Broadband cloaking and holography with exact boundary conditions Dirk-Jan van Manen, Marlies Vasmel, Stewart Greenhalgh, and Johan O. A. RobertssonJL
Citation: The Journal of the Acoustical Society of America 137, EL415 (2015); doi: 10.1121/1.4921340 View online: http://dx.doi.org/10.1121/1.4921340 View Table of Contents: http://asa.scitation.org/toc/jas/137/6 Published by the Acoustical Society of America
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
Broadband cloaking and holography with exact boundary conditions Dirk-Jan van Manen, Marlies Vasmel, Stewart Greenhalgh, and Johan O. A. Robertsson Institute of Geophysics, ETH Z€ urich, Sonneggstrasse 5, CH-8092, Z€ urich, Switzerland [email protected], [email protected], [email protected], [email protected]
Abstract: Broadband cloaking and holography are achieved by creating an exact boundary condition on a surface enclosing an object or free space. A time-recursive, discrete version of the Kirchhoff–Helmholtz integral predicts the wavefield impinging on the surface, as well as its transmission through an arbitrary embedding or replacement medium. Surface source distributions proportional to the predicted wavefield cancel the incident waves and radiate the desired response. The fields inside and outside the surface can be controlled independently. A two-dimensional numerical example shows that cloaking and holography can be achieved to within numerical precision across the frequency range of the incident radiation. C 2015 Acoustical Society of America V
[JL] Date Received: October 21, 2014
Date Accepted: May 8, 2015
1. Introduction After a decade of research into metamaterials and transformation optics, the idea of making objects appear optically or acoustically transparent is no longer dubitable. Following Pendry’s proposal of a perfect flat lens via negative refraction (Pendry, 2000), electromagnetic metamaterials have become ubiquitous in modern photonics research (Cai and Shalaev, 2010). Similarly, in acoustics, locally resonant structures have been found that display an effective macroscopic behaviour, such as negative density, beyond Newton’s second law (Craster and Guenneau, 2013). Nevertheless, most of the results obtained to-date with metamaterial cloaking achieve shielding only in quasi-monochromatic experiments. We present an approach that is not based on metamaterials but rather on creating an exact boundary condition on a surface enclosing the object to be cloaked. This exact boundary condition cancels arbitrary, broadband incident wavefields as well as radiates the field that would be transmitted through the embedding if the object was not present. The method is closely related to reciprocity theory (de Hoop, 1995; Fokkema and van den Berg, 1993; Wapenaar, 2007). Some work on suppressing scattering from objects using Kirchhoff–Helmholtz type expressions has been carried out previously (Malyuzhinets, 1964; Uosukainen, 2003; Friot et al., 2006). In these works the field scattered from the object is recorded on a soundtransparent surface surrounding the scatterer and extrapolated outward to a second soundtransparent surface of monopole and dipole sources which radiate the extrapolated scattered field in counterphase, thus canceling it. Our work differs significantly from these approaches in that the field is controlled before it reaches the scatterer and ultimately can be prevented from reaching it completely. Thus, if an incident field is detected in the enclosure, one knows that the cloaking is compromised. Furthermore, the sources can be placed on the surface of the object, to take advantage of the boundary conditions of the scattering object and requiring only the receiver surface to be implemented transparently. Exact boundary conditions (EBCs) were introduced by van Manen et al. (2007) for the purpose of re-computing full waveform seismograms after arbitrarily strong, localized model perturbations. Instead of repeating the computations on the entire model domain, full-waveform computations are only done on the part that is
J. Acoust. Soc. Am. 137 (6), June 2015
C 2015 Acoustical Society of America EL415 V
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
perturbed. The interaction of the wavefield scattered by the perturbations with the background model is fully accounted for by the EBC. More recently, a novel wave propagation laboratory for immersive experimentation has been proposed based on EBCs (Vasmel et al., 2013). By identifying a finite-size laboratory with a truncated computational domain, and a numerical extension of the physical domain with the background model, EBCs can be used to immerse the physical experiment in the larger numerical simulation in real-time. This enables low-frequency experimentation on targets of arbitrary size and complexity. Finally, EBCs can be used to remove freesurface related multiple reflections from marine seismic data (Vasmel et al., 2014). 2. Exact boundary conditions for general internal boundaries Consider two states1 A and B that may be present in the same volume, V, bounded by surfaces S1 and S2 with inward normals n1 and n2 [Fig. 1(A)]. It can be shown (Fokkema and van den Berg, 1993; Wapenaar, 2007) that a wavefield representation for the pressure in state A in terms of the sources and wavefields in states A and B is ð pA ðx; tÞ ¼ Ct fGBq ðx; x0 ; tÞ; qA ðx0 ; tÞg þ Ct fGBf ;i ðx; x0 ; tÞ; fi;A ðx0 ; tÞgdV0 V ð þ ½Ct fGBq ðx; x0 ; tÞ; vi;A ðx0 ; tÞg þ Ct fGBf ;i ðx; x0 ; tÞ; pA ðx0 ; tÞgni dS0 ; (1) fS1 ;S2 g
