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A nano-scaled coupled-line coupler based on the guidance of surface plasmon-polaritons (SPPs) is proposed, designed and simulated at optical frequencies. The coupler comprises layered dielectric materials and silver, which serve as two stacked nano-transmission lines to achieve broadside coupling. The key property of this coupler is that it operates based on the principle of contra-directional coupling between a forward and a backward wave giving rise to supermodes that are characterized by complex-conjugate eigenvalues (even when the materials are assumed lossless). The resulting exponential attenuation along the coupler leads to dramatically reduced coupling lengths compared to previously reported co-directional SPP couplers (e.g. from millimeters to submicrons). The effect of material losses and finite coupler width are also analyzed.
©2006 Optical Society of America
1. Introduction
The concept of utilizing low-dimensional waves to overcome the transverse diffraction limit of optical guided-wave structures has received considerable attention in recent years. In particular, surface plasmon-polaritons (SPPs) guided at the interface of media with positive and negative permittivities are used to illustrate such a concept [1]. Some interesting structures that support SPPs include the negative-dielectric-film (ND-film) and the negative-dielectric-gap (ND-gap), where a thin layer of ND material is sandwiched between positive dielectric media or vice versa (see Fig. 1). Noble metals, such as gold and silver, are well known candidates for the ND material because they exhibit small negative electric permittivities at optical frequencies. Some polaritonic materials, such as SiC, can also be employed as the ND medium for mid/long infrared applications [2]. The theoretical analysis of SPPs has continued for more than three decades, during which the characteristics of SPPs guided at the interface of a semi-infinite dielectric medium and metal, as well as those guided by thin metal films with various dimensions have been explored ([1], [3], [4] and [5]). One particular interesting concept in [4] is to recognize the ND-film and the ND-gap structures as backward-wave (BW-wave) and forward-wave (FW-wave) transmission lines at optical frequencies, respectively. The reader is reminded that backward-wave propagation is equivalent to left-handed propagation (E̅, H̅ and K̅ form a left-handed triplet), for which the Poynting vector and the phase velocity have opposite directions. Therefore, this concept could potentially open up the possibility of extending many applications derived from left-handed transmission-line metamaterials at microwave frequencies ([6]) to the optical domain. Indeed, the concept of the short coupler at optical frequencies presented in this paper is inspired by the microwave transmission-line couplers reported in [7] and [10]. In [7] and [10], the couplers consist of a forward transmission-line edge-coupled to a left-handed (backward) transmission-line. This leads to contra-directional coupling between the two lines and the resulting two coupled eigenmodes have propagation constants that form a complex-conjugate pair. The corresponding exponential attenuation along the coupler (which is valid even for lossless lines) indicates a rapid coupling rate, therefore leading to very short coupling lengths. In this paper, we explain how to extend these short couplers to the optical domain using plasmonic films (e.g. silver).
Fig. 1. Geometry and possible materials for the contra-directional SPP coupler featuring the stacked ND-film and ND-gap topology. (The arrows represent the power flow.)
The physical realization of discrete plasmonic components at the nanometer scale is made possible with recent developments in fabrication technologies. Some previous works, such as [8] and [9], have been carried out on co-directional plasmonic couplers. The operation of those couplers depends on the small difference between the propagation constants of the two coupled modes, which leads to long coupling lengths on the millimeter scale. In contrast, couplers that operate under the contra-directional coupling condition exhibit much shorter coupling lengths. At microwave frequencies, the two transmission line branches in [7] are implemented by employing microstrip technology in conjunction with loading lumped L-C elements. At optical frequencies, this paper proposes to replace the microstrip lines with their nano-transmission line counterparts. In addition, the FW-wave and the BW-wave branches are stacked to achieve broadside coupling (instead of edge coupling) as shown in Fig. 1. As predicted by contra-directional coupled-mode theory, this coupler supports complex coupled modes, whose attenuation constant increases drastically at a certain frequency range to form a leaky stopband ([7] and [10]). The very short coupling length exhibited by the coupler is a direct result of the large attenuation constant which holds true even when the materials are completely lossless. This attenuation constant physically signifies that the energy carried by the FW-wave line is efficiently redirected (coupled) into the BW-wave line or vise versa at a high exponential rate. The plasmonic coupler proposed in this paper features almost 0dB coupling within 1μm for ideal (lossless) metal, and -3.5dB coupling when losses are accounted for.
