Abstract

We propose what we believe to be a novel, refined model of the angular sensitivity function of artificial apposition compound eyes. Compared with the formerly used Gaussian approximation that was derived for natural compound eyes, our model is better suited to describe the resolution capacity of artificial compound eyes accounting for the cylindrical sensitivity function of technical receptors. It is shown that this analytic model is valid over a broad range of parameters of the optical system, which was not fulfilled by one of the previous models. Finally, an analytic approach is used to derive the modulation transfer function of these multichannel imaging systems.

© 2007 Optical Society of America

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References

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  1. R. Völkel, M. Eisner, and K. J. Weible, Microelectron. Eng. 67-68, 461 (2003).
    [CrossRef]
  2. J. Duparré, P. Dannberg, P. Schreiber, A. Bräuer, and A. Tünnermann, Appl. Opt. 43, 4303 (2004).
    [CrossRef] [PubMed]
  3. J. Kim, K.-H. Jeong, and L. P. Lee, Opt. Lett. 30, 5 (2005).
    [CrossRef] [PubMed]
  4. K. G. Götz, Biol. Cybern. 2, 215 (1965).
  5. A. W. Snyder, J. Comp. Physiol., A 116, 161 (1977).
    [CrossRef]
  6. J. S. Sanders and C. E. Halford, Opt. Eng. 34, 222 (1995).
    [CrossRef]
  7. S. Viollet and N. Franceschini, Proc. SPIE 3839, 144 (1999).
    [CrossRef]
  8. T. Wilson and A. R. Carlini, Opt. Lett. 12, 227 (1987).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

2005 (1)

2004 (1)

2003 (1)

R. Völkel, M. Eisner, and K. J. Weible, Microelectron. Eng. 67-68, 461 (2003).
[CrossRef]

1999 (1)

S. Viollet and N. Franceschini, Proc. SPIE 3839, 144 (1999).
[CrossRef]

1995 (1)

J. S. Sanders and C. E. Halford, Opt. Eng. 34, 222 (1995).
[CrossRef]

1987 (1)

1977 (1)

A. W. Snyder, J. Comp. Physiol., A 116, 161 (1977).
[CrossRef]

1965 (1)

K. G. Götz, Biol. Cybern. 2, 215 (1965).

Bräuer, A.

Carlini, A. R.

Dannberg, P.

Duparré, J.

Eisner, M.

R. Völkel, M. Eisner, and K. J. Weible, Microelectron. Eng. 67-68, 461 (2003).
[CrossRef]

Franceschini, N.

S. Viollet and N. Franceschini, Proc. SPIE 3839, 144 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Götz, K. G.

K. G. Götz, Biol. Cybern. 2, 215 (1965).

Halford, C. E.

J. S. Sanders and C. E. Halford, Opt. Eng. 34, 222 (1995).
[CrossRef]

Jeong, K.-H.

Kim, J.

Lee, L. P.

Sanders, J. S.

J. S. Sanders and C. E. Halford, Opt. Eng. 34, 222 (1995).
[CrossRef]

Schreiber, P.

Snyder, A. W.

A. W. Snyder, J. Comp. Physiol., A 116, 161 (1977).
[CrossRef]

Tünnermann, A.

Viollet, S.

S. Viollet and N. Franceschini, Proc. SPIE 3839, 144 (1999).
[CrossRef]

Völkel, R.

R. Völkel, M. Eisner, and K. J. Weible, Microelectron. Eng. 67-68, 461 (2003).
[CrossRef]

Weible, K. J.

R. Völkel, M. Eisner, and K. J. Weible, Microelectron. Eng. 67-68, 461 (2003).
[CrossRef]

Wilson, T.

Appl. Opt. (1)

Biol. Cybern. (1)

K. G. Götz, Biol. Cybern. 2, 215 (1965).

J. Comp. Physiol., A (1)

A. W. Snyder, J. Comp. Physiol., A 116, 161 (1977).
[CrossRef]

Microelectron. Eng. (1)

R. Völkel, M. Eisner, and K. J. Weible, Microelectron. Eng. 67-68, 461 (2003).
[CrossRef]

Opt. Eng. (1)

J. S. Sanders and C. E. Halford, Opt. Eng. 34, 222 (1995).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

S. Viollet and N. Franceschini, Proc. SPIE 3839, 144 (1999).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (5)

Fig. 1
Fig. 1

Schematic section of an artificial apposition compound eye. It consists of a microlens array (MLA) on top of a glass substrate with a pinhole array of different pitch in the MLA’s focal plane. The optical system is attached to an image sensor with matching pixel pitch.

Fig. 2
Fig. 2

Geometrical layout of one optical channel consisting of a microlens (ML) and a pinhole with a diameter d in the image plane. Drawing not to scale.

Fig. 3
Fig. 3

Analytic ASF compared with the Gaussian approximation, diffraction limited PSF of the microlens, and a numerical simulation for different pinhole diameters.

Fig. 4
Fig. 4

Comparison between a measured ASF of one channel and the analytic model.

Fig. 5
Fig. 5

Comparison between the analytic, the Gaussian and the numerically calculated MTF for different pinhole diameters. Angular frequencies are normalized to the cutoff frequency of the diffraction limited lens ν S = D λ .

Equations (10)

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I ( x , y ) = ( 1 M ) R ( x M ξ , y M η ) O ( ξ , η ) .
R ( x , y ) = PSF L ( x u , y v ) A ( u , v ) ,
ASF ( ϕ ) exp [ 4 ln 2 ( ϕ Δ φ ) 2 ] ,
Δ φ = ( λ D ) 2 + ( d f ) 2 .
R ( r ) = c 1 PSF L ( r ρ ) A ( ρ ) d ρ = c 2 [ 2 J 1 ( π f 0 ( r ρ ) ) π f 0 ( r ρ ) ] 2 circ ( ρ ) d ρ = c 2 d 2 d 2 [ 2 J 1 ( π f 0 ( r ρ ) ) π f 0 ( r ρ ) ] 2 d ρ .
R a ( r ) = [ c ( π f 0 2 ( 4 r 2 d 2 ) ) ] [ 8 π 2 f 0 2 r 3 ( G 0 2 G 0 2 + G 1 2 G 1 2 ) + 4 π 2 f 0 2 d r 2 ( G 0 2 + G 1 2 + G 0 2 + G 1 2 ) 8 π f 0 r 2 ( G 0 G 1 + G 0 G 1 ) + 2 π 2 f 0 2 d 2 r ( G 0 2 G 0 2 + G 1 2 G 1 2 ) π 2 f 0 2 d 3 ( G 0 2 + G 0 2 + G 1 2 + G 1 2 ) + 2 π f 0 d 2 ( G 0 G 1 + G 0 G 1 ) + 2 d ( G 1 2 + G 1 2 ) + 4 r ( G 1 2 G 1 2 ) ] .
G ± 0 , ± 1 ( r ) = J 0 , 1 [ π f 0 ( ± 2 r + d ) 2 ] .
ASF a ( ϕ ) = R a ( r = f tan ϕ ) R 0 .
MTF a = 4 f π 2 d ν [ arccos ( ν ν S ) ν ν S 1 ( ν ν S ) 2 ] × J 1 ( π d ν f ) .
ν C o = { D λ ν S ν B 1.220 f d ν S > ν B .

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