Abstract

We suggest using the theory of linear programming to design diffractive superresolution elements if the upper bound of the intensity distribution on the input plane is restricted, and using variation theory of functional or wide-sense eigenvalue theory of matrix if the upper bound of the radiation flux through the input plane is restricted. Globally optimal solutions can be obtained by each of these theories. Several rules of the structure and the superresolution performance of diffractive superresolution elements are provided, which verify the validity of these theories and set some limits of optical superresolution.

© 2002 Optical Society of America

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References

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  1. T. R. M. Sales, G. M. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. 22, 582–584 (1997).
    [CrossRef] [PubMed]
  2. I. J. Cox, “Increasing the bit packing densities of optical disk systems,” Appl. Opt. 23, 3260–3261 (1984).
    [CrossRef] [PubMed]
  3. M. Shinoda, K. Kime, “Focusing characteristics of an optical head with superresolution using a high-aspect-ratio red laser diode,” Jpn. J. Appl. Phys. 35, 380–383 (1996).
    [CrossRef]
  4. T. Wilson, B. R. Masters, “Confocal microscopy,” Appl. Opt. 33, 565–566 (1994).
    [CrossRef] [PubMed]
  5. K. Carlsson, P. E. Danielsson, R. Lenz, A. Liljeborg, L. Majlof, N. Aslund, “Three-dimensional microscopy using a confocal laser scanning microscope,” Opt. Lett. 10, 53–55 (1985).
    [CrossRef] [PubMed]
  6. S. Yibing, Y. Guoguang, H. Xiyun, “Research on phenomenon of the super-resolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).
  7. T. R. M. Sales, G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637–1646 (1997).
    [CrossRef]
  8. M. R. Wang, X. G. Huang, “Subwavelength-resolvable focused non-Gaussian beam shaped with a binary diffractive optical element,” Appl. Opt. 38, 2171–2176 (1999).
    [CrossRef]
  9. R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
    [CrossRef]
  10. 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]
  11. D. Zhihua, T. Weijian, B. Zhengkang, “Superresolution with high throughput via irradiance redistribution element,” Acta Opt. Sin. 20, 701–706 (2000).
  12. H. Anyun, “Wavefront engineering and optical superresolution,” Semicond. Inf. 35, 1–17 (1998).
  13. J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, New York, 1989), Chap. 2.
  14. L. E. Elsgolc, Calculus of Variations (Pergamon, Oxford, UK, 1961), Chap. I-2.
  15. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, UK, 1965), Chap. 5.
  16. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chaps. 3 and 8.

2000 (1)

D. Zhihua, T. Weijian, B. Zhengkang, “Superresolution with high throughput via irradiance redistribution element,” Acta Opt. Sin. 20, 701–706 (2000).

1999 (2)

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

S. Yibing, Y. Guoguang, H. Xiyun, “Research on phenomenon of the super-resolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

1998 (1)

H. Anyun, “Wavefront engineering and optical superresolution,” Semicond. Inf. 35, 1–17 (1998).

1997 (2)

1996 (1)

M. Shinoda, K. Kime, “Focusing characteristics of an optical head with superresolution using a high-aspect-ratio red laser diode,” Jpn. J. Appl. Phys. 35, 380–383 (1996).
[CrossRef]

1994 (1)

1985 (1)

1984 (1)

1982 (1)

1980 (1)

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

Anyun, H.

H. Anyun, “Wavefront engineering and optical superresolution,” Semicond. Inf. 35, 1–17 (1998).

Aslund, N.

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, 5th ed. (Pergamon, New York, 1975), Chaps. 3 and 8.

Carlsson, K.

Cox, I. J.

Danielsson, P. E.

Elsgolc, L. E.

L. E. Elsgolc, Calculus of Variations (Pergamon, Oxford, UK, 1961), Chap. I-2.

Guoguang, Y.

S. Yibing, Y. Guoguang, H. Xiyun, “Research on phenomenon of the super-resolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

Huang, X. G.

Kime, K.

M. Shinoda, K. Kime, “Focusing characteristics of an optical head with superresolution using a high-aspect-ratio red laser diode,” Jpn. J. Appl. Phys. 35, 380–383 (1996).
[CrossRef]

Lenz, R.

Liljeborg, A.

Majlof, L.

Masters, B. R.

Morris, G. M.

Sales, T. R. M.

Sheppard, C. J. R.

