Abstract

By geometrical optics and the Rayleigh–Sommerfeld diffraction formula, theories for the design of a hybrid refractive–diffractive superresolution lens (HRDSL) with high numerical aperture are constructed. Differences between the profile of the diffractive superresolution element (DSE) with high numerical aperture and that with low numerical aperture are indicated. Optimization theory can obtain a globally optimal solution through a linear programming much more simplified than the corresponding one in Liu et al. [ J. Opt. Soc. Am. A 19, 2185 ( 2002)]. The rules of the structure of the designed DSE are both theoretically proved and numerically verified. Comparison of this optimization theory with the other design theories and examples of designing the HRDSL with high numerical aperture are provided. Last, some limits of optical superresolution with high numerical aperture are set and compared with those for low numerical aperture.

© 2003 Optical Society of America

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References

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2002 (1)

1999 (2)

M. Testorf, “Perturbation theory as a unified approach to describe diffractive optical elements,” J. Opt. Soc. Am. A 16, 1115–1123 (1999).
[CrossRef]

Y. Shen, G. Yang, X. Hou, “Research on phenomenon of the superresolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

1998 (1)

1997 (2)

1995 (2)

1994 (2)

1985 (1)

1984 (1)

1982 (1)

1981 (1)

1980 (1)

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Aslund, N.

Blyth, T. S.

T. S. Blyth, E. F. Robertson, Further Linear Algebra (Springer, London, 2002), Chap. 1.

Boivin, A.

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Boivin, R.

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 3.

Brenner, K. H.

Carlsson, K.

Chugui, Y.

Cox, I. J.

Crosignani, B.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986), Chap. II.

Danielsson, P. E.

Delisle, C. A.

DiPorto, P.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986), Chap. II.

Elsgolc, L. E.

L. E. Elsgolc, Calculus of Variations (Pergamon, Oxford, UK, 1961), Chap. I.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

Grann, E. B.

Han, Y.

Hazra, L. N.

Hou, X.

Y. Shen, G. Yang, X. Hou, “Research on phenomenon of the superresolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9.

Jin, G.

Koronkevitch, V. P.

Krivenkov, B. E.

Lenz, R.

Liljeborg, A.

Liu, H.

Majlof, L.

Masters, B. R.

Mikhlyaev, S. V.

Moharam, M. G.

Morris, G. M.

Pommet, D. A.

Robertson, E. F.

T. S. Blyth, E. F. Robertson, Further Linear Algebra (Springer, London, 2002), Chap. 1.

Sales, T. R. M.

Shen, Y.

Y. Shen, G. Yang, X. Hou, “Research on phenomenon of the superresolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

Sheppard, C. J. R.

Singer, W.

Solimeno, S.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986), Chap. II.

Strayer, J. K.

J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, New York, 1989), Chap. 2.

Tan, Q.

Testorf, M.

Tiziani, H.

Wilson, T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 3.

Xu, Duanyi

Duanyi Xu, Principle and Design of Optical Storage Systems (National Defence Industry, Beijing, 2000), Chap. 1.

Yan, Y.

Yang, G.

Y. Shen, G. Yang, X. Hou, “Research on phenomenon of the superresolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

Acta Opt. Sin. (1)

Y. Shen, G. Yang, X. Hou, “Research on phenomenon of the superresolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Opt. Lett. (2)

Other (9)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 3.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986), Chap. II.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9.

T. S. Blyth, E. F. Robertson, Further Linear Algebra (Springer, London, 2002), Chap. 1.

J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, New York, 1989), Chap. 2.

L. E. Elsgolc, Calculus of Variations (Pergamon, Oxford, UK, 1961), Chap. I.

Duanyi Xu, Principle and Design of Optical Storage Systems (National Defence Industry, Beijing, 2000), Chap. 1.

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Figures (11)

Fig. 1
Fig. 1

Scheme of the HRDSL.

Fig. 2
Fig. 2

Computation of the profile of the aspheric surface by the principle of equal optical path.

Fig. 3
Fig. 3

Profile of the aspheric surface of an AAPL with NA t = 1 .

