Abstract

The vector Kirchhoff approach to thermal lensing in solids is investigated. The relation of the thermal rise in the sample to the optics of lensing is elucidated. Various analytical results and series representations for the transmitted intensity are derived for the case of Gaussian beams incident on circular samples, in various special limits. An expression for the time-dependent shift in the diffraction focus is obtained. Numerical computations illustrating lensing effects both at small times and in the steady state are presented. Effects of stress-induced birefringence, including the oscillation of the maximum intensity and diffraction focus as a function of time, are illustrated. Similar effects are also shown to occur when the incident power or sample thickness is varied, for solids characterized by large stress-induced contributions.

© 1973 Optical Society of America

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References

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  1. See, for example, J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
    [CrossRef]
  2. A. Hordvik, in Conference on High Power Infrared Laser Window Materials, C. S. Sahagian, C. A. Pitha, Eds. (AFCRL-71-0592, 1971), p. 389.
  3. M. Sparks, J. Appl. Phys. 42, 5029 (1971); F. A. Horrigan, T. F. Deutsch, Raytheon Research Division Reports on Contract DAAH01-70-C-1251 (1970, unpublished); J. R. Jasperse, P. D. Gianino, J. Appl. Phys. 43, 1686 (1972); P. D. Gianino, J. R. Jasperse, AFCRL-72-0202(1972).
    [CrossRef]
  4. J. Marburger, M. Flannery, in Conference on High Power Infrared Laser Window Materials, C. S. Sahagian, C. A. Pitha, Eds. (AFCRL-71-0592, 1971), p. 11; D. A. Holmes, J. E. Korka, P. V. Avizonis, Appl. Opt. 11, 565 (1972).
    [CrossRef] [PubMed]
  5. B. Bendow, J. R. Jasperse, P. D. Gianino, Optics Commun. 5, 98 (1972).
    [CrossRef]
  6. B. Bendow, P. D. Gianino, Bull. Am. Phys. Soc. 17, 672 (1972) and AFCRL-72-0322 (1972).
  7. M. Born, E. Wolf, Principles of Optics (Macmillan Co., New York, 1964), Chap. 5.
  8. See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Chap. 9.
  9. J. P. Campbell, L. G. DeShazer, J. Opt. Soc. Am. 59, 1427 (1969).
    [CrossRef]
  10. B. Bendow, P. D. Gianino, AFCRL-72-0322 (1972).
  11. F. W. Quelle, Appl. Opt. 5, 633 (1966).
    [CrossRef] [PubMed]
  12. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), 2d ed., Chap. 1; N. Y. Ölcer, Brit. J. Appl. Phys. 18, 89 (1967).
    [CrossRef]
  13. M. Abramowitz, I. Stegun, Eds. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).
  14. G. O. Olaofe, J. Opt. Soc. Am. 60, 1654 (1970).
    [CrossRef]

1972

B. Bendow, J. R. Jasperse, P. D. Gianino, Optics Commun. 5, 98 (1972).
[CrossRef]

B. Bendow, P. D. Gianino, Bull. Am. Phys. Soc. 17, 672 (1972) and AFCRL-72-0322 (1972).

1971

M. Sparks, J. Appl. Phys. 42, 5029 (1971); F. A. Horrigan, T. F. Deutsch, Raytheon Research Division Reports on Contract DAAH01-70-C-1251 (1970, unpublished); J. R. Jasperse, P. D. Gianino, J. Appl. Phys. 43, 1686 (1972); P. D. Gianino, J. R. Jasperse, AFCRL-72-0202(1972).
[CrossRef]

1970

1969

1966

1965

See, for example, J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Bendow, B.

B. Bendow, J. R. Jasperse, P. D. Gianino, Optics Commun. 5, 98 (1972).
[CrossRef]

B. Bendow, P. D. Gianino, Bull. Am. Phys. Soc. 17, 672 (1972) and AFCRL-72-0322 (1972).

B. Bendow, P. D. Gianino, AFCRL-72-0322 (1972).

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan Co., New York, 1964), Chap. 5.

Campbell, J. P.

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), 2d ed., Chap. 1; N. Y. Ölcer, Brit. J. Appl. Phys. 18, 89 (1967).
[CrossRef]

DeShazer, L. G.

Flannery, M.

J. Marburger, M. Flannery, in Conference on High Power Infrared Laser Window Materials, C. S. Sahagian, C. A. Pitha, Eds. (AFCRL-71-0592, 1971), p. 11; D. A. Holmes, J. E. Korka, P. V. Avizonis, Appl. Opt. 11, 565 (1972).
[CrossRef] [PubMed]

Gianino, P. D.

