Abstract

A new numerical approach to efficiently reconstruct the profile of a grating from measurements of reflection coefficients is demonstrated. The problem is posed in the mathematical framework of an inverse scattering problem and solved using gradient-based algorithms. The gradient is computed efficiently using adjoint equations, which amounts to an extra scattering computation per iteration. For symmetric profiles it is shown that only knowledge of the scattered field is sufficient to compute the gradient. As a result, complex profiles can be reconstructed rapidly, and the method can be potentially used in metrology applications in semiconductors. The technique is demonstrated for the case of TE polarization.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2007 (1)

2004 (2)

H.-T. Huang and F. L. Terry, Jr., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455, 828-836 (2004).
[CrossRef]

G. R. Feijóo, A. A. Oberai, and P. M. Pinsky, “An application of shape optimization in the solution of inverse acoustic scattering problems,” Inverse Probl. 20, 199-228 (2004).
[CrossRef]

2003 (1)

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel-Dekker, 2003).

2001 (1)

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97-111 (2001).
[CrossRef]

2000 (3)

T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (Dover, 2000).

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, 2000).

O. Dorn, E. L. Miller, and C. M. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119-1156 (2000).
[CrossRef]

1998 (1)

1996 (3)

1995 (3)

1994 (1)

G. E. Jellison and F. A. Modine, “Optical functions of silicon at elevated temperatures,” J. Appl. Phys. 76, 3758-3761 (1994).
[CrossRef]

1989 (1)

J. B. Keller and D. Givoli, “Exact non-reflecting boundary conditions,” J. Comput. Phys. 82, 172-192 (1989).
[CrossRef]

1988 (1)

A. Jameson, “Aerodynamic design via control theory,” J. Sci. Comput. 3, 233-260 (1988).
[CrossRef]

1987 (1)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

1986 (1)

F.-X. Le Dimet and O. Talagrand, “Variational algorithms for analysis and as similation of meteorological observations: Theoretical aspects,” Tellus 38A, 97-110 (1986).
[CrossRef]

1981 (1)

P. Gill, W. Murray, and M. Wright, Practical Optimization (Academic, 1981).

1979 (1)

J.-P. Zolesio, “Identification de domains par déformations,” Thèse d'etat (Université de Nice, 1979).

1976 (1)

F. Murat and S. Simon, Etudes de problèmes d'optimal design (Springer-Verlag, Berlin, 1976), J.Cea, ed., Vol. 40 of Lecture Notes in Computer Science, pp. 54-62.

Al-Assaad, R. M.

Arridge, S.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bao, J.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97-111 (2001).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Byrne, D. M.

Chew, W. C.

W. C. Chew and J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microw. Guid. Wave Lett. 5, 439-441 (1995).
[CrossRef]

Dorn, O.

O. Dorn, E. L. Miller, and C. M. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119-1156 (2000).
[CrossRef]

Feijóo, G. R.

G. R. Feijóo, A. A. Oberai, and P. M. Pinsky, “An application of shape optimization in the solution of inverse acoustic scattering problems,” Inverse Probl. 20, 199-228 (2004).
[CrossRef]

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, 2000).

Gaylord, T. K.

Gill, P.

P. Gill, W. Murray, and M. Wright, Practical Optimization (Academic, 1981).

Givoli, D.

J. B. Keller and D. Givoli, “Exact non-reflecting boundary conditions,” J. Comput. Phys. 82, 172-192 (1989).
[CrossRef]

Granet, G.

Grann, E. B.

Guizal, B.

Huang, H.-T.

H.-T. Huang and F. L. Terry, Jr., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455, 828-836 (2004).
[CrossRef]

Hughes, T. J. R.

T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (Dover, 2000).

Jakatdar, N.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97-111 (2001).
[CrossRef]

Jameson, A.

A. Jameson, “Aerodynamic design via control theory,” J. Sci. Comput. 3, 233-260 (1988).
[CrossRef]

Jellison, G. E.

G. E. Jellison and F. A. Modine, “Optical functions of silicon at elevated temperatures,” J. Appl. Phys. 76, 3758-3761 (1994).
[CrossRef]

Keller, J. B.

J. B. Keller and D. Givoli, “Exact non-reflecting boundary conditions,” J. Comput. Phys. 82, 172-192 (1989).
[CrossRef]

Lalanne, P.

