Abstract

For the application of optical data storage, theories that we present can be used to design a diffractive superresolution element (DSE) with the highest sidelobe suppressed. A globally optimal solution among general hybrid-type filters can be solved through linear programming. The obtained globally optimal performances set the exact performance limits of a general hybrid-type DSE with the highest sidelobe suppressed. A comparison of our design theories and the previous design methods shows the advantages of the former.

© 2005 Optical Society of America

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References

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  1. I. J. Cox, “Increasing the bit packing densities of optical disk systems,” Appl. Opt. 23, 3260–3261 (1984).
    [CrossRef] [PubMed]
  2. Z. S. Hegedus, V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [CrossRef]
  3. M. A. A. Neil, R. Juskaitis, T. Wilson, Z. J. Laczik, V. Sarafis “Optimized pupil-plane filters for confocal microscope point-spread function engineering,” Opt. Lett. 25, 245–247 (2000).
    [CrossRef]
  4. T. R. M. Sales, G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637–1646 (1997).
    [CrossRef]
  5. I. J. Cox, C. J. R. Sheppard, T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. 72, 1287–1291 (1982).
    [CrossRef]
  6. R. Boivin, 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]
  7. H. Liu, Y. Yan, Q. Tan, G. Jin, “Theories for the design of diffractive superresolution elements and limits of optical superresolution,” J. Opt. Soc. Am. A 19, 2185–2193 (2002).
    [CrossRef]
  8. N. Wei, M. Gong, R. Cui, “Numerical solution of sidelobe intensity for phase-shifting apodizers,” Jpn. J. Appl. Phys., Part 1 42, 104–108 (2003).
    [CrossRef]
  9. J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, New York, 1989), Chap. 2.
    [CrossRef]
  10. R. K. Luneberg, The Mathematical Theory of Optics (Cambridge U. Press, Cambridge, UK, 1966), pp. 344–359.
  11. T. R. M. Sales, G. M. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. 22, 582–584 (1997).
    [CrossRef] [PubMed]
  12. I. Leiserson, S. G. Lipson, V. Sarafis, “Superresolution in far-field imaging,” Opt. Lett. 25, 209–211 (2000).
    [CrossRef]
  13. I. Leizerson, S. G. Lipson, V. Sarafis, “Superresolution in far-field imaging,” J. Opt. Soc. Am. A 19, 436–443 (2002).
    [CrossRef]
  14. I. Leizerson, S. G. Lipson, V. Sarafis, “Improvement of optical resolution in far-field imaging by optical multiplication,” Micron 34, 301–307 (2003).
    [CrossRef] [PubMed]
  15. N. Wei, M. Gong, R. Cui, “Resolution limit of diffractive superresolution elements,” manuscript in preparation, available from author by email: nman@263.net.
  16. N. Wei, “Aspherics design and phase-shifting superresolution properties of the objective lens for optical disk data storage,” Ph.D. dissertation (Tsinghua University, Beijing, 2003), Chap 3.
  17. H. Liu, Y. Yan, D. Yi, G. Jin, “Theories for the design of a hybrid refractive-diffractive superresolution lens with high numerical aperture,” J. Opt. Soc. Am. A 20, 913–924 (2003).
    [CrossRef]
  18. H. Liu, “Investigations of design methods of diffractive optical elements to implement optical superresolution,” Ph.D. dissertation (Tsinghua University, Beijing, 2004), Chap 2.
  19. T. S. Blyth, E. F. Robertson, Further Linear Algebra (Springer, London, 2002), Chap. 1.
  20. L. E. Elsgolc, Calculus of Variations (Pergamon, Oxford, UK, 1961), Chap. I.

