Abstract

This work presents an analytic treatment for photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. Focusing initially on the steady-state condition, the photon diffusion in these two geometries is solved in cylindrical coordinates by using modified Bessel functions and by applying the extrapolated boundary condition. For large cylinder diameter, the analytic solutions may be simplified to a format employing the physical source and its image source with respect to a semi-infinite geometry and a radius-dependent term to account for the shape and dimension of the cylinder. The analytic solutions and their approximations are evaluated numerically to demonstrate qualitatively the effect of the applicator curvature—either concave or convex—and the radius on the photon fluence rate as a function of the source–detector distance, in comparison with that in the semi-infinite geometry. This work is subjected to quantitative examination in a coming second part and possible extension to time-resolved analysis.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2009

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

2007

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266-274 (2007).
[CrossRef] [PubMed]

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

2006

S. T. Cui, “Electrostatic potential in cylindrical dielectric media using the image charge method,” Mol. Phys. 104, 2993-3001 (2006).
[CrossRef]

D. Piao, H. Xie, W. Zhang, J. S. Kransinski, G. Zhang, H. Dehghani, and B. W. Pogue, “Endoscopic, rapid near-infrared optical tomography,” Opt. Lett. 31, 2876-2878 (2006).
[CrossRef] [PubMed]

2001

1998

K. S. Fine and C. F. Driscoll, “The finite length diocotron mode,” Phys. Plasmas 5, 601-607 (1998).
[CrossRef]

1997

1994

1992

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

1989

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Harcourt, 2005).

Arridge, S. R.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Bartels, K. E.

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

Boutet, J.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Bunting, C. F.

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

Carpenter, C.

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

Cheung, R.

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266-274 (2007).
[CrossRef] [PubMed]

Choi, H.

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266-274 (2007).
[CrossRef] [PubMed]

Contini, D.

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Cui, S. T.

S. T. Cui, “Electrostatic potential in cylindrical dielectric media using the image charge method,” Mol. Phys. 104, 2993-3001 (2006).
[CrossRef]

Debourdeau, M.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Dehghani, H.

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Dinten, J.-M.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Driscoll, C. F.

K. S. Fine and C. F. Driscoll, “The finite length diocotron mode,” Phys. Plasmas 5, 601-607 (1998).
[CrossRef]

Duboeuf, F.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Fantini, S.

Feng, T.

Fine, K. S.

K. S. Fine and C. F. Driscoll, “The finite length diocotron mode,” Phys. Plasmas 5, 601-607 (1998).
[CrossRef]

Franceschini, M. A.

Gratton, E.

Guyon, L.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Haskell, R. C.

Herve, L.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Holyoak, G. R.

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

Ishimaru, A.

Jackson, J. D.

J. D. Jackson, “Expansion of Green functions in cylindrical coordinates,” in Classical Electrodynamics, 3rd ed. (Wiley, 1998), pp. 125-126.

Jiang, S.

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

Jiang, Z.

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

Kaufman, P. A.

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

Kransinski, J. S.

Li, C.

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266-274 (2007).
[CrossRef] [PubMed]

Liengsawangwong, R.

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266-274 (2007).
[CrossRef] [PubMed]

Martelli, F.

McAdams, M. S.

Moler, C.

C. Moler, “Floating points: IEEE standard unifies arithmetic model,” Cleve's Corner, The MathWorks, Inc., 1996.

Ntziachristos, V.

V. Ntziachristos, “Concurrent diffuse optical tomography, spectroscopy and magnetic resonance imaging of breast cancer,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pennsylvania, 2000).

Patterson, M. S.

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]

Paulsen, K. D.

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

Peltie, P.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Piao, D.

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

D. Piao, H. Xie, W. Zhang, J. S. Kransinski, G. Zhang, H. Dehghani, and B. W. Pogue, “Endoscopic, rapid near-infrared optical tomography,” Opt. Lett. 31, 2876-2878 (2006).
[CrossRef] [PubMed]

Pogue, B. W.

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

D. Piao, H. Xie, W. Zhang, J. S. Kransinski, G. Zhang, H. Dehghani, and B. W. Pogue, “Endoscopic, rapid near-infrared optical tomography,” Opt. Lett. 31, 2876-2878 (2006).
[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]

Poplack, S. P.

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

Ritchey, J. W.

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

Saroul, L.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Sassaroli, A.

Slobodov, G.

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

Srinivasan, S.

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T.

Vray, D.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

Wang, L. V.

L. V. Wang and H. Wu, Biomedical Optics, Principles and Imaging (Wiley, 2007).

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Harcourt, 2005).

