Abstract

A generalization of the thermal lens model formalism is proposed to introduce and characterize the thermodiffusion phenomenon. The theory treats the case in which local heating generates a temperature gradient in a single-phase binary sample (a colloid, for instance) that yields, through thermodiffusion, concentration gradients in an initially homogeneous sample. The treatment generalizes the concept of a thermal lens to a material lens as a result of the coupling of a concentration variation with the optical properties of the medium. This formalism permits the use of the Z-scan technique to determine the Soret coefficient of samples. Applying this theory to the results of a Z-scan experiment with an ionic ferrofluid sample gives values that agree with those obtained from forced Rayleigh scattering measurements of the same material.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2002

J. Lenglet, A. Bourdon, J. C. Bacri, and G. Demouchy, “Thermodiffusion in magnetic colloids evidenced and studied by forced Rayleigh scattering experiments,” Phys. Rev. E 65, 031408-1-14 (2002).
[CrossRef]

F. Cuppo, A. M. Figueiredo Neto, S. Gomez, and P. Palffy-Muhoray, “Thermal-lens model compared with the Sheik-Bahae formalism in interpreting Z-scan experiments on lyotropic liquid crystals,” J. Opt. Soc. Am. B 19, 1342-1348 (2002).
[CrossRef]

1999

S. L. Gómez, F. L. S. Cuppo, A. M. Figueiredo Neto, T. Kosa, M. Muramatsu, and R. J. Horowicz, “Z-scan measurement of the nonlinear refractive indices of micellar lyotropic liquid crystals with and without the ferrofluid doping,” Phys. Rev. E 59, 3059–3063 (1999).
[CrossRef]

1994

L. C. Oliveira and S. C. Zilio, “Single-beam time-resolved z-scan measurements of slow absorbers,” Appl. Phys. Lett. 65, 2121-2123 (1994).
[CrossRef]

1993

W. Ko¨hler, “Thermodiffusion in polymer solutions as observed by forced Rayleigh scattering,” J. Chem. Phys. 98, 660-668 (1993).
[CrossRef]

1991

P. Palffy-Muhoray, H. J. Yuan, L. Li, and M. A. Lee, “Measurements of third order optical nonlinearities on nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 207, 291-305 (1991).
[CrossRef]

1990

O. Ecenarro, J. A. Madariaga, J. Navarro, C. M. Santamaria, J. A. Carrion, and J. M. Saviron, “Fickian and thermal diffusion coefficients from liquid thermogravitational columns,” J. Phys.: Condens. Matter 2, 2289-2296 (1990).

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

1989

1987

S. A. Akbar, M. Kaburagi, and H. Sato, “Soret effect in solids,” J. Phys. Chem. Solids 48, 579-586 (1987).
[CrossRef]

1985

K. Wada, A. Suzuki, H. Sato, and R. Kikuchi, “Soret effect in solids,” J. Phys. Chem. Solids 46, 1195-1205 (1985).
[CrossRef]

1984

1981

R. Massart, “Preparation of aqueous magnetic liquids in alkaline and acidic media,” IEEE Trans. Magn. MAG-17, 1247-1248 (1981).
[CrossRef]

1974

J. R. Whinnery, “Laser measurement of optical-absorption in liquids,” Acc. Chem. Res. 7, 225-231 (1974).
[CrossRef]

1973

1965

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]

1964

R. C. C. Leite, R. S. Moore, and J. R. Whinnery, “Low absorption measurements by means of thermal lens effect using He-Ne laser (absorption 10−3 to 10−5 cm−1 stimulated Raman scattering E-T),” Appl. Phys. Lett. 5, 141-143 (1964).
[CrossRef]

1880

Ch. Soret, “Sur l’état d’équilibre que prend au point de vue de sa concentration une dissolution saline primitivement homoge`ne dont deux parties sont portées a des températures différentes (deuxie`me note),” Arch. Sci. Phys. Nat. 4, 209-213 (1880).

1879

Ch. Soret, “Sur l’état d’équilibre que prend au point de vue de sa concentration une dissolution saline primitivement homoge`ne dont deux parties sont portées a` des températures différentes,” Arch. Sci. Phys. Nat. 2, 48-61 (1879).

Akbar, S. A.

