Abstract

Heating of surfaces by optical beams is investigated theoretically and compared with experimental results in the context of infrared imaging with vanadium dioxide thin films. Using known solutions for the diffusion of point heat sources at the interface between two semi-infinite media, the theory is extended to beams of Gaussian and flat profiles, for steady-state and dynamic regimes. Parameters relevant to imaging, such as spatial resolution and response time, are linked to thermal diffusivity, beam dimensions, and intensity.

© 2012 Optical Society of America

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  1. B. C. Forget, I. Barbereau, D. Fournier, S. Tuli, and A. B. Battacharyya, “Electronic diffusivity measurement in silicon by photothermal microscopy,” Appl. Phys. Lett. 69, 1107–1109 (1996).
    [CrossRef]
  2. S. Hirschi, A. C. Boccara, F. Lepoutre, and Z. Bozoki, “Interferometric polarization microscope for thermoelastic imaging of polycrystalline materials: experiments and model,” J. Opt. 28, 142–150 (1997).
    [CrossRef]
  3. A. C. Boccara, D. Fournier, W. Jackson, and N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
    [CrossRef]
  4. W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Method of determining thermal diffusivity, heat capacity and thermal conductivity,” J. App. Phys. 32, 1679–1684 (1961).
    [CrossRef]
  5. A. Passian, A. Lereu, E. Arakawa, A. Wig, T. Thundat, and T. Ferrell, “Modulation of multiple photon energies by use of surface plasmons,” Opt. Lett. 30, 41–43 (2005).
    [CrossRef]
  6. A. Passian, A. L. Lereu, E. T. Arakawa, R. H. Ritchie, T. Thundat, and T. L. Ferrell, “Opto-electronic versus electro-optic modulation,” Appl. Phys. Lett. 85, 2703–2705 (2004).
    [CrossRef]
  7. L. Tetard, A. Passian, R. H. Farahi, B. Davison, and T. Thundat, “Optomechanical spectroscopy with broadband interferometric and quantum cascade laser sources,” Opt. Lett. 36, 3251–3253 (2011).
    [CrossRef]
  8. L. Tetard, A. Passian, R. H. Farahi, B. H. Davison, A. L. Lereu, and T. Thundat, “Optical and plasmonic spectroscopy with cantilever shaped materials,” J. Phys. D 44, 445102 (2011).
    [CrossRef]
  9. R. Farahi, A. Passian, T. Ferrell, and T. Thundat, “Marangoni forces created by surface plasmon decay,” Opt. Lett. 30, 616–618 (2005).
    [CrossRef]
  10. R. H. Farahi, A. Passian, L. Tetard, and T. Thundat, “Pump–probe photothermal spectroscopy using quantum cascade lasers,” J. Phys. D 45, 125101 (2012).
    [CrossRef]
  11. D. B. Chrisey and G. K. Hubler, eds., Pulsed Laser Deposition of Thin Films (Wiley, 1994).
  12. M. Lax, “Temperature rise induced by a laser beam,” J. Appl. Phys. 48, 3919–3926 (1977).
    [CrossRef]
  13. S. Bonora, U. Bortolozzo, S. Residori, R. Balu, and P. V. Ashrit, “Mid-IR to near-IR image conversion by thermally induced optical switching in vanadium dioxide,” Opt. Lett. 35, 103–105 (2010).
    [CrossRef]
  14. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford University, 1995), pp. 353–386.
  15. K. D. Cole, J. V. Beck, A. Haji-Sheikh, and B. Litkouhi, Heat Conduction using Green’s Functions, 2nd ed. (CRC Press, 2011).
  16. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), Chap. 3.
  17. R. Balu and P. V. Ashrit, “Near-zero IR transmission in the metal-insulator transition of VO2 thin films,” Appl. Phys. Lett. 92, 021904 (2008).
    [CrossRef]

2012

R. H. Farahi, A. Passian, L. Tetard, and T. Thundat, “Pump–probe photothermal spectroscopy using quantum cascade lasers,” J. Phys. D 45, 125101 (2012).
[CrossRef]

