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] (英語:Dispersion )是光波 的相速度 隨着頻率而改變的現象。[ 2] 我們将擁有这种特性的介质称为色散 ( sàn ) 介质 (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] 复合消色差透镜 的设计在极大消除了色差,并通过阿贝数 V {\displaystyle V} 量化玻璃的色散程度,低阿贝数即对应较大的可见光谱 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 ,即頻率与波包群速度的关系。
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 ;这不是色散,尽管有时在紧密间隔的阻抗边界 (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] 在光纤中,材料和波导色散可以有效地相互抵消以产生零色散波长 ,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 r > λ y > λ b {\displaystyle \lambda _{\rm {r}}>\lambda _{\rm {y}}>\lambda _{\rm {b}}} , 那麼
1 < n ( λ r ) < n ( λ y ) < n ( 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 < 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] ),則介質被稱為擁有反常頻散 。
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 ) = B + C λ 2 + D 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 ,然而这仅仅定义了一种频率的速度。当含有不同频率成分的波叠加在一起,比如一个信号或者脉冲,我们更关心群速度 。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 v p d v p d 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 c d 2 n d 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 c λ 2 d 2 k d 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] 这个定义和群速度色散有关,不可以和前一节相混淆。一般来说这两者没有必然联系,读者必须从上下文推断含义。
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 会和色散相互作用,从而导致色散难以消除。
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 作为棱镜和光栅的替代得到发展。这种介电反射镜具有镀层,不同波长能透过的长度不同,因此具有不同的群延迟。这些镀层可以设计为形成净负色散。
高階色散的廣義公式 – Lah-Laguerre 光學[ 编辑 ] 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 ( ω ) = φ | ω 0 + ∂ φ ∂ ω | ω 0 ( ω − ω 0 ) + 1 2 ∂ 2 φ ∂ ω 2 | ω 0 ( ω − ω 0 ) 2 + … + 1 p ! ∂ p φ ∂ ω p | ω 0 ( ω − ω 0 ) p + … {\displaystyle {\begin{array}{c}\varphi \mathrm {(} \omega \mathrm {)} =\varphi \left.\ \right|_{\omega _{0}}+\left.\ {\frac {\partial \varphi }{\partial \omega }}\right|_{\omega _{0}}\left(\omega -\omega _{0}\right)+{\frac {1}{2}}\left.\ {\frac {\partial ^{2}\varphi }{\partial \omega ^{2}}}\right|_{\omega _{0}}\left(\omega -\omega _{0}\right)^{2}\ +\ldots +{\frac {1}{p!}}\left.\ {\frac {\partial ^{p}\varphi }{\partial \omega ^{p}}}\right|_{\omega _{0}}\left(\omega -\omega _{0}\right)^{p}+\ldots \end{array}}}
k ( ω ) = k | ω 0 + ∂ k ∂ ω | ω 0 ( ω − ω 0 ) + 1 2 ∂ 2 k ∂ ω 2 | ω 0 ( ω − ω 0 ) 2 + … + 1 p ! ∂ p k ∂ ω p | ω 0 ( ω − ω 0 ) p + … {\displaystyle {\begin{array}{c}k\mathrm {(} \omega \mathrm {)} =k\left.\ \right|_{\omega _{0}}+\left.\ {\frac {\partial k}{\partial \omega }}\right|_{\omega _{0}}\left(\omega -\omega _{0}\right)+{\frac {1}{2}}\left.\ {\frac {\partial ^{2}k}{\partial \omega ^{2}}}\right|_{\omega _{0}}\left(\omega -\omega _{0}\right)^{2}\ +\ldots +{\frac {1}{p!}}\left.\ {\frac {\partial ^{p}k}{\partial \omega ^{p}}}\right|_{\omega _{0}}\left(\omega -\omega _{0}\right)^{p}+\ldots \end{array}}}
波子 k ( ω ) = ω c n ( ω ) {\displaystyle k\mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}}n\mathrm {(} \omega \mathrm {)} } 的色散關係和階段 φ ( ω ) = ω c O P ( ω ) {\displaystyle \varphi \mathrm {(} \omega \mathrm {)} ={\frac {\omega }{c}}{\it {OP}}\mathrm {(} \omega \mathrm {)} } 可以表示為:
