Abstract

In following a suggestion of Sommerfeld, who was able to derive the paraxial properties for points on the optic axis from the existence and continuity of wave fronts satisfying the eikonal equation, it will be shown how the whole set of the third-order Seidel aberrations of a centered optical system made of refracting surfaces of revolution can be obtained if the series expansion in the radial coordinates is continued up to the fourth order.

© 2003 Optical Society of America

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References

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  1. G. Pozzi, “Sommerfeld revisited: a simple introduction to imaging and spherical aberration in refracting surfaces of revolution and thin lenses,” Eur. J. Phys. 22, 1–8 (2001).
    [CrossRef]
  2. A. Sommerfeld, Optics (Academic, New York, 1964).
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).
  4. A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
  5. D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 1987).
  6. S. Wolfram, The Mathematica Book (Wolfram Media/Cambridge University, Champaign, Ill., 1999).
  7. A. Maréchal, “Optique géométrique générale,” in Handbuch der Physik, S. Flügge, ed., Vol. XXIV (Springer, Berlin, 1956), pp. 44–170.

2001

G. Pozzi, “Sommerfeld revisited: a simple introduction to imaging and spherical aberration in refracting surfaces of revolution and thin lenses,” Eur. J. Phys. 22, 1–8 (2001).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).

Maréchal, A.

A. Maréchal, “Optique géométrique générale,” in Handbuch der Physik, S. Flügge, ed., Vol. XXIV (Springer, Berlin, 1956), pp. 44–170.

Pozzi, G.

G. Pozzi, “Sommerfeld revisited: a simple introduction to imaging and spherical aberration in refracting surfaces of revolution and thin lenses,” Eur. J. Phys. 22, 1–8 (2001).
[CrossRef]

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 1987).

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1964).

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

Wolfram, S.

S. Wolfram, The Mathematica Book (Wolfram Media/Cambridge University, Champaign, Ill., 1999).

Eur. J. Phys.

G. Pozzi, “Sommerfeld revisited: a simple introduction to imaging and spherical aberration in refracting surfaces of revolution and thin lenses,” Eur. J. Phys. 22, 1–8 (2001).
[CrossRef]

Other

A. Sommerfeld, Optics (Academic, New York, 1964).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).

D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 1987).

S. Wolfram, The Mathematica Book (Wolfram Media/Cambridge University, Champaign, Ill., 1999).

A. Maréchal, “Optique géométrique générale,” in Handbuch der Physik, S. Flügge, ed., Vol. XXIV (Springer, Berlin, 1956), pp. 44–170.

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Equations (120)

