JP2006350759A - Design method of optical system - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To shorten computation time necessary for optical design. <P>SOLUTION: A design method of an optical system for designing an optical system by differentiating the aberration of the optical system with respect to a parameter (such as radius of curvature, distance or refractive index) defining the optical system to calculate the rate of change corrects the differential rate of change with the rate of change of a variation in a Gaussian image plane due to a slight variation in the parameter defining the optical system. <P>COPYRIGHT: (C)2007,JPO&INPIT

Description

  The present invention relates to a method for designing an optical system used in various optical fields such as a semiconductor exposure apparatus, a liquid crystal exposure apparatus, a camera, and a microscope.

  Conventional optical systems are composed of refractive optical elements or reflective optical elements on the optical axis, or a combination of a plurality of them, and the parameters (curvature radius, spacing, refractive index, etc.) that make up those optical systems. Optimum design corrects aberrations and achieves the desired performance.

  In order to obtain the optimum parameter value, the aberration value at the time of the current design and the change rate of the aberration when the parameter of the design value is slightly changed are obtained, and the least square method or the attenuation least square method is obtained from these values. A better design solution was obtained by (DLS method). However, since the aberration value is extremely non-linear with respect to the parameter value, it is rare that the optimum value by the least square method or the attenuated least square method is obtained at one time. Was getting performance. In addition, what kind of optical system is constituted by what kind of optical element requires considerable trial and error, and as a result, considerable time is required for designing the optical system.

  However, Feder showed that differentiation by parameters is possible by differentiating and storing the ray movement equation and refraction equation between each surface of the lens data by the ray passing position and the direction cosine of the ray ( J. Opt. Soc. Am. 58, 1494 (1968), see Non-Patent Document 1). According to this, the calculation of the change rate according to the aberration parameter is remarkably quick, and the time required for the entire optical system design can be greatly shortened.

J. Opt. Soc. Am. 58, 1494 (1968)

  However, Non-Patent Document 1 does not disclose a method for differentially obtaining a change in aberration due to a paraxial Gaussian image plane movement caused by minutely changing a parameter or a change rate due to an astigmatism parameter. . Therefore, an object of the present invention is to provide a method for obtaining these amounts necessary for designing an optical system by differentiation.

  In addition, in an optical system that is rotationally asymmetric with respect to the optical axis due to the eccentricity of the optical axis, the astigmatism ray tracing method cannot be used, and the principal ray that enters the optical system and the principal ray that enters the optical system. Astigmatism was calculated from the position of the intersection where the light beams with slightly changed angles intersect near the image plane. However, in this calculation method, in order to obtain a highly accurate astigmatism position, the angle must be changed very slightly. However, if it is too small, there is a problem of deterioration in calculation accuracy due to a digit loss. Furthermore, when it was differentiated by parameters, there was a problem in calculation accuracy. In view of this, the present invention has an object to solve this problem by a method of differentiating the intersection position intersecting in the vicinity of the image plane with the direction cosine of the light ray incident on the optical system from the principal ray.

  The first means for solving the above problem is to differentiate the aberration caused by the optical system with parameters (curvature radius, interval, refractive index, etc.) constituting the optical system, and calculate the rate of change thereof. In the designing method, the differentiated rate of change is corrected by the rate of change of the Gaussian image plane caused by a minute change in parameters constituting the optical system.

  The second means for solving the above problem is to design the optical system by calculating the rate of change of astigmatism due to the optical system depending on the parameters (curvature radius, spacing, refractive index, etc.) constituting the optical system. In this method, the rate of change due to the astigmatism parameter of the optical system is obtained by differentiating an astigmatism ray tracing equation.

  According to a third means for solving the above problem, in the method of designing an optical system by calculating astigmatism by the optical system, the principal ray incident on the optical system and the angle incident on the optical system are determined as the principal ray. The position of the intersection where the light beam slightly changed from 1 intersects in the vicinity of the image plane is represented by the arrival position of the principal ray on the image plane and the direction cosine on the image plane, and the direction of the ray incident on the optical system from the principal ray In this optical system design method, astigmatism is obtained by differentiating with a cosine.

  According to a fourth means for solving the above problem, in the method of designing an optical system by calculating astigmatism by the optical system, the principal ray incident on the optical system and the angle incident on the optical system are determined as the principal ray. The position of the intersection where the light beam slightly changed from 1 intersects in the vicinity of the image plane is represented by the arrival position of the principal ray on the image plane and the direction cosine on the image plane, and the direction of the ray incident on the optical system from the principal ray In this optical system design method, astigmatism is obtained by differentiating with the cosine, and the rate of change due to the parameters constituting the optical system is obtained by differentiating the astigmatism equation with the parameter.

  With the optical system design method of the present invention, it is possible to obtain the aberration variation due to the paraxial Gaussian image plane movement necessary for the optical system design and the rate of change due to the astigmatism parameter by differentiation. It is possible to shorten the calculation time required for the design (first means). In addition, astigmatism can be accurately calculated even in an optical system that is asymmetric with respect to the optical axis (third means). Furthermore, by differentiating this calculation formula with parameters, it becomes possible to significantly reduce the calculation time required for optical design even in an optical system that is asymmetric with respect to the optical axis (second means, third means).

  First, in order to calculate a change in aberration due to a paraxial Gaussian image plane movement due to a minute change in a parameter, it is necessary to calculate a derivative by a paraxial ray tracing type parameter. First, the Feder method described in Non-Patent Document 1 is introduced so that the method can be easily understood.

  Let us consider a ray tracing formula when a light beam passes from the (i-1) th surface to the ith surface of the lens with the optical axis of the optical system as the x-axis and the meridian axis as the z-axis. In this specification, if vector symbols are used in text, image data must be used, which is cumbersome. Therefore, in the case of a vector, the fact that it is a vector is clearly indicated before that. . In formulas, vectors are represented by bold characters. The matrix is indicated by [] in the text and is indicated by bold characters in the formula.

