Abstract

We report on a pump–probe mode-mismatched thermal-lens Z-scan method for the measurement of small absorption coefficients. In this method the pump light beam is focused into the sample to induce a thermal lens, which is tested by a collimated probe-light beam. Comparison between mode-matched and mode-mismatched Z-scan schemes is performed by use of a Fresnel-diffraction approximation model. This method is used to measure the absorption of distilled water and optical glass in the near-infrared and visible regions of the spectrum.

© 2002 Optical Society of America

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References

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  1. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity single beam n2 measurement,” Opt. Lett. 14, 995–957 (1989).
    [CrossRef]
  2. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Van Stryland, “Sensitivity measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  3. H. Ma, A. S. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
    [CrossRef]
  4. M. Sheik-Bahae, J. Wang, R. De Salvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two-color Z-scan,” Opt. Lett. 17, 258–260 (1992).
    [CrossRef] [PubMed]
  5. J. Castillo, V. P. Kozich, and A. Marcano O., “Thermal lensing due to one- and two-photon absorption studied with two-color time-resolved Z-scan,” Opt. Lett. 19, 171–173 (1994).
    [CrossRef]
  6. V. P. Kozich, A. Marcano O., F. E. Hernandez, and 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]
  7. N. J. Dovichi, “Thermo-optical spectrophotometries in analytical chemistry,” Crit. Rev. Anal. Chem. 17, 357–423 (1987).
    [CrossRef]
  8. R. D. Snook and R. D. Lowe, “Thermal lens spectrometry, a review,” Analyst 120, 2052–2068 (1995).
    [CrossRef]
  9. M. Terazima and N. Hirota, “Population lens in thermal lens spectroscopy,” J. Phys. Chem. 96, 7147–7150 (1992).
    [CrossRef]
  10. M. Terazima, T. Hara, and N. Hirota, “Population lens in thermal lens spectrometry. 2. Probe wavelength dependence and a new method for subtracting the transient absorption from the thermal lens signal,” J. Chem. Phys. 97, 10554–10560 (1993).
    [CrossRef]
  11. 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, 52–63 (1990).
    [CrossRef] [PubMed]
  12. J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurement of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
    [CrossRef]
  13. J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
    [CrossRef]
  14. J. Shen, A. J. Soroka, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam profile image detection,” J. Appl. Phys. 78, 700–708 (1995).
    [CrossRef]
  15. A. Marcano O., C. Loper, and N. Melikechi, “High sensitivity absorption measurement in water and glass samples using a mode mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78, 3415–3417 (2001).
    [CrossRef]
  16. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 717.
  17. R. M. Pope and E. S. Fry, “Absorption spectrum (380–700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36, 8710–8723 (1997).
    [CrossRef]
  18. L. Levi, Applied Optics, Vol. 2 of Wiley Series in Pure and Applied Optics (Wiley, New York, 1980).
  19. N. J. Dovichi and J. M. Harris, “Time-resolved thermal lens calorimetry,” Anal. Chem. 53, 106–109 (1981).
    [CrossRef]
  20. N. J. Dovichi and J. M. Harris, “Laser induced thermal lens effect for calorimetric trace analysis,” Anal. Chem. 51, 728–731 (1979).
    [CrossRef]

2001 (1)

A. Marcano O., C. Loper, and N. Melikechi, “High sensitivity absorption measurement in water and glass samples using a mode mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78, 3415–3417 (2001).
[CrossRef]

1997 (1)

1995 (2)

R. D. Snook and R. D. Lowe, “Thermal lens spectrometry, a review,” Analyst 120, 2052–2068 (1995).
[CrossRef]

J. Shen, A. J. Soroka, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam profile image detection,” J. Appl. Phys. 78, 700–708 (1995).
[CrossRef]

1994 (3)

1993 (1)

M. Terazima, T. Hara, and N. Hirota, “Population lens in thermal lens spectrometry. 2. Probe wavelength dependence and a new method for subtracting the transient absorption from the thermal lens signal,” J. Chem. Phys. 97, 10554–10560 (1993).
[CrossRef]

1992 (3)

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

M. Terazima and N. Hirota, “Population lens in thermal lens spectroscopy,” J. Phys. Chem. 96, 7147–7150 (1992).
[CrossRef]

