Abstract

The observation of stable self-pulsing of a single-transverse-mode alexandrite laser with a Fabry–Perot cavity is reported. This process occurs many times above threshold and depends on the pump wavelength. The novel theoretical model, in which the dynamics of the host lattice phonons is taken into account, is shown to explain this phenomenon. The border of the Hopf bifurcation is found in the plane of two parameters: the intensity and the frequency of the pump laser. In the region where self-pulsations occur they are proved to be the result of the photon–phonon energy pulling.

© 1998 Optical Society of America

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  1. Review article by N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1 (1988), and all references.
    [CrossRef]
  2. W. Gadomski, S. Hodges, and M. G. Raymer, “Dynamics of a multimode, short-cavity dye laser,” in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1990), p. 337.
  3. F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110 (1997).
    [CrossRef]
  4. J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120 (1980).
    [CrossRef]
  5. L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756 (1978).
    [CrossRef]
  6. M. G. Raymer, Z. Deng, and M. Beck, “Strong-field dynamics of a multimode, standing-wave dye laser,” J. Opt. Soc. Am. B 5, 1588 (1988).
    [CrossRef]
  7. C. L. Tang, H. Statz, and G. deMars, “Spectral output and spiking behavior of solid-state lasers,” J. Appl. Phys. 34, 2289 (1963).
    [CrossRef]
  8. F. Sanchez and A. Kellou, “Laser dynamics with excited-state absorption,” J. Opt. Soc. Am. B 14, 209 (1997).
    [CrossRef]
  9. J. L. A. Chilla and O. Martinez, “Spatiotemporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10, 638 (1993).
    [CrossRef]
  10. E. A. Victorov, I. B. Vitrishchak, G. E. Novikov, O. A. Orlov, A. A. Mak, V. A. Sokolov, V. I. Ustyugov, and M. M. Khalev, “Instabilities and chaos in solid-state lasers as a result of mode coupling,” in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1990), p. 410.
  11. M. L. Shand and J. C. Walling, “Excited-state absorption in the lasing wavelength region of alexandrite,” IEEE J. Quantum Electron. QE-18, (1982).
  12. S. K. Gayen, W. B. Wang, V. Petricevic’, and R. R. Alfano, “Nonradiative transition dynamics in alexandrite,” Appl. Phys. Lett. 49, (1986).
    [CrossRef]
  13. S. K. Gayen, W. B. Wang, V. Petricevic’, K. M. Yoo, and R. R. Alfano, “Picosecond excite-and-probe absorption measurement of the intra-2EgE3/2-state vibrational relaxation time in Ti3+:Al2O3,” Appl. Phys. Lett. 50, 1494 (1987); S. K. Gayen, W. B. Wang, V. Petricevic’, R. Dorsinville, and R. R. Alfano, “Picosecond excite-and-probe absorption measurement of the 4T2 state nonradiative lifetime in ruby,” Appl. Phys. Lett. 47, 454 (1985).
    [CrossRef]
  14. S. G. Demos, J. M. Buchert, and R. R. Alfano, “Time resolved nonequilibrium phonon dynamics in the nonradiative decay of photoexcited forsterite,” Appl. Phys. Lett. 61, 660 (1992).
    [CrossRef]
  15. S. G. Demos and R. R. Alfano, “Subpicosecond time-resolved Raman investigation of optical phonon modes in Cr-doped Forsterite,” Phys. Rev. B 52, 987 (1995).
    [CrossRef]
  16. S. T. Lai and M. L. Shand, “High efficiency CW laser-pumped tunable alexandrite laser,” J. Appl. Phys. 54, 5642 (1983).
    [CrossRef]
  17. J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985); J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
    [CrossRef]
  18. H. Haken, Light (North-Holland, New York, 1981), Vol. 1.
  19. R. Englman, Nonradiative Decay of Ions and Molecules in Solids (North-Holland, New York, 1979).
  20. W. Gadomski and B. Ratajska-Gadomska, Phys. Rev. A 34, 1277 (1986).
    [CrossRef] [PubMed]
  21. M. Sargent III and M. O. Scully, “Theory of laser operation,” in Laser Handbook, F. T. Arecchi and E. O. Schilz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, p. 45.

