Abstract

The efficiency of thermal nonlinear refractive effects in the nanosecond regime is discussed. A single-beam Z-scan and a two-color time-resolved Z-scan experiments demonstrate the influence of the geometric configuration and the pulse duration on the photoinduced index variation. We present a theoretical analysis of thermal-lens formation that takes into account acoustic wave propagation and thermal diffusive effects. It permits us to quantify the index variation during the pulse and to define a new figure of merit for thermal nonlinear refractive effects.

© 1997 Optical Society of America

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References

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  1. L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron 17, 299 (1993).
  2. R. C. C. Leite, S. P. S. Porto, and T. C. Damen, “The thermal lens effect as a power-limiting device,” Appl. Phys. Lett. 10, 100 (1967).
    [CrossRef]
  3. G. A. Swatzlander, C. T. Law, and A. J. Campillo, “Improving the self-defocusing optical limiter,” in Nonlinear and Electro-Optic Materials for Optical Switching, M. Soileau, ed. Proc. SPIE1692, 15 (1992).
    [CrossRef]
  4. G. A. Swartzlander, B. L. Justus, A. L. Huston, and A. J. Campillo, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577 (1993).
    [CrossRef]
  5. B. L. Justus, A. L. Huston, and A. J. Campillo, “Broadband thermal optical limiter,” Appl. Phys. Lett. 63, 1483 (1993).
    [CrossRef]
  6. Jian-Guo Tian, Chungping Zhang, Guangyin Zhang, and Jiangwei Li, “Position dispersion and optical limiting resulting from thermally induced nonlinearities in chinese tea liquid,” Appl. Opt. 32, 6628 (1993).
    [CrossRef]
  7. B. L. Justus, Z. H. Kafafi, and A. L. Huston, “Excited-state absorption-enhanced thermal optical limiting in C60,” Opt. Lett. 18, 1603 (1993).
    [CrossRef] [PubMed]
  8. B. L. Justus, A. J. Campillo, and A. L. Huston, “Thermal-defocusing/scattering optical limiter,” Opt. Lett. 19, 673 (1994).
    [CrossRef] [PubMed]
  9. J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009 (1994).
    [CrossRef]
  10. J. Castillo, V. P. Kozich, and A. Marcano O., “Thermal lensing resulting from one- to two-photon absorption studied with a two-color time resolved z-scan,” Opt. Lett. 19, 171 (1994).
    [CrossRef]
  11. J. A. Riddick and W. B. Bunger, “Physical properties and methods of purification,” in Organic Solvents, Vol. 2 of Techniques of Chemistry (Wiley Interscience, New York, 1970), pp. 109, 165–167.
  12. R. C. Weast, ed., Handbook of Chemistry and Physics, 61st ed. (CRC Press, Boca Raton, Fla., 1980).
  13. J. Stone, “Measurements of absorption of light in low-loss liquids,” J. Opt. Soc. Am. 62, 327 (1972).
    [CrossRef]
  14. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
    [CrossRef]
  15. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955 (1989).
    [CrossRef] [PubMed]
  16. C. K. N. Patel and A. C. Tam, “Pulsed photoacoustic spectroscopy of condensed matter,” Rev. Mod. Opt. 53, 517 (1981).
  17. M. Terazima, “Transient lens spectroscopy in a fast time scale. Photoexcitation of Rhodamine 6G and Methyl Red solution,” Chem. Phys. Lett. 230, 87 (1994).
    [CrossRef]
  18. S. E. Braslvsky and G. E. Heibel, “Time-resolved photothermal and photoacoustic methods applied to photo-induced processes in solutions,” Chem. Rev. 92, 1381 (1992).
  19. J. H. Bechtel, “Heating of solid targets with laser pulses,” J. Appl. Phys. 46, 1585 (1975).
    [CrossRef]
  20. 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 (1969).
    [CrossRef]
  21. S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photoacoustic and photo-refractive detection of small absorptions in liquids,” Opt. Commun. 34, 199 (1980).
    [CrossRef]
  22. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), Chap. 4, p. 163.

1994 (4)

1993 (5)

Jian-Guo Tian, Chungping Zhang, Guangyin Zhang, and Jiangwei Li, “Position dispersion and optical limiting resulting from thermally induced nonlinearities in chinese tea liquid,” Appl. Opt. 32, 6628 (1993).
[CrossRef]

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron 17, 299 (1993).