Fig. 4
Fig. 4

Coordinate systems to obtain the dependence of the field on the angular coordinate with the rotational symmetry of the system.

Fig. 5
Fig. 5

Plots (a), (b), and (c) show, respectively, the x component, the y component, and the z component of the slowly varying amplitude of the transmissive field on the planar surface of the AAPL with NA t = 1 without the DSE.

Fig. 6
Fig. 6

Transformation of the field near the focus of the AAPL resulting from a phase-only DSE.

Fig. 7
Fig. 7

Curve of h ( ρ 1 ) / h L ( ρ 1 ) with ψ ( ρ 1 ) = ψ L ( ρ 1 ) for a HRDSL with NA t = 1 .

Fig. 8
Fig. 8

Intensity distribution on the focal plane of the designed HRDSLs with G = 0.68966 (for N = 1 , 2, and 3) and NA t = 1 . Here N = 0 means no DSE.

Fig. 9
Fig. 9

Intensity distribution on the optical axis of the designed HRDSLs with G = 0.68966 (for N = 1 , 2, and 3) and NA t = 1 . Here N = 0 means no DSE.

Fig. 10
Fig. 10

The dashed curve and the dotted curve correspond to S u ( G ) , the solid curve corresponds to S eu ( G ) with NA t 1 , and the curve with circles corresponds to S eu ( G ) with NA t = 1 . The curves in the inset are in logarithmic coordinates to magnify those to linear coordinates.

Fig. 11
Fig. 11

Curves of K m ( a ) ( G ) .

Tables (2)

Tables Icon

Table 1 Simulation Results of Four Theories with the Same Zero Constraints as Those in Ref. 7

Tables Icon

Table 2 Parameters of the Designed HRDSLs with NA t = 1

Equations (134)

