Ogled vira Disperzija (optika)

[[Slika:Prism-rainbow.svg|frame|right|350px|Po prehodu [[bela|bele]] [[svetloba|svetlobe]] skozi [[optična prizma|optično prizmo]] se zaradi odvisnosti [[lomni količnik|lomnega količnika]] od [[valovna dolžina|valovne dolžine]] na drugi strani pojavi cel [[spekter elektromagnetnega valovanja|spekter]] [[barva|barv]] ([[mavrica|mavrico]]).]]
{{drugi pomeni|disperzija (razločitev)}}

'''Disperzíja''' ali '''razklòn''' je v [[optika|optiki]] [[fizikalni pojav|pojav]], ki se kaže v tem, da je [[fazna hitrost]] [[valovanje|valovanja]] (v splošnem [[elektromagnetno valovanje|elektromagnetnega valovanja]] in tudi poljubnega valovanja) odvisna od [[frekvenca|frekvence]].<ref>{{sktxt|Breuer|1993|pp=249}}.</ref> Včasih se ta vrsta odvisnosti imenuje tudi kromatična disperzija. Nastane zaradi tega, ker je [[lomni količnik]] odvisen od [[valovna dolžina|valovne dolžine]] [[svetloba|svetlobe]] ([[frekvenca|frekvence]]). Opazi se lahko tudi disperzijo skupinske hitrosti.

Pojav je odkril [[Isaac Newton]] (1642–1727) okoli leta [[1672 v znanosti|1672]], pojasnili pa so ga mnogo let pozneje.

Najenostavnješi zgled za prikaz disperzije je prehod [[bela|bele]] svetlobe preko [[optična prizma|optične prizme]] (glej sliko na desni strani). Najbolj znan pojav, ki nastane zaradi disperzije svetlobe, je [[mavrica]]. Posledica disperzije je tudi [[kromatična aberacija]], ki je ena izmed [[optična napaka|napak]] optičnih naprav.

== Disperzija fazne hitrosti ==

[[Fazna hitrost]] je določena kot <math>v_{\rm p}=\frac{\omega}{k}</math>. Pokazatelj odvisnosti lomnega količnika od valovne dolžine je <math> \frac {\mathrm{d}n(\lambda)} {\mathrm{d}\lambda}\!</math> ali disperzija [[fazna hitrost|fazne hitrosti]]. Kadar je ta vrednost negativna, je disperzija ''normalna''. Kadar pa je pozitivna, se govori o ''nenormalni'' disperziji. Pri normalni disperziji je torej:

: <math> \frac{\mathrm{d}n}{\mathrm{d}\lambda} < 0 \!\, , </math>

To pomeni, da vrednost lomnega količnika pada kadar valovna dolžina svetlobe raste. To pomeni, da je lomni količnik [[rdeča|rdeče]] svetlobe manjši od lomnega količnika [[rumena|rumene]] svetlobe, ta pa je manjši od lomnega količnika [[modra|modre]] svetlobe. Fazna hitrost svetlobe v poljubnem sredstvu je podana z:

: <math>v = \frac{c}{n} \!\, , </math>
kjer je: 
* <math>v \!</math> [[fazna hitrost]] svetlobe
* <math>c \!</math> [[hitrost svetlobe]]
* <math>n \!</math> [[lomni količnik]] sredstva

V splošnem je lomni količnik funkcija frekvence <math> n = n(\nu) \!</math> oziroma valovne dolžine <math> n = n( \lambda) \!</math>. Oblika funkcije je odvisna od snovi. Opiše se s [[Cauchyjeva enačba|Cauchyjevo]] in tudi s [[Sellmeierjeva enačba|Sellmeierjevo enačbo]].

[[Skupinska hitrost]] pove kako hitro se gibljejo spremembe v [[amplituda|amplitudi]] valovanja.

== Disperzija skupinske hitrosti ==

Če se skupinsko hitrost označi z <math>v_{\rm g} \!</math>, potem velja naslednja zveza med fazno in skupinsko hitrostjo:

: <math> v_{\rm g} = v_{\rm p} - \lambda \frac{\mathrm{d} v_{\rm p}}{\mathrm{d} \lambda} \!\, , </math>

kjer je:
* <math> v_{\rm p} \!</math> [[fazna hitrost]] svetlobe
* <math> \lambda \!</math> valovna dolžina svetlobe v [[vakuum]]u

Skupinska hitrost je določena kot <math>v_{\rm g} = \frac{\partial \omega}{\partial k}</math>. 

