Abstract

A general solution is presented for the temperature rise produced by the absorption of a scanning Gaussian laser beam in a solid target. In normalized coordinates, the temperature rise is found to depend only on the ratio of the scan speed to the rate of heat diffusion in the solid and the ratio of the beam radius to the absorption depth. For slow scan speeds the solution simplifies to the steady-state approximation in which the power input is balanced by heat conduction into the solid. For fast scan speeds the solution approaches the energy density limit in which the temperature rise is proportional to the integrated beam intensity. For highly absorbing materials the solution simplifies to the surface absorption approximation. The general solution demonstrates the conditions under which each approximation can be used. Similar solutions are found for the related case of pulsed exposure by a stationary beam. The solution is demonstrated experimentally by exposing thermal paper with a CO2 laser.

© 1984 Optical Society of America

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References

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  1. For a review of materials processing by lasers, see J. F. Ready, Proc. IEEE 70, 533 (1982).
    [CrossRef]
  2. For a review of laser annealing, see A. E. Bell, RCA Rev. 40, 295 (1979).
  3. D. Maydan, Bell Syst. Tech. J. 50, 1761 (1971).
  4. J. M. O’Reilly, R. A. Mosher, W. L. Goffe, Photogr. Sci. Eng. 23, 314 (1979).
  5. R. S. Braudy, J. Appl. Phys. 45, 3612 (1974).
    [CrossRef]
  6. C. A. Bruce, J. T. Jacobs, J. Appl. Photogr. Eng. 3, 40 (1977).
  7. Y. H. Wong, R. L. Thomas, G. F. Hawkins, Appl. Phys. Lett. 32, 538 (1978).
    [CrossRef]
  8. M. Lax, J. Appl. Phys. 48, 3919 (1977).
    [CrossRef]
  9. M. Lax, Appl. Phys. Lett. 33, 786 (1978).
    [CrossRef]
  10. M. Bertolotti, C. Sibilia, IEEE J. Quantum Electron. QE-17, 1980 (1981).
    [CrossRef]
  11. I. D. Calder, R. Sue, J. Appl. Phys. 53, 7545 (1982).
    [CrossRef]
  12. H. E. Cline, T. R. Anthony, J. Appl. Phys. 48, 3895 (1977).
    [CrossRef]
  13. Y. I. Nissim, A. Lietoila, R. B. Gold, J. F. Gibbons, J. Appl. Phys. 51, 274 (1980).
    [CrossRef]
  14. J. E. Moody, R. H. Hendel, J. Appl. Phys. 53, 4364 (1982).
    [CrossRef]
  15. M. L. Burgener, R. E. Reedy, J. Appl. Phys. 53, 4357 (1982).
    [CrossRef]
  16. M. Noguchi, Appl. Opt. 21, 2665 (1982).
    [CrossRef] [PubMed]
  17. D. B. Congleton, M. R. Smith, A. S. Diamond, J. Appl. Photogr. Eng. 3, 97 (1977).
  18. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., Oxford, 1959).
  19. K. Mashio, in Proceedings, TAPPI 1979 Printing Reprography Testing Conference (TAPPI Press, Atlanta, 1979), p. 133.
  20. L. A. Kirk, C. Tatlicibasi, Tappi 55, 1697 (1972).
  21. R. J. Kerekes, Tappi 63, 137 (1980).
  22. D. J. Sanders, R. C. Forsyth, Rev. Sci. Instrum. 54, 238 (1983).
    [CrossRef]
  23. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1964).

1983 (1)

D. J. Sanders, R. C. Forsyth, Rev. Sci. Instrum. 54, 238 (1983).
[CrossRef]

1982 (5)

For a review of materials processing by lasers, see J. F. Ready, Proc. IEEE 70, 533 (1982).
[CrossRef]

I. D. Calder, R. Sue, J. Appl. Phys. 53, 7545 (1982).
[CrossRef]

J. E. Moody, R. H. Hendel, J. Appl. Phys. 53, 4364 (1982).
[CrossRef]

M. L. Burgener, R. E. Reedy, J. Appl. Phys. 53, 4357 (1982).
[CrossRef]

M. Noguchi, Appl. Opt. 21, 2665 (1982).
[CrossRef] [PubMed]

1981 (1)

M. Bertolotti, C. Sibilia, IEEE J. Quantum Electron. QE-17, 1980 (1981).
[CrossRef]

1980 (2)

Y. I. Nissim, A. Lietoila, R. B. Gold, J. F. Gibbons, J. Appl. Phys. 51, 274 (1980).
[CrossRef]

R. J. Kerekes, Tappi 63, 137 (1980).

1979 (2)

For a review of laser annealing, see A. E. Bell, RCA Rev. 40, 295 (1979).

