Abstract

Ultrathin metallic films have an interesting electromagnetic behavior as the frequency of the incident field is varied over several orders of magnitude, because of the dramatic dispersion exhibited by the metal permittivity. We study a finite multilayer of periodically placed planar conducting films for frequencies varying from the dc limit to the far ultraviolet. We provide the optimized reflectivity and transmittivity of the system for the various frequency regimes involved. Further, we produce the dispersion diagrams of the corresponding photonic bandgap structures, which clearly show the transition of the system from a metallic (low frequencies) to a dielectric (optical frequencies) behavior. In addition, simple design formulas for maximum reflectivity of finite film number N are presented in terms of film thickness and film spacing in each of the representative frequency ranges.

© 1999 Optical Society of America

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References

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  1. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 9, pp. 227–259 and references therein.
  2. A. J. Ward, J. B. Pendry, W. J. Stewart, “Photonic dispersion surfaces,” J. Phys. Condens. Matter 7, 2217–2224 (1995).
    [CrossRef]
  3. H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
    [CrossRef]
  4. E. Spiller, Soft X-Ray Optics (SPIE Optical Engineering Press, Bellingham, Wash., 1994).
  5. M. Scalora, M. Bloemer, A. Manka, A. Pethel, J. Dowling, C. Bowden, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 1–7 (1998);M. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
    [CrossRef]
  6. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 2, pp. 128–142 and references therein.
  7. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), Chap. 1, pp. 66–70 and references therein.
  8. H. Contopanagos, N. Alexopoulos, E. Yablonovitch, “High-Q rectangular cavities and waveguide filters using periodic metalo-dielectric slabs,” in IEEE Microwave Theory and Techniques International Microwave Sympo-sium Digest, June 7–12, 1998 (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1539–1542.
  9. H. Ehrenreich, H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128, 1622–1629 (1962);H. Ehrenreich, H. R. Philipp, B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
    [CrossRef]
  10. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10, 283–295 (1993);C. M. Soukoulis, ed., Photonic Band Gaps, Vol. 315 of NATO ASI Series E (Kluwer Academic, Dordrecht, The Netherlands, 1996).
    [CrossRef]
  11. A. E. Kaplan, “On the reflectivity of metallic films at microwave and radio frequencies,” Radio Eng. Electron. Phys. 9, 1476–1481 (1964).

1998 (2)

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

M. Scalora, M. Bloemer, A. Manka, A. Pethel, J. Dowling, C. Bowden, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 1–7 (1998);M. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

1995 (1)

A. J. Ward, J. B. Pendry, W. J. Stewart, “Photonic dispersion surfaces,” J. Phys. Condens. Matter 7, 2217–2224 (1995).
[CrossRef]

1993 (1)

1964 (1)

A. E. Kaplan, “On the reflectivity of metallic films at microwave and radio frequencies,” Radio Eng. Electron. Phys. 9, 1476–1481 (1964).

1962 (1)

H. Ehrenreich, H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128, 1622–1629 (1962);H. Ehrenreich, H. R. Philipp, B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

Alexopoulos, N.

H. Contopanagos, N. Alexopoulos, E. Yablonovitch, “High-Q rectangular cavities and waveguide filters using periodic metalo-dielectric slabs,” in IEEE Microwave Theory and Techniques International Microwave Sympo-sium Digest, June 7–12, 1998 (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1539–1542.

Alexopoulos, N. G.

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

Bloemer, M.

M. Scalora, M. Bloemer, A. Manka, A. Pethel, J. Dowling, C. Bowden, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 1–7 (1998);M. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 9, pp. 227–259 and references therein.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), Chap. 1, pp. 66–70 and references therein.

Bowden, C.

M. Scalora, M. Bloemer, A. Manka, A. Pethel, J. Dowling, C. Bowden, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 1–7 (1998);M. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

Contopanagos, H.

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

H. Contopanagos, N. Alexopoulos, E. Yablonovitch, “High-Q rectangular cavities and waveguide filters using periodic metalo-dielectric slabs,” in IEEE Microwave Theory and Techniques International Microwave Sympo-sium Digest, June 7–12, 1998 (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1539–1542.

