Abstract

The feasibility of tuning the optical response of a dipole nanoantenna using plasmonic core-shell particles is demonstrated. The proposed scheme consists of a two-step tuning process. First, it is demonstrated that when the gap between the nanodipole arms is loaded by a homogeneous dielectric sphere, the configuration exhibits effective material properties that can be described by a mixing rule, thereby allowing it to be effectively mapped onto an equivalent circuit topology. An additional degree of tunability is introduced by substituting the load consisting of a homogeneous sphere with one represented by a two-layer core-shell particle. An electrically small core-shell particle offers the advantage that it functions as a tunable nanocircuit element with properties that depend on the material constitution of the particle as well as its volume fraction. Effective medium theory is employed, and through rigorous analysis the resulting core-shell nanoparticle properties are then mapped onto an equivalent circuit topology. The complete derivation is presented for the total equivalent circuit model that corresponds to this two-step tuning process. Full-wave numerical predictions of the nanoantenna’s extinction cross section are presented that validate the results obtained through the equivalent circuit-based representation of the loaded antenna. The proposed methodology illustrates the inherent tuning capabilities that core-shell particles can offer. Additionally, a novel compact and efficient scheme is introduced in order to map general nanoantenna loads onto equivalent circuit representations. The proposed approach permits the examination of the nanodipole antenna’s performance in its transmitting mode; therefore it completely eliminates the need for time-consuming full-wave simulations of loaded nanoantenna structures. At the same time, it provides the optical designer with the capability to custom-engineer the nanoantenna’s response through fast and accurate circuit-based analysis.

© 2013 Optical Society of America

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  5. A. Alu and N. Engheta, “Wireless at the nanoscale: optical interconnects using matched nanoantennas,” Phys. Rev. Lett. 104, 213902 (2010).
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  6. N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10, 631–636 (2011).
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2013 (2)

A. Alu and N. Engheta, “Theory, modeling and features of optical nanoantennas,” IEEE Trans. Antennas Propag. 61, 1508–1517 (2013).
[CrossRef]

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N. J. Halas, and A. Alù, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[CrossRef]

2011 (6)

F. Arquer, V. Volski, N. Verellen, G. Vandenbosch, and V. Moshchalkov, “Engineering the input impedance of optical nanodipole antennas: material, geometry, and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[CrossRef]

Y. Zhao, N. Engheta, and A. Alu, “Effects of shape and loading of optical nanoantennas on their sensitivity and radiation properties,” J. Opt. Soc. Am. B 28, 1266–1274 (2011).
[CrossRef]

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[CrossRef]

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10, 631–636 (2011).
[CrossRef]

M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetic design problems using the covariance matrix adaptation evolutionary strategy,” IEEE Trans. Antennas Propag. 59, 1275–1285 (2011).
[CrossRef]

M. Alam, Y. Massoud, and G. V. Eleftheriades, “A time-varying approach to circuit modeling of plasmonic nanospheres using radial vector wave functions,” IEEE Trans. Microwave Theor. Tech. 59, 2595–2611 (2011).
[CrossRef]

2010 (2)

A. Alu and N. Engheta, “Wireless at the nanoscale: optical interconnects using matched nanoantennas,” Phys. Rev. Lett. 104, 213902 (2010).
[CrossRef]

A. Locatelli, “Analysis of the optical properties of wire antennas with displaced terminals,” Opt. Express 18, 9504–9510 (2010).
[CrossRef]

2009 (3)

2008 (4)

T. H. Taminau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16, 9144–9154 (2008).
[CrossRef]

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101, 043901 (2008).
[CrossRef]

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2, 307–310 (2008).
[CrossRef]

2007 (2)

T. H. Taminiau, F. B. Segerink, R. J. Moerland, L. Kuipers, and N. F. van Hulst, “Near-field driving of a optical monopole antenna,” J. Opt. A 9, S315–S321 (2007).
[CrossRef]

H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9, 1–12 (2007).
[CrossRef]

2006 (1)

A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagn. Res. 62, 317–331, 2006.
[CrossRef]

2005 (1)

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

1998 (1)

R. E. Diaz, W. M. Merrill, and N. G. Alexopoulos, “Analytic framework for the modeling of effective media,” J. Appl. Phys. 84, 6815–6826 (1998).
[CrossRef]

1981 (1)

G. W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981).
[CrossRef]

1978 (1)

D. J. Bergman, “The dielectric constant of a composite material: a problem in classical physics,” Phys. Rep. 43, 377–407 (1978).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Aizpurua, J.

