Abstract

We use the calculus of variations to give a general theory for finding optimal square and hexagonal phase gratings for doing beam splitting.

© 2007 Optical Society of America

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References

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  1. L. A. Romero and F. M. Dickey, "Theory for optimal beam splitting by phase gratings. I. One-dimensional gratings," J. Opt. Soc. Am. A 24, 2280-2295 (2007).
    [CrossRef]
  2. 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]
  3. R. Borghi, G. Cincotti, and M. Santarsiero, "Diffractive variable beam splitter: optimal design," J. Opt. Soc. Am. A 17, 63-67 (2000).
    [CrossRef]
  4. 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]
  5. 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]
  6. J. N. Mait, "Design for two dimensional non-separable array generators," Proc. SPIE 1555, 43-52 (1991).
    [CrossRef]
  7. J. N. Mait, "Fourier array generators," in Micro-optics, Elements, Systems, and Applications, H.P.Herzig, ed. (Taylor Francis, 1997).
  8. R. Courant and D. Hilbert, Methods of Mathematical Physics (Springer, 1937).
  9. A. Papoulis, Systems and Transforms with Applications to Optics, McGraw-Hill Series in System Science (McGraw-Hill, 1968).
  10. H. Weyl, Symmetry (Princeton U., 1952).
  11. J. V. Smith, Geometrical and Structural Crystallography (Wiley, 1982).
  12. F. Gori, "Diffractive optics and introduction," in Diffractive Optics and Optical Microsystems, S.Martellucci and A.N.Chester, eds. (Plenum, 1997).

2007 (1)

2000 (1)

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]

1992 (1)

1991 (1)

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

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]

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]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Springer, 1937).

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]

Dandliker, R.

Dickey, F. M.

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]

Gale, M. T.

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]

Herzig, H. P.

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Springer, 1937).

Mait, J. N.

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

J. N. Mait, "Fourier array generators," in Micro-optics, Elements, Systems, and Applications, H.P.Herzig, ed. (Taylor Francis, 1997).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications to Optics, McGraw-Hill Series in System Science (McGraw-Hill, 1968).

Prongue, D.

Romero, L. A.

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]

Smith, J. V.

J. V. Smith, Geometrical and Structural Crystallography (Wiley, 1982).

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]

Weyl, H.

H. Weyl, Symmetry (Princeton U., 1952).

Appl. Opt. (1)

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

Opt. Commun. (2)

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]

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]

Proc. SPIE (1)

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

Other (6)

J. N. Mait, "Fourier array generators," in Micro-optics, Elements, Systems, and Applications, H.P.Herzig, ed. (Taylor Francis, 1997).

R. Courant and D. Hilbert, Methods of Mathematical Physics (Springer, 1937).

A. Papoulis, Systems and Transforms with Applications to Optics, McGraw-Hill Series in System Science (McGraw-Hill, 1968).

H. Weyl, Symmetry (Princeton U., 1952).

J. V. Smith, Geometrical and Structural Crystallography (Wiley, 1982).

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

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Figures (3)

Fig. 1
Fig. 1

Portion of the tiling with hexagons, along with the rhombus that can be used as a unit cell. Note that we can divide both the rhombus and the hexagon into six “kites.” This shows that either the rhombus or the hexagon can be used as the unit cell.

Fig. 2
Fig. 2

Region of the unit cell where ϕ is 0 (inside the smaller square) and ϕ = π (outside the smaller square) for the problem of symmetric four-beam splitting using a square grating.

Fig. 3
Fig. 3

Region Ω S 2 that consists of two unit cells of the tiling with squares. The region Ω S 2 is the square that has been rotated by 45 deg and that has the vertices (0,1), (1,0), ( 0 , 1 ) , and ( 1 , 0 ) . Integrating over this region is equivalent to integrating over two unit cells.

