Chapter 6.
Laser: Theory and Applications
Reading: Sigman, Chapter 6, 7, and 26
Bransden & Joachain, Chapter 15
Laser Basics
Light Amplification by Stimulated Emission of Radiation
E1
Stimulated emission
hν = E1 − E0
hν
E0
Population inversion (N1 > N0) ⇒ laser, maser
2
4π 2 ⎛ e 2 ⎞ I (ω01 )
Transition rate
⎜
⎟
W01 = 2 ⎜
M 01 (ω01 ) ∝ I (ω01 )
2
⎟
for stimulated emission
m c ⎝ 4πε 0 ⎠ ω01
Incident light intensity
Pumping (optical, electrical, etc.) for population inversion
High reflector
Gain medium
Optical cavity
Out coupler
Longitudinal Modes in an Optical Cavity
EM wave in a cavity
:
:
Boundary condition:
cτ
λ
πc
L= m=
m = m m = 1, 2, 3, …
2
2
ω
πc
2L
c
⇒ λ=
,ν =
m, ω =
m
m
2L
L
2L
= mτ
Round-trip time of flight: T =
c
Typical laser cavity: L = 1.5 m, λ = 0.75 µm
T=
2L
3m
−8
=
=
10
sec = 10 nsec
8
c
3 × 10 m / sec
1
= 108 Hz = 100 MHz
T
2L
3m
6
m=
=
=
4
×
10
= 4 milion !!
−6
λ 0.75 × 10 m
⇒ νR =
L
Single mode
I (t ) = cos ω0t = cos t
2
Intensity
Spectrum
1
Intensity
Frequency
-100
-50
0
Time
50
100
2
Cavity Quality Factors, Qc
L
End mirror
Out coupler
T=1~5%
R≈1
Energy loss by reflection, transmission, etc.
∞
E (ω ) = ∫ E0 e −δ ct / T e −iω0t dt
E0 e − δ c t / T e − iω 0 t
0
t
Emission spectrum
E02
E (ω ) =
(ω − ω0 ) 2 + δ c2 / T 2
2
E0
=
i (ω − ω0 ) + δ c / T
E (ω )
Lorentzian
2
~
δc
T
=
ω0
ω
Qc
ω
Energy of circulating EM wave, Icirc(t)
t⎤
⎡
2L
I circ (t ) = I circ (0) × exp⎢− δ c ⎥ , T =
: round-trip time of flight
c
T⎦
⎣
Number of round trips in t
⎡ ω ⎤
⇒ I circ (t ) = I circ (0) × exp⎢− t ⎥
⎣ Qc ⎦
Q-factor of a RLC circuit
ω ωL
ωT 4π L 1
=
=
Q
Q
=
=
where c
R
∆ω
δc
λ δc
Typical laser cavity: L = 1 m, λ = 0.8 µm, δc = 0.01 (~1% loss/RT)
4π
1
9
6
Qc =
1
.
