Abstract

We analyze the effects of partial coherence in the image formation of a digital in-line holographic microscope (DIHM). The impulse response is described as a function of cross-spectral density of the light used in the space-frequency domain. Numerical simulation based on the applied model shows that a reduction in coherence of light leads to broadening of the impulse response. This is also validated by results from experiments wherein a DIHM is used to image latex beads using light with different spatial and temporal coherence.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
    [CrossRef] [PubMed]
  2. U. Schnars, H.-J. Hartman, and W. Jüptner, Digital Holography (Springer-Verlag, 2004).
  3. J. J. Barton, “Removing multiple scattering and twin images from holographic images,” Phys. Rev. Lett. 67, 3106-3109 (1991).
    [CrossRef] [PubMed]
  4. T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
    [CrossRef] [PubMed]
  5. G. Situ, J. P. Ryle, U. Gopinathan, and J. T. Sheridan, “Generalised in-line digital holographic technique based on intensity measurements at two different planes,” Appl. Opt. 47, 711-717 (2008).
    [CrossRef] [PubMed]
  6. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13, 9629-9635 (2005).
    [CrossRef] [PubMed]
  7. J. J. Barton, “Photoelectron holography,” Phys. Rev. Lett. 61, 1356-1359 (1988).
    [CrossRef] [PubMed]
  8. H.-W. Fink, W. Stocker, and H. Schmid, “Holography with low-energy electrons,” Phys. Rev. Lett. 65, 1204-1206 (1990).
    [CrossRef] [PubMed]
  9. 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]
  10. W. Xu, M. J. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography of microspheres,” Appl. Opt. 41, 5367-5375 (2002).
    [CrossRef] [PubMed]
  11. W. Xu, M. J. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett. 28, 164-166 (2003).
    [CrossRef] [PubMed]
  12. L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Commun. 29, 1132-1134 (2004).
  13. L. Repetto, R. Chittofrati, E. Piano, and C. Pontiggia, “Infrared lensless holographic microscope with a vidicon camera for inspection of metallic evaporations on silicon wafers,” Opt. Commun. 251, 44-50 (2005).
    [CrossRef]
  14. B. Javidi, I. Moon, S. Yeom, and E. Carapezza, “Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography,” Opt. Express 13, 4492-4506 (2005).
    [CrossRef] [PubMed]
  15. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
    [CrossRef] [PubMed]
  16. D. Gao, S. W. Wilkins, D. J. Parry, T. E. Gureyev, P. R. Miller, and E. Hansen, “X-ray ultramicroscopy using integrated sample cells,” Opt. Express 14, 7889-7894 (2006).
    [CrossRef] [PubMed]
  17. G. Pedrini, F. Zhang, and W. Osten, “Digital holographic microscopy in the deep (193 nm) ultraviolet,” Appl. Opt. 46, 7829-7835 (2007).
    [CrossRef] [PubMed]
  18. J. Pomarico, U. Schnars, H.-J. Hartman, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: a new method for displaying light in flight,” Appl. Opt. 34, 8095-8099 (1995).
    [CrossRef] [PubMed]
  19. G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. 41, 4489-4496 (2002).
    [CrossRef] [PubMed]
  20. L. Martínez-León, G. Pedrini, and W. Osten, “Applications of short-coherence digital holography in microscopy,” Appl. Opt. 44, 3977-3984 (2005).
    [CrossRef] [PubMed]
  21. T. Kozacki and R. Jóźwiki, “Near field hologram registration with partially coherent illumination,” Opt. Commun. 237, 235-242 (2004).
    [CrossRef]
  22. T. Kozacki and R. Jóźwiki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt. Commun. 252, 188-201 (2005).
    [CrossRef]
  23. J. Cheng and S. Han, “On x-ray in-line Gabor holography with a partially coherent source,” Opt. Commun. 172, 17-24 (1999).
    [CrossRef]
  24. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  25. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  26. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343-351 (1982).
    [CrossRef]
  27. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923-928 (1982).
    [CrossRef]

2008 (1)

2007 (2)

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

G. Pedrini, F. Zhang, and W. Osten, “Digital holographic microscopy in the deep (193 nm) ultraviolet,” Appl. Opt. 46, 7829-7835 (2007).
[CrossRef] [PubMed]

