Patent Publication Number: US-2023152517-A1

Title: Thermally modulated photonic switch and associated methods

Description:
TECHNICAL FIELD 
     The subject matter described herein relates in general to photonic switches and, more specifically, to a thermally modulated photonic switch and associated methods. 
     BACKGROUND 
     Photonic switches—devices that route a light beam from an input waveguide to a specific output waveguide in response to a control input—are used in a variety of applications such as communications, optical computing, analog photonics, photonic processing, and optical neural networks. In some applications, it is particularly important for a photonic switch to be compact in size, but many conventional photonic switches do not meet the target size specifications for those applications. 
     SUMMARY 
     Embodiments of a thermally modulated photonic switch are presented herein. In one embodiment, a thermally modulated photonic switch comprises a topology-optimized structure that includes dispersed silicon and silicon dioxide. This topology-optimized structure includes an input waveguide, a first output waveguide, and a second output waveguide. The topology-optimized structure routes a light beam from the input waveguide to the first output waveguide, when the topology-optimized structure is at a first predetermined temperature that causes a refractive index of the silicon in the topology-optimized structure to assume a first predetermined value, and the topology-optimized structure routes a light beam from the input waveguide to the second output waveguide, when the topology-optimized structure is at a second predetermined temperature that causes the refractive index of the silicon in the topology-optimized structure to assume a second predetermined value that is distinct from the first predetermined value. 
     Another embodiment of a thermally modulated photonic switch comprises a topology-optimized structure that includes dispersed silicon and silicon dioxide. This topology-optimized structure includes an input waveguide and N output waveguides. The topology-optimized structure routes a light beam from the input waveguide to a particular one of the N output waveguides, when the topology-optimized structure is at a corresponding one of N distinct predetermined temperatures that causes a refractive index of the silicon in the topology-optimized structure to assume a corresponding one of N distinct predetermined values. 
     Another embodiment is a method of thermally modulated photonic switching. The method comprises inputting a light beam to an input waveguide of a topology-optimized structure that includes dispersed silicon and silicon dioxide, wherein the topology-optimized structure includes N output waveguides. The method also includes routing the light beam from the input waveguide to a particular one of the N output waveguides by adjusting a temperature of the topology-optimized structure to a corresponding one of N distinct predetermined temperatures that causes a refractive index of the silicon in the topology-optimized structure to assume a corresponding one of N distinct predetermined values. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate various systems, methods, and other embodiments of the disclosure. It will be appreciated that the illustrated element boundaries (e.g., boxes, groups of boxes, or other shapes) in the figures represent one embodiment of the boundaries. In some embodiments, one element may be designed as multiple elements or multiple elements may be designed as one element. In some embodiments, an element shown as an internal component of another element may be implemented as an external component and vice versa. Furthermore, elements may not be drawn to scale. 
         FIG.  1 A  illustrates a topology-optimized (TO) structure in a thermally modulated photonic switch, in accordance with an illustrative embodiment of the invention. 
         FIG.  1 B  illustrates a TO structure in a thermally modulated photonic switch when the TO structure is in a first condition associated with a first temperature and a corresponding first refractive index that routes a light beam to a first output waveguide, in accordance with an illustrative embodiment of the invention. 
         FIG.  1 C  illustrates a TO structure in a thermally modulated photonic switch when the TO structure is in a second condition associated with a second temperature and a corresponding second refractive index that routes a light beam to a second output waveguide, in accordance with an illustrative embodiment of the invention. 
         FIG.  2    is a cross-sectional side view of a thermally modulated photonic switch, in accordance with an illustrative embodiment of the invention. 
         FIG.  3 A  illustrates a cascaded 1 × 8 thermally modulated photonic switch in a first illustrative combination of conditions of the three stages, in accordance with an illustrative embodiment of the invention. 
         FIG.  3 B  illustrates a cascaded 1 × 8 thermally modulated photonic switch in a second illustrative combination of conditions of the three stages, in accordance with an illustrative embodiment of the invention. 
         FIG.  4    illustrates a cascaded 1 × 9 thermally modulated photonic switch, in accordance with an illustrative embodiment of the invention. 