∂ p ∂ ω p k ( ω ) = 1 c ( p ∂ p − 1 ∂ ω p − 1 n ( ω ) + ω ∂ p ∂ ω p n ( ω ) ) {\displaystyle {\begin{array}{c}{\frac {{\partial }^{p}}{\partial {\omega }^{p}}}k\mathrm {(} \omega \mathrm {)} ={\frac {1}{c}}\left(p{\frac {{\partial }^{p-1}}{\partial {\omega }^{p-1}}}n\mathrm {(} \omega \mathrm {)} +\omega {\frac {{\partial }^{p}}{\partial {\omega }^{p}}}n\mathrm {(} \omega \mathrm {)} \right)\ \end{array}}} , ∂ p ∂ ω p φ ( ω ) = 1 c ( p ∂ p − 1 ∂ ω p − 1 O P ( ω ) + ω ∂ p ∂ ω p O P ( ω ) ) ( 1 ) {\displaystyle {\begin{array}{c}{\frac {{\partial }^{p}}{\partial {\omega }^{p}}}\varphi \mathrm {(} \omega \mathrm {)} ={\frac {1}{c}}\left(p{\frac {{\partial }^{p-1}}{\partial {\omega }^{p-1}}}{\it {OP}}\mathrm {(} \omega \mathrm {)} +\omega {\frac {{\partial }^{p}}{\partial {\omega }^{p}}}{\it {OP}}\mathrm {(} \omega \mathrm {)} \right)\end{array}}(1)}
任何可微函數 f ( ω | λ ) {\displaystyle f\mathrm {(} \omega \mathrm {|} \lambda \mathrm {)} } 在波長或頻率空間的導數通過 Lah 變換指定為:
∂ p ∂ ω p f ( ω ) = ( − 1 ) p ( λ 2 π c ) p ∑ m = 0 p A ( p , m ) λ m ∂ m ∂ λ m f ( λ ) {\displaystyle {\begin{array}{l}{\frac {\partial ^{p}}{\partial {\omega }^{p}}}f\mathrm {(} \omega \mathrm {)} ={}{\left(\mathrm {-} \mathrm {1} \right)}^{p}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{p}\sum \limits _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\lambda }^{m}{\frac {{\partial }^{m}}{\partial {\lambda }^{m}}}f\mathrm {(} \lambda \mathrm {)} }\end{array}}} , {\displaystyle ,} ∂ p ∂ λ p f ( λ ) = ( − 1 ) p ( ω 2 π c ) p ∑ m = 0 p A ( p , m ) ω m ∂ m ∂ ω m f ( ω ) ( 2 ) {\displaystyle {\begin{array}{c}{\frac {{\partial }^{p}}{\partial {\lambda }^{p}}}f\mathrm {(} \lambda \mathrm {)} ={}{\left(\mathrm {-} \mathrm {1} \right)}^{p}{\left({\frac {\omega }{\mathrm {2} \pi c}}\right)}^{p}\sum \limits _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\omega }^{m}{\frac {{\partial }^{m}}{\partial {\omega }^{m}}}f\mathrm {(} \omega \mathrm {)} }\end{array}}(2)}
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 p G D D ( ω ) = ( − 1 ) p ( λ 2 π c ) p ∑ m = 0 p A ( p , m ) λ m ∂ m ∂ λ m G D D ( λ ) {\displaystyle {\begin{array}{c}{\frac {{\partial }^{p}}{\partial {\omega }^{p}}}GDD\mathrm {(} \omega \mathrm {)} ={}{\left(\mathrm {-} \mathrm {1} \right)}^{p}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{p}\sum \limits _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\lambda }^{m}{\frac {{\partial }^{m}}{\partial {\lambda }^{m}}}GDD\mathrm {(} \lambda \mathrm {)} }\end{array}}}
將表示為折射率 n {\displaystyle n} 或光路 O P {\displaystyle OP} 的等式(2)代入等式(1),得到色散階的封閉式表達式。 一般來說, p t h {\displaystyle p^{th}} 階色散 POD 是負二階的拉蓋爾型變換:
P O D = d p φ ( ω ) d ω p = ( − 1 ) p ( λ 2 π c ) ( p − 1 ) ∑ m = 0 p B ( p , m ) ( λ ) m d m O P ( λ ) d λ m {\displaystyle POD={\frac {d^{p}\varphi (\omega )}{d\omega ^{p}}}=(-1)^{p}({\frac {\lambda }{2\pi c}})^{(p-1)}\sum _{m=0}^{p}{\mathcal {B(p,m)}}(\lambda )^{m}{\frac {d^{m}OP(\lambda )}{d\lambda ^{m}}}} , {\displaystyle ,} P O D = d p k ( ω ) d ω p = ( − 1 ) p ( λ 2 π c ) ( p − 1 ) ∑ m = 0 p B ( p , m ) ( λ ) m d m n ( λ ) d λ m {\displaystyle POD={\frac {d^{p}k(\omega )}{d\omega ^{p}}}=(-1)^{p}({\frac {\lambda }{2\pi c}})^{(p-1)}\sum _{m=0}^{p}{\mathcal {B(p,m)}}(\lambda )^{m}{\frac {d^{m}n(\lambda )}{d\lambda ^{m}}}}
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 k ( ω ) = 1 c ( n ( ω ) + ω ∂ n ( ω ) ∂ ω ) = 1 c ( n ( λ ) − λ ∂ n ( λ ) ∂ λ ) = v g r − 1 {\displaystyle {\begin{array}{l}{\boldsymbol {\it {GD}}}={\frac {\partial }{\partial \omega }}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(n\mathrm {(} \omega \mathrm {)} +\omega {\frac {\partial n\mathrm {(} \omega \mathrm {)} }{\partial \omega }}\right)={\frac {\mathrm {1} }{c}}\left(n\mathrm {(} \lambda \mathrm {)} -\lambda {\frac {\partial n\mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}\right)=v_{gr}^{\mathrm {-} \mathrm {1} }\end{array}}}
