ADVANCED REFRACTIVE OPTICS FOR IMMERSIVE VIRTUAL REALITY

Abstract

A display device has a display, operable to generate a real image, and an optical system, comprising one or more lenslets, arranged to generate a virtual sub-image from a partial real image on the display, by each lenslet projecting light from the display to an eye position. The sub-images combine to form a virtual image viewable from the eye position. At least one lenslet is symmetric with respect to a plane, and the display surface is cylindrical with its axis perpendicular to that plane.

Claims

1. A display device comprising: a display, operable to generate a real image; and an optical system, comprising one or more lenslets, arranged to generate a virtual sub-image from a respective partial real image on the display, by each lenslet projecting light from the display to an eye position; wherein the sub-images combine to form a virtual image viewable from the eye position; wherein at least one said lenslet is symmetric with respect to a plane and the display surface is cylindrical with its axis perpendicular to said plane.

2. The display device of claim 1, wherein each said lenslet is symmetric with respect to said plane.

3. The display device of claim 2, wherein said one or more lenslets comprise at least two said symmetric lenslets.

4. The display device of claim 1, wherein said one or more lenslets comprise at least two lenslets, and wherein said respective partial real images on the display are so arranged that the sub-images combine to form said viewable virtual image as an uninterrupted virtual image of a given image.

5. The display device of claim 1, wherein at least one said lenslet is arranged to produce a respective said partial virtual image that contains a part projected by an eye onto a 1.5 mm fovea of said eye when said eye is at the eye position with its pupil within a pupil range, said part of each said virtual image having a higher resolution than a peripheral part.

6. A headgear comprising the display device of claim 1, with a mount for positioning the display device on a human head with the eye position of the display device coinciding with an eye of the human.

7. The headgear of claim 5, further comprising a second display device according claim 1, mounted with the eye position of the second display device coinciding with a second eye of the human.

8. The headgear of claim 7, wherein said display of said display device and said display of said second display device are parts of a single display.

9. A display device comprising: a display, operable to generate a real image; and an optical system comprising a plurality of lenslets each arranged to generate a virtual sub-image from a respective partial real image on the display, by each lenslet projecting light from the display to an eye position; wherein the virtual sub-images combine to form a virtual image viewable from the eye position; wherein each lenslet comprises first and second freeform optical lenses so arranged that light from the display passes through both the first freeform optical lens and the second freeform optical lens to the eye position.

10. The display device of claim 9, comprising a freeform lens that is continuous at least in surface and slope, and that forms the second freeform optical lenses of at least two said lenslets that form distinct virtual sub-images.

11. The display device of claim 10, wherein the first lenses of said at least two said lenslets that form distinct virtual sub-images are parts of a single lens separated by at least a break in slope.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0033] FIG. 1 is a schematic view of an optical configuration composed of 2 rotationally symmetric lenses and one large flat display (prior art).

[0034] FIG. 2 is a schematic view of a motion picture projecting system (prior art).

[0035] FIG. 3 is a schematic view of ipixels to opixels mapping (prior art).

[0036] FIG. 4 is a schematic view of an optical configuration that consists of a single lenslet freeform embodiment per eye and one cylindrical digital display.

[0037] FIG. 5 is an isometric schematic view of the device whose horizontal cross section is shown in FIG. 4.

[0038] FIG. 6 shows the angular resolution of a typical human eye as a function of the peripheral angle.

[0039] FIG. 7 is a schematic view of an optical configuration of two 2-fold freeform lenses with one cylindrical digital display.

[0040] FIG. 8 is a schematic view of an optical system composed of one cylindrical digital display with two lenses per eye: one 2-fold freeform lens and one rotationally symmetric lens.

[0041] FIG. 9 is a schematic view of an optical system composed of one cylindrical digital display with two 2-fold lenses per eye placed one in front of the other.

[0042] FIG. 10 is a schematic view of a variant of the optical system shown in FIG. 4 with two cylindrical digital displays and two freeform single-lenslet optical systems.

[0043] FIG. 11 is a schematic view of a variant of the optical system shown in FIG. 4 with two tilted flat digital displays and two freeform single lenslets.

[0044] FIG. 12 is an isometric schematic view of the device whose horizontal cross section is shown in FIG. 11.

[0045] FIG. 13 describes the local coordinate system used to represent the surfaces of one detailed design.

[0046] FIG. 14 is the cross-sectional view of the detailed design of FIG. 13, showing some characteristic rays.

[0047] FIG. 15 is another cross-sectional view of the same design in FIG. 14.

[0048] FIG. 16 is a diagram for defining several angles.

