Abstract

We give an analytical basis for the theory of optimal beam splitting by one-dimensional gratings. In particular, we use methods from the calculus of variations to derive analytical expressions for the optimal phase function.

© 2007 Optical Society of America

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References

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  1. G. H. Dammann and K. Görtler, "High efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
    [CrossRef]
  2. L. P. Boivin, "Multiple imaging using various types of phase gratings," Appl. Opt. 11, 1782-1792 (1972).
    [CrossRef] [PubMed]
  3. H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
    [CrossRef]
  4. T. Lizotte, R. Rosenberg, and O. Obar, "Actual performance vs. modeled performance of diffractive beam splitters," Proc. SPIE 5876, 505-515 (2005).
  5. W. H. Lee, "High efficiency multiple beam gratings," Appl. Opt. 18, 2152-2158 (1979).
    [CrossRef] [PubMed]
  6. J. Jahns, M. M. Downs, M. E. Price, N. Streibl, and S. J. Walker, "Dammann gratings for laser beam shaping," Opt. Eng. (Bellingham) 28, 1267-1275 (1989).
  7. J. N. Mait, "Design for two dimensional non-separable array generators," Proc. SPIE 1555, 43-52 (1991).
    [CrossRef]
  8. J. Turunen, A. Vasara, J. Westrholm, G. Jin, and A. Salin, "Optimization and fabrication of beam splitters," J. Phys. D 21, s202-s205 (1988).
    [CrossRef]
  9. M. P. Dames, R. J. Dowling, P. McKee, and D. Wood, "Efficient optimization elements to generate intensity weighted spot arrays: design and fabrication," Appl. Opt. 30, 2685-2691 (1991).
    [CrossRef] [PubMed]
  10. D. C. O'Shea, "Reduction of the zero-order intensity in binary Dammann gratings," Appl. Opt. 34, 6533-6537 (1995).
    [CrossRef] [PubMed]
  11. F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Technique (CRC Press, 2000).
    [CrossRef]
  12. F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping: Applications (CRC Press, 2005).
    [CrossRef]
  13. F.M.Dickey and D.L.Shealy, eds., Laser Beam Shaping VI, Proc. SPIE 5876, 380 pp. (2005).
  14. T. E. Lizotte, "Beam shaping for microvia drilling," Printed Circuit Fabr. 26, 28-33 (2003).
  15. F. M. Dickey, "Laser beam shaping," Opt. Photonics News 14, 30-35 (2003).
    [CrossRef]
  16. I. M. Gelfand and S. V. Fomin, The Calculus of Variations (Prentice Hall, 1963).
  17. L. A. Romero and F. M. Dickey, "Theory for optimal beam splitting by phase gratings. II. Square and hexagonal gratings," J. Opt. Soc. Am. A 24, 2296-2312 (2007).
    [CrossRef]
  18. J. Tervo and J. Turunen, "Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings," Opt. Lett. 25, 785-786 (2000).
    [CrossRef]
  19. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, and E. DiFabrizio, "Analytical derivation of the optimum triplicator," Opt. Commun. 157, 13-16 (1998).
    [CrossRef]
  20. R. Borghi, G. Cincotti, and M. Santarsiero, "Diffractive variable beam splitter: optimal design," J. Opt. Soc. Am. A 17, 63-67 (2000).
    [CrossRef]
  21. D. Prongue, H. P. Herzig, R. Dandliker, and M. T. Gale, "Optimized kinoform structures for highly efficient fan-out elements," Appl. Opt. 31, 5706-5711 (1992).
    [CrossRef] [PubMed]
  22. F. Wyrowski, "Upper bound of the diffractive efficiency of diffractive phase elements," Opt. Lett. 16, 1915-1917 (1991).
    [CrossRef] [PubMed]
  23. U. Krackhardt, J. N. Mait, and N. Streibl, "Upper bound on the diffraction efficiency of phase-only fan out elements," Appl. Opt. 31, 27-37 (1992).
    [CrossRef] [PubMed]
  24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  25. F. Gori, "Diffractive optics and introduction," in Diffractive Optics and Optical Microsystems, S.Martellucci and A.N.Chester, eds. (Plenum, 1997).
  26. E. Hecht and A. Zajac, Optics (Addison Wesley, 1974).

