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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 12677–12687
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Vertical coupling between gap plasmon waveguides

Galen B. Hoffman and Ronald M. Reano  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12677-12687 (2008)
http://dx.doi.org/10.1364/OE.16.012677


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Abstract

This work examines vertical coupling between gap plasmon waveguides for use in high confinement power transfer and power splitting applications at 1.55 µm free space wavelength. The supermode interference method is used to obtain key coupler performance parameters such as coupling length, extinction ratio, net coupled output power, radiated power, and reflected power as a function of waveguide center-to-center spacing, core refractive index, and gap width. The initial power distribution among the two coupler supermodes is obtained via the mode matching method for a single input waveguide feed. Excellent agreement with three-dimensional finite difference time domain simulations is observed for the case of square 50 nm gaps with core refractive indices of 2.50 and a center-to-center spacing of 112 nm. Local maxima in the net coupled output power are found to coincide with local minima in the coupling length. An increase in the core refractive index from 1.00 to 2.5 increases the local maximum net coupled output power from 6.4% to 49% but decreases the extinction ratio from 12.7 to 6.94. A sweep of the width of the core from 25 to 100 nm increases the net coupled output power from 43.7% to 52.0%, but increases the coupling length from 1.58 to 3.19 µm and decreases the extinction ratio from 7.39 to 6.57.

© 2008 Optical Society of America

1. Introduction

Plasmonic waveguides are particularly promising for microelectronic circuits because they have the potential to replace conventional wires, using light as an information carrier with deep subwavelength confinement instead of electrons [1

1. J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, cross-talk and energy costs,” Opt. Express 15, 4474–4484 (2007). [CrossRef] [PubMed]

]. The plasmonic waveguiding structures that have been explored to date fall into three major categories: nanoparticle chains [2

2. K. Jung, F. L. Teixeira, and R. M. Reano, “Au/SiO2 nanoring plasmon waveguides at optical communication band,” IEEE J. Lightwave Tech. 25, 2757–2765 (2007). [CrossRef]

], metal stripes [3

3. P. Berini, “Air gaps in metal stripe waveguides supporting long-range surface plasmon polaritons,” J. Appl. Phys. 102, 33112 (2007). [CrossRef]

], and the gap waveguides [4–6

4. G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30, 3359–3361 (2005). [CrossRef]

]. The most important performance parameters of these guiding structures are their propagation length and the degree of confinement they can obtain. The propagation length (Lp) is defined as the length over which the component of the time-average power in the direction of propagation of a mode decays to 1/e of its original value. The confinement is less strictly defined and often depends on the waveguide geometry, though usually it is stated to be the width or area of the region in which the power density is greater than some proportion of its maximum value [7

7. R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15, 12174–12182 (2007). [CrossRef] [PubMed]

]. The nanoparticle chains have propagation lengths and confinements on the order of hundreds of nanometers [2

2. K. Jung, F. L. Teixeira, and R. M. Reano, “Au/SiO2 nanoring plasmon waveguides at optical communication band,” IEEE J. Lightwave Tech. 25, 2757–2765 (2007). [CrossRef]

], and the metal stripes, supporting long-range surface plasmon polariton (LRSPP) modes [8

8. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

], have propagation lengths of centimeters, but exhibit a confinement of microns [9

9. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mat. Today 9, 20–27 (2006). [CrossRef]

]. Various geometries of gap waveguides have been explored, such as the v-groove [10

10. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 46802 (2005). [CrossRef]

,11

11. M. Yan and M. Qiu, “Guided plasmon polariton at 2D metal corners,” J. Opt. Soc. Am. B 24, 2333–2342 (2007). [CrossRef]

], and slot or gap waveguides [4

4. G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30, 3359–3361 (2005). [CrossRef]

]. Gap waveguides provide a balanced tradeoff between confinement and propagation length, having confinements of hundreds of nanometers and a propagation length of several microns [4

4. G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30, 3359–3361 (2005). [CrossRef]

]. This paper will thus concern the gap plasmon waveguide (GPW), also known as the plasmonic slot waveguide [4

4. G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30, 3359–3361 (2005). [CrossRef]

].