DETAILED DESCRIPTION

[0049] FIG. 4 shows a schematic view of the top view of one embodiment with a single lenslet freeform design per eye and one cylindrical digital display. The cylinder axis of the digital display is perpendicular to the drawing plane. The digital display is designed to present a cylindrical configuration around the user's head, as suggested by 401. This geometry allows for a more ergonomic and compact device than an analogous design with a flat digital display, because the lateral parts of the digital display are closer to the user's face. Freeform lenses 402, one for each eye, each produce two consecutive refractions of light coming from the digital display. Each of the freeform lenses 402 has a plane of symmetry parallel to the plane of the paper in FIG. 4. Dashed lines 403 indicate the virtual rays traveling from the virtual screen towards the eye sphere, while continuous lines 404 represent the real rays traveling from the digital display towards the eye sphere, which are refracted by both surfaces of the respective lens 402. Freeform lenses shown herein are designed with a similar method to the one described in Section 6.14.1 Annex A and Section 6.14.2 Annex B of PCT1. However, in PCT1 the embodiments using a cylindrical digital display had the cylinder axis parallel to the plane of symmetry of the freeform surfaces, whereas in FIG. 4 the cylinder axis is perpendicular to that plane.

[0050] FIG. 5 is a perspective view of the optical configuration previously shown in 2D in FIG. 4. The digital display 502 is placed frontwards and clearly curve-shaped in the horizontal direction. Line 503 indicates the symmetry axis of the digital display, which divides the display 502 into two halves. Each half of the display 502 works with one of the eyes 501 and its corresponding lens 504. Notice that both refractive lenses are freeform (i.e. no rotational symmetry). This is indicated by the irregular curves of the mesh plotted on the surface S1 of each lens.

[0051] An example of this design configuration for a focal length of 45 mm and a curved digital display with a selected circular profile of radius 100 mm (the design not being restricted to this profile) achieves a field of view of 100°.

[0052] Another embodiment has one 2-fold freeform lens per eye and a single cylindrical display, and is shown in horizontal cross section in FIG. 7. We extend here the single lenslet case shown in FIG. 4 to a 2-fold lens. In general, we may extend it to optical devices that are k-fold (k>2). In contrast to the embodiments disclosed in PCT1, the 2-fold lens in FIG. 7 has a plane of symmetry which coincides with the plane of the cross section, and the axis of the cylindrical display is perpendicular to the symmetry plane of the freeform optics. Alternatively, two separate curved or tilted displays could be used.

[0053] Rays exiting the cylindrical digital display 701 suffer two refractions on 2-fold freeform optical devices 702. Dashed lines 704 indicate the virtual rays traveling from the virtual screen towards the eye sphere 705, while continuous lines 703 represent the rays traveling from the digital display 701 towards the eye sphere 705, which are refracted by both surfaces of the optical devices 702 as already mentioned. In general, the two halves of the lens 702 are not symmetric one to the other.

[0054] As shown in FIG. 7, surface S2 nearer to the eye (labeled as 706) is designed with a discontinuity in the derivative where the two lenslets meet. An alternative design can be developed with surface S1 further from the eye (labeled as 707) with the cusp (i.e. a derivative discontinuity), while surface S2 is continuous and differentiable. Both surfaces S1 and S2 may be non-differentiable at the same time in an alternative embodiment as well).

[0055] FIG. 8 shows a 2D cross section of an embodiment with a curved digital display 801 and two lenses per eye: a 2-fold freeform lens 809 and a rotationally symmetric lens 810. Human eye sphere is labeled as 812 and user's nose as 813. As shown in FIG. 8, rays 806, 807 and 808 emitted by the digital display 801 undergo 4 refractions: first they are refracted by both surfaces (802 and 803) of the 2-fold freeform lens 809, afterwards by both surfaces (804 and 805) of the rotationally symmetric lens 810 and finally they are directed towards the eye sphere center 811.

[0056] This configuration has 4 surfaces to design, 2 freeform (802 and 803) and 2 preferably rotationally symmetric (804 and 805). The additional degrees of freedom of this configuration provided by the 2 new rotational symmetric surfaces in comparison with the previous embodiment shown in FIG. 7 may be used to obtain a design with optimized resolution, i.e. a design whose focal length varies with the angular direction of the ipixels on the virtual screen. This variation is performed in order to match the human resolution as described in the reference PCT1.

