Abstract:
Photonic integrated circuits required connection to germanium doped silica cored optical fibers or silica cored and fluorine doped silica cladding optical fibers which have low index contrast and large mode field diameters. However, the optical waveguide within a photonic integrated circuit such as formed using silicon-on-insulator or compound semiconductors tends to be high index contrast and small mode field diameter. Accordingly, it is necessary to implement adiabatic waveguide tapers with a high coupling efficiency and small footprint in order to couple into the photonic integrated circuits to/from the optical fiber. Prior art tapers have been generally high loss and absorb valuable die footprint. In contrast the inventors demonstrate a small low loss waveguide taper designed using a methodology they refer to a “constant loss”.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This patent application claims the benefit of priority from U.S. Provisional Patent Application 62/163,498 filed May 19, 2015, currently pending, entitled “Methods and Systems Relating to Optical Waveguide Tapers”, the entire contents of which are incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to optical waveguide tapers and more particularly to a new class of optical waveguide tapers that exhibit constant loss per taper segment with reduced loss to prior art tapers. 
     BACKGROUND OF THE INVENTION 
     Silicon photonics exploiting silicon photonic integrated circuits (PICs) relates to the application of photonic systems which use silicon as an optical medium. The silicon is usually patterned with sub-micrometer precision, into photonic components and photonic integrated circuits which operate in the infrared, most commonly at the 1550 nm (1.55 μm) wavelength window used by most fiber optic telecommunication systems as it is the lowest attenuation region for singlemode silica optical fibers. In reality the silicon typically lies on top of a layer of silica in what by analogy with a similar construction in microelectronics for transistors is known as silicon on insulator (SOI). 
     Silicon photonic devices exploit existing semiconductor fabrication techniques, and because silicon is already used as the substrate for most integrated circuits, it is possible to create hybrid devices in which the optical and electronic components are integrated onto a single microchip. Consequently, silicon photonics is being actively researched by many electronics manufacturers as well as by academic research groups. 
     Accordingly, within SOI waveguides the high index contrast between silicon (n SI =3.47), the underlying silica (silicon dioxide, SiO 2 , n SILICA =1.6), and either air (n AIR =1) or silica cladding, does not allow light, at 1550 nm, to extend into the silicon oxide beyond a fraction of a micron. The use of a rib loaded waveguide structure on the top surface of a silicon chip guides the optical power in the plane of the silicon as the effective refractive index in the region with the rib is higher than the adjacent regions without the rib. The resulting SOI waveguide is tightly confined vertically and loosely confined laterally and due to the index contrast Δn˜57% SOI waveguides have a small mode field diameter (MFD≈0.3 μm). 
     In contrast singlemode fiber (SMF) with a germanium doped silica core with silica cladding or silica core with fluorine doped silica cladding have a low index contrast, Δn≈0.036%, such that the typical MFD≈10.4 μm. Accordingly, when a silicon photonic device is to be interfaced to SMF optical fiber then the ratio of mode area ratio between SOI and SMF is approximately 1:1000 (SOI˜0.09 μm 2 :SMF˜85 μm 2 ). Accordingly, it is necessary to implement silicon-on-insulator-based adiabatic waveguide tapers with a high coupling efficiency and small footprint in order to couple into the silicon photonic integrated circuits from the SMF optical fiber or vice-versa. 
     Within the prior art linear and parabolic tapers have been described and implemented to provide such transitions onto and off of the silicon photonic integrated circuit. However, these tend to be high loss and absorb valuable die footprint such that it would be beneficial to provide SOI PIC designers with a taper geometry that offers lower insertion loss and smaller footprint than the prior art but maintains compatibility with standard manufacturing methodologies. 
     Other aspects and features of the present invention will become apparent to those ordinarily skilled in the art upon review of the following description of specific embodiments of the invention in conjunction with the accompanying figures. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to mitigate limitations in the prior art relating to optical waveguide tapers and more particularly to a new class of optical waveguide tapers that exhibit constant loss per taper segment with reduced loss to prior art tapers. 
     In accordance with an embodiment of the invention, there is provided a method comprising designing an optical waveguide taper by segmenting the optical waveguide taper into a number of segments such that the incremental loss between each consecutive pair of segments is approximately constant. 
     In accordance with an embodiment of the invention there is provided a waveguide comprising:
     an input waveguide having a first predetermined width;   an output waveguide having a second predetermined width; and   an optical waveguide taper tapering in width from the a first predetermined width to the second predetermined width, the optical waveguide taper comprising a number of segments of waveguide; wherein   the incremental loss between each consecutive pair of segments is approximately constant.   

