JP3589774B2 - High-precision planar shape measurement method - Google Patents
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【0001】
【発明の属する技術分野】
本発明は高精度平面の形状測定方法に関し、特に干渉計の基準面等として用いられる平面の表面形状を求める高精度平面の形状測定方法に関するものである。
【0002】
【従来の技術】
平面度を測定する手法として、フィゾー型干渉計等の干渉計による測定が知られている。このような干渉計は高精度で被検面の平面度を測定できるものの、その測定は基準面に対する相対測定であって絶対測定ではない。したがって基準面として極めて高精度な平面が必要とされ、そのためこのような高精度の平面を有する基準面の測定方法が求められている。
このような基準面を測定する手法として、3枚の基準板を作成し、この3枚の中から選択された3つの基準板ペアの組み合わせ各々について基準面相対変位を測定し、この測定結果に基づき連立方程式を解くことで、各基準面の形状を測定する3面合わせ方法が知られている。
【0003】
以下、この3面合わせ方法について説明する。
3枚の基準板ガラスをA,B,Cとする。各基準板ガラスについて図1に示すような座標をとると、これらA,B,Cのガラス面の形状はx,yの関数で表すことができるので、それぞれをA(x,y),B(x,y),C(x,y)とする。なお、図1に示すようなZ座標をとったのは、ガラス面が凸面のときプラスで表し、凹面のときマイナスで表すようにしたためである。
ここで、例えば基準面をA(x,y)とし、被検面をB(x,y)とし、これら2つの面を図8(a)の如く対向させ、フィゾー型干渉計の所定位置にセットする。
【0004】
次に、この干渉計で測定された両面の相対的形状をφAB(x,y)とすれば、
φAB(x,y)=A(x,y)+B(x′,y′)
となる。
また、被検面の座標を基準面の座標で表すと、B(x′,y′)はB(x,−y)と置き換えられるので、
φAB(x,y)=A(x,y)+B(x,−y)
となる。
【0005】
同様に図8(b)、(c)に示す如き、他の組み合わせについては、
φCA(x,y)=C(x,y)+A(x,−y)
φBC(x,y)=B(x,y)+C(x,−y)
となる。
y=0のラインについては、
φAB(x,0)=A(x,0)+B(x,0)
φCA(x,0)=C(x,0)+A(x,0)
φBC(x,0)=B(x,0)+C(x,0)
となり、実際の測定によりφAB(x,0)、φCA(x,0)、φBC(x,0)が求まっているので、A(x,y),B(x,y),C(x,y)の各形状はこれら3つの関係式について連立方程式を解くことで求められる。
【0006】
【発明が解決しようとする課題】
しかしながら、上記方法によって求められた形状は面形状ではなく1ラインの断面形状である。基準面が球面対称であれば1ラインの断面形状を求めることで全体の表面形状を知ることができるが、一般には基準面は球面対象とはなっていないので、全体の表面形状を高精度で特定する手法が必要となる。
本発明は上記事情に鑑みなされたもので、高精度平面の全体の表面形状を高精度で特定することができる簡易な高精度平面の測定方法を提供することを目的とするものである。
【0007】
【課題を解決するための手段】
本発明の高精度平面の形状測定方法は、所定の3枚の平板のうち互いに異なる2枚のペアを順次3回選択し、この選択操作を行う度に、選択されたペアの平板の被測定面が互いに所定の間隔をおいて対向するように配置してこれら対向する被測定面の相対変位を2次元的に測定し、これら3回の測定結果を演算して前記各平板の被測定面の形状を求める方法において、
前記ペアの被測定面の相対変位は、一方の平板に対し他方の平板を所定の角度だけ回転する毎に測定し、
この所定の回転角度毎の測定結果と、前記平板の被測定面形状を近似する所定のベキ級数多項式とを対応させた関係式を作成し、
次に、前記各ペアの被測定面毎に作成した前記関係式同志を互いに演算して、前記各平板の被測定面の形状を求めることを特徴とするものである。
【0008】
上記所定の角度は90度とする。
また、上記の所定のベキ級数多項式は例えばツェルニケ多項式を用いる。
また、上記の相対変位を2次元的に測定する装置としてフィゾー型干渉装置を用いるのが好ましい。
【0009】
【発明の実施の形態】
