Coupled Wave Analysis of Two-Dimensional Second Order

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COUPLED WAVE ANALYSIS OF TWO-DIMENSIONAL

SECOND ORDER SURFACE-EMITTING DISTRIBUTED

FEEDBACK LASERS

Thesis

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Master of Science in Electro-Optics

Yangfei Shen

UNIVERSITY OF DAYTON

Dayton, Ohio

May, 2016

COUPLED WAVE ANALYSIS OF TWO-DIMENSIONAL

SECOND ORDER SURFACE-EMITTING DISTRIBUTED

FEEDBACK LASERS

Name: Shen, Yangfei

APPROVED BY:

______Andrew Sarangan, Ph.D. ,P.E. Partha Banerjee, Ph.D. Advisory Committee Chairman Committee Member Professor Director and Professor Electro-Optics Program Electro-Optics Program

______Qiwen Zhan, Ph.D. Committee Member Professor Electro-Optics Program

______John G. Weber, Ph.D. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean Dean, School of Engineering School of Engineering

Yangfei Shen

2016

iii

ABSTRACT

COUPLED WAVE ANALYSIS OF TWO-DIMENSIONAL

SECOND ORDER SURFACE-EMITTING DISTRIBUTED

FEEDBACK LASERS

Name: Shen, Yangfei University of Dayton

Advisor: Dr. Andrew Sarangan

A distributed feedback laser (DFB) is a type of semiconductor laser where the cavity is periodically structured as a diffraction grating. This allows the same grating region to act as a resonator, out-coupler and the gain region simultaneously. While conventional DFB lasers emit light along its edges, by modifying the grating structure, it is possible to make the output appear from the top surface. This is known as the surface- emitting DFB laser.

This thesis will discuss the case in which a two-dimensional grating (which consists of a 2D grid of holes or pillars) etched into the top cladding surface of the waveguide. The thesis consists of four main parts. For the first part, we give the background principles and the coupled wave equations with radiation modes. In the

iv second part, we derive the equations for a waveguide that utilizes a grating strip for guidance. The effective index of this guide is then used in the 2D DFB crossed grating structure. The third part begins with calculations of a particular GaAs:GaAlAs waveguide geometry with a grating. We present the fundamental resonant modes by utilizing the numerical shooting method. Then, the matrix solution method is applied in the calculation of the higher order guided and radiation modes. In the last part, we solve the transcendental equations for the eigenvalues to get the threshold gain and near field radiation pattern from the DFB surface.

Dedicated to my parents

ACKNOWLEDGEMENTS

I am greatly indebted to the people who have helped me over the years. Above all,

I would like to acknowledge my advisor, Dr. Andrew Sarangan, for his tremendous amount of help and support through my master period. I couldn’t thank him enough. I would like to thank him for his paper 《the novel waveguide utilizing a grating strip for guidance 》 which contributes the key point in my thesis. He also provides very pragmatic methods to calculate random shape of gratings and higher order grating field.

Additionally, I am grateful to Jack O'Gorman who enthusiastically helps me in the library searching system, which enables me to search related resources much effectively. Last but not least, I would like to thank my family. My parents are always there for me. None of this could have happened without their support.

vii

TABLE OF CONTENTS

ABSTRACT ...... iv

DEDICATION ...... vii

ACKNOWLEDGEMENTS ...... vii

LIST OF FIGURES ...... x

CHAPTER 1 INTRODUCTION OF COUPLE MODE THEORY FOR DISTRIBUTED

FEEDBACK ( DFB) STRUCTURES ...... 1

CHAPTER 2 LATERAL WAVEGUIDING IN A PERIODIC STRUCTURE ...... 11

2.1 Introduction of lateral grating waveguide ...... 11

2.2 Analysis...... 12

2.3 Applicaion for 2D-DFB ...... 21

CHAPTER 3 COUPLED MODE DESCRIPTION OF THE TWO-DIMENSIONAL

DFB CAVITY ...... 27

3.1 Utilizing numerical shooting method to calculate 퐸0 ...... 27

3.2 Calcultion of Higher order of 퐸푚using matrix solution method ...... 30

3.3 Calculation of coupling coeficient and other constants ...... 33

3.4 Summary & Discussions ...... 34

viii

CHAPTER 4 NEAR FIELD PROFILE AND THRESHOLD GAIN OF 2-D DFB

STRUCTURES ...... 35

4.1 Near field and threshold gain ...... 35

4.2 Comparison of 1-D and 2-D ...... 41

4.3 Summary & Discussions ...... 44

CHAPTER 5 THE CURRENT AND FUTURE RESEARCH ...... 46

REFERENCES ...... 48

LIST OF FIGURES

Figure 1-1 3-D schematic view of a 2-D surface-emitting laser and second-order DFB gratings with GaAs-Au gratings ...... 3

Figure 1-2 A guided-wave DFB structure with a grating ...... 5

Figure 1-3 Relationship between amplitude threshold gain and detuning factor coefficient of a mirrorless first order indux-coupled DFB laser diode ...... 10

Figure 2-1 Lateral Grating waveguide structure ...... 12

Figure 2-2 Modes of the grating waveguide at 1.0 휇푚, with 푊 = 100 휇푚, a grating

−3 period of 0.5 휇푚, 푛0 = 3.2, ∆푛 = 3.2 × 10 , and for various of 휅푊 ...... 18

Figure 2-3 First two modes of the grating waveguide for different values of 휅푊 ...... 19

Figure 2-4 Holes etched on the top of cladding surface in lab ...... 21

Figure 2-5 Cosine function simulation of the holes on cladding surface ...... 22

Figure 2-6 A guided-wave laser with “cosine” shape grating, where g is the grating height while t is the outer guide thickness...... 22

+ − Figure 2-7 Real part of 푛 (푥) and 푛 (푥) vs. depth x with 푛1, 푛3 = 3.2, 푛2 = 3.3, 휆 =

1.0 휇푚, Λ = 0.5휇푚, 𝑔 = 1 휇푚, 푡 = 1.5 휇푚 ...... 24

Figure 2-8 Differ from conventional DFB, 푛+ and 푛− change with z not only x...... 24

Figure 2-9 Real part of 푛+...... 25

Figure 2-10 Real part of 푛− ...... 25

Figure 2-11 Imaginary part of 푛+ and 푛− ...... 26

Figure 3-1 퐸0 in 2-D DFB waveguide (we seperate x from -2.5-3.5 into 9999 steps) .... 29

