Abstract

In a semiconductor illuminated by a strong terahertz (THz) field, the electron–hole pairs excited by linear polarized near-infrared (NIR) laser can recombine to emit high-order THz sideband. Previous experimental results have shown, under the same condition of excitation intensity, the polarization direction of the NIR laser could affect the sideband intensity. In this letter, we theoretically investigate the effect of the NIR laser polarization direction on high-order terahertz sideband generation in bulk GaAs.

© 2015 Optical Society of America

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References

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  1. R.-B. Liu and B.-F. Zhu, “Adiabatic stabilization of excitons in an intense terahertz laser,” Phys. Rev. B 66(3), 033106 (2002).
    [Crossref]
  2. J.-Y. Yan, R.-B. Liu, and Z. Bang-fen, “Exciton absorption in semiconductor superlattices in a strong longitudinal THz field,” New J. Phys. 11(8), 083004 (2009).
    [Crossref]
  3. Y. Jie-Yun, “Dynamics of a pair of electron and hole in semiconductor superlattice under an intense electric field,” Chin. Phys. B 17(12), 4640–4644 (2008).
    [Crossref]
  4. F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
    [Crossref]
  5. F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
    [Crossref]
  6. B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
    [Crossref] [PubMed]
  7. R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
    [Crossref]
  8. J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B 78(7), 075204 (2008).
    [Crossref]
  9. J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5, 4854 (2014).
    [Crossref] [PubMed]
  10. X.-T. Xie, B.-F. Zhu, and R.-B. Liu, “Effects of excitation frequency on high-order terahertz sideband generation in semiconductors,” New J. Phys. 15(10), 105015 (2013).
    [Crossref]
  11. J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B 89(12), 121202 (2014).
    [Crossref]
  12. B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
    [Crossref]
  13. H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
    [Crossref] [PubMed]

2014 (3)

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5, 4854 (2014).
[Crossref] [PubMed]

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B 89(12), 121202 (2014).
[Crossref]

2013 (4)

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

X.-T. Xie, B.-F. Zhu, and R.-B. Liu, “Effects of excitation frequency on high-order terahertz sideband generation in semiconductors,” New J. Phys. 15(10), 105015 (2013).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

2012 (1)

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

2009 (1)

J.-Y. Yan, R.-B. Liu, and Z. Bang-fen, “Exciton absorption in semiconductor superlattices in a strong longitudinal THz field,” New J. Phys. 11(8), 083004 (2009).
[Crossref]

2008 (2)

Y. Jie-Yun, “Dynamics of a pair of electron and hole in semiconductor superlattice under an intense electric field,” Chin. Phys. B 17(12), 4640–4644 (2008).
[Crossref]

J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B 78(7), 075204 (2008).
[Crossref]

2007 (1)

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

2002 (1)

R.-B. Liu and B.-F. Zhu, “Adiabatic stabilization of excitons in an intense terahertz laser,” Phys. Rev. B 66(3), 033106 (2002).
[Crossref]

Bang-fen, Z.

J.-Y. Yan, R.-B. Liu, and Z. Bang-fen, “Exciton absorption in semiconductor superlattices in a strong longitudinal THz field,” New J. Phys. 11(8), 083004 (2009).
[Crossref]

Banks, H.

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Crosse, J. A.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5, 4854 (2014).
[Crossref] [PubMed]

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B 89(12), 121202 (2014).
[Crossref]

Gossard, A. C.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Jie-Yun, Y.

Y. Jie-Yun, “Dynamics of a pair of electron and hole in semiconductor superlattice under an intense electric field,” Chin. Phys. B 17(12), 4640–4644 (2008).
[Crossref]

Liu, R.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Liu, R. B.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5, 4854 (2014).
[Crossref] [PubMed]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

Liu, R.-B.

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B 89(12), 121202 (2014).
[Crossref]

X.-T. Xie, B.-F. Zhu, and R.-B. Liu, “Effects of excitation frequency on high-order terahertz sideband generation in semiconductors,” New J. Phys. 15(10), 105015 (2013).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

J.-Y. Yan, R.-B. Liu, and Z. Bang-fen, “Exciton absorption in semiconductor superlattices in a strong longitudinal THz field,” New J. Phys. 11(8), 083004 (2009).
[Crossref]

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

R.-B. Liu and B.-F. Zhu, “Adiabatic stabilization of excitons in an intense terahertz laser,” Phys. Rev. B 66(3), 033106 (2002).
[Crossref]

Mack, S.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Sherwin, M. S.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5, 4854 (2014).
[Crossref] [PubMed]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

Xie, X.-T.