     In accordance with an embodiment of the invention there is provided a device comprising a silicon-on-insulator (SOI) optical waveguide comprising an input of a first predetermined width and an output of a second predetermined width wherein the SOI optical waveguide varies according to a predetermined profile between the input and the output. 
     In accordance with an embodiment of the invention there is provided method of using a design rule for an optical waveguide structure in the design of said optical waveguide structures wherein the design rule relates to an optical waveguide taper such the optical loss versus propagation distance within the optical waveguide taper is approximately constant. 
     Other aspects and features of the present invention will become apparent to those ordinarily skilled in the art upon review of the following description of specific embodiments of the invention in conjunction with the accompanying figures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments of the present invention will now be described, by way of example only, with reference to the attached Figures, wherein: 
         FIG. 1A  depicts the effective mode area and effective index (n eff ) of the fundamental quasi-TE mode at λ=1550 nm for varying waveguide core widths at a core thickness of 220 nm; 
         FIGS. 1B to 1D  respectively depict the major component of the electric field of the guided mode is shown at w=150 nm, w=450 nm, and w=1000 nm respectively; 
         FIG. 2A  depicts the overlap integral between the modes of a 4.9 μm MFD fiber and the taper tip of varying widths for quasi-TE and -TM polarizations; 
         FIGS. 2B to 2D  depict photomicrographs for the fabricated 15 μm-long constant loss taper optimized for TE coupling, the distinctive feature of the constant-loss taper (CLT) that abruptly expands near the final 450 nm width, and the taper tip; 
         FIGS. 3A to 3C  depict pictograms showing various monotonically increasing tapering functions that present the limit cases of potential taper functions; 
         FIG. 3D  depicts a method of deriving the constant-loss taper according to an embodiment of the invention; 
         FIG. 4A  depicts the width of the linear, parabolic and constant-loss taper as calculated using the constant-loss methodology; 
         FIG. 4B  depicts the power in the guided mode versus propagation distance for the different taper functions; 
         FIGS. 4C to 4F  depicts the major component of the electric field at  4  different positions wherein  FIGS. 4C and 4F  represent the input (155 nm) and output (450 nm) respectively; 
         FIG. 5  depicts coupler efficiency as a function of the taper length with a 4.9 μm MFD fiber using a 3D FDTD simulation and measured efficiency for 15 μm long tapers; 
         FIG. 6  depicts the distribution of the coupler loss efficiency given roughness perturbations for a linear, parabolic, and constant-loss tapers respectively where the vertical annotated line show the reference coupler efficiency of an unperturbed taper. 
         FIGS. 7 and 8  depict the basis of coupled mode theory as applied in modelling a linear taper; 
         FIG. 9  depicts prior art linear and parabolic tapers together with a segmented constant loss taper according to an embodiment of the invention; 
         FIG. 10  depicts results of modelling prior art linear and parabolic tapers together with a segmented constant loss taper according to an embodiment of the invention; 
         FIGS. 11 and 12  depict cumulative loss and taper geometry versus position for prior art linear and parabolic tapers together with a segmented constant loss taper according to an embodiment of the invention; 
         FIGS. 13 and 14  depict taper width versus position for prior art linear and parabolic tapers together with a segmented constant loss taper according to an embodiment of the invention; 
         FIG. 15  depicts the results of finite difference time domain (FDTD) modeling for prior art linear and parabolic tapers together with a segmented constant loss taper according to an embodiment of the invention; 
         FIG. 16  depicts exploitation of constant loss tapers according to embodiments of the invention within waveguide bends and turning mirrors; 
         FIG. 17  depicts the exploitation of waveguide tapers for optical routing between optical waveguides with or without intermedia waveguides absent waveguide crossings and/or directional couplers; and 
         FIG. 18  depicts TE and TM loss surfaces for varying w i  versus w i+1  together with total loss and loss difference which may employed in establishing low polarization dependent tapers according to embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     The present invention is directed to optical waveguide tapers and more particularly to a new class of optical waveguide tapers that exhibit constant loss per taper segment with reduced loss to prior art tapers. 
     The ensuing description provides exemplary embodiment(s) only, and is not intended to limit the scope, applicability or configuration of the disclosure. Rather, the ensuing description of the exemplary embodiment(s) will provide those skilled in the art with an enabling description for implementing an exemplary embodiment. It being understood that various changes may be made in the function and arrangement of elements without departing from the spirit and scope as set forth in the appended claims. 
     A: Constant Loss Tapers 
     A1. Introduction 
     The large index of refraction contrast of a silicon-on-insulator (SOI) platform enables high density, cost-effective photonics on a chip. The typical interconnecting channel waveguide on 220 nm SOI has a 450 nm width resulting in a tightly confined mode profile. This type of waveguide is often used to connect integrated devices together since it minimizes the mode area of the waveguide, thereby enabling high-density optical circuitry while keeping the propagation loss small over short distances. However, at the junction between the interconnecting waveguide and the chip external coupling interface there is an intrinsic mode mismatch requiring mode conversion. Tapers are used to gradually convert the mode area and the effective index of the guided mode, typically by varying the waveguide width. In  FIG. 1A , the effective mode area (EMA) of a 220 nm thick SOI waveguide is shown as a function of the waveguide width (w) using a full-vectorial mode solver (FULL, solid line) at λ=1550 nm. 
     The top-left inset in  FIG. 1A  depicts the waveguide geometry with the main regions of interest: core, side, and corner. It is observed that the EMA is minimized for the single-mode condition at w=450 nm as evident in  FIG. 1A . Here, the EMA is defined by Equation (1) where S Z =(E×H)·{circumflex over (z)} is the Poynting vector, {circumflex over (z)} is the unitary vector along the waveguide&#39;s propagation axis. 
     
