Abstract

We report on a transition in random lasers that is induced by the geometrical confinement of the emitting material. Different dye doped paper devices with controlled geometry are fabricated by soft lithography and show two distinguished behaviors in the stimulated emission: in the absence of boundary constraints, the energy threshold decreases for larger laser volumes showing the typical trend of diffusive nonresonant random lasers, while when the same material is lithographed into channels, the walls act as cavity and the resonant behavior typical of standard lasers is observed. The experimental results are consistent with the general theories of random and standard lasers and a clear phase diagram of the transition is reported.

© 2013 Optical Society of America

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  1. D. S. Wiersma, Nat. Photonics 7, 188 (2013).
  2. N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
    [CrossRef]
  3. S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, Opt. Express 20, 27369 (2012).
    [CrossRef]
  4. C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 97, 123903 (2006).
    [CrossRef]
  5. V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
    [CrossRef]
  6. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).
  7. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
    [CrossRef]
  8. M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).
  9. T. Sperling, W. Buhrer, C. M. Aegerter, and G. Maret, Nat. Photonics 7, 148 (2013).
  10. D. S. Wiersma, Nat. Phys. 4, 359 (2008).
  11. K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 98, 143901 (2007).
    [CrossRef]
  12. N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, and C. Conti, Laser Photon. Rev. 7, 432 (2013).
  13. X. Wu and H. Cao, Phys. Rev. A 77, 013832 (2008).
    [CrossRef]
  14. M. Leonetti, C. Conti, and C. Lopez, Nat. Photonics 5, 615 (2011).
  15. S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, Phys. Rev. Lett. 93, 053903 (2004).
    [CrossRef]
  16. C. Vanneste and P. Sebbah, Phys. Rev. Lett. 87, 183903 (2001).
    [CrossRef]
  17. P. Sebbah and C. Vanneste, Phys. Rev. B 66, 144202 (2002).
  18. C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
    [CrossRef]
  19. C. Vanneste, P. Sebbah, and H. Cao, Phys. Rev. Lett. 98, 143902 (2007).
    [CrossRef]
  20. G. van Soest, M. Tomita, and A. Lagendijk, Opt. Lett. 24, 306 (1999).
    [CrossRef]
  21. V. S. Letokhov, Sov. Phys. JETP 26, 835 (1968).
  22. A. C. Siegel, S. T. Phillips, M. D. Dickey, N. Lu, Z. Suo, and G. M. Whitesides, Adv. Funct. Mater. 20, 28 (2010).
    [CrossRef]
  23. A. W. Martinez, S. T. Phillips, and G. M. Whitesides, Proc. Natl. Acad. Sci. 105, 19606 (2008).
  24. M. Leonetti and C. Lopez, Appl. Phys. Lett. 101, 251120 (2012).
    [CrossRef]
  25. H. Cao, Waves Random Media 13, R1 (2003).
    [CrossRef]
  26. A. E. Siegman, Lasers (University Science Books, 1987).
  27. B. N. S. Bhaktha, N. Bachelard, X. Noblin, and P. Sebbah, Appl. Phys. Lett. 101, 151101 (2012).
    [CrossRef]

2013 (4)

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).

T. Sperling, W. Buhrer, C. M. Aegerter, and G. Maret, Nat. Photonics 7, 148 (2013).

N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, and C. Conti, Laser Photon. Rev. 7, 432 (2013).

D. S. Wiersma, Nat. Photonics 7, 188 (2013).

2012 (4)

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

M. Leonetti and C. Lopez, Appl. Phys. Lett. 101, 251120 (2012).
[CrossRef]

B. N. S. Bhaktha, N. Bachelard, X. Noblin, and P. Sebbah, Appl. Phys. Lett. 101, 151101 (2012).
[CrossRef]

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, Opt. Express 20, 27369 (2012).
[CrossRef]

2011 (1)

M. Leonetti, C. Conti, and C. Lopez, Nat. Photonics 5, 615 (2011).

2010 (3)

A. C. Siegel, S. T. Phillips, M. D. Dickey, N. Lu, Z. Suo, and G. M. Whitesides, Adv. Funct. Mater. 20, 28 (2010).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).