Le Dimet, F.-X.

F.-X. Le Dimet and O. Talagrand, “Variational algorithms for analysis and as similation of meteorological observations: Theoretical aspects,” Tellus 38A, 97-110 (1986).
[CrossRef]

Li, L.

Lin, J. H.

W. C. Chew and J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microw. Guid. Wave Lett. 5, 439-441 (1995).
[CrossRef]

Miller, E. L.

O. Dorn, E. L. Miller, and C. M. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119-1156 (2000).
[CrossRef]

Modine, F. A.

G. E. Jellison and F. A. Modine, “Optical functions of silicon at elevated temperatures,” J. Appl. Phys. 76, 3758-3761 (1994).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Murat, F.

F. Murat and S. Simon, Etudes de problèmes d'optimal design (Springer-Verlag, Berlin, 1976), J.Cea, ed., Vol. 40 of Lecture Notes in Computer Science, pp. 54-62.

Murray, W.

P. Gill, W. Murray, and M. Wright, Practical Optimization (Academic, 1981).

Nevière, M.

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel-Dekker, 2003).

Niu, X.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97-111 (2001).
[CrossRef]

Oberai, A. A.

G. R. Feijóo, A. A. Oberai, and P. M. Pinsky, “An application of shape optimization in the solution of inverse acoustic scattering problems,” Inverse Probl. 20, 199-228 (2004).
[CrossRef]

Pinsky, P. M.

G. R. Feijóo, A. A. Oberai, and P. M. Pinsky, “An application of shape optimization in the solution of inverse acoustic scattering problems,” Inverse Probl. 20, 199-228 (2004).
[CrossRef]

Pommet, D. A.

Popov, E.

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel-Dekker, 2003).

Rappaport, C. M.

O. Dorn, E. L. Miller, and C. M. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119-1156 (2000).
[CrossRef]

Schweiger, M.

Simon, S.

F. Murat and S. Simon, Etudes de problèmes d'optimal design (Springer-Verlag, Berlin, 1976), J.Cea, ed., Vol. 40 of Lecture Notes in Computer Science, pp. 54-62.

Spanos, C. J.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97-111 (2001).
[CrossRef]

Talagrand, O.

F.-X. Le Dimet and O. Talagrand, “Variational algorithms for analysis and as similation of meteorological observations: Theoretical aspects,” Tellus 38A, 97-110 (1986).
[CrossRef]

Terry, F. L.

H.-T. Huang and F. L. Terry, Jr., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455, 828-836 (2004).
[CrossRef]

Wright, M.

P. Gill, W. Murray, and M. Wright, Practical Optimization (Academic, 1981).

Zolesio, J.-P.

J.-P. Zolesio, “Identification de domains par déformations,” Thèse d'etat (Université de Nice, 1979).

IEEE Microw. Guid. Wave Lett. (1)

W. C. Chew and J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microw. Guid. Wave Lett. 5, 439-441 (1995).
[CrossRef]

IEEE Trans. Semicond. Manuf. (1)

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97-111 (2001).
[CrossRef]

Inverse Probl. (2)

O. Dorn, E. L. Miller, and C. M. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119-1156 (2000).
[CrossRef]

G. R. Feijóo, A. A. Oberai, and P. M. Pinsky, “An application of shape optimization in the solution of inverse acoustic scattering problems,” Inverse Probl. 20, 199-228 (2004).
[CrossRef]

J. Appl. Phys. (1)

G. E. Jellison and F. A. Modine, “Optical functions of silicon at elevated temperatures,” J. Appl. Phys. 76, 3758-3761 (1994).
[CrossRef]

J. Comput. Phys. (1)

J. B. Keller and D. Givoli, “Exact non-reflecting boundary conditions,” J. Comput. Phys. 82, 172-192 (1989).
[CrossRef]

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

J. Sci. Comput. (1)

A. Jameson, “Aerodynamic design via control theory,” J. Sci. Comput. 3, 233-260 (1988).
[CrossRef]

Opt. Express (1)

Tellus (1)

F.-X. Le Dimet and O. Talagrand, “Variational algorithms for analysis and as similation of meteorological observations: Theoretical aspects,” Tellus 38A, 97-110 (1986).
[CrossRef]

Thin Solid Films (1)

H.-T. Huang and F. L. Terry, Jr., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455, 828-836 (2004).
[CrossRef]

Other (7)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel-Dekker, 2003).