X.-T. Xie, B.-F. Zhu, and R.-B. Liu, “Effects of excitation frequency on high-order terahertz sideband generation in semiconductors,” New J. Phys. 15(10), 105015 (2013).
[Crossref]

Xu, X.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5, 4854 (2014).
[Crossref] [PubMed]

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

Yan, J.-Y.

J.-Y. Yan, R.-B. Liu, and Z. Bang-fen, “Exciton absorption in semiconductor superlattices in a strong longitudinal THz field,” New J. Phys. 11(8), 083004 (2009).
[Crossref]

J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B 78(7), 075204 (2008).
[Crossref]

Yang, F.

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Zaks, B.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

Zhu, B.-F.

X.-T. Xie, B.-F. Zhu, and R.-B. Liu, “Effects of excitation frequency on high-order terahertz sideband generation in semiconductors,” New J. Phys. 15(10), 105015 (2013).
[Crossref]

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

R.-B. Liu and B.-F. Zhu, “Adiabatic stabilization of excitons in an intense terahertz laser,” Phys. Rev. B 66(3), 033106 (2002).
[Crossref]

AIP Conf. Proc. (1)

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

Appl. Phys. Lett. (1)

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

Chin. Phys. B (1)

Y. Jie-Yun, “Dynamics of a pair of electron and hole in semiconductor superlattice under an intense electric field,” Chin. Phys. B 17(12), 4640–4644 (2008).
[Crossref]

Nat. Commun. (1)

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5, 4854 (2014).
[Crossref] [PubMed]

Nature (1)

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

New J. Phys. (4)

X.-T. Xie, B.-F. Zhu, and R.-B. Liu, “Effects of excitation frequency on high-order terahertz sideband generation in semiconductors,” New J. Phys. 15(10), 105015 (2013).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

J.-Y. Yan, R.-B. Liu, and Z. Bang-fen, “Exciton absorption in semiconductor superlattices in a strong longitudinal THz field,” New J. Phys. 11(8), 083004 (2009).
[Crossref]

Phys. Rev. B (3)

R.-B. Liu and B.-F. Zhu, “Adiabatic stabilization of excitons in an intense terahertz laser,” Phys. Rev. B 66(3), 033106 (2002).
[Crossref]

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B 89(12), 121202 (2014).
[Crossref]

J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B 78(7), 075204 (2008).
[Crossref]

Phys. Rev. Lett. (1)

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz Electron-Hole Recollisions in GaAs/AlGaAs Quantum Wells: Robustness to Scattering by Optical Phonons and Thermal Fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

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Figures (1)

Fig. 1
Fig. 1 The sideband intensity when the polarization direction of the NIR laser is parallel and perpendicular to the THz field. We have chosen γ 2 =2.5meV , d=6.7e A , the effective mass of electrons of conduction band (mc = 0.067me), heavy hole band (mh = 0.45me) and light hole band (ml = 0.082me), the photon energy of the THz field is 2.4meV and its strength is 11kV cm−1, and the NIR laser detuning is Ω E g =0 .

Equations (16)

