Abstract

We point out inconsistencies in previous analyses of multiple-channel nonlinear directional couplers based on Kerr-type media and present a modified set of coupled-mode equations for a three-channel nonlinear coupler that indicate a mode of power switching between channels with increased input power that is both qualitatively and quantitatively different from that predicted previously.

© 1992 Optical Society of America

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References

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  1. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1585 (1982).
    [CrossRef]
  2. G. L. Stegeman, R. M. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
    [CrossRef]
  3. R. Jin, C. L. Chuang, H. M. Gibbs, S. W. Koch, J. N. Polky, G. A. Pubanz, “Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers,” Appl. Phys. Lett. 53, 1791–1793 (1988).
    [CrossRef]
  4. C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).
  5. R. T. Deck, C. Mapalagama, “Improved theory of nonlinear directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).
  6. Y. Chen, A. W. Snyder, D. J. Mitchell, “Ideal optical switching by multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
    [CrossRef]
  7. N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities and chaos in three waveguide nonlinear coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
    [CrossRef]
  8. C. Schmidt-Hattenberger, U. Trutschel, F. Lederer, “Nonlinear switching in multiple-core couplers,” Opt. Lett. 16, 294–296 (1991).
    [CrossRef] [PubMed]
  9. F. J. Fraile-Palaez, G. Assanto, “Coupled mode equations for nonlinear directional couplers,” Appl. Opt. 29, 2216–2217 (1990).
    [CrossRef]
  10. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  11. A. Hardy, W. Striefer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
    [CrossRef]
  12. After adjustment of notational differences it is easily shown that in the simpler case of a two-channel coupler Eqs. (9) and (10) reduce identically to the equations of Ref. 9.
  13. This approximation can be eliminated by expansion of the coupler field E in terms of the supermodes of the total structure rather than in terms of the fields of the separate channels as in Eq. (2). On the other hand, it is shown in Ref. 5 for the case of a two-channel coupler that the nonlinear power switching curves obtained with and without this approximation are in reasonably good agreement.
  14. The two conditions are not identical because of the differences in the regions of integration in the definitions of the coefficients where the integrands of the defining integrals are large. As a consequence of these differences it is possible for the coefficient Q∼n to exceed the coefficient kn while the coefficients R∼n,n′ and T∼n,n′ remain less than the coefficient kn,n′.
  15. These parameter values approximately correspond to the values that characterize the two-channel coupler described in Ref. 4.
  16. This conclusion is consistent with the above analysis of Eqs. (9) and with the conclusions arrived at in Ref. 9.
  17. Further decrease in the coupling between the channels can cause the coupling length L to exceed the limits allowed in the design of a practical coupler.

1991

1990

F. J. Fraile-Palaez, G. Assanto, “Coupled mode equations for nonlinear directional couplers,” Appl. Opt. 29, 2216–2217 (1990).
[CrossRef]

Y. Chen, A. W. Snyder, D. J. Mitchell, “Ideal optical switching by multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[CrossRef]

N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities and chaos in three waveguide nonlinear coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
[CrossRef]

1989

1988

R. Jin, C. L. Chuang, H. M. Gibbs, S. W. Koch, J. N. Polky, G. A. Pubanz, “Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers,” Appl. Phys. Lett. 53, 1791–1793 (1988).
[CrossRef]

1985

A. Hardy, W. Striefer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

1982

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1585 (1982).
[CrossRef]

Assanto, G.

Chen, Y.

Y. Chen, A. W. Snyder, D. J. Mitchell, “Ideal optical switching by multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[CrossRef]

Chuang, C. L.

R. Jin, C. L. Chuang, H. M. Gibbs, S. W. Koch, J. N. Polky, G. A. Pubanz, “Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers,” Appl. Phys. Lett. 53, 1791–1793 (1988).
[CrossRef]

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Deck, R. T.

R. T. Deck, C. Mapalagama, “Improved theory of nonlinear directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Finlayson, N.

N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities and chaos in three waveguide nonlinear coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
[CrossRef]

Fraile-Palaez, F. J.

Fu, R.

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Gibbs, H. M.

