Optical see-through free-form head-mounted display

Abstract

A see-through free-form head-mounted display including a wedge-shaped prism-lens having free-form surfaces and low F-number is provided.

Claims

1. A free-form prism-lens for use in an optical see-through head-mounted display, comprising: a first free-form surface configured to receive light from a micro-display and configured to transmit the received light into the body of the prism-lens; a second free-form surface configured to receive the light transmitted into the body of the prism-lens from the first free-form surface and configured to totally internally reflect the received light at the second surface; and a third free-form surface configured to receive the light reflected by the second free-form surface and configured to reflect the light out of the prism-lens, wherein the prism-lens has an f-number less than 2.

2. The free-form prism-lens according to claim 1, wherein the first free-form surface is described by $z=\frac{c\left(x^2+y^2\right)}{1+\mathrm{sqrt}\left(1-\left(1+k\right)c^2\left(x^2+y^2\right)\right)}+\underset{j=2}{\overset{66}{.Math.}}{C}_j{x}^m{y}^n,j=\left[{\left(m+n\right)}^2+m+3n\right]\text{/}2+1,$ where the z is the sag of the first free-form surface measured along the z-axis of a local x, y, z coordinate system, c is the vertex curvature (CUY), k is the conic constant, and C.sub.j is the coefficient for x.sup.my.sup.n.

3. The free-form prism-lens according to claim 2, wherein the second free-form surface is described by $z=\frac{c\left(x^2+y^2\right)}{1+\mathrm{sqrt}\left(1-\left(1+k\right)c^2\left(x^2+y^2\right)\right)}+\underset{j=2}{\overset{66}{.Math.}}{C}_j{x}^m{y}^n,j=\left[{\left(m+n\right)}^2+m+3n\right]\text{/}2+1,$ where the z is the sag of the second free-form surface measured along the z-axis of a local x, y, z coordinate system, c is the vertex curvature (CUY), k is the conic constant, and C.sub.j is the coefficient for x.sup.my.sup.n.

4. The free-form prism-lens according to claim 3, wherein the third free-form surface is described by $z=\frac{c\left(x^2+y^2\right)}{1+\mathrm{sqrt}\left(1-\left(1+k\right)c^2\left(x^2+y^2\right)\right)}+\underset{j=2}{\overset{66}{.Math.}}{C}_j{x}^m{y}^n,j=\left[{\left(m+n\right)}^2+m+3n\right]\text{/}2+1,$ where the z is the sag of the third free-form surface measured along the z-axis of a local x, y, z coordinate system, c is the vertex curvature (CUY), k is the conic constant, and C.sub.j is the coefficient for x.sup.my.sup.n.

5. The free-form prism-lens according to claim 1, wherein the second free-form surface is described by $z=\frac{c_x{x}^2+c_y{y}^2}{1+\mathrm{sqrt}\left(1-\left(1+k_x\right)c_x{x}^2-\left(1+k_y\right)c_y{y}^2\right)}+\underset{j=1}{\overset{37}{.Math.}}{C}_j{x}^{2m}{y}^n$ $2m+n10,m=0,1,2,3,4,5,n=0,1,\mathrm{.Math.},10,$ where the z is the sag of the at least one surface measured along the z-axis of a local x, y, z coordinate system, c.sub.x is the radius of curvature of the at least one surface in the sagittal direction, c.sub.y is the radius of curvature of at least one surface in the tangential direction, and C.sub.j is the coefficient for x.sup.2my.sup.n.

6. The free-form prism-lens according to claim 1, wherein the third free-form surface is partially mirrored to permit the internally reflected light to be reflected by the second free-form surface and to permit light from a real-world view to be transmitted through the third free-form surface to the exit pupil.

7. The free-form prism-lens according to-claim 1, wherein second and third free-form surfaces are configured to provide a wedge-shaped prism lens.

8. The free-form prism-lens according to-claim 1, wherein the z-axis is parallel to the optical axis at the exit pupil, and the prism lens is symmetric about the y-z plane and asymmetric about the x-z plane.

9. The free-form prism-lens according to claim 1, wherein the diagonal field of view is at least 40 degrees.

10. The free-form prism-lens according to claim 1, wherein the exit pupil diameter is at least 6 mm.

11. The free-form prism-lens according to claim 1, wherein the modulation transfer function is at least 10%30 lps/mm.

12. The free-form prism-lens according to claim 1, wherein the eye clearance is at least 16 mm.

13. The free-form prism-lens according to claim 1, comprising an auxiliary lens disposed proximate the third free-form surface, the auxiliary lens configured to minimize the shift and distortion of rays from a real-world scene by the second and third surfaces of the prism-lens.

14. The free-form prism-lens according to claim 13, wherein the auxiliary lens has a surface with the same shape as the third free-form surface of the prism-lens and is disposed in optical contact with the third free-form surface of the prism-lens.

15. The free-form prism-lens according to claim 14, wherein at least one surface of the auxiliary lens is described by $z=\frac{c\left(x^2+y^2\right)}{1+\mathrm{sqrt}\left(1-\left(1+k\right)c^2\left(x^2+y^2\right)\right)}+\underset{j=2}{\overset{66}{.Math.}}{C}_j{x}^m{y}^n,j=\left[{\left(m+n\right)}^2+m+3n\right]\text{/}2+1,$ where the z is the sag of the at least one free-form surface measured along the z-axis of a local x, y, z coordinate system, c is the vertex curvature (CUY), k is the conic constant, and C.sub.j is the coefficient for x.sup.my.sup.n.

16. The free-form prism-lens according to claim 15, wherein the modulation transfer function is at least 10%30 lps/mm across the entire visual field.

17. The free-form prism-lens according to claim 13, wherein at least one surface of the auxiliary lens is described by $z=\frac{c_x{x}^2+c_y{y}^2}{1+\mathrm{sqrt}\left(1-\left(1+k_x\right)c_x{x}^2-\left(1+k_y\right)c_y{y}^2\right)}+\underset{j=1}{\overset{37}{.Math.}}{C}_j{x}^{2m}{y}^n$ $2m+n10,m=0,1,2,3,4,5,n=0,1,\mathrm{.Math.},10,$ where the z is the sag of the at least one surface measured along the z-axis of a local x, y, z coordinate system, c.sub.x is the radius of curvature of the at least one surface in the sagittal direction, c.sub.y is the radius of curvature of at least one surface in the tangential direction, and C.sub.j is the coefficient for x.sup.2my.sup.n.

18. The free-form prism-lens according to claim 17, wherein the modulation transfer function is at least 10%30 lps/mm across the entire visual field.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The foregoing summary and the following detailed description of the preferred embodiments of the present invention will be best understood when read in conjunction with the appended drawings, in which:

(2) FIG. 1 schematically illustrates a layout of an exemplary optical see-through head-mounted display system in accordance with the present invention;

(3) FIG. 2 schematically illustrates the layout of FIG. 1 showing the local coordinate system at each optical surface;

(4) FIG. 3 schematically illustrates the optical paths of the rays of different object fields and different pupil positions in an exemplary free-form-surface prism-lens see-through head-mounted display, with the incident angles of the rays on surfaces 1 and 1 depending on their field and pupil positions and controlled to satisfy TIR conditions and avoid stray light;

(5) FIGS. 4A-4D schematically illustrate a starting point for an exemplary design of the present invention, with FIG. 4A showing the optical layout in the YZ plane, FIG. 4B showing MTF plots, FIG. 4C showing ray fan plots of center fields, and FIG. 4D showing ray fan plots of marginal fields;

(6) FIG. 5 schematically illustrates the system layout and sampled fields definition during different design stages of an exemplary optical see-through head-mounted display;

(7) FIG. 6 schematically illustrates the layout of an exemplary free-form surface prism-lens system of the present invention having three free-form surfaces;

(8) FIG. 7 illustrates a distortion plot of the free-form surface prism-lens system of FIG. 6;

(9) FIGS. 8A-8D illustrate the performance of the free-form surface prism-lens system of FIG. 6, with FIG. 8A showing the polychromatic MTF plot of the center field of the virtual imaging system, FIG. 8B showing the polychromatic MTF plot of marginal fields of the virtual imaging system, FIG. 8C showing the ray fan plots of the center fields, and FIG. 8D showing the ray fan plots of the marginal fields;

