Abstract

We report design theories of a diffractive superresolution element (DSE) to implement optical superresolution of focused partially spatially coherent laser beams. The design problem of the DSE can be transformed into a problem of linear programming to obtain a globally optimal solution. By using the design theories, some fundamental limits of optical superresolution of focused partially spatially coherent laser beams are proposed, and several design examples are provided. As expected, both the fundamental limits and the design examples show that worse spatial coherence will cause worse superresolution performance. The design theories provide a design approach with partially coherent beams and may be useful for other design problems under partially coherent illumination.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. R. Wang and X. G. Huang, "Subwavelength-resolvable focused non-Gaussian beam shaped with a binary diffractive optical element," Appl. Opt. 38, 2171-2176 (1999).
    [CrossRef]
  2. Y. Shen, G. Yang, and X. Hou, "Research on phenomenon of the super-resolution in laser lithography," Acta Opt. Sin. 19, 1512-1517 (1999).
  3. I. J. Cox, "Increasing the bit packing densities of optical disk systems," Appl. Opt. 23, 3260-3261 (1984).
    [CrossRef] [PubMed]
  4. Z. S. Hegedus and V. Sarafis, "Superresolving filters in confocally scanned imaging systems," J. Opt. Soc. Am. A 3, 1892-1896 (1986).
    [CrossRef]
  5. T. R. M. Sales and G. M. Morris, "Diffractive superresolution elements," J. Opt. Soc. Am. A 14, 1637-1646 (1997).
    [CrossRef]
  6. J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley,1997).
  7. I. J. Cox, C. J. R. Sheppard, and T. Wilson, "Reappraisal of arrays of concentric annuli as superresolving filters," J. Opt. Soc. Am. 72, 1287-1291 (1982).
    [CrossRef]
  8. R. Boivin and A. Boivin, "Optimized amplitude filtering for superresolution over a restricted field: I. Achievement of maximum central irradiance under an energy constraint," Opt. Acta 27, 587-610 (1980).
    [CrossRef]
  9. N. Wei, M. Gong, and R. Cui, "Numerical solution of sidelobe intensity for phase-shifting apodizers," Jpn. J. Appl. Phys., Part 1 42, 104-108 (2003).
    [CrossRef]
  10. H. Liu, Y. Yan, and G. Jin, "Design theories and performance limits of diffractive superresolution elements with the highest sidelobe suppressed," J. Opt. Soc. Am. A 22, 828-838 (2005).
    [CrossRef]
  11. F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1980).
    [CrossRef]
  12. R. Gase, "The multimode laser radiation as a Gaussian-Schell model beam," J. Mod. Opt. 38, 1107-1116 (1991).
    [CrossRef]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  14. Dirk Schafer, "Design concept for diffractive elements shaping partially coherent laser beams," J. Opt. Soc. Am. A 18, 2915-2922 (2001).
    [CrossRef]
  15. J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.
  16. G. P. Agrawal and E. Wolf, "Propagation-induced polarization changes in partially coherent optical beams," J. Opt. Soc. Am. A 17, 2019-2023 (2000).
    [CrossRef]
  17. J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, 1989), Chap. 2.
    [CrossRef]
  18. L. E. Elsgolc, Calculus of Variations (Pergamon, 1961), Chap. 1.
  19. T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
    [CrossRef]

2005 (1)

2003 (1)

N. Wei, M. Gong, and R. Cui, "Numerical solution of sidelobe intensity for phase-shifting apodizers," Jpn. J. Appl. Phys., Part 1 42, 104-108 (2003).
[CrossRef]

2001 (1)

2000 (1)

1999 (2)

M. R. Wang and X. G. Huang, "Subwavelength-resolvable focused non-Gaussian beam shaped with a binary diffractive optical element," Appl. Opt. 38, 2171-2176 (1999).
[CrossRef]

Y. Shen, G. Yang, and X. Hou, "Research on phenomenon of the super-resolution in laser lithography," Acta Opt. Sin. 19, 1512-1517 (1999).

1997 (1)

1991 (1)

R. Gase, "The multimode laser radiation as a Gaussian-Schell model beam," J. Mod. Opt. 38, 1107-1116 (1991).
[CrossRef]

1986 (1)

1984 (1)

1982 (1)

1980 (2)

R. Boivin and A. Boivin, "Optimized amplitude filtering for superresolution over a restricted field: I. Achievement of maximum central irradiance under an energy constraint," Opt. Acta 27, 587-610 (1980).
[CrossRef]

F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1980).
[CrossRef]

Agrawal, G. P.

