Study of Slow Light Surface Plasmons
by
Abraham Vázquez Guardado
A Dissertation Submitted to the Program in Optics, Optics Department
in Partial Fullfillment of the Requierements for the Degree of
Master of Science in Optics
at the
National Institute for Astrophysics, Optics and Electronics
July 2012 Santa María Tonantzintla, Puebla
Advisors: Prof. José Javier Sánchez Mondragón1 Prof. Miguel Torres Cisneros2
1 INAOE 2 University of Guanajuato
© INAOE 2012 All rights reserved The author hereby grants to INAOE permission to reproduce and to distribute copies of this thesis document in whole o in part.
Study of Slow Light Surface Plasmons
Estudio de Plasmones Superficiales Lentos
Ing. Abraham Vázquez Guardado INAOE Coordinación de Óptica
Tesis de Maestría en Ciencias Especialidad en Óptica
Asesores: Dr. José Javier Sánchez Mondragón Dr. Miguel Torres Cisneros
Sta. María Tonantzintla, Pue. México Julio 2012
Summary INAOE 2012
Summary
In this thesis we study and analyze the Slow Light (SL) property of Surface Plasmon– Polaritions (SPPs) for three cases, for lossless metals, for lossy metals, and for this latter but considering an active dielectric.
In the first case, we study the qualitative property of SL in SPPs in a metal/dielectric and dielectric/metal/dielectric waveguide. Despite it is an unrealistic approach, the study is suitable to understand the general problem.
Then we proceed with the following case when metal losses are considered in both waveguides. It is here where we observe how the SL property of SPP is affected. In addition, propagation losses in the SL regime are considerably high such that even when a moderate high group index can be achieved the propagation length never surpass one micron. It is why the active medium is opted in order to reduce abortion losses in the metal.
Finally, in the third case we study the effects an active dielectric medium has in the SL performance of SPPs in the metal/dielectric waveguide. We study two approaches, when the gain response is constant and when it is a gain distribution function. In the first one we can see that the SL property of SPP does not improve but only the propagation length. It means that the natural SL regime of SPPs does not change. On the second method, however, we observe that, due to the finiteness of the gain response, a new SL region is induced in the SPP dispersion curve even far from the natural SL region.
In general, this theoretical study is with the aim to understand the property of SL in SPPs. In this work we introduce a new analysis for the dielectric gain medium that predicts that, if the gain response is considerably narrow, the SPPs can experience an induced SL behavior. This approach was not considered before in the study of amplified SL SPPs.
Abraham Vázquez–Guardado i INAOE 2012 Summary Resumen
En este trabajo de tesis estudiamos y analizamos la propiedad de Luz Lenta (LL) en Plasmones–Polaritones Superficiales (PPS). Tres casos de estudio se consideran: para metales sin pérdidas, para metales con pérdidas y para estos últimos pero considerando un dieléctrico activo.
En el primer caso, estudiamos cualitativamente la propiedad de LL en PPS en las guías de ondas, metal/dieléctrico y dieléctrico/metal/dieléctrico. A pesar que éste es un enfoque no realista, el estudio es adecuado para entender el problema.
Posteriormente, continuamos con el caso de un metales con pérdidas en ambas guías de ondas. Es aquí donde observamos cómo la propiedad de LL en PPS y la distancia de propagación se ven afectadas. Incluso cuando un índice de grupo moderadamente alto se puede conseguir la distancia de propagación nunca rebasa una micra. Es por ello que un medio activo es considerado para reducir las pérdidas por absorción en el metal.
Finalmente, en el tercer case estudiamos los efectos que tiene un medio activo en el desempeño de PPS lentos en la guía de ondas metal/dieléctrico. Estudiamos dos enfoques, modelo de ganancia constante y modelo de ganancia finita. En el primer caso podemos observar que la propiedad de LL en PPS no mejora, sólo lo hace la distancia de propagación. Esto significa que el régimen natural de LL del PPS no se ve afectado. Sin embargo, en el segundo caso observamos que una nueva región de LL se induce en la curva de dispersión del PPS. Esta región puede estar incluso lejos de la región natural de LL.
En general, este estudio teórico es con el objetivo de entender la propiedad de LL de los PPS. En este trabajo introducimos un análisis de ganancia en el dieléctrico nuevo que predice que si la respuesta de ganancia es lo suficientemente angosta, el PPS puede experimentar un comportamiento lento inducido, algo que no se había tocado antes. ii Abraham Vázquez–Guardado Acknowledgements INAOE 2012
Agradecimentos
En el marco de apoyo y cooperación para la realización de este trabajo participaron distintas personas e instituciones.
Quiero agradecer a mis padres y hermanos por haber sido un apoyo incondicional mientras realizaba mis estudios de maestría.
Al Dr. José Javier Sánchez Mondragón, por su apoyo académico en el transcurso de mis estudios de maestría y trabajos adyacentes que se generaron.
Al Instituto Nacional de Astrofísica Óptica y Electrónica, por ofrecer el posgrado de Maestría en Ciencias con Especialidad en Óptica, por medio del Departamento de Óptica, y por formarme y guiarme en esta travesía.
De igual forma, agradezco al Consejo Nacional de Ciencia y Tecnología por varios motivos. Primero, por ser el órgano promotor de la ciencia y tecnología en el país a través de centros de investigación como el INAOE. Segundo, por brindarme beca de maestría. Finalmente, por brindarme la oportunidad de realizar una estancia de investigación en el extranjero, para la cual me proporcionó una beca mixta.
Además, quiero agradecer al Prof. Robert W. Boyd, líder del Canada Excellence Research Chair in Quantum Nonlinear Optics, por brindarme la oportunidad de trabajar en su prestigiado grupo de investigación en la Universidad de Ottawa. De lo cual se forjo el tema principal de mi trabajo de tesis. Abraham Vázquez Guardado Sta. María Tonantzintla, Puebla Julio 2012
Abraham Vázquez–Guardado iii INAOE 2012 Acknowledgements
iv Abraham Vázquez–Guardado Content INAOE 2012
Index
Preface. 1!
Chapter 1 Introduction. 3!
Summary 3!
Introduction 3!
1.1–Maxwell’s equations. 4!
1.2–The wave equations. 5!
1.3–Constitutive relations. 6!
1.4–Spectral representation of time–dependent fields. 8!
1.5–Time–harmonic fields. 9!
1.6–The space notation. 10!
1.7–Electric and magnetic field components. 10!
1.8–Boundary conditions. 12!
1.9–Complex permittivity function. 14!
1.10–Conservation of energy. 15!
1.11–The permittivity function of the free electron gas. 17!
1.12–Free electron gas dispersion and volume plasmons. 19!
1.13–Real metals and interband transitions. 21!
1.14–Group and phase velocities. 23!
1.15–Slow, fast, backward and stop light. 26!
1.16–Slow light mechanisms. 28!
1.16.1–The Kamers–Kronig relations and the susceptibility. 28!
1.16.2–Resonance in materials. 30!
References. 34!
Chapter 2 Slow Light Surface Plasmons. 37!
Summary 37!
Introduction 37!
Abraham Vázquez–Guardado v INAOE 2012 Content
2.1–The wave equation for inhomogeneous media. 38!
2.2–Two–layer planar interface. 41!
2.2.1–Analysis for a generic Drude–like metal. 43!
2.3–Three–layer planar interface. 46!
2.4–Metal losses in SPPs performance. 50!
References. 53!
Chapter 3 Slow Surface Plasmons in amplifying media. 55!
Summary 55!
Introduction 55!
3.1–Dielectric with active media. 57!
3.2–Condition for bound modes. 59!
3.3–Constant gain model and condition for SPP lossless propagation. 62!
3.4–Finite gain model. 66!
3.4.1–Gaussian gain response model. 68!
References. 74!
Chapter 4 Results, discussions and further work. 77!
4.1–Results and discussions 77!
4.2–Gain requirements 81!
4.3–Future work 82!
References 83!
Appendix. 85!
A.1–Equations of Chapter 1. 85!
A1.1–Macroscopic Maxwell’s equations and constitutive equations. 85!
A.1.2–The wave equations. 85!
A.1.3–Electric and magnetic field components. 86!
A.1.4–Boundary conditions. 88!
A.1.5–Complex permittivity functions. 90!
A.1.6–The permittivity function of the free electron gas. 91!
A.1.7–The permittivity function of a metal with interband transitions. 92! vi Abraham Vázquez–Guardado Content INAOE 2012
A.1.8–Dispersion of volume Plasmon. 93!
A.2–Equations of Chapter 2. 94!
A.2.1–TE wave equation for the Two–Layer waveguide. 94!
A.2.2–TM wave equation for the two–layer waveguide. 96!
A.2.3–TE non–existence condition of SPPs. 97!
A.2.4–SPP dispersion relation. 98!
A.2.5–Dispersion relation in a three–layer system. 100!
A.3–Equations of Chapter 3. 104!
A.3.1–The constant gain model. 104!
A.3.2–Condition for bound solutions. 105!
A.3.3–Condition for lossless SPP propagation. 109!
A.3.4–Gaussian model for the active medium. 112!
A.3.4.1–Hilbert transform of the Gaussian. 113!
A.3.4.2–Calculation of the group index 116!
List of figures and tables 121!
Abraham Vázquez–Guardado vii INAOE 2012 Content
viii Abraham Vázquez–Guardado Preface INAOE 2012
Preface.
In this master thesis we will study Surface Plasmon–Polaritons [SPPs] and its Slow Light [SL] property, the group index, in planar waveguides. First, we review the passive planar waveguide without considering losses in the metal. This represents an ideal case. Then the lossy case is studied. Both are carried out for the two and three–layer systems. Under this point of view, we can study and analyze the natural SPP resonance and its SL region and the effects losses have to the SL property of SPPs.
Then, we analyze the active case. The theoretical study is for the two–layer systems since its mathematical expression is simpler. First, in order to study the SL property of SPPs with an active material there has been proposed and used the constant gain model. It consists in adding a constant negative permittivity imaginary part to the dielectric medium over a whole frequency range. Since SPPs losses are diminished or eliminated, the SPP mode can propagate for longer distances with high group index, if we are near the surface plasmon asymptote, but it is not the case when we are far from the surface plasmon resonance.
After that, a more realistic model is proposed to model the active response in frequency. It consists of a finite distribution gain model that accounts the maximum gain and the spectral distribution width or a given active medium. This model, besides reducing losses in the SPP propagation mode, induces a SL region around the gain distribution, which was not observed in the constant gain model. The magnitude of this value depends on the maximum gain and the width of the distribution. Hence the induced SL region in SPPs that can be even far from the natural SPP SL region.
Abraham Vázquez–Guardado 1 INAOE 2012 Preface Finally there is a discussion of results. There is also an analysis of gain media that is currently available. This is with the aim of analyzing how physically possible is to have SL SPPs with the current gain systems.
2 Abraham Vázquez–Guardado Chapter 1 INAOE 2012
Chapter 1 Introduction.
Summary
In this introduction chapter we make a bibliographic review [1–6] with the aim of studying the basic concepts leading to Surface Plasmon–Polaritons [SPPs] and Slow Light [SL]. We review Maxwell’s equations and the wave equation. We also review the material properties of metals and we introduce the basic understanding of plasmonics. Finally we review the SL phenomenon, the physics point of view and the circumstances it can exist.
Introduction
The mathematical representation of Maxwell’s equations is a powerful tool to analyze the electromagnetic fields for any problem under study, where electromagnetic fields interact with matter. This is the case of optical fields propagation in a medium with certain electromagnetic properties. The physical properties of this interaction are basically based on the interaction of propagating fields and how the medium reacts on it. Therefore we will do a review of Maxell’s equations, as well as the electromagnetic properties of materials.
SPPs are a special kind of surface waves that, due to their intrinsic bound nature, they are rich in applications. Nevertheless, some requirements are strictly necessary in order for a given systems to support surface waves. One of them, and the most important, is the electric permittivity of the medium. For this reason, we will discuss the fundamentals of material properties.
Abraham Vázquez–Guardado 3 INAOE 2012 Chapter 1 Finally, we will review the SL phenomenon. We treat the mathematical analysis and the physical implications of such property. As it will be described later, SL, is a property that is an intrinsic property of a material or a system, hence we will review both.
For most equation their derivation can be found in appendix 1.
1.1–Maxwell’s equations.
The macroscopic set of Maxwell’s equations governs the electromagnetic response of the electromagnetic field in whatever it is interacting with. It consists on a set of two vectorial fields, in case of free space, and three more when a material is considered instead
[7]. These equations are given in dependence of space r and time t . In addition, they can be complex, which implies that they have phase relative to each other and among the other components of the vectors. In SI unit system the macroscopic Maxwell’s equations have the form
$ ! " E(r,t) = # B(r,t) , (1.1) $t
# ! " H(r,t) = D(r,t) + J(r,t) , (1.2) #t
! "D(r,t) = #(r,t) , (1.3)
! "B(r,t) = 0 . (1.4)
In this group of four equations E stands for the electric field, D the electric displacement,
H the magnetic field, B the magnetic induction, J the current density, and ! the charge density. In total the components of these equation make a set of sixteen unknowns, which depend on the specific problem under study; nevertheless, it is possible to reduce the count.
4 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 Another important relation is the conservation of charge [1], which can be straightforwardly obtained by taking the divergence of equation (1.2), it is considered that ! "! # H equals zero, and then substituting equation (1.3) we get
$ ! " J(r,t) = # %(r,t) . (1.5) $t The electromagnetic properties for a given medium are usually treated in terms of the macroscopic polarization P and magnetization M according to the following equations
D(r,t) = !0E(r,t) + P(r,t) , (1.6)
!1 H(r,t) = µ0 B(r,t) ! M(r,t) , (1.7) where and are the permittivity and permeability of vacuum, respectively. These !0 µ0 equations are always valid due to they do not impose any conditions on the medium [1].
The electric polarization P describes the electric dipole momentum per unit volume that an external field exerts on inside the material. It is related to the internal charge density by
! "P(r,t) = #$int (r,t) and, along with the charge conservation equation leads to
! Jint (r,t) = P(r,t) . (1.8) !t
This description accounts for the nature of the internal current density Jint that is the time rate change in the polarization P [2]. In the following discussion, the space and time dependency will be omitted unless it is necessary to recall them.
1.2–The wave equations.
The general wave equations are obtained by applying vectorial analysis to Maxwell’s equations. After further substitutions one can get the following set of equations
1 #2 E # % #P ( ! " ! " E + 2 2 = $µ0 ' J + + ! " M* , (1.9) c #t #t & #t )
Abraham Vázquez–Guardado 5 INAOE 2012 Chapter 1
1 #2 H #P #M ! " ! " H + 2 2 = ! " J + ! " + µ0 . (1.10) c #t #t #t
The constant c = 1 !0µ0 stands for the speed of light in vacuum. In equation (1.9) the term in parenthesis can be associated to the total current density
! Jt = Js + Jc + + " # M (1.11) !t where we have divided J into the source current density J and the induced conduction s current density J . In addition, P / t and M relates the polarization and the c ! ! ! " magnetization current densities, respectively. Wave equations (1.9) and (1.10) do not impose any condition on the media at all, hence their general validity [1]. Nevertheless, when considering a specific problem under study, most of the pervious equation can be further simplified and the handling of such comes to ease the analysis.
1.3–Constitutive relations.
