Classifications

• • B—PERFORMING OPERATIONS; TRANSPORTING
  • B82—NANOTECHNOLOGY
  • B82Y—SPECIFIC USES OR APPLICATIONS OF NANOSTRUCTURES; MEASUREMENT OR ANALYSIS OF NANOSTRUCTURES; MANUFACTURE OR TREATMENT OF NANOSTRUCTURES
  • B82Y20/00—Nanooptics, e.g. quantum optics or photonic crystals
• • G—PHYSICS
  • G02—OPTICS
  • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
  • G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
  • G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
  • G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
  • G02B6/122—Basic optical elements, e.g. light-guiding paths
• • G—PHYSICS
  • G02—OPTICS
  • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
  • G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
  • G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
  • G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
  • G02B6/122—Basic optical elements, e.g. light-guiding paths
  • G02B6/1225—Basic optical elements, e.g. light-guiding paths comprising photonic band-gap structures or photonic lattices
• • G—PHYSICS
  • G02—OPTICS
  • G02F—OPTICAL DEVICES OR ARRANGEMENTS FOR THE CONTROL OF LIGHT BY MODIFICATION OF THE OPTICAL PROPERTIES OF THE MEDIA OF THE ELEMENTS INVOLVED THEREIN; NON-LINEAR OPTICS; FREQUENCY-CHANGING OF LIGHT; OPTICAL LOGIC ELEMENTS; OPTICAL ANALOGUE/DIGITAL CONVERTERS
  • G02F1/00—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics
  • G02F1/01—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics for the control of the intensity, phase, polarisation or colour 
  • G02F1/19—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics for the control of the intensity, phase, polarisation or colour  based on variable-reflection or variable-refraction elements not provided for in groups G02F1/015 - G02F1/169
  • G02F1/195—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics for the control of the intensity, phase, polarisation or colour  based on variable-reflection or variable-refraction elements not provided for in groups G02F1/015 - G02F1/169 by using frustrated reflection
• • G—PHYSICS
  • G02—OPTICS
  • G02F—OPTICAL DEVICES OR ARRANGEMENTS FOR THE CONTROL OF LIGHT BY MODIFICATION OF THE OPTICAL PROPERTIES OF THE MEDIA OF THE ELEMENTS INVOLVED THEREIN; NON-LINEAR OPTICS; FREQUENCY-CHANGING OF LIGHT; OPTICAL LOGIC ELEMENTS; OPTICAL ANALOGUE/DIGITAL CONVERTERS
  • G02F1/00—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics
  • G02F1/01—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics for the control of the intensity, phase, polarisation or colour 
  • G02F1/21—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics for the control of the intensity, phase, polarisation or colour  by interference
  • G02F1/225—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics for the control of the intensity, phase, polarisation or colour  by interference in an optical waveguide structure
• • G—PHYSICS
  • G02—OPTICS
  • G02F—OPTICAL DEVICES OR ARRANGEMENTS FOR THE CONTROL OF LIGHT BY MODIFICATION OF THE OPTICAL PROPERTIES OF THE MEDIA OF THE ELEMENTS INVOLVED THEREIN; NON-LINEAR OPTICS; FREQUENCY-CHANGING OF LIGHT; OPTICAL LOGIC ELEMENTS; OPTICAL ANALOGUE/DIGITAL CONVERTERS
  • G02F1/00—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics
  • G02F1/29—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics for the control of the position or the direction of light beams, i.e. deflection
  • G02F1/31—Digital deflection, i.e. optical switching
  • G02F1/313—Digital deflection, i.e. optical switching in an optical waveguide structure
  • G02F1/3132—Digital deflection, i.e. optical switching in an optical waveguide structure of directional coupler type
• • G—PHYSICS
  • G02—OPTICS
  • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
  • G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
  • G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
  • G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
  • G02B2006/12133—Functions
  • G02B2006/12145—Switch
• • G—PHYSICS
  • G02—OPTICS
  • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
  • G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
  • G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
  • G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
  • G02B2006/12133—Functions
  • G02B2006/12147—Coupler
• • G—PHYSICS
  • G02—OPTICS
  • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
  • G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
  • G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
  • G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
  • G02B2006/12133—Functions
  • G02B2006/1215—Splitter
• • G—PHYSICS
  • G02—OPTICS
  • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
  • G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
  • G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
  • G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
  • G02B2006/12133—Functions
  • G02B2006/12159—Interferometer
• • G—PHYSICS
  • G02—OPTICS
  • G02F—OPTICAL DEVICES OR ARRANGEMENTS FOR THE CONTROL OF LIGHT BY MODIFICATION OF THE OPTICAL PROPERTIES OF THE MEDIA OF THE ELEMENTS INVOLVED THEREIN; NON-LINEAR OPTICS; FREQUENCY-CHANGING OF LIGHT; OPTICAL LOGIC ELEMENTS; OPTICAL ANALOGUE/DIGITAL CONVERTERS
  • G02F2203/00—Function characteristic
  • G02F2203/10—Function characteristic plasmon

Definitions

• the invention relates to optical devices and is especially applicable to waveguide structures and integrated optics.
• optical radiation embraces electromagnetic waves having wavelengths in the infrared, visible and ultraviolet ranges.
• the terms "finite” and “infinite” as used herein are used by persons skilled in this art to distinguish between waveguides having “finite” widths in which the actual width is significant to the performance of the waveguide and the physics governing its operation and so-called “infinite” waveguides where the width is so great that it has no significant effect upon the performance and physics or operation.
• the electromagnetic properties of some metals closely resemble those of an electron gas, or equivalently of a cold plasma.
• Metals that resemble an almost ideal plasma are commonly termed "noble metals" and include, among others, gold, silver and copper.
• noble metals include, among others, gold, silver and copper.
• Numerous experiments as well as classical electron theory both yield an equivalent negative dielectric constant for many metals when excited by an electromagnetic wave at or near optical wavelengths [1,2].
• the dielectric function of silver has been accurately measured over the visible optical spectrum and a very close correlation between the measured dielectric function and that obtained via the electron gas model has been demonstrated [3]-
• TM Transverse Magnetic
• TM modes In general, only two purely bound TM modes, each having three field components, are guided by an infinitely wide metal film waveguide.
• the electric field of the modes In the plane perpendicular to the direction of wave propagation, the electric field of the modes is comprised of a single component, normal to the interfaces and having either a symmetric or asymmetric spatial distribution across the waveguide. Consequently, these modes are denoted s b and a b modes, respectively.
• the s b mode can have a small attenuation constant and is often termed a long-range surface plasmon-polariton.
• the fields related to the a b mode penetrate further into the metal than in the case of the s,, mode and can be much lossier by comparison.
• Infinitely wide metal film structures are of limited practical interest since they offer one-dimensional (1-D) field confinement only, with confinement occurring along the vertical axis perpendicular to the direction of wave propagation, implying that modes will spread out laterally as they propagate from a point source used as the excitation.
• Metal films of finite width have recently been proposed in connection with polarizing devices [12], but merely as a cladding.
• plasmon-polariton waves guided by a metal-dielectric interface are in general quite lossy. Even long-range surface plasmons guided by a metal film can be lossy by comparison with dielectric waveguides.
• Known devices exploit this high loss associated with surface plasmons for the construction of plasmon-polariton based modulators and switches.
• known plasmon-polariton based modulator and switch devices can be classified along two distinct architectures. The first architecture is based on the phenomenon of attenuated total reflection (ATR) and the second architecture is based on mode coupling between a dielectric waveguide and a nearby metal. Both architectures depend on the dissipation of optical power within an interacting metal structure.
• ATR based devices depend on the coupling of an optical beam, which is incident upon a dielectric-metal structure placed in optical proximity, to a surface plasmon- polariton mode supported by the metal structure. At a specific angle of incidence, which depends on the materials used and the particular geometry of the device, coupling to a plasmon mode is maximised and a drop in the power reflected from the metal surface is observed.
• ATR based modulators make use of this attenuated reflection phenomenon along with means for varying electrically or otherwise at least one of the optical parameters of one of the dielectrics bounding the metal structure in order to shift the angle of incidence where maximum coupling to plasmons occurs. Electrically shifting the angle of maximum coupling results in a modulation of the intensity of the reflected light. Examples of devices that are based on this architecture are disclosed in references [23] to [36].
• Mode coupling devices are based on the optical coupling of light propagating in a dielectric waveguide to a nearby metal film placed a certain distance away and in parallel with the dielectric waveguide.