群折射率 n g {\displaystyle n_{g}} 定義為: n g = c v g r − 1 {\displaystyle n_{g}=cv_{gr}^{\mathrm {-} \mathrm {1} }} .
G D D = ∂ 2 ∂ ω 2 k ( ω ) = 1 c ( 2 ∂ n ( ω ) ∂ ω + ω ∂ 2 n ( ω ) ∂ ω 2 ) = 1 c ( λ 2 π c ) ( λ 2 ∂ 2 n ( λ ) ∂ λ 2 ) {\displaystyle {\begin{array}{l}{\boldsymbol {\it {GDD}}}={\frac {{\partial }^{2}}{\partial {\omega }^{\mathrm {2} }}}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(\mathrm {2} {\frac {\partial n\mathrm {(} \omega \mathrm {)} }{\partial \omega }}+\omega {\frac {{\partial }^{2}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {2} }}}\right)={\frac {\mathrm {1} }{c}}\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)\left({\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}\right)\end{array}}}
T O D = ∂ 3 ∂ ω 3 k ( ω ) = 1 c ( 3 ∂ 2 n ( ω ) ∂ ω 2 + ω ∂ 3 n ( ω ) ∂ ω 3 ) = − 1 c ( λ 2 π c ) 2 ( 3 λ 2 ∂ 2 n ( λ ) ∂ λ 2 + λ 3 ∂ 3 n ( λ ) ∂ λ 3 ) {\displaystyle {\begin{array}{l}{\boldsymbol {\it {TOD}}}={\frac {{\partial }^{3}}{\partial {\omega }^{\mathrm {3} }}}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(\mathrm {3} {\frac {{\partial }^{2}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {2} }}}+\omega {\frac {{\partial }^{3}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {3} }}}\right)={-}{\frac {\mathrm {1} }{c}}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {2} }{\Bigl (}\mathrm {3} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+{\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}{\Bigr )}\end{array}}}
F O D = ∂ 4 ∂ ω 4 k ( ω ) = 1 c ( 4 ∂ 3 n ( ω ) ∂ ω 3 + ω ∂ 4 n ( ω ) ∂ ω 4 ) = 1 c ( λ 2 π c ) 3 ( 12 λ 2 ∂ 2 n ( λ ) ∂ λ 2 + 8 λ 3 ∂ 3 n ( λ ) ∂ λ 3 + λ 4 ∂ 4 n ( λ ) ∂ λ 4 ) {\displaystyle {\begin{array}{l}{\boldsymbol {\it {FOD}}}={\frac {{\partial }^{4}}{\partial {\omega }^{\mathrm {4} }}}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(\mathrm {4} {\frac {{\partial }^{3}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {3} }}}+\omega {\frac {{\partial }^{4}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {4} }}}\right)={\frac {\mathrm {1} }{c}}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {3} }{\Bigl (}\mathrm {12} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {8} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+{\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}{\Bigr )}\end{array}}}
F i O D = ∂ 5 ∂ ω 5 k ( ω ) = 1 c ( 5 ∂ 4 n ( ω ) ∂ ω 4 + ω ∂ 5 n ( ω ) ∂ ω 5 ) = − 1 c ( λ 2 π c ) 4 ( 60 λ 2 ∂ 2 n ( λ ) ∂ λ 2 + 60 λ 3 ∂ 3 n ( λ ) ∂ λ 3 + 15 λ 4 ∂ 4 n ( λ ) ∂ λ 4 + λ 5 ∂ 5 n ( λ ) ∂ λ 5 ) {\displaystyle {\begin{array}{l}{\boldsymbol {\it {FiOD}}}={\frac {{\partial }^{5}}{\partial {\omega }^{\mathrm {5} }}}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(\mathrm {5} {\frac {{\partial }^{4}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {4} }}}+\omega {\frac {{\partial }^{5}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {5} }}}\right)={-}{\frac {\mathrm {1} }{c}}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {4} }{\Bigl (}\mathrm {60} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {60} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {15} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+{\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}{\Bigr )}\end{array}}}
