Abstract

In this paper, a generalized shift-rotation absolute measurement method is proposed to measure the absolute surface shape of high-numerical-aperture spherical surfaces. Based on the wavefront difference method, the high order misalignment aberrations can be removed from the measurements. Our generalized shift-rotation absolute measurement process only needs one rotational measurement position and one translational measurement position. A wavefront reconstruction method based on the self-adaptive differential evolution algorithm is proposed to calculate the Zernike polynomials coefficient ai of the absolute surface shape Wtest(x,y), the rotation angle Δθ, the translation δx along the x axis, and the translation δy along the y axis. The translation error and rotation error in other absolute measurement methods are avoided using our generalized shift-rotation absolute measurement method. Experimental absolute results of the test surface and reference surface are given and the difference of reference surface shapes between two testings in experiments is 0.12 nm root mean square.

© 2017 Optical Society of America

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References

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  1. G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” (Lawrence Livermore National Lab., CA (United States), 1996).
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  3. N. I. Chkhalo, A. Y. Klimov, D. G. Raskin, V. V. Rogov, N. N. Salashchenko, and M. N. Toropov, “A new source of a reference spherical wave for a point diffraction interferometer,” in Proc. of SPIE Vol(2008), pp. 702506.
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    [PubMed]
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    [PubMed]
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    [PubMed]
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    [PubMed]
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2016 (2)

C. Tian and S. Liu, “Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization,” Opt. Express 24(4), 3202–3215 (2016).
[PubMed]

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

2015 (2)

Z. Yang, Z. Gao, S. Wang, X. Wang, and Q. Shi, “Sub-nanometer misalignment aberrations calibration of the Fizeau interferometer for high-numerical-aperture spherical surface ultra-precision measurement,” Optik-International Journal for Light and Electron Optics 126, 1865–1871 (2015).

Z. Yang, Z. Gao, D. Zhu, and Q. Yuan, “Absolute ultra-precision measurement of high-numerical-aperture spherical surface by high-order shift-rotation method using Zernike polynomials,” Opt. Commun. 355, 191–199 (2015).

2013 (1)

2012 (2)

2011 (2)

2010 (1)

2009 (1)

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13, 398–417 (2009).

2006 (1)

G. M. Dai, “Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms,” J. Refract. Surg. 22(9), 943–948 (2006).
[PubMed]

2002 (1)

K. Otaki, K. Ota, I. Nishiyama, T. Yamamoto, Y. Fukuda, and S. Okazaki, “Development of the point diffraction interferometer for extreme ultraviolet lithography: Design, fabrication, and evaluation,” J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 20, 2449–2458 (2002).

1997 (1)

R. Storn and K. Price, “Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces,” J. Glob. Optim. 11, 341–359 (1997).

1992 (1)

1980 (1)

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” JOSA 70, 998–1006 (1980).

Bloemhof, E. E.

Bon, P.

Chen, C.

Chkhalo, N. I.

N. I. Chkhalo, A. Y. Klimov, D. G. Raskin, V. V. Rogov, N. N. Salashchenko, and M. N. Toropov, “A new source of a reference spherical wave for a point diffraction interferometer,” in Proc. of SPIE Vol(2008), pp. 702506.

Creath, K.

Dai, G. M.

G. M. Dai, “Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms,” J. Refract. Surg. 22(9), 943–948 (2006).
[PubMed]

Fukuda, Y.

K. Otaki, K. Ota, I. Nishiyama, T. Yamamoto, Y. Fukuda, and S. Okazaki, “Development of the point diffraction interferometer for extreme ultraviolet lithography: Design, fabrication, and evaluation,” J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 20, 2449–2458 (2002).

Gao, Z.

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Z. Yang, Z. Gao, S. Wang, X. Wang, and Q. Shi, “Sub-nanometer misalignment aberrations calibration of the Fizeau interferometer for high-numerical-aperture spherical surface ultra-precision measurement,” Optik-International Journal for Light and Electron Optics 126, 1865–1871 (2015).