• the coupling coefficient between the optical mode propagating in the waveguide and the plasmon-polariton mode supported by the nearby metal film is adjusted via the materials selected and the geometrical parameters of the device.
• Means is provided for varying, electrically or otherwise, at least one of the optical parameters of one of the dielectrics bounding the metal. Varying an optical parameter (the index of refraction, say) varies the coupling coefficient between the optical wave propagating in the dielectric waveguide and the lossy plasmon-polariton wave supported by the metal. This results in a modulation in the intensity of the light exiting the dielectric waveguide.
• References [37] to [40] disclose various device implementations based upon this phenomenon. Reference [41] further discusses the physical phenomenon underlying the operation of these devices.
• Reference [42] discusses an application of the ATR phenomenon for realising an optical switch or bistable device.
• the present invention seeks to eliminate, or at least mitigate, one or more of the disadvantages of the prior art.
• a waveguide structure comprising a thin strip having finite width and thickness with dimensions such that optical radiation having a wavelength in a predetermined range couples to the strip and propagates along the length of the strip as a plasmon-polariton wave.
• the strip may comprise a material having a relatively high free charge carrier density, for example a conductor or certain classes of highly-doped semiconductor.
• the surrounding material may have a relatively low free charge carrier density, i.e. an insulator or undoped lightly doped semiconductor.
• Such a strip of finite width offers two-dimensional (2-D) confinement in the transverse plane, i.e. perpendicular to the direction of propagation, and, since suitable low-loss waveguides can be fabricated from such strip, it may be useful for signal transmission and routing or to construct components such as couplers, power splitters, interferometers/ modulators, switches and other typical components of integrated optics.
• different sections of the strip serving different functions, in some cases in combination with additional electrodes.
• the strip sections may be discrete and concatenated or otherwise interrelated, or sections of one or more continuous strips.
• the optical radiation has a free-space wavelength of 1550 nm
• the waveguide is made of a strip of a noble metal surrounded by a good dielectric, say glass
• suitable dimensions for the strip are thickness less than about 0.1 microns, preferably about 20 nm, and width of a few microns, preferably about 4 microns.
• the strip could be straight, curved, bent, tapered, and so on.
• the dielectric material may be inhomogeneous, for example a combination of slabs, strips, laminae, and so on.
• the conductive or semiconductive strip may be inhomogeneous, for example a gold layer sandwiched between thin layers of titanium.
• the plasmon-polariton wave which propagates along the structure may be excited by an appropriate optical field incident at one of the ends of the waveguide, as in an end- fire configuration, and/or by a different radiation coupling means.
• the low free-carrier density material may comprise two distinct portions with the strip extending therebetween, at least one of the two distinct portions having at least one variable electromagnetic property, and the device then may further comprise means for varying the value of said electromagnetic property of said one of the portions so as to vary the propagation characteristics of the waveguide structure and the propagation of the plasmon-polariton wave.
• propagation of the plasmon-polariton wave is supported and, for another value of said electromagnetic property, propagation of the plasmon-polariton wave is at least inhibited.
• Such embodiments may comprise modulators or switches.
• Different embodiments of the invention may employ different means of varying the electromagnetic property, such as varying the size of at least one of said portions, especially if it comprises a fluid.
• the at least one variable electromagnetic property of the material may comprise permittivity, permeability or conductivity.
• variable electromagnetic property will be permittivity, which may be varied by applying an electric field to the portion, or changing an electric field applied thereto, using suitable means.
• variable electromagnetic property will be permittivity which may be varied by applying a magnetic field to the portion or changing a magnetic field applied thereto, using suitable means.
• the electromagnetic property may be permittivity and be varied by changing the temperature of the material.
• the portion comprises an acousto-optical (photoelastic) material
• the electromagnetic property may be permittivity and be varied by changing mechanical strain in the material.
• the electromagnetic property will be permeability and may be varied by applying a magnetic field to the material or changing a magnetic field applied thereto, by suitable means.
• the electromagnetic property will be conductivity or permittivity and may be varied by changing free charge carrier density in said portion, using suitable means.
• the propagation of the plasmon-polariton wave may be varied by varying an electromagnetic property of the strip.
• the permittivity of the strip may be varied by changing the free charge carrier density or by changing or applying a magnetic field through the strip.
• Figures 1(a) and 1(b) are a cross-sectional illustration and a plan view, respectively, of a symmetric waveguide structure embodying the present invention in which the core is comprised of a lossy metal film of thickness t, width w, length / and permittivity e 2 embedded in a cladding or background comprising an "infinite" homogeneous dielectric having a permittivity ei;
• the waveguide cross-section is located in the x - y plane and the metal is bounded by the region -0.5 ⁇ x ⁇ 0.5 ⁇ m and -0.05
• the waveguide cross-section is located in the x - y plane and the metal is bounded by the region -0.5 ⁇ x ⁇ 0.5 ⁇ m and -0.05 ⁇ y ⁇ 0. 5 ⁇ m, outlined as the rectangular dashed contour.
• the waveguide cross-section is located in t & x - y plane and the metal is bounded by the region -0.5 ⁇ x ⁇ 0.5 ⁇ m and -0.05 ⁇ y ⁇ 0.05 ⁇ m, outlined as the rectangular dashed contour.
• the power confinement factor cf is also given in all cases, and is computed via equation (12) with the area of the waveguide core A,, taken as the area of the metal region. In all cases, the outline of the metal film is shown as the rectangular dashed contour;
• the waveguide cross-section is located in the x - y plane and the metal film is bounded by the region -0.5 ⁇ x ⁇ 0.5 ⁇ m and -0.01 ⁇ y ⁇ O.Ol m, outlined as the rectangular dashed contour;
• the waveguide cross-section is located in the x - y plane and the metal film is bounded by the region - 0.5 ⁇ x ⁇ 0.5 ⁇ m and -0.05 ⁇ y ⁇ 0.05 ⁇ m, outlined as the rectangular dashed contour;
• Figures 11(a) and (b) illustrate dispersion characteristics with thickness of the ss mode supported by symmetric metal film waveguides of various widths.
• the power confinement factor cf is also given in all cases, and is computed via equation (12) with the area of the waveguide core A,, taken as the area of the metal region. In all cases, the outline of the metal film is shown as the rectangular dashed contour;
• the normalized phase constant is plotted on the left axis and the normalized attenuation constant is plotted on the right one;
• the outline of the metal film is shown as the rectangular dashed contour;
• Figures 17(a) and 17(b) are a cross-sectional view and a plan view, respectively, of a second embodiment of the invention in the form of an asymmetric waveguide structure formed by a core comprising a lossy metal film of thickness , width w and permittivity e 2 supported by a homogeneous semi-infinite substrate of permittivity ei and with a cover or superstrate comprising a homogeneous semi-infinite dielectric of permittivity e 3 ;
• the waveguide cross-section is located in the x-y plane and the metal region is outlined as the rectangular dashed contour.
• the field distributions are normalized such that m ⁇ x
• 9 ⁇ E y ⁇ ] 1;
• the waveguide cross-section is located in the x-y plane and the metal region is outlined as the rectangular dashed contour.
• the field distributions are normalized such that max j 3 ⁇ E y ⁇
• 1;
• the waveguide cross-section is located in the x-y plane and the metal region is outlined as the rectangular dashed contour.
• the field distributions are normalized such that max
• 9 ⁇ E y ⁇ j 1;
• the waveguide cross-section is located in the x-y plane and the metal region is outlined as the rectangular dashed contour.