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z=zRi+x2+y22Ri+(1+δi) (x2+y2)28Ri3,
g(r)=nir
|g(r)|=ni
S0(r-r0)=n0[(x-x0)2+(y-y0)2+(z-z0)2]1/2+a0,
S0=S0(0)+S0(2)+S0(4),
S0(0)=n0z+a0,
S0(2)=n0(x-x0)2+(y-y0)22(z-z0),
S0(4)=-n0[(x-x0)2+(y-y0)2]28(z-z0)3.
r2=x2+y2,
r02=x02+y02,
k02=xx0+yy0.
S0(2)=n0r2+r02-2k022(z-z0),
S0(4)=-n0[r2+r02-2k02]28(z-z0)3.
Si=Si(0)+Si(2)+Si(4),
Si(0)=niai(z),
Si2=ni[bi(z)r2+ci(z)k02+di(z)r02],
Si4=ni[Bi(z)r4+Fi(z)r2k02+Di(z)r2r02+Ci(z)k04+Ei(z)k02r02+Ai(z)r04].
ai(z)2=1,
ai(z)=1.
4bi(z)2+2bi(z)ai(z)=0,
4bi(z)ci(z)+2ci(z)ai(z)=0,
ci(z)2+2di(z)ai(z)=0.
16bi(z)Bi(z)+bi(z)2+2Bi(z)ai(z)=0,
8Bi(z)ci(z)+12bi(z)Fi(z)+2bi(z)ci(z)
+2Fi(z)ai(z)=0,
2ci(z)Fi(z)+8bi(z)Di(z)+2bi(z)di(z)
+2Di(z)ai(z)=0,
4ci(z)Fi(z)+8bi(z)Ci(z)+ci(z)2
+2ai(z)Ci(z)=0,
4ci(z)Di(z)+4ci(z)Ci(z)+4bi(z)Ei(z)+2ci(z)di(z)
+2ai(z)Ei(z)=0,
2ci(z)Ei(z)+di(z)2+2ai(z)Ai(z)=0.
ai(z)=z.
bi(z)=12z-const.
bi(z)=12(z-zi).
ci(z)=constz-zi=-Miz-zi,
di(z)=Mi22(z-zi)+const.
Si(2)=ni2(z-zi) [(x-Mix0)2+(y-Mi y0)2],
SiSW(4)=-ni[r2+(Mir0)2-2Mik02]28(z-zi)3.
Bi(z)=const(z-zi)4-z8(z-zi)4.
Bi(z)=cBi(z-zi)4-18(z-zi)3.
Fi(z)=-4MicBi(z-zi)4+const(z-zi)3.
Fi(z)=-4MicBi(z-zi)4+cFiMi(z-zi)3+Mi2(z-zi)3.
Ci(z)=cCiMi2(z-zi)2-2cFiMi2(z-zi)3+4Mi2cBi(z-zi)4-Mi22(z-zi)3,
Di(z)=cDiMi2(z-zi)2-cFiMi2(z-zi)3+2Mi2cBi(z-zi)4-Mi24(z-zi)3,
Ei(z)=cEiMi3(z-zi)-2Mi3(cCi+cDi)(z-zi)2+3Mi3cFi(z-zi)3-4Mi3cBi(z-zi)4+Mi32(z-zi)3.
Ai(z)=cAiMi4-cEiMi4(z-zi)+Mi4(cCi+cDi)(z-zi)2-Mi4cFi(z-zi)3+Mi4cBi(z-zi)4-Mi48(z-zi)3.
Si=Si(0)+Si(2)+SiSW(4)+Φi(4),
Φi(4)=ni[B˜i(z)r4+F˜i(z)r2k02+C˜i(z)k04+D˜i(z)r2r02+E˜i(z)k02r02+A˜i(z)r04],
B˜i(z)=cBi(z-zi)4,
F˜i(z)=cFiMi(z-zi)3-4MicBi(z-zi)4,
C˜i(z)=cCiMi2(z-zi)2-2cFiMi2(z-zi)3+4Mi2cBi(z-zi)4,
D˜i(z)=cDiMi2(z-zi)2-cFiMi2(z-zi)3+2Mi2cBi(z-zi)4,
E˜i(z)=cEiMi3(z-zi)-2Mi3(cCi+cDi)(z-zi)2+3Mi3cFi(z-zi)3-4Mi3cBi(z-zi)4,
A˜i(z)=cAiMi4-cEiMi4(z-zi)+Mi4(cCi+cDi)(z-zi)2-Mi4cFi(z-zi)3+Mi4cBi(z-zi)4.
ni-11Ri-1zi-1-zRi=ni1Ri-1zi-zRi=Ki,
ni-1Mi-1zi-1-zRi=niMizi-zRi,
cBi-1ni-1(zi-1-zRi)4+ni-1(1+δi)8Ri3+ni-18(zi-1-zRi)3
-ni-1Ri(zi-1-zRi)2
=cBini(zi-zRi)4+ni(1+δi)8Ri3+ni8(zi-zRi)3
-niRi(zi-zRi)2.
cFi-1Mi-1ni-1(zi-1-zRi)3+4cBi-1Mi-1ni-1(zi-1-zRi)4+Mi-1ni-12(zi-1-zRi)3
-Mi-1ni-12Ri(zi-1-zRi)2
=cFiMini(zi-zRi)3+4cBiMini(zi-zRi)4+Mini2(zi-zRi)3
-Mini2Ri(zi-1-zRi)2,
cCi-1Mi-12ni-1(zi-1-zRi)2+2cFi-1Mi-12ni-1(zi-1-zRi)3+4cBi-1Mi-12ni-1(zi-1-zRi)4
+Mi-12ni-12(zi-1-zRi)3