B. Bendow, J. R. Jasperse, P. D. Gianino, Optics Commun. 5, 98 (1972).
[CrossRef]

B. Bendow, P. D. Gianino, Bull. Am. Phys. Soc. 17, 672 (1972) and AFCRL-72-0322 (1972).

B. Bendow, P. D. Gianino, AFCRL-72-0322 (1972).

Gordon, J. P.

See, for example, J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Hordvik, A.

A. Hordvik, in Conference on High Power Infrared Laser Window Materials, C. S. Sahagian, C. A. Pitha, Eds. (AFCRL-71-0592, 1971), p. 389.

Jackson, J. D.

See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Chap. 9.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), 2d ed., Chap. 1; N. Y. Ölcer, Brit. J. Appl. Phys. 18, 89 (1967).
[CrossRef]

Jasperse, J. R.

B. Bendow, J. R. Jasperse, P. D. Gianino, Optics Commun. 5, 98 (1972).
[CrossRef]

Leite, R. C. C.

See, for example, J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Marburger, J.

J. Marburger, M. Flannery, in Conference on High Power Infrared Laser Window Materials, C. S. Sahagian, C. A. Pitha, Eds. (AFCRL-71-0592, 1971), p. 11; D. A. Holmes, J. E. Korka, P. V. Avizonis, Appl. Opt. 11, 565 (1972).
[CrossRef] [PubMed]

Moore, R. S.

See, for example, J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Olaofe, G. O.

Porto, S. P. S.

See, for example, J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Quelle, F. W.

Sparks, M.

M. Sparks, J. Appl. Phys. 42, 5029 (1971); F. A. Horrigan, T. F. Deutsch, Raytheon Research Division Reports on Contract DAAH01-70-C-1251 (1970, unpublished); J. R. Jasperse, P. D. Gianino, J. Appl. Phys. 43, 1686 (1972); P. D. Gianino, J. R. Jasperse, AFCRL-72-0202(1972).
[CrossRef]

Whinnery, J. R.

See, for example, J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan Co., New York, 1964), Chap. 5.

Appl. Opt.

Bull. Am. Phys. Soc.

B. Bendow, P. D. Gianino, Bull. Am. Phys. Soc. 17, 672 (1972) and AFCRL-72-0322 (1972).

J. Appl. Phys.

See, for example, J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

M. Sparks, J. Appl. Phys. 42, 5029 (1971); F. A. Horrigan, T. F. Deutsch, Raytheon Research Division Reports on Contract DAAH01-70-C-1251 (1970, unpublished); J. R. Jasperse, P. D. Gianino, J. Appl. Phys. 43, 1686 (1972); P. D. Gianino, J. R. Jasperse, AFCRL-72-0202(1972).
[CrossRef]

J. Opt. Soc. Am.

Optics Commun.

B. Bendow, J. R. Jasperse, P. D. Gianino, Optics Commun. 5, 98 (1972).
[CrossRef]

Other

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), 2d ed., Chap. 1; N. Y. Ölcer, Brit. J. Appl. Phys. 18, 89 (1967).
[CrossRef]

M. Abramowitz, I. Stegun, Eds. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

B. Bendow, P. D. Gianino, AFCRL-72-0322 (1972).

M. Born, E. Wolf, Principles of Optics (Macmillan Co., New York, 1964), Chap. 5.

See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Chap. 9.

J. Marburger, M. Flannery, in Conference on High Power Infrared Laser Window Materials, C. S. Sahagian, C. A. Pitha, Eds. (AFCRL-71-0592, 1971), p. 11; D. A. Holmes, J. E. Korka, P. V. Avizonis, Appl. Opt. 11, 565 (1972).
[CrossRef] [PubMed]

A. Hordvik, in Conference on High Power Infrared Laser Window Materials, C. S. Sahagian, C. A. Pitha, Eds. (AFCRL-71-0592, 1971), p. 389.

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Figures (7)

Fig. 1
Fig. 1

Coordinate systems at an aperture (origin O) and in the observation region (origin O′ = Gaussian prefocus) with P the observation point and R0 ≡ |x0|.

Fig. 2
Fig. 2

Schematic of an annular-shaped sample.

Fig. 3
Fig. 3

Intensity distribution I(u, v) vs u and v at successive values of abstract time C≡|kC1ρ|, for α2 = 1: (a) Ge and (b) KCl. The translation of C to real time may be accomplished through Table I. Part (b) of figure is on the next page.

Fig. 4
Fig. 4

On-axis intensity vs u, for parameters indicated, at ten successive abstract times starting at 0 and progressing by the value ΔC indicated. When the decrease in maximum intensity for successive curves is not monotonic, the curves are numbered in sequence. (a) Ge and (b) KCl. Part (b) of figure is on next page.

Fig. 5
Fig. 5

Maximum intensity vs abstract time C for (a) Ge and (b) KCl, for parameters indicated.

