Patent Application: US-62865509-A

Abstract:
a low - loss waveguide that can be curved aggressively , that is , curved with a radius of curvature that is substantially zero , in the plane of propagation , without radiating , is formed by a slab of dielectric material having four metal plates , two on each opposite surface of the slab and mutually spaced to define in the dielectric slab between the four metal plates a confinement zone . in use , electromagnetic radiation injected in one end of the zone by suitable input means will propagate throughout the zone to an extraction means . lower loss and better confinement of the radiation may be obtained by providing plugs of dielectric material adjacent the inwardly - facing edge of each of the metal plates . embodiments of the invention can be used to implement integrated optical devices and circuits for routing or processing light signals .

Description:
throughout this disclosure , and as a matter of convenience , waveguides embodying the present invention are referred to as metallo - dielectric waveguides ( mdws ), given the prevalence of metals and dielectrics in their construction . fig2 a through 2e illustrate in the cylindrical coordinate system ( ρ - φ - z ) preferred embodiments of mdws . fig2 a illustrates a front cross - sectional view of the mdw . the mdw comprises a dielectric slab 10 of thickness s and refractive index n 1 . the dielectric slab is clad in region i by dielectrics d of lower refractive index n 2 ( n 2 & lt ; n 1 ), and in regions ii by parallel metal plates 12 l and 12 r of thickness t and refractive index n m . the pair of metal plates 12 l on opposite sides of slab 10 ( shown at the left hand side in fig2 a ) are in register with each other . likewise , the pair of metal plates 12 r ( shown at the right hand side in fig2 a ) on opposite sides of the slab 10 are in register with each other . region i is referred to as the “ dielectric waveguide ” region , and regions h as the “ parallel - plate waveguide ” regions . the mdw is a composite waveguide formed by combining a dielectric waveguide ( region i ) with two parallel - plate waveguides ( regions ii ) separated laterally by w ; w and s are henceforth referred to as the mdw width and thickness , respectively . four additional dielectric regions 14 , each of width w b , thickness t and refractive index n b are added near the metal plates in the locations shown ; these regions are henceforth referred to as “ dielectric plugs ”. the top and bottom claddings d are shown here as having the same refractive index n 2 , but could in fact comprise different materials each having a different refractive index as long as they are both lower than n 1 , the refractive index of the dielectric slab 10 . likewise , the metal plates 12 l and 12 r could each comprise a different metal , each having a different refractive index . the dielectric plugs 14 could each comprise a different dielectric , each having a different refractive index . fig2 b illustrates in top view a 90 ° curved section of the mdw having an arbitrary radius of curvature r 0 . fig2 c illustrates in top view a 90 ° curved section of the mdw having an aggressive radius of curvature r 0 → 0 . fig2 d illustrates in top view a straight section ( r 0 →∞) of mdw . fig2 e illustrates in isometric view a 90 ° curved section of mdw with an arbitrary radius of curvature r 0 . any radius of curvature ( 0 ≦ r 0 ≦∞) and any curve angle between 0 ° and 90 ° can be implemented using the mdw . fig1 a through 15e illustrate alternative embodiments of mdws , similar to those sketched in fig2 a through 2e respectively , except without the dielectric plugs 14 of the latter . in order to provide insight on the operation of the mdw , the dielectric waveguide ( region i ) and the parallel - plate waveguide ( regions ii ) were analyzed independently as the 1 d waveguides sketched in fig3 a and 3b . the computations were performed using the transfer matrix method [ 24 ] at a free - space operating wavelength of λ 0 = 1550 nm . the dielectric slab 10 used as the core of the dielectric waveguide ( fig3 a ) and as the core of the parallel - plate waveguide ( fig3 b ) was assumed to be an isotropic dielectric of index n 1 = 2 . 1375 , representative of si 3 n 4 ( n ˜ 2 . 0 ) and other high index dielectrics and crystals . the upper and lower claddings d of the dielectric waveguide ( fig3 a ) were assumed to be isotropic dielectrics of index n 2 = 1 . 444 , representative of sio 2 , optical polymers , and other dielectrics . the metal plates 12 r of the parallel - plate waveguide ( fig3 b ) were assumed to be highly conductive , comprising , for example , au , having an index n m =( 131 . 95 − j12 . 65 ) 0 . 