In our case, we will treat materials that are linear, nonmagnetic and isotropic [2]. These conditions lead to the following constitutive equations
D = !0!E, (1.12)
B = µ0µH . (1.13)
The term ! is known as the dielectric constant or the relative permittivity and µ as the relative permeability that, for nonmagnetic media, equals one. The linear relationship between E and D in equation (1.12) stands for the linear response of the material to the electric field and such is also described by the electric susceptibility ! , which relates E and
P in the form of
P E . (1.14) = !0 "
6 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 By using this relation and equation (1.12) in (1.6) we get the dielectric constant in function of the electric susceptibility,
! = 1+ " (1.15) Finally, another important constitutive equation that needs to be mentioned is the Ohm’s law
J = !E , (1.16) where ! defines the conductivity. Then, there can be a relationship between ! and ! which both can account for the electromagnetic behavior of the material and its use can be indistinctly depending on the problem [3]. For example, if the material is inhomogeneous ! will be a function of space. Furthermore, the material can be spatially dispersive if the constitutive equations are convolution over space and temporally dispersive when the material parameters are function of frequency [3]. Therefore a more general representation of equations (1.12) and (1.16) are
D r,t = ! dt dr ! r # r ,t # t E r ,t , (1.17) ( ) 0 $ " " ( " ") ( " ") J(r,t) = $$ dt!dr!" (r # r!,t # t!)E(r!,t!) . (1.18) where functions ! and ! describe the impulse response of the respective linear relationship. These equations consider the non–locality in time and space of the optical response of materials, taking into account the dependence on frequency and space. It must be remarked that all lengths are significantly larger than the physical parameters of the material, for example the lattice spacing is such that ensures homogeneity. This means that the impulse response function do not depend on absolute spatial and temporal coordinates, but on their differences. In the most general approach, where the response function can be a !–function, then equations (1.16) and (1.12) are straightforwardly recovered.
Abraham Vázquez–Guardado 7 INAOE 2012 Chapter 1 1.4–Spectral representation of time–dependent fields.
The spectrum, Eˆ r,! , of any time–dependent electric field, E r,t , can be ( ) ( ) obtained by the Fourier transformation [1] as
1 $ Eˆ (r,! ) = E(r,t)ei!t dt . (1.19) 2" %#$
The only restriction is that the field E r,t has to be real for which the following relation ( ) must be satisfied
Eˆ (r,-! ) = Eˆ * (r,! ) . (1.20)
Under this approach, we can restate Maxwell’s equation as
! " Eˆ r,# = i#Bˆ r,# , (1.21) ( ) ( )
! " Hˆ r,# = $i#Dˆ r,# + Jˆ r,# , (1.22) ( ) ( ) ( )
! "Dˆ r,# = $ˆ r,# , (1.23) ( ) ( )
! "Bˆ r,# = 0 . (1.24) ( )
In certain problems, it might be easy to solve Eˆ r,! , then the time–dependent field ( ) can be calculated from its inverse Fourier transform as
# E(r,t) = Eˆ (r,! )e"i!t d! . (1.25) $"#
This tool is quite helpful when our time dependent electromagnetic field is not harmonic. Then, by means of the Fourier transform we extract every spectral component and treat them as separately monochromatic filed [1].
8 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 1.5–Time–harmonic fields.
Another technique to easily analyze electromagnetic fields is to consider that the field time dependence can be split off from the space dependency [1]. The mathematical representation of a monochromatic field is
1 E(r,t) = Re E(r)e!i"t = #E(r)e!i"t + E* (r)ei"t % . (1.26) { } 2 $ &
The same methodology can be applied to the other fields. It is worthwhile to notice that while E r,t is real, the spatial part E r might be complex. Then, applying this to ( ) ( ) Maxwell’s equation we get the following set:
! " E(r) = i#B(r), (1.27)
! " H(r) = #i$D(r) + J(r), (1.28)
! "D(r) = #(r) , (1.29)
! "B(r) = 0 . (1.30)
The previous set of equations is straight equivalent to those equations (1.1) – (1.4) but for the spectra of an arbitrary time–dependent field [1]. Therefore, the solution of E r is fully ( ) equivalent to the spectrum Eˆ r,! of any arbitrary time–dependent field. In addition, the ( ) complex field amplitudes, E r for instance, depend on the angular frequency as well; this ( ) means that E r = E r,! , though ! is not included in the argument. Therefore, the ( ) ( ) material parameters ! , µ and ! are functions of space and frequency as well; ! = ! r," , ( ) µ = µ r,! and ! = ! r," . ( ) ( )
Abraham Vázquez–Guardado 9 INAOE 2012 Chapter 1 1.6–The space notation.
It is convenient to treat the propagating fields in term of its space dependency, in addition to the time–harmonic representation [3]. This scheme is quite useful, not to say necessary, when threating piecewise materials, such as planar waveguides. Then, taking the representation of equation (1.26) we can write the field as
i(k!r"#t) E(r,t) = E0 (r)e . (1.31)
The term r is the position vector, k the complex wave vector perpendicular to the plane of constant phase of a propagating field and E the space–dependent and time–independent 0 vector field. This latter is strictly determined by the problem under study. The complex wave vector can be decomposed in Cartesian components as
k = k xˆ + k yˆ + k zˆ . (1.32) x y z
Therefore, the propagating field along the k direction can be written as
i k y t i(kxx!"t) ( y !" ) i(kzz!"t) E(r,t) = Ex (r)e xˆ + Ey (r)e yˆ + Ez (r)e zˆ . (1.33)
Furthermore, when a different coordinate system is used the k vector is decomposed in each corresponding component, such as cylindrical or spherical. It is widely used the notation ! for representing the component of the k vector along the propagation direction. This notation will be used in the analysis to come.
1.7–Electric and magnetic field components.
The vectorial curl definition is applied to Maxwell’s equations for the time harmonic electric and magnetic fields [3]. Under this representation, equations (1.27) and (1.28) read as
10 Abraham Vázquez–Guardado Chapter 1 INAOE 2012
xˆ yˆ zˆ # # # ! " E = = iµ µ$H , (1.34) #x #y #z 0
Ex Ey Ez
xˆ yˆ zˆ # # # ! " H = = $i% %&E . (1.35) #x #y #z 0
H x H y H z
The first one leads to the magnetic field components,
i $ # # ' H x = ! & Ez ! Ey ) , (1.36) µ0µ" % #y #z (
i $ # # ' H y = ! & Ex ! Ez ) , (1.37) µ0µ" % #z #x (
i $ # # ' H z = ! & Ey ! Ex ) . (1.38) µ0µ" % #x #y (
In the same fashion the components for the electric field are
i % # # ( Ex = ' H z $ H y * , (1.39) !0!" & #y #z )
i % # # ( Ey = ' H x $ H z * , (1.40) !0!" & #z #x )
i % # # ( Ez = ' H y $ H x * . (1.41) !0!" & #x #y )
Depending on the problem and its conditions, this set of eight equations reduces considerable, for example, when treating Transverse Magnetic (TM) or Transverse Electric (TE) fields.
Abraham Vázquez–Guardado 11 INAOE 2012 Chapter 1 1.8–Boundary conditions.
In an inhomogeneous problem, such as waveguides, Maxwell’s equations are only valid within subdomains. Hence we must complement our equation repository with the so– called continuity equations [1,2]. This group, also called boundary conditions, imposes some restrictions on the electromagnetic fields at an abrupt interface separating two media.
In order to define such boundary condition we use Maxwell’s equations in their integral form; instead of the differential representation, since surfaces and volumes are considered instead of differences [2]. Gauss and Stokes theorems are applied to the corresponding equation and leads to their integral form:
% " E(r,t)!ds = $' B(r,t)!ns da , (1.42) "S # &S "t
* $ " ' H(r,t)!ds = J(r,t) + D(r,t) !ns da (1.43) #"S , & ) +S % "t (
D(r,t)!ns da = $(r,t)dV (1.44) #"V #V
B(r,t)!ns da = 0 . (1.45) #"V
In this set of equations, da denotes the surface element, n the normal unit vector to the s surface, ds a line element, dV the surface of the volume V and !S the border of the surface S . The desired boundary conditions are obtained when we restrict the analysis to a sufficiently small part of the boundary. Under this condition, the surface looks flat and the fields homogeneous on both sides.
Consider a very small rectangular closed path !S along the boundary !D , as ij depicted in Figure 1.1–a, as the area S gets smaller, thus the electric and magnetic fluxes through S vanish; this does not apply for the source current since a surface current density
12 Abraham Vázquez–Guardado Chapter 1 INAOE 2012
j might be present. For the two first Maxwell’s equations we get the boundary conditions of the tangential fields components as
n ! E " E = 0 , (1.46) ( i j )
n ! H " H = j . (1.47) ( i j )
n is a unit vector normal to the boundary.
ab D Dij ij D j Dj
Di Di hj2 hi2
n S3 n l S j S1 S2 l i V S V hj1 hi1
Figure 1.1 – Graphical representation of integration for an interface !D between media D and D , which ij i j lead to deduce the boundary conditions. a) Surface integration, b) Volume integration. On the other hand, as depicted in Figure 1.1–b, an expression for the normal field components can be obtained by considering an infinitesimal cylindrical volume V and surface !V . If the fields are considered homogeneous on both sides and a finite surface charge density ! , then from Maxwell’s equation (1.3) and (1.4) we obtain on the interface
!D ij
n! D " D = # , (1.48) ( i j )
n! B " B = 0. (1.49) ( i j )
Abraham Vázquez–Guardado 13 INAOE 2012 Chapter 1 In the most practical situations there are no sources in the individual domains, hence the terms j and ! become zero. These equations are not independent of each other since the fields on both sides of !D are linked by Maxwell’s equations [1]. ij
1.9–Complex permittivity function.
Light and matter interaction is a well–known phenomenon in several areas. Among the main characteristics and properties is the distinct behavior at different frequencies [3]. Lets consider a linear, isotropic, spatially nondispersive and temporarily dispersive media, as a metal for instance, then we can write the constitutive equations (1.12) and (1.16) as
D ! = " " ! E ! , (1.50) ( ) 0 ( ) ( )
J ! = " ! E ! . (1.51) ( ) ( ) ( ) Then, we take the corresponding Fourier representation of equations (1.6) and (1.8), which leads to the following expression
# (" ) ! (" ) = 1+ i (1.52) !0" In the interaction of light with metals, which is considered spatially local, this representation is valid as long as the wavelength ! is longer that all characteristic dimensions. For example the size of the unit cell or the mean free path of electrons; this is in general valid even at ultraviolet frequencies though [3]. In general, the permittivity function is complex, ! (" ) = !#(" ) + i!##(" ) , where the prime and double prime terms refer to the real and imaginary part, respectively. The refractive index n! of the material is related with the permittivity function as
n! ! = " ! . (1.53) ( ) ( )
14 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 It also follows the dependency of the wavelength. Since the permittivity function is complex then the refractive index is complex too n! ! = n ! + i" ! . Combining both ( ) ( ) ( ) expressions we arrive to the following set of equations
2 2 !" = n #$ , (1.54)
!"" = 2n# , (1.55)
!" 1 n2 = + !"2 + !""2 , (1.56) 2 2
"## ! = . (1.57) 2n The term ! is called the extinction coefficient and determines the optical absorption of electromagnetic waves propagating through the medium. In addition, the following equation links this term with the absorption coefficient ! of Beer’s law [3], which describes the intensity attenuation of a beam that propagates in a given medium I x = I e!"x . 0 ( ) 0
2# (" ) ! (" ) = (1.58) c Therefore !"" determines the absorption inside the material. It is a good indicative parameter for determining the quality of a material. On the other hand, for materials with
"! "!! , the real part of the refractive index is mainly governed by !" . !
1.10–Conservation of energy.
So far, the equations mentioned above mainly describe the behavior of the electric and magnetic fields. They are closely related to Maxwell’s equations and the properties of matter. Maxwell’s equations were initially postulated to explain the forces in Coulomb’s and Ampère’s laws, but they fail in providing any information about the forces or energy within a system. The Poynting’s theorem provides a more reliable relationship between the electromagnetic field and the energy it is carrying [1].
Abraham Vázquez–Guardado 15 INAOE 2012 Chapter 1
It follows from the scalar product of the field E and equation (1.2), and subtracting it from the scalar product of the field H and equation (1.1) results in
%B %D H!(" # E) $ E!(" # H) = $H! $ E! $ j!E . (1.59) %t %t
The LHS of this equation is equivalent to ! " E # H . Then, following the integration over ( ) space and using Gauss’s theorem we get
, & #B #D ) (E ! H)"nda = % H" + E" + j"E dV . (1.60) $#V . ( + -V ' #t #t * Furthermore, in spite of its straightforward representation of the Poynting’s theorem, this equation can provide additional physical information when substituting B and D from their general form, equations (1.6) and (1.7). Then equation (1.60) follows to be
1 # 1 , & #P #E ) 1 , & #M #H ) (E ! H)"nda + D"E + B"H dV = % j"E dV % E" % P" dV % H" % M" dV . (1.61) $#V $V [ ] $#V . ( + . ( + 2 #t 2 -#V ' #t #t * 2 -#V ' #t #t *
This result is a consequence of Maxwell’s equation, hence its validity. The Poynting’s theorem is an interpretation of this equation. The first term is equal to the net energy flow in or out the volume V , the second term is the time rate of change of electromagnetic energy inside V and the right side of the equal sign are the rate of energy dissipation inside V [1]. Then, in accordance with this interpretation, the Poynting’s vector reads as
S = E ! H , (1.62) and represents the energy flux density. In addition, the density of electromagnetic energy is
1 W = #"D!E + B !H%$ . (1.63) 2 In circumstances when the medium is linear, the dot product of the last two terms of equation (1.61) is zero and the remaining term is only j!E . These vanishing terms can be associated to the nonlinear losses in the material [1].
16 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 Electromagnetic field magnitudes vary very fast in time, and thus the Poynting’s vector does. Then the mean time value of S is of great interest. It describes the net power flux and is required for the evaluation of radiation patterns. If we assume that all fields are harmonic in time and the material is linear, then equation (1.61) becomes
1 S !nda = $ Re j% !E dV . (1.64) #"V 2 #V { }
The RHS term represents the mean energy dissipation within the volume V . The time average Poynting’s vector is then
1 S = Re E ! H* . (1.65) 2
1.11–The permittivity function of the free electron gas.
The electric properties of a material, particularly metals, are well explained by the Drude–Sommerfeld model. This model can explain their optical properties on a wide frequency range [3]. The model assumes a gas of free electrons of number density n that e moves against a fixed background of positive ion core. The lattice potential and electron– electron interactions are not taken into account but it is assumed that some aspects of the band structure are within the effective optical mass m of each electron. Since electrons oscillate under an external electromagnetic field, then their motion is damped due to collisions; with a characteristic collision frequency ! = 1 " , ! is known as the relaxation time of the free electron gas. This last parameter typically receives values of the order 10!14 s at room temperature.
The equation of motion of electrons in the plasma sea, under external stimuli E, is
m!x! + m! x! = "eE . (1.66)
Abraham Vázquez–Guardado 17 INAOE 2012 Chapter 1
2
1.5 p ω
/ 1 ω
0.5
0 0 0.5 1 1.5 2 kc ω / p Figure 1.2 – Dispersion relation graph of the free electron gas. The solid line represent the plasma dispersion and the dashed line the light line. As it can be observed electromagnetic wave propagation is not allowed for ! < ! p .
!i"t For a harmonic time dependence of the driving field, E(t) = E0e , there is a particular
!i"t solution which describes a harmonic oscillation of electrons, x(t) = x0e . The complex term x0 includes any phase shifts between the driving field and its response. In a general form the response of the electrons is given by
e x(t) = E. (1.67) m(! 2 + i" ! )
The displaced electrons contribute to the macroscopic polarization in the form of P n ex . It follows that this equation along with equation (1.12) gives the result for the = ! e frequency dependent permittivity function.