(10) FIGS. 9A-9D illustrate the incident angle on the TIR surface, with FIG. 9A showing the incident angle on surface 1 as the ray pupil position varies from the bottom to the top, FIG. 9B showing the incident angle on surface 1 as the field of the ray changes from the lowermost to the uppermost in the meridian plane, FIG. 9C showing the incident angle on surface 1 as the ray pupil position varies from the bottom to the top, and FIG. 9D showing the incident angle on surface 1 as the field of the ray changes from the lowermost to the uppermost position in the tangential plane;

(11) FIGS. 10A-10C schematically illustrates the design of an auxiliary lens to be used with the free-form surface prism-lens system of FIG. 6, with FIG. 10A showing see-through by the FFS prism-lens, FIG. 10B showing distortion caused by the FFS prism-lens, and FIG. 10C showing the design layout of the see-through system;

(12) FIG. 11 illustrates a distortion plot of the optical see-through system of FIG. 10;

(13) FIGS. 12A-12B illustrate polychromatic MTF plots of the optical see-through system of FIG. 10 with an ideal lens;

(14) FIG. 13 schematically illustrates the layout of the see-through HMD by coupling the FFS prism-lens system and auxiliary FFS lens of FIG. 10; and

(15) FIG. 14A illustrates a photo without pre-warping the input image, and FIG. 14B a photo after pre-warping the input image taken through a fabricated prototype of the FFS prism-lens of FIG. 6.

DETAILED DESCRIPTION

(16) The desire to achieve an optical see-through head-mounted display having a compact, light-weight, and nonintrusive form factor argues for a design having as few optical elements as possible. Accordingly, exemplary designs of the present invention provide a single-element prism-lens 110, 710 which has sufficient optical power on its own to deliver light from a micro-display 130 to a user, FIGS. 1, 6. However, providing a single optical element, such as the prism-lens 710, in which all the optical power resides can lead to greatly increased aberrations with accompanying loss in resolution and image quality, especially for low F/# systems. Despite these challenges, as a result of the lens design procedures and work described below, the present invention provides a single-element prism-lens 710 based on a 0.61 microdisplay 130, which offers a diagonal FOV of 53.5, an F/# of 1.875, an exit pupil diameter of 8 mm, and an eye relief of 18.25 mm. In addition, in order to maintain a non-distorted see-through view of a real-world scene, a cemented auxiliary lens 120, 720 may be provided for use in conjunction with the prism-lens 110, 710.

(17) Display System Specifications

(18) Turning first to the design of the wedge-shaped free-form prism-lens 110, design began with development of the display system specifications. An optical see-through HMD 100 typically consists of an optical path for viewing a displayed virtual image and a path for viewing a real-world scene directly. As shown in FIG. 1, the optical system 100 of our OST-HMD design may include a wedge-shaped free-form prism-lens 110 cemented to an auxiliary free-form lens 120. The prism-lens 110 serves as the near-eye viewing optics that magnifies the image displayed through a microdisplay 130 while the auxiliary free-form lens 120 is an auxiliary element attached to the prism-lens 110 in order to maintain a non-distorted see-through view of a real-world scene.

(19) As shown in FIG. 1, the wedge shaped free-form prism-lens 110 may include three surfaces labeled as 1, 2, and 3, respectively. For the sake of convenience, the surface adjacent to the exit pupil is labeled as 1 in the refraction path and as 1 in the reflection path. We set the center of the exit pupil as the origin of the global coordinate system and the rest of the surfaces were specified with respect to this global reference. We further adopted the convention of tracing the system backward, namely from the eye position to the microdisplay 130.

(20) The overall system was set to be symmetric about the YOZ plane, but not the XOZ plane. A ray emitted from a point on the microdisplay 130 is first refracted by the surface 3 next to the microdisplay 130. After two consecutive reflections by the surfaces 1 and 2, the ray is transmitted through the surface 1 and reaches the exit pupil of the system 100. The first surface (i.e., 1 and 1) of the prism-lens 110 is required to satisfy the condition of total internal reflection for rays reflected by this surface F. The rear surface 2 of the prism-lens 110 is coated as a half mirror in order to facilitate the optical see-through capability. The rays from the microdisplay 130 will be reflected by the rear surface 2 while the rays from a real-world scene will be transmitted. An auxiliary lens 120 may be cemented to the wedge-shaped prism-lens 110 in order to counteract the ray shift and distortion caused by the prism-lens 110. The front surface of the auxiliary free-form lens 120 may match the shape of the rear surface 2 of the prism-lens 110. The back surface 4 of the auxiliary free-form lens 120 may be optimized to minimize the shift and distortion introduced to the rays from a real-world scene when the auxiliary free-form lens 120 is combined with the prism-lens 110.

(21) TABLE-US-00001 TABLE 1 Specifications of FFS Prism-lens HMD System Prameter Specification LCD Size 0.61 in (15.5 mm) diagonally Active display area 12.7 mm 9.0 mm Resolution 800 600 pixels Virtual imaging system Type folded FFS prism-lens Effective focal length 15 mm Exit pupil diameter 8 mm Eye relief >17 (18.25) mm F/# 1.875 Number of free-form 3 surfaces Augmented viewing system Type Free-form lens Number of free-form 2 surfaces Other parameters Wavelength 656.3-486.1 nm Field of view 45 H 32 V Vignetting 0.15 for top and bottom fields Distortion <12% at the maximum field Image quality MTF > 10% at 30 lps/mm

(22) The overall specifications of the system are summarized in Table 1. Our goal was to achieve a very compact, lightweight, and wide FOV design using a wedge-shaped free-form prism-lens 110. A small size microdisplay 130 with high resolution was thus preferred. Based on the size, resolution, availability and cost, a pair of 0.61-inch Emagin OLED displays were selected, with a resolution of 800600 pixels and a 15 m pixel size. We further targeted an HMD system 100 with a diagonal full FOV of at least 50, which corresponds to a focal length no more than 16.6 mm. A 15 mm focal length was selected, which offers a reasonable balance between FOV (53.5 diagonally) and angular resolution (3.2 arc minutes per pixel). In the design of visual instruments, especially binocular HMDs, a large exit pupil is typically preferred to account for the swiveling of the eyes in their sockets without causing vignetting or loss of image. A large pupil offers better tolerance of the interpupilary distances (IPD) among different users without the need to mechanically adjust the IPD of the binocular optics. A large pupil, however, often not only compromises the compactness and weight of the optical system 100, but also imposes limitations on the FOV due to the dramatically increased challenge of designing low F/# systems. Taking into account these factors, we set the exit pupil diameter to be 8 mm, which leads to a system 100 with a F/# of 1.875. In designing HMD systems, a large eye relief is desired to accommodate users wearing eyeglasses, but it affects the compactness of the viewing optics. A minimum of a 18 mm eye relief was set to accommodate users wearing low-profile eyeglasses. Balancing between image uniformity and system compactness, we set the limit of the vignetting to be less than 15% at the top and bottom of the visual fields.

(23) Among the aberrations of an optical system, distortion causes the warping of the displayed image without reducing image sharpness, which allows computational or electronic correction. In designing conventional HMDs it is common to optimize the system 100 to minimize the optical aberrations that reduce image quality and cannot be compensated electronically or computationally. In a free-form optical system 100, however, the distortion can be very large and irregular if it is left without any constraints. We thus set a distortion limit of 12% at the maximum field angle and planned to correct the residual distortion using computational methods. In terms of other types of aberrations, the modulation transfer function (MTF) was selected to evaluate the overall image sharpness and was set to be no less than 10% across the entire visual field at a spatial frequency of 30 lps/mm. With the specifications established, development continued with design of the free-form elements 110, 120.

(24) Design of Free-Form Elements

(25) Free-form optical surfaces offer more degrees of freedom to optical designers than conventional rotationally symmetric optical surfaces, such as a spherical or aspherical surface, and achieve usually lower wavefront errors and distortion than that achievable with the same number of rotationally symmetric surfaces. A significant benefit in our OST-HMD design lies in its ability to yield display optics with an eyeglass-like form factor. An optical design using free-form surfaces, however, may cause a dramatic increase in the complexity of the design and optimization process. An inadequate method of representing and optimizing a free-form surface may lead to discouraging and unpredictable results. Key issues in the process of designing a FFS HMD include 1) a free-form surface representation and design strategy; 2) total internal reflection condition; and 3) structure constraints to form a valid prism-lens 110.