       
         
           
             
               
                 
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     Between points (w=450 nm)→(w=1000 nm), the EMA undergoes a direct, linear expansion as shown in the bottom inset of  FIG. 1A . During this process, the guided mode remains primarily confined in the silicon core ( FIGS. 1C and 1D ; FULL) since its effective index remains substantially larger than the cladding index, n CLAD =1.46. Because of this core-confined mode profile, the mode area can be well approximated by the dimensions of the core itself, i.e. EMA≅w×h, where h is the waveguide thickness and w is the waveguide width; thus explaining the linear mode expansion relationship. As a result of this linear behaviour, tapers used for a direct expansion almost exclusively use a linear tapering shape, other shapes have been considered showing only marginal improvements. Furthermore, from the evaluated mode profiles ( FIGS. 1C and 1D ; FULL) it can be seen that the amount of power located in the corner regions of the waveguide is negligible. This allows the use of the effective medium theory (EMT), which separates the field of the guided mode into Cartesian coordinates: ψ=ψ x ψ y . Using the EMT simplifies the structure to two simple one-dimensional slabs and reduces three-dimensional finite difference time-domain (FDTD) simulations to a less computational intensive two-point-five dimension (2.5D) FDTD. In  FIG. 1A , the EMT approximation (broken line) is applied in the direct expansion regime, with the calculated effective indices and mode profiles well approximated by the effective medium theory ( FIGS. 1C and 1D ; FULL vs EMT), thus confirming the validity of the approximation in this region. 
     From point (c) of  FIG. 1A  the mode area can also be increased through an indirect expansion by reducing the width of the waveguide. As w is reduced and reaches sub-wavelength dimensions, &lt;λ/2n, the guided mode can no longer be tightly localized. As a result, the effective index of the waveguide abruptly drops, approaching that of the cladding material, leading to a strong, non-linear expansion of the mode area as seen in the (c)→(b) region. This mode expansion mechanism is often used to design nanophotonic mode converters for fiber-waveguide coupling. However, the EMT can no longer be applied in this sub-wavelength regime since the amount of power in the waveguide corner regions exceeds 80%, as seen at w=150 nm in  FIG. 1B ; FULL vs EMT. Although the EMT has been used in the inverse expansion regime its use seriously compromises the results. Here, using a full vectorial solver, an inverse taper is designed for maximum coupling efficiency between a 4.9 μm mode-field diameter (MFD) fiber and a silicon waveguide (450 nm×220 nm) for TE polarized light at λ=1550 nm. 
     A2. Taper Tip Optimization 
     The coupling efficiency between an optical fiber and a SOI waveguide is determined by two main factors: the overlap integral of the electromagnetic fields between the fiber and the SOI waveguide at the tip of the taper, and the efficiency of the mode conversion process between the taper tip and the interconnecting waveguide.  FIG. 2A  shows the overlap integral between an optical fiber (MFD=4.9 μm; E i , H i ) and a silicon waveguide of different widths (E j ,H j ). The overlap integral, h, is defined by Equation (2).  FIG. 2B  depicts the taper as implemented together with the waveguide and tip ends in  FIGS. 2C and 2D  respectively. 
     
