Does a theory of semiconducting laser line width exist? B. Spivak UW, S. Luryi SUNY.
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Transcript of Does a theory of semiconducting laser line width exist? B. Spivak UW, S. Luryi SUNY.
A cartoon picture of a laser
Active medium >0
Semi-transparent mirrors
Pumping
n1
n2
A requirement for lasing : the inverse populationn=n1-n2 >0
laser light
laser light
Characteristic parameters:
Laser line width:
Spectral width of a cavity : C
Uniform broadening of electron eigenstates 1/
Inverse life time with respect to photon 1/ph
radiation
The width of the excitation spectrum I
The spectral width of the amplification
The frequency of Rabi oscillations R(I)CI,phhR<< h
An example: semiconducting injection lasers
electrons
holes
Why the laser’s line width is so narrow ?
I
KEKT
widthsspectral
sticcharacteriotherallthansmallermuchislinewidthLaser
F
phCI
)1000300(;sec103~300
sec10/1;sec1010/1;sec10,sec10,sec10
!!!!
sec1010:
113
1911312113110114
176
holes
electrons
P N
light
lasing mode
tierZtrE 0)(),(
Z is a complex number
is an eigenfunction of Maxwell equations with appropriate boundary conditions at the cavity mirrors
(Strictly speaking incorrect) rate equationsdescription of the laser kinetics
)(
h
NaNndt
dN
naNnI
dt
dn
he
nr
N+1 >>1
N~|Z|2 is the number of photons,n=n1 –n2 is the electron population difference, characterizes loss of photons as they leave the cavity through mirrors,I is the injection intensity,nr is a characteristic time of non-radiative recombination
)(
/
/
c
c
nrc
IIN
andIontindependenisan
ionconcentrattheIIAt
aI
relaxation oscillations of the laser intensity
t
N(t)
Perturbations of the number of the photons (or |Z|) decay in time, and N(t) approaches it’s equilibrium value
Intensity fluctuations: one can introduce delta-correlatedin time random Langevin sources in the rate equations
NntttJtI
NaNntttJtJ
In
aNntttItI
tJNNnadt
Nd
tIn
nNaIdt
nd
LL
LL
nrLL
L
Lnr
)'()'()(
])['()'()(
])['()'()(
)()(
)()(
N+1
ND
PNDN
PNANt
tNP
2
2,
Full statistics of the intensity fluctuation (C.H. Henry, P.S. Henry, M. Lax, 1984)
NdN
N
NdP
00
exp1
The frequency of the relaxation oscillations is smaller than the spontaneous emission rate.
Schawlow-Townes theory of laser’s line width
Z
.1
;)()0(,)()0(
),()(,,,h~ dZ
2
)()0(*
02/1
0
formulaTownesandSchawlowI
D
eetEEtDt
ttteEz
z
sp
tt
i
an(
Z<<Z
a photon
Z
tierZtrE 0)(),(
Questions:1.What is the frequency interval in which
spontaneous emission of photon determines the value of
2. More importantly: Is it correct that at the mean field level (before the spontaneous emission is taken into account) a single frequency
generation takes place? If not, then …….
3. What is the relation of the problem with the problem of turbulent plasma?
Another question which will help us to understand what the problem is: What is the number of lasing modes, N, and how does N depend on I?
In the case of semiconductor lasers nobody really knowsfor sure, but it looks like if no precautions are taken(no distributed feedback) the number of modes firstincreases with the injection intensity I (N<100) and then, at larger I, it decreases with I.
A simple (and probably, not entirely correct) model.
NnaNdt
dN
nInaNI
dt
dn
nrst
This model can explain the increase of N with I,but can not explain its subsequent decrease with I
electrons
Ist[n] is a scattering integral describing electron and electron-phonon scattering which conserve the total number of electronsAnd holes and redistribute them in energy. ( The characteristic rate is 1/
Back to the problem of laser line width
Does the line width increase with the injection intensity ?
Solution of the kinetic equations gives us
region of applicability of the kinetic equation :
3/1
I
electrons
Why the laser line width is so narrow?Because is short?What is wrong with previous arguments?
h/
22 /1
/1)()()(
hKGGh
he
ARhe
The electron distribution function should be calculated witha precision better than the broadening of the electron levels.This poses a very difficult theoretical problem, which does not have precedents in the kinetic theory
a. The limit corresponds to the model of two level system where at the mean field level (Is the semiconducting laser line width narrow because is short?)
b. Any way in the framework of the model the line width increases with I !
.......
)'(
][')'(
'
'
Iusgiveswhich
NNnKadt
dN
nnIdNnKaI
dt
dn
nrst
Ignoring this fact we can write
#2
A spatial hole burning n(r), (r) leads to both competition between modes, and to a competition between harmonics with different frequencies within one mode.
#3
......))cos((|||||| 212122
12
2121
tEEEbEbn
nEbItd
nd
eEeEE
nr
titi
Beating in time: Consider for example two lines(R. Kazarinov, C. Henry)
There is no stationary solution of this problem!n(t) depends on time, which leads to a competitionbetween modes and a competition of EMF oscillationsat different frequencies with a mode
c
)(0
)(0 ; tkritkri eHHeEE
linear analysis of Maxwell equations at :
c
Re
Im
At I>Ic there are no stationary solutions of the problem !
G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, M. Aoki, IEEE J. Quantum Electron. QE-27, 1782 (1991)
Experiments are not that impressive either
Other side of the problem:A model of two level system couples with EMF
.fieldtheofabsencethein
ofvaluemequilibriuthetoequalconstantais
,frequencyRabitheis
)(
2
2
143
2222
211
rr
E
rccdt
d
rrrcdt
dr
rcrdt
dr
R
R
R
R
This system of equations is equivalent to that exhibiting chaos and the Lorenz strange attractor!(Haken)