Abstract

The Abbe sine condition and the recently developed pupil astigmatism conditions provide a powerful set of relationships for describing imaging systems that are free from aberrations that have linear and quadratic dependence on field, to all orders in the pupil. We have proved both of these conditions and applied them to axisymmetric imaging systems. We now extend our approach to plane-symmetric systems. Still using Hamilton’s characteristic functions, we derive the general sine conditions and the pupil astigmatism conditions that describe plane-symmetric systems that are free of all aberrations with linear and quadratic field dependence.

© 2002 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (MacMillan, New York, 1964).
  2. H. A. Buchdahl, An Introduction To Hamiltonian Optics (Cambridge U. Press, New York, 1970).
  3. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1966).
  4. C. Zhao, J. H. Burge, “Criteria for correction of quadratic field-dependent aberrations,” J. Opt. Soc. Am. A 19, 2313–2321 (2002).
    [CrossRef]
  5. W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986).
  6. S. A. Comastri, J. M. Simon, R. Blendowske, “Generalized sine condition for image-forming systems with centering errors,” J. Opt. Soc. Am. A 16, 602–612 (1999).
    [CrossRef]
  7. J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
    [CrossRef]
  8. B. D. Stone, “Analog of third-order methods in the design of asymmetric systems,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 361–366.
  9. See Sec. 34 of Ref. 3.
  10. C. Zhao, J. H. Burge, “Application of the pupil astigmatism criteria in optical design,” Appl. Opt. (to be published).
  11. L. Mertz, Excursions in Astronomical Optics (Springer, New York, 1996), Chap. 1.

2002

1999

1994

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Blendowske, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (MacMillan, New York, 1964).

Buchdahl, H. A.

H. A. Buchdahl, An Introduction To Hamiltonian Optics (Cambridge U. Press, New York, 1970).

Burge, J. H.

C. Zhao, J. H. Burge, “Criteria for correction of quadratic field-dependent aberrations,” J. Opt. Soc. Am. A 19, 2313–2321 (2002).
[CrossRef]

C. Zhao, J. H. Burge, “Application of the pupil astigmatism criteria in optical design,” Appl. Opt. (to be published).

Comastri, S. A.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1966).

Mertz, L.

L. Mertz, Excursions in Astronomical Optics (Springer, New York, 1996), Chap. 1.

Sasian, J. M.

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Simon, J. M.

Stone, B. D.

B. D. Stone, “Analog of third-order methods in the design of asymmetric systems,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 361–366.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (MacMillan, New York, 1964).

Zhao, C.

C. Zhao, J. H. Burge, “Criteria for correction of quadratic field-dependent aberrations,” J. Opt. Soc. Am. A 19, 2313–2321 (2002).
[CrossRef]

C. Zhao, J. H. Burge, “Application of the pupil astigmatism criteria in optical design,” Appl. Opt. (to be published).

J. Opt. Soc. Am. A

Opt. Eng.

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Other

B. D. Stone, “Analog of third-order methods in the design of asymmetric systems,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 361–366.

See Sec. 34 of Ref. 3.

C. Zhao, J. H. Burge, “Application of the pupil astigmatism criteria in optical design,” Appl. Opt. (to be published).

L. Mertz, Excursions in Astronomical Optics (Springer, New York, 1996), Chap. 1.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986).

M. Born, E. Wolf, Principles of Optics (MacMillan, New York, 1964).

H. A. Buchdahl, An Introduction To Hamiltonian Optics (Cambridge U. Press, New York, 1970).

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1966).

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Figures (8)

Fig. 1
Fig. 1

Schematic definitions of mixed and angle characteristic functions.

Fig. 2
Fig. 2

Unobstructed telescope: an example of a plane-symmetric system.

Fig. 3
Fig. 3

Illustration of the definition of coordinate systems for a plane-symmetric system. The ray that originates from the center of the field and goes through the center of the pupil is called the optical axis ray (OAR). The object plane is x0-y0 and the image plane is x1-y1. The x0 and x1 axes point into the plane of the figure.

