Abstract

In confocal microscopy, spherical aberration is introduced when one is focusing deep within the specimen. This can be compensated for by altering the effective tube length at which the objective is operated. The limitations of this approach are investigated.

© 1991 Optical Society of America

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References

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  1. C. J. R. Sheppard, C. J. Cogswell, “3-D image formation in confocal microscopy,” J. Microsc. (Oxford) 159, 179–194 (1990).
    [CrossRef]
  2. C. J. R. Sheppard, “Aberrations in high aperture conventional and confocal imaging systems,” Appl. Opt. 27, 4782–4786 (1986).
    [CrossRef]
  3. C. J. R. Sheppard, C. J. Cogswell, “Effects of aberrating layers and tube length on confocal imaging properties,” Optik (Stuttgart) 87, 34–38 (1991).
  4. C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the sacnning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
    [CrossRef]
  5. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), p. 91.
  6. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), p. 467.

1991 (1)

C. J. R. Sheppard, C. J. Cogswell, “Effects of aberrating layers and tube length on confocal imaging properties,” Optik (Stuttgart) 87, 34–38 (1991).

1990 (1)

C. J. R. Sheppard, C. J. Cogswell, “3-D image formation in confocal microscopy,” J. Microsc. (Oxford) 159, 179–194 (1990).
[CrossRef]

1986 (1)

1981 (1)

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the sacnning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), p. 467.

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, “Effects of aberrating layers and tube length on confocal imaging properties,” Optik (Stuttgart) 87, 34–38 (1991).

C. J. R. Sheppard, C. J. Cogswell, “3-D image formation in confocal microscopy,” J. Microsc. (Oxford) 159, 179–194 (1990).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, C. J. Cogswell, “Effects of aberrating layers and tube length on confocal imaging properties,” Optik (Stuttgart) 87, 34–38 (1991).

C. J. R. Sheppard, C. J. Cogswell, “3-D image formation in confocal microscopy,” J. Microsc. (Oxford) 159, 179–194 (1990).
[CrossRef]

C. J. R. Sheppard, “Aberrations in high aperture conventional and confocal imaging systems,” Appl. Opt. 27, 4782–4786 (1986).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the sacnning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), p. 91.

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the sacnning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), p. 467.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the sacnning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

J. Microsc. (Oxford) (1)

C. J. R. Sheppard, C. J. Cogswell, “3-D image formation in confocal microscopy,” J. Microsc. (Oxford) 159, 179–194 (1990).
[CrossRef]

Optik (Stuttgart) (1)

C. J. R. Sheppard, C. J. Cogswell, “Effects of aberrating layers and tube length on confocal imaging properties,” Optik (Stuttgart) 87, 34–38 (1991).

Other (2)

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), p. 91.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), p. 467.

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Figures (10)

Fig. 1
Fig. 1

Geometry of a ray passing through a slab of dielectric.

Fig. 2
Fig. 2

Geometry of focusing by a lens.

Fig. 3
Fig. 3

Axial response intensity for focusing through a slab of dielectric for (a) various values of the coefficient A and (b) a system used at an incorrect tube length specified by the coefficient B. Numerical aperture 0.866.

Fig. 4
Fig. 4

Effect of the coefficient A on 1, the Strehl intensity; 2, the width of the response at half of the maximum height (half-width) measured in units of kz; 3, the strength of the first subsidiary maximum relative to the peak intensity; 4, the strength of the first minimum relative to the peak intensity.

Fig. 5
Fig. 5

Effect of the coefficient B on 1, the Strehl intensity; 2, the width of the response at half of the maximum height, (half-width) measured in units of kz; 3, the strength of the first subsidiary maximum relative to the peak intensity; 4, the strength of the first minimum relative to the peak intensity.

Fig. 6
Fig. 6

Axial response intensity for primary spherical aberration of strength given by the coefficient C: (a) for a system satisfying the sine condition with numerical aperture 0.866, (b) for a system of uniform angular illumination.

Fig. 7
Fig. 7

Axial response intensity for various values of coefficient A for a fixed value of B = 20.

Fig. 8
Fig. 8

Effect of the coefficient A, for B = 20, on 1, the Strehl intensity; 2, the width of the response at half of the maximum height (half-width) measured in units of kz; 3, the strength of the first subsidiary maximum relative to the peak intensity; 4, the strength of the first minimum relative to the peak intensity.

Fig. 9
Fig. 9

Axial response intensity for pairs of values of the coefficients A and B that approximately satisfy the condition for optimum performance.

Fig. 10
Fig. 10

Effect of the coefficient B, together with a coefficient A ≈ −3.8B, on 1, the Strehl intensity; 2, the width of the response at half of the maximum height (half-width) measured in units of kz; 3, the strength of the first subsidiary maximum relative to the peak intensity.

Equations (27)

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ϕ = kt ( n 2 cos θ 2 n cos θ ) ,
n 2 = n + Δ n ,
n sin θ = n 2 sin θ 2 ,
cos θ 2 = cos θ ( 1 + Δ n n tan 2 θ ) .
ϕ = kt Δ n sec θ
ϕ = A sec θ ,
A = kt Δ n .
ϕ = k ( l l 2 + h 2 + nd n d 2 + h 2 ) .
Δ ϕ = k [ l 2 Δ ( 1 l ) l 3 h 2 + l 2 Δ ( 1 l ) n Δ d + nd Δ d d 2 + h 2 ] .
h = d tan θ ,
l = Md ,
ϕ = k [ ( M 2 d 2 M 3 d 3 h 2 + M 2 d 2 ) Δ 1 l n Δ d ( 1 cos θ ) ] .
ϕ = 1 2 kd 2 Δ ( 1 l ) tan 2 θ + k n Δ d ( 1 cos θ ) .
ϕ = B tan 2 θ ,
B = 1 / 2 k d 2 Δ ( I / l ) .
I ( z ) = | 0 α exp [ 2 i ( ϕ + kz cos θ ) ] sin θ cos θ d θ | 2 .
I ( z ) = | cos α 1 exp [ 2 i ( A c + kzc ) ] c d c | 2 ,
I ( z ) = | cos α 1 exp [ 2 i B ( 1 c 2 1 ) + kzc ] c d c | 2 .
I ( z ) = | cos α 1 exp [ 2 i ( C c 2 + kzc ) ] cdc | 2 .
I ( z ) = | cos α 1 exp [ 2 i ( C c 2 + kzc ) ] d c | 2 ,
A < kz < A sec 2 α
2 B < kz < 2 B sec 3 α ,
2 C cos α < kz < 2 C .
sec θ = 1 + 2 s 2 + 4 s 4 + 8 s 6 + + 2 n s 2 n +
tan 2 θ = 3 ( 4 s 2 / 3 + 4 s 4 + 32 s 6 / 3 + 2 n ( n + 1 ) s 2 n / 3 + ) ,
s = sin ( θ / 2 ) .
8 A + 32 B 20 = 4 A + 12 B 30 = 2 A + 4 B 2 kz 12 ,

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