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1.INTRODUCTIONThere is a current need for communications systems to be smaller, faster, increased bandwidth, and with more robustness. There are fundamental limitations on electronic and optical technologies such as material fabrication and diffraction effects. Nanotechnology research can address these issues in particular plasmonics. Optical fields coupled to electron oscillations that are limited at a metal/dielectric interface are called surface plasmons (SPs). These fields are squeezed light that are confined to the subwavelength. The SPs exist in other structures besides waveguides such as triangular grooves, slot waveguides, spheres, cones, and arrays[3,5,22,25]. SPs contribute to electric field enhanced in Surface Enhanced Raman Scattering (SERS) [3]. The current research can lead to development of SP based components, devices, and circuits such as waveguides, surface enchanced Raman scattering (SERS) sensors, nanoantennas, resonator structures, integrated platform electronic/optical structures. Even soliton propagation is approximated based on plasmonic based equation scalar models such as the NLSE (or Nonlinear Schordinger Equation). Spatial soliton propagation is established by opposing phenonmena Kerr or Kerr-like effect (self focusing or self trapping) and diffraction. These solitons also exist below the diffraction limit [30-32] or subwavelength spatial solitons at the dielectric\metallic interface. Now, with the well known features of plasmonic waveguides with dielectric\metallic interface, can waveguides propagate with a Kerr or Kerr-like layer (or nonlinear medium)? It turns out that surface waves or SPs exist for dielectric\metallic\nonlinear medium. Recently, graphene layers or thin layers have been an area of interest for researchers in waveguides due to its similiarities to metal properties. Although SPs do not exist for the TE mode in a dielectic\metallic waveguide these surface waves can exist with graphene waveguides. Graphene layers that are very thin compared to incident wavelength can be approximated to boundary conditions in Maxwell's Equations to caluculate dispersion relations and transmission/reflection coefficients[4,21,23]. In this work, surface waveguide physics for dielectric\nonlinear interface is briefly presented to solve analytical solutions to nonlinear equations for tranverse electric(TE) and transverse magnetic (TM) with conductivity considering a thin graphene layer in the boundary conditions. Then, using the first integral approach, a dispersion relation is calculated for the TM mode with a derived equation for the power in the TM mode. Lastly, the single layer dielectric\thin graphene\nonlinear medium and the multilayered (dielectric\thin graphene\dielectric\thin graphene\nonlinear medium) waveguide are studied to calculate reflection and transmission coefficients but a different approach is taken for the intensity dependent index of refraction not normally encountered coefficient calculations. 2.NONLINEAR SURFACE WAVES IN WAVEGUIDESWaveguides have been studied typically in nanostructures such as dielectric\metallic, Kerr media\dielectric, dielectric\metallic\dielectric, metallic\dielectric\metallic or, recently dielectric/graphene interface. In a Kerr medium the change in the index of refraction or dielectric constant is driven by the electric field intensity source for isotropic medium and 2 or 1 dimensional electric field components for an anisotropic medium. Kerr-like media has been long associated with polarization field enhancement in nonlinear optics. This nonlinearity could be useful in field enhancement in plasmonics. The nonlinear media contributes to creation of temporal or spatial soliton propagation [1,27]. The models are approximated from Maxwell's or Helmholtz vector equations to a scalar equation called the Nonlinear Schrödinger Equation (NLSE). In the Kerr medium, the coefficient of the intensity of the electric field has to be considered self-focusing or defocusing. The metallic/Kerr-like and dielectric/Kerr-like have been solved previously by G.I. Stegeman, J. Ariyasu, K. M. Leung, J.J. Burke, Jian-Guo Ma, Liu, Bing-Can, and others [7-14, 17, 18]. Recently, graphene has been material considered to behave similar to metal at certain temperatures, chemical potential, electron energy and incident wavelengths that can support creation of SPs at not only TM modes but also TE modes unlike metals cannot support creation SPs [16, 25, and 26]. 2.1TheoryThe initial step of the derivation is to solve for surface waves in a semi-infinite wave guide with a Kerr/graphene/dielectric interface. Starting with Maxwell's equations, the next step to is to solve for the TM (or Transverse Electric) modes of the fields. Also, steps were taken to solve for the TE (or Transverse Electric) modes knowing the plasmonic waves also exist for similar to metal/dielectric interface due to the graphene layer. This layer is much thinner compared to the thickness of the nonlinear Kerr and dielectric media and can be approximated to boundary conditions including surface current density. Using the first integral approach, steps were taken to derive the dispersion relation and calculate the energy flux. In order to describe Kerr-like media propagation it is assumed the normalized Maxwell's Equations in the frequency domain to The constants μ0, , ɛ0, and are the magnetic permeability, relative permittivity tensor, electric conductivity tensor. Expanding Maxwell's equations into scalar equations and with gives with , x′ = k0x, y′ = k0y, and z′ = k0z. With the normalized Maxwell equations, we take the prime away from the spatial for convenience (x′ → x, y′ → y z′ → z). We consider solving for the