Abstract

A numerical investigation of pulse propagation in a quantum dot structure in the regime of electromagnetically induced transparency is reported. The quantum dot is described as a cone on top of a wetting layer and the calculated energy levels and dipole moments are used in an effective three-level model. Pulse propagation characteristics such as degree of slowdown, absorption, and pulse distortion are investigated with respect to their dependence on the dephasing rates and pulse width. It is seen how Rabi oscillations can seriously distort the pulse when the spectral width of the pulse becomes too large compared to the width of the EIT window.

© 2007 Optical Society of America

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References

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  1. L. V. Hau,  et al., "Light speed reduction to 17 meters per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
    [CrossRef]
  2. P. C. Ku, C. J. Chang-Hasnain, J. Kim, and S. L. Chuang. "Variable optical buffer using slow light in semiconductor nanostructures," Proc. of the IEEE 91, 1884-1897 (2003).
  3. J. Kim, S. L. Chuang, P. C. Ku, and C. J. Chang-Hasnain, "Slow light using semiconductor quantum dots," J. Phys.: Condens Matter 16, 3727-3735 (2004).
    [CrossRef]
  4. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
    [CrossRef]
  5. R. N. Shakhmuratov and J. Odeurs. "Pulse transformation and time-frequency filtering with electromagnetically induced transparency," Phys. Rev. A 71, 013819 (2005).
    [CrossRef]
  6. P. Jänes, J. Tidstrom, and L. Thylén. "Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media," J. Lightwave Technol. 23, 3893-3899 (2005).
    [CrossRef]
  7. R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Slow-light optical buffers: Capabilities and fundamental limitations," J. Lightwave Technol. 23, 4642-4654 (2005).
    [CrossRef]
  8. P. Borri,  et al., "Ultralong dephasing time in InGaAs quantum dots," Phys. Rev. Lett. 87, 157401 (2001).
    [CrossRef] [PubMed]
  9. J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, "Slow light in a semiconductor waveguide at gigahertz frequencies," Opt. Express 13, 8136-8145 (2005).
    [CrossRef] [PubMed]
  10. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005).
    [CrossRef]
  11. R. V. N. Melnik and M. Willatzen, "Bandstructures of conical quantum dots with wetting layers," Nanotechnology 15, 1-8 (2004).
    [CrossRef]
  12. T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, "Influence of wetting layer wave functions on phonon mediated carrier capture into self-assembled quantum dots, Phys. Rev. B 74, 195342 (2006).
    [CrossRef]
  13. L. A. Coldren and S. W. Corzine, "Diode Lasers and Photonic Integrated Circuits," (Wiley, 1995), pp. 119-123.
  14. P. Arve, P. Jänes, and L. Thylén, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 61, 063809 (2004).
    [CrossRef]
  15. R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Delay-bandwidth product and storage density in slow-light optical buffers," Electron. Lett. 41, (2005).
    [CrossRef]

2006

T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, "Influence of wetting layer wave functions on phonon mediated carrier capture into self-assembled quantum dots, Phys. Rev. B 74, 195342 (2006).
[CrossRef]

2005

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Delay-bandwidth product and storage density in slow-light optical buffers," Electron. Lett. 41, (2005).
[CrossRef]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

R. N. Shakhmuratov and J. Odeurs. "Pulse transformation and time-frequency filtering with electromagnetically induced transparency," Phys. Rev. A 71, 013819 (2005).
[CrossRef]

P. Jänes, J. Tidstrom, and L. Thylén. "Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media," J. Lightwave Technol. 23, 3893-3899 (2005).
[CrossRef]

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Slow-light optical buffers: Capabilities and fundamental limitations," J. Lightwave Technol. 23, 4642-4654 (2005).
[CrossRef]

J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, "Slow light in a semiconductor waveguide at gigahertz frequencies," Opt. Express 13, 8136-8145 (2005).
[CrossRef] [PubMed]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

2004

R. V. N. Melnik and M. Willatzen, "Bandstructures of conical quantum dots with wetting layers," Nanotechnology 15, 1-8 (2004).
[CrossRef]

J. Kim, S. L. Chuang, P. C. Ku, and C. J. Chang-Hasnain, "Slow light using semiconductor quantum dots," J. Phys.: Condens Matter 16, 3727-3735 (2004).
[CrossRef]

P. Arve, P. Jänes, and L. Thylén, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 61, 063809 (2004).
[CrossRef]

2003

P. C. Ku, C. J. Chang-Hasnain, J. Kim, and S. L. Chuang. "Variable optical buffer using slow light in semiconductor nanostructures," Proc. of the IEEE 91, 1884-1897 (2003).

