在三棱镜 中，材料色散效应（折射率 与波长 有关的现象）使不同颜色的光以不同角度折射 ，将白光分成光谱 。 透过阿米西棱镜 观察一体式荧光灯 的色散。 在光学 中，色散 （ sàn ） [ 1] Dispersion ）是光波 的相速度 隨着頻率而改變的現象。[ 2] 色散 （ sàn ） （英語：dispersive medium ）。
尽管色散这一术语在光学领域用于描述光波 和其他[电磁辐射|电磁波]]，但相同意义上的“散失”适用于任何类型的波，例如可产生频散 的声波和地震波，以及海浪中的重力波 。光学中的散失还可以描述输电线 信号（如同轴电缆 中的微波 ）或光纤 中脉冲的特性；而物理能量上的散失是指动能被吸收的现象。
在光学中，色散的主要現象是不同顏色的光在透過三棱镜 或有色差 的透鏡时因折射角 不同，而产生光谱。[ 3] 消色差透镜 的设计在极大消除了色差，并通过阿贝数
V
{\displaystyle V}
可见光谱 色散。在电信应用中，波包 或“脉冲”的传输往往比波的绝对相位更重要，此时就需要考虑并计算波包的群速度色散
所有常见的传输介质 的衰减 （归一化为传输长度）也随频率而变化，从而导致衰减失真 阻抗边界 （例如电缆中的压接段）处的反射会产生信号失真，并进一步加剧在信号带宽上观察到的不一致的传输时间。
彩虹 可能是最常见的色散现象。色散导致太阳光 在空间上分离成不同波长 （不同颜色 ）的部分。然而，色散在许多其他情况下也会产生影响：例如，群速度色散 导致脉冲 在光纤 中扩散，使长距离的信号衰减；此外，群速度色散和非线性 效应之间的抵消会导致孤波 产生。
大部分情況下，色散研究的是块状材料的色散。然而，在波导管 中也存在着波导色散 （英語：waveguide dispersion ），在这种情况下，波在结构中的相速度 取决于其频率，这仅仅是由于结构的几何形状。更广泛地说，波导色散可以发生在通过任何不均匀结构（如光子晶体 ）传播的波中，无论这些波是否被限制在某些区域。[可疑 在波导管中，两种 类型的色散通常都会存在，尽管它们不是严格意义上的相加。[來源請求] 零色散波长 ，这有助于光纤通信 速度的提高。
不同玻璃，真空折射率与波长的关系。可见光范围以灰色区域表示。 在光学上，材料色散有优点也有缺点。透过三棱鏡，光的色散为制作光谱仪 以及分光辐射计 的基础。有時候也会透过全像 光柵，來达成更显著的分光效果。然而，在透镜中的色散效应造成影像品质低落，在显微镜、望远镜及其他成像技术上可见一斑。
在均匀介质中，波传递的相速度 为
v
=
c
n
{\displaystyle v={\frac {c}{n}}}
其中，c 為真空中的光速，而 n 為介質的折射率。
对于不同波长 的光，介质 的折射率 n (λ ) 也不同。這個關係式通常由阿贝数 可以計算出，或是由柯西等式 或Sellmeier等式 的係數求得。
由克拉莫-克若尼關係式 ，波長與實部折射率的關係與材料的吸收率有關，此吸收率由折射率的虛部（或稱消光係數 ）。在非磁性物質中，克拉莫-克若尼關係式的χ 為電極化率χ e = n 2 − 1.
