A
Study o f Active Mode-Locking o f External Ccϟy
Semiconductor Lasers
by
M ih a ilo M ilovanovic
A thesis subm itted to the U n ive rsity o f London fo r the Degree
o f D octor o f Philosophy in E lectronic E ngineering
D epartm ent o f E lectronic and E le ctrica l E ngineering
U n iv e rs ity College London
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Abstract
T h is thesis describes an experim ental and th e o re tic a l in v e s tig a tio n o f
a ctive m ode-locking o f sem iconductor lasers. The lasers used in th is
p ro je ct belong to the class o f long w avelength, b u rie d h e te ro stru ctu re
devices.
In te g ra te d o p tica l m odulators used in th is p roject were designed and
fa b rica te d a t U C L. The waveguides were form ed by tita n iu m in d ifh is io n
in to a lith iu m niobate substrate. The tra v e llin g -w a v e electrodes were
devised as coplanar waveguides and asym m etric coplanar lines.
S hort optical pulse generation using composite ca vity InG aAsP - LiN bO g
lasers was dem onstrated. Three d iffe re n t configurations were tested and
m ode-locking by laser c u rre n t m o du la tio n , A M and F M m ode-locking
were dem onstrated. These stru ctu re s were com pared to b u lk e x te rn a l
cavities in co rp o ra tin g a plane m irro r or a g ratin g . The best perform ance
was achieved from the external g ra tin g laser follow ed by a sem iconductor
o p tica l a m p lifie r. O p tica l pulses sh o rte r th a n lOps and h a vin g a peak
power o f 5mW in the fib re were generated.
The th e o re tica l w ork was based on the frequency-dom ain fo rm u la tio n o f
m ode-locking o f m o n o lith ic e xte rn a l c a v ity sem iconductor lasers. The
e q u a tio n s w h ic h describe m o d e -lo ckin g b y c u rre n t m o d u la tio n ,
in tra c a v ity phase and am plitude m odulation were derived.
N u m e rica l sim u la tio ns were perform ed fo r g a in -sw itch in g o f a s o lita ry
la se r and the equivalent m o no lith ic exte rn a l ca vity laser. The frequency
dom ain fo rm u la tio n o f active m ode-locking was te ste d n u m e ric a lly
u tilis in g tw e n ty five modes and a parabolic loss p ro file . M ode-locking by
c u rre n t m odulation was system atically in ve stig a te d and A M , F M mode-
Zivki Momcilu
Acknowledgem ents
I w ish to express m y sincere g ra titu d e to Professor M G F W ilson fo r
p ro vid in g me w ith th is project and g u id in g me tow ards its com pletion. I
w ould also lik e to th a n k Professors D E N D avies, G P a rry and M r J W
A rte rto n who supported m y case and allow ed me to stay a t U C L d u rin g
tim es o f u n ce rta in ty.
T h is p ro je c t was sponsored by N P L, T e d d in g to n who p ro vid e d the
fin a n c ia l support fo r three and a h a lf years and the laboratory fa c ilitie s in
the fin a l stages. I am especially g rate fu l to D r D A H um phreys fo r ta k in g
a v e ry active role in the project and p ro v id in g help w henever i t was
needed. I w ould also lik e to th a n k D r A Roddie, D r D Henderson, M r A
G iffo rd and the re st o f the Laser and F ibre-O ptic M easurem ent Section fo r
th e ir h o s p ita lity du ring m y stay at NPL.
The lasers used in th is project were supplied by B TR L, Ipsw ich. I w ould
lik e to express m y g ra titu d e to D r M Robertson and D r J D e vlin fo r th e ir
help. Also, I w ould lik e to th a n k D r R F a ta h fo r several s tim u la tin g
discussions and a dozen o f lensed fibres. I w ould lik e to acknowledge M r I
M a rs h a ll who showed us the state o f the a rt o f e xp e rim e n ta l mode-
locking. L a st, b u t by no means least, I w ish to th a n k D r M Adam s fo r a
very constructive discussion on rate equations and m ode-locking.
R egarding the theoretical w ork, I would lik e to th a n k D r M Federighi and
D r P Radmore from U C L fo r several consultations. I also w ish to express
m y g ra titu d e to Professor J New from Im p e ria l College fo r a s tim u la tin g
dialogue concerning the tim e and frequency dom ain approaches to mode-
locking.
I am indebted to C hris W atson fo r proofreading th is thesis and m any
docum ents th a t I have w ritte n d u rin g m y stay a t U C L. I also have to
acknowledge his help in the fa b rica tio n o f in te g ra te d optic m odulators. I
fo r teaching me both in te g ra te d optics and physics in general. F in a lly , I
w ould lik e to m ention m y fo rm e r and present colleagues D id ie r Erasm e,
Tony O verbury, P eter Liang, M ike Plessey and R ayer Z w iggelaar who
have co n trib u te d to a very enjoyable research environm ent and provided
Table of Contents
A b stra ct 2
A cknow ledgm ents 4
Table o f Contents 6
L is t o f Figures and Tables 10
L is t o f Symbols 13
Chapter 1 Introduction
18
Chapter 2
Sem iconductor Laser
24
2.1 Basic Concepts 25
2.1.1 Classical F ield D escription 25
2.1.2 Threshold C ondition and L o n g itu d in a l Modes 27
2.1.3 G ain and S tim ulated Em ission 31
2.1.4 W aveguide Modes 33
2.1.5 Em ission C haracteristics 34
2.2 Rate Equations 37
2.3 E xte rn a l C avity Laser 40
Chapter 3
Optical Pulse Generation U sing Sem iconductor
44
Lasers
3.1 M ethods o f S hort Pulse G eneration 44
3.1.1 G ain-S w itching 44
3.1.3 M ode-Locking 51
3.2 Selected Review o f Reported E xperim ental Results 55
3.2.1 G ain-S w itching 55
3.2.2 Q -Sw itching 56
3.2.3 A ctive M ode-Locking 57
3.2.4 Passive and H yb rid M ode-Locking 59
3.2.5 M ode-locking o f M o no lith ic Sem iconductor Lasers 60
3.3 D iscussion 61
Chapter 4
lith iu m M obate Modulators
64
4.1 Waveguides 64
4.2 Electro-O ptic Effect in LiNhO g gg
4.3 Phase and In te n s ity M odulators 69
4.4 H igh-Frequency O peration 72
Chapter 5
Short Pulse Generation Using Composite Cavity
77
InGaAsF-liNbOg Lasers
5.1 L ith iu m Niohate E xternal C avity Laser 77
5.1.1 Continuous Wave O peration 78
5.1.2 AM , FM M ode-Locking and FM -Laser O peration 80
5.1.3 Technical Aspects o f the Composite C a vity 83
5.1.3.1 O ptical Coupling 84
5.1.3.2 A n tire fle ctive Coatings 87
5.1.3.3 M irro rs 88
5.2 E xperim ental W ork 90
5.2.1 E xperim ental Set-Up 91
5.2.2.1 E stim ation o f the In tra c a v ity O ptical Feedback S tre n g th 93
5.2.2.2 E stim ation o f the Q u a lity o f AR C oating 94
5.2.3 M ode-Locking U tilis in g Laser C u rre n t M o d u la tio n 96
5.2.4 A M M ode-Locking E xperim ent 100
C h a p te r 6 A c tiv e M ode-Locking U sing B u lk E x te rn a l 106
C avities
6.1. E x te rn a l M irro r C onfiguration 107
6.1.1 F irs t E xperim ental C onfiguration 107
