Abstract

An analytical expression for the degenerate four-wave mixing signal produced in a medium composed of two-level atoms by pump and probe lasers of arbitrary intensities and parallel linear polarization, suffering negligible absorption and depletion, has been derived. The Abrams–Lind approach [Opt. Lett. 2, 94 and 3, 205 (1978)] for deriving the phase-matched polarization has been extended to find all the other higher-order polarization terms contributing to the signal. The signal field amplitude is expressed as an infinite series whose convergence rate depends on the intensity of the input fields and their detuning from resonance. The solution is successfully verified by comparison with a full numerical nonperturbative calculation and is found to be equivalent to the Abrams–Lind result in the limit of weak probe intensity. The result is used to calculate the saturation effects on spectral line shape that are relevant to simulation of molecular spectra obtained by degenerate four-wave mixing with saturating pump and probe fields. The result is extended to treat atomic motion in the case of forward geometry of the input beams.

© 1999 Optical Society of America

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References

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  1. R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
  2. R. L. Abrams and R. C. Lind, Opt. Lett. 2, 94–96 and 3, 205 (1978).
  3. D. Bloch and M. Ducloy, J. Opt. Soc. Am. 73, 635–646 (1983) and 73, 1844–1845 (1983).
    [CrossRef]
  4. G. Grynberg, M. Pinard, and P. Verkerk, Opt. Commun. 50, 261–264 (1984).
    [CrossRef]
  5. R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, J. Opt. Soc. Am. B 10, 1508–1520 (1993).
    [CrossRef]
  6. R. G. Caro and M. C. Gower, IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
    [CrossRef]
  7. W. P. Brown, J. Opt. Soc. Am. 73, 629–634 (1983).
    [CrossRef]
  8. P. Ewart and S. V. O’Leary, J. Phys. B 17, 4595–4608 (1984).
    [CrossRef]
  9. M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, J. Opt. Soc. Am. 2, 1117–1121 (1985).
    [CrossRef]
  10. J. Cooper, A. Charlton, D. Meacher, P. Ewart, and G. Alber, Phys. Rev. A 40, 5705–5715 (1989).
    [CrossRef] [PubMed]
  11. D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, Phys. Rev. A 46, 2718–2725 (1992).
    [CrossRef] [PubMed]
  12. M. Ducloy, F. A. M. de Oliveira, and D. Bloch, Phys. Rev. A 32, 1614–1623 (1985).
    [CrossRef] [PubMed]
  13. P. Verkerk, P. Pinard, and G. Grynberg, Phys. Rev. A 35, 4679–4695 (1987).
    [CrossRef]
  14. S. Williams, R. N. Zare, and L. A. Rahn, J. Chem. Phys. 101, 1093–1107 (1994).
    [CrossRef]
  15. D. G. Steel, R. C. Lind, and J. F. Lam, Phys. Rev. A 23, 2513–2524 (1981).
    [CrossRef]
  16. P. M. Danehy, E. J. Friedmanhill, R. P. Lucht, and R. L. Farrow, Appl. Phys. B: Photophys. Laser Chem. 57, 243–248 (1993).
    [CrossRef]
  17. S. M. Wandzura, Opt. Lett. 4, 208–210 (1979).
    [CrossRef] [PubMed]
  18. B. Attal-Tretout, H. Bervas, J. P. Taran, S. Le Boiteux, P. Kelley, and T. K. Gustafson, J. Phys. B 30, 497–522 (1997).
    [CrossRef]
  19. R. J. Knize, Opt. Lett. 18, 1606–1608 (1993).
    [CrossRef] [PubMed]
  20. B. Ai and R. J. Knize, J. Opt. Soc. Am. B 13, 2408–2419 (1996).
    [CrossRef]
  21. R. W. Boyd, Nonlinear Optics (Academic, New York, 1995), p. 59.
  22. Ref. 21, p. 201.
  23. I. S. Gradstein and I. M. Ryzik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

1997 (1)

