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- Herausgeber: Wiley-VCH GmbH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Microcavity Semiconductor Lasers
Explore this thorough overview of integrable microcavity semiconductor lasers and their applications from two leading voices in the field
Attracting a great deal of attention over the last decades for their promising applications in photonic integration and optical interconnects, microcavity semiconductor lasers continue to develop via advances in fundamental physics, theoretical analysis, and numerical simulations. In a new work that will be of interest to researchers and practitioners alike, Microcavity Semiconductor Lasers: Principles, Design, and Applications delivers an application-oriented and highly relevant exploration of the theory, fabrication, and applications of these practical devices.
The book focuses on unidirectional emission microcavity lasers for photonic integrated circuits, including polygonal microresonators, microdisk, and microring lasers. After an introductory overview of optical microcavities for microlasers and detailed information of the lasers themselves, including mode structure control and characteristics, and lasing properties, the distinguished authors discuss fabrication and applications of different microcavity lasers. Prospects for future research and potential new applications round out the book.
Readers will also benefit from the inclusion of:
- A thorough introduction to multilayer optical waveguides, the FDTD Method, and Padé Approximation, and deformed, chaos, and unidirectional emission microdisk lasers
- An exploration of mode analysis for triangle and square microresonators similar as FP Cavity
- Practical discussions of mode analysis and control for deformed square microlasers
- An examination of hexagonal microcavity lasers and polygonal microcavities, along with vertical radiation loss for 3D microcavities
Perfect for laser specialists, semiconductor physicists, and solid-state physicists, Microcavity Semiconductor Lasers: Principles, Design, and Applications will also earn a place in the libraries of materials scientists and professionals working in the semiconductor and optical industries seeking a one-stop reference for integrable microcavity semiconductor lasers.
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Seitenzahl: 566
Veröffentlichungsjahr: 2021
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Table of Contents
Cover
Title Page
Copyright
Preface
1 Introduction
1.1 Whispering‐Gallery‐Mode Microcavities
1.2 Applications of Whispering‐Gallery‐Mode Microcavities
1.3 Ultra‐High Q Whispering‐Gallery‐Mode Microcavities
1.4 Mode Q Factors for Semiconductor Microlasers
1.5 Book Overview
References
2 Multilayer Dielectric Slab Waveguides
2.1 Introduction
2.2 TE and TM Modes in Slab Waveguides
2.3 Modes in Symmetric Three‐Layer Slab Waveguides
2.4 Eigenvalue Equations for Multilayer Slab Complex Waveguides
2.5 Eigenvalue Equations for One‐Dimensional Multilayer Waveguides
2.6 Mode Gain and Optical Confinement Factor
2.7 Numerical Results of Optical Confinement Factors
2.8 Effective Index Method
References
3 FDTD Method and Padé Approximation
3.1 Introduction
3.2 Basic Principle of FDTD Method
3.3 Padé Approximation for Time‐Domain Signal Processing
3.4 Examples of FDTD Technique and Padé Approximation
3.5 Summary
References
4 Deformed and Chaotic Microcavity Lasers
4.1 Introduction
4.2 Nondeformed Circular Microdisk Lasers
4.3 Deformed Microcavity Lasers with Discontinuous Boundary
4.4 Chaotic Microcavity Lasers with Smoothly Deformed Boundary
4.5 Summary
References
5 Unidirectional Emission Microdisk Lasers
5.1 Introduction
5.2 Mode Coupling in Waveguide‐Connected Microdisks
5.3 Waveguide‐Connected Unidirectional Emission Microdisk Lasers
5.4 Unidirectional Emission Microring Lasers
5.5 Unidirectional Emission Hybrid Deformed‐Microring Lasers
5.6 Wide‐Angle Emission and Multiport Microdisk Lasers
5.7 Summary
References
6 Equilateral‐Triangle‐Resonator Microlasers
6.1 Introduction
6.2 Mode Analysis Based on the ETR Symmetry
6.3 Mode‐Field Distributions
6.4 Far‐Field Emission and Waveguide‐Output Coupling
6.5 Mode Analysis Using Reflected Phase Shift of Plane Wave
6.6 Mode Characteristics of ETR Microlasers
6.7 Summary
References
7 Square Microcavity Lasers
7.1 Introduction
7.2 Analytical Solution of Confined Modes
7.3 Symmetry Analysis and Mode Coupling
7.4 Mode Analysis for High Q Modes
7.5 Waveguide‐Coupled Square Microcavities
7.6 Directional‐Emission Square Semiconductor Lasers
7.7 Dual‐Mode Lasing Square Lasers with a Tunable Interval
7.8 Application of Dual‐Mode Square Microlasers
7.9 Lasing Spectra Controlled by Output Waveguides
7.10 Circular‐Side Square Microcavity Lasers
7.11 Summary
References
8 Hexagonal Microcavity Lasers and Polygonal Microcavities
8.1 Introduction
8.2 Mode Characteristics of Regular Polygonal Microcavities
8.3 WGMS in Hexagonal Microcavities
8.4 Unidirectional Emission Hexagonal Microcavity Lasers
8.5 Octagonal Resonator Microlasers
8.6 Summary
References
9 Vertical Loss for 3D Microcavities
9.1 Introduction
9.2 Numerical Method for the Simulation of 3D Microcavities
9.3 Control of Vertical Radiation Loss for Circular Microcavities
9.4 Verical Radiation Loss for Polygonal Microcavities
9.5 Summary
References
10 Nonlinear Dynamics for Microcavity Lasers
10.1 Introduction
10.2 Rate Equation Model with Optical Injection
10.3 Dynamical States of Rate Equations with Optical Injection
10.4 Small Signal Analysis of Rate Equations
10.5 Experiments of Optical Injection Microdisk Lasers
10.6 Microwave Generation in Microlaser with Optical Injection
10.7 Integrated Twin‐Microlaser with Mutually Optical Injection
10.8 Discussion and Conclusion
References
11 Hybrid‐Cavity Lasers
11.1 Introduction
11.2 Reflectivity of a WGM Resonator
11.3 Mode Q‐Factor Enhancement for Hybrid Modes
11.4 Hybrid Mode‐Field Distributions
11.5 Fabrication of Hybrid Lasers
11.6 Q‐Factor Enhancement and Lasing Characteristics
11.7 Robust Single‐Mode Operation
11.8 Optical Bistability for HSRLS
11.9 All‐Optical Switching
11.10 All‐Optical Logic Gates
11.11 Hybrid Square/Rhombus‐Rectangular Lasers (HSRRLS)
11.12 Summary
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 The mode index,
G
/
g
, and optical confinement factor for TE
0
, TE
1
, an...
Chapter 6
Table 6.1 Character table of the point group
C
3
v
.
Chapter 7
Table 7.1 Character table of the point group
C
4
v
.
Table 7.2 The mode frequencies and
Q
‐factors obtained by FEM in a square micro...
Chapter 8
Table 8.1 Character table of the point group
C
Nv
for even
N
.
Table 8.2 Character table of the point group
C
Nv
for odd
N
.
Table 8.3 Mode frequencies and
Q
factors of TM
9,1
, TM
8,1
, TM
6,1
, and TM
8,1
mod...
Chapter 9
Table 9.1 Variation of effective refractive index
n
eff
with the cladding refra...
