Abstract

An analytical model of the thermal effects of cylindrically symmetric heating in laser rods is presented. We discuss two different methods for calculating the temperature distribution and thermal effects such as the thermally induced lens and thermal birefringence. One method is based on Taylor series; the other, on the integration of differential heating distributions. As an example, we discuss the effect of thermally induced birefringence in a Nd:YAG rod and compare the analytical solutions with finite-element simulations and experimental data. Compared with the numerical simulations, the calculations with the analytical expression are faster by several orders of magnitude and are therefore an excellent tool for optimization of the parameters related to heating and cooling of laser rods.

© 2000 Optical Society of America

Full Article  |  PDF Article

Errata

Marc Schmid, Th. Graf, and H. P. Weber, "Analytical model of the temperature distribution and the thermally induced birefringence in laser rods with cylindrically symmetric heating: erratum," J. Opt. Soc. Am. B 18, 1751-1751 (2001)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-18-11-1751

References

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  1. R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34, 1046–1053 (1998).
    [CrossRef]
  2. P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubled Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
    [CrossRef]
  3. A. Lucianetti, Th. Graf, R. Weber, and H. P. Weber, “Thermo-optical properties of transversally pumped composite YAG rods with Nd-doped core,” IEEE J. Quantum Electron. 36, 220–227 (2000).
    [CrossRef]
  4. R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11, 245–254 (1999).
    [CrossRef]
  5. M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36, 620–626 (2000).
    [CrossRef]
  6. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, Berlin (1996), pp. 393–409.
  7. St. C. Tidwell, J. F. Seasmans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
    [CrossRef]
  8. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, Oxford, 1959).
  9. D. C. Brown, “Ultrahigh-average-power diode-pumped Nd:YAG and Yb:YAG lasers,” IEEE J. Quantum Electron. 33, 861–873 (1997).
    [CrossRef]
  10. T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
    [CrossRef]

2000

A. Lucianetti, Th. Graf, R. Weber, and H. P. Weber, “Thermo-optical properties of transversally pumped composite YAG rods with Nd-doped core,” IEEE J. Quantum Electron. 36, 220–227 (2000).
[CrossRef]

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36, 620–626 (2000).
[CrossRef]

1999

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11, 245–254 (1999).
[CrossRef]

1998

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34, 1046–1053 (1998).
[CrossRef]

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubled Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

1997

D. C. Brown, “Ultrahigh-average-power diode-pumped Nd:YAG and Yb:YAG lasers,” IEEE J. Quantum Electron. 33, 861–873 (1997).
[CrossRef]

1993

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

1992

St. C. Tidwell, J. F. Seasmans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

Bowers, M. S.

St. C. Tidwell, J. F. Seasmans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

Brown, D. C.

D. C. Brown, “Ultrahigh-average-power diode-pumped Nd:YAG and Yb:YAG lasers,” IEEE J. Quantum Electron. 33, 861–873 (1997).
[CrossRef]

Clarkson, W. A.

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubled Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

Cousins, A. K.

St. C. Tidwell, J. F. Seasmans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

Fan, T. Y.

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

Graf, Th.

A. Lucianetti, Th. Graf, R. Weber, and H. P. Weber, “Thermo-optical properties of transversally pumped composite YAG rods with Nd-doped core,” IEEE J. Quantum Electron. 36, 220–227 (2000).
[CrossRef]

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36, 620–626 (2000).
[CrossRef]

Hanna, D. C.

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubled Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

Hardman, P. J.

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubled Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

Lucianetti, A.

A. Lucianetti, Th. Graf, R. Weber, and H. P. Weber, “Thermo-optical properties of transversally pumped composite YAG rods with Nd-doped core,” IEEE J. Quantum Electron. 36, 220–227 (2000).
[CrossRef]

Mac Donald, M.

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34, 1046–1053 (1998).
[CrossRef]

Neuenschwander, B.

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11, 245–254 (1999).
[CrossRef]

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34, 1046–1053 (1998).
[CrossRef]

Roos, M.

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36, 620–626 (2000).
[CrossRef]

Roos, M. B.

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34, 1046–1053 (1998).
[CrossRef]

Schmid, M.