Blyth, T. S.

T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
[CrossRef]

Boivin, A.

R. Boivin and A. Boivin, "Optimized amplitude filtering for superresolution over a restricted field: I. Achievement of maximum central irradiance under an energy constraint," Opt. Acta 27, 587-610 (1980).
[CrossRef]

Boivin, R.

R. Boivin and A. Boivin, "Optimized amplitude filtering for superresolution over a restricted field: I. Achievement of maximum central irradiance under an energy constraint," Opt. Acta 27, 587-610 (1980).
[CrossRef]

Cox, I. J.

Cui, R.

N. Wei, M. Gong, and R. Cui, "Numerical solution of sidelobe intensity for phase-shifting apodizers," Jpn. J. Appl. Phys., Part 1 42, 104-108 (2003).
[CrossRef]

Elsgolc, L. E.

L. E. Elsgolc, Calculus of Variations (Pergamon, 1961), Chap. 1.

Gase, R.

R. Gase, "The multimode laser radiation as a Gaussian-Schell model beam," J. Mod. Opt. 38, 1107-1116 (1991).
[CrossRef]

Gong, M.

N. Wei, M. Gong, and R. Cui, "Numerical solution of sidelobe intensity for phase-shifting apodizers," Jpn. J. Appl. Phys., Part 1 42, 104-108 (2003).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.

Gori, F.

F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1980).
[CrossRef]

Hegedus, Z. S.

Hou, X.

Y. Shen, G. Yang, and X. Hou, "Research on phenomenon of the super-resolution in laser lithography," Acta Opt. Sin. 19, 1512-1517 (1999).

Huang, X. G.

Jin, G.

Liu, H.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Morris, G. M.

Robertson, E. F.

T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
[CrossRef]

Sales, T. R. M.

Sarafis, V.

Schafer, Dirk

Shen, Y.

Y. Shen, G. Yang, and X. Hou, "Research on phenomenon of the super-resolution in laser lithography," Acta Opt. Sin. 19, 1512-1517 (1999).

Sheppard, C. J. R.

Strayer, J. K.

J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, 1989), Chap. 2.
[CrossRef]

Turunen, J.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley,1997).

Wang, M. R.

Wei, N.

N. Wei, M. Gong, and R. Cui, "Numerical solution of sidelobe intensity for phase-shifting apodizers," Jpn. J. Appl. Phys., Part 1 42, 104-108 (2003).
[CrossRef]

Wilson, T.

Wolf, E.

Wyrowski, F.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley,1997).

Yan, Y.

Yang, G.

Y. Shen, G. Yang, and X. Hou, "Research on phenomenon of the super-resolution in laser lithography," Acta Opt. Sin. 19, 1512-1517 (1999).

Acta Opt. Sin. (1)

Y. Shen, G. Yang, and X. Hou, "Research on phenomenon of the super-resolution in laser lithography," Acta Opt. Sin. 19, 1512-1517 (1999).

Appl. Opt. (2)

J. Mod. Opt. (1)

R. Gase, "The multimode laser radiation as a Gaussian-Schell model beam," J. Mod. Opt. 38, 1107-1116 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Jpn. J. Appl. Phys., Part 1 (1)

N. Wei, M. Gong, and R. Cui, "Numerical solution of sidelobe intensity for phase-shifting apodizers," Jpn. J. Appl. Phys., Part 1 42, 104-108 (2003).
[CrossRef]

Opt. Acta (1)

R. Boivin and A. Boivin, "Optimized amplitude filtering for superresolution over a restricted field: I. Achievement of maximum central irradiance under an energy constraint," Opt. Acta 27, 587-610 (1980).
[CrossRef]

Opt. Commun. (1)

F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1980).
[CrossRef]

Other (6)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, 1989), Chap. 2.
[CrossRef]

L. E. Elsgolc, Calculus of Variations (Pergamon, 1961), Chap. 1.

T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
[CrossRef]

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley,1997).

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

DSE to implement optical superresolution of a focused laser beam.

Fig. 2
Fig. 2

Illustration of the selection of ϵ and η i ( i = 1 , 2 , , N ) .

Fig. 3
Fig. 3

Linearization of nonlinear constraints (20b) with K 0 ( η i ) = 2 . X = d 1 ( η i ) k = 1 K A k q k , 1 ( η i ) ; Y = d 2 ( η i ) k = 1 K A k q k , 2 ( η i ) ; R = ϵ d 1 ( 0 ) k = 1 K A k q k , 1 ( 0 ) ; κ ( p ) = ( p 1 ) π / ( 2 P ) ; p = 1 , 2 , , P ; and P = 4 in this scheme.

