Dispersia luminii este fenomenul de descompunere prin refracție a luminii albe în fascicule de lumină colorate diferit. Aceste culori alcătuiesc spectrul luminii albe și sunt: roșu, oranj, galben, verde, albastru, indigo și violet. Ea constă în variația indicelui de refracție n al unei substanțe în funcție de lungimea de undă
λ λ -->
{\displaystyle \lambda }
Relația matematica pentru dispersie este:
n
=
n
(
λ λ -->
)
{\displaystyle n=n\left(\lambda \right)}
Vidul este nedispersiv. Pentru vid viteza de propagare a undei electromagnetice este constanta:
c
=
1
ϵ ϵ -->
0
μ μ -->
0
{\displaystyle c={\frac {1}{\sqrt {\epsilon _{0}\mu _{0}}}}}
Pentru un mediu oarecare:
n
=
c
v
=
ϵ ϵ -->
r
μ μ -->
r
{\displaystyle n={\frac {c}{v}}={\sqrt {\epsilon _{r}\mu _{r}}}}
fenomenul de descompunere prin refracție a luminii albe în fascicule de lumină colorate diferit. Aceste culori alcătuiesc spectrul luminii albe și sunt: roșu, oranj, galben, verde, albastru, indigo și violet.
Lumina provenită de la soare este albă. Isaac Newton a descoperit acum 300 de ani, cu ajutorul unei prisme, că lumina albă este formată din mai multe fascicule colorate diferit.
Prisma optică este un mediu omogen și transparent, mărginit de două fețe plane și neparalele.
La trecerea prin prismă, lumina se descompune în fascicule colorate.
Fasciculele colorate trec prin prismă cu viteze diferite, de aceea ies din prismă sub unghiuri diferite.
Curcubeul
Curcubeul poate fi observat vara , după ploaie. El apare datorită fenomenelor de refracție , reflexie și dispersia luminii provenită de la Soare prin picăturile de apă din atmosferă.
Curcubeu.
Descrierea dispersiei cromatice într-o manieră perturbativă prin intermediul coeficienților Taylor este avantajoasă pentru problemele de optimizare în care dispersia din mai multe sisteme diferite trebuie echilibrată. De exemplu, în amplificatoarele laser cu impulsuri chirp, impulsurile sunt mai întâi întinse în timp de un dispozitiv de întindere pentru a evita deteriorarea optică. Apoi, în procesul de amplificare, impulsurile acumulează inevitabil faza liniară și neliniară trecând prin materiale. Și, în cele din urmă, impulsurile sunt comprimate în diferite tipuri de compresoare. Pentru a anula orice ordine superioare reziduale în faza acumulată, de obicei ordinele individuale sunt măsurate și echilibrate. Cu toate acestea, pentru sistemele uniforme, o astfel de descriere perturbativă nu este adesea necesară (de exemplu, propagarea în ghiduri de undă).
Ordinele de dispersie au fost generalizate într-o manieră ușor de calculat, sub forma unor transformări de tip Lah-Laguerre.[ 1] [ 2]
Ordinele de dispersie sunt definite prin expansiunea Taylor a fazei sau a vectorului de undă.
φ φ -->
(
ω ω -->
)
=
φ φ -->
|
ω ω -->
0
+
∂ ∂ -->
φ φ -->
∂ ∂ -->
ω ω -->
|
ω ω -->
0
(
ω ω -->
− − -->
ω ω -->
0
)
+
1
2
∂ ∂ -->
2
φ φ -->
∂ ∂ -->
ω ω -->
2
|
ω ω -->
0
(
ω ω -->
− − -->
ω ω -->
0
)
2
+
… … -->
+
1
p
!
∂ ∂ -->
p
φ φ -->
∂ ∂ -->
ω ω -->
p
|
ω ω -->
0
(
ω ω -->
− − -->
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 ω -->
)
=
k
|
ω ω -->
0
+
∂ ∂ -->
k
∂ ∂ -->
ω ω -->
|
ω ω -->
0
(
ω ω -->
− − -->
ω ω -->
0
)
+
1
2
∂ ∂ -->
2
k
∂ ∂ -->
ω ω -->
2
|
ω ω -->
0
(
ω ω -->
− − -->
ω ω -->
0
)
2
+
… … -->
+
1
p
!
