MXPA00010820A - Method for the design of multifocal optical elements - Google Patents

Abstract

A method for designing a surface of a multifocal optical element, such as a progressive spectacle lens, includes the steps of partitioning a region into triangles, (&tgr;i) defining a set of functions, each representing a portion of the surface (li) over one of the triangles (&tgr;i), and optimizing the functions. A further embodiment includes the steps of partitioning a region into polygons (200), and defining a set of functions, each representing a portion of the surface over one of the polygons (202). Continuity constraints for values of the functions and values for the first derivatives of the functions at boundaries of the polygons are dictated. Continuity constraints for values of the second derivatives of the functions at boundaries of the polygons are not required. The functions are optimized subject to the dictated continuity constraints.

Description

METHOD FOR THE DESIGN OF MULT1FOCAL OPTICAL ELEMENTS FIELD OF THE INVENTION The present invention relates to a method for designing multifocal optical elements.

BACKGROUND OF THE INVENTION The optical elements are pieces of substantially transparent material having surfaces that reflect or refract light, such as mirrors, lenses, spacers and collimators. Optical elements are used in a variety of applications, including telescopes, microscopes, cameras and goggles. The optical elements can be characterized by their optical properties and their optical surface properties. Optical properties such as astigmatism, optical power and prism describe how a wavefront of incident light is deformed as it passes through the optical element. Optical surface properties such as surface astigmatism, surface optical power and gradients of surface astigmatism and surface optical power describe geometric properties of a surface of the optical element. The optical properties and the optical properties of the surface are closely related, but they are not identical. The multifocal optical elements have more than one optical power. For example, bifocal lenses have two subregions, each with a different optical power. Lenses for bifocal glasses can be used, for example, to correct myopia (short view) in one subregion and presbyopia (the loss of the eye's ability to change the shape of its lens) in the other subregion. Unfortunately, many people find it uncomfortable to wear glasses for bifocal glasses, due to the abrupt change in optical power from one subregion to the other. This led to the development of progressive glasses lenses, which are multifocal lenses in which the optical power varies slightly from one point to another on the lens. The optical surface power at any point on the surface of an optical element is defined by the average curvature of the surface. A progressive lens has variable optical power, so it has variable curvature, and is by definition spherical. However, since the surface of the progressive lens, or at least a substantial part of it, is by definition spherical, it has two different main curvatures ?? and? 2 on many points. The surface astigmatism at any point on the surface of an optical element is defined by the absolute value of the difference in the major curvatures? \ And?: 2.

The definitions of the average curvature H, Gaussian curvature G and major curvatures? and * - of a surface / at the point (x, y) are given in equations 1A-1D: From the first days of progressive lens design, the main design goals have been to: a) vary the optical power slightly; b) minimal astigmatism; c) reduction of a variety of optical aberrations such as oblique distortion, binocular imbalance, etc. Many different methods have been proposed to achieve these goals. The patent of E.U.A. 3,687,528 to Maitenaz describes a technique in which a base curve (meridian) runs from the top of the lens to its bottom. The lens surface is defined along the meridian so that the curvature varies gradually (and consequently the optical power also varies). Along the meridian itself, the principal curvatures? \ And K = satisfy? GI =? -2. The lens surface extends from the meridian horizontally in several different methods. Explicit formulas are given for extensions from the meridian. Maitenaz obtains an area in the upper part of the lens, and another area in the lower part of the lens in which there is a much more stable optical power. In addition, the astigmatism in the vicinity of the meridian is relatively small. Many designs for progressive lenses explicitly divide the progressive lens into three zones: an upper zone for far vision, a lower zone for near vision and an intermediate zone that forms a bridge between the first two zones. The upper and lower zones essentially provide clear vision. Many designs use spherical surfaces for the upper and lower zones. An important effort in the design process is to determine a suitable intermediate zone. The patent of E.U.A. 4,315,673 to Guilino and the patent of E.U.A. 4,861,153 to Winthrop describe a method that achieves a smooth transition area through the use of an explicit formula for the intermediate zone. The patent of E.U.A. 4,606,622 to Furter and G. Furter, "Zeiss Gradal HS - The progressive addition lens with maximum wearing comfort", Zeiss Information 97, 55-59, 1986, describes a method in which the lens designer defines the value of the surface of the lens. lens in the intermediate zone in a number of special points. The entire surface is then generated by the slot method. The designer adjusts the value of the lens surface at special points to improve the properties of the generated surface. The patent of E.U.A. 4,838,675 to Barkan et al., Describes yet another method. A progressive lens having an upper zone for far vision, a lower zone for near vision and an intermediate zone is described by a base surface function. An improved progressive lens is calculated by optimizing a defined function on a sub-region of the lens, where the optimized function will be added to the base surface function. A different technique is described by J. Loos, G. Greiner and H.P. Seidel, "A variational approach to progressive lens design", Computer Aided Design 30. 595-602, 1998 and by M. Tazeroulti, "Designing a progressive lens", in the book edited by P.J. Laurent et al., Curves and Surfaces in Geometric Pesian. AK Peters, 1994, pp. 467-474. The surface of the lens is defined as a combination of slot functions, and therefore the surface should be considered on a rectangle that is divided into smaller rectangles. This method is not natural for lenses that have to be defined on a shape other than a rectangle. A cost function is defined, and the slot coefficients are determined in such a way that the surface reduces the cost function as much as possible. This method does not impose boundary conditions on the surface, and therefore lenses that require a specific form at the boundary can not be designed using this method. Using cubic birranuras, this method provides a precision of 4, where h is the iB-tt-nl-l-Mi-i ratio of the diagonal of the smallest rectangle to the diagonal of the large rectangle. European patent application EP744646 to Kaga et al., Describes a method in which the surface of a progressive lens is divided into rectangles. At the limits of the rectangles, the surface must be continuous, differentially continuous and twice differentially continuous. Since the optical power is related to the curvature, and that the curvatures are determined by the second derivatives of the surface, it seems natural to impose continuity constraints on the second derivatives. Of course, many methods for designing progressive lenses include continuity constraints in the second derivatives of the lens surface.