(26) Free-Form Surface Representation and Design Strategy

(27) Selecting a suitable method for a free-form surface representation is very important. Different representation methods not only have different impacts on the ray tracing speed and the convergence of optimization, but also offer different degrees of design freedom. A suitable representation method shall 1) provide adequate degrees of freedom; 2) require a reasonable amount of ray tracing time; and 3) offer reliable convergence in the optimization process. Ray tracing speed is a particular concern in designing a free-form prism-lens 110, as a larger number of fields need to be sampled when optimizing a free-form optical system than need to be sampled in a rotationally symmetrical optical system. Speed becomes a more serious problem when a global optimization is necessary. Although most of the commercially available optical design software, such as CODE V (Optical Research Associates, Pasadena, Calif.), offers the ability to model free-form surfaces in user-defined methods, the ray tracing speed of user-defined representations typically is much slower than the standard methods available in the software packages.

(28) By taking into account the speed and convergence factors, the following design strategy was adopted in our design process. In the case when we lacked a starting point for an FFS surface, we started to optimize the surface with a spherical type to obtain the correct first-order parameters. The spherical surface was then converted to an aspheric type by adding a conic constant and a 4.sup.th order or higher aspheric coefficients. Following an intermediate state of optimization, the ASP-type surface was then converted to an AAS-type surface for better correction by directly adding asymmetric coefficients up to the 10.sup.th order. To avoid loss of information, use of aspheric terms higher than the 10.sup.th order was not pursued, because the AAS surface has only up to the 10.sup.th order of rotationally symmetric coefficients in CODE V. Optimization with the AAS type surface helped to create a good starting point. The AAS surface was then converted to the XYP-type through a fitting algorithm (e.g., a least square fitting method) for final stage of optimization. High precision was required for the fitting algorithm to avoid a significant deviation from the starting design produced by the AAS surface type.

(29) Total Internal Reflection Constraint

(30) As mentioned above, all the rays striking the first surface 1 of the prism-lens 110 from inside should be totally reflected off. The first surface 1 cannot be coated with a reflective film, because it is shared by both a refractive and reflective path of the same rays. Therefore, the incident angles of all the rays striking the first surface 1 from the microdisplay 130 should be larger than the critical angle, .sub.c, set by the TIR condition
.sub.c=arcsin(1/n)(1)
where n is the refractive index of the material for the FFS prism-lens 110. For example, if the index of the material is equal to 1.5, all the incident angles should be larger than 41.82. Rays incident on the first surface 1 of the prism-lens 110 at a smaller angle may be transmitted through the prism-lens 110 without the benefit of reflection off the rear surface 2 (and subsequent refraction at the first surface 1) and may directly enter the eye, which leads to stray light and a reduction in the image contrast observed by the user. If the TIR condition is met, however, after two consecutive reflections by the front and rear surfaces 1 and 2, respectively, the same ray is returned back and to be transmitted through the front surface 1. To ensure transmission of the ray after the two consecutive reflections, the incident angle of the ray should be smaller than the critical angle set by Eqn. (1) to avoid the TIR effect.

(31) It was impractical to constrain the incident angle of every ray incident on the surface of interest during the optimization process. An adequate and practical control method was required. Without loss of generality, we made two assumptions: (1) the local departure of the surface 1 from a spherical surface was sufficiently small compared to the primary radius of curvature of the surface so that the surface normal of every point on surface 1 could be adequately approximated by a line passing through to the center of the primary curvature of the surface (as shown in FIG. 3); and (2) the primary curvature of the surface 1 is concave, as shown in FIG. 3. Under these assumptions, we could prove that the top marginal ray, R.sub.1u, which corresponds to the ray from the maximum object field in the positive Y-direction (i.e. P.sub.1) passing through the top edge of the pupil, had the smallest incident angle among all the rays striking the surface 1 from the microdisplay 130 side. As shown in FIG. 3, the incident angle on surface 1 increased gradually as the ray from the same object field shifted from the top to the bottom of the pupil (e.g. from R.sub.1u to R.sub.1b); the angle also increased as the ray intersecting the same pupil position shifted from the top to the bottom of the object fields (e.g., from R.sub.1u to R.sub.2u). Therefore, the constraint on the incident angle was written as
.sub.1b1>arcsin(1/n)(2)
where .sub.1b is the incident angle of the top marginal ray, R.sub.1u, on surface 1 from the maximum object field in tangential plane of the microdisplay 130.

(32) We could further prove that after the two consecutive reflections the top marginal ray, R.sub.2u, of the maximum object field in the negative Y-direction (i.e. P.sub.2) had the largest incident angle on the surface 1 when the surface 1 was tilted counterclockwise about the X-axis (i.e., the tilt angle, .sub.1>0); otherwise the bottom marginal ray R.sub.1b, of the maximum object field in the positive Y-direction (P.sub.1) has the largest incident angle when the surface 1 was tilted clockwise. Therefore, the constraint used to avoid TIR condition on surface 1 was written as:

(33) $\begin{array}{cc}{}_{1b}=\{\begin{array}{cc}{}_{1b1}<\arcsin \left(1\text{/}n\right),& {}_{1}0\\ {}_{2u1}<\arcsin \left(1\text{/}n\right),& {}_{1}0\end{array}& \left(3\right)\end{array}$
where .sub.1b1 is the incident angle of the bottom marginal ray, R.sub.1b, striking the surface 1; and .sub.2u1 is the incident angle of the top marginal ray, R.sub.2, on surface 1, and .sub.1 is the tilt angle of surface 1 about the X-axis.

(34) The simplified constraints in Eqns. (2) and (3) were important in making the optimization practical in designing the FFS prism-lens 110. Increasing the refractive index of the material could help to relax the ray angle constraints and ease the design task. However, high refractive index materials can increase the color aberrations (due to lower Abbe number) and fabrication cost. Furthermore, our goal in this design was to achieve light weight by using plastic materials, which usually have a moderately low range of refractive indices.

(35) Structure Constraints

(36) Designing the wedge-shaped free-form prism-lens 110 required optimizing the shapes of individual surfaces to minimize wavefront errors under the ray angle constraints set by Eqns. 2 and 3. It further required additional structure constraints in order to ensure that the three surfaces together formed a valid prism-lens shape, that all the rays across the fields could be traced without obstruction or early escaping from a surface, and that the prism-lens 110 maintained desirable center and edge thickness. FIG. 3 illustrates the structure control method we employed. During each step of the optimization, we traced two rays: the top marginal ray, R.sub.1u, of the maximum field in the positive Y-direction and the bottom marginal ray, R.sub.2b, of the maximum field in the negative Y-direction. As shown in FIG. 3, P.sub.a, P.sub.a, P.sub.a and P.sub.b denote the intersection points of the ray R.sub.2b with surfaces 1, 2, 1 and 3, respectively; and P.sub.b, P.sub.c and P.sub.c label the intersection points of the ray R.sub.1u, with surfaces 2, 1 and 3, respectively. The coordinates of these ray-surface intersections were then used to define the constraints for optimizing the FFS HMD prism-lens 110. Based on the requirements of the physical structure, the constraints were defined as

(37) $\begin{array}{cc}\{\begin{array}{c}{Y}_{{\mathrm{Pa}}^{}}-Y_{\mathrm{Pa}}<0\\ Y_{{\mathrm{Pa}}^{}}-Y_{\mathrm{Pa}}>0\\ 0.5<Z_{{\mathrm{Pa}}^{}}-Z_{\mathrm{Pa}}<2\end{array}& \left(4\right)\\ \{\begin{array}{c}{Y}_{{\mathrm{Pb}}^{}}-Y_{\mathrm{Pb}}>0\\ -1.5<Z_{{\mathrm{Pb}}^{}}-Z_{\mathrm{Pb}}<-0.2\end{array}& \left(5\right)\\ \{\begin{array}{c}-2<Y_{\mathrm{Pc}}-Y_{{\mathrm{Pc}}^{}}0\\ 0<Z_{\mathrm{Pc}}-Z_{{\mathrm{Pc}}^{}}<1\end{array}& \left(6\right)\\ \{\begin{array}{c}{Z}_{\mathrm{pa}}\mathrm{the}\mathrm{eye}\mathrm{clearance}\mathrm{distance},i.e.,17\\ Z_{\mathrm{pc}}\mathrm{the}\mathrm{eye}\mathrm{clearance}\mathrm{distance},i.e.,17\end{array}& \left(7\right)\end{array}$
where all the Y, Z coordinates in the equations are referenced to the global coordinate system with the origin located at the center of the exit pupil.

