JP3670894B2 - Optical curved surface design method - Google Patents

Description

ここで積分定数 ｑ(ＸK,ＹL) は

である。
【００２５】
そして、上記のｑ(Ｘ,Ｙ) ≡ｈ／√(１＋ｈ²)の関係に基づき、関数ｑ(ＸK，Ｙ)から背骨直線上の格子点におけるＹ方向の傾きＢＹKjを滑らかにつなぐ関数ｈ(ＸK，Ｙ)を求める。
ｈ(ＸK,Ｙ)＝ｑ(ＸK,Ｙ)／√{１−ｑ(ＸK,Ｙ)²}
ステップＳ4の第２の積分段階では、基準格子点におけるサグ量ＺKLを積分定数として関数ｈ(ＸK，Ｙ)をＹで積分し、関数ｆ(ＸK，Ｙ)を求める。
【数２】

ステップＳ4の操作により、ＢＹKj＝ｈ(ＸK,Ｙj)、ＺKj＝ｆ(ＸK,Ｙj)が定まり、曲面の背骨直線に沿った形状が求められる。
【００２６】
ステップＳ5では、Ｘ軸に平行で格子点を通る各直線Ｙ＝Ｙjの上の格子点(Ｘi，Ｙj)におけるＸ方向の曲率ＣＸijをつなぐ関数ｕ(Ｘ，Ｙj)を求め、これを２回の積分を含む計算をすることにより、直線Ｙ＝Ｙjの上の格子点におけるサグ量Ｚijを滑らかにつなぐ関数ｆ(Ｘ，Ｙj)を各直線毎に求める。このステップＳ5の操作は、図４(Ｂ)の演算に対応する。この操作により曲面の直線Ｙ＝Ｙjに沿った形状が求められる。
【００２７】
ステップＳ5は、各直線毎に２回の積分段階を含む。第１の積分段階では、背骨直線上の格子点(ＸK，Ｙj)のＸ方向の傾きに対応する定数ｐ(ＸK，Ｙj)を積分定数として関数ｕ(Ｘ，Ｙj)をＸで積分し、Ｘ軸に平行な各直線上の格子点におけるＸ方向の傾きＢＸijに対応する分布ｐ(Ｘi，Ｙj)を滑らかにつなぐ関数ｐ(Ｘ，Ｙj)を求める。
【数３】

ここで積分定数 ｐ(ＸK,Ｙj) は

である。
【００２８】
そして、上記のｐ(Ｘ,Ｙ) ≡ｇ／√(１＋ｇ²)の関係に基づき、関数ｐ(Ｘ，Ｙj)からＸ軸に平行な直線Ｙ＝Ｙjの上の各格子点におけるＸ方向の傾きＢＸijを滑らかにつなぐ関数ｇ(Ｘ，Ｙj)を直線毎に求める。
ｇ(Ｘ,Ｙj)＝ｐ(Ｘ,Ｙj)／√{１−ｐ(Ｘ,Ｙj)²}
ステップＳ5の第２の積分段階では、関数ｇ(Ｘ，Ｙj)を背骨直線上の格子点におけるサグ量ＺKjを積分定数としてＸで積分し、関数ｆ(Ｘ，Ｙj)を直線毎に求める。
【数４】

ステップＳ5の操作により、ＢＸij＝ｇ(Ｘi,Ｙj)、Ｚij＝ｆ(Ｘi,Ｙj)が定まり、曲面全体の形状が求められる。
【００２９】
ステップＳ6,Ｓ7は、Ｙ方向の傾きＢＹijを求めるための処理であり、Ｘ方向の並びで求められたＸ方向の傾きＢＸijをＹ方向の並びと見てこれらの間を滑らかにつなぐ関数ｇ(Ｘi，Ｙ)を求め、これをＹで微分した後にＸで積分することにより、Ｙ方向の傾きＢＹijを求める。
ステップＳ6では、上記の傾きＢＸijを滑らかにつなぐ関数ｇ(Ｘi，Ｙ)をＹで微分し、各格子点(Ｘi，Ｙj)における微分値Ｗijを求める。
ｗ(Ｘi,Ｙ)＝∂ｇ(Ｘi,Ｙ)/∂Ｙ
Ｗij＝ｗ(Ｘi,Ｙj)
この操作は、図４(Ｃ)の演算に対応する。記号「※」が各格子点の微分値Ｗijを表す。
【００３０】
続くステップＳ7では、Ｘ軸に平行で格子点を通る各直線Ｙ＝Ｙjの上の格子点(Ｘi，Ｙj)における微分値Ｗijを滑らかにつなぐ関数ｗ(Ｘ，Ｙj)をＸで積分し、各格子点(Ｘi，Ｙj)におけるＹ方向の傾きＢＹijを求める。この操作は、図４(Ｄ)の演算に対応する。
【数５】

【００３１】
以上のステップにより、全格子点（０≦ｉ≦I，０≦ｊ≦Ｊ）において面形状を表す４種類のパラメータＺij、ＢＸij、ＢＹij、Ｗij が定まった。ステップＳ8では、矩形の各領域Ｒij＝{(Ｘ,Ｙ)｜Ｘ_i-1≦Ｘ≦Ｘi,Ｙ_j-1≦Ｙ≦Ｙj}内をＸ,Ｙの以下の双三次式で表現する。
【数６】

【００３２】
式に含まれる１６個の係数は、領域を囲む４つの格子点の４つのパラメータ、すなわち１６のパラメータを用いた連立方程式により決定される。(Ｘi-1,Ｙj-1)の格子点と(Ｘi,Ｙj)の格子点とを対角とするij番目の領域Ｒijにおける係数γab^(ij)は、以下の行列の演算により求められる。
【数７】

