Abstract

This paper is the first of two dealing with light diffusion in a turbid cylinder. The diffusion equation was solved for a homogeneous finite cylinder that is illuminated at an arbitrary location. Three solutions were derived for an incident δ-light source in the steady-state, frequency, and time domains, respectively, applying different integral transformations. The performance of these solutions was compared with respect to accuracy and speed. Excellent agreement between the solutions, of which some are very fast (< 10ms), was found. Six of the nine solutions were extended to a circular flat beam which is incident onto the top side. Furthermore, the validity of the solutions was tested against Monte Carlo simulations.

© 2010 Optical Society of America

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References

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  1. F. Voit, J. Schäfer, and A. Kienle, “Light Scattering by Multiple Spheres: Comparison between Maxwell Theory and Radiative-Transfer-Theory Calculations,” Opt. Lett. 34, 2593–2595 (2009).
    [CrossRef] [PubMed]
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic Press, New York, 1978).
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    [CrossRef] [PubMed]
  4. T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A Diffusion Theory Model of Spatially Resolved, Steady-State Diffuse Reflectance for the Noninvasive Determination of Tissue Optical Properties in Vivo,” Med. Phys. 19, 879–888 (1992).
    [CrossRef] [PubMed]
  5. D. Contini, F. Martelli, and G. Zaccanti, “Photon Migration through a Turbid Slab Described by a Model Based on Diffusion Approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
    [CrossRef] [PubMed]
  6. S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Path length in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [CrossRef] [PubMed]
  7. B. W. Pogue, and M. S. Patterson, “Frequency Domain Optical Absorption Spectroscopy of Finite Tissue Volumes Using Diffusion Theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
    [CrossRef] [PubMed]
  8. A. Kienle, “Light Diffusion through a Turbid Parallelepiped,” J. Opt. Soc. Am. A 22, 1883–1888 (2005).
    [CrossRef]
  9. A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. 44, 2747–2763 (1999).
    [CrossRef] [PubMed]
  10. A. Liemert, and A. Kienle, “Light diffusion in a turbid cylinder. II. Layered case,” accepted (2010).
  11. http://www.uni-ulm.de/ilm/index.php?id=10020200.
  12. A. Kienle, and M. S. Patterson, “Improved Solutions of the Steady-State and the Time-Resolved Diffusion Equations for Reflectance from a Semi-Infinite Turbid Medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
    [CrossRef]
  13. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  14. E. Meissel, and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics, (Bibliobazaar, 2008).
  15. A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van den Bergh, “Noninvasive Determination of the Optical Properties of Two-Layered Turbid Media,” Appl. Opt. 37, 779–791 (1998).
    [CrossRef]
  16. H. S. Carslaw, and J. C. Jaeger, Conduction of Heat in Solids, (Clarendon, Oxford, 1959).
  17. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1971).
  18. A. Liemert, and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt. 15, 025002 (2010).
    [CrossRef] [PubMed]

2010 (1)

A. Liemert, and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt. 15, 025002 (2010).
[CrossRef] [PubMed]

2009 (1)

2005 (1)

1999 (1)

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. 44, 2747–2763 (1999).
[CrossRef] [PubMed]

1998 (1)

1997 (2)

1994 (2)

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

B. W. Pogue, and M. S. Patterson, “Frequency Domain Optical Absorption Spectroscopy of Finite Tissue Volumes Using Diffusion Theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef] [PubMed]

1992 (2)

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A Diffusion Theory Model of Spatially Resolved, Steady-State Diffuse Reflectance for the Noninvasive Determination of Tissue Optical Properties in Vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Path length in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

1989 (1)

Arridge, S. R.

S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Path length in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Bays, R.

Chance, B.

Contini, D.

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Path length in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Path length in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Dögnitz, N.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A Diffusion Theory Model of Spatially Resolved, Steady-State Diffuse Reflectance for the Noninvasive Determination of Tissue Optical Properties in Vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Feng, T. C.

