JP3777659B2

JP3777659B2 - Method and apparatus for determining refractive index and birefringence by measuring interference spectrum

Info

Publication number: JP3777659B2
Application number: JP16706396A
Authority: JP
Inventors: 仁新田
Original assignee: 仁新田
Priority date: 1996-06-27
Filing date: 1996-06-27
Publication date: 2006-05-24
Anticipated expiration: 2016-06-27
Also published as: JPH1010041A

Description

【０００１】
【産業上の利用分野】
本発明は干渉スペクトルから物質の屈折率及び複屈折率を決定する方法並びにその装置に関する。
【０００２】
【従来の技術】
従来の干渉法による決定法（例えばＷ．ＫＵＣＺＹＮＳＫＩらの方法：Ｍｏｌ．Ｃｒｙｓｔ．Ｌｉｑ．Ｃｒｙｓｔ．，１９７５，Ｖｏｌ．３１，ｐｐ．２６７−２７３）では、ピ−クとなる条件式に含まれる整数値ｍを消去することによって複屈折率を決定している。この方法をもう少し詳しく述べると、干渉法において、数式７：
【数７】
の様にピ−クとなる条件式にはパラメータとして、任意な値である整数値ｍが含まれている。従って、条件式から得られる複屈折率の値はパラメータである任意の整数値ｍの数だけ存在することになる。言い換えるならば、整数値ｍは一意的な値ではない。しかし前記の論文の筆者らは２つのピ−クから条件式を導き出して、複屈折率と整数値ｍを２つの未知数として、連立方程式を解くことによって複屈折率を一意的に決定することを提案している。確かに二つの未知数を二つの連立方程式を解けば、二つの未知数は一意的に決定できるが、数式７における整数値ｍはパラメータでしかないので一意的には決定されているものではない。本来、方程式の解は元の方程式を同値変形することによって導き出されるものであるので、方程式として表現されている時点ではパラメータである任意の整数値ｍが、方程式を変形して解いた時に一意的な整数値ｍになることは矛盾する。従って、この方法では原理的に複屈折率の値を決定することは不可能である。
【０００３】
まず、この事実を反例に挙げて説明する。ここでの例では複屈折率の波長分散特性を考慮するので、未知数は４つとなり、用いるピ−ク数も４つ必要とするが、任意の整数値ｍを消去して複屈折率を求めるという操作自体は同じである。まず数式５：
【数５】
と数式６：
【数６】
において、Ａ＝３．２２８×１０９，Ｂ＝−２．９７８×１０３，Ｃ＝０．２１１，ｄ＝１０（μｍ）とすると、図１の様な干渉スペクトルが得られる。図１におけるピ−ク値は４１０、４３７、４７５、５３４、６３８（ｎｍ）である。数式５においてピ−ク値になるのは数式７：
【数７】
の場合である。数式６と数式７から数式８：
【数８】
が得られて、数式８における未知数はＡ，Ｂ，Ｃ，ｍの４つである。従って図１の様な干渉スペクトルが得られたら４つのピ−クを選んで、
数式９、
【数９】
数式１０、
【数１０】
数式１１、
【数１１】
数式１２
【数１２】
なる連立方程式が得られる。４つのピ−クとして４３７、４７５、５３４、６３８（ｎｍ）を選んで、前記の連立方程式を解くと、数式６は図２の▲２▼の様に決定された。図２の▲１▼が図１の干渉スペクトルを得るのに用いられたものであり、数式６においてＡ＝３．２２８×１０９，Ｂ＝−２．９７８×１０３，Ｃ＝０．２１１として得られるものであるから、図２の▲２▼は明らかに異なる値となってしまう。この様に従来の方法では正確な複屈折率は求められない。このことは図３により更にはっきりする。次に示す実例では、同じ図１の干渉ピ−クを用いるが、先に整数値ｍの値を決めてから連立方程式を解く。すなわち未知数は３つとなるので、解くべき連立方程式の数は３つとなり、この時選択したピ−クは前記の４つの中から４３７、４７５、５３４（ｎｍ）の３つを選択した。図３において、▲１▼はｍ＝１としてから数式９から数式１１までの連立方程式を解いた場合であり、▲３▼はｍ＝２としてから前記の連立方程式を解いた場合、▲４▼はｍ＝５とした場合である。▲２▼は図２の▲２▼を比較のために載せたものである。図３における▲４▼は図２における▲１▼に一致する。図３から分かることは、整数値ｍを未知数のままで連立方程式を解いても正確な値は求まらないが、整数値ｍを決めてから連立方程式を解けば正確な値が求められる。従って、ｍを決定せずには正確な複屈折率は求められないので、従来の干渉法からの決定法では複屈折率の値は求められないことになる。
【０００４】
そもそも従来の方法で、正確な値が求められないのは、数式７から分かるように干渉スペクトルがピ−クとなる条件には、任意な整数値ｍが含まれている為に、本来物理的には唯一である複屈折率の値は、その候補が無数に存在することになる。それは用いるピ−クの数を増やして連立方程式を立てた場合でも、整数値ｍを含んでいれば同じである。つまり、整数値ｍは任意な値であるから、ｍの値を決める度に数式９から数式１１の定数項が変化する。すなわち解くべき連立方程式が無数に存在することになる。ところが従来の方法では整数値ｍが一意のものであるとして連立方程式を解いている。言い換えると、連立方程式を解くと、一意的に整数値ｍが決定できるとしている。これは数式７における任意な整数値ｍの導入に矛盾している。その為に現実とは異なる値が求まってしまうと言う問題が生じる。先に述べたように、方程式の解は元の方程式を同値変形することによって導き出されるものであるので、方程式を解く前はパラメータである任意の整数値ｍが、方程式を解いた時に一意的な整数値ｍになることは矛盾するのである。
【０００５】
【発明が解決しようとする課題】
従来の干渉スペクトルを用いた複屈折率等の決定法では正確な値は得られないという問題点があった。それは、条件式として、単なるパラメータである任意の整数値ｍに付随して変化する複屈折率という関係式から物理的に一意的な値を決定しようとしたからである。
【０００６】
本発明は、本来測定法としては非常に簡単であり、干渉スペクトルが得られさえすれば、どんな材料にでも適用できるという利点を持つ干渉法により、物質の複屈折率等を正確に決定する方法を提供することと、その方法を実行するための装置を得ることを目的としている。
【０００７】
【課題を解決するための手段】
上記目的を達成するために、本発明ではある波長域において、厚さＬ１とＬ２の透明な物質がそれとは異なる種類の透明な物質に上下から挟まれたサンドイッチ構造の２つの試料である試料１と試料２があり、前記サンドイッチ構造の試料に前記波長域に渡って連続なスペクトルを有する光が照射されて干渉スペクトルが得られた場合、前記干渉スペクトルのピ−クに短波長側から順に整数値ｎ＝１，２，３・・・を割り当てることと、さらに適当な整数値ｍを決めて数式１：

