Abstract

It has been experimentally shown in recent papers that the surface profile of a thermocapillary liquid submitted to heating by a nonuniform light beam undergoes a time-dependent deformation. The profiles corresponding to illumination with (a) a Gaussian laser beam, (b) a point source, (c) a fringe system, and (d) a circular Gaussian speckle pattern are calculated. This study sets some theoretical bases for practical utilization of highly viscous liquid films as transient photographic receivers.

© 1980 Optical Society of America

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References

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  1. G. Da Costa, J. Calatroni, Appl. Opt. 17, 2381 (1978).
    [CrossRef]
  2. G. Da Costa, J. Calatroni, Appl. Opt. 18, 233 (1979).
    [CrossRef]
  3. G. Da Costa, J. Calatroni, Proceedings, International Commission for Optics Conference (ICO-11), Madrid, Spain (1978), pp. 779–782.
  4. L. Landau, E. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading, Mass., 1969; Mir, Moscow (1971), pp. 296–297).
  5. B. Gebhart, Heat Transfer (McGraw-Hill, New York, 1971).
  6. J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975).

1979

1978

Calatroni, J.

G. Da Costa, J. Calatroni, Appl. Opt. 18, 233 (1979).
[CrossRef]

G. Da Costa, J. Calatroni, Appl. Opt. 17, 2381 (1978).
[CrossRef]

G. Da Costa, J. Calatroni, Proceedings, International Commission for Optics Conference (ICO-11), Madrid, Spain (1978), pp. 779–782.

Da Costa, G.

G. Da Costa, J. Calatroni, Appl. Opt. 18, 233 (1979).
[CrossRef]

G. Da Costa, J. Calatroni, Appl. Opt. 17, 2381 (1978).
[CrossRef]

G. Da Costa, J. Calatroni, Proceedings, International Commission for Optics Conference (ICO-11), Madrid, Spain (1978), pp. 779–782.

Gebhart, B.

B. Gebhart, Heat Transfer (McGraw-Hill, New York, 1971).

Goodman, J. W.

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975).

Landau, L.

L. Landau, E. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading, Mass., 1969; Mir, Moscow (1971), pp. 296–297).

Lifshitz, E.

L. Landau, E. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading, Mass., 1969; Mir, Moscow (1971), pp. 296–297).

Appl. Opt.

Other

G. Da Costa, J. Calatroni, Proceedings, International Commission for Optics Conference (ICO-11), Madrid, Spain (1978), pp. 779–782.

L. Landau, E. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading, Mass., 1969; Mir, Moscow (1971), pp. 296–297).

B. Gebhart, Heat Transfer (McGraw-Hill, New York, 1971).

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975).

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Figures (6)

Fig. 1
Fig. 1

Dimensionless temperature distribution in a thin sample heated by a Gaussian laser beam, γ is the dimensionless time t/t0.

Fig. 2
Fig. 2

Power distribution for numerical simulation of heating by a fringe system.

Fig. 3
Fig. 3

Sample divided by a rectangular grid to perform the numerical calculation of temperature at different time instants; Δx is the grid dimension, and y0 is the sample thickness. In the image method the sample graph is of thickness 2y0.

Fig. 4
Fig. 4

Temperature distribution corresponding to heating with the power distribution of Fig. 2. Only the left-hand side of the graph is represented. It is seen that for small values of the time t the temperature distribution closely resembles the power distribution.

Fig. 5
Fig. 5

Surface profile corresponding to heating with the power distribution of Fig. 2. Only the left-hand side of the graph is represented. For small values of time t the surface depth is almost null in the optical axis of the system (that is, on the extreme right side of this figure), and the modulation ratio is high. For higher values of t, lateral heat conduction from the illuminated regions to the dark regions gives rise to a temperature increase in the central point. The depth at this point also diminishes and so does the modulation ratio of the written fringe system.

Fig. 6
Fig. 6

Laser beam L (wavelength λ) illuminates a random diffuser D placed a distance l from the liquid sample. Mean dimension of the speckle grains over the liquid surface is a = λ1/D. Each grain digs a hole in the sample. Then the latter behaves as a transient diffuser whose statistical properties are studied in the text.

