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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 22 — Nov. 4, 2013
  • pp: 27460–27480
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Plasmonic enhancement of the third order nonlinear optical phenomena: Figures of merit

Jacob B. Khurgin and Greg Sun  »View Author Affiliations


Optics Express, Vol. 21, Issue 22, pp. 27460-27480 (2013)
http://dx.doi.org/10.1364/OE.21.027460


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Abstract

Recent years have seen increased interest in the plasmonic enhancement of nonlinear optical effects, yet there remains an uncertainty as to the limits of this enhancement. We present a simple and physically transparent theory for the plasmonic enhancement of third order nonlinear optical processes and show that while a huge enhancement of the effective nonlinear index can be attained, the most relevant figure of merit, the phase shift per one absorption length, remains very low. This suggests that while nonlinear plasmonic materials are not suitable for applications requiring high efficiency, for example in all-optical switching and wavelength conversion, they can be very useful for applications where overall high efficiency is not critical, such as in sensing.

© 2013 Optical Society of America

1. Introduction

Nonlinear optical phenomena have been a scientific community focus ever since scientists gained access to intense optical fields with the invention of the laser in 1960 [1

1. T. H. Maiman, “Stimulated Optical Radiation in Ruby,” Nature 187(4736), 493–494 (1960). [CrossRef]

]. Indeed, shortly after this invention second and third order nonlinear optical effects were demonstrated [2

2. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7(4), 118–119 (1961). [CrossRef]

4

4. J. A. Giordmaine, “Mixing of light beams in crystals,” Phys. Rev. Lett. 8(1), 19–20 (1962). [CrossRef]

] and the theory of nonlinear optics developed [5

5. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]

7

7. N. Bloembergen and Y. R. Shen, “Quantum-theoretical comparison of nonlinear susceptibilities in parametric media, lasers, and Raman Lasers,” Phys. Rev. 133(1A), A37–A49 (1964). [CrossRef]

]. Today, a clear understanding of nonlinear optical effects in various media exists [8

8. Y. R. Shen, Principles of Nonlinear Optics (Wiley, 1984).

, 9

9. R. W. Boyd, Nonlinear Optics (Academic Press, 1992).

]. The fascinating promise of nonlinear optics has always been based on the fact that nonlinear optical phenomena allow one, in principle, to manipulate photons with other photons without relying on electronics. Yet, while there have been some spectacular success stories that have led to practical products (such as for example, frequency converters and Optical Parametric Oscillators to name a few), the majority of nonlinear optical effects have not been utilized for any practical applications.

The reason nonlinear optical phenomenon has not been more widely exploited can be explained as follows: all nonlinear optical phenomena can be divided into two broad classes, slow and ultra-fast. The slow nonlinear phenomena are generally classified as such by the fact that the optical fields do not interact directly, but through various “intermediaries”, such as electrons excited when the photons get absorbed, or through the temperature rise caused by the release of energy associated with the absorbed photons. As long as these “intermediaries” exist, i.e. while the electrons stay in the excited state or until the heat dissipates, their effect on the optical fields accumulates, hence these phenomena, such as saturable absorption, photo-refractive effects, or thermal nonlinearities, can be quite strong. This fact, however, makes them slow, as their temporal response is limited by a time constant associated with the appropriate relaxation, recombination, or heat diffusion times.

Ultrafast nonlinearities on the other hand, do not involve excitation of matter to a real excited state as there exists no transition between states that is resonant with the incident photon energy, hence they are often referred to as “virtual” transitions. When the non-energy-conserving “virtual” excitation does take place, its duration is determined by the uncertainty principle, and thus can be as short as a few femtoseconds or even a fraction of femtosecond It is precisely this fact that the excitation lasts for such a short time interval that makes the ultra-fast nonlinearities relatively weak. For example, the nonlinear refractive index, n2 that characterizes third order nonlinearities, ranges from n2~5×1016cm2/W for fused silica that is transparent all the way to UV, to perhaps n2~1×1013cm2/W for chalcogenide glasses transparent only in the IR range [10

10. L. B. Fu, M. Rochette, V. G. Ta’eed, D. J. Moss, and B. J. Eggleton, “Investigation of self-phase modulation based optical regeneration in single mode As2Se3 chalcogenide glass fiber,” Opt. Express 13(19), 7637–7644 (2005). [CrossRef] [PubMed]

].Therefore, very strong optical power densities, on the order of GW/cm2 are required in order to produce appreciable ultrafast nonlinear optical phenomena. The average optical power available from a compact laser rarely exceeds a few hundred milliwatts, furthermore, if one wants to envision all optical integrated circuits, the power dissipation requirements constrain the power to even lower levels, possibly less than a milliwatt. Hence early on it was understood that in order to make nonlinear optical phenomena practical, one must concentrate power in both space and time. Concentration in space usually implies coupling light into a tightly-confining optical waveguide or an optical fiber. In this case, the diffraction limit bounds the attainable concentration to roughly a single wavelength in the medium. One could consider the resonant concentration of optical energy through the use of micro-cavities [11

11. S. X. Qian, J. B. Snow, H. M. Tzeng, and R. K. Chang, “Lasing droplets: Highlighting the liquid-air interface by laser emission,” Science 231(4737), 486–488 (1986). [CrossRef] [PubMed]

,12

12. H. B. Lin and A. J. Campillo, “CW nonlinear optics in droplet microcavities diplaying enhanced gain,” Phys. Rev. Lett. 73(18), 2440–2443 (1994). [CrossRef] [PubMed]

], ring resonators [13

13. J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24(12), 847–849 (1999). [CrossRef] [PubMed]

], photonic bandgap structures [14

14. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998). [CrossRef]

] and/or slow light devices [15

15. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102(20), 203902 (2009). [CrossRef] [PubMed]

,16

16. C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12(10), 104003 (2010). [CrossRef]

], but all resonant effects inevitably limit bandwidth [17

17. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: Comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005). [CrossRef]

]. It is the concentration of optical power in the time domain, provided by pulsed sources, particularly by Q-switched [18

18. R. W. Hellwarth, “Control of fluorescent pulsations,” in Advances in Quantum Electronics, R. Singer, ed. (Columbia University, 1961), p. 334.

] and mode-locked lasers [19

19. L. Hargrove, R. L. Fork, and R. L. Pollack, “Locking of HeNe laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. 5(1), 4–5 (1964). [CrossRef]

,20

20. A. J. DeMaria, D. A. Stetson, and H. Heyma, “Mode locking of a Nd3+‐doped glass laser,” Appl. Phys. Lett. 8(1), 22–24 (1966). [CrossRef]

], that has proven to be the winning technique in nonlinear optics. In a low duty cycle mode-locked pulse, the peak power exceeds the average power by many orders of magnitude, hence the use of ultra-short, low duty cycle pulses has become the ubiquitous method used to exploit both the second and especially third order nonlinear optical phenomena such as in the generation of optical frequency combs and in continuum generation. If one is thinking of nonlinear optical applications in information processing however, the switches are expected to operate at the same modulation rate and duty cycle as the data stream. Therefore one needs other methods of concentrating the energy and one is inevitably drawn back to the space domain and the question arises: can one transfer the mode-locking techniques from time to space, i.e. to create a low duty cycle high peak power distribution of optical energy in space, rather than in time and to use it to effectively enhance nonlinear optical effects.

In the last decade researchers have observed enhancement of both linear (absorption, luminescence) [22

22. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006). [CrossRef] [PubMed]

,23

23. P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15(21), 14266–14274 (2007). [CrossRef] [PubMed]

] and non-linear (Raman) phenomena [24

24. M. Moskovits, L. Tay, J. Yang, and T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–227 (2002). [CrossRef]

26

26. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997). [CrossRef] [PubMed]

] in the vicinity of small metal nanoparticles. Experimentally, Raman scattering has shown plasmon mediated enhancement of many orders of magnitude [24

24. M. Moskovits, L. Tay, J. Yang, and T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–227 (2002). [CrossRef]

26

26. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997). [CrossRef] [PubMed]

], while the enhancement associated with luminescence and absorption has been more modest. To explain these differing effects we have developed a rigorous, yet physically transparent theory explaining the enhancements produced by single [27

27. G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94(10), 101103 (2009). [CrossRef]

] or coupled [28

28. G. Sun, J. B. Khurgin, and A. Bratkovsky, “Coupled-mode theory of field enhancement in complex metal nanostructures,” Phys. Rev. B 84(4), 045415 (2011). [CrossRef]

,29

29. G. Sun and J. B. Khurgin, “Theory of optical emission enhancement by coupled metal nanoparticles: An analytical approach,” Appl. Phys. Lett. 98(11), 113116 (2011). [CrossRef]

