Patent Publication Number: US-2023152608-A1

Title: Integrated-Optics Phase Controller Having Improved Electrode Configuration

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Patent Application Ser. No. 63/280,877, filed Nov. 18, 2021, entitled “Integrated-Optics Phase Controller Having Improved Electrode Configuration,” (Attorney Docket: 142-043PR2), which is incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to integrated optics in general and, more specifically, to devices for controlling the phase of an optical signal propagating in a surface waveguide of a planar waveguide circuit. 
     BACKGROUND 
     A Planar Lightwave Circuit (PLC) is an optical system comprising one or more integrated-optics-based waveguides that are integrated on the surface of a substrate, where the waveguides are typically combined to provide complex optical functionality. These “surface waveguides” typically include a core of a first material that is surrounded by a cladding material (or materials) having a refractive index that is lower than that of the first material. The change in refractive index at the interface between the core and cladding gives rise to internal reflection of light propagating through the core, thereby guiding the light along the length of the surface waveguide. 
     PLC-based devices and systems have made significant impact in many applications, such as optical communications systems, sensor platforms, solid-state projection systems, and the like. Surface-waveguide technology satisfies a need in these systems for small-sized, reliable optical circuit components that can provide functional control over a plurality of optical signals propagating through a system. Examples include simple devices (e.g., 1×2 and 2×2 optical switches, Mach-Zehnder interferometer-based sensors, etc.), as well as more complex, matrix-based systems having multiple surface waveguide elements and many input and output ports (e.g., wavelength add-drop multiplexers, cross-connects, wavelength combiners, etc.). 
     Common to many such systems is a need for a switching element. Historically, the most common switching elements suitable for use in a PLC are based on a device known as a thermo-optic (TO) phase controller. A TO phase controller takes advantage of the fact that the refractive index (i.e., the speed of light in a material) is temperature-dependent (referred to as the thermo-optic effect) by including a thin-film heater that is disposed on the top of the upper cladding of a surface waveguide. Electric current passed through the heater generates heat that propagates into the cladding and core materials, changing their temperature and, thus, their refractive indices. TO phase controllers have demonstrated induced phase changes greater than 2π (i.e., 360 degrees). 
     A TO phase controller can be included in a surface waveguide element, such as a Mach-Zehnder interferometer (MZI), to form an optical switching element. In an MZI switch arrangement, an input optical signal is split into two equal parts that propagate down a pair of substantially identical surface-waveguide paths (i.e., arms) to a junction where they are then recombined into an output signal. One of the arms incorporates a TO phase controller that controls the phase of the light in that arm. By imparting a phase difference of π between the light-signal parts in the arms, the two signals destructively interfere when recombined, thereby canceling each other out to result in a zero-power output signal. When the phase difference between the light-signal parts is 0 (or n*2π, where n is an integer), the two signals recombine constructively resulting in a full-power output signal. 
     Unfortunately, TO phase controllers are too slow for many applications because waveguide materials are normally not highly thermally conductive (i.e., they typically have low thermal-conductivity coefficients). As a result, the time required to heat or cool a surface waveguide structure can be relatively long (for example, 250 microseconds for a glass-based waveguide). In addition, the power consumption of TO phase controllers is high (&gt;100 mW in a static situation) which requires cooling control elements in the product or limits their usage in low-power environments. 
     More recently, stress-optic-based phase-tuning capability exploiting the photo-elastic effect has been demonstrated by incorporating a piezoelectric element disposed on a surface waveguide structure. By virtue of the photo-elastic effect, a stress-optic (SO) phase controller can induce a change in the refractive index of the materials of a waveguide with which it is operatively coupled by inducing a stress in the materials, as discussed in, for example, U.S. Pat. Nos. 9,221,074, 9,764,352, and 10,241,352, each of which is incorporated herein by reference. 
     While SO phase controllers are capable of inducing a 2π phase shift on an optical signal in as little as a few microseconds and exhibit relatively low static-power consumptions (less than &lt;1 μW, for example), they require higher voltages than thermo-optic phase controllers and significantly greater length over which the stress must be induced in a surface waveguide. For instance, while a thermo-optic phase controller might require an interaction length of approximately 1 mm to induce a 2π phase shift, the required interaction length for a comparable prior-art SO phase controller might be 2 cm or more. 