(0) Figure 3-2 The partial waves 퐸푚 for m=-8-5 (m ≠ 0, −2) ...... 32

Figure 4-1 Intensity of multiple modes as a function of L for the case of 푛+ ...... 37

Figure 4-2 Intensity of multiple modes as a function of L for the case of 푛− ...... 38

Figure 4-3 Threshold gain for different L for both 푛+ (circle and diamond points) and

푛− (rectangular and star points), red lines represent symmetric modes while blue lines represent antisymmetric modes...... 39

Figure 4-4 Graphic representation of the relationship between the wave vectors and the guided mode, the radiation modes, and the grating of a simple thin-film wave guide ..... 40

Figure 4-5 Intensity (1D-DFB) of different modes in different cavity length ...... 42

Figure 4-6 Threshold gain (1D-DFB) for different modes with the same L as 2-D case

...... 43

CHAPTER 1

INTRODUCTION OF COUPLED MODE THEORY FOR

DISTRIBUTED FEEDBACK (DFB) STRUCTURES

Distributed feedback lasers today are the workhorse of fiber communication applications because they are able to produce single frequency, single spatial mode emission. They are also dynamically stable, meaning they maintain their single frequency behavior under varying operating conditions, which is important in a fiber communication system to reduce dispersion. However, for higher power applications, semiconductor lasers are still inferior to solid state lasers, such as YAG, Ti-Sapphire, etc.

But there are still a large number of advantages to semiconductor DFB lasers. They are compact, robust and can be integrated into portable systems.

For high power applications (greater than 1W), the normally used approach is to incoherently combine the outputs of separate semiconductor lasers. This is acceptable in applications where the beam quality is not important. This can then be used to pump another laser system (such as a YAG). However, high-power direct emission from a semiconductor laser is still a desired goal because they can be compact, portable and simpler. The main difficulty is cavity volume. To reduce nonlinear effects and increase heat dissipation, the cavity (gain) volume has to be increased. But this results in multimode oscillation, both spatially and in frequency domain, which results in a poor

beam quality. Periodic gratings offer the possibility to maintain single mode oscillation even in a large cavity. While this is used in a conventional 1D DFB laser, the cavity volume is still small, so the power output is small. By using two gratings, one along the longitudinal direction, and another along the lateral direction, it becomes possible to maintain a single mode in a large volume cavity. This is the 2D DFB laser.

Additionally, by employing a second-order grating in one direction (or both directions), it is possible to extract the power from the top surface, similar to a Vertical Cavity Surface

Emitting Laser (VCSEL). Therefore, 2D surface-emitting DFB (SE-DFB) combines the advantages of edge-emitters and VCSELs in a single 2D platform for potential high power applications.

In this thesis, we develop the coupled-wave equations for such a 2D SE-DFB, by treating the lateral and longitudinal directions separately. In the lateral direction, the grating structure is treated as a “grating waveguide” similar to a dielectric waveguide.

The mode profile and effective index of this structure is then used in the longitudinal

DFB model to predict the threshold gain and emission profiles.

Figure 1-1. 3-D schematic view of a 2-D surface-emitting laser and second-order DFB gratings with GaAs- Au gratings [1]

Coupled-mode theory is a very useful method for the analysis of coupled optical waveguides as well as waveguides with integrated grating structures. Through perturbation analysis, the interaction between different eigen modes of a weakly-coupled structure can be calculated. The original coupled mode theory of DFB structures were presented in a series of papers by Streifer [2]-[3], which later generalized as the rigorous coupled wave analysis [4].

We start from the scalar wave equation [2]:

ퟐ 2 2 훁 퐸푦(푥, 푧) + 푘0 푛 (푥, 푧)퐸푦(푥, 푧) = 0

휕2퐸 휕2퐸 푦 + 푦 + 푘2푛2(푥, 푧)퐸 (푥, 푧) = 0 (1-1) 휕푥2 휕푧2 0 푦

2 where 푛 (x, z) is the complex-valued dielectric function representing the layered

waveguide structure. k0 = 2π/λ is the vacuum wave vector at the resonant wavelength λ.

For a periodic structure, this refractive index distribution can be expressed as an infinite

Fourier series as:

2 2 ∞ −푖2푚휋푧/훬 푛 (푥, 푧) = 푛̃0(푥) + ∑푚=−∞ 퐴푚(푥)푒 (1-2) 푚≠0

2 2 2 2iα̃(x)n0(x) α̃ (x) ñ0(x) = n0(x) − − 2 (can be neglected) (1-3) k0 k0

2 n0(x) is the zeroth-order refractive index profile in the structure for which the grating layer is represented by the spatial average dielectric constant. α̃(x) represents the gain

(α̃> 0) or loss (α̃< 0) in each region [3] and the refractive index of the waveguide is a function of the periodic profile of the grating inside the waveguide. A generic periodic profile can be described as shown in Figure 1-2, and its refractive index distribution function can be expressed as equation (1-4).

Figure 1-2: A guided-wave DFB structure with a grating [3]

2 푛1, 푥 < 0 2 2 2 {푛1[푤2(푥) − 푤1(푥)] + 푛2[Λ + 푤1(푥) − 푤2(푥)]}/Λ, 0 < 푥 < 𝑔 푛0(푥) = 2 푛2, 𝑔 < 푥 < 푡 2 { 푛3, 푡 < 푥

(1-4) where the Fourier coefficients are [6]

0, 푥 < 0, 𝑔 < 푥

Λ 1 2 푖2휋푚푧 ∫ 푛2(푥,푧) exp (− ) 푑푧 Λ Λ − Λ Am(푥) = 2 푛2 − 푛2 푖2휋푚푤 (푥) = 2 1 {exp [− 2 − 푖2휋푚 Λ { exp [−푖2휋푚푤1(푥)/Λ]}, 0 < 푥 < 𝑔

(1-5)

The Fourier expansion terms Am(x) are expressed in terms of the grating profile, where Λ is the grating period and m is the wave order.

Now, assume the grating is infinite in length. The solutions to (1-1) can be found subject to the periodic boundary conditions. We express the Ey(x, z) in a form of Floquet’s theorem:

∞ (푚) 푖훽푚푧 퐸푦(푥, 푧) = ∑푚=−∞ 퐸푦 (푥, 푧)푒 (1-6) with

2πm β = β + (1-7) m 0 Λ where m=0 is the zeroth order case in which there is no grating.

