# Simultaneous existence of amplified and attenuated surface-plasmon-polariton waves

## Abstract

The propagation of surface-plasmon-polariton (SPP) waves guided by the planar interface of (1) an isotropic metal and (2) an active uniaxial dielectric material was theoretically investigated, under the assumption that the optic axis of the uniaxial partnering material lies wholly in the interface plane. The uniaxial partnering material was conceptualized as a homogenized composite material (HCM) arising from both non-dissipative and active component materials. For a certain range of the volume fraction of the active component material, the SPP waves propagating in certain directions in the interface plane are amplified, but the SPP waves propagating in other directions are attenuated. In a critical propagation direction, the SPP wave is neither amplified nor attenuated. This critical propagation direction was found to be acutely sensitive to the composition of the HCM.

## Keywords

Surface plasmon polariton Loss Gain## Introduction

In recent times, engineered materials with exotic optical properties, as typified by metamaterials [1], have provided the backdrop for substantial developments in nanotechnology, at least at the conceptualization stage, and sometimes beyond [2]. In particular, when active and dissipative component materials are judiciously combined to form composite materials, the role of the active component material can surpass the simple act of overcoming the debilitating effects of dissipation [3, 4, 5, 6]. Homogenization theory predicts that a random mixture of aligned spheroidal particles, made from both active and dissipative materials, can function optically as a homogeneous uniaxial dielectric continuum in which plane waves propagating in certain propagation directions are amplified, but plane waves propagating in other propagation directions are attenuated [7]. In a similar vein, a stack of alternate active and dissipative layers can function as a homogeneous birefringent continuum capable of controlling plane-wave polarization states [8]. Certain mixtures of active and dissipative component materials are, in effect, homogenized composite materials (HCMs) which (a) amplify circularly polarized light of one handedness but attenuate circularly polarized light of the other handedness [9], and other mixtures are HCMs that (b) amplify incident light of one linear polarization state but attenuate incident light of the orthogonal polarization state [10].

*xy*plane at an angle \(\psi \) with respect to \(\hat{\underline{u}}_{\,x}\). Since material \({\mathcal {A}}\) is uniaxial, both (1)

*ordinary*plane waves governed by \(\epsilon _{\mathcal {A}}^\mathrm{s}\) and (2)

*extraordinary*plane waves governed jointly by \(\epsilon _{\mathcal {A}}^\mathrm{t}\) and \(\epsilon _{\mathcal {A}}^\mathrm{s}\) can propagate in it [18]. The metal occupying the half-space \(z<0\) is assigned the label \({\mathcal {B}}\). Its relative permittivity is written as \(\epsilon _{\mathcal {B}}=n^2_{\mathcal {B}}\). A schematic illustration of the regions occupied by materials \({\mathcal {A}}\) and \({\mathcal {B}}\) is provided in Fig. 1.

Let an SPP wave propagate parallel to \(\hat{\underline{u}}_{\,x}\) in the *xy* plane, with *q* being the surface wavenumber. There is no loss of generality accrued by this assumption, since the angle \(\psi \) of the optic axis is not fixed. Because of fourfold symmetry in the *xy* plane, we only consider \(\psi \in [0,\pi /2]\).

Thus, the SPP wave is amplified as it propagates if \(\mathrm{Im}\left\{ {\tilde{q}}\right\} < 0\), whereas it is attenuated if \(\mathrm{Im}\left\{ {\tilde{q}}\right\} > 0\). If \(\mathrm{Im}\left\{ {\tilde{q}}\right\} = 0\) then the SPP wave propagates without amplification or attenuation.

## Numerical studies

### Partnering materials

A non-dissipative dielectric material was chosen for component material \({\mathcal {A}}a\). Specifically, \(\epsilon _{{\mathcal {A}}a} = 4.426\), which is the relative permittivity of hafnium dioxide–yttrium oxide at a free-space wavelength of 650 nm [21]. An active dielectric material was chosen for component material \({\mathcal {A}}b\). Specifically, \(\epsilon _{{\mathcal {A}}b} = 2 - 0.1 i \), which lies comfortably within the range of relative permittivities typically encountered for active components of metamaterials at wavelengths in the visible regime. For example, across the frequency range 440–500 THz, a mixture of Rhodamine 800 and Rhodamine 6G has a relative permittivity with imaginary part in the range \(\left( -0.15, -0.02 \right) \) and real part in the range \(\left( 1.8, 2.3 \right) \), depending upon the relative concentrations and the external pumping rate [6].