Table 9.2 Mode wavelengths and
Q
‐factors obtained by 3D FDTD simulation and th...
Table 9.3 Mode wavelengths of WGMs and cut‐off wavelength of radiation modes ...
Table 9.4 Mode
Q
‐factors and wavelengths for the equilateral‐triangular microc...
Table 9.5 Mode
Q
‐factors and wavelengths for the square microcavity with
a
= 2...
Chapter 10
Table 10.1 Parameters used in rate equations for an AlGaInAs/InP microlaser a...
List of Illustrations
Chapter 1
Figure 1.1 Schematic diagrams of (a) circular, (b) triangular, (c) square, a...
Figure 1.2 Scanned electron micrograph image of a microdisk connected with a...
Figure 1.3 Laser‐mode linewidth and output power vs. injection current for s...
Figure 1.4 (a) Output power coupled into a multiple‐mode fiber vs. the injec...
Chapter 2
Figure 2.1 A three‐layer slab waveguide with a thickness of
d
and refractive...
Figure 2.2 Wave functions propagating in layers
j −
1,
j
, and
j
...
Figure 2.3 Schematic for zigzag propagating mode light ray and the effective...
Figure 2.4 One‐dimensional refractive index distribution for a vertical‐cavi...
Figure 2.5 Normalized squared electric field at (a) center region and (b) ex...
Figure 2.6 Lasing spectrum of a 1550‐nm Fabry–Perot semiconductor laser with...
Figure 2.7 One‐dimensional model for external cavity semiconductor lasers.
Figure 2.8 Lasing spectrum of a DFB laser with a stop band.
Figure 2.9 Comparison of mode gain and standard optical confinement factors ...
Figure 2.10 The squared electric fields for the multiple quantum well laser....
Figure 2.11 (a) Optical confinement factor and (b) mode wavelength vs. the a...
Figure 2.12 A simplified three‐dimensional waveguide guide of a ridge wavegu...
Chapter 3
Figure 3.1 2D Yee's grid for TE mode.
Figure 3.2 2D Yee's grid for TM mode.
Figure 3.3 3D Yee's grids in Cartesian coordinate.
Figure 3.4 3D Yee's grids in cylindrical coordinate. The 3D system is simpli...
Figure 3.5 Schematic diagram of reflection and transmission of light at the ...
Figure 3.6 Spectra of a two‐oscillator signal containing noise (
A
= 0.01) ob...
Figure 3.7 Spectra obtained by Padé approximation at different decimation ra...
Figure 3.8 Schematic of a microring channel filter. The reference and detect...
Figure 3.9 Transmission spectra
T
calculated by DFT from 2
17
‐item (dashed li...
Figure 3.10 Transmission spectra
T
and drop spectra
D
calculated as the rati...
Figure 3.11 The embedded ring resonators of a racetrack and inside ring coup...
Figure 3.12 (a) Transmission and (b) time‐delay spectra of the through port ...
Figure 3.13 Transmission coefficients vs. time with the time from 0 to (a) 7...
Figure 3.14 Intensity spectra obtained by the Padé approximation from 8000‐,...
Chapter 4
Figure 4.1 (a) An illustration of a 2D circular microdisk, and (b) light ray...
Figure 4.2 Analytical magnetic field distributions of TE standing wave modes...
Figure 4.3 Variation of (a) wavelengths and (b)
Q
factors with the angular m...
Figure 4.4 (a) Schematic of the double‐disk structure of the InGaAs/InGaAsP ...
Figure 4.5 (a) Real‐space false color plot of the modulus of the electric fi...
Figure 4.6 (a and b)Simulated intensity distributions of nearly degenerate q...
Figure 4.7 (a) Top‐view schematic of a spiral‐ring microcavity. (b) Simulate...
Figure 4.8 (a) Internal‐field amplitude distributions normalized to the peak...
Figure 4.9 (a) Regular and chaotic ray trajectories in the circular microdis...
Figure 4.10 (a) Simulated electric field intensity distribution of the TM mo...
Figure 4.11 (a) Chaotic ray trajectory in the limaçon cavity with
n
= 3.3 an...
Figure 4.12 Calculated intensity distribution of an even‐parity TE mode (a) ...
Figure 4.13 (a) Comparison of the FFPs obtained by wave‐optics and ray‐optic...
Figure 4.14 (a) Measured laser emission spectrum of a GaAs‐modified limaçon ...
Figure 4.15 (a) Calculated spatial intensity distribution of the CW (left) a...
Figure 4.16 (a) Poincaré SOS for a quadrupolar‐shaped microcavity with
ε
...
Chapter 5
Figure 5.1 The triangular‐shaped mode‐field patterns constructed by the supe...
Figure 5.2 The intensity spectra for TM modes obtained by FDTD simulation an...
Figure 5.3 The field distributions of (a) high
Q
and (b) low
Q
TM coupled mo...
Figure 5.4 Mode
Q
factors and output coupling efficiencies vs. the width of ...
Figure 5.5 The spectra of TE modes obtained by FDTD simulation and Padé appr...
Figure 5.6 (a) The intensity spectrum of TE modes obtained by FDTD simulatio...
Figure 5.7 Mode wavelengths vs. the radial mode number for high
Q
TE WGMs in...
Figure 5.8 Laser spectrum for a microdisk laser with a radius of 10 μm and a...
Figure 5.9 (a) Single‐mode optical fiber–coupled powers vs. continuous injec...
Figure 5.10 Schematic diagram of the apparatus used for small‐signal respons...
Figure 5.11 (a) Small‐signal modulation responses of the 7‐μm radius microdi...
Figure 5.12 Eye diagrams at the modulation bit rates of 20, 25, and 30 Gbit/...
Figure 5.13 Intensity patterns of |
H
z
|
2
for antisymmetric TE modes with mode...
Figure 5.14 Mode
Q
factors vs. the radius of inner wall for the coupled mode...
Figure 5.15 Lasing spectra of (a) microdisk laser and (b) microring laser wi...
Figure 5.16 (a) Schematic diagram of a deformed microring. (b) Internal fiel...
Figure 5.17 Internal field intensity distribution and normalized intensity v...
Figure 5.18 (a) Schematic diagram and (b) cross‐sectional view SEM image of ...
Figure 5.19 Schematic diagram of a 2D deformed‐circular resonator with a fla...
Figure 5.20 Mode
Q
factors and wavelengths vs. the width of the flat side wi...
Figure 5.21 Field patterns of the magnetic field components for (a) symmetri...
Figure 5.22 (a) The microscope image of a microdisk laser with two output wa...
Figure 5.23 Lasing spectra of the microdisk laser at room temperature with a...
Figure 5.24 Normalized far‐field patterns of (a) the microlaser with two out...
Chapter 6
Figure 6.1 (a) Mode light rays ED, DF, and FE inside an ETR confined by tota...
Figure 6.2 Schematic diagram of an ETR marked by three vertices
A
,
B
, and
C
,...
Figure 6.3 Analytical and FDTD simulated mode‐field distributions of
and
Figure 6.4 Analytical and simulated mode‐field distributions
and
for
TE0
...
Figure 6.5 Mode
Q
‐factors vs. longitudinal mode number for the fundamental t...
Figure 6.6 Mode
Q
‐factors vs. longitudinal mode number for (a) TE modes and ...
Figure 6.7 Mode
Q
factor and output efficiency vs. the width of output waveg...