Substituting (1-2) and (1-6) into (1-1) and collecting all terms with the z-dependence we can obtain

2 (푚) 휕 퐸푦 2 2 2 (푚) 2 ∞ (푞) 2 + [푘0푛0(푥) − 훽푚]퐸푦 = −푘0 ∑푞=−∞ 퐴푚−푞(푥)퐸푦 (1-8) 휕푥 푞≠푚

Now consider the resonant case with two counter-propagating waves in the grating structure. Assuming m equals to a particular p, such that the propagation vectors of two coupled counter-propagating modes are exactly equal, the forward and backwards propagation vectors are related through:

2휋푝 훽 = 훽 + = −훽 (1-9) 푝 0 Λ 0

Now, describing the forward and backward propagating field amplitudes as R and

S, we can write the cavity field distribution of these modes as:

(0) 퐸푦 (푥, 푧) = 푅(푧)ℰ0(푥) (1-10)

and

(푝) 퐸푦 (푥, 푧) = 푆(푧)ℰ0(푥) (1-11)

where ℰ0(x) satisfies the simple slab waveguide equation:

휕2ℰ 0 + (푘2푛2(푥) − 훽2)ℰ = 0 (1-12) 휕푥2 0 0 0 0

R(z) and S(z) are the axial dependences of the waves travelling in the +z and –z directions, respectively in the DFB cavity.

Then we use ℰ0 to derive the other higher order ℰm modes on the right side of (1-

8). These are basically the radiation modes that are excited by the fundamental guided mode (ℰ0 ) of the waveguide. Substitution of (1-8) into (1-12) results in:

( ) 휕2ℇ 푖 (푥) 푚 + [푘2푛2(푥) − 훽2 ]ℇ(푖)(푥) = −푘2퐴 (푥)ℰ (푥),∀푚,푚 ≠ 0,푝 휕푥2 0 0 푚 푚 0 푚−푖 0

(1-13)

Then after substituting (1-9), (1-10), (1-11) into (1-1) and quite a bit of simplifying we will get the well-known coupled-mode equations of a DFB cavity [3]:

′ 푅 + (−훼 − 푖훿 − 푖휉1)푅 = 푖(휅−푝 + 휉2)푆 (1-14)

′ −푆 + (−훼 − 푖훿 − 푖휉3)푆 = 푖(휅푝 + 휉4)푅 (1-15)

where α is the mode gain/loss, 훿 = 훽 − 훽0 is the detuning factor, and the coupling coefficient is

2 푘0 𝑔 2 휅푝 = ∫ 퐴푝(푥)ℇ0(푥)푑푥 (1-16) 2훽0푃 0

All the other terms in the coupled equations are:

∞ 2( ) 푃 = ∫−∞ ℇ0 푥 푑푥 (1-17)

∞ (0) 휉1 = ∑푞=−∞ 휂푞,−푞 (1-18) 푞≠0,푝

∞ (푝) 휉2 = ∑푞=−∞ 휂푞,−푞 (1-19) 푞≠0,푝

∞ (푝) 휉3 = ∑푞=−∞ 휂푞,푝−푞 (1-20) 푞≠0,푝

∞ (0) 휉4 = ∑푞=−∞ 휂푞,푝−푞 (1-21) 푞≠0,푝 and

2 (푖) 푘0 𝑔 (푖) 휂푟,푠 = ∫ 퐴푠(푥)ℰ0(푥)ℇ푟 (푥)푑푥 (1-22) 2훽0푃 0

0 Since 퐸푚(푥) is generated by the wave in the +z direction, 휉1 represents the first-

0 0 order effect of 퐸푚(푥) producing waves with 푚 ≠ 푝 and those waves reacting on 퐸푚(푥)

(푝) (0) itself. Similarly, 휉3 represents the reaction of all partial waves, 퐸푝+푚(푥), (except 퐸푦 (푥)) excited by the –z wave back to itself. So, in any teeth, 휉1 = 휉3. Their imaginary parts must be positive, because the radiating waves always carry power away from the laser or waveguide. Their real parts act to shift the wavelength. [5]

The quantities 휉2 and 휉4 differ somewhat from 휉1. 휉2 represents the sum effects of

(푝) all partial waves generated by 퐸푦 (푥) on the +z wave, and 휉4 plays the same role for

(0) waves generated by 퐸푦 (푥) on the -z wave. Thus 휉2 and 휉4 are coupling terms via the partial waves and can be considered as “corrections” to the coupling coefficient. For symmetrically shaped teeth, 휉2 = 휉4.

Equations (1-14) and (1-15) form the basis of the DFB coupled wave analysis. In the absence of radiation out-coupling, both 휉1 and 휉3 are real, and the coupled wave equation boils down to a simple system consisting of two counter-propagating waves. In many cases, the partial wave coupling terms 휉 are simply dropped and the coupled wave equations treated with just one coupling coefficient 휅푝. This is sufficient in most edge- emitting DFB structures. For surface-emission, a second-order or higher order grating has to be used, and hence the radiation modes and all partial waves have to be included in the analysis.

Figure 1-3: Relationship between threshold gain L and detuning coefficient L of a mirrorless (anti-

reflection coated) first order index-coupled DFB laser

Figure 1-3 shows our calculated results of the lasing threshold gain L vs the detuning factor L for various values of coupling coefficient L. As expected, the threshold gain decreases as the coupling coefficient increases on both sides of the L=0 line. The slight asymmetry between the two sides are due to the partial wave coupling terms 휉푖. This is consisted with other published results. In the absence of these terms partial wave coupling, the cavity would be exactly symmetric about the L=0 line.

CHAPTER 2

LATERAL WAVEGUIDING IN A PERIODIC STRUCTURE

2.1 Introduction of lateral grating waveguide

A one-dimensional DFB laser typically consists of a dielectric waveguide

(typically a ridge) with a longitudinally etched grating (which may be buried below the dielectric surface, or simply on the surface as a metal grating). We will model the 2D

DFB as a lateral waveguide first, and then consider the longitudinal grating separately. In a 2D DFB cavity, there is no dielectric waveguide in the conventional sense. However, guidance is still present because the gratings provide a confining effect similar to that of a dielectric waveguide

Before analyzing 2-D DFB case, let’s see the case in which only lateral guiding via a grating exists. If the propagation of light is parallel to the grating teeth, that structure can be treated as a waveguide in which the edges can be replaced by a DBR reflector. We will refer to this structure as a “grating waveguide”. This is shown in

Figure 2-1. The derivation presented here is based on [6] and [7].

Figure 2-1. Lateral grating waveguide structure [6]

In this Chapter, we will derive the guided wave equations from fundamental principles and get the equivalent effective indices of the grating waveguide, then use them in the analysis with a longitudinal grating to model the 2-D DFB resonator.

