## Numerical modeling of thermal refraction in liquids in the transient regime

Optics Express, Vol. 4, Issue 8, pp. 315-327 (1999)

http://dx.doi.org/10.1364/OE.4.000315

Acrobat PDF (272 KB)

### Abstract

We present the results of modeling of nanosecond pulse propagation in optically absorbing liquid media. Acoustic and electromagnetic wave equations must be solved simultaneously to model refractive index changes due to thermal expansion and/or electrostriction, which are highly transient phenomena on a nanosecond time scale. Although we consider situations with cylindrical symmetry and where the paraxial approximation is valid, this is still a computation-intensive problem, as beam propagation through optically thick media must be modeled. We compare the full solution of the acoustic wave equation with the approximation of instantaneous expansion (steady-state solution) and hence determine the regimes of validity of this approximation. We also find that the refractive index change obtained from the photo-acoustic equation overshoots its steady-state value once the ratio between the pulsewidth and the acoustic transit time exceeds a factor of unity.

© Optical Society of America

## 1. Introduction

16. D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High Dynamic Range Passive Optical Limiters,” Int. J. Nonlinear Opt. Phys. **2**, 483–501 (1993). [CrossRef]

17. P. Miles, “Bottleneck optical limiters: the optimal use of excited-state absorbers,” Appl. Opt. **33**, 6965–6979 (1994). [CrossRef] [PubMed]

^{2}), requiring numerical beam propagation algorithms to model limiter response. Modeling the nonlinear absorption itself is a nontrivial task; for instance, a system of rate equations must be solved to compute the excited-state absorption. Inclusion of thermally induced nonlinear index change makes the model even more complicated.

20. Jian-Gio Tian et al, “Position dispersion and optical limiting resulting from thermally induced nonlinearities in Chinese tea,” Appl. Opt. **32**, (1993). [CrossRef] [PubMed]

13. S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photo-acoustic and photo-refractive detection of small absorptions in liquids,” Opt. Comm. **34**, 199–204 (1980). [CrossRef]

15. P. Brochard, V. Grolier-Mazza, and R. Cabanel, “Thermal nonlinear refraction in dye solutions: a study of the transient regime,” J. Opt. Soc. Am. B **14**, 405–414 (1997) [CrossRef]

## 2. Thermally-induced index change

*n*) follow the temperature change, Δ

*T*. Hence, it is the shape of the beam, coupled with thermal diffusion, which dictates the temperature gradient. Heating the material in this case can be described by the following equation [23]:

*ρ*is the density of the medium,

*c*

_{p}is the specific heat at constant pressure and

*κ*the thermal conductivity. The source for the temperature change is

*Q*, the absorbed power of the laser beam per unit volume. In the simplest case where linear absorption is the primary absorption mechanism

*Q*is the product of the beam Irradiance,

*I*, and linear absorption coefficient,

*α*

_{L}, (

*Q*=

*α*

_{L}

*I*). The refractive index change is, in general, a function of temperature and density changes inside the material [14

14. J. -M. Heritier, “Electrostrictive limit and focusing effects in pulsed photoacoustic detection,” Opt. Comm. **44**, 267–272 (1983). [CrossRef]

*∂n*/

*∂ρ*)

_{T}describes the index changes due to thermal expansion or electrostriction, while (

*∂n*/

*∂T*)

_{ρ}is due to other temperature-dependent changes in index, which are of less importance in liquids [24]. On a time scale much longer than the acoustic transit time,

*τ*

_{ac}(here defined as

*w*/

*C*

_{s}where

*w*is the HW1/e

^{2}M beam size and

*C*

_{s}is the sound velocity), refractive index changes become linearly proportional to the change in temperature with the coefficient of proportionality called the thermo-optic coefficient. This occurs due to the fact that for later times Δ

*ρ*= (∂

*ρ*/∂

*T*)

_{p}Δ

*T*and (∂

*ρ*/∂

*T*)

_{p}becomes constant. In this regime the refractive index change is determined by thermal diffusion (Eq. 1.1). (Generally, the density changes due to electrostriction must be included, but in this work we concentrate on the thermally induced index change, as discussed later). The same long-term effect may also be observed for shorter pulses if the input laser has a high repetition rate (e.g. a modelocked laser pulse train). In this case the source term in Eq. (1.1) becomes a series of impulses causing local heating which subsequently undergoes thermal diffusion.

