Patent Publication Number: US-11650423-B2

Title: Eyepieces for augmented reality display system

Description:
INCORPORATION BY REFERENCE TO ANY PRIORITY APPLICATIONS 
     This application claims priority to U.S. Provisional Patent Application 62/863,871, filed Jun. 20, 2019, and entitled “EYEPIECES FOR AUGMENTED REALITY DISPLAY SYSTEM,” and to U.S. Provisional Patent Application 63/024,343, filed May 13, 2020, and entitled “EYEPIECES FOR AUGMENTED REALITY DISPLAY SYSTEM.” Each of these applications, and any and all other applications for which a foreign or domestic priority claim is identified in the Application Data Sheet as filed with the present application, are hereby incorporated by reference under 37 CFR 1.57. 
    
    
     BACKGROUND 
     Field 
     This disclosure relates to eyepieces for virtual reality, augmented reality, and mixed reality systems. 
     Description of the Related Art 
     Modern computing and display technologies have facilitated the development of virtual reality, augmented reality, and mixed reality systems. Virtual reality, or “VR,” systems create a simulated environment for a user to experience. This can be done by presenting computer-generated image data to the user through a head-mounted display. This image data creates a sensory experience which immerses the user in the simulated environment. A virtual reality scenario typically involves presentation of only computer-generated image data rather than also including actual real-world image data. 
     Augmented reality systems generally supplement a real-world environment with simulated elements. For example, augmented reality, or “AR,” systems may provide a user with a view of the surrounding real-world environment via a head-mounted display. However, computer-generated image data can also be presented on the display to enhance the real-world environment. This computer-generated image data can include elements which are contextually-related to the real-world environment. Such elements can include simulated text, images, objects, etc. Mixed reality, or “MR,” systems are a type of AR system which also introduce simulated objects into a real-world environment, but these objects typically feature a greater degree of interactivity. The simulated elements can often times be interactive in real time. 
       FIG.  1    depicts an example AR scene  1  where a user sees a real-world park setting  6  featuring people, trees, buildings in the background, and a concrete platform  20 . In addition to these items, computer-generated image data is also presented to the user. The computer-generated image data can include, for example, a robot statue  10  standing upon the real-world platform  20 , and a cartoon-like avatar character  2  flying by which seems to be a personification of a bumblebee, even though these elements  2 ,  10  are not actually present in the real-world environment. 
     SUMMARY 
     In some embodiments, an augmented reality display system comprises: a first eyepiece waveguide comprising a first optically transmissive substrate; a first input coupling grating (ICG) region formed on or in the first eyepiece waveguide, the first ICG region being configured to receive a set of input beams of light corresponding to an input image having a corresponding field of view, and to couple a first subset of the input beams into the substrate as a first set of guided beams, the first subset of the input beams corresponding to a first sub-portion of the field of view of the input image; a second eyepiece waveguide comprising a second optically transmissive substrate; and a second input coupling grating (ICG) region formed on or in the second eyepiece waveguide, the second ICG region being configured to receive at least a second subset of the input beams of light corresponding to the input image, and to couple the second subset of input beams into the substrate as a second set of guided beams, the second subset of the input beams corresponding to a second sub-portion of the field of view of the input image, wherein the first and second sub-portions of the field of view are at least partially different but together include the complete field of view of the input image. 
     In some embodiments, an augmented reality display system comprises: a first eyepiece waveguide comprising a first optically transmissive substrate; a first input coupling grating (ICG) region formed on or in the first eyepiece waveguide, the first ICG region being configured to receive a set of input beams of light, the set of input beams being associated with a set of k-vectors in k-space corresponding to an input image, and to translate the set of k-vectors to a location in k-space such that a first subset of the k-vectors lies inside a first k-space annulus associated with the first eyepiece waveguide, the first k-space annulus corresponding to a region in k-space associated with guided propagation in the first eyepiece waveguide; a second eyepiece waveguide comprising a second optically transmissive substrate; and a second input coupling grating (ICG) region formed on or in the second eyepiece waveguide, the second ICG region being configured to receive at least a portion of the set of input beams of light, and to translate the set of k-vectors to a location in k-space such that a second subset of the k-vectors lies inside a second k-space annulus associated with the second eyepiece waveguide, the second k-space annulus corresponding to a region in k-space associated with guided propagation in the second eyepiece waveguide; wherein the first and second subsets of the k-vectors are at least partially different but together include the complete set of k-vectors corresponding to the input image. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    illustrates a user&#39;s view of augmented reality (AR) through an AR device. 
         FIG.  2    illustrates an example of a wearable display system. 
         FIG.  3    illustrates a conventional display system for simulating three-dimensional image data for a user. 
         FIG.  4    illustrates aspects of an approach for simulating three-dimensional image data using multiple depth planes. 
         FIGS.  5 A- 5 C  illustrate relationships between radius of curvature and focal radius. 
         FIG.  6    illustrates an example of a waveguide stack for outputting image information to a user in an AR eyepiece. 
         FIGS.  7 A- 7 B  illustrate examples of exit beams outputted by a waveguide. 
         FIG.  8    illustrates an example of a stacked waveguide assembly in which each depth plane includes images formed using multiple different component colors. 
         FIG.  9 A  illustrates a cross-sectional side view of an example of a set of stacked waveguides that each includes an in-coupling optical element. 
         FIG.  9 B  illustrates a perspective view of an example of the plurality of stacked waveguides of  FIG.  9 A . 
         FIG.  9 C  illustrates a top-down plan view of an example of the plurality of stacked waveguides of  FIGS.  9 A and  9 B . 
         FIG.  10    is a perspective view of an example AR eyepiece waveguide stack. 
         FIG.  11    is a cross-sectional view of a portion of an example eyepiece waveguide stack with an edge seal structure for supporting eyepiece waveguides in a stacked configuration. 
         FIGS.  12 A and  12 B  illustrate top views of an eyepiece waveguide in operation as it projects an image toward a user&#39;s eye. 
         FIG.  13 A  illustrates a k-vector which can be used to represent the propagation direction of a light ray or a light beam. 
         FIG.  13 B  illustrates a light ray within a planar waveguide. 
         FIG.  13 C  illustrates the permissible k-vectors for light of a given angular frequency, co, propagating in an unbounded homogenous medium with refractive index, n. 
         FIG.  13 D  illustrates the permissible k-vectors for light of a given angular frequency, co, propagating in a homogenous planar waveguide medium with refractive index, n. 
         FIG.  13 E  illustrates an annulus in k-space which corresponds to k-vectors of light waves which can be guided within a waveguide having a refractive index, n 2 . 
         FIG.  13 F  shows a k-space diagram and an eyepiece waveguide which illustrate the relationship between a k-vector and the density of interactions between a guided beam corresponding to that k-vector and a diffraction grating formed on or in the waveguide. 
         FIG.  13 G  illustrates a top view of a diffraction grating and some of its associated k-space diffraction grating vectors (G− 2 , G− 1 , G 1 , G 2 ). 
         FIG.  13 H  illustrates a transverse view of the diffraction grating and its effect, in k-space, on a k-vector corresponding to a normally-incident ray or beam of light. 
         FIG.  13 I  illustrates a transverse view of the diffraction grating shown in  FIG.  13 G  and its effect, in k-space, on a k-vector corresponding to an obliquely-incident ray or beam of light. 
         FIG.  13 J  is a k-space diagram which illustrates the field of view of an image that is projected into an AR eyepiece waveguide. 
         FIG.  13 K  is a k-space diagram which shows the translational shift, in k-space, of the FOV rectangle which is caused by an input coupling grating (ICG) located at the entrance pupil of an eyepiece waveguide. 
         FIG.  14 A  illustrates an example eyepiece waveguide with an ICG region, an orthogonal pupil expander (OPE) region, and an exit pupil expander (EPE) region. 
         FIG.  14 B  illustrates the k-space operation of the eyepiece waveguide shown in  FIG.  14 A . 
         FIG.  14 C  illustrates the optical operation of the OPE region shown in  FIGS.  14 A and  14 B . 
         FIG.  14 D  illustrates a technique for determining the sizes and shapes of the OPE region and the EPE region. 
         FIG.  15 A  illustrates an example embodiment of a waveguide eyepiece in which the OPE region is tilted and located such that its lower border is parallel to the upper border of the EPE region. 
         FIG.  15 B  includes k-space diagrams which illustrate the operation of the eyepiece waveguide shown in  FIG.  15 A . 
         FIG.  15 C  is another k-space diagram which illustrates the operation of the eyepiece waveguide shown in  FIG.  15 A . 
         FIG.  15 D  is a diagram of the first generation of interactions between an input beam and the OPE region of the eyepiece waveguide embodiment shown in  FIG.  15 A . 
         FIG.  15 E  is a diagram of the second generation of interactions between an input beam and the OPE region of the eyepiece waveguide embodiment shown in  FIG.  15 A . 
         FIG.  15 F  is a diagram of the third generation of interactions between an input beam and the OPE region of the eyepiece waveguide embodiment shown in  FIG.  15 A . 
         FIG.  15 G  is a diagram which illustrates how a single input beam from the ICG region is replicated by the OPE region and redirected toward the EPE region as a plurality of beams. 
         FIG.  16 A  illustrates an example eyepiece waveguide that has a multi-directional pupil expander (MPE) region rather than an OPE region. 
         FIG.  16 B  illustrates a portion of an example 2D grating, along with its associated grating vectors, which can be used in the MPE region shown in  FIG.  16 A . 
         FIG.  16 C  is a k-space diagram which illustrates the k-space operation of the MPE region of the eyepiece waveguide shown in  FIG.  16 A . 
         FIG.  16 D  is a k-space diagram which further illustrates the k-space operation of the MPE region of the eyepiece waveguide shown in  FIG.  16 A . 
         FIG.  16 E  is a k-space diagram which illustrates the k-space operation of the eyepiece waveguide shown in  FIG.  16 A . 
         FIG.  16 F  is a diagram of the first generation of interactions between an input beam and the MPE region of the eyepiece waveguide embodiment shown in  FIG.  16 A . 
         FIG.  16 G  is a diagram of the second generation of interactions between an input beam and the MPE region of the eyepiece waveguide embodiment shown in  FIG.  16 A . 
         FIG.  16 H  is a diagram of the third generation of interactions between an input beam and the MPE region of the eyepiece waveguide embodiment shown in  FIG.  16 A . 
         FIG.  16 I  is a diagram of the fourth generation of interactions between an input beam and the MPE region of the eyepiece waveguide embodiment shown in  FIG.  16 A . 
         FIG.  16 J  is a diagram which illustrates various paths which beams may follow through the MPE region and ultimately to the EPE region according to the eyepiece waveguide embodiment shown in  FIG.  16 A . 
         FIG.  16 K  is a diagram which illustrates how a single input beam from the ICG region is replicated by the MPE region and redirected toward the EPE region as a plurality of beams. 
         FIG.  16 L  is a side-by-side comparison which illustrates the performance of an eyepiece waveguide with an OPE region versus that of an eyepiece waveguide with an MPE region. 
         FIG.  16 M  further illustrates the performance of an eyepiece waveguide with an MPE region versus others with OPE regions. 
         FIG.  17 A  illustrates a portion of an example 2D grating, along with its associated grating vectors, which can be used in the MPE region of an eyepiece waveguide. 
         FIG.  17 B  is a k-space diagram which illustrates the k-space operation of the MPE region of an eyepiece waveguide. 
         FIG.  17 C  is a k-space diagram which illustrates the k-space operation of an eyepiece waveguide with an MPE region. 
         FIG.  17 D  is a diagram of the first generation of interactions between an input beam and the MPE region of an eyepiece waveguide. 
         FIG.  17 E  is a diagram of the second generation of interactions between an input beam and the MPE region of an eyepiece waveguide. 
         FIG.  17 F  is a diagram of the third generation of interactions between an input beam and the MPE region of an eyepiece waveguide. 
         FIG.  17 G  is a diagram of the fourth generation of interactions between an input beam and the MPE region of an eyepiece waveguide. 
         FIG.  18 A  illustrates an example eyepiece waveguide with an ICG region, two orthogonal pupil expander regions, and an exit pupil expander region. 
         FIGS.  18 B and  18 C  illustrate top views of the EPE region of the eyepiece waveguide shown in  FIG.  18 A . 
         FIG.  19    illustrates an embodiment of an eyepiece waveguide with an expanded field of view. 
         FIG.  20 A  illustrates an embodiment of an expanded FOV eyepiece waveguide with an MPE region which is overlapped by an EPE region. 
         FIG.  20 B  illustrates a portion of an example 2D grating, along with its associated grating vectors, which can be used in the MPE region of the eyepiece waveguide in  FIG.  20 A . 
         FIG.  20 C  is a k-space diagram which illustrates the k-space operation of the ICG region of the eyepiece waveguide in  FIG.  20 A . 
         FIG.  20 D  is a k-space diagram which illustrates part of the k-space operation of the MPE region of the eyepiece waveguide in  FIG.  20 A . 
         FIG.  20 E  is a k-space diagram which illustrates another part of the k-space operation of the MPE region of the eyepiece waveguide in  FIG.  20 A . 
         FIG.  20 F  is similar to  FIG.  20 E , except that it shows the k-space operation of the MPE region on the FOV rectangle from  FIG.  20 D  which was translated to the 9 o&#39;clock position (instead of the 3 o&#39;clock position, as illustrated in  FIG.  20 E ). 
         FIG.  20 G  is a k-space diagram which illustrates the k-space operation of the EPE region in the eyepiece waveguide in  FIG.  20 A . 
         FIG.  20 H  is a k-space diagram which summarizes the k-space operation of the eyepiece waveguide in  FIG.  20 A . 
         FIG.  20 I  is a diagram which illustrates how beams of light spread through the eyepiece waveguide shown in  FIG.  20 A . 
         FIG.  20 J  illustrates how the diffractive efficiency of the MPE region in the eyepiece waveguide in  FIG.  20 A  can be spatially varied so as to enhance the uniformity of luminance in the waveguide. 
         FIG.  20 K  illustrates how the diffractive efficiency of the EPE region in the eyepiece waveguide in  FIG.  20 A  can be spatially varied so as to enhance the uniformity of luminance in the waveguide. 
         FIG.  20 L  illustrates an embodiment of the eyepiece waveguide in  FIG.  20 A  which includes one or more diffractive mirrors around the peripheral edge of the waveguide. 
         FIG.  20 M  illustrates an example embodiment of eyeglasses which incorporate one or more instances of the eyepiece waveguide in  FIG.  20 A . 
         FIG.  20 N  illustrates another example embodiment of eyeglasses which incorporate one or more instances of the eyepiece waveguide in  FIG.  20 A . 
         FIG.  21 A  illustrates another embodiment of an eyepiece waveguide with an MPE region which is overlapped by an EPE region. 
         FIG.  21 B  is a k-space diagram which illustrates the k-space operation of the eyepiece waveguide in  FIG.  20 A  on the first set of input beams corresponding to the first sub-portion of the FOV of an input image. 
         FIG.  21 C  is a k-space diagram which illustrates the k-space operation of the eyepiece waveguide in  FIG.  21 A  on the second set of input beams corresponding to the second sub-portion of the FOV of the input image. 
         FIG.  21 D  is a k-space diagram which summarizes the k-space operation of the eyepiece waveguide in  FIG.  21 A . 
         FIG.  21 E  illustrates an example embodiment of eyeglasses which incorporate one or more instances of the eyepiece waveguide in  FIG.  21 A . 
         FIG.  21 F  illustrates example FOVs corresponding to the eyeglasses in  FIG.  21 E . 
         FIG.  21 G  illustrates the k-space operation of another embodiment of the eyepiece waveguide shown in  FIG.  21 A . 
         FIG.  22 A  illustrates an embodiment of an eyepiece waveguide that can project an FOV which is expanded in two directions. 
         FIG.  22 B  illustrates the opposite side of the eyepiece waveguide shown in  FIG.  22 A . 
         FIG.  22 C  illustrates the k-space operation of the ICG regions and the OPE regions in the eyepiece waveguide embodiment in  FIG.  22 A . 
         FIG.  22 D  illustrates the k-space operation of the MPE region in the eyepiece waveguide embodiment in  FIG.  22 A . 
         FIG.  22 E  illustrates the k-space operation of the EPE region in the eyepiece waveguide embodiment in  FIG.  22 A . 
         FIG.  23    illustrates an example embodiment of an eyepiece waveguide designed to function with an angled projector. 
         FIG.  24 A  is an edge view of an example eyepiece waveguide that has a multiple combined pupil expander-extractor (CPE) regions. 
         FIG.  24 B  illustrates the operation of the first and second CPE regions in both physical space and in k-space according to a first type of main pathway of light through the eyepiece waveguide. 
         FIG.  24 C  illustrates the operation of the first and second CPE regions in both physical space and in k-space according to a second type of main pathway of light through the eyepiece waveguide. 
         FIG.  24 D  illustrates the operation of the first and second CPE regions in both physical space and in k-space according to both the first and second types of main pathways of light through the eyepiece waveguide. 
         FIG.  24 E  is a diagram of the first generation of interactions between an input beam and the CPE regions of the eyepiece waveguide embodiment shown in  FIG.  24 A . 
         FIG.  24 F  is a diagram of the second generation of interactions between the input beam and the CPE regions of the eyepiece waveguide embodiment shown in  FIG.  24 A . 
         FIG.  24 G  is a diagram of the third generation of interactions between the input beam and the CPE regions of the eyepiece waveguide embodiment shown in  FIG.  24 A . 
         FIG.  24 H  is a diagram of the fourth generation of interactions between the input beam and the CPE regions of the eyepiece waveguide embodiment shown in  FIG.  24 A . 
         FIG.  24 I  is a diagram of the fifth generation of interactions between the input beam and the CPE regions of the eyepiece waveguide embodiment shown in  FIG.  24 A . 
         FIG.  24 J  illustrates, in k-space, higher-order pathways of light through the eyepiece waveguide shown in  FIG.  24 A . 
         FIG.  24 K  is a diagram which illustrates how beams of light spread through the eyepiece waveguide shown in  FIG.  24 A . 
         FIG.  25 A  is an edge view of an example eyepiece waveguide that has a single 2D combined pupil expander-extractor (CPE) grating region. 
         FIG.  25 B  illustrates the operation of the 2D CPE region in both physical space and in k-space. 
         FIG.  26 A  is an edge view of an example eyepiece waveguide that has a 2D combined pupil expander-extractor (CPE) grating region on each of its sides. 
         FIG.  26 B  illustrates the so-called “screen door effect” which is an image artifact that is related to the density of output beams from an eyepiece waveguide. 
         FIG.  26 C  illustrates input coupling grating re-bounce, which is an effect that can cause light to be disadvantageously lost from an eyepiece waveguide. 
         FIG.  26 D  illustrates how the double-sided 2D CPE gratings in  FIG.  26 A  increase the density of output beams from the eyepiece waveguide. 
         FIG.  26 E  illustrates the density of output beams for the eyepiece waveguides shown in  FIG.  24 A  (double-sided 1D CPE gratings),  FIG.  25 A  (single-sided 2D CPE grating), and  FIG.  26 A  (double-sided 2D CPE gratings). 
         FIG.  26 F  shows example simulated images produced by eyepiece waveguides with 2D CPE gratings; images for both the case of the single-sided embodiment of  FIG.  25 A  and the double-sided embodiment of  FIG.  26 A  are shown. 
         FIG.  27 A  illustrates an example embodiment of an eyepiece waveguide stack with enhanced FOV. 
         FIG.  27 B  illustrates another example embodiment of an eyepiece waveguide stack with enhanced FOV. 
         FIGS.  27 C- 27 E  include k-space diagrams that illustrate the k-space operation of the example embodiments of the eyepiece waveguide stacks shown in  FIGS.  27 A and  27 B  for three different refractive indexes. 
         FIGS.  27 F- 27 H  illustrate the sub-portions of the FOV of each color component of the input image which can be in-coupled into each of the eyepiece waveguides in the stack shown in  FIGS.  27 A and  27 B , according to certain embodiments where each color component is partially carried in two of the eyepiece waveguides. 
         FIGS.  27 I- 27 K  are similar to  FIGS.  27 F- 27 H  in that they illustrate the sub-portions of the FOV of each color component of the input image which can be in-coupled into each of the eyepiece waveguides in the stack, except that  FIGS.  27 I- 27 K  illustrate certain embodiments where each color component is partially carried in three of the eyepiece waveguides rather than two. 
         FIGS.  27 L- 27 N  illustrate the sub-portions of the FOV of each color component of the input image which can be in-coupled into single-color-component-per-layer eyepiece waveguides. 
         FIG.  28 A  illustrates another example embodiment of an eyepiece waveguide stack with enhanced FOV and an in-line pupil ICG configuration. 
         FIG.  28 B  illustrates another example embodiment of an eyepiece waveguide stack with enhanced FOV and a split pupil ICG configuration. 
         FIG.  29    is a graph which plots the FOV values in Table 1 as a function of refractive index. 
         FIG.  30    illustrates an example of improved output image uniformity using the eyepiece waveguide stacks shown in  FIGS.  27 A- 27 B  and  FIGS.  28 A- 28 B . 
     
    
    
