Abstract

A simple method for calculating the effects of optical geometry on photothermal lens signals is shown. This method is based on calculating cumulative electric-field phase shifts produced by a series of Gaussian refractive-index perturbations produced by the photothermal effect. Theoretical results are found for both pulsed-laser and continuous Gaussian laser excitation sources and both single- and two-laser apparatuses commonly employed in photothermal lens spectroscopy. The effects of apparatus geometry on the resulting signal are shown. Analytical time-dependent signal results are found for small signals. Analytical pump–probe focus geometry results allow direct optimization for certain conditions. The calculations indicate that the photothermal lens signal is, in general, optimized for near-field detection-plane geometries.

© 1997 Optical Society of America

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  1. J. P. Gordon, R. C. Leite, R. S. Moore, S. P. 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 thermo-optical method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973).
    [Crossref] [PubMed]
  3. 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]
  4. C. A. Carter, J. M. Harris, “Comparison of models describing the thermal lens effect,” Appl. Opt. 23, 476–481 (1984).
    [Crossref] [PubMed]
  5. T. Higashi, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry with argon laser excitation source for nitrogen dioxide determination,” Anal. Chem. 55, 1907–1910 (1983).
    [Crossref]
  6. D. Weaire, B. S. Wherrett, D. A. B. Miller, S. D. Smith, “Effect of low power nonlinear refraction on laser-beam propagation in InSb,” Opt. Lett. 4, 331–333 (1979).
    [Crossref] [PubMed]
  7. M. Sheik-Bahae, A. A. Said, E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
    [Crossref] [PubMed]
  8. S. E. Bialkowski, “Photothermal lens aberration effects in two laser thermal lens spectrophotometry,” Appl. Opt. 24, 2792–2796 (1985).
    [Crossref] [PubMed]
  9. J. Castillo, V. P. Kozich, A. Marcano, “Thermal lensing resulting from one- and two-photon absorption studied with a two-color time-resolved Z scan,” Opt. Lett. 19, 171–173 (1994).
    [Crossref] [PubMed]
  10. V. P. Kozich, A. Marcano, F. E. Hernandez, J. A. Castillo, “Dual-beam time-resolved Z-scan in liquids to study heating due to linear and nonlinear light absorption,” Appl. Spectrosc. 48, 1506–1512 (1994).
    [Crossref]
  11. J. Slaby, “Calculation of probe beam diffraction at laser induced thermal lens,” Opt. Commun. 60, 133–138 (1986).
    [Crossref]
  12. J. Slaby, “Background illumination filtering in thermal lens spectroscopy,” Anal. Chem. 61, 2496–2499 (1989).
    [Crossref]
  13. J. F. Power, “Pulsed mode thermal lens effect detection in the near field via thermally induced probe beam spatial phase modulation: a theory,” Appl. Opt. 29, 52–62 (1990).
    [Crossref] [PubMed]
  14. J. Shen, R. D. Lowe, R. D. Snook, “A model for CW laser induced mode-mismatched dual beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992); J. Shen, A. J. Soroka, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam image detection,” J. Appl. Phys. 78, 700–708 (1995).
    [Crossref]
  15. S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis (Wiley, New York, 1996), Chap. 4.
  16. A. J. Twarowski, D. S. Kliger, “Multiphoton absorption spectra using thermal blooming. I. Theory,” Chem. Phys. 20, 253–258 (1977).
    [Crossref]
  17. H. S. Carlaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).
  18. A. Yariv, Introduction to Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985).
  19. T. Berthoud, N. Delorme, P. Mauchien, “Beam geometry optimization in dual-beam thermal lensing spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
    [Crossref]
  20. N. J. Dovichi, J. M. Harris, “Laser induced thermal lens effect for calorimetric trace analysis,” Anal. Chem. 51, 728 (1979).
    [Crossref]
  21. K. Mori, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry based on pulsed laser excitation,” Anal. Chem. 54, 2034–2038 (1982).
    [Crossref]

1994 (2)

1992 (1)

J. Shen, R. D. Lowe, R. D. Snook, “A model for CW laser induced mode-mismatched dual beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992); J. Shen, A. J. Soroka, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam image detection,” J. Appl. Phys. 78, 700–708 (1995).
[Crossref]

1990 (1)

1989 (2)

J. Slaby, “Background illumination filtering in thermal lens spectroscopy,” Anal. Chem. 61, 2496–2499 (1989).
[Crossref]

M. Sheik-Bahae, A. A. Said, E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[Crossref] [PubMed]

1986 (1)

