Abstract

In the usual model of an imaging system, only the effects of the aperture stop are considered in determining diffraction-limited system performance. In fact, diffraction at other stops—those associated with different lens elements, for example—can also affect system performance and cause the imaging to be space variant, even in the absence of vignetting in the conventional ray optics sense. For the 4-f imaging system investigated in this paper, the severity of the space variance depends on the relative sizes of the two lens stops and the aperture stops. If the diameters of the lenses are equal, the aperture of the first lens has a greater effect on system performance than does that of the second.

© 2007 Optical Society of America

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References

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  1. E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, 1962).
  2. R. N. Bracewell, The Fourier Transform and Its Applications, Int. ed. (McGraw-Hill, 1986).
  3. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).
  4. E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1989).
  5. W. T. Rhodes, Lecture Series: Fourier Optics and Holography (Imaging Technology Center, Florida Atlantic University, 2006).
  6. R. W. Dichtburn, Light (Dover, 1965).
  7. M. Gu, Advanced Optical Imaging Theory, 1st ed. (Springer-Verlag, 2000).
  8. T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).
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    [CrossRef]
  10. D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis," Opt. Commun. 263, 171-179 (2006).
    [CrossRef]
  11. D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part II: 3-D system analysis," Opt. Commun. 263, 180-188 (2006).
    [CrossRef]
  12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  13. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).
  14. S. Wolf, The Mathematica Book, 4th ed. (Cambridge U. Press, 1999).
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    [CrossRef]
  16. G. W. Forbes, "Validity of the Fresnel approximation in the diffraction of collimated beams," J. Opt. Soc. Am. A 13, 1816-1826 (1996).
    [CrossRef]
  17. D. P. Kelly, "Linear quadratic phase systems and their application to speckle photography based metrology systems," Ph.D. dissertation (National University College Dublin, Ireland, 2006).
  18. P. S. Considine, "Effects of coherence on imaging systems," J. Opt. Soc. Am. 20, 661-667 (1966).
  19. K. Dutton, S. Thompson, and B. Barraclough, The Art of Control Engineering (Addison-Wesley, 1997).
  20. D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. 45, 088201 (2006).
    [CrossRef]
  21. E. L. Shirley, "Intuitive diffraction model for multistaged optical systems," Appl. Opt. 43, 735-743 (2004).
    [CrossRef] [PubMed]

2006 (5)

W. T. Rhodes, Lecture Series: Fourier Optics and Holography (Imaging Technology Center, Florida Atlantic University, 2006).

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis," Opt. Commun. 263, 171-179 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part II: 3-D system analysis," Opt. Commun. 263, 180-188 (2006).
[CrossRef]

D. P. Kelly, "Linear quadratic phase systems and their application to speckle photography based metrology systems," Ph.D. dissertation (National University College Dublin, Ireland, 2006).

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. 45, 088201 (2006).
[CrossRef]

2005 (1)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

2004 (1)

2000 (1)

M. Gu, Advanced Optical Imaging Theory, 1st ed. (Springer-Verlag, 2000).

1999 (1)

S. Wolf, The Mathematica Book, 4th ed. (Cambridge U. Press, 1999).

1997 (1)

K. Dutton, S. Thompson, and B. Barraclough, The Art of Control Engineering (Addison-Wesley, 1997).

1996 (1)

1994 (2)

G. W. Forbes, "Scaling properties in the diffraction of focused waves and an application to scanning beams," Am. J. Phys. 62, 434-443 (1994).
[CrossRef]

J. T. Sheridan and C. J. R. Sheppard, "Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory," Opt. Commun. 105, 367-378 (1994).
[CrossRef]

1989 (1)

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1989).

1986 (1)

R. N. Bracewell, The Fourier Transform and Its Applications, Int. ed. (McGraw-Hill, 1986).

1984 (1)

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

1980 (1)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

1970 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

1966 (1)

1965 (1)

R. W. Dichtburn, Light (Dover, 1965).

1962 (1)

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, 1962).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Barraclough, B.

K. Dutton, S. Thompson, and B. Barraclough, The Art of Control Engineering (Addison-Wesley, 1997).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, Int. ed. (McGraw-Hill, 1986).

Considine, P. S.

Dichtburn, R. W.

R. W. Dichtburn, Light (Dover, 1965).

Dutton, K.

