Abstract

A criticism of the focus wave mode (FWM) solution for localized pulses is that it contains backward propagating components that are difficult to generate in many practical situations. We describe a form of FWM where the strength of the backward propagating components is identically zero and derive special cases where the field can be written in an analytic form. In particular, a free-space version of “backward light” pulse is considered, which moves in the opposite direction with respect to all its spectral constituents.

© 2008 Optical Society of America

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  1. J. N. Brittingham, "Focus wave modes in homogeneous Maxwell's equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
    [CrossRef]
  2. P. A. Bélanger, "Packetlike solutions of the homogeneous-wave equation," J. Opt. Soc. Am. A 1, 723-724 (1984).
    [CrossRef]
  3. A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985).
    [CrossRef]
  4. R. W. Ziolkowski, "Exact solutions of the wave equation with complex source locations," J. Math. Phys. 26, 861-863 (1985).
    [CrossRef]
  5. E. Heyman and L. B. Felsen, "Propagating pulsed beam solutions by complex source parameter substitution," IEEE Trans. Antennas Propag. AP-34, 1062-1065 (1986).
    [CrossRef]
  6. E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987).
    [CrossRef]
  7. E. Heyman and B. Z. Steinberg, "Spectral analysis of complex-source pulsed beams," J. Opt. Soc. Am. A 4, 3-10 (1987).
    [CrossRef]
  8. E. Heyman and L. B. Felson, "Complex-source pulsed-beam fields," J. Opt. Soc. Am. A 6, 806-817 (1989).
    [CrossRef]
  9. E. Heyman, "Focus wave modes: a dilemma with causality," IEEE Trans. Antennas Propag. AP-37, 1604-1608 (1989).
    [CrossRef]
  10. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
    [CrossRef]
  11. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, "Aperture realizations of exact solutions to homogeneous-wave equations," J. Opt. Soc. Am. A 10, 75-87 (1993).
    [CrossRef]
  12. R. W. Ziolkowski, "Localized transmission of electromagnetic energy," Phys. Rev. A 39, 2005-2033 (1989).
    [CrossRef] [PubMed]
  13. P. L. Overfelt, "Bessel-Gauss pulses," Phys. Rev. A 44, 3941--3947 (1991).
    [CrossRef] [PubMed]
  14. A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, "On the evanescent fields and the causality of the focus wave modes," J. Math. Phys. 36, 5565-5587 (1995).
    [CrossRef]
  15. E. Heyman and T. Melamed, "Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation," IEEE Trans. Antennas Propag. 42, 518-525 (1994).
    [CrossRef]
  16. C. J. R. Sheppard and X. Gan, "Free-space propagation of femto-second light pulses," Opt. Commun. 133, 1-6 (1997).
    [CrossRef]
  17. Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, "Space-time profiles of an ultrashort pulsed Gaussian beam," IEEE J. Quantum Electron. 33, 566-573 (1997).
    [CrossRef]
  18. M. A. Porras, "Ultrashort pulsed Gaussian light beams," Phys. Rev. E 58, 1086-1093 (1998).
    [CrossRef]
  19. C. J. R. Sheppard, "Bessel pulse beams and focus wave modes," J. Opt. Soc. Am. A 18, 2594-2600 (2001).
    [CrossRef]
  20. C. J. R. Sheppard, "Generalized Bessel pulse beams," J. Opt. Soc. Am. A 19, 2218-2222 (2002).
    [CrossRef]
  21. P. Saari and K. Reivelt, "Generation and classification of localized waves by Lorentz transformations in Fourier space," Phys. Rev. E 69, 036612 (2004).
    [CrossRef]
  22. K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
    [CrossRef]
  23. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).
  24. B. R. A. Nijboer, "The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations," Physica 13, 605-620 (1947).
    [CrossRef]
  25. J. Boersma "On the computation of Lommel's functions of two variables" Math. Comput. 16, 232-238 (1962).
  26. P. Saari, "Superluminal localized waves of electromagnetic field in vacuo," in Time's Arrows, Quantum Measurement and Superluminal Behavior, D. Mugnai, A. Ranfagni, L. S. Schulman eds. (Scientific Monographs: Physics Sciences Series, Italian CNR, Rome, 2001) http://xxx.lanl.gov/abs/physics/0103054.
  27. P. Saari, M. Menert, and H. Valtna, "Photon localization barrier can be overcome," Opt. Commun. 246, 445-450 (2005).
    [CrossRef]
  28. M. A. Porras, "Pulsed light beams in vacuum with superluminal and negative group velocities," Phys. Rev. E 67, 066604 (2003).
    [CrossRef]
  29. P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135 (1997).
    [CrossRef]
  30. R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, "Generation and characterization of spatially and temporally localized few-cycle optical wave packets," Phys. Rev. A 67, 063820 (2003).
    [CrossRef]
  31. H. Valtna, K. Reivelt, and P. Saari, "Methods for generating wideband localized waves of superluminal group velocity," Opt. Commun. 278, 1-7 (2007).
    [CrossRef]