Fig. 4
Fig. 4

Dependence of A c on σ.

Fig. 5
Fig. 5

S e u ( G ) with various σ or A c . The inset curves are of logarithmic vertical coordinate to magnify those of linear vertical coordinate.

Fig. 6
Fig. 6

Curve of G e l ( A c ) .

Fig. 7
Fig. 7

(a)–(d) show the designed intensity distributions of examples 1–4 in Table 1, respectively.

Tables (1)

Tables Icon

Table 1 Design Examples of the DSE

Equations (77)

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

E f ( x , y ) = ( i λ f ) 1 exp ( i 2 π f / λ ) exp [ i π ( x 2 + y 2 ) / ( λ f ) ] × E i ( u , v ) t ( u , v ) exp [ i 2 π ( u x + v y ) / ( λ f ) ] d u d v ,
i f ( x , y ) = E f ( x , y ) E f * ( x , y ) = ( λ f ) 2 W i ( u 1 , v 1 ; u 2 , v 2 ) t ( u 1 , v 1 ) t * ( u 2 , v 2 ) × exp { i 2 π [ ( u 1 u 2 ) x + ( v 1 v 2 ) y ] / ( λ f ) } d u 1 d v 1 d u 2 d v 2 ,
W i ( u 1 , v 1 ; u 2 , v 2 ) = [ i i ( u 1 , v 1 ) i i ( u 2 , v 2 ) ] 1 / 2 μ i ( Δ u , Δ v ) ,
i f ( x , y ) = ( λ f ) 2 [ ( i i 1 / 2 t ) ( i i 1 / 2 t ) ] ( Δ u , Δ v ) × μ i ( Δ u , Δ v ) exp { i 2 π ( x Δ u + y Δ v ) / ( λ f ) } d Δ u d Δ v ,
[ ( i i 1 / 2 t ) ( i i 1 / 2 t ) ] ( Δ u , Δ v ) = [ i i 1 / 2 ( u ¯ + Δ u / 2 , v ¯ + Δ v / 2 ) t ( u ¯ + Δ u / 2 , v ¯ + Δ v / 2 ) ] [ i i 1 / 2 ( u ¯ Δ u / 2 , v ¯ Δ v / 2 ) t ( u ¯ Δ u / 2 , v ¯ Δ v / 2 ) ] * d u ¯ d v ¯
i i ( u , v ) = i i ( r ) ,
μ i ( Δ u , Δ v ) = μ i ( ρ ) ,
t ( u , v ) = t ( r ) circ ( r / R ) , circ ( r / R ) = { 1 , if r R 0 , if r > R } ,
[ ( i i 1 / 2 t ) ( i i 1 / 2 t ) ] ( Δ u , Δ v ) = [ ( i i 1 / 2 t ) ( i i 1 / 2 t ) ] ( ρ ) = { 4 D ( ρ ) 2 r 1 r 2 4 r 1 2 r 2 2 ( r 1 2 + r 2 2 ρ 2 ) 2 [ i i ( r 1 ) i i ( r 2 ) ] 1 / 2 Re [ t ( r 1 ) t * ( r 2 ) ] d r 1 d r 2 , if 0 ρ 2 R 0 , if ρ > 2 R } ,
D ( ρ ) = { ( r 1 , r 1 ) 0 r 1 R , 0 r 2 r 1 , r 1 r 2 ρ r 1 + r 2 } .
i f ( x , y ) = i f ( r f ) = 8 π ( λ f ) 2 0 R d r 1 0 r 1 d r 2 [ i i ( r 1 ) i i ( r 2 ) ] 1 / 2 Re [ t ( r 1 ) t * ( r 2 ) ] × r 1 r 2 r 1 + r 2 μ i ( ρ ) J 0 ( 2 π λ f ρ r f ) 2 r 1 r 2 4 r 1 2 r 2 2 ( r 1 2 + r 2 2 ρ 2 ) 2 ρ d ρ ,
I ( η ) = 8 π 0 1 d α 0 α d β [ I i ( α ) I i ( β ) ] 1 / 2 Re [ T ( α ) T * ( β ) ] α β α + β f ( α , β , γ , η ) d γ ,
f ( α , β , γ , η ) = γ μ ( γ ) J 0 ( x J γ η ) 2 α β 4 α 2 β 2 ( α 2 + β 2 γ 2 ) 2 ,
