## Imaging a point with two spherical mirrors

JOSA A, Vol. 15, Issue 12, pp. 3045-3056 (1998)

http://dx.doi.org/10.1364/JOSAA.15.003045

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### Abstract

We investigate unobstructed, plane-symmetric systems of two spherical mirrors that are intended for imaging a single point. Low-order imaging constraints are determined that eliminate all but two of the seven parameters that specify the configuration of such a system. It is found that when these constraints are applied, the object point, the center of curvature of each mirror, and the image point necessarily lie along a single line. The associated merit function space is mapped as a function of the two independent parameters. Selected systems from various regions of the configuration space are illustrated.

© 1998 Optical Society of America

**OCIS Codes**

(080.2740) Geometric optics : Geometric optical design

(110.0110) Imaging systems : Imaging systems

(220.3620) Optical design and fabrication : Lens system design

(230.4040) Optical devices : Mirrors

**History**

Original Manuscript: May 5, 1998

Revised Manuscript: August 28, 1998

Manuscript Accepted: July 7, 1998

Published: December 1, 1998

**Citation**

Joseph M. Howard and Bryan D. Stone, "Imaging a point with two spherical mirrors," J. Opt. Soc. Am. A **15**, 3045-3056 (1998)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-12-3045

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### References

- B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–109 (1992). [CrossRef]
- B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992). [CrossRef]
- B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994). [CrossRef]
- B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994). [CrossRef]
- O. N. Stavroudis, “Two-mirror systems with spherical reflecting surfaces,” J. Opt. Soc. Am. 57, 741–748 (1967). [CrossRef]
- S. Rosin, “Inverse Cassegrainian systems,” Appl. Opt. 7, 1483–1497 (1968). [CrossRef] [PubMed]
- P. G. Hannan, “General analysis of two-mirror relay systems,” Appl. Opt. 31, 513–518 (1992). [CrossRef] [PubMed]
- J. Pan, X. Li, “Design of a tilted two-mirror system,” Opt. Rev. 1, 246–247 (1994). [CrossRef]
- B. M. Boshnyak, A. N. Korolev, “Synthesis of astigmatism-free condensers consisting of spherical mirrors,” Opt. Spectrosc. 43, 204–207 (1977).
- B. M. Boshnyak, “Meridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 40, 518–521 (1976).
- B. M. Boshnyak, “Extrameridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 41, 386–390 (1976).
- B. M. Boshnyak, “Summation of meridional aberrations of a spherical surfaces with arbitrary decentering angles,” Opt. Spectrosc. 41, 632–634 (1976).
- B. M. Boshnyak, “Addition of extrameridional aberrations of a spherical surfaces with large angles of decentering,” Opt. Spectrosc. 42, 106–110 (1977).
- For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2. In that reference the point-angle mixed characteristic is denoted by W2.
- For a description of the imaging properties that correspond to the terms of degree two in the Taylor expansion of Hamilton’s point-angle mixed characteristic see, for example, B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and the references cited therein. [CrossRef]
- The geometrical interpretation associated with Hamilton’s characteristic functions can be found in Chapter 2 of the reference cited in note 14, or alternatively in J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, UK, 1937), Sec. 6.
- A detailed example of a Taylor expansion of a characteristic function can be found in Section 2 of the reference cited in note 15.
- See Chapter 2 of the reference cited in note 14.
- For a description of Coddington’s equations, see, for example, Rudolf Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978), Sec. 10.1.
- A discussion of the Schwarzschild configuration for both infinite and finite conjugates can be found in W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, 2nd ed. (McGraw-Hill, New York, 1990), Sec. 13.2.The original paper on the subject discusses the infinite-conjugate case: K. Schwarzschild, “Untersuchungen zur geometrischen Optik, II: Theorie der Spiegeltelescope,” Abh. Königl. Ges. Wiss. Göttingen, Math.-Phys. Klasse 9, Neue Folge, Bd. IV, No. 2, 2–28 (1905).
- D. Shafer, “Optical design with only two surfaces,” in International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 256–261 (1980). [CrossRef]
- Although the systems of this paper are designed for imaging a single point, a magnification can be defined for these systems by considering the angle that a ray makes with the axis and taking the ratio of the object- and image-space angles of this ray. In the limit as the ray angles approach zero, this ratio approaches the transverse magnification. The axial NA is defined as the sine of the maximum angle (in object space) between the axis and the rays that are passed by the system.
- J. M. Howard, B. D. Stone, “Imaging a point to a line with a single spherical mirror,” Appl. Opt. 37, 1826–1834 (1998). [CrossRef]
- See Figs. 6 and 8 of Ref. 6.

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