" 2 1 p ! (" ) = # 2 . (1.68) " + i$"
2 ! p = nee m"0 is the plasma frequency of the free electron gas. This equation is also known as the Drude model of metal optical responses. This last equation can be further decomposed in real and imaginary parts of the permittivity function ! (" ) = !#(" ) + i!##(" ) as
18 Abraham Vázquez–Guardado Chapter 1 INAOE 2012
# 2% 2 "!(# ) = 1$ p , (1.69) 1+# 2% 2
# 2$ "!!(# ) = p . (1.70) # (1+# 2$ 2 )
– – – – – – – – – – – – – – – – –
+ + + + + + + + + + + + + + + +
Figure 1.3 – Representation of volume Plasmons, which are longitudinal collective oscillations of the conduction electrons of a certain metal.
1.12–Free electron gas dispersion and volume plasmons.
In the transparency regime, where ! > ! , we have the following characteristic p dispersion equation
! 2 = ! 2 + k 2c2 (1.71) p where k is the wave vector.
There are two implications that are worthwhile to discuss when considering a generic free electron metal, as seen in Figure 1.2. First, for ! < ! , the propagation of p transverse electromagnetic waves is forbidden; meanwhile, for ! > ! , the plasma supports p transverse waves [3].
Abraham Vázquez–Guardado 19 INAOE 2012 Chapter 1 The impact of the plasma frequency can be explained by recognizing that in the small damping limit, where ! (" p ) = 0 , the excitation corresponds to a collective longitudinal mode. In this case, D = 0 = !0E + P " E = # P !0 which implies that at the plasma frequency the electric field turns to be a pure depolarization field [3].
The physical meaning of the excitation at the plasma frequency can be further understood by the following example. Consider a collective longitudinal oscillation of the conduction electron gas versus a fixed positive background of the ion cores in a plasma slab, which is depicted in Figure 1.3. In this image a collective displacement of the electron cloud by a distance u leads to a surface charge density ! = ±neu at the slab boundaries.
Consequently, a homogeneous electric field E = neu !0 is formed within the slab. Therefor, the displaced electrons experience a restoring force whose movement can be described by the equation of motion nmu!! = !neE [3]. Inserting the last electric field expression into the equation of motion we obtain
u!!+! 2u = 0 . (1.72) p In this expression the plasma frequency is recognized as the natural frequency of a free oscillation of the electron sea under the assumption that electrons move in phase. The quanta of these charge oscillations are called plasmons, and more specifically, volume plasmons [3]. Their longitudinal nature does not allow them to couple to transverse electromagnetic waves and they are only excited by particle impact.
Nevertheless, when the charge oscillation couples to transverse electromagnetic field in system of two mediums, a conductor and a dielectric, two other similar modes might appear depending on the system. They are localized plasmons in nanoparticles and surface plasmons in flat interfaces. Both exhibit evanescently decaying field transversely to the interface. This is a signature of the strong confinement to the interface, or the waveguide for instance [3].
20 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 Localized surface plasmons are non–propagating excitations of the conduction electrons of metallic nanostructures coupled to the electromagnetic field. This kind of modes arise as a fundamental solution to a scattering problem of a small, sub–wavelength conductive nanoparticle in presence of an external oscillating electromagnetic field [3].
Surface plasmons, on the other hand, are two dimension propagation surface electromagnetic waves between a conductor and a dielectric; for example in an interface or a metal and a dielectric medium. Confinement can be in one or two transverse direction. And they only couple to transverse magnetic waves [3]. This kind of solution also called Surface Plasmon–Polariton is the main topic of this thesis and it will be study in the following chaptes.
1.13–Real metals and interband transitions.
The validity of Drude–Sommerfeld model provides an accurate representation of the permittivity function of metals in the infrared regime. However, this model needs to be supplemented in the visible range, where the response of bound electrons is considered. This behavior can be observed in gold, for instance, where at wavelengths shorter that 550nm, the measured imaginary part increases faster as predicted by the Drude– Sommerfeld theory. The cause is that photons at higher energies can promote electrons of lower–lying bans into the conduction band, which in metals exist in lower–lying shells of the metal atoms [3]. Analogously, the motion equation for this case is
mx m x m 2x eE (1.73) !! + ! ! + " 0 = where m is now the effective mass of the bounds electrons and 2 the spring constant of ! 0 the potential that keeps the electron in place. Using the same time harmonic behavior for the oscillation response we arrive to
Abraham Vázquez–Guardado 21 INAOE 2012 Chapter 1
"! 2 ! (" ) = 1+ p . (1.74) (" 2 #" 2 ) # i$ " o
The term !! 2 = n e2 / m" stands for the plasma frequency but with a different physical p b 0 meaning than before, and n the density of the bound electrons. This last equation can be b separated into its real and imaginary part as
#! 2 (# 2 $# 2 ) "!(# ) = 1+ p 0 , (1.75) (# 2 $# 2 )2 +% 2# 2 0 $ #! 2# "!!(# ) = p . (1.76) (# 2 %# 2 )2 +$ 2# 2 0 In Figure 1.4 the real and imaginary parts of the permittivity function are plotted for Gold. A resonant behavior is observed for the imaginary part, blue graph, and dispersion–like for the real part, red graph. Nonetheless, this approximation fits very well experimental data for a certain regime it fails in other whose more interband transitions are not included in the model.
6
5
4
3
2 interband
ε 1
0
−1
−2 200 300 400 500 600 700 800 900 1000 λ [nm] Figure 1.4 – Contribution made by bound electrons to the permittivity function of gold with these parameters: !! = 45 "1014 s#1 , ! = 9 "1014 s#1 , and ! = 2"c # with ! = 450 nm. The red line is the real part p 0 and the blue line the imaginary part of the permittivity function.
22 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 1.14–Group and phase velocities.
A temporal pulse of light is a mathematically composition of a group of monochromatic waves at different frequencies [3]. Each component weight is modulated by an envelope function. If we consider that function a Gaussian envelope, then the temporal optical pulse is graphically represented as that observed in Figure 1.5.
When the pulse travels in a dispersive media each component separately travels at different speed. For example the dispersion observed in a prism when white light travels across. However, when viewed as a whole, its apparent velocity depends on the extent of the spread of individual monochromatic wave velocities. While each monochromatic wave travels at its own phase velocity ! , the entire pulse does at a group velocity ! [4]. i g
1
0.5
0 E [a.u.]
−0.5
−1 −5 −4 −3 −2 −1 0 1 2 3 4 5 t [a.u.] Figure 1.5 – Graphic representation of a mathematical temporal pulse. The carrier wave with, solid line, is modulated in amplitude by the dashed lines, a Gaussian envelope. The finite property of this temporal pulse produces the appearance of a, theoretically, finite frequency spectrum. Consider now a complex electric field of a monochromatic plane wave of amplitude E propagating in the z direction of the form 0
i kz t E z,t = E e ( !" ) . (1.77) ( ) 0 When we speak of the propagation of the plane wave, we mean the motion of the wave’s phase fronts, or surfaces of constant phase. Lets define ! = kz "#t and rewrite equation
Abraham Vázquez–Guardado 23 INAOE 2012 Chapter 1
(1.77) as E z,t = E ei! . We see that the phase fronts are defined by constant values of ! . ( ) 0 At z = 0 and t = 0 we observe the motion of the phase font, which gives the relation
kz !"t = 0 . The wavevector can also be defined as k = n! c . If we use this last definition and substitute it into the previous equation, dispersion in k and n is considered but not absorption, we arrive to
! c z = t = t " # t . (1.78) k ! n ! p ( ) ( )
The term ! p is the phase velocity [5], which is the speed of propagation of the phase front of a monochromatic propagating wave, which is then expressed as
" c ! p (" ) = . (1.79) k " n " ( ) ( ) On the other hand, the group velocity in an optical pulse can be calculated by requiring that the pulse shape remains unchanged while in propagation, i.e. every plane– wave Fourier component must maintain its relative amplitude and phase. For this reason absorption has to be considered. Lets define the complex frequency dependent wavevector as
" (! ) n(! )! " (! ) kc (! ) = k(! ) + i = + i . (1.80) 2 c 2
Dispersion is considered in the frequency dependency of the refractive index. The term ! stands for the linear intensity absorption per unit length experienced by a plane wave,
2 2 i nz c t 2 z 2 I z ! E z,t = E e "( # )e#($ ) = E e#$z . Now lets assume that all Fourier components ( ) ( ) 0 0 are in a narrow bandwidth around a center frequency , also known as the carrier !" ! 0 frequency. Then from equation (1.80) we expand the real and imaginary parts in a Taylor series around leading to ! 0
24 Abraham Vázquez–Guardado Chapter 1 INAOE 2012
1 1 # k k(! ) = k + k "! + k "! 2 + k "! 3 +! = m ("! )m , (1.81) 0 1 2 2 6 3 $ m! m=0 1 1 $ ! !(" ) = ! +! #" + ! #" 2 + ! #" 3 +! = n (#" )n . (1.82) 0 1 2 2 6 3 % n! n=0
m m n n We have adopted the definitions km = d k d! , ! m = d ! d" and !" = " #" 0 to ! =!0 " ="0 ease the mathematical handling. Now a wave packet consisting in a sum of plane–waves Fourier components is defined as
$ i([k(! )+i" (! ) 2]z#!t) E(z,t) = E0 (! )e d! ; (1.83) %#$ by substituting equations (1.81) and (1.82) and using !" = " #" 0 and d! = d"! we arrive to
% % k z # % iz m ($" )m ! n ($" )n #0z & m! 2& n! i(k0z!"0t ) ! 2 ' i$" (k1z!"t ) m=2 n=1 E(z,t) = e e ) E(" 0 + $" )e e e d$" (1.84) (!%
For an unchanging shape propagation pulse, the relative amplitudes and phases of plane– wave Fourier components must remain constant. However, changes in the phase and amplitude are allowed. It can be reflected in an arbitrary complex constant rei! , hence the field at time t and position z is
$ i! i! E(z,t) = re E(0,0) = re E0 (" )d" . (1.85) %#$ Therefore, equation (1.85) must satisfy and has to be equivalent to (1.84). This condition can be possible when the following conditions satisfy ass well.
!"0z 2 r = e and ! = [k0z "# 0t], (1.86)
t = k1z , (1.87)
k k 0 and 0 . (1.88) 2 = 3 = ! = !1 = ! 2 = ! 3 = ! =
Abraham Vázquez–Guardado 25 INAOE 2012 Chapter 1 In first instance, equations in (1.86) are right away fulfilled since they account for change in amplitude and phase. Equations in (1.88) have direct implication in pulse propagation and distortion. Finally, equation (1.87) describes the group delay
dk ! " t = k z = z (1.89) g 1 d# or the time t required for a certain pulse to propagate a distance z , where the derivative is evaluated at ! = ! 0 . Consequently, the group velocity or the speed at which the pulse propagates is defined as
+1 z % dk ( !g " = ' * . (1.90) # g & d$ )
If we introduce the equivalence k = n! c in (1.90) then it follows that dk d! = 1 c(n + dn d! ) , which after substituting it in equation (1.90) the group velocity gives
)1 # dn(" )& c !g = c% n(" ) + ( = . (1.91) $ d" ' ng
ng = n(! ) + dn(! ) d! is known as the group index. This first term, as explained above, is the refractive index or the phase index. The second term is a measurement factor of dispersion; normal when it is positive and anomalous when negative. Commonly, most materials are non dispersive, which is reflected in ng ! n and !g " ! p .
1.15–Slow, fast, backward and stop light.
When light travels at very slow group velocities compared to the phase velocity,
! << ! , or large group index, then we refer as slow light [7]. In this case, the term g p
26 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 accounted for dispersion in equation (1.91) is relatively large for certain materials that can induce large group index [8].
Fast light occurs when !g > c or 0 ! ng < 1. This kind of propagation is also called “superluminal” since the propagation of the pulse is greater that the speed of light in vacuum [9]. In the limit, when group index is nearly zero, the term critically anomalous dispersion regime is used; in such a regime the pulse appears to leaves the medium exactly at the same time it enters it [10]. However, this phenomenon, contrary as expected, is within the bounds of causality and special relativity, hence its physical viability. Fast light can also be accounted when !g > ! p or 0 ! ng < n .
In the case of backward light ng < 0 a pulse traveling in a medium with !g < 0 gives the appearance of backward propagation. Under this regime, as depicted in Figure 1.6, the pulse appears to leave the medium before it enters. The peak of the pulse inside the medium propagates backward from the time the outbound pulse leaves to the time the inbound pulse enters; energy still flows in the forward direction though [11]. Causality is not violated; the effect can be seen as a pulse reshaping phenomenon in which the long leading edge of a Gaussian pulse is amplified and attenuated the original peak.
ng < 0 region
Figure 1.6 – Time slotted representation of backward pulse propagation in a medium that can support
ng < 0 [7].
Abraham Vázquez–Guardado 27 INAOE 2012 Chapter 1 Finally, stopped light or stored light is a special case of slow light. Here the light pulse is stopped or trapped inside the medium for some amount of time. The pulse, as it enters the medium, slows down considerably and, by some means, the medium is altered or stimulated so the pulse can be retrieved again. This can be possible if we induce a zero group velocity, or infinite group index; however there are other ways to do this, by mapping the pulse into the spin coherence or a coherently prepared atomic medium for instance [12].
In general, these four regimes of operation required strong material dispersion, which can be induced by material resonance, structured systems or another external means as it will be introduced shortly.
1.16–Slow light mechanisms.
1.16.1–The Kamers–Kronig relations and the susceptibility.
The susceptibility of a medium !(" ) gives direct implications in the refractive index n(! ) and the absorption coefficient !(" ) [5] Then, when an electric field is applied to a medium, electrons and protons shift positions in response to the field. This shift, consequently, produces another electric field, which macroscopically is called the polarization density P . As one would expect, some materials are more susceptible that others to be polarized by an incident electric field. This property is known as the electric susceptibility ! introduced in equation (1.14) and led the relation ! = 1+ " . Since
ñ = n + i! and ! = 2"# c , then the refractive index and the absorption coefficients become
n(! ) = Re{ " } = Re{ 1+ #(! )} , (1.92)
2" 2" !(" ) = Im # = Im 1+ $(" ) . (1.93) c { } c { } There can be made an approximation when the susceptibility is small, which leads to
28 Abraham Vázquez–Guardado Chapter 1 INAOE 2012
1 n(! ) " 1+ Re{#(! )} (1.94) 2
" !(" ) # Im{$(" )} (1.95) c Here we can see that, under this approximation, the susceptibility real part is the main contributor to the refractive index and the imaginary part to the absorption coefficient. On the other hand, when the susceptibility is high enough, this approximation fails and the more general form of equations (1.92) and (1.93) should be used.
In addition, the response of the medium to an electromagnetic field must be causal. It means that, any change in the polarization must be caused by changes in the electric field that happen before time. This is an important requirement since it is a physical system that is governed by physical laws.
Furthermore, the electric susceptibility of any causal material also follows the Kramers–Kronig relations [5], which are:
& 2" ' Re{!("%)} Im{!(" )} = # ) 2 2 d"% (1.96) $ (0 "% #"
& 2 ' "$Im{!("$)} Re{!(" )} = ) 2 2 d"$ (1.97) # (0 "$ %"
These equations are a special case of the Hilbert transform, where the holomorphic property of optical constant was considered [13]. They give us some important physical results. First, for materials that exhibit absorption they must also exhibit dispersion. Hence, a material that is dispersive must also possess some spectral variation in absorption. In addition, these expressions dictates that the refractive index will be nearly linear at the neighborhood of a smooth peak or valley in the absorption spectrum and its derivative can be negative or positive according to the concavity of the peak.
Abraham Vázquez–Guardado 29 INAOE 2012 Chapter 1 1.16.2–Resonance in materials.
