1. Introduction

Recent years have seen increased (or, more properly, revived) interest in the properties of surface plasmon polaritons (SPP's) [1, 2] existing on the metal/dielectric interfaces or in metal dielectric composites. This interest has been propelled by the fact that in structures incorporating metals the optical fields get confined on sub-wavelength scale and a variety of effects, linear and especially nonlinear, get enhanced. When combined with rapidly improving capabilities of nano-fabrication, plasmonic enhancement carries a promise of on-chip integration of efficient opto-electronic devices, including those using various nonlinear optical effects, such as switches, modulators and optical frequency converter. It is also well known that sub-wavelength confinement in nano-plasmonic structures is always accompanied by substantial losses, associated with field penetrating the metal [3, 4]. For given wavelength these losses cannot be reduced by engineering the shape of plasmonic structures [5], hence the figure of merit of nonlinear device, loosely defined here as strength of nonlinearity divided by the attenuation coefficient of nonlinear plasmonic devices is not high enough to justify their use in applications that require power efficiency for all intents and purposes, restricting nonlinear plasmonics to sensing. It is the goal of this work to explore whether there are circumstances in which despite being inherent lossy, nonlinear plasmonics can become competitive in terms of efficiency. For that we consider frequency conversion based on second order optical nonlinearity, χ⁽²⁾ since these are the lowest order (i.e. the strongest) nonlinear phenomena, and one can hope that satisfactory frequency conversion efficiency of a few percent per watt can be achieved over the distances less than one absorption length.

As mentioned above, SPP's can be divided into two broad classes: propagating SPP's on the metal/dielectric interface, and localized SPP's in the metal nanoparticles embedded into the dielectrics. The most ubiquitous of χ⁽²⁾ phenomena, second harmonic generation (SHG) has been observed by numerous groups in the media incorporating localized SPP's which may make plasmonic SHG into a diagnostic tool, but the power efficiency of the generation was quite small [6–10]. This follows from the fact that, as has been known for many decades [11–15], the metals themselves have tiny χ⁽²⁾, mostly due to the quadrupole interaction with the field, so the nonlinear polarization can only come from the surface [16] or the intrinsic nonlinearity of the surrounding dielectric. Then, first of all, while the fields do get locally enhanced in the vicinity of nanoparticles, the density of nanoparticles must be kept low (to avoid excessive absorption and overheating). Second, SHG is a three-wave process, and its efficiency depends on the overlap between the three interacting vector fields (two for fundamental frequency and one for second harmonic) and the tensor of χ⁽²⁾. This overlap is typically quite small for all geometries. Third, there is no provision for phase-matching. The combination of these three factors makes efficiency of optical nonlinearity using metals negligibly small [17] for most applications other than diagnostic, even though the effective nonlinearity per unit volume can be quite high.

This leaves one with propagating SPP’s and there has been an effort to study second-order nonlinearity of metal films early on [18], but, once again due to the inversion symmetry considerations, the strength of the interaction is weak, and, furthermore Kretchman excitation geometry [19–21] leads to very short interaction length, meaning that if one uses intrinsic χ⁽²⁾ of the dielectric the efficiency of nonlinear generation will be small. Once the inevitable absorption loss in the metal is accounted for, it appears that propagating SPP's are not capable of facilitating high efficiency χ⁽²⁾ processes, unless (a) the losses are kept to the minimum and (b) SPP offers additional benefits besides improved confinement. In this work we show that gap plasmons [22] in the metal-insulator-metal (MIM) waveguides [23] offer the added advantage of facilitating phase-matching in the isotropic nonlinear crystals, such as GaAs, and that by using gap plasmons only for the longest wavelength waves one can keep the losses within reasonable limit allowing efficient nonlinear generation.

2. Modal overlap and phase matching in MIM waveguides

Many of the III-V and some of II-VI semiconductors crystallize in the zinc-blend lattice that has no inversion symmetry, allowing for very large χ⁽²⁾, on the order of 100's of pm/V (compared to about 30 for LiNbO₃) [24], but this lattice is also perfectly isotropic which prevents conventional birefringence phase-matching. While quasi-phase-matching in GaAs has been explored [25] its implementation is difficult as in the absence of ferro-electricity the material cannot be poled. In the absence of material birefringence, the artificial waveguide birefringence can be employed using layered structures [26] or slot waveguides [27], but the fabrication is difficult, and the efficiency is greatly reduced by the fact that it is difficult to achieve good modal overlap between three waves with significantly different wavelengths in the same waveguide. This factor becomes particularly detrimental in the case of nonlinear generation of mid-IR radiation, which is of course the most desirable application of χ⁽²⁾ processes.