13. S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photo-acoustic and photo-refractive detection of small absorptions in liquids,” Opt. Comm. **34**, 199–204 (1980). [CrossRef]

14. J. -M. Heritier, “Electrostrictive limit and focusing effects in pulsed photoacoustic detection,” Opt. Comm. **44**, 267–272 (1983). [CrossRef]

*ρ*/∂

*T*)

_{p}is usually an order of magnitude smaller than in liquids so the effect is often masked by the electrostrictive effect [14

14. J. -M. Heritier, “Electrostrictive limit and focusing effects in pulsed photoacoustic detection,” Opt. Comm. **44**, 267–272 (1983). [CrossRef]

*n*/∂

*T*)

_{ρ}[24]. In liquids the effect of electrostriction is generally one or several orders of magnitude smaller than the thermal expansion (as long as the absorption of the liquid is significant), and therefore can often be neglected. From now on we concentrate on the index changes in liquids that occur on the nanosecond scale. This analysis is appropriate for the heating caused by, for example, single nanosecond pulses generated by a Q-switched Nd:YAG laser.

*μm*, for typical values of the sound velocity in liquids (1~2 μm/ns) the acoustic wave generated by the front of the pulse traverses the beam and creates an index change affecting the tail end of the pulse. If the pulse width,

*τ*

_{p}, is longer than the acoustic transit time

*τ*

_{p}>

*w*/

*C*

_{s}, we simplify the numerical modeling of the photo-acoustic effect by parameterizing the index change close to the propagation axis [7

7. P. R. Longaker and M. M. Litvak, “Perturbation of the refractive index of absorbing media by a pulsed laser beam,” J. Appl. Phys. **40**, 4033–4041 (1969) [CrossRef]

*dn*/

*dT*) =

*γ*

^{e}

*β*/(2

*n*) is the thermo-optic coefficient. Equation (1.7) is a commonly used approximation called the thermal lensing effect [2

2. S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-action of laser beams,” IEEE J. Quantum Electron. QE-**4**, 568–575 (1968). [CrossRef]

*dn*/

*dT*) was calculated in this approximation for different liquids [20–22

20. Jian-Gio Tian et al, “Position dispersion and optical limiting resulting from thermally induced nonlinearities in Chinese tea,” Appl. Opt. **32**, (1993). [CrossRef] [PubMed]

## 3. Beam Propagation Algorithm

*E*⃗(

*r*

_{⊥},

*z*,

*t*) and

*P*⃗(

*r*

_{⊥},

*z*,

*t*) are the electric field and the medium polarization. Making the Slowly Varying Envelope Approximation and assuming that group velocity dispersion can be neglected, this equation can be greatly simplified; and for slow (i.e. large F-number) systems rewritten in a scalar paraxial form:

*E*(

*r*

_{⊥},

*z*,

*t*) = ψ(

*r*

_{⊥},

*z*,

*t*)

*e*

^{jωt-jkz}. Here

*r*

_{⊥}denote the transverse Laplace operator and radial spatial coordinate, while

*k*=

*n*

_{0}

*k*

_{0}=

*n*

_{0}

*ω*/

*c*is the wave vector in the media with linear index of refraction

*α*

_{L}= -(

*k*

_{0}/

*n*

_{0})· Im{

*χ*

_{L}}.

*χ*

_{NL}(

*r*

_{⊥},

*z*,

*t*) is the nonlinear susceptibility of the material, which may consist of instantaneous and cumulative parts:

*z*, using the formal solution to Eq. (2.2). The unitary, finite difference Crank-Nicholson scheme was chosen [26] for its high efficiency when dealing with propagation through distances of more than a few Rayleigh ranges [27

27. D. Kovsh, S Yang, D. J. Hagan, and E. W. Van Stryland “Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements,” Proc. SPIE **3472**, 163–177 (1998). [CrossRef]

## 4. Results and discussion

^{-4}K

^{-1}[15

15. P. Brochard, V. Grolier-Mazza, and R. Cabanel, “Thermal nonlinear refraction in dye solutions: a study of the transient regime,” J. Opt. Soc. Am. B **14**, 405–414 (1997) [CrossRef]