Fig. 6
Fig. 6

Steady-state on-axis intensity vs u as a function of incident power P0, for (a) Ge and (b) KCl, for α2 = 1, ha = 0.1, η = 0.1. The units of P0 in the figure are in W/cm2, for the specific choice of λ = 10.6 μm and a = 25 cm.

Fig. 7
Fig. 7

Induced-birefringence parameters for a variety of materials at 10.6 μm.

Tables (1)

Tables Icon

Table I Thermal Lensing Parameters at Small Time for 10.6 μma

Equations (36)

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E ( x ) = - i k 2 π a 2 x e i k x U ( x ) , U ( x ) = x ^ X d ρ ρ 0 2 π d θ [ n ^ X E a ( ρ , θ ) ] exp ( - i v ρ cos θ - 1 2 i u ρ 2 ) ,
u k a 2 ( x 0 - 1 - x - 1 ) , v k a ρ / x .
E ρ = ρ ^ E ρ e i k Φ ρ , E θ = θ ^ E θ e i k Φ θ .
U = U 1 ^ 1 + U 2 ^ 2 ,
U 1 ( u , v ) 2 π σ 1 d ρ ρ E 1 ( ρ ) [ J 0 ( ρ v ) e i k Φ ρ - J 1 ( ρ v ) ρ v ( e i k Φ ρ - e i k Φ θ ) ] e - 1 2 i u ρ 2 , U 2 ( u , v ) 2 π σ 1 d ρ ρ E 2 ( ρ ) [ J 0 ( ρ v ) e i k Φ θ + J 1 ( ρ v ) ρ v ( e i k Φ ρ - e i k Φ θ ) ] e - 1 2 i u ρ 2 ,
U i = π σ 1 d ρ E i ( ρ ) [ J 0 ( ρ v ) ( e i k Φ ρ + e i k Φ θ J 2 ( ρ v ) ( e i k Φ ρ - e i k Φ θ ) ] e - 1 2 i k u ρ 2 ,
I ( u , v , t ) = U 1 ( u , v , t ) 2 + U 2 ( u , v , t ) 2 2 π 0 1 d ρ ρ E 1 ( ρ ) 2 + 2 π 0 1 d ρ ρ E 2 ( ρ ) 2 .
I = U ¯ 1 2 cos 2 ψ + U ¯ 2 2 sin 2 ψ ) / 2 π d ρ ρ E ( ρ ) 2
Φ γ ( r , t ) = 0 1 d z Δ n γ ( r , t ) + Δ L b ( r s , t ) ( n - 1 ) ,
Φ γ = a S 1 γ Δ T ¯ + 4 a S 2 γ ρ - 2 σ ρ d x x Δ T ¯ ,
Δ T ¯ ( ρ , t ) = - η η d z Δ T ( ρ , z , t )
S 1 ρ = ( n / T ) + α ¯ ( n 3 / 2 ) [ ( 1 - ν ) p 12 - ν p 11 ] + α ¯ ( 1 + ν ) ( n - 1 ) , S z ρ = α ¯ ( n 3 / 8 ) ( 1 + ν ) ( p 11 - p 12 ) = - S 2 , S 1 θ = S 1 ρ [ ( 1 - ν ) p 12 - ν p 11 p 11 - 2 ν p 12 ] .
[ ( 2 / r 2 ) + ( / τ ) ] Δ T = f ( r , τ ) , [ ( / n i ) + h i ] Δ T = 0 on the surfaces , Δ T = 0 at τ = 0 and Δ T = ( K / P 0 β a 2 ) Δ T at τ = ( κ / a 2 ) t .
Φ γ = C 1 γ e - 2 α 2 ρ 2 + C 2 γ [ e - 2 α 2 σ 2 - ( 1 - σ 2 ) e - 2 α 2 ρ 2 - σ 2 e - 2 α 2 ] / α 2 ρ 2 ( 1 - σ 2 ) .
C i γ = S i γ L P 0 β t / c ,
I = f - 2 ( α ) ( cos 2 ψ F 1 2 + sin 2 ψ F 2 2 )
f ( α ) = 1 2 ( 1 - e - α 2 ) / α 2 ,
F 1 ( u , v ) = σ 1 d ρ ρ e - ( 1 2 i u + α 2 ) ρ 2 [ J 0 ( ρ v ) e i k Φ ρ - J 1 ( ρ v ) ρ v ( e i k Φ ρ - e i k Φ θ ) ] , F 2 = F 1 ( Φ ρ Φ θ ) .
