Abstract

We describe the mapping of the optical transfer function (OTF) of an incoherent imaging system into a geometrical representation. We show that for defocused traditional and wavefront-coded systems the OTF can be represented as a generalized Cornu spiral. This representation provides a physical insight into the way in which wavefront coding can increase the depth of field of an imaging system and permits analytical quantification of salient OTF parameters, such as the depth of focus, the location of nulls, and amplitude and phase modulation of the wavefront-coding OTF.

© 2005 Optical Society of America

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References

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  1. S. Mezouari and A. R. Harvey, J. Opt. A Pure Appl. Opt. 5, S86 (2003).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 548.
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  7. J. M. Ali, R. M. Tookey, J. V. Ball, and A. A. Ball, J. Comput. Appl. Math. 102, 37 (1999).
    [CrossRef]

2003

S. Mezouari and A. R. Harvey, J. Opt. A Pure Appl. Opt. 5, S86 (2003).
[CrossRef]

S. Mezouari and A. R. Harvey, Opt. Lett. 28, 771 (2003).
[CrossRef] [PubMed]

2002

1999

J. M. Ali, R. M. Tookey, J. V. Ball, and A. A. Ball, J. Comput. Appl. Math. 102, 37 (1999).
[CrossRef]

1995

Ali, J. M.

J. M. Ali, R. M. Tookey, J. V. Ball, and A. A. Ball, J. Comput. Appl. Math. 102, 37 (1999).
[CrossRef]

Ball, A. A.

J. M. Ali, R. M. Tookey, J. V. Ball, and A. A. Ball, J. Comput. Appl. Math. 102, 37 (1999).
[CrossRef]

Ball, J. V.

J. M. Ali, R. M. Tookey, J. V. Ball, and A. A. Ball, J. Comput. Appl. Math. 102, 37 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 548.

Cathey, T. W.

Cathey, W. T.

Dowski, E.

Dowski, E. R.

Goodman, J. W.

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

Harvey, A. R.

S. Mezouari and A. R. Harvey, Opt. Lett. 28, 771 (2003).
[CrossRef] [PubMed]

S. Mezouari and A. R. Harvey, J. Opt. A Pure Appl. Opt. 5, S86 (2003).
[CrossRef]

Mezouari, S.

S. Mezouari and A. R. Harvey, J. Opt. A Pure Appl. Opt. 5, S86 (2003).
[CrossRef]

S. Mezouari and A. R. Harvey, Opt. Lett. 28, 771 (2003).
[CrossRef] [PubMed]

Tookey, R. M.

J. M. Ali, R. M. Tookey, J. V. Ball, and A. A. Ball, J. Comput. Appl. Math. 102, 37 (1999).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 548.

Appl. Opt.

J. Comput. Appl. Math.

J. M. Ali, R. M. Tookey, J. V. Ball, and A. A. Ball, J. Comput. Appl. Math. 102, 37 (1999).
[CrossRef]

J. Opt. A Pure Appl. Opt.

S. Mezouari and A. R. Harvey, J. Opt. A Pure Appl. Opt. 5, S86 (2003).
[CrossRef]

Opt. Lett.

Other

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 548.

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

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Figures (1)

Fig. 1
Fig. 1

(a)–(e) Traditional defocused OTFs and MTFs depicted as arc circles for α = 0 and v = 0.5 . (f)–(j) WC OTFs and MTFs depicted as GCSs for α = 2 and v = 0.5 . Note the absence of both nulls and phase reversal for the WC MTFs (solid curves), in contrast to the defocused OTFs (dashed curves).

Equations (11)

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L ( f ) = P ( x + R f f max ) P * ( x R f f max ) d x P ( x ) 2 d x ,
H ( r , v ) = 0 r h ( r , v ) d r 1 1 h ( r , 0 ) d r ,
h ( r , v ; w 20 ) = exp ( i 8 π w 20 v r ) ,
P ( r , v ; w 20 , α ) = p ( r ) exp [ i 2 π ( w 20 r 2 + α r 3 ) ] ,
h ( r , v ; w 20 , α ) = exp [ i 4 π v ( 3 α r 2 + 2 w 20 r + α v 2 ) ] .
H ( r , v ; w 20 , α ) = 1 2 0 r exp [ i 4 π v ( 3 α r 2 + 2 w 20 r + α v 2 ) ] d r .
L ( v ; w 20 , α ) ( 1 + i ) 4 6 α v exp [ i 4 π v ( α v 2 + w 20 2 3 α ) ] .
κ ( r , v ; w 20 , α ) = 16 π v ( w 20 + 3 α r ) ,
w 20 max = 3 α ( 1 v ) .
M ( v ; w 20 , α ) = 1 16 π v ( cos { ( 4 π v 3 α ) [ w 20 3 α ( 1 v ) ] 2 + ( π 4 ) } w 20 3 α ( 1 v ) cos { ( 4 π v 3 α ) [ w 20 + 3 α ( 1 v ) ] 2 + ( π 4 ) } w 20 + 3 α ( 1 ν ) ) ,
Δ θ = 3 α v 2 π v [ w 20 3 α ( 1 v ) ] sin { 4 π v 3 α [ w 20 3 α ( 1 v ) ] 2 + π 4 } .

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