BRIEF DESCRIPTION OF THE INVENTION 15 An object of the present invention is to provide a novel method for designing multifocal optical elements. Therefore, a method for designing a surface of a multifocal optical element is provided in accordance with a preferred embodiment of the present invention. The method includes the steps of dividing a region into triangles, defining a set of functions, each of which represents a portion of the surface on one of the triangles, and optimizing the functions, thus generating the surface.

In addition, according to a preferred embodiment of the present invention, the optical element is a lens. Moreover, according to a preferred embodiment of the present invention, the lens is a progressive lens for glasses. Further, according to a preferred embodiment of the present invention, the region is a two-dimensional region and the functions are polynomials of the fifth order in the two dimensions. According to another preferred embodiment of the present invention, the method also includes the steps of simulating the optical behavior of the optical element based on the generated surface, and if the generated surface is not satisfactory, adjust parameter values of the functions and repeat the steps of optimizing and simulating. In a further preferred embodiment of the present invention, the step of optimizing further includes the steps of defining a function of the function set and optimizing the function. Furthermore, according to a preferred embodiment of the present invention, the function is a cost function and the step of optimizing the function is the step of minimizing the function. Moreover, according to a preferred embodiment of the present invention, the method further includes the step of choosing values that will be included in the function. Furthermore, according to a preferred embodiment of the present invention, the values vary according to at least one location on the surface, the set of functions and the first derivatives of the function set. A method for designing a multifocal optical element surface is also provided according to a further preferred embodiment of the present invention. The method includes the steps of dividing a region into polygons, and defining a set of functions, each of which represents a portion of the surface on one of the polygons. Continuity constraints are dictated for values of the functions and values of the first derivatives of the functions in polygon boundaries. I dont know require continuity constraints for values of the second derivatives of the functions at the boundaries of the polygons. The functions are optimized subject to the constraints of continuity dictated, thus generating the surface. In addition, according to a preferred embodiment of the present In the invention, the method also includes the step of dictating continuity constraints for values of the second derivatives of the vertex functions of the polygons. Moreover, according to a preferred embodiment of the present invention, the polygons are triangles. A method for designing a surface of a multifocal optical element is also provided according to another preferred embodiment of the present invention. The method includes the steps of dividing a region into polygons and defining a set of functions, each of which "- -" • * - ^ - - ^ represents a portion of the surface on one of the polygons.A plurality of curves is determined along a subset of polygon boundaries.Values of at least one of the polygons. sets comprising the functions, first derivatives of the functions and second derivatives of the functions, or any combination thereof, are prescribed in selected vertices of polygons that coincide with the plurality of curves.The functions are optimized subject to the prescribed values, In addition, in accordance with a preferred embodiment of the present invention, the polygons are triangles, Moreover, according to a preferred embodiment of the present invention, the polygons are rectangles. a preferred embodiment of the present invention, a multifocal optical element having a plurality of surfaces, wherein at least one of the surfaces Cies is designed by one of the methods described hereinabove.