A 1 γ = 2 C 1 γ ( 2 g 1 - 3 g 2 ) + 2 C 2 γ ( 2 g 3 - 3 g 4 ) , A 2 γ = 6 C 1 γ ( 2 g 2 - g 1 ) + 6 C 2 γ ( 2 g 4 - g 3 ) ,
g 1 1 2 α - 2 ( 1 - e - 2 α 2 ) ; g 2 = g 1 - 1 2 α - 2 e - 2 α 2 , g 3 α - 2 [ E 1 ( 2 α 2 ) + l n ( 2 α 2 ) + 0.57722 ] , g 4 α - 2 ( 1 - g 1 ) ,
z 0 = - 2 A 2 R 0 2 / 1 + 2 A 2 R 0 .
0 1 d ρ ρ J 0 ( ρ v ) e - A ρ 2 = { 1 2 A e - A l = 1 ( 2 A v ) l J l ( v )             v 2 A , 1 2 A [ e - v 2 / 4 A - e - A l = 0 ( - v 2 A ) l J l ( v ) ]         v < 2 A
0 1 d ρ ρ J 1 ( ρ v ) ρ v e - A ρ 2 = 1 v 2 [ 1 - e - v 2 / 4 A + 1 2 l = 1 l E l ( A ) ( - v 2 A ) l J l ( v ) ]
( 1 - e - 2 α 2 ρ 2 ) / 2 α 2 ρ 2 e - α 2 ρ 2 .
F 1 = n = 0 m = 0 { ( i k C 1 ρ ) n ( 2 i k C 2 ρ ) m n ! m ! [ H 1 ( n m ) - H 2 ( n m ) ] + ( i k C 1 θ ) n ( 2 i k C 2 θ ) m n ! m ! H 2 ( n m ) } ,
H 1 0 1 d ρ ρ J 0 ( ρ v ) e - B n m ρ 2 , H 2 0 1 d ρ ρ [ J 1 ( ρ v ) / ρ v ] e - B n m ρ 2 , B n m ( 1 + 2 n + m ) α 2 + 1 2 i u .
H 1 ( 1 / 2 B n m ) e - v 2 / 4 B n m , H 2 ( 1 / v 2 ) ( 1 - e - v 2 / 4 B n m ) .
F 1 = n = 0 1 n ! [ ( i k C 1 ρ ) n ( G 1 + i k C 2 ρ α - 2 G 2 - v - 1 G 3 - i k C 2 ρ v - 1 α - 2 G 4 ) + ( i k C 1 θ ) n ( v - 1 G 3 + i k C 2 θ v - 1 α - 2 G 4 ) ] ,
G 1 = exp ( - 1 4 v 2 / B n 0 ) / 2 B n 0 , G 2 = 1 2 [ E 1 ( 1 4 v 2 / B n 2 ) - E 1 ( 1 4 v 2 / B n 0 ) - l n ( B n 0 / B n 2 ) ] , G 3 = v - 1 [ 1 - exp ( - 1 4 v 2 / B n 0 ) ] , G 4 = v 4 l = 1 ( - v 2 4 ) l 1 ( l + 1 ) ! l ( B n 0 - l - B n 2 - l ) ,
I ( u ) = 2 P ρ + P θ 2 f - 2 ( α ) , P γ = 0 1 d ρ ρ exp ( - 1 / 2 i u ρ 2 - α 2 ρ 2 + i k Φ γ ) .
P γ 1 4 ( π / d 2 γ ) 1 2 exp { i k ( C 1 γ + 2 C 2 γ ) + [ ( d 1 γ ) 2 / 4 d 2 γ ] } { erf [ ( d 1 γ ) / 2 d 2 γ ) + d 2 γ ] - erf ( d 1 γ / 2 d 2 γ ) } ,
d 1 γ 1 / 2 i u + α 2 [ 1 + 2 i k ( C 1 γ + C 2 γ ) ] , d 2 γ - 2 i k α 4 ( C 1 γ + 2 3 C 2 γ ) ,
P γ = 1 2 l = 1 ( i k C 1 γ ) l l ! { 1 - e - B n 0 B n 0 + i k C 2 γ α 2 × [ E 1 ( B n 0 + 2 α 2 ) - E 1 ( B n 0 ) + l n ( 1 + 2 α 2 / B n 0 ) ] } ,
P ρ P θ 1 4 α - 2 Γ ( Q ) γ * ( Q , - i k C 1 ρ ) , Q 1 2 ( 1 / 2 i u + α 2 ) α - 2 ,
P γ = 1 4 α - 2 ( π / a 1 γ ) 1 / 2 exp [ ( a 2 γ ) 2 / a 1 γ ] { erf [ ( a 2 γ / a 1 γ ) + a 1 γ ] - erf [ ( a 2 γ / a 1 γ ) + a 1 γ e - α 2 ] } ; a 1 γ i k C 1 γ , a 2 γ - i k C 2 γ .
t u c t / C = P 0 - 1 L - 1 [ k ( β / c ) S 1 ρ ] - 1 A / P 0 L .

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