Po prehodu bele svetlobe skozi optično prizmo se zaradi odvisnosti lomnega količnika od valovne dolžine na drugi strani pojavi cel spekter barv (mavrico ). Disperzíja ali razklòn je v optiki pojav , ki se kaže v tem, da je fazna hitrost valovanja (v splošnem elektromagnetnega valovanja in tudi poljubnega valovanja) odvisna od frekvence .[ 1] lomni količnik odvisen od valovne dolžine svetlobe (frekvence ). Opazi se lahko tudi disperzijo skupinske hitrosti.
Pojav je odkril Isaac Newton (1642–1727) okoli leta 1672 , pojasnili pa so ga mnogo let pozneje.
Najenostavnješi zgled za prikaz disperzije je prehod bele svetlobe preko 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 napak optičnih naprav.
Fazna hitrost je določena kot
v
p
=
ω
k
{\displaystyle v_{\rm {p}}={\frac {\omega }{k}}}
d
n
(
λ
)
d
λ
{\displaystyle {\frac {\mathrm {d} n(\lambda )}{\mathrm {d} \lambda }}\!}
fazne hitrosti . Kadar je ta vrednost negativna, je disperzija normalna . Kadar pa je pozitivna, se govori o nenormalni disperziji. Pri normalni disperziji je torej:
d
n
d
λ
<
0
,
{\displaystyle {\frac {\mathrm {d} n}{\mathrm {d} \lambda }}<0\!\,,}
To pomeni, da vrednost lomnega količnika pada kadar valovna dolžina svetlobe raste. To pomeni, da je lomni količnik rdeče svetlobe manjši od lomnega količnika rumene svetlobe, ta pa je manjši od lomnega količnika modre svetlobe. Fazna hitrost svetlobe v poljubnem sredstvu je podana z:
v
=
c
n
,
{\displaystyle v={\frac {c}{n}}\!\,,}
kjer je:
V splošnem je lomni količnik funkcija frekvence
n
=
n
(
ν
)
{\displaystyle n=n(\nu )\!}
n
=
n
(
λ
)
{\displaystyle n=n(\lambda )\!}
Cauchyjevo in tudi s Sellmeierjevo enačbo .
Skupinska hitrost pove kako hitro se gibljejo spremembe v amplitudi valovanja.
Če se skupinsko hitrost označi z
v
g
{\displaystyle v_{\rm {g}}\!}
v
g
=
v
p
−
λ
d
v
p
d
λ
,
{\displaystyle v_{\rm {g}}=v_{\rm {p}}-\lambda {\frac {\mathrm {d} v_{\rm {p}}}{\mathrm {d} \lambda }}\!\,,}
kjer je:
v
p
{\displaystyle v_{\rm {p}}\!}
fazna hitrost svetlobe
λ
{\displaystyle \lambda \!}
vakuumu Skupinska hitrost je določena kot
v
g
=
∂
ω
∂
k
{\displaystyle v_{\rm {g}}={\frac {\partial \omega }{\partial k}}}
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)
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.[ 2] [ 3]
Disperzijski redi so določeni s Taylorjevim razvojem faze ali valovnega vektorja.
φ
(
ω
)
=
φ
|
ω
0
+
∂
φ
∂
ω
|
ω
0
(
ω
−
ω
0
)
+
1
2
∂
2
φ
∂
ω
2
|
ω
0
(
ω
−
ω
0
)
2
+
…
+
1
p
!
∂
p
φ
∂
ω
p
|
ω
0
(
ω
−
ω
0
)
p
+
…
{\displaystyle {\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}}}
k
(
ω
)
=
k
|
ω
0
+
∂
k
∂
ω
|
ω
0
(
ω
−
ω
0
)
+
1
2
∂
2
k
∂
ω
2
|
ω
0
(
ω
−
ω
0
)
2
+
…
+
1
p
!