] nanoparticles in which we have traced the relatively weak enhancement associated with luminescence to large absorption in the metal. This metal absorption cannot be reduced in truly subwavelength mode in which the field is concentrated [30

30. J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011). [CrossRef]

,31

31. J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012). [CrossRef]

]. In that work [30

30. J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011). [CrossRef]

] it was shown that the decay rate of the electric field in the sub-wavelength mode is always on the order of the scattering time in the metal LSP host, i.e. 10-20 fs in noble metals. This is the natural consequence of the aforementioned fact that half of the time all the energy is contained in the kinetic motion of carriers in the metal where it dissipates with the scattering rate. As a result, a significant fraction of the SP’s simply dissipates inside the metal rather than radiating away. The net result is that only very inefficient emitters [32

32. J. B. Khurgin, G. Sun, and R. A. Soref, “Electroluminescence efficiency enhancement using metal nanoparticles,” Appl. Phys. Lett. 93(2), 021120 (2008). [CrossRef]

] and also absorbers [33

33. J. B. Khurgin, G. Sun, and R. A. Soref, “Practical limits of absorption enhancement near metal nanoparticles,” Appl. Phys. Lett. 94(7), 071103 (2009). [CrossRef]

] can be enhanced by plasmonic effects, such as, of course, the Raman process which is known to be extremely inefficient [34

34. G. Sun and J. B. Khurgin, “Origin of giant difference between fluorescence, resonance and non-resonance Raman scattering enhancement by surface plasmons,” Phys. Rev. A 85(6), 063410 (2012). [CrossRef]

], while the relatively efficient devices, such as LEDs [35

35. K. Okamoto, I. Niki, A. Scherer, Y. Narukawa, T. Mukai, and Y. Kawakami, “Surface plasmon enhanced spontaneous emission rate of InGaN/GaN quantum wells probed by time-resolved photoluminescence spectroscopy,” Appl. Phys. Lett. 87(7), 071102 (2005). [CrossRef]

], solar cells [36

36. S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells,” J. Appl. Phys. 101(9), 093105 (2007). [CrossRef]

], and detectors [37

37. S. C. Lee, S. Krishna, and S. R. J. Brueck, “Quantum dot infrared photodetector enhanced by surface plasma wave excitation,” Opt. Express 17(25), 23160–23168 (2009). [CrossRef] [PubMed]

] do not exhibit any significant plasmonic enhancement relative to what can be obtained without the metal by purely dielectric means [38

38. M. B. Dühring, N. Asger Mortensen, and O. Sigmund, “Plasmonic versus dielectric enhancement in thin-film solar cells,” Appl. Phys. Lett. 100(21), 211914 (2012). [CrossRef]

].

Based on the above argument, it is only natural to investigate what plasmonic enhancement can do for inherently weak nonlinear processes, and, although the first works along this direction are over 30 years old [39

39. M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974). [CrossRef]

44

44. A. Wokaun, J. G. Bergman, J. P. Heritage, A. M. Glass, P. F. Liao, and D. H. Olson, “Surface second-harmonic generation from metal island films and microlithographic structures,” Phys. Rev. B 24(2), 849–856 (1981). [CrossRef]

], interest has picked up significantly in the last decade [45

45. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012). [CrossRef]

,46

46. M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19(3), 2064–2078 (2011). [CrossRef] [PubMed]

]. There are a number of ways where nonlinear optical effects can be enhanced by surface plasmons. One is the coupling of the excitation field to form a much stronger localized field near the surface of metal structure which leads to enhancement of optical processes [47

47. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005). [CrossRef]

]. Such strong near-field effects are responsible for the experimental observations of significant Raman enhancement that has resulted in single molecule detection [24

24. M. Moskovits, L. Tay, J. Yang, and T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–227 (2002). [CrossRef]

26

26. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997). [CrossRef] [PubMed]

,48

48. B. Sharma, R. R. Frontiera, A. Henry, E. Ringe, and R. P. van Duyne, “SERS: Materials, applications, and the future,” Mater. Today 15(1-2), 16–25 (2012). [CrossRef]

], surface plasmon enhanced wave mixing like SHG on random [49

49. I. I. Smolyaninov, A. V. Zayats, and C. C. Davis, “Near-field second harmonic generation from a rough metal surface,” Phys. Rev. B 56(15), 9290–9293 (1997). [CrossRef]

51

51. C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, “Local second-harmonic generation enhancement on gold nanostructures probed by two-photon microscopy,” Opt. Lett. 28(9), 713–715 (2003). [CrossRef] [PubMed]

] and defined plasmonic structures [52

52. J. L. Coutaz, M. Nevière, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B Condens. Matter 32(4), 2227–2232 (1985). [CrossRef] [PubMed]

58

58. A. Lesuffleur, L. K. S. Kumar, and R. Gordon, “Enhanced second harmonic generation from nanoscale double-hole arrays in a gold film,” Appl. Phys. Lett. 88(26), 261104 (2006). [CrossRef]

], as well as the enhancement of linear processes such as optical absorption and luminescence [22

22. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006). [CrossRef] [PubMed]

,23

23. P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15(21), 14266–14274 (2007). [CrossRef] [PubMed]

]. Surface plasmon resonances are also ultra-sensitive to the dielectric properties of the metal plasmon host and its surrounding medium – a minor modification in the refractive index surrounding the metal surface can lead to a large shift of the plasmonic resonance [59

59. J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008). [CrossRef] [PubMed]

]. Such a phenomenon brings about the prospect of controlling light with light where the latter optical field induces optical property changes in the plasmonic structure, which in turn modifies the propagation of the original light. Motivated by this promise, researchers around the world have been pursuing the goal of practical all optical modulation or switching based on Kerr nonlinearities in either unconfined plasmonic materials [60

60. S. Link and M. A. El-Sayed, “Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods,” J. Phys. Chem. 103(40), 8410–8426 (1999). [CrossRef]

63

63. I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88(18), 187402 (2002). [CrossRef] [PubMed]

] or waveguides [64

64. A. V. Krasavin and N. I. Zheludev, “Active plasmonics: controlling signals in Au/ Ga waveguide using nanoscale structural transformations,” Appl. Phys. Lett. 84(8), 1416–1418 (2004). [CrossRef]

68

68. A. V. Krasavin, S. Randhawa, J.-S. Bouillard, J. Renger, R. Quidant, and A. V. Zayats, “Optically-programmable nonlinear photonic component for dielectric-loaded plasmonic circuitry,” Opt. Express 19(25), 25222–25229 (2011). [CrossRef] [PubMed]

], the demonstration of which has remained elusive to date.

Before embarking on the detailed calculation, we should perhaps mention that there exists more than one way to define the figure of merit for the enhancement of nonlinearity. Many scientists would consider the increase in nonlinear susceptibility to be a reliable measure of the enhancement, without taking into account increase in loss that often accompanies it and effectively negates all the benefits. To include the loss, the ratio of nonlinearity to absorption is often accepted as a more consistent figure of merit. Yet, even this figure of merit only takes into account what happens at low light intensities, and does not account for the inevitable saturation of nonlinearity, and disregarding saturation can read to wrong conclusions. From the engineering point of view what matters is whether a given outcome (full switching, high efficiency frequency conversion) can be achieved at all before the optical power deteriorates by many decibels. And it is in this “engineering criterion” where the plasmonic enhanced nonlinearity exhibits inherent weakness (obviously due to high loss in metal) as we explain in this work.

Consider the structure shown in Figs. 1(a)
Fig. 1 (a) Spherical Ag nanoparticle with the electric field distribution. (b) Elliptical nanoparticle with resonance at 1320 nm and its associated electric field distribution. (c) Extinction spectrum of the elliptical nanoparticle.
and 1(b) in which a nonlinear dielectric surrounds a metal nanoparticle. The goal of our treatment is to evaluate the enhancement of the third order nonlinear polarizability of this metamaterial, or, “artificial dielectric” whereby a metal nanoparticle is used to enhance the local field. In the course of this work we shall introduce figures of merit relevant to practical applications and see how the plasmonically enhanced nonlinear materials stack up against more conventional materials. To make our treatment both general and physically transparent we shall fully rely on analytical derivations, which, of course, will require certain simplifications that are justified as long as one is only looking for an order of magnitude estimate of the enhancement. We shall consider only spherical or elliptical (or spheroidal) nanoparticles, alone or coupled, but we shall indicate how this treatment can be expanded to other shapes of nanoparticles, such as nanoshells [69

69. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A Hybridization Model for the Plasmon Response of Complex Nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef] [PubMed]

], with all structures defined by just three parameters: resonant SP frequencyω0, quality factor Q, and effective SP mode volume Veff. We begin with Fig. 1(a) where is shown a spherical nanoparticles and Fig. 1(b) where is shown an elliptical nanoparticle resonant at the telecommunication wavelength of 1320 nm along with their numerically calculated field distributions. Also shown in Fig. 1(c) is the extinction spectrum of the elliptical particle where the LSP resonance can be observed.

