Abstract

Fiber optical light diffusers that enable interstitial light delivery have become a useful tool for various illumination tasks, such as in photodynamic therapy. However, existing methods based on light diffusing fiber tips are not applicable for spatially selective light delivery in more complex structures. Here, we employ femtosecond laser induced scattering centers without mechanical manipulation and removal of the outer coatings for generating customized emission patterns. Tailoring of the cumulative emission profile is achieved through controlling the step-width between modification spots. An in-depth evaluation shows that the side-emission pattern is the result of an interplay of several scattering mechanisms that interact with cladding and core modes.

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References

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  1. B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
    [Crossref]
  2. A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
    [Crossref]
  3. L. Vesselov, W. Whittington, and L. Lilge, “Design and performance of thin cylindrical diffusers created in ge-doped multimode optical fibers,” Appl. Opt. 44(14), 2754–2758 (2005).
    [Crossref]
  4. T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
    [Crossref]
  5. A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
    [Crossref]
  6. A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.
  7. K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
    [Crossref]
  8. L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
    [Crossref]
  9. L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
    [Crossref]
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  16. F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
    [Crossref]
  17. Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
    [Crossref]
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2019 (1)

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

2018 (2)

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
[Crossref]

2016 (1)

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

2015 (1)

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

2014 (1)

T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
[Crossref]

2008 (2)

B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
[Crossref]

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

2006 (2)

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006).
[Crossref]

2005 (1)

1998 (1)

1991 (1)

Andrehs, G.

G. Andrehs, H. Beyer. Theorie und Praxis der Interferenzmikroskopie (Akademische Verlagsgesellschaft Geest & Portig K.-G., 1974).

Baran, T. M.

T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
[Crossref]

Beck, J.

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Beck, J. C.

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

Betz, V.

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

Brakhage, A. A.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Calo, J.

Clare, J. F.

Driscoll, T.

Foster, T. H.

T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
[Crossref]

Gottschaldt, M.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Itoh, K.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Köhler, A.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Küsel, K.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Lasser, T.

Lawandy, N. M.

Leitgeb, R. A.

Leutenegger, M.

Lilge, L.

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

L. Vesselov, W. Whittington, and L. Lilge, “Design and performance of thin cylindrical diffusers created in ge-doped multimode optical fibers,” Appl. Opt. 44(14), 2754–2758 (2005).
[Crossref]

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Love, J.

A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012), pp. 135–140.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), pp. 376–387.

D. Marcuse, Principles of Optical Fiber Measurements (Academic Press, 1981), pp. 11–68.

Méndez-Ramos, J.

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Müller, F. A.

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Nolte, S.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Okawa, J.

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Pan, Z.

Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
[Crossref]

Patterson, M. S.

B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
[Crossref]

Plech, A.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Pohnert, G.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Rao, R.

Rendon, A.

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Richter, S.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Schacher, F. H.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Schaffer, C. B.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Schubert, U. S.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Snyder, A. W.

A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012), pp. 135–140.

Tunnermann, A.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Tyystjärvi, E.

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Vesselov, L.

Watanabe, W.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Weersink, R.

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

Whittington, W.

Wilson, B. C.

B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
[Crossref]

Wondraczek, L.

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
[Crossref]

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Yassine, A.-A.

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

Zhang, Q.

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Zimmermann, F.

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Adv. Mater. (1)

L. Wondraczek, G. Pohnert, F. H. Schacher, A. Köhler, M. Gottschaldt, U. S. Schubert, K. Küsel, and A. A. Brakhage, “Artificial microbial arenas: Materials for observing and manipulating microbial consortia,” Adv. Mater. 2019, 1900284 (2019).
[Crossref]

Adv. Sci. (1)

L. Wondraczek, E. Tyystjärvi, J. Méndez-Ramos, F. A. Müller, and Q. Zhang, “Shifting the sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis,” Adv. Sci. 2(12), 1500218 (2015).
[Crossref]

Appl. Opt. (1)

J. Biophotonics (1)

A.-A. Yassine, L. Lilge, and V. Betz, “Optimizing interstitial photodynamic therapy with custom cylindrical diffusers,” J. Biophotonics 12(1), e201800153 (2018).
[Crossref]

J. Opt. Soc. Am. A (1)

Laser Photonics Rev. (1)

F. Zimmermann, A. Plech, S. Richter, A. Tunnermann, and S. Nolte, “The onset of ultrashort pulse-induced nanogratings,” Laser Photonics Rev. 10(2), 327–334 (2016).
[Crossref]

Med. Phys. (1)

T. M. Baran and T. H. Foster, “Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy,” Med. Phys. 41(2), 022701 (2014).
[Crossref]

MRS Bull. (1)

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(08), 620–625 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (2)

B. C. Wilson and M. S. Patterson, “The physics, biophysics and technology of photodynamic therapy,” Phys. Med. Biol. 53(9), R61–R109 (2008).
[Crossref]

A. Rendon, J. C. Beck, and L. Lilge, “Treatment planning using tailored and standard cylindrical light diffusers for photodynamic therapy of the prostate,” Phys. Med. Biol. 53(4), 1131–1149 (2008).
[Crossref]

Sci. Rep. (1)

Z. Pan and L. Wondraczek, “Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission,” Sci. Rep. 8(1), 9527 (2018).
[Crossref]

Other (5)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), pp. 376–387.