J. M. O’Reilly, R. A. Mosher, W. L. Goffe, Photogr. Sci. Eng. 23, 314 (1979).

1978 (2)

Y. H. Wong, R. L. Thomas, G. F. Hawkins, Appl. Phys. Lett. 32, 538 (1978).
[CrossRef]

M. Lax, Appl. Phys. Lett. 33, 786 (1978).
[CrossRef]

1977 (4)

C. A. Bruce, J. T. Jacobs, J. Appl. Photogr. Eng. 3, 40 (1977).

M. Lax, J. Appl. Phys. 48, 3919 (1977).
[CrossRef]

H. E. Cline, T. R. Anthony, J. Appl. Phys. 48, 3895 (1977).
[CrossRef]

D. B. Congleton, M. R. Smith, A. S. Diamond, J. Appl. Photogr. Eng. 3, 97 (1977).

1974 (1)

R. S. Braudy, J. Appl. Phys. 45, 3612 (1974).
[CrossRef]

1972 (1)

L. A. Kirk, C. Tatlicibasi, Tappi 55, 1697 (1972).

1971 (1)

D. Maydan, Bell Syst. Tech. J. 50, 1761 (1971).

Anthony, T. R.

H. E. Cline, T. R. Anthony, J. Appl. Phys. 48, 3895 (1977).
[CrossRef]

Bell, A. E.

For a review of laser annealing, see A. E. Bell, RCA Rev. 40, 295 (1979).

Bertolotti, M.

M. Bertolotti, C. Sibilia, IEEE J. Quantum Electron. QE-17, 1980 (1981).
[CrossRef]

Braudy, R. S.

R. S. Braudy, J. Appl. Phys. 45, 3612 (1974).
[CrossRef]

Bruce, C. A.

C. A. Bruce, J. T. Jacobs, J. Appl. Photogr. Eng. 3, 40 (1977).

Burgener, M. L.

M. L. Burgener, R. E. Reedy, J. Appl. Phys. 53, 4357 (1982).
[CrossRef]

Calder, I. D.

I. D. Calder, R. Sue, J. Appl. Phys. 53, 7545 (1982).
[CrossRef]

Carslaw, H. S.

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

Cline, H. E.

H. E. Cline, T. R. Anthony, J. Appl. Phys. 48, 3895 (1977).
[CrossRef]

Congleton, D. B.

D. B. Congleton, M. R. Smith, A. S. Diamond, J. Appl. Photogr. Eng. 3, 97 (1977).

Diamond, A. S.

D. B. Congleton, M. R. Smith, A. S. Diamond, J. Appl. Photogr. Eng. 3, 97 (1977).

Forsyth, R. C.

D. J. Sanders, R. C. Forsyth, Rev. Sci. Instrum. 54, 238 (1983).
[CrossRef]

Gibbons, J. F.

Y. I. Nissim, A. Lietoila, R. B. Gold, J. F. Gibbons, J. Appl. Phys. 51, 274 (1980).
[CrossRef]

Goffe, W. L.

J. M. O’Reilly, R. A. Mosher, W. L. Goffe, Photogr. Sci. Eng. 23, 314 (1979).

Gold, R. B.

Y. I. Nissim, A. Lietoila, R. B. Gold, J. F. Gibbons, J. Appl. Phys. 51, 274 (1980).
[CrossRef]

Hawkins, G. F.

Y. H. Wong, R. L. Thomas, G. F. Hawkins, Appl. Phys. Lett. 32, 538 (1978).
[CrossRef]

Hendel, R. H.

J. E. Moody, R. H. Hendel, J. Appl. Phys. 53, 4364 (1982).
[CrossRef]

Jacobs, J. T.

C. A. Bruce, J. T. Jacobs, J. Appl. Photogr. Eng. 3, 40 (1977).

Jaeger, J. C.

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

Kerekes, R. J.

R. J. Kerekes, Tappi 63, 137 (1980).

Kirk, L. A.

L. A. Kirk, C. Tatlicibasi, Tappi 55, 1697 (1972).

Lax, M.

