Abstract

We describe what are to our knowledge the first measurements of the propagation of coherent, single-cycle pulses of terahertz radiation in a scattering medium. By measuring the transmission as a function of the length L of the medium, we extract the scattering mean free path lsω over a broad bandwidth. We observe variations in ls ranging over nearly 2 orders of magnitude and covering the entire thin sample regime from L/ls1 to L/ls10. We also observe scattering-induced dispersive effects, which can be attributed to the additional path traveled by photons scattered at small angles.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  2. P. A. Robinson, Philos. Mag. B 80, 2087 (2000).
    [CrossRef]
  3. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
    [CrossRef]
  4. F. Scheffold, R. Lenke, R. Tweer, and G. Maret, Nature 398, 206 (1999).
    [CrossRef]
  5. R. Graaff and J. J. T. Bosch, Opt. Lett. 25, 43 (2000).
    [CrossRef]
  6. X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, and A. G. Yodh, Phys. Rev. E 61, 4295 (2000).
    [CrossRef]
  7. D. E. Hyde, T. J. Farrell, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
    [CrossRef] [PubMed]
  8. A. A. Chabanov and A. Z. Genack, Phys. Rev. E 56, 1338 (1997).
    [CrossRef]
  9. R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 79, 4369 (1997).
    [CrossRef]
  10. M. Boguñá, J. M. Porrà, and J. Masoliver, Phys. Rev. E 59, 6517 (1999).
    [CrossRef]
  11. S. A. Ramakrishna and N. Kumar, Phys. Rev. E 60, 1381 (1999).
    [CrossRef]
  12. Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
    [CrossRef]
  13. J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, Phys. Rev. E 63, 46613 (2001).
    [CrossRef]
  14. P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
    [CrossRef]
  15. M. van Exter and D. Grischkowsky, IEEE Trans. Microwave Theory Tech. 38, 1684 (1990).
    [CrossRef]
  16. J. R. Birch, J. D. Dromey, and J. Lesurf, Infrared Phys. 21, 225 (1981).
    [CrossRef]
  17. D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
    [CrossRef]
  18. R. A. Cheville, R. W. McGowan, and D. Grischkowsky, Phys. Rev. Lett. 80, 269 (1998).
    [CrossRef]
  19. M. T. Reiten, D. Grischkowsky, and R. A. Cheville, Appl. Phys. Lett. 78, 1146 (2001).
    [CrossRef]
  20. S. Fraden and G. Maret, Phys. Rev. Lett. 65, 512 (1990).
    [CrossRef] [PubMed]
  21. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

2001

D. E. Hyde, T. J. Farrell, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, Phys. Rev. E 63, 46613 (2001).
[CrossRef]

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, Appl. Phys. Lett. 78, 1146 (2001).
[CrossRef]

2000

P. A. Robinson, Philos. Mag. B 80, 2087 (2000).
[CrossRef]

R. Graaff and J. J. T. Bosch, Opt. Lett. 25, 43 (2000).
[CrossRef]

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, and A. G. Yodh, Phys. Rev. E 61, 4295 (2000).
[CrossRef]

1999

M. Boguñá, J. M. Porrà, and J. Masoliver, Phys. Rev. E 59, 6517 (1999).
[CrossRef]

S. A. Ramakrishna and N. Kumar, Phys. Rev. E 60, 1381 (1999).
[CrossRef]

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, Nature 398, 206 (1999).
[CrossRef]

1998

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, Phys. Rev. Lett. 80, 269 (1998).
[CrossRef]

1997

A. A. Chabanov and A. Z. Genack, Phys. Rev. E 56, 1338 (1997).
[CrossRef]

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 79, 4369 (1997).
[CrossRef]

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

1996

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[CrossRef]

1990

S. Fraden and G. Maret, Phys. Rev. Lett. 65, 512 (1990).
[CrossRef] [PubMed]

M. van Exter and D. Grischkowsky, IEEE Trans. Microwave Theory Tech. 38, 1684 (1990).
[CrossRef]

1988

P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

1981

J. R. Birch, J. D. Dromey, and J. Lesurf, Infrared Phys. 21, 225 (1981).
[CrossRef]

Auston, D. H.

P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Bartolini, P.

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

Birch, J. R.

