大面積面射型雷射的螢光光譜與時空動力之研究: 量子渾沌的顯現

(1)

:

Investigation of Spontaneous Emission and Spatial-Temporal Dynamics

in Broad-Area Vertical-Cavity Surface-Emitting Lasers:

Manifestation of Quantum Chaos

:

(2)

:

Investigation of Spontaneous Emission and Spatial-Temporal Dynamics in Broad-Area

Vertical-Cavity Surface Emitting Lasers: Manifestation of Quantum Chaos

Student

Yan-Ting Yu

Advisor

Yung-Fu Chen

A Thesis

Submitted to Department of Electrophysics College of Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in Electrophysics

May 2012

(3)

(broad閾area VCSELs)

(Helmholtz equation) (time閾independent Schrödinger equation)

(paraxial

wave equation) (time閾dependent Schrödinger

equation)

(diffraction in time)

(chaotic mode) (scarred

(4)

Spatial-Temporal Dynamics in Broad-Area

Vertical-Cavity Surface Emitting Lasers:

Manifestation of Quantum Chaos

Student: Yan-Ting Yu Advisor: Prof. Yung-Fu Chen

Institute and Department of Electrophysics

National Chiao-Tung University

Abstract

Basic properties as well as applications in physics research of broad-area VCSELs from below to far above threshold are thoroughly investigated. Firstly, the high-resolution measurement demonstrates that over a thousand cavity modes with a narrow linewidth can be perfectly exhibited in the subthreshold emission spectra. Further analyses of the obtained spectra confirm that the subthreshold emission spectra of broad-area VCSELs can be exploited to analogously investigate the energy spectra of the 2D quantum billiards. In addition, due the analogy between Helmholtz equation and time-independent Schrödinger equation, transverse wave patterns of oxide-confined VCSELs enable us to visualize the wave functions of the 2D quantum billiards with the same lateral shapes. Therefore, typical lasing modes for VCSELs with various cavity shapes are explored. Furthermore, since paraxial wave equation and time-dependent Schrödinger equation are highly equivalent in mathematics, free space propagation of lasing modes for VCSELs can be use to analogously study diffraction in time effect which is one of the most relevant quantum transient phenomena in matter waves. Mode selection mechanism for stadium-shaped VCSELs is also experimentally studied. We then perform statistical analysis to quantitatively understand characteristics of experimental chaotic and scarred wave patterns. Finally,

(5)

(6)

Acknowledgement

(7)

Abstract (Chinese) i

Abstract ii

Acknowledgement iv

Contents v

List of Figures

viii

Chapter 1 Introduction...1

1.1 Optical Mechanical Analogy ...2

1.2 Quantum Chaos ...5

1.3 Wave Chaos ...9

1.4 Vertical-Cavity Surface-Emitting Lasers (VCSELs) ...10

1.5 Scope of the Thesis ...15

Reference...17

Chapter 2 Analysis of Energy Level and Wave Functions in Quantum Billiards 26 2.1 Eigenstates in Integrable Quantum Billiards ...27

2.1.1 Square Billiards...28

2.1.2 Equilateral-Triangular Billiards...30

2.2 Superscarred Wave States in Integrable Quantum Billiards...33

2.2.1 Superscarred Wave States in Square Billiards...33

2.2.2 Superscarred Wave States in Equilateral Triangular Billiards ....36

2.3 Wave Function in Chaotic Billiards ...39

2.3.1 Eigenfunctions in Chaotic Quantum Billiards ...39

2.3.2 Chaotic Wave Patterns ...47

2.3.3 Scarred Wave Patterns ...48

2.4 Statistical Analysis of Wave functions ...50

2.4.1 Statistics of Chaotic Wave Patterns...50

2.4.2 Statistics of Eigenfunction in 2D Square-Shaped Billiards ...53

2.4.3 Statistics of Superscarred Wave Patterns ...55

(8)

2.5.2 Energy Spectra and Gutzwiller Trace Formula ...68

Reference...73

Chapter 3 Analogous Experiments for Quantum Billiard Energy Level from Amplified Spontaneous Emission Spectra of VCSELs...80

3.1 Amplified Spontaneous Emission (ASE)...83

3.2 Experimental setup ...83

3.3 Spontaneous Emission Spectra in Square-Shaped VCSELs...85

3.3.2 Extracting POs from Spontaneous Emission Spectra ...88

3.3.3 Analysis of Angle Resolved Spontaneous Emission Spectra ...91

3.3.4 Analysis of Calculated Spontaneous Emission Spectra ...94

3.4 Spontaneous Emission Spectra in Equilateral Triangular VCSELs and Stadium-Shaped VCSELs ...96

3.4.1 Energy Level Statistics of Spontaneous Emission Spectra ...96

3.4.2 Extracting POs from Spontaneous Emission Spectra ...99

Reference...102

Chapter 4 Analogous Experiments for Quantum Billiard Wave Functions from Lasing Modes of VCSELs ...105

4.1 Determined Factor for Transverse Order of Lasing Modes...108

4.2 Experimental Setup for Near-Field Pattern Measurement... 110

4.3 Typical Lasing Modes in Integrabe Shaped VCSEL ... 111

4.3.1 Square-Shaped VCSEL ... 111

4.3.2 Comparison of Typical Lasing Modes in Square-Shaped VCSELs with Different Aperture Size ... 115

4.3.3 Equilateral-Triangular VCSEL ... 119

4.4 Typical Lasing Modes in Chaotic VCSEL ...122

4.4.1 Ripple Square-Shaped VCSELs ...122

4.4.2 Stadium-Shaped VCSELs ...127

4.5 Mode Selection Mechanism in Stadium-Shaped VCSELs ...131

4.6 Statistical Analysis of Wave Patterns ...138

4.7 Observation on Momentum-Space Distribution in Quantum billiards from VCSELs ...141

4.7.1 Experimental Setup ...141

4.7.2 Far-Field of Low Order Transverse Modes ...142

4.7.3 Far-Field of High Order Transverse Modes ...149 4.8 Analogous Investigation on Transient Dynamics of Released Coherent

