Abstract

We formulated a boundary-value problem for the excitation of Tamm waves guided by the periodically corrugated interface of a homogeneous material and a rugate filter. Both partnering materials are isotropic and dielectric. After solving the problem using the rigorous coupled-wave approach, the total transmittance, total reflectance, and absorptance were calculated as functions of the angle of incidence of a plane wave that could be either p or s polarized. The excitation of a Tamm wave was inferred for every absorptance peak that is independent of the thicknesses of both partnering materials. Therefore, multiple Tamm waves can be simultaneously excited using a converging or diverging incident beam instead of an incident plane wave. Even though the absorptance peaks are very low when a Tamm wave is excited, the corresponding spikes in the total-reflectance curves are much larger and guarantee experimental detection, as well as application to optical sensing.

© 2012 Optical Society of America

Full Article  |  PDF Article

Corrections

Drew Patrick Pulsifer, Muhammad Faryad, and Akhlesh Lakhtakia, "Grating-coupled excitation of Tamm waves: erratum," J. Opt. Soc. Am. B 30, 177-177 (2013)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-30-1-177

References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011 (4)

2007 (1)

A. Namdar, “Tamm states in one-dimensional photonic crystals containing left-handed materials,” Opt. Commun. 278, 194–198 (2007).
[CrossRef]

2006 (1)

J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A Pure Appl. Opt. 8, 630–638 (2006).
[CrossRef]

2005 (2)

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface bodes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72, 233102 (2005).
[CrossRef]

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Wilson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing and other techniques,” Chem. Rev. 105, 1171–1196 (2005).
[CrossRef]

2004 (1)

F. Wang, M. W. Horn, and A. Lakhtakia, “Rigorous electromagnetic modeling of near-field phase-shifting contact lithography,” Microelectron. Eng. 71, 34–53 (2004).
[CrossRef]

1999 (1)

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

1997 (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1978 (1)

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105(1978).
[CrossRef]

1968 (1)

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).

1959 (1)

T. Turbadar, “Complete absorption of light by thin metal films,” Proc. Phys. Soc. London 73, 40–44 (1959).
[CrossRef]

1932 (1)

I. Tamm, “Über eine mögliche Art der Elektronenbindung an Kristalloberflächen,” Z. Phys. A 76, 849–850 (1932).

Arie, A.

Bovard, B. G.

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, 1972).

Chateau, N.

Cho, A. Y.

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105(1978).
[CrossRef]

Dolev, I.

Dudley, D. G.

D. G. Dudley, Mathematical Foundations for Electromagnetic Theory (IEEE, 1994).

Faryad, M.

Friesem, A. A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Gates, B. D.

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Wilson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing and other techniques,” Chem. Rev. 105, 1171–1196 (2005).
[CrossRef]

Grann, E. B.

Horn, M. W.

F. Wang, M. W. Horn, and A. Lakhtakia, “Rigorous electromagnetic modeling of near-field phase-shifting contact lithography,” Microelectron. Eng. 71, 34–53 (2004).
[CrossRef]

Hugonin, J.-P.

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, 1972).

Kavokin, A. V.

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface bodes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72, 233102 (2005).
[CrossRef]

Kretschmann, E.

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).

Lakhtakia, A.

M. Faryad and A. Lakhtakia, “Grating-coupled excitation of multiple surface plasmon-polariton waves,” Phys. Rev. A 84, 033852 (2011).
[CrossRef]

J. A. Polo and A. Lakhtakia, “Surface electromagnetic waves: a review,” Laser Photon. Rev. 5, 234–246 (2011).
[CrossRef]

H. Maab, M. Faryad, and A. Lakhtakia, “Surface electromagnetic waves supported by the interface of two semi-infinite rugate filters with sinusoidal refractive-index profiles,” J. Opt. Soc. Am. B 28, 1204–1212 (2011).
[CrossRef]

F. Wang, M. W. Horn, and A. Lakhtakia, “Rigorous electromagnetic modeling of near-field phase-shifting contact lithography,” Microelectron. Eng. 71, 34–53 (2004).
[CrossRef]

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).

Maab, H.

Mackay, T. G.

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).

Malpuech, G.

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface bodes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72, 233102 (2005).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, 1991).

Martorell, J.

J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A Pure Appl. Opt. 8, 630–638 (2006).
[CrossRef]

May, M. S.

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

Moharam, M. G.

Morozov, G. V.

J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A Pure Appl. Opt. 8, 630–638 (2006).
[CrossRef]

Namdar, A.

A. Namdar, “Tamm states in one-dimensional photonic crystals containing left-handed materials,” Opt. Commun. 278, 194–198 (2007).
[CrossRef]

Polo, J. A.

J. A. Polo and A. Lakhtakia, “Surface electromagnetic waves: a review,” Laser Photon. Rev. 5, 234–246 (2011).
[CrossRef]

Pommet, D. A.

Porat, G.

Raether, H.

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).

Robertson, W. M.

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

Rosenblatt, D.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Ryan, D.