1997 (2)

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110 (1997).
[CrossRef]

F. Sanchez and A. Kellou, “Laser dynamics with excited-state absorption,” J. Opt. Soc. Am. B 14, 209 (1997).
[CrossRef]

1995 (1)

S. G. Demos and R. R. Alfano, “Subpicosecond time-resolved Raman investigation of optical phonon modes in Cr-doped Forsterite,” Phys. Rev. B 52, 987 (1995).
[CrossRef]

1993 (1)

1992 (1)

S. G. Demos, J. M. Buchert, and R. R. Alfano, “Time resolved nonequilibrium phonon dynamics in the nonradiative decay of photoexcited forsterite,” Appl. Phys. Lett. 61, 660 (1992).
[CrossRef]

1988 (2)

Review article by N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1 (1988), and all references.
[CrossRef]

M. G. Raymer, Z. Deng, and M. Beck, “Strong-field dynamics of a multimode, standing-wave dye laser,” J. Opt. Soc. Am. B 5, 1588 (1988).
[CrossRef]

1986 (2)

S. K. Gayen, W. B. Wang, V. Petricevic’, and R. R. Alfano, “Nonradiative transition dynamics in alexandrite,” Appl. Phys. Lett. 49, (1986).
[CrossRef]

W. Gadomski and B. Ratajska-Gadomska, Phys. Rev. A 34, 1277 (1986).
[CrossRef] [PubMed]

1983 (1)

S. T. Lai and M. L. Shand, “High efficiency CW laser-pumped tunable alexandrite laser,” J. Appl. Phys. 54, 5642 (1983).
[CrossRef]

1982 (1)

M. L. Shand and J. C. Walling, “Excited-state absorption in the lasing wavelength region of alexandrite,” IEEE J. Quantum Electron. QE-18, (1982).

1980 (1)

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120 (1980).
[CrossRef]

1978 (1)

L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756 (1978).
[CrossRef]

1963 (1)

C. L. Tang, H. Statz, and G. deMars, “Spectral output and spiking behavior of solid-state lasers,” J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Abraham, N. B.

Review article by N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1 (1988), and all references.
[CrossRef]

Alfano, R. R.

S. G. Demos and R. R. Alfano, “Subpicosecond time-resolved Raman investigation of optical phonon modes in Cr-doped Forsterite,” Phys. Rev. B 52, 987 (1995).
[CrossRef]

S. G. Demos, J. M. Buchert, and R. R. Alfano, “Time resolved nonequilibrium phonon dynamics in the nonradiative decay of photoexcited forsterite,” Appl. Phys. Lett. 61, 660 (1992).
[CrossRef]

S. K. Gayen, W. B. Wang, V. Petricevic’, and R. R. Alfano, “Nonradiative transition dynamics in alexandrite,” Appl. Phys. Lett. 49, (1986).
[CrossRef]

Beck, M.

Buchert, J. M.

S. G. Demos, J. M. Buchert, and R. R. Alfano, “Time resolved nonequilibrium phonon dynamics in the nonradiative decay of photoexcited forsterite,” Appl. Phys. Lett. 61, 660 (1992).
[CrossRef]

Casperson, L. W.

L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756 (1978).
[CrossRef]

Chilla, J. L. A.

deMars, G.

C. L. Tang, H. Statz, and G. deMars, “Spectral output and spiking behavior of solid-state lasers,” J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Demos, S. G.

S. G. Demos and R. R. Alfano, “Subpicosecond time-resolved Raman investigation of optical phonon modes in Cr-doped Forsterite,” Phys. Rev. B 52, 987 (1995).
[CrossRef]

S. G. Demos, J. M. Buchert, and R. R. Alfano, “Time resolved nonequilibrium phonon dynamics in the nonradiative decay of photoexcited forsterite,” Appl. Phys. Lett. 61, 660 (1992).
[CrossRef]

Deng, Z.