以下、本発明の一実施形態に係る高精度平面の形状測定方法について詳細に説明する。
まず、前述した従来技術と同様に、フィゾー型干渉計用の3枚の基準板ガラスを作成し、これらをA,B,Cとし、これら各基準板ガラスについて図1に示すような座標を適用し、各々の基準面をA(x,y)、B(x,y)、C(x,y)とする。
【0010】
次に、これら基準面の表面形状をツェルニケ多項式により近似して表す。
ツェルニケ多項式は、収差解析によく用いられる関数で、最小二乗法により干渉計で測定した形状に近似した形状を x,y の関数で表すことができる。ここでは、ツェルニケ多項式を6次まで用いた。
【0011】
なお、下表に6次までのツェルニケ多項式の各項を具体的に示す。
p ツェルニケ多項式Zp(x,y)
0 1
1 x
2 y
3 2(x2+y2)−1 回転対称
4 x2−y2 回転非対称
5 2xy 回転非対称
6 3(x3+xy2)−2x 回転非対称
7 3(x2y+y3)−2y 回転非対称
8 6(x4+2x2y2+y4−x2−y2)+1 回転対称
9 x3−3xy2 回転非対称
10 3x2y−y3 回転非対称
11 4(x4−y4)−3(x2−y2) 回転非対称
12 8(x3y+xy3)−6xy 回転非対称
13 10(x5+2x3y2+xy4)−12(x3+xy2)+3x 回転非対称
14 10(x4y+2x2y3+y5)−12(x2y+y3)+3y 回転非対称
15 20(x6+3x4y2+3x2y4+y6)−30(x4+2x2y2+y4)+12(x2+y2)−1 回転対称
【0012】
この6次のツェルニケ多項式でA(x,y),B(x,y),C(x,y)の形状に対する近似
形状を考えると、
【数1】
となる。
【0013】
ただし、上記式(1)〜(3)においてAp,Bp,Cpは近似したときの係数を示す。
ここで、図7に示す如きフィゾー型干渉計を用意し、基準板5の位置にAを、被検板6の位置にBをセットする。そして、基準面5aをA(x,y)、被検面6aの
表面形状をB(x,y)とし、次に、この干渉計で測定された両面の相対的形状をφAB(x,y)とする。
同様に基準面5aをC(x,y)、被検面6aをA(x,y)としたときの両面の相対的形状をφBC(x,y)、基準面5aをB(x,y)、被検面6aをC(x,y)としたときの両面の相対的形状をφBC(x,y)とする。
【0014】
そして、干渉計で測定した形状を近似式で表すと、
【数2】
となる。
【0015】
次に、被検体6位置のガラスを基準板5位置のガラスに対し、0,90度,180度,270度の各角度だけ光軸を中心として回転させ、各回転操作終了毎にその位置で両面の相対的な表面形状差測定を行う。
ここで、基準板5位置にA、被検体6位置にBをセットしたとき、
この測定によって得られた結果は、下式(7)〜(10)で表される。
【0016】
【数3】
【0017】
これら4つの式(7)〜(10)を足し合わせると上記表1から分かるように、回転非対称である、近似関数B項のp=4、5、6、7、9、10、11、12、13、14の項は相殺されてしまう。また、p=0の項は高さを表すものであり、p=1、2の項は、それぞれx、y方向の傾きを表すものであるので計算結果からはずしてもよい。
一方、p=3、8、15の項に関しては、回転対称であるから相殺されない。
【0018】
したがって、
【数4】
となる。
【0019】
これにより、A(x,y)とB(x,y)の近似式のうち、回転対称な項の係数を足し合わせたものと、A(x,y)の近似式のうち回転非対称な係数が求まる。
また、同様にして、基準板5位置にCを、被検体6位置にAをセットした場合の測定により、C(x,y)とA(x,y)の近似式のうち回転対称な項の係数を足し合わせたものと、C(x,y)の近似式のうち回転非対称な項の係数が求まり、さらに、基準板位置にBを、被検体位置にCをセットした場合の測定により、B(c,y)とC(x,y)の近似式のうち回転対称な項の係数の足し合わせたものと、B(x,y)の近似式の回転非対称な項の係数が求まる。
【0020】
【数5】
【0021】
すなわち、回転非対称な項の係数Aj,Bj,Cj(j=4、5、6、7、9、10、11、12、13、14)と、回転対称な項の係数を足し合わせたものAi+Bi、Ci+Ai、Bi+Ci(i=3、8、15)が求まる。この回転対称な係数の足し合わせた値に基づき、連立方程式を解くことにより、回転対称な係数Ai、Bi、Ci(i=3、8、15)が求まり、各基準面A(x,y),B(x,y),C(x,y)の近似形状が求まる。