       
         
           
             
               
                 
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     The overlap integral is maximized for w 0  of 155 nm and 100 nm at η=0.92 (0.36 dB) for the quasi-TE and -TM modes, respectively. The overlap integral sets the upper limit of coupling efficiency since additional loss is expected as the mode propagates through the taper. In the calculation of the overlap integral, the magnetic field should not be simplified as a direct function of its electric counterpart because the SOI waveguide modes are not well approximated by propagating plane waves as they are quasi-TE/TM modes. Accounting for the magnetic components of the guided modes leads to the calculation of optimal tip widths that are 20-30 nm larger than those previously reported. 
     A3. Constant-Loss Framework 
     Now that the overlap integral is maximized at the taper&#39;s tip, the tapering function needs to be optimized for maximum light coupling efficiency between the tip and the interconnecting waveguide. The range of different, monotonically increasing taper functions to connect the taper tip (width, w 0 ) to the waveguide (width, w out ) in a given taper length, L, is bounded by the two taper functions shown in  FIGS. 3A and 3C . These piecewise functions are equivalent to each other, under the assumption that untapered waveguides are lossless. Given that these functions are abrupt; they represent the least efficient taper functions possible. In-between these cases an optimal taper function, that maximizes the efficiency can be inferred. Among the possible taper functions, the linear taper ( FIG. 3B ) or the parabolic functions are often used because of their simplicity but not necessarily optimal efficiency. To determine a taper shape with the highest possible efficiency, let us first define the total loss, L{w}, incurred by a taper function, w(z) and the instantaneous taper loss at point z as α(w,{dot over (w)},z). The total loss of the taper can be expressed by Equation (3). 
     The optimal taper function w(z) that minimizes L{w} can be derived by solving the Euler-Lagrange differential equation given by Equation (4). 
     Assuming that α does not depend explicitly on z, which is a reasonable hypothesis under the assumption that an untapered waveguide is lossless, e.g. the taper is not within an absorbing waveguide, and that w(z) is a monotonically varying function, then simplified Equation (5) is obtained. 
     
       
         
           
             
               
                 
                   
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     This implicit equation provides a clear guideline for designing an optimal taper function: the instantaneous loss at any point z within w(z) should be constant. Equation (5) establishes a general design principle that can be applied to engineer different types of photonic devices with varying shape functions. For an inverse taper, α(w,{dot over (w)}) depends on the coupling between the guided mode and the radiation modes. Given the infinite number of radiation modes, this calculation is computationally intensive. However, because at any point z along the taper only a single guided mode is supported, the instantaneous transmitted (T) and reflected (R) powers between the guided modes at z− and z+ can be tracked. This approach is complementary to computing the instantaneous power loss to the radiation modes, i.e. α=1−R−T.  FIG. 3D  schematically shows the inventive taper design approach of the inventors where the taper function is discretized into steps. The width of the first step, w 0 , is selected to maximize the overlap integral with the optical fiber mode ( FIG. 2A ); then, recursively, the taper is constructed by adding steps widths, δ i−1,i =|w i−1 −w i |, selected such that the loss penalty is constant, α i−1,i =α threshold . To compute the step loss, a coupled-mode theory is employed, leading to Equations ( 6 A) to ( 6 C) respectively. In practice, the reflection coefficient is several orders of magnitude smaller than the transmission coefficient and can often be neglected. 
     
       
         
           
             
               
                 
