Local field enhancement on demand based on hybrid plasmonic-dielectric directional coupler

Kholod Adhem; Ivan Avrutsky

doi:10.1364/OE.24.005699

1. Introduction

Massive size mismatch between dimensions of integrated photonic components and integrated electronics has been one of the greatest challenges in integrated optics for years. Plasmonics provides a solution to this problem. Plasmonic waveguides could be used to guide signals that propagate at optical frequencies. Unlike the dielectric waveguides, plasmonic waveguides can confine light far beyond the diffraction limit of light. However, plasmonic waveguides suffer from high losses due to absorption in metal. In contrast, conventional dielectric waveguides, at least in principle, are low-loss waveguides. Modern technology requires a structure that offers both high confinement and low loss. Thus, the integration of plasmonic-dielectric waveguide directional couplers has been a subject of attention to many researchers.

Signal routing between plasmonic modes and dielectric modes in plasmonic integrated circuits (PIC) has been reported in [1–5]. Currently, strong confined modes are achieved by many plasmonic waveguide structures such as the slot waveguide [6], the dielectric-loaded waveguide [7], the groove waveguide [8] and the hybrid waveguide [9]. Many technical solutions have been proposed to develop hybrid dielectric-plasmonic structures that would provide high confinement and moderate optical losses. As an example, a horizontal directional coupler between a plasmonic waveguide and a silicon dielectric waveguide has been reported in [10].The study was to couple light from dielectric mode of size 297 nm×340 nm to a plasmonic mode of a size 200 nm×40 nm. The propagation loss is found to be 0.052 dB/μm and the associated propagation length is 83 μm at a communication wavelength of λ=1550 nm. Vertical directional coupling has also been reported, in which signal routing between metal-insulator-metal (MIM) plasmonic waveguide and silicon dielectric waveguide was analyzed [11]. It was found that the hybrid coupler was able to couple light from the silicon dielectric waveguide with a dimension of 220 nm×260 nm to the upper arm MIM plasmonic waveguide with dimensions of 200 nm×150 nm. The propagation loss was found to be 0.36 dB/μm and the corresponding propagation length was 12 μm at 1550 nm wavelength.

Overall, these studies indicate that the tradeoff between the degree of confinement and optical losses can be shifted in one or the other direction depending on what is more important for a particular application. Some applications for instance require that optical signals propagate for relatively large distances, on the order of few millimeters on a chip, while critical optical interactions are only feasible when the mode size is squeezed far beyond the diffraction limit. An example is an on-chip optical spectrometer that would ultimately measure absorption/emission spectra of individual quantum objects, such as large molecules or quantum dots. As of now, there is no technical solution for an optical waveguide that would meet such demanding requirements. We propose here a concept of local optical confinement on demand. In this concept, light mostly is guided by nearly lossless dielectric waveguides, but once needed, optical power of dielectric waveguide is transferred, almost completely, to a highly confined plasmonic mode, and then back to the dielectric mode. The concept is illustrated by numerical simulations using a commercial finite-element package from COMSOL Multiphysics.

The idea of the vertical directional coupler presented in this paper is inspired by the integrated optical polarizer device [12]. The functionality of the polarizer is to enable one polarization state (transverse magnetic (TM) or transverse electric (TE)) to propagate while the other polarization state is eliminated due to high propagation loss. Numerous types of waveguide polarizers have been realized over many years, including, for example, metal-clad waveguides [12,14]. We only deal with TM mode in our analysis throughout this paper. In contrast to polarizers, the directional coupler used here is designed in such a way that losses are relatively low due to short interaction length.

2. Description of structure and method of analysis

The structure considered in Fig. 1 consists of following layers: conductor (metal) cover at the top, followed by a low-index dielectric gap, high-index dielectric film, and dielectric substrate. Thickness of the gap layer is t_g, and thickness of the dielectric film is t_d. The permittivities of the conductor, gap, dielectric film and substrate are ε_c, ε_g, ε_d, and ε_s. Cover and substrate are semi-infinite layers. The structure is only variant in the z-direction. The layers can be for instance: silver (Ag), gap filled with air, silica-titania (SiO₂-TiO₂) film, and fused quartz SiO₂ substrate.

Fig. 1 (a) Schematic view of the conductor-gap-dielectric-substrate (CGDS) structure. (b) (CGDS) structure with a sketch of the electric field profile for the dielectric and plasmonic modes.