I ( η ) = 8 π k = 2 K m = 1 k 1 ( I k I m ) 1 / 2 Re ( T k T m * ) α k 1 α k d α α m 1 α m d β α β α + β f ( α , β , γ , η ) d γ + 8 π k = 1 K I k T k T k * α k 1 α k d α α k 1 α d β α β α + β f ( α , β , γ , η ) d γ ,
α k 1 α k d α α m 1 α m d β α β α + β f ( α , β , γ , η ) d γ = { ψ ( k , m , η ) , if m = 1 ψ ( k , m , η ) ψ ( k , m 1 , η ) , if m 2 } ,
α k 1 α k d α α k 1 α d β α β α + β f ( α , β , γ , η ) d γ = { φ ( k , η ) , if k = 1 φ ( k , η ) ψ ( k , k 1 , η ) , if k 2 } ,
ψ ( k , m , η ) = α k 1 α m α k α m μ ( γ ) J 0 ( x J γ η ) [ π α m 2 / 4 g ( α k 1 , γ , α m ) ] γ d γ + α k α m α k 1 + α m μ ( γ ) J 0 ( x J γ η ) [ g ( α k , γ , α m ) g ( α k 1 , γ , α m ) ] γ d γ + α k 1 + α m α k + α m μ ( γ ) J 0 ( x J γ η ) [ g ( α k , γ , α m ) + π α m 2 / 4 ] γ d γ ,
g ( α , γ , α m ) = α 2 2 arccos α 2 + γ 2 α m 2 2 α γ + α m 2 2 arcsin α 2 γ 2 α m 2 2 α m γ 1 4 4 γ 2 α m 2 ( α 2 γ 2 α m 2 ) 2 ,
φ ( k , η ) = 0 2 α k 1 μ ( γ ) J 0 ( x J γ η ) [ g 0 ( α k , γ ) g 0 ( α k 1 , γ ) ] γ d γ + 2 α k 1 2 α k μ ( λ ) J 0 ( x J γ η ) g 0 ( α k , γ ) γ d γ ,
g 0 ( α , γ ) = α 2 2 arccos γ 2 α γ 8 4 α 2 γ 2 .
T 1 , k = T k , H ̃ ( η ) k , m = { 8 π ( I k I m ) 1 / 2 α k 1 α k d α α m 1 α m d β α β α + β f ( α , β , γ , η ) d γ , if m < k 8 π I k α k 1 α k d α α k 1 α d β α β α + β f ( α , β , γ , η ) d γ , if m = k 0 , if m > k } ,
I ( η ) = Re [ T H ̃ ( η ) T + ] = T H ( η ) T + ,
max { T k } I ( 0 ) subject to I ( η i ) / I ( 0 ) ϵ ; T k 1 ;
k = 1 , 2 , , K ;
Q ( η ) + H ( η ) Q ( η ) = diag [ d 1 ( η ) , d 2 ( η ) , , d K ( η ) ] ,
I ( η ) = m = 1 K d m ( η ) k = 1 K T k q k , m ( η ) 2 ,
d m ( 0 ) d 1 ( 0 ) , if m 2 .
d m ( η ) θ d 1 ( η ) , θ 1 , if m > K 0 ( η ) ,
I ( 0 ) d 1 ( 0 ) k = 1 K T k q k , 1 ( 0 ) 2 ,
I ( η ) m = 1 K 0 ( η ) d m ( η ) k = 1 K T k q k , m ( η ) 2 , η > 0 .
max { A k , B k } { [ k = 1 K A k q k , 1 ( 0 ) ] 2 + [ k = 1 K B k q k , 1 ( 0 ) ] 2 }
m = 1 K 0 ( η i ) d m ( η i ) { [ k = 1 K A k q k , m ( η i ) ] 2 + [ k = 1 K B k q k , m ( η i ) ] 2 } ϵ d 1 ( 0 ) { [ k = 1 K A k q k , 1 ( 0 ) ] 2 + [ k = 1 K B k q k , 1 ( 0 ) ] 2 } ,
A k 2 + B k 2 1 , k = 1 , 2 , , K ,
max { A k } k = 1 K A k q k , 1 ( 0 )