In Section 2, the ω-β dispersion relations for SPPs guided by the metal-gap and the metal-film, as well as for the complex coupled modes are derived. All derivations assume the metal is lossless and the structure is infinite in one of the transverse directions. In Section 3, the full-wave simulations of the dispersion diagrams, wave propagation and coupling, and the S-parameter analysis of the coupler are conducted using Comsol Multiphysics simulation package. Finally, the effect of the metal loss (described by the Drude model) and the finite transverse confinement are presented in Section 4 and 5 respectively.
2. Theory
It is well known that noble metals, such as gold and silver, have negative electric permittivities below their plasma frequencies. Such materials can be treated as the ND medium at optical frequencies. The frequency dispersion characteristic of the relative electric permittivity of metals is described by Drude-Sommerfeld’s theorem in Eq. (1).
where the plasma frequency ωp = 1.29×1016 rad/s, which corresponds to a free-space wavelength λp of 146nm for silver.
2.1. Dispersion Characteristics of SPPs Guided by the Metal-film and the Metal-gap
Both the metal-film and the metal-gap in Fig. 2 support two TMz SPP modes with even and odd H-field distributions in the transverse direction. The ω-β dispersion relations of the SPP modes are derived from Maxwell’s equations and the consistency conditions.
We first start with the metal-film structure. For the TMz SPP modes, the transverse H-fields in the three layers are expressed in Eq. (2).
where k y,d and k y,m are the transverse wave-numbers in the dielectric medium and the metal respectively, and β is the phase constant. The four unknowns A, B, C and D represent the field amplitudes. From Maxwell’s equations, the electric field that is tangential to the interface can be expressed through Eq. (3) for each layer,
The boundary conditions imply that Hx and Ez for each layer must be continuous at the interfaces, from which we obtain a 4×4 homogeneous linear system of equations. The eigenvalue solution reveals that the dispersion relations of the even and the odd SPPs for the metal-film are described in Eqs. (4) and (5) respectively,
where h is the film thickness. Finally, the consistency conditions in Eq. (6) relate the transverse wave numbers k y,d and k y,m to the phase constant β.
The closed form expressions of the ω-β dispersion relations can be found through direct substitution. Finally, the dispersion characteristic for the metal-gap structure can be represented in exactly the same manner when interchanging k y,d with k y,m and εd with εm.
Fig. 2. Dispersion diagrams (The relative electric permittivity of the dielectric media are: εair = 1,εsilica-glass = 2.09,εglass = 4.2,εsilicon = 12.1. The ND-film guides BW even modes and the ND-gap guides FW even modes.)
The corresponding ω-β dispersion diagrams with various dielectric materials are illustrated in Fig. 2. The results match with the SPP dispersion behaviors described in [3]. In summary, we can see that the metal-film and the metal-gap can indeed act as surface-wave mode guiding structures at optical frequencies. More specifically, the metal-film guides backward even modes while the metal-gap guides forward even modes. However, the two even modes reside in different frequency bands for the same dielectric material. It should be noted that the even mode is of interest here because this mode profile facilitates its excitation with a Laser via the end-fire coupling technique [9]. Therefore, from now on, we will focus on the even mode only. The horizontal asymptotes that separate the backward and forward modes represent the surface resonance frequencies, at which εd = -εm. Note that the waves are tightly bounded at the metal-dielectric interfaces near the surface resonances. In this regime, the phase constant β becomes very large, which indicates that the effective wavelength becomes very small. This is particularly desirable for designing electrically small devices. However, this advantage is compromised when material losses are taken into consideration since the attenuation increases with β.
2.2. Dispersion Characteristics of the Coupled Modes
Since the surface resonance frequency decreases when εd increases as indicated in Fig. 2, the approach of stacking metal-film and metal-gap structures consisting of different dielectrics (see Fig. 1) renders the possibility of producing both FW and BW even modes at the same frequency. One interesting application of having both modes co-existing at the same frequency has been presented in [7] and [10], where an extended contra-directional coupled-mode theory was derived to demonstrate that short couplers comprising a BW-wave and a FW-wave transmission line can produce co-directional phase but contra-directional power flow in the two branches. Such anomalous coupling is due to the fact that this structure supports two supermodes having complex-conjugate propagation constants (γ= α±jβ), of which the real part α increases drastically around the intersection of the ω-β dispersion curves of the isolated BW and FW modes.
The dispersion relations for the coupled-mode system (see Fig. 3) can be derived in a similar fashion as we did for the individual metal-film or metal-gap structure. Assuming the coupled modes are also TMz modes, the transverse H-fields in the six layers are expressed in Eq. (7).
where there are ten unknowns (A,B,C, etc.) that are associated with the field amplitude, and the transverse wave numbers (ky’s) are related to the phase constant b through the consistency conditions. Together with the tangential Ez’s derived from Eq. (3), the boundary conditions at five interfaces render a 10×10 homogeneous system of equations. Since it is awkward to obtain the closed-form expressions for the ω-β dispersion equations, the corresponding eigenvalue solutions are now obtained by finding the roots of the 10×10 determinant numerically.