2008 (4)

X. Wu and H. Cao, Phys. Rev. A 77, 013832 (2008).
[CrossRef]

D. S. Wiersma, Nat. Phys. 4, 359 (2008).

A. W. Martinez, S. T. Phillips, and G. M. Whitesides, Proc. Natl. Acad. Sci. 105, 19606 (2008).

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

2007 (2)

C. Vanneste, P. Sebbah, and H. Cao, Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef]

K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 98, 143901 (2007).
[CrossRef]

2006 (1)

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 97, 123903 (2006).
[CrossRef]

2004 (1)

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, Phys. Rev. Lett. 93, 053903 (2004).
[CrossRef]

2003 (1)

H. Cao, Waves Random Media 13, R1 (2003).
[CrossRef]

2002 (1)

P. Sebbah and C. Vanneste, Phys. Rev. B 66, 144202 (2002).

2001 (1)

C. Vanneste and P. Sebbah, Phys. Rev. Lett. 87, 183903 (2001).
[CrossRef]

1999 (1)

1997 (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

1968 (1)

V. S. Letokhov, Sov. Phys. JETP 26, 835 (1968).

Aegerter, C. M.

T. Sperling, W. Buhrer, C. M. Aegerter, and G. Maret, Nat. Photonics 7, 148 (2013).

Angelani, L.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Bachelard, N.

B. N. S. Bhaktha, N. Bachelard, X. Noblin, and P. Sebbah, Appl. Phys. Lett. 101, 151101 (2012).
[CrossRef]

Barbarella, G.

N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, and C. Conti, Laser Photon. Rev. 7, 432 (2013).

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Bhaktha, B. N. S.

B. N. S. Bhaktha, N. Bachelard, X. Noblin, and P. Sebbah, Appl. Phys. Lett. 101, 151101 (2012).
[CrossRef]

Buhrer, W.

T. Sperling, W. Buhrer, C. M. Aegerter, and G. Maret, Nat. Photonics 7, 148 (2013).

Cao, H.

X. Wu and H. Cao, Phys. Rev. A 77, 013832 (2008).
[CrossRef]

C. Vanneste, P. Sebbah, and H. Cao, Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef]

H. Cao, Waves Random Media 13, R1 (2003).
[CrossRef]

Christodoulides, D. N.

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).

Conti, C.

N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, and C. Conti, Laser Photon. Rev. 7, 432 (2013).

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, Opt. Express 20, 27369 (2012).
[CrossRef]

M. Leonetti, C. Conti, and C. Lopez, Nat. Photonics 5, 615 (2011).

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 97, 123903 (2006).
[CrossRef]

DelRe, E.

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, Opt. Express 20, 27369 (2012).
[CrossRef]

Di Maria, F.

N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, and C. Conti, Laser Photon. Rev. 7, 432 (2013).

Dickey, M. D.

A. C. Siegel, S. T. Phillips, M. D. Dickey, N. Lu, Z. Suo, and G. M. Whitesides, Adv. Funct. Mater. 20, 28 (2010).
[CrossRef]

Folli, V.

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

Fratalocchi, A.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Gentilini, S.

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, Opt. Express 20, 27369 (2012).
[CrossRef]

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

Ghofraniha, N.

N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, and C. Conti, Laser Photon. Rev. 7, 432 (2013).

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, Opt. Express 20, 27369 (2012).
[CrossRef]

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 97, 123903 (2006).
[CrossRef]

Gigli, G.

N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, and C. Conti, Laser Photon. Rev. 7, 432 (2013).

Lagendijk, A.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).

K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 98, 143901 (2007).
[CrossRef]

G. van Soest, M. Tomita, and A. Lagendijk, Opt. Lett. 24, 306 (1999).
[CrossRef]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Leonetti, M.

M. Leonetti and C. Lopez, Appl. Phys. Lett. 101, 251120 (2012).
[CrossRef]

M. Leonetti, C. Conti, and C. Lopez, Nat. Photonics 5, 615 (2011).

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Letokhov, V. S.