2007 (1)

2005 (1)

T. Lizotte, R. Rosenberg, and O. Obar, "Actual performance vs. modeled performance of diffractive beam splitters," Proc. SPIE 5876, 505-515 (2005).

2003 (2)

T. E. Lizotte, "Beam shaping for microvia drilling," Printed Circuit Fabr. 26, 28-33 (2003).

F. M. Dickey, "Laser beam shaping," Opt. Photonics News 14, 30-35 (2003).
[CrossRef]

2000 (2)

1998 (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, and E. DiFabrizio, "Analytical derivation of the optimum triplicator," Opt. Commun. 157, 13-16 (1998).
[CrossRef]

1995 (1)

1992 (2)

1991 (3)

1989 (1)

J. Jahns, M. M. Downs, M. E. Price, N. Streibl, and S. J. Walker, "Dammann gratings for laser beam shaping," Opt. Eng. (Bellingham) 28, 1267-1275 (1989).

1988 (1)

J. Turunen, A. Vasara, J. Westrholm, G. Jin, and A. Salin, "Optimization and fabrication of beam splitters," J. Phys. D 21, s202-s205 (1988).
[CrossRef]

1979 (1)

1977 (1)

H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
[CrossRef]

1972 (1)

1971 (1)

G. H. Dammann and K. Görtler, "High efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Boivin, L. P.

Borghi, R.

R. Borghi, G. Cincotti, and M. Santarsiero, "Diffractive variable beam splitter: optimal design," J. Opt. Soc. Am. A 17, 63-67 (2000).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, and E. DiFabrizio, "Analytical derivation of the optimum triplicator," Opt. Commun. 157, 13-16 (1998).
[CrossRef]

Cincotti, G.

R. Borghi, G. Cincotti, and M. Santarsiero, "Diffractive variable beam splitter: optimal design," J. Opt. Soc. Am. A 17, 63-67 (2000).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, and E. DiFabrizio, "Analytical derivation of the optimum triplicator," Opt. Commun. 157, 13-16 (1998).
[CrossRef]

Dames, M. P.

Dammann, G. H.

G. H. Dammann and K. Görtler, "High efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Dammann, H.

H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
[CrossRef]

Dandliker, R.

Dickey, F. M.

L. A. Romero and F. M. Dickey, "Theory for optimal beam splitting by phase gratings. II. Square and hexagonal gratings," J. Opt. Soc. Am. A 24, 2296-2312 (2007).
[CrossRef]

F. M. Dickey, "Laser beam shaping," Opt. Photonics News 14, 30-35 (2003).
[CrossRef]

F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping: Applications (CRC Press, 2005).
[CrossRef]

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Technique (CRC Press, 2000).
[CrossRef]

DiFabrizio, E.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, and E. DiFabrizio, "Analytical derivation of the optimum triplicator," Opt. Commun. 157, 13-16 (1998).
[CrossRef]

Dowling, R. J.

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Price, N. Streibl, and S. J. Walker, "Dammann gratings for laser beam shaping," Opt. Eng. (Bellingham) 28, 1267-1275 (1989).

Fomin, S. V.

I. M. Gelfand and S. V. Fomin, The Calculus of Variations (Prentice Hall, 1963).

Gale, M. T.

Gelfand, I. M.

I. M. Gelfand and S. V. Fomin, The Calculus of Variations (Prentice Hall, 1963).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gori, F.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, and E. DiFabrizio, "Analytical derivation of the optimum triplicator," Opt. Commun. 157, 13-16 (1998).
[CrossRef]

F. Gori, "Diffractive optics and introduction," in Diffractive Optics and Optical Microsystems, S.Martellucci and A.N.Chester, eds. (Plenum, 1997).

Görtler, K.