Equations (219)

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tan ( ϕ ( x , y ) ) = Q CO ( x , y ) P CO ( x , y ) ,
P CO ( x , y ) = 1 + 2 μ 1 cos ( α 1 ) cos ( x ) + 2 μ 2 cos ( α 2 ) cos ( y ) ,
Q CO ( x , y ) = 2 μ 1 sin ( α 1 ) cos ( x ) + 2 μ 2 sin ( α 2 ) cos ( y ) .
x ̱ = ( x y )
m ̱ = ( m n )
p ̱ m ̱ = m p ̱ 1 + n p ̱ 2 .
f ( x ̱ + p ̱ m ̱ ) = f ( x ̱ )
p ̱ 1 = ( 2 π 0 ) ,
p ̱ 2 = ( 0 2 π ) .
f ( x ̱ ) = m ̱ a m ̱ e i q ̱ m ̱ x ̱ ,
q ̱ m ̱ = m q ̱ 1 + n q ̱ 2 ,
q ̱ 1 = ( 1 0 ) ,
q ̱ 2 = ( 0 1 ) .
a m ̱ = 1 A S Ω S f ( x ̱ ) e i q ̱ m ̱ x ̱ d x ̱ ,
x ̱ Ω S iff x π , y π ,
A S = 4 π 2
1 A S Ω S f ( x ̱ ) 2 d x ̱ = m ̱ a m ̱ 2 .
Ω S f ( x ̱ x ̱ 0 ) d x ̱ = Ω S f ( x ̱ ) d x ̱
a m ̱ = 1 A S Ω S f ( x ̱ ) e i q ̱ m ̱ x ̱ d x ̱ .
a m ̱ = 1 A S Ω S f ( x ̱ x ̱ 0 ) e i q ̱ m ̱ ( x ̱ x ̱ 0 ) d x ̱ = e i q ̱ m ̱ x ̱ 0 1 A S Ω S f ( x ̱ x ̱ 0 ) e i q ̱ m x ̱ d x ̱ = e i q ̱ m ̱ x ̱ 0 a ̂ m ̱ .
R 90 = ( 0 1 1 0 ) .
Ω S f ( x ̱ ) d x ̱ = Ω S f ( R 90 x ̱ ) d x ̱ .
f ( x ̱ ) = f ( R 90 x ̱ ) .
a m ̱ = 1 A S Ω S f ( x ̱ ) e i q ̱ m x ̱ d x ̱ = 1 A S Ω S f ( R 90 x ̱ ) e i q ̱ m ̱ ( R 90 x ̱ ) d x ̱ .
d = 2 π 3 .
R 60 = [ 1 2 3 2 3 2 1 2 ]
z ̱ k = R 60 k z ̱ 0 , k = 0 , 5 ,
z ̱ 0 = ( 2 π 3 0 ) .
p ̱ 1 = 2 π ( 3 2 1 2 ) ,
p ̱ 2 = 2 π ( 3 2 1 2 ) .
x ̱ Ω R iff x ̱ = s p ̱ 1 + t p ̱ 2 , s 1 2 , t 1 2 .
A H = A R = 2 π 2 3 .
f ( x ̱ ) = f ( x ̱ + p ̱ m ̱ ) integers m ̱ = ( m , n ) .
q ̱ 1 = ( 1 3 1 ) ,
q ̱ 2 = ( 1 3 1 ) .
q ̱ 1 p ̱ 2 = q ̱ 2 p ̱ 1 = 0 ,
q ̱ 1 p ̱ 1 = q ̱ 2 p ̱ 2 = 2 π .
x ̱ = 1 2 π ( ξ p ̱ 1 + η p ̱ 2 ) .
ξ ̱ T = ( ξ , η ) = ( q ̱ 1 x ̱ , q ̱ 2 x ̱ ) .
ξ ̱ = T Q x ̱ ,
T Q = ( q ̱ 1 T q ̱ 2 T ) = [ 1 3 1 1 3 1 ] ,
T Q 1 = 1 2 π T P = [ 3 2 3 2 1 2 1 2 ] ,
T P = ( p ̱ 1 , p ̱ 2 ) = 2 π [ 3 2 3 2 1 2 1 2 ] .
f ( x ̱ ) = m ̱ a m ̱ e i ( q ̱ m ̱ x ̱ ) ,
a m ̱ = 1 A H Ω f ( x ̱ ) e i ( q ̱ m ̱ x ̱ ) d x ̱ .
g ( ξ ̱ ) = m ̱ a m ̱ e i ( m ̱ ξ ̱ ) ,
