Abstract

The effects of the saturation of absorption on the propagation of Gaussian laser beams in an absorber are investigated analytically. Using a dimensionless parameter quantifying the level of saturation, we calculate the modifications in the beam parameters as perturbations from known solutions. The perturbation expansions are carried out in two directions, covering both low- and high-saturation cases. A formula is given that estimates the power transmission for all saturation levels. Numerical results are provided showing the effects of saturation on the spatial beam-spot size variation and the power transmission.

© 1997 Optical Society of America

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References

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  1. N. B. Abraham and W. J. Firth, “Overview of transverse effects in nonlinear optical systems,” J. Opt. Soc. Am. B 7, 951–962 (1990).
    [CrossRef]
  2. J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  3. M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusingof optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  4. G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B 7, 1328–1335 (1990).
    [CrossRef]
  5. R. Jin, M. Liang, G. Khitrova, H. M. Gibbs, and N. Peyghambarian, “Compression of bright optical pulses by dark solitons,” Opt. Lett. 18, 494–496 (1993).
    [CrossRef] [PubMed]
  6. G. S. McDonald and W. J. Firth, “Switching dynamics of spatial solitary wave pixels,” J. Opt. Soc. Am. B 10, 1081–1089 (1993).
    [CrossRef]
  7. D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Numerical calculations of spatially localized wave emission from anonlinear waveguide: two-level saturable media,” J. Opt. Soc. Am. B 7, 990–997 (1990).
    [CrossRef]
  8. A. Sennaroglu, A. Askar, and F. M. Atay, “Quantitative study of laser beam propagation in a thermally loadedabsorber,” J. Opt. Soc. Am. B 14, 356–363 (1997).
    [CrossRef]
  9. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 2.
  10. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 204.
  11. M. Abramowitz and I. A. Stegun, eds., Handbookof Mathematical Functions (Dover, New York, 1972), p. 229.

1997

1993

1990

1988

1976

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Abraham, N. B.

Askar, A.

Atay, F. M.

Feit, M. D.

M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusingof optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Firth, W. J.

Fleck , Jr., J. A.

M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusingof optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Gibbs, H. M.

Heatley, D. R.

Jin, R.

Khitrova, G.

Liang, M.

McDonald, G. S.

Morris, J. R.

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Peyghambarian, N.

Sennaroglu, A.

Stegeman, G. I.

Wright, E. M.

Appl. Phys.

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Other

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 2.

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 204.

M. Abramowitz and I. A. Stegun, eds., Handbookof Mathematical Functions (Dover, New York, 1972), p. 229.

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Figures (8)

Fig. 1
Fig. 1

Graphs of |pd(L)| versus δs as calculated with the low- and high-saturation solutions, plotted together with the numerical solution.

Fig. 2
Fig. 2

Same as Fig. 1 but for |1/qd(L)| versus δs.

Fig. 3
Fig. 3

Beam power transmission τ(L) across a 2-cm length absorber as a function of the saturation level δs, calculated with the two solutions for low and high saturation, plotted together with the numerical solution.

Fig. 4
Fig. 4

Comparison of the approximate analytical power transmission formula [Eq. (31)] with numerical calculations.

Fig. 5
Fig. 5

Beam-spot size ω(ζ) plotted against the normalized distance ζ for various values of the incident pump power (ω0 =100µ, b=0.75).

Fig. 6
Fig. 6

Beam power transmission τ(ζ) plotted against the normalized distance ζ for various values of the incident pump power. Curves from bottom to top correspond to powers of 0.01, 1, 2.5, 5, and 10 W (ω0=100µ, b=0).

Fig. 7
Fig. 7

Beam power transmission τ(L) across absorber length plotted against the incident pump power Pi for various values of the unperturbed beam waist ω0 (b=0). Note that since high-saturation expansions are used in this calculation, for Pi <2 W, the numerical values read off from the graph are erroneous.

Fig. 8
Fig. 8

Beam power transmission τ(L) plotted against b, the (normalized) focus location of the unperturbed beam, for various values of the incident pump power (ω0=100µ).

