Abstract

Spherical aberration arising from deviations of the thickness of an optical disc substrate from a nominal value can be compensated to a great extent by illuminating the scanning objective lens with a slightly convergent or divergent beam. The optimum conjugate change and the amount and type of residual aberration are calculated analytically for an objective lens that satisfies Abbe’s sine condition. The aberration sensitivity is decreased by a factor of 25 for numerical aperture values of approximately 0.85, and the residual aberrations consist mainly of the first higher-order Zernike spherical aberration term A60. The Wasserman–Wolf–Vaskas method is used to design biaspheric objective lenses that satisfy a ray condition that interpolates between the Abbe and the Herschel conditions. Requirements for coma by field use allow for only small deviations from the Abbe condition, making the analytical theory a good approximation for any objective lens used in practice.

© 2005 Optical Society of America

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Corrections

Sjoerd Stallinga, "Finite conjugate spherical aberration compensation in high numerical-aperture optical disc readout: erratum," Appl. Opt. 45, 2279-2279 (2006)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-45-10-2279

References

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  1. S. Stallinga, “Compact description of substrate-related aberrations in high numerical-aperture optical disk readout,” Appl. Opt. 44, 849–858 (2005).
    [CrossRef] [PubMed]
  2. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).
  3. M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge University Press, 1980).
  4. M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).
    [CrossRef]
  5. C. J. R. Sheppard, M. Gu, “Aberration compensation in confocal microscopy,” Appl. Opt. 30, 3563–3568 (1991).
    [CrossRef] [PubMed]
  6. C. J. R. Sheppard, “Confocal imaging through weakly aberrating media,” Appl. Opt. 39, 6366–6368 (2000).
    [CrossRef]
  7. C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
    [CrossRef]
  8. J. J. M. Braat, “Abbe sine condition and related imaging conditions in geometrical optics,” in Fifth International Topical Meeting on Education and Traning in Optics, C. H. F. Velzel, ed., Proc. SPIE3190, 59–64 (1997).
    [CrossRef]
  9. J. J. M. Braat, “Design of an optical system toward some isoplanatic prescription,” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 76–84 (1998).
    [CrossRef]
  10. G. D. Wasserman, E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62, 2–8 (1949).
    [CrossRef]
  11. E. M. Vaskas, “Note on the Wasserman–Wolf method for designing aspherical surfaces,” J. Opt. Soc. Am. 47, 669–670 (1957).
    [CrossRef]
  12. J. J. M. Braat, P. F. Greve, “Aplanatic optical system containing two aspheric surfaces,” Appl. Opt. 18, 2187–2191 (1979).
    [CrossRef] [PubMed]
  13. T. W. Tukker, “The design of flat intensity lenses for optical pick-up units,” in Technical Digest of International Conference on Optics 2004, Y. Ichioka, K. Yamamoto, eds.(Optical Society of Japan and International Commission for Optics, 2004), pp. 245–246.
  14. http://www.zemax.com .
  15. S. Wolfram, Mathematica: a System for Mathematics by Computer, 2nd ed. (Addison-Wesley, 1991).

2005 (1)

2000 (1)

1993 (1)

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

1991 (1)

1979 (1)

1957 (1)

1949 (1)

G. D. Wasserman, E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62, 2–8 (1949).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge University Press, 1980).

Braat, J. J. M.

J. J. M. Braat, P. F. Greve, “Aplanatic optical system containing two aspheric surfaces,” Appl. Opt. 18, 2187–2191 (1979).
[CrossRef] [PubMed]

J. J. M. Braat, “Abbe sine condition and related imaging conditions in geometrical optics,” in Fifth International Topical Meeting on Education and Traning in Optics, C. H. F. Velzel, ed., Proc. SPIE3190, 59–64 (1997).
[CrossRef]

J. J. M. Braat, “Design of an optical system toward some isoplanatic prescription,” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 76–84 (1998).
[CrossRef]

Greve, P. F.

Gu, M.

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Aberration compensation in confocal microscopy,” Appl. Opt. 30, 3563–3568 (1991).
[CrossRef] [PubMed]

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).
[CrossRef]

Sheppard, C. J. R.

Stallinga, S.

Tukker, T. W.

T. W. Tukker, “The design of flat intensity lenses for optical pick-up units,” in Technical Digest of International Conference on Optics 2004, Y. Ichioka, K. Yamamoto, eds.(Optical Society of Japan and International Commission for Optics, 2004), pp. 245–246.

Vaskas, E. M.

Wasserman, G. D.

G. D. Wasserman, E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62, 2–8 (1949).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

Wolf, E.

G. D. Wasserman, E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62, 2–8 (1949).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge University Press, 1980).

Wolfram, S.

S. Wolfram, Mathematica: a System for Mathematics by Computer, 2nd ed. (Addison-Wesley, 1991).

Appl. Opt. (4)

J. Mod. Opt. (1)

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. Phys. Soc. B (1)

G. D. Wasserman, E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62, 2–8 (1949).
[CrossRef]

Other (8)

T. W. Tukker, “The design of flat intensity lenses for optical pick-up units,” in Technical Digest of International Conference on Optics 2004, Y. Ichioka, K. Yamamoto, eds.(Optical Society of Japan and International Commission for Optics, 2004), pp. 245–246.

http://www.zemax.com .

S. Wolfram, Mathematica: a System for Mathematics by Computer, 2nd ed. (Addison-Wesley, 1991).

J. J. M. Braat, “Abbe sine condition and related imaging conditions in geometrical optics,” in Fifth International Topical Meeting on Education and Traning in Optics, C. H. F. Velzel, ed., Proc. SPIE3190, 59–64 (1997).
[CrossRef]

J. J. M. Braat, “Design of an optical system toward some isoplanatic prescription,” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 76–84 (1998).
[CrossRef]

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge University Press, 1980).

