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EE 232: Lightwave Devices
Lecture #2 – Optical gain and laser
cavities
Instructor: Seth A. Fortuna
Dept. of Electrical Engineering and Computer Sciences
University of California, Berkeley
Gain Medium
Gain medium(𝐺)𝐼0 𝐼0𝑒
𝐺𝐿
𝐿
(
)( (
) ( ) (
)
)
)(
z
I I L I
I
L
z z
z G L
G
I
I
z
z
I
+
+
= −
= −
=
1G
I
LI=
Gain is the fractional increasein light intensity per unit length(Units are 𝑐𝑚−1)
Gain Medium (with internal loss)
Gain medium(𝐺, 𝛼𝑖)
𝐼0 𝐼0𝑒𝐺𝐿
𝐿
)( (
( ) ( )(
)
)
)
)
(
(
( )
i
i
I I L I
I L I
I G L
z z
z I z G z
z
= −
= −
=
+
+ −
−
1( )i
I
LG
I
−
=
Internal loss (𝛼𝑖) is the fractional decreasein light intensity per unit length (unrelated to fundamental absorption)(Units are 𝑐𝑚−1)
Gain Medium (with internal loss)
Gain medium(𝐺, 𝛼𝑖)
𝐼0 𝐼0𝑒𝐺𝐿
𝐿
1 1( )i
dIG
I
I
d IL z− →
=
(
0
)( ) iG z
I z I e−
=
Gain with cavityGain medium (𝐺, 𝛼𝑖)
𝐿
((o))Mirror 1 Mirror 2
2 1
)2(( 01
2 )e
( 0 )th iG LL
RI
I zR
z
−
+
+== =
+
=
Round-trip gain
(Threshold conditionfor self-sustainingoscillation)
𝑧
Gain with cavityGain medium (𝐺, 𝛼𝑖)
𝐿
((o))Mirror 1 Mirror 2
(Threshold conditionfor self-sustainingoscillation)
𝑧
2 1
1 1ln
2th i
m i
GRL R
=
= +
+
Photon lifetime (τ𝐩)Gain medium (𝐺, 𝛼𝑖)
𝐿
Mirror 1 Mirror 2
𝑧
𝑁𝑝
After one round trip: 1 2(1 ) pR R N− photons are lost from the cavity(ignoring 𝛼𝑖)
1 2( ) ( ) (1 )RT
R
p p P p
RTT
N t N t RN N
t
R
+ −=
−= −
/
0pt
p pN N e−
= where 1 2
2 /
1p
nL c
R R =
−(photon lifetime)
Quality factor (Q)
Energy stored in cavity
Energy lost per cycle2Q =
0
N2
2p
pp
p
p
QN T
Q
T
=
=
=
Gain medium (𝐺, 𝛼𝑖)
𝐿
Mirror 1 Mirror 2
𝑧
𝑁𝑝
Alternative expression for 𝑮𝒕𝒉Gain medium (𝐺, 𝛼𝑖)
𝐿
((o))Mirror 1 Mirror 2
𝑧
cG
nis the fractional increase in photons per second
01p
p
th p
p
th th
NcG N
n
c nG G
n Q c
=
= → =
Semiconductor laser
𝐿
Cleavedfacet
𝑧
21
~ 30% for 3.51
nR n
n
− = =
+
𝑛2
𝑛2
𝑛1Cleavedfacet
𝑛1(gain)
Semiconductor laser
𝐿
Cleavedfacet
𝑧
𝑛2
𝑛2
Cleavedfacet
Power of optical mode in gain regionConfinement factor:
Total power of optical mode =
𝑛1(gain)
01th
nG
Q c
=
Fabry-Perot cavity modes
𝐿
Mirror 1 (𝑟1, 𝑡1) Mirror 2 (𝑟2, 𝑡2)
𝑧
𝐸𝑖𝑛𝑐𝐸𝑡
Field just to the right of Mirror 1 and propagating in +z direction:
2 4 2 3
1 1 2 1 2 1 2 1 2(1 ( ) ...)jk L jk L jk L
incE E t r r e r r r r e r r e+ − − −= + + + +
1k
k aar
r
=−
1 2
1 2
1 where
1nc jiE LE t
r r ek
+
−=
−=
Evaluatefield here
𝐸+ is the field traveling to the right (+z direction)
Fabry-Perot cavity modes2
2
2
0
2
0
2
2
1 22
1 2
2
2 2
1 2
22
1 2
2 1 2 1
1 0
22
1 2
2 2
1
22
2 0
2
1 2
2Transmission
2
1
1
1
(1 ) (1 )
1
(1 )(1 )
1
v
tt
inc inc inc
j
inc
j
cavi
inc
ca ityty
j
j
E t
SIT
I S E
E t tr r e
E
t t
r r e
r r
r r e
r r
r r e
+
−
−
− −
−
−
= = =
−=
=−
− −=
−
− −=
−
0
1
1
2
: Impedance of medium outsid
o
e cavity
: Impedance of m
: Time-average Poynting vector (intensity)
: transmission of mirror 1
: reflectivity of mirror 1
: transmission of mi
