Abstract

The scattering of a magnetodielectric multilayer has been studied by a first-order method. The model reported in this manuscript relies on the equivalence between heterogeneities of the medium and fictitious electric and magnetic sources. Types of inhomogeneities considered are roughness and bulk inhomogeneities and concern both permittivity and permeability. The numerical results are compared to those given in previous papers for optical scattering. It is shown in the microwave spectra that angle-resolved scattering allows identification of the scattering origins (permittivity or permeability spectra). The cases of isotropic films and metamaterials are presented and discussed.

© 2013 Optical Society of America

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References

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  1. M. Matsumoto and Y. Miyata, “Thin electromagnetic wave absorber for quasi-microwave band containing aligned thin magnetic metal particles,” IEEE Trans. Magn. 33, 4459–4464 (1997).
    [CrossRef]
  2. J. Neige, T. Lepetit, A.-L. Adenot-Engelvin, N. Malléjac, A. Thiaville, and N. Vukadinovic, “Microwave permeability of FeNiMo flakes-polymer composites with and without an applied static magnetic field,” IEEE Trans. Magn. 49, 1005–1008 (2013).
    [CrossRef]
  3. P. Ikonen and S. Tretyakov, “On the advantages of magnetic materials in microstrip antenna miniaturization,” Microw. Opt. Technol. Lett. 50, 3131–3134 (2008).
    [CrossRef]
  4. M. Saillard and A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
    [CrossRef]
  5. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces (Artech House, Inc., 1987).
  6. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  7. A. Sentenac and J. J. Greffet, “Mean-field theory of light scattering by one-dimensional rough surfaces,” J. Opt. Soc. Am. A 15, 528–532 (1998).
    [CrossRef]
  8. P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  9. J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
    [CrossRef]
  10. C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10, 365–374 (1993).
    [CrossRef]
  11. T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
    [CrossRef]
  12. K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
    [CrossRef]
  13. C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32, 5492–5503 (1993).
    [CrossRef]
  14. http://www.comsol.com/ .
  15. W. M. Merrill, R. E. Diaz, M. M. LoRe, M. C. Squires, and N. G. Alexopoulos, “Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum,” IEEE Trans. Antennas Propag. 47, 142–148 (1999).
    [CrossRef]
  16. I. V. Melchakova, E. A. Yankovskaya, P. A. Belov, and C. R. Simovski, “Material parameters of optical metamaterials formed by nanofishnet structures,” Proc. SPIE 7754, 77541V1–V14 (2010).
    [CrossRef]

2013 (1)

J. Neige, T. Lepetit, A.-L. Adenot-Engelvin, N. Malléjac, A. Thiaville, and N. Vukadinovic, “Microwave permeability of FeNiMo flakes-polymer composites with and without an applied static magnetic field,” IEEE Trans. Magn. 49, 1005–1008 (2013).
[CrossRef]

2010 (1)

I. V. Melchakova, E. A. Yankovskaya, P. A. Belov, and C. R. Simovski, “Material parameters of optical metamaterials formed by nanofishnet structures,” Proc. SPIE 7754, 77541V1–V14 (2010).
[CrossRef]

2008 (1)

P. Ikonen and S. Tretyakov, “On the advantages of magnetic materials in microstrip antenna miniaturization,” Microw. Opt. Technol. Lett. 50, 3131–3134 (2008).
[CrossRef]

2004 (1)

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

2001 (2)

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

M. Saillard and A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[CrossRef]

1999 (1)

W. M. Merrill, R. E. Diaz, M. M. LoRe, M. C. Squires, and N. G. Alexopoulos, “Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum,” IEEE Trans. Antennas Propag. 47, 142–148 (1999).
[CrossRef]

1998 (1)

1997 (1)

M. Matsumoto and Y. Miyata, “Thin electromagnetic wave absorber for quasi-microwave band containing aligned thin magnetic metal particles,” IEEE Trans. Magn. 33, 4459–4464 (1997).
[CrossRef]

1993 (2)

1984 (1)

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

1981 (1)

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Adenot-Engelvin, A.-L.

J. Neige, T. Lepetit, A.-L. Adenot-Engelvin, N. Malléjac, A. Thiaville, and N. Vukadinovic, “Microwave permeability of FeNiMo flakes-polymer composites with and without an applied static magnetic field,” IEEE Trans. Magn. 49, 1005–1008 (2013).
[CrossRef]

Alexopoulos, N. G.

W. M. Merrill, R. E. Diaz, M. M. LoRe, M. C. Squires, and N. G. Alexopoulos, “Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum,” IEEE Trans. Antennas Propag. 47, 142–148 (1999).
[CrossRef]

Amra, C.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces (Artech House, Inc., 1987).

Belov, P. A.

