CN100426038C - Method for designing non-spherical lens - Google Patents

Abstract

A method for designing an aspheric lens, the main steps of which include: setting the refractive index, refractive index and diameter of the aspheric lens; adopting a spherical design, and designing the two radii of curvature of the aspheric lens so that it reaches the above setting Refractive index, and use the two radii of curvature to minimize the distortion of 0.5 field of view, 0.7 field of view and 1.0 field of view; use the lens aspheric surface design formula to design its first aspheric surface and second aspheric surface, the lens aspheric The spherical surface design formula is the right formula, Z(r) is the depth of the mirror surface, C _v is the curvature of the apex of the aspheric surface, r is the vertical height from any point on the mirror surface to the optical axis, P is the constant value of the quadratic curve, B, C, D, E is the high-order term coefficient of the aspheric surface; the damped least square method is used to optimize the design of the first aspheric surface and the second aspheric surface; wherein, when optimizing the design, the aspheric lens is used to eliminate aberrations in at least five fields of view, And the aspheric coefficients P, B, C, D, E of the first aspheric surface and the second aspheric surface are obtained.

Description

Design method of aspheric lens

【Technical field】

The invention relates to a design method of an optical lens, in particular to a design method of an aspherical lens.

【Background technique】

At present, the lenses used in optical devices such as optical reading heads, camera lenses, and spectacle lenses generally adopt spherical or aspheric designs.

When the lens adopts a spherical design, since the two refracting surfaces of the lens are designed to be spherical, it is easy to manufacture and process. However, a spherical diverging lens with a thin center and a thick edge or a spherical converging lens with a thick center and a thin edge will thicken significantly at the edge or center of the lens as the refractive index (also called power) of the lens increases. The refractive index of the lens refers to the back vertex value of the lens, and its value is equal to the reciprocal of the paraxial rear focal length of the lens measured in meters (the distance from the vertex of the lens' image-side surface to the focus of the image-side). It can be calculated by formula (1):

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; v</span>}}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="f"\; filenames="C20051003447400151.png"\; state="\{\{state\}\}">f</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; t</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="f"\; filenames="C20051003447400151.png,C20051003447400162.png"\; state="\{\{state\}\}">f</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; t</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Among them, f _v is the paraxial back focal length of the lens, the radius of curvature of the spherical surface not close to the image side is R ₁ , the radius of curvature of the spherical surface close to the image side is R ₂ , the thickness of the lens center is t, and the lens material refracts The refractive index of the two spherical surfaces can be F ₁ =(n-1)/R ₁ and F ₂ =(1-n)/R ₂ respectively.

The unit of lens refractive index is usually represented by D (diopter). Usually, the lens power is the value of F _v multiplied by 100, that is, 1D = 100 degrees. From formula (1), when the lens material is constant, the refractive index of the lens is determined by R ₁ , R ₂ and t value.

When designing a spherical diverging lens, if t is fixed, the difference between R ₁ and R ₂ will be larger as the refractive index of the lens increases, which not only makes the edge of the spherical diverging lens thicker obviously, but also if the lens diameter If it is fixed, the thickness of the entire lens will increase.

When designing a spherical converging lens, since the central thickness t of the spherical converging lens depends on the values of R ₁ and R ₂ , the refractive index of the lens can also be adjusted by adjusting the values of R ₁ and R ₂ . It can be seen from the formula (1) that as the refractive index of the spherical converging lens increases, the greater the difference between _R1 and _R2 , the greater the central thickness t of the lens, making the volume of the entire lens larger.

Therefore, if the spherical diverging or converging lens designed by the spherical design method is used for a camera lens, it will increase the volume of the camera lens; if the spherical diverging or converging lens is a myopia or hyperopia lens, it will affect the appearance. , causing discomfort to the wearer.

In addition, when designing a lens, in addition to considering the ease of lens manufacturing and processing and the thinning of the lens, the imaging quality of the lens must also be considered.