[0057] FIG. 9 shows a schematic view of a horizontal cross section of an embodiment that consists of two 2-fold freeform lenses per eye labeled as 902 and 903 and placed consecutively between the human eye 911 and curved display 901. Lens 902 is made of a different material from that of lens 903 in order to decrease the chromatic aberration of the whole device. Since there are 4 freeform surfaces to design, this configuration has sufficient degrees of freedom, so they can be employed to control the focal distances, to widen the field of view and to improve the image formation quality in comparison with designs that only present two freeform surfaces per eye, and thus fewer degrees of freedom. Alternatively, surfaces 924 and 915 are made to coincide with 925 and 916, so they can be cemented and freeform doublets are obtained. The cylindrical digital display is labeled as 901. The user's nose 918 and two eye spheres 911 in FIG. 9 are shown for orientation purposes. A 2-fold design is illustrated in FIG. 9, but this configuration can be obviously extended to k-fold optical devices, with k>2.

[0058] Ray 906 exits the digital display 901 near its border. Then it is refracted by surfaces 923 and 924 of the freeform lens 902 and later by surfaces 925 and 926 of the freeform lens 903. After 4 refractions this ray is redirected towards the center of the eye pupil 910 when the eye rests looking forward. This ray schematically represents the outer border of the field of view. Ray 919 schematically illustrates the inner border of the field of view. In general, the lenslets of the 2-fold freeform lenses 902 or 903 are not identical, also because the inner angle 922 and the outer angle 921 of the field of view are not symmetrical with respect to the front direction 920. As in the preceding designs, we must guarantee that no optical cross-talk occurs. Rays 904 and 905 coming from the edges of the clusters 912 and 913 (which do not coincide in a single point because a tolerance guard is assumed herein), respectively, refract on surfaces 914, 915 and 916 and then impinge on the cusp of the surface 917 of lens 903. Afterwards, they are redirected towards edges 907 and 908 of the pupil range, respectively, which is defined by angle 909. This design condition ensures that every ray reaching any point inside the pupil range region will not experience optical cross-talk.

[0059] FIG. 10 shows a schematic view of a 2D cross section of another variant of the configuration shown in FIG. 4. Instead of one curved digital display 401 for both eyes, two separated curved digital displays 1001 and 1002 are shown herein, one per eye. This splitting of the curved digital display can also be applied to embodiments shown in FIG. 7, FIG. 8 and FIG. 9.

[0060] In FIGS. 4 and 7, the cylindrical display was a circular cylinder. However, the cylindrical displays 1001, 1002 in FIG. 10, are cylinders in the more general sense, partially flat and partially concave towards the eye, with a fairly smooth curve in the curved parts. The axis of the cylinder may be taken as any convenient reference line parallel to the generant.

[0061] Thus, we may approximate a cylindrical display architecture to two tilted flat digital displays in order to allow for a more ergonomic and compact device since the lateral sides of the digital display are closer to the user's face, as presented in FIG. 11. Apart from ergonomics, tilting the displays or making them curved can increase the horizontal field of view, increase the image quality, or both at the same time.

[0062] FIG. 11 shows a schematic view of a 2D cross section of another variant of embodiments shown in FIG. 4 and FIG. 10. FIG. 4 shows one freeform lens per eye with one large curved digital display. FIG. 10 shows the same lens with two separated curved digital displays (one per eye). FIG. 11 shows a 2D horizontal cross section of two single lenslet freeform designs with two separate flat digital displays tilted inboard-outboard with respect to the front direction when the eye pupil rests looking forward. Dashed lines 1104 indicate the virtual rays coming from the virtual screen towards the eye pupil 1106 on the eye sphere 1105, while continuous lines 1103 represent the rays coming from the digital display 1101 towards the eye pupil 1106, which are refracted by both surfaces of the freeform lens 1102.

[0063] FIG. 12 is a perspective view of the same optical device shown in FIG. 11, where digital displays 1201 are clearly tilted inwards-outwards with respect to the front direction when the eye pupil rests looking forward. Line 1203 belongs to the plane of symmetry of this configuration. Each digital display works with one of the eyes 1204 and its corresponding lens 1202. Both refractive lenses are freeform. This is indicated by means of the irregular shape of the mesh plotted on the surface S1 of each lens. The lenses may have a plane of symmetry, in particular the cross section with a horizontal plane passing through the eye spheres' centers and containing the 0,0 direction (i.e. the center of the virtual screen).

[0064] The field of view of the embodiments shown herein may be enlarged by partial binocular overlapping. For example, the optical system (both lens and display) for each eye in FIG. 10 and FIG. 11 can be rotated outboard an angle with respect to the eye sphere center, for instance, about 5 to 15 degrees. This makes the whole HMD present a better ergonomics, and increases the horizontal field of view by two times that angle. On the other hand, the binocular portion of the field of view is also reduced by two times that angle.

