Vacuum Ultraviolet Generation via Wave-Mixing
Nonlinear medium
- crystals
- metal vapors
- noble gases
Again Maxwell’s framework:
coupled wave equations
2 q 2 NL
d
Eq  i
Pq
dz
ckq
PqNL
Eq and
Fourier components of fields at
frequency q
Fields:
General issues in
four wave mixing or third harmonic generation:
Susceptibility of the medium :
involves transition dipoles in atoms and resonances
Transparency : problems at short wavelengths
Phase-matching :
including the effects of focused laser beams
Practical limitation: optical breakdown
 
Eq  Êq  z , t exp ik q z
Nonlinear polarization has several terms:
P3NL 
N
4
3 33 E E E   33 E E 2   3 ,3 E E 2 
1 1 1
3 3
3 1 
S
S
 T

THG
Lecture Notes Non Linear Optics; W. Ubachs
Phase-matching and focusing
Three processes of four-wave mixing:
I
1  2  3  4
II
1  2  3  4
III
1  2  3  4
Electric fields:
Er , t   ReE1r exp i1t   E2 r exp i2t   E3 r exp i3t 
Define optical beams as lowest order Gaussians
for each frequency 
Gaussian beam TM00
( higher order modes give differing results !)
  k x2  y 2
expikn z 
exp  n
En r   En0 r 
1  i
 b1  i 

Confocal parameter b
b
2w02 2w02n 20


 kw02
2

0
n
Lecture Notes Non Linear Optics; W. Ubachs


w0
beam waist radius
n index of refraction
 far-field diffraction angle

2 z  f 
b
normalized coordinate along z
Nonlinear Polarization
Newly generated field via processes I, II, III
P4 r  
3
N 3 4 ;1, 2 , 3 E1r E2 r E3 r 
2
P4 r  
3
N 3 4 ;1, 2 ,3 E1r E2 r E3 r 
2
P4 r  
3
N 3 4 ;1,2 ,3 E1r E2 r E3 r 
2
 the polarizability (per atom) and
N density of medium
Note degeneracy factors (field amplitudes);
those may differ for degenerate fields.
Insert fields with their mode structure:
  k ' x2  y 2 
3
expik ' z 
3

P4 r   N  4 ;1, 2 , 3 E10 E20 E30
exp

2
1  i 3  b 1  i 2 



k '  k1  k2  k3  k "

  k "ik ' x 2  y 2 
3
expik ' z 

3
P4 r   N  4 ;1, 2 ,3 E10 E20 E30
exp 

2
2
1  i 2 1  i  
b 1 

  k "ik ' x 2  y 2
3
expik ' z 
3

P4 r   N  4 ;1,2 ,3 E10 E20 E30
exp 
2
1  i 1  i 2 
b 1  2





Lecture Notes Non Linear Optics; W. Ubachs


Process I
k '  k1  k2  k3 Process II


k '  k1  k2  k3
Process III
Further steps; Maxwell’s equations
Derive amplitudes in a Fourier decomposition;
amplitude with wave vector K






Field E4 r  to be calculated, integration
over x and y, and z to z=L.

P4 K   2 3  dx"  dy"  dz"P4 r"exp iK  r"
 2R E4 R 
2
dR
R  x2  y 2
0
Insert in Maxwell’s equation:
    E4K r   k42 E4K r   4k02 P4 K expiK  r 
medium
vacuum
Result, generally:
- Modes cylindrically symmetric
around z-axis
-Only for Process I the generated beam
has lowest order Gaussian
For: k  k4  k '
Solutions, process I (similar equations for II and III)
E4 r   i
3 N 2 3
expik ' z 
k0 b  4 ;1, 2 , 3 E10 E20 E30
2k 4
1  i 
  k ' x 2  y 2   exp ib / 2 k  ' 
exp 
d '
 
1  i '2
 b1  i   


With integration boundaries:
2 z  f 
  2 f /b

b
Lecture Notes Non Linear Optics; W. Ubachs
The phase-matching can be quantified:
Phase-matching in (3) Processes
Define the phase-matching integrals:

b f k "  8 k42k '
1

2
F j  b k , , ,  
2R E4 R  dR

L L k '  9  3k04 b3  2 E E E 2 0

10 20 30
This integral can be evaluated
for various conditions,
for example in the
“tight-focusing limit”, b<<L
and integrating
For all three processes, j=I,II,III
 
 
Dimensionless parameters, defining geometry
and phases.
Then generated Power at P4 is
P4  
k04k1k2k3
k42k '
b f k" 