Haskell, R. C.

Imai, D.

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. 44, 2747–2763 (1999).
[CrossRef] [PubMed]

Kienle, A.

Liemert, A.

A. Liemert, and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt. 15, 025002 (2010).
[CrossRef] [PubMed]

Martelli, F.

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. 44, 2747–2763 (1999).
[CrossRef] [PubMed]

D. Contini, F. Martelli, and G. Zaccanti, “Photon Migration through a Turbid Slab Described by a Model Based on Diffusion Approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
[CrossRef] [PubMed]

McAdams, M.

Patterson, M. S.

Pogue, B. W.

B. W. Pogue, and M. S. Patterson, “Frequency Domain Optical Absorption Spectroscopy of Finite Tissue Volumes Using Diffusion Theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef] [PubMed]

Sassaroli, A.

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. 44, 2747–2763 (1999).
[CrossRef] [PubMed]

Schäfer, J.

Svaasand, L. O.

Tromberg, B. J.

Tsay, T. T.

van den Bergh, H.

Voit, F.

Wagnières, G.

Wilson, B. C.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A Diffusion Theory Model of Spatially Resolved, Steady-State Diffuse Reflectance for the Noninvasive Determination of Tissue Optical Properties in Vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time Resolved Reflectance and Transmittance for the Noninvasive Measurement of Tissue Optical Properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Yamada, Y.

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. 44, 2747–2763 (1999).
[CrossRef] [PubMed]

Zaccanti, G.

Appl. Opt. (3)

J. Biomed. Opt. (1)

A. Liemert, and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt. 15, 025002 (2010).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A Diffusion Theory Model of Spatially Resolved, Steady-State Diffuse Reflectance for the Noninvasive Determination of Tissue Optical Properties in Vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Med. Biol. (3)

A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. 44, 2747–2763 (1999).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Path length in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

B. W. Pogue, and M. S. Patterson, “Frequency Domain Optical Absorption Spectroscopy of Finite Tissue Volumes Using Diffusion Theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef] [PubMed]

Other (6)

E. Meissel, and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics, (Bibliobazaar, 2008).

H. S. Carslaw, and J. C. Jaeger, Conduction of Heat in Solids, (Clarendon, Oxford, 1959).

M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1971).

A. Liemert, and A. Kienle, “Light diffusion in a turbid cylinder. II. Layered case,” accepted (2010).

http://www.uni-ulm.de/ilm/index.php?id=10020200.

A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic Press, New York, 1978).

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Figures (8)

Fig. 1.
Fig. 1.

Scheme of the cylinder used in the calculations. Exemplarily, the beam is incident perpendicular onto the cylinder top at r⃗ = (ρ 0, ϕ 0,0). The isotropic point source is located at r⃗ = (ρ 0, ϕ, z 0)

Fig. 2.
Fig. 2.

Steady-state radial transmittance around the cylinder barrel at z = 10mm and ρ = a. The cylinder is illuminated perpendicular to the cylinder barrel at z 0 = 8mm, see inset. The optical parameters are μ s = 0.9mm-1, μa = 0.005 (black),0.01 (green),0.015-1(red)mm , n 0 = 1.0, and n 1 = 1.4. The geometrical data of the cylinder are a = 5mm and lz = 20mm.

Fig. 3.
Fig. 3.

Relative difference of two steady-state solutions for the cylinder considered in Fig. 2.

Fig. 4.
Fig. 4.

Time resolved transmittance from a cylinder that is illuminated at the cylinder barrel at z 0 = 9mm. The transmittance is detected in z-direction at z = lz , ρ = 0mm (red curve) and in radial direction at z = 9mm, ρ = a (black curve). The optical and geometrical parameters are μ s = 0.8mm-1, μa = 0.03mm-1, n 0 = 1.0, n 1 = 1.4, a = 9mm, and lz = 12mm. The dashed curves show the transmittance from a homogeneous cube.

Fig. 5.
Fig. 5.

