Abstract

Conventionally, for modeling in-line lensless holograms of systems with high numerical apertures and diverging spherical illumination, the samples are considered as an ensemble of secondary point sources. On following Huygens’s principle, the in-line hologram is the result of the amplitude superposition of the secondary spherical wavefronts with the wavefront originated in the point source. Albeit simple, this approach limits the shapes of the objects that can be modeled and the computation time rises with the complexity of the sample. In this work, we present a diffraction-based approach to model in-line lensless holograms. Samples with any shape or size can be modeled for in-line holographic systems with numerical apertures up to 0.57. The method is successfully applied to model objects of intricate submicrometer structures and/or multiple samples lying within a unique sample volume.

© 2011 OpticalSociety of America

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References

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  1. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
    [CrossRef] [PubMed]
  2. W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
    [CrossRef] [PubMed]
  3. S. K. Jericho, P. Klages, J. Nadeau, E. M. Dumas, M. H. Jericho, and H. J. Kreuzer, “In-line digital holographic microscopy for terrestrial exobiological research,” Planet. Space Sci. 58, 701–705 (2010).
    [CrossRef]
  4. S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
    [CrossRef]
  5. J. Garcia-Sucerquia, W. Xu, S. M. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluidic flow with digital in-line holographic microscopy,” Optik (Jena) 119, 419–423(2008).
    [CrossRef]
  6. H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
    [CrossRef]
  7. A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Riesenberg, “Packed domain Rayleigh–Sommerfeld wavefield propagation for large targets,” Opt. Express 18, 27036–27047 (2010).
    [CrossRef]
  8. J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  9. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH Verlag, 2005).
  10. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. 29, 854–856 (2004).
    [CrossRef] [PubMed]
  11. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).
    [CrossRef] [PubMed]
  12. V. Nascov and P. C. Logofătu, “Fast computation algorithm for the Rayleigh–Sommerfeld diffraction formula using a type of scaled convolution,” Appl. Opt. 48, 4310–4319 (2009).
    [CrossRef] [PubMed]
  13. X. Zeng, C. Liang, and Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 2042–2047(1997).
    [CrossRef] [PubMed]
  14. Y. M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperture beams,” J. Opt. Soc. Am. A 21, 2135–2145 (2004).
    [CrossRef]
  15. L. Bluestein, “Linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
    [CrossRef]
  16. H. J. Kreuzer, “Holographic microscope and method of hologram reconstruction,” U.S. patent 6411406 B1 (25 June 2002).
  17. S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
    [CrossRef]
  18. D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. A 197, 454–487 (1949).
    [CrossRef]
  19. M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

2010 (2)

S. K. Jericho, P. Klages, J. Nadeau, E. M. Dumas, M. H. Jericho, and H. J. Kreuzer, “In-line digital holographic microscopy for terrestrial exobiological research,” Planet. Space Sci. 58, 701–705 (2010).
[CrossRef]

A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Riesenberg, “Packed domain Rayleigh–Sommerfeld wavefield propagation for large targets,” Opt. Express 18, 27036–27047 (2010).
[CrossRef]

2009 (1)

2008 (1)

J. Garcia-Sucerquia, W. Xu, S. M. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluidic flow with digital in-line holographic microscopy,” Optik (Jena) 119, 419–423(2008).
[CrossRef]

2006 (3)

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef] [PubMed]

2004 (3)

2001 (1)

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef] [PubMed]

1997 (1)

1992 (1)

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

1970 (1)

L. Bluestein, “Linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

1949 (1)

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. A 197, 454–487 (1949).
[CrossRef]

Alfieri, D.

An, Y.

Bluestein, L.

L. Bluestein, “Linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

Coppola, G.

De Nicola, S.

Dumas, E. M.

S. K. Jericho, P. Klages, J. Nadeau, E. M. Dumas, M. H. Jericho, and H. J. Kreuzer, “In-line digital holographic microscopy for terrestrial exobiological research,” Planet. Space Sci. 58, 701–705 (2010).
[CrossRef]

Engelberg, Y. M.

Ferraro, P.

Finizio, A.

Fink, H.-W.

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Gabor, D.

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. A 197, 454–487 (1949).
[CrossRef]

Garcia-Sucerquia, J.

J. Garcia-Sucerquia, W. Xu, S. M. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluidic flow with digital in-line holographic microscopy,” Optik (Jena) 119, 419–423(2008).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef] [PubMed]

Goodman, J.

