Disperso (optiko)

*Pri la aliaj sencoj de la vorto vidu apartigilon *disperso*.*
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[eta|dekstra|250px|Disperso de blanka lumo trans [prismo.]]
En optiko, **disperso**  estas malkomponiĝo de lumo laŭ koloroj. Rezulte el ne tro dika radio de blanka lumo fariĝas diverskolora bendo. En ĝia unua ekstremo videblas ruĝa koloro, poste oranĝkolora, flava, verda, blua kaj violkolora en la alia ekstremo. Ekster limoj de la videbla parto estas areoj de transruĝa lumo kaj transviolkolora lumo.

Disperso okazas kiam lumo trairas optikajn aparatojn, kie estas refrakto pro tio, ke indico de refrakto de la refraktanta medio dependas de frekvenco (aŭ longo de ondo) de la lumo.

Tia disperso estas kaŭzo de kolora aberacio en objektivoj kaj lensoj. Ĝi ankaŭ kaŭzas ĉielarkojn pro multaj akvogutetoj en humidaj atmosferoj.

Disperso okazas ankaŭ kiam lumo trairas difraktan kradon aŭ estas respegulata de ĝi.

## Ĝeneraligita formuliĝo de la altaj ordoj de disperso - Lah-Laguerre-optiko 
La priskribo de la kromata disperso en perturba maniero tra Taylor-koeficientoj estas avantaĝa por optimumigaj problemoj kie la disperso de pluraj malsamaj sistemoj devas esti ekvilibra. Ekzemple, en pepa pulsaj laseramplifiloj, la pulsoj unue estas streĉitaj en tempo per brankardo por eviti optikan difekton. Tiam en la plifortiga procezo, la pulsoj akumuliĝas neeviteble lineara kaj nelinia fazo pasanta tra materialoj. Kaj laste, la pulsoj estas kunpremitaj en diversaj specoj de kompresoroj. Por nuligi iujn ajn restajn pli altajn ordojn en la amasigita fazo, kutime individuaj ordoj estas mezuritaj kaj ekvilibrigitaj. Tamen, por unuformaj sistemoj, tia perturba priskribo ofte ne estas necesa (t.e., disvastigo en ondgvidiloj).
La disvastigordoj estis ĝeneraligitaj en komputile amika maniero, en la formo de Lah-Laguerre-specaj transformoj.

La disvastigordoj estas difinitaj per la Taylor-vastigo de la fazo aŭ la ondovektoro.

  \begin{array}{c}\varphi \mathrm{(}\omega\mathrm{)} = \varphi\left.\  \right|_{\omega_{0}} + \left. \ \frac{\partial\varphi}{\partial\omega} \right|_{\omega_{0}}\left(\omega - \omega_{0} \right) + \frac{1}{2}\left. \ \frac{\partial^\varphi}{\partial\omega^{2}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{2}\  + \ldots + \frac{1}{p!}\left. \ \frac{\partial^\varphi}{\partial\omega^{p}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{p} + \ldots  \end{array} 

 \begin{array}{c}k\mathrm{(}\omega\mathrm{)} = k\left.\  \right|_{\omega_{0}} + \left. \ \frac{\partial k}{\partial\omega} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right) + \frac{1}{2}\left. \ \frac{\partial^k}{\partial\omega^{2}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{2}\  + \ldots + \frac{1}{p!}\left. \ \frac{\partial^k}{\partial\omega^{p}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{p} + \ldots  \end{array}

La disvastigrilatoj por la ondaktoro k \mathrm{(}\omega\mathrm{)} = \frac{\omega}{c}n\mathrm{(}\omega\mathrm{)} kaj la fazo
\varphi \mathrm{(}\omega\mathrm{)} = \frac{\omega}{c}{\it OP}\mathrm{(}\omega\mathrm{)} povas esti esprimita kiel:

 \begin{array}{c}\frac{ {\partial }^}{\partial {\omega }^p}k  \mathrm{(}\omega \mathrm{)}=\frac{1}{c}\left(p\frac{ {\partial }^{p-1}}{\partial {\omega }^{p-1}}n  \mathrm{(}\omega \mathrm{)}+\omega \frac{ {\partial }^}{\partial {\omega }^p}n  \mathrm{(}\omega \mathrm{)}\right)\  \end{array},
\begin{array}{c}\frac{ {\partial }^}{\partial {\omega }^p}\varphi    \mathrm{(}\omega \mathrm{)} = \frac{1}{c}\left(p\frac{ {\partial }^{p-1}}{\partial {\omega }^{p-1}}{\it OP}   \mathrm{(}\omega \mathrm{)}+\omega \frac{ {\partial }^}{\partial {\omega }^p}{\it OP}   \mathrm{(}\omega \mathrm{)}\right)   \end{array} (1)
 

