Abstract

A Fresnel diffraction model for dual beam pulsed mode laser thermal lens effect detection is described. The model accommodates the effects of aberrations in the lens element introduced by departures in the sample’s thermally induced refractive index profile from the ideal parabolic approximation. The model also accommodates probe and irradiation beams of arbitrary complex radius at the sample, and permits the computation of probe beam intensity profiles observed at arbitrary cell–detector distances. The theoretical basis for a new method of thermal lens effect detection is demonstrated in which the radial dimensions of the lens element are much smaller than the probe beam. Detection of the thermal blooming effect is achieved by a Fourier transform method which uses spatial frequency domain detection to measure thermally induced departures in the probe beam’s intensity profile from the TEM(0,0) Gaussian mode structure, as the lens element forms. This strategy, combined with near field detection predicts a sensitivity enhancement by a factor of 60 relative to the conventional far field beam center measurement.

© 1990 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, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
    [CrossRef]
  2. C. Hu, J. R. Whinnery, “New Thermooptical Measurement Method and a Comparison with Other Methods,” Appl. Opt. 12, 72–79 (1973).
    [CrossRef] [PubMed]
  3. N. J. Dovichi, J. M. Harris, “Laser Induced Thermal Lens Effect for Calorimetric Trace Analysis,” Anal. Chem. 51, 728–731 (1979).
    [CrossRef]
  4. J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
    [CrossRef]
  5. R. C. C. Leite, R. S. Moore, J. R. Whinnery, “Low Absorption Measurements by Means of the Thermal Lens Effect Using an He–Ne Laser,” Appl. Phys. Lett. 5, 141–143 (1964).
    [CrossRef]
  6. K. Miyaishi, T. Imasaka, N. Ishibashi, “Thermal Lens Spectrophotometry based on Image Detection of a Probe Laser Beam,” Anal. Chem. 54, 2039–2044 (1982).
    [CrossRef]
  7. K. L. Jansen, J. M. Harris, “Thermal Lens Measurements by Optical Computation of the Laser Beam Spot Size,” Anal. Chem. 57, 1698–1703 (1985).
    [CrossRef]
  8. A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming: I. Theory,” Chem. Phys. 20, 253–258 (1977).
    [CrossRef]
  9. S. J. Sheldon, L. V. Knight, J. M. Thorne, “Laser-Induced Thermal Lens Effect: a New Theoretical Model,” Appl. Opt. 21, 1663–1669 (1982).
    [CrossRef] [PubMed]
  10. M. E. Long, R. L. Swofford, A. C. Albrecht, “Thermal Lens Technique: a New Method of Absorption Spectroscopy,” Science 191, 183–184 (1976).
    [CrossRef] [PubMed]
  11. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.
  12. T. Berthoud, N. Delorme, P. Mauchien, “Beam Geometry Optimization in Dual Beam Thermal Lensing Spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
    [CrossRef]
  13. J. F. Power, E. D. Salin, “Mode-Mismatched Laser Induced Thermal Lens Effect Detection via Spatial Fourier Analysis of Beam Profiles,” Anal. Chem., 60, 838–842 (1988).
    [CrossRef]
  14. S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukorukov, R. V. Khoklov, “E-12-Thermal Self-Actions of Laser Beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
    [CrossRef]
  15. F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, “Thermally Self-Induced Phase Modulation of Laser Beams,” Appl. Phys. Lett. 16, 362–365 (1970).
    [CrossRef]
  16. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 9.
  17. G. N. Watson, A Treatise on the Theory of Bessel Functions, Second Edition, (Cambridge U.P., London, 1944), p. 393.

1988 (1)

J. F. Power, E. D. Salin, “Mode-Mismatched Laser Induced Thermal Lens Effect Detection via Spatial Fourier Analysis of Beam Profiles,” Anal. Chem., 60, 838–842 (1988).
[CrossRef]

1985 (2)

T. Berthoud, N. Delorme, P. Mauchien, “Beam Geometry Optimization in Dual Beam Thermal Lensing Spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
[CrossRef]

K. L. Jansen, J. M. Harris, “Thermal Lens Measurements by Optical Computation of the Laser Beam Spot Size,” Anal. Chem. 57, 1698–1703 (1985).
[CrossRef]

1982 (2)

