Abstract

Three analytical models have been established for superresolution parameters GAe, GTe, and Se related to transmission function A(ρ), phase function of ϕ(ρ), and the structural parameters with fabrication errors of an N-zone circular-symmetrical superresolution pupil filter. These new models established, directly relate the superresolution parameters of an N-zone super-resolution pupil filter to its fabrication errors to make the quantitative analyses of the effect of fabrication errors easier, thereby providing a theoretical basis for the analysis, design, and fabrication of an N-zone super-resolution pupil filter. The models established for GAe, GTe, and Se have been used to analyze the effect of the fabrication errors of a three-zone phase-only pupil filter on its superresolution property, to verify their validities.

© 2006 Optical Society of America

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References

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2005

M. Y. Yun, L. R. Liu, J. F. Sun, and D. A. Liu, "Three-dimension superresolution by three-zone complex pupil filters," Opt. Soc. Am. A 22, 272-277 (2005).
[CrossRef]

2004

2003

2002

H. T. Liu, Y. B. Yan, Q. F. Tan, and G. F. Jin, "Theories for the design of diffractive superresolution elements and limits of optical superresolution," Opt. Soc. Am. A 19, 2185-2193 (2002).
[CrossRef]

1999

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999).
[CrossRef]

1997

1988

Andres, P.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Campos, J.

Canales, V. F.

Caqiqal, M. P.

Choudhury, A.

de Juana, D. M.

Escalera, J. C.

Hegedus, Z. S.

Jin, G. F.

H. T. Liu, Y. B. Yan, D. Yi, and G. F. Jin. "Design of three-dimensional superresolution filters and limits of axial optical supperresolution," Appl. Opt. 42, 1463-1476 (2003).
[CrossRef] [PubMed]

H. T. Liu, Y. B. Yan, Q. F. Tan, and G. F. Jin, "Theories for the design of diffractive superresolution elements and limits of optical superresolution," Opt. Soc. Am. A 19, 2185-2193 (2002).
[CrossRef]

Kowalczyk, M.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Ledesma, S.

Liu, D. A.

M. Y. Yun, L. R. Liu, J. F. Sun, and D. A. Liu, "Three-dimension superresolution by three-zone complex pupil filters," Opt. Soc. Am. A 22, 272-277 (2005).
[CrossRef]

Liu, H. T.

H. T. Liu, Y. B. Yan, D. Yi, and G. F. Jin. "Design of three-dimensional superresolution filters and limits of axial optical supperresolution," Appl. Opt. 42, 1463-1476 (2003).
[CrossRef] [PubMed]

H. T. Liu, Y. B. Yan, Q. F. Tan, and G. F. Jin, "Theories for the design of diffractive superresolution elements and limits of optical superresolution," Opt. Soc. Am. A 19, 2185-2193 (2002).
[CrossRef]

Liu, L. R.

M. Y. Yun, L. R. Liu, J. F. Sun, and D. A. Liu, "Three-dimension superresolution by three-zone complex pupil filters," Opt. Soc. Am. A 22, 272-277 (2005).
[CrossRef]

Martinez-Corral, M.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Morris, G. M.

Oti, J. E.

Qiu, L. R.

Sales, T. R. M.

Sheppard, C. J. R.

Sun, J. F.

M. Y. Yun, L. R. Liu, J. F. Sun, and D. A. Liu, "Three-dimension superresolution by three-zone complex pupil filters," Opt. Soc. Am. A 22, 272-277 (2005).
[CrossRef]

Tan, J. B.

Tan, Q. F.

H. T. Liu, Y. B. Yan, Q. F. Tan, and G. F. Jin, "Theories for the design of diffractive superresolution elements and limits of optical superresolution," Opt. Soc. Am. A 19, 2185-2193 (2002).
[CrossRef]

Yan, Y. B.

H. T. Liu, Y. B. Yan, D. Yi, and G. F. Jin. "Design of three-dimensional superresolution filters and limits of axial optical supperresolution," Appl. Opt. 42, 1463-1476 (2003).
[CrossRef] [PubMed]

H. T. Liu, Y. B. Yan, Q. F. Tan, and G. F. Jin, "Theories for the design of diffractive superresolution elements and limits of optical superresolution," Opt. Soc. Am. A 19, 2185-2193 (2002).
[CrossRef]

Yi, D.

