## Abstract

We present a novel approach for achieving tightly concentrated optical field by a hybrid photonic-plasmonic device in an integrated platform, which is a triangle-shaped metal taper mounted on top of a dielectric waveguide. This device, which we call a plasmomic light concentrator (PLC), can achieve vertical coupling of light energy from the dielectric waveguide to the plasmonic region and light focusing into the apex of the metal taper(at the scale ∼ 10nm) at the same time. For a demonstration of the PLCs presented in this paper, we numerically investigate the performance of a gold taper on a Si_{3}N_{4} waveguide at working wavelengths around 800nm. We show that three major effects (mode beat, nanofocusing, and weak resonance) interplay to generate this light concentration phenomenon and govern the performance of the device. Combining these effects, the PLC can be designed to be super compact while maintaining high efficiency over a wide band. In particular, we demonstrate that under optimized size parameters and wavelength a field concentration factor (FCF), which is the ratio of the norm of the electric field at the apex over the average norm of the electric field in the inputting waveguide, of about 13 can be achieved with the length of the device less than 1*μ*m for a moderate tip radius 20nm. Moreover, we show that a FCF of 5 – 10 is achievable over a wavelength range of 700 – 1100nm with the length of the device further reduced (to about 400nm).

© 2013 Optical Society of America

## 1. Introduction

Plasmonic metallic structures are well-known for the ability of breaking the diffraction limit and achieving large field enhancements or field confinements in small volumes [1]. One mainstream of the approaches is to focus surface plasmon polariton (SPP) modes, which can be further divided into two categories. One is interference-induced plasmonic nanofocusing. For this type of techniques, the plasmonic excitation sites on a metal film are carefully tuned so that the propogation of SPP originated at the sites can get focused into a hot spot on the metal film. Typical planar patterns for plasmonic excitation on the metal film include individual circular rings [2], multiple concentric rings [3], curved arrays of nanoholes [4], and certain coaxial structures [5]. The other is geometry-induced plasmonic nanofocusing. In principle, tapering the cross section of the SPP propagation gradually will increase the intensity of the SPP and the slow down its group velocity. At the tip end of the taper, the group velocity becomes zero which means the surface plasmon becomes localized and the energy is highly concentrated [6–24]. This type of techniques can accurately control the focus point and usually have a higher concentrating ability.

Conventionally, the nanofocusing techniques have the SPP excited from free space. Very recently, several approaches of realizing plasmonic nanofocusing on integrated-optics platforms have been reported [25–30]. Using these on-chip light concentration techniques, the light energy is coupled from a photonic waveguide to a plasmonic structure (usually a metallic taper) and then concentrating the light energy to the tip of the taper. The coupling from the photonic waveguide to the plasmonic structure is efficient and robust, while the plasmonic part is usually compact since it only concentrates light energy from a “∼ *λ*” scale to a “< *λ*/10” scale. Therefore, the lossy nature of metals has less impacts on the concentration efficiencies. Moreover, many of such structures can be integrated on the same chip in series or in parallel in a very small footprint. This opens up a new avenue for multiplex nanofocusing in applications such as on chip trapping and sensing.

In this paper, we will present the design of a novel compact on-chip plasmonic light concentrator (PLC), and show that there are three major effects (mode beat, nanofocusing, and weak resonance) that govern this light concentration phenomenon and affect the efficiency of the device. By coordinating these effects, this proposed structure can maintain very high concentrating efficiency while the size dimensions (width∼ 400nm and length< 1*μ*m) are the smallest among the all the non-resonant on-chip plasmonic-light-concentrators that have been reported.