One easy way to account for the slow light feature of a material is to analyze the system from the inside, the atomic structure [4]. Consider the motion of bound charged particles in a certain material, such as electrons bound to atoms or molecules or nuclei within a crystal lattice. This system can be described as a damped harmonic oscillator, whose equation of motion can be written as
2 eE !x! +! x! +" r x = . (1.98) m The term x is the particle’s displacement from its equilibrium position, e the particle charge, E the magnitude of the electric field, m the charged particle’s mass, ! r the damping coefficient and ! r the resonance frequency. Under these conditions, the susceptibility of the medium comes to be
1 !(" ) # 2 2 . (1.99) " r $" $ i2"% r
a b
Figure 1.7 – Dispersive and absorptive features of a resonant material. a) Dispersion, green line, and absorption, blue line, or a resonant material with resonant frequency ! r and linewidth ! r . b) Group index of such resonant medium. Pay attention in frequencies around ! r where fast light behavior is exhibited while around ! r ±" r slow light is observed [4].
According to equation (1.93) the absorption spectrum shows a Lorentzian line shape centered at ! r with linewidth ! r . The last equation can be expressed graphically, in Figure
30 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 1.7–a where dispersion and absorption are plotted as blue and green graphs, respectively. The group index is displayed in Figure 1.7–b, where fast and slow regions are present at the neighborhood of the resonance frequency.
In general, for any system there might be more that one resonance spread in the absorption spectrum, each one with its corresponding center frequency, linewidth and relative strength. This property is intrinsically in a material; however there are other systems that present Lorentzian or other lineshape resonances, lasing for instance, whose behavior can be modeled with the aforementioned phenomenon.
One direct implication of a slow light medium in an optical pulse is spatial compression with a factor equal to the group index. Consider a pulse of duration ! that enters in a slow light medium. Its velocity reduces from c to c ng as well as its length from
L = c! to L! = L ng . Energy conservation principia dictate that if the pulse energy distributed over L compress down to L! then the energy density u must increase by the same factor, i.e. u! = ngu . However, the intensity of such a pulse travels unchanged in the slow light medium due to the increase in u .
Similarly, the electric field magnitude remains unaffected. Therefore, even when there is spatial compression in such slow light medium, the electric field peak keeps the same [14].
There are other several SL systems whose behavior is due to resonance in the optical response. For example, quantum–mechanical optical interaction models such as the two– level and three–level models. The optical field energy is such that it can induce resonance between the atomic energy levels. In a two level system this resonant or quasi–resonant field is such that can exert a transition between the states 1 and 2 , Figure 1.8–a. In a three
level system, the fields !13 and ! 23 are responsible for the transitions between 1 ! 3 and
Abraham Vázquez–Guardado 31 INAOE 2012 Chapter 1
2 ! 3 respectively, Figure 1.8–b. In both cases, the resonance in the absorption peak propitiates the appearance of a slow light regime.
a b
Figure 1.8 – Energy–level models of an atom or any other quantum–mechanical system. a) Two–level model, b) Three–level model. The energy of an incoming wave with frequency ! , ! p or ! c , is or nearly is resonant with energy levels 1 and 2 , 1 and 3 or 3 and 2 , respectively.
Electromagnetically induced transparency is another phenomenon that can propitiate slow light behavior in a system [15]. It is the case of a three–level ! –type model of an atom. First the atom is at the ground state 1 when a strong pump field at frequency
! c is applied to the 2 ! 3 transition such that it introduces coherence between 1 ! 3
and 2 ! 3 transitions. If a weak probe at ! p is applied to the 1 ! 3 transition, then it will undergo little or no absorption, whereas it would ordinarily be absorbed right away.
Since the hole created in the probe absorption spectrum at ! p is very narrow, then by
Kramers–Kronig relations the medium will have a large value of dn d! , thus a large group index at such frequency.
Coherently population oscillation is another effect occurring in a two level system [16]. It happens when a pump beam at ! and a probe beam at ! +" are applied to the same transition, i.e. 1 ! 2 . If both ! and ! +" are contained within the natural linewidth 1 T1 of the transition, then a portion of the atomic population will oscillate between levels 1 and 2 at the beat frequency ! . Consequently the oscillation produces a narrow hole in the absorption line centered at ! yielding to a rapid index variation, hence producing slow light.
32 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 Another effects that can induce a resonance, and hence slow light behavior, are stimulated Raman scattering (SRS) [17], see Figure 1.9, and stimulated Brillouin scattering (SBS) [18]. A pump field at ! is applied to the system and it scatters off a vibrational wave of frequency ! , then a scattered field is generated at the Stokes frequency ! S = ! " # . In the case of SBS ! can be several Gigahertz while in SRS it can reach up to some Terahertz.
Consequently, if a probe signal at ! S is applied to the medium, it will experience gain as energy is scattered from the pump into the Stokes frequency. Therefore, according to Kramers–Kronig relations, SBS or SRS gain line induces a slow light dispersion curve around ! S .
Figure 1.9 – Energy–level model representing the stimulated Raman scattering (SRS). A pump laser at frequency ! gets into the system and couples to molecules (or atoms) from the ground vibrational state and takes them up to an excited state. Consequently they emit Stokes photons at ! s = ! " # when they eventually decay to the lower intermediate state producing gain at the Stokes frequency. The presence of the pump, in the absence of four wave mixing effects, anti–Stokes photons at ! a = ! + " can be observed by an stimulated Raman transition in ground–state molecules, which is translated to losses in the anti–Stokes frequency. Finally, there is another phenomenon that can exhibit slow light around its natural resonance frequency, which is the main topic of this work. Surface Plasmon–Polaritons (SPP) present around the plasmon resonance a region whose group index can reduce considerably and will be the material for the following chapter.
Abraham Vázquez–Guardado 33 INAOE 2012 Chapter 1 References.
[1] L. Novotny and B. HechtT, Principles of Nano–Optics, 1st ed. (Cambridge University Press, 2006), p. 539. [2] Jackson, Classical Electrodynamics, 3rd Ed, 3rd ed. (John Wiley & Sons, 2007), p. 832. [3] S. A. Maier, Plasmonics, 1st ed. Fundamentals and Applications (Springer Verlag, 2007), p. 223. [4] J. E. Vornehm Jr and R. W. Boyd, Tutorials in Complex Photonic Media, 1st ed. (SPIE, 2009). [5] V. Lucarini, Kramers–Kronig Relations in Optical Materials Research, 1st ed. (Springer Verlag, 2005), p. 160. [6] J. C. Maxwell, "A dynamical theory of the electromagnetic field," Phil.Trans.Roy.Soc.Lond. 155, 459–512 (1865). [7] R. W. Boyd, "Slow and fast light: fundamentals and applications," Journal of Modern Optics 56, 1908–1915 (2009). [8] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594–598 (1999). [9] L. J. Wang, A. Kuzmich, and A. Dogariu, "Gain–assisted superluminal light propagation," Nature 406, 277–279 (2000). [10] P. G. Eliseev, H. Cao, C. Liu, G. A. Smolyakov, and M. Osi"ski, “Nonlinear mode interaction as a mechanism to obtain slow/fast light in diode lasers,” in Physics and Simulation of Optoelectronic Devices XIV, M. Osi"ski, F. Henneberger, and Y. Arakawa, Eds., Proc. SPIE 6115, 61150U (2006). [11] G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backwards pulse propagation through a medium with a negative group velocity,” Science, 312, 895–897 (2006). [12] A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001). [13] F. W. King, Hilbert Transforms: Volume 2, 1st ed. (Cambridge University Press, 2009), p. 698. [14] S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999). [15] S. E. Harris, J. E. Field, and A. Imamo#lu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107‒ 1110 (1990).
34 Abraham Vázquez–Guardado Chapter 1 INAOE 2012 [16] M.S. Bigelow, N.N.Lepeshkin and R.W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [17] F. L. Kien, J. Q. Liang, and K. Hakuta, “Slow light produced by far–off– resonance Raman scattering,” J. Sel. Topics Quantum Electron. 9, 93–101 (2003). [18] R.Y. Chiao, C.H. Townes, B.P. Stoicheff, “Stimulated brillouin scattering and coherent generation of intense hypersonic waves,”Phys. Rev. Lett. 12, 592 (1964). [19] M. Sandtke and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photon. 1, 573–576 (2007).
Abraham Vázquez–Guardado 35 INAOE 2012 Chapter 1
36 Abraham Vázquez–Guardado Chapter 2 INAOE 2012
Chapter 2 Slow Light Surface Plasmons.
Summary
In the development of this chapter we will review [1,2] and analyze, the planar two and three–layer structure that support SPPs and their SL property [3]. We start with the calculation of the wave equation for planar interfaces, from the full complex wave equation to the problem–specific equation. First the planar two–layer planar interface is treated and the SL property analyzed. After this, the planar three–layer is analyzed too. In both cases metal losses are not considered so that the study is mainly qualitative. Then losses are considered in the analysis of both systems. Under this more realistic approach we observe and analyze the implication metal losses have in the performance of SL property of SPPs.
Introduction
Surface Plasmon–Polaritons (SPPs) are TM polarized coupled states between a plasmon and a photon. Different from localized plasmons, they can propagate along an interface between two materials whose real part of electric permittivity is sign opposite [4]. The magnitude of the electromagnetic field is maximum right at the interface and exponentially decays perpendicularly. This characteristic exposes an intrinsic strong confinement in the interface transverse direction [5]. Therefore, this property is quite attractive in fields such as nanophotonics and subwavelength lasers [5,6], imaging [7], biosensors [8] and so on.
SPPs exhibit resonance behavior on its dispersion curve. Around the surface plasmon resonance the SPP group velocity slows down, theoretically to zero. Then a Slow Light (SL) regime appears [9].
Abraham Vázquez–Guardado 37 INAOE 2012 Chapter 2 One of the main challenges in subwavelength plasmonics are metal losses imposed by short lifetimes at visible and near infrared wavelengths. They are caused by energy dissipation, mainly due to electron collisions, interface roughness, and some others [1]. In addition, a SL regime means that light will spend more time interacting with the medium and, therefore, losses in that regime increase too. As a consequence, the predicted high group index achievable in a planar interface is then mitigated in the aforementioned circumstances.
Most equations are deduced from the fundamental equations whose development can be found in the appendix.
2.1–The wave equation for inhomogeneous media.
For a more insightful investigation of the physical properties of this kind of surface electromagnetic waves, we shall apply Maxwell’s equations to an inhomogeneous media composed by a flat interface between a metal and a dielectric. Then the appropriate boundary conditions are imposed. First of all we recast the general form of the wave equation (1.9). Under the lack of external stimuli, for non–magnetic media and considering the unchanging variation of the dielectric constant in the order of one optical wavelength [5] the RHS of that equation equals zero and becomes
# $2 E !2E " = 0 . (2.1) c2 $t 2 Then we consider the harmonic time dependence for an electromagnetic field, i.e. equation (1.31) is inserted into (2.1) that yields
2 2 ! E + k0 "E = 0 , (2.2)
where k0 = ! c is the wave number of the propagating wave in vacuum.
38 Abraham Vázquez–Guardado Chapter 2 INAOE 2012 This equation is also known as the Helmholtz equation. Solutions that satisfy this equation are harmonic and depend on the geometry of the system. For example, dielectric waveguides such as the slab has solutions that fulfill this equation at each subdomain. Then we match each subdomain solution at the interfaces to form a general waveguide solution or modes. Depending on the waveguide dielectric characteristics the system can support guided, radiative o leaky modes [10]. All of them having field distribution in the waveguide that is physically possible. Nevertheless, guided modes are the desired modes in waveguide problems and we try to avoid radiative and leaky modes.
In dielectric waveguides such as the slab, guiding is by means of total internal reflection [11] since the input field confines itself in the core domain. This is a widely used guiding mechanism and need at least three subdomains, whose refractive index has to be greater in the core than in the claddings.
In waveguide that do not have three subdomains to propitiate light confinement by total internal reflection other confinement mechanism can be opted. For example, in the two–subdomain system the electromagnetic field confines right at the interface and exponentially decays transversely at each subdomain. The solution at each interface is then matched to form a waveguide solution or mode that satisfies Helmholtz equation, in the harmonic case, or Maxwell’s equations, in the more general case. These solutions dictate strict conditions in the electric permittivity value in both subdomains; the real part has to be sign opposite [4].
When both subdomains satisfy the permittivity condition and they are only real values, then the propagation constant is real. It means that the propagating mode does not experience losses. On the other hand, when one or both electric permittivity is complex, then the surface wave propagation constant is complex as well. Depending on the real part the mode might experience losses or amplification.
Abraham Vázquez–Guardado 39 INAOE 2012 Chapter 2 Lets consider the two–layer waveguide. This is the simplest planar waveguide that can support surface waves solutions. We assume that the permittivity function is only x dependent. This means that the permittivity function changes in the x direction and shows no spatial variation in the perpendicular direction, yz planes, see Figure 2.1. According to electromagnetic surface problems at the interface plane, x = 0 , the interface supports the surface wave whose electromagnetic field is maximum and decays transversely at each subdomain [4]. This solution can be represented by E(r) = E(x)ei!z where the x dependency of the electric field includes the decay nature. The parameter ! = kz is called the propagation constant and corresponds the propagation direction component of the wave vector. This term can be a real or complex value. The wave equation for this specific case can be obtained as follows. Using this last equation and inserting it into equation (2.2) we obtain the diffusion equation for the electric field, and in the same way for the magnetic field.
y
1( ) 2 ( )
z x
Figure 2.1 – Two–layer system containing a flat interface at x = 0 . For x < 0 the material has permittivity ( ) and, for x > 0 is ( ) where both are homogeneous in y and z directions. Propagation is in the z !1 " ! 2 " direction and the propagation constant is ! = kz .
!2 E(x) + k 2" # $ 2 E(x) = 0 (2.3) !x2 ( 0 )
!2 H(x) k 2 2 H(x) 0 (2.4) 2 + ( 0 " # $ ) = !x
40 Abraham Vázquez–Guardado Chapter 2 INAOE 2012 These wave equations are the starting point for analyzing the guided electromagnetic modes supported by planar waveguides.
Only TM polarized waves can couple to plasma oscillation to form SPPs. This conclusion comes from the boundary condition for both transverse polarized waves. The TE polarization requires a condition that is no physically possible, which is not the case for the TM polarization, see appendix I. From the set of equations (1.39)–(1.41) the TM equations are
" Ex = ! H y , (2.5) # $0$
i $H y Ez = ! . (2.6) " #0# $z
Finally the governing wave equation is then,
!2 H + k 2" # $ 2 H = 0 . (2.7) !x2 y ( 0 ) y Complete physical information of the propagating SPP modes can be retrieved by solving this latter equation and using equations (2.5) and (2.6) [2].
2.2–Two–layer planar interface.
The geometry of a simple two–layer planar interface metal–dielectric is shown in
Figure 2.1. For x > 0 the material corresponds to a non–dispersive dielectric !2 = !d . For
x < 0 the materials is a metal !1 (" ) = ! m (" ) = ! m with the characteristic that Re[! m ] < 0 in some part of the electromagnetic spectrum. In analogy to the dielectric case, modes that propagate in the z direction and present an evanescent behavior transversely, i.e. in x direction, correspond to SPPs, therefore the solution at which we will look for are those having the following profile
Abraham Vázquez–Guardado 41 INAOE 2012 Chapter 2
i!z "kd x H y (x) = A2e e (2.8) for x > 0 and
i!z kmx H y (x) = A1e e (2.9)
for x < 0 . The term ki ! kx,i corresponds to the perpendicular component of the wave vector, where the index i is m for the metal and d for the dielectric. This value is the inverse of the evanescent decay length of the field transverse to the interface and quantitatively defines the wave confinement, xˆ = 1 kx . The continuity condition for the tangential electromagnetic fields, H y and Ex at the interface, x = 0 , leads us to have
A1 = A2 and
k " d = ! d . (2.10) km " m
For every medium, equation (2.7) has to be fulfilled, which yields
2 2 2 km = ! " k0 # m (2.11)
2 2 2 kd = ! " k0 #d (2.12) Finally, combining these last two equations and (2.10) we come with the dispersion relation of the SPP mode propagating along the flat interface
" m"d ! = k0 . (2.13) " m + "d
This equation is valid for real and complex values of metal and dielectric permittivities. Then the SPP propagation constant ! can be real or complex. In the most general form it
1 can be expressed as ! = !" # i 2 $ . The real part determines how the SPP spatial phase changes as it propagates along the interface for a given wavelength while the imaginary part gives the attenuation factor per distance unit. In chapter 1 the group and phase velocity was
42 Abraham Vázquez–Guardado Chapter 2 INAOE 2012 introduced. For this case, the group velocity is related to the real part of the propagation constant ! [3] then from equation (1.91) the SPP group velocity is
+1 % d#"( c !g = ' * = . (2.14) & d$ ) ng
There is another quality of SPP that is worthwhile to analyze. This is the propagation length L , which is the length at which the power decreases a factor of e!1 the launching signal.