Skupinska hitrost običajno tudi določa, kako hitro se giblje [[energija]] vzdolž smeri gibanja valovanja. Lahko se tudi reče, da pove, kako hitro se giblje sprememba amplitude (znana tudi kot ovojnica valovanja)

== Splošna formulacija visokih redov disperzije – Lah-Laguerrova optika ==
Opis kromatične disperzije na perturbativen način s Taylorjevimi koeficienti je koristen za optimizacijske probleme, pri katerih je treba uravnotežiti disperzijo iz več različnih sistemov. Na primer, v laserskih ojačevalnikih s kirpskimi impulzi se impulzi najprej časovno raztegnejo z raztegovalnikom, da se preprečijo optične poškodbe. Nato se v procesu ojačitve impulzi neizogibno kopičijo v linearni in nelinearni fazi, ki prehaja skozi materiale. In nazadnje se impulzi stisnejo v različnih vrstah kompresorjev. Za izničenje morebitnih preostalih višjih redov v nakopičeni fazi se običajno posamezni redi izmerijo in uravnotežijo. Vendar pri enotnih sistemih takšen perturbacijski opis pogosto ni potreben (npr. širjenje v valovodih). 
Disperzijski redi so bili posplošeni na računsko prijazen način v obliki transformacij tipa Lah-Laguerre.<ref>{{Cite journal|last1=Popmintchev|first1=Dimitar|last2=Wang|first2=Siyang|last3=Xiaoshi|first3=Zhang |last4=Stoev|first4=Ventzislav|last5=Popmintchev|first5=Tenio|date=2022-10-24|title=Analytical Lah-Laguerre optical formalism for perturbative chromatic dispersion|journal=[[Optics Express]]|language=EN|volume=30|issue=22|pages=40779–40808|doi=10.1364/OE.457139|pmid=36299007 |bibcode=2022OExpr..3040779P|doi-access=free}}</ref><ref>{{Cite arXiv|last1=Popmintchev|first1=Dimitar|last2=Wang|first2=Siyang|last3=Xiaoshi|first3=Zhang |last4=Stoev|first4=Ventzislav|last5=Popmintchev|first5=Tenio|date=2020-08-30|title= Theory of the Chromatic Dispersion, Revisited |class=physics.optics|language=EN|eprint=2011.00066}}</ref>

Disperzijski redi so določeni s Taylorjevim razvojem faze ali valovnega vektorja.

<math>  \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^{{2}}\varphi}{\partial\omega^{2}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{2}\  + \ldots + \frac{1}{p!}\left. \ \frac{\partial^{{p}}\varphi}{\partial\omega^{p}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{p} + \ldots  \end{array} </math>  

<math> \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^{{2}}k}{\partial\omega^{2}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{2}\  + \ldots + \frac{1}{p!}\left. \ \frac{\partial^{{p}}k}{\partial\omega^{p}} \right|_{\omega_{0}} \left(\omega - \omega_{0} \right)^{p} + \ldots  \end{array}</math>

Disperzijska razmerja za valovanje <math>k \mathrm{(}\omega\mathrm{)} = \frac{\omega}{c}n \mathrm{(}\omega\mathrm{)}</math> in fazo
<math>\varphi \mathrm{(}\omega\mathrm{)} = \frac{\omega}{c}{\it OP} \mathrm{(}\omega\mathrm{)}</math> lahko izrazimo kot:

<math> \begin{array}{c}\frac{ {\partial }^{{p}}}{\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 }^{{p}}}{\partial {\omega }^p}n  \mathrm{(}\omega \mathrm{)}\right)\  \end{array}</math>, 
<math>\begin{array}{c}\frac{ {\partial }^{{p}}}{\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 }^{{p}}}{\partial {\omega }^p}{\it OP}   \mathrm{(}\omega \mathrm{)}\right)   \end{array} (1)
 </math>

Odvodi vsake diferencirane funkcije <math>f\mathrm{(}\omega \mathrm{|}\lambda \mathrm{)}</math> v prostoru valovne dolžine ali frekvence so določeni z Lahovo transformacijo kot:

<math xmlns="http://www.w3.org/1998/Math/MathML">
\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 }^{{m}}}{\partial {\lambda }^m}f  \mathrm{(}\lambda \mathrm{)}}\end{array}   
</math>  <math>,</math>   
<math >
  \begin{array}{c} 
\frac{ {\partial }^{{p}}}{\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 }^{{m}}}{\partial {\omega }^m}f   \mathrm{(}\omega \mathrm{)}}\end{array}   

