Biased total thickness variations in waveguide display substrates

Abstract

To provide suitable waveguide display substrates.SOLUTION: A plurality of waveguide display substrates is provided, each waveguide display substrate having: a cylindrical portion having a diameter and a planar surface; a curved portion opposite the planar surface, defining a nonlinear change in thickness across the substrate and having a maximum height D with respect to the cylindrical portion; and a wedge portion between the cylindrical portion and the curved portion, defining a linear change in thickness across the substrate and having a maximum height W with respect to the cylindrical portion. A target maximum height Dt of the curved portion is 10-7 to 10-6 times the diameter, D is in range of about 70-130% of Dt, and W is less than about 30% of Dt.SELECTED DRAWING: Figure 5

Description

[Technical field]

[0001] (CROSS REFERENCE TO RELATED APPLICATIONS) This application claims the benefit of U.S. Patent Application No. 62 / 805,832, filed February 14, 2019, and U.S. Patent Application No. 62 / 820,769, filed March 19, 2019, both of which are incorporated by reference in their entireties.

[0002] (Technical field) The present invention is concerned with biased total thickness variations in a waveguide display substrate. [Background technology]

[0003] (background) Optical imaging systems, such as wearable head-mounted systems, typically include one or more eyepieces that present a projected image to a user. The eyepiece may be constructed using thin layers of one or more types of refractive materials. By way of example, the eyepiece may be constructed from one or more layers of highly refractive glass, silicon, metal, or polymer substrates.

[0004] In some cases, the eyepiece layer may be patterned (e.g., with one or more optically diffractive nanostructures) so that it displays the received light that is coupled in from an external projector. Additionally, multiple eyepiece layers (or "waveguides") may be used together to project a simulated three-dimensional image. For example, multiple waveguides, each with a specific pattern, may be layered, each of which may relay specific optical information (e.g., wavelength or focal length) of a portion of the volumetric image, such that the entire coherent volumetric image is visible in the aggregate of the specific optical information from each of the waveguides. Thus, the eyepieces may collectively present a full-color volumetric image across three dimensions to the user. This may be useful, for example, in presenting a "virtual reality" environment to the user.

[0005] Unintended variations in the eyepiece can reduce the quality of the projected image. Examples of such unintended variations include wrinkles, uneven thickness, and other physical distortions that can adversely affect the performance of the eyepiece. Summary of the Invention [Means for solving the problem]

[0006] (overview) A first general aspect includes a plurality of waveguide display substrates, each having a cylindrical portion having a diameter and a flat surface, a curved portion opposite the flat surface defining a non-linear change in thickness across the substrate and having a maximum height D relative to the cylindrical portion, and a wedge portion between the cylindrical portion and the curved portion defining a linear change in thickness across the substrate and having a maximum height W relative to the cylindrical portion. The target maximum height D of the curved portion t is 10 of the diameter -7 From 10 -6 D is double, D is double t W is between about 70% and about 130% of D t The average D for multiple waveguide display substrates is D mean and the maximum D for multiple waveguide display substrates is D max and the minimum D for multiple waveguide display substrates is D min and the maximum W for multiple waveguide display substrates is W max It is.

[0007] A second general aspect includes fabricating a plurality of the waveguide display substrates of the first general aspect.

[0008] Implementations of the first and second general aspects may include one or more of the following features.

[0009] In some implementations, the nonlinear change in thickness is a quadratic change in thickness. The curved portion can be in the form of a dome. In some cases, the dome is spherical.

[0010] The average thickness of the waveguide display substrates is typically between about 200 microns and about 2000 microns. The average diameter of the waveguide display substrates is typically between about 2 centimeters and about 50 centimeters. max / D mean is typically less than about 0.3. (D mean -D min ) / D mean is typically less than about 0.3. (D max -D min ) / D mean is typically less than about 0.3. D is typically in the range of about 0.1 microns to about 5 microns. W is typically in the range of 0 to about 1.5 microns. The average total thickness variation for multiple substrates is typically between about 0.1 microns and about 6.5 microns.

[0011] In some implementations, the waveguide display substrate comprises a molded polymer, hi some implementations, the waveguide display substrate comprises a polished glass, silicon, or metal substrate.

[0012] Implementations of the second general aspect can include one or more of the following features.

[0013] In some cases, creating the plurality of waveguide display substrates can include polishing a waveguide display substrate, where the waveguide display substrate is formed from glass, metal, or silicon. In certain cases, creating the plurality of waveguide display substrates includes molding a polymeric waveguide display substrate.