N 2  2P1P2P3 F j  bk , , , 
L L k' 

Define:
Tight-focus
b
Degeneracy factor
L
Phase-matching integral for Process I

b f k" 
exp ib / 2 k '

FI  bk , , ,   
d '
L L k '  

1  i '2
Plane wave limit
b
L
Lecture Notes Non Linear Optics; W. Ubachs
Phase-matching integrals Fj for processes and geometries
For “tight-focus”
“half-way” the cell
b L  1
f L  0.5
General conclusion:
THG (Process I) only possible,
in the tight-focusing limit if
Process I:
FI bk ,0,0.5,1 
 2 bk 2 ebk / 2 
0
k  0
k  0
k  0
So: only if the medium has
NEGATIVE DISPERSION !!
This also holds for sum-frequency mixing
Similarly for Process II and III
(if k '  k " assumed; lowest order mode)
Difference frequency mixing for any k
FII bk ,0,0.5,1   2e
b k 
0
FIII bk ,0,0.5,1  2
 bk 2 ebk / 2 
Lecture Notes Non Linear Optics; W. Ubachs
k  0
k  0
Numerical approach evaluating Fj ; all near tight focus
Phase-matching integral including
1)
Dispersion
2)
Phase evolution through focal region
b f k" 

F j  bk , , , 
L L k' 

FII
FIII
FI versus bk for b/L < 0.1 and f/L=0.5
Varying the tightness of focus b/L
and the mode function k”/k’
Lecture Notes Non Linear Optics; W. Ubachs
Results in the “plane-wave” limit
b
L
b
k "  4 L2

 ÄkL 
FI  bk , ,0.5,  
sinc2 

b / L  
L
k '  b2
 2 
lim
Note the similarity with analysis of
frequency-doubling in crystal;
This was done for plane parallel fields !
At b/L ~3 the peak has shifted to the
real plane-wave limit already, peaking at
k=0.
Note also the decrease in intensity
for the plane-wave limit case.
Tight-focusing is more efficient !
Lecture Notes Non Linear Optics; W. Ubachs
Shifting the focus; playing with the phase build-up
b f k" 

F j  bk , , , 
L L k' 

For near-tight-focusing of b/L=0.1
Focal positions f/L=0.5 toward f/L=1.5
In latter case focus shifts outside the cell
And effectively plane-wave limit is reached.
Lecture Notes Non Linear Optics; W. Ubachs
Near the true plane-wave limit at b/L=10
Only small changes to F-integral as long as
focus is inside the cell.
At f/L=20 the focus is twice the distance
of the confocal parameter b/L=10.
Physical interpretation of phase-matching integrals
1) Non-linear polarization peaks at the region
of the beam WAIST
 intensity is highest in focus
2) Phase-matching relates to phase-overlap
of all beams involved
For a Gaussian beam:
  k x2  y 2 
expikn z 
exp  n
En r   En0 r 

1  i
b

1

i





1

1  i
1
2
e i arctan 
1   
For tight-focusing:
  2
z  f b
 2
z  f b
Boundaries of focus at windows.
Lecture Notes Non Linear Optics; W. Ubachs

Lowest order Gaussian beam undergoes phase shift
arctan 
when propagating through the focus (Gouy phase),
adding up to  for
 ,  
Driving polarization undergoes
3 arctan 
Process I
arctan 
Process II
 arctan 
Process III
Optimum conversion if generated field is in
phase with driving fields.
Generated beam undergoes phase slip: arctan 
In-phase for Process II, destructive for I and III
Phase-slips compensated by DISPERSION
 2.2 / b
kopt  0
2.2 / b
I
II
III
Optimizing density – dispersion and phase-matching
Nonlinear generation:
b f k" 

P4  P1P2P3 N 2  2 F j  bk , , , 
L L k' 

N 2F j
can be optimized by varying
N, bk, b/L, f/L
if N varies – macroscopic effect on medium
- index of refraction changes
- k varies
Phase-mismatch is a dispersion effect and
k  N
One can define a dimensionless quantity:
b f k" 
b f k" 


G j  bk , , ,   bk 2 F j  bk , , , 
L L k' 
L L k' 


Lecture Notes Non Linear Optics; W. Ubachs
GI vs. bk for tight-focusing; always OK as
long as focus inside the cell.
THG production and dispersion in the noble gases
Third harmonic produced:
PTHG 
In region of bound states; index of refraction
(with fi the oscillator strength of transitions)
2
b