により前記干渉ピ−クの各波長に於ける試料１から得られたＮの値の集合Ｆ１（λ）：

と試料２から得られたＮの値の集合Ｆ２（λ）：

を求めて、前記Ｆ１（λ）とＦ２（λ）の値から波長と屈折率の関係、すなわち波長分散を示す式である数式３：

に従うように最小二乗法によって試料１の干渉スペクトルのピークから未知の係数Ａ，Ｂ，Ｃが決定されて得られた波長分散曲線Ｇ１（ｐ）と、同様に試料２の干渉スペクトルのピークから得られた波長分散曲線Ｇ２（ｑ）を適当な整数値ｐとｑをそれぞれ適当に変化させた毎に得ることと、前記波長分散曲線Ｇ１（ｐ）とＧ２（ｑ）の試料１と試料２の干渉スペクトルのピーク各波長における距離の総和が最小となる場合の組み合わせであるＧ１（ｐ１：整数値）とＧ２（ｑ２：整数値）を選び出すことと、選び出された前記波長分散曲線Ｇ１（ｐ１）とＧ２（ｑ２）の中間値によって波長と屈折率の関係を示す式である前記数式３を決定することを特徴とする干渉スペクトル測定による屈折率決定方法と、この屈折率決定方法の実行及びデ−タ処理するためのコンピュ−タと光源装置と分光装置とにより構成されていることを特徴とする干渉スペクトル測定による屈折率決定装置。
【０００８】
また、ある波長域において、厚さＬ１とＬ２の透明な物質がそれとは異なる種類の透明な物質に上下から挟まれた２つのサンドイッチ構造の２つの試料である試料１と試料２があり、さらに前記サンドイッチ構造の試料を偏光方向が互いに垂直となっている２つの偏光子の間に配置して、前記波長域に渡って連続なスペクトルを有する光を照射して干渉スペクトルが得られた場合において、前記干渉スペクトルのピ−クに短波長側から順に整数値ｎを割り当てることと、さらに適当な整数値ｍを決めて数式２：