• Figure 27 is a plan view of a waveguide with opposite sides stepped to provide different widths;
• Figure 28 is a plan view of a waveguide which is tapered and slanted
• Figure 29 is a plan view of a trapezoidal waveguide
• Figure 30 is a plan view of a waveguide having curved side edges and suitable for use as a transition piece
• Figure 31 is a plan view of a curved waveguide section suitable for interconnecting waveguides at a corner;
• Figure 32 is a plan view of a two-way splitter/combiner formed by a combination of three straight waveguide sections and one tapered waveguide section;
• Figure 33 is a plan view of an angled junction using a slanted section;
• Figure 34 is a plan view of a power divider formed by a trapezoidal section and pairs of concatenated bends
• Figure 35 is a plan view of a Mach-Zehnder interferometer formed using a combination of the waveguide sections;
• Figure 36(a) is a schematic plan view of a modulator using the Mach-Zehnder waveguide structure of Figure 35;
• Figures 36(b) and 36(c) are inset diagrams illustrating alternative ways of applying a modulation control voltage
• Figure 37 is a . plan view of a modulator using the Mach-Zehnder waveguide structure of Figure 35 and illustrating magnetic field control;
• Figure 38 is a plan view of a periodic structure formed by a series of unit cells each comprising two waveguide sections having different widths and lengths;
• Figure 39 is a plan view of a periodic waveguide structure formed by a series of unit cells each comprising two opposed trapezoidal waveguide sections;
• Figure 40(a) is a plan view of an edge coupler formed by two parallel strips of straight waveguide with various other waveguides for coupling signals to and from them;
• Figure 40(b) is an inset diagram illustrating a way of applying a modulation control voltage
• Figure 41(a) is a perspective view of an edge coupler in which the parallel strips are not co-planar;
• Figure 41(b) is an inset diagram illustrating a way of applying a modulation control voltage
• Figure 42 is a plan view of an intersection formed by four sections of waveguide;
• Figures 43(a) and 43(b) are a schematic front view and corresponding top plan view of an electro-optic modulator employing the waveguide structure of Figure 17(a);
• Figures 44(a) and 44(b) are a schematic front view and corresponding top view of an alternative electro-optic modulator also using the waveguide structure of Figure 17(a);
• Figure 44(c) illustrates an alternative connection arrangement of the modulator of Figure 44(a);
• Figure 45 is a schematic front view of a third embodiment of electro-optic modulator also using the waveguide structure of Figure 17(a);
• Figure 46 is a schematic front view of a magneto-optic modulator also using the waveguide structure of Figure 17(a);
• Figure 47 is a schematic front view of a thermo-optic modulator also using the waveguide structure of Figure 17(a);
• Figure 48 is a schematic perspective view of an electro-optic switch also using the waveguide structure of Figure 17(a);
• Figure 49 is a schematic perspective view of a magneto-optic switch also using the waveguide structure of Figure 17(a);
• Figure 50 is a schematic perspective view of a thermo-optic switch also using the waveguide structure of Figure 17(a);
• Figure 51 gives the mode power attenuation for metal film waveguides of various widths and thicknesses.
• the metal used is Au and the background dielectric is SiO 2 .
• Figure 52 gives the mode power attenuation for metal film waveguides of various widths and thicknesses.
• the metal used is Al and the background dielectric is SiO 2 .
• the core can be embedded in an "infinite" homogeneous dielectric medium as shown in Figure 1(a) or supported by a semi-infinite homogeneous dielectric substrate and covered by a different semi-infinite homogeneous dielectric superstrate as shown in Figure 17(a).
• the description is organized as follows. Section II summarizes the physical basis and numerical technique used to analyze the structures of interest. Sections III through VI describe the modes supported by symmetric structures as shown in Figure 1(a) and sections VII through X describe the modes supported by asymmetric structures as shown in Figure 17(a). Concluding remarks are given in section XI.
• a symmetric structure embodying the present invention is shown in Figures 1(a) and 1(b). It comprises a lossy metal film of thickness t, width w and equivalent permittivity ⁇ 2 , surrounded by a cladding or background comprising an infinite homogeneous dielectric of permittivity ⁇ x .
• Figure 17(a) shows an asymmetric structure ( ⁇ x ⁇ ⁇ _) embodying the present invention.
• the Cartesian coordinate axes used for the analysis are also shown with propagation taking place along the z axis, which is out of the page.
• ⁇ is the excitation frequency
• ⁇ p is the electron plasma frequency
• ⁇ is homogeneous and taken as the permeability of free space ⁇ 0 .
• the power confinement factor is defined as the ratio of mode complex power carried through a portion of a waveguide's cross-section with respect to the mode complex power carried through the entire waveguide cross-section. Formally it is expressed as:
• a c is usually taken as the area of the waveguide core and A ⁇ implies integration over the entire waveguide cross-section (which can be all cross-sectional space for an open structure) or the entire cross-sectional computational domain.
• S z refers to the z component of the Poynting vector:
• H * denotes the complex conjugate of H x y .
• the spatial distribution of a component of the Poynting vector is easily computed from the spatial distribution of the relevant electric and magnetic mode field components.
• the boundary value problem governed by equations (4) to (11) is solved by applying the Method of Lines (MoL).
• MoL is a well-known numerical technique and its application to various electromagnetic problems, including optical waveguiding, is well-established [14].
• the MoL is rigorous, accurate and flexible. It can handle a wide variety of waveguide geometries, including the structures at hand.
• the method is not known to generate spurious or non-physical modes.
• the MoL formulation used herein is based on the formulation reported in [15], but simplified for isotropic media, as prescribed by equations (4) - (11) and reported in [16]. Except for a 1-D spatial discretization, the method is exact.
• the x axis and the function e(x) are discretized using two shifted non- equidistant line systems, parallel to the y axis.
• the discretized wave equations are diagonalized using appropriate transformations matrices.
• the diagonalization procedure yields in the transform domain two systems of uncoupled one-dimensional (1-D) differential equations along the remaining dimension (in this case along the y axis).
• a mode power confinement factor can be computed by first computing the spatial distribution of S z which is then integrated according to Equation
• the remaining horizontal boundary conditions are placed at infinity and the remaining lateral boundary condition is either placed far enough from the guide to have a negligible effect on the mode calculation, or a lateral absorbing boundary condition is used to simulate infinite space, depending on the level of confinement observed in the resulting mode.
• the use of numerical methods to solve differential equations inevitably raises questions regarding the convergence of computed results and their accuracy.
• the propagation constant of a mode computed using the method of lines converges in a monotonic or smooth manner with a reduction in the discretization interval (which increases the number of lines in the calculation and thus the numerical effort). This suggests that extrapolation can be used to generate a more accurate value for the propagation constant, and this value can then be used to compute the error in values obtained using the coarser discretizations [17].
• This anticipated error does not correspond to the actual error in the propagation constant as the latter could only be known if the analytic or exact value were available.
• the anticipated error still provides a useful measure of accuracy since it must tend toward zero as more accurate results
• the propagation constants of the a b and s b modes have been computed as a function of film thickness and the normalized phase and attenuation constants are plotted in Figures 2(a) and 2(b), respectively. From Figures 2(a) and 2(b), it is observed that the a,, and s b modes become degenerate with increasing film thickness. As the separation between the top and bottom interfaces increases, the a b and s b modes begin to split into a pair of uncoupled plasmon- polariton modes localized at the metal-dielectric interfaces. The propagation constants of the a b and s b modes thus tend towards that of a plasmon-polariton mode supported by the interface between semi-infinite metallic and dielectric regions, which is given via the following equations [6]:
• the phase and attenuation constants of the a b mode increase, becoming very large for very thin films. This is due to the fact that the fields of this mode penetrate progressively deeper into the metal as its thickness is reduced. In the case of the s b mode, a decreasing film thickness causes the opposite effect, that is, the fields penetrate progressively more into the top and bottom dielectric regions and less into the metal.
• the propagation constant of this mode thus tends asymptotically towards that of a TEM (Transverse ElectroMagnetic) wave propagating in an infinite medium having the same permittivity as the top and bottom dielectric regions. In this case, the attenuation constant decreases asymptotically towards zero since losses were neglected in these regions.
• TEM Transverse ElectroMagnetic
• Table 1 Vertical-Horizontal wall combinations used along the axes of symmetry and proposed mode nomenclature: ew - electric wall, mw - magnetic wall.
• a superscript is then used to track the number of extrema observed in the spatial distribution of this field component along the largest dimension (usually along the* axis) between the corners.
• a second superscript n could be added to track the extrema along the other dimension (the y axis) if modes exhibiting them are found.
• a subscript b o ⁇ I is used to identify whether the mode is bound or leaky. Leaky modes are known to exist in metal film slab structures and though a search for them has not been made at this time, their existence is anticipated. Table 1 relates the proposed mode nomenclature to the corresponding vertical and horizontal wall combinations used along the axes of symmetry.
• the ssl, sa l > ⁇ l an d aa l modes are the first modes generated (one for each of the four possible quarter-symmetries listed in Table 1, and having the largest phase constant) and thus may be considered as the fundamental modes supported by the structure.
• the main transverse electric field component is the E component and the symmetries in the spatial distribution of this component are reflected in the mode nomenclature.
• the outline of the metal is clearly seen in the distribution of the E y component on all of these plots.
• Figures 2(a) and 2(b) suggests that the dispersion curves for these first four modes converge with increasing film thickness toward the propagation constant of a plasmon-polariton mode supported by an isolated corner (though pairs of corners in this case remain weakly coupled along the top and bottom edges due to the finite width of the film, even if its thickness goes to infinity). If both the film thickness and width were to increase further, the four fundamental modes would approach degeneracy with their propagation constant tending towards that of a plasmon-polariton mode supported by an isolated corner, and their mode fields becoming more localized near the corners of the structure with maxima occurring at all four corners and fields decaying in an exponential-like manner in all directions away from the corners.