2.2 Analysis

A grating waveguide structure is shown in Fig 1. The width of the waveguide is

W, and the field propagation direction is along z. We begin from the two-dimensional wave equations (2-1) :

휕2퐸 휕2퐸 + + 푘2푛2(푦,푧)퐸(푦,푧) = 0 (2-1) 휕푦2 휕푧2 0

We assume the grating and any loss or gain in the material to be a small perturbation to an average background index 푛0 , such as

푛(푦, 푧) = 푛0 + ∆푛푐표푠(퐾푦) + 푗∆푛i (2-2)

12 where K is the grating constant (K = 2π/Λ), ∆n is the amplitude fluctuation of refractive index, ∆ni is the imaginary part of the refractive index which may include a gain or loss modulation (gain-coupled or loss-coupled DFB). For this case, we will ignore all radiation mode coupling terms, and only keep the coupling coefficient .

Representing the counter-propagating waves along ±y directions as 푅(푦, 푧) and

푆(푦, 푧) , we will write the field inside the waveguide as a summation of two-fold degeneracy for each 푚-th order mode as:

∞ −푗푘푏푦 푗푘푏푦 −푗훽푧 퐸(푦, 푧) = ∑푚=−∞[푅푚(푥, 푧)푒 + 푆푚(푥, 푧)푒 ]푒 (2-3)

Substituting (2-3) into (2-1) and simplifying for Rm(y, z) and Sm(y, z) we can get the following expressions:

휕2푅 푒−푗푘푏푦 휕2푅 휕푅 푚 = [ 푚 − 2푗푘 푚 − 푘2푅 ]푒−푗푘푏푦 (2-4) 휕푦2 휕푦2 푏 휕푦 푏 푚

휕2푆 푒−푗푘푏푦 휕2푆 휕푅 푚 = [ 푚 + 2푗푘 푚 − 푘2푆 ]푒−푗푘푏푦 (2-5) 휕푦2 휕푦2 푏 휕푦 푏 푚

휕2푅 푒−푗푘푏푦 휕2푅 휕푅 푚 = [ 푚 − 2푗훽 푚 − 훽2푅 ]푒−푗훽푧 (2-6) 휕푧2 휕푧2 휕푧 푚

휕2푆 푒−푗푘푏푦 휕2푆 휕푆 푚 = [ 푚 − 2푗훽 푚 − 훽2푆 ]푒−푗훽푧 (2-7) 휕푧2 휕푧2 휕푧 푚

∂2R ∂2S ∂2R m m m Then we use the slowly varying envelope approximation to neglect ∂y2 , ∂y2 , ∂z2 ,

2 ∂ Sm ∂z2 . Then substituting (2-2), (2-3) into (2-1), and assuming a first-order Bragg grating such that 푘푏 = 퐾/2, we can obtain

휕푅 휕푅 (−2푗푘 푚 − 푘2푅 ) + (−2푗훽 푚 − 훽2푅 ) = −푘2푛2푅 − 푘2푛 Δ푛푆 푏 휕푦 푏 푚 휕푧 푚 0 0 푚 0 0 푚 (2-8) and

휕푅 휕푆 (2푗푘 푚 − 푘2푆 ) + (−2푗훽 푚 − 훽2푆 ) = −푘2푛2푆 − 푘2푛 Δ푛푅 푏 휕푦 푏 푚 휕푧 푚 0 0 푚 0 0 푚 (2-9)

Finally, they can be simplified to produce the following two coupled wave equations:

휕푅 (푦,푧) 휕푅 (푦,푧) 푘 푚 + 훽 푚 = 푘훼푅 (푦, 푧) − 푗푘휅푆 (푦, 푧) (2-10) 푏 휕푦 휕푧 푚 푚

휕푆 (푦,푧) 휕푆 (푦,푧) −푘 푚 + 훽 푚 = 푘훼푆 (푦, 푧) − 푗푘휅푅 (푦, 푧) (2-11) 푏 휕푦 휕푧 푚 푚 where the wave vector k, propagation constant 훽, coupling coefficient κ and the gain/loss coefficient α are defined as:

푘 = 푘0푛0 (2-12)

2 2 훽 = √푘 − 푘푏 (2-13)

휅 = 푘0Δ푛/2 (2-14)

훼 = 푘0∆푛i (2-15)

The coupled wave equations (2-10) and (2-11) can be solved by substituting one into the other and by writing them as second order differential equations involving only

푅(푦, 푧) and 푆(푦, 푧). The solution becomes:

∗ + −푗푏푚푧 − 푗푏푚푧 푅(푦, 푧) = 퐴푚(푦)푒 + 퐴푚(푦)푒 (2-16)

∗ + −푗푏푚푧 − 푗푏푚푧 푆(푦, 푧) = 퐵푚(푦)푒 + 퐵푚(푦)푒 (2-17)

Generally, the longitudinal parameter bm is complex valued. The 푧 dependence of

푅(푦, 푧) or 푆(푦, 푧) can be separated out and the solution expressed in terms of lateral modes which are independent of 푧, such as:

휅훼 ∗ ( −푗훽)푧 + 푗푏푚푧 − −푗푏푚푧 훽 퐸(푦, 푧) = ∑푚[푀푚(푦)푒 + 푀푚(푦)푒 ] 푒 (2-18)

+ − where 푀푚(푦) and 푀푚(y) can be considered as the 푚-th order lateral ‘waveguide-like’ modes of the grating waveguide.

+ − The expressions for 푀푚(푦) and 푀푚(y) can be obtained as:

+ + −푗푘푏푦 + 푗푘푏푦 푀푚(푦) = 퐴푚(푦)푒 + 퐵푚(푦)푒 (2-19)

− − −푗푘푏푦 − 푗푘푏푦 푀푚(푦) = 퐴푚(푦)푒 + 퐵푚(푦)푒 (2-20) where

+ −푗푎푚(푦+푊) 푗푎푚푦 퐴푚(푦) = 푗푟푚퐴푚[푒 − 푒 ] (2-21)

+ 푗푚푦 2 −푗푎푚(푦+푊) 퐵푚(푦) = 퐴푚[푒 − 푟푚푒 ] (2-22)

∗ ∗ − ∗ −푗푎푚푦 푗푎푚(푦+푊) 퐴푚(푦) = 푗푟푚퐵푚[푒 − 푒 ] (2-23)

∗ ∗ − −푗푎푚푦 ∗2 푗푎푚(푦+푊) 퐵푚(푦) = 퐵푚[푒 − 푟푚 푒 ] (2-24)

and Am and Bm are the amplitudes which depend on the initial excitation by the incident input field, and W is the width of grating waveguide.