Wells, W. A.

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

Wu, H.

L. V. Wang and H. Wu, Biomedical Optics, Principles and Imaging (Wiley, 2007).

Xie, H.

Xu, G.

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

Yamada, Y.

Zaccanti, G.

Zhang, G.

Zhang, W.

Antioxid. Redox. Signal.

S. Srinivasan, B. W. Pogue, C. Carpenter, S. Jiang, W. A. Wells, S. P. Poplack, P. A. Kaufman, and K. D. Paulsen, “Developments in quantitative oxygen-saturation imaging of breast tissue in vivo using multispectral near-infrared tomography,” Antioxid. Redox. Signal. 9, 1143-1156 (2007) (review).
[CrossRef] [PubMed]

Appl. Opt.

J. Biomed. Opt.

J. Boutet, L. Herve, M. Debourdeau, L. Guyon, P. Peltie, J.-M. Dinten, L. Saroul, F. Duboeuf, and D. Vray, “Bimodal ultrasound and fluorescence approach for prostate cancer diagnosis,” J. Biomed. Opt. 14, 064001 (2009).
[CrossRef]

J. Innovative Opt. Health Sciences

D. Piao, Z. Jiang, K. E. Bartels, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “In vivo trans-rectal ultrasound-coupled near-infrared optical tomography of intact normal canine prostate,” J. Innovative Opt. Health Sciences 2, 215-225 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Med. Phys.

C. Li, R. Liengsawangwong, H. Choi, and R. Cheung, “Using a priori structural information from magnetic resonance imaging to investigate the feasibility of prostate diffuse optical tomography and spectroscopy: a simulation study,” Med. Phys. 34, 266-274 (2007).
[CrossRef] [PubMed]

Mol. Phys.

S. T. Cui, “Electrostatic potential in cylindrical dielectric media using the image charge method,” Mol. Phys. 104, 2993-3001 (2006).
[CrossRef]

Opt. Lett.

Phys. Med. Biol.

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]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Phys. Plasmas

K. S. Fine and C. F. Driscoll, “The finite length diocotron mode,” Phys. Plasmas 5, 601-607 (1998).
[CrossRef]

Other

V. Ntziachristos, “Concurrent diffuse optical tomography, spectroscopy and magnetic resonance imaging of breast cancer,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pennsylvania, 2000).

J. D. Jackson, “Expansion of Green functions in cylindrical coordinates,” in Classical Electrodynamics, 3rd ed. (Wiley, 1998), pp. 125-126.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Harcourt, 2005).

L. V. Wang and H. Wu, Biomedical Optics, Principles and Imaging (Wiley, 2007).

IEEE 754-2008 Standard for Floating-Point Arithmetic (IEEE, 2008).

C. Moler, “Floating points: IEEE standard unifies arithmetic model,” Cleve's Corner, The MathWorks, Inc., 1996.

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Figures (11)

Fig. 1
Fig. 1

(a) Semi-infinite geometry [2]. The diffuse medium is to the right of the physical boundary, and the light is incident from the left. (b) The two cylindrical geometries in comparison with the semi-infinite geometry. The convex boundary represents that of a cylindrical applicator enclosed by the diffuse medium (e.g., imaging the prostate by a trans-rectal probe), and the concave boundary represents that of a cylindrical applicator enclosing the diffuse medium (e.g., imaging the breast by a ring probe).

Fig. 2
Fig. 2

(a) Details of the concave geometry indicating the equivalent isotropic source and the extrapolated boundary. The image source of the isotropic source with respect to the extrapolated boundary is located along the radial direction of the isotropic source due to symmetry. (b) The concave geometry and the “semi-infinite” image source that is the image source of the isotropic source with respect to a planar boundary tangential to the concave boundary at the location of the physical source.

Fig. 3
Fig. 3

(a) Details of the convex geometry indicating the equivalent isotropic source and the extrapolated boundary. (b) The convex geometry and the “semi-infinite” image source that is the image source of the isotropic source with respect to a planar boundary tangential to the concave boundary at the location of the physical source.

Fig. 4
Fig. 4

(a) Comparison of the contributions of the k terms when evaluating the cylindrical-coordinate solution to the steady-state photon diffusion in the homogeneous infinite medium. (b) Comparison between the solutions in spherical coordinates and cylindrical coordinates to the steady-state photon diffusion in the homogeneous infinite medium.

Fig. 5
Fig. 5

Concave and convex geometries with the source and the detector located at the same azimuthal plane.

Fig. 6
Fig. 6

(a) Outcome of applying pre-enlarge and pre-reduce methods for R 0 = 2 cm in convex geometry. (b) Outcome of applying “repeated averaging” for R 0 = 8 cm in concave geometry.