【0022】
なお、この後、上記のようにして、基準面の近似形状が求められた基準板A,B,Cの中から最も平面度の高いものを選択し、それを図7に示すフィゾー型干渉計の基準板5の位置にセットし、次に、測定したい被検面を有する被測定物を被検板6の位置にセットし、該被検面の形状を干渉縞観察によって行うようにする。
【0023】
この場合において、上記方法で得られた基準面の近似形状をも考慮して該被検面の形状を求めるのが好ましい。すなわち、干渉計での測定形状から該近似形状を差し引けば実際の形状により近いものが得られる。
なお、本発明の実施形態としては上記のものに限られるものではなく、種々の態様の変更が可能である。例えば、上記実施形態のものでは被検面を基準面に対して90度ずつ4回回転させ、その各回転操作後測定するようにしているが、4n回(n=1,2,3…)回転させ、そのうちの4つを1組とし、その4つのうち任意の1つに対して他の3つが90度、180度、270度の角度の差を有するようにすることが可能である(例えば 0、48゜、90゜、138゜、180゜、228゜、270゜、318゜)。
【0024】
また、上記実施形態では被測定面形状を近似する多項式として6次のツェルニケ多項式を用いているが、これに代えてその他の種々のベキ級数多項式を用いることができ、また、多項式の項も6次迄の項をとる代わりに、5次以下もしくは7次以上の項までとることも可能である。
さらに、上述した実施形態では実際の被測定物の表面形状を測定するフィゾー型干渉計によって基準面の表面形状を測定しているが、実際の被測定物の表面形状を測定する干渉計とは異なる種々の2面間相対変位測定手段を用いて基準面の表面形状を測定することも可能である。
【0025】
【実施例】
以下、具体的な実施例を用いて本発明方法をさらに詳細に説明する。
図2に示す如き横置のフィゾー型干渉計の基準板53として使用する平面ガラス板の表面形状を、このフィゾー型干渉計を用いて測定した。なお、図2中で51は干渉計本体、52は縞を動かすための(フリンジスキャニング用の)駆動装置、54は被検板、55および56は5軸調整台である。
【0026】
まず、基準板とし得る3枚の平面ガラス板A,B,Cを用意し、各々について、図3に示す如く、基準面61側に配される円環状のガラスホルダ62の0、90度、180度、270度部分にV溝63を切り、その間に細い糸64aを十字に張設し、各糸64aをV溝63内でねじ64bにより固定することで、ガラス板の中心と0、90度、180度、270度の位置が分かるようにした。なお、図3中で原点を識別するためのマーク65を形成した。
【0027】
このようにして準備されたガラス板A,B,Cのうち任意の1つを図2に示す如く5軸調整台56にセットし、他のガラス板A,B,Cのうちいずれかを駆動装置52を介して5軸調整台55にセットした。
このとき、セットされた2つのガラス板A,B,Cの基準面同志は互いに対向する。3つのガラス板A,B,Cの中から、2つを選択し、一方を基準板位置に、他方を被検板位置にセットする場合の組合せは6通り存在し、これら各組合せについて、セットされた2つの基準面同志の相対変位(相対形状)を測定した。
【0028】
すなわち、ガラス板Bを基準板位置に、ガラス板Aを被検板位置にセットしたときの組合せをAS−BTとし、ガラス板Cを基準板位置に、ガラス板Aを被検板位置にセットしたときの組合せをAS−CTとし、ガラス板Aを基準板位置に、ガラス板Bを被検板位置にセットしたときの組合せをBS−ATとし、ガラス板Cを基準板位置に、ガラス板Bを被検板位置にセットしたときの組合せをBS−CTとし、ガラス板Aを基準板位置に、ガラス板Cを被検板位置にセットしたときの組合せをCS−ATとし、ガラス板Bを基準板位置に、ガラス板Cを被検板位置にセットしたときの組合せをCS−BTとした。また、これら各組合せにおいて、被検板位置にセットされたガラス板を基準板位置にセットされたガラス板に対し、0、90度、180度、270度だけ回転させる度に形状測定を行った。また、該形状測定は各回転角度位置について5回ずつ繰り返し行った。
【0029】
なお、回転角度が0とは2つのガラス板のマーク65の位置が同一方向に配された場合で、いずれの回転角度位置においても2つの基準板における糸のラインが光軸方向に重なるようにセットした。