ここでΔＸ_i-1＝Ｘi−Ｘ_i-1、ΔＹ_j-1＝Ｙj−Ｙ_j-1 である。
【００３３】
係数γab^(ij)が求まると、領域Ｒij内の任意の位置でＺ＝ｆij(Ｘ,Ｙ)、 およびそのＸ,Ｙによる微分値が計算できる。したがって、上記の係数の演算を全ての領域で行うことにより、全定義域での曲面を表す方程式Ｚ＝ｆ(Ｘ,Ｙ)が、ｆij(Ｘ,Ｙ)[0≦i≦I,0≦j≦J]の集合として求まったことになる。Ｚ＝ｆ(Ｘ,Ｙ) は全定義域内で２次微分まで連続であることが保証される。ステップＳ8の操作は、図４(Ｅ)の計算に対応する。
【００３４】
ステップＳ9では、上記で求められた面形状に基づいて性能のシミュレーションが行われる。そして、ステップＳ10では、シミュレーション結果に基づいて性能を評価し、これが満足できるものであれば、その段階で設計を終了し、満足できるものでなければ、ステップＳ11でパラメータとして曲率ＣＸij、ＣＹKjを調整し、ステップＳ4からの処理を繰り返す。調整されるパラメータとしては、曲率のみでなく、背骨直線上の格子点に与えられた傾きＢＸKjを含めることができる。
【００３５】
上記のようにレンズ面を矩形の領域に区切り、各領域内をそれぞれ別の関数で表現することにより、パラメータの変更の影響がほぼ当該領域近傍に限定されるため、局所的な性能改善が容易であり、複雑な形状の非球面の設計に適した設計方法を提供できる。また、パラメータとして収差とほぼ直線的な関係を持つ曲率ＣＸij、ＣＹKjを用いることにより、性能評価に基づくパラメータ変更の際に、いずれのパラメータをどの程度修正すれば収差が改善されるかが直接的に把握できるため、設計の見通しが良好である。例えば、本実施形態のようなｆθレンズの場合、ｆθ特性および主走査断面の像面湾曲にはＣＹKj 、副走査断面の像面湾曲にはＣＸKj、ＢＯＷにはＢＸKj、というように着目する特性あるいは収差に対して支配的なパラメータが明確である。
【００３６】
次に、この発明の設計方法を累進眼鏡レンズの設計に適用した第２の実施形態について説明する。第２の実施形態の設計方法は、レンズ面をピッチが大きい第１格子と、ピッチが小さい第２格子との二種類の格子で分割して形状を設定している。パラメータの設定は第１格子の格子点について行われ、面形状は第１格子のパラメータから求められる第２格子のパラメータを経て決定される。図５は第２の実施形態にかかる光学曲面の設計方法を示すフローチャート、図６は第１格子の格子点でレンズ面に与えられるパラメータを示す説明図、図７は第１格子の格子点で与えられるパラメータの数値を示す図表、図８は第２格子の格子点でレンズ面に与えられるパラメータを示す説明図、図９〜１１は第２格子の格子点で与えられるパラメータの数値を示す図表、図１２はフローチャートの各計算段階を示す説明図である。
【００３７】
以下の説明では、第１格子の格子点を第１の実施形態と同様に(Ｘi,Ｙj)で表し、第２格子の格子点を(Ｘ~m,Ｙ~n)で表す。また、第１格子の基準格子点(ＸK,ＹL)と第２格子の基準格子点(Ｘ~P,Ｙ~Q)は等しく、したがって両格子の背骨直線も一致するものとする。レンズ面を含む１００ｍｍ×１００ｍｍの矩形範囲を、第１格子は２２×２２で分割し、第２格子は５０×５０に分割する。ここで、Ｋ＝１１，Ｌ＝１２，Ｉ＝Ｊ＝２２，Ｐ＝Ｑ＝２５，Ｍ＝Ｎ＝５０である。ただし、第１格子の分割は均等ピッチではない。曲率の変化が急な部分では、格子のピッチが大きいと複雑な形状を十分に表現することができない。そこで、第１格子は、曲率の変化が急激な累進部(レンズ中心部よりやや下側)での分割ピッチを他の部分よりも細かくしてある。第２格子の分割は均等ピッチである。
【００３８】
設計に当たって、図５のステップＳ21で第１格子の基準格子点(ＸK,ＹL)にパラメータとしてＺ方向のサグ量ＺKL、Ｘ方向の傾きＢＸKL、Ｙ方向の傾きＢＹKL、Ｘ方向の曲率ＣＸKL、Ｙ方向の曲率ＣＹKLを与える。ステップＳ22では、Ｙ軸に平行で基準格子点(ＸK,ＹL)を通る背骨直線Ｘ＝ＸKの上の基準格子点以外の第１格子の格子点(ＸK,Ｙj)[j≠L]にＸ方向の傾きＢＸKj、Ｘ方向の曲率ＣＸKj、Ｙ方向の曲率ＣＹKjを与える。ステップＳ23では、背骨直線上以外の第１格子の格子点(Ｘi,Ｙj)[i≠K]にＸ方向の曲率ＣＸijを与える。図６は、第１格子の格子点に与えられたパラメータを示す。図中の記号の意味は図２におけるのと同一である。
【００３９】
以上の３つのステップでパラメータの設定が終了する。具体的には、図７に示すような数値が設定される。図７の図表ではレンズの屈折力との対応関係が良いように、各格子点における曲率ＣＸij,ＣＹKjに代えて断面屈折力ＤＸij、ＤＹKjを与えている。曲率は、レンズ素材の屈折率をＩn(この例ではＩn=1.5)として、ＣＸij＝ＤＸij／(Ｉn−１)、ＣＹKj＝ＤＹKj／(Ｉn−１)により求められる。背骨直線上の格子点(ＸK，Ｙj)には表の左側のＤＹKj，ＢＸKjが与えられ、また各格子点には表の右側のＤＸijが与えられる。さらに基準格子点(ＸK,ＹL)には以下の値が与えられる。
ＢＹKL= 0.00E+00 ＺKL= 0.00E+00
【００４０】
なお、レンズの平面形状は円形であるため、設定された２２×２２の領域の全ての格子点(０≦ｉ≦２２、０≦ｊ≦２２)についてＤＸijを与える必要はない。この例ではレンズの有効径をφ８０ｍｍとし、それより一回り大きなφ９０ｍｍ以内の格子点についてパラメータの値を与える。これにより、パラメータの数が不必要に多くなるのを避けることができる。図７では、Ｘ軸に平行なｊ番目の直線上で有効な格子点の番号ｉの最小値ｉjmin、最大値ｉjmaxをｊの値と共に示している。表中の「#####」で示される欄は、パラメータを設定していない格子点を示す。
【００４１】
第２の実施形態においても、第１の実施形態と同様に第１格子の格子点に与えられたパラメータから関数ｆ(Ｘ,Ｙ)を求めることも可能である。しかし、第１格子のピッチは周辺部では５ｍｍ以上となり、求められる関数ｆ(Ｘ,Ｙ)の曲率の変化が折れ線的であることに鑑みると、眼鏡レンズとしては曲率を十分に滑らかに変化させることができない。そこで第２の実施形態では、第１格子の格子点に与えられたパラメータから直接ｆ(Ｘ,Ｙ)を求めずに、よりピッチが小さい第２格子を設定してその格子点におけるパラメータを求め、これに基づいてｆ(Ｘ,Ｙ)を求めるようにしている。
【００４２】
ステップＳ24では、レンズ面をピッチの細かい第２格子により分割し、第１格子の格子点(Ｘi,Ｙj)に与えられたパラメータＺKL、ＢＹKL、ＢＸKj、ＣＹKj、ＣＸijから第２格子の格子点(Ｘ~m,Ｙ~n)のパラメータＺ~PQ、ＢＹ~PQ、ＢＸ~Pn、ＣＹ~Pn、ＣＸ~mnを求める。ステップＳ24における第２格子の格子点のパラメータを求める手順について以下に説明する。
まず、Ｘ~P＝ＸK、Ｙ~Q＝ＹLとなるように設定したので、Ｚ~PQ＝ＺKL、ＢＹ~PQ＝ＢＹKLである。第２格子の背骨直線上の格子点(Ｘ~P,Ｙ~n)でのＹ方向の曲率ＣＹ~Pn、傾きＢＸ~Pnは、それぞれ第１格子の背骨直線Ｘ＝ＸKに沿って曲率ＣＹKj、傾きＢＸKjを１次元スプライン関数により補間して求める。
【００４３】
また、Ｘ軸と平行で第１格子の格子点を通る各直線Ｙ＝Ｙj に沿って曲率ＣＸijを１次元スプライン関数により補間して、第２格子の格子点を通る直線Ｘ＝Ｘ~mと前記の直線Ｙ＝Ｙjとの交点での曲率ＣＸ~mj を求める。そして、Ｙ軸と平行で第２格子の格子点を通る直線Ｘ＝Ｘ~mに沿って曲率ＣＸ~mjを１次元スプライン関数により補間し、第２格子の格子点(Ｘ~m,Ｙ~n)での曲率ＣＸ~mnを求める。第１格子の格子点の曲率ＣＸijから第２格子の格子点の曲率ＣＸ~mnを求める２つのステップは、２次元スプライン関数により同時に行うことも可能である。なお、ステップＳ24における１次元または２次元スプライン関数は、少なくとも一次微分が連続であるように選ぶ。
【００４４】
ステップＳ24により得られた第２格子の格子点でのパラメータの種類は図８の説明図に示される。図中の記号の意味は図２におけるのと同一である。具体的な数値は、図９〜図１１の図表に示される。なお、図９〜図１１では、パラメータが設定されない周辺部分(０≦ｎ≦２，４８≦ｎ≦５０，０≦ｍ≦２，４８≦ｍ≦５０)を除き、３≦ｍ≦１３の範囲を図９、１４≦ｍ≦３０の範囲を図１０、３１≦ｍ≦４７の範囲を図１１に分割して示している。
【００４５】
ステップＳ25〜Ｓ29は、第１の実施形態のフローチャートにおけるステップＳ4〜Ｓ8と同様であり、ここでは第２格子の格子点(Ｘ~m,Ｙ~n)に与えられたパラメータに基づいて第２格子の各領域を双三次式で表現する。図１２は、簡単のため分割数を６×６に減らして、双三次式を求めるまでの手順を示した図であり、(Ａ)〜(Ｅ)はそれぞれステップＳ25〜Ｓ29に対応している。
【００４６】
ステップＳ30では、上記で求められた面形状に基づいて性能のシミュレーションが行われる。そして、ステップＳ31では、シミュレーション結果に基づいて性能を評価し、これが満足できるものであれば、その段階で設計を終了し、満足できるものでなければ、ステップＳ32でパラメータとして第１格子の傾きＢＸKj、曲率ＣＸij、ＣＹKjを調整し、ステップＳ24からの処理を繰り返す。
【００４７】
上記のようにピッチの細かい第２格子を用いて面形状を表現することにより、第１格子のみを用いた場合より曲率の変化を滑らかにすることができる。また、ピッチの大きな第１格子を用いてパラメータを設定し、それに基づいて第２格子のパラメータを求めることにより、当初からピッチの細かい第２格子を用いてパラメータを設定する場合と比較して、設定すべきパラメータの数を削減し、設計の負担を軽減することができる。
【００４８】
第２の実施形態により設計された累進面も局所制御性に優れているので累進面の部分的な修正が容易である。また、主注視線に沿った屈折力および非点収差にはＤＹKjとＤＸKj、各部位の屈折力にはＤＸij、近用部を内寄せするためにＹ座標に関して左右非対称な累進面を定義するのにはＢＸKj、レンズの上下のサグ量をバランスさせるのにはＢＹKL、というように着目する特性あるいは収差に対して支配的なパラメータが明確であるため、いずれのパラメータをどの程度修正すれば収差が改善されるかが直接的に把握でき、設計の見通しが良好である。
なお、図１３(Ａ)は、第２の実施形態で求められた累進レンズの面アス(非点収差)分布、(Ｂ)は平均面屈折力分布を計算した結果を示す。
【００４９】
次に、第２の実施形態の累進眼鏡レンズの設計方法の最適化能力について説明する。ここでは、図５のフローチャートに示される設計方法の最適化の能力を従来の方法と比較するため、初期値として最適値から離れた値を設定した場合の最適化の過程について説明する。なお、ここでは比較を簡略化するため、背骨直線に沿った断面形状の最適化に限定して説明する。
【００５０】
まず、図５のステップＳ21〜Ｓ23において、累進面のパラメータとして第１格子の格子点でのＺKL、ＢＹKL、ＣＹKjに最適値から離れた適当な初期値を与える。その他のパラメータとして、非累進面側の形状、レンズ中心厚、素材の屈折率などを含めてもよい。ステップＳ24〜Ｓ29で第２格子の領域毎に断面形状が三次式で表現される。
ステップＳ30では、眼鏡の装用条件を考慮に入れて、光学性能および形状性能をシミュレーションする。ここで光学性能のシミュレーションとしては、微分幾何学的手法による断面屈折力の解析、光線追跡によるレンズを透過した光束の収差の計算等が行われる。形状性能としてはレンズの縁厚などが計算される。
【００５１】
ステップＳ31では、性能評価として予め設定された性能目標値とシミュレーション性能との比較評価を行う。評価結果が満足できるものでなければ、パラメータを変更して性能を改善する。パラメータの調整には減衰最小二乗法を利用できる。
【００５２】
透過屈折力の目標値を図１４のように設定し、パラメータの調整に減衰最小二乗法を利用して第２の実施形態の方法(以下、ＰＳＩ(Power Spline Integral)法という)により最適化した過程を図１５に示す。初期値として与えられるパラメータは、図１５(ａ)に示す３mm毎の離散的な面屈折力(ＤＹKj)である。この例では球面の値が初期値として与えられており、全ての格子点で２[Diopter](素材の屈折率1.60として曲率半径は300mm、曲率は3.33/ｍ)である。離散的な面屈折力に基づいて連続的な形状(ｂ)、連続的な面屈折力(ｃ)、連続的な透過屈折力(ｄ)が求められる。図１５(ｅ)は、図１５(ｄ)に示される透過屈折力の図１４に示す目標値との誤差を示す。図１５(ａ)では、「○」が初期値、「△」が最適化１回目、「●」が最適化２回目の値を示す。また、図１５(ｂ)〜(ｅ)では、ピッチの大きい破線が初期値、ピッチの小さい破線が最適化１回目、実線が最適化２回目の値を示す。
【００５３】
図１５に示されるように、ＰＳＩ法では、２回の最適化で目標値に到達できる。なお、初期値の球面でもレンズの上側と下側での性能が異なっているのは、評価する物体距離が異なるからである。
【００５４】
比較のため、離散的形状からスプライン補間して連続形状を求める方法(以下、ＳＳＤ(Shape Spline Differential)法という)と、多項式により累進面形状を表現する方法(以下、ＰＬＹ(Polynomial)法という)とを用いて、減衰最小二乗法により光学性能の最適化した例を以下に示す。
図１６は、ＳＳＤ法による最適化の過程を示す。ＳＳＤ法では、３mm毎の面のサグ量(ＺKj)を設計のパラメータとする。初期値はＰＳＩ法とは別の表現形式をとっているため、図１６(ａ)は横軸がサグ量で、図１５(ａ)とは異なる。ただし、初期形状は曲率半径300mmの球面であり、ＰＳＩ法の例と同一である。初期値、最適化１回目後、最適化１０回目後のパラメータを図１６(a)に、面形状、面屈折力、透過屈折力、目標値からの誤差を図１６(b)(c)(d)(e)に示す。ＳＳＤ法では、１０回の最適化でも目標に達していない。
【００５５】
図１７は、ＰＬＹ法による最適化の過程を示す。ここで用いたＰＬＹ法は、Ｙを変数として０次、１次を除く２６次までの項でサグ量Ｚを求める以下の多項式を立て、その係数Ａiをパラメータとする。
【数８】