Aberration is one of the main factors affecting the imaging quality of the lens. Generally, the aberrations that affect the imaging quality of the lens include oblique astigmatism, field curvature and distortion. Among them, the oblique astigmatism is due to the fact that the light beam emitted from a small light source at an off-axis object point has different focal points in the meridian field and the sagittal field, so that there is astigmatism in the imaging and the image is not clear. The curvature of field of view means that when a plane object perpendicular to the optical axis is imaged, the center of the image and the edge of the image are inconsistent on the image surface, so that the field of view of the image surface becomes a curved surface, which can also be called the average refractive index error, usually called the refractive index. error. Distortion is due to the difference in magnification between the near optical axis region and the far optical axis region, resulting in barrel distortion or pincushion distortion in the image. This aberration causes geometric changes in the imaging but does not affect the clarity of the imaging.

For example, if the refractive index of the imaging surface in the meridian field is F′ _T (D), the refractive index of the imaging surface in the sagittal field is T′ _S (D), the image height of the ideal image is MQ′, and the image height of the real image is MQ", then:

Oblique astigmatism = F' _T - F' _S

It can be seen from formula (1) that when the lens is designed with a spherical surface, the three aberrations of oblique astigmatism, refractive index error and distortion can only be eliminated by adjusting the radius of curvature of the two spherical surfaces. But in fact, if the aberrations are eliminated only by the curvature radii of the two spherical surfaces, the elimination of one aberration will increase the other two aberrations.

Therefore, the spherical design of the lens cannot make the lens both thin and effectively eliminate the three aberrations of oblique astigmatism, refractive index error and distortion at the same time.

In order to solve this problem, currently more lenses are designed with aspheric surface, wherein aspheric lens means that at least one refraction surface is aspherical.

The lens adopts aspherical design, which can effectively eliminate three kinds of aberrations including oblique astigmatism, refractive index error and distortion. For example, in the Chinese patent application publication No. CN1212766 published on March 31, 1999, a design of an aspheric lens is disclosed, which changes the curvature at each point of the lens by introducing a high-order term, thereby reducing the curvature at each point. However, the high-order items introduced in this technology include both odd-order items and even-order items, which will lead to asymmetry of the refractive surface of the lens and easily form the above-mentioned three kinds of aberrations, so it is difficult to design and process Lenses that meet the requirements.

In the open specification of Chinese patent application No. CN1412604 published on April 23, 2003, another aspheric lens design is disclosed, wherein at least one refraction surface of the lens is aspherical, and the aspheric calculation formula (2) Takes the following form:

$\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="cr"\; filenames="C20051003447400151.png,C20051003447400162.png"\; state="\{\{state\}\}">cr</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; c</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="r"\; filenames="C20051003447400151.png,C20051003447400162.png"\; state="\{\{state\}\}">r</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="4"\; label="r"\; filenames="C20051003447400162.png"\; state="\{\{state\}\}">r</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 8</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 10</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 5</span>}}{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

In the formula, z(r) represents the depth of the mirror surface, c represents the curvature of the apex of the aspheric surface, r represents the vertical distance from a certain point on the surface of the lens to the optical axis, a ₁ , a ₂ , a ₃ , a ₄ , and a ₅ represent the height of the aspheric surface Subterm coefficient.

In the lens aspheric design formula, although an even-order term is introduced to design the aspheric lens so that the refractive surface is symmetrical. However, the constant value of the conic curve in this design is 1, which indicates that the thickness of the aspheric lens cannot be effectively reduced. If r has the power of 12, it is difficult to effectively eliminate oblique astigmatism and refractive index error at the same time. and distortion three aberrations.

【Content of invention】

In view of this, it is necessary to provide a method for designing an aspheric lens so that the aspheric lens can effectively eliminate aberrations.

A method for designing an aspheric lens, the main steps of which include:

Set the refractive index, refractive index and diameter of the aspheric lens;

Spherical design is adopted, and the two radii of curvature of the aspheric lens are designed to achieve the above-mentioned set refractive index, and the distortions of 0.5 field of view, 0.7 field of view and 1.0 field of view are minimized by using the two radii of curvature;

The first aspheric surface and the second aspheric surface of the aspheric lens are designed by using the lens aspheric surface design formula, and the lens aspheric surface design formula is:

$\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; v</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="r"\; filenames="C20051003447400151.png,C20051003447400162.png"\; state="\{\{state\}\}">r</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; PC</span>}}_{\mathrm{<span\; class="notranslate">\; v</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="r"\; filenames="C20051003447400151.png,C20051003447400162.png"\; state="\{\{state\}\}">r</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="4"\; label="Br"\; filenames="C20051003447400162.png"\; state="\{\{state\}\}">Br</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Cr</span>}}^{\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Dr.</span>}}^{\mathrm{<span\; class="notranslate">\; 8</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Er</span>}}^{\mathrm{<span\; class="notranslate">\; 10</span>}}$

