Patent Grant
8035898
The present invention relates generally to image-forming optics.
The design of imaging optics is a centuries-old art with literally thousands of inventions in the prior art. Nearly all of these are combinations of flats and spherical surfaces, due to their relative ease of manufacture. Surfaces with substantial departure from sphericity are known as aspherics, but aside from known quartic shapes such as the ellipsoid and the paraboloid there was little use of aspheres until the twentieth century, when advances both in theory and fabrication technology brought them into prominence. Aspheric surfaces can advantageously substitute for multiple spherical surfaces, resulting in less costly devices, even if a single asphere is more costly than a sphere. With the advent of injection-molded plastic optics and precision molded glass optics, aspheres entered the optical-technology mainstream.
Applications of aspheric optics fall into two categories, imaging and non-imaging. Non-imaging optics is concerned with illumination and the distribution of optical power, with the defining constraint being the behavior of only the outermost rays (called edge rays) of a flux distribution. Imaging optics, however, is concerned with the spatial modulation of flux, with the goal of reproducing a particular flux distribution (the object) at another location (the image). Nonimaging optical design need only take care of the edge rays, a relatively small portion of all rays, but an imaging system must send all rays to their appropriate destination, parameterized by the system magnification m. A ray originating at coordinate point (x,y) on the object must arrive at the image coordinate (mx, my), known as the image-mapping requirement. In the real world of imaging each point on the object typically radiates flux in all forward directions (i.e., nearly hemispherically) and practicality demands that a significant percentage of this flux reach the proper image point, with little or none going anywhere else on the image plane, once it enters the optical system aperture. In the world of theory, however, many aspheric design procedures hold only when rays hitting a surface subtend only a small solid angle, near the surface normal, and relatively parallel to the optical axis. This enables approximate aberration coefficients to be rapidly calculated for aspheric surfaces in design optimization.
Attaining perfect image mapping for every object point would theoretically require an infinite number of surfaces, but with a limited number of surfaces, state of the art techniques define a merit function to evaluate deviations from ideal imaging in order to minimize loss of image formation over sampled points of the image plane.
A general difficulty with aspherics is that they generally do not have a closed form solution for ray intersections, unlike the algebraic ease with which intersections are calculated for flats, spheres, and the other quadric surfaces (torus, cylinder, ellipsoid, paraboloid, hyperboloid), enabling optimal designs to be derived just with a formula. Aspheres in general, however, generally require a computationally intense iterative search that closes in on the precise intersection.
State of the art imaging optics design is done via optimization techniques using a parametric representation of a selected group and type of optical surfaces. A merit function of those parameters is defined and the search for the optimum of the merit function is done by a computer-aided multiple-parameter algorithm. The implementation of this algorithm may be based on different techniques, as binned least-square methods, simulated annealing, genetic algorithms, etc. However, the differentiation between local and global optimum is not guaranteed, and the optimization depends for its success upon the particular mathematical representation chosen to specify the surfaces. Moreover, usually the optimum found is not too far from the initial guess of the designer, so solutions far from that guess are not accessible in practice. In this application some of the embodiments refer to wide angle projection optics using one mirror. Devices for such an application also including mirrors are disclosed in prior art patents as U.S. Pat. Nos. 6,771,427 and 6,612,704 and US Patent Application No. 2001/0050758 A1. All of them have been obtained by the use of standard optimization procedures.
The only cases in prior art where no optimization is done are based on problems stated in terms of ordinary differential equations. This is the case of the single surface designs to provide axial stigmatism (that is, correction of spherical aberration of all orders), as Cartesian ovals or Schmidt correctors, and the case of the two aspheric surface aplanatic designs, as those by Schwarzschild in 1905 for 2 mirrors (see Born & Wolf, Principles of Optics, p. 168).
Earlier versions of a simultaneous multiple surface (SMS) method for designing optical devices were disclosed in U.S. Pat. Nos. 6,639,733 and 6,896,381 and US Patent Application No. 2005/0086032. This design method generates surfaces locally, based on their refraction of the relevant rays incident upon them. However, the earlier versions of the SMS method disclosed in those patent documents were directed to non-imaging optics and for rotationally symmetrical optics they are restricted to the use of meridional rays, while the present application discloses constructive methods using skew rays.
SMS methods for designing imaging optics were discussed in published reports on two projects: “Televisión por proyección ultra-delgada de pantalla grande y alta resolución” (Television by ultra-thin, high-resolution projection on a large screen) (Financer Entity: Ministerio de Ciencia y Tecnología [Ministry of Science and Technology of the Kingdom of Spain] referencia TIC2001-3617-C02) and “Sistemas ópticos avanzados para displays de proyección (Advanced optical systems for projection displays)” (Financer Entity: Ministerio de Ciencia y Tecnología [Ministry of Science and Technology of the Kingdom of Spain] referencia TEC2004-04316), as well as the doctoral thesis “Sistemas ópticos avanzados de gran compactibilidad con aplicaciones en formación de imagen y en iluminación” (Ultra-compact advanced optical systems for image formation and illumination applications), all by: Fernando Muñoz and Pablo Benítez. These concepts comprise procedures to design optical devices for imaging applications, particularly wide angle projectors.
