Metal-dielectric metamaterials for guided wave silicon photonics A. Lupu,1,2,* N. Dubrovina,1 R. Ghasemi,1 A. Degiron,1,2 and A. de Lustrac1 1

Univ. Paris-Sud, Institut d'Electronique Fondamentale, UMR 8622, 91405 Orsay Cedex, France 2 CNRS, Orsay, F-91405, France * [email protected]

Abstract: The aim of the present paper is to investigate the potential of metallic metamaterials for building optical functions in guided wave optics at 1.5µm. A significant part of this work is focused on the optimization of the refractive index variation associated with localized plasmon resonances. The minimization of metal related losses is specifically addressed as well as the engineering of the resonance frequency of the localized plasmons. Our numerical modeling results show that a periodic chain of gold cut wires placed on the top of a 100nm silicon waveguide makes it possible to achieve a significant index variation in the vicinity of the metamaterial resonance and serve as building blocks for implementing optical functions. The considered solutions are compatible with current nano-fabrication technologies. ©2011 Optical Society of America OCIS codes: (0350.4238) Nanophotonics; (160.3918) Metamaterials; (160.1190) Anisotropic optical materials; (160.4760) Optical properties; 240.6680 Surface plasmons; (260.5740) Resonance; (130.3120) Integrated optics devices.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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1. Introduction The recent advent of transformation optics to mold the light flow opened the door to the design of devices with unusual optical properties [1, 2]. The most intriguing application conceived remains the invisibility cloak whose designs have been presented in microwave [3– 5] and optical regimes [6–8]. Other interesting wave manipulation applications such as

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concentrators [9], rotators [10], wormholes [11], waveguide transitions and bends [12–17] have also been proposed. Transformation optical devices are typically implemented with metamaterials (MM) because they require sophisticated anisotropic material parameters that cannot be readily found in conventional media [3–8]. This is especially true for designs based of nonconformal transformations which often exhibit permittivities ε and permeabilities µ much smaller and/or larger than 1. To obtain such parameters, it is often necessary to use MMs made of subwavelength metallic resonators featuring dramatic variations of ε and µ near the resonance wavelength. However, resonant MMs have a number of drawbacks such as a bandpass limited to a narrow frequency range in the vicinity of the resonance [3, 18]. In addition, the use of metals comes at a cost for MMs operating at optical wavelengths because they are extremely lossy in this spectral range, damping the MM response and saturating the resonance [19–24]. The problem of absorption in metals has generated an intense effort to develop active MMs and plasmonic components where losses are compensated by a gain medium [25–30]. Another promising approach consists in creating hybrid photonic structures in which metallic parts are coupled with dielectric (and almost lossless) waveguides [31–37]. In this configuration, useful functionalities are obtained by allowing just enough light to interact with the metallic parts of the system. The remaining part of the energy propagates in the dielectric waveguide, thereby considerably mitigating the losses. To date, the potential of hybrid structures has been mostly explored in the context of integrated plasmonic circuits [31–37]. There is much less literature on hybrid MMs combining subwavelength metallic inclusions and dielectric waveguides, except in the THz regime where it was shown that thin dielectric films incorporating metallic split ring resonators (SRRs) can support resonant THz surface modes [38]. In this article, we discuss the potential of hybrid MMs in guided wave optics in the near infrared range. We consider the case of a composite guiding structure made of a single sheet of metamaterial above a silicon waveguide (Fig. 1). In such a configuration most of the light is essentially confined in the silicon slab while only the evanescent tail interacts with the MM layer—in other words, the MM layer acts mostly as a perturbation. Its role is to modify the effective index of the composite waveguide structure. Such a solution allows to significantly reduce the propagation losses since the main part of the electromagnetic energy do not interact directly with the metallic part of the metamaterial. The ability to control the energy flow in a silicon waveguide based on the interaction of the evanescent tail with the MM layer constitutes a real opportunity to design a novel class of photonic devices.

Fig. 1. Silicon waveguide topped by a metamaterial layer. The light flow propagates mainly in the silicon layer.

2. Permittivity engineering with a transverse periodic chain of cut wires The aim of the present study is to achieve an efficient control over the flow of light in the waveguide underneath the metamaterial layer. By means of this MM layer we intend to obtain a significant variation of the effective refraction index in order to efficiently mold the flow of light in the waveguide. Here we will focus on a hybrid MM made of an array of subwavelength metallic cut-wires on top of a Si slab. As will be shown below, it is possible to

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approximate the entire structure—that is, the Si layer and the metallic cut wires on top of it— as a single dielectric slab guiding the signal by total internal reflection. We note that a similar treatment has already been performed in the THz regime for an array of SRR embedded in a very thin dielectric slab [38]. We start the study by considering the case of a hybrid MM with only one unit cell along the propagation direction. In this case, the cut wires form a chain transverse to the direction of propagation, as illustrated in Fig. 2(a). The equivalent effective model for this configuration is a thin slab of dielectric medium inserted in the Si slab, as shown in Fig. 2(b).

Fig. 2. (a) Sketch of the slab waveguide with a chain of MM on the top. Light propagation direction along Z axis. The E field is parallel to the X axis. (b) Equivalent model for a slab waveguide loaded with a MM.

In a first step we consider the case of a periodic chain of Au cut wires. We model the phase and amplitude of transmission and reflection for a single chain of cut wires placed on the top of a slab waveguide by means of HFSS software from Ansoft [39]. We assume that the length, width and height of the metal wires are 200×50×50nm respectively. The separation of two adjacent cut wires in the chain along the X axis is set to 100nm (Fig. 2). This corresponds to a moderate coupling between the chain elements. The considered dimensions, while being relatively small, are well mastered with existing nano-fabrication technologies. Again, we assume for the time being that the hybrid MM has only one unit cell along the propagation direction. For the sake of simplicity we consider a slab waveguide with a core refractive index of 3.45 surrounded by air on both sides. This index value is that of the silicon at 1.5µm. The dielectric permittivity of Au used for numerical modeling is that given by Palik [40]. The resulting transmission, reflection and loss spectra calculated for 100nm, 150nm and 200nm Si slab thickness are shown on the Figs. 3(a), 3(b) and 3(c), respectively. The loss is determined by subtracting the transmission and reflection from the incident power and represents the sum of absorption and dipole emission in the studied case. In all three cases, a broad peak appears in the spectra, indicating that the signal propagating in the Si waveguide couples to a surface plasmon resonance localized on the metal cut wires. The inspection of the obtained results shows that the position of the resonance frequency depends on the slab thickness. The position of the minimum of transmission shifts from 169THz to 158THz as the slab thickness is increased from 100nm to 200nm. These results indicate that the localized surface plasmons are sensitive to the mode effective index of the slab and not only to its index of refraction.

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Fig. 3. (a) to (c) Intensity of transmission (red), reflection (blue) and loss (black) for a 200x50x50nm wire placed on a 100nm, 150nm and 200nm Si slab respectively.

