Sensitivity of anomalous localized resonance phenomena with respect to dissipation
Authors:
Taoufik Meklachi, Graeme W. Milton, Daniel Onofrei, Andrew E. Thaler and Gregory Funchess
Journal:
Quart. Appl. Math. 74 (2016), 201-234
MSC (2010):
Primary 35Q60
DOI:
https://doi.org/10.1090/qam/1408
Published electronically:
March 16, 2016
MathSciNet review:
3505601
Full-text PDF Free Access
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Abstract: We analyze cloaking due to anomalous localized resonance in the quasistatic regime in the case when a general charge density distribution is brought near a slab superlens. If the charge density distribution is within a critical distance of the slab, then the power dissipation within the slab blows up as certain electrical dissipation parameters go to zero. The potential remains bounded far away from the slab in this limit, which leads to cloaking due to anomalous localized resonance. On the other hand, if the charge density distribution is farther than this critical distance from the slab, then the power dissipation within the slab remains bounded and cloaking due to anomalous localized resonance does not occur. The critical distance is shown to strongly depend on the rate at which the dissipation outside the slab goes to zero.
References
- H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, Anomalous localized resonance using a folded geometry in three dimensions, Proceedings of the Royal Society A 469 (2013), no. 2154, 20130048.
- Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Hyundae Lee, and Graeme W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal. 208 (2013), no. 2, 667–692. MR 3035988, DOI 10.1007/s00205-012-0605-5
- Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Hyundae Lee, and Graeme W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking by anomalous localized resonance II, Inverse problems and applications, Contemp. Math., vol. 615, Amer. Math. Soc., Providence, RI, 2014, pp. 1–14. MR 3221596, DOI 10.1090/conm/615/12244
- D. J. Bergman, Perfect imaging of a point charge in the quasistatic regime, Physical Review A 89 (2014), 015801.
- Guy Bouchitté and Ben Schweizer, Cloaking of small objects by anomalous localized resonance, Quart. J. Mech. Appl. Math. 63 (2010), no. 4, 437–463. MR 2738456, DOI 10.1093/qjmam/hbq008
- O. P. Bruno and S. Lintner, Superlens-cloaking of small dielectric bodies in the quasistatic regime, Journal of Applied Physics 102 (2007), 124502.
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- F. G. Friedlander, Introduction to the theory of distributions, 2nd ed., Cambridge University Press, Cambridge, 1998. With additional material by M. Joshi. MR 1721032
- D. J. Griffiths, Introduction to Electrodynamics, third ed., Prentice-Hall, Inc., Upper Saddle River, NJ, 1999.
- Wilfred Kaplan, Advanced calculus, 3rd ed., Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. MR 750297
- R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, A variational perspective on cloaking by anomalous localized resonance, Comm. Math. Phys. 328 (2014), no. 1, 1–27. MR 3196978, DOI 10.1007/s00220-014-1943-y
- U. Leonhardt and T. G. Philbin, General relativity in electrical engineering, New Journal of Physics 8 (2006), 247.
- U. Leonhardt and T. Tyc, Broadband invisibility by non-Euclidean cloaking, Science 323 (2009), no. 5910, 110–112.
- Graeme W. Milton and Nicolae-Alexandru P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2074, 3027–3059. MR 2263683, DOI 10.1098/rspa.2006.1715
- G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, Opaque perfect lenses, Physica B 394 (2007), 171–175.
- G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, Solutions in folded geometries, and associated cloaking due to anomalous resonance, New Journal of Physics 10 (2008), 115021.
- Graeme W. Milton, Nicolae-Alexandru P. Nicorovici, Ross C. McPhedran, and Viktor A. Podolskiy, A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2064, 3999–4034. MR 2186014, DOI 10.1098/rspa.2005.1570
- Hoai-Minh Nguyen, Superlensing using complementary media, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 2, 471–484. MR 3325246, DOI 10.1016/j.anihpc.2014.01.004
- N.-A. Nicorovici, R. C. McPhedran, and G. W. Milton, Optical and dielectric properties of partially resonant composites, Physical Review B 49 (1994), no. 12, 8479–8482.
- N.-A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, Relative local density of states and cloaking in finite clusters of coated cylinders, Waves Random Complex Media 21 (2011), no. 2, 248–277. MR 2795374, DOI 10.1080/17455030.2010.547885
- N.-A. P. Nicorovici, R. C. McPhedran, L. C. Botten, and G. W. Milton, Cloaking by plasmonic resonance among systems of particles: cooperation or combat?, Comptes Rendus Physique 10 (2009), 391–399.
