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]. Indeed, shortly after this invention second and third order nonlinear optical effects were demonstrated [2–4] and the theory of nonlinear optics developed [5–7]. Today, a clear understanding of nonlinear optical effects in various media exists [8, 9]. 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 $n_2~5\times {10}^{-16}$cm²/W for fused silica that is transparent all the way to UV, to perhaps $n_2~1\times {10}^{-13}$cm²/W for chalcogenide glasses transparent only in the IR range [10].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 [11,12], ring resonators [13], photonic bandgap structures [14] and/or slow light devices [15,16], but all resonant effects inevitably limit bandwidth [17]. It is the concentration of optical power in the time domain, provided by pulsed sources, particularly by Q-switched [18] and mode-locked lasers [19,20], 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.

Considering the time/space analogy, let us look at what limits energy concentration in both time and space. In the time domain, it is the dispersion of the group velocity, while in the space domain it is diffraction. While there are obvious equivalences between the mathematical description of dispersion and diffraction, there are also stark differences – the group velocity dispersion can be minimized by a number of techniques because it can be either positive or negative, while diffraction is always positive and there exists a hard diffraction limit to optical confinement in all dielectric medium. The diffraction limit however is applicable only to the all-dielectric structures with positive real parts of their dielectric constant. In all-dielectric structures, the energy oscillates between the electric and magnetic fields, and if the volume in which one tries to confine the optical energy is much less than a wavelength the magnetic field essentially vanishes (so-called quasi-static limit) and without this energy “reservoir” for storage every alternative quarter-cycle the energy simply radiates away. But, if the structure contains a medium with a negative real part of the dielectric constant, an alternative reservoir for energy opens up – the kinetic motion of these free carriers in the metal or the semiconductor, and the diffraction limit ceases being applicable. The optical energy can be then concentrated into tightly confined sub-wavelength modes through the use of tiny metallic particles. These modes, which combine the oscillating electric field with charge oscillation are called localized surface plasmons (LSP) and in the last decade the new field of plasmonics and the closely related field of metamaterials have arisen with the ultimate goal of taking advantage of the unprecedented degree of optical energy concentration now offered on sub-wavelength scales [21].

In the last decade researchers have observed enhancement of both linear (absorption, luminescence) [22,23] and non-linear (Raman) phenomena [24–26] in the vicinity of small metal nanoparticles. Experimentally, Raman scattering has shown plasmon mediated enhancement of many orders of magnitude [24–26], 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] or coupled [28,29] 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 [30,31]. In that work [30] 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] and also absorbers [33] can be enhanced by plasmonic effects, such as, of course, the Raman process which is known to be extremely inefficient [34], while the relatively efficient devices, such as LEDs [35], solar cells [36], and detectors [37] do not exhibit any significant plasmonic enhancement relative to what can be obtained without the metal by purely dielectric means [38].

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 [39–44], interest has picked up significantly in the last decade [45,46]. 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]. Such strong near-field effects are responsible for the experimental observations of significant Raman enhancement that has resulted in single molecule detection [24–26,48], surface plasmon enhanced wave mixing like SHG on random [49–51] and defined plasmonic structures [52–58], as well as the enhancement of linear processes such as optical absorption and luminescence [22,23]. 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]. 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 [60–63] or waveguides [64–68], the demonstration of which has remained elusive to date.

At this point it is important to differentiate between the source of the nonlinearity, associated with either the metallic element or the dielectric element. The nonlinear susceptibility of metals can be due to either free carriers or band-to-band transitions. The nonlinearity associated with a band-to-band transition (typically involving d-bands in noble metals) is no different than the interband transition related nonlinearity of both dielectrics and semiconductors. In metals, however, this nonlinearity is always accompanied by enhanced free carrier absorption and is strongest in the blue region of spectrum, while we prefer to focus on the telecommunication region of 1300-1500nm. On the other hand, oscillatory free electron related LSP nonlinearities are extremely weak because LSP’s (at least when there are only a few of them per nanoparticle) are nearly perfect harmonic oscillators. At this time there has been no reports of full optical switching with low loss using metal nonlinearities. For this work, we shall consider structures in which metallic nanoparticles are embedded in material with large nonlinearity and low loss. We shall consider only third order nonlinearities because they lead to optical switching and other interesting phenomena without concern for phase matching. Further, we shall also limit ourselves to only the nonlinear modulation of the refractive index (the real part of susceptibility) neglecting absorption effects (the imaginary part of the susceptibility). The reason being, that for the amplitude modulation of optical signals it is desirable to maintain the “zero” bit level as close as possible to real zero, which can only be done through interference (as in, for instance, via a Mach Zehnder interferometer) suggesting the use of refractive index modulation. In addition, through refractive index modulation one can take advantage of advanced phase–modulation formats, such as for example quadrature phase-shift keying (QPSK) and quadrature amplitude modulation (QAM). Index modulation is typically broadband and, in addition to simple modulation and switching, can be used for frequency conversion, while absorption modulation is an inherently narrow band, resonant phenomenon. Finally, changes in absorption are typically associated with real excitations hence they are not truly ultrafast as discussed previously.

