- Email: [email protected]
Time reversed wave propagation experiments in chaotic micro-structured cavities
Ultrasonics 42 (2004) 775–779 www.elsevier.com/locate/ultras
Time reversed wave propagation experiments in chaotic micro-structured cavities Rudolf Sprik a
a,*
, Arnaud Tourin
b
Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65-67, 1018 XE Amsterdam, The Netherlands b Laboratoire Ondes et Acoustique, Ecole Suprieure de Physique et de Chimie Industrielles, Paris, France
Abstract The elastic wave propagation in strongly scattering solid-state cavity consisting of a thin micro-patterned silicon wafer is studied experimentally. The chaotic behavior is induced by the irregular boundary of the cavity and/or by fabricating patterns of small holes in the wafer by laser machining. The pattern and hole size are designed with length scales matching the wavelength 6 1 MHz and induce multiple scattering within the wafer bounds. Regular patterns of holes add phononic band like dispersion properties to the system. Elastic waves obey under very general conditions time reversal and reciprocity symmetry. In the cavity system the strong mode mixing in the wafer between shear and compression waves, the strong anisotropy of silicon, and even small dissipation is not destroying the time reversal and reciprocity symmetry in the system. We present a systematic study of the quality of the time-reversal signal as function of the chosen time interval and the position in the recorded signal. Also the influence of time dilatation, i.e. stretching and compressing the time scale of the re-emitted signal, is studied. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Time reversal; Reciprocity; Chaotic cavity; Elastic wave propagation
The propagation of waves through systems with strong scatterers is ubiquitous in nature [1]. Examples are the propagation of electro-magnetic waves in turbid media, acoustic and elastic waves in seismology, and also in the quantum mechanical wave description of electrons in condensed matter. If the characteristic length scales of the inhomogeneity in the medium match the wavelength and the mean-free-path is smaller than the sample size, multiple scattering, interference and diffraction are essential in the wave propagation. Recently studied phenomena associated with the wave propagation in strongly scattering systems include strong and weak localization in optical wave propagation [1,2]. Also in systems where the scatterers are placed on regular lattices, multiple scattering may be dominant. The formation of band gaps in the energy spectrum where no wave propagation is possible is caused by the interference of the multiple scattered waves. Recently this has been studied in photonic band gap crystals and the acoustical pendent of phononic band gap systems *
Corresponding author. Tel.: +31-20-525-5645; fax: +31-20-5255788. E-mail address: [email protected] (R. Sprik). 0041-624X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2004.01.081
[2]. Multiple scattering is also present in the scattering from boundaries of enclosed spaces. In particular the cross-over behavior from a well defined spectrum in highly symmetric cavities to classical and wave chaotic behavior has been studied in the context of quantum mechanics, and classical wave mechanics [3]. The understanding of the fundamental properties of wave propagation in strongly scattering systems stimulated many applications including such diverse topics as medical imaging through turbid media to ultrasonic underwater communication. In strongly scattering environments waves still obey under very general conditions time reversal and reciprocity symmetry. Many fundamental aspects of these symmetry properties and applications have been demonstrated in acoustics by Fink et al. in Paris [4]. In particular, robust time-reversal reconstruction of ultrasonic pulsed signals in a 2D chaotic cavity system by a single transmitter–receiver pair has been realized [5–8]. The process of time-reversal reconstruction in a cavity using one channel is similar to the process of refocusing by time-reversal mirrors where the wavefront is reconstructed by multiple transducers. Here we study the effect on time reversal reconstruction of scattering from
776
R. Sprik, A. Tourin / Ultrasonics 42 (2004) 775–779
Fig. 1. Schematic overview of the ultrasonic set up. Arbitrary modulated waves at ’1 MHz are generated by an ultrasonic transducer. The transmitted waves are detected by a similar ultrasonic transducer and recorded. The height modulation can also be monitored by an optical interferometer.