2. Isolated metal nanoparticles embedded in the dielectric: linear properties

Consider a rather general scheme for determining the plasmoncially enhanced nonlinearity of the “artificial dielectric” shown in Fig. 2(a)
Fig. 2 Fields and polarizations in the plasmonically enhanced nonlinear metamaterials: (a) average E¯ω and local Eω electric fields, and dipole pω at the pump frequency ω ; (b) local nonlinear field Eω', dipole moment pnlω' and average nonlinear polarization P¯nlω' at the nonlinear output signal frequency ω'.
consisting of a concentration of Ns nanospheres, each of radius a, surrounded by a nonlinear dielectric with relative permittivityεd and nonlinear susceptibility tensor χ(3). In the most general case, χ(3)implies four wave interactions, with some of the waves being pumps (or switching signals) and some being the nonlinear output signals. In many practical cases, such as cross- and self-phase modulation, there is degeneracy and the number of interacting waves is reduced. In Fig. 2(a) we show just one pump (or switching) wave of frequency ω and one signal wave of frequency ω'.

As the pump wave propagates through the material, the average electric field is E¯ω and in this field the nanospheres become polarized, i.e. they acquire the dipole moment [70

70. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999) p.158.

]
pω=εmεdεm+2εd4πε0εda3E¯ω
(1)
as shown in Fig. 2(a). Using Drude model for the dielectric constant of metal εm=1ωp2/(ω2+jωγ)with plasma frequency ωpand scattering rate γ we can obtain
pω=αω02E¯ωω02ω2jωγαQL(ω)E¯ω
(2)
where ω0=ωp/1+2εd is the LSP resonant frequency [32

32. J. B. Khurgin, G. Sun, and R. A. Soref, “Electroluminescence efficiency enhancement using metal nanoparticles,” Appl. Phys. Lett. 93(2), 021120 (2008). [CrossRef]

], Q=ω0/γ is the Q-factor of the mode L(ω)=Q(1ω2/ω02)j is the resonant Lorentzian denominator, α=3ε0εdVβ is the polarizability of the nanoparticle and β=3εd/(2εd+1). For particles of different shapes, β will be somewhat different and polarization-dependent, yet will still be of the same order of magnitude as presented here. Similarly, although the value of the resonant frequency will change, because we are interested only in the order of magnitude results in this work, all the conclusions obtained here for spherical particles and their combinations will hold for particles of different shapes. It should also be noted that the Q-factor for different shapes depends only on the resonant frequency ω0since the decay rate γis shape independent. The only requirement is that the particle dimensions be much smaller than the incident wavelength in order to avoid scattering and diffraction effects.

The Q-factors for nanospheres of gold and silver, the two lowest loss plasmonic materials, are shown in Fig. 3
Fig. 3 Dispersions of Q-factors for gold and silver nanoparticles.
as functions of frequency. Near 1320 nm the Q-factor of bulk gold is about 12 and for bulk silver it is closer to 30 according to Johnson and Christy [71

71. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

], although for the silver nanoparticles interface scattering usually decreases the Q factor, consistent with measured Q-factors in visible and near infrared wavelengths [72

72. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]

76

76. E. J. R. Vesseur, R. de Waele, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Surface plasmon polariton modes in a single-crystal Au nanoresonator fabricated using focused-ion-beam milling,” Appl. Phys. Lett. 92(8), 083110 (2008). [CrossRef]

]. Although silver appears to be a superior material, gold is easier to work with as it does not readily oxidize. Because of this, in practice, the majority of researchers use gold when considering telecom wavelengths. In this work, however, we consider the best case scenario and hence use silver as an example with a Q of 20.

Equation (2) can be construed as the solution of the equation of motion of the harmonic oscillator, or the LSP mode characterized by the dipole moment p
d2pdt2+γdpdt=ω02p+ω02αE¯
(3)
and consisting of coupled oscillations of the free electron current insider inside the nanoparticle, and the electric field [32

32. J. B. Khurgin, G. Sun, and R. A. Soref, “Electroluminescence efficiency enhancement using metal nanoparticles,” Appl. Phys. Lett. 93(2), 021120 (2008). [CrossRef]

]
Eω(r)={p4πε0εda3r<a14πε0εdr3[3(pr^)r^p]r>a
(4)
inside and outside the nanoparticle, respectively, with the maximum field near the surface of nanoparticle equal to
Emax,ω2βQL(ω)E¯ω
(5)
Hence, near resonance the local field is enhanced by roughly a factor of 2Q relative to the average field.

If the nanoparticles are much smaller than the wavelength of light in the dielectric, one can apply a classical polarizability theory in which each nanoparticle is treated as polarizable atom. In this case, the “effective” dielectric constant of the composite medium (or a metamaterial if one wants to use a more modern, de rigueur terminology) can be found as the sum of the original dielectric constant and the susceptibility of the Ns density of nanoparticles,
εeff=εd+Nsαε0QL(ω)=εd(1+3fβQL(ω))
(6)
where we have introduced the “effective” filling factor f=NsV<<Q1.The latter condition is almost always satisfied in a medium with Q~20 and is required to avoid having to take into account the dipole-dipole interaction effects that would change the LSP resonant frequency according to the Lorentz-Lorentz formula. However, again, even for a very dense medium, frequency renormalization is not going to change the main conclusions of this work. In this approximation we can obtain an “effective” index of refraction
neff=εeff1/2nd[1+3fβ2Q2(1ω2/ω02)|L(ω)|2+3fβ2jQ|L(ω)|2],
(7)
wherend=εd. Similarly, the “effective” absorption coefficient is
αa=2πndλ3fβQ|L(ω)|2,
(8)
which also shows a Q-factor resonant enhancement.

3. Isolated metal nanoparticles embedded in a dielectric: third order non-linear properties

3.1 Nonlinear polarization and effective susceptibility

Let us now turn our attention to Fig. 2(b) where the local nonlinear microscopic polarization at the frequencyω',
pnl(r,t)=Pmax,nlω'G(r)ejω't
(9)
is established near the nanoparticle due to the presence of strong local pump field. As mentioned above, ω' could be the same as, or different then, the pump frequency ω that drives the nonlinear polarization. The maximum nonlinear polarization, usually occurring at the same location where the local pump field reaches maximum, is given by |pnl(rmax)|=Pmax,nlω'with G(r)being the normalized shape of the nonlinear polarization. The nonlinear polarization can now drive the LSP oscillations at the same frequency ω'according to the wave equation for the electric field of the LSP mode

2E(r,t)εr(r)c2д2дt2E(r,t)=1ε00c2д2дt2Pnl(r,t).
(10)

We seek solutions of the form
E(r,t)=lEmax,lω'Fl(r)ejω't
(11)
where Fl(r) is the normalized electric field of the l-th LSP eigen-mode with l=1being the dipole mode described by (4), whose amplitude Emax,1 we are trying to determine. Substituting Eq. (11) into Eq. (10) and using mode orthogonality, we obtain Emax,1for the steady state amplitude of the l=1 dipole mode driven by the nonlinear polarization at frequency ω' as
Emaxω'=Pmax,nlωκε0εdQL(ω')
(12)
where the overlap coefficient κ assuming that dielectric is non-dispersive and non-lossy, is given by
κ=εdF1(r)G(r)dV/(εr'ω)ωF12(r)dV.
(13)
According to Eq. (2) we can find the amplitude of the dipole mode as
pnlω'=32Vε0εdEmax,nlω'=32VκQL(ω)Pmax,nlω'.
(14)
and thus, the overall “effective” nonlinear polarization of the metamaterial will be given by
P¯nlω=32fκQL(ω)Pmax,nlω
(15)
As one can see, the local nonlinear polarization gets enhanced by being in resonance with the nanoparticle dipole mode and the enhancement is once again proportional to the Q-factor of this resonance.

It is instructive to re-cap the chain of events that leads to establishment of the enhanced nonlinear polarization illustrated in Fig. 2:

  • i. The average pump field E¯ωpolarizes the nanoparticles producing a linear dipole momentpωin each of them;
  • ii. Dipole oscillations are coupled with the linear local field Eω(r) in the vicinity of each nanoparticle. This field is resonantly enhanced by a factor on the order of Q relative to E¯ω;
  • iii. A local nonlinear polarization pnlω(r)is established in the vicinity of each nanoparticle. Since this polarization is proportional to the third order of the field, it is enhanced roughly by a factor of Q3;
  • iv. This polarization resonantly couples into the dipole LSP mode of the nanoparticle thus establishing the local nonlinear field Eω(r) and dipole moment pnlω. This resonance causes enhancement by another Q-factor;
  • v. Finally the localized dipoles pnlωcombine to establish the average nonlinear polarization pnlω.

