Plasmonics (2014) 9:409–413 DOI 10.1007/s11468-013-9637-4
Plasmonic Slot Waveguides with Core Nonlinearity Mohamed A. Swillam · Sherif A. Tawfik
Received: 21 July 2013 / Accepted: 7 November 2013 / Published online: 22 November 2013 © Springer Science+Business Media New York 2013
Abstract We analytically study the interplay between group velocity dispersion and material dispersion due to femtosecond ultrafast pulse inside plasmonic slot waveguides with nonlinear dielectric core. The analytic investigation of the role of the core nonlinearity on pulse propagation has been investigated. Interestingly, our model shows that the focusing and defocusing effects of the material can be revered if the material is confined inside the core of a plasmonic slot. We confirm our analytical results with nonlinear finite difference time domain (FDTD) simulations. Keywords Plasmon propagation · Kerr nonlinearity · Slot waveguide
Introduction Surface plasmon polariton (SPP) waveguides have attracted intense research efforts in the last decade due to the great variety of applications owing to their unique ability to guide light at subwavelength scales, as they make it possible to overcome the diffraction limit of light and confine light into M. A. Swillam · S. A. Tawfik () The American University in Cairo, New Cairo, Cairo Governorate, Egypt e-mail:
[email protected] M. A. Swillam e-mail:
[email protected] M. A. Swillam University of Toronto, Toronto, ON, Canada Present Address: S. A. Tawfik School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia
a width of few nanometers [1]. The ability to guide the optical field on the metal surface or between two metal surfaces provides distinctive and unique sensing capabilities for these waveguides [2, 3], as well as enhancing nonlinear effects such as second harmonic generation [4]. These SPP waveguides are utilized in a number of chemical and biological sensing applications as they provide compactness and high sensitivity [3]. Introducing three-layer system helps to increase substantially the nonlinear effect due to the field intensity of the field in the core between the two metal surface interfaces [2]. In general, structures based on metal– insulator–metal have been subject to many research efforts [3, 5–8]. The problem of spatial propagation of plasmonic pulses in media that incorporate nonlinearity has attracted some attention in the literature [9–11], especially for the analytical characterization of the effect of nonlinearity on nanofocusing in terms of the nonlinear Schrodinger equation (NSE) [2, 12, 13]. This work is a detailed analytical study for the full spatiotemporal pulse propagation. That is, unlike the purely spacial approach in, e.g., [2], we are concerned with the temporal, rather than spacial, evolution of the pulse shape along the plasmon propagation direction. We demonstrate the possibility of simplifying the time-dependent Maxwell problem into a spatiotemporal NSE incorporating the nonlinearity of the dielectric core by assuming an effective transverse electric fields acting on the structure. We test the validity of this model by performing extensive simulations on a range of system parameters. We start by describing our model in Section “Analytical Model”. Our model is compared with the nonlinear finite difference time domain (FDTD) simulations in Section “Application and Numerical Verification”. In this section also, the different model is tested and the effect of
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temporal focusing and defocusing has been demonstrated. Finally, the conclusion is given in Section “Conclusion”.
Analytical Model The objective of this work is to examine the process of pulse reshaping in a 2D plasmonic slot waveguide [14] (Fig. 1) . The nonlinear effect due to single plasmonic interface has been recently demonstrated in [15]. In this work, the temporal shaping of 2D pulse entering into the Kerr nonlinear core region is presented. Hence, the generalized NSE is derived for the first time as an extension to the standard 1D NSE derived in [16, 17]. Subsequently, a procedure similar to that in [1] is followed to reduce the two component systems of coupled equations into a single equation, and integrating out the x-axis dependence. Finally, we add the temporal dispersion terms following the method in [17]. We begin with the form of the TM wave [1] E = (A(z, t)Ex0 (x)x + iB(z, t)Ez0 (x)z)ei(ω0 t −β0 z)
(1)
and substitute it into the Maxwell equation ∇ × (∇ × E) = −μ0 E − μ0 PN L Given μ0 0 = 1/c2 , and PSW (x) is given by d + α |E|2 0d d d Sd m (11)
(3)
(4)
2 i Ex0 2β0 + Ex0 ∂x Ez0 − Ez0 ∂x Ex0 Az − l − β02 |E0 |2 A 2 + 2γ (x)i |E0 |2 − β02 Ez0 + β0 (Ex0 ∂x Ez0 − Ez0 ∂x Ex0 ) − Ez0 ∂xx Ez0 A + α(x)|A|2 |E0 |4 A = 0 (12)
− ∂zz Ex + ∂zx Ez = μ0 Ex in addition to Gauss’s law (∇ · E = 0) which is naturally satisfied. Then, assuming that the envelop functions A and B are slowly varying with time and ignoring the y-axis dependence, − Ex0 Azz − 2iβ0 Az − β02 A + ∂x Ez0 (β0 B + iBz ) 2 2 Ex0 A = −k02 l (x) + iγ (x) + α(x) |A|2 Ex0 + |B|2 Ez0
(5)
− ∂xx Ez0 B − ∂x Ex0 (β0 A + iAz ) 2 2 Ez0 B + |B|2 Ez0 = −k02 l (x) + iγ (x) + α(x) |A|2 Ex0
(6)
Fig. 1 Schematic of the 2D plasmonic slot waveguide structure
Note that β0 is the eigenmode of the waveguide structure without Kerr nonlinearity which can obtained from the following Eigen equation [5]: 2 2 2 tanh(dSd ) Sm d + Sd2 m + 2d m Sd Sm = 0 (8)
The objective is to arrive at an effective formulation across the x-axis, so we multiply Eq. 5 by Ex0 and Eq. 6 by Ez0 , then summing both equations and assuming that A ≈ B and ∂zz A = 0, we get
The propagation equations are then given by −∂xx Ez + ∂zx Ex = μ0 Ez
where k0 = ω0 /c, l is the linear part of the dielectric function, d 0
