Nonlinear Optics Lab. Hanyang Univ. Chapter 11. Laser Oscillation : Power and Frequency Power &...

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

Chapter 11. Laser Oscillation : Power and Frequency

Power & Frequency of Single mode continuous wave (cw) laser ?

11.2 Output Intensity : Uniform-Field Approximation

Assumptions : (1) Homogeneous broadened gain medium (2) Mean-field (uniform-field) approximation (3) All loss processes are independent of the cavity intensity (4) Steady-state (temporal) or CW operation

1) Gain

(10.5.8) , => IrrL

cIg

L

cl)1(

2)( 210I tgrr

lg )1(

2

1)( 21

121 rr

: The gain is clamped on the threshold gain for cw(steady-state) oscillation.


2) Output intensity

(10.12.10) or (10.11.14) => tgIII

gg

sat)()(

0

/1

)()(

1

)(0sat)()(

tg

gIII

Output intensity : )(1

out ItI

If =>121 rr )()( I

1

)(

2

1 0satout

tg

gItI

)(2

1)1(

2

1st

lr

lgt

1

)(2

2

1 0satout

st

lgItI

=>: A given medium, laser intensity depends on how the laser cavity (t or s) is chosen.


11.3 Optimal Output Coupling

1) Optimal mirror transmission coefficient

slsgt )(2 0opt

2

0satou

maxou 2/)( slgIII

opttttt

0

optt

out

t

I 0)(

2

2

11

2

2

12

00

opttt

satsat

st

lgtI

st

lgI

2) Maximum output intensity

If

l

sg

2)(0 : Lasing is possible when : scattering loss coefficient

l

sgst

lr

lg optoptoptt 2

)()(

2

1)1(

2

1)( 0

,2

)(0 l

sg lIgI sat

210maxout )( : theoretical upper limit of laser intensity


2) Maximum output intensity – another approach

INNdt

dNh )()( 12

emissionstimulated

2

(10.12.8) =>

)()(

emissionstimulated

2 )(

IIgdt

dNh

)()(2)()( ,sin2 IIIkzIII : uniform field approx.

(11.2.5) => tt

tt

ggIg

gIg

g

gIgIIg

)(1

)(1

)()()( 0

sat0sat0sat)()(

If tgg )(0 sat0

)()( )()(, IgIIg : maximum intensity per unit volume

Maximum intensity extracted from the medium of length l ;

lIgI out sat210max 2121

)( : (11.3.5a)


3) Input-to-output power conversion efficiency

3-level system, (10.7.12) =>131

sat210

max

)(

NPh

Ige

(10.11.12) =>21

2121210

))(()(

P

NPg T

(10.11.7) =>)(2

)(

21

212121

PhI sat 131

2121max 2

)(

NPh

hNPe T

In the case of strongly saturated case, TNNN2

121

31

21

31

2121max

P

Pe : quantum efficiency


11.4 Effect of Spatial Hole Burning

Standing wave inside the cavity, (10.2.9) => kzII

gg

2sat)(0

sin/21

)()(

(11.3.6) =>

kzII

kzIg

kzIgdt

dNh

2sat

20

2

emissionstimulated

2

sin)/2(1

sin)(2

sin)(2

: Power per unit volume, at the point z, extracted from the medium by stimulated emission.

The rate at which the field gains energy should equal the rate at which it losses energy ;

ll

kzII

dzkzIgdzgI

0 2sat

2

00 sin)/2(1

sin2 IlgIstIst t )()( 2

1)(

For ,1kl

sat

sat

0 2sat

2

/21

11

2sin)/2(1

sin

III

Il

kzII

dzkzl

sat0

sat/21

11

Ig

Ig

IIt


Put,

satI

Ix 2

1 xg

gx

t

02

4

12

2

1221 00

sat

tt g

g

g

g

I

Ix : Disired solution is the one with minus

sign since should be equal to 1 when g0/gt=1.

x

4

12

2

1221 00

sat

tt g

g

g

g

I

I

16

1

24

1 00sat

tt g

g

g

gII

16

1

2

)(

4

1)(

200satout

tt g

g

g

gI

tI

Output intensity : ItI out )2/(

: The effect of spatial hole burning is to reduce the output intensity


11.5 Large Output Coupling

Our analysis of output power thus far has assume that the output coupling is small( ),

and we have also assumed time averaged intensity I(+) and I(-) are independent of z.

We will now allow arbitrary output coupling and therefore allow the possibility I(+) and I(-) may vary with z.