The basic structure of the device is a small triangle-shaped metal taper on top of a dielectric waveguide, as shown in Fig. 1(a). In our analysis of this structure, the material for the metal tri-angle is gold (Au), and the material for the dielectric ridge waveguide is silicon nitride (Si_{3}N_{4}). The substrate is silicon dioxide (SiO_{2}). In addition, a SiO_{2} buffer layer is used to separate the metal triangle and the dielectric ridge waveguide. The ambient material is water. The dimensions of the ridge waveguide are chosen to support only the fundamental TM-like (vertical polarization) mode and the fundamental TE-like (horizontal polarization) mode. Only the fundamental TM-like mode can be used to generate light concentration. We set up the coordinate system with *X*-axis in the horizontal direction, *Y*-axis in the vertical direction and *Z*-axis in the propagation direction of the input light.

Figure 1(b) shows the top view of the metal triangle, which is isosceles. We use *W* to denote the length of the base of the isosceles triangle, and *L* to denote the the length of a perpendicular from the center of the curved tip to the middle point of the base side. For simplicity, we will call *W* the (maximum) width of the triangle and *L* the length of the triangle throughout this paper. Since the apex of the metal triangle is always rounded up in real fabrications, we introduce the radius of curvature *a* at the tip as a parameter.

## 2. Mode analysis of the hybrid photonic-plasmonic waveguide

To analyze the behavior of the PLC, we first need to find the modes of the corresponding hybrid photonic-plasmonic waveguide. Several techniques of combining plasmonic and dielectric waveguiding with different geometric settings have been reported [31–36]. The hybrid waveguide we analyze here is formed by placing a metal (Au) strip on top of a dielectric ridge waveguide with a buffer layer as shown in Fig. 2(a). The dimensions of different layers in the the Si_{3}N_{4} ridge waveguide have been selected for single-mode operation (for each polarization) at wavelength *λ* of 800nm with the ambient material being water. The Au strip has thickness of 40nm and width *w* as a variable. For *λ* = 800nm, the refractive indices of Si_{3}N_{4}, SiO_{2} and water are assumed to be 2.00, 1.46, and 1.33, respectively, and the permittivity of Au is assumed to be −24.02 + *j*1.18 (computation based on data from [37]).

The analysis technique is graphically shown in Figs. 2(b) and 2(c). We first analyze the modes of the corresponding purely photonic waveguide (no Au layer) and of the corresponding purely plasmonic waveguide (by replacing Si_{3}N_{4} with SiO_{2}). In the next step, we find the modes of the hybrid photonic-plasmonic waveguide by using the supermode analysis through adding and subtracting the modes of the two structures as shown in shown in Figs. 2(b) and 2(c).

Based on the designed dimensions, the purely photonic waveguide only support the fundamental TM-like mode (denoted by TM_{0}) and the fundamental TE-like mode (denoted by TE_{0}). The TM_{0} mode can only couple with symmetric modes of the purely plasmonic waveguide, and the TE_{0} mode can only couple with the asymmetric modes of the purely plasmonic waveguide. We only consider the fundamental symmetric (S_{0}) and asymmetric (A_{0}) modes of the purely plasmonic waveguide, since higher order modes are cut off with the dimensions of our structure at the wavelength *λ* of 800nm. Figure 2(b) shows two supermodes, H_{TM,0} and H_{TM,1}, are derived from the superposition of the TM_{0} mode and the S_{0} mode. Analogously, Fig. 2(c) shows two supermodes, H_{TE,0} and H_{TE,1}, are derived from the superposition of the TE_{0} mode and the A_{0} mode.

To give more details about the mode-coupling processes, we use finite element method (FEM, COMSOL Multiphysics) to compute the eigenmodes at the wavelength *λ* of 800nm. Note that the effective indices *n*_{eff} of the propagating modes must have real parts greater than 1.46, which is the refractive index of the SiO_{2} substrate.

Figure 3(a) shows the normalized electric field profiles for the TM_{0}, S_{0}, H_{TM,0} and H_{TM,1} modes, with the width *w* of the Au strip set to 620nm. The electric field lines are plotted. The effective indices of the modes are also listed in Fig. 3(a). Figure 3(b) illustrates similar data for the modes TE_{0}, A_{0}, H_{TE,0} and H_{TE,1}. The electric field lines in Fig. 3 clearly show the validity of the supermode analysis approach as shown in Fig. 2.