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

Jericho, M. H.

S. K. Jericho, P. Klages, J. Nadeau, E. M. Dumas, M. H. Jericho, and H. J. Kreuzer, “In-line digital holographic microscopy for terrestrial exobiological research,” Planet. Space Sci. 58, 701–705 (2010).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. M. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluidic flow with digital in-line holographic microscopy,” Optik (Jena) 119, 419–423(2008).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef] [PubMed]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef] [PubMed]

Jericho, S. K.

S. K. Jericho, P. Klages, J. Nadeau, E. M. Dumas, M. H. Jericho, and H. J. Kreuzer, “In-line digital holographic microscopy for terrestrial exobiological research,” Planet. Space Sci. 58, 701–705 (2010).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef] [PubMed]

Jericho, S. M.

J. Garcia-Sucerquia, W. Xu, S. M. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluidic flow with digital in-line holographic microscopy,” Optik (Jena) 119, 419–423(2008).
[CrossRef]

Kanka, M.

Klages, P.

S. K. Jericho, P. Klages, J. Nadeau, E. M. Dumas, M. H. Jericho, and H. J. Kreuzer, “In-line digital holographic microscopy for terrestrial exobiological research,” Planet. Space Sci. 58, 701–705 (2010).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef] [PubMed]

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH Verlag, 2005).

Kreuzer, H. J.

S. K. Jericho, P. Klages, J. Nadeau, E. M. Dumas, M. H. Jericho, and H. J. Kreuzer, “In-line digital holographic microscopy for terrestrial exobiological research,” Planet. Space Sci. 58, 701–705 (2010).
[CrossRef]

A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Riesenberg, “Packed domain Rayleigh–Sommerfeld wavefield propagation for large targets,” Opt. Express 18, 27036–27047 (2010).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. M. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluidic flow with digital in-line holographic microscopy,” Optik (Jena) 119, 419–423(2008).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef] [PubMed]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef] [PubMed]

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

H. J. Kreuzer, “Holographic microscope and method of hologram reconstruction,” U.S. patent 6411406 B1 (25 June 2002).

Liang, C.

Logofatu, P. C.

Meinertzhagen, I. A.

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef] [PubMed]

Nadeau, J.

S. K. Jericho, P. Klages, J. Nadeau, E. M. Dumas, M. H. Jericho, and H. J. Kreuzer, “In-line digital holographic microscopy for terrestrial exobiological research,” Planet. Space Sci. 58, 701–705 (2010).
[CrossRef]

Nakamura, K.

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Nascov, V.

Pierattini, G.

Riesenberg, R.

Ruschin, S.

Schmid, H.

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Wierzbicki, A.

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

Wuttig, A.

Xu, W.

J. Garcia-Sucerquia, W. Xu, S. M. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluidic flow with digital in-line holographic microscopy,” Optik (Jena) 119, 419–423(2008).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef] [PubMed]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef] [PubMed]

Yamaguchi, I.

Yaroslavsky, L. P.

Zeng, X.

Zhang, F.

Appl. Opt. (3)

IEEE Trans. Audio Electroacoust. (1)

L. Bluestein, “Linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (2)

Optik (Jena) (1)

J. Garcia-Sucerquia, W. Xu, S. M. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluidic flow with digital in-line holographic microscopy,” Optik (Jena) 119, 419–423(2008).
[CrossRef]

Planet. Space Sci. (1)

S. K. Jericho, P. Klages, J. Nadeau, E. M. Dumas, M. H. Jericho, and H. J. Kreuzer, “In-line digital holographic microscopy for terrestrial exobiological research,” Planet. Space Sci. 58, 701–705 (2010).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef] [PubMed]

Proc. R. Soc. A (1)

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. A 197, 454–487 (1949).
[CrossRef]

Rev. Sci. Instrum. (2)

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, “Submersible digital in-line holographic microscope,” Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

Ultramicroscopy (1)

H. J. Kreuzer, K. Nakamura, A. Wierzbicki, H.-W. Fink, and H. Schmid, “Theory of the point source electron microscope,” Ultramicroscopy 45, 381–403 (1992).
[CrossRef]

Other (4)

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

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH Verlag, 2005).

