Abstract

The theory underlying the use of the phase fluctuation optical heterodyne (PFLOH) technique in studies of molecular relaxation, thermal conduction, and extremely weak absorptions in the gas phase is presented. Several new solutions to the heat conduction equation, with and without mass diffusion, which are appropriate to pulsed, cw, and modulated-cw laser modulation of weakly absorbing gases, are presented. Experiments employing pulsed and modulated-cw excitation have borne out the essential predictions of the theory. PFLOH systems for use in a trace detection application have been built with a demonstrated sensitivity, in terms of measurable absorption by the trace material, below 10−10 cm−1.

© 1981 Optical Society of America

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Errata

Christopher C. Davis and Samuel J. Petuchowski, "Phase fluctuation optical heterodyne spectroscopy of gases: errata," Appl. Opt. 20, 4151-4151 (1981)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-20-24-4151

References

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  1. C. C. Davis, M. L. Swicord, at Open Symposium on the Biological Effects of Electromagnetic Waves, Nineteenth General Assembly of URSI, Helsinki, Finland (Aug. 1978);to be published in Radio Sci.
  2. M. L. Swicord, C. C. Davis, at Second Annual Meeting of the Bioelectromagnetic Society, San Antonio, Tex. (Sept. 1980).
  3. C. C. Davis, in Digest of Conference on Laser Engineering and Applications (Optical Society of America, Washington, D.C., 1979).
  4. C. C. Davis, Appl. Phys. Lett. 36, 515 (1980).
    [CrossRef]
  5. S. J. Petuchowski, C. C. Davis, Opt. Commun. (in press).
  6. A. J. Campillo, H-B. Lin, C. J. Dodge, C. C. Davis, Opt. Lett. 5, 424 (1980).
    [CrossRef] [PubMed]
  7. J. Stone, J. Opt. Soc. Am. 62, 327 (1972);Appl. Opt. 12, 1828 (1973).
    [CrossRef] [PubMed]
  8. F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 20, 129 (1972).
    [CrossRef]
  9. L. Sica, Appl. Phys. Lett. 27, 396 (1973);Appl. Opt. 12, 2848 (1973).
    [CrossRef] [PubMed]
  10. H. Aung, M. Katayama, Chem. Phys. Lett. 33, 502 (1975).
    [CrossRef]
  11. B. M. Oliver, Proc. IRE 49, 1960 (1961).
  12. M. Knudsen, The Kinetic Theory of Gases (Methuen, London, 1934).
  13. P. R. Longaker, M. M. Litvak, J. Appl. Phys. 40, 4033 (1969).
    [CrossRef]
  14. D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
    [CrossRef]
  15. J. I. Masters, J. Chem. Phys. 23, 1865 (1955).
    [CrossRef]
  16. G. W. C. Kaye, T. H. Laby, Tables of Physical and Chemical Constants (Longman, London, 1973).
  17. G. M. Burnett, A. M. North, Eds., Transfer and Storage of Energy by Molecules, Vol. 2, Vibrational Energy (Wiley-Inter-science, London, 1969).
  18. H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1956).
  19. E. K. Rideal, J. Tadayon, Proc. R. Soc. London Ser. A: 225, 357 (1954).
    [CrossRef]
  20. J. Crank, The Mathematics of Diffusion (Oxford U. P., London, 1956).
  21. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. P., London, 1959).
  22. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1956).
  23. G. Renner, M. Maier, Chem. Phys. Lett. 35, 226 (1975).
    [CrossRef]
  24. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).

1980

1975

G. Renner, M. Maier, Chem. Phys. Lett. 35, 226 (1975).
[CrossRef]

H. Aung, M. Katayama, Chem. Phys. Lett. 33, 502 (1975).
[CrossRef]

1973

L. Sica, Appl. Phys. Lett. 27, 396 (1973);Appl. Opt. 12, 2848 (1973).
[CrossRef] [PubMed]

1972

1969

P. R. Longaker, M. M. Litvak, J. Appl. Phys. 40, 4033 (1969).
[CrossRef]

D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
[CrossRef]

1961

B. M. Oliver, Proc. IRE 49, 1960 (1961).

1955

J. I. Masters, J. Chem. Phys. 23, 1865 (1955).
[CrossRef]

1954

E. K. Rideal, J. Tadayon, Proc. R. Soc. London Ser. A: 225, 357 (1954).
[CrossRef]

Aung, H.

H. Aung, M. Katayama, Chem. Phys. Lett. 33, 502 (1975).
[CrossRef]

Campillo, A. J.

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. P., London, 1959).

Crank, J.

J. Crank, The Mathematics of Diffusion (Oxford U. P., London, 1956).

Davis, C. C.