[0014] The second general aspect can further include forming one or more waveguides on each of the waveguide display substrates. The one or more waveguides can include at least two waveguides, and the waveguides can be positioned in a radial pattern on each waveguide display substrate.

[0015] The details of one or more embodiments of the presently disclosed subject matter are set forth in the accompanying drawings and description. Other features, aspects, and advantages of the present subject matter will become apparent from the description, drawings, and claims. The present invention provides, for example, the following: (Item 1) A plurality of waveguide display substrates, each of the waveguide display substrates comprising: a cylindrical portion having a diameter and a flat surface; a curved portion opposite the planar surface defining a non-linear variation in thickness across the substrate and having a maximum height D relative to the cylindrical portion; a wedge portion between the cylindrical portion and the curved portion, the wedge portion defining a linear variation in thickness across the substrate and having a maximum height W relative to the cylindrical portion; having The maximum target height D of the curved part t is 10 of the diameter -7 From 10 -6 D is double, D is double t Between 70% and 130% of D t 30% lower than The average of D for the plurality of waveguide display substrates is D mean and the maximum D for the plurality of waveguide display substrates is D max and the minimum D for the plurality of waveguide display substrates is D min and the maximum W for the plurality of waveguide display substrates is W max That is, A plurality of waveguide display substrates. (Item 2) Item 2. The multiple waveguide display substrate of item 1, wherein the nonlinear change in thickness is a quadratic change in thickness. (Item 3) Item 2. The multiple waveguide display substrate of item 1, wherein the curved portion is in the form of a dome. (Item 4) Item 2. The multiple waveguide display substrate of item 1, wherein the dome is spherical. (Item 5) Item 1 . The plurality of waveguide display substrates of item 1 , wherein an average thickness of the plurality of waveguide display substrates is between about 200 microns and about 2000 microns. (Item 6) Item 1 . The plurality of waveguide display substrates of item 1 , wherein an average diameter of the plurality of waveguide display substrates is between about 2 centimeters and about 50 centimeters. (Item 7) W max / D mean Item 2. The multiple waveguide display substrate of item 1, wherein (Item 8) (D mean -D min ) / D mean Item 2. The multiple waveguide display substrate of item 1, wherein (Item 9) (D max -D min ) / D mean Item 2. The multiple waveguide display substrate of item 1, wherein (Item 10) Item 2. The multiple waveguide display substrate of item 1, wherein D is in the range of about 0.1 microns to about 5 microns. (Item 11) Item 2. The multiple waveguide display substrate of item 1, wherein W is in the range of 0 to about 1.5 microns. (Item 12) Item 1 . The plurality of waveguide display substrates of item 1 , wherein the average total thickness variation of the plurality of substrates is between about 0.1 microns and about 6.5 microns. (Item 13) Item 10. The plurality of waveguide display substrates of item 1, wherein the waveguide display substrate comprises a molded polymer. (Item 14) Item 1 . The plurality of waveguide display substrates of item 1 , wherein the waveguide display substrate comprises a polished glass, silicon, or metal substrate. (Item 15) 1. A method for making a plurality of waveguide display substrates, each of which comprises: a cylindrical portion having a diameter and a flat surface; a curved portion opposite the planar surface defining a non-linear variation in thickness across the substrate and having a maximum height D relative to the cylindrical portion; a wedge portion between the cylindrical portion and the curved portion, the wedge portion defining a linear variation in thickness across the substrate and having a maximum height W relative to the cylindrical portion; having The maximum target height D of the curved part t is 10 of the diameter -7 From 10 -6 D is double, D is double t Between 70% and 130% of D t 30% lower than The average of D for the plurality of polished waveguide display substrates is D mean and the maximum D for the plurality of polished waveguide display substrates is D max and the minimum D for the plurality of polished waveguide display substrates is D min and the maximum W for the plurality of polished waveguide display substrates is W max That is, method. (Item 16) Item 16. The method of item 15, wherein creating the plurality of waveguide display substrates comprises polishing the waveguide display substrates, the waveguide display substrates being formed from glass, metal, or silicon. (Item 17) Item 16. The method of item 15, wherein fabricating the plurality of waveguide display substrates comprises molding a polymeric waveguide display substrate. (Item 18) Item 16. The method of item 15, further comprising forming one or more waveguides on each of the waveguide display substrates. (Item 19) Item 19. The method of item 18, wherein the one or more waveguides comprise at least two waveguides positioned in a radial pattern on each waveguide display substrate. [Brief description of the drawings]

[0016] BRIEF DESCRIPTION OF THE DRAWINGS [Figure 1] FIG. 1 depicts a sample waveguide display substrate with a substrate area.