The ray passing points on the (i-1) plane and the i plane, respectively, and the direction cosines of the rays on the (i-1) plane and the i plane, respectively (X _i-1 , Y _i-1 , Z _i-1 ), (X _i , Y ₁ , Z).

From the shape of the surface through which the light passes and the relational expression of the direction cosine, x _i is obtained from y _i and z _i , and Xi is obtained from Y _i and Z _i .
Here, the equation of movement from the (i-1) plane to the i plane is

ｉ面での屈折の式（または反射の式）を、

The refraction formula (or reflection formula) on the i-plane is

とする。そして、(1-1)式,(1-2)式をparameter：ｐで微分する場合のベクトルＹ_ｉ、ベクトルＹ_ｉ’を以下のように定義する。

And Then, the vector Y _i and the vector Y _i ′ when the equations (1-1) and (1-2) are differentiated by parameter: p are defined as follows.

すると、ベクトルＹ_ｉ、ベクトルＹ_ｉ'の微分は、次のようにベクトルＹ_ｉ−１、ベクトルＹ_ｉ−１'の微分で表わされる。

Then, the differentiation of the vector Y _i and the vector Y _i ′ is expressed by the differentiation of the vector Y _i−1 and the vector Y _i−1 ′ as follows.

ただし、

However,

となる。

It becomes.

Next, the image plane is the (F + 1) plane, [C _{F + 1} ] is the unit matrix, [C _F '] = [C _{F + 1} ] [T _{F + 1} ], [C _F ] = [C _F '] [R _F ], [C _i '] = [C _{i + 1} ] [T _i ], [C _i ] = [C _i '] [R _i ] ... (1-11)
And If p is a parameter on the i-plane, even if p is shifted, the ray tracing is not affected before the i-plane. That is, when 1 ≦ j ≦ i−1,

ゆえに、

therefore,

（1-11）より（1-13）は、

From (1-11) to (1-13)

pがi面での曲率半径ｒ_ｉまたは非球面係数Ｃ_ｉｎならば、

If p is the radius of curvature r _{i on} the i-plane or the aspherical coefficient C _in ,

以外はゼロになる。従って（1-14）式は、

Otherwise, it becomes zero. Therefore, equation (1-14) is

pが（i−１）面とi面での間隔Ｄ_ｉ−１ならば、

If p is the distance D _i−1 between the (i−1) plane and the i plane,

以外はゼロになる。従って（1-14）式は

Otherwise, it becomes zero. Therefore, equation (1-14) is

となる。このように、parameterに依存しない微分行列をあらかじめまとめて記憶しておいて計算することにより、計算時間の節約が可能になる。以上がFederによる方法の概略である。

It becomes. In this manner, calculation time can be saved by storing and calculating differential matrices that do not depend on parameters in advance. The above is the outline of the method by Feder.

Next, each partial differential element represented by (1-7) to (1-10) will be described with respect to the case of the aspherical non-eccentric ray tracing type.
First, the curvature radius and the aspherical coefficient at x _i is i-th surface,

となる。ただし、

It becomes. However,

である。従って、

It is. Therefore,

となり、

And

と置くと、

And put

となる。また、

It becomes. Also,

とする。これらを用いて、移動式の微分要素を表わすと、

And Using these, the mobile differential element is expressed as follows:

となる。移動式のパラメータによる微分は、パラメータが曲率半径ｒ_ｉの場合、

It becomes. The differentiation by the moving parameter is, when the parameter is the radius of curvature r _i ,

となる。パラメータが曲率半径ｒ_ｉ−１の場合、

It becomes. If the parameter is the radius of curvature r _i−1 ,

となり、パラメータが非球面係数の場合

And the parameter is an aspheric coefficient

となる。またパラメータが間隔Ｄ_ｉ−１の場合、

It becomes. If the parameter is the interval D _i−1 ,

となる。
次に屈折式の微分要素を示す。光線と屈折面の交点での屈折面の法線の方向余弦を（α_ｉ、β_ｉ、γ_ｉ）とすると、

It becomes.
Next, the differential element of the refraction formula is shown. If the direction cosine of the normal of the refracting surface at the intersection of the ray and the refracting surface is (α _i , β _i , γ _i ),

となる。従って、

It becomes. Therefore,

となる。ただし、

It becomes. However,

で与えられる。同様にして、

Given in. Similarly,

となる。次に、入射光線ベクトルと屈折面の法線の内積をＩ_ｉ、出射光線ベクトルと屈折面の法線の内積をＩ’_ｉ
とすると、

It becomes. Next, the inner product of the incident ray vector and the normal of the refracting surface is I _i , and the inner product of the outgoing ray vector and the refracting surface of the normal is I ′ _i.
Then,

となり、屈折式は、

And the refraction formula is

となる。これらのｙ_ｉによる微分は、

It becomes. These y _i derivatives are

となる。同様にして、

It becomes. Similarly,

また、方向余弦による微分は、

Also, the differentiation by direction cosine is

となる。さらに、

It becomes. further,

と書き換えることができるので、

Can be rewritten as

となり、屈折式のパラメータによる微分は、

And the differentiation by the parameters of the refraction formula is

となる。

It becomes.

Next, a calculation method that is an embodiment of the first means for solving the problem will be described. For this purpose, first, a method for calculating the derivative by the parameter of the paraxial ray tracing equation will be described. The paraxial ray tracing formula here is as follows. That is, if the paraxial position of the intersection of the i-plane and the ray is h _i and the angle between the optical axis of the ray on the i-plane is α _i ,

ただし、最終面Ｉ＝Ｆでは(２-２)を計算しない。そして、最終面では、光学系の倍率βが指定されている場合、α_Ｆを(２-１)の代わりに、
α_Ｆ＝α_０／β …(2-3)
とする。ただし、α_０は最初の面に入射する前の近軸的な光線と光軸のなす角度である。
また、光学系の倍率が指定されていない場合、(2-3)を計算しない。

However, (2-2) is not calculated for the final plane I = F. And on the final surface, when the magnification β of the optical system is specified, α _F is replaced with (2-1),
α _F = α ₀ / β (2-3)
And Here, α ₀ is an angle formed by the paraxial light beam before entering the first surface and the optical axis.
If the optical system magnification is not specified, (2-3) is not calculated.