Let us consider how employing MIM rather than all-dielectric waveguides can simultaneously provide good modal overlap and phase-matching. Consider the MIM waveguide shown schematically in Fig. 1(a) and consisting of the core layer of GaAs or related zinc blende crystal of thickness d surrounded by metal layers on both sides sitting on top of a wider bandgap dielectric with lower index of refraction. This waveguide is capable of supporting of both TE and TM modes. The TE modes are characterized by the electric field polarized along y-axis,$E_y=E_{y0}\exp \left(j{\beta }_{TE}z\right)\cos \left(q_{TE}x\right)$, inside the core layer and experience cut-off at long wavelength, as can be seen from the dispersion curve for the lowest order TE₀ wave Fig. 1(b). Note that the propagation constant ${\beta }_{TE}\left(\omega \right)$ is always smaller than the wave-vector $k_{GaAs}\left(\omega \right)=n_{GaAs}\left(\omega \right)\cdot \omega /c$of the plane wave propagating in unconstrained GaAs. The dispersion curve for the latter (often referred to as the light line) is also shown in Fig. 1(b) as dashed line, and its deviation from the straight line indicates strong material dispersion impeding phase-matching without the metal. At the same time the lowest order TM₀ wave inside GaAs can be described as $E_x=E_{x0}\exp \left(j{\beta }_{TM}z\right)\cos h\left(q_{TM}x\right)$. In microwave engineering this mode is regularly referred to as transverse-electro-magnetic (TEM) mode, while in optics it is frequently called gap SPP to indicate the fact that this mode does penetrate the metal much more than TE mode. Because of this the propagation constant ${\beta }_{TEM}\left(\omega \right)$is always larger than the wave-vector $k_{GaAs}\left(\omega \right)$as also shown in Fig. 1(b) and this metal-imposed birefringence can be sufficient to compensate the intrinsic dispersion of GaAs or other non-birefringent nonlinear crystals used as the core layer in the MIM waveguide.

 

Fig. 1 (a) The lowest order TE and TM modes in MIM waveguide, (b) their dispersion, (c) power densities of fundamental TM and second harmonic TE waves in the MIM waveguide designed for SHG at 775 nm, (d) effective indices, and (e) loss of these modes as a function of waveguide width d.

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Let us consider how one can practically take advantage of this phase-matching scheme. First of all the only non-vanishing component of second order susceptibility is ${\text{χ}}_{14}^{(2)}$ which requires three interacting fields to be polarized along three crystal axes of zinc blende lattice. Therefore, the propagation direction should be along one of the 110 axes (that happens to take advantage of natural cleavage planes in zinc blende lattice) with metal applied to the sides of the waveguide as shown in Fig. 1(c). To provide vertical confinement the waveguide is grown on top of AlAs cladding layer. The height of the waveguide h is chosen to be fixed and the width d is the adjustable parameter that provides phase-matching.

3. Second order nonlinear processes

For the ubiquitous case of SHG the fundamental wave of frequency ω is then coupled into TEM (gap SPP) mode and the second harmonic frequency 2ω is generated in the TE₀ mode. The field distributions are shown in Fig. 1(c). The phase-matching condition ${\beta }_{TE}\left(2\omega \right)=2{\beta }_{TM}\left(\omega \right)$ can also be written as equality of effective indices of refraction, $n_{TE}\left(2\omega \right)={\beta }_{TE}\left(2\omega \right)\cdot c/2\omega$ and $n_{TM}\left(\omega \right)={\beta }_{TM}\left(\omega \right)\cdot c/\omega$. As an example we use the fundamental radiation of wavelength ${\lambda }_{\omega }=1550$nm and the second harmonic${\lambda }_{2\omega }=755$nm in the MIM waveguide with Ag side walls, consisting of Al_0.2Ga_0.8As as the core layer of height h = 1µm and width d sitting on top of a AlAs layer of 1µm on a GaAs substrate. The Al_0.2Ga_0.8As core layer is chosen here to avoid optical absorption at λ_2ω = 775nm, and its indices of refraction are n = 3.3 at${\lambda }_{\omega }=1550$nm and n = 3.6 at ${\lambda }_{2\omega }=755$nm. We have numerically calculated the effective indices versus the width d using a FDTD software from Lumerical. The results are shown in Fig. 1(d) and indicate that phase-matching is achieved at d = 0.57µm.