(9)

4.8.2 Transient Dynamics of One Dimensional Infinite Potential Well

...159

4.8.3 Transient Dynamics of Released Coherent Waves from...162

Reference...176

Chapter 5 Influence of Lateral Confinement Shape on Polarization Dynamics of VCSELs...182

5.1 Experimental Setup ...187

5.2 Relation between L-I Curves and its Time Domain Dynamics...189

5.3 L-I Curves of Square-Shaped VCSELs with Different Aperture Size ..194

5.4 Comparison of L-I Curves between Square-Shaped and Stadium-Shaped VCSELs...198

5.5 Polarization Switching for very High-Order Single Modes ...201

Reference...205

Chapter 6 Summary and Future Works ...208

6.1 Summary...209

6.2 Future Works...212

Reference...213

Appendix A ...214

A.1 Employing Eigenfunctions of a Rectangular Billiard as Basis...215

A.2 Employing Sinc Functions as Basis Functions ...216

Appendix B ...221

Appendix C ...224

Reference for Appendix ...226

Curriculum Vitae ...227

(10)

List of Figures

Chapter 1

Fig. 1.1 Some members of stable, nonisolated POs in square billiard.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

8

Fig. 1.2 Two trajectories in stadium billiard: One formed a PO with diamond shape

(red line), and the other is the trajectory which slightly shifts from the diamond-shaped PO (blue line).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

8

Fig. 1.3 Schematic diagram for Vertical-Cavity Surface-Emitting Lasers device

structure.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

14

Chapter 2

Fig. 2.1 Some intensity patterns of eigenstates in a square billiard.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅

29

Fig. 2.2 Some intensity patterns of eigenstates corresponding to

ψ

_{m n}( )1_,

( )

x y, and

( )2

_{( )}

, , m n x y

ψ

, respectively.

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

32

Fig. 2.3 Some intensity patterns of high order eigenstates with morphology of

honeycomb corresponding to

ψ

m n( )1,

( )

x y, and

( )2

_{( )}

, , m n x y

ψ

, respectively.

⋅

32

Fig. 2.4 The intensity patterns of non-degenerate eigenstates ( )

( )

2 2

2 ,n n x y,

ψ

.

⋅⋅⋅⋅⋅⋅

32

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 in square billiards.

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

35

Fig. 2.6 Some standing wave representations of superscar modes corresponding to

POs with different ( , )p q .

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅

35

Fig. 2.7 Some SU(2) coherent states with traveling and standing wave

(11)

with ( , )p q =

( )

1,1 in an equilateral-triangular billiard.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

38

Fig. 2.8 Some standing wave representations of superscarred wave states

corresponding to POs with different ( , )p q .

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

38

Fig. 2.9 Intensity patterns of the first nine calculated eigenstates in Sinai billiard. 42

Fig. 2.10 Intensity patterns of the first nine calculated eigenstates in stadium-shaped

billiard.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

42

Fig. 2.11 Scarred wave patterns of the 534th, 581st, 586th, and 602nd eigenstates in

Sinai billiard.

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

43

Fig. 2.12 Scarred wave patterns of the 78th, 113th, 125th, and 126th eigenstates in

stadium-shaped billiard.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

43

Fig. 2.13 Chaotic wave patterns of the 192nd, 277th, 369th, and 434th eigenstates in

nonconcentric Sinai billiard.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

44

Fig. 2.14 Chaotic wave patterns of the 166th, 191st, 228th, and 233rd eigenstates in

stadium billiard with symmetry breaking.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

44

Fig. 2.15 The schematic diagram of the billiard with arbitrary shape.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅

45

Fig. 2.16 Intensity patterns of the first nine calculated eigenstates in the billiard with

arbitrary shape as shown in Fig. 2.15.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

45

Fig. 2.17 Chaotic wave patterns of the (a) 112th, (b) 167th, (c) 184th, (d) 237th, (e)

256th, and (f) 311th eigenstates in stadium billiard with symmetry breaking.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

46

Fig. 2.18 Chaotic wave patterns calculated in terms of Eq. (2.15) with the restriction

of 482 <m2 +n2 ≤502.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

47

Fig. 2.19 Calculated scarred wave patterns which correspond to the (a) 271st, (b)

272nd, (c) 332nd, (d) 425th, (e) 665th, and (f) 868th eigenstates for a stadium billiard.

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

49

Fig. 2.20 Amplitude distribution and intensity distribution of the calculated chaotic

wave patterns shown in Fig. 2.18.

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

52

Fig. 2.21 Calculated eigenfunctions by using Eq. (2.3) and the corresponding

amplitude statistics and intensity statistics. The quantum number

( )

m n,

(12)

Fig. 2.22 Plots of superscarred wave states via Eq. (2.9) and the corresponding amplitude statistics and intensity statistics. Parameters

(

p q N, , ,

φ

)

is fixed to be

(

1 1 40, , ,

π

2

)

and M are selected to be 5, 9, 13, and 17.

⋅⋅⋅⋅⋅⋅⋅⋅

56

Fig. 2.23 Plots of Eq. (2.21) and Eq. (2.22) with different parameters as well as

Porter-Thomas distribution.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

59

Fig. 2.24 Amplitude and intensity statistics of some distinct scarred wave patterns and

fitting curves by using Eqs. (2.21)–(2.23).

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

60

Fig. 2.25 Statistics of energy level for the first 10000 eigenvalues in (a) square

quantum billiard and (b) equilateral-triangular quantum billiard.

⋅⋅⋅⋅⋅⋅⋅⋅⋅

64

Fig. 2.26 Statistics of energy level for the first 10000 eigenvalues in rectangular

quantum billiards. The side length ratios of rectangles are chosen to be irrational ratio of a

b =

π

and rational ratio of ab =2, 3 , and 5 .