### Surface-wave analysis

Now we turn to SPP waves guided by the planar interface at \(z=0\). For this purpose, the metal occupying the half-space \(z<0\) was selected to be silver. Thus, \(\epsilon _{\mathcal {B}} = -19.440 + 0.461 i\) which represents the relative permittivity of silver at a free-space wavelength of 650 nm [22]. In Fig. 3, the real and imaginary parts of the normalized wavenumber \({\tilde{q}}\), as calculated from Eq. (3) using Eq. (4), are plotted versus the propagation angle \(\psi \) for \(f_{{\mathcal {A}}b} \in \left\{ 0.110, 0.116, 0.122 \right\} \). Unlike the case for Dyakonov surface waves [23, 24], for example, SPP-wave propagation is possible for all values of \(\psi \). When \(f_{{\mathcal {A}}b} = 0.110\), the imaginary part of \({\tilde{q}}\) is positive, regardless of the value of \(\psi \). (The same is true when \(0< f_{{\mathcal {A}}b} < 0.110\), but these results are not presented here). When \(f_{{\mathcal {A}}b} = 0.122\), the imaginary part of \({\tilde{q}}\) is negative, regardless of the value of \(\psi \). (The same is true when \(0.122< f_{{\mathcal {A}}b} < 1\), but these results are not presented here). Therefore, SPP waves are attenuated for small values of \(f_{{\mathcal {A}}b}\) but amplified at large values of \(f_{{\mathcal {A}}b}\), regardless of the direction of propagation.

Most interestingly, when \(f_{{\mathcal {A}}b}\) lies between 0.110 and 0.122, the imaginary part of \({\tilde{q}}\) is positive for small values of \(\psi \in [0,\pi /2]\) but negative for large values of \(\psi \in [0,\pi /2]\). For example, in the case of \(f_{{\mathcal {A}}b} = 0.116\) represented in Fig. 3, \(\text{ Im } \left\{ {\tilde{q}}\right\} > 0\) for \(\psi < 42.7^\circ \), whereas \(\text{ Im } \left\{ {\tilde{q}}\right\} < 0\) for \(\psi > 42.7^\circ \) . Thus, at intermediate values of \(f_{{\mathcal {A}}b}\), SPP waves are attenuated for certain propagation directions but amplified for other propagation directions.

Further light is shed on this matter by considering the electric field phasor \(\underline{E} (\underline{r}) = \underline{{\mathcal {E}}} \exp \left( i \underline{k} {\,{^\bullet }\, }\underline{r} \right) \), with complex-valued amplitude vector \(\underline{{\mathcal {E}}}\) and wave vector \(\underline{k}\), in the *xz* plane for the regions above and below the interface \(z=0\). The magnitudes of the Cartesian components of \(\underline{E} (x \hat{\underline{u}}_{\,x}+ z \hat{\underline{u}}_{\,z})\) are mapped in the *xz* plane in Fig. 4 for the case where \(\psi = 25.0^\circ \) and \(f_{{\mathcal {A}}b} = 0.116\); then \({\tilde{q}} = 2.26616 + 0.00020 i\) according to Fig. 3 and the SPP wave attenuates in the \(+ \hat{\underline{u}}_{\,x}\) direction. From Fig. 4, the attenuation is much less strong in the \(+\hat{\underline{u}}_{\,x}\) direction than it is in the \(\pm \hat{\underline{u}}_{\,z}\) directions, with attenuation in the \(-\hat{\underline{u}}_{\,z}\) direction being much stronger than in the \(+\hat{\underline{u}}_{\,z}\) direction. Furthermore, it is clear from Fig. 4 that the maximum of \( | \underline{E} (x \hat{\underline{u}}_{\,x}+ z \hat{\underline{u}}_{\,z}) | \) is not concentrated at the interface \(z=0\) but instead in the region slightly above the interface. This is particularly conspicuous in the case of the *y*-directed component of \(\underline{E} (x \hat{\underline{u}}_{\,x}+ z \hat{\underline{u}}_{\,z})\).

The field maps for \(\psi = 42.7^\circ \) are provided in Fig. 5. At this value of \(\psi \) we see from Fig. 3 that \({\tilde{q}}\) is purely real valued, i.e., \({\tilde{q}} = 2.26453\). Consequently, in Fig. 5 there is no attenuation or amplification observable in the \(\hat{\underline{u}}_{\,x}\) direction.

*y*-directed component of \(\underline{E} (x \hat{\underline{u}}_{\,x}+ z \hat{\underline{u}}_{\,z})\).

