INTRODUCTION

I .I INTRODUCTION

Recent progress in optoelectronics, such as in fibre-optic

communication systems is depending primarily on progress in opt0 electronic

semiconductor devices. Among those devices that have been showing the

most remarkable development are Optical power sources such as LEDs and

laser diodes, Photo detectors and Integrated optical devices. In this chapter,

a brief overview is presented on technological trends in these devices that

are at the front edge of the optoelectronic semiconductor devices.

In any case, light source is very essential to have knowledge of the

light characteristics both electrical and optical, before using the sources for a

particular application. The maximum optical power output, wavelength of

emission, linewidth radiation pattern and modulators bandwidths are some of

the basic optical characteristics that one need to know. The electrical

connections, operating voltage, current and their maximum ratings must be

known precisely before switching on the same.

As mentioned in the beginning of this introduction, a light and a photo

detector are the two basic components of an optical fibre system and it is

very essential to know the characteristics of the source and the detector as

well. In the field of optoelectronics and optical communications, a broad

knowledge of different types of sources and detectors, transmitter and

receiver etc., are necessary to deal with the different types of signals, coding,

modulation, de-modulation and receiver sensitivity for basics of

communication engineering.

LED'S and semiconductor laser diodes are used widely as optical

power sources in optoelectronic systems. Recently, so called dynamic single

mode laser diodes which emit a single wavelength spectrum such as

distributed feed back (DFB) type laser diode and distributed Bragg reflector

(DBR) type laser diodes are used in practical system in addition to

conventional Febry- Perot type laser diodes.

1. I . I Gallium Arsenide on Silicon Technology

Gallium arsenide, an increasingly important semiconductor material,

does not exist in nature and must therefore be synthesised for high speed

and opt0 electronic devices. The technology of GaAs has developed very

rapidly in the past decade and 3" diameter substrates of much improved

quality (semi-insulting and conducting) are now available for digital, analog

and optical applications. Vapour phase epitaxy (VPE) and vacuum deposition

techniques, molecular beam epitaxy (MBE) In particular are responsible for

many of the recent breakthroughs in novel thin film semiconductor structures.

e.g., superlattice and quantum well devices.

On the applications side, GaAs and related compounds are being

used andior explored for space electronics (due to their radiation hardness),

high speed digital and analog circuits and many facets of optoelectronics.

Employing GaAslAlGaAs modulation doped FET's switching speeds of about

5ps (77K) have been obtained with 0.3 pm gate lengths. Quarter-micron

gate GaAsllnGaAslAIGaAs FET's have current gain cut-off frequencies of

about 100 GHz with operating frequencies (for power gain) over 200 GHz.

Heterojunction GaAsIAlGaAs bipolar transistor has also shown outstanding

performance with gate delays of about 12 ps and current gain cut-off

frequencies of about 100GHz. In integrated circuits, compound

semiconductor static memories, shift registers, frequency dividers, multipliers

and millimeter wave amplifiers have already been realised. Optoelectronic

devices, particularly lasers are being explored rigorously for their high power

and high speed potential. The low threshold lasers (0.55 mA) provide direct

modulation for digital signals in the gigabitlsec range.

1.7.2 Hybrid Integration Technology

One of the greatest promises of GaAslSi is the partitioning of functions

between GaAs and Si devices on the same chip to optimise overall circuit

operation. For example, clocks, shift registers, and low-densityllow-power

fast cache memories can be made of GaAs while LSI type circuits can be

built mature Si technology, Individual components of Si and GaAs devices

have already been demonstrated in GaAslSi. Among them are the Si

MOSFETs with GaAs LEDs [I] and Si MOSEFTs with GaAs MODFETs 121.

GaAs LEDs are in fact driven by the Si MOSFETs at 27 MHz, a rate that

has limited by the speed of the large size MOSFETs driven are employed.

Recent advances in the performance of electronic and optical devices

fabricated in GaAs on Si substrates have led to the consideration of this

hybrid technology for novel applications. These range from GaAs substrates

with large area, lightweight, high strength Si substrates to high desirable

integration of GaAs and Si devices. Other applications include the use of

GaAs as an interlayer for subsequent growth of long wavelength compound

semiconductors (Ill-V and Il-VI) for focal plane arrays with built-in Si signal

orocessors.

Multifunction-multimaterial integration is also under study for other

applications. For example, current focal plane array detector technology

operating at wavelengths greater than 3 p n utiiises HgCdTe photovoltaic

detectors which are In bumped to Si CCDs for signal processing. Room

temperature alignment for In bumping can cause alignment problems at the

80 K operating temperature even for medium size arrays. Using GaAs as an

interlayer on the Si CCD circuitry for the subsequent deposition of HgCdTe

detectors may produce great cost savings for large area focal plane arrays.

Significant progress has already been made in demonstrating individual

components of this approach. For example, HgCdTe had been demonstrated

on Si substrates following a process that is amenable to Si CCDs [3]. Very

recently, HgCdTe photovoltaic detectors have been operated on GaAslSi

substrates which represents a significant milestone toward large area focal

plane in array [4].

Most recently, improved GaAs growth on Si has led to pn junctions to

allow the investigation of GaAs and Si junction's electrical characteristic [5].

With the additional improvements obtained through furnace annealing at

850°C for 20 min, ideality factors of 1.5 have already been achieved. A Si

super-self-aligned bipolar process actively integrated over GaAsISi self-

aligned bipolar transistors may yield better performance over their GaAs and

Al counterparts [6]. Even without annealing, transistor action between

AlGaAs (emitter)lGaAs (base) and Si (collector) npn transistors has been

obtained [7].

Recent progress in GaAsiSi technology invites considerations of its

imminent applications. It appears that the first serious applications of this

technology wiil be large area focal plane arrays and lightweight solar cells for

space applications where silicon's low density and high strength allow the

use of thinner substrates. Radiation-hard solar cells, already demonstrated

In GaAslSi technology, have great potential because of the availability of

large area substrates. This composite technology can find applications in

GaAs ICs and MMlCs - (Multifunction Multirnaterial Integrated Circuits), opt0

electronics for Si ICs and the integration of GaAs fast circuits with high-

density Si circuits.

For some material systems, it is now possible to form heterojunctions

and interfaces by expitaxial growth techniques (virtually unconstrained by the

lattice parameter of the host semiconductor). This enables electrical and

optical properties to be tailored with unprecedented flexibility and refinement.

These techniques have been used to fabricate transistors with current gain

cut-off frequencies of about 100 GHz using InxGa~.xAslAlxGal.xAs

! Electric field (KVIcm)

Fig. 1.1 A graph of electron drift velocity vs. electric field for three

materials at 300 K.

(high peak electron velocity in Ino,53Gao,riAs as the donar concentrat~on in all

cases is about i015 ~ r n ' ~ )

modulation-doped strained layer quantum wells. The expected frequency

range where the devices exhibit power gain is as high as 400 GHz.

The smaller transistors operate much faster and more economically

and have led to a many thousand-fold increase in speed, a point of

enormous importance to the telecommunication and computing industries.

The speed of transistors has resulted mainly from the reduction of

transit time between terminal. Short transit times s~mply imply that input

signals can reach the output with minimal delay. Choosing semiconductor

structures and materials in which signals propagate at a faster rate can also

shorten transit times very effectively. Since picosecond type transit times are

also possible, the mobility (i.e., acceleration to the ultimate carrier velocity

once a field is applied) is very important.

The semiconductor material determines the electron velocity and

mobility. For example, GaAs has mobility about 6 times larger than that of Si

for a donor concentration of 10" ~ m . ~ . In addition, the peak electron drift

velocity in GaAs is almost twice that in Si and occurs at much lower electric

fields as shown in Fig. 1 .l. Hence GaAs devices can operate faster and at

smaller voltages. Another very important advantage of GaAs is that the

ternary material AI,Ga,.,As that has a larger bandgap, still has a good lattice

match with GaAs and together they form heterojunctions that are the basis

for modern devices. The barrier is formed in the conduction band between

GaAs and AlxGal.,As. The abrupt change in energy bandgap is used to

confine electrons, to control electrons and limit the injection of electrons

across it.

1.1.3 High Speed Semiconductor Laser Technology

Although optical sources for fibre optic communicetion systems are

available for almost hvo decades, high-speed (multigigahertz) "light" sources

Current (mA)

Fig. 1.2 Optical output power vs. input current characteristics and

Voltage-current characteristics of an injecticn laser.

based on semiconductor injection lasers are a recent reality. A GaAlAs

semiconductor laser emits at a wavelength of around 0.85 pm while lasers

constructed from inGaAsP emit at 1.2 to 1.6 pm. These wavelengths are

well matched to the two most common optical fibre types produced today:

muitimode fibre designed for 0.8pm and single mode fibre at 1.3pm. These

present day materials offer a loss below IdBlkm and bandwidths from 1

GHz-km to over 20 GHz-km, depending on the type.

In a semiconductor laser, the mirrors that form the optical resonator

are constructed by cleaving two parallel facets of the semi-conductor crystal..

