Abstract

This paper addresses the possibility of realizing fixed as well as variable electric circuit elements at infrared and visible frequencies using a gyroelectric nanosphere biased with a static magnetic field. With a proper choice of port designation, one might exercise field control over the impedance offered by the nanoparticle. It is shown that although the driving-point impedance looking into a pair of terminals chosen in some directions remains fixed, it can vary significantly in other directions with respect to the magnetic field biasing the particle. When combined with other isotropic nanocircuit elements, more complex tunable nanocircuits can be designed. This paves the way for adaptive nanosystems for smarter applications.

© 2012 Optical Society of America

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References

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  1. R. F. Harrington, Time Harmonic Electromagnetic Fields(Wiley-Interscience IEEE Press, 2001), Chap. 1.
  2. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1966), Chap. 4.
  3. N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
    [CrossRef]
  4. N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
    [CrossRef]
  5. Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
    [CrossRef]
  6. J. L. Li and W. Ong, “A new solution for characterizing electromagnetic scattering by a gyroelectric sphere,” IEEE Trans. Antennas Propag. 59, 3370–3378 (2011).
    [CrossRef]
  7. J. D. Jackson, Classical Electrodynamics (Wiley, 1999), Chap. 4.
  8. H. C. Chen, Theory of Electromagnetic Waves—A Coordinate-Free Approach (McGraw-Hill, 1992), Chap. 1.
  9. I. V. Lindell, Methods for Electromagnetic Field Analysis (IEEE Press, 1992), Chap. 2.
  10. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Low-frequency scattering by an imperfectly conducting sphere immersed in a dc magnetic field,” Int. J. Infrared Millim. Waves 12, 1253–1264 (1991).
    [CrossRef]
  11. J. R. Gillies and P. Hlawiczka, “TE and TM modes in gyrotropic waveguides,” J. Phys. D 9, 1315–1322 (1976).
    [CrossRef]

2012 (1)

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[CrossRef]

2011 (1)

J. L. Li and W. Ong, “A new solution for characterizing electromagnetic scattering by a gyroelectric sphere,” IEEE Trans. Antennas Propag. 59, 3370–3378 (2011).
[CrossRef]

2007 (1)

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[CrossRef]

2005 (1)

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef]

1991 (1)

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Low-frequency scattering by an imperfectly conducting sphere immersed in a dc magnetic field,” Int. J. Infrared Millim. Waves 12, 1253–1264 (1991).
[CrossRef]

1976 (1)

J. R. Gillies and P. Hlawiczka, “TE and TM modes in gyrotropic waveguides,” J. Phys. D 9, 1315–1322 (1976).
[CrossRef]

Alù, A.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[CrossRef]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef]

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves—A Coordinate-Free Approach (McGraw-Hill, 1992), Chap. 1.

Collin, R. E.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1966), Chap. 4.

Edwards, B.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[CrossRef]

Engheta, N.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[CrossRef]

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[CrossRef]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef]

Gillies, J. R.

J. R. Gillies and P. Hlawiczka, “TE and TM modes in gyrotropic waveguides,” J. Phys. D 9, 1315–1322 (1976).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time Harmonic Electromagnetic Fields(Wiley-Interscience IEEE Press, 2001), Chap. 1.

Hlawiczka, P.

J. R. Gillies and P. Hlawiczka, “TE and TM modes in gyrotropic waveguides,” J. Phys. D 9, 1315–1322 (1976).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), Chap. 4.

Lakhtakia, A.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Low-frequency scattering by an imperfectly conducting sphere immersed in a dc magnetic field,” Int. J. Infrared Millim. Waves 12, 1253–1264 (1991).
[CrossRef]

Li, J. L.

J. L. Li and W. Ong, “A new solution for characterizing electromagnetic scattering by a gyroelectric sphere,” IEEE Trans. Antennas Propag. 59, 3370–3378 (2011).
[CrossRef]

Lindell, I. V.

I. V. Lindell, Methods for Electromagnetic Field Analysis (IEEE Press, 1992), Chap. 2.

Ong, W.

J. L. Li and W. Ong, “A new solution for characterizing electromagnetic scattering by a gyroelectric sphere,” IEEE Trans. Antennas Propag. 59, 3370–3378 (2011).
[CrossRef]

Salandrino, A.