初期値として与えられる形状は、上記の例と同様に曲率半径300mmの球面である。図１７(a)は、Ｙ＝30の位置での各次数による非球面成分Ｚi＝Ａi・30ⁱの値(係数Ａiの値に対応)、図１７(b)(c)(d)(e)はそれぞれ面形状、面屈折力、透過屈折力、目標値からの誤差を示す。最適化１０回目後でもまだわずかに誤差が残っている。
【００５６】
上記のＰＳＩ法、ＳＳＤ法、ＰＬＹ法について最適化回数と、メリット関数値の減少の様子を図１８に示す。ＰＳＩ法と比較すると、ＳＳＤ法は形状をパラメータとしているために透過屈折力による評価では調整が困難であること、そして、ＰＬＹ法は単一関数でレンズ面全体を表現するために局所的な調整が困難であること、が主因となって最適化スピードが遅い。これに対してＰＳＩ法では、ほぼ１回の最適化で目標値に達してメリット関数値が０近くまで減少しており、他の２つの方法に対して明らかに優れた結果を示している。
【００５７】
【発明の効果】
以上説明したように、請求項１の方法によれば、各格子点に与えられた曲率等をパラメータとし、背骨直線に沿った形状を求め、これに基づいて背骨直線に直交する方向の形状を求めるというステップを経由することにより、背骨直線以外の格子点では一方向の曲率のみ与えれば形状を算出でき、与えられたパラメータから効率よく形状を特定することができる。また、各領域内はそれぞれ別の関数で表現されているため、パラメータを変更する場合にも、その影響がほぼ当該領域近傍に限定され、局所的な性能改善が容易であり、複雑な形状の非球面の設計に適する。
【００５８】
請求項２の方法では、パラメータとして収差とほぼ直線的な関係を持つ曲率や傾きを用いているため、性能評価に基づくパラメータ変更の際に、いずれのパラメータをどの程度修正すれば収差が改善されるかが直接的に把握でき、設計の見通しが良好であり、効率よく最適な形状を求めることができる。
請求項３の方法では、曲率に基づいて２回の積分を含む計算をすることにより、形状を正確に求めることができる。
請求項４、７の方法では、曲率の変化が大きい部分のピッチを他の部分より小さくすることにより、複雑な形状でも十分に表現できる自由度を与えることができる。
【００５９】
請求項５、９の方法では、ピッチの細かい第２格子を用いて面形状を表現することにより、第１格子のみを用いた場合より曲率の変化を滑らかにすることができる。また、ピッチの大きな第１格子を用いてパラメータを設定し、それに基づいて第２格子のパラメータを求めることにより、当初からピッチの細かい第２格子を用いてパラメータを設定する場合と比較して、設定すべきパラメータの数を削減し、設計の負担を軽減することができる。
【００６０】
請求項６の方法では、第２格子により求められた形状に基づいて性能を評価し、設計を変更する際には第１格子の格子点のパラメータを調整することにより、正確、厳密な評価と、設計変更の容易さという一見相反する要求を共に満たすことができる。また、第１格子点のパラメータである曲率や傾きは収差に対してほぼ直線的な関係を持つため、見通しよく効率的な設計変更が可能となる。
【００６１】
請求項８の方法では、第２格子はもともと第１格子よりピッチが細かいため、部位によってピッチを変更する必要性が少なく、等ピッチで設定することにより、容易に設定でき、後の計算も容易となる。
【００６２】
請求項１０の方法では、曲率に代えて断面屈折力をパラメータとして用いることにより、レンズの屈折力との対応関係が良く、目標性能が屈折力として与えられる眼鏡レンズを設計する場合等には最適化の効率を向上させることができる。なお、請求項１１、１２の方法は、それぞれ請求項１、５の方法を具体的に示したものであり、同項と同様の効果を有する。
【図面の簡単な説明】
【図１】 第１の実施形態の設計方法を示すフローチャート。
【図２】 図１の設計方法の実施例としてのｆθレンズのレンズ面を表すパラメータの説明図。
【図３】 図２のレンズ面のパラメータの具体的な数値を示す図表。
【図４】 図１の設計方法の計算手順を示す説明図。
【図５】 第２の実施形態の設計方法を示すフローチャート。
【図６】 図５の設計方法の実施例としての累進眼鏡レンズのレンズ面を表す第１格子のパラメータの説明図。
【図７】 図６に示す第１格子のパラメータの具体的な数値を示す図表。
【図８】 図５の設計方法の実施例としての累進眼鏡レンズのレンズ面を表す第２格子のパラメータの説明図。
【図９】 図８に示す第２格子のパラメータの具体的な数値を３≦ｍ≦１３の範囲で示す図表。
【図１０】 図８に示す第２格子のパラメータの具体的な数値を１４≦ｍ≦３０の範囲で示す図表。
【図１１】 図８に示す第２格子のパラメータの具体的な数値を３１≦ｍ≦４７の範囲で示す図表。
【図１２】 図５の設計方法の計算手順を示す説明図。
【図１３】 第２の実施形態の設計方法により設計された累進眼鏡レンズのレンズ面の面アスおよび面屈折力を示す分布図。
【図１４】 最適化の性能目標値を示す透過屈折力のグラフ。
【図１５】 図５の設計方法で最適化した際の経過を示すグラフ。
【図１６】 従来のＳＳＤ法による最適化の過程を示すグラフ。
【図１７】 従来のＰＬＹ法による最適化の過程を示すグラフ。
【図１８】 図５の設計方法とＳＳＤ法、ＰＬＹ法とによる最適化スピードを比較して示すグラフ。
【符号の説明】
(ＸK,ＹL) 基準格子点
Ｘ＝ＸK 背骨直線
Ｚ サグ量
Ｂ 傾き
Ｃ 曲率[0001]
BACKGROUND OF THE INVENTION
The present invention relates to an optical curved surface design method suitable for designing a rotationally asymmetric lens surface such as a progressive lens or an fθ lens.
[0002]
[Prior art]
When designing an aspherical surface, especially an aspherical surface with a complex shape, the performance of the shape set as the initial value is evaluated, and some parameters that define the shape are changed based on the evaluation result and re-evaluated. To obtain an aspherical shape that satisfies the desired performance. When expressing an aspherical surface by some function, a method of expressing the entire surface with one higher-order polynomial or the like, and a method of dividing the entire surface into small lattice-like regions and expressing each region with a relatively low-order expression There is. When the aspherical surface is expressed by the former method, if the parameter is changed to improve the performance of a specific area based on the performance evaluation, the effect of the change will affect the entire surface. In order to optimize the whole, the number of trials and errors increases.
[0003]
In the latter method, that is, the method of setting a function for each divided area, the effect of parameter change is almost limited to the vicinity of the area. Therefore, local performance improvement is easy, and an aspherical surface with a complicated shape. Suitable for design. A lens surface design method using this method is disclosed in, for example, Japanese Patent Laid-Open No. 55-146212, or re-published patent WO 96/11421.
[0004]
Japanese Patent Application Laid-Open No. 55-146212 discloses a method for designing an aspheric surface of a progressive multifocal spectacle lens, in which the entire surface is divided into a rectangular region in a lattice shape, and the shape of the surface is expressed by a bicubic equation for each region. A method for determining each coefficient so that the entire surface is continuous up to the second derivative is disclosed. That is, the invention described in Japanese Patent Application Laid-Open No. 55-146212 is intended to provide a new prism refractive power distribution on a progressive surface, and a prism refractive power distribution is first given, on the basis of which It is described that a shape is obtained, and this is expressed by a bicubic equation for each region to evaluate the performance.
[0005]
In WO96 / 11421, first, the radius of curvature at the main points of the lens surface is set, and the lens performance is evaluated by ray tracing for each region at the stage where the whole is divided into grid-like regions. Then, the radius of curvature is corrected, and then the bicubic equation of each area is determined, the performance of the determined surface shape is checked, and if there is room for improvement, the curvature is reset and the necessary area is determined. A method is disclosed in which a bicubic equation is reset, and a check and reset cycle is repeated until an optimal shape is obtained.
[0006]
[Problems to be solved by the invention]
However, the above-mentioned Japanese Patent Application Laid-Open No. 55-146212 only describes evaluating the performance of the expressed lens surface, and does not disclose a design change method based on the evaluation. Also, the prism refractive power (that is, the slope of the surface of each point of the lens) is set first, and based on this, the differential value etc. is obtained to obtain the bicubic equation, so even if the surface shape is changed based on the evaluation Even so, the inclination of the surface must be a design parameter. However, the relationship between the lens shape represented by the surface tilt and the refractive power and astigmatism, which are important evaluation items for progressive multifocal lenses, is not straightforward. Since it is difficult to determine whether the correction is effective or not, the number of trials and errors for optimization, that is, the calculation amount increases.
[0007]
On the other hand, performance evaluation by ray tracing using a computer is possible only after the shape of the lens surface is expressed by a function, and cannot be actually evaluated by the method disclosed in WO96 / 11421. In the case of a rotationally asymmetric aspherical surface such as a progressive surface, the radius of curvature of a point on the lens surface should be different between the X direction and the Y direction, but there is no distinction about this, and further, on the lens surface It is not disclosed how to determine the lens surface shape from the radius of curvature given to the principal points.
[0008]
The present invention has been made in view of the above-mentioned problems of the prior art, and is directly related to aberrations on the premise of dividing the lens surface into small lattice areas and representing each area as a function. Providing an optical curved surface design method that can efficiently determine the shape of the entire lens surface and the shape in the rectangular area based on the parameters such as the curvature (power) of the lens surface with The challenge
(Purpose)
[0009]
[Means for Solving the Problems]
In order to achieve the above object, the optical curved surface design method according to the present invention divides the optical curved surface into a predetermined rectangular area in a lattice shape to give curvature to each lattice point, and based on this curvature, the optical curved surface In the design method for determining the shape of a grid, a grid point other than the reference grid point on the spine straight line passing through the reference grid point is set by using a specific grid point as a reference grid point, giving a sag amount and inclination in addition to the curvature as a parameter to the reference grid point A slope is given to the point in addition to the curvature, and a curved surface shape along the backbone line is obtained based on the sag amount of the reference lattice point, the inclination and the curvature of the lattice point on the backbone line, and on the backbone line determined by the surface shape. The surface shape in the direction orthogonal to the spine straight line is obtained based on the sag amount of the lattice points, the given inclination, and the curvature of each lattice point, and each area is expressed by a different function based on the obtained surface shape. It is characterized by expressing.
[0010]