T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (Dover, 2000).

P. Gill, W. Murray, and M. Wright, Practical Optimization (Academic, 1981).

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, 2000).

F. Murat and S. Simon, Etudes de problèmes d'optimal design (Springer-Verlag, Berlin, 1976), J.Cea, ed., Vol. 40 of Lecture Notes in Computer Science, pp. 54-62.

J.-P. Zolesio, “Identification de domains par déformations,” Thèse d'etat (Université de Nice, 1979).

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Figures (13)

Fig. 1
Fig. 1

Schematic representation of the scattering problem for a grating pattern.

Fig. 2
Fig. 2

Domain of definition of the variational problem.

Fig. 3
Fig. 3

Discretization of a grating in the RCWA method. For each layer, the position of its center x l , the width Δ l , and the thickness t l uniquely define the profile.

Fig. 4
Fig. 4

Geometry in the first example: Thin film on a substrate.

Fig. 5
Fig. 5

Geometry of the grating used to produce the results in Table 1. Dimensions are in meters. The period is 560 nm . The structure is discretized into 16 layers in the RCWA computation.

Fig. 6
Fig. 6

Profiles of target structures used in the first (a) and second (b) reconstruction experiments. Dimensions are in meters.

Fig. 7
Fig. 7

Index of refraction of the materials used in the reconstruction problems. (a) Index of refraction of grating lines (imaginary part is zero); (b) Index of refraction of silicon substrate.

Fig. 8
Fig. 8

Evolution of the profile and adjacent field (shown at an incident wavelength of 300 nm ) in the first reconstruction experiment. Panels (a)–(f) show the initial guess for the profile in the reconstruction subproblem corresponding to the indicated wavelength range.

Fig. 9
Fig. 9

Profiles and specular reflection coefficients corresponding to the initial guess (gray curves; green online), converged solution (thick black curves; red online), and the target (thin black curve; blue online) in the first reconstruction experiment. The thick black (red online), and thin black (blue online) curves for the reflection coefficients almost coincide.

Fig. 10
Fig. 10

Evolution of the profile and adjacent field (shown at an incident wavelength of 300 nm ) in the second reconstruction experiment. Panles (a)–(f) show the initial guess for the profile in the reconstruction subproblem corresponding to the indicated wavelength range.

Fig. 11
Fig. 11

Profiles and specular reflection coefficients corresponding to the initial guess (gray curves; green online), converged solution (thick black curves; red online), and the target (thin black curve; blue online) in the second reconstruction experiment. The thick black (red online) and thin black (blue online) almost coincide.

Fig. 12
Fig. 12

Evolution of the profile during the reconstruction of the target seen in Fig. 6a using noisy data. Captions indicate the wavelength range used in each stage of the reconstruction. The profile of the target structure is shown by thin black curves (blue online). The initial profile for each stage is shown by gray curves (green online), and the final profile is shown by thick black curves (red online).

Fig. 13
Fig. 13

Evolution of the profile during the reconstruction of the target seen in Fig. 6b using noisy data and the Gauss–Newton method. Captions indicate the wavelength range used in each stage of the reconstruction. The profile of the target structure is shown by thin black curves (blue online). The initial profile for each stage is shown by gray curves (green online), and the final profile is shown by thick black curves (red online).

Tables (1)

Tables Icon

Table 1 Derivatives of the Cost Functional for the First Eight Layers of the Structure Shown in Fig. 5

Equations (82)