high concentration of scatterers inside a nearly two dimensional enclosure. Two similar set ups were used to study the wave propagation (see Fig. 1 for a schematic overview of the components). Arbitrary modulated waves in the wafer were generated and detected by ultrasonic transducers connected to sharp metallic cones that act as point-like transducers with a size smaller than the ultrasonic wavelength [5,6]. Set up I based in Paris [6], France used transverse wave excitation and non-contact detection of flexural modes on the wafer by a heterodyne optical interferometer [9]. Set up II based in Enschede, The Netherlands used compressional wave excitation and detection with an equivalent needle transducer. Set up I used a dedicated multi-channel processor (Corelec Corp.), while Setup II was based on a HP33120A arbitrary wave generator for excitation and a EG&G 5113 amplifier and a Tektronix TDS440 oscilloscope for detection. Samples were made of 76 mm diameter, 0.35 mm thick mono-crystalline Silicon wafers. Untreated wafers have a small flat side to identify crystal orientation that make the wafers act as disk-like cavities with an irregular boundary (see Fig. 2). Further patterning of the Si wafers with holes was done by CNC (Computer Numerical Control) laser machining with a 150W YAG laser system at 1064 nm (Rohm and Haas). The patterns were generated with the use of a CAD (computer aided design) program and saved in a drawing format that is suitable to control the laser machining. The method produces high quality 1 mm diameter holes or other shapes with a better precision than 0.1 mm. In wave propagation studies (set up I) a wave is launched from the tip of the needle and the propagating wave field is mapped by the interferometer as function of time and place by sampling on a spatial grid. Fig. 3 shows an example of a map of the recorded signal at a fixed time. In a movie representation of the signal the propagation and scattering of the wavefronts originating from the source can be visualized. A typical time trace is shown at the top. Post-processing of the recorded traces allows first time arrival analysis, fourier transforming etc. The reverberation time for the signals
Fig. 2. Patterns of holes in laser machined silicon wafers. The 76 mm diameter, 0.35 mm thick mono-crystalline Silicon wafers are laser machined with a pattern of 1 mm diameter holes. (a) An ordered pattern of holes on a hexagonal lattice with a hole distance of 2 mm. The holes cover 22.7% of the surface area. (b) Same as (a) but with a random placement of the holes around the lattice position.
launched and detected by compressional transducers is in the ms range. More detailed analysis reveals that the relaxation has a fast component in the ms range and component ’3 times slower. The power spectra of the transmitted waves that can be derived from the recorded curves are not significantly different for wafers with disordered, ordered hole patterns, or unpatterned wafers. Hence, we did not obtained direct spectral evidence of e.g. phononic band gap formation.
R. Sprik, A. Tourin / Ultrasonics 42 (2004) 775–779
Fig. 3. Map of the wave height in a disordered sample using point excitation at the location marked by Ôx’ near the (10,8) pixel and detected by interferometer. The gray scale gives the amplitude of the wave. The holes in the wafer are indicated and show up as areas without signal. The time trace at pixel (8,8) is shown at the top (one point ¼ 0.5 ls).
The details of the elastic wave propagation in the thin wafers and the interaction with the boundary and the patterns of holes are quite involved. An additional complication is the anisotropy of wave velocities in Si; the longitudinal (vl ¼ 8433 m/s along [1 0 0] direction) and transverse (vt ¼ 5844 m/s along [1 0 0]) velocities vary as much as 20% in different crystal orientations in bulk material. The main propagating modes in the unpatterned wafer are the lowest symmetric (S0), antisymmetric (A0) Lamb waves, and the shear horizontal (SH 0) mode [10,11]. For the Si wafer the velocity of the (SH 0) mode is vt , of the S0 mode near vl , and of the A0 mode in the 500–3000 m/s range for frequencies between 0.1 and 3 MHz. In particular for the velocity of the A0 mode the associated wavelength matches the distance of the holes in the wafer, which should be favorable to observe band gap formation. The problem of scattering the Lamb waves from crack’s, holes and boundaries has been studied in the context of non-destructive materials testing and diag-
777
nostics. The resulting cross-section for the scattering of Lamb waves indicates efficient scattering with a mean-free-path of <1 mm for the A0 mode [12]. The small effect of the holes on the spectral features of the wave propagation can be readily explained. In the measurements the waves are transmitted and detected using point-like transducers that couple partially to the propagating waves. To observe with this point-to-point method e.g. band gap formation implies that the locally available density-of-states at the transmitter and/or receiver position should be considerably suppressed. This is true for a material with a full phononic band gap. If only a gap in certain directions in the ordered structure exists, the remaining available modes can still be excited and will support the wave propagation. Mode conversion at the boundary of the wafer will obscure the effect further. The time reversed wave propagation experiments between two transducers in contact with the wafer start by recording the multiple scattered response on one transducer (R) due to a short excitation on the other transducer (S). Part of the recorded signal is reversed in time and replayed through R and causes a while later a time-reversed reconstruction of the original signal at T . Also replaying through T and detecting at R shows a reconstruction as a consequence of reciprocity in the system. Time-reversal reconstruction is a very robust process (Fig. 4). Only a part of the response suffices to obtain a discernable signal. The quality of the reconstructed signal is quantified here as the maximum peakto-peak voltage (Vpp ) and studied as function of start position and recorded length (Fig. 5). As argued by Draeger and Fink [5], the reconstructed signal SDT ðtÞ from a segment DT can be represented by an expansion in a cosine series associated with the eigen modes wn ðrÞ in the cavity with an eigenfrequency xn :
Fig. 4. Time reversal reconstruction using the recorded signal. Trace (c) the transmitted time-reversed signal, (b) the reconstructed signal, and (a) a detail near the maximum of the reconstruction.
