Abstract

We conjecture that the lateral shift invariance of an imaging system must be limited if axial imaging capability is desired. We develop shift-invariance and depth-resolution metrics and demonstrate the trade-off in simple representative systems.

© 2002 Optical Society of America

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References

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  1. M. Minsky, “Microscopy apparatus,” U.S. patent3,013,467 (19December1961).
  2. M. Minsky, Confocal Microscopy, J. K. Stevens, L. R. Mills, J. E. Trogadis, eds. (Academic, San Diego, Calif., 1990).
  3. M. Minsky, Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Systems (Academic, San Diego, Calif., 1994).
  4. M. Levene, G. J. Steckman, D. Psaltis, “Method for controlling the shift invariance of optical correlators,” Appl. Opt. 38, 394–398 (1999).
    [CrossRef]
  5. G. Barbastathis, M. Balberg, D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett. 24, 811–813 (1999).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, New York, 1998).
  7. W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940–947 (1947).
    [CrossRef]

1999

1947

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940–947 (1947).
[CrossRef]

Balberg, M.

Barbastathis, G.

Bates, W. J.

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940–947 (1947).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, New York, 1998).

Brady, D. J.

Levene, M.

Minsky, M.

M. Minsky, Confocal Microscopy, J. K. Stevens, L. R. Mills, J. E. Trogadis, eds. (Academic, San Diego, Calif., 1990).

M. Minsky, Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Systems (Academic, San Diego, Calif., 1994).

M. Minsky, “Microscopy apparatus,” U.S. patent3,013,467 (19December1961).

Psaltis, D.

Steckman, G. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, New York, 1998).

Appl. Opt.

Opt. Lett.

Proc. Phys. Soc. London

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940–947 (1947).
[CrossRef]

Other

M. Minsky, “Microscopy apparatus,” U.S. patent3,013,467 (19December1961).

M. Minsky, Confocal Microscopy, J. K. Stevens, L. R. Mills, J. E. Trogadis, eds. (Academic, San Diego, Calif., 1990).

M. Minsky, Three-Dimensional Confocal Microscopy: Volume Investigation of Biological Systems (Academic, San Diego, Calif., 1994).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, New York, 1998).

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Figures (13)

Fig. 1
Fig. 1

Confocal microscope.

Fig. 2
Fig. 2

Shift-invariant versus shift-variant systems.

Fig. 3
Fig. 3

Demonstration of the response of a pinhole camera as a point object is shifted in the lateral direction. Top, impulse response (H 0). Middle, response due to the shifted object (H 1). Bottom, sum of the shifted impulse response and H 1. Note that since there is no overlap between the two images, S is just the sum of the two individual images.

Fig. 4
Fig. 4

Shift invariance for a confocal microscope. r is the radial coordinate normalized to the numerical aperture of the system, i.e., r = (v 2 + w 2)1/2.

Fig. 5
Fig. 5

Confocal microscope trade-off. (a) Shift invariance versus (b) depth resolution. Δr and Δz are normalized to the numerical aperture of the system.

Fig. 6
Fig. 6

Binocular system.

Fig. 7
Fig. 7

How the PSF shifts in the binocular system.

Fig. 8
Fig. 8

Shift invariance for a binocular system.

Fig. 9
Fig. 9

Binocular system trade-off. The x axis is the angle (θ) between the two optical axes of the cameras. Top, shift invariance; bottom two, depth resolution that was calculated with two different methods: triangulation and PSF multiplication.

Fig. 10
Fig. 10

PSF multiplication.

Fig. 11
Fig. 11

Shear interferometer.

Fig. 12
Fig. 12

Ambiguity of the shear interferometer. In both pictures, the solid disk represents the actual position of the point source and the unfilled disks show other possible object locations that would have yielded identical patterns. A thin slab gives a low-frequency interference pattern with poor depth resolution but has low lateral ambiguity, whereas a thick slab gives a high-frequency interference pattern but has high lateral ambiguity.

Fig. 13
Fig. 13

Symbol definitions for using the lens law.

Equations (19)

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S x ,   y =   - + h x ,   y ;   x ,   y - h 0 ,   0 ;   x - x ,   y - y 2 d x d y   - +   | h 0 ,   0 ;   x - x ,   y - y | 2 d x d y .
  - +   | h x ,   y ;   x ,   y | 2 d x d y   - +   | h 0 ,   0 ;   x ,   y | 2 d x d y ,
lim x , y   - +   | h x ,   y ;   x ,   y | 2 d x d y = 0 ,
h v ,   v ,   w ,   w = 2 J 1 v - v 2 + w - w 2 1 / 2 v - v 2 + w - w 2 1 / 2 4 circ r a n ; v = π λ NA x , v = π λ NA x , w = π λ NA y , w = π λ NA y , r = v 2 + w 2 1 / 2 , r = v 2 + w 2 1 / 2 , a n = π λ NA a .
1 / Δ z = M   tan θ / 2 / p .
I x ,   z = 1 + cos π b 2 x - x 0 - h λ z ,
M = - s 0 s 1 ,
1 s 0 + 1 s 1 = 1 f .
1 s 0 + 1 s 1 = 1 f ,
s 0 = s 0 - δ z 1 ,
s 1 = s 1 + ,
M = - s 0 s 1 .
1 s 0 - δ z 1 s 0 2 + 1 s 1 + s 1 2 = 1 f .
= s 0 2 / s 1 2 δ z 1 .
M = - s 0 - δ z 1 s 1 + M 2 δ z 1 .
M = - s 0 s 1 1 - δ z 1 s 0 - s 0 2 / s 1 2 δ z 1 s 1 .
A = - 1 s 0 + M 2 s 1 ,
Δ M = M - M = A δ z 1 .
h x ,   x = rect M x W   cos θ / 2 - Δ M x .

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