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Changing the axial focal position can be achieved by changing the conjugate of the objective lens (top) or by changing the free working distance (bottom).

Fig. 2
Fig. 2

Residual aberration sensitivity as a function of NA (solid curve) and approximation by the single Zernike term A60 (long-dashed curve) and by the Seidel term proportional to NA6 (short-dashed curve).

Fig. 3
Fig. 3

Parameter βcon that describes the relative change of conjugate (solid curve), the parameter βfwd that describes the relative change in free working distance (long-dashed curve), and the sum of the two parameters βcon + βfwd (short-dashed curve) as a function of NA.

Fig. 4
Fig. 4

Numerically determined parameters βcon (squares) and βfwd (triangles) for a lens with 0.85 NA as a function of the q parameter that interpolates between the Abbe and the Herschel conditions.

Fig. 5
Fig. 5

Sensitivity of coma for field use of the lens as a function of the q parameter that interpolates between the Abbe and the Herschel conditions (squares) and a fit proportional to 1 − q−2 (solid curve).

Equations (38)

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Δ d = Δ z con + Δ z fwd + Δ z ref .
n 0 sin θ 0 = M n 1 sin θ 1 ,
ρ = n 0 sin θ 0 NA 0 = n 1 sin θ 1 NA 1 ,
W con = n 1 Δ z 1 ( cos θ 1 1 ) n 0 Δ z 0 ( cos θ 0 1 ) .
Δ z 1 = M 2 n 1 n 0 Δ z 0 ,
W con = n 1 Δ z 1 { ( 1 ρ 2 NA 1 2 n 1 2 ) 1 / 2 1 + n 0 2 n 1 2 M 2 [ 1 ( 1 ρ 2 NA 0 2 n 0 2 ) 1 / 2 ] } = Δ z 1 [ ( n 1 2 ρ 2 NA 1 2 ) 1 / 2 n 1 + ρ 2 NA 1 2 n 1 + ( n 1 2 ρ 2 NA 1 2 M 2 n 1 2 / n 0 2 ) 1 / 2 ] .
W con = Δ z con [ ( n 2 ρ 2 NA 2 ) 1 / 2 n + p 2 NA 2 2 n ] ,
W fwd = Δ z fwd { ( n 2 ρ 2 NA 2 ) 1 / 2 n [ ( 1 ρ 2 NA 2 ) 1 / 2 1 ) ] .
W ref = Δ z ref [ ( n 2 ρ 2 NA 2 ) 1 / 2 n ] .
W = W con + W fwd + W ref = Δ d [ ( n 2 ρ 2 NA 2 ) 1 / 2 n ] + Δ z con ρ 2 NA 2 2 n Δ z fwd [ ( 1 ρ 2 NA 2 ) 1 / 2 1 ] .
W Δ d = f i = 1 , 2 β i g i ,
f = ( n 2 ρ 2 NA 2 ) 1 / 2 ,
g 1 = ρ 2 NA 2 2 n ,
g 2 = 1 n ( 1 ρ 2 NA 2 ) 1 / 2 ,
β 1 = β con = Δ z con Δ d ,
β 2 = β fwd = n Δ z fwd Δ d .
W rms 2 ( Δ d ) 2 = W 2 W 2 ( Δ d ) 2 = f rms 2 2 i = 1 , 2 υ i β i + i , j = 1 , 2 R i j β i β j ,
f rms 2 = f 2 f 2 ,
υ i = f g i f g i ,
R i j = g i g j g i g j ,
A = 1 π P d 2 ρ A ( ρ ) ,
β i = j = 1 , 2 R i j 1 υ j ,
W rms 2 ( Δ d ) 2 = f rms 2 i , j = 1 , 2 R i j 1 υ i υ j ,
W rms Δ d = ( n 2 1 ) NA 6 320 7 n 5 + ( n 2 1 ) ( 2 n 2 + 5 ) NA 8 1280 7 n 7 .
β con = n 2 1 n 2 + 3 ( n 2 1 ) NA 2 4 n 4 + 3 ( n 4 51 n 2 + 50 ) NA 4 280 n 6 ,
β fwd = 1 n 2 3 ( n 2 1 ) NA 2 4 n 4 + 3 ( n 4 + 22 n 2 25 ) NA 4 140 n 6 ,
β con + β fwd = 1 + 3 ( n 2 1 ) NA 4 40 n 6 .
Δ z 0 = Δ z con n M 2 = β con Δ d n NA 2 NA 0 2 .
n 0 sin ( θ 0 q ) = M n 1 sin ( θ 1 q ) ,
f = 2 3 NA 2 [ n 3 ( n 2 NA 2 ) 3 / 2 ] ,
f 2 = n 2 1 2 NA 2 ,
g 1 = NA 2 4 n ,
g 2 = 2 3 n NA 2 [ 1 ( 1 NA 2 ) 3 / 2 ] ,
f g 1 = 2 n 5 + ( 3 NA 4 n 2 NA 2 2 n 4 ) ( n 2 NA 2 ) 1 / 2 15 n NA 2 ,
f g 2 = 1 4 n NA 2 { n 3 + n ( n 2 + 1 2 NA 2 ) × ( n 2 NA 2 ) 1 / 2 ( 1 NA 2 ) 1 / 2 + ( n 2 1 ) 2 × log [ ( n 2 NA 2 ) 1 / 2 + ( 1 NA 2 ) 1 / 2 n + 1 ] } ,
g 1 2 = NA 4 12 n 2 ,
g 2 2 = 2 NA 2 2 n 2 ,
g 1 g 2 = 2 + ( 3 NA 4 NA 2 2 ) ( 1 NA 2 ) 1 / 2 15 n 2 NA 2 .

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