KR101425780B1 - Reconstruction and method for unknown aspheric surface equations from measurement of aspheric surfaces - Google Patents

Abstract

A method of restoring an aspherical surface equation, comprising the steps of: measuring an aspheric surface shape of a lens; A spherical equation data deriving step; Deriving an error curve data; An aspherical surface coefficient derivation step of obtaining an aspherical surface coefficient from an error curve data by an inverse matrix method; An iterative calculation step of calculating until the data difference between the measurement data of the measurement step and the data derived from the newly obtained aspherical surface equation is within a predetermined range; A parameter derivation step of deriving an aspherical equation parameter at that time when the data difference falls within a certain range; And restoring the aspherical surface equation by applying the parameters derived in the parameter deriving step. The present invention relates to a principle of restoring an aspheric surface equation and a restoration method by surface measurement of an arbitrary aspherical surface.

Description

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an aspheric surface equation for an aspheric surface,

In the present invention, the surface of an arbitrary aspherical lens is measured so that an aspherical lens whose aspherical surface equation is not known can be manufactured by preparing the same lens or supplementing optical characteristics. From the measured value, the radius of curvature of the lens, And restoring and restoring the aspheric equations by inversely calculating the constant and aspheric coefficient values.

Generally, aspherical lenses eliminate spherical aberration and astigmatism to improve the light condensing ability and simplify the construction of complex optical systems. Therefore, the aspherical lens can be used in scientific and industrial fields such as terahertz imaging, diagnosis system and night vision system , Has attracted much attention in military optical systems.

Particularly, when such an aspherical shape is applied to a spectacle lens, various advantages are obtained. If the spectacle lens is manufactured with a spherical surface, the focal distance at which the light passing through the paraxial portion of the lens is caused by the spherical aberration moves toward the lens toward the peripheral portion, thereby increasing the refractive power. As the focal length becomes shorter than the paraxial axis, the spherical aberration that the image is blurred can be removed to the aspherical shape, and the oblique astigmatism generated when the light is incident on the lens at an angle is eliminated, And the thickness of the spectacle lens can be made thinner than that of the spherical lens of the same power, so that it is possible to satisfy both light and aesthetic aspects at the same time.

However, aspheric lenses with these various advantages are difficult to design and manufacture than spherical lenses. Also, aspheric lenses are always manufactured by aspheric equations, and most of the important parameters included in the aspheric equations for the aspheric lenses are not disclosed.

Korean Patent No. 10-0490345 is known as a prior art relating to an aspherical spectacle lens and a method of designing the aspherical spectacle lens, and aspherical surface equations for deriving a general aspheric surface equation and the following formula of a meridional surface power are described.

That is, the conventional technique does not derive an aspherical surface equation by measuring the shape of an aspherical surface that has already been manufactured, and forms an aspherical surface curve by using an equation using a predetermined coefficient to form an aspherical surface. If such an important parameter such as aspherical surface coefficient applied to the aspherical surface equation is not known, there is a problem that the aspheric surface can be formed only by the aspherical surface equation or the aspheric surface can not be corrected to have complementary optical characteristics.

That is, in the case of any aspherical lens which does not know the parameters of the aspheric equation, it is not easy to measure the exact optical characteristics that it possesses. Therefore, existing aspheric lenses in which the aspheric equation is unknown are redesigned to complement optical performance or function The difficult thing is the reality.

SUMMARY OF THE INVENTION The present invention has been made to solve the above-mentioned problems, and it is an object of the present invention to provide a principle and a method for restoring an aspherical surface equation by measuring the surface of an arbitrary aspherical surface shape and calculating inversely curvature radii, conic constants, .

According to an aspect of the present invention, there is provided a method of restoring an aspheric surface equation and a method of restoring an aspheric surface equation, the method comprising: measuring an aspheric surface shape; A spherical equation data deriving step; Deriving an error function data; An aspherical surface coefficient derivation step of obtaining an aspherical surface coefficient from an error function data by an inverse matrix method; An iterative calculation step of calculating the data until the data difference between the measurement data measured in the measurement step and the data derived from the newly obtained aspherical surface equation is within a predetermined range; A parameter derivation step of deriving an aspherical equation parameter at that time when the data difference falls within a certain range; And restoring the aspherical surface equation by applying the parameters derived in the parameter deriving step.

