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Nonlinear Optics Lab. Hanyang 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 n propagating in the +z direction ;
zA
z z+Dz
un
The rate at which electromagnetic energy
passes through a plane of cross-sectional
area A at z is AzI )(n
The flux difference ;
zAIz
AIzzI D
D )(])([ nnn
Nonlinear Optics Lab. Hanyang Univ.
This difference gives the rate at which electromagnetic energy leaves the volume ADz,
zAIz
zAut
D
D
)()( nn
01
nn I
zI
tc cIu /nn
: 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 nnnn
)()(1
12 NNIIztc
nn n
Nonlinear Optics Lab. Hanyang Univ.
(7.4.19) : )()/()( nnn BSch
n
n
n
nn
n
n
nn
nn
INg
gN
ISNg
gN
A
ISNg
gNB
c
h
ISNNBc
hI
ztc
1
1
22
1
1
22
2
1
1
22
12
)(
)(8
)(
)()(1
(no degeneracies)
(degeneracies included)
Define, gain coefficient, g(n)
1
1
22
1
1
22
2
)(
)(8
)(
Ng
gN
SNg
gN
Ag
n
n
n
Nonlinear Optics Lab. Hanyang Univ.
nnn n Igt
I
cz
I)(
1
Temporal steady state : 0
t nn n Ig
dz
dI)(
zgeIzI )()0()( nnn : valid only for low intensity ( saturation effect, g=g(I))
* g>0 when1
2
12 N
g
gN : population inversion
* If ,1
2
12 N
g
gN
)()(
)(8
)(
1
2
1
2
2
1
21
1
21
1
22
nn
n
n
ag
gN
NSg
gAg
Ng
gN
g
gN
g
gN
The gain coefficient is identical, except
for its sign, to the absorption coefficient.
Nonlinear Optics Lab. Hanyang Univ.
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.
Nonlinear Optics Lab. Hanyang Univ.
<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 str
where, r : reflection coefficient
t : transmission coefficient
s : fractional loss
At the mirror at z=L and 0 ;
)()( )(
2
)( LIrLI nn
)0()0( )(
1
)( nn IrI
Nonlinear Optics Lab. Hanyang Univ.
In steady state (or CW operation), g(n) : constant
)()(
)(
nn n Ig
dz
dI )()(
)(
nn n Ig
dz
dI
zgeIzI )()()( )0()( nnn
))(()()( )()( zLgeLIzI nnn
Amplification condition : after one round trip must be higher than the initial)0()(
nI )0()(
nI
)0()0(][
])0([
)]([
)]([
)0()0(
)()()(2
21
)()()(
21
)(
2
)(
1
)()(
1
)(
1
)(
nnn
nn
n
nn
nn
nn
IIerr
eIerr
LIrer
LIer
IrI
Lg
LgLg
Lg
Lg
12
21 gLerr
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
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
)( 1
1
222
2
n
n SN
g
gN
n
Ag
)()(
82
2
1
1
22
nn
ttt
g
AS
gnN
g
gNN
D
nm 632.8at sec104.1 16 A
MHz1500MHzM
T11015.2amu20M,K400~
2/1
6
D
nNeT
39
101628
14
cm/atoms106.1
sec)103.6)(sec104.1()cm106328(
)cm102.2)(8(
D
tN
6
31
15
9
10108.4
106.1
D
N
Nt (very small !)
Nonlinear Optics Lab. Hanyang Univ.
10.5 Rate Equations for Photons and Populations
Time-dependent phenomena ?
1) Gain term (stimulated emission or absorption)
)()()(
)(1
n
nn n Igt
I
cz
I
)()()(
)(1
n
nn n Igt
I
cz
I
))(()(1
)( )()()()()()(
nnnnnn n IIgII
tcII
z
In many lasers there is very little gross variation of either or with z. =>)(
nI)(
nI 0
z
* l<L (l : gain medium length, L : cavity length)
))(()( )()()()( nnnn n IIcgIIdt
d
))(()( )()()()( nnnn n IIgL
clII
dt
d(10.5.4b)
Nonlinear Optics Lab. Hanyang Univ.
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
)()( nnnn IIrrL
cII
dt
d(10.5.8)
From (10.5.4b), (10.5.8), and put : total intensity)()( nnn III
nnn n Irr
L
cIg
L
cl
dt
dI)1(
2)( 21
or photon number, nn Iq
nn
nnn
n
n
qgL
clqg
L
cl
qrrL
cqg
L
cl
dt
dq
t
)(
)1(2
)( 21
Nonlinear Optics Lab. Hanyang Univ.