J. R. Birch, J. D. Dromey, and J. Lesurf, Infrared Phys. 21, 225 (1981).
[CrossRef]

Boguñá, M.

M. Boguñá, J. M. Porrà, and J. Masoliver, Phys. Rev. E 59, 6517 (1999).
[CrossRef]

Bosch, J. J. T.

Chabanov, A. A.

A. A. Chabanov and A. Z. Genack, Phys. Rev. E 56, 1338 (1997).
[CrossRef]

Chance, B.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, and A. G. Yodh, Phys. Rev. E 61, 4295 (2000).
[CrossRef]

Cheville, R. A.

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, Appl. Phys. Lett. 78, 1146 (2001).
[CrossRef]

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, Phys. Rev. Lett. 80, 269 (1998).
[CrossRef]

Culver, J. P.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, and A. G. Yodh, Phys. Rev. E 61, 4295 (2000).
[CrossRef]

de Vries, P.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 79, 4369 (1997).
[CrossRef]

Dromey, J. D.

J. R. Birch, J. D. Dromey, and J. Lesurf, Infrared Phys. 21, 225 (1981).
[CrossRef]

Durduran, T.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, and A. G. Yodh, Phys. Rev. E 61, 4295 (2000).
[CrossRef]

Farrell, T. J.

D. E. Hyde, T. J. Farrell, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

Fraden, S.

S. Fraden and G. Maret, Phys. Rev. Lett. 65, 512 (1990).
[CrossRef] [PubMed]

Genack, A. Z.

A. A. Chabanov and A. Z. Genack, Phys. Rev. E 56, 1338 (1997).
[CrossRef]

Gomez Rivas, J.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, Phys. Rev. E 63, 46613 (2001).
[CrossRef]

Graaff, R.

Grischkowsky, D.

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, Appl. Phys. Lett. 78, 1146 (2001).
[CrossRef]

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, Phys. Rev. Lett. 80, 269 (1998).
[CrossRef]

M. van Exter and D. Grischkowsky, IEEE Trans. Microwave Theory Tech. 38, 1684 (1990).
[CrossRef]

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

Hyde, D. E.

D. E. Hyde, T. J. Farrell, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Jacobsen, R. H.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[CrossRef]

Jones, I. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Kop, R. H. J.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 79, 4369 (1997).
[CrossRef]

Kumar, N.

S. A. Ramakrishna and N. Kumar, Phys. Rev. E 60, 1381 (1999).
[CrossRef]

Lagendijk, A.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, Phys. Rev. E 63, 46613 (2001).
[CrossRef]

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 79, 4369 (1997).
[CrossRef]

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

Lenke, R.

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, Nature 398, 206 (1999).
[CrossRef]

Lesurf, J.

J. R. Birch, J. D. Dromey, and J. Lesurf, Infrared Phys. 21, 225 (1981).
[CrossRef]

Li, X.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, and A. G. Yodh, Phys. Rev. E 61, 4295 (2000).
[CrossRef]

Maret, G.

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, Nature 398, 206 (1999).
[CrossRef]

S. Fraden and G. Maret, Phys. Rev. Lett. 65, 512 (1990).
[CrossRef] [PubMed]

Masoliver, J.

M. Boguñá, J. M. Porrà, and J. Masoliver, Phys. Rev. E 59, 6517 (1999).
[CrossRef]

McGowan, R. W.

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, Phys. Rev. Lett. 80, 269 (1998).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

Mittleman, D. M.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[CrossRef]

Noordam, L. D.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, Phys. Rev. E 63, 46613 (2001).
[CrossRef]

Nuss, M. C.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[CrossRef]

P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Page, J. H.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Pattanayak, D. N.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, and A. G. Yodh, Phys. Rev. E 61, 4295 (2000).
[CrossRef]

Patterson, M. S.

D. E. Hyde, T. J. Farrell, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

Porrà, J. M.

M. Boguñá, J. M. Porrà, and J. Masoliver, Phys. Rev. E 59, 6517 (1999).
[CrossRef]

Ramakrishna, S. A.

S. A. Ramakrishna and N. Kumar, Phys. Rev. E 60, 1381 (1999).
[CrossRef]

Reiten, M. T.

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, Appl. Phys. Lett. 78, 1146 (2001).
[CrossRef]

Rella, C. W.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, Phys. Rev. E 63, 46613 (2001).
[CrossRef]

Righini, R.

