finite element solutions of metal-clad dielectric ... · reported a similar low-loss and flexible...

INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL TECHNOLOGY

VOL. 1, NO. 1, JUNE 2006

IJMOT-2006-5-50 © 2006 ISRAMT

Finite Element Solutions of Metal-Clad Dielectric Waveguides at Optical and THz Frequencies

C.Themistos, B. M. A. Rahman*, M. Rajarajan, and K.T.V. Grattan

School of Engineering and Mathematical Sciences, City University,

Northampton Square, London EC1V 0HB, United Kingdom. Tel: +44-20-7040-8123 Fax: +44-20-7040-8568 Email: [email protected]

Abstract- Finite element analysis, based on the full-vectorial H-field formulation and incorporating the perturbation technique, is used to calculate the complex propagation characteristics of metal-coated dielectric waveguides for both the optical and terahertz frequency ranges. The propagation and attenuation characteristics of the surface plasmon modes at the metal/dielectric interfaces are presented. Index Terms- Optical waveguides, Terahertz waveguides, metal-clad dielectric waveguides, surface plasmon modes, modal solutions, finite element method.

I. INTRODUCTION The surface plasmon mode (SPM) is essentially the electromagnetic wave, which is located at the metal-dielectric interface because of the interaction with the free electrons of the conductor. Surface plasmon resonances in a metallic layer incorporated inside optical waveguide structures [1] have been extensively used for various fiber-optic and optoelectronic devices, such as the optical polarizers [2], the electro-optic modulators [3], the fiber-optic sensors [4], biosensors [5] and the scanning microscopy. Only recently, it has also been proposed that such surface plasmon modes in a nanometer-size metal structure could be used in the design very compact optical bend and power splitter. Besides their application at the optical frequencies, a new research frontier has opened up in the terahertz (THz) frequency range. This THz region occupies a large portion of the electromagnetic spectrum located between the microwave and optical frequencies and normally

is defined as the band from 0.1 to 10 THz. In recent years, this intermediate THz radiation band has attracted a lot of interest, because it offers significant scientific and technological potential for applications in many fields, such as in sensing [6], in imaging [7] and in spectroscopy [8]. Recent progress in sources [9] and receivers of THz waves has generated much interest in studying the waveguiding properties of these waves, both as a part of active or passive components, such as lasers, detectors, or filters, and also to connect various components in a system. Suitable waveguides could be used direct these beams to the correct locations, for example around bends or corners and such guided-wave technologies offer the possibility of compact sensor systems. However, waveguiding in this intermediate spectral region is a major challenge. Amongst the various THz waveguides that have been suggested, the metal-clad waveguides supporting surface plasmon modes show the greatest promise as low-loss waveguides for use both in active components and as passive waveguides. In 1999, McGowan et al. [10] reported on the use hypodermic stainless steel needles with 240 to 300 µm inner diameter for guiding THz waves. These waveguides supported more than one guided mode over the 0.8 to 3.5 THz frequency range with a loss value of 3 dB/cm. More recently, Hikada et al. [11] have reported a novel THz waveguide using ferroelectric polyvinylidene fluoride as the inner material of a hollow waveguide and the very high dielectric constant present reduces the modal loss

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significantly and a figure of 0.5 dB/cm has been reported. Subsequently, Harrington et al. [12] reported a similar low-loss and flexible hollow polycarbonate waveguide with copper and dielectric inner coatings, which were deposited by using a liquid chemistry approach. In the present work, initially the propagation and attenuation characteristics of metal-clad dielectric waveguides with infinite cladding have been studied in order to identify key waveguiding properties, such as the existence of surface plasmon modes, knowledge of which is important in the design of the various practical structures. The study further extends to the propagation and attenuation characteristics of metal-clad waveguides with finite cladding thickness, both at optical and terahertz frequencies, in the presence of other surrounding materials, which are vital to the effective design of polarization-dependent guided-wave structures.