R. Jin, C. L. Chuang, H. M. Gibbs, S. W. Koch, J. N. Polky, G. A. Pubanz, “Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers,” Appl. Phys. Lett. 53, 1791–1793 (1988).
[CrossRef]

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Hardy, A.

A. Hardy, W. Striefer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

Harten, P. A.

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Hong, C. S.

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1585 (1982).
[CrossRef]

Jin, R.

R. Jin, C. L. Chuang, H. M. Gibbs, S. W. Koch, J. N. Polky, G. A. Pubanz, “Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers,” Appl. Phys. Lett. 53, 1791–1793 (1988).
[CrossRef]

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Khitrova, G.

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Koch, S. W.

R. Jin, C. L. Chuang, H. M. Gibbs, S. W. Koch, J. N. Polky, G. A. Pubanz, “Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers,” Appl. Phys. Lett. 53, 1791–1793 (1988).
[CrossRef]

Lederer, F.

Lee, S. G.

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Mapalagama, C.

R. T. Deck, C. Mapalagama, “Improved theory of nonlinear directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Mitchell, D. J.

Y. Chen, A. W. Snyder, D. J. Mitchell, “Ideal optical switching by multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[CrossRef]

Peyghambarian, N.

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Polky, J. N.

R. Jin, C. L. Chuang, H. M. Gibbs, S. W. Koch, J. N. Polky, G. A. Pubanz, “Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers,” Appl. Phys. Lett. 53, 1791–1793 (1988).
[CrossRef]

Pubanz, G. A.

R. Jin, C. L. Chuang, H. M. Gibbs, S. W. Koch, J. N. Polky, G. A. Pubanz, “Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers,” Appl. Phys. Lett. 53, 1791–1793 (1988).
[CrossRef]

Schmidt-Hattenberger, C.

Snyder, A. W.

Y. Chen, A. W. Snyder, D. J. Mitchell, “Ideal optical switching by multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[CrossRef]

Sokoloff, J. P.

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Stegeman, G. I.

N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities and chaos in three waveguide nonlinear coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
[CrossRef]

Stegeman, G. L.

Stolen, R. M.

Striefer, W.

A. Hardy, W. Striefer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

Trutschel, U.

Xu, J.

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Appl. Opt.

Appl. Phys. Lett.

N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities and chaos in three waveguide nonlinear coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
[CrossRef]

R. Jin, C. L. Chuang, H. M. Gibbs, S. W. Koch, J. N. Polky, G. A. Pubanz, “Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers,” Appl. Phys. Lett. 53, 1791–1793 (1988).
[CrossRef]

Electron. Lett.

Y. Chen, A. W. Snyder, D. J. Mitchell, “Ideal optical switching by multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[CrossRef]

IEEE J. Quantum Electron.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1585 (1982).
[CrossRef]

J. Lightwave Technol.

A. Hardy, W. Striefer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Other

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

C. L. Chuang, R. Jin, J. Xu, P. A. Harten, G. Khitrova, H. M. Gibbs, S. G. Lee, J. P. Sokoloff, N. Peyghambarian, R. Fu, C. S. Hong, “GaAs/AlGaAs multiple quantum well nonlinear optical directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

R. T. Deck, C. Mapalagama, “Improved theory of nonlinear directional coupler,” Int. J. Nonlinear Opt. Phys. (to be published).

After adjustment of notational differences it is easily shown that in the simpler case of a two-channel coupler Eqs. (9) and (10) reduce identically to the equations of Ref. 9.

This approximation can be eliminated by expansion of the coupler field E in terms of the supermodes of the total structure rather than in terms of the fields of the separate channels as in Eq. (2). On the other hand, it is shown in Ref. 5 for the case of a two-channel coupler that the nonlinear power switching curves obtained with and without this approximation are in reasonably good agreement.

The two conditions are not identical because of the differences in the regions of integration in the definitions of the coefficients where the integrands of the defining integrals are large. As a consequence of these differences it is possible for the coefficient Q∼n to exceed the coefficient kn while the coefficients R∼n,n′ and T∼n,n′ remain less than the coefficient kn,n′.

These parameter values approximately correspond to the values that characterize the two-channel coupler described in Ref. 4.

This conclusion is consistent with the above analysis of Eqs. (9) and with the conclusions arrived at in Ref. 9.