B. Attal-Tretout, H. Bervas, J. P. Taran, S. Le Boiteux, P. Kelley, and T. K. Gustafson, J. Phys. B 30, 497–522 (1997).
[CrossRef]

1996 (1)

1994 (1)

S. Williams, R. N. Zare, and L. A. Rahn, J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

1993 (3)

R. J. Knize, Opt. Lett. 18, 1606–1608 (1993).
[CrossRef] [PubMed]

P. M. Danehy, E. J. Friedmanhill, R. P. Lucht, and R. L. Farrow, Appl. Phys. B: Photophys. Laser Chem. 57, 243–248 (1993).
[CrossRef]

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, J. Opt. Soc. Am. B 10, 1508–1520 (1993).
[CrossRef]

1992 (1)

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

1989 (1)

J. Cooper, A. Charlton, D. Meacher, P. Ewart, and G. Alber, Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

1987 (1)

P. Verkerk, P. Pinard, and G. Grynberg, Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef]

1985 (2)

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, J. Opt. Soc. Am. 2, 1117–1121 (1985).
[CrossRef]

1984 (2)

P. Ewart and S. V. O’Leary, J. Phys. B 17, 4595–4608 (1984).
[CrossRef]

G. Grynberg, M. Pinard, and P. Verkerk, Opt. Commun. 50, 261–264 (1984).
[CrossRef]

1983 (1)

1982 (1)

R. G. Caro and M. C. Gower, IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
[CrossRef]

1981 (1)

D. G. Steel, R. C. Lind, and J. F. Lam, Phys. Rev. A 23, 2513–2524 (1981).
[CrossRef]

1979 (1)

Ai, B.

Alber, G.

J. Cooper, A. Charlton, D. Meacher, P. Ewart, and G. Alber, Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

Attal-Tretout, B.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. Le Boiteux, P. Kelley, and T. K. Gustafson, J. Phys. B 30, 497–522 (1997).
[CrossRef]

Bervas, H.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. Le Boiteux, P. Kelley, and T. K. Gustafson, J. Phys. B 30, 497–522 (1997).
[CrossRef]

Bloch, D.

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

Boyd, R. W.

M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, J. Opt. Soc. Am. 2, 1117–1121 (1985).
[CrossRef]

Brown, W. P.

Caro, R. G.

R. G. Caro and M. C. Gower, IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
[CrossRef]

Charlton, A.

J. Cooper, A. Charlton, D. Meacher, P. Ewart, and G. Alber, Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

Cooper, J.

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

J. Cooper, A. Charlton, D. Meacher, P. Ewart, and G. Alber, Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

Danehy, P. M.

P. M. Danehy, E. J. Friedmanhill, R. P. Lucht, and R. L. Farrow, Appl. Phys. B: Photophys. Laser Chem. 57, 243–248 (1993).
[CrossRef]

de Oliveira, F. A. M.

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

Ducloy, M.

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

Ewart, P.

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

J. Cooper, A. Charlton, D. Meacher, P. Ewart, and G. Alber, Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

P. Ewart and S. V. O’Leary, J. Phys. B 17, 4595–4608 (1984).
[CrossRef]

Farrow, R. L.

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, J. Opt. Soc. Am. B 10, 1508–1520 (1993).
[CrossRef]

P. M. Danehy, E. J. Friedmanhill, R. P. Lucht, and R. L. Farrow, Appl. Phys. B: Photophys. Laser Chem. 57, 243–248 (1993).
[CrossRef]

Friedmanhill, E. J.

P. M. Danehy, E. J. Friedmanhill, R. P. Lucht, and R. L. Farrow, Appl. Phys. B: Photophys. Laser Chem. 57, 243–248 (1993).
[CrossRef]

Gaeta, A. L.

M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, J. Opt. Soc. Am. 2, 1117–1121 (1985).
[CrossRef]

Gower, M. C.

R. G. Caro and M. C. Gower, IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
[CrossRef]

Gruneisen, M. T.

M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, J. Opt. Soc. Am. 2, 1117–1121 (1985).
[CrossRef]

Grynberg, G.