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E ( i ) ( ρ ,   θ ,   z ) = y E ( i ) ( ρ ,   θ ,   z ) = y e ( i ) exp ( - ik 0 z ) ,
α ( i ) ( ρ 1 ) = arcsin ρ 1 n ρ 1 2 + f a 2 ,
z a ( ρ 1 ) + n [ t - z a ( ρ 1 ) ] / cos   α ( i ) ( ρ 1 ) + ρ 1 2 + f a 2
= [ ABCF a ] = [ ODF a ] = nt + f a ,
z a ( ρ 1 ) = nt [ 1 / cos   α ( i ) ( ρ 1 ) - 1 ] + ρ 1 2 + f a 2 - f a n / cos   α ( i ) ( ρ 1 ) - 1 .
ρ a ( ρ 1 ) = ρ 1 + [ t - z a ( ρ 1 ) ] tan   α ( i ) ( ρ 1 ) .
z a ( ρ 1 , M ) = nt { ( 1 + NA t 2 ) / [ 1 + ( 1 - 1 / n 2 ) NA t 2 ] - 1 } + ρ 1 , M 2 + f a 2 - f a n ( 1 + NA t 2 ) / [ 1 + ( 1 - 1 / n 2 ) NA t 2 ] - 1 .
t = ρ 1 , M ( 1 + 1 / NA t 2 - 1 / NA t ) ( n - β ) - n ( 1 - β ) ( 1 + NA t 2 ) / [ 1 + ( 1 - 1 / n 2 ) NA t 2 ] .
E ( ρ ,   0 ,   z ) = y E y ( ρ ,   0 ,   z ) ,
E ( ρ ,   π / 2 ,   z ) = y E y ( ρ ,   π / 2 ,   z ) + z E z ( ρ ,   π / 2 ,   z ) .
E ( ρ ,   θ ,   z ) = y E y ( ρ ,   0 ,   z ) ,
E ( ρ ,   θ ,   z ) = x E y ( ρ ,   π / 2 ,   z ) + z E z ( ρ ,   π / 2 ,   z ) .
E ( ρ ,   θ ,   z ) = E ( ρ ,   θ ,   z ) cos   θ + E ( ρ ,   θ ,   z ) sin   θ .
E x ( ρ ,   θ ,   z ) = - 0.5 [ E y ( ρ ,   0 ,   z ) - E y ( ρ ,   π / 2 ,   z ) ] sin   2 θ ,
E y ( ρ ,   θ ,   z ) = 0.5 [ E y ( ρ ,   0 ,   z ) + E y ( ρ ,   π / 2 ,   z ) ] + 0.5 [ E y ( ρ ,   0 ,   z ) - E y ( ρ ,   π / 2 ,   z ) ] cos   2 θ ,
E z ( ρ ,   θ ,   z ) = E z ( ρ ,   π / 2 ,   z ) sin   θ ,
x E x ( ρ ,   θ ,   z ) + y E y ( ρ ,   θ ,   z ) + z E z ( ρ ,   θ ,   z )
= E ( ρ ,   θ ,   z ) .
E ( ρ ,   θ ,   z ) = e ( ρ ,   θ ,   z ) exp [ - ik 0 ϕ ( ρ ,   θ ,   z ) ] ,
ϕ ( P 2 ) = ϕ ( P 1 ) + [ P 1 P 2 ] ,
e ( P 2 ) = e ( P 1 ) exp - 1 2 n P 1 P 2 2 ϕ ( P ) d s ,
