Abstract

In a general optical system, spherical aberration will arise when the on-axis position of an object is changed from its optimum position to another point. Induced spherical aberration can be used to compensate the aberration caused by inserting or removing a medium plate with any thickness that has a refractive index that differs from that of the original. To generate a degree of adequate aberration to balance the aberration from a thin layer, it is necessary to estimate the amount of arising aberration correctly when a point object deviates from its aberration-free position. We analytically induce the exact form of an arising spherical aberration with an on-axis object position for general optical systems that satisfy the Abbe sine condition and express a fourth-order approximation of that form using simple parameters that are conventionally used for the aberration of a thin lens. To verify the correctness of the proposed formula, a comparison between this analysis and simulation results is applied to several sample optical systems using commercial lens-design software.

© 2009 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. T. D. Milster, R. S. Upton, and H. Luo, “Objective lens design for multiple-layer optical data storage,” Opt. Eng. 38, 295-301 (1999).
    [CrossRef]
  7. Y. Zhang, T. D. Milster, J. S. Kim, and S. K. Park, “Advanced lens design for bit-wise volumetric optical data storage,” Jpn. J. Appl. Phys. 43, 4929-4936 (2004).
    [CrossRef]
  8. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).
  9. M. Mansuripur, Classical Optics and its Applications (Cambridge, 2002).
  10. V. N. Mahajan, Aberration Theory Made Simple (SPIE, 1991).
    [CrossRef]
  11. F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976).
  12. V. N. Mahajan, Optical Imaging and Aberrations (SPIE, 1998).
    [CrossRef]
  13. J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85-88 (1964).
    [CrossRef]
  14. M. Y. Asoma, “Microscope objective,” U.S. patent 4, 505,553 (19 March 1985).
  15. M. Laikin, Lens Design (Marcel Dekker, 2001).

2005 (2)

2004 (1)

Y. Zhang, T. D. Milster, J. S. Kim, and S. K. Park, “Advanced lens design for bit-wise volumetric optical data storage,” Jpn. J. Appl. Phys. 43, 4929-4936 (2004).
[CrossRef]

2000 (1)

1999 (1)

T. D. Milster, R. S. Upton, and H. Luo, “Objective lens design for multiple-layer optical data storage,” Opt. Eng. 38, 295-301 (1999).
[CrossRef]

1994 (1)

1991 (1)

1964 (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85-88 (1964).
[CrossRef]

Asoma, M. Y.

M. Y. Asoma, “Microscope objective,” U.S. patent 4, 505,553 (19 March 1985).

Brain, K.

Gu, M.

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976).

Kim, J. S.

Y. Zhang, T. D. Milster, J. S. Kim, and S. K. Park, “Advanced lens design for bit-wise volumetric optical data storage,” Jpn. J. Appl. Phys. 43, 4929-4936 (2004).
[CrossRef]

Laikin, M.

M. Laikin, Lens Design (Marcel Dekker, 2001).

Luo, H.

T. D. Milster, R. S. Upton, and H. Luo, “Objective lens design for multiple-layer optical data storage,” Opt. Eng. 38, 295-301 (1999).
[CrossRef]

Mahajan, V. N.

V. N. Mahajan, Aberration Theory Made Simple (SPIE, 1991).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations (SPIE, 1998).
[CrossRef]

Mansuripur, M.

M. Mansuripur, Classical Optics and its Applications (Cambridge, 2002).

Milster, T. D.

Y. Zhang, T. D. Milster, J. S. Kim, and S. K. Park, “Advanced lens design for bit-wise volumetric optical data storage,” Jpn. J. Appl. Phys. 43, 4929-4936 (2004).
[CrossRef]

T. D. Milster, R. S. Upton, and H. Luo, “Objective lens design for multiple-layer optical data storage,” Opt. Eng. 38, 295-301 (1999).
[CrossRef]

Park, S. K.

Y. Zhang, T. D. Milster, J. S. Kim, and S. K. Park, “Advanced lens design for bit-wise volumetric optical data storage,” Jpn. J. Appl. Phys. 43, 4929-4936 (2004).
[CrossRef]

Rayces, J. L.

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85-88 (1964).
[CrossRef]

Sheppard, C. J. R.

Stallinga, S.

Upton, R. S.

T. D. Milster, R. S. Upton, and H. Luo, “Objective lens design for multiple-layer optical data storage,” Opt. Eng. 38, 295-301 (1999).
[CrossRef]

Welford, W. T.

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

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976).

Zhang, Y.

Y. Zhang, T. D. Milster, J. S. Kim, and S. K. Park, “Advanced lens design for bit-wise volumetric optical data storage,” Jpn. J. Appl. Phys. 43, 4929-4936 (2004).
[CrossRef]

Zhou, H.

Appl. Opt. (5)

Jpn. J. Appl. Phys. (1)

Y. Zhang, T. D. Milster, J. S. Kim, and S. K. Park, “Advanced lens design for bit-wise volumetric optical data storage,” Jpn. J. Appl. Phys. 43, 4929-4936 (2004).
[CrossRef]

Opt. Acta (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85-88 (1964).
[CrossRef]

Opt. Eng. (1)

T. D. Milster, R. S. Upton, and H. Luo, “Objective lens design for multiple-layer optical data storage,” Opt. Eng. 38, 295-301 (1999).
[CrossRef]

Other (7)

M. Y. Asoma, “Microscope objective,” U.S. patent 4, 505,553 (19 March 1985).

M. Laikin, Lens Design (Marcel Dekker, 2001).

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

M. Mansuripur, Classical Optics and its Applications (Cambridge, 2002).

V. N. Mahajan, Aberration Theory Made Simple (SPIE, 1991).
[CrossRef]

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976).