The additional parameters in the above expressions were defined as:

푘 2 2 푎푚 = √휍푚(휅 + 훼 ) (2-25) 푘푏

푘 푏 = √(1 + 휍 )휅2 + 휍 훼2) (2-26) 푚 훽 푚 푚

푘휅 푟푚 = (2-27) 푎푚푘푏+푏푚훽

and ςm is the 푛-th order solution of the lateral resonance condition

2 −푗푎푚푊 푟푚푒 = 1 (2-28)

Equation (2-28) is the resonant condition of the grating waveguide. The solution of this equation results in the two quantities rm and am. From this, the third quantity bm can be evaluated which then allows us to calculate the field distribution of the lateral mode from expressions (21 - 8) – (2-24).

Even though we have referred to these as ‘modes’, there are important differences between conventional dielectric waveguides and grating waveguides. Conventional dielectric waveguides are based on total internal reflection, and can be perfectly losses, at least in theory. Grating waveguides are always lossy. This loss is a function of the grating

16 structure and also the order of the mode. This loss increases with increasing order of the modes. In the above derivation we have assumed that there are anti-reflection coatings on both edges of the cavity and that the waveguide structure is perfectly symmetric. As a

+ − result, both 푀푚(푦) and 푀푚(푥) will have identical longitudinal losses. If the cavity has some asymmetry (which could arise from different termination reflectivities), these modes will experience slightly different longitudinal losses. This is similar in behavior to conventional DFB cavities [7]. Furthermore, although are referred to them as degenerate

+ modes, the propagation constants 훽 of these two degenerate modes are not equal. 푀푚(푦)

− will propagate with 훽 + Re{푏푚}, while 푀푚(푦) propagates with 훽 − Re{푏푚}. Therefore, we can define the complex effective indices of these modes as:

+ 훽+푅푒{푏푚} 퐼푚{푏푚} 푛푚 = + 푗 (2-29) 푘0 푘0

− 훽−푅푒{푏푚} 퐼푚{푏푚} 푛푚 = + 푗 (2-30) 푘0 푘0

The following figure shows the calculated effective indices and the longitudinal losses bm of several modes in a 0.5 μm period grating waveguide for several values of κW between 1.0 and 10.0.

Figure 2-2. Modes of the grating waveguide at 1.0 휇푚, with 푊 = 100 휇푚, a grating period of 0.5 휇푚, −3 푛0 = 3.2, 훥푛 = 3.2 × 10 , and for various of 휅푊. [6] The primary design consideration in high power semiconductor lasers is to reduce mode loss and increase the discrimination of threshold gain so that all of the gain goes towards a single spatial mode during lasing. Otherwise, the possibility of multi-mode oscillation through filamentation becomes enhanced, reducing the brightness of the output. In the above figure, for a given W. the difference in loss between the adjacent modes represent the threshold gain discrimination. Higher W has lower loss, but the modes are closer together in loss. Therefore, κW represents a design compromise between the efficiency and the brightness of the laser output.

Figure 2-3. First two modes of the grating waveguide for different values of 휅푊. [6]

Figure 2-3 shows the envelopes of the first two modes for different values of κW

+ − in a 0.5 μm period grating waveguide. It should be noted that Mm(y) and Mm(x) have the same magnitude but differ only in their phase. (We have only shown the amplitudes in the figure). As expected, the fields are strongly confined for larger values of κW, which results in a smaller longitudinal propagation loss. (We assumed the initial conditions to be Am = 1 and Bm = 1).

It is also important to discuss the validity of the slowly varying envelope approximation in the above case. We invoked this approximation to derive the coupled

2 2 wave equations (2-4), (2-5), (2-6), (2-7) by dropping ∂ Rm(y, z)/ ∂y in favor of kb ∂Rm(y, z)/ ∂y. This approximation is quite valid when the grating period is much smaller than the device geometry, which is the case in conventional DFB cavities. The

2 2 magnitude of ∂ Rm(y, z)/ ∂y could become comparable in magnitude to kb ∂Rm(y, z)/

∂y if the grating period is large enough, so that it will change the effective indices and loss. Therefore, care must be taken to ensure that the grating period is not too large to violate this approximation.

2.3 Application for 2D-DFB

Figure 2-4. Holes etched on the top of cladding surface in lab

Figure 2-4 shows an example 2D DFB cavity fabricated using interference lithography. Although the pattern shows a repeating array of circles, it is in fact created from two crossed one-dimensional periodic lines. For our study, we will use a cosine variation in refractive index to simulate the hole on the top of waveguide cladding surface.

Figure 2-5.Cosine function simulation of the hole on cladding surface

We treat this model as waveguide grating first, which means longitudinal grating is neglected in order to calculate the effective indices in the lateral direction. Then we apply those effective indices into the longitudinal resonant case.

Figure 2-6. A guided-wave laser with “cosine” shape grating, where g is the grating height while t is the outer guide thickness A grating shape shown in Figure 2-6 can be written as

(푛2−푛2) 2푥−𝑔 푛2(푥) = 푛2 + 1 2 cos−1 (0 < x < g) (2-31) 0 2 휋 𝑔

Λ Λ 2푥−𝑔 푤 (푥) = − cos−1 (0 < x < g) (2-32) 1 2 2휋 𝑔

Λ Λ 2푥−𝑔 푤 (푥) = + cos−1 (0 < x < g) (2-33) 2 2 2휋 𝑔

Substitute (2-32) and (2-33) into (1-5), we will get

푛2−푛2 2푥−𝑔 Δ푛(푥) = 2 1 sin(cos−1 ) (0 < x < g) (2-34) 휋푛0(푥) 𝑔

Use (2-31) and (2-34) in (2-2), we can get n+(푥) and 푛−(푥) from (2-29) and (2-30) respectively. Figure 2-7 shows the effective index of the grating as a function of depth.

The n+(푥) and 푛−(푥) correspond to the two degenerate modes of the grating waveguide.

+ − Figure 2-7. Real parts of 푛 (푥) and 푛 (푥) vs. depth x with 푛1, 푛3 = 3.2, 푛2 = 3.3, 휆 = 1.0휇푚,

W=100휇푚, 훬 = 0.5휇푚, g=1 휇푚, 푡 = 1.5휇푚

When considering the longitudinal direction, g will be different with z in a periodic manner.

