I
r
- rlrt b
:s
o
Assume
that the center wavelength is 5000A
yo=6 X 1O1a Hz, AY- 1 Ghz
Votume of the medium is 10 cm3 n=1
can give up its
ditferent modes in whch the atom
There are (*nu2Au/c;jv'=i.ssrroib
field'
internal energy to this electromagnetic
field close
during the initial stage,-those resonant
inltance,
for
Few are toward tne mirror,
ThLn it will be amplified by stimulated
to line center were excited spontaneouslv.
cavity
a f"* round-trips through the
Lrni.rion process. Aft-",
quite
large'
field
Sufficient to make the
'
ExamPle
TEMs6q(O,O,q)modehasinitialintensityoflpw/cm2andlets^aythenetgain(RtR')t''
06 times (assuming
grow, 4'o-1
'Osxl
trips, tne intenlitv will
exp(yo(yo)l)=4. ntter nve round
(RtRz)"2 exp(yo(uo)l)=2 with
mayOe-9ain
modes,
another
The
the gain weff is
g'
after 5 round trip = smwlcm '
.on[ir.
initialintensity =0'5pWcm2'
Buttheintensitycannotkeepgrowingindefinitelythroughmoreandmorecavitytransit
times.
o\Nhenthestimulatingfieldissolargeastocausetheatomtogiveuptheirenergyasfast
--->equilibrium'
as they are being pumped
'Gainofthesystemmustchangetoalowervalueuntiltherateoforoductionof
theexcessinvertedpopulationisu,t"ncebythedestructionratebystimulation
emission
)oain
saturation'
.Thelasergaincoefiicienthasdecreased(orsaturated)tothelosscoefficientat
the frequency of laser oscillation'
will oscillate
Eventualty onty the centre mode
broadenino'
homoqeneous
of
Consequence
transition
Broadened
Gain saturation in a homogeneous
generalized 2-level model'
of gain saturation using
Need matnematiclldescription
'
'
o
\
stmulated smission and
aQorp!on
,t/\
\ r(,0
1.
5itr
reservior of atom in ground state
oRrincludesdirectexcitationfromthegroundstatetostatel(includeindirect
emission to 1' R2
states followed by spontaneous
nigne,
to
e*cltatioi
routes)such
rountes)'
o
to.state 2 (include indirect
pumping directly t'o' giouiO state
hv12 spontaneously
state 2 can radiate pnlton of energy
: i:#"'f:Tilil
o
Rates equation'
compared to that in 0.
popuration in 2 and 1 are very smail
f,
(10'1)
# = n,-Y-'* (Nz - Nr) """"
d&
\ *'!(Nz-N,).'..'...(10.2)
=R. *'' -
dt"'t^t1hY
. *, = spontaneous decaY 2)1
. atzo = loss os state 2 to ground state
. Total decaY rate of state
*
'i=*j causes
o Decay rate of state 1 due to all
I
Stimulated emission
I-.
.
f.g1u)Ay
Av
=1c3A21/8nhv3n3 )g(u)
of light in medium
N;te: we have written c'=c/n velocity
=
+
,n{$4
g(v) lv
and Ay=l/c therefore Bzrg(v) =1r02/8nhun2 )Azr
a
From
a
lead to so many important consequences'
10.1 and 10.2 areso fundamental
a
fr
Rz is in the
pumping to state 2 onry (i.e' Rr=O) and
r,=0 (no stimurated emission)
( for t<0 and
*n"'""'tti=Heaviside step function =0
form of a step function' R2(t)=R2su61
Rzo wJrich has a solution'
equation 1 0' 1 becomes f,+N2/t'7 =
= 1 for t>0 ) therefore
Nz(0)=0 lead to
particular, Rzotz using*ihe initial condition,
homogeneous, Aexp(-utJ
Nz(t)= Rzotz(1- exptUU)
.^-* Nr,/+-- anrr finire
oooulation of state
finite population
due to source term Nz/tzr and
Dynamic of state t'isiompticated,
1 (no direct PumPing)
From eqn 10.2
3"tiJ;rr"
""10'3
.
.
4& * &
dt q
\Nhe re Q zt=lzl lzt=bra n ch
i
n
g
Nz(t)
(10'2)
o21R2s(1 - exp(-tltr))""'
tzt =
rati
o
(1
/h
=
1
/t zt+ 1.l.lzo)
Ootn sides by a factor exp(Utz)
Using inteqratino faclorcpil.*tn'rn"ftipty
*j-;O
!
!
"-,
eJ
= 1N,(t) "*p
ii.*,co =i*r(lll,,,iJ
(i)r''
Nz(t)
dirrerentiar
.:'"t""t
exp()"
.
.10.1 , t>>tr' must
and thus Nr(t) sJ Nr(t)dt , for
term
=1
exponentiat
the
t<<tr
For times
as given by 10'3
evaluate the integralwith Nz(t)
N1(t)=oz1R2ot1{1+4*o(-'^,)-fr*n(-%,) to'
As
t)'
N2(')=Rzrtz and N1(-)=o21R2et1
steady state is ieacfteA with
2.5
2.O
R,T,
1.5
1.0
f,'n"t{ttt)
(a) t^"J,:rlf
the system
greater than that of state 1 and
density in state 2 is arways
ratio
g"i. for all values of t =>Favorable lifetime possible for only yhr short initial
ratio then gaii is
(b) ""iiiit.
for t2lt1<1 =,rnfauo'ab'l-r'it'i'"
interval of roughly tz
';"
2.
N,
1.5
R,(t)
1.0
Nz
0.5
(--
Rcgion vhere
2
NrN'
3 ttme
t/t,
Excitationmustbeintheformofpulse.ExampleN2laserwithwavelength33Tnm(uv)
10ns
unfavorable lifetime tr-10ps't2'
But
with tz=255ps and tr=30ns
YAG) favorable lifetime ratio
rate (t1=0)
1 decay at infinite fast
intensity but assume that state
finite
a
for
l
case
case 2 :modified
ideal laser sYstem
o
o
o
o
to decouPle 10'2 and 10'1
if t'=Q implies Nr=o trenceril
ifl:tlr,)Nz
dt tz' hy '
simPlifYing
*.
i(1
+
Rzou(t)""' (11'1)
?)N, =
where lr=saturation intensitY
o
(11.1)
= R,ou(t).....
&(t)
it has a solution in the form
=
t' - *o(t-'L')n.b)"""""""t'''
?f
1*j;
10'3
equation 11.2 is similar to
o
stimulatea emission
Uy
these quantities'
the intemsity luaffect both
I..
ExamPle for
l=1'5
-
1.5
_
N, (1,=0)
R,(t)
1.0
-
1".-{- \ 2
Nr (t.=1 51.)
3tim6%,
.',rnponrn,
The quantity try/to(y)tzl ,,
2)
intensity
(i.e.J'flL
=>saturation
Notes
'
if
lv<<1,
the population
N2 is
with dimension of watts/area
collection of atomic constants
wave
barely affected by the stimulating
.
N'?(t)=R21tllof N' is one-half of the value given bv
if ly=|,, the steady state peak amplitude
exp[-tltJ)
.notes:thetimeconstantcontrollingtheapproachtoequilibriumisreduces,i.e.
.
.
case
t2'4rllrrrvll,l
if t,>>1"
tie
peak population in
N'? is
to 1
Sreatly reduces compare
3 d/dt
o
=0
to simple form
zero reduces the differential equation
assuming all time derivatives equat to
*
=o=
r,- & """' (11 3)
-Y'"*
^,