6
10
≈
×
∆
ω
≈
10
Hz
−6
0.8 ×10 0.01
Two Level Rate Equations and Saturation
N2
E2
W21 N 2 γ 21 N 2
E1
N1
Two Level Rate Equation
W12 N1
1
γ 21 =
: non-radiative decay rate
T1
dN1 (t )
dN (t )
=− 2
dt
dt
= −W12 N1 (t ) + (W21 + γ 21 ) N 2 (t )
= −W12 [N1 (t ) − N 2 (t )] + γ 21 N 2 (t )
Total number of atoms: N = N1 (t ) + N 2 (t ) = constant
Population difference:
∆N (t ) = N1 (t ) − N 2 (t )
d
∆N (t ) = −2W12 [N1 (t ) − N 2 (t )] + 2γ 21 N 2 (t )
dt
= −2W12 [N1 (t ) − N 2 (t )] − γ 21 [N1 (t ) − N 2 (t ) − N1 (t ) − N 2 (t )]
d
∆N (t ) − N
∆N (t ) = −2W12 ∆N (t ) −
dt
T1
Steady-State Atomic Response: Saturation
d
∆N (t ) − N
∆N (t ) = 0 = −2W12 ∆N (t ) −
dt
T1
∆N ss
1
1
N
, Wsat ≡
=
∆N = ∆N ss =
1+ 2W12T
N
1 + W12 / Wsat
2T1
1.2
Gain coefficient
in laser materials
1.0
∆N ss
N
0.8
α m ∝ ∆N
0.6
α m0
α m (I ) =
0.4
Saturation
Intensity, I=Isat
0.2
0.0
0.01
0.1
1
Signal strength, 2W12T
10
1 + I / I sat
Transient Two-Level Solutions
⎛
d
1⎞
N
∆N (t ) − N
⎟
⎜
∆N (t ) = −2W12 ∆N (t ) −
= −⎜ 2W12 + ⎟∆N (t ) +
dt
T1
T1 ⎠
T1
⎝
⎡ ⎛
1⎞⎤
N
∆N (t ) = ∆N ss + A exp ⎢− ⎜⎜ 2W12 + ⎟⎟t ⎥, ∆N ss =
T1 ⎠ ⎦
1+ 2W12T
⎣ ⎝
Initial population difference at t = 0: ∆N (0) = ∆N ss + A
⎤
⎡
⎡
N
N
t⎤
∆N (t ) =
+ ⎢∆N (0) −
⎥ exp ⎢− (1 + 2W12T1 ) ⎥
1 + 2W12T ⎣
1 + 2W12T ⎦
T1 ⎦
⎣
1.0
∆N (t )
N
2W12T1 = 0
0.5
1
2
4
0.5
0.0
0
1
t / T1
Transient saturation behavior
following sudden turn-on of an
applied signal
2
3
Two-Level Systems with Degeneracy
Stimulated transition rates: g1W12 = g 2W21
N2
g2
E2
E1
Rate equation:
g1
N1
dN1 (t )
dN (t )
= − 2 = −W12 N1 (t ) + (W21 + γ 21 ) N 2 (t )
dt
dt
⎛ g2 ⎞
Population
∆N (t ) ≡ ⎜⎜ ⎟⎟ N1 (t ) − N 2 (t )
difference:
g
⎝
1
⎠
Effective signal-stimulated transition probability: Weff ≡
Rate equation:
1
(W12 + W21 )
2
d
∆N (t ) − N
∆N (t ) = −2Weff ∆N (t ) −
dt
T1
Atomic time constants: T1 and T2
T1 : longitudinal (on-diagonal) relaxation time
population recovery or energy decay time
T2 : dephasing time, transverse (off-diagonal) relaxation time
time constant for dephasing of coherent macroscopic polarization
Steady-State Laser Pumping and Population Inversion
Four-level pumping analysis
τ43 (fast)
E4
pump transition probability, W14 = W41 = W p
E3
pumping
Wp
τ32
(slow)
E2
τ21 (fast)
E1
Rate equation for level 4
dN 4
= W p ( N1 − N 4 ) − (γ 43 + γ 42 + γ 41 ) N 4
dt
= W p ( N1 − N 4 ) − N 4 / τ 4 ,
1
≡ γ 4 = γ 43 + γ 42 + γ 41
where
τ4
Steady state population:
N4 =
W pτ 4
1 + W pτ 4
N1 ≈ W pτ 4 N1 ,
if W pτ 4 << 1
Normalized pumping rate
Rate equations for level 2 and 3
at steady state
N
dN 3
N
= γ 43 N 4 − (γ 32 + γ 31 ) N 3 = 4 − 3
τ 43 τ 3
dt
N3 =
τ3
N4
τ 43
In a good laser system, τ 3 >> τ 43 so that N 3 >> N 4
dN 2
N 4 N3 N 2
= γ 42 N 4 + γ 32 N 3 − γ 21 N 2 =
+
−
dt
τ 42 τ 32 τ 21
⎛ τ 21 τ 43τ 21 ⎞
⎟⎟ N 3 = β N 3
+
N 2 = ⎜⎜
⎝ τ 32 τ 42τ 3 ⎠
where β ≡
τ43 (fast)
E4
pumping
Wp
E1
E3
τ32
(slow)
E2
τ21 (fast)
τ 21 τ 43τ 21
+
τ 32 τ 42τ 3
If β < 1, N 2 < N 3 : population inversion on the 3 → 2 transition
In a good laser system, γ 42 ≈ 0 so that the level 4 will relax primarily into the level 3.
