Abstract

The focusing of light into a turbid medium was studied with Monte Carlo simulations. Focusing was found to have a significant effect on the absorption distribution in turbid media when the depth of the focal point (the distance between the focal point and the surface of the turbid media) was less than or comparable with the transport mean free path. Focusing could significantly increase the peak absorption and narrow the absorption distribution. As the depth of the focal point increased, the peak absorption decreased, and the depth of peak absorption increased initially but quickly reached a plateau that was less than the transport mean free path. A refractive-index-mismatched boundary between the ambient medium and the turbid medium deteriorated the focusing effect, increased the absorption near the boundary, lowered the peak absorption, and broadened the absorption distribution.

© 1999 Optical Society of America

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  1. A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
    [CrossRef] [PubMed]
  2. M. Schweiger, S. R. Arridge, “The finite-element method for the propagation of light in scattering media—frequency domain case,” Med. Phys. 24, 895–902 (1997).
    [CrossRef] [PubMed]
  3. A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
    [CrossRef]
  4. L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
    [CrossRef] [PubMed]
  5. A. Singh, K. P. Gopinathan, “Confocal microscopy—a powerful technique for biological research,” Curr. Sci. 74, 841–851 (1998).
  6. B. C. Wilson, G. A. Adam, “Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef] [PubMed]
  7. S. T. Flock, B. C. Wilson, D. R. Wyman, M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues I: model predictions and comparison with diffusion-theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
    [CrossRef] [PubMed]
  8. S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. Vol. IS5 of SPIE Institute Series (SPIE, Bellingham, Wash., 1989), pp. 102–111.
  9. S. L. Jacques, L.-H. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds., (Plenum, New York, 1995), pp. 73–100.
    [CrossRef]
  10. L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. Prog. Biomed. 47, 131–146 (1995). The MCML/CONV software package may be downloaded from URL: http://people.tamu.edu/~lwang .
    [CrossRef]
  11. A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1997).
    [CrossRef]
  12. E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
    [CrossRef] [PubMed]
  13. L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comp. Meth. Prog. Biomed. 54, 141–150 (1997).
    [CrossRef]
  14. I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).
  15. C. Sturesson, S. Andersson-Engels, “Mathematical modeling of dynamic cooling and pre-heating, used to increase the depth of selective damage to blood vessels in laser treatment of port wine stains,” Phys. Med. Biol. 41, 413–428 (1996).
    [CrossRef] [PubMed]
  16. C. G. A. Hoelen, F. F. M. Demul, R. Pongers, A. Dekker, “Three-dimensional photoacoustic imaging of blood vessels in tissue,” Opt. Lett. 23, 648–650 (1998).
    [CrossRef]

1998

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

A. Singh, K. P. Gopinathan, “Confocal microscopy—a powerful technique for biological research,” Curr. Sci. 74, 841–851 (1998).

C. G. A. Hoelen, F. F. M. Demul, R. Pongers, A. Dekker, “Three-dimensional photoacoustic imaging of blood vessels in tissue,” Opt. Lett. 23, 648–650 (1998).
[CrossRef]

1997

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1997).
[CrossRef]

M. Schweiger, S. R. Arridge, “The finite-element method for the propagation of light in scattering media—frequency domain case,” Med. Phys. 24, 895–902 (1997).
[CrossRef] [PubMed]

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comp. Meth. Prog. Biomed. 54, 141–150 (1997).
[CrossRef]

1996

C. Sturesson, S. Andersson-Engels, “Mathematical modeling of dynamic cooling and pre-heating, used to increase the depth of selective damage to blood vessels in laser treatment of port wine stains,” Phys. Med. Biol. 41, 413–428 (1996).
[CrossRef] [PubMed]

1995

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. Prog. Biomed. 47, 131–146 (1995). The MCML/CONV software package may be downloaded from URL: http://people.tamu.edu/~lwang .
[CrossRef]

1994

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

1993

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[CrossRef]

1989

S. T. Flock, B. C. Wilson, D. R. Wyman, M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues I: model predictions and comparison with diffusion-theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

1983

B. C. Wilson, G. A. Adam, “Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Adam, G. A.

B. C. Wilson, G. A. Adam, “Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Alianelli, L.

Andersson-Engels, S.