Dowling, J.

M. Scalora, M. Bloemer, A. Manka, A. Pethel, J. Dowling, C. Bowden, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 1–7 (1998);M. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

Ehrenreich, H.

H. Ehrenreich, H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128, 1622–1629 (1962);H. Ehrenreich, H. R. Philipp, B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 9, pp. 227–259 and references therein.

Kaplan, A. E.

A. E. Kaplan, “On the reflectivity of metallic films at microwave and radio frequencies,” Radio Eng. Electron. Phys. 9, 1476–1481 (1964).

Manka, A.

M. Scalora, M. Bloemer, A. Manka, A. Pethel, J. Dowling, C. Bowden, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 1–7 (1998);M. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

Pendry, J. B.

A. J. Ward, J. B. Pendry, W. J. Stewart, “Photonic dispersion surfaces,” J. Phys. Condens. Matter 7, 2217–2224 (1995).
[CrossRef]

Pethel, A.

M. Scalora, M. Bloemer, A. Manka, A. Pethel, J. Dowling, C. Bowden, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 1–7 (1998);M. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

Philipp, H. R.

H. Ehrenreich, H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128, 1622–1629 (1962);H. Ehrenreich, H. R. Philipp, B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

Scalora, M.

M. Scalora, M. Bloemer, A. Manka, A. Pethel, J. Dowling, C. Bowden, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 1–7 (1998);M. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

Spiller, E.

E. Spiller, Soft X-Ray Optics (SPIE Optical Engineering Press, Bellingham, Wash., 1994).

Stewart, W. J.

A. J. Ward, J. B. Pendry, W. J. Stewart, “Photonic dispersion surfaces,” J. Phys. Condens. Matter 7, 2217–2224 (1995).
[CrossRef]

Ward, A. J.

A. J. Ward, J. B. Pendry, W. J. Stewart, “Photonic dispersion surfaces,” J. Phys. Condens. Matter 7, 2217–2224 (1995).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), Chap. 1, pp. 66–70 and references therein.

Yablonovitch, E.

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10, 283–295 (1993);C. M. Soukoulis, ed., Photonic Band Gaps, Vol. 315 of NATO ASI Series E (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

H. Contopanagos, N. Alexopoulos, E. Yablonovitch, “High-Q rectangular cavities and waveguide filters using periodic metalo-dielectric slabs,” in IEEE Microwave Theory and Techniques International Microwave Sympo-sium Digest, June 7–12, 1998 (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1539–1542.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 2, pp. 128–142 and references therein.

IEEE Trans. Microwave Theory Tech. (1)

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

J. Appl. Phys. (1)

M. Scalora, M. Bloemer, A. Manka, A. Pethel, J. Dowling, C. Bowden, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 1–7 (1998);M. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

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

J. Phys. Condens. Matter (1)

A. J. Ward, J. B. Pendry, W. J. Stewart, “Photonic dispersion surfaces,” J. Phys. Condens. Matter 7, 2217–2224 (1995).
[CrossRef]

Phys. Rev. (1)

H. Ehrenreich, H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128, 1622–1629 (1962);H. Ehrenreich, H. R. Philipp, B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

Radio Eng. Electron. Phys. (1)

A. E. Kaplan, “On the reflectivity of metallic films at microwave and radio frequencies,” Radio Eng. Electron. Phys. 9, 1476–1481 (1964).

Other (5)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 9, pp. 227–259 and references therein.

E. Spiller, Soft X-Ray Optics (SPIE Optical Engineering Press, Bellingham, Wash., 1994).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 2, pp. 128–142 and references therein.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), Chap. 1, pp. 66–70 and references therein.

H. Contopanagos, N. Alexopoulos, E. Yablonovitch, “High-Q rectangular cavities and waveguide filters using periodic metalo-dielectric slabs,” in IEEE Microwave Theory and Techniques International Microwave Sympo-sium Digest, June 7–12, 1998 (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1539–1542.

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Figures (14)

Fig. 1
Fig. 1

Plane-wave incidence on a multilayer of metal films.