BRIEF DESCRIPTION OF THE DRAWINGS The present invention will be more fully understood and appreciated from the following detailed description taken in conjunction with the accompanying drawings, in which: Figure 1 is a schematic flow diagram illustration of a design procedure for a surface of an element optical, according to a preferred embodiment of the present invention; Figure 2 is a schematic flow diagram illustration of a portion of the design procedure shown in Figure 1, in accordance with a preferred embodiment of the present invention. Figure 3 is a schematic illustration of a surface representation of a lens, according to a preferred embodiment of the present invention and Figure 4 is a schematic illustration of a region divided into triangles, useful for understanding the present invention.

DETAILED DESCRIPTION OF THE PRESENT INVENTION The present invention is a novel method for designing multifocal optical elements. The method makes it possible for the designer to emphasize specific areas of a surface of the optical element where a desired optical behavior is most crucial. The method also makes it possible for the designer to control optical properties along and in the vicinity of curves on the surface. In the particular case of designing a progressive lens, the present invention does not require the division of the lens into three zones: an upper zone for far vision, a lower zone for near vision and an intermediate zone that establishes a bridge between the first two zones. Instead, the entire lens surface is optimized as a whole. The method of the present invention includes a design method with feedback. This is shown in Figure 1, to which reference is now made. The designer specifies (step 100) the values of various parameters that will be used in the design procedure. Then the parameter values are used to calculate (step 102) the surface. In step 104, the designer evaluates the surface resulting from the calculation of step 102. The designer checks whether the optical properties of the surface are satisfactory. The optical properties of the surface include the optical power of the surface, defined as the average curvature of the surface, the astigmatism of the surface, defined as the absolute value of the difference between the principal curvatures of the surface, and the gradients of the power Optical surface and surface astigmatism. In addition, the designer uses a simulator to simulate the passage of at least one wavefront through the optical element, resulting in the determination of the optical properties of the optical element. If the calculated surface is not satisfactory (step 106), the designer adjusts (step 108) the values of the different parameters. The surface is then recalculated (step 102). This procedure is repeated until the designer is satisfied with the calculated surface. The specification of the parameter values (step 100) and the calculation of the surface (step 102) are shown in more detail in figure 2, to which reference is now made, which is a schematic flow diagram illustration of a portion of the design procedure. In addition, reference is made to Figure 3, which is a schematic illustration of a lens. The lens has a known surface g and an unknown surface. The surface of the lens / is considered on a region D in the plane (x, y). Region D is divided (step 200) into a collection of polygons. In a preferred embodiment of the present invention, the polygons are triangles t, i = 1, ...., N, where N is the total number of triangles. The number of triangles is not limited, and the triangles do not have to be identical to each other, thus providing the necessary flexibility to arbitrarily divide a D region. The surface / is a patch of a plurality of "lenses" / ,, where each lens / is defined on a triangle t. The height of the surface / on the point (x, y) in the triangle t, is given by l. { x, y) as expressed in equation 2: f (x, y) = U (x, y), (x, y) e (2) Since the triangles do not have to be identical to each other, they can use small contact lenses for those areas of the lens surface where fine details are required, and large lenses can be used for those areas of the lens surface where a coarse detail is sufficient. The surface of each lens /, is represented (step 202) on its triangle t, by a polynomial of fifth order, namely a polynomial in x and y l - «------ i - r- which includes all combinations of the form x'y" 1 with y + m less than or equal to 5. The height of the surface of a lens /, - on the point (x, y) in the triangle tj is given by k (xy) in equation 3: + a / ° x + ai "and + af ° ¿+ a / Jxy + a? ' 1? +. To "30 x 3 +, to" ¡21 x 2 y +, to "¡12 x y 2 +,", 03 y 3 +, to "¡40 x 4 - + L. D ~ i 31 x 3 y ,, +, a "¡22 x 2 y 2 +, a? ¡73 x y 3 +, a" ¿04 ?, 4 +, a ", - 50 5 j +. a¡ * x 4 y, - + a¡ 32 x 3 y,, 2 - + L a¡ 23 x 2 y 3 +, a "¡14 y +, a" ¡05 y, J Equation 3 rewrite more compactly as in equation 4: l, (x y) =? afxJym ,. { x, y) = ti, (4) j, m j + = 5 where ajm are the coefficients of the lens, .. On the surface / is thus completely determined by the coefficients a and m of the set of local polynomials. It will be appreciated that if there are, for example, 1,000 triangles in the division of region D, then the determination of the surface / implies determining the value of 21,000 coefficients. Without additional restrictions, there are infinitely many possible solutions for the surface /. As explained hereinabove, the most natural restriction is that the surfaces of the lens // are patched together so that the resulting surface / is continuous, differentially continuous and twice differentially continuous along the boundaries of the triangles v ,. Since the second derivative of / is the curvature, which is related to the power optical, and that the optical power should vary slightly, it is natural to demand that the surface / is twice differentially continuous along the boundaries.