According to the above method, the shape on the spine straight line is obtained by integration based on the reference lattice point based on the curvature or the like given to each lattice point, and then the spine on the basis of the point on the spine straight line. The shape on a straight line orthogonal to the straight line is obtained by integration. The surface shape in each rectangular region is expressed by a function such as a bicubic equation, and the performance can be evaluated based on this.
When the design is changed based on the evaluation result, the design can be efficiently changed with a high visibility by adjusting the curvature of each lattice point having a substantially linear relationship with the aberration as a variable parameter. As a variable parameter, a sag amount or inclination given only to a specific lattice point may be used in addition to the curvature given to all lattice points. The step of obtaining the surface shape from the given parameters includes the first integration for obtaining the distribution corresponding to the slope by integrating the curvature with the slope as an integration constant, and the shape corresponding to the slope with the distribution corresponding to the slope as the integration constant. And a second integral for obtaining.
[0011]
When the lens surface is divided into grid-like rectangular regions, the entire surface may be divided into regions of equal pitch, but the pitch may be changed depending on the location on the lens. When the number of divisions is limited, for example, it is desirable to finely divide the portion where the change in curvature is relatively large and to divide the portion where the change in curvature is relatively small into large portions.
[0012]
Further, when the optical curved surface is divided into a lattice shape, it is divided into two types of lattices, a first lattice having a large pitch and a second lattice having a smaller pitch than the first lattice, and is given to lattice points of the first lattice. The parameters of the lattice points of the second lattice are obtained by interpolating the obtained parameters, and the surface shape along the spine straight line and the surface shape in the direction perpendicular to the spine straight line for the lattice points of the second lattice based on the obtained parameters And may be obtained in order. In this case, the performance can be evaluated based on the obtained shape, and the parameters given to the lattice points of the first lattice can be adjusted based on the evaluation result.
[0013]
DETAILED DESCRIPTION OF THE INVENTION
Hereinafter, two embodiments of the optical curved surface design method according to the present invention will be described.
The first embodiment is an example in which the design method of the present invention is applied to an aspherical design of a rotationally asymmetric fθ lens. 1 is a flowchart showing a method for designing an optical curved surface according to the first embodiment, FIG. 2 is an explanatory diagram showing parameters given to an aspheric surface, FIG. 3 is a chart showing numerical values of the given parameters, and FIG. 4 is a flowchart. It is explanatory drawing which shows each calculation step.
[0014]
First, setting of coordinate axes and rectangular areas will be described with reference to FIG. The direction of the reference axis perpendicular to the paper surface of FIG. 2 is defined as the Z axis, and the curved surface is divided into a rectangular area as indicated by a broken line in the drawing on the XY plane orthogonal to the Z axis. The number of divided areas is I × J as a general formula, and 4 × 8 in this example. The intersections of the broken lines are lattice points, and the XY coordinates thereof are represented by (Xi, Yj) [0 ≦ i ≦ I, 0 ≦ j ≦ J, i, j are integers]. Further, a specific lattice point is set as a reference lattice point, and its XY coordinates are (XK, YL). In this example, the reference lattice point is located at the center of the effective area of the aspheric surface, and K = 2 and L = 4. K and L are constants, i. Since j is a discrete variable, for example, when the X coordinate is taken as an example, XK is one constant indicating the X coordinate of the reference lattice point, and Xi is the X coordinate of any lattice point of 0 ≦ i ≦ I. The discrete variable shown, X, represents a continuous variable indicating the X coordinate of an arbitrary point.
[0015]
In designing, the sag amount ZKL in the Z direction, the slope BXKL in the X direction, the slope BYKL in the Y direction, the curvature CXKL in the X direction, the curvature CYKL in the Y direction as parameters at the reference grid point (XK, YL) in step S1 of FIG. give. In FIG. 2, the symbol “•” represents Zij, “−” represents BXij, “|” represents BYij, “⌒” represents CXij, and “(” represents CYij.
[0016]
In step S2, grid points (XK, Yj) [j ≠ L] other than the reference grid points on a straight line X = XK (referred to as spine straight line) passing through the reference grid points (XK, YL) parallel to the Y axis. Are given an inclination BXKj in the X direction, a curvature CXKj in the X direction, and a curvature CYKj in the Y direction. In step S3, a curvature CXij in the X direction is given to lattice points (Xi, Yj) [i ≠ K] other than on the spine straight line.
[0017]
The parameter setting is completed in the above three steps. That is, the curvature CXij in the X direction is given to all the lattice points (Xi, Yj), and the inclination BXKj in the X direction and the curvature CYKj in the Y direction are given only to the lattice points (XK, Yj) on the backbone line. The sag amount ZKL and the inclination BYKL in the Y direction are given only to the reference grid point (XK, YL).
[0018]
Specifically, numerical values as shown in FIG. 3 are set. In this example, the curvatures CYKj and CXij are represented by the differences ΔCYKj and ΔCXij from the values CYKL and CXKL at the reference lattice point. The range of ± 60.00 mm is divided into 8 regions in the Y direction, the range of ± 6.00 mm is divided into 4 regions in the X direction, and the grid points (XK, Yj) on the spine straight line are displayed. ΔCYKj and BXKj on the left side of the table are given, and ΔCXij on the right side of the table is given to each lattice point. Each value given to the reference grid point (XK, YL) is as follows.
CXKL = -8.28E-03 BYKL = 0.00E + 00
CYKL = -3.73E-03 ZKL = 0.00E + 00
The notation E represents a power with 10 as the radix and the number to the right of E as the exponent. For example, the value “-8.28E-03” of the above CXKL means “−0.00828”.
[0019]
Subsequently, the surface shape is obtained as a function based on the parameters given as described above. Here, functions representing the sag amount Z, the slopes BX, BY, and the curvatures CX, CY at an arbitrary point (X, Y) are defined as follows.
Z = f (X, Y)
BX = g (X, Y)
BY = h (X, Y)
CX = u (X, Y)
CY = v (X, Y)
[0020]
The values at each grid point are as follows.
Zij = f (Xi, Yj)
BXij = g (Xi, Yj)
BYij = h (Xi, Yj)
CXij = u (Xi, Yj)
CYij = v (Xi, Yj)
[0021]
Basically, the functions g and h indicating the gradient distribution are obtained by integrating the functions u and v indicating the curvature distribution, and the function f indicating the shape distribution is obtained by further integrating the functions g and h. However, since the slope cannot be obtained from the curvature only by simple integration, the following functions p and q are defined in the middle.
p (X, Y) ≡g / √ (1 + g ² )
q (X, Y) ≡h / √ (1 + h ² )
[0022]
As a result, the following relationship is established.
g (X, Y) ≡∂f / ∂X
h (X, Y) ≡∂f / ∂Y
u (X, Y) ≡∂p / ∂X
v (X, Y) ≡∂q / ∂Y
Further, a function w (X, Y) ≡∂ that differentiates the shape f (X, Y) by X and Y ² Define f / ∂X∂Y.
[0023]
In step S4, a function v (XK, Y) for connecting the curvature CYKj in the Y direction at the lattice point (XK, Yj) on the spine straight line is obtained, and this is calculated including two integrations to obtain the spine straight line. A function f (XK, Y) for smoothly connecting the sag amount ZKj at the upper lattice point is obtained. The operation in step S4 corresponds to the calculation in FIG. In FIG. 4, for the sake of simplicity, the curved surface is divided into 2 × 4 regions, but the processing in each step is the same regardless of the number of divisions.
[0024]
Step S4 includes two integration stages. In the first integration stage, the function v (XK, Y) is integrated with Y using the constant q (XK, YL) corresponding to the gradient BYKL in the Y direction at the reference grid point (XK, YL) as an integration constant. As a result, a function q (XK, Y) that smoothly connects the distribution q (XK, Yj) corresponding to the inclination BYKj in the Y direction at the lattice point (XK, Yj) on the spine straight line is obtained.
[Expression 1]

