Abstract

A geometrical algorithm for optical ray tracing is used to trace the trajectories of probe laser beam rays in pulsed photothermal deflection spectroscopy (PDS) of a static gas. Rays are traced as a function of various parameters such as time, absorbed pump beam energy, radius, and pulse duration. The exit angle is also computed and compared with the deflection angle calculated analytically with very good agreement for a pump pulse duration shorter than the thermal diffusion time. For a long pump pulse duration, the analytical expression is inaccurate due to the neglect of thermal diffusion during the pump pulse. Probe beam ray crossing is examined and found to be unimportant for most common PDS situations. The saturation of the PD signal is found for high absorbed pump pulse energy; this is due to excessive heating of the absorber. Finally, the maximum probe beam deflection occurs when the deflection is measured at a time corresponding to the end of the pump pulse duration.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. A. Weimer, N. J. Dovichi, “Time-Resolved Crossed-Beam Thermal Lens Measurement as a Nonintrusive Probe of Flow Velocity,” Appl. Opt. 24, 2981 (1985).
    [CrossRef] [PubMed]
  2. H. Sontag, A. Tarn, “Time Resolved Flow-Velocity and Concentration Measurements using a Traveling Thermal Lens,” Opt. Lett. 10, 436 (1985).
    [CrossRef] [PubMed]
  3. J. A. Sell, “Quantitative Photothermal Deflection Spectroscopy in a Flowing Stream of Gas,” Appl. Opt. 23, 1586 (1984).
    [CrossRef] [PubMed]
  4. J. A. Sell, “Gas Velocity Measurements Using Photothermal Deflection Spectroscopy,” Appl. Opt. 24, 3725 (1985).
    [CrossRef] [PubMed]
  5. J. A. Sell, R. J. Cattolica, “Linear Imaging of Gas Velocities Using the Photothermal Deflection Effect,” Appl. Opt. 25, 1420 (1986).
    [CrossRef] [PubMed]
  6. C. J. Dasch, J. A. Sell, “Velocimetry in Laminar and Turbulence & Flows Using the Photothermal Deflection Effect with a Transient Grating,” Opt. Lett. 11, 603 (1986).
    [CrossRef] [PubMed]
  7. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal Deflection Spectroscopy and Detection,” Appl. Opt. 20, 1333 (1981).
    [CrossRef] [PubMed]
  8. L. Montagnino, “Ray Tracing in Inhomogeneous Media,” J. Opt. Soc. Am. 58, 1667 (1982).
    [CrossRef]
  9. D. T. Moore, J. M. Stagaman, “Ray Tracing in Anamorphic Gradient-Index Media,” Appl. Opt. 21, 999 (1982).
    [CrossRef] [PubMed]
  10. N. C. Schoen, “Ray Tracing Analysis for Media with Nonhomogeneous Indices of Refraction,” Appl. Opt. 21, 3329 (1982).
    [CrossRef] [PubMed]
  11. W. H. Southwell, “Ray Tracing in Gradient-Index Media,” J. Opt. Soc. Am. 72, 908 (1982).
    [CrossRef]
  12. R. W. Lewis, GM Research Laboratories; personal communication.
  13. R. W. Lewis, “A Quantitative Shadow Method for Measuring a One-Dimensional Index of Refraction Field,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 1 (1982).
  14. G. P. Montgomery, D. L. Reuss, “Effect of Refraction on Axisymmetric Flame Temperatures Measured by Holographic Interferometry,” Appl. Opt. 21, 1373 (1982).
    [CrossRef] [PubMed]
  15. M. V. Klein, Optics (Wiley, New York, 1970).
  16. A. C. Boccara, D. Fournier, W. Jackson, N. M. Aber, “Sensitive Photothermal Deflection Technique for Measuring Absorption in Optically Thin Media,” Opt. Lett. 5, 377 (1980).
    [CrossRef] [PubMed]
  17. L. C. Aamodt, J. C. Murphy, “Thermal Effects in Photothermal Spectroscopy and Photothermal Imaging,” J. Appl. Phys. 54, 581 (1983).
    [CrossRef]
  18. G. R. Long, S. E. Bialkowski, “Error Reduction in Pulsed Laser Photothermal Deflection Spectrometry,” Anal. Chem. 58, 80 (1986).
    [CrossRef] [PubMed]
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 30.
  20. R. W. Lewis, R. E. Teets, J. A. Sell, T. A. Seder, “Temperature Measurements in a Laser Heated Gas by Quantitative Shadowgraphy,” Appl. Opt. submitted.