C. Sturesson, S. Andersson-Engels, “Mathematical modeling of dynamic cooling and pre-heating, used to increase the depth of selective damage to blood vessels in laser treatment of port wine stains,” Phys. Med. Biol. 41, 413–428 (1996).
[CrossRef] [PubMed]

Arridge, S. R.

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Blumetti, C.

Bonner, R. F.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[CrossRef]

Contini, D.

Cope, M.

Dekker, A.

Delpy, D. T.

Demul, F. F. M.

Feld, M. S.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Firbank, M.

Flock, S. T.

S. T. Flock, B. C. Wilson, D. R. Wyman, M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues I: model predictions and comparison with diffusion-theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Gandjbakhche, A. H.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[CrossRef]

Gopinathan, K. P.

A. Singh, K. P. Gopinathan, “Confocal microscopy—a powerful technique for biological research,” Curr. Sci. 74, 841–851 (1998).

Hielscher, A. H.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Hoelen, C. G. A.

Ismaelli, A.

Itzkan, I.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Jacques, S. L.

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comp. Meth. Prog. Biomed. 54, 141–150 (1997).
[CrossRef]

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. Prog. Biomed. 47, 131–146 (1995). The MCML/CONV software package may be downloaded from URL: http://people.tamu.edu/~lwang .
[CrossRef]

S. L. Jacques, L.-H. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds., (Plenum, New York, 1995), pp. 73–100.
[CrossRef]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. Vol. IS5 of SPIE Institute Series (SPIE, Bellingham, Wash., 1989), pp. 102–111.

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. Vol. IS5 of SPIE Institute Series (SPIE, Bellingham, Wash., 1989), pp. 102–111.

Koblinger, L.

I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

Lux, I.

I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

Martelli, F.

Nossal, R.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[CrossRef]

Okada, E.

Patterson, M. S.

S. T. Flock, B. C. Wilson, D. R. Wyman, M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues I: model predictions and comparison with diffusion-theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Perelman, L. T.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Pongers, R.

Prahl, S. A.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. Vol. IS5 of SPIE Institute Series (SPIE, Bellingham, Wash., 1989), pp. 102–111.

Sassaroli, A.

Schweiger, M.

Singh, A.

A. Singh, K. P. Gopinathan, “Confocal microscopy—a powerful technique for biological research,” Curr. Sci. 74, 841–851 (1998).

Sturesson, C.

C. Sturesson, S. Andersson-Engels, “Mathematical modeling of dynamic cooling and pre-heating, used to increase the depth of selective damage to blood vessels in laser treatment of port wine stains,” Phys. Med. Biol. 41, 413–428 (1996).
[CrossRef] [PubMed]

Wang, L.-H.

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comp. Meth. Prog. Biomed. 54, 141–150 (1997).
[CrossRef]

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. Prog. Biomed. 47, 131–146 (1995). The MCML/CONV software package may be downloaded from URL: http://people.tamu.edu/~lwang .
[CrossRef]

S. L. Jacques, L.-H. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds., (Plenum, New York, 1995), pp. 73–100.
[CrossRef]

Weiss, G. H.

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[CrossRef]

Welch, A. J.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. Vol. IS5 of SPIE Institute Series (SPIE, Bellingham, Wash., 1989), pp. 102–111.

Wilson, B. C.

S. T. Flock, B. C. Wilson, D. R. Wyman, M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues I: model predictions and comparison with diffusion-theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

B. C. Wilson, G. A. Adam, “Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Wu, J.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Wyman, D. R.

S. T. Flock, B. C. Wilson, D. R. Wyman, M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues I: model predictions and comparison with diffusion-theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Zaccanti, G.

Zheng, L.-Q.

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comp. Meth. Prog. Biomed. 54, 141–150 (1997).
[CrossRef]

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. Prog. Biomed. 47, 131–146 (1995). The MCML/CONV software package may be downloaded from URL: http://people.tamu.edu/~lwang .
[CrossRef]

Appl. Opt.

Comp. Meth. Prog. Biomed.

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comp. Meth. Prog. Biomed. 47, 131–146 (1995). The MCML/CONV software package may be downloaded from URL: http://people.tamu.edu/~lwang .
[CrossRef]

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comp. Meth. Prog. Biomed. 54, 141–150 (1997).
[CrossRef]

Curr. Sci.

