Abstract

A simple analytical theory for finding eigensolutions for plane electromagnetic waves propagating along an axis in infinite regular arrays of small dipole particles is presented. The spacing between dipoles in every plane is assumed to be smaller than the wavelength; separation between the planes is arbitrary. The influence of evanescent modes is taken into account. This theory gives a model for an effective propagation constant that can be applied in a wide frequency range from the quasi-static regime to the Bragg reflection (photonic bandgap) region.

© 2000 Optical Society of America

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References

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  1. A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE Publishing, London, 1999).
  2. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, Mass, 1994).
  3. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1984; first Russian edition, Gostechizdat, Moscow, 1957).
  4. J. Slater, Insulators, Semiconductors, and Metals (McGraw-Hill, New York, 1967).
  5. H. F. Contopanagos, C. A. Kyriazidou, W. M. Merrill, “Effective response functions for photonic bandgap materials,” J. Opt. Soc. Am. A 16, 1682–1699 (1999).
    [CrossRef]
  6. C. A. Moses, N. Engheta, “An idea for electromagnetic ‘feedforward-feedbackward’ media,” IEEE Trans. Antennas Propag. 47, 918–928 (1999).
    [CrossRef]
  7. G. D. Mahan, G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1961).
    [CrossRef]
  8. R. C. McPhedran, D. R. McKenzie, “The conductivity of lattices of spheres. 1. Simple cubic lattice,” Proc. R. Soc. London, Ser. A 359, 45–63 (1978).
    [CrossRef]
  9. A. P. Vinogradov, Yu. N. Dmitriev, V. E. Romanenko, “Transition from planar to bulk properties in multi-layer system,” Electromagnetics 17, 563–571 (1997).
    [CrossRef]
  10. C. R. Simovski, S. A. Tretyakov, A. H. Sihvola, M. M. Popov, “On the surface effect in thin molecular or composite layers,” Eur. Phys. J. Appl. Phys. 9, 195–204 (2000).
    [CrossRef]
  11. S. A. Tretyakov, A. J. Viitanen, S. I. Maslovski, I. E. Saarela, “Impedance boundary conditions for regular dense arrays of dipole scatterers,” , Electromagnetics Laboratory Report Series (Helsinki U. Technology, Helsinki Finland, 1999).
  12. S. I. Maslovski, S. A. Tretyakov, “Full-wave interaction field in two-dimensional arrays of dipole scatterers,” Int. J. Electron. Commun. 53, 135–139 (1999).
  13. R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE Press, Piscataway, N.J., 1991).
  14. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, Tokyo, 1966).
  15. V. Yatsenko, S. Maslovski, S. Tretyakov, “Electromagnetic interaction of parallel arrays of dipole scatterers,” in Progress in Electromagnetics Research (EMW, Cambridge, Mass., 2000, pp. 285–307. [Abstract also in J. Electromagn. Waves Appl. 14, 79–80 (2000)].
    [CrossRef]
  16. J. I. Peltoniemi, “Variational volume integral equation method for electromagnetic scattering by irregular grains,” J. Quant. Spectrosc. Radiat. Transf. 55, 637–647 (1996).
    [CrossRef]
  17. A. Sihvola, R. Sharma, “Scattering corrections for the Maxwell Garnett mixing rule,” Microwave Opt. Technol. Lett. 22, 229–231 (1999).
    [CrossRef]

2000 (1)

C. R. Simovski, S. A. Tretyakov, A. H. Sihvola, M. M. Popov, “On the surface effect in thin molecular or composite layers,” Eur. Phys. J. Appl. Phys. 9, 195–204 (2000).
[CrossRef]

1999 (4)

S. I. Maslovski, S. A. Tretyakov, “Full-wave interaction field in two-dimensional arrays of dipole scatterers,” Int. J. Electron. Commun. 53, 135–139 (1999).

A. Sihvola, R. Sharma, “Scattering corrections for the Maxwell Garnett mixing rule,” Microwave Opt. Technol. Lett. 22, 229–231 (1999).
[CrossRef]

H. F. Contopanagos, C. A. Kyriazidou, W. M. Merrill, “Effective response functions for photonic bandgap materials,” J. Opt. Soc. Am. A 16, 1682–1699 (1999).
[CrossRef]

C. A. Moses, N. Engheta, “An idea for electromagnetic ‘feedforward-feedbackward’ media,” IEEE Trans. Antennas Propag. 47, 918–928 (1999).
[CrossRef]

1997 (1)

A. P. Vinogradov, Yu. N. Dmitriev, V. E. Romanenko, “Transition from planar to bulk properties in multi-layer system,” Electromagnetics 17, 563–571 (1997).
[CrossRef]

1996 (1)

J. I. Peltoniemi, “Variational volume integral equation method for electromagnetic scattering by irregular grains,” J. Quant. Spectrosc. Radiat. Transf. 55, 637–647 (1996).
[CrossRef]

1978 (1)

R. C. McPhedran, D. R. McKenzie, “The conductivity of lattices of spheres. 1. Simple cubic lattice,” Proc. R. Soc. London, Ser. A 359, 45–63 (1978).
[CrossRef]

1961 (1)

G. D. Mahan, G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1961).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE Press, Piscataway, N.J., 1991).

