Abstract

For a multilayer structure illuminated by a laser beam, absorption of optical energy in the absorptive layers and the diffusion of the resultant heat throughout the structure are studied. Analytical and numerical procedures for this study are described, and, as a specific example, the profiles of temperature distribution during recording on a magnetooptical disk are presented. The technique is also expected to be of value for studies of thermal marking and laser annealing.

© 1982 Optical Society of America

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References

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  1. D. Chen, G. N. Otto, F. M. Schmit, IEEE Trans. Magn. MAG-9, 66 (1973).
    [CrossRef]
  2. M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal and Noise in Magneto-Optical Readout,” to be published in J. Appl. Phys.May1982.
    [CrossRef]
  3. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1967).
  4. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. P., London, 1954).
  5. G. Birkhoff, R. S. Varga, D. Young, “Alternation Direction Implicit Methods,” in Advances in Computers, F. L. Alt, M. Rubinoff, Eds. (Academic, New York, 1962).
    [CrossRef]
  6. W. F. Ames, Numerical Methods for Partial Differential Equations (Barnes & Noble, New York, 1969).

1973 (1)

D. Chen, G. N. Otto, F. M. Schmit, IEEE Trans. Magn. MAG-9, 66 (1973).
[CrossRef]

Ames, W. F.

W. F. Ames, Numerical Methods for Partial Differential Equations (Barnes & Noble, New York, 1969).

Birkhoff, G.

G. Birkhoff, R. S. Varga, D. Young, “Alternation Direction Implicit Methods,” in Advances in Computers, F. L. Alt, M. Rubinoff, Eds. (Academic, New York, 1962).
[CrossRef]

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. P., London, 1954).

Chen, D.

D. Chen, G. N. Otto, F. M. Schmit, IEEE Trans. Magn. MAG-9, 66 (1973).
[CrossRef]

Connell, G. A. N.

M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal and Noise in Magneto-Optical Readout,” to be published in J. Appl. Phys.May1982.
[CrossRef]

Goodman, J. W.

M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal and Noise in Magneto-Optical Readout,” to be published in J. Appl. Phys.May1982.
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1967).

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. P., London, 1954).

Mansuripur, M.

M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal and Noise in Magneto-Optical Readout,” to be published in J. Appl. Phys.May1982.
[CrossRef]

Otto, G. N.

D. Chen, G. N. Otto, F. M. Schmit, IEEE Trans. Magn. MAG-9, 66 (1973).
[CrossRef]

Schmit, F. M.

D. Chen, G. N. Otto, F. M. Schmit, IEEE Trans. Magn. MAG-9, 66 (1973).
[CrossRef]

Varga, R. S.

G. Birkhoff, R. S. Varga, D. Young, “Alternation Direction Implicit Methods,” in Advances in Computers, F. L. Alt, M. Rubinoff, Eds. (Academic, New York, 1962).
[CrossRef]

Young, D.

G. Birkhoff, R. S. Varga, D. Young, “Alternation Direction Implicit Methods,” in Advances in Computers, F. L. Alt, M. Rubinoff, Eds. (Academic, New York, 1962).
[CrossRef]

IEEE Trans. Magn. (1)

D. Chen, G. N. Otto, F. M. Schmit, IEEE Trans. Magn. MAG-9, 66 (1973).
[CrossRef]

Other (5)

M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal and Noise in Magneto-Optical Readout,” to be published in J. Appl. Phys.May1982.
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1967).

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. P., London, 1954).

G. Birkhoff, R. S. Varga, D. Young, “Alternation Direction Implicit Methods,” in Advances in Computers, F. L. Alt, M. Rubinoff, Eds. (Academic, New York, 1962).
[CrossRef]

W. F. Ames, Numerical Methods for Partial Differential Equations (Barnes & Noble, New York, 1969).

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Figures (11)

Fig. 1
Fig. 1

Multilayer in the cylindrical coordinate system (r,θ,z). The number of layers is N, and the surface is in the z = 0 plane. A circularly symmetric Gaussian beam of light, propagating in the z direction, illuminates the surface. The axis of symmetry for the beam is r = 0.

Fig. 2
Fig. 2

Structure of a quadrilayer magnetooptical disk.

Fig. 3
Fig. 3

Average rate of flow of optical energy through the quadrilayer for a plane wave with unit intensity.

Fig. 4
Fig. 4

Laser pulses used in the numerical calculations of temperature distribution in the quadrilayer: (a) 60-nsec pulse with peak power of 2 mW and total energy of 96 pJ used in calculations pertaining to Figs. 510; (b) 40-nsec pulse with peak power of 8/3 mW and total energy of 64 pJ used in calculations pertaining to Fig. 11.

Fig. 5
Fig. 5

Quadrilayer’s temperature (above ambient) vs z at the center of the beam (r = 0) for several instants of time. At t = 0 the quadrilayer has been at ambient temperature. The laser pulse is shown in Fig. 4(a), and the radius of the beam at the exp(−1) point is 3500 Å. For these calculations the heat flow from the surface has been neglected (i.e., γ = 0).