2003 (3)

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

I. Leizerson, S. G. Lipson, V. Sarafis, “Improvement of optical resolution in far-field imaging by optical multiplication,” Micron 34, 301–307 (2003).
[CrossRef] [PubMed]

H. Liu, Y. Yan, D. Yi, G. Jin, “Theories for the design of a hybrid refractive-diffractive superresolution lens with high numerical aperture,” J. Opt. Soc. Am. A 20, 913–924 (2003).
[CrossRef]

2002 (2)

2000 (2)

1997 (2)

1986 (1)

1984 (1)

1982 (1)

1980 (1)

R. Boivin, 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]

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: I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Boivin, R.

R. Boivin, 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, R. Cui, “Numerical solution of sidelobe intensity for phase-shifting apodizers,” Jpn. J. Appl. Phys., Part 1 42, 104–108 (2003).
[CrossRef]

N. Wei, M. Gong, R. Cui, “Resolution limit of diffractive superresolution elements,” manuscript in preparation, available from author by email: nman@263.net.

Elsgolc, L. E.

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

Gong, M.

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

N. Wei, M. Gong, R. Cui, “Resolution limit of diffractive superresolution elements,” manuscript in preparation, available from author by email: nman@263.net.

Hegedus, Z. S.

Jin, G.

Juskaitis, R.

Laczik, Z. J.

Leiserson, I.

Leizerson, I.

I. Leizerson, S. G. Lipson, V. Sarafis, “Improvement of optical resolution in far-field imaging by optical multiplication,” Micron 34, 301–307 (2003).
[CrossRef] [PubMed]

I. Leizerson, S. G. Lipson, V. Sarafis, “Superresolution in far-field imaging,” J. Opt. Soc. Am. A 19, 436–443 (2002).
[CrossRef]

Lipson, S. G.

Liu, H.

Luneberg, R. K.

R. K. Luneberg, The Mathematical Theory of Optics (Cambridge U. Press, Cambridge, UK, 1966), pp. 344–359.

Morris, G. M.

Neil, M. A. A.

Robertson, E. F.

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

Sales, T. R. M.

Sarafis, V.

Sheppard, C. J. R.

Strayer, J. K.

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

Tan, Q.

Wei, N.

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

N. Wei, M. Gong, R. Cui, “Resolution limit of diffractive superresolution elements,” manuscript in preparation, available from author by email: nman@263.net.

N. Wei, “Aspherics design and phase-shifting superresolution properties of the objective lens for optical disk data storage,” Ph.D. dissertation (Tsinghua University, Beijing, 2003), Chap 3.

Wilson, T.

Yan, Y.

Yi, D.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

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

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

Micron (1)

I. Leizerson, S. G. Lipson, V. Sarafis, “Improvement of optical resolution in far-field imaging by optical multiplication,” Micron 34, 301–307 (2003).
[CrossRef] [PubMed]

Opt. Acta (1)

R. Boivin, 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. Lett. (3)

Other (7)

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

R. K. Luneberg, The Mathematical Theory of Optics (Cambridge U. Press, Cambridge, UK, 1966), pp. 344–359.

N. Wei, M. Gong, R. Cui, “Resolution limit of diffractive superresolution elements,” manuscript in preparation, available from author by email: nman@263.net.

N. Wei, “Aspherics design and phase-shifting superresolution properties of the objective lens for optical disk data storage,” Ph.D. dissertation (Tsinghua University, Beijing, 2003), Chap 3.

H. Liu, “Investigations of design methods of diffractive optical elements to implement optical superresolution,” Ph.D. dissertation (Tsinghua University, Beijing, 2004), Chap 2.

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

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

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Figures (9)

Fig. 1
Fig. 1

A filter placed at the exit pupil of an imaging system as a DSE.

Fig. 2
Fig. 2

For the solved GOS of problem P L ( S 0 , G 0 , η FOV ) with η FOV η FOV min ( S 0 , G 0 ) and various S 0 and G 0 , PSF intensity I ( η ) and intensity upper bound I u ( η ) are shown; the vertical coordinate of the horizontal lines is M, and the horizontal coordinate of the vertical lines is η u . (a) S 0 = 0.1 , G 0 = 0.6 , η FOV min ( S 0 , G 0 ) 2.2 . (b) S 0 = 0.05 , G 0 = 0.6 , η FOV min ( S 0 , G 0 ) 4 . (c) S 0 = 0.1 , G 0 = 0.7 , η FOV min ( S 0 , G 0 ) 5 . (d) S 0 = 0.05 , G 0 = 0.7 , η FOV min ( S 0 , G 0 ) 7.7 .