In the formula, Z(r) represents the depth of the mirror surface, C _v represents the curvature of the apex of the aspheric surface, r represents the vertical height from any point on the mirror surface to the optical axis, P represents the constant value of the quadratic curve, and B, C, D, E represent the aspheric surface of

Optimal design of the first aspheric surface and the second aspheric surface by damping least square method;

Wherein, when optimizing the design, the aspheric lens is used to eliminate aberrations in at least five fields of view, and obtain the aspheric coefficients P, B, C, D, and E of the first aspheric surface and the second aspheric surface.

Compared with the prior art, in the design method of the aspheric lens, two aspheric surfaces are simultaneously designed and at least five fields of view are used for optimal design. Compared with the design method of the aspheric lens with less than five fields of view for optimal design, it can further eliminate the aberration maximum value that exists in some fields of view, so that the aspheric lens designed by the design method of the aspheric lens of the present invention Spherical lens, oblique astigmatism, refractive index error and distortion are small in the entire field of view range from 0 field of view to 1.0 field of view, where 1.0 field of view refers to the distance between the refracted beam on the image side and the optical axis of the lens when it passes through the focal point of the image side The included angle is 30 degrees.

【Description of drawings】

FIG. 1 is a design flowchart of a preferred design method of an aspheric lens according to the present invention.

FIG. 2 is a schematic structural diagram of an aspheric diverging lens designed by the aspheric lens design method of the present invention.

FIG. 3A is a graph showing oblique astigmatism and refractive index error curves of an aspheric diverging lens designed by the aspheric lens design method of the present invention in five optimally designed fields of view.

FIG. 3B is a graph of distortion curves of five optimally designed aspheric diverging lenses designed by the aspheric lens design method of the present invention.

FIG. 4 is a schematic structural diagram of an aspheric converging lens designed by the aspheric lens design method of the present invention.

FIG. 5A is a graph showing oblique astigmatism and refractive index error curves of seven optimally designed aspheric converging lenses designed by the aspheric lens design method of the present invention.

FIG. 5B is a graph of distortion curves of seven optimally designed aspheric converging lenses designed by the aspheric lens design method of the present invention.

【Detailed ways】

As shown in Figure 1, it is a design flowchart of a preferred design method of aspheric lens of the present invention, and its main steps include:

The first step is to set the refractive index, refractive index and diameter of the aspheric lens to be designed.

The second step is to use the spherical design to design the two radii of curvature of the aspheric lens to make it reach the above-mentioned refractive index, and use the two radii of curvature to make the distortion in the field of view of 0.5, 0.7 and 1.0 be min.

The third step is to define the first aspheric surface and the second aspheric surface of the aspheric lens by using a lens aspheric surface design formula.

The lens aspheric design formula (3) is:

$\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="C\; v\; r"\; filenames="C20051003447400151.png,C20051003447400162.png"\; state="\{\{state\}\}">C</figure-callout></span><figure-callout\; id="2"\; label="C\; v\; r"\; filenames="C20051003447400151.png,C20051003447400162.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; v</span>}}{}^{}{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; PC</span>}}_{\mathrm{<span\; class="notranslate">\; v</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="r"\; filenames="C20051003447400151.png,C20051003447400162.png"\; state="\{\{state\}\}">r</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="4"\; label="Br"\; filenames="C20051003447400162.png"\; state="\{\{state\}\}">Br</figure-callout></span>}}^{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Cr</span>}}^{\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Dr.</span>}}^{\mathrm{<span\; class="notranslate">\; 8</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Er</span>}}^{\mathrm{<span\; class="notranslate">\; 10</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

Among them, Z(r) in the formula represents the depth of the mirror surface, C _v represents the curvature of the apex of the aspheric surface, r represents the vertical height from any point on the mirror surface to the optical axis, P represents the constant value of the quadratic curve, B, C, D, E represent the non-spherical Higher order coefficients for the sphere.

The fourth step is to use the damped least square method to optimize the design, so that the aspheric lens can eliminate oblique astigmatism, refractive index error and distortion in multiple fields of view.