6.1.2 Second E xperim ental C onfiguration 111
6.2. E xte rn a l G ra tin g C onfiguration 115
6.2.1 In itia l Experim ents 115
6.2.2 F in a l C onfiguration 118
6.2.2.1 D escription o f the System 118
6.2.2.2 C haracterization o f the System 119
6.2.2.2.14GHz E xternal C avity 120
6.2.2.2.2 2GHz E xternal C avity 124
C hapter 7 Theory o f A ctive M ode-Locking o f S em iconductor 128
Lasers
7.1. Theory o f Active Mode-Locking 129
7.1.1 Tim e and Frequency D escriptions o f A ctive 129
M ode-Locking
7.1.2 Development o f the Theory o f Active M ode-Locking 131
o f Sem iconductor Lasers
7.2. Frequency-Dom ain F orm ulation o f A ctive M ode-Locking 133
o f Sem iconductor Lasers
7 ’2.1 A ctive Mode-Locking by C u rre n t M odulation 134
7.2.2 FM M ode-Locking 147
7.2.3 A M M ode-Locking 150
7.2.4 M odulator Effects 154
C hapter 8 N u m e rica l S im ulations o f A c tiv e M ode-Locking o f 157
M o n o lith ic E xte rn a l C a vity S em iconductor Lasers
8.1.1 G ain-S w itching o f the S o lita ry Laser 158
8.1.2 G ain-S w itching o f the E xte rn a l C a vity Laser 159
8.2 M ode-Locking by C u rre n t M odulation 163
8.2.1 N um erical S im ulations 164
8.2.2 E vo lu tio n o f M ode-Locking 169
8.2.3 D etuning o f the M odulating Frequency 169
8.2.4 V a ria tio n o f the Bias C u rre n t 174
8.2.5 V a ria tio n o f the RF C u rre n t A m p litu d e 177
8.2.6 Effects o f the L in e w id th Enhancem ent Factor 177
8.3.FM M ode-Locking 179
8.3.1 N um erical S im ulation In c lu d in g the L in e w id th 181
Enhancem ent Factor
8.3.2 N um erical S im ulation E xcluding the L in e w id th 184
Enhancem ent Factor
8.3.3 FM -Laser O peration 187
8.4. A M M ode-Locking 189
8.4.1 N um erical S im ulation In c lu d in g the L in e w id th 189
Enhancem ent Factor
8.4.2 N um erical S im ulation E xcluding the L in e w id th 192
Enhancem ent Factor
8.5 Sum m ary 192
Chapter 9
Conclusions and Future Work
195
References 202
A ppendix 1 221
lis t o f Figures and Tables
Chapter 1
Chapter 2
Fig. 2.1
Fig. 2.2
Fig. 2.3
Fig. 2.4
Schematic illu s tra tio n o f a sem iconductor laser and its 28
Fabry-Perot cavity
G ain p ro file and lo n g itu d in a l modes o f a sem iconductor 28
la se r
Schem atic diagram o f a b u ried sem iconductor laser 35
Schematic lig h t - cu rre n t curve 35
Chapter 3
Fig. 3.1 R elaxation oscillations
Fig. 3.2 G ain-sw itching o f a s o lita ry laser
Fig. 3.3 Q -sw itching o f a m onolithic external ca vity laser
Fig. 3.4 Concept o f mode-locking
Table 3.1 Comparison between a gain-sw itched D FB and
a mode-locked ECL
Table 3.2 State o f the a rt performance a t re p e titio n rates
below lOGHz
45
46
49
52
62
62
Chapter 4
Fig. 4.1 Electrode and waveguide com binations 68
Fig. 4.2 Standard m odulator configurations 70
Fig. 4.3 Standard electrode structures 73
Fig. 4.4 N orm alised frequency response o f a tra v e llin g wave 76
Chapter 5
Fig. 5.1 In tra c a v ity optical coupling schemes 85
Fig. 5.2 M ode-locking experim ental arrangem ent 92
Fig. 5.3 Schematic diagram o f an external ca vity laser 95
Fig. 5.4 E stim a tio n o f the optical feedback 95
Fig. 5.5 Three InG aAsP-LiN bO g m ode-locking con fig u ra tio n s 97
Fig. 5.6 Sam pling oscilloscope traces - cu rre n t m odulation 99
Fig. 5.7 Sam pling oscilloscope trace - in tra c a v ity loss 102
m o d u la tio n
Fig. 5.8 Sam pling oscilloscope trace - in tra c a v ity phase 104
m o d u la tio n
Chapter 6
Fig. 6.1 M ode-locking experim ental arrangem ent
Fig. 6.2 A u tocorrelation trace
Fig. 6.3 M ode-locking experim ental arrangem ent I I
Fig. 6.4 Sam pling oscilloscope / scanning FP traces
Fig. 6.5 E xperim ental arrangem ent a t N P L
Fig. 6.6 L ig h t - cu rre n t characteristics
Fig. 6.7 A utocorrelation trace - am plifie d o u tp u t
Fig. 6.8 HP lightw ave analyser traces
Fig. 6.9 Frequency detuning characteristic
Fig. 6.10 Sam pling oscilloscope trace
108
110
112 114 117 121 122 123 125 126Chapter 7
Fig. 7.1 Schematic diagram s o f the m odelled m o n o lith ic
external cavity lasers
Fig. 7.2 In tra c a v ity m odulation efficiencies fo r a lum ped
and travelling-w ave m odulator
135
C hapter 8
Fig. 8.1 G ain-sw itching o f the so lita ry laser a t 1 GHz 160
Fig. 8.2 G ain-sw itching o f the m onolithic E C L a t IG H z 161
Fig. 8.3 G m n-sw itching o f the m o nolithic EC L a t lOGHz 162
Fig. 8.4 M ode-locking by cu rre n t m odulation 166
Fig. 8.5 E volution o f inode-locking 170
Fig. 8.6 D etuning o f the m odulating frequency 172
Fig. 8.7 'Sideband flip p in g ' effect 175
Fig. 8.8 V a ria tio n o f the bias cu rre n t 176
Fig. 8.9 V a ria tio n o f the RF cu rre n t am plitude 178
Fig. 8.10 Effects o f the lin e w id th enhancement factor 180
Fig. 8.11 FM mode-locking w ith Pg=5 182
Fig. 8.12 FM mode-locking w ith pg=0 185
Fig. 8.13 FM -laser operation 188
Fig. 8.14 A M m ode-locking w ith P^=5 190
Fig. 8.15 A M mode-locking w ith Pg=0 193
List of Symbols
Sem iconductor Laser
electric fie ld vector
co n d u ctivity o f the m edium a
vacuum p e rm ittiv ity Eq
speed o f lig h t in vacuum c
induced e le ctric p o la risa tio n fP
complex envelope o f the fie ld E
complex envelope o f the polarisation P
optical a n g u la r frequency co
optical frequency v
vacuum w avelength X
vacuum wave num ber
s u sce p tib ility o f the m edium %
s u s c e p tib ility in absence o f pum ping Xo
s u s c e p tib ility due to external pum ping %p
complex d ie le ctric constant e
complex propagation constant P
complex index o f re fra ctio n p
refi*active index o f the m edium p
background re fra ctive index \i^
re fra ctive index co n trib u tio n due to pum ping App
n e t gain coefficient g
net absorption coefficient a
confinem ent fa cto r T
m irro r re fle c tiv ity R
m irro r loss
in te rn a l loss
le n g th o f the laser L
ca vity resonance frequency Vj
lo n g itu d in a l mode spacing Ü
g ain coefficient a
c a rrie r density n
c a rrie r de n sity fo r transparency 1^0
constant (App) b
lin e w id th enhancem ent factor Pc
c a rrie r d iffu s io n D
c u rre n t density J
charge o f electron Qe
laser active area volum e V
thickness o f the active area d
w id th o f the active area w
to ta l recom bination rate R (n)
n o n -ra d ia tive recom bination factor ^ n r
ra d ia tiv e recom bination factor B
A uger recom bination factor C
photon density ^ p h
stim u la te d recom bination rate ^ s t
complex envelope o f the fie ld E
lo n g itu d in a l ca vity resonance ^ 0
gain constant G
loss constant Y
photon life tim e
envelope o f the fie ld (real) A
phase o f the fie ld *