Figure 6.8 The unfolded plane wave and the corresponding cavity of one‐perio...
Figure 6.9 Mode
Q
‐factors vs. longitudinal mode number
l
2
for (a) TM modes a...
Figure 6.10 Schematic diagram of the reflection of plane wave on a multilaye...
Figure 6.11 The power reflectivity of (a) TM and (b) TE plane waves on the m...
Figure 6.12 Scanning electron microscope images of (a) an ETR formed just af...
Figure 6.13 (a) Output power vs. continuous and pulsed injection current at ...
Figure 6.14 (a) Lasing spectrum of an ETR microlaser with a side length of 3...
Figure 6.15 Schematic diagram of an ETR microlaser with sidewalls surrounded...
Chapter 7
Figure 7.1 (a) The light ray of WG‐like modes in square microcavities and (b...
Figure 7.2 The analytical field distributions of modes (a)
, (b)
, (c)
,
...
Figure 7.3 Mode frequencies and
Q
‐factors for coupled modes in a rectangular...
Figure 7.4 Magnetic field amplitude (|
H
z
|) patterns obtained by FEM for the ...
Figure 7.5 (a) Intensity spectrum calculated by the FDTD technique and Padé ...
Figure 7.6 Schematic diagram of the square cavity with an output waveguide c...
Figure 7.7 (a) Mode‐intensity spectrum of TE modes and (b) detailed spectrum...
Figure 7.8 Mode‐field patterns of (a)
and (b)
obtained by FDTD simulatio...
Figure 7.9 (a) Mode‐intensity spectrum and (b) magnified spectrum of a group...
Figure 7.10 Field patterns of (a)
and (b)
obtained by FDTD simulation, a...
Figure 7.11 SEM images for square microresonators with output waveguide conn...
Figure 7.12 (a) Lasing spectra at the continuous currents of 17, 35, 47, and...
Figure 7.13 lasing spectra at the CW currents of 13, 33, 53, 80 mA, a micros...
Figure 7.14 Mode‐field patterns |
H
z
| for (a) the fundamental and (b) the fir...
Figure 7.15 (a) Microscopic image of a microsquare laser, (b) the wavelength...
Figure 7.16 (a) Experimental setup for THz wave generation using a dual‐mode...
Figure 7.17 (a) Experimental setup for OFC generation based on a dual‐mode s...
Figure 7.18 Mode
Q
factors vs. the output waveguide width for
,
,
, and
Figure 7.19 lasing spectra at different currents for the square microlaser w...
Figure 7.20 (a) Output powers coupled into single‐mode fiber and multiple‐mo...
Figure 7.21 (a) Schematic diagram of waveguide‐coupled circular‐side square ...
Figure 7.22 The distributions of |
H
z
| for the 0th (upper) and 1st (lower) sy...
Figure 7.23 (a) Lasing spectra for the circular‐side square microcavity lase...
Figure 7.24 (a) SEM image of a circular‐side square microresonator with a ta...
Figure 7.25 (a) Lasing spectrum of the circular‐side square microlaser with
Chapter 8
Figure 8.1 An illustration of a 2D regular polygonal microcavity. The operat...
Figure 8.2 The electric field patterns for (a) TM
9,1
, (b) TM
8,1
, (c) TM
6,1
, ...
Figure 8.3 Angular component distributions of TM
11,1
traveling wave modes in...
Figure 8.4 Schematic diagram of two adjacent sides of a CSPM.
Figure 8.5 Periodic orbits of the CSPMs with
N
= 3, 4, 5, 6, and 8. The orbi...
Figure 8.6 Poincaré SOSs of the CSPMs with
N
= (a) 3, (b) 4, (c) 11, and (d)...
Figure 8.7 (a) Schematic diagram and (b) symmetry operators of a 2D hexagona...
Figure 8.8 (a) Simulated TE modes in hexagonal microcavity with refractive i...
Figure 8.9 (a) Mode
Q
factors vs. normalized frequency for TE modes in the h...
Figure 8.10 Mode
Q
factors of (a) TE and (b) TM modes in the wavelength‐scal...
Figure 8.11 Mode
Q
factors of (a) TE and (b) TM modes in the wavelength‐scal...
Figure 8.12 Mode‐field distributions of |
H
z
| for symmetric TE modes of (a) t...
Figure 8.13 (a) Applied voltage and fiber‐coupled output power vs. the conti...
Figure 8.14 Mode
Q
factors and wavelengths of symmetric and antisymmetric fu...
Figure 8.15 (a) Schematic diagram of a CSHR, (b) cross‐sectional, and (c) la...
Figure 8.16 Intensity spectra for symmetric TE mode in circular resonator wi...
Figure 8.17 (a) SEM image of a deformed hexagonal microlaser after ICP etchi...
Figure 8.18 (a) The simulated intensity distribution of |
H
y
|
2
for the fundam...
Figure 8.19 (a) Lasing spectra at injection currents of 8, 9, 10, 11, 12, 13...
Chapter 9
Figure 9.1 Schematic cross‐section of a 3D microcavity.
Figure 9.2 Schematic cross‐section of a 3D circular microcavity.
Figure 9.3 Mode wavelengths of TE
7,1
and TM
7,1
WGMs obtained from 2D eigenva...
Figure 9.4 Cross section of a microcylinder with the vertical refractive ind...
Figure 9.5 Mode wavelengths (
Q
‐factors) of TE
7,1
(Mode A) and TM
7,1
(Mode B)...
Figure 9.6 (a) Schematic diagram for the fabrication processes of microcylin...
Figure 9.7 Applied voltage and fiber coupled power vs. the continuous inject...
Figure 9.8 Lasing spectra for the microcylinder lasers with (a)
R
= 3.75 μm ...
Figure 9.9 Measured threshold current and density vs. the radius of microcyl...
Figure 9.10 3D FDTD simulated mode
Q
factors of the first radial‐order modes...
Figure 9.11 (a) Horizontal and (b) vertical plane field distributions of Mod...
Figure 9.12 Lasing spectra of a 5‐μm‐radius microdisk laser at 288 K. The in...
Figure 9.13 Mode
Q
factor of TE
9,1
mode vs. the thickness of upper cladding ...
Figure 9.14 Field distributions of
E
z
for TE
9,1
mode at the upper cladding l...
Figure 9.15 The
Q
factors of modes TE
m
,1
with the wavelengths near 1.55 μm v...
Figure 9.16 The mode
Q
factor of TE
11,2
mode vs. the thickness of upper clad...
Figure 9.17 The schematic diagram of EH and HE mode light rays along the ang...
Figure 9.18 The analytical phase difference and mode wavelength vs. the thic...
Figure 9.19 Mode
Q
factors and wavelengths of TE
9,1
mode vs. the thickness o...
Figure 9.20 Field distributions of
H
z
for TE
9,1
mode at core layer thickness...
Figure 9.21 Schematic diagrams of (a) equilateral‐triangular and (b) square ...
Figure 9.22 Intensity spectra obtained by 3D FDTD simulation and Padé approx...
Figure 9.23 Intensity spectra of (a) TE modes and (b) TM modes obtained by t...
Chapter 10
Figure 10.1 Schematic of a resonator with an output waveguide.
E
0
is the las...
Figure 10.2 Calculated optical spectra, microwave spectra, optical intensity...