The steady state propagation characteristics of the GPW have already been thoroughly treated [4

4. G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30, 3359–3361 (2005). [CrossRef]

,12

12. G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” IEEE J. Lightwave Tech. 25, 2511–2521 (2007). [CrossRef]

]. One of the most important devices compatible with this waveguide is the directional coupler. Because this waveguide is not closed from above and below by metal, the field evanescently tails out of the gap vertically, allowing for crosstalk between two such waveguides in this direction.

The rest of this paper is organized as follows. Section two introduces our geometry, restrictions of the parameter space, simulation method, and design methodology. Section three presents simulation results, and finally section four concludes the paper.

2. Design approach

2.1 Geometry specification

Figure 1(a) is a schematic of the geometry of the single GPW. It consists of a rectangular gap etched into a metal film and later backfilled by a dielectric. In this work, the core region may not necessarily be of the same material as the cladding or substrate. The gap has a width w and a height h. Propagation of the wave occurs in the +z-direction, in which the waveguide geometry is considered to be invariant for the purpose of mode solving. The refractive indices of the core, metal, and cladding are denoted via ncore, nmetal, and nclad, respectively.

Figure 1(b) depicts the vertically-coupled geometry where one GPW is placed directly over the other with a center-to-center (CTC) spacing denoted by s. The refractive index of the middle spacer layer between the two metal films is denoted nmid. This structure is referred to as a vertical gap plasmon coupler (VGPC). We take the geometry to be vertically-symmetric. To quantify how the total power is distributed between the two waveguides as a function of propagation distance along the coupler, we define the channel 1 region as being that above the line midway between the two gaps, and the channel 2 region as being below it, and thus this line is the boundary between the two semi-infinite waveguide cross-section domains over which the integration of time average power density is done (dashed line in Fig. 1(b)).

Fig. 1. Schematic cross-section of the (a) gap plasmon waveguide (GPW) and (b) vertical gap plasmon coupler (VGPC).

2.2 Simulation method

For waveguides one can make the assumption that the field dependence in the propagation direction (z-direction here) goes as e -γz for forward propagating waves, where γ=α+ is the complex propagation constant, α is the attenuation constant and β is the phase constant. Thus the propagation length is Lp=1/(2α). The real part of the effective propagation index (neff) is obtained by dividing β by k 0, where k 0 is the free space phase constant. The real part of the effective propagation index and the propagation length of the fundamental gap plasmon mode are shown in Fig. 2(a) and Fig. 2(b) calculated via the finite difference frequency domain (FDFD) method [15

15. J. A. Pereda, A. Vegas, and A. Prieto, “An improved compact 2D full-wave FDFD method for general guided wave structures,” Microwave Opt. Technol. Lett. 38, 331–335 (2003). [CrossRef]

] and compared with both Reference [12

12. G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” IEEE J. Lightwave Tech. 25, 2511–2521 (2007). [CrossRef]

] and finite element frequency domain (FEFD) results using COMSOL Multiphysics™ (for consistency with Ref. 12

12. G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” IEEE J. Lightwave Tech. 25, 2511–2521 (2007). [CrossRef]

, Fig. 2 uses 1.55 µm free-space wavelength, w=h=50 nm, nmetal=(-143-10i)0.5, refractive index of 1.44 for core, cladding, and substrate) [16]. For both FDFD and FEFD formulations, γ is obtained as the eigenvalue of the respective mode. Our FDFD implementation uses a uniform orthogonal grid whose points sit on the geometry boundaries. We enforce homogeneous Dirichlet boundary conditions at the simulation edges and keep them far enough away from the waveguide so as not to have a numerically significant effect. Both the electric (E-FDFD) and magnetic (H-FDFD) field formulations were implemented as a cross-check. For the COMSOL simulations, we use the scattering boundary condition to truncate the boundary, use cubic Lagrangian elements, and keep the maximum element size along the vertical sidewalls of the GPW equal to 0.1h. There is good agreement among the four approaches. The FEFD method was found to converge on a solution to a given accuracy with fewer unknowns and computational effort compared to our FDFD implementation with a uniform grid. Further numerical computation in this paper is accomplished via FEFD.