により前記干渉ピ−クの各波長に於ける試料１から得られたΔｎの値の集合Ｐ１（λ１）：

と試料２から得られたΔｎの値の集合Ｐ２（λ２）：

が求まり、前記Ｐ１（λ１）とＰ２（λ２）の値から波長と複屈折率の関係を示す式、すなわち波長分散式である数式４：

に従うように最小二乗法によって試料１の干渉スペクトルのピークから未知の係数Ａ，Ｂ，Ｃが決定されて得られた波長分散曲線Ｑ１（ｐ）と同様に試料２から得られたＱ２（ｑ）を適当な整数値ｐとｑをそれぞれ適当に変化させた毎に得ることと、前記波長分散曲線Ｑ１（ｐ）とＱ２（ｑ）の試料１と試料２の干渉スペクトルのピーク各波長における距離の総和が最小となる場合の組み合わせであるＱ１（ｐ１：整数値）とＱ２（ｑ２：整数値）を選び出すことと、選び出された前記波長分散曲線Ｑ１（ｐ１）とＱ２（ｑ２）の中間値によって波長と複屈折率の関係を示す式である数式４を決定することを特徴とする干渉スペクトル測定による複屈折率決定方法と、この複屈折率決定方法の実行及びデ−タ処理するためのコンピュ−タと光源装置と分光装置とにより構成されていることを特徴とする干渉スペクトル測定による複屈折率決定装置。
【０００９】
また、２つ以上の試料を用意して屈折率あるいは複屈折率の決定方法の操作を繰り返して、その結果の平均を取ることによっても干渉スペクトル測定による屈折率あるいは複屈折率の決定方法が提供される。
【００１０】
さらに、前記サンドイッチ構造の試料の両端の前記透明な物質間の距離が可変になっていていれば、中間の透明な物質が液体などの場合には、試料を１つだけ用意すれば良い様な干渉スペクトル測定による屈折率及び複屈折率の決定方法が提供される。
【００１１】
【作用】
上記の方法では、干渉スペクトルを測定するだけで物質の屈折率及び複屈折率の波長依存性が正確に決定できる。従来のアッベの屈折計で何種類かの単色光源で屈折角の違いを精密に測定しようとすると装置的に複雑になるが、本発明では連続スペクトルの白色光源の光強度変化だけ測定すれば良く、自動測定の装置を実現するには有利である。
【００１２】
【実施例】
以下、本発明の１実施例を図によって詳細に説明する。
【００１３】
まず干渉法における測定装置の基本構成を図４に示す。良好な測定を行うためのレンズなどの部品は省略してある。１が光源装置で、２が試料台、３が分光装置であり、これらは４のコンピュ−タによって制御されている。試料台２は２つの試料を次々に測定したり、試料を偏光子で挟んだ状態で、あるいはそのままの状態で測定することがコンピュ−タ４の制御で自動で行われる。また測定操作を手動で行う様な従来の測定装置の組み合わせの場合は、デ−タ処理のみをフロピ−デスクの様な情報媒体５をコンピュ−タに組み込むことにより行うこともできる。
【００１４】
測定される試料としては、例えば液晶材料がある。図５（ａ）は試料の構成を示している。６が液晶材料であり、７は液晶セルを作成する基板である。例えば液晶材料にはネマティック液晶のチッソ社製ＧＲ−４１を用いることができ、液晶セルを作成する基板にはガラスなどを用いることができる。もちろん液晶セルを作成する基板は液晶材料が均一な配向をするような処理が施されている。図５（ａ）の様にホモジニアス配向とすると液晶材料の複屈折率、常光屈折率及び異常光屈折率を決定できるので都合がよい。次にそれらについて述べる。
【００１５】
さて試料を図５（ｂ）の様なクロスニコルの配置の偏光子８に挟んで、液晶の光軸を偏光子の偏光軸に対して４５°の位置に設定して干渉スペクトルを測定すると、液晶材料の複屈折率を導出することができる。その方法を順を追って説明する。まずセル厚の異なる２つの液晶セルを用意して干渉法により、可視光領域で図６の様な干渉スペクトルが得られたとする。図６（ａ）はセル厚が１２．０８μｍの液晶セル（１）の干渉スペクトル（ピーク：４１５、４４０、４７２、５２２、５９７ｎｍ）であり、図６（ｂ）はセル厚が２８．２μｍの液晶セル（２）の干渉スペクトル（ピ−ク：４１０、４２１、４３３、４４７、４６３、４８３、５０７、５３７、５７４、６２１、６８２ｎｍ）である。測定温度は２５℃である。図６の干渉スペクトルには短波長側からそれぞれ１から始まる整数値を付ける（例として図６（ａ）に付けてある）。そして各干渉スペクトルのピ−クに対して数式２を適用して各ピ−クにおける△ｎの値を算出する。この時のｍは△ｎ＞０となる様な適当な値を代入する。次に各ピ−クの△ｎの値から波長と複屈折率の関係を表す一般式である数式４を決定する。通常は数式４にはコ−シ−の分散式である数式６が適用される。数式６は第３項までを採用している。本実施例では、各ピ−クの△ｎの値を用いて最小二乗法により数式６の係数Ａ、Ｂ、Ｃを決定することによって複屈折率の波長分散曲線を算出した。
【００１６】
次はｍの値を適当に選んで、前記と同様な操作を行うと図７のような波長分散曲線のグル−プができる。図７（ａ）は液晶セル（１）のｍ＝７〜１１までの波長分散曲線を▲１▼から▲５▼に対応するように示してある。図７（ｂ）は液晶セル（２）のｍ＝１６〜２３までの波長分散曲線が▲１▼から▲５▼に対応するように同様に示してある。
【００１７】
そして液晶セル（１）と液晶セル（２）の波長分散曲線のグル−プを重ね合わせて、数式１７：
【数１７】
に従って液晶セル（１）のそれぞれの波長分散曲線に対する液晶セル（２）のそれぞれの波長分散曲線間の各波長に対する距離の総和を算出する。その過程の様子を図に示すと、図８の様になっている。▲１▼〜▲３▼は液晶セル（１）のｍ＝８〜１０の波長分散曲線であり、▲４▼〜▲６▼は液晶セル（２）のｍ＝１９〜２１の波長分散曲線である。その中から液晶セル（１）からのｍ＝９の波長分散曲線（図８の▲２▼）と液晶セル（２）からのｍ＝２０の波長分散曲線（図８の▲５▼）の距離の総和が最小なので、この両者を抜き出す。その結果が図９の▲１▼と▲２▼である。
【００１８】
液晶セル（１）と液晶セル（２）からそれぞれの代表の波長分散曲線が選び抜かれたので、数式１８：
【数１８】
に従って両者の中間値であるような波長分散曲線を算出する。その結果が図１０である。従来用いられているアッベの屈折計（波長：５８９．３ｎｍ）での値は０．２４であり、図１０から求めた値は０．２３（波長：５８９ｎｍ）であるので、両者はほぼ一致している。
【００１９】
さらに液晶材料の常光屈折率は偏光子を用いずに、図５（ａ）の様に液晶セルを単独のままで干渉スペクトルを測定すればよい。手順は複屈折率を求めたときと同じであるが、干渉ピ−クの条件式はもちろん偏光子で挟んだ場合とは異なるので、単独のままの条件式である数式１を用いなくてはならない。さらに波長と屈折率の関係を表す一般式は数式３であり、具体的にコ−シ−の分散式とすると数式１３：
【数１３】
となる。そして異常光屈折率は常光屈折率と複屈折率の和で表されるので、複屈折率と常光屈折率が測定で求められれば、液晶材料の複屈折率、常光屈折率及び異常光屈折率が決定されることになる。
【００２０】
ここでは本決定方法の原理について述べる。原理の説明には複屈折率の決定について説明するが、同様にして屈折率の決定も説明できる。さて、異なるセル厚の２つの液晶セルの干渉スペクトルを同じ座標軸に書くと何箇所かの交点が存在する。今、スペクトルの一つの交点に着目する。この干渉スペクトルの交点において、複屈折率はピ−クからの位相差φを用いると数式１４：
【数１４】
で表される。従って、数式１４をセル厚の異なる２つの干渉スペクトルに適用すると数式１５：
【数１５】
と数式１６：
【数１６】
と表現できる。数式１５と数式１６の表現で重要なのは共通のｍで表現されていることである。セル厚は異なっていても物理的に唯一なΔｎは、共通のｍと適当な位相差φ１、φ２により表すことができるはずである。ここで△ｎは複屈折率、λは交点での光の波長、ｄ１、ｄ２はそれぞれの液晶セルのセル厚、さらにφ１はセル厚ｄ１、φ２はセル厚ｄ２の場合の干渉スペクトルのピ−ク点からの位相差である。さて、数式１５と数式１６は共通の整数値ｍに関する直線であることが分かる。従って２つの直線の交点はただ一つであるので、その時のｍの値が真の複屈折率を与える。その他のｍの値における両者の直線の距離は、真の複屈折率を与えるｍから離れるに従ってどんどん大きくなる。この性質は波長分散曲線の場合にも当てはまる。なぜならピ−ク値から求めた波長分散曲線上には、当然交点に相当する波長での複屈折率も含まれているので、波長分散曲線同士の距離を比較することは、数式１５と数式１６を用いて説明した交点における複屈折率の距離を比較することと同じである。従って距離が最小となる２つの波長分散曲線が真の波長分散曲線に最も近いのである。完全に一致しないのは、測定誤差などの影響があるからである。
【００２１】
さらに、本決定方法の実施例として、強誘電性液晶に当てはめた場合について述べる。強誘電性液晶や反強誘電性液晶は通常作成される液晶セルの場合は、その捻れ構造と２値動作性により複屈折率等の測定が困難である。しかし本決定方法を用いるとネマテック液晶同様に決定することができる。例として、強誘電性液晶の場合について述べる。強誘電性液晶材料にはチッソ社製のＣＳ−１０２４を用いて、測定温度は２５℃である。液晶セルには周期が１ｍｓ、電圧が１０Ｖの方形波を印可する。この時液晶セルは捻れ構造は消失して、等価的に図５（ａ）の様なホモジニアス配向となる。セル厚を２４μｍとして、クロスニコル下での干渉スペクトルを図１１に示す。ネマテック液晶と同様な干渉スペクトルが得られる。更にもう一つセル厚の異なる試料を用意して、ネマテック液晶の場合と同じ手順によって求められた複屈折率の波長分散曲線を図１２に示す。常光屈折率と異常光屈折率の決定も既に述べたようにすれば決定できる。
【００２２】
ここまでの実施例では、試料を２つ用意してきたが、図１３の様に液晶セルを作成する基板間７の距離を可変できるようなセル厚可変装置９を設けることによって１つの試料で屈折率等を決定できる。さらに、測定値の信頼性を上げるには、できるだけ多くの試料を用意して、それぞれの結果の平均を取ればよい。
【００２３】
以上、本決定方法の手順等を液晶材料に当てはめて示したが、光源の波長域の選択により干渉スペクトルが得られるならば通常の光学材料は勿論のこと、半導体などの材料にでも薄膜化するなどして適用できるはずである。最後に用いた数式を列挙するが本文中の数式番号は以下の【数番号】表記に対応させてある。
【００２４】
【数１】