Fig. 3
Fig. 3

Contours of M min ( S 0 , G 0 ) with various S 0 and G 0 . Dashed curve, S 0 = S eu ( G 0 ) ; dotted curves, contours of M min ( S 0 , G 0 ) = 10 2 , 10 1 , , 10 4 with G 0 decreasing from 1 to 0; solid curves, contours of M min ( S 0 , G 0 ) = 2 × 10 n , 3 × 10 n , , 9 × 10 n ( n = 3 , 2 , , 3 ) with G 0 decreasing. The insert with S 0 in logarithmic coordinates to magnify the figure with S 0 in linear coordinate.

Fig. 4
Fig. 4

On the S G coordinate plane. Dashed curve, S = S eu ( G ) or G = G el ( S ) ; solid curve, M = M min ( S , G ) , S = S G M ( G , M ) , or G = G S M ( S , M ) . G = G M ( M ) and G = G S M ( 0 + , M ) correspond to two points on the G axis, and S = S M ( M ) corresponds to a point on S axis.

Fig. 5
Fig. 5

Intuitive explanation of the solution of problem (2). (a) S l S M ( M u ) , (b) S l > S M ( M u ) .

Fig. 6
Fig. 6

Curves of G M el ( M u ) and G l ( M u ) .

Fig. 7
Fig. 7

Intuitive explanation of the solution of problem (3). (a) G M el ( M u ) G u G M ( M u ) , (b) G u > G M ( M u ) .

Fig. 8
Fig. 8

Intuitive explanation of the solution of problem (4). Here S l S eu ( G u ) .

Fig. 9
Fig. 9

PSF intensity distributions of the five design examples in Table 1.

Tables (4)

Tables Icon

Table 1 Five Design Examples of General Design Problems (2, 3, 4)

Tables Icon

Table 2 Comparison of the Solutions of Problem (2) and Method [ 8 ]

Tables Icon

Table 3 Comparison of the Solutions of Problem (3) and Method [ 7 ]

Tables Icon

Table 4 Comparison of the Solution of Problem (4) and Method [ 4 ]

Equations (67)