上記干渉計による測定値にツェルニケ多項式(6次の項まで採用)を対応させて、上記演算を行い、これにより得られたガラス板A,B,Cの近似形状についての各等高線図を図4に示す。
なお、各図の上部に記載されている数値はP−V値(高低差)をWAVE(波長)で表したものである。
【0030】
図5は、上記各組合せについて上述した如き測定を行った場合の評価を示すグラフである。すなわち、各組合せについて各回転角度位置毎に、干渉計で得られた測定形状のRMS(a;実線)、上記測定と演算によって分離した近似形状A、B、Cの、測定に対応する各組合せ、各回転角毎にした2つの近似形状のたし合せを測定形状から差し引き補正したもののRMS(b;破線)、および測定形状をツェルニケ多項式(6次項まで)で近似しその近似形状を該測定形状から差し引いた場合のRMS(c;一点鎖線)(良好な測定を行えばbはcに近づく)を示す。
なお、グラフの縦軸はP−V値(高低差)をWAVE(波長)単位で示すものであり、また、横軸の1目盛中には各角度の5回分の測定データが示されている。
【0031】
また、図6の(a)、(b)、(c)は、上記ガラス板A,B,Cを各々基準板とし、被検板Dの被検面を上記フィゾー型干渉計により測定した場合の値に基づき得られた、対向面の相対形状の等高線図を示すものである。
すなわち、(a)はガラス板Aの基準面と被検板Dの被検面の相対形状変化A+Dを示すものであり、(b)はガラス板Bの基準面と被検板Dの被検面の相対形状変化B+Dを示すものであり、(c)はガラス板Cの基準面と被検板Dの被検面の相対形状変化C+Dを示すものである。
【0032】
また、図6の(a′)、(b′)、(c′)は、上記(a),(b),(c)に示される相対形状変化から、上記手法により求められた、対応するガラス基準板の基準面の近似形状を差し引いた形状の等高線図を示すものである。
すなわち、(a′)は上記A+Dからガラス板Aの基準面形状を差し引いた被検板Dの被検面形状を示すものであり、(b′)は上記B+Dからガラス板Bの基準面形状を差し引いた被検板Dの被検面形状を示すものであり、(c′)は上記C+Dからガラス板Cの基準面形状を差し引いた被検板Dの被検面形状を示すものである。
【0033】
以上に示すように近似形状をデータ測定と解析により求め、それにより補正することで、標準偏差1σ(=0.005wave)程度の平面を作成可能である。
【0034】
【発明の効果】
以上説明したように、本発明の高精度平面の形状測定方法によれば、3面合わせ方法を前提技術とし、これに2平面の相対形状をベキ級数多項式で近似する手法を採用しているので、簡易な演算で高精度平面全体の近似表面形状を求めることが可能である。特に、2つの被検面を互いに90度ずつ回転させ、その操作毎にこれらの相対変位を測定するようにし、これら4つの測定結果のそれぞれについて上記ベキ級数多項式を対応させて演算すると、各演算多項式の多くの項を互いに相殺することができ、演算処理を極めて簡易なものとすることができる。
【図面の簡単な説明】
【図1】本発明の一実施例形態に係る方法を説明するための図
【図2】本発明の実施例において用いられる装置を示す概略図
【図3】本発明の実施例において用いられる、ガラス板の回転方向位置決め用部材を示す概略図
【図4】本発明の実施例による測定結果に基づく等高線図
【図5】本発明の実施例による測定結果を示すグラフ
【図6】本発明の実施例による測定結果に基づく等高線図
【図7】従来から用いられているフィゾー型干渉計の一般的構成を示す概略図
【図8】従来の3面合わせ方法を説明するための概略図
【符号の説明】
5、53 基準板
5a、61 基準面
6、54 被検板
6a 被検面
51 干渉計本体
55、56 5軸調整台
62 ホルダ
63 V溝
64a 糸
65 マーク[0001]
TECHNICAL FIELD OF THE INVENTION
The present invention relates to a method for measuring the shape of a high-precision plane, and more particularly to a method for measuring the shape of a high-precision plane which determines the surface shape of a plane used as a reference plane of an interferometer.