The spherical equation data deriving step derives the data of the spherical equation while changing the radius of curvature by setting the cone constant to 0 in the spherical equation, and deriving the error function data by deriving the spherical equation data from the measurement data.

Wherein the aspheric surface data deriving step derives the aspherical surface data by substituting the curvature radius and the aspheric surface coefficient derived in the step of deriving the aspheric surface coefficient into a newly obtained aspheric surface equation, and in the parameter deriving step, the aspheric surface equation parameter is a radius of curvature, To derive the aspherical surface coefficient

The aspherical surface equation can be restored by measuring the surface of an arbitrary aspherical surface shape through the method comprising the steps described above.

The present invention has the effect of restoring the original aspherical surface equation of the aspherical surface shape using the inverse matrix method from the data of surface measurement of an unknown aspherical surface whose parameters of the aspherical surface equation are unknown.

Also, by using each parameter of the restored aspherical surface equation, it is possible to apply to the redesign of the aspherical shape having the improved performance.

1 is a view illustrating a method of restoring an aspheric surface equation by surface measurement of an arbitrary aspherical lens according to the present invention.
2 is a diagram showing a state in which the shape of an aspherical lens is measured by a three-dimensional measuring instrument.
Fig. 3 is a diagram showing a screen for measuring a shape of an aspherical lens and showing measured values by a CALYPSO program. Fig.
Fig. 4 is a diagram showing an error curve showing the difference between an aspherical shape and a spherical shape and between these two shapes.
5 is a view showing a shape error between an aspherical equation curve and a curve determined from measurement data.
6 is a view showing an aspherical shape realized by an aspheric surface equation restored by the inverse matrix method of the present invention.

Hereinafter, the restoration principle of the aspherical surface equation and the method of restoring the aspheric surface equation according to the present invention will be described in detail.

The restoration principle of the aspherical surface equation and the method of restoring the aspheric surface equation according to the present invention are as shown in FIG. 1,

An aspherical surface measuring step S1 for measuring the shape of the aspherical lens;

A spherical equation data deriving step (S2);

An error curve data deriving step (S3);

An aspherical surface coefficient deriving step (S4) of obtaining an aspherical surface coefficient from an error curve data by an inverse matrix method;

An iterative calculation step (S5) of calculating the data until the data difference between the measurement data measured in the measurement step and the data derived from the newly obtained aspherical surface equation is within a predetermined range;

A parameter derivation step (S6) of deriving an aspherical equation parameter at that time when the data difference falls within a certain range;

And restoring the aspherical surface equation by applying the parameters derived in the parameter deriving step (S7).

사이를 3.5mm 간격으로 측정한다. 또한 상기 3차원 측정기(2)는 최대 분해능이 0.2㎛이고 반복정밀도는 1.8㎛인 성능을 가지는 것을 사용하였고, 측정 프로프램은 3차원 측정이 가능한 공지의 CALYPSO 프로그램(3)을 사용하였다. More specifically, as shown in Fig. 2, the aspheric surface of the aspheric lens is measured by the three-dimensional measuring device 2 with the aspherical lens 1 having an arbitrary shape. For example, in the case of the aspherical surface lens 1 having a diameter of 63 mm,

Are measured at intervals of 3.5 mm. In addition, the above-mentioned three-dimensional measuring instrument 2 has a performance with a maximum resolution of 0.2 탆 and a repetition accuracy of 1.8 탆, and the measurement program uses a known CALYPSO program (3) capable of three-dimensional measurement.