II. NUMERICAL SIMULATIONS In the design of any waveguide, the key modal parameters are their propagation constants, loss coefficients, the modal field profiles and the dispersion properties. First of all, it is essential to develop a modal solution approach, which can provide this information for practical optical and THz waveguides with arbitrary shape, size, and material profiles. Practical metallic elements are not perfect conductors, but suffer a small amount of loss and therefore the modelling of loss [13] in the analysis of such waveguiding structures incorporating metallic films and the interaction of the metallic films with dielectric materials is considered to be important for the accurate design of various photonic and THz devices. Cao and Jahns [14] have used a field expansion approach, such as the Bessel function expansion and matching the field continuity at the metal dielectric interface of axially symmetrical copper wire. Gallot et al. [15] have applied the classical Sommerfeld waveguiding principle to metal waveguides operating in the millimetre wave

region, as well as mode expansion and field matching techniques and also a more versatile finite-difference time-domain (FDTD) technique for surface plasmon modes. General field expansion techniques are not sufficiently versatile and cannot be used for waveguides with irregular shapes. The alternative FDTD approach is computationally very expensive. More recently, Deibel et al. reported [16] a three-dimensional time-harmonic simulation by using the finite element method, but being a three-dimensional problem this approach requires very large computational resources, such as 20 hours of CPU time on a 64-bit dual processor with 16 GB of RAM. The finite element method (FEM) has emerged as one of the most successful numerical methods for the analysis of high frequency optical waveguide problems. In this method the problem domain is suitably divided into a patchwork of finite number of subregions called ‘elements’, each of which can have different shapes and sizes and by using many elements a complex problem can be accurately represented. A wide range of guided-wave devices can be modeled as each element can also have different material parameters such as refractive index, anisotropic tensors, nonlinearity, loss or gain factors. Most of the formulations used in the FEM, such as H-field formulations [17] and the scalar [18] are restricted to structures without modal loss or gain. Due to the necessity for the analysis of practical waveguides which suffer modal loss or gain, various alternative approaches have been developed, such as the FEM solution in terms of the transverse magnetic field, Ht, formulation [19], but this formulation generates a complex eigenvalue equation, and so is computationally more expensive. On the other hand, the full-vectorial H-field formulation used in conjunction with the perturbation technique [13], which is computationally more efficient, can be used for waveguides with low or medium modal loss values. In using the FEM method, the field distribution in the transverse plane is obtained by the application of the variational formulation in the region. The usual time and axial

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dependencies are given by exp(jωt) and exp(-γz) respectively, for angular frequency ω and time t, where γ = α + jβ is the complex propagation constant in the z direction, and β and α are the phase and attenuation constants respectively. The FEM H-field formulation utilizing the perturbation technique has been successfully applied to the estimation of loss/gain parameters in several optical waveguide problems with small to medium loss or gain, such as the analysis of surface-plasmon modes [13], semiconductor lasers [20] and sub-micron metal-clad optical fibers for near-field scanning microscopy [21]. In such cases, the perturbed fields were approximated by the fields obtained from the solution of the variational formulation, using only the real part of the dielectric constant. The attenuation constant was then calculated from the results obtained from the loss-free system by using simple matrix multiplication, thus reducing the slow calculation time and the large memory requirements necessary in other formulations, such as the Ht formulation. In the present work, the H-field FEM based full-vector formulation; in conjunction with the perturbation technique, has been used for the solution of the metal-clad waveguide modes, where the transverse and longitudinal magnetic field components are analyzed with respect to rectangular coordinates. Therefore, the waveguide modes are initially presented in terms of the transverse magnetic field components, Hx

mn and Hymn, as commonly used for integrated optical waveguide problems, where the m and n subscripts denote the field maxima along the x- and the y-axes, respectively. In this notation, as an example, for the Hy

mn mode (also known as the quasi-TE mode), the Hy (or Ex) field is dominant compared to the non-dominant Hx (or Ey) field component.

III. RESULTS Initially, a metal clad optical fiber with a silica (SiO2) core and an infinite copper (Cu) cladding is considered, and shown in Fig. 1. In this case, the refractive index of the silica core is taken as nco=1.44 and the complex refractive index of the

copper cladding is given by ncl=0.76-j10.36 at an operating wavelength of λ = 1.55 µm. A two-fold symmetry has been employed in the present analysis, where only a quarter of the fibre cross section has been divided into 100 and 120 azimuthal and radial divisions, respectively, thus forming a mesh of 23900 first order triangular elements.The polar coordinate discretization used here matched very accurately the circular cross sectional area of the fiber, with the percentage error in the fiber area being only 0.004%. A single modal solution takes about 28 seconds of CPU time on a 3.4 GHz Pentium processor.