In the optimal design of aspheric lenses to eliminate aberrations, aberrations are generally eliminated at certain specific viewing angles, where the viewing angle refers to the angle between the refracted beam on the image side and the optical axis of the lens when it passes through the focal point of the image side. Therefore, when designing an aspheric lens in this process, we began to eliminate oblique astigmatism, refractive index error and distortion in five fields of view.

The optimal design of the design method of this aspherical lens can adopt damping least square method, and its performance function (4) can be defined as:

$\mathrm{<span\; class="notranslate">\; \&phi;</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; e</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

Among them, w _i is the weight factor in the formula, and its value is taken as w _i > 0. The value of the weight factor is determined according to the importance of the corresponding aberration of the item. If the aberration to be eliminated by the item is very strict, then The value of which weight factor can be obtained is larger; m is the number of optimized items, and its value is an integer greater than or equal to 1; e _i is the correction item considered, that is, a certain aberration, and the number of items considered e _i is The value m; t _i is the target value of a certain aberration, and the value of the target value t _i depends on the situation of e _i .

Eliminate the three aberrations of oblique astigmatism, refractive index error and distortion in the five fields of view, that is, in the five fields of view, make the target value t _i of the three aberrations in the merit function (4) be zero .

The fifth step is to calculate the aspheric coefficients P, B, C, D, E of the first aspheric surface and the second aspheric surface of the aspheric lens from the solution of the damped least square method.

When the target value t _i of the oblique astigmatism, refractive index error and distortion in the performance function (4) is zero, the first aspheric surface sum of the aspheric lens is obtained by the solution of the damped least square method The aspheric coefficient of the second aspheric surface is used to obtain the design result of the aspheric lens.

The sixth step is to judge whether the oblique astigmatism, refractive index error and distortion of the aspheric lens in the entire field of view range from 0 field of view to 1.0 field of view can meet the requirements.

It can be judged by drawing oblique astigmatism, refractive index error and distortion from 0 field of view to 1.0 field of view, and judging the relative oblique astigmatism, refractive index error and distortion of these graphs from 0 field of view to 1.0 field of view In the entire field of view, whether the maximum value of aberration meets the requirements.

The seventh step is to determine whether it is necessary to increase the field of view for optimal design based on the judgment results. If the three aberrations cannot meet the requirements in a certain field of view, the field of view that cannot meet the requirements can be optimized one by one, that is, to eliminate the existing image. The maximum value of the difference exceeds the required oblique astigmatism, refractive index error and distortion of a certain field of view until the requirement is met.

The design method of the above-mentioned aspheric lens eliminates oblique astigmatism, refractive index error and distortion in at least five fields of view, so that the aspheric lens designed by the design method of the aspheric lens has a field of view of 0 to 1.0 All three aberrations, oblique astigmatism, refractive index error, and distortion, are small across the entire field of view. Therefore, the design method of the aspheric lens can design the aspheric lens with better optical performance.

As shown in Figure 2, it is a schematic diagram of the structure of the aspheric diverging lens designed by the design method of the above-mentioned aspheric lens, the aspheric diverging lens includes a first aspheric surface 1 and a second aspheric surface 2, the The curvature radius is R ₁ , the curvature radius of the second aspheric surface 2 is R ₂ , the lens center thickness is t ₁ , and the lens diameter is DA ₁ .

The design method of the aspheric diverging lens can be designed by using the lens aspheric surface design formula (3), wherein the aspheric coefficients of the first aspheric surface 1 are set to P ₁ , B ₁ , C ₁ , D ₁ , E ₁ , The aspheric coefficients P ₂ , B ₂ , C ₂ , D ₂ , E ₂ of the second aspheric surface 2, where P ₁ and P ₂ are the conic constant values of the first aspheric surface 1 and the second aspheric surface 2 respectively, B ₁ , C ₁ , D ₁ , E _{1 ,} and B ₂ , C ₂ , D ₂ , and E ₂ are high-order coefficients of the first aspheric surface 1 and the second aspheric surface 2, respectively.

The design of the aspheric diverging lens is optimized by the damped least squares method, and the performance function of the damped least squares method is shown in formula (4). The design of the aspheric diverging lens adopts at least five fields of view to eliminate aberrations, that is, the target value t _i of the merit function (4) is zero in at least five fields of view.

