Summary Kramers-Kronig Relation (KK relation) Gain Saturation in Homogeneous Laser Media Gain Saturation in Inhomogeneous Laser Media Gain constant uniformly

Summary

Kramers-Kronig Relation (KK relation)

Gain Saturation in Homogeneous Laser Media

Gain Saturation in Inhomogeneous Laser Media

)(/1

)()( 0

sII

00 /1)2/(|| sII

Gain constant uniformly drops in

homogeneous broadening case

sII /1

)()( 0

Spectral Hole Burning Effect

00 /1)2/(|| sII

Lecture 8

Chapter V Theory of Laser Oscillation and Its Control in Continuous and

Pulsed Regimes

Highlights

1. Fabry-Perot Laser

2. Oscillation Frequency

4. Mode Locking

5. Q-Switch Laser

3. Optimum Output Coupling

§5.1 Fabry-Perot Laser

iE iEt1lik

ieEt'

1 lik

ieEtt'

21

likieErt

'21

likieErt

'221

likieErrt

'2211

likieErrt

'3211

likieErrtt

'32121

l

Mirror 1 Mirror 2

Input Plane Ouput Plane

22

)("

2

)(''

22

i

nik

nkkk

: distributed passive losses of the medium le

]1[ '422

21

'221

'21 lkilkilik

it errerreEttE

2/ik propogate constant far away from resonant

"')( iComplex dielectric susceptibility from laser transition

llkki

llkki

ilki

lik

it eerr

eettE

err

ettEE

)()(221

2/)()(21

'221

'21

11


)(8

)()("

2

2

122

gtn

NNn

k

sp

22

)('

nkk

)2/(sin4)1(

)1(22

2

RR

R

I

I

i

t

In population inversion medium: 0 it EE

However, in a Fabry-Perot etalon

it II

1)()(221 llkki eerr

Oscillation


1)(21 lerr

Threshold Condition (gain becomes equal to the losses)

Phase condition mlkk 2)(2

Gain condition 21ln1

)( rrl

)ln1

()(

8)( 212

2

12 rrlg

tnNNN sp

tt

)(

8)(

3

22

12 gtc

tnNNN

c

sptt

Other form

Population Inversion condition

21ln

11rr

ln

c

tc

tc is call decay time of light intensity


Numerical Example: Population Inversion

He-Ne Laser nm8.632ns100sec10 7

spt

cm12l

1GHzHz10)(

1 9 g

Doppler Broadening Width

MHz10/1 spH t

0 98.021 RR

Calculate )cm10( -39tN

Transmission Reflectivity


§5.2 Oscillation Frequency

mlkk 2)(2 m

nkl

22

)('1

220

03

321

/)(41

/)(2

8

)()('

SPn

NN

])/()(4[1

1

8

)()("

220

3

321

SPn

NN)("

)(2)(' 0

Many frequencies exist, we want to know the frequency also satisfied gain condition :

2

)(")(

n

k

0 ( )1 mk

nl

mcm 2

mth resonance frequency of the passive Fabry-Perot etalon


0is the center frequency of the atomic lineshape function

Assume that the cavity length is adjusted so that one of its resonance frequencies is very close near . We hope that the oscillation frequency will also be close to .

m 0 m

is a slowly varying function of when )( 0

0 ( )1 m m

mk

n

c

km

mmm

mm 2

)()(

)()( 00

cnnk /2/2

Oscillation Frequency


l

R

1)(21ln

1)( rr

l

Rrr 21

1R0

xx ln1

Passive resonator linewidth :nl

Rc

R

R

nl

c

2

)1(1

22/1

2/10 )( mm

Frequency Pulling

0 m 0

0 m 0 towardspulling is

Rou

nd

tri

p p

hase s

hif

t

§5.3 Three- And Four-Level Lasers

Pumptransition

Ground state1

2 E2

t2

323

laserPumptransition

Energ

y

Ground state0

1

2

E1

E2

t2

t1

E1 >> kT

323

laser

Four-level System Three-level System

In four-level laser system

Lifetime t2 is much longer than the lifetime t1

Oscillation starts as long as tNN

Population on level 1 can be neglected

In three-level laser system

Population on level 1 cannot be neglected

Oscillation starts at 2/2/02 tNNN

2/2/01 tNNN


So that the pump rate at threshold in a three-level laser must exceed that of a four-level laser

