Abstract

A new technique for the generation of optical reference signals with optimal properties is presented. In grating measurement systems a reference signal is needed to achieve an absolute measurement of the position. The optical signal is the autocorrelation of two codes with binary transmittance. For a long time, the design of this type of code has required great computational effort, which limits the size of the code to 30 elements. Recently, the application of the dividing rectangles (DIRECT) algorithm has allowed the automatic design of codes up to 100 elements. Because of the binary nature of the problem and the parallel processing of the genetic algorithms, these algorithms are efficient tools for obtaining codes with particular autocorrelation properties. We design optimum zero reference codes with arbitrary length by means of a genetic algorithm enhanced with a restricted search operator.

© 2005 Optical Society of America

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References

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  1. X. Yang and C. Yin, J. Phys. E 19, 34 (1986).
    [CrossRef]
  2. L. Yajun, J. Mod. Opt. 34, 1571 (1987).
    [CrossRef]
  3. L. Yajun, Optik (Stuttgart) 79, 67 (1988).
  4. J. Sáez-Landete, J. Alonso, and E. Bernabeu, Opt. Express 13, 195 (2005).
    [CrossRef]
  5. D. R. Jones, in Encyclopedia of Optimization, Vol. 1 (Kluwer Academic, 2001), pp. 431–440.
    [CrossRef]
  6. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1988).
  7. S. Salcedo-Sanz, G. Camps-Vals, F. Pérez-Cruz, J. Sepúlveda-Sanchís, and C. Bousoño-Calzón, IEEE Trans. Syst. Man Cybern. 34, 749 (2005).

2005 (2)

J. Sáez-Landete, J. Alonso, and E. Bernabeu, Opt. Express 13, 195 (2005).
[CrossRef]

S. Salcedo-Sanz, G. Camps-Vals, F. Pérez-Cruz, J. Sepúlveda-Sanchís, and C. Bousoño-Calzón, IEEE Trans. Syst. Man Cybern. 34, 749 (2005).

1988 (1)

L. Yajun, Optik (Stuttgart) 79, 67 (1988).

1987 (1)

L. Yajun, J. Mod. Opt. 34, 1571 (1987).
[CrossRef]

1986 (1)

X. Yang and C. Yin, J. Phys. E 19, 34 (1986).
[CrossRef]

Alonso, J.

Bernabeu, E.

Bousoño-Calzón, C.

S. Salcedo-Sanz, G. Camps-Vals, F. Pérez-Cruz, J. Sepúlveda-Sanchís, and C. Bousoño-Calzón, IEEE Trans. Syst. Man Cybern. 34, 749 (2005).

Camps-Vals, G.

S. Salcedo-Sanz, G. Camps-Vals, F. Pérez-Cruz, J. Sepúlveda-Sanchís, and C. Bousoño-Calzón, IEEE Trans. Syst. Man Cybern. 34, 749 (2005).

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1988).

Jones, D. R.

D. R. Jones, in Encyclopedia of Optimization, Vol. 1 (Kluwer Academic, 2001), pp. 431–440.
[CrossRef]

Pérez-Cruz, F.

S. Salcedo-Sanz, G. Camps-Vals, F. Pérez-Cruz, J. Sepúlveda-Sanchís, and C. Bousoño-Calzón, IEEE Trans. Syst. Man Cybern. 34, 749 (2005).

Sáez-Landete, J.

Salcedo-Sanz, S.

S. Salcedo-Sanz, G. Camps-Vals, F. Pérez-Cruz, J. Sepúlveda-Sanchís, and C. Bousoño-Calzón, IEEE Trans. Syst. Man Cybern. 34, 749 (2005).

Sepúlveda-Sanchís, J.

S. Salcedo-Sanz, G. Camps-Vals, F. Pérez-Cruz, J. Sepúlveda-Sanchís, and C. Bousoño-Calzón, IEEE Trans. Syst. Man Cybern. 34, 749 (2005).

Yajun, L.

L. Yajun, Optik (Stuttgart) 79, 67 (1988).

L. Yajun, J. Mod. Opt. 34, 1571 (1987).
[CrossRef]

Yang, X.

X. Yang and C. Yin, J. Phys. E 19, 34 (1986).
[CrossRef]

Yin, C.

X. Yang and C. Yin, J. Phys. E 19, 34 (1986).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

S. Salcedo-Sanz, G. Camps-Vals, F. Pérez-Cruz, J. Sepúlveda-Sanchís, and C. Bousoño-Calzón, IEEE Trans. Syst. Man Cybern. 34, 749 (2005).

J. Mod. Opt. (1)

L. Yajun, J. Mod. Opt. 34, 1571 (1987).
[CrossRef]

J. Phys. E (1)

X. Yang and C. Yin, J. Phys. E 19, 34 (1986).
[CrossRef]

Opt. Express (1)

Optik (Stuttgart) (1)

L. Yajun, Optik (Stuttgart) 79, 67 (1988).

Other (2)

D. R. Jones, in Encyclopedia of Optimization, Vol. 1 (Kluwer Academic, 2001), pp. 431–440.
[CrossRef]

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1988).

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Figures (4)

Fig. 1
Fig. 1

Procedure for generation of the autocorrelation signal. We consider the gap between ZRCs close with regard to the width of the code slits and the width of the code slits greater than wavelength of the illuminating light.

Fig. 2
Fig. 2

Example of crossover operation, in which the initial couple swaps its genes from position 3.

Fig. 3
Fig. 3

Comparison of the second maximum reached with the DIRECT algorithm, the GA, and the theoretical lower bound. The code length is 200, and n 1 varies from 1 to 199.

Fig. 4
Fig. 4

Autocorrelation signal generated with an optimum code. The length of the code is 1000, and n 1 = 100 . The value of the second maximum is 13.

Equations (5)

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S k = S k = i = 1 n k c j c j + k , k = 0 , 1 , , n 1 ,
σ ( 2 n + 1 ) ( 2 n + 1 ) 2 4 n 1 ( n 1 1 ) 2 .
f ( c ) min c f ( c ) = max { S 1 , , S n 1 } , S k = j = 1 n k c j c j + k ,
k = 1 , , n 1 ,
( n n 1 ) .

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