Chapter 1 Introduction
1.5 Scope of the Thesis
The works of this thesis is organized as follows:
In chapter 2, we theoretically discussed characteristics of quantum billiards including wave functions and energy levels. In section 2.1–2.2, eigenstates and wave states with localization along classical PO of integrable billiards are illustrated. In section 2.3, typical wave states of chaotic billiards are demonstrated in detail. In section 2.4, we quantitatively study behaviors of wave states in terms of amplitude and intensity statistics. In the last section of chapter 2, we analyze energy spectra in both integrable and chaotic billiard systems.
In chapter 3, the investigation on spontaneous emission spectra for VCSELs with various cavity shapes are presented. In section 3.3, we perform very-high-resolution measurements to realize the significance of spontaneous emission spectra of square shaped VCSELs. In section 3.4, we develop similar procedure to further explore behaviors of spontaneous emission spectra of equilateral-triangular and stadium-shaped VCSELs.
In chapter 4, typical lasing modes of broad-area VCSELs with various shapes, mode selection mechanism of stadium-shaped VCSELs, statistcal analyses of experimental wave patterns, and free-space propagation of lasing modes are studied.
In section 4.3–4.4, typical lasing modes in integrable and chaotic VCSELs with varied temperature are illustrated. The mode selection mechanism for distinct types of wave states in stadium-shaped VCSELs is demonstrated in section 4.5. In section 4.6, we carry out statistical analyses to quantitatively understand the distinction between chaotic waves and localized wave states. In section 4.7, we discuss how to use far-field transverse patterns of VCSELs to analogously explore the momentum-space wave functions of a 2D quantum billiard. In the final section of chapter 4, due to the mathematical equivalence between paraxial wave equation and time-dependent Schrödinger equation, we utilize VCSELs to analogously observe transient dynamics of coherent waves released from quantum billiards.
In chapter 5, we experimentally explore polarization dynamics of broad-area VCSELs with different aperture size and cavity geometry far above threshold
condition. In section 5.2, we discuss typical characteristics of VCSELs when polarization switching takes place. In section 5.3, the comparison of polarization resolved L-I curves in square-shaped VCSELs with different area are investigated. In section 5.4–5.5, polarization behaviors of stadium-shaped VCSELs far above threshold operation are studied.
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Chapter 2
Analysis of Energy Level and Wave Functions in
Quantum Billiards
In this chapter, we theoretically investigated characteristics of quantum billiard systems. The analyses were concentrated on two main features of quantum systems:
energy eigenvalues and wave functions. Firstly, wave functions in integrable and chaotic billiard systems were illustrated. We then theoretically calculated amplitude and intensity distribution of the wave patterns in order to quantitatively distinguish chaotic and localized wave patterns. Furthermore, we analyzed features of energy spectra which is the most important quantity for realizing behaviors of quantum systems.
2.1 Eigenstates in Integrable Quantum Billiards
In section 1.3, the governing equation of 2D quantum billiard is 2D time-independent Schrödinger equation which is given by
( ) ( ) ( )
2
2 , , ,
2 V x y x y E x y
M ψ ψ
− ∇ + =
ℏ , (2.1)
with
( ) ( )
( )
0, ,
, , ,
if x y V x y
if x y
∈Ω
=
∞ ∉Ω
,
where M is the mass of the particle and Ω is determined by the boundary of the billiard. For the particle inside the quantum billiard, Eq. (2.1) can be rewritten as
(
∇ +2 k2)
ψ( )
x y, =0, (2.2)with k= 2ME/ℏ2 . Obviously, it can be seen that Eq. (2.2) is completely equivalent to 2D Helmholtz equation in optics by replacing ψ
( )
x y, with electromagnetic field.We now solve the eigenstates and eigenenergy of this equation.
2.1.1 Square Billiards
The 2D square billiard is one of the simplest billiards in classical mechanics [1].