TM mode of the fields due to the fact plasmonic propagation existence found in this mode. Also, we consider propagation along the x axis so with and the TM mode meaning ∂y = 0 and Here, since magnetic material is not considered, the relative permeability μyy = 1. In the above equations the relative permittivity ɛxx in the medium has an Kerr effect or The constant ɛ0x is the permittivity of the Kerr medium and α is the coefficient of the electric field intensity. The purpose of the constant +α (-α) is for focusing (defocusing) Kerr medium. Equation (9) is the uniaxial approximation [1]. The dielectric variable ɛzz is approximated ɛzz ≈ ɛ0x [1,7-14]. We assume no conductivity z direction or , but assume . A more complete form of equations (6-8) is These equations can also be combined into a scalar wave equation along with equation (9) Equation (13) is the scalar equation with a Kerr effect with focusing. A proposed ansatz for the solution The trigonometry identity of this function has terms such that as z → ∞, Ex → 0, ∂zEz → 0. Next, substitute equation (16) into equation (13) which results are The conditions to satisfy equation (17) with λ=1 are The term must be greater than zero so for Ex as z → ∞ Ex (z) → 0. It also cannot be purely imaginary. The amplitude of the electric field can be complex and depend on the nonlinear coefficient, dielectric constant, and conductivity. Using equations (10-12), one can solve for the other field components Alternatively, the wave equation for defocusing in the TM mode is with the solution being We can solve for the TE (or Transverse Electric) modes knowing the plasmonic waves also exist for similar to metal/dielectric interface due to the graphene layer. Again, this layer is much thinner compared to the thickness of the nonlinear Kerr and dielectric media and can be approximated to boundary conditions including surface current density. Also, the existence of TE mode solutions does not necessarily mean existence of plasmons at the interface like a metallic/dielectric interface does not in the TE mode. This depends on graphene conductivity calculation is more intricate than frequency dependent metal. In this mode, , so the based on equation (5) the electric field Ey wave equation is with solution being The coupled equations in the TE mode are The uniaxial approximation to the nonlinear dielectric material would be The solutions and the coupled Maxwell relations can be used along with boundary conditions to derive dispersion relations at an interface such a dielectric/very thin graphene/Kerr medium. 2.2Boundary conditions in the waveguideIf we first consider the TM mode equation (23) taking the first integral (multiplying ∂zEx and integrating as function of z) the results are Here, C=0. At z<0, the wave is ɛs is the dielectric constant at z<0. Assuming the graphene is sufficiently thin compared to the Kerr and dielectric media, it is approximated to boundary conditions of normal and tangential components which and ρs, is the surface current density and charge density with is oriented recalling . With equations (35a-d), at z=0 considering the TM mode, assuming and . In the case of the TM mode there is no Bjn, j=1,2 field to be considered in this case. Using equations (20 and 21), we can calculate the power in the nonlinear medium using the equation for the Polynting vector and The nonlinear equations for the TM and TE modes have analytical solutions that have surface wave characteristics. There solutions are similar to homogenous NLSE. These functions also account for conductivity which is a complex quantity. The above nonlinear equations accounted for self-focusing and defocusing. A dispersion relation for nonlinear surface waves with conductivity was derived but one has to match nonlinear intensity enhancement with SP excitation with graphene conductivity which is was not part of this study. 3.OPTICAL BISTABILITYSwitching states in nonlinear optic devices are a subject of intense investigation recently in optics[6,15,21,24,32]. Optical devices are being studied and engineered to eventually replace electronic devices and carry out the same function but with better performance. This research has been further explored in nanotechnology or nanostructure devices at the subwavelength scale. Optical nonlinearity which is mostly studied at the microscale is applicable at the nanoscale. Electric Field enhanced dielectrics such as the Kerr effect can be useful in creating switching states in a dielectric/nonlinear Kerr medium, dielectric/metallic/nonlinear Kerr medium, and even a dielectric/thin graphene/nonlinear Kerr medium interface. These interfaces have optical bistable effects because of abrupt discontinious jumps in solutions of the and fields or reflection (transmission) coefficients. Here, the purpose is to show bistable states and hysteresis with the excitation of surface plasmons establish a basis for a optical switching. First in nonlinear optics, nonlinear polarization displacement is and For an isotropic medium ignoring the other tensor terms which are zeros, so polarization now is with X1 and X3 being susceptibilities of nonlinearities of polarization. The constants in equation (47) such that The constants ɛ, ɛ0 and α are the nonlinear dielectric, dielectric, and nonlinear index of refraction of the Kerr medium. Considering figure (1), the simple dielectric/nonlinear interface using Maxwell's Equations in (cgs units), the fields are represented in Table 1. The fields are incident, reflected, and transmitted waves with directional k1 k2 wave vectors. All fields are assumed to be exp(-iœt) time dependent. The magnetic fields H1y and H2y calculated in medium 2 is based on a component of Faraday's Law in the frequency domain Table 1.