*w*

_{0}is (i) 6 μm and then (ii) 30 μm (HW1/e

^{2}M of irradiance). The acoustic transit times in cases (i) and (ii) are

*τ*

_{ac}= 4 ns and 20 ns respectively. The sample is chosen to be 1 mm thick in each case. Figure 1 shows an animation of the dynamics of the thermally induced refraction index change in case (i) as the laser pulse passes through the sample. We first include modeling of the acoustic equation (1.6) coupled with propagation and then apply the approximation (1.7), i.e. thermal lensing. Although the approximation (1.7) ignores the small index disturbances on the wings of the pulse, which are due to the acoustic wave propagation, it correctly predicts the changes of the refractive index close to the axis where most of the beam energy is concentrated. Moreover, the thickness of the sample was chosen to be several Rayleigh ranges, thus complicating the modeling since beam diffraction is included along with the effect of nonlinear self-action. In order to compare the results obtained with and without the approximation (1.7) we also look at the near and far field profiles of the beam. As seen in Fig. 2 the near and far field radial fluence distributions remain the same if computed with and without the approximation (1.7). This is as expected, since the pulse width in case (i) is equal to the acoustic transit time.

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

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

*τ*

_{p}= 10 ns (HW1/eM) and beam waists of 6 μm and 30 μm, we define the ratio between the pulsewidth and transit acoustic time (

*τ*

_{p}/

*τ*

_{ac}) to be 2.5 (Fig. 5) and 0.5 (Fig. 6) respectively. Although the use of thermal approximation results in a good agreement for the former case it clearly produces the erroneous result for the later.

*τ*

_{p}/

*τ*

_{ac}comparing the peak-to-valley change in transmittance Δ

*T*

_{P-V}obtained with and without this approximation. For two values of the pulse width (

*τ*

_{p}= 4 and 8 ns HW1/eM), we varied

*τ*

_{p}/

*τ*

_{ac}by changing the sound velocity. Changing

*τ*

_{p}/

*τ*

_{ac}in this way is convenient as it allows us to keep the geometry of the experiment the same. For both pulse widths the value of on-axis fluence at focus and therefore the closed-aperture Z-scan signal are the same [29

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

*τ*

_{p}/

*τ*

_{ac}. This dependence remains unchanged even after we varied the value of input energy to have Δ

*T*

_{P-V}in the range of 4–25% producing the same curve (Fig. 7). The estimation of the thermally induced refractive index change using Eq. (1.7) produces accurate results for values of the ratio between the HW1/eM pulse width and acoustic transit time larger than unity (or > 1.6 if FWHM value of the pulse width is used). Hence,

*τ*

_{p}/

*τ*

_{ac}= 1 defines the limit of validity of Eq. (1.7) beyond which the acoustic wave propagation must be included in the analysis. Figure 8 shows the on-axis distribution of the thermally induced refractive index change as a function of time. Note that with the increase of the parameter

*τ*

_{p}/

*τ*

_{ac}the value of the index change (and consequently Z-scan signal in Fig. 7) first reaches the value predicted by the steady state solution (1.7) and then surpasses this value (Fig. 8). This is because the index change initially overshoots the steady state solution of this equation. This can also be seen in Fig. 3. Although the solution of Eq. (1.6) reaches the steady state value of Δ

*n*for later times, it gives a larger index change for the central parts of the pulse where the irradiance is the largest. This effect was mentioned in the Ref. [15

15. P. Brochard, V. Grolier-Mazza, and R. Cabanel, “Thermal nonlinear refraction in dye solutions: a study of the transient regime,” J. Opt. Soc. Am. B **14**, 405–414 (1997) [CrossRef]

20. Jian-Gio Tian et al, “Position dispersion and optical limiting resulting from thermally induced nonlinearities in Chinese tea,” Appl. Opt. **32**, (1993). [CrossRef] [PubMed]

13. S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photo-acoustic and photo-refractive detection of small absorptions in liquids,” Opt. Comm. **34**, 199–204 (1980). [CrossRef]

16. D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High Dynamic Range Passive Optical Limiters,” Int. J. Nonlinear Opt. Phys. **2**, 483–501 (1993). [CrossRef]

17. P. Miles, “Bottleneck optical limiters: the optimal use of excited-state absorbers,” Appl. Opt. **33**, 6965–6979 (1994). [CrossRef] [PubMed]

16. D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High Dynamic Range Passive Optical Limiters,” Int. J. Nonlinear Opt. Phys. **2**, 483–501 (1993). [CrossRef]

## 5. Conclusions

*τ*

_{p}/

*τ*

_{ac}> 1), the thermally induced refractive index changes essentially instantaneously throughout the pulse. However, our results show that even in this case, the approximation may yield small (~10%) errors in estimating the index change due to an “overshoot” of the thermally induced density change. This knowledge helps us to understand the results of experiments performed to measure the thermal optical and absorptive characteristics of liquid materials.