Of course all the steps outlined above occur simultaneously and instantly, but in our view tracing the process step by step provides the clarity of the physical picture. We now turn our attention specifically to third order processes. Consider the third order nonlinearity in which the interaction of electromagnetic waves at three different frequencies described by the general local third order susceptibility is given by.

Pnlω1ω2+ω3(r)=ε0χ(3)(ω3,ω2,ω1)Eω1(r)Eω2*(r)Eω3(r).
(16)

In general, when all four frequencies ω1, ω2, ω3, and ω4=ω1ω2+ω3are different (but typically close to each other) the nonlinear process described by (16) is four wave mixing (FWM), whenω3=ω1,ω4=2ω1ω2 Eq. (16) describes optical parametric generation (OPG), when ω1=ω2and ω3=ω4it describes cross-phase modulation (XPM) and for the case when all frequencies are equal, Eq. (16) describes self-phase modulation (SPM). FWM and OPG are both of great interest in wavelength conversion while both XPM and SPM are important for optical switching.

In a composite medium the local fields Eωk(r)in Eq. (16) in the vicinity of the nanoparticle are all locally enhanced relative to the mean fields E¯ωk according to Eq. (5), i.e.
Eωk(r)=2βQL(ωk)E¯ωkF1(r).
(17)
Hence the local third-order nonlinear polarization is
Pnlω1ω2+ω3(r)=Pmax,nlω1ω2+ω3G(r)
(18)
where the amplitude is
Pmax,nlω1ω2+ω3=ε0|χ(3)(ω3,ω2,ω1)|(2βQ)3L(ω1)L*(ω2)L(ω3)E¯ω1E¯ω2*E¯ω3,
(19)
while the shape function is given by
G(r)=χ(3)/|χ(3)|F1(r)F1(r)F1(r),
(20)
and χ(3)/|χ(3)| is the normalized fourth-order nonlinear susceptibility tensor. Substituting Eq. (20) into Eq. (15) we obtain
Pnlω4=ε0χeff(3)(ω3,ω2,ω1)E¯ω1E¯ω2*E¯ω3
(21)
where for the typical case of all frequencies being close to each other, leading to the effective” nonlinear susceptibility as
χeff(3)32fκ3(2β)3Q4L2(ω)|L(ω)|2χ(3)
(22)
and the coupling coefficient for the third-order nonlinearity as

κ3=εdVeff,1r>aF1(r)χ(3)|χ(3)|F1(r)F1(r)F1(r)d3r.
(23)

The nonlinear susceptibility thus gets enhanced by a factor proportional to Q4. This is an outstanding result, exciting enough to attract attention of both plasmonic and nonlinear optics communities to this topic, which has witnessed an upsurge of research efforts and publications as described above. Indeed, even with f~0.001 filling factor one can expect more than a 100-fold enhancement of the susceptibility and the nonlinear refractive index, suggesting that one can achieve the same efficiency of nonlinear phase modulation in less than 1/100 of the length of a conventional device, and, more dramatically, the same efficiency for wavelength conversion in less than 1/10,000 of the length. It is these results that are often quoted as justification for using nanoplasmonics to enhance nonlinearity, yet one needs to maintain caution when it comes to reporting these results. Our prior research concerning the plasmonic enhancement of various emission processes including photoluminescence [27

27. G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94(10), 101103 (2009). [CrossRef]

], electroluminescence [32

32. J. B. Khurgin, G. Sun, and R. A. Soref, “Electroluminescence efficiency enhancement using metal nanoparticles,” Appl. Phys. Lett. 93(2), 021120 (2008). [CrossRef]

] and Raman scattering [34

34. G. Sun and J. B. Khurgin, “Origin of giant difference between fluorescence, resonance and non-resonance Raman scattering enhancement by surface plasmons,” Phys. Rev. A 85(6), 063410 (2012). [CrossRef]

] has shown that large enhancements are feasible only for those processes that have very low original efficiency (such as Raman scattering) but are far more modest for efficient processes such as fluorescence and electroluminescence. It is therefore reasonable to expect that there must exist an upper limit for plasmonic enhanced nonlinear effects.

3.2 Effective nonlinear index and maximum phase shift

To understand the limitations of the enhancement we shall first consider the XPM (or SPM) case for which the nonlinear polarization in Eq. (16) can be written as Pnlω2(r)=2ε0ndn2Iω1(r)Eω2(r), where n2=χ(3)η0/εdis the nonlinear index of the dielectric, and Iω1(r)=|Eω1(r)|2nd/2η0 is the local intensity Similarly, we now introduce the “effective” nonlinear index as n2,eff=χeff(3)η0/εdand write the average nonlinear polarization as P¯nlω2=2ε0ndn2,effI¯ω1E¯ω2 According to Eq. (22), the effective nonlinear index gets enhanced by the same giant factor proportional to Q4,

n2,eff32fκ3(2β)3Q4L2(ω2)|L(ω1)|2n2.
(24)

ΔΦmax=|L(ω1)|23fβQn2,effndI¯0κ3(2β)2Q3L2(ω2)n2ndI¯0.
(26)

Achieving the π-phase shift required to achieve photonic switching would then require at resonanceI¯π~πnd[κ3n2(2β)2Q3]1. If we assumeβ1.45(estimated numerically for the actual ellipsoid resonant at 1320 nm of Fig. 1(b), Q20and a large nonlinear index characteristic of chalcogenide glass n2=1013cm2/W, the required switching intensity is then on the order of I¯π~1.6×109W/cm2, which is quite high. This means that the giant nonlinear index enhancement given by Eq. (24) can only be used to reduce the length of the device, while the switching intensity remains quite high – requiring peak powers of about tens of W into a 1µm2 waveguide.

To elucidate these effects further, let us define the maximum local nonlinear index change attainable in a given material asΔnmaxand obtain the maximum change of effective index

Δneff,max=n2,effI¯3fκ3βQ2L2(ω2)Δnmax.
(27)

As we can see, the enhancement is now only proportional to Q2. This result makes perfect sense if we recognize that the local change of the dielectric constant Δεd,max=2ndΔnmax simply causes a shift of the LSP resonant frequency ω0=ωp/1+2εd, which in turn changes the effective dielectric constant of the metamaterial εeff,maxaccording to (6) which is proportional to Q2obtained via the differentiate of the Lorentzian, L(ω), in Eq. (6). It is crucial to note that this factor of Q2in Eq. (27) is applicable not just to an isolated nanoparticle but also to more sophisticate structures, like dimers and nano-antennae – in each case the local change of index causes the shift of the plasmonic resonance proportional to the same factor of Q2.

It follows then that maximum obtainable phase shift Eq. (26) can be put in terms of the maximum change of the effective index as

ΔΦmax=2πλαaΔneff,max=κ3Q|L(ω1)|2L2(ω2)Δnmaxnd.
(28)

The simple meaning of Eq. (28) is that, even if we assume an enormous local nonlinear index change of 1% (i.e. a local intensity of 1011 W/cm2), we cannot expect to get a phase shift higher than 0.1, almost two orders of magnitude less than required for π-phase shift switching. Notice also that for closely spaced frequencies of pump and signal, the maximum phase shift does not depend strongly on the position relative to the LSP resonance as an increase in the nonlinearity is balanced by an increase in absorption. It should be also noted that Eq. (28) can be used independent of the origin of the index change, i.e. it does not have to be all optical but can also be electro-optical or thermo-optical.

3.3 Efficiency of Frequency conversion

It is easy to see that small maximum phase shift for XPM or SPM corresponds to even smaller efficiency of the frequency conversion for FWM or OPG. Indeed the growth of the idler E¯ω3(z)in the presence of pump I¯ω1(z)=I¯0eαaz and signal Eω2*(z)=Eseαa2z can be found as
E¯i(z)=2πλαan2,effI¯0Es[1eαaz]eαaz/2
(29)
with a maximum near z=αa1ln3 equal to
E¯i,max=2πλαa233/2n2,effI¯0Es=233/2ΔΦmaxEs.
(30)
Therefore, the maximum conversion efficiency from the signal to the idler is
Ii/Is~0.15ΔΦmax2
(31)
and under no conceivable conditions can it exceed −30dB.

Here we also mention that one could use modulation of the refractive index of the metal itself, but it is difficult to see how one can change the index of a metal by more than 1% unless one operates near the interband transitions where the Q-factor is greatly reduced which defeats the whole purpose of plasmonic enhancement.