S i O D = ∂ 6 ∂ ω 6 k ( ω ) = 1 c ( 6 ∂ 5 n ( ω ) ∂ ω 5 + ω ∂ 6 n ( ω ) ∂ ω 6 ) = 1 c ( λ 2 π c ) 5 ( 360 λ 2 ∂ 2 n ( λ ) ∂ λ 2 + 480 λ 3 ∂ 3 n ( λ ) ∂ λ 3 + 180 λ 4 ∂ 4 n ( λ ) ∂ λ 4 + 24 λ 5 ∂ 5 n ( λ ) ∂ λ 5 + λ 6 ∂ 6 n ( λ ) ∂ λ 6 ) {\displaystyle {\begin{array}{l}{\boldsymbol {\it {SiOD}}}={\frac {{\partial }^{6}}{\partial {\omega }^{\mathrm {6} }}}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(\mathrm {6} {\frac {{\partial }^{5}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {5} }}}+\omega {\frac {{\partial }^{6}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {6} }}}\right)={\frac {\mathrm {1} }{c}}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {5} }{\Bigl (}\mathrm {360} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {480} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {180} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+\mathrm {24} {\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}+{\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}{\Bigr )}\end{array}}}
S e O D = ∂ 7 ∂ ω 7 k ( ω ) = 1 c ( 7 ∂ 6 n ( ω ) ∂ ω 6 + ω ∂ 7 n ( ω ) ∂ ω 7 ) = − 1 c ( λ 2 π c ) 6 ( 2520 λ 2 ∂ 2 n ( λ ) ∂ λ 2 + 4200 λ 3 ∂ 3 n ( λ ) ∂ λ 3 + 2100 λ 4 ∂ 4 n ( λ ) ∂ λ 4 + 420 λ 5 ∂ 5 n ( λ ) ∂ λ 5 + 35 λ 6 ∂ 6 n ( λ ) ∂ λ 6 + λ 7 ∂ 7 n ( λ ) ∂ λ 7 ) {\displaystyle {\begin{array}{l}{\boldsymbol {\it {SeOD}}}={\frac {{\partial }^{7}}{\partial {\omega }^{\mathrm {7} }}}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(\mathrm {7} {\frac {{\partial }^{6}n\mathrm {(} \omega \mathrm {)} }{{\partial \omega }^{\mathrm {6} }}}+\omega {\frac {{\partial }^{7}n\mathrm {(} \omega \mathrm {)} }{{\partial \omega }^{\mathrm {7} }}}\right)={-}{\frac {\mathrm {1} }{c}}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {6} }{\Bigl (}\mathrm {2520} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {4200} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {2100} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+\mathrm {420} {\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}+\mathrm {35} {\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}+{\lambda }^{\mathrm {7} }{\frac {{\partial }^{7}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {7} }}}{\Bigr )}\end{array}}}
E O D = ∂ 8 ∂ ω 8 k ( ω ) = 1 c ( 8 ∂ 7 n ( ω ) ∂ ω 7 + ω ∂ 8 n ( ω ) ∂ ω 8 ) = 1 c ( λ 2 π c ) 7 ( 20160 λ 2 ∂ 2 n ( λ ) ∂ λ 2 + 40320 λ 3 ∂ 3 n ( λ ) ∂ λ 3 + 25200 λ 4 ∂ 4 n ( λ ) ∂ λ 4 + 6720 λ 5 ∂ 5 n ( λ ) ∂ λ 5 + 840 λ 6 ∂ 6 n ( λ ) ∂ λ 6 + + 48 λ 7 ∂ 7 n ( λ ) ∂ λ 7 + λ 8 ∂ 8 n ( λ ) ∂ λ 8 ) {\displaystyle {\begin{array}{l}{\boldsymbol {\it {EOD}}}={\frac {{\partial }^{8}}{\partial {\omega }^{\mathrm {8} }}}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(\mathrm {8} {\frac {{\partial }^{7}n\mathrm {(} \omega \mathrm {)} }{{\partial \omega }^{\mathrm {7} }}}+\omega {\frac {{\partial }^{8}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {8} }}}\right)={\frac {\mathrm {1} }{c}}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {7} }{\Bigl (}\mathrm {20160} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {40320} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {25200} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+\mathrm {6720} {\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}+\mathrm {840} {\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}+\\+\mathrm {48} {\lambda }^{\mathrm {7} }{\frac {{\partial }^{7}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {7} }}}+{\lambda }^{\mathrm {8} }{\frac {{\partial }^{8}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {8} }}}{\Bigr )}\end{array}}}
