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Optics Letters

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  • Editor: Alan E. Willner
  • Vol. 34, Iss. 19 — Oct. 1, 2009
  • pp: 3018–3019
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Novel aplanatic designs

Roland Winston and Weiya Zhang  »View Author Affiliations


Optics Letters, Vol. 34, Issue 19, pp. 3018-3019 (2009)
http://dx.doi.org/10.1364/OL.34.003018


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Abstract

We have discovered an aplanatic design that contributes to a celebrated problem in classical optics in a novel way. In so doing new devices are envisioned with applications to illumination, concentration, and imaging.

© 2009 Optical Society of America

It is remarkable that an on-axis condition that can be simply formulated ensures good off-axis performance. The condition is that rays parallel to the axis intersect rays converging to the focus on the surface of a sphere, called “Abbe Sphere” [1

1. M. Born and E. Wolf, Principles of Optics (Macmillan, 1964).

]. (For recent advances in general sine conditions, see [2

2. C. Zhao and J. H. Burge, J. Opt. Soc. Am. A 19, 2467 (2002). [CrossRef]

].) Requiring both an on-axis image of infinity and the aplanatic condition clearly requires at least two surfaces (except for special configurations of high symmetry, like the Luneburg lens [3

3. R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964).

]). The method of designing a two-surface reflecting aplanat is illustrated by the classic monographs by Luneburg [3

3. R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964).

], Korsch [4

4. D. Korsch, Reflective Optics (Academic, 1991).

], and Mertz [5

5. L. N. Mertz, Excursions in Astronomical Optics (Springer, 1996). [CrossRef]

]. Analytical solutions have been given by Schawarzchild [6

6. K. Schawarzchild, Mitt. Sternw. Gottingen , 10, 11 (1905).

], Puryayev and Gontcharov [7

7. D. T. Puryayev and A. V. Gontcharov, Opt. Eng. 37, 2334 (1998). [CrossRef]

], and more recently by Bell [8

8. D. Lynden-Bell, Mon. Not. R. Astron. Soc. 334, 787 (2002). [CrossRef]

]. Following Luneburg and assuming rotational symmetric around optical axis as shown in Fig. 1 , the entire two surfaces can be constructed by successive approximation starting from the optical axis. The key is to adjust the slope of both surfaces properly such that the aplanatic condition is satisfied. The starting positions of the two surfaces along the optical axis are free parameters. A family of two-mirror designs can be generated by changing these two parameters or their equivalent [8

8. D. Lynden-Bell, Mon. Not. R. Astron. Soc. 334, 787 (2002). [CrossRef]

, 9

9. R. V. Willstrop and D. Lynden-Bell, Mon. Not. R. Astron. Soc. 342, 33 (2003). [CrossRef]

].

Fortunately, nature does offer something close to a “one-way” mirror. It is well known that for light incident on an interface between two refractive media, light from the lower refractive index is transmitted (except for Fresnel reflection loss), while light from the higher refractive index at incident angles higher than the critical angle is totally reflected as if the interface is a mirror, an effect called total internal reflection (TIR). This suggests that the problem can be addressed by filling the space between the two mirror surfaces with a refractive medium and reflective coating the rear surface. Then, over portions of the internal surface where the incidence angle exceeds the critical angle, we indeed have a one-way mirror. So the design strategy will be to treat the refractive medium as a one-way mirror and discard the portions where TIR fails or simply coat with a reflective surface. There is, however, a more vexing obstacle to overcome, causality! The front surface transmits light but also introduces a disturbance in the horizontal ray direction by refraction. Unfortunately, this disturbance is indeterminate, because it depends on the slope, which is not determined until future steps in the iteration. We address this problem by a new prescription, which includes two keys: 1) a self-consistent initial configuration and 2) building the two surfaces from edge to center (optical axis) such that the slope of the front surface is determined before ray tracing is required at the location.

Figure 3 shows a solution for an angular aperture of 120 deg (θ = 60 deg), which is a very fast system. The sharp focusing of both on-axial and off-axial (2 deg) rays indicates the aplanatic nature of the design. The central region of the front surface is coated with reflective material, but the obscuration of the input light is less than 4%. In other words, over most area of the front surface we have a “one-way” mirror. The aspect (height:diameter) of this aplanat is approximately 1:3. To our knowledge, this is a remarkably fast and compact system. We conceived of this as a high concentration device; therefore by the sine law of concentration [10

10. R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

] it has inherently small field of view. As a result, the higher-order Seidel aberrations [11

11. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974).