where Ct{} denotes temporal convolution and qA(x0 ) and fA(x0 ) denote source densities of volume injection rate and force, respectively. Quantities GBq ðx; x0 Þ and GBf ;i ðx; x0 Þ denote the Green’s functions for pressure in state B at location x due to point sources of volume injection, q, and unidirectional force, f, in the i direction, at location x0 . Finally, quantities pA(x0 ) and vA(x0 ) denote the pressure and particle velocity in state A, respectively. Considering the limiting case of an unbounded domain with transparent boundary conditions on S2 [Fig. 1(B)], de Hoop (1995) showed that provided all states are physical, the wavefields in Eq. (1) are causal and their far-field representations apply (on S2) for sufficiently large radius of S2. Therefore, the time convolutions in the integrands are also causal and for any finite value of time, t, the radius can be chosen large enough so that the contribution of S2 to the integrand vanishes. Thus, Eq. (1) applies to the configuration in Fig. 1(C), with internal surface S only. This representation is general because the boundary conditions for states A and B on S, affecting the behaviour of the wavefields on/across S [i.e., the actual values of pA(x0 ), vA(x0 ), GBq ðx; x0 Þ, and GBf ;i ðx; x0 Þ], have not yet been specified, and neither have the source distributions qA and fi,A. In the following, let us assume that the boundary conditions for state B and the source distributions qA and fi,A are fixed. Note that this still leaves the freedom to specify the boundary condition for state A on S. However, also note that the volume integral in the right hand side of Eq. (1) does not depend on the boundary
Fig. 1. (A) Volume V enclosed by external surface S2 and internal surface S1. (B) Beyond r, V is homogeneous and in the limit of S2 receding to infinity, the far-field representations for the Green’s functions apply to S2. (C) Configuration for cloaking and holography.
EL416 J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
condition in state A on S. Hence, it is possible to write down two representations (e.g., for state A0 in terms of states A0 and B, and for state A00 in terms of states A00 and B) where only the boundary conditions corresponding to the A-states vary and for which, consequently, the volume integral is identical. By differencing two such representations, as done below, the volume integrals cancel and we arrive at a completely general formulation for exact boundary conditions. Thus, defining fi;A0 ðx0 ; tÞ ¼ fi;A00 ðx0 ; tÞ ¼ fi;A ðx0 ; tÞ; qA0 ðx0 ; tÞ ¼ qA00 ðx0 ; tÞ ¼ qA ðx0 ; tÞ; pA0 ðx0 ; tÞ 6¼ pA00 ðx0 ; tÞ ðdue to differing BCsÞ; vi;A0 ðx0 ; tÞ 6¼ vi;A00 ðx0 ; tÞ ðdue to differing BCsÞ; and substituting the A0 quantities in Eq. (1) to obtain a first representation, a second representation follows by substituting the A00 quantities in Eq. (1). Taking the difference between the resulting representations and rearranging terms, we find pA0 ðx; tÞ ¼ pA00 ðx; tÞ þ pEBC ðx; tÞ;
(2)
where pEBC ðx; tÞ ¼
ð S
½Ct fGBq ; ðvi;A0 vi;A00 Þg þ Ct fGBf ;i ; ðpA0 pA00 Þgni dS0 :
(3)
Equations (2) and (3) show that the pressure in state A0 can be made equal to the pressure in state A00 by adding a pressure pEBC caused by specific boundary conditions on S proportional to inter-state wavefield differences. However, the simplicity of Eqs. (2) and (3) is deceptive since those boundary conditions to turn the pressure in state A0 into the pressure in state A00 depend on the actual pressures and particle velocities in states A0 and A00 themselves. As shown in van Manen et al. (2007) and Vasmel et al. (2013), these pressures and particle velocities have to be predicted by extrapolation from a transparent recording surface R enclosing S. This can be done using timerecursive, discrete versions of Green’s second identity, ð h ^ q 0 00 ðx; l n; x0 ; 0Þ^v j;A0 jA00 ðx0 ; nÞ ^p A0 jA00 ðx; l; nÞ ¼ ^p A0 jA00 ðx; l; n 1Þ þ G A jA R i f ;j ^ 0 00 ðx; l n; x0 ; 0Þ^p A0 jA00 ðx0 ; nÞ nj dS0 (4) þG A jA and ð h
^ q 0 00 ðx; l n; x0 ; 0Þ^v j;A0 jA00 ðx0 ; nÞ C i;A jA i ^ f ;j 0 00 ðx; l n; x0 ; 0Þ^p A0 jA00 ðx0 ; nÞ nj dS0 ; þC i;A jA
^v i;A0 jA00 ðx; l; nÞ ¼ ^v i;A0 jA00 ðx; l; n 1Þ þ
R
(5)
^ q 0 00 , evaluated for x 2 S and where ^ denotes discrete time sampling. Quantities G A jA ^ f ;j0 00 , C ^ q 0 00 , and C ^ f ;j 0 00 are the (components of the) pressure and particle velocity G A jA i;A jA i;A jA Green’s functions due to point sources of volume injection rate and force on R for the fields in state A0 and A00 , respectively. These functions are computed in advance through wave simulations on the models defining states A0 and A00 . Note that in the derivation of Eqs. (2) and (3) we have assumed that the sources of the incident field are inside V. We do not consider here any other cases. 3. Cloaking Consider the case in which an arbitrary object is present inside S in Fig. 1(C) and that the goal is to make this object invisible to a broadband incident wavefield. In this case,
J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography EL417
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
Table 1. States for cloaking. State A0 (emb)
State A00 (sct)
State B (sct)
pA0 ¼ pemb vA0 ¼ vemb transparent
pA00 ¼ psct vA00 ¼ vsct transparent
q GBq ¼ Gsct f G B ¼ G fsct transparent
Field BCs on S