Z. Yang, Z. Gao, D. Zhu, and Q. Yuan, “Absolute ultra-precision measurement of high-numerical-aperture spherical surface by high-order shift-rotation method using Zernike polynomials,” Opt. Commun. 355, 191–199 (2015).

Hou, X.

Huang, V. L.

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13, 398–417 (2009).

Huang, Y.

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Klimov, A. Y.

N. I. Chkhalo, A. Y. Klimov, D. G. Raskin, V. V. Rogov, N. N. Salashchenko, and M. N. Toropov, “A new source of a reference spherical wave for a point diffraction interferometer,” in Proc. of SPIE Vol(2008), pp. 702506.

Liu, M.

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Liu, S.

Ma, J.

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Miao, E.

Monneret, S.

Nishiyama, I.

K. Otaki, K. Ota, I. Nishiyama, T. Yamamoto, Y. Fukuda, and S. Okazaki, “Development of the point diffraction interferometer for extreme ultraviolet lithography: Design, fabrication, and evaluation,” J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 20, 2449–2458 (2002).

Okazaki, S.

K. Otaki, K. Ota, I. Nishiyama, T. Yamamoto, Y. Fukuda, and S. Okazaki, “Development of the point diffraction interferometer for extreme ultraviolet lithography: Design, fabrication, and evaluation,” J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 20, 2449–2458 (2002).

Osten, W.

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Ota, K.

K. Otaki, K. Ota, I. Nishiyama, T. Yamamoto, Y. Fukuda, and S. Okazaki, “Development of the point diffraction interferometer for extreme ultraviolet lithography: Design, fabrication, and evaluation,” J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 20, 2449–2458 (2002).

Otaki, K.

K. Otaki, K. Ota, I. Nishiyama, T. Yamamoto, Y. Fukuda, and S. Okazaki, “Development of the point diffraction interferometer for extreme ultraviolet lithography: Design, fabrication, and evaluation,” J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 20, 2449–2458 (2002).

Price, K.

R. Storn and K. Price, “Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces,” J. Glob. Optim. 11, 341–359 (1997).

Pruss, C.

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Qin, A. K.

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13, 398–417 (2009).

Raskin, D. G.

N. I. Chkhalo, A. Y. Klimov, D. G. Raskin, V. V. Rogov, N. N. Salashchenko, and M. N. Toropov, “A new source of a reference spherical wave for a point diffraction interferometer,” in Proc. of SPIE Vol(2008), pp. 702506.

Rogov, V. V.

N. I. Chkhalo, A. Y. Klimov, D. G. Raskin, V. V. Rogov, N. N. Salashchenko, and M. N. Toropov, “A new source of a reference spherical wave for a point diffraction interferometer,” in Proc. of SPIE Vol(2008), pp. 702506.

Salashchenko, N. N.

N. I. Chkhalo, A. Y. Klimov, D. G. Raskin, V. V. Rogov, N. N. Salashchenko, and M. N. Toropov, “A new source of a reference spherical wave for a point diffraction interferometer,” in Proc. of SPIE Vol(2008), pp. 702506.

Shi, Q.

Z. Yang, Z. Gao, S. Wang, X. Wang, and Q. Shi, “Sub-nanometer misalignment aberrations calibration of the Fizeau interferometer for high-numerical-aperture spherical surface ultra-precision measurement,” Optik-International Journal for Light and Electron Optics 126, 1865–1871 (2015).

Song, W.

Southwell, W. H.

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” JOSA 70, 998–1006 (1980).

Storn, R.

R. Storn and K. Price, “Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces,” J. Glob. Optim. 11, 341–359 (1997).

Su, D.

Suganthan, P. N.

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13, 398–417 (2009).

Sui, Y.

Sun, W.

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Tian, C.

Toropov, M. N.