Further decrease in the coupling between the channels can cause the coupling length L to exceed the limits allowed in the design of a practical coupler.

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Figures (7)

Fig. 1
Fig. 1

Coordinate system and geometry for a description of a symmetric three-channel directional coupler.

Fig. 2
Fig. 2

Contributions to the total dielectric function of a symmetric three-channel directional coupler.

Fig. 3
Fig. 3

Graphs of normalized power exiting from channels 1, 2, and 3 of coupler versus input power into channel 1, derived by use of coupled-mode theory with all nonlinear terms in equations retained and parameters consistent with experimental values in Ref. 4.

Fig. 4
Fig. 4

Same quantities as in Fig. 3 derived by use of coupled-mode theory with only the nonlinear self-phase modulation terms in equations retained.

Fig. 5
Fig. 5

Same quantities as in Fig. 3 (present theory) for case in which step size of dielectric increment Δεn is increased by 50%.

Fig. 6
Fig. 6

Same quantities as in Fig. 5 derived with only the self-phase modulation terms in the equations retained.

Fig. 7
Fig. 7

Same quantities as in Fig. 5 (present theory) for case in which separation distance between channels is increased from 2 to 2.5 μm.

Tables (2)

Tables Icon

Table 1 Coefficients in Eqs. (10) and (14) for Two Values of Δεn

Tables Icon

Table 2 Dominant Self- and Cross-Phase Coefficients in Eq. (10) for w = 2.0 μm and d = 2.5 μm

Equations (25)

