Function curved lens for accurately focusing parallel light beams through secondary refraction

Abstract

The invention relates to a function curved-surface lens which accurately focuses parallel light beams through secondary refraction, and is also called a C-shaped lens, the C-shaped lens is a function curved-surface lens with two non-spherical curved surfaces, and compared with a function curved-surface lens given in a previous invention patent (patent number: 201811072337.8), the C-shaped lens not only can realize accurate focusing, but also can realize accurate focusing, and can realize accurate focusing. And the C-shaped lens has higher efficiency and shorter focal length, so that the C-shaped lens is particularly suitable for manufacturing a large solar condensing lens, and the C-shaped lens can also be used in an optical instrument which needs a shorter focal length and is convenient to carry. The tangent slope of any point on the two function curved surfaces of the C-type lens can be calculated by a YZQ tangent formula, and coordinate values of a series of points on the two function curved surface curves of the C-type lens can be accurately calculated by applying a calculation method called a micro-decomposition equation calculation method. If the coordinate values exist, the C-shaped lens can be conveniently and accurately machined by utilizing a numerical control machine tool.

Description

Technical Field

[0001] The present invention relates to a novel aspherical optical lens, both of the two curved surfaces of which are functional surfaces, and this lens can accurately focus a parallel light beam after two refractions. Background Art

[0002] The present invention is relatively close in features to the invention patent with the patent number 201811072337.8 and the patent name "A Functional Surface Lens Capable of Accurate Focusing". They both belong to functional surface lenses. In this article, the patent invention with the patent number 201811072337.8 will be simply referred to as the "previous invention".

[0003] As used in this specification, the "parallel light beam" refers to a beam composed of parallel light rays to each other, and the "focused beam" as used in this specification refers to a beam emitted from a certain position and directed towards a certain focus.

[0004] In the specification of the previous invention, three types of functional surface lenses, namely type A, type B, and type AA, are given. Among them, only type A and type B functional surface lenses are relatively close to the present invention. For the convenience of description, the type A functional surface lens and the type B functional surface lens in the previous invention are respectively simply referred to as type A lens and type B lens, and the functional surface lens given by the present invention is simply referred to as type C lens. When using a type A lens to make a large solar concentrator lens or other optical instruments, a problem often encountered is how to minimize the focal length to the greatest extent, because the shorter the focal length, the lower the cost of the entire optical device, and it will also be more convenient to carry and use. An effective way to shorten the focal length is to combine the type A lens and the type B lens in the previous invention to form a compound lens. As shown in Figure 1 shown, Figure 1 the lens numbered 01 in is the same as the type A lens in the previous invention, and the lens numbered 02 is the same as the type B lens in the previous invention. The function of the type A lens is to convert a parallel light (sunlight) beam into a focused beam, and the function of the type B lens is to further shorten the focal length of the focused beam. In this way, not only can the parallel light beam be focused, but also a shorter focal length can be achieved. However, this approach brings a new problem, that is, it is more troublesome to make two lenses. Can one lens achieve the functions that require two lenses in the previous invention? The type C lens given by the present invention can achieve this idea, that is, the type C lens can achieve it with only one lens, which can both focus the parallel light beam and have a shorter focal length. How does the type C lens achieve this? Because both the type A and type B lenses complete one refraction of the incident light, while the type C lens can complete two refractions of the incident light with only one lens, so the type C lens has higher efficiency. For the optical path diagram of the type C lens focusing the parallel light beam, see Figure 3. Since the C-type lens has basically the same function as the compound lens composed of the A-type and B-type lenses, but the C-type lens has only one lens, under the condition of achieving the same function, in the case where a shorter focal length is required, the C-type lens has a lower cost than the compound lens composed of the A-type and B-type lenses, and low cost is an important factor for a product to be recognized by people.

[0005] Next, compare the C-type lens with the single A-type lens: Both types of function-curved lenses can achieve precise focusing of parallel light beams. However, under the condition of the same lens radius, the C-type lens can achieve a shorter focal length than the A-type lens. Therefore, in the case where a shorter focal length is required, the C-type lens is more practical than the A-type lens. To prove that the C-type lens can achieve a shorter focal length than the A-type lens, a quantitative analysis of the two types of A-type and C-type lenses is still needed. Now, a quantitative analysis of the two types of A-type and C-type lenses is carried out. Under extreme conditions, what determines the shortest focal length that the A-type lens can reach? It is determined by whether total internal reflection occurs at the edge points of the lens. Refer to Figure 2 , Figure 2 is the optical path diagram of the A-type lens focusing parallel light beams. Figure 2 The parallel lines with arrows marked from bottom to top in Figure 2 are incident light rays. In Figure 3 , point P is an arbitrary point near the edge on the surface of the A-type lens, PT is the normal line at point P, PQ is the tangent line at point P, PH is the incident light ray at point P, Pf is the refracted light ray at point P, PE is the extension line of the incident light ray at point P, angle a is the incident angle of the incident light ray at point P, and angle G is the refraction angle of the refracted light ray at point P. The G angle of the A-type lens cannot be greater than 90 degrees. When it is greater than 90 degrees, there will be no refracted light ray at point P here, that is, total internal reflection occurs. Taking the lens radius of 50 mm and the refractive index n of 1.5 as an example, applying the refractive index formula gives: sin(G) / sin(a) = n = 1.5. Also, since the G angle is equal to 90 degrees, that is, sin(G) = 1, so: sin(a) = sin(G) / n = 1 / 1.5 = 0.667. Looking up the inverse trigonometric function gives that the a angle is about 42 degrees. In the extreme case where the G angle is 90 degrees, the refracted light ray at point P coincides with the tangent line at point P. The included angle between the extension line PE of the incident light ray PH at point P and the refracted light ray Pf at point P is the b angle. The b angle = G angle - a angle. At this time, the b angle reaches the maximum value of 90 - 42 = 48 degrees, and the M angle is equal to the b angle (alternate interior angles of parallel lines are equal), which is also 48 degrees. The focal length obtained when the M angle is 48 degrees is the shortest focal length. At this time, the shortest focal length is close to equal to the radius of the lens, which is slightly less than 50 mm. Now look at Figure 3 is the optical path diagram of the C-type lens focusing parallel light beams. Figure 3 The parallel lines with arrows marked from bottom to top in Figure 3Point P in the middle is an arbitrary point near the edge on the second surface of the C-type lens. PT is the normal line of point P, and PQ is the tangent line of point P. The C-type lens realizes the secondary refraction of the parallel light beam. HJ is the incident light ray before the first refraction, and JP is the refracted light ray obtained after the first refraction. JP is also the incident light ray before the second refraction at the same time. Pf is the refracted light ray obtained after the second refraction. PF is the extension line of the straight line JP. Angle a is the incident angle during the second refraction. The included angle between the extension line PF of the incident light ray at point P and the refracted light ray Pf at point P is angle b. When the radius of the C-type lens is also 50 mm and the refractive index n is 1.5, it can be similarly proved that Figure 3 in it, the maximum value of angle a is 42 degrees, and the maximum value of the sum of angle a and angle b is 90 degrees. Therefore, the maximum value of angle b is also 48 degrees, and Figure 3 in it, angle M is not equal to angle b. Angle M is equal to angle b plus angle L (the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles), and angle L is not zero. Therefore, under the limit conditions, Figure 3 in it, angle M is greater than Figure 2 in it, angle M. Under the condition that the radii of the two lenses are equal, the lens that can achieve a larger angle M has a shorter focal length. Therefore, Figure 3 the C-type lens in it is shorter than Figure 2 the A-type lens in it in terms of focal length under the conditions that the lens radii are equal and the refractive index n is equal.