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The structure is a hybrid waveguide that can be thought of as a combination of two waveguides. The first waveguide is a plasmonic waveguide formed by a conductor and gap layers, while the second waveguide is a high index contrast dielectric waveguide formed by the film sandwiched between the gap and the substrate. For a small gap thickness, coupling between these two waveguides is strong. The plasmonic mode considered in this paper is a TM mode in nature. In a structure with flat interfaces made of isotropic materials, it can only couple to TM modes of the dielectric film. Theoretical analysis of hybrid plasmonic waveguides has been extensively studied in [15].

Modes supported by the structure can be obtained by extending an earlier theoretical work on conductor-gap-dielectric waveguide [16,17] by adding one more layer serving as a substrate for the film. The structure thus, becomes conductor-gap-dielectric-substrate system (CGDS). Assuming propagation in the x-direction, the expressions for the magnetic H_y field component in all layers from top to bottom can be written as follows:

H_{y} (x, z) = A \exp (- r (z - t_{g} - t_{d})) \exp (i k_{x} x), if z \geq t_{g} + t_{d} .

H_{y} (x, z) = [B \exp (q (z - t_{d})) + C \exp (- q (z - t_{d}))] \exp (i k_{x} x), if t_{d} \leq z \leq t_{g} + t_{d} .

H_{y} (x, z) = [D \exp (p z) + E \exp (- p z)] \exp (i k_{x} x), if 0 \leq z \leq t_{d} .

H_{y} (x, z) = F \exp (s z) \exp (i k_{x} x), if z < 0 .

Similar equations can be obtained for the x-component of the electric field using

E_{x} ~ \frac{1}{ε (z)} \frac{\partial H_{y} (z)}{\partial z}

. In the above equations r, q, p and s are related to the z-components of the wave-vector, and are given by

r = k_{o} \sqrt{{n^{*}}^{2} - ε_{c}}, q = k_{o} \sqrt{{n^{*}}^{2} - ε_{g}}, p = k_{o} \sqrt{{n^{*}}^{2} - ε_{d}}, s = k_{o} \sqrt{{n^{*}}^{2} - ε_{s}},

where n^* is the modal index, and

k_{o} = \frac{2 π}{λ}

is the wave-vector in free space. In case of n^*2 < ε_d, the p variable in Eq. (5) is pure imaginary quantity and Eq. (3) turns in a combination of sin and cos functions. By applying appropriate boundary conditions (continuity of H_y(z) and

\frac{1}{ε (z)} \frac{\partial H_{y} (z)}{\partial z}

across the interfaces at z = t_d and z = t_g + t_d), the dispersion equation for the modes in the CGDS system is found as

\tanh (q t_{g}) (C_{1} + C_{2}) - (A_{1} + A_{2}) = 0,

where

A_{1} = (\frac{r}{ε_{c}} + \frac{p}{ε_{d}}) \cdot (\frac{p}{ε_{d}} + \frac{s}{ε_{s}})

A_{2} = (\frac{r}{ε_{c}} - \frac{p}{ε_{d}}) \cdot (\frac{p}{ε_{d}} - \frac{s}{ε_{s}}) \cdot \exp (- 2 p \cdot t_{d})

C_{1} = - (\frac{q}{ε_{g}} + \frac{p r ε_{g}}{ε_{d} ε_{c} q}) \cdot (\frac{p}{ε_{d}} + \frac{s}{ε_{s}})

C_{2} = - (\frac{q}{ε_{g}} - \frac{p r ε_{g}}{ε_{d} ε_{c} q}) \cdot (\frac{p}{ε_{d}} - \frac{s}{ε_{s}}) \exp (- 2 p \cdot t_{d}) .

The CGDS structure with film thickness t_d = 500 nm supports two guided modes, the fundamental (m = 0), and the first order (m = 1) mode. At large values of t_g (greater than 1 μm), the guided mode in the CGDS structure becomes the mode of the thin film waveguide with gap/substrate claddings. Thus, the modal indices for (m = 0), and (m = 1) satisfy the dispersion equation of the thin film waveguide. Using the same notations in Eq. (5), the dispersion equation for the TM mode in a thin film waveguide [18] can be rewritten as:

p_{d} \cdot t_{d} - \tan^{- 1} (\frac{ε_{d} \cdot q}{ε_{g} \cdot p_{d}}) - \tan^{- 1} (\frac{ε_{d} \cdot s}{ε_{s} \cdot p_{d}}) - m π = 0,

where

p_{d} = k_{o} \sqrt{ε_{d} - {n^{*}}^{2}}

is a pure real quantity, and can be written using notation in Eq. (5) as p_d = −i · p.

