Abstract

For finite-thickness media with convective surface losses, the three-dimensional temperature distributions and thermal deformations of mirror substrates in laser resonators that are due to absorption of laser light with a Gaussian power-density profile are calculated by use of the well-known Green’s function methods. Some expressions and theoretical profiles of the temperature distributions and thermal deformations as functions of the radius and the thickness of a mirror substrate are obtained. The results of the calculations show that the rise in temperature is closely related to the absorption coefficient of the medium as well as to the convective heat-transfer coefficient, that the initial thermal deformations of mirror surfaces increase quickly at the beginning of laser heating and that then the thermal deformations are insensitive to laser heating times. Meanwhile, thermal deformations of a silicon mirror are experimentally demonstrated by use of CO2 laser irradiation. The experimental trends of thermal deformations are in agreement with the theoretical profiles.

© 2001 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. J. H. Torres, M. Motamedi, J. A. Pearce, A. J. Welch, “Experimental evaluation of mathematical models for predicting the thermal response of tissue to laser irradiation,” Appl. Opt. 32, 597–606 (1993).
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    [CrossRef]

1998 (1)

1997 (1)

1994 (2)

O. W. Shih, “A multilayer heat conduction solution for magneto-optical disk recording,” J. Appl. Phys. 75, 4382–4395 (1994).
[CrossRef]

P. Loza, D. Kouznetsov, R. Ortega, “Temperature distribution in a uniform medium heated by absorption of a Gaussian light beam,” Appl. Opt. 33, 3831–3836 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

W. A. McGahan, K. D. Cole, “Solutions of the heat conduction equation in multilayers for photothermal deflection experiments,” J. Appl. Phys. 72, 1362–1373 (1992).
[CrossRef]

1989 (2)

P. Grosse, R. Wynands, “Simulation of photoacoustic IR spectra of multilayer structures,” Appl. Phys. B 48, 59–65 (1989).
[CrossRef]

M. R. Madison, T. W. McDaniel, “Temperature distributions produced in an N-layer film structure by static or scanning laser or electron beam with application to magneto-optical media,” J. Appl. Phys. 66, 5738–5748 (1989).
[CrossRef]

1987 (1)

A. N. Burgess, K. E. Evans, M. Mackay, S. J. Abbott, “Comparison of transient thermal conduction in tellurium and organic dye based digital optical storage media,” J. Appl. Phys. 61, 74–80 (1987).
[CrossRef]

1985 (1)

A. Abtahi, P. F. Bräunlich, P. Kelly, J. Gasiot, “Laser stimulated thermoluminescence,” J. Appl. Phys. 58, 1626–1639 (1985).
[CrossRef]

1984 (1)

1982 (1)

1980 (2)

Abbott, S. J.

A. N. Burgess, K. E. Evans, M. Mackay, S. J. Abbott, “Comparison of transient thermal conduction in tellurium and organic dye based digital optical storage media,” J. Appl. Phys. 61, 74–80 (1987).
[CrossRef]

Abtahi, A.

A. Abtahi, P. F. Bräunlich, P. Kelly, J. Gasiot, “Laser stimulated thermoluminescence,” J. Appl. Phys. 58, 1626–1639 (1985).
[CrossRef]

Bräunlich, P. F.

A. Abtahi, P. F. Bräunlich, P. Kelly, J. Gasiot, “Laser stimulated thermoluminescence,” J. Appl. Phys. 58, 1626–1639 (1985).
[CrossRef]

Burgess, A. N.

A. N. Burgess, K. E. Evans, M. Mackay, S. J. Abbott, “Comparison of transient thermal conduction in tellurium and organic dye based digital optical storage media,” J. Appl. Phys. 61, 74–80 (1987).
[CrossRef]

Cole, K. D.

W. A. McGahan, K. D. Cole, “Solutions of the heat conduction equation in multilayers for photothermal deflection experiments,” J. Appl. Phys. 72, 1362–1373 (1992).
[CrossRef]

Connell, G. A. N.

Evans, K. E.

A. N. Burgess, K. E. Evans, M. Mackay, S. J. Abbott, “Comparison of transient thermal conduction in tellurium and organic dye based digital optical storage media,” J. Appl. Phys. 61, 74–80 (1987).
[CrossRef]

Gasiot, J.

A. Abtahi, P. F. Bräunlich, P. Kelly, J. Gasiot, “Laser stimulated thermoluminescence,” J. Appl. Phys. 58, 1626–1639 (1985).
[CrossRef]

Goodman, J. W.