         FIG.  5    is a flowchart of a method of thermally modulated photonic switching, in accordance with an illustrative embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     In various embodiments disclosed herein, a thermally modulated photonic switch (hereinafter sometimes referred to as a “photonic switch”) is constructed through inverse-design techniques. The desired dimensions and performance characteristics are input to a computerized inverse-design algorithm to produce a topology-optimized structure that satisfies the desired dimensions and performance characteristics. In one embodiment, a 1 × 2 (one-input, two-output) thermally modulated photonic switch comprises a topology-optimized (TO) structure that includes dispersed silicon (Si) and silicon dioxide (SiO 2 ) distributed in a nontrivial manner as a result of the inverse-design process. In this embodiment, the TO structure includes an input waveguide, a first output waveguide, and a second output waveguide. The topology-optimized structure routes a light beam from the input waveguide to the first output waveguide, when the topology-optimized structure is at a first predetermined temperature that causes the refractive index of the silicon in the topology-optimized structure to assume a first predetermined value, and the topology-optimized structure routes a light beam from the input waveguide to the second output waveguide, when the topology-optimized structure is at a second predetermined temperature that causes the refractive index of the silicon in the topology-optimized structure to assume a second predetermined value that is distinct from the first predetermined value. In some embodiments, one of the two predetermined temperatures is room temperature (e.g., 293 K). 
     In other embodiments, a thermally modulated photonic switch has one input waveguide and three output waveguides (a 1 × 3 configuration). In such an embodiment, three distinct predetermined temperatures are chosen, each of which causes the refractive index of the silicon in the TO structure to assume one of three distinct predetermined values. The corresponding refractive index of the silicon, in response to a particular one of the three distinct predetermined temperatures of the TO structure, routes a light beam from the input waveguide to a specific one of the three output waveguides. 
     The above concepts can be generalized to the construction of a 1 × N thermally modulated photonic switch, where N is a natural number greater than or equal to 2. In such an embodiment, N distinct predetermined temperatures are chosen, each of which causes the refractive index of the silicon in the TO structure to assume one of N corresponding distinct predetermined values. The corresponding refractive index of the silicon, in response to a particular one of the N distinct predetermined temperatures of the TO structure, routes a light beam from the input waveguide to a specific one of the N output waveguides. 
     In still other embodiments, a plurality of alike or similar 1 × N thermally modulated photonic switches can act as 1 × N switching subunits that are optically interconnected in a cascaded fashion to form a 1 × M photonic switch having k stages and N k  outputs (i.e., M = N k ). In such an embodiment, a particular 1 × N switching subunit at each stage can be thermally controlled to ultimately route a light beam from an input waveguide of the first stage to a specific one of the M output waveguides of the final (output) stage. 
     In some embodiments, a thermally modulated photonic switch operates with a light beam that lies within the infrared portion of the spectrum. In one embodiment, the wavelength of the infrared light beam is 1550 nanometers (nm). This specific wavelength is merely one example, however. 
     The foregoing and additional concepts are discussed in greater detail below. 
     Referring to  FIG.  1 A , it illustrates a TO structure  100  in a thermally modulated photonic switch, in accordance with an illustrative embodiment of the invention. As explained above, the TO structure  100  is obtained through use of a computerized inverse-design (topology-optimization) process. As shown in  FIG.  1 A , TO structure  100  includes an input waveguide  110 , an output waveguide  120 , and an output waveguide  130 . As also shown in  FIG.  1 A , TO structure  100  is made up of dispersed silicon  140  (black) and silicon dioxide  150  (white). The silicon  140  acts as a waveguide to channel/propagate a light beam, whereas the silicon dioxide  150  acts as a cladding to help confine the light beam within the silicon  140 . As illustrated in  FIG.  1 A , the dispersed silicon and silicon dioxide structure produced by the inverse-design process is nontrivial (i.e., it is not a simple geometric structure). 