2011

L. Tetard, A. Passian, R. H. Farahi, B. Davison, and T. Thundat, “Optomechanical spectroscopy with broadband interferometric and quantum cascade laser sources,” Opt. Lett. 36, 3251–3253 (2011).
[CrossRef]

L. Tetard, A. Passian, R. H. Farahi, B. H. Davison, A. L. Lereu, and T. Thundat, “Optical and plasmonic spectroscopy with cantilever shaped materials,” J. Phys. D 44, 445102 (2011).
[CrossRef]

2010

2008

R. Balu and P. V. Ashrit, “Near-zero IR transmission in the metal-insulator transition of VO2 thin films,” Appl. Phys. Lett. 92, 021904 (2008).
[CrossRef]

2005

2004

A. Passian, A. L. Lereu, E. T. Arakawa, R. H. Ritchie, T. Thundat, and T. L. Ferrell, “Opto-electronic versus electro-optic modulation,” Appl. Phys. Lett. 85, 2703–2705 (2004).
[CrossRef]

1997

S. Hirschi, A. C. Boccara, F. Lepoutre, and Z. Bozoki, “Interferometric polarization microscope for thermoelastic imaging of polycrystalline materials: experiments and model,” J. Opt. 28, 142–150 (1997).
[CrossRef]

1996

B. C. Forget, I. Barbereau, D. Fournier, S. Tuli, and A. B. Battacharyya, “Electronic diffusivity measurement in silicon by photothermal microscopy,” Appl. Phys. Lett. 69, 1107–1109 (1996).
[CrossRef]

1980

1977

M. Lax, “Temperature rise induced by a laser beam,” J. Appl. Phys. 48, 3919–3926 (1977).
[CrossRef]

1961

W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Method of determining thermal diffusivity, heat capacity and thermal conductivity,” J. App. Phys. 32, 1679–1684 (1961).
[CrossRef]

Abbott, G. L.

W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Method of determining thermal diffusivity, heat capacity and thermal conductivity,” J. App. Phys. 32, 1679–1684 (1961).
[CrossRef]

Amer, N. M.

Arakawa, E.

Arakawa, E. T.

A. Passian, A. L. Lereu, E. T. Arakawa, R. H. Ritchie, T. Thundat, and T. L. Ferrell, “Opto-electronic versus electro-optic modulation,” Appl. Phys. Lett. 85, 2703–2705 (2004).
[CrossRef]

Ashrit, P. V.

S. Bonora, U. Bortolozzo, S. Residori, R. Balu, and P. V. Ashrit, “Mid-IR to near-IR image conversion by thermally induced optical switching in vanadium dioxide,” Opt. Lett. 35, 103–105 (2010).
[CrossRef]

R. Balu and P. V. Ashrit, “Near-zero IR transmission in the metal-insulator transition of VO2 thin films,” Appl. Phys. Lett. 92, 021904 (2008).
[CrossRef]

Balu, R.

S. Bonora, U. Bortolozzo, S. Residori, R. Balu, and P. V. Ashrit, “Mid-IR to near-IR image conversion by thermally induced optical switching in vanadium dioxide,” Opt. Lett. 35, 103–105 (2010).
[CrossRef]

R. Balu and P. V. Ashrit, “Near-zero IR transmission in the metal-insulator transition of VO2 thin films,” Appl. Phys. Lett. 92, 021904 (2008).
[CrossRef]

Barbereau, I.

B. C. Forget, I. Barbereau, D. Fournier, S. Tuli, and A. B. Battacharyya, “Electronic diffusivity measurement in silicon by photothermal microscopy,” Appl. Phys. Lett. 69, 1107–1109 (1996).
[CrossRef]

Battacharyya, A. B.

B. C. Forget, I. Barbereau, D. Fournier, S. Tuli, and A. B. Battacharyya, “Electronic diffusivity measurement in silicon by photothermal microscopy,” Appl. Phys. Lett. 69, 1107–1109 (1996).
[CrossRef]

Beck, J. V.

K. D. Cole, J. V. Beck, A. Haji-Sheikh, and B. Litkouhi, Heat Conduction using Green’s Functions, 2nd ed. (CRC Press, 2011).

Boccara, A. C.

