Double-negative acoustic metamaterials
Authors:
Habib Ammari, Brian Fitzpatrick, Hyundae Lee, Sanghyeon Yu and Hai Zhang
Journal:
Quart. Appl. Math. 77 (2019), 767-791
MSC (2010):
Primary 35C20, 74J20
DOI:
https://doi.org/10.1090/qam/1543
Published electronically:
June 3, 2019
MathSciNet review:
4009331
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The aim of this paper is to provide a mathematical theory for understanding the mechanism behind the double-negative refractive index phenomenon in bubbly fluids. The design of double-negative metamaterials generally requires the use of two different kinds of subwavelength resonators, which may limit the applicability of double-negative metamaterials. Herein we rely on media that consists of only a single type of resonant element, and show how to turn the acoustic metamaterial with a single negative effective property obtained in [SIAM J. Math. Anal., 49 (2017), pp. 3252–3276] into a negative refractive index metamaterial, which refracts waves negatively, hence acting as a superlens. Using bubble dimers made of two identical bubbles, it is proved, under the assumption that their volume fraction is small and their orientation is uniformly distributed, that both the effective mass density and the bulk modulus of the bubbly fluid can be negative at frequencies slightly higher than the dipole hybridized frequency for a single constituent bubble dimer.
References
- Habib Ammari, Youjun Deng, and Pierre Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal. 220 (2016), no. 1, 109–153. MR 3458160, DOI https://doi.org/10.1007/s00205-015-0928-0
- H. Ammari, B. Fitzpatrick, H. Lee, E. Orvehed Hiltunen, and S. Yu, Honeycomb-lattice Minnaert bubbles, arXiv:1811.03905.
- Habib Ammari, Brian Fitzpatrick, David Gontier, Hyundae Lee, and Hai Zhang, Minnaert resonances for acoustic waves in bubbly media, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 7, 1975–1998. MR 3906861, DOI https://doi.org/10.1016/j.anihpc.2018.03.007
- Habib Ammari, Brian Fitzpatrick, Hyeonbae Kang, Matias Ruiz, Sanghyeon Yu, and Hai Zhang, Mathematical and computational methods in photonics and phononics, Mathematical Surveys and Monographs, vol. 235, American Mathematical Society, Providence, RI, 2018. MR 3837172
- H. Ammari, K. Hamdache, and J.-C. Nédélec, Chirality in the Maxwell equations by the dipole approximation, SIAM J. Appl. Math. 59 (1999), no. 6, 2045–2059. MR 1709796, DOI https://doi.org/10.1137/S0036139998334160
- Habib Ammari, Brian Fitzpatrick, Hyundae Lee, Sanghyeon Yu, and Hai Zhang, Subwavelength phononic bandgap opening in bubbly media, J. Differential Equations 263 (2017), no. 9, 5610–5629. MR 3688425, DOI https://doi.org/10.1016/j.jde.2017.06.025
- Habib Ammari, Brian Fitzpatrick, David Gontier, Hyundae Lee, and Hai Zhang, A mathematical and numerical framework for bubble meta-screens, SIAM J. Appl. Math. 77 (2017), no. 5, 1827–1850. MR 3717823, DOI https://doi.org/10.1137/16M1090235
- Habib Ammari, Brian Fitzpatrick, David Gontier, Hyundae Lee, and Hai Zhang, Sub-wavelength focusing of acoustic waves in bubbly media, Proc. A. 473 (2017), no. 2208, 20170469, 17. MR 3749985, DOI https://doi.org/10.1098/rspa.2017.0469
- Habib Ammari and Hyeonbae Kang, Polarization and moment tensors, Applied Mathematical Sciences, vol. 162, Springer, New York, 2007. With applications to inverse problems and effective medium theory. MR 2327884