An important feature of the present invention is that it is sufficient to require that the surfaces of the lens /, - be patched together so that the surface / resultant is continuous and differentially continuous to along the boundaries of triangles tl. This is expressed in the equations 5A-5C: or l¡ (? > y) = (* 'y) Á? > y) e t > n tj »(5A) dl, (x, y). { ?, y), (x, y) e t, n tj, (5B) dx dx dl ,. { x, y) (?, y), (x, y) e t¡ ntJ t (5C) dy where the intersection of the triangles t and t is the common limit for both triangles According to a preferred alternative embodiment of the present invention, the need for the contact lens surfaces to be • patched together so that the surface / resultant is twice 0 differentially continuous at the vertices of the triangles tl is added to the continuity constraints of equations 5A-5C. This is expressed in equations 5D-5F: _., ^. ^ -_.- ^^ - ^ _ i,, -i ,, | - | i, -. ",, Item ,,". ? . .ni i i nti i nni d%. { x, y) d2lj (x, y) d2lk (x, y), (x, y) e ti tj nt ,. (5D) dx2 dx 'dx2 d%. { x, y) 2lj. { x, y)% (x, y), (x, y) e t¡ nr n rk! (5F) dy2 dy2 dy2 where the intersection of triangles tt} and x is the vertex common to three triangles It will be appreciated that in certain extreme cases, where a vertex is common only to two triangles, the continuity constraints of equations 5D-5F are still maintained at those vertices. It will be appreciated that the continuity constraints of equations 5D-5F are significantly less restrictive than the prior art continuity constraints for the second derivative, which give along the full limits. Since the goal is a progressive lens with variable optical power, it is reasonable to restrict the surface / so that its average curvature matches as much as possible with a predetermined function P (x, y) which specifies (step 204) the desired optical power at each point (x, y). Also, since another goal is to reduce to the maximum the astigmatism of the progressive lens, it is reasonable restrict the surface / so that the difference between its main curvatures be reduced to the maximum. In addition, as mentioned above, the designer may wish to emphasize specific areas of the lens surface. This is achieved by choosing (step 204) weighting coefficients wu and w (2 which characterize the relative "strength" of each lens /, - and the relative importance of the astigmatism and the optical power for each lens /, -. An emphasized lens will be greater than the weighting coefficients of a de-emphasized lens For a particular lens l in which the optical power is more important than the astigmatism, the weighting coefficient w 2 will be greater than the weighting coefficient w. Similarly, for a lens /, - particular in which the astigmatism is more important than the optical power, the weighting coefficient wu will be greater than the weighting coefficient WJ.Therefore, due to the constraints of the equations 5A- 5F, the coefficients a, Jm representing the surface / must reduce to a maximum a cost function E given in equation 6: where? \ (x, y) and? z (x, y) are the principal curvatures of the surface / at the point (x, y), and dxdy is a surface area element of the triangle tj. The predetermined weighting coefficients wu and w depend on x, y, and possibly on / and the first derivatives of /. This dependency on / and the first derivatives of / may arise from a number of situations. For example, if the integral is calculated not on the triangle? But on the surface of the lens, -, then the surface area element dxdy is replaced by ^ + (dfl x) 2 + (df / dy) 2 dxdy. The factor -Jl + (df ¡dx) 2 +. { df / dy) 2 can be absorbed in the weighting coefficients, leaving the surface area element as dxdy. The factor ___ Í_ÍM --- K_ jl + (df / dx) 2 + (df / dy) 2 effectively increases the value of contact lenses whose surface area is large relative to other lenses, perhaps to compensate for the fact that more light through large contact lenses. Another example of weighting coefficients that depend on / and the first derivatives of / is when the designer wishes to penalize abrupt changes in the surface /. The surface / which minimizes the cost function E will not necessarily have the optical power P (x, y) at point (x, y). If the designer wishes to force the lens to have a particular optical shape and / or behavior at certain points, then the designer determines (step 208) a finite set of curves Cm, m = l, NC, where Nc is the total number of curves, along a subset of the boundaries of triangles t ,. The designer specifies (step 208) curve conditions, which dictate any or some of the values of the surface / and its first and second derivatives in the vertices V of the Cm curves. This has the effect of controlling the shape and optical properties in the vertices V and in the vicinity of the Cm curves. It will be appreciated that Cm curves do not necessarily define a closed subregion on the surface. According to a well-known technique in plate mechanics, the surface / can be expressed as a patch of fifth order polynomials defined on the triangles r ,, so the surface / will satisfy the continuity constraints of equations 5A-5F. This technique is described in the article The TUBA Family of Piet Elements for the Matrix Displacement Method "by JH Argyris, I. Fried and DW Scharpf, published in The Aeronautical Journal, 1968, vol 72, pp. 701-709, which is incorporated herein by reference, a number Ns of fifth-order polynomials S c, y) satisfying the continuity constraints of equations 5A-5F is constructed according to a preferred embodiment of the present invention, surface / is represented (step 202) as a linear combination of these polynomials of Sk form (x, y), as given in equation 7: f (x, y) =? dkSk (x, y), (7) where are the unknown coefficients in the linear combination. Then, by construction, the surface / of equation 7 satisfies the continuity constraints of equations 5A-5F. As will be explained below, the curve constraints prescribed by the designer effectively determine the value of some of the unknown coefficients. 4. When the surface / of equation 7 is replaced in the cost fusion E of equation 6, the result is an expression (step 206) of the cost function E in terms of the coefficients ¿4, some of which are unknown . The minimization of the cost function E can now be carried out regardless of the continuity constraints of equations 5A-5F and the curve constraints, to determine the unknown coefficients 4 remaining.