⋅ ⋅

64

Fig. 2.27 Statistics of energy level for the first 600 eigenvalues in a stadium quantum

billiard.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

65

Fig. 2.28 Statistics of energy spectra for the first 600 eigenvalues in stadium billiards

with different degree of symmetry breaking: (a)

3

π

θ

= , (b) 4

π

θ

= , (c) 6

π

θ

= , and (d) θ =0.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

66

Fig. 2.29 Statistics of energy level for the first 600 eigenvalues in stadium billiard

with a small obstacle.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

67

Fig. 2.30 Statistics of energy level for the first 600 eigenvalues in the billiard with

arbitrary shape shown in Fig. 2.15.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

67

Fig. 2.31 Numerical calculated length spectrum for square quantum billiard model. 71

Fig. 2.32 Numerical calculated length spectrum for equilateral-triangular quantum

billiard model.

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

71

Fig. 2.33 Numerical calculated length spectrum for stadium quantum billiard model.

(13)

Chapter 3

Fig. 3.1 Schematic diagram of the experimental setup for measuring spontaneous

emission spectra of VCSELs.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

84

Fig. 3.2 Schematic diagram of the square-shaped VCSELs device structure.

⋅⋅⋅⋅

85

Fig. 3.3 Emission spectra measured for several injection currents. (a)-(d): below

lasing threshold; (e) just above lasing threshold.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

87

Fig. 3.4 Fourier transformed spectra

ρ

~_SE(L), corresponding to the spontaneous

emission spectra shown in Figs. 3.3(a)–3.3(e), respectively.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅

90

Fig. 3.5 Schematic of the experimental setup for measuring angle resolved

spontaneous emission spectra.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

92

Fig. 3.6 The subthreshold emission spectra at angles of 0o, 45o, and 90o under

threshold condition and the corresponding Fourier-transformed spectra

( )

SE L

ρɶ _.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

₉₃

Fig. 3.7 (a)–(d) Calculated energy spectra g kɶ

( )

and the corresponding

Fourier-transformed spectra gɶ_FT( )L ((a’)–(d’)) with parameters of 0.03

γ = , γ =0.01, γ =0.007, and γ =0.004.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

95

Fig. 3.8 Schematic of the equilateral-triangular and stadium-shaped VCSELs device

structure.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

97

Fig. 3.9 The meas ured s ubthreshold emission sp ectrum

ρ

_SE

( )

K in (a)

equilateral-triangular and (b) stadium-shaped VCSELs.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

98

Fig. 3.10 The experimental statistics for the nearest-neighbor eigenvalue spacing

distribution p s

( )

in the form of a histogram for (a) equilateral-triangular and (b) stadium-shaped device. The curves represent (a) Poisson distribution and (b) Wigner distribution.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

98

Fig. 3.11 Fourier transformed spectrum

ρ

ɶ_SE

( )

L 2. The experimental and numerical

results are displayed as mirror images.

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

101

(14)

results are displayed as mirror images.

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

101

Chapter 4

Fig. 4.1 Temperature dependence of emission wavelength (red star) and longitudinal

wavelength (black star) for a stadium-shaped VCSEL with area of

2

60 30 m×

µ

.

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

109

Fig. 4.2 Schematic view of experimental setup for near-field pattern measurement.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

110

Fig. 4.3 Intensity patterns of transverse near-field patterns at temperatures of (a)

300K (room temperature), (b) 280 K, (c) 250 K, (d) 230 K, and (e) 210 K.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

113

Fig. 4.4 The calculated patterns by using representation of quantum coherent states

in square billiards corresponding to Figures 4.3(b)–4.3(e), respectively.

⋅

114

Fig. 4.5 Schematic diagram of device structure, temperature dependence of

threshold current and experimental near-field patterns for square-shaped VCSELs with area of 20 20 m×

µ

2.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

116

Fig. 4.6 Schematic diagram of device structure, temperature dependence of

threshold current and experimental near-field patterns for square-shaped

VCSELs with area of 2

30 30 m×

µ

.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

117

Fig. 4.7 Schematic diagram of device structure, temperature dependence of

threshold current and experimental near-field patterns for square-shaped VCSELs with area of 40 40 m×

µ

2.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

118

Fig. 4.8 Intensity patterns of transerse near-field patterns at temperatures of (a) 300

(room temperature), (b) 260, (c) 240, (d) 210, and (e) 150.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅

120

Fig. 4.9 (a) Numerical wave pattern S_32,7+

(

x y, ;1, 0, 0.22

π

)

2 corresponds to the

experimental superscar mode which impings on lateral sides vertically. (b) Numerical wave pattern

ψ

_6,60( )1

( )

x y, 2 corresponds to the experimental

(15)

honeycomb pattern. (c) Numerical wave pattern S_23,6+

(

x y, ;1,1, 0.35

π

)

2 corresponds to the experimental superscar mode with

(

p q,

) ( )

= 1, 1 .

⋅ ⋅⋅

121

Fig. 4.10 Schematic diagram of ripple square shape

.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

123

Fig. 4.11 Plots of top view of device structure, threshold current and the typical

near-field patterns in the operating temperature range of 200 K to 300 K for ripple square-shaped VCSEL (device 1).

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

124

Fig. 4.12 Plots of top view of device structure, threshold current and the typical

near-field patterns in the operating temperature range of 200 K to 300 K for ripple square-shaped VCSEL (device 2).

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

125

Fig. 4.13 Polarization resolved near-field patterns and emission spectra shown in Fig.

4.11(e).

