Method for calculating curvature of arbitrarily defined range of optical element, device for calculating curvature of arbitrarily defined range of optical element, and recording medium

Abstract

This method for calculating curvature of an arbitrarily defined range of an optical element includes: a step (a) for obtaining a lateral vector s of an element number n representing a local curvature distribution in an arbitrarily defined range of the optical element; and a step (b) for, when an average spherical power in a prescribed light flux range within an arbitrarily defined range is defined as a macro curvature c, calculating the macro curvature c without using wavefront information in the arbitrarily defined range by using a weighting vector w with which an approximate expression of c ≈ wsT is satisfied.

Description

Method for calculating curvature of an arbitrary range of an optical element, device for calculating curvature of an arbitrary range of an optical element, and recording medium

[0001] The present invention relates to a method for calculating the curvature of an optical element within an arbitrary range, an apparatus for calculating the curvature of an optical element within an arbitrary range, and a recording medium.

[0002] There are two methods for evaluating optical elements such as lenses: surface measurement and transmission measurement. The former is useful for understanding the quality of each front and rear surface, while the latter is useful for managing the quality of the lens as a whole. A typical transmission measurement is wavefront measurement using an interferometer or similar. Wavefront measurement can directly obtain wavefront information including aberrations up to higher orders, but it is only applicable to lenses that are close to aberration-free. One method that can be applied to lenses with large aberrations is power distribution measurement, which examines the wavefront curvature of a small beam of light by passing many small beams of light through the lens. The disadvantage of power distribution measurement is that it can only obtain wavefront information as a distribution of local curvature.

[0003] In the case of eyeglass lenses and contact lenses, power distribution measurement is often used because the lens as a whole has a structure that intentionally adds so-called aberrations such as astigmatism correction and addition, and the aberrations of interest are spherical power error and astigmatism, which are quantities that directly correspond to the wavefront curvature, when viewed in units of a light beam with a diameter of about 4 mm that passes through the pupil. For example, the device described in Patent Document 1 measures local power per minute light beam with a diameter of 0.5 mm.

[0004] Japanese Patent Application Laid-Open No. 2004-205438

[0005] However, recently emerging eyeglass lenses (or contact lenses) that have the effect of inhibiting the progression of myopia are systems that contain ultra-high-order aberrations, in which areas with prescribed power and areas with defocus are discretely scattered, even within a light beam passage range of about 4 mm in diameter that passes through the pupil, and there is a possibility that sufficient information cannot be obtained by measuring the power distribution.

[0006] In optical elements, the location of the focus per luminous flux can be determined from the radius of curvature when the wavefront within the luminous flux range (hereinafter referred to as the macroscopic wavefront) is approximated as a sphere. The refractive power (mean spherical power in ophthalmic lenses), which is the reciprocal of the focal length, can also be determined from the curvature. Hereinafter, the curvature corresponding to the macroscopic wavefront will be referred to as the "macroscopic curvature." The approximation error corresponds to the degree of reduction in the peak intensity of the spot. On the other hand, the location of the caustic and the type of pattern it creates can be determined from the local curvature distribution of the wavefront. Since power distribution measurement is equivalent to measuring the local curvature distribution of the wavefront, and it is impossible to reconstruct the wavefront from the local curvature distribution of the wavefront, it can be said that it is not suitable for obtaining information about the focus (such as the macroscopic curvature).

[0007] An object of one embodiment of the present invention is to provide a technique for calculating macro-curvature from local curvature distribution.

[0008] A first aspect of the present invention includes the steps of: (a) obtaining a horizontal vector s having n elements representing a local curvature distribution in an arbitrary range of an optical element; and (b) determining a macroscopic curvature c, where c is a mean spherical power in a predetermined light flux range within the arbitrary range, such that c≈ws Ｔ and (b) calculating the macroscopic curvature c without using wavefront information of the arbitrary range, using a weighting vector w such that the following approximate formula holds.

[0009] A second aspect of the present invention is a method for calculating the curvature of an arbitrary range of an optical element according to the first aspect, wherein the sum of the elements of the weighting vector w is 1, and a cross section passing through the center of a two-dimensional plot of the weighting vector w has an upwardly convex shape.

[0010] In a third aspect of the present invention, in the step (b), the unknown wavefront aberration distribution in the arbitrary range is defined as a horizontal vector z having n elements, and c=mz Ｔ The horizontal vector for calculating the macroscopic curvature c from the wavefront aberration distribution is defined as m, and the measurement device for obtaining the local curvature distribution is defined as s Ｔ = Lz Ｔ It is considered as a linear system that performs the operation wLLＴ = mL Ｔ The method for calculating the curvature of an arbitrary range of an optical element according to the first aspect, wherein the weighting vector w is calculated from the following relational expression:

[0011] In a fourth aspect of the present invention, in the step (b), the weighting vector w is expressed by a rotationally symmetric polynomial with k degrees of freedom, and the unknown wavefront aberration distribution in the arbitrary range is a horizontal vector z with n elements, and c=mz Ｔ The horizontal vector for calculating the macroscopic curvature c from the wavefront aberration distribution is defined as m, and the measurement device for obtaining the local curvature distribution is defined as s Ｔ = Lz Ｔ The diameter of the light beam range is considered as the small light beam diameter φ when calculating the local curvature. １ The calculation range is a range reduced by the value of ｐ Let F be a matrix in which k degrees of freedom are arranged vertically, let a coefficient vector of the rotationally symmetric polynomial be a row vector a with k elements, let e be a row vector with n elements whose values ​​are 1 within the calculation range and 0 outside the calculation range, and let a be the reciprocal of the number of measurement points included in the calculation range. ０ Let a(FL)(FL) Ｔ = (m-a ０ eL) (FL) Ｔ The method for calculating the curvature of an arbitrary range of an optical element according to the first aspect, wherein the weighting vector w is calculated from the following relational expression:

