Chapter 6.
Laser: Theory and Applications
Reading: Sigman, Chapter 6, 7, and 26Bransden & Joachain, Chapter 15
Laser Basics
Stimulated emission01 EEh −=ν
1E
0Eνh
Population inversion (N1 > N0) ⇒ laser, maser
Light Amplification by Stimulated Emission of Radiation
Transition rate for stimulated emission
)()()(01
201012
01
01
0
2
2
2
01 44 ωω
ωω
πεπ IMIe
cmW ∝⎟⎟
⎠
⎞⎜⎜⎝
⎛=
Incident light intensity
Gain mediumHigh reflector Out coupler
Optical cavity
Pumping (optical, electrical, etc.) for population inversion
Longitudinal Modes in an Optical Cavity
Boundary condition:
::
L
EM wave in a cavity
mcmcmLωπτλ
===22
mLcm
Lc
mL πωνλ ===⇒ ,,
22
m = 1, 2, 3, …
Round-trip time of flight: τmcLT ==
2
Typical laser cavity: L = 1.5 m, λ = 0.75 µm
nsec secsec/m
m 1010103
32 88 ==
×== −
cLT
!! milion m.
m 410410750
32 66 =×=
×== −λ
Lm
MHz Hz 100101 8 ===⇒TRν
Single mode22
0 tttI coscos)( == ω
-100 -50 0 50 100Time
Inte
nsity
Frequency1
Inte
nsity
Spectrum
Cavity Quality Factors, Qc
End mirrorR ≈ 1
Out couplerT = 1 ~ 5 %
L
Energy loss by reflection, transmission, etc.
t
tiTt eeE c 0/0
ωδ −−
TiE
dteeEE
c
tiTtc
/)(
)(
0
0
0
/0
0
δωω
ω ωδ
+−=
= ∫∞ −−
2220
202
/)()(
TEE
cδωωω
+−=
Emission spectrum
ωω0
c
c
QTωδ
=~
2)(ωE Lorentzian
Energy of circulating EM wave, Icirc(t)
,exp)()( ⎥⎦⎤
⎢⎣⎡−×=
TtItI ccirccirc δ0
Number of round trips in t
: round-trip time of flightcLT 2
=
⎥⎦
⎤⎢⎣
⎡−×=⇒ t
QItI
ccirccirc
ωexp)()( 0
wherecc
cLTQ
δλπ
δω 14
==
Q-factor of a RLC circuit
RLQ ω
ωω
=∆
=
Typical laser cavity: L = 1 m, λ = 0.8 µm, δc = 0.01 (~1% loss/RT)9
6 10610101
10804
×≈×
= − ...
πcQ Hz610≈∆ω
[ ] )()()()()()(
)()(
2212112
22121112
21
tNtNtNWtNWtNW
dttdN
dttdN
γγ+−−=
++−=
−=
Two Level Rate Equation
Two Level Rate Equations and Saturation
1E
2E
221NW 221Nγ
2N
1N
112NW
: non-radiative decay rate1
211T
=γ
Total number of atoms: constant)()( 21 =+= tNtNN
Population difference: )()()( 21 tNtNtN −=∆
[ ][ ] [ ])()()()()()(2
)(2)()(2)(
2121212112
2212112
tNtNtNtNtNtNW
tNtNtNWtNdtd
−−−−−−=
+−−=∆
γ
γ
112
)()(2)(T
NtNtNWtNdtd −∆
−∆−=∆
Steady-State Atomic Response: Saturation
112
)()(20)(T
NtNtNWtNdtd −∆
−∆−==∆
TWNNN ss
1221+=∆=∆
112 21,
/11
TW
WWNN
satsat
ss ≡+
=∆
0.01 0.1 1 100.0
0.2
0.4
0.6
0.8
1.0
1.2
NNss∆
TW122Signal strength,
SaturationIntensity, I=Isat
Gain coefficientin laser materials
Nm ∆∝α
sat
mm II
I/1
)( 0
+=
αα
Transient Two-Level Solutions
1112
112 )(12)()(2)(
TNtN
TW
TNtNtNWtN
dtd
+∆⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
−∆−∆−=∆
,12exp)(1
12 ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+−+∆=∆ t
TWANtN ss
TWNNss
1221+=∆
Initial population difference at t = 0: ANN ss +∆=∆ )0(
( ) ⎥⎦
⎤⎢⎣
⎡+−⎥
⎦
⎤⎢⎣
⎡+
−∆++
=∆1
1121212
21exp21
)0(21
)(TtTW
TWNN
TWNtN
0 1 2 30.0
0.5
1.0
NtN )(∆
1/Tt
Transient saturation behavior following sudden turn-on of an applied signal
02 112 =TW
5.0124
Two-Level Systems with Degeneracy
1E
2E 2g
1N1g
2N
)()()( 211
2 tNtNggtN −⎟⎟
⎠
⎞⎜⎜⎝
⎛≡∆Population
difference:
Stimulated transition rates: 212121 WgWg =
)()()()()(22121112