a m ̱ = 1 4 π 2 π π π π g ( ξ ̱ ) e i ( m ̱ ξ ̱ ) d ξ ̱ .
f ( x ̱ ) = m ̱ a m ̱ e i m ̱ T T Q x ̱ .
a m ̱ = det ( T Q ) 4 π 2 Ω R f ( x ̱ ) e i m ̱ T T Q x ̱ d x ̱ .
1 A H Ω f ( x ̱ ) 2 d x ̱ = m ̱ a m ̱ 2 .
Ω S f ( x ̱ x ̱ 0 ) d x ̱ = Ω S f ( x ̱ ) d x ̱
S = ( T Q R 60 T Q 1 ) T = ( 0 1 1 1 ) .
Ω H f ( x ̱ ) d x ̱ = Ω H f ( R 60 x ̱ ) d x ̱ .
f ( x ̱ ) = f ( R 60 x ̱ ) .
a m ̱ = 1 A S Ω S f ( x ̱ ) e i q ̱ m x ̱ d x ̱ = 1 A S Ω S f ( R 60 x ̱ ) e i q ̱ m ̱ ( R 60 x ̱ ) d x ̱ .
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 ) .
m ̱ a m ̱ 2 = 1 ,
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 q ̱ m ̱ k x e i α k ,
f ( x ̱ ) 2 = 1 A Ω f ( x ̱ ) 2 d x ̱ ,
η LS = e LS ( ϕ LS , λ LS , α ̱ LS ) .
M = [ m 1 n 1 1 m 2 n 2 1 m 3 n 3 1 ]
s ( x ̱ , α ̱ ) = P LS ( x ̱ , α ̱ ) + i Q LS ( x ̱ , α ̱ ) ,
P LS ( x ̱ , α ̱ ) = k = 1 N γ k cos ( q ̱ m ̱ k x ̱ + α k ) ,
Q LS ( x ̱ , α ̱ ) = k = 1 N γ k sin ( q ̱ m ̱ k x ̱ + α k ) .
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 ̱ ) .
Γ ( α ̱ ) = 1 A Ω P LS 2 ( x ̱ , α ̱ ) + Q LS 2 ( x ̱ , α ̱ ) d x ̱ .
ϕ LS ( x ̱ ) = ψ LS ( x ̱ , α ̱ max ) ,
λ = Γ ( α ̱ max ) S γ ,
η LS = Γ 2 ( α ̱ max ) S γ .
J ( ϕ ) = μ k ( A k 2 + B k 2 ) ,
A k + i B k = 1 A Ω e i ϕ ( x ̱ ) e i q ̱ m ̱ k x ̱ d x ̱ .
tan ( ϕ CO ( x ̱ ) ) = Q CO ( x ̱ , α ̱ , μ ̱ ) P CO ( x ̱ , α ̱ , μ ̱ ) ,
P CO ( x ̱ , α ̱ , μ ̱ ) = k = 1 N μ k γ k cos ( q ̱ m ̱ k x ̱ + α k ) ,
Q CO ( x ̱ , α ̱ , μ ̱ ) = k = 1 N μ k γ k sin ( q ̱ m ̱ k x ̱ + α k ) .
cos ( ϕ CO ( x ̱ ) ) = P CO ( x ̱ , α ̱ , μ ̱ ) P CO 2 ( x ̱ , α ̱ , μ ̱ ) + Q CO 2 ( x ̱ , α ̱ , μ ̱ ) ,
sin ( ϕ CO ( x ̱ ) ) = Q CO ( x ̱ , α ̱ , μ ̱ ) P CO 2 ( x ̱ , α ̱ , μ ̱ ) + Q CO 2 ( x ̱ , α ̱ , μ ̱ ) .
a m ̱ k = γ k γ 1 a m ̱ 1 for k = 2 , N .
I m ( a m ̱ k e i α k ) = 0 for k = 4 , N .
Γ CO ( α ̱ , μ ̱ ) α k 0 ,
Γ CO ( α ̱ , μ ̱ ) = 1 A Ω P CO 2 ( x ̱ , α ̱ , μ ̱ ) + Q CO 2 ( x ̱ , α ̱ , μ ̱ ) d x ̱ .