2006 (2)

2005 (5)

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13, 9629-9635 (2005).
[CrossRef] [PubMed]

L. Repetto, R. Chittofrati, E. Piano, and C. Pontiggia, “Infrared lensless holographic microscope with a vidicon camera for inspection of metallic evaporations on silicon wafers,” Opt. Commun. 251, 44-50 (2005).
[CrossRef]

B. Javidi, I. Moon, S. Yeom, and E. Carapezza, “Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography,” Opt. Express 13, 4492-4506 (2005).
[CrossRef] [PubMed]

T. Kozacki and R. Jóźwiki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt. Commun. 252, 188-201 (2005).
[CrossRef]

L. Martínez-León, G. Pedrini, and W. Osten, “Applications of short-coherence digital holography in microscopy,” Appl. Opt. 44, 3977-3984 (2005).
[CrossRef] [PubMed]

2004 (2)

T. Kozacki and R. Jóźwiki, “Near field hologram registration with partially coherent illumination,” Opt. Commun. 237, 235-242 (2004).
[CrossRef]

L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Commun. 29, 1132-1134 (2004).

2003 (2)

2002 (2)

1999 (1)

J. Cheng and S. Han, “On x-ray in-line Gabor holography with a partially coherent source,” Opt. Commun. 172, 17-24 (1999).
[CrossRef]

1995 (1)

1991 (1)

J. J. Barton, “Removing multiple scattering and twin images from holographic images,” Phys. Rev. Lett. 67, 3106-3109 (1991).
[CrossRef] [PubMed]

1990 (1)

H.-W. Fink, W. Stocker, and H. Schmid, “Holography with low-energy electrons,” Phys. Rev. Lett. 65, 1204-1206 (1990).
[CrossRef] [PubMed]

1988 (1)

J. J. Barton, “Photoelectron holography,” Phys. Rev. Lett. 61, 1356-1359 (1988).
[CrossRef] [PubMed]

1982 (2)

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Barton, J. J.

J. J. Barton, “Removing multiple scattering and twin images from holographic images,” Phys. Rev. Lett. 67, 3106-3109 (1991).
[CrossRef] [PubMed]

J. J. Barton, “Photoelectron holography,” Phys. Rev. Lett. 61, 1356-1359 (1988).
[CrossRef] [PubMed]

Carapezza, E.

Cheng, J.

J. Cheng and S. Han, “On x-ray in-line Gabor holography with a partially coherent source,” Opt. Commun. 172, 17-24 (1999).
[CrossRef]

Chittofrati, R.

L. Repetto, R. Chittofrati, E. Piano, and C. Pontiggia, “Infrared lensless holographic microscope with a vidicon camera for inspection of metallic evaporations on silicon wafers,” Opt. Commun. 251, 44-50 (2005).
[CrossRef]

Davis, T.

Duan, Z.

Fink, H.-W.

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

H.-W. Fink, W. Stocker, and H. Schmid, “Holography with low-energy electrons,” Phys. Rev. Lett. 65, 1204-1206 (1990).
[CrossRef] [PubMed]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Gao, D.

Garcia-Sucerquia, J.

Gopinathan, U.

Gureyev, T.

Gureyev, T. E.

Han, S.

J. Cheng and S. Han, “On x-ray in-line Gabor holography with a partially coherent source,” Opt. Commun. 172, 17-24 (1999).
[CrossRef]

Hansen, E.

Hartman, H.-J.

Javidi, B.

Jericho, M. H.

Jericho, M. J.

Jericho, S. K.

Józwiki, R.

T. Kozacki and R. Jóźwiki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt. Commun. 252, 188-201 (2005).
[CrossRef]

T. Kozacki and R. Jóźwiki, “Near field hologram registration with partially coherent illumination,” Opt. Commun. 237, 235-242 (2004).
[CrossRef]

Jüptner, W.

Klages, P.

Kozacki, T.

T. Kozacki and R. Jóźwiki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt. Commun. 252, 188-201 (2005).
[CrossRef]

T. Kozacki and R. Jóźwiki, “Near field hologram registration with partially coherent illumination,” Opt. Commun. 237, 235-242 (2004).
[CrossRef]

Kreuzer, H. J.