To calculate the effective permittivity εeff and permeability µ eff from the complex reflection and transmission coefficients r and t, respectively, of the slab loaded with a transverse cut wire chain, we use the retrieval method proposed by Smith et al. [41]. This method was initially developed to find the effective permittivity εeff and permeability µ eff of a bulk metamaterial illuminated by a plane wave in free space but it can also be applied to a guided wave configuration provided that the mode has planar wavefronts. The extension to determine the effective complex refractive index neff of a planar waveguide loaded with a metamaterial is presented in Appendix A. The difference with respect to the formula provided by Smith et al. in [41] is that the effective index of the waveguide slab, Neff slab, has to be introduced in the expression of the impedance z:

(1 + r ) − t 2 2 N eff2 slab (1 − r ) − t 2 2

z=±

(1)

which is used for calculating εeff = neff/z and µ eff = neff·z . The frequency variation of the cut wire chain real ε’eff and imaginary ε”eff components of the effective complex permittivity for the case of 100nm, 150nm and 200nm slab thickness are represented in Figs. 4(a), 4(b) and 4(c), respectively. As it can be observed, between 100THz and 300THz there is a strong variation of the ε”eff with the frequency ν that can be approximated by a Lorentzian type dispersion formula.

Fig. 4. (a) to (c) The real (blue) and imaginary (red) components of the complex effective permittivity of the 200×50×50nm wire on a 100nm, 150nm and 200nm Si slab respectively.

As expected, for a Lorentzian type dispersion the refractive index is high just below the resonance frequency and low (eventually even below 1, meaning strong reflection) just above the resonance. This opens up the ability, as a function of the target index value, to engineer the effective refractive index by an appropriate positioning of the cut wire resonance with respect to the operating frequency. At the same time it is highly desirable to avoid the adverse effects of losses inherent to the absorption of metal cut wires which can be very high at the resonance frequency. To minimize the losses, it is preferable to operate not at the exact

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resonance but in its near vicinity and find an optimal trade-off between index variation and loss. Due to the aspect ratio of the cut wires, it is expected that the hybrid MM structure exhibits a strong anisotropy [42]. We have verified that the response of a periodic chain of 200×50×50nm cut wires rotated by 90° with respect to the current configuration does not produce any noticeable resonance and that the effective index of the rotated structure is similar to that of a Si slab (results not shown here). This example shows that the response of the MM strongly depends on the orientation of the individual cut-wires on top of the Si waveguide. The degree of anisotropy can therefore be controlled through the orientation of the cut wires with respect to the propagation direction. The presence of the energy dissipation mechanism related to the poor conductivity of gold at such high frequencies prevents one from attaining high resonance quality factors. The quality factors for the investigated cut wires, determined in a conventional way—ν0 divided by the half-width of ε”eff(ν)—are typically around 10.

Fig. 5. (a) and (b) Intensity of transmission (red), reflection (blue) and loss (black) for a 100×50×50nm cut-wire chain placed on a 100nm Si slab and 150nm Si slab, respectively. (c) and (d) The real (blue) and imaginary (red) components of effective index of the 100×50×50nm cut-wire chain on a 100nm slab and 150nm Si slab, respectively. The green curve corresponds to the effective index of the silicon slab alone.

To reduce losses it is necessary to avoid the strong absorption region and operate either below or above the resonance frequency ν0. To achieve a low loss operation at 200THz, corresponding to the 1.5µm telecommunication wavelength, the resonance frequency should be shifted out of this spectral domain by a few half-widths of the absorption line. For example, if we wish to attain a resonance frequency of 250THz on a silicon slab, one evident #155009 - $15.00 USD

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solution is to decrease the length of the cut wires. Setting the cut wire length to 100nm, while leaving the rest of the parameters unchanged, indeed allows one to shift the resonance frequency toward 250THz. The spectral characteristics of the transmission, reflection and loss calculated for such 100nm length cut wires on 100nm and 150nm thick Si slabs are represented in Figs. 5(a) and 5(b), respectively. The corresponding complex effective index is drawn in Figs. 5(c) and 5(d). At 200THz the contrast between the MM loaded waveguide effective index and a 100nm Si slab effective index is around 0.45. The index contrast is smaller, around 0.24, in the case of a 150nm Si slab. As mentioned previously, this is related to the stronger field localization inside the slab when its thickness is increased. Although the solution of using 100nm length cut wires is in principle viable, an accurate control of the dimensions of such small structures is technologically challenging. To circumvent this issue and relax the fabrication tolerances, we explored another approach that leads to an increase of the cut wire length by a factor of two. The idea consists in modifying the confinement of the localized surface plasmons by introducing an intermediate dielectric layer between the cut wires and the Si slab, in much the same way as in [43]. Due to the strong localization of the surface plasmons near the metal-dielectric interface, their properties strongly depend on the permittivity of the adjacent dielectric to the metal. As known, the resonance frequency of surface plasmons is higher in the presence of a low index dielectric. This opens up the possibility for engineering the cut wires resonance frequency.

Fig. 6. A gold cut wire on a silicon substrate topped with SiO2 layer.

For this purpose we introduce an intermediary layer of 10nm of silica between the silicon slab and the metal (Fig. 6). Figure 7 shows the result obtained for a chain of 200 × 50 × 50nm cut wires on the top of a 10nm silicon dioxide layer deposited over a silicon slab with a thickness of either 100nm or 150nm. As for the previous cases the separation between the adjacent cut wires is set to 100nm.

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Fig. 7. (a) and (b) Intensity of transmission (red), reflection (blue) and loss (black) for a 200×50×50nm cut-wire chain placed on a 10nm silicon dioxide on the top of silicon slab of 100nm and 150nm, respectively. (c) and (d) The real (blue) and imaginary (red) components of effective index of the 200×50×50nm cut-wire chain on a on a 10nm silicon dioxide on the top of silicon slab of 100nm and 150nm, respectively. The green curve corresponds to the effective index of the silicon slab alone.

The HFSS numerical modeling results show that for the 100nm thick silicon slab, the effective index varies between 1.2 and 4 in the vicinity of the resonance frequency (Fig. 7c). The considered solution thus allows not only to get a higher resonance frequency but also to increase the contrast of the effective index at the operating frequency. At 200THz the contrast between the MM loaded waveguide effective index and Si slab effective index is now approximately 0.96. This index contrast is more than twice higher than compared to the case of shorter cut wires on the same thickness Si slab but without the silicon dioxide interlayer. A similar behavior is observed for a 150nm thick Si slab. Although the contrast between the MM effective index and Si slab effective index is slightly lower, around 0.7. It is almost three times higher as compared to the case of shorter cut wires without the silicon dioxide interlayer. In the first case, for a 100nm Si slab with a 10nm silica interlayer, the resonance frequency occurs near 270THz. The transmission level at 200THz is 82%, equivalent to −0.84dB. For the second case with 150nm Si slab and 10nm silica interlayer, the resonance frequency is around 240THz. The transmission is slightly higher, around 88% (−0.55dB), because of the weaker interaction between the slab confined wave and the chain of cut wires. 3. Array of coupled periodic chains of cut wires The next step of our analysis consists in expanding the developed approach to the case of a MM system with several unit cells along the propagation direction. In this case, the cut wires #155009 - $15.00 USD

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form a 2D array on top of the Si slab, as shown in Fig. 8. The main questions that we are attempting to answer are: • Does a 2D array of cut wires on top of a Si waveguide behave like an effective dielectric slab? • Do the results related to the permittivity engineering for a single periodic chain remain essentially valid? • What are the main guidelines for an optimal design of a coupled wire assembly? To answer these questions, we consider the case of a MM structure with 10 unit cells along the propagation direction. As before, a 10 nm thick silica layer is inserted between the Si slab and the cut wires and their dimensions are 200×50×50nm. We performed HFSS numerical simulations for cut wires separated by d=50nm, d=100nm and d=150nm. The transmission, absorption and reflection obtained in the case of 100nm and 150nm thick Si slab are represented in the Figs. 9(a) and Fig. 9(b), respectively.