【００２５】
【数２】

【００２６】
【数３】

【００２７】
【数４】

【００２８】
【数５】

【００２９】
【数６】

【００３０】
【数７】

【００３１】
【数８】

【００３２】
【数９】

【００３３】
【数１０】

【００３４】
【数１１】

【００３５】
【数１２】

【００３６】
【数１３】

【００３７】
【数１４】

【００３８】
【数１５】

【００３９】
【数１６】

【００４０】
【数１７】

【００４１】
【数１８】

【００４２】
【発明の効果】
本発明の効果は次のようになる。干渉法による複屈折率等の波長分散曲線の正しい値が得られる。干渉法も含めて、屈折角の測定などの従来の方法では求めることが困難であった強誘電性液晶や反強誘電性液晶の複屈折率等の波長分散曲線を決定できる。これによって強誘電性液晶や反強誘電性液晶の配向特性などを光学的に解析する場合、４×４行列法を用いるが、常光屈折率と異常光屈折率の波長分散曲線が分かるので、配向特性などの解析がより正確に分かるようになる。また強誘電性液晶や反強誘電性液晶を用いたデバイスの光学的設計に適用できる。これは高速な応答特性を有する強誘電性液晶及び反強誘電性液晶の応用分野の拡大に大きな意味を持つと考えられる。さらに干渉法は一軸性材料の液晶に限らず等方性材料など薄膜化して干渉スペクトルが得られさえすればよいので、光学材料のさらなる応用範囲を開くことが期待できる。
【図面の簡単な説明】
【図１】計算で求めた干渉スペクトルの図
【図２】元の複屈折率の波長分散曲線と連立方程式を解いて求めた複屈折率の波長分散曲線の比較の図
【図３】元の複屈折率の波長分散曲線と整数値ｍを決めてから連立方程式を解いて求めた複屈折率の波長分散曲線との関係を表した図
【図４】干渉スペクトルの測定装置の概略と情報媒体を示した図
【図５】液晶セルの構成及び偏光子との配置を示した図
【図６】実験により求められた干渉スペクトルの図
【図７】整数値ｍを変化させて求められた複屈折率の波長分散曲線のグル−プの図
【図８】２つの液晶セルから求められた複屈折率の波長分散曲線のグル−プを重ね合わせた図
【図９】２つの液晶セルから求められた複屈折率の波長分散曲線のグル−プの中から最も距離が近いもの同士を抜き出した図
【図１０】最終的に求められた複屈折率の波長分散曲線の図
【図１１】強誘電性液晶セルから実験で求められた干渉スペクトルの図
【図１２】強誘電性液晶の複屈折率の波長分散曲線の図
【図１３】セル厚可変型の液晶セルの構成図
【符号の説明】
１光源装置
２試料台
３分光装置
４コンピュ−タ
５情報媒体
６液晶材料
７液晶セルを作成する基板
８偏光子
９セル厚可変装置[0001]
[Industrial application fields]
The present invention relates to a method and apparatus for determining the refractive index and birefringence of a substance from an interference spectrum.
[0002]
[Prior art]
In the conventional determination method using the interference method (for example, the method of W. KUCZYNSKI et al .: Mol. Cryst. Liq. Cryst., 1975, Vol. 31, pp. 267-273) The birefringence is determined by erasing the numerical value m. To describe this method in more detail, in the interferometry, Equation 7:
[Expression 7]
Thus, the conditional expression which becomes a peak includes an integer m which is an arbitrary value as a parameter. Accordingly, there are as many birefringence values obtained from the conditional expression as the number of an arbitrary integer value m which is a parameter. In other words, the integer value m is not a unique value. However, the authors of the above paper derive a conditional expression from the two peaks, and determine that the birefringence is uniquely determined by solving the simultaneous equations with the birefringence and the integer value m as two unknowns. is suggesting. If two unknowns are solved by solving two simultaneous equations, the two unknowns can be uniquely determined. However, since the integer value m in Equation 7 is only a parameter, it is not uniquely determined. Originally, the solution of the equation is derived by equivalently transforming the original equation, and therefore, when the equation is expressed as an equation, an arbitrary integer value m that is a parameter is unique when the equation is transformed and solved. It is contradictory to become a large integer value m. Therefore, in principle, it is impossible to determine the birefringence value by this method.
[0003]
First, this fact will be described as a counterexample. In this example, since the wavelength dispersion characteristic of the birefringence is taken into consideration, the number of unknowns is four and the number of peaks to be used is four, but the birefringence is obtained by erasing an arbitrary integer value m. The operation itself is the same. First, Formula 5:
[Equation 5]
And Equation 6:
[Formula 6]
If A = 3.228 × 109, B = −2.978 × 103, C = 0.221, d = 10 (μm), an interference spectrum as shown in FIG. 1 is obtained. The peak values in FIG. 1 are 410, 437, 475, 534, and 638 (nm). The peak value in Formula 5 is Formula 7:
[Expression 7]
This is the case. Formula 6 and Formula 7 to Formula 8:
[Equation 8]
Is obtained, and there are four unknowns A, B, C, and m in Expression 8. Therefore, if the interference spectrum shown in Fig. 1 is obtained, select four peaks,
Formula 9,
[Equation 9]
Formula 10,
[Expression 10]
Formula 11,
[Expression 11]
Formula 12
[Expression 12]
The following simultaneous equations are obtained. When 437, 475, 534, and 638 (nm) were selected as the four peaks and the simultaneous equations were solved, Equation 6 was determined as shown in (2) of FIG. (1) in FIG. 2 is used to obtain the interference spectrum of FIG. 1, and is obtained as A = 3.228 × 109, B = −2.978 × 103, C = 0.221 in Equation 6. Therefore, (2) in FIG. 2 is obviously a different value. Thus, an accurate birefringence cannot be obtained by the conventional method. This is further clarified in FIG. In the following example, the same interference peak of FIG. 1 is used, but the simultaneous equations are solved after the integer value m is determined first. That is, since there are three unknowns, the number of simultaneous equations to be solved is three, and three peaks 437, 475, and 534 (nm) are selected from the above four peaks. In FIG. 3, (1) is a case where simultaneous equations from Equation 9 to Equation 11 are solved after m = 1, and (3) is a case where the simultaneous equations are solved after m = 2. Where m = 5. (2) is a comparison of (2) in FIG. 2 for comparison. (4) in FIG. 3 corresponds to (1) in FIG. It can be seen from FIG. 3 that an accurate value cannot be obtained by solving the simultaneous equations while the integer value m is unknown, but an accurate value can be obtained by solving the simultaneous equations after determining the integer value m. Therefore, since an accurate birefringence cannot be obtained without determining m, the birefringence value cannot be obtained by the conventional determination method from the interferometry.
[0004]
In the first place, the reason why an accurate value cannot be obtained by the conventional method is that, since an arbitrary integer value m is included in the condition for the peak of the interference spectrum as can be seen from Equation 7, it is inherently physical. There are an infinite number of candidates for the birefringence value that is unique to. Even if the simultaneous equations are established by increasing the number of peaks to be used, the same is true if the integer value m is included. That is, since the integer value m is an arbitrary value, each time the value of m is determined, the constant terms of Equation 9 to Equation 11 change. In other words, there are innumerable simultaneous equations to be solved. However, the conventional method solves the simultaneous equations on the assumption that the integer value m is unique. In other words, the integer value m can be uniquely determined by solving the simultaneous equations. This contradicts the introduction of an arbitrary integer value m in Equation 7. Therefore, there arises a problem that a value different from the actual value is obtained. As mentioned earlier, since the solution of the equation is derived by equivalently transforming the original equation, any integer value m that is a parameter is unique when the equation is solved before the equation is solved. The integer value m is contradictory.
[0005]
[Problems to be solved by the invention]
The conventional method of determining the birefringence index using the interference spectrum has a problem that an accurate value cannot be obtained. This is because, as a conditional expression, an attempt is made to determine a physically unique value from a relational expression of a birefringence index that changes accompanying an arbitrary integer value m, which is a mere parameter.
[0006]
The present invention is very simple as a measuring method by nature, and it can be applied to any material as long as an interference spectrum is obtained. And an apparatus for carrying out the method.
[0007]
[Means for Solving the Problems]
In order to achieve the above object, in the present invention, in a certain wavelength range, sample 1 is a sample having two sandwich structures in which transparent materials having thicknesses L1 and L2 are sandwiched from above and below by different types of transparent materials. When the interference spectrum is obtained by irradiating the sandwich structure sample with light having a continuous spectrum over the wavelength range, the peak of the interference spectrum is adjusted in order from the short wavelength side. Assign numerical values n = 1, 2, 3,..., And determine an appropriate integer value m.