P. Verkerk, P. Pinard, and G. Grynberg, Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef]

G. Grynberg, M. Pinard, and P. Verkerk, Opt. Commun. 50, 261–264 (1984).
[CrossRef]

Gustafson, T. K.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. Le Boiteux, P. Kelley, and T. K. Gustafson, J. Phys. B 30, 497–522 (1997).
[CrossRef]

Kelley, P.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. Le Boiteux, P. Kelley, and T. K. Gustafson, J. Phys. B 30, 497–522 (1997).
[CrossRef]

Knize, R. J.

Lam, J. F.

D. G. Steel, R. C. Lind, and J. F. Lam, Phys. Rev. A 23, 2513–2524 (1981).
[CrossRef]

Le Boiteux, S.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. Le Boiteux, P. Kelley, and T. K. Gustafson, J. Phys. B 30, 497–522 (1997).
[CrossRef]

Lind, R. C.

D. G. Steel, R. C. Lind, and J. F. Lam, Phys. Rev. A 23, 2513–2524 (1981).
[CrossRef]

Lucht, R. P.

P. M. Danehy, E. J. Friedmanhill, R. P. Lucht, and R. L. Farrow, Appl. Phys. B: Photophys. Laser Chem. 57, 243–248 (1993).
[CrossRef]

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, J. Opt. Soc. Am. B 10, 1508–1520 (1993).
[CrossRef]

Meacher, D.

J. Cooper, A. Charlton, D. Meacher, P. Ewart, and G. Alber, Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

Meacher, D. R.

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

O’Leary, S. V.

P. Ewart and S. V. O’Leary, J. Phys. B 17, 4595–4608 (1984).
[CrossRef]

Pinard, M.

G. Grynberg, M. Pinard, and P. Verkerk, Opt. Commun. 50, 261–264 (1984).
[CrossRef]

Pinard, P.

P. Verkerk, P. Pinard, and G. Grynberg, Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef]

Rahn, L. A.

S. Williams, R. N. Zare, and L. A. Rahn, J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

Rakestraw, D. J.

Smith, P. G. R.

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Steel, D. G.

D. G. Steel, R. C. Lind, and J. F. Lam, Phys. Rev. A 23, 2513–2524 (1981).
[CrossRef]

Taran, J. P.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. Le Boiteux, P. Kelley, and T. K. Gustafson, J. Phys. B 30, 497–522 (1997).
[CrossRef]

Verkerk, P.

P. Verkerk, P. Pinard, and G. Grynberg, Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef]

G. Grynberg, M. Pinard, and P. Verkerk, Opt. Commun. 50, 261–264 (1984).
[CrossRef]

Wandzura, S. M.

Williams, S.

S. Williams, R. N. Zare, and L. A. Rahn, J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

Zare, R. N.

S. Williams, R. N. Zare, and L. A. Rahn, J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem. (1)

P. M. Danehy, E. J. Friedmanhill, R. P. Lucht, and R. L. Farrow, Appl. Phys. B: Photophys. Laser Chem. 57, 243–248 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. G. Caro and M. C. Gower, IEEE J. Quantum Electron. QE-18, 1376–1380 (1982).
[CrossRef]

J. Chem. Phys. (1)

S. Williams, R. N. Zare, and L. A. Rahn, J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

W. P. Brown, J. Opt. Soc. Am. 73, 629–634 (1983).
[CrossRef]

M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, J. Opt. Soc. Am. 2, 1117–1121 (1985).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. B (2)

P. Ewart and S. V. O’Leary, J. Phys. B 17, 4595–4608 (1984).
[CrossRef]

B. Attal-Tretout, H. Bervas, J. P. Taran, S. Le Boiteux, P. Kelley, and T. K. Gustafson, J. Phys. B 30, 497–522 (1997).
[CrossRef]

Opt. Commun. (1)

G. Grynberg, M. Pinard, and P. Verkerk, Opt. Commun. 50, 261–264 (1984).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (5)

