Abstract

Aberrations of imaging systems can be described by using a polynomial expansion of the dependence on field position, or the off-axis distance of a point object. On-axis, or zero-order, aberrations can be calculated directly. It is well-known that aberrations with linear field dependence can be calculated and controlled by using the Abbe sine condition, which evaluates only on-axis behavior. We present a new set of relationships that fully describe the aberrations that depend on the second power of the field. A simple set of equations is derived by using Hamilton’s characteristic functions and simplified by evaluating astigmatism in the pupil. The equations, which we call the pupil astigmatism criteria, use on-axis behavior to evaluate and control all aberrations with quadratic dependence on the field and arbitrary dependence on the pupil. These relations are explained and are validated by using several specific optical designs.

© 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. R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, Calif., 1966).
  3. R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).
  4. J. Dyson, “Unit magnification optical system without Seidel aberrations,” J. Opt. Soc. Am. 49, 713–716 (1959).
    [CrossRef]
  5. A. Offner, “New concepts in projection mask aligners,” Opt. Eng. 14, 130–132 (1975).
    [CrossRef]
  6. ZEMAX is an optical design code from Focus Software, Inc., Tucson, Arizona.
  7. L. Mertz, Excursions in Astronomical Optics (Springer, New York, 1996), Chap. 5, pp. 99–100.
  8. C. Zhao, J. H. Burge, “Application of the pupil astigmatism criteria in optical design,” Appl. Opt. (to be published).
  9. M. Herzberger, Strahlenoptik (Springer-Verlag, Berlin, 1931), pp. 157–161.  

1975 (1)

A. Offner, “New concepts in projection mask aligners,” Opt. Eng. 14, 130–132 (1975).
[CrossRef]

1959 (1)

Born, M.

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

Burge, J. H.

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

Dyson, J.

Herzberger, M.

M. Herzberger, Strahlenoptik (Springer-Verlag, Berlin, 1931), pp. 157–161.  

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).

Luneburg, R. K.

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

Mertz, L.

L. Mertz, Excursions in Astronomical Optics (Springer, New York, 1996), Chap. 5, pp. 99–100.

Offner, A.

A. Offner, “New concepts in projection mask aligners,” Opt. Eng. 14, 130–132 (1975).
[CrossRef]

Wolf, E.

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

Zhao, C.

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

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

A. Offner, “New concepts in projection mask aligners,” Opt. Eng. 14, 130–132 (1975).
[CrossRef]

Other (7)

ZEMAX is an optical design code from Focus Software, Inc., Tucson, Arizona.

L. Mertz, Excursions in Astronomical Optics (Springer, New York, 1996), Chap. 5, pp. 99–100.

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

M. Herzberger, Strahlenoptik (Springer-Verlag, Berlin, 1931), pp. 157–161.  

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

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

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).

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Figures (10)

Fig. 1
Fig. 1

A rotationally symmetric optical system images the point O to the point I.

Fig. 2
Fig. 2

Thin bundle of parallel rays traced through the optical system. T is the tangential focus, and S is the sagittal focus. (a) Three-dimensional illustration, (b) side view, (c) top view.

Fig. 3
Fig. 3

Interpretation of Hamilton’s characteristic functions when the initial and final media are homogeneous.

Fig. 4
Fig. 4

Illustration of how to get p12W2(ρ)+W3(ρ) in (a) the tangential plane and (b) the sagittal plane.

Fig. 5
Fig. 5

Three-dimensional illustration of an optical system with object at infinity. A thin bundle of rays is traced from a point A in an arbitrary plane that is perpendicular to the optical axis.

Fig. 6
Fig. 6

(a) Dyson system. (b) Demonstration that the Dyson system is retroreflective, which means that t=s=. (c) Astigmatism is shown to be proportional to the fourth power of the field height.

Fig. 7
Fig. 7

(a) Offner relay. (b) Demonstration that the Offner relay is retroreflective, which means that t=s=. (c) Astigmatism is shown to be proportional to the fourth power of the field height.

Fig. 8
Fig. 8

(a) Modified Bouwers system. (b) Demonstration that the Bouwers system is retroreflective, which means that t=s=. (c) Astigmatism is shown to be proportional to the fourth power of the field height for a moderate field of view.