## Acknowledgments

## References and links

1. | J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. |

2. | S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-action of laser beams,” IEEE J. Quantum Electron. QE- |

3. | C. K. N. Patel and A. C. Tam, “Pulsed optoacoustic spectroscopy of condensed matter,” Rev. Mod. Phys. |

4. | J. N. Hayes, “Thermal blooming of laser beams in fluids,” Appl. Opt. |

5. | A. J. Twarowski and D. S. Kliger, “Multiphoton absorption spectra using thermal blooming. I. Theory,” Chem. Phys. |

6. | S. J. Sheldon, L. V. Knight, and J. M. Thorne, “Laser-induced thermal lens effect: a new theoretical model,” Appl. Opt. |

7. | P. R. Longaker and M. M. Litvak, “Perturbation of the refractive index of absorbing media by a pulsed laser beam,” J. Appl. Phys. |

8. | Gu Liu, “Theory of the photoacoustic effect in condensed matter,” Appl. Opt. |

9. | C. A. Carter and J. M. Harris, “Comparison of models describing the thermal lens effect,” Appl. Opt. |

10. | A. M. Olaizola, G. Da Costa, and J. A. Castillo, “Geometrical interpretation of a laser-induced thermal lens,” Opt. Eng. |

11. | F. Jurgensen and W. Schroer, “Studies on the diffraction image of a thermal lens,” Appl. Opt. |

12. | S. Wu and N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. |

13. | S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photo-acoustic and photo-refractive detection of small absorptions in liquids,” Opt. Comm. |

14. | J. -M. Heritier, “Electrostrictive limit and focusing effects in pulsed photoacoustic detection,” Opt. Comm. |

15. | P. Brochard, V. Grolier-Mazza, and R. Cabanel, “Thermal nonlinear refraction in dye solutions: a study of the transient regime,” J. Opt. Soc. Am. B |

16. | D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High Dynamic Range Passive Optical Limiters,” Int. J. Nonlinear Opt. Phys. |

17. | P. Miles, “Bottleneck optical limiters: the optimal use of excited-state absorbers,” Appl. Opt. |

18. | T. Xia, D. J. Hagan, A. Dogariu, A. A. Said, and E. W. Van Stryland, “Optimization of optical limiting devices based on excited-state absorption,” Appl. Opt. |

19. | T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B |

20. | Jian-Gio Tian et al, “Position dispersion and optical limiting resulting from thermally induced nonlinearities in Chinese tea,” Appl. Opt. |

21. | Y. M. Cheung and S. K. Gayen, “Optical nonlinearities of tea studied by Z-scan and four-wave mixing techniques,” J. Opt. Soc. Am. B |

22. | J. Castillo and V. P. Kozich et al, “Thermal lensing resulting from one- and two-photon absorption studied with a two-color time-resolved Z-scan,” Opt. Lett. |

23. | D. Landau and E. M. Lifshitz, |

24. | T. Xia, “Modeling and experimental studies of nonlinear optical self-action,” Ph.D. thesis, Univ. of Central Florida (1994). |

25. | R. W. Boyd, |

26. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

27. | D. Kovsh, S Yang, D. J. Hagan, and E. W. Van Stryland “Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements,” Proc. SPIE |

28. | D. Kovsh, S. Yang, D. Hagan, and E. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” submitted to Applied Optics. |

29. | M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. |

**OCIS Codes**

(190.4870) Nonlinear optics : Photothermal effects

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.5940) Nonlinear optics : Self-action effects

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 15, 1999

Published: April 12, 1999

**Citation**

Dmitriy Kovsh, David Hagan, and Eric Van Stryland, "Numerical modeling of thermal refraction inliquids in the transient regime," Opt. Express **4**, 315-327 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-8-315

