Abstract

We report numerical results for the spatiotemporal dynamics of circular-grating distributed-feedback devices with an intensity-dependent refractive index. For the specific geometry and range of variables considered, we find that modulational instabilities do not propagate in these structures. This is due to the 1/r intensity distribution associated with cylindrical waves, which reduces the effectiveness of the self- and cross-phase-modulation nonlinearities.

© 1999 Optical Society of America

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References

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  1. H. Winful, J. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
    [CrossRef]
  2. H. Winful and G. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
    [CrossRef]
  3. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  4. H. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
    [CrossRef]
  5. T. Erdogan and D. G. Hall, “Circularly-symmetric distributed-feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [CrossRef]
  6. T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
    [CrossRef]
  7. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [CrossRef]
  8. C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
    [CrossRef]
  9. S. Radic, N. George, and G. P. Agrawal, “Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures,” J. Opt. Soc. Am. B 12, 671–680 (1995).
    [CrossRef]
  10. A. E. Siegman, Lasers (University Science, Mill Valley, Calif. 1986), p. 647.
  11. C. M. de Sterke and J. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
    [CrossRef] [PubMed]
  12. D. G. Hall, “Coupled-amplitude equations via a Green’s-function technique,” Am. J. Phys. 61, 44–49 (1993).
    [CrossRef]
  13. N. W. Carlson, Monolithic Diode-Laser Arrays (Springer-Verlag, New York, 1997), Chap. 2.
  14. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995), Section 10.6.
  15. C. M. de Sterke, K. Jackson, and B. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
    [CrossRef]
  16. K. J. Kasunic and M. Fallahi, “Gain saturation in circular-grating distributed-feedback semiconductor lasers,” J. Opt. Soc. Am. B 14, 2147–2152 (1997).
    [CrossRef]
  17. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1962), Appendix D.
  18. C. H. Henry, “Performance of distributed feedback lasers designed to favor the energy gap mode,” IEEE J. Quantum Electron. QE-21, 1913–1918 (1985).
    [CrossRef]
  19. P. K. Milsom, A. Miller, and D. Herbert, “The effect of end reflections and mirror positioning on the optical response of a nonlinear DFB device,” Opt. Commun. 69, 319–324 (1989).
    [CrossRef]

1997 (1)

1995 (1)

1993 (2)

D. G. Hall, “Coupled-amplitude equations via a Green’s-function technique,” Am. J. Phys. 61, 44–49 (1993).
[CrossRef]

C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

1992 (2)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

1991 (2)

H. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

C. M. de Sterke, K. Jackson, and B. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
[CrossRef]

1990 (2)

T. Erdogan and D. G. Hall, “Circularly-symmetric distributed-feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

C. M. de Sterke and J. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

1989 (1)

P. K. Milsom, A. Miller, and D. Herbert, “The effect of end reflections and mirror positioning on the optical response of a nonlinear DFB device,” Opt. Commun. 69, 319–324 (1989).
[CrossRef]

1987 (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

1985 (1)

C. H. Henry, “Performance of distributed feedback lasers designed to favor the energy gap mode,” IEEE J. Quantum Electron. QE-21, 1913–1918 (1985).
[CrossRef]

1982 (1)

H. Winful and G. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

1979 (1)

H. Winful, J. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Agrawal, G. P.

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Cooperman, G.

H. Winful and G. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

de Sterke, C. M.

C. M. de Sterke, K. Jackson, and B. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
[CrossRef]

C. M. de Sterke and J. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly-symmetric distributed-feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Fallahi, M.

K. J. Kasunic and M. Fallahi, “Gain saturation in circular-grating distributed-feedback semiconductor lasers,” J. Opt. Soc. Am. B 14, 2147–2152 (1997).
[CrossRef]

C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

Feldman, S.

H. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

Garmire, E.

H. Winful, J. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

George, N.

Glinski, J.

C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

Hall, D. G.