(38) Here by constraining the Y coordinates of the points P.sub.a, P.sub.a, and P.sub.a, Eqn. (4) ensured that the surfaces 1 and 2 intersected properly so that the bottom marginal ray could be traced through the prism-lens 110 without obstruction. Equation 4 further set the upper and lower limits (e.g. 2 and 0.5 mm, respectively) on the edge thickness of the prism-lens 110 by constraining the Z coordinates of the points P.sub.a and P.sub.a. By constraining the Y and Z coordinates of the points P.sub.b and P.sub.b, Eqn. (5) avoided the escape of the top marginal ray after reflection by the surface 1 and helped to control the thickness of the prism-lens 110. By controlling the Y and Z coordinates of the points Pc and Pc, Eqn. (6) ensured that the surfaces 1 and 3 intersected properly so that the top marginal ray could be traced through the prism-lens 110 without obstruction or escaping from the prism-lens 110. It further helped control the height of the prism-lens 110. Eqns. 4 through 6 together ensured the three surfaces formed a valid prism-lens shape. These relationships further set limits on the tilt angles of the surfaces 1 and 2, which helped to limit the off axis aberrations. By limiting the Z coordinates of the points P.sub.a and P.sub.c, Eqn. (7) set the minimal value for the eye clearance distance.

(39) Optimization of the Free-Form Prism

(40) We selected a patented design by Takahashi (U.S. Pat. No. 5,959,780) as a starting point. The original prism design of Takahashi included two free-form surfaces 501, 502 and one planar surface 503. Based on a 1.3 inch microdisplay 530, the Takahashi design offered a full FOV of the system 500 of 57.834.6, with an exit pupil diameter of 4 mm and effective focal length of about 27.4 mm. The F/# of the system 500 was only 6.85. To meet our specifications, we scaled the effective focal length to 15 mm, reduced the horizontal FOV to 45, and increased the exit pupil diameter from 4 mm to 8 mm, yielding a system 500 with an F/# of 1.875. In the scaled system 500, the eye relief was reduced to 15.5 mm. The significantly reduced F/# imposed a critical challenge on system performance and invalidated several critical conditions of the prism-lens structure.

(41) For instance, the incident angles of the rays on the TIR surface 501 were far smaller than the critical angle and a part of the rays from the top and bottom fields escaped from the prism 510 before completing their paths. We thus had to set considerably large vignetting for the top and bottom fields to obtain a valid starting design. FIGS. 4A and 4B show the layout and the polychromatic MTF plots of the scaled starting system 500, respectively. The MTF of the starting design was evaluated at an exit pupil diameter of 8 mm with vignetting, was no higher than 0.1 at a spatial frequency of 10 lps/mm across the entire visual field. The rayfan plots were evaluated at a 3 mm pupil, shown in FIGS. 4C and 4D. We thus needed a better starting point that met our first-order specifications before performing a comprehensive optimization.

(42) The system of FIGS. 4A-4D was optimized with rays traced from the eye position to the microdisplay 530 in CODE V. During the optimization process, four representative wavelengths, 486.1, 546.1, 587.6, and 656.3 nm, were set with the weights of 1, 1, 2, and 1, respectively. The TIR constraints and structural constraints as well as the basic optical definitions, such as the effective focal length, were always applied. The effective focal lengths in both tangential and sagittal planes were constrained to be 15 mm. We further set the following parameters as variables: all the primary curvatures of all surfaces 501, 502, 503 in both tangential and sagittal planes, aspherical coefficients, decenter in both Y and Z directions, and tilt about the X axis. Although we did not directly set a constraint on distortion, we limited the height of the rays striking the image plane to avoid large and irregular distortion.

(43) Due to its single-plane symmetry, the free-form prism-lens design had to be optimized over half of the full FOV sampled in a rectangular grid, as opposed to a linear sample in the radial direction in a rotationally symmetric system. It was difficult, however, to start the optimization across the entire FOV in a densely-sampled grid given the low performance of the starting point. Instead, we adopted a progressive optimization strategy by gradually increasing field samples as the system performance improved during the optimization process. The weighting factors of the sampled fields were inversely proportional to their distance from the center of the field. The decenter and tilt parameters were set as variables during the entire optimization process. FIG. 5 illustrates the field sampling strategy during the different stages of optimization. In the initial stage, as illustrated in FIG. 5(a), we sampled five fields along the vertical direction with the sagittal field angle being zero. It was important to optimize the system to meet the physical requirements such as eye clearance and TIR condition in this stage. During the optimization, we set the curvatures of surfaces 601 and 602 and the aspherical coefficients on rear surface 602 of the prism-lens 610 as variables. We also added curvature to surface 603 (which was a flat surface 503 in the starting Takahashi system 500) as a variable, and this surface 603 was later turned into a free-form surface to help limit the ray heights of the marginal fields with respect to the center field and improve the overall optical performance. The surface layout of the optimized system 600 on the XZ plane is shown in FIG. 5(a). After the first stage of optimization, we expanded the field samples by adding a fraction of the field angles along the sagittal direction. This stage of optimization was done by converting the surface type to AAS-type from ASP-type, and then the curvatures of the three surfaces in XZ plane were set as variables. The layout of the optimized system on the XZ plane is shown in FIG. 5(b). We continued to expand the field horizontally until the maximum field met our specification, and optimized the system repetitively by gradually adding the asymmetric coefficients as variables. A good starting point was finally achieved after the re-optimization. FIG. 5(c) shows the layout of the system on the XZ plane. It is worth pointing out that this optimization strategy can reduce the dependence on the performance of the initial starting point.

(44) Following the design strategy above, we optimized the free-form surfaces using aspherical-type representations during the above steps for obtaining a good starting point. We then furthered the optimization by converting the ASP-type surfaces to AAS-type of surfaces and adding asymmetric coefficients up to the 10.sup.th order as variables. To further optimize the system 600, we converted the AAS-type surfaces to XYP representations through a least-square fitting algorithm and carried out a global optimization. We found that this step of optimization was very effective in optimizing the FFS prism-lens system 700. The layout of the final FFS prism-lens design 700 is shown in FIG. 6. The distance from the bottom 712 to the top 714 of the left edge of the prism-lens 710 was 22 mm, the width along X-direction was 25 mm, the thickness along Z-axis was 12 mm, and the weight was 5 grams. The optical material of the prism-lens 710 was PMMA having a refractive index of 1.492 and Abbe number of 57.2. The locations, and effective areas, of the surfaces 701, 702, 703 relative to the global coordinate system having its origin at the exit pupil are given in Tables 2 and 3, respectively, below. The SPS XYP surfaces 701, 702, 703 were 10.sup.th-order polynomial surfaces added to a base conic. The polynomial was expanded into monomials of x.sup.my.sup.n, where m+n10. The equation used was:

(45) $z=\frac{c\left(x^2+y^2\right)}{1+\mathrm{sqrt}\left(1-\left(1+k\right)c^2\left(x^2+y^2\right)\right)}+\underset{j=2}{\overset{66}{.Math.}}{C}_j{x}^m{y}^n,j=\left[{\left(m+n\right)}^2+m+3n\right]\text{/}2+1$
where z is the sag of the surface along the local z-axis, x and y are the coordinates in the local coordinate system, c is the vertex curvature (CUY), k is the conic constant, and C.sub.j is the coefficient for x.sup.my.sup.n. FIG. 2 illustrates the local, right-handed coordinate system at each surface 701, 702, 703 to show the local x-, y-, and z-axes and tilt, (the x-axis is perpendicular to the plane of the figure). The FFS polynomial coefficients are provided in Table 8 at the end of the Detailed Description.