Modal solutions of the fundamental Hx

11 mode, for a fiber diameter, D, of 4 µm are obtained, where the effective index, defined as ne = β/k0, and the normalized attenuation constant were calculated to be ne=1.4394 and α/k0= 0.0016232902, respectively. For this mode, the dominant Hx field at the top of the etal-clad fiber is tangential to the dielectric/metal interface, which satisfies the electric wall boundary condition, n.H=0, and supports a surface plasmon mode (SPM) along that interface. However, at the right of the metal-clad fiber, when the same electric wall boundary condition is imposed, this forces the Hx field to be zero at the dielectric/metal interface and no SPM exists. The variations of the Hx field along the x- and the y-axes for the above Hx

11 mode are presented in Fig.2. It can be observed that the field profile

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along the y-axis, as shown by a solid line, exhibits its maximum field intensity at the dielectric/metal interface at the radial distance of 2 µm and rapidly decays in the metal when the radial distance is greater than the radius of the fiber. The optical field profile along the x-axis has the maximum field intensity at the center of the fiber, shown by a dashed line and gradually decreases along the radial distance reaching a zero value at the silica/metal interface (r = a), and thus resembles more a typical field profile of a silica fiber mode. The optical field profiles of the above mode along the y- and the x-axes have similar characteristics to the TM/TE field intensity in planar metal dielectric waveguides, where the optical field is mainly supported by the dielectric/metal interfaces for the TM mode and there is no interaction of the optical field at the same interfaces for the TE mode [13].

The contour profile of the Hx field for the Hx

11 mode is presented in Fig.3a. The maximum field intensities at the upper and lower interfaces have an even-shaped profile along the y-axis and decay rapidly in the metal cladding, clearly exhibiting SPM-like properties. The maximum field intensity of the above mode gradually reduces in the horizontal direction from the y-axis and finally approaches zero at the SiO2/Cu interface at a distance r = a. Further study on the effect of the fiber diameter in the field profile of the Hx

11 mode (not presented here) has shown

that as the diameter of the fiber decreases below 2 µm and the field intensity at the centre of the fiber eventually becomes slightly higher than the field intensity at the metal/dielectric interface.

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The Hy11 mode of the above structure for a core

diameter, D = 4 µm, has also been obtained and the field distribution of the above mode, as shown in Fig. 3b, exhibits similar characteristics to those obtained for the Hx

11 mode (as shown in Fig. 3a), but rotated by 90°. These properties are found to be in agreement with the electric-wall boundary condition, imposed at the horizontal (top and bottom) and the vertical (left and right) dielectric/metal boundaries, where the Hy field is zero and tangential, respectively. Although rotational symmetry exists for this structure, however due to the non-identical boundary conditions in the vertical and horizontal directions, the Hy

11 modal field profile is also not circularly symmetric. As mentioned earlier, weakly guiding fiber hybrid modes having the same propagation constant can be grouped together and the amplitude of the radially polarized mode is shown in Fig. 3c, by the superimposition of the field profiles of Hx

11 and the Hy

11 modes. The above mode exhibits maximum field intensity along the circular dielectric/metal interface and the optical field decays rapidly in the metal cladding, but with a slower rate inside the core area, since the decay parameter for a given mode depends on the refractive index, and decays faster in the metal cladding region with a negative relative permittivity value. For this mode the direction of the magnetic field follows the radial direction and can be called the radially polarized, RP01 mode. The effective index and the normalized attenuation constant for the Hx

13 mode for a fiber diameter of 4 µm were calculated to be ne=1.3836 and α/k0=0.0014044, respectively. The above mode exhibits maximum field intensity at the center of the waveguide and a lower inverse peak along the y-axis at the dielectric/metal interface. But along the x-direction, the above mode exhibits only one maximum at the center of the fiber and the optical field reduces smoothly along the radius of the fiber with near zero field at the dielectric/metal interface. Another mode, the Hy

31, with the same equivalent index possesses field distribution characteristics similar to the Hx

13 mode, but rotated by 90°, and

therefore these two degenerate modes can be superimposed to form a radially polarized RP02 optical mode with fiber mode-like properties in the center of the fiber and surface plasmon-like behaviour with an inverse peak field intensity at the dielectric/metal interface along the circumference. Similarly, the Hx

