Pri la aliaj sencoj de la vorto vidu apartigilon disperso .
Disperso de blanka lumo trans prismo . En optiko , disperso [ 1] lumo laŭ koloroj . Rezulte el ne tro dika radio de blanka lumo fariĝas diverskolora bendo. En ĝia unua ekstremo videblas ruĝa koloro, poste oranĝkolora, flava, verda, blua kaj violkolora en la alia ekstremo. Ekster limoj de la videbla parto estas areoj de transruĝa lumo kaj transviolkolora lumo .
Disperso okazas kiam lumo trairas optikajn aparatojn, kie estas refrakto pro tio, ke indico de refrakto de la refraktanta medio dependas de frekvenco (aŭ longo de ondo ) de la lumo.
Tia disperso estas kaŭzo de kolora aberacio en objektivoj kaj lensoj . Ĝi ankaŭ kaŭzas ĉielarkojn pro multaj akvogutetoj en humidaj atmosferoj .
Disperso okazas ankaŭ kiam lumo trairas difraktan kradon aŭ estas respegulata de ĝi.
La priskribo de la kromata disperso en perturba maniero tra Taylor-koeficientoj estas avantaĝa por optimumigaj problemoj kie la disperso de pluraj malsamaj sistemoj devas esti ekvilibra. Ekzemple, en pepa pulsaj laseramplifiloj, la pulsoj unue estas streĉitaj en tempo per brankardo por eviti optikan difekton. Tiam en la plifortiga procezo, la pulsoj akumuliĝas neeviteble lineara kaj nelinia fazo pasanta tra materialoj. Kaj laste, la pulsoj estas kunpremitaj en diversaj specoj de kompresoroj. Por nuligi iujn ajn restajn pli altajn ordojn en la amasigita fazo, kutime individuaj ordoj estas mezuritaj kaj ekvilibrigitaj. Tamen, por unuformaj sistemoj, tia perturba priskribo ofte ne estas necesa (t.e., disvastigo en ondgvidiloj).
La disvastigordoj estis ĝeneraligitaj en komputile amika maniero, en la formo de Lah-Laguerre-specaj transformoj.[ 2] [ 3]
La disvastigordoj estas difinitaj per la Taylor-vastigo de la fazo aŭ la ondovektoro.
φ
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{\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
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{\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}
La disvastigrilatoj por la ondaktoro
k
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ω
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c
n
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{\displaystyle k\mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}n\mathrm {(} \omega \mathrm {)} }
φ
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=
ω
c
O
P
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ω
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{\displaystyle \varphi \mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}{\it {OP}\mathrm {(} \omega \mathrm {)} }
∂
p
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ω
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{\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
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1
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{\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)}
La derivaĵoj de iu diferencigebla funkcio
f
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{\displaystyle f\mathrm {(} \omega \mathrm {|} \lambda \mathrm {)} }
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{\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 ,}
∂
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λ
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{\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)}
La matricaj elementoj de la transformo estas la Lah-koeficientoj:
A
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{\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)!} }
Skribita por la GDD ĉi-supra esprimo deklaras ke konstanto kun ondolongo GGD, havos nul pli altajn ordojn. La pli altaj ordoj taksitaj de la GDD estas:
∂
p
∂
ω
p
G
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∂
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G
D
D
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{\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}
Anstataŭigi ekvacion (2) esprimitan por la refrakta indekso
n
{\displaystyle n}
O
P
{\displaystyle OP}
P
O
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=
d
p
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{\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
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{\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}
La matricaj elementoj de la transformoj estas la sensignaj Laguerre-koeficientoj de ordo minus 2, kaj estas donitaj kiel:
B
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p
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m
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=
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m
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m
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p
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{\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)!} }
La unuaj dek disvastigordoj, eksplicite skribitaj por la ondovektoro, estas:
G
D
=
∂
∂
ω
k
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ω
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c
(
n
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ω
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ω
∂
n
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∂
ω
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=
1
c
(
n
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λ
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−
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n
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=
v
g
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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}
La grupa refrakta indekso
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
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ω
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1
c
(
2
∂
n
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∂
ω
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ω
∂
2
n
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ω
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∂
ω
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1
c
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λ
2
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λ
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∂
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n
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λ
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∂
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{\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
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3
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{\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
=
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4
∂
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(
4
∂
3
n
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ω
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n
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ω
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∂
ω
4
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=
1
c
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λ
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π
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3
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12
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n
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{\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
=
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5
∂
ω
5
k
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ω
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(
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n
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ω
∂
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n
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∂
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π
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4
(
60
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∂
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{\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
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ω
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=
1
c
(
6
∂
5
n
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ω
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∂
ω
5
+
ω
∂
6
n
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ω
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∂
ω
6
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=
1
c
(
λ
2
π
c
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5
(
360
λ
2
∂
2
n
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λ
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∂
λ
2
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480
λ
3
∂
3
n
(
λ
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∂
λ
3
+
180
λ
4
∂
4
n
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λ
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∂
λ
4
+
24
λ
5
∂
5
n
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∂
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
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∂
λ
3
+
2100
λ
4
∂
4
n
(
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)
∂
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
(
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∂
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λ
8
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(
λ
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∂
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)
{\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
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9
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c
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181440
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423360
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317520
λ
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4
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λ
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105840
λ
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17640
λ
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+
1512
λ
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63
λ
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∂
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
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ω
9
+
ω
∂
10
n
(
ω
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∂
ω
10
)
=
1
c
(
λ
2
π
c
)
9
(
1814400
λ
2
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2
n
(
λ
)
∂
λ
2
+
4838400
λ
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λ
)
∂
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3
+
4233600
λ
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∂
4
n
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∂
λ
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1693440
λ
5
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λ
)
∂
λ
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+
+
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}
Eksplicite, skribita por la fazo \varphi , la unuaj dek dispersordoj povas esti esprimitaj kiel funkcio de ondolongo uzante la Lah-transformojn (ekvacio (2)) kiel:
∂
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
)
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∑
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
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∂
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
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ω
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∂
∂
ω
(
∂
φ
(
ω
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∂
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(
λ
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)
2
(
2
λ
∂
φ
(
λ
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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
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ω
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−
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λ
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c
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3
(
6
λ
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6
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λ
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+
λ
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∂
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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
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(
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(
24
λ
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{\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}
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φ
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−
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λ
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(
120
λ
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240
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2
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120
λ
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20
λ
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λ
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φ
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λ
)
∂
λ
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)
{\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}
∂
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(
λ
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720
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1200
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φ
(
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)
{\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}
∂
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φ
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7
=
−
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λ
2
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c
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7
(
5040
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15120
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2
φ
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∂
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2
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12600
λ
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3
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4200
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630
λ
5
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λ
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42
λ
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)
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+
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7
∂
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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
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6
+
56
λ
7
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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
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φ
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λ
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λ
6
+
+
2016
λ
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φ
(
λ
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72
λ
8
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λ
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λ
8
+
λ
9
∂
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φ
(
λ
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∂
λ
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{\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}
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{\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}
Referencoj
