Download Linear pulse propagation

Ultrafast Laser Physics
Ursula Keller / Lukas Gallmann
ETH Zurich, Physics Department, Switzerland
www.ulp.ethz.ch
Linear pulse propagation
Ultrafast Laser
Physics
ETH Zurich
Superposition of many monochromatic waves
U. Keller, ULP, Chap. 2, p. 1
Fourier Transformation
Plane wave, monochromatic
E(z,t) = E 0 e i ωt – kz
Inverse Fourier Transformation (for every position in space)
1 
+iω t
E
z,
E ( z,t ) = F −1 E ( z, ω ) =
ω
e
dω
(
)
2π ∫
{
E (t ) =
}
E (ω 1 ) +
E (ω 2 ) +
…
E (t ) =
1 
iω t
E
ω
e
dω
(
)
∫
2π
Fourier Transformation
E ( z, ω ) ≡ F { E ( z,t )} = ∫ E ( z,t ) e−iω t dt
U. Keller, ULP, Chap. 2, p. 3
Superposition of monochromatic waves
~
Spektrum A(Δω)
(a)
Frequenz Δω
Überlagerung der ebenen Wellen
Ebene Wellen
(b)
(c)
t
t
t
t
Light pulse
(d)
t
Wave packet
t
E (t ) =
1 
iω t
E
ω
e
dω
(
)
∫
2π
U. Keller, ULP, Chap. 2, p. 8-10
Monochromatic plane wave
E ( z, t ) = E0 ei(ω t−kn z )
Vacuum
Frequency:
Period:
Dispersive material
ν
ν
T =1 ν
Phase
velocity:
υp = c
Wave number: k = ω
c
2π
k=
λ
Wavelength:
λ
T =1 ν
υ p = cn = c n
ω ω
kn =
= n = kn
υp c
2π
kn =
= kn
λn
λ
λn =
n
U. Keller, ULP, Chap. 1
Optical Dispersion
U. Keller, ULP, Chap. 1
Helmholtz equation
E(z,t) = E 0 e i ωt – kz
Fourier Transformation
E ( z, ω ) ≡ F { E ( z,t )} = ∫ E ( z,t ) e−iω t dt
∂2
∂2
1 ∂2
E ( z,t ) − 2 2 E ( z,t ) = µ0 2 P ( z,t )
∂ z2
∂t
c ∂t
∂2
⇔ −ω 2
∂t 2
1 
E ( z, ω ) e+iω t dω
E ( z,t ) = F −1 E ( z, ω ) =
∫
2π
∂ 2 E ( z, ω ) ω 2 
2 
+
ω
=
−
µ
ω
E
z,
P ( z, ω )
(
)
0
∂ z2
c2
∂
∂t
2
2
∂
∂
⎛
⎞
ω 2 ⇔ − 2 = ⎜ −i ⎟
⎝ ∂t ⎠
∂t
P ( z, ω ) = χ (ω ) ε 0 E ( z, ω ) = ⎡⎣ε (ω ) − 1⎤⎦ ε 0 E ( z, ω )
{
}
ω ⇔ −i
∂3 ⎛ ∂ ⎞
ω ⇔ i 3 = ⎜ −i ⎟
⎝ ∂t ⎠
∂t
3
3
kn (ω ) =
ω
n (ω )
c
∂ 2 E ( z, ω )
2
E ( z, ω ) = 0
+
k
ω
(
)
⎡
⎤
n
⎣
⎦
2
∂z
E ( z, ω ) = E 0+ (ω ) e−ikn (ω ) z + E 0– (ω ) eikn (ω ) z
U. Keller, ULP, Chap. 2, p. 5
Analogy to the Schrödinger equation
Helmholtz equation:
Time independent Schrödinger equation:
Wave equation in the spectral domain
free particle
∂ E ( z, ω )
2
+
k
ω
⎡⎣ n ( ) ⎤⎦ E ( z, ω ) = 0
2
∂z
2
∂ 2ψ
2
+
k
ψ =0
2
∂x
particle in a potential field
∂ 2ψ 2m
2 +
2 E − E p ( x) ψ = 0

∂x
[
]
U. Keller, ULP, Chap. 2, p. 5
Dispersion for quantum mechanical particles
Photon in vacuum:
E = cp ⇔ ω ( k ) = ck
Photon in medium:
E = cn pn