까지 3.5 mm 간격으로 측정한 총 10개의 측정데이터를 사용한다.As shown in FIG. 3, when the aspheric lens having a diameter of 63 mm is measured using the above-mentioned three-dimensional measuring instrument, 19 points on the r axis are measured using the CALY PSO program, the shape of the aspherical lens can be measured. In this case, the above shape is axisymmetric

And the total of 10 measurement data measured at 3.5 mm intervals.

Next, in the spherical equation, deriving the data of the spherical equation while changing the radius of curvature by setting the cone constant at 0, deriving the error function data for deriving the spherical equation data from the measurement data, deriving the error function data, and deriving the aspherical surface coefficient A step of deriving aspherical surface data by substituting the obtained radius of curvature and the aspheric surface coefficient into the newly obtained aspherical surface equation and the step of deriving the aspherical surface equation parameters such as the radius of curvature and the conic constant and aspheric surface coefficient will be described.

In general, a two-dimensional shape with arbitrary curves is represented by various coordinates (x ₀ , y ₀ ), (x ₁ , y ₁ ), ... , and (x _n , y _n ), respectively. Such data can be represented by an n-th order polynomial as shown in the following equation (1).

(1)

(One)

In the equation (1), a ₀ , a ₁ , ... , a _n are polynomial coefficients, and n + 1 (x, y) is substituted into the equation (1), and n + 1 equations are obtained as shown in the following equation (2).

(2)

(2)

Equation (2) represents the unknown coefficients a ₀ , a ₁ , ... , and a _n . However, equation (2) can be thought of as a linear simultaneous equations for a _i coefficients (where i = 0, ..., n) since the values of (x, y) can be determined through measurements. Therefore, this equation can be expressed as a matrix multiplication expressed as Xa = b. Here, X denotes a variable matrix, a denotes a coefficient vector, and b denotes a function vector, which can be expressed by a matrix equation as shown in the following equation (3).

(3)

(3)

Since the value to be obtained from the matrix equation of equation (3) is the elements of coefficient vector a, the coefficient vector a can be obtained by multiplying by the inverse matrix X ^-1 of X, which can be expressed by the following equation ).

(4)

(4)

To calculate the inverse of X in Eq. (4), X must be invertible. The determinant det (X) must not be zero. In fact, if a is obtained by Eq. (4), the original equation can be reconstructed from the coefficients that are elements of vector a. If you measure the curve given by a particular equation, the measured value will be the value of y for x. One important point here is that even if you do not know the equation of the curve, the measure y will be the product of the appropriate exponent of x multiplied by the coefficient. Therefore, the coefficients are the elements of vector a obtained from equation (4). Hereinafter, the above method is referred to as an inverse matrix method.

In general, the aspherical equation is given by the following equation (5).

(5)

(5)

In Equation (5), K is a conic constant, r is a lateral coordinate, R is a radius of curvature, and a _2i is an aspherical coefficient.

To demonstrate the validity of the inverse matrix method, we assume an arbitrary aspheric equation z _as . The radius of curvature at the apex of this aspheric equation is R = 89 mm, diameter φ = 70 mm, cone constant K = 0, aspherical coefficients a ₄ = -5 × 10 ^-9 , a ₆ = -2.075 × 10 ^-9 , Assuming that a ₈ = 7.65625 × 10 ^-13 and a ₁₀ = -1.67481 × 10 ^-16 , the aspherical equation must be restored from the coordinate data (r, z _as ) of the equation (5) having known parameters. In the case of K = 0 in the equation (5) and the aspherical coefficient is omitted, the aspherical equation of the equation (5) will be a spherical equation.

를 계산해야 한다. 이 오차곡선에서 z_as은 수식(5)의 비구면방정식을 말하는 것이고, z_s는 수식(5)의 비구면방정식에서 고차항을 제외한 이차항만을 말하는 것이다.Therefore, the radius of curvature R, which is the radius of the circle, must be found. In order to obtain the radius of curvature R,

. In this error curve, z _as refers to the aspheric equation of Equation (5), and z _s refers to the secondary term except the high order in the aspheric equation of Equation (5).