N O D = ∂ 9 ∂ ω 9 k ( ω ) = 1 c ( 9 ∂ 8 n ( ω ) ∂ ω 8 + ω ∂ 9 n ( ω ) ∂ ω 9 ) = − 1 c ( λ 2 π c ) 8 ( 181440 λ 2 ∂ 2 n ( λ ) ∂ λ 2 + 423360 λ 3 ∂ 3 n ( λ ) ∂ λ 3 + 317520 λ 4 ∂ 4 n ( λ ) ∂ λ 4 + 105840 λ 5 ∂ 5 n ( λ ) ∂ λ 5 + 17640 λ 6 ∂ 6 n ( λ ) ∂ λ 6 + + 1512 λ 7 ∂ 7 n ( λ ) ∂ λ 7 + 63 λ 8 ∂ 8 n ( λ ) ∂ λ 8 + λ 9 ∂ 9 n ( λ ) ∂ λ 9 ) {\displaystyle {\begin{array}{l}{\boldsymbol {\it {NOD}}}={\frac {{\partial }^{9}}{\partial {\omega }^{\mathrm {9} }}}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(\mathrm {9} {\frac {{\partial }^{8}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {8} }}}+\omega {\frac {{\partial }^{9}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {9} }}}\right)={-}{\frac {\mathrm {1} }{c}}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {8} }{\Bigl (}\mathrm {181440} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {423360} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {317520} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+\mathrm {105840} {\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}+\mathrm {17640} {\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}+\\+\mathrm {1512} {\lambda }^{\mathrm {7} }{\frac {{\partial }^{7}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {7} }}}+\mathrm {63} {\lambda }^{\mathrm {8} }{\frac {{\partial }^{8}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {8} }}}+{\lambda }^{\mathrm {9} }{\frac {{\partial }^{9}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {9} }}}{\Bigr )}\end{array}}}
T e O D = ∂ 10 ∂ ω 10 k ( ω ) = 1 c ( 10 ∂ 9 n ( ω ) ∂ ω 9 + ω ∂ 10 n ( ω ) ∂ ω 10 ) = 1 c ( λ 2 π c ) 9 ( 1814400 λ 2 ∂ 2 n ( λ ) ∂ λ 2 + 4838400 λ 3 ∂ 3 n ( λ ) ∂ λ 3 + 4233600 λ 4 ∂ 4 n ( λ ) ∂ λ 4 + 1693440 λ 5 ∂ 5 n ( λ ) ∂ λ 5 + + 352800 λ 6 ∂ 6 n ( λ ) ∂ λ 6 + 40320 λ 7 ∂ 7 n ( λ ) ∂ λ 7 + 2520 λ 8 ∂ 8 n ( λ ) ∂ λ 8 + 80 λ 9 ∂ 9 n ( λ ) ∂ λ 9 + λ 10 ∂ 10 n ( λ ) ∂ λ 10 ) {\displaystyle {\begin{array}{l}{\boldsymbol {\it {TeOD}}}={\frac {{\partial }^{10}}{\partial {\omega }^{\mathrm {10} }}}k\mathrm {(} \omega \mathrm {)} ={\frac {\mathrm {1} }{c}}\left(\mathrm {10} {\frac {{\partial }^{9}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {9} }}}+\omega {\frac {{\partial }^{10}n\mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {10} }}}\right)={\frac {\mathrm {1} }{c}}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {9} }{\Bigl (}\mathrm {1814400} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {4838400} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {4233600} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+{1693440}{\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}+\\+\mathrm {352800} {\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}+\mathrm {40320} {\lambda }^{\mathrm {7} }{\frac {{\partial }^{7}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {7} }}}+\mathrm {2520} {\lambda }^{\mathrm {8} }{\frac {{\partial }^{8}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {8} }}}+\mathrm {80} {\lambda }^{\mathrm {9} }{\frac {{\partial }^{9}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {9} }}}+{\lambda }^{\mathrm {10} }{\frac {{\partial }^{10}n\mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {10} }}}{\Bigr )}\end{array}}}