However, those previous publications by Dr. Muñoz and Benitez do not disclose good calculation procedures for the application of the SMS-imaging design method, particularly no constructional algorithm for skew rays has hitherto been published. Therefore, a skilled person in the field cannot reproduce the designs from the aforementioned publications by Dr. Muñoz and Pablo Benitez. Additionally, the designs in those previous publications did not consider the possibility of both the object and the input pupil being decentered relative to the optical axis of the optical surfaces to be designed, while the present application does address that possibility. This case is valuable, particularly in mirror based designs, to solve problems of shading and obstruction in practical cases.
The present invention relates generally to image-forming optics and more particularly to external aspheric optical mirrors to be adapted to an existing commercial projector. The preferred embodiments disclosed herein of the present invention were generated by a new version of the simultaneous multiple surface (SMS) method, which is capable to calculate directly multiple rotational aspheric surfaces without restriction on their asphericity.
Embodiments of the present invention allow designing with the SMS not only with meridian rays but with skew rays. An advantage of designing with skew rays or a combination of meridian and skew rays is that the design ray bundles can be equispaced in the phase space, which allows a higher control of the imaging quality.
Embodiments of the present invention make it possible to design imaging devices with the following specific configurations among others (where R denotes a refractive surface and X denotes a reflective surface):
XX off axis (application: wide angle projection);
RRX off axis (application: wide angle projection);
RXXR off axis (application: binoculars);
XX on axis (application: compact lens with large focal length);
RXXR on axis (application: compact lens with large focal length).
According to an embodiment of the invention, there is provided a method of calculating an optical surface. A meridional optical line of the surface is calculated by repeatedly: selecting a ray passing a known point defining an end of a part of the optical line already calculated; extrapolating the optical line from the known point to meet the ray using a polynomial with at least one degree of freedom; adjusting the polynomial as necessary so that the selected ray is deflected at the extrapolated optical line to a desired target point; adding the polynomial to the optical line up to the point where the selected ray is deflected; and using said point where the selected ray is deflected as the known point in the next repetition.
According to another of the invention, there is provided a method of calculating first and second circularly symmetrical optical surfaces centered on an optical axis, comprising repeatedly: selecting a skew ray passing a part of the first optical surface already calculated; extrapolating the first optical surface to meet the ray at a first point; selecting a surface direction for the extrapolated first optical surface at the first point having a normal in a meridional plane; continuing the skew ray, deflected as implied by the selected normal, to a second point on the second optical surface, extrapolated as necessary; finding a surface direction at the second point so that the selected ray is deflected at the second point to a desired target point; adjusting the surface direction at the first point as necessary so that the normal to the surface direction at the second point is in a meridional plane; adding the extrapolated parts as far as the first and second points to the first and second optical surfaces with the determined surface directions; and using said added parts as the known parts in the next repetition.
According to a further embodiment of the invention, there is provided a method of designing a plurality N of optical surfaces of an optical system without requiring subsequent optimization methods, wherein the N optical surfaces are aspheric surfaces of rotation without restriction on their asphericity, comprising: simultaneously calculating the N surfaces point-by-point; using N uniparametric ray bundles previously selected at a starting position of the system; and applying the condition that the N uniparametric ray bundles are imaged without ray-aberration.
According to other embodiments of the invention, there are provided pluralities of optical surfaces designed or designable by the methods of the invention, optical systems comprising such pluralities of surfaces, and projectors, projection systems, and other apparatuses including such optical systems.
According to another embodiment of the invention, there are provided methods of manufacturing optical systems and apparatuses including such optical systems, comprising designing a plurality of optical surfaces in accordance with the methods of the invention and manufacturing one or more elements having optical surfaces in accordance with the design.
The above and other aspects, features and advantages of the present methods will be more apparent from the following more particular description thereof, presented in conjunction with the following drawings wherein:
A better understanding of the features and advantages of the present methods will be obtained by reference to the following detailed description of the invention and accompanying drawings, which set forth illustrative embodiments in which certain principles are used. They show various optical surfaces delineating transparent solids of one or more refractive indices, all in these embodiments known beforehand. These solids and their indices are implied in these Figures, though for the sake of clarity not all shown in full. In addition to the usual reference numerals some, but in the interests of clarity not all, of the alphanumeric algebraic symbols used in the present specification are shown in the drawings.