4. Enhancement of nonlinearity in more complex structures: dimers or nanolens

4.1 Field Enhancement

Consider two spherical nanoparticles of radii a1and a2separated by a vectorr12 as shown in Fig. 4(a)
Fig. 4 (a) Spherical nanoparticle dimer with the electric field distribution. (b) Elliptical nanoparticle dimer resonant at 1320 nm and associated electric field distribution. (c) Extinction spectrum of the elliptical dimer.
. The dipole oscillation Eq. (3) is augmented by the dipole-dipole interaction between the two dipoles associated with the two, coupled nanoparticles,

d2p1(2)dt2+γdp1(2)dt=ω02p1(2)+ω02α1(2)E¯ω+ω02α1(2)2p2(1)4πε0εdr123.
(32)

Following our prior work [79

79. G. Sun and J. B. Khurgin, “Comparative study of field enhancement between isolated and coupled metal nanoparticles: an analytical approach,” Appl. Phys. Lett. 97(26), 263110 (2010). [CrossRef]

,80

80. G. Sun and J. B. Khurgin, “Optimization of the nanolens consisting of coupled metal nanoparticles: an analytical approach,” Appl. Phys. Lett. 98(15), 153115 (2011). [CrossRef]

] we obtain the expression for the maximum fields near the nanoparticles
Emax,1(2)ω=2QL(ω)β+2β2Q(a2(1)r12)3L2(ω)4β2Q2(a1a2r122)3E¯ω.
(33)
In the limit of a20,a1r12, one gets

Emax,1ω2βQL(ω)E¯ω;Emax,2ω[2βQL(ω)]2E¯ω.
(34)

As one can see in Fig. 4(a) the field is greatly enhanced in the vicinity of the smaller particle. In our prior work [80

80. G. Sun and J. B. Khurgin, “Optimization of the nanolens consisting of coupled metal nanoparticles: an analytical approach,” Appl. Phys. Lett. 98(15), 153115 (2011). [CrossRef]

], using more precise calculations, we have shown that the simple analytical result of Eq. (34) can be used as an upper bound on the field enhancement in the nanolens, or, in fact, in the nano-gap between any two particles. In Fig. 4(b) we show the dimer of elliptical nanoparticles resonating at 1320 nm, as well as its extinction spectrum shown in Fig. 4(c) obtained using numerical calculations. Enhancement on the order of Q2 for the asymmetric dimer is realistic and motivates effects on nonlinearities associated with a dimer.

4.2 Effective nonlinearity of dimer

The high field in the vicinity of nanoparticle 2, the smaller nanoparticle associated with a dimer, will cause a nonlinear polarization similar to Eq. (9) as
Pnl,2ω(r,t)=Pmax,2ωG2(r)ejωt
(35)
where G2(r)is the normalized distribution of nonlinear polarization near the smaller particle. According to Eq. (14) this polarization will induce the nonlinear dipoles of two particles as

pnl,1ω=2πa23Q22β(a1r12)3κPmax,2ωL2(ω)4β2Q2(a1a2r122)3pnl,2ω=2πa23QL(ω)κPmax,2ωL2(ω)4β2Q2(a1a2r122)3.
(36)

One can see by comparing Eq. (36) to Eq. (14), the nonlinear dipole of the larger nanoparticle 1 experiences additional enhancement relative to the dipole of the smaller nanoparticle 2. But note that now, the volume of the smaller nanoparticle is present in the numerator of Eq. (36), hence the situation that is optimum for external field enhancement in a nanolens, i.e. the limit of a20,a1r12, is far from being optimal for the enhancement of nonlinear polarization.

Let us now estimate the effective nonlinear susceptibility of the nanolens. Finding from Eq. (19) and Eq. (33) the maximum nonlinear polarization near the smaller nanoparticle for the case of FWM and substituting it into Eq. (36), we obtain the nonlinear dipole of the larger particle 1 and the effective third order susceptibility becomes

χeff(3)=24fχ(3)κ3βQ5[L(ω)(a2r12)3+2βQ(a1a2r122)3]|L(ω)+2βQ(a1r12)3|2[L2(ω)4β2Q2(a1a2r122)3]2|L2(ω)4β2Q2(a1a2r122)3|2
(37)

So, what is the maximum attainable nonlinearity enhancement? According to Eq. (34) the local field gets enhanced by a factor proportional to Q2 instead of Q for a single nanoparticle. For Raman scattering, which is also a third-order nonlinear process, the enhancement afforded by a nanolens system can be Q8 instead of Q4 for a single nanoparticle, a tremendous improvement. But we cannot expect similar improvement for the FWM and other third order nonlinear processes because the largest enhancement of the local field is always attained when the volume of the smaller particle becomes negligibly small. The key characteristic of Eq. (37), already noted above, is the presence of the volume of the smaller nanoparticle in the numerator, hence the optimum condition for maximum effective χ(3)will not coincide with the condition of maximum local field enhancement and hence the overall enhancement will be much less than Q8.

Optimizing Eq. (37) we find out that χ(3)enhancement reaches its maximum when4β2Q2(a1a2r122)3=1/3 and is then given by
χeff(3)5fχ(3)κ3β2Q6.
(38)
As seen, the enhancement of χ(3) and the nonlinear index n2, provided by the nanolens system is only proportional to Q6. This is rather easy to interpret. The local intensity in the nanolens gets enhanced by a factor proportional to Q4, but then, the nonlinear refractive index change gets enhanced by the same additional factor Q2, whether it is a single particle, nanolens, dimer, or nano-antenna. The additional enhancement provided by coupled particle composites Eq. (38) compared to the isolate nanoparticle composite Eq. (22) is aboutχeff,2(3)/χeff,1(3)0.5Q2/β i.e a factor on the order of 200. Overall enhancement for the previously considered case of β~1.35, Q~20and f=0.001 in chalcogenide glass can be as high as 3 × 105, but the relevant question is what does it mean in terms of the maximum phase shift that can be obtain.

4.3 Limitations of maximum phase shift with dimers

This maximum phase shift for a dimer can be obtained in a manner similar to that used to obtain Eq. (26). Namely,
ΔΦmax1.7κ3βQ5n2ndI¯0.
(39)
Therefore, the pump optical intensity required to achieve a π-phase shift is I¯π~1.5×107W/cm2, i.e. only about 150 mW of peak power into a 1µm2 waveguide. This appears to be a reasonable power, but, of course the problem is that the local intensity is enhanced according to Eq. (33) roughly by Imax/I¯π~9β4Q45×106, indicating that the local intensity will be on the scale ofImax1014W/cm2which is far beyond the optical damage threshold. If we introduce once again the maximum local nonlinear index as Δnmax=n2I¯max=9β4Q4n2I¯0, Eq. (39) can be re-written for the case of a nano-lens as

ΔΦmax0.2κ3Qβ3Δnmaxnd.
(40)

This result for the nano-lens is even worse (by a factor of about 10) than the result in Eq. (28) obtained for the isolated nanoparticles. Clearly, the dependence κ3Qis common to any type of nanostructure, monomer, dimer, trimer, or nano-antenna. The maximum achievable index of refraction Δnmaxchanges the resonant LSP frequency, which provides enhancement by the factor of Q2 but the absorption coefficient also gets enhanced by a factor of Q leaving only a single factor of Q enhancement. The factor in front of κ3Qis reduced in dimers and more complicated structures relative to the monomers simply because a smaller fraction of the mode energy is contained in the region where the index change is maximal. Hence, one should not expect any improvement in the maximum obtainable nonlinear phase shiftΔΦmaxbeyond a single factor of Q in more complex structures like trimers, bowtie antennae and so on, even if the effective nonlinear index can be enhanced beyond the already huge enhancement seen in Eq. (38). The giant enhancement of nonlinearity will only mean that the nonlinear phase shift will saturate at a much shorter distance, but at essentially the same value given by Eq. (28) or less, suggesting to the best of our knowledge, that with existing materials it is impossible to achieve true all-optical switching using only plasmonic enhancement.

5. Results and discussion

In this section we illustrate the main results of our derivations and discuss conclusions. Consider first Fig. 5(a)
Fig. 5 Nonlinear phase shift in the chalcogenide waveguide doped with isolated Ag spheroids (a-c) and the dimers (d-f) with different filling factors f at various intensities. The phase shift of undoped waveguides is shown by dashed line.
where the nonlinear phase shift ΔΦ(z)induced by either XPM or SPM, is shown as a function of the propagation length in a chalcogenide glass waveguide (nonlinear index n2=1013cm2/W) doped with the isolated Ag spheroids of Fig. 1(b). The incident pump power density is I¯(0)=107W/cm2 (corresponding to 100 mW of 1320 nm pump power into 1 µm2 waveguide). The results are shown for four different filling factors from f = 10−6 to f = 10−3 as well as for the pure chalcogenide waveguide without Ag spheroids (dashed line). As one can see, almost a three order of magnitude enhancement is achieved at short propagation lengths for a densely filled waveguide with f = 10−3 but the nonlinear phase shift saturates also at a very short distance of about 5µm with maximum phase shift of only 0.01 rad. For lower filling factors, the initial enhancement is less but the saturation distance is also longer due to reduced absorption and in the end the maximum phase shift saturates at the same value of 0.01 rad. For the un-enhanced waveguide saturation does not occur (the absorption length in the waveguide is longer than 1cm) and as one can see for long propagation distances the un-enhanced structure outperforms all the plasmon-enhanced ones.