The transmittance of a sinusoidally oscillating source incident at z = 12mm, ρ = a and detected at the same height at ϕ = 3π/4. The optical and geometrical parameters are μ s = 0.7mm-1, μa = 0.02mm-1, n 0 = 1.0, n 1 = 1.4, a = 5,6,7mm, lz = 22mm, z 0 = 11mm. The modulation frequency is ω 0 = 2π · 500MHz.

Fig. 6.
Fig. 6.

Comparison of the spatially resolved reflectance calculated with the solution of the diffusion equation and with the Monte Carlo method for a turbid cylinder illuminated with a pencil beam in the middle of the cylinder top. The optical and geometrical parameters of the cylinders are μ s = 1.3mm-1, μa = 0.008mm-1, a = 14mm, n 1 = n 0 = 1.0, lz = 5 and 10mm.

Fig. 7.
Fig. 7.

Comparison of the spatially resolved transmittance calculated with the solution of the diffusion equation and with Monte Carlo simulations for a turbid cylinder illuminated with a pencil beam in the middle of the cylinder top. The optical and geometrical parameters of the cylinders are μ s = 1.3mm-1, μa = 0.008mm-1, a = 14mm, n 1 = n 0 = 1.0, lz = 5 and 10mm.

Fig. 8.
Fig. 8.

Spatially resolved reflectance from a homogeneous cylinder that is illuminated by a flat beam with ρw = 0,5,10mm. The optical and geometrical parameters are μ s = 1.2mm-1, μa = 0.01mm-1, a = 20mm, l2 = 5mm, and n 1 = n 0 = 1.0.

Equations (84)