A set F1 (λ) of N values obtained from the sample 1 at each wavelength of the interference peak by:

And a set F2 (λ) of N values obtained from the sample 2:

Is calculated from the values of F1 (λ) and F2 (λ), which is an expression showing the relationship between wavelength and refractive index, ie, wavelength dispersion:

And the chromatic dispersion curve G1 (p) obtained by determining the unknown coefficients A, B, C from the peak of the interference spectrum of the sample 1 by the least square method and the peak of the interference spectrum of the sample 2 as well. To obtain the obtained chromatic dispersion curve G2 (q) every time the appropriate integer values p and q are appropriately changed, and the

samples

1 and 2 of the chromatic dispersion curves G1 (p) and G2 (q) G1 (p1: integer value) and G2 (q2: integer value), which are combinations when the sum of the distances at the respective wavelengths of the peak of the interference spectrum is minimized, and the selected chromatic dispersion curve G1 (p1) ) And G2 (q2) are determined by the intermediate value between the wavelength and the refractive index, and the refractive index determination method based on interference spectrum measurement, and the execution of the refractive index determination method. - computer for data processing - data and refractive index determination apparatus according to interference spectrum measurement, characterized in that it is constituted by a light source device and a spectrometer.
[0008]
Further, in a certain wavelength region, there are Sample 1 and Sample 2, which are two samples of two sandwich structures in which transparent materials of thickness L1 and L2 are sandwiched from above and below by different types of transparent materials, When the sample having the sandwich structure is arranged between two polarizers whose polarization directions are perpendicular to each other and irradiated with light having a continuous spectrum over the wavelength range, an interference spectrum is obtained. , Assigning an integer value n to the peak of the interference spectrum in order from the short wavelength side, and further determining an appropriate integer value m:

A set P1 (λ1) of values of Δn obtained from the sample 1 at each wavelength of the interference peak:

And a set P2 (λ2) of Δn values obtained from the sample 2:

Equation 4 representing the relationship between wavelength and birefringence from the values of P1 (λ1) and P2 (λ2), that is, a wavelength dispersion equation:

Q2 (q) obtained from the sample 2 in the same manner as the wavelength dispersion curve Q1 (p) obtained by determining the unknown coefficients A, B, C from the peak of the interference spectrum of the sample 1 by the least square method. Is obtained every time the appropriate integer values p and q are appropriately changed, and the distance at each wavelength of the peak of the interference spectrum of the sample 1 and sample 2 of the chromatic dispersion curves Q1 (p) and Q2 (q) is obtained. Selecting Q1 (p1: integer value) and Q2 (q2: integer value), which are combinations when the sum is minimized, and an intermediate value between the selected wavelength dispersion curves Q1 (p1) and Q2 (q2) The method of determining the birefringence by the interference spectrum measurement characterized by determining the formula 4 which is an expression showing the relationship between the wavelength and the birefringence by the method, and for executing the birefringence determination method and data processing Computer and light Device and spectroscopic apparatus and that is constituted by a birefringence determination apparatus according to interference spectrum measurement according to claim.
[0009]
Also, a method for determining the refractive index or birefringence by interference spectrum measurement is provided by preparing two or more samples, repeating the operation of the method for determining the refractive index or birefringence, and taking the average of the results. Is done.
[0010]
Further, if the distance between the transparent substances at both ends of the sample having the sandwich structure is variable, only one sample may be prepared when the intermediate transparent substance is a liquid or the like. A method for determining refractive index and birefringence by interference spectrum measurement is provided.
[0011]
[Action]
In the above method, the wavelength dependence of the refractive index and birefringence of a substance can be accurately determined only by measuring the interference spectrum. Although it would be complicated to use a conventional Abbe refractometer to accurately measure the difference in refraction angle with several types of monochromatic light sources, in the present invention, it is only necessary to measure the light intensity change of a white light source with a continuous spectrum. It is advantageous to realize an automatic measurement device.
[0012]
【Example】
Hereinafter, an embodiment of the present invention will be described in detail with reference to the drawings.
[0013]
First, FIG. 4 shows a basic configuration of a measuring apparatus in the interferometry. Parts such as a lens for performing good measurement are omitted. Reference numeral 1 denotes a light source device, 2 denotes a sample stage, 3 denotes a spectroscopic device, and these are controlled by a computer 4. The sample stage 2 automatically measures two samples one after another, or in a state where the sample is sandwiched between polarizers or as it is, under the control of the computer 4. In the case of a combination of conventional measuring apparatuses in which the measurement operation is performed manually, only data processing can be performed by incorporating an information medium 5 such as a floppy desk in the computer.
[0014]
An example of a sample to be measured is a liquid crystal material. FIG. 5A shows the configuration of the sample. Reference numeral 6 denotes a liquid crystal material, and reference numeral 7 denotes a substrate for forming a liquid crystal cell. For example, nematic liquid crystal GR-41 manufactured by Chisso Corporation can be used as a liquid crystal material, and glass or the like can be used as a substrate for forming a liquid crystal cell. Of course, the substrate on which the liquid crystal cell is formed is treated so that the liquid crystal material is uniformly oriented. The homogeneous alignment as shown in FIG. 5A is advantageous because the birefringence, ordinary light refractive index and extraordinary light refractive index of the liquid crystal material can be determined. These are described next.
[0015]
Now, when the interference spectrum is measured by sandwiching the sample between the polarizers 8 having the crossed Nicols arrangement as shown in FIG. 5B and setting the optical axis of the liquid crystal at a position of 45 ° with respect to the polarization axis of the polarizer, The birefringence of the liquid crystal material can be derived. The method will be explained step by step. First, it is assumed that two liquid crystal cells having different cell thicknesses are prepared and an interference spectrum as shown in FIG. 6 is obtained in the visible light region by the interference method. 6A shows an interference spectrum (peaks: 415, 440, 472, 522, 597 nm) of the liquid crystal cell (1) having a cell thickness of 12.08 μm, and FIG. 6B shows a cell thickness of 28.2 μm. It is an interference spectrum (peaks: 410, 421, 433, 447, 463, 483, 507, 537, 574, 621, 682 nm) of the liquid crystal cell (2). The measurement temperature is 25 ° C. An integer value starting from 1 is added to the interference spectrum of FIG. 6 from the short wavelength side (as shown in FIG. 6A as an example). Then, the value of Δn at each peak is calculated by applying Formula 2 to the peak of each interference spectrum. At this time, an appropriate value such that Δn> 0 is substituted for m. Next, Formula 4 which is a general formula representing the relationship between the wavelength and the birefringence is determined from the value of Δn of each peak. Normally, Formula 6 is applied to Formula 4, which is a dispersion formula of a case. Formula 6 adopts up to the third term. In this example, the birefringence wavelength dispersion curve was calculated by determining the coefficients A, B, and C of Equation 6 by the method of least squares using the value of Δn of each peak.
[0016]
Next, if the value of m is appropriately selected and the same operation as described above is performed, a group of chromatic dispersion curves as shown in FIG. 7 can be obtained. FIG. 7A shows the chromatic dispersion curves of the liquid crystal cell (1) from m = 7 to 11 so as to correspond to (1) to (5). FIG. 7B similarly shows the chromatic dispersion curves of the liquid crystal cell (2) from m = 16 to 23 corresponding to (1) to (5).
[0017]
Then, the group of chromatic dispersion curves of the liquid crystal cell (1) and the liquid crystal cell (2) are overlapped to obtain Equation 17:
[Expression 17]
Thus, the sum of the distances for the respective wavelengths between the respective chromatic dispersion curves of the liquid crystal cell (2) with respect to the respective chromatic dispersion curves of the liquid crystal cell (1) is calculated. The state of the process is shown in FIG. (1) to (3) are chromatic dispersion curves of m = 8 to 10 of the liquid crystal cell (1), and (4) to (6) are chromatic dispersion curves of m = 19 to 21 of the liquid crystal cell (2). is there. The distance between the m = 9 wavelength dispersion curve ((2) in FIG. 8) from the liquid crystal cell (1) and the m = 20 wavelength dispersion curve ((5) in FIG. 8) from the liquid crystal cell (2). Since the sum of is the smallest, both are extracted. The results are (1) and (2) in FIG.
[0018]
Since the representative chromatic dispersion curves are selected from the liquid crystal cell (1) and the liquid crystal cell (2), Equation 18:
[Formula 18]
Then, a chromatic dispersion curve that is an intermediate value between the two is calculated. The result is shown in FIG. The value of Abbe's refractometer (wavelength: 589.3 nm) used in the past is 0.24, and the value obtained from FIG. 10 is 0.23 (wavelength: 589 nm). ing.
[0019]
Further, the ordinary refractive index of the liquid crystal material may be measured without using a polarizer and measuring the interference spectrum with the liquid crystal cell alone as shown in FIG. The procedure is the same as that for obtaining the birefringence, but the conditional expression of the interference peak is different from the case where it is sandwiched between the polarizers. Don't be. Further, the general formula representing the relationship between the wavelength and the refractive index is Formula 3, and specifically, the dispersion formula of the case is expressed by Formula 13:
[Formula 13]
It becomes. The extraordinary refractive index is represented by the sum of the ordinary refractive index and the birefringent index. Therefore, if the birefringent index and the ordinary refractive index are obtained by measurement, the birefringence index, the ordinary refractive index, and the extraordinary refractive index of the liquid crystal material are obtained. Will be determined.
[0020]
Here, the principle of this determination method will be described. In the explanation of the principle, the determination of the birefringence will be described, but the determination of the refractive index can be explained in the same manner. When the interference spectra of two liquid crystal cells having different cell thicknesses are written on the same coordinate axis, there are several intersections. Now, focus on one intersection of the spectra. At the intersection of the interference spectrum, the birefringence is expressed by the following equation 14 using the phase difference φ from the peak:
[Expression 14]
It is represented by Therefore, when Equation 14 is applied to two interference spectra having different cell thicknesses, Equation 15:
[Expression 15]
And Equation 16:
[Expression 16]
Can be expressed. What is important in the expressions of Expression 15 and Expression 16 is that they are expressed by a common m. Even if the cell thicknesses are different, the physically unique Δn can be expressed by a common m and appropriate phase differences φ1 and φ2. Where Δn is the birefringence, λ is the wavelength of light at the intersection, d1 and d2 are the cell thicknesses of the respective liquid crystal cells, φ1 is the cell thickness d1, and φ2 is the peak of the interference spectrum when the cell thickness is d2. The phase difference from the point. Now, it turns out that Numerical formula 15 and Numerical formula 16 are the straight lines regarding the common integer value m. Therefore, since the intersection of the two straight lines is only one, the value of m at that time gives a true birefringence. The distance between the two straight lines at other values of m becomes larger as the distance from m giving the true birefringence increases. This property also applies to the case of a chromatic dispersion curve. This is because the birefringence at the wavelength corresponding to the intersection is naturally included on the chromatic dispersion curve obtained from the peak value. This is the same as comparing the birefringence distances at the intersections described using. Therefore, the two chromatic dispersion curves having the smallest distance are closest to the true chromatic dispersion curve. The reason for not being completely in agreement is that there is an influence such as measurement error.
[0021]
Furthermore, as an example of this determination method, a case where the present invention is applied to a ferroelectric liquid crystal will be described. In the case of a liquid crystal cell in which a ferroelectric liquid crystal and an antiferroelectric liquid crystal are usually produced, it is difficult to measure a birefringence or the like due to its twisted structure and binary operability. However, when this determination method is used, it can be determined in the same manner as nematic liquid crystal. As an example, the case of a ferroelectric liquid crystal will be described. CS-1024 manufactured by Chisso Corporation is used as the ferroelectric liquid crystal material, and the measurement temperature is 25 ° C. A square wave having a period of 1 ms and a voltage of 10 V is applied to the liquid crystal cell. At this time, the twisted structure of the liquid crystal cell disappears, and equivalently the homogeneous alignment as shown in FIG. FIG. 11 shows an interference spectrum under crossed Nicols with a cell thickness of 24 μm. An interference spectrum similar to that of nematic liquid crystal is obtained. FIG. 12 shows the wavelength dispersion curve of the birefringence obtained by preparing another sample with a different cell thickness and obtaining the same procedure as in the case of nematic liquid crystal. The determination of the ordinary light refractive index and the extraordinary light refractive index can also be made as described above.
[0022]
In the embodiments so far, two samples have been prepared. However, as shown in FIG. 13, by providing a cell thickness variable device 9 that can change the distance between the substrates 7 on which the liquid crystal cell is formed, a single sample is refracted. Rate, etc. can be determined. Furthermore, in order to increase the reliability of the measured value, it is only necessary to prepare as many samples as possible and to average the results.
[0023]
In the above, the procedure of this determination method is applied to the liquid crystal material. However, if an interference spectrum can be obtained by selecting the wavelength range of the light source, it is possible to reduce the film thickness to a material such as a semiconductor as well as a normal optical material. Etc. should be applicable. The last used formulas are listed, but the formula numbers in the text correspond to the following [Numerical number] notation.
[0024]
[Expression 1]