β≈
τ 21
τ 32
condition for population inversion β ≡
N 2 τ 21
≈
<< 1
N 3 τ 32
Fluorescent quantum efficient
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
γ 43 γ rad τ 4 τ 3
η=
×
=
×
γ 4 γ 3 τ 43 τ rad
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 N = N1 + N 2 + N 3 + N 4
N
In a good laser system, τ rad >> τ 43 , β → 0.
ηW pτ rad
(1 − β )ηW pτ rad
N3 − N 2
≈
≈
N
1 + (1 + β )ηW pτ rad 1 + ηW pτ rad
∆N
(1 − β )ηW pτ rad
N3 − N 2
=
N
1 + [1 + β + 2τ 43 / τ rad ]ηW pτ rad
4-level system
∆Nth
0
3-level system
-N
0
1
Wpτrad
2
3
Laser gain saturation analysis
E3
N3
γ32
E2
N2
pumping
γ21
Wp
E1
E0
N0≈N
Pumping transition
dN 3
dt
stimulated
Wsig
N1
= Wp ( N0 − N3 ) ≈ Wp N0 ≈ Wp N
pump
Effective pumping rate:
quantum efficiency E3 → E2
γ1
Rate equations for laser levels 1 and 2:
Rp = η p Wp N0
γ 2 = γ 21 + γ 20
dN 2
= R p − Wsig ( N 2 − N1 ) − γ 2 N 2
dt
dN1
= Wsig ( N 2 − N1 ) + γ 21 N 2 − γ 1 N1
dt
N1 =
N2 =
Wsig + γ 21
Wsig (γ 1 + γ 20 ) + γ 1γ 2
Wsig + γ 1
Wsig (γ 1 + γ 20 ) + γ 1γ 2
Rp
Rp
Gain saturation behavior
⎛ γ 1 − γ 21 ⎞
1
⎟⎟ R p ×
∆N 21 = N 2 − N1 = ⎜⎜
1 + [(γ 1 + γ 20 ) / γ 1γ 2 ]Wsig
⎝ γ 1γ 2 ⎠
1
= ∆N 0
1 + Wsigτ eff
⎛ γ − γ 21 ⎞
⎛ τ1 ⎞
Small signal or unsaturated ∆N 0 = ⎜ 1
⎟⎟ R p = ⎜⎜1 −
⎟⎟ × R pτ 2
⎜
population inversion
⎝ γ 1γ 2 ⎠
⎝ τ 21 ⎠
Effective recovery time
If γ20 ≈ 0, γ2 ≈ γ21
1
τ eff
γ 1γ 2
=
γ 1 + γ 20
or
τ eff
⎛ τ1 ⎞
⎟⎟
= τ 2 ⎜⎜1 +
⎝ τ 20 ⎠
1
∆N 21 = R p (τ 2 − τ 1 ) ×
1 + Wsigτ 2
• Population inversion requires τ21 > τ1.
• ∆N 0 ∝ R p × τ 2 (1 − τ 1 / τ 2 )
• If τ1 → 0, τeff ≈ τ2.
• The saturation intensity of the inverted population is independent of Rp.