5 , and of thickness t much greater than the field penetration depth therein . fig4 a and 4b plot the computed effective index ( n eff ) of the first few te and tm modes that exist in the dielectric waveguide ( dark solid curves ), and in the parallel - plate waveguide ( dashed gray curves ), as a function of the dielectric slab thickness s . the effective index of the single interface spp guided along the au - n 2 interfaces of the parallel - plate waveguide is also plotted in fig4 b , for reference . the distribution of the main transverse electric field component of these modes is sketched onto fig3 a and 3b . from fig4 a and 4b it is noted that the effective index of the te 0 mode of the dielectric waveguide remains above that of all other modes shown ( the single interface spp at the au - n 2 interfaces and the parallel plate modes te 1 , te 2 , a b 0 , s b 1 ) for s & gt ; 37 nm , except for the parallel - plate s b 0 mode . the fundamental horizontally - polarized mode of the mdw ( fig2 a ) is denoted e ρ 11 , and resembles in character , polarization and effective index the te 0 mode of the dielectric slab , so the results plotted in fig4 a and 4b help elucidate its confinement mechanisms . the e ρ 11 mode is confined horizontally to the width w of region i by the parallel plate waveguides ( regions ii ), because its effective index is larger than that of all parallel - plate modes and the single - interface spp at the au - n 2 interfaces . the parallel - plate s b 0 mode is of no concern in this case because it is substantially orthogonal to the e ρ 11 mode . the e ρ 11 mode is confined vertically to the thickness s in region i by the steps in refractive index at the top and bottom interfaces between the slab ( 10 ) and the claddings ( d ). the e ρ 11 mode therefore occupies the area w × s ( roughly ) of region i . the mdw produces both vertical and horizontal confinement for this mode for any thickness s . the width of the parallel - plate waveguides ( regions ii ) is selected to be large enough for them to be optically infinite . these confinement mechanisms hold for straight ( fig2 d ) and curved ( fig2 b ) mdws in general , but if a thickness of s & lt ; 300 nm is chosen for the case modeled in fig4 a and 4b , then aggressive radii of curvature ( r 0 → 0 and r 0 = 0 , fig2 c ) can be used because all of the parallel - plate modes are cut off , leaving only the orthogonal s b 0 mode . in this case , the parallel plates effectively reduce the optical density of states in the plane of propagation such that there are no modes for the e ρ 11 mode to radiate into . thus , in general , it is particularly desirable to select the thickness s of the dielectric slab 10 to cut - off as many of the parallel - plate modes as possible . under this condition , the mdw is said to be “ substantially non - radiative ”. the straight mdw ( r 0 →∞, fig2 d ) was then modeled ; the computations were performed using the finite element method and the method of lines [ 7 ]. in an attempt to reduce the propagation loss of the e ρ 11 mode , small dielectric regions of relatively high or low refractive index were placed strategically at many different locations in the mdw cross - section . after much modeling , it was found that placing relatively small high - index plugs ( 14 ) along the edge of the metal plates ( 12 l and 12 r ), as shown in fig2 a , reduced the propagation loss by at least one order of magnitude . fig5 a and 5b give the effective index and propagation loss , respectively , of the e ρ 11 mode of the straight mdw as a function of the plug width w b . an index of n b = 3 . 4757 ( si ) was assumed for the plug . the other material parameters and the operating wavelength were set to the same values as in fig4 a and 4b . two dielectric slab thicknesses s = 300 , 500 nm , two metal plate thicknesses t = 100 , 200 nm , and two waveguide widths w = 1 . 7 , 4 μm are shown as examples . the high - index dielectric plugs ( 14 ) have a dramatic effect on the attenuation of the e ρ 11 mode , and when dimensioned properly , the plugs lower the propagation loss by more than one order of magnitude as shown in fig5 b . in the low loss region ( 0 . 1 ≦ w b ≦ 0 . 3 μm , approximately ), the dielectric plugs repel the fields away from the metal . when the plugs are too narrow ( w b ≦ 0 . 1 μm , approximately ), the e ρ 11 mode interacts strongly with the metal edges increasing its propagation loss and effective index . at the other extreme , when the plugs are too large ( w b ≧ 0 . 3 μm , approximately ), they support their own guided modes with fields confined within the plugs and interacting strongly with the metal edges , transforming the character of the e ρ 11 mode and increasing its propagation loss and effective index . fig6 a , 6 b and 6 c show the e ρ field component of the e ρ 11 mode for w b = 20 , 200 and 400 nm , respectively , with the other dimensions set to s = 500 nm , w = 1 . 