G. A. Swartzlander, B. L. Justus, A. L. Huston, and A. J. Campillo, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577 (1993).
[CrossRef]

B. L. Justus, A. L. Huston, and A. J. Campillo, “Broadband thermal optical limiter,” Appl. Phys. Lett. 63, 1483 (1993).
[CrossRef]

B. L. Justus, Z. H. Kafafi, and A. L. Huston, “Excited-state absorption-enhanced thermal optical limiting in C60,” Opt. Lett. 18, 1603 (1993).
[CrossRef] [PubMed]

1992 (1)

S. E. Braslvsky and G. E. Heibel, “Time-resolved photothermal and photoacoustic methods applied to photo-induced processes in solutions,” Chem. Rev. 92, 1381 (1992).

1990 (1)

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

1989 (1)

1981 (1)

C. K. N. Patel and A. C. Tam, “Pulsed photoacoustic spectroscopy of condensed matter,” Rev. Mod. Opt. 53, 517 (1981).

1980 (1)

S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photoacoustic and photo-refractive detection of small absorptions in liquids,” Opt. Commun. 34, 199 (1980).
[CrossRef]

1975 (1)

J. H. Bechtel, “Heating of solid targets with laser pulses,” J. Appl. Phys. 46, 1585 (1975).
[CrossRef]

1972 (1)

1969 (1)

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 (1969).
[CrossRef]

1967 (1)

R. C. C. Leite, S. P. S. Porto, and T. C. Damen, “The thermal lens effect as a power-limiting device,” Appl. Phys. Lett. 10, 100 (1967).
[CrossRef]

Bechtel, J. H.

J. H. Bechtel, “Heating of solid targets with laser pulses,” J. Appl. Phys. 46, 1585 (1975).
[CrossRef]

Belanger, L. J.

S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photoacoustic and photo-refractive detection of small absorptions in liquids,” Opt. Commun. 34, 199 (1980).
[CrossRef]

Boggess, T. F.

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron 17, 299 (1993).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), Chap. 4, p. 163.

Braslvsky, S. E.

S. E. Braslvsky and G. E. Heibel, “Time-resolved photothermal and photoacoustic methods applied to photo-induced processes in solutions,” Chem. Rev. 92, 1381 (1992).

Brueck, S. R. J.

S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photoacoustic and photo-refractive detection of small absorptions in liquids,” Opt. Commun. 34, 199 (1980).
[CrossRef]

Bunger, W. B.

J. A. Riddick and W. B. Bunger, “Physical properties and methods of purification,” in Organic Solvents, Vol. 2 of Techniques of Chemistry (Wiley Interscience, New York, 1970), pp. 109, 165–167.

Campillo, A. J.

B. L. Justus, A. J. Campillo, and A. L. Huston, “Thermal-defocusing/scattering optical limiter,” Opt. Lett. 19, 673 (1994).
[CrossRef] [PubMed]

G. A. Swartzlander, B. L. Justus, A. L. Huston, and A. J. Campillo, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577 (1993).
[CrossRef]

B. L. Justus, A. L. Huston, and A. J. Campillo, “Broadband thermal optical limiter,” Appl. Phys. Lett. 63, 1483 (1993).
[CrossRef]

G. A. Swatzlander, C. T. Law, and A. J. Campillo, “Improving the self-defocusing optical limiter,” in Nonlinear and Electro-Optic Materials for Optical Switching, M. Soileau, ed. Proc. SPIE1692, 15 (1992).
[CrossRef]

Castillo, J.

Damen, T. C.

R. C. C. Leite, S. P. S. Porto, and T. C. Damen, “The thermal lens effect as a power-limiting device,” Appl. Phys. Lett. 10, 100 (1967).
[CrossRef]

Hagan, D. J.

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009 (1994).
[CrossRef]

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

Heibel, G. E.

S. E. Braslvsky and G. E. Heibel, “Time-resolved photothermal and photoacoustic methods applied to photo-induced processes in solutions,” Chem. Rev. 92, 1381 (1992).

Huston, A. L.

B. L. Justus, A. J. Campillo, and A. L. Huston, “Thermal-defocusing/scattering optical limiter,” Opt. Lett. 19, 673 (1994).
[CrossRef] [PubMed]

B. L. Justus, Z. H. Kafafi, and A. L. Huston, “Excited-state absorption-enhanced thermal optical limiting in C60,” Opt. Lett. 18, 1603 (1993).
[CrossRef] [PubMed]

B. L. Justus, A. L. Huston, and A. J. Campillo, “Broadband thermal optical limiter,” Appl. Phys. Lett. 63, 1483 (1993).
[CrossRef]

G. A. Swartzlander, B. L. Justus, A. L. Huston, and A. J. Campillo, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577 (1993).
[CrossRef]

Justus, B. L.