photon num ber P
average ra te o f spontaneous emission ^ s p
o u tp u t power ^ o u t
c a rrie r num ber N
c u rre n t I
c a rrie r recom bination rate Ye
spontaneous c a rrie r life tim e '^e
E xte rn a l C a vity Laser
lin e w id th o f a single-mode laser ^^'sol
lin e w id th o f an external cavity laser ^^'ext
le n g th o f a s o lita ry laser Ll
length o f the cavity L
single-pass optical coupling efficiency ti
th re sh o ld c u rre n t
re fle c tiv ity o f AR coating
external re fle c tiv ity R ^^
m o d u la tin g frequency f
bias c u rre n t ly
RF power
M ode-Locking
complex am plitude o f the mode E[
mode am plitude (real)
photon num ber Pj
phase o f the mode ^
la sin g a n g u la r frequency cOq
m o d u la tin g angular frequency
lo n g itu d in a l mode spacing A
lo n g itu d in a l ca vity resonances Q j
central Fabry-P erot frequency Ag
la te ra l mode d is trib u tio n Ç
transverse mode d is trib u tio n Ç
s p a tia lly averaged d ielectric constant (e)
mode index \i
mode absorption index a
DC component o f the c a rrie r num ber Ny
F o u rie r component o f the c a rrie r num ber
coupling loss coefficient
length o f the cavity L
le n g th o f the lasing area
le n g th o f the m odulator L j^
m odified d is trib u te d m irro r loss yin
coupling coefficient Aq
c u rre n t I
m odified gain expression G ’
s e lf sa tu ra tio n , cross sa tu ra tio n coefficients Pgj
average ra te o f spontaneous em ission R^p
fi*equency s h ift due to the average rate o)gp
o f spontaneous em ission
phase m odulator co n trib u tio n to e
re fi'a ctive index p e rtu rb a tio n A p ^
m odulation depth o f the phase m odulator
F M coupling coefficient Apj^
am plitude m odulator co n trib u tio n to e
loss p e rtu rb a tio n due to m odulation
coupling loss A M ^AMO
coupling coefficient A M A ^j^
sp a tia l v a ria tio n o f the m odulating signal 6 ^
distance from the m irro r Zq
in tra c a v ity m odulating efficiency (m odulus)
in tra c a v ity m odulating efficiency (phase) \|/
L ith iu m N iobate M odulators
o rd in a ry index
e x tra o rd in a ry index p^
change o f the o rd in a ry index Ap^
change o f the e xtra o rd in a ry index Ap
electro-optic coefficients r^
applied fie ld E
d riv in g voltage
interelectrode gap Ç
overlap between el. fie ld and opt. mode
in te ra c tio n le n g th
induced phase s h ift
o u tp u t in te n s ity
m axim um lig h t in te n s ity (in phase) lout(max)
phase s h ift difference Aq>
d ire ctio n a l coupler fa cto r p
difference in eff. prop, constants Ap
e le ctrical ban d w id th A f
electrode resistance
electrode capacitance C
Chapter 1
Introduction
The fie ld o f sh o rt o p tica l pulse generation using sem iconductor lasers
was created in the la te sixties and e a rly seventies to e x p lo it th e m a in
advantages o f the sem iconductor m edium nam ely sm a ll size, e le c tric a l
pum ping and p o te n tia l fo r in te g ra tio n w ith other o p tica l and electronic
components. In the la te seventies, when re lia b le operation o f several laser
s tru c tu re s was achieved, the fie ld received a sudden im p e tu s b y the
d e m o n stra tio n o f active m ode-locking and g a in -s w itc h in g re s u ltin g in
opticEil pulses shorter th a n 20ps. Since then, considerable e ffo rt has been
directed tow ards short pulse generation using techniques o f m ode-locking
and ga in s w itch in g . U n til the end o f th e e ig h tie s , advances w ere
re s tric te d to im proved la se r s tru c tu re s (lo w p a ra s itic s s tru c tu re s ,
d is trib u te d feedback lasers and m u ltip le quantum w e ll devices) ra th e r
th a n ra d ica l im provem ent in the techniques o f sh o rt pulse generation.
The second m ajor development in the fie ld was made in the la te eighties
by the in tro d u c tio n o f fib re am plifiers. I t was realised th a t sem iconductor
la se r diodes can serve as o p tica l pulse generators in fu tu re so li ton
system s and also p la y a d o m ina n t ro le in o p tic a l p u m p in g o f fib re
a m p lifie rs. The concept o f short pulse generation was enriched because o f
th e in c lu s io n o f self-phase m o d u la tio n and disp e rsion in to p ra c tic a l
systems w h ich consist o f laser diodes, fib re a m p lifie rs and dispersion
shifted fibres.
The consequence o f th e s o lito n system race was th a t th e la rg e s t
te lecom m unication research in s titu te s (A T T & B e ll, N T T , B T R L ) have
c o n c e n tra te d on g e n e ra tio n o f tra n s fo rm lim ite d p u lse s u s in g
sem iconductor lasers. In itia l a ctivitie s were aim ed a t active m ode-locking
u s in g b u lk or h y b rid ca vitie s or a lte rn a tiv e ly a t g a in -s w itc h in g in
m o n o lith ic lasers u tilis in g concepts o f active and h y b rid m ode-locking, as
the in te g ra tio n o f lasers, waveguides and Bragg filte rs was already in an
advanced state because o f the w ork directed tow ards n a rro w lin e w id th
sources and frequency tunable lasers. A t present, the w o rk on active
m ode-locking o f m o n o lith ic lasers seems to be directed tow ards longer
stru ctu re s ( 5-10 GHz ro u n d -trip frequencies), high-frequency m odulation
c a p a b ility o f the laser section and the o p tim isatio n o f the Bragg section
R egarding passive and h yb rid m ode-locking the e ffo rt is concentrated on
the design o f saturable absorbers in m u ltip le quantum w e ll stru ctu re s^.
C o llid in g pulse m ode-locking has also been dem onstrated in m o n o lith ic
configurations and i t resulted in sub-picosecond optical pulses^.
The m a in disadvantage o f mode-locked sem iconductor lasers is th e ir
m oderate o u tp u t power. The power levels can be increased by the use o f
e x te rn a l sem iconductor lasers a m p lifie rs or fib re a m p lifie rs . A n
a lte rn a tiv e approach to increasing the o ptical power o f the m ode-locked
sem iconductor laser is the use o f a v e rtic a l ca vity sem iconductor laser
in ste a d o f edge-em itting laser stru ctu re . The o p tic a lly pum ped v e rtic a l
c a v ity sem iconductor lasers in an e x te rn a l c a v ity c o n fig u ra tio n can
produce sub-picosecond o p tic a l pulses. M oreover, a d d itio n a l pulse
s h o rte n in g can th e n be achieved u sin g sta n d a rd pulse com pression
techniques and m u ltip le fib re a m p lifie r stages. In th is m anner pulses o f
15fs a t optical w avelength o f 1.5pm have been produced and they represent
the shortest pulses achieved u tilis in g a sem iconductor laser m edium ^.
