Patent Publication Number: US-2022214502-A1

Title: Dual-Polarization Grating Coupler

Description:
CLAIM OF PRIORITY 
     This application claims priority under 35 U.S.C. 119(e) to U.S. Provisional Patent Application No. 63/133,416, filed on Jan. 4, 2021, the disclosure of which is incorporated herein by reference in its entirety for all purposes. 
    
    
     BACKGROUND 
     1. Field of the Invention 
     The present invention relates to optical data communication. 
     2. Description of the Related Art 
     Optical data communication systems operate by modulating laser light to encode digital data patterns. The modulated light is transmitted through an optical data network from a sending node to a receiving node. The modulated light having arrived at the receiving node is de-modulated to obtain the original digital data patterns. Therefore, implementation and operation of optical data communication systems is dependent upon having reliable and efficient mechanisms for transmitting light and detecting light at different nodes within the optical data network. It is within this context that the present invention arises. 
     SUMMARY 
     In an example embodiment, an optical grating coupler is disclosed. The optical grating coupler includes a primary layer formed of a material having a first refractive index. The optical grating coupler also includes a first plurality of scattering elements formed within the primary layer. The first plurality of scattering elements has a second refractive index that is different than the first refractive index. The optical grating coupler also includes a secondary layer formed over the primary layer. The secondary layer is formed of a material having a third refractive index. The optical grating coupler also includes a second plurality of scattering elements formed within the secondary layer. The second plurality of scattering elements has a fourth refractive index that is different than the third refractive index. The fourth refractive index is different than the second refractive index. At least some of the second plurality of scattering elements at least partially overlap corresponding ones of the first plurality of scattering elements. 
     In an example embodiment, a method is disclosed for combining light beams. The method includes having an optical grating coupler that includes a primary layer and a secondary layer. The primary layer includes a first plurality of scattering elements. The secondary layer includes a second plurality of scattering elements. At least some of the second plurality of scattering elements at least partially overlap corresponding ones of the first plurality of scattering elements. The method also includes directing a first beam of light into a first lateral side of the optical grating coupler. The method also includes directing a second beam of light into a second lateral side of the optical grating coupler. The second lateral side is adjacent to the first lateral side along a perimeter of the optical grating coupler. The first plurality of scattering elements and the second plurality of scattering elements collectively scatter both the first beam of light and the second beam of light into a third beam of light. 
     Other aspects and advantages of the invention will become more apparent from the following detailed description, taken in conjunction with the accompanying drawings, illustrating by way of example the present invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  shows a top view of a vertical grating coupler, in accordance with some embodiments. 
         FIG. 1B  shows a perspective view of the vertical grating coupler, in accordance with some embodiments. 
         FIG. 1C  shows a close-up view of the top of the vertical grating coupler, in accordance with some embodiments. 
         FIG. 2A  shows a vertical cross-section through a portion of the vertical grating coupler within the plane of symmetry, in accordance with some embodiments. 
         FIG. 2B  shows the same configuration of the vertical grating coupler of  FIG. 2A  with the light propagation directions reversed, in accordance with some embodiments. 
         FIG. 3A  shows a vertical cross-section through a portion of the vertical grating coupler within the plane of symmetry, in which the refractive indexes of materials is reversed with respect to the configuration of  FIGS. 2A and 2B , in accordance with some embodiments. 
         FIG. 3B  shows the same configuration of the vertical grating coupler of  FIG. 3A  with the light propagation directions reversed, in accordance with some embodiments. 
         FIG. 4A  shows an example configuration of the primary layer, in accordance with some embodiments. 
         FIG. 4B  shows a top view of the grid formed by gridlines, in accordance with some embodiments. 
         FIG. 4C  shows the scattering elements of the primary layer (level L 1 ) arranged in the v-shaped section S 1 , in accordance with some embodiments. 
         FIG. 4D  shows the scattering elements of the primary layer (level L 1 ) arranged in the v-shaped section S 2 , in accordance with some embodiments. 
         FIG. 4E  shows the scattering elements of the primary layer (level L 1 ) arranged in the v-shaped section S 3 , in accordance with some embodiments. 
         FIG. 4F  shows the scattering elements of the primary layer (level L 1 ) arranged in the polygonal-shaped section S 4 , in accordance with some embodiments. 
         FIG. 5A  shows an example configuration of the secondary layer, in accordance with some embodiments. 
         FIG. 5B  shows the scattering elements of the secondary layer (level L 2 ) arranged in the v-shaped section S 1 , in accordance with some embodiments. 
         FIG. 5C  shows the scattering elements of the secondary layer (level L 2 ) arranged in the v-shaped section S 2 , in accordance with some embodiments. 
         FIG. 5D  shows the scattering elements of the secondary layer (level L 2 ) arranged in the v-shaped section S 3 , in accordance with some embodiments. 
         FIG. 5E  shows the scattering elements of the secondary layer (level L 2 ) arranged in the polygonal-shaped section S 4 , in accordance with some embodiments. 
         FIG. 6  shows a close-up top view of the vertical grating coupler in which four example scattering element groups are identified, in accordance with some embodiments. 
         FIG. 7A  shows a perspective view of the first example scattering element group, in accordance with some embodiments. 
         FIG. 7B  shows a top view of the first example scattering element group, in accordance with some embodiments. 
         FIG. 8A  shows a perspective view of the second example scattering element group, in accordance with some embodiments. 
         FIG. 8B  shows a top view of the second example scattering element group, in accordance with some embodiments. 
         FIG. 9A  shows a perspective view of the third example scattering element group, in accordance with some embodiments. 
         FIG. 9B  shows a top view of the third example scattering element group, in accordance with some embodiments. 
         FIG. 10A  shows a perspective view of the fourth example scattering element group, in accordance with some embodiments. 
         FIG. 10B  shows a top view of the fourth example scattering element group, in accordance with some embodiments. 
         FIG. 11  shows a side view of the vertical grating coupler implemented within a chip to optically couple an off-chip light beam into an optical fiber, in accordance with some embodiments. 
         FIG. 12  shows a side view of the vertical grating coupler implemented within the chip to optically couple the off-chip light beam into an optical waveguide, in accordance with some embodiments. 
         FIG. 13  shows a side view of the vertical grating coupler within the chip to optically couple the off-chip light beam into another vertical grating coupler, in accordance with some embodiments. 
         FIG. 14  shows a side view of the vertical grating coupler implemented within the chip to direct the off-chip light beam toward a beam-turning device/assembly, in accordance with some embodiments. 
         FIG. 15A  shows a top view of an example scattering element as drawn in layout, in accordance with some embodiments. 
         FIG. 15B  shows a top view of an as-fabricated scattering element corresponding to the example scattering element drawn in layout in  FIG. 15A , in accordance with some embodiments. 
         FIG. 16  shows a flowchart of a method for combining light beams, in accordance with some embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     In the following description, numerous specific details are set forth in order to provide an understanding of the present invention. It will be apparent, however, to one skilled in the art that the present invention may be practiced without some or all of these specific details. In other instances, well known process operations have not been described in detail in order not to unnecessarily obscure the present invention. 
     Embodiments are disclosed herein for a vertical grating coupler that is configured to polarization-multiplex light from two on-chip optical waveguides into two polarizations of an off-chip beam of light. The vertical grating coupler embodiments disclosed herein are compatible with scalable silicon manufacturing processes, such as complementary metal oxide semiconductor (CMOS) manufacturing processes. The various vertical grating coupler embodiments disclosed herein include at least a first array of light scattering elements within a first layer and a second array of light scattering elements within a second layer located either above or below the first layer. In some embodiments, incoming light from the two on-chip optical waveguides interacts with the first array of light scattering elements within the first layer and the second array of light scattering elements within the second layer so as to be directed into the off-chip beam of light. In this manner, when light of a first polarization (TE or TM) is directed through the first on-chip optical waveguide into the vertical grating coupler and light of a second polarization (TE or TM) (different than the first polarization) is directed through the second on-chip optical waveguide into the vertical grating coupler, both the light of the first polarization and the light of the second polarization is scattered (turned) by the vertical grating coupler into the off-chip beam of light. 
     In various embodiments, the first and second layers that include the first and second arrays of light scattering elements, respectively, are independently patternable with respect to each other. In some embodiments, the first and second layers of the vertical grating coupler are formed as respective patterned silicon layers. In some embodiments, the first array of light scattering elements is formed of a material that has a lower refractive index relative to a material of the first layer within which the first array of light scattering elements is formed. And, the second array of light scattering elements is formed of a material that has a higher refractive index relative to a material of the second layer within which the second array of light scattering elements is formed. In some embodiments, the first array of light scattering elements is formed of a material that has a higher refractive index relative to a material of the first layer within which the first array of light scattering elements is formed. And, the second array of light scattering elements is formed of a material that has a lower refractive index relative to a material of the second layer within which the second array of light scattering elements is formed. 
     The various vertical grating coupler embodiments disclosed herein combine three features. A first feature of the vertical grating coupler embodiments is that light scattering elements in different layers are laterally offset with respect to each other in order to provide high directionality to the scattered light. A second feature of the vertical grating coupler embodiments is that light scattering elements in each of the layers are positioned in accordance with a regular array (or grid) that satisfies a phase-matching condition for scattering light from each of the two input optical waveguides into the off-chip beam of light. In some embodiments, a centerpoint of each of the scattering elements within a given layer is positioned at a gridpoint of a regular array (or grid) defined for the given layer to satisfy the phase-matching condition for scattering incoming light from each of the two input optical waveguides into the off-chip beam of light. A third feature of the various vertical grating coupler embodiments is that the light scattering elements within a given layer have apodized scattering strength as a function of position relative to the two input optical waveguides. Also, respective input optical waveguide tapers are provided to transition from the two input optical waveguides to the vertical grating coupler in order to produce a flat phase front of the light as it enters the vertical grating coupler. In some embodiments, the input optical waveguide tapers are configured as adiabatic tapers. 