ϕ ( t ) ( P ) = ϕ ( i ) ( P ) ,
e ( t ) ( P ) = T ( P ) e ( i ) ( P ) ,
e ( t ) ( ρ 1 ,   θ 1 ) = x e x ( t ) ( ρ 1 ,   θ 1 ) + y e y ( t ) ( ρ 1 ,   θ 1 ) + z e z ( t ) ( ρ 1 ,   θ 1 ) ,
ϕ ( ρ 1 ) = - ρ 1 2 + f a 2 + ϕ ( F a ) ,
E k ( ρ ,   θ ,   f   ) = x E x , k ( ρ ,   θ ,   f   ) + y E y , k ( ρ ,   θ ,   f   ) + z E z , k ( ρ ,   θ ,   f   )
| f - f a | λ / NA t 2 or | f - f a | < λ / NA t 2 ,
ρ λ / NA t or ρ < λ / NA t .
E x , k ( ρ ,   θ ,   f   )
= f i λ ρ 1 , k - 1 ρ 1 , k ρ 1 d ρ 1 0 2 π d θ 1 e x ( t ) ( ρ 1 ,   θ 1 ) exp [ - ik 0 ϕ ( ρ 1 ) ] × exp [ - ik 0 ρ 1 2 - 2 ρ 1 ρ cos ( θ 1 - θ ) + ρ 2 + f 2 ] ρ 1 2 - 2 ρ 1 ρ cos ( θ 1 - θ ) + ρ 2 + f 2 .
E x , k ( ρ ,   θ ,   f   )
f i λ ρ 1 , k - 1 ρ 1 , k d ρ 1 ρ 1 ρ 1 2 + f 2 exp { - ik 0 [ ϕ ( ρ 1 ) + ρ 1 2 + f 2 ] } × 0 2 π d θ 1 e x ( t ) ( ρ 1 ,   θ 1 ) × exp [ - ik 0 ρ 1 ρ / ρ 1 2 + f 2 cos ( θ 1 - θ ) ] ,
exp ( ix   cos   θ ) = n = - + J n ( x ) i n exp ( - in θ ) .
E x , k ( ρ ,   θ ,   f   )
f i λ ρ 1 , k - 1 ρ 1 , k d ρ 1 ρ 1 ρ 1 2 + f 2 exp { - ik 0 [ ϕ ( ρ 1 ) + ρ 1 2 + f 2 ] } × [ e y ( t ) ( ρ 1 ,   0 ) - e y ( t ) ( ρ 1 ,   π / 2 ) ] π J 2 ( k 0 ρ 1 ρ / ρ 1 2 + f 2 ) sin   2 θ .
E y , k ( ρ ,   θ ,   f   )
f i λ ρ 1 , k - 1 ρ 1 , k d ρ 1 ρ 1 ρ 1 2 + f 2 exp { - ik 0 [ ϕ ( ρ 1 ) + ρ 1 2 + f 2 ] } × { [ e y ( t ) ( ρ 1 , 0 ) + e y ( t ) ( ρ 1 ,   π / 2 ) ] π J 0 ( k 0 ρ 1 ρ / ρ 1 2 + f 2 ) - [ e y ( t ) ( ρ 1 ,   0 ) - e y ( t ) ( ρ 1 ,   π / 2 ) ] π J 2 ( k 0 ρ 1 ρ / ρ 1 2 + f 2 ) cos   2 θ } ,
E z , k ( ρ ,   θ ,   f   )
f λ ρ 1 , k - 1 ρ 1 , k d ρ 1 ρ 1 ρ 1 2 + f 2 exp { - ik 0 [ ϕ ( ρ 1 ) + ρ 1 2 + f 2 ] } × e z ( t ) ( ρ 1 ,   π / 2 ) 2 π J 1 ( k 0 ρ 1 ρ / ρ 1 2 + f 2 ) sin   θ .
 