Fig. 7
Fig. 7

Comparison of the contributions of k terms in the solution for source and detector located in the same azimuthal plane: (a) concave geometry; (b) convex geometry.

Fig. 8
Fig. 8

(a) Comparison of the solutions for concave and convex geometries with respect to the semi-infinite geometry, for source and detector located at the same azimuthal plane. (b) Comparison of the solutions for concave and convex geometries having large cylinder radius with respect to the semi-infinite geometry for source and detector located at the same azimuthal plane.

Fig. 9
Fig. 9

Concave and convex geometries with source and detector located longitudinally with the same azimuthal angle.

Fig. 10
Fig. 10

Comparison of the contributions of k terms in the solution for source and detector located longitudinally with the same azimuthal angle: (a) concave boundary; (b) convex boundary.

Fig. 11
Fig. 11

(a) Comparison of the solutions for concave and convex geometries with respect to the semi-infinite geometry for source and detector located longitudinally with the same azimuthal angle. (b) Comparison of the solutions for concave and convex geometries having large cylinder diameter with respect to the semi-infinite geometry for source and detector located longitudinally with the same azimuthal angle.

Equations (78)

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

2 Ψ ( r ) μ a D Ψ ( r ) = S ( r ) D ,
2 G ( r , r ) k 0 2 G ( r , r ) = δ ( r r ) ,
δ ( r r ) = ( 1 ρ ) δ ( ρ ρ ) δ ( φ φ ) δ ( z z ) ,
δ ( φ φ ) = 1 2 π m = e i m ( φ φ )
δ ( z z ) = 1 2 π 0 d k e i k ( z z ) = 1 π 0 d k cos [ k ( z z ) ] .
1 ρ ρ ( ρ G ( r , r ) ρ ) + 1 ρ 2 2 G ( r , r ) φ 2 + 2 G ( r , r ) z 2 k 0 2 G ( r , r ) = 1 2 π 2 ρ δ ( ρ ρ ) m = 0 d k e i m ( φ φ ) cos [ k ( z z ) ] .
G ( r , r ) = 1 2 π 2 m = 0 d k g m ( k , ρ , ρ ) e i m ( φ φ ) cos [ k ( z z ) ] ,
1 ρ ρ ( ρ g m ( k , ρ , ρ ) ρ ) ( k 2 + k 0 2 + m 2 ρ 2 ) g m ( k , ρ , ρ ) = 1 ρ δ ( ρ ρ ) .
k eff 2 = k 2 + k 0 2 or k eff = k 2 + k 0 2 .
1 ρ ρ ( ρ g m ( k , ρ , ρ ) ρ ) ( k eff 2 + m 2 ρ 2 ) g m ( k , ρ , ρ ) = 1 ρ δ ( ρ ρ ) .
G ( r , r ) = 1 2 π 2 0 d k { m = 0 ϵ m I m ( k eff ρ < ) K m ( k eff ρ > ) cos [ m ( φ φ ) ] } cos [ k ( z z ) ] ,
ϵ m = { 2 , m 0 1 , m = 0 } .
Ψ ( r , r ) = S 2 π 2 D 0 d k { m = 0 ϵ m I m ( k eff ρ < ) K m ( k eff ρ > ) cos [ m ( φ φ ) ] } cos [ k ( z z ) ] .
Ψ ( r , r ) = S 4 π D | r r | e k 0 | r r | .
ln ( Ψ d ) = k 0 d + ln ( S 4 π D ) .
Ψ 2 A D Ψ n = 0 ,
Ψ = Ψ real Ψ imag = S 4 π D l real e k 0 l real S 4 π D l imag e k 0 l imag ,
l real = d 2 + R a 2 , R a = 1 μ s ;
l imag = d 2 + ( 2 R b + R a ) 2 , R b = 2 A D .
ln ( Ψ d 2 ) = k 0 d + ln ( s 2 π D k 0 R b ( R a + R b ) ) ,
| Ψ real | extr = 1 2 π 2 D 0 d k cos [ k ( z z ) ] { m = 0 ϵ m S I m [ k eff ( R 0 R a ) ] K m [ k eff ( R 0 + R b ) ] cos [ m ( φ φ ) ] } ,