D. G. Steel, R. C. Lind, and J. F. Lam, Phys. Rev. A 23, 2513–2524 (1981).
[CrossRef]

J. Cooper, A. Charlton, D. Meacher, P. Ewart, and G. Alber, Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

D. R. Meacher, P. G. R. Smith, P. Ewart, and J. Cooper, Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

P. Verkerk, P. Pinard, and G. Grynberg, Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef]

Other (6)

R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

R. L. Abrams and R. C. Lind, Opt. Lett. 2, 94–96 and 3, 205 (1978).

D. Bloch and M. Ducloy, J. Opt. Soc. Am. 73, 635–646 (1983) and 73, 1844–1845 (1983).
[CrossRef]

R. W. Boyd, Nonlinear Optics (Academic, New York, 1995), p. 59.

Ref. 21, p. 201.

I. S. Gradstein and I. M. Ryzik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

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Figures (8)

Fig. 1
Fig. 1

Spatial arrangement of the four-wave vectors in the phase-matched geometry for DFWM and the location of the site of signal detection relative to the interaction region.

Fig. 2
Fig. 2

Relative deviation of the analytical result from that of the nonperturbative numerical solution of the DFWM signal intensity versus Ly.

Fig. 3
Fig. 3

Normalized signal spectrum for unsaturating fields, E3=Ep=0.1 (dashed curve) and saturating fields, E3=Ep=1 (solid curve) in units of the saturation field strength ESAT.

Fig. 4
Fig. 4

Normalized signal spectrum for strongly saturating fields, E3=Ep=10 in units of the saturation field strength ESAT.

Fig. 5
Fig. 5

Saturation surface representing the dependence of the signal intensity on the magnitude of both pump and probe fields. The maximum signal is obtained when pump and probe amplitudes are equal and have a value of 1.204 ESAT.

Fig. 6
Fig. 6

Spectral surface representing the line shape (signal amplitude versus normalized detuning δ) as a function of pump field strength in the case of a weak, unsaturating probe, E3=0.1.

Fig. 7
Fig. 7

Spectral surface for a saturating probe, E3=1.

Fig. 8
Fig. 8

Spectral surface for a strongly saturating probe, E3=6.

Equations (49)