(46) TABLE-US-00002 TABLE 2 Definition of the local surface references in the global coordinate system Orientation of the surface Origin of surface reference Rotation about X (mm) Y (mm) Z (mm) X-axis () Surface 1 0 0.305 18.25 1.7942 Origin: O.sub.1 (x.sub.1, y.sub.1, z.sub.1) Orientation: .sub.1 Surface 2 0 0 24.34 23.08 Origin: O.sub.2 (x.sub.2, y.sub.2, z.sub.2) Orientation: .sub.2 Surface 3 0 15.534 19.403 53.4547 Origin: O.sub.3 (x.sub.3, y.sub.3, z.sub.3) Orientation: .sub.3 Microdisplay 0 17.101 24.272 54.1888 Origin: O.sub.Im (x.sub.Im, y.sub.Im, z.sub.Im) Orientation: .sub.Im

(47) TABLE-US-00003 TABLE 3 Effective area of each surface (mm) surface 1 X 13 13 Y 8.3 15.9 surface 2 X 13 13 Y 9.25 12.25 surface 3 X 10 10 Y 8.55 2.75

(48) The optical performance of the optimized system 700 was assessed at the following representative field angles for the four design wavelengths: (0, 0), (0, 8), (7, 0), (14, 0), (016), (22.5, 0), (22.5, 16). FIGS. 7 through 8 demonstrate the optical performance of the microdisplay viewing path. As shown in FIG. 7, the distortion in the microdisplay path was up to 12% at the top left/right corners. Such large distortion was mainly due to the trapezoidal shape distortion caused by surface 702, which is very difficult to correct, as well as a small amount of barrel distortion. By fitting with the distortion grid, we chose to pre-warp the image displayed on the microdisplay 730 to balance the distortion of the virtual image. The polychromatic MTF plots shown in FIGS. 8A-8B were evaluated for a centered 3-mm pupil, at the spatial frequency of 30 cycles/mm, which corresponds to the threshold spatial frequency of the OLED microdisplay 730. The MTF was 0.7 for the central (0, 0) field, 0.2 for the (0, 16) field, and above 0.1 for the (22.5, 16) field. The rayfan plots of the system 100 were evaluated at a 3 mm pupil, shown in FIGS. 8C and 8D; the maximum error is the half of the starting system.

(49) To demonstrate the effectiveness of the TIR constraints, FIGS. 9A and 9B plot the incident angle of rays on the surface 701 as a function of the pupil position and field position of the rays, respectively. As the pupil position of the rays is shifted from bottom to top, the incident angle on surface 701 decreased from 59.13 to 42.98 for the top field, from 61.66 to 42.2 for the center field and from 65.02 to 46.70 for the bottom field. Given the refractive index of the material in our final design was 1.492, all these angles were well controlled to satisfy the Eqn. (2). FIGS. 9C and 9D plot the incident angle of rays on the surface 701 as a function of the pupil position and field position of the rays, respectively. As the pupil position of the rays was shifted from bottom to top, the incident angle on surface 701 decreased from 7.8 to 7.0 for the top field, from 3.15 to 2.84 for the center field and from 13.7 to 13.6 for the bottom field. All these angles were far smaller than the critical angle, so they were well controlled to satisfy the Eqn. (3).

(50) Design of the Auxiliary Free-Form Lens

(51) The free-form prism-lens 710 with curved surfaces produced optical power in the optical see-through path, causing a significant viewing axis deviation and undesirable distortion as well as other off-axis aberrations to the view of the real world scene. FIGS. 10A and 10B demonstrate the optical path and distortion grid of the real-world view through the free-form prism-lens 710. An auxiliary lens 720 was desired, to not only cancel the optical power in the see-through path, but also to correct the deviation of the optical axis and the off-axis aberrations introduced by the FFS prism-lens 710.

(52) We chose to trace rays from the real-world scene to the eye space, as shown in FIG. 10C. We flipped the optimized FFS prism-lens 710 (FIG. 6) along the X-axis so that the pupil was to the right of the prism-lens 710. The reflective mode of the concave mirror surface 702 was changed to refractive mode. We then inserted a plastic auxiliary lens 720 to the left of the prism-lens 710, and the lens surface adjacent to the prism-lens 710 was matched to the concave surface 702 of the prism-lens 710, which ensured that the auxiliary lens 720 and prism-lens 710 could be cemented accurately which simplified the design of the auxiliary free-form lens 720. As a result, the front surface 722 of the auxiliary lens 720 only needed to compensate for the optical power introduced by surface 701 of the prism-lens 710 of the FFS prism-lens 710. Although we could start the optimization of the lens with a planar front surface 722, a good approximation is to initialize the front surface 722 with the same shape as surface 701 of the prism-lens 710. The combination of the auxiliary lens and the prism-lens 710 should ideally form an afocal system 700 for a real-world scene at optical infinity as the object distance is considerably larger than the EFL of the system 700. Therefore, we inserted an ideal lens at the eye position with an effective focal length equivalent to the human eye to focus the collimated rays. During the optimization process, we only set the curvature and polynomial coefficients of the front surface 722 of the auxiliary lens 720 as variables. We set constraints on the distortion and aberrations. The specification for the front surface 722 of the auxiliary lens 720 is provided as Surface 4 in Table 8 below.

(53) FIGS. 11 through 12 demonstrate the optical performance of the optical see-through path. The polychromatic MTF plots were evaluated at a 3 mm pupil, shown in FIGS. 12A-12B. At the spatial frequency of 50 cycles/mm, the MTF was 0.6 for the central (0, 0) field, 0.4 for the (0, 8) field, and 0.2 for the marginal (22.5, 16) field, which suggested that the image quality of a real-world is well-balanced across the fields and well preserved. As shown in FIG. 11, the distortion in the see-through path is below 1.4% across the field, which is negligible. The distortion caused by the FFS prism-lens 710 is as high as 10%, as shown in FIG. 10B. The auxiliary free-form lens 720 effectively corrected the viewing axis deviation and the distortion.

(54) The final design of the auxiliary lens 720 combined with the FFS prism-lens 710 is shown in FIG. 13, which demonstrates excellent correction to the deviation of the see-through optical path. The overall thickness of the combined prism-lens system 700 is approximately the same as the prism-lens 710 alone. The optical material of the auxiliary lens 720 was PMMA having a refractive index of 1.492 and Abbe number of 57.2. The locations, and effective areas, of the surfaces relative to the global coordinate system having its origin at the exit pupil are give in Tables 4 and 5, respectively, below.

(55) TABLE-US-00004 TABLE 4 Definition of the local surface references in the global coordinate system OXYZ. Orientation of the surface Origin of surface reference Rotation about X (mm) Y (mm) Z (mm) X-axis () Surface 1 0 0.305 18.25 1.7942 Origin: O.sub.1 (x.sub.1, y.sub.1, z.sub.1) Orientation: .sub.1 Surface 2 0 0 24.34 23.08 Origin: O.sub.2 (x.sub.2, y.sub.2, z.sub.2) Orientation: .sub.2 Surface 3 0 15.534 19.403 53.4547 Origin: O.sub.3 (x.sub.3, y.sub.3, z.sub.3) Orientation: .sub.3 Surface 4 0 0 29 0 Origin: O.sub.4 (x.sub.4, y.sub.4, z.sub.4) Orientation: .sub.4 Microdisplay 0 17.101 24.272 54.1888 Origin: O.sub.Im (x.sub.Im, Y.sub.Im, z.sub.Im) Orientation: .sub.Im

(56) TABLE-US-00005 TABLE 5 Effective area of each surface (mm) Surface 1 X 13 13 Y 8.3 15.9 Surface 2 X 13 13 Y 9.25 12.25 Surface 3 X 10 10 Y 8.55 2.75 Surface 4 X 13 13 Y 10 12
Prototype and Experimental Results

(57) The FFS prism-lens 710 was fabricated through a molding approach. FIG. 14A shows a photo taken at the exit pupil of the system 700 to demonstrate the image quality of the microdisplay viewing optics. The distortion was noticeable and irregular. The upper portion of the displayed image seems nearer than the lower portion of the image. The distortion correction method of a free-form surface system 700 is different from rotationally symmetric systems where distortion can be corrected with sufficient accuracy by 3 radial and 2 tangential coefficients. To correct the distortion in the free-form system 700, a more complex model with more coefficients would be required. Alternatively, we calculated the mapping from the undistorted image to a distorted image using the distortion plot in FIG. 7 and then applied the mapping matrix on the undistorted image to pre-warp the image. As shown in FIG. 7, 11 by 11 grids were sampled to calculate the distortion mapping. Similarly, we divided the effective image plane into 800600 grids, then calculated the corresponding fields on the eye side from the height of the grid corners and implemented ray tracing in CODE V to find the chief rays' intersection points with the image plane of all the fields. The one to one mapping matrix from the undistorted image to the distorted image was calculated between ideal points (grid corners) and the ray traced points on the image plane. We could then generate the pre-warped image by applying the one to one mapping matrix on the undistorted image and displayed the pre-warped image on the microdisplay 730. FIG. 15A is a photo taken at the exit pupil position without pre-warping the input image, while FIG. 15B is a photo after pre-warping the input image.