Gadomski, W.

W. Gadomski and B. Ratajska-Gadomska, Phys. Rev. A 34, 1277 (1986).
[CrossRef] [PubMed]

Gayen, S. K.

S. K. Gayen, W. B. Wang, V. Petricevic’, and R. R. Alfano, “Nonradiative transition dynamics in alexandrite,” Appl. Phys. Lett. 49, (1986).
[CrossRef]

Kellou, A.

Lai, S. T.

S. T. Lai and M. L. Shand, “High efficiency CW laser-pumped tunable alexandrite laser,” J. Appl. Phys. 54, 5642 (1983).
[CrossRef]

Mandel, P.

Review article by N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1 (1988), and all references.
[CrossRef]

Martinez, O.

Morris, R. C.

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120 (1980).
[CrossRef]

Narducci, L. M.

Review article by N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1 (1988), and all references.
[CrossRef]

Peterson, O. G.

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120 (1980).
[CrossRef]

Petricevic’, V.

S. K. Gayen, W. B. Wang, V. Petricevic’, and R. R. Alfano, “Nonradiative transition dynamics in alexandrite,” Appl. Phys. Lett. 49, (1986).
[CrossRef]

Ratajska-Gadomska, B.

W. Gadomski and B. Ratajska-Gadomska, Phys. Rev. A 34, 1277 (1986).
[CrossRef] [PubMed]

Raymer, M. G.

Sanchez, F.

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110 (1997).
[CrossRef]

F. Sanchez and A. Kellou, “Laser dynamics with excited-state absorption,” J. Opt. Soc. Am. B 14, 209 (1997).
[CrossRef]

Shand, M. L.

S. T. Lai and M. L. Shand, “High efficiency CW laser-pumped tunable alexandrite laser,” J. Appl. Phys. 54, 5642 (1983).
[CrossRef]

M. L. Shand and J. C. Walling, “Excited-state absorption in the lasing wavelength region of alexandrite,” IEEE J. Quantum Electron. QE-18, (1982).

Statz, H.

C. L. Tang, H. Statz, and G. deMars, “Spectral output and spiking behavior of solid-state lasers,” J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Stephan, G.

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110 (1997).
[CrossRef]

Tang, C. L.

C. L. Tang, H. Statz, and G. deMars, “Spectral output and spiking behavior of solid-state lasers,” J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Walling, J. C.

M. L. Shand and J. C. Walling, “Excited-state absorption in the lasing wavelength region of alexandrite,” IEEE J. Quantum Electron. QE-18, (1982).

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120 (1980).
[CrossRef]

Wang, W. B.

S. K. Gayen, W. B. Wang, V. Petricevic’, and R. R. Alfano, “Nonradiative transition dynamics in alexandrite,” Appl. Phys. Lett. 49, (1986).
[CrossRef]

Appl. Phys. Lett. (2)

S. K. Gayen, W. B. Wang, V. Petricevic’, and R. R. Alfano, “Nonradiative transition dynamics in alexandrite,” Appl. Phys. Lett. 49, (1986).
[CrossRef]

S. G. Demos, J. M. Buchert, and R. R. Alfano, “Time resolved nonequilibrium phonon dynamics in the nonradiative decay of photoexcited forsterite,” Appl. Phys. Lett. 61, 660 (1992).
[CrossRef]

IEEE J. Quantum Electron. (3)

M. L. Shand and J. C. Walling, “Excited-state absorption in the lasing wavelength region of alexandrite,” IEEE J. Quantum Electron. QE-18, (1982).