K. Dutton, S. Thompson, and B. Barraclough, The Art of Control Engineering (Addison-Wesley, 1997).

Forbes, G. W.

G. W. Forbes, "Validity of the Fresnel approximation in the diffraction of collimated beams," J. Opt. Soc. Am. A 13, 1816-1826 (1996).
[CrossRef]

G. W. Forbes, "Scaling properties in the diffraction of focused waves and an application to scanning beams," Am. J. Phys. 62, 434-443 (1994).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Gu, M.

M. Gu, Advanced Optical Imaging Theory, 1st ed. (Springer-Verlag, 2000).

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1989).

Hennelly, B. M.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. 45, 088201 (2006).
[CrossRef]

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. 45, 088201 (2006).
[CrossRef]

D. P. Kelly, "Linear quadratic phase systems and their application to speckle photography based metrology systems," Ph.D. dissertation (National University College Dublin, Ireland, 2006).

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part II: 3-D system analysis," Opt. Commun. 263, 180-188 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis," Opt. Commun. 263, 171-179 (2006).
[CrossRef]

O'Neill, E. L.

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, 1962).

Rhodes, W. T.

W. T. Rhodes, Lecture Series: Fourier Optics and Holography (Imaging Technology Center, Florida Atlantic University, 2006).

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis," Opt. Commun. 263, 171-179 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part II: 3-D system analysis," Opt. Commun. 263, 180-188 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. 45, 088201 (2006).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Sheppard, C.

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

Sheppard, C. J. R.

J. T. Sheridan and C. J. R. Sheppard, "Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory," Opt. Commun. 105, 367-378 (1994).
[CrossRef]

Sheridan, J. T.

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis," Opt. Commun. 263, 171-179 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part II: 3-D system analysis," Opt. Commun. 263, 180-188 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. 45, 088201 (2006).
[CrossRef]

J. T. Sheridan and C. J. R. Sheppard, "Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory," Opt. Commun. 105, 367-378 (1994).
[CrossRef]

Shirley, E. L.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Thompson, S.

K. Dutton, S. Thompson, and B. Barraclough, The Art of Control Engineering (Addison-Wesley, 1997).

Wilson, T.

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

Wolf, S.

S. Wolf, The Mathematica Book, 4th ed. (Cambridge U. Press, 1999).

Am. J. Phys. (1)

G. W. Forbes, "Scaling properties in the diffraction of focused waves and an application to scanning beams," Am. J. Phys. 62, 434-443 (1994).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (3)

J. T. Sheridan and C. J. R. Sheppard, "Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory," Opt. Commun. 105, 367-378 (1994).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis," Opt. Commun. 263, 171-179 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, "Finite-aperture effects for Fourier transform systems with convergent illumination. Part II: 3-D system analysis," Opt. Commun. 263, 180-188 (2006).
[CrossRef]

Opt. Eng. (1)

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, "Analytical and numerical analysis of linear optical systems," Opt. Eng. 45, 088201 (2006).
[CrossRef]

Other (13)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

S. Wolf, The Mathematica Book, 4th ed. (Cambridge U. Press, 1999).

D. P. Kelly, "Linear quadratic phase systems and their application to speckle photography based metrology systems," Ph.D. dissertation (National University College Dublin, Ireland, 2006).

K. Dutton, S. Thompson, and B. Barraclough, The Art of Control Engineering (Addison-Wesley, 1997).

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, 1962).

R. N. Bracewell, The Fourier Transform and Its Applications, Int. ed. (McGraw-Hill, 1986).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1989).

W. T. Rhodes, Lecture Series: Fourier Optics and Holography (Imaging Technology Center, Florida Atlantic University, 2006).

R. W. Dichtburn, Light (Dover, 1965).

M. Gu, Advanced Optical Imaging Theory, 1st ed. (Springer-Verlag, 2000).

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

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Figures (11)

Fig. 1
Fig. 1

4 - f imaging system. OP, object plane; FP, Fourier (aperture) plane; IP, image plane; MR, marginal ray; LT, light tube; BR: α, bundle of rays (defined by cone angle α); OFT, optical transfer function.

Fig. 2
Fig. 2

Example of vignetting in the first OFT module of a 4 - f imaging system for off-axis PS VL and even farther off-axis point PS Q . OP, object plane; FP, Fourier (aperture) plane; LA, lens aperture; LT, light tube; BR:α, bundle of rays (defined by cone angle α).