Further Design Example

(58) The techniques described above where employed to provide a second exemplary design. Again, the optical material of the prism-lens was PMMA having a refractive index of 1.492 and Abbe number of 57.2. The locations, and effective areas, of the surfaces relative to the global coordinate system having its origin at the exit pupil are give in Tables 6 and 7, respectively, below. The FFS polynomial coefficients are provided in Table 9 below.

(59) TABLE-US-00006 TABLE 6 the local surface references in the global coordinate system OXYZ. Orientation of the surface Origin of surface reference Rotation about X (mm) Y (mm) Z (mm) X-axis () Surface 1 0 4 19.18 6.04937 Origin: O.sub.1 (x.sub.1, y.sub.1, z.sub.1) Orientation: .sub.1 Surface 2 0 3.2 23.65 26.4722 Origin: O.sub.2 (x.sub.2, y.sub.2, z.sub.2) Orientation: .sub.2 Surface 3 0 16.044 23.35 53.2281 Origin: O.sub.3 (x.sub.3, y.sub.3, z.sub.3) Orientation: .sub.3 Surface 4 0 0.514 29.968 6 Origin: O.sub.4 (x.sub.4, y.sub.4, z.sub.4) Orientation: .sub.4 Microdisplay 0 17.978 25.011 49.247 Origin: O.sub.Im (x.sub.Im, y.sub.Im, z.sub.Im) Orientation: .sub.Im

(60) TABLE-US-00007 TABLE 7 Effective area of each surface (mm) Surface 1 X 15 15 Y 6 22 Surface 2 X 15 15 Y 8 16 Surface 3 X 10 10 Y 6.5 6.5 Surface 4 X 15 15 Y 11.5 12.5

(61) These and other advantages of the present invention will be apparent to those skilled in the art from the foregoing specification. Accordingly, it will be recognized by those skilled in the art that changes or modifications may be made to the above-described embodiments without departing from the broad inventive concepts of the invention. For instance, other shapes of free-form surfaces may be utilized in the designs of the present invention. By way of example, if one wanted to vary the surface curvature independently in the x and y directions, the surface could be represented by

(62) $z=\frac{c_x{x}^2+c_y{y}^2}{1+\mathrm{sqrt}\left(1-\left(1+k_x\right)c_x{x}^2-\left(1+k_y\right)c_y{y}^2\right)}+\underset{j=1}{\overset{37}{.Math.}}{C}_j{x}^{2m}{y}^n$$2m+n10,m=0,1,2,3,4,5,n=0,1,\mathrm{.Math.},10$
where z is the sag along the local z-axis, x and y are the coordinates in the local coordinate system, k is the conic constant, c.sub.x is radius of curvature of surface in sagittal direction, c.sub.y is radius of curvature of surface in tangential direction, and C.sub.j is the coefficient for x.sup.2my.sup.n. It should therefore be understood that this invention is not limited to the particular embodiments described herein, but is intended to include all changes and modifications that are within the scope and spirit of the invention as set forth in the claims.

(63) TABLE-US-00008 TABLE 8 Coef- com- ficients ment Surface 1 Surface 2 cuy c 0.000285105 0.039092523 c67 normal- 1 1 ized radius c1 k 0 0 c2 x 0 0 c3 y 4.554727019060E02 0.000000000000E+00 c4 x.sup.2 9.529768572360E03 3.799883836350E03 c5 xy 0.000000000000E+00 0.000000000000E+00 c6 y.sup.2 8.880680850880E04 8.811637430275E03 c7 x.sup.3 0.000000000000E+00 0.000000000000E+00 c8 x.sup.2 y 2.274693243620E04 4.120156490413E05 c9 xy.sup.2 0.000000000000E+00 0.000000000000E+00 c10 y.sup.3 2.250128361500E05 1.162523138271E04 c11 x.sup.4 1.363374558440E05 8.274939784188E06 c12 x.sup.3 y 0.000000000000E+00 0.000000000000E+00 c13 x.sup.2 y.sup.2 3.835587301810E05 1.217474597638E05 c14 xy.sup.3 0.000000000000E+00 0.000000000000E+00 c15 y.sup.4 8.630487450540E06 1.240101301250E05 c16 x.sup.5 0.000000000000E+00 0.000000000000E+00 c17 x.sup.4 y 1.985502857350E07 3.848552333625E08 c18 x.sup.3 y.sup.2 0.000000000000E+00 0.000000000000E+00 c19 x.sup.2 y.sup.3 1.202547290150E06 4.252351660938E07 c20 xy.sup.4 0.000000000000E+00 0.000000000000E+00 c21 y.sup.5 2.259740420160E07 1.854520775606E08 c22 x.sup.6 2.190859953400E08 7.328932049703E10 c23 x.sup.5 y 0.000000000000E+00 0.000000000000E+00 c24 x.sup.4 y.sup.2 1.532516744660E09 3.774337431125E08 c25 x.sup.3 y.sup.3 0.000000000000E+00 0.000000000000E+00 c26 x.sup.2 y.sup.4 2.361733529370E08 9.746364392781E08 c27 xy.sup.5 0.000000000000E+00 0.000000000000E+00 c28 y.sup.6 1.116521684700E08 2.777925130281E08 c29 x.sup.7 0.000000000000E+00 0.000000000000E+00 c30 x.sup.6 y 6.000609985730E20 9.790340156094E10 c31 x.sup.5 y.sup.2 0.000000000000E+00 0.000000000000E+00 c32 x.sup.4 y.sup.3 5.043712711540E20 1.456155907344E10 c33 x.sup.3 y.sup.4 0.000000000000E+00 0.000000000000E+00 c34 x.sup.2 y.sup.5 5.117716418980E20 6.072571241477E09 c35 xy.sup.6 0.000000000000E+00 0.000000000000E+00 c36 y.sup.7 8.492659970480E18 9.635155206406E10 c37 x.sup.8 7.666995972280E11 1.071711944156E10 c38 x.sup.7 y 0.000000000000E+00 0.000000000000E+00 c39 x.sup.6 y.sup.2 7.767376892480E11 0.000000000000E+00 c40 x.sup.5 y.sup.3 0.000000000000E+00 0.000000000000E+00 c41 x.sup.4 y.sup.4 2.950846077350E11 0.000000000000E+00 c42 