Abstract

A simple and sensitive method of measuring the thermally induced index changes arising from absorption of a laser beam in a low-loss material is described. The sample is placed outside the laser cavity, but at the position of minimum radius of curvature of the wavefront, which is a confocal distance behind the waist. It is estimated that the method is sensitive enough to measure absorption coefficients of the order of 5 × 10−6 cm−1 and it is shown experimentally to have good accuracy on low-loss materials. A detailed comparison of sensitivity and accuracy estimates is given for the various published thermal-lens methods for measuring low absorption coefficients.

Although the method is illustrated for the thermal effect, it is applicable to other nonlinear index changes induced by laser beams.

© 1973 Optical Society of America

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  1. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
    [CrossRef]
  2. R. C. C. Leite, R. S. Moore, J. R. Whinnery, Appl. Phys. Lett. 5, 141 (1964).
    [CrossRef]
  3. D. Solimini, J. Appl. Phys. 37, 3314 (1966).
    [CrossRef]
  4. D. Solimini, App. Opt. 5, 1931 (1966).
    [CrossRef]
  5. Y. Kohanzadeh, D. H. Auston, IEEE J. Quantum Electron. QE-6(7), 475 (1970).
    [CrossRef]
  6. F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, P. L. Kelley, Appl. Phys. Lett. 16(9), 362 (1970).
    [CrossRef]
  7. K. W. Ma, “Measurements of Thermal Parameters by Thermally Self-Induced Phase Modulation,” M.S. Thesis, University of California, Berkeley, 1971.
  8. J. Stone, J. Opt. Soc. Am. 62(3), 327 (1972).
    [CrossRef]
  9. J. R. Whinnery, D. T. Miller, F. Dabby, IEEE Quantum Electron. QE-3, 382 (1967).
    [CrossRef]
  10. S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4(10), 568 (1968).
    [CrossRef]
  11. E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13(11), 369 (1968).
    [CrossRef]
  12. F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 4(2), 52 (1969).
    [CrossRef]
  13. D. C. Smith, IEEE J. Quantum Electron. QE-5(12), 600 (1969).
    [CrossRef]
  14. J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17(5), 220 (1970).
    [CrossRef]
  15. J. N. Hayes, Appl. Opt. 11(2), 455 (1972).
    [CrossRef] [PubMed]
  16. J. R. Whinnery, unpublished.
  17. K. Darée, Optics Commun. 4(3), 238 (1971).
    [CrossRef]
  18. S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp. Fiz. Nauk 93 (1968) [Sov. Phys. Usp. 10, 609 (1968)].
  19. P. Christinsen suggested that saturation of absorption is one of the possible effects.
  20. F. W. Dabby, R. W. Boyko, C. V. Shank, J. R. Whinnery, IEEE J. Quantum Electron. QE-5(10), 516 (1969).
    [CrossRef]
  21. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1965).
    [CrossRef]
  22. Y. Kohanzadeh, Electronics Research Lab., Univ. of California; private communication.
  23. J. Stone, J. Appl. Phys. 42(6), 2396 (1971).
    [CrossRef]
  24. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

1972 (2)

1971 (2)

K. Darée, Optics Commun. 4(3), 238 (1971).
[CrossRef]

J. Stone, J. Appl. Phys. 42(6), 2396 (1971).
[CrossRef]

1970 (3)

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17(5), 220 (1970).
[CrossRef]

Y. Kohanzadeh, D. H. Auston, IEEE J. Quantum Electron. QE-6(7), 475 (1970).
[CrossRef]

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, P. L. Kelley, Appl. Phys. Lett. 16(9), 362 (1970).
[CrossRef]

1969 (3)

F. W. Dabby, R. W. Boyko, C. V. Shank, J. R. Whinnery, IEEE J. Quantum Electron. QE-5(10), 516 (1969).
[CrossRef]

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 4(2), 52 (1969).
[CrossRef]

D. C. Smith, IEEE J. Quantum Electron. QE-5(12), 600 (1969).
[CrossRef]

1968 (3)

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4(10), 568 (1968).
[CrossRef]

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13(11), 369 (1968).
[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp. Fiz. Nauk 93 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

1967 (1)

J. R. Whinnery, D. T. Miller, F. Dabby, IEEE Quantum Electron. QE-3, 382 (1967).
[CrossRef]

1966 (2)

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

D. Solimini, App. Opt. 5, 1931 (1966).
[CrossRef]

1965 (3)

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

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1965).
[CrossRef]

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

1964 (1)

R. C. C. Leite, R. S. Moore, J. R. Whinnery, Appl. Phys. Lett. 5, 141 (1964).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4(10), 568 (1968).
[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp. Fiz. Nauk 93 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

Auston, D. H.