exp { - ik 0 [ ϕ ( ρ 1 ) + ρ 1 2 + f 2 ] }
exp [ - ik 0 ϕ ( F a ) ] exp [ - ik 0 f a ( f - f a ) / ρ 1 2 + f a 2 ] .
 
E x , k ( ρ ,   θ ,   f   )
f a i λ exp [ - ik 0 ϕ ( F a ) ] ρ 1 , k - 1 ρ 1 , k d ρ 1 ρ 1 ρ 1 2 + f a 2 × exp [ - ik 0 f a ( f - f a ) / ρ 1 2 + f a 2 ] × [ e y ( t ) ( ρ 1 ,   0 ) - e y ( t ) ( ρ 1 ,   π / 2 ) ] π J 2 ( k 0 ρ 1 ρ / ρ 1 2 + f a 2 ) sin   2 θ ,
E y , k ( ρ ,   θ ,   f   )
f a i λ exp [ - ik 0 ϕ ( F a ) ] ρ 1 , k - 1 ρ 1 , k d ρ 1 ρ 1 ρ 1 2 + f a 2 × exp [ - ik 0 f a ( f - f a ) / ρ 1 2 + f a 2 ] × { [ e y ( t ) ( ρ 1 ,   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ) ] × π J 0 ( k 0 ρ 1 ρ / ρ 1 2 + f a 2 ) - [ e y ( t ) ( ρ 1 ,   0 ) - e y ( t ) ( ρ 1 ,   π / 2 ) ] × π J 2 ( k 0 ρ 1 ρ / ρ 1 2 + f a 2 ) cos   2 θ } ,
E z , k ( ρ ,   θ ,   f   )
f a λ exp [ - ik 0 ϕ ( F a ) ] ρ 1 , k - 1 ρ 1 , k d ρ 1 ρ 1 ρ 1 2 + f a 2 × exp [ - ik 0 f a ( f - f a ) / ρ 1 2 + f a 2 ] × e z ( t ) ( ρ 1 ,   π / 2 ) 2 π J 1 ( k 0 ρ 1 ρ / ρ 1 2 + f a 2 ) sin   θ .
E x , k ( ρ ,   θ ,   f ;   h k )
f - h k i λ ρ 2 , k - 1 + ρ 2 , k - d ρ 2 ρ 2 ρ 2 2 + ( f - h k ) 2 × exp { - ik 0 [ ϕ ( ρ 2 ;   h k ) + ρ 2 2 + ( f - h k ) 2 ] } × [ e y ( t ) ( ρ 2 ,   0 ;   h k ) - e y ( t ) ( ρ 2 ,   π / 2 ;   h k ) ] × π J 2 ( k 0 ρ 2 ρ / ρ 2 2 + ( f - h k ) 2 ) sin   2 θ ,
E x , k ( ρ ,   θ ,   f ;   0 )
f i λ ρ 1 , k - 1 + ρ 1 , k - d ρ 1 ρ 1 ρ 1 2 + f 2 × exp { - ik 0 [ ϕ ( ρ 1 ;   0 ) + ρ 1 2 + f 2 ] } × [ e y ( t ) ( ρ 1 ,   0 ;   0 ) - e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ] × π J 2 ( k 0 ρ 1 ρ / ρ 1 2 + f 2 ) sin   2 θ ,
ρ 2 = ρ 1 - h k tan   α ( i ) ( ρ 1 ) ,
ρ 2 , k - 1 + = ρ 1 , k - 1 + - h k tan   α ( i ) ( ρ 1 , k - 1 + ) ,
ρ 2 , k - = ρ 1 , k - - h k tan   α ( i ) ( ρ 1 , k - ) .
ρ 1 , k - - ρ 1 , k - 1 + λ ,
ρ 1 , k - - ρ 1 , k - 1 + ρ 1 , M - .
h k λ or h k < λ .
ρ 1 , k - ρ 1 , k + , k = 0 , 1 , ,   M ,
ϕ ( ρ 2 ;   h k ) = ϕ ( ρ 1 ;   0 ) + nh k / cos   α ( i ) ( ρ 1 ) .
e y ( t ) ( ρ 2 ,   0 ;   h k ) e y ( t ) ( ρ 1 ,   0 ;   0 ) ,
e y ( t ) ( ρ 2 ,   π / 2 ;   h k ) e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) .
E x , k ( ρ ,   θ ,   f ;   h k )
f i λ ρ 1 , k - 1 ρ 1 , k d ρ 1 ρ 1 ρ 1 2 + f 2 exp { - ik 0 [ ϕ ( ρ 1 ;   0 ) + nh k / cos   α ( i ) ( ρ 1 ) + ρ 2 2 + ( f - h k ) 2 ] } × [ e y ( t ) ( ρ 1 ,   0 ,   ;   0 ) - e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ]
× π J 2 ( k 0 ρ 1 ρ / ρ 1 2 + f 2 ) sin   2 θ .
exp { - ik 0 [ ϕ ( ρ 1 ;   0 ) + nh k / cos   α ( i ) ( ρ 1 )
+ ρ 2 2 + ( f - h k ) 2 ] }