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### References

- J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples," J. Appl. Phys. 36, 3-8 (1965). [CrossRef]
- S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov and R. V. Khokhlov, "Thermal self-action of laser beams," IEEE J. Quantum Electron. QE-4, 568-575 (1968). [CrossRef]
- C. K. N. Patel and A. C. Tam, "Pulsed optoacoustic spectroscopy of condensed matter," Rev. Mod. Phys. 53, 517-550 (1981). [CrossRef]
- J. N. Hayes, "Thermal blooming of laser beams in fluids," Appl. Opt. 11, 455-461 (1972). [CrossRef] [PubMed]
- A. J. Twarowski and D. S. Kliger, "Multiphoton absorption spectra using thermal blooming. I. Theory," Chem. Phys. 20, 251-258 (1977).
- S. J. Sheldon, L. V. Knight and J. M. Thorne, "Laser-induced thermal lens effect: a new theoretical model," Appl. Opt. 21, 1663-1669 (1982). [CrossRef] [PubMed]
- P. R. Longaker and M. M. Litvak, "Perturbation of the refractive index of absorbing media by a pulsed laser beam," J. Appl. Phys. 40, 4033-4041 (1969). [CrossRef]
- Gu Liu, "Theory of the photoacoustic effect in condensed matter," Appl. Opt. 21, 955-960 (1982). [CrossRef] [PubMed]
- C. A. Carter and J. M. Harris, "Comparison of models describing the thermal lens effect," Appl. Opt. 23, 476-481 (1984). [CrossRef] [PubMed]
- A. M. Olaizola, G. Da Costa and J. A. Castillo, "Geometrical interpretation of a laser-induced thermal lens," Opt. Eng. 32, 1125-1130 (1993). [CrossRef]
- F. Jurgensen and W. Schroer, "Studies on the diffraction image of a thermal lens," Appl. Opt. 34, 41- 50 (1995). [CrossRef] [PubMed]
- S. Wu and N. J. Dovichi, "Fresnel diffraction theory for steady-state thermal lens measurements in thin films," J. Appl. Phys. 67, 1170-1182 (1990). [CrossRef]
- S. R. J. Brueck, H. Kildal and L. J. Belanger, "Photo-acoustic and photo-refractive detection of small absorptions in liquids," Opt. Comm. 34, 199-204 (1980). [CrossRef]
- J. -M. Heritier, "Electrostrictive limit and focusing effects in pulsed photoacoustic detection," Opt. Comm. 44, 267-272 (1983). [CrossRef]
- P. Brochard, V. Grolier-Mazza and R. Cabanel, "Thermal nonlinear refraction in dye solutions: a study of the transient regime," J. Opt. Soc. Am. B 14, 405-414 (1997) [CrossRef]
- D. J. Hagan, T. Xia, A. A. Said, T. H. Wei and E. W. Van Stryland, "High Dynamic Range Passive Optical Limiters," Int. J. Nonlinear Opt. Phys. 2, 483-501 (1993). [CrossRef]
- P. Miles, "Bottleneck optical limiters: the optimal use of excited-state absorbers," Appl. Opt. 33, 6965-6979 (1994). [CrossRef] [PubMed]
- T. Xia, D. J. Hagan, A. Dogariu, A. A. Said and E. W. Van Stryland, "Optimization of optical limiting devices based on excited-state absorption," Appl. Opt. 36, 4110-4122 (1997). [CrossRef] [PubMed]
- T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry and D. R. Coulter, "Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines," Appl. Phys. B 54, 46-51 (1992). [CrossRef]
- Jian-Gio Tian et al, "Position dispersion and optical limiting resulting from thermally induced nonlinearities in Chinese tea," Appl. Opt. 32, (1993). [CrossRef] [PubMed]
- Y. M. Cheung and S. K. Gayen, "Optical nonlinearities of tea studied by Z-scan and four-wave mixing techniques," J. Opt. Soc. Am. B 11, 636-643 (1994). [CrossRef]
- J. Castillo, V. P. Kozich et al, "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]
- D. Landau and E. M. Lifshitz, Course of theoretical physics. Volume 6. Fluid mechanics, (Pergamon Press).
- T. Xia, "Modeling and experimental studies of nonlinear optical self-action," Ph.D. thesis, Univ. of Central Florida (1994).
- R. W. Boyd, Nonlinear optics, (Academic Press, Inc. 1992).
- W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical recipes. The art of scientific computing, (Cambridge University Press, 1986).
- D. Kovsh, S Yang, D. J. Hagan and E. W. Van Stryland; "Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements," Proc. SPIE 3472, 163- 177 (1998). [CrossRef]
- D. Kovsh, S. Yang, D. Hagan and E. Van Stryland, "Nonlinear optical beam propagation for optical limiting," submitted to Applied Optics.
- M. Sheik-Bahae, A. A. Said and E. W. Van Stryland, "High-sensitivity, single-beam n2 measurements," Opt. Lett. 14, 955-957 (1989). [CrossRef] [PubMed]

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