[0012] In a fifth aspect of the present invention, in the step (b), when a generalized inverse matrix l is used to calculate the local curvature at one point with respect to z, ｐ and the two-dimensional representation of l ２ｄ The method for calculating the curvature of an arbitrary range of an optical element according to the fourth aspect, wherein FL is calculated without explicitly constructing L by the convolution operation of

[0013] A sixth aspect of the present invention is the method for calculating a curvature of an arbitrary range of an optical element according to the fourth aspect, wherein the weighting vector w is expressed by a rotationally symmetric polynomial including an odd degree.

[0014] A seventh aspect of the present invention is a method for manufacturing a light beam having a periodic microstructure, wherein the step (b) comprises: ｉ is changed within the periodic range of the fine structure, and the central position O ｉ The weight vector w corresponding to ｉ The method for calculating the curvature of an arbitrary range of an optical element according to the first aspect above, further comprising the steps of: calculating an average macroscopic curvature c' using a composite weighting vector w' obtained by adding up the above weighting vectors over the periodic range of the microstructure;

[0015] An eighth aspect of the present invention is the method for calculating a curvature of an arbitrary range of an optical element according to the first aspect, wherein in the step (a), the number of elements n of the horizontal vector s is set to 10,000 or more.

[0016] A ninth aspect of the present invention is the method for calculating a curvature of an arbitrary range of an optical element according to the fourth aspect, wherein in the step (b), the degree of freedom k of the weighting vector w is set to 1 or more and 20 or less.

[0017] In a tenth aspect of the present invention, in the step (a), a minute beam diameter φ when calculating the local curvature is １ is the diameter φ of the beam of light of the macro curvature c. ２ The method for calculating the curvature of an optical element in an arbitrary range according to the first aspect is to set the curvature to 0.7 times or less.

[0018] An eleventh aspect of the present invention provides a measurement unit that measures a local curvature distribution of n data points in an arbitrary range of an optical element, and a method for measuring a local curvature distribution of n data points in an arbitrary range of an optical element, the method comprising: Ｔ a calculation unit that calculates the macro curvature c without using wavefront information within the predetermined range, by using a weighting vector w such that the following approximate expression holds true; and

[0019] A twelfth aspect of the present invention is a curvature calculation device for an arbitrary range of an optical element according to the eleventh aspect, further comprising a recording unit that records the weighting vector w calculated in advance or polynomial coefficients for expressing the weighting vector w, and the calculation unit calculates the macrocurvature c from the recorded weighting vector w and the measured local curvature distribution.

[0020] A thirteenth aspect of the present invention is a computer-readable recording medium having recorded thereon a program for causing a computer to execute the curvature calculation method according to any one of the first to tenth aspects.

[0021] According to one embodiment of the present invention, it is possible to calculate the macro-curvature from the local curvature distribution.

[0022] FIG. 1 is a diagram showing an example of the wavefront and local curvature distribution of a spectacle lens that has a myopia progression suppression effect. FIG. 2 is a histogram of the local curvature distribution shown in FIG. 1. FIG. 3 is a diagram showing an example of the wavefront and local curvature distribution of a contact lens that has a myopia progression suppression effect. FIG. 4 is a histogram of the local curvature distribution shown in FIG. 3. FIG. 5 is a diagram showing a case in which only the wavefront gradient of the outer periphery of a light beam is considered, and the wavefront within the light beam is completely ignored. FIG. 6 is a diagram showing a case in which the wavefront of the outer periphery of a light beam has a gradient. FIG. 7 is a diagram showing a filter for calculating the local curvature distribution. FIG. 8 is a diagram showing a specific example of a weighting vector. FIG. 9 is a diagram showing an example of the wavefront of a spectacle lens that has a myopia progression suppression effect. FIG. 10 is a diagram showing the results of calculating the macro curvature while changing the center of the light beam. FIG. 11 is a diagram showing the local curvature distribution of a spectacle lens that has a myopia progression suppression effect, and a composite weighting vector. FIG. 12 is a profile showing the weighting function a. Fig. 13 is a diagram showing an example in which the weighting vector w oscillates in the peripheral area. Fig. 14 is a diagram showing the wavefront and local curvature distribution of samples 1 to 7 according to an embodiment of the present invention. Fig. 15 is a diagram showing the calculation results of the macro-curvature according to an embodiment of the present invention.

[0023] <Insights Gained by the Inventor> First, the insights gained by the inventor will be explained. Measuring power distribution is equivalent to measuring the local curvature distribution of a wavefront, and it is difficult to accurately determine the macrocurvature of a macro wavefront. For this reason, statistical values ​​of the local curvature distribution are often used instead. One method is the arithmetic mean of the local curvature distribution. Considering that the local curvature is constant in the case of a perfect spherical surface, this seems reasonable at first glance. Another method is to take the arithmetic mean of only frequently occurring data. This can be considered an improvement over the above method, as it reduces the influence of outliers. However, the inadequacy of these methods is demonstrated in the following examples.