778
R. Sprik, A. Tourin / Ultrasonics 42 (2004) 775–779
Fig. 5. The peak-to-peak voltage of the time reversal reconstruction as function of the width DT and the starting position in the pulse response (horizontal time axis). The vertical line indicates the section before the first arrival of signals.
SDT ðtÞ ¼
X 1 an cosðxn tÞ: x2n n
Fig. 6. Frequency dependence of the phase divided by the frequency and the amplitude of the reconstruction signal obtained by fourier transform of the time signal. The dashed line indicates /f =f ¼ 2ptR .
ð1Þ
The coefficients an are given by the value of the eigen modes at the source (rS ) and the receiver (rR ) position: 1 an ¼ DT w2n ðrS Þw2n ðrR Þ: 2
ð2Þ
The factor 1=2DT results from the cross-correlation integral between modes n and m over the interval from tb to tb þ DT : Imn ¼
Z
tb þDT
sinðxn ðt þ sÞÞ sinðxm sÞds:
ð3Þ
tb
The number of modes participating in the series depends on the properties of the cavity. Important are the frequency span Dx and the typical distance between modes dx. In a chaotic cavity the mode density and spacing at any location and frequency is sufficient to represent an arbitrary signal. Eq. (1) does not include effects of dissipation that cause the reduction of the reconstruction at later start times (Fig. 5). The reconstructed signal has a well determined phase dependence. The cosine terms in Eq. (1) should have a time dependence cosðxðt tR ÞÞ to have a maximum reconstruction of the input pulse at tR . This implies that the phase of the fourier transformed signal has a constant slope /f =f ¼ 2ptR . This feature is illustrated in Fig. 6 where the magnitude and the phase as function of frequency obtained by fourier transformation of the reconstructed signal is shown. Changes in the frequency Fa as set on of the arbitrary wave generator (Fig. 1) change the replay speed of the recorded signal. The time axis t0 of the replayed wave is stretched or compressed with respect to the original axis t if Fa is increased or decreased respectively. This acts
Fig. 7. Time reversal reconstruction for different replay frequencies of the arbitrary wave generator (Fig. 1). The horizontal axis specifies the time dilatation parameter d defined t0 ¼ tð1 þ dÞ. Open symbols: begin position at 1 ms; closed symbols: begin position at 2 ms. Up triangle DT ¼ 0:8 ms, down triangle DT ¼ 0:4 ms.
similar to time dilatation for the generated wave. As a result the reconstruction quality degrades rapidly (see Fig. 7). The effect of the dilatation can be estimated from the cross-correlation integral Eq. (3). To lowest order expansion in the dilatation parameter d defined by t0 ¼ tð1 þ dÞ, the maximum of the reconstructed signal scales as: 1 2 SDT ;d ’ 1 ð2pf DT Þ d2 : ð4Þ 6 Here f is the central frequency of the signal. For example for a f ¼ 1 MHz signal with DT ¼ 1 ms, the signal amplitude halves for d ’ 0:001. Fig. 7 shows the parabolic dependence on d in Eq. 4 for small d. Qualitatively the TR gave the same results for wafers without and with additional drilled holes. Work is in progress to further identify the interplay between cavity boundaries,
R. Sprik, A. Tourin / Ultrasonics 42 (2004) 775–779
and the scatterers, and is similar to the analysis of the multiple acoustic scattering in a tank with fish [13]. Acknowledgements The authors thank J. de Rosny from the ESPCI in Paris for assistance with the 2D acoustic experiments. Experiments were also performed at the University of Twente in the Netherlands as a guest of the group ÔComplex Photonic Systems’. This work is part of the research programme of the ÔStichting voor Fundamenteel Onderzoek der Materie (FOM)’, which is financially supported by the ÔNederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)’. References [1] P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, Academic Press, Boston, 1995. [2] C.M. Soukoulis (Ed.), Photonic Crystals and Light Localization in the 21st Century, Kluwer, Dordrecht, 2001. [3] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York, 1991.