Fig. 8. Sketch of the slab waveguide with several coupled chains of cut wires.

An inspection of the presented results shows that the transmission and loss of strongly coupled cut wires with 50nm of separation between adjacent wires are qualitatively similar to that of a single chain of cut wires. Nevertheless, when looking into details, it can be observed that at 200THz the transmission level is around 55% (−2.58dB) and 59% (−2.26dB) for the case of the structures on the top of 100nm or 150nm thick Si slab, respectively. The resulting transmission is accordingly roughly 3dB or 6dB higher than that could be expected from the transmission level of a single chain. This important difference is due namely to the great reduction of reflection and loss observed in the case of Fig. 9. There is a marked difference between the reflection spectra of the slab waveguide loaded with a single chain of cut wires and with a 2D array of strongly coupled cut wires. In the last case no pronounced maximum is observed in reflection. The reflection level is also much lower and does not exceed 11% and 3% for the case of structures on top of 100nm or 150nm thick Si slabs, respectively. This is approximately three times lower as compared to the reflection of single chain displayed in the Fig. 7. When the separation distance d between adjacent wires is increased to 100nm or 150nm, the coupling between them is decreased. This is manifested by a broadening and splitting of the resonance in the transmission and loss spectra. A maximum in reflection is progressively building in the vicinity of the resonance. The observed phenomena are more pronounced in the case of 100nm thick Si slab where the interaction with the chains of cut wires is stronger. The observed behavior presents greats similarities with that of a chain of coupled photonic crystal nano-cavities [44]. The transmission of strongly coupled cavities in the vicinity of the resonance is akin to an effective medium waveguide with a high group index. The reflection level is very low. When the coupling between the cavities becomes weaker, an interference

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effect occurs leading to the appearance of pronounced peaks in reflection and dips in transmission.

Fig. 9. (a) Intensity of transmission (red), reflection (blue) and loss (black) for 10 coupled 200×50×50nm periodic cut-wire chains separated by d=50nm, d=100nm and d=150nm placed on 10nm silicon dioxide on the top of 100nm silicon slab. (b) Same as in (a) for 150nm silicon slab thickness.

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Although in the case of the present study the involved interactions are more complex than for crystal nano-cavities, most of the observed results can be fairly explained by the model of coupled plasmonic nano-cavities. In the case of a slab waveguide loaded with strongly coupled cut wires, its behavior is mostly similar to that of an effective medium slab waveguide. The essential condition for the validity of the effective medium approximation is that the coupling between adjacent wires must be dominant with respect to other coupling in the system, namely with that between the chain elements and the wave confined in the silicon slab. This explains why for stronger field confinement the behavior of the waveguide loaded with coupled cut wires seems to be closer to that of an effective medium slab waveguide. This assertion for an effective medium behavior is also supported by the distribution of the electrical field in the structure provided by our numerical simulations. Figure 10 shows the snapshots of the wave propagation at 200THz in three cases for a chain of cut wires separated by 50nm (a), 100nm (b) and 150nm (c) on a 10nm layer of silicon dioxide on the top of a 100nm silicon slab. The field distribution across the coupled chain region is much more uniform in the case of a strong coupling corresponding to the 50nm of separation between the chains. Conversely, stronger field localization around the cut wires is observed when the separation between the adjacent chains is increased. The exchange of energy in this case preferentially occurs between the chain and the slab and not between the adjacent chains To retrieve the effective dielectric permittivity in the case of a slab loaded with chains of coupled cut wires, the application of the Smith retrieval method using Eq. (1) becomes impractical because the large optical thickness of the MM region does not allow to unambiguously determine the corresponding branch of the inverse cosine function. To find the effective index we use an alternative method based on the determination of spatial position of the electric field minima (or maxima) in the region of the cut wires. The results obtained for 10 unit cells with wires separated by 50nm, 100nm or 150nm in the case of 100nm and 150nm thick Si slabs are represented in the Fig. 11(a) and Fig. 11(b), respectively. The general behavior of the effective index variation is qualitatively similar to that of a single chain of cut wires though the index contrast with respect to that of a silicon slab is reduced by approximately a factor of two. The index contrast is higher in the case of stronger coupled wires, that is for smaller separation distance. For 50nm of separation distance the index contrast at 200THz is around 0.5 for 100nm Si slab and around 0.25 for 150nm Si slab. Such an index contrast can be sufficient to be used in integrated optics applications. The main issue with the considered system is related to the level of propagation losses, which is of the order of few dB per 10 unit cells. This corresponds to a propagation distance of around 1µm, implying that these structures are best suitable for extremely compact designs, which are the real targets of this technology.

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Fig. 10. Cross section view for the electric field distribution of the propagated wave at 200THz for 10 coupled 200×50×50nm periodic cut-wire chains separated by d=50nm (a), d=100nm (b) and d=150nm (c) placed on 10nm silicon dioxide on the top of 100nm silicon slab.

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Fig. 11. (a) The real components of effective index for a single (magenta) or coupled 200×50×50nm cut-wire chains with separation distance: d=50nm (blue), d=100nm (red), d=150nm (cyan) placed on a 10nm silicon dioxide on the top of silicon slab of 100nm. The green curve corresponds to the effective index of the silicon slab alone. (b) Same as in (a) for 150nm silicon slab thickness.

4. Summary and conclusions The aim of the present work was to investigate the potential of MMs for building optical functions in guided wave optics at 1.5µm (200THz). The undertaken approach consists in considering a composite guiding structure made of a metamaterial layer over a high index slab waveguide, as for instance silicon in our case. In such a configuration only the evanescent tail interacts with the MM layer which acts essentially as a perturbation. The effective permittivity εeff and permeability µ eff of such a composite slab was determined using a variant of the retrieval method described in [41]. The numerical simulations show that an array of gold coupled cut wires over a slab waveguide leads to a significant variation of the slab effective index in the vicinity of the resonance. To apply the effective medium approximation, the coupling between consecutive rows of cut wires must be dominant with respect to all other interactions in the system. This implies using small separation distances, below 100nm, between the cut wires. The resulting effective index contrast is around 0.5 for a 100nm thick Si slab and around 0.25 for a 150nm thick Si slab. Such an index contrast is high enough for implementing a variety of optical functions. In certain cases, such as in transformation optical devices, it is desirable to design MMs with a strong degree of anisotropy. We showed that the anisotropy of our hybrid MM waveguides can be increased by inserting a thin dielectric layer with a low index of refraction between the cut wires and the Si slab. This intermediate layer makes it possible to increase the length of the cut wires without shifting the resonance frequency, thus maximizing their aspect ratio and anisotropic response. A critical aspect of the considered problem is related to the MM absorption losses which are on the order of a few dB per microns. To minimize the absorption losses, we followed the usual procedure that consists in detuning the resonance frequency from the operating frequency. However, this approach reduces the range of effective indices that can be attained. One possible way to counteract this issue is to consider multi-layer structures of coupled cut wires. The obtained results represent a first step toward building optical function in guided wave optics using hybrid metallo - dielectric metamaterials. Apendix A: Generalization of the retrieval method The retrieval method initially developed by Smith et al. [41] is currently widely used to determine the effective permittivity ε and permeability µ from the transmission and reflection coefficients calculated for a wave normally incident on a finite length of metamaterial. This

