IOP PUBLISHING
JOURNAL OF OPTICS
J. Opt. 13 (2011) 075101 (4pp)
doi:10.1088/2040-8978/13/7/075101
Rugate-filter-guided propagation of multiple Fano waves Muhammad Faryad1 , Husnul Maab1,2,3 and Akhlesh Lakhtakia1,4 1
Nanoengineered Metamaterials Group (NanoMM), Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802-6812, USA 2 Department of Electronics, Quaid-i-Azam University, Islamabad 43210, Pakistan 3 Faculty of Electronics Engineering, GIK Institute of Engineering Sciences and Technology, Topi 23640, Pakistan E-mail:
[email protected]
Received 20 February 2011, accepted for publication 31 March 2011 Published 28 April 2011 Online at stacks.iop.org/JOpt/13/075101 Abstract The canonical boundary-value problem of surface-wave propagation guided by the planar interface of a rugate filter and a homogeneous medium with negative permittivity was solved numerically. Multiple Fano waves with differing phase speed and linear polarization state are guided by the interface, even when the magnitude of the permittivity of the negative-permittivity partnering medium is less than the minimum value of the permittivity in the rugate filter. The concepts of Fano waves and Tamm waves can be unified. Keywords: Fano wave, rugate filter, surface plasmon-polariton, Tamm wave
(Some figures in this article are in colour only in the electronic version)
permittivity can guide multiple surface plasmon–polariton waves at a specific frequency in the optical regime, we set out to investigate whether multiple Fano waves can also be supported by the interface of a positive-permittivity rugate filter and a negative-permittivity dielectric medium, both of which are isotropic and lossless. The theoretical formulation for tackling the underlying canonical boundary-value problem is exactly the same as in the predecessor paper [5], for which reason it is not repeated in this communication. We proceed directly to the presentation and discussion of illustrative numerical results in section 2. Concluding remarks are provided in section 3.
1. Introduction Fano waves are surface electromagnetic waves guided by the planar interface of two isotropic, lossless, homogeneous media with relative permittivities of opposite signs [1–3]. Like surface plasmon–polariton waves which arise when the negative-permittivity partnering medium is not lossless [3, 4], Fano waves have the following properties: (i) unattenuated propagation along the interface with a phase speed smaller than the phase speed of light in the positivepermittivity partnering medium; (ii) exponentially decaying field amplitudes on both sides of the interface; and (iii) a p polarization state.
2. Numerical results and discussion
The magnitude of the permittivity of the negative-permittivity partnering medium must exceed that of the permittivity of the positive-permittivity partnering medium. At a given frequency, at most one Fano wave can propagate in a specified direction along the interface. As we have shown elsewhere [5] that the planar interface of a metal and an isotropic, lossless, periodically nonhomogeneous, dielectric medium (a rugate filter) with positive
Let the half-space z < 0 be occupied by an isotropic and homogeneous medium with relative permittivity m < 0. The region z > 0 is occupied by a semi-infinite rugate filter with relative permittivity
r (z) =
4 Author to whom any correspondence should be addressed.
2040-8978/11/075101+04$33.00
1
2 nb − na z nb + na + sin π , 2 2
z > 0, (1)
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J. Opt. 13 (2011) 075101
M Faryad et al
Figure 1. Variation of the relative permittivity along the z axis for n a = 1.45, n b = 2.32, and m = −2. Although the semi-infinite rugate filter depicted here is a continuously non-homogeneous medium, it can also be piecewise homogeneous. Figure 2. Relative wavenumber κ/k0 versus m ∈ [−6, 0] for Fano-wave propagation when = λ0 = 633 nm, n a = 1.45, and n b = 2.32. The red circles represent s-polarized Fano waves, while the black triangles represent p-polarized Fano waves. The gap in one of the solution branches appears to be a numerical artifact.
where n b > n a > 0 and 2 is the period. The variation of the relative permittivity along the z axis is shown in figure 1. An exp(−iωt) time dependence is implicit, with ω denoting the angular frequency. Without loss of generality, surface-wave propagation is taken to occur along the x axis with an exp(iκ x) dependence. Field amplitudes must decay as z → ±∞. The free-space wavenumber and the free-space wavelength are √ denoted by k0 = ω 0 μ0 and λ0 = 2π/k0 , respectively, with μ0 and 0 being the permeability and permittivity of free space. Before proceeding to the results for the current boundaryvalue problem, let us consider the case when n b = n a , so that the rugate filter is replaced by a homogeneous medium. A ppolarized surface wave can then propagate with wavenumber
κ|n b =n a = k0 n a m /(n 2a + m ),
(2)
m < −n 2a .