N. I. Chkhalo, A. Y. Klimov, D. G. Raskin, V. V. Rogov, N. N. Salashchenko, and M. N. Toropov, “A new source of a reference spherical wave for a point diffraction interferometer,” in Proc. of SPIE Vol(2008), pp. 702506.

Wan, Y.

Wang, D.

Wang, S.

Z. Yang, Z. Gao, S. Wang, X. Wang, and Q. Shi, “Sub-nanometer misalignment aberrations calibration of the Fizeau interferometer for high-numerical-aperture spherical surface ultra-precision measurement,” Optik-International Journal for Light and Electron Optics 126, 1865–1871 (2015).

Wang, X.

Z. Yang, Z. Gao, S. Wang, X. Wang, and Q. Shi, “Sub-nanometer misalignment aberrations calibration of the Fizeau interferometer for high-numerical-aperture spherical surface ultra-precision measurement,” Optik-International Journal for Light and Electron Optics 126, 1865–1871 (2015).

Wattellier, B.

Wu, F.

Wyant, J. C.

Yamamoto, T.

K. Otaki, K. Ota, I. Nishiyama, T. Yamamoto, Y. Fukuda, and S. Okazaki, “Development of the point diffraction interferometer for extreme ultraviolet lithography: Design, fabrication, and evaluation,” J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 20, 2449–2458 (2002).

Yang, H.

Yang, Y.

Yang, Z.

Z. Yang, Z. Gao, S. Wang, X. Wang, and Q. Shi, “Sub-nanometer misalignment aberrations calibration of the Fizeau interferometer for high-numerical-aperture spherical surface ultra-precision measurement,” Optik-International Journal for Light and Electron Optics 126, 1865–1871 (2015).

Z. Yang, Z. Gao, D. Zhu, and Q. Yuan, “Absolute ultra-precision measurement of high-numerical-aperture spherical surface by high-order shift-rotation method using Zernike polynomials,” Opt. Commun. 355, 191–199 (2015).

Yuan, C.

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Yuan, Q.

Z. Yang, Z. Gao, D. Zhu, and Q. Yuan, “Absolute ultra-precision measurement of high-numerical-aperture spherical surface by high-order shift-rotation method using Zernike polynomials,” Opt. Commun. 355, 191–199 (2015).

Zhu, D.

Z. Yang, Z. Gao, D. Zhu, and Q. Yuan, “Absolute ultra-precision measurement of high-numerical-aperture spherical surface by high-order shift-rotation method using Zernike polynomials,” Opt. Commun. 355, 191–199 (2015).

Zhu, R.

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Zhuo, Y.

Appl. Opt. (5)

IEEE Trans. Evol. Comput. (1)

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13, 398–417 (2009).

J. Glob. Optim. (1)

R. Storn and K. Price, “Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces,” J. Glob. Optim. 11, 341–359 (1997).

J. Refract. Surg. (1)

G. M. Dai, “Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms,” J. Refract. Surg. 22(9), 943–948 (2006).
[PubMed]

J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. (1)

K. Otaki, K. Ota, I. Nishiyama, T. Yamamoto, Y. Fukuda, and S. Okazaki, “Development of the point diffraction interferometer for extreme ultraviolet lithography: Design, fabrication, and evaluation,” J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 20, 2449–2458 (2002).

JOSA (1)

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” JOSA 70, 998–1006 (1980).

Opt. Commun. (1)

Z. Yang, Z. Gao, D. Zhu, and Q. Yuan, “Absolute ultra-precision measurement of high-numerical-aperture spherical surface by high-order shift-rotation method using Zernike polynomials,” Opt. Commun. 355, 191–199 (2015).

Opt. Eng. (1)

Y. Huang, J. Ma, C. Yuan, C. Pruss, W. Sun, M. Liu, R. Zhu, Z. Gao, and W. Osten, “Absolute test for cylindrical surfaces using the conjugate differential method,” Opt. Eng. 55, 114104 (2016).