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method considers the metamaterial as a homogeneous layer for which it is possible to invert the scattering data to determine the refractive index n and impedance z. In view of the Smith retrieval method underlying assumptions, it is quite logically to expect that a similar kind of inversion can be performed for a single homogeneous dielectric film described by the characteristic matrix of a stratified optical medium. As will be shown below, the inversion of the characteristic matrix can be used to extend the retrieval method and find the ε, µ and z of a single layer in the most general case of oblique incidence and different input and output media. Incidentally this inversion will serve to derive Eq. (1) which is in fact just a particular case of the generalized retrieval solution. The propagation of an electromagnetic wave through a homogeneous film is sketched in Fig. A1. The expressions used for the characteristic matrix and reflection and transmission coefficients of the film are those provided by M Born and E. Wolf in [45]. For the sake of clarity we reproduce here these formula using the same notations as those employed by the authors.

Fig. A1. Propagation of an electromagnetic wave through a homogeneous film.

The expression of the characteristic matrix of a homogeneous film for a wave with electric field perpendicular to the plane of incidence (TE polarization in the authors [45] notation) is:

cos ( k0 nh cos θ ) M (h) = −ip sin ( k nh cos θ ) 0

−i sin ( k0 nh cos θ ) p cos ( k0 nh cos θ )

(A1)

where k0=2π/λ is the light wave vector in vacuum, n is the film refractive index, θ denotes the propagation angle and the quantity p is the characteristic admittance, i.e. the inverse of the medium impedance z:

p=

1 1 ε cos θ = cos θ = z zɶTE µ

(A2)

For a TM polarized wave, i.e. with the electric field in the plane of incidence, the same equation holds for the characteristic matrix, with p replaced by the characteristic impedance q:

q=

µ cos θ = z cos θ = zɶTM ε

(A3)

The reflection and transmission coefficients for a TE polarized wave are:

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r=

( m11 + m12 pl ) p1 − ( m21 + m22 pl ) ( m11 + m12 pl ) p1 + ( m21 + m22 pl )

(A4)

t=

2 p1 ( m11 + m12 pl ) p1 + ( m21 + m22 pl )

(A5)

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where the quantities p1 and pl given by Eq. (A2) correspond accordingly to the first and last semi-infinite media. The reflectivity and transmissivity coefficients of the film for the intensity of TE polarized wave are, respectively:

pl 2 t (A6) p1 At this step it is important to note that when using numerical modeling softwares like ANSOFT’s HFSS, the calculated transmission coefficient is the square root of the transmissivity and not Eq. (A5). In order to keep consistency it is necessary to renormalize the transmission coefficient as following: 2

R= r T =

2 p1 pl (A7) m m p + ( 11 12 l ) p1 + ( m21 + m22 pl ) The corresponding formulae for TM polarized wave are obtained from Eqs. (A4) and (A7) by replacing p1 and pl by q1 and ql given the Eq. A3. Denoting: t′ =

m11 = m22 = cos β ,

m12 = −izɶTE sin β ,

m21 = −

i zɶTE

sin β

(A8)

where:

β=

2π

n2 h cos θ 2

λ0

(A9)

we form the following system of equations: 1 r + = t′ t′

1 p1 pl

1 r − = t′ t′

( p1 cos β − ip1 pl zɶTE sin β )

1 i sin β p1 cos β − zɶTE p1 pl

(A10)

Solving this system of equations with respect to cosβ and z we obtain:

cos β =

(1 − r

p1 pl p1 (1 − r ) + pl (1 + r )

2

+ t′

2

)

t′

2 zɶTE

1 1+ r p1 pl t ′ (A11) = 1− r pl cos β − p1 pl t′ 1 cos β − pl

From this it is possible to determine the value of β:

β = ± arc cos ( cos β ) +

2π m k0 h

(A12)

where m is an integer. The explanation about the proper choice of the sign and value of the integer coefficient m can be found in [41]. Using the Snell law, the refractive index of the film and the refraction angle are: 2

β n 2 2 n2 = cos θ 2 = 1 − 1 sin 2 θ1 (A13) + n1 sin θ1 n2 k0 h Equations (A12), (A13) represent the generalized retrieval method for a single homogeneous film and a TE polarized wave. To obtain the expressions for TM polarization, it is sufficient to replace p by q in the systems of Eqs. A(10) and (A11). It follows that:

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1− r t′ (A14) zɶ = 1 1 1+ r cos β − ql q1ql t ′ The case of guided wave propagation is analogous to the case of normal incidence for a film sandwiched between two semi-infinite media with a refractive index corresponding to the effective index of the slab waveguide. Eq. (1) follows then straightforwardly from Eq. (A11). 2 TM

ql cos β − q1 ql

Acknowledgement This work was supported by the French Agence Nationale pour la Recherche (ANR Metaphotonique, contract number 7452RA09).

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Univ. Paris-Sud, Institut d'Electronique Fondamentale, UMR 8622, 91405 Orsay Cedex, France 2 CNRS, Orsay, F-91405, France * [email protected]