The dispersions of the four propagating supermodes (H_{TM,0}, H_{TM,1}, H_{TE,0} and H_{TE,1}) of the hybrid photonic-plasmonic waveguide are plotted in Fig. 4 in the form of the real and imaginary parts of the effective index versus the width *w* of the Au strip. The coordinate “Real(*n*_{eff})” has a minimum at 1.46 as only propagating modes are considered. The imaginary part of the effective index accounts for the propagation loss.

The Real(*n*_{eff}) of H_{TM,0} is greater than the Real(*n*_{eff}) of H_{TM,1}, and as *w* decreases, the difference between the two becomes larger. The imaginary parts follow the same trend. This is because as *w* decreases, the light energy becomes more concentrated around the lossy metal region for H_{TM,0}, while it becomes more concentrated in the dielectric region for H_{TM,1}. The mode profiles of both supermodes at *w* = 60nm are illustrated in Fig. 5. As *w* approaches 0, both the real and imaginary parts of the effective index of H_{TM,0} go to infinity. This means that the phase velocity goes to 0, and the H_{TM,0} mode becomes more and more like a localized mode. When *w* = 0, there is no metal strip, and the H_{TM,1} mode is actually the purely photonic mode TM_{0}. Note that as the width of the metal layer continuously and gradually shrinks down, the amplitude of the H_{TM,0} mode increases since its propagating velocity slows down. Such a process is common to most geometry-induced plasmonic nanofocusing techniques, and is one of the main mechanisms to generate light concentration in our triangle-shaped plasmonic device. Analogously, we observe that the H_{TE,0} mode evolves into the purely photonic mode TE_{0} as *w* approaches 0. Moreover, when *w* < 180nm, the H_{TE,1} mode is cut off. Therefore we can not take use of the supermodes H_{TE,0} and H_{TE,1} for our application of light concentration.

The supermodes in the hybrid waveguide can be excited by the modes in the corresponding bare (or purely photonic) waveguide. More specifically, the incident mode TM_{0} in the bare waveguide can excite two supermodes (H_{TM,0} and H_{TM,1}) of the hybrid waveguide. The super-position of these two supermodes at the input end has field in the dielectric region to match the incident TM_{0} mode. Since they have different effective indices, as can be seen from the dispersion diagram in Fig. 4, the two supermodes undergo a beat effect with light energy bouncing back and forth between the dielectric region and the metal region. For a specific width *w* of the Au strip, we have

*w*) is the beat length. In particular, at a distance about half the beat length from the input end, the light energy is vertically coupled efficiently into the metal region.

## 3. The performance of a PLC with specific size parameters

The PLCs presented in this paper (see Fig. 1) are designed as tapering the metal strip of a hybrid photonic-plasmonic waveguide. The performance of a PLC depends on the interplay of three effects, mode beat, nanofocusing and weak resonance: the mode beat effect comes from the interference of the two excited supermodes (H_{TM,0} and H_{TM,1}); the nanofocusing effect comes from the triangle taper; and the weak resonance effect comes from the reflections at the input end and the apex of the taper.

To clarify the roles of the those three effects, we consider the field profile in the triangular tapered structure shown in Fig. 6(a). The Si_{3}N_{4} ridge has a width of 620nm and a thickness of 200nm; the SiO_{2} buffer has a width of 620nm and a thickness of 100nm. The thickness of the Au triangle is 40nm. The ambient material is also water. These parameters are the same as the hybrid photonic-plasmonic waveguide we analyzed in the previous section. The coordinate system is chosen to be the same as that in Fig. 1. In particular, the coordinate origin is chosen such that the plane *X* = 0 cuts through the middle of the Si_{3}N_{4} ridge vertically, the plane *Y* = 0 coincides with the boundary between the Si_{3}N_{4} layer and the SiO_{2} substrate, and the plane *Z* = 0 goes through the center of the curved tip.