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36, 620–626 (2000).
[CrossRef]

Seasmans, J. F.

St. C. Tidwell, J. F. Seasmans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

Tidwell, St. C.

St. C. Tidwell, J. F. Seasmans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

Weber, H. P.

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36, 620–626 (2000).
[CrossRef]

A. Lucianetti, Th. Graf, R. Weber, and H. P. Weber, “Thermo-optical properties of transversally pumped composite YAG rods with Nd-doped core,” IEEE J. Quantum Electron. 36, 220–227 (2000).
[CrossRef]

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11, 245–254 (1999).
[CrossRef]

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34, 1046–1053 (1998).
[CrossRef]

Weber, R.

A. Lucianetti, Th. Graf, R. Weber, and H. P. Weber, “Thermo-optical properties of transversally pumped composite YAG rods with Nd-doped core,” IEEE J. Quantum Electron. 36, 220–227 (2000).
[CrossRef]

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36, 620–626 (2000).
[CrossRef]

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11, 245–254 (1999).
[CrossRef]

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34, 1046–1053 (1998).
[CrossRef]

IEEE J. Quantum Electron.

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34, 1046–1053 (1998).
[CrossRef]

A. Lucianetti, Th. Graf, R. Weber, and H. P. Weber, “Thermo-optical properties of transversally pumped composite YAG rods with Nd-doped core,” IEEE J. Quantum Electron. 36, 220–227 (2000).
[CrossRef]

M. Schmid, R. Weber, Th. Graf, M. Roos, and H. P. Weber, “Numerical simulation and analytical description of the thermally induced birefringence in laser rods,” IEEE J. Quantum Electron. 36, 620–626 (2000).
[CrossRef]

St. C. Tidwell, J. F. Seasmans, M. S. Bowers, and A. K. Cousins, “Scaling cw diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

D. C. Brown, “Ultrahigh-average-power diode-pumped Nd:YAG and Yb:YAG lasers,” IEEE J. Quantum Electron. 33, 861–873 (1997).
[CrossRef]

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

Opt. Commun.

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubled Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

Opt. Mater.

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11, 245–254 (1999).
[CrossRef]

Other

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, Oxford, 1959).

W. Koechner, Solid-State Laser Engineering (Springer-Verlag, Berlin (1996), pp. 393–409.

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Figures (8)

Fig. 1
Fig. 1

Relations of heating distribution Q(r) and differential heating q(rp).

Fig. 2
Fig. 2

Main axes of the indicatrix: nϕ,r, azimuthal and radial refractive indices, respectively; p, polarization of the light in the crystal.

Fig. 3
Fig. 3

Fluorescence of the inhomogeneously pumped Nd:YAG rod.

Fig. 4
Fig. 4

Radial heating distribution P(rp) measured from Fig. 3 (diamonds) and fitted by a polynomial of fourth order (solid curve).

Fig. 5
Fig. 5

Qualitative comparison of the depolarization at a pump power of 535 W. Left, experiment; right, analytical solution.

Fig. 6
Fig. 6

Depolarization Dbiref(r) in an inhomogeneously pumped rod with a pump power Ppump of 535 W. Dbiref is the total fractional depolarization in a beam with radius r. Filled circles, experimental result; solid curve, analytical solution.

Fig. 7
Fig. 7

Radial heating distribution. This heating function is a polynomial of second order.

Fig. 8
Fig. 8

Depolarization Dbiref(r) in an inhomogeneously pumped rod with a pump power Ppump of 1000 W. The length of the rod is 28.8 mm, and the rod radius is 2 mm. The heating distribution is the polynomial of second order shown in Fig. 7. Dbiref is the total fractional depolarization in a beam with radius r. Dashed curve, FE simulation; solid curve, analytical solution.