- Habib Ammari and Hyeonbae Kang, Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities, J. Math. Anal. Appl. 296 (2004), no. 1, 190–208. MR 2070502, DOI https://doi.org/10.1016/j.jmaa.2004.04.003
- Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Layer potential techniques in spectral analysis, Mathematical Surveys and Monographs, vol. 153, American Mathematical Society, Providence, RI, 2009. MR 2488135
- Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Asymptotic analysis of high-contrast phononic crystals and a criterion for the band-gap opening, Arch. Ration. Mech. Anal. 193 (2009), no. 3, 679–714. MR 2525115, DOI https://doi.org/10.1007/s00205-008-0179-4
- Habib Ammari, Hyundae Lee, and Hai Zhang, Bloch waves in bubbly crystal near the first band gap: a high-frequency homogenization approach, SIAM J. Math. Anal. 51 (2019), no. 1, 45–59. MR 3895120, DOI https://doi.org/10.1137/18M116722X
- Habib Ammari, Pierre Millien, Matias Ruiz, and Hai Zhang, Mathematical analysis of plasmonic nanoparticles: the scalar case, Arch. Ration. Mech. Anal. 224 (2017), no. 2, 597–658. MR 3614756, DOI https://doi.org/10.1007/s00205-017-1084-5
- H. Ammari, E. Orvehed Hiltunen, and S. Yu, A high-frequency homogenization approach near the Dirac points in bubbly honeycomb crystals, arXiv:1812.06178.
- Habib Ammari, Matias Ruiz, Sanghyeon Yu, and Hai Zhang, Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations, J. Differential Equations 261 (2016), no. 6, 3615–3669. MR 3527640, DOI https://doi.org/10.1016/j.jde.2016.05.036
- Habib Ammari, Wei Wu, and Sanghyeon Yu, Double-negative electromagnetic metamaterials due to chirality, Quart. Appl. Math. 77 (2019), no. 1, 105–130. MR 3897921, DOI https://doi.org/10.1090/qam/1516
- Habib Ammari and Hai Zhang, Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency, SIAM J. Math. Anal. 49 (2017), no. 4, 3252–3276. MR 3690650, DOI https://doi.org/10.1137/16M1078574
- Habib Ammari and Hai Zhang, Super-resolution in high-contrast media, Proc. A. 471 (2015), no. 2178, 20140946, 11. MR 3367310, DOI https://doi.org/10.1098/rspa.2014.0946
- S. A. Cummer, J. Christensen, and A. Alù, Controlling sound with acoustic metamaterials, Nature Rev., 1 (2016), 16001.
- R. Figari, G. Papanicolaou, and J. Rubinstein, Remarks on the point interaction approximation, Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986) IMA Vol. Math. Appl., vol. 9, Springer, New York, 1987, pp. 45–55. MR 914984, DOI https://doi.org/10.1007/978-1-4684-6347-7_4
- R. Figari, G. Papanicolaou, and J. Rubinstein, The point interaction approximation for diffusion in regions with many small holes, Stochastic methods in biology (Nagoya, 1985) Lecture Notes in Biomath., vol. 70, Springer, Berlin, 1987, pp. 202–209. MR 893647, DOI https://doi.org/10.1007/978-3-642-46599-4_16
- A. Figotin and P. Kuchment, Spectral properties of classical waves in high-contrast periodic media, SIAM J. Appl. Math. 58 (1998), no. 2, 683–702. MR 1617610, DOI https://doi.org/10.1137/S0036139996297249
- N. Kaina, F. Lemoult, M. Fink, and G. Lerosey, Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials, Nature, 525 (2015), 77–81L.
- M. Lanoy, R. Pierrat, F. Lemoult, M. Fink, V. Leroy, and A. Tourin, Subwavelength focusing in bubbly media using broadband time reversal, Phys. Rev. B, 91.22 (2015), 224202.
- J. Lekner, Capacitance coefficients of two spheres, Journal of Electrostatics, 69 (2011), 11–14
- F. Lemoult, N. Kaina, M. Fink, and G. Lerosey, Soda cans metamaterial: A subwavelength-scaled photonic crystal, Crystals, 6 (2016), 82.