N 2  3  P13 F j  bk , ,0.5,1
L


3 4



n  1lines  Nre 
fi
2 i i 2  
In the continuum ( ionization cross section)
In tight-focusing limit (b<<L):
n  1continuum 
 2 bk 2 ebk / 2 
FI bk ,0,0.5,1 
0
k  0
k  0
In plane-wave limit (b>>L):
FI bk , ,0.5,1 
4 L2
N

d i

2  i2   2
Cross sections and oscillator strength relate
to atomic structure, with C the phase mismatch
per atom;
k  CN 
2 n1  n3 

 ÄkL 
sinc2 

 2 
b2
We have seen : tight focus is more efficient,
but then negative dispersion required
Lecture Notes Non Linear Optics; W. Ubachs
Calculation of regions of positive and
negative dispersion
Negative and positive dispersion in the Kr and Xe
Shaded areas – “anomalous dispersion in the medium”
first resonance line
Regions of efficient THG
Lecture Notes Non Linear Optics; W. Ubachs
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
Lecture Notes Non Linear Optics; W. Ubachs
Experimental density effect on phase matching in Xe
Tight focusing at  = 118 nm in Xe
(negative dispersion);
b/L = 0.025
f/L =0.5 – centre of cell
Effect of G-integral and interplay
between N and k.
Lecture Notes Non Linear Optics; W. Ubachs
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.
Lecture Notes Non Linear Optics; W. Ubachs
Resonance enhanced VUV production
Non-linear susceptibility:
 3 
gri ir j jrk k r g
       2   3 
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
Advantage at two-photon-level
g ri ir J' J'r k k r g
 3  
      2  i   3 
gj
gk
ik terms gi
 = 212.5 nm resonance in Kr
“strongest two-photon resonance in nature”
Process I
Process II
Lecture Notes Non Linear Optics; W. Ubachs
Non-colinear Phase-matching for sum-frequency generation
Phase-vectors
Energy levels
1
 

k XUV  2k1  k2

k1
 

kVUV  2k1  k 2

k1
 XUV  21  2

k2

k2
1
VUV  21  2
1
Experimental realization
Volume considerations
Lecture Notes Non Linear Optics; W. Ubachs
Opt. Lett. 20 (2005) 1494
Polarization properties
Tensor nature of susceptibility
3 Ei E j Ek
P r ;i ,  j , k  
  


Dependent on matrix dipole moments
 3 
gri ir j jrk k r g
       2   3 
gj
gk
gijk terms gi
Note: a COHERENT sum should be taken
(as in multi-photon transitions)
Parametric and non-parametric processes;
relate to energy exchange
Use atomic physics framework: evaluate
Transition dipole moments with Wigner-Eckhart
theorem:
 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 
 J i 1 J g  J j 1 J i  J k 1 J j  J g 1 J k 




 3  








M
q
M

M
q
M

M
q
M

M
q
M
i
1
g 
j
2
i 
k
3
j 
g
4
k

The coherent sum requires M=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:
Lecture Notes Non Linear Optics; W. Ubachs
q1  q2  q3  1
Polarization in sum-frequency mixing; quantum levels
Well defined quantum level;
Real intermediate level
J2 = 0 or 2
Lecture Notes Non Linear Optics; W. Ubachs
Pulsed jets and differential pumping – the road to the windowless regime
A.H. Kung, Opt. Lett. 8, 24 (1983).
Lecture Notes Non Linear Optics; W. Ubachs
Some considerations
1)
Bandwidth effects
2)
Lasers and synchrotrons
3)
The perturbative regime and the non-perturbative -> recollision model
4)
Spectroscopy in VUV and XUV feasible
Lecture Notes Non Linear Optics; W. Ubachs
XUV-laser setup with PDA;
bandwidth ~250 MHz
 90 – 110 nm
Lecture Notes Non Linear Optics; W. Ubachs
Measurement of 11S – 21P resonance line of Helium
5 130 495 083 (45) MHz
1 1 S - 2 1 P at 58.4 nm
Intensity (arb.)
Results on Lamb shift in He ground state
41224 (45) MHz experiment (1997)
41233 (35-100) MHz; Drake (1993)
41223 (42) MHz theory; Korobow/Yelkovsky (2001)
6 00 M H z
2-electron QED effects
in He (1s2 1S0)
e ta lon
*
self-energy
I2 sa tu ra tion
5 84 n m
-40 0 0
-2 00 0
vacuum polarisation
e+
e-
e0
20 0 0
XUV fre qu e ncy (M Hz)
* P88(15-1) o component in I2
at 513049427.1(1.7) MHz
4 00 0
ep
n
Lecture Notes Non Linear Optics; W. Ubachs