Abstract

If the sampled diffraction pattern due to a planar object is used to reconstruct the object pattern by backpropagation, the obtained pattern is no longer the same as the original. The effect of such sampling on the reconstruction is analyzed. The formulation uses the plane-wave expansion, and therefore the provided solution is exact for wave propagation in media where scalar wave propagation is valid. In contrast to the sampling effects under the Fresnel approximation, the exact solution indicates that there are no modulated replicas of the original object in the reconstructed pattern. Rather, the distortion is in the form of modulated, translated, and dispersed versions of the original.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef]
  9. L. Onural, "Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in Fresnel diffraction simulations," Opt. Eng. 43, 2557-2563 (2004).
    [CrossRef]
  10. A. Stern and B. Javidi, "Analysis of practical sampling and reconstruction from Fresnel fields," Opt. Eng. 43, 239-250 (2004).
    [CrossRef]
  11. F. S. Roux, "Complex-valued Fresnel-transform sampling," Appl. Opt. 34, 3128-3135 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. F. Gori, "Fresnel transform and sampling theorem," Opt. Eng. 39, 293-297 (1981).
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    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  16. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).
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  18. M. Ayatollahi and S. Safavi-Naeini, "A new representation for the Green's function of multilayer media based on plane wave expansion," IEEE Trans. Antennas Propag. 52, 1548-1557 (2004).
    [CrossRef]
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    [CrossRef]
  20. L. Onural, "Projection-slice theorem as a tool for mathematical representation of diffraction," IEEE Signal Process. Lett. 14, 43-46 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  23. D. E. Dudgeon and R. M. Mersereau, Multidimensional Signal Processing (Prentice Hall, 1984).

2007 (1)

L. Onural, "Projection-slice theorem as a tool for mathematical representation of diffraction," IEEE Signal Process. Lett. 14, 43-46 (2007).
[CrossRef]

2006 (2)

L. Onural, "Impulse functions over curves and surfaces and their applications to diffraction," J. Math. Anal. Appl. 322, 18-27 (2006).
[CrossRef]

A. Stern and B. Javidi, "Improved resolution digital holography using generalized sampling theorem for locally band-limited fields," J. Opt. Soc. Am. A 23, 1227-1235 (2006).
[CrossRef]

2004 (4)

J. M. Coupland, "Holographic particle velocimetry: signal recovery from under-sampled CCD data," Meas. Sci. Technol. 15, 711-717 (2004).
[CrossRef]

M. Ayatollahi and S. Safavi-Naeini, "A new representation for the Green's function of multilayer media based on plane wave expansion," IEEE Trans. Antennas Propag. 52, 1548-1557 (2004).
[CrossRef]

L. Onural, "Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in Fresnel diffraction simulations," Opt. Eng. 43, 2557-2563 (2004).
[CrossRef]

A. Stern and B. Javidi, "Analysis of practical sampling and reconstruction from Fresnel fields," Opt. Eng. 43, 239-250 (2004).
[CrossRef]

2003 (1)

2002 (3)

2000 (1)

1999 (2)

1995 (1)

1987 (1)

L. Onural and P. D. Scott, "Digital decoding of in-line holograms," Opt. Eng. 26, 1124-1132 (1987).

1981 (1)

F. Gori, "Fresnel transform and sampling theorem," Opt. Eng. 39, 293-297 (1981).

1968 (1)

1967 (1)

An, Y.

Ayatollahi, M.

M. Ayatollahi and S. Safavi-Naeini, "A new representation for the Green's function of multilayer media based on plane wave expansion," IEEE Trans. Antennas Propag. 52, 1548-1557 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

Cai, L.

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Oxford U. Press and IEEE Press, reissue, 1996).

Coupland, J. M.

J. M. Coupland, "Holographic particle velocimetry: signal recovery from under-sampled CCD data," Meas. Sci. Technol. 15, 711-717 (2004).
[CrossRef]

Dudgeon, D. E.

D. E. Dudgeon and R. M. Mersereau, Multidimensional Signal Processing (Prentice Hall, 1984).

Goodman, J. W.

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

Gori, F.

F. Gori, "Fresnel transform and sampling theorem," Opt. Eng. 39, 293-297 (1981).

Javidi, B.

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Jüptner, W.

Lalor, É.

Li, W.

Mersereau, R. M.

D. E. Dudgeon and R. M. Mersereau, Multidimensional Signal Processing (Prentice Hall, 1984).

Milgram, J. H.

Ohzu, H.

Onural, L.

L. Onural, "Projection-slice theorem as a tool for mathematical representation of diffraction," IEEE Signal Process. Lett. 14, 43-46 (2007).
[CrossRef]

L. Onural, "Impulse functions over curves and surfaces and their applications to diffraction," J. Math. Anal. Appl. 322, 18-27 (2006).
[CrossRef]

L. Onural, "Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in Fresnel diffraction simulations," Opt. Eng. 43, 2557-2563 (2004).
[CrossRef]

L. Onural, "Sampling of the diffraction field," Appl. Opt. 39, 5929-5935 (2000).
[CrossRef]

L. Onural and P. D. Scott, "Digital decoding of in-line holograms," Opt. Eng. 26, 1124-1132 (1987).

Osten, W.