Detailed Example

[0065] We disclose herein a particular freeform optical design of a 2-fold lens that works with a curved digital display (i.e. a particular example of the configuration schematically shown in FIG. 7). There are two variants of this embodiment. As mentioned, one of them is illustrated in FIG. 7 where one large curved digital display works with a 2-fold lens and where the two lenslets of the lens are not identical. Another alternative configuration includes two 2-fold freeform lenses with two curved digital displays (one per eye). The two lenslets of the freeform lens are identical if the axis of symmetry of the cylindrical digital display (perpendicular to the cylinder axis) crosses the center of the eye rotation. The freeform lenses have a plane of symmetry perpendicular to the cylindrical display, such that the cylinder axis is perpendicular to the plane of symmetry of the freeform surfaces. Both freeform surfaces of each lenslet have been designed and are described with Forbes Q-polynomials (see reference G. Forbes, Characterizing the shape of freeform optics, Optics Express, Vol. 20, Issue 3, pp. 2483-2499, (2012)).

[0066] Any freeform surfaces in three dimensions can be well fitted with the standard equation:

[00001]$z\left(\rho ,\theta \right)=\frac{c.Math..Math.{\rho }^2}{1+\sqrt{1-c^2.Math.{\rho }^2}}+\frac{1}{\sqrt{1-c^2.Math.{\rho }^2}}.Math.\left\{u^2\left(1-u^2\right).Math.\underset{n=0}{\overset{N}{.Math.}}.Math.a_n^0.Math.Q_n^0\left(u^2\right)+\underset{m=1}{\overset{M}{.Math.}}.Math.u^m.Math.\underset{n=0}{\overset{N}{.Math.}}.Math.\left[a_n^m.Math.\cos .Math..Math.m.Math..Math.\theta +b_n^m.Math.\sin .Math..Math.m.Math..Math.\theta \right].Math.Q_n^m\left(u^2\right)\right\}$

where ρ, θ and z are cylindrical coordinates; u is defined by u=ρ/ρ.sub.max, so 0≦u≦1; c=1/R, R being a curvature radius; and Q.sup.m.sub.n (v) is the Forbes Q-polynomials of order n in v. All the freeform lenses in this embodiment have, at least, one plane of symmetry perpendicular to the cylinder axis of the digital display. Without loss of generality we can consider that this symmetry plane corresponds to θ=0 deg, implying that coefficients b.sub.n.sup.m=0 for all n and m.

[0067] Local coordinate systems of each surface used for the polynomial description are shown inthe yz-plane (i.e. x=0, where the z-axis points left and the y-axis points up) in FIG. 13. This figure illustrates one lenslet of the 2-fold refractive optics. Profiles of two surfaces (1303 and 1305) of the lenslet together with the half of curved digital display 1301 and eye sphere 1308 are shown in FIG. 13. We place the center of the global coordinate system 1307 in the center of the eye sphere 1308 (x,y,z)=(0,0,0). Surface S1 labeled as 1303 is described in local coordinate system x.sub.1y.sub.1z.sub.1 1304 that is centered at coordinates (x,y,z)=(0, 6.31953, 34.1892) of the global coordinate system and tilted 10.4312922648424 degrees clockwise around the x.sub.1-axis. Local coordinate system x.sub.2y.sub.2z.sub.2 1306 for the surface S2 (labeled as 1305) is centered at (x,y,z)=(0, 4.8118, 24.5392) and tilted 13.5352856790129 degrees clockwise around the x.sub.2-axis. As an example of a particular lenslet design with f=22 mm, FoV=100 degs, the non-null coefficients of the lens (all in mm, except c in min.sup.−1) shown in FIG. 14 and FIG. 15 are given in the following TABLE 1:

TABLE-US-00002 TABLE 1 Parameter surface S1 surface S2 c −0.036018769 0.039111809 ρ.sub.max 13.41314178 14.80122063 a.sub.0.sup.0 −0.226574179 1.081278496 a.sub.1.sup.0 0.282219769 0.587066393 a.sub.2.sup.0 −0.183934766 −0.490052211 a.sub.3.sup.0 0.062314163 0.2240385 a.sub.4.sup.0 −0.028306575 −0.05983387 a.sub.0.sup.1 0.29555848 0.665007299 a.sub.0.sup.2 −1.583095327 −4.806094782 a.sub.0.sup.3 −1.729840066 −4.655195451 a.sub.0.sup.4 −0.85130362 −1.065609344 a.sub.1.sup.1 −0.027108396 −1.838563691 a.sub.1.sup.2 −1.482754987 0.201614073 a.sub.1.sup.3 1.216452316 4.499133825 a.sub.1.sup.4 0.69377578 3.171464862 a.sub.2.sup.1 0.365700671 1.816523911 a.sub.2.sup.2 −0.033515118 −0.828786855 a.sub.2.sup.3 −0.09966736 −1.617233297 a.sub.2.sup.4 −0.474404363 −1.442214487 a.sub.2.sup.5 0.204159827 0.46673156 a.sub.3.sup.1 −0.122948506 −0.722297397 a.sub.3.sup.2 0.18898364 0.590836979 a.sub.3.sup.3 0.031629265 0.506895348 a.sub.3.sup.4 0.17950827 0.4734159 a.sub.3.sup.5 −0.078113732 −0.183213154 a.sub.4.sup.1 0.064421065 0.241900484 a.sub.4.sup.2 −0.069211879 −0.170613002 a.sub.4.sup.3 −0.041082485 −0.115088967 a.sub.4.sup.4 −0.030863378 −0.060368389 a.sub.4.sup.5 0.022121753 0.040395225