The comparison between the dispersion diagrams of the isolated FW (metal-gap) and the BW (metal-film) waves and of the coupled system is illustrated in Fig. 3. As shown, a stop-band forms for the two coupled modes close to the intersection of the isolated ω-β curves. By adopting the conventions used in [7], the two coupled modes are termed as the c-mode (e -(α-jβ)z) and the π-mode (e -(α+jβ)z). Figure 3 shows that the large attenuation in the stop-band is purely due to the coupling effect since no material losses are included in the analysis.
The corresponding coupled eigenmodes are characterized by complex-conjugate propagation constants.
The formation of the stopband at the intersection of the isolated ω-β curves can be understood more intuitively from the phase-matching condition, which requires that the fields across two stacked branches must incur the same phase when propagating along the z-direction. This condition is easiest to be satisfied when the two isolated modes exhibit the same phase constants. Furthermore, since one line supports a BW wave and the other supports a FW wave, the Poynting vectors along the two branches must point in opposite directions. In other words, this coupled system provides an immediate return path for the incident power (see the arrows in Fig. 1 that represent the power flow). Moreover, the large attenuation constant indicates that the energy leaks from the input to the coupled port at an exponential rate, hence manifesting a very short coupling length with a high coupling level. In addition, since the power decays exponentially along the direction of propagation, the coupling level increases as the length increases. Theoretically, 0dB coupling is achieved when the line is semi-infinite. This is different from the co-directional couplers, where the maximum power transfer occurs periodically along the line. Additionally, any power splitting ratio between the coupled and through ports can be achieved by adjusting the length and separation distance according to different design specifications. The dimensions and materials for the coupler presented in this paper are shown in Fig. 1 and show the feasibility of realizing this kind of a coupler. A side note is that the small dimension of the coupler may present a challenge in directly coupling light into the structure. However, these couplers can be used as interconnects in futuristic densely packed integrated optical circuits, where the neighbor components are on the same order of scale. In terms of characterizing the coupler through experimental measurements directly, one likely needs to design transitions that couple light in the structure utilizing near-field microscope tip devices at the nanometer scale.
3. Simulation Results
In order to verify the modal theory presented above, fullwave simulations have been carried out using Comsol Multiphysics software package (formerly known as FEMLAB). The dispersion diagrams are extracted through the application mode 2D Perpendicular Wave Eigenmode Analysis in Comsol’s Electromagnetics Module. Figure 3 shows that the results match with the theoretical dispersion diagrams very well.
The characteristics of the π-mode and the c-mode shown in Fig. 4(a) and 4(b) can be summarized as follows. The π-mode carries power forward in the metal-film layer and backward in the metal-gap layer, but the opposite is true for the c-mode. Finally, the two modes always exist simultaneously in the coupler. However, if the metal-gap is excited, then the c-mode dominates in order to satisfy the condition that the input power is flowing forward into the structure, similarly the π-mode dominates if the metal-film is excited.
To visualize the wave propagation and the coupling effect, simulations using application mode 2D TM In-Plane Harmonic Propagation in the Electromagnetics Module are conducted for both the passband and the stopband, and the results are presented in Fig. 5. We can see that the power flows from the input port to the through port directly in the passband, but rapidly leaks to the coupled port in the stopband.
The performance of the coupler is analyzed in terms of the coupling level and the coupling length. The coupling level can be interpreted as the coefficient S 21 in the S-parameter analysis of a four-port network. Figure 6 shows that almost 0dB coupling level with very high isolation is achieved in the stopband for the lossless metal. The coupling length can be determined qualitatively from Fig. 5 and more quantitatively from Fig. 3(b), where the power attenuation (dB per micron) is plotted. As shown, at the center of the stopband, the power attenuation is more than 70dB/μm. This suggests that the coupler acts almost as being semi-infinite even over a submicron scale.