the desired wavefield in state A0 equals the wavefield that would propagate in the embedding if the scattering object was absent. Hence, the scatterer needs to be defined in terms of a contrast with respect to an embedding medium. The wavefield in state A00 corresponds to the wavefield that exists with the scatterer present. It is convenient to take the injection surface S some distance outside the scatterer (but not coincident with surfaces of discontinuities in medium properties). In that case, the boundary conditions on S are transparent (i.e., non-scattering). This situation is summarised in Table 1. Substituting these states in Eqs. (2) and (3) yields pemb ðx; tÞ ¼ psct ðx; tÞ þ pEBC ðx; tÞ;
(6)
where pEBC ðx; tÞ ¼
ð h
i q f ;i ; ðvi;emb vi;sct Þg þ Ct fGsct ; ðpemb psct Þg ni dS0 : Ct fGsct
(7)
S
Thus, the EBC for cloaking corresponds to a combination of a distribution of monopole sources with strength (vemb vsct) n and a distribution of dipole sources with strength (pemb psct) on S. These quantities need to be predicted by extrapolation of the actual wavefield from the recording surface R using Eqs. (4) and (5). Thus, to predict the wavefield on S as it would propagate in the embedding medium, the Green’s functions for a medium with only the embedding present inside S have to be used. On the other hand, to predict the wavefield on S as it propagates with the scatterer present, the Green’s functions with the scatterer inside S have to be used. 4. Holography Another interesting case to consider is where there is no scatterer inside S, but where one wants to make it appear as if there is an arbitrary object present inside S. This corresponds to holography. In this case, state A0 equals the wavefield that would be present when the arbitrary object is present inside S and state A00 equals the actual wavefield as it propagates in the embedding medium. Taking the surface S larger than the holographic object ensures transparent boundary conditions on S. This is summarised in Table 2. Substituting in Eqs. (2) and (3) yields almost the same equations as before, except that the monopole and dipole source strengths are the opposite of the source strengths required for cloaking. In this sense cloaking is opposite to holography. 5. Numerical example Both cloaking and holography using EBCs will now be demonstrated with a numerical example using two-dimensional acoustic finite-difference modeling. First, a scattering Table 2. States for holography.
Field BCs on S
State A0 (sct)
State A00 (emb)
State B (emb)
pA0 ¼ psct vA0 ¼ vsct transparent
pA00 ¼ pemb vA00 ¼ vemb transparent
q GBq ¼ Gemb f G B ¼ G femb transparent
EL418 J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
object of dimensions 0.40 0.08 m is modeled inside in a 4 4 m fluid domain (mass density q ¼ 1000 kg/m3 and sound speed c ¼ 1500 m/s). A surface, S, of 0.5 0.5 m symmetrically surrounds the object and is covered with monopole and dipole sources in accordance with Eqs. (6) and (7). The receiver surface, R, is placed 0.1 m away from the surface S and completely surrounds it. Transparent boundary conditions exist on R. First, the extrapolation Green’s functions are computed between every (grid)point on the receiver surface and every (grid)point on the injection surface. As explained, this is done for both a homogeneous model with medium parameters equal to the host material and for the model with the scattering object present. Next, a point source comprising a 5 kHz Ricker wavelet is set off directly above the scattering object. The walls of the domain are modeled with rigid surfaces, so the reflected waves illuminate the cloaked region from many different angles over the course of the simulation. At every timestep of the simulation, the actual wavefield values are recorded on R and used to update the EBC at all future timesteps [using Eqs. (4) and (5) with the wavefields and Green’s functions for states A0 and A00 according to Table 1]. The current values of the boundary condition, which include contributions from all previous timesteps, are applied on S (i.e., the normal component of the difference between extrapolated particle velocities is applied as the source strength of the monopole sources whereas the difference between the extrapolated pressures is applied as the source strength of the dipole sources). Snapshots of the resulting wavefield taken every 375 ls are shown in Fig. 2 (first row) whereas the difference with respect to a reference solution (i.e., without the scattering object present) is shown in Fig. 2 (third row). Note that the difference has been amplified by a factor 1012 in order to reveal any potential reflections that would indicate deficiencies in the cloaking procedure. Apart from expected differences inside the cloaked region, no differences can be seen. Next, we illustrate holography. In this case, the scattering object is absent, but we want to make it appear as if present. The experiment is repeated, but with the extrapolation Green’s functions swapped. The result is shown in Fig. 2 (second row)
Fig. 2. (First row) The wavefield with a scatterer inside a cloaked region. (Second row) The wavefield without a scatterer inside a holographic region. (Third row) The difference between the field shown in the first row and a reference (not shown) without the object present, amplified by 1012. (Fourth row) The difference between the field shown in the second row and a reference (not shown) with the object present, amplified by 1012. Apart from expected differences inside S, no differences can be seen.