       FIG. 1A  shows a top view of a vertical grating coupler  100 , in accordance with some embodiments.  FIG. 1B  shows a perspective view of the vertical grating coupler  100 , in accordance with some embodiments.  FIG. 1C  shows a close-up view of the top of the vertical grating coupler  100 , in accordance with some embodiments. The vertical grating coupler  100  includes a region of light scattering elements  102 , as shown within the dashed line  101 . There are different types of light scattering elements  102  in multiple layers of the vertical grating coupler  100 . Therefore, the reference numeral  102  (typical) is used to refer to scattering elements in general. In some embodiments, the scattering elements  102  are positioned with respect to a regular arrays (or grid) within the two separate layers. A first optical waveguide  103  and a first optical waveguide taper  105  are provided to direct light into the vertical grating coupler  100 , as indicated by arrow  113 . A second optical waveguide  107  and a second optical waveguide taper  109  are also provided to direct light into the vertical grating coupler  100 , as indicated by arrow  115 . In some embodiments, the first optical waveguide taper  105  is connected (optically coupled) to a first lateral side  100 A of the vertical grating coupler  100 , and the second optical waveguide taper  109  is connected (optically coupled) to a second lateral side  100 B of the vertical grating coupler  100 , where the first lateral side  100 A and the second lateral side  100 B of the vertical grating coupler  100  are adjacent to each other along the perimeter of the vertical grating coupler  100 . In some embodiments, the first optical waveguide taper  105  and the second optical waveguide taper  109  are positioned and oriented in a substantially symmetric manner relative to a plane of symmetry  111  that extends vertically through the center of the vertical grating coupler  100  at a location between the first optical waveguide taper  105  and the second optical waveguide taper  109 . In some embodiments, light of a first polarization travels through the first optical waveguide  103 , through the first optical waveguide taper  105 , and into the vertical grating coupler  100 , as indicated by arrow  113 . And, light of a second polarization travels through the second optical waveguide  107 , through the second optical waveguide taper  109 , and into the vertical grating coupler  100 , as indicated by arrow  115 . In this manner, in some embodiments, the first optical waveguide taper  105  and the second optical waveguide taper  109  couple light of two polarizations, respectively, into the vertical grating coupler  100 , with the vertical grating coupler  100  functioning to redirect the light of the two polarizations into the off-chip beam of light. 
       FIG. 2A  shows a vertical cross-section through a portion of the vertical grating coupler  100  within the plane of symmetry  111 , in accordance with some embodiments. In the example embodiment of  FIG. 2A , the vertical grating coupler  100  includes a primary layer  205  formed on a buried oxide (BOX) layer  201 , and a secondary layer  207  formed above the primary layer  205 . A top layer  203  is formed over the secondary layer  207 . In some embodiments, the BOX layer  201  is formed of silicon dioxide. However, it should be understood that in various embodiments the BOX layer  201  can be formed of essentially any material used to formed a BOX layer in silicon-on-insulator (SOI) semiconductor device design and fabrication. In some embodiments, the top layer  203  is formed of an oxide material, such as silicon dioxide. However, it should be understood that in various embodiments the top layer  203  can be formed of essentially any type of dielectric material used in back-end-of-line (BEOL) semiconductor device design and fabrication. 
     In the example configuration of  FIG. 2A , the primary layer  205  is formed of a high refractive index (high-n) optical waveguide material  205 A, such as crystalline silicon (refractive index of about 3.97) or other similar material. The primary layer  205  includes low refractive index (low-n) scattering elements  102 A,  102 B, etc., formed/disposed within the optical waveguide material  205 A. In some embodiments, the primary layer  205  is formed by depositing/forming a continuous layer of the high refractive index optical waveguide material  205 A on the BOX layer  201 , followed by etching of cavities/holes into/through the optical waveguide material  205 A (where the cavities/holes correspond to shapes and sizes of the scattering elements  102 A,  102 B, etc.), followed by depositing low refractive index material within the cavities/holes within the optical waveguide material  205 A to form the low refractive index scattering elements  102 A,  102 B, etc. In some embodiments, a planarization process (etch planarization, plasma-based planarization, and/or chemical mechanical planarization (CMP)) is performed on the primary layer  205  to obtain high level of planarity across the top surface of the primary layer  205  prior to formation of the secondary layer  207 . In this manner, the high refractive index optical waveguide material  205 A forms a field of the primary layer  205 , with the low refractive index scattering elements distributed throughout the field of the primary layer  205 . 
     In the example configuration of  FIG. 2A , the secondary layer  207  is formed of a low refractive index (low-n) field material  207 A, such as silicon nitride (refractive index of about 2.02) or other similar material. The secondary layer  207  includes high refractive index (high-n) scattering elements  102 C,  102 D, etc., formed/disposed within the field material  207 A. In some embodiments, the secondary layer  207  is formed by depositing/forming a continuous layer of the low refractive index optical waveguide material  207 A on the primary layer  205 , followed by etching of holes through the field material  207 A (where the holes correspond to shapes and sizes of the scattering elements  102 C,  102 D, etc.), followed by depositing high refractive index material within the holes within the field material  207 A to form the high refractive index scattering elements  102 C,  102 D, etc. In some embodiments, a planarization process (etch planarization, plasma-based planarization, and/or chemical mechanical planarization (CMP)) is performed on the secondary layer  207  to obtain high level of planarity across the top surface of the secondary layer  207  prior to formation of the top layer  203 . In this manner, the high refractive index scattering elements  102 C,  102 D, etc., are formed and distributed within the low refractive index field material  207 A. 
     In some embodiments, the scattering elements  102  of the primary layer  205  (e.g., the low refractive index scattering elements  102 A,  102 B, etc.) are formed of an oxide material (such as silicon dioxide or other optically similar oxide material) in a layer otherwise made of silicon (such as crystalline silicon or polycrystalline silicon (polysilicon) or other optically similar material). Also, in some embodiments, the scattering elements  102  of the secondary layer  207  (e.g., the high refractive index scattering elements  102 C,  102 D, etc.) are formed of a silicon material (such as polycrystalline silicon (polysilicon) or other optically similar material) in a layer otherwise made of lower refractive index material (such as silicon nitride of other optically similar material). 
     In some embodiments, incoming light that enters the vertical grating coupler  100  (through the first optical waveguide  103  and first optical waveguide taper  105  and/or through the second optical waveguide  107  and second optical waveguide taper  109 ) is directed into the primary layer  205 , as indicated by arrow  209 . Also, in some embodiments, incoming light that enters the vertical grating coupler  100  (through the first optical waveguide  103  and first optical waveguide taper  105  and/or through the second optical waveguide  107  and second optical waveguide taper  109 ) is optionally directed into the secondary layer  207 , as indicated by the dashed arrow  211 . In some embodiments, the material used to form the first optical waveguide  103 , first optical waveguide taper  105 , the second optical waveguide  107 , and second optical waveguide taper  109  is the same material as the high refractive index optical waveguide material  205 A used to form the field of the primary layer  205 . For example, in some embodiments, crystalline silicon is used to form the first optical waveguide  103 , first optical waveguide taper  105 , the second optical waveguide  107 , and second optical waveguide taper  109 , and is the high refractive index optical waveguide material  205 A used to form the field of the primary layer  205  of the vertical grating coupler  100 . Also, in some embodiments, incoming light is directed into the secondary layer  207 , as indicated by dashed arrow  211 , through one or more polysilicon optical waveguides formed in a vertical space corresponding to the secondary layer  207 . For example, in some embodiments, polysilicon optical waveguides are formed vertically above crystalline silicon optical waveguides to direct the incoming light that enters the vertical grating coupler  100  into both the primary layer  205  and the secondary layer  207 , as indicated by arrows  209  and  211 , respectively. However, it should be understood that in some embodiments, the incoming light that enters the vertical grating coupler  100  is directed primarily into the primary layer  205 , as indicated by arrow  209 , but may not be substantially directed into the secondary layer  207 . 
     The incoming light that enters the vertical grating coupler  100  is scattered by the scattering elements  102 A,  102 B, etc., within the primary layer  205  and by the scattering elements  102 C,  102 D, etc., within the secondary layer  207 , such that the light that enters the vertical grating coupler  100  is directed out of the vertical grating coupler  100  in a controlled direction as the off-chip beam of light, as indicated by arrows  213 . In some embodiments, the first optical waveguide taper  105  and the second optical waveguide taper  109  are formed in a symmetric manner with respect to the vertical plane of symmetry  111 . Also, the scattering elements  102 A,  102 B, etc., within the primary layer  205  are formed in a symmetric manner with respect to the vertical plane of symmetry  111 . Also, the scattering elements  102 C,  102 D, etc., within the secondary layer  207  are formed in a symmetric manner with respect to the vertical plane of symmetry  111 . In this manner, the outgoing light that exits the vertical grating coupler  100  in the off-chip beam of light is directed at an angle  214  that falls substantially within the vertical plane of symmetry  111 . In various embodiments, the configuration of the scattering elements  102 A,  102 B, etc., within the primary layer  205  and the configuration of the scattering elements  102 C,  102 D, etc., within the secondary layer  207  are designed to control the angle  214  of the outgoing light that exits the vertical grating coupler  100  substantially within the vertical plane of symmetry  111 , where the angle  214  is controlled relative to a vector  212  normal to the bottom of the chip (normal to the bottom of the BOX layer  201 ). 
     In some embodiments, the incoming light that enters the vertical grating coupler  100  through the first optical waveguide taper  105  has a first polarization, and the incoming light that enters the vertical grating coupler  100  through the second optical waveguide taper  109  has a second polarization that is different than the first polarization. In these embodiments, the outgoing light in the off-chip beam of light that exits through the BOX layer  201 , as indicated by the arrows  213 , includes both light having the first polarization and light having the second polarization. In this manner, the vertical grating coupler  100  functions to polarization-multiplex light from the two on-chip optical waveguides  103  and  107  into two polarizations, respectively, within the off-chip beam of light, as indicated by arrows  213 . 
