Abstract

Repeated folding of the optical axis can be used to design space- and volume-efficient optical systems. We suggest that space-filling curves, such as the Peano and Hilbert curves, offer a useful way of realizing compact modular optics.

© 1994 Optical Society of America

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References

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  1. R. P. Bocker, H. J. Caulfield, K. Bromley, Appl. Opt. 22, 804 (1983).
    [CrossRef] [PubMed]
  2. Y. Ichioka, Department of Applied Physics, Faculty of Engineering, Osaka University, 2-1 Yamadaoka, Suita 565, Japan (personal communication, 1993).
  3. M. P. Schamschula, P. Reardon, H. J. Caulfield, C. F. Hester, “Regular geometries for folded optical modules,” Appl. Opt. (to be published).
  4. G. Peano, Math. Ann. 36, 157 (1890).
    [CrossRef]
  5. D. Hilbert, Math. Ann 38, 459 (1891).
    [CrossRef]

1983 (1)

1891 (1)

D. Hilbert, Math. Ann 38, 459 (1891).
[CrossRef]

1890 (1)

G. Peano, Math. Ann. 36, 157 (1890).
[CrossRef]

Bocker, R. P.

Bromley, K.

Caulfield, H. J.

R. P. Bocker, H. J. Caulfield, K. Bromley, Appl. Opt. 22, 804 (1983).
[CrossRef] [PubMed]

M. P. Schamschula, P. Reardon, H. J. Caulfield, C. F. Hester, “Regular geometries for folded optical modules,” Appl. Opt. (to be published).

Hester, C. F.

M. P. Schamschula, P. Reardon, H. J. Caulfield, C. F. Hester, “Regular geometries for folded optical modules,” Appl. Opt. (to be published).

Hilbert, D.

D. Hilbert, Math. Ann 38, 459 (1891).
[CrossRef]

Ichioka, Y.

Y. Ichioka, Department of Applied Physics, Faculty of Engineering, Osaka University, 2-1 Yamadaoka, Suita 565, Japan (personal communication, 1993).

Peano, G.

G. Peano, Math. Ann. 36, 157 (1890).
[CrossRef]

Reardon, P.

M. P. Schamschula, P. Reardon, H. J. Caulfield, C. F. Hester, “Regular geometries for folded optical modules,” Appl. Opt. (to be published).

Schamschula, M. P.

M. P. Schamschula, P. Reardon, H. J. Caulfield, C. F. Hester, “Regular geometries for folded optical modules,” Appl. Opt. (to be published).

Appl. Opt. (1)

Math. Ann (1)

D. Hilbert, Math. Ann 38, 459 (1891).
[CrossRef]

Math. Ann. (1)

G. Peano, Math. Ann. 36, 157 (1890).
[CrossRef]

Other (2)

Y. Ichioka, Department of Applied Physics, Faculty of Engineering, Osaka University, 2-1 Yamadaoka, Suita 565, Japan (personal communication, 1993).

M. P. Schamschula, P. Reardon, H. J. Caulfield, C. F. Hester, “Regular geometries for folded optical modules,” Appl. Opt. (to be published).

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Figures (4)

Fig. 1
Fig. 1

(a) First iteration in the creation of a Peano curve. In each step of construction a line segment is replaced by the nine line segments on the right. (b) The first and second stages of the Hilbert curve. Each square is successively replaced by the curve shown in the left-hand figure, oriented as shown in the right-hand figure.

Fig. 2
Fig. 2

Comparison of two curves: the area within the prism is covered twice, once by the incoming beam and once by the exiting beam. This occurs only in self-touching curves.

Fig. 3
Fig. 3

Correlator design based on the self-touching two-dimensional Peano curve. This example is an f/2 system with a collimated input.

Fig. 4
Fig. 4

4f, f/3 correlator based on the self-avoiding Hilbert curve.

Equations (3)

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D = log 9 log 3 = 2
GFM = ( AD + 3 V 3 6 .
Vol system = 4 f 3 ( f / # ) 3 ,

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