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n I sin ( θ m ) = n I sin ( θ ) + m λ d , m = 0 , ± 1 , ± 2 , ,
( 2 x 2 + 2 y 2 + k 2 ( x , y ) ) E z = 0 in R 2 Σ ,
E z = 0 on Σ ,
E z n = 0 on Σ .
E z ( x , y ) = E inc ( x , y ) + m R m exp i ( α m x + β I , m y ) for y > t ,
E inc ( x , y ) = exp i ( α 0 x β I , 0 y ) ,
E z ( x , y ) = m T m exp i ( α m x + β II , m y ) for y < 0 ,
α m 2 + ( β I , m ) 2 = k I 2 ,
α m 2 + ( β II , m ) 2 = k II 2 ,
α m = α 0 + m 2 π d ,
α 0 = k I sin ( θ ) .
a ( Σ ; w , E z ) = l ( w ) w V ,
a ( Σ ; w , E z ) Ω ( w * E z k 2 w * E z ) d Ω Γ I w * M I ( E z ) d Γ Γ II w * M II ( E z ) d Γ ,
l ( w ) Γ I w * [ E inc n M I ( E inc ) ] d Γ .
M I ( E z ) = m i β I , m d ( 0 d exp ( i α m x ) E z ( x , y I ) d x ) exp ( i α m x ) ,
M II ( E z ) = m i β II , m d ( 0 d exp ( i α m x ) E z ( x , y II ) d x ) exp ( i α m x ) .
ϵ ( x , y ) = m ϵ m exp ( i m 2 π x d ) , for 0 < y < t .
E z ( x , y ) = m E z , m ( y ) exp ( i α m x ) , for 0 < y < t .
d 2 E z d y 2 = k 0 2 A E z ,
A = W Q 2 W 1 ,
E z , m ( y ) = p ( w p ) m ( c p + exp ( + q p k 0 ( y t ) ) + c p exp ( q p k 0 y ) ) ,
j ( Σ ) J ( E z ( Σ ) ) 1 2 R 0 ( E z ( Σ ) ) R 0 m 2 ,
φ δ ( x ) = x δ = x + δ V , x Σ , δ R ,
D j ( Σ ) [ V ] = d d δ j ( Σ δ ) δ = 0 .
L ( Σ δ , λ , E z δ ) = j ( Σ δ ) + Re [ a ( Σ δ ; λ , E z δ ) l ( λ ) ] ,
a ( Σ δ ; w , E z δ ) = l ( w ) w V ,
D j ( Σ ) [ V ] = d d δ L ( Σ δ , λ , E z δ ) δ = 0 λ .
d d δ L ( Σ δ , λ , E z δ ) δ = 0 = Re [ D Σ a ( Σ ; λ , E z ) [ V ] ] + Re [ a ( Σ ; λ , E ̇ z ) ] + D E z J ( E z ) [ E ̇ z ] .
D Σ a ( Σ ; w , u ) [ V ] = d d δ a ( Σ δ ; w , u ) δ = 0 ,
E ̇ z = d d δ E z δ δ = 0 ,
Re [ a ( Σ ; λ , w ) ] = D E z J ( E z ) [ w ] w V .
D j ( Σ ) [ V ] = d d δ L ( Σ δ , λ , E z δ ) δ = 0 = Re [ D Σ a ( Σ ; λ , E z ) [ V ] ] ,
D Σ a ( Σ ; λ , E z ) [ V ] = d d δ [ Ω δ ( x δ λ * x δ E z k 2 λ * E z ) d Ω δ ] δ = 0 .
D Σ a ( Σ ; λ , E z ) [ V ] = Ω ( λ * E z k 2 λ * E z ) div V d Ω Ω λ * ( V + V T ) E z d Ω .
D j ( Σ ) [ V ] = Re [ Σ k 0 2 ( n I 2 n l 2 ) ( λ * E z ) V n d Γ ] ,
J ̂ ( E z ) = 1 2 Γ m E z ( x , y m ) E z m ( x ) 2 d Γ .
J ̂ ( E z ) = d 2 p = + R p R p m .
D E z J ̂ ( E z ) [ w ] = Re [ Γ m w ( E z E z m ) * d Γ ] .
Ω ( λ * w k 2 λ * w ) d Ω Γ I λ * M I ( w ) d Γ Γ II λ * M II ( w ) d Γ = Γ m w ( E z E z m ) * d Γ w V .
Γ I λ * M I ( w * ) d Γ = Γ I w * M I adj ( λ * ) d Γ ,