A. J. Campillo, H-B. Lin, C. J. Dodge, C. C. Davis, Opt. Lett. 5, 424 (1980).
[CrossRef] [PubMed]

C. C. Davis, Appl. Phys. Lett. 36, 515 (1980).
[CrossRef]

M. L. Swicord, C. C. Davis, at Second Annual Meeting of the Bioelectromagnetic Society, San Antonio, Tex. (Sept. 1980).

S. J. Petuchowski, C. C. Davis, Opt. Commun. (in press).

C. C. Davis, M. L. Swicord, at Open Symposium on the Biological Effects of Electromagnetic Waves, Nineteenth General Assembly of URSI, Helsinki, Finland (Aug. 1978);to be published in Radio Sci.

C. C. Davis, in Digest of Conference on Laser Engineering and Applications (Optical Society of America, Washington, D.C., 1979).

Dodge, C. J.

Gebhardt, F. G.

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 20, 129 (1972).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1956).

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. P., London, 1959).

Katayama, M.

H. Aung, M. Katayama, Chem. Phys. Lett. 33, 502 (1975).
[CrossRef]

Kaye, G. W. C.

G. W. C. Kaye, T. H. Laby, Tables of Physical and Chemical Constants (Longman, London, 1973).

Knudsen, M.

M. Knudsen, The Kinetic Theory of Gases (Methuen, London, 1934).

Laby, T. H.

G. W. C. Kaye, T. H. Laby, Tables of Physical and Chemical Constants (Longman, London, 1973).

Lin, H-B.

Litvak, M. M.

P. R. Longaker, M. M. Litvak, J. Appl. Phys. 40, 4033 (1969).
[CrossRef]

Longaker, P. R.

P. R. Longaker, M. M. Litvak, J. Appl. Phys. 40, 4033 (1969).
[CrossRef]

Maier, M.

G. Renner, M. Maier, Chem. Phys. Lett. 35, 226 (1975).
[CrossRef]

Margenau, H.

H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1956).

Masters, J. I.

J. I. Masters, J. Chem. Phys. 23, 1865 (1955).
[CrossRef]

Murphy, G. M.

H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1956).

Oliver, B. M.

B. M. Oliver, Proc. IRE 49, 1960 (1961).

Petuchowski, S. J.

S. J. Petuchowski, C. C. Davis, Opt. Commun. (in press).

Renner, G.

G. Renner, M. Maier, Chem. Phys. Lett. 35, 226 (1975).
[CrossRef]

Rideal, E. K.

E. K. Rideal, J. Tadayon, Proc. R. Soc. London Ser. A: 225, 357 (1954).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1956).

Sica, L.

L. Sica, Appl. Phys. Lett. 27, 396 (1973);Appl. Opt. 12, 2848 (1973).
[CrossRef] [PubMed]

Smith, D. C.

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 20, 129 (1972).
[CrossRef]

D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
[CrossRef]

Stone, J.

Swicord, M. L.

M. L. Swicord, C. C. Davis, at Second Annual Meeting of the Bioelectromagnetic Society, San Antonio, Tex. (Sept. 1980).

C. C. Davis, M. L. Swicord, at Open Symposium on the Biological Effects of Electromagnetic Waves, Nineteenth General Assembly of URSI, Helsinki, Finland (Aug. 1978);to be published in Radio Sci.

Tadayon, J.

E. K. Rideal, J. Tadayon, Proc. R. Soc. London Ser. A: 225, 357 (1954).
[CrossRef]

Appl. Phys. Lett.

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 20, 129 (1972).
[CrossRef]

L. Sica, Appl. Phys. Lett. 27, 396 (1973);Appl. Opt. 12, 2848 (1973).
[CrossRef] [PubMed]

C. C. Davis, Appl. Phys. Lett. 36, 515 (1980).
[CrossRef]

Chem. Phys. Lett.

H. Aung, M. Katayama, Chem. Phys. Lett. 33, 502 (1975).
[CrossRef]

G. Renner, M. Maier, Chem. Phys. Lett. 35, 226 (1975).
[CrossRef]

IEEE J. Quantum Electron.

D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
[CrossRef]

J. Appl. Phys.

P. R. Longaker, M. M. Litvak, J. Appl. Phys. 40, 4033 (1969).
[CrossRef]

J. Chem. Phys.

J. I. Masters, J. Chem. Phys. 23, 1865 (1955).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Proc. IRE

B. M. Oliver, Proc. IRE 49, 1960 (1961).

Proc. R. Soc. London Ser. A:

E. K. Rideal, J. Tadayon, Proc. R. Soc. London Ser. A: 225, 357 (1954).
[CrossRef]

Other

J. Crank, The Mathematics of Diffusion (Oxford U. P., London, 1956).

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. P., London, 1959).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1956).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).

C. C. Davis, M. L. Swicord, at Open Symposium on the Biological Effects of Electromagnetic Waves, Nineteenth General Assembly of URSI, Helsinki, Finland (Aug. 1978);to be published in Radio Sci.