121 rr

Ignoring the spatial hole burning, (11.2.4) => sat)()(0

)]()([1)(

IzIzI

gzg

(10.4.3) => )()( )()(

zIzgdz

dI

)()( )()(

zIzgdz

dI

0)(

)()(

)()()(

dz

dII

dz

dIIII

dz

d

CzIzIei constant)()(;, )()(

,/

1

1

sat

)()(0

)(

)(

IICI

g

dz

dI

I

sat

)()(0

)(

)( /1

1

IICI

g

dz

dI

I


2)(

)(

sat)(

sat)(

)(

)()(

)(sat)(0

1

11

1

I

dI

I

CdI

II

dI

dII

CI

IIdzg

)(

)0( 2)(

)(

sat

)(

)0(

)(sat

)(

)0( )(

)(

0 0

)(

)(

)(

)(

)(

)(

1

LI

I

LI

I

LI

I

L

I

dI

I

C

dIII

dIdzg

)0(

1

)(

1

)0()(1

)0(

)(ln

)()(sat

)()(sat)(

)(

0

ILII

C

ILIII

LIlg

)(

1

)0(

1

)()0(1

)0(

)(ln

)()(sat

)()(sat)(

)(

0

LIII

C

LIIII

LIlg

=>

)()(

)( )(2)(

)( LIrLI

CLI

2)(

2 )(LIrC

)0()0(

)0( )(1)(

)(

IrI

CI

)()0( )(12

)( LIrrI

)()0( )(21

)( LIrrI =>


1

22sat

)(

21sat

)(

21

0

)(

1)(1

ln

r

rr

I

LI

rrI

LI

rrlg=>

210

2121

sat1

21212

210sat

)(

ln)1()(

1

ln)(

rrlgrrrr

Ir

rrrrr

rrlgILI

Output intensity :

210

2121

1

1

22sat

)(2

)(1

out

ln)1()(

)()0(

rrlgrrrr

rt

r

rtI

LItItI

When r1=1, t1=s1=0, r=r2, t=t2, s=s2, and t+s<<1

1

2 0sat21out

st

lgItI (11.2.11) : small output coupling


<Total two-way intensity>

r

r

II

LILI

2

1

)0()0(

)()()()(

)()(

(11.5.10) => )()(

)( )(2)(

)( LIrLI

CLI

)()0()0( )(21

)(1

)( LIrrIrI (11.5.12) =>

(11.5.13) => )()0( )(12

)( LIrrI

rrr 21 ,1

Total intensities are comparable at the two mirrors for reflectivity as low as 50%.


11.6 Measuring Small-Signal Gain and Optimal Output CouplingEq. (11.2.11) for output intensity or its generalization (11.5. 18) has been shown experimentally to be quite accurate, because the spatial hole burning effect is usually negligible in gas lasers.

In general the small signal gain and the saturation intensity are difficult to calculate accurately, because the puping and decay rates of the relavant atomic levels may not be well known.

=> Experimentally measured !


<Maximal loss method to measure g0>

- The cavity loss is varied by inserting a reflecting knife-edge into the cavity

- The cavity loss is increased until the laser oscillation ceases.

- (11.2.4) => g0(=gt) : loss just when the laser oscillation ceases.

<Simultaneous measurement method>

- Scattering coefficient, s=Pin/P+

- Effective output coupling : t+s

- t=Pout/P+ : known => s=tPin/Pout

- Ptotal=Pin+Pout : total output power

- Determine s-value for which Ptotal is maximum

=> topt=sopt + t., Ptotal=(Pin+Pout)topt

- Small signal gain : s-value at which laser oscillation stops.


11.7 Inhomogeneously Broadened Laser MediaIn an inhomogeneously broadened gain medium the different active atoms have different central transition frequencies 21.

# Small signal gain : Doppler broadened lineshape

]/2ln)(4exp[)(

2ln4exp2ln41

8

)(8

)(

2221210

2221

2/1

0

2

0

2

0

D

DD

g

NA

SNA

g

The gain coefficient is obtained by integrating the contributions from the different frequency components, each of which saturates to a differnet degree depending on its detuning from the cavity mode frequency .

21221

221

0)/(1)/()(

1)()(

21

dgg sat


sat

0

1

)()(

II

gg

2)()(2* 21 zz dvvk

aax

dx

22

*

where,sat

sat hI21

The gain saturation set in more slowly as the intensity I is increased in the case of inhomogeneous broadening medium.

1

)(

2

2

0satout

tg

gI

tI

Output intensity :

cf) homogeneous medium, (11.2.9)

1

)(

20satout

tg

gIt

I


11.8 Spectral Hole Burning and the Lamb Dip<Spectral hole burning>

Spectral packets : The atom group with the central transition frequency of c 2121

The gain for spectral packets with frequency 21~(field frequency) is saturated more strongly than others : spectral packets with frequency detuned from by much more than the homogeneous linewidth, i.e., |21-|>>21, are hardly saturated at all.