21 tNWtNWdt
tdNdt
tdN γ++−=−=
Rate equation:
Effective signal-stimulated transition probability: ( )211221 WWWeff +≡
1
)()(2)(T
NtNtNWtNdtd
eff−∆
−∆−=∆Rate equation:
Atomic time constants: T1 and T2
T1 : longitudinal (on-diagonal) relaxation timepopulation recovery or energy decay time
T2 : dephasing time, transverse (off-diagonal) relaxation timetime constant for dephasing of coherent macroscopic polarization
Rate equation for level 4
pWWW == 4114
,/)(
)()(
4441
4414243414
τ
γγγ
NNNW
NNNWdt
dN
p
p
−−=
++−−=
41424344
1 γγγγτ
++=≡where
pump transition probability,
Steady-State Laser Pumping and Population Inversion
Four-level pumping analysis
E1
E2
E3
E4 τ43 (fast)
τ32(slow)
τ21 (fast)
Wp
pumping
Steady state population:
,1 141
4
44 NWN
WW
N pp
p ττ
τ≈
+= if 14 <<τpW
Normalized pumping rate
21
2
32
3
42
4221332442
2
τττγγγ NNNNNN
dtdN
−+=−+=
33342
2143
32
212 NNN β
ττττ
ττ
=⎟⎟⎠
⎞⎜⎜⎝
⎛+=
342
2143
32
21
ττττ
ττβ +≡where
If β < 1, 32 NN < : population inversion on the 3 → 2 transition
In a good laser system, 43 NN >>433 ττ >> so that
Rate equations for level 2 and 3
3
3
43
433132443
3 )(ττ
γγγ NNNNdt
dN−=+−=
at steady state
443
33 NN
ττ
=
E1
E2
E3
E4 τ43 (fast)
τ32(slow)
τ21 (fast)
Wp
pumping
In a good laser system, 042 ≈γ so that the level 4 will relax primarily into the level 3.
32
21
ττβ ≈ condition for population inversion 1
32
21
3
2 <<≈≡ττβ
NN
Fluorescent quantum efficient
rad
rad
ττ
ττ
γγ
γγη 3
43
4
34
43 ×=×=
The number of fluorescent photons spontaneously emitted on the laser transition divided by the number of pump photons absorbed on the pump transitions when the laser material is below threshold
Fraction of the total atoms excited to level 4 relax directly into the level 3
Fraction of the total decay out of level 3 is purely radiative decay to level2
Four level population inversion
[ ] radprad
radp
WW
NNN
τηττβτηβ
/211)1(
43
23
+++
−=
−
4321 NNNNN +++=
In a good laser system, 43ττ >>rad
radp
radp
radp
radp
WW
WW
NNN
τητη
τηβτηβ
+≈
++
−≈
−1)1(1
)1(23
, β → 0.3-level system
0 1 2 3
-N
0
N
∆N
Wpτrad
4-level system
∆Nth
Laser gain saturation analysis
E0
E1
E2
E3 γ32
γ21
γ1
Wp
pumping
N0≈N
N1
N2
N3
Wsig
stimulated
Pumping transition
NWNWNNWdt
dNppp
pump
≈≈−= 0303 )(
0NWR ppp η=Effective pumping rate:
quantum efficiency E3 → E2
22122 )( NNNWR
dtdN
sigp γ−−−=
Rate equations for laser levels 1 and 2:
11221121 )( NNNNW
dtdN
sig γγ −+−=
20212 γγγ +=
psig
sig RW
WN
21201
211 )( γγγγ
γ++
+=
psig
sig RW
WN
21201
12 )( γγγγ
γ++
+=
Gain saturation behavior
[ ]
effsig
sigp
WN
WRNNN
τ
γγγγγγγγ
+∆=
++×⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=−=∆
11
/)(11
0
2120121
2111221
Small signal or unsaturated population inversion
221
1
21
2110 1 τ
ττ
γγγγ
pp RRN ×⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=∆
Effective recovery time201
211γγ
γγτ +
=eff
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
20
12 1
ττττ effor
If γ20 ≈ 0, γ2 ≈ γ212
1221 11)(
τττ
sigp W
RN+
×−=∆
• Population inversion requires τ21 > τ1.