     DETAILED DESCRIPTION 
     Overview 
     This disclosure describes a variety of eyepiece waveguides which can be used in AR display systems to project images to a user&#39;s eye. The eyepiece waveguides are described both in physical terms and using k-space representations. 
     Example HMD Device 
       FIG.  2    illustrates an example wearable display system  60 . The display system  60  includes a display or eyepiece  70 , and various mechanical and electronic modules and systems to support the functioning of that display  70 . The display  70  may be coupled to a frame  80 , which is wearable by a display system user  90  and which is configured to position the display  70  in front of the eyes of the user  90 . The display  70  may be considered eyewear in some embodiments. In some embodiments, a speaker  100  is coupled to the frame  80  and is positioned adjacent the ear canal of the user  90 . The display system may also include one or more microphones  110  to detect sound. The microphone  110  can allow the user to provide inputs or commands to the system  60  (e.g., the selection of voice menu commands, natural language questions, etc.), and/or can allow audio communication with other persons (e.g., with other users of similar display systems). The microphone  110  can also collect audio data from the user&#39;s surroundings (e.g., sounds from the user and/or environment). In some embodiments, the display system may also include a peripheral sensor  120   a , which may be separate from the frame  80  and attached to the body of the user  90  (e.g., on the head, torso, an extremity, etc.). The peripheral sensor  120   a  may acquire data characterizing the physiological state of the user  90  in some embodiments. 
     The display  70  is operatively coupled by a communications link  130 , such as by a wired lead or wireless connectivity, to a local data processing module  140  which may be mounted in a variety of configurations, such as fixedly attached to the frame  80 , fixedly attached to a helmet or hat worn by the user, embedded in headphones, or removably attached to the user  90  (e.g., in a backpack-style configuration or in a belt-coupling style configuration). Similarly, the sensor  120   a  may be operatively coupled by communications link  120   b  (e.g., a wired lead or wireless connectivity) to the local processor and data module  140 . The local processing and data module  140  may include a hardware processor, as well as digital memory, such as non-volatile memory (e.g., flash memory or a hard disk drive), both of which may be utilized to assist in the processing, caching, and storage of data. The data may include data  1 ) captured from sensors (which may be, e.g., operatively coupled to the frame  80  or otherwise attached to the user  90 ), such as image capture devices (example e.g., cameras), microphones, inertial measurement units, accelerometers, compasses, GPS units, radio devices, gyros, and/or other sensors disclosed herein; and/or 2) acquired and/or processed using a remote processing module  150  and/or a remote data repository  160  (including data relating to virtual content), possibly for passage to the display  70  after such processing or retrieval. The local processing and data module  140  may be operatively coupled by communication links  170 ,  180 , such as via a wired or wireless communication links, to the remote processing module  150  and the remote data repository  160  such that these remote modules  150 ,  160  are operatively coupled to each other and available as resources to the local processing and data module  140 . In some embodiments, the local processing and data module  140  may include one or more of the image capture devices, microphones, inertial measurement units, accelerometers, compasses, GPS units, radio devices, and/or gyros. In some other embodiments, one or more of these sensors may be attached to the frame  80 , or may be standalone devices that communicate with the local processing and data module  140  by wired or wireless communication pathways. 
     The remote processing module  150  may include one or more processors to analyze and process data, such as image and audio information. In some embodiments, the remote data repository  160  may be a digital data storage facility, which may be available through the internet or other networking configuration in a “cloud” resource configuration. In some embodiments, the remote data repository  160  may include one or more remote servers, which provide information (e.g., information for generating augmented reality content) to the local processing and data module  140  and/or the remote processing module  150 . In other embodiments, all data is stored and all computations are performed in the local processing and data module, allowing fully autonomous use from a remote module. 
     The perception of an image as being “three-dimensional” or “3-D” may be achieved by providing slightly different presentations of the image to each eye of the user.  FIG.  3    illustrates a conventional display system for simulating three-dimensional image data for a user. Two distinct images  190 ,  200 —one for each eye  210 ,  220 —are output to the user. The images  190 ,  200  are spaced from the eyes  210 ,  220  by a distance  230  along an optical or z-axis that is parallel to the line of sight of the user. The images  190 ,  200  are flat and the eyes  210 ,  220  may focus on the images by assuming a single accommodated state. Such 3-D display systems rely on the human visual system to combine the images  190 ,  200  to provide a perception of depth and/or scale for the combined image. 
     However, the human visual system is complicated and providing a realistic perception of depth is challenging. For example, many users of conventional “3-D” display systems find such systems to be uncomfortable or may not perceive a sense of depth at all. Objects may be perceived as being “three-dimensional” due to a combination of vergence and accommodation. Vergence movements (e.g., rotation of the eyes so that the pupils move toward or away from each other to converge the respective lines of sight of the eyes to fixate upon an object) of the two eyes relative to each other are closely associated with focusing (or “accommodation”) of the lenses of the eyes. Under normal conditions, changing the focus of the lenses of the eyes, or accommodating the eyes, to change focus from one object to another object at a different distance will automatically cause a matching change in vergence to the same distance, under a relationship known as the “accommodation-vergence reflex,” as well as pupil dilation or constriction. Likewise, under normal conditions, a change in vergence will trigger a matching change in accommodation of lens shape and pupil size. As noted herein, many stereoscopic or “3-D” display systems display a scene using slightly different presentations (and, so, slightly different images) to each eye such that a three-dimensional perspective is perceived by the human visual system. Such systems can be uncomfortable for some users, however, since they simply provide image information at a single accommodated state and work against the “accommodation-vergence reflex.” Display systems that provide a better match between accommodation and vergence may form more realistic and comfortable simulations of three-dimensional image data. 
       FIG.  4    illustrates aspects of an approach for simulating three-dimensional image data using multiple depth planes. With reference to  FIG.  4   , the eyes  210 ,  220  assume different accommodated states to focus on objects at various distances on the z-axis. Consequently, a particular accommodated state may be said to be associated with a particular one of the illustrated depth planes  240 , which has an associated focal distance, such that objects or parts of objects in a particular depth plane are in focus when the eye is in the accommodated state for that depth plane. In some embodiments, three-dimensional image data may be simulated by providing different presentations of an image for each of the eyes  210 ,  220 , and also by providing different presentations of the image corresponding to multiple depth planes. While the respective fields of view of the eyes  210 ,  220  are shown as being separate for clarity of illustration, they may overlap, for example, as distance along the z-axis increases. In addition, while the depth planes are shown as being flat for ease of illustration, it will be appreciated that the contours of a depth plane may be curved in physical space, such that all features in a depth plane are in focus with the eye in a particular accommodated state. 
     The distance between an object and an eye  210  or  220  may also change the amount of divergence of light from that object, as viewed by that eye.  FIGS.  5 A- 5 C  illustrate relationships between distance and the divergence of light rays. The distance between the object and the eye  210  is represented by, in order of decreasing distance, R 1 , R 2 , and R 3 . As shown in  FIGS.  5 A- 5 C , the light rays become more divergent as distance to the object decreases. As distance increases, the light rays become more collimated. Stated another way, it may be said that the light field produced by a point (the object or a part of the object) has a spherical wavefront curvature, which is a function of how far away the point is from the eye of the user. The curvature increases with decreasing distance between the object and the eye  210 . Consequently, at different depth planes, the degree of divergence of light rays is also different, with the degree of divergence increasing with decreasing distance between depth planes and the user&#39;s eye  210 . While only a single eye  210  is illustrated for clarity of illustration in  FIGS.  5 A- 5 C  and other figures herein, it will be appreciated that the discussions regarding the eye  210  may be applied to both eyes  210  and  220  of a user. 
     A highly believable simulation of perceived depth may be achieved by providing, to the eye, different presentations of an image corresponding to each of a limited number of depth planes. The different presentations may be separately focused by the user&#39;s eye, thereby helping to provide the user with depth cues based on the amount of accommodation of the eye required to bring into focus different image features for the scene located on different depth planes and/or based on observing different image features on different depth planes being out of focus. 
     Example of a Waveguide Stack Assembly for an AR or MR Eyepiece 
       FIG.  6    illustrates an example of a waveguide stack for outputting image information to a user in an AR eyepiece. A display system  250  includes a stack of waveguides, or stacked waveguide assembly, 260 that may be utilized to provide three-dimensional perception to the eye/brain using a plurality of waveguides  270 ,  280 ,  290 ,  300 ,  310 . In some embodiments, the display system  250  is the system  60  of  FIG.  2   , with  FIG.  6    schematically showing some parts of that system  60  in greater detail. For example, the waveguide assembly  260  may be part of the display  70  of  FIG.  2   . It will be appreciated that the display system  250  may be considered a light field display in some embodiments. 
     The waveguide assembly  260  may also include a plurality of features  320 ,  330 ,  340 ,  350  between the waveguides. In some embodiments, the features  320 ,  330 ,  340 ,  350  may be one or more lenses. The waveguides  270 ,  280 ,  290 ,  300 ,  310  and/or the plurality of lenses  320 ,  330 ,  340 ,  350  may be configured to send image information to the eye with various levels of wavefront curvature or light ray divergence. Each waveguide level may be associated with a particular depth plane and may be configured to output image information corresponding to that depth plane. Image injection devices  360 ,  370 ,  380 ,  390 ,  400  may function as a source of light for the waveguides and may be utilized to inject image information into the waveguides  270 ,  280 ,  290 ,  300 ,  310 , each of which may be configured, as described herein, to distribute incoming light across each respective waveguide, for output toward the eye  210 . Light exits an output surface  410 ,  420 ,  430 ,  440 ,  450  of each respective image injection device  360 ,  370 ,  380 ,  390 ,  400  and is injected into a corresponding input surface  460 ,  470 ,  480 ,  490 ,  500  of the respective waveguides  270 ,  280 ,  290 ,  300 ,  310 . In some embodiments, the each of the input surfaces  460 ,  470 ,  480 ,  490 ,  500  may be an edge of a corresponding waveguide, or may be part of a major surface of the corresponding waveguide (that is, one of the waveguide surfaces directly facing the world  510  or the user&#39;s eye  210 ). In some embodiments, a beam of light (e.g. a collimated beam) may be injected into each waveguide and may be replicated, such as by sampling into beamlets by diffraction, in the waveguide and then directed toward the eye  210  with an amount of optical power corresponding to the depth plane associated with that particular waveguide. In some embodiments, a single one of the image injection devices  360 ,  370 ,  380 ,  390 ,  400  may be associated with, and inject light into, a plurality (e.g., three) of the waveguides  270 ,  280 ,  290 ,  300 ,  310 . 
     In some embodiments, the image injection devices  360 ,  370 ,  380 ,  390 ,  400  are discrete displays that each produce image information for injection into a corresponding waveguide  270 ,  280 ,  290 ,  300 ,  310 , respectively. In some other embodiments, the image injection devices  360 ,  370 ,  380 ,  390 ,  400  are the output ends of a single multiplexed display which may transmit image information via one or more optical conduits (such as fiber optic cables) to each of the image injection devices  360 ,  370 ,  380 ,  390 ,  400 . It will be appreciated that the image information provided by the image injection devices  360 ,  370 ,  380 ,  390 ,  400  may include light of different wavelengths, or colors. 
     In some embodiments, the light injected into the waveguides  270 ,  280 ,  290 ,  300 ,  310  is provided by a light projector system  520 , which includes a light module  530 , which may include a light source or light emitter, such as a light emitting diode (LED). The light from the light module  530  may be directed to, and modulated by, a light modulator  540  (e.g., a spatial light modulator), via a beamsplitter (BS)  550 . The light modulator  540  may spatially and/or temporally change the perceived intensity of the light injected into the waveguides  270 ,  280 ,  290 ,  300 ,  310 . Examples of spatial light modulators include liquid crystal displays (LCD), including a liquid crystal on silicon (LCOS) displays, and digital light processing (DLP) displays. 
     In some embodiments, the light projector system  520 , or one or more components thereof, may be attached to the frame  80  ( FIG.  2   ). For example, the light projector system  520  may be part of a temporal portion (e.g., ear stem  82 ) of the frame  80  or disposed at an edge of the display  70 . In some embodiments, the light module  530  may be separate from the BS  550  and/or light modulator  540 . 
     In some embodiments, the display system  250  may be a scanning fiber display comprising one or more scanning fibers to project light in various patterns (e.g., raster scan, spiral scan, Lissajous patterns, etc.) into one or more waveguides  270 ,  280 ,  290 ,  300 ,  310  and ultimately into the eye  210  of the user. In some embodiments, the illustrated image injection devices  360 ,  370 ,  380 ,  390 ,  400  may schematically represent a single scanning fiber or a bundle of scanning fibers configured to inject light into one or a plurality of the waveguides  270 ,  280 ,  290 ,  300 ,  310 . In some other embodiments, the illustrated image injection devices  360 ,  370 ,  380 ,  390 ,  400  may schematically represent a plurality of scanning fibers or a plurality of bundles of scanning fibers, each of which are configured to inject light into an associated one of the waveguides  270 ,  280 ,  290 ,  300 ,  310 . One or more optical fibers may transmit light from the light module  530  to the one or more waveguides  270 ,  280 ,  290 ,  300 , and  310 . In addition, one or more intervening optical structures may be provided between the scanning fiber, or fibers, and the one or more waveguides  270 ,  280 ,  290 ,  300 ,  310  to, for example, redirect light exiting the scanning fiber into the one or more waveguides  270 ,  280 ,  290 ,  300 ,  310 . 
     A controller  560  controls the operation of the stacked waveguide assembly  260 , including operation of the image injection devices  360 ,  370 ,  380 ,  390 ,  400 , the light source  530 , and the light modulator  540 . In some embodiments, the controller  560  is part of the local data processing module  140 . The controller  560  includes programming (e.g., instructions in a non-transitory medium) that regulates the timing and provision of image information to the waveguides  270 ,  280 ,  290 ,  300 ,  310 . In some embodiments, the controller may be a single integral device, or a distributed system connected by wired or wireless communication channels. The controller  560  may be part of the processing modules  140  or  150  ( FIG.  2   ) in some embodiments. 
     The waveguides  270 ,  280 ,  290 ,  300 ,  310  may be configured to propagate light within each respective waveguide by total internal reflection (TIR). The waveguides  270 ,  280 ,  290 ,  300 ,  310  may each be planar or have another shape (e.g., curved), with major top and bottom surfaces and edges extending between those major top and bottom surfaces. In the illustrated configuration, the waveguides  270 ,  280 ,  290 ,  300 ,  310  may each include out-coupling optical elements  570 ,  580 ,  590 ,  600 ,  610  that are configured to extract light out of a waveguide by redirecting the light, propagating within each respective waveguide, out of the waveguide to output image information to the eye  210 . Extracted light may also be referred to as out-coupled light and the out-coupling optical elements light may also be referred to light extracting optical elements. An extracted beam of light may be output by the waveguide at locations at which the light propagating in the waveguide strikes a light extracting optical element. The out-coupling optical elements  570 ,  580 ,  590 ,  600 ,  610  may be, for example, diffractive optical features, including diffractive gratings, as discussed further herein. While the out-coupling optical elements  570 ,  580 ,  590 ,  600 ,  610  are illustrated as being disposed at the bottom major surfaces of the waveguides  270 ,  280 ,  290 ,  300 ,  310 , in some embodiments they may be disposed at the top and/or bottom major surfaces, and/or may be disposed directly in the volume of the waveguides  270 ,  280 ,  290 ,  300 ,  310 , as discussed further herein. In some embodiments, the out-coupling optical elements  570 ,  580 ,  590 ,  600 ,  610  may be formed in a layer of material that is attached to a transparent substrate to form the waveguides  270 ,  280 ,  290 ,  300 ,  310 . In some other embodiments, the waveguides  270 ,  280 ,  290 ,  300 ,  310  may be a monolithic piece of material and the out-coupling optical elements  570 ,  580 ,  590 ,  600 ,  610  may be formed on a surface and/or in the interior of that piece of material. 
     Each waveguide  270 ,  280 ,  290 ,  300 ,  310  may output light to form an image corresponding to a particular depth plane. For example, the waveguide  270  nearest the eye may deliver collimated beams of light to the eye  210 . The collimated beams of light may be representative of the optical infinity focal plane. The next waveguide up 280 may output collimated beams of light which pass through the first lens  350  (e.g., a negative lens) before reaching the eye  210 . The first lens  350  may add a slight convex wavefront curvature to the collimated beams so that the eye/brain interprets light coming from that waveguide  280  as originating from a first focal plane closer inward toward the eye  210  from optical infinity. Similarly, the third waveguide  290  passes its output light through both the first lens  350  and the second lens  340  before reaching the eye  210 . The combined optical power of the first lens  350  and the second lens  340  may add another incremental amount of wavefront curvature so that the eye/brain interprets light coming from the third waveguide  290  as originating from a second focal plane that is even closer inward from optical infinity than was light from the second waveguide  280 . 
     The other waveguide layers  300 ,  310  and lenses  330 ,  320  are similarly configured, with the highest waveguide  310  in the stack sending its output through all of the lenses between it and the eye for an aggregate focal power representative of the closest focal plane to the person. To compensate for the stack of lenses  320 ,  330 ,  340 ,  350  when viewing/interpreting light coming from the world  510  on the other side of the stacked waveguide assembly  260 , a compensating lens layer  620  may be disposed at the top of the stack to compensate for the aggregate optical power of the lens stack  320 ,  330 ,  340 ,  350  below. Such a configuration provides as many perceived focal planes as there are available waveguide/lens pairings. Both the out-coupling optical elements of the waveguides and the focusing aspects of the lenses may be static (i.e., not dynamic or electro-active). In some alternative embodiments, either or both may be dynamic using electro-active features. 
     In some embodiments, two or more of the waveguides  270 ,  280 ,  290 ,  300 ,  310  may have the same associated depth plane. For example, multiple waveguides  270 ,  280 ,  290 ,  300 ,  310  may output images set to the same depth plane, or multiple subsets of the waveguides  270 ,  280 ,  290 ,  300 ,  310  may output images set to the same plurality of depth planes, with one set for each depth plane. This can provide advantages for forming a tiled image to provide an expanded field of view at those depth planes. 
     The out-coupling optical elements  570 ,  580 ,  590 ,  600 ,  610  may be configured to both redirect light out of their respective waveguides and to output this light with the appropriate amount of divergence or collimation for a particular depth plane associated with the waveguide. As a result, waveguides having different associated depth planes may have different configurations of out-coupling optical elements  570 ,  580 ,  590 ,  600 ,  610 , which output light with a different amount of divergence depending on the associated depth plane. In some embodiments, the light extracting optical elements  570 ,  580 ,  590 ,  600 ,  610  may be volumetric or surface features, which may be configured to output light at specific angles. For example, the light extracting optical elements  570 ,  580 ,  590 ,  600 ,  610  may be volume holograms, surface holograms, and/or diffraction gratings. In some embodiments, the features  320 ,  330 ,  340 ,  350  may not be lenses; rather, they may simply be spacers (e.g., cladding layers and/or structures for forming air gaps). 
     In some embodiments, the out-coupling optical elements  570 ,  580 ,  590 ,  600 ,  610  are diffractive features with a diffractive efficiency sufficiently low such that only a portion of the power of the light in a beam is re-directed toward the eye  210  with each interaction, while the rest continues to move through a waveguide via TIR. Accordingly, the exit pupil of the light module  530  is replicated across the waveguide to create a plurality of output beams carrying the image information from light source  530 , effectively expanding the number of locations where the eye  210  may intercept the replicated light source exit pupil. These diffractive features may also have a variable diffractive efficiency across their geometry to improve uniformity of light output by the waveguide. 
     In some embodiments, one or more diffractive features may be switchable between “on” states in which they actively diffract, and “off” states in which they do not significantly diffract. For instance, a switchable diffractive element may include a layer of polymer dispersed liquid crystal in which microdroplets form a diffraction pattern in a host medium, and the refractive index of the microdroplets may be switched to substantially match the refractive index of the host material (in which case the pattern does not appreciably diffract incident light) or the microdroplet may be switched to an index that does not match that of the host medium (in which case the pattern actively diffracts incident light). 
     In some embodiments, a camera assembly  630  (e.g., a digital camera, including visible light and IR light cameras) may be provided to capture images of the eye  210 , parts of the eye  210 , or at least a portion of the tissue surrounding the eye  210  to, for example, detect user inputs, extract biometric information from the eye, estimate and track the gaze direction of the eye, to monitor the physiological state of the user, etc. In some embodiments, the camera assembly  630  may include an image capture device and a light source to project light (e.g., IR or near-IR light) to the eye, which may then be reflected by the eye and detected by the image capture device. In some embodiments, the light source includes light emitting diodes (“LEDs”), emitting in IR or near-IR. In some embodiments, the camera assembly  630  may be attached to the frame  80  ( FIG.  2   ) and may be in electrical communication with the processing modules  140  or  150 , which may process image information from the camera assembly  630  to make various determinations regarding, for example, the physiological state of the user, the gaze direction of the wearer, iris identification, etc. In some embodiments, one camera assembly  630  may be utilized for each eye, to separately monitor each eye. 
       FIG.  7 A  illustrates an example of exit beams output by a waveguide. One waveguide is illustrated (with a perspective view), but other waveguides in the waveguide assembly  260  ( FIG.  6   ) may function similarly. Light  640  is injected into the waveguide  270  at the input surface  460  of the waveguide  270  and propagates within the waveguide  270  by TIR. Through interaction with diffractive features, light exits the waveguide as exit beams  650 . The exit beams  650  replicate the exit pupil from a projector device which projects images into the waveguide. Any one of the exit beams  650  includes a sub-portion of the total energy of the input light  640 . And in a perfectly efficient system, the summation of the energy in all the exit beams  650  would equal the energy of the input light  640 . The exit beams  650  are illustrated as being substantially parallel in  FIG.  7 A  but, as discussed herein, some amount of optical power may be imparted depending on the depth plane associated with the waveguide  270 . Parallel exit beams may be indicative of a waveguide with out-coupling optical elements that out-couple light to form images that appear to be set on a depth plane at a large distance (e.g., optical infinity) from the eye  210 . Other waveguides or other sets of out-coupling optical elements may output an exit beam pattern that is more divergent, as shown in  FIG.  7 B , which would require the eye  210  to accommodate to a closer distance to bring it into focus on the retina and would be interpreted by the brain as light from a distance closer to the eye  210  than optical infinity. 
     In some embodiments, a full color image may be formed at each depth plane by overlaying images in each of the component colors (e.g., three or more component colors, such as red, green, and blue).  FIG.  8    illustrates an example of a stacked waveguide assembly in which each depth plane includes images formed using multiple different component colors. The illustrated embodiment shows depth planes  240   a - 240   f , although more or fewer depths are also contemplated. Each depth plane may have three or more component color images associated with it, including: a first image of a first color, G; a second image of a second color, R; and a third image of a third color, B. Different depth planes are indicated in the figure by different diopter powers following the letters G, R, and B. The numbers following each of these letters indicate diopters (1/m), or inverse distance of the depth plane from a user, and each box in the figure represents an individual component color image. In some embodiments, to account for differences in the eye&#39;s focusing of light of different wavelengths, the exact placement of the depth planes for different component colors may vary. For example, different component color images for a given depth plane may be placed on depth planes corresponding to different distances from the user. Such an arrangement may increase visual acuity and user comfort or may decrease chromatic aberrations. 
     In some embodiments, light of each component color may be output by a single dedicated waveguide and, consequently, each depth plane may have multiple waveguides associated with it. In such embodiments, each box in the figure may be understood to represent an individual waveguide, and three waveguides may be provided per depth plane so as to display three component color images per depth plane. While the waveguides associated with each depth plane are shown adjacent to one another in this drawing for ease of illustration, it will be appreciated that, in a physical device, the waveguides may all be arranged in a stack with one waveguide per level. In some other embodiments, multiple component colors may be output by the same waveguide, such that, for example, only a single waveguide may be provided per depth plane. 
     With continued reference to  FIG.  8   , in some embodiments, G is the color green, R is the color red, and B is the color blue. In some other embodiments, other colors associated with other wavelengths of light, including yellow, magenta and cyan, may be used in addition to or may replace one or more of red, green, or blue. In some embodiments, features  320 ,  330 ,  340 , and  350  may be active or passive optical filters configured to block or selectively pass light from the ambient environment to the user&#39;s eyes. 
     References to a given color of light throughout this disclosure should be understood to encompass light of one or more wavelengths within a range of wavelengths of light that are perceived by a user as being of that given color. For example, red light may include light of one or more wavelengths in the range of about 620-780 nm, green light may include light of one or more wavelengths in the range of about 492-577 nm, and blue light may include light of one or more wavelengths in the range of about 435-493 nm. 
     In some embodiments, the light source  530  ( FIG.  6   ) may be configured to emit light of one or more wavelengths outside the visual perception range of the user, for example, IR or ultraviolet wavelengths. IR light can include light with wavelengths in a range from 700 nm to 10 μm. In some embodiments, IR light can include near-IR light with wavelengths in a range from 700 nm to 1.5 μm. In addition, the in-coupling, out-coupling, and other light redirecting structures of the waveguides of the display  250  may be configured to direct and emit this light out of the display towards the user&#39;s eye  210 , e.g., for imaging or user stimulation applications. 
     With reference now to  FIG.  9 A , in some embodiments, light impinging on a waveguide may need to be redirected so as to in-couple the light into the waveguide. An in-coupling optical element may be used to redirect and in-couple the light into its corresponding waveguide.  FIG.  9 A  illustrates a cross-sectional side view of an example of a set  660  of stacked waveguides that each includes an in-coupling optical element. The waveguides may each be configured to output light of one or more different wavelengths, or one or more different ranges of wavelengths. It will be appreciated that the stack  660  may correspond to the stack  260  ( FIG.  6   ) and the illustrated waveguides of the stack  660  may correspond to part of the plurality of waveguides  270 ,  280 ,  290 ,  300 ,  310 , except that light from one or more of the image injection devices  360 ,  370 ,  380 ,  390 ,  400  is injected into the waveguides from a position or orientation that requires light to be redirected for in-coupling. 
     The illustrated set  660  of stacked waveguides includes waveguides  670 ,  680 , and  690 . Each waveguide includes an associated in-coupling optical element (which may also be referred to as a light input area on the waveguide), with, for example, in-coupling optical element  700  disposed on a major surface (e.g., an upper major surface) of waveguide  670 , in-coupling optical element  710  disposed on a major surface (e.g., an upper major surface) of waveguide  680 , and in-coupling optical element  720  disposed on a major surface (e.g., an upper major surface) of waveguide  690 . In some embodiments, one or more of the in-coupling optical elements  700 ,  710 ,  720  may be disposed on the bottom major surface of the respective waveguide  670 ,  680 ,  690  (particularly where the one or more in-coupling optical elements are reflective optical elements). As illustrated, the in-coupling optical elements  700 ,  710 ,  720  may be disposed on the upper major surface of their respective waveguide  670 ,  680 ,  690  (or the top of the next lower waveguide), particularly where those in-coupling optical elements are transmissive optical elements. In some embodiments, the in-coupling optical elements  700 ,  710 ,  720  may be disposed in the body of the respective waveguide  670 ,  680 ,  690 . In some embodiments, as discussed herein, the in-coupling optical elements  700 ,  710 ,  720  are wavelength selective, such that they selectively redirect one or more wavelengths of light, while transmitting other wavelengths of light. While illustrated on one side or corner of their respective waveguide  670 ,  680 ,  690 , it will be appreciated that the in-coupling optical elements  700 ,  710 ,  720  may be disposed in other areas of their respective waveguide  670 ,  680 ,  690  in some embodiments. 
     As illustrated, the in-coupling optical elements  700 ,  710 ,  720  may be laterally offset from one another. In some embodiments, each in-coupling optical element may be offset such that it receives light without that light passing through another in-coupling optical element. For example, each in-coupling optical element  700 ,  710 ,  720  may be configured to receive light from a different image injection device  360 ,  370 ,  380 ,  390 , and  400  as shown in  FIG.  6   , and may be separated (e.g., laterally spaced apart) from other in-coupling optical elements  700 ,  710 ,  720  such that it substantially does not receive light from the other ones of the in-coupling optical elements  700 ,  710 ,  720 . 
     Each waveguide also includes associated light distributing elements, with, for example, light distributing elements  730  disposed on a major surface (e.g., a top major surface) of waveguide  670 , light distributing elements  740  disposed on a major surface (e.g., a top major surface) of waveguide  680 , and light distributing elements  750  disposed on a major surface (e.g., a top major surface) of waveguide  690 . In some other embodiments, the light distributing elements  730 ,  740 ,  750  may be disposed on a bottom major surface of associated waveguides  670 ,  680 ,  690 , respectively. In some other embodiments, the light distributing elements  730 ,  740 ,  750  may be disposed on both top and bottom major surface of associated waveguides  670 ,  680 ,  690  respectively; or the light distributing elements  730 ,  740 ,  750 , may be disposed on different ones of the top and bottom major surfaces in different associated waveguides  670 ,  680 ,  690 , respectively. 
     The waveguides  670 ,  680 ,  690  may be spaced apart and separated by, for example, gas, liquid, or solid layers of material. For example, as illustrated, layer  760   a  may separate waveguides  670  and  680 ; and layer  760   b  may separate waveguides  680  and  690 . In some embodiments, the layers  760   a  and  760   b  are formed of low refractive index materials (that is, materials having a lower refractive index than the material forming the immediately adjacent one of waveguides  670 ,  680 ,  690 ). Preferably, the refractive index of the material forming the layers  760   a ,  760   b  is at least 0.05, or at least 0.10, less than the refractive index of the material forming the waveguides  670 ,  680 ,  690 . Advantageously, the lower refractive index layers  760   a ,  760   b  may function as cladding layers that facilitate TIR of light through the waveguides  670 ,  680 ,  690  (e.g., TIR between the top and bottom major surfaces of each waveguide). In some embodiments, the layers  760   a ,  760   b  are formed of air. While not illustrated, it will be appreciated that the top and bottom of the illustrated set  660  of waveguides may include immediately neighboring cladding layers. 
     Preferably, for ease of manufacturing and other considerations, the material forming the waveguides  670 ,  680 ,  690  are similar or the same, and the material forming the layers  760   a ,  760   b  are similar or the same. In other embodiments, the material forming the waveguides  670 ,  680 ,  690  may be different between one or more waveguides, or the material forming the layers  760   a ,  760   b  may be different, while still holding to the various refractive index relationships noted above. 
     With continued reference to  FIG.  9 A , light rays  770 ,  780 ,  790  are incident on the set  660  of waveguides. Light rays  770 ,  780 ,  790  may be injected into the waveguides  670 ,  680 ,  690  by one or more image injection devices  360 ,  370 ,  380 ,  390 ,  400  ( FIG.  6   ). 
     In some embodiments, the light rays  770 ,  780 ,  790  have different properties (e.g., different wavelengths or different ranges of wavelengths), which may correspond to different colors. The in-coupling optical elements  700 ,  710 ,  720  each re-direct the incident light such that the light propagates through a respective one of the waveguides  670 ,  680 ,  690  by TIR. 
     For example, in-coupling optical element  700  may be configured to re-direct ray  770 , which has a first wavelength or range of wavelengths. Similarly, transmitted ray  780  impinges on and is re-directed by in-coupling optical element  710 , which is configured to re-direct light of a second wavelength or range of wavelengths. Likewise, ray  790  is re-directed by in-coupling optical element  720 , which is configured to selectively re-direct light of third wavelength or range of wavelengths. 
     With continued reference to  FIG.  9 A , light rays  770 ,  780 ,  790  are re-directed so that they propagate through a corresponding waveguide  670 ,  680 ,  690 ; that is, the in-coupling optical element  700 ,  710 ,  720  of each waveguide re-directs light into that corresponding waveguide  670 ,  680 ,  690  to in-couple light into that corresponding waveguide. The light rays  770 ,  780 ,  790  are re-directed at angles that cause the light to propagate through the respective waveguide  670 ,  680 ,  690  by TIR. The light rays  770 ,  780 ,  790  propagate through the respective waveguide  670 ,  680 ,  690  by TIR until interacting with the waveguide&#39;s corresponding light distributing elements  730 ,  740 ,  750 . 
     With reference now to  FIG.  9 B , a perspective view of an example of the plurality of stacked waveguides of  FIG.  9 A  is illustrated. As noted above, the light rays  770 ,  780 ,  790 , are in-coupled by the in-coupling optical elements  700 ,  710 ,  720 , respectively, and then propagate by TIR within the waveguides  670 ,  680 ,  690 , respectively. The light rays  770 ,  780 ,  790  then interact with the light distributing elements  730 ,  740 ,  750 , respectively. The light distributing elements  730 ,  740 ,  750  re-direct the light rays  770 ,  780 ,  790  so that they propagate towards the out-coupling optical elements  800 ,  810 , and  820 , respectively. 
     In some embodiments, the light distributing elements  730 ,  740 ,  750  are orthogonal pupil expanders (OPEs). In some embodiments, the OPEs both re-direct light to the out-coupling optical elements  800 ,  810 ,  820  and also expand the pupil associated with this light by sampling the light rays  770 ,  780 ,  790  at many locations across the light distributing elements  730 ,  740 ,  750  as they propagate to the out-coupling optical elements. In some embodiments (e.g., where the exit pupil is already of a desired size), the light distributing elements  730 ,  740 ,  750  may be omitted and the in-coupling optical elements  700 ,  710 ,  720  may be configured to re-direct light directly to the out-coupling optical elements  800 ,  810 ,  820 . For example, with reference to  FIG.  9 A , the light distributing elements  730 ,  740 ,  750  may be replaced with out-coupling optical elements  800 ,  810 ,  820 , respectively. In some embodiments, the out-coupling optical elements  800 ,  810 ,  820  are exit pupils (EPs) or exit pupil expanders (EPEs) that re-direct light out of the waveguides and toward a user&#39;s eye  210  ( FIG.  7   ). The OPEs may be configured to increase the dimensions of the eye box in at least one axis and the EPEs may be configured to increase the eye box in an axis crossing (e.g., orthogonal to) the axis of the OPEs. 
     Accordingly, with reference to  FIGS.  9 A and  9 B , in some embodiments, the set  660  of waveguides includes waveguides  670 ,  680 ,  690 ; in-coupling optical elements  700 ,  710 ,  720 ; light distributing elements (e.g., OPEs)  730 ,  740 ,  750 ; and out-coupling optical elements (e.g., EPEs)  800 ,  810 ,  820  for each component color. The waveguides  670 ,  680 ,  690  may be stacked with an air gap/cladding layer between each one. The in-coupling optical elements  700 ,  710 ,  720  direct incident light (with different in-coupling optical elements receiving light of different wavelengths) into a corresponding waveguide. The light then propagates at angles which support TIR within the respective waveguide  670 ,  680 ,  690 . Since TIR only occurs for a certain range of angles, the range of propagation angles of the light rays  770 ,  780 ,  790  is limited. The range of angles which support TIR may be thought of in such an example as the angular limits of the field of view which can be displayed by the waveguides  670 ,  680 ,  690 . In the example shown, light ray  770  (e.g., blue light) is in-coupled by the first in-coupling optical element  700 , and then continues to reflect back and forth from the surfaces of the waveguide while traveling down the waveguide, with the light distributing element (e.g., OPE)  730  progressively sampling it to create additional replicated rays which are directed toward the out-coupling optical element (e.g., EPE)  800 , in a manner described earlier. The light rays  780  and  790  (e.g., green and red light, respectively) will pass through the waveguide  670 , with light ray  780  impinging on and being in-coupled by in-coupling optical element  710 . The light ray  780  then propagates down the waveguide  680  via TIR, proceeding on to its light distributing element (e.g., OPE)  740  and then the out-coupling optical element (e.g., EPE)  810 . Finally, light ray  790  (e.g., red light) passes through the waveguides  670 ,  680  to impinge on the light in-coupling optical element  720  of the waveguide  690 . The light in-coupling optical element  720  in-couples the light ray  790  such that the light ray propagates to light distributing element (e.g., OPE)  750  by TIR, and then to the out-coupling optical element (e.g., EPE)  820  by TIR. The out-coupling optical element  820  then finally out-couples the light ray  790  to the user, who also receives the out-coupled light from the other waveguides  670 ,  680 . 
       FIG.  9 C  illustrates a top-down plan view of an example of the plurality of stacked waveguides of  FIGS.  9 A and  9 B . As illustrated, the waveguides  670 ,  680 ,  690 , along with each waveguide&#39;s associated light distributing element  730 ,  740 ,  750  and associated out-coupling optical element  800 ,  810 ,  820 , may be vertically aligned. However, as discussed herein, the in-coupling optical elements  700 ,  710 ,  720  are not vertically aligned; rather, the in-coupling optical elements may be non-overlapping (e.g., laterally spaced apart as seen in the top-down view). This non-overlapping spatial arrangement may facilitate the injection of light from different sources into different waveguides on a one-to-one basis, thereby allowing a specific light source to be uniquely optically coupled to a specific waveguide. In some embodiments, arrangements including non-overlapping spatially separated in-coupling optical elements may be referred to as a shifted pupil system, and the in-coupling optical elements within these arrangements may correspond to sub pupils. 
       FIG.  10    is a perspective view of an example AR eyepiece waveguide stack  1000 . The eyepiece waveguide stack  1000  may include a world-side cover window  1002  and an eye-side cover window  1006  to protect one or more eyepiece waveguides  1004  positioned between the cover windows. In other embodiments, one or both of the cover windows  1002 ,  1006  may be omitted. As already discussed, the eyepiece waveguides  1004  may be arranged in a layered configuration. The eyepiece waveguides  1004  may be coupled together, for instance, with each individual eyepiece waveguide being coupled to one or more adjacent eyepiece waveguides. In some embodiments, the waveguides  1004  may be coupled together with an edge seal (such as the edge seal  1108  shown in  FIG.  11   ) such that adjacent eyepiece waveguides  1004  are not in direct contact with each other. 
     Each of the eyepiece waveguides  1004  can be made of a substrate material that is at least partially transparent, such as glass, plastic, polycarbonate, sapphire, etc. The selected material may have an index of refraction above 1.4, for example, or above 1.6, or above 1.8, to facilitate light guiding. The thickness of each eyepiece waveguide substrate may be, for example, 325 microns or less, though other thicknesses can also be used. Each eyepiece waveguide can include one or more in-coupling regions, light distributing regions, image expanding regions, and out-coupling regions, which may be made up of diffractive features formed on or in each waveguide substrate  902 . 
     Although not illustrated in  FIG.  10   , the eyepiece waveguide stack  1000  can include a physical support structure for supporting it in front of a user&#39;s eyes. In some embodiments, the eyepiece waveguide stack  1000  is part of a head-mounted display system  60 , as illustrated in  FIG.  2   . In general, the eyepiece waveguide stack  1000  is supported such that an out-coupling region is directly in front of a user&#39;s eye. It should be understood that  FIG.  10    illustrates only the portion of the eyepiece waveguide stack  1000  which corresponds to one of the user&#39;s eyes. A complete eyepiece may include a mirror image of the same structure, with the two halves possibly separated by a nose piece. 
     In some embodiments, the eyepiece waveguide stack  1000  can project color image data from multiple depth planes into the user&#39;s eyes. The image data displayed by each individual eyepiece waveguide  1004  in the eyepiece  1000  may correspond to a selected color component of the image data for a selected depth plane. For example, since the eyepiece waveguide stack  1000  includes six eyepiece waveguides  1004 , it can project color image data (e.g., made up of red, green, and blue components) corresponding to two different depth planes: one eyepiece waveguide  1004  per color component per depth plane. Other embodiments can include eyepiece waveguides  1004  for more or fewer color components and/or more or fewer depth planes. 
       FIG.  11    is a cross-sectional view of a portion of an example eyepiece waveguide stack  1100  with an edge seal structure  1108  for supporting eyepiece waveguides  1104  in a stacked configuration. The edge seal structure  1108  aligns the eyepiece waveguides  1104  and separates them from one another with air space or another material disposed between. Although not illustrated, the edge seal structure  1108  can extend around the entire perimeter of the stacked waveguide configuration. In  FIG.  11   , the separation between each eyepiece waveguide is 0.027 mm, though other distances are also possible. 
     In the illustrated embodiment, there are two eyepiece waveguides  1104  designed to display red image data, one for a 3 m depth plane and the other for a 1 m depth plane. (Again, the divergence of the beams of light output by an eyepiece waveguide  1104  can make the image data appear to originate from a depth plane located at a particular distance.) Similarly, there are two eyepiece waveguides  1104  designed to display blue image data, one for a 3 m depth plane and the other for a 1 m depth plane, and two eyepiece waveguides  1104  designed to display green image data, one for a 3 m depth plane and the other for a 1 m depth plane. Each of these six eyepiece waveguides  1104  is illustrated as being 0.325 mm thick, though other thicknesses are also possible. 
     A world-side cover window  1102  and an eye-side cover window  1106  are also shown in  FIG.  11   . These cover windows can be, for example, 0.330 mm thick. When accounting for the thickness of the six eyepiece waveguides  1104 , the seven air gaps, the two cover windows  1102 ,  1106 , and the edge seal  1108 , the total thickness of the illustrated eyepiece waveguide stack  1100  is 2.8 mm. 
     K-Space Representations of AR Eyepiece Waveguides 
       FIGS.  12 A and  12 B  illustrate top views of an eyepiece waveguide  1200  in operation as it projects an image toward a user&#39;s eye  210 . The image can first be projected from an image plane  1207  toward an entrance pupil  1208  of the eyepiece waveguide  1200  using a projection lens  1210  or some other projector device. Each image point (e.g., an image pixel or part of an image pixel) has a corresponding input beam of light (e.g.,  1202   a ,  1204   a ,  1206   a ) which propagates in a particular direction at the entrance pupil  1208  (e.g., at a particular angle with respect to the optical axis of the projector lens  1210 ). Although illustrated as rays, the input beams of light  1202   a ,  1204   a ,  1206   a  may be, for example, collimated beams with diameters of a few millimeters or less when they enter the eyepiece waveguide  1200 . 
     In  FIGS.  12 A and  12 B , a middle image point corresponds to input beam  1204   a , which is illustrated with a solid line. A right-hand image point corresponds to input beam  1202   a , which is illustrated with a dashed line. And a left-hand image point corresponds to input beam  1206   a , which is illustrated with a dash-dot line. For clarity of illustration, only three input beams  1202   a ,  1204   a ,  1206   a  are shown at the entrance pupil  1208 , though a typical input image will include many input beams propagating at a range of angles, both in the x-direction and the y-direction, which correspond to different image points in a two-dimensional image plane. 
     There is a unique correspondence between the various propagation angles of the input beams (e.g.,  1202   a ,  1204   a ,  1206   a ) at the entrance pupil  1208  and the respective image points at the image plane  1207 . The eyepiece waveguide  1200  can be designed to in-couple the input beams (e.g.,  1202   a ,  1204   a ,  1206   a ), replicate them in a distributed manner through space, and guide them to form an exit pupil  1210 , which is larger than the entrance pupil  1208  and is made up of the replicated beams, all while substantially maintaining the correspondence between image points and beam angles. The eyepiece waveguide  1200  can convert a given input beam of light (e.g.,  1202   a ), which propagates at a particular angle, into many replicated beams (e.g.,  1202   b ) which are output across the exit pupil  1210  at an angle that is substantially uniquely correlated with that particular input beam and its corresponding image point. For example, the replicated output beams corresponding to each input beam can exit the eyepiece waveguide  1200  at substantially the same angle as their corresponding input beam. 
     As shown in  FIGS.  12 A and  12 B , the input beam of light  1204   a  corresponding to the middle image point at the image plane  1207  is converted into a set of replicated output beams  1204   b , shown with solid lines, which are aligned with an optical axis perpendicular to the exit pupil  1210  of the eyepiece waveguide  1200 . The input beam of light  1202   a  corresponding to the right-hand image point at the image plane  1207  is converted into a set of replicated output beams  1202   b , shown with dashed lines, which exit the eyepiece waveguide  1200  at a propagation angle such that they appear to have originated from a location in the right-hand portion of the user&#39;s field of view. Similarly, the input beam of light  1206   a  corresponding to the left-hand image point at the image plane  1207  is converted into a set of replicated output beams  1206   b , shown with dash-dot lines, which exit the eyepiece waveguide  1200  at a propagation angle such that they appear to have originated from a location in the left-hand portion of the user&#39;s field of view. The greater the range of input beam angles and/or output beam angles, the greater the field of view (FOV) of the eyepiece waveguide  1200 . 
     For each image, there are sets of replicated output beams (e.g.,  1202   b ,  1204   b ,  1206   b )—one set of replicated beams per image point—which are output across the exit pupil  1210  at different angles. The individual output beams (e.g.,  1202   b ,  1204   b ,  1206   b ) can each be collimated. The set of output beams corresponding to a given image point may consist of beams which propagate along parallel paths (as shown in  FIG.  12 A ) or diverging paths (as shown in  FIG.  12 B ). In either case, the specific propagation angle of the set of replicated output beams depends on the location of the corresponding image point at the image plane  1207 .  FIG.  12 A  illustrates the case where each set of output beams (e.g.,  1202   b ,  1204   b ,  1206   b ) consists of beams which propagate along parallel paths. This results in the image being projected so as to appear to have originated from optical infinity. This is represented in  FIG.  12 A  by the faint lines extending from the peripheral output beams  1202   b ,  1204   b ,  1206   b  toward optical infinity on the world-side of the eyepiece waveguide  1200  (opposite the side where the user&#39;s eye  210  is located).  FIG.  12 B  illustrates the case where each set of output beams (e.g.,  1202   b ,  1204   b ,  1206   b ) consists of beams which propagate along diverging paths. This results in the image being projected so as to appear to have originated from a virtual depth plane having a distance closer than optical infinity. This is represented in  FIG.  12 B  by the faint lines extending from the peripheral output beams  1202   b ,  1204   b ,  1206   b  toward points on the world-side of the eyepiece waveguide  1200 . 
     Again, each set of replicated output beams (e.g.,  1202   b ,  1204   b ,  1206   b ) has a propagation angle that corresponds to a particular image point at the image plane  1207 . In the case of a set of replicated output beams which propagate along parallel paths (see  FIG.  12 A ), the propagation angles of all the beams are the same. In the case of a set of replicated output beams which propagate along diverging paths, however, the individual output beams can propagate at different angles, but those angles are related to one another in that they create an aggregate diverging wavefront and appear to have originated from a common point along the axis of the set of beams (See  FIG.  12 B ). It is this axis which defines the angle of propagation for the set of diverging output beams and which corresponds to a particular image point at the image plane  1207 . 
     The various beams of light entering the eyepiece waveguide  1200 , propagating within the eyepiece waveguide, and exiting the eyepiece waveguide can all be described using one or more wave vectors, or k-vectors, which describe a beam&#39;s direction(s) of propagation. K-space is an analytical framework which relates k-vectors to geometrical points. In k-space, each point in space corresponds to a unique k-vector, which in turn can represent a beam or ray of light with a particular propagation direction. This allows the input and output beams, with their corresponding propagation angles, to be understood as a set of points (e.g., a rectangle) in k-space. The diffractive features which change the propagation directions of the light beams while traveling through the eyepiece can be understood in k-space as simply translating the location of the set of k-space points which make up the image. This new translated k-space location corresponds to a new set of k-vectors, which in turn represent the new propagation angles of the beams or rays of light after interacting with the diffractive features. 
     The operation of an eyepiece waveguide can be understood by the manner in which it causes a set of points, such as the points inside a k-space rectangle which correspond to a projected image, to move in k-space. This is in contrast to more complicated ray tracing diagrams which might otherwise be used to illustrate the beams and their propagation angles. K-space is therefore an effective tool for describing the design and operation of eyepiece waveguides. The following discussion describes the k-space representation of features and functions of various AR eyepiece waveguides. 
       FIG.  13 A  illustrates a k-vector  1302  which can be used to represent the propagation direction of a light ray or a light beam. The particular illustrated k-vector  1302  is representative of a plane wave with planar wavefronts  1304 . The k-vector  1302  points in the propagation direction of the light ray or beam which it represents. The magnitude, or length, of the k-vector  1302  is defined by a wavenumber, k. The dispersion equation, w=ck, relates the angular frequency, w, of the light, the speed of the light, c, and the wavenumber, k. (In a vacuum, the speed of the light is equal to the speed of light constant, c. In a medium, however, the speed of the light is inversely proportional to the refractive index of the medium. Thus, in a medium the equation becomes k=nω/c.) Note that by definition, k=2π/λ and ω=2πf, where f is the frequency of light (e.g. in units of Hertz). As is evident from this equation, light beams with higher angular frequencies, w, have larger wavenumbers, and thus larger-magnitude k-vectors (assuming the same propagation medium). For instance, assuming the same propagation medium, blue light beams have larger-magnitude k-vectors than red light beams. 
       FIG.  13 B  illustrates a light ray  1301  corresponding to the k-vector  1302  within a planar waveguide  1300 . The waveguide  1300  can be representative of any of the waveguides described herein and may be part of an eyepiece for an AR display system. The waveguide  1300  can guide light rays having certain k-vectors via total internal reflection (TIR). For example, as shown in  FIG.  13 B , the light ray  1301  illustrated by k-vector  1302  is directed toward the upper surface of the waveguide  1300  at an angle. If the angle is not too steep, as governed by Snell&#39;s law, then the light ray  1301  will reflect at the upper surface of the waveguide  1300 , at an angle equal to the angle of incidence, and then propagate down toward the lower surface of the waveguide  1300  where it will reflect again back towards the upper surface. The light ray  1301  will continue propagating in a guided fashion within the waveguide  1300 , reflecting back and forth between its upper and lower surfaces. 
       FIG.  13 C  illustrates the permissible k-vectors for light of a given angular frequency, ω, propagating in an unbounded homogenous medium with refractive index, n. The length, or magnitude, k, of the illustrated k-vector  1302  is equal to the refractive index, n, of the medium times the angular frequency, ω, of the light divided by the speed of light constant, c. For light rays or beams with a given angular frequency, ω, propagating in a homogeneous medium with refractive index, n, the magnitudes of all permissible k-vectors are the same. And for unguided propagation, all propagation directions are permitted. Therefore, the manifold in k-space which defines all permissible k-vectors is a hollow sphere  1306 , where the size of the sphere is dependent upon the angular frequency of the light and the refractive index of the medium. 
       FIG.  13 D  illustrates the permissible k-vectors for light of a given angular frequency, ω, propagating in a homogenous planar waveguide medium with refractive index, n. Whereas in an unbound medium, all permissible k-vectors lie on the hollow sphere  1306 , to determine the permissible k-vectors in a planar waveguide, we can project the sphere  1306  of permissible k-vectors onto a plane (e.g., the x-y plane). This results in a solid disk  1308  in projected k-space, which represents the k-vectors which can propagate within a planar waveguide. As shown in  FIG.  13 D , the k-vectors which can propagate within a planar waveguide in the x-y plane (e.g., waveguide  1300 ) are all those for which the component of the k-vector in the x-y plane is less than or equal to the refractive index, n, of the medium times the angular frequency, ω, of the light divided by the speed of light constant, c. 
     Every point within the solid disk  1308  corresponds to the k-vector of a wave which can propagate in the waveguide (though not all of these k-vectors result in guided propagation within the waveguide, as discussed below with respect to  FIG.  13 E ). At each point within the solid disk  1308 , there are two permitted waves: one with a z-component of propagation into the page, and another with a z-component of propagation out of the page. Therefore the out-of-plane component of the k-vector, k z , may be recovered using the equation k z =±√{square root over (|k| 2 −k x   2 −k y   2 )}, where the sign chosen determines whether the wave is propagating into or out of the page. Since all light waves of a given angular frequency, ω, propagating in a homogeneous medium with refractive index, n, have the same magnitude k-vector, light waves with k-vectors whose x-y components are closer in size to the radius of the solid disk  1308  have smaller z-components of propagation (resulting in the less steep propagation angles necessary for TIR, as discussed with respect to  FIG.  13 B ), while light waves with k-vectors whose x-y components are located closer to the center of the solid disk  1308  have larger z-components of propagation (resulting in steeper propagation angles which may not TIR). Henceforth, all mentions of k-space refer to the projected k-space (unless otherwise evident from context), in which the 2-dimensional k-plane corresponds to the plane of the waveguide; unless the propagation direction between surfaces of the waveguide is explicitly mentioned, the discussion and drawings generally only consider the directions parallel to the surfaces of the waveguide. Furthermore, when plotting k-space, it is typically most convenient to normalize the free-space disk radius to unity, so that plots are effectively normalized to co/c. 
       FIG.  13 E  illustrates an annulus  1310  in k-space which corresponds to k-vectors of light waves which can be guided within a waveguide having a refractive index, n 2  (e.g., n 2 =1.5). The waveguide is physically surrounded by a medium (e.g., air) having a lesser refractive index, n 1  (e.g., n 1 ≈1). As just discussed with respect to  FIG.  13 D , the k-vectors corresponding to permitted waves within a planar waveguide medium in the x-y plane are all those k-vectors whose respective x-y components lie in a solid disk  1308  in k-space. The radius of the solid disk  1308  is proportional to the refractive index of the waveguide medium. Thus, with reference back to  FIG.  13 E , the k-vectors which correspond to light waves which can propagate in a planar waveguide medium having refractive index n 2 =1.5 are those whose respective x-y components lie within the larger disk  1308   a . Meanwhile, the k-vectors which correspond to light waves which can propagate in the surrounding medium having refractive index n 1 =1 are those whose respective x-y components lie within the smaller disk  1308   b . All k-vectors whose respective x-y components lie inside the annulus  1310  correspond to those light waves which can propagate in the waveguide medium but not in the surrounding medium (e.g., air). These are the light waves which are guided in the waveguide medium via total internal reflection, as described with respect to  FIG.  13 B . Thus, light rays or beams can only undergo guided propagation within a waveguide of an AR eyepiece if they have k-vectors which lie in the k-space annulus  1310 . Note that propagating light waves having k-vectors outside of the larger disk  1308   a  are forbidden; there are no propagating waves whose k-vectors lie in that region (waves in that region have evanescently decaying, rather than constant, amplitude along their propagation direction). 
     The various AR eyepiece waveguides described herein can in-couple light by using diffractive features, such as diffractive structures, to direct the k-vectors of light beams propagating in free space (n 1 ≈1) (e.g., from a projector) into the k-space annulus  1310  of an eyepiece waveguide. Any light wave whose k-vector lies in the annulus  1310  can propagate in guided fashion within the eyepiece waveguide. The width of the annulus  1310  determines the range of k-vectors—and, hence, the range of propagation angles—which can be guided within the eyepiece waveguide. Thus, the width of the k-space annulus  1310  has typically been thought to determine the maximum field of view (FOV) which can be projected by the eyepiece waveguide. Since the width of the annulus  1310  depends on the radius of the larger disk  1308   a , which is itself partially dependent upon the refractive index, n 2 , of the eyepiece waveguide medium, one technique for increasing eyepiece FOV is to use an eyepiece waveguide medium with a larger refractive index (in comparison to the refractive index of the medium surrounding the eyepiece waveguide). There are, however, practical limitations on the refractive indexes of waveguide media which can be used in AR eyepieces, such as material cost. This in turn has been thought to place practical limitations on the FOV of AR eyepieces. But, as explained herein, there are techniques which can be used to overcome these limitations so as to allow for larger FOVs. 
     Although the radius of the larger disk  1308   a  in  FIG.  13 E  is also dependent on the angular frequency, ω, of the light, and the width of the annulus  1310  therefore depends on the color of the light, this does not imply that the FOV supported by the eyepiece waveguide is larger for light with higher angular frequencies, since any given angular extent corresponding to the FOV scales in direct proportion to the angular frequency as well. 
       FIG.  13 F  shows a k-space diagram similar to that depicted in  FIG.  13 E . The k-space diagram shows a smaller disk  1308   b  corresponding to permissible k-vectors in a first medium of refractive index n 1 , a larger disk  1308   a  corresponding to permissible k-vectors in a second medium of refractive index n 2  (n 2 &gt;n 1 ), and an annulus  1310  between the outer boundaries of smaller disk  1308   a  and larger disk  1308   b . Although all k-vectors in the width  1342  of the annulus  1310  correspond to guided propagation angles, it is possible that fewer than all of the k-vectors that lie within the width  1342  of the annulus  1310  may be satisfactory for use in displaying an image. 
       FIG.  13 F  also shows a waveguide  1350  with two guided beams shown in comparison to one another. The first light beam has a first k-vector  1344   a  near the outer edge of the annulus  1310 . The first k-vector  1344   a  corresponds to a first TIR propagation path  1344   b  shown in a cross-sectional view of the waveguide  1350  having refractive index n 2  surrounded by air of refractive index n 1 . A second light beam is also shown that has a second k-vector  1346   a  closer to the center of the k-space annulus  1310 . The second k-vector  1346   a  corresponds to a second TIR propagation path  1346   b  in the waveguide  1350 . The waveguide  1350  may include a diffraction grating  1352  on or within the waveguide  1350 . When a light beam encounters the surface of the waveguide  1350  with the diffraction grating  1352 , an interaction occurs which may send a sample of the light beam energy out of the waveguide while the beam continues to TIR in the waveguide. The angle at which a light beam propagates in TIR through the waveguide determines the density of reflection events, or the number of bounces per unit length against the surface of the waveguide  1350  with the diffraction grating  1352 . Returning to the example of the light beam comparison, the first light beam in the first TIR propagation path  1344   b  reflects from the waveguide surface with the diffraction grating  1352  four times to produce four exit pupils  1354  (illustrated with solid lines) over the length of the diffraction grating  1352 , while the second light beam in the second TIR propagation path  1346   b  reflects from the waveguide surface with diffraction grating  1352  ten times, over the same or similar distance, to produce ten exit pupils  1356  (illustrated with dashed lines) across the length of the diffraction grating  1352 . 
     In practice, it may be desirable to constrain the output beam, or exit pupil spacing, to be equal to, or within, a pre-selected range to ensure that a user will see the projected content from any position within the pre-defined eye box. With this information, it is possible to limit the width  1342  of the annulus  1310  to a subset  1344  of k-vectors for which this constraint holds, and to disqualify angles that are too grazing from being included in the design calculations. More or fewer angles than the subset  1344  may be acceptable depending on desired performance, diffraction grating design, and other optimization factors. Similarly, in some embodiments, k-vectors corresponding to propagation angles that are too steep with respect to the surface of the waveguide and provide too many interactions with the diffraction grating  1352  may also be disqualified from use. In such embodiments, the width  1342  of the annulus  1310  can be decreased by effectively moving the boundary of usable angles radially outward from the boundary between the larger disk  1308   a  and the smaller disk  1308   b . The designs of any of the eyepiece waveguides disclosed herein can be adjusted by constraining the width of the k-space annulus  1310  in this way. 
     As described above, k-vectors, within the annulus  1310 , corresponding to suboptimal TIR propagation pathways may be omitted from use in eyepiece design calculations. Alternatively, k-vectors corresponding to TIR propagation pathways with too grazing of an angle, and thus too low of a density of reflection events on the surface of the waveguide with a diffraction grating, may be compensated for using various techniques described herein. One technique is to use an in-coupling grating to direct portions of the field of view (FOV) of the incoming image to two different areas of the k-space annulus  1310 . In particular, it may be advantageous to direct the incoming image to a first side of the k-space annulus  1310 , represented by a first group of k-vectors, and to a second side of the k-space annulus  1310 , represented by a second group of k-vectors, where the first and second sides of the k-space annulus  1310  are substantially opposed from one another. For example, the first group of k vectors may correspond to an FOV rectangle of k-vectors on the left side of the annulus  1310  and the second group of k-vectors may correspond to an FOV rectangle of k-vectors on the right side of the annulus  1310 . The left FOV rectangle has its left edge near the outer edge of larger disk  1308   a , corresponding to near-grazing k-vector angles. Light at this edge would produce sparse exit pupils. However, the same left edge of the right FOV rectangle, located on the right side of the annulus  1310 , would be nearer to the center of the larger disk  1308   a . Light at the same left edge of the right FOV rectangle would have a high density of exit pupils. Thus, when the left and right FOV rectangles are rejoined exiting the waveguide toward the user&#39;s eye to produce an image, a sufficient number of exit pupils are produced at all areas of the field of view. 
     Diffractive features, such as diffraction gratings, can be used to couple light into an eyepiece waveguide, out of an eyepiece waveguide, and/or to change the propagation direction of light within the eyepiece waveguide. In k-space, the effect of a diffraction grating on a ray or beam of light represented by a particular k-vector is determined by vector addition of the k-vector component in the plane of the diffraction grating with a grating vector. The magnitude and direction of the grating vector depend on the specific properties of the diffraction grating.  FIGS.  13 G,  13 H, and  13 I  illustrate the operation of diffraction gratings on k-vectors in k-space. 
       FIG.  13 G  illustrates a top view of a diffraction grating  1320  and some of its associated k-space diffraction grating vectors (G −2 , G −1 , G 1 , G 2 ). The diffraction grating  1320  is oriented in the x-y plane and  FIG.  13 G  shows the view of the grating from the perspective of a light ray or beam which is incident upon it from the z-direction. The diffraction grating  1320  has an associated set of k-space diffraction grating vectors (e.g., G −2 , G −1 , G 1 , G 2 ) which are oriented in the same plane as the diffraction grating. The G 1  and G −1  grating vectors correspond to the ±1 diffractive orders, respectively, while the G 2  and G −2  grating vectors correspond to the ±2 diffractive orders, respectively. The grating vectors for the ±1 diffractive orders point in opposite directions (along the axis of periodicity of the grating) and have equal magnitudes which are inversely proportional to the period, Λ, of the diffraction grating  1320 . Thus, a diffraction grating with a finer pitch has larger grating vectors. The grating vectors for the ±2 diffractive orders also point in opposite directions and have equal magnitudes which are twice that of the grating vectors for the ±1 diffractive orders. There can also be grating vectors for additional higher diffractive orders, though they are not illustrated. For example, the magnitudes of the grating vectors for the ±3 diffractive orders are three times that of the grating vectors for the ±1 diffractive orders, and so on. Note that the fundamental grating vector G 1  is determined solely by the periodicity of the grating (direction and pitch), while the composition of the grating (e.g., surface profile, materials, layer structure) may affect other characteristics of the grating, such as diffraction efficiency and diffracted phase. Since all the harmonics of the fundamental grating vector (e.g., G −1 , G 2 , G −2 , etc.) are simply integer multiples of the fundamental G 1 , then all diffraction directions of the grating are solely determined by the periodicity of the grating. The action of the diffraction grating  1320  is to add the grating vectors to the in-plane component of the k-vector corresponding to the incident light ray or beam. This is shown in  FIG.  13 H . 
       FIG.  13 H  illustrates a transverse view of the diffraction grating  1320  and its effect, in k-space, on a k-vector  1302  corresponding to a normally-incident ray or beam of light. The diffraction grating  1320  diffracts the incident ray or beam of light into one or more diffractive orders. The new ray or beam of light in each of these diffractive orders is represented by a new k-vector (e.g.,  1302   a - e ). These new k-vectors (e.g.,  1302   a - e ) are determined by vector addition of the in-plane component of the k-vector  1302  with each of the grating vectors (e.g., G −2 , G −1 , G 1 , G 2 ). In the illustrated case of a normally-incident ray or beam of light, the k-vector  1302  has no component in the x-y plane of the diffraction grating. As such, the effect of the diffraction grating  1320  is to create one or more new diffracted rays or beams of light whose k-vectors (e.g.,  1302   a - e ) have x-y components equal to the corresponding grating vector. For example, the x-y components of the ±1 diffractive orders of the incident ray or beam of light become G 1  and G −1 , respectively. Meanwhile, the magnitudes of the new k-vectors are constrained to be 2π/ω), so the new k-vectors (e.g.,  1302   a - e ) all lie on a semi-circle, as shown in  FIG.  13 H . Since the in-plane component of the incoming k-vector  1302  is being added to grating vectors whose lengths are equal to a fundamental increment, or 2× the fundamental increment, etc., whereas the magnitude of each resulting k-vector is constrained, the angles between the k-vectors (e.g.,  1302   a - e ) for the various diffractive orders are not equal; rather the k-vectors (e.g.,  1302   a - e ) become more angularly sparse with increasing diffractive order. 
     In the case of diffraction gratings formed on or in a planar eyepiece waveguide, the in-plane components of the new k-vectors (e.g.,  1302   a - e ) may be of most interest because if they lie in the k-space annulus  1310  of the eyepiece waveguide, then the diffracted rays or beams of light will undergo guided propagation through the eyepiece waveguide. But if the in-plane components of the new k-vectors (e.g.,  1302   a - e ) lie in the central disk  1308   b , then the diffracted rays or beams of light will exit the eyepiece waveguide. 
       FIG.  13 I  illustrates a transverse view of the diffraction grating  1320  and its effect, in k-space, on a k-vector  1302  corresponding to an obliquely-incident ray or beam of light. The effect is similar to that described with respect to  FIG.  13 H . Specifically, the k-vectors of the diffracted rays or beams of light are determined by vector addition of the in-plane component of the incident k-vector with the grating vectors (G −2 , G −1 , G 1 , G 2 ). For an obliquely-incident k-vector  1302 , the component of the k-vector in the x-y plane of the diffraction grating  1320  is non-zero. This component is added to the grating vectors to determine the in-plane components of the new k-vectors for the diffracted rays or beams of light. The magnitudes of the new k-vectors are constrained to be 2π/ω). And, once again, if the in-plane components of the k-vectors of the diffracted rays or beams of light lie in the k-space annulus  1310  of the eyepiece waveguide, then the diffracted rays or beams of light will undergo guided propagation through the eyepiece waveguide. 
       FIG.  13 J  is a k-space diagram which illustrates the field of view (FOV) of an image that is projected into an AR eyepiece waveguide (e.g.,  1200 ,  1300 ). The k-space diagram includes a larger disk  1308   a , which defines the k-vectors of light beams or rays that can propagate within the eyepiece waveguide. The k-space diagram also includes a smaller disk  1308   b , which defines the k-vectors of light beams or rays which can propagate within a medium, such as air, that surrounds the eyepiece waveguide. And, as already discussed, the k-space annulus  1310  defines the k-vectors of light beams or rays that can undergo guided propagation within the eyepiece waveguide. 
     The input beams (e.g.,  1202   a ,  1204   a ,  1206   a ) which are projected into the entrance pupil of the eyepiece waveguide are shown in  FIGS.  12 A and  12 B . Each input beam has a propagation angle which is uniquely defined by the spatial location of a corresponding image point in the image plane. The set of input beams have a certain angular spread in both the x-direction and the y-direction. The angular spread in the x-direction can define a horizontal field of view, while the angular spread in the y-direction can define a vertical field of view. In addition, the angular spread of the input beams along, for example, the diagonal between the x-direction and the y-direction can define a diagonal field of view. 
     In k-space, the field of view of the input image can be approximated by an FOV rectangle  1330 . The FOV rectangle  1330  encloses a set of k-vectors which corresponds to the set of input light beams. The FOV rectangle  1330  has a dimension along the k x -axis which corresponds to the angular spread of the input beams in the x-direction. Specifically, the horizontal width of the FOV rectangle  1330  is 
               2   ⁢     n   ·     sin   ⁡   (       θ   x     2     )         ,         
where θ x  is the total horizontal FOV and n is the refractive index of the incident medium. The FOV rectangle  1330  also has a dimension along the k y -axis which defines the angular spread of the input beams in the y-direction. Similarly, the vertical height of the FOV rectangle  1330  is
 
               2   ⁢     n   ·     sin   ⁡   (       θ   y     2     )         ,         
where θ y  is the total vertical FOV. Although a rectangle is shown as representing the set of input beams, in some embodiments the set of input beams could be such that it would correspond to a different shape in k-space. But the k-space analyses herein which are generally shown using FOV rectangles or FOV squares can equally apply to other shapes in k-space as well.
 
     As shown in  FIG.  13 J , the FOV rectangle  1330  is centered on, and located completely within, the smaller disk  1308   b . This position of the FOV rectangle  1330  corresponds to the k-vectors of a set of input beams (e.g., in a configuration with on-axis, or telecentric, projection from the image source) or a set of output beams propagating generally in the ±z-direction (although the set of beams is centered on the z-axis, all of the beams—except those normal to the entrance pupil or exit pupil—have some amount of angular deviation relative to the ±z-direction). In other words, when the FOV rectangle  1330  is within the smaller disk  1308   b  in a k-space diagram, it can represent the input beams as they propagate from an image source, through free space, to the eyepiece waveguide. It can also represent the output beams as they propagate from the eyepiece waveguide to the user&#39;s eye. Each k-space point within the FOV rectangle  1330  corresponds to a k-vector which represents one of the input beam directions or one of the output beam directions. In order for the input beams represented by the FOV rectangle  1330  to undergo guided propagation within the eyepiece waveguide, the FOV rectangle  1330  must be translated to the k-space annulus  1310 . Conversely, in order for the output beams represented by the FOV rectangle  1330  to exit the eyepiece waveguide, the FOV rectangle  1330  must be translated from the k-space annulus  1310  back to the smaller disk  1308   b . In order to not introduce geometric and chromatic dispersion from propagation through the waveguide, the FOV rectangle  1330  of the input beams may coincide with the FOV rectangle of the output beams; in this configuration the eyepiece waveguide preserves beam angles from input to output. 
     The following equations describe the FOV which may be achieved in some eyepiece waveguides: 
                 θ   x     =     arcsin   ⁡   (         ❘   &#34;\[LeftBracketingBar]&#34;       k   x       ❘   &#34;\[RightBracketingBar]&#34;           ❘   &#34;\[LeftBracketingBar]&#34;     k     ❘   &#34;\[RightBracketingBar]&#34;         )       ⁢   
       FOV   x     =       max   ⁡   (     θ     x   ,   air       )     -     min   ⁡   (     θ     x   ,   air       )         ⁢   
       FOV   x     =       arcsin   ⁡   (       max   ⁡   (       ❘   &#34;\[LeftBracketingBar]&#34;       k     x   ,   air         ❘   &#34;\[RightBracketingBar]&#34;       )         ❘   &#34;\[LeftBracketingBar]&#34;       k   air       ❘   &#34;\[RightBracketingBar]&#34;         )     -     arcsin   ⁡   (       min   ⁡   (       ❘   &#34;\[LeftBracketingBar]&#34;       k     x   ,   air         ❘   &#34;\[RightBracketingBar]&#34;       )         ❘   &#34;\[LeftBracketingBar]&#34;       k   air       ❘   &#34;\[RightBracketingBar]&#34;         )               
If the FOV is horizontally centered at θ x =0, then a conventional eyepiece waveguide may have the following limit:
 
                 max   ⁡   (     FOV   x     )     =     2   ×     arcsin   ⁡   (       max   ⁡   (       ❘   &#34;\[LeftBracketingBar]&#34;       k     x   ,   air         ❘   &#34;\[RightBracketingBar]&#34;       )         ❘   &#34;\[LeftBracketingBar]&#34;       k   air       ❘   &#34;\[RightBracketingBar]&#34;         )         ⁢   
       max   ⁡   (     FOV   x     )     =     2   ×     arcsin   ⁡   (         1   2     ×     w   c     ⁢     (       n   2     -     n   air       )           w   c     ⁢     n   air         )         ⁢   
       max   ⁡   (     FOV   x     )     =     2   ×     arcsin   ⁡   (       1   2     ⁢     (       n   2     -   1     )       )               
The only dependence of max(FOV x ) on angular frequency is from the waveguide refractive index&#39;s dependence on angular frequency, which may be an important detail in some applications but often has a relatively small effect.
 