     As a result, a fast, space-efficient, low-power approach to phase control of a light signal propagating in a surface waveguide would be a welcome advance in the state of the art. 
     SUMMARY 
     The present disclosure is directed toward photonic systems that include stress-optic phase controllers that require low drive voltage and/or require small chip real estate. Embodiments in accordance with the present disclosure are particularly well suited for use in systems such as microwave photonics, LiDAR and the like. 
     The present disclosure provides an advance over the prior art by exploiting a heretofore unrecognized property of a stress-optic phase controller—namely, that it can impart stresses of opposite signs in a material such that a desired effect on the refractive index of an optical material can be accentuated and, in some cases, substantially optimized. As a result, a greater change in the refractive index of the material can be realized. 
     Like stress-optic-phase-controller-based photonic systems known in the prior art, a refractive index change in a waveguide is induced by imparting a stress on the waveguide material via a piezoelectric element disposed on top of the waveguide. However, in sharp contrast to the prior art, systems in accordance with the present disclosure are configured to induce stresses having opposite signs at different locations in a photonic system. 
     An illustrative embodiment is a photonics circuit comprising an integrated-optics-based asymmetric Mach-Zehnder Interferometer that is configured as an optical switch. The aMZI includes a pair of arms having different lengths, where each arm includes a stress-optic phase controller. One stress-optic phase controller is configured to induce positive stress in the waveguide material beneath it, while the other stress-optic phase controller is configured to induce negative stress in the waveguide material beneath it. As a result, the two stress-optic phase controllers operate in a “push-pull” operational mode that significantly reduces the amount of stress that must be induced to realize a relative 2π phase shift between light signals propagating in the two arms. 
     In some embodiments, a photonic system includes an MZI having arms of equal length, upon which are disposed stress-optic phase controllers configured to induce stress of opposite signs in the arms. 
     In some embodiments, a single stress-optic phase controller is operatively coupled with a single waveguide to induce a stress of a desired sign in the waveguide. 
     In some embodiments, a stress-optic phase controller is disposed on a planarized waveguide structure. 
     In some embodiments, a stress-optic phase controller is disposed on a waveguide structure characterized by a dome or projection included in its upper cladding, which gives rise to an enhanced stress level in the waveguide materials. 
     In some embodiments, a stress-optic phase controller comprises at least one electrode whose shape is configured to give rise to an enhanced stress level in the waveguide materials. 
     An embodiment in accordance with the present disclosure is an apparatus comprising a planar lightwave circuit ( 900 ) including: a first surface waveguide ( 910 B) that includes a first core ( 108 B) and a first cladding ( 506 ) that is at least partially disposed on the first core; and a phase-control module ( 904 ) that includes a first stress-optic phase-control (SOPC) element ( 916 B) comprising: (i) a first piezoelectric layer ( 512 ) disposed on the first cladding, the first piezoelectric layer having a first surface ( 516 - 1 ) that is proximal to the first cladding and a second surface ( 516 - 2 ) that is distal to the first cladding; (ii) a first electrode ( 918 - 1 ); and (iii) a second electrode ( 918 - 3 ); wherein the first electrode is in physical and electrical contact with the second surface; and wherein the second electrode is in physical and electrical contact with one of the first surface and second surface; wherein the phase-control module is configured to induce a first stress in the first surface waveguide in response to a first control signal (CSB) applied to the first and second electrodes. 
     Another embodiment in accordance with the present disclosure is a method comprising: providing a first surface waveguide ( 910 B) that includes a first core ( 108 B) and a first cladding ( 506 ) that is at least partially disposed on the first core; providing a phase-control module ( 904 ) that includes a first stress-optic phase-control (SOPC) element ( 916 B) that is operatively coupled with the first surface waveguide, the first SOPC element comprising: (i) a first piezoelectric layer ( 512 ) disposed on the first cladding, the first piezoelectric layer having a first surface ( 516 - 1 ) that is proximal to the first cladding and a second surface ( 516 - 2 ) that is distal to the first cladding; (ii) a first electrode ( 918 - 1 ); and (iii) a second electrode ( 918 - 3 ); wherein the first electrode is in physical and electrical contact with the second surface; wherein the second electrode is in physical and electrical contact with one of the first surface and second surface; and wherein phase-control module is configured to induce a first stress in the first surface waveguide in response to a first control signal (CSB) applied to the first and second electrodes; enabling propagation of a first light signal ( 914 B) through the first waveguide; and controlling a first phase of the first light signal by controlling the first control signal. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    depicts a cross-sectional view of a stress-optic (SO) phase controller that includes an SO phase element having a top-bottom electrode configuration in accordance with the present disclosure. 