     In TO structure  100 , the silicon dioxide  150  exhibits a constant refractive index n (e.g., 1.55, in one embodiment). However, the silicon  140  has a refractive index n that can vary by a relatively small factor with temperature. That property of the silicon  140  is exploited by the inverse-design process to produce a TO structure that has the desired characteristics to act as a thermally modulated photonic switch. More specifically, the TO structure  100  being at a first predetermined temperature causes the refractive index n of the silicon  140  in the TO structure  100  to assume a corresponding first predetermined value n 1  that routes a light beam entering the input waveguide  110  to a specific output waveguide (e.g., output waveguide  130  in  FIG.  1 A ). Similarly, the TO structure  100  being at a second predetermined temperature causes the refractive index n of the silicon  140  in the TO structure  100  to assume a corresponding second predetermined value n 2  that routes a light beam entering the input waveguide  110  to the other of the two output waveguides (e.g., output waveguide  120  in  FIG.  1 A ). Note that, throughout this description, these designations of “first” and “second” are arbitrary. 
     In one particular illustrative embodiment, the light beam is infrared light with a wavelength of 1550 nm. The first of the two predetermined temperatures is 293 K, at which the refractive index of the silicon  140  is n 1  = 3.4757. Those skilled in the art will recognize that 293 K corresponds to what is commonly called “room temperature.” In this embodiment, the second of the two predetermined temperatures is 700 K, at which the refractive index of the silicon  140  is n 2  = 3.5648. Though this change in refractive index n is relatively small, the inverse-design process nevertheless produces a structure in which this slight change in refractive index can be exploited to implement a thermally modulated photonic switch with small dimensions such as 2 microns in width and 4 microns in height. With such a design, a light beam can be routed to either output waveguide  130  or output waveguide  120  by controlling the temperature of the TO structure  100 , the selected temperature causing the refractive index of the silicon  140  to be the corresponding predetermined value. 
     The two fundamental states or conditions of a 1 × 2 thermally modulated photonic switch in accordance with the principles and techniques disclosed herein are illustrated in  FIGS.  1 B and  1 C . 
       FIG.  1 B  illustrates a TO structure  100  in a thermally modulated photonic switch when the TO structure  100  is in a first condition (“Condition 1”) associated with a first temperature T 1  and a corresponding first refractive index n 1  that routes a light beam from input waveguide  110  to a first output waveguide  130  (the lower output waveguide in  FIGS.  1 B and  1 C ), in accordance with an illustrative embodiment of the invention. Note that the regions of  FIG.  1 B  with contrasting shading denote the “high” and “low” portions of the light wave (crests and valleys). This particular condition, Condition 1, is referred to again below in connection with a discussion of  FIGS.  3 A and  3 B  (cascaded photonic switches). 
       FIG.  1 C  illustrates a TO structure  100  in a thermally modulated photonic switch when the TO structure  100  is in a second condition (“Condition 2”) associated with a second temperature T 2  and a corresponding second refractive index n 2  that routes a light beam to a second output waveguide  120  (the upper output waveguide in  FIGS.  1 B and  1 C ), in accordance with an illustrative embodiment of the invention. This particular condition, Condition 2, is also referred to again below in connection with a discussion of  FIGS.  3 A and  3 B  (cascaded photonic switches). 
       FIG.  2    is a cross-sectional side view of a thermally modulated photonic switch  200 , in accordance with an illustrative embodiment of the invention. As illustrated in  FIG.  2   , photonic switch  200  is an integrated silicon device that includes a plurality of layers made of different materials. Those layers include silicon handle  210 , buried thermal oxide layer  220 , plasma-enhanced chemical vapor deposition (PECVD) oxide cladding layer  230 , and PECVD oxide cladding layer  240 . In other embodiments, a type of oxide cladding other than PECVD can be used. As shown in  FIG.  2   , buried thermal oxide layer  220  is adjacent to and beneath the TO structure  100 . Silicon handle  210  is adjacent to and beneath buried thermal oxide layer  220 . PECVD oxide cladding layer  230  is adjacent to and on top of buried thermal oxide layer  220 , and PECVD oxide cladding layer  230  covers the TO structure  100  on its top and sides. PECVD oxide cladding layer  240  is adjacent to and on top of PECVD oxide cladding layer  230 . Note that PECVD oxide cladding layer  240  includes a TiW (titanium-tungsten) alloy heater  250  (a “micro-heater”) and a TiW/Al (titanium-tungsten/aluminum) routing layer  260 . In some embodiments, a micro-heater of a different type other than one made of TiW alloy can be used. Also, in some embodiments, the routing layer can be made of a material other than TiW/aluminum. 
     In the embodiment of  FIG.  2   , the micro-heater (TiW alloy heater  250 ) is used to control and adjust the temperature of the TO structure  100  and, ultimately, the temperature of the silicon  140  within the TO structure  100  (refer to the discussion of  FIG.  1 A ). As discussed above, controlling the temperature of the TO structure  100  controls the refractive index n of the silicon  140  in the TO structure  100  to route a light beam to the desired output waveguide of the thermally modulated photonic switch  200 . 