La derivaĵoj de iu diferencigebla funkcio f\mathrm{(}\omega \mathrm{|}\lambda \mathrm{)} en la ondolongo aŭ la frekvenca spaco estas precizigitaj tra Lah transformo kiel:

\begin{array}{l} \frac{ {\partial {p}}}{\partial {\omega }^p}f   \mathrm{(}\omega \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\lambda }^m\frac{ {\partial }^}{\partial {\lambda }^m}f  \mathrm{(}\lambda \mathrm{)}}\end{array}
 ,

  \begin{array}{c}
\frac{ {\partial }^}{\partial {\lambda }^p}f  \mathrm{(}\lambda \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\omega }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\omega }^m\frac{ {\partial }^}{\partial {\omega }^m}f   \mathrm{(}\omega \mathrm{)}}\end{array}

(2)

La matricaj elementoj de la transformo estas la Lah-koeficientoj: \mathcal{A}\mathrm{(}p,m\mathrm{)} = \frac{p\mathrm{!}}{\left(p\mathrm{-}m\right)\mathrm{!}m\mathrm{!}}\frac{\mathrm{(}p\mathrm{-}\mathrm{1)!}}{\mathrm{(}m\mathrm{-}\mathrm{1)!}}

Skribita por la GDD ĉi-supra esprimo deklaras ke konstanto kun ondolongo GGD, havos nul pli altajn ordojn. La pli altaj ordoj taksitaj de la GDD estas:
\begin{array}{c} \frac{ {\partial }^}{\partial {\omega }^p}GDD   \mathrm{(}\omega \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\lambda }^m\frac{ {\partial }^}{\partial {\lambda }^m}GDD  \mathrm{(}\lambda \mathrm{)}}
 \end{array}

Anstataŭigi ekvacion (2) esprimitan por la refrakta indekso n aŭ optika vojo OP en ekvacion (1) rezultigas fermitformajn esprimojn por la disvastigordoj. Ĝenerale la orda disvastigo POD estas Laguerre-speca transformo de negativa ordo du:

POD = \frac{d^p \varphi(\omega) }{d\omega^p}=(-1)^p(\frac{\lambda}{2\pi c})^{(p-1)}\sum_{m=0}^{p}\mathcal{B(p,m)} (\lambda)^m\frac{d^m OP(\lambda) }{d\lambda^m}  ,
POD = \frac{d^p k(\omega) }{d\omega^p}=(-1)^p(\frac{\lambda}{2\pi c})^{(p-1)}\sum_{m=0}^{p}\mathcal{B(p,m)} (\lambda)^m\frac{d^m n(\lambda) }{d\lambda^m} 

La matricaj elementoj de la transformoj estas la sensignaj Laguerre-koeficientoj de ordo minus 2, kaj estas donitaj kiel:  \mathcal{B}\mathrm{(}p,m\mathrm{)} = \frac{p\mathrm{!}}{\left(p\mathrm{-}m\right)\mathrm{!}m\mathrm{!}}\frac{\mathrm{(}p\mathrm{-}\mathrm{2)!}}{\mathrm{(}m\mathrm{-}\mathrm{2)!}}

La unuaj dek disvastigordoj, eksplicite skribitaj por la ondovektoro, estas:

\begin{array}{l}\boldsymbol = \frac{\partial }{\partial \omega }k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(n  \mathrm{(}\omega \mathrm{)}+\omega \frac{\partial n  \mathrm{(}\omega \mathrm{)}}{\partial \omega }\right) = \frac{\mathrm{1}}{c}\left(n \mathrm{(}\lambda \mathrm{)}-\lambda \frac{\partial n \mathrm{(}\lambda \mathrm{)}}{\partial\lambda }\right) = v^{\mathrm{-}\mathrm{1}}_{gr}\end{array}

La grupa refrakta indekso n_g estas difinita kiel: n_g   =   cv^{\mathrm{-}\mathrm{1}}_{gr}.

\begin{array}{l}\boldsymbol = \frac{ {\partial }^}{\partial {\omega }^{\mathrm{2}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{2}\frac{\partial n\mathrm{(}\omega \mathrm{)}}{\partial \omega }+\omega \frac{ {\partial }^n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}}\right)  =  \frac{\mathrm{1}}{c}\left(\frac{\lambda }{\mathrm{2}\pi c}\right)\left({\lambda }^{\mathrm{2}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}\right) \end{array}

\begin{array}{l}\boldsymbol = \frac{ {\partial }^}{\partial {\omega }^{\mathrm{3}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{3}\frac{ {\partial }^n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}}+\omega \frac{ {\partial }^n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}\right)    = {-} \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{2}}\Bigl(\mathrm{3}{\lambda }^{\mathrm{2}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +{\lambda }^{\mathrm{3}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}\Bigr) \end{array}

