Enhancement of 3rd-order nonlinearities in nanoplasmonic metamaterials: figures of merit

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Enhancement of 3rd-order nonlinearities in
nanoplasmonic metamaterials figures of merit

• Jacob B Khurgin
• Johns Hopkins University, Baltimore
• Greg Sun
• University of Massachusetts, Boston

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Scope

• Rationale
• Can one engineer nonlinearity in metal
  nanostructures?
• Coupled mode theory of enhancement
• Assessment of nonlinearity enhancement
• Conclusions

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Rationale
Nonlinear optical interactions are quite
interesting and important, yet are also very weak
how can one improve it?
It is well known that if one used pulsed
(mode-locked) laser and concentrate the same
average power into the high peak power with low
duty cycle (d.c) efficiency of nonlinear
processes will increase
Can we do the same in the space domain and
concentrate the same power into higher local
power density to increase the efficiency ?
Plasmonics as a silver bullet for nonlinear
optics
Mode-locking in space?
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Plasmonic concentrators
M. Stockman, P. Nordlander
But
In space there is an additional factor of modal
overlap k the field of pump(s) must overlap
with field of signal (conceptually similar to the
phase-matching)
Plasmonic concentration always brings loss
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Recent work
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Recent work
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Prior to the prior works
H. J. Simon et al, Optical Second-Harmonic
Generation with Surface Plasmons in Silver
Films, PRL, 1974
Hache, Flytzanis et al, Optical nonlinearities
of small metal resonance and quantum size
effects, JOSA B 1986
P. N. Butcher and T. P. MacLean, Proc. Phys. Soc.
81, 219 (1963).
S. H. Jha, Theory of Optical Harmonic Generation
at a Metal Surfaces Phys Rev 140, 1965
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Scope

• Rationale
• Can one engineer nonlinearity in metal
  nanostructures?
• Coupled mode theory of enhancement
• Assessment of nonlinearity enhancement
• Conclusions

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Can one engineer nonlinearity in metal?
Say we have 1 SPP per mode
How far do the carriers move?
In 30 nm sphereNV106 electrons Electrons
move less than 0.001A!!!!
In QW Electron moves up to a few nm
SPP modes analogy with giantatoms and molecules
is quite superficial
Conduction electrons do not move, see no
anharmonicity, and possess practically no
nonlinearity except for the very few ones at the
surface One must either use interband
transitions (no different from saturable absorber
except for much higher loss) or better revert to
nonlinear dielectrics
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Scope

• Rationale
• Can one engineer nonlinearity in metal
  nanostructures?
• Coupled mode theory of enhancement
• Assessment of nonlinearity enhancement
• Conclusions

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Four wave interactions
FWM (Four Wave Mixing)
c(3)
XPM (Cross Phase Modulation)
c(3)
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Practical figure of merit
For nonlinear switching using XPM or SPM
Maximum interaction length is determined by
absorption hence the ultimate figure of merit is
what is the a maximum phase shift achievable
And how close it is to 1
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Mechanism for the enhancement of nonlinearity
Stage 0
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Mechanism for the enhancement of nonlinearity
Stage 1
Nanopartcles get polarized at both pump and
signal frequencies
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Mechanism for the enhancement of nonlinearity
Stage 2
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Mechanism for the enhancement of nonlinearity
Stage 3
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Mechanism for the enhancement of nonlinearity
Stage 4
Third order nonlinear polarization does not
exactly match the mode
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Mechanism for the enhancement of nonlinearity
Stage 5
Third order nonlinear polarization does not
exactly match the mode
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Mechanism for the enhancement of nonlinearity
Stage 6
f filling factor
Introduce effective nonlinear susceptibility
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Scope

• Rationale
• Can one engineer nonlinearity in metal
  nanostructures?
• Coupled mode theory of enhancement
• Assessment of nonlinearity enhancement
• Conclusions

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Assessing nonlinearity enhancement
This sounds mighty good..
What about absorption?
Maximum phase shift
Enhanced as much as few hundreds times
This sounds really good..except
indicating that the input pump pump density must
be in excess of 10GW/cm2 in order to attain
switching or efficient frequency conversion,
meaning that while the length of the device can
get reduced manyfold, the switching power cannot
and remains huge.
and the things only go further downhill from here
on once it is realized that all of the
enhancement is achieved because the pump field is
really concentrated by a factor of Q2 gt100!
Local intensity is now in excess of 1000 GW/cm2
way past break down!
So, what is the real limit?
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A better figure of merit
Factor of Q2 makes perfect sense because SPP
mode is a harmonic oscillator with a given Q
changing local index shifts resonant frequency
and causes change in polarizability proportional
to Q2
the maximum phase shift is
There is no way to achieve either all-optical
switching or efficient frequency conversion!
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What if we use dimers or nano-lenses?
Field enhancement occurs in two steps first the
larger dipole mode gets excited then the gap mode
near smaller nanoparticle
But the relation between the average nonlinear
polarization and maximum index change is still
almost the same, therefore
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What does it mean?
At low powers and plasmonic enhancement allows
one to achieve still small nonlinear phase shift
at very short distance, but this shift always
saturates well below p.
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Scope

• Rationale
• Can one engineer nonlinearity in metal
  nanostructures?
• Coupled mode theory of enhancement
• Assessment of nonlinearity enhancement
• Conclusions

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Two ways to define figure of merit
Scientific approach
What is the maximum attainable enhancement of
nonlinear susceptibility?
For c(2) enhancement is kfQ3 102-103 For c(3)
enhancement is kfQ6 105-106
Engineering approach
What would be the overall maximum attainable
result at one absorption length?
DFmaxkQDnmax10-2ltltp
For the nonlinear index type process what is
the maximum phase shift attainable at 10dB loss?
Not enough for all-optical switch (or frequency
conversion)
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Why such a conflicting result ?
Scientific approach what matters is the
relative improvement
Take very weak process with efficiency
approaching 0.then if the end result is ltlt1
Engineering approach what matters is the end
result
Using metal nanoparticles for enhancement of
second order nonlinear processes may not be a
silver bullet we are looking for.
Plasmonic enhancement is an excellent technique
for study of nonlinear optical properties (the
higher order the better) and sensing using it,
but not for any type efficient switching,
conversion, gating etc.