779
[4] M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter, J.-L. Thomas, F. Wu, Time-reversed acoustics, Rep. Prog. Phys. 63 (2000) 1933–1995. [5] C. Draeger, M. Fink, One-channel time-reversal in chaotic cavities: Theoretical limits, J. Acoust. Soc. Am. 105 (1999) 611– 617. [6] C. Draeger, J.-C. Aime, M. Fink, One-channel time-reversal in chaotic cavities: Experimental results, J. Acoust. Soc. Am. 105 (1999) 618–625. [7] J. de Rosny, A. Tourin, M. Fink, Coherent backscattering in a 2D chaotic cavity, Phys. Rev. Lett. 84 (2000) 1693–1697. [8] J. de Rosny, M. Fink, Overcoming the diffraction limit in wave physics using a time-reversal mirror and a novel acoustic sink, Phys. Rev. Lett. 89 (2002) 124301-1. [9] D. Royer, O. Casula, A sensitive ultrasonic method for measuring transient motions of a surface, Appl. Phys. Lett. 67 (1995) 3248– 3250. [10] K.F. Graff, Wave Motion in Elastic Solids, Dover Publications, New York, 1991. [11] F.L. Degertekin, B.T. Khuri-Yakub, Lamb wave excitation by Hertzian contacts with applications in NDE, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 44 (1997) 769–779. [12] A.N. Norris, C. Vemula, Scattering of flexural waves on thin plates, J. Sound Vib. 181 (1995) 115–125. [13] J. de Rosny, P. Roux, Multiple scattering in a reflecting cavity: Applications to fish counting in a tank, J. Acoust. Soc. Am. 109 (2001) 2587–2597.
Time reversed wave propagation experiments in chaotic micro-structured cavities Rudolf Sprik a
a,*
, Arnaud Tourin
b
Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65-67, 1018 XE Amsterdam, The Netherlands b Laboratoire Ondes et Acoustique, Ecole Suprieure de Physique et de Chimie Industrielles, Paris, France
Abstract The elastic wave propagation in strongly scattering solid-state cavity consisting of a thin micro-patterned silicon wafer is studied experimentally. The chaotic behavior is induced by the irregular boundary of the cavity and/or by fabricating patterns of small holes in the wafer by laser machining. The pattern and hole size are designed with length scales matching the wavelength 6 1 MHz and induce multiple scattering within the wafer bounds. Regular patterns of holes add phononic band like dispersion properties to the system. Elastic waves obey under very general conditions time reversal and reciprocity symmetry. In the cavity system the strong mode mixing in the wafer between shear and compression waves, the strong anisotropy of silicon, and even small dissipation is not destroying the time reversal and reciprocity symmetry in the system. We present a systematic study of the quality of the time-reversal signal as function of the chosen time interval and the position in the recorded signal. Also the influence of time dilatation, i.e. stretching and compressing the time scale of the re-emitted signal, is studied. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Time reversal; Reciprocity; Chaotic cavity; Elastic wave propagation
The propagation of waves through systems with strong scatterers is ubiquitous in nature [1]. Examples are the propagation of electro-magnetic waves in turbid media, acoustic and elastic waves in seismology, and also in the quantum mechanical wave description of electrons in condensed matter. If the characteristic length scales of the inhomogeneity in the medium match the wavelength and the mean-free-path is smaller than the sample size, multiple scattering, interference and diffraction are essential in the wave propagation. Recently studied phenomena associated with the wave propagation in strongly scattering systems include strong and weak localization in optical wave propagation [1,2]. Also in systems where the scatterers are placed on regular lattices, multiple scattering may be dominant. The formation of band gaps in the energy spectrum where no wave propagation is possible is caused by the interference of the multiple scattered waves. Recently this has been studied in photonic band gap crystals and the acoustical pendent of phononic band gap systems *
Corresponding author. Tel.: +31-20-525-5645; fax: +31-20-5255788. E-mail address: [email protected] (R. Sprik). 0041-624X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2004.01.081