∂ ∂ -->
p
k
∂ ∂ -->
ω ω -->
p
|
ω ω -->
0
(
ω ω -->
− − -->
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}}}
Relațiile de dispersie pentru vectorul de undă
k
(
ω ω -->
)
=
ω ω -->
c
n
(
ω ω -->
)
{\displaystyle k\mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}}n\mathrm {(} \omega \mathrm {)} }
φ φ -->
(
ω ω -->
)
=
ω ω -->
c
O
P
(
ω ω -->
)
{\displaystyle \varphi \mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}}{\it {OP}}\mathrm {(} \omega \mathrm {)} }
∂ ∂ -->
p
∂ ∂ -->
ω ω -->
p
k
(
ω ω -->
)
=
1
c
(
p
∂ ∂ -->
p
− − -->
1
∂ ∂ -->
ω ω -->
p
− − -->
1
n
(
ω ω -->
)
+
ω ω -->
∂ ∂ -->
p
∂ ∂ -->
ω ω -->
p
n
(
ω 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 ω -->
p
φ φ -->
(
ω ω -->
)
=
1
c
(
p
∂ ∂ -->
p
− − -->
1
∂ ∂ -->
ω ω -->
p
− − -->
1
O
P
(
ω ω -->
)
+
ω ω -->
∂ ∂ -->
p
∂ ∂ -->
ω ω -->
p
O
P
(
ω 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)}
Derivatele oricărei funcții diferențiabile
f
(
ω ω -->
|
λ λ -->
)
{\displaystyle f\mathrm {(} \omega \mathrm {|} \lambda \mathrm {)} }
∂ ∂ -->
p
∂ ∂ -->
ω ω -->
p
f
(
ω ω -->
)
=
(
− − -->
1
)
p
(
λ λ -->
2
π π -->
c
)
p
∑ ∑ -->
m
=
0
p
A
(
p
,
m
)
λ λ -->
m
∂ ∂ -->
m
∂ ∂ -->
λ λ -->
m
f
(
λ 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 λ -->
p
f
(
λ λ -->
)
=
(
− − -->
1
)
p
(
ω ω -->
2
π π -->
c
)
p
∑ ∑ -->
m
=
0
p
A
(
p
,
m
)
ω ω -->
m
∂ ∂ -->
m
∂ ∂ -->
ω ω -->
m
f
(
ω 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)}
Elementele de matrice ale transformării sunt coeficienții Lah:
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)!} }}}
Scrisă pentru GDD, expresia de mai sus afirmă că o constantă cu lungimea de undă GGD, va avea ordinele superioare zero. Ordinele superioare evaluate din GDD sunt:
∂ ∂ -->
p
∂ ∂ -->
ω ω -->
p
G
D
D
(
ω ω -->
)
=
(
− − -->
1
)
p
(
λ λ -->
2
π π -->
c
)
p
∑ ∑ -->
m
=
0
p
A
(
p
,
m
)
λ λ -->
m
∂ ∂ -->
m
∂ ∂ -->
λ λ -->
m
G
D
D
(
λ 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}}}
Înlocuind ecuația (2) exprimată pentru indicele de refracție
n
{\displaystyle n}
O
P
{\displaystyle OP}
p
t
h
{\displaystyle p^{th}}
P
O
D
=
d
p
φ φ -->
(
ω ω -->
)
d
ω ω -->
p
=
(
− − -->
1
)
p
(
λ λ -->
2
π π -->
c
)
(
p
− − -->
1
)
∑ ∑ -->
m
=
0
p
B
(
p
,
m
)
(
λ λ -->
)
m
d
m
O
P
(
λ λ -->
)
d
λ 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 ω -->
)
d
ω ω -->
p
=
(
− − -->
1
)
p
(
λ λ -->
2
π π -->
c
)
(
p
− − -->
1
)
∑ ∑ -->
m
=
0
p
B
(
p
,
m
)
(
λ λ -->
)
m
d
m
n
(
λ λ -->
)
d
λ 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}}}}
Elementele de matrice ale transformărilor sunt coeficienții Laguerre fără semn de ordinul minus 2 și sunt date sub forma:
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)!} }}}
Primele zece ordine de dispersie, scrise explicit pentru vectorul de undă, sunt:
G
D
=
∂ ∂ -->
∂ ∂ -->
ω ω -->
k
(
ω ω -->
)
=
1
c
(
n
(
ω ω -->
)
+
ω ω -->
∂ ∂ -->
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
)
=
1
c
(
n
(
λ λ -->
)
− − -->
λ λ -->
∂ ∂ -->
n
(
λ λ -->
)
∂ ∂ -->
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}}}
Indicele de refracție de grup
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 ω -->
2
k
(
ω ω -->
)
=
1
c
(
2
∂ ∂ -->
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
+
ω ω -->
∂ ∂ -->
2
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
2
)
=
1
c
(
λ λ -->
2
π π -->
c
)
(
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
)
∂ ∂ -->
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 ω -->
3
k
(
ω ω -->
)
=
1
c
(
3
∂ ∂ -->
2
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
2
+
ω ω -->
∂ ∂ -->
3
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
3
)
=
− − -->
1
c
(
λ λ -->
2
π π -->
c
)
2
(
3
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
2
+
λ λ -->
3
∂ ∂ -->
3
n
(
λ λ -->
)
∂ ∂ -->
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 ω -->
4
k
(
ω ω -->
)
=
1
c
(
4
∂ ∂ -->
3
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
3
+
ω ω -->
∂ ∂ -->
4
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
4
)
=
1
c
(
λ λ -->
2
π π -->
c
)
3
(
12
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
2
+
8
λ λ -->
3
∂ ∂ -->
3
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
3
+
λ λ -->
4
∂ ∂ -->
4
n
(
λ λ -->
)
∂ ∂ -->
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 ω -->
5
k
(
ω ω -->
)
=
1
c
(
5
∂ ∂ -->
4
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
4
+
ω ω -->
∂ ∂ -->
5
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
5
)
=
− − -->
1
c
(
λ λ -->
2
π π -->
c
)
4
(
60
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
2
+
60
λ λ -->
3
∂ ∂ -->
3
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
3
+
15
λ λ -->
4
∂ ∂ -->
4
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
4
+
λ λ -->
5
∂ ∂ -->
5
n
(
λ λ -->
)
∂ ∂ -->
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 ω -->
6
k
(
ω ω -->
)
=
1
c
(
6
∂ ∂ -->
5
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
5
+
ω ω -->
∂ ∂ -->
6
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
6
)
=
1
c
(
λ λ -->
2
π π -->
c
)
5
(
360
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
2
+
480
λ λ -->
3
∂ ∂ -->
3
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
3
+
180
λ λ -->
4
∂ ∂ -->
4
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
4
+
24
λ λ -->
5
∂ ∂ -->
5
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
5
+
λ λ -->
6
∂ ∂ -->
6
n
(
λ λ -->
)
∂ ∂ -->
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
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2520
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4200
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2100
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420
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35
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7
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{\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
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7
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20160
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40320
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25200
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6720
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840
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48
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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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8
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181440
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423360
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317520
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105840
