Lasing modes in equilateral-triangular laser cavities H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo,* and A. Petrou Department of Physics, State University of New York at Buffalo, Buffalo, New York 14260 A. N. Cartwright Department of Electrical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received 26 August 1999; published 16 June 2000 We report the study of lasing modes in broad-area, equilateral-triangular laser cavities. An alternative approach is proposed to study optical modes in equilateral triangular cavities in an analytical form. The modes were obtained by examining the simplest optical paths inside the cavity, which yields the final solution with the boundary conditions. The cavities can be fabricated from semiconductor heterostructures grown on 111- oriented substrates, which can be easily cleaved into equilateral triangular shapes. Such a design takes advan- tage of total internal reflection at the cleaved facets of the cavity for circulating modes. Experimental results are obtained from cavities fabricated from a superlattice structure of In 0.13 Ga 0.87 As/GaAs grown on a 111 GaAs substrate. PACS numbers: 42.55.Px, 42.60.Da, 42.60.Jf I. INTRODUCTION Cleaved cavity semiconductor lasers are the simplest kind of laser structure because they are easy to fabricate and the cleaved facets have the highest degree of perfection. Thus they are widely used in commercial applications. Because of diverse performance needs, laser cavities of various shapes and dimensions have been intensely studied, including sur- face emitting lasers 1, microdisk lasers 2, micro-arc-ring lasers 3, triangular, L- and U-shaped ridge lasers 4–6, bow-tie lasers 7, and so on. The purpose of this study is to investigate a cavity of triangular shape, taking advantage of the crystalline symme- tries of the samples, namely those grown on 111 substrates, including GaAs- and GaN-based structures 8,9. From a ma- terial point of view, samples grown on 111-oriented sub- strates have shown excellent characteristics in the standard stripe laser geometry, which yield thresholds as low as 87 A/cm 2 10. Cavities of triangular shapes have been fabricated to ex- plore possible advantages of such laser structures. However, the mode structure inside a triangular cavity has not been theoretically analyzed and used in actual laser designs. The focus of this paper is the optical modes in an equilateral triangular cavity. An analysis of the modes is needed for understanding the properties of lasing characteristics in this configuration. Although there have been studies on cavities in this configuration, the results were qualitatively inter- preted 11,12. The lack of calculated results also led to less-than-optimum designs in the past. In Ref. 11, for in- stance, waveguides are created as a part of the cavity for extracting the light beam from the cavity. Such waveguides, however, are created at the corners of the cavity where the light intensity is the lowest, as indicated by our calculations of normal modes. It should be pointed out that the cavities studied here are different from the ring lasers having an etched ridge along the side of the triangle 4. Specifically, the configuration addressed here involves an equilateral triangular laser cavity in which the active medium covers the whole area of the triangle, cleaved from samples grown on 111 substrates. There are three cleavage planes perpendicular to 111, namely, (101 ¯ ), (11 ¯ 0), and (011 ¯ ), which form equilateral triangles. The cavities reported here have each side of the triangles equal to or greater than 75 m. Thus the optical modes can be calculated with the clas- sical treatment of light waves for our purpose. II. CAVITY NORMAL MODES We will first examine the optical modes in an equilateral triangular cavity, whose electrical and magnetic component obeys the Maxwell equations in the well-known form   2 +  k 2   =0, 1 after the time variable is separated out 13, where  can be either the electric or magnetic field. The problem of this second-order partial differential equation in an equilateral tri- angle was treated a long time ago and refined subsequently 14. In previous treatments, two wave vectors of the two- dimensional problem, k 1 and k 2 , are parallel to two sides of the triangle thus not orthogonal to each other and periodic boundary conditions were used. Because the two wave vec- tors are not linearly independent, the results have terms in- volving k 1 • k 2 . Such terms obscure the physical picture of normal modes inside the resonant cavity. In order to analyze lasing action in such triangular cavities, it is critical to find normal modes having linearly independent wave vectors. In order to find solutions with two wave vectors perpendicular to each other, the coordinate system shown in Fig. 1 was used to calculate the two-dimensional problem the x - y plane. The z direction, which is taken as the growth direc- tion, is identical to other edge-emitting semiconductor lasers, *Corresponding author. PHYSICAL REVIEW A, VOLUME 62, 013816 1050-2947/2000/621/0138166/$15.00 ©2000 The American Physical Society 62 013816-1