2007 (1)

H. Valtna, K. Reivelt, and P. Saari, "Methods for generating wideband localized waves of superluminal group velocity," Opt. Commun. 278, 1-7 (2007).
[CrossRef]

2005 (1)

P. Saari, M. Menert, and H. Valtna, "Photon localization barrier can be overcome," Opt. Commun. 246, 445-450 (2005).
[CrossRef]

2004 (1)

P. Saari and K. Reivelt, "Generation and classification of localized waves by Lorentz transformations in Fourier space," Phys. Rev. E 69, 036612 (2004).
[CrossRef]

2003 (2)

M. A. Porras, "Pulsed light beams in vacuum with superluminal and negative group velocities," Phys. Rev. E 67, 066604 (2003).
[CrossRef]

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, "Generation and characterization of spatially and temporally localized few-cycle optical wave packets," Phys. Rev. A 67, 063820 (2003).
[CrossRef]

2002 (2)

C. J. R. Sheppard, "Generalized Bessel pulse beams," J. Opt. Soc. Am. A 19, 2218-2222 (2002).
[CrossRef]

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

2001 (1)

1998 (1)

M. A. Porras, "Ultrashort pulsed Gaussian light beams," Phys. Rev. E 58, 1086-1093 (1998).
[CrossRef]

1997 (3)

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

C. J. R. Sheppard and X. Gan, "Free-space propagation of femto-second light pulses," Opt. Commun. 133, 1-6 (1997).
[CrossRef]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, "Space-time profiles of an ultrashort pulsed Gaussian beam," IEEE J. Quantum Electron. 33, 566-573 (1997).
[CrossRef]

1995 (1)

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, "On the evanescent fields and the causality of the focus wave modes," J. Math. Phys. 36, 5565-5587 (1995).
[CrossRef]

1994 (1)

E. Heyman and T. Melamed, "Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation," IEEE Trans. Antennas Propag. 42, 518-525 (1994).
[CrossRef]

1993 (1)

1991 (1)

P. L. Overfelt, "Bessel-Gauss pulses," Phys. Rev. A 44, 3941--3947 (1991).
[CrossRef] [PubMed]

1989 (4)

R. W. Ziolkowski, "Localized transmission of electromagnetic energy," Phys. Rev. A 39, 2005-2033 (1989).
[CrossRef] [PubMed]

E. Heyman and L. B. Felson, "Complex-source pulsed-beam fields," J. Opt. Soc. Am. A 6, 806-817 (1989).
[CrossRef]

E. Heyman, "Focus wave modes: a dilemma with causality," IEEE Trans. Antennas Propag. AP-37, 1604-1608 (1989).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

1987 (2)

1986 (1)

E. Heyman and L. B. Felsen, "Propagating pulsed beam solutions by complex source parameter substitution," IEEE Trans. Antennas Propag. AP-34, 1062-1065 (1986).
[CrossRef]

1985 (2)

A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985).
[CrossRef]

R. W. Ziolkowski, "Exact solutions of the wave equation with complex source locations," J. Math. Phys. 26, 861-863 (1985).
[CrossRef]

1984 (1)

1983 (1)

J. N. Brittingham, "Focus wave modes in homogeneous Maxwell's equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
[CrossRef]

1962 (1)

J. Boersma "On the computation of Lommel's functions of two variables" Math. Comput. 16, 232-238 (1962).

1947 (1)