x.sup.3 y.sup.5 0.000000000000E+00 0.000000000000E+00 c43 x.sup.2 y.sup.6 4.982363079300E12 0.000000000000E+00 c44 xy.sup.7 0.000000000000E+00 0.000000000000E+00 c45 y.sup.8 3.154691021580E13 0.000000000000E+00 c46 x.sup.9 0.000000000000E+00 0.000000000000E+00 c47 x.sup.8 y 1.654239598390E22 0.000000000000E+00 c48 x.sup.7 y.sup.2 0.000000000000E+00 0.000000000000E+00 c49 x.sup.6 y.sup.3 1.373514352470E22 0.000000000000E+00 c50 x.sup.5 y.sup.4 0.000000000000E+00 0.000000000000E+00 c51 x.sup.4 y.sup.5 3.089957605530E22 0.000000000000E+00 c52 x.sup.3 y.sup.6 0.000000000000E+00 0.000000000000E+00 c53 x.sup.2 y.sup.7 7.625247748780E22 0.000000000000E+00 c54 xy.sup.8 0.000000000000E+00 0.000000000000E+00 c55 y.sup.9 3.658104101010E20 0.000000000000E+00 c56 x.sup.10 5.304042934200E14 0.000000000000E+00 c57 x.sup.9 y 0.000000000000E+00 0.000000000000E+00 c58 x.sup.8 y.sup.2 2.253404112780E13 0.000000000000E+00 c59 x.sup.7 y.sup.3 0.000000000000E+00 0.000000000000E+00 c60 x.sup.6 y.sup.4 3.829417935900E13 0.000000000000E+00 c61 x.sup.5 y.sup.5 0.000000000000E+00 0.000000000000E+00 c62 x.sup.4 y.sup.6 3.253841807830E13 0.000000000000E+00 c63 x.sup.3 y.sup.7 0.000000000000E+00 0.000000000000E+00 c64 x.sup.2 y.sup.8 1.382388481590E13 0.000000000000E+00 c65 xy.sup.9 0.000000000000E+00 0.000000000000E+00 c66 y.sup.10 2.349220379980E14 0.000000000000E+00 Coef- com- ficients ment Surface 3 Surface 4 cuy c 0.047502239 4.62503916636415E03 c67 normal- 1 1 ized radius c1 k 0 0 c2 x 0 0 c3 y 3.713820097050E01 7.889190949566E02 c4 x.sup.2 7.602734138830E03 9.061467713679E03 c5 xy 0.000000000000E+00 0.000000000000E+00 c6 y.sup.2 1.328060538820E02 7.440925962039E04 c7 x.sup.3 0.000000000000E+00 0.000000000000E+00 c8 x.sup.2 y 6.162600900670E04 1.334980089604E04 c9 xy.sup.2 0.000000000000E+00 0.000000000000E+00 c10 y.sup.3 2.698297276700E03 1.082388324657E05 c11 x.sup.4 1.036808360720E05 6.869154882657E06 c12 x.sup.3 y 0.000000000000E+00 0.000000000000E+00 c13 x.sup.2 y.sup.2 6.395534320820E04 2.537076127696E05 c14 xy.sup.3 0.000000000000E+00 0.000000000000E+00 c15 y.sup.4 5.348289994560E04 2.872950078172E06 c16 x.sup.5 0.000000000000E+00 0.000000000000E+00 c17 x.sup.4 y 7.243323994940E06 3.579225277335E07 c18 x.sup.3 y.sup.2 0.000000000000E+00 0.000000000000E+00 c19 x.sup.2 y.sup.3 2.631914617550E05 7.674060114164E07 c20 xy.sup.4 0.000000000000E+00 0.000000000000E+00 c21 y.sup.5 1.207571795570E04 2.343303575169E07 c22 x.sup.6 6.925182707110E08 3.531736575015E08 c23 x.sup.5 y 0.000000000000E+00 0.000000000000E+00 c24 x.sup.4 y.sup.2 4.354972387950E06 1.047604139930E07 c25 x.sup.3 y.sup.3 0.000000000000E+00 0.000000000000E+00 c26 x.sup.2 y.sup.4 5.469927852330E06 7.816094559917E08 c27 xy.sup.5 0.000000000000E+00 0.000000000000E+00 c28 y.sup.6 1.348379393160E05 5.437264126834E08 c29 x.sup.7 0.000000000000E+00 0.000000000000E+00 c30 x.sup.6 y 0.000000000000E+00 2.345712968586E09 c31 x.sup.5 y.sup.2 0.000000000000E+00 0.000000000000E+00 c32 x.sup.4 y.sup.3 0.000000000000E+00 9.776159457326E10 c33 x.sup.3 y.sup.4 0.000000000000E+00 0.000000000000E+00 c34 x.sup.2 y.sup.5 0.000000000000E+00 3.747602576420E09 c35 xy.sup.6 0.000000000000E+00 0.000000000000E+00 c36 y.sup.7 0.000000000000E+00 3.134464841907E09 c37 x.sup.8 0.000000000000E+00 1.684012356810E10 c38 x.sup.7 y 0.000000000000E+00 0.000000000000E+00 c39 x.sup.6 y.sup.2 0.000000000000E+00 4.535953132119E10 c40 x.sup.5 y.sup.3 0.000000000000E+00 0.000000000000E+00 c41 x.sup.4 y.sup.4 0.000000000000E+00 7.837817283276E10 c42 x.sup.3 y.sup.5 0.000000000000E+00 0.000000000000E+00 c43 x.sup.2 y.sup.6 0.000000000000E+00 2.755761921660E10 c44 xy.sup.7 0.000000000000E+00 0.000000000000E+00 c45 y.sup.8 0.000000000000E+00 5.916877897125E10 c46 x.sup.9 0.000000000000E+00 0.000000000000E+00 c47 x.sup.8 y 0.000000000000E+00 4.770943033528E12 c48 x.sup.7 y.sup.2 0.000000000000E+00 0.000000000000E+00 c49 x.sup.6 y.sup.3 0.000000000000E+00 2.951188218903E13 c50 x.sup.5 y.sup.4 0.000000000000E+00 0.000000000000E+00 c51 x.sup.4 y.sup.5 0.000000000000E+00 3.371749455954E12 c52 x.sup.3 y.sup.6 0.000000000000E+00 0.000000000000E+00 c53 x.sup.2 y.sup.7 0.000000000000E+00 1.747159621915E11 c54 xy.sup.8 0.000000000000E+00 0.000000000000E+00 c55 y.sup.9 0.000000000000E+00 8.129325561197E12 c56 x.sup.10 0.000000000000E+00 2.312868445063E13 c57 x.sup.9 y 0.000000000000E+00 0.000000000000E+00 c58 x.sup.8 y.sup.2 0.000000000000E+00 7.302980007283E13 c59 x.sup.7 y.sup.3 0.000000000000E+00 0.000000000000E+00 c60 x.sup.6 y.sup.4 0.000000000000E+00 8.710198057552E13 c61 x.sup.5 y.sup.5 0.000000000000E+00 0.000000000000E+00 c62 x.sup.4 y.sup.6 0.000000000000E+00 2.578655721303E12 c63 x.sup.3 y.sup.7 0.000000000000E+00 0.000000000000E+00 c64 x.sup.2 y.sup.8 0.000000000000E+00 1.049427758427E13 c65 xy.sup.9 0.000000000000E+00 0.000000000000E+00 c66 y.sup.10 0.000000000000E+00 1.997616885026E12

