IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 10, NO. 4, APRIL 1998 475 Calculation and Measurement of Resonant-Mode Blueshifts in Oxide-Apertured VCSEL’s Michael J. Noble, Jae-Heon Shin, Kent D. Choquette, John P. Loehr, James A. Lott, and Yong-Hee Lee Abstract— As the aperture size of oxide-apertured vertical- cavity surface-emitting lasers (VCSEL’s) shrinks, the lasing wavelength blueshifts. We have calculated this effect using both a scalar effective index model and a full-vector weighted index model. Results were compared against experimental data for two different VCSEL designs emitting near 780 and 850 nm. We find that the full-vector weighted index calculation matches the data remarkably well, while the scalar effective index calculation underestimates the blueshift. Index Terms— Cavity resonators, laser modes, optical reso- nators, semiconductor lasers, surface-emitting lasers. I. INTRODUCTION R ECENTLY, it has been observed that the cavity res- onances in vertical-cavity surface-emitting lasers (VC- SEL’s) shift to shorter wavelengths as the transverse cavity dimensions are reduced [1]. This blueshift results because small cavities transform the lasing mode from a plane wave into a true three-dimensional mode [2], [3]. The amount of blueshift varies with aperture size and mode order: the smaller aperture and higher order modes shift the most. In this letter, we compare measured blueshifts for two different device designs against those calculated from two different models: a simple effective-index scalar model and a more comprehensive weighted-index vector model. Both models require only one adjustable parameter—the cavity thickness is adjusted to match the Fabry–Perot resonance of broad- area devices. We find that the weighted index calculation reproduces the dependence of lasing wavelength on aper- ture radius remarkably well, while the effective index model underestimates the blueshift. II. THEORY The blueshift can be understood by examining the lasing modes in cylindrically symmetric VCSEL structures. These modes must satisfy the scalar Helmholtz equation (1) Manuscript received September 26, 1997; revised December 15, 1997. M. J. Noble and J. A. Lott are with the Air Force Institute of Technology (AFIT/ENG), Wright-Patterson Air Force Base, OH 45433-7765 USA. J-H. Shin and Y.-H. Lee are with the Physics Department, Korea Advanced Institute of Science and Technology, Yusong-Ku, Taejon, Korea. K. D. Choquette is with the Center for Compound Semiconductor Science and Technology, Sandia National Laboratories, Albuquerque, NM 87185-0603 USA. J. P. Loehr is with Heterojunction Physics Branch, Avionics Directorate (WL/AADP), Wright-Patterson Air Force Base, OH 45433-7765 USA. Publisher Item Identifier S 1041-1135(98)02426-4. where is an integer, is the (cylindrically symmetric) refractive index profile of the device, and may represent a particular component of an electromagnetic field or vector potential. If the refractive index were a separable function, (1) could be solved by separation of variables. In this case, the solutions in the interior of the VCSEL would take the form (2) where is the th-order Bessel function, is a radial prop- agation constant, and is an axial propagation constant. These propagation constants are fixed by enforcing the appropriate boundary conditions on at the radial and axial interfaces. The separation condition leading to (2) also relates and to the resonant frequency via (3) As the transverse confinement increases, increases from zero while remains constant. Therefore, increases and the corresponding free-space wavelength blueshifts. Actual VCSEL geometries are not separable. Therefore, (2) and (3) cannot be used directly. Approximations are required to compute and . The simplest approximation is the effective index method. In this approach, an axial wavevector is calculated for each layer by propagating plane-waves through the dielectric layer stack, using the refractive indices at 0 for each layer. The resulting (scalar) electric field profile is then used to compute effective indexes (4) corresponding to the core and cladding indices for an “ef- fective” cylindrical dielectric waveguide. An effective axial wavevector is computed similarly. The radial wave equa- tion for this waveguide is then solved, taking regular Bessel functions inside the aperture and evanescent solutions outside. This procedure fixes the radial wavevector , and the resonant frequency may then be calculated from a weighted version of (3). Augmentations of this model and applications to oxide- apertured VCSEL’s are given in [4] and [5]. Although the effective index model is physically intuitive, it has two principle drawbacks: 1) it solves only for scalar fields and 2) the radial and axial problems are not treated on an equal footing. A more complete method is the full-vector weighted index method (WIM), in which both the radial and 1041–1135/98$10.00  1998 IEEE