2001

P. Borri,  et al., "Ultralong dephasing time in InGaAs quantum dots," Phys. Rev. Lett. 87, 157401 (2001).
[CrossRef] [PubMed]

1999

L. V. Hau,  et al., "Light speed reduction to 17 meters per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Arve, P.

P. Arve, P. Jänes, and L. Thylén, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 61, 063809 (2004).
[CrossRef]

Berg, T. W.

T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, "Influence of wetting layer wave functions on phonon mediated carrier capture into self-assembled quantum dots, Phys. Rev. B 74, 195342 (2006).
[CrossRef]

Borri, P.

P. Borri,  et al., "Ultralong dephasing time in InGaAs quantum dots," Phys. Rev. Lett. 87, 157401 (2001).
[CrossRef] [PubMed]

Boyd, R. W.

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Chang-Hasnain, C. J.

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Slow-light optical buffers: Capabilities and fundamental limitations," J. Lightwave Technol. 23, 4642-4654 (2005).
[CrossRef]

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Delay-bandwidth product and storage density in slow-light optical buffers," Electron. Lett. 41, (2005).
[CrossRef]

J. Kim, S. L. Chuang, P. C. Ku, and C. J. Chang-Hasnain, "Slow light using semiconductor quantum dots," J. Phys.: Condens Matter 16, 3727-3735 (2004).
[CrossRef]

P. C. Ku, C. J. Chang-Hasnain, J. Kim, and S. L. Chuang. "Variable optical buffer using slow light in semiconductor nanostructures," Proc. of the IEEE 91, 1884-1897 (2003).

Chuang, S. L.

J. Kim, S. L. Chuang, P. C. Ku, and C. J. Chang-Hasnain, "Slow light using semiconductor quantum dots," J. Phys.: Condens Matter 16, 3727-3735 (2004).
[CrossRef]

P. C. Ku, C. J. Chang-Hasnain, J. Kim, and S. L. Chuang. "Variable optical buffer using slow light in semiconductor nanostructures," Proc. of the IEEE 91, 1884-1897 (2003).

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

Gaeta, A. L.

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Gauthier, D. J.

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Hau, L. V.

L. V. Hau,  et al., "Light speed reduction to 17 meters per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

Jänes, P.

P. Jänes, J. Tidstrom, and L. Thylén. "Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media," J. Lightwave Technol. 23, 3893-3899 (2005).
[CrossRef]

P. Arve, P. Jänes, and L. Thylén, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 61, 063809 (2004).
[CrossRef]

Kim, J.

J. Kim, S. L. Chuang, P. C. Ku, and C. J. Chang-Hasnain, "Slow light using semiconductor quantum dots," J. Phys.: Condens Matter 16, 3727-3735 (2004).
[CrossRef]

P. C. Ku, C. J. Chang-Hasnain, J. Kim, and S. L. Chuang. "Variable optical buffer using slow light in semiconductor nanostructures," Proc. of the IEEE 91, 1884-1897 (2003).

Kjær, R.

Kristensen, P.

T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, "Influence of wetting layer wave functions on phonon mediated carrier capture into self-assembled quantum dots, Phys. Rev. B 74, 195342 (2006).
[CrossRef]

Ku, P. C.

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Slow-light optical buffers: Capabilities and fundamental limitations," J. Lightwave Technol. 23, 4642-4654 (2005).
[CrossRef]

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Delay-bandwidth product and storage density in slow-light optical buffers," Electron. Lett. 41, (2005).
[CrossRef]

J. Kim, S. L. Chuang, P. C. Ku, and C. J. Chang-Hasnain, "Slow light using semiconductor quantum dots," J. Phys.: Condens Matter 16, 3727-3735 (2004).
[CrossRef]

P. C. Ku, C. J. Chang-Hasnain, J. Kim, and S. L. Chuang. "Variable optical buffer using slow light in semiconductor nanostructures," Proc. of the IEEE 91, 1884-1897 (2003).

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

Markussen, T.

T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, "Influence of wetting layer wave functions on phonon mediated carrier capture into self-assembled quantum dots, Phys. Rev. B 74, 195342 (2006).
[CrossRef]

Melnik, R. V. N.

R. V. N. Melnik and M. Willatzen, "Bandstructures of conical quantum dots with wetting layers," Nanotechnology 15, 1-8 (2004).
[CrossRef]

Mørk, J.

T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, "Influence of wetting layer wave functions on phonon mediated carrier capture into self-assembled quantum dots, Phys. Rev. B 74, 195342 (2006).
[CrossRef]

J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, "Slow light in a semiconductor waveguide at gigahertz frequencies," Opt. Express 13, 8136-8145 (2005).
[CrossRef] [PubMed]

Odeurs, J.

R. N. Shakhmuratov and J. Odeurs. "Pulse transformation and time-frequency filtering with electromagnetically induced transparency," Phys. Rev. A 71, 013819 (2005).
[CrossRef]

Shakhmuratov, R. N.