M. Schnell, A. Garcia-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nat. Photonics 3, 287–291 (2009).
[CrossRef]

Alam, M.

M. Alam, Y. Massoud, and G. V. Eleftheriades, “A time-varying approach to circuit modeling of plasmonic nanospheres using radial vector wave functions,” IEEE Trans. Microwave Theor. Tech. 59, 2595–2611 (2011).
[CrossRef]

Alexopoulos, N. G.

R. E. Diaz, W. M. Merrill, and N. G. Alexopoulos, “Analytic framework for the modeling of effective media,” J. Appl. Phys. 84, 6815–6826 (1998).
[CrossRef]

Alivisatos, A. P.

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10, 631–636 (2011).
[CrossRef]

Alu, A.

A. Alu and N. Engheta, “Theory, modeling and features of optical nanoantennas,” IEEE Trans. Antennas Propag. 61, 1508–1517 (2013).
[CrossRef]

Y. Zhao, N. Engheta, and A. Alu, “Effects of shape and loading of optical nanoantennas on their sensitivity and radiation properties,” J. Opt. Soc. Am. B 28, 1266–1274 (2011).
[CrossRef]

A. Alu and N. Engheta, “Wireless at the nanoscale: optical interconnects using matched nanoantennas,” Phys. Rev. Lett. 104, 213902 (2010).
[CrossRef]

Alù, A.

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N. J. Halas, and A. Alù, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[CrossRef]

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101, 043901 (2008).
[CrossRef]

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2, 307–310 (2008).
[CrossRef]

Arquer, F.

F. Arquer, V. Volski, N. Verellen, G. Vandenbosch, and V. Moshchalkov, “Engineering the input impedance of optical nanodipole antennas: material, geometry, and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[CrossRef]

Bayraktar, Z.

M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetic design problems using the covariance matrix adaptation evolutionary strategy,” IEEE Trans. Antennas Propag. 59, 1275–1285 (2011).
[CrossRef]

Bergman, D. J.

D. J. Bergman, “The dielectric constant of a composite material: a problem in classical physics,” Phys. Rep. 43, 377–407 (1978).
[CrossRef]

Bharadwaj, P.

Boscolo, S.

Capobianco, A.-D.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Crozier, K.

M. Schnell, A. Garcia-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nat. Photonics 3, 287–291 (2009).
[CrossRef]

De Angelis, C.

Deutsch, B.

Diaz, R. E.

R. E. Diaz, W. M. Merrill, and N. G. Alexopoulos, “Analytic framework for the modeling of effective media,” J. Appl. Phys. 84, 6815–6826 (1998).
[CrossRef]

A. H. Panaretos and R. E. Diaz, “Subcell modeling of a plasmonic Drude nanosphere,” Proceedings of the International Conference on Electromagnetics in Advanced Applications (ICEAA), Torino, Italy, 2011, pp. 211–214.

Eisler, H.-J.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

Eleftheriades, G. V.

M. Alam, Y. Massoud, and G. V. Eleftheriades, “A time-varying approach to circuit modeling of plasmonic nanospheres using radial vector wave functions,” IEEE Trans. Microwave Theor. Tech. 59, 2595–2611 (2011).
[CrossRef]

Engheta, N.

A. Alu and N. Engheta, “Theory, modeling and features of optical nanoantennas,” IEEE Trans. Antennas Propag. 61, 1508–1517 (2013).
[CrossRef]

Y. Zhao, N. Engheta, and A. Alu, “Effects of shape and loading of optical nanoantennas on their sensitivity and radiation properties,” J. Opt. Soc. Am. B 28, 1266–1274 (2011).
[CrossRef]

A. Alu and N. Engheta, “Wireless at the nanoscale: optical interconnects using matched nanoantennas,” Phys. Rev. Lett. 104, 213902 (2010).
[CrossRef]

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2, 307–310 (2008).
[CrossRef]

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101, 043901 (2008).
[CrossRef]

Fischer, H.