I. V. Melchakova, E. A. Yankovskaya, P. A. Belov, and C. R. Simovski, “Material parameters of optical metamaterials formed by nanofishnet structures,” Proc. SPIE 7754, 77541V1–V14 (2010).
[CrossRef]

Bousquet, P.

Bruel, L.

Chew, W. C.

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Diaz, R. E.

W. M. Merrill, R. E. Diaz, M. M. LoRe, M. C. Squires, and N. G. Alexopoulos, “Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum,” IEEE Trans. Antennas Propag. 47, 142–148 (1999).
[CrossRef]

Elfouhaily, T. M.

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

Elson, J. M.

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

Flory, F.

Greffet, J. J.

Grèzes-Besset, C.

Guérin, C.-A.

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

Ikonen, P.

P. Ikonen and S. Tretyakov, “On the advantages of magnetic materials in microstrip antenna miniaturization,” Microw. Opt. Technol. Lett. 50, 3131–3134 (2008).
[CrossRef]

Lepetit, T.

J. Neige, T. Lepetit, A.-L. Adenot-Engelvin, N. Malléjac, A. Thiaville, and N. Vukadinovic, “Microwave permeability of FeNiMo flakes-polymer composites with and without an applied static magnetic field,” IEEE Trans. Magn. 49, 1005–1008 (2013).
[CrossRef]

LoRe, M. M.

W. M. Merrill, R. E. Diaz, M. M. LoRe, M. C. Squires, and N. G. Alexopoulos, “Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum,” IEEE Trans. Antennas Propag. 47, 142–148 (1999).
[CrossRef]

Malléjac, N.

J. Neige, T. Lepetit, A.-L. Adenot-Engelvin, N. Malléjac, A. Thiaville, and N. Vukadinovic, “Microwave permeability of FeNiMo flakes-polymer composites with and without an applied static magnetic field,” IEEE Trans. Magn. 49, 1005–1008 (2013).
[CrossRef]

Matsumoto, M.

M. Matsumoto and Y. Miyata, “Thin electromagnetic wave absorber for quasi-microwave band containing aligned thin magnetic metal particles,” IEEE Trans. Magn. 33, 4459–4464 (1997).
[CrossRef]

Melchakova, I. V.

I. V. Melchakova, E. A. Yankovskaya, P. A. Belov, and C. R. Simovski, “Material parameters of optical metamaterials formed by nanofishnet structures,” Proc. SPIE 7754, 77541V1–V14 (2010).
[CrossRef]

Merrill, W. M.

W. M. Merrill, R. E. Diaz, M. M. LoRe, M. C. Squires, and N. G. Alexopoulos, “Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum,” IEEE Trans. Antennas Propag. 47, 142–148 (1999).
[CrossRef]

Miyata, Y.

M. Matsumoto and Y. Miyata, “Thin electromagnetic wave absorber for quasi-microwave band containing aligned thin magnetic metal particles,” IEEE Trans. Magn. 33, 4459–4464 (1997).
[CrossRef]

Neige, J.

J. Neige, T. Lepetit, A.-L. Adenot-Engelvin, N. Malléjac, A. Thiaville, and N. Vukadinovic, “Microwave permeability of FeNiMo flakes-polymer composites with and without an applied static magnetic field,” IEEE Trans. Magn. 49, 1005–1008 (2013).
[CrossRef]

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Roche, P.

Saillard, M.

M. Saillard and A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[CrossRef]

Sentenac, A.

M. Saillard and A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[CrossRef]

A. Sentenac and J. J. Greffet, “Mean-field theory of light scattering by one-dimensional rough surfaces,” J. Opt. Soc. Am. A 15, 528–532 (1998).
[CrossRef]

Simovski, C. R.

I. V. Melchakova, E. A. Yankovskaya, P. A. Belov, and C. R. Simovski, “Material parameters of optical metamaterials formed by nanofishnet structures,” Proc. SPIE 7754, 77541V1–V14 (2010).
[CrossRef]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces (Artech House, Inc., 1987).

Squires, M. C.

W. M. Merrill, R. E. Diaz, M. M. LoRe, M. C. Squires, and N. G. Alexopoulos, “Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum,” IEEE Trans. Antennas Propag. 47, 142–148 (1999).
[CrossRef]

Thiaville, A.

J. Neige, T. Lepetit, A.-L. Adenot-Engelvin, N. Malléjac, A. Thiaville, and N. Vukadinovic, “Microwave permeability of FeNiMo flakes-polymer composites with and without an applied static magnetic field,” IEEE Trans. Magn. 49, 1005–1008 (2013).
[CrossRef]

Tretyakov, S.