(2)</math >    

Elementi matrike transformacije so Lahovi koeficienti: <math>\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)!}}</math> 

Zgornji izraz, zapisan za GDD, pravi, da ima konstanta z valovno dolžino GGD nič višjih redov. Višji redi, ocenjeni iz GDD, so:
<math>\begin{array}{c} \frac{ {\partial }^{{p}}}{\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 }^{{m}}}{\partial {\lambda }^m}GDD  \mathrm{(}\lambda \mathrm{)}}
 \end{array}
</math>   

Če enačbo (2), izraženo za lomni količnik <math>n</math> ali optično pot <math>OP</math>, nadomestimo z enačbo (1), dobimo zaprte izraze za disperzijske redove. Na splošno je <math>p^{th}</math> disperzijski red POD Laguerrova transformacija negativnega reda dva:

<math>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} </math> <math>,</math>   
<math>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} </math>       

Matrični elementi transformacij so nepodpisani Laguerrovi koeficienti reda minus 2 in so podani kot: <math> \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)!}}</math>    

Prvih deset disperzijskih redov, eksplicitno zapisanih za valovni vektor, je naslednjih:

<math>
\begin{array}{l}\boldsymbol{{\it GD}} = \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} 
</math>   
        
Skupinski lomni količnik <math>n_g</math> je definiran kot: <math>n_g   =   cv^{\mathrm{-}\mathrm{1}}_{gr}</math>.

<math>
\begin{array}{l}\boldsymbol{{\it GDD}} = \frac{ {\partial }^{{2}}}{\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 }^{{2}}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 }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}\right) \end{array} 
</math>

<math>
\begin{array}{l}\boldsymbol{{\it TOD}} = \frac{ {\partial }^{{3}}}{\partial {\omega }^{\mathrm{3}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{3}\frac{ {\partial }^{{2}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}}+\omega \frac{ {\partial }^{{3}}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 }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}\Bigr) \end{array} 
</math> 

<math>
\begin{array}{l}\boldsymbol{{\it FOD}} = \frac{ {\partial }^{{4}}}{\partial {\omega }^{\mathrm{4}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{4}\frac{ {\partial }^{{3}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}+\omega \frac{ {\partial }^{{4}}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 }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}    +\mathrm{8}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}\Bigr) \end{array} 
</math>   

<math>
\begin{array}{l}\boldsymbol{{\it FiOD}} = \frac{ {\partial }^{{5}}}{\partial {\omega }^{\mathrm{5}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{5}\frac{ {\partial }^{{4}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}+\omega \frac{ {\partial }^{{5}}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 }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{60}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{15}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\Bigr) \end{array} 
</math>  

<math>
\begin{array}{l}\boldsymbol{{\it SiOD}} = \frac{ {\partial }^{{6}}}{\partial {\omega }^{\mathrm{6}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{6}\frac{ {\partial }^{{5}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}+\omega \frac{ {\partial }^{{6}}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 }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +\mathrm{480}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{180}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{24}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}\Bigr) \end{array} 
</math>   

<math>    
\begin{array}{l}\boldsymbol{{\it SeOD}} = \frac{ {\partial }^{{7}}}{\partial {\omega }^{\mathrm{7}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{7}\frac{ {\partial }^{{6}}n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{6}}}+\omega \frac{ {\partial }^{{7}}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 }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{4200}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{2100}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{420}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+   \mathrm{35}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}\Bigr) \end{array} 
</math>   

<math>    
\begin{array}{l}\boldsymbol{{\it EOD}} = \frac{ {\partial }^{{8}}}{\partial {\omega }^{\mathrm{8}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{8}\frac{ {\partial }^{{7}}n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{7}}}+\omega \frac{ {\partial }^{{8}}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 }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +\mathrm{40320}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{25200}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{6720}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+ \mathrm{840}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}} +\\+\mathrm{48}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}\Bigr) \end{array} 
</math>  
   
<math>    
\begin{array}{l}\boldsymbol{{\it NOD}} = \frac{ {\partial }^{{9}}}{\partial {\omega }^{\mathrm{9}}}k \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{9}\frac{ {\partial }^{{8}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}+\omega \frac{ {\partial }^{{9}}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 }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{423360}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{317520}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{105840}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+  \mathrm{17640}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{1512}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{63}{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+{\lambda }^{\mathrm{9}}\frac{ {\partial }^{{9}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}\Bigr) \end{array} 
</math>        