P LS ( x ̱ ) = cos ( x ) + cos ( x ) + cos ( y ) + cos ( y ) = 2 cos ( x ) + 2 cos ( y ) ,
Q LS ( x , y ) = sin ( x ) + sin ( x ) + sin ( y ) + sin ( y ) = 0 .
cos ( ϕ LS ( x , y ) ) = sgn ( P LS ( x , y ) ) ,
sin ( ϕ LS ( x , y ) ) = 0 .
ϕ LS ( x , y ) = { 0 if cos ( x ) + cos ( y ) > 0 π if cos ( x ) + cos ( y ) < 0 } .
cos ( x ) + cos ( y ) = 2 cos ( x + y 2 ) cos ( x y 2 ) ,
x + y = π ,
x + y = π ,
x y = π ,
x y = π .
η CO = η LS .
( ξ , η ) = ( x y 2 , x + y 2 ) .
a 1 , 0 = 1 2 A S Ω S 2 e i ϕ ( x ̱ ) e i x d x ̱ .
ϕ ( ξ , η ) = ϕ 1 ( ξ ) + ϕ 1 ( η ) ,
ϕ 1 ( s ) = { 0 if cos ( s ) > 0 π if cos ( s ) < 0 } .
a 1 , 0 = 1 A S π π π π e i ξ e i ϕ 1 ( ξ ) e i η e i ϕ 1 ( η ) d ξ d η .
a 1 , 0 = ( 1 2 π π π e i ξ e i ϕ 1 ( ξ ) d ξ ) 2 = 4 π 2 .
η CO = 64 π 4 0.658 .
P LS ( x ̱ , α ) = cos ( x ) + cos ( x ) + cos ( y + α ) + cos ( y + α ) = 2 cos ( x ) + 2 cos ( α ) cos ( y ) ,
Q LS ( x ̱ , α ) = sin ( x ) + sin ( x ) + sin ( y + α ) + sin ( y α ) = 2 sin ( α ) cos ( y ) ,
Γ ( α ) = 2 A S Ω S cos 2 ( x ) + cos 2 ( y ) + 2 cos ( α ) cos ( x ) cos ( y ) d x ̱ .
η LS 0.9179 .
tan ( ϕ LS ( x , y ) ) = tan 1 ( cos ( y ) cos ( x ) ) .
cos ( ϕ LS ( x , y ) ) = cos ( x ) cos 2 ( x ) + cos 2 ( y ) ,
sin ( ϕ LS ( x , y ) ) = cos ( y ) cos 2 ( x ) + cos 2 ( y ) .
a 1 , 0 = 1 4 π 2 π π π π h ( x , y ) d x d y ,
h ( x , y ) = ( cos ( ϕ LS ( x ̱ ) ) cos ( x ) + sin ( ϕ LS ( x ̱ ) ) sin ( x ) ) + i ( sin ( ϕ LS ( x ̱ ) ) cos ( x ) cos ( ϕ LS ( x ̱ ) ) sin ( x ) ) ,
a 1 , 0 = 1 4 π 2 π π π π h ( y , x ) d x d y ,
h ( y , x ) = ( sin ( ϕ LS ( x ̱ ) ) cos ( y ) + cos ( ϕ LS ( x ̱ ) ) sin ( y ) ) + i ( cos ( ϕ LS ( x ̱ ) ) cos ( y ) sin ( ϕ LS ( x ̱ ) ) sin ( y ) ) .
η CO = η LS .
P LS ( x , y , α ) = cos ( α ) + cos ( x ) + cos ( x ) + cos ( y ) + cos ( y ) = cos ( α ) + 2 cos ( x ) + 2 cos ( y ) ,