The fields in the nonlinear medium are in the solution of the form with again, Faraday's Law in vector form The nonlinear Helmholtz wave equation is Equation (52) solved numerically but here the field will be approximated by equation (50). The equation can be approximated to model solitary propagation [27] and nonlinear surface waves (in previous section 2.1). The components of the wave vectors given by and According to figure (1) and table (1), we apply boundary conditions of the electric and fields at z=0 resulting the relations In equation (56), the second term on the right hand side is approximated to zero due to the slow vary electric field amplitude E2 as function of z. For a boundary condition, and Keeping in memory that the transmitted angle θ2 in the nonlinear medium can be complex so the k2 and k2z in equation have to be calculated. The critical angle θc for TIR (total internal reflection) is or Substituting equation (55) into equation (56) with using equations (57 and 58), the Fresnel relations are with . Equations (62-63) are in terms of the incident angle, the dielectric constant, and the nonlinear dielectric medium. Referring back to equations (36-37), equations (55-56) and figure (1), a very thin layer of graphene can be approximated to conductivity [16] such that for the TM mode. Boundary condition equation with conductivity (σ=σxx from the conductivity tensor) are This leads to the following Fresnel equations With σ=0, equations (67-68) return to equations (62-63). The conductivity for graphene media has to be carefully handled due its dependency on frequency, temperature, chemical potential, and electron energy. Graphene conductivity physics is a separate area popular research [16,26]. The integral for graphene conductivity [16] is The constants e, γ, Μc, ℏ, and T are electron charge, decay constant, chemical potential, electron energy, and temperature. The conductivity is characterized by interband and intraband transitions in the in the conduction and valance bands of graphene [28]. Equation (69) are results set in the complex domain so for certain frequencies the imaginary part of the conductivity becomes negative which means TE (Transverse Electric) surface waves can propagate along a waveguide graphene layer. At other frequencies only TM surface waves propagate along the graphene layer. However, this does not necessary account for a behavior of a graphene layer with a nonlinear Kerr medium. Metallic/Dielectric interfaces for waveguides do not support SPs TE modes [3,19]. Since the graphene conductivity is a complex number, the real and imaginary parts can be treated with care as dielectric equivalent similar to This dielectric can approximate to 1 atomic layer in a waveguide which is part of the boundary condition instead of another waveguide layer [29]. In figure(2), the is a comparison Reflection coefficients from Fresnel's Equations (62-63 and 67-68) of the single interface dielectric/dielectric medium interface with and without conductivity. The reflection coefficient is In figure (2), graphs (c) and (d) show total reflection greater 60 degrees with and without conductivity interface (equations 62-63, 67-68). The figures also include ɛ1> ɛ2 and ɛ1< ɛ2. The reflection equation (72) is not a good measure to handle nonlinear Kerr medium since the intensity is dependent on electric fields. This is similar problem with dispersion relations with nonlinear media [1, 6, and 15]. A better approach would to define the dimensionless intensity [4] which are incident, reflected, and transmitted nonlinear measures knowing Also the quantities can be converted back in terms of the Poynting vector For a Kerr-like medium equations (73-76) can be defined as which are incident, reflected, and transmitted nonlinear measures knowing The angle is the critical angle when nonlinear constant to the intensity is not present (α=0). The ensuing equations based on equations (73-77) for transmission, for TIR and with σ≠0 Now nonlinear reflection can be