Grosse, P.

P. Grosse, R. Wynands, “Simulation of photoacoustic IR spectra of multilayer structures,” Appl. Phys. B 48, 59–65 (1989).
[CrossRef]

Hartnett, J. P.

W. M. Rohsenow, J. P. Hartnett, Handbook of Heat Transfer (McGraw-Hill, New York, 1973).

Hauck, R.

Kelly, P.

A. Abtahi, P. F. Bräunlich, P. Kelly, J. Gasiot, “Laser stimulated thermoluminescence,” J. Appl. Phys. 58, 1626–1639 (1985).
[CrossRef]

Kortz, H. P.

Kouznetsov, D.

Loza, P.

Loze, M. K.

Mackay, M.

A. N. Burgess, K. E. Evans, M. Mackay, S. J. Abbott, “Comparison of transient thermal conduction in tellurium and organic dye based digital optical storage media,” J. Appl. Phys. 61, 74–80 (1987).
[CrossRef]

Madison, M. R.

M. R. Madison, T. W. McDaniel, “Temperature distributions produced in an N-layer film structure by static or scanning laser or electron beam with application to magneto-optical media,” J. Appl. Phys. 66, 5738–5748 (1989).
[CrossRef]

Mansuripur, M.

McDaniel, T. W.

M. R. Madison, T. W. McDaniel, “Temperature distributions produced in an N-layer film structure by static or scanning laser or electron beam with application to magneto-optical media,” J. Appl. Phys. 66, 5738–5748 (1989).
[CrossRef]

McGahan, W. A.

W. A. McGahan, K. D. Cole, “Solutions of the heat conduction equation in multilayers for photothermal deflection experiments,” J. Appl. Phys. 72, 1362–1373 (1992).
[CrossRef]

Motamedi, M.

Necati Özisik, M.

M. Necati Özişik, Heat Conduction (Wiley, New York, 1980).

Nowinsik, J. L.

J. L. Nowinsik, Theory of Thermoelasticity with Applications (Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1978).
[CrossRef]

Ortega, R.

Pearce, J. A.

Remo, J. L.

Rohsenow, W. M.

W. M. Rohsenow, J. P. Hartnett, Handbook of Heat Transfer (McGraw-Hill, New York, 1973).

Sanders, D. J.

Shih, O. W.

O. W. Shih, “A multilayer heat conduction solution for magneto-optical disk recording,” J. Appl. Phys. 75, 4382–4395 (1994).
[CrossRef]

Siegman, A. E.

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

Torres, J. H.

Weber, H.

Welch, A. J.

Wolf, H. F.

H. F. Wolf, Silicon Semiconduction Data (Signetics Corp., Albuquerque, N. Mex., 1969).

Wright, C. D.

Wynands, R.

P. Grosse, R. Wynands, “Simulation of photoacoustic IR spectra of multilayer structures,” Appl. Phys. B 48, 59–65 (1989).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. B (1)

P. Grosse, R. Wynands, “Simulation of photoacoustic IR spectra of multilayer structures,” Appl. Phys. B 48, 59–65 (1989).
[CrossRef]

J. Appl. Phys. (5)

M. R. Madison, T. W. McDaniel, “Temperature distributions produced in an N-layer film structure by static or scanning laser or electron beam with application to magneto-optical media,” J. Appl. Phys. 66, 5738–5748 (1989).
[CrossRef]

A. N. Burgess, K. E. Evans, M. Mackay, S. J. Abbott, “Comparison of transient thermal conduction in tellurium and organic dye based digital optical storage media,” J. Appl. Phys. 61, 74–80 (1987).
[CrossRef]

O. W. Shih, “A multilayer heat conduction solution for magneto-optical disk recording,” J. Appl. Phys. 75, 4382–4395 (1994).
[CrossRef]

A. Abtahi, P. F. Bräunlich, P. Kelly, J. Gasiot, “Laser stimulated thermoluminescence,” J. Appl. Phys. 58, 1626–1639 (1985).
[CrossRef]

W. A. McGahan, K. D. Cole, “Solutions of the heat conduction equation in multilayers for photothermal deflection experiments,” J. Appl. Phys. 72, 1362–1373 (1992).
[CrossRef]

Other (5)

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

J. L. Nowinsik, Theory of Thermoelasticity with Applications (Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1978).
[CrossRef]

H. F. Wolf, Silicon Semiconduction Data (Signetics Corp., Albuquerque, N. Mex., 1969).

W. M. Rohsenow, J. P. Hartnett, Handbook of Heat Transfer (McGraw-Hill, New York, 1973).

M. Necati Özişik, Heat Conduction (Wiley, New York, 1980).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of heat-conduction problems.