[0068] The profile of the curved digital display can be prescribed. In this example it has been fixed to the curve given by

z(y)=2ay.sup.6

[0069] Where a=−1.46434003091751e-008. The local coordinate system of digital display x.sub.3y.sub.3z.sub.3 is placed at (x,y,z)=(0, 0, 51.9457503) of the global coordinate system (see FIG. 13).

[0070] FIG. 14 shows the same yz-section of the system, x=0 (where the z-axis points left and the y-axis points up). Half of the complete system is shown in FIG. 14. The other half of the system can be made by a mirror image of the shown lenslet 1410 reflected in the xz-plane that contains axis 1404 (y=0). Another alternative would be that the other half of the system consists of the flat continuation of the display and lenslet that works with a flat display (for more detail see reference PCT1). Obviously, both lenslets should have similar optical characteristics (f, FoV).

[0071] We may examine design rays trajectories in FIG. 14. Reversed ray 1403 comes from the border of the pupil range 1411, impinges at the cusp of surface S2 labeled as 1409, refracts on the surface S1 labeled as 1408 and it is sent to the cluster edge 1402. Reverse ray 1405 exits the eye pupil center parallel to the axis y=0 when eye pupil rests looking forward. Reversed ray 1406 comes from the center 1413 of the eye sphere 1412 and it reaches the display. Reversed ray 1407 marks half field of view as it is the border peripheral ray that impinges on the border of the display from the eye pupil when it rests looking forward.

[0072] FIG. 15 illustrates the xz-section (plane y=0) of one lenslet belonging to the 2-fold lens configuration also shown in previously described FIG. 13 and FIG. 14. This section is perpendicular to the section shown in FIG. 14. We may appreciate the display labeled as 1501 and user's eye 1507. The lens is labeled as 1502, and it corresponds to lens 1410 in FIG. 14. Reversed ray 1504 marks half field of view as it is the border peripheral ray that impinges on the border of the display from the eye pupil center when the eye rests looking forward. Reversed ray 1505 leaves the eye pupil center when eye pupil gazes near the border of the pupil range 1508.

[0073] TABLE 2 and TABLE 3 show the root-mean-square (RMS) diameters of the polychromatic spots for some selected fields of the design in FIG. 14 and FIG. 15 for a pupil diameter of 4 mm. This design has a focal length of about 22 mm for the front direction and gradually decreases towards the edge of the field of view. Its horizontal field of view is 100 degrees and vertical field of view is 68 degrees for a 2.1″ (55 mm) 16:9 display, where the horizontal direction is the given by the y-z plane. Angles χ and γ are illustrated in FIG. 16. χ is defined as the angle 1601 between the projection 1602 on xz-plane of the peripheral angle's θ 1603 direction 1604 and the z-axis, while γ is the angle 1605 between the peripheral angle's θ projection 1606 on yz-plane of the same peripheral angle's θ direction 1604 and the z-axis.

[0074] FIG. 6 shows the angular resolution of a typical human eye as a function of the peripheral angle (according to J. J. Kerr, “Visual resolution in the periphery”, Perception & Psychophysics, Vol. 9 (3), 1971). Since the human eye resolution is much smaller in peripheral vision than close to the gazing direction, we can adjust the design conditions of any of the embodiments in this specification to make the image quality of the ipixels when they are directly gazed (the ipixels on the gazed region of the virtual screen) higher than the image quality of the ipixels when they are not gazed (particularly the ipixels in the outer region of the virtual screen).

[0075] TABLE 2 corresponds to the situation when the eye is gazing at each field, so the peripheral angle for the human eye perception is 0 for all the fields, and thus the optical resolution should be maximum. This table shows that opixels as small as 30-40 microns can be resolved well, although the RMS increases to some extent for the highest values of the angle x(deg). This is mainly caused by chromatic aberration.