H. J. Kreuzer, “Holographic microscope and method of hologram reconstruction,” U.S. patent 6411406 B1 (25 June 2002).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

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Figures (5)

Fig. 1
Fig. 1

Coordinate description for modeling DIHM. The source is located at the origin of coordinates, the sample plane is located at ( ξ , η ) coordinates with pitches ( Δ ξ , Δ η ) and whole indexes ( l , k ) . The digital recording plane is at the ( x , y ) coordinates ( Δ x , Δ y ) pixel sizes and whole indexes ( m , n ) . The distance between the sample and point source is z s . The recording plane is placed at a distance L from the point source.

Fig. 2
Fig. 2

First step for in-line hologram modeling. Panel a shows the drawing of a diatom for which in-line hologram is desired. Panel b is a schematic representation of the modeling process; at a distance z s from the point source is illustrated the amplitude of the point-to-point product of panels a and c. Panel c presents the phase of the illuminating spherical wavefront at the sample plane. The numerical aperture was set to 0.52 for a wavelength of 405 nm .

Fig. 3
Fig. 3

Obtaining the in-line contrast hologram in evenly spaced coordinates. Panel a shows the contrast in-line hologram given by the intensity obtained from Eq. (6). This intensity is represented in the coordinates illustrated by the circles of panel b. After interpolating panel a to the dot coordinates, the contrast in-line hologram shown in panel c is obtained.

Fig. 4
Fig. 4

Reconstruction of the modeled in-line contrast hologram. Panel a shows the reconstruction of contrast in-line hologram shown in panel c of Fig. 3. Panel b shows the enlargement of the delimited area by the white square in panel a. The profile illustrated in panel c is taken along the red line in panel b. Notice the achieved submicrometer resolution, which allows for correctly reproducing the whole honeycomb structure with cells in the range of 800 nm .

Fig. 5
Fig. 5

Model and reconstruction of a contrast in-line hologram that contains multiple specimens in the sample volume. Panel a shows the modeled contrast in-line hologram for two diatoms placed at different distances from the point source, 100 μm for panel b and 200 μm for panel c. The two latter panels show the reconstructions obtained from panel a for the corresponding distances.

Equations (15)

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U m ( z ) = 1 i λ l = 0 M 1 T l S l exp { i ( 2 π λ ) [ z 2 + ( m Δ x l Δ ξ ) 2 ] 1 / 2 } [ z 2 + ( m Δ x l Δ ξ ) 2 ] 1 / 2 .
[ z 2 + ( m Δ x l Δ ξ ) 2 ] 1 / 2 R [ 1 + 1 2 ( l Δ ξ ) 2 R 2 m Δ x l Δ ξ R 2 ] ,
U p ( z ) = 1 i λ R exp ( i 2 π λ R ) l = 0 M 1 T l S l exp [ i π ( l Δ ξ ) 2 λ R ] exp [ i 2 π p Δ f x l Δ ξ ] .
m Δ x = q Δ x [ 1 ( q Δ x z ) 2 ] 1 / 2 .
Δ x = λ R M Δ ξ .
U m ( z ) = 1 i λ R exp ( i 2 π λ R ) exp [ i π Δ x Δ ξ m 2 λ R ] l = 0 M 1 T l S l exp [ i π Δ ξ λ R ( Δ ξ Δ x ) l 2 ] exp [ i π Δ x Δ ξ λ R [ ( m l ) 2 ] ] .
NA 0.57 .
L min 2 2 M Δ x .
z s min = 2 2 M Δ ξ .
1 2 π ϕ l = 1 λ 2 l Δ ξ 2 [ z s min 2 + 2 ( l Δ ξ ) 2 ] 1 / 2 < 1 5 ,
z s min > M Δ ξ 2 λ [ 100 Δ ξ 2 2 λ 2 ] 1 / 2 .
I ( r ) = | A ref ( r ) + i = 1 N A scat ( i ) ( r ) | 2 ,
I ˜ ( r ) = [ A ref ( r ) i = 1 N A scat ( i ) * ( r ) + A ref * ( r ) i = 1 N A scat ( i ) ( r ) ] + i = 1 N j = 1 N A scat ( i ) ( r ) A scat ( j ) * ( r ) .
I ˜ ( r ) = [ A ref ( r ) i = 1 N A scat ( i ) * ( r ) + A ref * ( r ) i = 1 N A scat ( i ) ( r ) ] .
I ˜ ( r ) = i = 1 N [ A ref ( r ) A scat ( i ) * ( r ) + A scat ( i ) ( r ) A ref ( r ) * ] = i = 1 N I ˜ ( i ) ( r ) ,

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