M. L. Swicord, C. C. Davis, at Second Annual Meeting of the Bioelectromagnetic Society, San Antonio, Tex. (Sept. 1980).

C. C. Davis, in Digest of Conference on Laser Engineering and Applications (Optical Society of America, Washington, D.C., 1979).

M. Knudsen, The Kinetic Theory of Gases (Methuen, London, 1934).

G. W. C. Kaye, T. H. Laby, Tables of Physical and Chemical Constants (Longman, London, 1973).

G. M. Burnett, A. M. North, Eds., Transfer and Storage of Energy by Molecules, Vol. 2, Vibrational Energy (Wiley-Inter-science, London, 1969).

H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1956).

S. J. Petuchowski, C. C. Davis, Opt. Commun. (in press).

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Figures (13)

Fig. 1
Fig. 1

Basic PFLOH system for gas phase studies employing a Mach-Zehnder interferometer; BS, beam splitters; F, filter; M1,M2, total reflectors; P, beam defining apertures; PZT, piezoelectric transducer; ELB, excitation laser beam.

Fig. 2
Fig. 2

PFLOH system incorporating a modified Mach-Zehnder interferometer, which provides copropagation of excitation and probe beams only within sample cell.

Fig. 3
Fig. 3

PFLOH system incorporating a closely parallel double-beam interferometric system with provision for Stark modulation of a gaseous sample: A, beam defining aperture; CC, corner-cube retroreflector; M, totally reflecting surface; PR, partially reflecting surface; V, applied Stark voltage.

Fig. 4
Fig. 4

Time-dependent PFLOH signal resulting from coaxial short-pulse CO2 laser excitation of a 20.2:1 He–SF6 mixture at 644 mbar in a cylindrical cell of 25-mm radius. In the figure excitation occurs at a time of ≃125 μsec.

Fig. 5
Fig. 5

Early-time part of the data shown in Fig. 4, a 20.2:1 He–SF6 mixture at 644 mbar coaxially excited, together with a least squares fit (dashed line) to the function y = −a exp(−t/τ) + bct. Values of the fitted parameters in this case are 9.97 × 105 (a), 19.04 μsec (τ); 11.50 (b); 0.283 (μsec)−1 (c).

Fig. 6
Fig. 6

Early-time part of the PFLOH signal from pure CO2 at 455.00 mbar coaxially excited with a short-pulse CO2 laser together with a least squares fit to the function y = −a exp(−t/τ) + bct. The values of the fitted parameters in this case are 10047 (a); 17.1 μsec (τ); 1050 (b); 0.149 (μsec)−1 (c). In the figure laser excitation did not occur at t = 0.

Fig. 7
Fig. 7

Early-time part of the PFLOH signal from a 89.4:1 He–SF6 mixture at 351 mbar coaxially excited with a short-pulse CO2 laser together with a least squares fit to the function y = −a exp(−t/τ) + 6. Values of the fitted parameters in this case are 2913 (a); 11.2 μsec (τ): 1368 (b). In the figure laser excitation did not occur at t = 0.

Fig. 8
Fig. 8

Frequency dependence of the PFLOH signal from room air in the beam-chopped mode observed using the experimental arrangement shown in Fig. 3.

Fig. 9
Fig. 9

PFLOH signal as a function of trace methanol concentration in nitrogen observed in the beam-chopped mode using 9P(34) cw CO2 laser excitation.

Fig. 10
Fig. 10

Relative axial temperature modulation (M/ωmτc) of a cylindrical sample uniformly excited for ra by a sinusoidally modulated laser beam as a function of the dimensionless parameter 1/ωmτc. ωm = angular modulation frequency (rad/sec); τc = thermal conduction time.

Fig. 11
Fig. 11

Equilibrium radial temperature profiles produced in a weakly absorbing gas excited coaxially in a cylindrical cell of radius b by a cw TEM00 laser beam with spot size w.

Fig. 12
Fig. 12

Relative axial temperature modulation produced in a weakly absorbing gas by a sinusoidally modulated TEM00 laser beam of spot size w as a function of cylindrical container radius b.

Fig. 13
Fig. 13

Relative axial temperature modulation (w2F/8κ) of a cylindrical sample coaxially excited by a sinusoidally modulated TEM00 laser beam of spot size w as a function of the dimensionless parameter 1 / ω m τ c . ωm = angular modulation frequency (rad/sec); τ′ = thermal conduction time for a Gaussian-excited region.