7 μm and t = 200 nm . the outline of the mdw is also sketched in white on all three figures for reference . from fig5 b , w b = 20 and 400 nm are noted to be in the high loss regions , and fig6 a and 6c show the mode interacting strongly with the metal edges thus causing higher loss . from fig5 b w b = 200 nm is noted to be in the low loss region , and fig6 b shows the mode well centered and isolated from the metal edges by the plugs thus decreasing the loss . fig6 c also shows the mode localized to the four plugs , rather than having a single field maximum centered in the mdw as in the case of fig6 b . fig7 a and 7b give the effective index and propagation loss , respectively , of the e ρ 11 and e ρ 21 modes of the straight mdw as a function of the width w . the material parameters and operating wavelength were set to the same values as in fig5 a and 5b . two dielectric slab thicknesses s = 300 , 500 nm and two metal plate thicknesses t = 100 , 200 μn are shown as examples . the dielectric plug width was set to w b = 200 nm . the e ρ 21 mode is the first higher - order mode that is guided as the width w is increased . the propagation loss of the e ρ 11 and e ρ 21 modes increases drastically as the cut - off width is approached , at w ˜ 1 and 1 . 75 μm , respectively . the waveguide is observed to be single - mode ( e ρ 11 only ) for 1 & lt ; w & lt ; 1 . 75 μm ( approximately ). the thicker waveguides ( s = 500 nm ) have a lower propagation loss than the corresponding thinner ones ( s = 300 nm ), although they lack the non - radiative character as discussed above ( and further below ). the propagation loss of the e ρ 11 mode decreases to 1 . 2 db / mm for w = 5 μm . when designing components , wider straight sections could be used to propagate the mode over long distances , then a short aggressive non - radiative transition could be inserted to reduce the mdw width to a narrower single - mode section before , say , bending . fig8 shows the insertion loss of the e ρ 11 mode in 90 ° curved mdws ( fig2 b ) of various designs as a function of the radius of curvature r 0 . the material parameters and operating wavelength were set to the same values as in fig5 a and 5b . two dielectric slab thicknesses s = 300 , 500 nm and two metal plate thicknesses t = 100 , 200 nm are shown as examples . the dielectric plug width was set to w b = 200 nm in all cases except one where it was set to w b = 0 . the insertion loss of the e ρ 11 mode in a comparable 90 ° curved high - confinement single - mode rectangular dielectric waveguide was also computed for reference ; the dielectric waveguide had a thickness s = 300 nm and width w = 800 nm with core and cladding indices of n 1 = 2 . 1375 and n 2 = 1 . 444 , respectively . the computations were performed using the method of lines in cylindrical co - ordinates [ 6 ]. from fig8 , it is noted that the best performance is obtained with the mdw design having s = 300 nm , t = 100 nm and w b = 200 nm ; this design produces a relatively low insertion loss at a relatively aggressive bending radius r 0 , outperforming even the high - confinement dielectric waveguide , because all of the parallel - plate modes are cut - off except the orthogonal s b 0 mode . this design is capable of producing an insertion loss of only 0 . 61 db / 90 ° at r 0 = 150 nm . numerical limitations restricted r 0 from reaching zero but it is expected that the insertion loss will continue to decrease as r 0 → 0 ( fig2 c ), as the trend suggests . features 20 and 22 are noted in fig8 on the insertion loss curves of the 300 nm thick mdws at r 0 ˜ 2 μm . inspection of the fields ( not shown ) reveals that optical resonances occur involving the dielectric plugs at these radii . the other mdw designs were chosen with a thickness of s = 500 nm ; the minimum radius of curvature in these cases is significantly larger due to radiation into non cut - off parallel - plate modes . the effect of the dielectric plugs 14 ( fig2 a ) is apparent from fig8 illustrating the results for the mdw design having w b = 0 . as mentioned before , the straight case with no plugs ( r 0 →∞, w b = 0 ) has strong field localization to the edges of the metal plates . this localization increases in a curved structure as r 0 decreases . inspection of the mode fields ( not shown ) reveals radiation of the e ρ 11 mode into single - interface spps near r 0 = 20 μm . at approximately r 0 = 9 μm the insertion loss begins to decrease again , but the field concentration along the inner metal edges increases and the e ρ 11 mode changes appreciably in character . while including the dielectric plugs 14 is clearly preferred ( fig2 a to 2e ), the mdw without the dielectric plugs ( fig1 a to 15e ) still operates , albeit with greater loss . fig9 a - 9e show the e ρ 11 mode fields of various mdws , with the mdw cross - section sketched in white over the field distribution . the material parameters and operating wavelength were set to the same values as in fig5 a and 5b . fig9 a shows the main transverse electric field component e ρ for a straight ( r 0 →∞, fig2 d ) mdw ( w = 1 . 