[0006] Next, compare the C-type lens with a single B-type lens: Since the C-type lens focuses the parallel light beam, while the B-type lens changes the focal length of the focused light beam and then focuses it, there is a qualitative difference in the functions and uses between the C-type and B-type lenses. Therefore, the C-type lens and a single B-type lens are not comparable in terms of practicality.

[0007] The above compares the performance of the A-type, B-type, and C-type function surface lenses. The conclusion is that: The C-type lens can achieve a shorter focal length than the A-type lens, and has a lower cost than the compound lens composed of the A-type and B-type lenses. Therefore, when considering both a shorter focal length and a lower manufacturing cost, the C-type lens has significant progress in comprehensive performance compared with the A-type lens and the compound lens composed of the A-type and B-type lenses in the previous invention. A big difference between the C-type lens and the A-type and B-type lenses in the previous invention is that: Both the A-type lens and the B-type lens complete one refraction of the incident light ray, while the C-type lens completes two refractions of the incident light ray. This is a prominent substantive feature of the C-type lens. At the same time, the first surface of the C-type lens is a unique function surface. For a detailed introduction to the first surface of the C-type lens, see the description later in this article. This unique function surface of the C-type lens is completely different from the function surfaces of the A-type, B-type, and AA-type function surface lenses in the previous invention. This is another prominent substantive feature of the C-type lens. Summary of the Invention

[0008] As used in this specification, the "lens refraction surface" refers to the surface of the lens that causes light to refract.

[0009] As used in this specification, Lens refraction curved surface line " Lens refraction curved surface line the surface formed by rotating 360 degrees around the central axis of the lens is Lens refraction curved surface , and the mathematical analysis of Lens refraction curved surface can be decomposed into the mathematical analysis of Lens refraction curve Surface line . The mathematical analysis of Lens refraction curved surface line will yield the same result on Lens refraction curved surface .

[0010] The functions of a C-type lens are as follows: A C-type lens has two refraction surfaces. The function of one refraction surface is to refract the parallel light beam parallel to the central axis of the lens incident on the lens surface into a focused beam inside the lens. In this article, this surface is called the first surface of the C-type lens. The function of the other refraction surface of the C-type lens is to refract the focused beam that has entered the lens through the first surface for a second time, shorten the focal length of the focused beam and emit it from the lens. In this article, this surface is called the second surface of the C-type lens. In order to manufacture and process a C-type lens, it is necessary to first derive the mathematical function analytical formula of the refraction surface of the C-type lens. The specific derivation is as follows:

[0011] (01) Derive a deformation formula of the refractive index formula different from that in the previous invention from the optical refractive index formula:

[0012] Refer to Figure 4 , Figure 4 where light enters the transparent medium from the air. HP is the incident light, PB is the refracted light, PE is the extension line of the incident light, TS is the normal line passing through point P, angle a is the angle of incidence of the light, angle c is the angle of refraction of the light, and angle b is the angle between the extension line PE of the incident light and the refracted light PB. In this article, angle b is given a name: "deflection angle". Figure 4 Below in Figure 4 is the transparent medium, also known as the optically dense medium, and above is the air. n is the relative refractive index of light between the optically dense medium and the air. Since the value of n actually measured by people through experiments is greater than 1, and also because for

[0013] the experimental device shown, the actually observed phenomenon is that the angle of refraction c is less than the angle of incidence a. Considering these two factors and applying the optical refractive index formula, we get:

[0014] n = sin(a) / sin(c). By inverting both sides of this equation, the following equation can be obtained:

[0014] 1 / n = sin(c) / sin(a); (n is the relative refractive index of light, obtained from the refractive index formula)

[0015] = sin(a - b) / sin(a); (see Figure 4 , angle c = angle a - angle b, obtained)

[0016] = [sin(a)cos(b) - cos(a)sin(b)] / sin(a); (obtained from the trigonometric function formula)

[0017] = {[sin(a)cos(b)] / sin(a)} - {[cos(a)sin(b)] / sin(a)}

[0018] = cos(b) - [sin(b) / tan(a)]; (obtained from the tangent value of angle a, tan(a) = sin(a) / cos(a))

[0019] By moving the terms on both sides of the equation derived from 1 / n = cos(b) - [sin(b) / tan(a)] and rearranging and simplifying, the following formula can be obtained:

[0020] tan(a) = sin(b) / [cos(b) - (1 / n)] …………(1)

[0021] Formula (1) gives the mathematical relationship among the incident angle a, the deflection angle b, and the relative refractive index n. This formula is another deformation formula of the refractive index formula, which is different from the deformation formula of the refractive index formula given in the previous invention. Since both are deformation formulas of the refractive index formula, why are there two results? This is because in the previous invention, the situation is that light rays enter the air from an optically denser medium, while in this article, the situation is that light rays enter an optically denser medium from the air.

[0022] (02) Derive the third YZQ tangent formula:

[0023] See Figure 5 , Figure 5 is the optical path diagram in which the first curved surface of the C-type lens refracts the parallel light beam parallel to the lens central axis and transforms it into a focused beam inside the lens. Figure 5The curve in it is the refraction surface line of the first surface of the C-shaped lens, and the Y coordinate axis is the central axis of the lens refraction surface. The point (F1) on the Y coordinate axis is the focus of the focused light beam obtained inside the lens after the first surface of the C-shaped lens refracts a parallel light beam. In this specification, the point (F1) is called the virtual focus of the first surface of the C-shaped lens. The point (P) is an arbitrary point on the first surface of the C-shaped lens. The parallel line with an arrow marked from top to bottom is the incident light ray. The incident light ray is parallel to the Y coordinate axis. PH is the incident light ray of the point (P). The straight line PE is the extension line of the incident light ray of the point (P). The straight line TS is the normal line passing through the point (P). PQ is the tangent line of the point (P), perpendicular to the Y coordinate axis, and the straight line passing through the edge point (R) of the first surface and perpendicular to the Y coordinate axis is the X coordinate axis. The intersection point O of the Y coordinate axis and the X coordinate axis is the coordinate origin. This coordinate origin is also called: the coordinate origin of the first surface. The included angle between the refracted light ray P(F1) and the straight line PE is the angle b. Because the extension line PE of the incident light ray is parallel to the Y coordinate axis, the included angle between the refracted light ray P(F1) and the Y coordinate axis is also the angle b (the alternate interior angles of parallel lines are equal). The included angle between the incident light ray HP and the normal line TS is the incident angle a of the light ray. The angle d is the included angle between the tangent line PQ and the X axis. Because the two sides of the angle d are perpendicular to the two sides of the angle a respectively (the tangent line is perpendicular to the normal line, and the incident light ray is perpendicular to the X axis), so the angle d is equal to the angle a. The angle c is the included angle between the refracted light ray P(F1) and the normal line TS of the point (P). The angle c is the refraction angle of the refracted light ray at the point (P). Next, we will Figure 5 make a comparison between the angles a, b, and c in Figure 4 and the angles a, b, and c in Figure 4 . You will find that their physical properties are exactly the same, all being the incident angle, the deflection angle, and the refraction angle. The formula (1) deduced from Figure 5 is also correct in Figure 5 . Also, because in Figure 5 the angle a is equal to the angle d, the angle d is less than 90 degrees, and the angle d is the included angle between the tangent line of the point (P) and the X axis, so the absolute value |k| of the slope of the tangent line of the point (P) is |k| = tan(d) = tan(a). Substituting |k| in this equation into the formula (1), we get:

[0024] |k| = sin(b) / [cos(b) - (1 / n)] …………(2).

[0025] Formula (2) is the calculation formula for the absolute value of the slope of the tangent line at any point (P) on the first surface of the C-shaped lens. For the convenience of narration, give formula (2) a name: "YZQ Tangent Formula Three". In this formula, |k| is the absolute value of the slope of the tangent line at any point (P) on the first surface of the C-shaped lens, b is the included angle between the connection line of the corresponding tangent point (P) and the virtual focus (F1) point of the first surface of the C-shaped lens and the lens central axis, and n is the relative refractive index of the lens material with respect to air for light.

[0026] (03) Determine the tangent formula of the second surface of the C-type lens:

[0027] Figure 6 It is the optical path diagram in which the second surface of the C-type lens refracts the focused light beam that has entered the lens through the first surface for the second time, shortens the focal length of the focused light beam, and emits it from the lens. Figure 6 In it, the Y coordinate axis is the central axis of the C-type lens. The curve above the X coordinate axis is the refraction surface line of the second surface of the C-type lens, and the curve below the X coordinate axis is the refraction surface line of the first surface of the C-type lens. Figure 6 The point (F2) in it is the focus of the extension line of the light ray of the focused light beam that has entered the lens through the first surface. In this article, the point (F2) is called the virtual focus of the second surface of the C-type lens. Figure 6 The point (f) in it is the focus of the focused light beam that is refracted by the second surface of the C-type lens for the second time after the focused light beam that has entered the lens through the first surface is refracted, and the focal length is shortened and then emitted from the lens. In this article, the point (f) is called the focus of the C-type lens. Figure 6 The point (P) in it is any point on the second surface of the C-type lens, and PQ is the tangent at the point (P). The straight line perpendicular to the Y coordinate axis and passing through the edge point (R) of the second surface is the X coordinate axis. The focus O point of the Y coordinate axis and the X coordinate axis is the coordinate origin. This coordinate origin is also called: the coordinate origin of the second surface.

[0028] In the previous invention, a B-type lens was given. The B-type lens has 2 surfaces. One surface is a spherical surface, and the other surface is the unique refraction surface of the B-type lens. The function of the unique refraction surface of the B-type lens is: refract the focused light beam that has passed through the spherical surface and entered the lens, further shorten the focal length of the focused light beam, and emit it from the lens. The function of the second surface of the C-type lens is exactly the same as the function of the unique refraction surface of the B-type lens. They both refract the focused light beam that has entered the lens, further shorten the focal length of the focused light beam, and emit it from the lens. Therefore, the unique refraction surface of the B-type lens can be directly used on the second surface of the C-type lens. In this way, the second surface of the C-type lens and the unique refraction surface of the B-type lens are exactly the same functional surface. Therefore, the tangent formula of the second surface of the C-type lens is exactly the same as the tangent formula of the unique refraction surface of the B-type lens (i.e., the YZQ tangent formula two). Because there is a derivation and proof of the tangent formula of the unique refraction surface of the B-type lens (i.e., the YZQ tangent formula two) in the previous invention specification, it will not be repeated in this article. Therefore, the tangent formula of the second surface of the C-type lens is the "YZQ tangent formula two" given in the previous invention specification. The specific description of the YZQ tangent formula two is:

[0029] |k| = [W - tan(g)] / [1 + W * tan(g)] …………(3)

[0030] W in formula (3) is: W = sin(h - g) / [n - cos(h - g)]

[0031] Formula (3) was called the YZQ tangent formula II in the previous invention specification. This tangent formula is the calculation formula for the absolute value of the tangent slope of any point (P) on the unique refraction surface of the B-type lens, and it is also the calculation formula for the absolute value of the tangent slope of any point on the second surface of the C-type lens. For the meanings of the h-angle and g-angle parameters in formula (3), see Figure 6 。

[0032] For the second surface of the C-type lens, |k| in formula (3) is the absolute value of the tangent slope of any point (P) on the second surface of the C-type lens. h is the angle formed by the line connecting the corresponding tangent point (P) to the focal point (f) of the C-type lens and the central axis of the lens refraction surface. g is the angle formed by the line connecting the corresponding tangent point (P) to the virtual focal point (F2) of the second surface of the C-type lens and the central axis of the lens refraction surface. n is the relative refractive index of the lens material with respect to light in air.

[0033] (04) Explanation of the positional relationship between the first and second surfaces of the C-type lens:

[0034] The positions of the first and second surfaces of the C-type lens cannot be too close or too far apart. Their positional relationship needs to meet the following conditions: that is, the virtual focal point (F1) of the first surface coincides with the virtual focal point (F2) of the second surface at one point, and at the same time, the coordinate origin of the first surface coincides with the coordinate origin of the second surface at one point. Only in this way can the C-type lens achieve the set functions. These conditions that the positional relationship between the first and second surfaces of the C-type lens needs to meet are essential and indispensable when calculating the coordinate values of the refraction surface line.