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17640
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+
1512
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7
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63
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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
=
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10
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10
k
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1
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10
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9
(
1814400
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2
+
4838400
λ λ -->
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+
4233600
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+
1693440
λ λ -->
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+
352800
λ λ -->
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40320
λ λ -->
7
∂ ∂ -->
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n
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∂ ∂ -->
λ λ -->
7
+
2520
λ λ -->
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∂ ∂ -->
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∂ ∂ -->
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8
+
80
λ λ -->
9
∂ ∂ -->
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n
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∂ ∂ -->
λ λ -->
9
+
λ λ -->
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∂ ∂ -->
10
n
(
λ λ -->
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∂ ∂ -->
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}}}
În mod explicit, scrise pentru faza
φ φ -->
{\displaystyle \varphi }
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0
p
A
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p
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f
(
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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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∂ ∂ -->
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− − -->
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∂ ∂ -->
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f
(
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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}}}
∂ ∂ -->
φ φ -->
(
ω ω -->
)
∂ ∂ -->
ω ω -->
=
− − -->
(
2
π π -->
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ω ω -->
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)
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ω ω -->
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∂ ∂ -->
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=
− − -->
(
λ λ -->
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2
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)
∂ ∂ -->
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λ λ -->
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∂ ∂ -->
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 φ -->
(
ω ω -->
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∂ ∂ -->
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2
=
∂ ∂ -->
∂ ∂ -->
ω ω -->
(
∂ ∂ -->
φ φ -->
(
ω ω -->
)
∂ ∂ -->
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(
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)
2
(
2
λ λ -->
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λ λ -->
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λ λ -->
+
λ λ -->
2
∂ ∂ -->
2
φ φ -->
(
λ λ -->
)
∂ ∂ -->
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 φ -->
(
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=
− − -->
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λ λ -->
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(
6
λ λ -->
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6
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2
+
λ λ -->
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φ φ -->
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λ λ -->
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∂ ∂ -->
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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}}}
∂ ∂ -->
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φ φ -->
(
ω ω -->
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=
(
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24
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λ λ -->
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12
λ λ -->
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λ 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 φ -->
(
ω ω -->
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∂ ∂ -->
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5
=
− − -->
(
λ λ -->
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5
(
120
λ λ -->
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λ λ -->
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240
λ λ -->
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∂ ∂ -->
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λ λ -->
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120
λ λ -->
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∂ ∂ -->
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20
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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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(
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720
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1800
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1200
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300
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30
λ λ -->
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∂ ∂ -->
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+
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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}}}
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{\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}}}
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40320
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{\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}}}
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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}}}
Bibliografie
1.Manual de fizică pentru clasa a VII-a; D. Turcitu și M. Panaghianu
^ Popmintchev, Dimitar; Wang, Siyang; Xiaoshi, Zhang; Stoev, Ventzislav; Popmintchev, Tenio (24 octombrie 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 august 2020 ). „Theory of the Chromatic Dispersion, Revisited” (în engleză). arXiv :2011.00066 physics.optics ].