\begin{array}{l}\boldsymbol = \frac{ {\partial }^}{\partial {\omega }^{\mathrm{4}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{4}\frac{ {\partial }^n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}+\omega \frac{ {\partial }^n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}\right)    = \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{3}}\Bigl(\mathrm{12}{\lambda }^{\mathrm{2}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}    +\mathrm{8}{\lambda }^{\mathrm{3}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+{\lambda }^{\mathrm{4}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}\Bigr) \end{array}

\begin{array}{l}\boldsymbol = \frac{ {\partial }^}{\partial {\omega }^{\mathrm{5}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{5}\frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}+\omega \frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}\right)=  {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{4}} \Bigl(\mathrm{60}{\lambda }^{\mathrm{2}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{60}{\lambda }^{\mathrm{3}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{15}{\lambda }^{\mathrm{4}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+{\lambda }^{\mathrm{5}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\Bigr) \end{array}

\begin{array}{l}\boldsymbol = \frac{ {\partial }^}{\partial {\omega }^{\mathrm{6}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{6}\frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}+\omega \frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{6}}}\right) =   \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{5}}\Bigl(\mathrm{360}{\lambda }^{\mathrm{2}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +\mathrm{480}{\lambda }^{\mathrm{3}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{180}{\lambda }^{\mathrm{4}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{24}{\lambda }^{\mathrm{5}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+{\lambda }^{\mathrm{6}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}\Bigr) \end{array}

\begin{array}{l}\boldsymbol = \frac{ {\partial }^}{\partial {\omega }^{\mathrm{7}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{7}\frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{6}}}+\omega \frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{7}}}\right)    =   {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{6}} \Bigl(\mathrm{2520}{\lambda }^{\mathrm{2}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{4200}{\lambda }^{\mathrm{3}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{2100}{\lambda }^{\mathrm{4}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{420}{\lambda }^{\mathrm{5}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+   \mathrm{35}{\lambda }^{\mathrm{6}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+{\lambda }^{\mathrm{7}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}\Bigr) \end{array}

\begin{array}{l}\boldsymbol = \frac{ {\partial }^}{\partial {\omega }^{\mathrm{8}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{8}\frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{7}}}+\omega \frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}\right) =    \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{7}}\Bigl(\mathrm{20160}{\lambda }^{\mathrm{2}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +\mathrm{40320}{\lambda }^{\mathrm{3}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{25200}{\lambda }^{\mathrm{4}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{6720}{\lambda }^{\mathrm{5}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+ \mathrm{840}{\lambda }^{\mathrm{6}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}} +\\+\mathrm{48}{\lambda }^{\mathrm{7}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+{\lambda }^{\mathrm{8}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}\Bigr) \end{array}

\begin{array}{l}\boldsymbol = \frac{ {\partial }^}{\partial {\omega }^{\mathrm{9}}}k \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{9}\frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}+\omega \frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}\right) =   {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{8}}\Bigl(\mathrm{181440}{\lambda }^{\mathrm{2}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{423360}{\lambda }^{\mathrm{3}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{317520}{\lambda }^{\mathrm{4}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{105840}{\lambda }^{\mathrm{5}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+  \mathrm{17640}{\lambda }^{\mathrm{6}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{1512}{\lambda }^{\mathrm{7}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{63}{\lambda }^{\mathrm{8}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+{\lambda }^{\mathrm{9}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}\Bigr) \end{array}

\begin{array}{l}\boldsymbol = \frac{ {\partial }^}{\partial {\omega }^{\mathrm{10}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{10}\frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}+\omega \frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{10}}}\right) =   \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{9}}\Bigl(\mathrm{1814400}{\lambda }^{\mathrm{2}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{4838400}{\lambda }^{\mathrm{3}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{4233600}{\lambda }^{\mathrm{4}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+  {1693440}{\lambda }^{\mathrm{5}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\\+\mathrm{352800}{\lambda }^{\mathrm{6}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\mathrm{40320}{\lambda }^{\mathrm{7}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{2520}{\lambda }^{\mathrm{8}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+\mathrm{80}{\lambda }^{\mathrm{9}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}+  {\lambda }^{\mathrm{10}}\frac{ {\partial }^n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{10}}}\Bigr) \end{array}