Fig. 2. (a) Effective propagation index obtained by different mode solvers for the fundamental mode of the gap plasmon waveguide with h=w=50 nm, λ 0=1.55 µm, nclad=ncore=nsub=nsilica=1.44, nmetal=(-143-10i)0.5. b) Corresponding propagation length.

2.3 Parameter space reductions

In the analysis that follows, the materials are chosen for their ease of processing and low loss at room temperature. Silver is the metal, with optical parameters from Hagemann, Gudat, and Kunz [17

17. H. J. Hagemann, W. Gudat, and C. Kunz, “Optical constants from the far infrared to the x-ray region: Mg, Al, Cu, Ag, Au, Bi, C, and Al2O3,” J. Opt. Soc. Am. 65, 742–744 (1975). [CrossRef]

]. At 1.55 µm, nmetal≈(0.434-12.7i). The substrate, cladding and middle are composed of silica with a refractive index of 1.44. The gap cores can have refractive indices of 1.00, 1.44, 2.00, and 2.50. Materials that span this range of refractive indices include air, silicon dioxide, silicon nitride, and Group IV–VI amorphous glasses. The fabrication of air gaps has been feasibly demonstrated using multiple techniques [18

18. R. M. Reano and S. W. Pang, “Sealed three-dimensional nanochannels,” J. Vac. Sci. Technol. B 23, 2995–2999 (2005). [CrossRef]

,19

19. Bruce McConnel, “Using self-assembly to create airgap microprocessors” (IBM, 2007). http://www-03.ibm.com/press/us/en/presskit/21463.wss. Accessed 10-30-07.

].

2.4 Analysis method

The supermode interference (SI) method is used to evaluate the VGPC since our approach is in the steady-state frequency domain [21

21. K. Okamoto, Fundamentals of Optical Waveguides, (Academic Press, 2006).

]. In this method, the power oscillation between the waveguides is expressed in terms of the interference among the superimposed modes of the coupled structure. The interference pattern changes with propagation distance due to the different spatial phase constants of the modes. In general, many modes could be supported by these waveguides depending on the dimensions of the gap and the refractive index of the core; here we restrict ourselves to the case when the gaps are single mode. In this case, the vertically-coupled geometry supports two primary supermodes, both symmetric in the x-direction, and sometimes a third supermode, asymmetric horizontally and vertically. This third supermode is neglected in this analysis on the grounds that it would not be excitable by a horizontally-symmetric source. The two primary supermodes are either symmetric or anti-symmetric in the y-direction. The symmetry is determined based upon the polarization-defining field component, which is Ex for the coordinate system that we are considering. We denote the symmetric supermode with Es and the asymmetric supermode with Ea. The phase constants for the symmetric and asymmetric supermodes are donated βs and βa, respectively.

Of central importance is determining the initial power distribution among the modes given some input condition as this will affect the extinction ratio at the output plane. Taking this into account becomes significant when one of the supermodes is poorly matched to the input excitation (i.e. when the waveguides are close together) and this leads to a decreased power transfer from one waveguide to another. We assume that the input waveguide feeding the coupler is a single GPW whose dimensions and core refractive index are identical to the gaps that comprise the coupler. As shown in Fig. 3, the input waveguide feeds directly into the channel 1 gap. We label the input mode Ei. The input plane of the coupler is our phase reference point.

Fig. 3. Side-view schematic of our excitation condition for the VGPC along the yz-plane at x=0. The input mode of the single GPW (Ei) is used to excite the coupler.

Figure 4 shows Ei, Es, and Ea for w=h=50 nm and s=100 nm, with ncore=1.44. The white lines show the boundaries of the metal and the core, the surface plot shows the time average z-propagating power density (Savgz) profile of the modes, and the arrow plot shows the transverse electric field vector. All of the quantities are normalized to the peak value. From the power density profile, one can see that the wave’s power density peaks at the four corners of each of the gaps, demonstrating that the dominant guiding mechanism in this particular case is due to the coupled “wedge plasmon” mode [20

20. D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, “Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap,” J. Appl. Phys. 100, 13101 (2006). [CrossRef]

]. The polarization of the modes is shown by the arrow plot, which shows it being dominant in the x-direction. The arrow plot also shows the symmetry of the supermodes: when Es (Fig. 4(b)) and Ea (Fig. 4(c)) are as shown, they approximately superimpose to Ei (Fig. 4(a)). When the phase difference between them is 180°, the input mode is approximately replicated at the bottom waveguide position, at y=-50 nm.