The conversion efficiency of the SHG process depends strongly on the overlap between the fundamental and second harmonic modes and can be found as [24]

Of course, SHG by itself is of little practical interest, compared to the difference frequency generation (DFG) capable of converting near IR pumps into the important mid-IR radiation. The DFG efficiency is typically much lower than SHG efficiency, primarily because it is difficult to achieve a good overlap between the relatively short wavelength near IR pumps and long wave-length mid-IR signal. This is where MIM waveguide has distinct advantage as mid-IR TM wave that can be compressed way below diffraction limit without incurring excessive loss. As an example we consider the arrangement shown in Fig. 2(a) in which the TE₀ pump at 1.3 µm and TM signal at 2.0 µm combine in the In_0.2Ga_0.8As MIM waveguide with Ag side walls sitting on top of a GaAs substrate to produce DFG idler at 3.71 µm. The phase-matching condition ${\beta }_{TE}\left({\omega }_p\right)={\beta }_{TEM}\left({\omega }_s\right)+{\beta }_{TEM}\left({\omega }_i\right)$ as shown Fig. 2(b) is achieved at d = 0.84 µm when the effective indices of refraction at those wavelengths are $n_{TE}^{\left(pump\right)}=3.48$, $n_{TM}^{\left(sig\right)}=3.53$, and$n_{TM}^{\left(idl\right)}=3.37$. The conversion efficiency can be estimated as

 

Fig. 2 (a) Power densities of TE pump and TM signal and idler waves in the MIM waveguide designed for DFG at 3710 nm, (b) propagation constants, (c) loss of these modes as a function of waveguide width d, (d) power densities of TE pump and TM signal waves in the MIM waveguide designed for degenerate OPG at 3100 nm, (e) effective indices, and (f) loss of these modes as a function of waveguide width d.

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DFG is obviously not the best way to obtain mid-IR radiation since it requires two sources – it would be more advantageous to develop an optical parametric generator (OPG) as shown in Fig. 2(d) in which near IR pump at 1.55 µm can produce degenerate signal/idler at 3.1 µm. As one can see from Fig. 2(e) phase-matching is achieved with $n_{TE}^{\left(pump\right)}=n_{TM}^{\left(sig\right)}=3.41$when In_0.4Ga_0.6As MIM waveguide width is d = 0.85 µm and the waveguide loss for the signal propagating in TM mode is only 830 dB/cm (propagation length of 52 µm) as shown in Fig. 2(f). The OPG threshold [24] will occur when the parametric gain overcomes the waveguide loss, or

4. Summary

We have calculated the guided surface plasmon modes of silver nano-scale ridge plasmon waveguides. We find that only one guided plasmon mode is bounded on the top surface of the flat-top metal nano-ride. The nano-ridge plasmon mode is a TEM mode with very small longitudinal electromagnetic field components in the direction of the propagation. We calculated the dispersion of the nano-ridge plasmon mode and its propagation length as a function of the wavelength. We also calculated the figure-of-merit of the nano-ridge mode for different ridge widths. We find that as the nano-ridge width increases, the propagation length always increases, however, the figure of merit increases first as the ridge width increases and then reaches a maximum when the ridge width falls between 100 nm and 200 nm. The nano-ridge waveguide of 100 nm wide has a propagation length of 242.8 µm at an excitation wavelength of 1.55 µm. This propagation length is long enough for building plasmonic circuits. During the process of writing this paper, we have found that a round top single crystal gold nano-ridge waveguide has been fabricated using focused ion beam milling, and the appearance of a nano-ridge plasmon mode was confirmed by the experiment [26]. Metal nano-ridge surface plasmon waveguides can also be fabricated by metal filling nano-trenches etched in dielectric substrates. We believe the nano-ridge plasmon waveguide can be the building block for making plasmonic integrated circuits for sensor applications.