     It should be understood that the direction of light propagation through the vertical grating coupler  100  is reversable. For example,  FIG. 2B  shows the same configuration of the vertical grating coupler  100  of  FIG. 2A  with the light propagation directions reversed, in accordance with some embodiments. In  FIG. 2B  the incoming light enters the vertical grating coupler through the BOX layer  201 , as indicated by arrows  215 , and the outgoing light exits the vertical grating coupler  100  at the lateral sides  100 A and  100 B of the vertical grating coupler  100 , as indicated by arrows  217 , with a portion of the outgoing light traveling through the first optical waveguide taper  105  and into the optical waveguide  103 , and with a portion of the outgoing light traveling through the second optical waveguide taper  109  and into the optical waveguide  107 . 
     In some embodiments, the materials used to form the example vertical grating coupler  100  of  FIGS. 2A and 2B  can be reversed with respect to refractive index. For example,  FIG. 3A  shows a vertical cross-section through a portion of the vertical grating coupler  100  within the plane of symmetry  111 , in which the refractive indexes of materials is reversed with respect to the configuration of  FIGS. 2A and 2B , in accordance with some embodiments. In the example configuration of  FIG. 3A , the primary layer  205  is formed of a low refractive index (low-n) optical waveguide material  205 B, such as silicon nitride (refractive index of about 2.02) or other similar material. The primary layer  205  includes high refractive index (high-n) scattering elements  102 E,  102 F, etc., such as polysilicon (refractive index of about 3.93) or other similar material, formed/disposed within the optical waveguide material  205 B. In some embodiments, the primary layer  205  is formed by depositing/forming a continuous layer of the low refractive index optical waveguide material  205 B on the BOX layer  201 , followed by etching of cavities/holes into/through the optical waveguide material  205 B (where the cavities/holes correspond to shapes and sizes of the scattering elements  102 E,  102 F, etc.), followed by depositing high refractive index material within the cavities/holes within the optical waveguide material  205 B to form the high refractive index scattering elements  102 E,  102 F, etc. In this manner, the low refractive index optical waveguide material  205 B forms a field of the primary layer  205 , with the high refractive index scattering elements distributed throughout the field of the primary layer  205 . 
     In the example configuration of  FIG. 3A , the secondary layer  207  is formed of a high refractive index (high-n) field material  207 B, such as polysilicon or other similar material. The secondary layer  207  includes low refractive index (low-n) scattering elements  102 G,  102 H, etc., formed/disposed within the field material  207 B. In some embodiments, the secondary layer  207  is formed by depositing/forming a continuous layer of the high refractive index optical waveguide material  207 B on the primary layer  205 , followed by etching of holes through the field material  207 B (where the holes correspond to shapes and sizes of the scattering elements  102 G,  102 H, etc.), followed by depositing low refractive index material within the holes within the field material  207 B to form the low refractive index scattering elements  102 G,  102 H, etc. In this manner, the low refractive index scattering elements  102 G,  102 H, etc., are formed and distributed within the high refractive index field material  207 B. 
     The incoming light that enters the vertical grating coupler  100  is scattered by the scattering elements  102 E,  102 F, etc., within the primary layer  205  and by the scattering elements  102 G,  102 H, etc., within the secondary layer  207 , such that the light that enters the vertical grating coupler  100  is directed out of the vertical grating coupler  100  in a controlled direction in the off-chip beam of light, as indicated by arrows  213 . In some embodiments, the first optical waveguide taper  105  and the second optical waveguide taper  109  are formed in a symmetric manner with respect to the vertical plane of symmetry  111 . Also, the scattering elements  102 E,  102 F, etc., within the primary layer  205  are formed in a symmetric manner with respect to the vertical plane of symmetry  111 . Also, the scattering elements  102 G,  102 H, etc., within the secondary layer  207  are formed in a symmetric manner with respect to the vertical plane of symmetry  111 . In this manner, the outgoing light that exits the vertical grating coupler  100  is directed at the angle  214  that falls substantially within the vertical plane of symmetry  111 . In various embodiments, the configuration of the scattering elements  102 E,  102 F, etc., within the primary layer  205  and the configuration of the scattering elements  102 G,  102 H, etc., within the secondary layer  207  are designed to control the angle  214  of the outgoing light of the off-chip beam that exits the vertical grating coupler  100  substantially within the vertical plane of symmetry  111 . 
     As previously mentioned, the direction of light propagation through the vertical grating coupler  100  is reversable. For example,  FIG. 3B  shows the same configuration of the vertical grating coupler  100  of  FIG. 3A  with the light propagation directions reversed, in accordance with some embodiments. In  FIG. 3B  the incoming light enters the vertical grating coupler  100  through the BOX layer  201 , as indicated by arrows  215 , and the outgoing light exits the vertical grating coupler  100  at the lateral sides  100 A and  100 B of the vertical grating coupler  100 , as indicated by arrow  217 , with a portion of the outgoing light traveling through the first optical waveguide taper  105  and into the optical waveguide  103 , and with a portion of the outgoing light traveling through the second optical waveguide taper  109  and into the optical waveguide  107 . 
     In some embodiments, the scattering elements  102  are formed in two vertical layers of the vertical grating coupler  100 . In some embodiments, the scattering elements  102  of the vertical grating coupler  100  are formed in two vertical layers of independently patternable silicon.  FIG. 4A  shows an example configuration of the primary layer  205 , in accordance with some embodiments. Scattering elements  102 -L 1 S 1 ,  102 -L 1 S 2 ,  102 -L 1 S 3 , and  102 -L 1 S 4  are positioned on a grid  401  such as shown in  FIG. 4B .  FIG. 4B  shows a top view of the grid  401  formed by gridlines  401 A and  401 B, in accordance with some embodiments. The scattering elements  102 -L 1 S 1 ,  102 -L 1 S 2 ,  102 -L 1 S 3 , and  102 -L 1 S 4  are the “+” shapes (cross-shaped structures) shown in  FIG. 4A . In some embodiments, a centerpoint (or centroid) of each scattering element  102 -L 1 S 1 ,  102 -L 1 S 2 ,  102 -L 1 S 3 , and  102 -L 1 S 4  is positioned at a corresponding gridpoint of the grid  401 , where a given gridpoint of the grid  401  corresponds to a crossing location of any two perpendicularly oriented gridlines  401 A and  401 B of the grid  401 . However, in some embodiments, the centerpoint (or centroid) of one or more of the scattering elements  102 -L 1 S 1 ,  102 -L 1 S 2 ,  102 -L 1 S 3 , and  102 -L 1 S 4  is positioned at a location that is a specified distance and direction from a corresponding gridpoint of the grid  401 . 
     In the example vertical grating coupler  100  of  FIG. 4A , the scattering elements  102 -L 1 S 1  of the primary layer  205  (level L 1 ) are arranged in a v-shaped section S 1 .  FIG. 4C  shows the scattering elements  102 -L 1 S 1  of the primary layer  205  (level L 1 ) arranged in the v-shaped section S 1 , in accordance with some embodiments. Also, the scattering elements  102 -L 1 S 2  of the primary layer  205  (level L 1 ) are arranged in a v-shaped section S 2  that is located behind the v-shaped section S 1  relative to the propagation directions of the incoming light as indicated by arrows  113  and  115 .  FIG. 4D  shows the scattering elements  102 -L 1 S 2  of the primary layer  205  (level L 1 ) arranged in the v-shaped section S 2 , in accordance with some embodiments. Also, the scattering elements  102 -L 1 S 3  of the primary layer  205  (level L 1 ) are arranged in a v-shaped section S 3  that is located behind the v-shaped section S 2  relative to the propagation directions of the incoming light as indicated by arrows  113  and  115 .  FIG. 4E  shows the scattering elements  102 -L 1 S 3  of the primary layer  205  (level L 1 ) arranged in the v-shaped section S 3 , in accordance with some embodiments. Also, the scattering elements  102 -L 1 S 4  of the primary layer  205  (level L 1 ) are arranged in a polygonal-shaped section S 4  that is located behind the v-shaped section S 3  relative to the propagation directions of the incoming light as indicated by arrows  113  and  115 .  FIG. 4F  shows the scattering elements  102 -L 1 S 4  of the primary layer  205  (level L 1 ) arranged in the polygonal-shaped section S 4 , in accordance with some embodiments. In this manner, in some embodiments, the scattering elements  102 -L 1 S 1 ,  102 -L 1 S 2 ,  102 -L 1 S 3 , and  102 -L 1 S 4  in the respective sections S 1 , S 2 , S 3 , and S 4  are configured to implement apodization of light scattering strength along the incoming light propagation directions indicated by arrows  113  and  115 . 