         FIGS.  2 A-B  depict simulations showing the x- and y-components, respectively, of stress fields developed throughout the structure of waveguide  102  in response to a voltage applied to electrodes  110 - 1  and  110 - 2 . 
         FIG.  3    depicts a cross-sectional view of a stress-optic phase controller that includes an SOPC element having a top-top electrode configuration in accordance with the present disclosure. 
         FIGS.  4 A-B  depict simulations showing the x- and y-components, respectively, of stress fields developed throughout the structure of waveguide  102  in response to a voltage applied to electrodes  304 - 1  and  304 - 2 . 
         FIG.  5    depicts a schematic drawing of a cross-sectional view of a domed SO phase controller having a top-bottom electrode configuration in accordance with the present disclosure. 
         FIGS.  6 A-B  depict simulations showing the x- and y-components, respectively, of stress fields developed throughout the structure of waveguide  502  in response to a voltage applied to electrodes  510 - 1  and  510 - 2 . 
         FIG.  7    depicts a schematic drawing of a cross-sectional view of a domed SO phase controller having a top-top electrode configuration in accordance with the present disclosure. 
         FIGS.  8 A-B  show simulation results for the x- and y-components, respectively, of stress fields developed throughout the structure of waveguide  502  in response to a voltage applied to electrodes  510 - 1  and  510 - 2 . 
         FIG.  9 A  depicts a schematic drawing of a top view of a PLC-based switch comprising complimentary SO phase controllers in accordance with the present disclosure. 
         FIG.  9 B  depicts a schematic drawing of a sectional view of phase-control module  904 . 
         FIGS.  10 A-C  depict simulations showing the stress fields developed in arms  910 A and  910 B in response to different voltage configurations applied to electrodes  918 - 1  through  918 - 3 . 
         FIG.  10 D  depicts a simulation showing the phase shift induced on light portion  914 B in arm  910 B in response to a voltage applied between electrodes  918 - 1  and  918 - 2  as a function of the separation distance, d 1  between waveguides  910 A and  910 B. 
         FIG.  11 A  depicts a schematic drawing of a top view of an alternative PLC-based optical switch in accordance with the present disclosure. 
         FIG.  11 B  depicts a schematic drawing of a sectional view of phase-control module  1104 . 
         FIG.  12    depicts a plot of the phase changes in light portions  1114 A and  1114 B induced by the actuation of SO phase controllers  100 A and  110 B. 
         FIG.  13    depicts a schematic drawing of a sectional view of another alternative SO phase controller in accordance with the present disclosure. 
         FIG.  14    depicts a schematic drawing of top view of non-limiting examples of electrode shapes in accordance with the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     It is an aspect of the present disclosure that a piezoelectric element disposed on the top cladding of an integrated-optics waveguide creates different regions within the waveguide material that are characterized by stresses of opposite signs. For example, application of stress on the waveguide structure results not only in high compressive or tensile stress in the waveguide directly under the electrode, but also in a smaller, opposite stress at a certain distance away from the electrodes. 
     It is another aspect of the present disclosure that such a stress distribution can be exploited by arranging one or more waveguides relative to the distribution to induce different phase changes in the different waveguides. 
     Furthermore, it is yet another aspect of the present disclosure that the distribution of these stresses can be controlled by the configuration of the electrodes of the piezoelectric element. 
     As would be apparent to one skilled in the art, highly piezoelectric materials produce a single stress direction (compressive or tensile) independent of the sign of the applied electric field due to the operation above the coercive field. In some SO phase controllers in accordance with the present disclosure, however, opposite stress is realized by controlling the direction of the electric field through electrode design, from vertical in a top-bottom configuration to horizontal in a top-top configuration. 
       FIG.  1    depicts a cross-sectional view of a stress-optic (SO) phase controller that includes an SO phase element having a top-bottom electrode configuration in accordance with the present disclosure. 
     Phase controller  100  includes waveguide  102  and stress-optic phase-control (SOPC) element  104 , which is disposed on the top surface of the waveguide. As will be apparent to one skilled in the art, waveguide  102  is disposed on a suitable substrate (not shown), such as a silicon wafer, compound semiconductor wafer, glass substrate, or myriad alternative substrates suitable for use in planar-processing fabrication. 