이다. 비구면 계수는 상기 오차곡선으로부터 역행렬 방법을 이용하여 구할 수 있게 된다.As shown in FIG. 4, when the aspherical surface shape, the spherical shape, and the error curve indicating the difference between the two shapes are compared, the curve a shows the aspheric equation z _as And the curve (b) is the spherical equation z _s (C) is an error curve showing the difference between the aspherical shape and the spherical shape

to be. The aspherical surface coefficient can be obtained from the error curve using the inverse matrix method.

로부터 11개의 데이터 점들을 얻을 수 있다. 여기에 11개의 r 값들이 있기 때문에 11개의 비구면 계수들을 구할 수 있는 것이다. As shown in Fig. 4, when the coordinate r in the horizontal direction changes from 3.5 mm intervals from 0 to 35 mm

≪ / RTI > Since there are 11 r values, 11 aspheric coefficients can be obtained.

Among the aspheric coefficients of equation (5), the aspherical coefficients to be derived are the coefficients corresponding to the 4th, 6th, 8th, and 10th orders. Therefore, the constant term, the second order, and the odd order coefficients are omitted in the matrix calculation process of the error curve.

Since the calculation of the error curve is in the form of Xa = b as in equation (3), it can be expressed as the following equation (6).

(6)

(6)

In equation (6), X is an 11 × 4 matrix and a is a 4 × 1 vector, so b must be an 11 × 1 vector. To obtain the coefficient vector a, both sides of Eq. (6) must be multiplied by the inverse matrix X ^-1 . Here, since X ^-1 is a 4 × 11 matrix, the product of X ^-1 and b will be a 4 × 1 vector. Since this result is the same as a vector, a matrix operation is established and this relationship is shown in the following equation (7).

(7)

(7)

However, only the coordinate data (r, z _as ) can be known through the surface measurement of the aspherical surface, and the values for the radius of curvature R, the conical constant K, and the aspherical coefficient a included in the aspherical surface equation are unknown. Therefore, the radius of curvature R, the cone constant K, and the aspheric surface coefficient a can be obtained by the following equation (8).

(8)

(8)

는 오차곡선이고, z_N은 측정데이터로부터 결정되는 곡선이며, a_N4, …, a_N10은 역행렬 방법으로부터 계산된 비구면 계수이고, 그리고

는 원래의 곡선 z_as에서 벗어난 정도를 나타내는 형상오차이다. 비구면방정식에 필요한 각 파라미터를 구하기 위한 과정은 다음과 같다.In Equation (8), c and c _N represent the curvatures that are inverses of the curvature radii R and R _N. And

Is the error curve, z _N is the curve determined from the measurement data, a _N4 , ... , a _N10 is the aspherical coefficient calculated from the inverse matrix method, and

Is a shape error that indicates the degree of deviation from the original curve z _as . The procedure for obtaining each parameter required for the aspheric equation is as follows.

(1) Obtain the aspherical equation z _as when r changes from 0 to 35 mm at intervals of 3.5 mm.

(2) Set the cone constant K to 0, and change the radius of curvature R from 20 to 120 mm at intervals of 0.0001 mm.

를 구한다.(3) Error curves for each radius of curvature R

.

로부터 비구면 계수들을 구한다.(4) error curve using inverse matrix method

Aspherical surface coefficients.

(5) Substitute the curvature radius R obtained from equation (4) in the spherical equation z _N and the aspherical coefficients according to the new radius of curvature R _N and the aspherical coefficient a _N calculated from the matrix method, respectively.

가 임계값(threshold value) 내에 있는지를 확인한다.(6) Geometric error

Is within a threshold value.

가 어떤 임계값 안으로 들어올 때까지 반복적으로 수행되었다. 이 경우에는 임계값이 1×10^-5 mm (10 nm)이었고, 그 때 만족하는 새로운 곡률반경 R_N은 89.0013 mm이었다. The above process was run as a MATLAB program,

Was repeatedly carried out until it reached a certain threshold value. In this case, the threshold was 1 × 10 ^-5 mm (10 nm), and the new radius of curvature R _N satisfied at that time was 89.0013 mm.