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

Contopanagos, H. F.

Dmitriev, Yu. N.

A. P. Vinogradov, Yu. N. Dmitriev, V. E. Romanenko, “Transition from planar to bulk properties in multi-layer system,” Electromagnetics 17, 563–571 (1997).
[CrossRef]

Engheta, N.

C. A. Moses, N. Engheta, “An idea for electromagnetic ‘feedforward-feedbackward’ media,” IEEE Trans. Antennas Propag. 47, 918–928 (1999).
[CrossRef]

Kyriazidou, C. A.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1984; first Russian edition, Gostechizdat, Moscow, 1957).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1984; first Russian edition, Gostechizdat, Moscow, 1957).

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, Mass, 1994).

Mahan, G. D.

G. D. Mahan, G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1961).
[CrossRef]

Maslovski, S.

V. Yatsenko, S. Maslovski, S. Tretyakov, “Electromagnetic interaction of parallel arrays of dipole scatterers,” in Progress in Electromagnetics Research (EMW, Cambridge, Mass., 2000, pp. 285–307. [Abstract also in J. Electromagn. Waves Appl. 14, 79–80 (2000)].
[CrossRef]

Maslovski, S. I.

S. I. Maslovski, S. A. Tretyakov, “Full-wave interaction field in two-dimensional arrays of dipole scatterers,” Int. J. Electron. Commun. 53, 135–139 (1999).

S. A. Tretyakov, A. J. Viitanen, S. I. Maslovski, I. E. Saarela, “Impedance boundary conditions for regular dense arrays of dipole scatterers,” , Electromagnetics Laboratory Report Series (Helsinki U. Technology, Helsinki Finland, 1999).

McKenzie, D. R.

R. C. McPhedran, D. R. McKenzie, “The conductivity of lattices of spheres. 1. Simple cubic lattice,” Proc. R. Soc. London, Ser. A 359, 45–63 (1978).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, D. R. McKenzie, “The conductivity of lattices of spheres. 1. Simple cubic lattice,” Proc. R. Soc. London, Ser. A 359, 45–63 (1978).
[CrossRef]

Merrill, W. M.

Moses, C. A.

C. A. Moses, N. Engheta, “An idea for electromagnetic ‘feedforward-feedbackward’ media,” IEEE Trans. Antennas Propag. 47, 918–928 (1999).
[CrossRef]

Obermair, G.

G. D. Mahan, G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1961).
[CrossRef]

Peltoniemi, J. I.

J. I. Peltoniemi, “Variational volume integral equation method for electromagnetic scattering by irregular grains,” J. Quant. Spectrosc. Radiat. Transf. 55, 637–647 (1996).
[CrossRef]

Popov, M. M.

C. R. Simovski, S. A. Tretyakov, A. H. Sihvola, M. M. Popov, “On the surface effect in thin molecular or composite layers,” Eur. Phys. J. Appl. Phys. 9, 195–204 (2000).
[CrossRef]

Romanenko, V. E.

A. P. Vinogradov, Yu. N. Dmitriev, V. E. Romanenko, “Transition from planar to bulk properties in multi-layer system,” Electromagnetics 17, 563–571 (1997).
[CrossRef]

Saarela, I. E.

S. A. Tretyakov, A. J. Viitanen, S. I. Maslovski, I. E. Saarela, “Impedance boundary conditions for regular dense arrays of dipole scatterers,” , Electromagnetics Laboratory Report Series (Helsinki U. Technology, Helsinki Finland, 1999).

Sharma, R.

A. Sihvola, R. Sharma, “Scattering corrections for the Maxwell Garnett mixing rule,” Microwave Opt. Technol. Lett. 22, 229–231 (1999).
[CrossRef]

Sihvola, A.

A. Sihvola, R. Sharma, “Scattering corrections for the Maxwell Garnett mixing rule,” Microwave Opt. Technol. Lett. 22, 229–231 (1999).
[CrossRef]

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE Publishing, London, 1999).

Sihvola, A. H.

C. R. Simovski, S. A. Tretyakov, A. H. Sihvola, M. M. Popov, “On the surface effect in thin molecular or composite layers,” Eur. Phys. J. Appl. Phys. 9, 195–204 (2000).
[CrossRef]

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, Mass, 1994).