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅

126

Fig. 4.14 Schematic structure, temperature dependence of threshold current and

near-field patterns for a stadium-shaped VCSEL under threshold operation.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

129

Fig. 4.15 Transverse near-field patterns with different chaotic waves for

stadium-shaped VCSELs.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

130

Fig. 4.16 Transverse near-field patterns with different scarred morphology for

stadium-shaped VCSELs.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

130

Fig. 4.17 Calculated normalized carrier density in the active region of circular

VCSEL with small aperture size (solid line) and large aperture size (dash line).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

133

Fig. 4.18 Simulated carrier density distribution in stadium-shaped VCSELs with

different aperture size.

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

133

Fig. 4.19 Lasing modes at different operating temperature in the vicinity of the

threshold for the stadium-shaped VCSEL with the aperture size of

2

40 20 m×

µ

(device 1).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

134

2

40 20 m×

µ

(device 2).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

135

(16)

2

60 30 m×

µ

(device 1).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

136

2

60 30 m×

µ

(device 2).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

137

Fig. 4.23 Amplitude and intensity statistics for the experimental chaotic modes (red

stepped lines) shown in Figs. 4.15(a)–4.15(d), respectively.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

139

Fig. 4.24 Experimental scarred modes ((a)–(c)) and the corresponding amplitude and

intensity statistics ((a’)–(c’)).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

140

Fig. 4.25 Schematic view of the experimental setup for far-field pattern measurement.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

142

Fig. 4.26 (a) The near-field pattern and (b) the corresponding far-field pattern of the

superscar mode that is related to classical periodic orbits with ( , )p q =(1,1).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

145

Fig. 4.27 (a) Theoretical calculated coherent state S_22,5,1,1,0.55₊π( , )x y 2 and (b) the

corresponding momentum space.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

145

Fig. 4.28 Experimental near-field morphologies: (a) superscar with stick structure (b)

honeycomb eigenmode, (c) chaotic mode, and (d) superscarred mode. The far-field patterns (a’), (b’), (c’), and (d’) correspond to (a), (b), (c), and (d), respectively.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

146

Fig. 4.29 (a) Numerical wave pattern S_32,7+

(

x y, ;1, 0, 0.22

π

)

2 corresponds to the

experimental superscar with stick structure shown in Fig. 4.28(a). (b) Numerical wave pattern ( )

( )

2 1

6,60 x y,

ψ

corresponding to the experimental

honeycomb pattern shown in Fig. 4.28(b). (c) Reconstructed wave pattern corresponding to the experimental patterns shown in Fig. 4.28(c). (d) Numerical wave pattern S22,6

(

x y, ;1,1, 0.3

π

)

2

+

corresponding to the experimental patterns shown in Fig. 4.28(d). (a’)–(d’): Momentum-space distributions correspond to (a)–(d).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

146

(17)

corresponding far-field pattern.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

147

Fig. 4.31 The near-field chaotic pattern in stadium-shaped VCSEL and the

corresponding far-field pattern.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

147

Fig. 4.32 Numerically calculated momentum-space wave functions of (a)-(c) chaotic

states in nonconcentric Sinai billiards and (d) scarred waves in Sinai billiards.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

148

Fig. 4.33 Numerically calculated momentum-space wave functions of chaotic states

in stadium billiards with a small obstacle.

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

148

Fig. 4.34 (a)–(c) Calculated results for the near-field patterns S_{N M}p q, ,_, φ_,₊( , )x y 2 with

different orders of N =20, 30, and 40 and (a’)–(c’) the corresponding far-field patterns.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

152

Fig. 4.35 Experimental patterns of square shape VCSEL with high transverse order

(a): near field pattern (b): far-field pattern.

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

152

Fig. 4.36 Corrected high transverse order far field pattern corresponding to Fig.

4.35(b).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

153

Fig. 4.37 The sketch of the shutter problem.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

158

Fig. 4.38 The plot of the current density (red line) and classical particles in shutter

problem (blue dash line) at different time t .

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

158

Fig. 4.39 Schematic diagrams for transient dynamics of one dimensional (1D) infinite

potential well.

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

161

Fig. 4.40 Time evolution of particles with quantized wave vectors of n=1, n=2,

6

n= , and n=10 after suddenly removed confined potential well.

⋅⋅⋅⋅

162

Fig. 4.41 The numerical patterns for the wave states ( )

(

)

2 1,1,0.6 35,14 , , S + π x y t at (a) t =0, (b) t =0 06. , (c) T t =0 12. , (d) T t =0 18. , (e) T t =0 7. , and (f) T 10 t = .T

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

168

Fig. 4.42 The experimental transverse patterns of a square-shaped VCSEL for the

free-space propagation at the positions of (a)z z_d =0, (b)z z_d =0 06. , (c) 0 12.

d

z z = , (d)z z_d =0 18. , (e)z z_d =0 7. , and (f) the far-field regime, respectively.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

168

(18)

free-space propagation at the positions of (a) z z_d =0, (b) z z_d =0 025. , (c) z z_d =0 04. , (d) z z_d =0 08. , (e) z z_d =0 1. , (f) z z_d =0 14. , (g)

0 18.

d

z z = , (h) z z_d =0 3. , (i) z z_d =0 4. , (j) z z_d =0 6. , (k) z z_d =1 5. , and (f) the far-field regime, respectively.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

169

free-space propagation at the positions of (a) z z_d =0, (b) z z_d =0 025. , (c) z zd =0 04. , (d) z zd =0 08. , (e) z zd =0 1. , (f) z zd =0 14. , (g)

0 18.

d

z z = , (h) z z_d =0 3. , (i) z z_d =0 4. , (j) z z_d =0 6. , (k) z z_d =1 5. , and (f) the far-field regime, respectively.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

170

Fig. 4.45 The numerical patterns for the wave states Ψ₁

(

x y t, ,

)

2 at (a) t =0, (b)

0 025.