Free electron:
p2
E=
2m
⇔ ω ( k ) = cn kn
2 2
⇔ ω (k) =
k
2m
υp =
υg =
Electron in conduction band:
ω k
=
k 2m
dω k p mυ klass
=
= =
= υ klass
dk
m
m m
E ( k ) ≈ E at − 2Es cos ka ⇔ ω ( k ) ≈ ω at − 2ω s cos ka
E
±kmax = ± π a
k
-kmax
Phonon in 1-dimensional lattice:
ω (k) = 2
υ g ( ±kmax ) = 0
kmax
β
⎛1
⎞
sin ⎜ k a ⎟
⎝2
⎠
M
U. Keller, ULP, Chap. 2, Appendix
Linear system theory
Ein ( t )
Input
System
h(t)
Output
impulse response h(t)
Ein ( t ) = δ ( t )
Eout ( t )
Eout ( t ) = ∫ h ( t ') Ein ( t − t ') dt ' ≡ h ( t ) ∗ Ein ( t )
Eout ( t ) = ∫ h ( t ')δ ( t − t ')dt ' = h ( t )
Examples of linear systems:
• pulse propagation in dispersive media
• photo detector (impulse response links photo current with light power or intensity)
• active light modulator in the linear regime
• image propagation through a lens systems
• stochastic processes such as amplitude and phase noise (linearized as a perturbation
on a much stronger signal)
U. Keller, ULP, Chap. 2, p. 6
Linear system theory
Ein ( t )
Input
System
h(t)
Output
impulse response h(t)
Input
Eout ( t ) = ∫ h ( t ') Ein ( t − t ') dt ' ≡ h ( t ) ∗ Ein ( t )
Eout ( t ) = ∫ h ( t ')δ ( t − t ')dt ' = h ( t )
Ein ( t ) = δ ( t )
E in (ω )
Eout ( t )
System
h (ω )
Output
E out (ω ) = h (ω ) E in (ω )
transfer function h (ω )
U. Keller, ULP, Chap. 2, p. 6
Linear system theory
Ein ( t )
Input
System
h(t)
Output
impulse response h(t)
Eout ( t ) = ∫ h ( t ') Ein ( t − t ') dt ' ≡ h ( t ) ∗ Ein ( t )
Eout ( t ) = ∫ h ( t ')δ ( t − t ')dt ' = h ( t )
Ein ( t ) = δ ( t )
E in (ω )
Eout ( t )
Input
System
h (ω )
Output
E out (ω ) = h (ω ) E in (ω )
transfer function h (ω )
spectral density:
2
Pin (ω ) = E in (ω )
Input
System
h (ω )
Output
2

S (ω ) ≡ h (ω )
2

Pout (ω ) = Eout (ω )
Pout (ω ) = S (ω ) Pin (ω )
U. Keller, ULP, Chap. 2, p. 6
Linear System – Example: dispersion
What is the impulse reponse or transfer function?
What is the pulse form during propagation?
How is the spectrum of the pulse changing ?
Solution:
Chapter 2, page 7
Zeit
b
us
re
A
Time
domain
Zeitraum
n
itu
g
sr
t
ich
u
ng
z
Intensität
Intensität
Dispersive pulse broadening
Freq
uenz
b
us
A
Spectral
domain
Spektralraum
Linear pulse broadening:
Transform-limited pulse is broadening in the time domain
but its spectrum remains unchanged.