calculated In equations (79-82), U1r and U1 as functions of U2 can be determined with the incident angle θ1 and the dielectric ɛ fixed. Equations (79-80 and 84-85) are real (transmission) and (82-83 and 85-86) are imaginary (TIR). In equations (81-82) when R=1 (U1r = U1) In order for TIR mode to exist in the nonlinear medium, If U2 is equated in (equation 89) and solving for nonlinear critical intensity The TIR mode condition is more stringent with conductivity counted in equations (83-86). Assume conductivity to be imaginary σ=iσ', so In this case both expressions above have to be substituted back into equation (85-86) to get the nonlinear critical intensity with conductivity. For the Transverse Electric (TE) case, expressions for the intensities with conductivity are Since the waves are TE or (Ez = Ex = Hy = 0), conductivity is oriented Again, in metallic/dielectric interface surface plasmons do not exist in TE mode, but they exist for this mode on a graphene dielectric interface or a multilayer graphene/dielectric waveguide [16,26]. In order for bistability to occur for one value U1 there should be multiple values of U2 leading to different values of U1r. This expression for intensity can increase or decrease with no change in slope so a systematic approach to finding bistability is looking extremum points. In the case of conductivity being zero equations (81-82) bistability exist with the nonlinear coefficient being negative (or α<0). To check for slope change or switching points and the incident intensity approximated with the zero slope value and switching value (ɛ=1) If conductivity (with ɛ=1) is in the boundary conditions for TM mode the switching values are In order to deal the nonlinear dielectric that is intensity dependent the nonlinear coefficient and the dielectric constant of the nonlinear dielectric has to be dimensionless measure to properly handle reflection and transmission quantities. By calculating the dielectric constant integral for graphene based on frequency, chemical potential, temperature, electron energy the reflection and transmission for intensities can be calculated with nonlinear waveguide. Lastly, one has to be able to take advantage of conductivity and required quantity of intensity for the Kerr effect to study the full features of this waveguide. 4.MULTILAYER GRAPHENE AND ONE LAYER KERR MEDIUMIn the previous section the single layered dielectric/nonlinear Kerr dielectric with a thin graphene layer approximated to a boundary condition was presented to show optical bistability but here a multilayered configuration(figure 4 below), dielectric/dielectric/nonlinear Kerr dielectric with multiple thin graphene layers is considered. Similar incident, reflected, and transmitted waves functions for TM and TE modes are assumed. The approximated wave in the Kerr medium 3 is with the constant d being the distance between medium1 and 3. Again, for this case the derivative term in equation (101) is approximated to zero since it is slowly varying amplitude. The angles in medium 2 and 3 can be complex so the wave vectors relations are and the wave vectors relations are and Apply pertinent boundary conditions and z=0and z=d with continuity of electric and magnetic fields for TM modes with and without conductivity (σ) in matrix form The plasmon angle [2-5] is given by If we solve matrix equation (108) by reducing the matrix using Cramer's rule for calculating the determinants and solving for Without using known matrix methods the algebra can be quite difficult. The figure(5) below shows an example of switching state from R=1 to close R≈0 with a complex dielectric silver (1.06×10-6m) in medium 2 and nonlinear medium 3 with measures of dimensionless intensities. This approach can also be taken for TE surface waves setting boundary condition equations with correct conductivity constant and orientation. 