Tables (2)

Tables Icon

Table I Time-Dependent Radial Temperature Distributions Produced by Short-Pulse Laser Excitation Having Cylindrical Symmetry

Tables Icon

Table II Experimental Interferometer Sensitivities

Equations (154)

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V A ( t ) = A exp [ i ( ω t + ϕ A ) ] .
V B ( t ) = B exp { i [ ω t + ϕ B + Δ ϕ ( t ) ] } .
i ( t ) α ( V A + V B ) * ( V A + V B ) ,
i ( t ) α A 2 + B 2 + 2 A B [ cos ( ϕ B ϕ A ) sin ( ϕ B ϕ A ) Δ ϕ ( t ) ] ,
SNR = ( η P s ) / ( h ν Δ f ) ,
Δ ϕ ( t ) = m sin ω m t ,
m = 2 π l ν Δ n / c ,
P s = P m 2 / 2 ,
Δ n min = c 2 π l ν ( 2 h ν Δ f η P ) 1 / 2 .
Δ n = ( n 1 ) Δ T T abs ,
Δ T = 2 E α π w 2 ρ C p ,
I 0 α N 0 h ν vib / τ ,
A B ( υ = 1 ) + A B ( υ = 1 ) = A B ( υ = 0 ) + A B ( υ = 2 ) + Δ E
2 T r 2 + 1 r T r 1 κ T t = N ( r , t ) E vib ρ C p τ exp ( t / τ ) ,
υ c ( 2 α I 0 w 2 g ρ C p T ) 1 / 3 ,
υ c 3.8 ( α I 0 ) 1 / 3 cm sec 1 .
Δ T = ( 2 E α ) / ( π w 2 ρ C p ) ,
Δ ρ = ( 2 E α ) / ( π w 2 C p T ) .
f c ( 2 E α g ) / ( π w 2 ρ C p T ) ,
Δ x c = f c τ c 2 / 2 ,
i ( t ) 4 π 2 ( n 1 ) l E 0 2 α c T abs 0 r T ( r , t ) exp ( 2 r 2 / w p 2 ) d r ,
E ( r ) = E 0 exp ( r 2 / w p 2 ) .
i ( t ) T 0 [ 1 exp ( t / τ ) ] 0 r exp ( 2 r 2 / w p 2 ) P ( a 2 κ t , r 2 κ t ) d r ( t a 2 / 4 κ ) ,
i ( t ) T 0 w p 2 4 [ 1 exp ( t / τ ) ] P ( a 2 κ t + ω p 2 / 4 , 0 ) ( t a 2 / 4 κ ) .
i ( t ) T 0 w p 2 44 [ 1 exp ( t / τ ) ] { 1 exp [ 2 a 2 / ( 8 κ t + w p 2 ) ] } .
i ( t ) T 0 w 2 8 κ t + w 2 [ 1 exp ( t / τ ) ] × 0 r exp { 2 r 2 [ 1 / ( 8 κ t + w 2 ) + 1 / w p 2 ] } d r ,
i ( t ) T 0 w 2 w p 2 [ 1 exp ( t / τ ) ] 8 κ t + w 2 + w p 2 .
y = a exp ( t / τ ) + b c t ,
y = a exp ( t / τ ) + b c t .
y = a 1 exp ( t / τ 1 ) a 2 exp ( t / τ 2 ) + b
i ( t ) 1 exp [ 2 a 2 / ( 8 κ t + w p 2 ) ] .
y = A exp [ 1 / ( B t + C ) ] + D ,
B = 4 κ / a 2 .
i ( t ) 1 exp ( t / τ ) t + ( w 2 + w p 2 ) / 8 κ .
y = 1 A t + B + C ,
A / B = 8 κ / ( w 2 + w p 2 ) .
T ( r , t ) = constant T 0 r 2 8 κ + M T 0 2 ω m sin ( ω m t ϕ ) ,
i ( t ) M T 0 2 ω m sin ( ω m t ϕ ) 0 r exp ( 2 r 2 / w p 2 ) d r ,
N ( r , 0 ) = N 0 = 0 0 r a r > a
T ( 0 , t ) = T 0 [ 1 exp ( a 2 / 4 κ t ) ] t τ T ( r , t ) = T 0 [ 1 exp ( a 2 / 4 κ t ) ] t τ , r 2 / 4 κ T ( r , t ) = T 0 a 2 / 4 κ t t τ , r 2 / 4 κ , a 2 / 4 κ
T ( r , t ) = T 0 w 2 ( 8 κ t + w 2 ) exp [ 2 r 2 / ( 8 κ t + w 2 ) ]
