Lectures on
Spectroscopy and Metrology
1. How to do XUV precision spectroscopy
- XUV-light and how to make it: Harmonic generation
- Frequency chirping in lasers and harmonics
2. Frequency metrology with frequency comb lasers
- Concepts of frequency combs
- Use in calibration
- Direct frequency comb excitation
3. Precision spectroscopy of hydrogen
- Laboratory spectroscopy
- Spectroscopic observations of quasars
- Detecting a variation of the proton-electron mass ratio
4. Varying constants and a view on an evolutionary universe
Wim Ubachs
TULIP Summer School IV 2009
Noordwijk, April 15-18
XUV light
Lasers at very short wavelengths ?
competiton with spontaneous emission: A/B ~ ω3
no media for tunability
Transparancy of the atmosphere
vacuum ultraviolet : λ < 200 nm
No window materials
LiF cutoff : λ ~ 105 nm; MgF at 118 nm; CaF at ~130 nm
No optics for beam manilupation - focusing
No nonlinear crystals
Harmonic conversion only in gases
Gases have inversion symmetry; so third order
Re-absorption still important in some cases – differentail pumping
Optical breakdown and plasma formation
Typical conversion effeiciens < 10-6
Other sources
Gas discharges (Hopfield continua)
Synchrotron radiation
Nonlinear Optics and the Production of XUV light
The first non-linear optical laser experiment
P.A. Franken, A.E. Hill, C.W. Peters and G. Weinreich, Phys. Rev. Lett. 7 (1961) 118
Nicolaas Bloembergen
Nobel prize 1981
Further: Stimulated Raman, Stimulated Brillouin,
CARS, Photon echoes, Four-wave mixing
Surface sum-frequency-mixing
Graphical: Nonlinear response of a medium
Linear response:
induced polarization follows
the applied field on the medium
The medium becomes nonlinearly polarized
Polarization can be decomposed in
Fourier components
Non-linear response:
induced polarization cannot follow
the applied field on the medium
P = ∑ an sin(nωt + φn )
Other frequency
components
Nonlinear Optics and Maxwell’s equations
Nonperturbative
Nonlinear polarization induced in a medium:
In centrosymmetric media, such as gases, χ(2) =0:
Theory: Corkums 3-step model
(classical)
Lewenstein: Quantum-Mech. model
r
r
2
r
∂ E
∂ 2 r NL
∂E
2
− με 2 = μ 2 P
∇ E − μσ
∂t
∂t
∂t
E1 ( z , t ) = E1 ( z )exp(iω1t − ik1z )
E2 ( z, t ) = E2 ( z )exp(iω2t − ik2 z )
E3 ( z , t ) = E3 ( z )exp(iω3t − ik3 z )
A source term for Maxwell’s equation:
Input fields:
A new field E4(z,t) is created at frequencies:
At polarization:
ω4 = ±ω1 ± ω2 ± ω3
PNL ( z , t ) = dE1 ( z )E2 ( z )E3 ( z )exp[i(ω1 + ω2 + ω3 )t − i (k1 + k 2 + k3 )z ]
Coupled wave equations
Maxwell’s equation with nonlinear source term:
r
r
2
r
∂E4
∂ E4
∂ 2 r NL
2
∇ E4 − μσ
− με
=μ 2P
2
∂t
∂t
∂t
d2
dz
E ( z, t ) << 2ik4
2 4
d
E4 ( z , t )
dz
Variation of the amplitude of the distance of
a wavelength is small
Substitute left side:
use:
Slowly varying amplitude approximation
E4 ( z , t ) = E4 ( z )exp(iω4t − ik 4 z )
d2
dz
E ( z, t ) + 2ik4
2 4
d
E4 ( z, t ) − k42 E3 ( z, t )
dz
+ iω4 μσE4 ( z, t ) + μεω42 E4 ( z, t )
d2
d
d2
E ( z, t ) − μσ E4 ( z , t ) − με 2 E4 ( z, t ) =
2 4
dt
dz
dt
d
2
d
2
(
)
(
)
E
z
t
+
ik
E
z
t
−
k
E4 ( z , t )
,
2
,
4
4
4
4
2
dz
dz
+ iω4 μσE4 ( z, t )
+ μεω42 E4
(z, t )
For plane waves in a medium;
μεω42
− k42
=
ω42
c2
− k42 = 0
