Abstract

I show how computer algebra can be of material help in determining the potential quality of various zoom lens arrangements. One fortuitous result of this work is a proof that, when the object is kept at infinity, it is possible in principle to design zoom lenses for which all third-, fifth-, and seventh-order aberrations are corrected over a continuous range of zoom settings limited only by the restriction that the lens groups may not run into each other.

© 1999 Optical Society of America

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References

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  1. A. Walther, “Eikonal theory and computer algebra,” J. Opt. Soc. Am. A 13, 523–531 (1996).
    [CrossRef]
  2. A. Walther, “Eikonal theory and computer algebra II,” J. Opt. Soc. Am. A 13, 1763–1765 (1996).
    [CrossRef]
  3. J. C. Maxwell, “On the general laws of optical instruments,” Vol. 1 of Scientific papers (Cambridge U. Press, Cambridge, UK, 1890), pp. 271–285.
  4. For a general introduction to eikonal functions see Refs. 5-8.
  5. M. Herzberger, Strahlenoptik (Springer, Berlin, 1931).
  6. R. Luneburg, Mathematical Theory of Optics (University of California, Los Angeles, Calif., 1964).
  7. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, UK, 1970).
  8. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).
  9. We assume, as is usual in this context, that it is, at least in principle, possible to design a lens that realizes any specified eikonal. See also Ref. 8, Sec. 22.1.
  10. See, however, Sec. 7 and especially Ref. 18.
  11. See, for instance, R. Kingslake, A History of the Photographic Lens (Academic, New York, 1989), p. 159.
  12. R. Kingslake, A History of the Photographic Lens (Academic, New York, 1989), p. 161.
  13. This result agrees with the work described in Chap. 2 of J. B. Lasché, “Aberration correction in zoom systems: theoretical results,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1998).
  14. See Ref. 8, Sec. 6.4.
  15. Mock ray tracing is described in Ref. 8, Chap. 31. See also A. Walther, “Mock ray tracing,” J. Opt. Soc. Am. 60, 918–920 (1970), and “Irreducible aberrations of a lens used for a range of magnifications,” J. Opt. Soc. Am. A 6, 415–422 (1989).
    [CrossRef]
  16. See Ref. 8, p. 351, or W. H. Press, B. F. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1988), Sec. 9.6.
  17. OSLO SIX is a registered trademark of Sinclair Optics Inc., Fairport, New York.
  18. As pointed out by one of the referees, the separation between the design tasks is not complete. Section 4 shows that for the Donders system the coefficients Q1,Q4,Q7, and Q10 are still at our disposal. These coefficients affect the eikonals of both groups to be designed. So an iterative process involving the designs of both groups will be needed to find the optimum values of these coefficients.

1996

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, UK, 1970).

Herzberger, M.

M. Herzberger, Strahlenoptik (Springer, Berlin, 1931).

Kingslake, R.

See, for instance, R. Kingslake, A History of the Photographic Lens (Academic, New York, 1989), p. 159.

R. Kingslake, A History of the Photographic Lens (Academic, New York, 1989), p. 161.

Lasché, J. B.

This result agrees with the work described in Chap. 2 of J. B. Lasché, “Aberration correction in zoom systems: theoretical results,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1998).

Luneburg, R.

R. Luneburg, Mathematical Theory of Optics (University of California, Los Angeles, Calif., 1964).

Maxwell, J. C.

J. C. Maxwell, “On the general laws of optical instruments,” Vol. 1 of Scientific papers (Cambridge U. Press, Cambridge, UK, 1890), pp. 271–285.

Walther, A.

J. Opt. Soc. Am. A

Other

J. C. Maxwell, “On the general laws of optical instruments,” Vol. 1 of Scientific papers (Cambridge U. Press, Cambridge, UK, 1890), pp. 271–285.

For a general introduction to eikonal functions see Refs. 5-8.

M. Herzberger, Strahlenoptik (Springer, Berlin, 1931).

R. Luneburg, Mathematical Theory of Optics (University of California, Los Angeles, Calif., 1964).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, UK, 1970).

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).

We assume, as is usual in this context, that it is, at least in principle, possible to design a lens that realizes any specified eikonal. See also Ref. 8, Sec. 22.1.

See, however, Sec. 7 and especially Ref. 18.

See, for instance, R. Kingslake, A History of the Photographic Lens (Academic, New York, 1989), p. 159.

