Abstract

Adiabatic laser calorimetry, which is the most widely used method for studying the absorption coefficients of low-loss materials, can be adapted to study both the bulk and surface absorption by using a long rod sample geometry. In the limiting case of small heat losses, calculations of the thermal rise curves obtained in laser calorimetry indicate that two regions of constant slope can be expected. The first of these can be identified with the bulk absorption coefficient only and the second with the sum of the surface and bulk absorptions. Experimental data illustrating this two-slope behavior are presented.

© 1977 Optical Society of America

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References

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  1. L. H. Skolnik, “A Review of Techniques for Measuring Small Optical Losses in Infrared Transmitting Materials,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 405–434.
    [CrossRef]
  2. A. Hordvik, Appl. Opt. 16, 2827 (1977).
    [CrossRef] [PubMed]
  3. M. Hass, J. W. Davisson, H. B. Rosenstock, J. Babiskin, Appl. Opt. 14, 1128 (1975).
    [CrossRef] [PubMed]
  4. M. Hass, J. W. Davission, H. B. Rosenstock, J. A. Slinkman, J. Babiskin, “Improved Laser Calorimetric Techniques,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 435–442.
    [CrossRef]
  5. H. B. Rosenstock, D. A. Gregory, J. A. Harrington, Appl. Opt. 15, 2075 (1976).
    [CrossRef] [PubMed]
  6. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), p. 50.
  7. H. B. Rosenstock, in “High Energy Laser Windows,” Semi-Annual report 6, ARPA Order 2031 (Naval Research Laboratory, Washington, D.C., 30September1975), p. 15.
  8. M. Hass, J. W. Davisson, P. H. Klein, L. L. Boyer, J. Appl. Phys. 45, 3959 (1974).
    [CrossRef]
  9. E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945), p. 143.
  10. L. B. Jolley, Summation of Series (Dover, New York, 1961); Formula (573) reads∑1cos2nπθ/(2nπ)2=[θ2-θ+(1/6)]/4.Formula (559) reads2a2∑1cosnθ/(n2+a2)=-1+aπ cosha(π-θ)sinhaπ.

1977 (1)

1976 (1)

1975 (1)

1974 (1)

M. Hass, J. W. Davisson, P. H. Klein, L. L. Boyer, J. Appl. Phys. 45, 3959 (1974).
[CrossRef]

Babiskin, J.

M. Hass, J. W. Davisson, H. B. Rosenstock, J. Babiskin, Appl. Opt. 14, 1128 (1975).
[CrossRef] [PubMed]

M. Hass, J. W. Davission, H. B. Rosenstock, J. A. Slinkman, J. Babiskin, “Improved Laser Calorimetric Techniques,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 435–442.
[CrossRef]

Boyer, L. L.

M. Hass, J. W. Davisson, P. H. Klein, L. L. Boyer, J. Appl. Phys. 45, 3959 (1974).
[CrossRef]

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), p. 50.

Davission, J. W.

M. Hass, J. W. Davission, H. B. Rosenstock, J. A. Slinkman, J. Babiskin, “Improved Laser Calorimetric Techniques,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 435–442.
[CrossRef]

Davisson, J. W.

M. Hass, J. W. Davisson, H. B. Rosenstock, J. Babiskin, Appl. Opt. 14, 1128 (1975).
[CrossRef] [PubMed]

M. Hass, J. W. Davisson, P. H. Klein, L. L. Boyer, J. Appl. Phys. 45, 3959 (1974).
[CrossRef]

Emde, F.

E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945), p. 143.

Gregory, D. A.

Harrington, J. A.

Hass, M.

M. Hass, J. W. Davisson, H. B. Rosenstock, J. Babiskin, Appl. Opt. 14, 1128 (1975).
[CrossRef] [PubMed]

M. Hass, J. W. Davisson, P. H. Klein, L. L. Boyer, J. Appl. Phys. 45, 3959 (1974).
[CrossRef]

M. Hass, J. W. Davission, H. B. Rosenstock, J. A. Slinkman, J. Babiskin, “Improved Laser Calorimetric Techniques,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 435–442.
[CrossRef]

Hordvik, A.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), p. 50.