     Waveguide  102  is an integrated-optics-based waveguide that includes lower cladding  106 - 1 , core  108 , and upper cladding  106 - 2 . In the depicted example, each of lower cladding  106 - 1  and upper cladding  106 - 2  is a layer of silicon dioxide having thicknesses of 8 microns and 3 microns, respectively. It should be noted that a wide range of thicknesses for a lower and/or upper cladding can be used without departing from the scope of the present disclosure. In the depicted example, upper cladding  106 - 2  is planarized via a process such as chemical-mechanical polishing. As discussed below, however, the teachings of the present disclosure are also applicable to non-planarized waveguide structures. 
     In the depicted example, waveguide  102  is an asymmetric double-stripe (ADS) TriPleX™ waveguide comprising core  108 , which has width, w 1 . Core  108  comprises comprising lower core lc, central core cc, and upper core uc. In the depicted example, w 1  is approximately 1 micron, and lower core lc, central core cc, and upper core uc have thicknesses of 75 nm, 100 nm, and 175 nm, respectively. However, a wide range of widths and thicknesses can be used for any of lower core lc, central core cc, and upper core uc without departing from the scope of the present disclosure. 
     Although the depicted example is a multi-core ADS waveguide, the teachings of the present disclosure are applicable to virtually any waveguide structure, such as single-core waveguides comprising any suitable core material (e.g., silicon, doped silicon oxide, silicon oxynitride, silicon-nitride, compound semiconductor, etc.), multi-core symmetric waveguides comprising any combination of suitable core materials (e.g., silicon, doped silicon oxide, silicon oxynitride, silicon-nitride, compound semiconductor, etc.), and the like. Some non-limiting examples of waveguide structures particularly suitable for use in embodiments in accordance with the present disclosure are described in more detail in U.S. Pat. Nos. 7,146,087, 7,142,759, 9,221,074 and 9,764,352, each of which is incorporated herein by reference. 
     SOPC element  104  is a “top-bottom” stress-optic phase-control element comprising bottom electrode  110 - 1 , piezoelectric layer  112 , and top electrode  110 - 2 , where the bottom electrode is in physical and electrical contact with bottom surface  114 - 1  of piezoelectric layer  112  and the top electrode is in physical and electrical contact with top surface  114 - 2  of piezoelectric layer  112 . 
     In the depicted example, electrodes  110 - 1  and  110 - 2  comprise platinum and have thicknesses of 100 nm and 300 nm, respectively, while piezoelectric layer  112  comprises lead zirconate titanate (PZT) and has a thickness, t 1 , of 1.5 microns. The widths, w 2  and w 3 , of bottom and top electrodes  110 - 2  and  110 - 2  are 10 microns, and the electrodes are aligned with core  108 . In some embodiments, w 2  is much greater than w 3  and, in some cases, extends across the entire chip on which SO phase controller  100  resides. It should be noted that the materials and dimensions provided above are merely exemplary and that any suitable materials and thicknesses can be used for any of bottom electrode  110 - 1 , piezoelectric layer  112 , and top electrode  110 - 2  without departing from the scope of the present disclosure. Furthermore, many alternative materials are suitable for use in piezoelectric layer  112  such as, without limitation, barium titanate, lead titanate, lithium niobate, bismuth ferrite, sodium niobate, and the like. 
       FIGS.  2 A-B  depict simulations showing the x- and y-components, respectively, of stress fields developed throughout the structure of waveguide  102  in response to a control signal applied to electrodes  110 - 1  and  110 - 2 . 
     In response to a voltage (i.e., control signal CS) applied to electrodes  110 - 1  and  110 - 2 , piezoelectric layer  112  attempts to elongate in the direction of the resultant electric field (y-direction). In addition, due to volume conservation in the piezoelectric layer (with the Poisson ratio), it contracts perpendicular (x- and z-directions) to this electric field. In the depicted example, therefore, elongation of piezoelectric layer  112  in the y-direction gives rise to compressive stress in the waveguide material below it in all three directions. For the purposes of this Specification, a compressive stress in a material is designated as a negative stress, while a tensile stress in a material is designated as a positive stress. 
     As plots  200  and  202  show, an SOPC element having a top-bottom electrode configuration induces significant compressive stress into the upper cladding and core layers of a waveguide structure on which it is disposed. 
     On the other hand, it is an aspect of the present disclosure that, by changing the configuration of the electrodes of an SOPC element, a different stress configuration throughout a waveguide on which it is disposed can be achieved. 
       FIG.  3    depicts a cross-sectional view of a stress-optic phase controller that includes an SOPC element having a top-top electrode configuration in accordance with the present disclosure. 
     Phase controller  300  includes waveguide  102  and SOPC element  302 , which is disposed on upper cladding  106 - 2 . 