- V. Leroy, A. Bretagne, M. Fink, H. Willaime, P. Tabeling, and A. Tourin, Design and characterization of bubble phononic crystals, Appl. Phys. Lett., 95 (2009), 171904.
- V. Leroy, A. Strybulevych, M. Lanoy, F. Lemoult, A. Tourin, and J. H. Page, Superabsorption of acoustic waves with bubble metascreens, Phys. Rev. B, 91.2 (2015), 020301.
- J. Li and C. T. Chan, Double-negative acoustic meta-material, Phys. Rev. E, 70 (2004), p. 055602.
- G. Ma and P. Sheng, Acoustic metamaterials: From local resonances to broad horizons, Sci. Adv., 2 (2016), e1501595.
- G. W. Milton, N. A. P. Nicorovici, and R. C. McPhedran, Opaque perfect lenses, Phys. B, 394 (2007), 171–175.
- M. Minnaert, On musical air-bubbles and the sounds of running water, The London, Edinburgh, Dublin Philos. Mag. and J. of Sci., 16 (1933), 235–248.
- N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, Optical and dielectric properties of partially resonant composites, Phys. Rev. B, 49.12 (1994), 8479–8482.
- Shin Ozawa, Point interaction potential approximation for $(-\Delta +U)^{-1}$ and eigenvalues of the Laplacian on wildly perturbed domain, Osaka J. Math. 20 (1983), no. 4, 923–937. MR 727440
- Joseph B. Keller, David W. McLaughlin, and George C. Papanicolaou (eds.), Surveys in applied mathematics. Vol. 1, Surveys in Applied Mathematics, vol. 1, Plenum Press, New York, 1995. MR 1366206
- J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), 3966–3969.
- S. A. Ramakrishna, Physics of negative refractive index materials, Rep. Progr. Phys., 68 (2005), 449.
- R. Zhu, X. N. Liu, G. K. Hu, C. T. Sun, and G. L. Huang, Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial, Nature Comm., 5(2014), 5510.
References
- Habib Ammari, Youjun Deng, and Pierre Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal. 220 (2016), no. 1, 109–153. MR 3458160, DOI https://doi.org/10.1007/s00205-015-0928-0
- H. Ammari, B. Fitzpatrick, H. Lee, E. Orvehed Hiltunen, and S. Yu, Honeycomb-lattice Minnaert bubbles, arXiv:1811.03905.
- Habib Ammari, Brian Fitzpatrick, David Gontier, Hyundae Lee, and Hai Zhang, Minnaert resonances for acoustic waves in bubbly media, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 7, 1975–1998. MR 3906861, DOI https://doi.org/10.1016/j.anihpc.2018.03.007
- Habib Ammari, Brian Fitzpatrick, Hyeonbae Kang, Matias Ruiz, Sanghyeon Yu, and Hai Zhang, Mathematical and computational methods in photonics and phononics, Mathematical Surveys and Monographs, vol. 235, American Mathematical Society, Providence, RI, 2018. MR 3837172
- H. Ammari, K. Hamdache, and J.-C. Nédélec, Chirality in the Maxwell equations by the dipole approximation, SIAM J. Appl. Math. 59 (1999), no. 6, 2045–2059. MR 1709796, DOI https://doi.org/10.1137/S0036139998334160
- Habib Ammari, Brian Fitzpatrick, Hyundae Lee, Sanghyeon Yu, and Hai Zhang, Subwavelength phononic bandgap opening in bubbly media, J. Differential Equations 263 (2017), no. 9, 5610–5629. MR 3688425, DOI https://doi.org/10.1016/j.jde.2017.06.025
- Habib Ammari, Brian Fitzpatrick, David Gontier, Hyundae Lee, and Hai Zhang, A mathematical and numerical framework for bubble meta-screens, SIAM J. Appl. Math. 77 (2017), no. 5, 1827–1850. MR 3717823, DOI https://doi.org/10.1137/16M1090235
- Habib Ammari, Brian Fitzpatrick, David Gontier, Hyundae Lee, and Hai Zhang, Sub-wavelength focusing of acoustic waves in bubbly media, Proc. A. 473 (2017), no. 2208, 20170469, 17. MR 3749985, DOI https://doi.org/10.1098/rspa.2017.0469
