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[[File:Spherical refraction1.jpg|thumb|center|600px|光的球面折射]] '''球面像差'''的成因是经过透镜折射或面镜反射的光线，接近中心与靠近边缘的光线不能将影像聚集在一个点上的现象。 根据单球面的折射方程 > F1 := sin(J) = sin(U)*(L-r)/r; sin(U) (L - r) sin(J) = -------------- r > F2 := sin(Jp) = N*sin(J)/Np; N sin(J) sin(Jp) = -------- Np > F3 := U+J = Up+Jp; U + J = Up + Jp > F4 := Lp-r = sin(Jp)*r/sin(Up); sin(Jp) r Lp - r = --------- sin(Up) 可以导出单球面像差的准确方程式 EQten :=<math>Lp-lp = \frac{N*(l+L)*lp*Lp}{(L*l*Np)}+\frac{(2*(L-r))*N*sin((1/2)*Jp-1/2*J)*sin((1/2)*Jp-(1/2)*U)*lp*Lp}{L*cos((1/2)*Jp-(1/2)*Up)*r*Np}</math> <ref>Conrady, p74-80 (10)*</ref> 其中 *L 为 入射光与光轴交点到球面顶点的距离 *L<sub>p</sub> 折射光与光轴交点到球面顶点的距离 *J 为 入射角 *J<sub>p</sub> 为折射角 *U 为 入射光与光轴的夹角 *U<sub>p</sub> 折射光与光轴的夹角 *r 为球面半径 *N,N<sub>p</sub> 为球面两边的折射率 ==球面像差公式的推导== 先列出几个常用的子程序 > Pinvert := proc (epr) 1/op(1, epr) = 1/op(2, epr) end proc; > PoneMinusLH := proc (expres) simplify(1-lhs(expres) = 1-rhs(expres)) end proc; > > PsinAddsin := proc (a, b) sin(a)+sin(b) = 2*sin((1/2)*a+(1/2)*b)*cos((1/2)*a-(1/2)*b) end proc; > > PcosSubcos := proc (p, q) {cos(p)-cos(q) = 2*sin((1/2)*p+(1/2)*q)*sin((1/2)*q-(1/2)*p)} end proc; > > > > F1 := sin(J) = sin(U)*(L-r)/r; <math> sin(J) = \frac{sin(U) (L - r)}{r}</math> r > F2 := sin(Jp) = N*sin(J)/Np; <math> sin(Jp) = \frac{N sin(J)}{Np} </math> > F3 := U+J = Up+Jp; U + J = Up + Jp > F4 := Lp-r = sin(Jp)*r/sin(Up); <math> Lp - r =\frac{sin(Jp) r}{sin(Up)} </math> > > > sys1 := {F1}; / sin(U) (L - r)\ \ { sin(J) = -------------- } \ r / > > g2 := solve(sys1, L); / r (sin(J) + sin(U))\ { L = ------------------- } \ sin(U) / > G1 := op(1, g2); r (sin(J) + sin(U)) L = ------------------- sin(U) > > > sys4 := {F4}; / sin(Jp) r\ { Lp - r = --------- } \ sin(Up) / > > g4 := solve(sys4, Lp); / r (sin(Jp) + sin(Up))\ { Lp = --------------------- } \ sin(Up) / > > G4 := op(g4); r (sin(Jp) + sin(Up)) Lp = --------------------- sin(Up) > > > G1INV := Pinvert(G1); 1 sin(U) - = ------------------- L r (sin(J) + sin(U)) > G4INV := Pinvert(G4); 1 sin(Up) -- = --------------------- Lp r (sin(Jp) + sin(Up)) > > NG1 := r*G1INV; r sin(U) - = --------------- L sin(J) + sin(U) > NG4 := r*G4INV; r sin(Up) -- = ----------------- Lp sin(Jp) + sin(Up) > > NG1a := PoneMinusLH(NG1); L - r sin(J) ----- = --------------- L sin(J) + sin(U) > > NG4a := PoneMinusLH(NG4); Lp - r sin(Jp) ------ = ----------------- Lp sin(Jp) + sin(Up) > > > > > NG4aR := rhs(NG4a); NG1aR := rhs(NG1a); sin(Jp) ----------------- sin(Jp) + sin(Up) sin(J) --------------- sin(J) + sin(U) > h1a := r/L = 1-NG1aR; r sin(J) - = 1 - --------------- L sin(J) + sin(U) > h2a := r/Lp = 1-NG4aR; r sin(Jp) -- = 