J. Slaby, “Calculation of probe beam diffraction at laser induced thermal lens,” Opt. Commun. 60, 133–138 (1986).
[Crossref]

1985 (2)

S. E. Bialkowski, “Photothermal lens aberration effects in two laser thermal lens spectrophotometry,” Appl. Opt. 24, 2792–2796 (1985).
[Crossref] [PubMed]

T. Berthoud, N. Delorme, P. Mauchien, “Beam geometry optimization in dual-beam thermal lensing spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
[Crossref]

1984 (1)

1983 (1)

T. Higashi, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry with argon laser excitation source for nitrogen dioxide determination,” Anal. Chem. 55, 1907–1910 (1983).
[Crossref]

1982 (2)

K. Mori, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry based on pulsed laser excitation,” Anal. Chem. 54, 2034–2038 (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]

1979 (2)

1977 (1)

A. J. Twarowski, D. S. Kliger, “Multiphoton absorption spectra using thermal blooming. I. Theory,” Chem. Phys. 20, 253–258 (1977).
[Crossref]

1973 (1)

1965 (1)

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Berthoud, T.

T. Berthoud, N. Delorme, P. Mauchien, “Beam geometry optimization in dual-beam thermal lensing spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
[Crossref]

Bialkowski, S. E.

S. E. Bialkowski, “Photothermal lens aberration effects in two laser thermal lens spectrophotometry,” Appl. Opt. 24, 2792–2796 (1985).
[Crossref] [PubMed]

S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis (Wiley, New York, 1996), Chap. 4.

Carlaw, H. S.

H. S. Carlaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).

Carter, C. A.

Castillo, J.

Castillo, J. A.

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.

N. J. Dovichi, J. M. Harris, “Laser induced thermal lens effect for calorimetric trace analysis,” Anal. Chem. 51, 728 (1979).
[Crossref]

Gordon, J. P.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Harris, J. M.

C. A. Carter, J. M. Harris, “Comparison of models describing the thermal lens effect,” Appl. Opt. 23, 476–481 (1984).
[Crossref] [PubMed]

N. J. Dovichi, J. M. Harris, “Laser induced thermal lens effect for calorimetric trace analysis,” Anal. Chem. 51, 728 (1979).
[Crossref]

Hernandez, F. E.

Higashi, T.

T. Higashi, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry with argon laser excitation source for nitrogen dioxide determination,” Anal. Chem. 55, 1907–1910 (1983).
[Crossref]

Hu, C.

Imasaka, T.

T. Higashi, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry with argon laser excitation source for nitrogen dioxide determination,” Anal. Chem. 55, 1907–1910 (1983).
[Crossref]

K. Mori, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry based on pulsed laser excitation,” Anal. Chem. 54, 2034–2038 (1982).
[Crossref]

Ishibashi, N.

T. Higashi, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry with argon laser excitation source for nitrogen dioxide determination,” Anal. Chem. 55, 1907–1910 (1983).
[Crossref]

K. Mori, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry based on pulsed laser excitation,” Anal. Chem. 54, 2034–2038 (1982).
[Crossref]

Jaeger, J. C.

H. S. Carlaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).

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.

Kozich, V. P.

Leite, R. C.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Lowe, R. D.

J. Shen, R. D. Lowe, R. D. Snook, “A model for CW laser induced mode-mismatched dual beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992); J. Shen, A. J. Soroka, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam image detection,” J. Appl. Phys. 78, 700–708 (1995).
[Crossref]

Marcano, A.

Mauchien, P.

T. Berthoud, N. Delorme, P. Mauchien, “Beam geometry optimization in dual-beam thermal lensing spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
[Crossref]

Miller, D. A. B.

Moore, R. S.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Mori, K.

K. Mori, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry based on pulsed laser excitation,” Anal. Chem. 54, 2034–2038 (1982).
[Crossref]

Porto, S. P.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Power, J. F.

Said, A. A.

Sheik-Bahae, M.

Sheldon, S. J.

Shen, J.

J. Shen, R. D. Lowe, R. D. Snook, “A model for CW laser induced mode-mismatched dual beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992); J. Shen, A. J. Soroka, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam image detection,” J. Appl. Phys. 78, 700–708 (1995).
[Crossref]

Slaby, J.