G. H. Dammann and K. Görtler, "High efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Hecht, E.

E. Hecht and A. Zajac, Optics (Addison Wesley, 1974).

Herzig, H. P.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Technique (CRC Press, 2000).
[CrossRef]

F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping: Applications (CRC Press, 2005).
[CrossRef]

Jahns, J.

J. Jahns, M. M. Downs, M. E. Price, N. Streibl, and S. J. Walker, "Dammann gratings for laser beam shaping," Opt. Eng. (Bellingham) 28, 1267-1275 (1989).

Jin, G.

J. Turunen, A. Vasara, J. Westrholm, G. Jin, and A. Salin, "Optimization and fabrication of beam splitters," J. Phys. D 21, s202-s205 (1988).
[CrossRef]

Klotz, E.

H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
[CrossRef]

Krackhardt, U.

Lee, W. H.

Lizotte, T.

T. Lizotte, R. Rosenberg, and O. Obar, "Actual performance vs. modeled performance of diffractive beam splitters," Proc. SPIE 5876, 505-515 (2005).

Lizotte, T. E.

T. E. Lizotte, "Beam shaping for microvia drilling," Printed Circuit Fabr. 26, 28-33 (2003).

Mait, J. N.

McKee, P.

Obar, O.

T. Lizotte, R. Rosenberg, and O. Obar, "Actual performance vs. modeled performance of diffractive beam splitters," Proc. SPIE 5876, 505-515 (2005).

O'Shea, D. C.

Price, M. E.

J. Jahns, M. M. Downs, M. E. Price, N. Streibl, and S. J. Walker, "Dammann gratings for laser beam shaping," Opt. Eng. (Bellingham) 28, 1267-1275 (1989).

Prongue, D.

Romero, L. A.

Rosenberg, R.

T. Lizotte, R. Rosenberg, and O. Obar, "Actual performance vs. modeled performance of diffractive beam splitters," Proc. SPIE 5876, 505-515 (2005).

Salin, A.

J. Turunen, A. Vasara, J. Westrholm, G. Jin, and A. Salin, "Optimization and fabrication of beam splitters," J. Phys. D 21, s202-s205 (1988).
[CrossRef]

Santarsiero, M.

R. Borghi, G. Cincotti, and M. Santarsiero, "Diffractive variable beam splitter: optimal design," J. Opt. Soc. Am. A 17, 63-67 (2000).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, and E. DiFabrizio, "Analytical derivation of the optimum triplicator," Opt. Commun. 157, 13-16 (1998).
[CrossRef]

Shealy, D. L.

F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping: Applications (CRC Press, 2005).
[CrossRef]

Streibl, N.

U. Krackhardt, J. N. Mait, and N. Streibl, "Upper bound on the diffraction efficiency of phase-only fan out elements," Appl. Opt. 31, 27-37 (1992).
[CrossRef] [PubMed]

J. Jahns, M. M. Downs, M. E. Price, N. Streibl, and S. J. Walker, "Dammann gratings for laser beam shaping," Opt. Eng. (Bellingham) 28, 1267-1275 (1989).

Tervo, J.

Turunen, J.

J. Tervo and J. Turunen, "Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings," Opt. Lett. 25, 785-786 (2000).
[CrossRef]

J. Turunen, A. Vasara, J. Westrholm, G. Jin, and A. Salin, "Optimization and fabrication of beam splitters," J. Phys. D 21, s202-s205 (1988).
[CrossRef]

Vasara, A.

J. Turunen, A. Vasara, J. Westrholm, G. Jin, and A. Salin, "Optimization and fabrication of beam splitters," J. Phys. D 21, s202-s205 (1988).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, and E. DiFabrizio, "Analytical derivation of the optimum triplicator," Opt. Commun. 157, 13-16 (1998).
[CrossRef]

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Price, N. Streibl, and S. J. Walker, "Dammann gratings for laser beam shaping," Opt. Eng. (Bellingham) 28, 1267-1275 (1989).

Westrholm, J.

J. Turunen, A. Vasara, J. Westrholm, G. Jin, and A. Salin, "Optimization and fabrication of beam splitters," J. Phys. D 21, s202-s205 (1988).
[CrossRef]

Wood, D.