Next, the position d _F of the Gauss image plane can be set whether or not the magnification β of the optical system is specified.
d _F = h _F / α _F (2-4)
And calculate. Next, only when the magnification β of the optical system is not specified, the paraxial magnification β is changed to β = α ₀ / α _F (2-5)
And calculate. When the magnification β of the optical system is specified, the radius of curvature of the lens on the final surface is

と計算する。
次に、近軸光線追跡の微分について説明する。
α_ｉ, h_ｉ＋ｈのパラメータｐによる微分のベクトルを、

And calculate.
Next, the differentiation of paraxial ray tracing will be described.
The vector of differentiation by the parameter p of α _i , h _{i + h} is

とする。ただし、最終面では、(２-２)を計算しないので、

And However, since (2-2) is not calculated on the final surface,

と定義する。すると、漸化式

It is defined as Then the recurrence formula

が成立する。ただし[Ｇ_ｉ]は次のような２行２列の微分行列である。

Is established. [G _i ] is a differential matrix of 2 rows and 2 columns as follows.

ただし、最終面では、

However, on the final side,

となる。パラメータｐが、ｉ面の曲率半径ｒ_ｉまたは、間隔ｄ_ｉとすると、ｐがシフトしても、ｉ面以前の近軸光線追跡式には影響を及ぼさないので、

It becomes. If the parameter p is the radius of curvature r _i of the i-plane or the interval d _i , even if p is shifted, the paraxial ray tracing equation before the i-plane will not be affected.

となり、また、近軸光線追跡式においては、

In the paraxial ray tracing formula,

となるので、

So,

となる。(2-15)より、ｄα_Ｆ／ｄｐ、ｄｈ_Ｆ／ｄｐが求められると、光学系の倍率が指定されていない場合、(2-3),(2-4)より、Gauss像面のパラメータによる変化率ｄα_Ｆ／ｄｐ、近軸倍率のパラメータによる変化率ｄβ／ｄｐが

It becomes. From (2-15), when dα _F / dp and dh _F / dp are obtained, if the magnification of the optical system is not specified, the parameters of the Gauss image plane are obtained from (2-3) and (2-4). The rate of change dα _F / dp due to, and the rate of change dβ / dp due to the paraxial magnification parameter is

として求められる。また、光学系の倍率が指定されている場合、(2-3)より、α_Ｆはパラメータにより変動せず、そして、最終面の曲率半径を変動させて、倍率を一定に保持するので、

As required. If the magnification of the optical system is specified, from (2-3), α _F does not vary depending on the parameters, and the curvature radius of the final surface is varied to keep the magnification constant.

となる。次に、個々の偏微分要素を近軸光線追跡式(2-1),(2-2)から求める。先ず、(2-12)の要素は、(2-1),(2-2)より、

It becomes. Next, individual partial differential elements are obtained from paraxial ray tracing equations (2-1) and (2-2). First, the elements of (2-12) are from (2-1) and (2-2)

となる。次に

It becomes. next

の要素を求める。先ず、ｐがi面の曲率半径ｒ_ｉの場合、

Find the elements of. First, when p is the curvature radius r _i of the i-plane,

次にｐがi面と(i-1)面の間隔Ｄ_ｉの場合、

Next, when p is the distance D _i between the i plane and the (i-1) plane,

次に、ｐがi面の２次の非球面係数Ｃ_ｉ１の場合、

Next, when p is the second-order aspheric coefficient C _i1 of the i-plane,

以上のようにして、近軸光線式のパラメータによる微分が求められる。

As described above, the differentiation by the paraxial ray parameter is obtained.

Next, a method for correcting the aberration variation due to the movement of the Gauss image plane based on the differentiation based on the lateral aberration parameter obtained by the Feder method will be described. In FIG. 1, it is assumed that the light beam AB travels from the final plane F toward the Gauss image plane by the current lens data. The position of the Gauss image plane on the optical axis is at a distance from the final plane F to _DF , and the ray arrival position on the Gauss image plane is at a distance of GB = y _{F + 1} from the optical axis. Let A′B ′ be a light beam caused by a slight change in the parameter p. On the other hand, the Gauss image plane position moves from _DF to _{DF + 1} ′ due to the change of the parameter p. The light beam then extends to A'C '. The amount of aberration fluctuation due to the parameter p is not BB ′ = Δy _F , but C ₀ C ′ = Δy _F ′ due to the intersection C ₀ with the Gauss image plane that is moved parallel to the optical axis from B. As is clear from FIG. 1, BB ′ and CC ′ are substantially equal, and dy _F / dp and dy _F / dD _F obtained by differentiating the skew ray tracing equation by the Feder method and the paraxial ray tracing equation are differentiated. DD _F / dp

であるので、

So

となり、パラメータｐが微小変動したことによりGauss像面位置が移動したことによる補正された収差のパラメータによる微分は、

Then, the derivative by the parameter of the corrected aberration due to the movement of the Gauss image plane position by the minute change of the parameter p is

となり、同様にして、

となる。近軸光線追跡式の微分とSkew光線追跡式の微分とを区別する目安は、前者が最終面の間隔や曲率半径といったパラメータをパラメータで微分するのに対し、後者は像高や光線の方向余弦をパラメータで微分する量であるということである。

In the same way,

It becomes. The standard for distinguishing the derivative of the paraxial ray tracing equation from the derivative of the Skew ray tracing equation is that the former differentiates parameters such as the distance between the final surfaces and the radius of curvature by parameters, while the latter differentiates the image height and the direction cosine of the ray. Is a quantity that differentiates by a parameter.