1 - ----------------- Lp sin(Jp) + sin(Up) > > H1 := h1a*N/r; / sin(J) \ N |1 - ---------------| N \ sin(J) + sin(U)/ - = ----------------------- L r > H1EX := expand(H1); N N N sin(J) - = - - ------------------- L r r (sin(J) + sin(U)) > NG1aR; sin(J) --------------- sin(J) + sin(U) > H2 := h2a*Np/r; / sin(Jp) \ Np |1 - -----------------| Np \ sin(Jp) + sin(Up)/ -- = -------------------------- Lp r > > H2EX := expand(H2); Np Np Np sin(Jp) -- = -- - --------------------- Lp r r (sin(Jp) + sin(Up)) > > > H1H2 := H2EX-H1EX; Np N Np Np sin(Jp) N N sin(J) -- - - = -- - --------------------- - - + ------------------- Lp L r r (sin(Jp) + sin(Up)) r r (sin(J) + sin(U)) > > > > > K := algsubs(Np*sin(Jp) = N*sin(J), H1H2); Np N N sin(J) Np N N sin(J) -- - - = - --------------------- + -- - - + ------------------- Lp L r (sin(Jp) + sin(Up)) r r r (sin(J) + sin(U)) > > > KR := rhs(K); KL := lhs(K); N sin(J) Np N N sin(J) - --------------------- + -- - - + ------------------- r (sin(Jp) + sin(Up)) r r r (sin(J) + sin(U)) Np N -- - - Lp L > > > op(KR); N sin(J) Np N N sin(J) - ---------------------, --, - -, ------------------- r (sin(Jp) + sin(Up)) r r r (sin(J) + sin(U)) > > > KR1 := op(1, KR); N sin(J) - --------------------- r (sin(Jp) + sin(Up)) > > KR4 := op(4, KR); N sin(J) ------------------- r (sin(J) + sin(U)) > > KR1KR4 := KR1+KR4; N sin(J) N sin(J) - --------------------- + ------------------- r (sin(Jp) + sin(Up)) r (sin(J) + sin(U)) > > KR1KR4a := KR1KR4 = KR4*(1+X); N sin(J) N sin(J) - --------------------- + ------------------- = r (sin(Jp) + sin(Up)) r (sin(J) + sin(U)) N sin(J) (1 + X) ------------------- r (sin(J) + sin(U)) > > > KR1KR4RH := rhs(KR1KR4a); N sin(J) (1 + X) ------------------- r (sin(J) + sin(U)) > > > TM1 := subs(1+X = Z, KR1KR4RH); N sin(J) Z ------------------- r (sin(J) + sin(U)) > > NG1a; T1 := rhs(NG1a); T2 := lhs(NG1a); L - r sin(J) ----- = --------------- L sin(J) + sin(U) sin(J) --------------- sin(J) + sin(U) L - r ----- L > TM2 := subs(sin(J)+sin(U) = sinJplussinU, TM1); N sin(J) Z -------------- r sinJplussinU > TM3 := algsubs(sin(J)/sinJplussinU = (L-r)/L, TM2); N Z (L - r) ----------- r L > > > > > > Y := solve(KR1KR4a, X); sin(J) + sin(U) - ----------------- sin(Jp) + sin(Up) > kr1kr4 := KR4*(1+Y); / sin(J) + sin(U) \ N sin(J) |1 - -----------------| \ sin(Jp) + sin(Up)/ -------------------------------- r (sin(J) + sin(U)) > > T1 := algsubs(sin(J)+sin(U) = L*sin(J)/(L-r), KR1KR4RH); / L sin(J)\ N (1 + X) |-sin(U) + --------| (L - r) \ L - r / -------------------------------------- r L sin(J) > KR1and4 := subs(Z = 1+Y, TM3); / sin(J) + sin(U) \ N (L - r) |1 - -----------------| \ sin(Jp) + sin(Up)/ --------------------------------- r L > > > > > > > > s1 := PsinAddsin(Jp, Up); /1 1 \ /1 1 \ sin(Jp) + sin(Up) = 2 sin|- Jp + - Up| cos|- Jp - - Up| \2 2 / \2 2 / > op(s1); /1 1 \ /1 1 \ sin(Jp) + sin(Up), 2 sin|- Jp + - Up| cos|- Jp - - Up| \2 2 / \2 2 / > t4d := algsubs(s1, KR1and4); / -sin(J) - sin(U) \ N (L - r) |----------------------------------- + 1| | /1 1 \ /1 1 \ | |2 sin|- Jp + - Up| cos|- Jp - - Up| | \ \2 2 / \2 2 / / --------------------------------------------------- r L > s2 := PsinAddsin(J, U); /1 1 \ /1 1 \ sin(J) + sin(U) = 2 sin|- J + - U| cos|- J - - U| \2 2 / \2 2 / > t4e := algsubs(s2, t4d); / /1 1 \ /1 1 \ \ | sin|- J + - U| cos|- J - - U| | | \2 2 / \2 2 / | N (L - r) |- --------------------------------- + 1| | /1 1 \ /1 1 \ | | sin|- Jp + - Up| cos|- Jp - - Up| | \ \2 2 / \2 2 / / --------------------------------------------------- r L > t4f := algsubs(U+J = Up+Jp, t4e); / / 1 1 \ \ | cos|-U + - Up + - Jp| | | \ 2 2 / | N (L - r) |- --------------------- + 1| | /1 1 \ | | cos|- Jp - - Up| | \ \2 2 / / --------------------------------------- r L > t4g := simplify(t4f); / / 1 1 \ /1 1 \\ N (L - r) |cos|-U + - Up + - Jp| - cos|- Jp - - Up|| \ \ 2 2 / \2 2 // - ---------------------------------------------------- /1 1 \ r L cos|- Jp - - Up| \2 2 / > > > s3 := PcosSubcos((Jp-Up)*(1/2), -U+(1/2)*Up+(1/2)*Jp); / / 1 1 \ /1 1 \ /1 1 \ { -cos|-U + - Up + - Jp| + cos|- Jp - - Up| = -2 sin|- Jp - - U| \ \ 2 2 / \2 2 / \2 2 / /1 1 \\ sin|- U - - Up| } \2 2 // > s4 := op(s3); / 1 1 \ /1 1 \ /1 1 \ -cos|-U + - Up + - Jp| + cos|- Jp - - Up| = -2 sin|- Jp - - U| \ 2 2 / \2 2 / \2 2 / /1 1 \ sin|- U - - Up| \2 2 / > s5 := algsubs(s4, t4g); /1 1 \ /1 1 \ 2 N (L - r) sin|- Jp - - U| sin|- U - - Up| \2 2 / \2 2 / - ------------------------------------------- /1 1 \ r L cos|- Jp - - Up| \2 2 / > > s6 := subs(sin((1/2)*U-(1/2)*Up) = sin((Jp-J)*(1/2)), s5); /1 1 \ / 1 1 \ 2 N (L - r) sin|- Jp - - U| sin|- - Jp + - J| \2 2 / \ 2 2 / --------------------------------------------- /1 1 \ r L cos|- Jp - - Up| \2 2 / > > EQten := KL = op(2, KR)+op(3, KR)+s6; /1 1 \ / 1 1 \ 2 N (L - r) sin|- Jp - - U| sin|- - Jp + - J| Np N Np N \2 2 / \ 2 2 / -- - - = -- - - + --------------------------------------------- Lp L r r /1 1 \ r L cos|- Jp - - Up| \2 2 / EQten :=<math>Lp-lp = \frac{N*(l+L)*lp*Lp}{(L*l*Np)}+\frac{(2*(L-r))*N*sin((1/2)*Jp-1/2*J)*sin((1/2)*Jp-(1/2)*U)*lp*Lp}{L*cos((1/2)*Jp-(1/2)*Up)*r*Np}</math> <ref>Conrady, p74-80 (10)*</ref> ==参考文献== <references/> Alexander Eugen Conrady Applied Optics & Optical Design Dover
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