• the upper branch modes do not change in character as the film thickness decreases. Their field distributions remain essentially unchanged from those shown in Figures 4 and 6 with the exception that confinement to the metal region is increased thus causing an increase in their attenuation constant. This field behaviour is consistent with that of the a b mode supported by a metal film slab waveguide.
• the modes on the lower branch begin to split at a film thickness of about 80 «m, as shown in Figures 2(a) and 2(b).
• the ssl mo( l e follows closely the phase and attenuation curves of the S b mode supported by the metal film slab waveguide.
• the lower branch modes change in character with decreasing thickness, their fields evolving from being concentrated near the corners, to having Gaussian-like distributions along the waveguide width.
• the E y field component of the ss mo( le develops an extremum near the center of the top and bottom interfaces, while that of the asl mode develops two extrema, one on either side of the center. Since these modes change in character, they should be identified when the film is fairly thick.
• Figures 7(a) to 7(f) show the evolution of the ss m0Q fields with film thickness via contour plots of Re ⁇ S z ⁇ .
• S z is computed from the ssl m °de fields using Equation 13 and corresponds to the complex power density carried by the mode.
• the power confinement factor cf is also given in the figure for all cases, and is computed via equation (12) with the area of the waveguide core A_ taken as the area of the metal region.
• Figures 7(a) to 7(f) clearly show how the mode fields evolve from being confined to the corners of thick films to being distributed in a Gaussian-like manner laterally along the top and bottom edges, as the field coupling between these edges increases due to a reduction in film thickness.
• the confinement factor becomes smaller as the film thickness decreases, ranging from 14% confinement to 1.6% as the thickness goes from 80 «m to 20nm. This implies that fields become less confined to the metal, spreading out not only along the vertical dimension but along the horizontal one as well, as is observed by comparing Figures 7(a) and 7(b). This reduction in confinement to the lossy metal region explains the reduction in the attenuation constant of the mode with decreasing film thickness, as shown in Figure 2(b).
• Figure 8 shows that Re ⁇ S z ⁇ is negative in the metal film, implying that the mode real power is flowing in the direction opposite to the direction of mode propagation (or to the direction of phase velocity) in this region. It is clear however that the overall or net mode real power is flowing along the direction of propagation.
• the net mode real power can be made to flow in the direction opposite to that of phase velocity (as in metal film slab waveguides [10]) for values of e r l in the neighbourhood or greater than ⁇ Re ⁇ e ⁇ 2 ⁇ ⁇ .
• a metal film of finite width can support a number of higher order modes.
• the symmetries and number of extrema in the distributions of Re ⁇ E y ⁇ are reflected in the mode nomenclature.
• the evolution of the sal an ⁇ ⁇ l m0( les with film thickness is similar to the evolution of the sal anc aa l mo ⁇ es (and the a b mode supported by the metal film slab waveguide), in that their mode fields become more tightly confined to the metal as the thickness of the latter decreases, thereby causing an increase in the attenuation of the modes, as shown in Figure 2(b). Furthermore, the sal anc ac > mo ⁇ es ⁇ ° not change in character with film thickness, their field distributions remaining essentially unchanged in appearance from those computed at a thickness of lOOnm.
• the propagation constants of the sal a ⁇ ss l mo( ⁇ es converge to a single complex value as shown in Figures 2(a) and
• the aa b and sa " mo ⁇ e families do not have mode cutoff thicknesses. This is due to the fact that their confinement to the metal film increases with decreasing film thickness; thus the modes remain guided as t ⁇ 0.
• the as b and ss b mode families have cutoff thicknesses for all modes except the sl mo ⁇ le, which remains guided as t ⁇ 0, since it evolves into the TEM mode supported by the background.
• the other modes of these families, including the as mode cannot propagate as t ⁇ 0 because their mode fields do not evolve into a TEM mode. Rather, the modes maintain extrema in their field distributions and such variations cannot be enforced by an infinite homogeneous medium.
• the purely bound modes supported by a metal film of finite width appear to be formed from a coupling of modes supported by each metal-dielectric interface defining the structure.
• straight interfaces of finite length top, bottom, left and right edges
• corner interfaces are present. Since a straight metal-dielectric interface of infinite length can support a bound plasmon- polariton mode then so should an isolated corner interface and a straight interface of finite length bounded by corners (say the edge defined by a metal of finite width having an infinite thickness).
• Equations (14) and (15) A preliminary analysis of an isolated corner has revealed that a plasmon-polariton mode is indeed supported and that the phase and attenuation constants of this mode are greater than those of the mode guided by the corresponding infinite straight interface, as given by Equations (14) and (15). This is due to the fact that fields penetrate more deeply into the metal near the corner, to couple neighbouring perpendicular edges. All six field components are present in such a mode, having their maximum value at the corner and decreasing in an exponential-like manner in all directions away from the corner.
• a straight interface of finite length bounded by corners should support a discrete spectrum of plasmon-polariton modes with the defining feature in the mode fields being the number of extrema in their spatial distribution along the edge.
• a mode supported by a metal film of finite width may therefore be seen as being comprised of coupled 'corner modes' and 'finite length edge modes'.
• the s l mo ⁇ le could be used for optical signal transmission over short distances. Its losses decrease with decreasing film thickness in a manner similar to the s b mode supported by the metal film slab waveguide.
• the ssl mode does not have a cut-off thickness so losses could be made small enough to render it long-ranging, though a trade-off against confinement is necessary.
• the E y field component of the mode has a maximum near the center of the metal-dielectric interfaces, with a symmetric profile similar to that shown in Figure 8.
• the mode should be excitable using a simple end-fire technique similar to the one employed to excite surface plasmon-polariton modes [19,6]; this technique is based on maximizing the overlap between the incident field and that of the mode to be excited.
• a film width of 0.5 ⁇ m was selected in order to determine the impact of a narrowing film on the modes supported and to demonstrate that the structure can still function as a waveguide though the free-space optical wavelength is greater than both the width and thickness of the film.
• the ssl mode evolves with decreasing film thickness into the TEM wave supported by the background, but this evolution occurs more rapidly for a narrower width.
• Figures 12(a) to 12(d) clearly illustrate how the fields become less confined to the lossy metal as its width decreases, explaining the reduction in attenuation shown in Figure 11(b) at this thickness.
• the confinement factor ranges from 1.64% to 0.707% for the widths considered, further corroborating this fact.
• the fields are also seen to spread out farther, not only along the horizontal dimension but along the vertical one as well, as the film narrows. This indicates that the mode supported by a narrow film is farther along in its evolution into the TEM mode supported by the background, compared to a wider film of the same thickness. It is also clear from Figures 12(a) to 12(d) that the trade-off between mode confinement and attenuation must be made by considering not only the film thickness but its width as well.
• the changes in mode properties caused by varying the background permittivity as discussed above are consistent with the changes observed for the modes supported by a metal film slab waveguide and the observations are in general applicable to the other modes supported by a metal film of finite width.
• the higher order modes (m > 0) and those exhibiting a cutoff thickness (the as b modes for all and the ss b modes for m > 0) additional changes in the mode properties occur.
• the cut-off widths of the higher order modes increase as do all relevant cut-off thicknesses.
• the first is geometrical dispersion, which changes the optical or apparent size of the film
• the second is material dispersion, which is modeled for the metal region using Equation (1). If no material dispersion is present, then the geometrical dispersion renders the film optically smaller as the free-space wavelength is increased (an effect similar to reducing t and w) so, in the case of the ssl mo( le, confinement to the film is reduced and the mode spreads out in all directions away from the latter.
• Equation (1) it is clear that the magnitude of the real part of the film's permittivity
• reduces the penetration depth of the mode fields into the metal region and, combined with the geometrical dispersion, causes a net decrease in mode attenuation with increasing wavelength, even though the losses in the film increase in a 3 0 fashion.
• Figure 15(b) shows that mode power attenuation values in the range 10 to 0.1 dB/cm are possible near communications wavelengths ( ⁇ 0 ⁇ 1.5 ⁇ m) using structures of reasonable dimensions: w — l.O ⁇ m and t ⁇ 15nm. Such values of attenuation are low enough to consider the sl mode as being long-ranging, suggesting that these waveguides are practical for applications requiring propagation over short distances. As shown in the previous section, even lower attenuation values are possible if the background permittivity is lowered.