     SOPC element  302  is a “top-top” stress-optic phase-control element comprising electrodes  304 - 1  and  304 - 2 , which are in physical and electrical contact with top surface  114 - 2  of piezoelectric layer  112 . In the depicted example, each of electrodes  304 - 1  and  304 - 2  comprises platinum and has a width of 10 microns and a thickness of 300 nm. The separation, s 1 , between electrodes  304 - 1  and  304 - 2  is approximately 10 microns. 
       FIGS.  4 A-B  depict simulations showing the x- and y-components, respectively, of stress fields developed throughout the structure of waveguide  102  in response to a voltage applied to electrodes  304 - 1  and  304 - 2 . 
     As noted above, in response to the application of control signal CS to electrodes  304 - 1  and  304 - 2 , piezoelectric layer  112  attempts to elongate in the direction of the resultant electric field. In the depicted example, therefore, elongation of piezoelectric layer  112  in the x-direction gives rise to tensile stress in the waveguide material below it as it pulls the material along the x-direction. In addition, due to volume conservation in the piezoelectric layer (with the Poisson ratio), it contracts perpendicular (y- and z-directions) to this electric field. Therefore, the waveguide material is pulled along all three directions. 
     As evinced by plots  400  and  402 , a stress-optic phase-control element having a top-top electrode configuration induces significant tensile stress into the upper cladding and core layers of a waveguide structure on which it is disposed. 
     It should be noted that each of SOPC elements  104  and  302  is formed on a waveguide structure that has been planarized via chemical-mechanical polishing (CMP) such that its upper cladding is flat. As a result, electrodes  110 - 1 ,  110 - 2 ,  304 - 1 , and  304 - 2  are also flat. 
     In some embodiments, however, an SOPC element is disposed on a waveguide structure whose upper cladding includes a dome-like projection above its core. Waveguides having an upper cladding comprising a core, and their fabrication, are described in detail in U.S. Pat. No. 10,241,352, which is incorporated herein by reference. Such a configuration gives rise to a SOPC element whose electrodes and piezoelectric element are shaped such that they conform to the shape of the dome. As a result, the stresses induced in the underlying waveguide materials when the SOPC element is actuated are enhanced compared to the planar waveguide configuration. 
       FIG.  5    depicts a schematic drawing of a cross-sectional view of a domed SO phase controller having a top-bottom electrode configuration in accordance with the present disclosure. SO phase controller  500  comprises waveguide  502  and SOPC element  504 , which is disposed on the top surface of waveguide  502  (i.e., on upper cladding  506 ). 
     Waveguide  502  is analogous to waveguide  102 ; however, in waveguide  502 , upper cladding  506  includes dome  508 , which is aligned with core  108 . In the depicted example, height, h 1 , of dome  508  is equal to 850 nm. 
     SOPC element  504  is disposed on dome  508 , thereby creating curved piezoelectric layer  512  having arc  514 , which is between curved electrodes  510 - 1  and  510 - 2 , which are in physical and electrical contact with surfaces  516 - 1  and  516 - 2 , respectively, of piezoelectric layer  512 . 
     SOPC element  504  is analogous to SOPC element  104  described above; however, the shape of SOPC element  504  increases the effectiveness with which it creates stress in the waveguide layers of the waveguide on which it is disposed. 
     It should be noted that, although electrode  510 - 1  extends across the entire width of SO phase controller  500  in the depicted example, in some embodiments, electrode  510 - 1  has a width that is equal to or slightly greater than the width of electrode  510 - 2 . 
       FIGS.  6 A-B  depict simulations showing the x- and y-components, respectively, of stress fields developed throughout the structure of waveguide  502  in response to a voltage applied to electrodes  510 - 1  and  510 - 2 . 
     As see from plots  600  and  602 , the magnitude of stress induced in the layers of waveguide  502  by domed SOPC element  504  is significantly greater than that induced in the layers of waveguide  102  by SOPC element  100 . 
       FIG.  7    depicts a schematic drawing of a cross-sectional view of a domed SO phase controller having a top-top electrode configuration in accordance with the present disclosure. SO phase controller  700  comprises waveguide  502  and SOPC element  702 , which is disposed on the top surface of waveguide  502 . 
     SOPC element  702  is analogous to SOPC element  504 ; however, SOPC element  702  includes electrodes  704 - 1  and  704 - 2 , which are both disposed on the top surface of piezoelectric layer  512  and arranged adjacent to arc  514  such that the arc is located between them. 
     It should be noted that dome height, h 1 , has a significant impact on the magnitude of stress that an SOPC element imparts, for both the top-bottom electrode configuration and the top-top electrode configuration. Generally, for either electrode configuration, the greater the value of h 1 , the more effectively an SOPC element can induce stress in a waveguide structure beneath it, as indicated in Table 1 below. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Simulation results for the induced optical phase shifts 
               
