6
Lcc 9g- 4cl
Lecture 7
Consider the cavitY below
,1
1+ dx
Therefore,
the change of intensity in length dx,
dI
=
ftv (NzAzrdx
spontaneous
Note:
onrv
u small fraction of the Azr
+ NzWidx - NrWidx)...(6.3)
stimulated absorPtion
Nzdx photons liberated spontaneously will increase the
intensity along the x-direction
I
mirror
.1
i
Defining the acceptance angle dO (of the mirror),
Therefore amount of light adding to the process is dO
/
4n
Generally, we can omit this!
Therefore (6.3) becomes,
dI
=[N,-N,]Wiftvf
dx
IfNz>Nr
amplification
Inserting ,
Into above -_--;
Inserting,
'W,
.L
czlg{v)
= _Effi
=
iuto abotr
[ N,-Nr] c'zIg(v)
tls
Integrating'
or =[N-N,] crlgb) x
A
8n'r,
I(x) = Io (x=0) €
Or
I(x) = lo eT
,=
^t
*
[Nr-Nr] czlg(v)
W
Stimulated emission x-direction
2
o(x) = tur
?u
8an2
e(v)
I
For small signal, whereby I(v)
^:
0,
into above
y: To
(yo ---- used to denote small-signal gain )
coefficient (per unit length)
7.0 : Three and Four Level Lasers
Most laser can be classified as 3-level or 4-level system.
7.1:
3-Level system
.
Example : Ruby laser / E1a+ doPed
A three level system is one which lower laser level is either ground state or
whose separation
El
a level
from ground state is small compared to KT
L._._
2
OR
Er<
Ground state
Substantial fraction of the
population occuPies this level
&
I
i
hansitionrate
* l/trr
Er ( lifetime hr )
FUMF
trsnrlllon
, I/l"r*,
e round stete
At a pumping level that is strong strong enough to create,
Nt:Nr=
Optical gain
Y is
zero
-)
,]
-,
raken ['=
N'tN'= I{o
and assuming that the transition rate Wrz
Is very fast, so that no accumulation of atom in level 3
Oscillation condition,
N, - N, = Nt
by making
(threshold population)
I
ri
i
N!* & + Ht------'level 2
2X
Ht* N"-Nt
2t
( such
that N 2 - bf,,': Nt
level 1
)
Power needed to Produce this,
p -|.
**
N,l hvv
L2l
Most laser system
No)) N
\
volume
,
('):-reveiY
Minimum expenditure power needed to start a lasing action
r1
7.2
z 4 -level sYstem .
Wq:
very fast kansition rate
Nnv
flaser output)
t2
Ez>> kt
Note:
l)the separation (Ez - Er)>> kt
2)If the lifetime tz of atoms in level2 is short compared to ts (fu>> tz)
Can neglect the level2 population
-,
.'.Threshold condition is satisfied when
Nl=Nt
to
Laser oscillation begins when the upper laser level acquires a population density
the threshold value Nt
Minimum power required for 4- level system'
(,\
=Nthvv
\ / 4-Iever Zt,
lrl.l-
t
32
:
Iifetime ofthe uPPerlevel
,.
Comparing 3- and 4 levels
3Jevel
N , = I[o
(because No >> Nt)
7
4Jevel
Upper laser
levell*n
(*,)
,
* Nt
,-,*,*
(*r)- -level
Since No
N
-
No
2N;
>> Nt, the 3-level system required more pumping power than the 4-level'
8.0 Ontical Resonator
N
MI
M2
each other
optical resonator consist of two flat or curved mirrors set-up facing
mirror.
two
betlveen
forth
and
back
bounce
can
quari-ptur* optical wave
so that a
tt
(
mimor
Mt
-
mimor
1ooF6
Me
-
gs16
8.1 Rav anproach in optical svstem
Rav matrix
What is necessarT to specify everything about it?
1) Where is it with respect to some arbitrarily chosen axis?
2) In what direction is it heading?
Example
o
o
1
trajectory having r (z) transverse displacement away from the z-axis
small slope r'(z) = drldz with respect to z-axis
.rt
i
Examnle 2
(rr,rt
Lrr
(rr
L
n
Z
0
L
Zr
.
.
rr
ZE,
Zz 1=Zt + L)
Ray displacement and slope at the output plane Zz are related to the
-)
input plane atZr bY,
Ray propagating from
Zt
n
=
(From tan 0= drldzrtherefore rz =
and
Altenativelv
12'
rr +
ft * L
: tr'
Lrl'
tanO
,
........(1)
t2= rr + Lrt' )
""""""(2)
-l
Ifweknowtherayisatplanelandknowitsslopewithrespecttothe
know where it is going at exit plane2 '
Note:
tan O = sin
O=
r'
O
-O
axis, then we
(all rays are paraxial)
(slope)
Output parameter to input parameters
fz=lrft*d.ft'
4
fzt=0rfl1lrI1'
in matrix form
4
U
form,
tt =
+
f, = /c11'
I
dl[',
1
l"r,' I =f o
' ','.,l
I
]
t_
(8,,
_)[
In General,
:][:]
[:
]
[
is reduced to simple
Ray tracing through a sequence ofoptical components
2x2matrtx.
'i
.t
Example
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
In matrix
[:]
[:][:][:][:][:]
of,
[:]
m l :l[: ]
!
t
.,]
[:
[:
[:]
]
8.2 Some common raY matrices
next important examPle
thin lens of focal length ,f
consider ruy
rr':0
a
,
, outPut sloPe is
therefore
rz' a.: Cfr 6 *
or c= llf
Dr,'
a:
(- rr a) lf
= crr o t
D'0