Garcia-Etxarri, A.

M. Schnell, A. Garcia-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nat. Photonics 3, 287–291 (2009).
[CrossRef]

Giessen, H.

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10, 631–636 (2011).
[CrossRef]

Gregory, M. D.

M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetic design problems using the covariance matrix adaptation evolutionary strategy,” IEEE Trans. Antennas Propag. 59, 1275–1285 (2011).
[CrossRef]

Halas, N. J.

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N. J. Halas, and A. Alù, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[CrossRef]

Hansen, N.

N. Hansen, “The CMA evolution strategy: a comparing review,” in Towards a New Evolutionary Computation: Studies in Fuzziness and Soft Computing, J. Kacprzyk, ed. (Springer-Verlag, 2006), Vol. 192, pp. 75–102.

Hecht, B.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

Hentschel, M.

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10, 631–636 (2011).
[CrossRef]

Hillenbrand, R.

M. Schnell, A. Garcia-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nat. Photonics 3, 287–291 (2009).
[CrossRef]

Hofmann, H. F.

H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9, 1–12 (2007).
[CrossRef]

Huber, A. J.

M. Schnell, A. Garcia-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nat. Photonics 3, 287–291 (2009).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kadoya, Y.

H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9, 1–12 (2007).
[CrossRef]

Kosako, T.

H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9, 1–12 (2007).
[CrossRef]

Kuipers, L.

T. H. Taminiau, F. B. Segerink, R. J. Moerland, L. Kuipers, and N. F. van Hulst, “Near-field driving of a optical monopole antenna,” J. Opt. A 9, S315–S321 (2007).
[CrossRef]

Liu, N.

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N. J. Halas, and A. Alù, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[CrossRef]

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10, 631–636 (2011).
[CrossRef]

Locatelli, A.

Martin, O. J. F.

H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16, 9144–9154 (2008).
[CrossRef]

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

Massoud, Y.

M. Alam, Y. Massoud, and G. V. Eleftheriades, “A time-varying approach to circuit modeling of plasmonic nanospheres using radial vector wave functions,” IEEE Trans. Microwave Theor. Tech. 59, 2595–2611 (2011).
[CrossRef]

Merrill, W. M.

R. E. Diaz, W. M. Merrill, and N. G. Alexopoulos, “Analytic framework for the modeling of effective media,” J. Appl. Phys. 84, 6815–6826 (1998).
[CrossRef]

Midrio, M.

Milton, G. W.

G. W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981).
[CrossRef]

Modotto, D.

Moerland, R. J.

T. H. Taminiau, F. B. Segerink, R. J. Moerland, L. Kuipers, and N. F. van Hulst, “Near-field driving of a optical monopole antenna,” J. Opt. A 9, S315–S321 (2007).
[CrossRef]

Moshchalkov, V.

F. Arquer, V. Volski, N. Verellen, G. Vandenbosch, and V. Moshchalkov, “Engineering the input impedance of optical nanodipole antennas: material, geometry, and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[CrossRef]

Muhlschlegel, P.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

Nordlander, P.

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N. J. Halas, and A. Alù, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[CrossRef]

Novotny, L.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[CrossRef]

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009).
[CrossRef]

Panaretos, A. H.

A. H. Panaretos and R. E. Diaz, “Subcell modeling of a plasmonic Drude nanosphere,” Proceedings of the International Conference on Electromagnetics in Advanced Applications (ICEAA), Torino, Italy, 2011, pp. 211–214.

Pigozzo, F. M.

Pohl, D. W.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

Sacchetto, F.

Schnell, M.

M. Schnell, A. Garcia-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nat. Photonics 3, 287–291 (2009).
[CrossRef]

Segerink, F. B.

T. H. Taminau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

T. H. Taminiau, F. B. Segerink, R. J. Moerland, L. Kuipers, and N. F. van Hulst, “Near-field driving of a optical monopole antenna,” J. Opt. A 9, S315–S321 (2007).
[CrossRef]

Sihvola, A.

A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagn. Res. 62, 317–331, 2006.
[CrossRef]

Someda, C. G.

Stefani, F. D.

T. H. Taminau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

Taminau, T. H.

T. H. Taminau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

Taminiau, T. H.

