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