Equations (38)

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f ( β , γ ) = 1 + γ β 2 d θ θ e θ ,
f ( β , γ ) = f ( β , 0 ˙ ) + f ( β , γ ) γ / γ = 0 · γ + 2 f ( β , γ ) γ 2 / γ = 0 γ 2 2 ! + .
f ( β , γ ) = exp ( β 2 ) ( γ + β 2 1 2 ! γ 2 + β 4 4 β 2 + 2 3 ! γ 3 + ) .
H ( r ) = H 0 exp [ ( r a ) 2 ] = H 0 exp ( β 2 ) = P π a 2 · exp ( β 2 ) ,
f ( β , γ ) = γ H ( r ) H 0 + γ 2 ( r / a ) 2 1 2 H ( r ) H 0 + .
( n ) f ( β , γ ) γ ( n ) exp ( β 2 ) .
y = y 0 [ 1 + b f ( β , γ ) ] 1 / 2 .
y y 0 [ 1 + b 2 f ( β , γ ) ] ,
y = y 0 [ 1 + b 2 H 0 γ H ( r ) + b 2 H 0 ( r / a ) 2 1 2 γ 2 H ( r ) + ] .
0 2 π d φ 0 + r H ( r ) d r = P .
lim a 0 f ( β , γ ) = lim a 0 β 2 / 1 + γ β 2 d θ θ e θ = lim a 0 r 2 / a 2 + 4 κ t r 2 / a 2 d θ θ e θ = r 2 / 4 κ t + d θ θ e θ = E i ( r 2 4 κ t )
T T 0 = P 4 π k y 0 E i ( r 2 4 κ t ) .
T 0 , t + Δ t = F ( T 1 + T 2 + T 3 + T 4 ) + ( 1 4 F ) T 0
T s , t + Δ t = F ( 2 T 5 + T 6 + T 7 + 2 q Δ x k ) + ( 1 4 F ) T s .
k = 2 × 10 3 W / cm ° C ; ρ = 1 g / cm 3 ; heat capacity = 2 J / g ° C ; κ = 0.1 mm 2 / sec .
p ( H 0 ) = 1 H 0 exp ( H 0 H 0 ) .
y min = y 0 [ 1 H 0 H c f ( 0 , γ ) ] 1 / 2 ,
h = y 0 y min = y 0 { 1 [ 1 H 0 H c f ( 0 , γ ) ] 1 / 2 } ,
p h ( h ) = p H 0 [ H 0 ( h ) ] d H 0 d h = 1 H 0 exp { 1 H 0 · H c f ( 0 , γ ) · [ 1 ( 1 h y 0 ) 2 ] } · 2 H c ( 1 h / y 0 ) y 0 f ( 0 , γ ) = 2 y 0 F ( γ ) ( 1 h y 0 ) exp [ F ( γ ) h y 0 ( 2 h y 0 ) ] / ,
F ( γ ) = H c H 0 f ( 0 , γ ) / ;
p H ( H ) = 2 F ( γ ) ( 1 H ) exp [ F ( γ ) H ( 2 H ) ] .
f ( 0 , γ ) = lim β 0 β 2 / 1 + γ β 2 d θ θ e θ = ln ( 1 + γ )
F ( γ ) = H c H 0 ln ( 1 + γ ) .
0 + p ( H 0 ) d H 0 = 1 ,
0 1 p H ( H ) d H = 1 exp [ 2 F ( γ ) ] < 1.
p ( H > 1 ) = 1 + p H ( H ) d H = 1 0 1 p H ( H ) dH = exp [ 2 F ( γ ) ] .
H = 0 1 H p H ( H ) d H = 2 F ( γ ) 0 1 H ( 1 H ) × exp [ F ( γ ) ( H 2 2 H ) ] d H .
H = exp [ F ( γ ) ] + 1 F ( γ ) { [ F ( γ ) ] 1 / 2 · exp [ F ( γ ) ] · D [ F ( γ ) ] } ,
D ( x ) = 0 x exp ( y 2 ) d y
H 2 = 0 1 H 2 p ( H ) d H = 2 F ( γ ) 0 1 H 2 ( 1 H ) exp [ F ( γ ) ( H 2 2 H ) ] d H .
H 2 = [ 1 F ( γ ) 1 ] exp [ F ( γ ) ] + 2 F ( γ ) { [ F ( γ ) ] 1 / 2 · exp [ F ( γ ) ] · D [ F ( γ ) ] } 1 F ( γ ) .
α = F ( γ ) ,
H = exp ( α 2 ) + 1 α 2 α exp ( α 2 ) D ( α ) ,
H 2 = exp ( α 2 ) ( 1 α 2 1 ) + 2 α 2 α exp ( α 2 ) D ( α ) 1 α 2 .
D ( α ) = exp ( α 2 ) 2 α ( 1 + 1 2 α 2 + 3 4 α 3 + ) ,
H = 1 2 α 2 + 1 4 α 4 + ,
H 2 = 1 2 α 4 + 3 4 α 5 + ,
rms = ( H 2 H 2 ) 1 / 2 = 1 2 α 2 + .

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