If we increase the input power by a factor of 10 to I¯(0)=108W/cm2 the local power density near the metal surface will reach roughly 3×1011W/cm2which should, in principle, lead to a local index change of 3%. This index change however is unattainable. First, optical damage will most probably ensue at such power levels, but even in the absence of it, the index change will saturate at a value that is less than 1% [78

78. B. Borchers, C. Brée, S. Birkholz, A. Demircan, and G. Steinmeyer, “Saturation of the all-optical Kerr effect in solids,” Opt. Lett. 37(9), 1541–1543 (2012). [CrossRef] [PubMed]

]. To be optimistic, we disregard the possibility of optical damage but still consider the saturation of the nonlinear change with the results seen in Fig. 5(b). Once again, significant enhancement can be obtained at small propagation distances, but at saturation the maximum phase shift is still less than 0.05 rad.

When the input power is increased by yet another factor of 10 to I¯(0)=109W/cm2 (more than 10W of peak power) as shown in Fig. 5(c), saturation still prevents the nonlinear index change from exceeding 0.1rad although this change takes place over propagation distance of no more than 10 µm. Again, we stress here that in real structures optical damage most likely will occur for local power densities in excess of a TW/cm2, which would occur near the metal surface. Note that for the waveguides without nanoparticles the nonlinear phase shift of π radians required for switching is achieved with a few mm of propagation distance.

We now turn our attention to chalcogenide waveguides doped with optimized Ag dimers and first consider them at a relatively low input power density of just I¯(0)=104W/cm2with the results shown in Fig. 5(d). As one can see, at small propagation distances, the nonlinearity gets enhanced by more than 5 orders of magnitude and an appreciable phase change of 0.001 rad is achieved with a propagation distance of only 10 µm, after which saturation occurs.

Further increase of the input power density to I¯(0)=106W/cm2 [Fig. 5(e)] and I¯(0)=108W/cm2 [Fig. 5(f)] does not lead to a significant increase in the maximum nonlinear phase shift – it remains below 0.02rad and thus insufficient for optical switching.

Considering frequency conversion by means of FWM we first consider the same waveguide doped with isolated Ag spheroids and plot the conversion efficiency vs. distance in Fig. 6(a)
Fig. 6 Conversion efficiency in the FWM in the waveguide doped with isolated Ag spheroids (a) and the dimers (d-f) with different filling factors f. The phase shift of undoped waveguides is shown by dashed line.
for an input pump power of I¯(0)=107W/cm2. Once gain we obtain tremendous enhancement of conversion efficiency at short distances. In only a few micrometers one can attain conversion efficiency of nearly 0.01% (−40dB) which can be sufficient for some applications, but probably not for frequency conversion in optical communication schemes or for optical switching. At longer distances the conversion efficiency deteriorates due to absorption. Note that one can adjust the distance at which maximum conversion efficiency is achieved by varying f.

Going to the waveguides doped with dimers [Fig. 6(b)] allows one to reach conversion efficiency enhancement of nearly 10 orders of magnitude for very short structures, but the absolute value of conversion efficiency is predicted not to exceed 0.0001%. Perhaps this conversion efficiency is sufficient for some specialized applications, such as generating of entangled pairs of photons or autocorrelation measurements, but it is not enough for signal processing.

6. Conclusion

Thus we arrive at a rather dichotomous conclusion. On one hand, using waveguides impregnated with metallic monomers, dimers, and other constructions (one may call them plasmonic metamaterials) one can achieve huge enhancements of the effective nonlinear index on the order of 105 and more due to the high degree of field concentration associated with the “hot spots”. On the other hand, strong absorption in the metal causes saturation of the both the nonlinear phase shift for SPM and XPM applications or frequency conversion efficiency in the case of FWM and OPG at very short distances. Given the fact that maximum local index change is limited, generously, to about 1% due to optical damage, the nonlinear phase shift saturates at a very small value of a few tens of milli-radians – which is insufficient for any photonic switching operation. Similarly, conversion efficiency saturates at values of less than −30dB making use of plasmonic nonlinear metamaterials for this purpose highly inefficient. It is also clear that changes in Q by a factor of 2-3 that might be attainable in silver (although not demonstrated to date due to oxidation and surface scattering) will not change the results in any substantial way, and only assure earlier saturation of the nonlinear conversion. The one and only advantage of nonlinear plasmonic metamaterial is that nonlinear effects may be observable at very small propagation distances of a few micrometers with reasonable (but not low!) optical powers. At the moment, there is a giant chasm between being observed and being practical and at this point, with the presently available metals and nonlinear materials, one cannot see how nonlinear plasmonic metamaterials can bridge this chasm.

This conclusion is in line with our general conclusions about utility of plasmonic enhancement – the devices that have inherently low efficiency (e.g. Raman sensors) can be enhanced spectacularly, with important implications for sensing, but the devices that are already reasonably efficient (LED, Solar cell, etc.) will only see their performance deteriorate when metal is introduced. The nonlinear devices are no different – when propagation distance is short, very low efficiency can be enhanced significantly, but overall efficiency will still remain disappointingly low. For the longer devices the performance will deteriorate.

In retrospect, our rather unenthusiastic conclusion concerning the prospects of using plasmonic resonances to enhance nonlinearities does not appear to be surprising at all. Numerous resonant schemes for nonlinearity enhancement have been proposed and investigated at length [81

81. J. B. Khurgin, “Performance of nonlinear photonic crystal devices at high bit rates,” Opt. Lett. 30(6), 643–645 (2005). [CrossRef] [PubMed]

,82

82. J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24(12), 847–849 (1999). [CrossRef] [PubMed]

]. Some of the schemes rely upon intrinsic material resonances; others try to take advantage of photonic resonant structures, such as micro-resonators and photonic crystals. The Q-factor of the resonances ranges from a few hundreds to tens of thousands, and yet in the end, none of the resonant schemes have found any practical applications to date. This is due to the fact that in general, resonance is always associated with excessive absorption and dispersion. To this day optical fibers remain the nonlinear medium of choice in which low nonlinear coefficients are more than compensated for by long propagation lengths and a high degree of field confinement. All kinds of all-optical switching and frequency conversion techniques had been successfully demonstrated in fiber [83

83. G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2012).

]. The only other media in which all-optical switching has been consistently demonstrated is semiconductor optical amplifiers (SOAs) in which loss is simply offset by optical gain. Neither fibers nor SOAs rely upon any resonance despite its apparent appeal as one always tries to avoid loss and any excessive dispersion.

If the numerous relatively high Q resonant schemes for enhancing optical nonlinearities have failed to achieve practicality, it would have been naive to expect plasmonic resonances in metal nanoparticles with Q’s barely of the order of 10 to succeed where so many have failed. In retrospect, this work only confirms the obvious. And yet, this obvious fact has not been universally accepted by the community, and we hope that our effort has been useful as it has revealed the nature and limitations of the plasmonic enhancement of χ(3)in great detail and without reliance on excessive numerical modeling.

Not to end on entirely pessimistic note, we should mention that there exist broad classes of nonlinearities that rely on temperature change where the index change in excess of 1% can be achieved. That includes both conventional thermo optical effects in standard materials such as Si [84

84. F. G. Della Corte, M. Esposito Montefusco, L. Moretti, I. Rendina, and G. Cocorullo, “Temperature dependence analysis of the thermo-optic effect in silicon by single and double oscillator models,” J. Appl. Phys. 88(12), 7115–7119 (2000). [CrossRef]

] and the thermally induced metal-to-insulator transition in materials like VO2 [85

85. I. Karakurt, C. H. Adams, P. Leiderer, J. Boneberg, and R. F. Haglund Jr., “Nonreciprocal switching of VO2 thin films on microstructured surfaces,” Opt. Lett. 35(10), 1506–1508 (2010). [CrossRef] [PubMed]

]. The index change in the latter is on the order of 1! But the switching time is determined by the heat transfer and is typically very slow. In our opinion, this is where plasmonics can shift the whole paradigm since the local heating can be reduced to a nanometer scale (which is of course the case for nanometer scale particles) and the heat diffusion time would be on the order of picoseconds and one could talk about ultrafast thermal nonlinearities! We shall explore this idea as well as using non-metallic structures with negative permittivity and lower loss in future publications.