도 0으로 수렴하는 것을 알 수 있다. 도 5에서 적색선(4)은 원래의 비구면방정식 곡선 z_as이고, 하늘색 선(5)은 측정데이터로부터 역행렬 방법에 의해 결정되는 곡선 z_N이며, 초록색선(6)은 선 (4)와 선 (5) 사이의 형상오차

를 나타낸다. 또한 도 4의 (a), (b), (c), (d)에서 새로운 곡률반경은 (a) R_N=20.0000 mm, (b) R_N=60.0000 mm, (c) R_N=89.0013 mm, (d) R_N=120.0000 mm 이다.5 shows a process for obtaining a new radius of curvature R _N closest to the original radius of curvature R. FIG. As the new radius of curvature R _N approaches the original radius of curvature R,

0 < / RTI > 5, the red line 4 is the original aspheric equation curve z _{as the} sky blue line 5 is the curve z _N determined by the inverse matrix method from the measurement data and the green line 6 is the line 4 and line 5)

. In addition, (a), (b) in FIG. 4, (c), the new radius of curvature in (d) is _{(a) R N = 20.0000 mm} , (b) R N = 60.0000 mm, (c) R N = 89.0013 mm , (d) R _N = 120.0000 mm.

가 구해졌다. 그 후 새로운 곡률반경 R_N이 더욱 커질수록 형상오차

도 다시 커지기 시작했다. 아래 표 1은 새로운 곡률반경 R_N이 89.0013 mm일 때의 위치 r에 따른 다양한 형상오차

를 보여주고 있다. As shown in Fig. 5 (c), the shape error with the minimum value when the new radius of curvature R _N = 89.0013 mm

. Then, as the new radius of curvature R _N becomes larger,

Also began to grow again. Table 1 below shows various shape errors according to the position r when the new radius of curvature R _N is 89.0013 mm

Respectively.

R _N (mm) r (mm) δ z _d (mm) 89.0013 0 0.0000 3.5 9.2793 × 10 ^-7 7 2.8688 × 10 ^-6 10.5 3.8795 × 10 ^-6 14 2.3828 × 10 ^-6 17.5 1.2827 × 10 ^-6 21 4.3100 × 10 ^-6 24.5 2.8575 × 10 ^-6 28 4.1517 x 10 ^-6 31.5 9.7711 × 10 ^-6 35 4.6779 × 10 ^-6

The aspherical coefficients obtained from the inverse matrix method of equation (7) when the new radius of curvature R _N = 89.0013 mm are compared with the original coefficients in Table 2 below. As shown in Table 2, the aspheric coefficients obtained are very similar to the original aspheric coefficients. From this, it can be seen that the inverse matrix method of the present invention is a very effective method for deriving the aspherical surface coefficient.

Aspherical coefficient Original value Derived value a ₄ -5 × 10 ^-9 -4.465 × 10 ^-9 a ₆ -2.075 × 10 ^-9 -2.076 × 10 ^-9 a ₈ 7.65625 × 10 ^-13 7.6632 × 10 ^-13 a ₁₀ -1.67481 × 10 ^-16 -1.6763 × 10 ^-16

Also, in the case of K = 0 and K = -0.8 in Equation (5), the curvature radius R, the conical constant K, and the aspherical coefficient a can also be found using the inverse matrix method. This is because the curvature radius R was obtained when the conical constant K = 0 by the above-described method, and the value was 88.998 mm.

=1×10^-5 mm 보다 작은 값을 나타내는 원추상수 K를 구하였다. The difference between the curvature radius R (88.998) and the previous curvature radius R (89.0013) in this case is due to the difference in the conic constant K value. The cone constant K can now be determined in a manner similar to that for determining the radius of curvature R. [ To obtain the constant K, the radius of curvature R = 88.998 mm was applied to Eq. (8), and the cone constant K was changed from -1 to 1 at intervals of 0.001. Then, the shape error

= 1 × 10 ^-5 mm.