B. R. A. Nijboer, "The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations," Physica 13, 605-620 (1947).
[CrossRef]

IEEE J. Quantum Electron. (1)

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, "Space-time profiles of an ultrashort pulsed Gaussian beam," IEEE J. Quantum Electron. 33, 566-573 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

E. Heyman and T. Melamed, "Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation," IEEE Trans. Antennas Propag. 42, 518-525 (1994).
[CrossRef]

E. Heyman, "Focus wave modes: a dilemma with causality," IEEE Trans. Antennas Propag. AP-37, 1604-1608 (1989).
[CrossRef]

E. Heyman and L. B. Felsen, "Propagating pulsed beam solutions by complex source parameter substitution," IEEE Trans. Antennas Propag. AP-34, 1062-1065 (1986).
[CrossRef]

J. Appl. Phys. (1)

A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985).
[CrossRef]

J. Math. Phys. (3)

R. W. Ziolkowski, "Exact solutions of the wave equation with complex source locations," J. Math. Phys. 26, 861-863 (1985).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, "On the evanescent fields and the causality of the focus wave modes," J. Math. Phys. 36, 5565-5587 (1995).
[CrossRef]

J. Opt. Soc. Am. A (7)

J. Appl. Phys. (1)

J. N. Brittingham, "Focus wave modes in homogeneous Maxwell's equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
[CrossRef]

Math. Comput. (1)

J. Boersma "On the computation of Lommel's functions of two variables" Math. Comput. 16, 232-238 (1962).

Opt. Commun. (3)

P. Saari, M. Menert, and H. Valtna, "Photon localization barrier can be overcome," Opt. Commun. 246, 445-450 (2005).
[CrossRef]

H. Valtna, K. Reivelt, and P. Saari, "Methods for generating wideband localized waves of superluminal group velocity," Opt. Commun. 278, 1-7 (2007).
[CrossRef]

C. J. R. Sheppard and X. Gan, "Free-space propagation of femto-second light pulses," Opt. Commun. 133, 1-6 (1997).
[CrossRef]

Phys. Rev. A (3)

R. W. Ziolkowski, "Localized transmission of electromagnetic energy," Phys. Rev. A 39, 2005-2033 (1989).
[CrossRef] [PubMed]

P. L. Overfelt, "Bessel-Gauss pulses," Phys. Rev. A 44, 3941--3947 (1991).
[CrossRef] [PubMed]

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, "Generation and characterization of spatially and temporally localized few-cycle optical wave packets," Phys. Rev. A 67, 063820 (2003).
[CrossRef]

Phys. Rev. E (4)

M. A. Porras, "Pulsed light beams in vacuum with superluminal and negative group velocities," Phys. Rev. E 67, 066604 (2003).
[CrossRef]

P. Saari and K. Reivelt, "Generation and classification of localized waves by Lorentz transformations in Fourier space," Phys. Rev. E 69, 036612 (2004).
[CrossRef]

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

M. A. Porras, "Ultrashort pulsed Gaussian light beams," Phys. Rev. E 58, 1086-1093 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

Physica (1)

B. R. A. Nijboer, "The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations," Physica 13, 605-620 (1947).
[CrossRef]

Other (2)

P. Saari, "Superluminal localized waves of electromagnetic field in vacuo," in Time's Arrows, Quantum Measurement and Superluminal Behavior, D. Mugnai, A. Ranfagni, L. S. Schulman eds. (Scientific Monographs: Physics Sciences Series, Italian CNR, Rome, 2001) http://xxx.lanl.gov/abs/physics/0103054.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).

Supplementary Material (5)

» Media 1: MOV (1675 KB)     
» Media 2: MOV (2108 KB)     
» Media 3: MOV (1976 KB)     
» Media 4: MOV (1700 KB)     
» Media 5: MOV (1839 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Relations between k-space variables k, k ρ, k z and parameters k 0, k 1. For each plane wave constituent of the axisymmetric luminal localized wave the length k of its wave vector k depends linearly (see the solid line with slope 1 in the upper part of the plot) on the forward-direction axial projection k z of k, while the radial projection k ρ depends parabolically (solid curve in the lower part) on k z. Such relations between k, k ρ and k z of the plane wave constituents (or Bessel beam constituents) warrant equality to c of the axial group velocity of the wave packet, i. e. its propagation invariance. The straight line and the parabola become thick where the support of the spectrum starts, i. e. where the spectrum according to Eq. (6) becomes nonzero. The dashed straight lines correspond to strictly backward (θ=180°) and forward (θ=0) propagating plane waves with k ρ=0. The double-line arrow depicts the wave vector of length k 1 corresponding to the lowest-frequency plane-wave constituents that propagate under the maximum angle θmax with respect to the axis z. For general properties of supports of spectra of localized waves see [20, 21].