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef]

Sun, Y.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[CrossRef]

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Low-frequency scattering by an imperfectly conducting sphere immersed in a dc magnetic field,” Int. J. Infrared Millim. Waves 12, 1253–1264 (1991).
[CrossRef]

Varadan, V. V.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Low-frequency scattering by an imperfectly conducting sphere immersed in a dc magnetic field,” Int. J. Infrared Millim. Waves 12, 1253–1264 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. L. Li and W. Ong, “A new solution for characterizing electromagnetic scattering by a gyroelectric sphere,” IEEE Trans. Antennas Propag. 59, 3370–3378 (2011).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Low-frequency scattering by an imperfectly conducting sphere immersed in a dc magnetic field,” Int. J. Infrared Millim. Waves 12, 1253–1264 (1991).
[CrossRef]

J. Phys. D (1)

J. R. Gillies and P. Hlawiczka, “TE and TM modes in gyrotropic waveguides,” J. Phys. D 9, 1315–1322 (1976).
[CrossRef]

Nat. Mater. (1)

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[CrossRef]

Phys. Rev. Lett. (1)

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef]

Science (1)

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[CrossRef]

Other (5)

R. F. Harrington, Time Harmonic Electromagnetic Fields(Wiley-Interscience IEEE Press, 2001), Chap. 1.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1966), Chap. 4.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), Chap. 4.

H. C. Chen, Theory of Electromagnetic Waves—A Coordinate-Free Approach (McGraw-Hill, 1992), Chap. 1.

I. V. Lindell, Methods for Electromagnetic Field Analysis (IEEE Press, 1992), Chap. 2.

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Figures (5)

Fig. 1.
Fig. 1.

A gyroelectric nanoparticle in the presence of an incident wave E¯inc=E0z^ and a biasing DC-field B¯DC=BDC(sinαx^+cosαz^).

Fig. 2.
Fig. 2.

Electric field E¯res and the polarization vector P¯.

Fig. 3.
Fig. 3.

Lumped view of a nanoparticle with port designation with respect to a general u^-direction. We specialize u^ along x^-, y^-, and z^-directions and find the terminal quantities separately.

Fig. 4.
Fig. 4.

Variation in Csphzz with DC-field angle α.

Fig. 5.
Fig. 5.

Variation in Re and Im parts of Ysphzz with angle α.

Equations (53)