5.CONCLUSIONNonlinear surface wave propagation was presented in other order show that Kerr or Kerr-like material can support SP propagation especially with thin graphene layer between a dielectric and nonlinear dielectric. Using Maxwell's Equations and pertinent boundary conditions analytical solutions for TE and TM modes for self-focusing and self-defocusing, dispersion relation and expression for energy flux was derived. Next, reflection and transmission coefficients for a dielectric/nonlinear Kerr media presented to show that these quantities had to be calculated using dimensionless intensities that included the nonlinear coefficient and constant dielectric for that nonlinear medium. For the TM and TE mode reflection and transmission coefficients were presented that included a thin graphene layer. Briefly, the integral for dielectric of graphene presented. The incident frequency (among other quantities) impacts the conductivity of the graphene but the reaction of nonlinear Kerr material must be also considered. The approach to finding switching states for bistability would be to find the change in slope of the intensity equations that are used to calculate reflection and transmission coefficients. Lastly, an example for a multilayered configuration with multiple graphene thin layers was presented. The matrix equations were derived with and without conductivity in the TM mode. Reflection was calculated for multilayered configuration with silver in medium 2 (figure 5) to demonstrate switching in optical bistability. REFERENCESPonath, H.-E., and Stegeman, G. I.,
“Nonlinear Surface Electromagnetic Phenomena,”
324
–349 Elsevier Science Publishers, New York & Netherlands
(1991). Google Scholar
Heavens, O. C.,
“Optical Properties of Thin Solid Films,”
46
–73 Dover Publications, New York
(1955). Google Scholar
Bozhevolonyi, S. I.,
“Plasmonic Nanoguides and Circuits,”
1
–63 Pan Stanford Publishing, Singapore
(2009). Google Scholar
Wysin, G.M.,
“Optical Bistability with surface plasmons,”
Toledo, Ohio
(1980). Google Scholar
Vogel, M.W.,
“Theoretical and Numerical Investigation of Plasmon Nanofocusing in Metallic Tapered Rods and Grooves,”
Dissertation, Queensland University of Technology(2009). Google Scholar
Huang, J.-H., Railing, C., Leung, P.-T. and Tsai, D. P.,
“Nonlinear dispersion relation of surface plasmon at a metal-Kerr medium interface,”
Optics Communications, 282 1412
–1415
(2009). https://doi.org/10.1016/j.optcom.2008.12.025 Google Scholar
Ma, J.-G., Wolff, I.,
“Propagation Characteristics of TE-Waves Guided by Thin Films Bounded by Nonlinear Media,”
IEEE Transactions on Microwave Theory and Techniques, 43
(4), 790
–794
(1995). https://doi.org/10.1109/22.375225 Google Scholar
Stegeman G. I., Seaton, C. T., Ariyasu, J., Wallis R. F., Maradudin, A.A.,
“Nonlinear electromagnetic wave guided by a single interface,”
Journal of Applied Physics, 58 2453
–2459
(1985). https://doi.org/10.1063/1.335920 Google Scholar
Prade, B., Vinet, J. Y., Mysyowicz, A.,
“Guided optical waves in planar heterostructures with negative dielectric constant,”
Physical Review B, 44
(24), 13 556
–13 572
(1991). https://doi.org/10.1103/PhysRevB.44.13556 Google Scholar
Ariyasu, J., Seaton, C. T., Stegeman G. I., Wallis R. F., Maradudin, A.A.,
“Nonlinear surface polaritons guided by metal films,”
Journal of Applied Physics, 58
(7), 2460
–2466
(1985). https://doi.org/10.1063/1.335921 Google Scholar
Leung, K.M.,
“p-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions,”
Physical Review B, 32
(8), 5093
–5101
(1985). https://doi.org/10.1103/PhysRevB.32.5093 Google Scholar
Burke, J. J., Tamir, T., Stegeman G. I.,
“Surface-polariton-like waves guided by thin lossy metal films,”
Physical Review B, 33
(8), 5186
–5200
(1986). https://doi.org/10.1103/PhysRevB.33.5186 Google Scholar
Ma, J.-G., Chen, Z.,
“Nonlinear Surface Waves on the Interface of Two Non-Kerr-Like Nonlinear Media,”
IEEE Transactions on Microwave Theory and Techniques, 45
(6), 924
–930
(1997). https://doi.org/10.1109/22.588604 Google Scholar
Ma, J.-G., Wolff, I.,
“TE Wave Properties of Slab Dielectric Guide Bounded by Nonlinear Non-Kerr-Like Media,”