[2]. Multiple scattering is also present in the scattering from boundaries of enclosed spaces. In particular the cross-over behavior from a well defined spectrum in highly symmetric cavities to classical and wave chaotic behavior has been studied in the context of quantum mechanics, and classical wave mechanics [3]. The understanding of the fundamental properties of wave propagation in strongly scattering systems stimulated many applications including such diverse topics as medical imaging through turbid media to ultrasonic underwater communication. In strongly scattering environments waves still obey under very general conditions time reversal and reciprocity symmetry. Many fundamental aspects of these symmetry properties and applications have been demonstrated in acoustics by Fink et al. in Paris [4]. In particular, robust time-reversal reconstruction of ultrasonic pulsed signals in a 2D chaotic cavity system by a single transmitter–receiver pair has been realized [5–8]. The process of time-reversal reconstruction in a cavity using one channel is similar to the process of refocusing by time-reversal mirrors where the wavefront is reconstructed by multiple transducers. Here we study the effect on time reversal reconstruction of scattering from
776
R. Sprik, A. Tourin / Ultrasonics 42 (2004) 775–779
Fig. 1. Schematic overview of the ultrasonic set up. Arbitrary modulated waves at ’1 MHz are generated by an ultrasonic transducer. The transmitted waves are detected by a similar ultrasonic transducer and recorded. The height modulation can also be monitored by an optical interferometer.
high concentration of scatterers inside a nearly two dimensional enclosure. Two similar set ups were used to study the wave propagation (see Fig. 1 for a schematic overview of the components). Arbitrary modulated waves in the wafer were generated and detected by ultrasonic transducers connected to sharp metallic cones that act as point-like transducers with a size smaller than the ultrasonic wavelength [5,6]. Set up I based in Paris [6], France used transverse wave excitation and non-contact detection of flexural modes on the wafer by a heterodyne optical interferometer [9]. Set up II based in Enschede, The Netherlands used compressional wave excitation and detection with an equivalent needle transducer. Set up I used a dedicated multi-channel processor (Corelec Corp.), while Setup II was based on a HP33120A arbitrary wave generator for excitation and a EG&G 5113 amplifier and a Tektronix TDS440 oscilloscope for detection. Samples were made of 76 mm diameter, 0.35 mm thick mono-crystalline Silicon wafers. Untreated wafers have a small flat side to identify crystal orientation that make the wafers act as disk-like cavities with an irregular boundary (see Fig. 2). Further patterning of the Si wafers with holes was done by CNC (Computer Numerical Control) laser machining with a 150W YAG laser system at 1064 nm (Rohm and Haas). The patterns were generated with the use of a CAD (computer aided design) program and saved in a drawing format that is suitable to control the laser machining. The method produces high quality 1 mm diameter holes or other shapes with a better precision than 0.1 mm. In wave propagation studies (set up I) a wave is launched from the tip of the needle and the propagating wave field is mapped by the interferometer as function of time and place by sampling on a spatial grid. Fig. 3 shows an example of a map of the recorded signal at a fixed time. In a movie representation of the signal the propagation and scattering of the wavefronts originating from the source can be visualized. A typical time trace is shown at the top. Post-processing of the recorded traces allows first time arrival analysis, fourier transforming etc. The reverberation time for the signals
Fig. 2. Patterns of holes in laser machined silicon wafers. The 76 mm diameter, 0.35 mm thick mono-crystalline Silicon wafers are laser machined with a pattern of 1 mm diameter holes. (a) An ordered pattern of holes on a hexagonal lattice with a hole distance of 2 mm. The holes cover 22.7% of the surface area. (b) Same as (a) but with a random placement of the holes around the lattice position.
launched and detected by compressional transducers is in the ms range. More detailed analysis reveals that the relaxation has a fast component in the ms range and component ’3 times slower. The power spectra of the transmitted waves that can be derived from the recorded curves are not significantly different for wafers with disordered, ordered hole patterns, or unpatterned wafers. Hence, we did not obtained direct spectral evidence of e.g. phononic band gap formation.
