# 球面像差

## 球面像差公式

1. PA=0：物体和像与球面顶点重合；
2. I'=I：物体和物象在球面的曲率中心；
3. i=0；
4. I=U'或I'=U：在这种情形下的球面成为消球差曲面。

${\displaystyle L={\frac {r*(n+n')}{n}}}$

${\displaystyle L'={\frac {r*(n+n')}{n'}}}$

${\displaystyle BC=L-r=r*{\frac {n}{n'}}}$

${\displaystyle BC=L'-r=r*{\frac {n'}{n}}}$

${\displaystyle L=2.5*r}$

${\displaystyle L'=1.6667*r}$

LA'=trans+newsp

newsp= ${\displaystyle \sum _{k=1}^{k}({\frac {-2*PA*sin(-(1/2)*J'+(1/2)*J)*sin((1/2)*J'-(1/2)*U)*n*i}{(n'[k]*u'[k]*sin(U[k]))}}}$

## 球面像差展开式

LA'=${\displaystyle a*Y^{2}+b*Y^{4}+c*Y^{6}+}$ ………………[7][8]。其中Y是入射光线的在球面入射点到光轴的距离。

## 薄透镜组的球面像差

SC=${\displaystyle {\frac {y^{4}}{n_{0}'*u_{0}^{2}}}*\sum (G_{1}*c^{3}-G_{2}*c^{2}*c_{1}+G_{3}*c^{2}*v_{1}+G_{4}*c*c_{1}*v_{1}+G_{6}*c*v_{1}^{2})}$

${\displaystyle c={\frac {1}{f*(n-1)}}}$
${\displaystyle c={\frac {1}{r_{1}}}}$
${\displaystyle G_{1}={\frac {n^{2}*(n-1)}{2}}}$
${\displaystyle G_{2}={\frac {1}{2}}*(2*n+1)(n-1)}$
${\displaystyle G_{3}={\frac {1}{2}}*(3n+1)(n-1)}$
${\displaystyle G_{4}={\frac {1}{2*n}}*(n+2)(n-1)}$
${\displaystyle G_{5}{\frac {1}{2*n}}*(n^{2}-1)}$
${\displaystyle G_{6}={\frac {1}{2*n}}*(3*n+2)}$

## 薄透镜的球面像差

${\displaystyle {\frac {dLA'}{dc_{1}}}}$ =${\displaystyle -y^{2}*l'^{2}*(-G_{2}*c^{2}+2*G_{4}*c*c_{1}-G_{5}*c*v_{1})=0}$

${\displaystyle c_{1}={\frac {G_{2}c+G_{5}v_{1}}{2G_{4}}}}$ =${\displaystyle {\frac {0.5*n*(2*n+1)*c+2*(n+1)*v_{1}}{n+2}}}$ .

${\displaystyle {\frac {c_{2}}{c_{1}}}={\frac {r_{1}}{r_{2}}}={\frac {2n-n-4}{n*(2n+1)}}}$ [13]

n r_1/r_2
1.5 -6
1.518 -6.7374
1.6 -14
1.7 93.5
1.8 12.1765
2 5
3 1.9
4 1.5

## 参考文献

1. ^ Kingslake p104
2. ^ Rudolf Kingslake p104-105
3. Rudolf Kingslake p105
4. ^ Moritz von Rohr p244
5. ^ Rudolf Kingslake p106
6. ^ Rudolf Kingslake p104