In contrast to the prior art methods, design methods described herein are capable of directly calculating multiple rotational aspheric surfaces without restriction on their asphericity. The design procedure involves the simultaneous point-by-point calculation of N rotationally symmetric aspheric surfaces with the condition that N uniparametric ray bundles (previously selected by the designer at the input side) are perfectly imaged (no ray-aberration for those rays). Each bundle can be described by expressions such as x=x(t, ξ), y=y(t, ξ), z=z(t, ξ), where t is the parameter along the ray trajectory and ξ is the parameter of the bundle. For each value of ξ, this expression defines a straight line (a light ray trajectory).
One design strategy described herein consists in selecting the uniparametric ray bundles at the input so that a proper sampling of the phase space (i.e., spatial-angular space) at the object and entry pupil is produced, expecting that the perfect image quality for those rays will provide sufficient image quality of the remaining rays by proximity. In general, how the bundle of rays must be at the output side is not known in advance, but it is determined as they are calculated during the design, as a result of imposing the condition of zero ray-aberration for the selected rays. The selection of the ray bundles for the design is done by the designer to better suit a specific design problem.
For the more general case of bundle selection, it is useful to consider that a rotationally symmetric optical system defines the usual three independent variables (r,ρ,φ), and one independent ray can be represented as one point of that 3D-space r-ρ-φ, where r is the radius of the point at which the ray crosses the object plane, ρ is the radius of the point at which the ray crosses the entry pupil, and φ is the difference in azimuth about the optical axis of those two points. Each point in the r-ρ-φ space thus defines a ring of equivalent rays Additionally, due to the symmetry, the ray at (r,ρ,φ) is redundant with the ray at (r,ρ,−φ), one ray being a symmetrical image of the other in a meridional plane, In the case where the pupil is located at infinity (telecentric pupil), or the object is located at infinity, the same considerations apply, but the variable p of the ray at the pupil or the variable r of the ray at the object, respectively, must be replaced by the angle formed by the ray with the optical axis z.
In one implementation the object and entry pupil define a volumetric region of that r-ρ-φ space. Each uniparametric ray bundle describes a curve in that 3D space r-ρ-φ. The selection of the N uniparametric bundles for the proper sampling of the phase space referred before is done by choosing those curves so they fill the volume defined by the object and the entry pupil, and the distance between the curves is rather constant.
For the definition of such a distance, a metric in the r-ρ-φ space is needed, for instance, the one usually employed in ray-tracing through imaging optical systems, for which equally spaced object points (r variable) and equally spaced points in the pupil (for which ρ,φ are the polar coordinates) are considered evenly spaced. Therefore, this metric in the 3D space r-ρ-φ is given by the differential of length ds2=dr2+dρ2+ρ2dφ2. Two extreme cases of uniparametric ray bundle selections are of particular interest. One of them occurs when the object is small compared to the input pupil, and for this case the ray bundles can be selected such that the N uniparametric ray bundles depart from N different points rather evenly spaced inside the object. In the dual case in which the input pupil is small compared to the object, the ray bundles can be selected such that the N uniparametric ray bundles point towards N different points evenly spaced inside the pupil. These two cases can be referred to as object and pupil discretization, respectively.
For this purpose, a “small” pupil or object is one that can be well approximated by a low-order truncation of the power series expansion in the variable ρ or r, respectively of the Wavefront Aberration Function W. W is usually expressed with the variables r-ρ-φ. In the case of a pupil at infinity, ρ is not an appropriate variable, and W is studied using instead the variables r-θ-φ, where θ=arctan ρ/zpupil. A “small” pupil is then one for which W(r, θ, φ) can be well approximated by a low-order polynomial in θ, with r and φ appearing as coefficients. In the case of an object at infinity, W is similarly redefined as W(θ, ρ, φ) where θ=arctan r/zpupil, and a “small” object is one for which W(θ, ρ, φ) can be well approximated by a low-order polynomial in θ.
In general, the initial points P, Q, R, L may be chosen so as to sample the volume in the r-ρ-φ spatial-angular phase space defined by the object and the entry pupil, and so that the curves representing the selected bundles are spaced apart at rather constant increments of ds, where ds2=dr2+dρ2+ρ2dφ2. When selecting the ray bundles The user may assess how completely the selected ray bundles span the volume in the r-ρ-φ spatial-angular phase space defined by the object and the entry pupil, or how evenly the selected bundles are spaced apart in increments of ds, or both. In the 2004 Fernando Muñoz thesis, we made calculations for the optimum spacing among bundles and the optimum spacing from the bundles to the edge for the case of pupil discretization based on the Wavefront Aberration Function W(r, ρ, φ), which measures the image quality. Those calculations are incorporated herein by reference.
