# Maple/光的球面折射

AC=L1，AB=L2

${\displaystyle {\frac {sin(i1)}{OC}}={\frac {sin(U1)}{r}}}$

${\displaystyle {\frac {sin(i1)}{L1-r}}={\frac {sin(U1)}{r}}}$……………………(1)

${\displaystyle {\frac {sin(i2)}{OB}}={\frac {sin(U2)}{r}}}$

${\displaystyle {\frac {sin(i2)}{L2-r}}={\frac {sin(U2)}{r}}}$……………………(2)

sin(i1)*N1=sin(i2)*N2 …………………………………………(4)

## 球面折射的光路计算

Refraction of a thick lens

f1 := sin(i) = sin(U)*(L1-r)/r;

f2 := sin(i2) = N1*sin(i1)/N2

f3 := U1+i1 = U2+i2

f4 := L2-r = sin(i2)*r/sin(U2)

> sys := {f1, f2, f3, f4};

sys:={ L2 - r = sin(i2) r/sin(U2), U + i = U2 + i2, sin(i) = sin(U) (L - r)/r,sin(i2) =N1 sin(i)/N2 }

g:=proc(U,L,r,N1,N2) {L2 - r=sin(i2)*r/sin(U2),U+i=U2+i2,sin(i)=sin(U)*(L - r)/r,sin(i2)=N1*sin(i)/N2} end proc;

> d1 := evalf(convert(-degrees, radians));

-0.01745329252

> d2 := evalf(convert(-2*degrees, radians));

-0.03490658504

d3 := evalf(convert(-3*degrees, radians))

d3 := -0.5235987758e-1

> f5 := g(d1, -24, 10, 1, 1.5180);

f5:={ -0.01745329252 + i1 = U2 + i2, L2 - 10 = 10 sin(i2)/sin(U2),
sin(i) = 0.05933818190, sin(i2) = 0.6587615283 sin(i) }

> f6 := g(d2, -24, 10, 1, 1.5180);

f6:={-0.03490658504 + i1 = U2 + i2, L2 -10 =10 sin(i2)/sin(U2), sin(i) = 0.1186582888, sin(i2) = 0.6587615283 sin(i) }

f7 := {-0.5235987758e-1+i = U2+i2, L2-10 = 10*sin(i2)/sin(U2), sin(i) = .1779422513, sin(i2) = .6587615283*sin(i)}

> fsolve(f5);

{L2 = 148.6115927, U2 = 0.002820093360, i1 = 0.05937305901,i2 = 0.03909967313}

> fsolve(f6);

{L2 = 145.1313098, U2=0.0057845922, i1 = 0.1189385152,i2 = 0.07824733803}

> fsolve(f7);

{L2 = 139.6225687, U2 = 0.009043418012, i1 = 0.1788949332,i2 = 0.1174916376}

> f51 := g(0.2820093360e-2, 147.938, -5, 1.5180, 1);

： { 0.002820093360 + i = U2 + i2, L2 + 5 = -5 sin(i2)/ sin(U2),

sin(i) = -0.08625977332, sin(i2) = 1.5180 sin(i) }

> f61 := g(0.5784592167e-2, 144.551, -5, 1.5180, 1);

{0.0057845922 + i = U2 + i2, L2 + 5 = - 5 sin(i2)/ sin(U2),
sin(i) = -0.1730173437, sin(i2) = 1.5180 sin(i) }

> f71 := g(0.9043418012e-2, 139.046, -5, 1.5180, 1);

{ 0.009043418012 + i = U2 + i2, L2 + 5 = - 5 sin(i2) /sin(U2),
sin(i) = -0.2605300870, sin(i2) = 1.5180 sin(i) }

> fsolve(f51);

{L2 = 8.710016670, U2 = 0.04777242746, i = -0.08636710598,i2 = -0.1313194401}

> fsolve(f61);

{L2 = 8.469472346, U2 = 0.09764977648, i = -0.1738923957,i2 = -0.2657575800}

> fsolve(f71);

{L2 = 8.053825707, U2 = 0.1520676925, i = -0.2635712102,
i2 = -0.4065954847}

## 参考文献

• A.E.Conrady, Applied Optics and Optical Design p6-20 Dover（用八位对数表、三角函数表计算结果为8.7088，8.4701，8.0545）。
• 何青 王丽芬编著 Maple 教程 科学出版社
• Martha L. Bell &James Braselton, Maple V by Example