Nonlinear Optics Lab. Hanyang Univ. Chapter 10. Laser Oscillation : Gain and Threshold Detailed description of laser oscillation 10.2 Gain Consider a quasi-monochromatic

Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.

Chapter 10. Laser Oscillation : Gain and Threshold

Detailed description of laser oscillation

10.2 Gain

<Plane wave propagating in vacuum>Consider a quasi-monochromatic plane wave of frequency propagating in the +z direction ;

zA

z z+z

u

The rate at which electromagnetic energy passes through a plane of cross-sectional area A at z is AzI )(

The flux difference ;

zAIz

AIzzI

)(])([


This difference gives the rate at which electromagnetic energy leaves the volume Az,

zAIz

zAut

)()(

01

Iz

Itc

cIu /: Equation of continuity

<Plane wave propagating in medium>The change of electromagnetic energy due to the medium should be considered.=> The rate of change of upper(lower)-state population density due to both absorption and stimulated emission.

(7.3.4a) => energy flux added to the field : )()()()( 1212 NNIhNN

)()(1

12 NNIIztc


(7.4.19) : )()/()( BSch

INg

gN

ISNg

gN

A

ISNg

gNB

c

h

ISNNBc

hI

ztc

11

22

11

22

2

11

22

12

)(

)(8

)(

)()(1

(no degeneracies)

(degeneracies included)

Define, gain coefficient, g()

11

22

11

22

2

)(

)(8

)(

Ng

gN

SNg

gN

Ag


Igt

I

cz

I)(

1

Temporal steady state : 0t

Igdz

dI)(

zgeIzI )()0()( : valid only for low intensity ( saturation effect, g=g(I))

* g>0 when 12

12 Ng

gN : population inversion

* If ,12

12 N

g

gN

)()(

)(8

)(

1

2

1

22

1

21

1

21

1

22

ag

gN

NSg

gAg

Ng

gN

g

gN

g

gN

The gain coefficient is identical, except for its sign, to the absorption coefficient.


10.3 Feedback

In practice, g~0.01 cm-1 if the length of active medium is 1 m.=> A spontaneously emitted photon at one end of the active medium leads to a total of e0.01 x 100=e1=2.72 photons emerging at the other end.=> The output of such a laser is obviously not very impressive.=> Reflective mirrors at the ends of the active medium : Feedback.

10.4 Threshold

In a laser there is not only an increase in the number of cavity photons because of stimulated emission, but also a decrease because of loss effects.(Loss effects : scattering, absorption, diffraction, output coupling)

In order to sustain laser oscillation the stimulated amplification must be sufficient to overcome the losses.


<Threshold

condition>Absorption and scattering within the gain medium is quite small compared with the loss occurring at the mirrors of the laser. => Consider only the losses associated with the mirrors.

1 strwhere, r : reflection coefficient t : transmission coefficient s : fractional loss

At the mirror at z=L and 0 ;

)()( )(2

)( LIrLI

)0()0( )(1

)( IrI


In steady state (or CW operation), g() : constant

)()(

)(

Igdz

dI )()(

)(

Igdz

dI

zgeIzI )()()( )0()(

))(()()( )()( zLgeLIzI

Amplification condition : after one round trip must be higher than the initial)0()(I )0()(

I

)0()0(][

])0([

)]([

)]([

)0()0(

)()()(221

)()()(21

)(2

)(1

)()(1

)(1

)(

IIerr

eIerr

LIrer

LIer

IrI

Lg

LgLg

Lg

Lg

1221 gLerr


Threshold gain,

)ln(2

11ln

2

121

21

rrLrrL

gt

tg

* In the case of high reflectivity,

xxrr -)-ln(1 and ,121

0.9) ties,reflectivi(high)1(2

12121 rrrr

Lgt

* If the “distributed losses” (losses not associated with the mirrors) are included,

arrL

gt )ln(2

121

where, : effective loss per unit lengtha


Example) He-Ne laser, L=50 cm, r1=0.998, r2=0.980

14

1

102.2

)]980.0)(998.0ln[()50(2

1

cm

cmgt

Threshold population inversion ;

)(8

)( 11

222

2

SN

g

gN

n

Ag

)()(

82

2

11

22

ttt

g

AS

gnN

g

gNN

nm 632.8at sec104.1 16 A

MHz1500MHzM

T11015.2amu20M,K400~

2/16

D

NeT

39

101628

14

cm/atoms106.1

sec)103.6)(sec104.1()cm106328(

)cm102.2)(8(

tN

631

15

9

10108.4

106.1

N

N t (very small !)


10.5 Rate Equations for Photons and Populations

Time-dependent phenomena ?

1) Gain term (stimulated emission or absorption)

)()()(

)(1

Igt

I

cz

I

)()()(

)(1

Igt

I

cz

I

))(()(1

)( )()()()()()(

IIgIItc

IIz

In many lasers there is very little gross variation of either or with z. =>)(

I)(

I 0z

* l<L (l : gain medium length, L : cavity length)

))(()( )()()()( IIcgIIdt

d

))(()( )()()()( IIgL

clII

dt

d(10.5.4b)


2) Loss term (output coupling, absorption/scattering at the mirrors)

By the output coupling, a fraction 1-r1r2 of intensity is lost per round trip time 2L/c.

