Abstract

We present the results of degenerate four-wave mixing (DFWM) studies in three samples of p-type Ge with a wide range of absorptive properties. Three mechanisms for DFWM are observed in this material. A model, which includes the mechanisms of the inhomogeneously broadened saturable absorption of p-type Ge and the optical Kerr effect that is due to bound electrons in Ge but not the mechanism of bulk plasma formation that occurs only at extremely high pump intensities, is developed and compared with the experimental results. Pump-beam attenuation by the medium is shown to be an important effect in modeling DFWM. The model, which has no free parameters, fits the experimental data well for samples with small-signal transmission of greater than 20%.

© 1983 Optical Society of America

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Corrections

D. E. Watkins, C. R. Phipps, and W. W. Rigrod, "Degenerate four-wave mixing in p-type germanium: an absorbing medium: erratum," J. Opt. Soc. Am. 73, 1227-1227 (1983)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-73-9-1227

References

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  1. E. Bergmann, I. Bigio, B. Feldman, and R. Fisher, “High efficiency pulsed 10.6-μ m phase conjugate reflection via degenerate four-wave mixing,” Opt. Lett. 3, 82–84 (1978).
    [Crossref]
  2. D. E. Watkins, C. R. Phipps, and S. J. Thomas, “Observation of amplified reflection through degenerate four-wave mixing at CO2 laser wavelengths in germanium,” Opt. Lett. 6, 76–78 (1981).
    [Crossref] [PubMed]
  3. See F. Keilmann, “Infrared saturation spectroscopy in p-type germanium,” IEEE J. Quantum Electron. QE-12, 592–597 (1976); C. R. Phipps and S. J. Thomas, “Saturation behavior of p-type germanium at CO2laser wavelengths,” Opt. Lett. 1, 93–95 (1977).
    [Crossref] [PubMed]
  4. D. E. Watkins, C. R. Phipps, and S. J. Thomas, “Determination of the third-order nonlinear optical coefficients of Ge through ellipse rotation,” Opt. Lett. 5, 248–249 (1980).
    [Crossref] [PubMed]
  5. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); erratum 3, 205 (1978).
    [Crossref] [PubMed]
  6. See H. B. Briggs and R. C. Fletcher, “Absorption of infrared light by free carriers in germanium,” Phys. Rev. 91, 1342–1346 (1953); E. D. Capron and O. L. Brill, “Absorption coefficient as a function of resistance for optical germanium at 10 μ m,” Appl. Opt. 12, 569–572 (1973).
    [Crossref] [PubMed]
  7. D. E. Watkins, “A study of degenerate four-wave mixing in germanium and rhenate-doped potassium chloride at carbon dioxide laser wavelengths,” Ph.D. Thesis, catalog no. 1982-0-576-020/27 (U.S. Government Printing Office, Washington, D.C., 1982).
    [Crossref]
  8. D. E. Watkins, J. F. Figueira, and S. J. Thomas, “Observation of resonantly enhanced degenerate four-wave mixing in doped alkali halides,” Opt. Lett. 5, 169–171 (1980).
    [Crossref] [PubMed]
  9. R. W. Hellwarth, “Generation of time-reversed wavefronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
    [Crossref]

1981 (1)

1980 (2)

1978 (2)

1977 (1)

1976 (1)

See F. Keilmann, “Infrared saturation spectroscopy in p-type germanium,” IEEE J. Quantum Electron. QE-12, 592–597 (1976); C. R. Phipps and S. J. Thomas, “Saturation behavior of p-type germanium at CO2laser wavelengths,” Opt. Lett. 1, 93–95 (1977).
[Crossref] [PubMed]

1953 (1)

See H. B. Briggs and R. C. Fletcher, “Absorption of infrared light by free carriers in germanium,” Phys. Rev. 91, 1342–1346 (1953); E. D. Capron and O. L. Brill, “Absorption coefficient as a function of resistance for optical germanium at 10 μ m,” Appl. Opt. 12, 569–572 (1973).
[Crossref] [PubMed]

Abrams, R. L.

Bergmann, E.

Bigio, I.

Briggs, H. B.

See H. B. Briggs and R. C. Fletcher, “Absorption of infrared light by free carriers in germanium,” Phys. Rev. 91, 1342–1346 (1953); E. D. Capron and O. L. Brill, “Absorption coefficient as a function of resistance for optical germanium at 10 μ m,” Appl. Opt. 12, 569–572 (1973).
[Crossref] [PubMed]

Feldman, B.

Figueira, J. F.

Fisher, R.

Fletcher, R. C.

See H. B. Briggs and R. C. Fletcher, “Absorption of infrared light by free carriers in germanium,” Phys. Rev. 91, 1342–1346 (1953); E. D. Capron and O. L. Brill, “Absorption coefficient as a function of resistance for optical germanium at 10 μ m,” Appl. Opt. 12, 569–572 (1973).
[Crossref] [PubMed]

Hellwarth, R. W.

Keilmann, F.

See F. Keilmann, “Infrared saturation spectroscopy in p-type germanium,” IEEE J. Quantum Electron. QE-12, 592–597 (1976); C. R. Phipps and S. J. Thomas, “Saturation behavior of p-type germanium at CO2laser wavelengths,” Opt. Lett. 1, 93–95 (1977).
[Crossref] [PubMed]

Lind, R. C.

Phipps, C. R.

Thomas, S. J.

Watkins, D. E.