Wave Propagation in an Atomic Medium
Wave equation in a laser medium
[∇
2
]
+ ω µε (1 + χ at − iσ / ωε ) E ( x, y, z ) = 0
2
atomic
susceptibility
Ohmic loss
Plane wave approximation
⎡ d2
⎤
2
⎢ 2 + β (1 + χ at − iσ / ωε )⎥ E ( z ) = 0
⎣ dz
⎦
“Free-space” propagation constant: β = ω µε =
Propagation factor:
ω
c
n=
2π n
λ
E ( z ) = E 0 e − Γz
Γ 2 = − β 2 (1 + χ at − iσ / ωε )
Γ = iβ 1 + χ at − iσ / ωε = iβ 1 + χ ′(ω ) + iχ ′′(ω ) − iσ / ωε
Usually,
χ at , − iσ / ωε << 1
1
σ ⎤
⎡ 1
Γ ≈ iβ ⎢1 + χ ′(ω ) + i χ ′′(ω ) − i
2
2ωε ⎥⎦
⎣ 2
1
1
σ
= iβ + i βχ ′(ω ) − βχ ′′(ω ) +
2
2
2ε v
=a
χ ′(ω )
=
1
= iβ + i ∆β m (ω ) − α m (ω ) + α 0
2
Propagation of a +z traveling wave
a
χ at (ω ) =
ω 2 − ωa2 + iΓω
(ω
(ω
(ω
)
− ωa2 + iΓ ω
−ω
(
)
2 2
a
+ Γ 2ω 2
a ω 2 − ωa2
2
+i
χ ′′(ω )
2
2
−ω
(ω
)
2 2
a
)
+ Γ 2ω 2
aγ ω
2
−ω
)
2 2
a
+ γ 2ω 2
E ( z , t ) = Re E0 exp{iω t − i[β + ∆β m (ω )]z + [α m (ω ) − α 0 ]z}
Phase shift
by atomic transition
Gain by atomic transiton
+ ohmic loss
The effects of ohmic losses and atomic transition are included.
Propagation factors
φtot ( z , ω ) = [β + ∆β m (ω )]z
2π
“free-space”
Contribution
β(ω) z
total phase shift
∆β m ∝ χ ′(ω ) =
≈
atomic phase-shift contribution
α m (ω ) z
)
− ωa2 + Γ 2ω 2
2
∆β m (ω ) z
ω
ω
α m ∝ χ ′′(ω ) ≈
ωa
2
)
χ 0′ [2(ω − ωa ) / Γ]
2
1 + [2(ω − ωa ) / Γ ]
ωa
atomic gain
(ω
(
a ω 2 − ωa2
χ 0′′
2
1 + [2(ω − ωa ) / Γ ]
ω
Single-Pass Laser Amplification
z=0
Gain medium
z=L
Laser gain formulas
E ( L)
= exp{− i[β + ∆β m (ω )]L + [α m (ω ) − α 0 ]L}
Complex amplitude gain: g (ω ) ≡
E ( 0)
Total phase shift
Power or intensity gain: G (ω ) ≡
Lorenzian transition line shape:
amplitude
gain or loss
I ( L)
2
= g (ω ) = exp{2[α m (ω ) − α 0 ]L}
I ( 0)
1
= βχ ′′(ω )
2
χ 0′′ Midband value
χ ′′(ω ) =
2
1 + [2(ω − ωa ) / ∆ωa ]
⎫
⎧ω Lχ 0′′
1
Power gain: G (ω ) = exp ⎨
×
2⎬
1 + [2(ω − ωa ) / ∆ωa ] ⎭
⎩ v
Power gain in decibels (dB):
4.34ωa L