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Wilson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing and other techniques,” Chem. Rev. 105, 1171–1196 (2005).
[CrossRef]

Sharon, A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Shelykh, I. A.

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface bodes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72, 233102 (2005).
[CrossRef]

Sprung, D. W. L.

J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A Pure Appl. Opt. 8, 630–638 (2006).
[CrossRef]

Starzhinskii, V. M.

V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Wiley, 1975).

Stewart, M.

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Wilson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing and other techniques,” Chem. Rev. 105, 1171–1196 (2005).
[CrossRef]

Tamm, I.

I. Tamm, “Über eine mögliche Art der Elektronenbindung an Kristalloberflächen,” Z. Phys. A 76, 849–850 (1932).

Turbadar, T.

T. Turbadar, “Complete absorption of light by thin metal films,” Proc. Phys. Soc. London 73, 40–44 (1959).
[CrossRef]

Volodarsky, M.

Wang, F.

F. Wang, M. W. Horn, and A. Lakhtakia, “Rigorous electromagnetic modeling of near-field phase-shifting contact lithography,” Microelectron. Eng. 71, 34–53 (2004).
[CrossRef]

Whitesides, G. M.

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Wilson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing and other techniques,” Chem. Rev. 105, 1171–1196 (2005).
[CrossRef]

Wilson, C. G.

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Wilson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing and other techniques,” Chem. Rev. 105, 1171–1196 (2005).
[CrossRef]

Xu, Q.

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Wilson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing and other techniques,” Chem. Rev. 105, 1171–1196 (2005).
[CrossRef]

Yakubovich, V. A.

V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Wiley, 1975).

Yariv, A.

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105(1978).
[CrossRef]

Yeh, P.

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105(1978).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105(1978).
[CrossRef]

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

Chem. Rev. (1)

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Wilson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing and other techniques,” Chem. Rev. 105, 1171–1196 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

J. Martorell, D. W. L. Sprung, and G. V. Morozov, “Surface TE waves on 1D photonic crystals,” J. Opt. A Pure Appl. Opt. 8, 630–638 (2006).
[CrossRef]

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

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

Laser Photon. Rev. (1)

J. A. Polo and A. Lakhtakia, “Surface electromagnetic waves: a review,” Laser Photon. Rev. 5, 234–246 (2011).
[CrossRef]

Microelectron. Eng. (1)

F. Wang, M. W. Horn, and A. Lakhtakia, “Rigorous electromagnetic modeling of near-field phase-shifting contact lithography,” Microelectron. Eng. 71, 34–53 (2004).
[CrossRef]

Opt. Commun. (1)

A. Namdar, “Tamm states in one-dimensional photonic crystals containing left-handed materials,” Opt. Commun. 278, 194–198 (2007).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

M. Faryad and A. Lakhtakia, “Grating-coupled excitation of multiple surface plasmon-polariton waves,” Phys. Rev. A 84, 033852 (2011).
[CrossRef]

Phys. Rev. B (1)

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface bodes at the boundary between two periodic dielectric structures,” Phys. Rev. B 72, 233102 (2005).
[CrossRef]

Proc. Phys. Soc. London (1)

T. Turbadar, “Complete absorption of light by thin metal films,” Proc. Phys. Soc. London 73, 40–44 (1959).
[CrossRef]

Z. Naturforsch. A (1)

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).

Z. Phys. A (1)

I. Tamm, “Über eine mögliche Art der Elektronenbindung an Kristalloberflächen,” Z. Phys. A 76, 849–850 (1932).

Other (6)

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, 1972).

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, 1991).

V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Wiley, 1975).

J. Homola, ed., Surface Plasmon Resonance Based Sensors (Springer, 2006).

D. G. Dudley, Mathematical Foundations for Electromagnetic Theory (IEEE, 1994).

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Figures (10)

Fig. 1.
Fig. 1.

Schematic of the boundary-value problem involving the periodically corrugated interface of a homogeneous dielectric material and a periodic multilayered isotropic dielectric material. The incident field is a plane wave, whereas both the reflected and transmitted fields are discrete angular spectrums of plane waves. The label “0” is attached to the specular components of the reflected and transmitted plane waves, with the nonspecular components being labeled “±1,” etc.

Fig. 2.
Fig. 2.

Absorptance Ap versus the incidence angle θ when λ0=633nm, na=1.45, nb=2.32, nd=1.515(1+iδ), δ=104, Ω=λ0, Ld=λ0, and Lg=50nm. The red dashed curves are for d1=4Ω, the blue dashed–dotted curves for d1=6Ω, and the solid green curves for d1=8Ω. Panel (a) shows the plots of Ap over a wider angular domain, and panels (b)–(e) show the parts of the plots in panel (a) around the peaks that represent the excitation of p-polarized Tamm waves.

Fig. 3.
Fig. 3.

Same as Fig. 2, except that As is plotted versus θ.

Fig. 4.
Fig. 4.