Figure 10.3 Bifurcation diagrams for the microlaser under optical injection ...
Figure 10.4 Calculated stability diagrams of the microlaser under the optica...
Figure 10.5 Calculated stability diagrams as functions of
R
inj
and
Δ
f
wi...
Figure 10.6 Calculated modulation response curves for the microdisk laser at...
Figure 10.7 (a) Output power coupled into a single mode fiber and applied vo...
Figure 10.8 Experimental setup for studying nonlinear dynamics and small sig...
Figure 10.9 (a) Lasing spectra of the optically injected microdisk laser wit...
Figure 10.10 Measured stability diagrams of the microdisk laser at the biasi...
Figure 10.11 Measured P1 microwave frequency as functions of (a) the injecti...
Figure 10.12 Calculated stability diagrams as functions of
R
inj
and
Δ
f
...
Figure 10.13 (a) Measured small signal modulation responses and (b) optical ...
Figure 10.14 The injected and resonant mode wavelengths, and intensity diffe...
Figure 10.15 (a) Schematic of the experiment system setup. ML, master laser;...
Figure 10.16 (a) Multiple mode fiber coupled optical power and applied volta...
Figure 10.17 (a) Lasing spectra and (b) the corresponding microwave spectra ...
Figure 10.18 (a) The SEM image of a twin‐microdisk resonator after the
induc
...
Figure 10.19 (a) Lasing spectra map for the laser B at at
I
b
= 20 mA vs. inje...
Chapter 11
Figure 11.1 Simulated reflectivity spectra for (a) an FP end face with a wid...
Figure 11.2 Simulated reflectivity spectra at different gain levels in the s...
Figure 11.3 (a) Calculated mode
Q
factor vs. mode wavelength for the even TE...
Figure 11.4 Calculated mode
Q
factor vs. mode wavelength for the even TE mod...
Figure 11.5 Mode wavelengths and
Q
factors vs. the variations of (a)
ΔnFP
...
Figure 11.6 Mode‐intensity profiles of
|
H
z
|
2
for the high‐
Q
modes (a) M1 and...
Figure 11.7 (a) Energy distribution ratio in the FP cavity
R
vs. the variati...
Figure 11.8 (a) SEM image of a square‐rectangular coupled cavity after the I...
Figure 11.9 (a) Lasing spectra map exhibits the WG‐FP mode hybridization wit...
Figure 11.10 Continuous wavelength tuning for the HSRL. (a) Superimposed las...
Figure 11.11 (a) The output powers collected by an SMF and an integrated sph...
Figure 11.12 The lasing spectra of the device at (a)
I
FP
= 40 mA and
I
SQ
= 1...
Figure 11.13 Lasing characteristics with the variations of
I
FP
and
I
SQ
for t...
Figure 11.14 Threshold current density at
I
SQ
= 10 mA for the HSRLs with
a =
...
Figure 11.15 Schematic diagram of an HSRL with an isolation trench to ensure...
Figure 11.16 (a) Single‐mode fiber‐coupled output power vs. the FP‐cavity in...
Figure 11.17 Lasing spectra of the bistability states at
I
FP
= 23.5 mA for t...
Figure 11.18 (a) Lasing spectra of the upper and lower states at
I
FP
= 48 mA...
Figure 11.19 (a) Calculated output powers of modes M1and M2 and (b) carrier ...
Figure 11.20 Optical memory measured at (a)
λ
A
and (b)
λ
B
waveleng...
Figure 11.21 Oscilloscope traces showing the dynamic all‐optical flip‐flop o...
Figure 11.22 (a) Output powers coupled into an SMF vs.
I
FP
around threshold ...
Figure 11.23 Oscilloscope traces showing all‐optical flip‐flop with injected...
Figure 11.24 (a) Optical spectra for the HSRL at
I
SQ
= 15 mA and
I
FP
= 40 mA...
Figure 11.25 Extinction ratio vs. input wavelength (a) with input optical po...
Figure 11.26 High‐speed characteristic of all‐optical logic NOT gate of mode...
Figure 11.27 Output spectra for the HSRL at different optical injections at ...
Figure 11.28 (a) high‐speed characteristics of all‐optical logic NAND gate w...
Figure 11.29 Schematic diagram of the coupled‐cavity laser composed of an FP...
Figure 11.30 Squared
z
‐direction magnetic field
|
H
z
|
2
in the (a) HSRRL, (b) ...
Figure 11.31 Output powers coupled into a single‐mode fiber vs.
I
FP
as
I
SRM
...
Figure 11.32 Lasing spectra map vs. (a)
I
SRM
at
I
FP
= 64 mA and (b)
I
FP
at
IS
...
Guide
Cover
Table of Contents
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Microcavity Semiconductor Lasers
Principles, Design, and Applications
Yong-zhen Huang
Yue-de Yang
Authors
Prof. Yong‐zhen Huang
State Key Lab of Integrated Optoelectronics
Institute of Semiconductors
Chinese Academy of Sciences and College of Materials Sciences and Optoelectronic Technology
University of Chinese Academy of Sciences
No.A35, QingHua East Road
Haidian District
100083 Beijing
China
Prof. Yue‐de Yang
State Key Lab of Integrated Optoelectronics
Institute of Semiconductors
Chinese Academy of Sciences and College of Materials Sciences and Optoelectronic Technology
University of Chinese Academy of Sciences
No.A35, QingHua East Road
Haidian District
100083 Beijing
China
Cover Image: © Supphachai Salaeman/Shutterstock
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Print ISBN: 978‐3‐527‐34546‐5
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Preface
As typical whispering‐gallery microcavities, microdisks, and microrings have been studied for applications in integrated optics over a half century. The study of semiconductor microdisk lasers has become a distinct subject of optoelectronics, and deformed microdisk lasers have attracted great attention for realizing directional emission microlasers. In addition to circular microcavities, polygonal microcavities can also support high Q‐confined modes that rely on the total internal reflection, similar to whispering‐gallery modes in microdisks. We have investigated microcavity lasers for the past two decades, mainly focusing on mode analysis, design, processing technique, photonic integration, and applications of microcavity lasers. This book summarizes the research on semiconductor microcavity lasers based on whispering‐gallery modes. Although there are several books on optical microcavities, this book provides unique descriptions of directional emission microcavity lasers by directly connecting an output waveguide, mode behaviors based on group theory for polygonal microcavities, and hybrid‐cavity lasers with integrated microcavity and waveguide.
The book is organized into 11 chapters: introduction emphasized on mode Q factor, multilayer optical waveguides, FDTD method and Padé approximation, deformed and chaos microdisk lasers, unidirectional emission microdisk lasers, equilateral triangle resonator microlasers, square microcavity lasers, hexagonal microcavity lasers and polygonal microcavities, vertical loss for 3D microcavities, nonlinear dynamics for microcavity lasers, and hybrid‐cavity lasers.
We want to thank Huang's students, especially Dr. Wei‐Hua Guo, at Institute of Semiconductors; their works have contributed to main parts of this book.