P. Ikonen and S. Tretyakov, “On the advantages of magnetic materials in microstrip antenna miniaturization,” Microw. Opt. Technol. Lett. 50, 3131–3134 (2008).
[CrossRef]

Vukadinovic, N.

J. Neige, T. Lepetit, A.-L. Adenot-Engelvin, N. Malléjac, A. Thiaville, and N. Vukadinovic, “Microwave permeability of FeNiMo flakes-polymer composites with and without an applied static magnetic field,” IEEE Trans. Magn. 49, 1005–1008 (2013).
[CrossRef]

Warnick, K. F.

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Yankovskaya, E. A.

I. V. Melchakova, E. A. Yankovskaya, P. A. Belov, and C. R. Simovski, “Material parameters of optical metamaterials formed by nanofishnet structures,” Proc. SPIE 7754, 77541V1–V14 (2010).
[CrossRef]

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

W. M. Merrill, R. E. Diaz, M. M. LoRe, M. C. Squires, and N. G. Alexopoulos, “Effective medium theories for artificial materials composed of multiple sizes of spherical inclusions in a host continuum,” IEEE Trans. Antennas Propag. 47, 142–148 (1999).
[CrossRef]

IEEE Trans. Magn. (2)

M. Matsumoto and Y. Miyata, “Thin electromagnetic wave absorber for quasi-microwave band containing aligned thin magnetic metal particles,” IEEE Trans. Magn. 33, 4459–4464 (1997).
[CrossRef]

J. Neige, T. Lepetit, A.-L. Adenot-Engelvin, N. Malléjac, A. Thiaville, and N. Vukadinovic, “Microwave permeability of FeNiMo flakes-polymer composites with and without an applied static magnetic field,” IEEE Trans. Magn. 49, 1005–1008 (2013).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Microw. Opt. Technol. Lett. (1)

P. Ikonen and S. Tretyakov, “On the advantages of magnetic materials in microstrip antenna miniaturization,” Microw. Opt. Technol. Lett. 50, 3131–3134 (2008).
[CrossRef]

Phys. Rev. B (1)

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

Proc. SPIE (1)

I. V. Melchakova, E. A. Yankovskaya, P. A. Belov, and C. R. Simovski, “Material parameters of optical metamaterials formed by nanofishnet structures,” Proc. SPIE 7754, 77541V1–V14 (2010).
[CrossRef]

Waves Random Media (3)

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

M. Saillard and A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[CrossRef]

Other (2)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces (Artech House, Inc., 1987).

http://www.comsol.com/ .

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Figures (12)

Fig. 1.
Fig. 1.

Schematic representation of a rough multilayer. k is the wavevector; it is defined by the two angles θ (normal angle) and ϕ (polar angle) and its norm k. The functions hj are dependent on the roughness of the interface j, and the average of these functions is zj.

Fig. 2.
Fig. 2.

Schematic representation of an inhomogeneous bulk multilayer. k is the wavevector, defined by two angles θ (normal angle) and ϕ (polar angle) and its norm k. The interfaces j are located by zj.

Fig. 3.
Fig. 3.

Schematic representation of considered medium in SS polarization, for bulk scattering (in this case h(x,y)=cte). e is thickness, λ is wavelength, and θ is observation angle between 0 and 90.

Fig. 4.
Fig. 4.

Bulk and surface reflected scattering calculated at normal illumination (i0=0) for a low-index material: cryolite 20L, optical thickness equal to 20λ/4n with a refractive index of n=1.3.

Fig. 5.
Fig. 5.

Comparison between first-order method and exact calculation—p(x).

Fig. 6.
Fig. 6.

Comparison between first-order method and exact calculation—case a.

Fig. 7.
Fig. 7.

Comparison between first-order method and exact calculation—case b.

Fig. 8.
Fig. 8.

Bulk scattering calculated at normal illumination (i0=0) for a magnetodielectric material: εr=8+0.2ι, μr=1.6+0.5ι in SS polarization. It is important to notice that the magnitude of the correlation function γ is about the same as the square of the wavelength, so the magnitude of the scattered intensity is β·104.

Fig. 9.
Fig. 9.

Bulk scattering calculated under normal illumination (i0=0) for a fishnet material: εr=2, μr=3+ι in SS polarization. It is important to notice that the magnitude of the correlation function γ is about the same as the square of the wavelength, so the magnitude of the scattered intensity is β·1018.

Fig. 10.
Fig. 10.

Map of log|βε|.

Fig. 11.
Fig. 11.

Map of log|βμ|.

Fig. 12.
Fig. 12.

Map of relative difference log|(βεβμ)/βε|.