       FIG.  13 K  is a k-space diagram which shows the translational shift, in k-space, of the FOV rectangle  1330  which is caused by an input coupling grating (ICG) located at the entrance pupil of the eyepiece waveguide. The ICG has associated diffraction grating vectors (G −1 , G 1 ), as just discussed with respect to  FIGS.  13 G − 13 I. The ICG diffracts each of the input beams represented by the FOV rectangle  1330  into a +1 diffractive order and a −1 diffractive order. In k-space, the diffraction of the input beams into the +1 diffractive order is represented by the FOV rectangle  1330  being displaced in the k x -direction by the G 1  grating vector. Similarly, in k-space, the diffraction of the input beams into the −1 diffractive order is represented by the FOV rectangle  1330  being displaced in the −k x -direction by the G −1  grating vector. 
     For the particular example shown in  FIG.  13 K , the translated FOV rectangles are too large to fit entirely within the k-space annulus  1310 . This means that the eyepiece waveguide cannot support all of the input beams in the FOV in guided propagation modes, whether in the positive or negative diffractive order, because the angular spread between them is too large. The k-vectors corresponding to points in the translated FOV rectangles which lie outside the larger disk  1308   a  would not be diffracted at all by the ICG because those k-vectors are not permitted. (This would also prevent diffraction into the ±2 and higher diffractive orders in this case because the grating vectors associated with those orders are even longer and would therefore translate the k-vectors even further outside the larger disk  1308   a .) Meanwhile, if any part of the translated FOV rectangles were to still lie inside the smaller disk  1308   b  after translation by the ICG, then the light beams corresponding to those particular k-vectors would exit the eyepiece waveguide by transmitting through its planar face for failure to TIR and would not undergo guided propagation through the waveguide. 
     One possible modification which could be made in order to support more of the input beams of light represented by the translated FOV rectangles  1330  in guided modes may be to increase the difference between the refractive index of the eyepiece waveguide and that of the surrounding medium. This would increase the size of the larger disk  1308   a  and/or decrease the size of the smaller disk  1308   b  (a decrease in the size of the smaller disk  1308   b  is possible if the waveguide is not surrounded by air), thereby increasing the size of the k-space annulus  1310 . 
     Example AR Eyepiece Waveguides with Orthogonal Pupil Expanders 
       FIG.  14 A  illustrates an example eyepiece waveguide  1400  with an ICG region  1440 , an orthogonal pupil expander (OPE) region  1450 , and an exit pupil expander (EPE) region  1460 .  FIG.  14 B  includes k-space diagrams which illustrate the effect of each of these components of the eyepiece waveguide  1400  in k-space. The ICG region  1440 , OPE region  1450 , and EPE region  1460  of the eyepiece waveguide  1400  include various diffractive features which couple input beams into the eyepiece waveguide to propagate via guided modes, replicate the beams at multiple distributed locations in space, and cause the replicated beams to exit the eyepiece waveguide and be projected toward the user&#39;s eye. 
     Input beams corresponding to an input image can be projected into the eyepiece waveguide  1400  from one or more input devices. The input beams can be incident on the ICG region  1440 , which can coincide with the entrance pupil of the eyepiece waveguide  1400 . The input device used to project the input beams can include, for example, a spatial light modulator projector (located in front of, or behind, the eyepiece waveguide  1400  with respect to the user&#39;s face). In some embodiments, the input device may use liquid crystal display (LCD), liquid crystal on silicon (LCoS), fiber scanned display (FSD) technology, or scanned microelectromechanical systems (MEMS) mirror displays, though others can also be used. Input beams from the input device are projected into the eyepiece waveguide  1400 , generally in the illustrated −z-direction, at various propagation angles and are incident on the ICG region  1440  from outside the substrate of the eyepiece waveguide. 
     The ICG region  1440  includes diffractive features which redirect the input beams such that they propagate inside the eyepiece waveguide  1400  via total internal reflection. In some embodiments, the diffractive features of the ICG region  1440  may form a one-dimensionally periodic (1D) diffraction grating made up of many lines which extend vertically in the illustrated y-direction and periodically repeat horizontally in the illustrated x-direction. In some embodiments, the lines may be etched into the front or back surface of the eyepiece waveguide  1400  and/or they may be formed of material deposited onto the front or back surface. The period, duty cycle, depth, profile, blaze angle, etc. of the lines can be selected based on the angular frequency, ω, of light for which the eyepiece waveguide  1400  is designed, the desired diffractive efficiency of the grating, and other factors. In some embodiments, the ICG region  1440  is designed to primarily couple input light into the +1 and −1 diffractive orders. (The diffraction grating can be designed so as to reduce or eliminate the 0 th  diffractive order and higher diffractive orders beyond the first diffractive orders. This can be accomplished by appropriately shaping the profile of each line. In many practical ICGs in AR displays, however, all higher diffractive orders correspond to k-vectors which lie beyond the k-space annulus. Thus, those higher diffractive orders would be forbidden regardless of non-k-space attributes like grating duty cycle, depth, and profile.) The diffracted beams in one of the ±1 diffractive orders from the ICG region  1440  then propagate generally in the −x-direction toward the OPE region  1450 , while the diffracted beams in the other of the ±1 diffractive orders then propagate generally in the +x-direction and exit the eyepiece waveguide  1400 . 
     The OPE region  1450  includes diffractive features which can perform at least two functions: first, they can perform pupil expansion by spatially replicating each input beam of light at many new locations generally in the −x-direction; second, they can guide each replicated beam of light on a path generally toward the EPE region  1460 . In some embodiments, these diffractive features are lines formed on or in the substrate of the eyepiece waveguide  1400 . The period, duty cycle, depth, profile, blaze angle, etc. of the lines can be selected based on the angular frequency, ω, of light for which the eyepiece waveguide  1400  is designed, the desired diffractive efficiency of the grating, and other factors. The specific shape of the OPE region  1450  can vary, but in general it may be determined based on the fan out of the beams of light from the ICG region  1440  and on the size and location of the EPE region  1460 . This is discussed further with respect to  FIG.  14 D . 
     The diffraction grating of the OPE region  1450  can be designed with relatively low and/or variable diffractive efficiency. These properties can allow the OPE region  1450  to replicate each beam of light that arrives from the ICG region  1440  and/or to more evenly distribute the light energy in at least one dimension. Because of the relatively low diffractive efficiency, each interaction of a beam of light with the grating diffracts only a portion of the power in the light beam while the remaining portion continues to propagate in the same direction. (Some parameters that can be used to influence the diffractive efficiency of the grating are the height and width of the line features, or magnitude of refractive index difference between the line features and the background medium.) That is, when a beam interacts with the diffraction grating in the OPE region  1450 , a portion of its power will be diffracted toward the EPE region  1460  while the remaining portion will continue to transmit within the OPE region to encounter the grating again at a different spatial location, where another portion of the beam&#39;s power may be diffracted toward the EPE region  1460 , and so on. Since some portions of the power of each light beam travel further through the OPE region  1450  than others before being diffracted toward the EPE region  1460 , there are numerous copies of the incoming beam traveling towards the EPE region from different locations in the −x-direction. The spatial extent of the replicated beams, in the direction of propagation of the original incoming beam through the OPE region  1450 , therefore effectively increases, while the intensity of the incoming beam correspondingly decreases because the light which made up the input beam is now divided amongst many replicated beams. 
     The diffraction grating in the OPE region  1450  is obliquely oriented with respect to the beams arriving from the ICG region  1440  so as to diffract the beams generally toward the EPE region  1460 . The specific angle of the slant of the diffraction grating in the OPE region  1450  may depend upon the layout of the various regions of the eyepiece waveguide  1400  and can perhaps be seen more clearly in the k-space diagrams found and discussed later in  FIG.  14 B . In the eyepiece waveguide  1400 , the ICG region  1440  is located to the right of the OPE region  1450 , while the EPE region  1460  is located below the OPE region. Therefore, in order to re-direct light from the ICG region  1440  toward the EPE region  1460 , the diffraction grating of the OPE region  1450  may be oriented at about 45° with respect to the illustrated x-axis. 
       FIG.  14 C  is a three-dimensional illustration of the optical operation of the OPE region  1450  shown in  FIGS.  14 A and  14 B .  FIG.  14 C  shows the ICG region  1440  and the OPE region  1450 , both on the side of the waveguide that is closer to the viewer. The grating lines cannot be seen because they are microscopic. In this case, a single input beam  1401  is illustrated, but an image will be made up of many such input beams propagating through the eyepiece waveguide  1400  in slightly different directions. The input beam  1401  enters the OPE region  1450  from the ICG region  1440 . The input beam  1401  then continues to propagate through the eyepiece waveguide  1400  via total internal reflection, repeatedly reflecting back and forth between its surfaces. This is represented in  FIG.  14 C  by the zig-zagging in the illustrated propagation of each beam. 
     When the input beam  1401  interacts with the diffraction grating formed in the OPE region  1450 , a portion of its power is diffracted toward the EPE region, while another portion of its power continues along the same path through the OPE region  1450 . As already mentioned, this is due in part to the relatively low diffractive efficiency of the grating. Further, beams diffracted toward the EPE region may re-encounter the grating of the OPE region  1450  and diffract back into the original direction of propagation of the input beam  1401 . The paths of some of these beams are indicated in  FIG.  14 C  by arrows. The effect is that the spatial extent of the light is expanded since the input beam is replicated as it propagates through the OPE region  1450 . This is evident from  FIG.  14 C , which shows that the input beam  1401  is replicated into many light beams ultimately traveling generally in the −y-direction toward the EPE region. 
     The EPE region  1460  likewise includes diffractive features which can perform at least two functions: first, they can replicate beams along another direction (e.g, a direction generally orthogonal to the one in which beams are replicated by the OPE region  1450 ); second, they can diffract each beam of light out of the eyepiece waveguide  1400  towards the user&#39;s eye. The EPE region  1460  can replicate light beams in the same way as the OPE region  1450 . Namely, as a beam propagates through the EPE region  1460 , it repeatedly interacts with the diffraction grating and portions of its power diffract into the first diffractive order, thereby being out-coupled toward the user&#39;s eye. Other portions of the beam&#39;s power zero-order diffract and continue propagating in the same direction within the EPE region  1460  until later interacting with the grating again. The diffractive optical features of the EPE region  1460  may also impart a degree of optical power to the replicated output beams of light to make them appear as if they originated from a desired depth plane, as discussed elsewhere herein. This can be accomplished by imparting a curvature to the lines of the diffraction grating in the EPE region  1460  using a lens function. 
       FIG.  14 B  illustrates the operation of the eyepiece waveguide  1400  in k-space. Specifically,  FIG.  14 B  includes a k-space diagram (KSD) for each component of the eyepiece waveguide  1400  to illustrate the k-space effect of that component. The FOV rectangles in the k-space diagrams, and the arrows which show the corresponding directions of propagation of light through the eyepiece waveguide, have matching shading. The first k-space diagram, KSD 1 , shows the k-space representation of the input beams which are incident on the ICG region  1440  from an input device. As already discussed, the set of input beams can be represented in k-space by an FOV rectangle  1430  whose k x  and k y  dimensions correspond to the angular spread of the input beams in the x- and y-directions. Each specific point in the FOV rectangle in KSD 1  corresponds to the k-vector associated with one of the input beams, where the k x  component is indicative of the propagation angle of the input beam in the x-direction and the k y  component is indicative of the propagation angle of the input beam in the y-direction. More precisely, k x =sin(θ x ), where θ x  is the angle formed by the input beam and the y-z plane, and k y =sin(θ y ), where θ y  is the angle formed by the input beam and the x-z plane. The fact that the FOV rectangle in KSD 1  is centered on the k z -axis of the diagram means the represented input light beams have propagation angles centered about an input beam propagating in the −z-direction and therefore all the input beams are propagating generally in the −z-direction. (Although not illustrated here, any of the waveguide displays described herein can also be designed for an FOV that is off-axis with respect to the ±z-direction.) 
     The second k-space diagram, KSD 2 , shows the k-space operation of the ICG region  1440 . As already discussed, a diffraction grating has associated grating vectors (e.g., G 1 , G −1 ). KSD 2  shows the G 1  grating vector and the G −1  grating vector, which are equal in magnitude and opposite in direction along the axis of periodicity of the ICG. The ICG region  1440  diffracts the input beams into the ±1 diffractive orders. And, in k-space, this means that the ICG copies the FOV rectangle to two new locations by translating it using both the G 1  and G −1  grating vectors. In the illustrated instance, the ICG is designed with a period, A, based on the angular frequency, ω, of the input beams such that the magnitude of the grating vectors G 1 , G −1  places the copied FOV rectangles completely within the k-space annulus of the waveguide. Accordingly, all of the diffracted input beams enter guided propagation modes. 
     The copy of the FOV rectangle which is centered at a point on the −k x -axis (9 o&#39;clock position within the k-space annulus) indicates that the corresponding diffracted beams have propagation angles which are centered around a beam whose propagation component in the plane of the eyepiece waveguide  1400  is in the −x-direction. Thus, all of those beams propagate generally toward the OPE region  1450 , while reflecting back and forth between the front and back surfaces of the eyepiece waveguide  1400  via TIR. Meanwhile, the copy of the FOV rectangle which is centered at a point on the +k x -axis (3 o&#39;clock position within the k-space annulus) indicates that the corresponding diffracted beams have propagation angles which are centered around a beam whose propagation component in the plane of the eyepiece waveguide  1400  is in the +x-direction. Thus, all of those beams propagate generally toward the right edge of the eyepiece waveguide  1400 , while reflecting back and forth between the front and back surfaces of the eyepiece waveguide  1400  via TIR. In this particular eyepiece waveguide  1400 , those beams are generally lost and do not meaningfully contribute to projection of the image toward the eye of the user. 
     KSD 2  does not illustrate the higher-order grating vectors, which are multiples of the illustrated first-order grating vectors G 1 , G −1 . The ICG does not diffract light beams into those diffractive orders because doing so in this instance would translate the k-vectors which make up the FOV rectangle beyond the outer perimeter of the k-space disk which defines the permitted k-vectors. Accordingly, the higher diffractive orders do not occur in this embodiment. 
     The third k-space diagram, KSD 3 , shows the k-space operation of the OPE region  1450 . Once again, since the OPE region  1450  includes a diffraction grating, it has associated grating vectors (e.g., G 1 , G −1 ) which are equal in magnitude and opposite in direction along the axis of periodicity of the OPE grating. In this case, the axis of periodicity of the diffraction grating is at a 45° angle with respect to the x-axis. Accordingly, the grating vectors (e.g., G 1 , G −1 ) of the OPE diffraction grating point at 45° angles with respect to the k x -axis. As shown in KSD 3 , one of the grating vectors translates the FOV rectangle to a new location centered at a point located on the −k y -axis (6 o&#39;clock position within the k-space annulus). This copy of the FOV rectangle indicates that the corresponding diffracted beams have propagation angles which are centered around a beam whose propagation component in the plane of the eyepiece waveguide  1400  is in the −y-direction toward the EPE region  1460 . Meanwhile, the other illustrated OPE grating vector would place the FOV rectangle at a location outside the outer perimeter of the k-space disk. But k-vectors outside the disk are not permitted, so the OPE diffraction grating does not diffract beams into that diffractive order. The axis of periodicity of the diffraction grating in the OPE region  1450  need not necessarily be exactly 45°. For example, as seen by inspection of KSD 3 , the axis of periodicity could be at an angle somewhat more or less than 45° while still translating the FOV rectangle to a 6 o&#39;clock position where the FOV rectangle can fit entirely within the k-space annulus. This would place the FOV rectangle at a 6 o&#39;clock position but without the FOV rectangle necessarily being centered in the k-space annulus along the −k y -axis. 
     In the illustrated instance, the OPE diffraction grating is designed with a period, A, based on the angular frequency, ω, of the input beams such that one of the grating vectors G 1 , G −1  places the copied FOV rectangle completely within the k-space annulus of the waveguide at the 6 o&#39;clock position. Accordingly, all of the diffracted input beams remain in guided propagating modes. Since the k-space distance from the 9 o&#39;clock position in the k-space annulus to the 6 o&#39;clock position, which is the translation performed by the OPE grating, is greater than the distance from the origin of the k-space diagram to the annulus, which is the translation performed by the ICG, the OPE grating vectors must be different in magnitude than the ICG grating vectors. In particular, the OPE grating vectors are longer than the ICG grating vectors, which means the OPE grating therefore has a shorter period, A, than the ICG grating. 
     The fourth k-space diagram, KSD 4 , shows the k-space operation of the EPE region  1460 . Again, since the EPE region  1460  includes a diffraction grating, it has associated grating vectors (e.g., G 1 , G −1 ) which are equal in magnitude and opposite in direction along the axis of periodicity of the EPE grating. In this case, the axis of periodicity of the diffraction grating is along the y-axis of the eyepiece waveguide  1400 . Accordingly, the grating vectors (e.g., G 1 , G −1 ) of the EPE diffraction grating point in the ±k y -directions. As shown in KSD 4 , one of the grating vectors translates the FOV rectangle to a new location centered at the origin of the k-space diagram. This copy of the FOV rectangle indicates that the corresponding diffracted beams have propagation angles which are centered around a beam whose propagation component in the plane of the eyepiece waveguide  1400  is in the +z-direction toward the user&#39;s eye. Meanwhile, the other first order EPE grating vector would place the FOV rectangle at a location outside the outer perimeter of the k-space disk, so the EPE diffraction grating does not diffract beams into that diffractive order. One of the second order EPE grating vectors would, however, translate the FOV rectangle to the 12 o&#39;clock location in the k-space annulus. So, the EPE grating may diffract some of the light into one of the second diffractive orders. The second order diffraction direction can correspond to guided propagation directions along the +y-direction, and is typically an undesirable effect. For example, the second order diffraction can result in visual artifacts when the EPE grating is perturbed to introduce optical power, as discussed below, resulting in a flare or smearing effect in the image presented to the user. 
     In the illustrated instance, the EPE diffraction grating is designed with a period, A, based on the angular frequency, ω, of the input beams such that one of the grating vectors G 1 , G −1  places the copied FOV rectangle completely within the inner k-space disk of the waveguide. Accordingly, all of the beams diffracted by the EPE diffraction grating are no longer in guided propagation modes and therefore exit the eyepiece waveguide  1400 . Moreover, since the EPE diffraction grating translates the FOV rectangle back to the origin of the k-space diagram (where the FOV rectangle corresponding to the input beams was located), the output beams have the same propagation angles as their corresponding input beams. In the illustrated embodiment, the EPE diffraction grating has the same period, Λ, as the ICG because both of these diffraction gratings translate the FOV rectangle by the same k-space distance. This is not a requirement, however. If the k y  dimension of the FOV rectangle is less than the k y  dimension of the k-space annulus in the 6-o-clock position, then the FOV rectangle can have a range of possible 6-o-clock positions at different k y  locations in the annulus. Hence, there may be numerous engineering choices for the EPE grating vector—and in turn the OPE vector—to place the FOV rectangle at locations within the k-space annulus and/or near the origin of the k-space diagram. 
     In some embodiments, the lines of the EPE diffraction grating may be slightly curved so as to impart optical power to the output beams which exit the EPE region  1460 . For example, the lines of the diffraction grating in the EPE region  1460  can be bowed in the plane of the waveguide toward the OPE region to impart negative optical power. This can be used, for example, to make the output beams follow diverging paths, as shown in  FIG.  12 B . This causes the projected image to appear at a depth plane nearer than optical infinity. The specific curvature can be determined by a lens function. In k-space, this means that different spatial regions within the EPE region  1460  will have grating vectors that point in slightly different directions, depending on the curvature of the grating lines in that specific region. In these embodiments, this causes the FOV rectangle to be translated to a variety of different locations centered around the origin of the k-space diagram. This in turn causes the sets of output beams corresponding to each of the translated FOV rectangles to be centered around different propagation angles, which in turn causes the illusion of depth. 
       FIG.  14 D  illustrates a technique for determining the sizes and shapes of the OPE region  1450  and the EPE region  1460 .  FIG.  14 D  illustrates the same eyepiece waveguide  1400  shown in  FIGS.  14 A and  14 B , including the ICG region  1440 , the OPE region  1450 , and the EPE region  1460 .  FIG.  14 D  also includes simplified versions of the k-space diagrams KSD 1 , KSD 2 , and KSD 3 . With reference to the first k-space diagram, KSD 1 , the four corner k-vectors of the FOV rectangle are those which correspond to the input beams which are incident on the ICG at the most oblique angles from the corners of the image in the input plane (See  FIGS.  12 A and  12 B ). Since the propagation angles of these input beams are the most extreme of all those in the field of view, their k-vectors are located at the four corners of the FOV rectangle in k-space. 
       FIG.  14 D  shows rays which define the four diffracted beams from the ICG region  1440  which correspond to the four corners of the input image. In particular, the ray near the top of the OPE region  1450  defines the diffracted beam corresponding to the input beam which is incident on the ICG region  1440  at the most severe propagation angle in the direction upward and away from the OPE region (i.e., the k-vector located at the top right corner of the FOV rectangle). And the ray near the bottom of the OPE region  1450  defines the diffracted beam corresponding to the input beam which is incident on the ICG region  1450  at the most severe propagation angle downward and away from the OPE region (i.e., the k-vector located at the bottom right corner of the FOV rectangle). These two beams define the fan out of diffracted beams from the ICG region  1440 . In order to create replicated instances of these two beams, and all others in between, and project them toward the user&#39;s eye, the top and bottom boundaries of the OPE region should encompass the propagation paths of these two beams. Their specific propagation paths can be determined with reference to the second k-space diagram, KSD 2 . 
     KSD 2  shows the resulting k-vectors of the beams which diffract from the ICG region  1440  toward the OPE region  1450 . The arrow in KSD 2  shows the propagation angle of the beam corresponding to the k-vector located at the top right corner of the FOV rectangle. 
     The size, shape, and location of the EPE region  1460  can be determined by performing a backwards ray trace using the propagation angles which are evident from the k-vectors in the third k-space diagram, KSD 3 . As is evident from KSD 3 , the top left and right corner k-vectors of the FOV rectangle define the fan out of the propagation paths which beams follow while propagating in the direction from the OPE region  1450  toward the EPE region  1460 . By using these propagation angles to trace backwards from the portion of the EPE region  1460  which is located the furthest from the OPE region  1450  (i.e., the lower corners of the EPE region), one can determine the origination points in the OPE region of those light rays which would arrive at the lower corners of the EPE region with the propagation angles defined by the top left and right corner k-vectors. These origination points of those rays can be used to determine the remaining boundaries of the OPE region  1450 . For example, to direct the beams from the OPE region  1450  to the lower left corner of the EPE region  1460 , the worst-case propagation angle is the one indicated by the top right corner k-vector of the FOV rectangle. Thus, a propagation path with that angle can be used to define the left boundary of the OPE region  1450 . Similarly, to direct the beams from the OPE region  1450  to the lower right corner of the EPE region, the worst-case propagation angle is the one indicated by the top left corner k-vector of the FOV rectangle. Thus, a propagation path with that angle can be used to define the right boundary of the OPE region  1450 . 
     As shown in  FIG.  14 D , in the case of the illustrated eyepiece waveguide  1400 , the EPE region  1460  is located in the −x and −y-directions from the ICG region  1440 . And some of the diffracted beams fan out from the ICG region  1440  along paths in those same directions. In order to avoid these diffracted beams entering the EPE region before first having propagated through the OPE region  1450 , the ICG region  1440  can be located far enough away from the EPE region in the +y-direction such that the fan out of the diffracted beams does not intersect with the EPE region  1460 . This results in a gap between much of the lower border of the OPE region  1450  and the upper border of the EPE region  1460 . In some embodiments, it may be desirable to decrease the size of the eyepiece waveguide by removing or reducing this gap.  FIG.  15 A  illustrates an example embodiment which accomplishes these goals. 
       FIG.  15 A  illustrates an example embodiment of a waveguide eyepiece  1500  in which the OPE region  1550  is tilted and located such that its lower border is parallel to the upper border of the EPE region  1560 . In fact, the OPE region  1550  and the EPE region  1560  may actually share a border. According to this embodiment, the size of the waveguide eyepiece  1500  can be made more compact by reducing or eliminating the gap between the OPE and EPE regions in the eyepiece waveguide embodiment shown in  FIG.  14 A . 
     To accommodate the tilted orientation of the OPE region  1550 , the ICG region  1540  can be modified such that the fan out of diffracted beams from the ICG region is tilted to match the tilted orientation of the OPE region  1550 . For example, the grating lines of the ICG region  1540  can be oriented such that no diffracted beam exits the ICG region in a propagation direction that has a component in the −y-direction. In addition, the ICG region  1540  can be positioned near the shared border of the OPE region  1550  and the EPE region  1560  but such that no portion of the ICG region extends in the −y-direction beyond that shared border. The operation of the ICG region  1540  can be seen in the k-space diagrams shown in  FIG.  15 B . 
       FIG.  15 B  includes k-space diagrams which illustrate the operation of the eyepiece waveguide  1500  shown in  FIG.  15 A . The first k-space diagram, KSD 1 , shows the FOV rectangle corresponding to the input beams which are projected toward the ICG region  1540  from a projector located outside the eyepiece waveguide  1500 . In the illustrated embodiment, these input beams have propagation angles centered about the −z-direction. Therefore, in k-space, they can be represented by an FOV rectangle centered on the k z -axis at the origin of KSD 1 . 
     The second k-space diagram, KSD 2 , shows the operation of the ICG region  1540  on the input beams. The ICG region  1540  diffracts the input beams and redirects them toward the OPE region  1550 . In k-space, this corresponds to translating the FOV rectangle using the grating vector(s) associated with the ICG region  1540 . In this embodiment, the grating lines in the ICG region  1540  are oriented with an axis of periodicity which has a component in the +y-direction. This means that the grating vector associated with the ICG  1540  also has a component in the +k y -direction. The magnitude of this component in the +k y -direction can be greater than or equal to one half of the width of the FOV rectangle in the k y -direction. This means that no portion of the FOV rectangle, after being translated by the ICG region  1540 , extends below the horizontal axis of the k-space diagram KSD 2 . This in turn means that none of the diffracted beams from the ICG region  1540  has a propagation angle with a component in the −k y -direction. Accordingly, none of the diffracted beams travels downward toward the EPE region  1560  from the ICG region  1540 . And, therefore, none of the diffracted beams will enter the EPE region  1560  prior to having passed through the OPE region  1550 . 
     The third k-space diagram, KSD 3 , shows the operation of the OPE region  1550  on the diffracted beams from the ICG region  1540 . As illustrated, the diffraction grating of the OPE region  1550  can be oriented so as to redirect beams of light at angles which correspond to the FOV rectangle being translated to a position slightly displaced from the 6 o&#39;clock position in the k-space annulus. For example, the translated FOV rectangle in KSD 3  can be displaced from the 6 o&#39;clock position in the k-space annulus by the same angle as the translated FOV rectangle in KSD 2  is displaced from the 9 o&#39;clock position. In other words, the translated FOV rectangle in KSD 3  can be separated by 90° from the translated FOV rectangle in KSD 2 . This specific angular separation is not required, however; the specific location of each FOV rectangle can be dependent upon the layout of the various regions of the eyepiece waveguide with respect to one another. 
     Since the translated FOV rectangle in KSD 3  is centered around a k-vector which has a component in the −k x -direction, the beams of light from the OPE region  1550  generally travel toward the EPE region  1560  at angles which have components in the −x-direction. It can be seen from  FIG.  15 A  that, due to this angle, some of the light beams from the tip portion  1555  of the OPE region  1550  will not intersect with the EPE region  1560 . Since the tip portion  1555  of the OPE region  1550  may contribute a relatively small portion of light to the EPE region  1560 , the size advantages of eliminating the upper tip  1555  may outweigh any optical disadvantages. In some embodiments, the waveguide eyepiece  1500  can therefore be made even more compact by eliminating the upper tip  1555  of the OPE region  1550 . 
     Finally, the fourth k-space diagram, KSD 4 , shows that the EPE region  1560  has a diffraction grating designed to translate the FOV rectangle back to the origin of the k-space diagram. Since the starting location of the FOV rectangle in KSD 4  for the eyepiece waveguide embodiment shown in  FIG.  15 A  is slightly different from the starting location of the FOV rectangle in KSD 4  for the eyepiece waveguide embodiment shown in  FIG.  14 A , the design of the diffraction grating in the EPE region  1560  is also somewhat different. For example, the orientation of the grating lines of the diffraction grating in the EPE region  1560  can be tilted such that the associated grating vector has a component in the +k x -direction, so that the OPE region  1550  does not need to extend beyond the left edge of the EPE region  1560  (see the discussion of  FIG.  14 D  and compare the location of the top right corner k-vector in KSD 3  in  FIG.  14 D  with the location of the corresponding k-vector in KSD 3  in  FIG.  15 B ). This results in the FOV rectangle in KSD 4  of  FIG.  15 B  being translated back to the origin of the k-space diagram, which means the beams of light represented by the translated FOV rectangle are coupled out of the eyepiece waveguide  1500  toward the user&#39;s eye with the same propagation angles as their corresponding input beams, as has already been described herein (i.e., the FOV rectangle which represents the output beams is in the same location on the k-space diagram as the FOV rectangle which represents the input beams). 
       FIG.  15 C  is another k-space diagram which illustrates the operation of the eyepiece waveguide  1500  shown in  FIG.  15 A . The k-space diagram in  FIG.  15 C  is a superposition of all the k-space diagrams shown in  FIG.  15 B . And it also illustrates that light beams propagating through the OPE region  1550  can switch back and forth between propagation angles generally in the −k x -direction (as represented by the FOV rectangle located near the 9 o&#39;clock position of the k-space annulus) and propagation angles generally in the −k y -direction (as represented by the FOV rectangle located near the 6 o&#39;clock position of the k-space annulus). This is shown by the grating vector with the double-sided arrow between the FOV rectangle near the 9 o&#39;clock position of the k-space annulus and the FOV rectangle near the 6 o&#39;clock position.  FIGS.  15 D- 15 F  illustrate this behavior in more detail. 
       FIG.  15 D  is a diagram of the first generation of interactions between an input beam and the OPE region  1550  of the eyepiece waveguide embodiment shown in  FIG.  15 A . The OPE region  1550  of the eyepiece waveguide  1500  includes a diffraction grating made up of parallel grating lines which repeat in a direction of periodicity. The direction of periodicity determines the direction of the grating vectors associated with the diffraction grating. In this instance, the grating vector with the double-sided arrow in  FIG.  15 C  is the one which illustrates the operation of the OPE region  1550  and which points along the direction of periodicity of the grating lines shown in  FIGS.  15 D- 15 F . 
       FIG.  15 D  shows an input beam that enters the OPE region  1550  from the ICG region  1540 . The input beam is shown propagating in the direction which corresponds to the center point, or k-vector, of the FOV rectangle located near the 9 o&#39;clock position of the k-space annulus in  FIG.  15 C . As shown, the first generation of interactions between the input beam and the OPE region  1550  results in two diffracted output beams: some portion of the input beam&#39;s power simply reflects, as output 1 , from the top or bottom surface of the eyepiece waveguide  1500  and continues on in the same x-y direction as the input beam (i.e., the 0 th  order diffraction); and some portion of the input beam&#39;s power diffracts into the first order (e.g., by the first order grating vector, G 1 , of the OPE region), downward as output 2 . The output 2  beam is shown propagating in the direction which corresponds to the center point, or k-vector, of the FOV rectangle located near the 6 o&#39;clock position of the k-space annulus in  FIG.  15 C . After this first generation of interactions, the output 1  beam and the output 2  beam have different propagation angles, but they are both still propagating within the OPE region  1550  and may therefore have additional interactions with the OPE region, as shown in  FIGS.  15 E and  15 F . Although not illustrated, other input beams that enter the OPE region  1550  with different propagation angles will behave similarly but with slightly different input and output angles. 
       FIG.  15 E  is a diagram of the second generation of interactions between an input beam and the OPE region  1550  of the eyepiece waveguide embodiment shown in  FIG.  15 A . The beams related to the first generation of interactions are shown with dashed lines, while the beams related to the second generation of interactions are shown with solid lines. As shown in  FIG.  15 E , each of the output beams, output 1  and output 2 , from the first generation of interactions can now undergo similar interactions with the OPE region  1550  as occurred in the first generation. Namely, some portion of the power from the output 1  beam from  FIG.  15 D  simply continues on in the same x-y direction (i.e., the 0 th  order diffraction), while another portion of the power of that beam interacts with the grating and is redirected downward (e.g., by the first order grating vector, G 1 , of the OPE region). Similarly, some portion of the power from the output 2  beam from  FIG.  15 D  simply continues downward toward the EPE region  1560  (i.e., the 0 th  order diffraction), while another portion of the power of that beam interacts with the grating and is diffracted (e.g., by the negative first order grating vector, G −1 , of the OPE region), generally in the −x-direction, and continues propagating further into the OPE region  1550  in the same direction as the initial input beam. 
     After the second generation of interactions have occurred within the OPE region  1550 , there is an interference node  1556  where two of the resulting beams intersect. The optical paths followed by each of these beams to arrive at the interference node  1556  are substantially identical in length. Thus, the beams which leave the interference node  1556  propagating in the same direction may have the same or similar phases and may therefore undergo constructive or destructive wave interference with one another. This can result in image artifacts which are discussed below. 
       FIG.  15 F  is a diagram of the third generation of interactions between an input beam and the OPE region  1550  of the eyepiece waveguide embodiment shown in  FIG.  15 A . The beams related to the first and second generations of interactions are shown with dashed lines, while the beams related to the third generation of interactions are shown with solid lines. As shown in  FIG.  15 F , each of the output beams which resulted from the second generation of interactions can once more experience similar interactions with the OPE region  1550  as occurred in previous generations. Some portions of the power of those beams continue on in the same direction (i.e., the 0 th  order diffraction), while other portions of the power of those beams are redirected—some generally in the −x-direction and some generally in the −y-direction (i.e., by the first order grating vectors, G 1  and G −1 , of the OPE region). All of the beams propagating generally in the −x-direction are in the state represented by the FOV rectangle located near the 9 o&#39;clock position in the k-space annulus of the k-space diagram in  FIG.  15 C , while all of the beams propagating generally in the −y-direction are in the state represented by the FOV rectangle located near the 6 o&#39;clock position. As can be seen in  FIG.  15 C , for the case of an OPE region  1550  made up of a 1D periodicity diffraction grating, for any given input beam, the replicated beams of light corresponding to that input beam only travel in two directions within the OPE region (although the two directions will be different for different input beams which enter the OPE region at different propagation angles). 
     The third generation of interactions with the OPE region results in the creation of additional interference nodes  1556  where beams with the same or similar optical path lengths intersect with one another, possibly resulting in constructive or destructive wave interference. Each of the nodes  1556  serves as a source of light emitted toward the EPE region  1560 . In the case of an OPE region made up of a diffraction grating with 1D periodicity, the layout of these nodes  1556  forms a uniform lattice pattern and can therefore result in image artifacts, as shown in  FIG.  15 G . 
       FIG.  15 G  is a diagram which illustrates how a single input beam  1545  from the ICG region  1540  is replicated by the OPE region  1550  and redirected toward the EPE region  1560  as a plurality of beams  1565 . Each of the replicated beams  1565  shown propagating toward, or in, the EPE region  1560  originates from one of the interference nodes  1556 . These interference nodes have an ordered distribution and serve as a sparse, periodic array of sources. Due to the ordered distribution of the interference nodes  1556 , the replicated beams  1565  which illuminate the EPE region are all separated by the same spacing, although the beams may have non-monotonically varying intensity. And as a result, the replicated light beams  1565  from the OPE region  1550  may illuminate the EPE region  1560  with a relatively sparse, uneven distribution. In some embodiments, it may be advantageous if the replicated light beams which illuminate the EPE region of an eyepiece waveguide could be more evenly dispersed.  FIG.  16    illustrates such an embodiment. 
     Example AR Eyepiece Waveguides with Multi-Directional Pupil Expanders 
       FIG.  16 A  illustrates an example eyepiece waveguide  1600  that has a multi-directional pupil expander (MPE) region  1650  rather than an OPE region. On a macroscopic level, the illustrated embodiment of the eyepiece waveguide  1600  is similar to the eyepiece waveguide  1500  shown in  FIG.  15 A . Input beams are coupled into the eyepiece waveguide  1600  by the ICG region  1640 . The diffracted beams from the ICG region  1640  propagate toward and through the MPE region  1650 , which takes the place of an OPE region. Finally, the MPE region  1650  diffracts beams of light toward the EPE region  1660 , where they are out coupled toward the user&#39;s eye. The ICG region  1640  and the EPE region  1660  may be designed to function in the same way as the corresponding regions in the eyepiece waveguide  1500  described with respect to  FIGS.  15 A- 15 G . The MPE region  1650 , however, is distinct from the OPE region  1550  in that it diffracts light in more directions. This feature can advantageously decrease the periodic uniformity in the distribution of light beams in the EPE region  1660 , which in turn can cause the EPE region to be illuminated more evenly. 
     The MPE region  1650  is made up of diffractive features which exhibit periodicity in multiple directions. The MPE region  1650  may be composed of an array of scattering features arranged in a 2D lattice. The individual scattering features can be, for example, indentations or protrusions of any shape. The 2D array of scattering features has associated grating vectors, which are derived from the reciprocal lattice of that 2D lattice. As one example, the MPE region  1650  could be a 2D periodic diffraction grating composed of a crossed grating with grating lines that repeat along two or more distinct directions of periodicity. This can be accomplished by superimposing two 1D gratings with different directions of periodicity. 
       FIG.  16 B  illustrates a portion of an example 2D periodic grating, along with its associated grating vectors, which can be used in the MPE region  1650  shown in  FIG.  16 A . The 2D periodic grating  1650  can be a spatial lattice of diffractive features whose directions of periodicity are illustrated by the vectors u and v. Such a 2D periodic grating is associated with grating vectors. The two fundamental grating vectors, G and H, corresponding to the directions of periodicity, u and v, are mathematically defined by: 
               u   =     ⌊       u   x     ,     u   y       ⌋       ⁢   
     v   =     ⌊       v   x     ,     v   y       ⌋       ⁢   
     G   =         2   ⁢   π           u   x     ⁢     v   y       -       u   y     ⁢     v   x           [       v   y     ,     -     v   x         ]       ⁢   
     H   =         2   ⁢   π           u   x     ⁢     v   y       -       u   y     ⁢     v   x           [       -     u   y       ,     u   x       ]             
Mathematically, the vectors u and v define a spatial lattice, and G and H correspond to the fundamental dual, or reciprocal, lattice vectors. Note, that G is orthogonal to u, and H is orthogonal to v; however, u is not necessarily parallel to H, and v is not necessarily parallel to G.
 
     As one example, the 2D periodic grating can be designed or formed by superimposing two sets of 1D periodic grating lines, as shown in  FIG.  16 B  (though the 2D periodic grating could instead be made up of individual scattering features located at, for example, the intersection points of the grating lines shown in  FIG.  16 B ). The first set of grating lines  1656  can repeat along the direction of the fundamental grating vector G. The fundamental grating vector G can have a magnitude equal to 2π/a, where a is the period of the first set of grating lines  1656 . The 2D grating shown in  FIG.  16 B  is also associated with harmonics of the first fundamental grating vector G. These include −G and higher-order harmonics, such as 2G, −2G, etc. The second set of grating lines  1657  can repeat along the direction of the fundamental grating vector H. The fundamental grating vector H can have a magnitude equal to 2π/b, where b is the period of the second set of grating lines  1657 . The 2D grating shown in  FIG.  16 B  is also associated with harmonics of the second fundamental grating vector H. These include −H and higher-order harmonics, such as 2H, −2H, etc. 
     Any 2D periodic array of diffractive features will have associated grating vectors which correspond to the entire reciprocal lattice and point in directions determined by integer linear combinations (superpositions) of the basis grating vectors, G and H. In the illustrated embodiment, these superpositions result in additional grating vectors which are also shown in  FIG.  16 B . These include, for example, −G, −H, H+G, H−G, G−H, and −(H+G). Typically, these vectors are described with two indices: (±1,0), (0, ±1), (±1, ±1), (±2,0), etc. Although  FIG.  16 B  only illustrates the first order grating vectors, and their superpositions, associated with the 2D diffraction grating, higher-order grating vectors may also exist. 
     As already discussed elsewhere herein, the k-space operation of a grating on a set of light beams composing an image is to translate the FOV rectangle corresponding to the image using the grating vectors associated with the grating. This is shown in  FIGS.  16 C and  16 D  for the example 2D MPE diffraction grating shown in  FIG.  16 B . 
       FIG.  16 C  is a k-space diagram which illustrates the k-space operation of the MPE region  1650  of the eyepiece waveguide  1600  shown in  FIG.  16 A . The k-space diagram includes a shaded FOV rectangle located near the 9 o&#39;clock position of the k-space annulus. This is the location of the FOV rectangle after the ICG region  1640  has coupled the input beams into the eyepiece waveguide  1600  and redirected them toward the MPE region  1650 .  FIG.  16 C  shows how the 2D grating in the MPE region  1650  translates the FOV rectangle using the grating vectors shown in  FIG.  16 B . Since there are eight grating vectors (G, H, −G, −H, H+G, H−G, G−H, and −(H+G)), the MPE region  1650  attempts to translate the FOV rectangle to eight possible new k-space locations. Of these eight possible k-space locations, six fall outside the outer periphery of the k-space diagram. These are illustrated with unshaded FOV rectangles. Since k-vectors outside the bounds of the k-space diagram are not permitted, none of those six grating vectors results in diffraction. There are, however, two grating vectors (i.e., −G and −(H+G)) which do result in translations of the FOV rectangle to new positions within the bounds of the k-space diagram. One of these locations is near the 6 o&#39;clock position in the k-space annulus, and the other is near the 2 o&#39;clock position. Since k-vectors at these locations are permitted and do result in guided propagation modes, the FOV rectangles at these locations are shaded to indicate that beams of light are diffracted into those two states. Thus, the power of beams of light entering the MPE region  1650  with the propagation angles indicated by the FOV rectangle located near the 9 o&#39;clock position of the k-space annulus is partially diffracted into both of the states indicated by the other two shaded FOV rectangles (i.e., the FOV rectangle near the 2 o&#39;clock position and the FOV rectangle near the 6 o&#39;clock position). 
       FIG.  16 D  is a k-space diagram which further illustrates the k-space operation of the MPE region  1650  of the eyepiece waveguide  1600  shown in  FIG.  16 A . This particular k-space diagram illustrates the operation of the MPE region  1650  on beams of light which are in the propagation state illustrated by the FOV rectangle located near the 2 o&#39;clock position of the k-space annulus. Once again, the 2D diffraction grating in the MPE region  1650  attempts to diffract these beams of light into diffractive orders specified by its eight associated grating vectors. As shown, six of the grating vectors would translate the FOV rectangle to a position outside the bounds of the k-space diagram. Thus, those diffractive orders do not occur. These positions are illustrated with unshaded FOV rectangles. However, two of the grating vectors (i.e., H and H−G) translate the FOV rectangle to positions within the bounds of the k-space diagram. These are illustrated by the shaded FOV rectangles located near the 9 o&#39;clock position of the k-space annulus and near the 6 o&#39;clock position. Thus, the 2D diffraction grating in the MPE region  1650  partially diffracts the power of beams propagating in the directions indicated by the FOV rectangle located near the 2 o&#39;clock position of the k-space annulus into both of the states indicated by the other two shaded FOV rectangles (i.e., the FOV rectangle near the 9 o&#39;clock position and the FOV rectangle near the 6 o&#39;clock position). 
     Although not illustrated, a similar k-space diagram could be drawn to illustrate the k-space operation of the MPE region  1650  on beams of light traveling with the propagation angles indicated by the FOV rectangle located near the 6 o&#39;clock position of the k-space annulus. That k-space diagram would show that the 2D period diffraction grating in the MPE region  1650  partially diffracts the power of those beams into both of the states indicated by the two shaded FOV rectangles located near the 9 o&#39;clock position and near the 2 o&#39;clock position of the k-space annulus. 
       FIG.  16 E  is a k-space diagram which illustrates the k-space operation of the eyepiece waveguide  1600  shown in  FIG.  16 A . As already mentioned, the eyepiece waveguide  1600  can receive input beams of light which propagate generally in the −z-direction and are incident on the ICG region  1640  of the waveguide  1600  from an outside source. Those input beams are represented by the FOV rectangle centered on the k z -axis at the origin of the k-space diagram. The ICG region  1640  then diffracts the input beams such that they are guided and have propagation angles centered around a propagation direction which corresponds to the center point of the FOV rectangle located near the 9 o&#39;clock position of the k-space annulus. 
     The guided beams enter the MPE region  1650 , where they can have multiple interactions. During each generation of interactions, a portion of the power of each of the beams can zero-order diffract and continue propagating in the same direction through the MPE region  1650 . In the first generation of interactions, for example, this zero-order diffraction corresponds to that portion of the power of those beams staying in the state indicated by the FOV rectangle located near the 9 o&#39;clock position of the k-space annulus. Other portions of the power of the beams can be diffracted in new directions. Again, in the first generation of interactions, this creates respective diffracted beams that have propagation angles centered around a propagation direction which corresponds to the center point of the FOV rectangle located near the 2 o&#39;clock position of the k-space annulus and a propagation direction which corresponds to the center point of the FOV rectangle located near the 6 o&#39;clock position. 
     So long as the beams remain in the MPE region  1650 , they can experience additional interactions, each of which results in portions of the power of the beams zero-order diffracting and continuing on in the same direction, or being diffracted in new directions. This results in spatially distributed sets of diffracted beams that have propagation angles centered around each of the propagation directions indicated by the center points of the FOV rectangles in the k-space annulus shown in  FIG.  16 E . This behavior is represented by the double-sided arrows between each pair of FOV rectangles in the k-space annulus. 