• the modes have a symmetric-like or asymmetric-like field distribution with field localization along either the bottom or top metal-dielectric interface.
• the modes that have a symmetric-like distribution with respect to the horizontal dimension are localized along the metal-dielectric interface with the lowest dielectric constant, while modes that have an asymmetric-like distribution with respect to this axis are localized along the metal-dielectric interface with the highest dielectric constant. This behaviour is consistent with that observed for asymmetric metal slab waveguides.
• the mode nomenclature adopted for symmetric structures can be used without ambiguity to describe the modes supported by asymmetric structures as long as the modes are identified when the metal film is fairly thick, before significant coupling begins to occur through the metal film, and while the origin of the mode can be identified unambiguously. As the metal film thickness decreases, the modes (and their fields) can evolve and change considerably more in an asymmetric structure compared to a symmetric one. The number of extrema in the main transverse electric field component of the mode is counted along the lateral dimension at the interface where the fields are localized. This number is then used in the mode nomenclature.
• the modes supported by symmetric structures are in fact supermodes created from a coupling of "edge” and "corner” modes supported by each metal-dielectric interface defining the structure. As the thickness and width of the metal decrease, the coupling between these interface modes intensifies leading to dispersion and possibly evolution of the supermode.
• the bound modes are also supermodes created in a similar manner, except that dissimilar interface modes may couple to each other to create the supermode. For instance, a mode having one field extremum along the top interface (along the top edge bounded by the corners) may couple with a mode having three extrema along the bottom interface.
• the main selection criterion determining which interface modes will couple to create the supermode is similarity in the value of their propagation constants. For all modes supported by an asymmetric structure, an apparent symmetry or asymmetry with respect to the horizontal dimension can still be observed in the corner modes.
• the sal, aal, ssl m ⁇ as l modes are the fundamental modes supported by the structure.
• the sal an ⁇ W modes are comprised of coupled corner modes, resembling the corresponding modes in a symmetric structure, except that the fields are localized near the substrate. These two modes do not change in character as the thickness of the film decreases. A narrowing of the metal film would eventually break the degeneracy observed in Figures 18(a) and 18(b).
• Figures 19(a) to 19(d) show the evolution of the E y field component related to the s l mode as the thickness of the film ranges from 100 nm ( Figure 19(a)) to 40 nm ( Figure 19(d)).
• the mode evolves from a symmetric-like mode having fields localized near the superstrate to an asymmetric-like mode having fields localized along the substrate-metal interface.
• a similar evolution is observed for the asl mode - This change in character is also apparent in their dispersion curves: they follow the general behaviour of a symmetric-like mode for large thicknesses but then slowly change to follow the behaviour of an asymmetric-like mode as the thickness decreases. Since the substrate dielectric constant is larger than the superstrate dielectric constant, the mode is ' 'pulled" from a symmetric-like mode to an asymmetric-like mode (having field localization at the substrate-metal interface) as the metal film becomes thinner.
• Figures 20(a) to 20(d) show the E y field component related to the ssl and al mode s for two film thicknesses. From these Figures it is noted that the top and bottom edge modes comprising a supermode are different from each other. In Figure 20(a), for instance, it is seen that the bottom edge mode has three extrema and is of higher order than the top edge mode which has one extremum. A similar observation holds for Figure 20(c), where it can be seen that the bottom edge mode has one extremum while the top one has none.
• the substrate has a higher dielectric constant than the superstrate so the phase constant of a particular substrate- metal interface mode will be higher than the phase constant of the same mode at the metal-superstrate interface.
• the sal mode is seen to comprise a substrate-metal interface mode having one extremum for a film thickness of 100 nm, while for a thickness of 60 nm the substrate- metal interface mode has three extrema, as shown in Figure 20(d).
• edge modes causes a reduction in the phase constant of the S a mode in the neighbourhood of 60 nm, as shown in Figure 18(a).
• Another change occurs near 40 nm as the corner modes switch from being symmetric-like (as in Figures 20(c) and 20(d)) to being asymmetric-like with respect to the horizontal dimension.
• This change is again reflected in the dispersion curve of the sal mo( le as its phase constant is seen to increase with a further decrease in thickness.
• the changes in the edge and corner modes are consistent with the directions taken by the dispersion curves as the film thickness decreases, thus explaining the oscillations in the curves seen in Figures 18(a) and 18(b).
• the spatial distribution of the main transverse field component related to this mode evolves with decreasing thickness in the manner shown in Figure 20(a) and 20(b), such that near cutoff the spatial distribution has strong extrema along the top and bottom edges. These extrema render the mode less excitable using an end-fire technique, so coupling losses would be higher compared to the fundamental symmetric mode in symmetric waveguides.
• the fact that the mode would be operated near its cutoff thickness implies that very tight tolerances are required in the fabrication of structures. Nevertheless, it should be possible to observe propagation of this mode in a suitable structure using an end-fire experiment.
• the substrate-metal interface modes comprising the ssl and as l modes are of very high order. This is expected since the substrate dielectric constant is significantly higher than the superstrate dielectric constant and higher order modes have lower values of phase constant.
• the ssl an ⁇ as l modes shown in Figures 22(a) and 22(b) indeed have fields that are localized along the metal- superstrate interface, while the sal and l mo ⁇ les shown in Figures 22(c) and 22(d) have fields that are localized along the substrate-metal interface.
• edge modes having similar values of phase constant
• edge modes comprising a supermode are likely to change or switch as the thickness of the film is reduced, as shown in Figures 20(c) and 20(d).
• some of the higher order edge modes may be cutoff, thus rendering changes in edge modes impossible.
• the supermode may be forced to evolve in a smooth manner with decreasing film thickness.
• the ssl an l sal modes are of practical interest.
• the ssl mo( le is the main long-ranging mode supported by symmetric finite- width metal film structures, and, as demonstrated in the previous section, the sal mode can be the main long-ranging mode supported by asymmetric finite-width structures. In metal films of the right thickness, they are also the modes that are the most suitable to excitation in an end-fire arrangement.
• Figure 24(b) shows that near cutoff, the attenuation of the sal m °d supported by an asymmetric structure drops much more rapidly than the attenuation of the ssl mo( le supported by a symmetric structure.
• a means for range extension similar to that observed in asymmetric slab structures [7], exists for metal films of finite width, though the difficulties related to the excitation of a mode near its cutoff thickness, as described in Section VII B, also apply here.
• Figures 26(a) to 26(d) show contour plots of 9? ⁇ S Z ⁇ associated with the long- ranging modes for the four cases of superstrate permittivity considered.
• Figures 26(b), (c) and (d) show contours associated with the a mo( le for the three cases of structure asymmetry considered.
• the contour plots shown in Figures 26(b), (c) and (d) are computed for film thicknesses slightly above cutoff, representative of the thicknesses that would be used to observe these long-ranging modes experimentally. From those figures, it is noted that the contour plots become
• Asymmetric structures having superstrate dielectric constants that are slightly greater than that of the substrate were also analyzed.
• the purely bound optical modes supported by thin lossy metal films of finite width, embedded in an "infinite" homogeneous dielectric have been characterized and described.
• the modes supported by these symmetric structures are divided into four families depending on the symmetry of their mode fields and none of the modes are TM in nature (as they are in the metal film slab waveguide).
• numerous higher order modes are supported as well.
• a proposed mode nomenclature suitable for identifying them has been discussed.
• the dispersion of the modes with film thickness has been assessed and the behaviour in general terms found to be consistent with that of the purely bound modes supported by the metal film slab waveguide.
• it has been found that one of the fundamental modes and some higher order modes have cut-off thicknesses.
• Mode dispersion with film width has also been investigated and it has been determined that the higher order modes have a cut-off width, below which they are no longer propagated.
• the effect of varying the background permittivity on the modes has been investigated as well, and the general behaviour found to be consistent with that of the modes supported by a metal film slab waveguide.
• the cut-off width of the higher order modes decreases with decreasing background permittivity and that all cut-off thicknesses are increased.
• the ss mo ⁇ exhibits very interesting characteristics and is potentially quite useful.
• This mode evolves with decreasing film thickness towards the TEM wave supported by the background, (an evolution similar to that exhibited by the s_ mode in metal film slab waveguides), its losses and phase constant tending asymptotically towards those of the TEM wave.
• decreasing the film width can reduce the losses well below those of the s b mode supported by the corresponding metal film slab waveguide. Reducing the background permittivity further reduces the losses.