S. Hirschi, A. C. Boccara, F. Lepoutre, and Z. Bozoki, “Interferometric polarization microscope for thermoelastic imaging of polycrystalline materials: experiments and model,” J. Opt. 28, 142–150 (1997).
[CrossRef]

A. C. Boccara, D. Fournier, W. Jackson, and N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
[CrossRef]

Bonora, S.

Bortolozzo, U.

Bozoki, Z.

S. Hirschi, A. C. Boccara, F. Lepoutre, and Z. Bozoki, “Interferometric polarization microscope for thermoelastic imaging of polycrystalline materials: experiments and model,” J. Opt. 28, 142–150 (1997).
[CrossRef]

Butler, C. P.

W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Method of determining thermal diffusivity, heat capacity and thermal conductivity,” J. App. Phys. 32, 1679–1684 (1961).
[CrossRef]

Carslaw, H. S.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford University, 1995), pp. 353–386.

Chrisey, D. B.

D. B. Chrisey and G. K. Hubler, eds., Pulsed Laser Deposition of Thin Films (Wiley, 1994).

Cole, K. D.

K. D. Cole, J. V. Beck, A. Haji-Sheikh, and B. Litkouhi, Heat Conduction using Green’s Functions, 2nd ed. (CRC Press, 2011).

Davison, B.

Davison, B. H.

L. Tetard, A. Passian, R. H. Farahi, B. H. Davison, A. L. Lereu, and T. Thundat, “Optical and plasmonic spectroscopy with cantilever shaped materials,” J. Phys. D 44, 445102 (2011).
[CrossRef]

Farahi, R.

Farahi, R. H.

R. H. Farahi, A. Passian, L. Tetard, and T. Thundat, “Pump–probe photothermal spectroscopy using quantum cascade lasers,” J. Phys. D 45, 125101 (2012).
[CrossRef]

L. Tetard, A. Passian, R. H. Farahi, B. H. Davison, A. L. Lereu, and T. Thundat, “Optical and plasmonic spectroscopy with cantilever shaped materials,” J. Phys. D 44, 445102 (2011).
[CrossRef]

L. Tetard, A. Passian, R. H. Farahi, B. Davison, and T. Thundat, “Optomechanical spectroscopy with broadband interferometric and quantum cascade laser sources,” Opt. Lett. 36, 3251–3253 (2011).
[CrossRef]

Ferrell, T.

Ferrell, T. L.

A. Passian, A. L. Lereu, E. T. Arakawa, R. H. Ritchie, T. Thundat, and T. L. Ferrell, “Opto-electronic versus electro-optic modulation,” Appl. Phys. Lett. 85, 2703–2705 (2004).
[CrossRef]

Forget, B. C.

B. C. Forget, I. Barbereau, D. Fournier, S. Tuli, and A. B. Battacharyya, “Electronic diffusivity measurement in silicon by photothermal microscopy,” Appl. Phys. Lett. 69, 1107–1109 (1996).
[CrossRef]

Fournier, D.

B. C. Forget, I. Barbereau, D. Fournier, S. Tuli, and A. B. Battacharyya, “Electronic diffusivity measurement in silicon by photothermal microscopy,” Appl. Phys. Lett. 69, 1107–1109 (1996).
[CrossRef]

A. C. Boccara, D. Fournier, W. Jackson, and N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
[CrossRef]

Haji-Sheikh, A.

K. D. Cole, J. V. Beck, A. Haji-Sheikh, and B. Litkouhi, Heat Conduction using Green’s Functions, 2nd ed. (CRC Press, 2011).

Hirschi, S.

S. Hirschi, A. C. Boccara, F. Lepoutre, and Z. Bozoki, “Interferometric polarization microscope for thermoelastic imaging of polycrystalline materials: experiments and model,” J. Opt. 28, 142–150 (1997).
[CrossRef]

Hubler, G. K.

D. B. Chrisey and G. K. Hubler, eds., Pulsed Laser Deposition of Thin Films (Wiley, 1994).

Jackson, W.

Jaeger, J. C.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford University, 1995), pp. 353–386.

Jenkins, R. J.

W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Method of determining thermal diffusivity, heat capacity and thermal conductivity,” J. App. Phys. 32, 1679–1684 (1961).
[CrossRef]

Lax, M.