M. Lax, Appl. Phys. Lett. 33, 786 (1978).
[CrossRef]

M. Lax, J. Appl. Phys. 48, 3919 (1977).
[CrossRef]

Lietoila, A.

Y. I. Nissim, A. Lietoila, R. B. Gold, J. F. Gibbons, J. Appl. Phys. 51, 274 (1980).
[CrossRef]

Mashio, K.

K. Mashio, in Proceedings, TAPPI 1979 Printing Reprography Testing Conference (TAPPI Press, Atlanta, 1979), p. 133.

Maydan, D.

D. Maydan, Bell Syst. Tech. J. 50, 1761 (1971).

Moody, J. E.

J. E. Moody, R. H. Hendel, J. Appl. Phys. 53, 4364 (1982).
[CrossRef]

Mosher, R. A.

J. M. O’Reilly, R. A. Mosher, W. L. Goffe, Photogr. Sci. Eng. 23, 314 (1979).

Nissim, Y. I.

Y. I. Nissim, A. Lietoila, R. B. Gold, J. F. Gibbons, J. Appl. Phys. 51, 274 (1980).
[CrossRef]

Noguchi, M.

O’Reilly, J. M.

J. M. O’Reilly, R. A. Mosher, W. L. Goffe, Photogr. Sci. Eng. 23, 314 (1979).

Ready, J. F.

For a review of materials processing by lasers, see J. F. Ready, Proc. IEEE 70, 533 (1982).
[CrossRef]

Reedy, R. E.

M. L. Burgener, R. E. Reedy, J. Appl. Phys. 53, 4357 (1982).
[CrossRef]

Sanders, D. J.

D. J. Sanders, R. C. Forsyth, Rev. Sci. Instrum. 54, 238 (1983).
[CrossRef]

Sibilia, C.

M. Bertolotti, C. Sibilia, IEEE J. Quantum Electron. QE-17, 1980 (1981).
[CrossRef]

Smith, M. R.

D. B. Congleton, M. R. Smith, A. S. Diamond, J. Appl. Photogr. Eng. 3, 97 (1977).

Sue, R.

I. D. Calder, R. Sue, J. Appl. Phys. 53, 7545 (1982).
[CrossRef]

Tatlicibasi, C.

L. A. Kirk, C. Tatlicibasi, Tappi 55, 1697 (1972).

Thomas, R. L.

Y. H. Wong, R. L. Thomas, G. F. Hawkins, Appl. Phys. Lett. 32, 538 (1978).
[CrossRef]

Wong, Y. H.

Y. H. Wong, R. L. Thomas, G. F. Hawkins, Appl. Phys. Lett. 32, 538 (1978).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

M. Lax, Appl. Phys. Lett. 33, 786 (1978).
[CrossRef]

Y. H. Wong, R. L. Thomas, G. F. Hawkins, Appl. Phys. Lett. 32, 538 (1978).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Maydan, Bell Syst. Tech. J. 50, 1761 (1971).

IEEE J. Quantum Electron. (1)

M. Bertolotti, C. Sibilia, IEEE J. Quantum Electron. QE-17, 1980 (1981).
[CrossRef]

J. Appl. Photogr. Eng. (2)

D. B. Congleton, M. R. Smith, A. S. Diamond, J. Appl. Photogr. Eng. 3, 97 (1977).

C. A. Bruce, J. T. Jacobs, J. Appl. Photogr. Eng. 3, 40 (1977).

J. Appl. Phys. (7)

R. S. Braudy, J. Appl. Phys. 45, 3612 (1974).
[CrossRef]

M. Lax, J. Appl. Phys. 48, 3919 (1977).
[CrossRef]

I. D. Calder, R. Sue, J. Appl. Phys. 53, 7545 (1982).
[CrossRef]

H. E. Cline, T. R. Anthony, J. Appl. Phys. 48, 3895 (1977).
[CrossRef]

Y. I. Nissim, A. Lietoila, R. B. Gold, J. F. Gibbons, J. Appl. Phys. 51, 274 (1980).
[CrossRef]

J. E. Moody, R. H. Hendel, J. Appl. Phys. 53, 4364 (1982).
[CrossRef]

M. L. Burgener, R. E. Reedy, J. Appl. Phys. 53, 4357 (1982).
[CrossRef]

Photogr. Sci. Eng. (1)

J. M. O’Reilly, R. A. Mosher, W. L. Goffe, Photogr. Sci. Eng. 23, 314 (1979).

Proc. IEEE (1)

For a review of materials processing by lasers, see J. F. Ready, Proc. IEEE 70, 533 (1982).
[CrossRef]

RCA Rev. (1)

For a review of laser annealing, see A. E. Bell, RCA Rev. 40, 295 (1979).