                   
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     A4. Results 
     A4.1. Power Tracking 
       FIG. 4A  compares the width of three different, 15-mm-long taper functions: linear, parabolic, and constant-loss, as calculated using the methodology presented in the previous section.  FIG. 4B  shows the power in the guided mode using a 3D FDTD simulation. 
     In  FIG. 4B , at a given position z, the total power (P z ) is compared to the Gaussian, 4.9 μm MFD source power, Γ z,source =P z /P source . Subsequently, the overlap integral between the field distribution and the local guided mode at z is calculated to obtain the power in the guided mode, P mode =Γ z,source ·η z,mode . In  FIG. 4B , all three tapers start with 92% of the source power guided in the tip mode ( FIG. 2 ). In the linear taper, light is substantially dissipated near the tip as the mode propagates along the first 4 μm of the taper. The parabolic taper, whose width increases more slowly initially, yields an improved taper function. The best taper function is the constant-loss taper in which the mode conversion loss is more evenly distributed along the taper, in spite of the fast tapering rate in the 13-15 μm range ( FIG. 4A ). The not quite linear power decay characteristic of the Constant Loss Taper (CLT) in  FIG. 4B  is likely because a simple coupled-mode theory formalism was used to implement the constant-loss formalism; use of a finer 3D FDTD mesh could also improve the linearity of the loss profile. Halfway through the taper, the electric fields along the constant-loss taper and linear taper are compared in  FIGS. 4D and 4E , respectively. In this region, the constant-loss taper is in an intermediate state of field delocalization while the mode conversion process of a linear taper is mostly completed, indicating that the constant-loss taper shape more efficiently uses the conversion length L available. The electric fields at the tip (w=155 nm) and waveguide (w=450 nm) are depicted in  FIGS. 4C and 4F  respectively. 
     A4.2. Taper Length Dependency 
       FIG. 5  shows the coupler efficiency as a function of taper length simulated in 3D FDTD. One can observe that the efficiency of the couplers increases with increasing taper lengths; furthermore, the CLT is found to be more efficient than either the parabolic or linear taper shape, for any taper length. It is worth nothing that the simulations that were used to compute the curves in  FIG. 5  did not include a finite buried oxide layer (BOX) and therefore radiation losses to the substrate are neglected; i.e.  FIG. 5  shows how the coupler efficiency is impacted by the mode transformation loss only. In actuality, radiation losses to the substrate would limit the maximal achievable efficiency; as the tapers are made longer, the gain in efficiency that a smoother mode transformation process brings diminishes, as seen in  FIG. 5  at longer taper length, while the radiative losses would steadily increase with length. For example, a 15-mm-long constant-loss taper has approximately the same efficiency mode transformation loss than a 22-mm; but the parabolic taper, being longer, would suffer from greater radiation losses. An important goal in designing practical edge coupling structures is to expand the mode size to match fibers with large mode field diameters in order to relax the fiber alignment tolerances. For typical BOX layer thicknesses used (2-3 μm) loss by light leaking into the substrate becomes a limiting factor to the overall coupling efficiency. These substrate losses can be minimized for a given mode size by minimizing the taper length, the constant-loss taper quantifies the miniaturization that can be achieved by optimal design of the widely used inverse taper couplers. 
     A4.3. Taper Robustness to Fabrication 
     In  FIG. 5 , the linear (and to a lesser extent the parabolic) taper exhibits a pronounced point-to-point jaggedness (four points are specifically shown). This is attributed to the 3D FDTD grid approximating the taper. Because the CLT&#39;s shape is close to minimizing the coupler&#39;s loss at each point, shape distortions incurred by the mesh affect less its efficiency compared to the linear shape. The mesh distortions effect is roughly analogous to the distortions that can be introduce in fabrication; therefore, the CLT is expected to be more robust to fabrication. 