V. S. Letokhov, Sov. Phys. JETP 26, 835 (1968).

Lopez, C.

M. Leonetti and C. Lopez, Appl. Phys. Lett. 101, 251120 (2012).
[CrossRef]

M. Leonetti, C. Conti, and C. Lopez, Nat. Photonics 5, 615 (2011).

Lu, N.

A. C. Siegel, S. T. Phillips, M. D. Dickey, N. Lu, Z. Suo, and G. M. Whitesides, Adv. Funct. Mater. 20, 28 (2010).
[CrossRef]

Maret, G.

T. Sperling, W. Buhrer, C. M. Aegerter, and G. Maret, Nat. Photonics 7, 148 (2013).

Martinez, A. W.

A. W. Martinez, S. T. Phillips, and G. M. Whitesides, Proc. Natl. Acad. Sci. 105, 19606 (2008).

Mosk, A. P.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).

K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 98, 143901 (2007).
[CrossRef]

Mujumdar, S.

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, Phys. Rev. Lett. 93, 053903 (2004).
[CrossRef]

Noblin, X.

B. N. S. Bhaktha, N. Bachelard, X. Noblin, and P. Sebbah, Appl. Phys. Lett. 101, 151101 (2012).
[CrossRef]

Phillips, S. T.

A. C. Siegel, S. T. Phillips, M. D. Dickey, N. Lu, Z. Suo, and G. M. Whitesides, Adv. Funct. Mater. 20, 28 (2010).
[CrossRef]

A. W. Martinez, S. T. Phillips, and G. M. Whitesides, Proc. Natl. Acad. Sci. 105, 19606 (2008).

Ricci, M.

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, Phys. Rev. Lett. 93, 053903 (2004).
[CrossRef]

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Ruocco, G.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 97, 123903 (2006).
[CrossRef]

Sebbah, P.

B. N. S. Bhaktha, N. Bachelard, X. Noblin, and P. Sebbah, Appl. Phys. Lett. 101, 151101 (2012).
[CrossRef]

C. Vanneste, P. Sebbah, and H. Cao, Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef]

P. Sebbah and C. Vanneste, Phys. Rev. B 66, 144202 (2002).

C. Vanneste and P. Sebbah, Phys. Rev. Lett. 87, 183903 (2001).
[CrossRef]

Segev, M.

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).

Siegel, A. C.

A. C. Siegel, S. T. Phillips, M. D. Dickey, N. Lu, Z. Suo, and G. M. Whitesides, Adv. Funct. Mater. 20, 28 (2010).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1987).

Silberberg, Y.

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).

Sperling, T.

T. Sperling, W. Buhrer, C. M. Aegerter, and G. Maret, Nat. Photonics 7, 148 (2013).

Suo, Z.

A. C. Siegel, S. T. Phillips, M. D. Dickey, N. Lu, Z. Suo, and G. M. Whitesides, Adv. Funct. Mater. 20, 28 (2010).
[CrossRef]

Tjerkstra, R. W.

K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 98, 143901 (2007).
[CrossRef]

Tomita, M.

Torre, R.

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, Phys. Rev. Lett. 93, 053903 (2004).
[CrossRef]

Trillo, S.

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 97, 123903 (2006).
[CrossRef]

van der Molen, K. L.

K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 98, 143901 (2007).
[CrossRef]

van Soest, G.

Vanneste, C.

C. Vanneste, P. Sebbah, and H. Cao, Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef]

P. Sebbah and C. Vanneste, Phys. Rev. B 66, 144202 (2002).

C. Vanneste and P. Sebbah, Phys. Rev. Lett. 87, 183903 (2001).
[CrossRef]

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).

Viola, I.

N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, and C. Conti, Laser Photon. Rev. 7, 432 (2013).

Whitesides, G. M.

A. C. Siegel, S. T. Phillips, M. D. Dickey, N. Lu, Z. Suo, and G. M. Whitesides, Adv. Funct. Mater. 20, 28 (2010).
[CrossRef]

A. W. Martinez, S. T. Phillips, and G. M. Whitesides, Proc. Natl. Acad. Sci. 105, 19606 (2008).

Wiersma, D. S.