Fig. 6
Fig. 6

Radial temperature distribution of the magnetic film at several instants of time (conditions as in Fig. 5).

Fig. 7
Fig. 7

Time dependence of the magnetic film’s temperature at various radii (conditions same as in Fig. 5).

Fig. 8
Fig. 8

Quadrilayer’s temperature (above ambient) vs z at the center of the beam (r = 0) for several instants of time. At t = 0 the quadrilayer has been at ambient temperature. The laser pulse is shown in Fig. 4(a), and the radius of the beam at the exp(−1) point is 3500 Å. For these calculations the escape of heat from the surface has been taken into account by setting γ = 105 cm−1.

Fig. 9
Fig. 9

Radial temperature distribution of the magnetic film at several instants of time (conditions same as in Fig. 8).

Fig. 10
Fig. 10

Time dependence of the magnetic film’s temperature at various radii (conditions same as in Fig. 8).

Fig. 11
Fig. 11

Time dependence of the magnetic film’s temperature at various radii. At t = 0 the quadrilayer has been at ambient temperature. The laser pulse is shown in Fig. 4(b), and the radius of the beam at the exp(−1) point is 3500 Å. For these calculations the escape of heat from the surface has been taken into account by setting γ = 105 cm−1 (compare with Fig. 10).

Tables (2)

Tables Icon

Table I Basic Equations of Alternating Direction–Implicit Technique for Problem of Heat Transfer in Multilayers

Tables Icon

Table II Numerical Values for Optical and Thermal Parameters of the Quadrilayer

Equations (41)