T ( 0 , t ) = T 0 [ 1 exp ( t / τ ) ] [ 1 exp ( a 2 / 4 κ t ) ] τ a 2 / 4 κ , a 2 4 D T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] P ( a 2 κ t , r 2 κ t ) τ a 2 / 4 κ , a 2 / 4 D T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] ( 1 + r 2 / 4 κ t ) exp ( r 2 / 4 κ t ) t a 2 / 4 κ , r < a , τ a 2 / 4 κ T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] [ 1 exp ( a 2 / 4 κ t ) ] t r 2 / 4 κ T ( r , t ) = T 0 a 2 / 4 κ t t τ , r 2 / 4 κ , a 2 / 4 κ
T ( r , t ) = T 0 w 2 ( 8 κ t + w 2 ) [ 1 exp ( t / τ ) ] exp [ 2 r 2 / ( 8 κ t + w 2 ) ] τ w 2 / 8 κ T ( r , t ) = T 0 exp { 2 r 2 / w 2 [ 1 exp ( t / τ ) ] ( 1 8 κ t / w 2 ) } τ , t w 2 / 8 κ
T ( r , t ) = T 0 w 2 ( 8 κ t + w 2 ) [ 1 exp ( t / τ ) ] exp ( 2 r 2 / ( 8 κ t + w 2 ) τ w 2 / 8 κ , w 2 / 8 D T ( r , t ) = T 0 w 2 [ 1 ( 8 κ t + w 2 ) exp ( t / τ ) ( 8 D t + w 2 ) ] t τ and / or 8 κ + w 2 r 2 τ w 2 / 8 κ T ( r , t ) = T 0 w 2 / ( 8 κ t + w 2 ) t τ , 8 κ t + w 2 r 2
2 T r 2 + 1 r T r 1 κ T t = 0
T ( r , 0 ) = N ( r ) E vib ρ C p ,
T ( r , t ) = 0 0 T ( r , 0 ) J 0 ( k r ) J 0 ( k r ) exp ( κ k 2 t ) k r dkd r .
T ( r , t ) = a T 0 0 J 1 ( k a ) J 0 ( k r ) exp ( κ k 2 t ) d k ,
T ( r , t ) = T 0 2 κ t exp ( r 2 / 4 κ t ) 0 a exp ( r 2 / 4 κ t ) I 0 ( r r 2 κ t ) r d r = T 0 P ( a 2 κ t , r 2 κ t ) ,
T ( r , t ) = a 2 T 0 4 κ t m = 0 ( m + 1 ) m ! ( r 2 4 κ t ) m F ( m , m , 2 , a 2 / r 2 ) ,
T ( 0 , T ) = T 0 [ 1 exp ( a 2 / 4 κ t ) ] .
T = T 0 exp ( 2 r 2 / w 2 )
T ( r , t ) = T 0 w 2 ( 8 κ t + w 2 ) exp [ 2 r 2 / ( 8 κ t + w 2 ) ] .
κ / D = C υ / C p ,
d T = N ( r , t ) E vib exp ( t / τ ) τ ρ C p d t .
d T ( r , t ) = d T 2 κ ( t t ) r exp [ ( r 2 + r 2 ) / 4 κ ( t t ) ] I 0 [ r r 2 κ ( t t ) ] .
T ( r , t ) = T 0 τ 0 t 0 a exp ( t / t ) 2 κ ( t t ) r × exp [ ( r 2 + r 2 ) / 4 κ ( t t ) ] I 0 [ r r 2 κ ( t t ) ] d r d t ,
T ( r , t ) = T 0 τ 0 t exp ( t / τ P ) [ a 2 κ ( t t ) , r 2 κ ( t t ) ] d t ,
T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] P ( a 2 κ t , r 2 κ t ) .
T ( 0 , t ) = T 0 [ 1 exp ( t / τ ) ] [ 1 exp ( a 2 / 4 κ t ) ] .
I 0 ( r r 2 κ t ) 1 + ( r r 4 κ t ) 2
T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] exp ( r 2 / 4 κ t ) 2 κ t × 0 a exp ( r 2 / 4 κ t ) [ r + r ( r r 4 κ t ) ] 2 d r ,
T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] { ( 1 + r 2 4 κ t ) × [ 1 exp ( a 2 / 4 κ t ) ] exp ( r 2 / 4 κ t ) ( a r 4 κ t ) 2 exp [ ( a 2 + r 2 ) / 4 κ t ] } ,
T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] [ ( 1 + r 2 / 4 κ t ) exp ( r 2 / 4 κ t ) ] .
T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] [ 1 exp ( a 2 / 4 κ t ) ] .
T ( r , t ) = T 0 a 2 / 4 κ t .
N ( r , t ) exp ( t / τ ) = N 0 2 D t exp ( t / τ ) exp ( r 2 / 4 D t ) × 0 a exp ( r 2 / 4 D t ) I 0 ( r r 2 D t ) r d r ,