Q LS ( x , y , α ) = sin ( α ) + sin ( x ) + sin ( x ) + sin ( y ) + sin ( y ) = sin ( α ) .
Γ ( α ) = 2 A S Ω S 1 4 + ( cos ( x ) + cos ( y ) ) 2 + cos ( α ) ( cos ( x ) + cos ( y ) ) d x ̱ .
η LS 0.8206 .
a 0 , ± 1 2 = a ± 1 , 0 2 0.1236 ,
a 0 , 0 2 0.3695 .
P CO ( x , y , α , μ ) = μ cos ( α ) + 2 cos ( x ) + 2 cos ( y ) ,
Q CO ( x , y , α , μ ) = μ sin ( α ) .
μ = 0.40314 .
ϕ ( x , y ) = tan 1 ( μ 2 cos ( x ) + 2 cos ( y ) ) .
η CO = 0.7629 .
P LS ( x , y , α 1 , α 2 ) = cos ( α 2 ) + 2 cos ( x ) + 2 cos ( α 1 ) cos ( y ) ,
Q LS ( x , y , α 1 , α 2 ) = sin ( α 2 ) + 2 sin ( α 1 ) cos ( y ) .
ϕ LS ( x , y ) = tan 1 ( 2 cos ( y ) 1 + 2 cos ( x ) ) .
η LS 0.8639 .
P CO ( x , y , α 1 , α 2 , μ 1 , μ 2 ) = μ 2 cos ( α 2 ) + 2 cos ( x ) + 2 μ 1 cos ( α 1 ) cos ( y ) ,
Q CO ( x , y , α 1 , α 2 , μ 1 , μ 2 ) = μ 2 sin ( α 2 ) + 2 μ 1 sin ( α 1 ) cos ( y ) .
μ 1 1.1928 ,
μ 2 0.7192 ,
α 1 = π 2 ,
α 2 = 0 ,
η CO 0.8433 .
ϕ CO ( x , y ) = tan 1 ( 2 μ 1 cos ( y ) μ 2 + 2 cos ( x ) ) .
η sep = η 1 D 2 = 0.8456 .
P LS = 1 + P 1 ( x , y , α 1 ) + P 2 ( x , y , α 2 ) + P 3 ( x , y , α 3 ) ,
Q LS = Q 1 ( x , y , α 1 ) + Q 2 ( x , y , α 2 ) + Q 3 ( x , y , α 3 ) ,
P 1 ( x , y , α 1 ) = cos ( x + α 1 ) + cos ( x + α 1 ) + cos ( y + α 1 ) + cos ( y + α 1 ) = 2 cos ( α 1 ) ( cos ( x ) + cos ( y ) ) ,
P 2 ( x , y , α 2 ) = cos ( x + y + α 2 ) + cos ( x y + α 2 ) = 2 cos ( x + y ) cos ( α 2 ) ,
P 3 ( x , y , α 3 ) = cos ( x y + α 3 ) + cos ( x + y + α 3 ) = 2 cos ( x y ) cos ( α 3 ) ,
Q 1 ( x , y , α 1 ) = sin ( x + α 1 ) + sin ( x + α 1 ) + sin ( y + α 1 ) + sin ( y + α 1 ) = 2 sin ( α 1 ) ( cos ( x ) + cos ( y ) ) ,
Q 2 ( x , y , α 2 ) = sin ( x + y + α 2 ) + sin ( x y + α 2 ) = 2 cos ( x + y ) sin ( α 2 ) ,
Q 3 ( x , y , α 3 ) = sin ( x y + α 3 ) + sin ( x + y + α 3 ) = 2 cos ( x y ) sin ( α 3 ) .
α 1 = 0 ,
α 2 = 2.101 ,
α 3 = 4.182
η LS = 0.9393 .