B. L. Justus, A. J. Campillo, and A. L. Huston, “Thermal-defocusing/scattering optical limiter,” Opt. Lett. 19, 673 (1994).
[CrossRef] [PubMed]

B. L. Justus, Z. H. Kafafi, and A. L. Huston, “Excited-state absorption-enhanced thermal optical limiting in C60,” Opt. Lett. 18, 1603 (1993).
[CrossRef] [PubMed]

G. A. Swartzlander, B. L. Justus, A. L. Huston, and A. J. Campillo, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577 (1993).
[CrossRef]

B. L. Justus, A. L. Huston, and A. J. Campillo, “Broadband thermal optical limiter,” Appl. Phys. Lett. 63, 1483 (1993).
[CrossRef]

Kafafi, Z. H.

Kildal, H.

S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photoacoustic and photo-refractive detection of small absorptions in liquids,” Opt. Commun. 34, 199 (1980).
[CrossRef]

Kozich, V. P.

Law, C. T.

G. A. Swatzlander, C. T. Law, and A. J. Campillo, “Improving the self-defocusing optical limiter,” in Nonlinear and Electro-Optic Materials for Optical Switching, M. Soileau, ed. Proc. SPIE1692, 15 (1992).
[CrossRef]

Leite, R. C. C.

R. C. C. Leite, S. P. S. Porto, and T. C. Damen, “The thermal lens effect as a power-limiting device,” Appl. Phys. Lett. 10, 100 (1967).
[CrossRef]

Li, Jiangwei

Litvak, M. M.

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 (1969).
[CrossRef]

Longaker, P. R.

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 (1969).
[CrossRef]

Marcano O., A.

Patel, C. K. N.

C. K. N. Patel and A. C. Tam, “Pulsed photoacoustic spectroscopy of condensed matter,” Rev. Mod. Opt. 53, 517 (1981).

Porto, S. P. S.

R. C. C. Leite, S. P. S. Porto, and T. C. Damen, “The thermal lens effect as a power-limiting device,” Appl. Phys. Lett. 10, 100 (1967).
[CrossRef]

Riddick, J. A.

J. A. Riddick and W. B. Bunger, “Physical properties and methods of purification,” in Organic Solvents, Vol. 2 of Techniques of Chemistry (Wiley Interscience, New York, 1970), pp. 109, 165–167.

Said, A. A.

Sheik-Bahae, M.

Stone, J.

Swartzlander, G. A.

G. A. Swartzlander, B. L. Justus, A. L. Huston, and A. J. Campillo, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577 (1993).
[CrossRef]

Swatzlander, G. A.

G. A. Swatzlander, C. T. Law, and A. J. Campillo, “Improving the self-defocusing optical limiter,” in Nonlinear and Electro-Optic Materials for Optical Switching, M. Soileau, ed. Proc. SPIE1692, 15 (1992).
[CrossRef]

Tam, A. C.

C. K. N. Patel and A. C. Tam, “Pulsed photoacoustic spectroscopy of condensed matter,” Rev. Mod. Opt. 53, 517 (1981).

Terazima, M.

M. Terazima, “Transient lens spectroscopy in a fast time scale. Photoexcitation of Rhodamine 6G and Methyl Red solution,” Chem. Phys. Lett. 230, 87 (1994).
[CrossRef]

Tian, Jian-Guo

Tutt, L. W.

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron 17, 299 (1993).

Van Stryland, E. W.

Wang, J.

Wei, T. H.

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

Zhang, Chungping

Zhang, Guangyin

Appl. Opt. (1)

Appl. Phys. Lett. (2)

B. L. Justus, A. L. Huston, and A. J. Campillo, “Broadband thermal optical limiter,” Appl. Phys. Lett. 63, 1483 (1993).
[CrossRef]

R. C. C. Leite, S. P. S. Porto, and T. C. Damen, “The thermal lens effect as a power-limiting device,” Appl. Phys. Lett. 10, 100 (1967).
[CrossRef]

Chem. Phys. Lett. (1)

M. Terazima, “Transient lens spectroscopy in a fast time scale. Photoexcitation of Rhodamine 6G and Methyl Red solution,” Chem. Phys. Lett. 230, 87 (1994).
[CrossRef]

Chem. Rev. (1)

S. E. Braslvsky and G. E. Heibel, “Time-resolved photothermal and photoacoustic methods applied to photo-induced processes in solutions,” Chem. Rev. 92, 1381 (1992).