7 μm , s = 300 nm , t = 100 nm , w b = 200 nm ). fig9 b and 9c show e ρ and e z , respectively , for an aggressively curved mdw ( w = 1 . 7 s = 300 nm , t = 100 nm , w b = 200 nm , r 0 = 0 . 4 μm , fig2 b ) having s such that all parallel plate modes are cutoff except the s b 0 mode ( which is guided as s → 0 ). no radiation is observed inside the parallel plate sections for this design , and no coupling is observed with the parallel - plate se mode as expected because of orthogonality . only negligible leakage occurs into single interface spps at the upper / lower au - n 2 interfaces , as is apparent in fig9 c , and into plane waves above and below the metal plates . this mdw design is therefore substantially non - radiative . fig9 d and 9e show e ρ and e z , respectively , for a curved mdw ( w = 1 . 7 μm , s = 500 nm , t = 100 nm , w b = 200 nm , r 0 = 5 . 15 μm , fig2 b ) having s such that the a b 0 and s b 1 modes are not cut - off . in this case , radiation is clearly leaking into the a b 0 and s b 1 modes of the outer parallel - plate waveguide . comparing fig9 a and 9b reveals that the curved mdw mode ( fig9 b ) is de - centered and deformed compared to the straight one ( fig9 a ), suggesting that , in an end - to - end connection ( i . e . : a butt - coupling ), the waveguides should be offset laterally from each other in order to align the modes and minimize the coupling loss . more particularly , fig1 a shows in top view a straight ( r 0 →∞) mdw 30 butt - coupled to a curved mdw 32 with a centre - to - centre lateral offset of δx . fig1 b shows the transition loss computed between a straight ( r 0 →∞) and curved ( r 0 = 0 . 4 μm ) mdw as a function of the lateral offset δx ; both mdws are dimensioned with w = 1 . 7 μm , s = 300 nm , t = 100 nm and w b = 200 nm . the transition loss is calculated by computing the overlap integral between the main e ρ 11 mode field component ( e ρ ) of the straight and curved sections . fig1 b shows that a slight ( outward ) offset of δx ˜ 300 nm reduces the transition loss to & lt ; 1 db compared to 3 . 6 db with no offset . fig1 a and 11b show in top view some example components constructed with the mdw . a curved transition 40 can be implemented to join two straight mdw sections of different widths w 1 and w 2 by curving out the narrow width w 1 to the wide width w 2 over a short distance l . a linear transition 50 can be implemented to join two straight mdw sections of different widths , w 1 and w 2 , by flaring out the narrow width w 1 to the wide width w 2 over a short distance l . using substantially non - radiative mdws advantageously allows a short non - radiative transition of length l between wide and narrow mdw sections . an s - bend 60 , useful for redirecting optical radiation , is implemented by interconnecting straight mdws 63 and 64 through two oppositely curved mdw sections 61 and 62 . offsetting the sections 61 , 62 , 63 and 64 laterally ( not shown ), in the manner sketched in fig1 a , reduces the transition losses between the oppositely curved sections 61 / 62 and between the curved and straight sections 61 / 63 , and 62 / 64 . using substantially non - radiative mdws advantageously allows a short non - radiative s - bend length l . a y - junction 70 , useful for splitting or combining optical radiation , is implemented by connecting straight sections 71 and 72 to interconnected mirrored s - bends 73 and 74 , the s - bends themselves being connected to straight section 76 through short transition 75 . using substantially non - radiative mdws advantageously allows a short non - radiative y - junction length l . a coupler 80 , useful for splitting or combining optical radiation , is implemented by connecting straight parallel sections 85 and 86 to respective s - bends 81 / 83 and 82 / 84 , the length l c and separation s c of the parallel sections 85 and 86 determining the coupling ratio of the coupler . using substantially non - radiative mdws advantageously allows short non - radiative s - bends and thus a coupler of short overall length l . a mach - zehnder interferometer ( mzi ) 90 , useful for monitoring the interference of optical radiation , is implemented by connecting straight parallel sections 93 and 94 to a pair of y - junction splitters 91 and 92 , the length l m of the parallel sections 93 and 94 determining in part the difference in insertion phase between the modes propagating therealong . using substantially non - radiative mdws advantageously allows short non - radiative y - junctions . a multimode interferometer ( mmi ) 100 , useful for