Tables (1)

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Table 1 Physical Parameters Used in Calculations

Equations (44)

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α=α01+I/Is.
2E+k2E=0.
k=k0n0-i α2k0,
kc=k0n0-i α02,
2E+kc+i α02I/Is1+I/Is2E=0.
I=n02η0|E|2,
E0(r, z)=[e0 exp(-ikcz)]×exp-ip0(z)+r2 kc2q0(z),
q0(z)=z-b+ia,p0(z)=-i ln1+z-bia,
E(r, z)=[e0 exp(-ikcz)]exp-ip(z)+r2 kc2q(z),
δs=n0e022η0Is
ζ=α0z,qd(ζ)=α0q(ζ/α0),pd(ζ)=p(ζ/α0)
1qd(ζ)=u(ζ)u(ζ).
2Ez2kc2E,
pd(ζ)=p0(ζ)+δsp1(ζ)+,
u(ζ)=u0(ζ)+δsu1(ζ)+.
u0(ζ)=0,
u1(ζ)=ikcexp(-ζ)exp{2 Im[p0(ζ)]}×Imkc u0(ζ)u0(ζ)u0(ζ),
p0(ζ)=-i u0(ζ)u0(ζ),
p1(ζ)=-i u1(ζ)u0(ζ)-u1(ζ)u0(ζ)u02(ζ)+i2exp(-ζ)exp{2 Im[p0(ζ)]}.
u0(ζ)=ζ+c.
c=-b+ia,
p0(ζ)=-i ln1+ζ-bia.
kc=k0n0-i α02k0n0.
u1(ζ)=ak0n02kcIm{(ζ+c¯)[E1(c¯)-E1(ζ+c¯)]exp(c¯)}-a2k0n02kcζc¯-(ζ+c¯+1)×[E1(c¯)-E1(ζ+c¯)]exp(c¯)+1-exp(-ζ),
E1(ζ)=ζt-1 exp(-t)dt.
q(z)=1α0u0(ζ)+δsu1(ζ)1+δsu1(ζ),
p1(ζ)=-i u1(ζ)u0(ζ)+ia2(Im{[E1(c¯)-E1(ζ+c¯)]exp(c¯)}).
u1(ζ)=exp(-ζ)+ζ-1.
qd(ζ)=ζ-b+ia+δs[exp(-ζ)+ζ-1]1+δs[1-exp(-ζ)],
pd(ζ)=-i ln1+ζ-bia-iδsexp(-ζ)+ζ-1ζ-b+ia-12[1-exp(-ζ)].
2E+(k0n0)2E=0,
E(r, z)=[e0 exp(-ik0n0z)]×exp-ip0(ζ)+r2 k0n02q0(z),
E(r, z)=[e0 exp(-ikcz)]×exp-ii2α0z+p0(z)+r2 kc2q0(z).
pd(ζ)=i2ζ+p0(ζ)+δs-1p1(ζ)+ ,
u(ζ)=u0(ζ)+δs-1u1(ζ)+ .
u1(ζ)=-i6a[ζ3+3(-b+ia)ζ2],
p1(ζ)=-16aζ3+3(-b+ia)ζ2ζ-b+ia-i6a2[(ζ-b)3+3a2ζ+b3].
qd(ζ)=(ζ-b+ia)+12ζ2δs-11+ζδs-1,
pd(ζ)=i2ζ-i ln1+ζ-bia-i2δs-1ζ2ζ-b+ia+ζ.
τlow(L)=exp(-α0L){1+δs2[1-exp(-α0L)]2/2},
τhigh(L)=exp(-2α0L/δs).
τ(L)=exp(-α0L)+[1-exp(-α0L)]exp(-2α0L/δs),
τ(ζ)=ω2(ζ)ω2(0)exp(-ζ)exp{2 Im[pd(ζ)-pd(0)]},
1ω(ζ)2=Im-kcα02qd(ζ).

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