Opt. Express (1)

Opt. Lett. (2)

Optik-International Journal for Light and Electron Optics (1)

Z. Yang, Z. Gao, S. Wang, X. Wang, and Q. Shi, “Sub-nanometer misalignment aberrations calibration of the Fizeau interferometer for high-numerical-aperture spherical surface ultra-precision measurement,” Optik-International Journal for Light and Electron Optics 126, 1865–1871 (2015).

Other (4)

G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” (Lawrence Livermore National Lab., CA (United States), 1996).

U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” in Proc. SPIE(2005), p. 58690S.

N. I. Chkhalo, A. Y. Klimov, D. G. Raskin, V. V. Rogov, N. N. Salashchenko, and M. N. Toropov, “A new source of a reference spherical wave for a point diffraction interferometer,” in Proc. of SPIE Vol(2008), pp. 702506.

Q. Yuan, Z. Gao, and Z. Yang, “Simultaneously testing surface figure and radius of curvature for spheres by a point diffraction interferometer,” in Optical Fabrication and Testing (Optical Society of America, 2014), p. OM3C. 7.

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Figures (15)

Fig. 1
Fig. 1

Misalignment aberration introduced by the test spherical surface misalignment.

Fig. 2
Fig. 2

The measurement process: (a) Basic measurement, (b) Rotational measurement, (c) Translational measurement.

Fig. 3
Fig. 3

The process of Wavefront reconstruction process based on self-adaptive differential evolution algorithm.

Fig. 4
Fig. 4

The slope of the ideal surface (units: nm). (a) The coefficients of Zernike polynomials of the test surface and reference surface, (b) The test surface slope, (c) The reference surface slope.

Fig. 5
Fig. 5

The measurement results on the three positions (units: nm). (a) The original position, (b) The rotational position, (c) The translational position.

Fig. 6
Fig. 6

Wavefront difference data (units: nm). (a) Between rotational and original position, (b) Between translational and original position.

Fig. 7
Fig. 7

The simulation results (units: nm). (a) The Zernike polynomials coefficients of the test surface shape compared with the Zernike polynomials coefficients of simulated test surface, (b) Absolute test surface shape, (c) The comparison of the surface deviation of the simulated surface and test surface shape.

Fig. 8
Fig. 8

The measurement results of the first test spherical surface on the three positions in experiment (units: nm). (a) The original position, (b) The rotational position, (c) The translational position.

Fig. 9
Fig. 9

The wavefront difference data of the first test spherical surface in experiment (units: nm). (a) Between rotational and original position, (b) Between translational and original position.

Fig. 10
Fig. 10

The measurement results of the first test spherical surface in experiment (units: nm). (a) The absolute test surface shape, (b) The reference surface shape.

Fig. 11
Fig. 11

The measurement results of the second test spherical surface on the three positions in experiment (units: nm). (a) The original position, (b) The rotational position, (c) The translational position.

Fig. 12
Fig. 12

The wavefront difference data of the second test spherical surface in experiment (units: nm). (a) Wavefront difference data between rotational and original position, (b) Wavefront difference data between translational and original position.

Fig. 13
Fig. 13

The measurement results of the second test spherical surface in experiment (units: nm). (a) The absolute test surface shape, (b) The reference surface shape.

Fig. 14
Fig. 14

Deference of reference surface shapes between two testing (units: nm). (a) The Zernike polynomials coefficients of the reference surface shape, (b) The wavefront difference data between two testing.