| Ψ imag | extr = 1 2 π 2 D 0 d k { m = 0 ϵ m S m * I m [ k eff ( R 0 + R b ) ] K m [ k eff ρ i > ] cos [ m ( φ φ ) ] } cos [ k ( z z ) ] .
S m * K m ( k eff ρ i > ) = S m I m [ k eff ρ r < ] = S m I m [ k eff ( R 0 R a ) ] .
S m = S K m k eff ( R 0 + R b ) I m [ k eff ( R 0 + R b ) ] , m = 0 , 1 , 2 , .
Ψ = | Ψ real | phys | Ψ imag | phys = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ϵ m I m [ k eff ( R 0 R a ) ] K m ( k eff R 0 ) 1 I m ( k eff R 0 ) K m ( k eff R 0 ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] cos [ m ( φ φ ) ] } .
| Ψ imag semi | phys = S 2 π 2 D 0 d k cos [ k ( z z ) ] m = 0 ϵ m I m ( k eff R 0 ) K m [ k eff ( R 0 + R a + 2 R b ) ] cos [ m ( φ φ ) ] .
| Ψ imag | phys = S 2 π 2 D 0 d k cos [ k ( z z ) ] m = 0 ϵ m I m ( k eff R 0 ) K m [ k eff ( R 0 + R a + 2 R b ) ] η m cos [ m ( φ φ ) ] ,
η m = I m k eff ( R 0 R a ) I m [ k eff ( R 0 + R b ) ] K m k eff ( R 0 + R b ) K m [ k eff ( R 0 + R a + 2 R b ) ] .
η m = R 0 + R a + 2 R b R 0 R a .
| Ψ imag | phys = | Ψ imag semi | phys R 0 + R a + 2 R b R 0 R a .
Ψ = | Ψ real | phys | Ψ imag | phys = | Ψ real | phys | Ψ imag semi | phys R 0 + R a + 2 R b R 0 R a .
Ψ = S 4 π D e k 0 l r l r S 4 π D e k 0 l i l i R 0 + R a + 2 R b R 0 R a .
| Ψ real | extr = 1 2 π 2 D 0 d k cos [ k ( z z ) ] { m = 0 ϵ m S I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 + R a ) ] cos [ m ( φ φ ) ] } ,
| Ψ imag | extr = 1 2 π 2 D 0 d k { m = 0 ϵ m S m * I m [ k eff ρ i < ] K m [ k eff ( R 0 R b ) ] cos [ m ( φ φ ) ] } cos [ k ( z z ) ] .
S m * I m ( k eff ρ i < ) = S m K m [ k eff ρ r > ] = S m K m [ k eff ( R 0 + R a ) ] ,
S m = S I m k eff ( R 0 R b ) K m [ k eff ( R 0 R b ) ] m = 0 , 1 , 2 , .
Ψ = | Ψ real | phys | Ψ imag | phys = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ϵ m I m ( k eff R 0 ) K m [ k eff ( R 0 + R a ) ] 1 K m ( k eff R 0 ) I m ( k eff R 0 ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] cos [ m ( φ φ ) ] } .
| Ψ imag semi | phys = S 2 π 2 D 0 d k cos [ k ( z z ) ] m = 0 ϵ m I m [ k eff ( R 0 R a 2 R b ) ] K m ( k eff R 0 ) cos [ m ( φ φ ) ] .
| Ψ imag | phys = S 2 π 2 D 0 d k cos [ k ( z z ) ] m = 0 ϵ m I m [ k eff ( R 0 R a 2 R b ) ] K m ( k eff R 0 ) η m cos [ m ( φ φ ) ] ,
η m = I m k eff ( R 0 R b ) I m [ k eff ( R 0 R a 2 R b ) ] K m k eff ( R 0 + R a ) K m [ k eff ( R 0 R b ) ] .
η m = R 0 R a 2 R b R 0 + R a .
| Ψ imag | phys = | Ψ imag semi | phys R 0 R a 2 R b R 0 + R a .
Ψ = | Ψ real | phys | Ψ imag | phys = | Ψ real | phys | Ψ imag semi | phys R 0 R a 2 R b R 0 + R a .
Ψ = S 4 π D e k 0 l r l r S 4 π D e k 0 l i l i R 0 R a 2 R b R 0 + R a .
Ψ = S 2 π 2 D 0 d k cos [ k ( z z ) ] m = 0 ϵ m I m ( k eff ρ < ) K m ( k eff ρ > ) cos [ m ( φ φ ) ] .
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ϵ m I m [ k eff ( R 0 R a ) ] K m ( k eff R 0 ) 1 I m ( k eff R 0 ) K m ( k eff R 0 ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] cos [ m ( φ φ ) ] } ,