J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography EL419
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
whereas the difference with respect to a reference solution (i.e., with the scattering object present) is shown in Fig. 2 (fourth row). Again, the difference has been amplified by a factor 1012 but no differences outside of surface S can be observed. 6. Cloaking unknown objects The presented formulation is completely general and can be used to cloak arbitrary objects. Nevertheless, a drawback is that the Green’s functions of the cloaked object need to be measured or modeled beforehand. Fortunately, there is another way the Green’s functions can be chosen, which makes it possible to cloak objects whose Green’s functions are not known beforehand. Consider now the role of the different terms in the EBC in Eq. (3). The symmetry of the resulting expressions for cloaking and holography, combined with the fact that for the limiting case of a perfectly rigid/ free scatterer one of the first terms (i.e., vA0 n or pA0 ) remains unchanged (while the other three terms vanish on account of the rigid/free boundary condition) suggest that the role of the first term is to absorb incoming waves consistent with the desired incoming waves and to radiate outgoing waves consistent with the desired outgoing waves. On the other hand, the role of the second term appears to be to radiate incoming waves consistent with the actual incoming waves and absorb outgoing waves consistent with the actual outgoing waves. This is indeed the case. If the wave propagation inside S is not consistent with the wave propagation predicted by the second term then wavefields will “leak out” and interact with the rest of the model. Thus, by setting to zero the extrapolated wavefields corresponding to the second term, it is possible to create the correct wavefields outside S, while not creating any wavefields inside S. This makes it possible to cloak objects, whose internal structure is unknown: since no wavefield is generated within S, no wavefield will reflect or interact with the unknown structure. Similarly, it is possible to create holograms around unknown structures. 7. Discussion and conclusions Equations (2) and (3) can be compared with Eq. (4) of Vasmel et al. (2013), who consider impenetrable (rigid) exterior surfaces. As a result, only one type of source is needed on the surrounding surface (the other term disappears by virtue of the boundary condition and its effect on the Green’s function). Furthermore, in Vasmel et al. (2013) the wavefield in state A00 also disappears on the surface S, which means that only the wave-field of state A0 needs to be extrapolated and they were not dealing with wavefield differences. Equation (3) is more general and can be used for cloaking and holography of arbitrary objects. We note that the extrapolation of the four wavefields can be reduced to two by differencing the Green’s functions involved, since the same wavefield is being extrapolated. The approach was demonstrated for an arbitrary scatterer immersed in a fluid, with perfect cloaking and holography achieved across the frequency range of the source signal. Alternatives that do not require knowledge of the object being cloaked were also outlined. The presented formulations for cloaking and holography using EBCs extend to any wave-propagation problem (e.g., electromagnetic and elastodynamic). Acknowledgments This research was partially funded by SNF Grant No. 2-77532-12. We thank Andrew Curtis for comments which helped to improve the clarity of the paper. References and links 1
A state is a combination of wavefield values, source distributions and boundary conditions. Cai, W., and Shalaev, V. M. (2010). Optical Metamaterials (Springer, Berlin). Craster, R., and Guenneau, S. (2013). Acoustic Metamaterials (Springer, Berlin). de Hoop, A. T. (1995). Handbook of Radiation and Scattering of Waves (Academic, London).
EL420 J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
Fokkema, J. T., and van den Berg, P. M. (1993). Seismic Applications of Acoustic Reciprocity (Elsevier, Amsterdam). Friot, E., Guillermin, R., and Winniger, M. (2006). “Active control of scattered acoustic radiation: A realtime implementation for a three-dimensional object,” Acta Acust. Acust. 92, 278–288. Malyuzhinets, G. D. (1964). “On one theorem on analytical functions and its extension to the wave potentials,” in Proceedings of the 3rd All-Union Symposium on Wave Diffraction (Nauka, Moscow), pp. 113–116. Pendry, J. B. (2000). “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969. Uosukainen, S. (2003). “Active sound scatterers based on JMC method,” J. Sound Vib. 23, 383–390. van Manen, D.-J., Robertsson, J. O. A., and Curtis, A. (2007). “Exact wave field simulation for finitevolume scattering problems,” J. Acoust. Soc. Am. 122(4), EL115–EL121. Vasmel, M., Robertsson, J. O. A., and Amundsen, L. (2014). “A new solution to eliminate free surface related multiples in multicomponent streamer recordings,” in 76th EAGE Conference and Exhibition. Vasmel, M., Robertsson, J. O. A., van Manen, D.-J., and Curtis, A. (2013). “Immersive experimentation in a wave propagation laboratory,” J. Acoust. Soc. Am. 134(6), EL492–EL498. Wapenaar, K. (2007). “General representations for wavefield modeling and inversion in geophysics,” Geophysics 72(5), SM5–SM17.