S. A. Akbar, M. Kaburagi, and H. Sato, “Soret effect in solids,” J. Phys. Chem. Solids 48, 579-586 (1987).
[CrossRef]

Bacri, J. C.

J. Lenglet, A. Bourdon, J. C. Bacri, and G. Demouchy, “Thermodiffusion in magnetic colloids evidenced and studied by forced Rayleigh scattering experiments,” Phys. Rev. E 65, 031408-1-14 (2002).
[CrossRef]

Bourdon, A.

J. Lenglet, A. Bourdon, J. C. Bacri, and G. Demouchy, “Thermodiffusion in magnetic colloids evidenced and studied by forced Rayleigh scattering experiments,” Phys. Rev. E 65, 031408-1-14 (2002).
[CrossRef]

Carrion, J. A.

O. Ecenarro, J. A. Madariaga, J. Navarro, C. M. Santamaria, J. A. Carrion, and J. M. Saviron, “Fickian and thermal diffusion coefficients from liquid thermogravitational columns,” J. Phys.: Condens. Matter 2, 2289-2296 (1990).

Carter, C. A.

Cuppo, F.

Cuppo, F. L. S.

S. L. Gómez, F. L. S. Cuppo, A. M. Figueiredo Neto, T. Kosa, M. Muramatsu, and R. J. Horowicz, “Z-scan measurement of the nonlinear refractive indices of micellar lyotropic liquid crystals with and without the ferrofluid doping,” Phys. Rev. E 59, 3059–3063 (1999).
[CrossRef]

Demouchy, G.

J. Lenglet, A. Bourdon, J. C. Bacri, and G. Demouchy, “Thermodiffusion in magnetic colloids evidenced and studied by forced Rayleigh scattering experiments,” Phys. Rev. E 65, 031408-1-14 (2002).
[CrossRef]

Ecenarro, O.

O. Ecenarro, J. A. Madariaga, J. Navarro, C. M. Santamaria, J. A. Carrion, and J. M. Saviron, “Fickian and thermal diffusion coefficients from liquid thermogravitational columns,” J. Phys.: Condens. Matter 2, 2289-2296 (1990).

Gomez, S.

Gómez, S. L.

S. L. Gómez, F. L. S. Cuppo, A. M. Figueiredo Neto, T. Kosa, M. Muramatsu, and R. J. Horowicz, “Z-scan measurement of the nonlinear refractive indices of micellar lyotropic liquid crystals with and without the ferrofluid doping,” Phys. Rev. E 59, 3059–3063 (1999).
[CrossRef]

Gordon, J. P.

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]

Hagan, D. J.

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

Harris, J. M.

Horowicz, R. J.

S. L. Gómez, F. L. S. Cuppo, A. M. Figueiredo Neto, T. Kosa, M. Muramatsu, and R. J. Horowicz, “Z-scan measurement of the nonlinear refractive indices of micellar lyotropic liquid crystals with and without the ferrofluid doping,” Phys. Rev. E 59, 3059–3063 (1999).
[CrossRef]

Hu, C.

Kaburagi, M.

S. A. Akbar, M. Kaburagi, and H. Sato, “Soret effect in solids,” J. Phys. Chem. Solids 48, 579-586 (1987).
[CrossRef]

Kikuchi, R.

K. Wada, A. Suzuki, H. Sato, and R. Kikuchi, “Soret effect in solids,” J. Phys. Chem. Solids 46, 1195-1205 (1985).
[CrossRef]

Ko¨hler, W.

W. Ko¨hler, “Thermodiffusion in polymer solutions as observed by forced Rayleigh scattering,” J. Chem. Phys. 98, 660-668 (1993).
[CrossRef]

Kosa, T.

S. L. Gómez, F. L. S. Cuppo, A. M. Figueiredo Neto, T. Kosa, M. Muramatsu, and R. J. Horowicz, “Z-scan measurement of the nonlinear refractive indices of micellar lyotropic liquid crystals with and without the ferrofluid doping,” Phys. Rev. E 59, 3059–3063 (1999).
[CrossRef]

Lee, M. A.

P. Palffy-Muhoray, H. J. Yuan, L. Li, and M. A. Lee, “Measurements of third order optical nonlinearities on nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 207, 291-305 (1991).
[CrossRef]

Leite, R. C. C.