     As discussed above, in some embodiments, more than two distinct temperatures for control of the refractive index n of the silicon in the resulting TO structure are selected prior to the inverse-design process that produces a TO structure. In such an embodiment, N &gt; 2, and the TO structure routes a light beam from an input waveguide to a particular one of the N output waveguides, when the topology-optimized structure is at a corresponding one of N distinct predetermined temperatures that causes a refractive index n of the silicon in the topology-optimized structure to assume a corresponding one of N distinct predetermined values. For example, a 1 × 3 thermally modulated photonic switch can be constructed based on three distinct temperatures and three distinct corresponding values of the refractive index n of the silicon in the TO structure. In some embodiments, this 1 × 3 photonic switch can have similar dimensions to a 1 × 2 embodiment (e.g., a width of approximately 2 microns and a height of approximately 4 microns). An example of a 1 × 3 photonic switch being used as a switching subunit in a cascaded configuration is discussed below in connection with  FIG.  4   . In some embodiments in which N &gt; 2, one of the N distinct predetermined temperatures can be room temperature (e.g., 293 K). 
     As mentioned above, in some embodiments, a plurality of alike or similar 1 × N thermally modulated photonic switches in accordance with the principles and techniques described herein can act as 1 × N switching subunits that are optically interconnected in a cascaded fashion to form a 1 × M photonic switch having k stages and N k  outputs (i.e., M = N k ). In such an embodiment, a particular 1 × N switching subunit at each stage can be thermally controlled to ultimately route a light beam from an input waveguide of the first stage to a specific one of the M output waveguides of the final (output) stage. Examples of cascaded configurations are discussed below in connection with  FIGS.  3 A,  3 B, and  4   . 
       FIG.  3 A  illustrates a cascaded 1 × 8 thermally modulated photonic switch  300  in a first illustrative combination of conditions of the three stages, in accordance with an illustrative embodiment of the invention. Note that cascaded photonic switch  300  includes three stages (numbered within parentheses in  FIG.  3 A ). Thus, cascaded photonic switch  300  includes M = N k  = 2 3  = 8 outputs. In  FIG.  3 A , the TO structure  100  in one of the 1 × 2 switching subunits in each stage k has been set to a particular condition (“Condition 1” or “Condition 2,” as defined above in connection with the discussion of  FIGS.  1 B and  1 C ) to route a light beam from input waveguide  110   a  of TO structure  100   a  to output waveguide  120   f  of TO structure  100   f  in the last (third) stage. That is, selecting Condition 1 for TO structure  100   a  in Stage 1 routes the light beam to input waveguide  110   c  of TO structure  100   c  in Stage 2. Selecting Condition 2 in TO structure  100   c  in Stage 2 routes the light beam to input waveguide  110   f  of TO structure  100   f  in Stage 3. Selecting Condition 2 for TO structure  100   f  routes the light beam to output waveguide  120   f  of TO structure  100   f . Counting the outputs of the final stage from top to bottom, output waveguide  120   f  is Output No. 5 of the 1 × 8 cascaded photonic switch  300 . 
       FIG.  3 B  illustrates a cascaded 1 × 8 thermally modulated photonic switch  300  in a second illustrative combination of conditions of the three stages, in accordance with an illustrative embodiment of the invention. In this example, a light beam is routed from input waveguide  110   a  of TO structure  100   a  to output waveguide  130   d  of TO structure  100   d  in the third and final stage. That is, selecting Condition 2 for TO structure  100   a  in Stage 1 routes the light beam to input waveguide  110   b  of TO structure  110   b  in Stage 2. Selecting Condition 2 for TO structure  110   b  in Stage 2 routes the light beam to input waveguide  110   d  of TO structure  100   d  in Stage 3. Selecting Condition 1 for TO structure  100   d  routes the light beam to output waveguide  130   d  of TO structure  100   d . Counting the outputs of the final stage from top to bottom, output waveguide  130   d  is Output No. 2 of the 1 × 8 cascaded photonic switch  300 . 
     By selecting the correct combination of states (Condition 1 or Condition 2) at each stage, it is possible to route a light beam from input waveguide  110   a  of TO structure  100   a  to any of the 8 outputs of the 1 × 8 cascaded photonic switch  300 . 
