Abstract

A theoretical model for the laser-induced thermal lens effect in weakly absorbing media is derived. The model predicts the intensity variation in the far field of the laser beam in the presence of the lensing medium and takes into account the aberrant nature of the thermal lens. Some experimental results which support the validity of this approach are presented.

© 1982 Optical Society of America

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References

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  1. 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. J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974).
    [CrossRef]
  3. C. Hu, J. R. Whinnery, Appl. Opt. 12, 72 (1973).
    [CrossRef] [PubMed]
  4. R. Dennemeyer, Introduction to Partial Differential Equations and Boundary Value Problems (McGraw-Hill, New York, 1968), pp. 294–295.
  5. A. E. Siegman, Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), pp. 305–307.
  6. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 228–229.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 459–464.
  8. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1971), p. 67.
  9. R. E. Weast, Ed., CRC Handbook of Chemistry and Physics (CRC Press, Cleveland, 1977).
  10. D. Solimini, J. Appl. Phys. 37, 3314 (1966).
    [CrossRef]

1974

J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974).
[CrossRef]

1973

1966

D. Solimini, J. Appl. Phys. 37, 3314 (1966).
[CrossRef]

1965

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 459–464.

Dennemeyer, R.

R. Dennemeyer, Introduction to Partial Differential Equations and Boundary Value Problems (McGraw-Hill, New York, 1968), pp. 294–295.

Gordon, J. P.

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

Hu, C.

Leite, R. C. C.

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

Moore, R. S.

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

Porto, S. P. S.

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

Siegman, A. E.

A. E. Siegman, Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), pp. 305–307.

Solimini, D.

D. Solimini, J. Appl. Phys. 37, 3314 (1966).
[CrossRef]

Whinnery, J. R.

J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974).
[CrossRef]

C. Hu, J. R. Whinnery, Appl. Opt. 12, 72 (1973).
[CrossRef] [PubMed]

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 (Pergamon, New York, 1965), pp. 459–464.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1971), p. 67.

Acc. Chem. Res.

J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

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

D. Solimini, J. Appl. Phys. 37, 3314 (1966).
[CrossRef]

Other

R. Dennemeyer, Introduction to Partial Differential Equations and Boundary Value Problems (McGraw-Hill, New York, 1968), pp. 294–295.

A. E. Siegman, Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), pp. 305–307.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 228–229.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 459–464.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1971), p. 67.

R. E. Weast, Ed., CRC Handbook of Chemistry and Physics (CRC Press, Cleveland, 1977).

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Figures (8)

Fig. 1
Fig. 1

Components of the thermal lensing experiment.

Fig. 2
Fig. 2

Temperature distribution in the thermal lens at various times.

Fig. 3
Fig. 3

Symbols used in the diffraction integral.

Fig. 4
Fig. 4

Phase distribution at the input plane: (a) with the lensing medium absent; and (b) with the lensing medium present.

Fig. 5
Fig. 5

Thermal lens position data and best fit curve.

Fig. 6
Fig. 6

Optical layout and electronics block diagram.

Fig. 7
Fig. 7

[I(t)−I()]/I() vs t data and best fit curve for a sample of water −1 g/l K2HPO4 and bromothymol blue.

Fig. 8
Fig. 8

θ vs (bd/k)(dn/dT) data and best fit line for samples of water−1 g/l K2HPO4 and bromothymol blue.