The length over which this phase shift occurs is the coupling length (Lc=π/|βs-βa|) and is half the beat length between Es and Ea. This length sets the minimum required length of any coupler, and is itself a performance parameter. Given that GPW’s are highly lossy and that their advantage rests with their ability to confine radiation in a subwavelength geometry, it is desirable that this length be as short as possible for compact, efficient devices. We take the effective end of the coupler to be at the plane z=Lc and evaluate performance parameters at this location.

The fundamental mode of the input waveguide determines the initial power distribution of the supermodes by applying transverse field continuity at the input-coupler interface and determining the transmission and reflection coefficients for guided modes (i.e. mode matching method, MMM) [22

22. C. Pollock, Fundamentals of Optoelectronics, (Ceramic Book and Literature Service, 2003), Chap. 11.

], [23

23. W. C. Chew, Waves and Fields in Inhomogeneous Media, (IEEE Press, 1995), Chap. 6.

]. Since we are assuming that the core dimensions and material are the same for the input waveguide as the coupler, we ignore the radiation modes. We then use these transmission coefficients as scalar coefficients weighing the superposition of Es and Ea in the coupler, calling them ts and ta, respectively, normalizing each mode such that Pavgz is equal to one.

Fig. 4. (a) The computed transverse electric field vector (arrow plot) and time average power density, Savgz (surface plot) for the input mode, Ei, of a square silica-filled 50 nm gap, b) the symmetric supermode, Es, of the corresponding coupler, and c) the asymmetric supermode, Ea.

The fundamental mode of the input waveguide determines the initial power distribution of the supermodes by applying transverse field continuity at the input-coupler interface and determining the transmission and reflection coefficients for guided modes (i.e. mode matching method, MMM) [22

22. C. Pollock, Fundamentals of Optoelectronics, (Ceramic Book and Literature Service, 2003), Chap. 11.

], [23

23. W. C. Chew, Waves and Fields in Inhomogeneous Media, (IEEE Press, 1995), Chap. 6.

]. Since we are assuming that the core dimensions and material are the same for the input waveguide as the coupler, we ignore the radiation modes. We then use these transmission coefficients as scalar coefficients weighing the superposition of Es and Ea in the coupler, calling them ts and ta, respectively, normalizing each mode such that Pavgz is equal to one.

To determine the waveguide discontinuity scattering coefficients, we first consider the requirement for tangential continuity of the fields on either side of the input plane:

Eit(1+r)=tsEst+taEat
(1)
Hit(1r)=tsHst+taHat
(2)

Here the t superscript denotes the transverse components of the field. The parameter r denotes reflection coefficient. We then apply the unconjugated (applicable for lossy modes) orthonormality condition as the overlap integral using the output modes of the waveguide to reduce (1) and (2) to scalar equations [24

24. D. Marcuse, Light Transmission Optics, (Van Norstrand Reinhold Company, 1972), pp. 322–326.

]:

Ent×Hmtdxdy=Emt×Hntdxdy=δnm
(3)

The subscripts n and m denote modes of the same waveguide, and δnm is the Kronecker delta. Using Eq. (3), Eqs. (1) and (2) are transformed to a set of four equations that can be written in the following matrix form:

[ξis10ξia01ξsi10ξai01][rtsta]=[ξisξiaξsiξai]
(4)

The i, a, and s subscripts label the input, asymmetric, and symmetric supermodes, respectively. Since this is an over-determined system, we take the pseudo-inverse of the matrix on the left of (4) to find the scattering coefficients to obtain the best (least squares) approximation [25

25. G. Strang, Linear Algebra and Its Applications 3rd Ed., (Thomson Learning, 1988), pp. 448–449.

]. Each ξnm is the overlap between the electric field of mode n and the magnetic field of mode m, where the modes here are from different waveguides:

ξnm=Ent×Hmtdxdy
(5)