As mentioned in the article by Argyris et al., The continuity constraints of equations 5A-5F are equivalent to satisfying the continuity of /, their first derivatives and their second derivatives in the vertices of the triangles t ,, and the continuity of the derivative of / with respect to the normal in the midpoints of the sides of the triangles t ,. Therefore, for the complete set of triangles t »the continuity constraints of equations 5A-5F are equivalent to 6 continuity constraints for each vertex and 1 continuity constraint for each midpoint. The 6 exact values of /, their first derivatives and their second derivatives in each vertex and the exact value of the derivative of / with respect to the normal in each midpoint are free parameters, known as degrees of freedom. It will be appreciated that the number of vertices Nv in the set of triangles is less than 3 times the number of triangles N, since many vertices are common to more than one triangle. It will also be appreciated that the number of midpoints NM in the set of triangles is less than 3 times the number of triangles N, since most midpoints are common to two triangles. The total number of degrees of freedom for the complete set of triangles t, is given in equation 8: NS = 6NV + NM. (8) A polynomial of Sk (x, y) form is constructed for each degree of freedom, as shown in Figure 4, which is further referred to, which is a schematic illustration of a region divided into triangles. A triangle tl has three vertices V1, V2 and V3 and three midpoints M1, M2 and M3. A triangle t2 has three vertices V1, V2 and V4 and three midpoints M1, M4 and M5. The vertex V1 is common to six triangles, t} - t6. The midpoint M1 is common to triangles t} and t2, the midpoint M2 is common to the triangles tj and t7, and the midpoint M3 is common to the triangles tj and t6. One of the degrees of freedom Ns is related to the value of the surface / at the vertex V1. A polynomial of form Sj (x, y) is constructed so that its value in the vertex V1 is 1 and in each other vertex in the set of triangles is 0, its first and second derivatives in each vertex are 0, and its derivative with with respect to the normal in each middle point is 0. This is achieved by determining a polynomial of fifth order in each triangle, and establishing S (?, y) to be the collection by pieces of these polynomials of fifth order. The triangle tl has 21 degrees of freedom associated with it - the value of /, its first derivatives and their second derivatives in each of the vertices V1, V2 and V3, and the value of the derivative of / with respect to the normal at each of the midpoints M1, M2 and M3. Therefore, a polynomial of fifth order in xyy, having 21 coefficients, can be completely determined for triangle tj by adjusting the value in vertex V1 to 1, the value in vertices V2 and V3 to 0, the first and second derived in the vertices V1, V2 and V3 to 0, and the derivative with respect to the normal in the midpoints M1, M2 and M3 to 0. In a similar way, a polynomial of fifth order in x and y can be determined completely for triangle t2 by adjusting the value in the vertex V1 to 1, the value in the vertices V2 and V4 to 0, the first and second derivatives in the vertices V1, V2 and V4 a 0, and the derivative with respect to normal at the midpoints M1, M4 and M5 to 0. It will be appreciated that the fifth-order polynomial for triangle tj and the fifth-order polynomial for triangle t2 satisfy the constraints of continuity of equations 5A-5F. Similar fifth order polynomials in x and y are determined completely for the remaining triangles in the set, and the polynomial of form S (x, y) is defined as the piecewise collection of these fifth order polynomials. Similarly, a second polynomial of form S2 (?, Y) is constructed so that its value in each vertex is 0, its first derivative in the vertex V1 is 1 and in each other vertex is 0, its second derivatives in each vertex be 0 and its derivative with respect to normal at each midpoint is 0. It will be appreciated that each of the polynomials of form Sk (x, y) satisfies the constraints of continuity of equations 5A-5F by construction, and that by therefore the linear combination given in equation 7 satisfies them as well. It will also be appreciated that each of the fifth-order