Abstract

We have obtained expressions for the reflectance and transmittance of a scattering medium with weak absorption in terms of a diffusion model, where the source is an incoming beam, whose intensity exponentially decays along the propagation path. We have applied three experimental techniques, one of which is based on the developed model, to determine the transport mean-free-path in translucent samples of Nd:YAG ceramics.

© 2006 Optical Society of America

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References

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  1. N. T. Melamed, "Optical absorption coefficients and the absolute value of the diffuse reflectance," J. Appl. Phys. 34, 560-570 (1963).
    [CrossRef]
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, and Oxford U. Press, 1997).
  3. A. Z. Genack and J. M. Drake, "Relationship between optical intensity, fluctuations and pulse propagation in random media," Electron. Lett. 11, 331-336 (1990).
  4. N. Garcia, A. Z. Genack, and A. Lisyansky, "Measurement of the transport mean path of diffusing photons," Phys. Rev. B 46, 14475-14479 (1992).
    [CrossRef]
  5. J. Gomez-Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
    [CrossRef]
  6. M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in random medium," Phys. Rev. Lett. 55, 2693-2695 (1985).
  7. P. E. Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
    [CrossRef] [PubMed]
  8. P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, "Optical coherent backscattering by random media: an experimental study," J. Phys. (France) 49, 63-75 (1988).
    [CrossRef]
  9. M. B. van der Mark, M. P. van Albada, and A. Lagendijk, "Light scattering in strongly scattering media: multiple scattering and weak localization," Phys. Rev. B 37, 3575-3592 (1988).
    [CrossRef]
  10. J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
    [CrossRef] [PubMed]
  11. J. Lu, K. Ueda, H. Yagi, T. Yanagitani, Y. Akiyama, and A. Kaminskii, "Neodymium doped yttrium aluminum garnet (Y3Al5O12) nanocrystalline ceramics—a new generation of solid state laser and optical materials," J. Alloys Compd. 341, 220-225 (2002).
    [CrossRef]
  12. W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999).
  13. The reference absorption spectrum of a 1% Nd-doped Czochralski-grown YAG crystal was done by Michael Bass, College of Optics and Photonics, University of Central Florida.
  14. A. Z. Genack, "Optical transmission in disordered media," Phys. Rev. Lett. 58, 2043-2046 (1987).
    [CrossRef] [PubMed]
  15. M. Bahoura and M. A. Noginov, "Determination of the transport mean free path in a solid-state random laser," J. Opt. Soc. Am. B 20, 2389-2394 (2003).
    [CrossRef]
  16. K. Busch, C. M. Soukoulis, and E. N. Economou, "Transport and scattering mean free paths of classical waves," Phys. Rev. B 50, 93-98 (1994).
    [CrossRef]
  17. Yu. A. Barnakov, I. Veal, Z. Kabato, G. Zhu, M. Bahoura, and M. A. Noginov, "Simple route to Nd:YAG transparent ceramics," in Laser Source and System Technology for Defense and Security II, G.L.Wood and M.A.Dubinskii, eds., Proc. SPIE 6216, 62160Z (2006).

2003 (1)

2002 (1)

J. Lu, K. Ueda, H. Yagi, T. Yanagitani, Y. Akiyama, and A. Kaminskii, "Neodymium doped yttrium aluminum garnet (Y3Al5O12) nanocrystalline ceramics—a new generation of solid state laser and optical materials," J. Alloys Compd. 341, 220-225 (2002).
[CrossRef]

2001 (1)

J. Gomez-Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

1994 (1)

K. Busch, C. M. Soukoulis, and E. N. Economou, "Transport and scattering mean free paths of classical waves," Phys. Rev. B 50, 93-98 (1994).
[CrossRef]

1992 (1)

N. Garcia, A. Z. Genack, and A. Lisyansky, "Measurement of the transport mean path of diffusing photons," Phys. Rev. B 46, 14475-14479 (1992).
[CrossRef]

1991 (1)

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[CrossRef] [PubMed]

1990 (1)

A. Z. Genack and J. M. Drake, "Relationship between optical intensity, fluctuations and pulse propagation in random media," Electron. Lett. 11, 331-336 (1990).

