Abstract

A new approach towards the design of optimized distributed Bragg reflector (DBR) structures is proposed by taking advantage of recent developments related to the concept of parity-time (PT) in optics. This approach is based on using unidirectional gratings that provide coupling between co-propagating modes. Such couplers with PT symmetric gratings can provide co-directional mode coupling occurring only in one direction. This specific coupling property is achieved through a combined contribution of superimposed index and gain/loss modulations with same grating periodicity, but shifted with respect to one another by a quarter periods. Based on the transfer matrix approach, the transmission and reflection properties of the structure are modeled. One of the unique characteristics of the structure is very low lasing threshold. Such low threshold can be achieved by 100% reflectivity of the both Bragg grating mirrors, and by releasing the amplified signal in one single direction through a PT symmetric grating assisted co-directional coupler. Besides the lasing applications, the proposed structure can be implemented as an optical memory unit of replicating any input optical waveform.

© 2013 Optical Society of America

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  1. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32(17), 2632–2634 (2007).
    [CrossRef] [PubMed]
  2. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett.100(10), 103904 (2008).
    [CrossRef] [PubMed]
  3. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett.80(24), 5243 (1998).
    [CrossRef]
  4. E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A84(1), 013818 (2011).
    [CrossRef]
  5. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(3), 2963–2975 (1996).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2012

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature488(7410), 167–171 (2012).
[CrossRef] [PubMed]

M. A. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmtric laser amplifiers,” Opt. Lett.37, 764–766 (2012).

2011

E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A84(1), 013818 (2011).
[CrossRef]

2008

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett.100(10), 103904 (2008).
[CrossRef] [PubMed]

2007

2005

2004

1998

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett.80(24), 5243 (1998).
[CrossRef]

1996

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(3), 2963–2975 (1996).
[CrossRef] [PubMed]

1995

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

Akulova, Y.

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

Azaña, J.

Backbom, L.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

Barton, J. S.

Bélanger, N.

Bender, C. M.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett.80(24), 5243 (1998).
[CrossRef]

Berglind, E.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett.80(24), 5243 (1998).
[CrossRef]

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

Christodoulides, D. N.

M. A. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmtric laser amplifiers,” Opt. Lett.37, 764–766 (2012).

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature488(7410), 167–171 (2012).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett.100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Coldren, C. W.

Coldren, L. A.

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett.100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

Feng, L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

Fish, G. A.

Graefe, E. M.

E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A84(1), 013818 (2011).
[CrossRef]

Greenberg, M.

Johansson, L.

Jones, H. F.

E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A84(1), 013818 (2011).
[CrossRef]

Klinga, T.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

Kulishov, M.

Laniel, J. M.

LiKamWa, P.

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett.100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Miri, M. A.

M. A. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmtric laser amplifiers,” Opt. Lett.37, 764–766 (2012).

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett.100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Nilsson, S.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

Oliveira, J. E.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

Onishchukov, G.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Orenstein, M.

Peschel, U.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Plant, D. V.

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(3), 2963–2975 (1996).
[CrossRef] [PubMed]

Regensburger, A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Rigole, P.-J.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

Stalnacke, B.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

Stoltz, B.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

Wallin, J.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

Xu, Y.-L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

IEEE Photon. Technol. Lett.

P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett.7(7), 697–699 (1995).
[CrossRef]

J. Lightwave Technol.

Nat. Mater.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater.12(2), 108–113 (2012).
[CrossRef] [PubMed]

Nature

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature488(7410), 167–171 (2012).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. A

E. M. Graefe and H. F. Jones, “PT-symmetric sinusoidal optical lattices at the symmetry breaking threshold,” Phys. Rev. A84(1), 013818 (2011).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(3), 2963–2975 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett.100(10), 103904 (2008).
[CrossRef] [PubMed]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett.80(24), 5243 (1998).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Grating assisted codirectional coupler (GACC). The propagation constants of each waveguides are given by βi . The coupler is asynchronous 1 ≠ β2). The grating length is given by LG.

Fig. 2
Fig. 2

Transmission spectra of a traditional grating assisted coupler(a) where κα = 0 and (b) uni-directional grating assisted coupler where κn = κα GACCs for the bar-state (solid, red) and the cross-state (dash, blue) for κ12LG = π/2 and LG = 30 mm and the central wavelength of 1550 nm. The signal is launched into Port A.

Fig. 3
Fig. 3

Schematic structure of a DBR laser with (a) traditional index grating GACC and (b) PT-symmetric GACC as well as mode interaction pattern.

Fig. 4
Fig. 4

Schematic structure of a Fabry-Perot cavity based signal repeater with PT-symmetric co-directional coupler as well as mode interaction patterns.