J. Slaby, “Background illumination filtering in thermal lens spectroscopy,” Anal. Chem. 61, 2496–2499 (1989).
[Crossref]

J. Slaby, “Calculation of probe beam diffraction at laser induced thermal lens,” Opt. Commun. 60, 133–138 (1986).
[Crossref]

Smith, S. D.

Snook, R. D.

J. Shen, R. D. Lowe, R. D. Snook, “A model for CW laser induced mode-mismatched dual beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992); J. Shen, A. J. Soroka, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam image detection,” J. Appl. Phys. 78, 700–708 (1995).
[Crossref]

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]

Van Stryland, E. W.

Weaire, D.

Wherrett, B. S.

Whinnery, J. R.

C. Hu, J. R. Whinnery, “New thermo-optical method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973).
[Crossref] [PubMed]

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Yariv, A.

A. Yariv, Introduction to Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985).

Anal. Chem. (5)

T. Higashi, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry with argon laser excitation source for nitrogen dioxide determination,” Anal. Chem. 55, 1907–1910 (1983).
[Crossref]

J. Slaby, “Background illumination filtering in thermal lens spectroscopy,” Anal. Chem. 61, 2496–2499 (1989).
[Crossref]

T. Berthoud, N. Delorme, P. Mauchien, “Beam geometry optimization in dual-beam thermal lensing spectrometry,” Anal. Chem. 57, 1216–1219 (1985).
[Crossref]

N. J. Dovichi, J. M. Harris, “Laser induced thermal lens effect for calorimetric trace analysis,” Anal. Chem. 51, 728 (1979).
[Crossref]

K. Mori, T. Imasaka, N. Ishibashi, “Thermal lens spectrophotometry based on pulsed laser excitation,” Anal. Chem. 54, 2034–2038 (1982).
[Crossref]

Appl. Opt. (5)

Appl. Spectrosc. (1)

Chem. Phys. (2)

J. Shen, R. D. Lowe, R. D. Snook, “A model for CW laser induced mode-mismatched dual beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992); J. Shen, A. J. Soroka, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam image detection,” J. Appl. Phys. 78, 700–708 (1995).
[Crossref]

A. J. Twarowski, D. S. Kliger, “Multiphoton absorption spectra using thermal blooming. I. Theory,” Chem. Phys. 20, 253–258 (1977).
[Crossref]

J. Appl. Phys. (1)

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Opt. Commun. (1)

J. Slaby, “Calculation of probe beam diffraction at laser induced thermal lens,” Opt. Commun. 60, 133–138 (1986).
[Crossref]

Opt. Lett. (3)

Other (3)

H. S. Carlaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).

A. Yariv, Introduction to Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985).

S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis (Wiley, New York, 1996), Chap. 4.

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Figures (5)

Fig. 1
Fig. 1

Geometry that was used to define the theoretical photothermal lens signal. The probe laser enters from the left and is focused to a minimum spot radius of w0 at a distance z′ before the sample cell. It has a beam-waist radius of w1 at the sample and w2 at the pinhole aperture before the detector. The pinhole aperture is a distance d after the sample cell. Probe beams focused beyond the sample are indicated by negative z′.

Fig. 2
Fig. 2

Pulsed-laser-excited photothermal lens signal predicted from diffraction theory as a function of the probe-laser beam geometry. The excitation-laser beam waist was 20 µm in the sample. The minimum 632.8-nm probe-laser beam radius was 100 µm. The photothermal perturbation was small, and the signal was defined in the usual fashion.

Fig. 3
Fig. 3

(a) Far-field diffraction theory predictions for the pulsed-laser-excited photothermal lens signal as a function of the relative probe-laser beam-waist radius and focus position. The detector plane was at d = 10 m, the excitation beam radius was 100 µm in the sample, and the perturbation was small (10-5). (b) The conditions are the same as in (a). This view is given to allow inspection of the predicted surface. The line on the right-hand side is equivalent to that predicted from refractive optics.

Fig. 4
Fig. 4

Near-field diffraction theory predictions for the pulsed-laser-excited photothermal lens signal as a function of the relative probe-laser beam-waist radius and focus position. The detection plane is at d = 5 cm in this case. All other parameters are the same as those in Fig. 3.

Fig. 5
Fig. 5

Continuous-laser-excited photothermal lens signal predicted from diffraction theory. The signal magnitude is defined by [Φ(∞) - Φ(0)]/Φ(0). The excitation-laser beam waist is 20 µm and the 632.8-nm laser beam waist is 100 µm. The signal is apparently maximum at small detection-plane distances.