K. Miyaishi, T. Imasaka, N. Ishibashi, “Thermal Lens Spectrophotometry based on Image Detection of a Probe Laser Beam,” Anal. Chem. 54, 2039–2044 (1982).
[CrossRef]

S. J. Sheldon, L. V. Knight, J. M. Thorne, “Laser-Induced Thermal Lens Effect: a New Theoretical Model,” Appl. Opt. 21, 1663–1669 (1982).
[CrossRef] [PubMed]

1980 (1)

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
[CrossRef]

1979 (1)

N. J. Dovichi, J. M. Harris, “Laser Induced Thermal Lens Effect for Calorimetric Trace Analysis,” Anal. Chem. 51, 728–731 (1979).
[CrossRef]

1977 (1)

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming: I. Theory,” Chem. Phys. 20, 253–258 (1977).
[CrossRef]

1976 (1)

M. E. Long, R. L. Swofford, A. C. Albrecht, “Thermal Lens Technique: a New Method of Absorption Spectroscopy,” Science 191, 183–184 (1976).
[CrossRef] [PubMed]

1973 (1)

1970 (1)

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, “Thermally Self-Induced Phase Modulation of Laser Beams,” Appl. Phys. Lett. 16, 362–365 (1970).
[CrossRef]

1968 (1)

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukorukov, R. V. Khoklov, “E-12-Thermal Self-Actions of Laser Beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

1965 (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

1964 (1)

R. C. C. Leite, R. S. Moore, J. R. Whinnery, “Low Absorption Measurements by Means of the Thermal Lens Effect Using an He–Ne Laser,” Appl. Phys. Lett. 5, 141–143 (1964).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukorukov, R. V. Khoklov, “E-12-Thermal Self-Actions of Laser Beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Albrecht, A. C.

M. E. Long, R. L. Swofford, A. C. Albrecht, “Thermal Lens Technique: a New Method of Absorption Spectroscopy,” Science 191, 183–184 (1976).
[CrossRef] [PubMed]

Berthoud, T.

T. Berthoud, N. Delorme, P. Mauchien, “Beam Geometry Optimization in Dual Beam Thermal Lensing Spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
[CrossRef]

Dabby, F. W.

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, “Thermally Self-Induced Phase Modulation of Laser Beams,” Appl. Phys. Lett. 16, 362–365 (1970).
[CrossRef]

Delorme, N.

T. Berthoud, N. Delorme, P. Mauchien, “Beam Geometry Optimization in Dual Beam Thermal Lensing Spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
[CrossRef]

Dovichi, N. J.

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
[CrossRef]

N. J. Dovichi, J. M. Harris, “Laser Induced Thermal Lens Effect for Calorimetric Trace Analysis,” Anal. Chem. 51, 728–731 (1979).
[CrossRef]

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Gustafson, T. K.

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, “Thermally Self-Induced Phase Modulation of Laser Beams,” Appl. Phys. Lett. 16, 362–365 (1970).
[CrossRef]

Harris, J. M.

K. L. Jansen, J. M. Harris, “Thermal Lens Measurements by Optical Computation of the Laser Beam Spot Size,” Anal. Chem. 57, 1698–1703 (1985).
[CrossRef]

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
[CrossRef]

N. J. Dovichi, J. M. Harris, “Laser Induced Thermal Lens Effect for Calorimetric Trace Analysis,” Anal. Chem. 51, 728–731 (1979).
[CrossRef]

Hu, C.

Imasaka, T.

K. Miyaishi, T. Imasaka, N. Ishibashi, “Thermal Lens Spectrophotometry based on Image Detection of a Probe Laser Beam,” Anal. Chem. 54, 2039–2044 (1982).
[CrossRef]

Ishibashi, N.

K. Miyaishi, T. Imasaka, N. Ishibashi, “Thermal Lens Spectrophotometry based on Image Detection of a Probe Laser Beam,” Anal. Chem. 54, 2039–2044 (1982).
[CrossRef]

Jansen, K. L.

K. L. Jansen, J. M. Harris, “Thermal Lens Measurements by Optical Computation of the Laser Beam Spot Size,” Anal. Chem. 57, 1698–1703 (1985).
[CrossRef]

Khoklov, R. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukorukov, R. V. Khoklov, “E-12-Thermal Self-Actions of Laser Beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Kliger, D. S.