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Ej(r)exp(-iωt)=Aj exp[-i(ωt-kj·r)]
forj=1, 2, 3.
k4=k1+k2-k3.
××[E(r)exp(-iωt)]-1c22t2[E(r)exp(-iωt)]
=μ02t2[P(r)exp(-iωt)],
P(r)=kPk(r)exp(ik·r),
A4(s)=k24π0sΩPk4(r)dr3,
P(r)=[E1(r)+E2(r)+E3(r)]ESATΔN|μ|2(δ-i)T2/1+δ2+|E1(r)+E2(r)+E3(r)|2,
Ej(r)Ej(r)/ESATforj=1, 2, 3,
ESAT22/(4|μ|2T2T1);
δ(ω-ω0)T2
P(r)=P0n=0[E1(r)+E2(r)+E3(r)]B(r)×[E1(r)+E2(r)][E3(r)]*+c.c.-B(r)n,
B(r)1+δ2+E32+4Ep2 cos2(Δk·r/2),
Δkk1-k2,
E3A3/ESAT,
EpAp/ESAT,
P0ESATΔN|μ|2(δ-i)T2/.
-P022m(2m-1)!m!(m-1)!cos(Δk·r/2)B(r)2mE32m-1Ep2m×exp(ik4·r).
P022m(2m)!(m+1)!(m-1)!×[cos(Δk·r/2)]2m[B(r)]2m+1E32m+1Ep2m exp(ik4·r).
Pk4(r)=P0m=14m(2m-1)!E32m-1Ep2mm!(m-1)!×2mE32(m+1)B(r)-1cos(Δk·r/2)B(r)2m.
E4(s)A4(s)/ESAT=α0k(δ-i)2πsm=14m(2m-1)!E32m-1Ep2mm!(m-1)!×Ω2mE32(m+1)B(r)-1×cos(Δk·r/2)B(r)2md3r,
α0=|μ|2ΔNT2k20.
E4(s)=α0k(δ-i)VΔk2π2sm=14m(2m-1)!E32m-1Ep2mm!(m-1)!×0π/Δk2mE32(m+1)B(x)-1×cos(Δkx/2)B(x)2mdx,
E4(s)=α0kV(δ-i)2π2s×m=12m(2m-1)!E32m-1Ep2mm!(m-1)!B¯2m×-Φ1(f, m)+2mE32(m+1)B¯Φ2(f, m),
B¯(1+δ2)+2Ep2+E32,
f2Ep2/B¯
Φ1(f, m)0π(1+cos x)m(1+f cos x)2mdx,
Φ2(f, m)0π(1+cos x)m(1+f cos x)2m+1dx.
I4(s)=20cn|E4(s)|2ESAT2,
I4(s)=α020ck2V2(1+δ2)ISAT2ns2π4×m=12m(2m-1)!I3m-(1/2)Ipmm!(m-1)!B¯2m×-Φ1(f, m)+2mI3(m-1)B¯Φ2(f, m)2,
E4(s)=α0k(δ-i)V2πsm=122m(m-1)!m!(m-1)!E32m-1Ep2mB2m×2mE32(m+1)B-1,
B1+δ2+4Ep2+E32.
E4(s)2πsα0Vk(δ-i)2=1Ly0Ly[2Ep+E3 exp(ik3yy)]exp(ik3yy)1+δ2+|2Ep+E3 exp(ik3yy)|2dy2,
E4(s)=α0kV2π2sm=12m(2m-1)!E32m-1Ep2mm!(m-1)!2πkBT/M×-(δ+kv-i)exp-Mv22kBT-Φ1[f(v), m]+2mE32(m+1)B¯(v)Φ2[f(v), m][B¯(v)]2mdv,
B¯(v)[1+(δ+kv)2]+2Ep2+E32,
f(v)2Ep2/B¯(v).
P0[E1(r)+E2(r)+E3(r)](1+δ2)+|E1(r)+E2(r)|2+|E3(r)|2×1+|[E1(r)+E2(r)][E3(r)]*+c.c.|(1+δ2)+|E1(r)+E2(r)|2+|E3(r)|2-1.
|E1(r)+E2(r)|2+|E3(r)|2|[E1(r)+E2(r)][E3(r)]*+c.c.|.
(1+cos x)m=t=0mm!(m-t)!t!(cos x)t.
1(1+f cos x)a=n=0(-f)n(a+n-1)!(a-1)!n!(cos x)n.
(1+cos x)m(1+f cos x)a=n=0(-f)n(a+n-1)!(a-1)!n!(cos x)n×t=0mm!(m-t)!t!(cos x)t.
(1+cos x)m(1+f cos x)a=u=0(cos x)um!(a-1)!×t=0min(u, m)(-f)u-t(a+u-t-1)!(m-t)!t!(u-t)!,
0π(cos x)2u=π(2u-1)!!(2u)!!,
0π(1+cos x)m(1+f cos x)adx=u=0π(2u-1)!!(2u)!!m!(a-1)!×t=0min(2u, m)(-f)2u-t(a+2u-t-1)!(m-t)!t!(2u-t)!.
Φ1(f, m)=πm!(2m-1)!u=0(2u-1)!!(2u)!!×t=0min(2u, m)(-f)2u-t(2m+2u-t-1)(2u-t)!(m-t)!t!,
Φ2(f, m)=πm!(2m)!u=0(2u-1)!!(2u)!!×t=0min(2u, m)(-f)2u-t(2m+2u-t)(2u-t)!(m-t)!t!.
E4(s)=-α0kV(δ-i)E3Ep2π2sB¯20π1+cos x(1+f cos x)2dx.
40π/2cos2 w(α2 sin2 w+β2 cos2 w)2dw=παβ3,
E4(L)=-α0kV(δ-i)π2sE3Ep2([(1+δ2)+4Ep2]×{[(1+δ2)]2+4Ep2(1+δ2)}1/2)-1,

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