[0024] FIG. 1 is a diagram showing an example of the wavefront and local curvature distribution of a spectacle lens that has the effect of inhibiting the progression of myopia. For details of spectacle lenses that have the effect of inhibiting the progression of myopia, see, for example, U.S. Patent Application Publication No. 2017 / 0131567. Here, an example is shown in which convex regions (defocus regions) with a defocus power of +4D and a diameter of 0.8 mm are provided at 1.2 mm intervals on a surface with a prescription power of 0D. The plot in the upper left of FIG. 1 shows the wavefront, and the graph in the upper right of the page shows a cross-sectional view of the wavefront in the x direction (horizontal direction) passing through the center. Furthermore, the plot in the lower left of FIG. 1 shows the local curvature distribution (local curvature calculation diameter φ １ = 0.5 mm), and the graph at the bottom right of the page is a cross-sectional view of the local curvature in the x direction (horizontal direction) passing through the center.

[0025] In the example shown in FIG. 1, the beam diameter φ ２ When the mean spherical power in the range of φ = 4 mm was defined as macrocurvature c, the macrocurvature c was calculated from the wavefront and was found to be -0.01 D. Figure 2 is a histogram of the local curvature distribution shown in Figure 1. The arithmetic mean of this histogram was -0.2 D, and the mean using only the most frequently occurring data was -1.8 D. Therefore, the average value calculated from the local curvature distribution deviates from the macrocurvature c calculated from the wavefront, and the deviation is particularly large when only the most frequently occurring data is used.

[0026] FIG. 3 is a diagram showing an example of the wavefront and local curvature distribution of a contact lens that has the effect of inhibiting the progression of myopia. ２This shows an example in which the local power at a point 1.25 mm from the lens center is set to be +3D defocused with respect to the lens center power in a lens with a 0D (average power per 4 mm). The plot in the upper left of FIG. 3 shows the wavefront, and the graph in the upper right of the page shows a cross section of the wavefront in the x direction (horizontal direction) passing through the center. The plot in the lower left of FIG. 3 shows the local curvature distribution (local curvature calculation diameter φ １ = 0.5 mm), and the graph at the bottom right of the page is a cross-sectional view of the local curvature in the x direction (horizontal direction) passing through the center.

[0027] In the example shown in FIG. 3, the beam diameter φ ２ In the range of θ = 4 mm, the macrocurvature c calculated from the wavefront is 0 D. Figure 4 is a histogram of the local curvature distribution shown in Figure 3. The arithmetic mean of this histogram is +0.4 D, and the mean calculated using only the most frequently occurring data is +1.0 D. Therefore, the mean value calculated from the local curvature distribution deviates from the macrocurvature c calculated from the wavefront, and the deviation is particularly large when using only the most frequently occurring data.

[0028] Next, we will consider the case where the local curvature distribution is given as a continuous function. Let w(x, y) be the unknown wavefront at coordinates (x, y) on the lens. Hereinafter, we will assume that the coordinates (x, y) are continuous values, the local curvature calculation radius is sufficiently small, and the local curvature s(x, y) can be expressed as the second-order derivative of the wavefront as shown in the following equation.

[0029]

[0030] Here, the macroscopic curvature c is calculated by the simple average of the local curvatures. The average local curvature s (overline) for a light beam with a diameter φ and a radius u (corresponding to the pupil diameter) can be expressed by the following equation:

[0031]

[0032] According to Green's theorem, the surface integral of the second derivative of x can be replaced by a line integral. The same applies to the second derivative of y. Therefore, the average local curvature s (overline) can be expressed as follows:

[0033]

[0034] From the integral range of the above formula, it can be seen that the average local curvature s (overline) is determined only by the wavefront gradient at the outer periphery of the light beam. Therefore, as shown in FIG. 5, only the wavefront gradient at the outer periphery of the light beam is considered, and the wavefront within the light beam is completely ignored. Therefore, if the outer periphery of the light beam has a discontinuity or a similar structure, the value of the macroscopic curvature c will fluctuate significantly. For example, as shown in FIG. 6, in cases where the wavefront at the outer periphery of the light beam has a gradient, such as when the lens has spherical aberration (left side of FIG. 6) or when the light beam is subjected to a discontinuity (right side of FIG. 6), the simple average of the local curvature may overestimate the power.

[0035] The inventors have conducted extensive research into the above-mentioned problems. As a result, they have discovered a new calculation method that can calculate the macrocurvature c from the local curvature distribution without using wavefront information, even when the wavefront information is unknown. The present invention can calculate a macrocurvature that approximates a value determined from a wavefront, even for lenses with large aberrations, such as spectacle lenses that have the effect of inhibiting the progression of myopia, and is therefore particularly useful for quality control of such optical elements.

[0036] [Details of the embodiment of the present invention] Next, an embodiment of the present invention will be described below with reference to the drawings. Note that the present invention is not limited to these examples, but is defined by the claims, and is intended to include all modifications within the meaning and scope of the claims.

[0037] <First embodiment of the present invention> (1) Method for calculating curvature of an arbitrary range of an optical element A method for calculating curvature of an arbitrary range of an optical element according to this embodiment will be described. The method for calculating curvature of an arbitrary range of an optical element according to this embodiment includes, for example, a local curvature distribution acquisition step S100 and a macro curvature calculation step S200. In this embodiment, in an arbitrary range of an optical element such as a lens, when wavefront information is unknown, a predetermined luminous flux range (for example, luminous flux diameter φ) within the arbitrary range is calculated without using wavefront information. ２ A case where the macro curvature c, which is the mean spherical power at a point where the sphere is at a focal length of 4 mm (=4 mm), is calculated will be described.