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i ( r 2 ) = ( 2 π λ f ) 2 0 R u ( r 1 ) J 0 ( 2 π r 1 r 2 λ f ) r 1 d r 1 2 ,
I ( η ) = 4 0 1 U ( ρ ) J 0 ( x J η ρ ) ρ d ρ 2 ,
min U ( ρ ) G , subject to S S l , M M u , U ( ρ ) 1 ,
max U ( ρ ) S , subject to G G u , M M u , U ( ρ ) 1 ,
min U ( ρ ) M , subject to S S l , G G u , U ( ρ ) 1 .
min U ( ρ ) M , subject to S = S 0 , G = G 0 , U ( ρ ) 1 .
min { U ( ρ ) , M } M , subject to I ( 0 ) = S 0 , I ( G 0 ) = 0 ,
I ( η ) S 0 M
with η ( G 0 , + ) , U ( ρ ) 1 ,
min { A ( ρ ) , B ( ρ ) , M } M
subject to
[ 0 1 A ( ρ ) ρ d ρ ] 2 + [ 0 1 B ( ρ ) ρ d ρ ] 2 = S 0 4 ,
0 1 A ( ρ ) J 0 ( x J G 0 ρ ) ρ d ρ = 0 1 B ( ρ ) J 0 ( x j G 0 ρ ) ρ d ρ = 0 ,
[ 0 1 A ( ρ ) J 0 ( x J η ρ ) ρ d ρ ] 2 + [ 0 1 B ( ρ ) J 0 ( x j η ρ ) ρ d ρ ] 2
S 0 M 4 , η ( G 0 , + )
A ( ρ ) 2 + B ( ρ ) 2 1
min { A ( ρ ) , B ( ρ ) , M S 0 } M S 0
subject to
0 1 A ( ρ ) ρ d ρ = S 0 8 ,
0 1 A ( ρ ) J 0 ( x J G 0 ρ ) ρ d ρ = 0 ,
M S 0 0 1 A ( ρ ) J 0 ( x J η ρ ) ρ d ρ M S 0 , η ( G 0 , + ) ,
1 2 A ( ρ ) 1 2 ,
B ( ρ ) = A ( ρ ) ,
M min ( S 0 , G 0 ) = 8 M S 0 , min ( S 0 , G 0 ) 2 S 0 ,
min { A k , B k , M S 0 } M S 0
k = 1 K A k ( ρ k 2 ρ k 1 2 ) = S 0 2 ,
k = 1 K A k [ J 1 ( x J G 0 ρ k ) ρ k J 1 ( x J G 0 ρ k 1 ) ρ k 1 ] ( x J G 0 ) = 0 ,
M S 0 k = 1 K A k [ J 1 ( x J η i ρ k ) ρ k J 1 ( x J η i ρ k 1 ) ρ k 1 ] ( x J η i ) M S 0 , i = 2 , 3 , ... , N ,
1 2 A k 1 2 , B k = A k , k = 1 , 2 , ... , K ,
I ( η ) I u ( η ) = ( 1 + 2 i = 1 N b ρ i b ) 2 π ( 2 x J η ) 3 ,
M min ( S , G ) S > 0 ,
M min ( S , G ) G < 0 .
G ̃ min ( S l , M u ) = { G S M ( S l , M u ) if S l S M ( M u ) G el ( S l ) if S l > S M ( M u ) .
G M el ( M u ) = G S M ( 0 + , M u ) .
S ̃ max ( G u , M u ) = { No definition , if G u < G M el ( M u ) S G M ( G u , M u ) , if G M el ( M u ) G u G M ( M u ) S eu ( G u ) , if G u > G M ( M u ) .
M ̃ min ( S l , G u ) = { M min ( S l , G u ) , if S l S eu ( G u ) No definition , if S l > S eu ( G u ) .
A ( ρ ) = A 0 ( ρ ) , B ( ρ ) = B 0 ( ρ ) , M = M min ( S 0 , G 0 ) ,
A 0 ( ρ ) + i B 0 ( ρ ) = [ A 1 ( ρ ) + i B 1 ( ρ ) ] exp ( i ϕ 1 )
ϕ 1 = n 1 π + ψ 1 , ψ 1 = arctan { 0 1 [ A 1 ( ρ ) B 1 ( ρ ) ] ρ d ρ 0 1 [ A 1 ( ρ ) + B 1 ( ρ ) ] ρ d ρ } ,