(64) TABLE-US-00009 TABLE 9 Co- effi- com- cients ment Surface 1 Surface 2 cuy c 5.05744188235277E03 1.01290420358428E02 c67 normal- 1 1 ized radius c1 k 1.00000000000000E+00 1.00000000000000E+00 c2 x 0.00000000000000E+00 0.00000000000000E+00 c3 y 0.00000000000000E+00 0.00000000000000E+00 c4 x.sup.2 1.06569604270328E02 1.98880360585134E02 c5 xy 0.00000000000000E+00 0.00000000000000E+00 c6 y.sup.2 1.68658939805653E03 1.19385116890299E02 c7 x.sup.3 0.00000000000000E+00 0.00000000000000E+00 c8 x.sup.2 y 6.91090618400814E05 9.33625592200568E06 c9 xy.sup.2 0.00000000000000E+00 0.00000000000000E+00 c10 y.sup.3 2.20768710346282E04 2.29081872204714E04 c11 x.sup.4 6.83962391749639E06 2.78401802376246E06 c12 x.sup.3 y 0.00000000000000E+00 0.00000000000000E+00 c13 x.sup.2 y.sup.2 3.05668325239866E05 1.72926734056902E05 c14 xy.sup.3 0.00000000000000E+00 0.00000000000000E+00 c15 y.sup.4 1.36336411152319E05 1.96766859030307E05 c16 x.sup.5 0.00000000000000E+00 0.00000000000000E+00 c17 x.sup.4 y 8.58870602989718E07 3.48208722714691E07 c18 x.sup.3 y.sup.2 0.00000000000000E+00 0.00000000000000E+00 c19 x.sup.2 y.sup.3 2.59849465399784E07 2.19795381577235E06 c20 xy.sup.4 0.00000000000000E+00 0.00000000000000E+00 c21 y.sup.5 1.30586738289348E06 2.43633240996974E06 c22 x.sup.6 2.43871462953907E07 3.17005397026433E08 c23 x.sup.5 y 0.00000000000000E+00 0.00000000000000E+00 c24 x.sup.4 y.sup.2 2.17171575262769E07 8.43008544990865E09 c25 x.sup.3 y.sup.3 0.00000000000000E+00 0.00000000000000E+00 c26 x.sup.2 y.sup.4 1.09469718343971E07 3.80934245038872E08 c27 xy.sup.5 0.00000000000000E+00 0.00000000000000E+00 c28 y.sup.6 1.27848463353098E07 6.91418873061873E08 c29 x.sup.7 0.00000000000000E+00 0.00000000000000E+00 c30 x.sup.6 y 2.17215958916219E10 5.29841934213626E09 c31 x.sup.5 y.sup.2 0.00000000000000E+00 0.00000000000000E+00 c32 x.sup.4 y.sup.3 1.17052458860851E08 8.21084328918049E09 c33 x.sup.3 y.sup.4 0.00000000000000E+00 0.00000000000000E+00 c34 x.sup.2 y.sup.5 9.68888754082781E09 7.97369765852189E09 c35 xy.sup.6 0.00000000000000E+00 0.00000000000000E+00 c36 y.sup.7 3.27432053769373E09 1.04667540576694E08 c37 x.sup.8 7.15087135594710E10 1.58107927123443E10 c38 x.sup.7 y 0.00000000000000E+00 0.00000000000000E+00 c39 x.sup.6 y.sup.2 8.37241810688380E11 2.50870549731047E10 c40 x.sup.5 y.sup.3 0.00000000000000E+00 0.00000000000000E+00 c41 x.sup.4 y.sup.4 3.08448109642484E10 7.25055775891319E10 c42 x.sup.3 y.sup.5 0.00000000000000E+00 0.00000000000000E+00 c43 x.sup.2 y.sup.6 4.69822713757874E10 3.40656761505412E10 c44 xy.sup.7 0.00000000000000E+00 0.00000000000000E+00 c45 y.sup.8 3.74205140407221E11 9.66678413418157E10 c46 x.sup.9 0.00000000000000E+00 0.00000000000000E+00 c47 x.sup.8 y 6.76193659551364E11 1.12952194048511E11 c48 x.sup.7 y.sup.2 0.00000000000000E+00 0.00000000000000E+00 c49 x.sup.6 y.sup.3 2.52161603900163E11 7.04825907294106E11 c50 x.sup.5 y.sup.4 0.00000000000000E+00 0.00000000000000E+00 c51 x.sup.4 y.sup.5 1.74613643937042E10 6.33946672763876E11 c52 x.sup.3 y.sup.6 0.00000000000000E+00 0.00000000000000E+00 c53 x.sup.2 y.sup.7 4.02373533060283E11 6.52431095363973E11 c54 xy.sup.8 0.00000000000000E+00 0.00000000000000E+00 c55 y.sup.8 1.51494734534747E12 1.16444692674563E10 c56 x.sup.10 2.04692546379699E13 9.55272275007493E13 c57 x.sup.9 y 0.00000000000000E+00 0.00000000000000E+00 c58 x.sup.8 y.sup.2 1.27850873405047E12 1.26785440782157E12 c59 x.sup.7 y.sup.3 0.00000000000000E+00 0.00000000000000E+00 c60 x.sup.6 y.sup.4 3.62506934749027E12 2.65220449427970E12 c61 x.sup.5 y.sup.5 0.00000000000000E+00 0.00000000000000E+00 c62 x.sup.4 y.sup.6 6.19013257835136E12 2.60617377673244E12 c63 x.sup.3 y.sup.7 0.00000000000000E+00 0.00000000000000E+00 c64 x.sup.2 y.sup.8 5.47840265443679E13 1.51560477272646E12 c65 xy.sup.9 0.00000000000000E+00 0.00000000000000E+00 c66 y.sup.10 8.21677804323237E15 3.02669599955700E12 Co- effi- com- cients ment Surface 3 Surface 4 cuy c 1.36040070302022E01 3.57225273631067E03 c67 normal- 1 1 ized radius c1 k 1.00000000000000E+00 3.05442384082650E+02 c2 x 0.00000000000000E+00 0.00000000000000E+00 c3 y 0.00000000000000E+00 1.50268084206787E02 c4 x.sup.2 4.02243465863783E02 6.19425352271504E03 c5 xy 0.00000000000000E+00 0.00000000000000E+00 c6 y.sup.2 2.13980634999507E02 4.47303500982809E03 c7 x.sup.3 0.00000000000000E+00 0.00000000000000E+00 c8 x.sup.2 y 3.01138835536447E03 1.20873088719341E04 c9 xy.sup.2 0.00000000000000E+00 0.00000000000000E+00 c10 y.sup.3 2.02440549644019E02 4.88846955792256E05 c11 x.sup.4 4.47796835910435E04 3.55281378009418E07 c12 x.sup.3 y 0.00000000000000E+00 0.00000000000000E+00 c13 x.sup.2 y.sup.2 3.77675855322595E03 1.85583369659820E05 c14 xy.sup.3 0.00000000000000E+00 0.00000000000000E+00 c15 y.sup.4 4.80144970995529E03 9.25174463073967E06 c16 x.sup.5 0.00000000000000E+00 0.00000000000000E+00 c17 x.sup.4 y 4.02927533244600E05 3.50332571416944E07 c18 x.sup.3 y.sup.2 0.00000000000000E+00 0.00000000000000E+00 c19 x.sup.2 y.sup.3 5.39169467243012E05 2.04967612461798E07 c20 xy.sup.4 0.00000000000000E+00 0.00000000000000E+00 c21 y.sup.5 5.53544127253446E04 4.65892543560338E08 c22 x.sup.6 1.11280354729252E05 5.29354102911361E09 c23 x.sup.5 y 0.00000000000000E+00 0.00000000000000E+00 c24 x.sup.4 y.sup.2 4.80012193062278E05 4.02601860330999E09 c25 x.sup.3 y.sup.3 0.00000000000000E+00 0.00000000000000E+00 c26 x.sup.2 y.sup.4 1.26051539656191E04 2.49939246096273E08 c27 xy.sup.5 0.00000000000000E+00 0.00000000000000E+00 c28 y.sup.6 1.73588932255855E04 3.61819538534118E08 c29 x.sup.7 0.00000000000000E+00 0.00000000000000E+00 c30 x.sup.6 y 8.79779758365947E07 2.01195864259020E09 c31 x.sup.5 y.sup.2 0.00000000000000E+00 0.00000000000000E+00 c32 x.sup.4 y.sup.3 3.02147026981469E06 4.57346927308613E10 c33 x.sup.3 y.sup.4 0.00000000000000E+00 0.00000000000000E+00 c34 x.sup.2 y.sup.5 3.77643718632961E06 1.24553105773878E09 c35 xy.sup.6 0.00000000000000E+00 0.00000000000000E+00 c36 y.sup.7 3.76456044984959E06 1.79376358056304E09 c37 x.sup.8 1.15675211759524E07 2.21527691875407E10 c38 x.sup.7 y 0.00000000000000E+00 0.00000000000000E+00 c39 x.sup.6 y.sup.2 2.78655613640253E07 2.53421606734412E10 c40 x.sup.5 y.sup.3 0.00000000000000E+00 0.00000000000000E+00 c41 x.sup.4 y.sup.4 9.84349649743337E07 5.74780760494471E11 c42 x.sup.3 y.sup.5 0.00000000000000E+00 0.00000000000000E+00 c43 x.sup.2 y.sup.6 2.49702108607953E06 1.17433001848088E11 c44 xy.sup.7 0.00000000000000E+00 0.00000000000000E+00 c45 y.sup.8 2.58861741885522E06 3.74142357461756E10 c46 x.sup.9 0.00000000000000E+00 0.00000000000000E+00 c47 x.sup.8 y 6.03461822581042E09 9.50023944707191E12 c48 x.sup.7 y.sup.2 0.00000000000000E+00 0.00000000000000E+00 c49 x.sup.6 y.sup.3 1.70606513975964E08 7.11512617425205E12 c50 x.sup.5 y.sup.4 0.00000000000000E+00 0.00000000000000E+00 c51 x.sup.4 y.sup.5 1.33127271883197E08 1.18995002675616E12 c52 x.sup.3 y.sup.6 0.00000000000000E+00 0.00000000000000E+00 c53 x.sup.2 y.sup.7 4.80199259903842E08 3.34840697581670E13 c54 xy.sup.8 0.00000000000000E+00 0.00000000000000E+00 c55 y.sup.9 2.98879732404454E08 1.00015126063979E11 c56 x.sup.10 4.69852812284116E10 6.12912070190102E13 c57 x.sup.9 y 0.00000000000000E+00 0.00000000000000E+00 c58 x.sup.8 y.sup.2 5.85068948593321E10 1.03108702382589E12 c59 x.sup.7 y.sup.3 0.00000000000000E+00 0.00000000000000E+00 c60 x.sup.6 y.sup.4 3.00944950467783E09 2.49697011108537E13 c61 x.sup.5 y.sup.5 0.00000000000000E+00 0.00000000000000E+00 c62 x.sup.4 y.sup.6 5.78412894302924E09 5.08864307560358E14 c63 x.sup.3 y.sup.7 0.00000000000000E+00 0.00000000000000E+00 c64 x.sup.2 y.sup.8 2.34559082177138E08 2.18880110409132E13 c65 xy.sup.9 0.00000000000000E+00 0.00000000000000E+00 c66 y.sup.10 1.17892981297852E08 9.10176395465433E13