Fig. 2.
Fig. 2.

The frame t=0 of the animated 3D plot (1.6 MB) of the luminal wavepacket constituted from forward-propagating spectral components only (wavenumbers kk 1=2k 0), which has been computed according to Eq. (20) and previous Section. Shown are the real part (upper surface plot) and the modulus (pseudocoloured contour plot on the basal plane) of the wave function in dependence on the coordinate z (increasing from the left to the right) and on a transverse coordinate xρ. The value of the spectral decay parameter z 0=2/k 0. The distance Δ between the grid lines on the basal plane (x,z) is 2λ 0, λ 0=2π/k 0. Δ=1.25µm if we put k 0=105 cm-1, which corresponds to red light. To reveal better the complicated relief of the surface plot a blue diffuse and white specular “lighting”, as well as slight “transparency” and “shininess” have been applied to it. The plane of the contour plot has been slightly risen above the basal plane (x,z) in order to avoid obscuration of the narrow pulse image by the axial grid line, and due to corresponding “parallax” the centrum of the pulse seems to not coincide with the central crossing of the grid lines in the frame shown. [Media 1]

Fig. 3.
Fig. 3.

The frame t=0 of the animated plot (2.1 MB) of the FWM computed by Eq. (21). For explanatory details see the previous figure caption. [Media 2]

Fig. 4.
Fig. 4.

The frame t=0 of the animated plot (1.9 MB) of the truncated-spectrum FWM computed by Eq. (21). [Media 3]

Fig. 5.
Fig. 5.

The frame t=0 of the animated plot (1.8 MB) of the field computed by Eq. (24) with parameters k 1=2.25k 0, k 2=2.75k 0, z 0=0.3/k 0. [Media 4]

Fig. 6.
Fig. 6.

The same as Fig. 5, but k 1=1.25k 0, k 2=1.75k 0. [Media 5]

Equations (39)