     As any given input beam of light propagates within the MPE region  1650 , it is split into many diffracted beams which can only travel in three allowed directions—each direction being defined by the corresponding k-vector, or point, within the FOV rectangles in the annulus of the k-space diagram in  FIG.  16 E . (This is true for any input beam of light propagating within the MPE region  1650 . However, the three allowed directions will be slightly different depending on the propagation angle at which each initial input beam enters the MPE region  1650 .) And since portions of the power of any given input beam of light are diffracted into any of the same three propagation directions after any number of interactions with the MPE region  1650 , image information is preserved throughout these interactions. 
     There are advantages associated with the MPE region  1650  having three permissible propagation directions for each input beam—as opposed to the two permissible propagation directions of the OPE region  1550 . These advantages are discussed further below, but suffice it to say for now that the increased number of propagation directions in the MPE region  1650  can result in a more complicated distribution of interference nodes within the MPE region  1650 , which can in turn improve the evenness of illumination in the EPE region  1660 . 
     It should be understood that  FIG.  16 E  illustrates the k-space operation of one example embodiment of the MPE region  1650 . In other embodiments, the MPE region  1650  can be designed such that each input beam of light can diffract in more than three directions within the MPE region. For example, in some embodiments the MPE region  1650  may be designed to allow diffraction of each input beam of light in 4 directions, 5 directions, 6 directions, 7 directions, 8 directions, etc. As already discussed, the diffractive features in the MPE region  1650  can be designed to provide grating vectors which copy the FOV rectangle to locations in the k-space annulus corresponding to the selected diffraction directions. In addition, the diffractive features in the MPE region  1650  can be designed with periods corresponding to grating vector magnitudes which result in these copies of the FOV rectangle lying entirely inside the k-space annulus (and such that other attempted copies of the FOV rectangle lie entirely outside the outer periphery of the k-space diagram). 
     In some embodiments, the angular separation between each of the permitted propagation directions for a given beam of light inside the MPE region  1650  is at least 45 degrees. If the angular separation between any pair of the selected directions is less than this amount, then the diffractive features in the MPE region  1650  would need to be designed to provide grating vectors to make those angular transitions in the k-space annulus; and such grating vectors would be relatively short in comparison to the size of the k-space annulus due to the lesser angular separation. This could make it more likely that superpositions of the fundamental MPE grating vectors would create copies of the FOV rectangle which lie only partially inside the k-space annulus, which may result in the loss of image information (if not done carefully, as discussed further herein). In addition, if the angular separation between any pair of permitted propagation directions in the MPE region  1650  becomes too small, then the resulting relatively short grating vectors could also make it more likely that grating vector superpositions would create copies of the FOV rectangle which lie partially inside the central disk of the k-space diagram. This could be undesirable because it could result in light being out-coupled from the eyepiece waveguide  1600 , toward the user&#39;s eye, from a location outside the designated EPE region  1660 . 
     Various design guidelines can be followed when determining the permissible propagation directions within the MPE region  1650 . For example, the permissible propagation directions can be selected such that one corresponds to the direction from the ICG region  1640  to the MPE region  1650 . In addition, the permissible propagation directions can be selected such that only one would cause beams of light which propagate in that direction from a location inside the MPE region  1650  to intersect with the EPE region  1660 . This ensures that the replicated beams of light which correspond to each input beam enter the EPE region  1660  with the same propagation angle. In addition, the permissible propagation directions inside the MPE region  1650  can be selected such that the FOV rectangles do not overlap. Overlapping of FOV rectangles can result in mixing of image information from different image points and can cause ghost images. 
       FIG.  16 F  is a diagram of the first generation of interactions between an input beam and the MPE region  1650  of the eyepiece waveguide embodiment shown in  FIG.  16 A .  FIG.  16 F  shows an input beam that enters the MPE region  1650  from the ICG region  1640 . The input beam is shown propagating in the direction which corresponds to the center point, or k-vector, of the FOV rectangle located near the 9 o&#39;clock position of the k-space annulus in  FIG.  16 E . 
     The MPE region  1650  can include many sub-1 μm features. And at every interaction with the MPE region, an input ˜1 mm-diameter beam will split into 3 beams (with the same diameter but a fraction of the original power of the input beam) propagating in 3 different directions in TIR. One direction corresponds to zero-order diffraction and is the original propagation angle in the plane of the waveguide. The other two directions depend on the grating vectors G and H of the MPE region  1650 . As shown, the first generation of interactions between the input beam and the MPE region  1650  results in three beams: some portion of the power of the input beam simply reflects, as output 1 , from the top or bottom surface of the eyepiece waveguide  1600  and continues on in the same x-y direction as the input beam (i.e., the 0 th  order diffraction); some portion of the power of the input beam interacts with the 2D grating in the MPE region  1650  and is diffracted downward as output 2 ; and some portion of the power of the input beam interacts with the grating and is diffracted upward and to the right as output 3 . The output 2  beam is shown propagating in the direction which corresponds to the center point, or k-vector, of the FOV rectangle located near the 6 o&#39;clock position of the k-space annulus in  FIG.  16 E , while the output 3  beam is shown propagating in the direction which corresponds to the center point, or k-vector, of the FOV rectangle located near the 2 o&#39;clock position. After this first generation of interactions, the output 1  beam, the output 2  beam, and the output 3  beam have different propagation angles, but they are all still propagating within the MPE region  1650  and may therefore have additional interactions with the MPE region, as shown in  FIGS.  16 G − 16 I. Although not illustrated, other input beams that enter the MPE region  1650  with different propagation angles will behave similarly but with slightly different input and output angles. 
       FIG.  16 G  is a diagram of the second generation of interactions between an input beam and the MPE region  1650  of the eyepiece waveguide embodiment shown in  FIG.  16 A . The beams related to the first generation of interactions are shown with dashed lines, while the beams related to the second generation of interactions are shown with solid lines. As shown in  FIG.  16 G , each of the output beams, output 1 , output 2 , and output 3 , from the first generation of interactions can now undergo similar interactions with the MPE region  1650  as occurred in the previous generation. Namely, some portion of the power of the output 1  beam from  FIG.  16 F  simply continues on in the same x-y direction, while another portion of the power of that beam interacts with the grating and is diffracted in the direction corresponding to the FOV rectangle located near the 6 o&#39;clock position, and still another portion of the power of that beam interacts with the grating and is diffracted in the direction corresponding to the FOV rectangle located near the 2 o&#39;clock position. Similarly, some portion of the power of the output 2  beam from  FIG.  16 F  simply continues toward the EPE region  1660 , while another portion of the power of that beam interacts with the grating and is diffracted in the direction indicated by the FOV rectangle located near the 9 o&#39;clock position, and still another portion of the power of that beam interacts with the grating and is diffracted in the direction corresponding to the FOV rectangle located near the 2 o&#39;clock position. Further, some portion of the power of the output 3  beam from  FIG.  16 F  simply continues in the direction indicated by the FOV rectangle located near the 2 o&#39;clock position, while another portion of the power of that beam interacts with the grating and is diffracted in the direction indicated by the FOV rectangle located near the 9 o&#39;clock position, and still another portion of the power of that beam interacts with the grating and is diffracted in the direction corresponding to the FOV rectangle located near the 6 o&#39;clock position. 
       FIG.  16 H  is a diagram of the third generation of interactions between an input beam and the MPE region  1650  of the eyepiece waveguide embodiment shown in  FIG.  16 A . The beams related to the first and second generations of interactions are shown with dashed lines, while the beams related to the third generation of interactions are shown with solid lines. As shown in  FIG.  16 H , each of the output beams which resulted from the second generation of interactions can once more experience similar interactions with the MPE region  1650  as occurred in the previous generations. 
       FIG.  16 I  is a diagram of the fourth generation of interactions between an input beam and the MPE region  1650  of the eyepiece waveguide embodiment shown in  FIG.  16 A . The beams related to the first, second, and third generations of interactions are shown with dashed lines, while the beams related to the fourth generation of interactions are shown with solid lines. After all these interactions, all of the resulting beams are propagating in one of the three directions which are permitted inside the MPE region  1650  for any given input beam: the direction corresponding to the FOV rectangle located near the 9 o&#39;clock position; the direction corresponding to the FOV rectangle located near the 2 o&#39;clock position; or the direction corresponding to the FOV rectangle located near the 6 o&#39;clock position of the k-space annulus. Although there are nodes where some of these beams may intersect with one another while propagating through the MPE region  1650 , the locations of those nodes have a more complex distribution than in the case of the OPE region  1550  which was illustrated in  FIGS.  15 D- 15 G . Further, beams can arrive at each of these nodes via different paths and therefore will not necessarily be in phase with one another. Accordingly, image artifacts which may result from the ordered distribution of interference nodes can be reduced in the eyepiece waveguide embodiment  1600  which uses an MPE region  1650  instead of an OPE region (e.g.,  1550 ). This can be seen in  FIGS.  16 J and  16 K . 
       FIG.  16 J  is a diagram which illustrates various paths which beams may follow through the MPE region  1650  and ultimately to the EPE region  1660 . There are some paths which only include a single change in direction, while others include multiple changes in direction (though some of the longer, more complicated pathways will naturally carry less power). Due to the complexity introduced by the existence of another diffraction angle in the MPE region  1650 , there are many different spacings between the beams of light  1665  which ultimately illuminate the EPE region  1660 . And, in fact, any possible spacing between the light beams  1665  can be achieved through a sufficient number of interactions in the MPE region  1650 . As shown in  FIG.  16 K , this can result in more even illumination of the EPE region  1660 . 
       FIG.  16 K  is a diagram which illustrates how a single input beam  1645  from the ICG region  1640  is replicated by the MPE region  1650  and redirected toward the EPE region  1660  as a plurality of beams  1665 . Each of these beams  1665  originates from a dense grid of nodes. There may still be gaps between some of these replicated beams  1665 , but they are generally smaller and less regular than the gaps between the replicated beams which are output from an OPE region (e.g.,  1550 , as shown in  FIG.  15 G ). Since there are so many pathways toward the EPE region  1660 , all at different positions, the MPE region  1650  provides a complex exit pupil pattern which can more evenly illuminate the EPE region  1560 . 
       FIG.  16 L  is a side-by-side comparison which illustrates the performance of an eyepiece waveguide with an OPE region versus that of an eyepiece waveguide with an MPE region. On the left is shown the eyepiece waveguide  1500 , which includes an OPE region  1550  with a 1D periodic diffraction grating. As already discussed, the OPE region  1550  illuminates the EPE region  1560  with a sparse set of regularly spaced replicated light beams. Below the eyepiece waveguide  1500  is a simulated output image. This is the simulated output image which would be projected from the EPE region  1560  of the eyepiece waveguide  1500  in response to an input image made up of pixels that all have the same color and brightness. 
     On the right,  FIG.  16 L  shows the eyepiece waveguide  1600  which includes an MPE region  1650  with a 2D periodic diffraction grating. As can be seen in the figure, the MPE region  1650  illuminates the EPE region  1660  more evenly. Below the eyepiece waveguide  1600  is a simulated output image which is the result of the same input image used in the simulation for the eyepiece waveguide  1500  on the left. It is clear from the simulated image on the right that the eyepiece waveguide  1600  that uses the MPE region  1650  achieves a smoother, more uniform distribution of output light. In contrast, the image on the left, which is the simulated output of the eyepiece waveguide  1500  with the OPE region  1550 , has visible high spatial frequency striations which result from the sparse, ordered set of replicated light beams which illuminate its EPE region  1560 . 
       FIG.  16 M  further illustrates the performance of an eyepiece waveguide with an MPE region versus others with OPE regions. The top row of graphs in  FIG.  16 M  illustrate the performance of the eyepiece waveguide  1500  shown in  FIG.  15 A . The graph of the horizontal cross-section of a projected image from this eyepiece waveguide shows the relatively high spatial frequency variation which was visible as striations in the simulated output image shown in  FIG.  16 L .  FIG.  16 M  shows that the eyepiece waveguide  1500  has an eyebox efficiency of 1.2%. It also shows the point spread function associated with this eyepiece waveguide. The point spread function illustrates the output image obtained from the eyepiece waveguide in response to an input image of a single bright point. This shows that the eyepiece waveguide  1500  is quite sharp, as it only has blur of 2.5-5 arc minutes. 
     One approach to overcoming the high spatial frequency variation in output images from the eyepiece waveguide  1500  is to introduce some dithering in the OPE region  1550 . For example, small variations can be introduced in the orientation angle and/or grating period of the OPE region  1550 . This is done in an attempt to disrupt the ordered nature of the interference nodes which can be present in the OPE region  1550 . The second and third rows in  FIG.  16 M  illustrate the performance of the eyepiece waveguide  1500  with two different types of dithering. As can be seen in the horizontal cross-sections of the projected images for these waveguides, the high spatial frequency variations are still present. Further, the point spread functions for these dithered embodiments show a much larger amount of blur—in one case as much as 45 arc minutes. 
     The bottom row of  FIG.  16 M  illustrates the performance of the eyepiece waveguide  1600  with an MPE region  1650 . The cross-section of the projected image for this waveguide shows much less high spatial frequency variation. While there is still low frequency spatial variation, this can be corrected via software much more easily than can high spatial frequency variation. The eyebox efficiency of this eyepiece waveguide is slightly less, at 0.9%, than the others. This can be attributed to the fact that the MPE region  1650  redirects some of the input light in a general direction corresponding to the FOV rectangle located near the 2 o&#39;clock position in the annulus of the k-space diagram shown in  FIG.  16 E . Due to the macroscopic layout of the eyepiece waveguide  1600 , light which exits the MPE region  1650  with this propagation direction never enters the EPE region and is therefore not projected toward the user&#39;s eye; instead, it is lost out the edge of the waveguide  1600 . However, this loss of light results in only a relatively small decrease in eyebox efficiency. Meanwhile, the point spread function for the eyepiece waveguide  1600  shows that it is quite sharp, with a blur of only 2.5-5 arc minutes. 
       FIGS.  16 A- 16 M  illustrate an eyepiece waveguide  1600  with an MPE region  1650  that has three permissible propagation directions for each input beam. However, other embodiments of MPE regions can be designed to allow even more propagation directions for each input beam. One such example is illustrated in  FIGS.  17 A- 17 G . These figures illustrate an eyepiece waveguide  1700  that is identical in its macroscopic design to the eyepiece waveguide  1600 . Namely, the eyepiece waveguide  1700  includes an ICG region  1740 , an MPE region  1750 , and an EPE region  1760  which are all arranged in the same way as the corresponding regions in the eyepiece waveguide  1600  shown in  FIG.  16 A . However, the eyepiece waveguide  1700  differs in the microscopic design of its MPE region  1750 . 
       FIG.  17 A  illustrates a portion of an example 2D grating, along with its associated grating vectors, which can be used in the MPE region  1750  of the eyepiece waveguide  1700 . The 2D periodic grating  1750  can be a spatial lattice of diffractive features whose directions of periodicity are u and v. As already discussed, such a 2D periodic grating is associated with fundamental grating vectors, G and H. As one example, the 2D periodic grating  1750  can be designed or formed by superimposing two sets of 1D periodic grating lines (though the 2D periodic grating could instead be made up of individual scattering features located at, for example, the intersection points of the grating lines shown in  FIG.  17 A ). The first set of grating lines  1756  can repeat along the direction of the fundamental grating vector G. The fundamental grating vector G can have a magnitude equal to 2π/a, where a is the period of the first set of grating lines  1756 . The 2D grating shown in  FIG.  17 B  is also associated with harmonics of the first fundamental grating vector G. These include −G and higher-order harmonics, such as 2G, −2G, etc. The second set of grating lines  1757  can repeat along the direction of the fundamental grating vector H. The fundamental grating vector H can have a magnitude equal to 2π/b, where b is the period of the second set of grating lines  1657 . The 2D grating shown in  FIG.  17 B  is also associated with harmonics of the second fundamental grating vector H. These include −H and higher-order harmonics, such as 2H, −2H, etc. And, as already discussed, any 2D periodic array of diffractive features will have associated grating vectors which point in directions determined by integer linear combinations (superpositions) of the fundamental grating vectors. In this case, these superpositions result in additional grating vectors. These include, for example, −G, −H, H+G, H−G, G−H, and −(H+G). Although  FIG.  17 A  only illustrates the first order grating vectors, and their superpositions, associated with the 2D diffraction grating, higher-order grating vectors may also exist. 
       FIG.  17 B  is a k-space diagram which illustrates the k-space operation of the MPE region  1750  of the eyepiece waveguide  1700 . The k-space diagram includes a shaded FOV rectangle located near the 9 o&#39;clock position of the k-space annulus. This is the location of the FOV rectangle after the ICG region  1740  has coupled the input beams into the eyepiece waveguide  1700  and redirected them toward the MPE region  1750 .  FIG.  17 B  shows how the 2D grating in the MPE region  1750  translates the FOV rectangle using the grating vectors shown in  FIG.  17 A . Since there are eight grating vectors, the MPE region  1750  attempts to translate the FOV rectangle to eight possible new locations in the k-space diagram. Of these eight possible locations, five fall outside the outer periphery of the k-space diagram. These locations are illustrated with unshaded FOV rectangles. Since k-vectors outside the outer periphery of the k-space diagram are not permitted, none of those five grating vectors results in diffraction. There are, however, three grating vectors (i.e., −H, −G, and −(H+G)) which do result in translations of the FOV rectangle to new positions within the bounds of the k-space diagram. One of these locations is near the 6 o&#39;clock position in the k-space annulus, another is near the 12 o&#39;clock position, and the last is near the 3 o&#39;clock position. Since k-vectors at these locations are permitted and do result in guided propagation modes, the FOV rectangles at these locations are shaded to indicate that beams of light are diffracted into those three states. Thus, beams of light entering the MPE region  1750  with the propagation angles indicated by the FOV rectangle located near the 9 o&#39;clock position of the k-space annulus are diffracted into all of the states indicated by the other three shaded FOV rectangles (i.e., the FOV rectangle near the 12 o&#39;clock position, the FOV rectangle near the 3 o&#39;clock position, and the FOV rectangle near the 6 o&#39;clock position). 
     Although not illustrated, similar k-space diagrams could be drawn to illustrate the k-space operation of the MPE region  1750  on beams of light traveling with the propagation angles indicated by the FOV rectangles located near the 12 o&#39;clock position, near the 3 o&#39;clock position, and near the 6 o&#39;clock position of the k-space annulus. Those k-space diagrams would show that the 2D diffraction grating in the MPE region  1750  diffracts those beams into all of the remaining states indicated by the shaded FOV rectangles in the annulus of the k-space diagram in  FIG.  17 B . 
       FIG.  17 C  is a k-space diagram which illustrates the k-space operation of the eyepiece waveguide  1700 . The eyepiece waveguide  1700  can receive input beams of light which propagate generally in the −z-direction and are incident on an ICG region  1740  of the waveguide  1700  from an outside source. Those input beams are represented by the FOV rectangle centered on the k z -axis at the origin of the k-space diagram. The ICG region  1740  then diffracts the input beams such that they are guided and have propagation angles centered around a propagation direction which corresponds to the center point of the FOV rectangle located near the 9 o&#39;clock position of the k-space annulus. 
     The diffracted beams enter the MPE region  1750 , where they can have multiple interactions. During each generation of interactions, a portion of the power of each of the beams continues propagating in the same direction through the MPE region  1750 . In the first generation of interactions, for example, this would correspond to that portion of the power of those beams staying in the state indicated by the FOV rectangle located near the 9 o&#39;clock position. Other portions of the power of the beams can be diffracted in new directions. Again, in the first generation of interactions, this creates respective diffracted beams that have propagation angles centered around a propagation direction which corresponds to the center point of the FOV rectangle located near the 12 o&#39;clock position of the k-space annulus, the center point of the FOV rectangle located near the 3 o&#39;clock position, and the center point of the FOV rectangle located near the 6 o&#39;clock position. 
     The diffracted beams which still remain in the MPE region  1750  after each interaction can experience additional interactions. Each of these additional interactions results in some of the power of the beams zero-order diffracting and continuing on in the same direction, while some of the power of the beams is diffracted in new directions. This results in spatially distributed sets of diffracted beams that have propagation angles centered around each of the propagation directions indicated by the center points of the FOV rectangles in the k-space annulus shown in  FIG.  17 C . This is represented by the double-sided arrows between each pair of FOV rectangles in the k-space annulus. In other words, beams of light propagating in the MPE region  1750  can transition from any propagation state represented by one of the FOV rectangles in the k-space annulus to any other of these propagation states. 
     As any given input beam of light propagates within the MPE region  1750 , it is split into many diffracted beams which can only travel in four allowed directions—each direction being defined by the corresponding k-vector, or point, within the FOV rectangles in the annulus of the k-space diagram in  FIG.  17 C . (This is true for any input beam of light propagating within the MPE region  1750 . However, the four allowed directions will be slightly different depending on the propagation angle at which each initial input beam enters the MPE region  1750 .) And since portions of the power of any given input beam of light are diffracted into the same four propagation directions after any number of interactions with the MPE region  1750 , image information is preserved throughout these interactions. The additional propagation direction which is permitted in the MPE region  1750 , as compared to the MPE region  1650  described with respect to  FIGS.  16 A- 16 M , can result in even further improvements in the evenness of illumination in the EPE region  1760 . This can be seen in the diagrams shown in  FIGS.  17 D- 17 G . 
       FIG.  17 D  is a diagram of the first generation of interactions between an input beam and the MPE region  1750  of the eyepiece waveguide  1700 .  FIG.  17 D  shows an input beam that enters the MPE region  1750  from the ICG region  1740 . The input beam is shown propagating in the direction which corresponds to the center point, or k-vector, of the FOV rectangle located near the 9 o&#39;clock position of the k-space annulus in  FIG.  17 C . 
     The MPE region  1750  can include many sub-1 μm features. And at every interaction with the MPE region, a ˜1 mm-diameter beam will split into 4 beams (with the same diameter but a fraction of the original power of the input beam) propagating in 4 different directions in TIR. One direction corresponds to zero-order diffraction and is the original angle in the plane of the waveguide. The other three directions depend on the grating vectors G and H of the MPE region  1750 . As shown, the first generation of interactions between the input beam and the MPE region  1750  results in four beams: some portion of the power of the input beam simply reflects, as output 1 , from the top or bottom surface of the eyepiece waveguide  1700  and continues on in the same x-y direction as the input beam (i.e., the 0 th  order diffraction); some portion of the power of the input beam interacts with the grating and is diffracted downward as output 2 ; some portion of the power of the input beam interacts with the grating and is diffracted upward as output 3 ; and some portion of the power of the input beam interacts with the grating and is diffracted to the right as output 4 . The output 2  beam is shown propagating in the direction which corresponds to the center point, or k-vector, of the FOV rectangle located near the 6 o&#39;clock position of the k-space annulus in  FIG.  17 C , while the output 3  beam is shown propagating in the direction which corresponds to the center point, or k-vector, of the FOV rectangle located near the 12 o&#39;clock position, and the output 4  beam is shown propagating in the direction which corresponds to the center point, or k-vector, of the FOV rectangle located near the 3 o&#39;clock position. After this first generation of interactions, the output 1  beam, the output 2  beam, the output 3  beam, and the output 4  beam have different propagation angles, but they are all still propagating within the MPE region  1750  and may therefore have additional interactions with the MPE region, as shown in  FIGS.  17 E- 17 G . Although not illustrated, other input beams that enter the MPE region  1750  with different propagation angles will behave similarly but with slightly different input and output angles. 
       FIG.  17 E  is a diagram of the second generation of interactions between an input beam and the MPE region  1750  of the eyepiece waveguide  1700 . The beams related to the first generation of interactions are shown with dashed lines, while the beams related to the second generation of interactions are shown with solid lines. As shown in  FIG.  17 D , each of the output beams, output 1 , output 2 , output 3 , and output 4 , from the first generation of interactions can now undergo similar interactions with the MPE region  1750  as occurred in the previous generation. Namely, some portion of the power of the output 1  beam from  FIG.  17 D  simply continues on in the same x-y direction, while other portions of the power of that beam interact with the grating and are diffracted in the directions corresponding to the FOV rectangles located near the 12 o&#39;clock position, near the 3 o&#39;clock position, and near the 6 o&#39;clock position. Similarly, some portion of the power of the output 2  beam from  FIG.  17 D  simply continues toward the EPE region  1760 , while other portions of the power of that beam interact with the grating and are diffracted in the directions indicated by the FOV rectangles located near the 9 o&#39;clock position, near the 12 o&#39;clock position, and near the 3 o&#39;clock position. Further, some portion of the power of the output 3  beam from  FIG.  17 D  simply continues in the direction indicated by the FOV rectangle located near the 12 o&#39;clock position, while other portions of the power of that beam interact with the grating and are diffracted in the directions indicated by the FOV rectangles located near the 3 o&#39;clock position, near the 6 o&#39;clock position, and near the 9 o&#39;clock position. Finally, some portion of the power of the output 4  beam from  FIG.  17 D  simply continues in the direction indicated by the FOV rectangle located near the 3 o&#39;clock position, while other portions of the power of that beam interact with the grating and are diffracted in the directions indicated by the FOV rectangles located near the 6 o&#39;clock position, near the 9 o&#39;clock position, and near the 12 o&#39;clock position. 
       FIG.  17 F  is a diagram of the third generation of interactions between an input beam and the MPE region  1750  of the eyepiece waveguide embodiment  1700 . The beams related to the first and second generations of interactions are shown with dashed lines, while the beams related to the third generation of interactions are shown with solid lines. As shown in  FIG.  17 F , each of the output beams which resulted from the second generation of interactions can once more experience similar interactions with the MPE region  1750  as occurred in the previous generations. 
       FIG.  17 G  is a diagram of the fourth generation of interactions between an input beam and the MPE region  1750  of the eyepiece waveguide embodiment  1700 . The beams related to the first, second, and third generations of interactions are shown with dashed lines, while the beams related to the fourth generation of interactions are shown with solid lines. After all these interactions, all of the resulting beams are propagating in one of the four permitted propagation directions with the MPE region  1750  for any given input beam: the direction corresponding to the FOV rectangle located near the 9 o&#39;clock position; the direction corresponding to the FOV rectangle located near the 12 o&#39;clock position; the direction corresponding to the FOV rectangle located near the 3 o&#39;clock position; or the direction corresponding to the FOV rectangle located near the 6 o&#39;clock position of the k-space annulus. Although there are nodes where some of these beams may intersect with one another while propagating through the MPE region  1750 , the locations of those nodes have an even more complex distribution than in the case of the MPE region  1650  which was illustrated in  FIGS.  16 A- 16 M . Further, these nodes are even less likely to result in interference between two in-phase beams. Accordingly, this MPE region  1750  may result in an even more uniform illumination of the EPE region  1760 . 
     By way of summary, the MPE regions described herein are capable of some or all of the following advantages: MPE regions can expand an image pupil in multiple directions at once; MPE regions can create dense, non-periodic arrays of output pupils; MPE regions can reduce interference effects between light paths through the waveguide; MPE-based eyepiece waveguides can achieve improved luminance uniformity with reduced high-frequency striations and with high image sharpness. 
     Example AR Eyepiece Waveguides with Multiple Distinct Regions for Replicating Input Beams 
       FIG.  18 A  illustrates an example eyepiece waveguide  1800  with an ICG region  1840 , two orthogonal pupil expander (OPE) regions  1850   a ,  1850   b , and an exit pupil expander (EPE) region  1860 .  FIG.  18 A  also includes k-space diagrams which illustrate the effect of each of these components of the eyepiece waveguide  1800  in k-space. The ICG region  1840 , OPE regions  1850   a ,  1850   b , and EPE region  1860  of the eyepiece waveguide  1800  include various diffractive features which couple input beams into the eyepiece waveguide  1800  to propagate via guided modes, replicate the beams in a spatially distributed manner, and cause the replicated beams to exit the eyepiece waveguide and be projected toward the user&#39;s eye. In particular, the eyepiece waveguide  1800  includes multiple distinct and/or non-contiguous regions for replicating input beams. Replicated beams from these distinct regions can be re-combined in a common exit pupil region. 
     The eyepiece waveguide  1800  illustrated in  FIG.  18 A  is similar to the eyepiece waveguide  1400  illustrated in  FIG.  14 A  except that it includes two OPE regions  1850   a ,  1850   b  instead of one. Recall that the ICG region  1440  in the eyepiece waveguide  1400  diffracted input beams into the +1 and −1 diffractive orders but that the beams in one of these diffractive orders propagated away from the OPE region  1450  and were ultimately lost from the eyepiece waveguide. Accordingly, a portion of the light from the input beams was lost. The eyepiece waveguide  1800  shown in  FIG.  18 A  remedies this by including two OPE regions  1850   a ,  1850   b , one on either side of the ICG region  1840 . In this way, the eyepiece waveguide  1800  can make use of both the +1 and the −1 diffractive orders of the ICG  1840 . 
     The operation of the ICG region  1840  is similar to what has been described with respect to the ICG region  1440  in  FIGS.  14 A and  14 B . The same k-space diagram, KSD 1 , shown in  FIG.  14 B  is also illustrative of the FOV rectangle corresponding to the set of input beams that are incident on the ICG region  1840  in  FIG.  18 A . Namely, before the input beams are incident on the ICG region  1840 , the FOV rectangle is centered at the origin of the k-space diagram. 
     K-space diagram KSD 2  in  FIG.  18 A  illustrates the operation, in k-space, of the ICG region  1840 . Namely, as discussed with respect to the corresponding k-space diagram in  FIG.  14 B , the ICG region  1840  is associated with two grating vectors which respectively translate the FOV rectangle to the 3 o&#39;clock and 9 o&#39;clock positions inside the k-space annulus. The translated FOV rectangle located at the 3 o&#39;clock position represents diffracted beams which propagate toward the right OPE region  1850   b , while the translated FOV rectangle located at the 9 o&#39;clock position represents diffracted beams which propagate toward the left OPE region  1850   a.    
     The operation of the left OPE region  1850   a  is also similar to what has been described with respect to the OPE region  1450  in  FIGS.  14 A and  14 B . K-space diagram KSD 3   a  illustrates the k-space operation of the left OPE region  1850   a  and shows that its diffraction grating translates the FOV rectangle from the 9 o&#39;clock position in the k-space annulus to the 6 o&#39;clock position. The FOV rectangle located at the 6 o&#39;clock position represents diffracted beams which propagate in the −y-direction toward the EPE region  1860 . 
     The operation of the right OPE region  1850   b  is similar to that of the left OPE region  1850   a  except that its associated grating vectors are mirrored about a vertical line with respect to those of the left OPE region  1850   a . This is due to the fact that the lines of the diffraction grating in the right OPE region  1850   b  are mirrored about a vertical line with respect to those of the diffraction grating in the left OPE region  1850   a . As a result of this orientation of the lines of the diffraction grating in the right OPE region  1850   b , the effect of this grating in k-space is to translate the FOV rectangle from the 3 o&#39;clock position in the k-space annulus to the 6 o&#39;clock position, as shown in k-space diagram KSD 3   b . The translated FOV rectangles in KSD 3   a  and KSD 3   b  are in the same location at the 6 o&#39;clock position of the k-space annulus. Thus, although the power of each input beam is split into +1 and −1 diffractive orders by the ICG region  1840 , and those distinct diffractive orders travel different paths through the eyepiece waveguide  1800 , they nevertheless arrive at the EPE region  1860  with the same propagation angle. This means that the separate diffractive orders of each input beam which follow different propagation paths through the eyepiece waveguide  1800  ultimately exit the EPE region  1860  with the same angle and therefore represent the same point in the projected image. 
     Finally, the operation of the EPE region  1860  is also similar to what has been described with respect to the EPE region  1460  in  FIGS.  14 A and  14 B . K-space diagram KSD 4  illustrates the k-space operation of the EPE region  1860  and shows that its diffraction grating translates the FOV rectangle located at the 6 o&#39;clock position (which consists of light beams from both OPE regions  1850   a ,  1850   b ) of the k-space annulus back to the center of the k-space diagram. As already discussed elsewhere, this represents that the EPE region  1860  out-couples the beams of light generally in the z-direction toward the user&#39;s eye. 
       FIGS.  18 B and  18 C  illustrate top views of the EPE region  1860  of the eyepiece waveguide  1800  shown in  FIG.  18 A . The EPE region  1860  is supported directly in front of the user&#39;s eye  210 . As discussed elsewhere herein (see  FIGS.  12 A and  12 B ), the EPE region  1860  projects sets of replicated output beams, with each set of replicated output beams having a propagation angle which corresponds to one of the input beams which are projected into the eyepiece waveguide. 
       FIG.  18 B  illustrates one of these sets of replicated output beams. In this particular case, the replicated output beams  1861  exit the EPE region  1860  traveling from left to right. In other words, the replicated output beams  1861  have a propagation direction with a component in the +x-direction. This propagation angle of the replicated output beams  1861  results in some of them having a greater tendency to intersect with the user&#39;s eye  210  than others. In particular, the replicated output beams  1861  which exit from the left-hand portion of the EPE region  1860  have a greater tendency to intersect with the user&#39;s eye  210  due to the central position of the eye  210  and the left-to-right propagation of the light beams. These light beams are illustrated with solid lines. Meanwhile, the replicated output beams  1861  which exit from the right-hand portion of the EPE region  1860  have a greater tendency to miss the eye  210 . These light beams are illustrated with dashed lines. 
       FIG.  18 B  also includes a k-space diagram, KSD 5 , which illustrates the state of the output beams, in k-space, after the EPE region has translated the FOV rectangle back to the origin of the diagram. The FOV rectangle is illustrated with two halves. Each of the halves represents half of the horizontal field of view of the eyepiece waveguide  1800 . The shaded right half  1832  of the FOV rectangle includes the k-vectors with components in the +k x -direction. These are the k-vectors corresponding to the output beams  1861  which exit the EPE region  1860  with the type of left-to-right propagation illustrated in  FIG.  18 B . Although only one set of replicated output beams  1861  is illustrated exiting the EPE region  1860 , all of the output beams whose k-vectors lie in the shaded right half  1832  of the FOV rectangle would similarly exit the EPE region with propagation directions going left-to-right. Thus, it is true for all of the output beams whose k-vectors lie in the shaded right half  1832  of the FOV rectangle that those beams exiting the left-hand side of the EPE region  1860  will have a greater tendency to intersect with the eye  210  than those output beams which exit the right-hand side of the EPE region. 
       FIG.  18 C  illustrates another set of replicated light beams  1862  which exit the EPE region  1860  of the eyepiece waveguide  1800 . But in this case, the replicated output beams  1862  exit the EPE region  1860  traveling from right to left. In other words, the replicated output beams  1862  have a propagation direction with a component in the −x-direction. This propagation angle of the replicated output beams  1862  leads to the opposite observation of that which was drawn from  FIG.  18 B . Namely, for the right-to-left propagating output beams  1862 , the beams exiting from the right-hand portion of the EPE region  1860  (illustrated with solid lines) have a greater tendency to intersect with the eye  210 , while those light beams which exit from the left-hand portion of the EPE region (illustrated with dashed lines) have a greater tendency to miss the eye. 
     With reference to the k-space diagram, KSD 5 , included with  FIG.  18 C , the output beams whose k-vectors lie in the shaded left half  1831  of the FOV rectangle are those which exit the EPE region  1860  with the type of right-to-left propagation shown in  FIG.  18 C . Although all of the output beams whose k-vectors lie in the shaded left half  1831  of the FOV rectangle will have differing propagation angles, they all share the property that the beams exiting the right-hand side of the EPE region  1860  will have a greater tendency to intersect with the eye  210  than the output beams which exit from the left-hand side of the EPE region. 
     The conclusion which can be drawn from  FIGS.  18 B and  18 C  is that, based on the light beams which actually enter the user&#39;s eye  210 , half of the EPE region  1860  contributes predominantly to one half of the horizontal field of view, while the other half of the EPE region contributes predominantly to the other half of the horizontal field of view. Based on this observation, the field of view which can be projected by an eyepiece waveguide can be expanded in at least one dimension beyond the range of propagation angles supported by the eyepiece in guided modes because it is unnecessary to project the entire FOV rectangle from every portion of the EPE region  1960 . This is illustrated in  FIG.  19   . 
     Example AR Eyepiece Waveguides with Expanded Field of View 
       FIG.  19    illustrates an embodiment of an eyepiece waveguide  1900  with an expanded field of view. The eyepiece waveguide  1900  includes an ICG region  1940 , a left OPE region  1950   a , a right OPE region  1950   b , and an EPE region  1960 . At a macroscopic level, the eyepiece waveguide  1900  shown in  FIG.  19    can be identical to the eyepiece waveguide  1800  shown in  FIG.  18 A . However, some of the diffractive features in the eyepiece waveguide  1900  can be designed with characteristics which allow for increased field of view in at least one dimension. These features can be clearly understood based on the k-space operation of the eyepiece waveguide  1900 , which is illustrated by the k-space diagrams shown in  FIG.  19   . 
     The k-space diagrams shown in  FIG.  19    have larger FOV rectangles than those which are shown in  FIG.  18 A . This is because the FOV rectangles in the k-space diagrams in  FIG.  18 A  were constrained to not have any dimension larger than the width of the k-space annulus. This constraint ensured that those FOV rectangles could fit entirely in the k-space annulus, at any position around the annulus, and therefore that all of the beams represented by the k-vectors in the FOV rectangles could undergo guided propagation within the eyepiece waveguide  1800  while propagating in any direction in the plane of the eyepiece. In the example embodiment of  FIG.  19   , however, the FOV rectangles have at least one dimension (e.g., the k x  dimension) which is larger than the width of the k-space annulus. In some embodiments, one or more dimensions of the FOV rectangles can be up to 20%, up to 40%, up to 60%, up to 80%, or up to 100% larger than the width of the k-space annulus. 
     For the particular embodiment illustrated in the k-space diagrams of  FIG.  19   , the horizontal dimension of the FOV rectangles is wider than the k-space annulus. The horizontal dimension of the FOV rectangles corresponds to the horizontal spread in the propagation angles of the input beams which are projected into an eyepiece waveguide. Thus, since the eyepiece waveguide  1900  is illustrated as being capable of use with FOV rectangles having larger horizontal dimensions, this means that the horizontal field of view of the eyepiece waveguide is increased. For the case of an eyepiece waveguide (surrounded by air) with refractive index 1.8, whereas the eyepiece waveguide  1800  shown in  FIG.  18 A  is generally capable of achieving FOVs of 45° by 45°, the eyepiece waveguide  1900  shown in  FIG.  19    is capable of achieving FOVs of up to 90° by 45°, though some embodiments of the eyepiece waveguide may be designed for smaller FOVs of ˜60° by 45° so as to satisfy typical design constraints of eyebox volume—it may be advantageous to send some portion of the FOV to both sides of the eyepiece waveguide to provide an adequately-sized eyebox—and to avoid screen door artifacts resulting from sparsely spaced output beams. Although the techniques for expanding the field of view of the eyepiece waveguide  1900  are described in the context of expanded horizontal fields of view, the same techniques can also be used to expand the vertical field of view of the eyepiece waveguide  1900 . Moreover, in later embodiments, similar techniques are shown for expanding both the horizontal and vertical fields of view of an eyepiece waveguide. 
     It can be seen by inspection of the k-space diagrams in  FIG.  19    that although the illustrated FOV rectangles may not fit entirely within the k-space annulus when located at certain positions around the annulus, they can still fit entirely within the annulus when located at other positions. For example, if one dimension of the FOV rectangle is larger than the width of the k-space annulus, then the FOV rectangle may not fit entirely within the annulus when the FOV rectangle is located at or near the axis of the enlarged dimension: an FOV rectangle which is larger in the k x  dimension than the width of the k-space annulus may not fit entirely within the annulus when the FOV rectangle is located at or near the k x -axis (i.e., at or near the 3 o&#39;clock and 9 o&#39;clock positions); similarly, an FOV rectangle which is larger in the k y  dimension than the width of the k-space annulus may not fit entirely within the annulus when the FOV rectangle is located at or near the k y -axis (i.e., at or near the 12 o&#39;clock and 6 o&#39;clock positions). However, such an FOV rectangle may still fit entirely within the k-space annulus when it is located at or near the opposite axis: an FOV rectangle which is larger in the k x  dimension than the width of the k-space annulus may still fit entirely within the annulus when the FOV rectangle is located at or near the k y -axis (i.e., at or near the 12 o&#39;clock and 6 o&#39;clock positions); similarly, an FOV rectangle which is larger in the k y  dimension than the width of the k-space annulus may still fit entirely within the annulus when the FOV rectangle is located at or near the k x -axis (i.e., at or near the 3 o&#39;clock and 9 o&#39;clock positions). This is because there is more area in the k-space annulus in the azimuthal direction to accommodate larger FOV rectangles than in the radial direction. 
     The radial size of the k-space annulus corresponds to the range of propagation angles in the direction normal to the plane of the waveguide (i.e., the thickness direction) which support guided propagation modes. This range of propagation angles is constrained by Snell&#39;s Law and the requirements which must be satisfied for TIR to occur. In contrast, a spread of k-vectors in the azimuthal dimension of the k-space annulus corresponds to a spread of propagation angles in the in-plane direction of the planar waveguide. Since the spread of propagation angles within the plane of the planar waveguide is not limited by the same constraints as in the thickness direction, a wider range of beam propagation angles can be supported. 
     Moreover, it is possible to convert a spread of propagation angles in the thickness direction of an eyepiece waveguide to a spread of propagation angles in the in-plane direction, and vice versa. When a diffraction grating (or other group of diffractive features) translates an FOV rectangle from one position in the k-space annulus to another such that the set of beams represented by the FOV rectangle are then propagating in a new direction, this also causes some of the beams which were previously spread out in the thickness direction of the planar waveguide to instead be spread out in the in-plane direction, and vice versa. This can be seen when, for example, a diffraction grating translates an FOV rectangle from the 9 o&#39;clock position in the k-space annulus to the 6 o&#39;clock position. While in the 9 o&#39;clock position, the spread of beams in the k x  direction corresponds to a physical spread in the thickness direction of the waveguide since at that location the k x  direction corresponds to the radial direction of the k-space annulus. However, at the 6 o&#39;clock position, the spread of beams in the k x  direction corresponds to a physical spread in the in-plane direction of the waveguide since at that location the k x  direction corresponds to the azimuthal direction of the k-space annulus. 
     Using these observations, the FOV of an eyepiece waveguide can be increased by: dividing an FOV rectangle into multiple sub-portions; using diffractive features to replicate the beams, in a spatially distributed manner, belonging to the multiple sub-portions of the FOV; and using diffractive features to re-assemble the multiple sub-portions of the FOV at the exit pupil of the eyepiece waveguide such that the beams corresponding to each sub-portion of the FOV have the correct propagation angles to re-create the original image. For example, diffractive features can be used to translate each sub-portion of the FOV rectangle to one or more locations in k-space such that they ultimately have the same relative position with respect to the other sub-portions of the FOV rectangle as in the original image. 
     In some embodiments, the multiple sub-portions of the FOV can partially overlap one another (e.g., different pairs of FOV sub-portions can include some of the same input beams), as this can help ease the constraints for re-assembling the entire FOV at the exit pupil of the waveguide and can help to ensure that all of the beams are present. For example, in some embodiments, a pair of sub-portions of the input image FOV may overlap by no more than 10%, no more than 20%, no more than 30%, no more than 40%, no more than 50%, or more. 
     K-space diagram KSD 2  in  FIG.  19    illustrates the k-space operation of the ICG region  1940  on the input beams which are projected into the eyepiece waveguide  1900 . As discussed elsewhere herein, the input beams which are projected into the eyepiece waveguide  1900  can be represented by an FOV rectangle which is centered at the origin of the k-space diagram KSD 2 . The ICG region  1940  translates the location of this FOV rectangle in k-space based on its associated grating vectors. In the case of the ICG region  1840  illustrated in  FIG.  18 A , the ICG region was designed such that its associated grating vectors G 1 , G −1  had magnitudes equal to the distance from the origin of the k-space diagram to the midpoint of the k-space annulus. This caused the FOV rectangle to be centered within the k-space annulus. But the ICG region  1940  illustrated in  FIG.  19    can be designed to have larger grating vectors. And, as already discussed, the set of input beams which are projected into the eyepiece waveguide  1900  can have at least one dimension in k-space that is larger than the width of the k-space annulus. 
     In some embodiments, ICG region  1940  can be designed such that its grating vectors G 1 , G −1  translate the enlarged FOV rectangle far enough from the origin of the k-space diagram such that no portion of the enlarged FOV rectangle lies inside the inner disk of the k-space diagram. To achieve this goal in the case of an FOV rectangle whose horizontal dimension is twice as large as the width of the k-space annulus, the magnitude of the grating vectors G 1 , G −1  of the ICG  1940  would need to be approximately equal to the radius of the outer disk of the k-space diagram. Meanwhile, to achieve this goal in the case of an FOV rectangle whose horizontal dimension is just slightly larger than the width of the k-space annulus, the magnitude of the grating vectors G 1 , G −1  of the ICG region  1940  would need to be greater than the distance from the origin of the k-space diagram to the midpoint of the k-space annulus. Mathematically, this means 
                   n   2     ⁢   ω     c     ≥       ❘   &#34;\[LeftBracketingBar]&#34;         G   1     ,     G     -   1           ❘   &#34;\[RightBracketingBar]&#34;       &gt;       1   2     ⁢     (           n   2     ⁢   ω     c     +         n   1     ⁢   ω     c       )             
which gives
 