1988 (2)

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, "Optical coherent backscattering by random media: an experimental study," J. Phys. (France) 49, 63-75 (1988).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, "Light scattering in strongly scattering media: multiple scattering and weak localization," Phys. Rev. B 37, 3575-3592 (1988).
[CrossRef]

1987 (1)

A. Z. Genack, "Optical transmission in disordered media," Phys. Rev. Lett. 58, 2043-2046 (1987).
[CrossRef] [PubMed]

1985 (2)

M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in random medium," Phys. Rev. Lett. 55, 2693-2695 (1985).

P. E. Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[CrossRef] [PubMed]

1963 (1)

N. T. Melamed, "Optical absorption coefficients and the absolute value of the diffuse reflectance," J. Appl. Phys. 34, 560-570 (1963).
[CrossRef]

Akiyama, Y.

J. Lu, K. Ueda, H. Yagi, T. Yanagitani, Y. Akiyama, and A. Kaminskii, "Neodymium doped yttrium aluminum garnet (Y3Al5O12) nanocrystalline ceramics—a new generation of solid state laser and optical materials," J. Alloys Compd. 341, 220-225 (2002).
[CrossRef]

Akkermans, E.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, "Optical coherent backscattering by random media: an experimental study," J. Phys. (France) 49, 63-75 (1988).
[CrossRef]

Bahoura, M.

M. Bahoura and M. A. Noginov, "Determination of the transport mean free path in a solid-state random laser," J. Opt. Soc. Am. B 20, 2389-2394 (2003).
[CrossRef]

Yu. A. Barnakov, I. Veal, Z. Kabato, G. Zhu, M. Bahoura, and M. A. Noginov, "Simple route to Nd:YAG transparent ceramics," in Laser Source and System Technology for Defense and Security II, G.L.Wood and M.A.Dubinskii, eds., Proc. SPIE 6216, 62160Z (2006).

Barnakov, Yu. A.

Yu. A. Barnakov, I. Veal, Z. Kabato, G. Zhu, M. Bahoura, and M. A. Noginov, "Simple route to Nd:YAG transparent ceramics," in Laser Source and System Technology for Defense and Security II, G.L.Wood and M.A.Dubinskii, eds., Proc. SPIE 6216, 62160Z (2006).

Busch, K.

K. Busch, C. M. Soukoulis, and E. N. Economou, "Transport and scattering mean free paths of classical waves," Phys. Rev. B 50, 93-98 (1994).
[CrossRef]

Drake, J. M.

A. Z. Genack and J. M. Drake, "Relationship between optical intensity, fluctuations and pulse propagation in random media," Electron. Lett. 11, 331-336 (1990).

Economou, E. N.

K. Busch, C. M. Soukoulis, and E. N. Economou, "Transport and scattering mean free paths of classical waves," Phys. Rev. B 50, 93-98 (1994).
[CrossRef]

Garcia, N.

N. Garcia, A. Z. Genack, and A. Lisyansky, "Measurement of the transport mean path of diffusing photons," Phys. Rev. B 46, 14475-14479 (1992).
[CrossRef]

Genack, A. Z.

N. Garcia, A. Z. Genack, and A. Lisyansky, "Measurement of the transport mean path of diffusing photons," Phys. Rev. B 46, 14475-14479 (1992).
[CrossRef]

A. Z. Genack and J. M. Drake, "Relationship between optical intensity, fluctuations and pulse propagation in random media," Electron. Lett. 11, 331-336 (1990).

A. Z. Genack, "Optical transmission in disordered media," Phys. Rev. Lett. 58, 2043-2046 (1987).
[CrossRef] [PubMed]

Gomez-Rivas, J.

J. Gomez-Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, and Oxford U. Press, 1997).

Kabato, Z.

