1. Enhancement of 3rd-order nonlinearities in nanoplasmonic metamaterials: figures of merit
Jacob B Khurgin
Johns Hopkins University, Baltimore
Greg Sun
University of Massachusetts, Boston

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

3. 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 t P
Can we do the same in the space domain and concentrate the same power into higher local power density to increase the efficiency ? Ag
Plasmonics as a ”silver bullet” for nonlinear optics
“Mode-locking in space?”

4. 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

5. Recent work

6. Recent work F. B. P. Niesler et al , OPTICS LETTERS 34, 1997 (2009)
Palomba et al J. Opt. A: Pure Appl. Opt. 11 (2009)
Yu Zhang et al, Nano Lett., 2011, 11 (12), pp 5519–5523

7. “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

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

9. Can one engineer nonlinearity in metal?
In QW’s or QD’s….anharmonic potential-giant dipole of this “artifical atom” or “molecule”
How about electrons in SPP giant “artificial atoms” or “molecules” + +
Say we have 1 SPP per mode
Power dissipation is
Power density very high!
How far do the carriers move?
In 30 nm sphere…NV~106 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

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

11. Four wave interactions
FWM (Four Wave Mixing)
c(3)
Efficiency
XPM (Cross Phase Modulation)
c(3)
Nonlinear index
Nonlinear phase shift

12. Practical figure of meritDf
Switching
For nonlinear switching using XPM or SPM
For wavelength conversion
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…

13. Mechanism for the enhancement of nonlinearityAg
c(3)
Stage 0
Average values of fields

14. Mechanism for the enhancement of nonlinearity+ -
Stage 1
Nanopartcles get polarized at both pump and signal frequencies

15. Mechanism for the enhancement of nonlinearity+ -
c(3)
Stage 2
Locally enhanced field at both pump and signal frequencies

16. Mechanism for the enhancement of nonlinearity+ -
c(3)
Stage 3
Local nonlinear polarization is established

17. Mechanism for the enhancement of nonlinearity+ -
Stage 4
Local nonlinear field is established
Third order nonlinear polarization does not exactly match the mode

18. Mechanism for the enhancement of nonlinearity+ -
Stage 5
Accordingly, each nanoparticle acquires nonlinear dipole moment (at signal frequency)
Third order nonlinear polarization does not exactly match the mode

19. Mechanism for the enhancement of nonlinearity+ -
Stage 6
The whole medium then acquires average nonlinear polarization at the signal frequency
f – filling factor
Introduce effective nonlinear susceptibility

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

21. 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
still, assuming
(chalcogenide glass)
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 >100!
Local “intensity” is now in excess of 1000 GW/cm2 –way past break down!
So, what is the real limit?

22. 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
Assuming that maximum index change is limited by material properties to
the maximum phase shift is…
There is no way to achieve either all-optical switching or efficient frequency conversion!

23. What if we use dimers or “nano-lenses”?
c(3)
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

24. What does it mean? p P=8W 1mm2 P=0.8W P=0.8W P
Length (mm) 10 - 1 2 3 4 8 7 6 5
Nonlinear Phase Shift (rad) p
P=0.8W
P=0.8W
P=1.6mW
P=1.6mW
P=1.6mW
1mm2 P
1mm2 P
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.

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

26. Two ways to define figure of merit
Scientific approach
What is the maximum attainable enhancement of nonlinear susceptibility? + -
For c(2) enhancement is kfQ3 ~
For c(3) enhancement is kfQ6 ~
Engineering approach
What would be the overall maximum attainable result at ~one absorption length?
DFmax~kQDnmax~10-2<<p
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)

27. 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 <<1
Engineering approach: what matters is the end result Ag
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.