Equations (51)

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2T-1κ Tt=-Q(x, y, z, t)k,
2T(r)=2r2T(r)+1r rT(r)=-Q(r)k.
2 i Ti(r)=i 2Ti(r)=-1k i Qi(r)
2Ti(r)=-Qi(r)k,
Q(r)=i=0n Qi(r),T(r)=i=0n Ti(r).
q(rp)=dQ(r)drr=rp.
dTrp(r)=τ(r, rp)drp.
Q(r)=Q(rrod)+rrodrq(rp)drp,
T(r)=rrod0τ(r, rp)drp,
r2  τ(r, rp)drp= r2τ(r, rp)drp
=- q(rp)drpk
r2τ(r, rp)
=-q(rp)krrp0rrp.
Ti(r)=-1k ci(i+2)2ri+2.
Q(r)=i=0n Qi(r)=i=0nciri,
T(r)=Tb+i=0nTi(r)=Tb-1k i=0n ci(i+2)2ri+2,
T(rrod)=TF+PheatAh.
Pheat=0l02π0rrod Q(r)rdrdφdz=2πli=0n cii+2rrodi+2,
A=2πrrod l,
Tb=TF+1h i=1n cii+2rrodi+1+1k i=0n ci(i+2)2(rrodi+2).
T(r)=TF+1h i=1n cii+2rrodi+1+1k i=0n ci(i+2)2(rrodi+2-ri+2).
2T(r)=-Q(r)k+Q(rrod)k-Q(rrod)k=-Q(rrod)k-rrodr q(rp)k drp.
T(r)=Tho(r)+Tinh(r),
Tho(r)=TF+Q(rrod)4k(rrod2-r2).
r2τ(r, rp)=2r2τ (r, rp)+1r rτ(r, rp)=-q(rp)krrp0r>rp
τ (r, rp)=q(rp)rp22hrrod+q(rp)rp24k×-2 ln(rp/rrod)+1-(r/rp)2rpr-2 ln(r/rrod)rp<r.
T(r)=Tho(r)+Tinh(r)
=TF+Q(rrod)4k(rrod2-r2)+12hrrod rrod0q(rp)rp2drp-lnrrrod2k r0 q(rp)rp2drp-12k rrodrq(rp)rp2 lnrprroddrp+14k rrodr q(rp)rp2drp-r24k rrodr q(rp)drp.
dbiref=sin2(2ϕ)sin2δ2,  δ=2πλl(Δnϕ-Δnr),
nϕ-nr=n032(p11-p12)(εr-εϕ),
εz=1E[σz-ν(σr-σϕ)],
εϕ=1E[σϕ-ν(σr-σz)],
εr=1E[σr-ν(σϕ-σz)],
σr=αE1-ν[F-R(r)],
σϕ=αE1-ν[F+R(r)-T(r)],
σz=αE1-ν[2F-T(r)],
R(r)=1r2 0rT(r)rdr,
F=1rrod2 0rrodT(r)rdr.
F=TF2+rrod2h i=1n cii+2rrodi+rrod2k i=0n12-1i+4 ci(i+2)2rrodi,
R(r)=TF2+rrod2·h i=1ncii+2rrodi+1k i=0n ci(i+2)2 rrodi+22-ri+2i+4.
F=Fho+Finh=1rrod2 0rrodTho(r)rdr+Finh,
R(r)=Rho(r)+Rinh(r)=1r2 0rTho(r)rdr+Rinh(r).
Fho=Tf2+Q(rrod)rrod216k,
Rho(r)=Tf2+Q(rrod)rrod28k-Q(rrod)r216k.
Finh=1rrod2 0rrodrrod0τ(r,rp)drpdr,
Rinh(r)=1r2 0r rrod0τ(r, rp)drpdr.
F=Tf2+Q(rrod)16k+14hrrod rrod0q(rp)rp2drp+14k rrod0q(rp)rp2drp+116hrrod rrod0q(rp)rp4drp,
R(r)=Tf2+Q(rrod)rrod28k-Q(rrod)r216k+14hrrod rrod0q(rp)rp2drp+rrod24kr2 rrod0q(rp)rp2drp-[ln(r/rrod)-1/2]4k r0q(rp)rp2drp-116kr2 r0 q(rp)rp4drp-14k rrodrq(rp)rp2 lnrprroddrp+18k rrodr q(rp)rp2drp-r216k rrodr q(rp)drp.
Dbiref(r)=1πr2 0r02πdbiref(r, ϕ)rdϕdr
Q(r)=PpumpηconvηabspnP(r),
pn=0l02π0rrodP(r)rdrdφdz.

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