], which depend on higher powers of field angle, are small, making it useful for imaging. For example, the size of the spot diagram at 2 deg off-axis is ∼0.001 of the entrance aperture. For comparison, this ratio would be ∼0.025 for a parabola. Chromatic aberration is certainly present. For nonimaging optical application such as solar concentration, we found the effect is to decrease the acceptance angle by about 4%, which is negligible in most cases. For imaging applications, this sets a limit on the width of the spectral band.

Compact, fast aplanats have a broad vista of applications to solar energy, imaging, light collection, and illumination. As imaging devices, they are a compact telescope with large light-gathering power. For illumination, with an LED at the focal plane, they are a bright, collimated flash light. For solar energy applications, they are a high-concentration, optimal acceptance angle device suitable for highly efficient but costly multijunction solar cells.

We are grateful to one of our reviewers for bringing additional references to our attention, and we thank project Dedalos for support of this research.

Fig. 1 Luneburg method of constructing a two-mirror aplanat. The constructed surfaces are shown in the curves with dots.
Fig. 2 Method of constructing a two-surface aplanat with refractive media. The constructed surfaces are shown in the curves with dots.
Fig. 3 Example of two-surface aplanats with refractive media. (a) On-axial (solid) and off-axial (2 deg, dashed) ray tracing showing on the cross section of the aplanat, where the rays would have hit the central obscuration area are blocked by a screen for illustration purpose. Also shown is the close-up on the focal region, where sharp focusing of both on-axial and off-axial rays indicates the aplanatic nature of the design. (b) 3-D rendering of the aplanat.
1.

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

2.

C. Zhao and J. H. Burge, J. Opt. Soc. Am. A 19, 2467 (2002). [CrossRef]

3.

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964).

4.

D. Korsch, Reflective Optics (Academic, 1991).

5.

L. N. Mertz, Excursions in Astronomical Optics (Springer, 1996). [CrossRef]

6.

K. Schawarzchild, Mitt. Sternw. Gottingen , 10, 11 (1905).

7.

D. T. Puryayev and A. V. Gontcharov, Opt. Eng. 37, 2334 (1998). [CrossRef]

8.

D. Lynden-Bell, Mon. Not. R. Astron. Soc. 334, 787 (2002). [CrossRef]

9.

R. V. Willstrop and D. Lynden-Bell, Mon. Not. R. Astron. Soc. 342, 33 (2003). [CrossRef]

10.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

11.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974).

OCIS Codes
(080.0080) Geometric optics : Geometric optics
(220.0220) Optical design and fabrication : Optical design and fabrication

ToC Category:
Optical Devices

History
Original Manuscript: July 30, 2009
Revised Manuscript: September 2, 2009
Manuscript Accepted: September 3, 2009
Published: September 30, 2009

Citation
Roland Winston and Weiya Zhang, "Novel aplanatic designs," Opt. Lett. 34, 3018-3019 (2009)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-19-3018


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References

  1. M. Born and E. Wolf, Principles of Optics (Macmillan, 1964).
  2. C. Zhao and J. H. Burge, J. Opt. Soc. Am. A 19, 2467 (2002). [CrossRef]
  3. R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964).
  4. D. Korsch, Reflective Optics (Academic, 1991).
  5. L. N. Mertz, Excursions in Astronomical Optics (Springer, 1996). [CrossRef]
  6. K. Schawarzchild, Mitt. Sternw. Gottingen , 10, 11 (1905).
  7. D. T. Puryayev and A. V. Gontcharov, Opt. Eng. 37, 2334 (1998). [CrossRef]
  8. D. Lynden-Bell, Mon. Not. R. Astron. Soc. 334, 787 (2002). [CrossRef]
  9. R. V. Willstrop and D. Lynden-Bell, Mon. Not. R. Astron. Soc. 342, 33 (2003). [CrossRef]
  10. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).
  11. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974).

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