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming: I. Theory,” Chem. Phys. 20, 253–258 (1977).
[CrossRef]

Knight, L. V.

Kohanzadeh, Y.

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, “Thermally Self-Induced Phase Modulation of Laser Beams,” Appl. Phys. Lett. 16, 362–365 (1970).
[CrossRef]

Krindach, D. P.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukorukov, R. V. Khoklov, “E-12-Thermal Self-Actions of Laser Beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

R. C. C. Leite, R. S. Moore, J. R. Whinnery, “Low Absorption Measurements by Means of the Thermal Lens Effect Using an He–Ne Laser,” Appl. Phys. Lett. 5, 141–143 (1964).
[CrossRef]

Long, M. E.

M. E. Long, R. L. Swofford, A. C. Albrecht, “Thermal Lens Technique: a New Method of Absorption Spectroscopy,” Science 191, 183–184 (1976).
[CrossRef] [PubMed]

Mauchien, P.

T. Berthoud, N. Delorme, P. Mauchien, “Beam Geometry Optimization in Dual Beam Thermal Lensing Spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
[CrossRef]

Migulin, A. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukorukov, R. V. Khoklov, “E-12-Thermal Self-Actions of Laser Beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Miyaishi, K.

K. Miyaishi, T. Imasaka, N. Ishibashi, “Thermal Lens Spectrophotometry based on Image Detection of a Probe Laser Beam,” Anal. Chem. 54, 2039–2044 (1982).
[CrossRef]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

R. C. C. Leite, R. S. Moore, J. R. Whinnery, “Low Absorption Measurements by Means of the Thermal Lens Effect Using an He–Ne Laser,” Appl. Phys. Lett. 5, 141–143 (1964).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 9.

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Power, J. F.

J. F. Power, E. D. Salin, “Mode-Mismatched Laser Induced Thermal Lens Effect Detection via Spatial Fourier Analysis of Beam Profiles,” Anal. Chem., 60, 838–842 (1988).
[CrossRef]

Salin, E. D.

J. F. Power, E. D. Salin, “Mode-Mismatched Laser Induced Thermal Lens Effect Detection via Spatial Fourier Analysis of Beam Profiles,” Anal. Chem., 60, 838–842 (1988).
[CrossRef]

Sheldon, S. J.

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.

Sukorukov, A. P.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukorukov, R. V. Khoklov, “E-12-Thermal Self-Actions of Laser Beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Swofford, R. L.

M. E. Long, R. L. Swofford, A. C. Albrecht, “Thermal Lens Technique: a New Method of Absorption Spectroscopy,” Science 191, 183–184 (1976).
[CrossRef] [PubMed]

Thorne, J. M.

Twarowski, A. J.

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming: I. Theory,” Chem. Phys. 20, 253–258 (1977).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, Second Edition, (Cambridge U.P., London, 1944), p. 393.

Whinnery, J. R.

C. Hu, J. R. Whinnery, “New Thermooptical Measurement Method and a Comparison with Other Methods,” Appl. Opt. 12, 72–79 (1973).
[CrossRef] [PubMed]

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, “Thermally Self-Induced Phase Modulation of Laser Beams,” Appl. Phys. Lett. 16, 362–365 (1970).
[CrossRef]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

R. C. C. Leite, R. S. Moore, J. R. Whinnery, “Low Absorption Measurements by Means of the Thermal Lens Effect Using an He–Ne Laser,” Appl. Phys. Lett. 5, 141–143 (1964).
[CrossRef]

Anal. Chem. (6)

N. J. Dovichi, J. M. Harris, “Laser Induced Thermal Lens Effect for Calorimetric Trace Analysis,” Anal. Chem. 51, 728–731 (1979).
[CrossRef]

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
[CrossRef]

K. Miyaishi, T. Imasaka, N. Ishibashi, “Thermal Lens Spectrophotometry based on Image Detection of a Probe Laser Beam,” Anal. Chem. 54, 2039–2044 (1982).
[CrossRef]

K. L. Jansen, J. M. Harris, “Thermal Lens Measurements by Optical Computation of the Laser Beam Spot Size,” Anal. Chem. 57, 1698–1703 (1985).
[CrossRef]

T. Berthoud, N. Delorme, P. Mauchien, “Beam Geometry Optimization in Dual Beam Thermal Lensing Spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
[CrossRef]