12 optical mode also possesses surface plasmon properties for a fiber diameter of 4 µm, where the effective index and the normalized attenuation constant were calculated to be ne=1.4369 and α/k0=0.0023699, respectively. Using the hybrid mode notation, the above mode can be identified as the HE11 optical mode, with the Hx optical field being the dominant field component, having a pair of maxima and a single peak intensity in the azimuthal and radial dimensions, respectively. The radially polarized RP11 surface plasmon-like optical mode for a fiber diameter of 4 µm, formed by the constituent Hx and Hy optical modes with the same effective index that was calculated to be ne=1.4149, has also been obtained but not presented here. When two similar degenerate modes are superimposed, the optical mode exhibits one horizontally oriented pair and a single field maximum along both the azimuthal and the radial distance of the fiber. Variations of the effective indices with the fiber diameter for the SPMs discussed above are presented in Fig.4, where it can be seen that as the fiber diameter increases, the effective indices for all the modes examined also increase. As the fiber diameter approaches a value of 5 µm, the effective index of the HE11 mode, as shown by a dashed line, tends to reach the effective index of the fundamental RP01 optical mode, thus eventually contributing to the formation of the above mode, as discussed in related theory of optical guided modes in weakly guiding circular fibers [22]. Next the attenuation characteristics of the above modes have been calculated using the vector H-field FEM with perturbation and the variation of the normalized attenuation constant, α/k0 with the fiber diameter is presented in Fig.5. From this figure it can be seen that as the fiber diameter decreases, the normalized attenuation constants

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increase. The fundamental RP01 mode has the lowest attenuation for a fiber except when the fiber diameter is more than 3.5 µm, when the LP02 mode has the lowest attenuation. The accuracy of the normalized attenuation curves is considered to be within the accuracy limits of the perturbation approach, as discussed by Themistos et al. [23].

Further, the optical properties of metal-clad optical fibers with a finite cladding thickness, t, as shown in Fig.6, have also been studied and the effect of the outer cladding materials such as silica, air or acetone on the optical field have been considered. The variation of optical parameters of such structures due to the presence

of materials that affect the modal field distribution can be utilized in optical sensor applications. Initially the variation in the effective index of the fundamental LP01, for a fiber diameter of 4 µm, covered by a metal layer and further covered by silica, air or acetone as the outer cladding, has been studied and presented in Fig.7. From the above characteristics, it can be seen that for a larger metal thickness, the effective index of the LP01 optical mode with finite metal thickness is about the same as the effective index of the infinite cladding structure and is not affected by the materials in the outer cladding. For a large metal thickness, t, two SPMs exist at the two dielectric/metal circular boundaries and when the two dielectric materials are different, their propagation constants are different, and therefore they do not interact to form a supermode. However, as the metal thickness, t, is reduced, there is a stronger coupling between the SPMs and two supermodes, with odd- and even-like maximum field intensities at the two interfaces, being formed.

The first supermode has a higher propagation constant and is confined near the inner metal interface with the higher refractive index and has an odd-like field profile. On the other hand, the second supermode is confined near the outer metal interface with the lower refractive index and has an even-like field profile. The even-like second supermode is dominated by the effective index of the outer metal dielectric interface, and

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being close to cut-off, as the metal thickness decreases, the even-like mode becomes unbounded.

As can be seen from the effective index curves shown in Fig.7, when the metal thickness, t, decreases below 50 nm, the effective index of the odd-like supermode increases rapidly. For the waveguides with acetone or air as the outer cladding, the effective index of the even-type SPM is below the cut-off and therefore is not presented here. The variation of the optical properties with the metal thickness, t, and the outer cladding material can be better explained with the aid of the field distribution along the radial direction of the fiber and particularly near the dielectric (core)/metal and metal/dielectric (cladding) interfaces. Initially, a metal-clad optical fiber with its diameter, D = 4 µm and a metal cladding thickness, t = 0.01 µm, and with an infinite air cladding, has been examined. The field profile of the fundamental Hx odd-like optical supermode along the x- and the y-axes is presented in Fig.8. It can be observed that in the x-direction, as shown by a dashed line, the field profile decays smoothly near the dielectric (core)/metal interface, forcing the normal component of the H-field, Hx, to zero at the interface, and the field closely resembles that of a fiber mode. The Hx field profile along the y-axis, being transverse to

the interface, exhibits odd-like SPM properties shown by a solid line, where a high maximum positive peak at the silica/copper and lower negative peak at the copper/air interface can be observed. A vertical chained line shows the silica/copper interface.