     The apodization of light scattering strength is implemented by providing weaker light scattering elements closer to the first and second optical waveguide tapers  105  and  109 , and by providing progressively stronger light scattering elements along the light propagation directions (arrows  113 ,  115 ) moving away from the first and second optical waveguide tapers  105  and  109  toward a central region of the vertical grating coupler  100 . In various embodiments, the light scattering strength of a given scattering element  102  (e.g., providing weaker (less) light scattering versus providing stronger (more) light scattering) is controlled by controlling the shape and/or size and/or orientation and/or material of the given scattering element  102 . For example, smaller sized scattering elements  102  will have less interaction across the wavefront of the incoming light and will therefore scatter less of the incoming light. Conversely, larger sized scattering elements  102  will have more interaction across the wavefront of the incoming light and will therefore scatter more of the incoming light. In this manner, in some embodiments, the apodization of light scattering strength is implemented by configuring the light scattering elements  102 -L 1 S 1  in the v-shaped section S 1  to have smaller size than the light scattering elements  102 -L 1 S 2  in the v-shaped section S 2 ; and by configuring the light scattering elements  102 -L 1 S 2  in the v-shaped section S 2  to have smaller size than the light scattering elements  102 -L 1 S 3  in the v-shaped section S 3 ; and by configuring the light scattering elements  102 -L 1 S 3  in the v-shaped section S 3  to have smaller size than the light scattering elements  102 -L 1 S 4  in the polygonal-shaped section S 4 . Also, in some embodiments, the apodization of light scattering strength is supported by configuring the light scattering elements  102 -L 1 S 1  in the v-shaped section S 1  to have a smaller cross-section for interaction with the wavefront of the incoming light than the light scattering elements  102 -L 1 S 2  in the v-shaped section S 2 ; and by configuring the light scattering elements  102 -L 1 S 2  in the v-shaped section S 2  to have a smaller cross-section for interaction with the wavefront of the incoming light than the light scattering elements  102 -L 1 S 3  in the v-shaped section S 3 ; and by configuring the light scattering elements  102 -L 1 S 3  in the v-shaped section S 3  to have a smaller cross-section for interaction with the wavefront of the incoming light than the light scattering elements  102 -L 1 S 4  in the polygonal-shaped section S 4 . In various embodiments, the cross-section for interaction with the wavefront of the incoming light of a given scattering element ( 102 -L 1 S 1 ,  102 -L 1 S 2 ,  102 -L 1 S 3 ,  102 -L 1 S 4 ) is controlled by configuring the size and/or shape and/or orientation of the given scattering element ( 102 -L 1 S 1 ,  102 -L 1 S 2 ,  102 -L 1 S 3 ,  102 -L 1 S 4 ) with respect to the propagation direction ( 113 ,  115 ) of the wavefront of the incoming light. 
     Therefore, in some embodiments, the principle of apodization is implemented by providing weakly-scattering elements near the optical waveguide tapers  105  and  109  and more strongly-scattering elements near the center of the off-chip beam of light corresponding to the central region of the vertical grating coupler  100  along the plane of symmetry  111 . For example, the scattering elements  102 -L 1 S 1  in the v-shaped section S 1  closest to the optical waveguide tapers  105 ,  109  are sized to create small deviations from the nominal optical guiding structure of the primary layer  205  that would otherwise exist in the absence of the scattering elements  102 -L 1 S 1 . Increased deviation (in refractive index) from the nominal optical guiding structure of the primary layer  205  is provided in a progressive manner by the scattering elements  102 -L 1 S 2 ,  102 -L 1 S 3 , and  102 -L 1 S 4  in the sections S 2 , S 3 , and S 4  respectively. In the region near the center of the vertical grating coupler  100  along the plane of symmetry  111 , the scattering elements  102 -L 1 S 4  having the largest light scattering strength are disposed to create the largest deviation (in refractive index) from the nominal optical guiding structure of the primary layer  205 , which results in more light scattering into the off-chip beam of light. 
       FIG. 5A  shows an example configuration of the secondary layer  207 , in accordance with some embodiments. Scattering elements  102 -L 2 S 1 ,  102 -L 2 S 2 ,  102 -L 2 S 3 , and  102 -L 2 S 4  are positioned in reference to the grid  401  as shown in  FIG. 4B . The scattering elements  102 -L 2 S 1 ,  102 -L 2 S 2 ,  102 -L 2 S 3 , and  102 -L 2 S 4  are the “+” shapes (cross-shaped structures) shown in  FIG. 5A . In some embodiments, a centerpoint (or centroid) of a given scattering element  102 -L 2 S 1 ,  102 -L 2 S 2 ,  102 -L 2 S 3 ,  102 -L 2 S 4  is positioned at a corresponding gridpoint of the grid  401 . However, in some embodiments, the centerpoint (or centroid) of a given scattering element  102 -L 2 S 1 ,  102 -L 2 S 2 ,  102 -L 2 S 3 ,  102 -L 2 S 4  is positioned at a location that is a specified distance and direction from a corresponding gridpoint of the grid  401 . 
     In the example vertical grating coupler  100  of  FIG. 5A , the scattering elements  102 -L 2 S 1  of the secondary layer  207  (level L 2 ) are arranged in a v-shaped section S 1 .  FIG. 5B  shows the scattering elements  102 -L 2 S 1  of the secondary layer  207  (level L 2 ) arranged in the v-shaped section S 1 , in accordance with some embodiments. Also, the scattering elements  102 -L 2 S 2  of the secondary layer  207  (level L 2 ) are arranged in a v-shaped section S 2  that is located behind the v-shaped section S 1  relative to the propagation directions of the incoming light as indicated by arrows  113  and  115 .  FIG. 5C  shows the scattering elements  102 -L 2 S 2  of the secondary layer  207  (level L 2 ) arranged in the v-shaped section S 2 , in accordance with some embodiments. Also, the scattering elements  102 -L 2 S 3  of the secondary layer  207  (level L 2 ) are arranged in a v-shaped section S 3  that is located behind the v-shaped section S 2  relative to the propagation directions of the incoming light as indicated by arrows  113  and  115 .  FIG. 5D  shows the scattering elements  102 -L 2 S 3  of the secondary layer  207  (level L 2 ) arranged in the v-shaped section S 3 , in accordance with some embodiments. Also, the scattering elements  102 -L 2 S 4  of the secondary layer  207  (level L 2 ) are arranged in a polygonal-shaped section S 4  that is located behind the v-shaped section S 3  relative to the propagation directions of the incoming light as indicated by arrows  113  and  115 .  FIG. 5E  shows the scattering elements  102 -L 2 S 4  of the secondary layer  207  (level L 2 ) arranged in the polygonal-shaped section S 4 , in accordance with some embodiments. In this manner, in some embodiments, the scattering elements  102 -L 2 S 1 ,  102 -L 2 S 2 ,  102 -L 2 S 3 , and  102 -L 2 S 4  in the respective sections S 1 , S 2 , S 3 , and S 4  are configured to implement apodization of light scattering strength along the incoming light propagation directions indicated by arrows  113  and  115 . 
     As with the primary layer  205 , the apodization of light scattering strength in the secondary layer  207  is implemented by providing weaker light scattering elements closer to the first and second optical waveguide tapers  105  and  109 , and by providing progressively stronger light scattering elements along the light propagation directions (arrows  113 ,  115 ) moving away from the first and second optical waveguide tapers  105  and  109  toward the central region of the vertical grating coupler  100 . In various embodiments, the light scattering strength of the scattering elements  102 -L 2 S 1 ,  102 -L 2 S 2 ,  102 -L 2 S 3 ,  102 -L 2 S 4  is controlled by controlling the shape and/or size and/or orientation and/or material of the scattering elements  102 -L 2 S 1 ,  102 -L 2 S 2 ,  102 -L 2 S 3 ,  102 -L 2 S 4 . In some embodiments, the apodization of light scattering strength is implemented by configuring the light scattering elements  102 -L 2 S 1  in the v-shaped section S 1  to have smaller size than the light scattering elements  102 -L 2 S 2  in the v-shaped section S 2 ; and by configuring the light scattering elements  102 -L 2 S 2  in the v-shaped section S 2  to have smaller size than the light scattering elements  102 -L 2 S 3  in the v-shaped section S 3 ; and by configuring the light scattering elements  102 -L 2 S 3  in the v-shaped section S 3  to have smaller size than the light scattering elements  102 -L 2 S 4  in the polygonal-shaped section S 4 . Also, in some embodiments, the apodization of light scattering strength is supported by configuring the light scattering elements  102 -L 2 S 1  in the v-shaped section S 1  to have a smaller cross-section for interaction with the wavefront of the incoming light than the light scattering elements  102 -L 2 S 2  in the v-shaped section S 2 ; and by configuring the light scattering elements  102 -L 2 S 2  in the v-shaped section S 2  to have a smaller cross-section for interaction with the wavefront of the incoming light than the light scattering elements  102 -L 2 S 3  in the v-shaped section S 3 ; and by configuring the light scattering elements  102 -L 2 S 3  in the v-shaped section S 3  to have a smaller cross-section for interaction with the wavefront of the incoming light than the light scattering elements  102 -L 2 S 4  in the polygonal-shaped section S 4 . In various embodiments, the cross-section for interaction with the wavefront of the incoming light of a given scattering element ( 102 -L 2 S 1 ,  102 -L 2 S 2 ,  102 -L 2 S 3 ,  102 -L 2 S 4 ) is controlled by configuring the size and/or shape and/or orientation of the given scattering element ( 102 -L 2 S 1 ,  102 -L 2 S 2 ,  102 -L 2 S 3 ,  102 -L 2 S 4 ) with respect to the propagation direction ( 113 ,  115 ) of the wavefront of the incoming light. 
     Therefore, in some embodiments, the principle of apodization is implemented in the secondary layer  207  by providing weakly-scattering elements near the optical waveguide tapers  105  and  109  and more strongly-scattering elements near the center of the off-chip beam of light corresponding to the central region of the vertical grating coupler  100  along the plane of symmetry  111 . For example, the scattering elements  102 -L 2 S 1  in the v-shaped section S 1  closest to the optical waveguide tapers  105 ,  109  are sized to create small deviations (in refractive index) from the nominal optical guiding structure of the secondary layer  207  that would otherwise exist in the absence of the scattering elements  102 -L 2 S 1 . Increased deviation (in refractive index) from the nominal optical guiding structure of the secondary layer  207  is provided in a progressive manner by the scattering elements  102 -L 2 S 2 ,  102 -L 2 S 3 , and  102 -L 2 S 4  in the sections S 2 , S 3 , and S 4  respectively. In the region near the center of the vertical grating coupler  100  along the plane of symmetry  111 , the scattering elements  102 -L 2 S 4  having the largest light scattering strength are disposed to create the largest deviation (in refractive index) from the nominal optical guiding structure of the secondary layer  207 , which results in more light scattering into the off-chip beam of light. 