T ( r , t ) = T 0 τ 0 t 0 exp ( t / τ ) r exp [ ( r 2 + r 2 ) / 4 κ ( t t ) ] I 0 [ r r 2 κ ( t t ) ] exp ( r 2 / 4 D t ) 4 κ D t ( t t ) × 0 a exp ( r / 4 D t ) I 0 ( r r 2 D t ) r d r d r d t .
T ( r , t ) = T 0 τ 0 t 0 exp ( t / τ ) r 2 κ ( t t ) exp [ ( r 2 + r 2 ) / 4 κ ( t t ) ] × I 0 [ r r 2 κ ( t t ) ] P ( a 2 D t , r 2 D t ) d r d t .
T ( r , t ) = T 0 τ 0 t exp ( t / τ ) × P [ a 2 D t + 2 κ ( t t ) , r 2 D t + 2 κ ( t t ) ] d t .
T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] P [ a 2 κ t , r 2 κ t ] .
T ( 0 , t ) = T 0 [ 1 exp ( t / τ ) ] [ 1 exp ( a 2 / 4 κ t ) ] .
T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] [ 1 + r 2 / 4 κ t ) exp ( r 2 / 4 κ t ) ] .
N ( r , t ) = N 0 exp ( 2 r 2 / w 2 ) ,
T ( r , t ) = T 0 w 2 4 τ 0 t 0 exp ( t / τ ) r J 0 ( r r ) × exp { [ κ ( t t ) + w 2 / 8 ] r 2 } d r d t .
T ( r , t ) = T 0 w 2 4 τ 0 r J 0 ( r r ) exp ( w 2 r 2 / 8 ) ( κ r 2 1 / τ ) × [ exp ( t / τ ) exp ( κ r 2 t ) ] d r ,
T ( r , t ) = T 0 w 2 τ 0 t exp ( t / τ ) 8 κ ( t t ) + w 2 × exp { 2 r 2 / [ 8 κ ( t t ) + w 2 ] } d t .
T ( 0 , t ) = T 0 w 2 8 κ τ exp ( t τ + w 2 8 κ τ ) × [ E i ( t τ + w 2 8 κ τ ) E i ( w 2 8 κ τ ) ] ,
T ( r , t ) = T 0 w 2 ( 8 κ t + w 2 ) [ 1 exp ( t / τ ) ] exp [ 2 r 2 / ( 8 κ t + w 2 ) ] ,
T ( r , t ) = T 0 w 2 ( 8 κ t + w 2 ) [ 1 exp ( t / τ ) ] .
T ( r , t ) = T 0 exp ( 2 r 2 / w 2 ) [ 1 exp ( t / τ ) ] ( 1 8 κ t / w 2 ) .
N ( r , t ) = N 0 w 2 ( 8 D t + w 2 ) exp 2 r 2 / ( 8 D t + w 2 ) .
T ( r , t ) = T 0 w 2 τ 0 t exp ( t / τ ) [ 8 κ ( t t ) + 8 D t + w 2 ] × exp 2 r 2 / [ 8 κ ( t t ) + 8 D t + w 2 ] d t .
T ( r , t ) = T 0 w 2 8 τ ( D κ ) exp [ κ D κ ( t τ + w 2 8 κ τ ) ] × { E 1 [ κ D κ ( t τ + w 2 8 κ τ ) ] E 1 [ D D κ ( t τ + w 2 8 D τ ) ] } ,
κ D κ ( t τ + w 2 8 κ τ ) ,
T ( r , t ) = T 0 w 2 [ 1 ( 8 κ t + w 2 ) exp ( t / τ ) ( 8 D t + w 2 ) ] .
T ( r , t ) = T 0 w 2 / ( 8 κ t + w 2 ) .
T ( r , t ) = T 0 / ( 1 8 κ t / w 2 ) .
T ( r , t ) = T 0 [ 1 exp ( t / τ ) ] ( r a ) .
T ( r , t ) T 0 [ 1 f ( t ) ] ( r a ) .
T ( r , t ) T 0 0 t f ( t ) h ( t t ) d t ,
L T ( p ) = L f ( p ) L h ( p ) ;
τ / ( p τ + 1 ) = L T ( p ) / L h ( p ) ,
τ = L T ( p ) / L h ( p ) . lim p 0
i ( r , t ) = ½ I ( r ) ( 1 + sin ω m t ) ,
d N ( r , t ) = α I ( r ) 2 E vib ( 1 + sin ω m t ) d t ,
d T ( r , t ) T abs = T 0 b 2 [ 1 exp ( t / τ ) ] 0 a n = 1 × J 0 ( r x n ) J 0 ( r x n ) J 1 2 ( b x n ) r exp ( κ x n 2 t ) d r d t ,