splitting or combining optical radiation , is implemented by connecting straight sections 101 and 102 to a wider section 103 of length l and width w . the wider section 103 propagates the fundamental ( e ρ 11 ) and higher - order ( e ρ 21 , e ρ 31 , . . . ) modes , and its length l and width w are selected such that interference of these modes when excited by section 104 produces a prescribed splitting at sections 101 and 102 . alternatively , n sections could be connected in this manner to section 103 instead of 2 ( 101 , 102 ) as sketched . specific dimensions for these components ( fig1 a and 11b ) can be determined by modeling using the procedures described in [ 6 , 7 , 11 ] in order to achieve desired performance characteristics . fig1 shows in top view a mdw bragg grating 110 of length l g constructed by concatenating unit cell 120 n times . the length of the grating l g is then given by l g = nλ where λ is the grating period and length of a unit cell . fig1 a shows a magnified top view of unit cell 120 created by maintaining the width w of the mdw constant and stepping the width of the dielectric plugs 14 from w b1 to w b2 over lengths d 1 and d 2 , respectively , defining the period λ = d 1 + d 2 . fig1 b shows an alternative unit cell design where the width of the mdw is stepped from w 1 to w 2 and the width of the dielectric plugs 14 is stepped from w b1 to w b2 over lengths d 1 and d 2 , respectively , defining the period λ = d 1 + d 2 . fig1 c shows yet another alternative unit cell design where the width w b of the dielectric plugs 14 is maintained constant and the width of the mdw is stepped from w 1 to w 2 over lengths d 1 and d 2 , respectively , defining the period λ = d 1 + d 2 . referring to fig2 a , the grating designs depicted in fig1 , 13 a , 13 b and 13 c can be applied to the bottom dielectric plugs 14 and bottom metal plates 12 l and 12 r only , to the top dielectric plugs 14 and top metal plates 12 l and 12 r only , or to both the top and bottom dielectric plugs 14 and top and bottom metal plates 12 l and 12 r . specific grating dimensions can be determined by modeling using the procedures described in [ 25 ], in order to achieve desired performance characteristics . using an electro - optic material , such as an electro - optic crystal ( e . g . : linbo 3 , plzt ) or an electro - optic polymer ( or other ), as the dielectric slab 10 confers additional functionality to the mdw . the metal parallel - plates 12 l and 12 r are advantageously used to apply a transverse ( horizontal ) electric field e to the electro - optic dielectric slab 10 through connections to a voltage source 150 , as sketched in fig1 . voltage source 150 may have dc and ac components . as discussed above with reference to fig2 a , the fundamental horizontally - polarized mode ( e ρ 11 ) occupies the area w × s ( roughly ) of region i ( see fig2 a , 6 b and 9 a ), so the applied electric field e is polarization - aligned and overlaps almost perfectly with the e ρ 11 mode . the applied electric field modifies the refractive index n 1 of the electro - optic dielectric slab 10 through its electro - optic effect thus imparting a phase shift to the e ρ 11 mode . in the case of linbo 3 ( or a similar crystal ) using an x - cut layer as the electro - optic dielectric slab 10 is particularly advantageous because the strongest electro - optic coefficient ( r 33 ) is exploited ( i . e . : n 1 = n e − 0 . 5n e 3 r 33 e where n e is the extraordinary index ). in the case of an electro - optic polymer , the metal plates 12 l and 12 r provide a further advantage in that they are used to poll the material post - deposition to engender or enhance its electro - optic effect prior to use . any of the mdw components depicted in fig2 a - 2e , 11 a - 11 b , 12 , and 13 a - 13 c can be enhanced by using an electro - optic material as the dielectric slab 10 and connections to a voltage source as depicted in fig1 . in the case of the straight mdw ( fig2 d ), the resulting structure operates as a phase shifter . in the case of the directional coupler 80 ( fig1 b ), applying voltages to the straight parallel sections 85 and 86 creates a switch , a modulator or a wavelength filter . in the case of the mzi 90 ( fig1 b ), applying voltages to the straight parallel sections 93 and 94 creates a variable attenuator or an intensity modulator . combining electro - optic couplers ( 80 ) and mzis ( 90 ) creates variable multiplexers and tunable filters . in the case of the gratings 110 ( fig1 ) with unit cells 120 , 130 or 140 ( fig1 a - 13c ), applying a voltage creates a tunable reflection peak which in turn is used to create a tunable filter . in like manner , using a thermo - optic material , such as a polymer ( or other ), as the dielectric slab 10 confers additional functionality to the mdw . in this case , the