     In some embodiments, the size and shape of scattering elements within a given layer and a given section are substantially the same, where the given section includes at least two rows of scattering elements located at approximately the same distance from the nearest optical waveguide taper  105 ,  109 . For example, in some embodiments, the scattering elements  102 -L 1  S 1  in section S 1  of the primary layer  205  have a first size and a first shape, and the scattering elements  102 -L 2 S 1  in section S 1  of the secondary layer  207  have a second size and a second shape. Also, in some embodiments, the scattering elements  102 -L 1 S 2  in section S 2  of the primary layer  205  have a third size and a third shape, and the scattering elements  102 -L 2 S 2  in section S 2  of the secondary layer  207  have a fourth size and a fourth shape. Also, in some embodiments, the scattering elements  102 -L 1 S 3  in section S 3  of the primary layer  205  have a fifth size and a fifth shape, and the scattering elements  102 -L 2 S 3  in section S 3  of the secondary layer  207  have a sixth size and a sixth shape. Also, in some embodiments, the scattering elements  102 -L 1 S 4  in section S 4  of the primary layer  205  have a seventh size and a seventh shape, and the scattering elements  102 -L 2 S 4  in section S 4  of the secondary layer  207  have an eighth size and an eighth shape. 
     Because the sections S 1 , S 2 , and S 3  (of both the primary layer  205  and the secondary layer  207 ) are v-shaped, incoming light from the optical waveguide tapers  105  and  109  can be thought of as sequentially passing through the first section S 1 , then through the second section S 2 , then through the third section S 3 , and then into the fourth section S 4 . In some embodiments, in view of this sequential propagation of light through sections S 1 , S 2 , S 3 , and S 4 , apodization includes configuration/tuning of the scattering elements within a given section in a given layer to achieve a desired apodization effect. In some embodiments, sizes and shapes of scattering elements  102  within a given section in a given layer are symmetrized, and/or tuned to optimize efficiency, process compatibility, and/or other parameters. Therefore, in some embodiments, the size and/or shape and/or orientation of scattering elements  102  within a given section (S 1 , S 2 , S 3 , S 4 ) in a given layer ( 205 ,  207 ) are not identical, but are systematically varied to achieve a desired apodization effect. 
     In some embodiments, the vertical grating coupler  100  is approximately symmetric with respect to the plane of symmetry  111  (see  FIG. 1B ) that extends normal to the chip&#39;s surface. In these embodiments, the optical waveguide tapers  105  and  109  are placed (sized, shaped, positioned, and oriented) approximately symmetrically with respect to reflection across the plane of symmetry  111 . In these embodiments, the primary layer  205  provides for light reflection symmetry across a line of symmetry corresponding to intersection of the plane of symmetry  111  with the primary layer  205 . Also, in these embodiments, the secondary layer  207  provides for light reflection symmetry across a line of symmetry corresponding to intersection of the plane of symmetry  111  with the secondary layer  207 . Therefore, in these embodiments, the axis of the center of the outgoing beam of light (such as indicated by arrows  213  in  FIGS. 2A and 3A ) falls approximately within the plane of symmetry  111 . 
     In some embodiments, one or more scattering elements  102  within the primary layer  205  are grouped with one or more scattering elements  102  within the secondary layer  207  into a scattering element group to provide a prescribed light scattering effect. For example, in some embodiments, a given scattering element  102  within the primary layer  205  is grouped with a given scattering element  102  within the secondary layer  207  to form a scattering element group, such that the scattering element  102  within the scattering element group in the primary layer  205  and the scattering element  102  within the scattering element group in the secondary layer  205  work together to direct light into the outgoing beam of light as indicated by arrows  213  in  FIGS. 2A and 3A . For example, in some embodiments, a scattering element group includes one scattering element  102  in the primary layer  205  and one scattering element  102  in the secondary layer  207 . In some embodiments, the scattering element group is configured so that an offset exists between a position of the scattering element  102  in the primary layer  205  and a position of the scattering element  102  in the secondary layer  207 , where the offset is along the direction of the off-chip outgoing beam of light (along a direction that is parallel to both the plane of symmetry  111  and the chip, e.g., the bottom surface of the BOX layer  201  (see  FIGS. 2A, 3A )). 
     In some embodiments, highly directional light scattering is achieved through phase-matching. In some embodiments, the offset between the two scattering elements  102  of the scattering element group (the offset between the scattering element  102  in the primary layer  205  and the scattering element  102  in the secondary layer  207 ) is defined to lead to destructive interference in one of the vertical directions, so that light is more efficiently coupled into the other vertical direction, where the offset is in the direction parallel to both the plane of symmetry  111  and the chip. In some embodiments, where the two optical waveguide tapers  105  and  109  direct incoming light into the vertical grating coupler  100  in the directions  113  and  115 , respectively, on adjacent lateral sides  100 A and  100 B of the vertical grating coupler  100 , the offset between the two scattering elements  102  of the scattering element group in the different vertical layers (in the primary layer  205  and secondary layer  207 , respectively) is made along a direction that is substantially half-way between the two taper light propagation directions  113  and  115 . In this manner, in the example vertical grating coupler  100 , the offset between the two scattering elements  102  in the different vertical layers of the scattering element group is made along a direction that is substantially parallel to both the plane of symmetry  111  and the plane of the chip, e.g., bottom of the BOX layer  201 . 
       FIG. 6  shows a close-up top view of the vertical grating coupler  100  in which four example scattering element groups  601 ,  603 ,  605 , and  607  are identified, in accordance with some embodiments. A first example scattering element group  601  includes one scattering element  102 -L 1 S 1  in the primary layer  205  in the first section S 1 , and one scattering element  102 -L 2 S 1  in the secondary layer  207  in the first section S 1 . In the example vertical grating coupler  100 , the first example scattering element group  601  is repeated throughout the first v-shaped section S 1 . In this manner, each scattering element  102 -L 1 S 1  in the first section S 1  in the primary layer  205  belongs to a corresponding instance of the first example scattering element group  601 , and each scattering element  102 -L 2 S 1  in the first section S 1  in the secondary layer  207  also belongs to a corresponding instance of the first example scattering element group  601 .  FIG. 7A  shows a perspective view of the first example scattering element group  601 , in accordance with some embodiments.  FIG. 7B  shows a top view of the first example scattering element group  601 , in accordance with some embodiments. In the first example scattering element group  601 , a centerpoint (centroid)  703  of the scattering element  102 -L 2 S 1  in the secondary layer  207  is offset in the negative x-direction by a distance  705  relative to a centerpoint (centroid)  701  of the corresponding scattering element  102 -L 1 S 1  in the primary layer  205 , where the x-direction is parallel to both the plane of symmetry  111  and the plane of the chip. Also, in the first example scattering element group  601 , the centerpoint (centroid)  703  of the scattering element  102 -L 2 S 1  in the secondary layer  207  and the centerpoint (centroid)  701  of the corresponding scattering element  102 -L 1 S 1  in the primary layer  205  are co-located at a substantially same position in the y-direction, where the y-direction is perpendicular to the plane of symmetry  111  and parallel to the plane of the chip. In some embodiments, the centerpoint (centroid)  701  of the scattering element  102 -L 1 S 1  of the first example scattering element group  601  is located in the primary layer  205  at a gridpoint of the grid  401  as shown in  FIG. 4B . 
       FIG. 6  also shows a second example scattering element group  603  that includes one scattering element  102 -L 1 S 2  in the primary layer  205  in the second section S 2 , and one scattering element  102 -L 2 S 2  in the secondary layer  207  in the second section S 2 . In the example vertical grating coupler  100 , the second example scattering element group  603  is repeated throughout the second v-shaped section S 2 . In this manner, each scattering element  102 -L 1 S 2  in the second section S 2  in the primary layer  205  belongs to a corresponding instance of the second example scattering element group  603 , and each scattering element  102 -L 2 S 2  in the second section S 2  in the secondary layer  207  also belongs to a corresponding instance of the second example scattering element group  603 .  FIG. 8A  shows a perspective view of the second example scattering element group  603 , in accordance with some embodiments.  FIG. 8B  shows a top view of the second example scattering element group  603 , in accordance with some embodiments. In the second example scattering element group  603 , a centerpoint (centroid)  803  of the scattering element  102 -L 2 S 2  in the secondary layer  207  is offset in the positive x-direction by a distance  805  relative to a centerpoint (centroid)  801  of the corresponding scattering element  102 -L 1 S 2  in the primary layer  205 , where the x-direction is parallel to both the plane of symmetry  111  and the plane of the chip. Also, in the second example scattering element group  603 , the centerpoint (centroid)  803  of the scattering element  102 -L 2 S 2  in the secondary layer  207  and the centerpoint (centroid)  801  of the corresponding scattering element  102 -L 1 S 2  in the primary layer  205  are co-located at a substantially same position in the y-direction, where the y-direction is perpendicular to the plane of symmetry  111  and parallel to the plane of the chip. In some embodiments, the centerpoint (centroid)  801  of the scattering element  102 -L 1 S 2  of the second example scattering element group  603  is located in the primary layer  205  at a gridpoint of the grid  401  as shown in  FIG. 4B . 
       FIG. 6  also shows a third example scattering element group  605  that includes one scattering element  102 -L 1 S 3  in the primary layer  205  in the third section S 3 , and one scattering element  102 -L 2 S 3  in the secondary layer  207  in the third section S 3 . In the example vertical grating coupler  100 , the third example scattering element group  605  is repeated throughout the third v-shaped section S 3 . In this manner, each scattering element  102 -L 1 S 3  in the third section S 3  in the primary layer  205  belongs to a corresponding instance of the third example scattering element group  605 , and each scattering element  102 -L 2 S 3  in the third section S 3  in the secondary layer  207  also belongs to a corresponding instance of the third example scattering element group  605 .  FIG. 9A  shows a perspective view of the third example scattering element group  605 , in accordance with some embodiments.  FIG. 9B  shows a top view of the third example scattering element group  605 , in accordance with some embodiments. In the third example scattering element group  605 , a centerpoint (centroid)  903  of the scattering element  102 -L 2 S 3  in the secondary layer  207  is offset in the positive x-direction by a distance  905  relative to a centerpoint (centroid)  901  of the corresponding scattering element  102 -L 1 S 3  in the primary layer  205 , where the x-direction is parallel to both the plane of symmetry  111  and the plane of the chip. Also, in the third example scattering element group  605 , the centerpoint (centroid)  903  of the scattering element  102 -L 2 S 3  in the secondary layer  207  and the centerpoint (centroid)  901  of the corresponding scattering element  102 -L 1 S 3  in the primary layer  205  are co-located at a substantially same position in the y-direction, where the y-direction is perpendicular to the plane of symmetry  111  and parallel to the plane of the chip. In some embodiments, the centerpoint (centroid)  901  of the scattering element  102 -L 1 S 3  of the third example scattering element group  605  is located in the primary layer  205  at a gridpoint of the grid  401  as shown in  FIG. 4B . 