明確地,為相位 φ {\displaystyle \varphi } 編寫,前十個色散階可以使用 Lah 變換(等式(2))表示為波長的函數:
∂ p ∂ ω p f ( ω ) = ( − 1 ) p ( λ 2 π c ) p ∑ m = 0 p A ( p , m ) λ m ∂ m ∂ λ m f ( λ ) {\displaystyle {\begin{array}{l}{\frac {\partial ^{p}}{\partial {\omega }^{p}}}f\mathrm {(} \omega \mathrm {)} ={}{\left(\mathrm {-} \mathrm {1} \right)}^{p}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{p}\sum \limits _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\lambda }^{m}{\frac {{\partial }^{m}}{\partial {\lambda }^{m}}}f\mathrm {(} \lambda \mathrm {)} }\end{array}}} , {\displaystyle ,} ∂ p ∂ λ p f ( λ ) = ( − 1 ) p ( ω 2 π c ) p ∑ m = 0 p A ( p , m ) ω m ∂ m ∂ ω m f ( ω ) {\displaystyle {\begin{array}{c}{\frac {{\partial }^{p}}{\partial {\lambda }^{p}}}f\mathrm {(} \lambda \mathrm {)} ={}{\left(\mathrm {-} \mathrm {1} \right)}^{p}{\left({\frac {\omega }{\mathrm {2} \pi c}}\right)}^{p}\sum \limits _{m={0}}^{p}{{\mathcal {A}}\mathrm {(} p,m\mathrm {)} {\omega }^{m}{\frac {{\partial }^{m}}{\partial {\omega }^{m}}}f\mathrm {(} \omega \mathrm {)} }\end{array}}}
∂ φ ( ω ) ∂ ω = − ( 2 π c ω 2 ) ∂ φ ( ω ) ∂ λ = − ( λ 2 2 π c ) ∂ φ ( λ ) ∂ λ {\displaystyle {\begin{array}{l}{\frac {\partial \varphi \mathrm {(} \omega \mathrm {)} }{\partial \omega }}={-}\left({\frac {\mathrm {2} \pi c}{{\omega }^{\mathrm {2} }}}\right){\frac {\partial \varphi \mathrm {(} \omega \mathrm {)} }{\partial \lambda }}={-}\left({\frac {{\lambda }^{\mathrm {2} }}{\mathrm {2} \pi c}}\right){\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}\end{array}}}
∂ 2 φ ( ω ) ∂ ω 2 = ∂ ∂ ω ( ∂ φ ( ω ) ∂ ω ) = ( λ 2 π c ) 2 ( 2 λ ∂ φ ( λ ) ∂ λ + λ 2 ∂ 2 φ ( λ ) ∂ λ 2 ) {\displaystyle {\begin{array}{l}{\frac {{\partial }^{2}\varphi \mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {2} }}}={\frac {\partial }{\partial \omega }}\left({\frac {\partial \varphi \mathrm {(} \omega \mathrm {)} }{\partial \omega }}\right)={\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {2} }\left(\mathrm {2} \lambda {\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}+{\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}\right)\end{array}}}
∂ 3 φ ( ω ) ∂ ω 3 = − ( λ 2 π c ) 3 ( 6 λ ∂ φ ( λ ) ∂ λ + 6 λ 2 ∂ 2 φ ( λ ) ∂ λ 2 + λ 3 ∂ 3 φ ( λ ) ∂ λ 3 ) {\displaystyle {\begin{array}{l}{\frac {{\partial }^{3}\varphi \mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {3} }}}={-}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {3} }\left(\mathrm {6} \lambda {\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}+\mathrm {6} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+{\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}\right)\end{array}}}
∂ 4 φ ( ω ) ∂ ω 4 = ( λ 2 π c ) 4 ( 24 λ ∂ φ ( λ ) ∂ λ + 36 λ 2 ∂ 2 φ ( λ ) ∂ λ 2 + 12 λ 3 ∂ 3 φ ( λ ) ∂ λ 3 + λ 4 ∂ 4 φ ( λ ) ∂ λ 4 ) {\displaystyle {\begin{array}{l}{\frac {{\partial }^{4}\varphi \mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {4} }}}={\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {4} }{\Bigl (}\mathrm {24} \lambda {\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}+\mathrm {36} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {12} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+{\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}{\Bigr )}\end{array}}}