[0035] (05) Explanation of the tangent slope formula and the conditions to be met when the refraction surface of the C-type lens is made into a serrated shape (Fresnel lens form):

[0036] The refraction surface of the C-type lens can be composed of a single smooth surface or multiple smooth surfaces made into a serrated shape, Figure 7 are schematic diagrams of three forms when the refraction surface of the C-type lens is made into a serrated shape (Fresnel lens form). The three forms are respectively: Figure 7Among them, [a] and [b] are two of the forms, where the first surface or the second surface is made into a serrated shape, while the other surface remains a single smooth surface. [c] is the third form where both the first and second surfaces are made into serrated shapes. Since the formula for calculating the absolute value of the tangent slope of the two refracting surfaces of a C-type lens only relates to the angles (b angle, h angle, g angle) between the refracted light or the extension of the incident light and the central axis and the refractive index n, and has nothing to do with the number of segments into which the lens refracting surface line is divided. When calculating the tangent slope of each segment of the surface, only the corresponding b angle, h angle, and g angle of each segment of the surface need to be found, without considering the number of segments into which the surface is divided. Therefore, the analytical formula for the absolute value of the tangent slope when the refracting surface of a C-type lens is made into a serrated shape is the same as that of a single smooth surface. In addition, in order to make the virtual foci, foci, and the origin of coordinates of each segment of the surface of the C-type lens made into a serrated shape coincide, the following conditions need to be met for each segment of the surface made into a serrated shape: when the surface is the first surface, the virtual foci (F1) of each segment of the surface are located at the same point; when the surface is the second surface, the virtual foci (F2) of each segment of the surface are also located at the same point; the foci (f) of each segment of the second surface are also located at the same point; at the same time, the virtual foci (F1) of each segment of the first surface coincide with the virtual foci (F2) of each segment of the second surface at one point. In addition, the origin of coordinates of each segment of the first and second surfaces also coincides at the same point. These conditions that the C-shaped lens made into a serrated shape needs to meet are also essential and indispensable when calculating the coordinate values of the refracting surface line.

[0037] (06) Explanation of calculating the coordinate values of the refracting surface line of a C-type lens using the differential equation calculation method:

[0038] To draw and manufacture a C-type lens, only the tangent formula is not enough. It is also necessary to calculate the coordinate values of each point on the refracting surface lines of the two surfaces of the C-type lens. In the previous invention specification, a differential equation calculation method was given, and this calculation method is also applicable to calculating the coordinate values of the refracting surface lines of the two refracting surfaces of a C-type lens. The differential equation calculation method can be directly used, and the specific calculation details are as follows:

[0039] First, look at the calculation of the coordinate values of the refracting surface line of the first surface of a C-type lens:

[0040] The differential equation calculation method needs to convert the YZQ tangent formula three into the form of solving equations with X and Y coordinates. For the parameter marking of the refracting surface line of the first surface of a C-type lens, refer to Figure 8, first, calculate the coordinate values of the refraction surface line of the first surface of the C-type lens. Before the conversion formula, the refraction surface line of the lens needs to be segmented. First, define the coordinate system. Take the central axis of the refraction surface line of the lens as the Y coordinate axis, the edge point (R) of the lens is located on the X axis, and the O point is the origin of coordinates. The refraction surface line of the first surface is located in the first quadrant of the coordinate system. Then set an interval dx in the X direction, and divide the refraction surface line of the lens into many segments at intervals of dx in the X direction. The value of dx can be set to a suitable value according to needs. Each segment is defined as a straight line, and several key points on the straight line segment are defined as follows: The length of the projection of the straight line segment on the X axis is dx, and the length of the projection on the Y axis is dy. The straight line segment is divided into the starting point coordinates (x1, y1), the tangent point (P) coordinates (Xp, Yp), and the ending point coordinates (x2, y2) in the calculation order. Starting from the edge point (R) of the lens, when the refraction surface line of the lens is a single smooth curve interval, then x2 is always less than x1, and y2 is always greater than y1. Further, the following calculation formulas for x2 and y2 can be obtained:

[0041] x2 = x1 - dx …………(4)

[0042] y2 = y1 + dy …………(5)

[0043] The calculation formula for the coordinates of the (P) point by the equation-solving method is:

[0044] (P) point coordinate X value: Xp = x1 - dx / m …………(6)

[0045] (P) point coordinate Y value: Yp = y1 + dy / m …………(7)

[0046] In the calculation formula for the coordinates of the (P) point by the equation-solving method, m is a real number greater than 1, which is specified by humans. When the midpoint of the straight line segment is selected as the position where the (P) point is located, m in the calculation formula for the coordinates of the (P) point by the equation-solving method is m = 2. For the first surface of the C-type lens, according to Figure 8 the markings in, the following two trigonometric conversion equalities of the refraction surface line of the first surface of the C-type lens can be obtained:

[0047]

[0048]

[0049] In equations (8) and (9), F1 is the length from point O to point (F1). Substitute the coordinate calculation formulas (6) and (7) of point (P) in the equation-solving calculation method and the trigonometric function conversion equations (8) and (9) of the refraction surface line of the first surface of the C-type lens derived into the expression on the right side of the equal sign in YZQ tangent formula three. After simplification and arrangement, an absolute value expression of the tangent slope containing only variables x1, y1, dx, dy and constants F1, n, m can be obtained. This expression is simplified and represented as: T3(x1, y1, dx, dy). At the same time, the slope of the straight line segment |k| = dy / dx. Substitute |k| = dy / dx into the expression on the left side of the equal sign in YZQ tangent formula three, then YZQ tangent formula three can be converted into another expression form as follows:

[0050] dy / dx = T3(x1, y1, dx, dy) …………(10)

[0051] Write the expression (10) in the form of solving an equation as:

[0052] (dy / dx) - T3(x1, y1, dx, dy) = 0 …………(11)

[0053] In equation (11), dx is set artificially, so it is known. Since the coordinate calculation starts from the edge point (R) of the lens, and the coordinates of the edge point (R) of the lens do not need to be calculated, the coordinates of the edge point of the lens are (R, 0), where R is the radius of the lens. Then the coordinate value of the end point of each straight line segment is used as the coordinate value of the starting point of the next straight line segment. Therefore, in the continuous calculation process, for the straight line segment to be calculated, x1 and y1 are also known. In this way, there is only one unknown dy in equation (11). By solving equation (11), the value of dy can be obtained. Equation (11) is a multi-variable equation of one degree, and the common equation-solving methods cannot solve this equation. A computer program can be used to solve this equation. The program code for solving this equation with a VB program can be as follows:

[0054]

[0055]

[0056] This section of the program uses the trial method to solve equations, that is, let the value of dy increase gradually. When the value of dy increases to a certain value, the computer program calculates that (dy / dx) - T3(x1, y1, dx, dy) = 0 or the absolute value of (dy / dx) - T3(x1, y1, dx, dy) is less than 0.0001, and the program jumps out of the loop. At this time, the value of dy is the solution of the equation. The program continues to run. With the value of dy, the end point coordinates x2 and y2 of this line segment can be calculated according to equations (4) and (5). Then, let the starting point coordinates x1 and y1 of the next line segment be equal to the end point coordinates x2 and y2 of the previous line segment, and so on. By looping and calculating, the coordinate values of all the endpoints of the straight line segments on the first surface line of the C-shaped lens can be calculated. When we set the value of dx to be infinitely close to zero, a series of coordinate values calculated by the differential equation solving method are the accurate coordinate values of each point on the refraction surface line of the first surface of the C-shaped lens.