       FIG.  4    illustrates a cascaded 1 × 9 thermally modulated photonic switch  400 , in accordance with an illustrative embodiment of the invention. Thus, in this embodiment, M = N k  = 3 2  = 9 because there are two stages, and each switching subunit is a 1 × 3 thermally modulated photonic switch. As shown in  FIG.  4   , a light beam can be input to input waveguide  410   a  of TO structure  405   a . TO structure  405   a  includes output waveguide  420   a , output waveguide  430   a , and output waveguide  440   a . These are, respectively, connected optically with input waveguide  410   b  of TO structure  405   b , input waveguide  410   c  of TO structure  405   c , and input waveguide  410   d  of TO structure  405   d . TO structure  405   b  includes output waveguide  420   b , output waveguide  430   b , and output waveguide  440   b . TO structure  405   c  includes output waveguide  420   c , output waveguide  430   c , and output waveguide  440   c . TO structure  405   d  includes output waveguide  420   d , output waveguide  430   d , and output waveguide  440   d . In any given 1 × 3 switching subunit such as TO structure  405   a , the temperature can be controlled (e.g., via a micro-heater) to select one of three conditions, Condition 1, Condition 2, or Condition 3 corresponding to the respective refractive indexes and output waveguides of that TO structure. Selecting the correct combination of such conditions at each stage (e.g., via the micro-heaters) makes it possible to route a light beam from input waveguide  410   a  of TO structure  405   a  to any of the nine output waveguides ( 420   b - d ,  430   b - d , and  440   b - d ) in the second and final stage of cascaded photonic switch  400 . 
     One of the advantages of 1 × N thermally modulated photonic switches, as described herein, is that such structures can be scaled easily to create cascaded configurations such as those illustrated in  FIGS.  3 A,  3 B, and  4   . 
       FIG.  5    is a flowchart of a method  500  of thermally modulated photonic switching, in accordance with an illustrative embodiment of the invention. Method  500  will be discussed from the perspective of the thermally modulated photonic switches shown in  FIGS.  1 A- 4   . While method  500  is discussed in combination with these thermally modulated photonic switches, it should be appreciated that method  500  is not limited to being implemented within those specific thermally modulated photonic switches, but those thermally modulated photonic switches are instead examples of apparatuses that may implement method  500 . 
     At block  510 , a light beam is input to an input waveguide of a TO structure that includes dispersed silicon and silicon dioxide. As discussed above, such a TO structure includes N output waveguides (outputs), where N is a natural number greater than or equal to 2. In some embodiments N = 2 (e.g., the embodiment shown in  FIGS.  1 A- 1 C ), and in other embodiments N &gt; 2 (e.g., the 1 × 3 TO structures included in  FIG.  4   ). As also discussed above, the TO structure is produced by a computerized inverse-design process that receives, as input, the target performance characteristics of the structure. 
     At block  520 , the light beam is routed from the input waveguide to a particular one of the N output waveguides by adjusting the temperature of the TO structure to a corresponding one of N distinct predetermined temperatures that causes a refractive index n of the silicon in the TO structure to assume a corresponding one of N distinct predetermined values. As discussed above, in some embodiments the temperature of the TO structure is controlled using a micro-heater (refer to TiW alloy heater  250  in  FIG.  2   ). 
     In some embodiments, the method  500  shown in  FIG.  5    can be extended to include optically interconnecting, in a cascaded fashion, a plurality of alike 1 × N switching subunits (1 × N thermally modulated photonic switches as disclosed herein) to form a 1 × M photonic switch having k stages and N k  outputs. In such an embodiment, the individual topology-optimized structure discussed above in connection with method  500  forms a part of a particular 1 × N switching subunit among the plurality of alike 1 × N switching subunits (see  FIGS.  3 A,  3 B, and  4   ). 
     As discussed above, in some embodiments a given TO structure in a 1 × N thermally modulated photonic switch has a width of less than 2.1 microns (e.g., 2 microns) and a height of less than 4.1 microns (e.g., 4 microns). 
     This description next turns to an overview of the principles and mathematical techniques of inverse design that are used in the various embodiments disclosed herein to produce photonic structures such as TO structure  100 . The overview that follows is based on R. Christiansen and O. Sigmund, “Inverse Design in Photonics by Topology Optimization: Tutorial,”  Journal of the Optical Society of America B , Vol. 38, No. 2, February 2021, pp. 496-509. Additional details and examples regarding inverse design, as applied to photonics, can be found in that publication. 
     Solving a structural design problem via inverse design has, as its objective, the identification of a structure that maximizes one or more figures of merit without violating any of the constraints inherent in the problem to be solved. 
     In the discussion that follows, assume a Cartesian coordinate system to model space, such as r = {x, y, z} ∈ ℝ 3  in three dimensions and r = {x,y} ∈ ℝ 2  in two dimensions, where ℝ denotes the field of real numbers. To model the underlying physics, a spatially limited modeling domain Ω having an interior Ω I  and a boundary Γ can be defined. 
     In the embodiments disclosed herein, the inverse-design problems are treated as being time-harmonic, and any transient behavior is ignored. A time-harmonic exponential factor, e jωt , is used to model the time dependence, where t represents time, ω represents angular frequency, and j is the imaginary unit. 
     Given the above framework, the following field equations are used for the electric field ε and magnetic field  
     