[0038] (Local curvature distribution acquisition step S100) The local curvature distribution acquisition step S100 is a step of obtaining a row vector s with n elements representing the local curvature distribution, for example, in an arbitrary range of the optical element. The row vector s is a one-dimensional rearrangement of the local curvature distribution on the curved surface. The data of the local curvature distribution may be measured using a power distribution measurement device, or data prepared in advance may be used. In this embodiment, the data of the local curvature distribution is obtained by arranging n power (local curvature) values ​​in a row in a row, and the (x, y) coordinates indicating the measurement position in an arbitrary range of the optical element, and a row vector x=(x １ , x ２ …x ｎ ), y=(y １ , y ２ …y ｎ ), s=(s １ , s ２ …s ｎ ) In the following description, unless otherwise specified, a vector refers to a horizontal vector.

[0039] In the local curvature distribution acquisition step S100, it is preferable that the number of elements n of s is set to 10000 or more, which makes it possible to calculate a more accurate macro curvature c.

[0040] In the local curvature distribution acquisition step S100, the diameter φ of the minute beam of light when calculating the local curvature is １ is the diameter φ of the light beam with a macroscopic curvature c. ２ It is preferable that the value is 0.7 times or less. This makes it possible to calculate the macroscopic curvature c more accurately.

[0041] Before describing a method for calculating the macroscopic curvature c from the local curvature distribution, a method for calculating the macroscopic curvature c when wavefront information (for example, a quantified wavefront or wavefront aberration distribution) in an arbitrary range of the optical element is known will be described. A horizontal vector with n elements representing the wavefront aberration distribution in an arbitrary range of the optical element is defined as z=(z １ , z ２ …z ｎ ) Also, c = mz ＴLet m be the horizontal vector for calculating the macroscopic curvature c from the wavefront aberration distribution, which satisfies the following relationship: m corresponds to, for example, the generalized inverse matrix of spherical approximation by the least squares method (which is actually a horizontal vector because it is a 1 × n matrix), or a vectorized version of the defocus term of the normalized Zernike polynomial. Therefore, if the wavefront aberration distribution is known, the macroscopic curvature c can be easily calculated from z.

[0042] Next, a method for constructing m will be described in more detail. The central coordinate of the light beam is o=(o ｘ , o ｙ ), and the luminous flux radius is R, each element of vector u with the number of elements n, which expresses the case where the measurement point is included in the luminous flux range as 1 and the case where it is not included as 0, can be expressed by the following equation.

[0043]

[0044] Furthermore, each element of vector q, which has n elements that are quadratic within the luminous flux range and 0 outside the luminous flux range, can be expressed by the following equation using u.

[0045]

[0046] m can be constructed by orthogonalizing and normalizing this q. Orthogonalization is an operation for making the calculation result when calculating the macroscopic curvature c independent of the constant term (zeroth-order term) and tilt (first-order term) of the entire wavefront. For example, a vector q' with n elements obtained by orthogonalizing q with respect to the zeroth order can be expressed by the following equation. Note that in the case of a quadratic function, it is often automatically orthogonal to the tilt, but orthogonalization may be performed as necessary.

[0047]

[0048] Furthermore, normalization is an operation of dividing in advance the scalar quantity that is the denominator of c. Therefore, m can be expressed by the following equation using q'.

[0049]

[0050] (Macro-curvature calculation step S200) The macro-curvature calculation step S200 is a step of calculating the macro-curvature c without using wavefront information when, for example, wavefront information of an arbitrary range of the optical element is unknown. Below, a method of calculating the macro-curvature c from only s, which represents the local curvature distribution, without using z, which represents the wavefront aberration distribution, will be described.

[0051] The inventors have determined that c≒ws Ｔ In other words, the macroscopic curvature c can be obtained from the weighted sum of s, which represents the local curvature distribution, using the weighted vector w.

[0052] The sum of the elements of the weighting vector w is preferably 1. This is because, for example, in the case of a perfect spherical wave (with constant curvature), the local curvature of all measurement points is equal to c. Furthermore, from the viewpoint of assigning stronger weights toward the center, it is preferable that the cross section passing through the center of the two-dimensional plot of the weighting vector w has an upward convex shape, and it is more preferable that the maximum value is at the center. Note that the two-dimensional plot referred to here refers to a horizontal vector rearranged to match the two-dimensional coordinates representing the position on the curved surface.

[0053] A more detailed description will be given below of a method for calculating the weighting vector w. By using the following method, a more appropriate weighting vector w can be obtained, making it possible to calculate the macroscopic curvature c more accurately.

[0054] (Method 1) A measurement device (power distribution measurement device) for obtaining a local curvature distribution can be considered as a linear system that outputs the wavefront aberration distribution (unknown to the measurer) that an optical element gives to a light beam as a local curvature distribution. In other words, when an n×n matrix L is introduced, the power distribution measurement device can be expressed as s Ｔ = Lz Ｔ It can be regarded as a linear system that performs the following operations.

[0055] More specifically, the local curvature distribution can be said to be a local quadratic approximation of the wavefront aberration distribution at each point. For example, the power s ｉ is the central coordinate of the minute light beam oｉ = (o ｉｘ , o ｉｙ ), minute beam radius r ｉ If we assume that the wavefront is obtained from a quadratic function approximation of the wavefront, then the generalized inverse matrix l (actually a horizontal vector) of the spherical approximation at that point can be constructed in the same manner as when constructing m above. In other words, l can be said to be a generalized inverse matrix that calculates the local curvature at one point with respect to z. By preparing this for all n points, the n x n matrix L can be constructed.