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) 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], with all structures defined by just three parameters: resonant SP frequency${\omega }_0$, quality factor Q, and effective SP mode volume $V_{eff}$. 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.

 

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.

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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) consisting of a concentration of $N_s$ nanospheres, each of radius a, surrounded by a nonlinear dielectric with relative permittivity${\epsilon }_d$ and nonlinear susceptibility tensor ${\chi }^{(3)}$. In the most general case, ${\chi }^{(3)}$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 $\omega$ and one signal wave of frequency $\omega \text{'}$.

 

Fig. 2 Fields and polarizations in the plasmonically enhanced nonlinear metamaterials: (a) average ${\overline{E}}_{\omega }$ and local ${\text{E}}_{\omega }$ electric fields, and dipole ${\text{p}}_{\omega }$ at the pump frequency $\omega$ ; (b) local nonlinear field ${\text{E}}_{\omega \text{'}}$, dipole moment ${\text{p}}_{nl}^{\omega \text{'}}$ and average nonlinear polarization ${\overline{P}}_{nl}^{\omega \text{'}}$ at the nonlinear output signal frequency $\omega \text{'}$.

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As the pump wave propagates through the material, the average electric field is ${\overline{E}}_{\omega }$ and in this field the nanospheres become polarized, i.e. they acquire the dipole moment [70]

The Q-factors for nanospheres of gold and silver, the two lowest loss plasmonic materials, are shown in Fig. 3 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], although for the silver nanoparticles interface scattering usually decreases the Q factor, consistent with measured Q-factors in visible and near infrared wavelengths [72–76]. 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.

 

Fig. 3 Dispersions of Q-factors for gold and silver nanoparticles.

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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

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 $N_s$ density of nanoparticles,

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 frequency$\omega \text{'}$,

We seek solutions of the form

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

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.

In general, when all four frequencies ${\omega }_1$, ${\omega }_2$, ${\omega }_3$, and ${\omega }_4={\omega }_1-{\omega }_2+{\omega }_3$are different (but typically close to each other) the nonlinear process described by (16) is four wave mixing (FWM), when${\omega }_3={\omega }_1$,${\omega }_4=2{\omega }_1-{\omega }_2$ Eq. (16) describes optical parametric generation (OPG), when ${\omega }_1={\omega }_2$and ${\omega }_3={\omega }_4$it 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 $E_{{\omega }_k}(r)$in Eq. (16) in the vicinity of the nanoparticle are all locally enhanced relative to the mean fields ${\overline{E}}_{{\omega }_k}$ according to Eq. (5), i.e.

The nonlinear susceptibility thus gets enhanced by a factor proportional to $Q^4$. 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], electroluminescence [32] and Raman scattering [34] 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 $P_{nl}^{{\omega }_2}(\text{r})=2{\epsilon }_0{n}_d{n}_2{I}_{{\omega }_1}(\text{r})E_{{\omega }_2}(\text{r})$_, where $n_2={\chi }^{(3)}{\eta }_0/{\epsilon }_d$is the nonlinear index of the dielectric, and $I_{{\omega }_1}(\text{r})={\left|E_{{\omega }_1}(\text{r})\right|}^2{n}_d/2{\eta }_0$ is the local intensity Similarly, we now introduce the “effective” nonlinear index as $n_{2,eff}={\chi }_{eff}^{(3)}{\eta }_0/{\epsilon }_d$and write the average nonlinear polarization as ${\overline{P}}_{nl}^{{\omega }_2}=2{\epsilon }_0{n}_d{n}_{2,eff}{\overline{I}}_{{\omega }_1}{\overline{E}}_{{\omega }_2}$ According to Eq. (22), the effective nonlinear index gets enhanced by the same giant factor proportional to $Q^4$,

Next we estimate the nonlinear phase shift in the absorbing medium as

Achieving the π-phase shift required to achieve photonic switching would then require at resonance${\overline{I}}_{\pi }~\pi n_d{\left[{\kappa }_3{n}_2{\left(2\beta \right)}^2{Q}^3\right]}^{-1}$. If we assume$\beta \sim 1.45$(estimated numerically for the actual ellipsoid resonant at 1320 nm of Fig. 1(b), $Q\sim 20$and a large nonlinear index characteristic of chalcogenide glass $n_2={10}^{-13}{\text{cm}}^2/\text{W}$, the required switching intensity is then on the order of ${\overline{I}}_{\pi }~1.6\times {10}^9\text{W}/{\text{cm}}^2$, 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.

Next we ask the question – what is the actual local intensity near the metal surface? According to Eq. (5), the maximum local intensity approaches $I_{\max }={(2\beta Q)}^2{\overline{I}}_{\pi }$$\approx 5\times {10}^{12}\text{W}/{\text{cm}}^2$. In general, a local field in excess of 5 × 10⁷ V/cm, is significantly higher than the damage threshold of most materials. For this reason, if one searches through all the nonlinear materials, it is difficult to find one that is capable of achieving an ultrafast refractive index change larger than 0.1%. In addition to the limitations incurred due to overheating and optical damage, at higher powers the nonlinearities greater than third order, i.e. ${\chi }^{(5)}$, ${\chi }^{(7)}$become important. Because these higher order nonlinearities often have their sign opposites to ${\chi }^{(3)}$ [77] an actual decrease in the nonlinear index change at higher incident intensities often results [78].