The transition from \(\text{ Im } \left\{ {\tilde{q}}\right\} > 0\) to \(\text{ Im } \left\{ {\tilde{q}}\right\} < 0\) as \(\psi \) increases, as illustrated in Fig. 3 for \(f_{{\mathcal {A}}b} = 0.116\), is smooth. Consequently, there exists a critical value \(\psi _c\) of \(\psi \) which \(\text{ Im } \left\{ {\tilde{q}}\right\} = 0\). For example, \(\psi _c = 42.7^\circ \) for at \(f_{{\mathcal {A}}b} = 0.116\). At this critical propagation angle, the SPP wave propagates without amplification or attenuation. This matter is pursued in Fig. 7 wherein the critical angle \(\psi _c\) is plotted versus \(f_{{\mathcal {A}}b}\). Clearly, \(\psi _c\) is highly sensitive to changes in \(f_{{\mathcal {A}}b}\). For example, as \(f_{{\mathcal {A}}b}\) increases from 0.112 to 0.120, \(\psi _c\) decreases in an approximately linear manner from \(67.5^\circ \) to \(16.5^\circ \).

## Closing remarks

To conclude, SPP waves guided by the planar interface of an isotropic metal and a uniaxial dielectric HCM, arising from both non-dissipative and active component materials, were theoretically investigated for the case where the optic axis of the uniaxial HCM lies wholly in the interface plane. At low values of the volume fraction of the active component material, the SPP wave is attenuated, while amplification occurs at high values of the same volume fraction, regardless of the propagation direction. However, for a relatively small range of intermediate volume-fraction values, the SPP waves propagating in certain directions in the interface plane are amplified, but the SPP waves propagating in other directions are attenuated. Furthermore, in a critical propagation direction the SPP wave is neither amplified nor attenuated. The critical propagation direction is acutely sensitive to the composition of the HCM. These characteristics may be harnessed for applications involving optical sensing and/or optical communications.

Lossless solutions, arising from a combination of an isotropic dissipative material and an isotropic active material, appear in the context of PT-symmetric materials [25]. Thus, if \(\underline{\underline{\epsilon }}_{\mathcal {\,A}} = n^2_{\mathcal {A}} \underline{\underline{I}}\), then lossless surface waves *may* arise provided that \( n_{\mathcal {A}}\) and \( n_{\mathcal {B}}\) are complex conjugates. As an example, if \( n_{\mathcal {A}}=4-0.2i\) and \( n_{\mathcal {B}}=4+0.2i\), then \(\alpha _{{\mathcal {A}}1}=\alpha _{{\mathcal {A}}2}= 0.283+2.825i\), \(\alpha _{{\mathcal {B}}}=0.283-2.825i\), and \({\tilde{q}}=2.839\). However, no such symmetry relating the constitutive parameters of materials \({\mathcal {A}}\) and \({\mathcal {B}}\) is apparent when material \({\mathcal {A}}\) is a uniaxial dielectric material, the complexity of the dispersion Eq. (3) obscuring any such symmetry.

Lastly, let us note that this directional attenuation/amplification phenomenon stems from the anisotropy of the HCM: if the HCM were to be isotropic instead of anisotropic then the characteristics of the SPP wave would be the same for all propagation directions.

## Notes

### Acknowledgements

AL is grateful to the Charles Godfrey Binder Endowment at Penn State for ongoing support of his research.