P CO = 1 + 2 μ 1 cos ( α 1 ) ( cos ( x ) + cos ( y ) ) + 2 μ 2 cos ( α 2 ) cos ( x + y ) + 2 μ 3 cos ( α 3 ) cos ( x y ) ,
Q CO = 2 μ 1 sin ( α 1 ) ( cos ( x ) + cos ( y ) ) + 2 μ 2 sin ( α 2 ) cos ( x + y ) + 2 μ 3 sin ( α 3 ) cos ( x y ) .
α 1 = 0 ,
α 2 = 2.103 ,
α 3 = 4.1806 ,
μ 1 = 1.379 ,
μ 2 = 1.111 ,
μ 3 = 1.111 .
η CO = 0.9327 .
e CO = a 1 , 0 2 + a 0 , 1 2 + a 1 , 1 2 + a 1 , 0 2 + a 0 , 1 2 + a 1 , 1 2 .
q ̱ 1 , 0 x ̱ = q ̱ 1 x ̱ = x 3 y ,
q ̱ 1 , 1 x ̱ = q ̱ 1 x ̱ + q ̱ 2 x ̱ = 2 x 3 ,
q ̱ 0 , 1 x ̱ = q ̱ 2 x ̱ = x 3 + y .
P LS ( x , y ) = 2 cos ( 2 x 3 ) + 2 cos ( x 3 + y ) + 2 cos ( x 3 y ) ,
Q LS ( x , y ) = 0 .
cos ( ϕ LS ( x , y ) ) = P LS ( x , y ) P LS ( x , y ) ,
sin ( ϕ LS ( x , y ) ) = 0 .
ϕ LS ( x , y ) = { 0 if P LS ( x , y ) > 0 π if P LS ( x , y ) < 0 } .
η CO = η LS = 0.7107 .
P LS ( x , y ) = 2 cos ( x 3 y ) + 2 cos ( α 1 ) cos ( 2 x 3 ) + 2 cos ( α 2 ) cos ( x 3 + y ) ,
Q LS ( x , y ) = 2 sin ( α 1 ) cos ( 2 x 3 ) + 2 sin ( α 2 ) cos ( x 3 + y ) .
η LS 0.8623 .
P CO ( x , y ) = 2 cos ( x 3 y ) + 2 μ 1 cos ( α 1 ) cos ( 2 x 3 ) + 2 μ 2 cos ( α 2 ) cos ( x 3 + y ) ,
Q CO ( x , y ) = 2 μ 1 sin ( α ) cos ( 2 x 3 ) + 2 μ 2 sin ( α 2 ) cos ( x 3 + y ) .
μ 1 = 0.5671 ,
μ 2 = 1 ,
η CO 0.8338 .
ϕ CO ( x , y ) = tan 1 ( μ 1 cos ( 2 x 3 ) cos ( x 3 y ) + cos ( x 3 + y ) ) .
P LS ( x , y ) = cos ( α ) + 2 cos ( x 3 y ) + 2 cos ( 2 x 3 ) + 2 cos ( x 3 + y ) ,
Q LS ( x , y ) = sin ( α ) .
η LS 0.8380 ,
α max 2.13 .
P CO ( x , y ) = μ cos ( α ) + 2 ( cos ( x 3 y ) + cos ( 2 x 3 ) + cos ( x 3 + y ) ) ,
Q CO ( x , y ) = μ sin ( α ) .
μ 0.4455 ,
α 2.494 ,
η CO 0.8015 .
P LS ( x , y ) = cos ( α 3 ) + 2 ( cos ( x 3 y ) + cos ( α 1 ) cos ( 2 x 3 ) + cos ( α 2 ) cos ( x 3 + y ) ) ,
Q LS ( x , y ) = sin ( α 3 ) + 2 ( sin ( α 1 ) cos ( 2 x 3 ) + sin ( α 2 ) cos ( x 3 + y ) ) .
η LS 0.9067 ,
α 1 = α 2 = π 2 ,
α 3 = π .
P CO ( x , y ) = μ 3 cos ( α 3 ) + 2 ( cos ( x 3 y ) + μ 1 cos ( α 1 ) cos ( 2 x 3 ) + μ 2 cos ( α 2 ) cos ( x 3 + y ) ) ,