R. N. Shakhmuratov and J. Odeurs. "Pulse transformation and time-frequency filtering with electromagnetically induced transparency," Phys. Rev. A 71, 013819 (2005).
[CrossRef]

Thylén, L.

P. Jänes, J. Tidstrom, and L. Thylén. "Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media," J. Lightwave Technol. 23, 3893-3899 (2005).
[CrossRef]

P. Arve, P. Jänes, and L. Thylén, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 61, 063809 (2004).
[CrossRef]

Tidstrom, J.

Tromborg, B.

T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, "Influence of wetting layer wave functions on phonon mediated carrier capture into self-assembled quantum dots, Phys. Rev. B 74, 195342 (2006).
[CrossRef]

Tucker, R. S.

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Slow-light optical buffers: Capabilities and fundamental limitations," J. Lightwave Technol. 23, 4642-4654 (2005).
[CrossRef]

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Delay-bandwidth product and storage density in slow-light optical buffers," Electron. Lett. 41, (2005).
[CrossRef]

van der Poel, M.

Willatzen, M.

R. V. N. Melnik and M. Willatzen, "Bandstructures of conical quantum dots with wetting layers," Nanotechnology 15, 1-8 (2004).
[CrossRef]

Willner, A. E.

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Yvind, K.

Electron. Lett.

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Delay-bandwidth product and storage density in slow-light optical buffers," Electron. Lett. 41, (2005).
[CrossRef]

J. Lightwave Technol.

P. Jänes, J. Tidstrom, and L. Thylén. "Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media," J. Lightwave Technol. 23, 3893-3899 (2005).
[CrossRef]

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Slow-light optical buffers: Capabilities and fundamental limitations," J. Lightwave Technol. 23, 4642-4654 (2005).
[CrossRef]

J. Phys.: Condens Matter

J. Kim, S. L. Chuang, P. C. Ku, and C. J. Chang-Hasnain, "Slow light using semiconductor quantum dots," J. Phys.: Condens Matter 16, 3727-3735 (2004).
[CrossRef]

Nanotechnology

R. V. N. Melnik and M. Willatzen, "Bandstructures of conical quantum dots with wetting layers," Nanotechnology 15, 1-8 (2004).
[CrossRef]

Nature

L. V. Hau,  et al., "Light speed reduction to 17 meters per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Opt. Express

Phys. Rev. A

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

R. N. Shakhmuratov and J. Odeurs. "Pulse transformation and time-frequency filtering with electromagnetically induced transparency," Phys. Rev. A 71, 013819 (2005).
[CrossRef]

P. Arve, P. Jänes, and L. Thylén, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 61, 063809 (2004).
[CrossRef]

Phys. Rev. B

T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, "Influence of wetting layer wave functions on phonon mediated carrier capture into self-assembled quantum dots, Phys. Rev. B 74, 195342 (2006).
[CrossRef]

Phys. Rev. Lett.

P. Borri,  et al., "Ultralong dephasing time in InGaAs quantum dots," Phys. Rev. Lett. 87, 157401 (2001).
[CrossRef] [PubMed]

Proc. of the IEEE

P. C. Ku, C. J. Chang-Hasnain, J. Kim, and S. L. Chuang. "Variable optical buffer using slow light in semiconductor nanostructures," Proc. of the IEEE 91, 1884-1897 (2003).

Rev. Mod. Phys.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

Other

L. A. Coldren and S. W. Corzine, "Diode Lasers and Photonic Integrated Circuits," (Wiley, 1995), pp. 119-123.

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Figures (7)

Fig. 1.
Fig. 1.

(a) Schematic of the QD system. The dimensions of the QD and WL are indicated along with the outer numerical boundaries, R 0 and Lz . (b) Schematic of the three-level ladder scheme used for obtaining EIT. The TME for the two transitions and relevant semiconductor energies are indicated, see Table 1.

Fig. 2.
Fig. 2.

(color) Contour plot for the (a) interband and (b) intersubband transition matrix elements versus wetting layer thickness, d, and QD height, h, calculated from Eq. (5) and (6) using parameters from Table 1.

Fig. 3.
Fig. 3.

(a) Real, χ′, and imaginary, χ″, part of the complex susceptibility as a function of the normalized detuning -∆pc. (b) Plot of the group index n g (solid curves) and the imaginary part of the susceptibility χ″ (dashed curves), which is proportional to the absorption, as a function of the Rabi frequency Ωc. The dephasing rates have been put equal γ= γ12 = γ13 and are varied in the different series. Both plots are analytical results obtained for a CW probe field. Parameters are as in Table 1.

Fig. 4.
Fig. 4.