M. Sheik-Bahae, J. Wang, R. De Salvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two-color Z-scan,” Opt. Lett. 17, 258–260 (1992).
[CrossRef] [PubMed]

1991 (1)

H. Ma, A. S. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

1990 (2)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Van Stryland, “Sensitivity measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

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, 52–63 (1990).
[CrossRef] [PubMed]

1989 (1)

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity single beam n2 measurement,” Opt. Lett. 14, 995–957 (1989).
[CrossRef]

1987 (1)

N. J. Dovichi, “Thermo-optical spectrophotometries in analytical chemistry,” Crit. Rev. Anal. Chem. 17, 357–423 (1987).
[CrossRef]

1981 (1)

N. J. Dovichi and J. M. Harris, “Time-resolved thermal lens calorimetry,” Anal. Chem. 53, 106–109 (1981).
[CrossRef]

1979 (1)

N. J. Dovichi and J. M. Harris, “Laser induced thermal lens effect for calorimetric trace analysis,” Anal. Chem. 51, 728–731 (1979).
[CrossRef]

Baesso, M. L.

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurement of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[CrossRef]

Castillo, J.

Castillo, J. A.

de Araujo, C. B.

H. Ma, A. S. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

De Salvo, R.

Dovichi, N. J.

N. J. Dovichi, “Thermo-optical spectrophotometries in analytical chemistry,” Crit. Rev. Anal. Chem. 17, 357–423 (1987).
[CrossRef]

N. J. Dovichi and J. M. Harris, “Time-resolved thermal lens calorimetry,” Anal. Chem. 53, 106–109 (1981).
[CrossRef]

N. J. Dovichi and J. M. Harris, “Laser induced thermal lens effect for calorimetric trace analysis,” Anal. Chem. 51, 728–731 (1979).
[CrossRef]

Fry, E. S.

Gomes, A. S.

H. Ma, A. S. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

Hagan, D.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Van Stryland, “Sensitivity measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Hagan, D. J.

Hara, T.

M. Terazima, T. Hara, and N. Hirota, “Population lens in thermal lens spectrometry. 2. Probe wavelength dependence and a new method for subtracting the transient absorption from the thermal lens signal,” J. Chem. Phys. 97, 10554–10560 (1993).
[CrossRef]

Harris, J. M.

N. J. Dovichi and J. M. Harris, “Time-resolved thermal lens calorimetry,” Anal. Chem. 53, 106–109 (1981).
[CrossRef]

N. J. Dovichi and J. M. Harris, “Laser induced thermal lens effect for calorimetric trace analysis,” Anal. Chem. 51, 728–731 (1979).
[CrossRef]

Hernandez, F. E.

Hirota, N.

M. Terazima, T. Hara, and N. Hirota, “Population lens in thermal lens spectrometry. 2. Probe wavelength dependence and a new method for subtracting the transient absorption from the thermal lens signal,” J. Chem. Phys. 97, 10554–10560 (1993).
[CrossRef]

M. Terazima and N. Hirota, “Population lens in thermal lens spectroscopy,” J. Phys. Chem. 96, 7147–7150 (1992).
[CrossRef]

Kozich, V. P.

Loper, C.

A. Marcano O., C. Loper, and N. Melikechi, “High sensitivity absorption measurement in water and glass samples using a mode mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78, 3415–3417 (2001).
[CrossRef]

Lowe, R. D.

R. D. Snook and R. D. Lowe, “Thermal lens spectrometry, a review,” Analyst 120, 2052–2068 (1995).
[CrossRef]

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

Ma, H.

H. Ma, A. S. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

Marcano O., A.

Melikechi, N.

A. Marcano O., C. Loper, and N. Melikechi, “High sensitivity absorption measurement in water and glass samples using a mode mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78, 3415–3417 (2001).
[CrossRef]

Pope, R. M.

Power, J. F.

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Van Stryland, “Sensitivity measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity single beam n2 measurement,” Opt. Lett. 14, 995–957 (1989).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, J. Wang, R. De Salvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two-color Z-scan,” Opt. Lett. 17, 258–260 (1992).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Van Stryland, “Sensitivity measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity single beam n2 measurement,” Opt. Lett. 14, 995–957 (1989).
[CrossRef]

Shen, J.

J. Shen, A. J. Soroka, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam profile image detection,” J. Appl. Phys. 78, 700–708 (1995).
[CrossRef]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurement of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[CrossRef]

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

Snook, R. D.