[0017] [Diagram 2] 2A-2C depict polished waveguide display substrates having flat, convex and concave cross-sectional shapes, respectively.

[0018] [Diagram 3] 3A and 3B depict a total thickness variation (TTV) comparison of a waveguide display substrate.

[0019] [Figure 4] FIG. 4 depicts the TTV of a polished waveguide display substrate with a convex surface.

[0020] [Diagram 5] FIG. 5 shows a cross section of a waveguide display substrate with an offset TTV having linear (wedge) and nonlinear (dome) components.

[0021] [Figure 6]Figures 6A-6C show the yield vs. TTV, dome height, and wedge height of a waveguide display substrate for a very low TTV waveguide display substrate, and Figures 6D-6F show the yield vs. TTV, dome height, and wedge height of a waveguide display substrate for a biased TTV waveguide display substrate.

[0022] [Figure 7] FIG. 7 shows the waveguide eyebox efficiency versus the spherical waveguide display substrate “dome” height TTV (nm) on a 6 inch waveguide display substrate for a typical diffractive waveguide display with 300 μm average thickness and 0 nm “wedge” TTV. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0023] (Detailed Description) Total thickness variation (TTV) is one measure for improving the performance of light guides. As used herein, TTV generally refers to the difference between the maximum and minimum thickness of a waveguide or a waveguide display substrate on which the waveguide is formed. Since light typically travels through a light guide by total internal reflection, thickness variation changes the light propagation path(s). The angular difference of the light propagation path(s) can affect image quality with distortion of vision, blurring of images and loss of sharpness.

[0024] Waveguide preparation and processing is typically performed by arranging multiple waveguides in a designated area on a waveguide display substrate (e.g., a wafer). FIG. 1 depicts a waveguide display substrate 100 with a radial array of waveguides 102. TTV can be reduced by making a flat waveguide display substrate (i.e., a waveguide display substrate with zero TTV), for example, by polishing the substrate (e.g., a metal, glass, or silicon substrate) or shaping the substrate (e.g., a polymer substrate) with high precision from the beginning. On the other hand, polishing can create an amount of curvature on the waveguide display substrate and the resulting waveguides formed thereon. FIG. 2A depicts a polished flat waveguide display substrate 200. FIG. 2B and FIG. 2C depict a polished convex waveguide display substrate 202 and a polished concave waveguide display substrate 204, respectively. Polishing can impart a convex or concave curvature, although the embodiments described herein are described with reference to a convex curvature such as that depicted in FIG. 2B.

[0025] Fully flat polishing or shaping, such as that depicted in FIG. 2A, requires extensive and costly processing to achieve, so typically some amount of TTV is tolerated. For most low-TTV processes (e.g., 20 nm < TTV < 2 μm) for waveguide display substrates, there exists a thickness shape or profile of the substrate that varies (e.g., randomly) between portions. Here, “thickness shape” generally refers to a 3D mapping of the height difference between the top and bottom surfaces of the substrate. In one example, a typical plano-convex lens has a thickness shape that is a convex spherical or positive “dome”. In another example, a meniscus lens (e.g., non-prescription sunglasses) with the same radius of curvature on each surface has a thickness shape with near-zero flat TTV, but each surface itself is non-flat. When waveguides are stacked to form a multilayer multi-color waveguide display, random thickness shape differences can cause each of the red, green, and blue color channels to have different luminance uniformity patterns. Differences in luminance uniformity patterns can cause colors to vary across the field of view when the stacked waveguides are illuminated with a uniform white light image. These color non-uniformities can result in poorer image quality.

[0026] Referring to FIG. 3A, a waveguide display substrate 300 having a TTV is shown, where the TTV is measured as the highest point relative to the lowest point among the curved portions of the polished waveguide display substrate. The perfectly flat waveguide display substrate 302 depicted in FIG. 3B has zero TTV. FIG. 4 depicts a waveguide display substrate 400 having a minimum thickness X. That is, no portion of the waveguide display substrate is thinner than X, and any thickness variation is measured as the thickness in addition to X. The largest thickness is h, the TTV is mathematically Y, where Y = h - X. TTV specifications are typically expressed as a maximum allowable TTV (TTV max ), and the product specification is that the TTV max is below the TTV (Y) (i.e., 0 ≤ Y < TTV max) as used herein, a "very low TTV" waveguide display substrate refers to a waveguide display substrate that has a target TTV of zero (or as close to zero as practically possible).