Where the integral constant q (XK, YL) is

It is.
[0025]
And q (X, Y) = h / √ (1 + h) ² ) To obtain a function h (XK, Y) that smoothly connects the slopes BYKj in the Y direction at the lattice points on the spine straight line from the function q (XK, Y).
h (XK, Y) = q (XK, Y) / √ {1-q (XK, Y) ² }
In the second integration stage of step S4, the function h (XK, Y) is integrated with Y using the sag amount ZKL at the reference lattice point as an integration constant to obtain the function f (XK, Y).
[Expression 2]

By the operation of step S4, BYKj = h (XK, Yj) and ZKj = f (XK, Yj) are determined, and the shape along the backbone straight line of the curved surface is obtained.
[0026]
In step S5, a function u (X, Yj) that connects the curvature CXij in the X direction at the lattice point (Xi, Yj) on each straight line Y = Yj parallel to the X axis and passing through the lattice point is obtained, and this is calculated twice. Thus, a function f (X, Yj) for smoothly connecting the sag amounts Zij at the lattice points on the straight line Y = Yj is obtained for each straight line. The operation in step S5 corresponds to the calculation in FIG. By this operation, a shape along the straight line Y = Yj of the curved surface is obtained.
[0027]
Step S5 includes two integration steps for each straight line. In the first integration step, the function u (X, Yj) is integrated with X using the constant p (XK, Yj) corresponding to the X-direction inclination of the lattice point (XK, Yj) on the spine straight line as the integration constant, A function p (X, Yj) that smoothly connects the distribution p (Xi, Yj) corresponding to the gradient BXij in the X direction at the lattice point on each straight line parallel to the X axis is obtained.
[Equation 3]

Where the integral constant p (XK, Yj) is

It is.
[0028]
And p (X, Y) = g / √ (1 + g) ² ), The function g (X, Yj) for smoothly connecting the gradient BXij in the X direction at each lattice point on the straight line Y = Yj parallel to the X axis from the function p (X, Yj) for each straight line. Ask.
g (X, Yj) = p (X, Yj) / √ {1-p (X, Yj) ² }
In the second integration stage of step S5, the function g (X, Yj) is integrated with X using the sag amount ZKj at the lattice points on the spine straight line as an integration constant, and the function f (X, Yj) is obtained for each straight line.
[Expression 4]

By the operation of step S5, BXij = g (Xi, Yj) and Zij = f (Xi, Yj) are determined, and the shape of the entire curved surface is obtained.
[0029]
Steps S6 and S7 are processes for obtaining the Y-direction gradient BYij, and the function g () smoothly connects the X-direction gradient BXij obtained from the X-direction arrangement as a Y-direction arrangement. (Xi, Y) is obtained, and is differentiated by Y and then integrated by X to obtain the gradient BYij in the Y direction.
In step S6, the function g (Xi, Y) for smoothly connecting the gradient BXij is differentiated by Y to obtain a differential value Wij at each lattice point (Xi, Yj).
w (Xi, Y) = ∂g (Xi, Y) / ∂Y
Wij = w (Xi, Yj)
This operation corresponds to the calculation of FIG. The symbol “*” represents the differential value Wij of each grid point.
[0030]
In the following step S7, the function w (X, Yj) for smoothly connecting the differential values Wij at the lattice points (Xi, Yj) on each straight line Y = Yj passing through the lattice points parallel to the X axis is integrated with X. An inclination BYij in the Y direction at each lattice point (Xi, Yj) is obtained. This operation corresponds to the calculation of FIG.
[Equation 5]

[0031]
Through the above steps, four types of parameters Zij, BXij, BYij, and Wij representing the surface shape are determined at all lattice points (0 ≦ i ≦ I, 0 ≦ j ≦ J). In step S8, each rectangular area Rij = {(X, Y) | X _i-1 ≤X≤Xi, Y _j-1 ≦ Y ≦ Yj} is expressed by the following bicubic expression of X and Y.
[Formula 6]

[0032]
The 16 coefficients included in the equation are determined by simultaneous equations using four parameters of four grid points surrounding the region, that is, 16 parameters. The coefficient γab in the ij-th region Rij whose diagonal is the lattice point of (Xi-1, Yj-1) and the lattice point of (Xi, Yj) ^(ij) Is obtained by the following matrix operation.
[Expression 7]