Yu. A. Barnakov, I. Veal, Z. Kabato, G. Zhu, M. Bahoura, and M. A. Noginov, "Simple route to Nd:YAG transparent ceramics," in Laser Source and System Technology for Defense and Security II, G.L.Wood and M.A.Dubinskii, eds., Proc. SPIE 6216, 62160Z (2006).

Kaminskii, A.

J. Lu, K. Ueda, H. Yagi, T. Yanagitani, Y. Akiyama, and A. Kaminskii, "Neodymium doped yttrium aluminum garnet (Y3Al5O12) nanocrystalline ceramics—a new generation of solid state laser and optical materials," J. Alloys Compd. 341, 220-225 (2002).
[CrossRef]

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999).

Lagendijk, A.

J. Gomez-Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, "Light scattering in strongly scattering media: multiple scattering and weak localization," Phys. Rev. B 37, 3575-3592 (1988).
[CrossRef]

M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in random medium," Phys. Rev. Lett. 55, 2693-2695 (1985).

Lisyansky, A.

N. Garcia, A. Z. Genack, and A. Lisyansky, "Measurement of the transport mean path of diffusing photons," Phys. Rev. B 46, 14475-14479 (1992).
[CrossRef]

Lu, J.

J. Lu, K. Ueda, H. Yagi, T. Yanagitani, Y. Akiyama, and A. Kaminskii, "Neodymium doped yttrium aluminum garnet (Y3Al5O12) nanocrystalline ceramics—a new generation of solid state laser and optical materials," J. Alloys Compd. 341, 220-225 (2002).
[CrossRef]

Maret, G.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, "Optical coherent backscattering by random media: an experimental study," J. Phys. (France) 49, 63-75 (1988).
[CrossRef]

P. E. Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[CrossRef] [PubMed]

Maynard, R.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, "Optical coherent backscattering by random media: an experimental study," J. Phys. (France) 49, 63-75 (1988).
[CrossRef]

Melamed, N. T.

N. T. Melamed, "Optical absorption coefficients and the absolute value of the diffuse reflectance," J. Appl. Phys. 34, 560-570 (1963).
[CrossRef]

Noginov, M. A.

M. Bahoura and M. A. Noginov, "Determination of the transport mean free path in a solid-state random laser," J. Opt. Soc. Am. B 20, 2389-2394 (2003).
[CrossRef]

Yu. A. Barnakov, I. Veal, Z. Kabato, G. Zhu, M. Bahoura, and M. A. Noginov, "Simple route to Nd:YAG transparent ceramics," in Laser Source and System Technology for Defense and Security II, G.L.Wood and M.A.Dubinskii, eds., Proc. SPIE 6216, 62160Z (2006).

Noordam, L. D.

J. Gomez-Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Pine, D. J.

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[CrossRef] [PubMed]

Rella, C. W.

J. Gomez-Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Soukoulis, C. M.

K. Busch, C. M. Soukoulis, and E. N. Economou, "Transport and scattering mean free paths of classical waves," Phys. Rev. B 50, 93-98 (1994).
[CrossRef]

Sprik, R.

J. Gomez-Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Ueda, K.

J. Lu, K. Ueda, H. Yagi, T. Yanagitani, Y. Akiyama, and A. Kaminskii, "Neodymium doped yttrium aluminum garnet (Y3Al5O12) nanocrystalline ceramics—a new generation of solid state laser and optical materials," J. Alloys Compd. 341, 220-225 (2002).
[CrossRef]

van Albada, M. P.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, "Light scattering in strongly scattering media: multiple scattering and weak localization," Phys. Rev. B 37, 3575-3592 (1988).
[CrossRef]

M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in random medium," Phys. Rev. Lett. 55, 2693-2695 (1985).

van der Mark, M. B.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, "Light scattering in strongly scattering media: multiple scattering and weak localization," Phys. Rev. B 37, 3575-3592 (1988).
[CrossRef]

Veal, I.

Yu. A. Barnakov, I. Veal, Z. Kabato, G. Zhu, M. Bahoura, and M. A. Noginov, "Simple route to Nd:YAG transparent ceramics," in Laser Source and System Technology for Defense and Security II, G.L.Wood and M.A.Dubinskii, eds., Proc. SPIE 6216, 62160Z (2006).

