nonlinear optics lab. hanyang univ. chapter 11. laser oscillation : power and frequency power &...
TRANSCRIPT
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.
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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.
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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 =>
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
<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%.
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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 !
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
<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.
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
<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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
=> 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.
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
<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).
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
<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.
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
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 :
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Steady-state solution :
2
2
)(
)(
Ng
Nq
t
define,t
T
t N
Ny
N
Nx
and2
x
xq
1 satsat 11 qq
y
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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).
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
3) Fabry-Perot etalon / Grating / Prism
=> Selective transmission
Ex) Fabry-Perot etalon
- resonance frequency : ...,3,2,1,cos2
mnd
cmm
cos21 nd
cmm