## References

- 1.R.M. Walser, Electromagnetic metamaterials. Proc. SPIE
**4467**, 1 (2001). https://doi.org/10.1117/12.432921 ADSCrossRefGoogle Scholar - 2.R. Vajtai (ed.),
*Springer Handbook of Nanomaterials*(Springer, New York, 2013). https://doi.org/10.1007/978-3-642-20595-8 CrossRefGoogle Scholar - 3.S. Wuestner, A. Pusch, K.L. Tsakmakidis, J.M. Hamm, O. Hess, Overcoming losses with gain in a negative refractive index metamaterial. Phys. Rev. Lett.
**105**, 127401 (2010). https://doi.org/10.1103/PhysRevLett.105.127401 ADSCrossRefGoogle Scholar - 4.Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, X. Zhang, Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots. Appl. Phys. Lett.
**96**, 044104 (2010). https://doi.org/10.1063/1.3302409 ADSCrossRefGoogle Scholar - 5.G. Strangi, A. De Luca, S. Ravaine, M. Ferrie, R. Bartolino, Gain induced optical transparency in metamaterials. Appl. Phys. Lett.
**98**, 251912 (2011). https://doi.org/10.1063/1.3599566 ADSCrossRefGoogle Scholar - 6.L. Sun, X. Yang, J. Gao, Loss-compensated broadband epsilon-near-zero metamaterials with gain media. Appl. Phys. Lett.
**103**, 201109 (2013). https://doi.org/10.1063/1.4831768 ADSCrossRefGoogle Scholar - 7.T.G. Mackay, A. Lakhtakia, Dynamically controllable anisotropic metamaterials with simultaneous attenuation and amplification. Phys. Rev. A
**92**, 053847 (2015). https://doi.org/10.1103/PhysRevA.92.053847 ADSCrossRefGoogle Scholar - 8.A. Cerjan, S. Fan, Achieving arbitrary control over pairs of polarization states using complex birefringent metamaterials. Phys. Rev. Lett.
**118**, 253902 (2017). https://doi.org/10.1103/PhysRevLett.118.253902 ADSCrossRefGoogle Scholar - 9.T.G. Mackay, A. Lakhtakia, Simultaneous amplification and attenuation in isotropic chiral materials. J. Opt. (UK)
**18**, 055104 (2016). https://doi.org/10.1088/2040-8978/18/5/055104 ADSCrossRefGoogle Scholar - 10.T.G. Mackay, A. Lakhtakia, Polarization-state-dependent attenuation and amplification in a columnar thin film, J. Opt. (UK)
**19**, 12LT01 (2017). https://doi.org/10.1088/2040-8986/aa9127, erratum**20**, 019501 (2018). https://doi.org/10.1088/2040-8986/aa9ec3 - 11.J.A. Polo Jr., T.G. Mackay, A. Lakhtakia,
*Electromagnetic Surface Waves: A Modern Perspective*(Elsevier, Amsterdam, 2013)Google Scholar - 12.J. Homola (ed.),
*Surface Plasmon Resonance Based Sensors*(Springer, Heidelberg, 2006). https://doi.org/10.1007/b100321 CrossRefGoogle Scholar - 13.I. Abdulhalim, M. Zourob, A. Lakhtakia, Surface plasmon resonance for biosensing: a mini-review. Electromagnetics
**28**, 214–242 (2008). https://doi.org/10.1080/02726340801921650 CrossRefGoogle Scholar - 14.G. Borstel, H.J. Falge, Surface phonon–polaritons, in
*Electromagnetic Surface Modes, Chapter 6*, ed. by A.D. Boardman (Wiley, New York, 1982)Google Scholar - 15.R.F. Wallis, Surface magnetoplasmons on semiconductors, in
*Electromagnetic Surface Modes, Chapter 15*, ed. by A.D. Boardman (Wiley, New York, 1982)Google Scholar - 16.I. Abdulhalim, Surface plasmon TE and TM waves at the anisotropic film-metal interface. J. Opt. A Pure Appl. Opt.
**11**, 015002 (2009). https://doi.org/10.1088/1464-4258/11/1/015002 ADSCrossRefGoogle Scholar - 17.H.C. Chen,
*Theory of Electromagnetic Waves: A Coordinate-Free Approach, Chapter 1*(McGraw-Hill, New York, 1983)Google Scholar - 18.M. Born, E. Wolf,
*Principles of Optics 7th Expanded Edition, Chapter 15*(Cambridge University Press, Cambridge, 1999)Google Scholar - 19.W.S. Weiglhofer, A. Lakhtakia, J.C. Monzon, Maxwell–Garnett model for composites of electrically small uniaxial objects. Microw. Opt. Technol. Lett.
**6**, 681–684 (1993). https://doi.org/10.1002/mop.4650061205 ADSCrossRefGoogle Scholar - 20.T.G. Mackay, Towards metamaterials with giant dielectric anisotropy via homogenization: an analytical study. Photon. Nanostruct. Fundam. Appl.
**13**, 8–19 (2015). https://doi.org/10.1016/j.photonics.2014.10.005 ADSCrossRefGoogle Scholar - 21.D.L. Wood, K. Nassau, T.Y. Kometani, D.L. Nash, Optical properties of cubic hafnia stabilized with yttria. Appl. Opt.
**29**, 604–607 (1990). https://doi.org/10.1364/AO.29.000604 ADSCrossRefGoogle Scholar - 22.P.B. Johnson, R.W. Christy, Optical constants of the noble metals. Phys. Rev. B
**6**, 4370–4379 (1972). https://doi.org/10.1103/PhysRevB.6.4370 ADSCrossRefGoogle Scholar - 23.M.I. D’yakonov, New type of electromagnetic wave propagating at an interface. Sov. Phys. JETP
**67**, 714–716 (1988)Google Scholar - 24.O. Takayama, L. Crasovan, D. Artigas, L. Torner, Observation of Dyakonov surface waves. Phys. Rev. Lett.
**102**, 043903 (2009). https://doi.org/10.1103/PhysRevLett.102.043903 ADSCrossRefGoogle Scholar - 25.A.A. Zyablovsky, A.P. Vinogradov, A.A. Pukhov, A.V. Dorofeenko, A.A. Lisyansky, PT-symmetry in optics. Physics-Uspekhi
**57**, 1063–1082 (2014). https://doi.org/10.3367/UFNr.0184.201411b.1177 ADSCrossRefGoogle Scholar

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