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Chap.5 Optical Resonators Containing Amplifying Media
U D
• Objectives
– Interactions between resonator & amplifying
p y g media
– Mode behavior
– Output coupling
• Reading
R di
– Chapter 5 of Davis
• Outline
–
–
–
–
–
Laser oscillation
Threshold of laser oscillation
R
Resonant
t ffrequency
Multimode oscillations
Output power of a laser
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
1
U D
Resonator with Amplifying Media
Gain medium
z
2
k
'
k
[
1
'
(
)
/
2
n
]
Propagation constant
( ) k ' ' ( ) / n 2
Gain coefficient
Propagation of
plane wave
e
jk ' z
e
2
z
j(k
2
z
k '
jk ' '
j
) z
2
2
2
2 n
2 n
EOP 505: Introduction to Lasers
e
Copyright © Qiwen Zhan
jk
g
z
2
U D
Optical Oscillation
l
Gain medium
E 0 t1 r2 e
E0
j2kgl
E 0 t1 r2 e
E 0 t1 e
E 0 t1
jk g l
jk g l
E 0 t1 t 2 e
jk g l
( r2 , t 2 )
( r1 , t1 )
Total transmitted filed:
E t E 0 t1 t 2 e
jk g l
[1 r1 r2 e
j2kgl
( r1 r2 e
j2kgl
) 2 ...]
jk l
t1 t 2 e g
E0
j2kgl
1 r1 r2 e
Oscillation:
r1 r2 e
j2kgl
1 Finite input leads to infinite output
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
3
Threshold Condition
•
U D
Physically, the oscillation condition corresponds to a condition that
the amplitude after a round trip remains the same, which means the
gain per round trip is equal to the loss per round trip
trip. Meanwhile
Meanwhile, the
phase shift after a round trip is multiple of 2. These leads to the
threshold condition for oscillation.
r1 r2 e
j2kgl
1
( t )2l
2 2
1,
t iis threshold
th h ld gain
i
r1 r2 e
k '
(k
) 2 l 2 m , m is an integer
2
2n
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
4
U D
Threshold Population Inversion
Amplitude condition:
t
1
1
ln r1 r2 t l l ln R1 R 2
l
2
Gain per pass
Loss per pass
g2
c2 A21
g2
2 A21
( ) ( N2 N1) 2 g(v0 , v) ( N2 N1)
g(v0 , v)
g1
8v
g1
8
Threshold population inversion:
g2
1
8
8
( ln r1r2 )
( N2 N1)t
2
g1
g(v0 , v) A21
l
g(v0 , v)2 A21c 0
Homogeneous broadening:
Inhomogeneous broadening:
(N2
g2
N1 )t 2
g1
A21
(N2
g2
1
N 1 )t 3
g1
A21
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
Photon lifetime
in resonator
5
U D
Resonant Frequency: Mode Pulling
Phase condition:
k '
(k
)2l 2 m ,
2
2n
' ( )
2( 0 )
' ' ( )
m is an integer
2( 0 )
2( 0 ) ( )
(
( )
k
n2 )
mc
]
[1
m
k
2l
( )c
m ( 0 )
2
At threshold, ignore distributed loss:
m ( m 0 )
1/ 2
EOP 505: Introduction to Lasers
t
1 R
l
Copyright © Qiwen Zhan
6
U D
Resonant Frequency: Mode Pulling
If one of the resonator modes frequency coincides with the line center of the
line shape function, then the oscillation occurs at the line center frequency.
Otherwise the actual oscillation occurs at a frequency near m but is slightly
Otherwise,
shifted towards line center. This is the so-called “mode-pulling”.
A t l oscillation
Actual
ill ti ffrequency
c/2nl
0
EOP 505: Introduction to Lasers
m
m+1
Copyright © Qiwen Zhan
7
Multi Longitudinal Modes Oscillation: Homogeneous
()
U D
()
Loss line t
Loss line t
()
Loss line t
Final oscillation
frequency
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
8
Multi Longitudinal Modes Oscillation: Inhomogeneous
U D
1
Single mode:
0.9
0.8
0.7
Hole
Image
hole
0.6
0.5
0.4
0.3
Loss line t
0.2
0.1
0
-5
0
5
1
0 .9
0 .8
0 .7
Lamb Dip
0 .6
0 .5
0 .4
Loss line t
0 .3
0 .2
0 .1
0
-5
EOP 505: Introduction to Lasers
0
5
Copyright © Qiwen Zhan
9
U D
Multi Longitudinal Modes Oscillation: Inhomogeneous
Multi-mode:
1
0 .9
0 .8
0 .7
Hole
0 .6
0 .5
0 .4
Loss line0.t3
0 .2
0 .1
0
-5
0
5
If the homogeneous broadening is also significant, mode competition also
may occur in inhomogeneous broadened laser.