Q CO ( x , y ) = μ 3 sin ( α 3 ) + 2 ( μ 1 sin ( α 1 ) cos ( 2 x 3 ) + μ 2 sin ( α 2 ) cos ( x 3 + y ) ) .
η CO = 0.9003 ,
α 1 = α 2 = π 2 ,
α 3 = π ,
μ 1 = μ 2 = 1.3368 ,
μ 3 = 0.9811 .
F ( κ ) = Ω f ( x ̱ ) + κ g ( x ̱ ) d x ̱
F ( κ ) = 1 2 Ω g ( x ̱ ) f ( x ̱ ) + κ g ( x ̱ ) d x ̱
F ( κ ) = Ω f ( x ̱ ) + κ g ( x ̱ ) d x ̱ = Ω + f ( x ̱ ) + κ g ( x ̱ ) d x ̱ + Ω f ( x ̱ ) + κ g ( x ̱ ) d x ̱ .
F ( κ ) = Ω + ( f ( x ̱ ) + κ g ( x ̱ ) + f ( x ̱ ) κ g ( x ̱ ) ) d x ̱ .
Γ ( α ) = 2 A S Ω S cos 2 ( x ) + cos 2 ( y ) + 2 cos ( α ) cos ( x ) cos ( y ) d x ̱ .
Γ ( α ) = 2 A S Ω S sin 2 ( x ) + cos 2 ( y ) + 2 cos ( α ) sin ( x ) cos ( y ) d x ̱ .
Γ ( α ) = 2 A S Ω S 1 4 + ( sin ( x ) + sin ( y ) ) 2 + cos ( α ) ( sin ( x ) + sin ( y ) ) d x ̱ .
P LS ( x , y , α 1 , α 2 ) = cos ( α 2 ) + 2 cos ( x ) + 2 cos ( α 1 ) cos ( y ) ,
Q LS ( x , y , α 1 , α 2 ) = sin ( α 2 ) + 2 sin ( α 1 ) cos ( y ) ,
P LS 2 + Q LS 2 = F ( x , y , α 1 , α 2 ) = F 1 ( x , y ) + F 2 ( x , y , α 1 , α 2 ) ,
F 1 ( x , y ) = 1 + 4 cos 2 ( x ) + 4 cos 2 ( y ) ,
F 2 ( x , y , α 1 , α 2 ) = 8 cos ( α 1 ) cos ( x ) cos ( y ) + 4 cos ( α 1 α 2 ) cos ( y ) + 4 cos ( x ) cos ( α 2 ) .
F ( x , y π 2 , α 1 , α 2 ) = G ( x , y , α 1 , α 2 ) = G 1 ( x , y ) + G 2 ( x , y , α 2 , α 2 ) ,
G 1 ( x , y ) = 1 + 4 cos 2 ( x ) + 4 sin 2 ( y ) ,
G 2 ( x , y , α 1 , α 2 ) = 8 cos ( α 1 ) cos ( x ) sin ( y ) + 4 cos ( α 1 α 2 ) sin ( y ) + 4 cos ( x ) cos ( α 2 ) .
Γ ( α 1 , α 2 ) = 1 4 π 2 π π π π G ( x , y , α 1 , α 2 ) d x d y .
Γ α 1 = 1 4 π 2 π π π π 4 sin ( α 1 ) cos ( x ) sin ( y ) 2 sin ( α 1 α 2 ) sin ( y ) G ( x , y , α 1 , α 2 ) d x d y ,
Γ α 2 = 1 4 π 2 π π π π 2 sin ( α 1 α 2 ) sin ( y ) 2 sin ( α 2 ) cos ( x ) G ( x , y , α 1 , α 2 ) d x d y .
H = [ 2 Γ α 1 2 2 Γ α 1 α 2 2 Γ α 1 α 2 2 Γ α 2 2 ]

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