exp { - ik 0 [ ϕ ( ρ 1 ;   0 ) + ρ 1 2 + f 2 ] } exp - ik 0 h k n cos   α ( i ) ( ρ ¯ 1 , k ) - ρ ¯ 1 , k tan   α ( i ) ( ρ ¯ 1 , k ) + f a ρ ¯ 1 , k 2 + f a 2 ,
E x , k ( ρ ,   θ ,   f ;   h k ) E x , k ( ρ ,   θ ,   f ;   0 ) exp - ik 0 h k n cos   α ( i ) ( ρ ¯ 1 , k ) - ρ ¯ 1 , k tan   α ( i ) ( ρ ¯ 1 , k ) + f a ρ ¯ 1 , k 2 + f a 2 .
E y , k ( ρ ,   θ ,   f ;   h k ) E y , k ( ρ ,   θ ,   f ;   0 ) E z , k ( ρ ,   θ ,   f ;   h k ) E z , k ( ρ ,   θ ,   f ;   0 ) E x , k ( ρ ,   θ ,   f ;   h k ) E x , k ( ρ ,   θ ,   f ;   0 ) .
ψ k = - k 0 h k n cos   α ( i ) ( ρ ¯ 1 , k ) - ρ ¯ 1 , k tan   α ( i ) ( ρ ¯ 1 , k ) + f a ρ ¯ 1 , k 2 + f a 2 ,
ρ ¯ 1 , k [ ρ 1 , k - 1 ,   ρ 1 , k ] .
ψ ( ρ 1 ) ψ k , h ( ρ 1 ) h k ,
if ρ 1 [ ρ 1 , k - 1 ,   ρ 1 , k ] , k = 1 , , M .
ψ ( ρ 1 ) = - k 0 h ( ρ 1 ) n cos   α ( i ) ( ρ 1 ) - ρ 1 tan   α ( i ) ( ρ 1 ) + f a ρ 1 2 + f a 2 .
ψ L ( ρ 1 ) = - k 0 h L ( ρ 1 ) ( n - 1 ) ,
h ( ρ 1 ) / h L ( ρ 1 ) = ( n - 1 ) /
n cos   α ( i ) ( ρ 1 ) - ρ 1 tan   α ( i ) ( ρ 1 ) + f a ρ 1 2 + f a 2 .
I ( ρ ,   θ ,   f a ) = k = 1 M T k E x , k ( ρ ,   θ ,   f a ;   h k ) 2 + k = 1 M T k E y , k ( ρ ,   θ ,   f a ;   h k ) 2 + k = 1 M T k E x , k ( ρ ,   θ ,   f a ;   h k ) 2 .
max { T k , h k } I ( 0 ,   0 ,   f a )
I ( ρ j ( G ) ,   θ ,   f a ) = 0 , θ [ 0 ,   2 π ) , j = 1 , ,   N ,
| e y ( t ) ( ρ 1 ,   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ) |     | e y ( t ) ( ρ 1 ,   0 ) - e y ( t ) ( ρ 1 ,   π / 2 ) | ,
| e y ( t ) ( ρ 1 ,   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ) |     | 2 e z ( t ) ( ρ 1 ,   π / 2 ) | ,
k = 1 M T k exp ( i ψ k ) g k ( ρ j ( G ) ) = 0 ,
g k ( ρ j ( G ) ) = ρ 1 , k - 1 ρ 1 , k d ρ 1 ρ 1 ρ 1 2 + f a 2   [ e y ( t ) ( ρ 1 ,   0 ;   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ] J 0 ( k 0 ρ 1 ρ j ( G ) / ρ 1 2 + f a 2 ) .
A k = Re [ T k exp ( i ψ k ) ] ,
B k = Im [ T k exp ( i ψ k ) ] ;
max { A k , B k } k = 1 M A k g k ( 0 ) 2 + k = 1 M B k g k ( 0 ) 2
k = 1 M A k g k ( ρ j ( G ) ) = 0 , j = 1 , ,   N ,
k = 1 M B k g k ( ρ j ( G ) ) = 0 , j = 1 , ,   N ,
A k 2 + B k 2 1 , k = 1 , ,   M .
min { A k , B k } k = 1 M A k g k ( 0 )
k = 1 M A k g k ( ρ j ( G ) ) = 0 , j = 1 , ,   N ,
k = 1 M B k g k ( ρ j ( G ) ) = 0 , j = 1 , ,   N ,
k = 1 M A k g k ( 0 ) = k = 1 M B k g k ( 0 ) ,
A k 2 + B k 2 1 , k = 1 , ,   M .
A k = A k * , B k = B k *
k = 1 M A k * g k ( ρ j ( G ) ) = 0 , j = 1 , ,   N ,
k = 1 M B k * g k ( ρ j ( G ) ) = 0 , j = 1 , ,   N ,
k = 1 M A k * g k ( 0 ) = k = 1 M B k * g k ( 0 ) ,
A k * 2 + B k * 2 1 , k = 1 , ,   M ,
F min = k = 1 M A k * g k ( 0 ) .
A k = B k = ( A k * + B k * ) / 2