- Habib Ammari and Hyeonbae Kang, Polarization and moment tensors, Applied Mathematical Sciences, vol. 162, Springer, New York, 2007. With applications to inverse problems and effective medium theory. MR 2327884
- Habib Ammari and Hyeonbae Kang, Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities, J. Math. Anal. Appl. 296 (2004), no. 1, 190–208. MR 2070502, DOI https://doi.org/10.1016/j.jmaa.2004.04.003
- Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Layer potential techniques in spectral analysis, Mathematical Surveys and Monographs, vol. 153, American Mathematical Society, Providence, RI, 2009. MR 2488135
- Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Asymptotic analysis of high-contrast phononic crystals and a criterion for the band-gap opening, Arch. Ration. Mech. Anal. 193 (2009), no. 3, 679–714. MR 2525115, DOI https://doi.org/10.1007/s00205-008-0179-4
- Habib Ammari, Hyundae Lee, and Hai Zhang, Bloch waves in bubbly crystal near the first band gap: a high-frequency homogenization approach, SIAM J. Math. Anal. 51 (2019), no. 1, 45–59. MR 3895120, DOI https://doi.org/10.1137/18M116722X
- Habib Ammari, Pierre Millien, Matias Ruiz, and Hai Zhang, Mathematical analysis of plasmonic nanoparticles: the scalar case, Arch. Ration. Mech. Anal. 224 (2017), no. 2, 597–658. MR 3614756, DOI https://doi.org/10.1007/s00205-017-1084-5
- H. Ammari, E. Orvehed Hiltunen, and S. Yu, A high-frequency homogenization approach near the Dirac points in bubbly honeycomb crystals, arXiv:1812.06178.
- Habib Ammari, Matias Ruiz, Sanghyeon Yu, and Hai Zhang, Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations, J. Differential Equations 261 (2016), no. 6, 3615–3669. MR 3527640, DOI https://doi.org/10.1016/j.jde.2016.05.036
- Habib Ammari, Wei Wu, and Sanghyeon Yu, Double-negative electromagnetic metamaterials due to chirality, Quart. Appl. Math. 77 (2019), no. 1, 105–130. MR 3897921, DOI https://doi.org/10.1090/qam/1516
- Habib Ammari and Hai Zhang, Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency, SIAM J. Math. Anal. 49 (2017), no. 4, 3252–3276. MR 3690650, DOI https://doi.org/10.1137/16M1078574
- Habib Ammari and Hai Zhang, Super-resolution in high-contrast media, Proc. A. 471 (2015), no. 2178, 20140946, 11. MR 3367310, DOI https://doi.org/10.1098/rspa.2014.0946
- S. A. Cummer, J. Christensen, and A. Alù, Controlling sound with acoustic metamaterials, Nature Rev., 1 (2016), 16001.
- R. Figari, G. Papanicolaou, and J. Rubinstein, Remarks on the point interaction approximation, Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986) IMA Vol. Math. Appl., vol. 9, Springer, New York, 1987, pp. 45–55. MR 914984, DOI https://doi.org/10.1007/978-1-4684-6347-7_4
- R. Figari, G. Papanicolaou, and J. Rubinstein, The point interaction approximation for diffusion in regions with many small holes, Stochastic methods in biology (Nagoya, 1985) Lecture Notes in Biomath., vol. 70, Springer, Berlin, 1987, pp. 202–209. MR 893647, DOI https://doi.org/10.1007/978-3-642-46599-4_16
- A. Figotin and P. Kuchment, Spectral properties of classical waves in high-contrast periodic media, SIAM J. Appl. Math. 58 (1998), no. 2, 683–702. MR 1617610, DOI https://doi.org/10.1137/S0036139996297249
- N. Kaina, F. Lemoult, M. Fink, and G. Lerosey, Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials, Nature, 525 (2015), 77–81L.