J. F. Power, E. D. Salin, “Mode-Mismatched Laser Induced Thermal Lens Effect Detection via Spatial Fourier Analysis of Beam Profiles,” Anal. Chem., 60, 838–842 (1988).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

R. C. C. Leite, R. S. Moore, J. R. Whinnery, “Low Absorption Measurements by Means of the Thermal Lens Effect Using an He–Ne Laser,” Appl. Phys. Lett. 5, 141–143 (1964).
[CrossRef]

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, Y. Kohanzadeh, “Thermally Self-Induced Phase Modulation of Laser Beams,” Appl. Phys. Lett. 16, 362–365 (1970).
[CrossRef]

Chem. Phys. (1)

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming: I. Theory,” Chem. Phys. 20, 253–258 (1977).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukorukov, R. V. Khoklov, “E-12-Thermal Self-Actions of Laser Beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

J. Appl. Phys. (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Science (1)

M. E. Long, R. L. Swofford, A. C. Albrecht, “Thermal Lens Technique: a New Method of Absorption Spectroscopy,” Science 191, 183–184 (1976).
[CrossRef] [PubMed]

Other (3)

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 9.

G. N. Watson, A Treatise on the Theory of Bessel Functions, Second Edition, (Cambridge U.P., London, 1944), p. 393.

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Figures (13)

Fig. 1
Fig. 1

Schematic of the Fresnel diffraction model derived in this work for pulsed laser thermal lensing.

Fig. 2
Fig. 2

(a) Schematic of mode mismatched thermal lens detection and transverse intensity profiles computed from theory for the near field; (b) thermal defocusing; (c) thermal focusing. Mismatch between probe and pump beam radii is 10:1 in the cell. Sample’s thermooptic properties are given in the text. Other parameters are z2 = 1.08 m, ω1 = 6.5 × 10−4 m (at z = z1); z1 = 1.05 m; ω0 = 6.5 × 10−5 m, β = 0.05 cm−1.

Fig. 3
Fig. 3

Effect of varying ratio of probe: pump beam radius on near field intensity profile. All the profiles were computed assuming the following: ω1 = 1.020 × 10−3 m; z1 = 1.00 m; z2 = 1.25 m; t = 0 and (a) 2:1 spot radius mismatch with ω0 = 5 × 10−4 m and β = 5.00 cm−1; (b) 4:1 mismatch with ω0 = 2.5 × 10−4 m and β = 3.125 cm−1; (c) 10:1 mismatch with ω0 = 1 × 10−4 m and β = 0.05 cm−1. Irradiation conditions as in (a). Other optical and thermooptic parameters are described in the text.

Fig. 4
Fig. 4

Time dependence of near field intensity profile recovered at various delays past excitation. Time delays are as follows from (i) to (vi): 5, 10, 15, 20, 30, 40, and 64 ms, past excitation. ω1 = 1.020 × 10−3 m; ω0 = 1 × 10−4 m; z1 = 1.0 m; z2 = 1.25 m; β = 0.1 cm−1 (see text for other optical and thermooptic properties).

Fig. 5
Fig. 5

Equivalence of near field intensity profiles recorded as a function of time delay past excitation for varying pump beam radii ω01 and ω02: (a) ω0(1) = 1 × 10−4 m; t = 0; (b) ω0(1) = 1 × 10−4 m; t = 0.013125 s; ω0(2) = 2.5 × 10−4 m; t = 0 s. Beam profiles in the second case are identical, as predicted from theory. Other parameters are z1 = 1.00 m; z2 = 1.25 m; ω1 = 1.02 × 10−3 m (see text for further details).

Fig. 6
Fig. 6

Effect of sample absorption coefficient on near field transverse intensity profile of probe laser beam at t = 0 past excitation. (a) In the spatial domain curves vi–i had the following absorption coefficients: β = 0.025, 0.01, 0.005, 0.001, and 1 × 10−7 cm−1 and (b) curves i–vi had the same sequence of β as in (a). Absorption coefficients in (a) (bottom to top curve): β = 0.025, 0.01, 0.005, 0.0025, 0.001 and 1 × 10−7 cm−1 (curves vi–i). Signals calculated for z1 = 1.0 m; z2 = 1.125 m; ω1 = 1.02 × 10−3 m; ω0 = 1 × 10−4 m (with other optical and thermooptic properties described in the text).