Latychevskaia, T.

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Martínez-León, L.

Mayo, S.

Meinertzhagen, I. A.

Miller, P.

Miller, P. R.

Miyamoto, Y.

Moon, I.

Osten, W.

Paganin, D.

Parry, D. J.

Pedrini, G.

Piano, E.

L. Repetto, R. Chittofrati, E. Piano, and C. Pontiggia, “Infrared lensless holographic microscope with a vidicon camera for inspection of metallic evaporations on silicon wafers,” Opt. Commun. 251, 44-50 (2005).
[CrossRef]

L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Commun. 29, 1132-1134 (2004).

Pogany, A.

Pomarico, J.

Pontiggia, C.

L. Repetto, R. Chittofrati, E. Piano, and C. Pontiggia, “Infrared lensless holographic microscope with a vidicon camera for inspection of metallic evaporations on silicon wafers,” Opt. Commun. 251, 44-50 (2005).
[CrossRef]

L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Commun. 29, 1132-1134 (2004).

Repetto, L.

L. Repetto, R. Chittofrati, E. Piano, and C. Pontiggia, “Infrared lensless holographic microscope with a vidicon camera for inspection of metallic evaporations on silicon wafers,” Opt. Commun. 251, 44-50 (2005).
[CrossRef]

L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Commun. 29, 1132-1134 (2004).

Ryle, J. P.

Schmid, H.

H.-W. Fink, W. Stocker, and H. Schmid, “Holography with low-energy electrons,” Phys. Rev. Lett. 65, 1204-1206 (1990).
[CrossRef] [PubMed]

Schnars, U.

Sheridan, J. T.

Situ, G.

Starikov, A.

Stevenson, A.

Stocker, W.

H.-W. Fink, W. Stocker, and H. Schmid, “Holography with low-energy electrons,” Phys. Rev. Lett. 65, 1204-1206 (1990).
[CrossRef] [PubMed]

Takeda, M.

Tiziani, H. J.

Wang, W.

Wilkins, S.

Wilkins, S. W.

Wolf, E.

Xu, W.

Yeom, S.

Zhang, F.

Appl. Opt. (7)

J. Opt. Soc. Am. (2)

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Opt. Commun. (5)

T. Kozacki and R. Jóźwiki, “Near field hologram registration with partially coherent illumination,” Opt. Commun. 237, 235-242 (2004).
[CrossRef]

T. Kozacki and R. Jóźwiki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt. Commun. 252, 188-201 (2005).
[CrossRef]

J. Cheng and S. Han, “On x-ray in-line Gabor holography with a partially coherent source,” Opt. Commun. 172, 17-24 (1999).
[CrossRef]

L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Commun. 29, 1132-1134 (2004).

L. Repetto, R. Chittofrati, E. Piano, and C. Pontiggia, “Infrared lensless holographic microscope with a vidicon camera for inspection of metallic evaporations on silicon wafers,” Opt. Commun. 251, 44-50 (2005).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. Lett. (4)

J. J. Barton, “Photoelectron holography,” Phys. Rev. Lett. 61, 1356-1359 (1988).
[CrossRef] [PubMed]

H.-W. Fink, W. Stocker, and H. Schmid, “Holography with low-energy electrons,” Phys. Rev. Lett. 65, 1204-1206 (1990).
[CrossRef] [PubMed]

J. J. Barton, “Removing multiple scattering and twin images from holographic images,” Phys. Rev. Lett. 67, 3106-3109 (1991).
[CrossRef] [PubMed]

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Other (3)

U. Schnars, H.-J. Hartman, and W. Jüptner, Digital Holography (Springer-Verlag, 2004).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Schematic illustrating the notations used in the text. SF denotes spatial filter (see Fig. 4).

Fig. 2
Fig. 2

Impulse response evaluated at r x = r y = 0 for r z ranging from 0.1 to 0.1 mm for a Gaussian Schell source with (a) β = 5 , β = 0.2 and temporal FWHM bandwidth equal to 7 nm , (b) β = 5 , β = 0.2 and temporal FWHM bandwidth equal to 14 nm , (c) temporal FWHM bandwidth equal to 7 and 14 nm with β = 5 , and (d) temporal FWHM bandwidth equal to 7 and 14 nm with β = 0.2 . The X axis in all four plots indicates distance Δ r z in millimeters. The Y axis shows normalized amplitude.