Jahnke, E.

E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945), p. 143.

Jolley, L. B.

L. B. Jolley, Summation of Series (Dover, New York, 1961); Formula (573) reads∑1cos2nπθ/(2nπ)2=[θ2-θ+(1/6)]/4.Formula (559) reads2a2∑1cosnθ/(n2+a2)=-1+aπ cosha(π-θ)sinhaπ.

Klein, P. H.

M. Hass, J. W. Davisson, P. H. Klein, L. L. Boyer, J. Appl. Phys. 45, 3959 (1974).
[CrossRef]

Rosenstock, H. B.

H. B. Rosenstock, D. A. Gregory, J. A. Harrington, Appl. Opt. 15, 2075 (1976).
[CrossRef] [PubMed]

M. Hass, J. W. Davisson, H. B. Rosenstock, J. Babiskin, Appl. Opt. 14, 1128 (1975).
[CrossRef] [PubMed]

H. B. Rosenstock, in “High Energy Laser Windows,” Semi-Annual report 6, ARPA Order 2031 (Naval Research Laboratory, Washington, D.C., 30September1975), p. 15.

M. Hass, J. W. Davission, H. B. Rosenstock, J. A. Slinkman, J. Babiskin, “Improved Laser Calorimetric Techniques,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 435–442.
[CrossRef]

Skolnik, L. H.

L. H. Skolnik, “A Review of Techniques for Measuring Small Optical Losses in Infrared Transmitting Materials,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 405–434.
[CrossRef]

Slinkman, J. A.

M. Hass, J. W. Davission, H. B. Rosenstock, J. A. Slinkman, J. Babiskin, “Improved Laser Calorimetric Techniques,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 435–442.
[CrossRef]

Appl. Opt. (3)

J. Appl. Phys. (1)

M. Hass, J. W. Davisson, P. H. Klein, L. L. Boyer, J. Appl. Phys. 45, 3959 (1974).
[CrossRef]

Other (6)

E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945), p. 143.

L. B. Jolley, Summation of Series (Dover, New York, 1961); Formula (573) reads∑1cos2nπθ/(2nπ)2=[θ2-θ+(1/6)]/4.Formula (559) reads2a2∑1cosnθ/(n2+a2)=-1+aπ cosha(π-θ)sinhaπ.

L. H. Skolnik, “A Review of Techniques for Measuring Small Optical Losses in Infrared Transmitting Materials,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 405–434.
[CrossRef]

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), p. 50.

H. B. Rosenstock, in “High Energy Laser Windows,” Semi-Annual report 6, ARPA Order 2031 (Naval Research Laboratory, Washington, D.C., 30September1975), p. 15.

M. Hass, J. W. Davission, H. B. Rosenstock, J. A. Slinkman, J. Babiskin, “Improved Laser Calorimetric Techniques,” in Optical Properties of Highly Transparent Solids, S. S. Mitra, B. Bendow, Eds. (Plenum, New York, 1975), p. 435–442.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Calculated thermal rise curve for a long rod of KCl having both surface and bulk absorption. Note a change in slope at about 80 sec. The time axis intercept is indicated by an arrow.

Fig. 2
Fig. 2

Calculated thermal rise curves for a long rod of KCl having different amounts of surface absorption.

Fig. 3
Fig. 3

Calculated thermal rise curves for long rods of KCl of different lengths but the same bulk absorption coefficient and same surface absorption.

Fig. 4
Fig. 4

Calculated thermal rise curves for a long rod of KCl having only surface absorption. The cases of no heat losses and a large arbitrary heat loss are considered.