[0002]
[Prior art]
As a method of measuring flatness, measurement using an interferometer such as a Fizeau interferometer is known. Although such an interferometer can measure the flatness of the surface to be measured with high accuracy, the measurement is a relative measurement to a reference surface, not an absolute measurement. Therefore, an extremely high-precision plane is required as the reference plane. Therefore, a method of measuring a reference plane having such a high-precision plane is required.
As a method of measuring such a reference plane, three reference plates are created, and a reference plane relative displacement is measured for each combination of three reference plate pairs selected from the three. A three-plane matching method for measuring the shape of each reference plane by solving a simultaneous equation based on the same is known.
[0003]
Hereinafter, the three-plane alignment method will be described.
The three reference glass sheets are designated as A, B, and C. Taking the coordinates shown in FIG. 1 for each reference plate glass, the shapes of the glass surfaces of A, B, and C can be represented by a function of x and y, and therefore, A (x, y) and B ( x, y) and C (x, y). The Z coordinate as shown in FIG. 1 is used because the glass surface is represented by a plus sign when the glass surface is convex, and is represented by a minus sign when the glass surface is concave.
Here, for example, the reference surface is A (x, y), the test surface is B (x, y), and these two surfaces are opposed to each other as shown in FIG. 8A, and are located at predetermined positions of the Fizeau interferometer. set.
[0004]
Next, if the relative shape of both surfaces measured by this interferometer is φ AB (x, y),
φ AB (x, y) = A (x, y) + B (x ′, y ′)
It becomes.
When the coordinates of the test surface are represented by the coordinates of the reference surface, B (x ′, y ′) is replaced with B (x, −y).
φ AB (x, y) = A (x, y) + B (x, −y)
It becomes.
[0005]
Similarly, as shown in FIGS. 8B and 8C, for other combinations,
φ CA (x, y) = C (x, y) + A (x, −y)
φ BC (x, y) = B (x, y) + C (x, −y)
It becomes.
For the line with y = 0,
φ AB (x, 0) = A (x, 0) + B (x, 0)
φ CA (x, 0) = C (x, 0) + A (x, 0)
φ BC (x, 0) = B (x, 0) + C (x, 0)
Since φ AB (x, 0), φ CA (x, 0), and φ BC (x, 0) are obtained by actual measurement, A (x, y), B (x, y), C Each shape of (x, y) can be obtained by solving simultaneous equations for these three relational expressions.
[0006]
[Problems to be solved by the invention]
However, the shape obtained by the above method is not a planar shape but a cross-sectional shape of one line. If the reference surface is spherically symmetric, the entire surface shape can be known by calculating the cross-sectional shape of one line. However, in general, the reference surface is not a spherical object, so the entire surface shape can be determined with high accuracy. An identifying method is required.
The present invention has been made in view of the above circumstances, and an object of the present invention is to provide a simple high-precision plane measuring method capable of specifying the entire surface shape of a high-precision plane with high accuracy.
[0007]
[Means for Solving the Problems]
According to the method for measuring the shape of a high-precision flat plate of the present invention, two pairs different from each other are sequentially selected three times from predetermined three flat plates, and each time the selecting operation is performed, the measurement of the selected pair of flat plates is performed. The surfaces are arranged so as to face each other at a predetermined interval, the relative displacements of the facing surfaces to be measured are two-dimensionally measured, and the results of these three measurements are calculated to calculate the surface to be measured of each of the flat plates. In the method for determining the shape of
The relative displacement of the measured surface of the pair is measured each time the other flat plate is rotated by a predetermined angle with respect to one flat plate,
A measurement result for each predetermined rotation angle and a relational expression corresponding to a predetermined power series polynomial approximating the measured surface shape of the flat plate are created,
Next, the relational expressions created for each measured surface of each pair are calculated with each other to obtain the shape of the measured surface of each flat plate.
[0008]
The predetermined angle is 90 degrees.
The predetermined power series polynomial uses, for example, a Zernike polynomial.
Further, it is preferable to use a Fizeau interference device as a device for measuring the relative displacement two-dimensionally.