对于可见光 ，一般的透明材料：
如果
λ λ -->
r
>
λ λ -->
y
>
λ λ -->
b
{\displaystyle \lambda _{\rm {r}}>\lambda _{\rm {y}}>\lambda _{\rm {b}}}
那麼
1
<
n
(
λ λ -->
r
)
<
n
(
λ λ -->
y
)
<
n
(
λ λ -->
b
)
{\displaystyle 1<n(\lambda _{\rm {r}})<n(\lambda _{\rm {y}})<n(\lambda _{\rm {b}})}
或可用以下表达式表示：
d
n
d
λ λ -->
<
0
{\displaystyle {\frac {{\mathrm {d} }n}{{\mathrm {d} }\lambda }}<0}
在此狀況下，此介質擁有正常頻散 。然而，當折射率隨著波長增加而增加時（通常在紫外光區發現[ 4] 反常頻散 。
法国 数学家 柯西 发现折射率和光波长的关系，可以用一个级数 表示：
n
(
λ λ -->
)
=
B
+
C
λ λ -->
2
+
D
λ λ -->
4
+
⋯ ⋯ -->
{\displaystyle n(\lambda )=B+{\frac {C}{\lambda ^{2}}}+{\frac {D}{\lambda ^{4}}}+\cdots }
其中，B、C、D 是三个柯西色散係数，由物质的种类决定。只需测定三个不同波长的光的折射率 n (λ )，代入柯西色散公式中，便可得到三个联立方程式。解这组联立方程式就可以得到这种物质的三个柯西色散系数。有了三个柯西色散系数，就可以计算出其他波长的光的折射率，而不需要再进行测量。
除了柯西色散公式之外，还有其他的色散公式，如：Hartmann色散公式、Conrady色散公式、Hetzberger色散公式等。
在一种假想介质（k=ω²）中传播的短时脉冲的时间演化。这体现了长波成分比短波成分传播要更快（正群速度色散），产生啁啾和脉冲变宽。 色散的效应远不止是使得相速度随着波长变化，更重要的是它产生一种叫做群速度色散 的效应。相速度 v 被定义为 v = c / n 群速度 。群速度描述了一个脉冲或者信号中的信息随着波动传播的速度。在旁边的动图中，我们可以发现波动本身（橙色）以相速度移动，这个速度要比波包（黑色）代表的群速度更快。举个例子，这个脉冲可能是一个通讯信号，其内的信息只能以群速度传播，尽管它由速度更快的波前组成。
从折射率曲线 n (ω )，我们可以算出群速度。或者用一种更直接的计算方式。首先我们计算波数 k = ωn/c ，其中，ω =2πf 是角频率。这样，相速度的公式是vp =ω/k ，而群速度的计算公式可以用导数 v g =dω/dk 表示。或者，群速度也可以用相速度 vp 表示：
v
g
=
v
p
1
− − -->
ω ω -->
v
p
d
v
p
d
ω ω -->
.
{\displaystyle {\rm {v_{g}}}={\frac {\rm {v_{p}}}{1-{\frac {\omega }{\rm {v_{p}}}}{\frac {\rm {dv_{p}}}{d\omega }}}}.}
当存在色散的时候，群速度不但不等于相速度，它还会随着波长变化。这种现象被称作群速度色散（Group Velocity Dispersion, GVD），也导致一个脉冲会变宽，这是因为脉冲里含有多个频率的成分，它们的速度不同。群速度色散可以用群速度的倒数 对角频率的导数 d2 k/dω2 来定量描述。
如果一个光脉冲在介质中的传播具有正群速度色散，那么短波成分的群速度就小于长波成分的群速度，这个脉冲就是正啁啾 的 （up-chirped），它的频率随着时间升高。 反之，如果一个光脉冲在介质中的传播具有负群速度色散，那么短波成分的群速度就大于长波成分的群速度，这个脉冲就是负啁啾 的 （down-chirped），它的频率随着时间降低。
群速度色散参数 ：
D
=
− − -->
λ λ -->
c
d
2
n
d
λ λ -->
2
.
{\displaystyle D=-{\frac {\lambda }{c}}\,{\frac {{\rm {d}}^{2}n}{{\rm {d}}\lambda ^{2}}}.}
经常被用来定量描述群速度色散。D 和群速度色散的比值是一个负的系数：
D
=
− − -->
2
π π -->
c
λ λ -->
2
d
2
k
d
ω ω -->
2
.
