Abstract

A planar dielectric film waveguide with a substrate of a nearly-free-electron metal has an ordinary diffraction grating etched on its surface. The reflection of a light beam incident on the grating from the cover is investigated as a function of the angle of incidence. A perturbation procedure is used for the determination of the resulting absorption spectra. The angular location and the depth of the antiresonance in the angular response of the reflection coefficient are used to obtain estimates of the average thickness of the dielectric film and the relative amplitude of the grating.

© 1999 Optical Society of America

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References

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  1. K. G. Muller, M. Veith, S. Mittler-Neher, W. Knoll, “Plasmon surface polariton coupling with dielectric gratings and the thermal decomposition of these dielectric gratings,” J. Appl. Phys. 82, 4172–4176 (1997).
    [CrossRef]
  2. S. R. Seshadri, “Quasi-optics of a planar dielectric waveguide with a dispersive substrate,” J. Opt. Soc. Am. A 15, 1952–1958 (1998).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 624–627.
  4. S. R. Seshadri, “Leaky surface polariton,” J. Appl. Phys. 59, 1187–1195 (1986).
    [CrossRef]

1998

1997

K. G. Muller, M. Veith, S. Mittler-Neher, W. Knoll, “Plasmon surface polariton coupling with dielectric gratings and the thermal decomposition of these dielectric gratings,” J. Appl. Phys. 82, 4172–4176 (1997).
[CrossRef]

1986

S. R. Seshadri, “Leaky surface polariton,” J. Appl. Phys. 59, 1187–1195 (1986).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 624–627.

Knoll, W.

K. G. Muller, M. Veith, S. Mittler-Neher, W. Knoll, “Plasmon surface polariton coupling with dielectric gratings and the thermal decomposition of these dielectric gratings,” J. Appl. Phys. 82, 4172–4176 (1997).
[CrossRef]

Mittler-Neher, S.

K. G. Muller, M. Veith, S. Mittler-Neher, W. Knoll, “Plasmon surface polariton coupling with dielectric gratings and the thermal decomposition of these dielectric gratings,” J. Appl. Phys. 82, 4172–4176 (1997).
[CrossRef]

Muller, K. G.

K. G. Muller, M. Veith, S. Mittler-Neher, W. Knoll, “Plasmon surface polariton coupling with dielectric gratings and the thermal decomposition of these dielectric gratings,” J. Appl. Phys. 82, 4172–4176 (1997).
[CrossRef]

Seshadri, S. R.

Veith, M.

K. G. Muller, M. Veith, S. Mittler-Neher, W. Knoll, “Plasmon surface polariton coupling with dielectric gratings and the thermal decomposition of these dielectric gratings,” J. Appl. Phys. 82, 4172–4176 (1997).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 624–627.

J. Appl. Phys.

K. G. Muller, M. Veith, S. Mittler-Neher, W. Knoll, “Plasmon surface polariton coupling with dielectric gratings and the thermal decomposition of these dielectric gratings,” J. Appl. Phys. 82, 4172–4176 (1997).
[CrossRef]

S. R. Seshadri, “Leaky surface polariton,” J. Appl. Phys. 59, 1187–1195 (1986).
[CrossRef]

J. Opt. Soc. Am. A

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 624–627.

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Figures (5)

Fig. 1
Fig. 1

Geometry of a dielectric film waveguide with a dielectric cover and a substrate of a free-electron metal.

Fig. 2
Fig. 2

Dispersion curves of the surface polaritons. Normalized wave number β˜=βc/ωp and normalized frequency ω˜=ω/ωp. (a) β˜=ω˜c, (b) the dispersion curve of the surface polariton supported by the plane interface between the cover dielectric and the metal, (c) β˜=ω˜f, (d) the dispersion curve of the surface polariton supported by the plane interface between the film dielectric and the metal.

Fig. 3
Fig. 3

Dispersion curve of the guided wave. (a) β˜=ω˜c, (b) β˜=ω˜f, (c) the dispersion curve of the guided wave supported by the dielectric film with a dielectric cover and a metal substrate, c=1, f=(1.642)2, ωp=1.2984×1016 rad s-1, and 2a=100 nm.

Fig. 4
Fig. 4

(a) Excitation efficiency E, (b) absorption coefficient A, (c) reflection coefficient R as functions of the angle of incidence θi. Film parameters: c=1, f=(1.642)2, s=-18, si=0.47, ω=2.9788 P rad s-1, 2a=100 nm, Λ=300 nm, ηc=0.2, and L=0.2 mm.

Fig. 5
Fig. 5

Excitation efficiency E, absorption coefficient A, and reflection coefficient R as functions of the angle of incidence θi as in Fig. 4. Except as stated, film parameters are the same as in Fig. 4. (a) 2a=100 nm, (b) 2a=90 nm, (c) 2a=80 nm, and (d) 2a=70 nm.