[0025]
[Expression 2]

[0026]
[Equation 3]

[0027]
[Expression 4]

[0028]
[Equation 5]

[0029]
[Formula 6]

[0030]
[Expression 7]

[0031]
[Equation 8]

[0032]
[Equation 9]

[0033]
[Expression 10]

[0034]
[Expression 11]

[0035]
[Expression 12]

[0036]
[Formula 13]

[0037]
[Expression 14]

[0038]
[Expression 15]

[0039]
[Expression 16]

[0040]
[Expression 17]

[0041]
[Formula 18]

[0042]
【The invention's effect】
The effects of the present invention are as follows. The correct value of the wavelength dispersion curve such as the birefringence by the interferometry can be obtained. It is possible to determine a wavelength dispersion curve such as a birefringence of a ferroelectric liquid crystal or an antiferroelectric liquid crystal that has been difficult to obtain by a conventional method such as measurement of a refraction angle including an interference method. In this way, when optically analyzing the orientation characteristics of ferroelectric liquid crystal and antiferroelectric liquid crystal, the 4 × 4 matrix method is used, but the wavelength dispersion curves of ordinary light refractive index and extraordinary light refractive index can be understood. Analysis of characteristics etc. can be understood more accurately. It can also be applied to the optical design of devices using ferroelectric liquid crystals and antiferroelectric liquid crystals. This is considered to have a great significance in expanding the application fields of ferroelectric liquid crystals and antiferroelectric liquid crystals having high-speed response characteristics. Furthermore, the interference method is not limited to a liquid crystal of a uniaxial material, and it is only necessary to obtain an interference spectrum by forming a thin film such as an isotropic material.
[Brief description of the drawings]
FIG. 1 is a diagram of an interference spectrum obtained by calculation. FIG. 2 is a comparison between a wavelength dispersion curve of an original birefringence and a wavelength dispersion curve of a birefringence obtained by solving simultaneous equations. FIG. 4 is a diagram showing the relationship between the wavelength dispersion curve of the birefringence and the wavelength dispersion curve of the birefringence obtained by solving the simultaneous equations after determining the integer value m. FIG. FIG. 5 is a diagram showing a configuration of a liquid crystal cell and an arrangement with a polarizer. FIG. 6 is a diagram of an interference spectrum obtained by experiment. FIG. 7 is a diagram showing a complex obtained by changing the integer value m. FIG. 8 is a diagram showing a group of birefringence wavelength dispersion curves obtained from two liquid crystal cells. FIG. 9 is a diagram obtained from two liquid crystal cells. Select the closest birefringence group of chromatic dispersion curves. FIG. 10 is a diagram of a wavelength dispersion curve of a birefringence index finally obtained. FIG. 11 is a diagram of an interference spectrum obtained from an experiment using a ferroelectric liquid crystal cell. Diagram of wavelength dispersion curve of refractive index [FIG. 13] Configuration diagram of liquid crystal cell with variable cell thickness [Explanation of symbols]
DESCRIPTION OF SYMBOLS 1 Light source device 2 Sample stand 3 Spectrometer 4 Computer 5 Information medium 6 Liquid crystal material 7 Substrate 8 which produces a liquid crystal cell Polarizer 9 Cell thickness variable device