Tables (1)

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Table 1 Symbols Used in this Paper

Equations (40)

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Hr=2Qπw2exp-2r2w2,
Qr, t=2πδtαHrrdr
ρCpddtδTr, t-κ2δTr, t=Qr, t,
δTr, t=0t0Qr, t-tGr, r, tdr dt,
Gr, r, t=14πκtexp-r2+r24DTtI0rr2DTt
δTr, t=2αQYHπw2tρCpexp-2r2w2t,
tr, t=expiδϕr, texpikldndTδTr, t,  tr, texpkdndT2αlQYHπw2tρCpexp-2r2/w2t,
Er, z=E0q0qzexp-ikr22qz.
tr, t=expiδϕr=m=0iϕtmm!exp-2mr2w2t,  ϕt=-kdndT2αlQYHπρCpw2t.
Er, z=E0q0qzm=0iϕtmm!exp-ikr22qmt.
1qmt=1Rpz-i2kwm2,  1wm2=1wp2z+2mw2t,
Er, d=E0q0qzm=0iϕtmm!×qmtqmt+dexp-ikr22qmt+d.
Φt/Φ()1-ϕt8dkw2tz0,p2+z2+dzk2w4tz0,p2+z2+2dz+d2+8kw2td2z0,p+16d2z0,p2+z2,
limdΦt/Φ()1-ϕt×8kw2tzk2w4t+8kw2tz0,p+16z0,p2+z2.
Spulsedt=Φt-Φ()Φt2zfpulsed011+2w0,p2/w2+2t/tc2,
fpulsed10=dndT8αlYHQπw4ρCp.
Spulsed,optt=±dndT8αlQYHλpρCp×2w0,p2+w22w0,p2+w21+2t/tc2+2w0,p2+w22,
Spulsed,opt0=±dndT4αlQYHλpρCp2w0,p2+w2.
Spulsed,optt=dndT8αlQYHλpρCpw0,p2w22w0,p2+w21+2t/tc22w0,p4+2w2w0,p2+w4+22t/tc+1w2w0,p2+2w0,p4,
Spulsed,opt0=dndT8αlQYHλpρCpw0,p2w212w0,p2+w2.
limw0Spulsed,opt0=dndT4αlQYHλpρCpw2.
tr, t=expi0tδϕr, tdt=m=0imm!0tϕtexp-2r2w2tdtm.
Er, dE0q0qz+d+E0q0qz0tiϕtq1tq1t+d×exp-ikr22q1t+ddt.
E0, dE0q0qz+d+E0q0qztc2iϕ011+2t/tc×T-1qzT-i4/kw2qz+dT-i4/kw2dT.
ΦtΦ0-1tcϕ211+2t/tciqz+dqz×T-1qzT-i4/kw2qz+dT-i4/kw2+c.c.dT,
ΦtΦ01+tcϕ tan-1kw2z0,p2+z+d2+4d2z0,p4dz0,p2+z2+dz-tcϕ tan-1×kw21+2t/tcz0,p2+z+d2+4d2z0,p4dz0,p2+z2+dz.
Scw()=Φ0-Φ()Φ()11-tcϕ tan-14dz0,p2+z2+dzkw2z0,p2+z+d2+4d2z0,p-1.
-tcϕ=w2ρCp4κkdndT2αlΦ0YHπρCpw2=dndTαlΦ0YHλpκ=λλpz0fcw(),
limdΦtΦ01+tcϕ tan-1kw2+4z0,p4z-tcϕ tan-1kw21+2t/tc+4z0,p4z.
Φ()Φ0-1-tcϕ tan-14zkw2+2w0,p2.
Scwt=Φt-Φ()Φ()=tcϕ tan-14zkw21+2t/tc+4z01-tcϕ tan-14zkw2+4z0.
Scwt=tcϕ tan-12z/z0z/z02+3+2z/z02+1t/tc1-tcϕ tan-12z/z0z/z02+3.
Scw0=11-tcϕ tan-12z/z0z/z02+3-1.
Epulsed,opt=dndT4QYHλpρCp2w0,p2+w2.
EpulseddndT31/2QYHλw2ρCp.
Epulsed,opt=dndT8QYHλpρCpw0,p2w212w0,p2+w2.
Scw()=dndTαlΦ0YHλpκtan-14zk(w2+2w0,p2.
Ecw()=dndTπΦ0YH2λpκ.
Scw,max=Φ0-ϕ()ϕ()=π6tcϕ1-π6tcϕ.
Ecw=π6tcϕαl=dndTπΦ0YH6λκ,

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