Fig. 7
Fig. 7

(a) Conventional beam center signal dependence on absorption coefficient for near field profiles in Fig. 6. (b) β dependence of the near field signal in the spatial frequency domain. Signals correspond to the magnitude of the Fourier transform of the spatial frequency of the beam profile data and were evaluated at a spatial frequency of 2.602 × 103 m−1. All the optical and thermooptic conditions as per Fig. 6.

Fig. 8
Fig. 8

Comparison of conventional thermal lens effect detection at t = 0, (a) and (b), with results obtained using highly mismatched beams (c). (a) Conventional measurement with z1 = 0.02869 m (zc/√3); ω1(z = 0) = 1 × 10−4 m; ω0 = 1.1547 × 10−4 m; β = 0.025 cm−1 and far field detection z2 = 5.029m). Details in text (b) near to midfield signal corresponding to (a) with z2 = 0.0537 m; (c) Intensity profiles predicted for the near field of a defocused probe beam: ω1 (z = 0) = 1 × 10−3 m; z1 = 0.0286 m; z2 = 0.1536 m; all the other parameters as per (a).

Fig. 9
Fig. 9

Plots of the transverse phase variation of U2(r,z,t = 0) U(r,z) [Eq. (11)] at increasing offset distances z ¯ from a thermal lens element placed in the near field of the probe beam: (a) with spot radius mismatch of 10:1 in the cell (β = 0.005 cm−1 and pump beam radius 1 × 10−4 m); initial phase plot recorded at z ¯ = 0.05 m; (b) with spot radius mismatch of 3.33:1 in the cell (β = 0.4 cm−1 and pump beam radius of 3 × 10−4 m); initial phase plot recorded at z ¯ = 0.25 m. Other conditions are z1 = 1.00 m; ω1(z = 0) = 1 × 10−3 m. (All the other optical and thermooptical parameters are summarized in the text). Insets (a) and (b): intensity profiles computed at various distances z ¯ from the lens.

Fig. 10
Fig. 10

Plots of the transverse phase variation of U2(r,z,t = 0) observed when a thermal lens forms at the confocal point of the probe beam: (a) with spot radius mismatch of 10:1 in the cell (pump beam radius of 6.5 × 10−5 m and absorption coefficient β = 0.01 cm−1); (b) with spot radius mismatch of 3.33:1 in the cell (pump beam radius of 1.95 × 10−4 m and β = 0.09 cm−1). Other conditions for which these profiles were computed are z1 = 1.05 m; ω1 (z1 = 0) = 4.596 × 10−4 m (all the other optical and thermooptic parameters are summarized in the text.) Insets (a) and (b): intensity profiles computed at various distances z ¯ from the lens.

Fig. 11
Fig. 11

Plots of the thermal lens effect signal [as ΔI(t)]/[I(t)] vs z ¯ with placement of cell in the near field of the probe beam. (a) with spot mismatch of 10:1 and ω1(z1 = 0) = 1 × 10−3 m, ω0 = 1 × 10−4 m, t = 0, and β = 0.005 cm−1, z1 = 1.00 m. All the other optical and thermooptic properties are described in the text (b) study run with spot mismatch of 3.333:1 and ω1(z1 = 0) = 1 × 10−3 m, ω0 = 3 × 10−4 m, β = 0.4 cm−1, t = 0, z1 = 1.00 m. All the other optical and thermooptical parameters are described in the text.

Fig. 12
Fig. 12

Plots of the thermal lens effect signal [as ΔI(t)]/[I(t)] vs z with the placement of the cell at the confocal position of the probe beam (a) with 10:1 spot radius mismatch and ω1(z1 = 0 m) = 4.596 × 10−4 m, ω0 = 6.5 × 10−5 m, β = 0.01 cm−1 and t = 0; also z1 = 1.05 m (all the other optical and thermooptic parameters are summarized in the text); (b) with 3.333:1 spot radius mismatch, and ω1(z1 = 0 m) = 4.596 × 10−4 m, ω0 = 1.95 × 10−4 m, β = 0.09 cm−1; all the other parameters as per 11(a); (c) with 1:1 spot radius ratio, ω0 = 6.5 × 10−4 m, β = 1.0 cm−1 and all other conditions as in (a).

