## Imaging a point to a line with a single spherical mirror

Applied Optics, Vol. 37, Issue 10, pp. 1826-1834 (1998)

http://dx.doi.org/10.1364/AO.37.001826

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### Abstract

Methods for imaging a single point source to a line image are discussed, and a design study of single spherical mirror systems that form aberration-free line images is presented. An expression for the ray density along the line image is derived for such systems in the cases of (i) uniform beam profiles and (ii) Gaussian beam profiles. The resulting ray density profiles are illustrated for single spherical mirror systems over a wide range of design parameters.

© 1998 Optical Society of America

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(080.3620) Geometric optics : Lens system design

(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments

(230.4040) Optical devices : Mirrors

**History**

Original Manuscript: July 21, 1997

Revised Manuscript: November 19, 1997

Published: April 1, 1998

**Citation**

Joseph M. Howard and Bryan D. Stone, "Imaging a point to a line with a single spherical mirror," Appl. Opt. **37**, 1826-1834 (1998)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-37-10-1826

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### References

- D. W. Coutts, “Optimization of line-focusing geometry for efficient nonlinear frequency conversion from copper-vapor lasers,” IEEE J. Quantum Electron. 31, 2208–2214 (1995). [CrossRef]
- I. N. Ross, E. M. Hodgson, “Some optical designs for the generation of high quality line foci,” J. Phys. E 18, 169–173 (1985). [CrossRef]
- I. N. Ross, J. Boon, R. Corbett, A. Damerell, P. Gottfeldt, C. Hooker, M. H. Key, G. Kiehn, C. Lewis, O. Willi, “Design and performance of a new line focus geometry for x-ray laser experiments,” Appl. Opt. 26, 1584–1588 (1987). [CrossRef] [PubMed]
- L. V. L’vov, S. G. Merkulov, A. V. Ryadov, “Device for line focusing of radiation,” Sov. J. Quantum Electron. 24, 921–923 (1994). [CrossRef]
- For a general discussion of line foci see, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1989), Section 4.6.
- For a description of Coddington’s equations, see, for example, R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), Section 10.1.
- Methods for determining coefficients from which line image locations can be found appear in 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, 96–110 (1992). The use of these coefficients to determine line image locations is described in B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992).
- For a detailed discussion of second-order aberrations at a line focus see, for example, J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, England, 1937), Section 10.
- For details on the use of pairs of tilted plates in converging beams see S. Rosin, M. Amon, “Extending the stellar field of view of Ritchey-Chretien telescopes,” Appl. Opt. 11, 1623–1629 (1972).
- In this paper, the subscript 0 is used to identify quantities associated with the base ray.
- It is assumed here (and in Section 4) that the cone of rays does not include the axial ray, so that ω cannot equal zero (i.e., the pupil of the system is located completely off axis). In cases in which the axis is included in the cone, the end points of the line image are determined from the ray with angle ω0 + β and the paraxial image location of the object point.
- The R number is defined as the reciprocal of the product of the curvature and the clear aperture (CA) of the mirror: R number = (c CA)-1.
- If the cone of rays contains the axis (so that ω is zero for some ray in the system) then (dF/dω)|ω=0 and the ray density becomes infinite at the paraxial image location. In another example, if d0 = ∞ and the mirror is parabolic, all rays will merge to a single focal point on the axis, resulting in dω/dδ′ = 0, which also gives an infinite ray density.
- J. M. Howard, B. D. Stone are preparing the following paper for publication: “Imaging a point with two spherical mirrors.”

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