(color) Contour plots of field envelopes (left) and corresponding envelopes of the 1-2 polarization, Im((σ12), (right) for a pulse propagating through a 300 μm thick QD medium, which is situated in-between the two black lines. Parameter values are given in Table 1 and γ12 = γ13 = 10 × 109 s-1 . The two upper figures, (a-b), show a pulse which is spectrally narrow compared to the Rabi splitting (W ω = 2Ωc/10). The pulse propagates through the QD medium with high group index, without experiencing any dispersion. The two lower plots, (c-d), show a pulse which is spectrally wide (W ω = 2Ωc/0.7) and hence it interacts strongly with the split ∣2〉 state, which induces Rabi oscillations that break the pulse apart.

Fig. 5.
Fig. 5.

Dependence of pulse characteristics on dephasing rate for different spectral widths of the input pulse (left plots) and temporal pulse shapes after transmission through the EIT medium for specific operation points (right plots). The Rabi frequency of the coupling field is kept fixed at Ωc = 300 × 109 s-1, and the 1-3 dephasing rate is fixed at γ13 = 10 × 109 s-1; other parameters are given in Table 1. (a) shows the group index inferred from the simulations, with the group index for a CW probe shown for comparison. (b) shows the relative broadening, σrel = σoutin, of the pulse after passing through the active material. (c) shows the transmission of the pulse, T = ��out/��in. (d) and (e) show the output pulse shapes at the specific points indicated in (b).

Fig. 6.
Fig. 6.

(a-b) Calculated (CW) susceptibility for the parameters corresponding to points #2 and #4 in Fig. 5(b) where γ13 = 10 × 109 s-1. (a) illustrates the appearance of a “normal” window of EIT susceptibility and in (b) the line width has been increased to exceed the Rabi splitting. A dip in the absorption spectrum and a strong index dispersion are still seen and this leads to the long tails for pulse #3 and #4 in Fig. 5(d). (c) Probe temporal profiles for points #8 and #9 in Fig. 7(f).

Fig. 7.
Fig. 7.

Same as Fig. 5 (a)–(c), except that here γ13 = 1 × 109 s-1 in (a)-(c) and γ13 = 100 × 109 s-1 in(d)-(f).

Tables (1)

Tables Icon

Table 1. Parameter values used in the numerical simulations. In the table me is the free electron mass and e is the electron charge.

Equations (18)

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

t ρ nm = 1 ih̄ n [ Ĥ 0 + Ĥ I , ρ ˆ ] m γ nm ρ nm .
Ĥ I = d ˆ E = ey 2 { E ˜ p exp [ i ( k p x ω p t ) ] + E ˜ c exp [ i ( k c x ω c t ] + c . c } ,
ρ 12 = σ 12 exp [ i ( k p x ω p t ) ] , ρ 13 = σ 13 exp [ i [ ( k p + k c ) x ( ω p + ω c ) t ] ] , ρ 23 = σ 23 exp [ i ( k c x ω c t ) ] .
t ρ 11 = i Ω p * σ 12 i Ω p σ 12 * + γ 22 ρ 22 + γ 33 ρ 33 ,
t ρ 22 = i Ω p * σ 12 i Ω p σ 12 * + i Ω c * σ 23 - i Ω c * σ 23 - γ 22 ρ 22
t ρ 33 = i Ω c * σ 23 i Ω c * σ 23 γ 33 ρ 33 ,
t σ 12 = i Ω p ( ρ 11 ρ 22 ) + i Ω c * σ 13 ( γ 12 i Δ p ) σ 12 ,
t σ 13 = i Ω c σ 12 i Ω p σ 23 [ γ 13 i ( Δ p + Δ c ) ] σ 13 ,
t σ 23 = i Ω c ( ρ 22 ρ 33 ) i Ω p * σ 13 ( γ 23 i Δ c ) σ 23 ,
P = μ 12 N σ 12 exp [ i ( k p x ω p t ) ] + c . c . ,
( x + n b c t ) E ˜ p = i μ 0 ω p c μ 12 N Γ n b σ 12 * ,
[ 2 2 ( 1 m * ( r ) ) + V ( r ) ] F ( r ) = EF ( r ) ,
F nm j ( r ) = r | jnm = 1 2 π exp ( imφ ) f nm j r z ,
1 y 2 u v y u c F 10 v | F 10 c = u v y u c [ f 10 v ] * f 10 c rdrdz ,
2 y 3 u c | u c ( F 10 c y c 1,1 F 10 c y c 1 , 1 ) 1 i 2 = 1 2 [ f 10 c ] * f 11 c r 2 drdz ,
χ = μ 12 2 N ε 0 h ¯ i γ 13 Δ p ( γ 12 + i Δ p ) ( γ 13 + i Δ p ) + Ω c 2 .
E ˜ p 0 t = 𝓔 p exp ( 2 ln 2 t 2 W t 2 ) ,
σ = t 2 t 2 , t = 𝓝 1 t E ˜ p 2 dt , t 2 = 𝓝 1 t 2 E ˜ p 2 dt 𝓝 = E ˜ p 2 dt .

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