- N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, Finite wavelength cloaking by plasmonic resonance, New Journal of Physics 10 (2008), 115020.
- N.-A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance, Optics Express 15 (2007), no. 10, 6314–6323.
- J. B. Pendry, Negative refraction makes a perfect lens, Physical Review Letters 85 (2000), 3966–3969.
- J. B. Pendry and S. A. Ramakrishna, Near-field lenses in two dimensions, Journal of Physics: Condensed Matter 14 (2002), 8463–8479.
- Joel L. Schiff, The Laplace transform, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1999. Theory and applications. MR 1716143, DOI 10.1007/978-0-387-22757-3
- Andrew E. Thaler, Bounds on the volume of an inclusion in a body and cloaking due to anomalous localized resonance, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–The University of Utah. MR 3301720
- V. G. Veselago, The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$, Uspekhi Fizicheskikh Nauk 92 (1967), 517–526, English translation in Soviet Physics Uspekhi 10:509–514 (1968).
- M. Xiao, X. Huang, J.-W. Dong, and C. T. Chan, On the time evolution of the cloaking effect of a metamaterial slab, Optics Letters 37 (2012), 4594–4596.
- A. D. Yaghjian and T. B. Hansen, Plane-wave solutions to frequency-domain and time-domain scattering from magnetodielectric slabs, Physical Review E 73 (2006), 046608.
- M. Yan, W. Yan, and M. Qiu, Cylindrical superlens by a coordinate transformation, Physical Review B 78 (2008), no. 12, 125113.
- Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Springer-Verlag, New York, 1990. Linear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033497, DOI 10.1007/978-1-4612-0985-0
References
- H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, Anomalous localized resonance using a folded geometry in three dimensions, Proceedings of the Royal Society A 469 (2013), no. 2154, 20130048.
- Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Hyundae Lee, and Graeme W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal. 208 (2013), no. 2, 667–692. MR 3035988, DOI 10.1007/s00205-012-0605-5
- Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Hyundae Lee, and Graeme W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking by anomalous localized resonance II, Inverse problems and applications, Contemp. Math., vol. 615, Amer. Math. Soc., Providence, RI, 2014, pp. 1–14. MR 3221596, DOI 10.1090/conm/615/12244
- D. J. Bergman, Perfect imaging of a point charge in the quasistatic regime, Physical Review A 89 (2014), 015801.
- Guy Bouchitté and Ben Schweizer, Cloaking of small objects by anomalous localized resonance, Quart. J. Mech. Appl. Math. 63 (2010), no. 4, 437–463. MR 2738456 (2011k:35036), DOI 10.1093/qjmam/hbq008
- O. P. Bruno and S. Lintner, Superlens-cloaking of small dielectric bodies in the quasistatic regime, Journal of Applied Physics 102 (2007), 124502.
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943 (2011c:35002)
- F. G. Friedlander, Introduction to the theory of distributions, 2nd ed., with additional material by M. Joshi, Cambridge University Press, Cambridge, 1998. MR 1721032 (2000g:46002)
- D. J. Griffiths, Introduction to Electrodynamics, third ed., Prentice-Hall, Inc., Upper Saddle River, NJ, 1999.
- Wilfred Kaplan, Advanced calculus, 3rd ed., Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. MR 750297 (85e:00002)
- R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, A variational perspective on cloaking by anomalous localized resonance, Comm. Math. Phys. 328 (2014), no. 1, 1–27. MR 3196978, DOI 10.1007/s00220-014-1943-y
- U. Leonhardt and T. G. Philbin, General relativity in electrical engineering, New Journal of Physics 8 (2006), 247.
- U. Leonhardt and T. Tyc, Broadband invisibility by non-Euclidean cloaking, Science 323 (2009), no. 5910, 110–112.
- Graeme W. Milton and Nicolae-Alexandru P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2074, 3027–3059. MR 2263683 (2008e:78018), DOI 10.1098/rspa.2006.1715
- G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, Opaque perfect lenses, Physica B 394 (2007), 171–175.
- G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, Solutions in folded geometries, and associated cloaking due to anomalous resonance, New Journal of Physics 10 (2008), 115021.