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]

R. C. C. Leite, R. S. Moore, and J. R. Whinnery, “Low absorption measurements by means of thermal lens effect using He-Ne laser (absorption 10−3 to 10−5 cm−1 stimulated Raman scattering E-T),” Appl. Phys. Lett. 5, 141-143 (1964).
[CrossRef]

Lenglet, J.

J. Lenglet, A. Bourdon, J. C. Bacri, and G. Demouchy, “Thermodiffusion in magnetic colloids evidenced and studied by forced Rayleigh scattering experiments,” Phys. Rev. E 65, 031408-1-14 (2002).
[CrossRef]

Li, L.

P. Palffy-Muhoray, H. J. Yuan, L. Li, and M. A. Lee, “Measurements of third order optical nonlinearities on nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 207, 291-305 (1991).
[CrossRef]

Madariaga, J. A.

O. Ecenarro, J. A. Madariaga, J. Navarro, C. M. Santamaria, J. A. Carrion, and J. M. Saviron, “Fickian and thermal diffusion coefficients from liquid thermogravitational columns,” J. Phys.: Condens. Matter 2, 2289-2296 (1990).

Massart, R.

R. Massart, “Preparation of aqueous magnetic liquids in alkaline and acidic media,” IEEE Trans. Magn. MAG-17, 1247-1248 (1981).
[CrossRef]

Moore, R. S.

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]

R. C. C. Leite, R. S. Moore, and J. R. Whinnery, “Low absorption measurements by means of thermal lens effect using He-Ne laser (absorption 10−3 to 10−5 cm−1 stimulated Raman scattering E-T),” Appl. Phys. Lett. 5, 141-143 (1964).
[CrossRef]

Muramatsu, M.

S. L. Gómez, F. L. S. Cuppo, A. M. Figueiredo Neto, T. Kosa, M. Muramatsu, and R. J. Horowicz, “Z-scan measurement of the nonlinear refractive indices of micellar lyotropic liquid crystals with and without the ferrofluid doping,” Phys. Rev. E 59, 3059–3063 (1999).
[CrossRef]

Navarro, J.

O. Ecenarro, J. A. Madariaga, J. Navarro, C. M. Santamaria, J. A. Carrion, and J. M. Saviron, “Fickian and thermal diffusion coefficients from liquid thermogravitational columns,” J. Phys.: Condens. Matter 2, 2289-2296 (1990).

Neto, A. M. Figueiredo

F. Cuppo, A. M. Figueiredo Neto, S. Gomez, and P. Palffy-Muhoray, “Thermal-lens model compared with the Sheik-Bahae formalism in interpreting Z-scan experiments on lyotropic liquid crystals,” J. Opt. Soc. Am. B 19, 1342-1348 (2002).
[CrossRef]

S. L. Gómez, F. L. S. Cuppo, A. M. Figueiredo Neto, T. Kosa, M. Muramatsu, and R. J. Horowicz, “Z-scan measurement of the nonlinear refractive indices of micellar lyotropic liquid crystals with and without the ferrofluid doping,” Phys. Rev. E 59, 3059–3063 (1999).
[CrossRef]

Oliveira, L. C.

L. C. Oliveira and S. C. Zilio, “Single-beam time-resolved z-scan measurements of slow absorbers,” Appl. Phys. Lett. 65, 2121-2123 (1994).
[CrossRef]

Palffy-Muhoray, P.

F. Cuppo, A. M. Figueiredo Neto, S. Gomez, and P. Palffy-Muhoray, “Thermal-lens model compared with the Sheik-Bahae formalism in interpreting Z-scan experiments on lyotropic liquid crystals,” J. Opt. Soc. Am. B 19, 1342-1348 (2002).
[CrossRef]

P. Palffy-Muhoray, H. J. Yuan, L. Li, and M. A. Lee, “Measurements of third order optical nonlinearities on nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 207, 291-305 (1991).
[CrossRef]

Porto, S. P. S.

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]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive 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 measurements,” Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

Santamaria, C. M.

O. Ecenarro, J. A. Madariaga, J. Navarro, C. M. Santamaria, J. A. Carrion, and J. M. Saviron, “Fickian and thermal diffusion coefficients from liquid thermogravitational columns,” J. Phys.: Condens. Matter 2, 2289-2296 (1990).