Yong‐zhen Huang
Yue‐de Yang
State Key Lab of Integrated Optoelectronics
Institute of Semiconductors
Chinese Academy of Sciences and College of
Materials Sciences and Optoelectronic Technology
University of Chinese Academy of Sciences
Beijing, China
November 2020
1Introduction
1.1 Whispering‐Gallery‐Mode Microcavities
Optical resonant cavities, composed of two or more mirrors, are essential part of ordinary lasers and have been utilized in almost all branches of modern optics and photonics. Optical energy is recirculated inside the cavities due to the reflection on the mirrors, and one basic property of the optical cavities is the quality (Q) factor related to the mode lifetime for describing the light‐confining ability. Mode volume (V) is another important parameter of an optical cavity and a small V is of great importance for realizing a compact‐size integrated device. A suitable parameter, finesse, which is defined as the ratio of the free spectral range to the resonance linewidth, takes both the mode Q factor and the resonator size into account. For certain applications, high‐finesse microcavity with a large value of Q/V, which is also related to the electromagnetic field enhancement factor of an optical cavity, is very important. Compared with conventional lasers, microcavity lasers with a large Q/V can promise lower lasing threshold. Moreover, light–matter interactions can be greatly enhanced by storing optical energy in a small mode volume [1, 2]. The ability to concentrate light is important to both fundamental science studies and practical device applications [3], such as strong‐coupling cavity quantum electrodynamics, enhancement and suppression of spontaneous emission, high‐sensitivity sensors, low‐threshold light sources, and compact optical add‐drop filters in optical communication.
To obtain high Q modes in optical cavities with a small V, a high reflectivity close to unity is necessary, which can be realized by utilizing a periodic structure to construct a photonic forbidden band, such as that in vertical‐cavity surface‐emitting lasers and photonic crystal microcavities, or simply by total internal reflection (TIR) at the dielectric boundary with a high‐low refractive index contrast in whispering‐gallery (WG)‐mode optical microcavities [4]. The idea of WG mode was born out of the observation of acoustical phenomenon in [5] where sound waves were efficiently reflected with minimal diffraction and struck the wall again at the same angle and thereby traveled along the gallery surface. Similarly, classical electromagnetic waves can undergo reflection, refraction, and diffraction like the sound waves when the wavelengths of the waves are smaller than the bending radius of a reflection mirror. Among various kinds of optical microcavities, WG‐mode microcavities with simple cavity geometries and suitability for planar integration play an important role in photonics integration nowadays [6–8]. WG‐mode optical microcavities formed by various materials have been studied, such as liquid droplet, glass, crystal, polymer, and semiconductor [8].
Figure 1.1 Schematic diagrams of (a) circular, (b) triangular, (c) square, and (d) hexagonal microcavities with the confined TIR light rays.
The concept of WG mode was subsequently extended to the radiofrequency and optical domains for the electromagnetic waves. For WG‐mode microcavities, the optical modes will experience ultra‐low loss as the light rays are guided by continuous TIR at the boundaries. In fact, light guidance by continuous TIR is quite common in modern optics and photonics, such as the propagating optical modes in fibers and waveguides. The incident angles of the light rays in WG‐mode microcavities are greater than the TIR criticality at all boundaries, and the light rays are mostly like to propagate along the WG surface. There are various kinds of WG‐mode optical microcavities to maintain continuous TIR for the confined light rays. Figure 1.1 schematically shows circular, triangular, square, and hexagonal microcavities with the confined TIR light rays. As a natural choice, WG‐mode optical microcavities with circular shapes, which maintain a perfect rotational symmetry, have attracted most research interest for the demonstration of low‐power‐consumption compact‐size photonic devices, e.g. microdisk lasers [9]. In circular microcavities, the light ray confined inside the cavity has a conserved incident angle above the TIR critical angle resulting in ultrahigh Q factors and isotropic near‐ and far‐field patterns. However, for a practical device, efficient input or output coupling is crucial but can be hardly achieved in the circular‐shaped microcavities. By deforming the WG microcavity to a noncircular cavity shape, the far‐field emission patterns can be modulated while high Q factor mode is maintained [10, 11].
1.2 Applications of Whispering‐Gallery‐Mode Microcavities
WG‐mode optical microcavities with the light rays confined by continuous TIR at the cavity boundaries have unique mode properties, including high Q factors, small mode volume, and planar integration capability. Due to these mode properties, there has been a wide range of applications for WG‐mode microcavities in both fundamental physics and practical devices [12]. In this section, part of the applications, including photonic filters, sensors, and microlasers, based on both passive and active WG‐mode microcavities, is briefly summarized.
One basic application of microcavities is photonic filter based on the wavelength selectivity property of WG modes, including all‐pass and add‐drop filters. A common microcavity‐based filter includes a WG‐mode microcavity and optical coupler for coupling light into or out from the WG modes [13–15]. In an optical filter, the filtering response is relative to both the intrinsic mode Q factors and the coupling coefficients for the WG modes. The high‐order transverse WG modes have lower Q factors but stronger coupling, which is undesirable for most filtering applications. Hence, microrings are utilized for suppressing high‐order transverse modes and realizing high‐performance optical filter. Microrings have similar mode properties with the microdisks, as the modes are confined by the continuous TIR at the outer boundaries [14]. A typical on‐chip all‐optical four‐port add‐drop filter includes a microring cavity and two evanescent‐field coupled waveguides. The add‐drop filter exhibits pass‐band filtering characteristics with the on‐resonance light dropping through the drop port. The microring‐based filters and corresponding active devices have the advantages of compact size, narrow band, and a large free spectral range, and have been widely studied in the silicon photonics for on‐chip optical interconnection application. The pass‐band characteristics can be further improved by using higher‐order filter structure with multiple coupled microrings. Based on the microring photonic filters, cascaded microring‐based matrix switches have been demonstrated for on‐chip optical networks [16]. The networks‐on‐chip can be passive networks with fixed‐wavelength assignment and switching networks with the resonance wavelength tuning by thermal effect or carrier injection. With the structure of all‐pass microring filter, silicon‐based microring electro‐optic modulators were demonstrated with the carrier injection or depletion to change the resonance wavelength of the microring cavity [17].
Another important application of WG optical microcavities is photonic sensing. WG microcavities have been extensively investigated for their applications in chemicals and biosensing. Strong light–matter interactions and high optical energy intensity in the optical microcavities with a large value of Q/V can help to achieve ultrasensitive and label‐free detection. The sensing principle is to measure the spectral changes of a WG mode in response to changes in the environment, e.g. refractive index shift of surrounding media or nanoparticles onto the cavity surface [18, 19]. The key feature is the strong evanescent field of the WG mode that propagates along and extends from the surface of the microcavity leading to strong interaction between the internal field and the external environment. The measured transmission spectra will experience a wavelength shift and/or splitting for sensing. The resonance shift in a microcavity is a more direct detection scheme, but it can be easily perturbed by environmental noises resulting in a reduction of the sensing resolution. The environmental noises are minimized in the mode splitting–based detection scheme as the two split modes suffer the same noises. The variation of mode splitting carries the information of particles to be measured. The detection resolution for a passive microcavity‐based sensing is limited by the linewidth of the WG modes. In a microcavity laser–based active sensing devices, the stimulated emission will narrow down the linewidth, and hence the sensing resolution is greatly enhanced [20]. For a cold cavity with a Q factor of 108, the laser linewidth can be as narrow as a few Hertz allowing ultrasensitive detection [8].