Simovski, C. R.

C. R. Simovski, S. A. Tretyakov, A. H. Sihvola, M. M. Popov, “On the surface effect in thin molecular or composite layers,” Eur. Phys. J. Appl. Phys. 9, 195–204 (2000).
[CrossRef]

Slater, J.

J. Slater, Insulators, Semiconductors, and Metals (McGraw-Hill, New York, 1967).

Tretyakov, S.

V. Yatsenko, S. Maslovski, S. Tretyakov, “Electromagnetic interaction of parallel arrays of dipole scatterers,” in Progress in Electromagnetics Research (EMW, Cambridge, Mass., 2000, pp. 285–307. [Abstract also in J. Electromagn. Waves Appl. 14, 79–80 (2000)].
[CrossRef]

Tretyakov, S. A.

C. R. Simovski, S. A. Tretyakov, A. H. Sihvola, M. M. Popov, “On the surface effect in thin molecular or composite layers,” Eur. Phys. J. Appl. Phys. 9, 195–204 (2000).
[CrossRef]

S. I. Maslovski, S. A. Tretyakov, “Full-wave interaction field in two-dimensional arrays of dipole scatterers,” Int. J. Electron. Commun. 53, 135–139 (1999).

S. A. Tretyakov, A. J. Viitanen, S. I. Maslovski, I. E. Saarela, “Impedance boundary conditions for regular dense arrays of dipole scatterers,” , Electromagnetics Laboratory Report Series (Helsinki U. Technology, Helsinki Finland, 1999).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, Mass, 1994).

Viitanen, A. J.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, Mass, 1994).

S. A. Tretyakov, A. J. Viitanen, S. I. Maslovski, I. E. Saarela, “Impedance boundary conditions for regular dense arrays of dipole scatterers,” , Electromagnetics Laboratory Report Series (Helsinki U. Technology, Helsinki Finland, 1999).

Vinogradov, A. P.

A. P. Vinogradov, Yu. N. Dmitriev, V. E. Romanenko, “Transition from planar to bulk properties in multi-layer system,” Electromagnetics 17, 563–571 (1997).
[CrossRef]

Yatsenko, V.

V. Yatsenko, S. Maslovski, S. Tretyakov, “Electromagnetic interaction of parallel arrays of dipole scatterers,” in Progress in Electromagnetics Research (EMW, Cambridge, Mass., 2000, pp. 285–307. [Abstract also in J. Electromagn. Waves Appl. 14, 79–80 (2000)].
[CrossRef]

Electromagnetics (1)

A. P. Vinogradov, Yu. N. Dmitriev, V. E. Romanenko, “Transition from planar to bulk properties in multi-layer system,” Electromagnetics 17, 563–571 (1997).
[CrossRef]

Eur. Phys. J. Appl. Phys. (1)

C. R. Simovski, S. A. Tretyakov, A. H. Sihvola, M. M. Popov, “On the surface effect in thin molecular or composite layers,” Eur. Phys. J. Appl. Phys. 9, 195–204 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

C. A. Moses, N. Engheta, “An idea for electromagnetic ‘feedforward-feedbackward’ media,” IEEE Trans. Antennas Propag. 47, 918–928 (1999).
[CrossRef]

Int. J. Electron. Commun. (1)

S. I. Maslovski, S. A. Tretyakov, “Full-wave interaction field in two-dimensional arrays of dipole scatterers,” Int. J. Electron. Commun. 53, 135–139 (1999).

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (1)

J. I. Peltoniemi, “Variational volume integral equation method for electromagnetic scattering by irregular grains,” J. Quant. Spectrosc. Radiat. Transf. 55, 637–647 (1996).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

A. Sihvola, R. Sharma, “Scattering corrections for the Maxwell Garnett mixing rule,” Microwave Opt. Technol. Lett. 22, 229–231 (1999).
[CrossRef]

Phys. Rev. (1)

G. D. Mahan, G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1961).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

R. C. McPhedran, D. R. McKenzie, “The conductivity of lattices of spheres. 1. Simple cubic lattice,” Proc. R. Soc. London, Ser. A 359, 45–63 (1978).
[CrossRef]

Other (8)

S. A. Tretyakov, A. J. Viitanen, S. I. Maslovski, I. E. Saarela, “Impedance boundary conditions for regular dense arrays of dipole scatterers,” , Electromagnetics Laboratory Report Series (Helsinki U. Technology, Helsinki Finland, 1999).

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE Publishing, London, 1999).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, Mass, 1994).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1984; first Russian edition, Gostechizdat, Moscow, 1957).

J. Slater, Insulators, Semiconductors, and Metals (McGraw-Hill, New York, 1967).

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE Press, Piscataway, N.J., 1991).