So left side of wave equation;
2ik4
d
E4 ( z, t ) + iω4 μσE4 ( z, t )
dz
Coupled wave equations
r
r
2
r
∂
E
∂
E
∂ 2 r NL
2
4
4
∇ E4 − μσ
− με
=μ 2P
∂t
∂t
∂t 2
left side of wave equation;
2ik4
right side of wave equation;
d
E4 ( z, t ) + iω4 μσE4 ( z, t )
dz
reabsorption
∂ 2 r NL
d2
μ 2 P = μ 2 χE1 ( z )E2 ( z )E3 ( z )exp[i(ω1 + ω2 + ω3 )t − i(k1 + k2 + k3 )z ] =
∂t
dt
− μ (ω1 + ω2 + ω2 )2 χE1 ( z )E2 ( z )E3 ( z )exp[i(ω1 + ω2 + ω3 )t − i(k1 + k2 + k3 )z ]
Conservation of energy;
ω4 = ω1 + ω2 + ω3
This does not hold for the wave vectors, because;
Hence;
r r r r r
Δk = k4 − k1 − k2 − k3
Coupled equation;
Result
ωi =
cki
ki
=
με(ω i ) n(ω i )
d
E4 ( z ) ∝ χ (3) E1 ( z )E2 ( z )E3 ( z )exp[− iΔkz ]
dz
Perturbative analysis of THG
Result
Susceptibility
of the medium:
material property
(quantum resonances)
Input field intensities
High power;
Focused laser beams
Phases of input fields;
Geometry of setup
(confocal parameter,
Gouy phase, phase matching)
Susceptibility, quantum level structure,
reabsorption and resonance enhancement
Non-linear susceptibility:
χ (3) ∝
gri ir j jrk k r g
∑ ∑ (ω ± ω )(ω ± 2ω )(ω ± 3ω )
gi
gj
gk
Again level structure of the noble gases
is favorable:
gijk terms
Resonances possible at the
- One photon
- Two-photon
- Three photon levels (reabsorption)
Advantage at two-photon-level
χ (3) ∝ ∑
g ri ir J' J' r k k r g
∑ (ω ± ω )(ω − 2ω − iΓ )(ω ± 3ω )
gj
gk
ik terms gi
λ = 212.5 nm resonance in Kr
“strongest two-photon resonance in nature”
Process I
Process II
Non-colinear Phase-matching for sum-frequency generation
Phase-vectors
Energy levels
r r
v
k XUV = 2k1 + k2
v
k1
r r
v
kVUV = 2k1 − k2
v
k1
v
k2
v
k2
ω1
ω1
ω1
ω XUV = 2ω1 + ω2
ωVUV = 2ω1 − ω2
Experimental realization
Volume considerations
Opt. Lett. 20 (2005) 1494
Problems and solutions in colinear phase-matching in THG
Beams must be focused: this has an effect on the wave front
w0
beam waist radius
n index of refraction
θ far-field diffraction angle
ξ=
2( z − f )
b
normalized coordinate along z
Confocal parameter b
b=
Electromagnetic fields (for TEM00):
2πw02
λ
En (r ) = En0 (r )
=
2πw02n
λ0
=
2λ0
nθ
2
= kw02
(
⎡ − k x2 + y 2
exp(ikn z )
exp ⎢ n
1 + iξ
⎢⎣ b(1 + iξ )
Lowest order Gaussian beam undergoes phase shift
)⎤⎥
⎥⎦
arctan ξ
when propagating through the focus (Gouy phase), adding up to π for
(− ∞, ∞ )
Conditions for the phase-matching integral in THG
Solutions, process THG
E4 (r ) = i
3 N 2 (3)
exp(ik ' z )
πk0 bχ (− ω4 ; ω1 , ω2 , ω3 )E10 E20 E30
(1 + iξ )
2k 4
(
)
⎡ − k ' x 2 + y 2 ⎤ ξ exp[− (ib / 2)Δk (ξ '−ξ )]
dξ '
× exp ⎢
⎥∫
2
(
)
b
1
i
ξ
+
(
)
+
1
i
ξ
'
⎦ −ζ
⎣
Phases get into the integral: wave front + dispersion
Field
) be calculated, integration
E4 (rto
over x and y, and z to z=L.
Then generated Power at P4 is
With the phase-matching integral
P4 = η
∞
2
∫ 2πR E4 (R ) dR
R = x2 + y2
0
k04 k1k 2 k3
k 42 k '
b f k" ⎞
⎛
N 2 χ 2 P1P2 P3 F j ⎜ bΔk , , , ⎟
L L k' ⎠
⎝
b f k " ⎞ ξ exp[− (ib / 2)Δkξ ']
⎛
FI ⎜ bΔk , , , ⎟ = ∫
dξ '
2
L
L
k
'
⎠ −ζ
⎝
(1 + iξ ')
Influence of the geometry in THG
b
Tight-focus
L
Process THG – evaluation of integral for b<<L:
FI (bΔk ,0,0.5,1) =
2
π (bΔk ) e(bΔk / 2 )
Δk < 0
0
Δk ≥ 0
2
b
THG only possible,
in the tight-focusing limit if
Δk < 0
So: only if the medium has
NEGATIVE DISPERSION !!