Light sources, such as microlasers and quantum sources, are an extremely important research direction in optics and photonics. Compared to other optical cavities for laser application, the WG‐mode microcavities have extraordinarily high Q factor and small V, which lead to diverse applications in the study of laser physics and the realization of compact‐size microlasers. In WG‐mode microcavities, the optical density of states can be modulated by designing the cavity structure and matching the resonance wavelength to the emission wavelength of the active material. Thus, the Purcell factor can be enhanced greatly in optical microcavities [1]. Semiconductor quantum dot is a quasi‐atom gain material; the coupling between the quantum dot and the optical mode can be enhanced in the high Q microcavity with a ultrasmall V. Strong coupling of a single GaAs quantum dot to a WG mode of a microdisk has been observed, facilitating the investigations of cavity quantum electrodynamics and single photon source [21]. High Q factor and small V also allow the demonstration of conventional low‐threshold semiconductor microlasers [9]. The high Q factor of a WG mode guarantees a low‐threshold current density and a small V leads to a compact size for the microlaser for achieving low‐power consumption. Continuous wave lasing with a threshold of 40 μA was realized in an InGaAsP microdisk laser at room temperature [22]. However, the nearly perfect confinement of the mode light ray and the rotational symmetry of a circular microcavity led to low‐output power and isotropic emission to free space despite a low‐lasing threshold. This is a serious problem for most practical applications of WG microcavity lasers. Evanescent wave coupling of a waveguide is one traditional scheme to couple lasing light out from the circular microcavity lasers, but it has extremely high requirements for fabrication processing technology and parameter control. Experimental results show that a small variation in the coupling gap will reduce the output optical power by several orders of magnitude. In addition, the competition between the clockwise and counterclockwise modes in the circular microcavities will cause instability of the output optical power in the waveguide. To realize directional lasing emission, various deformed microcavities, such as adding local boundary defects or using smoothly deformed cavity shapes, have been proposed and demonstrated [3]. By carefully designing the cavity geometries, directional emission, or even unidirectional emission with low divergence angle in free space was achieved for deformed microcavity lasers while preserving high‐Q WG modes for low‐threshold lasing. However, the directional or unidirectional emission of asymmetric microcavities is still limited to free space, and the application to on‐chip photonic integration requires waveguide‐coupled output. Moreover, regular‐polygonal‐shaped microcavities have distinct mode properties, as the WG modes distribute nonuniformly along the cavity boundaries. A waveguide directly connecting to the position with weak mode field can be used for realizing a waveguide‐coupled microcavity laser without strong perturbation to the corresponding high Q WG mode. Especially, a quasi‐analytical solution can be obtained for the equilateral‐triangular and square microcavities with integrable internal dynamics, making them a reliable solution to demonstrate waveguide‐coupled unidirectional‐emission semiconductor microlasers [23].
1.3 Ultra‐High Q Whispering‐Gallery‐Mode Microcavities
Spherical optical microcavities of liquid droplets and highly transparent silica have been extensively investigated, which can have nearly perfect microspheres due to surface tension of liquid and fused silica [1]. Based on liquid droplet microcavities, cavity quantum electrodynamics with modified spontaneous and stimulated emission spectra were studied and ultralow threshold of nonlinear optical processes was observed with fluorescent dyes. The effects of droplet deformation on the resonance frequencies and Q factors were investigated experimentally and theoretically using first‐order perturbation theory, and the dye‐lasing spectra from liquid droplet optical microcavities were observed under perturbations. By using the CO2 laser fusion process, high‐Q silica microspheres were fabricated by fusing the end of a silica fiber. WG modes with Q factors up to 109 ∼ 1010 were observed, and low‐threshold microlasers based on silica microspheres with doped irons were realized. Droplet microlasers as easily replaced coherent light sources were investigated for potential applications in integrated lab‐on‐a‐chip systems [24]. The droplet‐based microlasers can be prepared in microfluidic chip with different active media, such as live bacteria. In addition, intracellular droplet microlasers were studied by injecting oil doped with a dye gain medium inside biological cells as luminescent probes [25].
Ultra‐high‐Q microcavities can also be fabricated on a silicon wafer using wafer‐scale processing, in the form of a microcavity on‐a‐chip suitable for photonic integrated circuits [2]. The fabrication processing is simply summarized in the following. First, silica circular patterns were transformed from photoresist layer to the thermally oxidized surface layer of a silicon wafer using lithography and etching technique processes, and then silica disks were used as an etching mask for selectively removing the underneath silicon. Finally, silica microdisks on a silicon post were fabricated with a vertical optical confined by air for avoiding a leaking loss into high‐index silicon substrate. The WG modes with Q‐factors up to 3 × 106 were measured for such microdisks under optimal processing conditions. As the mode‐field distributions located near the disk periphery, the Q‐factors were mainly limited by scattering loss due to disk roughness caused by lithography and etching. To further increase mode Q factors, a reflow process for the silica microdisks was applied under the surface‐normal irradiation of CO2 laser by improving surface smooth of the microdisks without affecting the underneath silicon post. The reflow process under the laser irradiation can lead to melting and collapse of the silica at the disk periphery and form silica microtoroid on a silicon chip. The ultra‐high Q factor based on linewidth measurement is a challenge as a loaded cavity Q factor is measured with coupled waveguide. In addition, WG mode splits into doublets caused by weak back scattering in the microcavity, and thermal effects due to input optical power induce distortion of the resonance peak. An intrinsic cavity Q factor of 4.3 × 108 was obtained by cavity ring‐down measurement for a microtoroid cavity mode. The WG mode loss is negligible for microtoroids with principal tori‐radii larger than 15 μm, and measured Q values are more than 108 in the wavelength of 1550‐nm band. The ultra‐high‐Q microcavities are especially suitable to study optical nonlinear processes, such as Raman and Kerr nonlinearities. Under fiber evanescently coupling with a low‐input power at resonant frequency, high‐mode field intensity can be stored inside an ultra‐high‐Q microcavity for ultra‐low‐threshold fiber‐compatible Raman lasers and parameter oscillators. Furthermore, optical frequency combs were realized through cascaded four‐wave mixing process in a high Q microresonator [26].
In addition, chemically etched wedge resonators on‐a‐chip were fabricated using conventional semiconductor processing, with a Q factor of 875 million surpassing microtoroids [27]. The smoothness of wedge resonators was improved using post exposure bake method to cure the roughness of photoresist patterns and extend the chemically etched time to form wedge profiles for the resonator perimeter. Without the reflow process of laser irradiation, the wedge resonators are of easy‐to‐control size and can be integrated with other photonic devices.
1.4 Mode Q Factors for Semiconductor Microlasers
1.4.1 Output Efficiency and Mode Q Factor
For a microcavity laser, as shown in Figure 1.2, with a passive cavity mode Q factor QR related to planar and vertical radiation losses αr and αv and an output coupling loss αo, we can have a modified mode lifetime varied with a mode gain Гg as:
where Г is the optical confinement factor, αi is an internal material absorption loss, vg = c/ng is the light group speed with a group index ng, and ω is the mode angular frequency. Mode QT factor, including the absorption loss, can be defined as:
Figure 1.2 Scanned electron micrograph image of a microdisk connected with an output waveguide.