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ΔΦ ( r , ω ) ( μ a D + i ω Dc ) Φ ( r , ω ) = 1 D δ ( r r 0 ) ,
2 ρ 2 Φ + 1 ρ ρ Φ + 1 ρ 2 2 ϕ 2 Φ + 2 z 2 Φ ( μ a D + i ω Dc ) Φ = 1 δ ( ρ ρ 0 ) δ ( ϕ ϕ 0 ) δ ( z z 0 ) ,
Φ ( ϕ , m ) = 0 2 π Φ ( ϕ ) cos ( m ( ϕ ϕ ) ) , m = 0,1,2 ,
2 ρ 2 Φ + 1 ρ ρ Φ m 2 ρ 2 Φ + 2 z 2 Φ ( μ a D + i ω Dc ) Φ = cos ( m ( ϕ ϕ 0 ) ) δ ( ρ ρ 0 ) δ ( z z 0 ) .
Φ ( s n ) = 0 a ρ Φ ( ρ ) J m ( s n ρ ) d ρ ,
z b = 1 + R eff 1 R eff 2 D ,
0 a ρ ( 2 ρ 2 Φ + 1 ρ ρ Φ m 2 ρ 2 Φ ) J m ( s n ρ ) d ρ = s n 2 Φ ( s n ) + a s n J m + 1 ( a s n ) Φ ( ρ = a )
2 z 2 Φ ( s n 2 + μ a D + i ω Dc ) Φ = 1 D J m ( s n ρ 0 ) cos ( m ( ϕ ϕ 0 ) ) δ ( z z 0 ) .
2 z 2 G ( s n , z , ω ) α 2 G ( s n , z , ω ) = 1 D δ ( z z 0 ) ,
Φ ( s n , ϕ , m , z , ω ) = G ( s n , z , ω ) J m ( s n ρ 0 ) cos ( m ( ϕ ϕ 0 ) ) .
Φ ( s n , ϕ , m ) = 0 2 π 0 a ρ Φ ( ρ , ϕ ) J m ( s n ρ ) cos ( m ( ϕ ϕ ) ) d ρ d ϕ ,
m = 0 Φ ( ϕ , m ) = 0 2 π Φ ( ϕ ) m = 0 cos ( m ( ϕ ϕ ) ) d ϕ .
m = 0 cos ( m ( ϕ ϕ ) ) = 1 2 + πδ ( ϕ ϕ )
0 2 π Φ ( ϕ ) [ 1 2 + πδ ( ϕ ϕ ) ] = Φ ( ϕ , 0 ) + π Φ ( ϕ )
Φ ( ϕ ) = 1 2 π m = 0 ( 2 δ m , 0 ) Φ ( ϕ , m ) .
n = 1 ρ J m ( s n ρ ) J m ( s n ρ 0 ) J m + 1 2 ( a s n ) = a 2 2 δ ( ρ ρ 0 ) .
Φ ( ρ , ϕ ) = 1 π a 2 m = ( 2 δ m , 0 ) n = 1 Φ ( s n , ϕ , m ) J m ( s n ρ ) J m + 1 2 ( a s n ) .
Φ ( r , ω ) = 1 π a 2 m = 0 ( 2 δ m , 0 ) cos ( ) n = 1 G ( s n , z , ω ) J m ( s n ρ 0 ) J m ( s n ρ ) J m + 1 2 ( a s n ) ,
J m + 1 ( a s n ) = 2 m a s n J m ( a s n ) J m 1 ( a s n ) = J m 1 ( a s n ) .
Φ ( r , ω ) = 1 π a 2 m = cos ( ) n = 1 G ( s n , z , ω ) J m ( s n ρ 0 ) J m ( s n ρ ) J m + 1 2 ( a s n ) .
Φ ( r , ω ) = 1 π a 2 n = 1 G ( s n , z , ω ) J 0 ( s n ρ ) J 1 2 ( a s n ) .
T ( ϕ , z , ω ) = D Φ ( ρ , ϕ , z , ω ) / ρ | ρ = a
T ( ϕ , z , ω ) = D π a 2 m = cos ( )
× n = 1 G ( s n , z , ω ) J m ( s n ρ 0 ) ( s n J m + 1 ( a s n ) m a J m ( a s n ) ) J m + 2 2 ( a s n ) .
T ( z , ω ) = D π a 2 n = 1 s n G ( s n , z , ω ) J 1 ( a s n ) J 1 2 ( a s n ) .
2 ρ 2 Φ + 1 ρ ρ Φ + 2 z 2 Φ ( μ a D + i ω Dc ) Φ = 1 D S ( r , ω ) ,
S ( r , ω ) = 1 π ρ w 2 circ ( ρ ρ w ) δ ( z z 0 ) ,
circ ( ρ ρ w ) = { 1 , ρ ρ w 0 , ρ > ρ w .
2 z 2 Φ ( μ a D + s n 2 + i ω Dc ) Φ = 1 D S ( s n , z , ω ) = 1 D J 1 ( s n ρ w ) π s n ρ w δ ( z z 0 ) .