In the paper Pablo Benítez, Juan C. Miñano, “Ultrahigh-numerical-aperture imaging concentrator”, Journal of the Optical Society of America A, 14, pp. 1988-1997, (1997)) an RX design for an imaging system, using 2 meridian ray bundles with object discretization was analyzed. In that paper we showed for a specific design that the optimum spacing was to place the 2 object point at r=+−⅔R, where R is the radius of the object. (In that calculation, the object is located at infinity, so r becomes the angle α, such as α6 of
In the present processes, the objective of uniform spacing ds may be violated when desired, for example, when a different approach is convenient for other reasons. Even when uniform ds is a starting objective, strict compliance may not be practical, because the curves are not necessarily parallel, indeed, some pairs of curves can meet at the starting points, so the spacing ds may not be well-defined. In the start condition for the described embodiments, all bundles coincide in the same equivalent ray. This configuration, although undesired with respect to the uniform spacing, allows us to create the surfaces without any iterative method, as described below. If equispaced bundles are forced from the beginning, it is believed to be difficult to conduct the surface calculations without inconvenient and time consuming iteration.
As mentioned above, conventional methods are based on the calculation of the minimum of a merit function using ray tracing and a parametric representation of a selected group and type of optical surfaces. In those methods, the ray-tracing sampling is discrete in all variables r, ρ, φ. (The sample rays correspond to isolated points in that space.) That is, no uniparametric ray-bundle families are used.
The processes described below may still require a trial and error optimization to find the best choice of the bundles. However, the design methods are radically different, in that the conventional methods need an initial guess of the optical surfaces to start optimization, while the processes described in the present specification need an initial guess of the design bundles to start optimization. The result is in general rather different, because the conventional initial guess is usually based on simple spherical, paraxial optics, while the first surfaces obtained with the presently described processes starting from initially guessed bundles are neither spherical nor paraxial.
In this case, rays (except ray ν1,0) are not contained in a meridional plane. Rays that belong to bundle N1 are parallel to plane XZ and exit from points (ζ, y0, 0), where ζ is the parameter, and y0 is a positive real number. Thus Cartesian coordinates will be used instead of cylindrical. The object plane 801 is at z=0 and the image plane 802 is at z=z′. Designate the two surfaces to be calculated as S1 and S2. There are n1 known surfaces (see feature 803 in
Define a line perpendicular to the normal vector to S1 at A1,0 and rotate it around the optical axis (z axis) to generate a cone, as shown in
Next rotate the line perpendicular to the normal vector to N2 at A2,0 around the optical axis to generate another cone. The intersection between that cone and the ray deflected at A1,1 defines point A2,1. The ray deflected at A2,1 must be redirected to point Q′1,1=(m·δx, m·y1,0, 0). Since we know the incident and deflected rays at point A2,1, we can calculate its normal. There is, however, no guarantee that this normal vector is perpendicular to a surface with circular symmetry. If not, then try different values of the normal vector at A1,1 until the normal vector at A2,1 is in a meridional plane.
It seems that with one design bundle (e.g. N1) we are able to calculate two optical surfaces, but, in fact, we are using at the same time the other and symmetric bundle N2 because we would have calculated the same portions of surfaces S1 and S2 if we had used the other bundle N2. So here and after, we are going to say that with one skew bundle of rays (and implicitly with its symmetrical bundles) we are able to design two optical surfaces.
In these types of designs there is no need for the existence of a previously known portion of surface as a basis to generate a new optical surface, since the two optical surfaces are generated at the same time. Therefore, it only takes the choice of two initial seed points to develop both optical surfaces. These two points are usually off-axis and sometimes it is necessary to define a portion of surface between each one of them and the optical axis. These portions of surface may be prefixed by an interpolated surface or generated using skew rays. When using skew rays, the method described for
Designate as T1, T2 and T3 the surfaces to be calculated, and consider that there may be known intermediate surfaces in the system: n1 surfaces 1303 (not shown in
Designate as H1, H2 and H3 three uniparametric bundles as shown in
Points R(x, y, 0) on the object plane 1301 having the values of y chosen as shown in
Since the skew rays are symmetrical relative to the plane XZ, a ray of H2 and its symmetrically corresponding ray in H3, in the following description, when a skew bundle is referred to, the reference also applies to its symmetrical correspondent, and the same is true for the rays of these bundles. The calculation of the surfaces T1, T2 and T3 is as follows:
First choose the initial seeds of the surfaces. Here we are to consider the case where one surface is initially seeded as a point.
Calculate the initial portions of two surfaces and choose a point on the third surface. This leads to three possibilities for starting the design, depending upon which of the three surfaces is chosen as the point.