Wyrowski, F.

Zajac, A.

E. Hecht and A. Zajac, Optics (Addison Wesley, 1974).

Appl. Opt. (6)

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

J. Turunen, A. Vasara, J. Westrholm, G. Jin, and A. Salin, "Optimization and fabrication of beam splitters," J. Phys. D 21, s202-s205 (1988).
[CrossRef]

Opt. Acta (1)

H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
[CrossRef]

Opt. Commun. (2)

G. H. Dammann and K. Görtler, "High efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, and E. DiFabrizio, "Analytical derivation of the optimum triplicator," Opt. Commun. 157, 13-16 (1998).
[CrossRef]

Opt. Eng. (Bellingham) (1)

J. Jahns, M. M. Downs, M. E. Price, N. Streibl, and S. J. Walker, "Dammann gratings for laser beam shaping," Opt. Eng. (Bellingham) 28, 1267-1275 (1989).

Opt. Lett. (2)

Opt. Photonics News (1)

F. M. Dickey, "Laser beam shaping," Opt. Photonics News 14, 30-35 (2003).
[CrossRef]

Printed Circuit Fabr. (1)

T. E. Lizotte, "Beam shaping for microvia drilling," Printed Circuit Fabr. 26, 28-33 (2003).

Proc. SPIE (2)

J. N. Mait, "Design for two dimensional non-separable array generators," Proc. SPIE 1555, 43-52 (1991).
[CrossRef]

T. Lizotte, R. Rosenberg, and O. Obar, "Actual performance vs. modeled performance of diffractive beam splitters," Proc. SPIE 5876, 505-515 (2005).

Other (7)

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Technique (CRC Press, 2000).
[CrossRef]

F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping: Applications (CRC Press, 2005).
[CrossRef]

F.M.Dickey and D.L.Shealy, eds., Laser Beam Shaping VI, Proc. SPIE 5876, 380 pp. (2005).

I. M. Gelfand and S. V. Fomin, The Calculus of Variations (Prentice Hall, 1963).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

F. Gori, "Diffractive optics and introduction," in Diffractive Optics and Optical Microsystems, S.Martellucci and A.N.Chester, eds. (Plenum, 1997).

E. Hecht and A. Zajac, Optics (Addison Wesley, 1974).

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Figures (7)

Fig. 1
Fig. 1

Optimal phase for three-beam splitting. This has α 1 = π 2 and μ 1 = 1.329 .

Fig. 2
Fig. 2

Optimal phase for five-beam splitting. This has α 1 = π 2 , α 2 = π , μ 1 = 0.459 , μ 2 = 0.899 .

Fig. 3
Fig. 3

Optimal phase for seven-beam splitting. This has α 1 = 0.99 , α 2 = 1.89 , α 3 = 7.03 , μ 1 = 1.28 , μ 2 = 1.45 , μ 3 = 1.24 .

Fig. 4
Fig. 4

Optimal phase for nine-beam splitting. This has α 1 = 0.720 , α 2 = 5.57 , α 3 = 3.03 , α 4 = 1.41 , μ 1 = 0.971 , μ 2 = 0.963 , μ 3 = 0.943 , μ 4 = 1.03 .

Fig. 5
Fig. 5

Optimal phase for eleven-beam splitting. This has α 1 = 0.311 , α 2 = 4.49 , α 3 = 2.85 , α 4 = 5.55 , α 5 = 4.41 , μ 1 = 1.21 , μ 2 = 1.30 , μ 3 = 1.48 , μ 4 = 1.43 , μ 5 = 1.28 .

Fig. 6
Fig. 6

Optimal phase for four-beam splitting. This is the optimized solution including the modes m = ± 1 , ± 2 .

Fig. 7
Fig. 7

Optimal phase for four-beam splitting. This is the optimized solution including the modes m = ± 1 , ± 3 .