Φ ( s n , z , ω ) = J 1 ( s n ρ w ) π s n ρ w G ( s n , z , ω ) .
Φ ( r , ω ) = 2 a ' 2 n = 1 Φ ( s n , z , ω ) J 0 ( s n ρ ) J 1 2 ( s n a ) .
Φ ( r , ω ) = 2 π a 2 ρ w n = 1 G ( s n , z , ω ) J 0 ( s n ρ ) J 1 ( s n ρ w ) s n J 1 2 ( s n a ) .
Φ ( ρ , ϕ , z = z b , ω ) = 0
Φ ( ρ , ϕ , z = l z + z b , ω ) = 0 .
G ( s n , z , ω ) = 1 2 k = exp ( α z z 1 k ) exp ( α z z 2 k ) ,
z 1 k = 2 k l z + 4 k z b + z 0
z 2 k = 2 k l z + ( 4 k 2 ) z b z 0
α = μ a D + s n 2 + i ω Dc .
Φ ( r , ω ) = 1 2 πD a 2 m = cos ( ) n = 1 J m ( s n ρ 0 ) J m ( s n ρ ) α J m + 1 2 ( a s n )
× k = exp ( α z z 1 k ) exp ( α z z 2 k ) .
Φ ( r , ω ) = 1 2 πD a 2 n = 1 J 0 ( s n ρ ) α J 1 2 ( a s n ) k = exp ( α z z 1 k ) exp ( α z z 2 k ) .
G ( s n , z , ω ) = exp ( α z z 0 ) exp ( α ( z + z 0 + 2 z b ) ) 2 sinh [ z 0 + z b ]
× sinh [ α ( z + z b ) ] exp ( α ( l z + 2 z b ) ) sinh [ α ( l z + 2 z b ) ] .
Φ ( λ k ) = z 1 z 2 Φ ( z ) sin ( λ k ( z z 1 ) ) dz , λ k = / ( z 2 z 1 ) , k = 1,2 , ,
z 1 z 2 2 z 2 Φ ( z ) sin ( λ k ( z z 1 ) ) d z = λ k [ Φ ( z 1 ) ( 1 ) k Φ ( z 2 ) ] λ k 2 Φ ( λ k )
( s n 2 + λ k 2 + μ a D + i ω Dc ) Φ = 1 D J m ( s n ρ 0 ) sin ( λ k ( z 0 + z b ) ) cos ( m ( ϕ ϕ 0 ) ) .
Φ ( s n , ϕ , m , λ k , ω ) = c J m ( s n ρ 0 ) sin ( λ k ( z 0 + z b ) ) cos ( m ( ϕ ϕ 0 ) ) Dc ( s n 2 + λ k 2 ) + μ a c .
exp ( at ) ε ( t ) exp ( iωt ) dt = 1 a + , a > 0 ,
Φ ( s n , ϕ , m , λ k , t ) = c J m ( s n ρ 0 ) sin ( λ k ( z 0 + z b ) ) cos ( m ( ϕ ϕ 0 ) )
× exp ( μ a ct ) exp ( Dc λ k 2 t ) exp ( Dc s n 2 t ) , t > 0 .
Φ ( z ) = 2 z 2 z 1 k = 1 Φ ( λ k ) sin ( λ k ( z z 1 ) ) .
Φ ( r , t ) = 2 π a 2 ( l z + 2 z b ) exp ( μ a ct ) k = 1 exp ( Dc λ k 2 t ) sin ( λ k ( z 0 + z b ) ) sin ( λ k ( z + z b ) )
× m = cos ( ) n = 1 exp ( Dc s n 2 t ) J m ( s n ρ ) J m ( s n ρ ) J m + 1 2 ( a s n ) .
Φ ( r , t ) = 2 c π a 2 ( l z + 2 z b ) exp ( μ a ct ) k = 1 exp ( Dc λ k 2 t ) sin ( λ k ( z 0 + z b ) ) sin ( λ k ( z + z b ) )
× n = 1 exp ( Dc s n 2 t ) J 0 ( s n ρ ) J 1 2 ( a s n ) .
1 πt exp ( at ) exp ( b 2 4 t ) ε ( t ) exp ( iωt ) d t = exp ( b a + ) a + , a > 0
Φ ( r , t ) = 1 2 π a 2 c Dπt exp ( μ a ct ) k = exp ( ( z z 1 k ) 2 4 Dct ) exp ( ( z z 2 k ) 2 4 Dct )