V. N. Mahajan, Optical Imaging and Aberrations (SPIE, 1998).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Rays of an optical system under the Abbe sine condition.

Fig. 2
Fig. 2

Rays in an off-axis system.

Fig. 3
Fig. 3

Induced aberration by the object position change.

Fig. 4
Fig. 4

On-axis system and its corresponding off-axis system.

Fig. 5
Fig. 5

Definition of the relationship between the transverse and the longitudinal spherical aberration.

Fig. 6
Fig. 6

(a) Exact wavefront aberration and its fourth-order approximation along the position change of a point object. (b) Normalized wavefront error ratio of ( W W 040 ) / W according to the object position for some optical systems with different NA values and the same focal length.

Fig. 7
Fig. 7

(a) Change in the maximum value of h according to the object position change when the entrance pupil is located some distance from the principal plane. (b) Ray tracing of marginal ray regarding pupils and principal planes.

Fig. 8
Fig. 8

Arising SA caused by the change of the object position for microscope objective lenses: aberration for (a) the infinity- corrected system and (b) the system corrected at the object position of 180 mm . The solid curve shows the data obtained from a commercial simulation program. The numerical results are represented by the dotted curve.

Fig. 9
Fig. 9

Arising SA caused by the change of the object position for a single lens with aspheric surfaces: aberration for (a) a system with NA = 0.85 and a small field size and (b) a system with NA = 0.15 and a large field size.

Fig. 10
Fig. 10

Arising SA caused by the change of the object position for the Petzval-type lens: aberration for (a) a system with NA = 0.25 and imperfectly corrected for a large field and (b) a system with NA = 0.4 and well corrected for a relatively small field.

Equations (23)

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n sin α 0 = M n sin α 0 ,
s 0 sin α 0 = s 0 sin α 0 h ,
sin α 0 = h s 0 ,
sin α 0 = h s 0 .
W = [ P 1 E 1 E 1 P 1 ] [ P 1 C C P 1 ] { [ P 0 E 1 E 1 P 0 ] [ P 0 V ] + [ P 0 V ] } { [ P 0 C C P 0 ] [ P 0 P 1 ] + [ P 0 P 1 ] } = { [ P 0 P 1 ] [ P 0 V ] } + { [ P 0 V ] [ P 0 P 1 ] } n Δ z ( 1 cos θ ) n Δ z ( 1 cos θ ) ,
α 0 + β = α 1 .
α 0 β = α 1 ,
sin α 1 = s 0 sin α 0 s 1 = h s 1 ,
sin α 1 = s 0 sin α 0 s 1 = h s 1 .
n α 1 + n α 1 = n α 0 + n α 0 .
α 1 + α 1 = α 0 + α 0 .
n · sin 1 ( h s 1 ) + n · sin 1 ( h s 1 ) = n · sin 1 ( h s 0 ) + n · sin 1 ( h s 0 ) .
LSA = s 1 s 1 p = h · { sin [ n n · sin 1 ( h s 0 ) + sin 1 ( h s 0 ) n n · sin 1 ( h s 1 ) ] } 1 ( n n · 1 s 0 + 1 s 0 n n · 1 s 1 ) 1 ,
W h n · s 1 p s 1 s 1 p tan α 1 .
tan α 1 = tan [ sin 1 ( h s 1 ) ] = h s 1 + h 3 2 s 1 3 + O ( h 5 ) ,
sin 1 x = x + 1 6 x 3 + O ( x 5 ) tan x = x + 1 3 x 3 + O ( x 5 ) .
1 s 1 = 1 h sin [ n n · sin 1 ( h s 0 ) + sin 1 ( h s 0 ) n n · sin 1 ( h s 1 ) ] = 1 h sin [ n n · ( h s 0 + h 3 6 s 0 3 ) + ( h s 0 + h 3 6 s 0 3 ) n n · ( h s 1 + h 3 6 s 1 3 ) + O ( h 5 ) ] = 1 s 1 p + h 2 6 ( n n · 1 s 0 3 + 1 s 0 3 n n · 1 s 1 3 1 s 1 p 3 ) + O ( h 4 ) .
sin x = x 1 3 ! x 3 + O ( x 5 ) .
W h n · s 1 p s 1 s 1 p tan α 1 = n · ( 1 s 1 1 s 1 p ) · [ h + h 3 2 s 1 2 + O ( h 5 ) ] = n · [ h 2 6 ( n n · 1 s 0 3 + 1 s 0 3 n n · 1 s 1 3 1 s 1 p 3 ) + O ( h 5 ) ] · [ h + h 3 2 s 1 2 + O ( h 5 ) ] = h 3 6 ( n s 0 3 + n s 0 3 n s 1 3 n s 1 p 3 ) + O ( h 5 ) .
W 040 = h 4 24 ( n s 0 3 + n s 0 3 n s 1 3 n s 1 p 3 ) .
p = s 1 p s 1 s 1 p + s 1 .
p 0 = s 0 s 0 s 0 + s 0 .
W 040 = h 4 24 { [ ( 1 s 0 + 1 s 0 ) 3 3 ( 1 s 0 + 1 s 0 ) 1 s 0 s 0 ] [ ( 1 s 1 + 1 s 1 p ) 3 3 ( 1 s 1 + 1 s 1 p ) 1 s 1 s 1 p ] } = h 4 8 f 3 [ s 0 s 0 ( s 0 + s 0 ) 2 s 1 s 1 p ( s 1 + s 1 p ) 2 ] = h 4 32 f 3 ( p 0 2 p 2 ) ,

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