Fig. 9
Fig. 9

(a) Layout of a four-element symmetric system. (b) Ray fans of the system. EX, EY, lateral aberration; PX, PY, normalized pupil coordinates.

Fig. 10
Fig. 10

Plot of the tangential and sagittal ray aberrations versus sin(θ). The discrete values are the ZEMAX ray-tracing result. The solid and dashed curves are the theoretical predictions.

Tables (2)

Tables Icon

Table 1 Surface Data of the Optical System Shown in Fig. 9(a)

Tables Icon

Table 2 Calculated s , t/cos2(θ), and Aberration Coefficients for the System Shown in Fig. 9(a)

Equations (58)

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W(h, ρ)=W2m+k,2n+k,k(h2)m(ρ2)n(h  ρ)k,
OPL[OI]=const.(independentofθ),
sin(α)sin(θ)=const.(independentofθ).
s(θ)=t(θ)cos2(θ)=const.
t(θ)cos2(θ)=const.
s(θ)=const.
s(θ)=t(θ)cos2(θ).
[AB]=OPLalongtherayfrompointAtopointB,
Pointcharacteristic:V(x0, y0, z0; x1, y1, z1)
=[P0P1],
Mixedcharacteristic:W(x0, y0, z0;p1, q1)
=[P0Q1],
Anglecharacteristic:T(p0, q0; p1, q1)=[Q0Q1].
p0=-Vx0,
p1=Vx1;
p0=-Wx0,
X1=-Wp1;
X0=Tp0,
X1=-Tp1.
h2=x02+y02,
ρ2=p12+q12,
h  ρ=x0 p1+y0q1.
W(x0, y0, p1, q1; z0, z1)=W0(ρ)+(x0 p1+y0q1)W1(ρ)+(x0 p1+y0q1)2W2(ρ)+(x02+y02)W3(ρ),
W(x0, y0, p1, q1; z0, z1)
=W0(ρ)+x0 p1W1(ρ)+x02[p12W2(ρ)+W3(ρ)].
x1=-W0(ρ)p1-x0p1 [p1W1(ρ)]-x02p1 [p12W2(ρ)+W3(ρ)],
y1=-W0(ρ)q1-x0q1 [p1W1(ρ)]-x02q1 [p12W2(ρ)+W3(ρ)].
p0=-p1W1(ρ)
M(ρ)=-W1(ρ).
x1=M(0)x0.
Δx=x1-x1=-W0(ρ)p1+x0p1 {p1[M(ρ)-M(0)]}-x02p1 [p12W2(ρ)+W3(ρ)],
δy=y1=-W0(ρ)q1+x0q1 [p1M(ρ)]-x02q1 [p12W2(ρ)+W3(ρ)].
M(ρ)=p0p1=const.,
p12W2(ρ)+W3(ρ)=const.,
[TI]=nit,
W(O)=[OT]+[TI],
W(A)+[AT]+[TB],
[AT]=[OT]-x0 p0,
[TB]=[TI]cos(δθ)[TI]-[TI](δθ)2/2.
W(A)=W(O)-x0 p0-[TI](δθ)2/2,
δθ=x1cos(θ)/t=x0M(ρ)cos(θ)/t,
W(A)=W(O)-x0p0-x02[niM2(ρ)cos2(θ)/(2t)].
p12W2(ρ)+W3(ρ)=-niM2(ρ)cos2(θ)2t
[SI]=nis.
W(O)=[OS]+[SI],
W(A)=[AS]+[SB],
[AS]=[OS],
[SB]=[SI]cos(δθ)[SI]-[SI](δθ)2/2.
W(A)=W(O)-[SI](δθ)2/2,
δθ=x1s=x0M(ρ)s.
W(A)=W(O)-x02niM2(ρ)2s.
W3(ρ)=-niM2(ρ)2s.
p12W2(ρ)=-niM2cos2(θ)2t+niM22s.
W2(ρ)=W222+W242ρ2+W262ρ4+ ,
W3(ρ)=W200+W220ρ2+W240ρ4+W260ρ6+ .
p12W2(ρ)=35.99 sin4 (θ),
W3(ρ)=-4.63+5.5 sin4 (θ).
W240=5.51m-1,W242=35.99 m-1,

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