First choose an axial point B2,0 on surface T2 and a first portion T1,0 of surface T1. Make T1,0 a surface that is generated from circular symmetry of a differentiable curve (e.g., an even polynomial). Now calculate the first portion T3,0 of surface T3 as the interpolated surface whose edge contains points B3,0, B3,00 and B*3,00. Define rays η1,0, η2,0 and η3,0 from bundles H1, H2 and H3 that are deflected at T1,0, pass through point B2,0 and arrive at B3,0, B3,00 and B*3,00, respectively. These rays are emitted from points R1,0, R2,0 and R3,0 on object plane 1301 and after deflecting on T3,0 arrive at points R′1,0, R′2,0 and R′3,0 on object plane 1302. Once the rays η1,0, η2,0 and η3,0 are chosen, and assuming the bundles H2 and H3 exit from lines parallel to but offset from the X axis as shown in
The new portions T2,1 and T3,1 will be the skew-imaging ovals between points B2,0 and B2,1 and B3,0 and B3,1, respectively, that transform the rays of bundle H2,1 emitted from points R(x, y, 0) in the range [R2,0, R2,1] on object plane 1301 into rays belonging to another bundle H′2,1 that cross the image plane 1302 at points R′(x′, y′, z′) in the range [R′2,0, R′2,1], such that x′=mx and y′=my. These imaging ovals are calculated as follows:
Define Δ as an incremental displacement normal to the optical axis (z axis) and to the y axis. Consider an offset parallel to the x axis relative to the point R2,0(x2,0, ∈, 0), where ∈ is the distance between the XZ plane and the plane of the points that emit bundle H2. The new point has the coordinates R(x2,0+Δ, ∈, 0). Consider the cone coaxial with the optical axis of the system and tangent to the second optical surface T2 at point B2,0. In this particular case the normal to B2,0 points in the direction of the z axis so that the cone degenerates into a plane perpendicular to the z axis.
A skew ray of bundle H2 emitted from point R propagates through the first n1 initial surfaces (see 1303 in
Next consider the cone tangent to the optical surface at point B3,0 (and coaxial with the optical axis of the system as defined above). The ray deflected at B2 propagates through n3 intermediate surfaces 1305, intersects this cone and defines point B3. The ray deflected at B3 must cross the image plane 1302 at R′(m(x2,0+Δ), m∈, z′). The directions of the incident and deflected rays at B3, together with the properties of the surface T3, are sufficient to defines the normal to the surface T3 at B3. There is, however, no guarantee that this normal vector is perpendicular to a rotationally symmetric surface. If not, alter the normal vector at B2 until the normal vector at the new B3 is also in a meridional plane (to ensure that the surfaces being generated have circular symmetry). The propagation of this skew ray through the system then defines the normal vectors at both B2 and B3.
Then continue incrementing Δ until we reach R2,1.
The new portion T1,1 will be the meridional imaging oval between points B1,0 and B1,1 that transforms the rays of bundle H1,1 emitted from points R(x, 0) in the range [R1,0, R1,1] on object plane 1301 into rays [η′1,0, η′1,1] belonging to another bundle H′1,1 crossing image plane 1302 at points R′(x′, z′) in the range [R′1,0, R′1,1], such that x′=mx. This imaging oval is calculated as follows:
Let δx be an increment of the x-coordinate. Define a point R as (x1,0+δx, 0). Let η be the ray emitted from point R, propagated through the n1 surfaces [not shown], that intersects surface T1 at a new point B1 to be calculated by means of an algorithm. Then, it will continue propagating as ray η′. This ray will propagate through the n2 surfaces [not shown] between T1 and T2, through the surface T2, through the n3 surfaces [not shown] between T2 and T3, through the surface T3 and cross the image plane at R′, defined as (m·(x1,0+δx), 0).
The portion of surface between points B1 (to be calculated) and B1,0 can be represented by a 2nd degree polynomial: F(x)=b2x2+b1x+b0 with coefficients b2, b1 and b0. The position of B1,0 imposes a condition F(xB10)=zB10=b2·x2B10+b1·xB10+a0 on the polynomial and the normal at B1,0 imposes another condition on its first derivative: F′(xB10)=z′B10=2·b2·xB10+b1. These two conditions determine two coefficients, leaving one free (e.g., b2). We may now choose a value for this free coefficient (e.g., b2), completely defining the polynomial at B1,0. This polynomial curve extends the surface beyond point B1,0. Ray μ intersects this polynomial curve at a point B*1 and is refracted there. Ray μ′ continues to propagate through the n2 surfaces [not shown], surface T2, n3 surfaces [not shown], surface T3 and reaches the image plane 1302 at a point R′. If R′=(m·(x1,0+δx), 0), then we make B1=B*1 and the normal at B1 is given by the normal to the polynomial curve at that point. If R′≠(m·(x1,0+δx), 0), then we must iterate, choosing different values for free polynomial coefficient (e.g. b2) until condition P′=(m·(x1,0+δx), 0) is met. This process defines another point B on T1 between points B1,0 and B1,1 based on point B1,0.
We may now define another point R on the object plane 1301 as, for example, R=(x1,0+2δx, 0) and repeat the process calculating a new point B1 on T1 between points B1,0 and B1,1, but now based on point B1 previously calculated. The process continues until the portion between points B1,0 and B1,1 on T1 is completely defined.
Continue calculating successively portions T1,k, T2,k and T3,k in accordance with
In
First, having to define the prescribed curve increases the number of initial parameters selected by the user. That increases the complexity of the design process, because of the task of finding a set of optimum initial parameters to start the design.