1 L = . (2.15) !
Equation (2.13) as its asymptote, which corresponds to the characteristic surface plasmon frequency, at
! p ! sp = . (2.16) 1+ "d
2.2.1–Analysis for a generic Drude–like metal.
A qualitative study of the SL property of SPPs in two–layer systems is to use the Drude model to describe the metal permittivity function without losses; consider the damping factor in equation (1.68) as zero. Then, the permittivity of the dielectric medium is fixed, a nondispersive medium [2]. Then the SPP propagation constant is calculated. Once the propagation constant is calculated, then equation (2.14) calculates the SPP group index.
These parameters are plotted in Figure 2.2 for two different dielectric mediums. The dispersion curve of a SPP mode is in Figure 2.2–a where two sets are observed; the red lines correspond to air ( 1) and the blue ones to fused silica ( 2.25) . In that picture the ! d = ! d = straight dashed lines corresponds to the light lines. That of greater slope is the light line in vacuum. The solid lines are the real part of the SPP propagation constant. Since the
Abraham Vázquez–Guardado 43 INAOE 2012 Chapter 2 permittivity of the metal is accounted to be lossless, then the propagation constant is real or imaginary. Hence the gap in the real part of the SPP between ! and ! , where ! is the sp p sp surface plasmon frequency. In this gap the imaginary part of the SPP mode is nonzero, doted lines.
a b
1 Air Silica 1
ω 0.8 sp,air 0.8 Air ω sp,air p p ω ω / / 0.6 0.6 ω ω ω ω sp,silica sp,silica Silica 0.4 0.4
0.2 0.2
0 0 0 1 2 3 4 0 10 20 30 40 βc/ω n p g Figure 2.2 – Propagation constant and group index of the SPP mode supported by a two–layer system. The Drude model computes the permittivity for a generic metal. The dielectric permittivities are constant for Air and Fused Silica. a) SPP dispersion curve [2], where the red plots correspond the system with Air and the blue ones that for Silica. The straight dashed lines are the light lines, the solid lines are the real part of the SPP propagation constant and the dotted ones the imaginary part. b) Group index for both systems, the red line is the group index for the SPP mode in Air and the blue one that for Silica. Notice that at the frequency asymptote, the group index goes to infinity. One physical characteristic of SPPs is their bound nature; it means that the real part of the dispersion curve lays in the right side of the light line of the dielectric, see Figure
2.2–a dashed and solid plots. As mentioned earlier in chapter 1 when ! > ! p , also known as transparency regime, radiation in the metal occurs meaning that transverse waves are supported. This region, however, is not of main interest, but it does the region for ! < ! sp , a region that supports bound guided modes.
In addition, in Figure 2.2–b the group index for both SPP modes is plotted. The red line is the group index of the SPP mode when the dielectric is air ( 1) and the blue line ! d = when it is (! = 2.25) . Observe that two SL region appear around ! and ! frequencies, d sp p which correspond to the edges of the gap. Around the plasma frequency, both curves share
44 Abraham Vázquez–Guardado Chapter 2 INAOE 2012 the same asymptote and they are overlapped. Nevertheless, each waveguide system, with air of fused silica as dielectrics, has its own surface plasmon asymptote, equation (2.16).
In the range of mid–infrared and lower frequencies, the SPP wavevector magnitude is small and propagates close to the light line. This means that the group index tends to n , i the dielectric refractive index; the group velocity is almost that of the phase velocity. SPPs that are in compliance with these characteristics are known as Sommerfeld–Zenneck waves [12] for their grazing–incidence light field nature.
On the other hand, for large propagation constants, the frequency of SPPs gets near the value of the characteristic surface plasmon frequency ! . It is the asymptote of sp equation (2.13), and is easily appreciated in this lossless case. While "! approaches ! sp the group index asymptotically tends to infinity. Therefore, this mode acquires electrostatic character and is known as the surface plasmon [2]. It is in this regime that high values of group index can be achieved, theoretically infinite values at the SPP asymptote.
ΙΙy Ι ΙΙΙ
2( ) 1 ( ) 3( )
z x −aa
Figure 2.3 – Three–layer sharing two flat interfaces that support bound SPPs on each one. The thickness of the middle layer is 2a . The cladding layers extend far enough to avoid interaction with whatever is beyond their limits. The SPP modes propagate in the z direction with propagation constant ! . The materials are homogenous in the yz plane but discontinuous in the x axis.
In addition, due to losses were not included in the analysis, the imaginary part of the
SPP propagation constant is always zero, for frequencies lower than ! sp . As a consequence,
Abraham Vázquez–Guardado 45 INAOE 2012 Chapter 2 the SPP propagation length is always infinite. Physically this asseveration is not possible due to real metals exhibit losses.
2.3–Three–layer planar interface.
Two–layer systems sharing a flat interface is the simplest system supporting propagating SPPs; however, multilayer systems such as the three–layer waveguide support, similarly, SPP modes [13]. In this system, each single interface can withstand bound SPP modes that propagate much like the aforementioned two–layer system. When the separation between both interfaces gets smaller than the penetration length xˆ in the middle medium, there is an interaction producing coupled modes. With the aim of describing the physical properties of this bound couple SPP modes we consider the structure depicted in Figure 2.3. The structure contains three dielectrics slabs where the one in the middle has a finite width. At the planes x = a there are two flat interfaces.
Among the modes supported by this structure, we are not interested in oscillatory ones along the x direction, radiative modes, but the one that exhibit the confinement signature of exponential decay transversely to the interface. We will focus on TM polarized fields since SPP modes are only present when this polarization is considered. The magnetic field in the y direction will be studied since the other component can be straightforwardly calculated from equations (2.5) and (2.6). For medium II, x < !a , and medium III, x > a , the magnetic field component are
i!z k2 (x+a) H y (x) = A2e e , (2.17)
i!z "k3 (x"a) H y (x) = A3e e , (2.18) respectively. As can be seen, the confinement behavior is expressed in the decaying field nature of the magnetic field in the cladding. Finally, for the medium I, where x < a we have
46 Abraham Vázquez–Guardado Chapter 2 INAOE 2012
i!z "k1(x+a) i!z k1(x"a) H y (x) = A11e e + A12e e . (2.19)
The continuity condition for the tangential component of the electromagnetic field at each interface leads to a linear system of four coupled equations. In addition, from the
2 2 2 governing wave equation (2.7) the wave number at each medium ki = ! " k0 #i must be satisfied. We apply the condition for a nontrivial solution of this linear system and the result is an implicit dispersion relation between ! and ! .
k " + k " k " + k " e!4 k1a = 1 1 2 2 1 1 3 3 (2.20) k1 "1 ! k2 "2 k1 "1 ! k3 " 3
One special case of this structure is that it dispersion equation becomes (2.13) when
a ! ". Under this condition the core is so thick, compared to the penetration length in the core, that two independent modes propagate at each interface without noticing the presence of the other. In the other case where the core is thin enough, 2a < 100 nm in the case of Silver, the physical modal interaction of both flat interfaces becomes two distinct coupled modes [2].
Ex a Ex b
x x
−a a −a a
Ez Ez
x x
−a a −a a Figure 2.4 – Electric field components for the SPP mode in the three–layer system. a) Symmetric mode or LRSPP and b) Antisymmetric mode or SRSPP. For simplicity in the analysis, lets consider that both mediums II and III are a dielectric, !2 = ! 3 = !d , implying that k2 = k3 = kd , Such as the IMI waveguide. From the
Abraham Vázquez–Guardado 47 INAOE 2012 Chapter 2 square root of (2.20) we take the positive and negative signs to end up in two transcendental equations.
kd" m tanh(akm ) = ! (2.21) km"d
km"d tanh(akm ) = ! (2.22) kd" m
These last to equations describe the wave vector of SPP modes with odd and even field parity, the first one describe the odd electric field function parity (odd Ex (x) and even
H y (x) and Ez (x) ), while the second the even electric field function parity (even Ex (x) and
odd H y (x) and Ez (x) ).
The propagation length is one of the key characteristics of these SPP modes and its symmetry dependence is one of the key physical properties. Let’s consider the electric field component in the transverse direction, E ; if the mode field distribution has an even x symmetry, then it is called a symmetric mode. On the other hand if the mode has an odd symmetry, then it is called antisymmetric mode, see Figure 2.4. The propagation length corresponding to the symmetric mode is larger than the antisymmetric mode; therefore it is called the Long Range Surface Plasmon–Polariton (LRSPP) [13]. The electromagnetic energy is located mainly within the dielectric medium, implying that the light interaction within the metal is small propitiating weak confinement to the core. Consequently, the SPP mode tends to have a plane wave behavior, for most of the SPP frequency regime, whose group velocity is quite close to that of the phase velocity of a plane wave traveling in the dielectric.
The other case is the Short Range Surface Plasmon–Polariton (SRSPP), for obvious reasons, and corresponds to the antisymmetric mode [13]. The reason for this behavior is the high confinement in the metal that produces more absorption losses in the propagating
48 Abraham Vázquez–Guardado Chapter 2 INAOE 2012 mode, and enhances other losses mechanisms such as roughness scattering. Thus, the imaginary part of the propagation constant is high.
In order to study the properties of the coupled SPP modes in these structures we analyze the IMI waveguide. It consists on a thin metal film of thickness 2a cladded by two dielectrics. Permittivity ! m (" ) represents that of metal and is frequency dependent, and !d the one for a non–dispersive and non–absorptive dielectric.
For example, lets consider a three–layer system with Silica as the dielectric, 2.25, and Silver as the metal with ( ) . Silver permittivity is approximated to ! d = ! m " experimental data [14] by the Drude model, equation (1.68), and, to describe qualitatively the physical properties of this system, losses will not be considered at this time. The thickness of the core is set at 20 nm.
300 a 200 b 150 350 Slow Light Region 100 LRSPP SRSPP 400 50 SRSPP g
450 n 0 [nm] LRSPP λ Slow −50 Light 500 Region −100 550 −150
600 −200 0 2 468 10 12 300 350 400 450 500 β /k λ[nm] r 0 Figure 2.5 – IMI waveguide with Silica and Silver neglecting losses. The waveguide core has a thickness 2a = 20 nm. a) Dispersion curve for the LRSPP and SRSPP modes. b) The corresponding group index for such modes. In both cases, the SL region near the flat side of the curves is exposed as a shaded rectangle. The LRSPP mode exhibits more interesting SL behavior than the SRSPP mode, Figure 2.5–a. Therefore it is of key importance to consider its SL capabilities. It can be graphically shown that the absolute group index of the LRSPP can be much higher that that of the SRSPP group index. This is shown in Figure 2.5–a, at the flat region within the
Abraham Vázquez–Guardado 49 INAOE 2012 Chapter 2 shaded rectangle just before following the asymptote toward infinity, where it exhibits higher absolute values of group index, as shown in Figure 2.5–b.
Even though group index as high as we desire, while we go far into the plasmon asymptote, can be achieved theoretically, in the real world this turns out to be but an inauspicious asseveration. The metal by itself has intrinsic losses increase considerably near the asymptote.
2.4–Metal losses in SPPs performance.
In the real world metal losses play an important role in the SPPs performance [15], and in SL is not the exception. Near the asymptote, SPP losses become large, i.e. the imaginary part of the SPP propagation constant is large. Consequently, due to the inverse relation between the propagation length and this value, the maximum length that the SPP mode can propagate makes it worthless for any important application involving guiding. In addition, the maximum achievable wave number that the SPP mode can reach easily degrades. It is appreciated as a bend–back behavior in the SPP dispersion curve. As a result, the maximum group index for a considerable propagation length is not but slightly higher than that for a plane wave propagating in the dielectric. This behavior, as it will be exposed shortly, shows in both the two and three–layer systems. Of course, the same behavior is shown in other phenomena where plasmon resonance is accounted for.
First of all, let’s analyze the two–layer system. We consider the same example as before, Silica and Silver but now with metal losses, i.e. "" 0 . The Silica Permittivity is ! m # 2.25 and the one for the Silver is computed using the Drude model. The dispersion ! d = curve, the group index and the propagation length are displayed on Figure 2.6. As a comparative frame, the red line represents the ideal lossless case. In the dispersion curve a bend–back behavior is observed in the real part of the propagation constant, Figure 2.6–a
50 Abraham Vázquez–Guardado Chapter 2 INAOE 2012 blue solid graph. Now, due to the metal lossy property, the imaginary part of the propagation constant is different to zero and it is well displayed by the blue dashed graph in the same figure. The group index graph, blue solid plot in Figure 2.6–b, clearly shows such degradation. It reaches a finite maximum value before falling off again. Finally, Figure 2.6–c shows the propagation length and it is finite, quite a difference with the infinitum propagation predicted in the lossless case. The plot is normalized with respect to L 1 m, 0 = µ that means that for log L / L > 0 the propagation length is greater that 1µm . ( 0 )
340 a 340 b 340 c
360 360 360 [nm] [nm] [nm] λ λ λ 380 380 380
400 400 400 0 5 10 0 500 1000 −2 0 2 β/k n log(L/L ) 0 g 0 Figure 2.6 – Representative graphs for the two–layer system with Silica and Silver including losses. The red plot is that for the lossless case. The horizontal dashed line is the plasmon asymptote and the vertical dashed line is the zero axis. a) Propagation constant, blue dashed line is the imaginary part while the blue solid line the real part. b) Group index, solid blue line. c) Propagation length normalized with L0 = 1µm .
Therefore, the two–layer system can support SL SPPs. Even in the lossy regime, a group index as much as about 500 is theoretically achievable. Nevertheless, for a propagation length much of about 1µm the group index is quite close that of a plane wave traveling in the dielectric. Then for propagation lengths of about 100 or 1000µm the group index is far from the SL regime.
The aforementioned property is observed in the three–layer system as well. Losses near the plasmon asymptote are too high, such that the dispersion curve suffers in the same way the two–layer system does. As a result, it degrades severely the group index that the SPP mode may reach. It can be seen in Figure 2.7 the propagation constant, the group index and the propagation length for the previous three–layer system but accounting losses in the
Abraham Vázquez–Guardado 51 INAOE 2012 Chapter 2 metal. The red line corresponds that of the SRSPP, and black and blue lines that of the LRSPP.
In Figure 2.7 –a, the real part of the dispersion curve is plotted. It can be observed that the SRSPP mode does not have that asymptotic behavior as it was observed in Figure 2.5–a. The same is observed in the LRSPP mode whose dispersion curve has split into two pieces. The group index is shown within Figure 2.7 –b. The aforesaid infinite tendency near the asymptote and in the upper region of the LRSPP mode of the group index is not displayed anymore.