J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography EL421
Citation: The Journal of the Acoustical Society of America 137, EL415 (2015); doi: 10.1121/1.4921340 View online: http://dx.doi.org/10.1121/1.4921340 View Table of Contents: http://asa.scitation.org/toc/jas/137/6 Published by the Acoustical Society of America
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
Broadband cloaking and holography with exact boundary conditions Dirk-Jan van Manen, Marlies Vasmel, Stewart Greenhalgh, and Johan O. A. Robertsson Institute of Geophysics, ETH Z€ urich, Sonneggstrasse 5, CH-8092, Z€ urich, Switzerland [email protected], [email protected], [email protected], [email protected]
Abstract: Broadband cloaking and holography are achieved by creating an exact boundary condition on a surface enclosing an object or free space. A time-recursive, discrete version of the Kirchhoff–Helmholtz integral predicts the wavefield impinging on the surface, as well as its transmission through an arbitrary embedding or replacement medium. Surface source distributions proportional to the predicted wavefield cancel the incident waves and radiate the desired response. The fields inside and outside the surface can be controlled independently. A two-dimensional numerical example shows that cloaking and holography can be achieved to within numerical precision across the frequency range of the incident radiation. C 2015 Acoustical Society of America V
[JL] Date Received: October 21, 2014
Date Accepted: May 8, 2015
1. Introduction After a decade of research into metamaterials and transformation optics, the idea of making objects appear optically or acoustically transparent is no longer dubitable. Following Pendry’s proposal of a perfect flat lens via negative refraction (Pendry, 2000), electromagnetic metamaterials have become ubiquitous in modern photonics research (Cai and Shalaev, 2010). Similarly, in acoustics, locally resonant structures have been found that display an effective macroscopic behaviour, such as negative density, beyond Newton’s second law (Craster and Guenneau, 2013). Nevertheless, most of the results obtained to-date with metamaterial cloaking achieve shielding only in quasi-monochromatic experiments. We present an approach that is not based on metamaterials but rather on creating an exact boundary condition on a surface enclosing the object to be cloaked. This exact boundary condition cancels arbitrary, broadband incident wavefields as well as radiates the field that would be transmitted through the embedding if the object was not present. The method is closely related to reciprocity theory (de Hoop, 1995; Fokkema and van den Berg, 1993; Wapenaar, 2007). Some work on suppressing scattering from objects using Kirchhoff–Helmholtz type expressions has been carried out previously (Malyuzhinets, 1964; Uosukainen, 2003; Friot et al., 2006). In these works the field scattered from the object is recorded on a soundtransparent surface surrounding the scatterer and extrapolated outward to a second soundtransparent surface of monopole and dipole sources which radiate the extrapolated scattered field in counterphase, thus canceling it. Our work differs significantly from these approaches in that the field is controlled before it reaches the scatterer and ultimately can be prevented from reaching it completely. Thus, if an incident field is detected in the enclosure, one knows that the cloaking is compromised. Furthermore, the sources can be placed on the surface of the object, to take advantage of the boundary conditions of the scattering object and requiring only the receiver surface to be implemented transparently. Exact boundary conditions (EBCs) were introduced by van Manen et al. (2007) for the purpose of re-computing full waveform seismograms after arbitrarily strong, localized model perturbations. Instead of repeating the computations on the entire model domain, full-waveform computations are only done on the part that is
J. Acoust. Soc. Am. 137 (6), June 2015
C 2015 Acoustical Society of America EL415 V
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
perturbed. The interaction of the wavefield scattered by the perturbations with the background model is fully accounted for by the EBC. More recently, a novel wave propagation laboratory for immersive experimentation has been proposed based on EBCs (Vasmel et al., 2013). By identifying a finite-size laboratory with a truncated computational domain, and a numerical extension of the physical domain with the background model, EBCs can be used to immerse the physical experiment in the larger numerical simulation in real-time. This enables low-frequency experimentation on targets of arbitrary size and complexity. Finally, EBCs can be used to remove freesurface related multiple reflections from marine seismic data (Vasmel et al., 2014). 2. Exact boundary conditions for general internal boundaries Consider two states1 A and B that may be present in the same volume, V, bounded by surfaces S1 and S2 with inward normals n1 and n2 [Fig. 1(A)]. It can be shown (Fokkema and van den Berg, 1993; Wapenaar, 2007) that a wavefield representation for the pressure in state A in terms of the sources and wavefields in states A and B is ð pA ðx; tÞ ¼ Ct fGBq ðx; x0 ; tÞ; qA ðx0 ; tÞg þ Ct fGBf ;i ðx; x0 ; tÞ; fi;A ðx0 ; tÞgdV0 V ð þ ½Ct fGBq ðx; x0 ; tÞ; vi;A ðx0 ; tÞg þ Ct fGBf ;i ðx; x0 ; tÞ; pA ðx0 ; tÞgni dS0 ; (1) fS1 ;S2 g