If the magnification β of the optical system is specified, if the parameter p varies, the radius of curvature of the lens on the final surface varies depending on (2-6) and (2-7). The differentiation is expressed by (2-33), (2-) by dy _F / dr _F obtained by differentiating the skew ray tracing equation by Feder's method and dr _F / dp obtained by differentiating the paraxial ray tracing equation. In addition to the equation of (34)

となる。

It becomes.

As described above, FIG. 2 shows an algorithm from the paraxial ray tracing and its differentiation to the correction based on the aberration parameter for correcting the movement of the Gauss image plane by the parameter. In FIG. 2, paraxial ray tracing such as (2-1) and (2-2) is first performed by calculation processing S1, and (2-20) to (2-) are calculated for each surface of ray tracing by calculation processing S2. 29) Calculate paraxial differential elements such as (2), (2-8), (2-9), (2-11), and (2-12). When the paraxial ray tracing is finished up to the final surface, the paraxial amount is calculated as (2-3) to (2-7) by the arithmetic processing S3, and the paraxial as (2-15) is calculated by the arithmetic processing S4. The derivative with respect to the parameter p _i is calculated. Next, the skew ray tracing is performed by the calculation processing S5 according to the schedule calculation schedule, and skew differential elements such as (1-17) to (1-85) are calculated by the calculation processing S6 for each surface of the ray tracing. When the Skew ray tracing is completed to the end, the skew differential matrix is calculated as shown in (1-11) by the calculation process S7, and the aberration parameter p is calculated as (1-15) and (1-16) by the calculation process S8. Calculate the derivative by _i . Finally, correction by Gaussian image plane movement such as (2-33) to (2-36) is performed by the arithmetic processing S9 from the differentiation by the aberration parameter and the paraxial differential amount. When all of the Skew ray tracing schedules are completed, the calculation of the differentiation by the aberration parameters is terminated.

Next, a method for obtaining the change rate by the astigmatism parameter by differentiation, which is an embodiment of the second means for solving the problem, will be described. When the chief ray is refracted by the i-plane, the distance between the intersection of the chief ray and the i-plane and the meridional (meridian) imaging position is M _i , and the intersection of the sagittal imaging position is S _i , (i-1 ) The length of the ray from the plane to the i plane is W _i , the cosine of the angle between the ray incident on the i plane and the normal to the i plane at the intersection of the i plane and I is the normal and i If the cosine of the angle formed by the ray refracted from the surface is I ′, and the reciprocals of the meridional and sagittal curvature radii at the intersection are ρ _M and ρ _S respectively,

となる。（3-1）および（3-2）を変形すると、

It becomes. When (3-1) and (3-2) are transformed,

となる。ただし、
Ｊ_ｉ＝ｎ_ｉＩ’−ｎ_ｉ−１Ｉ …（3-6）
である。従って、parameterｐ_ｉがｉ面の曲率半径や非球面係数であるとき、または、(i-1)面の間隔Ｄ_ｉ−１であるとき、ｐ_ｉを変化させても、(i-1)面以前の光線追跡には影響を及ぼさず、(i-1)面からｉ面までの光線の長さＷ_ｉには影響を及ぼす。従って、

It becomes. However,
J _i = n _i I′−n _i−1 I (3-6)
It is. Therefore, when parameter _pi is the radius of curvature of the i-plane or the aspherical coefficient, or when it is the distance D _i-1 of the (i-1) plane, even if _pi is changed, the (i-1) plane The previous ray tracing is not affected, and the ray length W _i from the (i-1) plane to the i plane is affected. Therefore,

となる。そして、Ｓ_ｉおよびＭ_ｉのＰ_ｉによる微分は、

It becomes. And the differentiation of S _i and M _i by P _i is

となる。
次に、ｄＳ_ｉ＋１／ｄｐ_ｉ及びｄＭ_ｉ＋１／ｄｐ_ｉは、ｄＳ_ｉ／ｄｐ_ｉ及びｄＭ_ｉ／ｄｐ_ｉを用いて、

It becomes.
Next, dS _{i + 1} / dp _i and dM _{i + 1} / dp _i use dS _i / dp _i and dM _i / dp _i ,

となるが、後述するように、微分要素を検討することにより、Ｗ_ｉやＩ、Ｉ’がｘ_ｉ、ｙ_ｉ＋１、Ｙ_ｉの関数であることがわかるので、

However, as will be described later, by examining the differential elements, it can be seen that W _i , I, I ′ are functions of x _i , y _{i + 1} , Y _i .

とすることができる。(3-1),(3-2)式は子午面内での式であるので、（3-13），（3-14）のようにｚ方向の微分を考慮する必要はない。ここで注意すべきことは、(1-22)と同様にして、

It can be. Since equations (3-1) and (3-2) are equations in the meridian plane, there is no need to consider z-direction differentiation as in (3-13) and (3-14). What should be noted here is the same as (1-22)

であるので、ｊ≧ｉ＋１の場合、

Therefore, if j ≧ i + 1,

であるが、(1-34),(1-35)から分かるように、ｊ＝ｉの場合は、(3-16)は成立しない。しかも、(1-34),(1-35)から、

However, as can be seen from (1-34) and (1-35), when j = i, (3-16) does not hold. Moreover, from (1-34), (1-35),

であるが、
Ｙ_ｉ−１＝０ …（3-18）
の場合は、(3-17)より、ｄｙ_ｉ／ｄｐ_ｉから、ｄｘ_ｉ／ｄｐ_ｉを求めることはできない。

In Although,
Y _i-1 = 0 (3-18)
For the (3-17) than _from dy i / dp _i, it can not be obtained dx _i / dp i.