                   n   2     ⁢   ω     c     ≥       ❘   &#34;\[LeftBracketingBar]&#34;         n   2     Λ       ❘   &#34;\[RightBracketingBar]&#34;       &gt;       1   2     ⁢       (           n   2     ⁢   ω     c     +         n   1     ⁢   ω     c       )     .             
(Note: This equation can also be applied to the other eyepiece waveguide embodiments described herein, such as, for example, those shown in  FIGS.  20 - 22    and described below.)
 
     In other words, this technique for expanding the field of view of the eyepiece waveguide  1900  means that the grating vectors G 1 , G −1  of the ICG region  1940  are designed to be longer than in embodiments where the field of view is constrained in all dimensions by the range of propagation angles which can fit within the radial dimension of the k-space annulus of a given eyepiece waveguide. Since the length of the grating vectors G 1 , G −1  is increased by decreasing the grating period, Λ, this means that the ICG region  1940  has a finer pitch than what would conventionally be used for light of a given angular frequency, ω, to ensure that all of the input beams can be diffracted into guided modes. 
     Of course, according to the embodiment illustrated in  FIG.  19   , the larger size of the FOV rectangle and the longer grating vectors G 1 , G −1  cause portions of the translated FOV rectangles, after diffraction by the ICG region  1940 , to extend beyond the outer perimeter of the larger disk in the k-space diagram. Since k-vectors outside this disk are not permitted, the input beams corresponding to those k-vectors are not diffracted by the ICG region  1940 . Instead, only the input beams corresponding to k-vectors in the shaded portions of the translated FOV rectangles in KSD 2  enter guided propagation modes within the eyepiece waveguide  1900 . The input beams which would diffract into the +1 order with k-vectors that would lie outside the outer disk of the k-space diagram are not permitted to diffract and are therefore lost. Similarly, the input beams which would diffract into the −1 order with k-vectors that would lie outside the outer disk of the k-space diagram are not permitted to diffract and are therefore lost. Fortunately, the beams which are lost from each of these diffractive orders are not the same ones. This allows the full field of view to be recovered at the EPE region  1960 . Even though neither the truncated FOV rectangle located at the 3 o&#39;clock position of the k-space diagram KSD 2 , nor the truncated FOV rectangle located at the 9 o&#39;clock position, includes the complete set of input beams, when these truncated FOV rectangles are appropriately recombined at the EPE region  1960 , the complete set of input beams can be recovered. 
     The k-space diagrams KSD 3   a  and KSD 3   b  respectively illustrate the k-space operation of the diffraction gratings in the left OPE region  1950   a  and the right OPE region  1950   b . As discussed with respect to  FIG.  18 A , these OPE regions can include diffraction gratings which are oriented so as to translate the FOV rectangles located at the 3 o&#39;clock and 9 o&#39;clock positions to the 6 o&#39;clock position. In the embodiment illustrated in  FIG.  19   , however, the orientations of the diffraction gratings in the OPE regions  1950   a ,  1950   b  may need to be adjusted in order to accomplish this aim. Specifically, since the grating vectors G 1 , G −1  associated with the ICG region  1940  may no longer terminate at the midpoint of the k-space annulus in the 3 o&#39;clock and 9 o&#39;clock positions, the magnitudes and directions of the grating vectors associated with the OPE regions may need to be adjusted in order to translate the FOV rectangles to a location at the 6 o&#39;clock position (e.g., one which is centered in the k-space annulus in the k y -direction). These adjustments can be accomplished by altering the orientations of the grating lines in the OPE regions  1950   a ,  1950   b  and/or by changing their grating periods, A, in comparison to the OPE regions in an embodiment without an expanded FOV. 
     The shaded right-hand portion of the FOV rectangles in KSD 3   a  represents a first sub-portion of the FOV, while the shaded left-hand portion of the FOV rectangles in KSD 3   b  represents a second sub-portion of the FOV. In the illustrated embodiment, these FOV sub-portions overlap in the central region of the FOV rectangles. 
     K-space diagram KSD 3   a  illustrates that when the FOV rectangle located at the 9 o&#39;clock position is translated to the 6 o&#39;clock position, only the beams corresponding to the shaded right-hand region of the FOV rectangle are present. K-space diagram KSD 3   b  shows the same phenomenon except that the absent beams are the ones whose k-vectors are located on the opposite side of the FOV rectangle. Finally, k-space diagram KSD 4  shows that when the two truncated FOV rectangles are superimposed at the 6 o&#39;clock position of the k-space annulus, the unshaded portions of the FOV rectangle are filled in, meaning that all of the beams which make up the complete FOV of the input image are now present and can be projected out of the eyepiece waveguide  1900  toward the user&#39;s eye by the diffraction grating in the EPE region  1960 . Similar to the embodiment in  FIG.  18 A , the EPE region  1960  translates the FOV rectangle back to the origin in k-space diagram KSD 4 . Importantly, the two truncated FOV rectangles from the 9 o&#39;clock and 3 o&#39;clock positions should be translated to the 6 o&#39;clock position in such a manner as to maintain the relative positions of the shaded regions within the original FOV rectangle. This ensures that the beams of light in each sub-portion of the FOV have the correct propagation angles so as to re-create the original image. 
     What this means in physical terms is that the eyepiece waveguide  1900  divides the image field of view into multiple parts. The light beams corresponding to each of these parts of the image field of view propagate through the eyepiece waveguide  1900  along different paths, where they may be replicated in a spatially distributed manner by different OPE regions  1950   a ,  1950   b . And ultimately the separate parts of the image field of view are recombined in the EPE region  1960  to be projected toward the user&#39;s eye. 
     In some embodiments, the various diffraction gratings of the eyepiece  1900  can be designed such that there is overlap between the subsets of beams which are supplied to the EPE region  1960  by the respective OPE regions  1950   a ,  1950   b . In other embodiments, however, the diffraction gratings can be designed such that each OPE region  1950   a ,  1950   b  supplies a unique subset of the beams which are required to fully re-create the input image. 
     Example AR Eyepiece Waveguides with Expanded Field of View and Overlapping MPE and EPE Regions 
     While  FIG.  19    illustrates an embodiment of an eyepiece waveguide with an expanded FOV which uses OPE regions to replicate the input beams, other embodiments can advantageously use MPE regions.  FIGS.  20 A- 20 L  illustrate one such example embodiment. 
       FIG.  20 A  illustrates an embodiment of an expanded FOV eyepiece waveguide  2000  with an MPE region  2050  which is overlapped by an EPE region  2060 . The eyepiece waveguide  2000  can achieve an expanded field of view which can be larger than the range of propagation angles that can be supported in guided propagation modes in the thickness direction of the waveguide. The eyepiece waveguide  2000  has a first surface  2000   a  and a second surface  2000   b . As discussed further below, different diffractive features can be formed on or in the opposite surfaces  2000   a ,  2000   b  of the eyepiece waveguide  2000 . The two surfaces  2000   a ,  2000   b  of the eyepiece waveguide  2000  are illustrated in  FIG.  20 A  as being displaced in the x-y plane with respect to one another. However, this is only for purposes of illustration to be able to show the different diffractive features formed on or in each surface; it should be understood that the first surface  2000   a  and the second surface  2000   b  are aligned with one another in the x-y plane. In addition, while the MPE region  2050  and the EPE region  2060  are illustrated as being the same size and exactly aligned in the x-y plane, in other embodiments they may have somewhat different sizes and may be partially misaligned. In some embodiments, the MPE region  2050  and the EPE region  2060  overlap one another by at least 70%, at least 80%, at least 90%, or at least 95%. 
     The eyepiece waveguide  2000  includes an ICG region  2040 , an MPE region  2050 , and an EPE region  2060 . The ICG region  2040  receives a set of input beams from a projector device. As described elsewhere herein, the input beams can propagate from the projector device through free space generally in the z-direction until they are incident upon the ICG region  2040 . The ICG region  2040  diffracts those input beams so that they all, or at least some, enter guided propagation modes within the eyepiece waveguide  2000 . The grating lines of the ICG region  2040  can be oriented so as to direct the diffracted beams in the −y-direction toward the MPE region  2050 . 
     The MPE region  2050  can include a plurality of diffractive features which exhibit periodicity along multiple axes. The MPE region  2050  may be composed of an array of scattering features arranged in a 2D lattice. The individual scattering features can be, for example, indentations or protrusions of any shape. The 2D array of scattering features has associated grating vectors, which are derived from the reciprocal lattice of that 2D lattice. As one example, the MPE region  2050  could be a 2D diffraction grating composed of a crossed grating with grating lines that repeat along two or more directions of periodicity. The diffractive features which make up the MPE region  2050  can have a relatively low diffractive efficiency (e.g., 10% or less). As discussed herein, this allows beams of light to be replicated in a spatially distributed manner in multiple directions as they propagate through the MPE region  2050 . 
       FIG.  20 B  illustrates a portion of an example 2D grating, along with its associated grating vectors, which can be used in the MPE region  2050  of the eyepiece waveguide  2000 . A crossed grating is illustrated, though the 2D periodic grating could instead be made up of individual scattering features located at, for example, the intersection points of the illustrated grating lines. The 2D grating has a first set of grating lines  2056  which repeat along a first direction of periodicity. These grating lines  2056  have an associated fundamental grating vector G which points along the direction of periodicity of the first set of grating lines  2056  and has a magnitude equal to 2π/a, where a is the period of the first set of grating lines  2056 . The 2D grating shown in  FIG.  20 B  is also associated with harmonics of the first fundamental grating vector G. These include −G and higher-order harmonics, such as 2G, −2G, etc. The 2D grating in the MPE region  2050  also has a second set of grating lines  2057  which repeat along a second direction of periodicity. In some embodiments, the first and second directions of periodicity are not perpendicular. The second set of grating lines  2057  have an associated fundamental grating vector H which points along the direction of periodicity of the second set of grating lines, with a magnitude equal to 2π/b, where b is the period of the second set of grating lines  2057 . The 2D grating shown in  FIG.  20 B  is also associated with harmonics of the second fundamental grating vector H. These include −H and higher-order harmonics, such as 2H, −2H, etc. Finally, any 2D array of diffractive features will also have associated grating vectors which point in directions determined by integer linear combinations (superpositions) of the basis grating vectors, G and H. In the illustrated embodiment, these superpositions result in additional grating vectors which are also shown in  FIG.  20 B . These include, for example, −G, −H, H+G, H−G, G−H, and −(H+G). Although  FIG.  20 B  only illustrates the first order grating vectors, and their superpositions, associated with the 2D diffraction grating, higher-order grating vectors may also exist. 
       FIG.  20 C  is a k-space diagram, KSD 1 , which illustrates the k-space operation of the ICG region  2040  of the eyepiece waveguide  2000 . The FOV rectangle centered at the origin of KSD 1  represents the set of input beams which are projected toward the ICG region  2040  by a projector device. The dimension of the FOV rectangle in the k x -direction represents the FOV of the input beams in the x-direction, while the dimension of the FOV rectangle in the k y -direction represents the FOV of the input beams in the y-direction. As illustrated, in this particular embodiment, the k x  dimension of the FOV rectangle is larger than the width of the k-space annulus. 
     Since the MPE region  2050  is located in the −y-direction from the ICG region  2040  according to the physical layout of the eyepiece waveguide  2000  shown in  FIG.  20 A , the diffraction grating in the ICG region  2040  can be designed so as to diffract input beams in that direction. Thus, KSD 1  in  FIG.  20 C  shows that the ICG region  2040  translates the FOV rectangle from the origin of the k-space diagram to a location on the −k y -axis at the 6 o&#39;clock position in the k-space annulus. At this particular position, the wider dimension of the FOV rectangle is oriented in the azimuthal direction of the k-space annulus and so the FOV rectangle fits entirely within the annulus. This means that all of the beams represented by the FOV rectangle enter guided propagation modes within the eyepiece waveguide  2000  and propagate generally in the −y-direction toward the MPE region  2050 . 
     Just as in other MPE regions discussed herein (e.g.,  1650 ,  1750 ), the MPE region  2050  expands the image pupil in multiple directions by replicating the input beams in a spatially distributed manner as they propagate through it.  FIGS.  20 D- 20 F and  20 H  illustrate this behavior of the MPE region  2050  in k-space. 
       FIG.  20 D  is a k-space diagram, KSD 2 , which illustrates part of the k-space operation of the MPE region  2050  of the eyepiece waveguide  2000 . The k-space diagram includes a shaded FOV rectangle located at the 6 o&#39;clock position of the k-space annulus. This is the location of the FOV rectangle after the ICG region  2040  has coupled the input beams into the eyepiece waveguide  2000  and diffracted them toward the MPE region  2050 .  FIG.  20 D  shows how the 2D grating in the MPE region  2050  translates the FOV rectangle using the grating vectors shown in  FIG.  20 B . Since there are eight grating vectors, the MPE region  2050  attempts to translate the FOV rectangle from the 6 o&#39;clock position in the k-space annulus to eight possible new locations in the k-space diagram. Of these eight possible locations, five fall completely outside the outer periphery of the k-space diagram. These locations are illustrated with unshaded FOV rectangles. Since k-vectors outside the outer periphery of the k-space diagram are not permitted, none of those five grating vectors results in diffraction. There are, however, three grating vectors (i.e., G, −H, and G−H) which do result in translations of the FOV rectangle to new positions at least partially within the bounds of the k-space diagram. One of these locations is at the 9 o&#39;clock position in the k-space annulus, another is at the 12 o&#39;clock position, and the last is at the 3 o&#39;clock position. Since k-vectors at these locations are permitted and do result in guided propagation modes, the FOV rectangles at these locations are shaded to indicate that beams of light are diffracted into those three states. 
     In the case of the 9 o&#39;clock and 3 o&#39;clock positions in the k-space annulus, the translated FOV rectangles do not fit completely within the annulus because their k x  dimension is larger than the width of the annulus. Thus, the translated FOV rectangles at these locations are truncated, meaning that the beams whose k-vectors fall outside the outer periphery of the k-space diagram are not guided. This is represented in KSD 2  by the unshaded portions of the translated FOV rectangles at the 9 o&#39;clock in 3 o&#39;clock positions. This means that the set of beams which are spreading through the MPE region  2050  in the +x and the −x directions, respectively, do not each include all of the original set of input beams. The set of beams propagating through the MPE region  2050  in the +x direction are missing the beams corresponding to the right-hand side of the FOV rectangle, while the set of beams propagating in the −x direction are missing the beams corresponding to the left-hand side of the FOV rectangle. Collectively, however, all of the beams which make up the FOV are still present. 
     The shaded right-hand portion of the translated FOV rectangle at the 9 o&#39;clock position represents a first sub-portion of the FOV, while the shaded left-hand portion of the FOV rectangle at the 3 o&#39;clock position represents a second sub-portion of the FOV. In the illustrated embodiment, these FOV sub-portions overlap in the central region of the FOV rectangles (though overlap is not necessarily required). 
     As already mentioned, in some embodiments the first and second axes of periodicity in the 2D grating of the MPE region  2050  are not orthogonal. This in turn means that the fundamental grating vectors G and H are likewise not orthogonal. This can allow the 2D grating in the MPE region  2050  to translate the FOV rectangles at the 3 o&#39;clock and 9 o&#39;clock positions such that the centers of those rectangles lie beyond the midpoint of the k-space annulus, whereas the centers of the FOV rectangles at the 6 o&#39;clock and 12 o&#39;clock positions can be located at, or closer to, the midpoint of the annulus. As a result, the translated FOV rectangles at the 3 o&#39;clock and 9 o&#39;clock positions are truncated, which results in the FOV being divided into first and second sub-portions. This is noteworthy in the illustrated embodiment because dividing the FOV into first and second sub-portions is part of the process for increasing the FOV of the eyepiece waveguide  2000 . 
       FIG.  20 E  is a k-space diagram, KSD 3 , which illustrates another part of the k-space operation of the MPE region  2050  of the eyepiece waveguide  2000 . KSD 3  includes a partially shaded FOV rectangle located at the 3 o&#39;clock position of the k-space annulus. This is the location of one of the translated FOV rectangles after a first interaction within the MPE region  2050 .  FIG.  20 E  shows how, during subsequent interactions, the 2D grating in the MPE region  2050  translates this FOV rectangle using the grating vectors shown in  FIG.  20 B . Once again, since there are eight grating vectors, the MPE region  2050  attempts to translate the FOV rectangle from the 3 o&#39;clock position in the k-space annulus to eight possible new locations in the k-space diagram. Of these eight possible locations, five again fall outside the outer periphery of the k-space diagram. These locations are illustrated with unshaded FOV rectangles. Since k-vectors outside the outer periphery of the k-space diagram are not permitted, none of those five grating vectors results in diffraction. There are, however, three grating vectors (i.e., G, H, and H+G) which do result in translations of the FOV rectangle to new positions at least partially within the bounds of the k-space diagram. One of these locations is at the 9 o&#39;clock position in the k-space annulus, another is at the 12 o&#39;clock position, and the last is back at the 6 o&#39;clock position. Since k-vectors at these locations are permitted and do result in guided propagation modes, the FOV rectangles at these locations are shaded to indicate that beams of light are diffracted into those three states (or zero-order diffracted beams can remain in the propagation state represented by the FOV rectangle at the 3 o&#39;clock position). 
     As shown in you  FIG.  20 E , the translated FOV rectangle at the 3 o&#39;clock position of the k-space annulus had already been truncated as a result of the first diffraction interaction in the MPE region  2050  which is shown in  FIG.  20 D . Thus, only the truncated FOV rectangle is translated to the 9 o&#39;clock, 12 o&#39;clock, and 6 o&#39;clock positions of the k-space annulus. In the case of the 9 o&#39;clock position, the FOV rectangle is further truncated, meaning that only the beams corresponding to the central shaded portion of that particular translated FOV rectangle are actually diffracted to this state. 
       FIG.  20 F  is similar to  FIG.  20 E , except that it shows the k-space operation of the MPE region  2050  on the FOV rectangle from  FIG.  20 D  which was translated to the 9 o&#39;clock position (instead of the 3 o&#39;clock position, as illustrated in  FIG.  20 E ). The operation of the MPE region  2050  on the beams in this state is a mirror image (about the k y -axis) of what is shown in  FIG.  20 E . 
     Although not illustrated, a similar k-space diagram could be drawn to illustrate the k-space operation of the MPE region  2050  on beams of light traveling with the propagation angles indicated by the FOV rectangle located at the 12 o&#39;clock position of the k-space annulus. That k-space diagram would show that the 2D diffraction grating in the MPE region  2050  would diffract those beams into the states represented by the FOV rectangles at the 3 o&#39;clock, 6 o&#39;clock, and 9 o&#39;clock positions in the annulus of the k-space diagrams in  FIGS.  20 D,  20 E, and  20 F . 
     As shown by the k-space diagrams in  FIGS.  20 D- 20 F , when the diffracted light beams from the ICG region  2040  arrive at the MPE region  2050 , many replicated beams are formed in a spatially distributed manner. And all of these replicated beams propagate in one of the directions indicated by the FOV rectangles at the 3 o&#39;clock, 6 o&#39;clock, 9 o&#39;clock, and 12 o&#39;clock positions in the k-space annulus. Light beams propagating through the MPE region  2050  may undergo any number of interactions with the diffractive features of the MPE region, resulting in any number of changes in the direction of propagation. In this way, the light beams are replicated throughout the MPE region  2050  along both the x-direction and the y-direction. This is represented by the arrows in the MPE region  2050  of the eyepiece waveguide  2000  in  FIG.  20 A   
     Since the EPE region  2060  overlaps the MPE region  2050  within the x-y plane of the eyepiece waveguide  2000 , the replicated light beams also interact with the EPE region  2060  as they spread through the waveguide, reflecting back and forth between the first surface  2000   a  and the second surface  2000   b  via total internal reflection. When one of the light beams interacts with the EPE region  2060 , a portion of its power is diffracted and exits the eyepiece waveguide toward the user&#39;s eye, as shown by the arrows in the EPE region  2060  of the eyepiece waveguide  2000  in  FIG.  20 A . 
     In some embodiments, the EPE region  2060  includes a diffraction grating whose lines are oriented perpendicularly with respect to the lines of the diffraction grating which makes up the ICG region  2040 . An example of this is shown in  FIG.  20 A , where the ICG region  2040  has grating lines which extend in the x-direction, and periodically repeat in the y-direction, whereas the EPE region  2060  has grating lines which extend in the y-direction, and periodically repeat in the x-direction. It is advantageous that the grating lines in the EPE region  2060  are oriented perpendicularly with respect to the grating lines in the ICG region  2040  because this helps to ensure that the light beams will interact with the MPE region  2050  before being coupled out of the eyepiece waveguide  2000  by the EPE region  2060 . This behavior is shown in k-space in  FIG.  20 G . 
       FIG.  20 G  is a k-space diagram, KSD 5 , which illustrates the k-space operation of the EPE region  2060  in the eyepiece waveguide  2000  shown in  FIG.  20 A . As already discussed, beams of light propagate through the MPE region  2050  in all of the directions indicated by the FOV rectangles located at the 12 o&#39;clock, 3 o&#39;clock, 6 o&#39;clock, and 9 o&#39;clock positions of the k-space annulus. And since the EPE region  2060  physically overlaps the MPE region  2050 , beams of light in all of these propagation states come into contact with the diffraction grating in the EPE region while spreading through the MPE region. 
     Since the axis of periodicity of the diffraction grating in the EPE region  2060  points in the ±k x -direction, the grating vectors associated with the EPE region likewise point in the same direction.  FIG.  20 G  shows how the EPE region  2060  attempts to translate the FOV rectangles at the 12 o&#39;clock, 3 o&#39;clock, 6 o&#39;clock, and 9 o&#39;clock positions using these grating vectors. Due to their orientation in the ±k x -direction, the grating vectors associated with the EPE region  2060  can only translate the FOV rectangles located at the 3 o&#39;clock and 6 o&#39;clock positions of the k-space annulus back to the origin of the k-space diagram. Thus, the EPE region  2060  can only out-couple beams of light which are in either of those two propagation states; the EPE region does not out couple beams of light which are propagating in the states corresponding to the FOV rectangles at the 12 o&#39;clock and 6 o&#39;clock positions of the k-space annulus. 
     It is important to note that if the axis of periodicity for the grating lines in the EPE region  2060  were parallel with, rather than perpendicular to, the axis of periodicity for the grating lines in the ICG region  2040 , then the grating vectors associated with the EPE region would point in the ±k y -direction. This would in turn allow light beams in the propagation states corresponding to the FOV rectangles at the 12 o&#39;clock and 6 o&#39;clock positions of the k-space annulus to be out coupled by the EPE region. Since input beams arrive at the MPE/EPE regions in the propagation state which corresponds to the 6 o&#39;clock position, this would mean that light beams could be out-coupled by the EPE region  2060  before interacting with, and being spread by, the MPE region  2050 , which would typically be undesirable. The fact that the axis of periodicity for the grating lines in the EPE region  2060  is perpendicular to that of the ICG region  2040  means that light beams will typically need to undergo at least one change of direction, and possibly many more, within the MPE region before being out coupled. This allows for enhanced spreading of the light beams within the MPE region  2050 . 
       FIG.  20 H  is a k-space diagram, KSD 6 , which summarizes the k-space operation of the eyepiece waveguide  2000  shown in  FIG.  20 A . It is essentially a superposition of the k-space diagrams shown in  FIGS.  20 C- 20 G . Again, the k-space diagram in  FIG.  20 H  shows FOV rectangles having at least one dimension that is larger than the width of the k-space annulus. In some embodiments, at least one dimension of the FOV rectangles can be up to approximately 2 times larger than the width of the k-space annulus. In the illustrated embodiment, the horizontal dimension of the FOV rectangles is larger than the width of the k-space annulus, but the same techniques can also be used to expand the vertical field of view. 
     KSD 6  includes an FOV rectangle centered at the origin of the diagram. Once again, this location of the FOV rectangle can describe either the input beams being projected into the eyepiece waveguide  2000  or the replicated output beams being projected out of the waveguide toward the user&#39;s eye. In the illustrated embodiment, the operation of the ICG region  2040  in k-space is to translate the FOV rectangle from the center of the k-space diagram down to the 6 o&#39;clock position. As illustrated, the ICG region  2040  can be designed such that one of its grating vectors is oriented in the −k y -direction. This causes the diffracted beams to propagate in the −y-direction toward the MPE region  2050 . Further, the ICG region  2040  can be designed such that the magnitude of its grating vectors causes the FOV rectangle to be copied to a position where it fits completely within the k-space annulus at the 6 o&#39;clock position. This can be done by, for example, designing the ICG region  2040  with a pitch such that the magnitude of its first-order grating vectors is equal to the distance from the origin of the k-space diagram to the midpoint of the k-space annulus. Since the FOV rectangle at the 6 o&#39;clock position lies completely within the k-space annulus, all of the diffracted beams enter guided modes of propagation. 
     As already discussed, the MPE region includes a plurality of diffractive features which exhibit periodicity along multiple different axes. This means that the MPE region has multiple associated grating vectors which can translate the FOV rectangle from the 6 o&#39;clock position to any of the 9 o&#39;clock, 12 o&#39;clock, and 3 o&#39;clock positions. During additional interactions with the MPE region  2050 , the FOV rectangles can be translated back and forth between any of the 12 o&#39;clock, 3 o&#39;clock, 6 o&#39;clock, and 9 o&#39;clock positions. This is represented by the double-sided arrows between those propagation states. As shown in  FIG.  20 H , the FOV rectangles at the 3 o&#39;clock and 6 o&#39;clock positions of the k-space annulus are truncated, meaning that not all of the beams of light associated with the full FOV are present in each of those propagation states. However, when those sub-portions of the FOV are considered collectively, all of the beams of light which make up the full FOV are present. Thus, when the FOV rectangles are eventually translated from the 3 o&#39;clock or 6 o&#39;clock position back to the origin of the k-space diagram, so as to out-couple the beams of light toward the user&#39;s eye, all of the beams which are required to make up the full FOV of the input image are present and are projected from the eyepiece waveguide  2000 . 
       FIG.  20 I  is a diagram which illustrates how beams of light spread through the eyepiece waveguide  2000  shown in  FIG.  20 A . A guided beam which enters the MPE region  2050  propagating in the −y-direction from the ICG region  2040  is replicated into many beams in a spatially distributed manner, some traveling in the ±y-directions (corresponding to the FOV rectangles at the 6 o&#39;clock and 12 o&#39;clock positions in the k-space annulus), and some traveling in the ±x-directions (corresponding to the FOV rectangles at the 3 o&#39;clock and 9 o&#39;clock positions in the k-space annulus). In this way, light beams spread laterally throughout the entire eyepiece waveguide  2000 . 
       FIG.  20 J  illustrates how the diffractive efficiency of the MPE region  2050  in the eyepiece waveguide  2000  can be spatially varied so as to enhance the uniformity of luminance in the waveguide. In the figure, darker shades within the MPE region  2050  represent higher diffractive efficiency, while lighter shades represent lower diffractive efficiency. The spatial variation in the diffractive efficiency of the MPE region  2050  can be accomplished by introducing spatial variation in grating characteristics, such as grating depth, duty cycle, blaze angle, slant angle, etc. 
     As seen in  FIG.  20 J , the uniformity of the luminance in the waveguide can be enhanced by designing portions of the MPE region  2050  which are closer to the ICG region  2040  to have higher diffractive efficiency. Since this is where light beams enter the MPE region  2050  from the ICG region  2040 , more light is present in this area and therefore diffractive efficiency can be higher here so as to more effectively spread the light to other portions of the MPE region  2050  where there is less light. In addition, or alternatively, multiple ICG regions can be provided at various angular locations around the periphery of the MPE region  2050  so as to input light at more locations and thereby improve uniformity of luminance in the waveguide. 
     The uniformity of the luminance can also be enhanced by designing the central portion of the MPE region  2050 , along the direction in which beams propagate from the ICG region  2040  into the MPE region  2050 , to have higher diffractive efficiency. Once again, more light is present in this area of the MPE region  2050  because it is located along the axis where the ICG region  2040  inputs light. Since there is more light present in this area, the diffractive efficiency can be higher so as to more effectively spread the light to other parts of the MPE region  2050 . 
       FIG.  20 K  illustrates how the diffractive efficiency of the EPE region  2060  in the eyepiece waveguide  2000  can be spatially varied so as to enhance the uniformity of luminance in the waveguide. Darker shades within the EPE region  2060  once again represent higher diffractive efficiency, while lighter shades represent lower diffractive efficiency. The EPE region  2060  can be designed to have higher diffractive efficiency in peripheral areas. The higher diffractive efficiency in the peripheral areas of the EPE region  2060  helps to out-couple light to the user&#39;s eye before the light is lost out of the edge of the waveguide. 
       FIG.  20 L  illustrates an embodiment of the eyepiece waveguide  2000  which includes one or more diffractive mirrors  2070  around the peripheral edge of the waveguide. The diffractive mirrors  2070  can receive light which propagates through the MPE/EPE regions and exits from the edge of the waveguide  2000 . The diffractive mirrors can then diffract that light back into the MPE/EPE regions so that it can be used to contribute to projection of the image from the eyepiece waveguide  2000 . As already discussed, the MPE region  2050  permits propagation of beams in four general directions: generally in the x-direction (i.e., as represented by the FOV rectangle at the 3 o&#39;clock position of the k-space annulus; generally in the −x-direction (i.e., as represented by the FOV rectangle at the 9 o&#39;clock position); generally in the y-direction (i.e., as represented by the FOV rectangle at the 12 o&#39;clock position); and generally in the −y-direction (i.e., as represented by the FOV rectangle at the 6 o&#39;clock position). The diffractive mirrors  2070  can be designed to diffract beams into one of these same propagation states. 
     For example, the diffraction mirror  2070  on the left side of the eyepiece waveguide  2000  can diffract beams which are incident generally from the −x-direction into the propagation state represented by the FOV rectangle at the 3 o&#39;clock position such that they travel back through the OPE region  2050  generally in the x-direction. Similarly, the diffraction mirror  2070  on the bottom of the eyepiece waveguide  2010  can diffract beams which are incident generally from the −y-direction into the propagation state represented by the FOV rectangle at the 12 o&#39;clock position such that they travel back through the OPE region  2050  generally in the y-direction. 
       FIG.  20 L  illustrates the k-space operation of the bottom diffractive mirror  2070 . As shown in the k-space diagram, the bottom diffractive mirror  2070  can be designed with a period that is half that of the grating in the ICG region  2040 . This finer period results in the bottom diffractive mirror having an associated grating vector which is twice as long as that of the ICG region  2040 . Accordingly, the bottom diffractive mirror can translate the FOV rectangle from the 6 o&#39;clock position in the k-space annulus to the 12 o&#39;clock position. Although illustrated with respect to the eyepiece waveguide  2000 , the same techniques (i.e., spatial variation in diffractive efficiency of an OPE, MPE, EPE region etc., and the usage of diffractive mirrors along peripheral edges) can also be used with any of the other embodiments described herein. 
       FIG.  20 M  illustrates an example embodiment of eyeglasses  70  which incorporate one or more instances of the eyepiece waveguide  2000 . A first instance of the eyepiece waveguide  2000  is integrated into the left viewing portion of the eyeglasses  70 , while a second instance of the eyepiece waveguide  2000  is integrated into the right viewing portion. In the illustrated embodiment, each of the waveguides  2000  is about 50×30 mm 2 , though many different sizes can be used. Each waveguide  2000  can be accompanied by a separate projector  2020  which projects images into the corresponding waveguide. Assuming that the eyepiece waveguide is made of a material with a refractive index of 1.8, some embodiments of the eyepiece waveguide  2000  are able to achieve an FOV of as much as 90° by 45°, though some embodiments of the eyepiece waveguide may be designed for smaller FOVs of ˜60° by 45° so as to satisfy typical design constraints of eyebox volume—it may be advantageous to send some portion of the FOV to both sides of the eyepiece waveguide to provide an adequately-sized eyebox—and to avoid screen door artifacts resulting from sparsely spaced output beams. 
       FIG.  20 N  illustrates another example embodiment of eyeglasses  70  which incorporate one or more instances of the eyepiece waveguide  2000 . This embodiment of the eyeglasses  70  is similar to that which is shown in  FIG.  20 M  except that the orientation of the waveguides  2000  and accompanying projectors  2020  have been rotated 90° towards the temples of the eyeglasses  70 . In this configuration, some embodiments of the eyepiece waveguide  2000  are able to achieve an FOV of as much as 45° by 90°, assuming that the eyepiece waveguide is made of a material with a refractive index of 1.8, though some embodiments may be designed for smaller FOVs of ˜45° by 60° to satisfy other design constraints. 
       FIG.  21 A  illustrates another embodiment of an eyepiece waveguide  2100  with an MPE region  2150  which is overlapped by an EPE region  2160 . Similar to the eyepiece waveguide  2000  shown in  FIG.  20 A , the eyepiece waveguide  2100  shown in  FIG.  21 A  can achieve an expanded field of view which can be larger than the range of propagation angles that can be supported in guided propagation modes in the thickness direction of the waveguide. The eyepiece waveguide  2100  has a first surface  2100   a  and a second surface  2100   b . As discussed further below, different diffractive features can be formed on or in the opposite surfaces  2100   a ,  2100   b  of the eyepiece waveguide  2100 . The two surfaces  2100   a ,  2100   b  of the eyepiece waveguide  2100  are illustrated in  FIG.  21 A  as being displaced in the x-y plane with respect to one another. However, this is only for purposes of illustration to be able to show the different diffractive features formed on or in each surface; it should be understood that the first surface  2100   a  and the second surface  2100   b  are aligned with one another in the x-y plane. In addition, while the MPE region  2150  and the EPE region  2160  are illustrated as being the same size and exactly aligned in the x-y plane, in other embodiments they may have somewhat different sizes and may be partially misaligned. In some embodiments, the MPE region  2150  and the EPE region  2160  overlap one another by at least 70%, at least 80%, at least 90%, or at least 95%. 
     Like the eyepiece waveguide  2000  shown in  FIG.  20 A , the eyepiece waveguide  2100  shown in  FIG.  21 A  includes an MPE region  2150  and an EPE region  2160 . Unlike the eyepiece waveguide  2000  shown in  FIG.  20 A , the eyepiece waveguide  2100  shown in  FIG.  21 A  includes two ICG regions  2140   a ,  2140   b , rather than a single ICG region, located on opposite sides of the MPE/EPE regions. Each of the ICG regions  2140   a ,  2140   b  can have its own associated projector. The two projectors can each input a sub-portion of the complete input image FOV into the eyepiece waveguide  2100 . Accordingly, each of the ICG regions  2140   a ,  2140   b  can likewise be used to in-couple input beams corresponding to a sub-portion of the FOV. Those sub-portions can then be combined at the exit pupil of the eyepiece waveguide  2100 . 
     The left ICG region  2140   a  receives a first set of input beams corresponding to a first sub-portion of the FOV from the first projector device, while the right ICG region  2140   b  receives a second set of input beams corresponding to a second sub-portion of the FOV from the second projector device. The first and second sub-portions of the FOV may be unique or they may partially overlap. The first set of input beams can be projected toward the left ICG region  2140   a  generally along the −z-direction but centered around an input beam which has a component of propagation in the −x-direction, while the second set of input beams can be projected toward the right ICG region  2140   b  generally along the −z-direction but centered around an input beam which has a component of propagation in the +x-direction. The left ICG region  2140   a  diffracts the first set of input beams so that at least some enter guided modes propagating in the +x-direction, and the right ICG region  2140   b  diffracts the second set of input beams so that at least some enter guided modes propagating in the −x-direction. In this way, both the first and second sets of input beams corresponding to the first and second sub-portions of the FOV are coupled into the eyepiece waveguide  2100  so that they propagate toward the MPE region  2150  located between the left and right ICG regions  2140   a ,  2140   b.    
     Similar to the eyepiece waveguide  2000  shown in  FIG.  20 A , the eyepiece waveguide  2100  shown in  FIG.  21 A  can also include an MPE region  2150  which is formed on or in a first side  2100   a  of the waveguide and an overlapping EPE region  2160  which is formed on or in the second side  2100   b  of the waveguide. The MPE region  2150  in the eyepiece waveguide  2100  shown in  FIG.  21 A  can be similar to the MPE region  2050  in the eyepiece waveguide  2000  shown in  FIG.  20 A . Namely, the MPE region  2150  can include a plurality of diffractive features which exhibit periodicity along multiple axes. Similarly, the EPE region  2160  in the eyepiece waveguide  2100  shown in  FIG.  21 A  can be similar to the EPE region  2060  in the eyepiece waveguide  2000  shown in  FIG.  20 A . Namely, the EPE region  2160  can include a diffraction grating whose axis of periodicity is orthogonal to that of the two ICG regions  2140   a ,  2140   b . The operation of the MPE region  2150  and the EPE region  2160  in  FIG.  21 A  can also be similar to that of the MPE region  2050  and the EPE region  2060  in  FIG.  20 A , as shown in  FIGS.  21 B- 21 D . 
       FIG.  21 B  is a k-space diagram, KSD 1 , which illustrates the k-space operation of the eyepiece waveguide  2100  on the first set of input beams corresponding to the first sub-portion of the FOV of an input image. The FOV rectangle centered at the origin of KSD 1  represents the beams of light which correspond to the complete input image FOV that is to be projected by the eyepiece waveguide  2100  toward the user&#39;s eye. The size of the FOV rectangle as a whole has a dimension which is up to approximately two times larger than the width of the k-space annulus. Hence, the eyepiece waveguide  2100  shown in  FIG.  21 A  is designed to have an enhanced FOV similar to the embodiments shown in  FIGS.  19  and  20 A . However, the first set of input beams which are projected toward the left ICG region  2140   a  correspond to only the shaded sub-portion of the FOV rectangle. As shown in  FIG.  21 B , the shaded portion of the FOV rectangle which corresponds to the first set of input beams is the left-hand portion of the FOV rectangle. Since the center of the shaded portion of the FOV rectangle is offset in the −k x -direction from the origin of the k-space diagram, the first set of input beams from the first projector are not centered about a beam propagating exactly in the −z-direction (which would be the case if the shaded portion of the FOV rectangle were centered about the origin of the k-space diagram) but rather about an oblique beam with a propagation component in the −x-direction. 
     The left ICG region  2140   a  can be designed such that its grating vectors are oriented in the ±k x -direction. The operation of the left ICG region  2140   a  in k-space is to translate the shaded left-hand portion of the FOV rectangle from the center of the k-space diagram to the 3 o&#39;clock position in the k-space annulus. This will cause the diffracted beams to propagate generally in the x-direction toward the MPE region  2150 . In some embodiments, the shaded left-hand portion of the FOV rectangle can constitute half of the FOV rectangle or more. And, in some embodiments, the left ICG region  2140   a  can be designed to translate the center of the FOV rectangle to any radial position from the midpoint of the k-space annulus to the outer boundary of the annulus. Further, the left ICG region  2140   a  can be designed such that the magnitude of its grating vectors causes the FOV rectangle to be copied to a position where the shaded portion fits completely within the k-space annulus at the 3 o&#39;clock position. This can be accomplished by, for example, setting the magnitude of the ICG grating vectors to be greater than the distance from the origin of the k-space diagram to the midpoint of the k-space annulus. Since the shaded portion of the FOV rectangle at the 3 o&#39;clock position lies completely within the k-space annulus, all of the first set of input beams corresponding to the first sub-portion of the FOV enter guided modes of propagation. Although the FOV rectangle at the 3 o&#39;clock position of the k-space annulus has a right-hand portion which extends outside the annulus, this portion of the FOV rectangle corresponds to input beams which are not necessarily part of the first sub-portion of the FOV provided to the left ICG region  2140   a  by its associated projector. 
     Although the left ICG region  2140   a  can also diffract a portion of the first set of input beams in the opposite direction (i.e., translation of the FOV rectangle to the 9 o&#39;clock position of the k-space annulus), in the illustrated embodiment of the eyepiece waveguide  2100  those particular diffracted beams would simply exit out the edge of the waveguide. 
     The MPE region  2150  includes a plurality of diffractive features which have multiple axes of periodicity. In some embodiments, the MPE region  2150  can be similar to the MPE region  2050  illustrated and discussed with respect to  FIGS.  20 A- 20 M . For example, the MPE region  2150  can have multiple associated grating vectors which can translate the FOV rectangle from the 3 o&#39;clock position to any of the 6 o&#39;clock, 9 o&#39;clock, and 12 o&#39;clock positions of the k-space annulus. As shown in  FIG.  21 B , the shaded portion of the FOV rectangle at the 9 o&#39;clock position of the k-space annulus is truncated, meaning that not all of the beams of light associated with the first sub-portion of the FOV are necessarily present in that particular propagation state. 
     During additional interactions with the MPE region  2150 , the FOV rectangles can be translated back and forth between any of the 12 o&#39;clock, 3 o&#39;clock, 6 o&#39;clock, and 9 o&#39;clock positions. This is represented by the double-sided arrows between those propagation states in KSD 1 . In this way, the first set of input beams can be replicated throughout the MPE region  2150  by undergoing multiple interactions with its diffractive features, as described herein. This is shown by the arrows in the OPE region  2150  of the eyepiece waveguide  2100  in  FIG.  21 A . 
     Since the EPE region  2160  overlaps the MPE region  2150  within the x-y plane of the eyepiece waveguide  2100 , the replicated light beams also interact with the EPE region  2160  as they spread through the waveguide, reflecting back and forth between the first surface  2100   a  and the second surface  2100   b  via total internal reflection. Each time one of the replicated light beams interacts with the EPE region  2160 , a portion of its power is diffracted and out-coupled toward the user&#39;s eye, as shown by the arrows in the EPE region  2160  of the eyepiece waveguide  2100  in  FIG.  21 A . 
     In some embodiments, the EPE region  2160  includes a diffraction grating whose lines are oriented perpendicularly with respect to the lines of the diffraction grating which makes up the ICG regions  2140   a ,  2140   b . In this particular example, since the ICG regions  2140   a ,  2140   b  have grating lines which extend in the y-direction, and periodically repeat in the x-direction, the EPE region  2160  has grating lines which extend in the x-direction, and periodically repeat in the y-direction. Once again, it is advantageous that the grating lines in the EPE region  2160  are oriented perpendicularly with respect to the grating lines in the ICG regions  2140   a    2140   b  because this helps to ensure that the light beams will interact with the MPE region  2150  before being coupled out of the eyepiece waveguide  2100  by the EPE region  2160 . 
       FIG.  21 B  also illustrates the k-space operation of the EPE region  2160  on the first set of beams corresponding to the first sub-portion of the FOV. As already discussed, beams of light can propagate through the MPE region  2150  in any of the directions indicated by the FOV rectangles located at the 12 o&#39;clock, 3 o&#39;clock, 6 o&#39;clock, and 9 o&#39;clock positions of the k-space annulus. And since the EPE region  2160  overlaps the MPE region  2150 , beams of light in any of these propagation states can interact with the EPE region and be out-coupled from the eyepiece waveguide  2100 . Since the axes of periodicity of the diffraction grating in the EPE region  2160  point in the ±k y -direction, the grating vectors associated with the EPE region likewise point in the same direction.  FIG.  21 B  shows how the EPE region  2160  therefore translates the FOV rectangles located at the 12 o&#39;clock and 6 o&#39;clock positions of the k-space annulus back to the origin of the k-space diagram. Thus, the EPE region  2160  can only out couple beams of light which are in either of those two propagation states. As shown in  FIG.  21 B , when the FOV rectangles are eventually translated back to the center of the k-space diagram KSD 1 , all of the first set of beams which make up the first sub-portion of the FOV are present and are projected toward the user&#39;s eye. 
       FIG.  21 C  is a k-space diagram, KSD 2 , which illustrates the k-space operation of the eyepiece waveguide  2100  on the second set of input beams corresponding to the second sub-portion of the FOV of the input image. Once again, the FOV rectangle centered at the origin of KSD 2  represents the beams of light which correspond to the complete input image that is to be projected by the eyepiece waveguide  2100  toward the user&#39;s eye. However, the second set of input beams which are projected toward the right ICG region  2140   b  correspond to only the shaded sub-portion of the FOV rectangle. As shown in  FIG.  21 C , the shaded portion of the FOV rectangle which corresponds to the second set of input beams is the right-hand portion of the FOV rectangle. Since the center of the shaded portion of the FOV rectangle is offset in the +k x -direction from the origin of the k-space diagram, the second set of input beams from the second projector are not centered about a beam propagating exactly in the −z-direction (which would be the case if the shaded portion of the FOV rectangle were centered about the origin of the k-space diagram) but rather about an oblique beam with a propagation component in the +x-direction. 
     In the illustrated embodiment, the operation of the right ICG region  2140   b  in k-space is to translate the right-hand shaded portion of the FOV rectangle from the center of the k-space diagram to the 9 o&#39;clock position. As illustrated, the right ICG region  2140   b  can be designed such that its grating vectors are oriented in the ±k x -direction. This will cause some of the diffracted beams to propagate in the −x-direction toward the MPE region  2150 . In some embodiments, the shaded right-hand portion of the FOV rectangle can constitute half of the FOV rectangle or more. And, in some embodiments, the right ICG region  2140   b  can be designed to translate the center of the FOV rectangle to any radial position from the midpoint of the k-space annulus to the outer boundary of the annulus. Further, the right ICG region  2140   b  can be designed such that the magnitude of its grating vectors causes the FOV rectangle to be copied to a position where the shaded portion fits completely within the k-space annulus at the 9 o&#39;clock position. This can be done by, for example, designing the ICG such that the magnitude of its grating vectors is greater than the distance from the origin of the k-space diagram to the midpoint of the k-space annulus. Since the shaded portion of the FOV rectangle at the 9 o&#39;clock position lies completely within the k-space annulus, all of the second set of input beams corresponding to the second sub-portion of the FOV enter guided modes of propagation. Although the FOV rectangle at the 9 o&#39;clock position of the k-space annulus has a left-hand portion which extends outside the annulus, this portion of the FOV rectangle corresponds to input beams which are not necessarily part of the second sub-portion of the FOV which are projected into the right ICG region  2140   b  by its associated projector. 
     Although the right ICG region  2140   b  can also diffract a portion of the second set of input beams in the opposite direction (i.e., translation of the FOV rectangle to the 3 o&#39;clock position of the k-space annulus), in the illustrated embodiment of the eyepiece waveguide  2100  those particular diffracted beams would simply exit out the edge of the waveguide. 
     As already discussed, the MPE region  2150  can have multiple associated grating vectors which can translate the FOV rectangle from the 9 o&#39;clock position to any of the 6 o&#39;clock, 3 o&#39;clock, and 12 o&#39;clock positions of the k-space annulus. As shown in  FIG.  21 C , the shaded portion of the FOV rectangle at the 3 o&#39;clock position of the k-space annulus is truncated, meaning that not all of the beams of light associated with the second sub-portion of the FOV are necessarily present in that particular propagation state. 
     During additional interactions with the MPE region  2150 , the FOV rectangles can be translated back and forth between any of the 12 o&#39;clock, 3 o&#39;clock, 6 o&#39;clock, and 9 o&#39;clock positions. This is represented by the double-sided arrows between those propagation states in KSD 2 . In this way, the second set of input beams can be replicated throughout the MPE region  2150  by undergoing multiple interactions with its diffractive features, as described herein. Once again, this is shown by the arrows in the OPE region  2150  of the eyepiece waveguide  2100  in  FIG.  21 A . 
       FIG.  21 C  also illustrates the k-space operation of the EPE region  2160  on the second set of beams which correspond to the second sub-portion of the FOV. As already discussed, the EPE region  2160  translates the FOV rectangles located at the 12 o&#39;clock and 6 o&#39;clock positions of the k-space annulus back to the origin of the k-space diagram. Thus, the EPE region  2160  can only out-couple beams of light which are in either of those two propagation states. As shown in  FIG.  21 C , when the FOV rectangles are eventually translated back to the center of the k-space diagram KSD 2 , all of the second set of beams which make up the second sub-portion of the FOV are present and are projected toward the user&#39;s eye. 
       FIG.  21 D  is a k-space diagram, KSD 3 , which summarizes the k-space operation of the eyepiece waveguide  2100  shown in  FIG.  21 A . It is essentially a superposition of the k-space diagrams shown in  FIGS.  21 B and  21 C . Again, the k-space diagram in  FIG.  21 D  shows FOV rectangles having at least one dimension that is larger than the width of the k-space annulus. In some embodiments, at least one dimension of the FOV rectangles can be up to approximately 2 times larger than the width of the k-space annulus. In the illustrated embodiment, the horizontal dimension of the FOV rectangles is larger than the width of the k-space annulus. Although the eyepiece waveguide  2100  is illustrated as providing an expanded horizontal field of view, the same techniques can also be used to expand the vertical field of view. 
     As shown in  FIG.  21 D , although the first and second sets of input beams are separately projected into the eyepiece waveguide  2100  using separate projectors and ICG regions  2140   a ,  2140   b , once the various FOV rectangles from the 12 o&#39;clock, 3 o&#39;clock, 6 o&#39;clock, and 9 o&#39;clock positions of the k-space annulus are translated back to the origin of the k-space diagram, and are therefore out-coupled toward the user&#39;s eye, all of the beams required to make up the complete image FOV are present. And the first and second sub-portions of the FOV are aligned in k-space with the same relative positions with respect to one another as in the complete input FOV. 
       FIG.  21 E  illustrates an example embodiment of eyeglasses  70  which incorporate one or more instances of the eyepiece waveguide  2100 .  FIG.  21 F  illustrates example FOVs corresponding to the eyeglasses  70  in  FIG.  21 E . A first instance of the eyepiece waveguide  2100  is integrated into the left viewing portion of the eyeglasses  70 , while a second instance of the eyepiece waveguide  2100  is integrated into the right viewing portion. In the illustrated embodiment, each of the eyepiece waveguides  2100  is about 50×30 mm 2 , though many different sizes can be used. Each eyepiece waveguide  2100  can be accompanied by two separate projectors  2120   a ,  2120   b  which each project sub-portions of the FOV into the corresponding waveguide, as just discussed. In some embodiments, the first projector  2120   a  for each of the waveguides  2100  can input light on the temple side of the eyepiece waveguide  2100 , while the second projector  2120   b  can input light on the nasal side of the eyepiece waveguide. For the case of an eyepiece waveguide made of a material having a refractive index of n=1.8, each of the projectors  2120   a ,  2120   b  can input a sub-portion of the FOV as large as 50° by 60°, or more depending on other design constraints such as eyebox size and screen door artifacts. And the complete FOV can be as large as 100° by 60°, or more. This is shown as the monocular eyepiece FOV configuration illustrated in  FIG.  21 F . As illustrated by matching shading, in this configuration the first projectors  2120   a  (temple side) can be used to project the nasal side of the complete FOV, and the second projectors  2120   b  (nasal side) can be used to project the temple side of the complete FOV. Note that the cross hair shows one possible pupil alignment, though others can also be used. 
     Alternatively, the two instances of the eyepiece waveguide  2100  and the eyeglasses  70  can be used jointly to provide a binocular FOV. For example, each of the eyepiece waveguides  2100  can project an FOV, as shown in the monocular eyepiece configuration. However, the FOVs projected by the two eyepiece waveguides  2100  can be at least partially overlapped.  FIG.  21 F  illustrates the case where the FOVs projected by the two eyepiece waveguides  2100  are overlapped by 50° in the horizontal direction and provide an overall binocular FOV of 150° by 60°. The binocular FOV can be even larger if less overlap is provided between the FOVs of the two eyepiece waveguides  2100 . As illustrated by matching shading, in the binocular FOV configuration, the first projectors  2120   a  (temple side) can be used to project the middle portion of the binocular FOV, and the second projectors  2120   b  (nasal side) can be used to project the sides of the binocular FOV. 
       FIG.  21 G  illustrates the k-space operation of another embodiment of the eyepiece waveguide  2100  shown in  FIG.  21 A . In this embodiment, the size of the FOV rectangle can exceed the width of the k-space annulus in both the k x  and the k y  dimensions. In  FIG.  21 G , the darker-shaded portions of the FOV rectangles correspond to the right portion of the FOV, while the lighter-shaded portions of the FOV rectangle correspond to the left portion of the FOV. The left and right ICG regions  2140   a ,  2140   b  can be designed with grating vectors to shift the FOV rectangles to the 3 o&#39;clock and 9 o&#39;clock positions, as already discussed. The magnitudes of the grating vectors of the ICG regions can be such that the center of the complete FOV rectangle is shifted to, for example, any radial position between the midpoint of the k-space annulus and the outer perimeter of the annulus. And the MPE region can be designed with grating vectors that shift the complete FOV rectangles to the 3 o&#39;clock, 6 o&#39;clock, 9 o&#39;clock and 12 o&#39;clock positions, as already discussed. But the magnitudes of the grating vectors of the MPE regions  2150  can also be designed such that the center of the complete FOV rectangle is shifted to, for example, any radial position between the midpoint of the k-space annulus and the outer perimeter of the annulus at those locations. Accordingly, even at the 12 o&#39;clock and 6 o&#39;clock positions, which are located along the axis of the shorter dimension of the FOV rectangle, a portion of the FOV rectangle may extend beyond the outer perimeter of the k-space annulus such that some portion of the rectangle is truncated. 
     Although the guided beams which correspond to the truncated portions of the FOV rectangles may be lost, all of the beams necessary to make up the complete FOV are still present in the waveguide when taking into consideration all the propagation states represented by the 3 o&#39;clock, 6 o&#39;clock, 9 o&#39;clock and 12 o&#39;clock positions. The left FOV (lighter-shaded rectangles) is preserved completely at the 9 o&#39;clock position, while the bottom portion is preserved at the 12 o&#39;clock position and the top portion is preserved at the 6 o&#39;clock position. Similarly, the right FOV (darker-shaded rectangles) is preserved completely at the 3 o&#39;clock position, while the bottom portion is preserved at the 12 o&#39;clock position and the top portion is preserved at the 6 o&#39;clock position. Thus, when the FOV rectangles are translated back to the origin of the k-space diagram, and are out-coupled toward the user&#39;s eye, all of the beams necessary to make up the complete FOV are present and the complete FOV can be re-created. The expansion of the FOV rectangle in multiple directions is discussed further in  FIGS.  22 A- 22 E . 
       FIG.  22 A  illustrates an embodiment of an eyepiece waveguide  2200  that can project an FOV which is expanded in two directions beyond the range of propagation angles which can be supported in guided propagation modes in the thickness direction of the eyepiece waveguide. The eyepiece waveguide  2200  includes a left ICG region  2240   a  provided between a first pair of top and bottom OPE regions  2250   a   1 ,  2250   a   2 . It also includes a right ICG region  2240   b  provided between a second pair of top and bottom OPE regions  2250   b   1 ,  2250   b   2 . Finally, an MPE region  2250   c  and an overlapping EPE region  2260  are provided between the first and second ICG regions  2240   a ,  2240   b  and their respective OPE regions. The MPE region  2250   c  can be provided on or in a first surface  2200   a  of the eyepiece waveguide  2200  (shown in  FIG.  22 A ), while the EPE region  2260  can be provided on or in a second surface of the waveguide (shown in  FIG.  22 B ). While the MPE region  2250   c  and the EPE region  2260  are illustrated as being the same size and exactly aligned in the x-y plane, in other embodiments they may have somewhat different sizes and may be partially misaligned. In some embodiments, the MPE region  2250   c  and the EPE region  2260  overlap one another by at least 70%, at least 80%, at least 90%, or at least 95%. 
     The left ICG region  2240   a  and the first pair of top and bottom OPE regions  2250   a   1 ,  2250   a   2  function similarly to what has been shown and described with respect to  FIG.  19   . Namely, a projector or other input device projects a set of beams corresponding to an input image FOV toward the left ICG region  2240   a  generally along the −z-direction. The left ICG region  2240   a  has grating lines which extend in the x-direction and periodically repeat in the y-direction. The left ICG region  2240   a  therefore couples input beams of light into a +1 diffractive order and a −1 diffractive order which propagate generally in the +y-direction toward the upper OPE region  2250   a   1  and in the −y-direction toward the lower OPE region  2250   a   2 . The first set of upper and lower OPE regions  2250   a   1 ,  2250   a   2  replicate those input beams, as discussed herein, and then guide the sets of replicated output beams generally in the x-direction toward the MPE/EPE regions. 
     The right ICG region  2240   b  and the second pair of top and bottom OPE regions  2250   a   1 ,  2250   a   2  function in the same way, but mirrored about the y-axis. Namely, a projector or other input device projects the same set of input beams toward the right ICG region  2240   b  generally along the −z-direction. The right ICG region  2240   b  also has grating lines which extend in the x-direction and periodically repeat in the y-direction. The right ICG region  2240   b  therefore also couples input beams of light into a +1 diffractive order and a −1 diffractive order which propagate generally in the +y-direction toward the upper OPE region  2250   b   1  and in the −y-direction toward the lower OPE region  2250   b   2 . The second set of upper and lower OPE regions  2250   b   1 ,  2250   b   2  replicate those input beams and then guide the sets of replicated output beams generally in the −x-direction toward the MPE/EPE regions. 
       FIG.  22 C  illustrates the k-space operation of the ICG regions  2240   a ,  2240   b  and the OPE regions  2250   a   1 ,  2250   a   2 ,  2250   b   1 ,  2250   b   2  in the eyepiece waveguide embodiment  2200  shown in  FIG.  22 A . Specifically, the left panel (KSD 1   a ) of  FIG.  22 C  illustrates the k-space operation of the left ICG region  2240   a  and its associated first set of top and bottom OPE regions  2250   a   1 ,  2250   a   2 , while the right panel (KSD 1   b ) of  FIG.  22 C  illustrates the k-space operation of the right ICG region  2240   b  and its associated second set of top and bottom OPE regions  2250   b   1 ,  2250   b   2 . 
     A set of input beams corresponding to the FOV of an input image is projected toward both the left ICG region  2240   a  and the right ICG region  2240   b . This set of input beams is illustrated in KSD 1   a  and KSD 1   b  as an FOV square centered at the respective origins of these k-space diagrams. Unlike previous illustrated enhanced FOV embodiments which showed only a single dimension of the FOV being larger than the width of the k-space annulus, both dimensions of the FOV square in KSD 1   a  and KSD 1   b  are larger than the width of the k-space annulus. In some embodiments, both dimensions of the FOV square can be up to approximately 2 times larger than the width of the k-space annulus. Although this embodiment is illustrated using an FOV square with equal horizontal and vertical FOVs, this is not a requirement, as the horizontal and vertical FOVs need not necessarily be equal. Embodiments of the eyepiece waveguide  2200  shown in  FIG.  22 A  may be capable of achieving FOVs as large as 100° by 60°, or more (e.g., 100° by 90°) depending on other design constraints such as eyebox size and screen door artifacts, assuming an eyepiece waveguide (surrounded by air) with refractive index 1.8. 
     In KSD 1   a , the FOV square is translated in the ±k y -direction in k-space by the grating vectors associated with the left ICG region  2240   a . Similarly, in KSD 1   b , the FOV square is translated in the ±k y -direction in k-space by the grating vectors associated with the right ICG region  2240   b . In both cases, after being in-coupled into the eyepiece waveguide  2200  by the ICG regions  2240   a ,  2240   b , the input beams are in propagation states represented by the translated FOV squares at the 12 o&#39;clock and 6 o&#39;clock positions of the k-space annulus. As shown in both KSD 1   a  and KSD 1   b , the FOV squares in these positions are truncated because they do not fit entirely within the k-space annulus. Only those beams corresponding to the shaded lower portion of the FOV square at the 12 o&#39;clock position enter guided propagation modes. Meanwhile, only those beams corresponding to the shaded upper portion of the FOV square at the 6 o&#39;clock position enter guided propagation modes. 
     KSD 1   a  also shows the k-space operation of the first set of top and bottom OPE regions  2250   a   1 ,  2250   a   2 . These OPE regions include diffraction gratings which are designed to have associated grating vectors which translate the FOV squares from the 12 o&#39;clock and 6 o&#39;clock positions to the 3 o&#39;clock position. Beams in the 3 o&#39;clock position propagate generally in the x-direction toward the MPE/EPE regions. 
     The beams corresponding to the upper portion of the FOV square at the 3 o&#39;clock position in k-space are provided by the FOV square which was previously located at the 6 o&#39;clock position, whereas the beams corresponding to the lower portion of the FOV square at the 3 o&#39;clock position are provided by the FOV square which was previously located at the 12 o&#39;clock position. However, the FOV square is once again too large to fit entirely within the k-space annulus at the 3 o&#39;clock position. The FOV square is therefore truncated, but this time the beams corresponding to the shaded left-hand portion of the FOV square remain in guided propagation modes, whereas the beams corresponding to the unshaded right-hand portion of the FOV square fall outside the k-space annulus and are lost. 
     The k-space operation of the second set of top and bottom OPE regions  2250   b   1 ,  2250   b   2  is a mirrored version (about the k y -axis) of the k-space operation of the first set of top and bottom OPE regions  2250   a   1 ,  2250   a   2 . Thus, as shown in KSD 1   b , the second set of top and bottom OPE regions  2250   b   1 ,  2250   b   2  ultimately produce a truncated FOV square at the 9 o&#39;clock position of the k-space annulus where the beams corresponding to the shaded right-hand portion of the square propagate in guided modes toward the MPE/EPE regions, while the beams corresponding to the unshaded left-hand portion of the FOV square fall outside the k-space annulus and are lost. 
       FIG.  22 D  illustrates the k-space operation of the MPE region  2250   c  in the eyepiece waveguide embodiment  2200  shown in  FIG.  22 A . Specifically, the left panel (KSD 2   a ) of  FIG.  22 D  illustrates the k-space operation of the MPE region  2250   c  on the beams received from the left ICG region  2240   a  and its associated first set of top and bottom OPE regions  2250   a   1 ,  2250   a   2 , while the right panel (KSD 2   b ) illustrates the k-space operation of the MPE region  2250   c  on the beams received from the right ICG region  2240   b  and its associated second set of top and bottom OPE regions  2250   b   1 ,  2250   b   2 . 
     The MPE region  2250   c  can operate similarly to what has been described with respect to the MPE regions  2050 ,  2150  in  FIGS.  20 A and  21 A . Namely, as already discussed, the MPE region  2250   c  can be composed of a 2D array of diffractive features which exhibit periodicity in multiple directions. The MPE region  2250   c  therefore has multiple associated grating vectors which can translate FOV square back and forth amongst the 3 o&#39;clock, 6 o&#39;clock, 9 o&#39;clock, and 12 o&#39;clock positions of the k-space annulus. This is represented by the double-sided arrows between those propagation states in KSD 2   a  and KSD 2   b . In this embodiment, the grating vectors G and H of the MPE region  2250   c  can be perpendicular to one another because the FOV is expanded beyond the width of the k-space annulus in both dimensions, and therefore the center of the FOV square can be translated to the same radial locations in the k-space annulus in both the k x  and k y  directions. 
     As already discussed, the beams which arrive at the MPE region  2250   c  from the left ICG region  2240   a  and the first set of top and bottom OPE regions  2250   a   1 ,  2250   a   2  are in the propagation state represented by the FOV square at the 3 o&#39;clock position of the k-space annulus. Only the beams corresponding to the shaded left-hand portion of the FOV square are present in this propagation state. As shown in KSD 2   a , when the MPE region  2250   c  diffracts these beams into the propagation state represented by the FOV square at the 12 o&#39;clock position, the FOV square is once again truncated and only the beams corresponding to the shaded lower left portion of the FOV square remain in guided propagation states. Meanwhile, when the MPE region  2250   c  diffracts beams from the propagation state represented by the FOV square at the 3 o&#39;clock position into the propagation state represented by the FOV square at the 6 o&#39;clock position, the FOV square is also truncated again; only the beams corresponding to the shaded upper left portion of the FOV square remain in guided propagation states. Finally, when the FOV squares are translated from either the 12 o&#39;clock position or the 6 o&#39;clock position of the k-space annulus to the 9 o&#39;clock position, the FOV square is yet again truncated, which may possibly not leave any of the beams in guided propagation states. This is shown by the unshaded FOV square at the 9 o&#39;clock position in KSD 2   a.    
     KSD 2   b  is a mirror image of KSD 2   a  about the k y -axis. KSD 2   b  shows the k-space operation of the MPE region  2250   c  on the beams of light which arrive from the right ICG region  2240   b  and the second set of top and bottom OPE regions  2250   b   1 ,  2250   b   2 . These beams are in the propagation state represented by the FOV square at the 9 o&#39;clock position of the k-space annulus. Only the beams corresponding to the shaded right-hand portion of the FOV square are present in this propagation state. As shown in KSD 2   b , when the MPE region  2250   c  diffracts these beams into the propagation state represented by the FOV square at the 12 o&#39;clock position, the FOV square is once again truncated and only the beams corresponding to the shaded lower right portion of the FOV square remain in guided propagation states. Meanwhile, when the MPE region  2250   c  diffracts beams from the propagation state represented by the FOV square at the 9 o&#39;clock position into the propagation state represented by the FOV square at the 6 o&#39;clock position, the FOV square is also truncated again; only the beams corresponding to the shaded upper right portion of the FOV square remain in guided propagation states. Finally, when the FOV squares are translated from either the 12 o&#39;clock position or the 6 o&#39;clock position of the k-space annulus to the 3 o&#39;clock position, the FOV square is yet again truncated, which may possibly not leave any of the beams in guided propagation states. This is shown by the unshaded FOV square at the 3 o&#39;clock position in KSD 2   b.    