[0076] TABLE 3 corresponds to the situation when the eye is gazing frontwards, so the peripheral angle for the human eye perception is not zero, but equal to θ given by equation

[00002]$\theta =\arccos .Math.\sqrt{\frac{1}{{\tan }^2.Math.\chi +\frac{1}{{\cos }^2.Math.\gamma }}}.$

Therefore, the optical resolution can be lower without affecting the human perception of optical quality. This design is adapted to the human eye resolution of FIG. 6 as discussed in PCT1. For this reason, the RMS values are in general much higher in TABLE 3 than in TABLE 2 for the same fields.

TABLE-US-00003 TABLE 2 χ (deg) γ (deg) RMS (μm) χ (deg) γ (deg) RMS (μm) 0 0 39.3 0 18 37.3 0 2 43.7 0 20 42.2 0 4 47.8 0 22 48.6 0 6 42.9 0 24 56.5 0 8 34.0 0 26 66.4 0 10 27.0 0 28 90.4 0 12 25.3 0 30 128.7 0 14 28.1 2 0 41.1 0 16 32.7 2 2 45.3 2 4 48.6 4 20 48.5 2 6 43.7 4 22 55.2 2 8 35.3 4 24 62.1 2 10 29.2 4 26 73.6 2 12 28.4 4 28 113.8 2 14 31.3 6 0 51.2 2 16 35.2 6 2 52.0 2 18 39.1 6 4 53.0 2 20 43.8 6 6 48.3 2 22 50.3 6 8 42.7 2 24 57.9 6 10 40.1 2 26 68.0 6 12 41.6 2 28 95.6 6 14 44.5 4 0 45.9 6 16 48.1 4 2 48.8 6 18 50.7 4 4 50.8 8 0 55.8 4 6 45.9 8 2 54.3 4 8 38.7 8 4 53.8 4 10 34.4 8 6 49.9 4 12 34.9 8 8 45.9 4 14 37.9 8 10 45.1 4 16 41.2 8 12 47.0 4 18 44.8 8 14 50.0 8 16 53.2 12 6 57.5 8 18 57.7 12 8 57.6 8 20 64.1 12 10 59.0 8 22 70.1 12 12 60.5 8 24 75.4 12 14 62.4 8 26 107.4 12 16 64.6 10 0 61.4 12 18 70.0 10 2 56.0 12 20 75.8 10 4 54.7 12 22 79.0 10 6 51.8 12 24 105.3 10 8 50.1 14 0 75.2 10 10 50.9 14 2 67.8 10 12 52.9 14 4 64.9 10 14 55.4 14 6 65.5 10 16 58.4 14 8 67.6 10 18 63.8 14 10 70.0 10 20 70.8 14 12 71.5 10 22 75.7 14 14 71.8 10 24 84.5 14 16 73.8 10 26 153.0 14 18 77.8 12 0 67.6 14 20 80.0 12 2 60.9 14 22 86.4 12 4 58.5 16 0 87.6 16 2 77.7 18 12 102.3 16 4 75.4 18 14 104.7 16 6 75.7 18 16 105.9 16 8 78.7 18 18 103.4 16 10 82.8 18 20 98.4 16 12 85.2 18 22 168.7 16 14 86.0 20 0 203.6 16 16 87.1 20 2 138.2 16 18 87.8 20 4 121.4 16 20 86.3 20 6 116.2 16 22 106.6 20 8 115.8 18 0 118.6 20 10 122.8 18 2 98.2 20 12 132.2 18 4 94.1 20 14 149.8 18 6 92.0 20 16 169.5 18 8 93.7 20 18 187.1 18 10 98.3