       
         
           
             H 
             = 
             
               1 
               
                 
                   μ 
                   0 
                 
               
             
             B 
             : 
           
         
       
     
     
       
         
           
             
               
                   
                   
                   
                   
                   
                   
                 ∇ 
                 ⋅ 
                 ε 
                 = 
                 
                   ρ 
                   
                     
                       ε 
                       r 
                     
                     
                       ε 
                       0 
                     
                   
                 
                 , 
                   
                   
                   
                 ∇ 
                 ⋅ 
                 H 
                 = 
                 0 
                 , 
                   
                   
                 ∇ 
                 × 
                 ε 
                 = 
                 − 
                 
                   μ 
                   0 
                 
                 
                   
                     ∂ 
                     H 
                   
                   
                     ∂ 
                     t 
                   
                 
                 , 
               
             
             
               
                 ∇ 
                 × 
                 H 
                 = 
                 
                   J 
                   f 
                 
                 = 
                 
                   ε 
                   r 
                 
                 
                   ε 
                   0 
                 
                 
                   
                     ∂ 
                     ε 
                   
                   
                     ∂ 
                     t 
                   
                 
                 , 
                   
                   
                   
                 ε 
                 = 
                 E 
                 
                   e 
                   
                     j 
                     ω 
                     t 
                   
                 
                 , 
                   
                   
                   
                 H 
                 = 
                 H 
                 
                   e 
                   
                     j 
                     ω 
                     t 
                   
                 
                 , 
               
             
           
         
       
     
      where J f  and ρ represent the free-current and free-charge densities; ε 0  and µ 0  represent the vacuum electric permittivity and the vacuum magnetic permeability, respectively; the symbol ε r  represents the relative electric permittivity of the medium through which the fields ε and ℌ propagate; and the symbols E and H represent the spatially dependent portion of the electric and magnetic fields, respectively. 
     In some embodiments, the current and charge densities are assumed to be zero in the interior of the model domain. This means that J f (r) = 0 and ρ(r) = 0 for r ∈ Ω I . Based on these assumptions, equations for E and H in Ω I  can be derived as follows:  
     
       
         
           
             ∇ 
             × 
             ∇ 
             × 
             E 
             
               r 
             
             − 
             
               
                 
                   ω 
                   2 
                 
               
               
                 
                   c 
                   2 
                 
               
             
             
               ε 
               r 
             
             
               r 
             
             E 
             
               r 
             
             = 
             0 
             , 
               
               
             r 
             ∈ 
             
               Ω 
               I 
             
             ⊂ 
             
               ℝ 
               3 
             
           
         
       
     
     
       
         
           
             ∇ 
             × 
             
               
                 
                   1 
                   
                     
                       ε 
                       r 
                     
                     
                       r 
                     
                   
                 
                 ∇ 
                 × 
                 H 
                 
                   r 
                 
               
             
             − 
             
               
                 
                   ω 
                   2 
                 
               
               
                 
                   c 
                   2 
                 
               
             
             H 
             
               r 
             
               
             = 
               
             0 
             , 
               
             r 
             ∈ 
             
               Ω 
               I 
             
             ⊂ 
             
               ℝ 
               3 
             
             . 
           
         
       
     
     In Eqs. (2) and (3) above, the speed of light in a vacuum is denoted as  
     
       
         
           
             c 
             = 
             
               1 
               
                 
                   
                     
                       μ 
                       
                         0 
                         
                           ε 
                           0 
                         
                       
                     
                   
                 
               
             
             . 
           