Picosecond o ptical pulses produced by m ode-locking o f va rio u s types o f
la s e r system s have fo und a p p lic a tio n s in th e s tu d y o f u ltra fa s t
phenomena in physics, chem istry and biology. The sm all size, low power
re qu ire m en t and ease o f use o f gain switched and m o n o lith ic mode-locked
sem iconductor lasers allows the p o s s ib ility o f applications th a t w ould not
be p ra ctica l using other laser systems. A c tiv e ly mode-locked lasers have
b e tte r phase noise performance than gain-sw itched lasers and th a t w ould
One o f the m ost p rom isin g applications o f m ode-locked sem iconductor
lasers is fo r high-speed tim e d iv isio n m u ltip le x e d telecom m unications
system s^. On the tra n s m ittin g end, a mode-locked sem iconductor laser is
used as a source o f n e a rly tra n s fo rm -lim ite d o p tica l pulses fo r so lito n
generation. The outputs o f several o f these mode-locked lasers can th e n be
interleaved to form a very high data rate channel w ith ve ry sm all am ount
o f tim in g jitte r . M oreover, th e re is the p o te n tia l o f in te g ra tin g the
m odulator and the optical pulse source on the same chip. A t the receiving
end, m ode-locked lasers can be used to d e m u ltip le x th e received data
stream using four-wave m ixin g effects in optical fib re s^ or in conjunction
w ith n o n lin e a r loop m irro r sv d tc h in g ^ .
M ode-locked sem iconductor lasers have also found ap p lica tio n s in opto
e le ctro n ic m easurem ent systems. These a p p lica tio n s in clu d e im p u lse
response te s tin g o f photodiodes, h ig h re s o lu tio n o p tic a l tim e -d o m a in
in te rfe ro m e try and electro-optic sam pling o f electronic signals®.
The project described in th is thesis was proposed in 1987 and i t sta rte d in
1988. The project was funded by N ational Physical Laboratory, Teddington
and our objective was to generate lOps optical pulses fo r the ca lib ra tio n o f
fa s t photodetectors. The proposed m ethod was active m ode-locking o f a
com posite c a v ity In G a A sP -L iN b O g la s e r u tilis in g an in tra c a v ity
m o d u la to r and the p ro je ct was e n title d 'M ode-Locked L a se r u sin g
In te g ra te d O ptics'. T his approach was in itia lly adopted because a t th a t
tim e U C L had fa b rica tio n fa c ilitie s fo r lith iu m niobate waveguide devices
and fa st sem iconductor lasers were not com m ercially available.
In the ja rg o n o f mode-locking the proposed project could be called A M and
FM m ode-locking o f the composite ca vity sem iconductor lasers (A M refers
to in tra c a v ity a m p litu d e m o d u la tio n , F M to in tr a c a v ity phase
m o d u la tio n ). The th e o ry o f A M and F M m o d e -lo c k in g o f th e
inhom ogeneously and hom ogeneously broadened la se rs w ith lo n g
re la x a tio n tim es was fo rm u la te d in th e la te s ix tie s , fo llo w in g the
W hen we started th is project there was no reported dem onstration o f FM
mode lo ckin g o f sem iconductor lasers. The best exam ple o f A M mode-
lo ckin g was a tw in-section AlG aAs diode laser w ith in an e xte rn a l ca vity
in co rp o ra tin g a b u lk g ra tin g generating 8ps pulses^. The oth e r reported
co n fig u ra tio n , on w hich our project proposal was based was a composite
c a v ity In G a A sP -L iN b O g la se r in c o rp o ra tin g a h ig h speed d ire c tio n a l
coupler sw itch , w h ich produced 22ps pulses The state o f the a rt o f
active m ode-locking by c u rre n t m o du la tio n was a fa s t InG aA sP la se r
w ith in a b u lk external ca vity producing m u ltip le subpicosecond pulses a t
the re p e titio n rate o f Ib G H z ^ .
In th e fir s t and second year o f the p ro je ct we concentrated on th e
experim ental w ork : setting up a b u lk external ca vity laser, fa b ric a tio n o f
tita n iu m in d iffu s e d lith iu m niobate waveguides and the fa b ric a tio n o f
both in te n s ity and phase m odulators. A t the beginning o f the th ird year
(Septem ber 1990) we had dem onstrated sh o rt pulse generation by both
in tra c a v ity in te n s ity and phase m odulation (pulses o f the order o f 45-50 ps)
and by m ode-locking o f a composite InG aAsP - LiN bO g laser by c u rre n t
m odulation (pulses in the range o f 30-35 ps). W ith the same laser, used in
a b u lk e xte rn a l ca vity, in co rp o ra tin g a plane m irro r we produced 27ps
pulses a t a m uch low er bias current. A t th a t tim e we made a decision to
stop fu rth e r fa b rica tio n o f lith iu m niobate m odulators and assemble an
e xternal g ra tin g laser in order to fu lfil our research contract. We planned
to use the re m a in in g tim e in order to develop a n u m e rica l m odel th a t
w ould u n ite our experim ental attem pts. U ltim a te ly , we concluded our
experim ental w ork by generating optical pulses shorter th a n 10 ps using
an external g ra tin g laser followed by a sem iconductor laser a m p lifie r. In
o u r th e o re tic a l in v e s tig a tio n , we perform ed n u m e rica l s im u la tio n s o f
a ctive m ode-locking o f m o n o lith ic e x te rn a l c a v ity lasers by c u rre n t,
a m plitud e and phase m odulation. The model was based on a frequency-
dom ain fo rm u la tio n using m ulti-m ode fie ld equations.
Because o f th e broad scope o f the m a te ria l presented in th is thesis,
C hapters 2, 3, and 4 are in tro d u cto ry in th e ir character.
The basic th e o ry concerning a sem iconductor la s e r is presented in
C hapter 2. The classical description o f the laser fie ld in conjunction w ith
the phenom enological gain model is given. E m ission ch a ra cte ristics o f a
sem iconductor laser are o u tlined. The ra te equations fo r the la se r fie ld
and c a rrie r population are introduced. F in a lly , the optical feedback effects
and the external ca vity operation o f a sem iconductor laser are discussed.
G eneral m ethods fo r sh o rt o p tica l pulse generation in sem iconductor
lasers are o u tlin e d in the fir s t section o f the C hapter 3. In th e second
section, the experim ental re su lts re la te d to d iffe re n t m ethods o f pulse
gen e ra tio n are review ed, w ith the em phasis placed on a ctive mode-
locking. In the la s t section, a b rie f discussion on short pulse generation is
presented.
A n in tro d u ctio n to integrated optic lith iu m niobate m odulators is given in
C h a p te r 4. W aveguides, the e le ctro -o p tic effect, d iffe re n t m o d u la to r
configurations and high-frequency design requirem ents are discussed.
The p o s s ib ilitie s o f co m p o site -ca vity In G a A sP -L iN b O g lasers are
discussed in C hapter 5. The technical requirem ents o f com posite-cavity
lasers such as optical coupling, a n tire fle c tiv e coatings, and m irro rs , are
considered. E xperim ental results on short o ptical pulse generation using
d iffe re n t m ode-locking configurations are presented in the second section
o f the chapter. O ur experim ents have included m o d u la tio n o f th e laser
cu rre n t as w ell as in tra c a v ity in te n s ity and phase m odulation.
O u r e xp e rim e n ta l w o rk concerning a ctive m ode-locking u s in g b u lk
e x te rn a l c a v itie s is presented in C h a p te r 6. The e x te rn a l m irro r
c o n fig u ra tio n s w ere te ste d a t U C L , w h ile th e m ore com plete
m easurem ents o f the exte rn a l g ra tin g laser follow ed by a sem iconductor
Tim e-dom ain and frequency-dom ain theories o f active m ode-locking are
exam ined in the fir s t section o f C hapter 7. This is follow ed by a review o f
the th e o ry o f active m ode-locking o f sem iconductor lasers. In th e second
section o f C hapter 7, the frequency-dom ain fo rm u la tio n o f active mode-
lo ckin g o f m o n o lith ic sem iconductor lasers is derived. We obtained three
s im ila r sets o f equations fo r m ode-locking by cu rre n t m odulation, A M and
F M m ode-locking. C hapter 7 is concluded w ith a com m ent on the
in tra c a v ity m o d u la tio n e ffic ie n c y o f lum ped and tra v e llin g -w a v e
m odulators.