The electric field patterns in the horizontal (Fig. 6(a) and 6(b)) and vertical (Fig. 6(a)) planes are calculated using a 3D FEM simulation. Here *Y* = 200nm+100nm+20nm, which is the sum of the thickness of the Si_{3}N_{4} layer, the thickness of the SiO_{2} buffer layer and half of the thickness of the Au layer. The Au triangle has dimensions *L* = 900nm, *W* = 400nm and *a* = 20nm. We let |*E*_{0}| be the average norm of the incident electric field in the Si_{3}N_{4} region, which is calculated from the the TM_{0} mode of corresponding purely photonic waveguide. We estimate the total power flow by a plane-wave approximation with an electric field *E*_{0} homogeneously distributed and restricted within the Si_{3}N_{4} region. In particular, the norm of *E*_{0} can be computed by the formula

*n*= 2 is the refractive index of Si

_{3}N

_{4},

*A*= 620nm × 200nm is the area of the cross section of the Si

_{3}N

_{4}slab, and ∫

_{D}**S̄**

*is the time-averaged power flow through a cross-section plane*

_{z}*D*. The electric field patterns shown in Fig. 6 are normalized to |

*E*

_{0}|. The field becomes highly concentrated at the metallic tip. To evaluate the efficiency of field concentration induced by the PLC, we introduce a field concentration factor (FCF) as the electric field norm at the apex point (point

*Q*in Fig. 6 (b) and (c)) normalized to |

*E*

_{0}|. Our calculations show that

*FCF*= 12.6 for the tapered structure in Fig. 6. Note that the FCF depends on the radius of curvature

*a*. While reducing

*a*can result in higher FCF, we should choose

*a*large enough to accommodate the fabrication requirement. Here we set

*a*= 20nm, and write FCF(

*a*= 20nm) = 12.6. (Many authors use the intensity ratio to evaluate the concentration efficiency, which corresponds to the square of the FCF in this paper.) Compared to other reported results [28, 30] of on-chip plasmonic light concentrators, the concentration efficiency (evaluated by intensity ratio) reported here is at least two times better while using a much smaller

*a*.

## 4. The three underlying effects of the device

As mentioned above, the field concentration is governed by three effects: mode beat, nanofocusing and weak resonance.

As discussed in Section 2, the incident mode TM_{0} in the bare photonic waveguide can excite two supermodes, H_{TM,0} and H_{TM,1}, in the corresponding hybrid waveguide with a metal (Au) strip integrated on top. The triangle-shaped PLC can be considered as a taper with the width of the Au strip gradually going down to 0. Therefore, we can still use the two supermodes for the Au taper in our analysis. The mode beat between the two supermodes can make a vertical coupling (or side-coupling), which transfers light energy quickly from the dielectric region to the metal region, and the triangular taper performs plasmonic nanofocusing at the same time. This simultaneous light coupling and focusing in a single triangle-shaped metal taper is a major advantage of our design, resulting in a very compact structure. More accurately speaking, the tapering also induces energy coupling between the two supermodes. In other words, the ratio of energy stored in the two modes will change at different locations along the metal taper, and only the H_{TM,0} mode becomes localized as the metallic tip. In our analysis based on a simplified model, we consider that the two supermodes have constant mode effective indices which are calculated using the average width of the taper.

The third effect that must be studied in our device is the weak resonance caused by the reflection of light the apex and the input end the the taper. This weak resonance also affects the efficiency of light concentration into the tip. By adjusting the size parameters of the Au taper, we can combine the three effects to optimize the concentration efficiency of the PLC.

Figure 7 shows how the FCF varies with respect to the length *L* of the Au triangle, while the radius of curvature *a* is fixed at 20nm. The wavelength *λ* is 800nm. The two curves correspond to two cases of the maximum width *W* of the triangular taper: the blue solid one is for the case *W* = 300nm, and the red dashed one is for the case *W* = 400nm. Two oscillatory behaviors can be seen in Fig. 7: the slow variations corresponding to the mode-beat effect and the ripples with high oscillation frequency corresponding to the weak resonance effect.

