Transcript Creation of Colloidal Periodic Structure
Chapter 10. Laser Oscillation : Gain and Threshold
Detailed description of laser oscillation
10.2 Gain
I
n (
z
)
A
The flux difference ; [
I
n (
z
D
z
)
I
n ]
A
z
(
I
n
A
) D
z
Nonlinear Optics Lab . Hanyang Univ.
This difference gives the rate at which electromagnetic energy leaves the volume A D z,
t
(
u
n
u
n
I
n /
c
A
D
z
)
z
(
I
n 1
c
t I
n
A
) D
z
z I
n 0 : Equation of continuity
=> 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 : ( n ) (
N
2
N
1 )
h
n ( n )
I
n (
N
2
N
1 ) 1
c
t
z I
n ( n )
I
n (
N
2
N
1 ) Nonlinear Optics Lab . Hanyang Univ.
(7.4.19) : ( n ) (
h
n /
c
)
BS
( n ) 1
c
t
z I
n
h
n
c B
(
N
2
N
1 )
S
( n )
I
n
h
n
c
B
N
2
g
2
g
1
N
1
S
( n )
I
n 2 8
A
N
2
g
2
g
1
N
1
S
( n )
I
n ( n )
N
2
g
2
g
1
N
1
I
n (no degeneracies) (degeneracies included) Define, gain coefficient, g( n )
g
( n ) 2 8
A
N
2
g
2
g
1
N
1
S
( n ) ( n )
N
2
g
2
g
1
N
1 Nonlinear Optics Lab . Hanyang Univ.
I
n
z
1
c
I
n
t
g
( n )
I
n Temporal steady state :
t
0
I
n (
z
)
I
n ( 0 )
e g
( n )
z
dI
n
g
( n )
I
n
dz
* g>0 when
N
2
g g
2 1
N
1 * If
N
2
g g
2 1
N
1 , : population inversion
N
2
g
2
g
1
N
1
g
2
g
1
g
( n ) 2 8
A g
2
g
1
N
1
g
2
g
1
NS
( n )
N
N g
2
g
1 ( n )
a
( n ) 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 e 0.01 x 100 =e 1 =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.
r
t
s
1 where, r : reflection coefficient t : transmission coefficient s : fractional loss At the mirror at z=L and 0 ;
I
n ( ) (
L
)
r
2
I
n ( ) (
L
)
I
n ( ) ( 0 )
r
1
I
n ( ) ( 0 ) Nonlinear Optics Lab . Hanyang Univ.
In steady state (or CW operation), g( n ) : constant
dI
n ( )
dz
g
( n )
I
n ( )
dI
n ( )
dz
g
( n )
I
n ( )
I
n ( ) (
z
)
I
n ( ) ( 0 )
e g
( n )
z I
n ( ) (
z
)
I
n ( ) (
L
)
e g
( n )(
L
z
)
I
n (
I
n ( ) ( 0 )
r
1
I
n ( ) ( 0 )
r
1 [
e g
( n )
L I
n ( ) (
L
)] ( 0 )
r
1
e g
( n )
L
[
r
2
I
n ( ) (
L
)]
r
1
r
2
e g
( n )
L
[
I
n ( ) ( 0 )
e g
( n )
L
] [
r
1
r
2
e
2
g
( n )
L
]
I
n ( ) ( 0 )
I
n ( ) ( 0 )
r
1
r
2
e
2
gL
1 Nonlinear Optics Lab . Hanyang Univ.
I
n ( ) ( 0 )
Threshold gain,
g t g t
1 2
L
ln 1
r
1
r
2 1 2
L
ln(
r
1
r
2 ) * In the case of high reflectivity,
r
1
r
2 1 , and ln(1 -
x
) -
x
g t
1 2
L
( 1
r
1
r
2 ) (high reflectivi ties,
r
1
r
2 0.9) * If the “distributed losses” (losses not associated with the mirrors) are included,
g t
2 1
L
ln(
r
1
r
2 )
a
Nonlinear Optics Lab . Hanyang Univ.