where Ct{} denotes temporal convolution and qA(x0 ) and fA(x0 ) denote source densities of volume injection rate and force, respectively. Quantities GBq ðx; x0 Þ and GBf ;i ðx; x0 Þ denote the Green’s functions for pressure in state B at location x due to point sources of volume injection, q, and unidirectional force, f, in the i direction, at location x0 . Finally, quantities pA(x0 ) and vA(x0 ) denote the pressure and particle velocity in state A, respectively. Considering the limiting case of an unbounded domain with transparent boundary conditions on S2 [Fig. 1(B)], de Hoop (1995) showed that provided all states are physical, the wavefields in Eq. (1) are causal and their far-field representations apply (on S2) for sufficiently large radius of S2. Therefore, the time convolutions in the integrands are also causal and for any finite value of time, t, the radius can be chosen large enough so that the contribution of S2 to the integrand vanishes. Thus, Eq. (1) applies to the configuration in Fig. 1(C), with internal surface S only. This representation is general because the boundary conditions for states A and B on S, affecting the behaviour of the wavefields on/across S [i.e., the actual values of pA(x0 ), vA(x0 ), GBq ðx; x0 Þ, and GBf ;i ðx; x0 Þ], have not yet been specified, and neither have the source distributions qA and fi,A. In the following, let us assume that the boundary conditions for state B and the source distributions qA and fi,A are fixed. Note that this still leaves the freedom to specify the boundary condition for state A on S. However, also note that the volume integral in the right hand side of Eq. (1) does not depend on the boundary
Fig. 1. (A) Volume V enclosed by external surface S2 and internal surface S1. (B) Beyond r, V is homogeneous and in the limit of S2 receding to infinity, the far-field representations for the Green’s functions apply to S2. (C) Configuration for cloaking and holography.
EL416 J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
condition in state A on S. Hence, it is possible to write down two representations (e.g., for state A0 in terms of states A0 and B, and for state A00 in terms of states A00 and B) where only the boundary conditions corresponding to the A-states vary and for which, consequently, the volume integral is identical. By differencing two such representations, as done below, the volume integrals cancel and we arrive at a completely general formulation for exact boundary conditions. Thus, defining fi;A0 ðx0 ; tÞ ¼ fi;A00 ðx0 ; tÞ ¼ fi;A ðx0 ; tÞ; qA0 ðx0 ; tÞ ¼ qA00 ðx0 ; tÞ ¼ qA ðx0 ; tÞ; pA0 ðx0 ; tÞ 6¼ pA00 ðx0 ; tÞ ðdue to differing BCsÞ; vi;A0 ðx0 ; tÞ 6¼ vi;A00 ðx0 ; tÞ ðdue to differing BCsÞ; and substituting the A0 quantities in Eq. (1) to obtain a first representation, a second representation follows by substituting the A00 quantities in Eq. (1). Taking the difference between the resulting representations and rearranging terms, we find pA0 ðx; tÞ ¼ pA00 ðx; tÞ þ pEBC ðx; tÞ;
(2)
where pEBC ðx; tÞ ¼
ð S
½Ct fGBq ; ðvi;A0 vi;A00 Þg þ Ct fGBf ;i ; ðpA0 pA00 Þgni dS0 :
(3)
Equations (2) and (3) show that the pressure in state A0 can be made equal to the pressure in state A00 by adding a pressure pEBC caused by specific boundary conditions on S proportional to inter-state wavefield differences. However, the simplicity of Eqs. (2) and (3) is deceptive since those boundary conditions to turn the pressure in state A0 into the pressure in state A00 depend on the actual pressures and particle velocities in states A0 and A00 themselves. As shown in van Manen et al. (2007) and Vasmel et al. (2013), these pressures and particle velocities have to be predicted by extrapolation from a transparent recording surface R enclosing S. This can be done using timerecursive, discrete versions of Green’s second identity, ð h ^ q 0 00 ðx; l n; x0 ; 0Þ^v j;A0 jA00 ðx0 ; nÞ ^p A0 jA00 ðx; l; nÞ ¼ ^p A0 jA00 ðx; l; n 1Þ þ G A jA R i f ;j ^ 0 00 ðx; l n; x0 ; 0Þ^p A0 jA00 ðx0 ; nÞ nj dS0 (4) þG A jA and ð h
^ q 0 00 ðx; l n; x0 ; 0Þ^v j;A0 jA00 ðx0 ; nÞ C i;A jA i ^ f ;j 0 00 ðx; l n; x0 ; 0Þ^p A0 jA00 ðx0 ; nÞ nj dS0 ; þC i;A jA
^v i;A0 jA00 ðx; l; nÞ ¼ ^v i;A0 jA00 ðx; l; n 1Þ þ
R
(5)
^ q 0 00 , evaluated for x 2 S and where ^ denotes discrete time sampling. Quantities G A jA ^ f ;j0 00 , C ^ q 0 00 , and C ^ f ;j 0 00 are the (components of the) pressure and particle velocity G A jA i;A jA i;A jA Green’s functions due to point sources of volume injection rate and force on R for the fields in state A0 and A00 , respectively. These functions are computed in advance through wave simulations on the models defining states A0 and A00 . Note that in the derivation of Eqs. (2) and (3) we have assumed that the sources of the incident field are inside V. We do not consider here any other cases. 3. Cloaking Consider the case in which an arbitrary object is present inside S in Fig. 1(C) and that the goal is to make this object invisible to a broadband incident wavefield. In this case,
J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography EL417
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
Table 1. States for cloaking. State A0 (emb)
State A00 (sct)
State B (sct)
pA0 ¼ pemb vA0 ¼ vemb transparent
pA00 ¼ psct vA00 ¼ vsct transparent
q GBq ¼ Gsct f G B ¼ G fsct transparent
Field BCs on S