Second, the geometry of the design may dictate a large design with a big initial zone defined by an optical surface that is not calculated for the design bundles.
One solution to these concerns could consist in designing this surface simultaneously with the other. We have seen that we can calculate two surfaces at the same time if we use skew bundles.
Referring now to
Let us suppose that, from the four design bundles K1, K2, K3 and K4 of skew rays originating from the object plane 2001, the two bundles with odd sub-index are symmetric and the two bundles with even sub-index are also symmetric. It will usually be enough to mention the first bundle of each pair when we are referring to the four bundles. The same considerations apply with the uniparametric, skew and initially unknown bundles K′1, K′2, K′3 and K′4 falling on the image plane 2102.
Given M meridian ray-bundles and S non-redundant skew ray bundles, it is possible to design N=M+2S surfaces. The factor of 2 comes from the fact that we are allocating the symmetric (redundant) bundles. It is conceptually clearer to state that when designing N surfaces, N bundles (counting both members of any symmetric pair) are used. In
For each point L of the object 2001, two rays κ1(L) and κ2(L), one ray from each of the design bundles K1, K2 will leave, pass through the system and exit converted into κ′1(L) and κ′2(L), rays that will arrive at point L1 in the image 2002 that is related to L by the expressions: x′=m·x and y′=m·y, where L(x, y, z) and L′(x′, y′, z′) are points of the object plane 2001 and image plane 2002, respectively.
Let us consider the most general case of an optical system of four surfaces U1, U2, U3 and U4, in which there may be some known intermediate surfaces, i.e., n1 surfaces 2003 between the object plane 2101 and U1, n2 surfaces 2004 between U1 and U2, n3 surfaces 2005 between U2 and U3, n4 surfaces 2006 between U3 and U4 and n5 surfaces 2007 between U4 and the image plane 2002. As in the previous embodiments, each of n1, n2, n3, n4, and n5 may independently be zero, one, or more. The surfaces 2003, 2004, 2005, 2006, 2007 are shown in
The process to calculate the surfaces U1, U2, U3 and U4 will be the following:
First, obtain initial sections of three surfaces and an initial fixed point of the fourth according to
Define initial sections U1,0 of the surface U1 between points C1,00, C1,0, C*1,00 and C*1,0 and U3,0 of the surface U3 between points C3,00, C3,0, C*3,00 and C*3,0 as two differentiable curves, e.g., even degree polynomials, and define C2,0 as the axial point that belongs to U2. Because of the rotational symmetry of the system, we may suppose that the surface U2 has zero derivative on axis, so the normal vector of U2 at C2,0 is parallel to the optical axis.
Let K0 be a uniparametric bundle of meridional rays that pass through the surfaces existing between the object and U2 and arrive at C2,0. Let κ1,0, κ2,0, κ3,0 and κ4,0 be four edge rays that leave from the points L1,0, L2,0, L3,0 and L4,0 in the object plane and whose intersection with U1 defines the points C1,0, C1,00, C*1,0 and C*1,00. If the radius to the points C1,0, C1,00, C*1,0 and C*1,00 and the gradient of surface U1 at those points has been defined as described above, this step determines the radius to the points L1,0, L2,0, L3,0 and L4,0 and the angle α7 of the rays when they exit those points. However, depending on the constraints of a specific design, different ones of these variables may be preselected, so that the remainder are derived at this step of the calculation. After exiting from C2,0 and passing through the n3 surfaces 2005 between U2 and U3, the edge rays κ1,0, κ2,0, κ3,0 and κ4,0 fall on U3 at points C3,0, C3,00, C*3,0 and C*3,00, which those edge rays define. The intersection of bundle K0 with surface U1 and U3 define the prescribed portions of such surfaces.
The rays of K0 then pass through the n4 surfaces [not shown] between U3 and U4, the edge rays κ1,0, κ2,0, κ3,0 and κ4,0 fall on U4 at points C4,0, C4,00, C*4,0 and C*4,00, which those edge rays define.
Calculate U4,0 between points C4,0, C4,00, C*4,0 and C*4,00 as the interpolated surface that fulfils the requirement that rays that leave points C4,0, C4,00, C*4,0 and C*4,00 arrive at L′1,0, L′2,0, L′3,0 and L′4,0.
Let K1,1 be a set of rays belonging to the bundle K1 that are emitted from points between L1,0 and L1,1 and arrive at surface U1 and surface U2 after propagating through the initial n1 surfaces 2003 and the intermediate n2 surfaces 2004.
The points of K1,1 from L1,0 to L1,1 are in a plane that is parallel to the XZ plane, at y=∈. Because K0 was defined uniparametrically, the azimuth θ of L1,0, and therefore the value of ∈, are not predetermined, and a convenient value, typically with θ for the starting points L1,0, L2,0, L3,0 and L4,0 about 45° to the X and Y axes, may be chosen. The values of y=∈ for L1,0 and y=η for L2,0 may be chosen independently.