Rev. Sci. Instrum. (1)

D. J. Sanders, R. C. Forsyth, Rev. Sci. Instrum. 54, 238 (1983).
[CrossRef]

Tappi (2)

L. A. Kirk, C. Tatlicibasi, Tappi 55, 1697 (1972).

R. J. Kerekes, Tappi 63, 137 (1980).

Other (3)

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1964).

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

K. Mashio, in Proceedings, TAPPI 1979 Printing Reprography Testing Conference (TAPPI Press, Atlanta, 1979), p. 133.

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Figures (7)

Fig. 1
Fig. 1

Time evolution of the normalized temperature rise θ at the origin for a geometric factor γ = 1 and various normalized scan velocities ν. The dashed line is the steady-state limit.

Fig. 2
Fig. 2

Time evolution of the temperature rise at the origin for γ = 1 and various scan velocities ν. The temperature axis has been renormalized to (ν/γ)θ. The dashed line is the energy density limit.

Fig. 3
Fig. 3

Horizontal temperature profiles along the y axis at t = 0 for γ = 1 and various scan velocities ν. The dashed lines are limiting cases.

Fig. 4
Fig. 4

Vertical temperature profiles along the z axis at t = 0 for γ = 1 and various scan velocities ν. The dashed lines are limiting cases.

Fig. 5
Fig. 5

Time evolution of the temperature rise at the origin for γ = 1 and various normalized pulse lengths τp. In each case the temperature decays after the pulse is terminated.

Fig. 6
Fig. 6

Maximum temperature rise at the origin vs pulse length τp for various geometric factors γ. The dashed lines are limiting solutions for γ = 10°.

Fig. 7
Fig. 7

Measured CO2 laser power vs exposure time to produce a 1.4-mm spot on thermal paper. The solid line is calculated from the general solution. The dashed line is the energy density limit, and the dash–dot line is the steady-state limit.

Equations (31)