In the performance function (4), the design of the aspheric diverging lens expresses the process of correcting the aberration to the target value as having 10 variable design parameters, that is, the performance function of 10 design variables. The aspheric coefficients P ₁ , B ₁ , C ₁ , D ₁ , E ₁ of the first aspheric surface 1 and the second aspheric surface are represented by x ₁ , x ₂ , x ₃ , ... x ₉ , x ₁₀ 2 aspheric coefficients P ₂ , B ₂ , C ₂ , D ₂ , E ₂ .

……1.0视场而定，再采用m/3项来消除折光率误差和m/3项来消除畸变。则可定义一个误差函数(5)为：According to the performance function (4), there are m optimization items, and the design method of the aspheric diverging lens can use m/3 items to eliminate oblique astigmatism, and the items are based on the size of the field of view

... 1.0 field of view, then use the m/3 term to eliminate refractive index error and the m/3 term to eliminate distortion. Then an error function (5) can be defined as:

f _i =W _i (e _i -t _i )=f _i (x ₁ , x ₂ , x ₃ ... x _n ), i=1, 2, ..., m(5)

To calculate the aspheric coefficients P ₁ , B ₁ , C ₁ , D ₁ , E ₁ of the first aspheric surface 1 and the aspheric coefficients P ₂ , B ₂ , C ₂ , D ₂ , E ₂ of the second aspheric surface 2 value. Let the variables before optimization be represented by x ₁₀ , x ₂₀ , x ₃₀ , ..., x _n0 , n=10; aberrations are represented by f ₁₀ , f ₂₀ , f ₃₀ , ..., f _m0 means, optimized variables are represented by x ₁ , x ₂ , x ₃ , ... x ₉ , x _n , n=10; aberrations are represented by f ₁ , f ₂ , f ₃ , ... ..f _m-1 and f _m represent.

The solution (6) of the damped least squares method is:

X＝(A ^T A+pI) ^-1 A ^T f ₀ (6)

The symbols in the formula can be defined by matrices:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 10</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 20</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 30</span>}}\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right)<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}\end{array}\right)<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 10</span>}}\\ {\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 20</span>}}\\ {\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 30</span>}}\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ {\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right)<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; <figure-callout\; id="3"\; label="f"\; filenames="C20051003447400162.png"\; state="\{\{state\}\}">f</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ {\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\end{array}\right)<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 10</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 20</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 30</span>}}\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}\right)$

A is an m×n matrix, where $A_{\mathrm{ij}}=\frac{{\&PartialD;f}_i}{{\&PartialD;x}_j},$ i=1, 2, . . . , m, j=1, 2, . . . , n, n=10. Where ^AT is the transposed matrix of A, p is the damping factor, I is the identity matrix, ( ^AT A+pI) ^-1 means to invert the matrix of ( ^AT A+pI), through the operation of the above matrix, we can get The calculated value of X, through x=x ₀ +X, can determine the value of x, and then can get the corrected P ₁ , B ₁ , C _{1 , D 1} , E ₁ , P ₂ , B _{2 ,} C ₂ , Values of D ₂ and E ₂ .

Assuming that the aspheric divergence lens designed by the design method of the above-mentioned aspheric lens is optimized in five fields of view of 0.3 field of view, 0.5 field of view, 0.7 field of view, 0.85 field of view and 1.0 field of view, wherein the aspheric divergence The material of the lens is a plastic with a refractive index of 1.586. The design parameters of the aspheric diverging lens are shown in Table 1.

Table 1

In performance function (4), weight factor w ₁ =w ₂ =...=w ₁₄ =w ₁₅ =1, target value t ₁ =t ₂ =...=t ₁₄ = t ₁₅ =0, through calculation, it can be obtained that Fig. 3A shows the oblique astigmatism and refractive index error curves of the aspheric diverging lens designed by the design method of the aspheric lens in five viewing fields, as shown in Fig. 3B Distortion curves of the aspheric diverging lens designed for the design method of the aspheric lens at five optimally designed fields of view.