metal parallel - plates 12 l and 12 r are advantageously used to heat the dielectric slab 10 through resistive heating by connecting each metal plate to its own ( or to a common ) current source . any of the mdw components depicted in fig2 a - 2e , 11 a - 11 b , 12 , and 13 a - 13 c can be enhanced by using a thermo - optic material as the dielectric slab 10 and connections to a current source . the devices enabled by an electro - optic material ( described above ) can be implemented as thermo - optic devices . fig1 illustrates in isometric view a 90 ° curved section of mdw having an arbitrary radius of curvature r 0 , butt - coupled to input and output means 160 and 165 , respectively , shown here as single - mode optical fibers . alternatively , the means 160 or 165 may be tapered single - mode optical fibers , single - mode polarization - maintaining optical fibers or tapered single - mode polarization - maintaining optical fibers , or any other suitable optical waveguide means . the cores 162 and 167 of the optical fibers 160 and 165 , respectively , are shown in fig1 aligned with the area w × s of region i , thus ensuring good overlap of the fiber modes with the e ρ 11 mode propagating in the mdw . the overlap can be optimized by offsetting the optical fibers 160 and 165 toward the outside of the curve in the manner sketched in fig1 a , and as discussed with regards to fig1 a and 10b . the e ρ 11 mode is horizontally polarized , so good polarization alignment between the input fiber mode ( 160 ) and the e ρ 11 mode is required . this can be readily achieved by controlling the polarization of the input mode using a polarization controller , or by coupling a polarized light source to a principle axis of a polarization - maintaining optical fiber and aligning this axis along the width of the mdw . the mdw ( fig2 a ) can be implemented using many combinations of materials . the general requirements are that the dielectric plugs 14 should have as high an index n b as possible , the dielectric slab 10 should have an index n 1 higher than that of the surrounding claddings d ( n 1 & gt ; n 2 ), and the metal plates should be implemented using a metal having the lowest possible optical loss . in the embodiments and calculations given throughout this disclosure , the free - space operating wavelength was set to λ 0 = 1550 nm , but embodiments of the invention may be arranged and configured to propagate radiation having any wavelength in the range encompassing the ultra - violet , through the visible , near infra - red , mid infra - red , far infra - red , down to millimeter - waves and microwaves . in the embodiments and calculations given throughout this disclosure the dielectric slab 10 had an index close to that of si 3 n 4 , the claddings d had an index close to that of sio 2 , the dielectric plugs 14 had an index close to that of si , and the metal parallel - plates 12 l and 12 r had an index close to that of au , but many other materials could be used . good choices for the metal parallel - plates 12 l and 12 r include au , ag , cu , al but other metals such as pt , pd , ti , ni , cr , mo could be used . good choices from which to choose for the dielectric slab 10 , claddings d and plugs 14 include si 3 n 4 , sio 2 , tio 2 , polymers , electro - optic materials ( linbo 3 , litao 3 , batio 3 , plzt , ktp , kdp , dkdp , adp , ad * p , plzt , pzt ), crystals , and semiconductors ( si , ge , gaas , infp and variants thereof ), the combination selected to fulfill the requirements described above . the mdw and components can be manufactured on standard si wafers ( for example ) using conventional fabrication techniques such as physical or chemical vapor deposition , evaporation , etching or lift - off , and optical or e - beam lithography . embodiments of the present invention provide a low - loss optical waveguide structure that can curve or bend aggressively in the plane of propagation without radiating substantially . they may be used to implement integrated optical components , devices and circuits for routing or processing light . advantageously , the waveguide and components ( devices , circuits ) are of relatively small size , at least in the context of conventional integrated optics , and can be manufactured by applying conventional fabrication techniques ; these advantages when taken together may greatly reduce cost . although embodiments of the invention have been described and illustrated in detail , it is to be clearly understood that the same are by way of illustration and example only and not to be taken by way of limitation , the scope of the present invention being limited only by the appended claims . r . l . espinola , r . u . ahmad , f . pizzuto , m . j . steel , and r . m . osgood , “ a study of high - 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