× m = cos ( ) n = 1 exp ( Dc s n 2 t ) J m ( s n ρ 0 ) J m ( s n ρ ) J m + 1 2 ( a s n ) .
Φ ( r , t ) = 1 2 π a ' 2 c Dπt exp ( μ a ct ) k = exp ( ( z z 1 k ) 2 4 Dct ) exp ( ( z z 2 k ) 2 4 Dct )
n = 1 exp ( Dc s n 2 t ) J 0 ( s n ρ 0 ) J 1 2 ( a ' s n ) .
2 ρ 2 Φ + 1 ρ ρ Φ + 1 ρ 2 2 ϕ 2 α 2 Φ = 1 D δ ( ρ ρ 0 ) δ ( ϕ ϕ 0 ) ,
Φ ( ρ , φ , ω ) = 1 2 πD [ K 0 ( α ρ 2 + ρ 0 2 2 ρ ρ 0 cos φ ) .
m = cos ( ) I m ( α ρ 0 ) K m ( αa ) I m ( αρ ) I m ( αa ) ] .
Φ ( r , ω ) = 1 4 πD k = exp ( μ eff r 1 ) r 1 exp ( μ eff r 2 ) r 2 1 πD ( l z + 2 z b )
× k = 1 sin ( λ k ( z 0 + z b ) ) sin ( λ k ( z + z b ) ) m = cos ( ) I m ( α ρ 0 ) K m ( αa ) I m ( αρ ) I m ( αa ) ,
r 1 = ρ 2 + ρ 0 2 2 ρ ρ 0 cos φ + ( z z 1 k ) 2
r 2 = ρ 2 + ρ 0 2 2 ρ ρ 0 cos φ + ( z z 2 k ) 2
μ eff = μ a / D + / Dc .
T ( φ , z , ω ) = a ρ 0 cos φ 4 π k = ( μ eff + 1 / r 1 ) exp ( μ eff r 1 ) r 1 2 ( μ eff + 1 / r 2 ) exp ( μ eff r 2 ) r 2 2
+ 1 π ( l z + 2 z b ) k = 1 sin ( λ k ( z 0 + z b ) ) sin ( λ k ( z + z b ) )
× m = cos ( ) I m ( α ρ 0 ) K m ( αa ' ) [ α I m + 1 ( αρ ) + m / a I m ( αa ) ] I m ( αa ' ) .
Φ ( r , ω ) = 1 4 πD k = exp ( μ eff r 1 ) r 1 exp ( μ eff r 2 ) r 2
1 πD ( l z + 2 z b ) k = 1 sin ( λ k ( z 0 + z b ) ) sin ( λ k ( z + z b ) ) K 0 ( αa ' ) I 0 ( αρ ) I 0 ( αa ' ) ,
T ( z , ω ) = a 4 π k = ( μ eff + 1 / r 1 ) exp ( μ eff r 1 ) r 1 2 ( μ eff + 1 / r 2 ) exp ( μ eff r 2 ) r 2 2
1 π ( l z + 2 z b ) k = 1 α sin ( λ k ( z 0 + z b ) ) sin ( λ k ( z + z b ) ) K 0 ( αa ) I 1 ( αa ) I 0 ( αa ) .
Φ ( r , t ) = 1 4 πD πDct k = exp ( ( z z 1 k ) 2 4 Dct ) exp ( ( z z 2 k ) 2 4 Dct )
× [ 1 2 t exp ( μ a ct ) exp ( ρ 2 ρ 0 2 2 ρ ρ 0 cos ρ 4 Dct )
1 2 π m = cos ( ) I m ( μ eff ρ 0 ) K m ( μ eff a ) I m ( μ eff ρ ) I m ( μ eff a ) exp ( iωt ) d ω ] ,
T ( φ , z , t ) = 1 4 π πDct k = exp ( ( z z 1 k ) 2 4 Dct ) exp ( ( z z 2 k ) 2 4 Dct )
× [ a ρ 0 cos φ 4 Dc t 2 exp ( μ a ct ) exp ( a 2 ρ 0 2 2 a ρ 0 cos ρ 4 Dct )
+ 1 2 π m = cos ( ) I m ( μ eff ρ 0 ) K m ( μ eff a ) I m ( μ eff a ) I m ( μ eff a ) exp ( iωt ) d ω ] ,
S ( r , t ) = δ ( r r 0 ) cos ( ω 0 t ) .
T ( r , t ) = Re { T ( r , ω = ω 0 ) e i ω 0 t } .
T ( r , ω 0 ) = 10 4 mm 2 × { 2.5774906387 1.2540621701 i 2.5774906366 1.2540621700 i .

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