re
n
ti u
g
ch
i
r
s
t
g
un
z
Light pulse
pulse envelope A(t)
electric field E(t)
E (t ) = A (t ) e
iω 0 t
U. Keller, ULP, Chap. 2, p. 10
Laser pulse
Pulse envelope A(t)
Electric Field E(t)
E (t ) =
E ( t ) = A ( t ) eiω 0t
1 
iω t
ω
e
dω
E
(
)
∫
2π
where A ( t ) =
1 
iΔω t
ω
e
dΔω
A
Δ
(
)
∫
2π
Laser pulse
Pulse envelope A(t)
Electric Field E(t)
E (t ) =
E ( t ) = A ( t ) eiω 0 t
1 
iω t
ω
e
dω
E
(
)
∫
2π
where A ( t ) =
1 
iΔω t
ω
e
dΔω
A
Δ
(
)
∫
2π
1 
1 iω 0 t 
i (ω 0 + Δω ) t
iΔω t
E
A
Δ
ω
+
Δ
ω
e
dΔ
ω
=
e
ω
e
dΔω
E (t ) =
(
)
(
)
0
∫
∫
2π
2π
~
E(ω)
Δω 0
0
ω0
ω
~
A(Δω)
0
Δω
A ( Δω ) = E (ω 0 + Δω )
U. Keller, ULP, Chap. 2, p. 8-11
Example: Gaussian pulse
Einhüllende
A(t)
pulse envelope
A(t)
t
E (t ) = e
− Γt 2
eiω 0 t
A (t ) = e
τp =
2 ln 2
Γ1
E(t)
electric field E(t)
− Γt 2
, Γ ≡ Γ1 − iΓ 2
dφtot ( t )
ω (t ) ≡
= ω 0 + 2Γ 2t
dt
U. Keller, ULP, Chap. 2, p. 12
Chirped Gaussian Pulse
(
)
E ( t ) = A ( t ) exp ( iω 0t ) = exp −Γt 2 exp ( iω 0t )
Γ ≡ Γ1 − iΓ 2
Γ2 = 0
Γ ≡ Γ1 − iΓ 2
Γ2 ≠ 0
φtot ( t ) ≡ ω 0t + Γ 2t 2
ω (t ) ≡
dφtot ( t )
= ω 0 + 2Γ 2t
dt
U. Keller, ULP, Chap. 2, p. 13
Time-Bandwidth Products
I (t )
τp τ
Δν p ⋅ τ p
2 ln2
0.4413
1.7627
0.3148
⎪⎧1, t ≤ τ 2
I (t ) = ⎨
⎪⎩ 0 , t > τ 2
1
0.8859
⎧⎪1 − x 2 , t ≤ τ 2
I (t ) = ⎨
⎩⎪ 0 , t > τ 2
1
0.7276
2
0.2206
ln2
0.1420
(x ≡ t τ )
1. Gaussian
I (t ) = e
− x2
2. Hyperbolic secant (soliton
pulse)
I (t ) = sech 2 x
3. Rectangle
4. Parabolic
5. Lorentzian
I (t ) =
1
1 + x2
6. Symmetric two-sided exponent
I (t ) = e−2 x
U. Keller, ULP, Chap. 2, p. 12
Spectral phase yielding shortest pulse
t : center of gravity
∞
t =
∫ t ⋅ I(t) dt
E (ω ) = E (ω ) exp ( iϕ (ω ))
−∞
∞
∫ I(t) dt
−∞
rms pulse duration Δt:
shortest pulse for:
2
Δt =
∂φ (ω )
=0
∂ω
(t − t
)
2
=
∞
∫ (t − t
−∞
)
2
I(t)dt
where φ (ω ) := ϕ (ω ) − ω t
U. Keller, ULP, Chap. 2, p. 15
Inappropriateness of FWHM definition
Temporal FWHM of pulse with nonlinear spectral phase is shorter than
“transform-limited” pulse
FWHM is the standard in the community – but one has to be aware of its
limitations
Optical Dispersion
Positive dispersion:
∂2 n
>0
2
∂ω
Negative dispersion:
∂2 n
<0
2
∂ω
Dispersive pulse broadening
τ p (0)
τ p (z)
t
t
Linear pulse propagation
Helmholtz equation:
∂ 2 E ( z, ω )