The system with 2D square shape is classically integrable and separable. We can solve Eq. (2.2) by representing ψ
( )
x y, as the multiplication of two one dimensional solutions. Consequently, the quantum eigenstates ψm n,( )
x y, for the vertices are at(
±a 2 , ±a 2)
and(
±a 2 , ∓a 2)
and are expressed as ,( )
, 2 sin sin
2 2
m n m n
a a
x y k x k y
ψ = a + +
, (2.3) where kn =nπ a
(
n=1, 2, 3,…)
and a is the length of the square boundary. The eigenenergy of square billiard is given byEm n, =2πMa2ℏ22
(
m2+n2)
. (2.4) Figure 2.1 shows some intensity patterns of eigenstates. It can be seen that these eigenstates present regular arrangement.((((m,n m,n m,n m,n) =(1,1) ) =(1,1) ) =(1,1) ) =(1,1) ((((m,n m,n m,n m,n) =(2,1) ) =(2,1) ) =(2,1) ) =(2,1) ((((m,n m,n m,n) =(1,2) m,n ) =(1,2) ) =(1,2) ) =(1,2)
((((m,n m,n m,n) =(2,2) m,n ) =(2,2) ) =(2,2) ) =(2,2) ((((m,n m,n m,n m,n) =(3,2) ) =(3,2) ) =(3,2) ) =(3,2) ((((m,n m,n m,n m,n) =(2,3) ) =(2,3) ) =(2,3) ) =(2,3)
((((m,n m,n m,n) =(3,3) m,n ) =(3,3) ) =(3,3) ) =(3,3) ((((m,n m,n m,n m,n) =(4,3) ) =(4,3) ) =(4,3) ) =(4,3) ((((m,n m,n m,n m,n) =(20,20) ) =(20,20) ) =(20,20) ) =(20,20) ((((m,n m,n m,n m,n) =(1,1) ) =(1,1) ) =(1,1) ) =(1,1) ((((m,n m,n m,n m,n) =(2,1) ) =(2,1) ) =(2,1) ) =(2,1) ((((m,n m,n m,n) =(1,2) m,n ) =(1,2) ) =(1,2) ) =(1,2)
((((m,n m,n m,n) =(2,2) m,n ) =(2,2) ) =(2,2) ) =(2,2) ((((m,n m,n m,n m,n) =(3,2) ) =(3,2) ) =(3,2) ) =(3,2) ((((m,n m,n m,n m,n) =(2,3) ) =(2,3) ) =(2,3) ) =(2,3)
((((m,n m,n m,n) =(3,3) m,n ) =(3,3) ) =(3,3) ) =(3,3) ((((m,n m,n m,n m,n) =(4,3) ) =(4,3) ) =(4,3) ) =(4,3) ((((m,n m,n m,n m,n) =(20,20) ) =(20,20) ) =(20,20) ) =(20,20)
Fig. 2.1 Some intensity patterns of eigenstates in a square billiard.
2.1.2 Equilateral-Triangular Billiards
The system with 2D equilateral triangular shape is classically integrable but nonseparable. We assume equilateral triangular billiard with vertices located at
( )
0, 0 ,(
a 2 , 3a 2)
, and(
−a 2 , 3a 2)
. Several groups have solved the eigenstates and eigenenergy of 2D equilateral triangular billiard [2-4]. According to the Ref. [3], for2
m≠ n, there are two degenerate states with different symmetry properties which can be written as
The eigenenergy of 2D equilateral triangular billiard is given by
Em n, =2Maℏ2 243π 2
(
m2+ −n2 mn)
. (2.8)Figure 2.2 depicts some intensity patterns of low order eigenstates corresponding to ψm n( )1,
( )
x y, and ψm n( )2,( )
x y, , respectively. We also calculated the intensity patterns of high order eigenstates, as seen in Fig. 2.3. It can be seen that these high order eigenstates display the morphology of honeycomb. The intensity patterns of non-degenerate eigenstates in 2D equilateral triangular billiard are shown in Fig. 2.4.( )1,
( )
, 22.2 Superscarred Wave States in Integrable Quantum Billiards
It is known from the section 2.1 that because of the high symmetry of systems, eigenstates in integrable billiards display regular nodal patterns even when the order of the eigenstate is very high. An important question arises: How can we describe the connection between quantum theory and classical mechanics in this system by using correspondence principle? Schrödinger first realized this problem [5]. He noticed that the motion of classical particles can be viewed as the propagation of wave packet state in space. The position of the particle corresponds to the position of central maximum of wave packet. Wave packet states can not be obtained by directly solving time-independent Schrödinger equation. Instead, superposition of many eigenstates can lead to wave packet states. According to this concept, Schrödinger defined the wave packet state whose motion is equivalent to classical particle as coherent states. It is a greatest triumph since we can describe the motion of classical particles by using the theory of quantum mechanics.