Tables (1)

Tables Icon

Table 1 Values of η LS and η CO for Splitting a Beam into N Modes Number of Beams

Equations (143)

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ϕ ( x ) = tan 1 ( 2 μ cos ( x ) ) ,
a k = 1 2 π π π e i ϕ ( x ) e i k x d x for k = 0 ± 1 , ± 2 , .
e CO ( ϕ ) = k = 1 N a m k 2 ,
a m k = K γ k for k = 1 , N .
η CO = e CO ( ϕ CO ) .
k = a k 2 = 1 .
e i ϕ ( x ) = λ s ( x ) ,
s ( x , α 1 , α 2 , ) = k = 1 N γ k e i m k x e i α k .
λ 2 = 1 k = 1 N γ k 2 .
f ( x ) 2 = 1 2 π π π f ( x ) 2 d x .
e LS ( ϕ , λ , α ̱ ) = 1 e i ϕ ( x ) λ s ( x , α ̱ ) 2 ,
s ( x , α ̱ ) = k = 1 N γ k e i m k x e i α k .
η LS = e LS ( ϕ LS , λ LS , α ̱ LS ) .
a m k = K γ k e i β k for 1 k N .
K = η CO k = 1 N γ k 2 .
λ s ( x , α ̱ ) = k = 1 N a m k e i m k x ,
e LS ( ϕ CO , K , β ̱ ) = 1 e i ϕ CO ( x ) k = 1 N a m k e i m k x 2 = k = 1 N a m k 2 = η CO .
π π f ( x x 0 ) d x = π π f ( x ) d x .
1 2 π π π e i ϕ ( x ) e i k x d x = 1 2 π π π e i ϕ ( x x 0 ) e i k ( x x 0 ) d x = e i k x 0 1 2 π π π e i ϕ ( x x 0 ) e i k x d x .
e i ϕ ( x ) λ s ( x , α ̱ ) 2 = 1 + λ 2 s ( x , α ̱ ) 2 2 λ Re ( e i ϕ ( x ) s ( x , α ̱ ) ¯ )
s ( x , α ̱ ) = P LS ( x , α ̱ ) + i Q LS ( x , α ̱ ) ,
P LS ( x , α ̱ ) = k = 1 N γ k cos ( m k x + α k ) ,
Q LS ( x , α ̱ ) = k = 1 N γ k sin ( m k x + α k ) .
s ( x , α ̱ ) = s ( x , α ̱ ) e i ψ LS ( x , α ̱ ) ,
s ( x , α ̱ ) = P LS 2 ( x , α ̱ ) + Q LS 2 ( x , α ̱ ) ,
ψ LS ( x , α ̱ ) = tan 1 ( Q LS ( x , α ̱ ) P LS ( x , α ̱ ) ) .
cos ( ψ LS ( x , α ) ) = P LS ( x , α ̱ ) P LS 2 ( x , α ̱ ) + Q LS 2 ( x , α ̱ ) ,
sin ( ψ LS ( x , α ) ) = Q LS ( x , α ̱ ) P LS 2 ( x , α ̱ ) + Q LS 2 ( x , α ̱ ) .
1 2 π π π s ( x , α ̱ ) 2 d x = S γ = k M ̱ γ k 2 .
e LS ( ϕ , λ , α ̱ ) = λ 2 S γ + 2 λ 1 2 π π π s ( x , α ̱ ) cos ( ϕ ( x ) ψ LS ( x , α ̱ ) ) d x .
ϕ ( x , α ̱ ) = ψ LS ( x , α ̱ ) .
Γ ( α ̱ ) = 1 2 π π π P LS 2 ( x , α ̱ ) + Q LS 2 ( x , α ̱ ) d x .
λ = Γ ( α ̱ ) S γ .
ϵ LS ( α ̱ ) = max λ , ϕ e LS ( λ , ϕ , α ̱ ) ,
ϵ LS ( α ̱ ) = Γ 2 ( α ̱ ) S γ .