Tables (1)

Tables Icon

Table I Thermooptic Constants of the Four Samples

Equations (39)

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c ρ t [ Δ T ( r , t ) ] = q ˙ ( r ) + k 2 [ Δ T ( r , t ) ] ; r < ; Δ T ( r , 0 ) = 0 .
Δ I ( r ) = I 0 ( r ) I ( r ) I 0 ( r ) bl ,
q ˙ ( r ) = Δ I ( r ) l = I 0 ( r ) b .
I 0 ( r ) = 2 ( 0.24 P ) π ω 2 exp ( 2 r 2 / ω 2 ) .
q ˙ ( r ) = 2 ( 0.24 P ) b π ω 2 exp ( 2 r 2 / ω 2 ) .
Δ T ( r , t ) = 2 ( 0.24 P ) b π c ρ ω 2 0 t ( 1 1 + 2 t / t c ) × exp ( 2 r 2 / ω 2 1 + 2 t / t c ) d t ,
t c = ω 2 c ρ 4 k = ω 2 4 κ .
Δ T ( r , t ) = 0.24 Pb 4 π k { ln ( 1 + 2 t t c ) + m = 1 ( 2 r 2 / ω 2 ) m mm ! × [ 1 ( 1 1 + 2 t / t c ) m ] } .
n ( r , t ) = n 0 dn dT Δ T ( r , t ) ,
U bc ( t ) = i λ 0 0 2 π U i ( r , t ) ( 1 + cos α 2 ) × exp [ i ( 2 π / λ ) | z 2 r | ] | z 2 r | rdrd θ ,
| z 2 r | z 2 ,
1 + cos α 2 1 ,
2 π λ | z 2 r | 2 π λ ( z 2 + r 2 2 z 2 ) .
U bc ( t ) = A 0 0 2 π U i ( r , t ) exp ( i π λ r 2 z 2 ) rdrd θ ,
| U i | = B exp ( r 2 / ω 2 ) ,
2 π λ L = 2 π λ ( R 2 + r 2 ) 1 / 2 ,
2 π λ ( R + r 2 / 2 R ) ,
( π r 2 ) / ( λ R ) .
Φ 0 = n 0 l .
Φ ( r , t ) = l [ n ( r , t ) n ( 0 , t ) ] .
2 π λ Φ ( r , t ) = 2 π λ dn dT l [ Δ T ( 0 , t ) Δ T ( r , t ) ] ,
U i ( r , t ) = B exp ( r 2 / ω 2 ) exp ( i ( π / λ ) ( r 2 / R + 2 Φ )
U bc ( t ) = C 0 exp { u + i [ 2 π λ Φ ( u , t ) + π ω 2 λ ( 1 R + 1 z 2 ) u ] } du
ω ( z 1 ) = ω 0 [ 1 + ( z 1 / z c ) 2 ] 1 / 2 ,
R ( z 1 ) = 1 z 1 ( z 1 2 + z c 2 ) ,
z c = π ω 0 2 λ .
i [ z 1 z c + z 1 z 2 ( z 1 z c + z c z 1 ) ] u .
exp [ i ( 2 π / λ ) Φ ] 1 i 2 π λ Φ ,
U bc ( t ) = C 0 ( 1 i 2 π λ Φ ) exp [ ( 1 + i ζ ) u ] du ,
2 π λ Φ ( u , t ) = θ t c 0 t τ [ 1 exp ( 2 τ u ) ] d t
θ = 0.24 pl λ b k dn dT ,
τ ( t ) = 1 1 + 2 t / t c .
U bc ( t ) = C 0 { 1 i θ t c 0 t τ [ 1 exp ( 2 τ u ) ] d t } × exp [ ( 1 + i ζ ) u ] du .
I ( t ) I ( ) I ( ) = 1 θ tan 1 [ 2 ζ 3 + ζ 2 + ( 9 + ζ 2 ) ( t c / 2 t ) ] 1 θ tan 1 ( 2 ζ 3 + ζ 2 ) 1 .
I I ( ) I ( ) = 1 1 θ tan 1 ( 2 ζ 3 + ζ 2 ) 1 .
I ( t ) I ( ) I ( ) = 1 θ tan 1 ( 0.577 1 + t c / t ) 1 θ ( 0.524 ) 1 .
b = b s + b d .
θ = ( 0.24 pl λ ) b d k dn dT + 0.24 P 1 b s k λ dn dT ,
t c = ( ω 2 4 ) 1 κ .

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