Abstract

The solution of the nondegenerate two-wave-mixing problem in an absorptive Kerr medium including the contribution from nonlinear absorption is obtained for the first time to our knowledge. This solution enables us to analyze the influence of nonlinear absorption on the two-wave-mixing gain and account for previously unexplained results reported for ruby and Cr3+:YAlO3. The effect of nonlinear absorption of a pump beam on the gain of a probe beam is shown to be minimized when incident intensities of the interacting beams are equal. The intensity dependence of the two-wave-mixing gain for dispersive (ruby and Cr3+:YAlO3) media and the gain measurements for absorptive (fluorescein-doped glass) media are also in agreement with the new theory.

© 1996 Optical Society of America

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References

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  1. Y. Silberberg and I. Bar-Joseph, “Optical instabilities in a nonlinear Kerr medium,” J. Opt. Soc. Am. B 1, 662 (1984).
    [CrossRef]
  2. M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear–optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A 34, 2026 (1986).
    [CrossRef] [PubMed]
  3. I. McMichael, P. Yeh, and P. Beckwith, “Nondegenerate two-wave mixing in ruby,” Opt. Lett. 13, 500 (1988).
    [CrossRef] [PubMed]
  4. P. Yeh, “Exact solution of a nonlinear model of two-wave mixing in Kerr media,” J. Opt. Soc. Am. B 3, 747 (1986).
    [CrossRef]
  5. S. A. Boothroyd, J. Chrostowski, and M. S. O’Sullivan, “Two-wave mixing by phase and absorption gratings in saturable absorbers,” J. Opt. Soc. Am. B 6, 766 (1989).
    [CrossRef]
  6. A. G. Skirtach, D. J. Simkin, and S. A. Boothroyd, “Nondegenerate two-wave mixing in Cr3+:Er3+:YAlO3,” J. Opt. Soc. Am. B 13, 546 (1996).
    [CrossRef]
  7. J. C. Penaforte, E. A. Gouveia, and S. C. Zilio, “Nondegenerate two-wave mixing in GdAlO3:Cr3+,” Opt. Lett. 16, 452 (1991).
    [CrossRef] [PubMed]
  8. R. Saxena, I. McMichael, and P. Yeh, “Dynamics of refractive-index changes and two-beam coupling in resonant media,” Appl. Phys. B 51, 243 (1990).
    [CrossRef]
  9. I. McMichael, R. Saxena, T. Chang, Q. Shu, S. Rand, J. Chen, and H. Tuller, “High-gain nondegenerate two-wave mixing in Cr:YAlO3,” Opt. Lett. 19, 1511 (1994).
    [CrossRef] [PubMed]
  10. S. A. Boothroyd, J. Chrostowski, and M. S. O’Sullivan, “Determination of the phase of the complex nonlinear refractive index by transient two-wave mixing in saturable absorbers,” Opt. Lett. 14, 946 (1989).
    [CrossRef] [PubMed]

1996 (1)

1994 (1)

1991 (1)

1990 (1)

R. Saxena, I. McMichael, and P. Yeh, “Dynamics of refractive-index changes and two-beam coupling in resonant media,” Appl. Phys. B 51, 243 (1990).
[CrossRef]

1989 (2)

1988 (1)

1986 (2)

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear–optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A 34, 2026 (1986).
[CrossRef] [PubMed]

P. Yeh, “Exact solution of a nonlinear model of two-wave mixing in Kerr media,” J. Opt. Soc. Am. B 3, 747 (1986).
[CrossRef]

1984 (1)

Bar-Joseph, I.

Beckwith, P.

Boothroyd, S. A.

Boyd, R. W.

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear–optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A 34, 2026 (1986).
[CrossRef] [PubMed]

Chang, T.

Chen, J.

Chrostowski, J.

Gouveia, E. A.

Kramer, M. A.

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear–optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A 34, 2026 (1986).
[CrossRef] [PubMed]

McMichael, I.