Example) He-Ne laser, L=50 cm, r 1 =0.998, r 2 =0.980
g t
1 2 ( 50 ) ln[( 0 .
998 )( 0 .
980 )]
cm
1 2 .
2 10 4
cm
1 Threshold population inversion ;
g
( n ) 2
A
8
n
2
N
2
g
2
g
1
N
1
S
( n ) D
N t
N
2
g
2
g
1
N
1 8 2
n AS
2
g
( n
t
)
g
( n
t
)
A
1 .
4 10 6 sec 1
T
~ D 400
N t
K , M
Ne
at 632.8
nm 20 ( 6328 10 8 amu ( 8 cm ) 2 n D 2 .
15 10 6 )( 2 .
2 10 4 cm 1 ) ( 1 .
4 10 6 sec 1 1 )( 6 .
3 10 10 T M 1 / 2 sec) MHz 1500 MHz 1 .
6 10 9 atoms / cm 3 D
N t N
1 .
6 10 9 4 .
8 10 15 1 3 10 6 (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)
I
n (
z
) 1
c
I
n (
t
)
g
( n )
I
n ( )
I
n (
z
)
z
(
I
n ( ) 1
c
I
n (
t
)
I
n ( )
g
) 1
c
t
( n )
I
n ( ) (
I
n ( )
I
n ( ) )
g
( n )(
I
n ( )
I
n ( ) )
d dt
(
I
n ( )
I
n ( ) )
cg
( n )(
I
n ( )
I
n ( ) ) *
l
l : gain medium length, L : cavity length) d dt ( I n ( ) I n ( ) ) cl L g ( n )( I n ( ) I n ( ) ) (10.5.4b) I n ( ) I n ( ) Nonlinear Optics Lab . Hanyang Univ. z 0 2) Loss term (output coupling, absorption/scattering at the mirrors) By the output coupling, a fraction 1-r 1 r 2 of intensity is lost per round trip time 2L/c. d dt ( I n ( ) I n ( ) ) c 2 L ( 1 r 1 r 2 )( I n ( ) I n ( ) ) (10.5.8) I n I n ( ) I n ( ) dI n dt cl L g ( n ) I n c 2 L ( 1 r 1 r 2 ) I n or photon number, q n I n dq n dt cl L g ( n ) q n c 2 L ( 1 r 1 r 2 ) q n cl L g ( n ) q n cl L g t q n Nonlinear Optics Lab . Hanyang Univ. If g 1 =g 2 , dI n dt cl L 2 8 A ( N 2 N 1 ) S ( n ) I n c 2 L ( 1 r 1 r 2 ) I n cl L ( n )( N 2 N 1 ) I n c 2 L ( 1 r 1 r 2 ) I n Coupled equations for the light and the atoms in the laser cavity including pumping effect : dN 1 dt dN 2 1 N 1 AN 2 g ( n ) n ( 2 A ) N 2 g ( n ) n dt d n dt cl L g ( n ) n K c 2 L ( 1 r 1 r 2 ) n Nonlinear Optics Lab . Hanyang Univ. 1) Two-level laser scheme is not possible. Neglecting 1 , 2 in (7.3.2) => N 2 ( ) N A 21 2 N 2 : can’t achieve the population inversion 2) Three-level laser scheme (10.5.14) => dN 1 dt PN 1 21 N 2 n ( N 2 N 1 ) dN 2 PN 1 21 N 2 n ( N 2 N 1 ) dt d n dt cl L g ( n ) n c 2 L ( 1 r 1 r 2 ) n P : pumping rate dN 1 dt dN 2 dt pumping PN 1 pumping dN 3 dt pumping dN 1 dt pumping PN 1 Nonlinear Optics Lab . Hanyang Univ. N 2 0 N 2 P 21 N 1 And, N 1 N 2 N T : total population is conserved N 1 P 21 21 N T N 2 P P 21 N T N 2 N 1 P P 21 21 N T # Positive (steady-state) population inversion : P 21 # Threshold pumping power : Pwr h n 31 P N 1 V h n 31 P 21 P 21 N T P min 21 1 2 21 N T h n 31 Nonlinear Optics Lab . Hanyang Univ. Lower laser level is not the ground level : The depletion of the lower laser level obviously enhances the population inversion. Population rate equations : dN 0 dt PN 0 10 N 1 dN 1 dt 10 N 1 21 N 2 ( n )( N 2 N 1 ) n dN 2 dt PN 0 21 N 2 ( n )( N 2 N 1 ) n Nonlinear Optics Lab . Hanyang Univ. N 0 N 1 N 2 constant N T Steady-state solutions : N N 0 1 10 10 21 21 10 21 10 P 21 P 10 P 21 P 21 P N 2 10 21 10 10 P P 21 P N N T N T T Population inversion : N 2 N 1 10 P ( 10 21 10 21 P ) N T 21 P # Positive inversion : 10 21 lower level decays more rapidly than the upper level. If 10 21 , P N 2 N 1 N 2 P P 21 N T Nonlinear Optics Lab . Hanyang Univ. 1) Threshold pumping rate, P t (10.7.9) => ( P t ) three level (10.8.7), (10.8.6) => ( P t laser ) N N T four level T D N t D N t laser 21 N T D N t D N t 21 ( P t ) four level laser ( P t ) three level laser N T D N t D N t 1 2) Threshold pumping power, (Pwr) t (10.7.15) => ( Pwr ) t V three level laser 1 2 h n 31 N T 21 ( Pwr ) t V four level laser h n 30 D N t 21 (( Pwr ) t / V ) four level laser (( Pwr ) t / V ) three level laser 2 n 30 D N t n 31 N T Nonlinear Optics Lab . Hanyang Univ. For the three-level laser scheme, steady-state solution including the stimulated emission term ; N 2 N 1 ( P P 21 21 ) N T 2 n Gain coefficient (assuming g 1 =g 2 ), g ( n ) P ( n )( 21 P 2 21 ( n ) N T ) n fn ( n 1 ) : 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.” N 1 N 2 Nonlinear Optics Lab . Hanyang Univ. g ( n ) 1 ( n )( P P 21 21 ) N T g 0 n ( n / ) sat n 1 [ 2 1 ( n ) n /( P 21 )] Where we define, Small signal gain as g 0 ( n ) ( n )( P P 21 21 ) N T Saturation flux as sat n P 2 21 ( n ) * 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. g n 21 (10.11.3) => g ( n ) g 0 ( n 21 ) ( n 21 n ) 2 1 /( n 21 ) 2 1 ( n / sat n 21 ) where, g 0 ( n 21 ) ( n 21 )( P P 21 21 ) N T : Line-center small signal gain * Power-broadened gain width : D n g n 21 1 n sat n 21 1 / 2 Nonlinear Optics Lab . Hanyang Univ. sat n 2 1 2 P 21 ( n 21 ) 4 2 2 n A 21 ( P 21 ) n 21 : Line-center saturation flux is directly proportional to the transition linewidth. Nonlinear Optics Lab . Hanyang Univ. 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 ; E ( z , t ) E 0 cos t sin kz 1 2 E 0 [sin( E ( z , t ) kz t ) E ( z , t ) sin( kz t )] where, E ( z , t ) 1 2 E 0 sin( kz t ) The time-averaged square of the electric field gives a field energy density : h n ( n ) c 0 8 E 0 2 => Total photon flux, n 2 [ ( n ) ( n ) ] sin 2 kz I n h n n I n 2 [ I n ( ) I n ( ) ] sin 2 kz Nonlinear Optics Lab . Hanyang Univ. (10.11.3) => g ( n ) 1 2 [( ( n ) g 0 ( n ( n ) ) ) / sat n ] sin 2 kz Nonlinear Optics Lab . Hanyang Univ.10.7 Three-Level Laser Scheme
10.8 Four-Level Laser Scheme
10.9 Comparison of Pumping Requirements for Three and Four-Level Lasers
10.11 Small-Signal Gain and Saturation
10.12 Spatial Hole Burning