Fig. 3
Fig. 3

Magnitude distribution of F ( x ap , K , ξ ) for K = 15.7 and ξ = 0.5 .

Fig. 4
Fig. 4

Magnitude deviations of F ( x ap , K , ξ ) with ξ = 0 . The deviations range between a minimum of 0.5 and a maximum of 1.2 and are marked with ten contour levels.

Fig. 5
Fig. 5

Intensity of image distribution for a N = 0.25 , Ω a = 0.0039 , and P a = 0.050 . LSI PSF is also plotted (see dashed curve). (b) Intensity of image distribution at the vignetting limit: ξ = 0.75 , a N = 0.25 , Ω b = 0.0357 , and P b = 0.045 . (c) Intensity of image distribution for ξ = 1 , a N = 0.25 , Ω c = 0.1491 , and P c = 0.017 .

Fig. 6
Fig. 6

Variation of Ω as a function of K and ξ.

Fig. 7
Fig. 7

H ( ϖ ) of LSI (dashed lines) and SPM (solid curves) with a N = 0.25 (left column) and Diff ( ϖ ) = H LSI ( ϖ ) H SPM ( ϖ ) (right column). (a) ξ = 0 , (b) ξ = 0.5 , (c) ξ = 0.75 .

Fig. 8
Fig. 8

LSI (solid curve) and SPM (dots) coherent distributions in the image plane for a 1-D step centered at ξ = 0 , K = 157 , and a N = 0.25 . (b) LSI (solid curve) and SPM (dots) distributions for a 1-D step centered at ξ = 0.5 .

Fig. 9
Fig. 9

LSI (solid curve) and SPM (dots) incoherent distributions in the image plane for a 1-D step centered at ξ = 0 , K = 157 , and a N = 0.25 . (b) LSI (solid curve) and SPM (dots) distributions for a 1-D step centered at ξ = 0.5 .

Fig. 10
Fig. 10

Variation of Ω with K = 15.7 , L = 10 cm , and a N = 0.25 as a function of Λ (solid curve). Dashed line indicates corresponding Ω when Λ .

Fig. 11
Fig. 11

I SPM Λ ( x im ) refers to the distribution in the image plane when diffraction from both L 1 and L 2 is considered. LSI PSF is also plotted (see dashed curves in plots). Top, Λ = L ; bottom, Λ = 1.5 L .

Equations (16)

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U im ( x im ) = h ( x im ; ς ) U g ( ς ) d ς ,
U g ( ς ) = 1 M U obj ( ς M ) ,
U im ( x im ) = h ( x im ψ ) U g ( ψ ) d ψ ,
h ( x im ) = A λ d ex p ex ( x ) exp ( j π x x im λ d ex ) d x .
p ( x ) = { 1 , x a 0 , otherwise ,
U L 1 ( x L , ξ ) = exp [ j π λ f 1 ( x L ξ ) 2 ] .
U pw ( x , ξ ) = exp ( j 2 π K x ξ ) ,
U ap ( x ap ) = 1 j λ f 1 L + L U L 1 ( x L , ξ ) exp [ j π λ f 1 ( x L x ap ) 2 ] d x L .
U ap ( x ap ) = K j π 1 1 exp { j K [ X 2 + x ap 2 2 X ( x ap + ξ ) ] } d X = 1 2 [ exp ( j 2 π K x ap ξ ) ] { erfi [ j K ( x ap + ξ 1 ) ] erfi [ j K ( x ap + ξ + 1 ) ] } ,
U ap ( x ap ) U pw ( x ap ) = F ( x ap , K , ξ ) exp ( j Δ ϕ ) = erfi [ j K ( x ap + ξ 1 ) ] erfi [ j K ( x ap + ξ + 1 ) ] ,
x s = N π z s 4 1 K .
U im ( x im ) = F { U ap ( x ap ) p ( x ap ) } ( u ) u x im λ f 2 .
Ω = R + ξ L R + ξ L I LSI ( a N , K ) I SPM ( a N , K ) 2 d x im ,
p g ( x ap ) = p ( x ap ) F ( x ap , K , ξ ) exp ( j Δ ϕ ) .
H SPM ( ϖ ) = p g ( ϖ ) * p g ( ϖ ) h SPM ( u ) 2 d u ,
step ( x ) = { 1 , x 0 0 , x < 0 .

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