The new portions U1,1 and U2,1 will be the skew imaging ovals between points C1,0 and C1,1 and between C2,0 and C2,1, respectively, that transform the rays of bundle K1,1 emitted from points L(x, ∈, 0) in the range [L1,0, L1,1] on the object plane 2001 onto rays belonging to another bundle K′1,1 crossing the image plane 2002 at points L′(x′, y′, z′) in the range [L′1,0, L′1,1], such that x′=mx and y′=m∈.
These imaging ovals are calculated as follows:
Let Δ be a differential increment. Let us consider an offset along the x axis relative to the point L1,0(x1,0, ∈, 0), where ∈ is the distance between XZ plane and the plane of the points that emit bundle K1. The new point has the coordinates L(x1,0+Δ, ∈, 0), but we only are going to represent its projection onto plane XZ.
We now consider the cone coaxial with the optical axis of the system and tangent to the optical surface at point C1. In this particular case the normal to C1 points in the direction of the z axis and therefore the cone degenerates into a plane perpendicular to the z axis.
A ray emitted from point L propagates through the first n1 surfaces 2003. It intersects the cone coaxial with the optical axis of the system and tangent to the optical surface at point C1,0 and defines point C1. We choose a normal vector to surface U1 at C1 on the meridional plane through C1.
Let us now consider the cone tangent to the optical surface at point C2,0 (and coaxial with the optical axis of the system as defined above). The ray deflected at C1 propagates through the n2 intermediate surfaces 2004, intersects this cone and defines point C2. The ray deflected at C2 must cross the image plane 2002 at L′(m·(x2,0+Δ), m·∈, 0), after being deflected through the subsequently intervening optical surfaces U3 and U4. The direction of the incident and deflected rays at C2 defines its normal. There is, however, no guarantee that this normal vector is perpendicular to a rotationally symmetric surface. We iterate the normal vector at C1 until the normal vector at the new C2 is also contained on a meridional plane (to ensure that the surfaces being generated have circular symmetry).
The propagation of this skew ray through the system then defines the normal vectors at both C1 and C2.
Let K2,1 be a set of rays belonging to the bundle K2 that are emitted from points between L2,0 and L2,1 and arrive at surface U3 and surface U4 after propagating through the n1 initial surfaces 2003, U1, n2 intermediate surfaces 2004, U2, and the n3 intermediate surfaces 2005.
The points of L2 from L2,0 to L2,1 are in a plane that is parallel to XZ plane at y=η.
The new portions U3,1 and U4,1 will be the skew imaging ovals between points C3,0 and C3,1 and between points C4,0 and C4,1, respectively, that transform the rays of bundle K2,1 emitted from points L(x, y, 0) in the range [L2,0, L2,1] on the object plane 2001 onto rays belonging to another bundle K′2,1 crossing the image plane 2002 at points L′(x′, y′, z′) in the range [L′2,0, L′2,1], fulfilling the requirement that x′=m·x and y′=m·y.
These imaging ovals are calculated as follows:
Let Δ be a differential increment. Let us consider an offset along the x axis relative to the point L2,0(x2,0, η, 0), where η is the distance between XZ plane and the plane of the points that emit bundle K2. The new point has the coordinates L(x1,0+Δ, η, 0), but we only are going to represent its projection onto plane XZ.
We now consider the cone coaxial with the optical axis of the system and tangent to the optical surface at point C3,0. A ray emitted from point L propagates through the first n1 surfaces 2003, U1 and U2, and the intermediate n2 surfaces 2004 and n3 surfaces 2005. It intersects the aforementioned cone tangent to the optical surface at point C3,0 and defines point C3. We choose a normal vector to surface U3 at C3 on the meridional plane through C3.
Let us now consider the cone tangent to the optical surface at point C4,0 (coaxial with the optical axis of the system as defined above). The ray deflected at C3 propagates through intermediate n4 surfaces 2006, intersects this cone and defines point C4. The ray deflected at C4 must cross the image plane 2002 at L′(m(x2,0+Δ), m η, 0). The direction of the incident and deflected rays at C4 defines its normal. There is, however, no guarantee that this normal vector is perpendicular to a rotationally symmetric surface. We iterate the normal vector at C3 until the normal vector at the new C4 is also contained on a meridional plane (to ensure that the surfaces being generated have circular symmetry).
The propagation of this skew ray through the system then defines the normal vectors at both C3 and C4.
The process continues, calculating successively portions U1,k, U2,k, U3,k and U4,k according to
So far we have described the detailed steps for design method using the particular case of pupil discretization (in which the starting ray bundles are distributed fairly evenly across the entry pupil). The described design methods can be analogously applied to the general case and to the dual case of object discretization. As an example,
The only significant change when comparing the object discretization and pupil discretization cases relates to the rays arriving at the image plane. The image plane 3701 in
The beam from physical source pupil 2805 with edge points at P1 and P2 corresponds to projector 2701 of
An example of a design similar to the one shown in
The surfaces are rotationally symmetric. The polynomials that generate them are given by the following polynomial, where x and z are given in millimeters:
In this example the values of the coefficients ai are listed in Table 2.