the desired wavefield in state A0 equals the wavefield that would propagate in the embedding if the scattering object was absent. Hence, the scatterer needs to be defined in terms of a contrast with respect to an embedding medium. The wavefield in state A00 corresponds to the wavefield that exists with the scatterer present. It is convenient to take the injection surface S some distance outside the scatterer (but not coincident with surfaces of discontinuities in medium properties). In that case, the boundary conditions on S are transparent (i.e., non-scattering). This situation is summarised in Table 1. Substituting these states in Eqs. (2) and (3) yields pemb ðx; tÞ ¼ psct ðx; tÞ þ pEBC ðx; tÞ;
(6)
where pEBC ðx; tÞ ¼
ð h
i q f ;i ; ðvi;emb vi;sct Þg þ Ct fGsct ; ðpemb psct Þg ni dS0 : Ct fGsct
(7)
S
Thus, the EBC for cloaking corresponds to a combination of a distribution of monopole sources with strength (vemb vsct) n and a distribution of dipole sources with strength (pemb psct) on S. These quantities need to be predicted by extrapolation of the actual wavefield from the recording surface R using Eqs. (4) and (5). Thus, to predict the wavefield on S as it would propagate in the embedding medium, the Green’s functions for a medium with only the embedding present inside S have to be used. On the other hand, to predict the wavefield on S as it propagates with the scatterer present, the Green’s functions with the scatterer inside S have to be used. 4. Holography Another interesting case to consider is where there is no scatterer inside S, but where one wants to make it appear as if there is an arbitrary object present inside S. This corresponds to holography. In this case, state A0 equals the wavefield that would be present when the arbitrary object is present inside S and state A00 equals the actual wavefield as it propagates in the embedding medium. Taking the surface S larger than the holographic object ensures transparent boundary conditions on S. This is summarised in Table 2. Substituting in Eqs. (2) and (3) yields almost the same equations as before, except that the monopole and dipole source strengths are the opposite of the source strengths required for cloaking. In this sense cloaking is opposite to holography. 5. Numerical example Both cloaking and holography using EBCs will now be demonstrated with a numerical example using two-dimensional acoustic finite-difference modeling. First, a scattering Table 2. States for holography.
Field BCs on S
State A0 (sct)
State A00 (emb)
State B (emb)
pA0 ¼ psct vA0 ¼ vsct transparent
pA00 ¼ pemb vA00 ¼ vemb transparent
q GBq ¼ Gemb f G B ¼ G femb transparent
EL418 J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
object of dimensions 0.40 0.08 m is modeled inside in a 4 4 m fluid domain (mass density q ¼ 1000 kg/m3 and sound speed c ¼ 1500 m/s). A surface, S, of 0.5 0.5 m symmetrically surrounds the object and is covered with monopole and dipole sources in accordance with Eqs. (6) and (7). The receiver surface, R, is placed 0.1 m away from the surface S and completely surrounds it. Transparent boundary conditions exist on R. First, the extrapolation Green’s functions are computed between every (grid)point on the receiver surface and every (grid)point on the injection surface. As explained, this is done for both a homogeneous model with medium parameters equal to the host material and for the model with the scattering object present. Next, a point source comprising a 5 kHz Ricker wavelet is set off directly above the scattering object. The walls of the domain are modeled with rigid surfaces, so the reflected waves illuminate the cloaked region from many different angles over the course of the simulation. At every timestep of the simulation, the actual wavefield values are recorded on R and used to update the EBC at all future timesteps [using Eqs. (4) and (5) with the wavefields and Green’s functions for states A0 and A00 according to Table 1]. The current values of the boundary condition, which include contributions from all previous timesteps, are applied on S (i.e., the normal component of the difference between extrapolated particle velocities is applied as the source strength of the monopole sources whereas the difference between the extrapolated pressures is applied as the source strength of the dipole sources). Snapshots of the resulting wavefield taken every 375 ls are shown in Fig. 2 (first row) whereas the difference with respect to a reference solution (i.e., without the scattering object present) is shown in Fig. 2 (third row). Note that the difference has been amplified by a factor 1012 in order to reveal any potential reflections that would indicate deficiencies in the cloaking procedure. Apart from expected differences inside the cloaked region, no differences can be seen. Next, we illustrate holography. In this case, the scattering object is absent, but we want to make it appear as if present. The experiment is repeated, but with the extrapolation Green’s functions swapped. The result is shown in Fig. 2 (second row)
Fig. 2. (First row) The wavefield with a scatterer inside a cloaked region. (Second row) The wavefield without a scatterer inside a holographic region. (Third row) The difference between the field shown in the first row and a reference (not shown) without the object present, amplified by 1012. (Fourth row) The difference between the field shown in the second row and a reference (not shown) with the object present, amplified by 1012. Apart from expected differences inside S, no differences can be seen.