Abstract: The aim of the present paper is to investigate the potential of metallic metamaterials for building optical functions in guided wave optics at 1.5µm. A significant part of this work is focused on the optimization of the refractive index variation associated with localized plasmon resonances. The minimization of metal related losses is specifically addressed as well as the engineering of the resonance frequency of the localized plasmons. Our numerical modeling results show that a periodic chain of gold cut wires placed on the top of a 100nm silicon waveguide makes it possible to achieve a significant index variation in the vicinity of the metamaterial resonance and serve as building blocks for implementing optical functions. The considered solutions are compatible with current nano-fabrication technologies. ©2011 Optical Society of America OCIS codes: (0350.4238) Nanophotonics; (160.3918) Metamaterials; (160.1190) Anisotropic optical materials; (160.4760) Optical properties; 240.6680 Surface plasmons; (260.5740) Resonance; (130.3120) Integrated optics devices.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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17. R. Ghasemi, P.-H. Tichit, A. Degiron, A. Lupu, and A. de Lustrac, “Efficient control of a 3D optical mode using a thin sheet of transformation optical medium,” Opt. Express 18(19), 20305–20312 (2010). 18. D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85(14), 2933– 2936 (2000). 19. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). 20. J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005). 21. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). 22. G. Dolling, C. Enkrich, M. Wegener, C. M. 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Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). 28. A. Fang, Th. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12(2), 024013 (2010). 29. Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96(4), 044104 (2010). 30. A. N. Lagarkov, V. N. Kisel, and A. K. Sarychev, “Loss and gain in metamaterials,” J. Opt. Soc. Am. B 27(4), 648–659 (2010). 31. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5836 (2004). 32. A. Degiron, C. Dellagiacoma, J. G. McIlhargey, G. Shvets, O. J. F. Martin, and D. R. Smith, “Simulations of hybrid long-range plasmon modes with application to 90 ° bends,” Opt. Lett. 32(16), 2354–2356 (2007). 33. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). 34. D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17(19), 16646–16653 (2009). 35. J. T. Kim, S. Park, J. J. Ju, S. Lee, and S. Kim, “Low bending loss characteristics of hybrid plasmonic waveguide for flexible optical interconnect,” Opt. Express 18(23), 24213–24220 (2010). 36. M. Wu, Z. Han, and V. Van, “Conductor-gap-silicon plasmonic waveguides and passive components at subwavelength scale,” Opt. Express 18(11), 11728–11736 (2010). 37. M. Fevrier, P. Gogol, A. Aassime, R. Megy, A. Bondi, J. M. Lourtioz, and B. Dagens, “Localized surface plasmon excitation through evanescent coupling from dielectric or SOI waveguide at telecom wavelength,” presented at the CLEO/QELS 2011, paper EJ1.1, Germany, Munich, 22–25 May 2011. 38. B. Reinhard, O. Paul, R. Beigang, and M. Rahm, “Experimental and numerical studies of terahertz surface waves on a thin metamaterial film,” Opt. Lett. 35(9), 1320–1322 (2010). 39. www.ansoft.com/products/hf/hfss/ 40. E. D. Palik, Handbook of optical constants of solids (Academic Press, 1998). 41. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002). 42. M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332(6030), 702–704 (2011). 43. D. J. Shelton, D. W. Peters, M. B. Sinclair, I. Brener, L. K. Warne, L. I. Basilio, K. R. Coffey, and G. D. Boreman, “Effect of thin silicon dioxide layers on resonant frequency in infrared metamaterials,” Opt. Express 18(2), 1085–1090 (2010). 44. S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. 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1. Introduction The recent advent of transformation optics to mold the light flow opened the door to the design of devices with unusual optical properties [1, 2]. The most intriguing application conceived remains the invisibility cloak whose designs have been presented in microwave [3– 5] and optical regimes [6–8]. Other interesting wave manipulation applications such as

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concentrators [9], rotators [10], wormholes [11], waveguide transitions and bends [12–17] have also been proposed. Transformation optical devices are typically implemented with metamaterials (MM) because they require sophisticated anisotropic material parameters that cannot be readily found in conventional media [3–8]. This is especially true for designs based of nonconformal transformations which often exhibit permittivities ε and permeabilities µ much smaller and/or larger than 1. To obtain such parameters, it is often necessary to use MMs made of subwavelength metallic resonators featuring dramatic variations of ε and µ near the resonance wavelength. However, resonant MMs have a number of drawbacks such as a bandpass limited to a narrow frequency range in the vicinity of the resonance [3, 18]. In addition, the use of metals comes at a cost for MMs operating at optical wavelengths because they are extremely lossy in this spectral range, damping the MM response and saturating the resonance [19–24]. The problem of absorption in metals has generated an intense effort to develop active MMs and plasmonic components where losses are compensated by a gain medium [25–30]. Another promising approach consists in creating hybrid photonic structures in which metallic parts are coupled with dielectric (and almost lossless) waveguides [31–37]. In this configuration, useful functionalities are obtained by allowing just enough light to interact with the metallic parts of the system. The remaining part of the energy propagates in the dielectric waveguide, thereby considerably mitigating the losses. To date, the potential of hybrid structures has been mostly explored in the context of integrated plasmonic circuits [31–37]. There is much less literature on hybrid MMs combining subwavelength metallic inclusions and dielectric waveguides, except in the THz regime where it was shown that thin dielectric films incorporating metallic split ring resonators (SRRs) can support resonant THz surface modes [38]. In this article, we discuss the potential of hybrid MMs in guided wave optics in the near infrared range. We consider the case of a composite guiding structure made of a single sheet of metamaterial above a silicon waveguide (Fig. 1). In such a configuration most of the light is essentially confined in the silicon slab while only the evanescent tail interacts with the MM layer—in other words, the MM layer acts mostly as a perturbation. Its role is to modify the effective index of the composite waveguide structure. Such a solution allows to significantly reduce the propagation losses since the main part of the electromagnetic energy do not interact directly with the metallic part of the metamaterial. The ability to control the energy flow in a silicon waveguide based on the interaction of the evanescent tail with the MM layer constitutes a real opportunity to design a novel class of photonic devices.

Fig. 1. Silicon waveguide topped by a metamaterial layer. The light flow propagates mainly in the silicon layer.

2. Permittivity engineering with a transverse periodic chain of cut wires The aim of the present study is to achieve an efficient control over the flow of light in the waveguide underneath the metamaterial layer. By means of this MM layer we intend to obtain a significant variation of the effective refraction index in order to efficiently mold the flow of light in the waveguide. Here we will focus on a hybrid MM made of an array of subwavelength metallic cut-wires on top of a Si slab. As will be shown below, it is possible to

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approximate the entire structure—that is, the Si layer and the metallic cut wires on top of it— as a single dielectric slab guiding the signal by total internal reflection. We note that a similar treatment has already been performed in the THz regime for an array of SRR embedded in a very thin dielectric slab [38]. We start the study by considering the case of a hybrid MM with only one unit cell along the propagation direction. In this case, the cut wires form a chain transverse to the direction of propagation, as illustrated in Fig. 2(a). The equivalent effective model for this configuration is a thin slab of dielectric medium inserted in the Si slab, as shown in Fig. 2(b).

Fig. 2. (a) Sketch of the slab waveguide with a chain of MM on the top. Light propagation direction along Z axis. The E field is parallel to the X axis. (b) Equivalent model for a slab waveguide loaded with a MM.

In a first step we consider the case of a periodic chain of Au cut wires. We model the phase and amplitude of transmission and reflection for a single chain of cut wires placed on the top of a slab waveguide by means of HFSS software from Ansoft [39]. We assume that the length, width and height of the metal wires are 200×50×50nm respectively. The separation of two adjacent cut wires in the chain along the X axis is set to 100nm (Fig. 2). This corresponds to a moderate coupling between the chain elements. The considered dimensions, while being relatively small, are well mastered with existing nano-fabrication technologies. Again, we assume for the time being that the hybrid MM has only one unit cell along the propagation direction. For the sake of simplicity we consider a slab waveguide with a core refractive index of 3.45 surrounded by air on both sides. This index value is that of the silicon at 1.5µm. The dielectric permittivity of Au used for numerical modeling is that given by Palik [40]. The resulting transmission, reflection and loss spectra calculated for 100nm, 150nm and 200nm Si slab thickness are shown on the Figs. 3(a), 3(b) and 3(c), respectively. The loss is determined by subtracting the transmission and reflection from the incident power and represents the sum of absorption and dipole emission in the studied case. In all three cases, a broad peak appears in the spectra, indicating that the signal propagating in the Si waveguide couples to a surface plasmon resonance localized on the metal cut wires. The inspection of the obtained results shows that the position of the resonance frequency depends on the slab thickness. The position of the minimum of transmission shifts from 169THz to 158THz as the slab thickness is increased from 100nm to 200nm. These results indicate that the localized surface plasmons are sensitive to the mode effective index of the slab and not only to its index of refraction.