Photons generated within the cavity by spontaneous recombination of

electron-hole pairs circulate within the laser cavity stimulating further

em~ssion of additional coherent photons (i.e., having the same wavelength

and phase). As injection current increases, optical gain eventually overcomes

the optical losses in the resonator and the device becomes as oscillator.

Under these conditions, emission with narrow spectral width is obtained. The

minimum current at which this phenomenon is observed is called the

threshold current of the laser (I,). A typical light versus current characteristic

of a sem~conductor laser is shown in Fig. 1.2 and current-voltage

characteristic of the laser diode is also shown in the same figure.

An important parameter that characterises the performance of a laser

diode is the slope (q) of the light output versus current curve above the

threshold current. This represents the quantum efficiency of the device, i.e.,

the number of photons generated per injected electron. High quality lasers

are characterised by low threshold current and high quantum efficiencies.

Above the lasing threshold, the optical output power from a

semiconductor laser is a linear function of injection current. Modulation of

the optical output can be accomplished easily at frequencies up to the

rnultigigahertz range by modulating the injection current into the laser diode

6

(which is prebiased above threshold). A theoretical analysis shows that the

intrinsic modulation response of a semiconductor laser behaves as a second

order low-pass network, exhibiting a response peak before rolling off at 40

dB1decade at high frequencies. The -3 dB modulation bandwidth of a laser

follows the relationship.

where A is a parameter that depends on the structure of the laser and Po,, is

the cw optical output power if the laser [a]. The value of A ranges from 1 to 4

GHz / ( m ~ ) " depending on device construction.

It is obvious from Equation (1.1) that a larger bandwidth can be

obtained by biasing the laser at a high cw optical power. However, the

maximum rated optical output power of the laser diode should not be

exceeded at any point. Now, there does not exist a universal definition of

what the maximum rated power of a laser should be. Obviously, when

different criteria are applied such as catastrophic damage, reliability, non-

linearity, etc, the results will be different. The most commonly used criterion

for maximum rated power and laser reliability under cw operation based on

statistical data obtained for a particular class of device. The maximum power

determines the dynamic rage of the laser transmitter.

In general, the sample can have multiple layers-even superlattices

and each layer can have a mixture of components. For material analysis, it

may be noted that light penetrates various semiconductors to different

depths, depending on wavelength shown in table 1.1. In the past few years,

Various Angle Spectroscopic Ellipsometry (VASE) are powerful tools for the

investigations of surfaces, chemical interface, semiconductor heterojunction

quantum well structures and opto electronic materials [9-111. Table 1.2

shows the bandgap, relative dielectric constant (E,), mobility of electron (I,),

mobility of hole ( l h ) and the ratio between effective mass in the

valencelconduction band to rest mass (m,'lm~) of some common

semiconductors for theoretical calculations.

Table 1.2 The bandgap, relative dielectric constant (E,), mobility of electron

Table 1.1 Light Penetration Depth for Various Semiconductors for

different wavelength, in A

(p,), mobility of hole (ph) and the ratio between effective mass in the

(A) 2100

Si

56

valenceiconduction band to rest mass (m,'lmo) of some common

semiconductors for theoretical calculations.

InAs(d)

InSb(d)

Ge

59

3000 i 57 64

144

342

1965

4000

6000

j 8000

I - mdlred, d - direct

0.36

0.17

GaAs

65

822

17,680

1 0'

InP

77

119

148

2067

7446

14.6

17.7

136

183

1417

3032

33000

80000

0.023

0.0145

460

1250

6.0584

6.4794

1.2 INTRODUCTION TO PLANAR MICROLENS

There has been great progress in optical fibre communications, and

many working systems have now been installed. Three types of optical

components have been considered for use in optical fibre communication

systems:

1. Micro-optics that consists of microlenses such as distributed index rod

lenses or tiny spherical lens.

2. Optical fibre circuits that are made from manufactured fibres.

3, Integrated Optics

Many problems such as optical alignment and fabrication process

have emerged in the first two schemes and integrated optical devices are still

far from practicable. A new stacked planar optics is proposed to overcome

these problems. This optics consists of planar and 2-D arrayed optical

components such as microlenses, filters, and mirrors, stacked in tandem to

achieve funct~onal optical components. For example, optical taps, branches,

directional couplers, wavelength rnultiplexersldemultiplexers and other

potential components including active devices.

1.2.1 Concept of Planar Technology

The following is a possible fabrication process for stacked planar

optics:

1. Design of planar optical devices (determination of thickness, design of

the mask shape, etc);

2. Fabrication of planar optical devices;

3. Optical alignment;

4. Adhesion;

5. Connection of optical fibres in the case of arrayed components and

6. Separation of individual components in the case of discrete

components and connection of optical fibres. 9

Features of stacked planar optics include

1. Mass production of standardised optical components or circuits is

possible since the planar devices are fabricated by planar technology.

2. Optical alignment is easy.

3. Optical components of different materials (glass, semiconductors,

electro-optical crystals, etc.) can be connected in tandem. This had

been though difficult in integrated optics consisting of planar optical

wave-guides which are formed on planar substrates where the

connection of different components requires high precision optical

adjustment since the light is transmitted through a thin wave-guide

only a few microns in thickness and width.

4, Coupling of optical fibres is easy, i.e., it could possible be done

without optical adjustment if precise fabrication of a 2-D array of holes

is available.

1.2.2 Recent Improvement in Planar Microlens

This planar microlens is fabricated by using an electro-migration

technique. The substrate IS a planar glass of 40 X 40 X 3 mm3 where planar

microlenses were formed as a 17 X 17 matrix with a 2mm pitch. The radius

of the mask is 50 pm, and the diameter of the resultant lens 0.9mm. The

depth of the d~ffused region is -0.45mm which is nearly equal to the radius of

the resulting lens. The improvement of focal length and NA is due to a better

choice in mask radius and migration time and the reduction of a current leak

through the edges. When the focused spot of the collimated He-Ne laser

beam (A = 0.63 p n ) is measured by using the planar microlens, an Airy-like

disc originating from diffraction and aberration can be observed. The spot

diameter is 3.8 pm. That is small enough in comparison with 50 pm of the

core diameter of a multimode fibre. When it is used in the long wavelength

region of h =I .3 - 1.6 pm.

1.2.3 Performance of Planar Microlens

In order to characterise planar microlens, an automatic interference

system using an interference microscope and a combination of In/ and

microcomputer is adopted. This system can be applied to the index profiling

and aberration testing of many micro-optic components such as rod lenses,

planar microlenses, optical fibres, preform rods, etc. In a shearing

interference measuring system, the image of a sample is divided into two

arms by the Mach - Zehnder interferometer and one of them is shifted by

distance S (we call this shearing distance) by means of a shearing prism.

The observed fringe shift expresses the difference of phases displaced by S

in the object plane [ IZ ] .

The fringe pattern is classified into types according to whether the

shearing distance S is smaller or larger than the sample. The former is called

a shearing (or differential) interferogram and is utilised to measure rather

thick samples such as rod lenses, preform rods and planar microienses. The

later is called a total shearing interferogram and is used to measure thin

samples such as optical fibres.

1.2.4 Index Profiling of Planar Microlenses

Since the planar microlenses has three dimensional index

distributions, sample must be sliced to form a thin plate containing the optical

axis perpendicular to the surface. In this case the shearing distance must be

larger than the size of the sample. The total shearing interference pattern of

the longitudinal thin plate of a planar microlens shows the interference

fringes correspond to the cross section of equi-index surfaces. Therefore, an

index profile in an arbitrary direction can be obtained by scanning in that

direction. The planar microlens has a 3-0 index distribution, but the

distribution can be represented by a 2-D function of the transverse distance

and the depth of symmetry around the optical axis can represent the

distribution.

1.2.5 Aberration Testing of Planar Microlens

Since the shearing interference method is based on the measurement

of the phase profile at the object plane of an interference microscope, this

method can also be applied to the measurement of wave aberration of

microlens. The phase profile at the subject plane of microscope can be

reconstructed from the fringe shift. The wave aberration is calculated by

comparing the measured phase profile and the reference phase profile of

Gaussian focus.

The fringe shift is observed by an In/ and is traced by using a

microcomputer. The collected data are sent to a main computer. The

calculated wave aberration is sent back to the microcomputer and is graphed

by an X-Y plotter. Figure 1.3 shows the resultant output of the X-Y plotter.

All components must have the same 2-D spatial relationship that can

be achieved by planar technology with the aid of photolithographic fabrication

of the integrated circuits. Once the optical axis has been aligned and all of

the stacked components adhered, 2-D arrayed components are realised, the

mass-production of axially aligned discrete components is also possible if

individual components are separated. This is the fundamental concept of the

stacked planar optics that may be considered a new type of integrated optics.

1.2.6 Planar Optical Devices

The following optical devices used in stacked planar optics are

discussed below. The actual component configurations are analogous to

discrete Dl (Distributed Index) lens systems but these have been adopted as

2-D arrayed devices that exactly match the lens arrays.

(i) Circular hole anav: The circular hole array in a planar substrate is used to

connect fibres with a stacked planar optical circuit. The diameter of the hole

is selected to equal the outer diameter of the fibre. The position of each hole

must be matched to that of the arrayed planar microlens in order to align it

with the optical axis. The alignment process, therefore can be achieved

automatically.