ϕ LS ( x ) = ψ LS ( x , α ̱ max ) ,
λ = Γ ( α ̱ max ) S γ ,
η LS = Γ 2 ( α ̱ max ) S γ .
η LS = k = 1 N a m k 2 .
e i ϕ LS ( x ) = κ k = 1 N γ k e i α k e i m k x + r ( x ) ,
e = 1 1 2 π π π ( κ λ ) k = 1 N γ k e i m k x + α k + r ( x ) 2 d x .
η LS = 1 1 2 π π π r ( x ) 2 d x = k = 1 N a m k 2 .
( F ( x ̱ ) k = 1 p λ k G k ( x ̱ ) ) = 0 .
H ( ϕ + δ ϕ ) = H ( ϕ ) + H ϕ δ ϕ + .
H ( ϕ ) = 0 1 F ( x , ϕ ( x ) ) d x .
H ( ϕ + δ ϕ ) = H ( ϕ ) + 0 1 F ( x , ϕ ( x ) ) y δ ϕ ( x ) d x + .
H ϕ ψ = 0 1 F ( x , ϕ ( x ) ) y ψ ( x ) d x .
A k ( ϕ ) + i B k ( ϕ ) = 1 2 π π π e i ϕ ( x ) e i m k x d x .
A k ( ϕ ) = 1 2 π π π ( cos ( ϕ ( x ) ) cos ( m k x ) + sin ( ϕ ( x ) ) sin ( m k x ) ) d x ,
B k ( ϕ ) = 1 2 π π π ( sin ( ϕ ( x ) ) cos ( m k x ) cos ( ϕ ( x ) ) sin ( m k x ) ) d x .
I ( ϕ ) = k = 1 N ( A k 2 ( ϕ ) + B k 2 ( ϕ ) ) ,
G k ( ϕ ) = A k 2 ( ϕ ) + B k 2 ( ϕ ) γ k 2 S γ I ( ϕ ) = 0 for k = 1 , N .
J ( ϕ ) = k = 1 N μ k ( A k 2 ( ϕ ) + B k 2 ( ϕ ) )
δ A k 2 = 2 A k δ A k ,
δ B k 2 = 2 B k δ B k ,
δ A k = 1 2 π π π ( sin ( ϕ ( x ) ) cos ( m k x ) + cos ( ϕ ( x ) ) sin ( m k x ) ) δ ϕ ( x ) d x ,
δ B k = 1 2 π π π ( cos ( ϕ ( x ) ) cos ( m k x ) + sin ( ϕ ( x ) ) sin ( m k x ) ) δ ϕ ( x ) d x .
δ J ( ϕ ) = 2 1 2 π π π ( cos ( ϕ ( x ) ) Q ( x , μ ̱ ) sin ( ϕ ( x ) ) P ( x , μ ̱ ) ) δ ϕ ( x ) d x ,
P ( x , μ ̱ ) = k = 1 N μ k ( A k ( ϕ ) cos ( m k x ) B k ( ϕ ) sin ( m k x ) ) ,
Q ( x , μ ̱ ) = k = 1 N μ k ( A k sin ( m k x ) + B k cos ( m k x ) ) .
A k ( ϕ ) + i B k ( ϕ ) = γ k A ( cos ( α k ) + i sin ( α k ) ) .
P CO ( x , α ̱ , μ ̱ ) = k = 1 N μ k γ k cos ( m k x + α k ) ,
Q CO ( x , α ̱ , μ ̱ ) = k = 1 N μ k γ k sin ( m k x + α k ) .
tan ( ϕ CO ( x ) ) = Q CO ( x , α ̱ , μ ̱ ) P CO ( x , α ̱ , μ ̱ ) .
cos ( ϕ ( x ) ) = P ( x , α ̱ , μ ̱ ) P 2 ( x , α ̱ , μ ̱ ) + Q 2 ( x , α ̱ , μ ̱ ) ,
sin ( ϕ ( x ) ) = Q ( x , α ̱ , μ ̱ ) P 2 ( x , α ̱ , μ ̱ ) + Q 2 ( x , α ̱ , μ ̱ ) .
a m k = γ k γ 1 a m 1 for k = 2 , N .
Im ( a m k e i α k ) = 0 for k = 3 , N .
Γ CO ( α ̱ , μ ̱ ) α k = 0 ,