[0056] Substantially, the minute beam radius r ｉ is constant, and each point is the central coordinate of the infinitesimal light beam o ｉ = (o ｉｘ , o ｉｙ In this case, the generalized inverse matrix l is converted into a filter l, which is a two-dimensional array of row vectors with n elements, as shown in FIG. ２ｄ Then, z is rearranged into two dimensions as z ２ｄ Ni ２ｄ The plot on the left side of the page in Figure 7 shows l ２ｄ The plot on the right side of the page is a two-dimensional plot of ２ｄ However, care must be taken when performing quadratic approximation across both ends of the measurement region, as this will result in an incorrect value for the local curvature.

[0057] Note that if the power distribution measurement device uses a microbeam that is not circular (for example, rectangular), l and L can be configured to match the conditions. Also, if the power distribution measurement device does not use quadratic function approximation (for example, if the device is configured to internally obtain measurements as local gradients in each direction of the wavefront and obtain a power distribution chart from the difference between these values), the operation of taking the difference between linear function approximations between two points can be expressed by l and L. Hereinafter, the range in which l has a value will be treated as the microbeam range.

[0058] As described above, the power distribution measuring device is Ｔ = Lz Ｔ However, when the range of the minute beam (for example, the diameter of the minute beam φ) is １Since one point of s is calculated from z (=0.5 mm), the number of s points is less than the number of z measurement points. Therefore, the equation m = wL cannot be solved, and in reality, an approximate solution is sought that minimizes the absolute value of |wL - m|. Therefore, by using Lagrange's method of undetermined multipliers, wLL Ｔ = mL Ｔ It is sufficient to calculate w that satisfies the following relational expression.

[0059] By calculating the weighting vector w using Method 1, the macro-curvature c can be accurately calculated from s, which represents the local curvature distribution. Ｔ Since is an n × n matrix, the order of computational complexity of Method 1 is O(n ６ Specifically, assuming n = 10,000, the order of computational complexity for Method 1 is O(10 ２４ ) is extremely large. In addition, storing and accessing the matrix L having 100,000,000 elements on a recording medium also imposes a burden.

[0060] (Method 2) Method 2 is a method that reduces the amount of calculation compared to Method 1 by expressing the weighting vector w as a rotationally symmetric polynomial with k degrees of freedom. The degrees of freedom k correspond to the number of independent coefficients in the polynomial, and is, for example, preferably between 1 and 20, and more preferably between 2 and 10. This makes it possible to accurately calculate the macroscopic curvature c while reducing the amount of calculation.

[0061] Here, when measuring the local curvature at the outer periphery of the light beam range, a part of the minute light beam range may extend outside the light beam range, and in order to avoid including information outside the light beam range, the range in which the power (local curvature) is determined only from the information of the light beam range on the optical element is referred to as the calculation range. In other words, the calculation range is defined as the range in which the diameter of the light beam range is calculated using the minute light beam diameter φ １ The diameter is φ ２ -φ １ Then, a horizontal vector f having n elements and the sum of all elements being 0, whose values ​​within the calculation range are determined by the position coordinates of each measurement point of a p-th order rotationally symmetric function and whose values ​​outside the calculation range are 0, is ｐLet F be a matrix in which k degrees of freedom are arranged vertically. Also, let a be the coefficient vector of the rotationally symmetric polynomial, and let e be the horizontal vector with n elements whose values ​​are 1 within the calculation range and 0 outside the calculation range. Let a be the inverse of the number of measurement points included in the calculation range (i.e., the number of elements in the horizontal vector e whose value is 1). ０ In this case, m = wL = (aF + a ０ e) The relational expression for L is obtained. That is, |(aF+a ０ e) An approximate solution is sought that minimizes the absolute value of L-m|. Therefore, by the Lagrange multiplier method, a(FL)(FL) Ｔ = (m-a ０ eL) (FL) Ｔ w can be calculated from the following relational expression.

[0062] By calculating the weighting vector w using Method 2, the macro-curvature c can be accurately calculated from s, which represents the local curvature distribution. In addition, since FL is the multiplication of a k×n matrix and an n×n matrix, the order of calculations in Method 2 is O(kn ２ ), which reduces the amount of calculation. Specifically, assuming n = 10000 and k = 10, the order of calculation amount for Method 2 is O(10 ９ ), which reduces the amount of calculation compared to Method 1.

[0063] In Method 2, in order to reduce the cost of storing L in a recording medium, it is preferable to use a method that does not explicitly configure L. Specifically, the two-dimensional f ｐ and the two-dimensional representation of l ２ｄ It is preferable to calculate FL by the convolution operation without explicitly configuring L. In the convolution operation, the amount of calculation can be reduced to O(kn log n) because the amount of calculation can be reduced by the Fourier transform. It is also preferable to calculate eL by the convolution operation in the same way.

[0064] In Method 2, it is preferable to express the weighting vector w as a rotationally symmetric polynomial including odd degrees, which can further reduce the amount of calculation.

[0065] As a specific example, the luminous flux diameter φ ２= 4 mm, local curvature calculation diameter (minimal luminous flux diameter) φ １ The results of calculating the weighting vector w under the conditions of = 0.5 mm, measurement pitch 0.05 mm (400 measurement points in 1 mm square), and weighting vector w being a polynomial with degrees of freedom k = 7 (2nd to 8th order) including odd orders are shown in Figure 8. The diameter of the calculation range, i.e., the range where w has a value other than 0, is φ ２ From φ １ The left side of FIG. 8 shows a two-dimensional plot of the weighting vector w, and the right side of FIG. 8 shows a plot of a cross section passing through the center. ０ ~a ８ The values ​​are as shown in the table below.

[0066]

[0067] (Method 3) Method 3 is an effective method when the optical element has a periodic fine structure (for example, a spectacle lens that has the effect of suppressing the progression of myopia). First, the problems that arise when the optical element has a periodic fine structure will be described.