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i ψ( p,t ) t ={ 1 2μ [ peA( t ) ] 2 + E g i γ 2 }ψ( p,t )+ d cv E( t ),
P( t )= d ^ ( t ) = 1 ( 2π ) 3 d cv ψ( p,t )dp ,
ψ( p,t )= i 0 e i( E g / i γ 2 )τi tτ t d t [ peA( t ) ] 2 / 2μ d cv E( tτ )dτ .
P( t )= i ( 2π ) 3 d cv * d cv E 0 e iS θ( τ )dτdp ,
S= tτ t d t [ peA( t ) ] 2 / 2μ ( E g / Ωi γ 2 )τΩt.
| +/ =|S| , | 3 2 ,± 1 2 = 2 3 | Z p | | X p ±i| Y p 6 | , | 3 2 ,± 3 2 = | X p ±i| Y p 2 | .
[ | X p | Y p | Z p ]=[ cos θ p cos ϕ p cos θ p sin ϕ p sin θ p sin ϕ p cos ϕ p 0 sin θ p cos ϕ p sin θ p sin ϕ p cos θ p ][ |X |Y |Z ].
d ch * d ch = d 2 [ e ^ x e ^ x ( cos 2 θ p cos 2 ϕ p + sin 2 ϕ p )+ e ^ y e ^ y ( cos 2 θ p sin 2 ϕ p + cos 2 ϕ p )            ( e ^ x e ^ y + e ^ y e ^ x ) sin 2 θ p sin ϕ p cos ϕ p ( e ^ x e ^ z + e ^ z e ^ x )sin θ p cos θ p cos ϕ p     ,             + e ^ z e ^ z sin 2 θ p ( e ^ y e ^ z + e ^ z e ^ y )sin θ p cos θ p sin ϕ p ]
d cl * d cl = 1 3 d 2 [ e ^ x e ^ x ( 1+3 sin 2 θ p cos 2 ϕ p )+ e ^ y e ^ y ( 1+3 sin 2 θ p sin 2 ϕ p )            +( e ^ x e ^ y + e ^ y e ^ x )3 sin 2 θ p sin ϕ p cos ϕ p +( e ^ x e ^ z + e ^ z e ^ x )3sin θ p cos θ p cos ϕ p .            + e ^ z e ^ z ( 1+3 cos 2 θ p ) +( e ^ y e ^ z + e ^ z e ^ y )3sin θ p cos θ p sin ϕ p ]
P ( t )= i d 2 E 0 e ^ z ( 2π ) 3 4 sin 2 θ p e i S ch θ( τ )dτ p 2 sin θ p dpd θ p d ϕ p          + i d 2 E 0 e ^ z 3 ( 2π ) 3 4 ( 1+3 cos 2 θ p ) e i S cl θ( τ )dτ p 2 sin θ p dpd θ p d ϕ p ,
S cj = tτ t d t [ p 2 2pcos θ p + e 2 A 2 ( t ) ] / 2 μ cj ( E g / Ωi γ 2 )τΩt.
P ( t )= i d 2 E 0 e ^ x ( 2π ) 3 4 ( cos 2 θ p cos 2 ϕ p + sin 2 ϕ p ) e i S ch θ( τ )dτ p 2 sin θ p dpd θ p d ϕ p          + i d 2 E 0 e ^ x 3 ( 2π ) 3 4 ( 1+3 sin 2 θ p cos 2 ϕ p ) e i S cl θ( τ )dτ p 2 sin θ p dpd θ p d ϕ p .
P (Ω+2N ω 0 )= i N D E p { μ ch 1 τ Θ ch J N ( Φ ch ) sin 3 θ p θ(τ)d θ p dτ            + U ch μ ch Θ ch γ 2 [ 2i J N ( Φ ch ) J N1 ( Φ ch )+ J N+1 ( Φ ch ) ]                   × sin 3 θ p cos 2 θ p θ(τ)d θ p dτ           + μ cl 3 1 τ Θ cl J N ( Φ cl )sin θ p ( 1+3 cos 2 θ p ) θ(τ)d θ p dτ           + U cl μ cl 3 Θ cl γ 2 [ 2i J N ( Φ cl ) J N1 ( Φ cl )+ J N+1 ( Φ cl ) ]                    ×sin θ p cos 2 θ p ( 1+3 cos 2 θ p )θ(τ)d θ p dτ },
P (Ω+2N ω 0 )     = i N D E p { μ ch 2 1 τ Θ ch J N ( Φ ch )sin θ p ( cos 2 θ p +1 )θ(τ)d θ p dτ     + U ch μ ch 2 Θ ch γ 2 [ 2i J N ( Φ ch ) J N1 ( Φ ch )+ J N+1 ( Φ ch ) ]                    ×sin θ p cos 2 θ p ( cos 2 θ p +1 )θ(τ)d θ p dτ     + μ cl 6 1 τ Θ cl J N ( Φ cl )sin θ p ( 2+3 sin 2 θ p )θ(τ)d θ p dτ     + U cl μ cl 6 Θ cl γ 2 [ 2i J N ( Φ cl ) J N1 ( Φ cl )+ J N+1 ( Φ cl ) ]                     ×sin θ p cos 2 θ p ( 2+3 sin 2 θ p )θ(τ)d θ p dτ }.
Φ cj = U cj τγ[ cos( ω 0 τ 2 ) cos 2 θ p γ ], Θ cj = π μ cj 2iτ e i( Λ cj +N ω 0 )τ ,D= d 2 (2π) 2 4 ,    Λ cj =Ω E g U cj +i γ 2 + U cj cos 2 θ p γ 2 , U cj = e 2 F 2 4 μ cj ω 0 2 ,γ= sin( ω 0 τ /2 ) ω 0 τ /2 .
χ 2N λ = P λ ( Ω+2N ) ε 0 E 0 ( λ=, ).

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