Equations (60)

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

×E(x,y,z)=ιωμH(x,y,z),
×H(x,y,z)=ιωεE(x,y,z).
ε(x,y,z)=ε1+jδεjH(zhj(x,y)),
δεj=εi+1εi.
E(x,y,z)=E1(x,y,z)+j=1nδEj(x,y,z)H(zhj(x,y)),
δEj(x,y,z)=Ei+1(x,y,z)Ei(x,y,z),
×Ei(x,y,z)=ιωμiHi(x,y,z),
×Hi(x,y,z)=ιωεiEi(x,y,z),
nhj×δEj(x,y,hj)=0,
nhj×δHj(x,y,hj)=0,
nhj=zhj.
(E,H)=(E0,H0)+(Ed,Hd).
×Eid(x,y,z)=ιωμiHid(x,y,z),
×Hid(x,y,z)=ιωεiEid(x,y,z),
nhj×δEjd(x,y,hj)=nhj×δEj0(x,y,hj),
nhj×δHjd(x,y,hj)=nhj×δHj0(x,y,hj).
δEid(x,y,z)|z=hj=δEid(x,y,z)|z=zj+hj(x,y)δEid(x,y,z)z|z=zj+O(hj2).
z×δEjd(x,y,zj)=hj(x,y)×δEj0(x,y,zj)hj(x,y)z×δEj0(x,y,z)z|z=zjdef=Mi,
z×δHjd(x,y,zj)=hj(x,y)×δHj0(x,y,zj)hj(x,y)z×δHj0(x,y,z)z|z=zjdef=Ji.
ΔEid+ki2Eid=0,
ΔHid+ki2Hid=0,
Eid(r,z)=σE^id(σ,z)eiσ·rdσ.
2z2E^id(σ,z)+αi2(σ)E^id(σ,z)=0,
2z2H^id(σ,z)+αi2(σ)H^id(σ,z)=0
E^id(σ,z)=Ai+(σ)eiαi(σ)z+Ai(σ)eiαi(σ)z.
H^i=1ωμi[σ×(Ai+(σ)eiαi(σ)z+Ai(σ)eιαi(σ)z)+αi(σ)z×(Ai+(σ)eιαi(σ)zAi(σ)eιαi(σ)z)].
eιαiziAi,x+eιαiziAi,x+eιαi+1ziAi+1,x++eιαi+1ziAi+1,x=Mi,y,
eιαiziAi,y+eιαiziAi,y+eιαi+1ziAi+1,y++eιαi+1ziAi+1,y=Mi,x,
Xi+Ai,x+Xix+Ai,z+XiAi,xXixAi,zXi+1+Ai+1,x++Xi+1x+Ai+1,z++Xi+1Ai+1,x+Xi+1xAi+1,z=Ji,x,
Xi+Ai,y+Xiy+Ai,z+XiAi,yXiyAi,zXi+1+Ai+1,y++Xi+1y+Ai+1,z++Xi+1Ai+1,y+Xi+1yAi+1,z=Ji,y,
Xj±=αjωμje±ιαjzi,Xjx±=σxωμje±ιαjzi,Xjy±=σyωμje±ιαjzi.
σxAi,x++σyAi,y++αiAi,z+=0,
σxAi,x+σyAi,yαiAi,z=0.
W¯¯A¯=S¯r.
E^1+=A1+(σ0)eια1(σ0)z=A0eια10z,
ε˜i=εi[1+pi(x,y,z)],
μ˜i=μi[1+qi(x,y,z)].
×Ei=ιωμ˜iHi,
×Hi=ιωε˜iEi,
z×δEi(x,y,zi)=0,
z×δHi(x,y,zi)=0.
×EidιωμiHid=ιωμiqiHi,
×Hid+ιωεiEid=ιωεipiEi.
×Eid=ιωμiHid+Mi,
×Hid=ιωεiEid+Ji,
Mi=ιωμiqiHi0,
Ji=ιωεipiEi0.
ΔEid+k2Eid=SiE,
ΔHid+k2Hid=SiH
SiE=ιωμiJi+1ιωεi(·Ji)×Mi,
SiH=ιωεiMi1ιωμi(·Mi)×Ji.
Eid=Eid+Eid*,
E^id(σ,z)=Ai+(σ)eιαi(σ)z+Ai(σ)eιαi(σ)z,
E^id*(σ,z)=12iαiz=0eiS^iE(σ,z)eιαi|zz|dz.
z×δEid(x,y,zi)=z×δEid*(x,y,zi),
z×δHid(x,y,zi)=z×δHid*(x,y,zi).
γh(σ)=γε(σ)=14π(δgLg)2exp[(σLg2)2]+12π(δeLe)2[1+(σLe)2]32.
δe=0.0171,Le=2000nm,δg=0.00856,Lg=200nm,
δe=1nm,Le=2000nm,δg=0.5nm,Lg=200nm,
Id=βεγε+βμγμ+βεμR(γεμ).

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