t = , (c) T t =0 04. , (d) T t=0 08. , (e) T t=0 1. , (f) T t=0 14. , T

(g) t =0 18. , (h) T t=0 3. , (i) T t =0 4. , (j) T t=0 6. , (k) T t =1 5. , T

and (f) t=10 , respectively.T

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

171

(

x y t, ,

)

2 at (a) t =0, (b)

0 025.

t = , (c) T t =0 04. , (d) T t=0 08. , (e) T t=0 1. , (f) T t=0 14. , T

(g) t =0 18. , (h) T t=0 3. , (i) T t =0 4. , (j) T t=0 6. , (k) T t =1 5. , T

and (f) t=10 , respectively.T

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

172

free-space propagation at the positions of (a) z z_d =0, (b) z z_d =0 02. , (c) 0 04. d z z = , (d) z z_d =0 068. , (e) z z_d =0 09. , (f) z z_d =0 15. , (g) 0 24. d z z = , (h) z zd =0 3. , (i) z zd =0 4. , (j) z zd =0 6. , (k) z zd =1,

and (f) the far-field regime, respectively.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

173

free-space propagation at the positions of (a) z z_d =0, (b) z z_d =0 01. , (c) 0 03. d z z = , (d) z zd =0 07. , (e) z zd =0 085. , (f) z zd =0 12. , (g) 0 14. d z z = , ( h ) z z_d =0 25. , ( i ) z z_d =0 3. , ( j ) z z_d =0 45. , ( k ) 0 55. d

(19)

(

x y t, ,

)

2 at (a) t =0, (b) 0 02.

t = , (c) T t =0 04. , (d) T t =0 068. , (e) T t=0 09. , (f) T t=0 15. , T

(g) t =0 24. , (h) T t=0 3. , (i) T t =0 4. , (j) T t=0 6. , (k) T t =1 , and T

(f) t =10 , respectively.T

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅

175

(

x y t, ,

)

2 at (a) t =0, (b)

0 01. t = , (c) T t=0 03. , (d) T t=0 07. , (e) T t =0 085. , (f) T t=0 12. , T ( g ) t=0 14. , ( h ) T t =0 25. , ( i ) T t=0 3. , ( j ) T t =0 45. , ( k ) T 0 55. t = , and (f) T t =10 , respectively.T

⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

176

Chapter 5

Fig. 5.1 Schematic diagram for (a) optical mode density of cavity in two orthogonal

polarized states and (b) selection of linearly polarized lasing mode.

⋅⋅⋅⋅⋅

186

Fig. 5.2 Relation between gain profile and two orthogonal polarized states with

varied operating condition.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

187

Fig. 5.3 Schematic view of experimental setup.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

189

Fig. 5.4 (a) Near-field patterns near threshold operation in both polarized states and

the L-I curve at the operating temperature of 298 K for (b) global view and (c) zoon-in view.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

192

Fig. 5.5 The near-field patterns and the corresponding emission spectrum in two

polarized states around switching current in L-I curve.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

193

Fig. 5.6 Intensity time trace of two polarized states at switching points of A – F in

L-I curve shown in Fig. 5.4(b).

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

194

Fig. 5.7 Temperature dependence of threshold current, near-field patterns near

threshold operation, and polarization resolved L-I curves for square-shaped VCSELs with area of 20 20×

µ

m2.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

197

Fig. 5.8 Temperature dependence of threshold current, near-field patterns near

threshold operation, and polarization resolved L-I curves for square-shaped VCSELs with area of 40 40×

µ

m2.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

198

(20)

Fig. 5.9 Temperature dependence of threshold current, near-field patterns near threshold operation, and polarization resolved L- I curves for

stadium-shaped VCSELs with area of 2

60 30 m×

µ

.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅

201

Fig. 5.10 Polarization resolved wave patterns and the corresponding wavelength

spectrum at the injected current of 1.14 Ith, 1.20 Ith, 1.26 Ith, and 1.32 Ith.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

204

Fig. 5.11 Intensity time trace of the two orthogonal polarized states at the switching

point A and B.

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

205

Appendix

Fig. A.1 Schematic diagram of a two dimensional quantum billiard.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

220

(21)

Chapter 1

(22)

1.1 Optical Mechanical Analogy

In 17th century, it was believed that the dynamics of particles and the propagation of wave were two entirely different physical phenomena. In 19th century, Plank and Einstein found that light has quantized energy of the value hν . The quantized light is described by photon which is one type of elementary particle. The observation inspired de Broglie to propose the famous wave-particle duality theory of matter. Based on de Broglie's suggestion, Schrödinger successfully established the wave equation of matter wave. Over the past several decades, many important physical phenomena in microscopic systems have been well explained by Schrödinger equation.

When the concept of matter wave was proposed, one may intuitively ask a question: Are the characteristics of matter waves similar to optical waves? In the early 19th century, though the notion of matter wave was unknown, Hamilton first realized that there exists highly analogy between motion of classical particles and waves in geometric optics [1, 2]. The statement can easily be seen from the Schrödinger equation and electromagnetic wave equation under semiclassical limit. In homogeneous isotropic medium, the scalar electromagnetic wave equation is written as

(

)

2 2 2 2 1 , , ; 0 A x y z t u t  _∂  ∇ − =   ∂   , (1.1) where u is the velocity of wave. Assume the representation of wave

(

, , ;

)

(

, , ;

)

exp 0

(

, , ;

)

A x y z t = Φ x y z t _ik Θ x y z t _{ in Eq. (1.1), we obtain}

( )

2 2 2 0 0 2 2 2 2 0 0 0 2 2 2 2 1 2 ik k ik ik k u t t t t t ∇ Φ + ∇Φ ⋅∇Θ − Φ ∇Θ _{∂ Φ} _{∂Φ ∂Θ} _{∂ Θ} _∂Θ      =_ _ + + Φ − Φ_ _  ∂ ∂ ∂ ∂ ∂  _   _ . (1.2)

(23)

and hence we have

( )

2 2 2 1 u t ∂Θ   ∇Θ = _ _ ∂   . (1.3)

For monochromatic wave with angular frequency

ω

, the phase term of representation of wave is Θ

(

x y z t, , ;

) (

=

φ

x y z, ,

) (

−

ω

k t₀

)

. Substitute Θ

(

x y z t, , ;

)

into Eq. (1.3) and we can obtain

( )

2 2 n

φ

∇ = , (1.4) where n is the refractive index of medium. Eqation (1.4) is the basic equation of geometrical optics in homogeneous isotropic medium which is so-called eikonal equation.