{\displaystyle D=-{\frac {2\pi c}{\lambda ^{2}}}\,{\frac {{\rm {d}}^{2}k}{{\rm {d}}\omega ^{2}}}.}
一些书的作者把折射率对波长的二阶导数 大于0（或小于0），也即D 小于0（或大于0），称为正常色散/反常色散。[ 5]
不论是负还是正的群速度色散，其最终结果皆为脉冲在时间上的扩展。这使得色散在管理在基于光纤的光学通讯系统中十分重要。因为如果色散过于强烈，对应于一组比特的一系列脉冲将在时域扩散开并相互混合，使得信号无法被解读。这限制了信号在光纤中传输的距离（如果没有进行信号重新生成）。
此问题的可能解法之一是在光纤中传输群速度色散为0的信号（例如，在硅纤维中 1.3–1.5 μm 的信号），此波长的信号在传输过程中的色散可以控制到最小。
然而，在实务上，这种做法引发的问题比其解决的问题要麻烦很多：群色散为0的信号放大了其他非线性效应（例如四波混频 ）。
另一种选项是在负色散区域使用孤子 脉冲，其特性是它利用非线性光学效应保持自身形状。然而，孤子的现实问题是它需要脉冲具有一定水平的功率以保证非线性光学的效应的强度正确。
目前，实际使用的方案是进行色散补偿，一般是将具有相反符号色散效应的光纤组合起来把色散效应抵消掉。这样的补偿受到非线性效应的限制，例如自相位调制
色散控制在超短脉冲 激光 中也十分重要。激光生成的总色散是评估激光脉冲的长度的重要因素。一对棱镜 可用于生成净负色散，从而用于抵消常用激光介质中的正色散。衍射光栅 亦可用于产生色散效应，并通常在高功率激光增幅系统中应用。
近年来，啁啾镜
通過泰勒係數以微擾方式描述色散對於需要平衡來自多個不同系統的色散的優化問題是有利的。 例如，在啁啾脈衝激光放大器中，脈衝首先由展寬器及時展寬，以避免光學損傷。 然後在放大過程中，脈沖不可避免地累積通過材料的線性和非線性相位。 最後，脈沖在各種類型的壓縮器中被壓縮。 為了在累積階段取消任何剩餘的更高訂單，通常會測量和平衡單個訂單。 然而，對於統一系統，通常不需要這種擾動描述（即在波導中傳播）。
色散階已以計算友好的方式推廣，以 Lah-Laguerre 類型變換的形式。[ 6] [ 7]
色散階數由相位或波矢量的泰勒展開式定義。
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}}}
波子
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)}
任何可微函數
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)}
變換的矩陣元素是 Lah 係數：
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)!} }}}
為 GDD 編寫的上述表達式表明，具有波長 GGD 的常數將具有零高階。 從 GDD 評估的更高階數是：
∂ ∂ -->
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}}}
將表示為折射率
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}}}}
變換的矩陣元素是負 2 階的無符號拉蓋爾係數，給出如下：
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)!} }}}
前十個色散階，明確地為波矢量編寫，是：
G
D
=
∂ ∂ -->
∂ ∂ -->
ω ω -->
k
(
ω ω -->
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1
c
(
n
(
ω ω -->
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+
ω ω -->
∂ ∂ -->
n
(
ω ω -->
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∂ ∂ -->
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=
1
c
(
n
(
λ λ -->
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− − -->
λ λ -->
∂ ∂ -->
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∂ ∂ -->
λ λ -->
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=
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}}}
群折射率
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 -->
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=
1
c
(
2
∂ ∂ -->
n
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ω ω -->
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∂ ∂ -->
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ω ω -->
∂ ∂ -->
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n
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ω ω -->
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∂ ∂ -->
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=
1
c
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λ λ -->
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π π -->
c
)
(
λ λ -->
2
∂ ∂ -->
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n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
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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
O
D
=
∂ ∂ -->
3
∂ ∂ -->
ω ω -->
3
k
(
ω ω -->
)
=
1
c
(
3
∂ ∂ -->
2
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
2
+
ω ω -->
∂ ∂ -->
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n
(
ω ω -->
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∂ ∂ -->
ω ω -->
3
)
=
− − -->
1
c
(
λ λ -->
2
π π -->
c
)
2
(
3
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
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∂ ∂ -->
λ λ -->
2
+
λ λ -->
3
∂ ∂ -->
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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
∂ ∂ -->
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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
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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
∂ ∂ -->
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n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
2
+
60
λ λ -->
3
∂ ∂ -->
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n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
3
+
15
λ λ -->
4
∂ ∂ -->
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n
(
λ λ -->
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∂ ∂ -->
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
e
O
D
=
∂ ∂ -->
7
∂ ∂ -->
ω ω -->
7
k
(
ω ω -->
)
=
1
c
(
7
∂ ∂ -->
6
n
(
ω ω -->
)
∂ ∂ -->
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6
+
ω ω -->
∂ ∂ -->
7
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
7
)
=
− − -->
1
c
(
λ λ -->
2
π π -->
c
)
6
(
2520
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
2
+
4200
λ λ -->
3
∂ ∂ -->
3
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