M. Lax, “Temperature rise induced by a laser beam,” J. Appl. Phys. 48, 3919–3926 (1977).
[CrossRef]

Lepoutre, F.

S. Hirschi, A. C. Boccara, F. Lepoutre, and Z. Bozoki, “Interferometric polarization microscope for thermoelastic imaging of polycrystalline materials: experiments and model,” J. Opt. 28, 142–150 (1997).
[CrossRef]

Lereu, A.

Lereu, A. L.

L. Tetard, A. Passian, R. H. Farahi, B. H. Davison, A. L. Lereu, and T. Thundat, “Optical and plasmonic spectroscopy with cantilever shaped materials,” J. Phys. D 44, 445102 (2011).
[CrossRef]

A. Passian, A. L. Lereu, E. T. Arakawa, R. H. Ritchie, T. Thundat, and T. L. Ferrell, “Opto-electronic versus electro-optic modulation,” Appl. Phys. Lett. 85, 2703–2705 (2004).
[CrossRef]

Litkouhi, B.

K. D. Cole, J. V. Beck, A. Haji-Sheikh, and B. Litkouhi, Heat Conduction using Green’s Functions, 2nd ed. (CRC Press, 2011).

Parker, W. J.

W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Method of determining thermal diffusivity, heat capacity and thermal conductivity,” J. App. Phys. 32, 1679–1684 (1961).
[CrossRef]

Passian, A.

R. H. Farahi, A. Passian, L. Tetard, and T. Thundat, “Pump–probe photothermal spectroscopy using quantum cascade lasers,” J. Phys. D 45, 125101 (2012).
[CrossRef]

L. Tetard, A. Passian, R. H. Farahi, B. H. Davison, A. L. Lereu, and T. Thundat, “Optical and plasmonic spectroscopy with cantilever shaped materials,” J. Phys. D 44, 445102 (2011).
[CrossRef]

L. Tetard, A. Passian, R. H. Farahi, B. Davison, and T. Thundat, “Optomechanical spectroscopy with broadband interferometric and quantum cascade laser sources,” Opt. Lett. 36, 3251–3253 (2011).
[CrossRef]

A. Passian, A. Lereu, E. Arakawa, A. Wig, T. Thundat, and T. Ferrell, “Modulation of multiple photon energies by use of surface plasmons,” Opt. Lett. 30, 41–43 (2005).
[CrossRef]

R. Farahi, A. Passian, T. Ferrell, and T. Thundat, “Marangoni forces created by surface plasmon decay,” Opt. Lett. 30, 616–618 (2005).
[CrossRef]

A. Passian, A. L. Lereu, E. T. Arakawa, R. H. Ritchie, T. Thundat, and T. L. Ferrell, “Opto-electronic versus electro-optic modulation,” Appl. Phys. Lett. 85, 2703–2705 (2004).
[CrossRef]

Residori, S.

Ritchie, R. H.

A. Passian, A. L. Lereu, E. T. Arakawa, R. H. Ritchie, T. Thundat, and T. L. Ferrell, “Opto-electronic versus electro-optic modulation,” Appl. Phys. Lett. 85, 2703–2705 (2004).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), Chap. 3.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), Chap. 3.

Tetard, L.

R. H. Farahi, A. Passian, L. Tetard, and T. Thundat, “Pump–probe photothermal spectroscopy using quantum cascade lasers,” J. Phys. D 45, 125101 (2012).
[CrossRef]

L. Tetard, A. Passian, R. H. Farahi, B. Davison, and T. Thundat, “Optomechanical spectroscopy with broadband interferometric and quantum cascade laser sources,” Opt. Lett. 36, 3251–3253 (2011).
[CrossRef]

L. Tetard, A. Passian, R. H. Farahi, B. H. Davison, A. L. Lereu, and T. Thundat, “Optical and plasmonic spectroscopy with cantilever shaped materials,” J. Phys. D 44, 445102 (2011).
[CrossRef]

Thundat, T.