For matter waves, the time-dependent Schrödinger equation is given by

( )

2 2 2m V r

ψ

i t

ψ

  _∂ − ∇ + =   _∂   ℏ ℏ . (1.5)

We write wave function as

ψ

=

ρ

exp

[

iS ℏ and substitute it into Eq. (1.5).

]

Straightforward differentiations lead to

( )

2 2 2 2 2 2 1 2 i i S S S V m i S i t t ρ ρ ρ ρ ρ ρ _ρ           − ∇ +  ∇ ⋅∇ −  ∇ +  ∇ +           _∂ _{ } _∂  =  +_{ }  ∂   ∂     ℏ ℏ ℏ ℏ ℏ ℏ . (1.6)

In the classical limit (ℏ→0 limit), we then obtain

2 1 0 2 S S V m t ∂ ∇ + + = ∂ . (1.7)

Equation (1.7) is Hamilton-Jacobi equation in classical mechanics. For stationary state, we can write S x y z t

(

, , ;

)

=S₀

(

x y z, ,

)

−Et and Eq. (1.7) can be rewrite as 2 0 2 1 1 S

λ

 _∇  ₌   ℏ  , (1.8)

whereλ is the wavelength of matter wave which is given by ℏ p=ℏ 2m E V

(

−

)

. Obviously, it can be seen that Eq. (1.8) is completely analogous to Eq. (1.4). In other words, the transition from quantum mechanics to classical mechanics is equivalent to the relation between wave optics and geometric optics.

(24)

The equivalence of Hamilton-Jacobi equation and eikonal equation tells us that the trajectory of classical particle is similar to a ray in geometric optics. On the other hand, in microscopic system, several experiments confirmed that matter waves can also refract, diffract, interfere, and scatter in the same manner as electromagnetic waves. These observation implies us that matter waves is highly similar to optical waves such that we can use optical waves to analogously explore many quantum laws or phenomena.

(25)

1.2 Quantum Chaos

Scale of systems is crucial for determining whether matter behave like particles or waves. Over the past several decades, it has been confirmed that the performance of electrical devices will be improved when the devices are fabricated to small size in which quantum effects become relevant. These devices are called mesoscopic devices since their scale are between microscopic and macroscopic systems. The world of mesoscopic systems consist of large number of particles with wave properties which can interfere and scatter with each other. Quantum chaos is one of the striking physical phenomena in mesoscopic systems.

After theory of quantum mechanics successfully described behaviors of particles in microscopic system, to further realize the correlation between quantum and classical mechanics is a meaningful task. In real worlds, most physical systems are nonlinear and their dynamical behaviors are chaotic. Quantum chaos is the field of physics that investigates the connection between the theory of quantum mechanics and the phenomena of classical chaos. Two dimensional (2D) billiard is a simple model for exploring quantum chaos.

2D Billiard systems contain a point particle which moves freely along straight trajectories inside the two dimensional region and reflects elastically on the boundary. The dynamical property of a billiard system strongly depends on its boundary shape. In the theory of classical mechanics, dynamical systems can be divided into integrable and nonintegrable type. Integrable systems have equal number of degrees of freedom and constant motion [3]. Billiards with highly symmetric boundary such as square, equilateral-triangle, and circle are integrable type. On the other hand, billiards with arbitrary shape or some perturbations are nonintegrable type.

In integrable billiard systems, the motions of classical particles are regular and the position of each trajectory may be precisely predicted for any long time [4]. In contrast, classical particles always exhibit chaotic dynamics in nonintegrable billiard systems and hence the motions of particles are unpredictable. Chaotic motions of particles sensitively depend on their initial condition. Lyapunov derived a formula that is named “Lyapunov exponent” to quantify the sensitive dependence on initial

(26)

condition for chaotic behavior [4]. The expression of “Lyapunov exponent” is given by ( )

( )

0 0 1 lim lim ln 0 t x x t t x

λ

→∞ ∆ →  _∆  =  _ _ ∆      , (1.9)

where ∆x t

( )

is the distance between two neighboring trajectories. Trajectories of particles in billiard system with positive Lyapunov exponent are called chaotic. The trajectory of particle is called periodic orbit (PO) when the particle can return to its initial position after some definite time. Periodic orbits can be classified into stable and unstable type. The definition of stable PO is that the distance between two POs departed from nearby initial points is always smaller than a constant number

ε

at any time. If these POs form a continuous family of orbits with same path length, we also call them nonsolated POs [5]. The stable, nonisolated POs always exist in the integrable billiard systems because of their high symmetry. Figure 1.1 shows members of stable, nonisolated POs in square billiard. On the other hand, two POs departed from nearby initial points diverge with time are called unstable. Unstable POs are not members of a continuous family of paths with the same length and thus unstable POs can often be called isolated POs. Figure 1.2 depicts two trajectories in stadium billiard. One formed a PO with diamond shape, and the other is the trajectory which slightly shifts from the diamond-shaped PO. Obviously, when the trajectory slightly shifts from the diamond-shaped PO, it can’t also form a PO with the same length of diamond shape. After a long time, two trajectories must be quite different. This is the apparent characteristic of motion in classical chaotic billiards.

