A FINITE-DIFFERENCE TIME-DOMAIN ELECTROMAGNETIC FIELD SOLVER FOR VERTICAL-CAVITY LASERS

ABSTRACT: We present a full- ¤ ector Maxwell equation sol ¤ er for theanalysis of ¤ ertical-ca ¤ ity surface-emitting lasers VCSELs with circular()symmetry using the ﬁnite-difference time-domain FDTD method. We()apply this model to the air-post index-guided 1.55 m m VCSEL structure,and determine the resonant wa ¤ elengths and the quality factor forselected lowest order modes. The results of this study enable quantitati ¤ eand qualitati ¤ e descriptions of the trans erse-mode competition andemission spectra of VCSELs. Q 1998 John Wiley & Sons, Inc.Microwave Opt Technol Lett 18: 385]387, 1998. Key words: electromagnetic ﬁeld sol ¤ er; ﬁnite-difference time-domainmethods; ¤ ertical-ca ity lasers Vertical-cavity surface-emitting lasers VCSELs demonstrate.many properties desirable for optical communications andinterconnects, such as low-beam divergence and single-modeoperation which result in superior coupling efﬁciency intosingle-mode ﬁbers 1 , the possibility of wafer-level testing,wxand ease for two-dimensional array fabrication 2 . In manywxapplications, and for long-haul optical communications, inparticular, single-transverse-mode operation is desired. Forthis reason, a number of theoretical analyses have beenrecently reported in an attempt to understand the transverse-mode competition and the emission spectra of these lasers.The scalar beam-propagation 4 and the effective index 5wx wxmethods have been successfully used, but since they neglectthe vector nature of the electromagnetic ﬁeld, these methodsare unable to model the full transverse-mode spectrum ofthese lasers. The transverse modes of vertical-cavity lasersare rather complicated, while their spatial shape and polar-ization properties depend on the type of mode deﬁnitionscheme. Index-guided, gain-guided, and oxide-apertured lasersshow distinctly different transverse modes and their depen-dence on injection current 6, 7 . Because these cavities arewxsmall on the order of a few wavelengths in all directions , the.application of beam-propagation methods and effective-indexmethods is unable to provide a sufﬁciently accurate descrip-tion of the optical ﬁeld in the cavity. Moreover, the beam-propagation method does not account for multiple reﬂectionsin quarter-wave mirrors. Particularly difﬁcult structures tomodel are the index-guided air-post devices and the oxide-apertured devices in which the large lateral refractive indexvariations severely add to the complexity of the problem 8 .wxFor this reason, it is necessary to include the vector nature ofthe electromagnetic ﬁeld to accurately analyze vertical-cavitylasers. In this paper, we present an analysis of the passivecavity of a 1.55 mm air-post index-guided VCSEL using afull-vector ﬁnite-difference time-domain FDTD method..The analyzed VCSEL structure is a 1550 nm double-fusedcavity that consists of an InGaAsP cavity sandwiched betweentwo AlGaAs AlGaAs.rGaAs quarter-wavelength distributedBragg reﬂectors DBR . A detailed description and the physi-.cal and optical structure parameters of this device are givenin 9 , while the simpliﬁed structure is shown in Figure 1 a .wx .The transverse modes of this resonator are determined byindex guiding in the top mirror, while the cavity loss occursdue to diffraction of the wave in the unguided sections of theactive layer and the bottom mirror. In a high-quality res-onator, such as the VCSEL, one may assume that the modesare not signiﬁcantly perturbed by free-space propagation, andthe approximate cavity spectrum may be obtained by model-ing the VCSEL as a perfect cylindrical dielectric resonatorwx10 . However, this model does not provide the quality factorof each individual mode, namely, the discrimination betweenthe modes. For this purpose, one has to resort to a moreinvolved analysis: we use a full-vector ﬁnite-difference time-domain FDTD applicable to cylindrical high-quality open.resonators with dielectric andror metal boundaries. The pro-gram is used to determine the resonance wavelengths, thequality factor, and the mode spatial distribution for selectlowest order modes in the VCSEL cavity.The principle of operation of the FDTD program is basedon determining the passive-resonator temporal response tonarrowband noise delivered by an antenna source within thecavity. The resonator temporal response, detected at several