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Fig. 3. (a) to (c) Intensity of transmission (red), reflection (blue) and loss (black) for a 200x50x50nm wire placed on a 100nm, 150nm and 200nm Si slab respectively.

To calculate the effective permittivity εeff and permeability µ eff from the complex reflection and transmission coefficients r and t, respectively, of the slab loaded with a transverse cut wire chain, we use the retrieval method proposed by Smith et al. [41]. This method was initially developed to find the effective permittivity εeff and permeability µ eff of a bulk metamaterial illuminated by a plane wave in free space but it can also be applied to a guided wave configuration provided that the mode has planar wavefronts. The extension to determine the effective complex refractive index neff of a planar waveguide loaded with a metamaterial is presented in Appendix A. The difference with respect to the formula provided by Smith et al. in [41] is that the effective index of the waveguide slab, Neff slab, has to be introduced in the expression of the impedance z:

(1 + r ) − t 2 2 N eff2 slab (1 − r ) − t 2 2

z=±

(1)

which is used for calculating εeff = neff/z and µ eff = neff·z . The frequency variation of the cut wire chain real ε’eff and imaginary ε”eff components of the effective complex permittivity for the case of 100nm, 150nm and 200nm slab thickness are represented in Figs. 4(a), 4(b) and 4(c), respectively. As it can be observed, between 100THz and 300THz there is a strong variation of the ε”eff with the frequency ν that can be approximated by a Lorentzian type dispersion formula.

Fig. 4. (a) to (c) The real (blue) and imaginary (red) components of the complex effective permittivity of the 200×50×50nm wire on a 100nm, 150nm and 200nm Si slab respectively.

As expected, for a Lorentzian type dispersion the refractive index is high just below the resonance frequency and low (eventually even below 1, meaning strong reflection) just above the resonance. This opens up the ability, as a function of the target index value, to engineer the effective refractive index by an appropriate positioning of the cut wire resonance with respect to the operating frequency. At the same time it is highly desirable to avoid the adverse effects of losses inherent to the absorption of metal cut wires which can be very high at the resonance frequency. To minimize the losses, it is preferable to operate not at the exact

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resonance but in its near vicinity and find an optimal trade-off between index variation and loss. Due to the aspect ratio of the cut wires, it is expected that the hybrid MM structure exhibits a strong anisotropy [42]. We have verified that the response of a periodic chain of 200×50×50nm cut wires rotated by 90° with respect to the current configuration does not produce any noticeable resonance and that the effective index of the rotated structure is similar to that of a Si slab (results not shown here). This example shows that the response of the MM strongly depends on the orientation of the individual cut-wires on top of the Si waveguide. The degree of anisotropy can therefore be controlled through the orientation of the cut wires with respect to the propagation direction. The presence of the energy dissipation mechanism related to the poor conductivity of gold at such high frequencies prevents one from attaining high resonance quality factors. The quality factors for the investigated cut wires, determined in a conventional way—ν0 divided by the half-width of ε”eff(ν)—are typically around 10.

Fig. 5. (a) and (b) Intensity of transmission (red), reflection (blue) and loss (black) for a 100×50×50nm cut-wire chain placed on a 100nm Si slab and 150nm Si slab, respectively. (c) and (d) The real (blue) and imaginary (red) components of effective index of the 100×50×50nm cut-wire chain on a 100nm slab and 150nm Si slab, respectively. The green curve corresponds to the effective index of the silicon slab alone.

To reduce losses it is necessary to avoid the strong absorption region and operate either below or above the resonance frequency ν0. To achieve a low loss operation at 200THz, corresponding to the 1.5µm telecommunication wavelength, the resonance frequency should be shifted out of this spectral domain by a few half-widths of the absorption line. For example, if we wish to attain a resonance frequency of 250THz on a silicon slab, one evident #155009 - $15.00 USD

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solution is to decrease the length of the cut wires. Setting the cut wire length to 100nm, while leaving the rest of the parameters unchanged, indeed allows one to shift the resonance frequency toward 250THz. The spectral characteristics of the transmission, reflection and loss calculated for such 100nm length cut wires on 100nm and 150nm thick Si slabs are represented in Figs. 5(a) and 5(b), respectively. The corresponding complex effective index is drawn in Figs. 5(c) and 5(d). At 200THz the contrast between the MM loaded waveguide effective index and a 100nm Si slab effective index is around 0.45. The index contrast is smaller, around 0.24, in the case of a 150nm Si slab. As mentioned previously, this is related to the stronger field localization inside the slab when its thickness is increased. Although the solution of using 100nm length cut wires is in principle viable, an accurate control of the dimensions of such small structures is technologically challenging. To circumvent this issue and relax the fabrication tolerances, we explored another approach that leads to an increase of the cut wire length by a factor of two. The idea consists in modifying the confinement of the localized surface plasmons by introducing an intermediate dielectric layer between the cut wires and the Si slab, in much the same way as in [43]. Due to the strong localization of the surface plasmons near the metal-dielectric interface, their properties strongly depend on the permittivity of the adjacent dielectric to the metal. As known, the resonance frequency of surface plasmons is higher in the presence of a low index dielectric. This opens up the possibility for engineering the cut wires resonance frequency.

Fig. 6. A gold cut wire on a silicon substrate topped with SiO2 layer.

For this purpose we introduce an intermediary layer of 10nm of silica between the silicon slab and the metal (Fig. 6). Figure 7 shows the result obtained for a chain of 200 × 50 × 50nm cut wires on the top of a 10nm silicon dioxide layer deposited over a silicon slab with a thickness of either 100nm or 150nm. As for the previous cases the separation between the adjacent cut wires is set to 100nm.

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Fig. 7. (a) and (b) Intensity of transmission (red), reflection (blue) and loss (black) for a 200×50×50nm cut-wire chain placed on a 10nm silicon dioxide on the top of silicon slab of 100nm and 150nm, respectively. (c) and (d) The real (blue) and imaginary (red) components of effective index of the 200×50×50nm cut-wire chain on a on a 10nm silicon dioxide on the top of silicon slab of 100nm and 150nm, respectively. The green curve corresponds to the effective index of the silicon slab alone.