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

Robinson, P. A.

P. A. Robinson, Philos. Mag. B 80, 2087 (2000).
[CrossRef]

Scheffold, F.

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, Nature 398, 206 (1999).
[CrossRef]

Schriemer, H. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Sheng, P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Smith, P. R.

P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Sprik, R.

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, Phys. Rev. E 63, 46613 (2001).
[CrossRef]

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 79, 4369 (1997).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

Tweer, R.

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, Nature 398, 206 (1999).
[CrossRef]

van Exter, M.

M. van Exter and D. Grischkowsky, IEEE Trans. Microwave Theory Tech. 38, 1684 (1990).
[CrossRef]

Weitz, D. A.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Wiersma, D. S.

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

Wilson, B. C.

D. E. Hyde, T. J. Farrell, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

Yodh, A. G.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, and A. G. Yodh, Phys. Rev. E 61, 4295 (2000).
[CrossRef]

Zhang, Z. Q.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

Appl. Phys. Lett.

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, Appl. Phys. Lett. 78, 1146 (2001).
[CrossRef]

IEEE J. Quantum Electron.

P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

M. van Exter and D. Grischkowsky, IEEE Trans. Microwave Theory Tech. 38, 1684 (1990).
[CrossRef]

Infrared Phys.

J. R. Birch, J. D. Dromey, and J. Lesurf, Infrared Phys. 21, 225 (1981).
[CrossRef]

Nature

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

F. Scheffold, R. Lenke, R. Tweer, and G. Maret, Nature 398, 206 (1999).
[CrossRef]

Opt. Lett.

Philos. Mag. B

P. A. Robinson, Philos. Mag. B 80, 2087 (2000).
[CrossRef]

Phys. Med. Biol.

D. E. Hyde, T. J. Farrell, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

Phys. Rev. E

A. A. Chabanov and A. Z. Genack, Phys. Rev. E 56, 1338 (1997).
[CrossRef]

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, and A. G. Yodh, Phys. Rev. E 61, 4295 (2000).
[CrossRef]

M. Boguñá, J. M. Porrà, and J. Masoliver, Phys. Rev. E 59, 6517 (1999).
[CrossRef]

S. A. Ramakrishna and N. Kumar, Phys. Rev. E 60, 1381 (1999).
[CrossRef]

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999).
[CrossRef]

J. Gomez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, Phys. Rev. E 63, 46613 (2001).
[CrossRef]

Phys. Rev. Lett.

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, Phys. Rev. Lett. 80, 269 (1998).
[CrossRef]

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 79, 4369 (1997).
[CrossRef]

S. Fraden and G. Maret, Phys. Rev. Lett. 65, 512 (1990).
[CrossRef] [PubMed]

Other

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Schematic of the transmission experiment.

Fig. 2
Fig. 2

(a) THz waveforms transmitted through a random medium consisting of a collection of Teflon spheres. Path lengths are as shown. The reference waveform has been reduced in amplitude by a factor of 2. Each vertical tick mark is 1  nA. (b) Amplitudes and (c) phases of the Fourier transforms of the reference waveform (dashed lines), and the first, third, fifth, and seventh waveforms in (a).

Fig. 3
Fig. 3

Scattering mean free path lsω versus free-space wavelength, on a log scale. The filled squares are for the smaller spheres, with diameter 0.794  mm, and the open circles are for the larger spheres, d=1.19 mm. The solid curve shows lsω for the smaller spheres, computed with Mie theory. The error bars represent the uncertainties associated with repeated measurements. The inset shows the measured values of kls versus wavelength for the smaller spheres.

Fig. 4
Fig. 4

The bottom solid curve shows the effective refractive index, which exhibits dispersive effects due toscattering. The error bars represent the uncertainties associated with repeated measurements. The dashed curve is the result of a Monte Carlo simulation of the effective index using the measured values of lsω (Fig.  2) and the long-wavelength limiting value of neffω as inputs. The arrows show two effective-medium approximations for the composite sample, obtained using the measured volume fraction. The uncertainty in the volume fraction leads to the error bars shown. The top portion of the figure shows the measured index of solid Teflon, which is not dispersive and agrees well with literature values.16

Metrics