J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography EL419
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
whereas the difference with respect to a reference solution (i.e., with the scattering object present) is shown in Fig. 2 (fourth row). Again, the difference has been amplified by a factor 1012 but no differences outside of surface S can be observed. 6. Cloaking unknown objects The presented formulation is completely general and can be used to cloak arbitrary objects. Nevertheless, a drawback is that the Green’s functions of the cloaked object need to be measured or modeled beforehand. Fortunately, there is another way the Green’s functions can be chosen, which makes it possible to cloak objects whose Green’s functions are not known beforehand. Consider now the role of the different terms in the EBC in Eq. (3). The symmetry of the resulting expressions for cloaking and holography, combined with the fact that for the limiting case of a perfectly rigid/ free scatterer one of the first terms (i.e., vA0 n or pA0 ) remains unchanged (while the other three terms vanish on account of the rigid/free boundary condition) suggest that the role of the first term is to absorb incoming waves consistent with the desired incoming waves and to radiate outgoing waves consistent with the desired outgoing waves. On the other hand, the role of the second term appears to be to radiate incoming waves consistent with the actual incoming waves and absorb outgoing waves consistent with the actual outgoing waves. This is indeed the case. If the wave propagation inside S is not consistent with the wave propagation predicted by the second term then wavefields will “leak out” and interact with the rest of the model. Thus, by setting to zero the extrapolated wavefields corresponding to the second term, it is possible to create the correct wavefields outside S, while not creating any wavefields inside S. This makes it possible to cloak objects, whose internal structure is unknown: since no wavefield is generated within S, no wavefield will reflect or interact with the unknown structure. Similarly, it is possible to create holograms around unknown structures. 7. Discussion and conclusions Equations (2) and (3) can be compared with Eq. (4) of Vasmel et al. (2013), who consider impenetrable (rigid) exterior surfaces. As a result, only one type of source is needed on the surrounding surface (the other term disappears by virtue of the boundary condition and its effect on the Green’s function). Furthermore, in Vasmel et al. (2013) the wavefield in state A00 also disappears on the surface S, which means that only the wave-field of state A0 needs to be extrapolated and they were not dealing with wavefield differences. Equation (3) is more general and can be used for cloaking and holography of arbitrary objects. We note that the extrapolation of the four wavefields can be reduced to two by differencing the Green’s functions involved, since the same wavefield is being extrapolated. The approach was demonstrated for an arbitrary scatterer immersed in a fluid, with perfect cloaking and holography achieved across the frequency range of the source signal. Alternatives that do not require knowledge of the object being cloaked were also outlined. The presented formulations for cloaking and holography using EBCs extend to any wave-propagation problem (e.g., electromagnetic and elastodynamic). Acknowledgments This research was partially funded by SNF Grant No. 2-77532-12. We thank Andrew Curtis for comments which helped to improve the clarity of the paper. References and links 1
A state is a combination of wavefield values, source distributions and boundary conditions. Cai, W., and Shalaev, V. M. (2010). Optical Metamaterials (Springer, Berlin). Craster, R., and Guenneau, S. (2013). Acoustic Metamaterials (Springer, Berlin). de Hoop, A. T. (1995). Handbook of Radiation and Scattering of Waves (Academic, London).
EL420 J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography
van Manen et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4921340]
Published Online 22 May 2015
Fokkema, J. T., and van den Berg, P. M. (1993). Seismic Applications of Acoustic Reciprocity (Elsevier, Amsterdam). Friot, E., Guillermin, R., and Winniger, M. (2006). “Active control of scattered acoustic radiation: A realtime implementation for a three-dimensional object,” Acta Acust. Acust. 92, 278–288. Malyuzhinets, G. D. (1964). “On one theorem on analytical functions and its extension to the wave potentials,” in Proceedings of the 3rd All-Union Symposium on Wave Diffraction (Nauka, Moscow), pp. 113–116. Pendry, J. B. (2000). “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969. Uosukainen, S. (2003). “Active sound scatterers based on JMC method,” J. Sound Vib. 23, 383–390. van Manen, D.-J., Robertsson, J. O. A., and Curtis, A. (2007). “Exact wave field simulation for finitevolume scattering problems,” J. Acoust. Soc. Am. 122(4), EL115–EL121. Vasmel, M., Robertsson, J. O. A., and Amundsen, L. (2014). “A new solution to eliminate free surface related multiples in multicomponent streamer recordings,” in 76th EAGE Conference and Exhibition. Vasmel, M., Robertsson, J. O. A., van Manen, D.-J., and Curtis, A. (2013). “Immersive experimentation in a wave propagation laboratory,” J. Acoust. Soc. Am. 134(6), EL492–EL498. Wapenaar, K. (2007). “General representations for wavefield modeling and inversion in geophysics,” Geophysics 72(5), SM5–SM17.
J. Acoust. Soc. Am. 137 (6), June 2015
van Manen et al.: Broadband cloaking and holography EL421