               
                 of an optical signal propagating through SO phase controllers 
               
               
                 500 and 700 having 1-cm electrode lengths, using applied 
               
               
                 actuation voltages of 40 V and 200 V, respectively. 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Device 
                 h1 = 0 nm (planar) 
                 h1 = 450 nm 
                 h1 = 950 nm 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 500 
                 1.7π 
                 2.3π 
                 2.7π 
               
               
                   
                 700 
                 1.6π 
                 1.8π 
                 2.0π 
               
               
                   
                   
               
            
           
         
       
     
       FIGS.  8 A-B  show simulation results for the x- and y-components, respectively, of stress fields developed throughout the structure of waveguide  502  in response to a voltage applied to electrodes  510 - 1  and  510 - 2 . 
     Comparing plots  800  and  802  to plots  400  and  402 , it is readily apparent that the magnitude of stress induced in the layers of waveguide  502  by domed SOPC element  702  is significantly greater than that induced in the layers of waveguide  102  by SOPC element  300 . 
     Although it is an aspect of the present disclosure that individual SOPC elements can be provided with an electrode configuration suitable for imparting whatever desired type of strain (i.e., tensile or compressive) into the materials of a single waveguide, it is yet another aspect that combinations of SOPC elements having different electrode configurations can be used in concert to generate complicated and, in some embodiments, complimentary strain fields in a planar lightwave circuit including one or more waveguides. 
       FIG.  9 A  depicts a schematic drawing of a top view of a PLC-based switch comprising complimentary SO phase controllers in accordance with the present disclosure. PLC  900  includes asymmetric Mach-Zehnder interferometer (aMZI)  902  and phase-control module  904 , which are arranged to define an integrated-optics waveguide switch in which the intensity of a light signal is controlled by phase-control module  904  as the light signal passes from the input port of aMZI  902  to the output port of aMZI  902 . 
       FIG.  9 B  depicts a schematic drawing of a sectional view of phase-control module  904 . The sectional view shown in  FIG.  9 B  is taken through line a-a shown in  FIG.  9 A . 
     aMZI  902  is a network of waveguides arranged to define input port  908 , arms  910 A and  910 B, and output port  912 . Each of the waveguides of aMZI  902  is analogous to waveguide  502  described above, having a domed upper cladding of height, h 1 . In addition, the centers of arms  910 A and  910 B are separated by distance, d 1 , at phase controller  906 B. In the depicted example, h 1  is 850 nm and d 1  is 10 microns; however, any suitable value can be used for one or both of h 1  and d 1 . 
     As noted above, although the examples described herein comprise multi-core ADS waveguides, embodiments in accordance with the present disclosure can include waveguides having any suitable waveguide structure. 
     Phase-control module  904  includes phase controllers  906 A and  906 B, which include arms  910 A and  910 B, respectively. 
     Phase controller  906 A is analogous to SO phase controller  500 , described above and with respect to  FIGS.  5  and  6 A -B. Phase controller  906 A includes SOPC element  916 A disposed on arm  910 A. SOPC element  916 A includes electrodes  918 - 1  and  918 - 2  and the portion of piezoelectric layer  512  that resides between them. In the depicted example, electrodes  918 - 1  and  918 - 2  have widths w 2  and w 3 , respectively, of 10 microns. 
     Phase controller  906 B is analogous to SO phase controller  700 , described above and with respect to  FIGS.  7  and  8 A -B. SO phase controller  906 B includes SOPC element  916 B disposed on arm  910 B. SOPC element  916 B includes electrodes  918 - 1  and  918 - 3  and the portion of piezoelectric layer  512  disposed beneath and between them. In the depicted example, the separation, s 1 , between electrodes  918 - 1  and  918 - 3  is 10 microns. 
     In operation light signal  914  is split into two equal light portions,  914 A and  914 B, which propagate through arms  910 A and  910 B, respectively. After travelling through arms  910 A and  90 B, light portions  914 A and  914 B recombine at output port  912 . As will be apparent to one skilled in the art, the optical power of recombined optical signal  914 ′ is based on the phase difference between light portions  914 A and  914 B when they recombine. 
     The phase difference between light portions  914 A and  914 B is determined by the difference in the lengths of arms  910 A and  910 B plus any phase changes induced on the light portions by phase controllers  906 A and  906 B in response to control signals CSA and CSB, respectively. 
     Refractive index is a function of material stress and, therefore, it can be increased or decreased by raising or lowering material stress. Since phase-control elements  906 A and  906 B are configured to induce opposite signs of stress, as discussed above, opposite changes in refractive index can be induced in arms  910 A and  910 B. Therefore, the teachings of the present disclosure enable a significantly greater phase difference to be imparted on light portions  914 A and  914 B by inducing opposite phase changes in the two arms. As a result, phase-control module  904  can induce up to a 2π phase change on light portions  914 A and  914 B required for full switching functionality in PLC  900  with significantly shorter interaction lengths and requires dramatically less chip real estate than prior-art systems. 
       FIGS.  10 A-C  depict simulations showing the stress fields developed in arms  910 A and  910 B in response to different voltage configurations applied to electrodes  918 - 1  through  918 - 3 . 
     Plot  1000  shows the composite stress fields (combined x-, y-, and z-components) when voltages of 0, 40, and 0 volts are applied to electrodes  918 - 1 ,  918 - 2 , and  918 - 3 , respectively. 
     Plot  1002  shows the composite stress fields when voltages of 0, 0, and 200 volts are applied to electrodes  918 - 1 ,  918 - 2 , and  918 - 3 , respectively. 
     Plot  1004  shows the composite stress fields when voltages of 0, 40, and 200 volts are applied to electrodes  918 - 1 ,  918 - 2 , and  918 - 3 , respectively. 
     As is clear from plot  1004 , large-magnitude stresses having opposite sign can be induced in arms  910 A and  910 B. 
     It should be noted that the stress-inducing effect of each SOPC element is substantially restricted to the waveguide arm for which it is intended, and with a 10-20% opposite stress at larger distances from the main electrode area. As a result, the phase changes induced in each arm can be increased by adding the two elements together. This is indicated in Table 2 below, where 0.4π can be added to yield a total simulated phase shift of 3.2π in the top-bottom configuration; or a −0.8π to the top-top configuration for a total of −2.9π. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Phase change induced on light portions 914A and 914B for 
               
