Abstract

The effect of thermal loading on the propagation of Gaussian laser beams in a solid-state absorber is modeled by a novel quantitative scheme. The zeroth-order Gaussian beam solution of the wave equation in a homogeneous, cylindrically symmetric absorbing medium is used as the source term in the heat equation to calculate the temperature field. Modifications in the beam parameters caused by the temperature dependence of the absorption coefficient and the index of refraction are then calculated as first-order corrections. The formulation identifies a dimensionless parameter that controls the strength of thermal effects. Numerical results that show the dependence of crystal transmission and the spatial beam spot-size variation on incident pump power are presented. In particular, the power transmission of the crystal is found to decrease with increasing incident power, and power-dependent thermal lensing is observed. The asymptotic behavior of the solutions yields explicit formulas for the focal length of the thermal lens and the power transmission of the crystal. These explicit formulas should prove useful as a rule of thumb for experimentalists.

© 1997 Optical Society of America

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References

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  1. A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
    [CrossRef]
  2. T. M. Baer and M. S. Keirstead, “Modeling of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. II of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CFMl.
  3. A. Sennaroglu, C. R. Pollock, and H. Nathel, “Generation of 48-fs pulses and measurement of crystal dispersion by using a regeneratively initiated self-mode-locked chromium-doped forsterite laser,” Opt. Lett. 18, 826–828 (1993).
    [CrossRef] [PubMed]
  4. A. Sennaroglu, C. R. Pollock, and H. Nathel, “Efficient continuous-wave chromium-doped YAG laser,” J. Opt. Soc. Am. B 12, 930–937 (1995).
    [CrossRef]
  5. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 2.
  6. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
    [CrossRef]
  7. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 229.

1995 (1)

1993 (1)

1992 (1)

A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
[CrossRef]

1990 (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Baer, T. M.

T. M. Baer and M. S. Keirstead, “Modeling of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. II of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CFMl.

Cousins, A. K.

A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
[CrossRef]

Fields, R. A.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Fincher, C. L.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Innocenzi, M. E.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Keirstead, M. S.

T. M. Baer and M. S. Keirstead, “Modeling of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. II of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CFMl.

Nathel, H.

Pollock, C. R.

Sennaroglu, A.

Yariv, A.

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

Yeh, P.

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

Yura, H. T.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Appl. Phys. Lett. (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Other (3)

T. M. Baer and M. S. Keirstead, “Modeling of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. II of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CFMl.

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

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

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Figures (6)

Fig. 1
Fig. 1

Sketch of the zeroth-order Gaussian beam solution with the relevant beam parameters.

Fig. 2
Fig. 2

Variation of the beam spot size ω(ζ) in micrometers as a function of the normalized propagation distance ζ for different levels of the incident pump power Pi (ω0=100 µm and zf=0.5 cm).

Fig. 3
Fig. 3

Variation of the transmission correction factor τ* as a function of the normalized propagation distance ζ for different levels of the incident pump power Pi (ω0=100 µm and zf=0).

Fig. 4
Fig. 4

Variation of the transmission correction factor τ* as a function of the incident pump power Pi for different values of the unperturbed beam waist ω0 (zf=0 and ζ=3.)

Fig. 5
Fig. 5

Variation of the transmission correction factor τ* as a function of the unperturbed beam waist location zf for different levels of the incident pump power Pi (ω0=100 µm and ζ=3.)

Fig. 6
Fig. 6

Comparison of the various perturbation schemes employed to solve Eq. (25) with the numerical solution ψ(ζ). ψ1(ζ), ψ2(ζ), and ψ3(ζ) are obtained by application of different perturbation expansions as described in Table 2. (Pi=5 W, ω0=100 µm, zf=0.5 cm, and |δT|=0.44.)

Tables (2)

Tables Icon

Table 1 Relevant Physical Parameters That Characterize the Absorber

Tables Icon

Table 2 Description of the Three Perturbation Schemes Employed to Solve Eq. (25)

Equations (44)

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Ex(r, z)=e0 ω0ω(z)exp[iΦ(r, z)]×exp-α0z2exp-rω(z)2,
ω(z)=ω01+z-zfz021/2,
z0=πω02n0λ,
τ=exp(-α0L),
2Ex+k2Ex=0.
k=kon-i α2k0,
n=n0+nT(T-Tr),
α=α0+αT(T-Tr).
S=12n0|Ex|2η0eˆz,
P(z)=S·dA.
S=eˆz 2Piπω2(z)exp-2r2ω(z)2exp(-α0z).
h=α0Sz.
1rrr Tr+2Tz2=-hκ,
T(r, z)=Tb-α0Pi4πκ2 lnrr0-E12r02ω2(z)+E12r2ω2(z)exp(-α0z).
E1(x)=x exp(-t)tdt.
T(r, z)-TbT0(z)-T1(z)r2.
T0(z)=α0Pi exp(-α0z)4πκ×γ+ln2r02ω2(z)+E12r02ω2(z),
T1(z)=α0P1 exp(-α0z)2πκω2(z),
E1(z)=-γ-ln z-n=1 (-1)nznnn!.
k2=kc2{1+A[Tb-Tr+T0(z)-T1(z)r2]}.
kc=k0n0(1-iβ0),β0=α02k0n0,
A=21-iβ0nTn0-iβ0 αTα0.
Ex(r, z)=e0 exp-ip(z)+kcr22q(z)exp(-ikcz).
2Ez2k2E.
1q2+ddz1q+AT1(z)=0,
dpdz=Akc2[Tb-Tr+T0(z)]-iq(z).
1q=1ududz,
ζ=α0z,
d2udζ2+δTf(ζ)u(ζ)=0,
f(ζ)=a2e-ζa2+(ζ-b)2,
δT=An0α0κPiπω02,
u(ζ)=u(0)(ζ)+δTu(1)(ζ)+O(δT2).
u(0)(ζ)=(ζ-b)+ia,
u(1)(ζ)=-0ζdζ0ζdζf(ζ)u(0)(ζ).
u(1)(ζ)=a2{(ζ-b-ia)exp(-b-ia)[E1(ζ-b-ia)-E1(-b-ia)]+[1-exp(-ζ)]}.
qT(ζ)=1α0u(0)(ζ)+δTu(1)(ζ)1+δT du(1)(ζ)dζ.
ωT(ζ)=1Im-kc2qT(ζ)1/2.
P(ζ)=π4n0η0E02ωT(ζ)2 exp{2 Im[pT(ζ)]-ζ}.
τ*(ζ)=τT/τ,
τ*(ζ)=ωT(ζ)ωT(0)2 exp{2 Im[pT(ζ)]}.
E1(x)e-xx1-1x+. . .,
u(1)(ζ)-(ζ-2)-ia(ζ-1).
fT=n0κnTπω02Pi.
τ*(ζ)=exp-PiαT4πκγ+ln2r02ω02

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