Absorptance Ap versus the incidence angle θ when λ0=633nm, na=1.45, nb=2.32, nd=1.515(1+iδ), δ=104, Ω=λ0, Lg=50nm, and d1=6Ω. The red dashed curves are for Ld=λ0, the blue dashed–dotted curves for Ld=2λ0, and the solid green curves for Ld=3λ0. Only those parts of the plots are shown that have Ap peaks representing the excitation of p-polarized Tamm waves.

Fig. 5.
Fig. 5.

Same as Fig. 4, except that As is plotted versus θ.

Fig. 6.
Fig. 6.

Variation of Px(x,z) when x=3L/4, λ0=633nm, na=1.45, nb=2.32, nd=1.515(1+iδ), δ=104, Ω=λ0, Ld=2λ0, Lg=50nm, and d1=6Ω. (a) The incident plane wave is p polarized and θ=11.976° (Tamm wave), (b) the incident plane wave is s polarized and θ=41.335° (Tamm wave), and (c) the incident plane wave is s polarized and θ=17.381° (waveguide mode). The solid green vertical line is the interface of the two partnering materials.

Fig. 7.
Fig. 7.

(a) Total transmittance Tp, (b) total reflectance Rp, and (c) absorptance Ap versus θ when λ0=633nm, na=1.45, nb=2.32, nd=1.515(1+iδ), δ=104, Ω=λ0, Ld=λ0, Lg=50nm, and d1=6Ω. The red spikes identify the Tamm waves.

Fig. 8.
Fig. 8.

(a) Total transmittance Ts, (b) total reflectance Rs, and (c) absorptance As versus θ when λ0=633nm, na=1.45, nb=2.32, nd=1.515(1+iδ), δ=104, Ω=λ0, Ld=λ0, Lg=50nm, and d1=6Ω. The red spikes identify the Tamm waves.

Fig. 9.
Fig. 9.

Absorptance Ap versus the incidence angle θ when λ0=633nm, na=1.45, nb=2.32, nd=nd(1+iδ), δ=104, Ω=λ0, Ld=λ0, Lg=50nm, and d1=6Ω. The red dashed curve is for nd=1, the blue dashed–dotted curve for nd=1.515, and the solid green curve for nd=1.7. Panels (a)–(d) show closeups of the plots around the Ap peaks representing the excitation of Tamm waves.

Fig. 10.
Fig. 10.

Same as Fig. 9, except that As is plotted versus θ.

Tables (1)

Tables Icon

Table 1. Angular Location θ and Relative Wavenumber kx(1)/k0 of the Floquet Harmonic of Order n=1 for All Four Ap Peaks in Fig. 2 and All Four As Peaks in Fig. 3 That Are Independent of the Thickness d1 Beyond Some Threshold Valuea

Equations (25)

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

ϵrug(z)=[(nb+na2)+(nbna2)sin(πd2zΩ)]2,
ϵg(x,z)={ϵd[ϵdϵrug(z)]U(d2zg(x)),x(0,L1)ϵrug(z),x(L1,L),
g(x)=(d2d1)sin(πxL1),L1(0,L),
U(ζ)={1,ζ0,0,ζ<0.
Einc(r)=nZ(as(n)u^y+ap(n)pn+)exp[i(kx(n)x+kz(n)z)],z0,
Eref(r)=nZ(rs(n)u^y+rp(n)pn)exp[i(kx(n)xkz(n)z)],z0,
Etr(r)=nZ(ts(n)u^y+rp(n)pn+)exp{i[kx(n)x+kz(n)(zd3)]},zd3.
kx(n)=k0sinθ+2πn/L
kz(n)=+k02(kx(n))2,
pn±=kz(n)k0u^x+kx(n)k0u^z.
ϵ(x,z)=nZϵ(n)(z)exp(i2πnx/L),z[0,d3],
ϵ(0)(z)={ϵrug(z),z[0,d1]1L0Lϵg(x,z)dx,z(d1,d2)ϵd,z[d2,d3],
ϵ(n)(z)={1L0Lϵg(x,z)exp(i2πnx/L)dx,z(d1,d2)0,otherwise;n0.
E(r)=nZE(n)(z)exp(ikx(n)x),z[0,d3],
ddz[f(z)]=i[(z)]·[f(z)],z(0,d3),
[f(z)]=[[Ex(z)]T,[Ey(z)]T,ηo[Hx(z)]T,ηo[Hy(z)]T]T,
[(z)]=[[][][]k0[]1k0[x]·[ϵ̳(z)]1·[x][][]k0[][][]1k0[x]2k0[ϵ̳(z)][][]k0[ϵ̳(z)][][][]].
[Xσ(z)]=[Xσ(Nt)(z),Xσ(Nt)(z),,Xσ(0)(z),,Xσ(Nt1)(z),Xσ(Nt)(z)]T,
Rp=n=NtNt|rp(n)|2Re(kz(n)kz(0))
Tp=n=NtNt|tp(n)|2Re(kz(n)kz(0))
Rs=n=NtNt|rs(n)|2Re(kz(n)kz(0)),
Ts=n=NtNt|ts(n)|2Re(kz(n)kz(0)),
Ap=1RpTp,
As=1RsTs,
P(x,z)=12Re[E(x,z)×H*(x,z)].

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