Figure 2-8. Differ from conventional DFB, n+ and n− change with z not only x

+ − Now, we know that 푛 (푥) and 푛 (푥) are numerical function of z. Assume 푛1, 3 =

3.2, 푛2 = 3.3, 휆 = 1.0휇푚, , 푡 = 1.5휇푚 ,W=100휇푚, 훬 = 0.5휇푚, 훬푧 = 0.32휇푚 (we will discuss it in the next chapter). We will get 푛+(푥, 푧) and 푛−(푥, 푧) as following: (only one period will be demonstrated)

Figure 2-9. Real part of n+

Figure 2-10. Real part of n−

Figure 2-11. Imaginary part of n+ and n−

Both n+(x, z) and n−(x, z) are symmetric in the longitudinal direction. We can assume their duty cycle to be 0.5.

For refractive indices in the cladding and substrates can be written as:

2 2 2 푘0푛푒1 = √푘0푛1 − 푘푏 + 푖푚푎𝑔푖푛푎푟푦 푝푎푟푡

2 2 2 푘0푛푒2 = √푘0푛2 − 푘푏 + 푖푚푎𝑔푖푛푎푟푦 푝푎푟푡

We use x → 0 and x → g to simulate the imaginary parts of cladding and substrate indices

ne1 =3.0397+0.005472i

ne2 =3.1448+0.005289i

We will use this data for the examples in following chapters.

CHAPTER 3

COUPLED MODE DESCRIPTION OF THE TWO-

DIMENSIONAL DFB CAVITY

In [2] and [8], rectangular grating and trapezoidal-shaped grating were analyzed already. The authors used waveguide and Airy functions to express the higher mode electric fields. However, when we convert the 2-D DFB to a 1-D cavity using the grating waveguide model that was described in Chapter 2, the effective index 푛(푥, 푧) is not shaped in a manner that can be described by an analytical function. It has to be described as a complex valued numerical function. So we need a new way to deal with this problem.

3.1 Utilizing numerical shooting method to calculate 푬ퟎ

We will derive the numerical shooting method for Maxwell’s equation and apply it to solve fundamental field 퐸0 of our waveguide mode.

Start with the 1-D plane wave propagation equation in 푥 direction. [9]

휕2퐸(푥) = −푘2푛2(푥)퐸(푥) + 훽2퐸(푥) (3-1) 휕푥2 0 0

The first step is to introduce a computational grid. Divide the intervals [a,b] into N equal- sized subintervals respectively , where N is a positive integer and h is the step size.

푏−푎 ℎ = (3-2) 푁

With the computational grid defined, next evaluate the governing differential equation at each interior point (those for which j=1,2,3,…, N-1) .

The second order derivative at the grid point 푗 can be expressed using infinite- difference discretization as:

휕2퐸(푥) 퐸(푥 )−2퐸(푥 )+퐸(푥 ) | = 푗−1 푗 푗+1 (3-3) 휕푥2 ℎ2 푥푗

Substituting (3-3) into (3-1) allows us to express the one-dimensional electromagnetic wave equation in finite-difference notation as:

퐸(푥 )−2퐸(푥 )+퐸(푥 ) 푗−1 푗 푗+1 = [−푘2푛2(푥 ) + 훽2]퐸(푥 ) (3-4) ℎ2 0 푗 0 푗

If we know the first two values of the electromagnetic field 퐸(푥푖−1) and 퐸(푥푖) we can express the subsequent value 퐸(푥푖+1)

2 2 2 2 퐸(푥푗+1) = (2 − 푛 (푥푗)푘0ℎ )퐸(푥푗) − 퐸(푥푗−1) + 훽0 퐸(푥푗) (3-5)

In the case of a waveguide, we make the first and second values of field 퐸(푥1) and 퐸(푥2) to be 0 and 1, respectively. Then, the Newton-Raphson method is applied to recursively modify 푅푒(훽0) so that the last value 퐸푁 approaches to 0. As a result, this method can be applied to any refractive index distribution, not just rectangular or trapezoidal shapes.

In our case, the two-dimensional refractive index distribution is reduced to a one-dimensional effective index which can be written as:

2 푛푒1, 푥 < 0 + 2 2 (푛 ) (푥), 0 < 푥 < 𝑔 푛 (푥) = 2 푛푒2, 𝑔 < 푥 < 푡 2 { 푛푒1, 푡 < 푥

(3-6)

where (푛+)2(푥) is the mean of (푛+)2(푥, 푧) with respect to a period of 푧.

After normalization (make 푃 in (1-17) equal to 1), we can get 퐸0. (Since it is difficult to evaluate when the indices are complex, as long as the imaginary parts are small we ignore this as an approximation).

Figure 3-1. 퐸0 in 2-D DFB waveguide (we separate x from -2.5-3.5 into 9999 steps)

3.2 Calculation of Higher order of 푬풎 using the matrix solution method

For second order gratings, we substitute 푝 = −2, 푅푒(훽0) into (1-9). It’s simple to calculate that the longitudinal grating period Λ푧 is supposed to be 0.32휇푚.

Let’s rewrite (1-13) as:

(i) ∂2E (x) (i) m + [k2n2(x) − β2 ]E (x) = −k2A (x)E (x),∀m,m ≠ 0,p (3-7) ∂x2 0 0 m m 0 m−i 0

We can still write 퐸푚(푥) as 퐸푚(푥푗), then

(푖) (푖) (푖) 퐸 (푥 )−2퐸 (푥 )+퐸 (푥 ) 푚 푗+1 푚 푗 푚 푗−1 + (푘2푛2(푥 ) − 훽2 )퐸(푖)(푥 ) = −푘2퐴 퐸 (푥 ), 푖 = 0, 푝 ℎ2 0 푗 푚 푚 푗 0 푚−푖 0 푗

(3-8)

We can collect all the coefficients and write (3-8) as

(i) (i) (i) (i) Bj,j−1Em (xj−1) + Bj,jEm (xj) + Bj,j+1Em (xj+1) = C (xj) (3-9)

When j=1 and j=N, where N is the last point in the one-dimensional grid. The field at the boundaries will be modeled as outgoing plane waves:

−푗푘푥ℎ 퐸(푥2) = 퐸(푥1)푒 (3-10)

′ 푗푘푥ℎ 퐸(푥푁) = 퐸(푥푁−1)푒 (3-11)

2 2 2 ′ 2 2 2 where 푘푥 = √푘0푛푒1 − 훽푚, 푘푥 = √푘0푛푒2 − 훽푚

푗푘푥ℎ 2 2 2 2 퐵1,1 = (푒 − 2)/ℎ − 훽푚 + 푘0푛 (푥1) (3-12)