[0009]
BEST MODE FOR CARRYING OUT THE INVENTION
Hereinafter, a method for measuring the shape of a highly accurate plane according to an embodiment of the present invention will be described in detail.
First, in the same manner as in the prior art described above, three reference plate glasses for a Fizeau-type interferometer were prepared, these were designated as A, B, and C, and the coordinates as shown in FIG. Let each reference plane be A (x, y), B (x, y), C (x, y).
[0010]
Next, the surface shapes of these reference planes are approximated and represented by Zernike polynomials.
The Zernike polynomial is a function often used for aberration analysis, and can express a shape approximated to a shape measured by an interferometer by the least squares method as a function of x, y. Here, Zernike polynomials are used up to the sixth order.
[0011]
The following table specifically shows each term of the Zernike polynomial up to the sixth order.
p Zernike polynomial Z p (x, y)
0 1
1 x
2 y
3 2 (x 2 + y 2 ) -1 rotational symmetry 4 x 2 -y 2 rotationally asymmetric 5 2xy rotationally asymmetric 6 3 (x 3 + xy 2 ) -2x rotationally asymmetrical 7 3 (x 2 y + y 3) -2y rotationally asymmetrical 8 6 (x 4 + 2x 2 y 2 + y 4 -x 2 -y 2) +1
Considering the approximate shape of A (x, y), B (x, y), C (x, y) in this 6th-order Zernike polynomial,
(Equation 1)
It becomes.
[0013]
However, in the above equations (1) to (3), A p , B p , and C p indicate coefficients when approximated.
Here, a Fizeau interferometer as shown in FIG. 7 is prepared, and A is set at the position of the reference plate 5 and B is set at the position of the
Similarly, when the reference surface 5a is C (x, y) and the surface 6a is A (x, y), the relative shape of both surfaces is φ BC (x, y), and the reference surface 5a is B (x, y). y), the relative shape of both surfaces when the test surface 6a is C (x, y) is φ BC (x, y).
[0014]
Then, when the shape measured by the interferometer is represented by an approximate expression,
(Equation 2)
It becomes.
[0015]
Next, the glass at the position of the subject 6 is rotated about the optical axis by 0, 90 degrees, 180 degrees, and 270 degrees with respect to the glass at the reference plate 5 position. A relative surface shape difference measurement of both surfaces is performed.
Here, when A is set at the reference plate 5 position and B is set at the subject 6 position,
The results obtained by this measurement are represented by the following equations (7) to (10).
[0016]
(Equation 3)
[0017]
When these four equations (7) to (10) are added, as can be seen from Table 1 above, p = 4, 5, 6, 7, 9, 10, 11, 12 of the rotationally asymmetric approximation function B term. , 13, and 14 are canceled out. The term of p = 0 represents the height, and the term of p = 1 and 2 represents the inclination in the x and y directions, respectively, and may be excluded from the calculation result.
On the other hand, the terms p = 3, 8, and 15 are not canceled because they are rotationally symmetric.
[0018]
Therefore,
(Equation 4)
It becomes.
[0019]
Accordingly, the sum of the coefficients of the rotationally symmetric terms in the approximate expressions of A (x, y) and B (x, y) and the rotationally asymmetric coefficient in the approximate expression of A (x, y) Is found.
Similarly, by measuring when C is set at the position of the reference plate 5 and A is set at the position of the subject 6, a rotationally symmetric term in the approximate expression of C (x, y) and A (x, y) is obtained. And the coefficient of a rotationally asymmetric term in the approximation formula of C (x, y) are obtained. Further, the measurement is performed by setting B at the reference plate position and setting C at the subject position. , B (c, y) and the sum of the coefficients of the rotationally symmetric terms in the approximate equations of C (x, y) and the coefficients of the rotationally asymmetric terms of the approximate equations of B (x, y) are obtained. .
[0020]
(Equation 5)
[0021]
That is, the coefficients A j , B j , and C j (j = 4, 5, 6, 7, 9, 10, 11, 12, 13, 14) of the rotationally asymmetric terms are added to the coefficients of the rotationally symmetric terms. Ai + Bi, Ci + Ai, and Bi + Ci (i = 3, 8, 15) are obtained. By solving simultaneous equations based on the sum of the rotationally symmetric coefficients, rotationally symmetric coefficients Ai, Bi, Ci (i = 3, 8, 15) are obtained, and each reference plane A (x, y) is obtained. , B (x, y) and C (x, y) are obtained.