Where ΔX _i-1 = Xi-X _i-1 , ΔY _j-1 = Yj-Y _j-1 It is.
[0033]
Coefficient γab ^(ij) Is obtained, Z = fij (X, Y) at an arbitrary position in the region Rij and its differential value by X, Y can be calculated. Therefore, by performing the above-described calculation of the coefficients in all regions, the equation Z = f (X, Y) representing the curved surface in the entire definition region becomes fij (X, Y) [0 ≦ i ≦ I, 0 ≦. j ≦ J]. Z = f (X, Y) is guaranteed to be continuous up to the second derivative within the entire domain. The operation in step S8 corresponds to the calculation in FIG.
[0034]
In step S9, performance simulation is performed based on the surface shape obtained above. In step S10, the performance is evaluated based on the simulation result. If this is satisfactory, the design is finished at that stage. If not satisfactory, the curvatures CXij and CYKj are adjusted as parameters in step S11. Then, the processing from step S4 is repeated. As the parameter to be adjusted, not only the curvature but also the inclination BXKj given to the lattice point on the spine straight line can be included.
[0035]
By dividing the lens surface into rectangular areas and expressing each area with a separate function as described above, the effect of parameter changes is limited to the vicinity of the area, making it easy to improve local performance. Thus, a design method suitable for designing an aspherical surface having a complicated shape can be provided. In addition, by using the curvatures CXij and CYKj that have a substantially linear relationship with the aberration as parameters, it is possible to directly determine which parameter is modified to what extent the aberration is improved when changing the parameter based on the performance evaluation. Therefore, the design prospects are good. For example, in the case of the fθ lens as in the present embodiment, the characteristics to be focused on such as fθ characteristics and CYKj for the field curvature of the main scanning section, CXKj for the field curvature of the sub-scanning section, and BXKj for the bow, or The dominant parameter for aberrations is clear.
[0036]
Next, a second embodiment in which the design method of the present invention is applied to the design of a progressive spectacle lens will be described. In the design method of the second embodiment, the shape is set by dividing the lens surface into two types of gratings, a first grating having a large pitch and a second grating having a small pitch. The parameters are set for the lattice points of the first lattice, and the surface shape is determined through the parameters of the second lattice obtained from the parameters of the first lattice. FIG. 5 is a flowchart showing a method of designing an optical curved surface according to the second embodiment, FIG. 6 is an explanatory diagram showing parameters given to the lens surface at the lattice points of the first lattice, and FIG. 7 is the lattice points of the first lattice. FIG. 8 is an explanatory diagram showing parameters given to the lens surface at the lattice points of the second lattice, and FIGS. 9 to 11 are diagrams showing numerical values of parameters given at the lattice points of the second lattice. FIG. 12 is an explanatory diagram showing each calculation stage of the flowchart.
[0037]
In the following description, the lattice point of the first lattice is represented by (Xi, Yj) as in the first embodiment, and the lattice point of the second lattice is represented by (Xm, Yn). Further, the reference lattice points (XK, YL) of the first lattice and the reference lattice points (X˜P, Y˜Q) of the second lattice are equal, and therefore the backbone straight lines of both lattices also coincide. A rectangular range of 100 mm × 100 mm including the lens surface is divided by 22 × 22 for the first grating and 50 × 50 for the second grating. Here, K = 11, L = 12, I = J = 22, P = Q = 25, and M = N = 50. However, the division of the first lattice is not a uniform pitch. In a portion where the change in curvature is steep, a complicated shape cannot be sufficiently expressed if the pitch of the lattice is large. Therefore, in the first grating, the division pitch at the progressive portion (slightly below the center of the lens) where the change in curvature is abrupt is made finer than other portions. The division of the second lattice is a uniform pitch.
[0038]
In designing, the sag amount ZKL in the Z direction, the inclination BXKL in the X direction, the inclination BYKL in the Y direction, and the curvature CXKL in the X direction as parameters at the reference lattice point (XK, YL) of the first lattice in step S21 of FIG. Gives the direction curvature CYKL. In step S22, X is set to the lattice point (XK, Yj) [j ≠ L] of the first lattice other than the reference lattice point on the backbone line X = XK parallel to the Y axis and passing through the reference lattice point (XK, YL). A direction inclination BXKj, a curvature CXKj in the X direction, and a curvature CYKj in the Y direction are given. In step S23, the curvature CXij in the X direction is given to the lattice point (Xi, Yj) [i ≠ K] of the first lattice other than the spine straight line. FIG. 6 shows the parameters given to the grid points of the first grid. The meaning of the symbols in the figure is the same as in FIG.
[0039]
The parameter setting is completed in the above three steps. Specifically, numerical values as shown in FIG. 7 are set. In the chart of FIG. 7, sectional refractive powers DXij and DYKj are given in place of the curvatures CXij and CYKj at each lattice point so that the correspondence with the refractive power of the lens is good. The curvature is obtained by CXij = DXij / (In-1) and CYKj = DYKj / (In-1) where In is the refractive index of the lens material (In = 1.5 in this example). The grid points (XK, Yj) on the spine straight line are given DYKj, BXKj on the left side of the table, and DXij on the right side of the table is given to each grid point. Further, the following values are given to the reference grid points (XK, YL).
BYKL = 0.00E + 00 ZKL = 0.00E + 00
[0040]
Since the planar shape of the lens is circular, it is not necessary to give DXij to all lattice points (0 ≦ i ≦ 22, 0 ≦ j ≦ 22) in the set 22 × 22 region. In this example, the effective diameter of the lens is set to φ80 mm, and parameter values are given for lattice points within φ90 mm that is slightly larger than that. Thereby, it is possible to avoid an unnecessarily large number of parameters. In FIG. 7, the minimum value ijmin and the maximum value ijmax of the number i of the effective grid point on the jth straight line parallel to the X axis are shown together with the value of j. The column indicated by “#####” in the table indicates grid points for which no parameter is set.
[0041]
Also in the second embodiment, the function f (X, Y) can be obtained from the parameters given to the lattice points of the first lattice as in the first embodiment. However, in view of the fact that the pitch of the first grating is 5 mm or more in the peripheral portion and the change in curvature of the required function f (X, Y) is a polygonal line, the curvature is changed sufficiently smoothly as a spectacle lens. I can't. Therefore, in the second embodiment, instead of directly obtaining f (X, Y) from the parameters given to the lattice points of the first lattice, a second lattice having a smaller pitch is set and the parameters at the lattice points are obtained. Based on this, f (X, Y) is obtained.
[0042]
In step S24, the lens surface is divided by a second grating having a fine pitch, and the lattice points of the second grating (from the parameters ZKL, BYKL, BXKj, CYKj, CXij given to the grating points (Xi, Yj) of the first grating ( X to m, Y to n) parameters Z to PQ, BY to PQ, BX to Pn, CY to Pn, CX to mn are obtained. The procedure for obtaining the parameters of the lattice points of the second lattice in step S24 will be described below.
First, since X˜P = XK and Y˜Q = YL are set, Z˜PQ = ZKL and BY˜PQ = BYKL. The curvatures CY to Pn and inclinations BX to Pn in the Y direction at the lattice points (X to P, Y to n) on the backbone line of the second lattice are the curvatures CYKj along the backbone line X = XK of the first lattice, respectively. The slope BXKj is obtained by interpolation using a one-dimensional spline function.
[0043]
Further, the curvature CXij is interpolated by a one-dimensional spline function along each straight line Y = Yj parallel to the X axis and passing through the lattice points of the first lattice, and a straight line X = X˜m passing through the lattice points of the second lattice is obtained. The curvature CX to mj at the intersection with the straight line Y = Yj is obtained. Then, the curvature CX˜mj is interpolated by a one-dimensional spline function along a straight line X = X˜m parallel to the Y axis and passing through the lattice points of the second lattice, and the lattice points (X˜m, Y˜) of the second lattice are interpolated. The curvature CX˜mn at n) is obtained. The two steps for obtaining the curvatures CX˜mn of the lattice points of the second lattice from the curvatures CXij of the lattice points of the first lattice can be simultaneously performed by a two-dimensional spline function. Note that the one-dimensional or two-dimensional spline function in step S24 is selected so that at least the first derivative is continuous.
[0044]
The types of parameters at the lattice points of the second lattice obtained in step S24 are shown in the explanatory diagram of FIG. The meaning of the symbols in the figure is the same as in FIG. Specific numerical values are shown in the charts of FIGS. 9 to 11, the range of 3 ≦ m ≦ 13 except for the peripheral portion (0 ≦ n ≦ 2, 48 ≦ n ≦ 50, 0 ≦ m ≦ 2, 48 ≦ m ≦ 50) where the parameter is not set. 9, the range of 14 ≦ m ≦ 30 is divided into FIG. 10, and the range of 31 ≦ m ≦ 47 is divided into FIG.
[0045]
Steps S25 to S29 are the same as steps S4 to S8 in the flowchart of the first embodiment. Here, the second step is based on the parameters given to the lattice points (X to m, Y to n) of the second lattice. Each region of the lattice is expressed by a bicubic equation. FIG. 12 is a diagram showing a procedure for obtaining a bicubic equation by reducing the number of divisions to 6 × 6 for simplicity, and (A) to (E) correspond to steps S25 to S29, respectively. .
[0046]
In step S30, performance simulation is performed based on the surface shape obtained above. In step S31, the performance is evaluated based on the simulation result. If this is satisfactory, the design is completed at that stage. If not satisfactory, the slope BXKj of the first lattice is set as a parameter in step S32. The curvatures CXij and CYKj are adjusted, and the processing from step S24 is repeated.
[0047]
By expressing the surface shape using the second grating with a fine pitch as described above, the change in curvature can be made smoother than when only the first grating is used. In addition, by setting the parameters using the first grating having a large pitch and obtaining the parameters of the second grating based thereon, compared to the case of setting the parameters using the second grating having a fine pitch from the beginning, The number of parameters to be set can be reduced, and the design burden can be reduced.
[0048]
Since the progressive surface designed according to the second embodiment is also excellent in local controllability, partial correction of the progressive surface is easy. Also, DYKj and DXKj are used for the refractive power and astigmatism along the main line of sight, DXij is used for the refractive power of each part, and a progressive surface that is asymmetrical with respect to the Y coordinate is defined in order to bring the near portion inward. BXKj and BYKL to balance the sag amount on the top and bottom of the lens are parameters that are dominant to the characteristic or aberration to be noticed. It is possible to directly grasp whether it will be improved, and the design prospect is good.
FIG. 13A shows the result of calculating the surface astigmatism distribution (astigmatism) of the progressive lens obtained in the second embodiment, and FIG. 13B shows the result of calculating the average surface power distribution.
[0049]
Next, the optimization capability of the design method of the progressive spectacle lens of the second embodiment will be described. Here, in order to compare the optimization capability of the design method shown in the flowchart of FIG. 5 with a conventional method, an optimization process when a value far from the optimal value is set as an initial value will be described. Here, in order to simplify the comparison, description is limited to optimization of the cross-sectional shape along the spine straight line.
[0050]
First, in steps S21 to S23 in FIG. 5, appropriate initial values apart from the optimum values are given to ZKL, BYKL, and CYKj at the lattice points of the first lattice as parameters of the progressive surface. Other parameters may include the shape on the non-progressive surface side, the lens center thickness, the refractive index of the material, and the like. In steps S24 to S29, the cross-sectional shape is expressed by a cubic equation for each region of the second lattice.
In step S30, the optical performance and the shape performance are simulated in consideration of the spectacle wearing conditions. Here, as a simulation of optical performance, analysis of cross-sectional refractive power by a differential geometric technique, calculation of aberration of a light beam transmitted through a lens by ray tracing, and the like are performed. As the shape performance, the edge thickness of the lens is calculated.
[0051]
In step S31, a comparative evaluation between a performance target value set in advance as a performance evaluation and the simulation performance is performed. If the evaluation result is not satisfactory, change the parameters to improve the performance. Attenuating least squares can be used to adjust the parameters.
[0052]
The target value of the transmission refractive power is set as shown in FIG. 14 and optimized by the method of the second embodiment (hereinafter referred to as PSI (Power Spline Integral) method) using the attenuation least square method for parameter adjustment. The process is shown in FIG. The parameter given as the initial value is the discrete surface power (DYKj) every 3 mm shown in FIG. In this example, the value of the spherical surface is given as an initial value, and is 2 [Diopter] (the refractive index of the material is 1.60, the radius of curvature is 300 mm, and the curvature is 3.33 / m) at all lattice points. Based on the discrete surface power, a continuous shape (b), a continuous surface power (c), and a continuous transmission power (d) are obtained. FIG. 15E shows an error of the transmission refractive power shown in FIG. 15D from the target value shown in FIG. In FIG. 15A, “◯” represents the initial value, “Δ” represents the first optimization value, and “●” represents the second optimization value. In FIGS. 15B to 15E, a broken line with a large pitch indicates an initial value, a broken line with a small pitch indicates a first optimization value, and a solid line indicates a second optimization value.
[0053]
As shown in FIG. 15, in the PSI method, the target value can be reached by two optimizations. It should be noted that the performance on the upper and lower sides of the lens is different even with the initial spherical surface because the object distance to be evaluated is different.
[0054]
For comparison, a method of obtaining a continuous shape by performing spline interpolation from a discrete shape (hereinafter referred to as SSD (Shape Spline Differential) method) and a method of expressing a progressive surface shape by a polynomial (hereinafter referred to as PLY (Polynomial) method) An example of optimizing the optical performance by the attenuation least square method is shown below.
FIG. 16 shows an optimization process by the SSD method. In the SSD method, the sag amount (ZKj) of the surface every 3 mm is used as a design parameter. Since the initial value takes an expression form different from that of the PSI method, FIG. 16A is different from FIG. However, the initial shape is a spherical surface with a curvature radius of 300 mm, which is the same as the example of the PSI method. Fig. 16 (a) shows the initial values, parameters after the first optimization, and parameters after the 10th optimization. Fig. 16 (b) (c) () shows the surface shape, surface refractive power, transmission refractive power, and errors from the target values. d) As shown in (e). In the SSD method, even the optimization of 10 times does not reach the target.
[0055]
FIG. 17 shows an optimization process by the PLY method. In the PLY method used here, the following polynomial for determining the sag amount Z is established with terms up to the 26th order excluding the 0th order and the 1st order using Y as a variable, and the coefficient Ai is used as a parameter.
[Equation 8]

