Dağılma

Elektromanyetizmada ve optikte **dağılma** ya da **dispersiyon**, elektromanyetik dalganın ilerlediği ortamdaki faz hızının frekansına bağlı olması durumudur. Kırılma indisinin frekansa bağlılığı olarak da tanımlanabilmektedir. Bu özelliğe sahip ortamlar dağıtıcı ortamlar olarak bilinir. Faz hızı ile grup hızının eşit olması durumunda dağılma sıfırlanır; grup hızının daha büyük olması anormal dağılma olarak bilinir. İletim hatları ve optik fiberler gibi dalga kılavuzlarında dalga yayılımını büyük ölçüde etkileyen dağılma, dalga denkleminin geçerliği olduğu diğer sistemlerde de gözlemlenebilmektedir.

Bir sistemdeki sinyal iletimi dağılma diyagramı ile gösterilebilir; bu diyagram, frekans ile dalga vektörünü ilişkilendirir. Farklı frekansların farklı faz hızlarının olması ortam veya sistemde ilerleyen sinyallerde bozunmaya yol açar; düşük ve yüksek frekanslı bileşenler farklı hızda hareket ettiği için sinyal zarfı ilerleme ile birlikte genişler. Dalga kılavuzlarındaki dağılma, dalganın ilerlediği etken kırılma indisinin hesaplanması ile anlaşılabilir.

Kayıpsız iletim hatlarında dağılmasız iletim koşulları telgrafçı denklemleri ile hesaplanabilmektedir.

## Galeri

Optical dispersion dynamics.gif|Bir ışık darbesinin dağılmalı bir ortamda bozunması
Alexander's band.svg|Dağılma ile gök kuşağı oluşumu
An incandescent light-bulb viewed through a transmissive diffraction grating.jpg|Bir kırınım ızgarasından geçen ampül ışığının dağılması


## Yüksek dağılım mertebelerinin genelleştirilmiş formülasyonu - [[Lah-Laguerre optiği]]
Kromatik dağılımın Taylor katsayıları aracılığıyla pertürbatif bir şekilde tanımlanması, birkaç farklı sistemden gelen dağılımın dengelenmesi gereken optimizasyon problemleri için avantajlıdır. Örneğin, chirp pulse lazer amplifikatörlerinde, optik hasarı önlemek için pulslar önce bir gerici tarafından zaman içinde gerilir. Daha sonra amplifikasyon sürecinde, darbeler kaçınılmaz olarak malzemelerden geçen doğrusal ve doğrusal olmayan faz biriktirir. Ve son olarak, darbeler çeşitli tipte kompresörlerde sıkıştırılır. Biriken fazda kalan yüksek mertebeleri iptal etmek için genellikle tek tek mertebeler ölçülür ve dengelenir. Bununla birlikte, düzgün sistemler için, bu tür pertürbatif tanımlamaya genellikle ihtiyaç duyulmaz (örneğin, dalga kılavuzlarında yayılma). 
Dağılım düzenleri, Lah-Laguerre tipi dönüşümler şeklinde hesaplama dostu bir şekilde genelleştirilmiştir.

Dağılım mertebeleri, fazın veya dalga vektörünün Taylor açılımı ile tanımlanır.

  \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}

Dalga vektörü k \mathrm{(}\omega\mathrm{)} = \frac{\omega}{c}n \mathrm{(}\omega\mathrm{)} ve faz için dağılım ilişkileri
\varphi \mathrm{(}\omega\mathrm{)} = \frac{\omega}{c}{\it OP} \mathrm{(}\omega\mathrm{)} olarak ifade edilebilir:

 \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)
 

Herhangi bir türevlenebilir fonksiyonun f\mathrm{(}\omega \mathrm{|}\lambda \mathrm{)} dalga boyu veya frekans uzayındaki türevleri bir Lah dönüşümü ile şu şekilde belirtilir:


\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)

Dönüşümün matris elemanları Lah katsayılarıdır: \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)!}}

GDD için yazılan yukarıdaki ifade, dalga boyu GGD olan bir sabitin sıfır yüksek mertebeye sahip olacağını belirtir. GDD'den değerlendirilen yüksek mertebeler şunlardır:
\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}


Kırılma indisi n veya optik yol OP için ifade edilen denklem (2)'nin denklem (1)'de yerine konması, dağılım mertebeleri için kapalı form ifadeleri ile sonuçlanır. Genel olarak p^{th} mertebeli dağılım POD, negatif iki mertebeli Laguerre tipi bir dönüşümdür:

POD = \frac{d^m \varphi(\omega) }{d\omega^m}=(-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^m k(\omega) }{d\omega^m}=(-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} 

Dönüşümlerin matris elemanları eksi 2 mertebesindeki işaretsiz Laguerre katsayılarıdır ve şu şekilde verilir:  \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)!}}

Dalga vektörü için açıkça yazılan ilk on dağılım mertebesi şunlardır:


\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} 
   
        
Grup kırılma indisi n_g olarak tanımlanır: 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{)}}}+\omega \frac{ {\partial }^n  \mathrm{(}\omega \mathrm{)}}}\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{)}}}+\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} 

    
Açıkça, \varphi fazı için yazılan ilk on dağılım derecesi, Lah dönüşümleri (denklem (2)) kullanılarak dalga boyunun bir fonksiyonu olarak şu şekilde ifade edilebilir:


\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}}\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}


## Ayrıca bakınız

* Faz hızı
* Grup hızı
* Kırınım
* Kramers-Kronig ilişkileri

## Kaynakça

;Bibliyografi
* 
* 
*