Fig. 3
Fig. 3

Impulse response evaluated in the plane r z = 0 for a Gaussian Schell source with (a) β = 5 , β = 0.2 and temporal FWHM bandwidth equal to 7 nm , (b) β = 5 , β = 0.2 and temporal FWHM bandwidth equal to 14 nm , (c) temporal FWHM bandwidth equal to 7 and 14 nm with β = 5 , and (d) temporal FWHM bandwidth equal to 7 and 14 nm with β = 0.2 . The X and Y axes in all four plots indicate r x and r y in pixels. Each pixel translates to a physical distance 3 μ m . The Z axis shows normalized amplitude.

Fig. 4
Fig. 4

Schematic of a lensless DIH setup. An SF, which acts as a secondary source, is illuminated via an imaging system by a primary source (S). The light scattered by the micro-object and the unscattered light forms an in-line hologram at the CCD plane.

Fig. 5
Fig. 5

Experimental result showing the reconstructed amplitude at Δ r x = Δ r y = 0 for Δ r z ranging from 0.1 to 0.1 mm for (a) source LD1 ( FWHM = 2 nm ) with spatial filters of 1 and 5 μ m in diameter, (b) source LD2 ( FWHM = 12 nm ) with spatial filters of 1 and 5 μ m in diameter, (c) source LD1 and LD2 with spatial filter of 1 μ m in diameter, and (d) source LD1 and LD2 with spatial filter of 5 μ m in diameter. The X axis in all the four plots indicates distance Δ r z in millimeters. The Y axis shows normalized amplitude.

Fig. 6
Fig. 6

Experimental result showing the reconstructed amplitude in the plane Δ r z = 0 for (a) source LD1 with spatial filter of 1 μ m in diameter, (b) source LD1 with spatial filter of 5 μ m in diameter, (c) source LD2 with spatial filter of 1 μ m in diameter, and (d) source LD2 with spatial filter of 5 μ m in diameter. The X and Y axes in all four plots indicate Δ r x and Δ r y in pixels. Each pixel corresponds to a distance of 3 μ m in the reconstruction space. The Z axis shows normalized amplitude.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