               
                 actuation of only SOPC element 916A, only SOPC element 
               
               
                 916B alone, and both of SOPC elements 916A and 916B. 
               
            
           
           
               
               
               
               
            
               
                   
                 Phase Change 
                 Phase Change 
                 Phase Change 
               
               
                   
                 V1 = V3 = 0 V, 
                 V1 = V2 = 0 V, 
                 V1 = 0 V, V2 = 40 V, 
               
               
                 Device 
                 V2 = 40 V 
                 V3 = 200 V 
                 V3 = 200 V 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                 914A 
                 2.8π 
                 0.4π 
                 3.2π 
               
               
                 914B 
                 −0.8π 
                 −2.1π 
                 −2.9π 
               
               
                   
               
            
           
         
       
     
     It is another aspect of the present disclosure that an SOPC element can develop fields of opposite stress at points that are separated by a medium-to-large distance. 
       FIG.  10 D  depicts a simulation showing the phase shift induced on light portion  914 B in arm  910 B in response to a voltage applied between electrodes  918 - 1  and  918 - 2  as a function of the separation distance, d 1  between waveguides  910 A and  9106 . 
     Plot  1006  shows that, for this simulation and with a separation of 10 microns between arms  910 A and  9106 , a phase shift of −0.25π develops on light portion  914 B in arm  910 B due solely to the effect of SOPC element  916 A, while actuation of SOPC element  916 B contributes an additional 2.6*pi phase shift on light portion  914 B. 
     In some embodiments, laterally displaced fields of opposite stress generated by a single SOPC element are exploited to enable phase control in a photonic circuit. In some embodiments, multiple such SOPC elements are used cooperatively such that the laterally displaced fields of opposite stress from each element collectively develop a desired composite stress configuration. 
     Furthermore, it is yet another aspect of the present disclosure that the material properties (e.g., Young&#39;s modulus, Poisson ratio, etc.) of the materials used in a photonic circuit and/or a SOPC element included in it, can be selected to realize or augment a desired stress configuration. Furthermore, etched or deposited features formed in or laterally displaced from an SOPC element can be used to tailor (or augment) the performance of the element. For example, simulations have shown a 10% increase in phase shift is realized when the piezoelectric material of an SOPC element is removed at a position 2-5 microns away from the waveguide core of a waveguide located under the SOPC element. Non-limiting examples of such configurations include:
         i. a capping layer that increases or decreases the effective stress generated by an SPOC element within one or more waveguides; or   ii. features (e.g., channels, vias, domes, etc.) formed in and/or laterally displaced from a piezoelectric layer of an SOPC element and/or features formed in one or more layers of a photonic circuit including the SOPC element by removing or adding material; or   iii. a combination of i and ii.       