IEEE J. Quantum Electron. (1)

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

Int. J. Nonlinear Opt. Phys. (1)

G. A. Swartzlander, B. L. Justus, A. L. Huston, and A. J. Campillo, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577 (1993).
[CrossRef]

J. Appl. Phys. (2)

J. H. Bechtel, “Heating of solid targets with laser pulses,” J. Appl. Phys. 46, 1585 (1975).
[CrossRef]

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 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

S. R. J. Brueck, H. Kildal, and L. J. Belanger, “Photoacoustic and photo-refractive detection of small absorptions in liquids,” Opt. Commun. 34, 199 (1980).
[CrossRef]

Opt. Lett. (4)

Prog. Quantum Electron (1)

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron 17, 299 (1993).

Rev. Mod. Opt. (1)

C. K. N. Patel and A. C. Tam, “Pulsed photoacoustic spectroscopy of condensed matter,” Rev. Mod. Opt. 53, 517 (1981).

Other (4)

G. A. Swatzlander, C. T. Law, and A. J. Campillo, “Improving the self-defocusing optical limiter,” in Nonlinear and Electro-Optic Materials for Optical Switching, M. Soileau, ed. Proc. SPIE1692, 15 (1992).
[CrossRef]

J. A. Riddick and W. B. Bunger, “Physical properties and methods of purification,” in Organic Solvents, Vol. 2 of Techniques of Chemistry (Wiley Interscience, New York, 1970), pp. 109, 165–167.

R. C. Weast, ed., Handbook of Chemistry and Physics, 61st ed. (CRC Press, Boca Raton, Fla., 1980).

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), Chap. 4, p. 163.

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Figures (14)

Fig. 1
Fig. 1

Absorption spectra of dimethyl 1,2 Indoaniline in toluene and of water-soluble Nigrosine in water.

Fig. 2
Fig. 2

Schematic of the single beam Z-scan experiment: D1, D2, detectors.

Fig. 3
Fig. 3

Comparison of experimental Z scans obtained with solutions of Indoaniline in toluene (×) and in ethanol (□) and with a solution of Nigrosine in water (●) in 1-mm-thick cells. w0=9 µm, F0=1.1J cm-2, S<2%. Each data point is an average of 32 shots taken with the aid of an automatic data-acquisition system.

Fig. 4
Fig. 4

Comparison of experimental Z scans performed with a 1-mm-thick cell of Indoaniline in toluene with w0 at the values indicated, F0=1.1J cm-2, S<2%, and the calculated Z scan (ΔΦ0=-2π, steady-state approximation).

Fig. 5
Fig. 5

Schematic of the two-color time-resolved Z-scan experiment: M1, M4, mirrors; M2, M3, dichroic beam splitters; OD's, optical densities; F(λ)'s, spike filters at λ; ϕ1–ϕ3, diaphragms; D1, D2, detectors; A.P.D., avalanche photodiode; L's, lenses.

Fig. 6
Fig. 6

Propagation of the acoustic wave in a solution of Indoaniline in toluene. Signals correspond to spatial shifts between pump and probe of 120 µm (dashed curve) and 280 µm (solid curve). The corresponding measured Δt is 160 ns, giving a propagation velocity of 1000 m s-1.

Fig. 7
Fig. 7

Thermal lens relaxation in a 1-mm-thick cell of Indoaniline in toluene for two beam sizes. Relaxation characteristic times are 2.5 ms for w0=16 µm and 10 ms for w0=32 µm.

Fig. 8
Fig. 8

(a) Formation of the thermoacoustic effect in a 1-mm-thick cell of Indoaniline in toluene. (b) Steady-state experimental Z scan obtained by displaying VT(z)/V0 as function of the z position of the sample. For (a) and (b), w0=32 µm (z0=6 mm) and F0=0.8J cm-2. The experimental curve (♦) is compared with the calculated Z scan obtained in the steady-state approximation (solid curve).

Fig. 9
Fig. 9

Probe intensity variation for three positions of the sample (z=-5 mm, z=-3 mm, and z=1.5 mm). w0=16 µm and z0=1.5 mm.

Fig. 10
Fig. 10

(a) Calculation of the acoustic part and the diffusive part (dashed curve) and the sum of the two contributions (solid curve) for r=0, m=5.33 (τac=16 ns, τ=3 ns). (b) Display of the on-axis index variation as function of time for three values of m corresponding to w0=8 µm (m=2.67), w0=16 µm (m=5.33), and w0=32 µm (m=16.7), and τ=3 ns. Comparison with a Gaussian shape (dashed curve). All calculations are normalized with the steady-state index variation.

