PHYSICAL REVIEW E 84, 056201 (2011) Wave pattern and weak localization of chaotic versus scarred modes in stadium-shaped surface-emitting lasers Y. T. Yu, P. H.Tuan, P. Y. Chiang, H. C. Liang, K. F. Huang, and Y. F. Chen * Department of Electrophysics, National Chiao Tung University, 1001 Ta-Huseh Road, Hsinchu, 30050, Taiwan (Received 17 May 2011; revised manuscript received 24 August 2011; published 3 November 2011) We explore the lasing mode selection between the chaotic and scarred modes in stadium-shaped vertical-cavity surface-emitting lasers (VCSELs). Experimental results reveal that the spatial gain distribution in the active layer of a VCSEL can be modified via the aperture size to favor the generation of either the chaotic or the scarred modes. Experimentally obtained chaotic and scarred modes are further employed to perform statistical analysis of wave function intensities for making a comparison with predictions based on the nonlinear σ model. We verify that the scarring effect can be quantitatively relevant to the weak-localization correction in the intensity probability distribution. DOI: 10.1103/PhysRevE.84.056201 PACS number(s): 05.45.Mt, 42.55.Px, 42.60.Jf, 73.20.Fz I. INTRODUCTION Current research in quantum chaos or wave chaos is mainly focused on wave function structure and transport in wave systems with a chaotic ray limit [1]. One of the most intriguing discoveries is the presence of scarred states that deviate significantly from a Gaussian random fluctuation by exhibiting a large excess of intensity near extraordinary unstable periodic orbits of the corresponding classical dynamics [2]. Since their theoretical discovery in a quantum framework, scars have been observed and predicted both experimentally and numerically in a wide variety of physical systems, including microwave cavities [3], Faraday surface waves [4], vibrating soap films [5], acoustic radiation [6], hydrogen atoms in a magnetic field [7], electrons in a resonant tunneling diode with a magnetic field [8], and molecular vibration [9]. The spatial structures of laser modes in broad-area res- onators have received much interest for a long time because they give a deep insight into the pattern formation of natural waves [10–17]. Scarred modes have also been observed in microdisk lasers [18,19] and have been confirmed to possess the highest quality factors and high directionality. However, only a few efficient scarred modes can be supported in microdisk lasers because boundary losses played a critical role in the mode selection mechanism [20]. Moreover, the reimaging of lasing wave patterns on the vertical surface is a thorny subject due to the lateral radiation of microdisk lasers. The spatial patterns of lasing modes have significant value for exploring the statistical properties of wave function intensities in position space, which can be compared with the related theoretical model for verifying the presence of the scarring phenomenon. In contrast to two-dimensional (2D) microdisk lasers, vertical-cavity surface-emitting lasers (VCSELs) have a dom- inant longitudinal wave vector k z that makes it quite feasible to measure the spatial patterns of lasing modes with simple optics. Recently, the transverse modes of oxide-confined large- aperture VCSELs have been identified as novel emulations of the wave functions of 2D quantum billiards with the same confinement [21–23]. Even though the spatial gain distribution * yfchen@cc.nctu.edu.tw has been verified to play an important role in the usual optical resonators for the selective excitation of modes [24,25], its influence on mode selection in large-aperture VCSELs is still an open issue [26]. The spatial gain distribution of VCSELs is mainly de- termined by the oxide aperture size for a given operation temperature. To illustrate the influence of the aperture size, we use a distributed resistance network [27] to numerically analyze the carrier density distribution for VCSEL devices with mesa diameter of 140 μm and aperture diameters of 40 and 60 μm, respectively. As shown in Fig. 1, the carrier density of the very-large-aperture (60 μm) VCSEL is concentrated more in the neighborhood of the aperture boundary, compared with the 40 μm VCSEL. This result indicates that the aperture size considerably affects the spatial gain distribution and may be a functional parameter in mode selection. Inspired by this finding, we fabricated stadium-shaped VCSELs of different sizes to explore the scarring effect on the resonant modes and to further analyze the position-space wave function statistics in real devices. The experimental results reveal that chaotic modes and scarred modes can be effectively selected by use of different aperture sizes. Furthermore, we analyze the mode intensity statistics to make a comparison with predictions based on the nonlinear σ model [28,29]. Numerical analyses indicate that the intensity statistics of experimental scarred modes agree very well with the theoretical predictions for the weak-localization phenomenon. II. EXPERIMENTAL RESULTS AND DISCUSSION To investigate the influence of the aperture size on the lasing mode, we fabricated two categories of VCSELs with the same stadium shape but with different aperture sizes of 30 × 60 and 20 × 40 μm 2 . The device structures of the oxide- confined VCSELs were similar to those described in [14]. The emission wavelengths of all VCSELs were approximately 808 nm. Figure 2 shows an optical microscope image of the device operated with an electric current under threshold current at room temperature. The bright region indicates the stadium-shaped pattern of spontaneous emission. The VCSEL device was placed in a cryogenic system with a temperature stability of 0.1 K in the range of 200–300 K. A power supply 056201-1 1539-3755/2011/84(5)/056201(4) ©2011 American Physical Society