edium inside cavity
rr
cavity
S
t
r
t
2
2 1 2
1 0 1
2 1 2
2 0 2
r 2
: reflectivity of mirror 2
1
1
cavity
cavity
r
t r
t r
−
−
+ =
+ =
Assume that𝑟1,2 and 𝑡1,2are real forsimplicity
Fabry-Perot cavity modes
2
2
2
2
2
2
2
2 2
1
22
1 2
2 2
1
2 2
1 2 1 2
2 2
1
2 2 2
1 2 1 2
2 2
1
2
1 2 1 2
2 2
1
2
1 2 1 2
2
1
(1 )(1 )
1
(1 )(1 )
(1 )(1 )
(1 )(1 )
1 (
(1 )(1 )
1 2 cos 2 (
(1 )(1 )
1 2 1 2sin (
(1 )(
)
)
)
1
j
j j
j j
r r
r r e
r r
r e r
r r
r r e e r
r r
r r r
r
r
r r e
r
r
r
r
r r
r r
−
−
−
− −=
−
− −=
− −
− −=
− + +
− −=
− +
− −=
− − +
− −=
2
2 2
1 2 1 2 1 2
2 2
1
2 2
1 2 1 2
2
)
1 2 4 sin (
(1 )(1 )
(1 ) 4 sin
)r r
r kL
r r r r
r rT
r r r
− + +
− −=
− +
0
1
1
2
: Impedance of medium outsid
o
e cavity
: Impedance of m
: Time-average Poynting vector (intensity)
: transmission of mirror 1
: reflectivity of mirror 1
: transmission of mi
edium inside cavity
rr
cavity
S
t
r
t
2
2 1 2
1 0 1
2 1 2
2 0 2
r 2
: reflectivity of mirror 2
1
1
cavity
cavity
r
t r
t r
−
−
+ =
+ =
Fabry-Perot cavity modes
𝐿
Mirror 1 Mirror 2
𝑧
𝐸𝑖𝑛𝑐
𝐸𝑡
Evaluatefield here
Resonance condition: 2
0 r
s 0
m = intege2
in kL m
mL
n
= → =
=
Fabry-Perot cavity modes
𝜆𝑚 𝜆𝑚+1 𝜆𝑚+2 𝜆𝑚+3 𝜆𝑚+4
Tra
nsm
issi
on
𝑟 = 0.5𝑟 = 0.7
𝑟 = 0.99
2
2nL
Fabry-Perot cavity modes
𝜆𝑚 𝜆𝑚+1 𝜆𝑚+2 𝜆𝑚+3 𝜆𝑚+4
Loss / G
ain(𝒄𝒎
−𝟏)
αm + α𝑖
Gain
e.g. with simple gain curveThreshold is achieved for one Fabry-Perot mode.
Tra
nsm
issi
on
Population inversion
2E
1E
Spontaneous emission Absorption Stimulated emission
2 21
2 1
21 1B WNWN B
N N
21
21
1
2
Spontaneous emission rate
Stimulated emission / absorption rate
Electromagnetic energy density
Number of electrons in ground state
N Number of electrons in excited state
N Number of photonsp
A
B W
W
N
=
=
=
=
=
=
Gain will occur if stimulated emission > absorptionwhich implies that number of electrons in the excited state is higher than the ground state (i.e. population inversion)
2-level system
2E
1E
21
2121 1 21 2
dNA N B WN B
dtWN= − + −
Not physical,population inversion not possiblein a 2-level system (at equilibrium)!
2 1
21 21 21
21 21 21 21
21 21 21
For population inversion
2
2
B W B W
B B
W A B W
N N
AN N
A W A W
B
+
+
+ +
2 211 2 1 21 2
At equilibrium,
0 A N B WN B WN= − + −
1 2
21 211
21 21
21
21 2
2
1
Let
2
2
N N N
AN
A
B W
WB
B W
N
N NA WB
+
+
+
=
=
+
=
3-level system
2E
1E
0E
pu
mp
τ small
Lasing transition
Population inversion (𝑁2 > 𝑁1) is possible if a state with energy smaller than 𝐸1 is added; providedelectrons quickly decay from state 1 into state 0.In other words, 𝐴10 must be larger than 𝐴21.
Gain in a semiconductor
𝐹𝑐
𝐹𝑣
𝑘𝑐
𝑘𝑣
) ) / ]
() ) / ]x
1( )
1 exp[( (
1)
1 e p[( ( v v
c c
c c
v v
kE k
f
fF kT
kF kTE k
−
−
=+
=+
(1 ) (1 )c v p v c p
pdNB f f N B f f N
dt = − − −
Stimulated emission Absorption
transition rate coeffic
sdensity of stat
ient
e
B
=
=
)0 for ( )1 (1p
c v v c
dNf f f f
dt − −
c vF F − Bernard-DuraffourgCondition
More on this as we progress in the course