N u m e rica l sim u la tio n s o f g a in -sw itch in g o f a s o lita ry and m o n o lith ic
e xte rn a l c a v ity sem iconductor laser are presented in the fir s t section o f
C hapter 8. The aim was to illu s tra te the behaviour o f the averaged, single
mode ra te equations. In the second section, th e re s u lts o f n u m e rica l
sim u la tio ns o f active m ode-locking by cu rre n t m o du la tio n are presented.
We have in ve stig a te d the evo lu tio n o f m ode-locking, d e tu n in g o f the
m odulating frequency, va ria tion s o f the bias and RF drive, and role o f the
lin e w id th enhancem ent factor. The results obtained fo r FM m ode-locking
presented in the th ird section o f the chapter, show tw o d is tin c t types o f
behaviour. F M -laser operation is also dem onstrated in o u r n u m e rica l
m odel fo r F M m ode-locking. F in a lly , a set o f sim u la tio n s o f A M mode-
locking is presented.
C onclusions fro m the w o rk as a w hole are described in C h a p te r 9,
Chapter 2
Sem iconductor Laser
Since the fir s t dem onstration o f la sin g in a sem iconductor m edium in
1962, th e re has been enormous progress in b o th the th e o ry and the
m anufacture o f sem iconductor lasers optim ised fo r d iffe re n t applications.
U sing d iffe re n t sem iconductor m aterials, la sin g has been achieved over a
v e ry broad range o f frequencies, from near u ltra v io le t to fa r in fra re d .
N um erous books have been w ritte n on the subject o f sem iconductor
lasers. The standard textbooks date fro m the la te seventies and e a rly
e i g h t i e s a n d deal w ith basic device physics. M ore recent books are
concerned w ith more specific topic such as long w avelength lasers^»^,
m odulation and noise® and single-frequency structures^.
In th is thesis the in tro d u ctio n to sem iconductor lasers w ill be presented
fro m the p o in t o f view o f laser user, ra th e r th a n th e la se r designer.
M oreover, the em phasis w ill be placed on tim e-dependent phenom ena,
w hich are p e rtin e n t to gain-sw itching and m ode-locking. The lasers used
in th is p ro je c t belong to the class o f devices designed fo r o p tic a l
com m unications systems - 1.55pm index-guiding (buried h e te ro stru ctu re )
InG aAsP lasers. The basic th eoretical fo rm u la tio n and n o ta tio n th a t we
adopt are th a t o f Agraw al's textbook^ on long w avelength sem iconductor
lasers.
The m ajor p a rt o f th is chapter (Section 2.1) is devoted to basic th e o ry o f
se m ico n d u cto r la se rs in tro d u c in g th e c la ssica l fie ld d e s c rip tio n ,
th re sh o ld condition, lo n g itu d in a l modes, gain and s tim u la te d em ission,
w aveguide modes, and the em ission ch a ra cte ristics. S ection 2.2 deals
w ith the single-mode rate equations and includes the d e fin itio n o f several
im p o rta n t e n titie s . In the fin a l section o f th is chapter the concept o f
2.1 Basic Concepts
In sem iconductor lasers a sem iconductor m a te ria l is e le c tric a lly pum ped
using a forw ard-biased p-n diode stru ctu re . Charge ca rrie rs, w h ich are
in je cte d in to a th in active la ye r, provide the necessary gain a t o p tica l
frequencies. A re so n a to r form ed by the tw o cleaved facets o f th e
sem iconductor gain m edium can produce a s u ffic ie n t am ount o f o p tica l
feedback. The s tre n g th o f the pum ping o f a sem iconductor la se r is
determ ined by the injected cu rre n t density. The th re sh o ld co n d itio n fo r
the b u ild -u p o f laser o scilla tio n is reached when the gain overcomes the
to ta l ca vity loss. A ny fu rth e r increase in injected c u rre n t density above its
th re s h o ld value leads to la se r ra d ia tio n d o m ina te d b y s tim u la te d
em ission.
In th is section, our in te n tio n is to provide a b rie f d e scrip tio n o f th e
elem entary concepts related to understanding o f the sem iconductor la se r
dynam ics and m ode-locking. We s ta rt w ith the classical fie ld description.
2.1.1 C lassical F ield D escription
The classical description o f optical phenomena is based upon M a xw e ll’s
equations. The p ropagation o f an electrom agnetic wave th ro u g h an
atom ic m edium is described by the wave equation. The a im o f th is
subsection is to introduce the wave equation and optical param eters such
as the d ie le c tric constant, the re fra c tiv e in d e x and th e a b so rp tio n
coefficient w hich determ ine the propagation characteristics..
S ta rtin g fro m M axw ell’s equations a wave equation o f the fo llo w in g form
can be derived:
v Ç - - 5 - — - — = (2.1)
In th is equation *E and T represent the e lectric fie ld vector and induced electric p o la risatio n vector respectively; Eq is the vacuum p e rm ittiv ity , c is
the speed o f lig h t and o is the co n d u ctivity o f the m edium . The wave
equation (2.1) is v a lid fo r a rb itra ry tim e -va ryin g fie ld s. F o r optical fie ld s
w ith harm onic tim e v a ria tio n we w rite
S (x,y,z,t) = Re [ E (x,y,z) e 'j°*] (2.2)
fP(x,y,z,t) = Re [ P (x,y,z) e-J“ ^] (2.3)
w here co = 2tcv is the a n g u la r frequency and v = c/A, is th e o s c illa tio n
frequency o f the optical fie ld a t the vacuum w avelength X. S u b s titu tin g
Eqs. 2.2 and 2.3 in Eq. 2.1 we obtain:
V Ï + k | ( l+ j5 . ) E = - ^ p (2.4)
e<p) 6o
where kQ=co/c= IkIX is the vacuum wave num ber.
U nder steady-state conditions the response o f the m edium to the e lectric
fie ld is governed by the s u s c e p tib ility % defined by P = £oX(co)E. The
s u s c e p tib ility can be decomposed in to tw o p a rts : Xq* th e m edium
s u s c e p tib ility in the absence o f external pum ping and %p rela te d to the
s tre n g th o f pum ping. B o th Xo Xp are com plex and frequency-
dependent.
E quation 2.4 can he re -w ritte n as the tim e-independent wave equation o f
the form :
V^E + ek?E = 0 (2.5)
where we have introduced the complex d ielectric constant defined as
e = e' + je" = 1 + Xo + Xp + j ^ (2.6)
The tim e-independent wave equation can be used to o b ta in th e s p a tia l
mode stru ctu re o f the optical fie ld in a p ra ctica l device. In the sim plest
case, the plane-wave solutions o f Eq. 2.5 can be treated. The propagation
ch a ra cte ristics o f a plane wave in a m edium are u s u a lly described in
term s o f tw o optical constants, the index o f refi*action \i and the absorption
constant a. I f we consider a plane wave propagating in the z d ire ctio n
E = Eoexp(jPz) (2.7)
where Eq is the constant am plitude and the complex propagation constant
is given as
P = k j/ë = k<p (2.8)
and where p is the complex refractive index u su a lly w ritte n as
= (2.9)
A ssum ing th a t a<<|xk^ using e = p? and equating the re a l and im a g in a ry
parts, fo llo w in g relations can be obtained:
p = VF = Vl+Re(xJ+Re(%j^ (2.10)
(2.11)
2.1.2 Threshold C ondition and L o n g itu d in a l Modes
The plane wave so lu tio n o f the wave equation can be used to o b ta in an
estim ate o f the laser frequency and the optical gain requ ire d fo r the onset
o f lasing. In a sem iconductor laser o f le n g th L , shown schem atically in
Fig. 2.1, the th in ce n tra l region provides the o p tica l g ain and cleaved
facets form a Fabry-P erot cavity to provide optical feedback. The o p tica l
C u rre n t In je ctio n
A ctive Region
Cleaved Facets
R1
G ain M edium
FABRY-PEROT C A VITY
R2
F ig . 2.1 Schematic illu s tra tio n o f a sem iconductor laser and its
Fabry-Perot cavity
gain
lasing mode loss
lo n g itu d in a l modes
►
optical frequency
F ig . 2.2 G ain p ro file and lo n g itu d in a l modes o f a sem iconductor
laser. The threshold is reached when the gain o f the
the complex propagation constant can be obtained by com bining Eq. 2.8
and 2.9:
P = ^iko+j|.