Shinoda, M.

M. Shinoda, K. Kime, “Focusing characteristics of an optical head with superresolution using a high-aspect-ratio red laser diode,” Jpn. J. Appl. Phys. 35, 380–383 (1996).
[CrossRef]

Strayer, J. K.

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

Wang, M. R.

Weijian, T.

D. Zhihua, T. Weijian, B. Zhengkang, “Superresolution with high throughput via irradiance redistribution element,” Acta Opt. Sin. 20, 701–706 (2000).

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, UK, 1965), Chap. 5.

Wilson, T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chaps. 3 and 8.

Xiyun, H.

S. Yibing, Y. Guoguang, H. Xiyun, “Research on phenomenon of the super-resolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

Yibing, S.

S. Yibing, Y. Guoguang, H. Xiyun, “Research on phenomenon of the super-resolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

Zhengkang, B.

D. Zhihua, T. Weijian, B. Zhengkang, “Superresolution with high throughput via irradiance redistribution element,” Acta Opt. Sin. 20, 701–706 (2000).

Zhihua, D.

D. Zhihua, T. Weijian, B. Zhengkang, “Superresolution with high throughput via irradiance redistribution element,” Acta Opt. Sin. 20, 701–706 (2000).

Acta Opt. Sin. (2)

D. Zhihua, T. Weijian, B. Zhengkang, “Superresolution with high throughput via irradiance redistribution element,” Acta Opt. Sin. 20, 701–706 (2000).

S. Yibing, Y. Guoguang, H. Xiyun, “Research on phenomenon of the super-resolution in laser lithography,” Acta Opt. Sin. 19, 1512–1517 (1999).

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

M. Shinoda, K. Kime, “Focusing characteristics of an optical head with superresolution using a high-aspect-ratio red laser diode,” Jpn. J. Appl. Phys. 35, 380–383 (1996).
[CrossRef]

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)

Semicond. Inf. (1)

H. Anyun, “Wavefront engineering and optical superresolution,” Semicond. Inf. 35, 1–17 (1998).

Other (4)

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-2.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, UK, 1965), Chap. 5.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chaps. 3 and 8.

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Figures (8)

Fig. 1
Fig. 1

(a) DSE with the UBIDIP restricted. (b) DSE with the UBRFIP restricted.

Fig. 2
Fig. 2

Constraint (19e) corresponding to an area surrounded by a circularity can be approximated to constraints (26) corresponding to an area surrounded by a regular polygon with 4P sides. κ(p)=(p-1)π/(2P), p=1 ,, P.

Fig. 3
Fig. 3

Curves of I1(η1) of the designed DSEs with η2G1=η2A/2 and with the UBRFIP restricted.

Fig. 4
Fig. 4

Curves of I2(η2) of the designed DSEs with η2G1=η2A/2.

Fig. 5
Fig. 5

Curves of the dependence of η1b, η1b1 or η1b2 on G, where η1b=0 or η1b1=η1b2=0 means no discontinuous point.

Fig. 6
Fig. 6

Curves of Seu(G). The insert shows the curves of the logarithmic coordinates to magnify the curves of the homogeneous coordinates.

Fig. 7
Fig. 7

Curves of K(G) of the globally optimal solutions.

Fig. 8
Fig. 8

Curve of the dependence of η1bη2G on G.

Equations (150)