tN

N

N

N

2~

)(

)( 0

level42

level32

tNN 0

Minimum expenditure power of three-level system

2

0level-3 2

)(t

VhNPs

Minimum expenditure power of four-level system

2level-4)(

t

VhNP ts

When t2=tsp, the above two equations are equal to the power

emitted through fluorescence by atoms within the volume at

threshold. It is referred as the critical fluorescence power.

In case of four-level laser2

3

3

2level-4

8)(

t

t

t

Vhn

t

VhNP sp

c

ts


Numerical Example: Critical Fluorescence Power of an Nd3+:Glass Laser

μm06.1m1006.1 6

ns10sec10)1(

8

cR

nltc

cm10l

Hz103 12

5.1n95.0ty)reflectivi(mirror R

3cm10V

Calculate: )cm105.8( 315 tN )watts150(sP

§5.4 Power in Laser Oscillators

When the pumping intensity is increased beyond threshold point,

laser will break into oscillation and emit power. We will derive the

relation of laser output power to the pumping intensity.

Rate Equations (four level system)

iWNNNRdt

dN)( 122122

2

iWNNNNRdt

dN)( 122121011

1

iW

RRRNN

21

211021212

)]/1)(/(1[

Steady state solution

Necessary condition for population inversion 1021


2

1

10

212 11

R

RRR

Effective pumping rate

iW

RNN

2112

In order to keep population inversion unchange, assume Wi is allowed to increase once R exceeds its threshold value

Below oscillation threshold 0iW RNN 12and

21tNR When R increase until

Gain > Loss and steady state assumption violate

21t

i N

RW 21tNR


1

)(

2121

tt

ite

N

RN

Vh

hWVN

Vh

P

In idealized model spt 121

1

/21c

te

tp

RN

Vh

P 3

23

03

23 8

)(

8

c

n

gc

np

)(

8

03

22

gtc

tnN

c

spt and

1

tse R

RPP sP Critical fluorescence power

tR Threshold pumping rate

Example:

§5.5 Optimum Output Coupling in Laser Oscillators

Total loss can be attributed to two different sources:

(a) The inevitable residual loss due to absorption and scattering in the laser media and mirrors, as well as diffraction losses in the finite diameter reflectors;

(b) The useful loss due to coupling of output power through the partially transmissvie reflector.

as small as possible

(1) Zero coupling (no transmission), threshold will be minimum, pe will be maximum, but no output;(2) Large coupling, threshold will be larger than pumping level, oscillation will cease, output is zero again;(3) Exist an optimum coupling value, where output is maximum.


21

21

2112 /1

/

ii W

R

W

RNN

21

0

/1 iW

12 NN

se PP /10

hWVNP ite )(

21)( hVNP ts

1)1()(21 Leerr ll

Recall oscillation conditionlerrL 211

In case of small loss Ll

)1( 0 L

gPP se

lg 00

Pe is the total power given off by the atoms due to stimulated emission


TLL i Total loss per pass

Output power is thus as

iis

ieo LT

T

TL

gP

LT

TPP

)1( 0

Recall)()1(

22

21 TLc

nl

cL

nl

eRRc

nlt

ilc

)/(

8

)/(

8

22

2

23

3

spcsps tt

ALhn

ttt

VhnP

lVA / Cross section

11)/(

8 00

22

2

is

ispo LT

gATI

LT

gT

tt

AhnP

22

2

)/(

8

sps tt

hnI


Maximizing Po with respect to T

iiopt LgLT 0

20

20 )()()( iisopto LgSLgAIP

Documents

Summary Kramers-Kronig Relation (KK relation) Gain Saturation in Homogeneous Laser Media Gain Saturation in Inhomogeneous Laser Media Gain constant uniformly