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

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? Ag 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 P t 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?”

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) 114030 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 merit Switching For nonlinear switching using XPM or SPM Df 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 nonlinearity c(3) Stage 0 Average values of fields Ag Ag Ag Ag Ag Ag Ag Ag

14. Mechanism for the enhancement of nonlinearity c(3) 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 c(3) Stage 4 + + + + + + + + - - - - - - - - Local nonlinear field is established Third order nonlinear polarization does not exactly match the mode

18. Mechanism for the enhancement of nonlinearity c(3) + + + - - - + + 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 c(3) + + + - - - + + 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 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…. still, assuming (chalcogenide glass) 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=8W 1 p 10 P=0.8W 0 10 1mm2 - 1 P=0.8W 10 P P P P=1.6mW - 2 10 P=1.6mW - 3 10 Nonlinear Phase Shift (rad) P=1.6mW - 4 10 1mm2 - 5 10 - 6 10 - 7 10 1mm2 - 8 10 - 1 0 1 2 3 4 10 10 10 10 10 10 Length (mm) 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 ~102-103 For c(3) enhancement is kfQ6 ~105-106 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 ? Ag 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 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.