At small values of t_g (less than 1 nm), the guided mode in the CGDS structure becomes the plasmonic mode of the metal-cladding waveguide with conductor/substrate claddings. The modal index of the fundamental (m = 0) mode becomes the modal index of the surface plasmon polariton (SPP) mode at the conductor/dielectric interface and can be obtained using [19]:

n^{*} = {(\frac{ε_{c} \cdot ε_{d}}{ε_{c} + ε_{d}})}^{1 / 2} .

The first order (m = 1) mode is the mode of the dielectric film with conductor/substrate claddings. Its modal index satisfies the dispersion equation of the metal-cladding waveguide. Following the same notations in Eq. (5), the dispersion equation for the TM mode for the metal-cladding waveguide [20] can be rewritten as:

p_{d} \cdot t_{d} - \tan^{- 1} (\frac{ε_{d} \cdot r}{ε_{c} \cdot p_{d}}) - \tan^{- 1} (\frac{ε_{d} \cdot s}{ε_{s} \cdot p_{d}}) - m π = 0 .

Note that modal indices found from Eqs. (11) and (13) match very well the limits at t_g → ∞ and t_g → 0 found from solving Eq. (6).

The plasmonic mode is a combination of the m = 0 and m = 1 supermodes. The optimal gap size is calculated such that the minimal modal indices difference between m = 0 and m = 1 occurs. The propagation length of the plasmonic mode at the optimal gap size can be estimated according to [22]:

L_{o} = \frac{λ}{4 π Im (\frac{n_{o}^{*} + n_{1}^{*}}{2})},

where

n_{o}^{*}

and

n_{1}^{*}

are the complex modal indices for m = 0 and m = 1 modes respectively.

Propagation of coupled guided modes is often considered in terms of supermodes which are eigenmodes of the entire multilayer structure. Coupling between the film mode and the plasmonic mode forms two types of supermodes which later will be referred to as TM-quasi-even and TM-quasi-odd modes. They propagate along the CGDS structure with different propagation constants. The interference between the supermodes results in the electromagnetic field confined mainly to the film or to the gap depending on the relative phase of the supermodes. At a distance equal to the coupling length L_c, almost all energy of the dielectric film is getting transferred to the gap mode. The coupling length is a measure of the beating length of the two eigenmodes, and it can be related to propagation constants by

L_{c} = \frac{π}{β_{e} - β_{o}},

where β_e and β_o are the propagation constants for the TM-quasi-even and odd eigenmodes respectively. The energy exchange between the two waveguides takes place every coupling length L_c. The propagation constant is related to the modal index as β = n^*k_o.

3. Modes analysis and local field enhancement

We numerically solve Eq. (6) with definitions in Eqs. (5), (7) and (10) to find the exact solution for the modal indices n^* of the guided modes for various gap thicknesses at 632.8 nm wavelength. Various programming languages can be used for this purpose, we used the ’root’ function of Mathcad as our choice. We compare the values obtained from analytical method as described above to those obtained by the finite-element code, Comsol Multiphysics. The effective indices predicted by the two methods matched well. Figure 2 provides the values of effective mode indices for the guided modes as a function of gap layer thickness t_g. The solid lines are obtained by the analytical method [solving Eq. (6) using Mathcad], and the points indicate the results from Comsol Multiphysics. Although the agreement between the analytical method and Comsol Multiphysics is also good in the whole range of gap layer thickness, we only show the range from [1 nm–100 nm] from Comsol. The dispersion curves in Fig. 2 show the anti-crossing behavior typical for coupled modes. The plasmonic mode as such has a dispersion curve with large negative slope. The anticrossing at t_g=17nm, where the difference between the indices takes on minimal value, indicates strong interaction between the mode of the film and the plasmonic mode. The minimum modal index difference is depicted in Fig. 2 with red dots for each mode. Therefore, t_g=17nm is the gap size to be used in the rest of our analysis.