For a silica microdisk without gain and αi ≈ 0, it is easy to measure mode Q factor from the transmission linewidth because QT = QR. However, semiconductor lasers usually have an absorption loss, which limited mode Q factors. Taking the absorption loss αi = 1 and 10 cm−1, which corresponds to the magnitude of the absorption loss for GaAs and InP system semiconductor lasers, respectively, we have mode QA factor of 1.4 × 105 and 1.4 × 104 at ng = 3.5 and mode wavelength of 1550 nm.
Accounting for the internal absorption loss αi related to QA, output coupling loss through the output waveguide αo, the vertical loss αv into the substrate, and the other radiation loss, including scattering loss due to rough perimeter αr, we can define an output efficiency as
The laser output efficiency will be very low for an ultra‐high Q microcavity with QR ≫ QA. The material of low absorption loss with a high QA is important for realizing high‐output efficiency for a microlaser.
1.4.2 Measurement of Mode Q Factor
The mode Q factors of a microlaser are usually measured as the ratio of mode wavelength to the full‐width at half maximum (FWHM) of the resonator peak at the threshold. The mode linewidth is described by the Schawlow–Townes linewidth formula below the threshold [28]:
Due to the gain‐refractive index coupling effect with a linewidth enhancement factor α and carrier density clamping above threshold, lasing mode linewidth above the threshold is given by the modified Schawlow–Townes linewidth formula [29]:
which is equal to (1.4) multiplying by a factor of (1 + α2)/2. The laser linewidth enhancement near threshold was observed experimentally for semiconductor microlasers [30].
The output characteristics of semiconductor microcavity lasers can be described by the following single‐mode rate equations
where s is the photon density, n is the carrier density, I is the injection current, ηi is the injection efficiency, q is the electron charge, Va is the volume of the active region, A, B, and C are the defect, bimolecular, and Auger recombination coefficients, respectively, β is the spontaneous emission factor, and τpc = QR/ω is the passive cavity mode lifetime. The threshold gain of semiconductor lasers is usually expressed as
which is only approached in the steady state. From Eq. (1.7), we can obtain the output term in steady state as
The mode lifetime is determined by the passive cavity QR factor as Γg = αi from (1.1), so the passive mode QR factor should be measured at the following condition
instead of threshold gain of (1.8). However, it is difficult to determine the condition of (1.10) from the curve of the output power vs. injection current.
In the following part of this section, we give numerical results of the rate Eqs. (1.6) and (1.7). The gain coefficient is assumed to be a logarithmic function as [31]
where g0 is the material gain parameter, Ntr is the transparency carrier density, and Ns is a gain parameter. Taking the parameters as λ = 1550 nm, Ntr = 1.2 × 1018 cm−3, Ns = 1.1 × 1018 cm−3, mode group index ng = 3.5, ηi = 0.8, Γ = 0.1, A = 1 × 108 s−1, B = 1 × 10−10 cm−3 s−1, C = 1 × 10−28 cm6 s−1, g0 = 1500 cm−1, we numerically calculate the steady solutions of Eqs. (1.6) and (1.7), and plot the output powers and linewidths of Eqs. (1.4) and (1.5) vs. the injection current in Figure 1.3 at QR = 60 000, 20 000, and 6000, and αi = 4 cm−1, for microdisk lasers with a circular radius of 10 μm, β = 10−3, and a linewidth enhancement factor α = 3. The output power is related to the mode photon density as
Figure 1.3 Laser‐mode linewidth and output power vs. injection current for semiconductor microdisk lasers with β = 10−3 at QR = 60 000, 20 000, and 6000, and αi = 4 cm−1. The three horizontal dashed lines correspond to the FWHMs determined by the QR factors, and vertical dotted line is at the condition of Γg = αi.
where hv is the photon energy. The FWHMs determined by the QR factors are marked by the horizontal dashed lines. The FWHMs vary rapidly around the threshold, and Q factor measured using the FWHM at the threshold will overestimate the passive mode QR factor greatly.
The FWHMs at the vertical dashed line in Figure 1.3 should be used to calculate QR factors, which is difficult to determine from measured curve of output power vs. injection current. If the lasing mode output power can be divided into the first term and the second term of the right side of Eq. (1.9) from lasing spectra, we can calculate the derivative of the first term with respect to the output power and determine the condition of Γg = αi by finding the zero of the derivative.
Finally, we present experimental results for a microdisk laser as shown in Figure 1.1, with a radius of 8 μm and a 2‐μm‐wide output waveguide. The output powers vs. injection currents at 288, 290, and 296 K are presented in Figure 1.4a with magnified curves around threshold current in Figure 1.4b, where extrapolated lines are plotted as dashed lines. The practical output powers are limited by heating effect; even a thermoelectric cooler was used to control the temperature. The lasing spectra at 288 K and 10 mA are plotted in Figure 1.4c with single‐mode operation at 1540.3 nm. The FWHM of the lasing mode vs. the injection current at 288 K is shown in Figure 1.4d, where the FWHMs in A and B regions divided by a vertical solid line are directly measured using an optical spectrum analyzer at the finest resolution of 0.02 nm, and estimated from the FWHM of the beating microwave signal by mixing the outputs of the microdisk laser and a tunable laser [32]. The measured FWHM of the beating microwave can be considered as the FWHM of the microdisk laser, as its linewidth is much larger than that of the tunable laser. The FWHMs of the lasing mode are 250, 166, 92.5, 20.8, 4.2, and 1.0 pm at 4, 4.5, 5, 5.5, 6, and 10 mA, respectively, which varies greatly around the threshold. We can estimate mode Q factors of 1.7 × 104, 7.4 × 104, and 3.7 × 105 from the FWHMs at the currents of 5, 5.5, and 6 mA. The mode Q factor of 1.7 × 104 measured at 5 mA, about the intercept of the extrapolated dashed line with the current axis in Figure 1.4b, is in agreement with that obtained by numerical simulation for two‐dimensional microcavity under effective index approximation. The agreement indicates that the choice of the intercept current is a better approximation for measuring passive mode Q factor than using the FWHM at the kink position of the power vs. current.
Figure 1.4 (a) Output power coupled into a multiple‐mode fiber vs. the injection current, (b) the power vs. the injection current around threshold, (c) lasing spectra at 10 mA, and (d) lasing‐mode linewidth vs. injection current, for a microdisk laser connected to a 2‐μm‐width output waveguide with a radius of 8 μm.
1.5 Book Overview
In this book, we present an overview of the principle, design, and application of semiconductor microlasers, especially for directional emission microlasers based on polygonal optical microcavities. In Chapter 2, we derive an eigenvalue equation for multilayer complex slab optical waveguides, give an optical confinement factor based on the relation between mode gain and material gain, and discuss the effective index method for reducing three‐dimensional (3D) waveguide to a two‐dimensional (2D) problem. In Chapter 3, we simply introduce finite‐difference time‐domain method and Padé approximation with Baker's algorithm for simulating optical microcavities, and give some numerical results for simulating mode frequencies and mode Q factors of microcavities and the transmission coefficient for optical microring add‐drop filters. Chapter 4 presents an eigenvalue equation for 2D microdisk and deformed and chaotic microcavity lasers for directional emission. In Chapter 5, we summarize unidirectional emission microdisk lasers based on mode coupling due to connection with an output waveguide, and propose to realize unidirectional emission hybrid microlaser on silicon wafer by wafer bonding using locally deformed microring resonator. In Chapter 6, we derive analytical mode solution for equilateral triangle resonator, and compare with numerical simulated results and lasing spectra of fabricated devices. In Chapter 7, we present analytical‐mode solution for square microcavities and discuss the formation of high‐Q coupled modes. Furthermore, the enhancement of mode Q factors by circular‐sided square microcavities is discussed, and dual‐mode lasing microlasers are designed and realized in addition to single‐mode microlasers. In Chapter 8, we discuss mode characteristics for polygonal microcavities based on group theory especially for hexagonal microcavity, and present lasing characteristics for hexagonal and octagonal and circular‐sided hexagonal microlasers. In Chapter 9, we consider the vertical radiation loss for 3D semiconductor microcavities with vertical semiconductor waveguiding, and discuss lateral size limit for such microcavities due to the vertical radiation loss. In Chapter 10, we summarize nonlinear dynamics for microlasers subject to optical injection and integrated microlasers with mutually optical injection. Finally, in Chapter 11, we demonstrate a hybrid cavity composed of a Fabry–Pérot (FP) cavity and a square microcavity for mode selection. Stable single‐mode operation with high coupling efficiency to a single mode fiber is realized, and controllable optical bistability is achieved for all‐optical signal processing.