Next, a metal-clad fiber with D = 4 µm and a finite metal thickness, t, with acetone material around the metal cladding has been considered. The field profiles of the fundamental Hx odd-like optical supermode along the y-axis, for different values of the metal thickness, t, have been investigated and are shown in Fig.9. As can be seen from this figure, for a metal thickness of 0.2 µm (as shown by a solid line), the field profile has similar properties to those obtained for an infinite cladding fiber, where the field has a maximum intensity at the dielectric/metal interface and decays rapidly in the metal region, except that a small negative peak is clearly visible at the copper/acetone interface due to the weak coupling between the two non- synchronous SPMs. As the metal thickness decreases, the maximum field intensity increases anti-symmetrically at both the interfaces, with the highest maximum being at the silica/copper interface, and exhibiting the properties of an odd-like bounded and strongly coupled SPM in a planar dielectric/metal/dielectric waveguide. The field profile is shown by a dotted line for a metal thickness, t = 0.01 µm.

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The superimposed field intensity from the Hx and the Hy fundamental odd-like supermodes that form the radially polarized RP01 mode, for a metal-clad silica fiber with a diameter, D = 4 µm and a metal thickness, t = 0.01 µm and air outside the metal-cladding is presented in Fig.10. From the above field intensity, it can be seen that the optical field decays in both the center of the fiber and the outer cladding, with the anti-symmetric peak field intensities at the inner silica/copper (where the field is positive) and a small negative peak (which is not visible) at the outer copper/air interfaces. Next, a copper coated silicon tube is considered operating at 1 THz , as shown in Fig.11. In this case D is the diameter of the inner air-core and a and t are thicknesses of the silicon tube and the thin metal layer, respectively. The refractive index of the air-core and the silicon tube are taken as ng=1.00 and na= 3.4205267 (εr = 11.7 [24]), respectively, and the complex refractive index of the copper cladding is given by nm=438-j494, [25] at an operating frequency of 1 THz. There are two metal/dielectric interfaces, one at the outer copper/silicon boundary and the other at the inner copper/air-core boundary. For

this guide, the dominant Hx field at the upper and lower metal-dielectric interfaces are tangential to these boundaries and supports a surface plasmon mode (SPM) along these metal/dielectric interfaces. The refractive index of the inner and outer cladding materials being very different, the two SPMs have widely different propagation constants, and they do not interact with each other. Variations of the Hx fields along the y-axis for both the SPMs are shown in Fig. 12.

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A solid line shows the first SPM at the outer copper/silicon interface, where the field reduces almost linearly inside the silicon layer and very rapidly inside the thin metal layer. In this case, the air-core diameter, D = 8 mm, the thickness of the silicon tube, a = 1 mm and the thickness of the copper coating, t = 0.5 µm. The inset on the right of the figure shows the rapid field decay near the outer metal/silicon interface. A dashed line shows the variation of the Hx field along the y-axis for the second SPM at the inner copper/air interface. It can be observed that the Hx field decays very slowing in the air-core region and then extremely rapidly inside the metal layer. The rapid decay of the Hx field in the inner metal/air interface is shown by another inset on the left hand side. It should be noted that in this example, the metal layer thickness, t, is only 0.5 µm, which is at least three orders of magnitude smaller than the other dimensions.

However, at the right and left hand sides of the metal/dielectric interfaces, when the same electric-wall boundary condition is imposed, the Hx field is forced to be zero at the metal boundary and no SPM exists. The modal field profile along the x-axis (not shown here) has its maximum field intensity at the center of the waveguide and gradually decreases along the radial distance, reaching a zero value at the copper boundary. The Hx field intensity for the first SPM mode has its maximum value at the

outer metal/silicon interface, where the electric-wall boundary condition allows the Hx component to exist. On the other hand, the same electric-wall boundary condition forces the Hx field to be zero on the left-hand and the right-hand side electric walls. Although for this structure a rotational symmetry exists, due to the non-identical boundary conditions in the vertical and horizontal directions, the modal field profile is no longer circularly symmetric.

The SPMs decrease slowly in the inner or outer cladding as shown in Fig.12, but more rapidly inside the metal layer. For a larger metal thickness, t, the field at one of the interfaces does not extend to the adjacent interface. So, in this case, although two SPMs exist at the two dielectric/metal circular boundaries, when the two dielectric materials are different, their propagation constants are also different, and therefore they do not easily interact to form a supermode. However, as the metal thickness, t, is reduced, the coupling between the SPMs increases and two supermodes, with odd- and even-like maximum field intensities at the inner and outer interfaces can be formed. On the other hand, identical SPMs exist at the upper and lower interfaces and form even and odd-type supermodes with respect to the x-axis.