Although Equation (1) is derived for calculating the beat length of a hybrid photonic-plasmonic waveguide with a fixed metallic width *w*, we can use it to estimate the beat length of the tapered structure by using the average metallic width over the tapered length for *w*. For a triangular taper, this corresponds to about 1/2 of the maximum width *W* of the taper. Using this approach, we can find *n*_{eff} for the hybrid H_{TM,0} and H_{TM,1} modes, and then compute the beat lengths for the two cases (*W* = 300nm and *W* = 400nm) shown in Fig. 7. For the narrower taper (*W* = 300nm and *w* = *W*/2 = 150nm), we obtain Real(*n*_{eff}(H_{TM,0}(150nm))) = 1.9694, Real(*n*_{eff}(H_{TM,1}(150nm))) = 1.5475 and BL(150nm) = 1896nm. For the wider taper (*W* = 400nm and *w* = *W*/2 = 200nm), we obtain Real(*n*_{eff}(H_{TM,0}(200nm))) = 1.9216, Real(*n*_{eff}(H_{TM,1}(200nm))) = 1.5413 and BL(200nm) = 2104nm. We can see that the computed beat length for the wider taper is a little larger than that for the narrower one.

The beat lengths can also be estimated directly from Fig. 7 by locating the the first minimum of the slow variation part of each curve. Using this method, the estimated beat length for the narrower taper is 1.8*μ*m and for the wider taper is 2.0*μ*m, agreeing well with the computations based on Equation (1).

When *L* is about half of the beat length (e.g. *L* ⋍ 1.0*μ*m for the wider taper with *W* = 400nm), the light energy transfers from the dielectric region to the metal region most efficiently, which results in a higher FCF, as shown in Fig. 7.

The effect of weak resonance can also be explained by our simple theoretical model. The fast ripples in Fig. 7 are due to the limited length of the taper. By approximating the triangular tapered structure with a Fabry-Pérot structure and using the value of *n*_{eff}(H_{TM,0}(*w*)) with *w* being the average width of the triangular taper, we can calculate the free spectral range (FSR) of these fast oscillation using

*W*= 400nm and

*w*= 200nm) in Fig. 7, we have Real(

*n*

_{eff}(H

_{TM,0}(200nm))) = 1.9216, and FSR(200nm) = 208nm. This corresponds to about 15 peaks for the red curve, matching well with an actual counting on the figure.

The FCF is usually the highest, when *L* is chosen to be at a resonance peak close to half the beat length. In an actual design, we might choose *L* smaller than its optimal value to reduce the device dimension while keeping *FCF* high. For example, by choosing *L* = 500nm, an FCF of about 10 can be obtained as seen from Fig. 7.

## 5. Analysis of the transmission, reflection and FCF spectra

The field concentration effect of a PLC can affect the transmission and generate some reflection in the underlying photonic waveguide. In this section, we investigate the relations among the amplitudes of the transmission, reflection and FCF by analyzing their spectra. We use the finite-difference-time-domain (FDTD, Lumerical Solutions) method for simulations.

Figure 8 shows the transmission *T*, reflection *R*, and *T* + *R* spectra for (a) *L* = 1*μ*m and (b) *L* = 2*μ*m for the triangular taper with *W* = 400nm and *a* = 20nm. The wavelength range is 600 – 1100nm.

Let us consider the *L* = 1*μ*m case first. As we have computed, when *W* = 400nm, the beat length is about 2.1*μ*m at the wavelength of 800nm. For wavelengths in a range around 800nm, the beat length does not change much. Therefore, 1*μ*m is around half the beat length, when the PLC is expected to be most efficient. The term 1 − *T* − *R* indicates the power consumption on the metal structure due to the internal material absorption and the radiative loss.