[0022]
After that, a plate having the highest flatness is selected from among the reference plates A, B, and C for which the approximate shape of the reference surface has been obtained as described above, and is selected from the Fizeau interferometer shown in FIG. Then, an object to be measured having a surface to be measured is set at the position of the
[0023]
In this case, it is preferable to determine the shape of the test surface in consideration of the approximate shape of the reference surface obtained by the above method. That is, a shape closer to the actual shape can be obtained by subtracting the approximate shape from the shape measured by the interferometer.
The embodiments of the present invention are not limited to those described above, and various modifications can be made. For example, in the above-described embodiment, the test surface is rotated four times by 90 degrees with respect to the reference surface, and measurement is performed after each rotation operation, but 4n times (n = 1, 2, 3,...) It is possible to rotate them so that four of them are a set, and the other three have an angle difference of 90 degrees, 180 degrees and 270 degrees with respect to any one of the four ( For example, 0, 48, 90, 138, 180, 228, 270, 318).
[0024]
In the above-described embodiment, a 6th-order Zernike polynomial is used as a polynomial approximating the shape of the surface to be measured. However, other various power series polynomials can be used instead. Instead of taking the terms up to the next, it is also possible to take the terms up to the fifth or lower or the seventh or higher.
Furthermore, in the above-described embodiment, the surface shape of the reference surface is measured by the Fizeau interferometer that measures the actual surface shape of the device under test, but the interferometer that measures the surface shape of the actual device under test is It is also possible to measure the surface shape of the reference surface using various different relative displacement measuring means between the two surfaces.
[0025]
【Example】
Hereinafter, the method of the present invention will be described in more detail with reference to specific examples.
The surface shape of a flat glass plate used as the
[0026]
First, three flat glass plates A, B, and C, which can be used as reference plates, are prepared, and as shown in FIG. The V-
[0027]
Any one of the glass plates A, B, and C prepared in this manner is set on the 5-axis adjustment table 56 as shown in FIG. 2, and any one of the other glass plates A, B, and C is driven. It was set on a 5-axis adjustment table 55 via the
At this time, the reference surfaces of the two set glass plates A, B, and C face each other. There are six combinations in which two are selected from the three glass plates A, B, and C, and one is set at the reference plate position and the other is set at the test plate position. The relative displacement (relative shape) of the two reference planes measured was measured.
[0028]
That is, the reference plate position the glass plate B, and combinations when setting the glass plate A to the subject plate position and A S -B T, the reference plate position the glass plate C, the test plate position the glass plate A combinations when set as a S -C T to the glass plate a to the reference plate position, the combination of when setting the glass plate B to the subject plate position and B S -A T, relative to the glass plate C a plate position, the combination of when setting the glass plate B to the subject plate position and B S -C T, the glass plate a to the reference plate position, the combination of when setting the glass plate C to the subject plate position and C S -A T, the reference plate position the glass plate B, and the combination of when setting the glass plate C to the subject plate position with C S -B T. In each of these combinations, the shape was measured every time the glass plate set at the test plate position was rotated by 0, 90 degrees, 180 degrees, and 270 degrees with respect to the glass plate set at the reference plate position. . The shape measurement was repeated five times for each rotation angle position.
[0029]
Note that the rotation angle of 0 means that the positions of the
The Zernike polynomial (up to the sixth order term) is made to correspond to the value measured by the interferometer, the above calculation is performed, and the contour maps of the approximate shapes of the glass plates A, B, and C obtained by this are shown in FIG. Shown in
In addition, the numerical value described in the upper part of each figure represents the PV value (difference in height) by WAVE (wavelength).
[0030]
FIG. 5 is a graph showing an evaluation when the above-described measurement is performed for each combination. That is, each combination corresponding to the measurement of the RMS (a; solid line) of the measured shape obtained by the interferometer and the approximate shapes A, B, and C separated by the above measurement and calculation for each rotation angle position for each combination RMS (b; dashed line) obtained by subtracting and correcting the sum of the two approximate shapes for each rotation angle from the measured shape, and approximating the measured shape by a Zernike polynomial (up to the sixth order) and measuring the approximate shape by the measurement. RMS (c; dash-dot line) when subtracted from the shape (b approaches c if good measurement is performed).