R. Kingslake, A History of the Photographic Lens (Academic, New York, 1989), p. 161.

This result agrees with the work described in Chap. 2 of J. B. Lasché, “Aberration correction in zoom systems: theoretical results,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1998).

See Ref. 8, Sec. 6.4.

Mock ray tracing is described in Ref. 8, Chap. 31. See also A. Walther, “Mock ray tracing,” J. Opt. Soc. Am. 60, 918–920 (1970), and “Irreducible aberrations of a lens used for a range of magnifications,” J. Opt. Soc. Am. A 6, 415–422 (1989).
[CrossRef]

See Ref. 8, p. 351, or W. H. Press, B. F. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1988), Sec. 9.6.

OSLO SIX is a registered trademark of Sinclair Optics Inc., Fairport, New York.

As pointed out by one of the referees, the separation between the design tasks is not complete. Section 4 shows that for the Donders system the coefficients Q1,Q4,Q7, and Q10 are still at our disposal. These coefficients affect the eikonals of both groups to be designed. So an iterative process involving the designs of both groups will be needed to find the optimum values of these coefficients.

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Tables (4)

Tables Icon

Table 1 Two-group Zoom Lens: Eikonal Coefficients Pi and Qi for Third-Order Correction of All Aberrations Except Spherical Aberration, at All Zoom Settings

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Table 2 Symmetric Donders Telescope: Eikonal Coefficients Pi and Qi for Perfect Correction of All Third- and Fifth-Order Aberrations at All Zoom Settings

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Table 3 Eikonal Coefficients Used in the Numerical Check

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Table 4 Meridional Ray Trace at Full Aperture and Field (Column 2) and with the Aperture and Field Reduced by a Factor 101/7=1.3895 (Column 3)a

Equations (41)

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t=1+fαf-1.
zF=1+fα.
a=(L2+M2)/2,
b=LL+MM,
c=(L2+M2)/2.
W1(a, b, c)=-(a-b+c)+a(P1a+P2b+P3c)+b(P4b+P5c)+P6c2.
W2=a-b+cα+a(Q1a+Q2b+Q3c)+b(Q4b+Q5c)+Q6c2.
W1W1+t1-2c-t.
W2W2+zF1-2c-zF.
Wid(a, b, c)=-f b1-2a
coma=C1f+C2f2+C3f3+C4f4,
C1=Q5α,
C2=-Q3α-4Q4α,
C3=3Q2α-P5+1,
C4=-2Q1α-2P6α+1-α.
Werr=2Q6α-f-12αc2.
Werr=2Q6α-f-1128α F4u4.
Werr=2Q6α-f-1768 α F4(6u4-6u2+1).
RMSW=2Q6α-f-17685αF4.
RMSWmax=(fmax-fmin)/27685αF4.
t1=f(α-1-G),
t2=f(α-1-1/G).
W1(a, b, c)=αf(a-b+c)+a(P1a+P2b+P3c)+b(P4b+P5c)+P6c2.
W2(a, b, c)=-f(a-b+c)+a(Q1a+Q2b+Q3c)+b(Q4b+Q2c)+Q1c2.
W3(a, b, c)=αf(c-b+a)+c(P1c+P2b+P3a)+b(P4b+P5a)+P6a2.
p=(x2+y2)/2,
v=xL+yM,
c=(L2+M2)/2.
LN=G LN,MN=G MN.
L=GL[1-2(1-G2)c]1/2,
Vperf(p, v, c)=-Gv[1-2(1-G2)c]1/2+F(c).
Vab(p, v, c)=V123(p, v, c)-Vperf(p, v, c)
ast2=-12α2f2(A5G5+A4G4+A3G3+A2G2),
A5=-f+4αP6+2αf+4Q2+4αQ1-2α2f,
A4=-2α2P6+2αf-6αQ2-α2f-2α2Q1-8Q4-2Q3+α3f+2P5,
A3=4Q2-α2f-2α P5+2α2Q2-f+8αQ4+2αQ3,
A2=-f+4 P4-2P5-2P6-2Q1+αf-2α2Q4-2αQ2.
x=W(L, M, L, M)L,y=W(L, M, L, M)M
x=-W(L, M, L, M)L,
y=-W(L, M, L, M)M.
t1=50(3-G),t2=50(3-1/G).

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