m = 1 K 0 ( η i ) d m ( η i ) [ k = 1 K A k q k , m ( η i ) ] 2 ϵ d 1 ( 0 ) [ k = 1 K A k q k , 1 ( 0 ) ] 2 ,
1 / 2 A k 1 / 2 , k = 1 , 2 , , K ,
ϵ d 1 ( 0 ) k = 1 K A k q k , 1 ( 0 ) d 1 ( η i ) k = 1 K A k q k , 1 ( η i ) ϵ d 1 ( 0 ) k = 1 K A k q k , 1 ( 0 ) .
[ d 1 ( η i ) k = 1 K A k q k , 1 ( η i ) ] ( 1 ) m cos θ ( p ) + [ d 2 ( η i ) k = 1 K A k q k , 2 ( η i ) ] ( 1 ) n sin θ ( p ) [ ϵ d 1 ( 0 ) k = 1 K A k q k , 1 ( 0 ) ] cos π 4 P ,
d 1 ( η i ) [ k = 1 K A k q k , 1 ( η i ) ] 2 + d 2 ( η i ) [ k = 1 K A k q k , 2 ( η i ) ] 2 R 1 2 ,
R 1 2 + d 3 ( η i ) [ k = 1 K A k q k , 3 ( η i ) ] 2 ϵ d 1 ( 0 ) [ k = 1 K A k q k , 1 ( 0 ) ] 2 ,
d 1 ( η i ) [ k = 1 K A k q k , 1 ( η i ) ] 2 + d 2 ( η i ) [ k = 1 K A k q k , 2 ( η i ) ] 2 R 1 2 , R n 2 + d n + 2 ( η i ) [ k = 1 K A k q k , n + 2 ( η i ) ] 2 R n + 1 2 , n = 1 , 2 , , K 0 ( η i ) 3 ,
R K 0 ( η i ) 2 2 + d K 0 ( η i ) ( η i ) [ k = 1 K A k q k , K 0 ( η i ) ( η i ) ] 2 ϵ d 1 ( 0 ) [ k = 1 K A k q k , 1 ( 0 ) ] 2 ,
A c = 2 π 0 2 μ ( γ ) 2 γ d γ 2 π 0 2 γ d γ = σ 2 4 [ 1 exp ( 4 σ 2 ) ] .
max { T k } I ( 0 ) subject to I ( G η L , 0.5 ) / I ( 0 ) = 0.5 ,
T k 1 , k = 1 , 2 , , K ,
I ( η ) = 0 1 d α 0 1 [ A ( α ) A ( β ) + B ( α ) B ( β ) ] h ( α , β , η ) d β ,
h ( α , β , η ) = 4 π [ I i ( α ) I i ( β ) ] 1 / 2 α β α + β f ( α , β , γ , η ) d γ ,
max { A ( α ) , B ( α ) } I ( 0 ) subject to I ( η i ) ϵ I ( 0 ) + ω i 2 = 0 ,
A ( α ) 2 + B ( α ) 2 1 + C ( α ) 2 = 0 ,
F [ A ( α ) , B ( α ) , ω i , C ( α ) , λ i , τ ( α ) ] = I ( 0 ) + i = 1 N λ i [ I ( η i ) ϵ I ( 0 ) + ω i 2 ] + 0 1 [ A ( α ) 2 + B ( α ) 2 1 + C ( α ) 2 ] τ ( α ) d α = 0 1 d α 0 1 [ A ( α ) A ( β ) + B ( α ) B ( β ) ] G ( α , β , λ i ) d β + i = 1 N λ i ω i 2 + 0 1 [ A ( α ) 2 + B ( α ) 2 1 + C ( α ) 2 ] τ ( α ) d α ,
G ( α , β , λ i ) = i = 1 N λ i h ( α , β , η i ) + h ( α , β , 0 ) ( 1 i = 1 N λ i ϵ ) .
δ A ( α ) F [ A ( α ) , B ( α ) , ω i , C ( α ) , λ i , τ ( α ) ] = 0 1 d α 0 1 [ A ( α ) δ A ( β ) + A ( β ) δ A ( α ) ] G ( α , β , λ i ) d β + 0 1 2 A ( α ) δ A ( α ) τ ( α ) d α = 2 0 1 [ A ( α ) τ ( α ) + 0 1 A ( β ) G ( α , β , λ i ) d β ] δ A ( α ) d α = 0 ,
δ B ( α ) F [ A ( α ) , B ( α ) , ω i , C ( α ) , λ i , τ ( α ) ] = 2 0 1 [ B ( α ) τ ( α ) + 0 1 B ( β ) G ( α , β , λ i ) d β ] δ B ( α ) d α = 0 ,