The shape given as the initial value is a spherical surface having a curvature radius of 300 mm as in the above example. FIG. 17A shows an aspherical component Zi = Ai · 30 according to each order at the position of Y = 30. ⁱ (Corresponding to the value of the coefficient Ai) and FIGS. 17 (b), (c), (d), and (e) show the surface shape, surface refractive power, transmission refractive power, and error from the target value, respectively. A slight error still remains after the 10th optimization.
[0056]
FIG. 18 shows the number of optimizations for the PSI method, SSD method, and PLY method and how the merit function value decreases. Compared to the PSI method, the SSD method uses the shape as a parameter, so adjustment is difficult by evaluation with transmission power. The PLY method is a local adjustment to express the entire lens surface with a single function. The optimization speed is slow mainly due to the difficulty. On the other hand, in the PSI method, the merit function value is reduced to nearly 0 by reaching the target value in almost one optimization, and clearly shows excellent results compared to the other two methods.
[0057]
【The invention's effect】
As described above, according to the method of claim 1, the shape along the spine straight line is obtained based on the curvature given to each lattice point as a parameter, and the shape in the direction orthogonal to the spine straight line is obtained based on this. By going through the step of obtaining, it is possible to calculate the shape if only the curvature in one direction is given at lattice points other than the spine straight line, and the shape can be efficiently identified from the given parameters. In addition, since each area is represented by a different function, even when parameters are changed, the effect is limited to the vicinity of the area, and local performance improvement is easy. Suitable for aspherical design.
[0058]
In the method of claim 2, since the curvature or inclination having a substantially linear relationship with the aberration is used as the parameter, the aberration can be improved by modifying which parameter to what extent when changing the parameter based on the performance evaluation. It is possible to grasp directly, the prospect of design is good, and an optimal shape can be obtained efficiently.
In the method of claim 3, the shape can be accurately obtained by performing a calculation including two integrations based on the curvature.
In the methods of claims 4 and 7, by making the pitch of the portion where the change in curvature is large smaller than that of the other portions, it is possible to give a degree of freedom to sufficiently express even a complicated shape.
[0059]
In the methods of claims 5 and 9, by expressing the surface shape using the second grating with a fine pitch, the change in curvature can be made smoother than when only the first grating is used. In addition, by setting the parameters using the first grating having a large pitch and obtaining the parameters of the second grating based on the parameters, compared to the case of setting the parameters using the second grating having a fine pitch from the beginning, The number of parameters to be set can be reduced, and the design burden can be reduced.
[0060]
According to the method of claim 6, the performance is evaluated based on the shape obtained by the second grid, and when changing the design, the parameters of the grid points of the first grid are adjusted. Both seemingly contradictory requirements for ease of design change can be met. Further, since the curvature and inclination, which are parameters of the first lattice point, have a substantially linear relationship with respect to the aberration, it is possible to change the design efficiently with a high visibility.
[0061]
In the method of claim 8, since the pitch of the second lattice is originally finer than that of the first lattice, there is little need to change the pitch depending on the part, and it can be easily set by setting the pitch at an equal pitch, and subsequent calculations are also easy. It becomes.
[0062]
In the method of claim 10, by using the cross-sectional refractive power as a parameter instead of the curvature, the correspondence relationship with the refractive power of the lens is good, and it is optimal when designing a spectacle lens in which the target performance is given as the refractive power. Efficiency can be improved. The methods of claims 11 and 12 specifically show the methods of claims 1 and 5, respectively, and have the same effects as those of the above claims.
[Brief description of the drawings]
FIG. 1 is a flowchart illustrating a design method according to a first embodiment.
FIG. 2 is an explanatory diagram of parameters representing a lens surface of an fθ lens as an example of the design method of FIG.
3 is a chart showing specific numerical values of parameters of the lens surface of FIG.
FIG. 4 is an explanatory diagram showing a calculation procedure of the design method of FIG. 1;
FIG. 5 is a flowchart showing a design method according to the second embodiment.
6 is an explanatory diagram of parameters of a first grating representing a lens surface of a progressive spectacle lens as an example of the design method of FIG. 5;
7 is a chart showing specific numerical values of parameters of the first lattice shown in FIG.
FIG. 8 is an explanatory diagram of parameters of a second grating representing a lens surface of a progressive spectacle lens as an example of the design method of FIG. 5;
9 is a chart showing specific numerical values of parameters of the second lattice shown in FIG. 8 in a range of 3 ≦ m ≦ 13.
FIG. 10 is a chart showing specific numerical values of parameters of the second lattice shown in FIG. 8 in a range of 14 ≦ m ≦ 30.
11 is a chart showing specific numerical values of parameters of the second lattice shown in FIG. 8 in a range of 31 ≦ m ≦ 47.
12 is an explanatory diagram showing a calculation procedure of the design method of FIG. 5;
FIG. 13 is a distribution diagram showing surface asperities and surface refractive powers of lens surfaces of a progressive spectacle lens designed by the design method according to the second embodiment.
FIG. 14 is a graph of transmission refractive power showing performance target values for optimization.
FIG. 15 is a graph showing the progress when the design method of FIG. 5 is optimized.
FIG. 16 is a graph showing a process of optimization by a conventional SSD method.
FIG. 17 is a graph showing a process of optimization by a conventional PLY method.
18 is a graph showing a comparison of optimization speeds between the design method of FIG. 5, the SSD method, and the PLY method.
[Explanation of symbols]
(XK, YL) Reference grid point
X = XK spine straight line
Z sag amount
B inclination
C Curvature