IEEE J. Quantum Electron. (1)

See F. Keilmann, “Infrared saturation spectroscopy in p-type germanium,” IEEE J. Quantum Electron. QE-12, 592–597 (1976); C. R. Phipps and S. J. Thomas, “Saturation behavior of p-type germanium at CO2laser wavelengths,” Opt. Lett. 1, 93–95 (1977).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

Opt. Lett. (5)

Phys. Rev. (1)

See H. B. Briggs and R. C. Fletcher, “Absorption of infrared light by free carriers in germanium,” Phys. Rev. 91, 1342–1346 (1953); E. D. Capron and O. L. Brill, “Absorption coefficient as a function of resistance for optical germanium at 10 μ m,” Appl. Opt. 12, 569–572 (1973).
[Crossref] [PubMed]

Other (1)

D. E. Watkins, “A study of degenerate four-wave mixing in germanium and rhenate-doped potassium chloride at carbon dioxide laser wavelengths,” Ph.D. Thesis, catalog no. 1982-0-576-020/27 (U.S. Government Printing Office, Washington, D.C., 1982).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Geometry for analysis of DFWM. Only plane-wave analysis is done; thus the symmetry is defined by the boundaries of the medium, which are parallel to the xy plane. The wave vectors are in the xz plane, both at an angle θ to the z axis. The wave vectors kb and k2 are antiparallel to kf and k1, respectively.

Fig. 2
Fig. 2

DFWM at 10.6 μm in a 6-mm, p-type sample of Ge. The small-signal transmission of the sample is 70%. All three mechanisms for DFWM in p-type Ge are clearly distinguishable in the data.

Fig. 3
Fig. 3

DFWM in two 3-mm samples of Ge, one intrinsic (squares) and the other p-type (circles) with small-signal transmission of 43%. Enhanced reflection that is due to saturable absorption in the p-type sample is observed below 30 MW/cm2.

Fig. 4
Fig. 4

DFWM in a 1-cm, p-type sample of Ge. The small-signal transmission of this sample is only 8%, causing severe attenuation of the pump beams at low intensity.

Equations (26)

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P = i n α r 2 π k E + i n α 0 2 π k E ( 1 + E 2 / E s 2 ) 1 / 2 + 3 χ 3 E 2 E .
P = P 0 + Δ P ,
P 0 = i n α r E s 2 π k E 0 + i α 0 E s 2 π k E 0 ( 1 + E 0 2 ) 1 / 2 + 3 χ 3 E s 3 E 0 2 E 0
Δ P = i n α r E s 2 π k Δ E + i α 0 E s 2 π k ( 2 + E 0 2 ) Δ E - E 0 2 Δ E * ( 1 + E 0 2 ) 3 / 2 + 6 χ 3 E s 3 E 0 2 Δ E + 3 χ 3 E s 3 E 0 2 Δ E * .
2 E 0 + n 2 k 2 E 0 = - 4 π k 2 E s P 0
2 Δ E + n 2 k 2 Δ E = - 4 π k 2 E s Δ P .
d E f d z = [ - α p + i β ( ρ f 2 + 2 ρ b 2 ) ] E f ,
d E b d z = [ α p - i β ( ρ b 2 + 2 ρ f 2 ) ] E b ,
d E 1 d z = α E 1 + i κ * E 2 * exp ( i ϕ + ) ,
d E 2 d z = - α E 2 - i κ * E 1 * exp ( i ϕ + ) ,
α p = α r cos ( θ ) + α 0 π cos ( θ ) [ 1 + ( ρ f + ρ b ) 2 ] 1 / 2 K ( b ) ,
β = 6 π k χ 3 E s 2 n cos ( ϕ ) ,
α = - α r cos ( θ ) - α 0 π cos ( θ ) [ 1 + ( ρ f + ρ b ) 2 ] 1 / 2 × [ K ( b ) + E ( b ) 1 + ( ρ f - ρ b ) 2 ] + 2 i β ( ρ f 2 + ρ b 2 ) ,
κ = i α 0 cos ( θ ) [ 1 + ( ρ f + ρ b ) 2 ] 1 / 2 { ρ f 2 + ρ b 2 2 ρ f ρ b [ K ( b ) - E ( b ) ] - ( ρ f - ρ b ) 2 1 + ( ρ f - ρ b ) 2 E ( b ) } + 2 i β ρ f ρ b .
d ρ f d z = - α p ρ f ,
d ρ b d z = α p ρ b ,
d ϕ + d z = β ( ρ b 2 - ρ f 2 ) .
E 1 2 = A 1 2 exp ± i 0 z α I ( x ) d x ,
d A 1 d z = α R A 1 + i κ * A 2 * exp ( i ϕ + )
d A 2 d z = - α R A 2 - i κ * A 1 * exp ( i ϕ + ) .
d ρ 1 d z = α R ρ 1 - [ κ R sin ( ϕ ) - κ I cos ( ϕ ) ] ρ 2 ,
d ρ 2 d z = - α R ρ 2 + [ κ R sin ( ϕ ) - κ I cos ( ϕ ) ] ρ 1 ,
ρ 1 d ϕ 1 d z = [ κ R cos ( ϕ ) + κ I sin ( ϕ ) ] ρ 2 ,
ρ 2 d ϕ 2 d z = - [ κ R cos ( ϕ ) + κ I sin ( ϕ ) ] ρ 1 .
tan [ ϕ ( 0 ) ] = - κ R ( 0 ) κ I ( 0 ) .
d ϕ d z = d ϕ + d z - [ κ R cos ( ϕ ) + κ I sin ( ϕ ) ] ( ρ 2 ρ 1 - ρ 1 ρ 2 ) .