∂
p
k
∂
ω
p
|
ω
0
(
ω
−
ω
0
)
p
+
…
{\displaystyle {\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}}}
Disperzijska razmerja za valovanje
k
(
ω
)
=
ω
c
n
(
ω
)
{\displaystyle k\mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}}n\mathrm {(} \omega \mathrm {)} }
φ
(
ω
)
=
ω
c
O
P
(
ω
)
{\displaystyle \varphi \mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}}{\it {OP}}\mathrm {(} \omega \mathrm {)} }
∂
p
∂
ω
p
k
(
ω
)
=
1
c
(
p
∂
p
−
1
∂
ω
p
−
1
n
(
ω
)
+
ω
∂
p
∂
ω
p
n
(
ω
)
)
{\displaystyle {\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}}}
∂
p
∂
ω
p
φ
(
ω
)
=
1
c
(
p
∂
p
−
1
∂
ω
p
−
1
O
P
(
ω
)
+
ω
∂
p
∂
ω
p
O
P
(
ω
)
)
(
1
)
{\displaystyle {\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)}
Odvodi vsake diferencirane funkcije
f
(
ω
|
λ
)
{\displaystyle f\mathrm {(} \omega \mathrm {|} \lambda \mathrm {)} }
∂
p
∂
ω
p
f
(
ω
)
=
(
−
1
)
p
(
λ
2
π
c
)
p
∑
m
=
0
p
A
(
p
,
m
)
λ
m
∂
m
∂
λ
m
f
(
λ
)
{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\lambda }^{m}{\frac {{\partial }^{m}}{\partial {\lambda }^{m}}}f\mathrm {(} \lambda \mathrm {)} }\end{array}}}
,
{\displaystyle ,}
∂
p
∂
λ
p
f
(
λ
)
=
(
−
1
)
p
(
ω
2
π
c
)
p
∑
m
=
0
p
A
(
p
,
m
)
ω
m
∂
m
∂
ω
m
f
(
ω
)
(
2
)
{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\omega }^{m}{\frac {{\partial }^{m}}{\partial {\omega }^{m}}}f\mathrm {(} \omega \mathrm {)} }\end{array}}(2)}
Elementi matrike transformacije so Lahovi koeficienti:
A
(
p
,
m
)
=
p
!
(
p
−
m
)
!
m
!
(
p
−
1
)
!
(
m
−
1
)
!
{\displaystyle {\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)!} }}}
Zgornji izraz, zapisan za GDD, pravi, da ima konstanta z valovno dolžino GGD nič višjih redov. Višji redi, ocenjeni iz GDD, so:
∂
p
∂
ω
p
G
D
D
(
ω
)
=
(
−
1
)
p
(
λ
2
π
c
)
p
∑
m
=
0
p
A
(
p
,
m
)
λ
m
∂
m
∂
λ
m
G
D
D
(
λ
)
{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\lambda }^{m}{\frac {{\partial }^{m}}{\partial {\lambda }^{m}}}GDD\mathrm {(} \lambda \mathrm {)} }\end{array}}}
Če enačbo (2), izraženo za lomni količnik
n
{\displaystyle n}
O
P
{\displaystyle OP}
p
t
h
{\displaystyle p^{th}}
P
O
D
=
d
p
φ
(
ω
)
d
ω
p
=
(
−
1
)
p
(
λ
2
π
c
)
(
p
−
1
)
∑
m
=
0
p
B
(
p
,
m
)
(
λ
)
m
d
m
O
P
(
λ
)
d
λ
m
{\displaystyle 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}}}}
,
{\displaystyle ,}
P
O
D
=
d
p
k
(
ω
)
d
ω
p
=
(
−
1
)
p
(
λ
2
π
c
)
(
p
−
1
)
∑
m
=
0
p
B
(
p
,
m
)
(
λ
)
m
d
m
n
(
λ
)
d
λ
m
{\displaystyle 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}}}}
Matrični elementi transformacij so nepodpisani Laguerrovi koeficienti reda minus 2 in so podani kot:
B
(
p
,
m
)
=
p
!
(
p
−
m
)
!
m
!
(
p
−
2
)
!
(
m
−
2
)
!