GdB (ω ) ≡ 10 log10 G (ω ) = 4.34 ln G (ω ) =
χ ′′(ω )
v
Amplification bandwidth and gain narrowing
3-dB amplifier bandwidth: ∆ω3 dB = ∆ωa
Amplifier phase shift
Total phase shift:
3
GdB (ωa ) − 3
φtot ( z , ω ) = [β + ∆β m (ω )]L =
ωL
v
+
βL
2
χ ′(ω )
Atomic transition phase shift:
⎛ ω − ωa ⎞
2(ω − ωa ) / ∆ωa
GdB (ωa )
⎜
⎟
∆β m (ω ) L = ⎜ 2
× α m (ω ) L =
×
2
⎟
∆
ω
20
log
e
1 + [2(ω − ωa ) / ∆ωa ]
10
a ⎠
⎝
Saturation Intensities in Laser Materials
Saturation of the population difference
Traveling wave:
dI
= 2α m I = ∆Nσ I
dz
Stimulated transition cross-section
Population difference:
1
1
∆N = ∆N 0 ×
= ∆N 0 ×
1 + Wτ eff
1 + I / I sat
2α m 0
1 dI ( z )
= 2α m ( z ) =
I ( z ) dz
1 + I ( z ) / I sat
∫
I = I out
I = I in
L
⎡1 1 ⎤
⎥ dI = 2α m 0 ∫0 dz
⎢ +
⎣ I I sat ⎦
⎛ I out
ln⎜⎜
⎝ I in
where 2α m 0 ≡ ∆N 0σ
unsaturated
power gain
⎞ I out − I in
⎟⎟ +
= 2α m 0 L = ln G0 where G0 ≡ exp(2α m 0 L)
I sat
⎠
Overall power gain:
⎡ I out − I in ⎤
I out
G≡
= G0 × exp ⎢−
⎥
I
I in
sat
⎣
⎦
⎡ (G − 1) I out ⎤
⎡ (G − 1) I in ⎤
= G0 × exp ⎢−
⎥
⎥ = G0 × exp ⎢−
GI
I
sat
sat
⎣
⎦
⎣
⎦
I out
1
⎛ G0 ⎞
ln⎜ ⎟
=
I sat G − 1 ⎝ G ⎠
and
I out
G
⎛ G0 ⎞
ln⎜ ⎟
=
I sat G − 1 ⎝ G ⎠
Power extraction and available power
I extr ≡ I out
I avail
⎛ G0 ⎞
− I in = I sat × ln⎜ ⎟
⎝G⎠
⎡
⎛ G0 ⎞⎤
≡ lim ⎢ I sat × ln⎜ ⎟⎥ = I sat × ln (G0 ) = 2α m 0 L × I sat
G →1
⎝ G ⎠⎦
⎣
Specific Laser Systems
Laser media:
gas, dye, chemical, excimer, solid-state, fiber, semiconductor, free-electron
http://en.wikipedia.org/wiki/Image:Laser_spectral_lines.png
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 nonradiative 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.
Energy level diagram of Ruby laser
http://web.phys.ksu.edu/vqm/laserweb/Ch-6/C6s2t1p2.htm
He-Ne Laser
Energy level diagram
Schematics
http://en.wikipedia.org/wiki/Helium-neon_laser
Excimer Lasers
Excimer Wavelenth
• 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.
