The phenomenon of super-resolution in time-reversal acoustics is analyzed theoretically and with numerical simulations. A signal that is recorded and then retransmitted by an array of transducers, propagates back though the medium, and refocuses approximately on the source that emitted it. In a homogeneous medium, the refocusing resolution of the time-reversed signal is limited by diffraction. When the medium has random inhomogeneities the resolution of the refocused signal can in some circumstances beat the diffraction limit. This is super-resolution. A theoretical treatment of this phenomenon is given, and numerical simulations which confirm the theory are presented.
REFERENCES
1.
Arnold, A. (1995). “Numerically Absorbing Boundary Conditions for Quantum Evolution Equations,” in Proceedings of the International Workshop on Computational Electronics, Tempe, AZ.
2.
Asch
, M.
, Kohler
, W.
, Papanicolaou
, G.
, Postel
, M.
, and White
, B.
(1991
). “Frequency content of randomly scattered signals
,” SIAM Rev.
33
, 519
–626
.3.
Bal, G., Papanicolaou, G., and Ryzhik, L. (2001). “Radiative transport in a time-dependent random medium,” J. Stat. Phys. (submitted).
4.
Bamberger
, A.
, Engquist
, B.
, Halpern
, L.
, and Joly
, P.
(1988
). “Parabolic wave equation approximations in heterogeneous media
,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
48
(1
), 99
–128
.5.
Barryman, J., Borcea, L., Papanicolaou, G., and Tsogka, C. (2001). “Imaging and time reversal in random media,” Inverse Problems (submitted).
6.
Bécache, E., Collino, F., and Joly, P. (1998). “Higher-order Numerical Schemes and Operator Splitting for Solving 3D Paraxial Wave Equations in Heterogeneous Media,” Technical Report 3497, INRIA: Institute National de Recherche en Informatique et en Automatique.
7.
Bouc
, R.
, and Pardoux
, E.
(1984
). “Asymptotic analysis of PDEs with wide-band noise disturbances and expansion of the moments
,” Stoch. Anal. Appl.
2
, 369
–422
.8.
Clouet
, J. F.
, and Fouque
, J. P.
(1997
). “A time-reversal method for an acoustical pulse propagating in randomly layered media
,” Wave Motion
25
, 361
–368
.9.
Dashen
, R.
, Flatté
, S. M.
, and Reynolds
, S. A.
(1985
). “Path-integral treatment of acoustic mutual coherence functions for rays in a sound channel
,” J. Acoust. Soc. Am.
77
, 1716
–1722
.10.
Dawson
, D. A.
, and Papanicolaou
, G.
(1984
). “A random wave process
,” Appl. Math. Opt.
12
, 97
–114
.11.
Derode
, A.
, Roux
, P.
, and Fink
, M.
(1995
). “Robust acoustic time reversal with high-order multiple scattering
,” Phys. Rev. Lett.
75
(23
), 4206
–4209
.12.
Dowling
, D. R.
, and Jackson
, D. R.
(1990
). “Phase conjugation in underwater acoustics
,” J. Acoust. Soc. Am.
89
, 171
–181
.13.
Dowling
, D. R.
, and Jackson
, D. R.
(1992
). “Narrow-band performance of phase-conjugate arrays in dynamic random media
,” J. Acoust. Soc. Am.
91
, 3257
–3277
.14.
15.
16.
Flatté
, S. M.
, Reynolds
, S. A.
, and Dashen
, R.
(1987
). “Path-integral treatment of intensity behavior for rays in a sound channel
,” J. Acoust. Soc. Am.
82
, 967
–972
.17.
Furutsu, K. (1993). Random Media and Boundaries: Unified Theory, Two-Scale Method, and Applications (Springer, Berlin).
18.
Hodgkiss
, W. S.
, Song
, H. C.
, Kuperman
, W. A.
, Akal
, T.
, Ferla
, C.
, and Jackson
, D. R.
(1999
). “A long-range and variable focus phase-conjugation experiment in shallow water
,” J. Acoust. Soc. Am.
105
, 1597
–1604
.19.
Kohler, W., and Papanicolaou, G. (1977). Wave Propagation in a Randomly Inhomogeneous Ocean, Lecture Notes in Physics 70, Wave Propagation and Underwater Acoustics (Springer, Berlin).
20.
Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations (Cambridge University Press, Cambridge).
21.
Kuperman
, W. A.
, Hodgkiss
, W.
, Song
, H. C.
, Akal
, T.
, Ferla
, C.
, and Jackson
, D. R.
(1997
). “Phase conjugation in the ocean
,” J. Acoust. Soc. Am.
102
, 1
–16
.22.
Papanicolaou, G., Ryzhik, L., and Solna, K. (2001). “The parabolic approximation and time reversal in a random medium,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (submitted).
23.
Porter, R. P. (1989). Generalized Holography with Application to Inverse Scattering and Inverse Source Problems, Number XXVII in Progress in Optics (Elsevier Science, New York).
24.
Ryzhik
, L. V.
, Papanicolaou
, G. C.
, and Keller
, J. B.
(1996
). “Transport equations for elastic and other waves in random media
,” Wave Motion
24
, 327
–370
.25.
Solna
, K.
, and Papanicolaou
, G.
(2000
). “Ray theory for a locally layered random medium
,” Random Media
10
, 155
–202
.26.
Tappert, F. D. (1977). The Parabolic Approximation Method, Lecture Notes in Physics 70, Wave Propagation and Underwater Acoustics (Springer, Berlin).
27.
Tatarskii, V. I., Ishimaru, A., and Zavorotny, V. U., editors (1993). Wave Propagation in Random Media (Scintillation) (SPIE and IOP, SPIE, Bellingham, WA; IOP, Philadelphia PA.)
28.
Tsogka, C., and Papanicolaou, G. (2001). “
Time reversal through a solid–liquid interface and super-resolution,” Inverse Problems (submitted).
29.
van Rossum
, M. C. W.
, and Nieuwenhuizen
, T. M.
(1999
). “Multiple scattering of classical waves: Microscopy, mesoscopy, and diffusion
,” Rev. Mod. Phys.
71
(1
), 313
–371
.
This content is only available via PDF.
© 2002 Acoustical Society of America.
2002
Acoustical Society of America
You do not currently have access to this content.