Fig. 11
Fig. 11

Numerical calculation of Δn0eff/Δn0ss in the hypothesis of a Gaussian temporal pulse. Points are fitted by an exponential function (solid curve).

Fig. 12
Fig. 12

Experimental Z scans of a 1-mm-thick cell of Indoaniline in toluene for w0=16 µm (○) and w0=32 µm (×); F0=1.1J cm-2. Comparison with the respective theoretical Z scans (solid curves) that take into account the transient regime. The steady-state Z scan calculated under the same conditions is also displayed (dashed curve).

Fig. 13
Fig. 13

Temporally averaged on-axis index variation displayed as a function of time duration (FWHM) for a 1-J cm-2 fluence. A1 and A2 represent fast nonlinear effects (as molecular reorientation) calculated with nonlinear index variations n2 equal to, respectively, 3×10-14cm2 W-1 (carbon disulfide) and 3×10-13cm2 W-1. Other curves represent thermally induced index variations calculated for dye solutions in carbon disulfide (B1 and B2), toluene (C1 and C2), and chloroform (D1 and D2). Values of physical parameters for each solvent are summarized in Table 1. Linear absorption is 3 cm-1 for curves B–D; it is null for the curves A. Calculations use w0=8 µm for B1, C1, and D1 and w0=16 µm for B2, C2, and D2.

Fig. 14
Fig. 14

Dynamic figure of merit as function of the steady-state figure of merit for the usual solvents calculated with τp=5 ns and (a) w0=10 µm and (b) w0=40 µm.

Tables (1)

Tables Icon

Table 1 Physical Parameters of Some Common Solventsa

Equations (33)

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

τresp<τP<τD,
Δn0ss=12α0F0ρ0cpnT.
FMss=dn/dTρ0cp.
ΔΦ(z, r)=ΔΦ0(z)exp-2 r2w2(z),
ΔΦ0(z)=k Δn01+z2z02Leff,
T(z)=PT[ΔΦ0(z, t)]dtSPi(t)dt,
Δn=nρTΔρ+nTρΔT.
Δn(r, t)=γe2n0ρ0Δρ(r, t),
-2t2+vs2γ+ηρ0t(Δρ)+νs2βρ0γ(ΔT)
=γe2n0cΔI
t-κρ0cν(ΔT)-(γ-1)βρ0(Δρ)t=α0ρ0cνI,
-2t2+νs2γ(Δρ)+νs2βρ0γ(ΔT)=0,
t-D(ΔT)=α0ρ0cpI.
ΔnIMP(r, t)=Δndiff(r, t)+Δnac(r, t),
Δndiff=Δn0ss 11+2 tτDexp-2 r2w21+2 tτD,
Δnac=-Δn0ss w22πγνstr dss2-r2×exp-2w2s-νstγ2-exp-2w2s+vstγ2,
Δn0ss=-γeβ2n0α0ρ0cp2Etπw2.
τac=wγ/νs.
Δn(t)=ΔnIMP(t)1τπexp-t2τ2,
Δn(r, u, m)=Δn0ssfdn(r, u, m),
fdn(r, u, m)=fdiff(r, u)+fac(r, u, m).
Δn0(z)= Δn(z, r=0, t)I(t)dt I(t)dt.
Δn0ss(z)=-12γeβ2n0α0ρ0cp2Etπw(z)2.
Δn0eff(m)=Δn0ss  fdn(r=0, m, u)exp(-u2)du exp(-u2)du.
Δn0eff(m)Δn0ssexp-m2.7.
ΔΦ0(x)=k Δn0eff(x)1+x2Leff,
Δn0eff(x)=Δn0ssexp-γνsw(x)2.7τ,
Δn0=n2F02τp.
FDYN=FMss exp-11.62γνsw0τP.
fdiff(r, u)=12erfc(-u)exp-2 r2w2,
fac(r, u, m)=m2[2π(2+m2)]1/2×r/w exp[-2s2+u2]s2-rw21/2u-sm2+m2×X(s, m, u)-u+sm2+m2Y(s, m, u)ds,
X(s, m, u)=exp(2s+mu)22+m2erfc-2s+mu2+m2,
Y(s, m, u)=exp(2s-mu)22+m2erfc2s-mu2+m2.

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