To elucidate these effects further, let us define the maximum local nonlinear index change attainable in a given material as$\Delta n_{\max }$and obtain the maximum change of effective index

As we can see, the enhancement is now only proportional to $Q^2$. This result makes perfect sense if we recognize that the local change of the dielectric constant $\Delta {\epsilon }_{d,\max }=2n_d\Delta n_{\max }$ simply causes a shift of the LSP resonant frequency ${\omega }_0={\omega }_p/\sqrt{1+2{\epsilon }_d}$, which in turn changes the effective dielectric constant of the metamaterial ${\epsilon }_{eff,\max }$according to (6) which is proportional to $Q^2$obtained via the differentiate of the Lorentzian, $L(\omega )$, in Eq. (6). It is crucial to note that this factor of $Q^2$in 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 $Q^2$.

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

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 ${\overline{E}}_{{\omega }_3}(z)$in the presence of pump ${\overline{I}}_{{\omega }_1}(z)={\overline{I}}_0{e}^{-{\alpha }_{a}z}$ and signal $E_{{\omega }_2}^{*}(z)=E_s{e}^{-\frac{{\alpha }_a}{2}z}$ can be found as

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

We have shown that while the nonlinear susceptibility and the nonlinear index of refraction do get enhanced significantly in simple nanostructures, strong absorption makes the maximum attainable phase shift less than desired. From our own previous work [79,80] as well as from others [81], we have established that local fields can be enhanced even further in more complicated nanoparticle structures. We have shown enhanced local fields in the gap between two identical nanoparticles (dimer) [79] or in the vicinity of a smaller nanoparticle coupled to a larger nanoparticle of the same shape (nanolens) [80]. We have shown that in both cases the maximum filed enhancement was proportional to Q² rather than Q for a single nanoparticle, hence much larger “cascaded” enhancements of absorption, Raman scattering, and in some cases photoluminescence could be achieved in these “hot spots”. Therefore, it is tempting to evaluate the possibility of using such hot spots to enhance nonlinear effects. Since we have shown that in either a dimer or a nanolens the field enhancement is similar, we shall limit our analysis to the case of a nanolens only, as it is easier to describe analytically.

Consider two spherical nanoparticles of radii $a_1$and $a_2$separated by a vector${\text{r}}_{12}$ as shown in Fig. 4(a). The dipole oscillation Eq. (3) is augmented by the dipole-dipole interaction between the two dipoles associated with the two, coupled nanoparticles,

 

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.

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Following our prior work [79,80] we obtain the expression for the maximum fields near the nanoparticles

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], 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

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 $a_2\approx 0,a_1\approx r_{12}$, 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

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 ${\chi }^{(3)}$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 ${\chi }^{(3)}$enhancement reaches its maximum when$4{\beta }^2{Q}^2{\left(a_1{a}_2{r}_{12}^{-2}\right)}^3=1/3$ and is then given by

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,

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 ${\kappa }_{3}Q$is common to any type of nanostructure, monomer, dimer, trimer, or nano-antenna. The maximum achievable index of refraction $\Delta n_{\max }$changes 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 ${\kappa }_{3}Q$is 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 shift$\Delta {\Phi }_{\max }$beyond 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) where the nonlinear phase shift $\Delta \Phi (z)$induced by either XPM or SPM, is shown as a function of the propagation length in a chalcogenide glass waveguide (nonlinear index $n_2={10}^{-13}{\text{cm}}^2/\text{W}$) doped with the isolated Ag spheroids of Fig. 1(b). The incident pump power density is $\overline{I}(0)={10}^7\text{W}/{\text{cm}}^2$ (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.

 

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.

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If we increase the input power by a factor of 10 to $\overline{I}(0)={10}^8\text{W}/{\text{cm}}^{\text{2}}$ the local power density near the metal surface will reach roughly ${\text{3×10}}^{\text{11}}{\text{W/cm}}^2$which 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]. 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 $\overline{I}(0)={10}^9\text{W}/{\text{cm}}^2$ (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 $\overline{I}(0)={10}^4\text{W}/{\text{cm}}^2$with 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 $\overline{I}(0)={10}^6\text{W}/{\text{cm}}^2$ [Fig. 5(e)] and $\overline{I}(0)={10}^8\text{W}/{\text{cm}}^2$ [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) for an input pump power of $\overline{I}{\text{(0)=10}}^{\text{7}}{\text{W/cm}}^{\text{2}}$. 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.

 

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.

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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 [81,82]. 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]. 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 ${\chi }^{(3)}$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] and the thermally induced metal-to-insulator transition in materials like VO₂ [85]. 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.