n 1 = { even number , if F 1 0 odd number , if F 1 < 0 ,
F 1 = [ 0 1 A 1 ( ρ ) ρ d ρ ] cos ψ 1 [ 0 1 B 1 ( ρ ) ρ d ρ ] sin ψ 1 .
0 1 A 0 ( ρ ) ρ d ρ = 0 1 B 0 ( ρ ) ρ d ρ 0 .
A ( ρ ) = B ( ρ ) = [ A 0 ( ρ ) + B 0 ( ρ ) ] 2 , M = M min ( S 0 , G 0 )
[ 0 1 A 0 ( ρ ) + B 0 ( ρ ) 2 ρ d ρ ] 2 + [ 0 1 A 0 ( ρ ) + B 0 ( ρ ) 2 ρ d ρ ] 2
= [ 0 1 A 0 ( ρ ) ρ d ρ ] 2 + [ 0 1 B 0 ( ρ ) ρ d ρ ] 2 = S 0 4 ,
[ 0 1 A 0 ( ρ ) + B 0 ( ρ ) 2 J 0 ( x J η ρ ) ρ d ρ ] 2 + [ 0 1 A 0 ( ρ ) + B 0 ( ρ ) 2 J 0 ( x J η ρ ) ρ d ρ ] 2 = 1 2 [ 1 × 0 1 A 0 ( ρ ) J 0 ( x J η ρ ) ρ d ρ + 1 × 0 1 B 0 ( ρ ) J 0 ( x J η ρ ) ρ d ρ ] 2 1 2 ( 1 2 + 1 2 ) { [ 0 1 A 0 ( ρ ) J 0 ( x J η ρ ) ρ d ρ ] 2 + [ 0 1 B 0 ( ρ ) J 0 ( x J η ρ ) ρ d ρ ] 2 } S 0 M min ( S 0 , G 0 ) 4 , η ( G 0 , + ) ,
[ A 0 ( ρ ) + B 0 ( ρ ) 2 ] 2 + [ A 0 ( ρ ) + B 0 ( ρ ) 2 ] 2
= [ 1 × A 0 ( ρ ) + 1 × B 0 ( ρ ) ] 2 2 ( 1 2 + 1 2 ) [ A 0 ( ρ ) 2 + B 0 ( ρ ) 2 ] 2 1 ,
M S 0 0 1 A ( ρ ) J 0 ( x J η i ρ ) ρ d ρ M S 0 , i = 2 , 3 , , N ,
0 1 A ( ρ ) J 0 ( x J η i ρ ) ρ d ρ = M S 0 cos β i ,
i = 2 , 3 , , N , M S 0 > 0 ,
A ( ρ ) = 1 2 cos θ ( ρ ) ,
min { θ ( ρ ) , β i , M S 0 } M S 0
0 1 cos θ ( ρ ) ρ d ρ = S 0 2 ,
0 1 cos θ ( ρ ) J 0 ( x J G 0 ρ ) ρ d ρ = 0 ,
0 1 cos θ ( ρ ) J 0 ( x J η i ρ ) ρ d ρ = 2 M S 0 cos β i ,
i = 2 , 3 , , N , M S 0 > 0 .
F [ θ ( ρ ) , β i , M S 0 , λ i ] = M S 0 + λ 0 [ 0 1 cos θ ( ρ ) ρ d ρ S 0 2 ] + λ 1 0 1 cos θ ( ρ ) J 0 ( x J G 0 ρ ) ρ d ρ + i = 2 N λ i [ 0 1 cos θ ( ρ ) J 0 ( x J η i ρ ) ρ d ρ 2 M S 0 cos β i ] = 0 1 [ i = 0 N λ i J 0 ( x J η i ρ ) ] cos θ ( ρ ) ρ d ρ + M S 0 ( 1 2 i = 2 N λ i cos β i ) λ 0 S 0 2 ,
δ θ ( ρ ) F [ θ ( ρ ) , β i , M S 0 , λ i ] = 0 1 [ i = 0 N λ i J 0 ( x J η i ρ ) ] ( 1 ) sin θ ( ρ ) ρ δ θ ( ρ ) d ρ = 0 ,
F [ θ ( ρ ) , β i , M S 0 , λ i ] M S 0 = 1 2 i = 2 N λ i cos β i = 0 ,
[ i = 0 N λ i J 0 ( x J η i ρ ) ] ( 1 ) sin θ ( ρ ) ρ = 0 , ρ [ 0 , 1 ] ,
sin θ ( ρ ) = 0 , ρ [ 0 , 1 ] .
A ( ρ ) { 1 2 , 1 2 } .
M min ( S l , G ) M min ( S , G ) M M u = M min ( S l , G S M ( S l , M u ) ) ,
G G S M ( S l , M u ) .
S eu ( G ) S S l = S eu ( G el ( S l ) ) ,
G G el ( S l ) .

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