2
E ( z, ω ) = 0
+
k
ω
(
)
⎡
⎤
n
⎣
⎦
2
∂z
E˜ ( z,ω 0 + Δω ) = A˜ ( z,Δ ω )e −ikn (ω 0 ) z
Δω ≡ ω − ω 0
A ( z, Δω ) = ∫ A ( z,t )e−iΔω t dt
2
2
∂2 
∂ 

⎡
⎤
⎡
⎤
A
z,
Δ
A
z,
Δ
A
z,
Δ
ω
−
2ik
ω
ω
−
k
ω
ω
+
k
ω
+
Δ
ω
(
)
(
) ⎣ n ( 0 )⎦ (
) ⎣ n( 0
)⎦ A ( z, Δω ) = 0
n( 0)
2
∂z
∂z
“slowly varying envelope approximation”
∂A
<< kn (ω 0 ) A ,
∂z
∂A
<< ω 0 A
∂t
∂ 
A ( z, Δω ) + i Δ kn A ( z, Δω ) = 0
∂z
Very simple equation of motion for the pulse envelope in the spectral domain
with a very easy solution (i.e. a phase shift Δknz for each frequency component)
−i ⎡ k ω + Δω −k ω ⎤ z
A ( z, Δω ) = A ( 0, Δω ) e−iΔkn z = A ( 0, Δω ) e ⎣ n ( 0 ) n ( 0 )⎦
U. Keller, ULP, Chap. 2, p. 17-18
First and second order dispersion
Taylor expansion around the center frequency ω0 : Δω = ω − ω 0
1
kn (ω ) ≅ kn (ω 0 ) + kn′ Δω + kn′′Δω 2 + ...
2
First order dispersion:
kn′ = dkn dω
Second order dispersion:
kn′′ = d 2 kn dω 2
U. Keller, ULP, Chap. 2, p. 18
Phase and group velocity
2
⎡
⎤
⎡
⎤
⎛
⎞
⎛
⎞
z
z
⎥
E ( z,t ) ∝ exp ⎢iω 0 ⎜ t −
⎥ × exp ⎢ − Γ ( Ld ) × ⎜ t −
⎟
⎟
⎢
⎝ υ p (ω 0 ) ⎠ ⎥⎦
⎝ υ g (ω 0 ) ⎠ ⎥⎦
⎢⎣
⎣
vg
vp
υ p (ω 0 ) ≡ cn =
ω
kn
ω =ω 0
⎛ dω ⎞
1
1
υ g (ω 0 ) ≡
=
=⎜
dk
kn′ (ω 0 ) ⎛ n ⎞
⎝ dkn ⎟⎠ ω = ω 0
⎜⎝
⎟
dω ⎠ ω = ω 0
U. Keller, ULP, Chap. 2, p. 19-20
Dispersive pulse broadening
τ p (0)
τ p (z)
t
t
⎛ 4 ln 2 d φ dω ⎞
τ p (z )
= 1+ ⎜
⎟
2
τ p ( 0)
τ p ( 0)
⎝
⎠
2
Gaussian pulse:
(initially unchirped pulse)
Approximation for
(strong pulse broadening)
dφ
2
>>
τ
p ( 0)
2
dω
2
2
2
d 2φ
τ p (z ) ≈ 2 Δω p
dω
U. Keller, ULP, Chap. 2, p. 21-23
Phase Velocity υ p
ω
kn
c
n
Group Velocity υg
dω
dkn
1
c
n 1− dn λ
dλ n
z dφ
Tg = =
, φ ≡ kn z
υ g dω
nz ⎛ dn λ ⎞
1−
⎝
c
dλ n ⎠
Dispersion 1. Order
(Group Delay, GD)
dφ
dω
nz ⎛ dn λ ⎞
1−
c ⎝ dλ n ⎠
Dispersion 2. Order
(Group Delay
Dispersion, GDD)
d 2φ
dω 2
λ3 z d 2 n
2πc2 dλ2
Dispersion 3. Order
(Third-Order
Dispersion, TOD)
d 3φ
dω 3
− λ4 z ⎛⎜ d 2 n
d 3 n ⎞⎟
2 3 3
2 +λ
4π c ⎝ dλ
dλ3 ⎠
Group Delay
Tg
U. Keller, ULP, Chap. 2, p. 27
Example: fused quartz
n⎝⎜ 0.8 µm⎠⎟ = 1.45332
⎛
⎞
1
∂n
= − 0.017 µm
∂λ 800 nm
∂kn
∂ω
∂2n
1
=
0.04
∂λ2 800 nm
µm 2
∂2kn
∂ω 2
Pulse broadening
1000
800 nm
3
1cm
Example:
What is the final pulse duration:
10cm 1m
10 fs pulse through 10 lenses, each
1 cm thick?
