利用量子球檯介觀波函數類比研究大面積面射型雷射之橫向模態

Abstract

近軸近似光學與二維量子力學之間的類比性是本篇論文的核心概念。基於Helmholtz方程式與非時變Schrödinger方程式之間的相似性，我們可以用量子球檯的介觀波函數解釋大面積面射型雷射的高階近場橫向模態。相反地，非可積量子球檯的波函數雖然沒有解析解，但我們可以利用混沌形狀的面射型雷射做類比觀察。除此之外，同調波突然從量子球檯釋放出來之後的演變也可以透過面射型雷射之橫向模態的傳遞做類比觀察。由於二維量子系統與近軸光學的類比性不僅止於波函數與橫向模態，還包括它們的軌道角動量，所以藉由計算同調波從量子球檯釋放出來之後的軌道角動量可以類比分析面射型雷射光束的軌道角動量。更有趣的是，面射型雷射的模態除了線性偏振之外，還可以形成一種偏振與空間糾纏的向量場。既然光的偏振乃是對應於量子波的自旋，分析面射型雷射之中的向量場可以提供重要的資訊給考慮電子自旋的量子球檯系統。

The analogies between paraxial optics and two-dimensional (2-D) quantum mechanics lie in the heart of this thesis. Based on the similarity between Helmholtz equation and time-independent Schrödinger equation, mesoscopic wave functions of quantum billiards are used to interpret the high-order transverse modes of broad-area Vertical-Cavity Surface-Emitting Lasers (VCSELs). Reversely, chaotic-shaped VCSELs can be employed to analogously observe the wave functions of non-integrable billiards which have no analytic solutions. In addition, the free-time evolution of coherent waves suddenly released from quantum billiards can be analogously observed from the free-space propagation of lasing modes emitted from VCSELs. Furthermore, the analogies are not restricted to quantum wave functions and optical transverse modes, the orbital angular momentum (OAM) density carried by a light beam emitted from VCSELs can be analogously analyzed by calculating the OAM of coherent waves abruptly set free from quantum billiards. More interestingly, the lasing modes of VCSELs are not only linearly-polarized but can form the vector fields, in which the polarization is spatially dependent. Since the polarization of light corresponds to the spin of quantum wave, the analyses of the vector fields in VCSELs can provide important information for quantum-billiard systems (such as ballistic quantum dots) with consideration on electronic spin.