Therefore, as shown in (3-13) and (3-14), they cannot be combined into derivatives of y _i and Y _i and must be expressed by derivatives of x _i , y _{i + 1} , and Y _i . When j ≧ i + 2, it becomes possible to combine y _i−1 and Y _i−1 into derivatives,

となる。
ｊ面でのｄＭ_ｊ／ｄｐ_ｉ、ｄＳ_ｊ／ｄｐ_ｉ、ｄｙ_ｊ／ｄｐ_ｉ、ｄＹ_ｊ／ｄｐ_ｉを一つのベクトルの成分とすると、(j-1)面でのベクトルとの関係は、次のような偏微分行列により、

It becomes.
_dM j _/ dp _i of the _{j-surface, dS j / dp i, dy} j / dp i, when the components of one vector dY j / dp _i, the relationship between the vector at (j-1) plane, From the partial differential matrix

となる。(3-21)における部分行列

It becomes. Submatrix in (3-21)

は、(1-7)、(1-8)でｙ方向成分のみを取った行列により、

Is a matrix that takes only the y-direction component in (1-7) and (1-8),

となる。

It becomes.

と置き、レンズデータの最終面をF面とすると、F面での微分ベクトルは、

If the final surface of the lens data is the F surface, the differential vector on the F surface is

となり、微分行列をＦ面から逆に(i+2)面までかける過程で、次々と非点収差のparameterによる微分が求められるので、計算時間の省略ができる。微分ベクトルのｊ＝ｉ＋１の場合は(3-13),(3-14)より、ｊ＝ｉの場合は(3-9),(3-10)より求められる。次に各微分要素を求める。
＜３−１＞
先ず(3-9),(3-10)のｊ＝ｉの場合の各微分要素を求める。(3-4),(3-5)においてそれぞれ、

Thus, in the process of multiplying the differential matrix from the F-plane to the (i + 2) plane, the differentiation by the astigmatism parameter is obtained one after another, so that the calculation time can be omitted. When j = i + 1 of the differential vector, it is obtained from (3-13) and (3-14), and when j = i, it is obtained from (3-9) and (3-10). Next, each differential element is obtained.
<3-1>
First, each differential element in (3-9) and (3-10) when j = i is obtained. In (3-4) and (3-5),

と置くと、(3-11),(3-12)における偏微分要素は、

Then, the partial differential elements in (3-11) and (3-12) are

および、

and,

となる。
(3-1-4),(3-1-7)において、

It becomes.
In (3-1-4) and (3-1-7),

は、(1-64)より、

From (1-64)

となるので、

So,

となる。一方、(3-11),(3-12)における全微分要素ｄＷ_ｉ／ｄｐ_ｉは、(3-3)より、

It becomes. On the other hand, (3-11), the total differential element _dW i / dp _i in (3-12) and (3-3),

となる。(3-1-11)において、ｄｘ_ｉ／ｄｐ_ｉは、ｐ_ｉがｉ面の曲率半径の場合(1-34)で、ｉ面の非球面係数の場合(1-37)で与えられる。(3-1-11d)において、

It becomes. In _{(3-1-11), dx i / dp} i is, when p _i is the radius of curvature of the i-th surface (1-34) is given in the case of non-spherical coefficient of i-th surface (1-37). (3-1-11d)

であるので、

So

となる。

It becomes.

  Since (3-1) and (3-2) are ray traces in the meridional plane, all z directions are zero.

  Therefore, from (1-58)

となり、(3-1-14)をｐ_ｉで微分して、

Next, by differentiating the (3-1-14) in p _i,

となる。（3-1-15）のｄα_ｉ／ｄｐ_ｉ、ｄβ_ｉ／ｄｐ_ｉは、

It becomes. Dα _i / _{dp i} of (3-1-15), I am dβ _i / dp _i,

であるが、

In Although,

はそれぞれ、(1-76),(1-51),(1-77),(1-52)である。
ｐ_ｉがｉ面の曲率半径や非球面係数であるとき、(1-49),(1-21)より、

Are (1-76), (1-51), (1-77), and (1-52), respectively.
When p _i is the radius of curvature and aspheric coefficients of the i-th surface, (1-49) and (1-21),

より、子午面内ではｙ_ｉ＝ｈ_ｉ、ｚ_ｉ＝０、γ_ｉ＝０であるので、

Since y _i = h _i , z _i = 0, and γ _i = 0 in the meridian plane,

となり、

And

となる。また、

It becomes. Also,

は、ｐ_ｉがｉ面の曲率半径の場合(1-35)により、ｉ面の非球面係数の場合(1-38)により、(i-1)面の間隔Ｄ_ｉ−１の場合(1-48)により与えられる。
なお、球面の場合、

Is optionally p _i is the radius of curvature of the i-th surface (1-35), optionally aspheric coefficient of i-th surface (1-38), when the distance D _i-1 of the (i-1) plane (1 -48).
In the case of a spherical surface,

である。
＜３−２＞
(3-11),(3-12)における

It is.
<3-2>
In (3-11) and (3-12)

は、（3-1-3）から（3-1-8）のｉに (i+1)を置き換えたものに等しい。一方

Is equal to (3-1-3) to (3-1-8) with i replaced by (i + 1). on the other hand

は、

Is

と置くと、

And put

となる。次に(3-11),(3-12)における全微分を求めると、
ｄＳ_ｉ／ｄｐ_ｉ、ｄＭ_ｉ／ｄｐ_ｉは既に求められているが、ｄＷ_ｉ＋１／ｄｐ_ｉは、

It becomes. Next, when calculating the total derivative in (3-11) and (3-12),
_{dS i / dp i, dM i} / dp i is already _{sought, dW i} + 1 / dp _i is,

より、

Than,

となる。（3-2-6）において、 (3-16)のように、ｄｘ_ｉ＋１／ｄｐ_ｉをｄｙ_ｉ＋１／ｄｐ_ｉで表わすことはできるが，ｄｘ_ｉ／ｄｐ_ｉをｄｙ_ｉ／ｄｐ_ｉで表わすことはできない。また、

It becomes. In (3-2-6), (3-16) as _in, although it is possible to represent the _{dx i} + 1 / dp _i in _{dy i} + 1 / dp _i, represent the dx i / dp _i in _dy i / dp _i I can't. Also,

はそれぞれ、

Respectively

となる。ρ_Ｓｉ＋１、ρ_Ｍｉ＋１はそれぞれ、(i+1)面のサジタル的およびメリジオナル的曲率半径の逆数なので、球面の場合は

It becomes. Since ρ _{Si + 1} and ρ _{Mi + 1} are the reciprocals of the sagittal and meridional curvature radii of the (i + 1) plane,

であるが、非球面の場合は＜３−４＞を参照されたい。
また、(1-5)および(1-6)より、さらに

However, in the case of an aspherical surface, refer to <3-4>.
From (1-5) and (1-6),

以外はゼロになることにより、

By becoming zero except

となる。従って、(3-2-6),(3-2-7),(3-2-8),(3-2-9)において、ｄｙ_ｉ＋１／ｄｐ_ｉ、ｄＹ_ｉ／ｄｐ_ｉは

It becomes. Therefore, (3-2-6), (3-2-7), (3-2-8), in _{(3-2-9), dy i + 1} / dp i, dY i / dp i is

となる。

It becomes.