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{ ( d 2 / d Z 2 ) + [ 2 π n ( k ) / λ 0 ] 2 } E k ( Z ) = 0 ;
( d / d Z ) E k ( Z ) + i ( 2 π / λ 0 ) H k ( Z ) = 0.
E k ( Z ) = A 1 ( k ) ( exp { - i [ 2 π n ( k ) / λ 0 ] ( Z - j = k + 1 N + 1 z j ) } + A 2 ( k ) exp { + i [ 2 π n ( k ) / λ 0 ] ( Z - j = k + 1 N + 1 z j ) } ) ,
H k ( Z ) = n ( k ) A 1 ( k ) ( exp { - i [ 2 π n ( k ) / λ 0 ] ( Z - j = k + 1 N + 1 z j ) } + A 2 ( k ) exp { + i [ 2 π n ( k ) / λ 0 ] ( Z - j = k + 1 N + 1 z j ) } ) .
A 2 ( k ) = [ n ( k ) - n ( k - 1 ) ] / [ n ( k ) + n ( k - 1 ) ] + A 2 ( k - 1 ) 1 + { [ n ( k ) - n ( k - 1 ) ] / [ n ( k ) + n ( k - 1 ) ] } A 2 ( k - 1 ) × exp { - i [ 4 π n ( k ) / λ 0 ] z k } ;             2 k N + 1.
A 1 ( k ) = ( exp { - i [ 2 π n ( k + 1 ) / λ 0 ] z k + 1 } + A 2 ( k + 1 ) exp { + i [ 2 π n ( k + 1 ) / λ 0 ] z k + 1 } ) 1 + A 2 ( k ) × A 1 ( k + 1 )             1 k N .
Y ( z ) = Re [ ½ E ( z ) H * ( z ) ] = ½ [ n r ( k ) ( exp { - [ 4 π n Im ( k ) / λ 0 ] ( Z - j = k + 1 N + 1 z j ) } - A 2 ( k ) 2 exp { + [ 4 π n Im ( k ) / λ 0 ] ( Z - j = k + 1 N + 1 z j ) } ) - 2 n Im ( k ) A 2 ( k ) sin { 4 π n r ( k ) / λ 0 ( Z - j = k + 1 N + 1 z j ) + Φ [ A 2 ( k ) ] } ] · A 1 ( k ) 2 ,
I ( r , t ) = P 0 ( t ) / ( π r 0 2 ) ] exp [ - ( r / r 0 ) 2 ] ,
C n ( / t ) Θ ( r , z , t ) - K n 2 Θ ( r , z , t ) = g ( r , z , t ) ;
( / z ) Θ ( r , z = 0 , t ) = γ Θ ( r , z = 0 , t ) ;
Θ ( r , z = , t ) = Θ ( r = , z , t ) = 0 ;
Θ ( r , z , t = 0 ) = f ( r , z ) .
[ C ( / t ) - K ( 2 / x 2 ) ] Θ ( x , t ) = g ( x , t ) ;             0 x L ; t 0 ,
( / x ) Θ ( x = 0 , t ) = γ Θ ( x = 0 , t ) ;
Θ ( x = L , t ) = 0 ;
Θ ( x , t = 0 ) = f ( x ) .
C [ ( Θ j k + 1 - Θ j k ) / Δ t ] - K [ ( Θ j + 1 k - 2 Θ j k + Θ j - 1 k ) / ( Δ x ) 2 ] = [ 1 / ( Δ t Δ x ) ] × [ k Δ t ( k + 1 ) Δ x d t ( j - 1 / 2 ) Δ x ( j + 1 / 2 ) Δ x g ( x , t ) d x ] ;             1 j j max - 1 ; k 0 ,
C [ ( Θ 0 k + 1 - Θ 0 k ) / Δ t ] - K [ ( 2 Θ 1 k - 2 ( 1 + γ Δ x ) Θ 0 k ) / ( Δ x ) 2 ] = [ 2 / ( Δ t Δ x ) ] × [ k Δ t ( k + 1 ) Δ t d t 1 1 / 2 Δ x g ( x , t ) d x ] ;             k 0
Θ j max k = 0 ;             k 0
Θ j 0 = f ( j Δ x ) ;             0 j j max
Θ j 0 = f ( j Δ x ) ;             0 j j max
Θ 0 k + 1 = Θ 0 k + [ D Δ t / ( Δ x ) 2 ] [ 2 Θ 1 k - 2 ( 1 + γ Δ x ) Θ 0 k ] + [ 2 / ( C Δ x ) ] [ k Δ t ( k + 1 ) Δ t d t 0 1 / 2 Δ x g ( x , t ) d x ] ;             k 0
Θ 0 k + 1 = Θ 0 k + [ D Δ t / ( Δ x ) 2 ] [ Θ j + 1 k - 2 Θ j k + Θ j - 1 k ] + [ 1 / ( C Δ x ) ] [ k Δ t ( k + 1 ) Δ t d t ( j - 1 / 2 ) Δ x ( j + 1 / 2 ) Δ x g ( x , t ) d x ] ;             1 j j max - 1 ; k 0
Θ j max k + 1 = 0 ;             k 0.
C [ ( Θ j k + 1 - Θ j k ) / Δ t ] - K [ ( Θ j + 1 k + 1 - 2 Θ j k + 1 + Θ j - 1 k + 1 ) / ( Δ x ) 2 ] = [ 1 / ( Δ t Δ x ) ] × [ k Δ t ( k + 1 ) Δ x d t ( j - 1 / 2 ) Δ x ( j + 1 / 2 ) Δ x g ( x , t ) d x ] ;             1 j j max - 1 ; k 0
C [ ( Θ 0 k + 1 - Θ 0 k ) / Δ t ] - K { [ 2 Θ 1 k + 1 - 2 ( 1 + γ Δ x ) Θ 0 k + 1 ) / ( Δ x ) 2 ] = [ 2 / ( Δ t Δ x ) ] × [ k Δ t ( k + 1 ) Δ t d t 0 1 / 2 Δ x g ( x , t ) d x ] ;             k 0
Θ j max k = 0 ;             k 0
Θ j 0 = f ( j Δ x ) ;             0 j j max .
Θ 0 k + 1 = A 1 ( 0 ) Θ 1 k + 1 + A 2 ( 0 ) ;             k 0
B 0 ( j ) Θ j - 1 k + 1 + B 1 ( j ) Θ j k + 1 + B 2 ( j ) Θ j + 1 k + 1 = B 3 ( j , k ) ;             1 j j max - 1 ; k 0
Θ j max k + 1 = 0 ;             k 0
Θ j 0 = f ( j Δ x ) ;             0 j j max ,
A 1 ( 0 ) = { [ ( Δ x ) 2 / ( 2 D Δ t ) ] + ( 1 + γ Δ x ) } - 1
A 2 ( 0 ) = { [ ( Δ x ) 2 / ( 2 D Δ t ) ] Θ 0 k + [ Δ x / ( K Δ t ) ] × [ k Δ t ( k + 1 ) Δ t d t 0 1 / 2 Δ x g ( x , t ) d x ] } × { [ ( Δ x ) 2 / ( 2 D Δ t ) ] + ( 1 + γ Δ x ) } - 1 ,
B 0 ( i ) = - [ D Δ t / ( Δ x ) 2 ] ,
B j ( i ) = 1 + 2 [ D Δ t / ( Δ x ) 2 ] ,
B 2 ( i ) = - [ D Δ t / ( Δ x ) 2 ] ,
B 3 ( j , k ) = Θ j k + [ 1 / ( C Δ x ) ] × [ k Δ t ( k + 1 ) Δ t d t ( j - 1 / 2 ) Δ x ( j + 1 / 2 ) Δ x g ( x , t ) d x ] .
Θ j k + 1 = A 1 ( j ) Θ j + 1 k + 1 + A 2 ( j ) ;             0 j j max - 1 ,
A 1 ( j ) = - B 2 ( j ) / [ B 1 ( j ) + B 0 ( j ) A 1 ( j - 1 ) ] ,
A 2 ( j ) = [ B 3 ( j , k ) - B 0 ( j ) A 2 ( j - 1 ) ] / [ B 1 ( j ) + B 0 ( j ) A 1 ( j - 1 ) ] .

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