D. Marcuse, Principles of Optical Fiber Measurements (Academic Press, 1981), pp. 11–68.

A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012), pp. 135–140.

G. Andrehs, H. Beyer. Theorie und Praxis der Interferenzmikroskopie (Akademische Verlagsgesellschaft Geest & Portig K.-G., 1974).

A. Rendon, J. Okawa, R. Weersink, J. Beck, and L. Lilge, “Conformal light delivery using tailored cylindrical diffusers,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy XVI, vol. 6427 (International Society for Optics and Photonics, 2007), p. 64270M.

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Figures (9)

Fig. 1.
Fig. 1. Schematics and microscope image of light scattering on femtosecond laser modifications. a) Creation of scattering centers through focused femtosecond laser irradiation of the fiber core. b) Light scattering on the laser modifications in a two stage process: Light is scattered out of the core into free space and into the cladding, where it is again guided by total internal reflection or, eventually, scattered out into the environment. c) Microscope image of light scattering on laser modifications.
Fig. 2.
Fig. 2. Three-level representation of the energy exchange between the core, the cladding and free space: At first all light is contained in the core and gets scattered into free space (green) and the cladding (red). After some distance this leads to a mixed case where light is also contained in the cladding and scattered into free space (blue) and back into the core (yellow). Because the latter effect is very small, the core is eventually depleted and only the remaining light in the cladding is scattered.
Fig. 3.
Fig. 3. a) Motorized linear stage: The integrating sphere is moved incrementally alongside the optical fiber; the spectrometer measures an emission spectrum for every position. b) Integrating sphere: Light emitted by the fiber segment $\Delta z$ - limited by the fiber guide - is homogeneously distributed on the sphere wall by multiple diffuse reflections. The irradiance on the detector port is proportional to the emitted flux.
Fig. 4.
Fig. 4. Microscope images of the laser modifications in the fiber core. Top view is in the direction of laser irradiation and side view is orthogonal to it. The contrast of the brightfield images is low, so additional phase contrast images are provided. Here a higher refractive index shows up darker e.g. the fiber core is the dark band in the center of the pictures.
Fig. 5.
Fig. 5. Wavelength-resolved transmission and emission plots for scattering centers with constant spacing. Transmission spectra are plotted as a function of the modification number and the emission spectra as a function of position starting from the initial maximum. The top plots show the average spectral flux in certain wavelength intervals. This shows a steady decline in transmitted power with consecutive modifications for in transmission and an overall decline but with local maximums and minimums in emission.
Fig. 6.
Fig. 6. Transmission spectra for increasing amounts of scattering centers (indicated by the labels) with their corresponding emission spectra, measured at the indicated positions.
Fig. 7.
Fig. 7. Transmission and emission spectra integrated over three different wavelength ranges and plotted as a function of the number of scattering centers or the position with their respective fits according to Eq. (7) and Eq. (9)
Fig. 8.
Fig. 8. Fit results of Eq. (7) and Eq. (9) to every wavelength of the emission and transmission data set. This yields scattering spectra $s_{23}$ of a modification in emission or transmission with its corresponding standard error (shaded area). The spectra show an increase in scattering for lower wavelength as well as a local maximum at 900 nm and the difference in magnitude $\Delta$. A scattering function $\propto \lambda ^{-4}$ was fitted to the transmission data in the range 600 nm to 840 nm with an $R^2=0.967$.
Fig. 9.
Fig. 9. a) Measured transmission with fits according to Eq. (7) at three selected wavelength ranges shows a second degree exponential decay. b) Comparison of the measured emission behavior and the calculated emission profile. The right-side maximum is caused by the decrease of the scattering center distance leading to an increase in emission per unit length.

Equations (14)

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

dϕdz=(α+σ)ϕ ϕ(z)=ϕ0exp{0zα(z)+σ(z)dz}.
dϕ1dn=(s12+s13)ϕ1+s21ϕ2,
dϕ2dn=s12ϕ1(s23+s21)ϕ2,
dϕ3dn=s13ϕ1+s23ϕ2.
ϕ1(n)=ϕ0exp{(s12+s13)n},
ϕ2(n)=ϕ0s12s12+s13s23[exp{s23n}exp{(s12+s13)n}].
T(n)=ϕ1+ϕ2ϕ0=(1As)exp{(s12+s13)n}+Asexp{s23n}.
As=s12s12+s13s23.
E(z)=1ϕ0dϕ3dz=1ϕ0dϕ3dndndz,
=[(s13s23As)exp{(s12+s13)n(z)}+s23Asexp{s23n(z)}]dndz.
M=ρϕA(1ρ(1f)).
MiM0=ϕiϕ0.
dn+1=0.002(60mmdn).
dndzΔnΔz=1dn.

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