[0057] For the calculation of the coordinate values of the refraction surface line of the second surface of the C-shaped lens, see Figure 9 , similar to the calculation of the coordinate values of the refraction surface line of the first surface, it is also necessary to convert the tangent formula of the refraction surface line of the second surface of the C-shaped lens, that is, the YZQ tangent formula two, into the form of solving equations with X and Y coordinates, and it is necessary to segment the refraction surface line and set the coordinates. Similarly, take the central axis of the lens refraction surface line as the Y coordinate axis, the edge point (R) of the lens is located on the X axis, and the O point is the coordinate origin. Place the refraction surface line of the second surface in the first quadrant of the coordinate system. Then set an interval dx in the X direction, divide the lens refraction surface line into many segments at intervals of dx in the X direction. The value of dx can be set to a suitable value according to needs. Each segment is defined as a straight line. There are several key points on the straight line segment defined as follows: the length of the projection of the straight line segment on the X axis is dx, and the length of the projection on the Y axis is dy. The straight line segment is divided into the starting point coordinates (x1, y1), the tangent point (P) coordinates (Xp, Yp), and the end point coordinates (x2, y2) in the calculation order. Starting from the edge point (R) of the lens, when the lens refraction surface line is a single smooth curve interval, according to Figure 9 the annotation in, equations (4), (5), (6), and (7) are still correct when calculating the coordinate values of the refraction surface line of the second surface. At the same time, according to Figure 9 the annotation in, the following 6 trigonometric function conversion equations can also be obtained:

[0058] tan(g) = Xp / (F2 - Yp)………(12)

[0059]

[0060]

[0061]

[0062]

[0063]

[0064] It should be noted that the value of F2 in equations (12), (16), and (17) is the length from point O to point (F2), and the value of F2 must be equal to the value of F1 in the first surface trigonometric conversion equations (8) and (9). The value of f in equations (14) and (15) is the length from point O to point (f). Substitute the conversion equations (12), (13), (14), (15), (16), (17) and the coordinate calculation formulas (6) and (7) of point (P) in the solution equation calculation method into the expression on the right side of the YZQ tangent formula II. After simplification and arrangement, an expression for the absolute value of the tangent slope containing only variables x1, y1, dx, dy and constants f, F2, n, m can be obtained. This expression is simplified and represented as: T2(x1, y1, dx, dy). At the same time, the slope of the straight line segment |k| = dy / dx. Substitute |k| = dy / dx into the expression on the left side of the YZQ tangent formula II, then the YZQ tangent formula II can be converted into another expression form as follows:

[0065] dy / dx = T2(x1, y1, dx, dy) …………(18)

[0066] Write the expression (18) in the form of a solution equation as:

[0067] (dy / dx) - T2(x1, y1, dx, dy) = 0 …………(19)

[0068] Because it also starts from the edge point (R) of the lens and then uses the coordinate value of the end point of each straight line segment as the starting point coordinate value of the next straight line segment, for the straight line segment to be calculated, its x1 and y1 are known, and since dx is set artificially, it is also known. Thus, there is only one unknown dy in equation (19). Equation (19) is a polynomial equation of one variable, and a computer program can be used to solve this equation. After obtaining the value of dy by solving the equation, calculate the values of x2 and y2 according to equations (4) and (5). Then, let the coordinate values x1 and y1 of the starting point of the next straight line segment be equal to the coordinate values x2 and y2 of the end point of the previous straight line segment, and so on to calculate the coordinate values of all the endpoints of the straight line segments. When we set the value of dx to be infinitely close to zero, the series of coordinate values calculated by the differential solution equation calculation method are the accurate coordinate values of each point on the refraction surface line of the second surface of the C-type lens.

[0069] When the coordinate values of the refraction surface lines of the first surface and the second surface calculated by the differential equation calculation method are combined into a complete lens coordinate value series in the same coordinate system, all the Y coordinate values of the refraction surface line of one of the surfaces need to be mirror-transformed. Note:

[0070] Because when calculating the coordinate values of the refraction surface lines of the first surface and the second surface by the differential equation calculation method, the refraction surface lines are placed in the first quadrant of the coordinate system, and the results calculated in this way overlap when placed in the same coordinate system. When the coordinate values of the refraction surface lines of the two calculated surfaces need to appear in the same coordinate system for drawing or other purposes, the coordinate values of the refraction surface line of one of the surfaces need to be mirror-transformed with the X coordinate axis as the mirror axis, that is, multiply all the calculated Y coordinate values of the refraction surface line of one of the two surfaces by (-1) to take negative values, and then place the coordinate values of the refraction surface lines of the two surfaces in the same coordinate system, so as to obtain a complete series of coordinate values of the refraction surface line of a C-type lens.

[0071] (08) YZQ tangent formula three and YZQ tangent formula two can have many different forms. Note:

[0072] The above gives the descriptions of YZQ tangent formula three and YZQ tangent formula two. Another thing to note is that YZQ tangent formula three and YZQ tangent formula two can have many different forms. For example, the YZQ tangent formula can be transformed by trigonometric identities to replace the sine function or cosine function with the tangent function, or describe YZQ tangent formula three or YZQ tangent formula two in the form of X and Y coordinates, etc. For these various tangent formulas that can be obtained only through simple pure mathematical identity transformations and have the same calculation results as YZQ tangent formula three or YZQ tangent formula two, it is impossible to list them all in this specification. These tangent formulas obtained through simple pure mathematical identity transformations, which have different forms from YZQ tangent formula three or YZQ tangent formula two but the same calculation results, are substantially the same as the YZQ tangent formula. To determine whether a tangent formula is substantially the same as YZQ tangent formula three or YZQ tangent formula two, it can be judged by verifying their calculation results, rather than just by comparing whether their forms are the same.