         
       
     
      In some embodiments, additional problem-specific boundary conditions in addition to Eqs. (2) and (3) can be imposed on the boundary of the model domain Γ to account for external fields and to appropriately truncate it. 
     In some embodiments, a two-dimensional (2D) model can be applied instead of the above three-dimensional model. That is, material invariance in the out-of-plane direction (i.e., the z direction) can be assumed. Further, in some embodiments, it can also be assumed that the E or H field is linearly polarized in the z direction so that the above relationships can be reduced to the following scalar Helmholtz equation in two dimensions:  
     
       
         
           
             
               L 
               
                 E 
                 M 
               
             
             
               ϕ 
             
             = 
             ∇ 
             ⋅ 
             
               
                 a 
                 ∇ 
                 ϕ 
               
             
             + 
             
               
                 
                   ω 
                   2 
                 
               
               
                 
                   c 
                   2 
                 
               
             
             b 
             ϕ 
             = 
             0 
             , 
               
               
             r 
             ∈ 
             
               Ω 
               I 
             
             ⊂ 
             
               ℝ 
               2 
             
             . 
           
         
       
     
      In embodiments in which it is necessary to model an E z -polarized field (E x  = E y  = 0) (hereinafter “TE”), ϕ = E z , a = 1, and b = ε r . In the case of a problem including an H z -polarized field (H x  = H y  = 0) (hereinafter “TM”), ϕ = H z ,  
     
       
         
           
             a 
             = 
             
               1 
               
                 
                   ε 
                   r 
                 
               
             
             , 
           
         
       
     
      and b = 1. As those skilled in the art will recognize, given the solution to Eq. (4) above, ε and ℌ (E andH) can be computed using Eq. (1). 
     To solve any structural design problem using inverse design, the problem is defined as a continuous constrained optimization problem, which can be express formally as follows:  
     
       
         
           
             
               
                 s 
                 .t 
                 . 
                   
                   
                   
                   
                 
                   c 
                   i 
                 
                 
                   ξ 
                 
                 = 
                 0 
                 , 
                   
                   
                 
                   c 
                   i 
                 
                 : 
                   
                 
                   
                     
                       0 
                       , 
                       1 
                     
                   
                   
                     
                       Ω 
                       d 
                     
                   
                 
                 → 
                 ℝ 
                 , 
                   
                 i 
                 ∈ 
                 
                   
                     0 
                     , 
                     1 
                     , 
                       
                     … 
                       
                     , 
                     
                       N 
                       i 
                     
                   
                 
                 , 
                   
                 
                   N 
                   i 
                 
                 ∈ 
                 
                   ℕ 
                   0 
                 
                 , 
               
             
             
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 
                   c 
                   i 
                 
                 
                   ξ 
                 
                 &lt; 
                 0 
                 , 
                   
                   
                 
                   c 
                   i 
                 
                   
                 : 
                   
                 
                   
                     
                       0 
                       , 
                       1 
                     
                   
                   
                     
                       Ω 
                       d 
                     
                   
                 
                 → 
                 ℝ 
                 , 
                   
                   
                 , 
                   
                   
                 j 
                 ∈ 
                 
                   
                     0 
                     , 
                     1 
                     , 
                       
                     … 
                       
                     , 
                     
                       N 
                       i 
                     
                   
                 
                 , 
                   
                   
                 
                   N 
                   i 
                 
                 ∈ 
                 
                   ℕ 
                   0 
                 
                 , 
               
             
           
         
       
     
     In problem definition of Eq. (5), ξ(r) ∈ [0,1] represents a continuous field sometimes referred to as the “design field” with respect to which the function Φ, the figure of merit (hereinafter “FOM”), is to be maximized. In Eq. (5), the c i (ξ) = 0 and c j (ξ) &lt; 0 relationships denote N i  equality constraints and N j  inequality constraints, respectively. In formulating an inverse design problem, it is important to select a FOM (e.g., ϕ(ξ) in Eq. (5)) that reliably measures the performance of the structure being designed. In the thermally modulated photonic switch example, the FOM can be described as maximizing the time averaged power flow from the input waveguide into one of the N output waveguides, depending on the N distinct predetermined temperatures. 
     Different FOMs could be employed in solving the above illustrative problem, but what they have in common is that they can be written as simple functions of the electric field, the magnetic field, or both evaluated with respect to points, lines, or areas. 
     Also, the state equation(s) such as Eq. (4) above can be conceptualized as a set of equality constraints as follows:  
     
       
         
           
             