Fig. 2
Fig. 2

Conductor skin depth versus frequency for various values of ωp, γ. The difference can be normalized away by transforming the axes as ωω/γ, δδ/λ(γ).

Fig. 3
Fig. 3

Dispersion diagrams for the PBG medium: (a), (b) at ω=0.001γ (microwave); (c), (d) at ω=γ (far IR); (e), (f) at ω=10γ (near IR); (g), (h) at ω=50γ (visible); (i), (j) at the plasma frequency ω=100γ (near UV).

Fig. 4
Fig. 4

Power balance of a single metal film, as a function of the film’s normalized thickness in terms of its skin depth δ, at frequency ω=2π×5 GHz. Pab, Ptr, and Pr refer to the absorbed, transmitted, and reflected power, respectively.

Fig. 5
Fig. 5

Normalized Q for fixed resonant air-gap thicknesses k0da=π-(1/Δτ) versus normalized metal film thickness xdc/δ. The curves (from left to right) correspond to τ=80, 70, 60, 50, 40, 30, 20, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1.

Fig. 6
Fig. 6

Normalized maximum QN (with continuously optimized air-gap thicknesses y=π-(x0.9965/Δ)] versus normalized metal film thickness xdc/δ.

Fig. 7
Fig. 7

Normalized QN, maximized in both dc and da, versus number of layers N.

Fig. 8
Fig. 8

Electric field distribution, normalized to incident field amplitude, inside the semi-infinite structure (z>0) for normal incidence and two film thicknesses and for two air-gap thicknesses: (a) quarter-wavelength (antiresonant) and (b) optimized according to Eq. (58) for maximum reflectivity.

Fig. 9
Fig. 9

(a) Reflectivity (solid curves) and transmittivity (dashed curves) of a single metal film, versus ω, for various film thicknesses parameterized in terms of conductor skin depth δ0 (evaluated at ω=γ); (b) transmittivity for fixed unit-cell thicknesses as a function of N and ω for N=1 (solid curve), N=2 (dashed curve), N=3 (dotted curve), and N=4 (dotted–dashed curve).

Fig. 10
Fig. 10

Multiplexing at optical frequencies for different air (dielectric) thicknesses da: (a) for film thickness δ0 and (b) for film thickness 2δ0.

Fig. 11
Fig. 11

Maximum reflectivity for the semi-infinite system under normal plane-wave incidence, versus normalized film thickness, at the plasma frequency (near UV).

Fig. 12
Fig. 12

Same as in Fig. 11, but at the far UV (ω=5ωp).

Fig. 13
Fig. 13

Convergence of the reflectivity at resonant spacing for ω=ωp (corresponding to Fig. 11) and two selected thicknesses: (a) film thickness δ0 and (b) film thickness 0.1δ0.

Fig. 14
Fig. 14

Same as in Fig. 13, but at the far UV (ω=5ωp) and for the two selected thicknesses in Fig. 12: (a) optimum film thickness 0.7δ0 and (b) film thickness 0.1δ0.

Equations (70)