はそれぞれ，(1-26)，(1-30)，(1-65)より、

Respectively from (1-26), (1-30), (1-65)

はそれぞれ、(1-35)または(1-38)，(1-82)，(1-41)または(1-44)より求められる。

Are obtained from (1-35) or (1-38), (1-82), (1-41) or (1-44), respectively.

は(1-58)より求められる。(3-13)、（3-14）において、ｄｘ_ｉ／ｄｐ_ｉは、ｐ_ｉが_ｉ面の曲率半径の場合(1-34)で、ｉ面の非球面係数の場合(1-37)で与えられ、(i-1)面の間隔Ｄ_ｉ−１の場合(３-1-12)で与えられる。(3-11),(3-12)に(3-2-6),(3-2-7),(3 -2-8),(3-2-9)を代入し、
ｄｙ_ｉ＋１／ｄｐ_ｉ、ｄＹ_ｉ／ｄｐ_ｉ、等でまとめると、

Is obtained from (1-58). (3-13), in _(3-14), dx i / dp _i is, when _{p i} is the radius of curvature of the _i-th surface (1-34), if the aspherical coefficients of the i-th surface (1-37) In the case of (i−1) plane distance D _i−1 , it is given by (3-1-12). Substituting (3-2-6), (3-2-7), (3-2-8), and (3-2-9) into (3-11) and (3-12)
_{dy i + 1 / dp i,} dY i / dp i, summarized the like,

となる。つまり、(3-13),(3-14)の偏微分要素は、ｊ＝ｉ＋１として、

It becomes. That is, the partial differential elements of (3-13) and (3-14) are j = i + 1,

となる。

＜３−３＞
ｊ≧ｉ＋２の場合、

It becomes.

<3-3>
If j ≧ i + 2,

において、

In

はそれぞれ、（3-1-3）から（3-1-8）のｉにｊを置き換えたものに等しい。

Are equivalent to (3-1-3) to (3-1-8) with j replaced by i.

はそれぞれ、(3-2-3),(3-2-4)のｉ＋１にｊを置き換えたものに等しい。また、(3-3),(3-16)より、

Are equivalent to i + 1 in (3-2-3) and (3-2-4) with j replaced. From (3-3) and (3-16),

となり、さらに

And then

となるので、(3-3-1),(3-3-2)に(3-3-3)から(3-3-7)を代入して、(3-2-17)〜(3-2-22)を用いて整理すると、

Therefore, by substituting (3-3-7) from (3-3-3) into (3-3-1) and (3-3-2), (3-2-17) to (3 -2-22)

となる。つまり、(3-19),(3-20)の微分要素は、

It becomes. In other words, the differential elements of (3-19) and (3-20) are

となる。

＜３−４＞
非球面の場合、サジタル的曲率半径の逆数は、

It becomes.

<3-4>
For aspheric surfaces, the inverse of the sagittal radius of curvature is

となる。σ_ｉは(1-20)で表わされる。
(3-4-1)のparameterによる微分は、(1-48)を用いて、

It becomes. σ _i is represented by (1-20).
The differentiation by the parameter of (3-4-1) uses (1-48),

となり、

And

となる。ただし、

It becomes. However,

より、

Than,

となる。ｚ方向の成分がゼロなので、

It becomes. Since the z-direction component is zero,

となり、これを代入すると、

And substituting this,

となる。

It becomes.

となり、

And

となる。ｄｙ_ｉ／ｄｒ_ｉは(1-35)により求められる。
ｐ_ｉがｉ面の非球面係数である場合、

It becomes. dy _i / dr _i is obtained by (1-35).
If p _i is the aspherical coefficient of the i-plane,

より、

Than,

となる。ｄｙ_ｉ／ｄＣ_１ｎは、(1-38)により求められる。なお、(3-2-17)における

It becomes. dy _i / dC _1n is obtained by (1-38). In (3-2-17)

は、

Is

となる。

＜３−５＞
メリジオナル的曲率半径の逆数を求めると、非球面の場合、メリジオナル的曲率半径の逆数は、

It becomes.

<3-5>
When the reciprocal of the meridional radius of curvature is obtained, in the case of an aspherical surface, the reciprocal of the meridional radius of curvature is

となる。

It becomes.

より、

Than,

となる。ここで(1-48)を用いると、

It becomes. Where (1-48) is used,

となる。また、

It becomes. Also,

となる。τ_ｉ、ν_ｉを

It becomes. τ _i , ν _i

と定義し、また(1-21)、(3-5-5)に代入すると、

And assigning to (1-21) and (3-5-5)

となる。ｐが曲率半径ｒ_ｉのとき、

It becomes. When p is the radius of curvature r _i ,

となり、(3-5-12),(3-5-13)を(3-5-11)に代入すれば、求める微分が得られる。(3-4-9)と同様に、ｄｙ_ｉ／ｄｒ_ｉは(1-35)により求められる。ｐが非球面係数Ｃ_ｉｎのとき、

Substituting (3-5-12) and (3-5-13) into (3-5-11) gives the desired derivative. Similarly to (3-4-9), dy _i / dr _i is obtained by (1-35). When p is the aspheric coefficient C _in ,

となり、(3-5-14),(3-4-15)を(3-5-11)に代入すれば、求める微分が得られる。(3-4-11)と同様に、ｄｙ_ｉ／ｄＣ_ｉｎは(1-38)により求められる。なお、(3-2-18)における

Substituting (3-5-14) and (3-4-15) into (3-5-11) gives the desired derivative. Similarly to (3-4-11), dy _i / dC _in is obtained from (1-38). In (3-2-18)

は、

Is

となる。

＜３−６＞
次に反射面の場合の非点収差のパラメータによる微分を求める。反射面の場合、非点収差の追跡式は、

It becomes.