R. D. Snook and R. D. Lowe, “Thermal lens spectrometry, a review,” Analyst 120, 2052–2068 (1995).
[CrossRef]

J. Shen, A. J. Soroka, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam profile image detection,” J. Appl. Phys. 78, 700–708 (1995).
[CrossRef]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurement of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[CrossRef]

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

Soroka, A. J.

J. Shen, A. J. Soroka, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam profile image detection,” J. Appl. Phys. 78, 700–708 (1995).
[CrossRef]

Terazima, M.

M. Terazima, T. Hara, and N. Hirota, “Population lens in thermal lens spectrometry. 2. Probe wavelength dependence and a new method for subtracting the transient absorption from the thermal lens signal,” J. Chem. Phys. 97, 10554–10560 (1993).
[CrossRef]

M. Terazima and N. Hirota, “Population lens in thermal lens spectroscopy,” J. Phys. Chem. 96, 7147–7150 (1992).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae, J. Wang, R. De Salvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two-color Z-scan,” Opt. Lett. 17, 258–260 (1992).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Van Stryland, “Sensitivity measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity single beam n2 measurement,” Opt. Lett. 14, 995–957 (1989).
[CrossRef]

Wang, J.

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Van Stryland, “Sensitivity measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Anal. Chem. (2)

N. J. Dovichi and J. M. Harris, “Time-resolved thermal lens calorimetry,” Anal. Chem. 53, 106–109 (1981).
[CrossRef]

N. J. Dovichi and J. M. Harris, “Laser induced thermal lens effect for calorimetric trace analysis,” Anal. Chem. 51, 728–731 (1979).
[CrossRef]

Analyst (1)

R. D. Snook and R. D. Lowe, “Thermal lens spectrometry, a review,” Analyst 120, 2052–2068 (1995).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

A. Marcano O., C. Loper, and N. Melikechi, “High sensitivity absorption measurement in water and glass samples using a mode mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78, 3415–3417 (2001).
[CrossRef]

H. Ma, A. S. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

Appl. Spectrosc. (1)

Chem. Phys. (1)

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser induced mode mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

Crit. Rev. Anal. Chem. (1)

N. J. Dovichi, “Thermo-optical spectrophotometries in analytical chemistry,” Crit. Rev. Anal. Chem. 17, 357–423 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Van Stryland, “Sensitivity measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

J. Appl. Phys. (2)

J. Shen, A. J. Soroka, and R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry based on probe beam profile image detection,” J. Appl. Phys. 78, 700–708 (1995).
[CrossRef]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurement of thin-film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[CrossRef]

J. Chem. Phys. (1)

M. Terazima, T. Hara, and N. Hirota, “Population lens in thermal lens spectrometry. 2. Probe wavelength dependence and a new method for subtracting the transient absorption from the thermal lens signal,” J. Chem. Phys. 97, 10554–10560 (1993).
[CrossRef]

J. Phys. Chem. (1)

M. Terazima and N. Hirota, “Population lens in thermal lens spectroscopy,” J. Phys. Chem. 96, 7147–7150 (1992).
[CrossRef]

Opt. Lett. (3)

Other (2)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 717.

L. Levi, Applied Optics, Vol. 2 of Wiley Series in Pure and Applied Optics (Wiley, New York, 1980).

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Figures (7)

Fig. 1
Fig. 1

Scheme of the pump–probe TL Z-scan experiment showing the probe beam p, the pump beam e, the sample S located at position z, the probe and the pump beam-waist positions ap and ae, respectively, and the aperture of radius ro located at the position L.

Fig. 2
Fig. 2

TL phase shift as a function of the transverse coordinate for different times. The calculation was performed with Eq. (5) and the parameters ap=0, z=ae=0, zo=0.2 cm, ωoe=2.3×10-3 cm, and D=1.431×10-7 m2/cm.

Fig. 3
Fig. 3

Mode-matched Z-scan signal calculated with Eq. (23) and the parameters zp=0.2 cm, ze=0.2 cm, ap=0, ae=0, L=1000 cm, ωop=2×10-3 cm, ωoe=2.3×10-3 cm, D=1.431×10-7 m2/cm, and times 0.01, 0.1, 1, and 10 s.

