本文證實了大面積面射型雷射（broad-area vertical cavity surface emitting lasers)可用於類比研究二維量子彈 子球楼系統（two-dimensional quantum billiards)。實驗結果顯現其近場及遠場分布可分別類比對應於二維 量子彈子球檯的位置空間及動量空間波函數，進一步研究大面積面射型雷射的光譜特性，結果顯示，當雷 射操作在接近閾值條件（threshold condition)時，放大自發輻射（amplified spontaneous emission)會使螢光光譜呈現上千個共振腔模態的峰值結構，深入分析發現，此光譜隱含相鄭模態的交互作用行為及雷射孔徑 的週期軌道資訊，這跟二維量子彈子球檯能譜密度（energy level density)的物理特徵一致，此實驗結果證 實大面積面射型雷射接近閾值的螢光光譜可完善類比對應二維量子彈子球檯的特徵能譜密度。

We experimentally confirmed that broad-area vertical cavity surface emitting lasers (VCSELs) can be utilized to analogously investigate characteristics of two-dimensional (2D) quantum billiards. We discovered that the near-field and far-field transverse patterns of a broad-area VCSEL evidently represent the coordinate-space and momentum-space wave functions of a 2D quantum billiard, respectively. For the optical spectrum of broad-area VCSELs, the high-resolution measurement clearly demonstrates that over a thousand cavity modes with a narrow linewidth can be perfectly exhibited in the subthreshold spontaneous emission spectrum. It is found that these subthreshold spontaneous emission spectra can manifest the signature of photon periodic orbits of the aperture shape, which is quite similar to the behaviors of the energy spectra of the 2D quantum billiards. Accordingly, it is concluded that the subthreshold emission spectra of VCSELs can be analogously used to manifest the quantum-billiard energy spectra.