[0027] In optical products, image quality and uniformity can be sensitive to the particular shape or profile of the polished waveguide display substrate as the TTV approaches zero. In one example, the comparative difference in image quality (as measured by uniformity) of a waveguide produced on a 20nm TTV and a 40nm TTV waveguide display substrate can be much higher than the comparative quality of a waveguide produced on a 100nm TTV substrate and a 120nm waveguide display substrate, even though both pairs only differ by 20nm. In other words, a 100nm TTV waveguide and a 120nm TTV waveguide can give a similarly uniform image than a 20nm TTV waveguide and a 40nm TTV waveguide (the former pair is more dome-shaped compared to the latter pair). This allows the 100nm TTV waveguide and the 120nm TTV waveguide to produce lower image variability and more consistent image uniformity across the product line.

[0028] The path length difference of the light propagation path(s) in the waveguide produced on the non-flat waveguide display substrate can also affect the image quality with brightness pattern non-uniformity and color non-uniformity. Sources of brightness pattern non-uniformity include electromagnetic interference patterns produced by the multiple paths through the pupil replica waveguide display substrate. A typical pupil replica waveguide unit cell resembles a Mach-Zender interferometer, where there are two paths per unit cell from the input to the location of the replicated pupil, which is the output. The path length difference between the two paths is affected by the path length through the thickness of the waveguide display substrate, which is defined by the TTV index and the thickness profile (whether the thickness varies linearly or quadratically with an angle for the light undergoing total internal reflection in the waveguide display). If the path lengths have equal or opposite phases, constructive or destructive interference can exist, respectively. Therefore, the thickness profile can affect the intensity in the pupil replicated copy and ultimately the output image coupled out by the waveguide display.

[0029] This may be advantageous to minimize thickness profile variation and TTV. Since perfect replication may not be achieved in the polishing or shaping process, some manufacturing distribution may occur. In a hypothetical distribution model, with a circular substrate shape defined by Zernike polynomials, a standard set of shape basis functions may be defined. There are even and odd Zernike polynomials. The even ones are

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[0030] Image quality, especially brightness uniformity, is nonlinearly sensitive to thickness profile as TTV approaches zero. In other words, as TTV decreases below a certain threshold, image quality becomes increasingly variable between waveguides exhibiting even small changes in thickness profile. To compensate for this anomaly, biased TTV with consistent thickness profile can be incorporated into substrate processing. As used herein, "biased" TTV generally refers to a TTV with a nonzero target. More specifically, "biased" TTV generally refers to a substrate thickness profile having one or more coefficients of a Zernike fit polynomial with a nonzero target and all remaining coefficients of the Zernike fit polynomial with a zero target. Consistency of thickness profile within multiple waveguide display substrates generally refers to multiple waveguide display substrates having low variation in the coefficients of the Zernike fit polynomial. In the case of a "biased" TTV, thickness profile consistency refers to a set of waveguide display substrates having (i) all Zernike coefficients targeted to non-zero that have minimal variation from the target magnitude, e.g., the non-zero targeted coefficients for all waveguide display substrates are about 70% to about 130% of the target magnitude, and (ii) all Zernike coefficients targeted to zero that have an absolute magnitude significantly lower than the non-zero targeted Zernike coefficients, e.g., the zero targeted coefficients are 0 to about 30% of the non-zero targeted coefficients.