Fig. 4
Fig. 4

Mode-mismatched Z-scan signal calculated with Eq. (23) and parameters zp=200 cm, ze=0.2 cm, ap=0, ae=0, L=1000 cm, ωop=6.3×10-2 cm, ωoe=2.3×10-3 cm, D=1.431×10-7 m2/cm, and times 0.01, 0.1, 1, and 10 s.

Fig. 5
Fig. 5

Thermal-lens signal of distilled water (a) and BK7 glass (b) when chopping the pump field at 7 Hz. The pump beam was generated by the Ti-sapphire laser at 871 nm with an average power of 50 mW.

Fig. 6
Fig. 6

Z-scan signal for distilled water and BK7 glass. The solid curve (a) is the result of a theoretical fitting obtained with Eqs. (20)–(22) and the parameters Φo=-0.13, zp=70 cm, ze=0.05 cm, L=2000 cm, ro=0.1 cm, ap=-20, ae=0, t=0.145 s, and D=1.431×10-7 m2/cm. The solid curve (b) was performed with Eqs. (20) through (22), Φo=0.00056, and D=8.71×10-7 m2/cm. The rest of the parameters are as in (a).

Fig. 7
Fig. 7

TL signal from distilled water in a 0.2-cm path-length glass cell. The pump light was generated by a multiline argon laser with an average power of 23 mW and an average wavelength of 490 nm. In the inset we show the noise detected under the same experimental conditions when the pump field is off.

Tables (1)

Tables Icon

Table 1 Diffusion Coefficient, Photothermal Parameter, Thermal Conductivity, and Photothermal-Enhancement Factor of Distilled Water, BK7 Glass, and CCL4

Equations (23)

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U(r, z, t)=C(z, t)0 exp{-[1+iν(z)]g-iΦ(g, z t)}Jo[Y(z)rg]dg,
C(z, r)=iωp(z)2πPp/(λpL)exp[-i(π/λp)×(2L+r2/L)+i arctan(z/zp)],
Y(z)=2πωp(z)/(λpL),
ν(z)=(z-ap)/zp+(zp/L)[1+(z-ap)2/zp2],
Φ(g, z, t)=-Φo2 (1/[1+2t/tc(z)])1 [1-exp(-2m(z)gτ)]τ dτ;
m(z)=ωp2(z)/ωe2(z),
ωp,e(z)=ωop,oe1+(z-ap,e)2/zop,oe2,
Φo=Peαl(ds/dT)/(κλp);
ds/dT=dn/dT+(n-l)/ldl/dT,
exp[-iΦ(z, r, t)]1-iΦ(z, r, t).
0 exp(-αx2)Jo(βx)dx=12α exp-β22α.
U(z, r, t)=C(z, r)A(z, r, 0)+iΦo21/(1+2t/tc)1[A(z, r, 0)-A(z, r, τ)]τ dτ,
A(z, r, τ)=H(z, τ)exp{-F(z, τ)Y2(z)r2+i[β(z, τ)Y2(z)r2+η(z, τ)]},
H(z, τ)=1ν2(z)+[1+2m(z)τ]2,
F(z, τ)=1+2m(z)τ4{ν2(z)+[1+2m(z)τ]2},
β(z, τ)=ν(z)4{ν2(z)+[1+2m(z)τ]2},
η(z, τ)=arctan-ν(z)1+2m(z)τ.
T(z, t)=2πoro|U(z, r, t)|2rdr,
S(z, t)=T(z, t)-ToTo,
S(z, t)=-2F(z ,0)Φo/(H(z, 0)×{1-exp[-2F(z, 0)Y(z)2ro2]})×1/(1+2t/tc)1H(z, τ)G(z, τ)/τdτ,
G(z, τ)=sin[Ψ(z, τ)]-exp[-F(z, τ)Y(z)2ro2]sin{Ψ(z, τ)+[β(z, 0)-β(z, τ)]Y(z)2ro2}{[F(z, 0)+F(z, τ)]2+[β(z, 0)-β(z, τ)]2}1/2,
Ψ(z, τ)=arctan{[β(z, 0)-β(z, τ)]/[F(z, 0)+F(z, τ)]}-η(z, 0)+η(z, τ).
S(z, t)=Φo arctan4m(z)ν(z)t/tc(z)ν(z)2+[1+2m(z)]2+[1+2m(z)+ν(z)2]2t/tc(z).

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