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Recently, Deibel et al. [16] have also reported similar horizontal or vertical ‘donut’ shaped field electric field profiles for the horizontally or vertically polarized waves in a metal wire waveguide. Results presented by Wang and Mittleman [26] also show a similar dipole shaped vertically polarized (Ey or Hx) wave, confined in the upper and lower interfaces of a solid metal wire guide. Similarly another mode, but with the dominant Hy field would also form SPMs, however at the left and the right interfaces, and this is not shown here. The Hy field profile of this mode is similar to the Hx field profile, but rotated by 90 degrees. These two modes have identical propagation constants and so they are degenerate. The superposition of their field profiles is shown in Fig.13, which indicates a rotationally symmetric profile. In this case the field maximum is at the outer silicon/metal interface and the direction of the vector H-field is along the radial direction. Similar circularly symmetric field profiles have been reported for dielectric clad metal waveguides supporting surface plasmon modes at optical frequencies [21, 27]. The effective index of the outer SPM reduces slowly with the diameter, D, and is slightly lower than na. As mentioned earlier, the second SPM exists at the inner metal/core interface, as shown in Fig.

14, but due to the lack of phase matching it would not form a supermode by interacting with the first SPM of the outer interface. The Hx field profile of this SPM, as shown in Fig.14, shows two field maxima at the upper and at the lower interfaces. However, the field changes very slowly along the y-axis and has a larger value at the center of the waveguide. The electrical-wall boundary condition forces the Hx field to be zero at the left and right hand side electric walls, so the Hx modal field is also no longer circularly symmetric. Similarly another degenerate mode, but with a dominant Hy field, also forms a SPM at the left and right-side inner metal/air interfaces. Its field profile is similar to the Hx field profile, but is rotated by 90 degrees. These two modes, being degenerate, can be superimposed to form a radially polarized RP01 mode. However, this mode with an almost constant field value inside the air-core region is quite different from the mode shown in Fig.13 confined at the outer interface. This mode would also be different from the linearly polarized LP01 mode in an optical fiber, with an almost uniform field profile inside the core region. It has been observed that for a larger core diameter, the field profile is more flat in the core region. Hence, the field profile of this mode inside the air-core can be controlled by choosing an appropriate diameter of the inner core, a feature which would be useful in the design of a scan tip for near field scanning microscopy. Since the variation of the inner core diameter changes with the waveguide dimensions, the effective index of the inner SPM increases with the core diameter, and this value is slightly lower than 1.0, with the mode being confined at the air/metal interface. The inner and outer SPMs interact and form SPSMs for lower values of t or for better phase matched SPMs. The outer SPSM, being associated with the higher refractive index, na, shows the features of an odd-type first supermode. Similarly, with the inner SPSM mostly being confined near the region with lower refractive index, its effective index is lower than nb and this supermode shows even-like second supermode properties. It has been reported earlier [13], that unlike the supermodes in a dielectric

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waveguide, the odd-type SPSM is the first supermode and its effective index is higher than that of the even-type second SPSMs, and the effective indices for the odd-type and the even-type SPSMs reduce and increase, respectively, as the metal thickness, t, is increased [13]. In the above example, the propagation constants of the two SPMs at the air/metal and metal/silicon interfaces are very different and do not form an effective supermode. In the next example given, a solid silicon core replaces the inner air core. In this case the two supermodes are nearly phase matched. The variations of the effective indices with the core diameter, D, for the two supermodes are shown in Fig.15 by a solid line and a dashed line for the odd-like and even-like SPSMs, respectively. However it can be clearly observed that these two lines do not cross each other. For coupled SPSMs, the odd-like SPSM has a higher effective index than the even-like SPSM. The effective index of the inner SPSM strongly depends on the core diameter and increases monotonically. For a lower core diameter, D, the odd SPSM is confined at the outer metal/cladding interface.

However, for a larger core diameter, the odd supermode is actually confined at the inner interface. Similarly, the even-like SPSM is confined at the inner interface for a lower core diameter and at the outer interface for a larger

core diameter. On the other hand, the effective index of the outer SPSM reduces with the core diameter and asymptotically reaches that of a parallel planar structure. When the two modes are nearly phase matched, there is the strongest interaction between the modes and they form truly even- and odd-type supermodes. In all these cases, the effective index of the outer SPSM is slightly higher than the cladding indices, na = nb = 3.4205267. The Hx field profile along the y-axis is shown in Fig.16 when the SPMs are nearly phase matched, at D = 10.8 mm. A solid line and a dashed line show the Hx field profile for the odd-like and the even-like supermodes, respectively.