- Graeme W. Milton, Nicolae-Alexandru P. Nicorovici, Ross C. McPhedran, and Viktor A. Podolskiy, A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2064, 3999–4034. MR 2186014 (2006h:78001), DOI 10.1098/rspa.2005.1570
- Hoai-Minh Nguyen, Superlensing using complementary media, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 2, 471–484. MR 3325246, DOI 10.1016/j.anihpc.2014.01.004
- N.-A. Nicorovici, R. C. McPhedran, and G. W. Milton, Optical and dielectric properties of partially resonant composites, Physical Review B 49 (1994), no. 12, 8479–8482.
- N.-A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, Relative local density of states and cloaking in finite clusters of coated cylinders, Waves Random Complex Media 21 (2011), no. 2, 248–277. MR 2795374 (2012c:78008), DOI 10.1080/17455030.2010.547885
- N.-A. P. Nicorovici, R. C. McPhedran, L. C. Botten, and G. W. Milton, Cloaking by plasmonic resonance among systems of particles: cooperation or combat?, Comptes Rendus Physique 10 (2009), 391–399.
- N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, Finite wavelength cloaking by plasmonic resonance, New Journal of Physics 10 (2008), 115020.
- N.-A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance, Optics Express 15 (2007), no. 10, 6314–6323.
- J. B. Pendry, Negative refraction makes a perfect lens, Physical Review Letters 85 (2000), 3966–3969.
- J. B. Pendry and S. A. Ramakrishna, Near-field lenses in two dimensions, Journal of Physics: Condensed Matter 14 (2002), 8463–8479.
- Joel L. Schiff, The Laplace transform, Theory and applications, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1999. MR 1716143 (2000k:44002), DOI 10.1007/978-0-387-22757-3
- Andrew E. Thaler, Bounds on the volume of an inclusion in a body and cloaking due to anomalous localized resonance, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–The University of Utah. MR 3301720
- V. G. Veselago, The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$, Uspekhi Fizicheskikh Nauk 92 (1967), 517–526, English translation in Soviet Physics Uspekhi 10:509–514 (1968).
- M. Xiao, X. Huang, J.-W. Dong, and C. T. Chan, On the time evolution of the cloaking effect of a metamaterial slab, Optics Letters 37 (2012), 4594–4596.
- A. D. Yaghjian and T. B. Hansen, Plane-wave solutions to frequency-domain and time-domain scattering from magnetodielectric slabs, Physical Review E 73 (2006), 046608.
- M. Yan, W. Yan, and M. Qiu, Cylindrical superlens by a coordinate transformation, Physical Review B 78 (2008), no. 12, 125113.
- Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Linear monotone operators, translated from the German by the author and Leo F. Boron, Springer-Verlag, New York, 1990. MR 1033497 (91b:47001), DOI 10.1007/978-1-4612-0985-0
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Additional Information
Taoufik Meklachi
Affiliation:
Department of Mathematics, Drexel University, 15 South 33rd Street, Room 206, Philadelphia, PA 19104
Email:
tmeklachi@gmail.com
Graeme W. Milton
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, Room 233, Salt Lake City, Utah 84112-0090
Email:
milton@math.utah.edu
Daniel Onofrei
Affiliation:
Department of Mathematics, University of Houston, 651 PGH, Houston, Texas 77204-3008
MR Author ID:
753389
Email:
onofrei@math.uh.edu
Andrew E. Thaler
Affiliation:
Institute for Mathematics and its Applications, 207 Church Street SE, Minneapolis, MN 55455
Email:
andythaler05@gmail.com
Gregory Funchess
Affiliation:
Department of Mathematics, University of Houston, 651 PGH, Houston, Texas 77204-3008
Email:
gfunchess@gmail.com
Received by editor(s):
June 19, 2014
Published electronically:
March 16, 2016
Additional Notes:
The work of the first author was supported by the Air Force through grant AFOSR YIP Early Career Award FA9550-13-1-0078
The work of the second author was supported by the National Science Foundation through grant DMS-1211359.
The work of the third author was supported under the Simons Collaborative Grant and the Air Force through grant AFOSR YIP Early Career Award FA9550-13-1-0078
The work of the fourth author was supported by the National Science Foundation through grant DMS-1211359.
The work of the fifth author was supported by the Air Force through grant AFOSR YIP Early Career Award FA9550-13-1-0078
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