As shown in Fig. 8(a), a resonance corresponds to the a drop in the transmission spectrum and a peak in the reflection spectrum. At resonance, the field is more concentrated in the metal region, resulting in an increased power consumption. This lowers the transmission *T* and the overall *T* + *R*, while inducing a larger reflection *R*. It is noticeable that the reflection part contributes less significantly to the overall *T* + *R* than the transmission part, and therefore the profile of the *T* + *R* spectrum is basically determined by the transmission spectrum. When the structure is off resonance, we have *T* + *R* about 0.8, which means a certain amount of power is still coupled to the metal region. This is quite different from the typical spectrum of a plasmonic resonator, for which *T* + *R* is very close to 1 when the structure is off resonance.

Figure 8(b) shows the spectra in the *L* = 2*μ*m case. The spacing between two successive drops in the transmission spectrum is clearly less than that of the *L* = 1*μ*m case. Moreover, since 2*μ*m is close to the beat length, the PLC is expected to be less efficient. As a result, the overall reflection is considerably smaller than that in the *L* = 1*μ*m case.

In Fig. 9, we compare transmission spectra and FCF spectra for three groups of metal triangle lengths. Those in the first group (Fig. 9(a)) are 0.4*μ*m, 0.425*μ*m and 0.45*μ*m, those in the second group (Fig. 9(b)) are 1*μ*m, 1.025*μ*m and 1.05*μ*m, and those in the third group (Fig. 9(c)) are 2*μ*m, 2.025*μ*m and 2.05*μ*m. We use *W* = 400nm and *a* = 20nm for all structures. Recall that by definition, the FCF(*a* = 20nm) is the normalized norm of the electric field at the point (0, 320nm, 20nm) in the coordinate system. When the wavelength is less than 700nm, the bare photonic waveguide supports higher order TM modes and the hybrid structure supports more modes in addition to H_{TM,0} and H_{TM,1}. Thus our analytic model using two supermodes is no longer valid in this case. The simulations show that the field concentration effect is tremendously suppressed in this case. For wavelengths larger than 700nm, the vertical dashed lines indicate that a local minima of the transmission corresponds almost exactly to a local maxima of the FCF, and correspondingly the plasmonic structure is at resonance.

## 6. Conclusion

In this paper, we have systematically analyzed the behavior and performance of a compact plasmonic light concentrator (PLC) which is a triangle-shaped Au taper integrated on a Si_{3}N_{4} waveguide. We use the fundamental TM-like mode for the incident light from a photonic waveguide to excite two supermodes in the hybrid structure. We have shown that three major effects (mode beat, nanofocusing, and weak resonance) interplay to generate the light concentration phenomenon and govern the performance of the device. By proper combination of these three effects, the overall structure can be optimized to obtain very large field concentration. In particular, we have demonstrated that after an optimization of the size parameters, a field concentration factor (FCF) of about 13 can be achieved with the length of the device less than 1*μ*m for a moderate tip radius of 20nm. Such a field concentration efficiency is about twice better than all reported results of on-chip plasmonic light concentrators in literature with much larger device sizes and an assumption of smaller tip radii. In addition, the dimensions of the device can be reduced further while maintaining a high concentration efficiency (a FCF of 5 – 10 is achievable with the length of triangle about 400nm).

In addition to realizing a larger field concentration with a smaller device size, our plasmonic device is mounted on top of a photonic waveguide and the waveguide does not terminate after the light concentration as in typical schemes reported earlier [25, 26, 28, 30]. This makes our device not only simpler in fabrication, but also more compatible for the integrated optical circuits. Finally, the three major effects commonly appear when different materials are used for the photonic waveguide and plasmonic layer while working at different wavelengths, and we can apply the same philosophy to optimize the design.

## Acknowledgment

The authors would like to thank Ali A. Eftekhar for helpful discussions. This work was supported by the Defense Advanced Research Projects Agency (DARPA) under Contract HR 0011 – 10 – 1 – 0075 through the DARPA CIPHER Project (S. Rodgers).

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