Y. Kohanzadeh, D. H. Auston, IEEE J. Quantum Electron. QE-6(7), 475 (1970).
[CrossRef]

Avizonis, P. V.

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17(5), 220 (1970).
[CrossRef]

Boyko, R. W.

F. W. Dabby, R. W. Boyko, C. V. Shank, J. R. Whinnery, IEEE J. Quantum Electron. QE-5(10), 516 (1969).
[CrossRef]

Dabby, F.

J. R. Whinnery, D. T. Miller, F. Dabby, IEEE Quantum Electron. QE-3, 382 (1967).
[CrossRef]

Dabby, F. W.

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, P. L. Kelley, Appl. Phys. Lett. 16(9), 362 (1970).
[CrossRef]

F. W. Dabby, R. W. Boyko, C. V. Shank, J. R. Whinnery, IEEE J. Quantum Electron. QE-5(10), 516 (1969).
[CrossRef]

Darée, K.

K. Darée, Optics Commun. 4(3), 238 (1971).
[CrossRef]

Gebhardt, F. G.

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 4(2), 52 (1969).
[CrossRef]

Glass, A. J.

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13(11), 369 (1968).
[CrossRef]

Gordon, J. P.

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

Gustafson, T. K.

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, P. L. Kelley, Appl. Phys. Lett. 16(9), 362 (1970).
[CrossRef]

Hayes, J. N.

Hogge, C. B.

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17(5), 220 (1970).
[CrossRef]

Kelley, P. L.

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, P. L. Kelley, Appl. Phys. Lett. 16(9), 362 (1970).
[CrossRef]

Kenemuth, J. R.

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17(5), 220 (1970).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4(10), 568 (1968).
[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp. Fiz. Nauk 93 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1965).
[CrossRef]

Kohanzadeh, Y.

Y. Kohanzadeh, D. H. Auston, IEEE J. Quantum Electron. QE-6(7), 475 (1970).
[CrossRef]

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, P. L. Kelley, Appl. Phys. Lett. 16(9), 362 (1970).
[CrossRef]

Y. Kohanzadeh, Electronics Research Lab., Univ. of California; private communication.

Krindach, D. P.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4(10), 568 (1968).
[CrossRef]

Leite, R. C. C.

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

R. C. C. Leite, R. S. Moore, J. R. Whinnery, Appl. Phys. Lett. 5, 141 (1964).
[CrossRef]

Li, T.

Ma, K. W.

K. W. Ma, “Measurements of Thermal Parameters by Thermally Self-Induced Phase Modulation,” M.S. Thesis, University of California, Berkeley, 1971.

McLean, E. A.

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13(11), 369 (1968).
[CrossRef]

Migulin, A. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4(10), 568 (1968).
[CrossRef]

Miller, D. T.

J. R. Whinnery, D. T. Miller, F. Dabby, IEEE Quantum Electron. QE-3, 382 (1967).
[CrossRef]

Moore, R. S.

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

R. C. C. Leite, R. S. Moore, J. R. Whinnery, Appl. Phys. Lett. 5, 141 (1964).
[CrossRef]

Porto, S. P.

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

Shank, C. V.

F. W. Dabby, R. W. Boyko, C. V. Shank, J. R. Whinnery, IEEE J. Quantum Electron. QE-5(10), 516 (1969).
[CrossRef]

Sica, L.

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13(11), 369 (1968).
[CrossRef]

Smith, D. C.

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 4(2), 52 (1969).
[CrossRef]

D. C. Smith, IEEE J. Quantum Electron. QE-5(12), 600 (1969).
[CrossRef]

Solimini, D.

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

D. Solimini, App. Opt. 5, 1931 (1966).
[CrossRef]

Stone, J.

J. Stone, J. Opt. Soc. Am. 62(3), 327 (1972).
[CrossRef]

J. Stone, J. Appl. Phys. 42(6), 2396 (1971).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp. Fiz. Nauk 93 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4(10), 568 (1968).
[CrossRef]

Whinnery, J. R.