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
10
U D
Mode Beating
Assume two longitudinal modes:
E i E 1e
j t
E 2e
j ( )t
Signal on square law detector is:
i(t ) Ei
2
[ E1e j t E 2 e j ( ) t ][ E1 e jt E 2 e j ( ) t ]
*
*
E1 E 2 E1 E 2* e j t E 2 E1* e j t
2
2
E1 E 2 2 E1 E 2 cos( t )
2
2
Beat signal
•
•
•
Optical oscillation is too fast to observe directly
Beat signal of optical signal frequencies normally in the RF range, thus can
be easily detected
If mode pulling is considered
considered, due to its nonlinearity
nonlinearity, beat signals
frequencies will split.
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
11
U D
Output Power of Laser
T1I4
T2I2
R1, A1 (loss
R1
(l
att mirror
i
or other
th
distributed loss)
R2 A2
R2,
I-+II+
I4
I1
II+
I2
I3
z
Total output is:
Ioutt=T
T1I4+T2I2
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
12
Output Power of Laser
U D
In the laser, intensity at both directions contribute to saturation
(z)
0
1 (I I) / Is
1 dI
(z)
I
1 dI
1 dI
dz
0
1 dI
I
dz
I
dz
(z)
I dz
d
d (II)
0 II C
dz
I3 I2R2
I1 I 4 R1
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
13
U D
Output Power of Laser
1
I
dI
dz
dI
I
1 (I
dI
I
d [ln
I
ln
I 2
I
I1
ln
I
I
4
I
2
4
2
I
I
(
s
3
R
s
(
C
I
I
s
0
dz
]
0
dz
s
I1
C
1
1
(
)
I s
I s
I 2
I1
I
I s
4
3
I
I
I
I
) / I
CdI
2
I I s
s
I
0
R
1
R
1
s
1
C
1
1
(
)
I s
I 4
I3
0
(
)
0
l ln
R 2 )( 1
R 2 ( 0 l ln
R
EOP 505: Introduction to Lasers
2
)( 1
R1R
2
0
l
l
R1R 2 )
R1R 2 )
R1R
2
)
Copyright © Qiwen Zhan
14
U D
Output Power of Laser
T1 1 R
T2 1 R
1
2
A1
A2
1. If A1=A2=A
I out I s
(1 A
1
R1R 2 )
R1R 2
( 0 l ln
l
R1R 2 )
2. If T1=0, R1=1
I out I 2 T 2 T 2 I s
[ 0 l
1
ln( 1 A 2 T 2 )]
2
A2 T2
3 Symmetric resonator: R1=R2=R
3.
R1=R2=R, A1=A2=A
A1=A2=A, output from each mirror:
I out
2
I s (1 A R )
( 0 l ln R )
2
1 R
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
15
Optimal Coupling
U D
Consider a symmetric mirror resonator, for maximal output:
dI out
dR
0
( 1 A R )( 0 l ln R )
(1 A R ) 1
0
2
R
1 R
(1 R )
1 R
1 A R
R
( 0 l ln R )
A
1 R
T opt
1 A T opt
[ 0 l ln( 1 A T opt )]
A
A T opt
0 l ln R
1 A T opt
A T opt
T opt (
[ 0 l ( A T opt )]
0l
A
1) A
0 lA A
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
16
Optimal Coupling
U D
With the optimal coupling, output power can be calculated:
Po I out Amod e Amod e I s
Ps
Ps
T opt A
T opt
T opt A
T opt
T opt A
T opt
[ 0 l ln( 1 T opt )]
[ 0 l ln( 1 T optt )]
[ 0 l T opt ]
Ps [ 0 l 2 A
0 lA
0 lA
0 lA A
Ps [ 0 l A 2 0 lA ] Ps [ 0 l
Ps [ g 0
]
A ]2
A ]2
EOP 505: Introduction to Lasers
Copyright © Qiwen Zhan
17

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