))(1(2

)( )()(21

)()( IIrrL

cII

dt

d (10.5.8)

From (10.5.4b), (10.5.8), and put : total intensity)()( III

Irr

L

cIg

L

cl

dt

dI)1(

2)( 21

or photon number, Iq

qgL

clqg

L

cl

qrrL

cqg

L

cl

dt

dq

t

)(

)1(2

)( 21


If g1=g2,

IrrL

cINN

L

cl

IrrL

cISNN

A

L

cl

dt

dI

)1(2

))((

)1(2

)()(8

2112

2112

2

Coupled equations for the light and the atoms in the laser cavity including pumping effect :

)1(2

)(

)()(

)(

21

222

2111

rrL

cg

L

cl

dt

d

KgNAdt

dN

gANNdt

dN


10.7 Three-Level Laser Scheme1) Two-level laser scheme is not possible.

22)(

212

N

A

NN

Neglecting 1, 2 in (7.3.2) => : can’t achieve the population inversion

2) Three-level laser scheme

P : pumping rate

1pumping

1 PNdt

dN

1pumping

1

pumping

3

pumping

2 PNdt

dN

dt

dN

dt

dN

(10.5.14) =>

)(

)(

1222112

1222111

NNNPNdt

dN

NNNPNdt

dN

)1(2

)( 21rrL

cgL

cl

dt

d


<Threshold pumping> for the steady-state operation i) Near threshold the number of cavity photons is small enough that stimulated

emission may be omitted from Eq. (10.7.4) ii) Steady-state : 021 NN

121

2 NP

N

TNNN 21

TNPN

21

211

TNP

PN

212

21P

: total population is conservedAnd,

# Positive (steady-state) population inversion :

# Threshold pumping power : 131

PwrNPh

V

TNP

PNN

21

2112

TNP

Ph

21

2131

21min P

312121 hN T


10.8 Four-Level Laser Scheme

Lower laser level is not the ground level : The depletion of the lower laser level obviously enhances the population inversion.

Population rate equations :

))((

))((

1222102

122211101

11000

NNNPNdt

dN

NNNNdt

dN

NPNdt

dN


TNNNN constant210

Steady-state solutions :

T

T

T

NPP

PN

NPP

PN

NPP

N

21102110

102

21102110

211

21102110

21100

Population inversion :

PP

NPNN T

21102110

211012

)(

# Positive inversion : 2110 lower level decays more rapidly than the upper level.If P,2110

TNP

PNNN

21212


10.9 Comparison of Pumping Requirements for Three and Four-Level Lasers

1) Threshold pumping rate, Pt

21laserlevelthree)(

tT

tTt NN

NNP

21laserlevelfour)(

tT

tt NN

NP

(10.7.9) =>

(10.8.7), (10.8.6) =>

1)(

)(

laserlevelthree

laserlevelfour

tT

t

t

t

NN

N

P

P

2) Threshold pumping power, (Pwr)t

2131laserlevelthree 2

1)Pwr(

T

t NhV

(10.7.15) =>

2130laserlevelfour

)Pwr(

t

t NhV

T

t

t

t

N

N

V

V

31

30

laserlevelthree

laserlevelfour 2

)/)Pwr((

)/)Pwr((


10.11 Small-Signal Gain and Saturation

For the three-level laser scheme, steady-state solution including the stimulated emission term ;

2

)(

21

2112 P

NPNN T

Gain coefficient (assuming g1=g2),

)(fn)(2

))(()( 1

21

21

P

NPg T

: A large photon number, and therefore a large stimulated emission rate, tends to equalize the populations and . In this case, the gain is said to be “saturated.”

1N 2N

<Microscopic view of the gain saturation>As the cavity photon number increased, the stimulated absorption as well as the stimulated emission increased. => The lower level’s absorption rate is exactly equal to the upper level’s emission rate in the extreme limit. The gain is

zero.


<Small signal gain / Saturation flux>

sat0

2121

21

/1

)(

)]/()(2[1

1))(()(

g

PP

NPg T

Where we define, Small signal gain as

21

210

))(()(

P

NPg T

Saturation flux as

)(221sat

P

* The larger the decay rates, the larger the saturation flux.* The saturation flux Stimulated emission rate : The average of the upper- and lower-level decay rates.


<Gain width>

When the absorption lineshape is Lorentzian,

1)/()(

1)()(

221

221

21

* Small signal gain width : 21 g

(10.11.3) =>

)/(1)/()(

1)()(

sat221

221

210

21

gg

where,21

2121210

))(()(

P

NPg T : Line-center small signal gain

* Power-broadened gain width : 2/1

sat21

21

1

g


2121221

2

21

21sat )(4

)(221

PA

P

: Line-center saturation flux is directly proportional to the transition linewidth.


10.12 Spatial Hole Burning

In most laser, we have standing waves rather than traveling waves.

=> Cavity standing wave field is the sum of two oppositely propagating traveling wave fields ;

),(),(

)]sin()[sin(

sincos),(

021

0

tzEtzE

tkztkzE

kztEtzE

)sin(),( 021 tkzEtzE where,

The time-averaged square of the electric field gives a field energy density : 20

0)(

8E

c

h

=> Total photon flux,

kz2)()( sin][2

hI kzIII 2)()( sin][2


(10.11.3) =>

kz

gg

2sat)()(0

sin]/)[(21

)()(

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Nonlinear Optics Lab. Hanyang Univ. Chapter 10. Laser Oscillation : Gain and Threshold Detailed description of laser oscillation 10.2 Gain Consider a quasi-monochromatic