k = 1 M g k ( 0 ) ( A k * + B k * ) / 2 = F min ,
A k * + B k * 2 2 + A k * + B k * 2 2
= ( A k * × 1 + B k * × 1 ) 2 2 ( A k * 2 + B k * 2 ) ( 1 2 + 1 2 ) 2 = A k * 2 + B k * 2 1 ,
min { A k } k = 1 M A k g k ( 0 )
k = 1 M A k g k ( ρ j ( G ) ) = 0 , j = 1 , ,   N ,
- 1 / 2 A k 1 2 , k = 1 , ,   M ,
B k = A k , k = 1 , ,   M .
A ( ρ 1 ) A k if ρ 1 [ ρ 1 , k - 1 ,   ρ 1 , k ] , k = 1 , ,   M .
min A ( ρ 1 ) 0 ρ 1 , M d ρ 1 A ( ρ 1 )   ρ 1 ρ 1 2 + f a 2   [ e y ( t ) ( ρ 1 ,   0 ;   0 )
+ e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ]
0 ρ 1 , M d ρ 1 A ( ρ 1 )   ρ 1 ρ 1 2 + f a 2   [ e y ( t ) ( ρ 1 ,   0 ;   0 )
+ e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ] J 0 ( k 0 ρ 1 ρ j ( G ) / ρ 1 2 + f a 2 ) = 0 ,
- 1 / 2 A ( ρ 1 ) 1 / 2 .
A ( ρ 1 ) = [ cos   θ ( ρ 1 ) ] / 2
min θ ( ρ 1 ) 0 ρ 1 , M d ρ 1 [ cos   θ ( ρ 1 ) ]   ρ 1 ρ 1 2 + f a 2
× [ e y ( t ) ( ρ 1 ,   0 ;   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ]
0 ρ 1 , M d ρ 1 [ cos   θ ( ρ 1 ) ]   ρ 1 ρ 1 2 + f a 2   [ e y ( t ) ( ρ 1 ,   0 ;   0 )
+ e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ] J 0 ( k 0 ρ 1 ρ j ( G ) / ρ 1 2 + f a 2 ) = 0 .
F [ θ ( ρ 1 ) ,   λ j ] = 0 ρ 1 , M d ρ 1 [ cos   θ ( ρ 1 ) ]   ρ 1 ρ 1 2 + f a 2   [ e y ( t ) ( ρ 1 ,   0 ;   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ] + j = 1 N λ j 0 ρ 1 , M d ρ 1 [ cos   θ ( ρ 1 ) ]   ρ 1 ρ 1 2 + f a 2 × [ e y ( t ) ( ρ 1 ,   0 ;   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ] × J 0 ( k 0 ρ 1 ρ j ( G ) / ρ 1 2 + f a 2 ) = 0 ρ 1 , M d ρ 1 [ cos   θ ( ρ 1 ) ]   ρ 1 ρ 1 2 + f a 2 × [ e y ( t ) ( ρ 1 ,   0 ;   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ]   × j = 0 N λ j J 0 ( k 0 ρ 1 ρ j ( G ) / ρ 1 2 + f a 2 ) ,
δ θ ( ρ 1 ) F [ θ ( ρ 1 ) ,   λ j ] = - 0 ρ 1 , M d ρ 1 δ θ ( ρ 1 ) × [ sin   θ ( ρ 1 ) ]   ρ 1 ρ 1 2 + f a 2   [ e y ( t ) ( ρ 1 ,   0 ;   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ] × j = 0 N λ j J 0 ( k 0 ρ 1 ρ j ( G ) / ρ 1 2 + f a 2 ) = 0 ,
[ sin   θ ( ρ 1 ) ]   ρ 1 ρ 1 2 + f a 2   [ e y ( t ) ( ρ 1 ,   0 ;   0 ) + e y ( t ) ( ρ 1 ,   π / 2 ;   0 ) ]
× j = 0 N λ j J 0 ( k 0 ρ 1 ρ j ( G ) / ρ 1 2 + f a 2 ) = 0 , ρ 1 [ 0 ,   ρ 1 , M ] .
j = 0 N λ j J 0 ( k 0 ρ 1 ρ j ( G ) / ρ 1 2 + f a 2 ) 0 , ρ 1 [ a ,   b ] ,
[ a ,   b ] [ 0 ,   ρ 1 , M ] .
sin   θ ( ρ 1 ) = 0 , ρ 1 [ 0 ,   ρ 1 , M ] .
A ( ρ 1 ) { - 1 / 2 ,   1 / 2 } , ρ 1 [ 0 ,   ρ 1 , M ] .
A k { - 1 / 2 ,   1 / 2 } , k = 1 , ,   M .
λ ρ 1 , M / M ρ 1 , M   1 M ρ 1 , M / λ .
ρ 1 , M / M P λ     M ρ 1 , M / ( P λ ) ,
S LP S NE > S LE > S VF ,

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