       FIG. 6  also shows a fourth example scattering element group  607  that includes one scattering element  102 -L 1 S 4  in the primary layer  205  in the fourth section S 4 , and one scattering element  102 -L 2 S 4  in the secondary layer  207  in the fourth section S 4 . In the example vertical grating coupler  100 , the fourth example scattering element group  607  is repeated throughout the polygonal-shaped section S 4 . In this manner, each scattering element  102 -L 1 S 4  in the fourth section S 4  in the primary layer  205  belongs to a corresponding instance of the fourth example scattering element group  607 , and each scattering element  102 -L 2 S 4  in the fourth section S 4  in the secondary layer  207  also belongs to a corresponding instance of the fourth example scattering element group  607 .  FIG. 10A  shows a perspective view of the fourth example scattering element group  607 , in accordance with some embodiments.  FIG. 10B  shows a top view of the fourth example scattering element group  607 , in accordance with some embodiments. In the fourth example scattering element group  607 , a centerpoint (centroid)  1003  of the scattering element  102 -L 2 S 4  in the secondary layer  207  is offset in the positive x-direction by a distance  1005  relative to a centerpoint (centroid)  1001  of the corresponding scattering element  102 -L 1 S 4  in the primary layer  205 , where the x-direction is parallel to both the plane of symmetry  111  and the plane of the chip. Also, in the fourth example scattering element group  607 , the centerpoint (centroid)  1003  of the scattering element  102 -L 2 S 4  in the secondary layer  207  and the centerpoint (centroid)  1001  of the corresponding scattering element  102 -L 1 S 4  in the primary layer  205  are co-located at a substantially same position in the y-direction, where the y-direction is perpendicular to the plane of symmetry  111  and parallel to the plane of the chip. In some embodiments, the centerpoint (centroid)  1001  of the scattering element  102 -L 1 S 4  of the fourth example scattering element group  607  is located in the primary layer  205  at a gridpoint of the grid  401  as shown in  FIG. 4B . 
     A phase-matching directionality mechanism is provided by the offset of scattering elements  102 -L 1 S 1  and  102 -L 2 S 1  within scattering element group  601 , and the offset of scattering elements  102 -L 1 S 2  and  102 -L 2 S 2  within scattering element group  603 , and the offset of scattering elements  102 -L 1 S 3  and  102 -L 2 S 3  within scattering element group  605 , and the offset of scattering elements  102 -L 1 S 4  and  102 -L 2 S 4  within scattering element group  607 . It should be appreciated that the phase-matching directionality mechanism discussed herein is intrinsic to the configuration of the primary layer  205  and the secondary layer  207 , and does not require a complex substrate or incorporation of additional reflecting layers. Therefore, the vertical grating coupler  100  disclosed herein does not require reflecting layers to achieve low light loss. Unlike the vertical grating coupler  100  disclosed herein, other previous grating designs have attempted to achieve low light loss by employing one or more reflecting layers, including a multi-layer substrate designed to reflect light “upward” (away from the wafer substrate), which causes challenges with regard to fabrication cost and incompatibility with standard CMOS fabrication processes. These challenges are obviated by the vertical grating coupler  100  disclosed herein. 
       FIG. 11  shows a side view of the vertical grating coupler  100  implemented within a chip  1105  to optically couple an off-chip light beam  1101  into an optical fiber  1103 , in accordance with some embodiments. The vertical grating coupler  100  scatters light from the first optical waveguide  103  and the second optical waveguide  107  into the off-chip light beam  1101 . In some embodiments, the off-chip light beam  1101  is the fundamental mode of the optical fiber  1103 . In some embodiments, the off-chip light beam  1101  is approximated as a gaussian beam. 
       FIG. 12  shows a side view of the vertical grating coupler  100  implemented within the chip  1105  to optically couple the off-chip light beam  1101  into an optical waveguide  1203 , in accordance with some embodiments. In some embodiments, the optical waveguide  1203  is implemented within a planar lightwave circuit  1201 . In some embodiments, the chip  1105  is attached to the planar lightwave circuit  1201 . 
       FIG. 13  shows a side view of the vertical grating coupler  100  within the chip  1105  to optically couple the off-chip light beam  1101  into another vertical grating coupler  100   x , in accordance with some embodiments. In some embodiments, the other vertical grating coupler  100   x  is implemented within another chip  1301 . In some embodiments, the chip  1105  is attached to the other chip  1301 . The other vertical grating coupler  100   x  is implemented to direct light from the light beam  1101  into one or more optical waveguide(s)  1303  formed within the other chip  1301 . 
       FIG. 14  shows a side view of the vertical grating coupler  100  implemented within the chip  1105  to direct the off-chip light beam  1101  toward a beam-turning device/assembly  1401 , in accordance with some embodiments. In some embodiments, the beam-turning device/assembly  1401  is an angled reflector. In some embodiments, the beam-turning device/assembly  1401  is implemented to redirects the light beam  1101  into an optical waveguide  1403 , in accordance with some embodiments. In some embodiments, the optical waveguide  1403  is an optical fiber. In some embodiments, the optical waveguide  1403  is formed within a planar lightwave circuit or other chip. 
     In some embodiments, the vertical grating coupler  100  includes some scattering elements  102  that are not part of a scattering element group. Also, in some embodiments, the vertical grating coupler  102  includes some scattering elements  102  that are positioned with a non-phase-matched offset distance and/or direction relative to another scattering element  102  within a scattering element group, where the non-phase-matched offset distance and/or direction is different (possibly substantially different) from that indicated by a light phase-matching condition for directionality of the off-chip light beam. For example,  FIGS. 6, 7A , and  7 B show that the scattering element  102 -L 1 S 1  in the primary layer  205  and the scattering element  102 -L 2 S 1  in the secondary layer  207  (within a given instance of the scattering element group  601  in section S 1  closest to the two optical waveguide tapers  105  and  109 ) have a very small offset  705  (nearly zero) in the x-direction with respect to each other, and have essentially zero offset in the y-direction with respect to each other. This very small offset in the x-direction between the scattering element  102 -L 1 S 1  in the primary layer  205  and the scattering element  102 -L 2 S 1  in the secondary layer  207  within a given instance of the scattering element group  601  in section S 1  closest to the two optical waveguide tapers  105  and  109  improves the apodization by lowering a total amount of light scattering from the section S 1  of the vertical grating coupler  100 . 
     In some embodiments, placement of scattering elements  102  so that their respective centerpoints (centroids) fall on a regular array (such as the grid  401  of  FIG. 4B , by way of example) within at least one vertical layer (within the primary layer  205 , the secondary layer  207 , or any other higher level layer) provides a number of advantages compared to scattering element  102  placement patterns that are not in an array configuration or that have a substantially distorted and/or curved array configuration. In some embodiments, placement of scattering elements  102  so that their respective centerpoints (centroids) fall on a regular array (such as the grid  401  of  FIG. 4B , by way of example) within at least one vertical layer (within the primary layer  205 , the secondary layer  207 , or any other higher level layer) provides for good light phase-matching between the flat phase-fronts and the off-chip light beam. Also, having a regular pattern of scattering elements  102  within a given layer, such as by placement of scattering elements  102  so that their respective centerpoints (centroids) fall on a regular array, can beneficially reduce the complexity of chip design, such as when the impact of process variation needs to be inferred from device measurements, by way of example. Also, with the scattering elements  102  disposed in a regular pattern in a given layer, each scattering element  102  has a known nearest-neighbor distance in all directions (at all locations around the scattering element  102 ), so that no scattering element  102  causes a non-compliance problem with regard to design rule and/or fabrication process constraints. In some embodiments, the scattering elements  102  within a given layer are disposed in a regular pattern such that a given scattering element  102  has essentially the same nearest-neighbor distance at all locations, so that no scattering element  102  causes an adverse impact with regard to design-rule and/or fabrication process constraints, such as with regard to minimum separation between scattering elements  102 , by way of example. 
     In some embodiments, the shape of individual scattering elements  102  is selected to maximize the worst-case transmission of light over any superposition of fields from the two optical waveguide modes of the optical waveguide tapers  105  and  109  into the off-chip light beam. In some embodiments, the relative phases and intensities of the input light beams from the two optical waveguide tapers  105  and  109  respectively map onto polarization, and the worst-case transmission of light is across the relative phase and relative intensity as well as wavelength. The worst-case transmission of light as mentioned herein is understood as the transmission of light for the worst relative phase and ratio of power between the fields of the two input optical waveguides, e.g., of the two optical waveguide tapers  105  and  109 . 
     In some embodiments, the various scattering elements  102  within the vertical grating coupler  100  have shapes such as shown by the scattering elements  102 -L 1 S 1  and  102 -L 2 S 1  in the scattering element group  601  (see  FIGS. 7A and 7B ), and such as shown by the scattering elements  102 -L 1 S 2  and  102 -L 2 S 2  in the scattering element group  603  (see  FIGS. 8A and 8B ), and such as shown by the scattering elements  102 -L 1 S 3  and  102 -L 2 S 3  in the scattering element group  605  (see  FIGS. 9A and 9B ), and such as shown by the scattering elements  102 -L 1  S 4  and  102 -L 2 S 4  in the scattering element group  607  (see  FIGS. 10A and 10B ). However, in other embodiments, any of the scattering elements  102  in the vertical grating coupler  100  (in any layer (primary layer  205 , secondary layer  207 , or other additional higher level layer) and/or in any section (S 1 , S 2 , S 3 , S 4 , or any other section) of any layer) can have a size and shape that is customized to provide a prescribed light scattering effect. 