2.5 Performance parameters

The purpose of a directional coupler is to transfer as much propagating power, defined as the time averaged-Poynting power in the z-direction (Pavgz), over from one waveguide to another. One performance parameter of the directional coupler is the extinction ratio (ER), which is the ratio of the output channel power (the time-averaged z-component of the power density, Savgz, integrated over the channel 2 region) to the input channel (channel 1) power and is ideally infinite so that the desired output channel is isolated from the other. Since we are interested in extracting as much power in channel 2 as possible, we define another performance parameter to be the net coupled output power, Pnet, which is the channel 2 power at the output plane, taking into account both the losses at the input of the coupler due to reflected and radiated power, and the absorption losses along the coupler. This intrinsically includes information about ER as well because a large ER coupler will have most of its power in channel 2. We evaluate both Pnet and ER at a length along the coupler equal to Lc. We have always found that the ER is maximized at this point, though Pnet is actually maximized at a slightly shorter distance. We also examine the guided reflected power (Prefl) and total estimated radiated power (Prad) in terms of the transmission and reflection coefficients:

Prad=1(Prefl+ts2+ta2)
(6)
Prefl=r2.
(7)

2.6 Design strategy

Fig. 5. a) Real part of the effective propagation indices (neff’s) for MIM-TM0, Es, Ea, and Ei for w=h=50 nm and ncore=1.44 in the target materials-geometry-excitation system note that the effective indices of the supermodes cross each other at s≈110 nm b) coupling length, Lc, versus vertical CTC gap spacing, s.

To determine analytically how the power oscillations occur, we examine the definition of Savgz in Eq. (8), where the “tot” suffix denotes a field quantity that is the weighted sum of the two supermodes:

Savgztot=12·Re{Etott×Htott*}=12·Re{Savgzsse2αsz+Savgzaae2αaz+Savgzsae2αz·ejβΔz+Savgzase2αz.ejβΔz}.
(8)

The average attenuation constant is denoted α′ and βΔβs-βa. The power densities, Savgz-nm, are given by

Savgznm=12·Re{Ent×Hmt*}.
(9)

The parameters Savgz-ss and Savgz-aa are the mode power densities of each supermode by itself. However, Savgz-sa and Savgz-as are the cross power densities. The power in each channel is obtained by integrating (8) over the regions specified in Fig. 1(b).

The cross power densities show how power can oscillate as a function of distance along the coupler. The mode power densities are symmetric in the y-direction, whereas the cross power densities are asymmetric. Thus, as the phase of the cross terms is incremented, the cross power densities interfere with the self power densities in a way such that power is subtracted from the top channel and added to the lower one.

3. Optimization

The purpose of this section is to optimize the coupler geometry with respect to net coupled output power. Here we explore the effects of changing ncore, the CTC gap spacing, s, and the gap width, w. We vary s from 60 to 500 nm, take w=25, 50, or 100 nm. All other parameters are as specified in section 2.3 above.

3.1 Square gaps

Figure 6 shows the results for the case where w=h=50 nm. The coupling length in general decreases as the waveguides are brought closer together, but there is a singularity (denoted sinf) where the supermode phase constants cross each other. There is a local minimum in Lc for s values larger than sinf. An infinite coupling length, however, does not necessarily imply complete isolation. Even when the phase constants are the same, power is transferred by the more rapid attenuation of one supermode versus another (Ea has always been observed to have a higher loss constant for these analyses), in which case, at the limit of large coupling length, almost half of the total power would be transferred. The ER is shown in Fig. 6(b). ER also decreases with s, exhibiting a local minimum near sinf where a close to unity extinction ratio is obtained. Strictly speaking the ER is not defined at sinf, so only ER values away from sinf are considered. Also shown is Prad in Fig. 6(c), which decreases monotonically as the waveguides are separated further apart, corresponding to a better match of the supermodes to the single input mode. In all cases tested it was found that Prefl was less than 0.5%.

Increasing ncore produces a smaller minimum Lc, larger ER, smaller sopt and greater Pnet. This can be explained by the fact that the increased ER and decreased Lc are able to overcome the decrease in the propagation length. Figure 6(d) shows that the increase in the maximum Pnet increases at a decreasing rate with ncore. Because increasing ncore also leads to more highly confined modes, Prad and Prefl also decrease.