′ 푗푘푥∆푥 2 2 2 2 퐵푁,푁 = (푒 − 2)/ℎ − 훽푚 + 푘0푛 (푥푁) (3-13)

2 퐵1,2 = 퐵푁,푁−1 = 1/ℎ (3-14)

When 1

2 퐵 = − − 훽2 + 푘2푛2(푥 ) (3-15) 푗,푗 ℎ2 푚 0 푗

2 퐵푗,푗−1 = 퐵푗,푗+1 = 1/ℎ (3-16)

(푖) 2 퐶 (푗) = −푘0 퐴푚−푖퐸0(푥푗), 푖 = 0, 푝 (3-17) where

훬 1 2 2 푗2휋푚푧 퐴푚(푥) = ∫ 훬 푛 (푥, 푧) 푒푥푝 (− ) 푑푧 (0 < 푥 < 𝑔) (3-18) 훬 − 훬 2

∗ 2 퐴푚 = 퐴−푚 when we ignore the imaginary part of 푛 for two reasons. If we consider the imaginary part, 퐴푚 = 퐴−푚, then the imaginary parts of 휉1 and 휉3are negative.

(We have discussed this point in chapter 1). Another reason is the imaginary part of

푛2 is much smaller than the real parts.

Through this equation, we can figure out that 퐴푚 is real if the shape of the grating is symmetric. (3-9) can be written as a matrix:

(푖) (푖) 퐵1,1 퐵1,2 0 ⋯0 퐸푚 (푥1) 퐶 (1) (푖) 퐵2,1 퐵2,2 퐵2,3 ⋯0 퐶(푖)(2) 퐸푚 (푥2) = (3-19) ⋮ ⋮ ⋮ 0 0 ⋯퐵푗,푗−1 퐵푗,푗 퐵푗,푗+1⋯0 ⋮ ⋮ ⋮ ⋮ ⋮ (푖) ( 0 0 ⋯퐵푁,푁−1 퐵푁,푁 ) (푖) (퐶 (푁)) (퐸푚 (푥푁))

(푖) (푖) 퐵 ∙ 퐸푚 = 퐶 (3-20)

(푖) (푖) B is a 푁 × 푁 matrix, 퐸푚 and 퐶 are 푁 × 1 matrices.

(0) (푝) Then, 퐸푚 (forward wave) and 퐸푚 (푏푎푐푘푤푎푟푑 푤푎푣푒) can be calculated by solving the above matrix equation. The resulting field distributions are shown below.

These are the partial waves excited by the fundamental guided mode (lasing mode) of the structure.

(0) Figure 3-2.The partial waves 퐸푚 for m=-8-5 (m ≠ 0, −2)

As we see in the case with the second order grating (푝 = −2), only the mode with

푚 = −1 becomes a radiation mode while all other modes are decaying modes in shape and magnitude. Therefore, for a second –order grating the m=-1 the mode will radiate and

32 will form the output of the laser cavity. For a third-order grating, two modes m=-1 and m=-2 will radiate (this case is not shown on the above plot).

3.3 Calculation of coupling coefficient and other constants

The summations in (1-16), (1-18)~(1-22) are truncated so that only −8 ≤ m ≤ 5 terms are included. In other words, only 14 partial waves are considered. The overall accuracy is approximately 1 percent, and the computational solution procedures rather than the truncation are responsible for most of this error.

For 푛+ case,

−1 휅푝 = 휅−푝 = −34.44 푐푚

−1 휉1 = 휉3 = −0.40 + 푖0.02 푐푚

−1 휉2 = 휉4 = −0.63 + 푖0.02 푐푚

For 푛− case,

−1 휅푝 = 휅−푝 = −158.35 푐푚

−1 휉1 = 휉3 = −2.73 + 푖0.03 푐푚

−1 휉2 = 휉4 = −4.40 + 푖0.03 푐푚

These parameters can now be used in the coupled mode equations, which are restated from equations (1-14) and (1-15)::

′ 푅 + (−훼 − 푖훿 − 푖휉1)푅 = 푖(휅푝 + 휉2)푆 (3-21)

′ −푆 + (−훼 − 푖훿 − 푖휉1)푆 = 푖(휅푝 + 휉2)푅 (3-22)

3.4 Summary & Discussions

In this chapter, we mainly discussed the numerical shooting method and Newton-

Raphson method to solve the fundamental mode of irregular (non-analytical) shaped refractive index distribution. Its precision depends on the computational grid step size.

Smaller the step is, more exact the result can be. Calculation for the integrals of 퐴푚 follows the same rule. In actual calculation, 휉1 and 휉3 are slightly different due to floating point overflow problems in the computer.

CHAPTER 4

NEAR FIELD PROFILE AND THRESHOLD GAIN OF 2D DFB

STRUCTURES

4.1 Near field and threshold gain

The concept of 2D DFB was proposed by Shyh Wang in 1973 [10] but they have never been studied thoroughly. On the other hand, there are a large number of papers on

1D surface-emitting DFB (SE-DFB). Multiple quantum well structures have been discussed in [11]; Emission characteristics have been given by [12], [13].

In our case, we assume no reflection from the edge of both ends of the DFB

퐿 퐿 R(− ) = S ( ) = 0 cavity: ( 2 2 ). In [3] and [14], the general solutions of (3-21) and (3-22) are in the form

훾푧 −훾푧 푅(푧) = 푟1푒 + 푟2푒 (4-1)

and

훾푧 −훾푧 푆(푧) = 푠1푒 + 푠2푒 (4-2) with the complex propagation constant γ obeying the dispersion relation

2 2 훾 = (훼 + 푖훿 + 푖휉1) + (휉2 + 휅−푝)(휉4 + 휅푝) (4-3)

Because of the assumed symmetry of the device we get symmetric E(−z) = E(z) and antisymmetric E(−z) = −E(z) field solutions (L is the length of the grating).

Combining this with the zero-reflective boundary condition, we can get

1 푅(푧) = sinh 훾(푧 + 퐿) (4-4) 2

1 푆(푧) = ±sinh 훾(푧 − 퐿) (4-5) 2 where the plus sign represents symmetric case and minus sign represents antisymmetric one. We can get a set of solutions which correspond to a discrete set of eigenvalues γ.