340 50 360 380 0 400
g −50 n [nm] 420 λ 440 −100 460 −150 480 a b 500 −200 0 5 10 350 400 450 500 β /k r 0 λ[nm]
340 2 360 1 380 ) 400 0 0
[nm] 420 −1 λ
440 log(L/L −2 460 480 cd−3 500 −4 −4 −2 0 2 4 350 400 450 500 β /k λ i 0 [nm] Figure 2.7 – Representative graphs for the three–layer system with Silica and Silver with losses with a 20nm thick core. The red graph is that for the SRSPP mode. Black and blue graphs stand for the split curves of the LRSPP. Horizontal back dashed line in a and c and vertical in b and d is the plasmon asymptote. The horizontal magenta dashed line in b and d is the zero reference. a) Real part of propagation constant. b)
Group index. c) Imaginary part of propagation constant. d) Propagation length normalized with L0 = 1µm .
52 Abraham Vázquez–Guardado Chapter 2 INAOE 2012 Furthermore, the imaginary part of the propagation constant and the propagation length, normalized to L0 = 1µm are exposed in Figure 2.7–c and Figure 2.7–d, respectively. It can easily be observed that, the propagation length for the LRSPP mode is about two orders of magnitude above that of the SRSPP mode, and here the reason for their names [16]. The analysis of this property is a key point in the evaluation of these SPP modes for guiding applications where SL properties are needed [17].
In this system, even though high group index are achievable, despite the metal losses, the propagation length is still under 1µm . Consequently further consideration needs to be engaged to overcome or mitigate this restriction, as it will be discussed in the following chapter.
References.
[1] H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, 1st ed. (Springer, 1988), p. 136. [2] S. A. Maier, Plasmonics, 1st ed. Fundamentals and Applications (Springer Verlag, 2007), p. 223. [3] J. E. Vornehm Jr and R. W. Boyd, Tutorials in Complex Photonic Media, 1st ed. (SPIE, 2009). [4] J. Crowell and R. H. Ritchie, "Surface–Plasmon Effect in the Reflectance of a Metal," Journal of the Optical Society of America 60, 794–799 (1970). [5] W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824–830 (2003). [6] R. F. Oulton, V. J. Sorger, T. Zentgraf, R.–M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, "Plasmon lasers at deep subwavelength scale," Nature 461, 629–632 (2009). [7] S. Kawata, Y. Inouye, and P. Verna, "Plasmonics for near–field nano–imaging and superlensing," Nature Photon 3, 388–394 (2009). [8] J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, "Biosensing with plasmonic nanosensors," Nature materials 7, 442–453 (2008). [9] J. B. Khurgin, Slow Light, 1st ed. Science and Applications (CRC, 2008), p. 404. [10] J. Hu and C. R. Menyuk, "Understanding leaky modes: slab waveguide revisited," Adv.
Abraham Vázquez–Guardado 53 INAOE 2012 Chapter 2 Opt. Photon. 1, 58 (2009). [11] K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis, 1st ed. Solving Maxwell's Equations and the Schrödinger Equation (Wiley–Interscience, 2001), p. 275. [12] G. Goubau, "Surface waves and their application to transmission lines," J. Applied Physics 21, 1119–1129 (1950). [13] E. N. Economou, "Surface Plasmons in Thin Films," Physical Review 182, 539–554 (1969). [14] P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370–4379 (1972). [15] J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface–polariton–like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186–5201 (1986). [16] P. Berini, "Long–range surface plasmon polaritons," Adv. Opt. Photon. 1, 484 (2009). [17] C. G. A. S. I. B. Thomas W Ebbesen, "Surface–plasmon circuitry," Physics Today 44– 50 (2008).
54 Abraham Vázquez–Guardado Chapter 3 INAOE 2012
Chapter 3 Slow Surface Plasmons in amplifying media.
Summary
In this chapter we analyze the Slow Light (SL) property of Surface Plasmon– Polaritons (SPPs) when the dielectric medium host an active material. In order to include gain to the dielectric, we add an imaginary part to the dielectric permittivity. First of all we review the condition for bound modes, characteristic property of surface waves. We find the condition for this imaginary part in order to keep the bound condition. Then, we also study the mode propagation constant and set the condition for it to be an amplified mode. Then, we review the constant gain model, widely used to model gain in plasmonics waveguides. Under this approach, we analyze the gain coefficient and find the gain condition for a mode lossless propagation. Finally, we introduce the finite spectral gain model. Whit this approach, we study the spectral dispersion in the dielectric and its SL implication in the SPP group index. For its mathematical simpleness we study and analyze the two–layer system.
Introduction
It was previously shown how the SPP SL property suffers from the intrinsic metal losses. It is also true that SPP modes in a three–layer system can benefit from the small thickness of the core [1]. Therein, this reduces the losses in the SPP mode because even when the mode confines to the metal core it can widely expand into the dielectric claddings, i.e. with large penetration length. It is the case of the LRSPP [1] that exhibit lower losses that the SRSPP, even without additional changes to the structure. In addition, as the thickness gets thinner, losses get low too and most of the energy gets stored in the lossless dielectric claddings. Nevertheless, losses are still high to achieve worthwhile
Abraham Vázquez–Guardado 55 INAOE 2012 Chapter 3 propagation lengths as those required in plasmon circuitry, for example, within the SL regimen.
A solution came to overcome the intrinsic losses in metals. It consists in adding gain to the dielectric medium, as it has been studied in different systems [2,3]. As a consequence, metal losses are reduced or even overcome. This technique, theoretically published, shows that losses can be completely eliminated at a point that the system can be seen as a plasmon amplifier [4], inducing stimulated emission [5], a phenomenon that has been experimentally proved [6].
Amplification of SPP modes has then been demonstrated. Nevertheless, all this experiments were carried out in a region far away from the natural plasmon resonance frequency. There, SPP mode propagation losses in this regime are considerably low and the group velocity is close to that of a plane wave. However, the real challenge comes when we want to amplify SPPs in the SL regime and that is the problem we want to address in this section.
Most of the work published until now has considered a constant gain model for the amplifying media [2,7]. This approach consists in adding up a negative constant in the permittivity imaginary part of the dielectric medium. Depending on such value, and the wavelength we are considering, the group index remains the same, but the propagation constant improves from several hundreds nanometers to even infinite propagation. In contrast, introducing a finite response in the dielectric medium induces slow or fast light around the central wavelength of maximum gain. In addition the propagation length improves too. It depends on the spectrum of the active gain response, its spectral width and the maximum optical gain. This approach is more realistic since it considers many physical aspects of active media.
For most equations their development from the basic equation are shown in appendix 1.
56 Abraham Vázquez–Guardado Chapter 3 INAOE 2012 3.1–Dielectric with active media.
The gain parameter is included in the dielectric and is represented as a permittivity imaginary part. First of all we find the condition for having an amplified SPP mode in the system and then we find the permittivity imaginary sign condition that will lead us to determine whether the material amplifies or absorbs.
Let’s consider a generic surface wave that propagates along certain planar interface and it is of the following general form,
i(k!r"#t) E(r,t) = E0 (r)e . (3.1)
Let’s assume that the propagation constant has two components, one in the propagation direction and the other one in the transverse direction. The propagation constant will be, then, of the form k k xˆ k zˆ . Each component is complex, k k! i 1 k!! and = x + z x = x + 2 x
! 1 !! kz = kz + i 2 kz . Under these assumptions, the intensity of the surface wave is
## 2 "kz z "kx##x I (x,z) ! E0 e e . Where the first exponential term accounts for amplification or
"" attenuation in the propagation direction and !kz is usually referred as ! , the attenuation coefficient. The second exponential term stands for the transverse decaying factor [8] and can be a measure of the penetration depth transversely from the interface. This factor represent the bound condition characteristic of surface waves, i.e. the field exponentially decays as we move away from the interface in both mediums. Hence, in order to keep this
!! bound behavior, the value kx has to be positive. The aforementioned intensity expression is,
2 "z #kx$$x I (x,z) ! E0 e e . (3.2)
This equation tells us the parameters needed to have an amplified SPP mode in the
!! waveguide, kx and ! . These parameters must satisfy the following conditions:
Abraham Vázquez–Guardado 57 INAOE 2012 Chapter 3
!! 1. For bound modes, kx > 0 .
2. For amplification, ! > 0 .
Once these conditions are stated, then we follow with the sign condition for the imaginary permittivity part of the dielectric medium. Let’s consider a wave plane electric field that propagates in the positive plane of z in a given dielectric with complex
i(k z t) permittivity, i , and has the form E(z,t) E e z !" ; is the permittivity real ! = !" + !"" = 0 "! part, is the permittivity imaginary part and k the propagation constant. We know that "!! z the propagation constant is related to the complex refractive index, k = k n! . The complex z 0
2 refractive index n! = n + i! , is related to the dielectric complex permittivity as n! = ! whose real and imaginary components can be determined by means of the set of equations (1.54)
1 # 2 2 2 & – (1.57). The propagation constant becomes k = k 1 "! + 1 "! + "!! + i"!! 2n . Then z 0 $%( 2 2 ) '( we substitute this parameter into the electric field, which leads
1 2 "!! ik 1"!+ 1 "!2 +"!!2 z #k z 0( 2 2 ) 0 2n #i$t E(z,t) = E0e e e . (3.3)
"!! > 0 "!! < 0
$## #"" "k z k z "! > 0 or 2 0 n 2 0 n I (z) ! E0 e I (z) ! E0 e
"! < 0 & "! < "!2 + "!!2 (3.4) (3.5)
$## #"" "k0 z 2 2 k0 z 2 2 2 n "k0 2 $# " $# +$## z 2 n $k0 2 #" $ #" +#"" z I (z) ! E0 e e I (z) ! E0 e e "! < 0 & "! > "!2 + "!!2
(3.6) (3.7)
Table 3.1 – Intensity function of a plane wave propagating in a dielectric medium with complex permittivity. Depending on the sign and magnitude of the complex permittivity components, i.e. "! and "!! , the intensity function can increase or decrease.
58 Abraham Vázquez–Guardado Chapter 3 INAOE 2012
Depending on the sign and magnitude of each dielectric permittivity component, i.e. "! and "!! , the plane wave might experience amplification or attenuation. Table 3.1 shows the different intensity functions for some condition of sign and magnitudes. As we can see in equations (3.4) and (3.6), they describe a plane wave that is getting attenuated. On the other hand, equation (3.5) indicates that the plane wave subject to amplification. In addition, equation (3.7) can experience both, attenuation and amplification depending o the argument of both exponential terms. First "! > "!2 + "!!2 must be valid then if
"!! n > 2 "! # "!2 + "!!2 then it is amplification otherwise it is attenuation.
For a generic dielectric medium, n and "! are always positive then from equations (3.4) and (3.5) we conclude that the dielectric medium produces attenuation when "!! > 0 and produces amplification when "!! < 0 . When the material is a metal "! < 0 and "!! > 0 , hence equation (3.4) applies, which implies that the propagating plane wave in the metal experiences attenuation. Therefore, in order to consider a dielectric with gain we have to include a negative imaginary part to its permittivity [2]. The function of this value is to diminish the contribution from the imaginary part for the metal permittivity and, as a consequence, to produce a lossless propagation and, if there is enough gain, to produce amplification [4].
3.2–Condition for bound modes.
The two–layer system is the first one to be analyzed under this scheme because it exhibits friendly mathematical equations. Let’s consider the structure composed by two semi–infinite planes, see Figure 2.1. For x > 0 the medium consists in a dielectric with a complex wavelength dependent permittivity and for x < 0 it consist in a metal with ! d complex permittivity , such as those defined in the previous section. ! m
Abraham Vázquez–Guardado 59 INAOE 2012 Chapter 3 Surface waves, which include SPPs, have the characteristic that the electromagnetic field is maximum at the interface and exhibits a transverse exponential decay away from it. In the previous section we considered a generic surface wave whose intensity is
2 "z #kx$$x I (x,z) ! E0 e e . If this expression describes the intensity of the SPP mode propagating
!! along the interface then kx > 0 for the dielectric semi–infinite plane whose domain is x > 0 in our waveguide geometry. This condition is the bound condition that we will find within this section.
As it was introduced previously, a negative imaginary part in the permittivity function must be introduced in the dielectric medium to take into account for the dielectric gain and then diminish the losses induced by the intrinsic metal losses. Nevertheless, as it will be analyzed, the condition for bound waves can be broken if excess gain or absorption is consider in the dielectric model [2]. Hence, our goal is to find the dielectric imaginary permittivity part, "d!! , that is within the bound condition.
Let’s consider a surface wave that propagates along the aforementioned flat interface.
The propagation is in the z direction and the surface mode is confined in the x direction. The mathematical form that describes this mode is
i k x z E r,t = E e ( x +! )e"i#t . (3.8) ( ) 0
The complex propagation constant along the z direction k is changed for the standard z
1 notation ! = !" # i 2 $ and is given by (2.13). We use the expression relating, mathematically, the propagation constants, k 2 + ! 2 = " k 2 . This expression must fulfill in x,i i 0 both mediums, i; in the dielectric it becomes k 2 + ! 2 = " k 2 and in the metal x,d d 0 k 2 + ! 2 = " k 2 . Then, the wavevector in the x direction in the dielectric can be obtained x,m m 0 by combining (2.13) and k 2 + ! 2 = " k 2 , which is, x,d d 0
60 Abraham Vázquez–Guardado Chapter 3 INAOE 2012
2 " "" !d + i!d 2 2 ( ) kx,d = k0 . (3.9) !" + i!"" + !" + i!"" ( m m ) ( d d )
The permittivity real and imaginary parts are considered.
!! The condition for bound mode has been already established, i.e. kx > 0 . This condition results in the following restriction for the imaginary part of the permittivity of the dielectric.
$ 2 ' $ 2 ' 1 ## ## # # # ## 1 ## ## # # # &!" m ! " m ! 8"d " m + "d ) < "d < &!" m + " m ! 8"d " m + "d ) (3.10) 2 % ( ) ( )( 2 % ( ) ( )(
We consider the next condition:
1. A metal with high plasmonic resonance, i.e. " 2 "" 1. ! m ! m ! 2. The absolute value of is greater than the other three components. " m!
The first condition is general true for good plasmonic metals such as Gold or Silver [3]. The permittivity real part of these two metals falls quickly to negative values. For a dielectric medium with refractive index of 1.5 its permittivity real part is roughly the square of that values, 2.25. Hence the second condition fulfills for the point when "!! > 2.25, which is the m range of SPPs [9].
Then, under these assumptions is large enough to guarantee that the condition for " m! bound mode, equation (3.10), will be within the real values bound. This result is a reference point that states that, if is out of the range of equation (3.10), then the " d!! propagation will be something but not an SPP mode.
Abraham Vázquez–Guardado 61 INAOE 2012 Chapter 3 3.3–Constant gain model and condition for SPP lossless propagation.
For certain dielectrics, we can assume that its refractive index is non–dispersive and active but not wavelength dependent. The plane wave propagation constant, in the z direction, is related with the complex refractive index n! = n + i! of the medium as ! = k n! = !" # i 1 $ . Then, comparing the complex propagation constant and the refractive 0 2 index, we arrive to the pair of equations
"! = k0n , (3.11)
2k . (3.12) ! = " o# Equation (3.11) gives us the real part of the propagation constant, which is the rate of phase change of the propagating wave in space. On the other hand, equation (3.12)
!1 stands for the amount of optical gain, in [cm ] , that the active dielectric medium can produce. We use equation (1.56) into (1.57) and if the real part of the dielectric permittivity is greater that its imaginary part, then equation (3.12) can be approximated by:
k #$$ ! = " 0 d . (3.13) $ #d
As it has been exposed previously, the detriment influence of losses in the metal degrades both, group index and propagation length. So we will study the gain coefficient, ! , require for SPPs lossless propagation. Hence, this value will be a reference point to analyze, then, the SPP mode in the amplification, lossless, and lossy propagation regimes.