R. Sprik, A. Tourin / Ultrasonics 42 (2004) 775–779
Fig. 3. Map of the wave height in a disordered sample using point excitation at the location marked by Ôx’ near the (10,8) pixel and detected by interferometer. The gray scale gives the amplitude of the wave. The holes in the wafer are indicated and show up as areas without signal. The time trace at pixel (8,8) is shown at the top (one point ¼ 0.5 ls).
The details of the elastic wave propagation in the thin wafers and the interaction with the boundary and the patterns of holes are quite involved. An additional complication is the anisotropy of wave velocities in Si; the longitudinal (vl ¼ 8433 m/s along [1 0 0] direction) and transverse (vt ¼ 5844 m/s along [1 0 0]) velocities vary as much as 20% in different crystal orientations in bulk material. The main propagating modes in the unpatterned wafer are the lowest symmetric (S0), antisymmetric (A0) Lamb waves, and the shear horizontal (SH 0) mode [10,11]. For the Si wafer the velocity of the (SH 0) mode is vt , of the S0 mode near vl , and of the A0 mode in the 500–3000 m/s range for frequencies between 0.1 and 3 MHz. In particular for the velocity of the A0 mode the associated wavelength matches the distance of the holes in the wafer, which should be favorable to observe band gap formation. The problem of scattering the Lamb waves from crack’s, holes and boundaries has been studied in the context of non-destructive materials testing and diag-
777
nostics. The resulting cross-section for the scattering of Lamb waves indicates efficient scattering with a mean-free-path of <1 mm for the A0 mode [12]. The small effect of the holes on the spectral features of the wave propagation can be readily explained. In the measurements the waves are transmitted and detected using point-like transducers that couple partially to the propagating waves. To observe with this point-to-point method e.g. band gap formation implies that the locally available density-of-states at the transmitter and/or receiver position should be considerably suppressed. This is true for a material with a full phononic band gap. If only a gap in certain directions in the ordered structure exists, the remaining available modes can still be excited and will support the wave propagation. Mode conversion at the boundary of the wafer will obscure the effect further. The time reversed wave propagation experiments between two transducers in contact with the wafer start by recording the multiple scattered response on one transducer (R) due to a short excitation on the other transducer (S). Part of the recorded signal is reversed in time and replayed through R and causes a while later a time-reversed reconstruction of the original signal at T . Also replaying through T and detecting at R shows a reconstruction as a consequence of reciprocity in the system. Time-reversal reconstruction is a very robust process (Fig. 4). Only a part of the response suffices to obtain a discernable signal. The quality of the reconstructed signal is quantified here as the maximum peakto-peak voltage (Vpp ) and studied as function of start position and recorded length (Fig. 5). As argued by Draeger and Fink [5], the reconstructed signal SDT ðtÞ from a segment DT can be represented by an expansion in a cosine series associated with the eigen modes wn ðrÞ in the cavity with an eigenfrequency xn :
Fig. 4. Time reversal reconstruction using the recorded signal. Trace (c) the transmitted time-reversed signal, (b) the reconstructed signal, and (a) a detail near the maximum of the reconstruction.
778
R. Sprik, A. Tourin / Ultrasonics 42 (2004) 775–779
Fig. 5. The peak-to-peak voltage of the time reversal reconstruction as function of the width DT and the starting position in the pulse response (horizontal time axis). The vertical line indicates the section before the first arrival of signals.
SDT ðtÞ ¼
X 1 an cosðxn tÞ: x2n n
Fig. 6. Frequency dependence of the phase divided by the frequency and the amplitude of the reconstruction signal obtained by fourier transform of the time signal. The dashed line indicates /f =f ¼ 2ptR .