D. S. Wiersma, Nat. Photonics 7, 188 (2013).

D. S. Wiersma, Nat. Phys. 4, 359 (2008).

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, Phys. Rev. Lett. 93, 053903 (2004).
[CrossRef]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Wu, X.

X. Wu and H. Cao, Phys. Rev. A 77, 013832 (2008).
[CrossRef]

Adv. Funct. Mater. (1)

A. C. Siegel, S. T. Phillips, M. D. Dickey, N. Lu, Z. Suo, and G. M. Whitesides, Adv. Funct. Mater. 20, 28 (2010).
[CrossRef]

Appl. Phys. Lett. (2)

M. Leonetti and C. Lopez, Appl. Phys. Lett. 101, 251120 (2012).
[CrossRef]

B. N. S. Bhaktha, N. Bachelard, X. Noblin, and P. Sebbah, Appl. Phys. Lett. 101, 151101 (2012).
[CrossRef]

Laser Photon. Rev. (1)

N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, and C. Conti, Laser Photon. Rev. 7, 432 (2013).

Nat. Photonics (5)

M. Leonetti, C. Conti, and C. Lopez, Nat. Photonics 5, 615 (2011).

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photonics 4, 320 (2010).

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).

T. Sperling, W. Buhrer, C. M. Aegerter, and G. Maret, Nat. Photonics 7, 148 (2013).

D. S. Wiersma, Nat. Photonics 7, 188 (2013).

Nat. Phys. (1)

D. S. Wiersma, Nat. Phys. 4, 359 (2008).

Nature (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

X. Wu and H. Cao, Phys. Rev. A 77, 013832 (2008).
[CrossRef]

Phys. Rev. B (1)

P. Sebbah and C. Vanneste, Phys. Rev. B 66, 144202 (2002).

Phys. Rev. Lett. (8)

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

C. Vanneste, P. Sebbah, and H. Cao, Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef]

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, Phys. Rev. Lett. 93, 053903 (2004).
[CrossRef]

C. Vanneste and P. Sebbah, Phys. Rev. Lett. 87, 183903 (2001).
[CrossRef]

C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 97, 123903 (2006).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 98, 143901 (2007).
[CrossRef]

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

Proc. Natl. Acad. Sci. (1)

A. W. Martinez, S. T. Phillips, and G. M. Whitesides, Proc. Natl. Acad. Sci. 105, 19606 (2008).

Sov. Phys. JETP (1)

V. S. Letokhov, Sov. Phys. JETP 26, 835 (1968).

Waves Random Media (1)

H. Cao, Waves Random Media 13, R1 (2003).
[CrossRef]

Other (1)

A. E. Siegman, Lasers (University Science Books, 1987).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Confocal microscopy image (3D reconstruction) showing the paper mash impregnated by dye solution; the scale bar is 50 μm. (b) Fluorescence image of the microfluidic channel with Ly=300μm. The dashed circle indicates the pump spot with diameter Lx.

Fig. 2.
Fig. 2.

Emission spectra at different pump energy density for the 300 μm thick sample (a) and corresponding peak intensity (blue diamonds) and line width (red circles) versus pump energy (b).

Fig. 3.
Fig. 3.

(a) Peaks of the emitted intensity spectra versus pump energy density for samples with different thickness showing different laser thresholds; solid lines are the linear fitting curves. (b) Energy density threshold Eth versus laser volume V phase diagram showing the resonant–nonresonant transition. Sketches of the illuminated volumes are reported: the green spot indicates the pumped volume. Solid lines are the fitting curves after Eqs. (4) and (5).

Tables (1)

Tables Icon

Table 1. Sample Parameters

Equations (5)

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

W(r,t)t=D2W(r,t)+vlgW(r,t),
W(r,t)=nanΨn(r)e(DBn2v/lg)t,
lgth=vDB12=3ltB12,
EthV2/3.
EthV.

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