(2.12)
I f Re(Xp) is sm all (Eq. 2.10), the re fra ctive index can be a p p ro xim a te ly
expressed as
p = Pb+A^ip (2.13)
where pb= Vl+Re(%() and App = ■
2|J-b
Pb is th e background in d e x o f the unpum ped m a te ria l and App is
c o n trib u tio n due to the presence o f charge ca rrie rs. A ppis negative and
a lth o u g h i t is often less th a n 1% o f pb i t s ig n ific a n tly affe cts th e
sem iconductor laser behaviour.
The a b so rp tio n co e fficie n t a has th re e c o n trib u tio n s : th e m a te ria l
absorption Im(%Q), its reduction w ith e xte rn a l pum ping Im(%p) and the
o/EqCO te rm w hich accounts fo r other in te rn a l losses such as fre e -c a rrie r
absorption and scattering. I t is often convenient to in troduce the n e t gain
g defined as
g = — Im(Xo+Xp) (2.14)
and to express the net absorption coefficient as
a = -Fg + (2.15)
In th is expression F represents the confinem ent fa c to r, w h ich is the
To o b ta in the th reshold condition, i t is re q u ire d th a t the o p tic a l fie ld
should reproduce its e lf a fte r each ro u n d -trip under steady-state condition.
The threshold condition is:
Y R % e W = i (2.16)
E q u a tin g the re a l and im a g in a ry p a rts o f Eq. 2.16 and using Eq. 2.12
fo llo w in g expressions can be obtained:
Y R ^ e - “ L = l (2.17)
sin(2|ikJL<) = 0 (2.18)
The fir s t expression (Eq. 2.17) gives the threshold gain. I t can be w ritte n
in the form :
+ “ in t (2.19)
where represents the m irro r loss expressed as
The physical m eaning o f the threshold gain equation (Eq. 2.19) is th a t the
gain due to external pum ping m ust balance the to ta l losses. The effect o f
spontaneous em ission has been ignored in th is sim ple analysis.
The im a g in a ry p a rt o f the threshold equation (Eq. 2.18) can be used to
obtain the lasing frequency. Eq. 2.18 has m u ltip le solutions o f the form
2 p k o L = 2i7c ( 2 . 2 1 )
where i is an integer. U sing kQ=27cv/c, the lasing frequency v is
v = V i = ^ (2.22)
The la s e r tends to o scilla te a t a frequency th a t coincides w ith a
lo n g itu d in a l mode supported by the FP cavity. In the case o f homogeneous
broadening only one lo n g itu d in a l mode, whose frequency n e a rly coincides
w ith th e peak gain, reaches th re sh o ld (F ig. 2.2) and the la s e r also
rem ains single-moded above threshold.
In a sem iconductor laser, the re fra ctive index p va rie s w ith frequency,
and the lo n g itu d in a l mode spacing Q is given as
(2.23) 2ngL
where |Xg = n + represents the group refractive index.
2.1.3 G ain and S tim ulated Em ission
In sem iclassical la se r theory the m edium response is governed b y the
p o la riz a tio n induced by the optical fie ld and leads to a s u s c e p tib ility %
defined by the expression P = eo%(co)E. The induced p o la riza tio n is given in
te rm s o f d e n s ity -m a trix and dipole-m om ent o p e ra to rs. The decay
m echanism s include in tra b a n d processes th a t occur a t a tim e scale o f
-O .lp s and in te rb a n d processes a t a tim e scale o f few ns. In tra b a n d
processes are electron-electron scattering and electron-phonon sca tte rin g
w h ile in te rb a n d processes include ra d ia tiv e recom bination (spontaneous
and s tim u la te d em ission) and n o n ra d ia tiv e re c o m b in a tio n (A u g e r
process). The a n a lysis o f a sem iconductor la s e r u s in g a rig o ro u s
sem iclassical approach is very complex.
A phenomenological approach, w hich is based on the observation th a t the
g a in is a lm o st a lin e a r fu n c tio n o f th e in je c te d c a rrie r d e n s ity , is
g e n e ra lly used fo r describing a sem iconductor laser. Therefore, th e gain
g(n) = a(n-ng) (2.24)
w here a is the gain coefficient and n^ is the c a rrie r d e n sity re q u ire d to
achieve transparency (an^ is the absorption coefficient o f the unpum ped
m a te ria l). The re fra ctive index is also assumed to v a ry lin e a rly w ith the
c a rrie r density
A|Xp=bn (2.25).
where b = ^ and is often determ ined experim entally
on
C om paring the Eqs. 2.24 and 2.25 to the Eqs. 2.13 and 2.14 i t can be seen
th a t the v a ria tio n o f the complex su sce p tib ility Xp is also a lin e a r fu n ctio n
o f the c a rrie r density
X p = ^ b |2 b - j^ n (2.26)
A ve ry im p o rta n t param eter fo r the sem iconductor la se r m o d e llin g is
defined as:
Im(Xp) (2.27)
9n
\9n/
and i t is called th e lin e w id th enhancem ent fa c to r o r a n tig u id in g
p a ra m e te r.
The c a rrie r density n is related to the pum p param eter, c u rre n t de n sity J,
th ro u g h the rate equation th a t incorporates a ll the m echanism s by w hich
the ca rrie rs are generated and a n n ih ila te d inside the active region:
^ = D ( V n ) + - i- - R ( n ) (2.28)
In the c a rrie r density ra te equation (Eq. 2.28) the fir s t te rm accounts fo r
c a rrie r d iffu s io n and D is the d iffu s io n co e fficie n t. The second te rm
describes e x te rn a l p u m p in g w h ich is in v e rs e ly p ro p o rtio n a l to the
m agnitude o f the electron charge q@ and the active la ye r thickness d. The
la s t te rm , R (n), represents th e c a rrie r loss due to ra d ia tiv e and
n o n ra d ia tive recom bination.
In the case o f stro n g ly index-g u id in g lasers the d iffu s io n te rm can be
neglected because the dim ensions o f the active region are often sm all
com pared to the d iffu s io n length. In th is case the steady-state can be
w ritte n as
J = qedR(n) (2.29)
The c a rrie r recom bination rate is often expressed expressed as
R(n) = A j^ n + B n^ + Cn^ + R gtN p^ (2.30)
The fir s t te rm represents n o n -ra d ia tiv e re co m b in a tio n such as the
recom bination a t defects. The quadratic te rm B n^ is due to spontaneous
ra d ia tiv e re c o m b in a tio n . The cubic te rm C n ^ is due to A uger
re co m b in a tio n and i t is p a rtic u la rly im p o rta n t fo r lo n g -w a ve le n g th
lasers. The la s t te rm R g t^ p h is due to stim u la te d recom bination w hich
leads to lasing action and i t is proportional to the photon density N p ^ w ith
the net rate
R s t = -^ g ( n ) ■ (2.31)
2.1.4 W aveguide Modes
The lig h t e m itte d by a sem iconductor la s e r has fin ite tra n sve rse
dim ensions because i t o rig in ate s from th e th in active re gion , w h ich
provides the gain via stim ulated emission. The o u tp u t is in the fo rm o f a
the beam is dependent on the laser structure.