U Aoerture and -1 freouencv filter: These devices work to eliminate

useless light and filter-out unwanted modes. The lithographic technique can

be used to fabricate these components on a substrate or back surface of the

planar microlens. Windows aperture arrays are used as aperture stops

opened on thin evaporated on a planar substrate. Special frequency filters

set on the Fourier plane to limit the spatial frequency.

Wavelenoth filter and mirror: A mirror and a wavelength filter on a planar

substrate are well known and widely used in conventional optics and optical

circuits in optical fibre communication systems. In the stacked planar optics,

these devices are prepared in a simple-batch process with many of the

arrayed components. A grating has the same functions but must be set

aslant which may require some contrivance for effective use.

(/vJ Polariser and D ~ ~ S Q && In a stacked planar optical circuit, a poiariser,

analyser, h12 phase plate and U4 phase plate can be easily stacked.

(vl Active O~t ica l Comaonents: Components with electro-optic effect and

magneto-optic effect can be applied to deflectors, switches, modulators and

unid~rectional wave-gu~des. Although the arrangemect of the necessary

electrodes may sometimes prevent construction of 2-D array at least a l -D

array with stacked configuration can be profitably mass-produced.

&J Since the radiating angle of light from a

semiconductor laser diode (LD) and a light emitting diode (LED) is large (40"

for LD and 90" for LED), a large numerical aperture lens is required to accept

the light effectively. An arrayed lens with a curved surface, formed from

13

precise cast plastic or glass and adhered semispheres on a substrate may

be used although it is not of planar structure.

1.2.7 Applications of Planar Technology

As summarised in table (1.3), many kinds of optical circuits can be

integrated in the form of stacked planar optics, where " denotes a circuit that

can be integrated in a 2-D array and ' denotes a circuit which can be

integrated in a I -D array. The coupler array is composed of two lens

elements. One element collimates the light from an input fibre and another

element focuses the collimated light into an output fibre. These two elements

have the same structure and the two-lens array assures sufficient NA. The

length of one element is 2 mm and designed to locate a fibre near the

surface. Multimode Dl fibres (50 pm core) with NA of 0.23 is commonly

used. The input fibre has uniformly excited by using a He-Ne laser (1.0.63

pm) and the output was monitored by Si-solar cells. The average coupling

loss has 0.51 dB with an optimum value of 0.46 dB [13]. This could be

improved when the modes are in a steady state and the fibres are carefully

located at an optimum pos~tion.

Since the accumulation of aberrat~on of lenses may bring about

coupling loss, the number of stacking is limited by the aberration of planar

microlenses. The reduction of aberration in the planar microlens is

important. Therefore, stacked planar optics can be applied to more complex

components with large number of stacks. By using stacked planar optics not

only the monolithic fabrication of optical circuits such as directional coupler,

wavelength demultiplexer etc. is possible but also constructions of 3-D

optical circuits by allowing coupling among individual components in the

array with a suitable design.

Table 1.3 basic opticai components and opticai circuit

Components Application

+++

I (iii) Non-coaxial imaging components I

(i) Coaxial imaging components

** Coupler

(ii) Non-coaxial imaging components

(transmission-type)

(reflection -type)

=I=!

" Branching circuit

" Directional coupler

Star coupler

" Wavelength demultiplexer

" Optical tap

" Wavelength demultiplexer

" Optical switch

I I J '1-D array "2-0 array

(iv) Collimating components

1.3 INTRODUCTION TO TYPES OF SEMICONDUCTOR LASERS

Remarkable process has recently been made in the technology and

understanding of semiconductor lasers. The operating lives of short

Wavelength AlGaAs and long wavelength InGaAsP lasers have been

prolonged by improved crystal growth and processing technologies. Laser

" Branching Insertion circuit

Optical switch

" Directional coupler

" Attenuater

performance has also been dramatically improved by transverse mode

stabilisation. Various kinds of mode-stabilised lasers that are now finding

practical applications in fibre optic communications and optical information

processing have been developed.

Semiconductor lasers have been anticipated as a "key" device in fibre

optic communications and optical information processing due to superior

features such as their small size, high efficiency and high-speed modulation

characteristics.

Recently, remarkable progress has been made in the technology and

performance of semiconductor lasers. Their operating life has been

drastically prolonged from a few seconds to more that l o 6 h at room

temperature by improved crystal growth and processing technology.

Laser performance has been dramatically improved by transverse

mode stabilisation. At present, two kinds of cw lasers viz., short wavelength

AlGaAs lasers and long wavelength InGaAsP lasers, are now finding

practical applications.

1.3, I Technological Progress in Types of Semiconductor Injection Lasers

The idea of lasing in a semiconductor was first proposed by

Nishizawa in 1957. From 1958 through 1961 there were further suggestions

1141 which demonstrated quantitatively how direct bandgap materials such as

ill-V compounds, might satisfy the necessary condition for stimulated

emission. The separation of the quasi-Fermi levels corresponding to the

non-equilibrium concentrations of electrons and holes must exceed the

energy of the emitted radiation. These considerations stimulated lasing in

semiconductor materials.

{i) GaAs homoiunction Lasing in semiconductors was achieved in 1962 by pulsing fonvard biased GaAs pn junctions at the temperature of

liquid nitrogen [15]. After the advent of the GaAs homojunction lasers, lasing

was reported in numerous direct bandgap materials of Ill-V, IV-VI and Il-VI

compounds with optical or electron beam excitation as well as with carrier

injection through a forward biased pn junction [16]. The carrier injection type

or injection laser has received the most attention because it is the simplest to

operate and the most compact of any known lasing device.

Substantial effort was made in reducing the threshold current density

of GaAs lasers in order to attain cw operation at room temperature. Liquid

phase epitaxy [17] and improved diffusion [I81 techniques has introduced to

fabricate good quality lasers. Threshold current density has reduced as low

as 3 X l o 4 ~ l c m ' at room temperature by a 3 layered p'pn structure by

optimising the doping level and thickness of the active p-layers [19].

However, further reduction of the threshold current density could not be

obtained. It has rewgnised that electro-magnetic penetration loss into a

heavily p' layer hampered the reduction of threshold current [20].

U AIGaAs/GaA$ heteroiunction To reduce the threshold current

density and to eliminate penetration loss, the heterojunction laser [21] was

introduced to replace the p' - GaAs layer with a p - AlGaAs layer which had

been known to be closely lattice - matched to GaAs. A good quality AlGaAs

hetero-epitaxial layer was grown by liquid phase epitaxy on GaAs substrate

with infrared light emitting diodes [22]. Further, the quality of the AlGaAs

hetero-epitaxial layer grown by liquid phase epitaxy for lasing material with

Zn - diffused homojunction structure was also investigated. A wide range of

lasing wavelengths ranging from 0.84 to 0.638 pm were obtained with the

corresponding Al content x of AI,G~I.~ As from 0 to 0.38 at the temperature of

liquid nitrogen. Room temperature lasing was also achieved from 0.9 to 0.78

(0<x<0.2) with threshold current density of 3-4 x lo4 A/cmZ, which was

comparable to that of good GaAs homojunction lasers in 1967 [23].

Stimulated emission from AlGaAslGaAs single heterojunction was also

observed with somewhat higher threshold current than that of homojunction

lasers that seemed to indicate the possibility of attaining further reduction of

threshold current in GaAs lasers if the junction structure is optimised.

Room temperature lasing with remarkable reduction of threshold

current around l o 4 Ncm2 was achieved in 1969 by the single heterojunction

structure [24]. Following the single heterojunction structure, a double

heterojunction structure capable of confining carriers in an extremely thin

layer thickness much less than the carrier diffusion length was introduced.

Drastic reduction of threshold current density of around l o 3 Ncm2 along with

cw operation at room temperature was achieved in 1970 [25].

After the advent of room temperature cw operating AIGaAsIGaAs DH

laser, various kinds of stripe geometry lasers have been devised to reduce

threshold current and to control transverse mode along the heterojunction.

The problem with early lasers was their short life expectancy that rarely

extended beyond a few minutes. An improvement in laser performance by

transverse mode control has achieved in 1974 when the problem of rapid

degradation of the laser was overcome.

It has been found that the rapid degradation of AIGaAsIGaAs DH

lasers is caused by the dark-line defects (DLD's) developed from the dark-

spot defects (DSD's) in the active layer [26]. The most obvious origin of

defects was shown to be the generation of stacking faults and dislocation in

the active layer, lshii 1271, et.al., have demonstrated that oxygen

contamination during liquid phase epitaxy can have a profound effect on

DSD formation while the use of a small amount of Al in the active layer to

slightly increase the lasing photon energy also improves reliability. The

success that has been achieved in drastically reducing the probability of DLD

formation has resulted in lifetimes of over l o 4 h in lasers even with a pure

GaAs active layer [28].

1.3.2 Transverse Mode Stabilisation

The transverse mode instability in conventional stripe geometry lasers

along the heterojunction was proved to be a problem in fibre-optic system.

The mode instability is the origin of non-linear light output with current "Kink"

which are accompanied by anamalous lasing behaviour such as beam

direction shifts, deterioration of modulation characteristics, excess noise and

SO on.