• a reduction in losses is always accompanied by a reduction in field confinement to the waveguide core, which means that both of these parameters must be traded-off one against the other.
• the film's thickness and width can make the S sl mode the only long-ranging mode supported. It has also been demonstrated that mode power attenuation values in the range of 10 to 0.1 dB/cm are achievable at optical communications wavelengths, with even lower values possible. Finally, evolved into its most useful form, the ss ° mode has a field distribution that renders it excitable using end-fire techniques.
• the waveguide offers two- dimensional field confinement in the transverse plane, rendering it useful as the basis of an integrated optics technology. Interconnects, power splitters, power couplers and interferometers could be built using the guides.
• the structures are quite simple and so should be inexpensive to fabricate.
• the long-ranging modes supported by asymmetric structures of finite width have a rapidly diminishing attenuation near their cutoff thickness (like asymmetric slab structures). The rate of decrease of the attenuation with decreasing thickness near cutoff is greater than the rate related to the S l mode in symmetric structures.
• the waveguide structure 100 shown in Figures 1(a) and 1(b) comprises a strip of finite thickness t and width w of a first material having a high free (or almost free) charge carrier density, surrounded by a second material which has a very low free carrier density.
• the strip material can be a metal or a highly doped semiconductor and the background material can be a dielectric.
• Suitable materials for the strip include (but are not limited to) gold, silver, copper, aluminium and highly n- or p-doped GaAs, InP or Si, while suitable materials for the surrounding material include (but are not limited to) glass, quartz, polymer and undoped or very lightly doped GaAs, InP or Si. Particularly suitable combinations of materials include Au for the strip and SiO 2 for the surrounding material.
• the thickness t and the width w of the strip are selected such that the waveguide supports a long-ranging plasmon-polariton mode at the free-space operating wavelength of interest.
• Suitable dimensions for Au/SiO 2 waveguides at an operating free-space wavelength of 1550 nm are about 10 to 30 nm for the thickness t and about 2 to 12 ⁇ m for the width w; a thickness of 20 nm and a width of 4 ⁇ m are good dimensions.
• Figures 51 and 52 illustrate mode power attenuation for waveguides constructed from strips of gold (Au) and aluminium (Al), respectively, each embedded in silicon dioxide (SiO 2 ) for various widths and thicknesses of the metal film.
• the plasmon-polariton field may be excited by optical radiation coupled to the strip in an end-fire manner from a fiber butt-coupled to the input of the waveguide.
• the output of the waveguide can also be butt-coupled to a fibre.
• the waveguide could be excited at an intermediate position by an alternative means, for example using the so-called attenuated total reflection method (ATR).
• ATR attenuated total reflection method
• the length / shown in Figure 1(b) is arbitrary and will be selected to implement a desired interconnection. It has been demonstrated that a straight waveguide 100 with the dimensions set out above is polarisation sensitive.
• the plasmon-polariton wave is highly linearly polarised in the vertical direction, i.e. perpendicular to the plane of the strip. Hence, it may serve as a polarisation filter, whereby substantially only a vertical polarised mode (aligned along the y-axis as defined in Figure 1(a)) of the incident light is guided.
• Figure 28 shows an angled section 104 which can be used as an interconnect. Its dimensions, W 1; W 2 and 1 and the angles ⁇ , and ⁇ 2 , are adjusted for a particular application as needed. Usually the angles are kept small, in the range of 1 to 15 degrees and the input and output widths are usually similar, about 4 ⁇ m. Although the sides of the angled section 104 shown in Figure 28 are tapered, they could be parallel. It should also be appreciated that the angle of the inclination could be reversed, i.e. the device could be symmetrical about the bottom right hand corner shown in Figure 28 or transposed about that axis if not symmetrical about it.
• Figure 29 shows a tapered waveguide section 106, which can be used to interconnect two waveguides of different widths.
• the length of the taper is usually adjusted such that the angles are small, usually in the range of 1 to 15 degrees.
• the taper angles at the two sides are not necessarily the same.
• Such a configuration might be used as an input port, perhaps as an alternative to the layout shown in Figure 27, or as part of another device, such as a power splitter. Any symmetry of the structure shown can be used.
• Figure 30 illustrates an alternative transition waveguide section 130 which has curved sides, rather than straight as in the trapezoidal transition section disclosed in Figure 29.
• the curved sides are shown as sections of circles of radius R x and R 2 , subtending angles ⁇ and ⁇ _ respectively, but it should be appreciated that various functions can be implemented, such as exponential or parabolic, such that the input and output reflections are minimised.
• Figure 31 shows a curved waveguide section 108 which can be used to redirect the plasmon-polariton wave.
• the angle ⁇ of the bend can be in the range of 0 to 360 degrees and the bending radius R can be in the range of a few microns to a few centimetres. For a 45-degree bend, a radius of 0.5 to 2 cm is appropriate.
• the critical dimensions are the radius R and the positions of the input and output straight sections 100.
• Figure 31 shows no gradual transition between the straight waveguides 100 at the input and output and the ends of the curved section 108, it is envisaged that, in practice, a more gradual offset could be provided so as to reduce edge effects at the corners.
• Figure 32 shows a two-way power splitter 110 formed from a trapezoidal section 106 with a straight section 100 coupled to its narrower end 112 and two angled sections 104 coupled side-by-side to its wider end 114.
• the angle between the output waveguides 104 is usually in the range of 0.5 to 10 degrees and their widths are usually similar.
• the offsets S t and S 2 between the output waveguides and the longitudinal centre line of the trapezoidal section 106 preferably are set to zero, but could be non-zero, if desired, and vary in size. Ideally, however, the output sections 104 should together be equal in width to the wider end 114.
• Offset S x need not be equal to offset S 2 but it is preferable that both are set to zero.
• the widths of the output sections 104 can be adjusted to vary the ratio of the output powers.
• the dimensions of the centre tapered section 106 are usually adjusted to minimise input and output reflections and radiation losses in the region between the output sections 104.
• the centre tapered section 106 could have angles that vary according to application and need not be symmetrical. It is envisaged that the tapered section 106 could be replaced by a rectangular transition section having a width broader than the width of the input waveguide 100 so that the transition section favoured multimode propagation causing constructive/destructive interference patterns throughout its length. The length could be selected so that, at the output end of the rectangular transition section, the constructive portions of the interference pattern would be coupled into the different waveguides establishing, in effect, a 1-to-N power split. Such a splitter then would be termed a multimode interferometer-based power divider.
• the device shown in Figure 32 could also be used as a combiner.
• the light would be injected into the waveguide sections 104 and combined by the tapered centre section 106 to form the output wave which would emerge from the straight waveguide section 100.
• the number of arms or limbs 104 at the output could be far more than the two that are shown in Figure 32.
• an angled waveguide section 104 may be used to form an intersection between two straight waveguide section 100, with the dimensions adjusted for the particular application. It should be noted that, as shown in Figure 32, the two straight sections 100 are offset laterally away from each other by the distances Oj and O 2 , respectively, which would be selected to optimise the couplings by reducing radiation and reflection losses, in the manner discussed with reference to Figure 31. The angle of the trapezoidal section 104 will be a factor in determining the best values for the offsets Oj and O 2 .
• the sections 100 and 104 need not be connected directly together, but could be spaced by the distances dj and d 2 and/or coupled by a suitable transition piece that would make the junction more gradual (i.e., the change of direction would be more gradual).
• Figures 31 and 32 illustrate a general principle of aligning optical fields, conveniently by offsets, wherever there is a transition or change of direction of the optical wave and an inclination relative to the original path, which can cause radiation and reflection if field extrema are misaligned. Such offsets would be applied whether the direction-changing sections were straight or curved.
• a power divider 116 can also be implemented using a pair of concatenated curved sections 108 instead of each of the angled sections 104 in the splitter 110 shown in Figure 32.
• the curved section nearest to the wider end 114 of the tapered section 106 curves outwards from the longitudinal centre line of the tapered section 106 while the other curved section curves oppositely so that they form an "S" bend.
• the curved sections in each pair are offset by distance O x or O 2 one relative to the other for the reasons discussed with respect to the bend 108 shown in Figure 31.
• Figure 35 illustrates a Mach-Zehnder interferometer 118 created by interconnecting two power splitters 110 as disclosed in Figure 32. Of course, either or both of them could be replaced by the power splitter 116 shown in Figure 34.
• the insertion phase along one or both arms of the device is modified, then destructive interference between the re-combined waves can be induced.
• This induced destructive interference is the basis of a device that can be used to modulate the intensity of an input optical wave.