The range of the x-variable, in millimeters, is: Surface #1: X∈[24, 87], Surface #2: x∈[50, 258]
As another example, a 3 surface design similar to that described in
The surfaces are rotationally symmetric. The coefficients of the polynomials that describe them are listed in Table 4.
The range of the x-variable is: Surface #1: x∈[0, 23], Surface #2: x∈[0, 24], Surface #3: x∈[55, 451]
The monochromatic performance of the XX can be much improved if more aspheric surfaces are added. This is the case of the RXXR device, which can be designed with the methods described herein, for instance, using 2-skew rays. This general case is shown.
The four surfaces of the RXXR can also be designed with three rays, one of them being the on-axis ray, and designing with the optical path length condition for the 3 rays and the Conrady condition for the central ray (since the path length inside the dielectric is equal for all the on-axis rays.
If the entry surface 3303 is restricted to be a flat and the exit surface 3302 to be a sphere centered at the on-axis image point, the system will fulfill the Conrady condition of achromatism for the on-axis point. For a finite field of view, achromatism can be improved if instead of a sphere as exit surface an ellipse with its two foci on the image plane is prescribed.
The physical source pupil 4003 corresponds to one of the projectors 3901 of
An example of a design similar to the one shown in
The surfaces are rotationally symmetric about the Z axis. Surface #1 (4006) is spherical, with radius=−205 mm, centre at z=−105 mm (so that the curve of surface #1, if extended, would cross the Z axis at z=+100 mm). The polynomials that generate surfaces #2 and #3 are given by the following polynomial, where x and z are given in millimeters:
In this example the values of the coefficients ai are listed in Table 6.
The range of the x-variable is: Surface #1: x∈[14, 80], Surface #2: x∈[14, 80], Surface #3: x∈[53, 300]
In the interests of clarity and simplicity, the mirror 4103 (corresponding to surface 4004) and lens 4104 (corresponding to surfaces 4005, 4006) are shown “in air” with the holders in which the lens and mirror are mounted omitted. The holders may be of conventional construction, and it is within the ordinary skill in the art to design and construct suitable holders once the shapes and positions of the optical components have been determined. The projector axis Z* is shown as a dashed line in
Although specific embodiments have been described, the skilled reader will understand how features of different embodiments may be combined and substituted, and such combinations are within the scope of the present invention.
For example, although the embodiments have been described with reference to light rays propagating in a specific direction, for example, left to right in
Although in the interests of clarity the described process embodiments are shown in
Although in the methods of construction shown in
Processes with two, three, and four optical surfaces to be calculated have been described. There is in principle no limit to the number of surfaces that can be calculated, provided a sufficient number of bundles of rays (one pair of skew bundles for two surfaces, and optionally a meridional bundle for one surface) are used. However, the complexity of the calculation, and the risk that the calculation fails and must be restarted choosing different initial values, increases with the number of surfaces. Moderation is therefore presently recommended.
As may be seen by comparing
Where the processes described in the embodiments start from points, line sections, or other constraints that are decided before the described calculation starts, described by terms such as “chosen,” “defined,” “known,” “prescribed,” or “predetermined,” those constraints may be dictated by existing considerations, or freely chosen by the user of the process as an initial or preliminary step, or may be a mixture.
The above examples have been described with calculations starting from known ray bundles at the “object” plane, and proceeding to derive initially unknown ray bundles at the “image” plane, implicitly assuming that the actual photons propagate from object to image. However, in some circumstances the actual image may be initially better constrained than the actual object, and because of the reversibility of non-diffusive optics the described methods may be applied proceeding against the direction of photon flow. It is also contemplated that rays may be traced in both directions. However, that may render the calculations more complicated without compensating advantage, and bidirectional tracing is presently not preferred.
The preceding description of the presently contemplated best mode of practicing the invention is not to be taken in a limiting sense, but is made merely for the purpose of describing the general principles of the invention. The full scope of the invention should be determined with reference to the Claims.
This application claims benefit of U.S. Provisional Patent Application No. 61/188,449, filed 8 Aug. 2008 by Miñano et al., which is incorporated herein by reference in its entirety.
| Number | Name | Date | Kind |
|---|---|---|---|
| 6639733 | Minano et al. | Oct 2003 | B2 |
| 6896381 | Benitez et al. | May 2005 | B2 |
| 7826142 | Harned et al. | Nov 2010 | B2 |
| 20050086032 | Benitez et al. | Apr 2005 | A1 |
| Number | Date | Country | |
|---|---|---|---|
| 20100042363 A1 | Feb 2010 | US |
| Number | Date | Country | |
|---|---|---|---|
| 61188449 | Aug 2008 | US |