Γ CO ( α ̱ , μ ̱ ) = 1 2 π π π P CO 2 ( x , α ̱ , μ ̱ ) + Q CO 2 ( x , α ̱ , μ ̱ ) d x .
e CO = a 1 2 + a 1 2 ,
a 1 = a 1 .
P LS ( x ) = cos ( x ) + cos ( x ) = 2 cos ( x ) ,
Q LS ( x ) = sin ( x ) + sin ( x ) = 0 .
tan ( ϕ LS ( x ) ) = 0 ,
cos ( ϕ LS ( x ) ) = cos ( x ) cos ( x ) ,
sin ( ϕ LS ( x ) ) = 0 .
ϕ LS ( x ) = { 0 for cos ( x ) > 0 π for cos ( x ) < 0 } .
Γ = 1 2 π π π 2 cos ( x ) d x = 4 π ,
S γ = 2 .
η LS = 8 π 2 0.8106 .
η CO = 8 π 2 0.8106 .
tan ( ϕ CO ( x ) ) = Q CO ( x ) P CO ( x ) ,
P CO ( x ) = 1 + a cos ( x ) ,
Q CO ( x ) = a sin ( x ) .
P LS ( x , α ) = 1 + cos ( x + α ) + cos ( x + α ) = 1 + 2 cos ( α ) cos ( x ) ,
Q LS ( x , α ) = sin ( x + α ) + sin ( x + α ) = 2 sin ( α ) cos ( x ) .
Γ ( α ) = 1 2 π π π P LS 2 ( x , α ) + Q LS 2 ( x , α ) d x .
η LS 0.9374 .
ϕ LS ( x ) = tan 1 ( 2 cos ( x ) ) ,
cos ( ϕ LS ( x ) ) = 1 1 + 4 cos 2 ( x ) ,
sin ( ϕ LS ( x ) ) = 2 cos ( x ) 1 + 4 cos 2 ( x ) .
P CO ( x ) = 1 + 2 μ cos ( α ) cos ( x ) ,
Q CO ( x ) = 2 μ sin ( α ) cos ( x ) .
Im ( a 1 e i α ) = 0 .
μ 1.32859 .
tan ( ϕ ( x ) ) = 2 μ cos ( x ) .
η CO = 0.92556 .
P LS ( x , α ̱ ) = 1 + k = 1 N ( cos ( k x + α k ) + cos ( k x + α k ) ) = 1 + 2 k = 1 N cos ( α k ) cos ( k x ) ,
Q LS ( x , α ̱ ) = k = 1 N ( sin ( k x + α k ) + sin ( k x + α k ) ) = 2 k = 1 N sin ( α k ) cos ( k x ) .
1 2 π π π e i ϕ ( x ) e i m x d x { e i π 4 e i m 2 ( 2 β ) 1 2 π β for m 2 < β 2 0 for m 2 > β 2 } .
P CO = 2 cos ( x ) + 2 μ cos ( α ) cos ( 2 x ) ,
Q CO = 2 μ sin ( α ) cos ( 2 x ) .
P CO = 2 cos ( x ) + 2 μ cos ( α ) cos ( 3 x ) ,
Q CO = 2 μ sin ( α ) cos ( 3 x ) .
P CO = 2 cos ( x ) + 2 μ 1 cos ( α 1 ) cos ( 3 x ) + 2 μ 2 cos ( α 2 ) cos ( 5 x ) ,
Q CO = 2 μ 1 sin ( α 1 ) cos ( 3 x ) + 2 μ 2 sin ( α 2 ) cos ( 5 x ) .
u ( x , y , z ) = 1 4 π 2 e i ( k x x + k y y ) F ( k x , k y ) e i z k 2 k x 2 k y 2 d k x d k y ,
f ( x , y ) = m = a m e i 2 π m x D .
F ( k x , k y ) = 4 π 2 δ ( k y ) m = a m δ ( k x m k 0 ) ,
k 0 = 2 π D .
u ( x , y , z ) = a m e i m k 0 x e i z k 2 m 2 k 0 2 .