{\displaystyle {\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)!} }}}
Prvih deset disperzijskih redov, eksplicitno zapisanih za valovni vektor, je naslednjih:
G
D
=
∂
∂
ω
k
(
ω
)
=
1
c
(
n
(
ω
)
+
ω
∂
n
(
ω
)
∂
ω
)
=
1
c
(
n
(
λ
)
−
λ
∂
n
(
λ
)
∂
λ
)
=
v
g
r
−
1
{\displaystyle {\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_{gr}^{\mathrm {-} \mathrm {1} }\end{array}}}
Skupinski lomni količnik
n
g
{\displaystyle n_{g}}
n
g
=
c
v
g
r
−
1
{\displaystyle n_{g}=cv_{gr}^{\mathrm {-} \mathrm {1} }}
G
D
D
=
∂
2
∂
ω
2
k
(
ω
)
=
1
c
(
2
∂
n
(
ω
)
∂
ω
+
ω
∂
2
n
(
ω
)
∂
ω
2
)
=
1
c
(
λ
2
π
c
)
(
λ
2
∂
2
n
(
λ
)
∂
λ
2
)
{\displaystyle {\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}}}
T
O
D
=
∂
3
∂
ω
3
k
(
ω
)
=
1
c
(
3
∂
2
n
(
ω
)
∂
ω
2
+
ω
∂
3
n
(
ω
)
∂
ω
3
)
=
−
1
c
(
λ
2
π
c
)
2
(
3
λ
2
∂
2
n
(
λ
)
∂
λ
2
+
λ
3
∂
3
n
(
λ
)
∂
λ
3
)
{\displaystyle {\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}}}
F
O
D
=
∂
4
∂
ω
4
k
(
ω
)
=
1
c
(
4
∂
3
n
(
ω
)
∂
ω
3
+
ω
∂
4
n
(
ω
)
∂
ω
4
)
=
1
c
(
λ
2
π
c
)
3
(
12
λ
2
∂
2
n
(
λ
)
∂
λ
2
+
8
λ
3
∂
3
n
(
λ
)
∂
λ
3
+
λ
4
∂
4
n
(
λ
)
∂
λ
4
)
{\displaystyle {\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}}}
F
i
O
D
=
∂
5
∂
ω
5
k
(
ω
)
=
1
c
(
5
∂
4
n
(
ω
)
∂
ω
4
+
ω
∂
5
n
(
ω
)
∂
ω
5
)
=
−
1
c
(
λ
2
π
c
)
4
(
60
λ
2
∂
2
n
(
λ
)
∂
λ
2
+
60
λ
3
∂
3
n
(
λ
)
∂
λ
3
+
15
λ
4
∂
4
n
(
λ
)
∂
λ
4
+
λ
5
∂
5
n
(
λ
)
∂
λ
5
)
{\displaystyle {\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}}}
S
i
O
D
=
∂
6
∂
ω
6
k
(
ω
)
=
1
c
(
6
∂
5
n
(
ω
)
∂
ω
5
+
ω
∂
6
n
(
ω
)
∂
ω
6
)
=
1
c
(
λ
2
π
c
)
5
(
360
λ
2
∂
2
n
(
λ
)
∂
λ
2
+
480
λ
3
∂
3
n
(
λ
)
∂
λ
3
+
180
λ
4
∂
4
n
(
λ
)
∂
λ
4
+
24
λ
5
∂
5
n
(
λ
)
∂
λ
5
+
λ
6
∂
6
n
(
λ
)
∂
λ
6
)
{\displaystyle {\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}}}
S
e
O
D
=
∂
7
∂
ω
7
k
(
ω
)
=
1
c
(
7
∂
6
n
(
ω
)
∂
ω
6
+
ω
∂
7
n
(
ω
)
∂
ω
7
)
=
−
1
c
(
λ
2
π
c
)
6
(
2520
λ
2
∂
2
n
(
λ
)
∂
λ
2
+
4200
λ
3
∂
3
n
(
λ
)
∂
λ
3
+
2100
λ
4
∂
4
n
(
λ
)
∂
λ
4
+
420
λ
5
∂
5
n
(
λ
)
∂
λ
5
+
35
λ
6
∂
6
n
(
λ
)
∂
λ
6
+
λ
7
∂
7
n
(
λ
)
∂
λ
7
)
{\displaystyle {\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}}}
E
O
D
=
∂
8
∂
ω
8
k
(
ω
)
=
1
c
(
8
∂
7
n
(
ω
)
∂
ω
7
+
ω
∂
8
n
(
ω
)
∂
ω
8
)
=
1
c
(
λ
2
π
c
)
7
(
20160
λ
2
∂
2
n
(
λ
)
∂
λ
2
+
40320
λ
3
∂
3
n
(
λ
)
∂
λ
3
+
25200
λ
4
∂
4
n
(
λ
)
∂
λ
4
+
6720
λ
5
∂
5
n
(
λ
)
∂
λ
5
+
840
λ
6
∂
6
n
(
λ
)
∂
λ
6
+
+
48
λ
7
∂
7
n
(
λ
)
∂
λ
7
+
λ
8
∂
8
n