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g^=(sinαx^+cosαz^).
ϵ¯¯=ϵag^g^+ϵt(I¯¯g^g^)iδg^×I¯¯.
E¯int=E¯0+E¯resforr<R,
E¯ext=E¯0+E¯dipforr>R.
P¯=3ϵ0γ¯¯·E¯0,
p¯=τP¯,
=4πϵ0R3γ¯¯·E¯0.
E¯res=ϑ=0π14ϵ0PP¯¯·r^sin2ϑdϑ,
=13ϵ0P¯,
=γ¯¯·E¯0,
E¯dip=3r^r^·p¯p¯4πϵ0r3,
E¯dip=(3r^r^·γ¯¯·E¯0γ¯¯·E¯0)R3r3.
E¯int=E¯0γ¯¯·E¯0,
E¯ext=E¯0+(3r^r^·γ¯¯·E¯0γ¯¯·E¯0)R3r3.
r^·ϵ¯¯·E¯int|r=R=r^·ϵ0I¯¯·E¯ext|r=R,
r^·ϵ¯¯·(E¯0γ¯¯·E¯0)=r^·ϵ0(I¯¯+3r^r^·γ¯¯γ¯¯)·E¯0,
=r^·(ϵ0I¯¯+2ϵ0I¯¯·γ¯¯)·E¯0,
r^·{(ϵ¯¯ϵ0I¯¯)(ϵ¯¯+2ϵ0I¯¯)·γ¯¯}·E¯0=0.
(ϵ¯¯ϵ0I¯¯)(ϵ¯¯+2ϵ0I¯¯)·γ¯¯=¯¯
γ¯¯=(ϵ¯¯+2ϵ0I¯¯)1·(ϵ¯¯ϵ0I¯¯),
(A¯¯+αI¯¯)1(α2I¯¯+α(A¯¯tI¯¯A¯¯)+adjA¯¯)/|A¯¯|,
γ¯¯=(ϵ¯¯ϵ0I¯¯)·(ϵ¯¯+2ϵ0I¯¯)1,
γ¯¯=[(ϵt+2ϵ0+ϵaϵt){(ϵt+2ϵ0)(ϵtϵ0)δ2}I¯¯3iδϵ0(ϵt+2ϵ0+ϵaϵt)g^×I¯¯+3ϵ0{δ2(ϵt+2ϵ0)(ϵaϵt)}g^g^]/[(ϵt+2ϵ0+ϵaϵt)((ϵt+2ϵ0)2δ2)]
γ¯¯=3(ϵ¯¯ϵ0I¯¯)·((ϵ¯¯+2ϵ0I¯¯)××(ϵ¯¯+2ϵ0I¯¯))T((ϵ¯¯+2ϵ0I¯¯)××(ϵ¯¯+2ϵ0I¯¯)):(ϵ¯¯+2ϵ0I¯¯),
γ¯¯·E¯0·x^=3ϵ0E0sin2α(δ2+(ϵaϵt)(2ϵa+ϵt))2(2ϵ0+ϵa)((2ϵ0+ϵt)2δ2),
γ¯¯·E¯0·y^=3iϵ0δE0sinα(2ϵ0+ϵt)2δ2,
γ¯¯·E¯0·z^=3ϵ0E0[13ϵ0cos2α2ϵ0+ϵa(2ϵ0+ϵt)sin2α(2ϵ0+ϵt)2δ2].
E¯dip|r=R·r^=2γ¯¯·E¯0·r^,
E¯res|r=R·r^=γ¯¯·E¯0·r^.
Ifringex=(iωϵ0)ϕ=0π/2θ=0π4γ¯¯·E¯0·r^R2dΩ,
Ifringey=(iωϵ0)ϕ=0πθ=0π2γ¯¯·E¯0·r^R2dΩ,
Ifringez=(iωϵ0)ϕ=02πθ=0π/22γ¯¯·E¯0·r^R2dΩ,
I¯fringe=(i2πR2ωϵ0)γ¯¯·E¯0.
Isphx=(iωϵ¯¯)·ϕ=0π/2θ=0π2γ¯¯·E¯0·r^R2dΩ,
Isphy=(iωϵ¯¯)·ϕ=0πθ=0πγ¯¯·E¯0·r^R2dΩ,
Isphz=(iωϵ¯¯)·ϕ=02πθ=0π/2γ¯¯·E¯0·r^R2dΩ,
Isphx=iπE0R2ωsin2α·[ϵaϵt2+3ϵ02{12ϵ0+ϵa2ϵ0+ϵt(2ϵ0+ϵt)2δ2}],
Isphy=E0πR2δωsinα[(2ϵ0+ϵt)2δ26ϵ02(2ϵ0+ϵt)2δ2],
Isphz=iE0πR2ω[3δ2sin2α(2ϵ0+ϵt)2δ2+3ϵ0(ϵaϵt)sin22α(δ2+(ϵaϵt)(2ϵ0+ϵt))4(2ϵ0+ϵa)((2ϵ0+ϵt)2δ2)+(ϵacos2α+ϵtsinα)·{13ϵ0(cos2α2ϵ0+ϵa+(2ϵ0+ϵt)sin2α(2ϵ0+ϵt)2δ2)}].
V¯fringe=γ¯¯·E¯0R.
Zfringexx=1(i2πRωϵ0),
Zfringeyy=1(i2πRωϵ0),
Zfringezz=1(i2πRωϵ0),
Zsphxx=3iϵ0(ϵt22ϵ0(ϵaϵt)ϵaϵtδ2)/[πRω{2ϵ02(ϵaϵt2ϵa2+3δ2+ϵt2)2ϵ0(ϵaϵt)(2ϵaϵtδ2+ϵt2)+ϵa(ϵaϵt)(δ2ϵt2)+4ϵ03(ϵaϵt)}],
Zsphyy=3iϵ0πRω(δ2+2ϵ024ϵ0ϵtϵt2),
Zsphzz=i(T1+ϵ0T2)2πRω(T3T4+ϵ02T5+ϵ0(T6+T7)),
T1=2ϵa(ϵt2δ2)8ϵ03+2ϵ02(ϵaϵt)(1+3cos2α),
T2=ϵaϵt(5+3cos2α)+(ϵt2δ2)(13cos2α),
T3=(ϵacos2α+ϵtsin2α),
T4=ϵa(ϵt2δ2)4ϵ03,
T5=(4ϵa2cos2αϵaϵt(3+cos2α)+2(ϵt2+3δ2)sin2α),
T6=2ϵt(ϵt2δ2)sin2α+4ϵa2ϵtcos2α,
T7=ϵa((2cos2α)δ2ϵt2cos2α).

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