The HFSS numerical modeling results show that for the 100nm thick silicon slab, the effective index varies between 1.2 and 4 in the vicinity of the resonance frequency (Fig. 7c). The considered solution thus allows not only to get a higher resonance frequency but also to increase the contrast of the effective index at the operating frequency. At 200THz the contrast between the MM loaded waveguide effective index and Si slab effective index is now approximately 0.96. This index contrast is more than twice higher than compared to the case of shorter cut wires on the same thickness Si slab but without the silicon dioxide interlayer. A similar behavior is observed for a 150nm thick Si slab. Although the contrast between the MM effective index and Si slab effective index is slightly lower, around 0.7. It is almost three times higher as compared to the case of shorter cut wires without the silicon dioxide interlayer. In the first case, for a 100nm Si slab with a 10nm silica interlayer, the resonance frequency occurs near 270THz. The transmission level at 200THz is 82%, equivalent to −0.84dB. For the second case with 150nm Si slab and 10nm silica interlayer, the resonance frequency is around 240THz. The transmission is slightly higher, around 88% (−0.55dB), because of the weaker interaction between the slab confined wave and the chain of cut wires. 3. Array of coupled periodic chains of cut wires The next step of our analysis consists in expanding the developed approach to the case of a MM system with several unit cells along the propagation direction. In this case, the cut wires #155009 - $15.00 USD

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form a 2D array on top of the Si slab, as shown in Fig. 8. The main questions that we are attempting to answer are: • Does a 2D array of cut wires on top of a Si waveguide behave like an effective dielectric slab? • Do the results related to the permittivity engineering for a single periodic chain remain essentially valid? • What are the main guidelines for an optimal design of a coupled wire assembly? To answer these questions, we consider the case of a MM structure with 10 unit cells along the propagation direction. As before, a 10 nm thick silica layer is inserted between the Si slab and the cut wires and their dimensions are 200×50×50nm. We performed HFSS numerical simulations for cut wires separated by d=50nm, d=100nm and d=150nm. The transmission, absorption and reflection obtained in the case of 100nm and 150nm thick Si slab are represented in the Figs. 9(a) and Fig. 9(b), respectively.

Fig. 8. Sketch of the slab waveguide with several coupled chains of cut wires.

An inspection of the presented results shows that the transmission and loss of strongly coupled cut wires with 50nm of separation between adjacent wires are qualitatively similar to that of a single chain of cut wires. Nevertheless, when looking into details, it can be observed that at 200THz the transmission level is around 55% (−2.58dB) and 59% (−2.26dB) for the case of the structures on the top of 100nm or 150nm thick Si slab, respectively. The resulting transmission is accordingly roughly 3dB or 6dB higher than that could be expected from the transmission level of a single chain. This important difference is due namely to the great reduction of reflection and loss observed in the case of Fig. 9. There is a marked difference between the reflection spectra of the slab waveguide loaded with a single chain of cut wires and with a 2D array of strongly coupled cut wires. In the last case no pronounced maximum is observed in reflection. The reflection level is also much lower and does not exceed 11% and 3% for the case of structures on top of 100nm or 150nm thick Si slabs, respectively. This is approximately three times lower as compared to the reflection of single chain displayed in the Fig. 7. When the separation distance d between adjacent wires is increased to 100nm or 150nm, the coupling between them is decreased. This is manifested by a broadening and splitting of the resonance in the transmission and loss spectra. A maximum in reflection is progressively building in the vicinity of the resonance. The observed phenomena are more pronounced in the case of 100nm thick Si slab where the interaction with the chains of cut wires is stronger. The observed behavior presents greats similarities with that of a chain of coupled photonic crystal nano-cavities [44]. The transmission of strongly coupled cavities in the vicinity of the resonance is akin to an effective medium waveguide with a high group index. The reflection level is very low. When the coupling between the cavities becomes weaker, an interference

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21 November 2011 / Vol. 19, No. 24 / OPTICS EXPRESS 24754

effect occurs leading to the appearance of pronounced peaks in reflection and dips in transmission.

Fig. 9. (a) Intensity of transmission (red), reflection (blue) and loss (black) for 10 coupled 200×50×50nm periodic cut-wire chains separated by d=50nm, d=100nm and d=150nm placed on 10nm silicon dioxide on the top of 100nm silicon slab. (b) Same as in (a) for 150nm silicon slab thickness.

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Although in the case of the present study the involved interactions are more complex than for crystal nano-cavities, most of the observed results can be fairly explained by the model of coupled plasmonic nano-cavities. In the case of a slab waveguide loaded with strongly coupled cut wires, its behavior is mostly similar to that of an effective medium slab waveguide. The essential condition for the validity of the effective medium approximation is that the coupling between adjacent wires must be dominant with respect to other coupling in the system, namely with that between the chain elements and the wave confined in the silicon slab. This explains why for stronger field confinement the behavior of the waveguide loaded with coupled cut wires seems to be closer to that of an effective medium slab waveguide. This assertion for an effective medium behavior is also supported by the distribution of the electrical field in the structure provided by our numerical simulations. Figure 10 shows the snapshots of the wave propagation at 200THz in three cases for a chain of cut wires separated by 50nm (a), 100nm (b) and 150nm (c) on a 10nm layer of silicon dioxide on the top of a 100nm silicon slab. The field distribution across the coupled chain region is much more uniform in the case of a strong coupling corresponding to the 50nm of separation between the chains. Conversely, stronger field localization around the cut wires is observed when the separation between the adjacent chains is increased. The exchange of energy in this case preferentially occurs between the chain and the slab and not between the adjacent chains To retrieve the effective dielectric permittivity in the case of a slab loaded with chains of coupled cut wires, the application of the Smith retrieval method using Eq. (1) becomes impractical because the large optical thickness of the MM region does not allow to unambiguously determine the corresponding branch of the inverse cosine function. To find the effective index we use an alternative method based on the determination of spatial position of the electric field minima (or maxima) in the region of the cut wires. The results obtained for 10 unit cells with wires separated by 50nm, 100nm or 150nm in the case of 100nm and 150nm thick Si slabs are represented in the Fig. 11(a) and Fig. 11(b), respectively. The general behavior of the effective index variation is qualitatively similar to that of a single chain of cut wires though the index contrast with respect to that of a silicon slab is reduced by approximately a factor of two. The index contrast is higher in the case of stronger coupled wires, that is for smaller separation distance. For 50nm of separation distance the index contrast at 200THz is around 0.5 for 100nm Si slab and around 0.25 for 150nm Si slab. Such an index contrast can be sufficient to be used in integrated optics applications. The main issue with the considered system is related to the level of propagation losses, which is of the order of few dB per 10 unit cells. This corresponds to a propagation distance of around 1µm, implying that these structures are best suitable for extremely compact designs, which are the real targets of this technology.

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21 November 2011 / Vol. 19, No. 24 / OPTICS EXPRESS 24756

Fig. 10. Cross section view for the electric field distribution of the propagated wave at 200THz for 10 coupled 200×50×50nm periodic cut-wire chains separated by d=50nm (a), d=100nm (b) and d=150nm (c) placed on 10nm silicon dioxide on the top of 100nm silicon slab.