R. H. Farahi, A. Passian, L. Tetard, and T. Thundat, “Pump–probe photothermal spectroscopy using quantum cascade lasers,” J. Phys. D 45, 125101 (2012).
[CrossRef]

L. Tetard, A. Passian, R. H. Farahi, B. H. Davison, A. L. Lereu, and T. Thundat, “Optical and plasmonic spectroscopy with cantilever shaped materials,” J. Phys. D 44, 445102 (2011).
[CrossRef]

L. Tetard, A. Passian, R. H. Farahi, B. Davison, and T. Thundat, “Optomechanical spectroscopy with broadband interferometric and quantum cascade laser sources,” Opt. Lett. 36, 3251–3253 (2011).
[CrossRef]

A. Passian, A. Lereu, E. Arakawa, A. Wig, T. Thundat, and T. Ferrell, “Modulation of multiple photon energies by use of surface plasmons,” Opt. Lett. 30, 41–43 (2005).
[CrossRef]

R. Farahi, A. Passian, T. Ferrell, and T. Thundat, “Marangoni forces created by surface plasmon decay,” Opt. Lett. 30, 616–618 (2005).
[CrossRef]

A. Passian, A. L. Lereu, E. T. Arakawa, R. H. Ritchie, T. Thundat, and T. L. Ferrell, “Opto-electronic versus electro-optic modulation,” Appl. Phys. Lett. 85, 2703–2705 (2004).
[CrossRef]

Tuli, S.

B. C. Forget, I. Barbereau, D. Fournier, S. Tuli, and A. B. Battacharyya, “Electronic diffusivity measurement in silicon by photothermal microscopy,” Appl. Phys. Lett. 69, 1107–1109 (1996).
[CrossRef]

Wig, A.

Appl. Phys. Lett.

B. C. Forget, I. Barbereau, D. Fournier, S. Tuli, and A. B. Battacharyya, “Electronic diffusivity measurement in silicon by photothermal microscopy,” Appl. Phys. Lett. 69, 1107–1109 (1996).
[CrossRef]

A. Passian, A. L. Lereu, E. T. Arakawa, R. H. Ritchie, T. Thundat, and T. L. Ferrell, “Opto-electronic versus electro-optic modulation,” Appl. Phys. Lett. 85, 2703–2705 (2004).
[CrossRef]

R. Balu and P. V. Ashrit, “Near-zero IR transmission in the metal-insulator transition of VO2 thin films,” Appl. Phys. Lett. 92, 021904 (2008).
[CrossRef]

J. App. Phys.

W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Method of determining thermal diffusivity, heat capacity and thermal conductivity,” J. App. Phys. 32, 1679–1684 (1961).
[CrossRef]

J. Appl. Phys.

M. Lax, “Temperature rise induced by a laser beam,” J. Appl. Phys. 48, 3919–3926 (1977).
[CrossRef]

J. Opt.

S. Hirschi, A. C. Boccara, F. Lepoutre, and Z. Bozoki, “Interferometric polarization microscope for thermoelastic imaging of polycrystalline materials: experiments and model,” J. Opt. 28, 142–150 (1997).
[CrossRef]

J. Phys. D

L. Tetard, A. Passian, R. H. Farahi, B. H. Davison, A. L. Lereu, and T. Thundat, “Optical and plasmonic spectroscopy with cantilever shaped materials,” J. Phys. D 44, 445102 (2011).
[CrossRef]

R. H. Farahi, A. Passian, L. Tetard, and T. Thundat, “Pump–probe photothermal spectroscopy using quantum cascade lasers,” J. Phys. D 45, 125101 (2012).
[CrossRef]

Opt. Lett.

Other

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford University, 1995), pp. 353–386.

K. D. Cole, J. V. Beck, A. Haji-Sheikh, and B. Litkouhi, Heat Conduction using Green’s Functions, 2nd ed. (CRC Press, 2011).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), Chap. 3.

D. B. Chrisey and G. K. Hubler, eds., Pulsed Laser Deposition of Thin Films (Wiley, 1994).

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Figures (8)

Fig. 1.
Fig. 1.

A layer with thickness Δz covers a semi-infinite substrate. From a heated area of radius R, heat may diffuse along the film, to the air (z<0), and into the substrate with fluxes ϕ0, ϕ1, and ϕ2, respectively. Rc is the thermal boundary resistance between the film and layer. In the substrate (or film), thermal conductivity is κ (or κ0) and volumetric heat capacity is Cv (or CV0).