<3-6>
Next, the differentiation by the parameter of astigmatism in the case of the reflecting surface is obtained. For reflective surfaces, the astigmatism tracking formula is

となる。

It becomes.

と置くと、

And put

となる。(3-6-5),(3-6-6)において、

It becomes. In (3-6-5) and (3-6-6),

はそれぞれ、(3-1-11)または
(3-1-11d),(3-1-15)により求められる。また、(3-11)における偏微分要素は、

Is (3-1-11) or
It is obtained by (3-1-11d) and (3-1-15). The partial differential element in (3-11) is

となる。また、(3-12)における偏微分要素は、

It becomes. The partial differential element in (3-12) is

＜３−７＞
最後に像面での非点収差のパラメータによる微分の計算方法を示す。像面を（F+1）面とすると、像面からの非点収差は、F面（レンズデータの最終面）での非点収差から、

<3-7>
Finally, the calculation method of the differentiation by the parameter of astigmatism on the image plane is shown. If the image plane is the (F + 1) plane, astigmatism from the image plane is derived from astigmatism on the F plane (the final surface of the lens data).

のように求められる。ただし、

It is required as follows. However,

である。(3-7-3)をパラメータｐ_ｉで微分すると、

It is. Differentiating this with respect to the parameters p _i a (3-7-3),

であるので、

So

となる。同様にして、

It becomes. Similarly,

となる。従って最終面での非点収差の偏微分行列は、(3-16)より、

It becomes. Therefore, the partial differential matrix of astigmatism on the final surface is (3-16),

となる。Ｆ≧ｉの場合、(3-24)より、

It becomes. When F ≧ i, from (3-24)

Ｆ＝ｉ以外の場合、

When F is not i,

と置くと、(3-24)より、

Then, from (3-24),

となる。以上、非点収差のパラメータによる微分の計算方法を図示すると、図３のようになる。図３において、各面の光線追跡中に、演算処理Ｓ１０は(3-9)，(3-10)により計算し、演算処理Ｓ１１は(3-2-17)〜(3-2-20)により、演算処理Ｓ１２は(3-2-21)，(3-2-22)により計算する。演算処理Ｓ１３は
(3-2-3),(3-2-4),(3-3-10)〜(3-3-13),(3-22)により計算する。光線追跡後に、演算処理Ｓ１４は(3-13)，(3-14)により計算し、演算処理Ｓ１５は、(3-7-9)により計算し、最後に演算処理Ｓ１６は(3-7-10)，(3-7-11)により計算する。

It becomes. The differential calculation method using the astigmatism parameters is illustrated in FIG. In FIG. 3, during the ray tracing of each surface, the calculation processing S10 is calculated by (3-9) and (3-10), and the calculation processing S11 is (3-2-17) to (3-2-20). Thus, the calculation processing S12 is calculated by (3-2-21) and (3-2-22). Arithmetic processing S13
Calculate according to (3-2-3), (3-2-4), (3-3-10) to (3-3-13), (3-22). After ray tracing, the calculation process S14 is calculated by (3-13) and (3-14), the calculation process S15 is calculated by (3-7-9), and finally the calculation process S16 is (3-7- Calculate with 10) and (3-7-11).

If Gauss image plane is moved by fluctuations in the parameters, derivative with astigmatism parameters than dD F _/ dp _i obtained by the differentiation of the paraxial ray tracing,

となる。

It becomes.

Next, a method for calculating astigmatism by a method for differentiating the astigmatism by the direction cosine of the light ray incident on the optical system from the principal ray, which is an embodiment of the third means for solving the invention, will be described. 4, when the direction cosines _{Y 0} of the incident light was varied by [Delta] Y _0, 'if the, AB and A'B' rays AB in Gauss image plane GB ₀ rays A'B an intersection and C To do. The intersection of the straight line and the Gauss image plane drawn parallel to the optical axis from the C (x-axis) and B _0. When the arrival position on the Gauss plane changes from GB = y _{F + 1} to GB ′ = y _{F + 1} + Δy _{F + 1} and changes from ∠B ₀ CB = θ to ∠B ₀ CB ′ = θ + Δθ, the Gauss image plane moves to the lens direction. 4 where m is the position of the meridional image plane,

となり、これを方向余弦Ｙ_Ｆで表わせば、

Next, if expressed this in direction cosine Y _F,

となる。

It becomes.

と表わすことができるので、(4-2)より、

From (4-2),

となる。

It becomes.

Further, as shown in FIG. 5, when there is a light beam AB inclined by θ from the optical axis in the meridian plane (xy plane) in the vicinity of the image plane, the axis parallel to the optical axis and intersecting with A is defined as x ′ axis. Let the intersection of the axis and the Gauss image plane be B ₀ and ∠B ₀ AB = θ. When the direction cosine of the incident ray is changed by ΔZ ₀ from the meridian plane, the change of the arrival position of the ray AB ′ on the Gauss image plane is BB ′ = Δz _{F + 1} , the change of the angle is ∠BAB ′ = Δφ, and the S image The position is A, and AB = s ′ and AB ₀ = s. As is clear from FIG.

ただし、

However,

となる。
Δφが微小であることから、

It becomes.
Since Δφ is very small,

と近似でき、(4-6)〜(4-8)より、

From (4-6) to (4-8),

となり、

And

と表わすことができるので、

Can be expressed as

となる。(4-5),(4-12)におけるＹ_０、Ｚ_０による微分量は、(1-7)、(1-8)、(1-11)より、

It becomes. In (4-5) and (4-12), the derivative by Y ₀ and Z ₀ is (1-7), (1-8) and (1-11)

となる。

It becomes.