polynomials Sk (x, y) are not zero only in triangles where a degree of freedom is 1 at a vertex or midpoint of that triangle. The curve constraints specified by the designer are prescribed values for / and possibly some of their first and second derivatives in the vertices Vc of the Cm curves. Each of these restrictions ^ B ^^ - ^ - alBa ^ á of fixed curve the value for / or one of its first or second vatives at a particular vertex in the set of triangles, and therefore completely determines (step 208) the value of a 4 particular that appears in the linear combination of equation 7. The number of coefficients 4 that remain unknown is given by the expression Ns -.number of curve constraints. Other techniques are known in plate mechanics to reconstruct polynomials of fifth orform that satisfy the continuity constraints of equations 5A-5F. For example, said technique is described in the article "Static and Dynamic Applications of a High-Precision Triangular Plate Bending Element" by G.R. Cowper, E. Kosko, G.M. Lindberg and M.D. Olson, published in AIAA Journal. 1969, vol. 7. No. 10. pp. 1957-1965, which is incorporated herein by reference. The polynomials of form according to Argyris provide a precision of h6, while the polynomials of form according to Cowper provide a precision of h5, where h is the ratio of an edge of a triangle to a dimension of region D. It will be appreciated that any fifth-orpolynomial that satisfies the continuity constraints of equations 5A-5F can be used as the basis of a linear combination for / in place of the function functions described hereinabove. It will be appreciated that if polygons other than triangles are used to divide the D region, the number of degrees of freedom associated with the polygon will be more than 21, and therefore the polynomials will have to be higher than the fifth orto compensate all degrees of freedom. As mentioned above, the linear combination of equation 7 is replaced in the cost function E of equation 6, and the result is an expression of the cost function E in terms of the coefficients ¿4, some of which are unknown. The minimization of the cost function E is carried out now (step 210) if the continuity constraints of equations 5A-5F and the curve constraints are imported, to determine the coefficients 4 unknown. It is well known from the calculation of variations that the minimization problem is equivalent to solving a particular equation, known as in the Euler-Lagrange equation, for the unknown surface /. This is explained in the book Calculus of Variations. by I.M. Gelfand and S.V. Fomin, Prentice Hall, 1963. For the cost function E given in equation 6, the Euler-Lagrange equation is nonlinear. In addition, the integral over the triangle r; contains a non-linear expression in the unknown coefficients ¿4 > and therefore it is difficult to calculate explicitly. In this way, an iterative method is used to solve (step 210) the Euler-Lagrange equation for the unknown coefficients. 4- In a preferred embodiment of the present invention the Newton method is used, but instead any adequate iterative method. The iterative method consists of producing a sequence of surfaces, / i, / 2, in such a way that the sequence converges to the necessary solution /. There are many ways to choose the initial surface 1 for the iteration. For example, you can choose a plane in x and y for the initial surface / 1. Each surface in the sequence solves a system of linear algebraic equations, the system characterized by a matrix. The construction of the matrices that appear in the iteration method is described in the finite element literature, for example, in the books by T.J.R. Hughes, The Finite Elements Method. Prentice, Hall, 1987, and by W.G. Strang and G. Fix, An Analysis to the Finite Element Method. Wellesley Cambridge, 1973, which are incorporated herein by reference. One benefit of using the Finite Element Method is that the matrices used in the iteration procedure have a special structure, in which all the elements of the matrix except for a narrow band near the diagonal are zero. These matrices are known as "band matrices". This feature greatly reduces the computational complexity of solving the system of equations, and therefore makes it possible to use many degrees of freedom when representing