A. Singh, K. P. Gopinathan, “Confocal microscopy—a powerful technique for biological research,” Curr. Sci. 74, 841–851 (1998).

IEEE Trans. Biomed. Eng.

S. T. Flock, B. C. Wilson, D. R. Wyman, M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissues I: model predictions and comparison with diffusion-theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Med. Phys.

B. C. Wilson, G. A. Adam, “Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, “The finite-element method for the propagation of light in scattering media—frequency domain case,” Med. Phys. 24, 895–902 (1997).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Med. Biol.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

C. Sturesson, S. Andersson-Engels, “Mathematical modeling of dynamic cooling and pre-heating, used to increase the depth of selective damage to blood vessels in laser treatment of port wine stains,” Phys. Med. Biol. 41, 413–428 (1996).
[CrossRef] [PubMed]

Phys. Rev. E

A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev. E 48, 810–818 (1993).
[CrossRef]

Phys. Rev. Lett.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Other

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. Vol. IS5 of SPIE Institute Series (SPIE, Bellingham, Wash., 1989), pp. 102–111.

S. L. Jacques, L.-H. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds., (Plenum, New York, 1995), pp. 73–100.
[CrossRef]

I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic of focusing an optical beam into a turbid medium, where z f was the depth of the focal point in the turbid medium, ρ f was the radius of the optical beam on the surface of the turbid medium, n a was the index of refraction of the ambient medium, and n, was the index of refraction of the turbid medium. Focusing is illustrated for both matched boundary condition (solid converging lines) and mismatched boundary condition (dashed lines with arrows). (b) Schematic of a skin model containing the epidermal and the dermal layers. A 0.006-cm- (60-µm-) diameter blood vessel was buried 0.015 cm deep in the dermis unless stated otherwise. The thickness of the epidermis was 0.006 cm (60 µm).

Fig. 2
Fig. 2

Two-dimensional distributions of the absorption density (inverse cubic centimeters) in the turbid medium under various conditions [see Fig. 1(a)]. The radius of the optical beam ρ f was 0.5 cm. (a) Matched indices of refraction between the ambient and the turbid mediums; z f = 0.05 cm. (b) Matched indices; z f = 0.1 cm. (c) Matched indices; z f = 0.2 cm. (d) Matched indices; z f = ∞ (unfocused). (e) Mismatched indices; z f = 0.1 cm without focus correction. (f) Mismatched indices; z f = 0.1 cm with focus correction.

Fig. 3
Fig. 3

One-dimensional depth distributions of the absorption density in the turbid medium under various conditions [see Fig. 1(a)]. (a) Comparison in the case of matched indices of refraction between the ambient and the turbid mediums. (b) Comparison among the absorption density curves showing the effects of mismatched indices and focus correction. (c) Comparison among the normalized absorption density curves showing the effects of mismatched indices and focus correction.

Fig. 4
Fig. 4

Two-dimensional distributions of the absorption density (inverse cubic centimeters) in the skin model under various conditions [see Fig. 1(b)]. The 0.006-cm-diameter blood vessel was buried 0.015 cm deep. The radius of the optical beam ρ f was 0.075 cm. (a) Matched indices of refraction between the ambient and the turbid mediums; z f = ∞ (unfocused). (b) Matched indices; z f = 0.015 cm. (c) Mismatched indices; z f = ∞. (d) Mismatched indices; z f = 0.015 cm without focus correction. (e) Mismatched indices; z f = 0.015 cm with focus correction.

Fig. 5
Fig. 5

One-dimensional depth distributions of the absorption density in the skin model under various conditions [see Fig. 1(b)]. (a) Effects of matched indices of both refraction and focusing were considered. The distance between the center of the cylinder and the surface of the turbid medium was 0.015 cm. (b) Effect of focusing under matched boundary conditions was studied when the distance between the center of the cylinder and the surface of the turbid medium was 0.030 cm. (c) Effect of focusing under matched boundary conditions was studied when the distance between the center of the cylinder and the surface of the turbid medium was 0.060 cm.

Equations (10)

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

ρ=ρfξρ,
θ=2πξθ,
x=ρ cosθ,
y=ρ sinθ.
ux=-x/ρ2+zf21/2,
uy=-y/ρ2+zf21/2,
uz=zf/ρ2+zf21/2.
R=-lnξR/μm,
rk=rk-1+Ruk-1,
μtede+μtddd,

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