O’Sullivan, M. S.

Penaforte, J. C.

Rand, S.

Saxena, R.

I. McMichael, R. Saxena, T. Chang, Q. Shu, S. Rand, J. Chen, and H. Tuller, “High-gain nondegenerate two-wave mixing in Cr:YAlO3,” Opt. Lett. 19, 1511 (1994).
[CrossRef] [PubMed]

R. Saxena, I. McMichael, and P. Yeh, “Dynamics of refractive-index changes and two-beam coupling in resonant media,” Appl. Phys. B 51, 243 (1990).
[CrossRef]

Shu, Q.

Silberberg, Y.

Simkin, D. J.

Skirtach, A. G.

Tompkin, W. R.

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear–optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A 34, 2026 (1986).
[CrossRef] [PubMed]

Tuller, H.

Yeh, P.

Zilio, S. C.

Appl. Phys. B (1)

R. Saxena, I. McMichael, and P. Yeh, “Dynamics of refractive-index changes and two-beam coupling in resonant media,” Appl. Phys. B 51, 243 (1990).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Lett. (4)

Phys. Rev. A (1)

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, “Nonlinear–optical interactions in fluorescein-doped boric acid glass,” Phys. Rev. A 34, 2026 (1986).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

NDTWM signal for Cr3+:YAlO3 for (a) pump beam I1 when I1(0) = 5I2(0), (b) pump beam I1 when I1(0) = I2(0), (c) probe beam I2 when I2 = I1, and (d) probe beam I2 when I2 = I1/5. Triangular voltage applied to a piezoelement to upshift (positive slope) and down shift (negative slope) the frequency of the beam I2 is shown in curve (e). These results are presented at λ = 514.5 nm for Δ = 1 and I1 = 0.25 kW/cm2.

Fig. 2
Fig. 2

I2(z)/I2(0) of Eq. (4) for different values of the frequency detuning Δ versus propagation distance z for n2 = 2 × 10−8 cm2/W, r = 0.1, I2(0)/I1(0) = 0.01, α = 0.85 cm−1, l = 1 cm, λ = 514.5 nm, and I1(0) = 0.45 kW/cm2. The parameters chosen in the analysis are close to those of ruby.3,5

Fig. 3
Fig. 3

Numerical solution of the coupled wave equations for the probe beam I2(z)/I2(0) (solid curves) together with the solution given by Eq. (4) (dashed curves) for different values of the frequency detuning Δ and the same parameters as in Fig. 2.

Fig. 4
Fig. 4

Numerical solution of the coupled wave equations for the probe beam I2(z)/I2(0) for different pump beam intensities: (a) I1(0) = 0.05Is and (b) I1(0) = 0.3Is. The parameters n2 = 22 × 10−8 cm2/W, r = 0.1, Is = 1.2 kW/cm2, I2(0)/I1(0) = 0.01, α = 1.8 cm−1, L0 = 0.2 cm, and λ = 514.5 nm are close to those of Cr3+:GdAlO3.7 I(±Δ) = ±[I(Δ = ±3) − I(Δ = 0)].

Fig. 5
Fig. 5

NDTWM signal of the probe beam I2 at Δ = ±1 normalized to I2(Δ = −1) at different ratios of the incident probe-to-pump beam intensities, C, for (a) ruby and (b) Cr3+:YAlO3, all at λ = 514.5 nm and I(0) = 0.3 kW/cm2.

Fig. 6
Fig. 6

Γsym = I2(+Δ)I2(−Δ)/I2(Δ = 0)2, for ruby at λ = 488 nm and λ = 580 nm and for Cr3+:YAlO3 at λ = 514.5 nm. (Data for ruby were digitized from Ref. 3.) Positive and negative detuning Δ corresponds, respectively, to the frequency upshift and downshift of the weak probe beam relative to the pump beam.