Claims

In a certain wavelength region, there are two samples of a sandwich structure in which a transparent material of thickness L1 and L2 is sandwiched from above and below by a different type of transparent material. When an interference spectrum is obtained by irradiating light having a continuous spectrum over the wavelength range, integer values n = 1, 2, 3,... In order from the short wavelength side to the peak of the interference spectrum. And an appropriate integer value m is determined, and Equation 1:

A set F1 (λ) of N values obtained from the sample 1 at each wavelength of the interference peak:

And a set F2 (λ) of N values obtained from the sample 2:

And the chromatic dispersion curve G1 (p) obtained by determining the unknown coefficients A, B, C from the peak of the interference spectrum of the sample 1 by the least square method and the peak of the interference spectrum of the sample 2 as well. To obtain the obtained chromatic dispersion curve G2 (q) every time the appropriate integer values p and q are appropriately changed, and the samples 1 and 2 of the chromatic dispersion curves G1 (p) and G2 (q) G1 (p1: integer value) and G2 (q2: integer value), which are combinations when the sum of the distances at the respective wavelengths of the peak of the interference spectrum is minimized, and the selected chromatic dispersion curve G1 (p1) ) And G2 (q2) are used to determine Formula 3 which is a formula indicating the relationship between wavelength and refractive index based on an intermediate value between G2 (q2) and a refractive index determination method by interference spectrum measurement.

In a certain wavelength region, there are two samples of two sandwich structures in which transparent materials of thickness L1 and L2 are sandwiched from above and below by different types of transparent materials, and sample 1 and sample 2, and the sandwich In the case where an interference spectrum is obtained by arranging a sample having a structure between two polarizers whose polarization directions are perpendicular to each other and irradiating light having a continuous spectrum over the wavelength range, Assigning an integer value n to the peak of the interference spectrum in order from the short wavelength side, and further determining an appropriate integer value m, Equation 2:

And a set P2 (λ2) of Δn values obtained from the sample 2:

Q2 (q) obtained from the sample 2 in the same manner as the wavelength dispersion curve Q1 (p) obtained by determining the unknown coefficients A, B, C from the peak of the interference spectrum of the sample 1 by the least square method. Is obtained every time the appropriate integer values p and q are appropriately changed, and the distance at each wavelength of the peak of the interference spectrum of the sample 1 and sample 2 of the chromatic dispersion curves Q1 (p) and Q2 (q) is obtained. Selecting Q1 (p1: integer value) and Q2 (q2: integer value), which are combinations when the sum is minimized, and an intermediate value between the selected wavelength dispersion curves Q1 (p1) and Q2 (q2) A method for determining a birefringence index by measuring an interference spectrum, wherein Formula 4 is determined to determine a relationship between a wavelength and a birefringence index.

And a set F2 (λ) of N values obtained from the sample 2:

And the chromatic dispersion curve G1 (p) obtained by determining the unknown coefficients A, B, C from the peak of the interference spectrum of the sample 1 by the least square method and the peak of the interference spectrum of the sample 2 as well. To obtain the obtained chromatic dispersion curve G2 (q) every time the appropriate integer values p and q are appropriately changed, and the samples 1 and 2 of the chromatic dispersion curves G1 (p) and G2 (q) G1 (p1: integer value) and G2 (q2: integer value), which are combinations when the sum of the distances at the respective wavelengths of the peak of the interference spectrum is minimized, and the selected chromatic dispersion curve G1 (p1) ) And G2 (q2) are determined by the intermediate value between the wavelength and the refractive index, and the refractive index determination method based on interference spectrum measurement, and the execution of the refractive index determination method. - computer for data processing - data and refractive index determination apparatus according to interference spectrum measurement, characterized in that it is constituted by a light source device and a spectrometer.

And a set P2 (λ2) of Δn values obtained from the sample 2:

Q2 (q) obtained from the sample 2 in the same manner as the wavelength dispersion curve Q1 (p) obtained by determining the unknown coefficients A, B, C from the peak of the interference spectrum of the sample 1 by the least square method. Is obtained every time the appropriate integer values p and q are appropriately changed, and the distance at each wavelength of the peak of the interference spectrum of the sample 1 and sample 2 of the chromatic dispersion curves Q1 (p) and Q2 (q) is obtained. Selecting Q1 (p1: integer value) and Q2 (q2: integer value), which are combinations when the sum is minimized, and an intermediate value between the selected wavelength dispersion curves Q1 (p1) and Q2 (q2) The method of determining the birefringence by the interference spectrum measurement characterized by determining the formula 4 which is an expression showing the relationship between the wavelength and the birefringence by the method, and for executing the birefringence determination method and data processing Computer and light Device and spectroscopic apparatus and that is constituted by a birefringence determination apparatus according to interference spectrum measurement according to claim.

In the method of determining refractive index and birefringence by interference spectrum measurement according to claim 1 to claim 4, two or more samples are prepared, and the operation of claim 1 or claim 2 is repeated, and the result A method of determining a refractive index and a birefringence by measuring an interference spectrum, wherein the refractive index or the birefringence is determined by taking an average of

6. The method for determining refractive index and birefringence by interference spectrum measurement according to claim 1, wherein the distance between the transparent materials at both ends of the sample having the sandwich structure is variable. A method for determining a refractive index and a birefringence by measuring a characteristic interference spectrum.