Spectral hole burning

(Bennet hole)


<Lamb dip>

Suppose the cavity mode frequency the center frequency of the Doppler gain profile

i) The traveling-wave field propagating in the +z direction will strongly saturate the

spectral packet of atoms with Doppler-shifted frequencies ’21=.

: The Doppler effect has brought these atoms into resonance with the wave. Therefore,

those atoms have the z component of velocity given by

c

v1

c

vor

ii) Similarly, the traveling-wave field propagating in the -z direction will strongly saturate those atoms with the z component of velocity given by

c

v


=> The standing wave cavity field will burn two holes in the Doppler line profile.

When the mode frequency is exactly at the center of the Doppler line, the two holes merge together. => The field can now strongly saturate only those atoms having no z component of velocity. => The output power exactly at resonance will be lower than slightly off resonance. : Lamb Dip.


11.9 Cavity Frequency and Frequency Pulling

Cavity mode frequency : mL

mc 2

In general,)()(

2

lLln

mc

where, l : gain medium length, n: refractive index.

or, mnL

l1 where,

L

mcm 2

: bare(passive) cavity mode frequency

(3.3.22), (3.3.25) =>

gn21

2121

41

for homogeneous broadening medium

21

2121

4)(

4)(

Llgc

Llgc m

put,L

lcgc

4

)( : cavity bandwidth

21

2121

c

mc

mc vvv 2121or : frequency pulling


<Frequency pulling and gain>

In most lasers, c 21

21

21

212121

21

2121 11

v

v

v

v

v

v

cmm

cm

c

c

mc

D

cmm

D

cmm

v

v

21

21

88.1

2ln4

(homogeneous broadening)

(inhomogeneous broadening)D 21

cm *tc g

L

cl

4and

: The larger the threshold gain gt, the greater the frequency pulling for fixed gain linewidth (21 or D).


<Mode spacing>

21

2121)(

c

mcm

21

21

211)()1(

/1

1

2

c

c

mmmm

L

c

: The effect of frequency pulling is to reduce the mode spacing from c/2L.


11.11 Laser Power at Threshold

Laser power near the threshold ? => spontaneous emission.

(10.5.7) qg

L

clqg

L

cl

dt

dqt )(

sat

212

1

)(2

qq

N

P

PNNNN

T

T

(10.11.12)

)(2

)(

21

2112 P

NPNN T

(Mean-field approx.)

21P

qcg

L

lqN

L

lc

dt

dqt 1)( 2

Including the spontaneous emission :


Steady-state solution :

2

2

)(

)(

Ng

Nq

t

define,t

T

t N

Ny

N

Nx

and2

x

xq

1 satsat 11 qq

y

qq

NNx tT

0)1( satsat2 yqqyqq

sat2

sat

4)1(

2

1)1(

2

1

q

yyy

q

q

tggy 0

(11.11.1) => TNN 2

<1 : below the threshold

>1 : above the threshold

1y (far above threshold)

10sat

tg

gqq

: (11.2.5), (11.2.9)

1y (near threshold)

: (11.2.5), (11.2.9)

(10.11.8), (10.11.10), P>>21

PVfe

mq 212

0sat 12


In many lasers, PV~103s-1, f~1, 21~10GHz => qsat~1010

1y 5satsatsat21

threshold 104 qqqq (very low power)

cf) 1

thermal 1

kTheq (ex) 6328 A He-Ne laser, 023.0thermal q

thermal)( qq t

<The rate of change of q with y>

sat2

satsat

21

/4)1(

/211

qyy

qyq

dy

qd

1021sat

21

threshold

10

q

dy

qd

Extremely rapid rise in the cavity photon number at the point y=1 (threshold).


11.12 Obtaining Single-Mode Oscillation1) Short cavity length

gL

c 2

g

Ex) g~1500 MHz (He-Ne laser) => c/2L>1500MHz => L<10 cm (low power)

2) Homogeneous broadening medium


3) Fabry-Perot etalon / Grating / Prism

=> Selective transmission

Ex) Fabry-Perot etalon

- resonance frequency : ...,3,2,1,cos2

mnd

cmm

cos21 nd

cmm

Nonlinear Optics Lab. Hanyang Univ. Chapter 11. Laser Oscillation : Power and Frequency Power &...

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Transcript of Nonlinear Optics Lab. Hanyang Univ. Chapter 11. Laser Oscillation : Power and Frequency Power &...