Fig. 4
Fig. 4

Three-dimensional illustration of the definitions of the object height h0 and the ray vector ν1.

Fig. 5
Fig. 5

Trace of a thin parallel bundle of rays centered on a ray from the field center with ray vector ν0 in the object space.

Fig. 6
Fig. 6

(a) Three-dimensional illustration of the sagittal ray of the parallel bundle in image space. The ray intersects the image plane at (x1, 0). The angle between the sagittal ray and the center ray of the bundle is Δθ. (b) Enlarged view of the triangle TsIH1.

Fig. 7
Fig. 7

Three-dimensional illustration of the tangential ray of the parallel bundle in image space. The ray intersects the image plane at (0, y1). The angle between the tangential ray and the center ray of the bundle is Δθ.

Fig. 8
Fig. 8

Three-dimensional illustration of the parallel bundle of rays in image space for an axisymmetric system.

Tables (2)

Tables Icon

Table 1 Aberrations with Linear Field Dependence

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Table 2 Aberrations with Quadratic Field Dependence

Equations (34)

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h0=x0x^0+y0y^0,h1=x1x^1+y1y^1.
ν0=p0x^0+q0y^0+m0z^0,
ν1=p1x^1+q1y^1+m1z^1.
ρ0=p0x^0+q0y^0,ρ1=p1x^1+q1y^1.
W(h0, ρ1)=optical path length along the ray from A to D.
ρ0=-h0W,h1=-ρ1W,
h0=x^0x0+y^0y0,ρ1=x^1p1+y^1q1.
T(ρ0, ρ1)=optical path length along the ray from C to D.
h0=ρ0T,h1=-ρ1T,
ρ0=x^0p0+y^0q0.
h0h0=x02+y02,ρ1ρ1=p12+q12.
h0ρ1=x0p1+y0q1,
h0i=y0,
ρ1i=q1.
W(h0, ρ1)=W2k+n+p,2m+n+q,n,p,q(h0h0)k×(ρ1ρ1)m(h0ρ1)n(ih0)p(iρ1)q.
W(h0, ρ1)=A(ρ1ρ1, iρ1)+(h0ρ1)B(ρ1ρ1, ρ1i)+(h0i)C(ρ1ρ1, ρ1i)+(h0h0)D(ρ1ρ1, ρ1i)+(h0ρ1)2E(ρ1ρ1, ρ1i)+(h0i)2F(ρ1ρ1, ρ1i)+(h0i)(h0ρ1)G(ρ1ρ1, ρ1i)+  .
p0x^0+q0y^0=-h0W(h0, ρ1)=-(p1x^0+q1y^0)B(ρ1ρ1, ρ1i)-y^0C(ρ1ρ1, ρ1i).
A(ρ1ρ1, ρ1i)=const.
h1=-ρ1W(h0ρ1)=-(x0x^1+y0y^1)B(ρ1ρ1, ρ1i)-(x0p1+y0q1)ρ1B(ρ1ρ1, ρ1i)-y0ρ1C(ρ1ρ1, ρ1i).
B=-mx,C=-a-b(ρ1i),
x1=mxx0,y1=myy0.
p0=mxp1,q0=myq1+a.
D=const.,E=G=0,F=const.
Ws(x0, ρ1)=W(0, ρ1)-x0p0-n1ts+n1tscos ΔθW(0, ρ1)-x0p0-n1ts(Δθ)2/2,
Δθ=|IF||ts|=mxx0ts|ν1×x^1|n1,
W(x0, ρ1)W(0, ρ1)-x0p0-mx2|ν1×x^1|22n1ts x02.
mx2|ν1×x^1|22n1ts=const.
my2|ν1×y^1|22n1t1=const.
|ν1×x^1|2ts=const.,
|ν1×y^1|2tt=const.
ν1=-y^1n1sin θ+z^1n1cos θ.
|ν1×x^1|2=n12,|ν1×y^1|2=n12cos2 θ.
1ts=const.,
cos2 θtt=const.

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