Fig. 5
Fig. 5

(a) Output transmission spectrum of the proposed Fabry-Perot structure with PT-symmetric coupling. (b) The reflection spectrum of two identical Bragg grating-reflectors (red, solid), cross-transmission spectrum of the PT-symmetric codirectional coupler (blue, dash) and the spectrum of the input Gaussian pulse 10 ps FWHM.

Fig. 6
Fig. 6

Temporal response of the proposed Fabry-Perot structure with PT-symmetric coupling to an input Gaussian pulse 10 ps FWHM. (a) First ten replicas and (b) 508th – 518th replicas.

Equations (23)

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Δn=Δ n 1 cos(kz)jΔ α 1 sin(kz),
[ D C ]=[ Т 11 Т 12 Т 21 Т 22 ][ A B ],
T 11 =[ cos( γ G L G )+j Δβ γ G sin( γ G L G ) ]exp[ j( β 1 Δβ ) L G ], T 12 =j κ 12 γ G sin( γ G L G )exp[ j( β 1 Δβ ) L G ], T 21 =j κ 21 γ G sin( γ G L G )exp[ j( β 2 +Δβ ) L G ], T 22 =[ cos( γ G L G )j Δβ γ G sin( γ G L G ) ]exp[ j( β + 2 Δβ ) L G ],
T 11 1 =[ cos( γ G L G )j Δβ γ G sin( γ G L G ) ]exp[ j( β 1 Δβ ) L G ], T 12 1 =j κ 12 γ G sin( γ G L G )exp[ j( β 2 +Δβ ) L G ], T 21 1 =j κ 21 γ G sin( γ G L G )exp[ j( β 1 Δβ ) L G ], T 22 1 =[ cos( γ G L G )+j Δβ γ G sin( γ G L G ) ]exp[ j( β 2 +Δβ ) L G ].
T=[ exp[ j β 1 L G ] 0 j κ 21 Δβ sin(ΔβL ) G exp[ j( β 1 + β 2 2 π Λ G ) L G ] exp[ j β 2 L G ] ].
T 1 =[ exp[ j β 1 L G ] 0 j κ 21 Δβ sin(Δβ L G )exp[ j( β 1 + β 2 2 + π Λ G ) L G ] exp[ j β 2 L G ] ].
| r 1 (ω)|| r 2 (ω)|exp(j( φ 1 (ω)+ φ 2 (ω))exp(2d( g 0 (ω) α 0 (ω)))exp(2jβ(ω)d)=1
g 0 ( ω і ) α 0 ( ω і )= 1 2d ln( 1 | r 1 ( ω і )|| r 2 ( ω і )| )
B (+) (L) B () (L) = r (+) = m 12 (1) m 22 (1) ,
B () (L+d) B (+) (L+d) = r () = m 21 (2) m 22 (2)
[ A (+) (L+ d 1 + L G ) B (+) (L+ d 1 + L G ) ]=[ Т 11 Т 12 Т 21 Т 22 ][ exp(j β 1 (L+ d 1 )) B (+) (L+ d 1 ) ],
[ A () (L+ d 1 ) B () (L+ d 1 ) ]=[ Т 11 1 Т 12 1 Т 21 1 Т 22 1 ][ 0 B () (L+ d 1 + L G ) ].
B (+) (L+ d 1 )= B (+) (L)exp(j β 2 d 1 ), B (+) (L+d)= B (+) (L+ d 1 + L G )exp(j β 2 ( d 1 L G )), B () (L+ L G + d 1 )= B (+) (L+d)exp(j β 2 ( d 2 L G )),
A () (L+ d 1 )= T 21 1 T 21 m 21 (2) m 22 (2) exp(j β 2 (d2 L G + d 2 )) 1 T 11 1 T 22 m 21 (2) m 12 (1) m 22 (2) m 22 (1) exp(2j β 2 (d L G )) .
Т 11 1 =exp(j β 2 L G ); T 22 =exp(j β 2 L G ),
A () (L+ d 1 )= κ 12 2 Δ β 2 r () sin 2 (Δβ L G )exp(j( β 1 β 2 ) L G )exp(2j β 2 d) 1 r () r (+) exp(2j β 2 d) ,
n=0 x n = 1 1x
A () (L+ d 1 )= κ 12 2 Δ β 2 r () sin 2 (Δβ L G )exp(j(2 β 2 d+( β 1 β 2 ) L G )) m=0 ( r () r (+) ) m exp(2jm β 2 d) ,
| r () r (+) exp(2jm β 2 d) |>1,
r (±) = j(κ/γ)sinh(γL) cosh(γL)j(σ/γ)sinh(γL)
a () (L+ d 1 ,t)= F 1 { H in (ω) A () (L+ d 1 ,ω)}
h in (t)=exp[ t 2 τ 0 2 2ln(2) ]exp(j ω 0 t)
H in (ω)= ( π τ 0 2 2ln(2) ) 1/2 exp[ τ 0 2 8ln(2) (ω ω 0 ) 2 ]

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