Fig. 5. Power flow along the 1μm coupler assuming the metal is lossless (Note: the power scale shown in the figure is normalized with respect to the incident power)
4. Effect of Loss
A more accurate Drude-Sommerfeld’s model that also takes loss into consideration is given in Eq. (8).
where the relaxation time τ = 1.25 × 10-14 s for silver. The effect of loss on the dispersion property of the isolated FW and BW SPP’s, as well as the coupled modes can be observed in Fig. 7. This figure shows that the coupled modes are no longer lossless in the passband; Instead, they follow the lossy pattern of the isolated FW and BW waves. In the stopband, the two coupled modes do not have the same phase constant β except at the center of the stopband, and the π-mode attenuates slightly faster than the c-mode. However, the coupling effect for both supermodes still dominates over the material loss in the stopband since the attenuation constants are very close to the ones extracted from the lossless model. The power attenuation due to material losses when the ND-gap is excited is illustrated in Fig. 8. In the passband, the power attenuates along the lower branch. In the stopband, we can also observe the contra-directional coupling phenomenon, but the intensity of the coupled power is diminished.
A more quantitative analysis is conducted by comparing Fig. 6 and 9, we can see that the material losses reduce the coupling level from 0dB to -3.5dB when the coupler operates at the center frequency of the stopband. The loss is stronger away from the stopband center due to the fact that the wave experiences more attenuation since it travels longer before coupling into the other branch. Figure 7(b) shows the loss-per-micron due to both the material loss and the coupling effect at each frequency, from which the power attenuation along the line and the coupling length can be estimated.
Fig. 8. Power flow along 1 μm coupler assuming the metal loss in Eq. (8) (Note: the power scale shown in the figure is normalized with respect to the incident power)
5. Effect of Finite Width
In practice, the metal film/gap has finite width as well as finite thickness. Therefore, the corresponding guided modes contain coupled components of the surface waves at four edges. The effect of the 2D confinement in the transverse direction on SPPs is discussed extensively in [5]. This discussion suggests that the fundamental even mode has a near Gaussian profile, therefore can be excited using a simple end-fire technique. It also suggests that the dispersion behavior of the fundamental even mode approximates the one of the infinite width when the aspect ratio of width/thickness is large. This section explores the effect of finite width on the coupled modes by comparing the dispersion diagram with the one for infinite width, and determines the minimum width necessary to match their dispersion properties for the chosen thickness and separation distance. Figure 10(a) shows that a decrease in the width results in smaller phase constants for both coupled modes, but it does not alter the frequency range where the stop-band occurs. Figure 10(b) also shows that the attenuation constant for the π-mode increases significantly as the width decreases below 100nm. For design purposes, Fig. 10 suggests that any width larger than 200nm is a good approximation of the one designed assuming an infinite width.
6. Conclusion
In conclusion, this paper presents the theory and simulation of a nano-scaled coupled-line coupler based on the guidance of surface plasmon-polaritons at optical frequencies. The coupler comprises layered dielectric materials and silver, which serve as two stacked nano-transmission lines to achieve broadside coupling. Comparing to conventional optical dielectric couplers, the guided modes are surface waves that can overcome the diffraction limit in the transverse direction of the guided-wave structures. Therefore it has the potential to be used in nano-scaled circuits at optical frequencies.
More importantly, this plasmonic coupler is based on the principle of the contra-directional coupling phenomenon that involves complex modes previously observed in a microstrip line coupler designed to operate at microwave frequencies. This leads to dramatically reduced coupling lengths. The characteristics of the coupled modes are analyzed from the theoretical derivations as well as simulations using Comsol Multiphysics finite-element solver. The ω-β dispersion diagrams generated by both approaches agree very well to each other. We observe that the power can efficiently couple between the FW-wave and the BW-wave branches giving rise to supermodes that are characterized by complex-conjugate eigenvalues, whose real part (attenuation constant) increases drastically for certain frequencies. This results in exponential attenuation along the coupler that leads to dramatically reduced coupling lengths compared to previously reported co-directional SPP couplers (e.g. from millimeters to submicrons).
The coupler shows 0dB coupling within submicron coupling length for the stopband if considering the lossless Drude model for silver. The coupling level reduces to -3.5dB when the effect of loss is considered. In addition, the effect of a finite coupler width is also investigated, and it is observed that any width that is greater than 200nm will approximate the coupled modes of the infinite width very well.
Finally, we would like to thank the reviewers who brought to our attention that the experimentally measured εr values of silver depart from the Drude model significantly at a certain frequency range. This has been investigated in several papers, including [11] and [12]. We have conducted some preliminary dispersion analysis on the ND-film and ND-gap structures based on the data in [12]. The results suggest that the operating frequency shifts to a lower value, but the condition that facilitates the concept of contra-directional coupling, that is the coupling between a forward and a backward wave as implied by Fig. 3(a), remains intact. Moreover, at this new operating frequency, the imaginary part of the permittivity of silver is well predicted by the Drude model. The ultimate characterization of this coupler can only be assessed in detail through an actual experiment, but this is beyond the scope of this present paper.
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