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2 T = ( 1 / D ) T / t - G / K ,
G = ( α P / π δ 2 ) exp [ - ( x - v t ) 2 / δ 2 - y 2 / δ 2 - α z ] ,
d T ( x , y , z , t ) = ( d q / { 8 ρ C [ π D ( t - t ) ] 3 / 2 } ) × exp { - [ ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 ] / 4 D ( t - t ) } .
T ( x , y , z , t ) = [ P / ( 2 π K δ ) ] θ ( χ , ξ , ζ , τ ; ν , γ ) ,
θ ( χ , ξ , ζ , τ ; ν , γ ) = ( γ / 2 π ) τ 0 τ + τ 1 [ exp ( γ 2 τ ) / ( τ + 1 ) ] × exp { - [ ( χ - ν ( τ - τ ) ) 2 + ξ 2 ] / ( τ + 1 ) } × [ exp ( ζ ) erfc ( γ τ + ζ / 2 γ τ ) + exp ( - ζ ) erfc ( γ τ - ζ / 2 γ τ ) ] d τ ,
τ 0 = 0 , - τ 1 < τ < τ 2 , = τ - τ 2 , τ > τ 2 .
θ ( 0 , 0 , 0 , τ ; ν , γ ) = ( γ / π ) 0 [ exp ( γ 2 τ ) / ( τ + 1 ) ] × exp [ - ν 2 ( τ - τ ) 2 / ( τ + 1 ) ] erfc ( γ τ ) d τ .
θ 0 ( η , ζ ; γ ) = ( 2 γ / π ) 0 J 0 ( 2 λ η ) exp ( - λ 2 ) × { [ γ exp ( - λ ζ / γ ) - λ exp ( - ζ ) ] / ( γ 2 - λ 2 ) } d λ ,
θ ( 0 , 0 , 0 , τ ; ν , γ ) ( γ / π ) 0 exp [ - ν 2 ( τ - τ ) 2 ] d τ = ( γ / 2 ν ) erfc ( - ν τ ) .
T ( 0 , 0 , 0 , t ) = [ P α / ( 2 π ρ C v δ ) ] erfc ( - v t / δ ) .
p ( x , y , t ) = ( P / π δ 2 ) exp [ - ( x - v t ) 2 / δ 2 - y 2 / δ 2 ] .
( 0 , 0 , t ) = - t p ( 0 , 0 , t ) d t = [ P / 2 π δ v ] erfc ( - v t / δ ) .
T ( 0 , 0 , 0 , t ) = α ( 0 , 0 , t ) / ρ C .
θ ( 0 , 0 , 0 , τ ; ν , γ ) ( 1 / π ) 0 exp [ - ν 2 ( τ - τ ) 2 / ( τ + 1 ) ] × [ τ ( τ + 1 ) ] - 1 d τ .
θ ( 0 , 0 , τ ; τ p , γ ) = ( γ / π ) τ 0 τ [ exp ( γ 2 τ ) / ( τ + 1 ) ] erfc ( γ τ ) d τ ,
τ 0 = 0 , 0 < τ < τ p = τ - τ p , τ > τ p .
θ ( 0 , 0 , τ p 0 ; γ ) γ τ p / π .
T ( 0 , 0 , t p ) = α P t p / ( π δ 2 ρ C ) ,
θ ( 0 , 0 , τ p ; γ ) ( 2 / π ) tan - 1 τ p ( 2 / π ) τ p for τ p 1.
P = ( 2 π K δ Δ T ) / θ ( 1 , 0 , τ p ; γ ) .
d T ( x , y , z , t ) = { d q / [ 8 ρ C π 3 / 2 ( D x D y D z ) 1 / 2 ( t - t ) 3 / 2 ] } × exp { - i ( i - i ) 2 / 4 D i ( t - t ) } .
G = [ α P / ( π δ x δ y ) ] exp [ - ( x - v t ) 2 / δ x 2 - y 2 / δ y 2 - α z ] .
T ( x , y , z , t ) = [ P / ( 2 π ( K ¯ K z ) 1 / 2 δ ¯ ) ] θ ( χ , ξ , ζ , τ ; ν , γ ) ,
χ = x / δ ¯ , ξ = y / δ ¯ , ζ = α z , τ = 4 D ¯ t / δ ¯ 2 , ν = v δ ¯ / 4 D ¯ , γ = ( K z / K ¯ ) 1 / 2 α δ ¯ / 2 , β = δ x / δ y , κ = ( K x / K y ) 1 / 2 ,
θ ( χ , ξ , ζ , τ ; ν , γ ) = ( γ / 2 π ) τ 0 τ + τ 1 exp ( γ 2 τ ) × [ ( β + κ τ ) 1 / 2 ( β - 1 + κ - 1 τ ) 1 / 2 ] - 1 × exp { - [ χ - ν ( τ - τ ) ] 2 / ( β + κ τ ) } × exp [ - ξ 2 / ( β - 1 + κ - 1 τ ) ] × [ exp ( ζ ) erfc ( γ τ + ζ / 2 γ τ ) + exp ( - ζ ) erfc ( γ τ - ζ / 2 γ τ ) ] d τ .
exp ( - k / p ) / p = 0 exp ( - p t ) J 0 ( 2 k t ) d t .
exp [ - η 2 / ( τ + 1 ) ] / ( τ + 1 ) = 0 J 0 ( 2 λ η ) exp [ - λ 2 ( τ + 1 ) ] 2 λ d λ .
f ( τ ) = [ exp ( γ 2 τ ) / 2 ] [ exp ( ζ ) erfc ( ζ / 2 γ τ + γ τ ) + exp ( - ζ ) erfc ( ζ / 2 γ τ - γ τ ) ] ,
θ ( η , ζ , τ ; 0 , γ ) = ( 2 γ / π ) 0 J 0 ( 2 λ η ) exp ( - λ 2 ) λ { exp ( - ζ ) 0 exp [ - ( λ 2 - γ 2 ) τ ] d τ - 0 exp ( - λ 2 τ ) f ( τ ) d τ } d λ .
0 exp ( - λ 2 τ ) f ( τ ) d τ = γ exp ( - λ ζ / γ ) / [ λ ( λ 2 - γ 2 ) ] .
θ ( η , ζ , τ ; 0 γ ) = ( 2 γ / π ) 0 J 0 ( 2 λ η ) exp ( - λ 2 ) × { [ γ exp ( - λ ζ / γ ) - λ exp ( - ζ ) ] / ( γ 2 - λ 2 ) } d λ = θ 0 ( η , ζ ; γ ) .

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