3
+
2100
λ λ -->
4
∂ ∂ -->
4
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
4
+
420
λ λ -->
5
∂ ∂ -->
5
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
5
+
35
λ λ -->
6
∂ ∂ -->
6
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
6
+
λ λ -->
7
∂ ∂ -->
7
n
(
λ λ -->
)
∂ ∂ -->
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 ω -->
8
k
(
ω ω -->
)
=
1
c
(
8
∂ ∂ -->
7
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
7
+
ω ω -->
∂ ∂ -->
8
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
8
)
=
1
c
(
λ λ -->
2
π π -->
c
)
7
(
20160
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
2
+
40320
λ λ -->
3
∂ ∂ -->
3
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
3
+
25200
λ λ -->
4
∂ ∂ -->
4
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
4
+
6720
λ λ -->
5
∂ ∂ -->
5
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
5
+
840
λ λ -->
6
∂ ∂ -->
6
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
6
+
+
48
λ λ -->
7
∂ ∂ -->
7
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
7
+
λ λ -->
8
∂ ∂ -->
8
n
(
λ λ -->
)
∂ ∂ -->
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 ω -->
9
k
(
ω ω -->
)
=
1
c
(
9
∂ ∂ -->
8
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
8
+
ω ω -->
∂ ∂ -->
9
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
9
)
=
− − -->
1
c
(
λ λ -->
2
π π -->
c
)
8
(
181440
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
2
+
423360
λ λ -->
3
∂ ∂ -->
3
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
3
+
317520
λ λ -->
4
∂ ∂ -->
4
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
4
+
105840
λ λ -->
5
∂ ∂ -->
5
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
5
+
17640
λ λ -->
6
∂ ∂ -->
6
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
6
+
+
1512
λ λ -->
7
∂ ∂ -->
7
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
7
+
63
λ λ -->
8
∂ ∂ -->
8
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
8
+
λ λ -->
9
∂ ∂ -->
9
n
(
λ λ -->
)
∂ ∂ -->
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 ω -->
10
k
(
ω ω -->
)
=
1
c
(
10
∂ ∂ -->
9
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
9
+
ω ω -->
∂ ∂ -->
10
n
(
ω ω -->
)
∂ ∂ -->
ω ω -->
10
)
=
1
c
(
λ λ -->
2
π π -->
c
)
9
(
1814400
λ λ -->
2
∂ ∂ -->
2
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
2
+
4838400
λ λ -->
3
∂ ∂ -->
3
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
3
+
4233600
λ λ -->
4
∂ ∂ -->
4
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
4
+
1693440
λ λ -->
5
∂ ∂ -->
5
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
5
+
+
352800
λ λ -->
6
∂ ∂ -->
6
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
6
+
40320
λ λ -->
7
∂ ∂ -->
7
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
7
+
2520
λ λ -->
8
∂ ∂ -->
8
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
8
+
80
λ λ -->
9
∂ ∂ -->
9
n
(
λ λ -->
)
∂ ∂ -->
λ λ -->
9
+
λ λ -->
10
∂ ∂ -->
10
n
(
λ λ -->
)
∂ ∂ -->
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}}}
明確地，為相位
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{\displaystyle \varphi }
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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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{\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}}}
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2
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{\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}}}
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2
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=
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2
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2
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)
{\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}}}
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6
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)
{\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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=
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24
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36
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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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5
(
120
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240
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120
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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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720
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300