There is a very thin metal region, t = 0.5 µm, sandwiched between the two silicon layers. The Hx field changes very rapidly in this thin metal region, and to illustrate its variation more clearly, this area is expanded and shown as an inset on this figure. It can be observed that the even-like supermode Hx field dips in the middle of the metal layer and the field amplitudes are positive in both the dielectric regions. On the other hand, for the odd supermode, the Hx field rapidly changes inside the metal layer and the field magnitudes are of opposite signs in the two adjacent dielectric regions.

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The Hx field of the first supermode is mostly confined at the inner metal/air interfaces but with a very small negative field also at the outer metal/silicon interfaces. Hence, this SPM can be identified as an odd-like SPSM, being odd-functional with respect to the inner and outer metal interfaces. Two such SPSMs at the upper and lower interfaces couple to form a more complex even-type SPSM, as the signs of the peak fields are identical at the upper and lower interfaces. On the other hand, the second SPSM has a larger positive peak Hx field at the outer metal/silicon interface, with a very small positive

field value at the inner metal/air interface. This can be labelled as an even-like SPSM, because of it being an even-shape functional with respect to the inner and outer metal interfaces. Again, two such SPSMs at the identical upper and lower interfaces again couple to form a more complex even-type SPSM, with respect to the symmetry plane along the x-axis, and with equal positive peaks at the upper and lower interfaces. However, the field at the center of the air-core is almost negligible. In the above example, for D = 16 mm, as it is far away from the phase matching condition, the SPMs at the inner and outer interfaces will predominantly be confined either at the inner or outer interfaces without much interaction between them. On the other hand, when D is close to 10.8 mm, the two SPSMs at the outer and inner interfaces are phase matched and confined around both the interfaces forming two odd-type and even-type supermodes. Again, similar supermodes at the upper and lower interfaces couple to form more complex supermodes. Another set of supermodes also exists at the left and right interfaces but with the dominant field being Hy rather than Hx. The superposition of these vertically and horizontally polarized degenerate supermodes form radially symmetric SPSMs. The odd and even-type SPSMs are shown in Fig.17 (a) and (b), respectively. It can be seen that the odd-type SPSM has a positive peak at the inner metal/core interface and a negative peak at the outer metal/cladding interface. Alternatively, the even-type SPSM shows positive peaks at both the inner and outer metal/dielectric interfaces, but with a slightly reduced field value inside the metal layer.

IV. CONCLUSIONS A finite-element approach based on a full-vectorial H-field formulation used, in conjunction with the perturbation technique, has been used to study the detailed modal properties, such as the mode field distribution, the effective index and the attenuation constant of a metal-clad dielectric

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waveguides at optical and THz frequency ranges. The origin of the SPM at the inner and outer metal/dielectric interfaces, their phase matching, and the formation of the odd-like and even-like supermodes at these metal/dielectric interfaces are discussed. Subsequently, such odd- or even-like SPSMs at the upper and lower interfaces also couple to forms a more complex coupled supermode. By adjusting the metal thicknesses and the refractive index values of the cladding layers, the odd and even-type coupled SPMs can be formed and potentially exploited for various sensing, imaging and communication systems. It has been shown here that a circularly symmetric metal-coated guide supports both vertically and horizontally polarized waves. However, due to the imposition of the electric-wall boundary conditions at the vertical or horizontal interfaces, the resultant Hy and Hx field profiles do not have rotational symmetry. However, these modes being degenerate, their superposition produces a typical RPmn mode, which has rotational symmetry. It is also shown here that for the air-core metal-clad silicon tube, the SPSM at the inner interface shows a uniform modal field profile inside the air-core region. By controlling the waveguide dimensions, a flat field profile with a sharper decay at the inner metal interface can be produced, thus being suitable for near field scanning microscopy. On the other hand, by controlling the metal thickness and adjacent dielectric cladding regions, phase matching can be controlled for various applications, such as optical or THz sensing. By using a segmented metal cladding, the rotational symmetry can be broken, creating a situation where the vertically and horizontally polarized waves would not degenerate, and a high modal birefringence or polarization-maintaining waveguide thus can be fabricated.

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