In 1995, Pollet et al. used wave function of the SU(2) coherent state for 2D quantum harmonic oscillation to successfully obtain the wave states that are well localized on the corresponding classical elliptical trajectory [6]. In integrable billiard systems, wave function of the coherent state can also localize on classical stable POs.
This wave states are called “superscars” in order to distinguish from “scars” that will be demonstrated in the following section. We now briefly discussed how to obtain superscarred wave states in square and equilateral triangular billiards.
2.2.1 Superscarred Wave States in Square Billiards
For a square billiard with the vertices at (±a/ 2, ±a/ 2) and (±a/ 2, ∓a/ 2), the quantum eigenstates ψm n,
( )
x y, are written as Eq. (2.3). Each family of classical periodic orbits in a square billiard can be denoted by with three parameters ( , , )p q φ ,where p and q are two positive integers describing the number of collisions with the horizontal and vertical walls, and the phase factor φ is in the range of −π to π that is related to the wall positions of specular reflection points [7]. The path lengths of POs can be written as L p q
( )
, =2a p2+q2 . It has been verified that with the Schwinger SU(2) representation the coherent states associated with periodic orbits( , , )p q φ can be analytically expressed as [8] weighting coefficient, and !
!( )!
Equation (2.9) is the representation of the traveling-wave state. The standing-wave representation of can be given by SN Mp q, ,, φ,±
( )
x y, = Ψ( )+ p qN M, ,, φ( , )x y ± Ψ( )−N Mp q, ,, φ( , )x y . Note that the coherent states obtained as a linear superposition of a few nearly degenerate eigenstates have been verified to be the persistent stationary states in real mesoscopic systems and to display the quantum interference features on the classical periodic orbits.Figure 2.5 displays some SU(2) coherent states with traveling and standing wave representations corresponding to a continuous family of nonisolated POs with
( )
( , )p q = 1,1 in square billiards. The calculated standing wave representations of superscarred wave states corresponding to POs with other ( , )p q are shown in Fig.
2.6.
φ = 0 φ π
= 4φ
=2π
4φ
=3π
4φ π =
Traveling wave
Standing wave
φ = 0 φ π
= 4φ
=2π
4φ
=3π
4φ π =
Traveling wave
Standing wave
Fig. 2.5 Some SU(2) coherent states with traveling and standing wave representations corresponding to a continuous family of nonisolated POs with ( , )p q =
( )
1,1 insquare billiards.
1 1 1 1
p p p p qqqq
1 1 1
1 2 2 2 2 3 3 3 3 4 4 4 4
2222
3333
4444 1 1 1 1
p p p p qqqq
1 1 1
1 2 2 2 2 3 3 3 3 4 4 4 4
2222
3333
4444
Fig. 2.6 Some standing wave representations of superscar modes corresponding to POs with different ( , )p q .
2.2.2 Superscarred Wave States in Equilateral Triangular Billiards
Classically, the formation of periodic orbits in equilateral-triangular billiard can be denoted by three parameters
(
p q, ,±φ)
, where parameters p and q are nonnegative integers with the restriction that p≥q; the parameter φ is in the range of 0 to π. The sign ± and parameters p and q correspond to the initial angle of the billiard ball by [9].( )
1Superscarred waves that are associated with classical POs can be analytically expressed with the representation of quantum coherent states which is written as [10]
,
( )
1 0 ,0 ( 1 )( )
where m0 =
(
p+2q N)
and n0 =(
2p+q N)
are the order of the coherent state. Aswhere m0 =