[0068] 9 is a diagram showing an example of the wavefront of a spectacle lens that has the effect of inhibiting the progression of myopia. Here, an example is shown in which convex regions (defocus regions) with a defocus power of +4D and a diameter of 0.8 mm are provided at 1.2 mm intervals on a surface with a prescription power of 0D. When an optical element has a periodic microstructure like this, even a slight change in the measurement position (center position of the light beam) will also cause the calculated macroscopic curvature c to change (vibrate). For example, at point O in FIG. １ Point O shifted 1.03 mm upward on the paper surface from ２ While changing the beam center, the beam diameter φ ２ Figure 10 shows the results of calculating the macroscopic curvature c in the range of θ = 6 mm. Note that in Figure 10, the macroscopic curvature c was calculated from the wavefront. As shown in Figure 10, the macroscopic curvature c fluctuates over a range of about 0.04 D (it changes in a mirror-symmetric manner after the 1.03 mm point). When measuring power distribution, it is difficult to align the measurement position, so the effects of vibration are easily apparent. Therefore, a robust evaluation method that eliminates the effects of vibration is required. Furthermore, in terms of quality control of such optical elements, values ​​that eliminate the effects of vibration are important.

[0069] To eliminate the influence of vibration, it is sufficient to repeatedly calculate the macroscopic curvature within the periodic range of the microstructure and add them up (since light beams do not interact with each other, simple addition is possible). However, if the macroscopic curvature is repeatedly calculated, the amount of calculation increases. Therefore, it is necessary to calculate the center position O of the light beam range. ｉ is changed within the periodic range of the fine structure, and the central position O ｉ The weight vector w corresponding to ｉ It is preferable to prepare a composite weighting vector w' by adding up the weights w' over the periodic range of the microstructure, and calculate the average macroscopic curvature c' by taking the weighted sum of w' with respect to s. This makes it possible to calculate the average macroscopic curvature c' in a single calculation.

[0070] A specific example is shown in Figure 11. The left side of Figure 11 is a plot showing the local curvature distribution of a spectacle lens that has the effect of suppressing the progression of myopia. The area surrounded by the dashed line in this plot (0 mm ≤ x ≤ 0.6 mm, 0 mm ≤ y ≤ 1.03 mm) is set as the periodic range, and the central positions O of 400 points divided into 20 in each of the x and y directions are １ …O ４００ and then assign a weighting vector w corresponding to each position by the above-mentioned method 1 or method 2. ｉ This weighting vector w ｉ The composite weight vector w' can be obtained by adding them up using the following formula: A two-dimensional plot of the composite weight vector w' is shown on the right side of the page in FIG.

[0071]

[0072] Using the composite weight vector w', c' = w's Ｔ The average macroscopic curvature c' can be calculated by the following formula: By using this method, even when the optical element has a periodic microstructure, the average macroscopic curvature c' can be easily calculated while eliminating the influence of vibration.

[0073] By applying this method, it is possible to calculate the magnitude of vibration (the difference in macroscopic curvature). In addition, since it can be applied to any linear calculation, not just averages and differences, it is also possible to evaluate the amplitude and period of macroscopic curvature vibration in the same way as a discrete Fourier transform.

[0074] (Method 4) Method 4 is a method for obtaining a weighting vector w from a continuous weighting function a when the local curvature distribution is given as a continuous function or when the number of data points for the local curvature distribution is very large. The method for obtaining the weighting function a will be described below.

[0075] Macro curvature s ｍａｃｒｏ can be expressed as a weighted average using a weighting function a(x, y) as shown in the following equation: where s(x, y) is the local curvature distribution, w(x, y) is the unknown wavefront, and u is the radius of the light beam.

[0076]

[0077] The conditions required for the weighting function a will be considered below.

[0078] (Condition 1) Since the weighting function a is a function for taking an average, the first condition is that the sum is 1, as shown in the following equation.

[0079]

[0080] (Condition 2) Macro curvature s ｍａｃｒｏ is assumed to be equal to the result of spherical approximation (hereinafter replaced by orthogonal quadratic approximation) of the unknown wavefront w(x, y) as shown in the following equation.

[0081]

[0082] Calculating the denominator of the above equation, we obtain the following equation:

[0083]

[0084] Also, as shown in the following equation, it is assumed that the integral of the product of the second derivative of the weighting function a and the wavefront w is equal to the integral of the product of the second derivative of the wavefront w and the weighting function a.

[0085]

[0086] From the above two equations, the second condition for the weighting function a is to satisfy the following equation:

[0087]

[0088] (Condition 3) The equation stated in Condition 2, where the integral of the product of the second derivative of the weighting function a and the wavefront w is equal to the integral of the product of the second derivative of the wavefront w and the weighting function a, is integrated partially with respect to x (the integral range is x ０ From x １ ) and rearrange to obtain the following equation:

[0089]

[0090] In order for the above formula to hold for all wavefronts w, the following formula should be satisfied.

[0091]

[0092] The same is true for y, so when expanded to two variables, the third condition for the weighting function a is that the value and gradient of a are 0 on the circumference of the light beam, as shown in the following equation.

[0093]

[0094] The weighting function a that satisfies the above three conditions is expressed by the following formula: Furthermore, Fig. 12 shows the profile of the weighting function a.

[0095]

[0096] In this method, by using such a continuous weighting function a, it is possible to reduce the calculation load even when, for example, the number of data points of the local curvature distribution is extremely large.