Transverse mode instability in conventional lasers has been attributed

to the deformation of the laser gain profile that determines the transverse

mode [29]. An effective method in stabilising the mode is to introduce a rigid

burlt-in-index change along the heterojunction for defining the transverse

mode.

The first mode-stabilised laser is the transverse junction stripe (TJS)

laser that is devised by forming a p'pn homojunction structure along the

heterojunctron. The transverse mode stabillsation of the TJS laser was

confined experimentally by Namizaki in 1975 [30]. The well-defined wave-

guide structures by the p'pn homojunction makes it possible to operate in a

single longitudinal mode oscillation as well as in a stable fundamental

transverse mode oscillation which had not been achieved in conventional

lasers. The single longitudinal mode oscillation is explained by theoretical

analysis assuming that the TJS laser is a homogeneously broadened laser.

TJS lasers exhibit improved characteristics such as low threshold and "Kink"

free light output with current by stable fundamental transverse mode

oscillation.

After the success of transverse mode stabilisation in TJS lasers,

varlous kinds of index-guided lasers have been developed. Improved

characteristics such as single longitudinal mode oscillation and "K~nk" free

light output current have been confirmed [31].

1.3.3 Operating Lifetime in Mode Stabilised Lasers

As laser performance has been improved by transverse mode

stabilisation, life expectancy of lasers has drastically improved. It was

recognised that slow degradation was caused by mirror surface deterioration

due to oxidation durlng the current flow which can be eliminated by mirror

surface coating with dielectric films such as Alz03 [32], SiOz 1331 and Si3N4

[34]. A Si submount has found to be useful in low threshold lasers to reduce

mechanical stress between the laser chip and the metal heat sink. It is

possible to make use of high temperature solder such as Au-Si, instead of In,

which causes solder degrsdation by forming a hard alloy of Au-In during

current flow and limits the laser operating life. Estimated mean time of failure

more than l o 6 h has been reported in single-mode junction-up TJS lasers

mounted on a Si submount and passivated with Si3N4 [35].

1.3.4 Long Wavelength lnGaAsP Lasers

In 1976, a new InGaAsPilnP DH laser capable of room temperature

cw operation was introduced 1361. The lasing wavelengths extend 1.1 to 1.67

pm with low threshold current density [37]. The lasing wavelength range of

the material coincides with low attenuation loss and low dispersion

wavelength range of the quartz fibre. Low threshold current around 20 mA

has been realised in the burried heterostructure fBH) laser at 1.3 and 1.55

pm band in the buried crescent (BC) laser at 1.3 Mm band [38]. Operating

life of more than l o 5 h at 50°C has been estimated [39]. These features are

suitable for future long distance and wide band fibre optic communication

systems.

1.3.5 Visible Lasers

As the performance and operating life on the infrared AlGaAs lasers

(7, = 0.8 - 0.9 pm) have been improved, visible lasing emission has received

attention to obtain from AlGaAs lasers with wider band gap active layer. Very

low threshold currents of 20 - 30 mA have been achieved in TJS lasers for

the wavelengths between 0.9 (infrared) - 0.75 (visible) Fm. Such low

threshold current has also been obtained with lasing wavelengths down to

0.74 Fm in TS lasers. However, the operating life of lasers with lasing

wavelengths less than 0.74 pm IS very short [40].

1.3.6 Transverse Mode Stabiiised Lasers

Transverse mode instability in conventional lasers has been attributed

to gain profile deformation by hole burning along the heterojunction. It is

necessary to cut off higher order transverse modes and to stabilise with the

fundamental mode for stable oscillation. There are two approaches for mode

stabillsation with fundamental modes: (1) introduction of a rigid "built-in"

wave-guide structure and (2) narrowing the wave-guide width.

Figure 1.4 shows typical stripe geometry lasers stabilised with the

fundamental mode. They are classified into two groups by wave-guiding

mechanism: (I) index-guided type and (2) gain-guided type.

1.3.7 Index-Guided Lasers

a [41]: The Index guided lasers has a "built-in" index wave-guide

along the heterojunction to guide only the fundamental mode. The TJS laser

has a p'pn homojunction structure along the heterojunction as shown in

figure 1.4 (a) in which a p region 2 pm wide, is the active region sandwiched

by a p' and n region. The p'pn structure forms a wave-guide provided by a

refractive index change produced by doping profile. For carrier densities

higher than loi8 cm4 in GaAs, the refractive index of the p-type material

becomes insignificantly larger than that of n-type material. Moreover, the

21

refractive index becomes smaller as the carrier density increases, the

effective index Step is 0.01 [42] which is sufficiently large to support a stable

fundamental mode.

@ k r 1431: In the burried heterostructure BH laser the wave-guide is

formed by an active filament totally embedded in low index material. The

index step is provided by a differences in material composition. In order to

obtain the fundamental mode, the stripe width should be very narrow

preferably less than 2 pm.

fld BG k r 1441: BH lasers can be fabricated by growing the DH structure

on a narrow channel grooved on a substrate. Under certain growth

conditions, the active region becomes a crescent-like cross-section and this

is known as a burried crescent lasers (BC). The effective refractive index is a

function of the active layer thickness that varies in a quasi-parabolic fashion

along the heterojunction. For stable fundamental mode oscillation, the active

region width should be iess than 2 pm for the maximum thickness of the

region of 0.1 pm as d~scussed in InGaAsPI inP BC lasers.

[45]: In channelled-substrate-planar (CSP) lasers which is

schematically shown in figure 1.4 (d), the transverse mode is stabilised by

the excess optical loss that the mode tail suffers because of the penetration

of the evanescent field into the highly absorbing substrate. An interesting

feature of the structure is that there is a refractive index step present. This is

because, in the presence of absorbing material across a thin cladding layer,

the mode tends to be pushed into the p-AIGaAs cladding layer, resulting in a

decrease in the effective refractive index.

&) IS [46]: The rib guide structure is also effective for devising for

mode stabilised lasers. A rib guide laser, called Terraced Substrate (TS)

laser in which the bent portion of the active layer near the channel shoulder

is slightly thicker than the planar regions, thus forming a rib guide.

1.3.8 Gah-Guided Laser

fl V-aroove h r [47]: The reduction of the wave-guide width in the

conventional stripe geometry laser without any built-in index profile along the

heterojunction has also proved useful in reducing hole-burning effect. This is

accomplished simply by reducing the stripe width and the width of the light-

emitting region below twice the carrier diffusion length. The v-groove laser

schematically shown in figure 1.4 (e), has a very narrow stripe for current

tnjection through V-channel, in which carrier diffusion can smooth out the

"hole" of the carrier density caused by local carrier consumption.

1.3.9 Characteristics of Mode Stabilised Lasers

fl Threshold current and wavelenath: Reduction of threshold current was

one of the goals to be attained for efficient operation. It is also important for

attaining high temperature operation with a Si heat sink that eliminates the

mechanical stress between the laser chip and the heat sink for prolonging

the lifetime. If is possible to attain low threshold current if a smaller cross-

section of the active region is obtained. Low threshold current of 10-20 mA

has been achieved in TJS [48], BH [49] and BC [50] lasers. However, there is

a problem with the rapid threshold increase at high temperature due to a

shunt current flow through outside of the active region that limits the highest

cw operation temperature to a lower level.

It is important, therefore, to eliminate the shunt current flow through

the outside of the active region and to confine current injection into the active

region. Modifying the structure to include a self-reversed-biased pn-junction

[51], a burried p-region in the GaAs substrate I521 and a semi-insulating

GaAs substrate, can eliminate the shunt current. Cw operation as high as

130°C were obtained in the simplest structure using the semi-insulating

substrate with a junction-up configuration mounted on a Si submount.

The threshold current of the InGaAsIlnP long wavelength laser is more

sensitive than that of the AlGaAslGaAs laser, with a lasing wavelength of 0.8

- 0.9 pm, which might be due to inherent properties of the material.

However, high temperature cw operation above 100°C has been realised in

1.3 pm BH lasers by eliminating the shunt current flow with a self-reverse-

biased pn junction (531.

In AlGaAs lasers, low threshold current has been obtained in the

lasing wavelength range from 0.9 to 0.75 pm with the corresponding Al

content x of AI,Gal.,As from 0 to 0.25. Threshold current Is almost constant

for the lasing wavelength between 0.9 and 0.75 pm and 20 - 30 mA at 300 K

(401. Rapid increase of threshold current below 0.75 pm is due to the thermal

excitation of injected electrons into the indirect conduction bands. Low-

threshold current was obtained in TS lasers with the wavelength as low as

0.74 pm. The shortest lasing wavelength attained at room temperature is

around 0.71 pm that has been confirmed in CSP lasers with several times

higher threshold current than those of longer wavelength AlGaAs CSP lasers

(541.

In GaAsP lasers, iofl threshold current density is obtained in the

wavelength range between 1.2 and 1.6 pm [55]. Low threshold BH lasers of

the lasing wavelength at 1.55 pm where the attenuation loss of the quartz

fibre is minimised [33].

The stable fundamental transverse mode is obtained in index-guided

type lasers, in which the mode is determined by the refractive index profile

and the wave-guide dimensions.