Yun, M. Y.

M. Y. Yun, L. R. Liu, J. F. Sun, and D. A. Liu, "Three-dimension superresolution by three-zone complex pupil filters," Opt. Soc. Am. A 22, 272-277 (2005).
[CrossRef]

Yzuel, M. J.

Zapata-Rodriguez, C. J.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Zhao, W. Q.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Soc. Am. A

H. T. Liu, Y. B. Yan, Q. F. Tan, and G. F. Jin, "Theories for the design of diffractive superresolution elements and limits of optical superresolution," Opt. Soc. Am. A 19, 2185-2193 (2002).
[CrossRef]

M. Y. Yun, L. R. Liu, J. F. Sun, and D. A. Liu, "Three-dimension superresolution by three-zone complex pupil filters," Opt. Soc. Am. A 22, 272-277 (2005).
[CrossRef]

Other

M. L. Melocchi, Phase apodization for resolution enhancement (Ph.D. Dissertation, University of Rochester, 2003).

T. R. M. Sales, Phase-only Superresolution Elements (University of Rochester. Ph.D. Dissertation, 1997).

T. Wilson, Confocal Microscopy (Academic Press. London, 1990).

<jrn>L. R. Qiu, W. Q. Zhao, Z. D. Feng, and X. M. Ding, "An approach to higher spatial resolution in a laser probe measurement system using a phase-only pupil filter," Opt. Eng. 45 (to be published).</jrn>

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Figures (6)

Fig. 1.
Fig. 1.

Circular-symmetrical pupil filter with radius R.

Fig. 2.
Fig. 2.

Schematic eccentricity of an N-zone pupil filter.

Fig. 3.
Fig. 3.

Variation of (a) ΔGA , (b) ΔGT , and (c) ΔS with concentricity error.

Fig. 4.
Fig. 4.

Variations of (a) ΔGA , (b) ΔGA and (c) ΔS with angle between concentricity errors.

Fig. 5.
Fig. 5.

Variations of (a) ΔGA , (b) ΔGT and (c) ΔS with radial error caused by etching line width.

Fig. 6.
Fig. 6.

Variations of (a) ΔGA , (b) ΔGT and (c) ΔS with phase error caused by etching depth.

Equations (39)