2
1.5
10
fs2
= 36.1 mm
Fused quartz @ 800 nm
2.5
100
800 nm
= 4.84 ns
m
10 fs pulse through 10 cm fused quartz?
1
0.5
1
0
-15
1fs
-14.5
-14
10fs
-13.5
-13
100fs
-12.5
-12
1ps
Incident pulse width (s)
U. Keller, ULP, Chap. 2, p. 22
Optical communication
λ0 [ m]
Losses [dB/km]
0.87
1.312
1.55
1.5
0.3
0.16
nd
2 order dispersion
[ps/km⋅nm]
-80
0
+17
1 dTg
ω 2 dTg
GVD : Dλ ≡ −
=
LF d λ 2π cLF dω
⇒ units :
ps
km ⋅ nm
2π cLd Dλ
d 2φ
τ p ( Ld ) =
Δ
ω
=
Δω = Dλ ⋅ Ld ⋅ Δλ
dω 2
ω2
U. Keller, ULP, Chap. 2, p. 31
Higher order dispersion
φ (ω ) = φ 0 +
∂φ
∂ω
(ω − ω ) +
0
1 ∂φ
2
2 ∂ω
2
(ω − ω )
0
2
+
1 ∂φ
3
6 ∂ω
3
(ω − ω )
+ ...
∂φ ∂ω
First order dispersion (group delay)
Second order dispersion (group delay dispersion - GDD)
Third order dispersion (TOD)
0
3
3
∂ φ ∂ω
3
∂ 2φ ∂ω
2
…
U. Keller, ULP, Chap. 2, p. 37
Wigner representation of ultrashort pulses
Wigner distribution:
W (t, ω ) =
⎛ t ′ ⎞ * ⎛ t ′ ⎞ −iω t ′
E
∫−∞ ⎜⎝ t + 2 ⎟⎠ E ⎜⎝ t − 2 ⎟⎠ e dt ′
∞
“window” function
W (t, ω ) =
1
2π
∞
ω′⎞ *⎛
ω ′ ⎞ iω ′t
⎛
E
ω
+
ω
−
E
∫−∞ ⎜⎝ 2 ⎟⎠ ⎜⎝ 2 ⎟⎠ e dω ′
Time-frequency representation /
windowed Fourier transform /
“instantaneous spectrum vs time”
Example:
Properties:
∞
2
• I(t) = E(t) = ∫ W (t,ω )dω
−∞
2
∞
• I(ω ) = E(ω ) = ∫ W (t,ω )dt
−∞
• Linear propagation in non-absorbing
medium conserves area (≘ minimum time-bandwidth product)
Effect of dispersion orders on Wigner trace
Everything calculated for an initially 10-fs long Gaussian pulse
After 100 fs2 of GDD:
ϕ (ω ) =
1
2
⋅100 fs 2 ⋅ (ω − ω 0 )
2
• Pulse remains Gaussian in time!