[0097] By using the calculation method described above, even if the wavefront information is unknown, the macro curvature c can be calculated from the local curvature distribution in any range of the optical element without using the wavefront information.

[0098] Depending on the calculation conditions, the weighting vector w may oscillate in the peripheral areas, as shown in Figure 13. This is likely to occur when the luminous flux range cannot be drawn smoothly due to discrete measurement data, resulting in a jagged appearance, or when the degree of the polynomial for the weighting vector w is selected inappropriately. When the weighting vector w oscillates, the calculation of the macroscopic curvature c is likely to become unstable. In such cases, it is preferable to reduce the weighting of the peripheral areas of the luminous flux range when creating the generalized inverse matrix m of the spherical approximation, or to use an odd polynomial, a Gaussian function, or the like, in combination with a high-order polynomial, rather than a high-order polynomial.

[0099] In the macro-curvature calculation step S200, for example, the macro-curvature c may be calculated using a weighting vector w calculated in advance. In this case, it is not necessary to calculate the weighting vector w each time, and the macro-curvature c can be calculated easily. Alternatively, a plurality of weighting vectors w calculated in advance may be prepared, and an appropriate weighting vector w may be selected in accordance with the measurement conditions.

[0100] (2) Curvature calculation device for an arbitrary range of an optical element The present invention can also be applied as a curvature calculation device for an arbitrary range of an optical element. The curvature calculation device for an arbitrary range of an optical element of this embodiment includes, for example, a measurement unit that measures a local curvature distribution of n data points in an arbitrary range of an optical element, and a calculation unit that calculates ... Ｔ and a calculation unit that calculates the macroscopic curvature c without using wavefront information when wavefront information within a predetermined range is unknown, using a weighting vector w such that the following approximation holds. A known power distribution measurement device can be used as the measurement unit. The calculation of the macroscopic curvature c by the calculation unit is as described in (1) Method for calculating the curvature of an arbitrary range of an optical element.

[0101] The curvature calculation device for an arbitrary range of an optical element according to this embodiment may further include, for example, a recording unit that records a pre-calculated weighting vector w or a polynomial coefficient for expressing the weighting vector w, and the calculation unit may calculate the macro-curvature c from the recorded weighting vector w and the measured local curvature distribution. In this case, the calculation unit does not need to calculate the weighting vector w each time, and therefore the macro-curvature c can be calculated easily.

[0102] (3) Recording Medium The present invention can also be applied as a computer-readable recording medium that records a program that causes a computer to execute the curvature calculation method described in (1) Method for Calculating Curvature of an Arbitrary Range of an Optical Element.

[0103] <Other embodiments of the present invention> Although the embodiments of the present invention have been specifically described above, the present invention is not limited to the above-described embodiments and can be modified in various ways without departing from the spirit of the present invention.

[0104] For example, in the above-described embodiment, the case of calculating macro curvature from a local curvature distribution was described, but the present invention also makes it possible to calculate macro x-direction power from a local x-direction power distribution, or to obtain macro astigmatism amount from a local astigmatism distribution, using similar procedures.

[0105] For example, in the above embodiment, the curvature of a circular area corresponding to a light beam is calculated, but the method can also be applied to an area with a missing part of a circle or a rectangular area. However, in this case, symmetry is reduced, and therefore the amount of calculation increases, such as when using a polynomial for the weighting vector w in Method 2, the number of parameters required increases, or when optimizing the weighting vector w without using a polynomial.

[0106] In the above embodiment, the weighting vector w (or the composite weighting vector w') is calculated by methods 1 to 4, but the methods for obtaining the weighting vector w are not limited to these. For example, it is possible to obtain a weighting vector w by machine learning or the like, where c≒ws Ｔ Alternatively, w may be directly calculated so that the following approximate expression holds.

[0107] Next, examples of the present invention will be described. These examples are merely examples of the present invention, and the present invention is not limited to these examples.

[0108] In this example, wavefront aberration distribution data for Samples 1 to 7 was created, and the macroscopic curvature values ​​obtained by the following calculation methods (Calculation Methods 1 to 4) were compared. The common calculation condition was the beam diameter φ ２ = 2mm, local curvature calculation diameter (minimal luminous flux diameter) φ １ = 0.5 mm, and the measurement pitch was 0.005 mm (40,000 measurement points per 1 mm square).

[0109] FIG. 14 shows the wavefronts (above the page) and local curvature distributions (below the page) of Samples 1 to 7. In FIG. 14, Sample 1 is arranged from the left on the page, followed by Sample 2. As shown in FIG. 14, Samples 1 to 7 had sinusoidally undulating wavefronts and corresponding frequency distributions. The waviness period of Sample 1 was 0.6 mm, the waviness period of Sample 2 was 1.2 mm, the waviness period of Sample 3 was 1.9 mm, the waviness period of Sample 4 was 2.6 mm, the waviness period of Sample 5 was 3.2 mm, the waviness period of Sample 6 was 3.8 mm, and the waviness period of Sample 7 was 4.5 mm. Note that for Samples 1 to 7, the amplitude was normalized so that the local curvature in a small area in the center was −2D.

[0110] In calculation method 1, the macrocurvature was calculated by spherical approximation of the wavefront aberration distribution. In calculation method 2, the macrocurvature was calculated by taking the average of the local curvatures within the range. In calculation method 3, the macrocurvature was calculated by taking the average of the top 5% of the most frequent values ​​in a histogram of the local curvatures within the range. In calculation method 4, the macrocurvature was calculated by technique 2 described in the above embodiment. In calculation method 4, the weighting vector w was a polynomial with k=7 degrees of freedom (2nd to 8th order), including odd orders. Naturally, except for calculation method 1, calculations were performed when only the local curvature distribution was known and the wavefront aberration distribution was unknown.