•• If τ1 → 0, τeff ≈ τ2.
• The saturation intensity of the inverted population is independent of Rp.
)/1( 2120 τττ −×∝∆ pRN
Wave Propagation in an Atomic Medium
Wave equation in a laser medium
( )[ ] 0),,(/122 =−++∇ zyxEiat ωεσχµεωOhmic lossatomic
susceptibility
( ) 0)(/122
2
=⎥⎦
⎤⎢⎣
⎡−++ zEi
dzd
at ωεσχβ
Plane wave approximation
“Free-space” propagation constant:λπωµεωβ nn
c2
===
Propagation factor:
( )ωεσχβ /122 iat −+−=Γ
zeEzE Γ−= 0)(
ωεσωχωχβωεσχβ /)()(1/1 iiiii at −′′+′+=−+=Γ
1/, <<− ωεσχ iatUsually,
0)()(21
2)(
21)(
21
2)(
21)(
211
αωαωββ
εσωχβωχββ
ωεσωχωχβ
+−∆+=
+′′−′+=
⎥⎦⎤
⎢⎣⎡ −′′+′+≈Γ
mmii
vii
iii
Propagation of a +z traveling wave
[ ] [ ]{ }zzitiEtzE mm 00 )()(expRe),( αωαωββω −+∆+−=
The effects of ohmic losses and atomic transition are included.
Phase shiftby atomic transition
Gain by atomic transiton+ ohmic loss
( )( )
( )( )
( ) 22222
22222
22
22222
22
22)(
ωγωωωγ
ωωωωω
ωωωωωω
ωωωωχ
+−+
Γ+−
−=
Γ+−
Γ+−=
Γ+−=
a
a
a
a
a
aat
ai
a
ia
ia
)(ωχ′
)(ωχ ′′
atomic phase-shift contribution zm )(ωβ∆
aωω
Propagation factors
[ ]zz mtot )(),( ωββωφ ∆+=
π2total phase shift
“free-space”Contributionβ(ω) z
ω
( )( )
[ ]( )[ ]2
0
22222
22
/21/)(2
)(
Γ−+Γ−′
≈
Γ+−
−=′∝∆
a
a
a
am
a
ωωωωχ
ωωωωωωχβ
aω
atomic gain zm )(ωα
ω( )[ ]2
0
/21)(
Γ−+
′′≈′′∝
am ωω
χωχα
Single-Pass Laser Amplification
Gain medium0=z Lz =
Laser gain formulas
Complex amplitude gain: [ ] [ ]{ }LLiE
LEg mm 0)()(exp)0()()( αωαωββω −+∆+−=≡
Total phase shift amplitude gain or loss
Power or intensity gain: ( ) [ ]{ }LgI
LIG m 02 )(2exp
)0()()( αωαωω −==≡
)(21 ωχβ ′′=
Lorenzian transition line shape: [ ]20
/)(21)(
aa ωωωχωχ
∆−+
′′=′′
Midband value
Power gain:[ ] ⎭
⎬⎫
⎩⎨⎧
∆−+×
′′= 2
0
/)(211exp)(
aavLG
ωωωχωω
Power gain in decibels (dB):
)(34.4)(ln34.4)(log10)( 10 ωχωωωω ′′==≡v
LGGG adB
Amplification bandwidth and gain narrowing
3-dB amplifier bandwidth:3)(
33 −
∆=∆adB
adB G ωωω
Amplifier phase shift
[ ] )(2
)(),( ωχβωωββωφ ′+=∆+=L
vLLz mtotTotal phase shift:
[ ]210 /)(21
/)(2log20
)()(2)(aa
aaadBm
a
am e
GLLωωω
ωωωωωαωωωωβ
∆−+∆−
×=×⎟⎟⎠
⎞⎜⎜⎝
⎛∆−
=∆
Atomic transition phase shift:
Saturation Intensities in Laser Materials
Saturation of the population difference
sateff IIN
WNN
/11
11
00 +×∆=
+×∆=∆
τ
INIdzdI