• the lengths of the arms 100 are usually adjusted such that the phase difference in the re-combined waves is 180 degrees for a particular relative change in insertion phase per unit length along the arms. The structure will thus be optically long if the mechanism used to modify the per unit length insertion phase is weak (or optically short if the mechanism is strong).
• Figure 36(a) illustrates a modulator 120 based on the Mach-Zehnder 118 disclosed in Figure 35.
• parallel plate electrodes 122 and 124 are disposed above and below, respectively, each of the strips 100 which interconnects two angled sections 104, and spaced from it, by the dielectric material, at a distance large enough that optical coupling to the electrodes is negligible.
• the electrodes are connected in common to one terminal of a voltage source 126, and the intervening strip 100 is connected using a minimally invasive contact to the other terminal. Variation of the voltage V applied by source 126 effects the modulating action.
• a change in the carrier density of the latter causes a change in its permittivity, which in turn causes a change in the insertion phase of the arm.
• the change induced in the permittivity is described by the plasma model representing the guiding strip 100 at the operating wavelength of interest. Such model is well known to those of ordinary skill in the art and so will not be described further herein. For more information the reader is directed to reference [21], for example.) This change is sufficient to induce destructive interference when the waves in both arms re-combine at the output combiner.
• Figure 36(c) illustrates an alternative connection arrangement in which the two plate electrodes 122 and 124 are connected to respective ones of the terminals of the voltage source 126.
• the dielectric material used as the background of the waveguide is electro-optic (LiNbO 3 , a polymer,).
• the applied voltage V effects a change in the permittivity of the background dielectric, thus changing the insertion phase along the arm. This change is sufficient to induce destructive interference when the waves in both arms re-combine at the output combiner.
• one voltage source supplies the voltage Vj while the other supplies the voltage V 2 .
• V x and V 2 may or may not be equal.
• Electrodes 122 and 124 and a source 126 for only one of the intervening strips 100 in order to provide the required interference. It should be appreciated that other electrode structures could be used to apply the necessary voltages.
• the electrodes 122 and 124 could be coplanar with the intervening strip 100, one on each side of it. By carefully laying out the electrodes as a microwave waveguide, a high frequency modulator (capable of modulation rates in excess of 10 Gbit/s) can be achieved.
• Figure 37 illustrates an alternative implementation of a Mach-Zehnder 128 which has the same set of waveguides as that shown in Figure 35 but which makes use of magnetic fields B applied to either or both of the middle straight section arms to induce a change in the permittivity tensor describing the strips.
• the change induced in the tensor is described by the plasma model representing the guiding strip at the operating wavelength of interest. Such model is well known to those of ordinary skill in the art and so will not be described further herein.
• the change induced in the permittivity tensor will induce a change in the insertion phase of either or both arms thus inducing a relative phase difference between the light passing in the arms and generating destructive interference when the waves re-combine at the output combiner. Modulating the magnetic field thus modulates the intensity of the light transmitted through the device.
• the magnetic field B can be made to originate from current-carrying wires or coils disposed around the arms 100 in such a manner as to create the magnetic field in the desired orientation and intensity in the optical waveguides.
• the magnetic field may have one or all of the orientations shown, B x , B y or B z or their opposites.
• the wires or coils could be fabricated using plated via holes and printed lines or other conductors in known manner. Alternatively, the field could be provided by an external source, such as a solenoid or toroid having poles on one or both sides of the strip.
• Figure 38 illustrates a periodic waveguide structure 132 comprising a series of unit cells 134, where each cell 134 comprises two rectangular waveguides 100 and 100' having different lengths l t and l 2 and widths w_ and w 2 , respectively.
• the dimensions of the waveguides in each unit cell 134, the spacing d ⁇ therebetween, the number of unit cells, and the spacings d 2 between cells are adjusted such that reflection occurs at a desired operating wavelength or over a desired operating bandwidth for an optical signal propagating along the grating axis, i.e. the longitudinal axis of the array of cells 134.
• the period of the periodic structure i.e.
• each unit cell, / ; + l 2 + d : + d 2 can be made optically long, such that a long-period periodic structure is obtained.
• the dimensions of the elements 100, 100' in each unit cell 134 can also be made to change along the direction of the periodic structure in order to implement a prescribed transfer function (like in a chirped periodic structure).
• each cell need not be rectangular, but a variety of other shapes could be used.
• Figure 39 illustrates a portion, specifically two unit cells 138 only, of an alternative periodic structure 136 in which each unit cell 138 comprises two of the trapezoidal waveguide sections 106, 106' like that described with reference to Figure 30, with their wider edges opposed.
• the trapezoidal waveguides 106/106' could be replaced by the transition sections 130, shown in Figure 30, with or without spacings d 1 and d 2 , to form a periodic structure having sinusoidally- varying sides.
• these periodic structures are merely examples and not intended to provide an exhaustive detailing of all possibilities; various other periodic structures could be formed from unit cells comprised of all sorts of different shapes and sizes of elements.
• voltages can be applied to some or all of the strips in order to establish charges on the strips of the unit cells, which would change their permittivity and thus vary the optical transfer function of the periodic structures. If the dielectric material surrounding the strip is electro-optic, then the applied voltages would also change the permittivity of the dielectric, which also contributes to changing of the optical transfer function of the periodic structure.
• Photonic bandgap structures can be created by placing two-dimensional arrays of unit cells (comprised of strips of various shapes and sizes) over numerous planes separated by dielectric material. The size and shape of the strips are determined such that stop bands in the optical spectrum appear at desired spectral locations.
• Figure 40(a) illustrates an edge coupler 139 created by placing two strips 100" parallel to each other and in close proximity over a certain length. The separation S c between the strips 100" could be from 1 ⁇ m (or less) to 20 ⁇ m and the coupling length L c could be in the range of a few microns to a few dozen millimeters depending on the separation S 0 , width and thickness of the strips 100", the materials used, the operating wavelength, and the level of coupling desired. Such a positioning of the strips 100" is termed "edge coupling".
• the gaps between the input and output of the waveguide sections shown would ideally be set to zero and a lateral offset provided between sections where a change of direction is involved. Curved sections could be used instead of the sections 104, 100 and 100" shown in Figure 40(a).
• FIG. 40(b) shows a voltage can be applied to the two edge-coupled sections 100" via minimally invasive electrical contacts.
• Figure 40(b) shows a voltage source 126 connected directly to the sections 100" but, if the sections 100, 104 and 100" in each arm are connected together electrically, the source 126 could be connected to one 5 of the other sections in the same arm. Applying a voltage in such a manner charges the arms of the coupler, which, according to the plasma model for the waveguide, changes its permittivity. If, in addition, the dielectric material placed between the two waveguides 100 is electro-optic, then a change in the background permittivity will also be effected as a result of the applied voltage. The first effect is sufficient to change the
• Figures 41(a) and 41(b) illustrate coupled waveguides similar to those shown in Figure 40(a) but placed on separate layers in a substrate having several layers 140/1,
• the strips could be placed one directly above the other with a thin region of dielectric of thickness d placed between them. Such positioning of the strips is termed "broadside coupling".
• the coupled guides can also be offset from broadside a distance S c , as shown in Figures 41(a)and 41(b).
• the separation S c could be in the range of -20 to +20 ⁇ m, depending on the width and thickness of the strips, the materials used and the level of coupling desired. (The negative indicates an overlap condition)
• 25 Gaps can be introduced longitudinally between the segments of strip if desired and a lateral offset between the straight and angled (or curved) sections could be introduced.
• a voltage source 126 could be connected directly or indirectly to the middle (coupled) sections 100" in a similar manner to that shown in Figure 40(b).
• an intersection 142 can be created by connecting together respective ends of four of the angled waveguide sections 104, their distal ends
• the waveguide structure shown in Figures 1(a) and 1(b), and implicitly those shown in other Figures have a single homogeneous dielectric surrounding a thin metal film, it would be possible to sandwich the metal film between two slabs of different dielectric material; or at the junction between four slabs of different dielectric material.
• the multilayer dielectric material(s) illustrated in Figure 41(a) could be used for other devices too.
• the thin metal film could be replaced by some other conductive material or a highly n- or p-doped semiconductor. It is also envisaged that the conductive film, whether metal or other material, could be multi-layered.
• Modulation and switching devices will now be described which make use of the fact that an asymmetry induced in optical waveguiding structures having as a guiding element a thin narrow metal film may inhibit propagation of the main long-ranging purely bound plasmon-polariton mode supported.
• the asymmetry in the structure can be in the dielectrics surrounding the metal film.