Pedrini, G.

Roux, F. S.

Safavi-Naeini, S.

M. Ayatollahi and S. Safavi-Naeini, "A new representation for the Green's function of multilayer media based on plane wave expansion," IEEE Trans. Antennas Propag. 52, 1548-1557 (2004).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Scott, P. D.

L. Onural and P. D. Scott, "Digital decoding of in-line holograms," Opt. Eng. 26, 1124-1132 (1987).

Seebacher, S.

Sherman, G. C.

Stern, A.

Takaki, Y.

Tiziani, H. J.

Wagner, C.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

Yu, L.

Zhang, Y.

Appl. Opt. (6)

IEEE Signal Process. Lett. (1)

L. Onural, "Projection-slice theorem as a tool for mathematical representation of diffraction," IEEE Signal Process. Lett. 14, 43-46 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

M. Ayatollahi and S. Safavi-Naeini, "A new representation for the Green's function of multilayer media based on plane wave expansion," IEEE Trans. Antennas Propag. 52, 1548-1557 (2004).
[CrossRef]

J. Math. Anal. Appl. (1)

L. Onural, "Impulse functions over curves and surfaces and their applications to diffraction," J. Math. Anal. Appl. 322, 18-27 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Meas. Sci. Technol. (2)

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

J. M. Coupland, "Holographic particle velocimetry: signal recovery from under-sampled CCD data," Meas. Sci. Technol. 15, 711-717 (2004).
[CrossRef]

Opt. Eng. (4)

L. Onural and P. D. Scott, "Digital decoding of in-line holograms," Opt. Eng. 26, 1124-1132 (1987).

F. Gori, "Fresnel transform and sampling theorem," Opt. Eng. 39, 293-297 (1981).

L. Onural, "Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in Fresnel diffraction simulations," Opt. Eng. 43, 2557-2563 (2004).
[CrossRef]

A. Stern and B. Javidi, "Analysis of practical sampling and reconstruction from Fresnel fields," Opt. Eng. 43, 239-250 (2004).
[CrossRef]

Opt. Express (1)

Other (4)

D. E. Dudgeon and R. M. Mersereau, Multidimensional Signal Processing (Prentice Hall, 1984).

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Oxford U. Press and IEEE Press, reissue, 1996).

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Figures (7)

Fig. 1
Fig. 1

Circular aperture.

Fig. 2
Fig. 2

Fresnel reconstruction from the sampled Fresnel diffraction pattern of the object given in Fig. 1. The real part of the field is shown.

Fig. 3
Fig. 3

Rayleigh–Sommerfeld reconstruction from the sampled Rayleigh–Sommerfeld diffraction pattern of the object given in Fig. 1. The distance is large, and therefore the pattern is similar to the Fresnel case. The real part of the field is shown.

Fig. 4
Fig. 4

Rayleigh–Sommerfeld reconstruction from the sampled Rayleigh–Sommerfeld diffraction pattern of the object given in Fig. 1. The distance is smaller than that of Fig. 3, and therefore the discussed dispersion effects are more visible. The real part of the field is shown.

Fig. 5
Fig. 5

Same simulation as in Fig. 4 but with a smaller diffraction distance.

Fig. 6
Fig. 6

Same simulation as in Fig. 5 but with a smaller diffraction distance.

Fig. 7
Fig. 7

Intensity pattern of the field whose real part is shown in Fig. 6.

Equations (35)