The SPP propagation constant in the two–layer system is given by equation (2.13). If we include the dielectric and metal complex permittivities, " i "" and " i "" !d = !d + !d ! m = ! m + ! m respectively, then the SPP propagation constant is
62 Abraham Vázquez–Guardado Chapter 3 INAOE 2012
# ## # ## "d + i"d " m + i" m 2 2 ( )( ) ! = k0 . (3.14) "# + i"## + "# + i"## ( d d ) ( m m ) This expression expanded and separated in real and imaginary becomes
# 2 # 2 ## 2 ## 2 " m "d + "d " m + i " m "d + "d " m 2 2 ( ) ( ) ! = k0 2 2 , (3.15) "# + "# + "## + "## ( d m) ( m d )
2 2 2 where ! = ! " + ! "" , and i is d for the dielectric or m for the metal. For SPPs lossless i ( i ) ( i ) propagation condition, the imaginary part of the propagation constant has to be zero. This condition leads us to find out the required condition for the dielectric imaginary part value
"d!! and, therein, the minimum optical power gain for lossless SPP mode propagation or the minimum for SPP amplification. This condition becomes a quadratic equation
2 ! "" 2 m "" " 2 !d + !d + !d = 0 whose solutions are: "" ! m
2 " 2 "" ! "" !d ! m m !d = # , (3.16) + 2 "" ! m ! m
# 2 ## ## !d ! m !d" = " 2 . (3.17) ! m
The first solution is very large and quite impossible to accomplish physically; however, the second one is lower in magnitude and physically possible to achieve [2]. Finally, combining equations (3.17) and (3.13) we get the overall null gain and therein the ! for the optical power minimum gain for lossless SPP mode propagation, and the minimum for SPP amplification, as given for the following expression
3 2 "## "# m ( d ) ! 0 = k0 2 . (3.18) " m
Abraham Vázquez–Guardado 63 INAOE 2012 Chapter 3 Then using equations (3.17) and (3.18) we can analyze quantitatively the gain requirement for a two–layer system.
We are going to graphically analyze the permittivity of Silver permittivity with respect to the active dielectric permittivity imaginary part for lossless propagation and that for bound mode limit. In addition, we will also analyze the corresponding gain for lossless SPP mode propagation and the bound modes maximum optical gain, Figure 3.1–b. For this reason we consider the two–layer system with Silver and a dielectric with " 2.25 , such as ! d = fused silica (glass), or a polymer.
In Figure 3.1–a we plot the real and imaginary parts of the Silver permittivity [9], red and black solid lines respectively. The last one is compared with the dielectric medium permittivity imaginary parts for a lossless propagation, as given in equation (3.17) and the pink dashed line plot, and the limit for bound modes, equation (3.10) and the solid blue line plot. We can observe here that the magnitude of the permittivity imaginary part for lossless propagation is compared to the Silver permittivity imaginary part, which is what we should expect since one is with the means to contrary the other. The dielectric permittivity imaginary part limit for bound mode is high in magnitude, and grows higher as we move toward the infrared.
In Figure 3.1–b we compare the optical gain at which the SPP mode propagates without losses, equation (3.17) red line plot, with the maximum optical gain allowed to have bound mods, equation (3.10) blue line plot. We can notice that the maximum optical gain for bound modes has reduced by a factor of ten in order to make it fit within the range of the optical gain for lossless propagation, which the latter is much lower. Fro reference purposes, we have plotted the 1000cm!1 optical gain valued, dashed black line plot.
64 Abraham Vázquez–Guardado Chapter 3 INAOE 2012
a 6 b 0 5 x10 4 4
−5 x10 ] 1 ε
− 3
εAg′ [cm 0 −10 2 εAg′′ ε′′ 1 lossless 1
ε′′bound −15 0 350 400 450 500 550 600 400 500 600 700 λ[nm] λ[nm] Figure 3.1 – We plot the Silver permittivity [9] and the active dielectric permittivity imaginary part as function of ! . In addition, we also plot the gain coefficient ! for lossless propagation comparing it against the limit gain for bound mode, a function of ! as well. a) Real part (black line) and imaginary part (red line) of Silver permittivity [9], and imaginary part of the dielectric function of the dielectric medium to have lossless propagation (dashed pink line) and limit for bound mode (blue line). b) Gain required to have lossless propagation (red line) and limit for bound mode (solid line), the broken line is a reference of !1 1000cm . Now, we select a point where to analyzed the system and the gain constant, for example lets position at a wavelength of 640nm, which is the central wavelength of the RHODAMINE 640 PERCHRCLORATE laser dye. The laser dye can be dissolved in ethylene glycol of 1.5 of refractive index, or " 2.25 as the dielectric. The permittivity of ! d = Silver is 18.7619 0.4727i at 640nm Therefore, for lossless propagation ! m = " + .
"" 0.0068 that corresponds to an optical gain of ! = 444.71cm"1 ; the limit for bound ! d = # 0 modes is determined as "" 8.8596 that corresponds to an optical gain of ! d = # 5.7986x105 cm"1 . !1 =
In Figure 3.2 three important parameters are plotted: the imaginary part of the SPP propagation constant ! , Figure 3.2–a, the imaginary part of the SPP transverse propagation constant in the dielectric k , Figure 3.2–b, and the propagation length in a logarithmic x,d scale normalized to L 1 m, Figure 3.2–c. 0 = µ
Abraham Vázquez–Guardado 65 INAOE 2012 Chapter 3
5 3 10 a bc 4 0.5 10 ]
3 2 1 = 444.71cm 0.4 3 0 ) 10 0 0 [x10 0.3 2 1 /k 0
1 10 m c x2 1 /k k 7
0.2 .
β 4 5 1 log(L/L 1 = 5.8x10 cm 4 4 1 10 444.71
Im = Im 0 0.1 0
0 10 0 -1 −1 −0.1 10 0 100 200 300 400 500 0 2 4 6 0 100 200 300 400 500 −1 −1 5 −1 [cm ] [cm ] x10 [cm ] Figure 3.2 – For a fixed point, 640nm for the central line of RHODAMINE640 PERCHLORATE laser dye, the permittivity of Silver is ! = "18.7619 + 0.4727i and ! " = 2.25 . The required gain for lossless propagation m d is ! = 444.71cm"1 which corresponds to ! "" = #0.0068 and the limit gain for bound mode solutions 0 d ! = 5.8x105cm"1 that corresponds to ! "" = #8.8596 . a) Imaginary part of the SPP propagation constant. b) 1 d Imaginary part of the transverse propagation constant. c) Propagation length in a logarithmic scale normalized to L = 1µm . 0 Finally, if we position at , Figure 3.2–a, we clearly see that the imaginary part of ! 0 the propagation constant is zero. As we approach from the left we can observe that the ! 0 propagation length increases tending to infinite right next to , Figure 3.2–c. Regarding ! 0 the gain requirement for having bound solutions, we can see that the imaginary part of the transverse propagation constant becomes zero for 5.8x105 cm"1 in this particular case, !1 = Figure 3.2–b. Therefore, there is plenty of range to consider amplification without worrying about the bound condition for the SPP mode.
3.4–Finite gain model.
In the previous discussion, we introduced and analyzed a model with constant gain. Hence there were missing some interesting phenomena that a finite gain model introduces to the system.
Let’s consider a dielectric medium with refractive index n and permittivity whose 0 ! relationship with the electric susceptibility is ! d
66 Abraham Vázquez–Guardado Chapter 3 INAOE 2012
! = ! 1+ " . (3.19) 0 ( d )
If that medium host an active material with electric susceptibility ! , then the total g permittivity functions will be ! = ! 1+ " , where ! = ! + ! represent the total electric d 0 ( T ) T d g susceptibility of both materials [8]. Combining equation (3.19) and the relation n2 1 0 = + ! d the total permittivity function becomes
! = ! n2 + " . (3.20) d 0 ( 0 g ) The complex propagation constant after using equation (3.20) turns to
1 " ! % 2 k = k n 1+ g . (3.21) 0 0 $ n2 ' # 0 &
If we assume that ! n2 << 1, which is valid for weak gain media [8] equation (3.21) can be g 0 approximated to
" ! % k ! k n + g . (3.22) 0 $ 0 2n ' # 0 &
" "" The gain medium susceptibility is complex, ! g = ! g + i! g . The complex propagation
1 constant, using the attenuation coefficient ! is k = k! " i 2 # ; hence comparing both the complex propagation constant and refractive index, ñd = nd! + ind!! , we arrive to the following expressions
"! n! = n + g , (3.23) d 0 2n 0
"!! n!! = g , (3.24) d 2n 0
Abraham Vázquez–Guardado 67 INAOE 2012 Chapter 3
n $ !"" = # 0 . (3.25) g k 0 The active characteristic of the entire medium is directly related to the imaginary part of the electric susceptibility of the active guest. Then, if we know the entire gain response of the system, ! , we can know the real part of the refractive index by means of Kramers–Kronig relations [10] we can know the contribution it has to the real part of the refractive index. For certain gain responses, , n! can severely be disturbed, which induces ! d SL behavior in the SPP dispersion curve, as it will be treated shortly.
If we treat a population of active particles, such as laser dyes dissolved in a material, in analogy to laser amplifying systems the gain coefficient, ! (" ) , is given by
# 2 ! (" ) = N 0 g(" ). (3.26) 8$tsp
In this equation N is the population difference of active particles, the wavelength at the !0 center of distribution, t the spontaneous lifetime and g ! the lineshape distribution sp ( ) function of the active medium [8]. Typically the lineshape distribution function can be represented by a Lorentzian or a Gaussian function and the terms next to it can be accounted as the maximum optical gain, g = N! 2 / 8"t . max 0 sp
3.4.1–Gaussian gain response model.
Since a dispersive medium can exhibit absorption or gain, this process applies either way and it is explicitly stated in Kramers–Kronig relations [10]. The total complex and frequency dependent refractive index of the dielectric, ñ n! in!! , in terms of the electric d = d + d susceptibility is
68 Abraham Vázquez–Guardado Chapter 3 INAOE 2012
!! " g! " g ñd = n0 + + i . (3.27) 2n0 2n0
The complex refractive index, n! , and the real and imaginary electric susceptibility parts, d " "" ! g and ! g , are frequency dependent. Then, using the Kramers–Kronig relation for the real part in function of the imaginary part of the electric susceptibility, equation (3.27) becomes
& $$ $$ 1 ' s# g (s) # g (! ) n!(! ) = n0 + ds + i , (3.28) "n ) s2 %! 2 2n 0 (0 0 when the imaginary part of the electric susceptibility is known and is given by equation (3.25).
A Gaussian distribution models the lineshape gain function g ! and the ( ) normalized distribution function is
1 2 2 g(! ) = e$(!$!0 ) 2# . (3.29) 2"#
Then, the gain coefficient, ! (" ) , is
2 ("#"0 ) # e 2$ 2 , (3.30) ! (" ) = !"0 where ! = ! " = g 2#$ , ! is the centered frequency and 0.5! the spectral width "0 ( 0 ) max ( ) o of the gain distribution, 0.5! = "# . The later is related to the spectral width in wavelength as c / 2 . In addition, it is common to measure the distribution full width at half !" = !# "0
! 1 maximum. For this purpose the parameter ! has to be multiplied for 2ln2 2 . g is the ( ) max maximum gain that the system can provide.
Abraham Vázquez–Guardado 69 INAOE 2012 Chapter 3 Finally, the imaginary part of the electric susceptibility becomes
2 (#$#0 ) $ n0c 2' 2 " g!!(# ) = $%# e . (3.31) 0 2&#
Kramers–Kronig relations are an approximation of the Hilbert transforms of a given function. They are deduced using the holomorphic property of optical constants. Hence, for a given function describing any optical response both, the Kramer–Kronig relation and the Hilbert transform can be used [11].
The Hilbert transforms that relates the real and imaginary parts of the electric susceptibility read as
& 1 ' !""(s) !" (# ) = P) ds (3.32) $ (%& s %#
& 1 ' "!(s) "!!(# ) = $ P) ds (3.33) % ($& s $# where P stands for the Cauchy principal value. Then the real part of the electric susceptibility of the active medium is
2 s$# ( $( 0 ) n c ) 1 e 2' 2 0 P+ ds . (3.34) " g! (# ) = $%# 2 0 2& + s $# s *$( The Hilbert transform for a Gaussian function is given by [12]
$ ! 1 % 1 #q2 2 #! 2 w2 f ! = P' e dq = # e e dw . (3.35) ( ) " q #! (0 &#$ " After doing some mathematical work on equation (3.34) the result is:
2 2 #%# 2& 2 2 %# 2& n c ( % #%# 2& ( 0 ) n2 %# 2& 0 m2 + "! # = g 0 e ( 0 ) e dn % e 0 e dm . (3.36) g ( ) max 2 * '0 '0 - $ # ) ,
70 Abraham Vázquez–Guardado Chapter 3 INAOE 2012 Both integrals in this result correspond to the Dawson function [12] evaluated at two different points, a constant and a variable. The constant is defined in terms of
X = ! 2" and the variable is basically defined in terms of the resonance 0 0
X = ! "! 2# . Then we restate equation (3.36) in order to reduce notation. ( 0 )
n c " # = g d &F X % F X ( (3.37) !( ) max 2 ' ( ) ( 0 )) $ # F X defines the Dawson function and X its argument. Finally, the group index of the ( ) SPP mode under this model gain approach is:
1 3 " ! %2 c " ! %2 1 n n m g m F X 2X 1 F X g = d $ 2 ' + max 2 $ 2 ' ) + ( 0 ) * ( ) + ) ( ) + ! + n 2( ! + n ) { } # m d & # m d & (3.38) 3 2 )*) 2 *( 0 ) c " ! % 1 1 2 ig $ m ' 2X) ++ e 2+ max 4( ! + n2 2 ) ( ) # m d & 2(+
As we can notice, this value is complex because in the analysis we took the full complex SPP propagation constant. Then it is the real part of equation (3.38) that describes the SPP group index.
Abraham Vázquez–Guardado 71 INAOE 2012 Chapter 3 This result analytically describes how the real part of the electric susceptibility changes due to the influence of the active medium, in particular near the largest value that occurs at a specific value of the resonant frequency in terms of the width ! of the gain profile. Then equation (3.28) gives the total complex refractive index. This value is, consequently, used in the analysis of SL in SPPs. We compute the electric permittivity of
2 the dielectric medium, ! = n! , and use it in the SPP dispersion equation, then we compute the group SPP group index.
Consider the two–layer system with an active media having a Gaussian gain response with a maximum gain, g in cm!1 with a spectral width of at FWHM. In order to max !" analyze its quantitative behavior the gain function will be modeled with distinct parameters.
First a maximum gain of 500cm!1 with a width !" of 1 and 10 nm, and a gain of 1000cm!1 with the same two widths. The Kramers–Kronig are computed using equation (3.28), paying special care on the principal value of the integral.
−3 x 10 2.258 0
−2 2.254 −4 | |
2 −6 2 ε 2.25 ε −8 Im| Re| −10 2.246 −12 g =500cm−1 &∆ =1nm max g =1000cm−1&∆ =10nm max g =500cm−1 &∆ =1nm −14 max g =1000cm−1&∆ =10nm a b max 2.242 −16 600 620 640 660 680 600 620 640 660 680 λ[nm] λ[nm]
Figure 3.3 – Active dielectric permittivity for four models; g = 500cm!1 plus !" = 1nm and max FWHM !" = 10nm , red line and blue dashed lines respectively, and g = 1000cm!1 plus !" = 1nm and FWHM max FWHM !" = 10nm , black line and pink dashed line respectively. a) Permittivity real part and, b) Permittivity FWHM imaginary part.
72 Abraham Vázquez–Guardado Chapter 3 INAOE 2012 In Figure 3.3 the real (a) and imaginary (b) parts of the active dielectric medium are plotted for the aforementioned values. The SPP group index (a) and the propagation length (b) are plotted in Figure 3.4.