[0031] An example of a biased TTV and conformal substrate thickness profile is a substrate that has been significantly polished (or shaped) into a spherical shape with a TTV many times larger than the typical minimum TTV range of the substrate polishing (or shaping) process. This substrate shape can be described as having a wedge (linear change in thickness) component and a dome (quadratic change in thickness) component. FIG. 5 depicts a waveguide display substrate 500 with a wedge component 502 having height W, a dome component 504 having height D, and a cylindrical component 506 with a thickness t and diameter d with a flat surface 508. Using the Zernike fit polynomial, the average thickness of the waveguide display substrate is Z 0 0 and the height of the "wedge" component of the TTV is sqrt(Z 1 -1 +Z 1 1 ), and the height of the convex "dome" component of the TTV may be defined as -2×Z 2 0 The average thickness of a waveguide display substrate is typically between 200 μm and 2000 μm. In a substrate polishing or shaping process that produces a "dome" biased TTV, the waveguide display substrate is typically 10 times the substrate diameter. -7 From 10 -6 Within the range of the target "dome" height D t Each substrate may have a "dome" height less than 30% of the target "dome" height and a "wedge" height less than 30% of the target "dome" height. In one example, a substrate may have a diameter of 150 mm and an average "dome" height (D mean ) and the range from 700 nm to 1300 nm (D min From D max ) and a maximum "wedge" height (W max) will have increased efficiency, brightness uniformity and color uniformity and reduced part-to-part variation in efficiency, brightness uniformity and color uniformity compared to waveguide display substrates with near-zero targeted TTV with "dome" and "wedge" heights in the range of 0 to 300 nm. With the radial display layout of FIG. 1 and the radially symmetric substrate thickness profile with the matching shape biased to a dome, a matching thickness variation from the waveguide display input coupler to the output coupler can be achieved among the many parts arranged on a single substrate and between substrates. Such an arrangement of matching thickness shapes shows improved color uniformity and image quality compared to the typical very low TTV with random thickness shapes used in conventional waveguide displays.

[0032] The biased TTV with conformal shape can be applied to the waveguide display substrate in a number of ways. For glass or quartz substrates, it can be applied by grinding into the conformal shape, or by applying a coating with a non-uniform thickness (of conformal shape and size) to a very low TTV waveguide display substrate with small but random thickness shape variations. For moldable polymeric materials, the biased thickness profile can be designed in the mold that produces the waveguide display substrate.

[0033] 6A-6C show the yield vs. TTV, yield vs. dome height, and yield vs. wedge height of a waveguide display substrate for a very low TTV waveguide display substrate. In these very low TTV waveguide display substrates, the magnitudes of the dome height and wedge height are typically similar to each other. Also, the range of dome height variation is similar in magnitude to the average dome height. FIG. 6D-6F shows the yield vs. TTV, yield vs. dome height, and yield vs. wedge height of a waveguide display substrate for a biased TTV waveguide display substrate. In these biased TTV waveguide display substrates, the magnitude of the dome height is greater than the magnitude of the wedge height, and the dome height variation is less than the average dome height. FIG. 6D-6F express and depict the biased relationships without reference to specific values. The specification for the biased TTV tolerance is: W max / D mean <X、 (D mean -D min ) / D mean <Y、 (D max -D mean ) / D mean <Z For these relationships, X, Y, and Z typically range from 0 to 10 between different substrate polishing or shaping processes. FIGS. 6A-6C show a set of waveguide display substrates that result in higher values ​​of X, Y, and Z than the set of waveguide display substrates shown in FIGS. 6D-6F. Rather than forcing the TTV itself to zero, as X, Y, and Z approach zero, the overall efficiency of the multiple waveguides produced on the multiple subsections of the multiple waveguide display substrates increases and has less variation. Also, as X, Y, and Z approach zero, the resulting luminance and color uniformity between the waveguides also increases and has less variation. In a typical substrate polishing or shaping process, X, Y, and Z are proportional to the targeted TTV. maxapproaches zero. Ultra-low TTV waveguide display substrates typically have X, Y, and Z in the ranges of 1 to 10, while biased TTV waveguide display substrates have X, Y, and Z in the ranges of 0 to 0.3, and thus biased (or non-zero target) TTV can contribute to improved image quality of waveguide displays.

[0034] FIG. 7 shows the waveguide eyebox efficiency versus the "dome" TTV (nm) of a spherical waveguide display substrate geometry on a 6-inch wafer for a typical diffractive waveguide display with an average thickness of 300 μm and a "wedge" TTV of 0 nm. As can be seen in FIG. 7, the eyebox efficiency is maximum between a TTV of 400 nm and a TTV of 600 nm. Here, the waveguide eyebox efficiency is the sum of the light incident on the input coupling grating of the waveguide display relative to the light incident on the input coupling grating of the waveguide display 15 mm from the eye side of the output grating of the diffractive waveguide display, as calculated by a typical diffractive waveguide simulation. 2 15×20mm apart 2 It refers to the sum of the light that falls into a rectangular area.

[0035] Although the present disclosure contains many specific embodiment details, these should not be interpreted as limitations on the scope of the present subject matter or the scope of the claims, but rather as descriptions of features that may be specific to a particular embodiment. Certain features described in the present disclosure in the context of separate embodiments may also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment may also be implemented in multiple embodiments, separately or in any suitable subcombination. Moreover, although the features described above may be described as acting in a combination, and may even be initially claimed as such, one or more features from the claimed combination may in some cases be separated from the combination, and the claimed combination may be directed to a subcombination or a variation of the subcombination.