Fe
157 nm
ArF
193 nm
KrF
248 nm
XeCl
308 nm
Applications
• Marking
• Micromachining
• Laser Ablation
• Laser Annealing
• Surface Structuring
• Laser Vision Correction
• Optical Testing and Inspection
• Pulsed Laser Deposition
• Fiber Bragg Gratings
http://tftlcd.khu.ac.kr/research/poly-Si/chapter4.html
Semiconductor Lasers – laser diodes
e
h
forward bias off
e
h
forward bias on
Band structure near a semiconductor p-n junction
Diode laser structure
Vertical cavity surface emitting lasers
(VCSEL) structure
en.wikipedia.org/wiki/Laser_diode, www.mtmi.vu.lt/pfk/funkc_dariniai/diod/led.htm
Laser Q-Switching
Q-switched laser output:
short and intense burst of laser output dumping all the accumulated
population inversion in a single short laser pulse (~10 ns)
loss
High initial cavity loss
loss
gain
loss
gain
loss
gain
t
Pumping process builds up a
large population inversion
t
Cavity loss is suddenly
“switched” to low value
t
laser
output
t
“giant pulse” laser action
takes place
Laser Q-switching techniques
Rotating mirror
Electro-optic
Acousto-optic
Saturable absorber
Active Q-switching: rate-equation analysis
N2
n(t )
E2
E1
N1 ≈ 0
Coupled rate equations
dn(t )
= KN (t )n(t ) − γ c n(t )
dt
dN (t )
= R p − γ 2 N (t ) − Kn(t ) N (t )
dt
N(t) = N2 − N1: inverted population
n(t) : cavity photon number
difference
γc = 1/τc : total cavity decay rate
Rp : pumping rate
γ2 : decay rate for N
K : coupling coefficient between photons and atoms
Pumping interval, and population build-up
loss
Pumping process builds up
a large population inversion
gain
t
n(t) = 0 and a pump pulse with constant intensity Rp is
turned on at t = 0.
dN (t )
= R p − N (t ) / τ 2
dt
⇒ N (t ) = R pτ 2 [1 − exp(−t / τ 2 )]
R pτ 2
Energy storage during
the pumping interval for
a fixed pumping rate
N(t)
2τ2
τ2
Pump Pulse Duration
Typical solid state
lasers: τ2 ≈ 200 µs
Pulse build-up time, Tb
loss
Tb
gain
Cavity loss is suddenly
“switched” to low value
t
Tb (~100 ns) << τ2 (∼100 µs)
N i Ni : population inversion just after switching
Initial inversion ratio: r ≡
N th Nth : threshold inversion just after switching
dn(t )
r −1
1
n( t )
= KN (t )n(t ) − n(t ) ≈ K [N i − N th ]n(t ) ≈
dt
τc
τc
⇒ n(t ) = ni exp[( r − 1)t / τ c ], 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
⎡ ( r − 1)Tb ⎤
nss
= exp⎢
⎥
τ
ni
c
⎣
⎦
⎛ nss ⎞
⇒ Tb =
ln⎜⎜ ⎟⎟
r − 1 ⎝ ni ⎠
τc
Ratio of initial to final photon numbers:
nss
≈ 108 to 1012
ni
Therefore, Tb
⎛ nss ⎞
±20 % in ln⎜⎜ ⎟⎟
⎝ ni ⎠
(
25 ± 5) T
≈
×
r −1
δc
where τ c =
T
2L
and T =
c
δc
Fractional power loss per round trip
Examples:
1. Flash-pumped Nd:YAG laser
L = 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 laser
L = 2 m (T = 12 ns), R=0.8 (δc = ln(1/R)=0.22), r = 1.5 ⇒ Tb ≈ 3 µs
Pulse output interval
gain
loss
dn(t )
= KN (t )n(t ) − γ c n(t ) = K [N (t ) − N th ]n(t )
dt
laser
output
t
dN (t )
= R p − γ 2 N (t ) − Kn(t ) N (t ) ≈ − Kn(t ) N (t )
dt
Initial conditions: N = N i = rN th and n = ni = 1 at the switching time t = ti
dn N th − N
=
dN
N
⎛ N th ⎞
⇒ ∫ dn = ∫ ⎜
− 1⎟dN
ni
Ni
⎝ N
⎠
N (t ) ⎛ N
N (t )
⎞
th
⇒ n(t ) ≈ ∫ ⎜
− 1⎟dN = N th ln
− ( N (t ) − N i )
Ni
Ni
⎝ N
⎠
Ni
Ni
Ni
where r =
⇒ n(t ) ≈ N i − N (t ) − ln
N th
r
N (t )
n(t )
N (t )
Time evolution of n(t) and N(t) during the pulse output interval
np
N (t )
Ni
n(t )
Nth
ni ≈ 0
nf ≈ 0
Nf
t
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 modelocked. Interference between these modes causes the laser light to
be produced as a train of pulses. Depending on the properties of
the 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
E (t ) = ∑ Em e
i ( ω m t +φm )
φm : initial phase of the mth mode
m
The laser is said to be “mode-locked” when the initial phases are equal.