(3)
Without that periodicity, only one Fano wave is possible, that too if −m is sufficiently large. The exponential decay rate normal to the interface in the half-space z < 0 is proportional to Im(αm ) =
Im( m k02 − κ 2 ) > 0. Since m < 0, αm is purely imaginary for real-valued κ , signifying a very high attenuation in the halfspace z < 0. Moreover, for fixed m , the attenuation rate in the half-space z < 0 is higher for a Fano wave with a higher κ . For SPP-wave propagation, αm is generally a complex number because both m and κ are complex-valued [5]. Spatial profiles of the magnitudes of the Cartesian components of the electric and magnetic field phasors along a line normal to the interface are given in figure 3 for two Fano waves, one p-polarized and the other s-polarized, when m = −6. The figure shows relatively strong localization of the p-polarized wave to the plane z = 0, as compared to that for the s-polarized wave, due to the higher value of κ for the former wave. Seven other Fano waves are also possible, per figure 2, and their spatial profiles are qualitatively similar to the ones presented. The spatial profiles in figure 4 are for two of the eight Fano waves possible when m = 0. One of the two spatial profiles is for a p-polarized Fano wave, the other for an s-polarized Fano wave. Since m = 0, all components of the magnetic field phasor vanish in the half-space z < 0 for the p-polarized Fano wave, which means that energy transport occurs only in the rugate filter. As noted previously, when n b = n a > 0, Fano-wave propagation can occur only if m is sufficiently negative; surface-wave propagation is impossible if m > 0. However, when n b > n a , surface-wave propagation can occur even if m is positive. Solutions of the dispersion equation are presented in figure 5 for the same parameters as for figure 2, except that m ∈ [0, 2]. Such surface waves have to be classified as Tamm waves [7]. Just a change in the sign of the relative permittivity of the homogeneous medium occupying the half-space
but only if
The situation changes dramatically when n b and n a are dissimilar—with r (z) > 0 ∀z > 0, of course. This becomes evident from figure 2, which contains solutions of the dispersion equation for surface-wave propagation [5] when m ∈ [−6, 0], = λ0 = 633 nm, n a = 1.45, and n b = 2.32. The minimum and maximum indices of refraction of the rugate filter were fixed from an example provided by Baumeister [6]. Nine solutions of the dispersion equation were found for m −n 2b and up to eight solutions were found for m ∈ (−n 2b , 0]; indeed, solutions exist even when m > −n 2a . Since κ/k0 is real-valued for all the solutions found, the wave propagation is lossless. Some solutions possess the p polarization state, the others being s-polarized. Clearly, the presence of periodic non-homogeneity in the positivepermittivity dielectric partnering medium has: (i) engendered multiple Fano waves of two different linear polarization states; and (ii) permitted Fano-wave propagation at low values of −m , even as low as 0. 2
J. Opt. 13 (2011) 075101
M Faryad et al
Figure 3. Variations of the magnitudes of the Cartesian components of electric and magnetic field phasors with z . The x -, y -, and z -directed components are represented by solid red, blue dashed, and black chain-dashed lines, respectively, for m = −6. Left: κ/k0 = 3.1283 and the p polarization state. Right: κ/k0 = 1.9885 and the s polarization state.
Figure 4. Same as figure 3, except that m = 0. Left: κ/k0 = 1.7145 and the p polarization state. Right: κ/k0 = 1.5161 and the s polarization state.
z < 0 leads to the propagation of Tamm/Fano waves instead of Fano/Tamm waves, the periodically non-homogeneous medium in the other half-space remaining unchanged. Thus, the hitherto different concepts of Fano waves and Tamm waves can be conceptually unified. Parenthetically, we also note that Fano waves are replaced by surface plasmon–polariton waves [5] if the homogeneous medium in the region z < 0 with Re(m ) < 0 is also dissipative.
is negative and (ii) the magnitude of the permittivity of the negative-permittivity partnering medium is sufficiently high. We have shown here that multiple Fano waves—with differing phase speed and polarization state—can propagate if the positive-permittivity partnering medium is periodically nonhomogeneous normal to the interface. No restriction exists on the magnitude of the permittivity of the negative-permittivity partnering medium. The additional Fano waves, whose creation can be attributed to the periodic non-homogeneity of the medium occupying the half-space z > 0, are not waveguide modes [8]. Furthermore, Fano waves transmute into Tamm waves when both partnering media have positive permittivities. These findings buttress the hypothesis that
3. Concluding remarks A surface wave, called a Fano wave, can be guided by the interface of two isotropic, homogeneous, lossless, dielectric media only (i) if the product of their relative permittivities 3
J. Opt. 13 (2011) 075101
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AL is grateful to Charles Godfrey Binder Endowment at the Pennsylvania State University for partial support of this work.
References [1] Fano U 1941 The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves) J. Opt. Soc. Am. 31 213–22 [2] Datsko V N and Kopylov A A 2008 On surface electromagnetic waves Phys. Usp. 51 101–2 [3] Agranovich V M 1978 Surface electromagnetic waves and Raman scattering on surface polaritons Sov. Phys. Usp. 21 995–7 [4] Pitarke J M, Silkin V M, Chulkov E V and Echenique P M 2007 Theory of surface plasmons and surface-plasmon polaritons Rep. Prog. Phys. 70 1–87 [5] Faryad M and Lakhtakia A 2010 On surface plasmon–polariton waves guided by the interface of a metal and a rugate filter with sinusoidal refractive-index profile J. Opt. Soc. Am. B 27 2218–23 [6] Baumeister P W 2004 Optical Coating Technology (Bellingham, WA: SPIE Optical Engineering Press) sec 5.3.3.2 [7] Polo J A Jr and Lakhtakia A 2011 Surface electromagnetic waves: a review Laser Photon. Rev. 5 234–46 [8] Faryad M and Lakhtakia A 2011 Propagation of surface waves and waveguide modes guided by a dielectric slab inserted in a sculptured nematic thin film Phys. Rev. A 83 013814
Figure 5. Same as figure 2, except that m ∈ [0, 2]. The waves represented by these solutions have to be classified as Tamm waves [7].
periodic non-homogeneity normal to the interface results in the possibility of multiple surface waves [5].
Acknowledgments MF thanks the Trustees of the Pennsylvania State University, HM thanks the Higher Education Commission of Pakistan, and
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