Fig. 7
Fig. 7

(a) NDTWM signal [I2(Δ) − I2(Δ = 0)]/I2(Δ = 0) for fluorescein-dye-doped glass at 5 mW/cm2 (solid squares) and 30 mW/cm2 (open circles) at λ = 457.9 nm and I2(0)/I1(0) = 0.05. The inset shows the NDTWM signal with large losses that are due to nonlinear absorption for both positive and negative detunings Δ = ±3.5 and the triangular voltage applied to a piezoelectric mirror to upshift (positive slope) and downshift (negative slope) the frequency of beam I2. (b) The NDTWM signal Γ = I2(Δ = 1, S)/I2(Δ = 1, S = 0) versus normalized pump intensity S = I1/Is for fluorescein at λ = 457.9 nm. The data points at S = 0.12 and S = 0.75 are determined from the measurements at these intensities in (a). The experimental results are shown together with the theory including the nonlinear absorption bias contribution [Eq. (8)] (solid curve) and that without this term3,5 (dashed curve).

Equations (19)

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d I 1 ( z ) d z = α I 1 + β n 2 ( I 1 + I 2 ) I 1 + β n 2 1 + Δ 2 I 1 I 2 + β n 2 Δ 1 + Δ 2 I 1 I 2 ,
d I 2 ( z ) d z = α I 2 + β n 2 ( I 1 + I 2 ) I 2 + β n 2 1 + Δ 2 I 1 I 2 β n 2 Δ 1 + Δ 2 I 1 I 2 ,
I 1 ( z ) = I 1 ( 0 ) H ( z ) [ H ( z ) D ( 1 C 1 ) + C 1 ] .
I 2 ( z ) = I 2 ( 0 ) H ( z ) [ H ( z ) D ( 1 C 2 ) + C 2 ] ,
H ( z ) = k I ( 0 ) exp ( α z ) [ k I ( 0 ) 1 ] .
Γ ant = 1 2 ln I 2 ( + Δ ) I 2 ( Δ ) = n 2 β I ( 0 ) 1 exp ( α L ) α Δ 1 + Δ 2 .
I 1 ( L ) I 1 ( 0 ) = ( 1 + A ) exp ( α L ) 1 + A   exp ( γ ) ,
I 2 ( L ) I 2 ( 0 ) = ( 1 + A 1 ) exp ( α L ) 1 + A 1   exp ( γ ) ,
γ = β ( 1 + A ) Γ 2 I 1 ( 0 ) 1 exp ( α L ) α ,
Γ 1 = n 2 Δ + r ( C 1 + 2 + C 1 Δ 2 + Δ 2 ) 1 + Δ 2 ,
Γ 2 = n 2 Δ r ( C + 2 + C Δ 2 + Δ 2 ) 1 + Δ 2 ,
d I ( z ) dz = α I ( z ) + β n 2 I 2 ( z ) ,
I 1 ( z ) + I 2 ( z ) = I ( 0 ) exp ( α z ) 1 + β n 2 α I ( 0 ) [ exp ( α z ) 1 ] ,
Γ ant = 1 2 ln I 2 ( + Δ ) I 2 ( Δ ) n 2 β I ( 0 ) 1 exp ( α L ) α Δ 1 + Δ 2 × 1 + n 2 β I ( 0 ) 1 exp ( α L ) α .
d I Γ 1 d z = α I Γ 1 + γ I Γ 1 I Γ 2 ,
d I Γ 2 d z = α I Γ 2 γ I Γ 1 I Γ 2 ,
I 1 ( L ) I 1 ( 0 ) = ( 1 + C ) exp ( α L ) 1 + C   exp ( γ ) ,
I 2 ( L ) I 2 ( 0 ) = ( 1 + C 1 ) exp ( α L ) 1 + C 1   exp ( γ ) .
γ = β n 2 [ I 1 ( 0 ) + I 2 ( 0 ) ] Δ 1 + Δ 2 1 exp ( α L ) α .

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