δ C ( α ) F [ A ( α ) , B ( α ) , ω i , C ( α ) , λ i , τ ( α ) ] = 0 1 2 C ( α ) δ C ( α ) τ ( α ) d α = 0 ,
F [ A ( α ) , B ( α ) , ω i , C ( α ) , λ i , τ ( α ) ] / ω i = 2 λ i ω i = 0 , i = 1 , 2 , , N ,
A ( α ) τ ( α ) + 0 1 A ( β ) G ( α , β , λ i ) d β = 0 , α [ 0 , 1 ] ,
B ( α ) τ ( α ) + 0 1 B ( β ) G ( α , β , λ i ) d β = 0 , α [ 0 , 1 ] ,
C ( α ) τ ( α ) = 0 , α [ 0 , 1 ] ,
C ( α ) 0 , α [ 0 , 1 ] ,
C ( α ) 0 , α [ a , b ] , [ a , b ] [ 0 , 1 ] .
G A ( α ) = 0 1 A ( β ) G ( α , β , λ i ) d β = 0 ,
α [ a , b ] , [ a , b ] [ 0 , 1 ] ,
G B ( α ) = 0 1 B ( β ) G ( α , β , λ i ) d β = 0 ,
α [ a , b ] , [ a , b ] [ 0 , 1 ] ,
G A ( α ) = 0 1 A ( β ) G ( α , β , λ i ) d β = 0 , α [ 0 , 1 ] ,
G B ( α ) = 0 1 B ( β ) G ( α , β , λ i ) d β = 0 , α [ 0 , 1 ] .
0 = i = 1 N λ i [ I ( η i ) ϵ I ( 0 ) + ω i 2 ] = 0 1 d α 0 1 [ A ( α ) A ( β ) + B ( α ) B ( β ) ] [ G ( α , β , λ i ) h ( α , β , 0 ) ] d β i = 1 N λ i ω i 2 = I ( 0 ) ,
C ( α ) = 0 , α [ 0 , 1 ] .
A ( α ) 2 + B ( α ) 2 = 1 , α [ 0 , 1 ] ,
A k + i B k = ( A k + i B k ) exp ( i ϕ 0 )
k = 1 K A k q k , 1 ( 0 ) = k = 1 K B k q k , 1 ( 0 ) ,
ϕ 0 = n π + arctan k = 1 K ( A k B k ) q k , 1 ( 0 ) k = 1 K ( A k + B k ) q k , 1 ( 0 ) ,
k = 1 K A ˜ k q k , 1 ( 0 ) = k = 1 K B ˜ k q k , i ( 0 ) .
A k = B k = ( A ˜ k + B ˜ k ) / 2 .
[ k = 1 K A ˜ k + B ˜ k 2 q k , 1 ( 0 ) ] 2 + [ k = 1 K A ˜ k + B ˜ k 2 q k , 1 ( 0 ) ] 2 = [ k = 1 K A ˜ k q k , 1 ( 0 ) ] 2 + [ k = 1 K B ˜ k q k , 1 ( 0 ) ] 2 ,
m = 1 K 0 ( η i ) d m ( η i ) { [ k = 1 K A ˜ k + B ˜ k 2 q k , m ( η i ) ] 2 + [ k = 1 K A ˜ k + B ˜ k 2 q k , m ( η i ) ] 2 } = m = 1 K 0 ( η i ) d m ( η i ) 1 2 [ 1 × k = 1 K A ˜ k q k , m ( η i ) + 1 × k = 1 K B ˜ k q k , m ( η i ) ] 2 m = 1 K 0 ( η i ) d m ( η i ) 1 2 ( 1 2 + 1 2 ) { [ k = 1 K A ˜ k q k , m ( η i ) ] 2 + [ k = 1 K B ˜ k q k , m ( η i ) ] 2 } 2 ϵ d 1 ( 0 ) { [ k = 1 K A ˜ k q k , 1 ( 0 ) ] 2 + [ k = 1 K B ˜ k q k , 1 ( 0 ) ] 2 } = ϵ d 1 ( 0 ) { [ k = 1 K A ˜ k + B ˜ k 2 q k , 1 ( 0 ) ] 2 + [ k = 1 K A ˜ k + B ˜ k 2 q k , 1 ( 0 ) ] 2 } ,
[ ( A ˜ k + B ˜ k ) / 2 ] 2 + [ ( A ˜ k + B ˜ k ) / 2 ] 2 = ( 1 × A ˜ k + 1 × B ˜ k ) 2 / 2 ( 1 2 + 1 2 ) ( A ˜ k 2 + B ˜ k 2 ) / 2 1 ,

Metrics