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{ G A = u 1 u 0 G T = v 1 v 0 S = h ( 0 , u F ) 2
φ k = 2 πnh k λ 2 π ( h k h 1 ) λ
φ k = 2 π ( n 1 ) h k λ
P k ( ρ ) = t k e k ( r k 1 < ρ < r k , k = 1,2,3 , , N )
h v u = 2 0 π 0 ρ 1 P 1 ( ρ ) e iu ρ 2 2 J 0 ( ρv ) ρdρdθ + + 2 0 π ρ k 1 ρ k P k ( ρ ) e i 2 π ( n 1 ) λ e iuρ 2 2 J 0 ( ρv ) ρdρdθ + + 2 0 π ρ N 1 1 P k ( ρ ) e i 2 π ( n 1 ) Δ h N λ e iuρ 2 2 J 0 ( ρv ) ρdρdθ
= k = 1 N 2 0 π ρ k 1 ρ k P k ( ρ ) e i 2 π ( n 1 ) Δ h k λ e iuρ 2 2 J 0 ( ρv ) ρdρdθ
{ ρ k = Δ ρ k cos ( θ ϕ k ) + Δ ρ k 2 cos 2 ( θ ϕ k ) Δ ρ k 2 + ( a k + σ k 1 + σ N ) 2 ρ 0 = 0 , ρ N = 1 ( k = 1 , , N ) θ [ 0 , π ]
h v u = 2 π M m = 1 M [ 0 ρ 1 m P 1 ( ρ ) e iuρ 2 2 J 0 ( ρv ) ρd ρ + + ρ ( k 1 , m ) ρ k m P k ( ρ ) e 2 π ( n 1 ) Δ h k λ e i 2 2 J 0 ( ρv ) ρd ρ + + ρ ( N 1 , m ) 1 P N ( ρ ) e 2 π ( n 1 ) Δ h N λ e iuρ 2 2 · J 0 ( ρv ) ρd ρ ]
= 2 π M m = 1 M k = 1 N ρ ( k 1 , m ) ρ k m P k ( ρ ) e 2 π ( n 1 ) Δ h k λ e iuρ 2 2 J 0 ( ρv ) ρd ρ
ρ k m = Δ ρ k cos ( m M π ϕ k ) + Δ ρ k 2 cos 2 ( m M π ϕ k ) Δ ρ k 2 + ( a k + σ k 1 + σ N ) 2
{ ρ 0 m 0 ρ N m 1 m = 1 , , M
I 0 u = h 0 u 2 = 2 π M m = 1 M k = 1 N ρ ( k 1 , m ) ρ k m P k ( ρ ) e i 2 π ( n 1 ) Δ h k λ e iuρ 2 2 ρd ρ 2
I ( 0 , u ) A e u 4 + B e u 3 + C e u 2 + D e u + E e
A e = [ k = 1 N m = 1 M ρ 6 k m ρ 6 ( k 1 , m ) 24 t k cos φ k ] 2 + [ k = 1 N m = 1 M ρ 6 k m ρ 6 ( k 1 , m ) 24 t k sin φ k ] 2
B e = 1 48 { k = 1 N m = 1 M [ ρ 6 k m ρ 6 ( k 1 , m ) ] t k sin φ k } { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k sin φ k } 1 48 { k = 1 N m = 1 M [ ρ 6 k m ρ 6 ( k 1 , m ) ] t k cos φ k } { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k sin φ k }
C e = 1 16 { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k sin φ k } 2 + 1 16 { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k cos φ k } 2 1 12 { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k sin φ k } { k = 1 N m = 1 M [ ρ 6 k m ρ 6 ( k 1 , m ) ] t k sin φ k } 1 12 { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k cos φ k } { k = 1 N m = 1 M [ ρ 6 k m ρ 6 ( k 1 , m ) ] t k cos φ k }
D e = 1 2 { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k cos φ k } { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k sin φ k } 1 2 { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k sin φ k } { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k cos φ k }
E e = { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k cos φ k } 2 + { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k sin φ k } 2
u Fe = D e 2 C e
{ φ k = φ k 0 + 2 π ( n 1 ) λ Δ h k t k = t k 0 + Δ t k k = 1 , N
I 0 u u = 0
G Ae = D e 2 4 C e E e 192 × C e 2
I ( v , u Fe ) = h v , u Fe ) 2 = 2 π M m = 1 M k = 1 N ρ ( k 1 , m ) ρ k m P k ( ρ ) e 2 π ( n 1 ) Δ h k λ e i u Fe ρ 2 2 J 0 ( ρv ) ρd ρ 2
I ( v , u Fe ) H e v 4 2 F e v 2 + X e
F e = F e 8 u Fe 24 + { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k cos φ k } { k = 1 N m = 1 M [ ρ 6 k m ρ 6 ( k 1 , m ) ] t k sin φ k }
u Fe 24 { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k sin φ k }{ k = 1 N m = 1 M [ ρ 6 k m ρ 6 ( k 1 , m ) ] t k cos φ k }
+ u Fe 2 192 { [ k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k sin φ k } { k = 1 N m = 1 M [ ρ 6 k m ρ 6 ( k 1 , m ) ] t k sin φ k }
+ u Fe 2 192 { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k cos φ k } { k = 1 N m = 1 M [ ρ 6 k m ρ 6 ( k 1 , m ) ] t k cos φ k }
H e = H e 64 + B e 2 u Fe + A e u Fe 2
H e = { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k cos φ k } 2 + { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k sin φ k } 2
F e = { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k cos φ k } { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k cos φ k ]
+ { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k sin φ k } { k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k sin φ k }
X e = { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k cos φ k + u Fe 4 k = 1 N m = 1 M [ ρ 4 k m ρ 4 ( k 1 , m ) ] t k sin φ k } 2
+ { k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k sin φ k u Fe 4 k = 1 N m = 1 M [ ρ 2 k m ρ 2 ( k 1 , m ) ] t k cos φ k } 2
I ( v , u F ) v = 0
G Te = F e 8 H e
S e = E e + D e 2 u Fe
{ Δ G A = G Ae G A 0 Δ G T = G Te G T 0 Δ S = S e S 0
r k = r k 0 + σ k 1 + σ 3 ( k = 1,2,3 )

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