     In this way, the beams which are replicated by propagation through the MPE region  2250   c  are divided into four sub-portions of the FOV: a first sub-portion corresponding to the upper left portion of the FOV square; a second sub-portion corresponding to the upper right portion of the FOV square; a third sub-portion corresponding to the lower left portion of the FOV square; and a fourth sub-portion corresponding to the lower right portion of the FOV square. Any pair of these sub-portions of the complete FOV can be partially overlapping. In other words, any pair of these sub-portions of the FOV can include beams which correspond to one or more of the same input beams. Alternatively, the sub-portions of the FOV could also be unique with no overlap. In either case, the sub-portions of the FOV are combined to re-create the complete FOV at the exit pupil of the eyepiece waveguide  2200 . This is shown in  FIG.  22 E . 
       FIG.  22 E  illustrates the k-space operation of the EPE region  2260  in the eyepiece waveguide embodiment  2200  shown in  FIG.  22 A . The EPE region  2260  can function similarly to what has been described with respect to the EPE regions  2060 ,  2160  in  FIGS.  20 A and  21 A . As discussed herein, since the EPE region  2260  overlaps the MPE region  2250   c , beams of light propagating in the MPE region can also interact with the EPE region and be out-coupled from the eyepiece waveguide  2200 . The EPE region  2260  includes a diffraction grating whose axis of periodicity is aligned with those of the left ICT region  2240   a  and the right ICG region  2240   b . In the illustrated embodiment, the axis of periodicity for the EPE region  2260  points in the ±k y -direction. The EPE region  2260  therefore has associated grating vectors which likewise point in the same direction and translate the FOV squares located at the 12 o&#39;clock and 6 o&#39;clock positions of the k-space annulus back to the origin of the k-space diagram.  FIG.  22 E  shows that when this occurs, the four sub-portions of the FOV are assembled to re-create the complete FOV. All of the beams required to make up the complete image FOV are present. And the four sub-portions of the FOV are aligned in k-space with the same relative positions with respect to one another as in the complete input FOV. 
     Eyepiece Waveguides Designed to Work with Angled Projectors 
     Many of the eyepiece waveguide embodiments described herein have been designed to work with a projector (or other image input device) whose optical axis intersects the ICG region at a perpendicular angle. In such embodiments, the center input beam (which corresponds to the center point of the input image) is perpendicularly incident on the ICG region, and the input beams corresponding to the top/bottom and left/right portions of the input image are incident on the ICG region at symmetrical angles. In some embodiments, however, an eyepiece waveguide may be designed to function with an angled projector (or other image input device).  FIG.  23    illustrates an example of such an embodiment. 
       FIG.  23    illustrates an example embodiment of an eyepiece waveguide  2300  designed to function with an angled projector. The eyepiece waveguide  2300  includes an ICG region  2340 , left and right OPE regions  2350   a ,  2350   b , and an EPE region  2360 . Input beams from a projector are incident on the ICG region  2340  and are coupled into the eyepiece waveguide  2300  in guided propagation modes. In this embodiment, the projector is oriented at a non-perpendicular angle with respect to the ICG region  2340 . The center input beam  2341  from the projector is therefore incident on the ICG region  2340  at an oblique angle (e.g., as illustrated in  FIG.  13 I ). This results in a shift in k-space of the k-vectors for the input beams, causing them to no longer be centered about the origin of a k-space diagram. As a result, the optical design of the ICG, OPE, and/or EPE regions may need to be altered, along with their physical shape (e.g., according to the principles described with reference to  FIG.  14 D ), and the placement of FOV rectangles in the k-space annulus may also change, as discussed below. 
     The positive and negative diffractive orders from the ICG region  2340  then propagate to the left and right OPE regions  2350   a ,  2350   b , respectively. The OPE regions  2350  replicate the input beams in a spatially distributed manner in the horizontal direction and direct them toward the EPE region  2360 . The EPE region  2360  then further replicates the beams in a spatially distributed manner in the vertical direction and out couples them toward the user&#39;s eye, as discussed elsewhere herein. 
       FIG.  23    includes a k-space diagram, KSD, which illustrates the k-space operation of the eyepiece waveguide  2300 . As described elsewhere herein, the FOV rectangle in the central portion of the k-space diagram corresponds to the input beams from the projector and the output beams from the eyepiece waveguide  2300 . The FOV rectangles near the 4 o&#39;clock and 8 o&#39;clock positions in the k-space annulus correspond to the beams of light propagating from the ICG region  2340  to the OPE regions  2350 . Lastly, the FOV rectangle at the 6 o&#39;clock position in the k-space annulus corresponds to the beams of light propagating from the OPE regions  2350  downward toward the EPE region  2360 . 
     Since the projector is angled with respect to the ICG region  2340 , the FOV rectangle corresponding to the input beams is not centered at the origin of the k-space diagram. Instead, in the illustrated embodiment, the FOV rectangle corresponding to the input beams is centered on the k y -axis but located below the k x -axis. This means that none of the input beams have propagation directions with components in the +y-direction. In other words, the input beams propagate downward from the projector toward the ICG region. The ICG region  2340  then translates the FOV rectangle horizontally into the k-space annulus in the ±k x -directions. 
     Since none of the guided light beams from the ICG region  2340  have k-vectors with a positive k y  component (i.e., the FOV rectangles are located below the k x -axis), the top edges of the OPE regions  2350  can be horizontal, as illustrated, since there is no need to accommodate beams of light fanning out upwardly in the +y-direction. This characteristic of the OPE regions  2350  may be advantageous in some embodiments because it may allow for a compact design. However, the horizontal top edge of the OPE regions  2350  is made practical by the angled image projector. The angled image projector may, however, be associated with some disadvantages. For example, since the eyepiece waveguide  2300  (including, for example, the optical design and/or physical layout of gratings) is designed to receive input light from an upward angle, light from overhead sources, such as the sun or overhead light fixtures, may likewise be coupled into the eyepiece waveguide. This may result in undesirable image features, such as ghost images of those light sources superimposed on the displayed virtual content, artifacts, reduced contrast, etc. Although light from overhead sources may be blocked by including a visor so as to shade the eyepiece waveguide  2300  from overhead light, such a visor may be bulky or aesthetically undesirable. Thus, eyepiece waveguides which are designed to function with perpendicular projectors may be preferred because the need for a visor can be reduced or eliminated. In addition, for upward or downward angled projector designs, the fact that output beams also exit the waveguide at an angle similar to the input beams means that the eyepiece waveguide may need to be tilted relative to the user&#39;s central gaze vector and/or it may need to be placed above or below—rather than directly in front of—the eye. 
     Example AR Eyepiece Waveguides with Combined Pupil Expander-Extractor Regions 
       FIG.  24 A  is an edge view of an example eyepiece waveguide  2400  that has multiple combined pupil expander-extractor (CPE) regions  2455 . The CPE regions  2455  take the place of the OPE, MPE, and/or EPE regions which are described herein with respect to other embodiments. The illustrated embodiment has first and second CPE regions  2455   a ,  2455   b  on opposing sides of the eyepiece waveguide  2400 . The first and second CPE regions  2455   a ,  2455   b  both spread light laterally inside the eyepiece waveguide  2400 , similar to an OPE region. They also both extract the light from the eyepiece waveguide  2400 , similar to an EPE region. 
     The eyepiece waveguide  2400  shown in  FIG.  24 A  can be formed using a substrate made of an optically transmissive material. The eyepiece waveguide  2400  has an eye-facing side  2400   a  and an outward-facing side  2400   b . In the illustrated embodiment of the eyepiece waveguide  2400 , an ICG region  2440  is provided at the top center of the eyepiece waveguide  2400 , and the first and second CPE regions  2455   a ,  2455   b  are provided below the ICG region  2440  on the eye-facing side  2400   a  and the outward-facing side  2400   b , respectively. 
     In some embodiments, the ICG region  2440  is a diffraction grating formed on or in a surface of the eyepiece waveguide  2400  (e.g., on the eye-facing side  2400   a ). The ICG region  2440  receives a set of input beams from an input device, such as a projector. As described elsewhere herein, the input beams can propagate from the input device generally in the ±z-direction until they are incident upon the ICG region  2440 . The ICG region  2440  diffracts those input beams so that at least some enter guided propagation modes within the eyepiece waveguide  2400 . 
     The illustrated embodiment of the diffraction grating inside the ICG region  2440  has one-dimensional periodicity (i.e., it is a 1D grating). The grating lines of the ICG region  2040  can be oriented so as to direct some of the diffracted beams in the −y-direction toward the first and second CPE regions  2455   a ,  2455   b . Thus, in the illustrated embodiment, the ICG region  2440  includes diffractive lines which extend in the ±x-direction and repeat periodically in the ±y-direction. As described elsewhere herein, the spacing between the diffractive lines which make up the ICG region  2440  can be set so as to couple the input beams of light into guided propagation modes inside the eyepiece waveguide  2400 . The diffracted beams from the ICG region  2440  then propagate via TIR toward the first and second CPE regions  2455   a ,  2455   b.    
     The first CPE region  2455   a  is formed on or in one side of the eyepiece waveguide (e.g., the eye-facing side  2400   a ) and the second CPE region  2455   b  is formed on or in the opposite side of the eyepiece waveguide (e.g., the outward-facing side  2400   b ). In the illustrated embodiment, the first and second CPE regions  2455   a ,  2455   b  are both 1D diffraction gratings. The first CPE region  2455   a  is illustrated as a 1D diffraction grating made up of diffractive lines oriented at an angle of −30° with respect to the y-axis (when viewed from the eye-facing side  2400   a ), and the second CPE region  2455   b  is illustrated as a 1D diffraction grating made up of diffractive lines oriented at an angle of +30° with respect to the y-axis (when also viewed from the eye-facing side  2400   b ). 
     In some embodiments of the eyepiece waveguide  2400 , the relative angle between the 1D grating of the first CPE region  2455   a  and the 1D grating of the second CPE region  2455   b  is substantially 60° (i.e., 60°±5°, or 60°±3°, or 60°±1°, or 60°±0.5°, or 60°±) 0.1°. In addition, in some embodiments, the relative angles between the 1D grating of the ICG region  2440  and the 1D gratings of both of the CPE regions  2455   a ,  2455   b  are also substantially 60° (i.e., 60°±5°, or 60°±3°, or 60°±1°, or 60°±0.5°, or 60°±0.1°). Other layouts for the eyepiece waveguide  2400  besides the specific example shown in  FIG.  24 A  are also possible. For example, the ICG region  2440  could instead be located on the temporal or medial side of the eyepiece waveguide  2400  and the orientations of the CPE regions  2455   a ,  2455   b  could be adjusted accordingly to maintain the relative angles between gratings. 
     As discussed further below, the relative angle of substantially 60° between each of the respective 1D gratings of the ICG region  2440 , the first CPE region  2455   a , and the second CPE region  2455   b  contributes to the characteristic that the CPE regions can both laterally spread light in the eyepiece waveguide  2400  and out-couple light towards the user&#39;s eye. 
     In some embodiments, the 1D gratings of the first and second CPE regions  2455   a ,  2455   b  are identical apart from their orientations. For example, the first and second CPE regions  2455   a ,  2455   b  can have the same line spacing, the same etch depth, etc. This can be advantageous because it permits both CPE regions  2455  to be manufactured from the same master template. In addition, in some embodiments, the 1D grating of the ICG region  2440  also has the same line spacing as the first and second CPE regions  2455   a ,  2455   b.    
     While the first CPE region  2455   a  and the second CPE region  2455   b  are illustrated as being the same size and exactly aligned in the x-y plane, in other embodiments they may have somewhat different sizes and/or they may be partially misaligned. In some embodiments, the first and second CPE regions  2455   a ,  2455   b  overlap one another by at least 70%, or by at least 80%, or by at least 90%, or by at least 95%. 
     As already mentioned, the guided beams of light from the ICG region  2440  propagate through the eyepiece waveguide  2400  via TIR, meaning they reflect back and forth between the respective surfaces of the eye-facing side  2400   a  and the outward-facing side  2400   b . As the guided beams propagate through the eyepiece waveguide  2400  in this manner, they alternately interact with the diffraction gratings of the first and second CPE regions  2455   a ,  2455   b . The operation of the first and second CPE regions  2455   a ,  2455   b  on the guided beams of light is discussed further with respect to  FIGS.  24 B- 24 K . 
       FIG.  24 B  illustrates the operation of the first and second CPE regions  2455   a ,  2455   b  in both physical space and in k-space according to a first type of main pathway of light through the eyepiece waveguide  2400 . A physical diagram of the eyepiece waveguide  2400  is shown on the left hand side of  FIG.  24 B . The eyepiece waveguide  2400  is shown as viewed from the eye-facing side  2400   a . A k-space diagram, KSD 1   a , of the operation of the ICG region  2440  and the first and second CPE regions  2455   a ,  2455   b  is shown on the right hand side of  FIG.  24 B . 
     As already discussed, a set of input beams is incident on the ICG region  2440  of the eyepiece waveguide  2400  from an input device, such as a projector. This set of input beams is represented by the FOV rectangle shown in the center of k-space diagram KSD 1   a . The diffraction grating in the ICG region  2440  has associated positive and negative grating vectors which point in the ±k y -directions. Thus, the k-space operation of the ICG region  2440  is to shift the central FOV rectangle to both the six o&#39;clock and 12 o&#39;clock positions on k-space diagram KSD 1   a . (The FOV rectangle at the 12 o&#39;clock position corresponds to light beams propagating in the +y-direction. Since those beams exit the eyepiece waveguide  2400  out of its top edge, that particular FOV rectangle is not illustrated and those beams are not discussed further.) The length of the grating vectors associated with the ICG region  2440  can be set, based on the spacing of the diffractive lines and the wavelength of the light, such that the translated FOV rectangle at the six o&#39;clock position lies completely within the k-space annulus. 
     For ease of illustration, the physical diagram on the left hand side of  FIG.  24 B  only shows one of the guided beams of light from the ICG region  2440  (i.e., guided beam  2441  corresponding to the center k-vector in the FOV rectangle located at the six o&#39;clock position of the k-space diagram KSD 1   a ). Guided beam  2441  from the ICG region  2440  propagates downward through the eyepiece waveguide  2400  in the −y-direction, reflecting back and forth in TIR between the surface of the eye-facing side  2400   a  and the surface of the outward-facing side  2400   b . Each time guided beam  2441  reflects from the eye-facing side  2400   a , it can interact with the first CPE region  2455   a . And each time guided beam  2441  reflects from the outward facing side  2400   b , it can interact with the second CPE region  2455   b . The diffractive efficiency of the first and second CPE regions  2455   a ,  2455   b  can be set so that only a portion of the power of each beam of light is diffracted with each of these interactions. For example, in some embodiments, the diffractive efficiency of the first and second CPE regions  2455   a ,  2455   b  is 10% or less. The diffractive efficiency of the first and second CPE regions  2455   a ,  2455   b  can be determined by, for example, the etch depth of the diffractive lines. 
     The physical diagram on the left hand side of  FIG.  24 B  shows the interactions of guided beam  2441  with the first CPE region  2455   a  which cause light to spread laterally in the −x-direction through the eyepiece waveguide  2400 . As guided beam  2441  propagates downward in the −y-direction through the eyepiece waveguide  2400 , a portion of its power is diffracted at a +120° angle with respect to the y-axis during each interaction with the first CPE region  2455   a . The remaining portion of the power of guided beam  2441  continues propagating downward in the −y-direction until the next interaction with the first CPE region  2455   a , where another portion of its power is diffracted at the same +120° angle. This process creates a plurality of spaced apart diffracted beams  2456   a  which propagate through the eyepiece waveguide  2400  at a +120° angle with respect to the y-axis. These diffracted beams  2456   a  are represented by the FOV rectangle located at the 8 o&#39;clock position in k-space diagram KSD 1   a  on the right hand side of  FIG.  24 B . 
     As with any 1D diffraction grating, there are positive and negative grating vectors associated with the first CPE region  2455   a . These grating vectors point along the direction of periodicity of the grating lines in the first CPE region  2455   a . Accordingly, one of the first-order grating vectors associated with the first CPE region  2455   a  points at +60° with respect to the y-axis (as shown in KSD 1   a ), while the other points in the opposite direction at −120° with respect to the y-axis. The same is true for the positive and negative higher-order grating vectors. The first-order grating vector which points at +60° with respect to the y-axis shifts the FOV rectangle from the six o&#39;clock position (which corresponds to the downward propagating guided beams from the ICG region  2440 ) to the eight o&#39;clock position (which corresponds to the diffracted beams  2456   a  propagating at the +120° angle with respect to the y-axis). (The first-order grating vector which points at −120° with respect to the y-axis would shift the FOV rectangle from the six o&#39;clock position to a location outside of the k-space annulus and therefore does not result in diffraction.) 
     Once guided beams from the ICG region  2440  interact with the first CPE region  2455   a  and are diffracted into the propagation states represented by the FOV rectangle at the eight o&#39;clock position of k-space diagram KSD 1   a , they then interact with the second CPE region  2455   b  on the next TIR bounce as they are guided through the eyepiece waveguide  2400 . The interaction of these beams  2456   a  with the second CPE region  2455   b  can result in them being out-coupled from the eyepiece waveguide  2400  toward the user&#39;s eye. The out-coupled beams  2457   a  are shown in the physical diagram of the eyepiece waveguide  2400  on the left hand side of  FIG.  24 B  as circled dots, indicating that those beams are propagating in the z-direction out of the page. The out-coupling of beams  2456   a  by the second CPE region  2455   b  can be understood by reference to k-space diagram KSD 1   a.    
     Just as there are positive and negative grating vectors associated with the first CPE region  2455   a , there are also positive and negative grating vectors associated with the second CPE region  2455   b . These grating vectors point along the direction of periodicity of the grating lines in the second CPE region  2455   b . Accordingly, one of the first-order grating vectors associated with the second CPE region  2455   b  points at −60° with respect to the y-axis (as shown in KSD 1   a ), while the other points in the opposite direction at +120° with respect to the y-axis. The same is true for the positive and negative higher-order grating vectors. The first-order grating vector which points at −60° with respect to the y-axis shifts the FOV rectangle from the eight o&#39;clock position (which corresponds to the diffracted beams  2456   a  propagating at a +120° angle with respect to the y-axis) to the center of k-space diagram KSD 1   a  (which corresponds to out-coupled beams of light which are no longer in guided propagation modes inside the eyepiece waveguide  2400 ). (The first-order grating vector which points at +120° with respect to the y-axis would shift the FOV rectangle from the eight o&#39;clock position to a location outside of the k-space annulus and therefore does not result in diffraction.) 
     The physical diagram on the left hand side of  FIG.  24 B  shows how the interactions of light beams  2456   a  with the second CPE region  2455   b  results in multiple spaced-apart out-coupled beams  2457   a . As light beams  2456   a  propagate at the +120° angle with respect to the y-axis, a portion of their power is out-coupled by each interaction with the second CPE region  2455   b . The remaining portion of the power of light beams  2456   a  continues propagating at the +120° angle with respect to the y-axis until the next interaction with the second CPE region  2455   b , where another portion of the power of those beams is out-coupled. This process creates a plurality of spaced-apart out-coupled beams  2457   a  which exit the eyepiece waveguide  2400  at different spatial locations and propagate toward the user&#39;s eye. As already noted, these out-coupled beams  2457   a  are represented by the FOV rectangle located at the center of k-space diagram KSD 1   a.    
     The passage of beams of light through the eyepiece waveguide  2400  in the manner shown in k-space diagram KSD 1   a  in  FIG.  24 B  is the first type of main pathway of light through the eyepiece waveguide. There is also a second type of main pathway of light through the eyepiece waveguide  2400  which is illustrated by k-space diagram KSD 1   b  in  FIG.  24 C . 
       FIG.  24 C  illustrates the operation of the first and second CPE regions  2455   a ,  2455   b  in both physical space and in k-space according to the second type of main pathway of light through the eyepiece waveguide  2400 . Once again, a physical diagram of the eyepiece waveguide  2400  is shown on the left hand side of  FIG.  24 C . The eyepiece waveguide  2400  is again shown as viewed from the eye-facing side  2400   a . A k-space diagram, KSD 1   b , of the operation of the ICG region  2440  and the first and second CPE regions  2455   a ,  2455   b  is shown on the right hand side of  FIG.  24 C . 
     The physical diagram on the left hand side of  FIG.  24 C  shows the same guided beam  2441  from the ICG region  2440  as is shown in  FIG.  24 B . But this time, the physical diagram shows the interactions of guided beam  2441  with the second CPE region  2455   b  which cause light to laterally spread in the +x-direction through the eyepiece waveguide  2400 . Namely, as guided beam  2441  propagates downward in the −y-direction through the eyepiece waveguide  2400 , a portion of its power is diffracted at a substantially −120° angle with respect to the y-axis during each interaction with the second CPE region  2455   b . The remaining portion of the power of guided beam  2441  continues propagating downward in the −y-direction until the next interaction with the second CPE region  2455   b , where another portion of its power is diffracted at the same −120° angle. This process creates a plurality of spaced apart diffracted beams  2456   b  which propagate through the eyepiece waveguide  2400  at a −120° angle with respect to the y-axis. These diffracted beams  2456   b  are represented by the FOV rectangle located at the 4 o&#39;clock position in k-space diagram KSD 1   b  on the right hand side of  FIG.  24 C . 
     As already discussed, one of the first-order grating vectors associated with the second CPE region  2455   b  points at −60° with respect to the y-axis (as shown in KSD 1   b ), while the other points in the opposite direction at +120° with respect to the y-axis. The first-order grating vector which points at −60° with respect to the y-axis shifts the FOV rectangle from the six o&#39;clock position (which corresponds to the downward propagating guided beams from the ICG region  2440 ) to the four o&#39;clock position (which corresponds to the diffracted beams  2456   b  propagating at the −120° angle with respect to the y-axis). (The first-order grating vector which points at +120° with respect to the y-axis would shift the FOV rectangle from the six o&#39;clock position to a location outside of the k-space annulus and therefore does not result in diffraction.) 
     Once guided beams from the ICG region  2440  interact with the second CPE region  2455   b  and are diffracted into the propagation states represented by the FOV rectangle at the four o&#39;clock position of k-space diagram KSD 1   b , they then interact with the first CPE region  2455   a  on the next TIR bounce as they are guided through the eyepiece waveguide  2400 . The interaction of these beams  2456   b  with the first CPE region  2455   a  can result in them being out-coupled from the eyepiece waveguide  2400  toward the user&#39;s eye. The out-coupled beams  2457   b  are shown in the physical diagram of the eyepiece waveguide  2400  on the left hand side of  FIG.  24 C  as circled dots, indicating that those beams are propagating in the z-direction out of the page. The out-coupling of beams  2456   b  by the first CPE region  2455   a  can be understood by reference to k-space diagram KSD 1   b.    
     As already discussed, one of the first-order grating vectors associated with the first CPE region  2455   a  points at +60° with respect to the y-axis (as shown in KSD 1   b ), while the other points in the opposite direction at −120° with respect to the y-axis. The first-order grating vector which points at +60° with respect to the y-axis shifts the FOV rectangle from the four o&#39;clock position (which corresponds to the diffracted beams  2456   b  propagating at a −120° angle with respect to the y-axis) to the center of k-space diagram KSD 1   a  (which corresponds to out-coupled beams of light which are no longer in guided propagation modes inside the eyepiece waveguide  2400 ). (The first-order grating vector which points at −120° with respect to the y-axis would shift the FOV rectangle from the four o&#39;clock position to a location outside of the k-space annulus and therefore does not result in diffraction.) 
     The physical diagram on the left hand side of  FIG.  24 C  shows how the interactions of light beams  2456   b  with the first CPE region  2455   a  results in multiple spaced-apart out-coupled beams  2457   b . As light beams  2456   b  propagate at the −120° angle with respect to the y-axis, a portion of their power is out-coupled by each interaction with the first CPE region  2455   a . The remaining portion of the power of light beams  2456   b  continues propagating at the −120° angle with respect to the y-axis until the next interaction with the first CPE region  2455   a , where another portion of the power of those beams is out-coupled. This process creates a plurality of spaced-apart out-coupled beams  2457   b  which exit the eyepiece waveguide  2400  at different spatial locations and propagate toward the user&#39;s eye. As already noted, these out-coupled beams  2457   b  are represented by the FOV rectangle located at the center of k-space diagram KSD 1   b.    
       FIG.  24 D  illustrates the operation of the first and second CPE regions  2455   a ,  2455   b  in both physical space and in k-space according to both the first and second types of main pathways of light through the eyepiece waveguide  2400 . Yet again, a physical diagram of the eyepiece waveguide  2400  is shown on the left hand side of  FIG.  24 D . The eyepiece waveguide  2400  is again shown as viewed from the eye-facing side  2400   a . A k-space diagram, KSD 2 , of the operation of the ICG region  2440  and the first and second CPE regions  2455   a ,  2455   b  is shown on the right hand side of  FIG.  24 D . 
     As already discussed, both types of main pathways of light through the eyepiece waveguide  2400  begin with a set of input light beams—corresponding to an input image—which are incident on the ICG region  2440 . The set of input light beams is represented by the FOV rectangle located at the center of k-space diagram KSD 2 . The ICG region  2440  couples the input light beams into guided propagation modes within the eyepiece waveguide  2400 . This is represented by the translation of the FOV rectangle—by one of the first-order grating vectors associated with the ICG region—from the center of k-space diagram KSD 2  to the 6 o&#39;clock position of the k-space annulus. The physical diagram on the left hand side of  FIG.  24 D  shows a single one of the resulting guided beams (i.e., guided beam  2441 ). It should be understood, however, that many guided input beams will be present, each of which will correspond to a different k-vector inside the FOV rectangle located at the 6 o&#39;clock position in the k-space annulus of KSD 2 . 
     The guided light beams from the ICG region  2440  then have multiple alternating interactions with the first and second CPE regions  2455   a ,  2455   b  as they TIR between the surface of the eye-facing side  2400   a  of the eyepiece waveguide  2400  and the surface of the outward-facing side  2400   b . During each generation of interactions, a portion of the power of each of the beams can zero-order diffract and continue propagating in the same direction in the x-y plane of the eyepiece waveguide  2400 , while another portion of the power of each of the beams can first-order diffract into a new propagation direction. 
     Some of the light beams in the propagation states represented by the FOV rectangle at the 6 o&#39;clock position in KSD 2  will first interact with the first CPE region  2455   a , while others will first interact with the second CPE region  2455   b . In the case of those light beams whose initial interaction is with the first CPE region  2455   a , a portion of the power of each of those beams will first-order diffract, thereby creating diffracted beams of light (e.g., diffracted beams  2456   a ) whose propagation states are represented by the FOV rectangle at the 8 o&#39;clock position of the k-space annulus in KSD 2 , and another portion of the power of each of those beams will zero-order diffract resulting in diffracted beams of light whose propagation states continue to be represented by the FOV rectangle at the 6 o&#39;clock position. All of those beams of light will then interact with the second CPE region  2455   b  on the subsequent TIR bounce as they propagate through the eyepiece waveguide  2400 . 
     During the interaction with the second CPE region  2455   b , a portion of the power of the beams whose propagation states are represented by the FOV rectangle at the 8 o&#39;clock position will first-order diffract, thereby creating out-coupled beams of light (e.g., beams  2457   a ) whose propagation states are represented by the FOV rectangle at the center of the k-space annulus in KSD 2 , and another portion of the power of each of those beams will zero-order diffract resulting in beams of light (e.g., beams  2456   a ) whose propagation states continue to be represented by the FOV rectangle at the 8 o&#39;clock position. Meanwhile, a portion of the power of the beams whose propagation states are represented by the FOV rectangle at the 6 o&#39;clock position will follow the second type of main pathway through the eyepiece waveguide  2400 . Namely, a portion of the power of the beams whose propagation states are represented by the FOV rectangle at the 6 o&#39;clock position will first-order diffract in the interaction with the second CPE region  2455   b , thereby creating beams of light (e.g., beams  2456   b ) whose propagation states are represented by the FOV rectangle at the 4 o&#39;clock position of the k-space annulus in KSD 2 , and another portion of the power of each of those beams will zero-order diffract resulting in beams of light whose propagation states continue to be represented by the FOV rectangle at the 6 o&#39;clock position. All of those beams of light will then interact with the first CPE region  2455   a  on the subsequent TIR bounce as they propagate through the eyepiece waveguide  2400 . 
     During the next interaction with the first CPE region  2455   a , a portion of the power of the beams whose propagation states are represented by the FOV rectangle at the 4 o&#39;clock position will first-order diffract, thereby creating out-coupled beams of light (e.g., beams  2457   b ) whose propagation states are represented by the FOV rectangle at the center of the k-space annulus in KSD 2 , and another portion of the power of each of those beams will zero-order diffract resulting in beams of light (e.g., beams  2456   b ) whose propagation states continue to be represented by the FOV rectangle at the 4 o&#39;clock position. Meanwhile, a portion of the power of the beams whose propagation states are represented by the FOV rectangle at the 6 o&#39;clock position will follow the first type of main pathway through the eyepiece waveguide  2400 . Namely, a portion of the power of the beams whose propagation states are represented by the FOV rectangle at the 6 o&#39;clock position will first-order diffract in the interaction with the first CPE region  2455   a , thereby creating beams of light (e.g., beams  2456   a ) whose propagation states are represented by the FOV rectangle at the 8 o&#39;clock position of the k-space annulus in KSD 2 , and another portion of the power of each of those beams will zero-order diffract resulting in beams of light whose propagation states continue to be represented by the FOV rectangle at the 6 o&#39;clock position. All of those beams of light will then interact with the second CPE region  2455   b  on the subsequent TIR bounce as they propagate through the eyepiece waveguide  2400  and the cycle will repeat. 
     As is evident from the k-space diagrams in  FIGS.  24 B- 24 D , the 1D diffraction gratings in the ICG region  2440 , the first CPE region  2455   a , and the second CPE region  2455   b  can be oriented such that their associated grating vectors are all at substantially 60° angles with respect to one another. In addition, the 1D diffraction gratings in the ICG region  2440 , the first CPE region  2455   a , and the second CPE region  2455   b  can all have the same line spacing such that their associated grating vectors all have the same magnitude. These properties, in combination with the fact that the first and second CPE regions  2455   a ,  2455   b  are on opposite sides of the eyepiece waveguide  2400 , and therefore light beams alternately interact with those gratings, causes light beams to propagate along paths in k-space which are defined by equilateral triangles. These equilateral triangular paths allow the first and second CPE regions  2455   a ,  2455   b  to both spread light laterally in the eyepiece waveguide  2400  and to both out-couple light from the eyepiece waveguide to the user&#39;s eye. 
       FIG.  24 E  is a diagram of the first generation of interactions between an input beam and the CPE regions  2455  of the eyepiece waveguide embodiment shown in  FIG.  24 A . In the illustrated case, the first generation of interactions is with the first CPE region  2455   a , though it could alternatively be with the second CPE region  2455   b .  FIG.  24 E  shows a guided input beam that enters the first CPE region  2455   a  from the ICG region  2440 . The input beam is shown propagating in a direction which corresponds to one of the k-vectors in the FOV rectangle located at the 6 o&#39;clock position of the k-space annulus in  FIGS.  24 B- 24 D . In some embodiments, the input beam has a diameter of ˜5 mm or less, or of ˜1 mm or less. 
     At every interaction with the first CPE region  2455   a , the input beam will split into 2 beams (each with the same diameter but a fraction of the original power of the input beam) propagating in 2 different directions in TIR. One direction corresponds to zero-order diffraction and is the original propagation angle in the x-y plane of the eyepiece waveguide  2400 . The other direction depends on the grating vectors associated with the first CPE region  2455   a . As shown, the first generation of interactions between the input beam and the first CPE region  2455   a  results in two beams: some portion of the power of the input beam simply reflects, as output 1 , from the surface of the eyepiece waveguide  2400  and continues on in the same x-y direction as the input beam (i.e., the 0 th  order diffraction); and some portion of the power of the input beam interacts with the 1D grating in the first CPE region  2455   a  and is diffracted as output 2 . The output 2  beam is shown propagating in a direction which corresponds to one of the k-vectors in the FOV rectangle located at the 8 o&#39;clock position of the k-space annulus in  FIGS.  24 B- 24 D . After this first generation of interactions, the output 1  beam and the output 2  beam may subsequently interact with the second CPE region  2455   b . Although not illustrated, other guided input beams that enter the first CPE region  2455   a  from the ICG region  2440  with different propagation angles will behave similarly but with slightly different input and output angles. 
       FIG.  24 F  is a diagram of the second generation of interactions between the input beam and the CPE regions  2455  of the eyepiece waveguide embodiment shown in  FIG.  24 A . In the illustrated case, the second generation of interactions is with the second CPE region  2455   b . The beams related to the first generation of interactions are shown with dashed lines, while the beams related to the second generation of interactions are shown with solid lines. As shown in  FIG.  24 F , each of the output beams, output 1  and output 2 , from the first generation of interactions can now interact with the second CPE region  2455   b . Some portion of the power of the output 1  beam from  FIG.  24 E  zero-order diffracts and continues on in the same x-y direction (corresponding to one of the k-vectors in the FOV rectangle at the 6 o&#39;clock position of the k-space diagrams in  FIGS.  24 B- 24 D ), while another portion of the power of that beam interacts with the grating in the second CPE region  2455   b  and is first-order diffracted in a direction corresponding to one of the k-vectors in the FOV rectangle located at the 4 o&#39;clock position of the k-space diagrams in  FIGS.  24 B- 24 D . Similarly, some portion of the power of the output 2  beam from  FIG.  24 E  zero-order diffracts and continues on in the same direction (corresponding to one of the k-vectors in the FOV rectangle located at the 8 o&#39;clock position of the k-space diagrams in  FIGS.  24 B- 24 D ), while another portion of the power of that beam interacts with the grating in the second CPE region  2455   b  and is first-order diffracted and out-coupled from the eyepiece waveguide  2400 . After this second generation of interactions, the output 1  beams and the output 2  beam may subsequently interact with the first CPE region  2455   a.    
       FIG.  24 G  is a diagram of the third generation of interactions between the input beam and the CPE regions of the eyepiece waveguide embodiment shown in  FIG.  24 A . In the illustrated case, the third generation of interactions is with the first CPE region  2455   a . The beams related to the first and second generations of interactions are shown with dashed lines, while the beams related to the third generation of interactions are shown with solid lines. As shown in  FIG.  24 G , each of the output beams, output 1  and output 2 , from the second generation of interactions can now interact with the first CPE region  2455   a . Some portion of the power of the output 1  beam from  FIG.  24 F  which belongs to the FOV rectangle located at the 8 o&#39;clock position of the k-space annulus zero-order diffracts and continues on in the same x-y direction, while another portion of the power of that beam is first-order diffracted in a direction corresponding to one of the k-vectors in the FOV rectangle located at the 6 o&#39;clock position. Some portion of the power of the output 1  beam from  FIG.  24 F  which belongs to the FOV rectangle located at the 6 o&#39;clock position of the k-space annulus zero-order diffracts and continues on in the same x-y direction, while another portion of the power of that beam is first-order diffracted in a direction corresponding to one of the k-vectors in the FOV rectangle located at the 8 o&#39;clock position. Finally, some portion of the power of the output 2  beam from  FIG.  24 F  zero-order diffracts and continues on in the same x-y direction, while another portion of the power of that beam is first-order diffracted and out-coupled from the eyepiece waveguide  2400 . After this third generation of interactions, the output 1  beams and the output 2  beams may subsequently interact with the second CPE region  2455   b.    
       FIG.  24 H  is a diagram of the fourth generation of interactions between the input beam and the CPE regions  2455  of the eyepiece waveguide embodiment shown in  FIG.  24 A . In the illustrated case, the fourth generation of interactions is with the second CPE region  2455   b . The beams related to the first, second, and third generations of interactions are shown with dashed lines, while the beams related to the fourth generation of interactions are shown with solid lines. In this generation of interactions, some of the beams of light are out-coupled from the eyepiece waveguide  2400  and each of the others is diffracted into a direction corresponding to a k-vector which belongs to one of the FOV rectangles at the 4 o&#39;clock, 6 o&#39;clock, or 8 o&#39;clock positions in the k-space annulus of the k-space diagrams in  FIGS.  24 B- 24 D . After this fourth generation of interactions, the output 1  beams and the output 2  beams may subsequently interact with the first CPE region  2455   a.    
       FIG.  24 I  is a diagram of the fifth generation of interactions between the input beam and the CPE regions  2455  of the eyepiece waveguide embodiment shown in  FIG.  24 A . In the illustrated case, the fifth generation of interactions is with the first CPE region  2455   a . The beams related to the first, second, third, and fourth generations of interactions are shown with dashed lines, while the beams related to the fifth generation of interactions are shown with solid lines. As in the previous generations of interactions, some of the beams of light are out-coupled from the eyepiece waveguide  2400  and each of the others is diffracted into a direction corresponding to a k-vector which belongs to one of the FOV rectangles at the 4 o&#39;clock, 6 o&#39;clock, or 8 o&#39;clock positions in the k-space annulus of the k-space diagrams in  FIGS.  24 B- 24 D . After this fifth generation of interactions, the output 1  beams and the output 2  beams may subsequently interact with the second CPE region  2455   b  and the cycle continues to repeat. 
       FIG.  24 J  illustrates, in k-space, higher-order pathways of light through the eyepiece waveguide  2400  shown in  FIG.  24 A . The k-space diagrams in  FIGS.  24 B- 24 D  show the first-order grating vectors associated with the ICG region  2440  and the CPE regions  2455 . The first-order grating vectors result in guided propagation modes represented by the FOV rectangles at the 4 o&#39;clock, 6 o&#39;clock, and 8 o&#39;clock positions in the k-space annulus. However, each of the CPE  2455  regions is also associated with positive and negative second-order grating vectors, some of which also result in guided propagation modes. 
     As already discussed herein, second-order grating vectors point in the same directions as the corresponding first-order grating vectors but have twice the magnitude. Thus, as shown in  FIG.  24 J , light beams in the propagation modes represented by the FOV rectangle at the 4 o&#39;clock position in the k-space annulus can be second-order diffracted by the first CPE region  2455   a  into propagation modes represented by the FOV rectangle at the 10 o&#39;clock position of the k-space annulus. Similarly, light beams in the propagation modes represented by the FOV rectangle at the 8 o&#39;clock position in the k-space annulus can be second-order diffracted by the second CPE region  2455   b  into propagation modes represented by the FOV rectangle at the 2 o&#39;clock position of the k-space annulus. From the 2 o&#39;clock and 10 o&#39;clock positions, first-order diffractions by the CPE regions  2455  can result in light beams in the propagation modes represented by the FOV rectangle at the 12 o&#39;clock position. 
     The propagation modes at the 10 o&#39;clock, 12 o&#39;clock, and 2 o&#39;clock positions in the k-space annulus, which are associated with second-order diffractions paths, can still be out-coupled to the user&#39;s eye. For example, light beams in the propagation modes represented by the FOV rectangle at the 10 o&#39;clock position in the k-space annulus can be first-order diffracted by the first CPE region  2455   a  as out-coupled beams represented by the FOV rectangle at the center of the k-space annulus. Similarly, light beams in the propagation modes represented by the FOV rectangle at the 2 o&#39;clock position in the k-space annulus can be first-order diffracted by the second CPE region  2455   b  as out-coupled beams represented by the FOV rectangle at the center of the k-space annulus. 
       FIG.  24 K  is a diagram which illustrates how beams of light spread through the eyepiece waveguide  2400  shown in  FIG.  24 A . A guided beam which enters the CPE regions  2455  propagating in the −y-direction from the ICG region  2440  is replicated into many beams, some traveling in the ±y-directions (corresponding to the FOV rectangles at the 6 o&#39;clock and 12 o&#39;clock positions in the k-space annulus), some traveling at ±60° with respect to the y-axis (corresponding to the FOV rectangles at the 2 o&#39;clock and 10 o&#39;clock positions in the k-space annulus), and some traveling at ±120° with respect to the y-axis (corresponding to the FOV rectangles at the 4 o&#39;clock and 8 o&#39;clock positions in the k-space annulus). In this way, light beams spread laterally throughout the entire eyepiece waveguide  2400 . 
       FIG.  25 A  is an edge view of an example eyepiece waveguide  2500  that has a single 2D combined pupil expander-extractor (CPE) grating region  2555 . The single 2D CPE region  2555  operates in a manner similar to the combined operation of the two 1D CPE regions  2455   a ,  2455   b  shown in  FIG.  24 A . For example, the CPE region  2555  spreads light laterally inside the eyepiece waveguide  2500 , similar to an OPE region, and it also extracts the light from the eyepiece waveguide  2500 , similar to an EPE region. 
     Although the single 2D CPE region  2555  in  FIG.  25 A  operates in a similar fashion as the two 1D CPE regions  2455   a ,  2455   b  in  FIG.  24 A  do collectively, it has a distinct structure in that it is made up of diffractive features that exhibit periodicity in two or more directions, whereas each of the 1D CPE regions  2455   a ,  2455   b  in  FIG.  24 A  is made up of diffractive features with periodicity in a single direction. Since the 2D CPE region  2555  in  FIG.  25 A  can perform the operations that are collectively performed by the two 1D CPE regions  2455   a ,  2455   b  in  FIG.  24 A , it can be formed on or in a single side of the eyepiece waveguide  2500 , whereas the CPE regions  2455   a ,  2455   b  in  FIG.  24 A  are respectively formed on or in both sides of the eyepiece waveguide  2400 . 
     The fact that the CPE region  2555  in  FIG.  25 A  is a single-sided 2D design—as opposed to the double-sided 1D design of  FIG.  24 A —may be advantageous in terms of fabrication, as an eyepiece waveguide (e.g.,  2500 ) with gratings on only one side may be less complicated to manufacture than an eyepiece waveguide (e.g.,  2400 ) with gratings on both sides. For example, manufacture of the double-sided design of  FIG.  24 A  may involve procedures to obtain precise angular alignment of grating  2455   a  with respect to grating  2455   b  on the opposite side, whereas manufacturer of the single-sided design of  FIG.  25 A  may omit those angular alignment procedures. Some embodiments of the single-sided design in  FIG.  25 A  may also offer certain advantages in optical performance because there is no risk of angular misalignment—and the degraded optical performance that can result therefrom—between gratings on opposite sides of the eyepiece waveguide. 
     The eyepiece waveguide  2500  shown in  FIG.  25 A  can be formed using a substrate made of an optically transmissive material. The eyepiece waveguide  2500  has an eye-facing side  2500   a  and an outward-facing side  2500   b . In the illustrated embodiment of the eyepiece waveguide  2500 , an ICG region  2540  is provided at the top center of the eyepiece waveguide  2500 , and the CPE region  2555  is provided below the ICG region  2540  on the eye-facing side  2400   a . However, other configurations are possible. For example, the CPE region  2555  and/or the ICG region  2540  may alternatively be provided on the outward-facing side  2500   b  of the eyepiece waveguide  2500  so that the ICG and CPE regions act in reflection or transmission modes. In addition, as in other embodiments, the ICG region could be positioned at other locations, such as the temporal or medial side of the eyepiece waveguide  2500 . 
     In some embodiments, the ICG region  2540  is a diffraction grating formed on or in a surface of the eyepiece waveguide  2500  (e.g., on the eye-facing side  2500   a ). The ICG region  2540  receives a set of input beams from an input device, such as a projector. As described elsewhere herein, the input beams can propagate from the input device generally in the ±z-direction until they are incident upon the ICG region  2540 . The ICG region  2540  diffracts those input beams so that at least some enter guided propagation modes within the eyepiece waveguide  2500 . 
     The illustrated embodiment of the diffraction grating inside the ICG region  2540  has one-dimensional periodicity (i.e., it is a 1D grating). The grating lines of the ICG region  2540  can be oriented so as to direct some of the diffracted beams in the −y-direction toward the CPE region  2555 . Thus, in the illustrated embodiment, the ICG region  2540  includes diffractive lines which extend in the ±x-direction and repeat periodically in the ±y-direction. As described elsewhere herein, the spacing between the diffractive lines which make up the ICG region  2540  can be set so as to couple the input beams of light into guided propagation modes inside the eyepiece waveguide  2500 . The diffracted beams from the ICG region  2540  then propagate via TIR toward the CPE region  2555 . 
     The CPE region  2555  in  FIG.  25 A  has two-dimensional periodicity (i.e., it is a 2D grating). The 2D grating  2555  has a corresponding set of k-space grating vectors that includes the grating vectors of both of the CPE regions  2455   a ,  2455   b  in the design of  FIGS.  24 A- 24 K . In some embodiments, the CPE region  2555  in  FIG.  25 A  consists of a crossed grating created by superposition of CPE region  2455   a  and CPE region  2455   b  from  FIGS.  24 A- 24 K . In some embodiments, the CPE region  2555  in  FIG.  25 A  consists of an array of diffractive features located at (e.g., centered on) the intersection points  2556  where the line gratings of CPE region  2455   a  and CPE region  2455   b  would cross if superimposed. 
     As already discussed above, CPE region  2455   a  in  FIGS.  24 A- 24 K  can be a 1D diffraction grating made up of diffractive lines oriented at an angle of −30° with respect to the y-axis. This 1D grating corresponds to a k-space grating vector that is labeled as grating vector G in  FIG.  25 B . Meanwhile, CPE region  2455   b  can be a 1D diffraction grating made up of diffractive lines oriented at an angle of +30° with respect to the y-axis. This 1D grating corresponds to a k-space grating vector that is labeled as grating vector H in  FIG.  25 B . The relative angle between the 1D grating of CPE region  2455   a  and the 1D grating of CPE region  2455   b , and between each of those gratings and the 1D grating of the ICG region  2440 , is substantially 60° (i.e., 60°±5°, or 60°±3°, or 60°±1°, or 60°±0.5°, or 60°±0.1°). Thus, the k-space grating vectors G, H for the CPE regions  2455   a ,  2455   b  in  FIGS.  24 A- 24 K  are likewise oriented at substantially 60° with respect to one another. The 2D grating of CPE region  2555  in  FIG.  25 A  likewise has these same first-order grating vectors G and H (in addition to higher-order grating vectors corresponding to the sums of ±G and ±H). 
     Besides being oriented at substantially 60° with respect to one another, the first-order grating vectors G, H of the 2D grating of the CPE region  2555  are also oriented at substantially 60° with respect to the grating vector of the ICG region  2540 . Furthermore, the 2D grating of the CPE region  2555  can be designed with spatial periodicities such that its first-order grating vectors G, H are substantially equal in magnitude to the first-order grating vector of the ICG region  2540 . The operation of the CPE region  2555  on the guided beams of light from the ICG region  2540  is described with respect to  FIG.  25 B . 
       FIG.  25 B  illustrates the operation of the 2D CPE region  2555  in both physical space and in k-space. A physical diagram of the eyepiece waveguide  2500  is shown at the top of  FIG.  25 B . A k-space diagram, KSD 1 , of the operation of the ICG region  2540  and the CPE region  2555  is shown at the bottom of  FIG.  25 B . 
     As already discussed, a set of input beams is incident on the ICG region  2540  of the eyepiece waveguide  2500  from an input device, such as a projector. This set of input beams is represented by the FOV rectangle shown in the center of k-space diagram KSD 1 . The diffraction grating in the ICG region  2540  has associated positive and negative grating vectors which point in the ±k y -directions. Thus, the k-space operation of the ICG region  2540  is to shift the central FOV rectangle to both the six o&#39;clock and 12 o&#39;clock positions on k-space diagram KSD 1 . (The FOV rectangle at the 12 o&#39;clock position corresponds to light beams propagating in the +y-direction. Since those beams exit the eyepiece waveguide  2500  out of its top edge, that particular FOV rectangle is not illustrated and those beams are not discussed further.) The length of the ICG grating vector can be set, based on the spacing of the diffractive lines and the wavelength of the light, such that the translated FOV rectangle at the six o&#39;clock position lies completely within the k-space annulus. 
     For ease of illustration, the physical diagram at the top of  FIG.  25 B  only shows one of the guided beams of light from the ICG region  2540  (i.e., guided beam  2541  corresponding to the center k-vector in the FOV rectangle located at the six o&#39;clock position of the k-space diagram KSD 1 ). It should be understood, however, that many guided input beams will be present, each of which will correspond to a different k-vector inside the FOV rectangle located at the 6 o&#39;clock position in the k-space annulus of KSD 1 . 
     Guided beam  2541  from the ICG region  2540  propagates downward through the eyepiece waveguide  2500  in the −y-direction, reflecting back and forth in TIR between the surface of the eye-facing side  2500   a  and the surface of the outward-facing side  2500   b . Each time guided beam  2541  reflects from the eye-facing side  2500   a , it can interact with the CPE region  2555 . The diffractive efficiency of the CPE region  2555  can be set so that only a portion of the power of each beam of light is diffracted with each of these interactions. For example, in some embodiments, the diffractive efficiency of the CPE region  2555  is 10% or less. The diffractive efficiency of the CPE region  2555  can be determined by, for example, the etch depth of the diffractive features. For example, in some embodiments, the heights of the diffractive features can range from about 5 nm up to about 200 nm. In some embodiments, the heights of the diffractive features can range from just greater than zero up to a half wavelength of guided beam  2541 . 
     The physical diagram at the top of  FIG.  25 B  shows the interactions of guided beam  2541  with the CPE region  2555  which cause light to spread laterally in both of the ±x-directions through the eyepiece waveguide  2500 . As guided beam  2541  propagates downward in the −y-direction through the eyepiece waveguide  2500 , portions of its power are diffracted at ±120° angles with respect to the y-axis during each interaction with the CPE region  2555 . The remaining portion of the power of guided beam  2541  continues propagating downward in the −y-direction until the next interaction with the CPE region  2555 , where portions of its power are again diffracted at the same ±120° angles. This process creates a plurality of spaced apart diffracted beams  2556   a ,  2556   b  which propagate through the eyepiece waveguide  2500  at a +120° angle and a −120° angle, respectively, with respect to the y-axis. Diffracted beams  2556   a , propagating at the +120° angle, are represented by the FOV rectangle located at the 8 o&#39;clock position in k-space diagram KSD 1 , while diffracted beams  2556   b , propagating at the −120° angle, are represented by the FOV rectangle located at the 4 o&#39;clock position. 
     With reference to k-space diagram KSD 1  at the bottom of  FIG.  25 B , the first-order grating vector G, which points at +60° with respect to the k y -axis, shifts the FOV rectangle from the six o&#39;clock position (which corresponds to the downward propagating guided beams from the ICG region  2540 ) to the eight o&#39;clock position (which corresponds to the diffracted beams  2556   a  propagating at the +120° angle with respect to the y-axis). Similarly, the first-order grating vector H, which points at −60° with respect to the k y -axis, shifts the FOV rectangle from the six o&#39;clock position (which corresponds to the downward propagating guided beams from the ICG region  2540 ) to the 4 o&#39;clock position (which corresponds to the diffracted beams  2556   b  propagating at the −120° angle with respect to the y-axis). 
     Once guided beams from the ICG region  2540  interact with the CPE region  2555  and are diffracted into the propagation states represented by the FOV rectangles at the 4 o&#39;clock and eight o&#39;clock positions of k-space diagram KSD 1 , they then interact again with the CPE region  2555  on a subsequent TIR bounce as they are guided through the eyepiece waveguide  2500 . This subsequent interaction of beams  2556   a  and  2556   b  with the CPE region  2555  can result in them being out-coupled from the eyepiece waveguide  2500  toward the user&#39;s eye. The out-coupled beams  2557  are shown in the physical diagram of the eyepiece waveguide  2500  at the top of  FIG.  25 B  as circled dots, indicating that those beams are propagating in the z-direction out of the page. The out-coupling of beams  2556   a ,  2556   b  by the CPE region  2555  can be understood by reference to k-space diagram KSD 1 . 
     The first-order grating vector H, which points at −60° with respect to the y-axis, shifts the FOV rectangle from the eight o&#39;clock position (which corresponds to the diffracted beams  2556   a  propagating at a +120° angle with respect to the y-axis) to the center of k-space diagram KSD 1  (which corresponds to out-coupled beams of light  2557  which are no longer in guided propagation modes inside the eyepiece waveguide  2500 ). Similarly, the first-order grating vector G, which points at +60° with respect to the y-axis, shifts the FOV rectangle from the four o&#39;clock position (which corresponds to the diffracted beams  2556   b  propagating at a −120° angle with respect to the y-axis) to the center of k-space diagram KSD 1  (which corresponds to out-coupled beams of light  2557  which are no longer in guided propagation modes inside the eyepiece waveguide  2500 ). 
     The physical diagram at the top of  FIG.  25 B  shows how the subsequent interactions of light beams  2556   a ,  2556   b  with the CPE region  2555  results in multiple spaced-apart out-coupled beams  2557 . As light beams  2556   a ,  2556   b  propagate at the ±120° angles with respect to the y-axis, portions of their power are out-coupled by each subsequent interaction with the CPE region  2555 . The remaining portions of the power of light beams  2556   a ,  2556   b  continue propagating at the ±120° angles with respect to the y-axis until the next interaction with the CPE region  2555 , where another portion of the power of those beams is out-coupled. This process creates a plurality of spaced-apart out-coupled beams  2557  which exit the eyepiece waveguide  2500  at different spatial locations and propagate toward the user&#39;s eye. As already noted, these out-coupled beams  2557  are represented by the FOV rectangle located at the center of k-space diagram KSD 1 . 
     In addition, although not illustrated in  FIG.  25 B , light can also spread through the eyepiece waveguide  2500  in the manner shown in  FIG.  24 J . That is, due to higher-order diffractions, light can also spread in directions represented by FOV rectangles at 2 o&#39;clock, 10 o&#39;clock, and 12 o&#39;clock positions of the k-space annulus. 
     As shown in k-space diagram KSD 1  in  FIG.  25 B , light beams propagate through the eyepiece waveguide  2500  along paths in k-space which are substantially similar to equilateral triangles. These substantially-equilateral triangular paths allow the CPE region  2555  to both spread light laterally in the eyepiece waveguide  2500  and to out-couple light from the eyepiece waveguide to the user&#39;s eye. 
       FIG.  26 A  is an edge view of an example eyepiece waveguide  2600  that has a 2D combined pupil expander-extractor (CPE) grating region  2655  on each of its sides. Each of the 2D CPE regions  2655   a ,  2655   b  can be similar to the 2D CPE region  2555  of  FIGS.  25 A- 25 B . For example, CPE region  2655   a  can be an instance of CPE region  2555  located on the eye-facing side  2600   a  of the eyepiece waveguide  2600 , while CPE region  2655   b  can be an instance of CPE region  2555  located on the outward-facing side  2600   b . The two 2D CPE regions  2655   a ,  2655   b  can partially or wholly overlap in the x- and y-directions, and can be angularly aligned with one another. The double-sided embodiment with 2D CPE regions  2655  on both sides of the eyepiece waveguide functions similarly to the single-sided embodiment of  FIGS.  25 A- 25 B . In k-space, the operation of the double-sided embodiment of  FIG.  26 A  is the same as that of the single-sided embodiment in  FIG.  25 A . However, the double-sided embodiment does increase the density of output beams as compared to the single-sided embodiment. The increased density of output beams can be useful in addressing the design complications shown in  FIGS.  26 B and  26 C . 
       FIG.  26 B  illustrates the so-called “screen door effect” which is an image artifact that is related to the density of output beams from an eyepiece waveguide. The top panel in  FIG.  26 B  shows an eyepiece waveguide  2600  with a diffraction grating on the top surface. A guided light beam  2656  is shown propagating through the eyepiece waveguide via TIR. At the location of each interaction of the guided light beam  2656  with the diffraction grating, an output beam  2657  is out-coupled from the eyepiece waveguide  2600 . If the entrance pupil of the user&#39;s eye happens to be aligned with one of the output beams  2657 , as shown in the top panel of  FIG.  26 B , then the user will see a bright spot. (Note: the respective dimensions of the eyepiece waveguide  2600 , the light beams  2656 ,  2657 , and the entrance pupil of the eye are not necessarily drawn to scale.) 
     The bottom panel of  FIG.  26 B  shows the same eyepiece waveguide  2600 , but this time the guided beam  2656  and the output beams  2657  correspond to a different region of the field of view of the output image being displayed. The output beams  2657  therefore exit the eyepiece waveguide at a different angle such that the entrance pupil of the user&#39;s eye is not aligned with any of the output beams  2657 . In this instance, the user will see a dark spot. 
     As the density of the output beams  2657  increases, so does the likelihood that one or more will always intersect with the entrance pupil of the eye, for all regions of the FOV of the output image. Therefore, eyepiece waveguide designs with higher densities of output beams  2657  may be advantageous. 
     The severity of the screen door effect is dependent on multiple factors, including the diameter of the light beams and the thickness of the eyepiece waveguide  2600 . One technique for increasing the density of the output beams  2657  is to decrease the thickness of the eyepiece waveguide. As is evident from  FIG.  26 B , if the thickness of the eyepiece waveguide  2600  were smaller, the guided beam  2656  would travel a shorter distance in the x-direction between interactions with the diffraction grating and the density of the output beams  2657  would increase. If a beam diameter of about 1 mm is assumed, it may be advantageous for the thickness of the eyepiece waveguide  2600  to be 325 μm or smaller so as to avoid an unacceptable degree of screen door effect. However, decreasing the thickness of the eyepiece waveguide  2600  can cause other difficulties, as shown in  FIG.  26 C . 
       FIG.  26 C  illustrates input coupling grating re-bounce, which is an effect that can cause light to be disadvantageously lost from an eyepiece waveguide.  FIG.  26 C  illustrates an eyepiece waveguide  2600  with an input coupling grating (ICG). An input beam  2602  is incident on the ICG and is coupled into a guided propagation mode by the ICG. The resulting guided beam  2656  then propagates through the eyepiece waveguide  2600  via TIR. Depending on a variety of factors, including the size of the ICG, the thickness of the eyepiece waveguide  2600 , and the light beam diameter, the guided beam  2656  may interact with the ICG after having reflected from the opposite surface of the eyepiece waveguide  2600 . This situation is illustrated in  FIG.  26 C . The region where this interaction occurs between the guided beam  2656  and the ICG is labeled as the re-bounce region. 
     In the re-bounce region, some of the power of the guided beam  2656  may be out-coupled from the eyepiece waveguide  2600 . For example, if the input beam  2602  is coupled into the eyepiece waveguide  2600  by the +1 diffractive order of the ICG, then the −1 diffractive order will out-couple the beam if it subsequently interacts with the ICG in the re-bounce region. The ICG is typically designed with a high diffractive efficiency in order to in-couple as much light as possible, but that high diffractive efficiency also results in strong out-coupling in the re-bounce region. Thus, ICG re-bounce results in lost light and reduced efficiency. 
     The ICG re-bounce effect can be lessened by increasing the thickness of the eyepiece waveguide. As is evident from  FIG.  26 C , if the thickness of the eyepiece waveguide  2600  were larger, the guided beam  2656  would travel a larger distance in the x-direction after diffracting from the ICG and before returning to the surface of the eyepiece waveguide  2600  that the ICG is located on. This would reduce the size of the re-bounce region, or even eliminate it completely. If a beam diameter of about 1 mm is assumed, it may be advantageous for the thickness of the eyepiece waveguide  2600  to be 650 μm or larger so as to avoid ICG re-bounce. 
     As illustrated by  FIGS.  26 B and  26 C , the thickness of the eyepiece waveguide  2600  affects the severity of both the screen door effect and the ICG re-bounce effect but in opposite ways. Decreasing the thickness of the eyepiece waveguide  2600  lessens the screen door effect but worsens ICG re-bounce. Increasing the thickness of the eyepiece waveguide  2600  lessens ICG re-bounce but worsens the screen door effect. Thus, in some embodiments, it would be advantageous to size the thickness of the eyepiece waveguide  2600  large enough to avoid ICG re-bounce while still limiting the screen door effect to an acceptable degree. This can be accomplished by increasing the density of output beams  2657  that are supported by an eyepiece waveguide of a given thickness. And this is precisely what is accomplished by the double-sided embodiment of the eyepiece waveguide  2600  that is shown in  FIG.  26 A . 
       FIG.  26 D  illustrates how the double-sided 2D CPE gratings in  FIG.  26 A  increase the density of output beams from the eyepiece waveguide  2600 . The top panel in FIG. 26D shows how the screen door effect is reduced for the central portion of the FOV of the output image, while the bottom panel shows how the screen door effect is reduced for a peripheral portion of the FOV of the output image. 
     The top panel in  FIG.  26 D  shows a guided beam  2656  propagating through the eyepiece waveguide  2600 . In the top panel in  FIG.  26 D , guided beam  2656  corresponds to a k-vector located at the center of the FOV rectangle for the image being displayed by the eyepiece waveguide  2600 . A first 2D CPE grating  2655   a  is provided on the top surface of the eyepiece waveguide, and a second 2D CPE grating  2655   b  is provided on the bottom surface. Output beams  2657   a  result from interactions between guided beam  2656  and CPE grating  2655   a  on the top surface of the eyepiece waveguide  2600 , while output beams  2657   b  result from interactions between guided beam  2656  and CPE grating  2655   b  on the bottom surface. Since the output beams  2655   a ,  2655   b  correspond to the center of the FOV of the output image, they exit the eyepiece waveguide normal to its surface. As shown in  FIG.  26 D , output beams  2657   a  and  2657   b  exit the eyepiece waveguide  2600  at alternating positions in the x-direction. Thus the density of output beams is increased. 
     The bottom panel in  FIG.  26 D  also shows a guided beam  2656  propagating through the eyepiece waveguide  2600 . In the bottom panel in  FIG.  26 D , guided beam  2656  corresponds to a k-vector located at the periphery of the FOV rectangle for the image being displayed by the eyepiece waveguide  2600 . A first 2D CPE grating  2655   a  is provided on the top surface of the eyepiece waveguide, and a second 2D CPE grating  2655   b  is provided on the bottom surface. Output beams  2657   a  result from interactions between guided beam  2656  and CPE grating  2655   a  on the top surface of the eyepiece waveguide  2600 , while output beams  2657   b  result from interactions between guided beam  2656  and CPE grating  2655   b  on the bottom surface. Since the output beams  2655   a ,  2655   b  correspond to the periphery of the FOV of the output image, they exit the eyepiece waveguide at an angle. As shown in  FIG.  26 D , output beams  2657   a  and  2657   b  exit the eyepiece waveguide  2600  at alternating positions in the x-direction. Thus the density of output beams is increased. 
       FIG.  26 E  illustrates the density of output beams  2657  for the eyepiece waveguides shown in  FIG.  24 A  (double-sided 1D CPE gratings),  FIG.  25 A  (single-sided 2D CPE grating), and  FIG.  26 A  (double-sided 2D CPE gratings). The solid lines represent light beams propagating via TIR from surface A (e.g., the eye-facing surface) of the eyepiece waveguide to surface B (e.g., the outward-facing surface), while the dashed lines represent light beams propagating from surface B to surface A. Each point where a solid line turns to a dashed line, or vice versa, represents an interaction of a light beam with one of the surfaces of an eyepiece waveguide. 
     The left panel shows the density of output beams  2457  for the double-sided embodiment of  FIG.  24 A , which uses 1D CPE gratings  2455   a ,  2455   b . In that embodiment, the 1D CPE gratings  2455   a ,  2455   b  divide a guided beam  2441  into branches of spaced-apart diffracted beams  2456 , but this occurs only with every other surface interaction. A part of each of those diffracted beams  2456  is then out-coupled as an output beam  2457  with every other surface interaction. 
     The middle panel shows the density of output beams  2557  for the single-sided embodiment of  FIG.  25 A , which uses a 2D CPE grating  2555  on one side of an eyepiece waveguide  2500 . In that embodiment, the 2D CPE grating  2555  divides a guided beam  2541  into branches of spaced-apart diffracted beams  2556 , and two branches are created with every other surface interaction. A part of each of those diffracted beams  2556  is then out-coupled as an output beam  2557  with every other surface interaction. 
     The right panel shows the density of output beams  2657  for the double-sided embodiment of  FIG.  26 A , which uses 2D CPE gratings  2655   a ,  2655   b  on both sides of an eyepiece waveguide  2600 . In that embodiment, the CPE gratings  2655   a ,  2655   b  divide a guided input beam  2641  into branches of spaced-apart diffracted beams  2656 , and two branches are created with every surface interaction, rather than every other surface interaction. In addition, a part of each of those diffracted beams  2656  is then out-coupled as an output beam  2557  with every surface interaction, rather than every other surface interaction. The double-sided embodiment of  FIG.  26 A  therefore doubles the density of output beams  2557  in the x-direction and in the y-direction. This results in a 4X increase in the density of output beams  2557  per unit area when compared with the single-sided design in  FIG.  25 A . 
     Due to the increased density of output beams  2557  from the double-sided eyepiece waveguide  2600  with 2D CPE gratings  2655   a ,  2655   b , this design can be used to limit the severity of the screen door effect while still allowing for the eyepiece waveguide  2600  to be thick enough to reduce or eliminate ICG re-bounce. For example, in some embodiments, the eyepiece waveguide  2600  may be as thick as approximately one third (e.g., ±10%, or ±20%, or ±30%) of the diameter of the input beams of light. 