• the permittivity, permeability or the dimensions of the dielectrics surrounding the strip can be different.
• the dielectrics above and below the metal strip have different permittivities, in a manner similar to that shown in Figure 17(a).
• a dielectric material exhibiting an electro-optic, magneto-optic, thermo-optic, or piezo-optic effect can be used as the surrounding dielectric medium.
• the modulation and switching devices make use of an external stimulus to induce or enhance the asymmetry in the dielectrics of the structure.
• Figures 43(a) and 43(b) depict an electro-optic modulator comprising two metal strips 110 and 112 surrounded by a dielectric 114 exhibiting an electro-optic effect.
• a dielectric has a permittivity that varies with the strength of an applied electric field. The effect can be first order in the electric field, in which case it is termed the Pockels effect, or second order in the electric field (Kerr effect), or higher order.
• Figure 43(a) shows the structure in cross-sectional view and Figure 43(b) shows the structure in top view.
• the lower metal strip 110 and the surrounding dielectric 114 form the optical waveguide.
• the lower metal strip 110 is dimensioned such that a purely bound long- ranging plasmon-polariton wave is guided by the structure at the optical wavelength of interest. Since the "guiding" lower metal strip 110 comprises a metal, it is also used as an electrode and is connected to a voltage source 116 via a minimally invasive electrical contact 118 as shown.
• the second metal strip 112 is positioned above the lower metal strip 110 at a distance large enough that optical coupling between the strips is negligible. It is noted that the second strip also be placed below the waveguiding strip instead of above.
• the second strip acts as a second electrode.
• the intensity of the optical signal output from the waveguide can be varied or modulated by varying the intensity of the voltage V applied by the source 116.
• the waveguiding structure When no voltage is applied, the waveguiding structure is symmetrical and supports a plasmon- polariton wave.
• an asymmetry in the waveguiding structure When a voltage is applied, an asymmetry in the waveguiding structure is induced via the electro-optic effect present in the dielectric 114, and the propagation of the plasmon-polariton wave is inhibited.
• the degree of asymmetry induced may be large enough to- completely cut-off the main purely bound long-ranging mode, thus enabling a very high modulation depth to be achieved.
• a high frequency modulator capable of modulation rates in excess of 10 G bit/s
• Figures 44(a) and 44(b) show an alternative design for an electro-optic modulator which is similar to that shown in Figure 43(a) but comprises electrodes 112A and 112B placed above and below, respectively, the metal film 110 of the optical waveguide at such a distance that optical coupling between the strips is negligible.
• Figure 44(a) shows the structure in cross-sectional view and Figure 44(b) shows the structure in top view.
• a first voltage source 116A connected to the metal film 110 and the upper electrode 112 A applies a first voltage V x between them.
• a second voltage source 116B connected to metal film 110 and lower electrode 112B applies a voltage V 2 between them.
• the voltages V x and V 2 which are variable, produce electric fields E x and E 2 in portions 114 A and 114B of the dielectric material.
• the dielectric material used exhibits an electro-optic effect.
• the waveguide structure shown in Figure 44(c) is similar in construction to that shown in Figure 44(a) but only one voltage source 116C is used.
• the positive terminal (+) of the voltage source 116C is shown connected to metal film 110 while its negative terminal (-) is shown connected to both the upper electrode 112A and the lower electrode 112B.
• Figure 45 shows in cross-sectional view yet another design for an electro-optic modulator.
• the metal film 110 is embedded in the middle of dielectric material 114 with first portion 114D above it and second portion 114E below it.
• Electrodes 112D and 112E are placed opposite lateral along opposite lateral edges, respectively, of the upper portion 114D of the dielectric 114 as shown and connected to voltage source 116E which applies a voltage between them to induce the desired asymmetry in the structure.
• the electrodes 112D, 112E could be placed laterally along the bottom portion 114E of the dielectric 114, the distinct portions of the dielectric material still providing the asymmetry being above and below the strip.
• Figure 46 shows an example of a magneto-optic modulator wherein the waveguiding strip 110 and overlying electrode 112F are used to carry a current 7 in the opposite directions shown.
• the dielectric material surrounding the metal waveguide strip 110 exhibits a magneto-optic effect or is a ferrite.
• the magnetic fields generated by the current I add in the dielectric portion between the electrodes 110 and 112F and essentially cancel in the portions above the top electrode 112F and below the waveguide 110.
• the applied magnetic field thus induces the desired asymmetry in the waveguiding structure.
• the electrode 112F is placed far enough from the guiding strip 110 that optical coupling between the strips is negligible.
• Figure 47 depicts a thermo-optic modulator wherein the waveguiding strip 110 and the overlying electrode 112G are maintained at temperatures T 2 and T 7 respectively.
• the dielectric material 114 surrounding the metal waveguide exhibits a thermo-optic effect.
• the temperature difference creates a thermal gradient in the dielectric portion 114G between the electrode 112G and the strip 110.
• the variation in the applied temperature thus induce the desired asymmetry in the waveguiding structure.
• the electrode 112G is placed far enough from the guiding strip 110 that optical coupling between the strips is negligible.
• the modulator devices described above with reference to Figures 43(a) to 47 may also serve as variable optical attenuators with the attenuation being controlled via the external stimulus, i.e. voltage, current, temperature, which varies the electromagnetic property.
• Figures 48, 49 and 50 depict optical switches that operate on the principle of "split and attenuate".
• the input optical signal is first split into N outputs using a power divider; a one-to-two power split being shown in Figures 48, 49 and 50.
• the undesired outputs are then "switched off” or highly attenuated by inducing a large asymmetry in the corresponding output waveguides.
• the asymmetry must be large enough to completely cut-off the main purely bound long-ranging mode supported by the waveguides.
• the asymmetry is induced by means of overlying electrodes as in the waveguide structures of Figures 43, 46 or 47, respectively.
• the basic waveguide configuration is the same and comprises an input waveguide section 120 coupled to two parallel branch sections 122A and 122B by a wedge-shaped splitter 124. All four sections 120, 122A, 122B and 124 are co-planar and embedded in dielectric material 126.
• the thickness of the metal film is d 3 .
• Two rectangular electrodes 128A and 128B, each of thickness d x are disposed above branch sections 122 A and 122B, respectively, and spaced from them by a thickness d 2 of the dielectric material 126 at a distance large enough that optical coupling between the strips is negligible.
• Each of the electrodes- 128A and 128B is wider and shorter than the underlying metal film 122A or 122B, respectively.
• the asymmetry is induced electro- optically by means of a first voltage source 130A connected between metal film 122A and electrode 128A for applying voltage V x therebetween, and a second voltage source 130B connected between metal film 122B and electrode 128B, for applying a second voltage V 2 therebetween.
• the asymmetry is induced magneto-optically by a first current source 132A connected between metal film 122A and electrode 128A, which are connected together by connector 134A to complete the circuit, and a second current source 132B connected between metal film 122B and electrode 128B, which are connected together by connector 134B to complete that circuit.
• the asymmetry is induced thermo-optically by maintaining the metal strips 122A and 122B at temperature T 0 and the overlying electrodes 128A and 128B at temperatures T x and T 2 , respectively.
• the dielectric surrounding the metal strip will be electro-optic, magneto-optic, or thermo- optic, or a magnetic material such as a ferrite, as appropriate.
• any of the sources may be variable.
• the switches shown in Figures 48, 49 and 50 are 1 x 2 switches, the invention embraces 1 x N switches which can be created by adding more branch sections and associated electrodes, etc.
• the external stimulus used to induce or enhance the asymmetry could be determined by analogy.
• a structure similar to that shown in Figure 45 could be used with the electro-optic material replaced by acousto-optic material and the electrodes 112D and 112E used to apply compression or tension to the upper portion 114D.
• the various devices embodying the invention have been shown and described as comprising several separate sections of the novel waveguide structure. While it would be feasible to construct devices in this way, in practice, the devices are likely to comprise continuous strips of metal or other high charge carrier density material, i.e. integral strip sections, fabricated on the same substrate.
• Embodiments, of the invention may be useful for signal transmission and routing or to construct components such as couplers, power splitters/combiners, interferometers, modulators, switches, periodic structures and other typical components of integrated optics.
• BERINI, P. "Plasmon-Polariton Modes Guided by a Metal Film of Finite Width Bounded by Different Dielectrics", Optics Express, Vol. 7, No. 10, pp. 329-335.
• BERINI, P. "Plasmon-Polariton waves guided by thin lossy metal films of finite width: Bound Modes of Asymmetric Structures", Physical Review B, in Press. Not yet published.

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