In Fig. 6(f) we validate the supermode interference/mode matching method (SI-MMM) by observing the power oscillations along the length of a coupler using the finite-difference time-domain method (FDTD), as implemented by RSoft [28]. We choose the case where ncore=2.5 and s=sopt=112 nm. The power in channel 1 and 2 is denoted Pch1 and Pch2, respectively. The FDTD simulation consists of an input waveguide feeding into the coupled structure, with the input mode field profile generated by COMSOL as the excitation. Our simulation setup was exactly as depicted in Fig. 3, using the input mode as a CW source. We extracted the power values along the coupler length by projecting the total FDTD field onto the weighted sum of the coupler supermodes obtained from COMSOL, using overlap integrals (similar to those used to obtain Eq. (4) from Eqs. (1) and (2)) to obtain the weights and normalizing the weights to the input mode power at coupler input plane. For this case, |ts|=0.8024 and |ta|=0.5687 (both with negligible imaginary parts). Our results show that not only does the SI method given here give the correct propagation length, but also that the MMM, using the pseudo-inverse of the over-determined matrix to find the scattering parameters at the discontinuity, correctly calculates initial power distribution of the supermodes at the coupler input. A benefit of the SI-MMM approach is that it transforms a 3D problem into a 2D one, allowing one to analyze the structure in a matter of minutes instead of days (on a desktop computer).

Fig. 6. Performance parameters for the VGPC versus ncore and s: in Figs. (a)–(e), the thick solid line is for ncore=1.00, the thin solid line is for ncore=1.44, the thick dotted line is for ncore=2.00, and the thin dotted line is for ncore=2.50. a) Coupling Length, Lc, which tends to decrease with increasing ncore and decreasing s, but reaches a singularity where the supermode phase constants cross each other b) Extinction Ratio, ER, which increases when either ncore or s is increased c) Radiated power, Prad, which monotonically decreases with increasing s, d) Net coupled output power, Pnet, which is large where Lc is small, and reaches a maximum where s=112 nm for ncore=2.5 e) Average supermode propagation length, Lp-avg, which is a weak function of s, f) Comparison of supermode interference with the mode matching method (SI-MMM) and FDTD in predicting the power oscillations along the coupler for the case where ncore=2.50. Excellent agreement is observed.

It is important to note that Pch2 is actually maximized a little bit before the coupling length. This is due to the high loss of the waveguide causing the total propagating power of the waveguide to decay before the amount transferred to channel 2 has maximized. However, one can see that the difference in powers and coupler lengths is small, so we simply evaluate Pch2 at Lc. In all cases that we studied, we found that the ER along the length of the coupler was maximized at z=Lc as well, and this could be advantageous for noise reduction.

3.2 Comparison of square gaps to rectangular gaps

In this section, we compare the optimal performance parameters of the square gap with gaps with w=25 and 100 nm (h=50 nm). The relationships of the performance parameters with respect to ncore and s, shown in Fig. 6, mainly carry over to the cases where the gap is no longer square. Because we have found that Pnet is maximized at larger values for larger ncore, we summarize the results obtained in Table 1, with ncore=2.50.

Increasing w causes the modes to be less-highly confined in the gaps, and so Ea is more easily perturbed by the proximity of another metal layer. Thus, the sopt values tend to increase with w. The coupling length also increases, doubling from about 1.5 µm to 3 µm with w increasing from 25 to 100 nm, this being due to the magnitude of the propagation constants being smaller. The ER does decrease by a smaller amount, and with the Lp-avg increasing significantly, this causes Pnet to increase by 9 %, even with the longer coupling length. This means that a designer may decide to use a wider gap to increase the output power at the small expense of a decreased extinction ratio. The radiated power stays around 3%, and the reflected power is negligible.

Table 1. Performance parameters for ncore=2.50 for different core widths, w.

table-icon
View This Table

4. Conclusions

References and links

1.

J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, cross-talk and energy costs,” Opt. Express 15, 4474–4484 (2007). [CrossRef] [PubMed]

2.

K. Jung, F. L. Teixeira, and R. M. Reano, “Au/SiO2 nanoring plasmon waveguides at optical communication band,” IEEE J. Lightwave Tech. 25, 2757–2765 (2007). [CrossRef]

3.