• Chirp is linear
Effect of dispersion orders on Wigner trace
Everything calculated for an initially 10-fs long Gaussian pulse
After 1000 fs3 of TOD:
ϕ (ω ) =
1
3
⋅1000 fs 3 ⋅ (ω − ω 0 )
6
• “Beating of simultaneous frequencies”
causes post-(pre-)pulses
Effect of dispersion orders on Wigner trace
Everything calculated for an initially 10-fs long Gaussian pulse
After 10000 fs4 of FOD:
ϕ (ω ) =
1
4
⋅10000 fs 4 ⋅ (ω − ω 0 )
24
• Even dispersion orders yield symmetric
pulse distortions in time
(for a symmetric spectrum)
Pulse after 3 mm of fused silica
Everything calculated for an initially 10-fs long Gaussian pulse
After 3 mm of fused silica (800 nm center wavelength):
ϕ (ω ) = 3 mm ⋅ k ⋅ n(ω )
• Chirp is dominantly linear – however,
influence of higher orders clearly visible
• Pulse is not Gaussian in time anymore!
Higher order dispersion
Material
Fused quartz
Refractive index n ( λ )
At 800 nm
Propagation constant kn (ω )
At 800 nm
∂n
1
= −0.017
∂λ 800 nm
µm
∂ kn
∂ω
n ( 800 nm ) = 1.45332
1
∂ n
= 0.04
2
µm 2
∂λ 800 nm
∂ kn
∂ω 2
1
∂ n
= −0.24
3
µm 3
∂λ 800 nm
∂ kn
∂ω 3
∂n
1
= −0.0496
∂λ 800 nm
µm
∂ kn
∂ω
2
3
SF10-glass
n ( 800 nm ) = 1.71125
1
∂ n
= −0.997
3
µm 3
∂λ 800 nm
∂ kn
∂ω 3
∂n
1
= −0.0268
∂λ 800 nm
µm
∂ kn
∂ω
n ( 800 nm ) = 1.76019
800 nm
3
∂ kn
∂ω 2
3
Sapphire
2
1
∂ n
= 0.176
2
µm 2
∂λ 800 nm
2
800 nm
800 nm
800 nm
800 nm
3
800 nm
800 nm
∂ kn
∂ω 2
1
∂ 3n
= −0.377
3
µm 3
∂λ 800 nm
∂ 3 kn
∂ω 3
800 nm
800 nm
fs 2
s2
= 36.1
m
mm
= 2.74 × 10
fs3
s3
= 27.4
m
mm
−41
s
m
= 1.59 × 10 −25
fs2
s2
= 159
m
mm
= 1.04 × 10 −40
fs 3
s3
= 104
m
mm
= 5.87 × 10 −9
2
s
m
= 3.61 × 10 −26
= 5.70 × 10 −9
2
1
∂ n
= 0.064
2
µm 2
∂λ 800 nm
2
= 4.84 × 10 −9
s
m
= 5.80 × 10 −26
fs 2
s2
= 58
m
mm
= 4.21 × 10 −41
fs3
s3
= 42.1
m
mm
U. Keller, ULP, Chap. 2, p. 38
Higher order dispersion
φ (ω ) = φ 0 +
∂φ
∂ω
(ω − ω ) +
0
1 ∂φ
2
2 ∂ω
2
(ω − ω )
0
2
+
1 ∂φ
3
6 ∂ω
3
(ω − ω )
+ ...
∂φ ∂ω
First order dispersion (group delay)
Second order dispersion (group delay dispersion - GDD)
Third order dispersion (TOD)
0
3
3
∂ φ ∂ω
3
∂ 2φ ∂ω
2
…
GDD becomes important, when: GDD > τ 02
TOD becomes important, when: TOD > τ 03
Dispersion lengths:
Gaussian pulse
Soliton pulse
τ2
τ3
LD ≡
, LD′ ≡
kn′′
kn′′′
⎛ t2 ⎞
E ( t ) ∝ exp ⎜ − 2 ⎟ ⇒ τ p = 2 ln2 τ = 1.665 τ
⎝ 2τ ⎠
⎛t⎞
E ( t ) ∝ sech ⎜ ⎟ ⇒ τ p = 2ln 1 + 2 τ = 1.763 τ
⎝τ⎠
(
)
τ p (z)
1
1
≅ z 1+ 2 +
LD 4 LD′ 2
τ p (0)
U. Keller, ULP, Chap. 2, p. 37