[0073] (09) How to define whether a certain lens is the same as a C-type lens. Judgment method. Note:

[0074] This specification provides a calculation method for calculating the coordinate values of a series of points on the refraction surface line of a C-type lens using the differential equation calculation method. The calculated coordinate data is used as a template value and compared with the coordinate measurement values of the actually effective points on the refraction surface of a certain lens to be detected. (When measuring the coordinate values, the refraction surface of the lens to be measured needs to be placed in the first quadrant of the coordinate system). If the coordinate measurement values of the two refraction surfaces of the lens to be detected are all in agreement with the calculated template values, then this lens is considered to be the same as the C-type lens. The term "in agreement" here has two meanings. One is that when comparing the measurement values and the template values, those points that are significantly deviated from the smooth curve due to measurement or processing errors need to be excluded, and those points that are intentionally set to deviate significantly from the smooth curve for other requirements (such as for fixing screws) also need to be excluded. On this basis, the template value and the measurement value are 100% the same within the allowable error range. The other is that when the physical measurement value is compared with the template value and the error is within the allowable range, it is considered to be in agreement. This allowable error range can be within one-tenth of a millimeter or within one-hundredth of a millimeter or within one-thousandth of a millimeter. Specifically, which allowable error range level to choose is based on being able to clearly explain the facts, and choose the larger allowable error range level.

[0075] (10) Uses and advantages of the C-type lens:

[0076] The C-type lens made into a serrated shape can be used to make large (with a diameter exceeding 1 meter) solar concentrator lenses for the development and utilization of solar energy because it has an accurate focusing function and a short focal length. In addition, the C-type lens can also be used in other optical instruments to replace spherical lenses to achieve higher focusing accuracy and a shorter focal length, etc. Also, because the differential equation calculation method can accurately calculate the coordinate values of a series of points on the refraction surface line of the C-type lens, and the coordinate value data of these points can be conveniently copied into the computer of the numerical control machine tool, it is thus possible to conveniently machine the C-type lens or machine the mold for making the C-type lens with the numerical control machine tool. Brief description of the drawings

[0077] Figure 1 It is the optical path diagram of focusing a parallel light beam with a compound lens composed of the A-type lens and the B-type lens in the previous invention.

[0078] Figure 2 It is the optical path diagram of focusing a parallel light beam with the A-type lens in the previous invention.

[0079] Figure 3 It is the optical path diagram of focusing a parallel light beam with the C-type lens.

[0080] Figure 4 It is the optical path diagram of light refraction when light enters a transparent medium from the air.

[0081] Figure 5 It is the optical path diagram in which the first curved surface of the C-type lens converts a parallel light beam into a focused beam inside the lens.

[0082] Figure 6 It is the optical path diagram in which the second curved surface of the C-type lens refracts the focused beam that has entered the lens through the first curved surface for a second time, shortens the focal length of the focused beam and emits it from the lens.

[0083] Figure 7 It is the schematic diagram of three forms in which the refraction curved surface of the C-type lens is made into a serrated shape (Fresnel lens form).

[0084] Figure 8 It is the schematic diagram for marking the calculation parameters of the coordinate values of the refraction curved surface line of the first curved surface of the C-type lens.

[0085] Figure 9 It is the schematic diagram for marking the calculation parameters of the coordinate values of the refraction curved surface line of the second curved surface of the C-type lens.

[0086] Figure 10 It is the schematic reference diagram of the shape of the refraction curved surface line of the lens cross-section in the embodiment.

[0087] Figure 11 It is the VB program for calculating the coordinate values of the first curved surface in the embodiment, and the schematic diagram of the form layout with labels, text boxes and command buttons placed in the form respectively.

[0088] Figure 12 It is the VB program for calculating the coordinate values of the second curved surface in the embodiment, and the schematic diagram of the form layout with labels, text boxes and command buttons placed in the form respectively. Detailed implementation mode

[0089] In a specific embodiment, a C-type lens with a diameter of 1 meter will be designed to focus sunlight and develop and utilize solar energy. The first and second curved surfaces of the lens are both made into serrated shapes. Refer to Figure 10 , Figure 10 The shape of the refraction curved surface line of the lens cross-section in Figure 10The curve above the X-axis is the refraction surface line of the first surface, and the curve below the X-axis is the refraction surface line of the second surface. When calculating the coordinate values of the refraction surface line of the second surface, the refraction surface line of the second surface needs to be placed in the first quadrant of the coordinate system for calculation. Since sunlight is light that is very close to a parallel light beam, we can regard sunlight as a parallel light beam. A lens is made of glass, and the size of the selected glass processing material is: a cylindrical glass with a thickness of 90 mm and a diameter of 1 m. Looking up relevant information, the refractive index n of the glass is determined to be 1.5. The virtual focal length F1 of the first surface of the designed lens is 3 m long, and the virtual focal length F2 of the second surface is also 3 m long. The focal length f of the second surface is 1 m long. The refraction surface lines of the two surfaces of the lens are both made into three-segment sawtooth shapes, as Figure 10 shown. Figure 10 The unit of the values in

[0090] is millimeters. The X value of sawtooth 1 of the first surface ranges from 500 mm to 410 mm, the X value of sawtooth 2 ranges from 410 mm to 290 mm, and the X value of sawtooth 3 ranges from 290 mm to 0 mm. The X value of sawtooth 1 of the second surface ranges from 500 mm to 394.5 mm, the X value of sawtooth 2 ranges from 394.5 mm to 270 mm. The X value of sawtooth 3 ranges from 270 mm to 0 mm. Such a design is to make the Y coordinate values of the maximum heights of the three sawtooth shapes of the first surface and the second surface close to equal. The central axis of the refraction surface is used as the Y-axis. Next, the differential equation calculation method is written into a computer VB program to calculate the coordinate values of a series of points on the refraction surface lines of the two surfaces of the lens. In the VB program, one-thousandth of a millimeter is used as the unit of numbers. The maximum radius of the lens is 500 mm, and the value in the VB program is 500000. The lengths of the virtual focal lengths F1 and F2 are 3 m, and the values in the VB program are 3000000. The focal length f is 1 m long, and the value in the VB program is 1000000, etc. When calculating the tangent slope in the VB program, the position of the midpoint of each tiny straight line segment is specified as the position of the tangent point (P).