T. H. Taminiau, F. B. Segerink, R. J. Moerland, L. Kuipers, and N. F. van Hulst, “Near-field driving of a optical monopole antenna,” J. Opt. A 9, S315–S321 (2007).
[CrossRef]

Tang, M. L.

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10, 631–636 (2011).
[CrossRef]

van Hulst, N.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[CrossRef]

van Hulst, N. F.

T. H. Taminau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

T. H. Taminiau, F. B. Segerink, R. J. Moerland, L. Kuipers, and N. F. van Hulst, “Near-field driving of a optical monopole antenna,” J. Opt. A 9, S315–S321 (2007).
[CrossRef]

Vandenbosch, G.

F. Arquer, V. Volski, N. Verellen, G. Vandenbosch, and V. Moshchalkov, “Engineering the input impedance of optical nanodipole antennas: material, geometry, and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[CrossRef]

Verellen, N.

F. Arquer, V. Volski, N. Verellen, G. Vandenbosch, and V. Moshchalkov, “Engineering the input impedance of optical nanodipole antennas: material, geometry, and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[CrossRef]

Volski, V.

F. Arquer, V. Volski, N. Verellen, G. Vandenbosch, and V. Moshchalkov, “Engineering the input impedance of optical nanodipole antennas: material, geometry, and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[CrossRef]

von Hippel, A.

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Figures (8)

Fig. 1.
Fig. 1.

(a) Nanodipole geometry. (b) Silver nanodipole ECS when the center gap is filled with silver and with free space.

Fig. 2.
Fig. 2.

(a) Susceptance corresponding to the intrinsic admittance of the nanodipole, along with minus the susceptance of the gap when it is filled with free space and silver. (b) Nanodipole impedance when the gap is air filled. (c) Nanodipole impedance when the gap is filled with silver.

Fig. 3.
Fig. 3.

(a) Partially filled capacitor model for the representation of the effective permittivity of the dielectric sphere loaded capacitor. (b) Effective permittivity of the loaded capacitor computed by solving Laplace’s equation and using Eq. (4), compared against the partially filled capacitor model in Eq. (5).

Fig. 4.
Fig. 4.

(a), (b) Nanodipole intrinsic susceptance along with minus the effective susceptance of the load. (c), (d) Input impedance of the nanodipole loaded with a silver and a golden sphere. (e), (f) ECS of the nanodipole loaded by the two different metallic spheres.

Fig. 5.
Fig. 5.

(a) 3D geometry of a core-shell particle. (b) Cross section of the same core-shell particle.

Fig. 6.
Fig. 6.

(a), (b) Nanodipole intrinsic susceptance along with the susceptance of the load. (c), (d) Total input impedance of the loaded nanodipole. (e), (f) Extinction cross section of the loaded nanodipole.

Fig. 7.
Fig. 7.

(a) Partially filled capacitor model for the CM mixing rule. (b) Equivalent circuit of the CM partially filled capacitor model.

Fig. 8.
Fig. 8.

(a) Equivalent circuit representation of the core-shell material implementation presented in Section 4. (b) Equivalent circuit representation of the mixing rule characterized by Eq. (5).

Equations (13)

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Yload=jωε0εrπa2h,
Im{Yint+Yload}=0
εeεh=εhn=1NAnuun,uεhεhεf.
VPzdr3=Vε0(εe1)Ezdr3εe=1+VPzdr3ε0VEzdr3,
{w1,w2,w3}{0.2911,0.0205,0.0007}and{h1,h2,h3}{0.7718,0.5956,0.2681}.
εeεh=εh[w1h1u(1h1)+w2h2u(1h2)+w3h3u(1h3)].
α=4πa23ε21+f(2ε2+1)ζε2+2+2f(ε21)ζ,ζε1ε2ε1+2ε2
α=4πa23ε21+2fζ1fζ1ε21+2fζ1fζ+2.
εe=ε21+2fζ1fζ.
C0=ε2ε0(1w1),C11=ε2ε0w11h1,C12=ε1ε0w1h1
w1h1=f,h1=2+f3,
εrε0=εε0+ε0ωp2ω(jvω)εrε0=C+1Lω(jRLω).
C0=εhε0(1i=13wi),Ci1=εhε0wi1hi,Ci2=εfε0wihi.

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