W ( p 1 , p 2 , ω ) = U * ( p 1 , ω ) U ( p 2 , ω ) ω .
V 1 ( r , ω ) = Θ ( r i , ω ) U ( p 1 , ω ) e j k r i p 1 r i p 1 e j k r r i r r i + U ( p 1 , ω ) e j k r p 1 r p 1 .
V 1 ( r s , ω ) = Θ ( r i , ω ) U ( p 1 , ω ) e j k r i r i e j k r r e j k ( s i p 1 + s r i ) + U ( p 1 , ω ) e j k r r e j k s p 1 .
V 2 ( r s , ω ) = Θ ( r i , ω ) U ( p 2 , ω ) e j k r i r i e j k r r e j k ( s i p 2 + s r i ) + U ( p 2 , ω ) e j k r r e j k s p 2 .
S ( r s , ω ) = σ σ V 1 * ( r s , ω ) V 2 ( r s , ω ) d 2 p 1 d 2 p 2 .
S ( s , ω ) = σ σ W ( p 1 , p 2 , ω ) [ Θ ( r i , ω ) 2 r i 2 e j k s i ( p 2 p 1 ) + e j k s ( p 2 p 1 ) + e j k r i r i Θ ( r i , ω ) e j k ( s i p 2 + s r i s p 1 ) + e j k r i r i Θ * ( r i , ω ) e j k ( s i p 1 + s r i s p 2 ) ] d 2 p 1 d 2 p 2 .
W ( p 1 , p 2 , ω ) = n α n ( ω ) ϕ n * ( p 1 , ω ) ϕ n ( p 2 , ω ) ,
σ W ( p 1 , p 2 , ω ) ϕ n ( p 1 , ω ) d 2 p 1 = α n ( ω ) ϕ n ( p 2 , ω ) .
S ( s , ω ) = Θ ( r i , ω ) 2 r i 2 n α n ( ω ) ϕ ̃ n ( k s i , ω ) 2 + n α n ( ω ) ϕ ̃ n ( k s , ω ) 2 + e j k r i r i Θ ( r i , ω ) e j k s r i n α n ( ω ) ϕ ̃ n ( k s i , ω ) ϕ ̃ n * ( k s , ω ) + e j k r i r i Θ * ( r i , ω ) e j k s r i n α n * ( ω ) ϕ ̃ n * ( k s i , ω ) ϕ ̃ n ( k s , ω ) ,
ϕ ̃ n ( . , ω ) = ϕ n ( p , ω ) e j p ( . ) d 2 p .
H ( s , . ) = S ( s , ω ) e j ω . d ω .
U ( r c , τ ) = Λ H ( s , τ ) e j k s r c d 2 s .
U i ( r c , τ ) = Θ ( r i , ω ) e j k r i r i n α n ( ω ) ϕ ̃ n ( k s i , ω ) Λ ϕ ̃ n * ( k s , ω ) e j k s ( r c r i ) d 2 s e j ω τ d ω .
U i ( r c , τ ) = n U i n ( r c , τ ) ,
U i n ( r c , τ ) = Θ ( r i , ω ) e j k r i r i α n ( ω ) ϕ ̃ n ( k s i , ω ) Λ ϕ ̃ n * ( k s , ω ) e j k s ( r c r i ) d 2 s e j ω τ d ω
U ( r c , τ ) = Ω Θ ( r i , ω ) e j k r i r i n α n ( ω ) ϕ ̃ n ( k s i , ω ) Λ ϕ ̃ n * ( k s , ω ) e j k s ( r c r i ) d 2 s e j ω τ d ω d 3 r i .
U ( r c , τ ) = Ω Θ ( r i , ω ) e j k r i r i h i ( r c r i , ω ) e j ω τ d ω d 3 r i ,
h i ( . , ω ) = n α n ( ω ) ϕ ̃ n ( k s i , ω ) Λ ϕ ̃ n * ( k s , ω ) e j k s ( . ) d 2 s .
h i coh ( . , ω ) = α 0 ( ω ) ϕ ̃ 0 ( k s i , ω ) Λ ϕ ̃ 0 * ( k s , ω ) e j k s ( . ) d 2 s .
h i nb ( . , ω ) = T ( ω ) n α n ( ω 0 ) ϕ ̃ n ( k 0 s i , ω 0 ) Λ ϕ ̃ n * ( k 0 s , ω 0 ) e j k 0 s ( . ) d 2 s .
W ( p 1 , p 2 , ω ) = S ( p 1 , ω ) S ( p 2 , ω ) μ ( p 1 p 2 , ω ) ,
S ( p , ω ) = A ( ω ) e p 2 2 σ s 2 ( ω ) ,
μ ( p 1 p 2 , ω ) = e p 1 p 2 2 2 σ μ 2 ( ω ) .
ϕ n ( x ) = ( 2 c π ) 1 4 1 2 n n ! H n ( x 2 c ) e c x 2 ,
α n = A ( π a + b + c ) ( b a + b + c ) n ,
ϕ ̃ n ( ν ) = ( 2 c π ) 1 4 1 2 n n ! H ̃ n ( ν 2 c ) e ν 2 4 c .
h i Schell ( . , ω ) = T ( ω ) n ( 2 c π ) 1 2 α n ( ω 0 ) 2 n n ! H ̃ n * ( k 0 2 c s i ) Λ H ̃ n ( k 0 2 c s ) e k 0 2 ( s 2 + s 0 2 ) 4 c e j k 0 s ( . ) d 2 s .
h i Schell ( . , τ ) = ψ ( τ ) n ( 2 c π ) 1 2 α n ( ω 0 ) 2 n n ! H ̃ n * ( k 0 2 c s i ) Λ H ̃ n ( k 0 2 c s ) e k 0 2 ( s 2 + s 0 2 ) 4 c e j k 0 s ( . ) d 2 s ,
ψ ( τ ) = T ( ω ) e j ω τ d ω .

Metrics