     In order to further demonstrate the robustness of the constant-loss taper to fabrication, 15 μm-long tapers were perturbed using an exponential autocorrelation roughness model with a standard deviation of 25 nm and a autocorrelation length of 250 nm. A total of 100 perturbation profiles were generated for each coupler and then simulated in 3D FDTD.  FIG. 6  reports the normalized probability distribution of the coupler efficiency given our perturbation model. It can clearly be seen that not only is the constant-loss taper more efficient, but its distribution is more tightly grouped near the unperturbed taper&#39;s efficiency showing that the constant-loss taper is indeed more robust to fabrication than the linear or parabolic liner taper. The linear taper actually has some perturbed tapers (5 counts) that outperformed the unperturbed case; the efficiency of the linear taper is very sensitive to perturbation that occurs near its tip, as seen in  FIG. 4B , as some result perturbations that resulted in making the tip of the linear taper less abrupt resulted in a slightly better coupler. The median coupler efficiencies are 5.0, 4.6, and 3.5 dB respectively for a linear, parabolic, and constant-loss taper. In an industrial setting, producing components that have increased robustness to fabrication is key in improving one&#39;s fabrication yield, use of a constant-loss taper would result in improved taper fabrication robustness. 
     A4.4. Experimental Results 
     Inverted tapers were fabricated using a 220 nm silicon-on-insulator wafer with 3 mm buried oxide layer. The structures were defined using a hydrogen silsesquioxane (HSQ) resist with a inductively coupled, SF6-C4F8 plasma etching and covered in a 3 mm thick plasma enhanced chemical vapor deposition (PECVD) oxide. In  FIG. 5 , at an approximately 15 μm taper length, the expected efficiency of a constant-loss taper is noticeably higher compared to a linear or parabolic taper, therefore 15 μm long tapers were fabricated. Using a tunable laser and a polarization controller, light was coupled to a polarization-maintaining, 5 μm MFD or 3 μm MFD, lens fiber. The insertion loss of a single coupler was determined by measuring the total insertion loss to the chip, the total loss is composed of the loss from the input coupler, the waveguide, and the output coupler. The loss from the waveguide was de-embeded from the total insertion by measuring the waveguide loss through the cut-back method. Specifically, the waveguide loss was measured at 1.1 dB/cm or 0.53 dB for our 480 μm long waveguide. This low waveguide loss enables precise measurement of the coupler&#39;s loss given that only 0.53 dB had to be deducted from the total insertion loss measurement in order to receive the couplers&#39; loss. 
     The measured efficiencies are reported in  FIG. 5  for the 5 μm MFD fiber. The CLT is the best taper at 1.95 dB followed the parabolic (2.21 dB), and linear (3.25 dB). Measured efficiencies are slightly higher than simulated and this is attributed to the lens fiber used that collimates the beam to a 5 μm spot. Using a 3 μm MFD fiber, the measured efficiencies improved respectively to 0.56 dB, 0.84 dB, and 0.95 dB. Use of the smaller MFD fiber resulted in better efficiency since radiation losses to the substrate were not as dominant as they are with a larger MFD fiber. Specifically, the efficiency of the constant-loss taper measured at 0.56 dB is remarkably high given its short 15 μm length. 
     B. Segmented Constant Loss Tapers 
     Referring to  FIGS. 7 and 8  there are depicted the basis of exploiting coupled mode theory as applied to the modelling of a linear taper. However, it would be evident that coupled mode theory represents one formalism to calculate the losses throughout the tapers according to embodiments of the invention and design/model such tapers. Others may be employed as would be evident to one skilled in the art including, for example, finite-difference time domain (FDTD) and beam propagation method (BPM). Further, in some instances combinations of models may be employed. 
     As depicted the waveguide taper starts from an initial width W( 1 )=20 nm and over its length L increases in taper width to W( 2 )=250 nm. The resulting linear taper is then broken into a number of sections each of an intermediate width. At the transition from one waveguide section  710  to another waveguide section  720  then there are the following optical signals to consider:
         1) The optical signal propagating within the current section;   2) The optical signals coupled forward and guided by the next section;   3) The optical signals coupled forward but not guided by the next section;   4) The optical signal reflected at the segment boundary and guided by the current section; and   5) The optical signal reflected at the segment boundary but not guided by the current section.       