A near and a far field patterns of the TJS laser shows that the half

width of the near field pattern is about 2 pm and a half-angle of the far field

pattern is 14 deg. along the heterojunction. These values are consistent with

theoretical calculations [42].

In order to attain stable fundamental transverse mode oscillation in BH

and BC lasers, the wave-guide dimensions should be precisely deslgned to

propagate only the lowest order fundamental mode. The higher order mode

cut off condition is calculated by applying an equivalent refractive index

method to the InGaAsP BC laser [56].

The far field pattern along the heterojunction of the gain-guided laser

has a dip In the centre. Typical far field patterns for a V-grooved laser

perpendicular to and along the heterojunction shows that a double peak of

the far field pattern along the heterojunction is caused by the non-planar

phase fronts due to gain guiding. The virtual beam waist is observed 30 prn

behind the laser mirror. This beam is astigmatic because the beam waist is at

the mirror face for the field confined perpendicularly to the heterojunction

while the virtual beam waist for the field confined along the heterojunction is

behind the mirror face. This astigmatism can be an important consideration

when the light output is coupled into lenses.

Index-guided lasers exhibit single longitudinal mode oscillation as first

observed in the TJS laser. The single longitudinal mode oscillation is

accompanied by suppression of oscillation for competing adjacent modes,

which is explained by a multi-mode rate equation. Spontaneous emission of

the TJS laser uniformly saturates above the threshold both in spatial and

spectral distribution in the direction perpendicular to that of lasing itself. This

behaviour indicates that the TJS laser is a nearly homogeneously broadened

laser.

It is interesting to note that BH and CSP lasers have a hysteresis of

the longitudinal mode in the lasing wavelength-temperature characteristics

that might be a direct indication of strong mode coupling in semiconductor

lasers.

Using a carefully arranged Fabry-Perot system, the linewidth of the

lasing mode has been estimated to be 1-2 MHz in a CSP laser which is very

close to the theoretical limit 1571.

The gain-guided laser usually oscillates in multi-longitudinal modes as

observed in the narrow stripe laser and the V-groove laser. The multi-

longitudinal mode oscillation of these gain-guided lasers is attributed to the

large spontaneous emission factor into lasing modes [58].

1.3.10 Light Output

The "kink" in the light output vs, current characteristics due to

transverse mode instability has been eliminated by the advent of the mode-

stabilised lasers.

In the cuwe of light output vs. current characteristics of conventional

TJS lasers shows that the linear light output has been obtained more than 15

mW with a single longitudinal mode oscillation [52]. However, the maximum

light output is limited by the catastrophic optical damage (COD) of around 20

mW. In order to obtain higher light output, it is necessary to reduce the COD

failure.

It is recognised that COD is caused by local heating near the mirror

surface with absorption of intense super radiant light. The highly absorbing

region can exist near the mirror surface because of highly absorbing surface

recombination in AiGaAs lasers.

Recently, COD levels increased to 120-180 mW in pulsed operation

by a "crank" type laser. The light is emitted through low absorbing layers

where carriers are not injected which has been used in the "troughed"

homojunction GaAs laser and the AlGaAs DH window structure laser. Single

longitudinal mode cw operation to 30 mW can be obtained from a crank type

TJS laser [59].

In InGaAsP lasers, the surface recombination velocity is very low, and

a much higher COD level can be achieved compared with that of AlGaAs

lasers. However, the inherent temperature sensitive threshold current limits

the cw maximum power to lower levels.

1.3.11 Modulation Characteristics

High frequency modulation capabilities of lasers are of great

importance in applications such as wideband flbre optic cornmunlcation

systems. A requirement for the wldeband laser is the suppression of the

resonance-like peak in analogue modulation systems or of relaxation

oscillation in a pulse code modulation system. At data rates above 100 Mbls,

the relaxation oscillation can produce a serious deterioration of the pulse

shape. The resonance-like peak or relaxation oscillation is found to be

especially pronounced in wide stripe (z 10 gm) conventional DH lasers.

Suppression of the resonance-like peak or relaxation oscillation has

been observed in mode stabilised lasers and is believed to be due lateral

carrier diffusion and the feeding of spontaneous emission into the lasing

modes [60].

The TJS laser has a wide frequency modulation band width over 4

GHz with an almost-suppressed resonance-like peak. The modulation

characteristics agreed with a simple analytical formula derived from a small

signal analysis of the single mode rate equations for carriers and photons

Frequency. 4-14 MHZ

Bandwidth. 300 KHz

(Single Lon~lludlnal Mode)

Frequency.4-14 MHz

BandwldU1.300 KHz

(MUID-Lon~lludnai Mocs)

*

involving the carrier diffusion. The spontaneous emission rate into the

oscillating modes is too small to influence the modulation characteristics.

0 Small i%l!d and !WE 3dqLd bandwidth. The relationship between the

optical Power and modulation bandwidth of a laser given in Equation (1 .I), is

basically the result of small signal. i.e., the variation in the optical output is

only a small fraction of the average optical emission. This is a common way

of characterising the high speed behav~our of a laser and measurements. It can be performed with the microwave setup described by s-parameter

system used for measuring frequency response of injection lasers [61]. This

results are obtained for small signal modulation of injection lasers to various

bias optical power by an Ortel laser diode with a 10 GHz bandwidth [62].

Actually, to be qualified in the "small region" regime, the requirement is that

the product of the amplitudes of electron and photon modulation be smail.

Due to carrier clamping effect above lasing threshold, the fluctuation in

elections is indeed quite smaii. This impiies that the small signal regime will

hold for even large photon modulation as high as 80 percent. When the

electron density begins to show large deviations from its clamped value that

the small signal regime begins to break down and this Occurs when the laser

IS at a state close to or below threshold, i.e., the photon density is allowed to

drop to a low level.

1.3.12 Noise

Noise is another important factor in applications such as fibre optic

communication systems and optical data processing. The noise features in

fibre optic communication systems are discussed below.

It is interesting to compare noise behaviour between the index-guided

laser and the gain-guided laser. Figure 1.5 shows the temperature

dependence of the noise for three lasers: a single longitudinal mode TJS

laser (A), a few longitudinal mode crank type TJS laser (8) and a multi-

28

longitudinal mode V-groove laser (C) [63]. In the single mode TJS laser, the

noise level is very low (S BO dB), but it increases at critical temperatures to a

relatively high level (-60 dB) due to the simultaneous oscillation of the two

adjacent modes. In the multi-longitudinal mode crank type TJS and V-groove

lasers such rapid increase in noise levels have not been obtained. In multi-

mode lasers, however, the noise level is relatively high (-70 dB). The noise

features should be considered when lasers are used in such systems.

1.3.13 Operation Lifetime

Recently the operating life expectancy of Infrared AlGaAs lasers (1 =

0.8 - 0.9 kin) has been dramatically improved. The principal technological

improvements, which have resulted in the prolongation of life expectancies,

are as follows:

1. High quality crystal growth in oxygen-free ambient to eliminate the

DSD's 1271.

2, Improved laser chip bonding to reduce stress by using Si submount

3. Mirror surface coating for protecting mirrors from oxidation with Alz03,

Si02 and Si3N4 1341.

4. Elimination of solders degradation by using high-temperature solder,

such as Au-Si instead of In [35].

As a result, single mode operation with an estimated mean time of

more than l o 6 h at room temperature has now been obtained in infrared TJS

lasers with junction-up configuration mounted on Si submount.

1.3.14 Concluding Remarks

The progress of semiconductor injection lasers and present state-of-

the-art in transverse mode stabilised AlGaAsP lasers are discussed.

Elimination of rapid degradation due to DSD's has greatly prolonged the

operating life of AlGaAs lasers. Performance of lasers is drastically Improved

by transverse mode stabiiisation that has been achieved by various kinds of

laser structures. At present, long lived infrared AlGaAs and long wavelength

InGaAsP mode stabilised lasers have been developed.

Many important subjects remain to be investigated. One of them is

longitudinal mode control. Usual index-guided lasers produce relatively hlgh-

level noise at critical temperatures due to simultaneous oscillation of adjacent

modes. In order to eliminate the longitudinal mode instability, a distributed-

feedback or a distributed-Bragg-reflector laser capable of operating

continuously at temperature is needed. The mode instability caused by

external feedback and the mode degradation during ageing are important

problems to be solved. Another problem to be solved Is prolonging the

operating life of visible AlGaAs lasers. Higher cw light output lasers must be

developed with stable operation and long life.

The laser fabricated by liquid phase epitaxy is not productive. MOCVD

and MBE techniques are now being introduced to replace the liquid phase

epitaxy and low threshold current density comparable with the lasers

fabricated by liquid phase epitaxy are reported recently [64].

1.4 INTRODUCTION TO DENSITY MATRIX THEORY

Our main task is to determine how many electronics states (density of

states) are available to be filled by an electron in the conduction band or

emptied in the valence band. We have to proceed in two steps: first to

identify the number available and then to compute the probability that the

state is filled or+ernptied. Hence, it is the filled states in the conduction band

minus the filled states in the valence band, which correspond to N2 - N, of

general description of lasers.