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ψ ( x ) = B ( k ) exp ( j k T x ) d k ,
ψ ( x ) = k x 2 + k y 2 k 2 A ( k ) exp ( j k T x ) d k ,
A ( k ) = B ( k ) k ( k 2 k T k ) 1 2 ;
A ( k ) exp ( j k T x ) z = 0 = A ( k ) exp [ j k T x ] ,
A ( k ) exp ( j k T x ) z = Z = A ( k ) exp [ j k T x ] exp [ j ϕ Z ( k ) ] ,
ϕ Z ( k ) = Z ( k 2 k T k ) 1 2 ,
ψ 0 ( x ) ψ ( x , y , 0 ) = A ( k ) exp [ j k T x ] d k ,
ψ Z ( x ) ψ ( x , y , Z ) = A ( k ) exp [ j k T x ] exp [ j Z ( k 2 k T k ) 1 2 ] d k .
4 π 2 A ( k ) = F { ψ ( x ) } ,
Ψ Z ( k ) = F 1 { 4 π 2 A ( k ) H Z ( k ) } ,
H Z ( k ) = H Z ( k x , k y ) exp [ j Z ( k 2 k T k ) 1 2 ] = exp [ j Z ( k 2 k x 2 k y 2 ) 1 2 ] .
h Z ( x ) = h Z ( x , y ) = F 1 { H Z ( k ) } = 1 2 π Z exp [ ( j k ( x 2 + y 2 + Z 2 ) 1 2 ] ( x 2 + y 2 + Z 2 ) 1 2 = 1 2 π Z exp [ ( j k ( x T x + Z 2 ) 1 2 ] ( x T x + Z 2 ) 1 2 = 1 2 π Z exp ( j k r ) r .
ψ 0 ( x ) = F 1 { Ψ Z ( k ) H Z ( k ) } = ψ Z ( x ) h Z ( x )
H Z ( k ) H Z ( k ) = 1
h Z ( x ) h Z ( x ) = δ ( x ) ,
ψ Z s ( x ) ψ Z ( x ) n δ ( x V n ) = n ψ Z ( V n ) δ ( x V n ) ,
p ( x ) = n δ ( x V n ) .
Ψ Z s ( k ) = F { ψ Z s ( x ) } = 1 det V m Ψ Z ( k U m ) ,
Ψ R ( k ) = Ψ Z s ( k ) H Z ( k ) = 1 det V m Ψ Z ( k U m ) H Z ( k ) = 1 det V m 4 π 2 A ( k U m ) H Z ( k U m ) H Z ( k ) .
H Z ( k U m ) H Z ( k ) = exp { j Z [ k 2 ( k U m ) T ( k U m ) ] 1 2 } exp { j Z [ k 2 k T k ] 1 2 } ,
Ψ R ( k ) = 4 π 2 det V m A ( k U m ) exp ( j Z { [ k 2 ( k U m ) T ( k U m ) ] 1 2 [ k 2 k T k ] 1 2 } ) = 4 π 2 det V m A ( k U m ) exp ( j ϕ ( k U m ) ) exp ( j ϕ ( k ) ) ,
ϕ ( k ) = [ k 2 ( k ) T ( k ) ] 1 2 Z .
ψ R ( x ) = F 1 { Ψ R ( k ) } = 1 4 π 2 Ψ R ( k ) exp [ j k T x ] d k .
ψ R , m ( x ) F 1 { 4 π 2 det V A ( k U m ) exp ( j Z { [ k 2 ( k U m ) T ( k U m ) ] 1 2 [ k 2 k T k ] 1 2 } ) } ,
ψ R ( x ) = m ψ R , m ( x )
ϕ ( k ) = k Z 1 2 k k T k Z + higher - order terms .
ϕ m ( k ) = ϕ ( k U m ) ϕ ( k ) = 1 k Z ( U m ) T k + 1 2 k Z ( U m ) T ( U m ) + higher - order terms .
p ( x ) = m p m ( x ) = 1 det V m exp [ j ( k x , m x k y , m y ± k z , m ( z Z ) ) ] ,
p m ( x ) = 1 det V exp { j [ k x , m x + k y , m y + ( k 2 k x , m 2 k y , m 2 ) 1 2 ( z Z ) ] } ,
ψ Z ( x ) p m ( x ) = ψ Z ( x ) 1 det V exp { j [ k x , m x + k y , m y ] } = 1 det V A ( k x , k y ) exp ( j ( k 2 k x 2 k y 2 ) 1 2 Z ) exp ( j { ( k x + k x , m ) x + ( k y + k y , m ) y } ) d k x d k y .
ψ m ( x ) = 1 det V A ( k x , k y ) exp ( j ( k 2 k x 2 k y 2 ) 1 2 Z ) exp ( j { ( k x + k x , m ) x + ( k y + k y , m ) y + [ k 2 ( k x + k x , m ) 2 ( k y + k y , m ) 2 ] 1 2 ( z Z ) } ) d k x d k y .
ψ R , m ( x ) = 1 det V A ( k x , k y ) exp ( j { ( k 2 k x 2 k y 2 ) 1 2 [ k 2 ( k x + k x , m ) 2 ( k y + k y , m ) 2 ] 1 2 } Z ) exp ( j [ ( k x + k x , m ) x + ( k y + k y , m ) y ] ) d k x d k y .
ψ R ( x ) = 1 det V A ( k x k x , m , k y k y , m ) exp ( j { ( k 2 ( k x k x , m ) 2 ( k y k y , m ) 2 ) 1 2 [ k 2 k x 2 k y 2 ] 1 2 } Z ) exp ( j ( k x x + k y y ) ) d k x d k y .
ψ R ( x ) = 1 det V F 1 { A ( k x k x , m , k y k y , m ) exp ( j { [ k 2 ( k x k x , m ) 2 ( k y k y , m ) 2 ] 1 2 [ k 2 k x 2 k y 2 ] 1 2 } Z ) } .
ψ R ( x ) = m ψ R , m ( x ) ,

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