m σα ∆== 2Traveling wave:Stimulated transition cross-section
Population difference:
sat
mm IzI
zdz
zdIzI /)(1
2)(2)()(
1 0
+==
αα σα 002 Nm ∆≡where
∫∫ =⎥⎦
⎤⎢⎣
⎡+
=
=
L
m
II
IIsat
dzdIII
out
in 00211 α
00 ln2ln GLI
IIII
msat
inout
in
out ==−
+⎟⎟⎠
⎞⎜⎜⎝
⎛α where )2exp( 00 LG mα≡
unsaturatedpower gain
⎥⎦
⎤⎢⎣
⎡ −−×=⎥
⎦
⎤⎢⎣
⎡ −−×=
⎥⎦
⎤⎢⎣
⎡ −−×=≡
sat
out
sat
in
sat
inout
in
out
GIIGG
IIGG
IIIG
IIG
)1(exp)1(exp
exp
00
0
Overall power gain:
⎟⎠⎞
⎜⎝⎛
−=
GG
GII
sat
out 0ln1
1⎟⎠⎞
⎜⎝⎛
−=
GG
GG
II
sat
out 0ln1
and
Power extraction and available power
⎟⎠⎞
⎜⎝⎛×=−≡
GGIIII satinoutextr
0ln
( ) satmsatsatGavail ILGIGGII ×=×=⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛×≡
→ 000
12lnlnlim α
Specific Laser SystemsLaser media: gas, dye, chemical, excimer, solid-state, fiber, semiconductor, free-electron
http://en.wikipedia.org/wiki/Image:Laser_spectral_lines.png
Ruby Laser
http://web.phys.ksu.edu/vqm/laserweb/Ch-6/C6s2t1p2.htmEnergy level diagram of Ruby laser
• This system is a three level laser with lasing transitions between E2 and E1. • The excitation of the Chromium ions is done by light pulses from flash lamps (usually Xenon). • The Chromium ions absorb light at wavelengths around 545 nm (500-600 nm). As a result the ions are transferred to the excited energy level E3. • From this level the ions are going down to the metastable energy level E2 in a non-radiative transition. The energy released in this non-radiative transition is transferred to the crystal vibrations and changed into heat that must be removed away from the system. • The lifetime of the metastable level (E2) is about 5 msec.
He-Ne Laser
Energy level diagram
Schematics http://en.wikipedia.org/wiki/Helium-neon_laser
Excimer Lasers
Applications• Marking • Micromachining • Laser Ablation • Laser Annealing • Surface Structuring • Laser Vision Correction • Optical Testing and Inspection • Pulsed Laser Deposition • Fiber Bragg Gratings
• Gain medium: inert gas (Ar, Kr, Xe etc.) + halide (Cl, F etc.)
• Excited state is induced by an electrical discharge or high-energy electron beams.
• Laser action in an excimer molecule occurs because it has a bound (associative) excited state, but a repulsive (disassociative) ground state.
