Abstract

We investigate unobstructed, plane-symmetric imaging systems of four spherical mirrors. Fifteen parameters are necessary to specify the configuration of such a system. Constraints are determined that ensure that any resultant system possesses a given set of first-order properties. These constraints remove four parameters as available degrees of freedom. To illustrate the efficacy of this design approach, we present two example studies: one for a class of systems with the object at infinity and another for finite-conjugate projection systems. For each study a global optimizer is used as the primary search tool. Example systems from these studies are presented.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991), Chaps. 10 and 12.
  2. The original research can be found in K. Schwarzschild, “Untersuchungen zur geometrischen Optik. II. Theorie der Spiegelteleskope,” Abh. Koenigl. Ges. Wiss. Goettingen Math.-Phys. Klasse, Folge 9 IV(2), 1–28 (1905). A modern summary translated into English can be found in R. N. Wilson, “Karl Schwarzschild Lecture of the German Astronomical Society,” Rev. Mod. Astron. 7, 1–29 (1993).
  3. G. D. Wassermann, E. Wolf, “On the theory of aplanatic aspheric system,” Proc. Phys. Soc. B 62, 2–8 (1949).
    [CrossRef]
  4. D. R. Shafer, “Four-mirror unobscured anastigmatic telescopes with all-spherical surfaces,” Appl. Opt. 17, 1072–1074 (1977).
    [CrossRef]
  5. J. M. Howard, B. D. Stone, “Imaging with three spherical mirrors,” Appl. Opt. 39, 3216–3231 (2000).
    [CrossRef]
  6. For the mechanics of differential ray tracing through homogeneous media, see, for example, A. Cox, A System of Optical Design (Focal, London, 1964), pp. 112–121; D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
  7. Details of this global optimization method can be found in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optimization 6, 1–34 (1995).
  8. Details of imaging with a single spherical mirror can be found in J. M. Howard, B. D. Stone, “Imaging a point to a line with a single spherical mirror,” Appl. Opt. 37, 1826–1834 (1998).
  9. J. M. Howard, B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15, 3045–3056 (1998).
    [CrossRef]

2000 (1)

1998 (2)

1995 (1)

Details of this global optimization method can be found in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optimization 6, 1–34 (1995).

1977 (1)

1949 (1)

G. D. Wassermann, E. Wolf, “On the theory of aplanatic aspheric system,” Proc. Phys. Soc. B 62, 2–8 (1949).
[CrossRef]

1905 (1)

The original research can be found in K. Schwarzschild, “Untersuchungen zur geometrischen Optik. II. Theorie der Spiegelteleskope,” Abh. Koenigl. Ges. Wiss. Goettingen Math.-Phys. Klasse, Folge 9 IV(2), 1–28 (1905). A modern summary translated into English can be found in R. N. Wilson, “Karl Schwarzschild Lecture of the German Astronomical Society,” Rev. Mod. Astron. 7, 1–29 (1993).

Cox, A.

For the mechanics of differential ray tracing through homogeneous media, see, for example, A. Cox, A System of Optical Design (Focal, London, 1964), pp. 112–121; D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968).

Forbes, G. W.

Details of this global optimization method can be found in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optimization 6, 1–34 (1995).

Howard, J. M.

Jones, A. E. W.

Details of this global optimization method can be found in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optimization 6, 1–34 (1995).

Korsch, D.

D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991), Chaps. 10 and 12.

Schwarzschild, K.

The original research can be found in K. Schwarzschild, “Untersuchungen zur geometrischen Optik. II. Theorie der Spiegelteleskope,” Abh. Koenigl. Ges. Wiss. Goettingen Math.-Phys. Klasse, Folge 9 IV(2), 1–28 (1905). A modern summary translated into English can be found in R. N. Wilson, “Karl Schwarzschild Lecture of the German Astronomical Society,” Rev. Mod. Astron. 7, 1–29 (1993).

Shafer, D. R.

Stone, B. D.

Wassermann, G. D.

G. D. Wassermann, E. Wolf, “On the theory of aplanatic aspheric system,” Proc. Phys. Soc. B 62, 2–8 (1949).
[CrossRef]

Wolf, E.

G. D. Wassermann, E. Wolf, “On the theory of aplanatic aspheric system,” Proc. Phys. Soc. B 62, 2–8 (1949).
[CrossRef]

Abh. Koenigl. Ges. Wiss. Goettingen Math.-Phys. Klasse, Folge 9 (1)

The original research can be found in K. Schwarzschild, “Untersuchungen zur geometrischen Optik. II. Theorie der Spiegelteleskope,” Abh. Koenigl. Ges. Wiss. Goettingen Math.-Phys. Klasse, Folge 9 IV(2), 1–28 (1905). A modern summary translated into English can be found in R. N. Wilson, “Karl Schwarzschild Lecture of the German Astronomical Society,” Rev. Mod. Astron. 7, 1–29 (1993).

Appl. Opt. (3)

J. Global Optimization (1)

Details of this global optimization method can be found in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optimization 6, 1–34 (1995).

J. Opt. Soc. Am. A (1)

Proc. Phys. Soc. B (1)

G. D. Wassermann, E. Wolf, “On the theory of aplanatic aspheric system,” Proc. Phys. Soc. B 62, 2–8 (1949).
[CrossRef]

Other (2)

For the mechanics of differential ray tracing through homogeneous media, see, for example, A. Cox, A System of Optical Design (Focal, London, 1964), pp. 112–121; D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968).

D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991), Chaps. 10 and 12.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Schematic representation of a plane-symmetric system composed of four spherical mirrors. The cross section in the plane of symmetry is shown. The base ray and tilted object and image planes are also illustrated.