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120 (1980).
[CrossRef]

L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756 (1978).
[CrossRef]

J. Appl. Phys. (2)

C. L. Tang, H. Statz, and G. deMars, “Spectral output and spiking behavior of solid-state lasers,” J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

S. T. Lai and M. L. Shand, “High efficiency CW laser-pumped tunable alexandrite laser,” J. Appl. Phys. 54, 5642 (1983).
[CrossRef]

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

Phys. Rev. A (1)

W. Gadomski and B. Ratajska-Gadomska, Phys. Rev. A 34, 1277 (1986).
[CrossRef] [PubMed]

Phys. Rev. B (1)

S. G. Demos and R. R. Alfano, “Subpicosecond time-resolved Raman investigation of optical phonon modes in Cr-doped Forsterite,” Phys. Rev. B 52, 987 (1995).
[CrossRef]

Phys. Rev. E (1)

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110 (1997).
[CrossRef]

Prog. Opt. (1)

Review article by N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1 (1988), and all references.
[CrossRef]

Other (7)

W. Gadomski, S. Hodges, and M. G. Raymer, “Dynamics of a multimode, short-cavity dye laser,” in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1990), p. 337.

E. A. Victorov, I. B. Vitrishchak, G. E. Novikov, O. A. Orlov, A. A. Mak, V. A. Sokolov, V. I. Ustyugov, and M. M. Khalev, “Instabilities and chaos in solid-state lasers as a result of mode coupling,” in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1990), p. 410.

S. K. Gayen, W. B. Wang, V. Petricevic’, K. M. Yoo, and R. R. Alfano, “Picosecond excite-and-probe absorption measurement of the intra-2EgE3/2-state vibrational relaxation time in Ti3+:Al2O3,” Appl. Phys. Lett. 50, 1494 (1987); S. K. Gayen, W. B. Wang, V. Petricevic’, R. Dorsinville, and R. R. Alfano, “Picosecond excite-and-probe absorption measurement of the 4T2 state nonradiative lifetime in ruby,” Appl. Phys. Lett. 47, 454 (1985).
[CrossRef]

J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985); J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
[CrossRef]

H. Haken, Light (North-Holland, New York, 1981), Vol. 1.

R. Englman, Nonradiative Decay of Ions and Molecules in Solids (North-Holland, New York, 1979).

M. Sargent III and M. O. Scully, “Theory of laser operation,” in Laser Handbook, F. T. Arecchi and E. O. Schilz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, p. 45.

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Figures (9)

Fig. 1
Fig. 1

Output of the Kr-ion pumped alexandrite laser: (a) transient damped oscillations; (b) regular pulsations.

Fig. 2
Fig. 2

Transverse mode instability.

Fig. 3
Fig. 3

Output of the Ar-ion-pumped alexandrite laser.

Fig. 4
Fig. 4

(a) Square repetition frequency and (b) amplitude of pulses in Fig. 3 versus the pump intensity.

Fig. 5
Fig. 5

Five-level model of the alexandrite laser.

Fig. 6
Fig. 6

Results of the numerical calculations assuming the multimode laser operation obtained for the number of pump photons I0=8×1011: (a) Stable output achieved after the transient damped oscillations obtained for f=1. The enlarged part of the trace, shown by the arrow, displays the oscillatory character. The sampling rate is 1000 times smaller than the beat frequency. (b) Self-pulsing of the laser output I, the constant part of the population inversion W0, and the phonon number, obtained for f=1.5, which corresponds to the Ar-ion pump.

Fig. 7
Fig. 7

Laser output for different values of the parameter f and I0=8×1011.

Fig. 8
Fig. 8

Stability diagram for the alexandrite laser output.

Fig. 9
Fig. 9

Results of the numerical calculations assuming the one-mode laser operation: self-pulsing of the laser intensity I and of the population inversion W, obtained for f=1.5.

Equations (39)