     It should be noted that photonic circuits comprising cooperative SO phase controllers of different arrangements are within the scope of the present disclosure. For example, in some embodiments, a photonic circuit includes a plurality of SOPC elements having the same electrode configuration (e.g., top-top or top-bottom), where the SOPC elements are configured to operate cooperatively. In some embodiments, the electrodes of a top-bottom SOPC element are independent of the electrodes of a top-top SOPC element with which it cooperatively operates. For example, in some embodiments, piezoelectric layer  512  is not continuous between SOPC elements  916 A and  916 B and each of the SOPC elements includes a pair of electrodes, none of which is common to both SOPC elements. 
     As noted above, the application of stress on a waveguide structure via an SOPC element results in high compressive or tensile stress in the waveguide directly under the electrode of the SPOC element, as well as a smaller magnitude stress of the opposite type at a certain distance away from the electrodes. In some embodiments in accordance with the present disclosure, these regions of opposite stress are utilized in an aMZI configuration by locating one arm of the aMZI a stress field having a first sign (e.g., tensile) and the other arm of the aMZI in the stress field sign of the opposite sign (e.g., compressive). 
       FIG.  11 A  depicts a schematic drawing of a top view of an alternative PLC-based optical switch in accordance with the present disclosure. PLC  1100  includes asymmetric Mach-Zehnder interferometer (aMZI)  1102  and phase-control module  1104 , which are arranged to define an integrated-optics waveguide switch in which the intensity of a light signal is controlled by phase-control module  1104  as the light signal passes from the input port of aMZI  1102  to the output port of aMZI  1102 . 
       FIG.  11 B  depicts a schematic drawing of a sectional view of phase-control module  1104 . The sectional view shown in  FIG.  11 B  is taken through line b-b shown in  FIG.  11 A . 
     aMZI  1102  includes a network of waveguides that are analogous to those of aMZI  902 ; however, the waveguides aMZI  1102  have a planar top cladding (i.e., the top cladding does not include a dome). 
     Phase-control module  1104  includes SO phase controllers  1106 A and  1106 B, each of which is analogous to SO phase controller  100 , described above and with respect to  FIG.  1   . SO phase controllers  1106 A and  1106 B are optically coupled with arms  1110 A and  1110 B, respectively. 
       FIG.  12    depicts a plot of the phase changes in light portions  1114 A and  1114 B induced by the actuation of SO phase controllers  1106 A and  1106 B. 
     Plot  1200  demonstrates that a phase change of 0.8π can be realized for the optical signals propagating in each arm for an applied voltage of only 40 V. 
     In some embodiments, the voltages applied to SO phase controllers  1106 A and  1106 B are generated in sequence to account for any cancellation of stress effect caused by the voltage applied to the other SO phase controller. In other words, a voltage is first applied to SO phase controller  1106 A while SO phase controller  1106 B has no voltage applied, which enables a full phase shift differential to be realized on the signals propagating in the two arms. 
       FIG.  13    depicts a schematic drawing of a sectional view of another alternative SO phase controller in accordance with the present disclosure. SO phase controller  1300  includes waveguide  102  and SOPC element  1302 . It should be noted that SO phase controller  1300  is suitable for use in any embodiment in accordance with the present disclosure. 
     SOPC element  1302  is analogous to SOPC element  104 ; however, SOPC element  1302  includes one or more additional piezoelectric layers and interleaving electrodes that collectively define a piezoelectric stack disposed on waveguide  102 . In the depicted example, SOPC element  1302  includes one additional piezoelectric layer  1304  and one additional electrode (i.e., electrode  1306 ). It should be noted that, although SOPC element  1302  is disposed on planarized waveguide  102 , in some embodiments, SOPC element  1302  is disposed on a domed waveguide, such as waveguide  502 . 
     Piezoelectric layer  1304  includes bottom surface  1308 - 1  and top surface  1308 - 2  and is disposed on piezoelectric layer  112  and electrode  110 - 1 . As a result, electrode  110 - 1  is in physical and electrical contact with both of surfaces  114 - 2  and  1308 - 1 . 
     Electrode  1306  is disposed on piezoelectric layer  1304  such that it is in physical and electrical contact with surface  1308 - 2 . 
     As will be apparent to one skilled in the art, the piezoelectric effect works as displacement/applied voltage, regardless of the thickness of the piezoelectric layer. In other words, the amount of displacement, and by near linear scaling the stress on the waveguides, depends only on the voltage across the piezoelectric material. 
     It is an aspect of the present disclosure that the magnitude of stress induced in a waveguide by an SOPC element disposed on it can be multiplied by the number of piezoelectric layers it includes. In other words, the exemplary SOPC element  1302  includes a stack of two piezoelectric layers that includes substantially identical piezoelectric layers  112  and  1304 , so the amount of stress it can induce in waveguide  102  in response to control signals applied to electrodes  110 - 1 ,  110 - 2 , and  1306  is substantially twice that of SOPC element  104 . In some embodiments, more than two piezoelectric layers are included in an SOPC element. 
     It should be noted that the breakdown voltage for a piezoelectric layer is based on its thickness, which can limit the total amount of stress that can be induced by an SOPC element. 
     It is yet another aspect of the present disclosure that the design of the electrodes of an SOPC element can have dramatic effect on the effectiveness with which it can induce stress in an underlying waveguide structure. 
       FIG.  14    depicts a schematic drawing of top view of non-limiting examples of electrode shapes in accordance with the present disclosure. 
     Electrode patterns  1400  through  1410  are particularly well suited for use in SOPC elements having a top-top electrode configuration; however, SOPC elements having a top-bottom electrode configuration can also benefit from employing split electrodes in at least one electrode level. For example, a split electrode includes greater edge length, at which large stress can be induced since it is the boundary between where a piezoelectric layer can and cannot expand. 
     In some embodiments, the orientation of an electrode is not aligned with the axis of the waveguide beneath it. 
     It is to be understood that the disclosure teaches just one example of the illustrative embodiment and that many variations of embodiments in accordance with the present disclosure can easily be devised by those skilled in the art after reading this disclosure and that the scope of the present invention is to be determined by the following claims.