Wherein, in FIG. 3A, the axis of abscissa represents the magnitude of the refractive index, and the axis of ordinate represents the magnitude of the field of view; in FIG. 3B, the axis of abscissa represents the magnitude of distortion, and the axis of ordinate represents the magnitude of the field of view. In Fig. 3A, F' _T is the refractive index of the imaging surface in the meridian field, F' _S is the refractive index of the imaging surface in the sagittal field, FPS=F _v is the refractive index of the lens, and the curves in Fig. 3B are the distortion in each field of view size in , then:

Oblique astigmatism = F' _T - F'_S;

From Figure 3A and Figure 3B, it can be seen that the aspheric diverging lens designed by the aspheric lens design method has oblique astigmatism less than 0.044D in the entire field of view range from 0 field of view to 1.0 field of view, and the refractive index The error is less than 0.041D, and the distortion is less than 0.067%. These three aberration values are corrected very small, so the three aberrations of oblique astigmatism, refractive index error and distortion have been effectively eliminated.

As shown in Figure 4, the structural representation of the aspherical converging lens designed for the design method of the aspheric lens, the aspheric converging lens comprises a first aspheric surface 3 and a second aspheric surface 4, the first aspheric surface 3 The curvature radius is R ₃ , the curvature radius of the second aspheric surface 4 is R ₄ , the lens center thickness is t ₂ , and the lens diameter is DA ₂ .

The design method of the aspheric converging lens is basically the same as the design method of the above-mentioned aspheric diverging lens, the difference is that in the design method of the aspheric converging lens, oblique astigmatism is eliminated in at least 7 fields of view , Refractive index error and distortion.

The aspheric converging lens designed by the design method of the aspheric lens is carried out in seven fields of view of 0.3 field of view, 0.5 field of view, 0.7 field of view, 0.85 field of view, 0.9 field of view, 0.95 field of view and 1.0 field of view Optimal design, wherein the material of the aspheric converging lens is a kind of plastic, the refractive index is 1.586, and the design parameters of the aspheric converging lens are shown in Table 2.

Table 2

In the performance function (4), select w ₁ =2, w ₂ =4, w ₃ =5, w ₄ =8, w ₅ =9, w ₆ =9, w ₇ =10 as the oblique astigmatism weight factors, Choose w ₈ = 1, w ₉ = 2, w ₁₀ = 3, w ₁₁ = 10, w ₁₂ = 12, w ₁₃ = 12, w ₁₄ = 14 as the refractive index error weight factors, choose w ₁₅ = 0.2, w ₁₆ =0.5, w ₁₇ =0.5, w ₁₈ =0.8, w ₁₉ =1, w ₂₀ =1, w ₂₁ =1.8 are distortion weight factors, target value t ₁ =t ₂ =...=t ₂₀ = t ₂₁ =0, through calculation, it can be obtained that Fig. 5A shows the oblique astigmatism and refractive index error curves of the aspheric converging lens designed by the design method of this aspheric lens in seven field of view optimization designs, Fig. 5B Distortion curves of seven optimally designed aspheric converging lenses designed for this aspheric lens design method.

Wherein, in FIG. 5A, the axis of abscissa represents the magnitude of the refractive index, and the axis of ordinate represents the magnitude of the field of view; in FIG. 5B, the axis of abscissa represents the magnitude of distortion, and the axis of ordinate represents the magnitude of the field of view. In Fig. 5A, F' _T is the refractive index of the imaging surface in the meridian field, F' _S is the refractive index of the imaging surface in the sagittal field, FPS=F _v is the refractive index of the lens, and the curve in Fig. 5B is the distortion in each view The size of the field, then:

Oblique astigmatism = F' _T - F'_S;

From Figure 5A and Figure 5B, it can be seen that the aspheric converging lens designed by the design method of the aspheric lens has oblique astigmatism less than 0.0377D in the entire field of view range from 0 field of view to 1.0 field of view, and the refractive index The error is less than 0.0815D, and the distortion is less than 1.1493%. These three aberration values are all corrected very small, so oblique astigmatism, refractive index error and distortion have been effectively eliminated at the same time.

In summary, the design method of the aspheric lens can effectively eliminate aberrations. However, the above description is only the preferred implementation mode. For those who are familiar with the technology of this case, the equivalent modifications or changes made in accordance with the creative spirit of this case should be included in the scope of the following patent application.