Sato, H.

S. A. Akbar, M. Kaburagi, and H. Sato, “Soret effect in solids,” J. Phys. Chem. Solids 48, 579-586 (1987).
[CrossRef]

K. Wada, A. Suzuki, H. Sato, and R. Kikuchi, “Soret effect in solids,” J. Phys. Chem. Solids 46, 1195-1205 (1985).
[CrossRef]

Saviron, J. M.

O. Ecenarro, J. A. Madariaga, J. Navarro, C. M. Santamaria, J. A. Carrion, and J. M. Saviron, “Fickian and thermal diffusion coefficients from liquid thermogravitational columns,” J. Phys.: Condens. Matter 2, 2289-2296 (1990).

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive 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 measurements,” Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

Soret, Ch.

Ch. Soret, “Sur l’état d’équilibre que prend au point de vue de sa concentration une dissolution saline primitivement homoge`ne dont deux parties sont portées a des températures différentes (deuxie`me note),” Arch. Sci. Phys. Nat. 4, 209-213 (1880).

Ch. Soret, “Sur l’état d’équilibre que prend au point de vue de sa concentration une dissolution saline primitivement homoge`ne dont deux parties sont portées a` des températures différentes,” Arch. Sci. Phys. Nat. 2, 48-61 (1879).

Suzuki, A.

K. Wada, A. Suzuki, H. Sato, and R. Kikuchi, “Soret effect in solids,” J. Phys. Chem. Solids 46, 1195-1205 (1985).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive 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 measurements,” Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

Wada, K.

K. Wada, A. Suzuki, H. Sato, and R. Kikuchi, “Soret effect in solids,” J. Phys. Chem. Solids 46, 1195-1205 (1985).
[CrossRef]

Wei, T. H.

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

Whinnery, J. R.

J. R. Whinnery, “Laser measurement of optical-absorption in liquids,” Acc. Chem. Res. 7, 225-231 (1974).
[CrossRef]

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

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]

R. C. C. Leite, R. S. Moore, and J. R. Whinnery, “Low absorption measurements by means of thermal lens effect using He-Ne laser (absorption 10−3 to 10−5 cm−1 stimulated Raman scattering E-T),” Appl. Phys. Lett. 5, 141-143 (1964).
[CrossRef]

Yuan, H. J.

P. Palffy-Muhoray, H. J. Yuan, L. Li, and M. A. Lee, “Measurements of third order optical nonlinearities on nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 207, 291-305 (1991).
[CrossRef]

Zilio, S. C.

L. C. Oliveira and S. C. Zilio, “Single-beam time-resolved z-scan measurements of slow absorbers,” Appl. Phys. Lett. 65, 2121-2123 (1994).
[CrossRef]

Acc. Chem. Res.

J. R. Whinnery, “Laser measurement of optical-absorption in liquids,” Acc. Chem. Res. 7, 225-231 (1974).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

L. C. Oliveira and S. C. Zilio, “Single-beam time-resolved z-scan measurements of slow absorbers,” Appl. Phys. Lett. 65, 2121-2123 (1994).
[CrossRef]

R. C. C. Leite, R. S. Moore, and J. R. Whinnery, “Low absorption measurements by means of thermal lens effect using He-Ne laser (absorption 10−3 to 10−5 cm−1 stimulated Raman scattering E-T),” Appl. Phys. Lett. 5, 141-143 (1964).
[CrossRef]

Arch. Sci. Phys. Nat.

Ch. Soret, “Sur l’état d’équilibre que prend au point de vue de sa concentration une dissolution saline primitivement homoge`ne dont deux parties sont portées a` des températures différentes,” Arch. Sci. Phys. Nat. 2, 48-61 (1879).

Ch. Soret, “Sur l’état d’équilibre que prend au point de vue de sa concentration une dissolution saline primitivement homoge`ne dont deux parties sont portées a des températures différentes (deuxie`me note),” Arch. Sci. Phys. Nat. 4, 209-213 (1880).

IEEE J. Quantum Electron.

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

IEEE Trans. Magn.

