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

widthsspectralsticcharacteriotherallthansmallermuchislinewidthLaser

F

phCI

)1000300(;sec103~300

sec10/1;sec1010/1;sec10,sec10,sec10

!!!!sec1010:

113

1911312113110114

176

holes

electrons

P N

light

A very simple theory (model ?) of laser

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

NaNndtdN

naNnIdtdn

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

IINandIontindependenisan

ionconcentrattheIIAtaI

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

InaNntttItI

tJNNnadtNd

tInnNaIdtnd

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)

NdNN

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

ttteEzz

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

nInaNIdtdn

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

heARhe

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

nnIdNnKaIdtdn

nrst

Ignoring this fact we can write

#2A 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

tEEEbEbnnEbItdnd

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 !

Theory is not that impressiveWhat about experiment ?

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

rrE

rccdt

d

rrrcdtdr

rcrdtdr

R

R

R

R

This system of equations is equivalent to that exhibiting chaos and the Lorenz strange attractor!(Haken)

Conclusion:

We do not know why semiconducting laser line widths are narrow and which factors determine it