Nishimura, Koayashi, lkegami and Suematsu and Nishimura and

Nishimura did the first analysis of a semi conductor laser based on density

matrix formalisation [65,66]. This formalism has been followed up and further

developed by Yamada and Suematsu and by Asada and Suematsu [67]

resulting in obtaining the analytical expression of electronic polarisation

where the importance of taking into account the relaxation effect even in

calculation of the linear gain has been postulated. The relaxation dissipates

the electron energy instead of transferring the energy to the optical field,

resulting in a spectral broaden~ng and a threshold current increase.

Gain co-efficient in semiconductor lasers is expressed in terms of the

macroscopic polarisation formed by electron-hole pairs. The electronic dipole

moment that gives the optical transition probability in semiconductor lasers is

obtained in combination with semiconductor material parameters.

1.4.1 ~olahsat ion and Gain

The gain in a laser is expressed by the sum of the linear term and

non-linear terms with respect to the optical power. In order to analyse both

the linear gain and the non-linear gain (gain suppression) of semiconductor

lasers, it is necessary to take into account the phase relation between the

optical field and the polarisation formed by electron-hole pairs and the

electronic relaxation effect. Density matrix formalisation has the advantage of

being able to care for both linear and non-linear gains, because it takes into

account explicit information about the phase of the electronic polarisation. In

the present theory, the gain in the laser cavity is related to the macroscopic

polarisation formed by the electron hole pairs using the classical Maxwell

equation. Following this procedure, the polarisation is derived quantum-

mechanically by the electron density-matrix as follows:

The time evolution of the electric field E in the laser cavity is written as

where EO and p, are the dielectric constant and the permeability of the

vacuum respectively, n, is the refractive index, Gfh is the loss coefficient in

the cavity per unit time and P i s the polarisation formed by the electron-hole

pairs. E and P (component of P parallel to E) are expanded into series of

mode distribution functions in the laser cavity, Fp (r), as

where o, is the angular frequency of the mode p, Ep and P, are factors

varying very slowly compared with exp ow pt) and Fp ( r ) is normalised so

that the integral of the absolute square over all space is unity.

Subst~tuting equations (1.3) and (1.4) in equation (1.2) as

where G, is the gain coefficient per unit time given by

GP = cup I (conr2) In (Pp I E,)

Rewriting Equation (1.5) into an equation with respect to distance along the

light propagation as

where a, and ath are the gain and loss coefficients per unit length

respectively and are related to G, and Gth in the following way :

and

Appropriate application of the two gain coefficient G, and a, is

determined according to a particular case. The linear gain is discussed with

a,, while Gp is mainly used in the necessary rate equation for gain

suppression and mode competition phenomena.

The gain is thus expressed by the macroscopic polarisation. The

polarisation is obtained quantum-mechanically from the density matrix of

electrons as (681

P = RTr ( p R ) = N ZJpmnRnm + Rmnpnm)

where n and m are energy levels in the conduction and valence bands

respectively. Subscripts (n,m) and (m,n) indicate elements of the matrix. Fi is

the total number of electrons per unit volume including both the conduction

and the valence bands. R is the drpole moment operator.

7.4.2 Electronic Dipole Moment

The element of the dipole moment operator between an electron in the

conduction band and a hole in the heavy-hole band, appearing in Equation

(1.9), is given by

Rnm= <Vnkn I er / ymkm> = je himo) 1 (En - Em) nkn I ( - jhV) I ymkm> ... (1 . l o )

where Wnkn and Wmkm with subscript k denoting the wavevector, are the

wavefunctions of the electron and the hole respectively, e is the electron

charge, m, is the free-electron mass and En and Em are the different energy

levels of the corresponding wavefunctions. Since broadening of the gain

spectra is attributed to the intraband relaxation effect. The electron

wavefunctions are conventional Bloch functions in contrast with the band tail

model.

Using the K . P method [69] in caiculating the momentum matrix

element in Equation (1.10), each component of R,,, is obtained as follows in

an arbitrary co-ordinate system :

R(cos 0 sin g + j cos 4) (for the x - direction) ... 1.11 (a)

- R sin 0 (for the y - direction) . . , l , l l (b)

R(cos6 cos g - j sin g) (for the z- direction) ... 1.11 (c)

where 0 and g are the latitude and longitude of the wavevector (K K, = K,)

with the y-axis as the polar axis and

where E, is the bandgap energy, A is the spin-crbit splitting in the valence

band, m, is the effective electron mass in the conduction band, a, expresses

the proportion of the atomic s-like function included in y,rn which is well

approximated by unity for wide gap semiconductors and Enm = hmnm = En-Em.

)kSb

$ - n ~ s

GaSb

+-+- Gao<,lo,r&i

D 1-

I I > 2 5 '0 20 5 0 100

Equvalent Length of Dipole Moment c r > A

Fig. 1.6(b) Numer~cal value of tne equivalent length of the dipole moment

Vs Band gap wavelength for various semiconductors.

Assuming that the electric field is along the y-axis, the squared dipole

moment along the electric field is obtained by substituting Equations (1. l lb)

and (1.11) in Eq. (1.10) and by averaging over all directions with respect to 8

and $, as

Numerical examples of the equivalent length of the dipole moment at

the band edge, < r > =d(Rmn 1'1 e, vs. bandgap energy E, and bandgap

wavelength are shown n figures 1.6 (a) and 1.6 (b) for various Ill-V

semiconductors. From the figure, it is assumed that the dipole moment is

larger for longer wavelength semiconductors resulting in larger gain for

longer wavelength lasers

The analysis ment~oned here can be extended to the case for

quantum-well lasers with ultra thin active layers. It assumes that the y-axis is

perpendicular to the well interface and that light is propagating along the z-

axis. The electric field is along the x-axis for TE modes and nearly along the

y-axis for TM modes. Since the y-component of the wave vector and the

angle 8 are Rxed in each subband of quantum-well, the effective squared

dipole moment is obtained for TE modes and for TM modes averaging with

respect to the angle 0, i.e., within the plane parallel to the well interface:

/ R",/ = R~(I + cos2 8) I 4 (for TE modes) ...( 1.14a)

/ R,, / = R2 sin2 0 12 (for TM modes) ...( 1.14b)

where the selection rule for the y-component of the wave vector included in

R' is regarded as the subband number selection rule.

35

It should be pointed out that the effective dipole movement in

quantum-well lasers is polarisation-dependent and is equal to zero for TM

modes at the subband edges. Also pointed out is the magnitude of the

effective dipole moment for TE mode is 1.5 times larger that that of

conventional ones at the subband edges.

The density matrix p included in Equation (1.9) is obtained by the

following dynamic equation:

or in terms of each element,

dpnnldt = (1 1 jh) Z(pnR~n- Rnp1n)E- (pnn - Fn)irc - rPnn - pn) 17s + An ...( 1.15b)

and one more equation with n and c exchanged for m and v, respectively in

(1.15 b), pn is the electron distribution function at the thermal equilibrium, p,

is the distribution function at quasi-equilibrium determined by the intraband

relaxation effect, r, is the spontaneous carrier lifetime, 5 , (7,) and rln are the

intraband relaxation time of the diagonal and off-diagonal elements,

respectively and A, is the pumping by the current injection.

In contrast to the band tail model, the broadening of gain spectra in

semiconductor lasers is given by the intraband relaxation effect included in

Equation (1.15) in the present model. The Energy State in relaxation to

luminescence for the band tails models and for the relaxation broadening

model is compared schematically in Fig. 1.7.

Electron distribution In Equation (1.15a) IS illustrated in Fig 1.8. Since

the number of electrons in the conduction band never changes with the

intraband relaxation effect. Therefore,

where level n is restricted to within the conduction band while the spatial

integration is done over the whole area of the active region. A similar relation

also holds good In the valence band.

As mentioned earlier a, alone can be written as - up I" = nrd EO Gp ...( 1.17)

Density matrix p which is non-linear with respect to the field E as

shown in Equation (1.15) is analysed through a perturbation approach where

p is expanded into a power series with E:

The polarisation P n Equation (1.9) consists of only the off-diagonal

elements of the density matrix which are of the odd order with respect to the

field E. Therefore, :he gain coefficients G, and a, in Equations (1.6) and (1.8)

are expressed in odd order terms of the expanded series in Equation (1.17).

Using Equation (1.3), a, is expanded corresponding to Equation (1.17) as

According to each expanding order, the gain coefficients a,('), a,l,/3', ...I are

called the linear gain coemcient, third order gain (gain suppression)

coefficient and so on.

1.4.3 Relaxation Time

Relaxation times in (1.15) are very important parameters in forming

the energy distribution of both the linear and non-linear gain coefficients. 37

The relaxation t~mes, r, (and r,) in Equation (1.15) should have the

following relations with the relaxation times T-,,,~~,I and 7h-mobll estimated by the

value of the mobilities introduced in the electron transport analysis:

where re-, and 7h.h are determined by interactions among the electrons and

holes, respect~vely.