Fig. 2.
Fig. 2.

Temperature rise induced by a pointlike heat pulse incident on a surface at r=0 and t=0. Curves A, B, and C are temperatures at distances r2/4D=0.8, 1, and 1.2, respectively.

Fig. 3.
Fig. 3.

Time evolution of surface temperature with a continuous point heat source located at r=0. Curves A, B, and C are temperatures at distances r2D=0.5, 1, and 2, respectively.

Fig. 4.
Fig. 4.

Surface temperature profile with a continuous, Gaussian, surface heat source at times t=10tw, t=100tw, and 105tw (curves A, B, and C, respectively). The absorption layer is chosen so that z/w=103, and for visual clarity, results are plotted with ρ taking positive and negative values.

Fig. 5.
Fig. 5.

Steady-state surface temperature with Gaussian surface heat source (shown here with dashed line).

Fig. 6.
Fig. 6.

Profile of the surface temperature heated by a uniform disk of radius R. Curves A to D correspond to times such that the ratio Ld/R=2Dt/R=0.1, 1, 2, and 100, respectively. To better compare profiles, each curve is normalized to its own temperature at center.

Fig. 7.
Fig. 7.

Transmission pattern at 850 nm of a VO2 film heated by a 10×10 array of Gaussian laser beams at 1940 nm. Images are taken at the moment of phase transition, which occurs at a temperature near 70  °C. The gray scale color is inverted for better visual effect (i.e., bright spots correspond to low transmission).

Fig. 8.
Fig. 8.

Calculated transmission pattern of a VO2 film heated by a 10×10 array of Gaussian laser beams. Ld is the thermal diffusion length at the time of image capture. Quick capture times correspond to higher beam intensities.

Equations (34)

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

ϕ0=ΔTRκ0,
ϕ2=ΔTΔz2κ0+Rc+ζ2κ2,
ζ=ΔzCv0Cv2
Q˙0=ϕ02πRΔz,
Q˙2=ϕ2πR2.
R[2κ0Δz(Δz2κ0+Rc+ΔzCv02κCv)]1/2=rmin,
1DT(r,t)t=2T(r,t)1κg(r,t),
g(r,t)=Q0δ(rr)δ(tτ),
G(r,t|r,τ)=18rr(πD(tτ))12[exp((rr)24D(tτ))exp((r+r)24D(tτ))],
T(r,t)=Dκ0tg(r,t)G(r,t|r,τ)dτdV.
T(r,t)=QoCV(4πDt)3/2exp(r24Dt),
T(r,t)=2Q0CV(4πDt)3/2exp(r24Dt).
Tmax=2Q0CV(2eπ/3)3/2r3,
g(r,t)=Pδ(r),
0t(τ)32exp(aτ)dτ=πa[erf(at)1],
T(r,t)=Tmax(r)[1erf(r2Dt)],
Tmax=P2πrκ
T(r⃗,t)=2ϕ(r⃗)CV(4πDt)32exp(|r⃗r⃗|204Dt)dA.
T(r⃗,t)=I(r⃗)2πκ1|r⃗r⃗|[1erf(|r⃗r⃗|2Dt)]dA.
φ(ρ)=φ0exp(2ρ2w2),
T(ρ,z,t)=φ0w2cV(πDt)1/2(8Dt+w2)exp(2ρ28Dt+w2z24Dt).
tw=w28D
tz=z24D
T(ρ,z,t)=B0ttwe2(ρ/w)21+τz21w2ττ12(1+τ)dτ,
B=wI02κ2π.
T(0,0,t)=wI0κ2πtan1(ttw),
Tmax=πwI02κ2.
Tmax=Pκw2π.
T(r=0,t)=12πκI0(r)r[1erf(r2Dt)]dA=I0κ0R[1erf(rLd)]dr=I0κ[R+Ldπ(1eR2/Ld2RπLderf(RLd))],
Tmax=RI0κ=PπRκ,
t=0.231R2/D.
T(r⃗,t)=RI02πκf(r)|r⃗r⃗|[1erf(RLd|r⃗r⃗|)]dA,
Imin=κΔTminSp.
TmaxSiI¯κ,

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