       FIG. 15A  shows a top view of an example scattering element  102 -E 1 A as drawn in layout, in accordance with some embodiments. The example scattering element  102 -E 1 A includes a body section  1501 A having a substantially rectangular shape defined by a width dimension d 3  and a length dimension d 4 . The example scattering element  102 -E 1 A also includes corner sections  1501 B,  1501 C,  1501 D, and  1501 E that project outward from respective corners of the body section  1501 A. In the example of  FIG. 15A , each of the corner sections  1501 B,  1501 C,  1501 D,  1501 E has a substantially rectangular shape defined by a width dimension d 1  and a length dimension d 2 . The length dimension d 2  is referred to as a “corner emphasis” parameter/dimension. The scattering element  102 -E 1 A is configured as a union of the rectangular-shaped body section  1501 A and each of the rectangular-shaped corner sections  1501 B,  1501 C,  1501 D,  1501 E, where the lengthwise centerline of each rectangular-shaped corner section  1501 B,  1501 C,  1501 D,  1501 E bisects the ninety degree angle between the adjacent sides of the rectangular-shaped body section  1501 A that meet at the vertex (corner) of the rectangular-shaped body section  1501 A from which the corner section  1501 B,  1501 C,  1501 D,  1501 E projects outward from the body section  1501 A. For example, the corner section  1501 E has a lengthwise centerline  1503  that bisects the ninety degree angle  1504  between the adjacent sides  1505  and  1506  of the body section  1501 A that meet at the vertex (corner)  1507  of the body section  1501 A from which the corner section  1501 E projects outward from the body section  1501 A. Also, in some embodiments, the scattering element  102 -E 1 A is configured such that each rectangular-shaped corner section  1501 B,  1501 C,  1501 D,  1501 E (as drawn in layout) has two adjacent vertices coincident with adjacent sides of the body section  1501 A. For example, the corner section  1501 E has two adjacent vertices  1508  and  1509  coincident with the adjacent sides  1505  and  1506 , respectively, of the body section  1501 A. 
     It should be understood that the as-fabricated shape of a scattering element  102  can differ from the corresponding layout-drawn shape of the scattering element  102  due to fabrication processes and/or limitations. Therefore, scattering element  102  shapes drawn in layout may differ from scattering element  102  shapes that occur in actual devices. For example,  FIG. 15B  shows a top view of an as-fabricated scattering element  102 -E 1 B corresponding to the example scattering element  102 -E 1 A drawn in layout in  FIG. 15A , in accordance with some embodiments.  FIG. 15B  shows that the as-fabricated scattering element  102 -E 1 B has corner rounding effects due to fabrication processes and/or limitations. Specifically, the convex corners of the scattering element  102 -E 1 B (corresponding to the outer corners of the rectangular-shaped corner section  1501 B,  1501 C,  1501 D,  1501 E) are rounded to an extent, and the concave corners of the scattering element  102 -E 1 B (corresponding to the interfaces between the rectangular-shaped corner sections  1501 B,  1501 C,  1501 D,  1501 E and the body section  1501 A) are rounded to an extent. In some embodiments, scattering elements  102 -E 1 A/ 102 -E 1 B that have a smaller dimension d 2  are characterized as being more “round” than scattering elements  102 -E 1 A/E 1 B that have a larger dimension d 2 . The example scattering elements  102 -E 1 A/ 102 -E 1 B also demonstrate how a scattering element  102  can be elongated along a given direction (along the lengthwise direction of the body section  1501 A corresponding to the dimension d 4 ). In this manner, in some embodiments, individual scattering elements  102  are elongated along one direction in order to maximize the worst-case light transmission over any superposition of fields from the two optical waveguide modes of the optical waveguide tapers  105  and  109  into the off-chip light beam. In some embodiments, implementation of such elongated scattering elements  102  within the vertical grating coupler  100  provides for improved apodization (matching between scattered light and the off-chip light beam profile) by reducing or increasing light scattering strength at specified locations within the vertical grating coupler  100 . 
     In some embodiments, scattering element  102  shapes that are drawn in layout are subject to design rules imposed by a fabrication process and/or fabrication facility (foundry). For example, in some embodiments, the layout-drawn scattering element  1501 A of  FIG. 15A  has to satisfy a design rule that requires the dimension d 1  to be greater than a specified minimum size (width). In some embodiments, the scattering element  1501 A is defined to have the dimension d 1  be greater than about 100 nanometers in order to improve compatibility with standard data-preparation and fabrication process flows. However, in some embodiments, the scattering element  1501 A is defined to have the dimension d 1  be greater than about 70 nanometers. In some embodiments, if the scattering element  102 -E 1 A/ 102 -E 1 B has a small total area and a large corner emphasis dimension d 2 , then a minimum-spacing design rule and/or fabrication process constraint may direct an adjustment of the scattering element  102 -E 1 A/ 102 -E 1 B shape and/or size. For example, in some embodiments, due to design rule and/or fabrication process constraints, scattering elements  102 -E 1 A/ 102 -E 1 B having smaller total area are configured to have a smaller corner emphasis dimension d 2  so as to be more round-shaped, and scattering elements  102 -E 1 A/ 102 -E 1 B having larger total area are configured to have a larger corner emphasis dimension d 2  so as to be more spike-shaped. In some embodiments, scattering elements  102 -E 1 A/ 102 -E 1 B positioned near the input optical waveguide tapers  105  and  109  are configured to have the smallest total area and the smallest corner emphasis dimension d 2  of the scattering elements  102  within the vertical grating coupler  100 . In some embodiments, the smallest scattering elements  102 -E 1 A/ 102 -E 1 B positioned near the input optical waveguide tapers  105  and  109  are configured to have corner emphasis dimension d 2  near zero. 
     It should be understood that in various embodiments, a scattering element  102  (such as scattering elements  102 -L 1 S 1 ,  102 -L 2 S 1 ,  102 -L 1 S 2 ,  102 -L 2 S 2 ,  102 -L 1 S 3 ,  102 -L 2 S 3 ,  102 -L 1 S 4 ,  102 -L 2 S 4 ,  102 -E 1 A/ 102 -E 1 B) within the vertical grating coupler  100  can be a high refractive index scattering element  102  (such as silicon or material with similar optical properties) within an otherwise low refractive index layer (such as an oxide material, e.g., silicon dioxide, among others). Alternatively, in various embodiments, a scattering element  102  (such as scattering elements  102 -L 1 S 1 ,  102 -L 2 S 1 ,  102 -L 1 S 2 ,  102 -L 2 S 2 ,  102 -L 1 S 3 ,  102 -L 2 S 3 ,  102 -L 1 S 4 ,  102 -L 2 S 4 ,  102 -E 1 A/ 102 -E 1 B) within the vertical grating coupler  100  can be a low refractive index scattering element  102  (such as an oxide material, e.g., silicon dioxide, among others) within an otherwise high refractive index layer (such as silicon or other material with similar optical properties). Also, in various embodiments, a scattering element  102  (such as scattering elements  102 -L 1 S 1 ,  102 -L 2 S 1 ,  102 -L 1 S 2 ,  102 -L 2 S 2 ,  102 -L 1 S 3 ,  102 -L 2 S 3 ,  102 -L 1 S 4 ,  102 -L 2 S 4 ,  102 -E 1 A/ 102 -E 1 B) within the vertical grating coupler  100  can be formed as a empty/open region/volume (such as an air filled space) within an otherwise high refractive index layer (such as silicon or other material with similar optical properties). In this manner, the scattering element  102  (such as scattering elements  102 -L 1 S 1 ,  102 -L 2 S 1 ,  102 -L 1 S 2 ,  102 -L 2 S 2 ,  102 -L 1 S 3 ,  102 -L 2 S 3 ,  102 -L 1 S 4 ,  102 -L 2 S 4 ,  102 -E 1 A/ 102 -E 1 B) within the vertical grating coupler  100  can be formed by the absence of a high refractive index material in a layer otherwise formed by the high refractive index material. 
     Also, while the example vertical grating coupler  100  has been described as having scattering elements  102  formed within two light scattering layers (within the primary layer  205  and the secondary layer  207 ), it should be understood that in other embodiments the vertical grating coupler  100  can include scattering elements  102  formed in more than two light scattering layers. For example, another embodiment of the vertical grating coupler  100  can include scattering elements  102  formed within three or more layers, with a more complicated light directionality (phase-matching) condition than that of two layer vertical grating coupler  100 , such as described by way of example herein. In some embodiments, the vertical grating coupler  100  is extended vertically to include the primary layer  205 , the secondary layer  207 , and a tertiary layer formed above the secondary layer  207 . An example embodiment of the three-scattering layer vertical grating coupler  100  includes the primary layer  205  formed as a layer of body silicon (such as crystalline silicon), with scattering elements  102  formed within the primary layer  205  by either a partial etching of regions vertically through the primary layer  205  or a full etching of regions vertically through the primary layer  205 , and by an optional filling of the etched regions within the primary layer  205  with a material having a refractive index sufficiently different than the body silicon. Also, the example embodiment of the three-scattering layer vertical grating coupler  100  includes the secondary layer  207  formed as a layer of polysilicon above the body silicon of the primary layer  205 , with the scattering elements  102  formed within the secondary layer  207  by a full etching of regions vertically through the secondary layer  205 , and by an optional filling of the etched regions within the secondary layer  207  with a material having a refractive index sufficiently different than the polysilicon. Also, the example embodiment of the three-scattering layer vertical grating coupler  100  includes the tertiary layer formed as a layer of nitride material (such as silicon nitride, among others) above the polysilicon of the secondary layer  207 , with the scattering elements  102  formed within the tertiary layer by a full etching of regions vertically through the tertiary layer, and by an optional filling of the etched regions within the tertiary layer with a material having a refractive index sufficiently different than the nitride material. 