=cCiMi2ni(zi-zRi)2+2cFiMi2ni(zi-zRi)3+4cBiMi2ni(zi-zRi)4
+Mi2ni2(zi-zRi)3,
cDi-1Mi-12ni-1(zi-1-zRi)2+cFi-1Mi-12ni-1(zi-1-zRi)3+2cBi-1Mi-12ni-1(zi-1-zRi)4
+Mi-12ni-12(zi-1-zRi)3-Mi-12ni-14Ri(zi-1-zRi)2
=cDiMi2ni(zi-zRi)2+cFiMi2ni(zi-zRi)3+2cBiMi2ni(zi-zRi)4
+Mi2ni2(zi-zRi)3-Mi2ni4Ri(zi-zRi)2,
cEi-1Mi-13ni-1(zi-1-zRi)+2(cCi-1+cDi-1)Mi-13ni-1(zi-1+zRi)2
+3cFi-1Mi-13ni-1(zi-1-zRi)3-4cBi-1Mi-13ni-1(zi-1-zRi)4
+Mi-13ni-12(zi-1-zRi)3
=cEiMi3ni(zi-zRi)+2(cCi+cDi)Mi3ni(zi-zRi)2+3cFiMi3ni(zi-zRi)3
+4cBiMi3ni(zi-zRi)4+Mi3ni2(zi-zRi)3.
x=xPn+1nnSnxz=zPnt,
y=yPn+1nnSnyz=zPnt,
z=zPn+1nnSnzz=zPnt,
x˜n=xn+1nnΦn(4)xz=zPn(zn-zPn),
y˜n=yn+1nnΦn(4)yz=zPn(zn-zPn);
Δx0=x˜n-xnMn=(zn-zPn)nnMnΦn(4)xz=zPn,
Δy0=y˜n-ynMn=(zn-zPn)nnMnΦn(4)yz=zPn.
h(z0)=0,h(z0)=1,
h(z)=z-z0,
h(z)=h(zi)(z-zi)=hizRi-zi (z-zi),
h1=zR1-z0,hi+1zRi+1-zi=hizRi-zi.
niMihizRi-zi=n0.
x=αh(z),y=βh(z),
Δx0=(zn-zPn)nnMnh(zPn)Φn(4)αz=zPn=-(zRn-zn)nnMnhnΦn(4)αz=zPn,
Δy0=(zn-zPn)nnMnh(zPn)Φn(4)βz=zPn=-(zRn-zn)nnMnhnΦn(4)βz=zPn,
Δx0=-1n0Φn(4)αz=zPn,
Δy0=-1n0Φn(4)βz=zPn.
Φi(4)=Bi*(z)(α2+β2)2+Fi*(z)(α2+β2)(αx0+βy0)+ ,
Bi*(z)=niB˜i(z)hi(z)4,
Fi*(z)=niF˜i(z)hi(z)3.
Bi-1*(z)=ni-1B˜i-1(z) hi-14(zRi-1-zi-1)4 (z-zi-1)4=ni-1cBi-1hi-14(zRi-1-zi-1)4,
Bi-1*(zRi-1)=Bi-1*(zRi)=ni-1cBi-1hi4(zRi-zi-1)4,
Bi*(zRi)=nicBihi4(zRi-zi)4=ni-1cBi-1hi4(zRi-zi-1)4+Tihi4=Bi-1*(zRi-1)+Tihi4,
Ti=ni(1+δi)8Ri3+ni8(zi-zRi)3-niRi(zi-zRi)2-ni-1(1+δi)8Ri3-ni-18(zi-1-zRi)3-ni-1Ri(zi-1-zRi)2.
Ti=(ni-1-ni)δi8Ri3+Ki281ni-1(zi-1-zRi)-1ni(zi-zRi),
Bi*(zRi)=Bi-1*(zRi-1)+hi4Ti,
Bn*(zRn)=hi4Ti,
Fi-1*(z)=ni-1Mi-1cFi-1-4cBi-1z-zi-1×hi-13(zRi-1-zi-1)3,
Fi-1*(zRi)=Fi-1*(zRi-1)+4Mi-1Bi-1*(zRi-1)×(1/hi-1-1/hi).
Fi*(zRi)=Fi-1*(zRi)+hi3Ui,
Ui=Mi-1ni-12(zi-1-zRi)3-Mi-1ni-12Ri(zi-1-zRi)2+Mini2(zi-zRi)3-Mini2Ri(zi-1-zRi)2,
Ui=n0Ki2hi1ni-1(zi-1-zRi)-1ni(zi-zRi).
Fi*(zRi)=Fi-1*(zRi-1)+hi3Ui+4Mi-1Bi-1*(zRi-1)×(1/hi-1-1/hi),
g(z)=g(zPi)(z-zPi)=gizRi-zPi (z-zPi),
gi+1zRi+1-zPi=gizRi-zPi,
g(zi)=Mi.
Fi-1*(zRi)=Fi-1*(zPi-1)-4Mi-1Bi-1*(zPi-1)×(zPi-1-zRi)(zRi-1-zi)hi-1(zPi-1-zi-1)(zRi-zi).
Fi*(zPi)=Fi*(zRi)+4MiBi*(zRi)(zPi-zRi)hi(zPi-zi).
4(gi/hi)[Bi*(zRi)-Bi-1*(zRi)]=4giTihi3
ni-11Ri-1zPi-1-zRi=ni1Ri-1zPi-zRi=Li.
Fi*(zPi)=Fi-1*(zPi-1)+4gihi3(ni-1-ni)δi8Ri3+Ki281ni-1(zi-1-zRi)-1ni(zi-zRi)+n0Kihi221ni-1(zi-1-zRi)-1ni(zi-zRi),
Fi*(zPi)=Fi-1*(zPi-1)+gihi32(ni-1-ni)δi8Ri3+KiLi1ni-1(zi-1-zRi)-1ni(zi-zRi).

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