1
LaserOscillation:adifferentviewpoint
Before stimulated emission becomes important, we must obtain the initial photon from the “noise”.
The laser oscillation builds up from the spontaneous emission “noise” emitted from the upper state until the coherent photon flux saturates the gain.
(1) - rate of generation of spontaneous photons into all frequencies. Modes are separated by:
(2) - fraction of emission that appears in the interval
(3) only the generated has a high Q, but there are - modes in the volume
21A N V( )2v c ndD =
( )( ) ( ) 2g v v g v c ndD = é ùë û ( )2v c ndD =
0,0,qTEM3 2
3
8 n v V vcp
´D
The rate of increase of photons in the mode caused by spontaneous emission:0,0,qTEM
( )21 2 2 2.
2
1mode( )2 8#modes =
2
p
spont
dN cA N V g vdt nd n v c V
c ndp
é ù= ê ú æ öë û ´ç ÷è ø
2
2 21 2.
( )8
pSE
spont
dNN c A g v N c
dtl sp
é ù= =ê ú
ë ûStimulated emission cross-section
2
If , still increases but the output (caused by spontaneous emission) is extremely small:
Photons in the cavity mode bounce back and forth between the 2 mirrors, being amplified by G per pass, with some of them escaping by mirror transmission.
If - is starting # of photons, then returns after a round trip taking seconds.(i.e.
1 2 pGRGR N 2nd c21 2 )p p pN G R R N ND = - Þ
21 2
cavity withgain
12
pp
dN G R R Ndt nd c
-=
Gain + spontaneous emission:“works” in the beginning at small pN
21 2
21
2p
p SE
dN G R R N N cdt nd c
s-= + (1)
takes over at large pN
~ 0pN pN
For (large!!)(typical)
if , then (negligible)
12 2
12 32
3 1010
cmN cms -
-
= ´
=10 1
2 9 10SEN c ss -Þ = ´191 (1.6 10 )hv eV J-= ´ 914.4 10P W-= ´
If is large enough, # photons grows exponentially:pN21 2 1( ) (0)exp2p p
G R RN t N tnd c
é ù-= ê ú
ë ûThen, saturation becomes important and eventually 1st term becomes negative (but small) and exactly balances the positive 2nd term so that we have a steady-state laser.
pN
3
In the steady-state:
( ) ( )1 212
pout NP R Rhv nd c
= -
survived photons after a roundtrip
( )2 ( )1 2 2
20 see (1) 1p Sp SE
dN ndG R R N N cdt c
s= Þ ® - + =
( )21 2 2
1 2
111
outSE
PG R R N chv R R
s- =-
( )S saturationº
For typical values: 1( ) 12 3 12 22
10 1 0.9
1 10 3 10out L
SSE
P mW R Rhv eV N cm cms- -
= = =
= = = ´
( ) ( )2 ( )1 2 1 2 21 1 S
SEout
hvG R R R R N cP
s- = - ( )7~10- (2)
For any computational purpose 2
1 2
1GR R
=
The fact that the gain does saturate at a value slightly less than the loss, implies that the laser will have a finite, nonzero, spectral width caused by the “noise” contributed by the spontaneous emission. (minimum laser linewidth, as follows…)
4
at , has a peak (same as in (3) has a peak at )
MinimumLaserLinewidthFor Fabry-Perot cavity (see 6.3.3 in the text):
( )0 22
1 2 1 2
1
1 4 sinI I
R R R R q=
- +
2 ndvc
pq = (3)
For cavity with gain, using same approach as in 6.3.1-6.3.3, we obtain 21 2 1 2R R G R R®
( ) ( )21 1 22 2
1 2 1 2
( )21 4 sin ( )S S q
KP vdG R R G R R v vcp
=é ùé ù- + -ê úê úë û ë û
constant
qv v= ( )P v I aq p=
For FWHM: ( )2 211
221 2 1 2
24 1 ( )2osc
S SvdG R R G R R
cpé D ùæ ö é ù= -ç ÷ê ú ê úë ûè øë û
( )
12
1 21
1 22
1 2
1 ( )2
Sosc
S
G R R cvd
G R Rp
-D =
é ùê úë û
vD - very small
GS (R1R2 ) 1⇒
1−G2S (R1R2 ) = 1−GS (R1R2 )
12#
$%&'(1+GS (R1R2 )
12#
$%&'(= 2 1−GS (R1R2 )
12#
$%&'(2
( )
221 2
1 212
1 2
1 ( ) 1 ( )2 42
Sosc S
S
G R R c cv G R Rd dG R R pp
- é ùD = = -ë ûé ùê úë û
(*)
5
use ( ) ( )1 2 1 2 21 ( ) 1 S
S SEout
hvG R R R R N cP
s- = - (from (2)) (**)
1 2 1 2
2
1 1RT
p
dc
R R R Rtt = =- -
photon lifetime
( )1 1 22
1 12 2p
cv R Rdpt p
D = = - - see 6.4.5 (4)
( )1 22
Sosc SE
out
hvv v N cP
sD = D
gain coefficient ( ) ( )22 1
1
lossunit length
S SgN Ng
sæ ö
- =ç ÷è ø
N2(S )σ SE 1−
g2N1(S )
g1N2(S )
"
#$$
%
&''=
12dln 1R1R2
=12dln 11− 1− R1R2( )(
)**
+
,--1− R1R22d
( )1 2 1 2ln 1 1 1R R R R+ - » -é ùë û
N2(S )σ SE
1− R1R22d
1−g2N1
(S )
g1N2(S )
"
#$$
%
&''
−1
(5)
(6)
Using (4),(5),(6)
( )1 1( ) 22 1
1( ) 21 2
2 1S
osc Sout
g Nhvv vP g N
p-
æ öD = - Dç ÷
è øSchawlow – Townes formula
(see (4), (*) and (**))
- in cavity with gain!
6
use:
114
7
0 (ideal system)
2.4 105 1010
Nv HzQP mW
=
= ´
= ´=
12
5 (for passive cavity)vv MHzQ
ÞD = =
⇒Δvosc 10−3Hz
Even for (which is normally the case) is very small. (minimum laser linewidth) oscvD1 0N =
In practice, perturbations in the mirror separation completely overwhelm the foregoing limit.
A “ -function” for the spectral representation of the laser is an excellent approximation.
d