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U ( ρ , z , t ) = k min J 0 { k ρ sin [ θ ( k ) ] } exp { ikz cos [ θ ( k ) ] } exp ( ikct ) f ( k ) d k ,
k sin 2 ( θ ( k ) 2 ) = k 0
U ( ρ , z , t ) = exp [ ik 0 ( z + ct ) ] k 0 J 0 ( 2 k 0 ρ cot θ 2 ) exp ( ik 0 ct cot 2 θ 2 ) f ( k ) d k ,
p 2 = cot 2 θ 2 = k k 0 k 0 ,
U ( ρ , z , t ) = k 0 exp [ i k 0 ( z + ct ) ] 0 2 g ( p ) J 0 ( 2 k 0 ρ p ) exp ( i k 0 ct p 2 ) p d p .
f ( k ) = z 0 [ ( s + 1 ) ( k k 1 ) z 0 ] s exp [ ( s + 1 ) ( k k 1 ) z 0 ] , k k 1 ,
= 0 , k < k 1 ,
g ( p ) = z 0 [ ( s + 1 ) ( k 0 k 1 + k 0 p 2 ) z 0 ] s exp [ ( s + 1 ) ( k 1 k 0 ) z 0 ] exp [ ( s + 1 ) k 0 z 0 p 2 ] ,
p [ ( k 1 k 0 ) k 0 ] 1 2 ,
= 0 , p < [ ( k 1 k 0 ) k 0 ] 1 2 .
θ max = 2arc cot { [ ( k 1 k 0 ) k 0 ] 1 2 } = arccos [ ( k 1 2 k 0 ) k 1 ] .
U ( ρ , z , t ) = k 0 z 0 exp [ ( s + 1 ) ( k 1 k 0 ) z 0 ] exp [ ik 0 ( z + ct ) ]
× ( k 1 k 0 ) k 0 2 [ ( s + 1 ) ( k 0 p 2 k 1 + k 0 ) z 0 ] s J 0 ( 2 k 0 ρ p 2 )
× exp [ ( s + 1 ) k 0 z 0 p 2 ] exp ( ik 0 ct p 2 ) p d p .
ν = 2 ρ [ k 0 ( k 1 k 0 ) ] 1 2 ,
u = 2 ( k 1 k 0 ) [ ct i ( s + 1 ) z 0 ] ,
l = p ( k 0 k 1 k 0 ) 1 2 ,
U = ( ρ , z , t ) ( k 1 k 0 ) z 0 [ ( s + 1 ) ( k 1 k 0 ) z 0 ] s exp [ ( s + 1 ) ( k 1 k 0 ) z 0 ik 0 ( z + ct ) ]
× 1 2 ( l 2 1 ) s J 0 ( ν l ) exp ( 1 2 iul 2 ) l d l .
U ( ρ , z , t ) = ( k 1 k 0 ) z 0 exp [ ( k 1 k 0 ) z 0 ] exp [ ik 0 ( z + ct ) ]
× [ 2 iu exp ( iv 2 2 u ) 0 1 2 J 0 ( ν l ) exp ( 1 2 iul 2 ) l d l ] .
U ( ρ , z , t ) = 2 ( k 1 k 0 ) z 0 exp [ ( k 1 k 0 ) z 0 ] exp [ ik 0 ( z + ct ) ] e iu 2 iu L ( u , ν ) ,
L ( u , v ) = V 0 ( u , v ) iV 1 ( u , v ) = m = 0 ( 1 ) m ( v u ) 2 m [ J 2 m ( v ) + iv u J 2 m + 1 ( v ) ] .
U ( ρ , z , t ) = 2 ( k 1 k 0 ) z 0 exp [ ( k 1 k 0 ) z 0 ] exp [ ik 0 ( z + ct ) ]
× [ 1 iu exp ( iv 2 2 u ) e iu 2 u M ( u , v ) ] ,
M ( u , v ) = U 1 ( u , v ) + iU 2 ( u , v ) = m = 0 ( 1 ) m ( u v ) 2 m + 1 [ J 2 m + 1 ( v ) + iu v J 2 m + 2 ( v ) ] .
U ( ρ , z , t ) = 2 ( k 1 k 0 ) z 0 exp [ ( k 1 k 0 ) z 0 ik 0 ( z + ct ) ]
× [ 1 iu exp ( iv 2 2 u ) exp ( iu 4 ) S ( u , v ) ] ,
S ( u , v ) = m = 0 i m ( 2 m + 1 ) j m ( u 4 ) J 2 m + 1 ( v ) v ,
U k 1 , ( ρ , z , t ) = U k 0 , ( ρ , z , t ) U k 0 , k 1 ( ρ , z , t ) ,
U k 0 , ( ρ , z , t ) = exp [ ( k 1 k 0 ) z 0 ]
× { z 0 i ( ct z ) + z 0 exp [ ik 0 ( ct + z ) k 0 ρ 2 i ( ct z ) + z 0 ] }
U k 0 , k 1 ( ρ , z , t ) = exp { 1 2 [ ( k 1 k 0 ) z 0 + i ( k 1 3 k 0 ) z i ( k 1 + k 0 ) ct ] }
× 2 ( k 1 k 0 ) z 0 S ( u , v ) .
f ( k ) = 0 , k < k 1 ,
= z 0 exp [ ( k k 1 ) z 0 ] , k 1 k k 2 ,
= 0 = z 0 exp [ ( k k 1 ) z 0 ] z 0 exp [ ( k 1 k 2 ) z 0 ] exp [ ( k k 2 ) z 0 ] , k > k 2
U k 1 , k 2 ( ρ , z , t ) = U k 1 , ( ρ , z , t ) exp [ ( k 1 k 2 ) z 0 ] U k 2 , ( ρ , z , t ) =
= U k 0 , k 1 ( ρ , z , t ) + exp [ ( k 1 k 2 ) z 0 ] U k 0 , k 2 ( ρ , z , t )

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