D. G. Hall, “Coupled-amplitude equations via a Green’s-function technique,” Am. J. Phys. 61, 44–49 (1993).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly-symmetric distributed-feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Henry, C. H.

C. H. Henry, “Performance of distributed feedback lasers designed to favor the energy gap mode,” IEEE J. Quantum Electron. QE-21, 1913–1918 (1985).
[CrossRef]

Herbert, D.

P. K. Milsom, A. Miller, and D. Herbert, “The effect of end reflections and mirror positioning on the optical response of a nonlinear DFB device,” Opt. Commun. 69, 319–324 (1989).
[CrossRef]

Jackson, K.

Kasunic, K. J.

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Maciejko, R.

C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

Makino, T.

C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

Marburger, J.

H. Winful, J. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Miller, A.

P. K. Milsom, A. Miller, and D. Herbert, “The effect of end reflections and mirror positioning on the optical response of a nonlinear DFB device,” Opt. Commun. 69, 319–324 (1989).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Milsom, P. K.

P. K. Milsom, A. Miller, and D. Herbert, “The effect of end reflections and mirror positioning on the optical response of a nonlinear DFB device,” Opt. Commun. 69, 319–324 (1989).
[CrossRef]

Najafi, S.

C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

Radic, S.

Robert, B.

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Sipe, J.

C. M. de Sterke and J. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Svilans, M.

C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Winful, H.

H. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

H. Winful and G. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

H. Winful, J. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Wu, C.

C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

Zamir, R.

H. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

Am. J. Phys. (1)

D. G. Hall, “Coupled-amplitude equations via a Green’s-function technique,” Am. J. Phys. 61, 44–49 (1993).
[CrossRef]

Appl. Phys. Lett. (4)

H. Winful, J. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

H. Winful and G. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

H. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

IEEE J. Quantum Electron. (3)

C. Wu, T. Makino, S. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. 28, 612–623 (1992).
[CrossRef]

C. H. Henry, “Performance of distributed feedback lasers designed to favor the energy gap mode,” IEEE J. Quantum Electron. QE-21, 1913–1918 (1985).
[CrossRef]

J. Appl. Phys. (1)

T. Erdogan and D. G. Hall, “Circularly-symmetric distributed-feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

P. K. Milsom, A. Miller, and D. Herbert, “The effect of end reflections and mirror positioning on the optical response of a nonlinear DFB device,” Opt. Commun. 69, 319–324 (1989).
[CrossRef]

Phys. Rev. A (1)

C. M. de Sterke and J. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Other (4)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif. 1986), p. 647.

N. W. Carlson, Monolithic Diode-Laser Arrays (Springer-Verlag, New York, 1997), Chap. 2.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995), Section 10.6.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1962), Appendix D.

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Figures (8)

Fig. 1
Fig. 1

Schematic of circular-grating VCSEL.

Fig. 2
Fig. 2

Comparison of time-dependent solutions to the exact coupled-mode equations (7) and (8) with the large-radius-approximation (LRA) equations (9) and (10). Shown are the spatial distributions of the outward-going field A(r) for an active grating. The LRA results follow exactly those found with the steady-state software described in Ref. 16.

Fig. 3
Fig. 3

Coordinate system showing the relationship between the real (r and t) coordinates and the characteristic (ρ and τ) coordinates along which the nonlinear coupled-mode equations (9) and (10) are integrated. The dashed lines indicate characteristic directions.

Fig. 4
Fig. 4

Threshold gain versus detuning for Ω=π/2 modes (indicated by filled points) as the coupling coefficient κR is varied (solid lines). The dashed curves show the variation of gain with detuning for a given κR as Ω is varied from 0 (at δ=0) to multiples of 2π. The absence of linear modes at δ=0 is a necessary condition for the existence of gap (Bragg) solitons.