tan ( α m ) = m k 0 k 2 m 2 k 0 2 .
Γ α k = 0 , k = 1 , N 2 ,
2 Γ α k 2 < 0 , k = 1 , N 2 .
Γ α k = 0 , k = N 1 , N ,
2 Γ α k 2 < 0 , k = N 1 , N .
Γ α k = Im ( a m k e i α k ) .
Γ α k = 1 2 π π π P LS ( P LS α k ) + Q LS ( Q LS α k ) P LS 2 + Q LS 2 d x .
2 Γ α k 2 = Re ( a m k e i α k ) + S k ,
2 Γ α k 2 = 1 2 π π π P LS ( 2 P LS α k 2 ) + Q LS ( 2 Q LS α k 2 ) P LS 2 + Q LS 2 d x + S k ,
S k = 1 2 π π π ( ( P LS α k ) 2 + ( Q LS α k ) 2 ) ( P LS 2 + Q LS 2 ) ( P LS ( P LS α k ) + Q LS ( Q LS α k ) ) 2 P LS 2 + Q LS 2 3 d x .
Re ( a m k e i α k ) = 1 2 π π π P LS ( 2 P LS α k 2 ) + Q LS ( 2 Q LS α k 2 ) P LS 2 + Q LS 2 d x .
tan ( ϕ CO ( x ) ) = Q CO ( x , α ̱ , μ ̱ ) P CO ( x , α ̱ , μ ̱ ) .
cos ( ϕ CO ) = ± P CO ( x ) P CO 2 ( x ) + Q CO 2 ( x ) ,
sin ( ϕ CO ) = ± Q CO ( x ) P CO 2 ( x ) + Q CO 2 ( x ) .
J ( x ) = k = 1 N μ k ( A k ( x ) 2 + B k ( x ) 2 )
A k ( x ) = 1 2 π π x cos ( ϕ CO ( x , α ̱ , μ ̱ ) m k x ) d x + 1 2 π x π cos ( ϕ CO ( x , α ̱ , μ ̱ ) m k x ) d x ,
B k ( x ) = 1 2 π π x sin ( ϕ CO ( x , α ̱ , μ ̱ ) m k x ) d x + 1 2 π x π sin ( ϕ CO ( x , α ̱ , μ ̱ ) m k x ) d x ,
d A k d x = 1 π ( cos ( ϕ CO ( x ) m k x ) ) .
d B k d x = 1 π ( sin ( ϕ CO ( x ) m k x ) ) .
d J d x = 4 ( cos ( ϕ CO ( x ) ) Q CO ( x , α ̱ , μ ̱ ) + sin ( ϕ CO ( x ) ) P CO ( x , α ̱ , μ ̱ ) ) .
cos ( ϕ CO ( x ) Q CO ( x , α ̱ , μ ̱ ) ) + sin ( ϕ CO ( x 0 ) ) P CO ( x , α ̱ , μ ̱ ) = 0 .
1 + x + 1 x 2 ,
x 1 + x x 1 x 0 .
F ( κ ) = π π f ( x ) + κ g ( x ) d x
F ( κ ) = 1 2 π π g ( x ) f ( x ) + κ g ( x ) d x
F ( κ ) = 0 π ( f ( x ) + κ g ( x ) + f ( x ) κ g ( x ) ) d x .
F ( κ ) = 1 2 0 π ( g ( x ) f ( x ) + κ g ( x ) g ( x ) f ( x ) κ g ( x ) ) d x .
P LS 2 ( x , α ) + Q LS 2 ( x , α ) = 1 + 4 cos 2 ( x ) + 4 cos ( α ) cos ( x ) .
Γ ( α ) = π π 1 + 4 cos 2 ( x ) + 4 cos ( α ) cos ( x ) d x .
Γ ( α ) = π π 1 + 4 sin 2 ( x ) + 4 cos ( α ) sin ( x ) d x .
P LS 2 ( x , α ) + Q LS 2 ( x , α ) = 1 + 4 γ 2 cos 2 ( x ) + 4 γ cos ( α ) cos ( x ) .

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