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ΓNp(θ)=2u21(1-ξN)(u11-u22)(1-ξN)+ζ(u11+u22)(1+ξN),
ζ=u11-u22u11+u22 1+4u12u21(u11-u22)21/2,ξ=1-ζ1+ζ.
U=v×u,
u=1-Γa,c1;pΓa,d1;p-(Γa,d1;p+Γa,c1;pZa,dp)Γa,c1;p+Γa,d1;pZa,cp-Γa,c1;pΓa,d1;p+Za,cpZa,dp,
v=v1v2,
vj=1-(Γa, jp)2 exp(-2γjdj)1-(Γa, jp)2exp(γjdj),j=1, 2,
Γa, j1;p=Γa, jp1-exp(-2γjdj)1-(Γa,jp)2 exp(-2γjdj),
Za,jp=-(Γa,jp)2+exp(-2γjdj)1-(Γa,jp)2 exp(-2γjdj).
Γa,jp=1=(ηj/cos θj)-(1/cos θ)(ηj/cos θj)+(1/cos θ),
Γa,jp=2=ηj cos θj-cos θηj cos θj+cos θ,
γj=jk0nj cos θj,
nj=1nj,nj=j,
cos θj=(1-ηj2 sin2 θ)1/2.
TNp(θ)=ΓNp(θ)u11+u22u21×ζ2Γa,c1;pΓa,d1;p exp(-γcdc)exp(-γddd)Γa,cpΓa,dp(u11+u22)(1+ζ)N×(1-ξN)-1.
exp[±jk(dc+da)]=w1,2=Tr(U)±{[Tr(U)]2-4 Det(U)}1/22=v(u11+u22)(1±ζ)2.
ww1=v(u11+u22)(1+ζ)2.
kia=lnv(u11+u22)(1+ζ)2.
Γ=U21w-U22=vu21w-cz=2u21u11-u22+ζ(u11+u22)
ξ1-ζ1+ζ=exp(-2jka)|ξ|<1
limN= ΓNp=2u21u11-u22+ζ(u11+u22),
c=1-ωp2ω(ω-jγ),
7eV<hωp2π<15eV,0.05eV<hγ2π<0.13eV
hωp2π=7eVωp=1016Hz,γ=10-2ωp.
(ωγ)1-j (ωp2/γ)ωσ=0 ωp2γ
O(107S/m),
nc=cβ-jα,
δ(ω)=1k0α,
s0ωpγ,sωγ,
β(s, s0)=12 1-s021+s2+1-s021+s22+s02/s1+s221/21/2,
α(s, s0)=12 s02/s1+s21-s021+s2+1-s021+s22+s02/s1+s221/21/2.
δ(s, s0)=λ(γ)2π 1sα(s, s0),
γcdc=j β(s, s0)α(s, s0)+1 dcδ(s, s0).
xdcδ(s, s0).
γada=jk0da=jy,
yk0da=2πdaλ(ω)=s 2πdaλ(γ).
dc=x0δ0,δ0δ(s=1, s0),
QN=11-|ΓN|2,
Qmetal=11-|Γa,c|2,
ωp2γω1s02s1,
γcdc(1+j)x,
β(s, s0)α(s, s0)1+O(s/s02).
Γa,c-1-1+jΔ,Δs022s1.
δIFSET=20σμ0=2γωp2μ0.
y=π-2,
1+j2Δ1+j2 τ 2h 2,
1Δ=τ 2,
Γa,c=-1-h+h22 2,
Γa,c1=-1-h 1+χ1-χ +1+χ2+6χ2(1-χ)2 2,
χexp[-(1+j)2x].
Za,cZa,a(x, y)=-1+2h 1+χ1-χ-j2-2h 1+χ1-χ-j222.
1-exp(-2jy)2Γa,c11-Za,cZa,a1/2
=h2-jh 1+χ1-χ-141/2ψ,
ζ=1-Za,cZa,a1+Za,cZa,aψ,
Γ=2Γa,c11-z 11+ψ-1-ψ+j2+12 ψ+j222.
|Γ|2=1-2ψr+22ψr2,
|Γa,c|2=1-2hr+22hr2.
QQmetal=hrψr 1-hr1-ψr,
hr=τ2,
ψr=τ2 R-(x)τ-12τ2+R-(x)τ-12τ22+R+(x)τ-121/21/2,
R±(x)1-exp(-4x)±2 exp(-2x)sin 2x1+exp(-4x)-2 exp(-2x)cos 2x.
QQmetal1R-(x)τ-12τ2+R-(x)τ-12τ22+R+(x)τ-121/21/2.
yR(k0da)R=π-1Δτ.
QmaxQmetal=π δdc
yR(x)(k0da)R(x)=π-x0.9965Δ.
QNmaxQmetaldc,da=opt=N
2ρ<zn<2ρ+1z
=ρ(dc+da)+(zn-2ρ)dc,
2ρ+1<zn<2(ρ+1)z
=ρ(dc+da)+dc+(zn-2ρ-1)da.
QmaxQmetal=π δdc|Γmax|2=1-Qmetalπ dcδ.

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