1986 (3)

1985 (3)

1984 (1)

1983 (1)

L. C. Aamodt, J. C. Murphy, “Thermal Effects in Photothermal Spectroscopy and Photothermal Imaging,” J. Appl. Phys. 54, 581 (1983).
[CrossRef]

1982 (6)

1981 (1)

1980 (1)

Aamodt, L. C.

L. C. Aamodt, J. C. Murphy, “Thermal Effects in Photothermal Spectroscopy and Photothermal Imaging,” J. Appl. Phys. 54, 581 (1983).
[CrossRef]

Aber, N. M.

Amer, N. M.

Bialkowski, S. E.

G. R. Long, S. E. Bialkowski, “Error Reduction in Pulsed Laser Photothermal Deflection Spectrometry,” Anal. Chem. 58, 80 (1986).
[CrossRef] [PubMed]

Boccara, A. C.

Cattolica, R. J.

Dasch, C. J.

Dovichi, N. J.

Fournier, D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 30.

Jackson, W.

Jackson, W. B.

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970).

Lewis, R. W.

R. W. Lewis, “A Quantitative Shadow Method for Measuring a One-Dimensional Index of Refraction Field,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 1 (1982).

R. W. Lewis, R. E. Teets, J. A. Sell, T. A. Seder, “Temperature Measurements in a Laser Heated Gas by Quantitative Shadowgraphy,” Appl. Opt. submitted.

R. W. Lewis, GM Research Laboratories; personal communication.

Long, G. R.

G. R. Long, S. E. Bialkowski, “Error Reduction in Pulsed Laser Photothermal Deflection Spectrometry,” Anal. Chem. 58, 80 (1986).
[CrossRef] [PubMed]

Montagnino, L.

Montgomery, G. P.

Moore, D. T.

Murphy, J. C.

L. C. Aamodt, J. C. Murphy, “Thermal Effects in Photothermal Spectroscopy and Photothermal Imaging,” J. Appl. Phys. 54, 581 (1983).
[CrossRef]

Reuss, D. L.

Schoen, N. C.

Seder, T. A.

R. W. Lewis, R. E. Teets, J. A. Sell, T. A. Seder, “Temperature Measurements in a Laser Heated Gas by Quantitative Shadowgraphy,” Appl. Opt. submitted.

Sell, J. A.

Sontag, H.

Southwell, W. H.

Stagaman, J. M.

Tarn, A.

Teets, R. E.

R. W. Lewis, R. E. Teets, J. A. Sell, T. A. Seder, “Temperature Measurements in a Laser Heated Gas by Quantitative Shadowgraphy,” Appl. Opt. submitted.

Weimer, W. A.

Anal. Chem. (1)

G. R. Long, S. E. Bialkowski, “Error Reduction in Pulsed Laser Photothermal Deflection Spectrometry,” Anal. Chem. 58, 80 (1986).
[CrossRef] [PubMed]

Appl. Opt. (8)

J. Appl. Phys. (1)

L. C. Aamodt, J. C. Murphy, “Thermal Effects in Photothermal Spectroscopy and Photothermal Imaging,” J. Appl. Phys. 54, 581 (1983).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (3)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. W. Lewis, “A Quantitative Shadow Method for Measuring a One-Dimensional Index of Refraction Field,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 1 (1982).

Other (4)

M. V. Klein, Optics (Wiley, New York, 1970).

R. W. Lewis, GM Research Laboratories; personal communication.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 30.

R. W. Lewis, R. E. Teets, J. A. Sell, T. A. Seder, “Temperature Measurements in a Laser Heated Gas by Quantitative Shadowgraphy,” Appl. Opt. submitted.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Optical rays traced as a function of time. The initial rays are horizontal at y0 = 0.01 cm, the pump pulse radius is 0.01 cm, the absorbed pump pulse energy is 3.0 × 10−6 cm−1 J and lasts for 1 ms, and the value of δ0 [see Eq. (9)] is 3.0 × 10−4.

Fig. 2
Fig. 2

Magnified view of one of the ray trajectories from Fig. 1. Here t = 1 ms, and all other parameters are the same as in the caption for Fig. 1. Note the more sensitive scales in this plot.

Fig. 3
Fig. 3

Ray trajectories for three rays which initially are horizontal at y0 = 0.01, 0.02, and 0.03 cm. The pump beam radius is 0.01 cm, the absorbed pump pulse energy is 3.0 × 10−6 cm−1 J and lasts for 1 ms, and the time at which the rays are observed (observation time) is 1 ms. Note that the y offset for the rays at 0.02 and 0.03 cm is arbitrary in this figure so that the position of the ray crossing is best determined from Eq. (21) in the text rather than graphically.