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maxu1(r1) i2(0)
i2(r2Gi)=0,i=1,N,
|u1(r1)|2iiu(r1),
i2(r2)=2πλf20Ru1(r1)J02πr1r2λfr1dr12
E1u=2π0Ri1u(r1)r1dr1,
I1u(η1)=i1u(r1)/[E1u/(πR2)],
U1(η1)=u1(r1)/E1u/(πR2),
I2(η2)=4(ra/R)2i2(r2)/[E1u/(πR2)],
I2(η2)=401U1(η1)J0(η1η2)η1dη12,
S=I2(0),
maxu1(r1) I2(0)
I2(η2Gi)=0,
|U1(η1)|2I1u(η1).
maxA1(η1),B1(η1)01A1(η1)η1dη12+01B1(η1)η1dη12
01A1(η1)J0(η1η2Gi)η1dη1=0,
01B1(η1)J0(η1η2Gi)η1dη1=0,
A1(η1)2+B1(η1)2I1u(η1).
U1(η1)=U1m(η1)
U1(η1)=U1m(η1)exp(iϕ0)
01Re[U1(η1)]η1dη1=01Im[U1(η1)]η1dη1.
ϕ0=nπ+arctanσm-μmσm+μm,
σm=01Re[U1m(η1)]η1dη1,
μm=01Im[U1m(η1)]η1dη1,
01A1(η1)η1dη1=01B1(η1)η1dη1.
minA1(η1),B1(η1) min01A1(η1)η1dη1, -01A1(η1)η1dη1,
minA1(η1),B1(η1)01A1(η1)η1dη1,
01A1(η1)J0(η1η2Gi)η1dη1=0,
01B1(η1)J0(η1η2Gi)η1dη1=0,
01A1(η1)η1dη1=01B1(η1)η1dη1,
A1(η1)2+B1(η1)2I1u(η1),
min-A1(η1),-B1(η1)01[-A1(η1)]η1dη1
01[-A1(η1)]J0(η1η2Gi)η1dη1=0,
01[-B1(η1)]J0(η1η2Gi)η1dη1=0,
01[-A1(η1)]η1dη1=01[-B1(η1)]η1dη1,
[-A1(η1)]2+[-B1(η1)]2I1u(η1).
U1m±(η1)=U1m+(η1)ifFm+Fm-U1m-(η1)ifFm-Fm+.
A1(η1)cos γ(p)+B1(η1)sin γ(p)
=I1u(η1)cos[π/(4P)],
-A1(η1)cos γ(p)+B1(η1)sin γ(p)
=I1u(η1)cos[π/(4P)],
-A1(η1)cos γ(p)-B1(η1)sin γ(p)
=I1u(η1)cos[π/(4P)],
A1(η1)cos γ(p)-B1(η1)sin γ(p)
=I1u(η1)cos[π/(4P)].
(-1)mA1(η1)cos γ(p)+(-1)nB1(η1)sin γ(p)
I1u(η1)cos[π/(4P)],
min{Ak,Bk}k=1kmAkαk(0),
k=1kmAkαk(η2Gi)=0
k=1kmBkαk(η2Gi)=0
k=1km(Ak-Bk)αk(0)=0
(-1)mAkcos γ(p)+(-1)nBksin γ(p)Ikucos[π/(4P)],k=1, ,km,
αk(η2)
=[J1(ηkη2)ηk-J1(ηk-1η2)ηk-1]/η2 ifη20(ηk2-ηk-12)/2, ifη2=0.
maxu1(r1) i2(0)
i2(r2Gi)=0,i=1,,N,
2π0R|u1(r1)|2r1dr1E1u,
maxU1(η1) I2(0)
I2(η2Gi)=0,i=1,,N,
201|U1(η1)|2η1dη11,
maxA1(η1),B1(η1)01A1(η1)η1dη12+01B1(η1)η1dη12
01A1(η1)J0(η1η2Gi)η1dη1=0,
01B1(η1)J0(η1η2Gi)η1dη1=0,
201[A1(η1)2+B1(η1)2]η1dη11,
A1(η1)=A1m(η1),
B1(η1)=B1m(η1)
Fie[A1(η1),B1(η1),τi,ωi]
=01A1(η1)η1dη12+01B1(η1)η1dη12
+i=1Nτi01A1(η1)J0(η1η2Gi)η1dη1
+i=1Nωi01B1(η1)J0(η1η2Gi)η1dη1,
δA1(η1)Fie=δB1(η1)Fie=0,
F/τi=F/ωi=0,
δA1(η1)Fie=012cA+i=1NτiJ0(η1η2Gi)η1δA1(η1)dη1,
δB1(η1)Fie=012cB+i=1NωiJ0(η1η2Gi)η1δB1(η1)dη1,
Fie/τi=01A1(η1)J0(η1η2Gi)η1dη1,