A method of further differentiating by a parameter, a method for calculating astigmatism by differentiating with a direction cosine of a light ray incident on an optical system from a principal ray, which is an embodiment of a fourth means for solving the problem Will be described. (4-5) Differentiating the parameter p _i (4-12), respectively,

となる。(5-1),(5-2)における1次微分は前記の方法で求められるが、2次微分

It becomes. The first derivative in (5-1) and (5-2) can be obtained by the above method, but the second derivative

は次のようにして求められる。すなわち、パラメータp_iがi面の曲率半径または非球面係数の場合、

Is obtained as follows. That is, if the parameter p _i is the radius of curvature of the i-plane or the aspheric coefficient,

となる。(5-3)は行列Ｔ_ｉから行列Ｔ_Ｆ＋１まで[2(F-i+1)+1]個の行列のｐ_ｉによる全微分を含む項の和である。次に(5-3)における微分行列のパラメータによる全微分を偏微分と漸化式で表わす。その前に、(1-3),(1-4)と同様にして、次のようにベクトルを定義する。

It becomes. (5-3) is the sum of terms containing all derivative with p _i of the matrix T _{F + 1} to [2 (F-i + 1 ) +1] pieces of matrix from the matrix T _i. Next, the total differentiation by the parameter of the differentiation matrix in (5-3) is expressed by partial differentiation and recurrence formula. Before that, vectors are defined as follows in the same manner as (1-3) and (1-4).

すると、微分行列のパラメータによる全微分は以下のようになる。

Then, the total differentiation based on the parameters of the differentiation matrix is as follows.

In (5-7)

と,(5-8)における

And in (5-8)

は、２階テンソルの行列をベクトルで微分しているので、３階テンソルになる。(5-6)から(5-10)を(5-3)に代入し、(5-1),(5-2)に代入することにより，非点収差の追跡式を微分するのとは別の方法で、非点収差のパラメータによる微分が得られる。

Since the second-order tensor matrix is differentiated by a vector, it becomes a third-order tensor. What is differentiating the astigmatism tracking equation by substituting (5-6) through (5-10) into (5-3) and substituting into (5-1) and (5-2)? In another way, differentiation with astigmatism parameters is obtained.

  By using the algorithm as shown in FIGS. 2 and 3 by the above calculation method, differentiation based on the parameters of Feder's aberration becomes more practical. Examples thereof are shown below.

For the calculation time of automatic correction (DLS method), in order to investigate how much the calculation time differs depending on whether the derivative of the aberration parameter is approximated or differentiated, projection of 40 lens surfaces When the number of parameters is increased from 1 to 40 with a lens, the number of target values is 50 (the target value of the paraxial ray tracing is 2 and the target value of astigmatism is 6). The calculation time for each cycle of automatic correction by the above method was measured. The result is shown in FIG. In FIG. 6, (1) is the difference and (2) is the differentiation. In the case of the difference, it can be seen that the calculation time increases almost in proportion to the number of parameters, whereas in the case of differentiation, the calculation time hardly increases.
Table 1 shows the calculation time when the parameter number is increased by 10. It was found that if the number of parameters is 5 or more, differentiation is faster than approximation by difference, and if the number of parameters is 40, differentiation takes about 1/5 of the difference. There is no significant difference in merit function between difference and differentiation,
In many cases, the derivative is slightly smaller.
(Table 1)

It is a figure which shows the method of correct | amending the differentiation by the parameter of the aberration when a Gauss image plane moves. FIG. 6 is an algorithm diagram of calculation for correcting differentiation by an aberration parameter when a Gauss image plane moves. It is a figure which shows the method of calculating the differentiation by the parameter of astigmatism. It is a figure which shows the method of calculating a meridional image surface movement position by changing the direction cosine of the light ray which injects into an optical system minutely. It is a figure which shows the method of calculating the sagittal image plane movement position by changing the direction cosine of the light ray which injects into an optical system minutely. It is a figure which shows the difference in the calculation time by whether the differentiation by the parameter of an aberration is a difference approximation or differentiation in automatic correction.

Claims (4)

In the method of designing an optical system by differentiating the aberration due to the optical system with the parameters (curvature radius, spacing, refractive index, etc.) constituting the optical system and calculating the rate of change, the differentiated rate of change is calculated using the optical system. A method for designing an optical system, wherein correction is made with the rate of change of Gaussian image plane variation caused by minute fluctuations in the parameters constituting the lens. In the method of designing an optical system by calculating the rate of change of astigmatism due to the optical system due to the parameters (curvature radius, spacing, refractive index, etc.) constituting the optical system, the change due to the astigmatism parameters of the optical system A method for designing an optical system, characterized in that a rate is obtained by differentiating a ray tracing formula of astigmatism. In a method of designing an optical system by calculating astigmatism due to the optical system, a principal ray incident on the optical system intersects with a light beam that slightly changes the angle of incidence on the optical system from the principal ray in the vicinity of the image plane. The astigmatism is obtained by differentiating the position of the intersecting point by the arrival position of the principal ray on the image plane and the direction cosine of the image on the image plane, and differentiating them by the direction cosine of the ray incident on the optical system from the principal ray. An optical system design method characterized by the above. In a method of designing an optical system by calculating astigmatism due to the optical system, a principal ray incident on the optical system intersects with a light beam that slightly changes the angle of incidence on the optical system from the principal ray in the vicinity of the image plane. The astigmatism is obtained by differentiating the position of the intersecting point by the arrival position of the principal ray on the image plane and the direction cosine of the image on the image plane and differentiating them by the direction cosine of the ray incident on the optical system from the principal ray. A method for designing an optical system, characterized in that a rate of change due to a parameter constituting the system is obtained by differentiating the astigmatism equation with a parameter.

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