d T ( r , t ) T abs = T 0 b 2 a [ 1 exp ( t / τ ) ] n = 1 × J 0 ( r x n ) J 1 ( a x n ) x n J 1 2 ( b x n ) exp ( κ x n 2 t ) d t ,
2 π r k T r = π a 2 I 0 α / 2 ,
T ( r ) T abs = T 0 a 2 ln ( b / r ) 4 κ .
2 π r k T r = π r 2 I 0 α / 2 ,
T ( a ) = T 0 a 2 ln ( b / a ) 4 κ , T ( r ) T abs = T 0 a 2 ln ( b / a ) 4 κ + T 0 8 κ ( a 2 r 2 ) ,
T ( r , t ) T abs = T 0 a b 2 n = 1 J 0 ( r x n ) J 1 ( a x n ) x n J 1 2 ( b x n ) × t { 1 exp [ ( t t ) / τ ] } × exp [ κ x n 2 ( t t ) ] sin ω m t d t ,
T ( r , t ) T abs = T 0 a b 2 n = 1 J 0 ( r x n ) J 1 ( a x n ) x n J 1 2 ( b x n ) { sin ( ω m t ϕ n ) [ ( κ x n 2 ) 2 + ω m 2 ] 1 / 2 sin ( ω m t ϕ n ) [ ( κ x n 2 + 1 / τ ) 2 + ω m 2 ] 1 / 2 } ,
T ( r , t ) T abs = T 0 a b n = 1 J 0 ( r β n / b ) J 1 ( a β n / b ) β n J 1 2 ( β n ) × { sin ( ω m t ϕ n ) [ ( κ β n 2 / b 2 ) 2 + ω m 2 ] 1 / 2 } .
T ( r , t ) T abs = T 0 a b κ sin ω m t n = 1 J 0 ( r β n / b ) J 1 ( a β n / b ) β n 3 J 1 2 ( β n ) ,
T ( r , t ) a = b T abs = T 0 8 κ ( a 2 r 2 ) sin ω m t .
T ( r , t ) = T 0 2 t { 1 exp [ ( t t ) / τ ] } × P [ a 2 κ ( t t ) , × r 2 κ ( t t ) ] sin ω m t d t .
T ( r , t ) = T 0 2 t { 1 exp [ ( t t ) / τ ] } × { 1 exp [ a 2 / 4 κ ( t t ) ] } sin ω m t d t .
T ( 0 , t ) = T 0 2 t sin ω m t { 1 exp [ a 2 / 4 κ ( t t ) ] } d t .
T ( 0 , t ) = T 0 a 2 8 κ ω m ( A 2 + B 2 ) 1 / 2 sin ( ω m t ϕ ) ,
A = 2 i ( ω m κ ) 1 / 2 a [ exp ( i π / 4 ) K 1 [ a exp ( i π / 4 ) ( ω m / κ ) 1 / 2 ] exp ( i π / 4 ) K 1 [ a exp ( i π / 4 ) ( ω m / κ ) 1 / 2 ] ,
B = 2 ( ω m κ ) 1 / 2 a { exp ( i π / 4 ) K 1 [ a exp ( i π / 4 ) ( ω m / κ ) 1 / 2 ] + exp ( i π / 4 ) K 1 [ a exp ( i π / 4 ) ( ω m / κ ) 1 / 2 ] } , tan ϕ = B / A
T ( 0 , t ) = T 0 a 2 ( ω m κ ) 1 / 2 N 1 [ a ( ω m κ ) 1 / 2 ] sin ( ω m t ϕ ) ,
N 1 ( x ) = [ k e r 1 2 ( x ) + k e i 1 2 ( x ) ] 1 / 2 .
τ c = a 2 / 4 κ ln 2 .
T ( 0 , t ) = T 0 ( ln 2 ) 1 / 2 ω m ( ω m τ c ) 1 / 2 N 1 [ 2 ( ln 2 ) 1 / 2 ( ω m τ c ) 1 / 2 ] × sin ( ω m t ϕ ) .
K ν ( z ) = ( π / 2 z ) 1 / 2 exp ( z ) ,
T ( 0 , t ) = T 0 π 1 / 2 2 ω m [ a ( ω m / κ ) 1 / 2 ] 1 / 2 exp [ a ( ω m / 2 κ ) 1 / 2 ] sin ( ω m t ϕ ) .
K 1 ( z ) ~ 1 / z
T ( 0 , t ) ~ T 0 2 ω m cos ω m t .
M / ω m τ c = 2 ( ω m τ c ln 2 ) 1 / 2 N 1 [ 2 ( ω m τ c ln 2 ) 1 / 2 ] / ω m τ c
I ( r ) = I 0 exp ( 2 r 2 / w 2 ) 2 ,
d T ( r , t ) T abs = T 0 [ 1 exp ( t / τ ) ] b 2 n = 1 J 0 ( r x n ) exp ( κ x n 2 t ) J 1 2 ( b x n ) × 0 b J 0 ( r x n ) exp ( 2 r 2 / w 2 ) r d r d t ,
d T ( r , t ) T abs = T 0 w 2 4 b 2 [ 1 exp ( t / τ ) ] n = 1 J 0 ( r x n ) J 1 2 ( b x n ) × exp ( x n 2 w 2 / 8 ) exp ( κ x n 2 t ) d t .