Fie/ωi=01B1(η1)J0(η1η2Gi)η1dη1,
cA=01A1(η1)η1dη1
cB=01B1(η1)η1dη1
2cAJ0(η1η2G0)+i=1NτiJ0(η1η2Gi)=0,
2cBJ0(η1η2G0)+i=1NωiJ0(η1η2Gi)=0,
cA=cB=0,
τi=ωi=0,
Fe[A1(η1),B1(η1),τi,ωi,θ]
=01A1(η1)η1dη12+01B1(η1)η1dη12+i=1Nτi01A1(η1)J0(η1η2Gi)η1dη1+i=1Nωi01B1(η1)J0(η1η2Gi)η1dη1+θ201[A1(η1)2+B1(η1)2]η1dη1-1,
δA1(η1)Fe=δB1(η1)Fe=0,
Fe/τi=Fe/ωi=0,
Fe/θ=0,
δA1(η1)Fe=012cA+i=1NτiJ0(η1η2Gi)+4θA1(η1)η1δA1(η1)dη1,
δB1(η1)Fe=012cB+i=1NωiJ0(η1η2Gi)+4θB1(η1)η1δB1(η1)dη1,
Fe/τi=01A1(η1)J0(η1η2Gi)η1dη1,
Fe/ωi=01B1(η1)J0(η1η2Gi)η1dη1,
Fe/θ=201[A1(η1)2+B1(η1)2]η1dη1-1.
A1(η1)=c0A1+i=1NdiAJ0(η1η2Gi),
B1(η1)=c0B1+i=1NdiBJ0(η1η2Gi),
c0A=-cA/(2θ),
c0B=-cB/(2θ),
diA=τi/(2cA),
diB=ωi/(2cB).
cA=cB
c0A=c0B=c00,
011+i=1NdiAJ0(η1η2Gi)J0(η1η2Gj)η1dη1=0,
011+i=1NdiBJ0(η1η2Gi)J0(η1η2Gj)η1dη1=0,
[diA]=[diB]=[di]=-[Tji]-1[qj],
Tji=01J0(η1η2Gi)J0(η1η2Gj)η1dη1=η2GjJ0(η2Gi)J1(η2Gj)-η2GiJ0(η2Gj)J1(η2Gi)(η2Gj)2-(η2Gi)2 ifji[J0(η2Gj)2+J1(η2Gj)2]/2ifj=i,
qj=01J0(η1η2Gj)η1dη1=J1(η2Gj)/η2Gj.
A1(η1)=B1(η1)=c01+i=1NdiJ0(η1η2Gi).
c0=±0.5011+i=1NdiJ0(η1η2Gi)2η1dη1-1/2.
011+i=1NdiJ0(η1η2Gi)2η1dη1
=0.5+2i=1Ndi01J0(η1η2Gi)η1dη1+i=1Nj=1Ndidj01J0(η1η2Gi)J0(η1η2Gj)η1dη1
=0.5+2[di]T[qi]+[di]T[Tij][dj]=0.5-[qj]T[Tij]-1[qi].
c0=±0.5(0.5-[qj][Tij]-1[qi])-1/2.
Seu=1-2[qj]T[Tij]-1[qi].
[Tij]=[J0(η2G)2+J1(η2G)2]/2,
[qi]=Ji(η2G)/η2G,
Seu=1-4J0(η2G)2+J1(η2G)2J1(η2G)η2G2,
Seu(G=0)=0,
Seu(G=1)=1.
minA1(η1),B1(η1) -I2(0)+i=1NνiI2(η2Gi)/4
201[A(η1)2+B(η1)2]η1dη11,
minx xTHx
xTLx1,
xT=(x1T, x2T),
H=diag(h,h),
L=diag(β, β),
[x1]k,1=Ak,
[x2]k,1=Bk
h=-α(0)α(0)T+i=1Nνiα(η2Gi)α(η2Gi)T,
α(η2)k,1=αk(η2),
βk,l=0ifkl(ηk2-ηk-12)ifk=l.
x=xm
F(x, λ)=xTHx-λ(xTLx-1)
F/x=F/λ=0,
(F/x)T=2(Hx-λLx),
F/λ=-(xTLx-1).
Hx=λLx,
xTLx=1.
minx λ,
Hx=λminLx,
xTLx=1,
[(xTHx)/x]T=2Hx=0.
U1(η1)=1 ifη1[0, η1b]-1 ifη1(η1b, 1].
I2(η2)=4[2J1(η1bη2)η1b-J1(η2)]2/η22.
2J1(η1bη2G)η1b-J1(η2G)=0,
η1bη2G[0, 1.2356],
J1(η1bη2G)η1bη2G/2-(η1bη2G)3/16.
η1b[1-1-0.5η2GJ1(η2G)]1/2/(0.5η2G),
η1b(G=0)=2/2,
η1b(G=1)=0.
Seu=[1-2(η1b)2]2,
Seu(G=0)=0,
Seu(G=1)=1,

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