의 최대값이 8.6nm보다 크지 않으므로, 본 발명에 따른 역행열 방법으로 비구면 계수를 만족스럽게 도출할 수 있다는 것을 확인할 수 있다.The cone constant K determined by this procedure was -0.732, and the largest shape error at that time was 8.6152 × 10 ^-6 mm, which was about 8.6 nm. Finally, the aspheric coefficients at that time were a ₄ = -1.7652 × 10 ^-8 , a ₆ = -2.0746 × 10 ^-9 , a ₈ = 7.6537 × 10 ^-13 , and a ₁₀ = -1.6752 × 10 ^-16 . The values of the conical constant K and the aspherical coefficient thus derived are slightly different from the original values, but the shape error

Is not larger than 8.6 nm, it can be confirmed that the aspherical surface coefficient can be satisfactorily derived by the retrograde heating method according to the present invention.

Further, the inverse matrix method according to the present invention is applied to actual situations. The surface of the lens with unknown parameters was measured with the CONTURA G2 (Karl Zeiss) model, a three-dimensional measuring instrument. The parameters derived from the measured coordinate values using the inverse matrix method are as follows: curvature radius R = 121.33 mm, conical constant K = 0, aspherical coefficient a ₄ = 1.4193 × 10 ^-6 , a ₆ = -3.4706 × 10 ^-9 , a ₈ = 3.8349 × 10 ^-12 , and a ₁₀ = -1.5313 × 10 ^-15 .

의 최대값은 2.328×10^-3mm (약 2.3μm)이다. 이 경우에 있어 형상오차

가 이론적인 경우보다 높게 나타나는 주요한 원인은 측정기기의 정확도의 한계 (반복정밀도 1.8 μm)와 어떠한 측정 결과에서도 존재할 수 있는 의심, 즉 측정의 불확정성 때문인 것으로 생각된다. At this time, a shape error indicating the difference between the measured data and the data obtained from the restored aspherical surface equation

Is about 2.328 × 10 ^-3 mm (about 2.3 μm). In this case,

Is considered to be due to the limit of accuracy of the measuring instrument (repetition accuracy: 1.8 μm) and the suspicion that may exist in any measurement result, ie, the measurement uncertainty.

Therefore, this limitation can be sufficiently overcome by reducing the uncertainty of the measurement by using a measurement device having a higher precision and by repeatedly measuring a number of times. As shown in FIG. 2, the surface of the aspherical lens is measured, The aspherical surface shape can be realized as shown in FIG. 6 by an aspheric surface equation obtained by restoring the aspherical surface shape using the inverse matrix method.

1: Aspherical lens
2: 3D measuring instrument
3: CALYPSO program screen

Claims (5)

A method for restoring an aspherical surface equation,
An aspherical surface measuring step of measuring the shape of the aspherical lens;
A spherical equation data deriving step;
Deriving an error curve data;
An aspherical surface coefficient derivation step of obtaining an aspherical surface coefficient from an error curve data by an inverse matrix method;
An iterative calculation step of calculating the difference between the measured data of the measuring step and the data derived from the following aspherical surface equation until the difference in the shape errors is within a predetermined range;
A parameter derivation step of deriving an aspherical equation parameter at that time when the shape error difference is within a certain range;
And restoring the aspherical surface equation by applying the parameters derived in the step of deriving the parameter. The method for restoring aspheric surface equations according to any of the preceding claims, wherein the aspherical surface equations are restored.
[bottom]

The method according to claim 1,
The spherical equation data deriving step is a step of deriving the data of the spherical equation while varying the radius of curvature by setting the cone constant at 0 in the spherical equation to restore the aspherical equation by surface measurement of arbitrary aspherical shape The method according to claim 1,
Wherein the error curve derivation step derives the spherical equation data from the measurement data by subtracting the spherical equation data from the measured data. delete The method according to claim 1,
Wherein the aspheric equation parameter in the parameter derivation step is a radius of curvature, a cone constant, and an aspheric surface coefficient.

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