 Göte- borg, Sweden (e-mail: johan.gustavsson@mc2.chalmers.se). Digital Object Identifier 10.1109/JQE.2004.833211 the achievable bit-error rate. In analog fiber optical links, laser relative intensity noise (RIN) is typically the dominant source of noise which limits the signal-to-noise ratio and the dynamic range. Due to its relatively large transverse dimensions, the VCSEL normally supports multiple transverse modes. The strong com- petition between the excited modes for carriers from a common reservoir causes a complex behavior of the VCSEL. For example, the individual modes exhibit a higher RIN at lower frequencies than does the total output, i.e. the summed modal powers. This phenomenon is known as mode-partition-noise (MPN). Theoretical studies on RIN in multimode VCSELs have been presented. A cold-cavity model was then applied, where a Langevin noise term was added to the rate equation for the photon number (or field amplitude) of a mode. The numerical results showed that a larger degree of spatial overlap between the modes will enhance the MPN, due to a more intense carrier competition [1]. Further, it has also been observed that MPN can cause multiple resonance peaks in the RIN spectrum of both the individual modes and the total output [2]. The peaks are referred to as “mode-partition frequencies” [3]. Apart from fluctuations in intensity, the VCSEL also exhibits fluctuations in the phase of the output lightwave. This phase noise (PN) causes a limited spectral purity of the laser, i.e. a broadening of the linewidth. Depending on the viewpoint, it is sometimes more convenient to interpret phase fluctuations as fluctuations in the instantaneous frequency, i.e. as frequency noise (FN). In this work, PN is used in the derivation of the noise terms whereas the results are given as FN. To our knowledge, very little work has been made on studying the characteristics of this type of noise in VCSELs. The extensive noise model for VCSELs presented here is a time-domain model that considers the stochastic fluctuations in the local carrier density and in the intensity and phase of the transverse modes. The noise model is based on a deterministic quasithree-dimensional dynamic model that is described in de- tail in [4]. It consists of a number of interdependent submodels that treat the optical fields and carrier and heat transport in the device. The optical model utilizes a scalar effective index method [5], [6], in which the VCSEL structure is assumed to only depend upon the longitudinal position in a number of trans- verse regions. The electrical field can then be described by a time-independent longitudinal and a nearly harmonic time-de- pendent transverse component. This transforms the wave equa- tion into one longitudinal and one transverse eigenvalue equa- tion. The former is solved once for each transverse region. The solution gives the longitudinal field and an eigenvalue, whose real and imaginary parts are related to the effective index and 0018-9197/04$20.00 © 2004 IEEE