Fig. 4
Fig. 4

Exit angle as a function of time for a ray which initially is horizontal at y0 = 0.01 cm. The pump radius is 0.01 cm, and the absorbed pump pulse energy is 3.0 × 10−6 cm−1 J and lasts for 1 μs. The deflection angle (solid line) computed from Eq. (13) times 2 is indistinguishable from the exit angle (dotted line).

Fig. 5
Fig. 5

Exit angle as a function of pump beam radius for a ray which initially is horizontal at y0 = 0.01 cm. The absorbed pump pulse is 3.0 × 10−6 cm−1 J and lasts for 1 μs. The observation time is 1 ms. The deflection angle (solid line) computed from Eq. (13) times 2 is indistinguishable from the exit angle (dotted line).

Fig. 6
Fig. 6

Exit angle (dotted line) and deflection angle (solid line) as a function of absorbed pump pulse energy for a ray which initially is horizontal at y0 = 0.01 cm. The pump radius is 0.01 cm and lasts for 1 μs. The observation time is 1 ms.

Fig. 7
Fig. 7

Exit angle as a function of time for three rays which initially are horizontal at y0 = 0.01, 0.02, and 0.03 cm. The pump radius is 0.01 cm, and the absorbed pump pulse energy is 3.0 × 10−6 cm−1 J and lasts for 1 ms.

Fig. 8
Fig. 8

Exit angle as a function of pump beam radius for three rays which initially are horizontal at y0 = 0.01, 0.02, and 0.03 cm. The absorbed pump pulse energy is 3.0 × 10−6 cm−1 J and lasts for 1 ms. The observation time is 1 ms.

Fig. 9
Fig. 9

Exit angle as a function of absorbed pump pulse energy for three rays which initially are horizontal at y0 = 0.01, 0.02, and 0.03 cm. The pump radius is 0.01 cm and lasts for 1 ms. The observation time is 1 ms.

Fig. 10
Fig. 10

Exit angle as a function of pump pulse duration for four rays which initially are horizontal at y0 = 0.005, 0.01, 0.02, and 0.03 cm. The pump radius is 0.01 cm, and the absorbed pump pulse energy is 3.0 × 10−6 cm−1 J. The observation time is 1 ms.

Fig. 11
Fig. 11

Same as Fig. 10, but the observation time is 0.1 ms.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

n = d d s ( n τ ) ,
τ = d r d s ,
n = d d s ( n d r d s ) .
Δ r = r 2 - r 1 = τ 1 Δ s .
r 2 = r 1 + τ 1 Δ s .
n d s = d ( n τ ) ,
n Δ s = Δ ( n τ ) = n ( r 2 ) τ 2 - n ( r 1 ) τ 1 ,
τ 2 = n ( r 1 ) n ( r 2 ) τ 1 + Δ s n ( r 2 ) n .
n = 1 + δ = 1 + δ 0 ( T 0 T ) .
T = T 0 + 2 α c D E 0 π λ ( w 0 2 + 8 D t ) exp ( - 2 r 2 w 0 2 + 8 D t )
n x = n T T x = δ T T r r x ,
n x = - δ 0 ( T 0 T 2 ) T r x r ,
ϕ = - 1 n d n d T 4 α c D E 0 ( v t - a ) 2 π λ ( w 0 2 + 8 D t ) 3 / 2 exp [ - 2 ( v t - a ) 2 ( w 0 2 + 8 D t ) ]
T r = α c E 0 π λ Δ t 2 r [ exp ( - 2 r 2 w 0 2 + 8 D t ) - exp ( - 2 r 2 w 2 ) ] ,
w 2 = { w 0 2 if 0 t Δ t ; w 0 2 + 8 D ( t - Δ t ) , t > Δ t .
T = T 0 + α c E 0 π λ Δ t 2 r 0 [ exp ( - β r 2 ) - exp ( - γ r 2 ) ] d r r ,
β = 2 w 0 2 + 8 D t ,
γ = 2 w 2 .
η = β r 2 ,
x 0 exp ( - η ) d η η .
l = - ( d ϕ d y ) - 1 .
ρ C [ T ( r , t ) t ] = λ 2 T ( r , t ) + Q ( r , t ) ,
ρ t = - ( · ρ v ) .

Metrics