IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 3, MARCH 2001 353 Effects of Radiation on Circular-Grating DFB Lasers—Part I: Coupled-Mode Equations Pamela L. Greene and Dennis G. Hall Abstract—We derive near-threshold coupled-mode equa- tions that include the effects of radiation for TE fields in circular-grating surface-emitting distributed feedback (DFB) lasers with second-order gratings. The analysis uses an exact description of the periodic grating that is valid for any grating shape and depth. For low-order azimuthal modes, approximate equations are found that are the same as those for a linear DFB laser, though the boundary conditions differ. We solve the cou- pled-mode equations for a weakly guiding laser numerically for low- and higher-order radial and azimuthal modes and compare these numerical results to previous experimental observations and to the results found for linear gratings. For sufficiently strong coupling, radiative losses have a significant impact on the mode structure of the laser, breaking the radial (longitudinal) mode symmetry otherwise seen for the lowest order azimuthal modes and improving mode selectivity. Index Terms—Distributed feedback lasers, laser modes, numer- ical analysis, semiconductor lasers, surface-emitting lasers. I. INTRODUCTION S EMICONDUCTOR lasers are among the most widely used sources of coherent light. They are small, inexpensive, reli- able, and easy to power and modulate; these attributes have con- tributed to their popularity in a variety of commercial applica- tions. However, the standard edge-emitting semiconductor laser, although in widespread use, is not ideal for applications that re- quire high power, a narrow spectrum, array configurations, or a narrow or circularly symmetric beam. For some time, con- siderable effort has been devoted to the development of a sur- face-emitting laser that has these properties without sacrificing the considerable advantages of edge-emitting devices. Distributed feedback (DFB) and distributed Bragg reflector (DBR) lasers, in which a Bragg grating provides longitudinal confinement, were developed and studied beginning in the early 1970s in response to a desire for compactness, stability, and im- proved mode discrimination [1]–[4]. Because the emission aper- ture can be large, the angle of diffraction of the emitted beam can be made much narrower than that emitted from a conventional edge-emitting laser. Spreading the output over a larger emission aperture also allows greater total emitted power, which is ulti- mately limited by facet damage in edge emitters. Grating-cou- Manuscript received March 23, 2000; revised October 26, 2000. This work was supported by the U.S. Army Research Office and the National Science Foundation. P. L. Greene is with The Institute of Optics, University of Rochester, Rochester, NY 14627-0186 USA. D. G. Hall is with The Institute of Optics and The Rochester Theory Center for Optical Science and Engineering, University of Rochester, Rochester, NY 14627-0186 USA. Publisher Item Identifier S 0018-9197(01)01620-7. Fig. 1. Cut-away diagram of a concentric-circle-grating DFB semiconductor laser designed for optical pumping. The cover and substrate extend to , respectively. The phase of the grating is shifted by at the center, corresponding to a distance . Typical dimensions for an AlGaAs–GaAs single-quantum-well structure are nm, m (300 periods), and Å. pled lasers, unlike vertical-cavity devices, can also be easily fabricated in a variety of material systems. However, as with an edge-emitting device, the resonant cavity and, therefore, the emitted beam, of a linear grating-coupled laser is still highly asymmetric about the beam axis, producing an elliptical beam with significant eccentricity. The concentric-circle-grating surface-emitting (CCGSE) semiconductor laser, shown in a cut-away view in Fig. 1, combines the advantages of the grating-coupled DFB laser with circular symmetry and an even larger emission aperture. Though experimental observation of stimulated emission from the side of an uncorrugated cylindrical GaAs diode was reported in the early 1960s, [5] most analyses of cylindrical resonators concentrated on their passive characteristics [6]–[8]. Kerner et al. extended the theory to corrugated waveguides, deriving coupled-mode equations for the inward- and outward-traveling circular waves [9]. Zheng and Lacroix used the coupled-mode formalism in an analysis of in-plane coupling to optical tapers inserted radially into a circular resonator [10], and Wu et al. incorporated TE–TM cross-coupling in a vector formulation [11], [12]. Analyses of an active concentric-circle-grating device, i.e., a disk-shaped DFB or DBR laser, began with Toda’s calculation of the reflectivity and resonance conditions for circular DFB lasers [13] and continued with Erdogan and Hall’s scalar [14] and vector [15] treatments of TE–TE mode coupling in first-order-grating lasers and Gong’s investigation of radial mode discrimination in a second-order DBR laser [16]. Threshold gain calculations for circular DBR and DFB devices have also been presented [17], [18], as have mode analyses incorporating gain saturation [19], [20]. Emission from a CCGSE laser has been observed experimentally from 0018–9197/01$10.00 © 2001 IEEE