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21 November 2011 / Vol. 19, No. 24 / OPTICS EXPRESS 24757

Fig. 11. (a) The real components of effective index for a single (magenta) or coupled 200×50×50nm cut-wire chains with separation distance: d=50nm (blue), d=100nm (red), d=150nm (cyan) placed on a 10nm silicon dioxide on the top of silicon slab of 100nm. The green curve corresponds to the effective index of the silicon slab alone. (b) Same as in (a) for 150nm silicon slab thickness.

4. Summary and conclusions The aim of the present work was to investigate the potential of MMs for building optical functions in guided wave optics at 1.5µm (200THz). The undertaken approach consists in considering a composite guiding structure made of a metamaterial layer over a high index slab waveguide, as for instance silicon in our case. In such a configuration only the evanescent tail interacts with the MM layer which acts essentially as a perturbation. The effective permittivity εeff and permeability µ eff of such a composite slab was determined using a variant of the retrieval method described in [41]. The numerical simulations show that an array of gold coupled cut wires over a slab waveguide leads to a significant variation of the slab effective index in the vicinity of the resonance. To apply the effective medium approximation, the coupling between consecutive rows of cut wires must be dominant with respect to all other interactions in the system. This implies using small separation distances, below 100nm, between the cut wires. The resulting effective index contrast is around 0.5 for a 100nm thick Si slab and around 0.25 for a 150nm thick Si slab. Such an index contrast is high enough for implementing a variety of optical functions. In certain cases, such as in transformation optical devices, it is desirable to design MMs with a strong degree of anisotropy. We showed that the anisotropy of our hybrid MM waveguides can be increased by inserting a thin dielectric layer with a low index of refraction between the cut wires and the Si slab. This intermediate layer makes it possible to increase the length of the cut wires without shifting the resonance frequency, thus maximizing their aspect ratio and anisotropic response. A critical aspect of the considered problem is related to the MM absorption losses which are on the order of a few dB per microns. To minimize the absorption losses, we followed the usual procedure that consists in detuning the resonance frequency from the operating frequency. However, this approach reduces the range of effective indices that can be attained. One possible way to counteract this issue is to consider multi-layer structures of coupled cut wires. The obtained results represent a first step toward building optical function in guided wave optics using hybrid metallo - dielectric metamaterials. Apendix A: Generalization of the retrieval method The retrieval method initially developed by Smith et al. [41] is currently widely used to determine the effective permittivity ε and permeability µ from the transmission and reflection coefficients calculated for a wave normally incident on a finite length of metamaterial. This

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method considers the metamaterial as a homogeneous layer for which it is possible to invert the scattering data to determine the refractive index n and impedance z. In view of the Smith retrieval method underlying assumptions, it is quite logically to expect that a similar kind of inversion can be performed for a single homogeneous dielectric film described by the characteristic matrix of a stratified optical medium. As will be shown below, the inversion of the characteristic matrix can be used to extend the retrieval method and find the ε, µ and z of a single layer in the most general case of oblique incidence and different input and output media. Incidentally this inversion will serve to derive Eq. (1) which is in fact just a particular case of the generalized retrieval solution. The propagation of an electromagnetic wave through a homogeneous film is sketched in Fig. A1. The expressions used for the characteristic matrix and reflection and transmission coefficients of the film are those provided by M Born and E. Wolf in [45]. For the sake of clarity we reproduce here these formula using the same notations as those employed by the authors.

Fig. A1. Propagation of an electromagnetic wave through a homogeneous film.

The expression of the characteristic matrix of a homogeneous film for a wave with electric field perpendicular to the plane of incidence (TE polarization in the authors [45] notation) is:

cos ( k0 nh cos θ ) M (h) = −ip sin ( k nh cos θ ) 0

−i sin ( k0 nh cos θ ) p cos ( k0 nh cos θ )

(A1)

where k0=2π/λ is the light wave vector in vacuum, n is the film refractive index, θ denotes the propagation angle and the quantity p is the characteristic admittance, i.e. the inverse of the medium impedance z:

p=

1 1 ε cos θ = cos θ = z zɶTE µ

(A2)

For a TM polarized wave, i.e. with the electric field in the plane of incidence, the same equation holds for the characteristic matrix, with p replaced by the characteristic impedance q:

q=

µ cos θ = z cos θ = zɶTM ε

(A3)

The reflection and transmission coefficients for a TE polarized wave are:

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r=

( m11 + m12 pl ) p1 − ( m21 + m22 pl ) ( m11 + m12 pl ) p1 + ( m21 + m22 pl )

(A4)

t=

2 p1 ( m11 + m12 pl ) p1 + ( m21 + m22 pl )

(A5)

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where the quantities p1 and pl given by Eq. (A2) correspond accordingly to the first and last semi-infinite media. The reflectivity and transmissivity coefficients of the film for the intensity of TE polarized wave are, respectively:

pl 2 t (A6) p1 At this step it is important to note that when using numerical modeling softwares like ANSOFT’s HFSS, the calculated transmission coefficient is the square root of the transmissivity and not Eq. (A5). In order to keep consistency it is necessary to renormalize the transmission coefficient as following: 2

R= r T =

2 p1 pl (A7) m m p + ( 11 12 l ) p1 + ( m21 + m22 pl ) The corresponding formulae for TM polarized wave are obtained from Eqs. (A4) and (A7) by replacing p1 and pl by q1 and ql given the Eq. A3. Denoting: t′ =

m11 = m22 = cos β ,

m12 = −izɶTE sin β ,

m21 = −

i zɶTE

sin β

(A8)

where:

β=

2π

n2 h cos θ 2

λ0

(A9)

we form the following system of equations: 1 r + = t′ t′

1 p1 pl

1 r − = t′ t′

( p1 cos β − ip1 pl zɶTE sin β )

1 i sin β p1 cos β − zɶTE p1 pl

(A10)

Solving this system of equations with respect to cosβ and z we obtain:

cos β =

(1 − r

p1 pl p1 (1 − r ) + pl (1 + r )

2

+ t′

2

)

t′

2 zɶTE

1 1+ r p1 pl t ′ (A11) = 1− r pl cos β − p1 pl t′ 1 cos β − pl

From this it is possible to determine the value of β:

β = ± arc cos ( cos β ) +

2π m k0 h

(A12)

where m is an integer. The explanation about the proper choice of the sign and value of the integer coefficient m can be found in [41]. Using the Snell law, the refractive index of the film and the refraction angle are: 2

β n 2 2 n2 = cos θ 2 = 1 − 1 sin 2 θ1 (A13) + n1 sin θ1 n2 k0 h Equations (A12), (A13) represent the generalized retrieval method for a single homogeneous film and a TE polarized wave. To obtain the expressions for TM polarization, it is sufficient to replace p by q in the systems of Eqs. A(10) and (A11). It follows that:

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1− r t′ (A14) zɶ = 1 1 1+ r cos β − ql q1ql t ′ The case of guided wave propagation is analogous to the case of normal incidence for a film sandwiched between two semi-infinite media with a refractive index corresponding to the effective index of the slab waveguide. Eq. (1) follows then straightforwardly from Eq. (A11). 2 TM

ql cos β − q1 ql

Acknowledgement This work was supported by the French Agence Nationale pour la Recherche (ANR Metaphotonique, contract number 7452RA09).

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