∂ 5 φ ( ω ) ∂ ω 5 = − ( λ 2 π c ) 5 ( 120 λ ∂ φ ( λ ) ∂ λ + 240 λ 2 ∂ 2 φ ( λ ) ∂ λ 2 + 120 λ 3 ∂ 3 φ ( λ ) ∂ λ 3 + 20 λ 4 ∂ 4 φ ( λ ) ∂ λ 4 + λ 5 ∂ 5 φ ( λ ) ∂ λ 5 ) {\displaystyle {\begin{array}{l}{\frac {{\partial }^{\mathrm {5} }\varphi \mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {5} }}}={-}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {5} }{\Bigl (}\mathrm {120} \lambda {\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}+\mathrm {240} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {120} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {20} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+{\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}{\Bigr )}\end{array}}}
∂ 6 φ ( ω ) ∂ ω 6 = ( λ 2 π c ) 6 ( 720 λ ∂ φ ( λ ) ∂ λ + 1800 λ 2 ∂ 2 φ ( λ ) ∂ λ 2 + 1200 λ 3 ∂ 3 φ ( λ ) ∂ λ 3 + 300 λ 4 ∂ 4 φ ( λ ) ∂ λ 4 + 30 λ 5 ∂ 5 φ ( λ ) ∂ λ 5 + λ 6 ∂ 6 φ ( λ ) ∂ λ 6 ) {\displaystyle {\begin{array}{l}{\frac {{\partial }^{6}\varphi \mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {6} }}}={\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {6} }{\Bigl (}\mathrm {720} \lambda {\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}+\mathrm {1800} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {1200} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {300} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+\mathrm {30} {\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}\mathrm {\ +} {\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}{\Bigr )}\end{array}}}
∂ 7 φ ( ω ) ∂ ω 7 = − ( λ 2 π c ) 7 ( 5040 λ ∂ φ ( λ ) ∂ λ + 15120 λ 2 ∂ 2 φ ( λ ) ∂ λ 2 + 12600 λ 3 ∂ 3 φ ( λ ) ∂ λ 3 + 4200 λ 4 ∂ 4 φ ( λ ) ∂ λ 4 + 630 λ 5 ∂ 5 φ ( λ ) ∂ λ 5 + 42 λ 6 ∂ 6 φ ( λ ) ∂ λ 6 + λ 7 ∂ 7 φ ( λ ) ∂ λ 7 ) {\displaystyle {\begin{array}{l}{\frac {{\partial }^{7}\varphi \mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {7} }}}={-}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {7} }{\Bigl (}\mathrm {5040} \lambda {\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}+\mathrm {15120} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {12600} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {4200} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+\mathrm {630} {\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}+\mathrm {42} {\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}+{\lambda }^{\mathrm {7} }{\frac {{\partial }^{7}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {7} }}}{\Bigr )}\end{array}}}
∂ 8 φ ( ω ) ∂ ω 8 = ( λ 2 π c ) 8 ( 40320 λ ∂ φ ( λ ) ∂ λ + 141120 λ 2 ∂ 2 φ ( λ ) ∂ λ 2 + 141120 λ 3 ∂ 3 φ ( λ ) ∂ λ 3 + 58800 λ 4 ∂ 4 φ ( λ ) ∂ λ 4 + 11760 λ 5 ∂ 5 φ ( λ ) ∂ λ 5 + 1176 λ 6 ∂ 6 φ ( λ ) ∂ λ 6 + 56 λ 7 ∂ 7 φ ( λ ) ∂ λ 7 + + λ 8 ∂ 8 φ ( λ ) ∂ λ 8 ) {\displaystyle {\begin{array}{l}{\frac {{\partial }^{8}\varphi \mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {8} }}}={\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {8} }{\Bigl (}\mathrm {40320} \lambda {\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}+\mathrm {141120} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {141120} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {58800} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+\mathrm {11760} {\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}+\mathrm {1176} {\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}+\mathrm {56} {\lambda }^{\mathrm {7} }{\frac {{\partial }^{7}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {7} }}}+\\+{\lambda }^{\mathrm {8} }{\frac {\partial ^{8}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {8} }}}{\Bigr )}\end{array}}} ∂ 9 φ ( ω ) ∂ ω 9 = − ( λ 2 π c ) 9 ( 362880 λ ∂ φ ( λ ) ∂ λ + 1451520 λ 2 ∂ 2 φ ( λ ) ∂ λ 2 + 1693440 λ 3 ∂ 3 φ ( λ ) ∂ λ 3 + 846720 λ 4 ∂ 4 φ ( λ ) ∂ λ 4 + 211680 λ 5 ∂ 5 φ ( λ ) ∂ λ 5 + 28224 λ 6 ∂ 6 φ ( λ ) ∂ λ 6 + + 2016 λ 7 ∂ 7 φ ( λ ) ∂ λ 7 + 72 λ 8 ∂ 8 φ ( λ ) ∂ λ 8 + λ 9 ∂ 9 φ ( λ ) ∂ λ 9 ) {\displaystyle {\begin{array}{l}{\frac {{\partial }^{9}\varphi \mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {9} }}}={-}{\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {9} }{\Bigl (}\mathrm {362880} \lambda {\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}+\mathrm {1451520} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {1693440} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {846720} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+\mathrm {211680} {\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}+\mathrm {28224} {\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}+\\+\mathrm {2016} {\lambda }^{\mathrm {7} }{\frac {{\partial }^{7}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {7} }}}+\mathrm {72} {\lambda }^{\mathrm {8} }{\frac {{\partial }^{8}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {8} }}}+{\lambda }^{\mathrm {9} }{\frac {\partial ^{\mathrm {9} }\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {9} }}}{\Bigr )}\end{array}}}
∂ 10 φ ( ω ) ∂ ω 10 = ( λ 2 π c ) 10 ( 3628800 λ ∂ φ ( λ ) ∂ λ + 16329600 λ 2 ∂ 2 φ ( λ ) ∂ λ 2 + 21772800 λ 3 ∂ 3 φ ( λ ) ∂ λ 3 + 12700800 λ 4 ∂ 4 φ ( λ ) ∂ λ 4 + 3810240 λ 5 ∂ 5 φ ( λ ) ∂ λ 5 + 635040 λ 6 ∂ 6 φ ( λ ) ∂ λ 6 + + 60480 λ 7 ∂ 7 φ ( λ ) ∂ λ 7 + 3240 λ 8 ∂ 8 φ ( λ ) ∂ λ 8 + 90 λ 9 ∂ 9 φ ( λ ) ∂ λ 9 + λ 10 ∂ 10 φ ( λ ) ∂ λ 10 ) {\displaystyle {\begin{array}{l}{\frac {{\partial }^{10}\varphi \mathrm {(} \omega \mathrm {)} }{\partial {\omega }^{\mathrm {10} }}}={\left({\frac {\lambda }{\mathrm {2} \pi c}}\right)}^{\mathrm {10} }{\Bigl (}\mathrm {3628800} \lambda {\frac {\partial \varphi \mathrm {(} \lambda \mathrm {)} }{\partial \lambda }}+\mathrm {16329600} {\lambda }^{\mathrm {2} }{\frac {{\partial }^{2}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {2} }}}+\mathrm {21772800} {\lambda }^{\mathrm {3} }{\frac {{\partial }^{3}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {3} }}}+\mathrm {12700800} {\lambda }^{\mathrm {4} }{\frac {{\partial }^{4}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {4} }}}+\mathrm {3810240} {\lambda }^{\mathrm {5} }{\frac {{\partial }^{5}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {5} }}}+\mathrm {635040} {\lambda }^{\mathrm {6} }{\frac {{\partial }^{6}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {6} }}}+\\+\mathrm {60480} {\lambda }^{\mathrm {7} }{\frac {{\partial }^{7}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {7} }}}+\mathrm {3240} {\lambda }^{\mathrm {8} }{\frac {{\partial }^{8}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {8} }}}+\mathrm {90} {\lambda }^{\mathrm {9} }{\frac {{\partial }^{9}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {9} }}}+{\lambda }^{\mathrm {10} }{\frac {{\partial }^{10}\varphi \mathrm {(} \lambda \mathrm {)} }{\partial {\lambda }^{\mathrm {10} }}}{\Bigr )}\end{array}}}
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 0521642221 . OCLC 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) .