If g1=g2,
nn
nnn
n
n
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 :
nnn
n
n
n
n
n
)1(2
)(
)()(
)(
21
222
2111
rrL
cg
L
cl
dt
d
KgNAdt
dN
gANNdt
dN
Nonlinear Optics Lab. Hanyang Univ.
10.7 Three-Level Laser Scheme
1) Two-level laser scheme is not possible.
22)(
21
2
N
A
NN
Neglecting 1, 2 in (7.3.2) => : can’t achieve the population inversion
2) Three-level laser scheme
P : pumping rate
1
pumping
1 PNdt
dN
1
pumping
1
pumping
3
pumping
2 PNdt
dN
dt
dN
dt
dN
(10.5.14) =>
)(
)(
1222112
1222111
NNNPNdt
dN
NNNPNdt
dN
n
n
nnn n
)1(
2)( 21rr
L
cg
L
cl
dt
d
Nonlinear Optics Lab. Hanyang Univ.
<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
1
21
2 NP
N
TNNN 21
TNP
N21
211
TNP
PN
21
2
21P
: total population is conservedAnd,
# Positive (steady-state) population inversion :
# Threshold pumping power :131
PwrNPh
Vn
TNP
PNN
21
2112
TNP
Ph
21
2131
n21min P
312121 nhNT
Nonlinear Optics Lab. Hanyang Univ.
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 :
n
n
n
n
))((
))((
1222102
122211101
11000
NNNPNdt
dN
NNNNdt
dN
NPNdt
dN
Nonlinear Optics Lab. Hanyang Univ.
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
21
212
Nonlinear Optics Lab. Hanyang Univ.
10.9 Comparison of Pumping Requirements for Three and
Four-Level Lasers
1) Threshold pumping rate, Pt
21laserlevelthree)( D
D
tT
tTt
NN
NNP
21laserlevelfour)( D
D
tT
tt
NN
NP
(10.7.9) =>
(10.8.7), (10.8.6) =>
1)(
)(
laserlevelthree
laserlevelfour
D
D
tT
t
t
t
NN
N
P
P
2) Threshold pumping power, (Pwr)t
2131
laserlevelthree 2
1)Pwr(
Tt Nh
Vn(10.7.15) =>
2130
laserlevelfour
)Pwr(D
tt Nh
Vn
T
t
t
t
N
N
V
V
31
30
laserlevelthree
laserlevelfour 2
)/)Pwr((
)/)Pwr((
n
n D
Nonlinear Optics Lab. Hanyang Univ.
10.11 Small-Signal Gain and Saturation
For the three-level laser scheme, steady-state solution including the
stimulated emission term ;
n
2
)(
21
2112
P
NPNN T
Gain coefficient (assuming g1=g2),
)(fn)(2
))(()(
1
21
21
n
nn
nn
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.
Nonlinear Optics Lab. Hanyang Univ.
<Small signal gain / Saturation flux>
sat
0
2121
21
/1
)(
)]/()(2[1
1))(()(
nn
n
n
n
nn
g
PP
NPg T
Where we define, Small signal gain as
21
210
))(()(
P
NPg Tn
n
Saturation flux as
)(2
21sat
nn
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.
Nonlinear Optics Lab. Hanyang Univ.
<Gain width>
When the absorption lineshape is Lorentzian,
1)/()(
1)()(
2
21
2
21
21
nnn
nn
* Small signal gain width : 21nn D g
(10.11.3) =>
)/(1)/()(
1)()(
sat2
21
2
21
210
21nnnnnnn
gg
where,21
2121210
))(()(
P
NPg Tn
n : Line-center small signal gain
* Power-broadened gain width :
2/1
sat21
21
1
D
n
nnn g
Nonlinear Optics Lab. Hanyang Univ.
21212
21
2
21
21sat )(4
)(221n
n
nn
P
A
P
: Line-center saturation flux is directly
proportional to the transition linewidth.
Nonlinear Optics Lab. Hanyang Univ.
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 :2
00)(
8E
c
h nn
=> Total photon flux,
kz2)()( sin][2 nnn
nn nhI kzIII 2)()( sin][2 nnn
Nonlinear Optics Lab. Hanyang Univ.
(10.11.3) =>
kz
gg
2sat)()(
0
sin]/)[(21
)()(
nnn
nn