               L 
               
                   
                 k 
               
             
             
               
                 
                   x 
                   k 
                 
               
             
             = 
             
               f 
               k 
             
             , 
               
               
             k 
             ∈ 
             
               
                 1 
                 , 
                 2 
                 , 
                   
                 … 
                   
                 , 
                 
                   N 
                   k 
                 
               
             
             , 
               
               
               
             
               N 
               k 
             
             ∈ 
             ℕ 
             , 
           
         
       
     
      where the L k  operator applies the characteristics of the physical system to the state field x k  for a given excitation f k . 
     In solving an optimization problem in the form shown above in Eq. (5), the continuous design field ξ(r) is used to interpolate the material parameters modeled by the state equation between the background material(s) and the material(s) constituting the structure under design. Which material interpolation techniques are used depends on the particular problem. In the thermally modulated photonic switch example, the following interpolation functions can be used to interpolate between silicon dioxide and silicon at N distinct predetermined temperatures:  
     
       
         
           
             
               ε 
               r 
               N 
             
             
               
                 ξ 
                 
                   r 
                 
               
             
             = 
             
               ε 
               
                 r 
                 , 
                 S 
                 i 
               
               N 
             
             + 
             ξ 
             
               r 
             
             
               
                 
                   ε 
                   
                     r 
                     , 
                     S 
                     i 
                     
                       O 
                       2 
                     
                   
                 
                 − 
                 
                   ε 
                   
                     r 
                     , 
                     S 
                     i 
                   
                   N 
                 
               
             
             , 
           
         
       
     
      where  
     
       
         
           
             
               ε 
               
                 r 
                 , 
                 S 
                 i 
               
               N 
             
           
         
       
     
      and ε r,SiO 2    represent the relative permittivity of silicon and silicon dioxide, respectively. In this case,  
     
       
         
           
             ξ 
             = 
             0 
             ⇔ 
             
               ε 
               r 
             
             = 
             
               ε 
               
                 r 
                 , 
                 S 
                 i 
               
               N 
             
             , 
           
         
       
     
      and ξ = 1 ⇔ ε r  = ε r,SiO 2   . 
     In some computer-software-based inverse-design implementations, gradient-based algorithms are employed. For example, in some embodiments a technique sometimes referred to in the literature as the “Method of Moving Asymptotes (MMA)” is used. MMA is a gradient-based method for solving constrained nonlinear optimization problems. To overcome the computational difficulties associated with finite differences, some embodiments make use of adjoint sensitivity analysis, which requires solving only one equation for the FOM and an additional equation for each constraint in the optimization problem, regardless of how large the design space happens to be. In one embodiment, COMSOL Multiphysics software (https://www.comsol.com/) is used to solve the physics equilibrium, and MATLAB (https://www.mathworks.com/) is used to perform the iterative optimization updates. 
     Detailed embodiments are disclosed herein. However, it is to be understood that the disclosed embodiments are intended only as examples. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a basis for the claims and as a representative basis for teaching one skilled in the art to variously employ the aspects herein in virtually any appropriately detailed structure. Further, the terms and phrases used herein are not intended to be limiting but rather to provide an understandable description of possible implementations. Various embodiments are shown in  FIGS.  1 A- 5   , but the embodiments are not limited to the illustrated structure or application. 
     Herein, designations such as “first” or “second” are arbitrary and do not signify priority or importance. Rather, they are used to refer to particular elements among a plurality of elements of the same type (e.g., a set of waveguides, a set of temperatures, a set of refractive indexes, etc.). 
     The terms “a” and “an,” as used herein, are defined as one or more than one. The term “plurality,” as used herein, is defined as two or more than two. The term “another,” as used herein, is defined as at least a second or more. The terms “including” and/or “having,” as used herein, are defined as comprising (i.e. open language). The phrase “at least one of ... and ....” as used herein refers to and encompasses any and all possible combinations of one or more of the associated listed items. As an example, the phrase “at least one of A, B, and C” includes A only, B only, C only, or any combination thereof (e.g. AB, AC, BC or ABC). 
     As used herein, “cause” or “causing” means to make, command, instruct, and/or enable an event or action to occur or at least be in a state where such event or action may occur, either in a direct or indirect manner. 
     Aspects herein can be embodied in other forms without departing from the spirit or essential attributes thereof. Accordingly, reference should be made to the following claims rather than to the foregoing specification, as indicating the scope hereof.