[0090] The specific programming method is: Start the VB6 programming software of Microsoft, enter the VB6 programming environment, first create a new standard project program file, and then appropriately increase the size of the form1 form with the mouse, and press Figure 11 and Figure 12 shown to arrange the labels, text boxes, and command buttons in the form1 form. Write the VB program for calculating the coordinate values of the refraction surface line of the first surface. Press Figure 11 shown in the layout diagram. First, place 8 labels Label1 to Label8 in the form1 form, then place 9 text boxes Text1 to Text9 and 3 command buttons Command1 to Command3. Write the VB program for calculating the coordinate values of the refraction surface line of the second surface. Press Figure 12Layout diagram: First, place 9 labels Label1 to Label9, then 10 text boxes Text1 to Text10 and 3 command buttons Command1 to Command3 in the form1 form. After placing these labels, text boxes and command buttons in the form1 form, the sizes of the labels, text boxes and command buttons need to be appropriately adjusted with the mouse so that the text to be displayed can be accommodated and they are aligned according to the layout diagram. Then, the code of the VB program can be started to be written. The following is the program code of two VB programs. The two VB programs need to be run separately and are used to calculate the coordinate values of the points on the refraction curved surface lines of the first and second curved surfaces respectively. Just input the following two sections of program code item by item into the program code areas of the two VB programs respectively. The text description parts marked with the symbol '#' do not need to be input. After the program code is written, the program can be run. In the VB6 programming environment, open the VB program to be run and click the "Run / / Start" command in the drop-down menu. The program will start running and display the placed labels, text boxes, command buttons and text in the form1 form. At this time, the command button marked with "Calculate" can be clicked with the mouse, and the program will start calculating the coordinate values of the curve. After waiting for a few seconds, the program calculation is completed, and the Y coordinate value at X = 0 is displayed in the "Corresponding Y value" text box. The program stores a series of calculated coordinate values in an array named zbsz in the memory. These coordinate data can be displayed one by one on the screen or stored in the hard disk. The method of displaying on the screen is: input an X coordinate value (any positive integer value between 0 and 500000) in the "Input X value" text box, and then click the command button marked with "Display Y value" with the mouse. At this time, the Y coordinate value corresponding to this X value will be displayed in the "Corresponding Y value" text box. Note: Both the X value and the Y value are in units of one-thousandth of a millimeter. The method of storing the coordinate data in the hard disk is: click the command button marked with "Save to disk" with the mouse. At this time, there will be an additional file named "abc01.txt" or "abc02.txt" in the D drive of the computer hard disk. The "abc01.txt" file stores the coordinate value data of the first curved surface, and the "abc02.txt" file stores the coordinate value data of the second curved surface. These two files can be opened separately with the Notepad program. Each line in the file has 2 numbers. The first number is the X coordinate value of a certain point, and the second number is the Y coordinate value of the same point. There are a total of 500000 coordinate values of the points on the lens refraction curved surface line, and each coordinate value is displayed and stored in the hard disk in units of one-thousandth of a millimeter. We can copy the two data files abc01.txt or abc02.txt to the computer of the numerical control machine tool, and the numerical control machine tool can process the lens according to the coordinate values. The following is the program code of the two VB programs.

[0091]

[0092]

[0093]

[0094]

[0095] When the above VB program runs normally and the "Calculate" command button is clicked, after the program finishes running, the value displayed in the "Corresponding Y Value" text box is 41966.6549021889. This value is the Y coordinate value at the origin of the first surface refraction surface line X = 0. Converting this value to millimeters is 41.966 millimeters.

[0096] The above is the program code of the VB program for calculating the coordinate values of the first surface refraction surface line. The following is the program code of the VB program for calculating the coordinate values of the second surface refraction surface line.

[0097]

[0098]

[0099]

[0100]

[0101]

[0102]

[0103] When the VB program for calculating the coordinate values of the second surface runs normally and the "Calculate" command button is clicked, after the program finishes running, the value displayed in the "Corresponding Y Value" text box is 35255.0064961808. This value is the Y coordinate value at the origin of the second surface refraction surface line X = 0. Converting this value to millimeters is 35.255 millimeters. Adding this value to the Y coordinate value of about 42 millimeters at the origin of the first surface X = 0 is less than 80 millimeters. At the same time, the sum of the maximum Y coordinate values of the serrations 1, 2, and 3 of the first surface and the maximum Y coordinate values corresponding to the three serrations of the second surface is also less than 80 millimeters. Therefore, it is feasible to choose a glass material with a thickness of 90 millimeters.

[0104] The above is the program code of two VB programs. The calculation precision for solving equations set in the two VB programs is 0.0001. In production practice, if this precision is considered not high enough, some VB programming techniques can be used, or wait for a longer running time, or repeat the calculation with higher precision on the basis of 0.0001 precision, etc., then it is possible to conveniently improve the calculation precision of solving equations to a higher level.

[0105] If it is necessary to display the coordinate values of the refraction surface lines of the first and second surfaces in the same coordinate system, it is necessary to perform a mirror transformation on the coordinate values of the refraction surface line of one of the first surface or the second surface, that is, multiply the Y value of all the coordinate values of the refraction surface line of one of the first surface or the second surface obtained by the VB program by (-1) to make it negative, and then place all the coordinate values of the refraction surface line of the other surface in the same coordinate system. In this way, a complete series of coordinate values of the lens cross-section shape curve on the right side of the Y coordinate axis as shown in Figure 10 is obtained. Figure 10 The lens cross-section shape curve on the left side of the Y coordinate axis in [Figure] is a mirror copy of the curve on the right side of the Y coordinate axis with the Y coordinate axis as the mirror axis. According to this method, the cross-section shape of the serrated C-shaped lens as shown in Figure 10 can be completely drawn.

Claims

1. A functional surface lens that precisely focuses a parallel light beam through secondary refraction, also known as a C-type lens. Both surfaces of this lens are aspherical surfaces. The C-type lens has two refracting surfaces, namely the first surface and the second surface. The function of the first surface is to refract the parallel light beam parallel to the lens central axis that irradiates on the surface of the C-type lens, and convert it into a focused beam inside the lens. The function of the second surface is to refract the focused beam that enters the lens through the refraction of the first surface for the second time, shorten the focal length of the focused beam and emit it from the lens. The first surface of the C-type lens is completely different from the functional surfaces of the A-type, B-type, and AA-type functional surface lenses given in the previous invention patent with the patent number 201811072337.

8. The second surface of the C-type lens is the same as that of the patent number 201811072337.The unique functional curved surfaces of the B-type functional curved surface lenses given in the previous invention patents of 8 are exactly the same functional curved surfaces. The two surfaces of the C-type lens can also be made into a serrated shape (Fresnel lens form), a C-type lens with a single smooth surface, and a C-type lens with a serrated lens surface (Fresnel lens form). The common feature is that the coordinate values of each point on the refraction curved surface lines of the first and second surfaces of the C-type lens coincide with the coordinate values of each point obtained by calculating the refraction curved surface lines of the first and second surfaces of the C-type lens respectively using the differential equation calculation method. When calculating the coordinate values of points on the refraction curved surface line of the first surface using the differential equation calculation method, the absolute value of the tangent slope is calculated according to the YZQ tangent formula three (or according to other forms of tangent formulas that are substantially the same as the YZQ tangent formula three and have the same calculation results). When calculating the coordinate values of points on the refraction curved surface line of the second surface using the differential equation calculation method, the absolute value of the tangent slope is calculated according to the YZQ tangent formula two (or according to other forms of tangent formulas that are substantially the same as the YZQ tangent formula two and have the same calculation results). The mathematical expression of the YZQ tangent formula three is |k| = sin(b) / [cos(b)-(1 / n)]. In the YZQ tangent formula three, |k| is the absolute value of the tangent slope of any point (P) on the first surface of the C-type lens. In the YZQ tangent formula three, b is the angle formed by the connection line between the corresponding tangent point (P) and the virtual focus (F1) of the first surface of the C-type lens and the central axis of the lens refraction curved surface. The mathematical expression of the YZQ tangent formula two is |k| = [W - tan(g)] / [1 + W*tan(g)]. In the YZQ tangent formula two, W = sin(h - g) / [n - cos(h - g)]. In the YZQ tangent formula two, |k| is the absolute value of the tangent slope of any point (P) on the second surface of the C-type lens. In the YZQ tangent formula two, h is the angle formed by the connection line between the corresponding tangent point (P) and the focus (f) of the C-type lens and the central axis of the lens refraction curved surface. In the YZQ tangent formula two, g is the angle formed by the connection line between the corresponding tangent point (P) and the virtual focus (F2) of the second surface of the C-type lens and the central axis of the lens refraction curved surface. In the YZQ tangent formula three and the YZQ tangent formula two, n is the relative refractive index of the lens material with respect to air for light.