φm = φ
: constant
Single mode
I (t ) = cos ω0t = cos t
2
Spectrum
Intensity
120
100
Intensity
2
80
1
Frequency
60
40
20
0
-100
-50
0
Time
50
100
Three modes in phase
I (t ) = cos 0.9t + cos t + cos1.1t
120
Spectrum
0.9 1.1
Intensity
100
Intensity
2
80
1
Frequency
60
δω = 0.1
2π
T=
= 20π
δω
40
20
0
-100
-50
0
Time
50
100
Eleven modes with random phases
I (t ) = cos(0.5t + φ0.5 ) + cos(0.6t + φ0.6 ) + ... cos(1.0t + φ1.0 ) + ... + cos(1.5t + φ1.5 )
120
Intensity
Spectrum
Intensity
100
80
0.5
1
Frequency
1.5
60
40
20
0
-100
-50
0
Time
50
100
2
Eleven modes with the same phase, φ = 0
I (t ) = cos 0.5t + cos 0.6t + ... cos1.0t + ... cos1.4t + cos1.5t
120
Intensity
Spectrum
Intensity
100
80
0.5
40
20
0
-50
0
Time
50
1
Frequency
60
-100
2
100
1.5
In the time domain, EM field is a periodic repetition: E (t ) = E (t + nT )
m
E (t + nT ) = ∑ Em e
m
mπ c
i
( t + nT )
L
m
mπ c
t
L
= ∑ Em e
i
mπ c ⎛ 2 nL ⎞
⎜ t+
⎟
L ⎝
c ⎠
m
1
Spectrum for one period: E (ω ) =
T
= ∑ Em e
i
=1
E ( t ) = ∑ E m e iω m t = ∑ E m e
i
mπ c
t
2π mni
L
e
= E (t )
m
∫
T
0
E (t )eiω t dt
Note Em = E (ω m )
Normalized spectrum for N periods:
1
E N (ω ) =
NT
∫
NT
0
1
E (t )eiω t dt =
NT
N −1 T
1
=
NT
∑∫
⎛1
= ⎜⎜
⎝N
N −1
n =0
0
∑e
n =0
E (t '+ nT )e
iω nT
N −1 ( n +1)T
∑∫
n =0
iω ( t ' + nT )
nT
E (t )e iω t dt
1
dt ' =
N
N −1
∑e
n =0
⎞
1 1 − e iω NT
E (ω )
⎟⎟ E (ω ) =
iω T
N 1− e
⎠
iω nT
1
T
∫
T
0
E (t ' )eiω t ' dt '
Normalized power spectrum for N periods:
2
I (ω ) = E (ω ) =
N
N
iω NT
1 1− e
N 2 1 − e iω T
2
E (ω )
2
1 sin 2 ( NωT / 2)
= 2
I (ω )
2
N sin (ωT / 2)
2π
T
1
δf ≈
NT
δω =
Periodic comb
sin 2 ( NωT / 2)
When N goes to infinity,
sin 2 (ωT / 2)
⇒
N →∞
2π n ⎞
⎛
δ ⎜ω −
⎟
∑
T ⎠
n = −∞ ⎝
Dirac comb
∞
Periodic repetition of EM wave in a laser cavity in the time domain
E (t ) = E (t + nT )
ω0
T
τp
Laser power spectrum I (ω )
δω =
2π π c
=
T
L
∆ω ≈
δf
ω0
1
τp
Spectrum
120
Intensity
Intensity
Single
mode
100
80
60
40
20
0
120
1
1.5
0.5
1
1.5
0.5
1
1.5
1
1.5
100
80
Intensity
Intensity
Three
modes
in phase
0.5
60
40
20
0
120
Intensity
Eleven modes
random phases
Intensity
100
80
60
40
20
0
120
80
Intensity
Eleven modes
same phase
Intensity
100
60
40
20
0
-100
-50
0
Time
50
100
0.5
Frequency
Net gain and cavity modes