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, P. L. Kelley, Appl. Phys. Lett. 16(9), 362 (1970).
[CrossRef]

F. W. Dabby, R. W. Boyko, C. V. Shank, J. R. Whinnery, IEEE J. Quantum Electron. QE-5(10), 516 (1969).
[CrossRef]

J. R. Whinnery, D. T. Miller, F. Dabby, IEEE Quantum Electron. QE-3, 382 (1967).
[CrossRef]

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

R. C. C. Leite, R. S. Moore, J. R. Whinnery, Appl. Phys. Lett. 5, 141 (1964).
[CrossRef]

J. R. Whinnery, unpublished.

App. Opt. (1)

D. Solimini, App. Opt. 5, 1931 (1966).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (5)

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17(5), 220 (1970).
[CrossRef]

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13(11), 369 (1968).
[CrossRef]

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 4(2), 52 (1969).
[CrossRef]

R. C. C. Leite, R. S. Moore, J. R. Whinnery, Appl. Phys. Lett. 5, 141 (1964).
[CrossRef]

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, P. L. Kelley, Appl. Phys. Lett. 16(9), 362 (1970).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

IEEE J. Quantum Electron. (4)

F. W. Dabby, R. W. Boyko, C. V. Shank, J. R. Whinnery, IEEE J. Quantum Electron. QE-5(10), 516 (1969).
[CrossRef]

Y. Kohanzadeh, D. H. Auston, IEEE J. Quantum Electron. QE-6(7), 475 (1970).
[CrossRef]

D. C. Smith, IEEE J. Quantum Electron. QE-5(12), 600 (1969).
[CrossRef]

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4(10), 568 (1968).
[CrossRef]

IEEE Quantum Electron (1)

J. R. Whinnery, D. T. Miller, F. Dabby, IEEE Quantum Electron. QE-3, 382 (1967).
[CrossRef]

J. Appl. Phys. (3)

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

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

J. Stone, J. Appl. Phys. 42(6), 2396 (1971).
[CrossRef]

J. Opt. Soc. Am. (1)

Optics Commun. (1)

K. Darée, Optics Commun. 4(3), 238 (1971).
[CrossRef]

Usp. Fiz. Nauk (1)

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp. Fiz. Nauk 93 (1968) [Sov. Phys. Usp. 10, 609 (1968)].

Other (4)

P. Christinsen suggested that saturation of absorption is one of the possible effects.

J. R. Whinnery, unpublished.

K. W. Ma, “Measurements of Thermal Parameters by Thermally Self-Induced Phase Modulation,” M.S. Thesis, University of California, Berkeley, 1971.

Y. Kohanzadeh, Electronics Research Lab., Univ. of California; private communication.

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Figures (5)

Fig. 1
Fig. 1

Expansion of a laser beam by a thin lens. Full line shows the incident and the defocused beam. Dotted line is the extrapolation of the defocused beam.

Fig. 2
Fig. 2

Fractional change of the far-field beam radius vs the position of the lens. Ww0′ assumed.

Fig. 3
Fig. 3

Arrangement of the experiment.

Fig. 4
Fig. 4

Oscilloscope trace of Ibc (inverted) vs time after the shutter opens. Horizontal scale is 500 ms/div. Laser power is 9.4 mW. (a) Cell filled with ethanol; (b) cell filled with CCl4; (c) same as (b) except that the vertical scale is expanded 50 times; (d) Ibc(t) calculated for Θ = −0.01 (line) and Θ = −0.3 (crosses).

Fig. 5
Fig. 5

Arrangements of the measurement methods. (a) Beam expansion by a thermal lens; (b) spot size change due to an intracavity thermal lens; (c) beat frequency shift between two transverse modes due to an intracavity thermal lens; (d) thermal self-frequency modulation; and (e) interferometry. C is the sample cell, D is detector, G is glass plate, K is knife edge, L is lens, M is mirror, P is pinhole, S is shutter, and T is laser tube.