M I adj ( λ * ) = m i β ̂ I , m d ( 0 d exp ( i α ̂ m x ) λ * ( x , y I ) d x ) exp ( i α ̂ m x ) ,
α ̂ m 2 + ( β ̂ I , m ) 2 = k I 2 ,
α ̂ m = α ̂ 0 + m 2 π d ,
α ̂ 0 = k I sin ( θ ) .
Ω ( w * λ * k 2 w * λ * ) d Ω Γ I w * M I adj ( λ * ) d Γ Γ II w * M II adj ( λ * ) d Γ = Γ m w * ( E z E z m ) * d Γ w V ,
( 2 x 2 + 2 y 2 + k 2 ( x , y ) ) λ * = ( E z E z m ) * δ Γ m ( x , y ) in R 2 Σ ,
λ * = 0 on Σ ,
λ * n = 0 on Σ .
1 2 i k I cos θ ( R 0 R 0 m ) *
λ * ( x , y ) = 1 2 i k I cos θ ( R 0 R 0 m ) * exp i ( α ̂ 0 x β ̂ I , 0 y ) + m A m exp i ( α ̂ m x + β ̂ I , m y ) for y > t ,
v n ( s ) = V n = i = 1 n ψ i ( s ) v i , s Σ
D j ( Σ ) [ V ] = i = 1 n G i v i ,
G i = Re [ Σ k 0 2 ( n I 2 n l 2 ) λ * ( s ) E z ( s ) ψ i ( s ) d s ] .
j Δ l = 1 2 Re [ y l 1 y l k 0 ( n I 2 n l 2 ) λ * ( x , y ) E z ( x , y ) d y + y l 1 y l k 0 ( n I 2 n l 2 ) λ * ( x + , y ) E z ( x + , y ) d y ] ,
x ± = x l c ± Δ l 2 .
j ( Σ ) = p = 1 N λ R 0 ( Σ , λ p ) R 0 m ( λ p ) 2 ,
R ( t ) = R 01 + R 12 γ 2 1 + R 01 R 12 γ 2 ,
γ 2 = exp ( 2 i β 1 t ) ,
α m = k m sin θ m ,
β m = k m cos θ m , m = 0 , 1 , 2 .
α 0 = α 1 = α 2 .
R 01 = β 0 β 1 β 0 + β 1 , R 12 = β 1 β 2 β 1 + β 2 .
j ( t ) = 1 2 R ( t ) 2 ,
d j d t = Re [ R * d R d t ] .
d R d t = R 12 ( 1 R 01 2 ) ( 1 + R 01 R 12 γ 2 ) 2 2 i β 1 γ 2 .
d j d t = lim L + Re [ L + L ( k 2 2 k 1 2 ) λ * ( x , t ) E z ( x , t ) d x ] .
E z ( x , t ) = T exp ( i α 2 x ) ,
T = γ ( 1 + R 12 ) 1 + R 12 γ 2 ( 1 + R ) .
λ * ( x , t ) = 1 2 L R * 2 i β 0 T exp ( i α 2 x ) .
d j / d t = Re [ R * k 2 2 k 1 2 2 i β 0 T 2 ] .
k 2 2 k 1 2 2 i β 0 T 2 = β 2 2 β 1 2 2 i β 0 T 2 .
k 2 2 k 1 2 2 i β 0 T 2 = R 12 ( 1 R 01 2 ) ( 1 + R 01 R 12 γ 2 ) 2 2 i β 1 γ 2 = d R d t ,
r p ( Σ , λ p ) = R 0 ( Σ , λ p ) R 0 m ( λ p ) , p = 1 , , N λ
F δ ( x ) = φ δ ( x ) = I + δ V ( x ) ,
x δ u = F δ T u ,
d Ω δ = det F δ d Ω .
d d δ ( F δ ) δ = 0 = V ,
d d δ ( x δ u ) δ = 0 = V T u ,
d d δ ( d Ω δ ) δ = 0 = div V d Ω .
Ω ( λ * E z ) div V d Ω = Σ λ * E z v n d Γ Ω i = { Ω I , Ω II , Ω l } [ Ω i λ , i j * E z , i V j d Ω + Ω i λ , i * E z , i j V j d Ω ] .
D Σ a ( Σ ; λ , E z ) [ V ] = Σ λ * E z k 2 λ * E z 2 ( λ * n ) ( E z n ) v n d Γ + Ω i = { Ω I , Ω II , Ω l } Ω i ( λ * V ) ( 2 E z + k 2 E z ) d Ω + Ω i = { Ω I , Ω II , Ω l } Ω i ( E z V ) ( 2 λ * + k 2 λ * ) d Ω .
D Σ a ( Σ ; λ , E z ) [ V ] = Σ k 2 ( λ * E z ) v n d Γ = Σ k 0 2 ( n I 2 n l 2 ) ( λ * E z ) v n d Γ ,

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