Excimer WavelenthFe 157 nm
ArF 193 nmKrF 248 nm
XeCl 308 nm
http://tftlcd.khu.ac.kr/research/poly-Si/chapter4.html
Semiconductor Lasers – laser diodes
Vertical cavity surface emitting lasers (VCSEL) structure
Band structure near a semiconductor p-n junction
e h e h
forward bias off forward bias on
Diode laser structure
en.wikipedia.org/wiki/Laser_diode, www.mtmi.vu.lt/pfk/funkc_dariniai/diod/led.htm
Laser Q-SwitchingQ-switched laser output: short and intense burst of laser output dumping all the accumulated population inversion in a single short laser pulse (~10 ns)
High initial cavity lossloss
gain
loss
loss
lossgain
gain
laseroutput
t
t
tPumping process builds up a large population inversion
t
Cavity loss is suddenly “switched” to low value
“giant pulse” laser action takes place
Laser Q-switching techniques
Rotating mirror
Electro-optic
Acousto-optic
Saturable absorber
Active Q-switching: rate-equation analysis
2E
1E01 ≈N
2N)(tn
Coupled rate equations
)()()()( tNtKntNRdt
tdNp −−= 2γ)()()()( tntntKN
dttdn
cγ−=
N(t) = N2 − N1: inverted population difference
Rp : pumping rateγ2 : decay rate for N
n(t) : cavity photon numberγc = 1/τc : total cavity decay rate
K : coupling coefficient between photons and atoms
Pumping interval, and population build-up
gaint
lossPumping process builds up a large population inversion
n(t) = 0 and a pump pulse with constant intensity Rp is turned on at t = 0.
[ ])/exp()(/)()(222 1 τττ tRtNtNR
dttdN
pp −−=⇒−=
Pump Pulse Durationτ2 2τ2
2τpR
N(t)
Energy storage during the pumping interval for a fixed pumping rate
Typical solid state lasers: τ2 ≈ 200 µs
Pulse build-up time, Tb
Initial inversion ratio:th
i
NNr ≡
Ni : population inversion just after switchingNth : threshold inversion just after switching
[ ]
[ ],/)(exp)(
)()()()()()(
ci
cthi
c
trntn
tnrtnNNKtntntKNdt
tdn
τττ
1
11
−=⇒
−≈−≈−=
loss
gaint
Cavity loss is suddenly “switched” to low value
Tb
Tb (~100 ns) << τ2 (∼100 µs)
ni ≈ 1 : initial spontaneous emission noise excitation
Steady state photon number, nss : • Photon number with continuous pumping at a pumping rate r times above its threshold
• Stimulate emission (KN(t)n(t)) becomes significant.• nss << np (photon number at the peak power)
Build-up time, Tb from the initial photon density ni to a photon density nss
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⇒⎥⎦
⎤⎢⎣
⎡ −=
i
sscb
c
b
i
ss
nn
rTTr
nn ln)(exp
11 τ
τ
Ratio of initial to final photon numbers:128 1010 to≈
i
ss
nn ±20 % in ⎟⎟
⎠
⎞⎜⎜⎝
⎛
i
ss
nnln
( )c
bT
rT
δ×
−±
≈1525
cc
Tδ
τ =wherecLT 2
=
Fractional power loss per round trip
Therefore, and
Examples:
1. Flash-pumped Nd:YAG laserL = 60 cm (T = 4 ns), R=0.35 (δc = ln(1/R)=1.05), r = 3 ⇒ Tb ≈ 50 ns
2. cw-pumped Nd:YAG laser or low-gain CO2 laserL = 2 m (T = 12 ns), R=0.8 (δc = ln(1/R)=0.22), r = 1.5 ⇒ Tb ≈ 3 µs
Pulse output interval
[ ] )()()()()()( tnNtNKtntntKNdt
tdnthc −=−= γ
loss
gainlaseroutput
t )()()()()()( tNtKntNtKntNRdt
tdNp −≈−−= 2γ
at the switching time t = tiInitial conditions: thi rNNN == 1== innand
)(ln)()(
))(()(ln)()(
)()(
tNN
rNtNNtn
NtNN
tNNdNN
Ntn
dNN
NdnN
NNdNdn
iii
ii
th
tN
Nth
tN
Nthtn
nth
i
ii
−−≈⇒
−−=⎟⎠⎞
⎜⎝⎛ −≈⇒
⎟⎠⎞
⎜⎝⎛ −=⇒
−=
∫
∫∫
1
1
th
i
NNr =where
Time evolution of n(t) and N(t) during the pulse output interval
Ni
Nf
Nth
ni ≈ 0 nf ≈ 0
np
t
)(tN
)(tn
Mode-Locking
Mode-locking is a technique in optics by which a laser can be made to produce ultrashort pulses with the pulse width of the order of subpicoseconds (< 10-12). The basis of the technique is to induce a fixed phase relationship between the modes of the laser's resonant cavity. The laser is then said to be phase-locked or mode-locked. Interference between these modes causes the laser light to be produced as a train of pulses. Depending on the properties ofthe laser, these pulses may be of extremely brief duration, as short as a few femtoseconds. - Wikipedia
General description of electric field in a laser cavity
∑ +=m
tim
mmeEtE )()( φω
The laser is said to be “mode-locked” when the initial phases are equal.