It is not our goal to make predictions of where this research will go in the future, our sole purpose was to provide a set of simple expressions and numbers for others so they can ascertain the prospects for using nonlinear plasmonic metamaterials for their own applications. Still, we can make a broad statement, that plasmonically enhanced structures in nonlinear optics might not find too many applications which require even modest efficiency, such as switching, wavelength conversion, etc., but may be of use in such applications where efficiency is not much of an issue such as in sensing and also in the fundamental studies of the optical properties of different materials under extremely high fields.

Acknowledgments

This work was supported by the Air Force Office of Scientific Research under the contract FA9550-10-1-0417 and Mid-InfraRed Technologies for Health and the Environment Research Center (National Science Foundation Grants MIRTHE NSF-ERC; EEC0540832)

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10.

L. B. Fu, M. Rochette, V. G. Ta’eed, D. J. Moss, and B. J. Eggleton, “Investigation of self-phase modulation based optical regeneration in single mode As2Se3 chalcogenide glass fiber,” Opt. Express 13(19), 7637–7644 (2005). [CrossRef] [PubMed]

11.

S. X. Qian, J. B. Snow, H. M. Tzeng, and R. K. Chang, “Lasing droplets: Highlighting the liquid-air interface by laser emission,” Science 231(4737), 486–488 (1986). [CrossRef] [PubMed]

12.

H. B. Lin and A. J. Campillo, “CW nonlinear optics in droplet microcavities diplaying enhanced gain,” Phys. Rev. Lett. 73(18), 2440–2443 (1994). [CrossRef] [PubMed]

13.

J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24(12), 847–849 (1999). [CrossRef] [PubMed]

14.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998). [CrossRef]

15.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102(20), 203902 (2009). [CrossRef] [PubMed]

16.

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12(10), 104003 (2010). [CrossRef]

17.

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: Comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005). [CrossRef]

18.

R. W. Hellwarth, “Control of fluorescent pulsations,” in Advances in Quantum Electronics, R. Singer, ed. (Columbia University, 1961), p. 334.

19.

L. Hargrove, R. L. Fork, and R. L. Pollack, “Locking of HeNe laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. 5(1), 4–5 (1964). [CrossRef]

20.

A. J. DeMaria, D. A. Stetson, and H. Heyma, “Mode locking of a Nd3+‐doped glass laser,” Appl. Phys. Lett. 8(1), 22–24 (1966). [CrossRef]

21.

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011). [CrossRef] [PubMed]

22.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006). [CrossRef] [PubMed]

23.

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15(21), 14266–14274 (2007). [CrossRef] [PubMed]

24.

M. Moskovits, L. Tay, J. Yang, and T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–227 (2002). [CrossRef]

25.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997). [CrossRef]

26.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997). [CrossRef] [PubMed]

27.

G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94(10), 101103 (2009). [CrossRef]

28.

G. Sun, J. B. Khurgin, and A. Bratkovsky, “Coupled-mode theory of field enhancement in complex metal nanostructures,” Phys. Rev. B 84(4), 045415 (2011). [CrossRef]

29.

G. Sun and J. B. Khurgin, “Theory of optical emission enhancement by coupled metal nanoparticles: An analytical approach,” Appl. Phys. Lett. 98(11), 113116 (2011). [CrossRef]

30.

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. 99(21), 211106 (2011). [CrossRef]

31.

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012). [CrossRef]

32.

J. B. Khurgin, G. Sun, and R. A. Soref, “Electroluminescence efficiency enhancement using metal nanoparticles,” Appl. Phys. Lett. 93(2), 021120 (2008). [CrossRef]

33.

J. B. Khurgin, G. Sun, and R. A. Soref, “Practical limits of absorption enhancement near metal nanoparticles,” Appl. Phys. Lett. 94(7), 071103 (2009). [CrossRef]

34.

G. Sun and J. B. Khurgin, “Origin of giant difference between fluorescence, resonance and non-resonance Raman scattering enhancement by surface plasmons,” Phys. Rev. A 85(6), 063410 (2012). [CrossRef]

35.

K. Okamoto, I. Niki, A. Scherer, Y. Narukawa, T. Mukai, and Y. Kawakami, “Surface plasmon enhanced spontaneous emission rate of InGaN/GaN quantum wells probed by time-resolved photoluminescence spectroscopy,” Appl. Phys. Lett. 87(7), 071102 (2005). [CrossRef]

36.

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells,” J. Appl. Phys. 101(9), 093105 (2007). [CrossRef]

37.

S. C. Lee, S. Krishna, and S. R. J. Brueck, “Quantum dot infrared photodetector enhanced by surface plasma wave excitation,” Opt. Express 17(25), 23160–23168 (2009). [CrossRef] [PubMed]

38.

M. B. Dühring, N. Asger Mortensen, and O. Sigmund, “Plasmonic versus dielectric enhancement in thin-film solar cells,” Appl. Phys. Lett. 100(21), 211914 (2012). [CrossRef]

39.

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974). [CrossRef]

40.

D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys. 78(9), 5324 (1983). [CrossRef]

41.

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57(3), 783–826 (1985). [CrossRef]

42.

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39, 375–377 (1974).

43.

C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett. 46(2), 145–148 (1981). [CrossRef]

44.

A. Wokaun, J. G. Bergman, J. P. Heritage, A. M. Glass, P. F. Liao, and D. H. Olson, “Surface second-harmonic generation from metal island films and microlithographic structures,” Phys. Rev. B 24(2), 849–856 (1981). [CrossRef]

45.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012). [CrossRef]

46.

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19(3), 2064–2078 (2011). [CrossRef] [PubMed]

47.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005). [CrossRef]

48.

B. Sharma, R. R. Frontiera, A. Henry, E. Ringe, and R. P. van Duyne, “SERS: Materials, applications, and the future,” Mater. Today 15(1-2), 16–25 (2012). [CrossRef]

49.

I. I. Smolyaninov, A. V. Zayats, and C. C. Davis, “Near-field second harmonic generation from a rough metal surface,” Phys. Rev. B 56(15), 9290–9293 (1997). [CrossRef]

50.

S. I. Bozhevolnyi, J. Beermann, and V. Coello, “Direct observation of localized second-harmonic enhancement in random metal nanostructures,” Phys. Rev. Lett. 90(19), 197403 (2003). [CrossRef] [PubMed]

51.

C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, “Local second-harmonic generation enhancement on gold nanostructures probed by two-photon microscopy,” Opt. Lett. 28(9), 713–715 (2003). [CrossRef] [PubMed]

52.

J. L. Coutaz, M. Nevière, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B Condens. Matter 32(4), 2227–2232 (1985). [CrossRef] [PubMed]

53.

S. Linden, F. B. P. Niesler, J. Förstner, Y. Grynko, T. Meier, and M. Wegener, “Collective effects in second-harmonic generation from split-ring-resonator arrays,” Phys. Rev. Lett. 109(1), 015502 (2012). [CrossRef] [PubMed]

54.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006). [CrossRef] [PubMed]

55.

N. Feth, S. Linden, M. W. Klein, M. Decker, F. B. P. Niesler, Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, J. V. Moloney, and M. Wegener, “Second-harmonic generation from complementary split-ring resonators,” Opt. Lett. 33(17), 1975–1977 (2008). [CrossRef] [PubMed]

56.

M. D. McMahon, R. Lopez, R. F. Haglund Jr, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401 (2006). [CrossRef]

57.

T. Xu, X. Jiao, G. P. Zhang, and S. Blair, “Second-harmonic emission from sub-wavelength apertures: effects of aperture symmetry and lattice arrangement,” Opt. Express 15(21), 13894–13906 (2007). [CrossRef] [PubMed]

58.

A. Lesuffleur, L. K. S. Kumar, and R. Gordon, “Enhanced second harmonic generation from nanoscale double-hole arrays in a gold film,” Appl. Phys. Lett. 88(26), 261104 (2006). [CrossRef]

59.

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008). [CrossRef] [PubMed]

60.

S. Link and M. A. El-Sayed, “Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods,” J. Phys. Chem. 103(40), 8410–8426 (1999). [CrossRef]

61.

H. Baida, D. Mongin, D. Christofilos, G. Bachelier, A. Crut, P. Maioli, N. Del Fatti, and F. Vallée, “Ultrafast nonlinear optical response of a single gold nanorod near its surface plasmon resonance,” Phys. Rev. Lett. 107(5), 057402 (2011). [CrossRef] [PubMed]

62.

M. Abb, P. Albella, J. Aizpurua, and O. L. Muskens, “All-optical control of a single plasmonic nanoantenna-ITO hybrid,” Nano Lett. 11(6), 2457–2463 (2011). [CrossRef] [PubMed]

63.