References
1 Chang, R.K. and Campillo, A.J. (eds.) (1996).
Optical Processes in Microcavities
. Singapore: World Scientific.
2 Vahala, K.J. (ed.) (2004).
Optical Microcavities
. Singapore: World Scientific.
3 Cao, H. and Wiersig, J. (2015). Dielectric microcavities: model systems for wave chaos and non‐Hermitian physics.
Rev. Mod. Phys.
87: 61–111.
4 Vahala, K.J. (2003). Optical Microcavities.
Nature
424: 839–846.
5 Rayleigh, L. (1910). The problem of the whispering gallery.
Philos. Mag.
20: 1001–1004.
6 Matsko, A.B. and Ilchenko, V.S. (2006). Optical resonators with whispering‐gallery modes – part I: basics.
IEEE J. Sel. Top. Quantum Electron.
12: 3–14.
7 Ward, J. and Benson, O. (2011). WG microresonators: sensing, lasing and fundamental optics with microspheres.
Laser Photonics Rev.
5: 553–570.
8 He, L.N., Ozdemir, S.K., and Yang, L. (2013). Whispering gallery microcavity lasers.
Laser Photonics Rev.
7: 60–82.
9 Mccall, S.L., Levi, A.F.J., Slusher, R.E. et al. (1992). Whispering‐gallery mode microdisk lasers.
Appl. Phys. Lett.
60: 289–291.
10 Nockel, J.U. and Stone, A.D. (1997). Ray and wave chaos in asymmetric resonant optical cavities.
Nature
385: 45–47.
11 Yang, Y.D. and Huang, Y.Z. (2016). Mode characteristics and directional emission for square microcavity lasers.
J. Phys. D: Appl. Phys.
49: 253001.
12 Ilchenko, V.S. and Matsko, A.B. (2006). Optical resonators with whispering‐gallery modes – part II: applications.
IEEE J. Sel. Top. Quantum Electron.
12: 15–32.
13 Little, B.E., Chu, S.T., Haus, H.A. et al. (1997). Microring resonator channel dropping filters.
J. Lightwave Technol.
15: 998–1005.
14 Schwelb, O. (2004). Transmission, group delay, and dispersion in single‐ring optical resonators and add/drop filters – a tutorial overview.
J. Lightwave Technol.
22: 1380–1394.
15 Heebner, J., Grover, R., and Ibrahim, T. (2008).
Optical Microresonators: Theory, Fabrication, and Applications
. London: Springer‐Verlag.
16 Poon, A.W., Luo, X.S., Xu, F., and Chen, H. (2009). Cascaded microresonator‐based matrix switch for silicon on‐chip optical interconnection.
Proc. IEEE
97: 1216–1238.
17 Xu, Q.F., Schmidt, B., Pradhan, S., and Lipson, M. (2005). Micrometre‐scale silicon electro‐optic modulator.
Nature
435: 325–327.
18 Vollmer, F. and Arnold, S. (2008). Whispering‐gallery‐mode biosensing: label‐free detection down to single molecules.
Nat. Methods
5: 591–596.
19 Zhu, J.G., Ozdemir, S.K., Xiao, Y.F. et al. (2010). On‐chip single nanoparticle detection and sizing by mode splitting in an ultrahigh‐
Q
microresonator.
Nat. Photonics
4: 46–49.
20 He, L.N., Ozdemir, K., Zhu, J.G. et al. (2011). Detecting single viruses and nanoparticles using whispering gallery microlasers.
Nat. Nanotechnol.
6: 428–432.
21 Peter, E., Senellart, P., Martrou, D. et al. (2005). Exciton‐photon strong‐coupling regime for a single quantum dot embedded in a microcavity.
Phys. Rev. Lett.
95: 067401.
22 Fujita, M., Ushigome, R., and Baba, T. (2000). Continuous wave lasing in GaInAsP microdisk injection laser with threshold current of 40 mu A.
Electron. Lett.
36: 790–791.
23 Yang, Y.D. and Huang, Y.Z. (2007). Symmetry analysis and numerical simulation of mode characteristics for equilateral‐polygonal optical microresonators.
Phys. Rev. A
76: 023822.
24 Jonaš, A., McGloin, D., and Kiraz, A. (2015). Droplet lasers.
Opt. Photonics News
26 (5): 36–43.
25 Humar, M. and Yun, S.H. (2015). Intracellular microlasers.
Nat. Photonics
9: 572–576.
26 Del'Haye, P., Schliesser, A., Arcizet, O. et al. (2007). Optical frequency comb generation from a monolithic microresonator.
Nature
450: 1214–1217.
27 Lee, H., Chen, T., Li, J. et al. (2012). Chemically etched ultrahigh‐
Q
wedge‐resonator on a silicon chip.
Nat. Photonics
6: 369–373.
28 Schawlow, A.L. and Townes, C.H. (1958). Infrared and optical masers.
Phys. Rev.
112: 1940–1949.
29 Henry, C.H. (1982). Theory of the linewidth of semiconductor‐lasers.
IEEE J. Quantum Electron.
18: 259–264.
30 Bagheri, M., Shih, M.H., Choi, S.J. et al. (2009). Microcavity laser linewidth close to threshold.
IEEE J. Quantum Electron.
45: 935–939.
31 Coldren, L.A. and Corzine, S.W. (1995).
Diode Lasers and Photonic Integrated Circuits
. New York: Wiley.
32 Zou, L.X., Huang, Y.Z., Liu, B.W. et al. (2015). Nonlinear dynamics for semiconductor microdisk laser subject to optical injection.
IEEE J. Sel. Top. Quantum Electron.
21 (6): 1800408.
2Multilayer Dielectric Slab Waveguides
2.1 Introduction
Semiconductor lasers are usually grown in a substrate with multiple‐layer semiconductor materials, such as AlGaAs/GaAs and InGaAsP/InP material systems, which can form a multilayer slab waveguide. The multilayer slab waveguide is the simplest case and has easy‐to‐analyze mode characteristics for guided and radiation modes [1–3]. The modes are solutions of Maxwell's equations under boundary conditions imposed on the mode fields at the dielectric interfaces of the waveguide. For a multiple‐layer slab waveguide with interfaces parallel to y–z plane, we can model z