T ( r , t ) T abs = T 0 w 2 4 b 2 n = 1 J 0 ( r x n ) J 1 2 ( b x n ) exp ( x n 2 w 2 / 8 ) × t { 1 exp [ ( t t ) / τ ] } × exp [ κ x n 2 ( t t ) ] d t ,
T ( r , t ) T abs = T 0 w 2 4 κ n = 1 J 0 ( r β n / b ) exp ( β n 2 ω 2 / 8 b 2 ) β n 2 ( 1 + β n 2 κ τ / b 2 ) J 1 2 ( β n ) .
T ( r , t ) T abs = T 0 w 2 4 κ n = 1 J 0 ( r β n / b ) exp ( β n 2 ω 2 / 8 b 2 ) β n 2 J 1 2 ( β n ) .
T ( r , t ) T abs = T 0 w 2 4 κ n = 1 ( 1 r 2 β n 2 / 4 b 2 ) β n 2 J 1 2 ( β n ) exp ( β n 2 w 2 / 8 b 2 ) ,
T ( r , t ) T abs = A T 0 w 2 r 2 16 κ b 2 n = 1 exp ( β n 2 w 2 / 8 b 2 ) J 1 2 ( β n ) ,
A = T 0 w 2 4 κ n = 1 exp ( β n 2 w 2 / 8 b 2 ) β n 2 J 1 2 ( β n ) .
W = π I 0 α 0 r r exp ( 2 r 2 / w 2 ) d r = π I 0 α w 2 4 [ 1 exp ( r 2 / w 2 ) ] .
2 π r k d T d r = π I 0 α w 2 4 [ 1 exp ( 2 r 2 / w 2 ) ]
d T d r = I 0 α w 2 8 k [ 1 r exp ( 2 r 2 / w 2 ) r ] ,
T = T 0 w 2 16 κ n = 1 ( 1 ) n x n n n ! + constant ,
n = 1 ( 1 ) x n n n n ! = γ ln x E 1 ( x ) ,
T ( r ) T abs = T 0 w 2 16 κ [ 2 ln ( b / r ) + E 1 ( 2 b 2 / w 2 ) E 1 ( 2 r 2 / w 2 ) ] ,
T ( r ) T abs = T 0 w 2 16 κ [ 2 ln ( b / r ) E 1 ( 2 r 2 / w 2 ) ] .
T ( r ) T abs = A T 0 r 2 8 κ ,
A = T 0 w 2 16 κ [ γ + ln 2 b 2 / w 2 + E 1 ( 2 b 2 / w 2 ) ] .
T ( r , t ) T abs = T 0 w 2 4 b 2 n = 1 J 0 ( r x n ) J 1 2 ( b x n ) exp ( x n 2 w 2 / 8 ) × t { 1 exp [ ( t t ) / τ ] } × exp [ 2 κ x n 2 ( t t ) ] sin ω m t d t ,
T ( r , t ) T abs = T 0 w 2 4 b 2 n = 1 J 0 ( r x n ) J 1 2 ( b x n ) × exp ( x n 2 w 2 / 8 ) sin ( ω m t ϕ n ) [ ( 2 κ x n 2 ) 2 + ω m 2 ] 1 / 2 ,
T ( r , t ) T abs = T 0 w 2 8 κ sin ω m t n = 1 J 0 ( r β n / b ) β 2 J 1 2 ( β n ) exp ( β n 2 w 2 / 8 b 2 ) .
Δ T n = 1 exp ( β n 2 w 2 / 8 b 2 ) β 2 J 1 2 ( β n ) ,
T ( r , t ) = T 0 w 2 2 t { 1 exp [ ( t t ) / τ ] } 8 κ ( t t ) + w 2 × exp [ 2 r 2 / ( 8 κ ( t t ) + w 2 ) ] sin ω m t d t .
T ( r , t ) = T 0 w 2 16 κ w 2 / 8 κ exp ( 2 r 2 8 κ x ) { 1 exp [ ( x w 2 / 8 κ ) / τ ] } × [ sin ω m ( t + w 2 8 κ ) cos ω m x cos ω m ( t + w 2 8 κ ) sin ω m x ] d x .
T ( r w , t ) = T 0 w 2 16 κ { sin ω m ( t + ω 2 8 κ ) [ C i ( w 2 ω m 8 κ ) ] cos ω m ( t + ω 2 8 κ ) [ π 2 S i ( w 2 ω m 8 κ ) ] } ,
T ( r w , t ) = T 0 w 2 F 16 κ sin ( ω m t ϕ ) ,
F = { [ C i ( w 2 ω m 8 κ ) ] 2 + [ π 2 S i ( w 2 ω m 8 κ ) ] 2 } 1 / 2
ϕ = arctan ( cot F w 2 ω m / 8 κ ) .
T ( r w , t ) = T 0 2 ω m sin ω m t .
T ( r w , t ) T 0 τ c 2 ln ( r / ω m τ c ) .
T ( r w , t ) = T 0 sin ( ω m t ϕ ) 2 π κ [ 2 r ( ω m / κ ) 1 / 2 ] 1 / 2 ,
ϕ = π / 8 + r ( ω m / 2 κ ) 1 / 2 .

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