       FIG.  26 F  shows example simulated images produced by eyepiece waveguides with 2D CPE gratings; images for both the case of the single-sided embodiment of  FIG.  25 A  and the double-sided embodiment of  FIG.  26 A  are shown. Images i) and ii) were produced by the single-sided embodiment of  FIG.  25 A . Image i) was created using an LED light source (with a spectrum of about 20 nm), whereas image ii) was created using a laser light source (with a spectrum of about 2 nm). The LED image has better uniformity than the laser image—due to a smearing effect from the broader bandwidth of the LED—but high-frequency screen door artifact is present in both images. 
     Images iii) and iv) were produced by the double-sided embodiment of  FIG.  26 A . Image iii) was created using the LED light source, while image iv) was created using the laser light source. There is a clear reduction in high-frequency screen door artifact in the images produced by the double-sided embodiment of  FIG.  26 A . This reduction in screen door artifact is attributable to the increased density of output beams for the double-sided embodiment. 
     FOV Expansion by Splitting the FOV of Each of One or More Color Components of an Image Amongst Multiple Eyepiece Waveguides 
     As already discussed above, the various AR eyepiece waveguides in this disclosure can use diffractive features to in-couple light from an external source (e.g., from a projector) such that the light propagates in the eyepiece waveguide via total internal reflection. According to a k-space description, the beams of light from the external source correspond to a set of k-vectors. The k-space dimensions of the this set of k-vectors are determined by the field of view (FOV) of the input image from the external source. The in-coupling of the input beams of light into the eyepiece waveguide by the diffractive features causes the k-vectors of the free-space input light beams to be translated into the k-space annulus corresponding to the eyepiece waveguide. Any light beam whose k-vector is translated by the diffractive features into the k-space annulus can propagate in guided fashion within the eyepiece waveguide. 
     The width of the eyepiece waveguide&#39;s k-space annulus—which is dependent upon the refractive index of the material from which the eyepiece waveguide is formed—determines the range of k-vectors—and, hence, the range of propagation angles—which can be guided within the eyepiece waveguide. Thus, the refractive index of the eyepiece waveguide, and the corresponding width of its k-space annulus, has typically been thought to place limitations on the maximum FOV which can be projected by the eyepiece waveguide. This disclosure describes techniques for increasing the FOV of images projected by an eyepiece waveguide, notwithstanding the limitations imposed by refractive index. 
     For example,  FIGS.  19 - 23 K  illustrate embodiments where the FOV of an input image is split into multiple different sub-portions in k-space. The light beams corresponding to each sub-portion of the input image FOV are routed along different paths through the eyepiece waveguide and then the sub-portions of the image FOV are appropriately re-combined at the output of the eyepiece waveguide. As already discussed, the techniques illustrated by  FIGS.  19 - 23 K  can be implemented in a single eyepiece waveguide. Now, this section of the disclosure will disclose techniques for enhancing the maximum image FOV of a stack of two or more eyepiece waveguides. The FOV of, for example, a color component of an input image to be displayed by the eyepiece waveguide stack can be split into multiple different sub-portions in k-space using, for example, diffractive features of input coupling grating (ICG) regions formed on or in the eyepiece waveguides in the stack. The sub-portions of the image FOV can then be routed through different eyepiece waveguides in the stack; each different k-space sub-portion of the image FOV can be output from one of the eyepiece waveguides in the stack, and the k-space sub-portions of the FOV can be appropriately re-combined at the output of the eyepiece waveguide stack to re-create the full FOV of the image. The techniques described in this section of the disclosure can be implemented alone or in combination with any of the techniques described elsewhere herein. 
       FIG.  27 A  illustrates an example embodiment of an eyepiece waveguide stack with enhanced FOV. The FOV enhancement is made possible by the split propagation of different sub-portions of the FOV of each color component of an input image through different eyepiece waveguides in the stack. The eyepiece waveguide stack in  FIG.  27 A  includes three separate eyepiece waveguides: a first eyepiece waveguide  2700   a , a second eyepiece waveguide  2700   b , and a third eyepiece waveguide  2700   c . The individual eyepiece waveguides  2700   a ,  2700   b ,  2700   c  can each have the same dimensions and, with reference to the coordinate axes shown in the figure, can be arranged in planes parallel to the x-y plane, generally aligned in the x- and y-directions. In other embodiments, however, the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  can have differing dimensions and/or alignments. The eyepiece waveguides  2700   a ,  2700   b ,  2700   c  are separated either by air gaps or lower-index materials in the z-direction. Although  FIG.  27 A  illustrates a stack of three eyepiece waveguides, in other embodiments two or more can be used (e.g., a stack of 2, 3, 4, 5, 6, 7, 8, 9 waveguides, etc.). 
     The first eyepiece waveguide  2700   a  is designated as the “first” because it is first in the optical path (generally along the z-axis) of the input beams  2702 ,  2704 ,  2706  of light from an external source (e.g., a projector such as projector system  520  described and illustrated herein). Similarly, the second eyepiece waveguide  2700   b  is designated as the “second” because it is second along the optical path of the input beams  2702 ,  2704 ,  2706 , and the third eyepiece waveguide  2700   c  is designated as the “third” because it is third along that optical path. 
     The first eyepiece waveguide  2700   a  includes an ICG region  2740   a . The ICG region  2740   a  of the first eyepiece waveguide  2700   a  is made up of diffractive features designed to in-couple at least a portion of the input beams for at least one color component of the input image provided from the external source. Similarly, the second eyepiece waveguide  2700   b  includes an ICG region  2740   b , and the third eyepiece waveguide  2700   c  includes an ICG region  2740   c , which likewise have diffractive features designed to in-couple at least a portion of the input beams for at least one color component of the input image into those respective eyepiece waveguides. In the embodiment illustrated in  FIG.  27 A , the ICG regions  2740   a ,  2740   b ,  2740   c  are laterally aligned with one another and each ICG region is designed to receive input light beams from all three color components of the input image. This type of configuration is well-suited for use in a system that employs an in-line projector pupil. 
     Similar to other ICG regions described herein, ICG regions  2740   a ,  2740   b ,  2740   c  can each be made up of diffractive features such as line gratings. Each of the ICG regions  2740   a ,  2740   b ,  2740   c  has at least one corresponding k-space grating vector (e.g., a first-order grating vector) that points in the grating&#39;s direction of periodicity. The magnitude of the first-order grating vector for each of the ICG regions  2740   a ,  2740   b ,  2740   c  is inversely proportional to the spatial period, Λ, of its diffractive features. Thus, the diffractive effect of each of the ICG regions  2740   a ,  2740   b ,  2740   c  is partially determined by its respective spatial periodicity. The diffractive effect of a given ICG region is also dependent in part on the wavelength of the input light, as discussed further below. For example, a given spatial period, Λ, will appear larger in comparison to higher-frequency light (e.g., blue light) with relatively short wavelengths and will therefore diffract such light at smaller angles than lower-frequency light (e.g., red light) with longer wavelengths. Conversely, the spatial period, Λ, will appear smaller in comparison to lower-frequency light (e.g., red light) with relatively long wavelengths and will therefore diffract such light at larger angles than higher-frequency light (e.g., blue light) with shorter wavelengths. 
     In some embodiments, the first eyepiece waveguide  2700   a  has a large first-order grating vector and can alternately be referred to as the large-grating-vector layer or eyepiece waveguide  2700   a . The second eyepiece waveguide  2700   b  can have a medium-sized first-order grating vector and can alternately be referred to as the medium-grating-vector layer or eyepiece waveguide  2700   b . The third eyepiece waveguide  2700   c  can have a small first-order grating vector and can alternately be referred to as the small-grating-vector layer or eyepiece waveguide  2700   c . In this context, the terms large grating vector, medium grating vector, and small grating vector are intended only to designate the relative magnitudes of the first-order grating vectors corresponding to the respective ICG regions  2740   a ,  2740   b ,  2740   c  of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c ; these terms are not intended to imply any specific grating vector magnitude for any of the ICG regions. 
     Since the first eyepiece waveguide  2700   a  has the largest grating vector, the diffractive features of its ICG region  2740   a  have the finest pitch (i.e., the smallest spatial periodicity). In contrast, since the third eyepiece waveguide  2700   c  has the smallest grating vector, the diffractive features of its ICG region  2740   c  have the largest pitch (i.e., the largest spatial periodicity). The spatial periodicity of the diffractive features of the ICG region  2740   b  of the second eyepiece waveguide  2700   b  is in between that of ICG region  2740   a  and ICG region  2740   c.    
       FIG.  27 A  shows examples of input beams  2702 ,  2704 ,  2706  of light which are normally-incident on the eyepiece waveguide stack from the external source. Each of the input beams  2702 ,  2704 ,  2706  represents a color component of the input image from the external source. An input beam  2702  of red light (e.g., 625 nm ±20 nm, with a bandwidth of 50 nm or less), an input beam  2704  of green light (e.g., 525 nm ±20 nm, with a bandwidth of 50 nm or less), and an input beam  2706  of blue light (e.g., 455 nm ±20 nm, with a bandwidth of 50 nm or less) are illustrated, though other color components can also be used. Although only a single input beam for each color component is illustrated, each beam is intended to represent all of the input beams for the entire FOV of the corresponding color component (e.g., red, green, blue) of the input image. For example, input beam  2702  represents all of the input beams for the red color component of the input image. Similarly, input beam  2704  represents all of the input beams for the green color component of the input image, and beam  2706  represents all of the input beams for the blue color component of the input image. 
     As illustrated, each of the input beams  2702 ,  2704 ,  2706  is first incident on the ICG region  2740   a  of the large-grating-vector eyepiece waveguide  2700   a . ICG region  2740   a  is formed on or in the first eyepiece waveguide  2700   a . In the illustrated embodiment, ICG region  2740   a  is a partially reflective, partially transmissive diffraction grating formed on a back surface of the large-grating-vector eyepiece waveguide  2700   a . Other types of diffraction gratings and locations (e.g., a front surface of eyepiece waveguide  2700   a ) can also be used for ICG region  2740   a.    
     As described further below with respect to  FIGS.  27 C- 27 E , ICG region  2740   a  can be designed such that it diffracts only some (or possibly even none) of the beams represented by the input beam  2702  of red light into guided propagation modes within the first eyepiece waveguide  2700   a . It is important to again note that input beam  2702  represents all of the input beams for the red color component of the input image. In physical space, each of those beams has a different propagation angle that corresponds to a point in the input image. In k-space, this means that each of those beams has a k-vector that lies inside a shape (e.g., a rectangle) that defines the FOV of the input image. The same is true of input beam  2704  with respect to the input beams of green light, and of input beam  2706  with respect to the input beams of blue light. ICG region  2740   a  can be designed such that it likewise diffracts only some (or possibly even none) of the beams represented by the input beam  2704  of green light into guided propagation modes within the first eyepiece waveguide  2700   a , and such that it diffracts only some (or possibly even none) of the beams represented by the input beam  2706  of blue light into guided propagation modes. 
     In some embodiments, the FOV of the input image can be large enough such that it has at least one k-space dimension (e.g., the k-space dimension in the direction of the grating vector of ICG region  2740   a ) that is larger than the width of the k-space annulus corresponding to the first eyepiece waveguide  2700   a  (i.e., the FOV of the input image is wider than the range of propagation angles in the thickness direction of the waveguide which can be supported in TIR). ICG region  2740   a  can be designed with a feature pitch that yields a first-order grating vector with a magnitude that translates some (or possibly even all) of the k-vectors corresponding to the beams respectively represented by input beam  2702 , input beam  2704 , and input beam  2706 , to positions that lie outside of the k-space annulus—e.g., beyond its outer perimeter—for the first eyepiece waveguide  2700   a . Those beams whose k-vectors are translated by the first-order grating vector of ICG region  2740   a  into the k-space annulus are diffracted by ICG region  2740   a  and enter guided propagation modes within the first eyepiece waveguide  2700   a . (It should be understood, however, that ICG region  2740   a  may, in some embodiments, have a diffractive efficiency less than 100% (e.g., 5-90%). Thus, even in the case of those input beams which ICG region  2740   a  can diffract into guided propagation modes within the first eyepiece waveguide  2700   a , some percentage of the power of those beams may be diffracted while the remaining percentage of the power of those beams may continue on toward the second eyepiece waveguide  2700   b .) Those input beams whose k-vectors are not translated by the first-order grating vector of ICG region  2740   a  into the k-space annulus for the first eyepiece waveguide  2700   a  are not diffracted by ICG region  2740   a  and instead continue on toward the second eyepiece waveguide  2700   b.    
     Thus, interaction between the red light beams (represented by input beam  2702 ) and ICG region  2740   a  can result in some first-order diffracted beams which enter guided propagation modes, as represented in  FIG.  27 A  by first-order diffracted red beam  2702   a . The red light beams which are not diffracted by ICG region  2740   a  simply continue on toward the second eyepiece waveguide  2700   b , as represented by the continuation of input beam  2702 . In a similar manner, interaction of the green light beams (represented by input beam  2704 ) and the blue light beams (represented by input beam  2706 ) with ICG region  2740   a  can likewise result in some first-order diffracted green and blue beams that enter guided propagation modes inside the first eyepiece waveguide  2700   a , as represented in  FIG.  27 A  by green and blue first-order diffracted beams  2704   a ,  2706   a , respectively. The green and blue light beams which are not diffracted by ICG region  2740   a  continue to propagate toward the second eyepiece waveguide  2700   b , as represented by the continuation of input beam  2704  and by the continuation of input beam  2706 , respectively. 
     After passing through the first eyepiece waveguide  2700   a , the input beams  2702 ,  2704 ,  2706  then propagate to the second eyepiece waveguide  2700   b  and are incident upon its ICG region  2740   b . ICG region  2740   b  is illustrated as a partially reflective, partially transmissive diffraction grating formed on or in a back surface of the medium-grating-vector eyepiece waveguide  2700   b , though other locations (e.g., a front surface of eyepiece waveguide  2700   b ) and types of gratings can also be used. 
     Similar to ICG region  2740   a , ICG region  2740   b  can be designed with a feature pitch that yields a first-order grating vector with a magnitude that translates some of the k-vectors corresponding to the respective sets of beams represented by input beam  2702 , input beam  2704 , and input beam  2706 , to positions that lie outside of the k-space annulus—e.g., beyond its outer perimeter or inside its inner perimeter—for the second eyepiece waveguide  2700   b . Thus, interaction between the red light beams (represented by input beam  2702 ) and ICG region  2740   b  can result in some first-order diffracted beams which enter guided propagation modes, as represented in  FIG.  27 A  by first-order diffracted red beam  2702   b . (Once again, however, ICG region  2740   b  may, in some embodiments, have a diffractive efficiency less than 100% (e.g., 5-90%) such that some percentage of the power—of those beams which can be in-coupled—is diffracted, while the remaining percentage of the power of those beams is not diffracted.) The red light beams which are not diffracted by ICG region  2740   b  simply continue on toward the third eyepiece waveguide  2700   c , as represented by the continuation of input beam  2702 . In a similar manner, interaction of the green light beams (represented by input beam  2704 ) and the blue light beams (represented by input beam  2706 ) with ICG region  2740   b  can likewise result in some first-order diffracted green and blue beams that enter guided propagation modes inside the second eyepiece waveguide  2700   b , as represented in  FIG.  27 A  by green and blue first-order diffracted beams  2704   b ,  2706   b , respectively. The green and blue light beams which are not diffracted by ICG region  2740   b  continue to propagate toward the third eyepiece waveguide  2700   c , as represented by the continuation of input beam  2704  and by the continuation of input beam  2706 , respectively. 
     For simplicity,  FIG.  27 A  shows the same red beam  2702 , green beam  2704 , and blue beam  2706  seemingly being diffracted into guided propagation modes in both the large-grating-vector eyepiece waveguide  2700   a  and the medium-grating-vector eyepiece waveguide  2700   b . But, once again, each of the input beams  2702 ,  2704 ,  2706  represents all of the beams in the FOV of the corresponding color component and, in actual operation, different subsets of the set of input beams for each color component may be diffracted into guided propagation modes in different eyepiece waveguides  2700   a ,  2700   b ,  2700   c  in the stack. 
     The undiffracted beams  2702 ,  2704 ,  2706  from the ICG region  2740   b  of the medium-grating-vector eyepiece waveguide  2700   b  continue on to an optical filter  2745 . The optical filter  2745  can be located between the ICG region  2740   b  of the medium-grating-vector layer  2700   b  and the ICG region  2740   c  of the small-grating-vector layer  2700   c  (e.g., in the gap between the second eyepiece waveguide  2700   b  and the third eyepiece waveguide  2700   c ). In some embodiments, the optical filter  2745  is a red filter which effectively absorbs at least 90% of the green light (represented by beam  2704 ) and at least 90% of the blue light (represented by beam  2706 ). In some embodiments, the −3 dB cutoff for the optical filter  2745  may be located within the range of 560-600 nm, though other locations can also be used. The purpose of the optical filter  2745  is to absorb green and blue light which could otherwise be second-order diffracted by ICG region  2740   c  into guided propagation modes within the small-grating-vector layer  2700   c , as discussed further with respect to the k-space diagrams shown in  FIGS.  27 C- 27 E . Such second-order diffractions could result in artifacts, such as ghost images, and thus it may be advantageous to prevent them with the use of the optical filter  2745 . In some embodiments, one or more optical dyes can be added to the substrate material for the third eyepiece waveguide  2700   c . The optical dye(s) can absorb green and blue light, similar to optical filter  2745 . Thus, the dye(s) added to the substrate material for the third eyepiece waveguide  2700   c  could be used in place of optical filter  2745  or in addition to it. Additionally and/or alternatively, second-order diffractions may be mitigated by designing ICG region  2740   c  to have low second-order diffraction efficiency (e.g., less than 50%, or 40%, or 30%, or 20%, or 10%) for green and blue light. 
     Since, in some embodiments, beams of green and blue light are blocked by the optical filter  2745 , only the red light beams (represented by beam  2702 ) are illustrated as being incident on the ICG region  2740   c  of the third eyepiece waveguide  2700   c . ICG region  2740   c  is illustrated as a reflective diffraction grating formed on a back surface of the third eyepiece waveguide  2700   c . (A highly reflective diffraction grating can be used for ICG region  2740   c  because there are no remaining eyepiece waveguides behind the third eyepiece waveguide  2700   c  in the illustrated embodiment.) For example, a metallized diffraction grating could be used. Other locations and types of gratings are also possible, however. 
     Similar to ICG regions  2740   a  and  2740   b , ICG region  2740   c  can be designed with a feature pitch that yields a first-order grating vector with a magnitude that translates some of the k-vectors corresponding to the set of red beams represented by input beam  2702  to positions that lie outside of the k-space annulus—e.g., inside its inner perimeter—for the third eyepiece waveguide  2700   c . Thus, interaction between the red light beams (represented by input beam  2702 ) and ICG region  2740   c  can result in some first-order diffracted beams which enter guided propagation modes, as represented in  FIG.  27 A  by first-order diffracted red beam  2702   c . Red light beams which are diffracted by ICG region  2740   c  but not into guided propagation modes (i.e., those beams whose k-vectors are translated to a position inside the inner perimeter of the k-space annulus) are diffracted into free space and may be blocked from reaching the user by, for example, appropriately-placed absorptive baffle material 
     In some embodiments, ICG region  2740   c  may have a diffractive efficiency less than 100% (e.g., 5-90%) such that some red light, green light, and/or blue light may reflect from ICG region  2740   c  back through the stack of eyepiece waveguides  2700   a ,  2700   b ,  2700   c . This will typically not substantially degrade the image provided there is good parallelism between the three eyepiece waveguides  2700   a ,  2700   b ,  2700   c . The reflected path from a surface located at  2740   c  can even be advantageous for efficiency in certain designs. 
     Once again, in some embodiments the system is intentionally designed with an input image FOV that has a k-space dimension—in the direction of the grating vectors of ICG regions  2740   a ,  2740   b ,  2740   c —that is larger than the width of the k-space annulus for the eyepiece waveguides  2700   a ,  2700   b ,  2700   c . This means that only a subset of the input beams for each color component of the input image is in-coupled into guided propagation modes within each of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c . In fact, in some cases, it may be true that none of the input beams for a particular color component of the input image are diffracted into guided propagation states within one or more of the waveguides. Furthermore, ICG regions  2740   a ,  2740   b ,  2740   c  are designed with feature pitches that yield first-order grating vectors with magnitudes that cause different subsets of the beams of light for each color component to be diffracted into guided propagation modes in the various different eyepiece waveguides  2700   a ,  2700   b ,  2700   c  which make up the stack. This is explained in detail below with reference to the k-space diagrams in  FIGS.  27 C- 27 E . 
       FIG.  27 A  illustrates a configuration of ICG regions  2740   a ,  2740   b ,  2740   c  which is well-suited for the situation where the input beams  2702 ,  2704 ,  2706  are provided to the eyepiece waveguide stack via an in-line projector pupil. In this type of embodiment, the input beams of light for multiple color components of the input image are incident on the same ICG regions. However, a split projector pupil configuration can also be used. This is shown in  FIG.  27 B . 
       FIG.  27 B  illustrates another example embodiment of an eyepiece waveguide stack with enhanced FOV. The operation of the eyepiece waveguide stack in  FIG.  27 B  is similar to the one in  FIG.  27 A  except that the stack in  FIG.  27 B  is well-suited to a split projector pupil configuration. In this type of embodiment, the input beams of light for each color component of the input image are incident on different ICG regions. For example, the ICG region  2740   a  for the first eyepiece waveguide  2700   a  can be spatially divided into a first section  2740   a   1  which receives the input beams  2702  of red light, a second section  2740   a   2  which receives the input beams  2704  of green light, and a third section  2740   a   3  which receives the input beams  2706  of blue light. The spatial period, Λ, of the diffractive features in the first section  2740   a   1 , the second section  2740   a   2 , and the third section  2740   a   3  can be the same. 
     As was the case in  FIG.  27 A , each input beam  2702 ,  2704 ,  2706  is intended to represent an entire set of input beams—with different propagation angles—for the FOV of a color component (e.g., red, green, blue) of the input image. ICG region  2740   a   1  may couple some (or possibly none) of the input beams of red light into guided propagation modes within the first eyepiece waveguide  2700   a  (as represented by first-order diffracted beam  2702   a ), while ICG region  2740   a   2  may couple some (or possibly none) of the input beams of green light into guided propagation modes (as represented by first-order diffracted beam  2704   a ), and ICG region  2740   a   3  may couple some (or possibly none) of the input beams of blue light into guided propagation modes (as represented by first-order diffracted beam  2706   a ). The input beams of red light, green light, and blue light which are not in-coupled into the first eyepiece waveguide  2700   a  can proceed on to the second eyepiece waveguide  2700   b . In some embodiments, ICG region  2740   a   1  is optional, as discussed with respect to  FIG.  27 F . 
     In the second eyepiece waveguide  2700   b , ICG region  2740   b  can be spatially divided into a first section  2740   b   1  which receives the red light, a second section  2740   b   2  which receives the green light, and a third section  2740   b   3  which receives the blue light. In some embodiments, the sections  2740   b   1 ,  2740   b   2 ,  2740   b   3  of ICG region  2740   b  can be aligned in the x- and/or y-directions with the respective sections  2740   a   1 ,  2740   a   2 ,  2740   a   3  of ICG region  2740   a . The first ICG section  2740   b   1  in the second eyepiece waveguide  2700   b  can be a partially reflective, partially transmissive grating located on or in the second eyepiece waveguide. In contrast, the second ICG section  2740   b   2  and the third ICG section  2740   b   3  can be, for example, highly reflective (e.g., metallized) gratings located on a back surface of the second eyepiece waveguide  2700   b ; the second and third ICG sections need not be transmissive because—as was the case in  FIG.  27 A —the green and blue light is to be prevented from reaching the third eyepiece waveguide  2700   c.    
     The input beams of red light which are not in-coupled into the second eyepiece waveguide  2700   b  can proceed on to the third eyepiece waveguide  2700   c . In the third eyepiece waveguide  2700   c , ICG region  2740   c   1  can in-couple some or all of the remaining red light beams into guided propagation states within the third eyepiece waveguide  2700   c.    
     The eyepiece waveguide stacks shown in  FIGS.  27 A and  27 B  can be used as near-eye displays, such as for AR systems. As described elsewhere herein, some near-eye displays are constructed of an eyepiece waveguide designed to support red light (˜625 nm, with full width at half maximum (FWHM)&lt;50 nm), a separate eyepiece waveguide designed to support green light (˜525 nm, with FWHM&lt;50 nm), and yet another separate eyepiece waveguide designed to support blue light (˜455 nm, with FWHM&lt;50 nm). In those single-color-component-per-layer designs, each separate eyepiece waveguide can use a set of diffraction gratings to propagate the entire FOV of an intended color component from a projector in a variety of directions in total-internal-reflection (TIR). Each layer can then replicate the projector pupil of a given color component to deliver a virtual image to the user&#39;s eye over a large eye box. The waveguide stacks shown in  FIGS.  27 A and  27 B  are distinct, however, in that they are designed such that different sub-portions of the FOV of each color component of the input image are supported by different eyepiece waveguides; in addition, one or more (or possibly even all) of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  in the stack can be designed to support sub-portions of the FOV of more than one color component. This can be achieved by changing the spatial periods of the ICG regions to allow coupling of each color component into multiple eyepiece waveguides (e.g., sub-portions of the FOV of each color component can be coupled into different waveguides). (In addition, the periods of CPE/EPE/OPE gratings can be correspondingly altered to ensure dispersion compensation.) For example, different sub-portions of the FOV of the red input image can be coupled, in some embodiments, into the large, and/or medium, and/or small-grating-vector layers  2700   a ,  2700   b ,  2700   c  (e.g., into all three grating vector layers, or into just the medium and small-grating-vector layers  2700   b ,  2700   c , as shown in  FIGS.  27 A and  27 B ). Similarly, different sub-portions of the respective FOVs of the green input image and the blue input image can be coupled into the large, and/or medium, and/or small-grating-vector layers  2700   a ,  2700   b ,  2700   c . Subsequently combining the outputs of the large, medium, and small-grating-vector eyepiece waveguides  2700   a ,  2700   b ,  2700   c  can result in an image FOV—for one or more of the color components—that is larger than what might otherwise be possible in a single-color-per-layer configuration. This is most easily illustrated using k-space diagrams. 
       FIGS.  27 C- 27 E  include k-space diagrams that illustrate the k-space operation of the example embodiments of the eyepiece waveguide stacks shown in  FIGS.  27 A and  27 B  for three different refractive indexes. In  FIG.  27 C , the refractive index of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  is n=1.5. In  FIG.  27 D , the refractive index of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  is n=1.75. In  FIG.  27 E , the refractive index of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  is n=2.0. The width of the k-space annulus for each eyepiece waveguide  2700   a ,  2700   b ,  2700   c  is determined by the refractive index of that waveguide. A larger refractive index corresponds to a wider k-space annulus. Thus, the annulus for the k-space diagrams in  FIG.  27 E  (n=2.0) is wider than the annulus for the k-space diagrams in FIG. 27D (n=1.75), which is wider than the annulus for the k-space diagrams in  FIG.  27 C  (n=1.5). 
     The left-hand k-space diagrams in  FIGS.  27 C- 27 E  illustrate the operation of the first eyepiece waveguide  2700   a , or large-grating-vector layer. The middle k-space diagrams in  FIGS.  27 C- 27 E  illustrate the operation of the second eyepiece waveguide  2700   b , or medium-grating-vector layer. The right-hand k-space diagrams in  FIGS.  27 C- 27 E  illustrate the operation the third eyepiece waveguide  2700   c , or small-grating-vector layer. 
     Each of the k-space diagrams in  FIGS.  27 C- 27 E  illustrates the k-space effect of one of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  on multiple color components of light (i.e., red, green, and blue). Therefore, each of the k-space diagrams actually represents a superposition of multiple k-space diagrams—one for each color component. Due to the wavelength-dependence of k-vectors (i.e., k=nω/c), all of the k-space diagrams are normalized (scaled by c/co) so as to maintain consistency between the superimposed k-space diagrams for the different color components. The result of the normalization is that each of the k-space diagrams in  FIGS.  27 C- 27 E  has a radius equal to the refractive index of the eyepiece waveguide. However, the normalization also causes the grating vector for the ICG region of each eyepiece waveguide to be scaled in inverse proportion to the angular frequency of each color component. This means that the grating vectors for ICG regions  2740   a ,  2740   b ,  2740   c  have different magnitudes for each color component. 
     In  FIGS.  27 C- 27 E , the scaled grating vectors for ICG regions  2740   a ,  2740   b ,  2740   c  are labeled GX R  for the red color component, GX G  for the green color component, and GX B  for the blue color component, where X is equal to the number of the eyepiece waveguide (X=1 for the first eyepiece waveguide  2700   a , X=2 for the second eyepiece waveguide  2700   b , and X=3 for the third eyepiece waveguide  2700   c ). As is evident from the k-space diagrams, an ICG with a given pitch translates beams of red light the greatest distance in k-space, followed by beams of green light, and then beams of blue light. The reason for this is that gratings with smaller pitches have a greater diffractive effect on light, and the pitch of the diffractive features in a given ICG is smaller in comparison to longer wavelengths such that the ICG diffracts light with longer wavelengths at greater angles (which corresponds to greater distances in k-space). 
     The center of each k-space diagram in  FIGS.  27 C- 27 E  includes a central FOV rectangle (shown with a solid line) that encompasses the k-vectors for the input light beams  2702 ,  2704 ,  2706  as they propagate generally in the z-direction from an external source (e.g., a projector) toward the stack of eyepiece waveguides  2700   a ,  2700   b ,  2700   c . The input light beams  2702 ,  2704 ,  2706  then interact with the ICG region  2740   a  of the large-grating-vector eyepiece waveguide  2700   a . In k-space, this interaction corresponds to the black FOV rectangle being translated by the grating vector of ICG region  2740   a.    
     The k-space operation of ICG region  2740   a —in the large-grating-vector layer  2700   a —is shown by the left-hand k-space diagram in each of  FIGS.  27 C- 27 E . A red light FOV rectangle (shown with a dashed line) shows the resulting k-space translation of the red input beams  2702  by G 1   R , which is the grating vector of ICG region  2740   a  as scaled for red light. In the embodiment illustrated by  FIG.  27 C , G 1   R  has a magnitude that causes all of the red light FOV rectangle to be shifted to a location beyond the outer periphery of the k-space annulus. In the embodiments illustrated by  FIGS.  27 D and  27 E , however, G 1   R  has a magnitude that causes part of the red light FOV rectangle to be shifted to a location inside the k-space annulus and the remaining part to be shifted beyond the outer periphery of the k-space annulus. In other words, in the  FIGS.  27 D and  27 E  embodiments, G 1   R  can be designed with a magnitude that causes the inner edge of the red light FOV rectangle to be shifted into the k-space annulus, while the outer edge is shifted beyond the outer periphery of the annulus. The k-vectors which are translated into the k-space annulus correspond to red light beams which are capable of being first-order diffracted into guided propagation modes inside the large-grating-vector layer  2700   a . The k-vectors which are translated beyond the outer periphery of the k-space annulus correspond to red light beams which are not able to be diffracted by ICG region  2740   a  and instead continue to propagate toward the medium-grating-vector layer  2700   b.    
     With continued reference to the k-space operation of ICG region  2740   a —in the large-grating-vector layer  2700   a —as shown by the left-hand k-space diagram in each of  FIGS.  27 C- 27 E , a green light FOV rectangle (shown with a dash-dot line) shows the resulting k-space translation of the green input beams  2704  by G 1G , which is the grating vector of ICG region  2740   a  as scaled for green light. In the illustrated embodiments, G 1G  has a magnitude that causes part of the green light FOV rectangle to be shifted to a location inside the k-space annulus and the remaining part to be shifted beyond the outer periphery of the k-space annulus. In other words, G 1G  can be designed with a magnitude that causes the inner edge of the green light FOV rectangle to be shifted into the k-space annulus, while the outer edge is shifted beyond the outer periphery of the annulus. The k-vectors which are translated into the k-space annulus correspond to green light beams which are capable of being first-order diffracted into guided propagation modes inside the large-grating-vector layer  2700   a . The k-vectors which are translated beyond the outer periphery of the k-space annulus correspond to green light beams which are not able to be diffracted by ICG region  2740   a  and instead continue to propagate toward the medium-grating-vector layer  2700   b.    
     With continued reference to the k-space operation of ICG region  2740   a —in the large-grating-vector layer  2700   a —as shown by the left-hand k-space diagram in each of  FIGS.  27 C- 27 E , a blue light FOV rectangle (shown with a dotted line) shows the k-space translation of the blue input beams  2706  by G 1   B , which is the grating vector of ICG region  2740   a  as scaled for blue light. In the illustrated embodiments, G 1   B  likewise has a magnitude that causes part of the blue light FOV rectangle to be shifted to a location inside the k-space annulus and the remaining part to be shifted beyond the outer periphery of the k-space annulus. In other words, G 1   B  can be designed with a magnitude that causes the inner edge of the blue light FOV rectangle to be shifted into the k-space annulus (e.g., to the inner peripheral edge of the annulus), while the outer edge is shifted beyond the outer periphery of the annulus. The k-vectors which are translated into the k-space annulus correspond to blue light beams which are capable of being first-order diffracted into guided propagation modes inside the large-grating-vector layer  2700   a , whereas those which are translated beyond the outer periphery of the k-space annulus correspond to blue light beams which are not able to be diffracted by ICG region  2740   a  and instead continue to propagate toward the medium-grating-vector layer  2700   b.    
     The k-space operation of ICG region  2740   b —in the medium-grating-vector layer  2700   b —is shown by the middle k-space diagram in each of  FIGS.  27 C- 27 E . A red light FOV rectangle (shown with a dashed line) shows the resulting k-space translation of the red input beams  2702  by G 2   R , which is the grating vector of ICG region  2740   b  as scaled for red light. In the illustrated embodiments, G 2   R  has a magnitude that causes part of the red light FOV rectangle to be shifted to a location inside the k-space annulus and the remaining part to be shifted beyond the outer periphery of the k-space annulus. In other words, G 2   R  can be designed with a magnitude that causes the inner edge of the red light FOV rectangle to be shifted into the k-space annulus, while the outer edge is shifted beyond the outer periphery of the annulus. The k-vectors which are translated into the k-space annulus correspond to red light beams which are capable of being first-order diffracted into guided propagation modes inside the medium-grating-vector layer  2700   b . The k-vectors which are translated beyond the outer periphery of the k-space annulus correspond to red light beams which are not able to be diffracted by ICG region  2740   b  and instead continue to propagate toward the small-grating-vector layer  2700   c.    
     With continued reference to the k-space operation of ICG region  2740   b —in the medium-grating-vector layer  2700   b —as shown by the middle k-space diagram in each of  FIGS.  27 C- 27 E , a green light FOV rectangle (shown with a dash-dot line) shows the k-space translation of the green input beams  2704  by G 2   G , which is the grating vector of ICG region  2740   b  as scaled for green light. In the illustrated embodiments, G 2   G  has a magnitude that causes the outer edge of the green light FOV rectangle to be shifted to a position inside the k-space annulus (e.g. to the outer peripheral edge of the k-space annulus). Since the FOV rectangle is larger in the k x -direction than the width of the k-space annulus, part of the green light FOV rectangle extends into the center region of the k-space diagram. The k-vectors which are translated into the k-space annulus correspond to green light beams which are capable of being first-order diffracted into guided propagation modes inside the medium-grating-vector layer  2700   b . The k-vectors which remain in the center region of the k-space diagram correspond to green light beams which may have already been diffracted into guided propagation modes inside the large-grating-vector layer  2700   a  by ICG region  2740   a  (but which if they were to arrive at the medium-grating-vector layer  2700   b  would be diffracted by ICG region  2740   b  but not into guided propagation modes—those green light beams may be either absorbed by the optical filter  2745  or prevented from reaching the small-grating-vector layer  2700   c  by the reflective grating  2740   b   2 ). 
     Lastly, with continued reference to the k-space operation of ICG region  2740   b —in the medium-grating-vector layer  2700   b —as shown by the middle k-space diagram in each of  FIGS.  27 C- 27 E , a blue light FOV rectangle (shown with a dotted line) shows the k-space translation of the blue input beams  2706  by G 2   B , which is the grating vector of ICG region  2740   b  as scaled for blue light. In the illustrated embodiments, G 2   B  has a magnitude that causes part of the blue light FOV rectangle to be shifted to a location inside the k-space annulus and the remaining part to remain in the center region of the k-space diagram. The k-vectors which are translated into the k-space annulus correspond to blue light beams which are capable of being first-order diffracted into guided propagation modes inside the medium-grating-vector layer  2700   b , whereas those which remain in the center region of the k-space diagram correspond to blue light beams which may have already been diffracted into guided propagation modes inside the large-grating-vector layer  2700   a  by ICG region  2740   a  (but which if they were to arrive at the medium-grating-vector layer  2700   b  would be diffracted by ICG region  2740   b  but not into guided propagation modes—those blue light beams may be either absorbed by the optical filter  2745  or prevented from reaching the small-grating-vector layer  2700   c  by the reflective grating  2740   b   3 ). 
     The k-space operation of ICG region  2740   c —in the small-grating-vector layer  2700   c —is shown by the right-hand k-space diagram in each of  FIGS.  27 C- 27 E . A red light FOV rectangle (shown with a dashed line) shows the resulting k-space translation of the red input beams  2702  by G 3   R , which is the grating vector of ICG region  2740   c  as scaled for red light. In the illustrated embodiments, G 3   R  has a magnitude that causes the outer edge of the red light FOV rectangle to be shifted into the k-space annulus (e.g., to the outer peripheral edge of the k-space annulus). Since the FOV rectangle is larger in the k x -direction than the width of the k-space annulus, part of the red light FOV rectangle extends into the center region of the k-space diagram. The k-vectors which are translated into the k-space annulus correspond to red light beams which are capable of being first-order diffracted into guided propagation modes inside the small-grating-vector layer  2700   c . The k-vectors which remain in the center region of the k-space diagram correspond to red light beams which may have already been diffracted into guided propagation modes inside the large-grating-vector layer  2700   a  by ICG region  2740   a , or inside the medium-grating-vector layer  2700   b  by ICG region  2740   b . To the extent any such red light beams have not been previously in-coupled into one of the eyepiece waveguides, e.g., due to less-than-100% diffractive efficiency, they can be blocked by baffles, etc. 
     With continued reference to the k-space operation of ICG region  2740   c —in the small-grating-vector layer  2700   c —as shown by the right-hand k-space diagram in each of  FIGS.  27 C- 27 E , a green light FOV rectangle (shown with a dash-dot line) shows the k-space translation of the green input beams  2704  by G 3   G , which is the grating vector of ICG region  2740   c  as scaled for green light. Similarly, a blue light FOV rectangle (shown with a dotted line) shows the k-space translation of the blue input beams  2706  by G 3   B , which is the grating vector of ICG region  2740   c  as scaled for blue light. As already discussed, an optical filter  2745  can be provided to prevent the green and blue light beams from reaching ICG region  2740   c . If, however, green and blue light beams were permitted to interact with ICG region  2740   c , some of the first-order diffracted beams would enter guided propagation modes because G 3   G  and G 3   B  would translate a portion of those FOV rectangles into the k-space annulus; the k-vectors which would remain in the center region of the k-space diagram correspond to green and blue light beams which may have already been diffracted into guided propagation modes inside the large-grating-vector layer  2700   a  by ICG region  2740   a , or inside the medium-grating-vector layer  2700   b  by ICG region  2740   b.    
     The right-hand k-space diagram in each of  FIGS.  27 C- 27 E  also shows the effect of second-order diffraction by ICG region  2740   c  in the small-grating-vector layer  2700   c . Second-order diffraction causes the central FOV rectangle (shown with a solid line) to be translated by amounts equal to two times each of the scaled grating vectors. A green light FOV rectangle (shown with a light-weight dash-dot line) shows the second-order diffraction of the green light  2704  by a grating vector of length 2·G 3   G , while a blue light FOV rectangle (shown with a light-weight dotted line) shows the second-order diffraction of the blue light  2706  by a grating vector of length 2·G 3   B . As illustrated, parts of the second-order diffracted green light and blue light FOV rectangles (shown with light-weight dash-dot and dotted lines, respectively) can be translated to locations inside the k-space annulus, thus resulting in some guided propagation modes. The reason green and blue light beams may be blocked by the optical filter  2745  from interacting with ICG region  2740   c  in the small-grating-vector layer  2700   c  is to prevent any such second-order modes which could cause image artifacts. In some embodiments, however, the optical filter  2745  can be omitted if the second-order diffraction efficiency of ICG region  2740   c  in the small-grating-vector layer  2700   c  is sufficiently low (e.g., below 30%, or below 20%, or below 10%, or below 5%). 
       FIGS.  27 C- 27 E  only show the in-coupling of light into the stack of eyepiece waveguides  2700   a ,  2700   b ,  2700   c . Once light has been in-coupled into the eyepiece waveguides  2700   a ,  2700   b ,  2700   c , it can be handled by OPE regions, MPE regions, CPE regions, and/or EPE regions according to any of the other embodiments described in this disclosure. In such embodiments, the periods of all diffraction gratings in a given eyepiece waveguide can be modified to maintain dispersion compensation. This can be achieved when the sum of the grating vectors in a given eyepiece waveguide equals zero. 
     According to the embodiments illustrated in  FIGS.  27 A- 27 E , the stack of eyepiece waveguides  2700   a ,  2700   b ,  2700   c  can project an image whose FOV is larger in at least one k-space dimension than the width of the k-space annulus for the waveguides. This is possible because the FOV of each color component of the input image is divided into multiple k-space sub-portions which are separately handled by multiple respective eyepiece waveguides. Using the techniques described in this section of the disclosure, relatively wide FOVs can be achieved: For a refractive index of n=1.5 ( FIG.  27 C ), an FOV of up to 41.5° in the direction corresponding to the input coupling grating vectors can be achieved (as compared to 29° in a single-color-component-per-layer eyepiece waveguide); for a refractive index of n=1.75 ( FIG.  27 D ), an FOV of up to 55.9° in the direction corresponding to the input coupling grating vectors can be achieved (as compared to 44° in a single-color-component-per-layer eyepiece waveguide); and for a refractive index of n=2.0 ( FIG.  27 E ), an FOV of up to 64.7° in the direction corresponding to the input coupling grating vectors can be achieved (as compared to 60° in a single-color-component-per-layer eyepiece waveguide). 
       FIGS.  27 F- 27 H  illustrate the sub-portions of the FOV of each color component of the input image which can be in-coupled into each of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  in the stack, according to certain embodiments where each color component is partially carried in two of the eyepiece waveguides. In these figures, the dimension of the FOV of the input image in the direction of the ICG vectors is larger than the width of k-space annulus corresponding to the eyepiece waveguides, but the FOV expansion techniques described in this section are implemented so as to allow the eyepiece waveguide stack to accommodate the entire image FOV. 
     In the illustrated embodiment, each of the color components of the input image are partially carried by two of the three eyepiece waveguides  2700   a ,  2700   b ,  2700   c .  FIG.  27 F  shows the sub-portions of the FOV of the red color component of the input image which can be in-coupled into each of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c , as well as the FOV of the red output image from the waveguide stack. As discussed further below, in the illustrated embodiment the red color component is partially carried in the medium-grating-vector layer  2700   b  and the small-grating-vector layer  2700   c , but not in the large-grating-vector layer  2700   a . This can be due, for example, to the large-grating-vector layer  2700   a  in  FIG.  27 B  not including the first ICG section  2740   a   1  which receives the input beams  2702  of red light. Alternatively, it may be due to the ICG region  2740   a  of the large-grating-vector layer  2700   a  in  FIG.  27 A  being designed to have a grating vector G 1   R  that is long enough to translate the red color component of the FOV rectangle completely beyond the outer peripheral edge of the k-space annulus, as illustrated in  FIG.  27 C . Meanwhile,  FIG.  27 G  and  FIG.  27 H  respectively show the sub-portions of the FOV of the green and blue color components of the input image which can be in-coupled into each of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c , as well as the FOV of the green and blue output images from the waveguide stack. As discussed further below, in the illustrated embodiment the green and blue color components are partially carried by the large-grating-vector layer  2700   a  and the medium-grating-vector layer  2700   b , but not in the small-grating-vector layer  2700   c . The small-grating-vector layer  2700   c  does not carry any portion of the green and blue color components due to the presence of the optical filter  2745 , which, in some embodiments, blocks green and blue light from reaching the small-grating-vector layer  2700   c.    
     Each of  FIGS.  27 F- 27 H  includes four columns of diagrams. Each of the diagrams in the first three columns (left to right) represents the FOV for one of the color components of the input image in one of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c . The first column of diagrams, starting on the left-hand side, corresponds to the large-grating-vector layer  2700   a . The second column of diagrams corresponds to the medium-grating-vector layer  2700   b . And the third column of diagrams corresponds to the small-grating-vector layer  2700   c . Regions of the FOV which are shown in black are not in-coupled, whereas regions of the FOV which are shown in gray are in-coupled. The fourth and final column of diagrams shows the FOV for the color component output images R out , G out , B out . Regions of the FOV which are shown in black are not present, whereas regions of the FOV which are shown in gray are present. 
     As shown in the first column of  FIG.  27 F , in the illustrated embodiment, no portion of the FOV of the red input image is in-coupled into the large-grating-vector layer. However, the first column of  FIG.  27 G  shows that a sub-portion G 1  of the green input image can be in-coupled into the large-grating-vector layer  2700   a . The first column of  FIG.  27 H  similarly shows that a sub-portion B 1  of the blue input image can be in-coupled into the large-grating-vector layer  2700   a . (Note: The first column of  FIGS.  27 G and  27 H  show the left-hand side of the green and blue color components of the input image being carried in the large-grating-vector layer  2700   a , whereas  FIGS.  27 C- 27 E  show the right-hand sides of the green and blue FOV rectangles having supported guided propagation modes due to being located in the k-space annulus of the large-grating-vector layer. The reason for this difference is that the k-space diagrams in  FIGS.  27 C- 27 E  show the direction light propagates, whereas  FIGS.  27 F- 27 H  are indicative of the perceived angles from which the light beams come. For example, light propagating to the right is perceived as light on the left side of the FOV. This results in an inversion of both axes between what is shown in the k-space diagrams and what is seen by the user.) 
     The second column of  FIG.  27 F  shows that a sub-portion R 1  of the red input image can be in-coupled into the medium-grating-vector layer  2700   b . The second columns of  FIGS.  27 G and  27 H  show that sub-portions G 2  and B 2 , respectively, can be in-coupled into the medium-grating-vector layer  2700   b.    
     The third column of  FIG.  27 F  shows that a sub-portion R 2  of the red input image can be in-coupled into the small-grating-vector layer  2700   c . The third columns of  FIGS.  27 G and  27 H  show that, for the illustrated embodiment, no portion of the green or blue input image is in-coupled into the small-grating-vector layer  2700   c . This is due to the fact that green and blue light beams are prevented from reaching the small-grating-vector layer  2700   c , as discussed above with respect to  FIGS.  27 A and  27 B . 
     The fourth column of  FIGS.  27 F- 27 H  shows the portion of the FOV which is present in the output image for the different color components. Regions of the FOV which are shown in black are not present in the output image, whereas regions of the FOV which are shown in gray are present in the output image. For each of  FIGS.  27 F- 27 H , the diagram in the fourth column is solid gray, meaning that the full FOV for each color component (R out , G out , B out ) is present in the output image from the stack of waveguides  2700   a ,  2700   b ,  2700   c . This is true because the union of the R 1  and R 2  sub-portions, the union of the G 1  and G 2  sub-portions, and the union of the B 1  and B 2  sub-portions each includes the complete FOV of the input image. 
     If the diffraction gratings in the ICG regions (e.g.,  2740   a ,  2740   b ,  2740   c ) were 100% efficient, then the respective sub-portions (R 2 /R 1 , G 2 /G 1 , B 2 /B 1 ) of each color component of the input image would not overlap with one another. For example, sub-portion R 2  of the red color component of the input image would not overlap with sub-portion R 1  because any red light beams in R 1  would not be available to be in-coupled at the small-grating-vector layer  2700   c  as they would have already been in-coupled at the medium-grating-vector layer  2700   b . Similarly, sub-portions G 2  and B 2  of the green and blue color components of the input image would not overlap with sub-portions G 1  and B 1 , respectively, because any green light beams in G 1 , and any blue light beams in B 1 , would have already been in-coupled at the large-grating-vector layer  2700   c  and would therefore not be available to be in-coupled at the medium-grating-vector layer  2700   b . In some embodiments, however, the diffraction gratings in the ICG regions (e.g.,  2740   a ,  2740   b ,  2740   c ) are less than 100% efficient. This means that some portion of the light that can be in-coupled at a given layer will transmit through that layer&#39;s ICG region and will have the opportunity to in-couple at the next layer&#39;s ICG region. This can provide a degree of overlap between the respective sub-portions (R 2 /R 1 , G 2 /G 1 , B 2 /B 1 ) of each color component to ensure there are no gaps in the FOV that is displayed to the user. In some embodiments, for example, the ICG regions (e.g.,  2740   a ,  2740   b ,  2740   c ) have a diffractive efficiency in the range of 5-90% 
       FIGS.  27 I- 27 K  are similar to  FIGS.  27 F- 27 H  in that they illustrate the sub-portions of the FOV of each color component of the input image which can be in-coupled into each of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  in the stack, except that  FIGS.  27 I- 27 K  illustrate certain embodiments where each color component is partially carried in three of the eyepiece waveguides rather than two. 
       FIG.  27 I  shows the sub-portions of the FOV of the red color component of the input image which can be in-coupled into each of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c , as well as the FOV of the red output image from the waveguide stack. As discussed further below, in the illustrated embodiment the red color component is partially carried in all three eyepiece waveguides  2700   a ,  2700   b ,  2700   c . This can be the case where, for example, the ICG region  2740   a  of the large-grating-vector layer  2700   a  is designed to have a grating vector G 1   R  that translates the red color component of the FOV rectangle to a k-space location at least partially inside the k-space annulus, as illustrated in  FIGS.  27 D and  27 E . Meanwhile,  FIG.  27 J  and  FIG.  27 K  respectively show the sub-portions of the FOV of the green and blue color components of the input image which can be in-coupled into each of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c , as well as the FOV of the green and blue output images from the waveguide stack. As discussed further below, in the illustrated embodiment the green and blue color components are likewise partially carried in all three eyepiece waveguides  2700   a ,  2700   b ,  2700   c . This can be the case where, for example, the second-order diffraction efficiency of ICG region  2740   c  in the small-grating-vector layer  2700   c  is sufficiently low such that the optical filter  2745  is not required in order to prevent image artifacts resulting from second-order diffraction in the small-grating-vector layer. 
     Each of  FIGS.  27 I- 27 K  includes four columns of diagrams. Each of the diagrams in the first three columns (left to right) represents the FOV for one of the color components of the input image in one of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c . The first column of diagrams, starting on the left-hand side, corresponds to the large-grating-vector layer  2700   a . The second column of diagrams corresponds to the medium-grating-vector layer  2700   b . And the third column of diagrams corresponds to the small-grating-vector layer  2700   c . Regions of the FOV which are shown in black are not in-coupled, whereas regions of the FOV which are shown in gray are in-coupled. The fourth and final column of diagrams shows the FOV for the color component output images R out , G out , B out . Regions of the FOV which are shown in black are not present, whereas regions of the FOV which are shown in gray are present. 
     As shown in the first column of  FIG.  27 I , in the illustrated embodiment, a portion of the FOV of the red input image is in-coupled into the large-grating-vector layer. Similarly, the first column of  FIG.  27 J  shows that a sub-portion G 1  of the green input image can be in-coupled into the large-grating-vector layer  2700   a , and the first column of  FIG.  27 K  shows that a sub-portion B 1  of the blue input image can be in-coupled into the large-grating-vector layer  2700   a.    
     The second column of  FIG.  27 I  shows that a sub-portion R 2  of the red input image can be in-coupled into the medium-grating-vector layer  2700   b . The second columns of  FIGS.  27 J and  27 K  show that sub-portions G 2  and B 2 , respectively, can be in-coupled into the medium-grating-vector layer  2700   b.    
     The third column of  FIG.  27 I  shows that a sub-portion R 3  of the red input image can be in-coupled into the small-grating-vector layer  2700   c . The third columns of  FIGS.  27 J and  27 K  show that sub-portions G 3  and B 3 , respectively, can be in-coupled into the small-grating-vector layer  2700   b.    
     The fourth column of  FIGS.  27 I- 27 K  shows the portion of the FOV which is present in the output image for the different color components. Regions of the FOV which are shown in black are not present in the output image, whereas regions of the FOV which are shown in gray are present in the output image. For each of  FIGS.  27 I- 27 K , the diagram in the fourth column is solid gray, meaning that the full FOV for each color component (R out , G out , B out ) is present in the output image from the stack of waveguides  2700   a ,  2700   b ,  2700   c . This is true because the union of the R 1 , R 2 , and R 3  sub-portions, the union of the G 1 , G 2 , and G 3  sub-portions, and the union of the B 1 , B 2 , and B 3  sub-portions each includes the complete FOV of the input image. 
     In contrast to  FIGS.  27 F- 27 H  and  FIGS.  27 I- 27 K ,  FIGS.  27 L- 27 N  illustrate the sub-portions of the FOV of each color component of the input image which can be in-coupled into single-color-component-per-layer eyepiece waveguides. The same refractive index, n=1.75, and image FOV are used in  FIGS.  27 L- 27 N  as in  FIGS.  27 F- 27 H  and  FIGS.  27 I- 27 K . However, since single-color-component-per-layer eyepiece waveguides are used for  FIGS.  27 L- 27 N , these figures illustrate the FOV limitations which are present when the FOV expansion techniques described with respect to  FIGS.  27 A- 27 K  are not implemented. 
       FIG.  27 L  shows the sub-portions of the FOV of the red color component of the input image which can be in-coupled into each of three eyepiece waveguides, as well as the FOV of the red output image from the waveguide stack. Meanwhile,  FIG.  27 M  and  FIG.  27 N  respectively show the sub-portions of the FOV of the green and blue color components of the input image which can be in-coupled into each of the eyepiece waveguides, as well as the FOV of the green and blue output images from the waveguide stack. 
     Each of  FIGS.  27 L- 27 N  includes four columns of diagrams. Each of the diagrams in the first three columns (left to right) represents the FOV for one of the color components of the input image in one of the eyepiece waveguides. The first column of diagrams, starting on the left-hand side, corresponds to an eyepiece waveguide that is designed to accommodate blue light. The second column of diagrams corresponds to an eyepiece waveguide that is designed to accommodate green light. And the third column of diagrams corresponds to an eyepiece waveguide that is designed to accommodate red light. Regions of the FOV which are shown in black are not in-coupled, whereas regions of the FOV which are shown in gray are in-coupled. The fourth and final column of diagrams shows the FOV for the color component output images R out , G out , B out . Regions of the FOV which are shown in black are not present, whereas regions of the FOV which are shown in gray are present. 
     As shown in the first column of  FIGS.  27 L and  27 M , no portion of the FOV of the red input image or the green input image is in-coupled into the blue light eyepiece waveguide. The first column of  FIG.  27 N  shows that a sub-portion B 1  of the blue input image can be in-coupled into the blue light eyepiece waveguide. However, some portions of the FOV of the blue input image are not in-coupled because the FOV has a k-space dimension that is larger than the width of the k-space annulus corresponding to the blue light eyepiece waveguide. 
     The second column of  FIGS.  27 L and  27 N  shows that no portion of the FOV of the red input image or the blue input image is in-coupled into the green light eyepiece waveguide. The second column of  FIG.  27 M  shows that a sub-portion G 1  of the green input image can be in-coupled into the green light eyepiece waveguide. However, some portions of the FOV of the green input image are not in-coupled because the FOV has a k-space dimension that is larger than the width of the k-space annulus corresponding to the green light eyepiece waveguide. 
     The third column of  FIGS.  27 M and  27 N  shows that no portion of the FOV of the green input image or the blue input image is in-coupled into the red light eyepiece waveguide. The third column of  FIG.  27 L  shows that a sub-portion R 1  of the red input image can be in-coupled into the red light eyepiece waveguide. However, some portions of the FOV of the red input image are not in-coupled because the FOV has a k-space dimension that is larger than the width of the k-space annulus corresponding to the red light eyepiece waveguide. 
     The fourth column of  FIGS.  27 L- 27 N  shows the portion of the FOV which is present in the output image for the different color components. Regions of the FOV which are shown in black are not present in the output image, whereas regions of the FOV which are shown in gray are present in the output image. For each of  FIGS.  27 L- 27 N , the diagram in the fourth column has multiple black regions, meaning that the full FOV for each color component (R out , G out , B out ) is not present in the output image from the stack of single-color-per-layer eyepiece waveguides. This shows that the techniques described with respect to  FIGS.  27 A- 27 K  can be used to implement waveguide stacks capable of displaying full color images with a larger FOV than the single-color-component-per-layer eyepiece waveguides illustrated in  FIGS.  27 L- 27 N . 
       FIG.  28 A  illustrates another example embodiment of an eyepiece waveguide stack  2800   a ,  2800   b ,  2800   c  with enhanced FOV and an in-line pupil ICG configuration. The waveguide stack in  FIG.  28 A  includes a large-grating-vector eyepiece waveguide  2800   a , a medium-grating-vector eyepiece waveguide  2800   b , and a small-grating-vector eyepiece waveguide  2800   c . Each of the eyepiece waveguides  2800   a ,  2800   b ,  2800   c  includes a respective ICG region  2840   a ,  2840   b ,  2840   c . The waveguide stack in  FIG.  28 A  is similar to the waveguide stack in  FIG.  27 A  in structure and operation except that it is designed to handle red light and green light—not just red light—in the small-grating-vector layer  2800   c . Accordingly, the red optical filter  2745  in the waveguide stack of  FIG.  27 A  is replaced by a yellow optical filter  2845  in the waveguide stack of  FIG.  28 A . 
     The waveguide stack of  FIG.  28 A  is designed to receive input beams of red light  2802 , green light  2804 , and blue light  2806 . Once again, input beam  2802  is intended to represent the entire set of input beams for the red color component of the input image, while input beam  2804  represents the entire set of input beams for the green color component of the input image and input beam  2806  represents the entire set of input beams for the blue color component of the input image. 
     The ICG region  2840   a  for the large-grating-vector layer  2800   a  can be designed to in-couple a sub-portion of the red input beams, a sub-portion of the green input beams, and/or a sub-portion of the blue input beams in guided propagation states within the first eyepiece waveguide  2800   a  (as represented by first-order diffracted beams  2802   a ,  2804   a ,  2806   a , respectively). The ICG region  2840   b  for the medium-grating-vector layer  2800   b  can likewise be designed to in-couple a sub-portion of the red input beams, a sub-portion of the green input beams, and/or a sub-portion of the blue input beams in guided propagation states within the second eyepiece waveguide  2800   b  (as represented by first-order diffracted beams  2802   b ,  2804   b ,  2806   b , respectively). 
     The undiffracted beams  2802 ,  2804 ,  2806  from the ICG region  2840   b  of the medium-grating-vector eyepiece waveguide  2800   b  continue on to the optical filter  2845 . The optical filter  2845  can be located between the ICG region  2840   b  of the medium-grating-vector layer  2800   b  and the ICG region  2840   c  of the small-grating-vector layer  2800   c  (e.g., in the gap between the second eyepiece waveguide  2800   b  and the third eyepiece waveguide  2800   c ). In some embodiments, the optical filter  2845  is a yellow filter which effectively absorbs at least 90% of the blue light (represented by beam  2706 ). In some embodiments, the −3 dB cutoff for the optical filter  2845  may be located within the range of 465-500 nm, though other locations can also be used. The purpose of the optical filter  2845  is to absorb blue light which could otherwise be second-order diffracted by ICG region  2840   c  into guided propagation modes within the small-grating-vector layer  2800   c , as such second-order diffractions could result in artifacts, such as ghost images. In some embodiments, one or more optical dyes can be added to the substrate material for the third eyepiece waveguide  2800   c . The optical dye(s) can absorb blue light, similar to optical filter  2845 . Thus, the dye(s) added to the substrate material for the third eyepiece waveguide  2800   c  could be used in place of optical filter  2845  or in addition to it. 
     Since beams of blue light are blocked by the optical filter  2845 , only red light beams (represented by beam  2802 ) and green light beams (represented by beam  2804 ) are incident on the ICG region  2840   c  of the third eyepiece waveguide  2800   c . ICG region  2840   c  can be designed to in-couple a sub-portion of the red input beams and/or a sub-portion of the green input beams in guided propagation states within the third eyepiece waveguide  2800   c  (as represented by first-order diffracted beams  2802   c  and  2804   c , respectively). 
       FIG.  28 B  illustrates another example embodiment of an eyepiece waveguide stack  2800   a ,  2800   b ,  2800   c  with enhanced FOV and a split pupil ICG configuration. The waveguide stack in  FIG.  28 B  includes a large-grating-vector eyepiece waveguide  2800   a , a medium-grating-vector eyepiece waveguide  2800   b , and a small-grating-vector eyepiece waveguide  2800   c . Each of the eyepiece waveguides  2800   a ,  2800   b ,  2800   c  includes a respective ICG region  2840   a ,  2840   b ,  2840   c . The ICG region  2840   a  of the first eyepiece waveguide  2800   a  is divided into three spaced apart sections  2940   a   1 ,  2940   a   2 ,  2940   a   3 . Similarly, the ICG region  2840   b  of the second eyepiece waveguide  2800   b  is divided into three spaced apart sections  2840   b   1 ,  2840   b   2 ,  2840   b   3 , and the ICG region  2840   c  of the third eyepiece waveguide  2800   c  is divided into two spaced apart sections  2840   c   1 ,  2840   c   2 . The waveguide stack in  FIG.  28 B  is similar to the waveguide stack in  FIG.  27 B  in structure and operation except that it is designed to handle red light and green light—not just red light—in the small-grating-vector layer  2800   c . Accordingly, the reflective ICG region  2740   b   2  in the waveguide stack of  FIG.  27 B  is replaced by a partially transmissive ICG region  2940   b   2  in the waveguide stack of  FIG.  28 B . 
     Although not shown, k-space diagrams for the eyepiece waveguide stacks shown in  FIGS.  28 A and  28 B  would be similar to those provided for the eyepiece waveguide stacks shown in  FIGS.  27 A and  27 B  except that they would show the FOV rectangle for green light being partially split amongst not just the large-grating vector layer  2800   a  and the medium-grating-vector layer  2800   b  but also the small-grating-vector layer  2800   c.    
     The example eyepiece waveguide stacks in  FIGS.  27 A- 27 B and  28 A- 28 B  can be designed, in some embodiments, using the following considerations: 1) That the union of the respective sub-portions of the FOV of each color component of the input image that are coupled into the eyepiece waveguides encompasses the full FOV of the input image (e.g., as shown in  FIGS.  27 F- 27 H  and in  FIGS.  27 I- 27 K ); and 2) That second-order ICG diffraction of any portion of the FOV of any color component is not permitted to couple into a guided propagation mode in any eyepiece waveguide, as those paths may result in image artifacts, such as ghost images. 
     The maximum FOV that can be achieved according to some embodiments of the example eyepiece waveguide stacks in  FIGS.  27 A- 27 B , subject to the foregoing design considerations, can be calculated using the following variables and equations:
         n is the refractive index of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c;      the radius of the outer peripheral edge of the k-space annulus for the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  is equal to n;   the radius of the inner peripheral edge of the k-space annulus for the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  is equal to 1;   Λ 1  is the period of the ICG region  2740   a  of the first eyepiece waveguide, or large-grating-vector layer,  2700   a;      Λ 2  is the period of the ICG region  2740   b  of the second eyepiece waveguide, or medium-grating-vector layer,  2700   b;      Λ 3  is the period of the ICG region  2740   c  of the third eyepiece waveguide, or small-grating-vector layer,  2700   c;      λ Red  is the center wavelength of the red input light  2702 ; Green is the center wavelength of the green input light  2704 ;   λ Blue  is the center wavelength of the blue input light  2706 ;   k FoV  is the k-space dimension of the input image in the direction of the ICG vectors;       