Plane wave limit
L
Process THG – evaluation of integral for b>>L:
b
k " ⎞ 4 L2
⎛
⎛ ΔkL ⎞
FI ⎜ bΔk , ,0.5, ⎟ =
sinc 2 ⎜
⎟
b / L →∞ ⎝
L
k ' ⎠ b2
⎝ 2 ⎠
lim
The known sinc-function
Also found in frequency-doubling
(for plane waves)
Less efficient than tight-focusing
Real example: Negative and positive dispersion in the Kr and Xe
2π (n1 − n3 )
Wave vector mismatch
Δk = CN =
Related to dispersion
λ
with
(n − 1)lines = Nre ∑
fi
2π i λi− 2 − λ
Shaded areas – “anomalous dispersion in the medium”
first resonance line
Regions of efficient THG
THG in Xe; experiment
Experiment THG production
Calculation of
Phase-matchingintegral
Conclusion: 1) theory of phase-matching works
2) THG effective at blue side of ns and nd resonances
Note : absolute intensities at the 10-6 – 10-7 scale
Density effect on THG in Xe
Wave vector mismatch
Related to dispersion
Δk = CN =
2π (n1 − n3 )
λ
Phase mismatch is macroscopic
(like the refractive index):
dependent on gas density
NB: use mixing of gases
with opposite dispersion
Polarization (orientation of the E-vector)
Nonlinear susceptibility:
χ (3) ∝
gri ir j jrk k r g
∑ ∑ (ω − ω )(ω − 2ω )(ω − 3ω )
gi
gj
gk
Note: parametric four-wave mixing:
no energy/angular momentum
exchange with the medium
gijk terms
Involves a sequence of four coherent photon interactions (no relaxation)
⎛ Ji 1 J g ⎞
⎟ J g r (1) J i
g r i = J g M g rq(1) J i M i = (− )J i − M i ⎜⎜
⎟
⎝ − Mi q M g ⎠
(Wigner-Eckhart)
Hence in four-wave mixing:
⎛
J i 1 J g ⎞⎛ J j 1 J i ⎞⎛ J k 1 J j ⎞⎛ J g 1 J k ⎞
⎟⎜
⎟⎜
⎟⎜
⎟
⎟
⎜
⎟
⎜
⎟
⎜
⎟
−
−
−
−
M
q
M
M
q
M
M
q
M
M
q
M
i
1
g ⎠⎝
j
2
i ⎠⎝
k
3
j ⎠⎝
g
4
k⎠
⎝
χ (3) ∝ ⎜⎜
The coherent sum requires ΔM=0 over four-photon cycle;
q=0, q=-1, q=1 projections of dipole moment on angular basis (polarizations)
Evaluate the four-product of Wigner-3j symbols
Results:
1) all polarizations linear is possible
q1 = q2 = q3 = q4 = 0
2) THG with circular light is NOT possible:
q1 = q2 = q3 = 1
Production of Polarized XUV
Well defined quantum level;
Real intermediate level
J2 = 0 or 2
Defeating the negative dispersion problem
All calculations performed for integral from –L to +L.
Cut-off the medium at f; stop the destructive interference.
Can be done in “forbidden region”.
Efficiency remains low; but not zero.
cutoff
b f k " ⎞ 0 exp[− (ib / 2)Δkξ ']
⎛
dξ '
Cutoff of the phase-matching integral FI ⎜ bΔk , , , ⎟ = ∫
2
L
L
k
'
⎠ −ζ
⎝
(1 + iξ ')
Qualitatively: stop the process when negative dispersion sets in
Pulsed jets and differential pumping – the road to the windowless regime
A.H. Kung, Opt. Lett. 8, 24 (1983).
Real life THG conversion at high laser powers
Features of the quantum
mechnical level structure
visible, via resonance
enhancement
Practical:
XUV production possible
at all wavelengths
XUV-laser setup with
Pulsed Dye Amplification;
bandwidth ~250 MHz
λ = 90 – 110 nm
For metrology: is fseed = fpulsed ?
Phenomena of frequency chirp and precision metrology
In harmonic conversion exact harmonics produced
for Gaussian beam profile
Bandwidths:
for Lorentzian beam profile
For non-FT-limited pulses, with possible chirp:
Phase variations possibly due to index modulation
Example: 5th harmonics (atom as a ruler)
Xe
C2H2
Ionization in the focus
modulates the index:
Frequency chirp in precision metrology of pulsed lasers
Spectroscopy performed with an injection seeded laser (+upconversion)
Calibration of the CW seed laser
Verification of chirp effect in pulsed systems
Chirp detection scheme
Chirp detection and chirp analysis
CW-pulse
beat detection
FT
Reverse FT’s