Fig. 2 Modal indices n^* of the two eigenmodes versus gap thickness t_g, materials from top to bottom are Ag, air, SiO₂-TiO₂, and SiO₂ at λ = 632.8 nm respectively. Film thickness=500nm.

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In order to be consistent with our numerical analysis of modes supported by the structure in Fig. 1, we will assume the conductor layer is of same width as the width of the gap-dielectric-substrate layers. The permittivities used in simulations are ε_c =−15.822+ 1.075i [21], ε_d =3.13, ε_s = 2.1 and ε_g =1.0 for Ag, SiO₂-TiO₂, SiO₂ and air at λ = 632.8 nm respectively. To guarantee a good coupling between the two waveguides, the following structural parameters are selected in the simulation: w_d = 500 nm, t_d = 500 nm, t_s = 760 nm, w_c = 200 nm, t_c = 304 nm, and t_g = 17 nm. Figures 3(a) and 3(c) depict the electric field component E_z of the two supermodes supported by the structure at t_g =17 nm. Red line in Figs. 3(a) and 3(c) at y = 0.25 μm corresponds to the middle of the structure. Figures 3(b) and 3(d) provide the E_z profile at y = 0.25 μm for the TM-quasi-even and the TM-quasi-odd modes.

Fig. 3 Electric field component E_z of (a) the TM-quasi-even and (c) the TM-quasi-odd modes supported by the hybrid coupler. (b) and (d) are corresponding electric field E_z at y = 0.25 μm in the y − z plane.

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It can be seen that for the TM-quasi-even mode, the electric field directions in the gap and in the film are the same while they are opposite in the TM-quasi-odd mode, as shown by black arrows.

Figure 4(a) provides the light intensity profile in the x − z plane for y = 0.25 μm as a function of the position x when the dielectric mode is excited first at x = 0. In the COMSOL model, this is achieved by a simple excitation of the two supermodes with appropriate relative phases. As x increases, the field transfers from the dielectric waveguide to the gap plasmonic waveguide gradually owing to the interference of the TM-quasi-even and TM-quasi-odd eigenmodes.

Fig. 4 (a) Light intensity profile along x − z plane corresponds to y = 0.25 μm. (b) Periodic energy exchange along the propagation length x.

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At a distance x equal to the coupling length L_c, almost all the fields transfer to the gap mode, then the field transfers back to the film mode at x = 2L_c. The coupling length 2.1 μm found from Eq. (15) matches well the Comsol results as depicted in Fig. 4. Figure 4(b) depicts periodic energy exchange along the propagation length x (light intensity in the gap at the surface of the film). Graduate decrease of energy from one period to another is consistent with 20 μm propagation length calculated from Eq. (14). The dotted line in Fig. 4(b) shows exponential decay of intensity corresponding to the loss factor of 0.2 dB/μm which can be evaluated from Eq. (14). The dashed line shows the level of light intensity at the surface of the dielectric film waveguide in absence of the metal tip. For local field enhancement concept, only one period is needed to illustrate the light intensity exchange from the dielectric film to the nano-gap and then back to the film.

4. Performance analysis

Figure 5 shows a plot for the 3-D version of the structure previously analyzed in section 3. In addition to the variables t_g (gap size) and t_d (film thickness) introduced earlier, Fig. 5 shows other dimensions: t_s (substrate thickness), w_d (width of the dielectric film and substrate), w_c (width of the conductor). The conductor (metal blade) is physically detached from the rest of the structure. It can be placed at the desirable location using AFM-style nano-positioners. The proposed system thus belongs to the category of tip-enhanced tools, which provides local, on demand, enhancement of the electromagnetic field at the location of this specially designed tip. Figures 6(a) and 6(b) depict the electric field component E_z of the silica-titania dielectric mode at x = 0 and the corresponding E_z profile at y = 0.25 μm. The simulation figures are obtained from the finite-element-method based commercial software Comsol Multiphysics. The obtained modal index n^* for the fundamental (m = 0) silica-titania dielectric mode is 1.685 at 632.8 nm. This value is slightly less compared to the modal index evaluated theoretically from Eq. (6) at t_g → ∞. The reason for this difference is the lateral confinement (finite size in the y direction) in the Comsol model.