After knowing the physical pictures of particles in classical billiard, we then briefly introduce quantum billiard systems. Quantum-billiard systems have been widely used to explore many striking mesoscopic phenomena such as electron transport phenomena [6], conductance fluctuations in quantum dots [7, 8], and diffraction in time effect [9]. The model of quantum billiard is that a particle with wave natural in two dimensional infinite potential well

( )

, 0, ,

( )

_{( )}

, , if x y V x y if x y ∈Ω  = ∞ ∉Ω  ,

(27)

2D quantum billiard is 2D time-independent Schrödinger equation which is given by

( ) ( )

( )

2 2 , , , 2M V x y

ψ

x y E

ψ

x y   − ∇ + =     ℏ , (1.10) where M is the mass of the particle. For the particle inside the quantum billiard, Eq. (1.10) can be rewritten as

(

2 2

)

( )

, 0

k ψ x y

∇ + = , (1.11) with k= 2ME/ℏ2 . Unlike to the classical billiard, there is no concept of trajectories in the world of quantum. We can’t use Lyapunov exponent to describe the dynamics of quantum billiards. Other physical quantities should be utilized to realize features of systems. In the field of quantum chaos, physicists focus on investigating properties of wave functions and analyzing the energy spectra in quantum billiards.

(28)

Fig. 1.1 Some members of stable, nonisolated POs in square billiard.

Fig. 1.2 Two trajectories in stadium billiard: One formed a PO with diamond shape

(red line), and the other is the trajectory which slightly shifts from the diamond-shaped PO (blue line).

(29)

1.3 Wave Chaos

The wave natural of particles convinces us that waves exist everywhere in the universe from macroscopic to microscopic scale. Investigation of quantum chaos from billiard systems not only enables us to study the connection between the theory of quantum and classical mechanics, but also provides an avenue to understand wave interaction phenomena in mesoscopic physics. Interference effect of matter waves will emerge when the size of the systems are smaller than the phase coherence length of particles. Semiconductor nanostructures are of this order which can be regarded as billiard systems for investigating quantum chaos. In the past two decades, several dynamics of quantum billiards have been experimentally observed from quantum dots [10-12] and quantum wells [13-15].

Nevertheless, it is usually hard to directly investigate features of quantum billiards from experiments owing to the difficulty of measurements in microscopic scale. The analogy between different physical systems is a new side of the coin in understanding these fundamental physical concepts. Characteristics of matter waves are similar to other waves such as surface waves and electromagnetic waves. It is confirmed that quantum phenomena are often possible to be described by the same mathematical equation as some other macroscopic systems. As mentioned in section 1.3, the mathematical model of 2D quantum billiard is time-independent Schrödinger equation which has the same form as 2D Helmholtz equation with Dirichlet boundary condition in optics. Accordingly, a vast of experiments such as microwave cavities [16-20], microdisk lasers [21, 22], and optical fibers [23, 24] have been used to analogously investigate quantum chaos. These researches are new filed that is termed “wave chaos”. In this thesis, we exploited broad-area oxide-confined VCSELs to study wave chaos.

(30)

1.4 Vertical-Cavity Surface-Emitting Lasers (VCSELs)

Semiconductor lasers played an important role for the applications in optical engineering for a long time. Edge emitting laser (EELs) and Vertical-cavity surface-emitting lasers (VCSELs) are two famous types of semiconductor lasers. Compared to the conventional EELs, VCSELs have advantages of single longitudinal mode emission, low-threshold operation [25], high efficiency [26, 27], low manufacturing costs, and high coupling efficiency for communication systems. VCSELs were invented by K. Iga and coworkers in the late 1970s [28, 29]. They proposed the first InGaAsP/InP VCSEL that was electrically operated by current pulses at the temperature of 77 K. InGaAsP/InP-based VCSEL has the problem of large nonradiative recombination in materials which can limit the objective of obtaining low threshold current. Instead, GaAs have distinctive properties of optical materials and are possible to grow structures of high-reflectivity distributed Bragg reflectors (DBRs), and hence most researchers focused on developing low-threshold and high-power GaAs-based VCSELs [30, 31]. During the past two decades, the widespread use of DBRs in VCSELs enabled us to achieve relatively low threshold current [32, 33]. It is proved that utilizing quantum well (QW) as gain medium provides a vital way to obtain optimal threshold current density which is significant for developing continuous-wave (CW) operation at room temperature [34].

Another crucial factor for determining the performance of VCSELs is optical and electrical confinement in active layer. In general, optical field can be confined inside active layer of VCSELs in terms of gain guiding and index guiding. Gain-guided VCSELs have drawbacks of poor transverse confinement of optical field and electrical current, leading to high threshold current. Such problem can be improved by using ion implantation into DBRs to definitely control the flow of injected current [35]. However, since the diffusion of carriers along transverse direction is hard to control, high optical absorption loss could be a serious problem for gain-guided VCSELs with ion-implanted structure. Additionally, large resistance resulted from ion implantation into DBRs can generate enormous heat inside the cavity, which strongly influences characteristics of VCSELs. In the early 1990s, the development of

(31)

index-guided VCSELs drastically ameliorated disadvantages of ion-implanted gain-guided VCSELs [36, 37]. For index-guided VCSELs, using oxide aperture to confine optical field and carriers is the most popular fabrication technique due to low production costs. Besides, oxide-confined VCSELs have extremely good transverse confinement, very low threshold current and high wallplug efficiency, which are attractive features for manufacturing commercialized VCSELs with high reliability [38-41]. My works is to utilize broad-area oxide-confined VCSELs to investigate their basic properties as well as applications in physics research. Figure 1.3 depicts schematic diagram of oxide-confined VCSELs device structure.