Tables (2)

Tables Icon

Table I Measured Absorption Coefficients b × 104 cm−1 and a Comparison with the Results of Other Measurement Methods

Tables Icon

Table II Sensitivities of the Thermooptical Measurement Methodsa

Equations (30)

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Q = ( λ f / λ e ) ( 1 - α th / α pm ) .
1 / F = P abs ( d n / d t ) / π k w 2 ( 1 + t c / 2 t ) .
θ max P abs ( d n / d T ) / 1.6 π k w < λ / π w 0 ,
P abs < 1.6 λ k / ( d n / d T ) ,
Δ Φ = [ P abs ( d n / d T ) / 2 π k ] 0 w ( 1 - e - 2 r 2 / w 2 ) ( d r / r ) ( 4 / 9 π ) P abs ( d n / d T ) / k .
w 0 2 = w 2 / [ 1 + ( π w 2 / λ R ) 2 ] ,
z = R / [ 1 + ( λ R / π u 2 ) 2 ] .
W = λ ( Z + z ) / π w 0 .
Δ W / W = Δ ( 1 / R ) [ ( W / z ) d z / d ( 1 / R ) + ( W / w 0 ) d w 0 / d ( 1 / R ) ] = Δ ( 1 / R ) w 2 { ( π w 0 / λ W ) [ ( 2 w 0 2 / w 2 ) - 1 ] ± ( π / λ ) ( w 0 / w ) [ 1 - ( w 0 2 / w 2 ) ] ½ }
Δ W / W = [ - Θ / ( 1 + t c / 2 t ) ] { ( w 0 / w ) [ ( 2 w 0 2 / w 2 ) - 1 ] ± ( w 0 / w ) [ 1 - ( w 0 2 / w 2 ) ] ½ } ,
Θ P abs ( d n / d T ) / λ k .
Δ W / W = - Θ / 2 ( 1 + t c / 2 t ) .
W 2 λ 2 Z ( Z + 2 z ) / π 2 w 0 2 = ( π 2 / Z ) w 2 / [ Z λ 2 + 2 w 4 π 2 ( 1 + Z / R ) / R ] .
W 2 ( t = ) / W 2 ( t = 0 ) = 1 - { [ 2 Θ λ π w 2 ( 1 + Z / R 0 ) + θ 2 Z λ 2 ] / [ Z λ 2 + π 2 ( w 2 / R 0 ) ( 2 w 2 + Z w 2 / R 0 ) ] } .
W 2 ( t = ) / W 2 ( t = 0 ) = 1 - Θ + Θ 2 / 2.
I bc ( t = 0 ) / I bc ( t = ) = 1 - Θ + Θ 2 / 2.
b = - [ Δ I bc / I bc ( t = ) ] λ k / P l ( d n / d T ) ,
I bc ( t ) = I bc ( 0 ) [ 1 - Θ / ( 1 + t c / 2 t ) + Θ 2 / 2 ( 1 + t c / 2 t ) 2 ] - 1 .
Δ w 2 / w 2 = - Θ / 4 ( 1 - g ) ( g 1 g 2 ) ½ .
Δ ν / ν = Θ / 2 ( 1 - g 1 ) ½ ,
Δ ϕ = t · 4 P abs ( d n / d T ) / λ w 2 ρ c = Θ t / t c
2 π f th = ( Δ ϕ ) / t = Θ / t c .
I ( t ) = I 0 ( 1 + Δ ϕ ) = I 0 ( 1 + Θ t / t c ) .
w 2 2 = ( λ d / π ) { ( g 1 - R 1 / F ) / g 2 [ 1 - ( g 1 - R 1 / F ) g 2 ] } ½ ( λ d / π ) { ( g 1 - λ R 1 θ g 1 / w 2 2 g 2 π ) / g 2 [ 1 - ( g 1 - λ R 1 θ g 1 / w 2 2 g 2 π ) ] } ½
Δ ( w 2 2 ) / w 2 2 - Θ / 2 ( 1 - g 1 ) ( g 1 g 2 ) ½ ,
Δ w 2 / w 2 = - Θ / 4 ( 1 - g 1 ) ( g 1 g 2 ) ½ .
ν = ( ν 0 / π ) cos - 1 ( 1 - d / R i { 1 + [ R 1 - d 1 / F ] } ) ½ ( ν 0 / π ) cos - 1 [ 1 - ( d / 2 R 1 ) { 1 + ( R 1 - d 1 ) / F } ]
ν = ( ν 0 / π ) { d [ ( 1 / R 1 ) + ( 1 / F ) ] } ½ = ( ν 0 / π ) { d [ ( 1 / R 1 ) + ( λ Θ / w 2 2 π ) ] } ½ .
Δ ν = ( ν 0 λ Θ / 2 π 2 ) [ ( d R 1 ) ½ / w 2 2 ] = ν 0 Θ / 2 π
Δ ν / ν = Θ / 2 ( d / R 1 ) ½ = Θ / 2 ( 1 - g 1 ) ½ .

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