Claims (12)

In the optical curved surface design method for dividing the optical curved surface into a predetermined rectangular region in a lattice shape and giving a curvature to each lattice point, and obtaining the shape of the optical curved surface based on the curvature,
A specific lattice point is set as a reference lattice point, and a sag amount and an inclination are given to the reference lattice point as a parameter in addition to the curvature, and in addition to the curvature to lattice points other than the reference lattice point on the spine straight line passing through the reference lattice point A slope is given, a curved surface shape along the backbone line is obtained based on a sag amount of the reference lattice point, an inclination and a curvature of the lattice point on the backbone line, and a lattice point on the backbone line determined by the surface shape The surface shape in the direction orthogonal to the spine straight line is obtained based on the sag amount, the given inclination, and the curvature of each lattice point, and the respective regions are represented by different functions based on the obtained surface shape. An optical curved surface design method characterized by the above. 2. The method according to claim 1, further comprising: evaluating the performance of the optical curved surface and adjusting at least one of a sag amount, an inclination, and a curvature given to the lattice point as a variable parameter based on an evaluation result. Design method of optical curved surface. The step of obtaining the shape from the parameters includes a first integration for obtaining a distribution corresponding to the slope by integrating the curvature with the slope as an integration constant, and a shape by integrating the distribution corresponding to the slope with the sag amount as an integration constant. The method of designing an optical curved surface according to claim 1, further comprising: a second integration for obtaining 2. The optical curved surface according to claim 1, wherein the lattice-shaped region is finely divided at a portion where the change in curvature is relatively large, and is largely divided at a portion where the change in curvature is relatively small. Design method. When dividing the optical curved surface into a lattice shape, the optical curved surface is divided into two types of lattices, a first lattice having a large pitch and a second lattice having a smaller pitch than the first lattice, and is given to the lattice points of the first lattice. The parameters of the lattice points of the second lattice are obtained by interpolating the obtained parameters, and the surface shape along the spine straight line for the lattice points of the second lattice based on the obtained parameters is orthogonal to the spine straight line 2. The method of designing an optical curved surface according to claim 1, wherein a surface shape in a direction is obtained in order. 6. The method of designing an optical curved surface according to claim 5, wherein the performance is evaluated based on the obtained shape, and the parameter given to the lattice point of the first lattice is adjusted based on the evaluation result. 6. The region divided by the first lattice is finely divided at a portion where the change in curvature is relatively large, and is largely divided at a portion where the change in curvature is relatively small. Design method of optical curved surface. 6. The method of designing an optical curved surface according to claim 5, wherein the region divided by the second lattice is divided equally evenly as a whole. The optical curved surface is divided into two types of lattices, a first lattice having a large pitch and a second lattice having a smaller pitch than the first lattice, parameters are given to the lattice points of the first lattice, and the given parameters are interpolated. Then, the parameters of the lattice points of the second lattice are obtained, the performance is evaluated based on the shape obtained from the obtained parameters of the lattice points of the second lattice, and the lattice of the first lattice is obtained based on the evaluation result. An optical curved surface design method comprising adjusting a parameter given to a point. 10. The optical curved surface design method according to claim 1, wherein a cross-sectional refractive power is given as a parameter instead of the curvature of each lattice point. The reference axis direction is the Z axis, and the curved surface is divided into a grid of I × J rectangular areas on the XY plane orthogonal to the Z axis, and the XY coordinates of each grid point are (Xi, Yj) [0 ≦ i ≦ I, 0 ≦ j ≦ J, where i, j are integers]
A specific lattice point is set as a reference lattice point (XK, YL), and a sag amount ZKL in the Z direction, an inclination BXKL in the X direction, an inclination BYKL in the Y direction, a curvature CXKL in the X direction with respect to the reference lattice point (XK, YL), Gives the curvature CYKL in the Y direction,
Inclination in the X direction BXKj, in the X direction to the lattice point (XK, Yj) [j ≠ L] other than the reference lattice point on the backbone line X = XK parallel to the Y axis and passing through the reference lattice point (XK, YL) Give the curvature CXKj, the curvature CYKj in the Y direction,
A function v () that gives the curvature CXij in the X direction to the lattice points (Xi, Yj) [i ≠ K] other than on the backbone line, and connects the curvature CYKj in the Y direction at the lattice points (XK, Yj) on the backbone line. XK, Y) is integrated with Y using a value obtained from the Y-direction gradient BYKL at the reference lattice point (XK, YL) as an integration constant, and the Y-direction gradient BYKj at the lattice point on the backbone line is smoothly connected. Find the function h (XK, Y)
The function h (XK, Y) is integrated by Y using the sag amount ZKL at the reference lattice point as an integration constant, and a function f (XK, Y) for smoothly connecting the sag amount ZKj at the lattice point on the spine straight line is obtained. ,
A function u (X, Yj) for connecting the curvature CXij in the X direction at a lattice point (Xi, Yj) on each straight line Y = Yj parallel to the X axis and passing through the lattice point is represented by a lattice point (XK) on the backbone line. , Yj) is a function g (X, Yj) that smoothly integrates the X-direction gradient BXij at each lattice point on the straight line Y = Yj by integrating the value obtained from the X-direction gradient BXKj of X, as an integration constant. Seeking
The function g (X, Yj) is integrated by X with the sag amount ZKj at the lattice point on the backbone straight line as an integration constant, and the function f (s) that smoothly connects the sag amount Zij at the lattice point on the straight line Y = Yj. X, Yj)
An inclination BXij in the X direction at each lattice point (Xi, Yj) is obtained by the function g (X, Yj),
A function g (Xi, Y) that smoothly connects the gradient BXij in the X direction on each straight line X = Xi parallel to the Y axis and passing through the lattice point is differentiated by Y, and the differential value at each lattice point (Xi, Yj). Find Wij,
A function w (X, Yj) that smoothly connects the differential values Wij at lattice points (Xi, Yj) on each straight line Y = Yj parallel to the X axis and passing through the lattice points is integrated by X, and each lattice point ( (Xi, Yj) to obtain the gradient BYij in the Y direction,
For each region, a curved surface in each region is expressed by a bicubic equation based on Zij, BXij, BYij, Wij at four lattice points surrounding the region, and based on the shape obtained as a set of the bicubic equation A method of designing an optical curved surface, wherein performance is evaluated and at least one of CXij, CYKj, and BXKj is adjusted as a variable parameter based on an evaluation result. The optical curved surface is divided into two types of lattices, ie, I × J first lattices and M × N second lattices having a finer pitch, and the XY coordinates of each lattice point by the second lattice are (X ~ m, Y ~ n) [0≤m≤M, 0≤n≤N, m, n are integers], reference lattice points by the second lattice that coincide with the reference lattice points (XK, YL) by the first lattice Is represented by (X˜P, Y˜Q), the sag amount ZKL given to each lattice point (Xi, Yj) of the first lattice, the inclination BXKj in the X direction, the inclination BYKL in the Y direction, the curvature in the X direction. CXij, Y-direction curvature CYKj is interpolated, and corresponding parameters Z to PQ, BX to Pn, BY to PQ, CX to mn, and CY to Pn for the lattice points (X to m, Y to n) of the second lattice The integration is performed for each grid point by the second grid instead of the integration for each grid point of the first grid, the performance is evaluated by expressing the area divided by the second grid as a function, and the evaluation result Based on the above Designing method of the optical curved surface according to claim 11, wherein adjusting the parameters of the lattice points of the grid.

Images