Fig. 2
Fig. 2

Systems A–H were found as part of the search of the constrained configuration space. The number next to the images represent the maximum RMS spot radii of the 15 field points evaluated. A scale is given next to each system. Systems A–C have the same scale, as do D–H. The sequence of + and - after each system label corresponds to the sign of the tilt angle for each of the four mirrors.

Fig. 3
Fig. 3

Systems I–P were found in the constrained configuration space by use of ASA. The numbers next to the images represent the maximum RMS spot radii of 15 field points evaluated across one half of a 2-cm-square object. Note that systems I, M, N, and P are drawn with the same scale, as are J, K, and O. The sequence of + and - after each system label corresponds to the sign of the tilt angle for each of the four mirrors.

Tables (2)

Tables Icon

Table 1 Parameters for the Systems Illustrated in Fig. 2

Tables Icon

Table 2 Parameters for the Systems Illustrated in Fig. 3

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

yp=abcdyp+O2.
a=a1100a22,  b=b1100b22.
abcd=T4R4T3R3T2R2T1R1T0,
Ti=cos θi sec θi+10di sec θi sec θi+10010di00cos θi+1 sec θi00001,
Ri=100001002ci cos θi01002ci cos θi01.
b11=0,  b22=0,
a11=m1,  a22=m2,
a11=0,  a22=0,
b11=f1,  b22=f2,
a11=cos θobj sec θimGsec θ1, sec θ2, sec θ3, sec θ4,
a22=Gcos θ1, cos θ2, cos θ3, cosθ4,
b11=sec θobj sec θimHsec θ1, sec θ2, sec θ3, sec θ4,
b22=Hcos θ1, cos θ2, cos θ3, cos θ4,
Gt1, t2, t3, t4=2c2q2t2+c3d3+d4t3+c4d4×2c3d3t3+1t4+c1t12c2q2t2d1+d1+d2+q1+2c3d1+d2q1t3+2c4d1+d2t4d4+1,
Ht1, t2, t3, t4=d0+d1+d2+d3+d4+2c2d0+d1q2t2+c3d0+d1+d2×d3+d4t3+c4d4d0+d1+d2+d3+2c3d0+d1+d2d3t3t4+c1d0t1q2+d12c2q2t2+2c3q1t3+2c4d4t4+1
q1=d3+d4+2c4d3d4t4,
q2=q1+d2+2c3d2q1t3+2c4d2d4t4.
c3=K1c1, c2, d0, d1, d2, m1, m2, θobj, θ1, θ2, θ3, θ4, θim,
d3=K2c1, c2, d0, d1, d2, m1, m2, θobj, θ1, θ2, θ3, θ4, θim,
c4=K3c1, c2, d0, d1, d2, m1, m2, θobj, θ1, θ2, θ3, θ4, θim,
d4=K4c1, c2, d0, d1, d2, m1, m2, θobj, θ1, θ2, θ3, θ4, θim.
c3=L1c1, c2, d1, d2, f1, f2, θ1, θ2, θ3, θ4, θim,
d3=L2c1, c2, d1, d2, f1, f2, θ1, θ2, θ3, θ4, θim,
c4=L3c1, c2, d1, d2, f1, f2, θ1, θ2, θ3, θ4, θim,
d4=L4c1, c2, d1, d2, f1, f2, θ1, θ2, θ3, θ4, θim.
c3=-cos θ3m2d22-sin2 θ4-m1d11 cos2 θ4 cos θim2m2b22 cos2 θ3-m1b11 cos2 θ4 cos θim,
d3=m2b22-m1b11 cos θimm2b22 cos2 θ3-m1b11 cos2 θ4 cos θimsin2 θ4m2b22 cos2 θ3m1d11 cos θim-1-m1b11 cos θimm2d22-1,
c4=-cos θ4m1d11+2m1b11c3 sec θ3-sec θim2m1d11d3+b111+2c3d3 sec θ3,
d4=-m1 cos θimd11d3+b111+2c3d3 sec θ3.
b11=d0+d1+d2+2c2d2d0+d1sec θ2+2c1d02c2d1d2+d1+d2cos θ2×sec θ1 sec θ2sec θobj,
b22=d0+d1+d2+2c2d2d0+d1cos θ2+2c1d0d1+d2+2c2d1d2 cos θ2cos θ1,
d11=2c2d0+cos θ2+2c2d1×1+2c1d0 sec θ1sec θ2 sec θobj,
d22=2c2d0 cos θ2+1+2c2d1 cos θ2×1+2c1d0 cos θ1.
abcd=T2R2T1R1T0,
c3=-cos θ3f2c22+sin2 θ4-f1c11 cos2 θ4 cos θim2f2a22 cos2 θ3-f1a11 cos2 θ4 cos θim,
d3=f2a22-f1a11 cos θimf2a22 cos2 θ3-f1a11 cos2 θ4 cos θimsin2 θ4f2a22 cos2 θ3f1c11 cos θim+1-f1a11 cos θimf2c22+1,
c4=-cos θ4f1c11+2f1a11c3 sec θ3-sec θim2f1c11d3+a111+2c3d3 sec θ3,
d4=f1 cos θimc11d3+a111+2c3d3 sec θ3.
a11=1+2c2d2 sec θ2+2c1d1+d2+2c2d1d2 sec θ2sec θ1,
a22=1+2c2d2 cos θ2+2c1d1+d2+2c2d1d2 cos θ2cos θ1,
c11=2c1+c22c1d1+cos θ1sec θ2sec θ1,
c22=2c1+c22c1d1+sec θ1cos θ2cos θ1.

Metrics