     In various embodiments, the vertical grating coupler  100  and variations thereof as described herein are compatible with CMOS fabrication processes and can be integrated within high-volume semiconductor device/chip production. Therefore, the vertical grating coupler  100  and variations thereof as described herein are suitable for implementation within semiconductor chips/devices and/or other types of devices that are fabricated using standard CMOS fabrication processes. 
     In accordance with the foregoing, in some embodiments, the optical grating coupler  100  includes the primary layer  205  formed of a material that has a first refractive index. The optical grating coupler  100  also includes the first plurality of scattering elements  102 A,  102 B, etc., formed within the primary layer  205 . The first plurality of scattering elements  102 A,  102 B, etc., has a second refractive index that is different than the first refractive index. The optical grating coupler  100  also includes the secondary layer  207  formed over the primary layer  205 . The secondary layer  207  is formed of a material having a third refractive index. The optical grating coupler  100  also includes the second plurality of scattering elements  102 C,  102 D, etc., formed within the secondary layer  207 . The second plurality of scattering elements  102 C,  102 D, etc., has a fourth refractive index that is different than the third refractive index. The fourth refractive index is also different than the second refractive index. At least some of the second plurality of scattering elements  102 C,  102 D, etc., at least partially overlap corresponding ones of the first plurality of scattering elements  102 A,  102 B, etc. In some embodiments, the first reactive index is greater than the second reactive index, and the third reactive index is less than the fourth reactive index. In some embodiments, the first reactive index is less than the second reactive index, and the third reactive index is greater than the fourth reactive index. 
     In some embodiments, the first plurality of scattering elements  102 A,  102 B, etc., are positioned symmetrically on each side of the vertical plane of symmetry  111  that bisects the optical grating coupler  100 , and the second plurality of scattering elements  102 C,  102 D, etc., are positioned symmetrically on each side of the vertical plane of symmetry  111 . The first plurality of scattering elements  102 A,  102 B, etc., and the second plurality of scattering elements  102 C,  102 D, etc., are collectively formed and positioned to scatter both a first incoming light beam and a second incoming light beam into an off-chip beam of light. The first incoming light beam is received through the first lateral side  100 A of the optical grating coupler  100 . The second incoming light beam is received through the second lateral side  100 B of the optical grating coupler  100 . The first lateral side  100 A and the second lateral side  100 B of the optical grating coupler  100  are adjacent sides along an outer perimeter of the optical grating coupler  100 . The first lateral side  100 A of the optical grating coupler  100  is on a first side of the vertical plane of symmetry  111 . The second lateral side  100 B of the optical grating coupler  100  is on a second side of the vertical plane of symmetry  111 . 
     In some embodiments, the first lateral side  100 A of the optical grating coupler  100  is optically coupled to the first optical waveguide taper  105 . The first incoming light beam is received through the first optical waveguide taper  105 . Also, the second lateral side  100 B of the optical grating coupler  100  is optically coupled to the second optical waveguide taper  109 . The second incoming light beam is received through the second optical waveguide taper  109 . 
     In some embodiments, the first plurality of scattering elements  102 A,  102 B, etc., and the second plurality of scattering elements  102 C,  102 D, etc., are arranged to include the first section S 1  of scattering element groups  601  positioned along the first lateral side  100 A and the second lateral side  100 B of the optical grating coupler  100 . Also, the first plurality of scattering elements  102 A,  102 B, etc., and the second plurality of scattering elements  102 C,  102 D, etc., are arranged to include the second section S 2  of scattering element groups  603  positioned behind the first section S 1  of scattering element groups  601  in a direction away from the first lateral side  100 A and the second lateral side  100 B of the optical grating coupler  100 . Also, the first plurality of scattering elements  102 A,  102 B, etc., and the second plurality of scattering elements  102 C,  102 D, etc., are arranged to include the third section S 3  of scattering element groups  605  positioned behind the second section S 2  of scattering element groups  603  in the direction away from the first lateral side  100 A and the second lateral side  100 B of the optical grating coupler  100 . Also, the first plurality of scattering elements  102 A,  102 B, etc., and the second plurality of scattering elements  102 C,  102 D, etc., are arranged to include the fourth section S 4  of scattering element groups  607  positioned behind the third section S 3  of scattering element groups  605  in the direction away from the first lateral side  100 A and the second lateral side  100 B of the optical grating coupler  100 . 
     Each scattering element group  601 ,  603 ,  605 ,  607  is a pair of scattering elements  102  that includes a corresponding scattering element  102  of the first plurality of scattering elements  102 A,  102 B, etc., and a corresponding scattering element  102  of the second plurality of scattering elements  102 C,  102 D, etc. In some embodiments, the scattering elements  102 -L 1 S 2  of the first plurality of scattering elements  102 A,  102 B, etc., in the second section S 2  of scattering element groups  603  are larger than the scattering elements  102 -L 1 S 1  of the first plurality of scattering elements  102 A,  102 B, etc., in the first section S 1  of scattering element groups  601 . And, the scattering elements  102 -L 2 S 2  of the second plurality of scattering elements  102 C,  102 D, etc., in the second section S 2  of scattering element groups  603  are larger than the scattering elements  102 -L 2 S 1  of the second plurality of scattering elements  102 C,  102 D, etc., in the first section S 1  of scattering element groups  601 . 
     In some embodiments, the scattering elements  102 -L 1 S 3  of the first plurality of scattering elements  102 A,  102 B, etc., in the third section S 3  of scattering element groups  605  are larger than the scattering elements  102 -L 1 S 2  of the first plurality of scattering elements  102 A,  102 B, etc., in the second section S 2  of scattering element groups  603 . And, the scattering elements  102 -L 2 S 3  of the second plurality of scattering elements  102 C,  102 D, etc., in the third section S 3  of scattering element groups  605  are larger than the scattering elements  102 -L 2 S 2  of the second plurality of scattering elements  102 C,  102 D, etc., in the second section S 2  of scattering element groups  603 . 
     In some embodiments, the scattering elements  102 -L 1 S 4  of the first plurality of scattering elements  102 A,  102 B, etc., in the fourth section S 4  of scattering element groups  607  are larger than the scattering elements  102 -L 1 S 3  of the first plurality of scattering elements  102 A,  102 B, etc., in the third section S 3  of scattering element groups  605 . And, the scattering elements  102 -L 2 S 4  of the second plurality of scattering elements  102 C,  102 D, etc., in the fourth section S 4  of scattering element groups  607  are larger than the scattering elements  102 -L 2 S 3  of the second plurality of scattering elements  102 C,  102 D, etc., in the third section S 3  of scattering element groups  605 . 
     In some embodiments, some of the second plurality of scattering elements  102 C,  102 D, etc., are positioned to have a lateral offset  705 ,  805 ,  905 ,  1005  with respect to corresponding overlapped ones of the first plurality of scattering elements  102 A,  102 B, etc., in a direction parallel to the vertical plane of symmetry  111 . In some embodiments, the lateral offset  705 ,  805 ,  905 ,  1005  is defined to control the angle  214  of the off-chip beam of light (represented by arrows  213 ) as measured relative to the vector  212  that extends perpendicular to the primary layer  205 . Additionally, in some embodiments, the first plurality of scattering elements  102 A,  102 B, etc., are positioned on respective gridpoints of the grid  401 . 
       FIG. 16  shows a flowchart of a method for combining light beams, in accordance with some embodiments. The method includes an operation  1601  for having an optical grating coupler ( 100 ) that includes a primary layer ( 205 ) and a secondary layer ( 207 ), where the primary layer ( 205 ) includes a first plurality of scattering elements ( 102 A,  102 B, etc.), and where the secondary layer ( 207 ) includes a second plurality of scattering elements ( 102 C,  102 D, etc.), and where at least some of the second plurality of scattering elements ( 102 C,  102 D, etc.) at least partially overlap corresponding ones of the first plurality of scattering elements ( 102 A,  102 B, etc.). The method also includes an operation  1603  for directing a first beam of light into a first lateral side ( 100 A) of the optical grating coupler ( 100 ). The method also includes an operation  1605  for directing a second beam of light into a second lateral side ( 100 B) of the optical grating coupler ( 100 ), where the second lateral side ( 100 B) is adjacent to the first lateral side ( 100 A) along a perimeter of the optical grating coupler ( 100 ). The first plurality of scattering elements ( 102 A,  102 B, etc.) and the second plurality of scattering elements ( 102 C,  102 D, etc.) collectively scatter both the first beam of light and the second beam of light into a third beam of light. In some embodiments, the first beam of light has a first polarization and the second beam of light has a second polarization different than the first polarization. In these embodiments, the third beam of light includes both light of the first polarization from the first beam of light and light of the second polarization from the second beam of light. 
     In some embodiments, the first plurality of scattering elements ( 102 A,  102 B, etc.) and the second plurality of scattering elements ( 102 C,  102 D, etc.) are configured to implement apodization of light scattering strength along propagations directions of the first beam of light and the second beam of light. In some embodiments, the third beam of light is transmitted out of the optical grating coupler ( 100 ) in a non-coplanar direction ( 213 ) with respect to a plane of the optical grating coupler ( 100 ). In some embodiments, the first plurality of scattering elements ( 102 A,  102 B, etc.) are positioned symmetrically on each side of a vertical plane of symmetry ( 111 ) that bisects the optical grating coupler ( 100 ), and the second plurality of scattering elements ( 102 C,  102 D etc.) are positioned symmetrically on each side of the vertical plane of symmetry ( 111 ). In these embodiments, the third beam of light is transmitted within the vertical plane of symmetry ( 111 ). 
     The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the invention, and all such modifications are intended to be included within the scope of the invention. 
     Although the foregoing invention has been described in some detail for purposes of clarity of understanding, it will be apparent that certain changes and modifications can be practiced within the scope of the invention description. Accordingly, the present embodiments are to be considered as illustrative and not restrictive, and the invention is not to be limited to the details given herein, but may be modified within the scope and equivalents of the described embodiments.