     Accordingly, each of these is associated with a term in the equation depicted in  FIG. 7  defining the boundary between the i th  and (i+ 1 ) th  segments wherein the terms r and r(β) relate to the reflected portions and t and t(β) relate to the transmitted portions. Now using mode orthogonality then as depicted within  FIG. 8  the terms r and t can be defined in terms of propagation constants β and modal electric fields E(x) and the loss at each interface determined. 
     Referring to  FIG. 9  there are depicted prior art linear taper  900 A and parabolic taper  900 B together with a segmented constant loss taper  900 C according to an embodiment of the invention. Mathematically the prior art linear taper  900 A is defined such that the waveguide width w(x) is defined by w(x)=a·x+b such that for x=0 w( 0 )=w IN  and that for x=L w(L)=w OUT . In contrast the prior art parabolic taper  900 B is defined such that the waveguide width w(x) is defined by w(x)=a·x 2 +b·x+c such that again the same start and end boundary conditions of w( 0 )=w IN  and w(L)=w OUT  are met but now we also have dw/dx| w=0 =0. In contrast the Segmented Constant Loss Taper (SEG-CLT) taper  900 C is defined such that step_loss(w i ;w i+1 )=X, a constant and accordingly based upon the rate of loss with width δsteploss(w i ;w i+1 )/δw i+1  such that if this is large then the taper is implemented with a large number of small step transitions but if this is small then the taper is implemented with a small number of larger step transitions. 
     Now referring to  FIG. 10  there are depicted 2D simulation results of modelling prior art linear taper  1010  and parabolic taper  1020  together with a SEG-CLT taper  1030  according to an embodiment of the invention wherein the number of steps within the tapers were set to 45 in each case. Accordingly, it can be seen that the SEG-CLT taper loss versus taper section is constant as expected whilst that from the linear taper follows an approximately inverse exponential curve where in the initial segment transitions are very lossy but the later transitions are low loss. In contrast to these the parabolic taper starts and ends with low losses for segment transitions but in the first half rises in loss per segment before reducing. Accordingly, the accumulated loss after each segment is displayed in  FIG. 11  wherein the accumulated loss of the linear taper  1010 , parabolic taper  1020  and SEG-CLT taper  1030  are 4.2% (0.19 dB), 2.1% (0.09 dB) and 0.16% (0.07 dB) respectively. The resulting profiles of the three different waveguide tapers are depicted in  FIG. 12  wherein it can be seen that whilst the loss per segment transition is constant,  FIG. 11 , the width rate of change varies substantially with the largest rate at the larger widths towards the end of the taper. These are depicted as the evolution of half-width of the waveguide for linear taper  1010  and parabolic taper  1020  in  FIG. 13  and the SEG-CLT taper  1030  in  FIG. 14 . 
     The inventors have also modelled the different tapers using finite difference time domain (FDTD) modeling for prior art linear and parabolic tapers together with a SEG-CLT according to an embodiment of the invention. The resulting modelled geometry for a SEG-CLT taper is depicted in  FIG. 15  in first graph  1500 A whilst second graph  1500 B depicts the FDTD modelling results for the three tapers wherein the resulting losses for the linear taper  1010 , parabolic taper  1020  and SEG-CLT taper  1030  were 37.6% (2.05 dB), 31.6% (1.65 dB) and 22.3% (1.1 dB) respectively. 
     C: Extensions of Constant Loss Tapers and Segmented Constant Loss Tapers 
     It would be evident to one skilled in the art that whilst the embodiments of the invention have been presented based upon SOI waveguides that alternate embodiments may be implemented with the same waveguide material system and other material systems. For example, other material systems including, but not limited to, SiO 2 , Si 3 N 4 , SiO X N Y , Si, LiNbO 3 , InP, InGaAsP, GaAs, AlGaAs, GaInN, AlInGaP, GaInNAs, and polymers may be employed. 
     It would be evident to one skilled in the art that whilst the embodiments of the invention have been presented based upon optical waveguides within the near infrared (λ=1550 nm) that alternate embodiments may be implemented at other optical wavelengths according to the waveguide material(s) from ultra-violet to far infra-red. Further embodiments of the invention may be implemented for microwave waveguides and other electromagnetic waveguides. 
     It would be evident to one skilled in the art that the optical waveguides exploiting embodiments of the invention may be formed through a range of techniques including, but not limited to, material composition, rib-loading, ridges, doping, ion-implantation, and ion-exchange. 
     It would be evident to one skilled in the art that optical waveguides exploiting X-on-insulator may include, but not be limited to, silicon, germanium, silicon nitride—silicon, intrinsic BOX layers, fabricated BOX layers, and silicon-oxide clad silicon. 
     Further, the embodiments of the invention may be exploited to provide short low loss waveguide tapers in order to provide enhanced small footprint bends and 90° bends. Accordingly, a SEG-CLT or CLT may be employed as depicted in  FIG. 16  wherein in first schematic  1600  an input waveguide  1610  is coupled to an output waveguide  1650  via an etched reflector  1630  wherein input and output tapers  1620  and  1640  respectively, which may be SEG-CLT and/or CLT, provide mode expansion/reduction to/from the reflector  1630  thereby reducing the impact of surface roughness and increasing tolerances on the formation of the reflector  1630 . Optionally, the reflector  1630  may be metallized on its vertical surface adjacent the waveguides to increase reflectivity. Further, whilst it is known to employ waveguide bends that are dual-circular arc segments these present increased losses due to the discontinuous radius at each straight waveguide interface and at their junction in the middle. Whilst it is also known to employ waveguide bends which exploit a mathematical function which either continuous in first differential of radius with position or continuous in first and second differentials in radius with position the inclusion of waveguide tapers such as SEG-CLT and CLT has not been considered such that the waveguide width can be also varied with position to provide increased/reduced confinement as desired around the waveguide bend. For example, in the transition around the bend increased confinement may be desired whilst at the straight waveguide interfaces reduced confinement so that low losses result, in fact the straight waveguide may taper as well. If two waveguides are intended to cross, then reduced confinement may be beneficial and accordingly SEG-CLT/CLT tapers may provide short transitions at the crossing as well. 