ð1Þ
The coefficients an are given by the value of the eigen modes at the source (rS ) and the receiver (rR ) position: 1 an ¼ DT w2n ðrS Þw2n ðrR Þ: 2
ð2Þ
The factor 1=2DT results from the cross-correlation integral between modes n and m over the interval from tb to tb þ DT : Imn ¼
Z
tb þDT
sinðxn ðt þ sÞÞ sinðxm sÞds:
ð3Þ
tb
The number of modes participating in the series depends on the properties of the cavity. Important are the frequency span Dx and the typical distance between modes dx. In a chaotic cavity the mode density and spacing at any location and frequency is sufficient to represent an arbitrary signal. Eq. (1) does not include effects of dissipation that cause the reduction of the reconstruction at later start times (Fig. 5). The reconstructed signal has a well determined phase dependence. The cosine terms in Eq. (1) should have a time dependence cosðxðt tR ÞÞ to have a maximum reconstruction of the input pulse at tR . This implies that the phase of the fourier transformed signal has a constant slope /f =f ¼ 2ptR . This feature is illustrated in Fig. 6 where the magnitude and the phase as function of frequency obtained by fourier transformation of the reconstructed signal is shown. Changes in the frequency Fa as set on of the arbitrary wave generator (Fig. 1) change the replay speed of the recorded signal. The time axis t0 of the replayed wave is stretched or compressed with respect to the original axis t if Fa is increased or decreased respectively. This acts
Fig. 7. Time reversal reconstruction for different replay frequencies of the arbitrary wave generator (Fig. 1). The horizontal axis specifies the time dilatation parameter d defined t0 ¼ tð1 þ dÞ. Open symbols: begin position at 1 ms; closed symbols: begin position at 2 ms. Up triangle DT ¼ 0:8 ms, down triangle DT ¼ 0:4 ms.
similar to time dilatation for the generated wave. As a result the reconstruction quality degrades rapidly (see Fig. 7). The effect of the dilatation can be estimated from the cross-correlation integral Eq. (3). To lowest order expansion in the dilatation parameter d defined by t0 ¼ tð1 þ dÞ, the maximum of the reconstructed signal scales as: 1 2 SDT ;d ’ 1 ð2pf DT Þ d2 : ð4Þ 6 Here f is the central frequency of the signal. For example for a f ¼ 1 MHz signal with DT ¼ 1 ms, the signal amplitude halves for d ’ 0:001. Fig. 7 shows the parabolic dependence on d in Eq. 4 for small d. Qualitatively the TR gave the same results for wafers without and with additional drilled holes. Work is in progress to further identify the interplay between cavity boundaries,
R. Sprik, A. Tourin / Ultrasonics 42 (2004) 775–779
and the scatterers, and is similar to the analysis of the multiple acoustic scattering in a tank with fish [13]. Acknowledgements The authors thank J. de Rosny from the ESPCI in Paris for assistance with the 2D acoustic experiments. Experiments were also performed at the University of Twente in the Netherlands as a guest of the group ÔComplex Photonic Systems’. This work is part of the research programme of the ÔStichting voor Fundamenteel Onderzoek der Materie (FOM)’, which is financially supported by the ÔNederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)’. References [1] P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, Academic Press, Boston, 1995. [2] C.M. Soukoulis (Ed.), Photonic Crystals and Light Localization in the 21st Century, Kluwer, Dordrecht, 2001. [3] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York, 1991.
779
[4] M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter, J.-L. Thomas, F. Wu, Time-reversed acoustics, Rep. Prog. Phys. 63 (2000) 1933–1995. [5] C. Draeger, M. Fink, One-channel time-reversal in chaotic cavities: Theoretical limits, J. Acoust. Soc. Am. 105 (1999) 611– 617. [6] C. Draeger, J.-C. Aime, M. Fink, One-channel time-reversal in chaotic cavities: Experimental results, J. Acoust. Soc. Am. 105 (1999) 618–625. [7] J. de Rosny, A. Tourin, M. Fink, Coherent backscattering in a 2D chaotic cavity, Phys. Rev. Lett. 84 (2000) 1693–1697. [8] J. de Rosny, M. Fink, Overcoming the diffraction limit in wave physics using a time-reversal mirror and a novel acoustic sink, Phys. Rev. Lett. 89 (2002) 124301-1. [9] D. Royer, O. Casula, A sensitive ultrasonic method for measuring transient motions of a surface, Appl. Phys. Lett. 67 (1995) 3248– 3250. [10] K.F. Graff, Wave Motion in Elastic Solids, Dover Publications, New York, 1991. [11] F.L. Degertekin, B.T. Khuri-Yakub, Lamb wave excitation by Hertzian contacts with applications in NDE, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 44 (1997) 769–779. [12] A.N. Norris, C. Vemula, Scattering of flexural waves on thin plates, J. Sound Vib. 181 (1995) 115–125. [13] J. de Rosny, P. Roux, Multiple scattering in a reflecting cavity: Applications to fish counting in a tank, J. Acoust. Soc. Am. 109 (2001) 2587–2597.