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

V. Yatsenko, S. Maslovski, S. Tretyakov, “Electromagnetic interaction of parallel arrays of dipole scatterers,” in Progress in Electromagnetics Research (EMW, Cambridge, Mass., 2000, pp. 285–307. [Abstract also in J. Electromagn. Waves Appl. 14, 79–80 (2000)].
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Effective refractive index for a three-dimensional array of dielectric spheres with cubic cells. Permittivity of spheres, =200; volume fraction of inclusions, f=0.1; distance between cubic cells, d=a.

Fig. 2
Fig. 2

Effective refractive index for a periodic arrangement of lattices of dielectric spheres. Sphere permittivity, =200; sphere radius, r=0.4a; distance between arrays, d=10a.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

E=χJ,χ=a2jω1α-jη0μ0ω36π-Re β(0).
p=αEloc,Eloc=Eext+β(0)p,
β(0)=ωa2η4cos kR0kR0-sin kR0+jη0μ0ω36π-ηω2a2,
Im1α=η0μ0ω36π,
cos qd=cos kd+jη2χsin kd,
p(0)=αEloc=αn=-β(nd)p(nd).
p(nd)=exp(-jqnd)p(0),
β(nd)=Re-jωη4a21+(nd)2R02+(nd)2-1-(nd)2R02+(nd)21jkR02+(nd)2×exp[-jkR02+(nd)2]-14π01(|n|d)3+jk(nd)2-k2|n|d×exp(-jk|n|d)-jηω2a2cos knd.
Im[β(nd)]=ηω2a2cos knd.
Re[β(nd)]=ωη4a21-(nd)2R02+(nd)2cos kR02+(nd)2kR02+(nd)2-1+(nd)2R02+(nd)2×sin kR02+(nd)2 - 14π01(|n|d)3-k2|n|d×cos knd+k(nd)2sin k|n|d.
Re[β(nd)]-ηω2a2sin k|n|d.
β(nd)-jηω2a2exp(-jk|n|d).
Eloc=ωa2η4cos kR0kR0-sin kR0+jημω36πp(0)-jn=-η2ωa2exp(-jk|n|d)exp(-jqnd)p(0).
p(0)=αEloc,
1α=ωa2η4cos kR0kR0-sin kR0+jη0μ0ω36π-jn=-η2ωa2exp(-jk|n|d)exp(-jqnd).
n=-exp(-jk|n|d)exp(-jqnd)
=1+ n=1exp[-j(q+k)nd]+n=1exp[j(q-k)nd]=1-exp[-j(q+k)d]exp[-j(q+k)d]-1-exp[j(q-k)d]exp[j(q-k)d]-1=j sin kdcos kd-cos qd.
Re1α-β(0)=η2ωa2sin kd(cos kd-cos qd).
χ=a2jωn=- β(nd)=-η2j sin kd(cos kd-cos qd),
cos qd=cos kd+jη2χsin kd.
χ=-η2j sin kd(cos kd-cos qd)+a2jω×n=-,n0Re[β(nd)]+η2ωa2sin k|n|dexp(-jqnd).
χχeq=χ+ja2ωn=12Re[β(nd)]+ηωa2sin k|n|dcos qnd
cos qd=cos kd+jη2χeqsin kd,
cos qd=cos kd-sin kd(2a2/ηω)1α-j(η0μ0ω3/6π)-Re[β(0)]-n=1{(4a2/ηω)Re[β(nd)]+2 sin knd}cos qnd.
4a2ηωRe[β(nd)]-2 sin knd+(ka)2πkndcos knd+O(knd)-2.
(qa)22=(ka)22+ka/2a2ηωRe(1α)-12kR0-n=1R02k[R02+(na)2]3/2-1πkan3×cos(qna).
eff=1+1/0a3Re(1α)-a4R0-12n=1R02a[R02+(na)2]3/2-1πn3×cos(kaeffn).
eff=1+1/0a3Re(1α)-0.3595-12n=10.4836[0.4836+n2]3/2-1πn3×cos(kaeffn).
eff=1+α/(a30)1-αC,
C=β(0)ω=0+2n=1 β(nd)ω=0.
eff=1+1/0a3Re1α-0.3595-12n=10.4836[0.4836+n2]3/2-1πn3.
α=3V0-0+20,
eff=1+10a3 Re(1/α)-1/3,
eff=1+3f(-0)+20-f(-0),
a4R0+12n=1R02a(R02+n2a2)3/2-1πn3=13,
α=3V0-0+20×1-3 -0+20G1(kr)+0G2(kr)-1,
G1(kr)=23[(1+jkr)exp(-jkr)-1],
G2(kr)=1+jkr-715(kr)2-j215(kr)3×exp(-jkr)-1.

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