Fig. 13
Fig. 13

Plots of the thermal lens effect signal [as ΔI(t)]/[I(t)] vs z with the placement of the cell in the far field. For this study: ω1(z1 = 0 m) = 1 × 10−5 m, ω0 = 6.5 × 10−5 m, z1 = 0.032307 mΨ, β = 0.01 cm−1, and t = 0. All the other optical and thermooptic parameters are described in the text). Insets give transverse intensity profiles at z = 0.5 m and z = 32.0 m.

Equations (21)

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U 2 ( x , y , z ) = U 1 ( x , y , z ) * * j exp ( - j k z ¯ ) λ z ¯ exp [ j k 2 z ¯ ( x 2 + y 2 ) ] ,
exp [ - j k 2 z ¯ ( x 2 + y 2 ) ]
U 0 ( x , y , z ) = 2 π exp ( - j k z 1 ) ω 1 exp [ - j k 2 q 1 ( x 2 + y 2 ) ] ,
1 q 1 = 1 R 1 - j λ π ω 1 2 ,
U 1 ( x , y , z , t ) = 2 π 1 ω 1 exp [ - j k ( x 2 + y 2 ) 2 q 1 ] exp [ - j k Δ n ( r , t ) l ] ,
Δ n ( r , t ) = n ( r , t ) - n ( 0 , t ) .
n ( r , t ) = n 0 + ( n T ) Δ T ( r , t ) .
Δ T ( r , t ) = α A 0 π K ( ω 0 2 + 4 α t ) exp [ - r 2 / ( ω 0 2 + 4 α t ) ] ,
E pulse = h ν p P 0 δ ( t ) d t ,
U 2 ( x , y , z , t ) = j exp ( - j k z ¯ ) 2 z ¯ λ ω 1 2 π - exp [ - j k ( x 0 2 + y 0 2 ) 2 q 1 ] × exp { - j k n T [ Δ T ( r , t ) - Δ T ( 0 , t ) ] l } × exp { - j k 2 z ¯ [ ( x - x 0 ) 2 + ( y - y 0 ) 2 ] } d x 0 d y 0 .
U 2 ( r , z , t ) = j exp ( - j k z ¯ ) exp [ j k n T Δ T ( 0 , t ) l ] z ¯ λ ω 1 2 π exp ( - j k r 2 2 z ¯ ) × 0 exp [ - j k 2 ( 1 q 1 + 1 z ¯ ) r 0 2 ] × exp [ - j k n T Δ T ( r 0 , t ) ] J 0 ( k ¯ r 0 ) r 0 d r 0 .
exp [ - j k n T Δ T ( r 0 , t ) ] = n = 0 [ - j k n t A 0 α π K ( ω 0 2 + 4 α t ) ] n × exp [ - n r 0 2 ( ω 0 2 + 4 α t ) ] n ! .
U 2 ( r , z , t ) = A 1 n = 0 Λ n exp ( - j k r 2 2 z ¯ ) n ! 0 exp ( - j k r 0 2 ξ 2 ) × J 0 ( k ¯ r 0 ) r 0 d r 0 .
A 1 = j exp ( - j k z ¯ ) exp [ j k n T Δ T ( 0 , t ) l ] z ¯ λ ω 1 π 2 , Λ = - j k n T α A 0 π K ( ω 0 2 + 4 α t ) , ξ = 1 q 1 + 1 z ¯ + 2 n j k ( ω 0 2 + 4 α t ) .
0 exp ( - a 2 r 2 ) J 0 ( k r ) r d r = 1 2 a 2 exp ( - k 2 / 4 a 2 ) .
U 2 ( r , z , t ) = 2 π A 1 n = 0 Λ n n ! 1 j k ( 1 q p n + 1 z ¯ ) × exp [ - j k r 2 2 ( q p n + z ¯ ) ] ,
1 q p n = 1 q 1 + 2 n j k ( ω 0 2 + 4 α t ) .
r < ( λ z ¯ 3 25 ) 1 / 4 .
α = 1 × 10 - 6 m 2 / s , K = 1.5 W / m K , n T = - 1 × 10 - 5 K - 1 .
Δ I ( t ) I ( t ) = [ I ( r = 0 , t = ) - I ( r = 0 , t ) ] / I ( r = 0 , t ) ,
exp [ - j k r 2 R eff ] ,

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