In general, the relaxation time of the off-diagonal elements of the

density matrix in Equation (1.15) Tin, is related to the decay rate of the

macroscopic polarisation and IS not identical with that in the diagonal

elements of the density matrix. However, when the major scattering process

is regarded as adiabatic, relaxation time of the off-diagonal elements is

represented by the following simple relationship with the relaxation time of

the diagonal elements:

The estimated value of the relaxation time is shown in Fig. 1.9 along

with tne variation ot acceptor concentration NA of Zn in GaAs material, r,, in

the figure is an intraband relaxation time in the gain due tc transition from the

conduction band to the acceptor levels in p-type semiconductor material.

1.4.4 Linear Gain Coefficient

The linear gain coefficient per unit length is obtained by calculating the

matrix element of the first order terms, ~,n")and pnm(')as follows:

ap(') = a(') / F, (r) / dr = @a(') ... (1.23a) V

with

where the integral is made within the active region, k ~ , is the optical

confinement factor and pn,'O) and pm,(o' are zero order components of the

density matrix whose distributions are given by the Fermi-Dirac functions, fc

and fv, in which potential levels are characterised by quasi-Fermi levels EFc

and E F ~ . Due to the wave-number selection rule for electronic transition,

energy levels n in the conduction band and m in the valence band have a

one-to-one correspondence. Accordingly, the Fermi-Dirac distributions and

the state density are represented as a function of Enm = En - Em instead of En

or Em, result~ng in the following representation of the linear gain coefficient:

x (hi 7," ) i [(hw - ~ n m ? + (h i dl dEnm ...( 1.24)

where

fc (Enm) = 1 I [I + exp {[(Enm - Eg) mv 1 (mc + mv) - E F ~ + Ec] I K ~ V ] ...( 1 .25)

f v (Enm) = l ib + exp {[- (En,- Eg) mc I (m, + my)- E F ~ + Ev] I KBT)] ...( 1.26)

gcv(Enm)dEnm = [(2mcmv) I (m, + mv)la 1 / (2nzh3) (Enm - Eg )''d~nm ... (1 .27)

The linear gain is controlled in principle by the quasi-Fermi levels EFc

and EFV, as given in the above equations and not by injection current. The

difference between the two quasi-Fermi levels in electron volt is equal to the

applied voltage at the p-n or p-i-n junction plane in a laser.

The injected electron density is also given with the quasi-Fermi levels

in the Fermi-Dirac functions as

where?, is the distribution at thermal equilibrium. Numerical examples of the

linear gain spectra in undoped GaAs (Ep = 1.43eV) and Ga 0.47, In 0.53As (Eg

= 0.75 eV) for several values of injected electron density are given in [70].

The gain profile reveals a tail-like shape expanding into the band gap

due to the relaxation effect in spite of having adopted parabolic state density

with no energy state in the band gap Equation (1.28). The gain magnitude is

also reduced due to the relaxation effect. When the relaxation time is less

that lui4 sec, the linear gain coefficient oddly become positive.

The linear gain magnitude is large for longer wavelength lasers since

the effective mass is smaller for longer wavelength materials.

The linear gain for quantum-well lasers is also obtained by using the

following step-like state density [71] for each subband instead of Equation

(1.28):

where w is the well thickness, u(E) is the unit-step function and EC, and Ev,

are the quantised energy levels in conduction-band and valence-band well,

respectively.

40

Due to the relaxation effect, the spectra for the gain spectra for 100A-

wide Ga0.47 In053 As / InP quantum-well lasers are broadened in spite of the

step-like state density. The gain difference between TE and TM modes due

to the polarisation-dependent dipole moment is also obtained.

1.4.5 Injection Current and the Threshold

The carrier density N, Eq.(1.26) is combined with the injection current I

as

I = e JZ .znd3r = e [ (N,) 1 (7,) ] V ..(1,30)

where V is the volume of the active region and r, is the spontaneous carrier

lifetime that consists of the radiative (spontaneous emission) and non-

radiative recombination times ( T ~ and r,,) as

The radiative recombination time rr is obtained from the transition

probability by spontaneous emission into one mode multiplied by the

radiation mode density, resulting in the following equation

Numerical calculation of this equation gives the following approximate

relationship with the injected electron density NI :

where 0, is the radiative recombination coefficient. 41

I I I I

k 50 100 200 300 500

1 1

Temperature T (K) ~~ I

'I 11 Fig 1 10 Exper~mentai and theoret~cal values of the threshold

Current of a GaAs laser w~ th var~atlon of temperature 11 I I I

The non-radiative recombination time r,, consists mainly of the three

effects: the Auger effect whose recombination rate is approximately

proportional to NI', the carrier leakage over the heterobarrier and non-

radiative recombination centres. Expanding l/rnr into power series of NI up to

the third order. It can be expressed by an approximated formula

The laser threshold is the condition where the gain is equal to the loss,

i.e.,a, = a,"' = avl"' in Equation (1.7), giving the threshold carrier density

according to equations (1.24) and (1.28). This loss is mainly due to free

absorption and the intervalence band absorption in particular In long

wavelength lasers. The threshold current Ith is obtained by substituting the

threshold carrier density into Equations (1.30) - (1.34). The non-radiative

recombinations are not negligible in long wavelength lasers such as

GalnAsPllnP lasers and are thus responsible for the increase in the

threshold current and the decrease in the spontaneous emission efficiency.

As an example, the spontaneous emission efficiency q,,, = ?s 1 r, of '12%

has been obtained in 1.58 prn wavelength GalnAsPllnP at the laser

threshold level in room temperature [72]. In contrast, for short wave length

lasers such as GsAs lasers, the non-radiative recombination is negligibly

small. The carrier lifetime rJ is nearly equal to the radiative recombination

time rr. The temperature dependence of the threshold current of a GaAs

laser with the temperature variation of the intraband relaxation time is shown

in Fig. 1.10.

1.4.6 Gain Suppression and Mode-Competition Phenomena

Gain suppression and mode-competition behaviour under laser

oscillation are analysed with the help of the higher order terms of the density

matrix in Equation (1.17). The gain coefficients are calculated up to the third

order terms, resulting in the following rate equations of photon number S, of

a cavity mode and the injected carrler density N, as

where G,"' and G,(,/~) are the linear gain coefficient and the third order gain

(gain suppression) coefficient of the mode p, corresponding to a$'' and

aP(,j3' in Equation (1.18), given by,

Gpcqi(3i = ho 1 [(2nr3 E~ ( ~ ~ p ~ ) " ~ ] ap(q/3)

= ( I +&,,).' I G I Fp (r) Fq(r) 1 'dr V

with

G "' = (o) 1 n,%, I{[R~z~, ( ~ ~ r n ) Vs(Enrn) - fv(Enm)l (h I rin)} I E9

[(E - En,)' + (h I rln7]dEnm r a[M - Ng - b (X - b)'] ...(I ,391

43

where coefficient (1 + s,,)-' in Equation (1.35) rises from the difference in

phases between vibrations of polarisation and the optical field. The

coefficient a and b in Equation (1.39) are introduced to represent the gain

coefficients with simply approximated equations including the injected

electron density N instead of the quasi-Fermi levels. N, and N, are the

electron densities at which the gain-coefficients acquire a positive value.

The coefficient C in Equation (1.35) is the ratio of the spontaneous

emission going into the mode p, and the term D V ~ Ni in Equation (1.36)

expresses the spatial diffusion of carriers obtained with the help of the quasi-

equilibrium distribution characteristics ;.

Equations (1.35) and (1.36) differ from ordinary rate equations for

semiconductor laser in the gain suppression coefficient terms. Ordinary rate

equations include only the gain saturation due to the saturation of the carrier

density N, above threshold. This saturation is homogenous over all resonant

modes. Equations (1.35) and (1.36) include the gain suppression

independent of the gain saturation, in particular the gain suppression of one

mode due to the output of the other modes which explains the decrease of

the nonlasing mode above threshold.

The gain profile above threshold with gain suppression am = (GA') - G ~ ( ~ / ~ ) Sp) nr ( E ~ P ~ ) " ~ and the increase in injection current where a mode p

oscillating and all existing modes are restricted to being transverse

fundamental modes. The gain in the non-oscillating modes is suppressed

because the relaxation time in most undoped semiconductor Is estimated to

be in the order of 10 -13 s and thus the relation G,(,/~' a 4/3 G~~,,'~' exists.

Such an effect has actually been observed as the decrease in the output

power of the non-oscillation modes with the increase in injection current.

If any transverse higher order mode exists in the cavity, the gain

suppression is less effective due to the relation G~(,,(~' < G ~ ~ ~ ) ( ~ ~ , resulting in

muitimode operation in both the transverse and longitudinal modes.

Using Equations (1.39) and (1.40), the ratio of the third order gain

coefficient to the linear gain coefficient is given by:

G i3) 1 G (')a (2ho,) 1 (n;~,) (t,, I h)'+ R,' [(N, - Nr) I (N, - Np)] ... (1.41)

As discussed above, the dipole moment and the linear gain is larger

for longer wavejength lasers. Therefore, G,(,)'~' and thus the gain

suppression are larger for longer wavelength lasers and more stable single

mode operation is expected for longer wavelength lasers. Comparison of the

gain suppression between a long wavelength GaAsPiinP (h = I .56 km) and

a short wavelength GaAs ( h = 0.85 pm) lasers is shown theoretically and

experimentally in Fig. 1.11. It is interesting to note that the experimental

results are in good agreement with the theory mentioned here.