Fig. 15
Fig. 15

The simulate results with noise (units: nm). (a) The Zernike polynomials coefficients of the test surface shape compared with the Zernike polynomials coefficients of simulated test surface, (b) Absolute test surface shape, (c) The comparison of the surface deviation of the simulated surface and test surface shape.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

T(x,y)= W test (x,y)+ W reference (x,y)+ W adjustment (x,y)+ τ noise (x,y)
T(x,y)= W test (x,y)+ W reference (x,y)+ W adjustment (x,y)
W adjustment (x,y)= a 1 Z 1 1 4R ( a 1 Z 1 ) 2 + 1 8 R 2 ( a 1 Z 1 ) 3 + a 2 Z 2 1 4R ( a 2 Z 2 ) 2 + 1 8 R 2 ( a 2 Z 2 ) 3 + a 3 Z 3 + a 8 ' Z 8 + a 15 ' Z 15 + a 24 ' Z 24 + a 35 ' Z 35 + a 48 ' Z 48
{ T 0 (x,y)= W test (x,y)+ W reference (x,y)+ W adjustment0 (x,y) T 0 '(x,y)= W test (x,y)+ W reference (x,y)+ W adjustment0' (x,y)
T 0 (x,y) T 0 '(x,y)= W adjustment0 (x,y) W adjustment0' (x,y) =Δ a 1 Z 1 1 4R (Δ a 1 Z 1 ) 2 + 1 8 R 2 (Δ a 1 Z 1 ) 3 +Δ a 1 Z 1 1 4R (Δ a 1 Z 1 ) 2 + 1 8 R 2 (Δ a 1 Z 1 ) 3 +Δ a 3 Z 3 + r 1 Δ a 3 Z 8 + r 2 Δ a 3 Z 15 + r 3 Δ a 3 Z 24 + r 4 Δ a 3 Z 35 + r 5 Δ a 3 Z 48
T 1 (r,θ)= W test (r,θ+Δθ)+ W reference (r,θ)+ W adjustment1 (r,θ)
T 2 (x,y)= W test (θ+Δθ;x+ δ x ,y+ δ y )+ W reference (x,y)+ W adjustment2 (x,y)
{ T 0 (x,y)= W test (x,y)+ W reference (x,y) T 1 (r,θ)= W test (r,θ+Δθ)+ W reference (r,θ) T 2 (x,y)= W test (θ+Δθ;x+ δ x ,y+ δ y )+ W reference (x,y)
{ Δ W 1 (r,θ)= W test (r,θ+Δθ) W test (r,θ)= T 1 (x,y) T 0 (x,y) Δ W 2 (x,y)= W test (θ+Δθ;x+ δ x ,y+ δ y ) W test (x,y)= T 2 (x,y) T 0 (x,y)
{ W test (x,y)= i=1 N a i Z i (x,y) W test (x,y)= i=1 N a i Z i (r,θ)
{ Δ W 1 (r,θ)= i=1 N a i [ Z(r,θ+Δθ)Z(r,θ) ] Δ W 2 (x,y)= i=1 N a i [ Z(θ+Δθ,x+ δ x ,y+ δ y )Z(x,y) ]
f( a 1 , a 2 ,, a N ,Δθ, δ x , δ y ) = (x,y)S { { [ T 1 (x,y) T 0 (x,y) ] i=1 N a i [ Z(r,θ+Δθ)Z(r,θ) ] } + + { [ T 2 (x,y) T 0 (x,y) ] i=1 N a i [ Z(θ+Δθ,x+ δ x ,y+ δ y )Z(x,y) ] } }
X( a 1 , a 2 ,, a N ,Δθ, δ x , δ y )=argminf( X )
X i,G ={ x i,G 1 , x i,G 2 ,, x i,G D } ={ a 1i,G 1 , a 2i,G 2 ,, a Ni,G N ,Δ θ i,G N+1 δ xi,G N+2 , δ yi,G N+3 }
{ X max ={ x max 1 , x max 2 ,, x max D } X min ={ x max 1 , x min 2 ,, x min D }
x i,0 j = x min j +rand(0,1)×( x max j x min j )
u i,j ={ x r 1 ,j +F( x r 2 ,j x r 3 ,j ),ifrand[0,1]<CRorj= j rand x i,j ,otherwise
X i,G+1 ={ U i,G ,iff( U i,G )f( X i,G ) X i,j ,otherwise