Weitz, D. A.

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[CrossRef] [PubMed]

Wolf, P. E.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, "Optical coherent backscattering by random media: an experimental study," J. Phys. (France) 49, 63-75 (1988).
[CrossRef]

P. E. Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[CrossRef] [PubMed]

Yagi, H.

J. Lu, K. Ueda, H. Yagi, T. Yanagitani, Y. Akiyama, and A. Kaminskii, "Neodymium doped yttrium aluminum garnet (Y3Al5O12) nanocrystalline ceramics—a new generation of solid state laser and optical materials," J. Alloys Compd. 341, 220-225 (2002).
[CrossRef]

Yanagitani, T.

J. Lu, K. Ueda, H. Yagi, T. Yanagitani, Y. Akiyama, and A. Kaminskii, "Neodymium doped yttrium aluminum garnet (Y3Al5O12) nanocrystalline ceramics—a new generation of solid state laser and optical materials," J. Alloys Compd. 341, 220-225 (2002).
[CrossRef]

Zhu, G.

Yu. A. Barnakov, I. Veal, Z. Kabato, G. Zhu, M. Bahoura, and M. A. Noginov, "Simple route to Nd:YAG transparent ceramics," in Laser Source and System Technology for Defense and Security II, G.L.Wood and M.A.Dubinskii, eds., Proc. SPIE 6216, 62160Z (2006).

Zhu, J. X.

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[CrossRef] [PubMed]

Electron. Lett. (1)

A. Z. Genack and J. M. Drake, "Relationship between optical intensity, fluctuations and pulse propagation in random media," Electron. Lett. 11, 331-336 (1990).

J. Alloys Compd. (1)

J. Lu, K. Ueda, H. Yagi, T. Yanagitani, Y. Akiyama, and A. Kaminskii, "Neodymium doped yttrium aluminum garnet (Y3Al5O12) nanocrystalline ceramics—a new generation of solid state laser and optical materials," J. Alloys Compd. 341, 220-225 (2002).
[CrossRef]

J. Appl. Phys. (1)

N. T. Melamed, "Optical absorption coefficients and the absolute value of the diffuse reflectance," J. Appl. Phys. 34, 560-570 (1963).
[CrossRef]

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

J. Phys. (France) (1)

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, "Optical coherent backscattering by random media: an experimental study," J. Phys. (France) 49, 63-75 (1988).
[CrossRef]

Phys. Rev. A (1)

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (3)

K. Busch, C. M. Soukoulis, and E. N. Economou, "Transport and scattering mean free paths of classical waves," Phys. Rev. B 50, 93-98 (1994).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, "Light scattering in strongly scattering media: multiple scattering and weak localization," Phys. Rev. B 37, 3575-3592 (1988).
[CrossRef]

N. Garcia, A. Z. Genack, and A. Lisyansky, "Measurement of the transport mean path of diffusing photons," Phys. Rev. B 46, 14475-14479 (1992).
[CrossRef]

Phys. Rev. E (1)

J. Gomez-Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

M. P. van Albada and A. Lagendijk, "Observation of weak localization of light in random medium," Phys. Rev. Lett. 55, 2693-2695 (1985).

P. E. Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[CrossRef] [PubMed]

A. Z. Genack, "Optical transmission in disordered media," Phys. Rev. Lett. 58, 2043-2046 (1987).
[CrossRef] [PubMed]

Other (4)

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999).

The reference absorption spectrum of a 1% Nd-doped Czochralski-grown YAG crystal was done by Michael Bass, College of Optics and Photonics, University of Central Florida.

Yu. A. Barnakov, I. Veal, Z. Kabato, G. Zhu, M. Bahoura, and M. A. Noginov, "Simple route to Nd:YAG transparent ceramics," in Laser Source and System Technology for Defense and Security II, G.L.Wood and M.A.Dubinskii, eds., Proc. SPIE 6216, 62160Z (2006).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, and Oxford U. Press, 1997).