The vertical axis of the graph indicates the PV value (difference in height) in WAVE (wavelength) units, and one scale on the horizontal axis indicates measurement data for five times at each angle. .
[0031]
6 (a), (b) and (c) show the case where the above-mentioned glass plates A, B and C are each used as a reference plate and the surface to be measured of the plate to be tested D is measured by the above Fizeau interferometer. FIG. 3 is a contour diagram of the relative shape of the facing surface obtained based on the value of.
That is, (a) shows the relative shape change A + D between the reference surface of the glass plate A and the test surface of the test plate D, and (b) shows the test result of the reference surface of the glass plate B and the test plate D. FIG. 7C shows the relative shape change B + D of the surface, and FIG. 7C shows the relative shape change C + D between the reference surface of the glass plate C and the test surface of the test plate D.
[0032]
(A '), (b'), and (c ') in FIG. 6 correspond to the relative shape changes shown in (a), (b), and (c), which are obtained by the above method. FIG. 3 shows a contour map of a shape obtained by subtracting an approximate shape of a reference surface of a glass reference plate.
That is, (a ') shows the shape of the test surface of the test plate D obtained by subtracting the shape of the reference surface of the glass plate A from the above A + D, and (b') shows the shape of the reference surface of the glass plate B from the above B + D. (C ') shows the shape of the test surface of the test plate D obtained by subtracting the reference surface shape of the glass plate C from the above C + D. .
[0033]
As described above, a plane having a standard deviation of about 1σ (= 0.005 wave) can be created by obtaining an approximate shape by data measurement and analysis and correcting the approximate shape.
[0034]
【The invention's effect】
As described above, according to the high-precision plane shape measuring method of the present invention, the technique of approximating the relative shape of the two planes with a power series polynomial is adopted as the premise technique based on the three-plane matching method. The approximate surface shape of the entire high-precision plane can be obtained by a simple calculation. In particular, when the two test surfaces are rotated by 90 degrees with respect to each other and their relative displacements are measured each time the operation is performed, and each of these four measurement results is calculated in correspondence with the above power series polynomial, each calculation is performed. Many terms of the polynomial can be canceled each other, and the arithmetic processing can be made extremely simple.
[Brief description of the drawings]
FIG. 1 is a diagram for explaining a method according to an embodiment of the present invention. FIG. 2 is a schematic diagram showing an apparatus used in an embodiment of the present invention. FIG. 3 is a diagram used in an embodiment of the present invention. FIG. 4 is a schematic diagram showing a member for positioning a glass plate in a rotation direction. FIG. 4 is a contour diagram based on measurement results according to an embodiment of the present invention. FIG. 5 is a graph showing measurement results according to an embodiment of the present invention. FIG. 7 is a schematic diagram showing a general configuration of a conventionally used Fizeau interferometer. FIG. 8 is a schematic diagram for explaining a conventional three-plane alignment method. Description]
5, 53
Claims (4)
前記ペアの被測定面の相対変位は、一方の平板に対し他方の平板を所定の角度だけ回転する毎に測定し、
この所定の回転角度毎の測定結果と、前記平板の被測定面形状を近似する所定のベキ級数多項式とを対応させた関係式を作成し、
次に、前記各ペアの被測定面毎に作成した前記関係式同志を互いに演算して、前記各平板の被測定面の形状を求めることを特徴とする高精度平面の形状測定方法。Two different pairs of the predetermined three flat plates are sequentially selected three times, and each time the selecting operation is performed, the measurement target surfaces of the selected pair of flat plates face each other with a predetermined interval therebetween. And measuring the relative displacement of these opposed surfaces to be measured two-dimensionally and calculating the results of these three measurements to determine the shape of the surface to be measured of each of the flat plates.
The relative displacement of the measured surface of the pair is measured each time the other flat plate is rotated by a predetermined angle with respect to one flat plate,
A measurement result for each predetermined rotation angle and a relational expression corresponding to a predetermined power series polynomial approximating the measured surface shape of the flat plate are created,
Next, a method of measuring the shape of a high-precision plane, wherein the relational expressions created for the surfaces to be measured of each pair are calculated with each other to obtain the shape of the surface to be measured of each flat plate.
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