(
λ
)
∂
λ
8
)
{\displaystyle {\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}}}
N
O
D
=
∂
9
∂
ω
9
k
(
ω
)
=
1
c
(
9
∂
8
n
(
ω
)
∂
ω
8
+
ω
∂
9
n
(
ω
)
∂
ω
9
)
=
−
1
c
(
λ
2
π
c
)
8
(
181440
λ
2
∂
2
n
(
λ
)
∂
λ
2
+
423360
λ
3
∂
3
n
(
λ
)
∂
λ
3
+
317520
λ
4
∂
4
n
(
λ
)
∂
λ
4
+
105840
λ
5
∂
5
n
(
λ
)
∂
λ
5
+
17640
λ
6
∂
6
n
(
λ
)
∂
λ
6
+
+
1512
λ
7
∂
7
n
(
λ
)
∂
λ
7
+
63
λ
8
∂
8
n
(
λ
)
∂
λ
8
+
λ
9
∂
9
n
(
λ
)
∂
λ
9
)
{\displaystyle {\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}}}
T
e
O
D
=
∂
10
∂
ω
10
k
(
ω
)
=
1
c
(
10
∂
9
n
(
ω
)
∂
ω
9
+
ω
∂
10
n
(
ω
)
∂
ω
10
)
=
1
c
(
λ
2
π
c
)
9
(
1814400
λ
2
∂
2
n
(
λ
)
∂
λ
2
+
4838400
λ
3
∂
3
n
(
λ
)
∂
λ
3
+
4233600
λ
4
∂
4
n
(
λ
)
∂
λ
4
+
1693440
λ
5
∂
5
n
(
λ
)
∂
λ
5
+
+
352800
λ
6
∂
6
n
(
λ
)
∂
λ
6
+
40320
λ
7
∂
7
n
(
λ
)
∂
λ
7
+
2520
λ
8
∂
8
n
(
λ
)
∂
λ
8
+
80
λ
9
∂
9
n
(
λ
)
∂
λ
9
+
λ
10
∂
10
n
(
λ
)
∂
λ
10
)
{\displaystyle {\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}}}
Eksplicitno, zapisano za fazo
φ
{\displaystyle \varphi }
∂
p
∂
ω
p
f
(
ω
)
=
(
−
1
)
p
(
λ
2
π
c
)
p
∑
m
=
0
p
A
(
p
,
m
)
λ
m
∂
m
∂
λ
m
f
(
λ
)
{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\lambda }^{m}{\frac {{\partial }^{m}}{\partial {\lambda }^{m}}}f\mathrm {(} \lambda \mathrm {)} }\end{array}}}
,
{\displaystyle ,}
∂
p
∂
λ
p
f
(
λ
)
=
(
−
1
)
p
(
ω
2
π
c
)
p
∑
m
=
0
p
A
(
p
,
m
)
ω
m
∂
m
∂
ω
m
f
(
ω
)
{\displaystyle {\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 _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\omega }^{m}{\frac {{\partial }^{m}}{\partial {\omega }^{m}}}f\mathrm {(} \omega \mathrm {)} }\end{array}}}
∂
φ
(
ω
)
∂
ω
=
−
(
2
π
c
ω
2
)
∂
φ
(
ω
)
∂
λ
=
−
(
λ
2
2
π
c
)
∂
φ
(
λ
)
∂
λ
{\displaystyle {\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}}}
∂
2
φ
(
ω
)
∂
ω
2
=
∂
∂
ω
(
∂
φ
(
ω
)
∂
ω
)
=
(
λ
2
π
c
)
2
(
2
λ
∂
φ
(
λ
)
∂
λ
+
λ
2
∂
2
φ
(
λ
)
∂
λ
2
)
{\displaystyle {\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}}}
∂
3
φ
(
ω
)
∂
ω
3
=
−
(
λ
2
π
c
)
3
(
6
λ
∂
φ
(
λ
)
∂
λ
+
6
λ
2
∂
2
φ
(
λ
)
∂
λ
2
+
λ
3
∂
3
φ
(
λ
)
∂
λ
3
)
{\displaystyle {\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}}}
∂
4
φ
(
ω
)
∂
ω
4
=
(
λ
2
π
c
)
4
(
24
λ
∂
φ
(
λ
)
∂
λ
+
36
λ
2
∂
2
φ
(
λ
)
∂
λ
2
+
12
λ
3
∂
3
φ
(
λ
)
∂
λ
3
+
λ
4
∂
4
φ
(
λ
)
∂
λ
4
)
{\displaystyle {\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}}}
∂
5
φ
(
ω
)
∂
ω
5
=
−
(
λ
2
π
c
)
5
(
120
λ
∂
φ
(
λ
)
∂
λ
+
240
λ
2
∂
2
φ
(