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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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(
5040
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15120
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12600
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4200
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42
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+
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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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141120
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141120
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+
58800
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11760
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1176
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+
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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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(
362880
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1451520
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+
1693440
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846720
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+
211680
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+
28224
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+
+
2016
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+
72
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+
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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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φ φ -->
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)
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=
(
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)
10
(
3628800
λ λ -->
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+
16329600
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2
+
21772800
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∂ ∂ -->
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3
+
12700800
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4
∂ ∂ -->
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(
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)
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4
+
3810240
λ λ -->
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∂ ∂ -->
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φ φ -->
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λ λ -->
)
∂ ∂ -->
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+
635040
λ λ -->
6
∂ ∂ -->
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φ φ -->
(
λ λ -->
)
∂ ∂ -->
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6
+
+
60480
λ λ -->
7
∂ ∂ -->
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φ φ -->
(
λ λ -->
)
∂ ∂ -->
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7
+
3240
λ λ -->
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∂ ∂ -->
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φ φ -->
(
λ λ -->
)
∂ ∂ -->
λ λ -->
8
+
90
λ λ -->
9
∂ ∂ -->
9
φ φ -->
(
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)
∂ ∂ -->
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9
+
λ λ -->
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∂ ∂ -->
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φ φ -->
(
λ λ -->
)
∂ ∂ -->
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}}}
在日光下使用一桶水和一片鏡子就可以观察光的色散現象了。为了便于观察现象，实验中光路需要较大的出射角 来增大色散角度。此演示实验中镜子起到调整日光出射水面角度的作用。
^ 辞海网络版 - 色散 . www.cihai.com.cn. [2024-02-29 ] . （原始内容存档 于2024-02-29）. ^ 1882-1970., Born, Max,. Principles of optics : electromagnetic theory of propagation, interference and diffraction of light. 7th expanded. Cambridge: Cambridge University Press https://web.archive.org/web/20080620012317/http://www.worldcat.org/oclc/40200160 . 1999 [2019-01-28 ] . ISBN 0521642221OCLC 40200160 原始内容 存档于2008-06-20）. ^ Dispersion Compensation （页面存档备份 ，存于互联网档案馆 ） Retrieved 25-08-2015.^ Born, M. and Wolf, E. (1980) "Principles of Optics, 6th ed." pg. 93. Pergamon Press.
^ Saleh, B.E.A. and Teich, M.C. Fundamentals of Photonics (2nd Edition) Wiley, 2007.
^ Popmintchev, Dimitar; Wang, Siyang; Xiaoshi, Zhang; Stoev, Ventzislav; Popmintchev, Tenio. Analytical Lah-Laguerre optical formalism for perturbative chromatic dispersion. Optics Express . 2022-10-24, 30 (22): 40779–40808. Bibcode:2022OExpr..3040779P PMID 36299007 doi:10.1364/OE.457139 （英语） . ^ Popmintchev, Dimitar; Wang, Siyang; Xiaoshi, Zhang; Stoev, Ventzislav; Popmintchev, Tenio. Theory of the Chromatic Dispersion, Revisited. 2020-08-30. arXiv:2011.00066 physics.optics ] （英语） .