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88(18), 187402 (2002). [CrossRef] [PubMed]

64.

A. V. Krasavin and N. I. Zheludev, “Active plasmonics: controlling signals in Au/ Ga waveguide using nanoscale structural transformations,” Appl. Phys. Lett. 84(8), 1416–1418 (2004). [CrossRef]

65.

D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007). [CrossRef]

66.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011). [CrossRef] [PubMed]

67.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and M. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3(1), 55–58 (2009). [CrossRef]

68.

A. V. Krasavin, S. Randhawa, J.-S. Bouillard, J. Renger, R. Quidant, and A. V. Zayats, “Optically-programmable nonlinear photonic component for dielectric-loaded plasmonic circuitry,” Opt. Express 19(25), 25222–25229 (2011). [CrossRef] [PubMed]

69.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A Hybridization Model for the Plasmon Response of Complex Nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef] [PubMed]

70.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999) p.158.

71.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

72.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]

73.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]

74.

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]

75.

J.-C. Weeber, A. Bouhelier, G. Colas de Francs, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett. 7(5), 1352–1359 (2007). [CrossRef] [PubMed]

76.

E. J. R. Vesseur, R. de Waele, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Surface plasmon polariton modes in a single-crystal Au nanoresonator fabricated using focused-ion-beam milling,” Appl. Phys. Lett. 92(8), 083110 (2008). [CrossRef]

77.

S. Wu, X. C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett. 61(8), 919–921 (1992). [CrossRef]

78.

B. Borchers, C. Brée, S. Birkholz, A. Demircan, and G. Steinmeyer, “Saturation of the all-optical Kerr effect in solids,” Opt. Lett. 37(9), 1541–1543 (2012). [CrossRef] [PubMed]

79.

G. Sun and J. B. Khurgin, “Comparative study of field enhancement between isolated and coupled metal nanoparticles: an analytical approach,” Appl. Phys. Lett. 97(26), 263110 (2010). [CrossRef]

80.

G. Sun and J. B. Khurgin, “Optimization of the nanolens consisting of coupled metal nanoparticles: an analytical approach,” Appl. Phys. Lett. 98(15), 153115 (2011). [CrossRef]

81.

J. B. Khurgin, “Performance of nonlinear photonic crystal devices at high bit rates,” Opt. Lett. 30(6), 643–645 (2005). [CrossRef] [PubMed]

82.

J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24(12), 847–849 (1999). [CrossRef] [PubMed]

83.

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2012).

84.

F. G. Della Corte, M. Esposito Montefusco, L. Moretti, I. Rendina, and G. Cocorullo, “Temperature dependence analysis of the thermo-optic effect in silicon by single and double oscillator models,” J. Appl. Phys. 88(12), 7115–7119 (2000). [CrossRef]

85.

I. Karakurt, C. H. Adams, P. Leiderer, J. Boneberg, and R. F. Haglund Jr., “Nonreciprocal switching of VO2 thin films on microstructured surfaces,” Opt. Lett. 35(10), 1506–1508 (2010). [CrossRef] [PubMed]

OCIS Codes
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(190.4223) Nonlinear optics : Nonlinear wave mixing
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Plasmonics

History
Original Manuscript: August 15, 2013
Revised Manuscript: October 16, 2013
Manuscript Accepted: October 16, 2013
Published: November 4, 2013

Virtual Issues
Surface Plasmon Photonics (2013) Optics Express

Citation
Jacob B. Khurgin and Greg Sun, "Plasmonic enhancement of the third order nonlinear optical phenomena: Figures of merit," Opt. Express 21, 27460-27480 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-27460


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References

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  11. S. X. Qian, J. B. Snow, H. M. Tzeng, and R. K. Chang, “Lasing droplets: Highlighting the liquid-air interface by laser emission,” Science231(4737), 486–488 (1986). [CrossRef] [PubMed]
  12. H. B. Lin and A. J. Campillo, “CW nonlinear optics in droplet microcavities diplaying enhanced gain,” Phys. Rev. Lett.73(18), 2440–2443 (1994). [CrossRef] [PubMed]
  13. J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett.24(12), 847–849 (1999). [CrossRef] [PubMed]
  14. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett.81(19), 4136–4139 (1998). [CrossRef]
  15. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102(20), 203902 (2009). [CrossRef] [PubMed]
  16. C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt.12(10), 104003 (2010). [CrossRef]
  17. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: Comparative analysis,” J. Opt. Soc. Am. B22(5), 1062–1074 (2005). [CrossRef]
  18. R. W. Hellwarth, “Control of fluorescent pulsations,” in Advances in Quantum Electronics, R. Singer, ed. (Columbia University, 1961), p. 334.
  19. L. Hargrove, R. L. Fork, and R. L. Pollack, “Locking of HeNe laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett.5(1), 4–5 (1964). [CrossRef]
  20. A. J. DeMaria, D. A. Stetson, and H. Heyma, “Mode locking of a Nd3+‐doped glass laser,” Appl. Phys. Lett.8(1), 22–24 (1966). [CrossRef]
  21. M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express19(22), 22029–22106 (2011). [CrossRef] [PubMed]
  22. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett.97(1), 017402 (2006). [CrossRef] [PubMed]
  23. P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express15(21), 14266–14274 (2007). [CrossRef] [PubMed]
  24. M. Moskovits, L. Tay, J. Yang, and T. Haslett, “SERS and the single molecule,” Top. Appl. Phys.82, 215–227 (2002). [CrossRef]
  25. K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett.78(9), 1667–1670 (1997). [CrossRef]
  26. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science275(5303), 1102–1106 (1997). [CrossRef] [PubMed]
  27. G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett.94(10), 101103 (2009). [CrossRef]
  28. G. Sun, J. B. Khurgin, and A. Bratkovsky, “Coupled-mode theory of field enhancement in complex metal nanostructures,” Phys. Rev. B84(4), 045415 (2011). [CrossRef]
  29. G. Sun and J. B. Khurgin, “Theory of optical emission enhancement by coupled metal nanoparticles: An analytical approach,” Appl. Phys. Lett.98(11), 113116 (2011). [CrossRef]
  30. J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett.99(21), 211106 (2011). [CrossRef]
  31. J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett.100(1), 011105 (2012). [CrossRef]
  32. J. B. Khurgin, G. Sun, and R. A. Soref, “Electroluminescence efficiency enhancement using metal nanoparticles,” Appl. Phys. Lett.93(2), 021120 (2008). [CrossRef]
  33. J. B. Khurgin, G. Sun, and R. A. Soref, “Practical limits of absorption enhancement near metal nanoparticles,” Appl. Phys. Lett.94(7), 071103 (2009). [CrossRef]
  34. G. Sun and J. B. Khurgin, “Origin of giant difference between fluorescence, resonance and non-resonance Raman scattering enhancement by surface plasmons,” Phys. Rev. A85(6), 063410 (2012). [CrossRef]
  35. K. Okamoto, I. Niki, A. Scherer, Y. Narukawa, T. Mukai, and Y. Kawakami, “Surface plasmon enhanced spontaneous emission rate of InGaN/GaN quantum wells probed by time-resolved photoluminescence spectroscopy,” Appl. Phys. Lett.87(7), 071102 (2005). [CrossRef]
  36. S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells,” J. Appl. Phys.101(9), 093105 (2007). [CrossRef]
  37. S. C. Lee, S. Krishna, and S. R. J. Brueck, “Quantum dot infrared photodetector enhanced by surface plasma wave excitation,” Opt. Express17(25), 23160–23168 (2009). [CrossRef] [PubMed]
  38. M. B. Dühring, N. Asger Mortensen, and O. Sigmund, “Plasmonic versus dielectric enhancement in thin-film solar cells,” Appl. Phys. Lett.100(21), 211914 (2012). [CrossRef]
  39. M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett.26(2), 163–166 (1974). [CrossRef]
  40. D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys.78(9), 5324 (1983). [CrossRef]
  41. M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys.57(3), 783–826 (1985). [CrossRef]
  42. S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP39, 375–377 (1974).
  43. C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett.46(2), 145–148 (1981). [CrossRef]
  44. A. Wokaun, J. G. Bergman, J. P. Heritage, A. M. Glass, P. F. Liao, and D. H. Olson, “Surface second-harmonic generation from metal island films and microlithographic structures,” Phys. Rev. B24(2), 849–856 (1981). [CrossRef]
  45. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics6(11), 737–748 (2012). [CrossRef]
  46. M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011). [CrossRef] [PubMed]
  47. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep.408(3-4), 131–314 (2005). [CrossRef]
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