Eksplicite, skribita por la fazo \varphi , la unuaj dek dispersordoj povas esti esprimitaj kiel funkcio de ondolongo uzante la Lah-transformojn (ekvacio (2)) kiel:

\begin{array}{l} \frac{ {\partial {p}}}{\partial {\omega }^p}f   \mathrm{(}\omega \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\lambda }^m\frac{ {\partial }^}{\partial {\lambda }^m}f  \mathrm{(}\lambda \mathrm{)}}\end{array}
 ,

  \begin{array}{c}
\frac{ {\partial }^}{\partial {\lambda }^p}f  \mathrm{(}\lambda \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\omega }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\mathrm{(}p,m\mathrm{)}{\omega }^m\frac{ {\partial }^}{\partial {\omega }^m}f   \mathrm{(}\omega \mathrm{)}}\end{array}

\begin{array}{l}\frac{\partial \varphi\mathrm{(}\omega \mathrm{)}}{\partial \omega }= {-}\left(\frac{\mathrm{2}\pi c}{{\omega }^{\mathrm{2}}}\right)\frac{\partial \varphi  \mathrm{(}\omega \mathrm{)}}{\partial \lambda } = {-}\left(\frac{ {\lambda }^{\mathrm{2}}}{\mathrm{2}\pi c}\right)\frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }\end{array}

\begin{array}{l}\frac{ {\partial }^\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}} = \frac{\partial }{\partial \omega }\left(\frac{\partial \varphi  \mathrm{(}\omega \mathrm{)}}{\partial \omega }\right) = {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{2}}\left(\mathrm{2}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+{\lambda }^{\mathrm{2}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}\right) \end{array}

\begin{array}{l}\frac{ {\partial }^\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}=  {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{3}}\left(\mathrm{6}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{6}{\lambda }^{\mathrm{2}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+{\lambda }^{\mathrm{3}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}\right) \end{array}
 

\begin{array}{l}\frac{ {\partial }^\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}=  {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{4}}\Bigl(\mathrm{24}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{36}{\lambda }^{\mathrm{2}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{12}{\lambda }^{\mathrm{3}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}
+{\lambda }^{\mathrm{4}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}\Bigr) \end{array}

\begin{array}{l}\frac{ {\partial
}^{\mathrm{5}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}= {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{5}}\Bigl(\mathrm{120}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{240}{\lambda }^{\mathrm{2}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}
+\mathrm{120}{\lambda }^{\mathrm{3}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{20}{\lambda }^{\mathrm{4}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+{\lambda }^{\mathrm{5}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\Bigr) \end{array}

\begin{array}{l}\frac{ {\partial }^\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{6}}}= {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{6}}\Bigl(\mathrm{720}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{1800}{\lambda }^{\mathrm{2}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{1200}{\lambda }^{\mathrm{3}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{300}{\lambda }^{\mathrm{4}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{30}{\lambda }^{\mathrm{5}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\mathrm{\ +}{\lambda }^{\mathrm{6}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}\Bigr) \end{array}

\begin{array}{l}\frac{ {\partial }^\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{7}}}= {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{7}} \Bigl(\mathrm{5040}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{15120}{\lambda }^{\mathrm{2}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+ \mathrm{12600}{\lambda }^{\mathrm{3}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{4200}{\lambda }^{\mathrm{4}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{630}{\lambda }^{\mathrm{5}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{42}{\lambda }^{\mathrm{6}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+{\lambda }^{\mathrm{7}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}} \Bigr) \end{array}

\begin{array}{l}\frac{ {\partial }^\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}=  {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{8}}\Bigl(\mathrm{40320}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{141120}{\lambda }^{\mathrm{2}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+ \mathrm{141120}{\lambda }^{\mathrm{3}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{58800}{\lambda }^{\mathrm{4}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{11760}{\lambda }^{\mathrm{5}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{1176}{\lambda }^{\mathrm{6}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\mathrm{56}{\lambda }^{\mathrm{7}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\\  +{\lambda }^{\mathrm{8}}\frac{\partial^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial{\lambda }^{\mathrm{8}}}\Bigr) \end{array}

\begin{array}{l}\frac{ {\partial }^\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}=  {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{9}}\Bigl(\mathrm{362880}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{1451520}{\lambda }^{\mathrm{2}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+ \mathrm{1693440}{\lambda }^{\mathrm{3}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{846720}{\lambda }^{\mathrm{4}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{211680}{\lambda }^{\mathrm{5}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{28224}{\lambda }^{\mathrm{6}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{2016}{\lambda }^{\mathrm{7}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{72}{\lambda }^{\mathrm{8}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+{\lambda }^{\mathrm{9}}\frac{\partial ^{\mathrm{9}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}\Bigr) \end{array}

\begin{array}{l}\frac{ {\partial }^\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{10}}}=  {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{10}}\Bigl(\mathrm{3628800}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{16329600}{\lambda }^{\mathrm{2}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{21772800}{\lambda }^{\mathrm{3}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{12700800}{\lambda }^{\mathrm{4}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{3810240}{\lambda }^{\mathrm{5}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{635040}{\lambda }^{\mathrm{6}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{60480}{\lambda }^{\mathrm{7}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}  +\mathrm{3240}{\lambda }^{\mathrm{8}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+\mathrm{90}{\lambda }^{\mathrm{9}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}+{\lambda }^{\mathrm{10}}\frac{ {\partial }^\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{10}}}\Bigr) \end{array}

## Referencoj 
Kategorio:Optiko