λ
)
∂
λ
2
+
120
λ
3
∂
3
φ
(
λ
)
∂
λ
3
+
20
λ
4
∂
4
φ
(
λ
)
∂
λ
4
+
λ
5
∂
5
φ
(
λ
)
∂
λ
5
)
{\displaystyle {\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}}}
∂
6
φ
(
ω
)
∂
ω
6
=
(
λ
2
π
c
)
6
(
720
λ
∂
φ
(
λ
)
∂
λ
+
1800
λ
2
∂
2
φ
(
λ
)
∂
λ
2
+
1200
λ
3
∂
3
φ
(
λ
)
∂
λ
3
+
300
λ
4
∂
4
φ
(
λ
)
∂
λ
4
+
30
λ
5
∂
5
φ
(
λ
)
∂
λ
5
+
λ
6
∂
6
φ
(
λ
)
∂
λ
6
)
{\displaystyle {\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}}}
∂
7
φ
(
ω
)
∂
ω
7
=
−
(
λ
2
π
c
)
7
(
5040
λ
∂
φ
(
λ
)
∂
λ
+
15120
λ
2
∂
2
φ
(
λ
)
∂
λ
2
+
12600
λ
3
∂
3
φ
(
λ
)
∂
λ
3
+
4200
λ
4
∂
4
φ
(
λ
)
∂
λ
4
+
630
λ
5
∂
5
φ
(
λ
)
∂
λ
5
+
42
λ
6
∂
6
φ
(
λ
)
∂
λ
6
+
λ
7
∂
7
φ
(
λ
)
∂
λ
7
)
{\displaystyle {\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}}}
∂
8
φ
(
ω
)
∂
ω
8
=
(
λ
2
π
c
)
8
(
40320
λ
∂
φ
(
λ
)
∂
λ
+
141120
λ
2
∂
2
φ
(
λ
)
∂
λ
2
+
141120
λ
3
∂
3
φ
(
λ
)
∂
λ
3
+
58800
λ
4
∂
4
φ
(
λ
)
∂
λ
4
+
11760
λ
5
∂
5
φ
(
λ
)
∂
λ
5
+
1176
λ
6
∂
6
φ
(
λ
)
∂
λ
6
+
56
λ
7
∂
7
φ
(
λ
)
∂
λ
7
+
+
λ
8
∂
8
φ
(
λ
)
∂
λ
8
)
{\displaystyle {\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}}}
∂
9
φ
(
ω
)
∂
ω
9
=
−
(
λ
2
π
c
)
9
(
362880
λ
∂
φ
(
λ
)
∂
λ
+
1451520
λ
2
∂
2
φ
(
λ
)
∂
λ
2
+
1693440
λ
3
∂
3
φ
(
λ
)
∂
λ
3
+
846720
λ
4
∂
4
φ
(
λ
)
∂
λ
4
+
211680
λ
5
∂
5
φ
(
λ
)
∂
λ
5
+
28224
λ
6
∂
6
φ
(
λ
)
∂
λ
6
+
+
2016
λ
7
∂
7
φ
(
λ
)
∂
λ
7
+
72
λ
8
∂
8
φ
(
λ
)
∂
λ
8
+
λ
9
∂
9
φ
(
λ
)
∂
λ
9
)
{\displaystyle {\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}}}
∂
10
φ
(
ω
)
∂
ω
10
=
(
λ
2
π
c
)
10
(
3628800
λ
∂
φ
(
λ
)
∂
λ
+
16329600
λ
2
∂
2
φ
(
λ
)
∂
λ
2
+
21772800
λ
3
∂
3
φ
(
λ
)
∂
λ
3
+
12700800
λ
4
∂
4
φ
(
λ
)
∂
λ
4
+
3810240
λ
5
∂
5
φ
(
λ
)
∂
λ
5
+
635040
λ
6
∂
6
φ
(
λ
)
∂
λ
6
+
+
60480
λ
7
∂
7
φ
(
λ
)
∂
λ
7
+
3240
λ
8
∂
8
φ
(
λ
)
∂
λ
8
+
90
λ
9
∂
9
φ
(
λ
)
∂
λ
9
+
λ
10
∂
10
φ
(
λ
)
∂
λ
10
)
{\displaystyle {\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}}}
↑ Breuer (1993) , str. 249.↑ Popmintchev, Dimitar; Wang, Siyang; Xiaoshi, Zhang; Stoev, Ventzislav; Popmintchev, Tenio (24. oktober 2022). »Analytical Lah-Laguerre optical formalism for perturbative chromatic dispersion« . Optics Express 30 (22): 40779–40808. Bibcode :2022OExpr..3040779P . doi :10.1364/OE.457139 PMID 36299007 . ↑ Popmintchev, Dimitar; Wang, Siyang; Xiaoshi, Zhang; Stoev, Ventzislav; Popmintchev, Tenio (30. avgust 2020). »Theory of the Chromatic Dispersion, Revisited« (v angleščini). arXiv :2011.00066 physics.optics ].