Fig. 5 Schematic diagram of the hybrid directional coupler. The green region denotes SiO₂, the red region denotes (SiO₂-TiO₂), and the yellow region denotes Ag. x is the light propagation direction.

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As soon as the silver blade is approaching the top of the structure to form what we call CGDS system, the film mode couples to the plasmonic mode supported by the bottom surface of the silver blade. Figures 6(c) and 6(d) depict the electric field component E_z of the coupled modes at x = 0 along the y − z plane, and the corresponding E_z profile at y = 0.25 μm respectively.

Fig. 6 Electric field component E_z of (a) the silica-titania dielectric mode in the y − z plane. (b) the corresponding E_z profile at y = 0.25 μm along x = 0. (c) depict the electric field component E_z of the coupled modes at t_g = 17 nm and (d) depicts the corresponding E_z profile for y = 0.25 μm.

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Although the concept of energy exchange is well illustrated by the interaction between the two guided modes in section 3, it does not account for scattering in the CGDS structure with finite dimensions of the conductor tip. Below we consider the thin film mode (m = 0) launched at x = 0, while the 4.2 μm wide tip is centered at x = 17.1 μm. This film mode has a modal index n^* = 1.685 as was mentioned earlier in the text. It propagates freely in the film and once it reaches the metal tip, it interacts with the CGDS structure. Not only it couples with the gap mode, but it also scatters in all possible directions owing to the sharp corners of the metal tip. Figure 7(a) depicts the relative intensity of light in the x − z plane for y = 0.25 μm as a function of the position x when the thin film mode (m = 0) is launched at x = 0. Figure 7(b) depicts the local light intensity in the x − z plane at z = 1.5 μm. It is obvious that the intensity in the gap is more than one order of magnitude stronger than at the film surface. This is what we refer to as local field enhancement on demand in the CGDS structure.

Fig. 7 (a) Relative intensity of light along x − z plane corresponds to y = 0.25 μm. (b) Light intensity in the gap along the propagation length x. (c) Power flux for the dielectric film mode at the input port (x = 0 μm) and at the output port (x = 34.2 μm).

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The interference of guided modes propagating in the waveguide forward and backward results in periodical variations in intensity with period equals to λ/2n^* ≃ 0.2 μm. Periodicity of intensity pattern is visible in Figs. 7(a) and 7(b). Figure 7(b) shows a small depth of intensity modulation in the interference pattern which indicates weak reflection in the CGDS structure. Figure 7(a) shows a drop in the overall intensity level by approximately 1 dB. This total attenuation is due to both, absorption in metal and scattering by the metal tip. Attenuation due to absorption in metal while guided mode propagates under 4.2 μm wide tip is estimated to be 0.2 dB/μm×4.2 μm ≃ 0.84 dB. This leaves about 0.16 dB for overall scattering. Interference with other scattered waves results in some distortion in the modal field also visible in Fig. 7(b). Figure 7(c) shows power flux for the dielectric film mode at the input port (x = 0 μm) and at the output port (x = 34.2 μm). The units are arbitrary and the power flux for the input and output ports are normalized. The value of the power flux at the output port drops by 1dB is consistent with Fig. 7(a). Port boundary conditions are used to launch the mode into the structure. Perfect Electric Conductor boundary conditions (PEC) are applied to the top and bottom boundaries.

5. Conclusion

Local field enhancement on demand based on vertical directional coupling between film mode and gap plasmonic mode in conductor-gap-dielectric-substrate system (CGDS) has been proposed and investigated using an analytical model confirmed by finite element method simulations. The dispersion equation has been derived analytically and solved numerically. The structure provides strong on-demand field enhancement at the surface of the dielectric film waveguide. Such a hybrid structure can be potentially exploited for developing photonic-plasmonic hybrid functional components for signal routing, power splitting, etc. in PICs. This structure is well suitable for applications which require light-matter enhanced interactions. In particular, it is very useful for biomedical applications where the visible range of wavelength is used.

Acknowledgments

This work was partly supported by CPHOM, NSF center for Photonics and Multiscale Nanomaterials DMR 1120923.

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Local field enhancement on demand based on hybrid plasmonic-dielectric directional coupler

Abstract

1. Introduction

2. Description of structure and method of analysis

3. Modes analysis and local field enhancement

4. Performance analysis

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (15)

Optics Express