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ϵ m I m [ k eff R 0 ] K m [ k eff ( R 0 + R a ) ] 1 K m ( k eff R 0 ) I m ( k eff R 0 ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] cos [ m ( φ φ ) ] } .
Ψ = S 4 π D e k 0 l r l r S 4 π D e k 0 l i l i R 0 + R a + 2 R b R 0 R a ,
Ψ = S 4 π D e k 0 l r l r S 4 π D e k 0 l i l i R 0 R a 2 R b R 0 + R a ,
Ψ = ( S 4 π D d ) e k 0 d ,
Ψ ( ρ , φ ) = S 2 π 2 D 0 d k m = 0 ϵ m I m ( 0 ) K m ( k eff ρ > ) .
I ( k eff ρ < ) K ( k eff ρ > ) = 1 2 k eff ρ < ρ > e k eff ( ρ > ρ < ) ,
Ω = m = 0 ϵ m I m ( 0 ) K m ( k eff ρ > ) .
Ψ = S 2 π 2 D 0 d k { m = 0 ϵ m I m [ k eff ( R 0 R a ) ] K m ( k eff R 0 ) 1 I m ( k eff R 0 ) K m ( k eff R 0 ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] cos [ m ( φ φ ) ] } ,
Ψ = S 2 π 2 D 0 d k { m = 0 ϵ m I m ( k eff R 0 ) K m [ k eff ( R 0 + R a ) ] 1 K m ( k eff R 0 ) I m ( k eff R 0 ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] cos [ m ( φ φ ) ] } .
I m [ k eff ( R 0 R a ) ] K m ( k eff R 0 ) 1 I m ( k eff R 0 ) K m ( k eff R 0 ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] = e k eff R a 2 k eff R 0 ( R 0 R a ) ( 1 e 2 k eff R b ) ,
I m ( k eff R 0 ) K m [ k eff ( R 0 + R a ) ] 1 K m ( k eff R 0 ) I m ( k eff R 0 ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] = e k eff R a 2 k eff R 0 ( R 0 + R a ) ( 1 e 2 k eff R b ) .
l r = [ R a 2 + d 2 ( R a d 2 R 0 ) ] 1 2 ,
l i = [ ( R a + 2 R b ) 2 + d 2 + ( R a + 2 R b ) ( d 2 R 0 ) ] 1 2
l r = [ R a 2 + d 2 + ( R a d 2 R 0 ) ] 1 2 ,
l i = [ ( R a + 2 R b ) 2 + d 2 ( R a + 2 R b ) ( d 2 R 0 ) ] 1 2
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ϵ m I m [ k eff ( R 0 R a ) ] K m ( k eff R 0 ) 1 I m ( k eff R 0 ) K m ( k eff R 0 ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] } ,
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ϵ m I m ( k eff R 0 ) K m [ k eff ( R 0 + R a ) ] 1 K m ( k eff R 0 ) I m ( k eff R 0 ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] } .
l r = ( R a 2 + d 2 ) 1 2 ,
l i = [ ( R a + 2 R b ) 2 + d 2 + ] 1 2
1 ρ ρ ( ρ g m ( k , ρ , ρ ) ρ ) ( k eff 2 + m 2 ρ 2 ) g m ( k , ρ , ρ ) = 1 ρ δ ( ρ ρ ) .
g m ( k , ρ , ρ ) = ψ 1 ( k eff ρ < ) ψ 2 ( k eff ρ > ) ,
| d g m d ρ | + | d g m d ρ | = 1 ρ ,
| d g m d ρ | + | d g m d ρ | = k eff ( ψ 1 ψ 2 ψ 2 ψ 1 ) = k eff W [ ψ 1 , ψ 2 ] ,
d d x [ p ( x ) d y d x ] + g ( x ) y = 0 ,
W [ ψ 1 ( x ) , ψ 2 ( x ) ] = 1 x ,
ψ 1 ( k eff ρ < ) = Ω I m ( k eff ρ < ) and ψ 2 ( k eff ρ > ) = k m ( k eff ρ > ) ,
Ω W [ I m ( x ) , K m ( x ) ] = 1 x ,
W [ I m ( x ) , K m ( x ) ] = 1 x ,
g m ( k , ρ , ρ ) = I m ( k eff ρ < ) K m ( k eff ρ > ) .
G ( r , r ) = 1 2 π 2 m = 0 d k e i m ( φ φ ) [ I m ( k eff ρ < ) K m ( k eff ρ > ) ] cos [ k ( z z ) ] .
G ( r , r ) = 1 2 π 2 0 d k { m = 0 ϵ m I m ( k eff ρ < ) K m ( k eff ρ > ) cos [ m ( φ φ ) ] } cos [ k ( z z ) ] ,
where ϵ m = { 2 , m 0 1 , m = 0 } .

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