the lens surface. In principle, there are infinitely many steps to the iteration procedure. Therefore, it is necessary to determine a criterion to stop the interaction procedure at any particular point, and use the resulting surface as the solution. An example of such a criterion is to compare a surface fM with the previous surface fM -? - An example of such a comparison is to take the difference of the coefficients of linear combination ¿4 for the surface fM and the surface fM- - If the sum of the squares of this difference is smaller than a predetermined threshold, then the sequence is said to have converged to the surface fM- The designer may find that the convergence to a solution is very slow, for example, that the difference between the surface fM and the anterior surface fM-? It is too big. According to a preferred embodiment of the present invention, the convergence of the sequence of the surfaces can be increased by replacing the cost function E given in equation 6 by a variant cost function E for one or more steps in the iteration, before return to the original problem. The variant cost function E is the cost function E of equation 6 with the principal curvatures ?? and KZ of equations 1C and 1D, respectively, replaced by? and? '2 as given in equations 9C and 9D, respectively: ? [(ß) = H ß) + ^ H ß) 2 - G \ ß), (9C) (9D) It will be appreciated that when β has the value 1, equations 9A-9D are identical to equations 1A-1D, and therefore the variant cost function E is identical to the cost function E given in equation 6. When ß has the value 0, the variant cost function E is a linearization of the cost function E given in equation 6. As mentioned above, the convergence of the sequence of the surfaces can be increased using the variant cost function E when ß has a value between 0 and 1, including 0 but excluding 1, for one or more steps in the iteration, before returning to the original problem. It will also be appreciated that when ß has a value other than 1,? and? 'z lose their geometric meaning. It will be appreciated that the surface calculation described hereinabove can be used to calculate a single unknown surface of any optical element, which could consist of several refractory surfaces. According to a preferred embodiment of the present invention, the parameters specified in step 100 of Figure 1, adjusted in step 108 and used in the calculation of step 102, are the division of D into triangles t "the power function desired optical P (x, y), the weighting coefficients wj, j and wi? 2, the Cm curves and the curve constraints. In addition, as mentioned hereinabove with respect to step 104 of Figure 1, once the lens surface / has been calculated, the designer reviews the surface properties of the calculated surface and simulates optical properties of the lens as a all. The revised surface properties include the optical power of the surface, the astigmatism of the surface and the gradients of optical power of the surface and surface astigmatism. For example, the surface astigmatism of the calculated surface may change too fast, and cause discomfort for the user of a lens based on the calculated surface. According to a preferred embodiment of the present invention, the simulation simulates the optical properties of the lens as seen from one eye. One or more wave fronts propagate towards the lens. In a preferred embodiment of the present invention, a plurality of spherical wave fronts are propagated to the lens. In general, the wave fronts deform when passing through the lens. The resulting deformed wave fronts on the other side of the lens can be characterized by a variety of properties, including astigmatism, optical power, prism and the like. The calculation of deformed wavefronts can be achieved by the well-known beam tracking technique. The article "Testing and centering of lenses by means of a Hartmann test with four holes" by D. Malacara and Z. Malacara, published in Qptical Engineering. 1992, vol. 31 no. 7, pp. 1551-1555, describes a technique for determining the properties of deformed wave fronts. It will be appreciated by those skilled in the art that the present invention is not limited to what has been particularly shown and described herein. Rather, the scope of the invention is defined by the following claims.