In h e te ro s tru c tu re sem iconductor lasers the fie ld confinem ent in the
transverse d irection, perpendicular to the ju n c tio n plane, occurs th ro u g h
d ie le ctric w aveguiding due to the re fra ctive index d is c o n tin u ity between
the active and cladding layers. F ield confinem ent in the la te ra l d ire ctio n ,
p a ra lle l to the ju n c tio n plane depends on the la se r s tru c tu re . In gain-
guided lasers, the v a ria tio n o f the optical gain (due to c a rrie r d iffu s io n )
confines the optical mode. Index-guiding lasers have a b u ilt-in re fra ctive
index step th a t confines the mode. A schematic cross-section o f a b u rie d
h e te ro s tru c tu re laser, w hich is a ty p ic a l exam ple o f a s tro n g ly in d e x
g u id in g la se r, is shown in F ig . 2.3. Besides p ro v id in g th e o p tic a l
confinem ent, the buried heterostructure laser incorporates p-n-p blocking
structures w hich create the cu rre n t confinem ent.
The m ost common m ethod fo r so lvin g tra n sve rse and la te ra l fie ld
d is trib u tio n s o f an index guided sem iconductor laser is the effective index
m ethod. Instead o f solving the tim e-independent tw o-dim ensional wave
equation (Eq. 2.5), the problem is s p lit in to tw o one-dim ensional p a rts
whose solutions are re la tiv e ly easy to obtain. The effects o f gain and loss
are tre a te d as p e rtu rb a tio ns o f the lossless case. The mode propagation
constant is given by
P = ^ o + j | (2.32)
w here |i and a represent the re fra c tiv e in d e x and th e a b so rp tio n
coefficient o f the mode supported by re cta n g u la r waveguide o f w id th w
and thickness d.
2.1.5 Em ission C haracteristics
E m ission c h a ra cte ristics used to characterise th e perform ance o f a
sem iconductor laser are the lig h t-c u rre n t, spatial-m ode, sp e ctra l and
p - InGaAsP
n - In P p - InP
n - InP
InGaAsP active area
p - In P p - InP
n - InP
n - InP Substrate
F ig . 2.3 Schematic diagram o f a buried heterostructure laser. The
re fra ctive index d isco n tin u ity between the active and cladding layers is
p ro vid in g mode confinem ent through the to ta l in te rn a l re flectio n .
power saturation
lineair regime
5
th re sh o l
C u rre n t (mA)
F ig . 2.4 Schematic lig h t-c u rre n t curve o f a buried hete ro stru ctu re
laser. C h a racte ristic regions are threshold, lin e a r region and pow er
T yp ica l lig h t-c u rre n t ch a ra cte ristic o f an index-guided long-w avelength
la se r is shown in F ig 2.4. Three regions o f the lig h t-c u rre n t curve
correspond to below th re sh o ld o p e ra tio n dom inated b y spontaneous
em ission, lin e a r a b ove-threshold re g io n d o m in a te d b y s tim u la te d
em ission and the power sa tu ra tio n region. The o u tp u t power s a tu ra tio n
can be caused by the increase o f leakage currents, A uger recom bination
and in te rn a l loss. The slope o f dP Q ^^/dl is a m easure o f th e device
efficiency.
The dim ensions o f the e llip tic a l spot and its divergence angles, both
p a ra lle l and p e rp e n d icu la r to the ju n c tio n plane, are im p o rta n t beam
param eters associated w ith the laser mode. In the case o f th e index-
guided laser the near fie ld d is trib u tio n can be expressed as a G aussian,
and from th is the fa r-fie ld p a tte rn can be easily obtained. The s itu a tio n is
more com plicated fo r n a rro w -strip e gain-guided lasers where the la te ra l
fa r fie ld takes the form o f a tw in-lobe pattern.
In o p tic a l co m m u n ica tio n a p p lic a tio n s , th e pow er sp e ctru m o f a
sem iconductor la se r is an im p o rta n t device c h a ra c te ris tic . B elow
th re sh o ld , the o u tp u t is determ ined b y spontaneous em ission w ith a
spectrum o f the order o f 30nm. Above threshold, the lo n g itu d in a l mode
closest to the gain peak increases in power, w h ile th e pow er in th e
re m a in in g side peaks saturates. The gain p ro file o f th e sem iconductor
la s e r is hom ogeneously broadened and the m u lti-m o d e b e h a vio u r is
a ttrib u te d to spatial-hole b u rning, spectral-hole b u rn in g and a h ig h ra te
o f spontaneous em ission e n tering the laser mode. A p a ram eter used to
describe the spectral p u rity o f a sem iconductor la s e r is th e mode-
suppression ra tio (MSR). MSR is defined as a ra tio o f the m ain-m ode
pow er to the power ca rrie d by the m ost intense side mode. F o r some
o p tica l com m unication applications, MSRs in excess o f 20dB are needed,
and they can be achieved w ith distributed-feedback sem iconductor laser
configurations. A nother q u a n tity o f p ractica l in te re s t is the spectral w id th
flu c tu a tio n s associated w ith the process o f spontaneous em ission.
S pectral lin e w id th is ty p ic a lly o f the order o f lOOMHz and is in v e rs e ly
prop o rtio n a l to the laser power.
In m any applications, p a rtic u la rly in optical fib re com m unications, the
device c u rre n t is m odulated p e rio d ica lly and the la se r o u tp u t takes a
pulsed form . The laser responses to a step increase o f c u rre n t w ith a
tu m -o n delay o f few nanoseconds, follow ed by the ra p id increase in laser
power and oscillations o f several nanoseconds before a tta in in g a steady-
state value. The frequency o f re la xa tio n oscilla tio n s is governed by the
n o n lin e a r dynam ics o f the p h o to n -ca rrie r in te ra c tio n s . The re la x a tio n
o s c illa tio n frequency is ty p ic a lly in the low er GHz range and increases
w ith an increase o f the laser output power.
U nder high-frequency d ire ct m odulation, the spectral side modes are
considerably enhanced compared to those in the CW case. M oreover, the
lin e w id th o f an in d iv id u a l lo n g itu d in a l mode increases. The lin e
broadening is related to the carrier-induced re fra ctive index change and
leads to frequency chirping.
2.2 Rate Equations
Rate equations provide the standard way o f in v e s tig a tin g sta tic, spectral
and dynam ic characteristics o f a sem iconductor laser. In general, they
are m ulti-m ode equations, b u t in m ost cases a s in g le -lo n g itu d in a l mode
tre a tm e n t is su fficie n t. Rate equations can be fo rm u la te d e ith e r fo r the
optical fie ld , in cases when both the am plitude and phase in fo rm a tio n are
needed, or ju s t fo r the photon density or number.
The fie ld ra te equation can be derived from the wave equation u sin g the
a d iabatic approxim ation. The m eaning o f th is appro xim a tio n is th a t the
m a te ria l response is in sta n ta n e o u s, w h ich is c e rta in ly v a lid on
tim escales o f Ip s or longer. The optical fie ld can be expanded in term s o f
to e lim in a te second o rde r d e riv a tiv e s and p ro d u cts o f fir s t o rd e r
d e riv a tiv e s . T h is procedure is em ployed in C h a p te r 7 to d e rive the
equations o f m ode-locking. A fu rth e r a p p ro xim a tio n w h ich is u s u a lly
a pplied is to replace localised m irro r losses by a d is trib u te d m irro r loss.