R. Massart, “Preparation of aqueous magnetic liquids in alkaline and acidic media,” IEEE Trans. Magn. MAG-17, 1247-1248 (1981).
[CrossRef]

J. Appl. Phys.

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]

J. Chem. Phys.

W. Ko¨hler, “Thermodiffusion in polymer solutions as observed by forced Rayleigh scattering,” J. Chem. Phys. 98, 660-668 (1993).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem. Solids

K. Wada, A. Suzuki, H. Sato, and R. Kikuchi, “Soret effect in solids,” J. Phys. Chem. Solids 46, 1195-1205 (1985).
[CrossRef]

S. A. Akbar, M. Kaburagi, and H. Sato, “Soret effect in solids,” J. Phys. Chem. Solids 48, 579-586 (1987).
[CrossRef]

J. Phys.: Condens. Matter

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Figures (2)

Fig. 1
Fig. 1

Schematic of the Z-scan setup: Ch, chopper; L1L3 lenses; BS, beam splitter; D1, D2, detectors; PC, computer; S, sample.

Fig. 2
Fig. 2

Typical z-scan experimental results [ΓN exp(z)] from the magnetic colloidal sample investigated in this research. Solid curve, best-fit curve with Eq. (32).

Equations (41)

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Jm=-Dm(φ+STT),
Jth=TSTμφP,T-TμTP,φ+μJm-κT,
γ=z/z0,
Po(t)P(z, t)=ω0ωdZz021-2γ z0f+(1+γ2)z0f2,
l(r, t)=bn0+nT δT(r, t)+nφ δφ(r, t)+nI I(r, t),
1f(t)=-d2l(r, t)dr2r=0.
1fT(t)=-b nTd2dr2 δT(r, t)r=0.
Tt-DΔT=q˙ρcP,
d2dr2 δT(r, t)r=0=-αPoπω02κ11+γ2tt+tc,
tc=(1+γ2)tco,tco=ω02/8D.
z0fT(t)=CT11+γ2tt+tc,
CT=bz0αPoπω02κnT.
C=z0fS+z0fN.
Γexp(z, t)=P(z, t)P(z, 0).
Γth(z, t)=1+[(1+γ2)C-γ]21+[CTt/(t+tc)+(1+γ2)C-γ]2.
1fS=-b nφd2dr2 δφ(r, t)r=0.
Jm=0.
δφ=-STδT,
d2dr2 δφ(r, t)r=0=-STd2dr2 δT(r, t)r=0,
d2dr2 δT(r, t)r=0ON
=-αPoπω02κ11+γ21-2tcτln1+τ2tc.
d2dr2 δT(r, t)r=0
=-αPoπω02κ11+γ2τ/2tc+τ/2-t-τ/2tc+t-τ/2.
d2dr2 δT(r, t)r=0=-αPo2πω02κ11+γ2τ2tc+τ.
z0fS=bSTnφd2dr2 δT(r, t)r=0=CS(1+s+sγ2)(1+γ2),
CS=-bz0STαPo2πκω02nφ,s=2 tc0τ.
z0fN=CN(1+γ2)2,
CN=8bz0Poπω04nI.
ΓN exp(z)=P(z, τ/2)-P(z, 0)P(z, 0).
ΓN exp(z)=Γexp(z, τ/2)-1.
ΓNth(z)=CTm(1+γ2) 2γ-(1+γ2)(CTm+2C)1+[γ-(1+γ2)(CTm+C)]2.
ΓNth(z)
=CT1+s+sγ2×2γ-[(CT+2CS)/(1+s+sγ2)+(2CN)/(1+γ2)]1+[γ-(CT+CS)/(1+s+sγ2)+(CN)/(1+γ2)]2.
 
CT/2+CS1+s+sγ02+CN1+γ02=γ0,
ΓNth(γ)=4(γ-γ0)p1[1+s+sγ02+sγ0γ+sγ2+p2(γ0+γ)/(1+γ2)](1+s+sγ2)2+{p1-(γ-γ0)[1+s+sγ02+sγ0γ+sγ2+p2(γ0+γ)/(1+γ2)]}2,
p1=CT2,p2=CN(1+γ02),
CT=2p1,
CN=p2(1+γ02),
CS=(1+s+sγ02)γ0-CN1+γ02-CT/2,
ST=2NFCSCT,NF=-nΦ/nT.

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