P. Berini, “Air gaps in metal stripe waveguides supporting long-range surface plasmon polaritons,” J. Appl. Phys. 102, 33112 (2007). [CrossRef]

4.

G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30, 3359–3361 (2005). [CrossRef]

5.

L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31, 2133–2135 (2006). [CrossRef] [PubMed]

6.

D. Gramotnev, “Adiabatic nanofocusing of plasmons by sharp metallic grooves: Geometrical optics approach,” J. Appl. Phys. 98, 104302 (2005). [CrossRef]

7.

R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15, 12174–12182 (2007). [CrossRef] [PubMed]

8.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

9.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mat. Today 9, 20–27 (2006). [CrossRef]

10.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 46802 (2005). [CrossRef]

11.

M. Yan and M. Qiu, “Guided plasmon polariton at 2D metal corners,” J. Opt. Soc. Am. B 24, 2333–2342 (2007). [CrossRef]

12.

G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” IEEE J. Lightwave Tech. 25, 2511–2521 (2007). [CrossRef]

13.

G. Veronis and S. Fan, “Crosstalk between three-dimensional plasmonic slot waveguides,” Opt. Express 16, 2129–2140 (2008). [CrossRef] [PubMed]

14.

H. S. Won, K. C. Kim, S. H. Song, C. Oh, P. S. Kim, S. Park, and S. Kim, “Vertical coupling of long-range surface plasmon polaritons,” Appl. Phys. Lett. 88, 11110 (2006). [CrossRef]

15.

J. A. Pereda, A. Vegas, and A. Prieto, “An improved compact 2D full-wave FDFD method for general guided wave structures,” Microwave Opt. Technol. Lett. 38, 331–335 (2003). [CrossRef]

16.

http://www.comsol.com

17.

H. J. Hagemann, W. Gudat, and C. Kunz, “Optical constants from the far infrared to the x-ray region: Mg, Al, Cu, Ag, Au, Bi, C, and Al2O3,” J. Opt. Soc. Am. 65, 742–744 (1975). [CrossRef]

18.

R. M. Reano and S. W. Pang, “Sealed three-dimensional nanochannels,” J. Vac. Sci. Technol. B 23, 2995–2999 (2005). [CrossRef]

19.

Bruce McConnel, “Using self-assembly to create airgap microprocessors” (IBM, 2007). http://www-03.ibm.com/press/us/en/presskit/21463.wss. Accessed 10-30-07.

20.

D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, “Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap,” J. Appl. Phys. 100, 13101 (2006). [CrossRef]

21.

K. Okamoto, Fundamentals of Optical Waveguides, (Academic Press, 2006).

22.

C. Pollock, Fundamentals of Optoelectronics, (Ceramic Book and Literature Service, 2003), Chap. 11.

23.

W. C. Chew, Waves and Fields in Inhomogeneous Media, (IEEE Press, 1995), Chap. 6.

24.

D. Marcuse, Light Transmission Optics, (Van Norstrand Reinhold Company, 1972), pp. 322–326.

25.

G. Strang, Linear Algebra and Its Applications 3rd Ed., (Thomson Learning, 1988), pp. 448–449.

26.

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A. 21, 2442–2446 (2004). [CrossRef]

27.

S. Collin, F. Pardo, and J. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15, 4310–4320 (2007). [CrossRef] [PubMed]

28.

http://www.rsoftdesign.com

OCIS Codes
(130.1750) Integrated optics : Components
(130.2790) Integrated optics : Guided waves
(130.3120) Integrated optics : Integrated optics devices
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Integrated Optics

History
Original Manuscript: July 14, 2008
Revised Manuscript: August 4, 2008
Manuscript Accepted: August 5, 2008
Published: August 6, 2008

Citation
Galen B. Hoffman and Ronald M. Reano, "Vertical coupling between gap plasmon waveguides," Opt. Express 16, 12677-12687 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12677