[0111] Fig. 15 shows the results of calculating the macro curvature of samples 1 to 7 using calculation methods 1 to 4. As shown in Fig. 15, calculation method 4, which uses technique 2 described in the above embodiment, produced values ​​that were almost identical to those of calculation method 1, which is a spherical approximation of the wavefront. On the other hand, calculation method 2, which took the average value of the local curvature, and calculation method 3, which took the average of the most frequent values ​​of the local curvature, showed significant deviations from calculation method 1.

[0112] From the above, it has been confirmed that by using an appropriate weighting vector w, it is possible to accurately calculate the macroscopic curvature without using wavefront information, even when the wavefront information is unknown.

[0113] S100: Local curvature distribution acquisition step S200: Macro curvature calculation step

Claims

1. Step (a) of obtaining a horizontal vector s with n elements representing a local curvature distribution in an arbitrary range of an optical element; and, when the mean spherical power in a predetermined light flux range within the arbitrary range is defined as c, c ≒ ws Ｔ and (b) calculating the macroscopic curvature c without using wavefront information in the arbitrary range, using a weighting vector w such that the following approximate formula holds.

2. A method for calculating the curvature of an arbitrary range of an optical element according to claim 1, wherein the sum of the elements of the weighting vector w is 1, and a cross section passing through the center of a two-dimensional plot of the weighting vector w has an upwardly convex shape.

3. In the step (b), the unknown wavefront aberration distribution in the arbitrary range is defined as a horizontal vector z having n elements, and c=mz Ｔ The horizontal vector for calculating the macroscopic curvature c from the wavefront aberration distribution is defined as m, and the measurement device for obtaining the local curvature distribution is defined as s Ｔ = Lz Ｔ It is considered as a linear system that performs the operation wLL Ｔ = mL Ｔ 2. The method for calculating the curvature of an arbitrary range of an optical element according to claim 1, wherein the weighting vector w is calculated from the following relational expression:

4. In the step (b), the weighting vector w is expressed by a rotationally symmetric polynomial with k degrees of freedom, and the unknown wavefront aberration distribution in the arbitrary range is a horizontal vector z with n elements, and c=mz Ｔ The horizontal vector for calculating the macroscopic curvature c from the wavefront aberration distribution is defined as m, and the measurement device for obtaining the local curvature distribution is defined as s Ｔ = Lz Ｔ The diameter of the light beam range is considered as the small light beam diameter φ when calculating the local curvature. １ The calculation range is a range reduced by the value of ｐ Let F be a matrix in which k degrees of freedom are arranged vertically, let a coefficient vector of the rotationally symmetric polynomial be a row vector a with k elements, let e be a row vector with n elements whose values ​​are 1 within the calculation range and 0 outside the calculation range, and let a be the reciprocal of the number of measurement points included in the calculation range. ０ Let a(FL)(FL) Ｔ = (m-a ０ eL) (FL) Ｔ 2. The method for calculating the curvature of an arbitrary range of an optical element according to claim 1, wherein the weighting vector w is calculated from the following relational expression:

5. In the step (b), when the generalized inverse matrix l is used to calculate the local curvature at one point with respect to z, the two-dimensional f ｐ and the two-dimensional representation of l ２ｄ 5. The method for calculating the curvature of an arbitrary range of an optical element according to claim 4, wherein FL is calculated without explicitly constructing L by a convolution operation of 6. The method for calculating the curvature of an arbitrary range of an optical element according to claim 4, wherein the weighting vector w is expressed by a rotationally symmetric polynomial including odd degrees.

7. The optical element has a periodic microstructure, and in step (b), the central position O of the light beam range is ｉ is changed within the periodic range of the fine structure, and the central position O ｉ The weight vector w corresponding to ｉ 2. The method for calculating the curvature of an arbitrary range of an optical element according to claim 1, wherein an average macroscopic curvature c' is calculated using a composite weighting vector w' obtained by adding up the above weighting vectors over the periodic range of the microstructure.

8. The method for calculating the curvature of an arbitrary range of an optical element according to claim 1, wherein in step (a), the number of elements n of the horizontal vector s is set to 10,000 or more.

9. The method for calculating the curvature of an arbitrary range of an optical element according to claim 4, wherein in step (b), the degree of freedom k of the weighting vector w is set to 1 or more and 20 or less.

10. In the step (a), the small beam diameter φ １ is the diameter φ of the beam of light of the macro curvature c. ２ 2. The method for calculating the curvature of an optical element in an arbitrary range according to claim 1, wherein the curvature is set to 0.7 times or less.

11. A measurement unit that measures a local curvature distribution of n data points in an arbitrary range of an optical element, and when the local curvature distribution is a horizontal vector s with n elements and the mean spherical power in a predetermined light flux range within the arbitrary range is c, then c ≒ ws Ｔ a calculation unit that calculates the macroscopic curvature c without using wavefront information within the predetermined range, using a weighting vector w such that the following approximate expression holds true; 12. A device for calculating the curvature of an arbitrary range of an optical element according to claim 11, further comprising a recording unit that records the weighting vector w calculated in advance or polynomial coefficients for expressing the weighting vector w, and the calculation unit calculates the macro-curvature c from the recorded weighting vector w and the measured local curvature distribution.

13. A computer-readable recording medium having recorded thereon a program for causing a computer to execute the method for calculating the curvature of an arbitrary range of an optical element according to any one of claims 1 to 10.

Images