JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 13, JULY 1, 2008 1935 Analysis and Design of Antiresonant Reflecting Optical Waveguide Vertical-Cavity Surface-Emitting Lasers for Above-Threshold Operation X. F. Li, S. H. Tsang, S. F. Yu, Senior Member, IEEE, and W. J. Fan Abstract—Modal characteristics of antiresonant reflecting op- tical waveguide (ARROW) vertical-cavity surface-emitting lasers (VCSELs) are studied under high current injection. It is found that the influence of carrier spatial hole burning increases the radia- tion loss margin of ARROW at low injection current, while the thermal lensing effect significantly reduces radiation loss margin of ARROW at high injection current. Hence, the excitation of mul- tiple-transverse modes in VCSELs is inevitable under high injec- tion current even if the dimensions of the ARROW have been opti- mized to maximum radiation loss margin. Therefore, it is proposed to modify the refractive index profile of the ARROW to compen- sate against the influence of thermal lensing in VCSELs. It can be shown that the radiation loss margin of the modified ARROW can remain unchanged under the influence of thermal lensing if the di- mensions and refractive index profile of the ARROW are properly designed. Hence, stable single-mode operation can be obtained in ARROW VCSELs at high injection current. Index Terms—Antiresonant reflecting optical waveguide, spatial hole burning, thermal lensing effect, transverse leaky mode, ver- tical-cavity surface-emitting lasers. I. INTRODUCTION R ECENTLY, antiresonant reflecting optical waveguides (ARROWs) have attracted considerable attention to realize high-power stable single-mode operation in ver- tical-cavity surface-emitting lasers (VCSELs) [1]–[6]. This is because high-order transverse leaky modes (LMs), which are slightly detuned from the oscillation conditions of ARROW, will exhibit radiation losses much higher than those of the fundamental mode. Therefore, it is expected that only the fundamental mode will be supported inside ARROW VC- SELs [1]–[8]. However, it was observed experimentally that high-order transverse LMs can still be excited in ARROW VC- SELs at high injection current [2]. This is due to the reduction of radiation loss margin (i.e., radiation loss difference between fundamental and high-order transverse LMs) by the influ- ence of nonlinear effects such as carrier spatial hole burning (SHB) [9] and thermal lensing [10]. Currently, self-consistent Manuscript received September 18, 2007; revised October 20, 2007. Pub- lished August 29, 2008 (projected). This work was supported by the Defense Advanced Research Projects Agency under Grant HR0011-07-2-0002. X. F. Li is with the School of Information Science and Technology, Southwest Jiaotong University, Chengdu 610031, China (e-mail: xfl_79@yahoo.com.cn). S. H. Tsang, S. F. Yu, and W. J. Fan are with the School of Electrical and Electronic Engineering, Nanyang Technological University, 639798 Singapore (e-mail: esfyu@ntu.edu.sg). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2007.913736 electrooptothermal models have been developed to study the multiple-transverse-mode operation of ARROW VCSELs [5], [11]. It was shown that the radiation losses of transverse LMs are sensitive to the effective index profile of ARROW. At high injection current, the radiation loss margin can be reduced by the nonlinear effects so that the excitation of high-order transverse LMs is unavoidable. Therefore, it is a challenge to realize much higher power single-mode lasing emission from VCSELs by using ARROW. In this paper, a simple self-consistent model, in which all the necessary nonlinear effects have been taken into consideration, is developed to study the modal characteristics of ARROW VC- SELs. Hence, the influence of carrier SHB and thermal lensing on the radiation losses of LMs is investigated. It is found that the reduction of radiation loss margin at high injection cur- rent cannot be improved only by optimizing the dimensions of ARROW. Therefore, a new ARROW structure with a step index profile is proposed to sustain stable single-mode opera- tion in VCSELs. It is verified that single-mode operation can be achieved in VCSELs at high injection current if the dimensions and refractive index profile of ARROW are selected appropri- ately. This design may lead to even higher single mode power ( 10 mW) in ARROW VCSELs. II. LASER MODEL The structure of ARROW VCSELs under consideration is shown in Fig. 1 [5]. The center low-index core region with di- ameter is surrounded by two cladding layers. The first cladding layer has thickness of and the second one has thickness of . In the conventional design of ARROW, the re- fractive indexes of the core and second cladding layers are set to be the same. The refractive indexes of the first and outer cladding layers are also set to be the same but slightly higher than those of the core and cladding layers. In the investi- gation, only two transverse leaky modes LM and LM with the lowest radiation losses will be considered [5]. The emission characteristics of ARROW VCSELs can be modeled by the following rate equations of photon density and carrier concentration [5], [11], [12]: (1) (2) 0733-8724/$25.00 © 2008 IEEE