2. The C-type lens of the function surface lens according to claim 1, wherein the positional relationship between the first surface and the second surface is characterized in that: the condition to be satisfied by the positional relationship between the first surface and the second surface is that the virtual focus (F1) point of the first surface coincides with the virtual focus (F2) point of the second surface at one point, and at the same time, the coordinate origin of the first surface coincides with the coordinate origin of the second surface at one point.

3. The C-type lens of the function surface lens according to claim 1, wherein the C-type lens is made into a serrated (Fresnel lens form) function surface lens Characterized in that: The C-type lens can be made into a serrated (Fresnel lens form) in three forms. The three forms are: two of the forms are that the first surface or the second surface is made into a serration, while the other surface remains a single smooth surface. The third form is that both the first and second surfaces are made into serrations. The calculation formula for the absolute value of the tangent slope of each section of the surface made into a serration is the same as the calculation formula for the absolute value of the tangent slope of a single smooth surface. The conditions that each section of the surface made into a serration needs to satisfy are: when the surface is the first surface, the virtual focus (F1) points of each section of the surface are located at the same point; when the surface is the second surface, the virtual focus (F2) points of each section of the surface are also located at the same point; the focus (f) points of each section of the second surface are also located at the same point. At the same time, the virtual focus (F1) points of each section of the first surface coincide with the virtual focus (F2) points of each section of the second surface at one point. In addition, the coordinate origins of each section of the first and second surfaces coincide at the same point.

4. The C-type lens of the function surface lens according to claim 1, the differential equation calculation method Characterized in that: Place the first and second surfaces of the C-shaped lens in the first quadrant of the coordinate system respectively and calculate them separately. The specific calculation method is as follows: Take the central axes of the first and second surfaces of the lens as the Y coordinate axes respectively. The edge point (R) of the lens is located on the X axis. Divide the lens refraction surface line into many segments at intervals of dx in the X axis direction. The value of dx can be set to a suitable value according to needs. Define each segment as a straight line. The length of the projection of the straight line segment on the X axis is dx, and the length of the projection on the Y axis is dy. Starting from the edge point (R) of the lens refraction surface, calculate in order. Set the starting coordinates of the straight line segment as (x1, y1), the coordinates of the tangent point (P) as (Xp, Yp), and the ending coordinates of the straight line segment as (x2, y2). When calculating the coordinate values of the refraction surface line of the first surface, applying the (P) point coordinate calculation formulas (6), (7) of the equation-solving calculation method and the derived first-surface trigonometric function conversion equalities (8), (9), the expression on the right side of the equal sign of the YZQ tangent formula can be converted into a mathematical expression that only contains the variables x1, y1, dx, dy and necessary constants, and is denoted as T3(x1, y1, dx, dy). At the same time, because the absolute value of the slope of the straight line segment |k| = dy / dx, |k| can be expressed as dy / dx. Thus, the YZQ tangent formula three can be converted into the form of solving equations for X and Y coordinates as (dy / dx) - T3(x1, y1, dx, dy) = 0. When calculating the coordinate values of the refraction surface line of the second surface, applying the (P) point coordinate calculation formulas (6), (7) of the equation-solving calculation method and the derived second-surface trigonometric function conversion equalities (12), (13), (14), (15), (16), (17), the expression on the right side of the equal sign of the YZQ tangent formula two can be converted into a mathematical expression that only contains the variables x1, y1, dx, dy and necessary constants, and is denoted as T2(x1, y1, dx, dy). At the same time, because the absolute value of the slope of the straight line segment |k| = dy / dx, |k| can be expressed as dy / dx. Thus, the YZQ tangent formula two can be converted into the form of solving equations for X and Y coordinates as (dy / dx) - T2(x1, y1, dx, dy) = 0. Since the calculation of the coordinate values of the refraction surface lines of the first and second surfaces both starts from the edge point (R) of the lens, and then the ending coordinate value of each straight line segment is used as the starting coordinate value of the next straight line segment, for the straight line segment to be calculated, both x1 and y1 are known. Also, because dx is set artificially, it is also known. Therefore, there is only one unknown dy in the equations (dy / dx) - T3(x1, y1, dx, dy) = 0 and (dy / dx) - T2(x1, y1, dx, dy) = 0. A computer program can solve for the unknown dy by solving the equations (dy / dx) - T3(x1, y1, dx, dy) = 0 and (dy / dx) - T2(x1, y1, dx,Solve the equation \(dy = 0\) to obtain the value of \(dy\). After obtaining the value of \(dy\), then apply the calculation formulas (4) and (5) of \(x2\) and \(y2\) to calculate the coordinate values \((x2, y2)\) of the end point of this line segment. Take the coordinate values of the end point of each calculated line segment as the starting coordinate values of the next line segment, and continuously repeat the calculation until the coordinate values of all points with an interval of \(dx\) on the refraction surface lines of the first surface and the second surface that need to be calculated are calculated.

5. The C-type lens of the function surface lens according to claim 1, when the coordinate values of the refraction surface lines of the first surface and the second surface calculated by the differential equation calculation method are combined into the coordinate value series of the complete lens in the same coordinate system, all the Y coordinate values of the refraction surface line of one of the surfaces need to be mirror-converted. Characterized in that When the coordinate values of the refraction surface lines of the two calculated surfaces are combined into the coordinate value series of the complete lens in the same coordinate system, it is necessary to mirror-convert all the Y coordinate values of the refraction surface line of one of the surfaces with the X coordinate axis as the mirror axis, that is, multiply all the calculated Y coordinate values of the refraction surface line of one of the two surfaces by (-1), take the negative value, and then place the coordinate values of the refraction surface lines of the two surfaces in the same coordinate system.

Images