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of the transmission measurements in an integrating sphere, and (b) schematic of the CBS measurement.

Fig. 2
Fig. 2

Transmission spectra of plates with different thicknesses (sample A). 1, 0.015 cm ; 2, 0.028 cm ; 3, 0.071 cm ; 4, 0.094 cm ; and 5, 0.147 cm .

Fig. 3
Fig. 3

Transmission measured at 415 nm as a function of the plate thickness (sample A). The solid curve is the exponential fit ( L 0.7 mm ) , and the dashed curve is the curve obtained from the general expression for transmittance, Eq. (7).

Fig. 4
Fig. 4

CBS profile measured in sample A at λ = 514.5 nm .

Tables (2)

Tables Icon

Table 1 Characteristic Scattering and Absorption Lengths Determined in Sample A Based on the Results of the Transmission Measurements

Tables Icon

Table 2 Transport Mean-Free-Paths in Three Samples Studies Measured by Different Techniques

Equations (20)

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l t = l s 1 cos θ ,
2 U d z 2 U d l abs 2 = 3 J 0 4 π l s [ 1 l t + μ ¯ l * ] exp ( z l * ) ,
J d ( z ) = J 0 l t l s μ ¯ exp ( z l * ) z ̂ 4 π 3 l t U d z z ̂ .
U d ( z = 0 ) 2 3 l t h 0 U d z ( z = 0 ) + h 0 J 0 l t μ ¯ 2 π l s = 0 ,
U d ( z = L ) + 2 3 l t h L U d z ( z = L ) h L J 0 l t μ ¯ 2 π l s exp ( L l * ) = 0 ,
r = 3 R 2 + 2 R 1 3 R 2 2 R 1 + 2 ,
R 1 = 0 π 2 R ( θ ) cos θ sin θ d θ , R 2 = 0 π 2 R ( θ ) ( cos θ ) 2 sin θ d θ .
T = a + b exp ( L l * ) cosh ( L l abs ) + c exp ( L l * ) sinh ( L l abs ) d cosh ( L l abs ) + e sinh ( L l abs ) ,
R = f exp ( L l * ) + f cosh ( L l abs ) + g sinh ( L l abs ) d cosh ( L l abs ) + e sinh ( L l abs ) ,
T = 2 ( 2 h + 3 ) l t 4 h l t + 3 l abs exp ( L l abs ) ,
R = 3 l abs + ( 2 h 3 ) l t 4 h l t + 3 l abs .
l t 0.7 λ 2 π W ,
I out = I in exp ( L l * ) = I in exp ( L ( 1 l s + 1 l i ) ) .
a = l i 2 l t l abs [ 3 l i l s + ( 3 μ ¯ ( l i + l s ) + 2 h ( l i + 3 μ ¯ l s + l s ) ) l t ] ,
b = l i l t l abs [ 3 l i l s ( l i + 4 h l s ) + ( 3 μ ¯ l i ( l i + l s ) + 2 h ( l i 2 3 ( μ ¯ 1 ) l i l s + 2 l s 2 ) ) l t ] ,
c = ( 9 l i 2 l s 2 3 l i l s ( ( 2 h + 3 μ ¯ 1 ) l i + ( 4 h 2 1 ) l s ) l t + 2 h ( l i + l s ) ( 2 h ( l i + l s ) 3 μ ¯ l i ) l t 2 ) ,
d = 12 h l abs ( ( l i + l s ) 2 l t 3 l i l s 2 ) ,
e = ( ( l i + l s ) 2 l t 3 l i l s 2 ) ( 4 l t h 2 + 3 l i ) ,
f = 3 l abs [ l i ( 3 l i l s + ( 3 μ ¯ ( l i + l s ) + 2 h ( l i + 3 μ ¯ l s + l s ) ) l t ) ] ,
g = 9 l abs 2 [ l i ( l i 2 h l s + 3 μ ¯ l s + l s ) 2 h μ ¯ l t ( l i + l s ) ] .

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