Net gain spectrum
2π π c
δω =
=
T
L
∆ω =
2 2 ln 2 1
π
τp
… ω− 4 ω−3 ω− 2 ω−1 ω0 ω1 ω2 ω3 ω4 …
ωn = ω0 + nδω
Total electric field with phase-locked modes:
ω
E ( t ) = ∑ E n e iω n t
n
If the gain spectrum is a Gaussian distribution,
2
2
⎡
⎤
⎡
−
ω
ω
δω
n
⎛ n
⎛
⎞ ⎤
0 ⎞
En = E (ωn ) = E0 exp⎢− 4 ln 2⎜
⎟ ⎥ = E0 exp⎢− 4 ln 2⎜
⎟ ⎥
⎝ ∆ω ⎠ ⎥⎦
⎝ ∆ω ⎠ ⎥⎦
⎢⎣
⎢⎣
2
⎡
⎛ t − mT ⎞ ⎤
iω t
iω t
inδω t
⎟ ⎥
E (t ) = e ∑ E (ωn )e
= E0 e ∑ exp⎢− 2 ln 2⎜
⎟ ⎥
⎜ τ
⎢
n
m
p
⎠ ⎦
⎝
⎣
0
0
ω0
T
τp
t
Net gain spectrum
δω =
ωn = ω0 + nδω
2π π c
=
T
L
∆ω =
2 2 ln 2 1
π
τp
… ω− 4 ω−3 ω− 2 ω−1 ω0 ω1 ω2 ω3 ω4 …
For a laser cavity, L = 1.5m:
m
2
λ
2π π c
∆ω
δω =
=
= 6.28 ×108 Hz = 0.628 GHz , ∆λ ≈
2π c
T
L
Gain Medium
(wavelength)
Bandwidth
∆λ (nm)
Number of modes
m = ∆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
ω
Passive and Hybrid Mode-Locking
Passive mode-locking is accomplished by interplay between saturable
absorber and gain medium without any external modulation.
Saturable
gain medium
1
G0
Gain (G)
Transmission (T)
Saturable
absorber
1/2
G0/2
1
T0
Isabs
Intensity (I)
Isamp
Intensity (I)
Pulse shape modification by saturable absorber and gain medium
Saturable
absorber
Before
After
Saturable
gain medium
Before
Leading edge
absorption
Leading edge
amplification
After
Pulse shortening process due to saturation of the absorption and the gain
Loss
net gain > 0
Gain
Pulse peak is amplified
Mode-locked
pulse
Pulse tail is attenuated
t
Ar+ laser, 515 nm
Colliding-pulse mode-locked dye laser
Pulse
compressor
symmetric
amplification
G
SA
Gain
Rh6G
absorber
Saturable absorber
DODCI
gain
620 nm
~50 fs
dispersion
t
Kerr-Lens Mode-Locking (KLM)
Ti:Sapphire femtosecond oscillator
Band width
~ 200 nm
76 MHz
100 fs
13 nJ
M7
Ti:Sapphire
Low power Slit
High power
self-focusing: n = n0 + n2 I
OC
Low power
Nonlinear
media
Hard aperture
nonlinear medium + hard aperture
= Instantaneous saturable absorber :
High power
Transmission is lower at low intensity
compared to high intensity.
Intensity
Intensity
Intensity
Note: 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.
120
100
80
60
40
20
0
120
100
80
60
40
20
0
120
100
80
60
40
20
0
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.
Time