Claims (32)

NOVELTY OF THE INVENTION CLAIMS

1. - A method for designing a surface of a multifocal optical element, the method comprising the steps of: dividing a region into triangles; defining a set of functions, each of which represents a portion of said surface on one of said triangles; and optimizing said functions, thus generating said surface.

2. A method according to claim 1, further characterized in that said optical element is a lens.

3. A method according to claim 2, further characterized in that said lens is a lens for progressive glasses.

4. A method according to claim 1, further characterized in that said region is a region of two dimensions and wherein said functions are polynomials of fifth order in said two dimensions.

5. A method according to claim 1, further comprising the steps of: simulating the optical behavior of said optical element based on said generated surface; and if said generated surface is not satisfactory, adjust parameter values of said functions and repeat said steps of optimizing and simulating.

6. - A method according to claim 1, further characterized in that said step of optimizing further comprises the steps of: defining a function of said set of functions and optimizing said function.

7 - A method according to claim 6, further characterized in that said function is a cost function and said step of optimizing said function is the step of minimizing said function.

8. A method according to claim 6, further comprising the step of: choosing values that will be included in said function.

9. A method according to claim 8, further characterized in that said values vary according to at least one location on said surface, said set of functions and first derivatives of said set of functions.

10. A method for designing a surface of a multifocal optical element, the method comprising the steps of: dividing a region into polygons; defining a set of functions, each of which represents a portion of said surface on one of said polygons; dictate continuity restrictions for values of said functions and values of the first derivatives of said functions in limits of said polygons; not require continuity restrictions for values of the second derivatives of said functions in the limits of said polygons; and optimize said .Á-a - ^ - ú --- functions subject to said dictated continuity restrictions, thus generating said surface.

11. A method according to claim 10, further characterized in that said optical element is a lens.

12. A method according to claim 11 further characterized in that said lens is a progressive lens for spectacles.

13. A method according to claim 10, which further comprises the step of: dictating continuity constraints for values of the second derivatives of said functions in vertices of said 10 polygons.

14. A method according to claim 10, further characterized in that said polygons are triangles.

15. A method according to claim 14, further characterized in that said region is a region of two dimensions 15 and where said functions are polynomials of fifth order in said two dimensions.

16. A method according to claim 10, further comprising the steps of: simulating the optical behavior of said optical element based on said generated surface and if said generated surface 20 is not satisfactory, adjust parameter values of said functions and repeat those steps to optimize and simulate.

17. A method according to claim 10, further characterized in that said optimization step further comprises t-i-a-a-J - i-ii-t-U-H > - * _- a ^ j ^. the steps of: defining a function of said set of functions and optimizing said function.

18. A method according to claim 17, further characterized in that said function is a cost function and said step of optimizing said function is the step of reducing said function as much as possible.

19. A method according to claim 17, further comprising the step of: choosing values that will be included in said function.

20. A method according to claim 19, further characterized in that said values vary according to at least one location on said surface, said set of functions and first derivatives of said set of functions.

21. A method to design a surface of a multifocal optical element, the method comprises the steps of: dividing a region into 15 polygons; define a set of functions, each of which represents a portion of said surface on one of said polygons; determining a plurality of curves along a subset of boundaries of said polygons; prescribe values of at least one of the set comprising said functions, first derivatives of said functions and second derivatives of said functions, or any combination thereof, in selected vertices of polygons that coincide with said plurality of curves and optimize said functions subject to said predetermined values, thus generating said surface. a ___ ^ u ____ ^ __ ^ * lÉ_ .... > ", I, to ** 8 *. * I.

22. - A method according to claim 21, further characterized in that said optical element is a lens.

23. A method according to claim 22 further characterized in that said lens is a lens for progressive glasses.

24. A method according to claim 21, further characterized in that said polygons are triangles.

25. A method according to claim 21, further characterized in that said polygons are rectangles.

26. A method according to claim 24, further characterized in that said region is a region of two dimensions and wherein said functions are polynomials of fifth order in said two dimensions.

27. A method according to claim 21, further comprising the steps of: simulating the optical behavior of said optical element based on said generated surface, and if said generated surface is not satisfactory, adjust parameter values of said functions and repeat those steps to optimize and simulate.

28. A method according to claim 21, further characterized in that said step of optimizing further comprises the steps of: defining a function of said set of functions and optimizing said function.

29. - A method according to claim 28, further characterized in that said function is a cost function and said step of optimizing said function is the step of minimizing said function.

30. A method according to claim 28, further comprising the step of: choosing values that will be included in said function.

31.- A method according to claim 30, further characterized in that said values vary according to at least one location on said surface, said set of functions and first derivatives of said set of functions.

32. A multifocal optical element having a plurality of surfaces, further characterized in that at least one of said surfaces is designed by the method according to claim 1. 33.- A multifocal optical element having a plurality of surfaces, further characterized in that at least one of said surfaces is designed by the method according to claim 10. 34.- A multifocal optical element having a plurality of surfaces, further characterized in that at least one of said surfaces is designed by the method according to claim 21.