[0036] Particular embodiments of the present subject matter have been described. Other embodiments, alterations, and permutations of the described embodiments are within the scope of the following claims, as will be apparent to those skilled in the art. Although operations are depicted in a particular order in the figures and claims, this should not be understood as requiring such operations to be performed in the particular order or sequence shown, or that all illustrated operations be performed (although some operations may be considered optional) to achieve desirable results.

[0037] Accordingly, the exemplary embodiments described above do not define or constrain the present disclosure, and other modifications, substitutions, and alterations are also possible without departing from the spirit and scope of the present disclosure.

Claims

1. 1. A waveguide display substrate, comprising: a cylindrical portion having a flat surface, the area and shape of the flat surface being the area and shape of the waveguide display substrate when viewed in a thickness direction of the waveguide display substrate; a curved portion opposite the plane that defines a nonlinear variation in thickness across the substrate; a wedge portion between the cylindrical portion and the curved portion that defines a linear variation in thickness across the substrate; and Equipped with A waveguide display substrate, wherein the dome height of the bent portion ranges between 700 nm and 1300 nm, and the maximum height of the wedge portion is less than 300 nm.

2. The waveguide display substrate of claim 1, wherein D is a maximum height of the curved portion relative to the cylindrical portion, and D is between 70% and 130% of a target value Dt for D of the curved portion.

3. The waveguide display substrate of claim 1, wherein W is a maximum height of the wedge portion relative to the cylindrical portion, and W is less than 30% of a target value Dt of D of the curved portion.

4. 10. The waveguide display substrate of claim 1, wherein the non-linear variation in thickness is a quadratic variation in thickness.

5. The waveguide display substrate of claim 1 , wherein the curved portion is in the form of a dome.

6. The waveguide display substrate of claim 5 , wherein the dome is spherical.

7. 10. The waveguide display substrate of claim 1, wherein the average thickness of the waveguide display substrate is between about 200 microns and about 2000 microns.

8. 10. The waveguide display substrate of claim 1, wherein the average diameter of the waveguide display substrate is between about 2 centimeters and about 50 centimeters.

9. 10. The waveguide display substrate of claim 1, wherein the waveguide display substrate comprises a molded polymer.

10. 10. The waveguide display substrate of claim 1, wherein the waveguide display substrate comprises at least one of a polished glass, silicon, or metal substrate.

11. 1. A method, comprising: Making a waveguide display substrate, the substrate comprising: a cylindrical portion having a flat surface, the area and shape of the flat surface being the area and shape of the waveguide display substrate when viewed in a thickness direction of the waveguide display substrate; a curved portion opposite the plane and defining a nonlinear variation in thickness across the substrate; a wedge portion between the cylindrical portion and the curved portion that defines a linear variation in thickness across the substrate; and To ensure that Including, The method wherein the dome height of the curved portion ranges between 700 nm and 1300 nm and the maximum height of the wedge portion is less than 300 nm.

12. The method of claim 11, wherein D is a maximum height of the curved portion relative to the cylindrical portion, and the waveguide display substrate is fabricated such that D is between 70% and 130% of a target value Dt for D of the curved portion.

13. The method of claim 11, wherein W is the maximum height of the wedge portion relative to the cylindrical portion, and the waveguide display substrate is fabricated such that W is less than 30% of a target value Dt of D of the curved portion.

14. The method of claim 11 , wherein the waveguide display substrate is fabricated such that the nonlinear variation in thickness is a quadratic variation in thickness.

15. The method of claim 11 , wherein the waveguide display substrate is fabricated such that the curved portion is in the form of a dome.

16. The method of claim 15 , wherein the waveguide display substrate is fabricated such that the dome is spherical.

17. 12. The method of claim 11, wherein the waveguide display substrate is fabricated such that the average thickness of the waveguide display substrate is between about 200 microns and about 2000 microns.

18. 12. The method of claim 11, wherein the waveguide display substrate is fabricated such that an average diameter of the waveguide display substrate is between about 2 centimeters and about 50 centimeters.

19. The method of claim 11 , wherein the waveguide display substrate is fabricated using a molded polymer.

20. The method of claim 11 , wherein the waveguide display substrate is fabricated using at least one of a polished glass, silicon, or metal substrate.

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