Claims (9)

1. the method for designing of an aspherical lens, the method for designing key step of this aspherical lens comprises: index of refraction, refractive index and the diameter of setting this aspherical lens;

The employing sphere designs, and designs two radius-of-curvature of this aspherical lens, makes its index of refraction that reaches above-mentioned setting, and utilizes these two radius-of-curvature to make the distortion of 0.5 visual field, 0.7 visual field and 1.0 visual fields be minimum value;

Adopt eyeglass aspheric surface design formula to design first aspheric surface and second aspheric surface of this aspherical lens, this eyeglass aspheric surface design formula is:

$Z\left(r\right)=\frac{C_v{r}^2}{1+\sqrt{1-{\mathrm{PC}}_v^2{r}^2}}+{\mathrm{Br}}^4+{\mathrm{Cr}}^6+{\mathrm{Dr}}^8+{\mathrm{Er}}^{10}$

In the formula, Z (r) the expression minute surface degree of depth, C _vThe curvature on expression aspheric surface summit, r represents that any point is to the vertical height of optical axis on the minute surface, and P represents the conic constant value, and B, C, D, E represent aspheric high-order term coefficient;

Adopt damped least square method that first aspheric surface and second aspheric surface are optimized design;

It is characterized in that: be optimized when design, make this aspherical lens eliminate aberrations, and draw above-mentioned first aspheric surface and the second aspheric asphericity coefficient P, B, C, D, E at least five visual fields.

2. the method for designing of aspherical lens as claimed in claim 1, it is characterized in that: this aspherical lens is an aspherical focusing lens, then optimal design is to eliminate aberrations at least seven visual fields.

3. the method for designing of aspherical lens as claimed in claim 1 or 2 is characterized in that: this optimal design adopts the performance function of damped least square method to be:

$\mathrm{\&phi;}=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\left[w_i(e_i-t_i)\right]}^2$

Wherein, w in the formula _iBe weight factor, the item number of m for optimizing, e _iFor the correction term of being considered is certain aberration, t _iDesired value for certain aberration.

4. the method for designing of aspherical lens as claimed in claim 3, it is characterized in that: among the item number m of this optimization, the m/3 item is used to eliminate the oblique fire astigmatism, and the m/3 item is used to eliminate the index of refraction error, and the m/3 item is used for eliminating distortion.

5. the method for designing of aspherical lens as claimed in claim 1 or 2, it is characterized in that: this optimal design is eliminated aberration in 0 visual field to 1.0 field range, and wherein 1.0 visual fields are meant that field angle is the position of 30 degree.

6. the method for designing of aspherical lens as claimed in claim 1, it is characterized in that: these five visual fields are 0.3 visual field, 0.5 visual field, 0.7 visual field, 0.85 visual field and 1.0 visual fields.

7. the method for designing of aspherical lens as claimed in claim 6, it is characterized in that: this aspherical lens is that aspheric surface is dispersed eyeglass, the index of refraction of dispersing eyeglass when this aspheric surface of setting is-14.00D, refractive index is 1.586, optic diameter is 30mm, center thickness is 1mm, the first aspheric radius-of-curvature is 36.07206cm, the second aspheric radius-of-curvature is 19.26785cm, and in this 0.3 visual field, 0.5 visual field, 0.7 visual field, 0.85 the oblique fire astigmatism is eliminated in these five visual fields of visual field and 1.0 visual fields, during the distortion of index of refraction sum of errors, this aspheric surface is dispersed first aspheric surface and the second aspheric asphericity coefficient such as the following table of eyeglass:

8. the method for designing of aspherical lens as claimed in claim 2, it is characterized in that: these seven visual fields are 0.3 visual field, 0.5 visual field, 0.7 visual field, 0.85 visual field, 0.9 visual field, 0.95 visual field and 1.0 visual fields.

9. the method for designing of aspherical lens as claimed in claim 8, it is characterized in that: when the index of refraction of setting this aspherical focusing lens is 14.00D, refractive index is 1.586, optic diameter is 32mm, center thickness is 9.2mm, the first aspheric radius-of-curvature is 26.27803cm, the second aspheric radius-of-curvature is 50.45961cm, and in this 0.3 visual field, 0.5 visual field, 0.7 visual field, 0.85 visual field, 0.9 visual field, 0.95 the oblique fire astigmatism is eliminated in seven visual fields in visual field and 1.0 visual fields, during the distortion of index of refraction sum of errors, first aspheric surface of this aspherical focusing lens and second aspheric asphericity coefficient such as the following table:

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