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

E(t, z)=iλeλωλ20 (aλ-aλ+)sin kλz,
H=HE+HP+HF+HEF+HEP,
HE=n0(z)iEibi+bi
HP=αΩαAα+Aα+12+α,α,α[H(3)(α,α,α)Aα+AαAα+c.c.]
HF=λωλaλ+aλ+ω0a0+a0
HEF=n0(z)[V510b5+b1a0-V150b1+b5a0++λ(V42λb4+b2aλ-V24λb2+b4aλ+)]
Vijλ=-iωλ20 μijeλ sin kλz,
HPE=n0(z)nβ,nβ[F3,5(ββ)(Aβ+)nβ(Aβ+)nβb3+b5+F5,3(ββ)(Aβ)nβ(Aβ)nβb5+b3]+[F4,3(β)Aβb4+b3+F3,4(β)Aβ+b3+b4]
ddt bi+bj=i ΔEij-γijbi+bj+i λ,k(Vkiλbk+bj-Vjkλbi+bk)(aλ-aλ+)+i nβ,nβk(Fkibk+bj-Fjkbi+bk)[(Aβ)nβ(Aβ)nβ+(Aβ+)nβ(Aβ+)nβ],
i,j=1, 2, 3, 4, 5,
ddtaλ+aλ=iΔλλaλ+aλ-κλaλ+aλ+i p(V2p,4λaλ+b2p+b4+V4,2pλb4+b2paλ),
ddt N=-Γ[N-N(0)]+i [F4,3(β)b4+b3Aβ-F3,4(β)b3+b4Aβ+]+i nβ[F53(ββ)b5+b3(Aβ)nβ(Aβ)nβ-F35(ββ)b3+b5(Aβ+)nβ(Aβ+)nβ],
ddt Aβ=-iΩβAβ-ΓAβ-i nβF3,5(ββ)×b3+b5(Aβ+)nβ(Aβ+)nβ-1+Φ(t),
Φβ(t)Φβ*(t)=ΩβΓβ|Aβ(0)|2exp[iΩβ(t-t)]
B53=1Γ |F53|2[Aβ+Aβ(0)+1]nβf(ΔE53-nβωβ-nβωβ),
B35=1Γ |F53|2[Aβ+Aβ(0)]nβf(ΔE53-nβωβ-nβωβ),
γ43NR=B43(N+1),γ34NR=B34N,
withB43=B34B=1Γ |F43|2f(ΔE43-ωβ).
b2p+b2p=1γ2p1 γ12pb1+b1+i λ(V42pb4+b2paλ+V2p4b2p+b4aλ+),
ddt b5+b5=-[γ5+B53(N+1)nβ]b5+b5+AI0b1+b1+B35Nnβb3+b3,
ddt b3+b3=-(γ3+BN+B35Nnβ)b3+b3+B(N+1)b4+b4+B53(N+1)nβb5+b5,
ddt b4+b4=-[γ4+B(N+1)]b4+b4+BNb3+b3-i pλ(V2p,4λb2p+b4aλ++V4,2pλb4+b2paλ),
ddt b2p+b4=i ΔE2p4-γ4-γ21NR-BN-λλCλλpaλ+aλb2p+b4+i λV4,2pλ(b4+b4-gb1+b1)aλ,
ddt b1+b1=-AI0b1+b1+γ3b3+b3+γ4b4+b4+i pλ(V4,2pλb4+b2paλ+V2p,4λb2p+b4aλ+),
ddt N=-Γ[N-N(0)]+B(N+1)b4+b4-(BN+nβB35Nnβ)b3+b3+nβB53(N+1)nβb5+b5,
ddt n3=-Γ3n3+K3WW+K30n0,
ddt W=-ΓWW-2λλ CλλIλλW1+λCλλIλλ/ΓP+KW3n3+KW0n0,
ddt Iλλ=-iΔλλIλλ-2κλIλλ+λ (CλλIλλ+CλλIλλ)1+λCλλIλλ/ΓP+BN/ΓP W,
ddt N=-ΓP[N-N(0)]-KN3n3+KNWW+KN0n0,
ΓW=(2γ4+BN+AI0)/(1+g),
Γ3=γ3+[AI0+(2g+1)BN]/(1+g),
KW3=[(2g+1)BN-AI0+2gγ4]/(1+g)-γ3,
K3W=(BN-AI0)/(1+g),
KW0=(AI0-gBN-2γ4)/(1+g),
K30=(AI0+gBN)/(1+g),
KN3=[(2g+1)BN+fAI0]/(1+g),
KNW=(BN-fAI0)/(1+g),
KN0=[fAI0+gBN)/(1+g).
f=nβ B53(N+1)nβγ5+B53(N+1)nβnβ

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