TABLE-US-00004 TABLE 3 χ (deg) γ (deg) RMS (μm) χ (deg) γ (deg) RMS (μm) 0 0 39.5 0 40 128.0 0 2 50.8 0 42 170.6 0 4 65.5 0 44 241.5 0 6 69.2 0 46 325.6 0 8 63.2 0 48 412.6 0 10 51.1 0 50 532.5 0 12 37.0 2 0 41.0 0 14 24.7 2 2 51.8 0 16 17.1 2 4 66.1 0 18 14.4 2 6 69.7 0 20 14.1 2 8 63.8 0 22 15.0 2 10 52.1 0 24 18.7 2 12 38.7 0 26 27.0 2 14 27.6 0 28 40.2 2 16 21.1 0 30 56.9 2 18 18.7 0 32 74.2 2 20 17.5 0 34 89.2 2 22 17.1 0 36 99.7 2 24 19.6 0 38 109.3 2 26 27.3 2 28 40.4 4 22 22.6 2 30 57.2 4 24 22.4 2 32 74.4 4 26 28.7 2 34 89.3 4 28 41.5 2 36 99.5 4 30 58.3 2 38 108.8 4 32 75.3 2 40 127.4 4 34 89.6 2 42 170.0 4 36 98.9 2 44 240.9 4 38 107.4 2 46 324.7 4 40 125.6 2 48 411.7 4 42 168.3 2 50 533.8 4 44 238.9 4 0 45.0 4 46 321.8 4 2 54.6 4 48 409.0 4 4 67.7 4 50 538.2 4 6 71.1 6 0 50.8 4 8 65.6 6 2 58.9 4 10 54.9 6 4 70.1 4 12 43.3 6 6 73.0 4 14 34.7 6 8 68.1 4 16 30.2 6 10 58.9 4 18 27.8 6 12 49.7 4 20 25.3 6 14 43.8 6 16 40.7 6 36 98.0 6 18 38.3 6 38 104.9 6 20 34.8 6 40 122.5 6 22 30.4 6 42 165.2 6 24 27.9 6 44 235.3 6 26 32.3 6 46 317.0 6 28 43.7 6 48 404.9 6 30 59.9 6 50 548.0 6 32 76.3 8 0 57.3 6 34 90.3 8 2 63.2 6 36 98.0 8 4 72.7 6 38 104.9 8 6 75.1 6 40 122.5 8 8 70.9 6 16 40.7 8 10 63.5 6 18 38.3 8 12 57.0 6 20 34.8 8 14 53.5 6 22 30.4 8 16 51.8 6 24 27.9 8 18 49.6 6 26 32.3 8 20 45.6 6 28 43.7 8 22 40.4 6 30 59.9 8 24 36.8 6 32 76.3 8 26 39.7 6 34 90.3 8 28 49.6 8 30 64.7 10 24 49.5 8 32 79.6 10 26 51.7 8 34 91.3 10 28 60.7 8 36 97.6 10 30 72.5 8 38 101.5 10 32 85.0 8 40 117.8 10 34 93.3 8 42 160.4 10 36 96.6 8 44 229.0 10 38 97.0 8 46 310.0 10 40 111.3 8 48 400.3 10 42 154.6 8 50 567.3 10 44 221.2 10 0 63.0 10 46 301.0 10 2 67.2 10 48 397.2 10 4 75.1 10 50 603.7 10 6 77.1 12 0 68.2 10 8 73.9 12 2 70.8 10 10 68.7 12 4 77.0 10 12 65.2 12 6 78.9 10 14 64.0 12 8 77.1 10 16 63.5 12 10 74.7 10 18 62.0 12 12 74.4 10 20 58.3 12 14 75.4 10 22 53.1 12 16 76.8 12 18 76.4 14 12 85.0 12 20 73.5 14 14 89.2 12 22 69.2 14 16 92.4 12 24 66.5 14 18 93.6 12 26 68.0 14 20 92.0 12 28 75.1 14 22 88.9 12 30 85.3 14 24 87.4 12 32 94.4 14 26 88.6 12 34 97.5 14 28 93.8 12 36 95.8 14 30 100.9 12 38 91.8 14 32 106.1 12 40 103.0 14 34 105.7 12 42 145.5 14 36 97.2 12 44 211.1 14 38 87.9 12 46 290.9 14 40 94.4 12 48 399.6 14 42 134.1 12 50 669.6 14 44 199.4 14 0 72.0 14 46 281.6 14 2 73.4 14 48 415.4 14 4 78.4 14 50 784.5 14 6 80.6 16 0 75.8 14 8 81.2 16 2 74.5 14 10 82.1 16 4 78.4 16 6 81.9 18 0 80.5 16 8 85.6 18 2 75.6 16 10 90.6 18 4 77.2 16 12 97.3 18 6 82.1 16 14 104.7 18 8 89.5 16 16 110.7 18 10 99.4 16 18 114.0 18 12 111.0 16 20 114.3 18 14 122.4 16 22 113.7 18 16 131.7 16 24 112.7 18 18 137.8 16 26 113.8 18 20 141.7 16 28 117.3 18 22 142.5 16 30 121.2 18 24 142.9 16 32 122.2 18 26 144.1 16 34 117.0 18 28 146.1 16 36 104.1 18 30 147.2 16 38 86.5 18 32 144.5 16 40 86.5 18 34 135.1 16 42 123.6 18 36 117.7 16 44 188.5 18 38 95.1 16 46 277.9 18 40 88.5 16 48 460.7 18 42 120.0 16 50 975.1 18 44 184.0 18 46 289.6 20 42 134.1 18 48 568.8 20 44 194.9 20 0 89.8 20 46 339.3 20 2 77.6 20 48 788.2 20 4 75.8 22 0 105.0 