Fig. 5
Fig. 5

Experimental thermal rise curve for KCl. Note a two slope behavior with approximately equal contributions to the heat from surface and bulk effects. The characteristic time (½L)2/6α is about 50 sec for this sample. A small jump occurs on laser turnon and turnoff due to scattered light. The bulk absorption coefficient in this sample is 9 × 10−5 cm−1, and the surface absorption is 1.1 times the bulk absorption.

Equations (35)

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m c ( d T / d t ) = A P ,
T - T s = P m c { β L ( t - b 2 8 α ) + 2 S [ t - b 2 4 α - ( ½ L ) 2 6 α ] }
T - T s = P m c { β L t short times , β L t + 2 S [ t - ( ½ L ) 2 6 α ] long times
T - T s = P m c ( β L + 2 S ) [ t - 2 S β L + 2 S ( ½ L ) 2 6 α ]
T ( r , z , t ) = ( 2 π P / k ) m n [ R n 2 Z m 2 / γ m n J 0 2 ( n b ) ] · u ( m , z ) J 0 ( n r ) ( β P 1 m + S P 2 m ) · [ 1 - exp ( - γ m n α t ) ] .
R n - 2 = 2 π 2 b 2 [ ( H / n ) 2 + 1 ] Z m - 2 = H + ( L / 2 ) ( m 2 + H 2 )
u ( m , z ) = m cos m z + H sin m z ,
tan m L = 2 H m / ( m 2 - H 2 ) , H J 0 ( n b ) = n J 1 ( n b ) .
m P 1 m = H ( 1 - cos m L ) + m sin m L , P 2 m = m , and γ n n = m 2 + n 2
2 T + g ( z , r , t ) k = 1 α T t
g = P δ ( r ) ( 2 π r ) - 1 [ β + S δ ( z ) ]
k T / n + h T = 0
P 2 m = m ( 1 + cos m L ) + H sin m L ,
m = m π / L ,             m 0 , 1 , 2 , ,
u ( m , z ) = m cos m z ;
T ( r , z , t ) = ( P / V K ) { β L F ( r , t ) + S [ F ( r , t ) + 2 G ( z , r , t ) ] } ,
F ( r , t ) = α t + n = 1 J 0 ( n r ) n 2 J 0 2 ( n b ) [ 1 - exp ( - α n 2 t ) ] ,
G ( z , r , t ) = m = 1 cos m z × n = 0 J 0 ( m r ) ( m 2 + m 2 ) J 0 2 ( n b ) { 1 - exp [ - α ( m 2 + m 2 ) t ] } .
F ( b , ) = α t - b 2 / 8.
G ( z , b , ) = n = 0 H n ( z ) / J 0 ( n b ) ,
H n ( z ) = m = 1 cos ( m π z / L ) / [ ( m π / L ) 2 + n 2 ] .
G ( z , b , ) = - b 2 16 + L 2 12 ( 3 z ¯ 2 - 6 z ¯ + 2 ) + b L 2 n = 1 B n ( z ¯ ) ,
z ¯ = z / L ,
B ( z ¯ ) = exp ( - p n z ¯ ) x n J 0 ( x n ) { 1 + exp [ - 2 p n ( 1 - z ¯ ) ] 1 - exp ( - 2 p n ) } ,
p n = L x n / b ,
G ( z , b , ) + G ( L - z , b , ) = - b 2 8 + L 2 2 [ z ¯ ( 1 - z ¯ ) + 1 6 ] + b L 2 n = 1 [ B n ( z ¯ ) + B n ( 1 - z ¯ ) ] ,
D 2 T = T / α t .
T = f ( r D , t ) = ( π τ ) - D / 2 exp ( - r D 2 / τ ) ,
τ = 4 α t
r D 2 = 1 D r i 2
D 2 = 1 D 2 / x i 2 .
F ( r D , t ) = 0 t f ( r D , t - t ) d t = 0 t d s f ( r D , s ) .
t A = r D 2 / 2 α D .
1cos2nπθ/(2nπ)2=[θ2-θ+(1/6)]/4.
2a21cosnθ/(n2+a2)=-1+aπcosha(π-θ)sinhaπ.

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