Equations (70)

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xc(z)=a(1+δηc cos Kz),
2x2+2z2+ω2μ00˜νHyν(x, z)=0,
xˆExν(x, z)+zˆEzν(x, z)=1iω0˜νxˆz-zˆx×Hyν(x, z).
˜s=s+δ2isi=1-ωp2ω2+δ2iωp2νω3.
Hyf(x, z)=Hyc(x, z)forx=xc(z),
Ezf(x, z)+ddzxc(z)Exf(x, z)
=Ezc(x, z)+ddzxc(z)Exc(x, z)
forx=xc(z),
Hyf(x, z)=Hys(x, z)forx=-a,
Ezf(x, z)=Ezs(x, z)forx=-a.
Hyν(x, z)=Hyν0(x, z)+δHyν1(x, z)+δ2Hyν2(x, z).
Hyc0(x, z)=AjNgj exp[-αcj(x-a)]exp(iβjz)
fora<x<,
Hyf0(x, z)=AjNgjcos[kj(x-a)+ϕcj]cos ϕcjexp(iβjz)
for-a<x<a,
Hys0(x, z)=AjNgj(-1)(j-1)cos ϕsjcos ϕcj×exp[αsj(x+a)]exp(iβjz)
for-<x<-a,
ανj=(βj2-ω2μ00ν)1/2forν=c,s,
kj=(ω2μ00f-βj2)1/2,
tan ϕνj=fανj/kjνforν=c,s.
4kja-2ϕcj-2ϕsj=2π(j-1),
Ngj=4ω0f cos2 ϕcjwβj(2a)eff,j1/2,
(2a)eff,j=2a+1/qcjαcj+1/qsjαsj,
qνj=(1/f+1/ν)βj2/ω2μ00-1forν=c, s.
P=|Aj|2.
K<βj+ω(μ00c)1/2,2K>βj+ω(μ00c)1/2,
Hyc1(x, z)=N1c{A-1ji exp[-ik-1c(x-a)]+A-1jr exp[ik-1c(x-a)]}exp[i(βj-K)z]+A+1j exp[-α+1c(x-a)]exp[i(βj+K)z]
fora<x<,
Hyf1(x, z)={B-1j exp[-ik-1f(x-a)]+C-1j exp[ik-1f(x-a)]}exp[i(βj-K)z]+{B+1j exp[α+1f(x-a)]+C+1j exp[-α+1f(x-a)]}×exp[i(βj+K)z]for-a<x<a,
Hys1(x, z)=D-1j exp[α-1s(x+a)]exp[i(βj-K)z]+D+1j exp[α+1s(x+a)]exp[i(βj+K)z]
for-<x<-a,
k-1ν=[ω2μ00ν-(βj-K)2]1/2forν=c, f,
α+1ν=[(βj+K)2-ω2μ00ν]1/2forν=c, f,
α-1s=[(βj-K)2-ω2μ00s]1/2,
α+1s=[(βj+K)2-ω2μ00s]1/2.
N1c=2ω0cwk-1c1/2.
A-1jr=CccA-1ji+CcgAgj,
Ccc=Dn,-1*/Dn,-1,
Dn,-1=fk-1ck-1fccos(2k-1fa-ϕ-1s)-i sin(2k-1fa-ϕ-1s),
tan ϕ-1s=fα-1sk-1fs,
Ccg=i2aηcNgjN1c(f-c)cNu,-1Dn,-1,
Nu,-1=αcj sin(2k-1fa-ϕ-1s)+βj(βj-K)k-1f×cos(2k-1fa-ϕ-1s).
ddzAgj(z)+αgjAgj(z)=CgcA-1ji(z)+CggAgj(z),
αgj=sif cos2 ϕsjβj(2a)eff,js2αsjαsj2+12ω2μ00s,
Cgc=Ccg
Cgg=-cos2 ϕcja2ηc24βj(2a)eff,j(f-c)c×(f-c)fc-iαcjk-1cc+βj(βj-K)fNu,-1Dn,-1+i(f-c)fcαcjα+1cc+βj(βj+K)fNu,+1Dn,+1+i2αcjβj2(f+c)c-ω2μ00f,
Nu,+1=1α+1ffsα+1sαcj-βj(βj+K)cosh(2α+1fa)+αcj-fα+1sβj(βj+K)α+1f2ssinh(2α+1fa),
Dn,+1=fα+1fα+1cc+α+1sscosh(2α+1fa)+1+f2α+1cα+1sα+1f2cssinh(2α+1fa).
Cgg+Cgg*=-CgcCgc*.
ddz|Agj(z)|2+2αgj|Agi(z)|2=|A-1ji(z)|2-|A-1jr(z)|2,
E+A+R=1,
E=|Agj(z=L)|20L|A-1ji(z)|2dz,
A=2αgj0L|Agj(z=L)|2dz0L|A-1ji(z)|2dz,
R=0L|A-1jr(z)|2dz0L|A-1ji(z)|2dz.
A-1ji(z)=Ai0 exp[i(1+h)Cgg,imz],
Cgg=Cgg,re+iCgg,im;
θi(h)=sin-1{[(βj-K)+(1+h)Cgg,im]/ω(μ00c)1/2}.
E(h)=-2Cgg,reL×|exp(ihCgg,imL)-exp[-(αgj-Cgg,re)L]|2Q,
A(h)=-4αgjCgg,reQ1-2(αgj-Cgg,re)LQ+2 exp[-(αgj-Cgg,re)L]LQ×[(αgj-Cgg,re)cos(hCgg,imL)-hCgg,im sin(hCgg,imL)]+exp[-(αgj-Cgg,re)L]L(αgj-Cgg,re)×sinh[(αgj-Cgg,re)L],
Q=(αgj-Cgg,re)2+h2Cgg,im2.
Ccc=-0.5268-i0.8500,
cωp Ccg=cωp Cgc=-0.0185-i0.0103,
cωpCgg=(-2.2318-i4.8231)×10-4.
cωpαgj=0.0014.
βj=ω(μ00c)1/2 sin θi(h=0)+K-Cgg,im.
ω(μ00c)1/2 sin θi(h=0)=βj-K.
cωpαgj=14.5×10-4,
cωpCggηc-2=-0.0056-i0.0119.
cωpCgg,re=-1.56×10-4.
cωpCgg,re=-2.0747×10-4.

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