IEEE Transactions on Microwave Theory and Techniques, 44
(5), 730
–738
(1996). https://doi.org/10.1109/22.493927 Google Scholar
Can, L. B., Li Y., Xin, L.Z., Kai, Z,
“The dispersion relations for surface plasmon in a nonlinear-metal-nonlinear dielectric structure,”
Chinese. Phys. B, 19
(9), 097303-1
–097303-5
(2010). Google Scholar
Buslaev, P. I., Iorsh, I.V, Shadrivov, I. V., Belov, P.A., and Kivshar, Yu. S.,
“Plasmons in Waveguide Structures Formed by Two Graphene Layers,”
JETP Letters, 97
(9), 535
–539
(2013). https://doi.org/10.1134/S0021364013090063 Google Scholar
Stegeman G. I., Seaton, C. T.,
“Nonlinear surface plasmons guided by thin metal films,”
Optics Letters, 9
(6), 235
–237
(1984). https://doi.org/10.1364/OL.9.000235 Google Scholar
Bloembergen, N., Pershan, P. S.,
“Light Waves at the Boundary of Nonlinear Media,”
Physical Review, 128
(2), 606
–622
(1962). https://doi.org/10.1103/PhysRev.128.606 Google Scholar
Pitarke, J. M., Silkin, V.M.,, Chulkov, E. V., and Echenique, P. M.,
“Theory of surface plasmons and surface-plasmon polaritons,”
Rep. Prog. Phys., 70 1
–87
(2007). https://doi.org/10.1088/0034-4885/70/1/R01 Google Scholar
Simon, H. J., Mitchell, D. E., Watson, J. G.,
“Surface plasmons in silver films-a novel undergraduate experiment,”
American Journal of Physics., 43
(7), 630
–636
(1975). https://doi.org/10.1119/1.9764 Google Scholar
Balkhanov, V.K., Angarkhaeva, L. Kh., Bashkuev, Yu. B., and Gantimurov, A.G.,
“The Transmission and Reflection Coefficients of an Electromagnetic Wave for a Gradient Dielectric Layer,”
Journal of Communications Technology and Electronics, 57
(11), 1160
–1165
(2012). https://doi.org/10.1134/S1064226912080128 Google Scholar
Maksymov, I. S., Davoyan, A. R., Miroshnichenko, A.E., Simovske, C., Belov, P., Kivshar, Y. S.,
“Multifrequency tapered plasmonic nanoantennas,”
Optical Communications, 285 821
–824
(2012). https://doi.org/10.1016/j.optcom.2011.11.050 Google Scholar
Rani, G.R., Raju, G. S. N.,
“Transmission and Reflection Characteristics of Electromagnetic Energy in Biological Tissues,”
International Journal of Electronics and Communication Engineering, 6
(1), 119
–129
(2013). Google Scholar
Attiya, A. M.,
“Reflection and Transmission of Electromagnetic wave due to a quasi-fractional space slab,”
Progress In Electromagnetic Research Letters, 24 119
–128
(2011). https://doi.org/10.2528/PIERL11051105 Google Scholar
Davoyan, A. R., Shadrivov, I. V., Kivshar, Y. S.,
“Nonlinear plasmonic slot waveguides,”
Optics Express, 16 21209
–21214
(2008). https://doi.org/10.1364/OE.16.021209 Google Scholar
Bludov, Yu. V., Ferreira, A., Perses, N. M. R., and Vasilevskiy, M. I.,
“A Primer on surface plasmon-polaritons in Graphene,”
International Journal of Modern Physics B, 27
(10), 1341001(1
–74)
(2013). https://doi.org/10.1142/S0217979213410014 Google Scholar
Hasegawa, A., Mathsumoto M.,
“Optical Solitons in Fibers,”
45
–59 3Springer, New York
(2003). Google Scholar
Hill, A, Mikhailov A., and Ziegler, K.,
“Dielectric function and plasmons in graphene,”
EPL A Letters Journal Exploring the Frontiers of Physics, 87 27005(1
–5
(2009). Google Scholar
Hanson, G.,
“Dyadic Greens functions and guided surface waves for a surface conductivity model of graphene,”
Journal of Applied Physics, 106
(6), 064302-1
–064302-8
(2008). Google Scholar
Gisin, B., Malomed, B. A.,
“Subwavelength spatial solitons in optical media with non-Kerr nonlinearities,”
Journal of Optics A:Plasmons, 3 284
–290
(2001). https://doi.org/10.1088/1464-4258/3/4/309 Google Scholar
Gisin, B., Malomed, B. A.,
“One and two dimensional subwavelength solitons in saturable media,”
Journal of Optical Society of America B, 18 1356
–1361
(2001). https://doi.org/10.1364/JOSAB.18.001356 Google Scholar
Crutcher S., Osei, A.,
“Derivation of the Effective Nonlinear Schrodinger Equations for dark and power law spatial plasmon-polariton solitons using nano self-focusing,”
Progress In Electromagnetic Research B, 29 83
–103
(2011). https://doi.org/10.2528/PIERB11020306 Google Scholar
|