20 6 81.0 22 2 83.5 20 8 92.7 22 4 75.6 20 10 108.3 22 6 80.8 20 12 125.4 22 8 96.5 20 14 141.7 22 10 118.1 20 16 155.3 22 12 141.3 20 18 166.6 22 14 163.5 20 20 171.8 22 16 184.1 20 22 175.5 22 18 197.9 20 24 177.7 22 20 208.9 20 26 179.5 22 22 216.5 20 28 180.5 22 24 222.0 20 30 180.9 22 26 225.0 20 32 175.4 22 28 226.0 20 34 163.2 22 30 223.8 20 36 142.6 22 32 216.8 20 38 120.6 22 34 203.8 20 40 110.9 22 36 185.2 22 38 166.5 24 36 248.5 22 40 158.1 24 38 234.9 22 42 173.1 24 40 228.6 22 44 231.7 24 44 262.7 22 46 468.5 24 46 539.9 24 0 122.9 26 0 131.4 24 2 92.9 26 2 107.2 24 4 81.6 26 4 100.3 24 6 86.6 26 6 111.4 24 8 106.5 26 8 134.6 24 10 134.9 26 10 165.9 24 12 164.4 26 12 198.5 24 14 193.1 26 14 233.0 24 16 217.6 26 16 264.9 24 18 238.9 26 18 292.6 24 20 255.6 26 20 315.4 24 22 268.0 26 22 333.2 24 24 276.7 26 24 346.1 24 26 282.0 26 26 353.8 24 28 283.9 26 28 357.9 24 30 281.9 26 30 357.4 24 32 274.7 26 32 349.6 24 34 263.2 26 34 337.8 26 36 324.5 28 36 441.3 26 38 313.4 28 38 439.4 26 40 309.3 28 40 440.8 26 42 310.8 28 42 494.7 26 44 421.3 28 44 763.2 28 0 167.2 30 0 277.1 28 2 140.9 30 2 215.1 28 4 138.9 30 4 196.5 28 6 154.4 30 6 209.8 28 8 181.6 30 8 240.8 28 10 216.8 30 10 279.7 28 12 255.0 30 12 322.9 28 14 293.7 30 14 367.4 28 16 330.7 30 16 409.9 28 18 364.1 30 18 448.8 28 20 391.7 30 20 482.0 28 22 414.8 30 22 510.5 28 24 432.3 30 24 533.3 28 26 444.4 30 26 550.9 28 28 451.7 30 28 564.4 28 30 454.5 30 30 575.1 28 32 453.3 30 32 586.8 28 34 446.1 30 34 602.9 30 36 631.7 32 10 336.9 30 38 690.6 32 12 376.2 30 40 669.7 32 14 422.5 30 42 661.6 32 16 471.7 30 44 888.8 32 18 519.1 30 46 1570.5 32 20 562.4 32 0 615.1 32 22 600.2 32 2 476.2 32 24 635.1 32 4 384.6 32 26 668.7 32 6 334.9 32 28 706.3 32 8 320.5 32 30 756.5 32 10 336.9 32 32 837.8 32 12 376.2 32 34 980.0 32 14 422.5 32 36 979.4 32 16 471.7 32 38 923.8 32 18 519.1 32 40 807.1 32 20 562.4 32 42 779.5 32 22 600.2 32 44 864.6 32 0 615.1 34 0 1371.4 32 2 476.2 34 2 1223.6 32 4 384.6 34 4 1076.0 32 6 334.9 34 6 634.9 32 8 320.5 34 8 532.2 34 10 494.6 34 26 933.4 34 12 483.7 34 28 1145.3 34 14 488.9 34 30 1211.4 34 16 516.6 34 32 1209.0 34 18 561.1 34 34 1181.2 34 20 620.9 34 36 1095.2 34 22 695.3 34 38 989.5 34 24 792.3 34 40 888.8

[0077] Although specific embodiments have been described, the preceding description of presently contemplated modes of practicing the invention is not to be taken in a limiting sense, but is made merely for the purpose of describing certain general principles of the invention. Variations are possible from the specific embodiments described. For example, the patents and applications cross-referenced above describe systems and methods that may advantageously be combined with the teachings of the present application. Although specific embodiments have been described, the skilled person will understand how features of different embodiments may be combined.

[0078] The full scope of the invention should be determined with reference to the claims, and features of any two or more of the claims may be combined.