                     λ   Blue       Λ   2       +       k   FoV     2       &lt;   n     ,         
which provides that the outer edge of the blue FOV rectangle propagates in TIR (i.e., lies in the k-space annulus) in the second eyepiece waveguide  2700   b ;
 
                 n   -       λ   Green       Λ   1         &gt;     1   -       λ   Green       Λ   2           ,         
which provides that the inner portion of the green FOV rectangle that propagates in TIR (i.e., lies in the k-space annulus) in the first eyepiece waveguide  2700   a  is larger than the outer portion of the green FOV rectangle that does not propagate in TIR (i.e., does not lie in the k-space annulus) in the second eyepiece waveguide  2700   b ;
 
                 n   -       λ   Red       Λ   1         &gt;     1   -       λ   Red       Λ   2           ,         
which provides that inner portion of the red FOV rectangle that propagates in TIR (i.e., lies in the k-space annulus) in the first eyepiece waveguide  2700   a  is larger than the outer portion of the red FOV rectangle that does not propagate in TIR (i.e., does not lie in the k-space annulus) in the second eyepiece waveguide  2700   b ;
 
                 Λ   1     &lt;       λ   Blue       1   +       k   FoV     2           ,         
which provides that the maximum period of the first eyepiece waveguide  2700   a  can be the maximum period at which the inner edge of the blue FOV rectangle propagates in TIR (i.e., lies in the k-space annulus) in the large grating vector layer (any larger grating period in the first eyepiece waveguide  2700   a  would result in a shorter grating vector which would not translate the inner edge of the blue FOV rectangle into the k-space annulus);
 
                 Λ   3     &gt;       λ   Red       n   -       k   FoV     2           ,         
which provides that the minimum period of the ICG region  2740   c  of the third eyepiece waveguide  2700   c  can be the minimum period at which the outer edge of the red FOV rectangle propagates in TIR (i.e., lies in the k-space annulus) in the small-grating-vector layer (any smaller grating period in the third eyepiece waveguide  2700   c  would result in a longer grating vector which would translate the outer edge of the red FOV rectangle beyond the k-space annulus); additional constraints may be necessary to set a maximum period for Λ 3  to enforce that the full red FOV rectangle is preserved;
 
                     2   ⁢     λ   Blue         Λ   2       -       k   FoV     2       &gt;   n     ,         
which provides that the second-order diffraction of blue light from the ICG region  2740   b  on the second eyepiece waveguide  2700   b  does not propagate in TIR (i.e., does not lie in the k-space annulus) over the full FOV;
 
                 Λ   2     &gt;       λ   Green       n   -       k   FoV     2           ,         
which provides that the minimum period of the ICG region  2740   b  of the second eyepiece waveguide  2700   b  is the minimum period at which the outer edge of the green FOV rectangle propagates in TIR (i.e., lies in the k-space annulus) in the medium-grating-vector layer.
 
     The maximum FOV that can be achieved according to some embodiments of the example eyepiece waveguide stacks in  FIGS.  28 A- 28 B , subject to the foregoing design considerations, can be calculated using the following variables and equations:
         n is the refractive index of the eyepiece waveguides  2800   a ,  2800   b ,  2800   c;      the radius of the outer peripheral edge of the k-space annulus for the eyepiece waveguides  2800   a ,  2800   b ,  2800   c  is equal to n;   the radius of the inner peripheral edge of the k-space annulus for the eyepiece waveguides  2800   a ,  2800   b ,  2800   c  is equal to 1;   Λ 1  is the period of the ICG region  2740   a  of the first eyepiece waveguide, or large-grating-vector layer,  2800   a;      Λ 2  is the period of the ICG region  2840   b  of the second eyepiece waveguide, or medium-grating-vector layer,  2800   b;      Λ 3  is the period of the ICG region  2840   c  of the third eyepiece waveguide, or small-grating-vector layer,  2800   c;      λ Red  is the center wavelength of the red input light  2802 ;   λ Green  is the center wavelength of the green input light  2804 ;   λ Blue  is the center wavelength of the blue input light  2806 ;   k FoV  is the k-space dimension of the input image in the direction of the ICG vectors;       

                     λ   Blue       Λ   2       +       k   FoV     2       &lt;   n     ,         
which provides that the outer edge of the blue FOV rectangle propagates in TIR (i.e., lies in the k-space annulus) in the second eyepiece waveguide  2800   b ;
 
                 n   -       λ   Green       Λ   1         &gt;     1   -       λ   Green       Λ   2           ,         
which provides that the inner portion of the green FOV rectangle that propagates in TIR (i.e., lies in the k-space annulus) in the first eyepiece waveguide  2800   a  is larger than the outer portion of the green FOV rectangle that does not propagate in TIR (i.e., does not lie in the k-space annulus) in the second eyepiece waveguide  2800   b ;
 
                 n   -       λ   Red       Λ   1         &gt;     1   -       λ   Red       Λ   2           ,         
which provides that inner portion of the red FOV rectangle that propagates in TIR (i.e., lies in the k-space annulus) in the first eyepiece waveguide  2800   a  is larger than the outer portion of the red FOV rectangle that does not propagate in TIR (i.e., does not lie in the k-space annulus) in the second eyepiece waveguide  2800   b ;
 
                     λ   Green       Λ   3       -   1     &gt;         λ   Green       Λ   2       -   n       ,         
which provides that the outer portion of the green FOV rectangle that propagates in TIR (i.e., lies in the k-space annulus) in the third eyepiece waveguide  2800   c  is larger than the outer portion of the green FOV rectangle that does not propagate in TIR (i.e., does not lie in the k-space annulus) in the second eyepiece waveguide  2800   b ;
 
                 Λ   1     &lt;       λ   Blue       1   +       k   FoV     2           ,         
which provides that the maximum period of the first eyepiece waveguide  2800   a  can be the maximum period at which the inner edge of the blue FOV rectangle propagates in TIR (i.e., lies in the k-space annulus) in the large grating vector layer (any larger grating period in the first eyepiece waveguide  2800   a  would result in a shorter grating vector which would not translate the inner edge of the blue FOV rectangle into the k-space annulus);
 
                 Λ   3     &gt;       λ   Red       n   -       k   FoV     2           ,         
which provides that the minimum period of the ICG region  2840   c  of the third eyepiece waveguide  2800   c  can be the minimum period at which the outer edge of the red FOV rectangle propagates in TIR (i.e., lies in the k-space annulus) in the small-grating-vector layer (any smaller grating period in the third eyepiece waveguide  2800   c  would result in a longer grating vector which would translate the outer edge of the red FOV rectangle beyond the k-space annulus); additional constraints may be necessary to set a maximum period for Λ 3  to enforce that the full red FOV rectangle is preserved;
 
                     2   ⁢     λ   Blue         Λ   2       -       k   FoV     2       &gt;   n     ,         
which provides that the second-order diffraction of blue light from the ICG region  2840   b  on the second eyepiece waveguide  2800   b  does not propagate in TIR (i.e., does not lie in the k-space annulus) over the full FOV;
 
                     2   ⁢     λ   Green         Λ   3       -       k   FoV     2       &gt;   n     ,         
which provides that the second-order diffraction of green light from the ICG region  2840   c  on the third eyepiece waveguide  2800   c  does not propagate in TIR over the full FOV.
 
     The maximum FOV achieved using the example eyepiece waveguide stacks in  FIGS.  27 A- 27 B and  28 A- 28 B , subject to the foregoing design considerations, is a function of the refractive index of the eyepiece waveguides. The maximum FOV—in the direction of the ICG grating vector—as a function of refractive index is shown in Table 1. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Maximum FOV in the ICG grating vector direction 
               
               
                 as a function of substrate index. 
               
            
           
           
               
               
            
               
                   
                 Maximum FOV in ICG Grating Vector Direction (°) 
               
            
           
           
               
               
               
               
            
               
                 Substrate 
                 Single 
                 3 Layers w/Multiple 
                 3 Layers w/Multiple 
               
               
                 Refractive 
                 Color 
                 Colors Per Layer 
                 Colors Per Layer 
               
               
                 Index 
                 Per Layer 
                 (FIGS. 27A-27B) 
                 (FIGS. 28A-28B) 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                 1.5 
                 29.0 
                 41.5 
                 44.5 
               
               
                 1.6 
                 34.9 
                 50.4 
                 47.3 
               
               
                 1.7 
                 41.0 
                 54.2 
                 50.9 
               
               
                 1.8 
                 47.2 
                 57.5 
                 53.6 
               
               
                 1.9 
                 53.5 
                 60.9 
                 57.3 
               
               
                 2.0 
                 60.0 
                 64.7 
                 60.9 
               
               
                 2.1 
                 66.7 
                 68.0 
                 63.6 
               
               
                 2.2 
                 73.7 
                 71.9 
                 67.3 
               
               
                 2.3 
                 81.1 
                 75.8 
                 70.9 
               
               
                 2.4 
                 88.9 
                 79.6 
                 74.6 
               
               
                   
               
            
           
         
       
     
       FIG.  29    is a graph  2900  which plots the FOV values in Table 1 as a function of refractive index. The FOV values for single-color-per-layer eyepiece waveguides (i.e., no “cross-talk” between layers) are shown with diamonds. The FOV values for the embodiments of  FIGS.  27 A- 27 B  are shown with triangles. The FOV values for the embodiments of  FIGS.  28 A- 28 B  are shown with squares. 
     As seen in the graph  2900 , for lower refractive indexes below about 2.0-2.1, the embodiments which utilize “cross-talk” between eyepiece waveguides (e.g., the embodiments shown in  FIGS.  27 A- 27 B  and in  FIGS.  28 A- 28 B ) are able to achieve a higher FOV (in the direction of the ICG grating vectors) than single-color-per-layer waveguides which do not utilize “cross-talk” between layers. However, for higher refractive indexes above about 2.0-2.1, the FOV of embodiments which utilize “cross-talk” may become limited by second-order diffractions. 
     The data in Tables 2A and 2B show example ICG grating periods for maximum FOV in the ICG grating vector direction according to the embodiments shown in  FIGS.  27 A- 27 B  and in  FIGS.  28 A- 28 B . The grating periods are compared to those for single-color-per-layer embodiments. The values in the table were calculated using 455 nm, 525 nm, and 625 nm wavelengths for red, green, and blue light, respectively. In the case of narrow line-width (&lt;5 nm) light sources, like lasers, the values and ranges describe the limits of grating periods to hit the maximum FOV. In the case of broader line-width light sources, like LEDs, the values and ranges have more flexibility because of the range of wavelengths available to fill the FOV for each color channel. 
     Table 2A illustrates example periods for the ICG regions  2740   a ,  2740   b ,  2740   c  of the eyepiece waveguides  2700   a ,  2700   b ,  2700   c  according to the embodiments illustrated in  FIGS.  27 A- 27 B . These grating periods can be calculated using the equations discussed above. 
     
       
         
           
               
             
               
                 TABLE 2A 
               
             
            
               
                   
               
               
                 Example ICG grating periods for maximum FOV in the ICG grating vector direction. The values given are 
               
               
                 calculated using 455 nm, 525 nm, and 625 nm wavelengths for Red, Green, and Blue light, respectively. 
               
            
           
           
               
               
               
            
               
                   
                   
                 3 Layers with Multiple Colors Per Layer 
               
               
                   
                 Single Color Per Layer 
                 (FIGS. 27A-27B) 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                   
                 Max FOV in 
                 Blue Layer 
                 Green 
                 Red Layer 
                 Max FOV in 
                 Large Grating 
                 Medium Grating 
                 Small Grating 
               
               
                 Substrate 
                 ICG Grating 
                 ICG 
                 Layer ICG 
                 ICG 
                 ICG Grating 
                 Vector Layer 
                 Vector Layer 
                 Vector Layer 
               
               
                 Refractive 
                 Vector 
                 Period 
                 Period 
                 Period 
                 Vector 
                 ICG Period 
                 ICG Period 
                 ICG Period 
               
               
                 Index 
                 Direction (°) 
                 (nm) 
                 (nm) 
                 (nm) 
                 Direction (°) 
                 (nm) 
                 (nm) 
                 (nm) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 1.5 
                 29 
                 364 
                 420 
                 500 
                 41.5 
                 ≤336 
                 ≥458 
                 ≥546 
               
               
                 1.6 
                 34.9 
                 350 
                 404 
                 481 
                 50.4 
                 ≤319 
                 ≥447 
                 ≥532 
               
               
                 1.7 
                 41 
                 337 
                 389 
                 463 
                 54.2 
                 ≤313 
                 ≥422 
                 ≥502 
               
               
                 1.8 
                 47.2 
                 325 
                 375 
                 446 
                 57.5 
                 ≤307 
                 ≥398 
                 ≥474 
               
               
                 1.9 
                 53.5 
                 314 
                 362 
                 431 
                 60.9 
                 ≤302 
                 ≥377 
                 ≥449 
               
               
                 2.0 
                 60 
                 303 
                 350 
                 417 
                 64.7 
                 ≤296 
                 ≥358 
                 ≥427 
               
               
                 2.1 
                 66.7 
                 294 
                 339 
                 403 
                 68 
                 ≤292 
                 ≥341 
                 ≥406 
               
               
                 2.2 
                 73.7 
                 284 
                 328 
                 391 
                 71.9 
                 ≤287 
                 ≥325 
                 ≥387 
               
               
                 2.3 
                 81.1 
                 276 
                 318 
                 379 
                 75.8 
                 ≤282 
                 ≥311 
                 ≥371 
               
               
                 2.4 
                 88.9 
                 268 
                 309 
                 368 
                 79.6 
                 ≤277 
                 ≥298 
                 ≥355 
               
               
                   
               
            
           
         
       
     
     Table 2B illustrates example periods for the ICG regions  2840   a ,  2840   b ,  2840   c  of the eyepiece waveguides  2800   a ,  2800   b ,  2800   c  according to the embodiments illustrated in  FIGS.  28 A- 28 B . These grating periods can be calculated using the equations discussed above. 
     
       
         
           
               
             
               
                 TABLE 2B 
               
             
            
               
                   
               
               
                 Example ICG grating periods for maximum FOV in the ICG grating vector direction. The values given are 
               
               
                 calculated using 455 nm, 525 nm, and 625 nm wavelengths for Red, Green, and Blue light, respectively. 
               
            
           
           
               
               
               
            
               
                   
                   
                 3 Layers with Multiple Colors Per Layer 
               
               
                   
                 Single Color Per Layer 
                 (FIGS. 28A-28B) 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                   
                 Max FOV in 
                 Blue Layer 
                 Green 
                 Red Layer 
                 Max FOV in 
                 Large Grating 
                 Medium Grating 
                 Small Grating 
               
               
                 Substrate 
                 ICG Grating 
                 ICG 
                 Layer ICG 
                 ICG 
                 ICG Grating 
                 Vector Layer 
                 Vector Layer 
                 Vector Layer 
               
               
                 Refractive 
                 Vector 
                 Period 
                 Period 
                 Period 
                 Vector 
                 ICG Period  
                 ICG Period 
                 ICG Period 
               
               
                 Index 
                 Direction (°) 
                 (nm) 
                 (nm) 
                 (nm) 
                 Direction (°) 
                 (nm) 
                 (nm) 
                 (nm) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 1.5 
                 29 
                 364 
                 420 
                 500 
                 44.5 
                 ≤330 
                 408-447 
                 ≥557 
               
               
                 1.6 
                 34.9 
                 350 
                 404 
                 481 
                 47.3 
                 ≤325 
                 380-454 
                 ≥521 
               
               
                 1.7 
                 41 
                 337 
                 389 
                 463 
                 50.9 
                 ≤318 
                 359-427 
                 ≥492 
               
               
                 1.8 
                 47.2 
                 325 
                 375 
                 446 
                 53.6 
                 ≤314 
                 338-403 
                 ≥463 
               
               
                 1.9 
                 53.5 
                 314 
                 362 
                 431 
                 57.3 
                 ≤308 
                 321-382 
                 ≥440 
               
               
                 2.0 
                 60 
                 303 
                 350 
                 417 
                 60.9 
                 ≤302 
                 306-362 
                 ≥419 
               
               
                 2.1 
                 66.7 
                 294 
                 339 
                 403 
                 63.6 
                 ≤298 
                 298-346 
                 ≥397 
               
               
                 2.2 
                 73.7 
                 284 
                 328 
                 391 
                 67.3 
                 ≤293 
                 293-330 
                 ≥380 
               
               
                 2.3 
                 81.1 
                 276 
                 318 
                 379 
                 70.9 
                 ≤288 
                 288-315 
                 ≥363 
               
               
                 2.4 
                 88.9 
                 268 
                 309 
                 368 
                 74.6 
                 ≤283 
                 283-302 
                 ≥348 
               
               
                   
               
            
           
         
       
     
     To this point, the limit to the FOV in the direction perpendicular to the ICG grating vector has not been described with respect to the embodiments of  FIGS.  27 A- 27 B  and  FIGS.  28 A- 28 B . It is largely a factor of the eyepiece grating architecture used to expand the exit pupil. A number of eyepiece grating architectures are described with respect to  FIGS.  14 A- 26 F . Any of these eyepiece waveguide architectures are compatible with the embodiments shown in  FIGS.  27 A- 27 B  and  FIGS.  28 A- 28 B . The maximum FOV in the direction perpendicular to the ICG grating vector will depend on which layer level grating architecture is used. 
     As an example, the embodiments shown in  FIGS.  24 A- 26 F  can be used with the stack configuration shown in  FIG.  27 B . Table 3 provides example CPE grating periods and angles for the case where the eyepiece waveguides have a refractive index of 1.75. 
     
       
         
           
               
             
               
                 TABLE 3 
               
               
                   
               
               
                 Example Grating Periods and Angles used in example eyepiece stacks  
               
               
                 with and without cross-talk (e.g., color component images being split 
               
               
                 amongst multiple eyepiece waveguides) on index 1.75 substrates 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Cross- 
                 ICG  
                 ICG  
                 CPE1 
                 CPE1  
                 CPE2 
                 CPE2 
               
               
                 Talk 
                 Period 
                 Angle 
                 Period 
                 Angle 
                 Period 
                 Angle 
               
               
                 Stack 
                 (nm) 
                 (°) 
                 (nm) 
                 (°) 
                 (nm) 
                 (°) 
               
               
                   
               
               
                 Layer 1 
                 310 
                 0 
                 294 
                 61.61 
                 294 
                 −61.61 
               
               
                 Layer 2 
                 410 
                 0 
                 349 
                 64.80 
                 349 
                 −64.80 
               
               
                 Layer 3 
                 500 
                 0 
                 418 
                 65.32 
                 418 
                 −65.32 
               
               
                   
               
               
                 No Cross- 
                 ICG  
                 ICG  
                 CPE1 
                 CPE1  
                 CPE2 
                 CPE2  
               
               
                 Talk 
                 Period 
                 Angle 
                 Period  
                 Angle 
                 Period 
                 Angle 
               
               
                 Stack 
                 (nm) 
                 (°) 
                 (nm) 
                 (°) 
                 (nm) 
                 (°) 
               
               
                   
               
               
                 Layer 1 
                 331 
                 0 
                 299 
                 63.35 
                 299 
                 −63.35 
               
               
                 Layer 2 
                 382 
                 0 
                 345 
                 63.35 
                 345 
                 −63.35 
               
               
                 Layer 3 
                 455 
                 0 
                 410 
                 63.35 
                 410 
                 −63.35 
               
               
                   
               
            
           
         
       
     
     Besides being capable of enhancing FOV, the techniques described herein for utilizing multiple eyepiece waveguides to project different sub-portions of the FOV of each color component can also be utilized to improve the uniformity of the luminance of the output light from the eyepiece waveguides across the FOV of the output image. The additional degrees of freedom available when coupling a color component into multiple eyepiece waveguides allow for a tradeoff between these two benefits of increased FOV and improved uniformity. 
     When a color component is coupled into two eyepiece waveguides with different grating vectors, each angle in the FOV takes a different path through the eyepiece waveguide before eventually being diffracted out and toward the user&#39;s eye. This diversity of optical paths of the same angle in the FOV can be used to the designer&#39;s advantage in order to compensate for regions in the FOV that would otherwise be darker, thus improving the luminance uniformity of the virtual image projected to the user. 
       FIG.  30    illustrates an example of improved output image uniformity using the eyepiece waveguide stacks shown in  FIGS.  27 A- 27 B  and  FIGS.  28 A- 28 B . The left-hand image shows a 45°×55° FOV uniformity pattern for the green color channel in a single-color-per-layer design as compared to the right-hand image, which shows a 45°×55° FOV uniformity pattern for the green color channel in a design where sub-portions of the green FOV are propagated through different eyepiece waveguides (e.g., as in  FIGS.  27 A- 27 B  and  FIGS.  28 A- 28 B ). As can be seen in  FIG.  30   , the right-hand image has a more uniform distribution of light. 
     Example Embodiments 
     An augmented reality display system comprising: a first eyepiece waveguide comprising a first optically transmissive substrate; a first input coupling grating (ICG) region formed on or in the first eyepiece waveguide, the first ICG region being configured to receive a set of input beams of light corresponding to an input image having a corresponding field of view, and to couple a first subset of the input beams into the substrate as a first set of guided beams, the first subset of the input beams corresponding to a first sub-portion of the field of view of the input image; a second eyepiece waveguide comprising a second optically transmissive substrate; and a second input coupling grating (ICG) region formed on or in the second eyepiece waveguide, the second ICG region being configured to receive at least a second subset of the input beams of light corresponding to the input image, and to couple the second subset of input beams into the substrate as a second set of guided beams, the second subset of the input beams corresponding to a second sub-portion of the field of view of the input image, wherein the first and second sub-portions of the field of view are at least partially different but together include the complete field of view of the input image. 
     The augmented reality display system of any preceding embodiment, wherein the field of view of the input image is larger in at least one dimension than a range of total internal reflection propagation angles in a thickness direction of the first and second eyepiece waveguides. 
     The augmented reality display system of any preceding embodiment, further comprising: a third eyepiece waveguide comprising a third optically transmissive substrate; and a third input coupling grating (ICG) region formed on or in the third eyepiece waveguide, the third ICG region being configured to receive at least a third subset of the input beams of light corresponding to the input image, and to couple the third subset of input beams into the substrate as a third set of guided beams, the third subset of the input beams corresponding to a third sub-portion of the field of view of the input image, wherein the first, second, and third sub-portions of the field of view are at least partially different but together include the complete field of view of the input image. 
     The augmented reality display system of any preceding embodiment, wherein: the first ICG region comprises periodic diffractive features with a first spatial period, the second ICG region comprises periodic diffractive features with a second spatial period, the third ICG region comprises periodic diffractive features with a third spatial period, the first spatial period is smaller than the second spatial period, and the second spatial period is smaller than the third spatial period. 
     The augmented reality display system of any preceding embodiment, wherein: for a refractive index n≤1.5, the first spatial period is &lt;336 nm, the second spatial period is 408-458 nm, and the third spatial period &gt;546 nm; for a refractive index 1.5&lt;n≤1.6, the first spatial period is &lt;325 nm, the second spatial period is 380-454 nm, and the third spatial period &gt;521 nm; for a refractive index 1.6&lt;n≤1.7, the first spatial period is &lt;318 nm, the second spatial period is 359-427 nm, and the third spatial period &gt;492 nm; for a refractive index 1.7&lt;n≤1.8, the first spatial period is &lt;314 nm, the second spatial period is 338-403 nm, and the third spatial period &gt;463 nm; for a refractive index 1.8&lt;n≤1.9, the first spatial period is &lt;308 nm, the second spatial period is 321-382 nm, and the third spatial period &gt;440 nm; for a refractive index 1.9&lt;n≤2.0, the first spatial period is &lt;302 nm, the second spatial period is 306-362 nm, and the third spatial period &gt;419 nm; or for a refractive index 2.0&lt;n≤2.1, the first spatial period is &lt;298 nm, the second spatial period is 298-346 nm, and the third spatial period &gt;397 nm. 
     The augmented reality display system of any preceding embodiment, further comprising a projector system configured to project the set of input beams corresponding to the input image toward the first, second, and third eyepiece waveguides, wherein the first eyepiece waveguide is located before the second eyepiece waveguide along the optical path of the set of input beams, and wherein the second eyepiece waveguide is located before the third eyepiece waveguide along the optical path of the set of input beams. 
     The augmented reality display system of any preceding embodiment, wherein: the first and second sub-portions of the field of view partially overlap, and the second and third sub-portions of the field of view partially overlap. 
     The augmented reality display system of any preceding embodiment, wherein the first, second, and third ICG regions are laterally aligned. 
     The augmented reality display system of any preceding embodiment, wherein: the first, second, and third ICG regions are configured to receive input beams of light corresponding to a plurality of color components of the input image; the first ICG region is configured to couple first subsets of the input beams for two or more of the color components into the first substrate as first sets of guided beams, the first subsets of the input beams corresponding to first sub-portions of the field of view of the color components of the input image; the second ICG region is configured to couple second subsets of the input beams for two or more of the color components into the second substrate, the second subsets of the input beams corresponding to second sub-portions of the field of view of the color components of the input image; the third ICG region is configured to couple third subsets of the input beams for two or more of the color components into the third substrate, the third subsets of the input beams corresponding to third sub-portions of the field of view of the color components of the input image; and the first, second, and third sub-portions of the field of view of the respective color components of the input image are at least partially different but together include the complete field of view of the color components of the input image. 
     The augmented reality display system of any preceding embodiment, wherein: the first ICG region comprises a plurality of spatially separated sub-sections, each corresponding to one of the color components; the second ICG region comprises a plurality of spatially separated sub-sections, each corresponding to one of the color components; and the third ICG region comprises a plurality of spatially separated sub-sections, each corresponding to one of the color components. 
     The augmented reality display system of any preceding embodiment, wherein the third ICG region has a second-order diffractive efficiency less than 10%. 
     The augmented reality display system of any preceding embodiment, further comprising an optical filter positioned after the second eyepiece waveguide along an optical path of the input beams, the optical filter being configured to selectively absorb input beams for the color component having the shortest wavelength. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is a yellow filter which absorbs at least 90% of blue light. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is configured to selectively absorb input beams for the two color components having the two shortest wavelengths. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is a red filter which absorbs at least 90% of green and blue light. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is a component provided between the second eyepiece waveguide and the third eyepiece waveguide. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is a dye provided in the third eyepiece waveguide. 
     The augmented reality display system of any preceding embodiment, wherein the first and second ICG regions have a first-order diffractive efficiency of 5-90%. 
     The augmented reality display system of any preceding embodiment, wherein the first ICG region is configured such that the second subset of input beams passes through without being diffracted. 
     The augmented reality display system of any preceding embodiment, wherein the first and second eyepiece waveguides have a refractive index of 1.5-2.1. 
     The augmented reality display system of any preceding embodiment, wherein: for a refractive index n≤1.5, the field of view of the input image in at least one direction is greater than 29.0° but less than or equal to 44.5°; for a refractive index 1.5&lt;n≤1.6, the field of view of the input image in at least one direction is greater than 34.9° but less than or equal to 50.4°; for a refractive index 1.6&lt;n≤1.7, the field of view of the input image in at least one direction is greater than 41.0° but less than or equal to 54.2°; for a refractive index 1.7&lt;n≤1.8, the field of view of the input image in at least one direction is greater than 47.2° but less than or equal to 57.5°; for a refractive index 1.8&lt;n≤1.9, the field of view of the input image in at least one direction is greater than 53.5° but less than or equal to 60.9°; for a refractive index 1.9&lt;n≤2.0, the field of view of the input image in at least one direction is greater than 60.0° but less than or equal to 64.7°; or for a refractive index 2.0&lt;n≤2.1, the field of view of the input image in at least one direction is greater than 66.7° but less than or equal to 68.0°. 
     The augmented reality display system of any preceding embodiment, wherein the first and second eyepiece waveguides have the same dimensions. 
     The augmented reality display system of any preceding embodiment, wherein the first and second eyepiece waveguides are laterally aligned and longitudinally separated by a gap. 
     The augmented reality display system of any preceding embodiment, further comprising: a first orthogonal pupil expander (OPE) region, or a first multi-directional pupil expander (MPE) region, or a first combined pupil expander-extractor (CPE) region formed on or in the first eyepiece waveguide, the first OPE, MPE, or CPE region being configured to receive the first set of guided beams and to replicate them over a spatially-distributed portion of the first eyepiece waveguide; and a second OPE region, or a second MPE region, or a second CPE region formed on or in the second eyepiece waveguide, the second OPE, MPE, or CPE region being configured to receive the second set of guided beams and to replicate them over a spatially-distributed portion of the second eyepiece waveguide. 
     The augmented reality display system of any preceding embodiment, further comprising: a first output coupling grating region formed on or in the first eyepiece waveguide, the first output coupling grating region being configured to output the first set of guided beams from the first eyepiece waveguide as a first set of output beams; and a second output coupling grating region formed on or in the second eyepiece waveguide, the second output coupling grating region being configured to output the second set of guided beams from the second eyepiece waveguide as a second set of output beams, wherein the first and second set of output beams together include the complete field of view of the input image. 
     An augmented reality display system comprising: a first eyepiece waveguide comprising a first optically transmissive substrate; a first input coupling grating (ICG) region formed on or in the first eyepiece waveguide, the first ICG region being configured to receive a set of input beams of light, the set of input beams being associated with a set of k-vectors in k-space corresponding to an input image, and to translate the set of k-vectors to a location in k-space such that a first subset of the k-vectors lies inside a first k-space annulus associated with the first eyepiece waveguide, the first k-space annulus corresponding to a region in k-space associated with guided propagation in the first eyepiece waveguide; a second eyepiece waveguide comprising a second optically transmissive substrate; and a second input coupling grating (ICG) region formed on or in the second eyepiece waveguide, the second ICG region being configured to receive at least a portion of the set of input beams of light, and to translate the set of k-vectors to a location in k-space such that a second subset of the k-vectors lies inside a second k-space annulus associated with the second eyepiece waveguide, the second k-space annulus corresponding to a region in k-space associated with guided propagation in the second eyepiece waveguide; wherein the first and second subsets of the k-vectors are at least partially different but together include the complete set of k-vectors corresponding to the input image. 
     The augmented reality display system of any preceding embodiment, wherein the set of k-vectors corresponding to the input image has at least one dimension in k-space that is larger than a width of the first and second k-space annuli. 
     The augmented reality display system of any preceding embodiment, further comprising: a third eyepiece waveguide comprising a third optically transmissive substrate; and a third input coupling grating (ICG) region formed on or in the third eyepiece waveguide, the third ICG region being configured to receive at least a portion of the set of input beams of light, and to translate the set of k-vectors to a location in k-space such that a third subset of the k-vectors lies inside a third k-space annulus associated with the third eyepiece waveguide, the third k-space annulus corresponding to a region in k-space associated with guided propagation in the third eyepiece waveguide; wherein the first, second, and third subsets of the k-vectors are at least partially different but together include the complete set of k-vectors corresponding to the input image. 
     The augmented reality display system of any preceding embodiment, wherein the display system satisfies the equation 
                     λ   Blue       Λ   2       +       k   FoV     2       &lt;   n     ,         
where n is a refractive index of the first, second, and third eyepiece waveguides, λ Blue  is a center wavelength of blue input light, Λ 2  is a period of the second ICG region, and k FoV  is a k-space dimension of the input image in a direction of an ICG vector.
 
     The augmented reality display system of any preceding embodiment, wherein the display system satisfies the equation 
                 n   -       λ   Green       Λ   1         &gt;     1   -       λ   Green       Λ   2           ,         
where n is a refractive index of the first, second, and third eyepiece waveguides, λ Green  is a center wavelength of green input light, Λ 1  is a period of the first ICG region, and Λ 2  is a period of the second ICG region.
 
     The augmented reality display system of any preceding embodiment, wherein the display system satisfies the equation 
                 n   -       λ   Red       Λ   1         &gt;     1   -       λ   Red       Λ   2           ,         
where n is a refractive index of the first, second, and third eyepiece waveguides, λ Red  is a center wavelength of red input light, Λ 1  is a period of the first ICG region, and Λ 2  is a period of the second ICG region.
 
     The augmented reality display system of any preceding embodiment, wherein the display system satisfies the equation 
                 Λ   1     &lt;       λ   Blue       1   +       k   FoV     2           ,         
where n is a refractive index of the first, second, and third eyepiece waveguides, λ Blue  is a center wavelength of blue input light, Λ 1  is a period of the first ICG region, and k FoV  is a k-space dimension of the input image in a direction of an ICG vector.
 
     The augmented reality display system of any preceding embodiment, wherein the display system satisfies the equation 
                 Λ   3     &gt;       λ   Red       n   -       k   FoV     2           ,         
where n is a refractive index of the first, second, and third eyepiece waveguides, λ Red  is a center wavelength of red input light, Λ 3  is a period of the third ICG region, and k FoV  is a k-space dimension of the input image in a direction of an ICG vector.
 
     The augmented reality display system of any preceding embodiment, wherein the display system satisfies the equation 
                     2   ⁢     λ   Blue         Λ   2       -       k   FoV     2       &gt;   n     ,         
where n is a refractive index of the first, second, and third eyepiece waveguides, λ Blue  is a center wavelength of blue input light, Λ 2  is a period of the second ICG region, and k FoV  is a k-space dimension of the input image in a direction of an ICG vector.
 
     The augmented reality display system of any preceding embodiment, wherein the display system satisfies the equation 
                 Λ   2     &gt;       λ   Green       n   -       k   FoV     2           ,         
where n is a refractive index of the first, second, and third eyepiece waveguides, λ Green  is a center wavelength of green input light, Λ 2  is a period of the second ICG region, and k FoV  is a k-space dimension of the input image in a direction of an ICG vector.
 
     The augmented reality display system of any preceding embodiment, wherein the display system satisfies the equation 
                     λ   Green       Λ   3       -   1     &gt;         λ   Green       Λ   2       -   n       ,         
where n is a refractive index of the first, second, and third eyepiece waveguides, λ Green  is a center wavelength of green input light, Λ 2  is a period of the second ICG region, and Λ 3  is a period of the third ICG region.
 
     The augmented reality display system of any preceding embodiment, wherein the display system satisfies the equation 
                     2   ⁢     λ   Green         Λ   3       -       k   FoV     2       &gt;   n     ,         
where n is a refractive index of the first, second, and third eyepiece waveguides, λ Green  is a center wavelength of green input light, Λ 3  is a period of the third ICG region, and k FoV  is a k-space dimension of the input image in a direction of an ICG vector.
 
     The augmented reality display system of any preceding embodiment, wherein: the first ICG region has a first-order grating vector with a first magnitude; the second ICG region has a first-order grating vector with a second magnitude; the third ICG region has a first-order grating vector with a third magnitude; the first magnitude is larger than the second magnitude; and the second magnitude is larger than the third magnitude. 
     The augmented reality display system of any preceding embodiment, further comprising a projector system configured to project the set of input beams toward the first, second, and third eyepiece waveguides, wherein the first eyepiece waveguide is located before the second eyepiece waveguide along the optical path of the set of input beams, and wherein the second eyepiece waveguide is located before the third eyepiece waveguide along the optical path of the set of input beams. 
     The augmented reality display system of any preceding embodiment, wherein: the first and second subsets of the k-vectors partially overlap, and the second and third subsets of the k-vectors partially overlap. 
     The augmented reality display system of any preceding embodiment, wherein the first, second, and third ICG regions are laterally aligned. 
     The augmented reality display system of any preceding embodiment, wherein: the first, second, and third ICG regions are configured to receive input beams of light for a plurality of color components of the input image, the input beams for each color component being associated with a set of k-vectors in k-space; the first ICG region is configured to translate the sets of k-vectors to locations in k-space such that first subsets of the k-vectors for two or more of the color components lie inside the first k-space annulus; the second ICG region is configured to translate the sets of k-vectors to locations in k-space such that second subsets of the k-vectors for two or more of the color components lie inside the second k-space annulus; the third ICG region is configured to translate the sets of k-vectors to locations in k-space such that third subsets of the k-vectors for two or more of the color components lie inside the third k-space annulus; and the first, second, and third subsets of the k-vectors for the respective color components of the input image are at least partially different but together include the complete sets of k-vectors for the color components of the input image. 
     The augmented reality display system of any preceding embodiment, wherein: the first ICG region comprises a plurality of spatially separated sub-sections, each corresponding to one of the color components; the second ICG region comprises a plurality of spatially separated sub-sections, each corresponding to one of the color components; and the third ICG region comprises a plurality of spatially separated sub-sections, each corresponding to one of the color components. 
     The augmented reality display system of any preceding embodiment, wherein the third ICG region has a second-order diffractive efficiency less than 10%. 
     The augmented reality display system of any preceding embodiment, further comprising an optical filter positioned after the second eyepiece waveguide along an optical path of the input beams, the optical filter being configured to selectively absorb input beams for the color component having the shortest wavelength. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is a yellow filter which absorbs at least 90% of blue light. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is configured to selectively absorb input beams for the two color components having the two shortest wavelengths. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is a red filter which absorbs at least 90% of green and blue light. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is a component provided between the second eyepiece waveguide and the third eyepiece waveguide. 
     The augmented reality display system of any preceding embodiment, wherein the optical filter is a dye provided in the third eyepiece waveguide. 
     The augmented reality display system of any preceding embodiment, wherein the first and second ICG regions have a first-order diffractive efficiency of 5-90%. 
     The augmented reality display system of any preceding embodiment, wherein the first ICG region is configured such that the second subset of k-vectors passes through without being diffracted. 
     The augmented reality display system of any preceding embodiment, wherein the first and second eyepiece waveguides have a refractive index of 1.5-2.1. 
     The augmented reality display system of any preceding embodiment, wherein the first and second eyepiece waveguides have the same dimensions. 
     The augmented reality display system of any preceding embodiment, wherein the first and second eyepiece waveguides are laterally aligned and longitudinally separated by a gap. 
     The augmented reality display system of any preceding embodiment, further comprising: a first orthogonal pupil expander (OPE) region, or a first multi-directional pupil expander (MPE) region, or a first combined pupil expander-extractor (CPE) region formed on or in the first eyepiece waveguide, the first OPE, MPE, or CPE region being configured to receive beams of light corresponding to the first subset of the k-vectors and to replicate them over a spatially-distributed portion of the first eyepiece waveguide; and a second OPE region, or a second MPE region, or a second CPE region formed on or in the second eyepiece waveguide, the second OPE, MPE, or CPE region being configured to receive beams of light corresponding to the second subset of the k-vectors and to replicate them over a spatially-distributed portion of the second eyepiece waveguide. 
     The augmented reality display system of any preceding embodiment, further comprising: a first output coupling grating region formed on or in the first eyepiece waveguide, the first output coupling grating region being configured to translate the first subset of the k-vectors to a position in the center of the first k-space annulus; and a second output coupling grating region formed on or in the second eyepiece waveguide, the second output coupling grating region being configured to translate the second subset of the k-vectors to a position in the center of the second k-space annulus, wherein the first and second subsets of the k-vectors together include the complete set of k-vectors corresponding to the input image. 
     ADDITIONAL CONSIDERATIONS 
     Any of the features described herein with respect to any eyepiece waveguide can alternatively be implemented with any other eyepiece waveguide described herein. 
     Unless the context clearly requires otherwise, throughout the description and the claims, the words “comprise,” “comprising,” “include,” “including,” “have” and “having” and the like are to be construed in an inclusive sense, as opposed to an exclusive or exhaustive sense; that is to say, in the sense of “including, but not limited to.” The word “coupled”, as generally used herein, refers to two or more elements that may be either directly connected, or connected by way of one or more intermediate elements. Likewise, the word “connected”, as generally used herein, refers to two or more elements that may be either directly connected, or connected by way of one or more intermediate elements. Depending on the context, “coupled” or “connected” may refer to an optical coupling or optical connection such that light is coupled or connected from one optical element to another optical element. Additionally, the words “herein,” “above,” “below,” “infra,” “supra,” and words of similar import, when used in this application, shall refer to this application as a whole and not to any particular portions of this application. Where the context permits, words in the above Detailed Description using the singular or plural number may also include the plural or singular number, respectively. The word “or” in reference to a list of two or more items is an inclusive (rather than an exclusive) “or”, and “or” covers all of the following interpretations of the word: any of the items in the list, all of the items in the list, and any combination of one or more of the items in the list, and does not exclude other items being added to the list. In addition, the articles “a,” “an,” and “the” as used in this application and the appended claims are to be construed to mean “one or more” or “at least one” unless specified otherwise. 
     As used herein, a phrase referring to “at least one of” a list of items refers to any combination of those items, including single members. As an example, “at least one of: A, B, or C” is intended to cover: A, B, C, A and B, A and C, B and C, and A, B, and C. Conjunctive language such as the phrase “at least one of X, Y and Z,” unless specifically stated otherwise, is otherwise understood with the context as used in general to convey that an item, term, etc. may be at least one of X, Y or Z. Thus, such conjunctive language is not generally intended to imply that certain embodiments require at least one of X, at least one of Y and at least one of Z to each be present. 
     Moreover, conditional language used herein, such as, among others, “can,” “could,” “might,” “may,” “e.g.,” “for example,” “such as” and the like, unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain embodiments include, while other embodiments do not include, certain features, elements and/or states. Thus, such conditional language is not generally intended to imply that features, elements, and/or states are in any way required for one or more embodiments or whether these features, elements, and/or states are included or are to be performed in any particular embodiment. 
     While certain embodiments have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of the disclosure. Features of any one of the embodiments can be combined and/or substituted with features of any other one of the embodiments. Certain advantages of various embodiments have been described herein. But not all embodiments necessarily achieve each of these advantages. 
     Embodiments have been described in connection with the accompanying drawings. However, the figures are not drawn to scale. Distances, angles, etc. are merely illustrative and do not necessarily bear an exact relationship to actual dimensions and layout of the devices illustrated. 
     The foregoing embodiments have been described at a level of detail to allow one of ordinary skill in the art to make and use the devices, systems, methods, etc. described herein. A wide variety of variation is possible. Components, elements, and/or steps may be altered, added, removed, or rearranged. While certain embodiments have been explicitly described, other embodiments will become apparent to those of ordinary skill in the art based on this disclosure.