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References

  1. J. A. Conway, S. Sahni, and T. Szkopek, "Plasmonic interconnects versus conventional interconnects: a comparison of latency, cross-talk and energy costs," Opt. Express 15, 4474 - 4484 (2007). [CrossRef] [PubMed]
  2. K. Jung, F. L. Teixeira, and R. M. Reano, "Au/SiO2 nanoring plasmon waveguides at optical communication band," IEEE J. Lightwave Tech. 25, 2757 - 2765 (2007). [CrossRef]
  3. P. Berini, "Air gaps in metal stripe waveguides supporting long-range surface plasmon polaritons," J. Appl. Phys. 102, 33112 (2007). [CrossRef]
  4. G. Veronis and S. Fan, "Guided subwavelength plasmonic mode supported by a slot in a thin metal film," Opt. Lett. 30, 3359 - 3361 (2005). [CrossRef]
  5. L. Chen, J. Shakya, and M. Lipson, "Subwavelength confinement in an integrated metal slot waveguide on silicon," Opt. Lett. 31, 2133 - 2135 (2006). [CrossRef] [PubMed]
  6. D. Gramotnev, "Adiabatic nanofocusing of plasmons by sharp metallic grooves: Geometrical optics approach," J. Appl. Phys. 98, 104302 (2005). [CrossRef]
  7. R. Buckley and P. Berini, "Figures of merit for 2D surface plasmon waveguides and application to metal stripes," Opt. Express 15, 12174 - 12182 (2007). [CrossRef] [PubMed]
  8. P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10484 - 10503 (2000). [CrossRef]
  9. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, "Plasmonics: the next chip-scale technology," Mat. Today 9, 20 - 27 (2006). [CrossRef]
  10. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 46802 (2005). [CrossRef]
  11. M. Yan and M. Qiu, "Guided plasmon polariton at 2D metal corners," J. Opt. Soc. Am. B 24, 2333 - 2342 (2007). [CrossRef]
  12. G. Veronis and S. Fan, "Modes of subwavelength plasmonic slot waveguides," IEEE J. Lightwave Tech. 25, 2511 - 2521 (2007). [CrossRef]
  13. G. Veronis and S. Fan, "Crosstalk between three-dimensional plasmonic slot waveguides," Opt. Express 16, 2129 - 2140 (2008). [CrossRef] [PubMed]
  14. H. S. Won, K. C. Kim, S. H. Song, C. Oh, P. S. Kim, S. Park, and S. Kim, "Vertical coupling of long-range surface plasmon polaritons," Appl. Phys. Lett. 88, 11110 (2006). [CrossRef]
  15. J. A. Pereda, A. Vegas, and A. Prieto, "An improved compact 2D full-wave FDFD method for general guided wave structures," Microwave Opt. Technol. Lett. 38, 331 - 335 (2003). [CrossRef]
  16. http://www.comsol.com
  17. H. J. Hagemann, W. Gudat, and C. Kunz, "Optical constants from the far infrared to the x-ray region: Mg, Al, Cu, Ag, Au, Bi, C, and Al2O3," J. Opt. Soc. Am. 65, 742-744 (1975). [CrossRef]
  18. R. M. Reano and S. W. Pang, "Sealed three-dimensional nanochannels," J. Vac. Sci. Technol. B 23, 2995 - 2999 (2005). [CrossRef]
  19. Bruce McConnel, "Using self-assembly to create airgap microprocessors" (IBM, 2007), http://www-03.ibm.com/press/us/en/presskit/21463.wss, Accessed 10-30-07.
  20. D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, "Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap," J. Appl. Phys. 100, 13101 (2006). [CrossRef]
  21. K. Okamoto, Fundamentals of Optical Waveguides, (Academic Press, 2006).
  22. C. Pollock, Fundamentals of Optoelectronics, (Ceramic Book and Literature Service, 2003), Chap. 11.
  23. W. C. Chew, Waves and Fields in Inhomogeneous Media, (IEEE Press, 1995), Chap. 6.
  24. D. Marcuse, Light Transmission Optics, (Van Norstrand Reinhold Company, 1972), pp. 322-326.
  25. G. Strang, Linear Algebra and Its Applications 3rd Ed., (Thomson Learning, 1988), pp. 448 - 449.
  26. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A. 21, 2442 - 2446 (2004). [CrossRef]
  27. S. Collin, F. Pardo, and J. Pelouard, "Waveguiding in nanoscale metallic apertures," Opt. Express 15, 4310 - 4320 (2007). [CrossRef] [PubMed]
  28. http://www.rsoftdesign.com

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