     Within silicon microphotonics, optical interconnection almost exclusively occurs in plane, since Si microphotonics are based on the low-loss Si layer of an SOI substrate. As optical devices and interconnection becomes denser it becomes necessary to build efficient cross links in order to transport an optical signal across another optical signal. Currently, this is achieved through X-crossings or directional couplers. However, in many instances an optical waveguide must cross multiple waveguides to achieve the desired circuit layout which requires multiple waveguide crossings or multiple directional couplers which also move the intermediate optical signals to different optical waveguides. In contrast, the inventors have demonstrated the use of waveguide tapers, e.g. CLT tapers, to achieve optical coupling over larger distances between two optical waveguides than previously achieved. Further this optical coupling between the optical waveguides can be achieved over an intermediate optical waveguide or waveguides without physically crossing the waveguides or using multiple directional couplers. 
     Referring to  FIG. 17  with first and second schematics  1700 A and  1700 B then the waveguide tapers are exploited within the first waveguide  1710  and second waveguide  1720  wherein it is sought to couple from the first waveguide  1710  to the second waveguide  1720  and/or vice-versa. As, by tapering the waveguide, we are able to expand the effective mode area of the guided mode by up to 3 orders of magnitude then we can enable the optical modes of the coupled lines to strongly overlap despite the large separating distance. Moreover, by having the tapers of both coupled lines aligned we keep the optical modes phase matched, which could result in near 100% of coupling efficiency. However, these large effective modes will only couple minimally to a central waveguide  1730  or waveguides  1740 (A) to  1740 (N) in the center since the effective mode area of the center waveguide is so tightly confined and can be designed such that it is not phase matched. Further, as depicted in third schematic  1750  this coupling can be employed to couple between an upper waveguide  1760  and lower waveguide  1780  and/or vice-versa through the intermediate lower cladding  1770  of the multiple stack waveguide geometry. The lower waveguide  1780  having also a lower cladding  1790  between it and the substrate  1795 . As depicted in third schematic  1750  the upper and lower waveguides  1760  and  1780  are depicted with uniform thickness as the tapers are implemented within their width. However, in other embodiments of the invention the waveguide thickness and/or composition may be varied discretely or in combination with lateral waveguide geometry variations. 
     Within many optical circuits it is important that the circuit have equal TE and TM polarization dispersion and/or performance at least in part or fully. However, this is difficult to achieve in many instances as TE and TM polarized light travel at different velocities within the waveguide and is seen within Si microphotonic waveguides as well as those in compound semiconductors, ferroelectrics, etc. However, the inventors have established that it is possible to engineer a dispersionless taper. For example, within a SOI waveguide when the width (w) of the on-chip waveguide is smaller than the thickness of the SOI (h SOI ), i.e. w&lt;h SOI , then the effective index of the TM mode is larger than the TE mode. However, when w&gt;h SOI  the effective index of the TE mode is then larger than the TM. Thus, by carefully engineering the shape of a taper that spans waveguide widths that are both smaller and larger than h SOI  it is possible to engineer the taper in such a manner that the TE and TM modes exit the taper phase matched or with a predetermined phase shift that counters the phase shift introduced by a following polarisation splitter/rotator or any other TE/TM phase shifting element. 
     However, in addition to the phase matching it is important that the TE and TM losses are also balanced for ensuring polarisation independent performance. Accordingly, referring to  FIG. 18  there are depicted in first and second graphs  1800 A and  1800 B the TE and TM step losses are presented as calculated using the overlap integral of the electromagnetic fields between the fiber and the SOI waveguide at the tip of the taper, and the efficiency of the mode conversion process between the taper tip and the interconnecting waveguide. Basically, these are the losses (z-axis) as a function of the width (w i ) in nm and the difference in the width (w i+1 ) in nm. For example, in the TE step losses, a taper of initial width w i =60 nm, is to change by 10 nm then the loss that would incurred would be that value on the surface for w i =60 nm and w i+1 =70 nm. As evident from first and second graphs  1800 A and  1800 B the TE losses are higher for the lower widths whilst the TM losses are higher for larger widths. This is also evident in fourth graph  1800 D where the difference between TE and TM losses is plotted. Third graph  1800 C depicts the total step loss. 
     Accordingly, in order to design a low-loss dispersionless taper, i.e. total TE losses=total TM losses, then the taper shape should be established such that when the total TE and total TM losses are calculated they are equal. As such this means limiting designs to taper shapes in which the TE and TM losses are approximately equal and then choosing those taper shapes which have a low total loss. One method to achieving such a design goal would be to use a Monte Carlo approach to generate different taper shapes which will traverse the TE/TM curves such as depicted in first and second graphs  1800 A and  1800 B in  FIG. 18  and keep those taper profiles which have a low loss and have approximately equal TE and TM losses. It would be evident that other computational algorithms exploiting repeated random sampling may be employed to perform the optimization process. 
     It would be evident to one skilled in the art that embodiments of the invention may be derived from an iterative design and simulation process wherein an initial design is iterated in a number of segments and/or loss per segment and/or width variation per segment until a constant loss per segment transition is achieved within a predetermined tolerance. 
     The foregoing disclosure of the exemplary embodiments of the present invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Many variations and modifications of the embodiments described herein will be apparent to one of ordinary skill in the art in light of the above disclosure. The scope of the invention is to be defined only by the claims appended hereto, and by their equivalents. 
     Further, in describing representative embodiments of the present invention, the specification may have presented the method and/or process of the present invention as a particular sequence of steps. However, to the extent that the method or process does not rely on the particular order of steps set forth herein, the method or process should not be limited to the particular sequence of steps described. As one of ordinary skill in the art would appreciate, other sequences of steps may be possible. Therefore, the particular order of the steps set forth in the specification should not be construed as limitations on the claims. In addition, the claims directed to the method and/or process of the present invention should not be limited to the performance of their steps in the order written, and one skilled in the art can readily appreciate that the sequences may be varied and still remain within the spirit and scope of the present invention.