A p p lyin g these approxim ations to the case o f a single-m ode laser, a fie ld
ra te equation can be derived as
# = j-^co - Oo)E + i(G - yXl - jPc)E (2.33)
a t |Lig It
In th is e q u a tio n E represents th e s lo w ly v a ry in g com plex mode
am plitude. Pg is is the group index corresponding to the mode index p.
is the cavity-resonance frequency and co is the la sin g frequency. Pq is the
lin e w id th enhancem ent factor. The net ra te o f s tim u la te d em ission and
photon decay rate are defined as
G=rvgg (2.34)
7 = V g (c C jj^ + C 4 j^ ^ )= l/X p ( 2 .3 5 )
where Xp represents the photon life tim e and Vg is the group velocity.
The a m p litu d e and phase ra te equations can be obtained i f th e fie ld
equation is separated in to its real and im a g in a ry p a rts using
E = Ae-j4> (2.36)
The am plitude and phase rate equations are
^ = 1 (G -y)A (2.37)
co; 2t
— - — (co - Qq) + & Pc(G - y) (2.38)
C/t Pg z
The am plitude rate equation is more often w ritte n in term s o f the photon
P = ^ l N " d V (2.39)
where hco is the photon energy and V is the volum e o f the active layer. For
the photon num ber we can w rite follow ing ra te equation
^ = ( G - Y ) A + Rsp (2.40)
at
The la s t te rm Rgp represents the ra te a t w hich spontaneously e m itte d
photons are added to the to ta l photon population. The o u tp u t power from a
facet, Pout, is related to the photon num ber P by the fo llo w in g re la tio n
Pout = ^ hcovgamP (2.41)
where v ^ a ^ represents the ra te a t w hich photons o f energy hco escape
thro u g h the tw o facets.
The c a rrie r ra te equation (Eq. 2.28) can be expressed in te rm s o f the
c a rrie r num ber inside the active layer defined as
N = I ndV = nV (2.42)
where the volum e V is V=Lw d (length x w id th x thickness). I f the effect o f
c a rrie r d iffu sio n is neglected, the ca rrie r num ber ra te equation becomes
f - i - n n - G P (2.43)
w here
Yg represents the c a rrie r recom bination ra te and is used to define the
spontaneous c a rrie r life tim e Xg.
The la s t te rm in the photon num ber ra te equation (Eq. 2.40), Rgp can he
defined as
Rsp=PspB^ (2.45)
where pgp is called the spontaneous em ission factor and B is the ra d ia tiv e
recom bination coefficient.
Equations 2.40, 2.38 and 2.43 are the single-mode ra te equations. They are
d e te rm in is tic equations because the spontaneous em ission noise is
represented by an average value. These d e te rm in is tic equations can be
used fo r studying the steady-state characteristics and tra n s ie n t behaviour
o f single-m ode or m ulti-m oded sem iconductor lasers. H ow ever, to model
spontaneous em ission effects such as in te n s ity noise and lin e w id th i t is
necessary to in clu d e stochastic noise sources in th e ra te equations.
R egarding dynam ic behaviour, spontaneous em ission also p la ys an
im p o rta n t role in m ode-locking and is the source o f tim in g jit t e r in gain-
sw itched lasers.
2.3 E xte rn a l C avity Semiconductor Laser
The behaviour o f continuous-wave and m odulated sem iconductor lasers is
affected by re fle c tio n feedback. M uch research has been devoted to
s tu d y in g b o th the m e rits and dem erits o f e xte rn a l o p tic a l feedback.
Im proved mode selection o f Fabry-Perot lasers®, lin e w id th n a rro w in g and
tu n in g ^ and reduced ch irp in g 10 can be m entioned as advantages o f the
e x te rn a l c a v ity stru ctu re s. W hereas, in a c e rta in feedback re gion , an
in s ta b ility named coherence collapse' can o c c u rH ; m oreover, tiltin g o f
em ission^^.
The operation o f a sem iconductor laser w ith an e xte rn a l re fle cto r depends
on b o th the m agnitude o f the re fle ctio n and the le n g th o f the e xte rn a l
ca vity. The CW perform ance o f a distributed-feedback la se r exposed to
re fle ctio n feedback has been classified in to five regimes
1) A t the low est levels o f feedback the laser operation is stable b u t the
am ount o f lin e n a rro w in g depends on the phase o f the re fle cte d lig h t.
L in e w id th reduction has been observed fo r a feedback level o f -80dB.
2) A t a feedback level w hich depends on the distance to the e xte rn a l
re fle cto r (-70dB fo r a 40cm cavity), mode-hopping between exte rn a l ca vity
modes has been detected.
3) In the range from -45dB to -40dB stable operation and lin e n a rro w in g
have been observed, independent o f the feedback phase.
4) F o r optical feedback levels from -40dB to -lO dB lin e w id th broadening,
by several orders o f m agnitude, has been observed. T his phenom enon is
called coherence collapse' and i t is accompanied by a la rg e increase o f
in te n s ity noise.
5) F o r strong feedback levels stable single-mode operation w ith a n arrow
lin e w id th has been observed. In th is region, the s ta b ility o f operation is not
affected by changes in the external cavity length.
G enerally, e xternal ca vity devices should be designed to operate in the
strong feedback regim e. In the case o f b u lk e xte rn a l ca vitie s an a n ti-
re fle ctive coating should be deposited on one sem iconductor laser facet in
order to im prove the coupling efficiency.
One o f th e m ost im p o rta n t c rite ria fo r sem iconductor lasers in h ig h
The lin e w id th o f a single-mode laser diode is described by the m odified
Schawlow-Townes fo rm u la ^^:
A V s o l= ^ ( l+ p / ) (2.46)
where Rgp is the average spontaneous em ission ra te in to the la sin g mode,
P is the to ta l lasing photon num ber in the sem iconductor ca vity and is
the lin e w id th enhancem ent factor. The lin e w id th o f an e xte rn a l c a v ity
laser can be expressed as^"^:
AVecl = AVsol / (1+11^^) (2.47)
L and are the optical lengths o f the external ca vity and s o lita ry laser, T)
is the effective single-pass coupling efficiency. In re a lity , the lin e w id th o f
an exte rn a l ca vity laser is inversely proportional below a ce rta in value
(-1 5 c m ) above w h ich i t reaches a s a tu ra te d va lu e . One possible
explanation o f lin e w id th saturation is the effect o f 1 /f noise
The lo n g itu d in a l mode separation in long external ca vity lasers is o f the
order o f IG H z and i t is necessary to include a frequency selective elem ent
in the ca vity in order to prevent mode-hopping. The m ost common way is
to use a g ra tin g in ste a d o f the plane m irro r. The e x te rn a l g ra tin g
c o n fig u ra tio n has the a d d itio n a l b e n e fit o f a broad tu n in g range. The
lin e w id th o f 2kH z and a coarse tu n in g o f 90nm have been reported fo r the
18cm long c a v ity^^. A more compact, fixed w avelength co n fig u ra tio n can
be form ed usin g a d istributed-feedback la se r w ith th e fibre-extended
c a v ity ^7. F or such a device the lin e w id th o f 70kHz has been reported fo r a
5.5cm c a v ity . M o n o lith ic e x te rn a l c a v ity la se rs h a v in g sub-M H z
l i n e w i d t h h a v e been dem onstrated usin g sem iconductor stru ctu re s
shorter th a n 2mm.
The th e o re tic a l a n a lysis o f e x te rn a l c a v ity sem iconductor lasers is
reported in the lite ra tu re deal w ith the case o f w eak feedback, w hich
s tric tly speaking, is n ot v a lid fo r an exte rn a l c a v ity laser. In the weak
feedback approxim ation a single re flectio n is added to the sem iconductor
laser fie ld rate-equation^^. The effects o f strong feedback can be tre a te d by
in tro d u c in g a frequency dependent am plitude re fle c tiv ity ^ ^ in to the ra te
equation fo r the laser fie ld . A nother approach is to tre a t a composite ca vity
system in w hich separate fie ld equations fo r active and passive c a v ity are