<math>    
\begin{array}{l}\boldsymbol{{\it TeOD}} = \frac{ {\partial }^{{10}}}{\partial {\omega }^{\mathrm{10}}}k  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{10}\frac{ {\partial }^{{9}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}+\omega \frac{ {\partial }^{{10}}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 }^{{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{4838400}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{4233600}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+  {1693440}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\\+\mathrm{352800}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\mathrm{40320}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{2520}{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+\mathrm{80}{\lambda }^{\mathrm{9}}\frac{ {\partial }^{{9}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}+  {\lambda }^{\mathrm{10}}\frac{ {\partial }^{{10}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{10}}}\Bigr) \end{array} 
</math>
    
Eksplicitno, zapisano za fazo <math>\varphi</math>, lahko prvih deset disperzijskih redov izrazimo kot funkcijo valovne dolžine z uporabo Lahove transformacije (enačba (2)) kot:

<math xmlns="http://www.w3.org/1998/Math/MathML">
\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 }^{{m}}}{\partial {\lambda }^m}f  \mathrm{(}\lambda \mathrm{)}}\end{array}   
</math>  <math>,</math>   
<math >
  \begin{array}{c} 
\frac{ {\partial }^{{p}}}{\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 }^{{m}}}{\partial {\omega }^m}f   \mathrm{(}\omega \mathrm{)}}\end{array}   

</math >    


<math>
\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}
</math>

<math>
\begin{array}{l}\frac{ {\partial }^{{2}}\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 }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}\right) \end{array}
</math>

<math>
\begin{array}{l}\frac{ {\partial }^{{3}}\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 }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}\right) \end{array}
 </math>

<math>
\begin{array}{l}\frac{ {\partial }^{{4}}\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 }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{12}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}
+{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}\Bigr) \end{array}
</math>

<math>
\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 }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}   
+\mathrm{120}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{20}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\Bigr) \end{array}
</math>

<math>
\begin{array}{l}\frac{ {\partial }^{{6}}\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 }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{1200}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{300}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{30}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\mathrm{\ +}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}\Bigr) \end{array}
</math>

<math>
\begin{array}{l}\frac{ {\partial }^{{7}}\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 }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+ \mathrm{12600}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{4200}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{630}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{42}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}} \Bigr) \end{array} 
</math>

<math>
\begin{array}{l}\frac{ {\partial }^{{8}}\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 }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+ \mathrm{141120}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{58800}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{11760}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{1176}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\mathrm{56}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\\  +{\lambda }^{\mathrm{8}}\frac{\partial^{{8}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial{\lambda }^{\mathrm{8}}}\Bigr) \end{array} 
</math>
<math>
\begin{array}{l}\frac{ {\partial }^{{9}}\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 }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+ \mathrm{1693440}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{846720}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{211680}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{28224}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{2016}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{72}{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}\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}
</math>

<math>
\begin{array}{l}\frac{ {\partial }^{{10}}\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 }^{{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{21772800}{\lambda }^{\mathrm{3}}\frac{ {\partial }^{{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{12700800}{\lambda }^{\mathrm{4}}\frac{ {\partial }^{{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{3810240}{\lambda }^{\mathrm{5}}\frac{ {\partial }^{{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{635040}{\lambda }^{\mathrm{6}}\frac{ {\partial }^{{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{60480}{\lambda }^{\mathrm{7}}\frac{ {\partial }^{{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}  +\mathrm{3240}{\lambda }^{\mathrm{8}}\frac{ {\partial }^{{8}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+\mathrm{90}{\lambda }^{\mathrm{9}}\frac{ {\partial }^{{9}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}+{\lambda }^{\mathrm{10}}\frac{ {\partial }^{{10}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{10}}}\Bigr) \end{array}
</math>

== Glej tudi ==

* [[optična prizma]]
* [[Optično vlakno#Disperzija|disperzija v optičnem vlaknu]]

== Sklici ==

{{sklici|1}}

== Viri ==

* {{citat|last1= Breuer|first1= Hans|title= Atlas klasične in moderne fizike|location= Ljubljana|publisher= DZS|date= 1993|isbn= 86-341-1105-9|cobiss= 35693056}}

{{normativna kontrola}}
[[Kategorija:Optika]]
[[Kategorija:Steklo]]
[[Kategorija:1672 v znanosti]]