This set corresponds to a structure with given length and a given coupling coefficient. To determine the eigenvalues, we insert (4-4), (4-5) into (3-21), (3-22), form the sum and the difference of the resulting equations and drop the common factors. After sorting out, we can get

훾 = ±i√(휉2 + 휅−푝)(휉4 + 휅푝)sinh 훾퐿 (4-6)

Substituting (4-6) into (4-3):

2 훼 + 푖훿 + 푖휉1 = √훾 − (휉2 + 휅−푝)(휉4 + 휅푝) (4-7)

Figure 4-1. Intensity of multiple modes as a function of L for the case of 푛+

Figure 4-2. Intensity of multiple modes as a function of L for the case of 푛−

Figure 4-3. Threshold gain for different L for both 푛+(circle and diamond points) and 푛−(rectangular and

star points), red lines are symmetric modes while blue lines are antisymmetric modes

Threshold gains are virtually unaffected by the inclusion of 휉푖 because the coupling |ξ4 + κp|is only slightly different than |κp| for 푝 = −2. The curve will change dramatically when the grating order is higher. It is important to notice that those points are not symmetric because radiation is considered.

In conventional DFB structures, the second order grating enables the radiation mode to emit along both directions, into the cladding and into the substrate [15].

At present, first-order gratings are used in most DFB lasers. However, second- order gratings are sometimes preferred because they are usually easier to make, since their grating period is twice as large as the of first-order gratings.

Figure 4-4. Graphic representation of the relationship between the wave vectors of the guided mode, the

radiation modes, and the grating vector of a simple thin-film wave guide

+ − + However, in our case, there are 푛 and 푛 . It demonstrates that for 푛 , 훽−1 = 0,

− ′ which guarantees surface emission. But for 푛 , 훽−1 is slightly different, its angles to cladding surface and substrate are respectively:

′ ′ 2휋 훽0 − −1 훽−1 −1 Λ 휃−1|푐푙푎푑푑푖푛𝑔 = cos = cos = 90.078° |푘0푛푒1| |푘0푛푒1|

′ ′ 2휋 훽0 − ′ −1 훽−1 −1 Λ 휃−1|푠푢푏푠푡푟푎푡푒 = cos = cos = 90.075° |푘0푛푒2| |푘0푛푒2|

4.2 Comparison of 1-D and 2-D

Repeating the process in chapter 3 for 1-D case (no refractive indices variation in lateral direction, using the cosine variation along the longitudinal direction like in (2-31)), we can calculate the coupling constants to be as:

−1 휅푝 = 휅−푝 = −236.31 푐푚

−1 휉1 = 휉3 = −4.52 + 푖0.18 푐푚

−1 휉2 = 휉4 = −7.34 + 푖0.18 푐푚

Its period for second order grating is 0.3μm

Using these parameters we can get near the field radiation pattern and threshold gain as shown in the following plots:

Figure 4-5. Intensity (1-D DFB) of different modes in different cavity length

Figure 4-6. Threshold gain (1-D DFB) for different modes with the same L as 2-D case

Comparing their properties, for the same 휅푝퐿 (eigen-values γL are the same), their field profiles are the same and the differences in their threshold gains are small. We found 휅푝 in the 1-D case is much larger than 2-D case. Thus, to reach the same 휅푝퐿, 퐿 in

2-D is longer than 1-D’s.

4.3 Summary & Discussions

In this study, we have considered a DFB cavity with anti-reflection coatings on both sides (no reflection). However, in practical cases, the reflection from the ends can be used to induce an asymmetry in the cavity which then breaks the degeneracy of the two

DFB modes. This will then result in single frequency lasing [16]. Inserting a phase shift in the middle of the DFB also serves a similar purpose [1],[17]. Filamentation, instability and methods to alleviate them have been discussed in [18].

In fact, not only can reflection coefficient and phase shifts be changed, but other parameters such as grating height 𝑔, duty cycle 푤/Λ, lateral width W, difference of

2 2 square of refractive indice 푛2 − 푛1 or refractive index 푛3 even outer guide thickness 푡 can be controlled. Many of these parameters will affect the lateral and the longitudinal

2 2 resonance properties. For instance, 푛2 − 푛1, W or 푛3 will affect the effective index in the lateral direction (discussed in chapter 2). We have not presented those results here.

There’s another factor we need to pay attention to. In our case we assumed, 훬푥 = 0.5휇푚 and 훬푧 = 0.32휇푚. In order to make the radiation emit perpendicular to the cladding surface, the “holes” (at least in this example) have to be elliptical, and not round as shown in the SEM image Figure 2-4. We have shown that, for the 2-D case, the emission downward the substrate is close to 90°. To increase its emitting power, we can also reflect the downward radiation back to the top. Then, the radiation power of each mode becomes [2]

1 2 2 2 2 2 2 2 2 푃푚 = 푅푒 {√푘0푛1 − 훽푚|ℰ푚(0)| + √푘0푛3 − 훽푚|ℰ푚(푡)| } (4-8) 2휔휇0

Here the first and second bracketed terms are the parts radiated into cladding and

2 2 2 2 2 substrate respectively, and if 훽푚 exceeds 푘0 푛1 and/or 푘0푛3 , imaginary terms do not contribute to 푃푚.

The process is much complicated in practice. It includes the electrical process: electrical carrier transport, the electro-optic process: the carrier-photon interactions, the optical process: optical wave propagation, the thermal process: heat transport. In our model, we did not consider many of these parameters and limited ourselves to the optical

DFB cavity only [19]. Even in coupled wave equations, we only used the scalar wave equation. We ignored the time derivative terms and spontaneous emission effect

(Longevin force) [20]. So there is still much left for other researchers to do.

CHAPTER 5

THE CURRENT AND FUTURE RESEARCH

Semiconductor lasers still lag behind other laser types in terms of beam quality and power output. High power semiconductor lasers today are achieved by incoherently combining multiple lasers. A high power output from a single cavity in a single spatial mode at a single frequency has yet to be realized. Such lasers could potentially replace solid state lasers with much power higher efficiency, portability and robustness.

In [21] and [22], they incorporated a photonic-crystal structure into a VCSEL

(vertical-cavity surface-emitting laser), referred to as a PCSEL. In this structure, vertical asymmetric air holes are formed during MOCVD growth (metal-organic chemical vapor deposition). This is a good way for high power CW operation in two-dimensional SE-

DFB [23]. Its superlattice structures also provide a possibility of multilayer [24]. What’s more, photonic crystals can control the propagation of light. The light can be diffracted normal to the surface when the standing wave of the Γ point of photonic band is utilized.

By using this property, emission angle can be narrowed under room temperature. [25]

From researches above, higher output power under room-temperature, CW operation has been reached up with narrow divergence angle. That represents an important milestone in the field of laser because it highly improves the quality of the beam and overcome the current application bottlenecks. As the lower of threshold gain

46 and the wider tunable wavelength range are achieved, more applications the DFB will contribute to the world.

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