Fig. 5
Fig. 5

Grating transmission versus detuning for both linear [B(R)=10-5] and nonlinear [B(R)=1.0 and 2.0] regimes. The parameters are κR=2.5, Ω=π/2, and γR=0.1.

Fig. 6
Fig. 6

Time-dependent field as measured at the approximate center of a passive grating for (a) δR=0.0 and (b) δR=-0.9π. The parameters are κR=2.5, Ω=π/2, and B(R)=1.0.

Fig. 7
Fig. 7

Time-dependent response for the conditions shown in Fig. 6(b). The dashed curve shows the effects of setting the 2/πβr term equal to unity, thus removing the effects of the 1/r intensity distribution.

Fig. 8
Fig. 8

Field distribution over the grating radius at t=50tR for δR=-0.9π. The parameters are the same as in Fig. 5.

Equations (29)

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1vgAt+Ar=iπβr2KmHm(2)[AHm(1)+BHm(2)]+iγπβr2Hm(1)Hm(2)[|AHm(1)|2+2|BHm(2)|2]A,
1vgBt-Br=iπβr2KmHm(1)[AHm(1)+BHm(2)]+iγπβr2Hm(1)Hm(2)[|BHm(2)|2+2|AHm(1)|2]B,
Km=i(αg-gm)+2κ cos(2β0r-Ω),
gm=gom1+|AHm(1)|2+|BHm(2)|2,
ρ=22(vgt+r),
τ=22(vgt-r),
2dAdρ=iπβr2KmHm(2)[AHm(1)+BHm(2)]+iγπβr2Hm(1)Hm(2)[|AHm(1)|2+2|BHm(2)|2]A,
2dBdτ=iπβr2KmHm(1)[AHm(1)+BHm(2)]+iγπβr2Hm(1)Hm(2)[|BHm(2)|2+2|AHm(1)|2]B,
r=ρ-τ2,
t=ρ+τvg2.
2dAdρ=i(δ+iαg-igm)A-(-1)mκB exp(-iΩ)+iγ2πβr[|A|2+2|B|2]A,
2dBdτ=i(δ+iαg-igm)B+(-1)mκA exp(+iΩ)+iγ2πβr[|B|2+2|A|2]B
gm=gom1+2(|A|2+|B|2)/πβr
1rrrEr+1r22Eθ2+β2E
=-4πω2c2PNL-4πf(r, θ),
PNL=14πn0n2|E|2E.
f(r, θ)=14πω2c2n0n2|E|2E.
1rrrGr+1r22Gθ2+β2G
=-4πrδ(r-r)δ(θ-θ).
G(r, r)=iπH0(1)(β|r-r|).
E(r, θ)=02π0G(r, r)f(r, θ)rdrdθ.
E(r, θ)=[Am(r)Hm(1)(βr)+Bm(r)Hm(2)(βr)]exp(imθ).
dAm(r)dr=iπr2Hm(2)(βr)02π exp(-imθ)f(r, θ)dθ,
dBm(r)dr=-iπr2Hm(1)(βr)02π exp(-imθ)f(r, θ)dθ.
|E|2E=Am(r)Hm(1)[|Am(r)Hm(1)|2+2|Bm(r)Hm(2)|2]exp(imθ)+Bm(r)Hm(2)[|Bm(r)Hm(2)|2+2|Am(r)Hm(1)|2]exp(imθ),
dAm(r)dr=iγπβr2Hm(1)Hm(2)[|Am(r)Hm(1)|2+2|Bm(r)Hm(2)|2]Am(r),
dBm(r)dr=-iγπβr2Hm(1)Hm(2)[|Bm(r)Hm(2)|2+2|Am(r)Hm(1)|2]Bm(r),
1vgAt+Ar=iγπβr2Hm(1)Hm(2)[|AHm(1)|2+2|BHm(2)|2]A,
1vgBt-Br=iγπβr2Hm(1)Hm(2)[|BHm(2)|2+2|AHm(1)|2]B,

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