- M. Lanoy, R. Pierrat, F. Lemoult, M. Fink, V. Leroy, and A. Tourin, Subwavelength focusing in bubbly media using broadband time reversal, Phys. Rev. B, 91.22 (2015), 224202.
- J. Lekner, Capacitance coefficients of two spheres, Journal of Electrostatics, 69 (2011), 11–14
- F. Lemoult, N. Kaina, M. Fink, and G. Lerosey, Soda cans metamaterial: A subwavelength-scaled photonic crystal, Crystals, 6 (2016), 82.
- V. Leroy, A. Bretagne, M. Fink, H. Willaime, P. Tabeling, and A. Tourin, Design and characterization of bubble phononic crystals, Appl. Phys. Lett., 95 (2009), 171904.
- V. Leroy, A. Strybulevych, M. Lanoy, F. Lemoult, A. Tourin, and J. H. Page, Superabsorption of acoustic waves with bubble metascreens, Phys. Rev. B, 91.2 (2015), 020301.
- J. Li and C. T. Chan, Double-negative acoustic meta-material, Phys. Rev. E, 70 (2004), p. 055602.
- G. Ma and P. Sheng, Acoustic metamaterials: From local resonances to broad horizons, Sci. Adv., 2 (2016), e1501595.
- G. W. Milton, N. A. P. Nicorovici, and R. C. McPhedran, Opaque perfect lenses, Phys. B, 394 (2007), 171–175.
- M. Minnaert, On musical air-bubbles and the sounds of running water, The London, Edinburgh, Dublin Philos. Mag. and J. of Sci., 16 (1933), 235–248.
- N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, Optical and dielectric properties of partially resonant composites, Phys. Rev. B, 49.12 (1994), 8479–8482.
- Shin Ozawa, Point interaction potential approximation for $(-\Delta +U)^{-1}$ and eigenvalues of the Laplacian on wildly perturbed domain, Osaka J. Math. 20 (1983), no. 4, 923–937. MR 727440
- Joseph B. Keller, David W. McLaughlin, and George C. Papanicolaou (eds.), Surveys in applied mathematics. Vol. 1, Surveys in Applied Mathematics, vol. 1, Plenum Press, New York, 1995. MR 1366206
- J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), 3966–3969.
- S. A. Ramakrishna, Physics of negative refractive index materials, Rep. Progr. Phys., 68 (2005), 449.
- R. Zhu, X. N. Liu, G. K. Hu, C. T. Sun, and G. L. Huang, Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial, Nature Comm., 5(2014), 5510.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35C20,
74J20
Retrieve articles in all journals
with MSC (2010):
35C20,
74J20
Additional Information
Habib Ammari
Affiliation:
Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
MR Author ID:
353050
Email:
habib.ammari@math.ethz.ch
Brian Fitzpatrick
Affiliation:
Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
MR Author ID:
1228712
Email:
brian.fitzpatrick@sam.math.ethz.ch
Hyundae Lee
Affiliation:
Department of Mathematics, Inha University, 253 Yonghyun-dong Nam-gu, Incheon 402-751, Republic of Korea
Email:
hdlee@inha.ac.kr
Sanghyeon Yu
Affiliation:
Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
MR Author ID:
959778
Email:
sanghyeon.yu@sam.math.ethz.ch
Hai Zhang
Affiliation:
Department of Mathematics, HKUST, Clear Water Bay, Kowloon, Hong Kong
MR Author ID:
890053
Email:
haizhang@ust.hk
Keywords:
Acoustic wave propagation,
double-negative metamaterial,
Minnaert resonance,
hybridization
Received by editor(s):
February 6, 2019
Published electronically:
June 3, 2019
Additional Notes:
The work of the third author was supported by National Research Fund of Korea (NRF-2015R1D1A1A01059357, NRF-2017R1A4A1014735).
The work of the fifth author was partially supported by Research Grant Council of Hong Kong (GRF grant 16304517) and startup fund R9355 from HKUST.
Article copyright:
© Copyright 2019
Brown University