φm : initial phase of the mth mode
φφ =m : constant
Single mode22
0 tttI coscos)( == ω
-100 -50 0 50 1000
20406080
100120
Time
Inte
nsity Frequency
1
Inte
nsity
Spectrum
Three modes in phase21190 ttttI .coscos.cos)( ++=
-100 -50 0 50 1000
20406080
100120
Time
Inte
nsity Frequency
1
Inte
nsity
Spectrum0.9 1.1
πδω
πδω
20210
==
=
T
.
Eleven modes with random phases2
51016050 51016050 ).cos(...).cos(...).cos().cos()( .... φφφφ ++++++++= tttttI
-100 -50 0 50 1000
20406080
100120
Time
Inte
nsity Frequency
1
Inte
nsity
Spectrum
0.5 1.5
Eleven modes with the same phase, φ = 025141016050 ttttttI .cos.cos....cos....cos.cos)( ++++=
-100 -50 0 50 1000
20406080
100120
Time
Inte
nsity Frequency
1
Inte
nsity
Spectrum
0.5 1.5
In the time domain, EM field is a periodic repetition: )()( TntEtE +=
∑∑ ==m
tL
cmi
mm
tim eEeEtE m
πω)(
)()()(
tEeeEeEeEnTtE mni
m
tL
cmi
mm m
cnLt
Lcmi
m
nTtL
cmi
m ====+ ∑∑ ∑⎟⎠⎞
⎜⎝⎛ ++ π
πππ2
2 =1
dtetET
ET ti∫=0
1 ωω )()( )( mm EE ω=Spectrum for one period: Note
Normalized spectrum for N periods:
)()(
')'(')'(
)()()(
')'(
)(
ωω
ω
ω
ωω
ωωω
ωω
Eee
NEe
N
dtetET
eN
dtenTtENT
dtetENT
dtetENT
E
Ti
NTiN
n
nTi
N
n
T tinTiN
n
T nTti
N
n
Tn
nT
tiNT tiN
−−
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
=+=
==
∑
∑ ∫∑∫
∑∫∫
−
=
−
=
−
=
+
−
=
+
1111
111
11
1
0
1
00
1
00
1
0
1
0
Normalized power spectrum for N periods:
)()/(sin)/(sin
)()()(
ωωω
ωωω ω
ω
ITTN
N
Eee
NEI Ti
NTiNN
221
111
2
2
2
22
2
2
=
−−
==
Periodic comb
Tπδω 2
=
NTf 1
≈δ
When N goes to infinity, ∑∞
−∞=∞→⎟⎠⎞
⎜⎝⎛ −⇒
nN Tn
TTN πωδ
ωω 2
22
2
2
)/(sin)/(sin
Dirac comb
Periodic repetition of EM wave in a laser cavity in the time domain
)()( TntEtE +=
T
0ω
pτ
Laser power spectrum )(ωI
0ω
pτω 1
≈∆
Lc
Tππδω ==
2
fδ
-100 -50 0 50 100
Time
020406080
100120
Inte
nsity
020406080
100120
Inte
nsity
020406080
100120
Inte
nsity
020406080
100120
Inte
nsity
Frequency1
Inte
nsity
0.5 1.5
1
Inte
nsity
0.5 1.5
Inte
nsity
Inte
nsity
Spectrum
10.5 1.5
10.5 1.5
Eleven modesrandom phases
Eleven modessame phase
Singlemode
Threemodesin phase
Net gain and cavity modes
Total electric field with phase-locked modes:
If the gain spectrum is a Gaussian distribution,
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
∆−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
∆−
−==2
0
20
0 2424ω
δωωωωω nEEEE n
nn lnexplnexp)(