Fig. 2
Fig. 2

Laser-induced temperature rise T′ versus mirror radius r for absorption coefficients μ of the medium: (a) 3.3 m-1 and (b) 103 m-1.

Fig. 3
Fig. 3

Laser-induced temperature rise T′ versus axial distance z for absorption coefficients μ of the medium: (a) 3.3 m-1 and (b) 103 m-1.

Fig. 4
Fig. 4

Calculated thermal deformations versus mirror radius r for absorption coefficients μ of the medium: (a) 3.3 m-1 and (b) 103 m-1.

Fig. 5
Fig. 5

Calculated maximum thermal deformations versus laser irradiating times for absorption coefficients μ of the medium: (a) 3.3 m-1 and (b) 103 m-1.

Fig. 6
Fig. 6

Calculated maximum thermal deformations versus medium thickness d for absorption coefficients μ of the medium: (a) 3.3 m-1 and (b) 103 m-1.

Fig. 7
Fig. 7

Experimental maximum thermal deformations versus laser irradiating times. Symbols, experimental data; curves, fitted profiles.

Tables (1)

Tables Icon

Table 1 Parameters of the Silicon Mirrora

Equations (31)

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2Tr, z, tr2+1rTr, z, tr+2Tr, z, tz2+1κ gr, z=1αTr, z, tt, 0r<b, 0<z<d, t>0.
Tr+HT-T=0,  r=b, t>0,
Tz=0,  z=0, t>0,
Tz+HT-T=0,  z=d, t>0,
T=T,  t=0
2Tr, z, tr2+1rTr, z, tr+2Tr, z, tz2+1κ gr, z=1αTr, z, tt,  0r<b, 0<z<d, t>0.
Tr+HT=0,  r=b, t>0,
Tz=0,  z=0, t>0,
Tz+HT=0,  z=d, t>0,
T=0,  t=0
Tr, z, t=ακVτ=0t Gr, z, t|r, z, τ×gr, z2πrdrdzdτ,
2ψr, z, tr2+1rψr, z, tr+2ψr, z, tz2=1αψr, z, tt,  0r<b, 0<z<d, t>0.
ψr+Hψ=0,  r=b, t>0,
ψz=0,  z=0, t>0,
ψz+Hψ=0,  z=d, t>0,
ψ=Fr, z,  t=0
ψr, z, t=m=1p=1exp-αβm2+ηp2tNβmNηp R0βm, r×Zηp, z×r=0bz=0d R0βm, r×Zηp, zFr, z2πrdrdz.
R0βm, r=J0βmr,1Nβm=2J02βmbβm2b2H2+βm2,
βmJ1βmb=HJ0βmb,
Zηp, z=cos ηpz,1Nηp=2 ηp2+H2dηp2+H2+H,
ηp tan ηpd=H.
ψr, z, t=r=0bz=0d Gr, z, t|r, z, τ|τ=0×Fr, z2πrdrdz.
Gr, z, t|r, z, τ|τ=0=m=1p=1exp-αβm2+ηp2tNβmNηp×R0βm, rZηp, zR0βm, rZηp, z.
Gr, z, t|r, z, τ=m=1p=1exp-αβm2+ηp2t-τNβmNηpR0βm, rZηp, zR0βm, rZηp, z.
Tr, z, t=ακr=0bz=0dτ=0tm=1p=1×exp-αβm2+ηp2t-τNβmNηp R0βm, rZηp, z×R0βm, rZηp, zgr, z2πrdrdzdτ.
Tr, z, t=T+ακr=0bz=0dτ=0tm=1p=1×exp-αβm2+ηp2t-τNβmNηp R0βm, rZηp, z×R0βm, rZηp, zgr, z2πrdrdzdτ,
Ir=2Pπω2exp-2r2ω2
gr, z=2P1-Rfπω2 μ exp-2r2ω2-μz,
Δd=Δdf+Δdt,
Δdr, tΔdf=αl0d Tr, z, tdz.
xκTx+yκTy+zκTz+gx, y, z, t=ρCpTt.

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