VCSELs have been identified as a promising light source for a variety of practical applications in short distance communication, data transmission, high-density optical storage, and optical sensors and sensors [42-50]. However, VCSELs have the major drawback of polarization instability due to anisotropy in semiconductor materials. The dominant polarized state can switch to another orthogonal polarization when the operation condition is varied. In recent years, there has been intensive research on polarization dynamics of VCSEL [51-59]. Another popular research on VCSELs is to study the emission properties controlled by optical feedback. It is found that optical feedback has apparent impact on threshold condition, pattern morphology, emission spectrum, laser output dynamics, and polarization dynamics [60-80]. In addition, S. P. Hegarty reported that waves inside VCSEL cavity would undergo total internal reflection at the current-guiding oxide boundary due to large index discontinuities between the oxide layer and the active material. For this reason, VCSEL cavity can be considered as a planar waveguide with a dominant wave vector along the vertical direction [81].

T h e e l e c t r i c f i e l d o f o p t i c a l b e a m s o b e y s t h e w a v e e q u a t i o n

(

₂ 2

)

(

)

2 0 E x y z t, , , 0 t µ ε∂

∇ − _∂ = . According to the waveguide theory [82], the electric

fields with a predominantly z direction of propagation can be approximated as

(

_{, , ,}

)

( )

_, i k z( z t)

E x y z t =E x y e −ω , where k is the wave vector along z-direction and _z

ω

is the angular frequency [82]. Although VCSELs have highly symmetric structure, the existence of anisotropy can break the degeneracy of transverse modes in two orthogonal polarizations, resulting in a split in oscillation frequency. Therefore, total

(32)

electric field includes the two polarized states can be expressed as

(

)

( )

( , ) ˆ , , , , i kz xz xt x E x y z t =E x y e −ω i+

( )

( , ) ˆ , i kz yz yt y E x y e −ω , (1.12) j

where kz,α

(

α

=x y,

)

is the wave vector along z-direction and

ω α

α

(

=x y,

)

is the

angular frequency in

α

-polaried state. After separating the z component in the wave equation, we are left with a two-dimensional Helmholtz equation in the two polarized states

(

)

( )

(

)

( )

2 2 2 0 , 2 2 2 0 , , 0 , 0 t x z x x t y z y y k E x y k E x y

µ εω

_{∇ +} ₋  ₌    _{∇ +} ₋  ₌    , (1.13)

where ∇_t2 means the Laplacian operator operating on the transverse coordinates. Since the electric field in VCSELs experience total reflection on the lateral walls, transverse optical field at the boundary of cavity can be approximated as

( )

, x y, 0 E_α x y

∈ℑ= with

α

====x or y . Obviously, transverse electric fields of VCSELs

are thoroughly equivalent to eigenfunctions of 2D Schrödinger equation with hard wall boundaries of the same geometry. In other words, wave functions

ψ

( )

x y, in two-dimensional (2D) quantum billiard can be analogously investigated from transverse optical fields E_α

( )

x y, of VCSELs. This important theoretical prediction has been experimentally confirmed in our previous works [83]. Afterwards our group further exploited VCSELs to study properties of wave functions in 2D quantum billiards. Broad-area VCSELs with integrable cavity such as square and equilateral triangle have been systematically discussed [84-87]. However, the investigation on lasing modes in chaotic shaped VCSELs had not ever been explicated thoroughly. In this thesis, we concentrated on studying characteristics of typical lasing patterns in two famous chaotic shaped VCSELs: Ripple square shaped VCSELs and stadium-shaped VCSELs.

Under circumstance of paraxial approximation, the far-field pattern is the Fourier transform of the near-field pattern, which is equivalent to the relation between momentum-space and coordinate-space wavefunctions of quantum systems. That is, it is able to realize the information about momentum-space wavefunctions of 2D quantum billiards from far-field lasing modes of VCSELs. However, as the transverse

(33)

order of the lasing mode is so high that paraxial approximation is invalid, the far-field pattern may drastically influence by non-paraxial contributions. Therefore, it is essentially important to develop an appropriate correcting method for extracting the momentum-space wavefunctions from the experimental far-field patterns with the substantial non-paraxial contribution. In this thesis, we develop a reliable method to extract the momentum-space wavefunctions from the experimental far-field patterns with the substantial non-paraxial contribution.

Energy spectrum is another crucial quantity for understanding behaviors of quantum systems. According to Eq. (1.11) and Eq. (1.13), transverse wavenumber of VCSELs is able to analogously correspond to specific energy eigenvalue of 2D quantum billiard. However, since the mode-competition phenomena usually induce mode-hopping instability, VCSELs can only exhibit several tens of cavity modes in the lasing spectra. It is apparent that these lasing spectra can not reveal all information about energy spectra of quantum billiards. For this reason, so far, VCSEL devices have never been successfully employed to manifest the signatures of quantum energy spectra. In 1963, Nathan et al. experimentally observed that the GaAs injection laser presents apparent oscillations in spontaneous emission spectra and suggested that this modulation on spectra arises from the resonance modes in the cavity [88]. In addition, in the past decades, it has been verified that spontaneous emission spectra could be suppressed or enhanced due to the coupling between atoms and cavities [89]. Inspired by this concept of light-matter interaction between confining photons and optical cavity, in this thesis, we systematically investigated characteristics of spontaneous emission spectra for broad-area VCSELs, which is lack of discussions in the past.

In the past few decades, most research on polarization dynamics of VCSELs mainly focused on small-aperture devices in order to ameliorate the problem of polarization instability for the use of VCSELs. Broad-area VCSELs tend to generate high-order modes and cause complex spatial current distribution in active layer, which strongly affect polarization dynamics of lasers. These unexpected effects are difficult to analyze and thus the polarization behavior far above threshold condition for broad-area VCSELs is still an open issue. In this thesis, we concentrated on studying polarization behaviors of broad-area VCSELs with different aperture size and different cavity geometry.

(34)

p-type DBR

n-type DBR

Oxide layers

Active layer

Metal contact

p-type DBR

n-type DBR

Oxide layers

Active layer

Metal contact

Fig. 1.3 Schematic diagram for Vertical-Cavity Surface-Emitting Lasers device