∑∑ ⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−==
m p
ti
n
tinn
ti mTteEeEetE2
0 2200
τω ωδωω lnexp)()(
∑=n
tin
neEtE ω)(
0ω
pτπω 1222 ln
=∆Lc
Tππδω ==
2
ω1ω 3ω2ω 4ω1−ω2−ω4−ω 3−ω
Net gain spectrum
δωωω nn += 0……
T0ω
pτt
For a laser cavity, L = 1.5m:
pτπω 1222 ln
=∆Lc
Tππδω ==
2
ω
Net gain spectrum
δωωω nn += 0
……
m0ω 1ω 3ω2ω 4ω1−ω2−ω4−ω 3−ω
ωπλλ ∆≈∆
c2
2
Gain Medium(wavelength)
Bandwidth∆λ (nm)
Number of modesm = ∆ω / δω
Minimum Pulse duration, τp
Ar+-ion (520 nm) ~ 0.007 ~ 80 ~ 150000 fs
Rubi (694.3 nm) ~ 0.2 ~ 1300 ~ 6000 fs
Nd:YAG (1064 nm) ~ 10 ~ 25000 ~ 120 fs
Dye (620 nm) ~ 100 ~ 8×105 ~ 12 fs
Ti:Sapphire (800 nm) ~ 400 ~2 ×106 ~ 3 fs
,.. GHzHzLc
T6280102862 8 =×===
ππδω
Passive and Hybrid Mode-LockingPassive mode-locking is accomplished by interplay between saturable absorber and gain medium without any external modulation.
Saturableabsorber
Saturablegain medium
T0
1
Intensity (I)Isabs
1/2
Tran
smis
sion
(T)
1
G0
Intensity (I)Isamp
G0/2G
ain
(G)
Pulse shape modification by saturable absorber and gain medium
Saturable absorber
Leading edgeabsorption
Before After
Leading edgeamplificationSaturable
gain medium
Before After
Pulse shortening process due to saturation of the absorption and the gain
Loss
Gainnet gain > 0
Pulse peak is amplified
Pulse tail is attenuated
Mode-lockedpulse
t
Colliding-pulse mode-locked dye laser
GSA
Ar+
lase
r, 51
5 nm Pulse
compressor
GainRh6G
Saturable absorberDODCI
620 nm~50 fs
symmetric amplification
absorber gain dispersion
t
Kerr-Lens Mode-Locking (KLM)
Ti:Sapphire
High power
Low power
self-focusing:
SlitM7 OC
Ti:Sapphire femtosecond oscillator
76 MHz100 fs13 nJ
Band width~ 200 nm
Innn 20 +=
High power
Low power Hard aperturenonlinear medium + hard aperture= Instantaneous saturable absorber :Transmission is lower at low intensity compared to high intensity.Nonlinear
mediaNote: Sometimes a hard aperture is not necessary, because larger overlap between the pump beam and the cavity mode in the gain medium at high intensity leads to larger modal gain.
Time
020406080
100120
Inte
nsity
020406080
100120
Inte
nsity
020406080
100120
Inte
nsity 1. Mechanical shocks (e.g., vibrating starter,
jolting mirror) initiate mode-locking.
2. The pulse regime is favored over the continuous regime.
3. Spectral broadening by SPM supports the pulse shortening.