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[ 2 z 2 + 2 x 2 + ω 2 c 2 ɛ ( x , z ) ] E ( x , z ) = 0 ,
E ( x , z ) = n = 1 N E n ( x , z ) = n = 1 N a n ( z ) ɛ n ( x ) e i β z ,
[ d 2 d x 2 + ω 2 c 2 ɛ n ( x ) β 2 ] ɛ n ( x ) = 0 ,
d x ɛ n ( x ) ɛ n ( x ) = { 1 n = n Δ n = n ± 1 Δ < 1 , 0 otherwise
ɛ ( x ) = ɛ n ( x ) + Δ ɛ n ( x ) .
ɛ ( x , z ) = ɛ ( x ) + ɛ ( 2 ) | E ( x , z ) | 2 ,
E n ( x , z ) = [ a n ( z ) ɛ n ( x ) + a n + 1 ( z ) ɛ n + 1 ( x ) + a n 1 ( z ) ɛ n 1 ( x ) ] exp ( i β z ) .
n = 1 3 [ ɛ n ( x ) e i β z 2 i β d a n ( z ) d z + ω 2 c 2 Δ ɛ n ( x ) ɛ n ( x ) exp ( i β z ) a n ( z ) + ω 2 c 2 ɛ ( 2 ) | E n ( x , z ) | 2 ɛ n ( x ) e i β z a n ( z ) ] = 0 .
i d a 1 d z = k 1 a 1 + k 12 a 2 + ( Q 1 | a 1 | 2 + 2 Q 12 | a 2 | 2 ) a 1 + ( 2 R 12 | a 1 | 2 + T 21 | a 2 | 2 ) a 2 + Q 12 ( a 2 ) 2 a 1 * + R 12 ( a 1 ) 12 a 2 * , i d a 2 d z = k 2 a 2 + k 21 a 1 + k 23 a 3 + [ Q 2 | a 2 | 2 + 2 Q 21 | a 1 | 2 + 2 Q 23 | a 3 | 2 + 2 S 231 Re ( a 3 a 1 * ) ] a 2 + ( 2 R 23 | a 2 | 2 + T 32 | a 3 | 2 ) a 3 + ( 2 R 21 | a 2 | 2 + T 12 | a 1 | 2 ) a 1 + [ Q 21 ( a 1 ) 2 + Q 23 ( a 3 ) 2 ] a 3 * + R 21 ( a 2 ) 2 a 1 * + R 23 ( a 2 ) 2 a 3 * , i d a 3 d z = k 3 a 3 + k 32 a 2 + ( Q 3 | a 3 | 2 + 2 Q 32 | a 2 | 2 ) a 3 + ( 2 R 32 | a 3 | 2 + T 23 | a 2 | 2 ) a 2 + Q 32 ( a 2 ) 2 a 3 * + R 32 ( a 3 ) 2 a 2 * ,
k n = ω 2 c 2 1 2 β d x Δ ɛ n ( x ) ɛ n ( x ) 2 , k n , n ± 1 = ω 2 c 2 1 2 β d x Δ ɛ n ± 1 ( x ) ɛ n ( x ) ɛ n ± 1 ( x ) , Q n = ω 2 c 2 1 2 β d x ɛ ( 2 ) ɛ n ( x ) 4 , Q n , n ± 1 = ω 2 c 2 1 2 β d x ɛ ( 2 ) ɛ n ( x ) 2 ɛ n ± 1 ( x ) 2 , R n , n ± 1 = ω 2 c 2 1 2 β d x ɛ ( 2 ) ɛ n ( x ) 3 ɛ n ± 1 ( x ) , T n ± 1 , n = ω 2 c 2 1 2 β d x Δ ɛ ( 2 ) ɛ n ± 1 ( x ) 3 ɛ n ( x ) , S n , n + 1 , n 1 = ω 2 c 2 1 2 β d x ɛ ( 2 ) ɛ n ( x ) 2 ɛ n + 1 ( x ) ɛ n 1 ( x ) .
ã n ( z ) = a n ( z ) / a 1 ( 0 )
η = 8 π ω c 2 β [ P in ( 0 ) l y ] ,
ɛ ( 2 ) = η ɛ ( 2 ) ,
Q n = η Q n , Q n , n ± 1 = η Q n , n ± 1 , R n , n ± 1 = η R n , n ± 1 , T n ± 1 , n = η T n ± 1 , n , S n , n + 1 , n 1 = η S n , n ± 1 , n 1 ,
I ( x , z ) = c 2 8 π ω Re ( i E E * δ z ) = c 2 8 π ω β | E | 2 .
I ( x , z ) = c 2 8 π ω β [ | ɛ 1 | 2 | a 1 | 2 + | ɛ 2 | 2 | a 2 | 2 + | ɛ 3 | 2 | a 3 | 2 + 2 Re ( a 1 a 2 * ) ɛ 1 ɛ 2 + 2 Re ( a 2 a 3 * ) ɛ 2 ɛ 3 + 2 Re ( a 3 a 1 * ) ɛ 3 ɛ 1 ] .
P 1 ( z ) = l y ( w + d ) / 2 I ( x , z ) d x , P 2 ( z ) = l y ( w + d ) / 2 ( w + d ) / 2 I ( x , z ) d x , P 3 ( z ) = l y ( w + d ) / 2 I ( x , z ) d x .
L = 2 π / [ ( k 1 k 2 ) 2 + 4 ( k 23 k 32 + k 12 k 21 ) ] 1 / 2 .
P in ( 0 ) l y I ( x , 0 ) d x ,
P 1 ( L ) P in ( 0 ) = 8 π ω c 2 β ( w + d ) / 2 d x I ( x , L ) / | a 1 ( 0 ) | 2 P ̂ 1 ( L ) , P 2 ( L ) P in ( 0 ) = 8 π ω c 2 β ( w + d ) / 2 ( w + d ) / 2 d x I ( x , L ) / | a 1 ( 0 ) | 2 P ̂ 2 ( L ) , P 3 ( L ) P in ( 0 ) = 8 π ω c 2 β ( w + d ) / 2 d x I ( x , L ) / | a 1 ( 0 ) | 2 P ̂ 3 ( L ) .
n = n 0 + n ( 2 ) I .
n ɛ 0 1 / 2 + 1 2 ɛ ( 2 ) ɛ 0 1 / 2 | E 2 | = n 0 + 1 2 ɛ ( 2 ) ɛ 0 1 / 2 | E 2 | ,
n ɛ 0 1 / 2 ( 1 N n = 1 N ɛ n ) 1 / 2 .
n n 0 + ɛ ( 2 ) ɛ 0 1 / 2 4 π ω c 2 β I ,
ɛ ( 2 ) = c 2 β ɛ o 1 / 2 4 π ω n ( 2 ) .

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