The merit of the density matrix theory of semiconductor is able to take

into account the phase relation between the optical field and the polarisation

formed by electron-hole pairs and the relaxation effect of electron waves.

The energy profile of both linear gain and gain suppression coefficients

broaden with the intraband relaxation. The density matrix theory for

semiconductor lasers can be extended to quantum-well lasers and numerical

examples of calculated gains for both conventional and quantum-well lasers

have been discussed. Realistic gain profile for quantum-well is discussed.

Gain suppression and mode competition phenomena have been analysed by

an improved rate equation derived from the present theory satisfactorily

explains experimental results for both short and long wavelength lasers.

Further it shows stronger gain suppression and stable single mode operation

for longer wavelength lasers.

REFERENCES:

1. H. K. Choi, C.W. Turner, T.H. Windhorn and B.Y. Tsaur, IEEE, Electron

Dev. Lett., EDL-7, 500 (1986)

2. R. Fisher, et. al, ibid., 47, 983 (1985)

3. H. Morkoc, H. Unlu, H. Zabel and N. Otsuka, Solid State Technology, A

Pennwell publication, 71 (1 988)

4. T. Won, et al., J. Appl. Phys., 62, 3860 (1987)

5. J . Chen, et al., Tech. Digest of IDEM, 82 (1987)

6. G. W. Turner, H. K. Choi, B-Y. Tsaur, IEEE Electron Dev. Lett., Appl.

Phys. Lett, 12, 156 (1968)

7. 1 . D'Haenens, Hughes Research Labs., Private communication.

8. K. Y. Lau, Solid State Technology, A Pennwell publication, 91 (1988)

9. D. E. Aspens, in SPIE Symposium on Microlithography, SPIE,

Bellingham, WA (1986)

10. K. Vedam, MRS Bulletin, 12 (I) , 21 (1987)

11. Joseph T. Verdeyen, Laser Electronics, znd edition, 365 (1989)

12. Y. Kokubun and K. Iga: Appl. Opt. 21,1030 (1982)

13. K. Iga, M. Oikawa, S. Misawa, J. Banno and Y. Kokuban: Appl. Opt. 21,

3456 (1982)

14. P.W. Dumke: Phy. Rev., 127,1559 (1962)

15. M.I. Nathan, W.P. Dumke, G. Burns, F.H. Dill, Jr., and G. Lasher: Appl.

Phys. Lett, l , 6 2 (1962)

16. J. Geusic, et al.,: Proc JEEE, 58,58 (1970)

17. H. Nelson: RCA Rev., 24,603 (1963)

18. R.O. Carlson: J. Appl Phys., 38,661 (1967)

19. W. Susaki, T. 0k1.1 and T. Sogo; IEEE J. Quantum Electron., 4, 122

(1 968)

20. W.W. Anderson: lEEE J. Quantum Electron., 1, 228 (1965)

21. H. Kroemer: Proc. IEEE, 52, 1782 (1963)

22. H. Rupprecht, J.M. Woodall, and G.D. Pettit: Appl Phys, Lett., 11, 81

(1967)

23. W. Susaki, T. Sogo and T. Oku: IEEE J. Quantum Electron. 4, 442

(1 968)

24. 1 . Hayashi, M.B. Panish and P.W. Foy: IEEE' J. Quantum electron., 5,

21 1 (1 969)

25. 1 . Hayashi, M.B. Pan~sh, P.W. Foy and S. Sumaski: Appl. Phys. Lett., 17,

109 (1970)

26. B.C. De Loach, Jr., B.W. Hakki, R.L. Hartman and L.A. D'Asato: Proc.

IEEE, 61, 1042 (1973)

27. M. Ishii, H. Kan, W. Susaki and Y. Ogata: Appl. Phys. Lett., 28, 138

(1 975)

28. H. Kan, M. lsh~i and W. Susaki: Japan. J. Appl. Phys., 16, Suppl. 16-1,

461 (1977)

29. N. Chinone: J. Appl. Phys.. 48, 3237 (1977)

30. H. Nam~zaki: Trans, IECE Japan, 59,8 (1976)

31, Y. Ide, T. Furuse, I. Sakuma and K. Nishida: Appl. Phys. Lett., 34, 393

(1 979)

32. Ladany, M. Ettenberg, H.F. Lockwood and H. Kressel: Appl Phys. Lett.,

30,87 (1 977)

33. T. Yuasa. K. Endo, T. Torikai and H. Yonezu: Appl Phys. Lett., 34, 687

(1979)

34. H. Namizaki, S. Takamiya, M, lshii and W. Susaki: J. Appl. Phys., 50,

3743 (1979)

35. S. Nita, H. Namizaki, S. Takarniya and W. Susaki: IEEE J. Quantum

Electron., 15, 1208 (1 979)

36. J. Hsieh: Appl. Phys. Lett., 30, 429 (1977)

37. S. Arai, Y. Suematsu and Y. Itaya: Japan J. Appl. Phys., 18, 709 (1979)

38. M. Hirao, et al.: J. Appl. Phys., 51,4539 (1980)

39. S. Arai, et al.: Electron Lett., 16, 349 (1980)

40. W. Susaki and S. Takamiya: Japan J. Appl. Phys., 20, Suppl. 20-1, 205

(1981)

41. H. Namizaki, H. Kan, M. lshii and A. Ito: J. Appl. Phys., 45, 2785 (1974)

42. H. Namizaki: Trans. IECE Japan, 59,8 (1976)

43. T. Tsukada: J. Appl. Phys., 45,4899 (1974)

44. R.D. Burnham and D.R. Scifres: Appl. Phys. Lett., 27, 510 (1975)

45. K. Aiki, M. Nakarnura, T. Kuroda and J. Urneda: Appl. Phys. Lett., 30,

649 (1 977)

46. T. Sugino, M. Wada, H. Shimizu, K, ltoh and I, Teramoto: Appl. Phys.

Lett., 34,270 (1979)

47. P. Marshall, et al.: Electron. Lett., 15, 38 (1979)

48. H. Kumabe, T. Tanaka, H. Tamizaki, M. lshii and W. Susaki: Appl. Phys.

Lett., 33, 38 (1978)

49. K. Saito and R. Ito: IEEE J. Quantum Electron., 16, 205 (1980)

50. E. Oornura, H. Higuchi, R. Hirano, H. Narnizaki, T. Murotani and W.

Susaki: Technical Digest of 100C'81,44 (1981)

51. W. Susaki, T. Tanaka, H. Kan and M. Ishii: IEEE J. Quantum Electron.,

13, 587 (1977)

52. E. Oomura, R. Hirano, T. Tanaka, M. lshii and W. Susaki: IEEE J.

Quantum Electron., 14,460 (1978)

53. G.H.B. Thompson and G.D. Henshall: Electron. Lett., 16,42 (1980)

54. T. Kajimura, T. Kuroda, S . Yarnashita, M. Nakamura and J. Urneda:

Appl. Optics, 18, 1812 (1979)

55. S. Arai, Y. Suematsu and Y. Itaya: IEEE J. Quantum Electron., 16, 197

(1 980)

56. E. Oomura, T. Muraotani, H. Higuchi, H. Namizaki and W. Susaki: IEEE

J. Quantum Electron., 17, 646 (1981)

57. T. Takakura, K. Iga and T. Tako: Japan J. Appl. Phys., 19, 725 (1980)

58. K. Petermann: IEEE J. Quantum Electron., 15, 566 (1979)

59. H. Yonezu, M. Ueno, T. Kamejima and I. Hayashi: IEEE J. Quantum

Electron., 15, 566 (1979)

60. P.M. Boers, M.T. Vlaardngerbroek and M. Danielsen: Electron. Lett., 11,

206 (1 975)

61. K. Y. Lau, A. Yariv, IEEE J. Quantum Electron., 21, 121 (1985)

62. R.S. Tucker, IEEE J, Lightwave Tech., 3, 1180 (1985)

63. D. Botez and G. J. Herskowitz: Proc. IEEE, 68, 689 (1980)

64. W.T. Tsang, R.L. Hartmann, H.E. Elder and W.R. Holbrook: Appl. Phys.

Lett., 37, 141 (1980)

65. Y. Nishimura, K. Kobayashi, T. lkegami and Y. Suematsu: Technical

Report form IECE of Japan, 71,22 (1971)

66. Y. Nishimura and Y. luish~mura: IEEE J. Quantum Electron., 9, 1011

(1 973)

67. M. Yamada and Y. Suematsu: IEEE J. Quantum Electron., 15, 743

(1 979)

68. N. Bloembergen: W.A. Benjamin Inc., London, 20, (1965)

69. E. 0. Kane: J. Phys. Chem. Solids, 1. 249 (1957)

70. M. Asada and Y. Suematsu: Int. Quantum Electron. Conf., Mll-3

Anaheim, CA, USA.: to be published in IEEE J. Quantum Electron (1984)

71. M. Asada, A. Kameyama and Y. Suematsu: IEEE J. Quantum Electron.,

20,745 (1 984)

72. M. Asada and Y. Suematsu: IEEE J. Quantum Electron., 19, 917 (1983)

50