For very narrow spectral widths of the optical gain response the optical dispersion is narrow too and exhibits sharp changes around the central wavelength. In addition, the group index also benefits from the narrow spectral width, as it can be observed in Figure 3.4–a. For both models with narrow widths, 1nm solid lines, the maximum group !"FWHM = index the SPP mode can achieve is considerable higher than that for wider spectral widths, dashed lines. The propagation length, Figure 3.4–b, also increases accordingly. It can be seen that around the center wavelength the propagation is increasing until it become infinite. Then there is a gap in Figure 3.4–b in which the imaginary part of the propagation constant is negative; this implies gain in the SPP mode.
10 4.5 g =500cm−1 &∆ =1nm max g =1000cm−1&∆ =10nm a max b g =500cm−1 &∆ =1nm max 4 g =1000cm−1&∆ =10nm 8 max 3.5 )
6 0 3 g n 2.5 4 log(L/L
2 2 1.5
0 1 620 630 640 650 660 620 630 640 650 660 λ[nm] λ[nm] Figure 3.4 – For every corresponding values displayed in Figure 3.4 here is plotted the a) Group index and b) Propagation length normalized to L = 1µm . 0 On the other hand, if we want to induce, theoretically, a high group index value around the center wavelength two options can be considered. First, a dielectric medium with high gain can be considered. Nevertheless, the regime of bound modes has to be taken into account. The second option is to choose an active dielectric medium with narrow gain
Abraham Vázquez–Guardado 73 INAOE 2012 Chapter 3 response. This, as seen in Figure 3.4–a, induces a high group index just at the center wavelength.
In general, the advantage this model presents against the constant gain model is that it brings an implicit SL regime far from the natural SPP SL region. It is due to the finiteness of the gain model response. This is not observed when a constant gain was considered. It is true that for a very narrow range of frequency, the constant model can give us very good results; however, this model fails to predict what to expect when we position not at the center wavelength considered in the model but in a place next to it.
References.
[1] P. Berini, "Long–range surface plasmon polaritons," Adv. Opt. Photon. 1, 484 (2009). [2] M. P. Nezhad, K. Tetz, and Y. Fainman, "Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides," Optics Express 12, 4072–4079 (2004). [3] S. A. Maier, "Gain–assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides," Optics Communications 258, 295–299 (2006). [4] I. De Leon and P. Berini, "Theory of surface plasmon–polariton amplification in planar structures incorporating dipolar gain media," Phys. Rev. B 78, 161401 (2008). [5] M. Ambati, S. H. Nam, E. Ulin–Avila, D. A. Genov, G. Bartal, and X. Zhang, "Observation of Stimulated Emission of Surface Plasmon Polaritons," Nano Lett. 8, 3998–4001 (2008). [6] I. De Leon and P. Berini, "Amplification of long–range surface plasmons by a dipolar gain medium," Nature Photon 4, 382–387 (2010). [7] E.–P. Fitrakis, T. Kamalakis, and T. Sphicopoulos, "Slow light in insulator–metal– insulator plasmonic waveguides," J. Opt. Soc. Am. B, JOSAB 28, 2159 (2011). [8] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 1st ed. (Wiley, 2007), p. 1177. [9] P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370–4379 (1972). [10] V. Lucarini, Kramers–Kronig Relations in Optical Materials Research, 1st ed. (Springer
74 Abraham Vázquez–Guardado Chapter 3 INAOE 2012 Verlag, 2005), p. 160. [11] K. E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy: Classical Theory (Springer, 1999). [12] J. A. C. Weideman, "Computing the Hilbert transform on the real line," Math. Comp. 64, 745–762 (1995).
Abraham Vázquez–Guardado 75 INAOE 2012 Chapter 3
76 Abraham Vázquez–Guardado Chapter 4 INAOE 2012
Chapter 4 Results, discussions and further work.
4.1–Results and discussions
Material resonance is the natural manifestation of Slow Light (SL). This is the aforementioned case of Surface Plasmon–Polaritons (SPPs), which in their dispersion curve a flat region around the surface plasmon resonance means a SL regime. The basic two configuration waveguides that support SPPs modes, as it was discussed in chapter 2, a SL region around the surface plasmon resonance exhibit high values of group index.
The two–layer system can support a guided SPP mode right at the interface. First, for a phenomenological description the lossless case was studied. Under this approach the dispersion equation (2.13) exhibits a resonance at the surface plasmon resonance frequency
! . It is here where the SPP mode exhibits very slow group velocities, theoretically zero. sp The imaginary part of the propagation constant accounts for mode losses, which is translated in short propagation distances. However, since there were not losses included in the metal, then the imaginary part of the propagation constant is zero.
On the other hand, the three–layer system is analyzed under this approach as well. The structure, Insulator–Metal–Insulator was chosen in order to focus on the physical properties of its supported SPP modes and their SL property. From the dispersion relation, equation (2.20), two separated solutions are determined, which corresponds to two– coupled SPP modes supported by the waveguide. They can be categorized by their electric field distribution in the transverse direction, symmetric or antisymmetric. The antisymmetric mode dispersion curve is very similar to that of the two–layer system hence its SL behavior is then similar too. However, the symmetric mode is of quite interest because, as it can be observed in Figure 2.5, the real part of the propagation constant rises
Abraham Vázquez–Guardado 77 INAOE 2012 Chapter 4 until it reaches a maximum and then decays asymptotically toward the surface plasmon resonance. In this region, two SL regimes are present; one with negative value and other with positive value.
This approach helps us analyze qualitatively the SL performance of SPP in an ideal frame for both the two and three layer system; however, it fails to give a more quantitative description of the SPP propagating properties. The following study, then, focused on analyzing the SL and propagating properties of these two systems when losses are included in the metal.
It is exposed on Figure 2.6 the SPP propagation constant of the two–layer system with losses in the metal. It can be clearly seen that the dispersion curve does not display that asymptotic behavior at the surface plasmon resonance. As a consequence, the group index does not tend to infinity but reaches a maximum value. The imaginary part of the propagation constant, on the other hand, is not zero as before. The propagation length is then finite. In can be seen in Figure 2.6–a the propagation length is under 1µm for most of the SL region in the dispersion curve.
A similar behavior is presented in the three–layer system. Losses in the metal propitiate the imaginary propagation constant to be nonzero. First the dispersion curve for the antisymmetric mode does not tend to infinity at the surface plasmon resonance frequency and exhibit the same behavior that the two–layer system. On the other hand, the symmetric mode dispersion curve splits into two separate curves. It can be seen on Figure 2.7 that the flat region at the top of this dispersion curve is not present. Then the group index that once was infinite now limits to a finite value. The imaginary part also in nonzero that implies that the SPP mode will have a finite propagation length, also displayed on that figure. That plot shows the main property of these coupled SPP modes, the propagation length. As it can be observed, the symmetric mode exhibit up to two orders of magnitude larger propagation distance that the antisymmetric mode. It is why the term Long Range
78 Abraham Vázquez–Guardado Chapter 4 INAOE 2012 Surface Plasmon Polariton (LRSPP) was coined for the symmetric mode and Short Range Surface Plasmon Polariton (SRSPP) to the antisymmetric mode.
In general, in a SL SPP applications the most important parameters are the group index and the propagation length, large values for both. It is true that dispersion plays an important role in optical data transmission lines, but its importance comes in a second plane against the first two. Analyzing the propagation length versus the group index in both structures we can see that there are still group index values very high but the propagation length reduces to some hundreds of nanometers. For this reason the option to include a gain medium as a solution to mitigate losses and increase the propagation length and remain the natural high values of group index.
Two gain approaches are discussed for the gain model in the dielectric. One is the constant gain and the other the finite gain. The gain parameter is included as a negative imaginary part of the permittivity in the dielectric. In the metal this value is positive, which means losses. For given gain values, or absorption, the bound mode condition of the surface wave is fulfilled but it can become radiative if that gain value exceeds a given value. In addition, the lossless propagation condition was also treated. At this point the imaginary part of the propagation constant is zero, if it is negative the SPP mode experiences amplification. As it is observed in Figure 3.1 the optical gain required for lossless propagation is much lower than that to break the bound condition; hence amplification of SPP can be treated without worrying to become the SPP mode into a radiative mode.
As we move toward the infrared the gain required for lossless propagation decreases, see Figure 3.1–b. The explanation is tightly related to the interband transition in the metal. Silver, for example, exhibit high absorption around 340 nanometers. It is why gain to mitigate losses increase as we move close to this wavelength. The maximum optical gain to break the bound mode condition follows the inverse behavior, it decrease as we move toward the infrared.
Abraham Vázquez–Guardado 79 INAOE 2012 Chapter 4 This model of constant gain does not alter considerably the real part of the SPP propagation constant; hence there are not any changes in the group index. Nevertheless, it does alter the imaginary part provoking lossy, lossless and amplification propagation. Then, for given a wavelength at lossless propagation, the maximum group index the SPP mode can achieve is that as in the former analysis when no losses were considered. It is why this model is good on explaining the propagation length enhancement in the SPP mode but not the group index.
The finite model gain, on the other hand, is a more realistic approach to model the gain for a given active medium. In this case, the gain is not considered constant but is modeled by a distribution function representing the spectral response of certain active population, for example a laser dye. This cold population can be approximated by a Gaussian function, which exhibits a maximum gain and a limited spectral width. The optical gain, under this scheme, directly relates the imaginary part of the dielectric permittivity. However, by means of the Kramers–Kronig dispersion relations the real part can be affected.
For very narrow gain responses the dielectric real permittivity exhibit sharp changes around the center wavelength. In wide spectral responses the effect is not that strong but it is smooth instead, as observed in Figure 3.3. The effect this has on the SPP group index is impressing. At the center wavelength, the SPP group index curve is modified and a region of SL is induced. Then this model predicts a SL region around the gain resonance even far from the natural SPP SL regime.
It was observed that for having a large maximum group index there are two options. First, a gain medium with very large gain and wide spectral response or second, a gain medium with low gain but with narrow spectral width. In addition, the propagation length, as it was expected, becomes infinite or even amplified propagation within the neighborhood of the central wavelength.
80 Abraham Vázquez–Guardado Chapter 4 INAOE 2012 4.2–Gain requirements
In the real world it might me quite difficult to meet the gain requirements, high values. Whether it is for enhancing the propagation length at the natural SL region in the SPP dispersion curve or induce large group index far from the surface plasmon resonance.
For example, laser dyes are a typical active media choice. Their center wavelength of maximum gain spread from around 300nm to 1200nm . However the maximum optical gain is relatively low, for example 360cm!1 [1]. Even though it can be enough for lossless propagation or low lossy propagation, as it was experimentally proved but at a wavelength of operation far from the SL regime where the SPP group index is near that of a plane wave in the dielectric medium. If we look at the system with Silver and a dielectric, n = 1.5 , such as the one discussed within this work, for inducing a large maximum group index, as seen on Figure 3.4, a spectral width of about 1nm is needed. This requirement is far from being achieved in laser dyes, whose spectral widths range about 40nm .
If the choice is to move to the infrared, at 1550nm for example, laser dyes are not available at those wavelengths. However, quantum dots and quantum wells work very well at those wavelengths. The gain that can be achieved are in the range of 8000cm!1 but with wide spectral widths, larger that 15nm [2–5]. In addition the group index at those frequencies will be that of a plane wave.
In the three–layer system, despite the SL characteristic was not analyzed in Chapter 3; we can infer that the results are going to be very similar. In the constant gain model, the group index will have a maximum, such as that when metal losses were not included. On the other hand, in the finite gain model a SL region will be induced with a range of amplifying propagation.
Abraham Vázquez–Guardado 81 INAOE 2012 Chapter 4 4.3–Future work
As we observed that the SL induction by a narrow spectral gain response of amplifying is not a possibility for these simple plasmonic waveguides. Then there can be alternative plasmonic waveguides that exhibit low mode losses, such as the thin metal stripe. However, since the dispersion characteristics are quite similar to that of the LRSPP [6] in the three–layer system and the plasmonic metal is usually Silver or Gold, then the gain requirements near that surface plasmon resonance is considerably high for physically possible gain mediums.
On the other hand, structured or complex plasmonic waveguides such Moiré surfaces [7,8], metallic Bragg grating waveguide [9,10]. These systems predict high Plasmon modes group index but losses affect severely the propagation of the SL mode. It is mainly because there can appear a transmission band where the SPP mode can propagate in the system. However, an analysis of gain requirement for these systems can help us analyze the feasibility of Plasmonic SL propagation with the active medium.
Furthermore, if the choice is to look for alternative plasmonic material with lower losses semiconductors can be one good option [11,12]. It has been recently proposed and proved that highly doped semiconductors, such as the Aluminum–doped Zinc Oxide (AZO) [13,14] and the Indium Thin Oxide (ITO) [14,15], can support plasmonic waves. Their surface plasmon resonance frequency falls near the infrared, quite near the telecommunication frequency window, and losses are relative lower that those of Silver or Gold.
This master thesis opened a chance to further study in the field of SL SPP with active materials. We reviewed and introduced basic parameter for studying the SL properties of SPP modes. There is chance to keep searching for good SL SPP waveguide candidates whose gain requirement can be fulfilled by current gain materials.
82 Abraham Vázquez–Guardado Chapter 4 INAOE 2012 References
[1] I. De Leon and P. Berini, "Amplification of long–range surface plasmons by a dipolar gain medium," Nature Photon 4, 382–387 (2010). [2] P. Ramvall, Y. Aoyagi, A. Kuramata, P. Hacke, K. Domen, and K. Horino, "Doping– dependent optical gain in GaN," Appl. Phys. Lett. 76, 2994 (2000). [3] J. Chen, W. J. Fan, Q. Xu, X. W. Zhang, S. S. Li, and J. B. Xia, "Electronic structure and optical gain of truncated InAs1–xNx /GaAs quantum dots," Superlattices and Microstructures 46, 498–506 (2009). [4] P. Seoung–Hwan, "Optical Gain Characteristics of Long–wavelength Type–II InGaAs/GaPAsSb Quantum Wells Grown on GaAs Substrates," J. Korean Phys. Soc. 58, 434 (2011). [5] N. Kirstaedter, O. Schmidt, N. Ledentsov, D. Bimberg, V. Ustinov, A. Y. Egorov, A. Zhukov, M. Maximov, P. Kop’ev, and Z. I. Alferov, "Gain and differential gain of single layer InAs/GaAs quantum dot injection lasers," Appl. Phys. Lett. 69, 1226 (1996). [6] P. Berini, "Long–range surface plasmon polaritons," Adv. Opt. Photon. 1, 484 (2009). [7] A. Kocabas, S. Senlik, and A. Aydinli, "Slowing Down Surface Plasmons on a Moiré Surface," Phys. Rev. Lett. 102, 063901 (2009). [8] S. Balci, A. Kocabas, C. Kocabas, and A. Aydinli, "Slowing surface plasmon polaritons on plasmonic coupled cavities by tuning grating grooves," Appl. Phys. Lett. 97, 131103 (2010). [9] J. Zhang, L. Cai, W. Bai, and G. Song, "Flat Surface Plasmon Polariton Bands in Bragg Grating Waveguide for Slow Light," J. Lightwave Technol. 28, 2030–2036 (2010). [10] L. Yang, C. Min, and G. Veronis, "Guided subwavelength slow–light mode supported by a plasmonic waveguide system," Opt Lett 35, 4184–4186 (2010). [11] G. V. Naik and A. Boltasseva, "Semiconductors for plasmonics and metamaterials," phys. stat. sol. (RRL) 4, 295–297 (2010). [12] P. West, S. Ishii, G. Naik, N. Emani, and A. Boltasseva, "Identifying low–loss plasmonic materials," SPIE Newsroom (2010). [13] K. H. Kim, K. C. Park, and D. Y. Ma, "Structural, electrical and optical properties of AZO films prepared," J. Appl. Phys. 81, 7764–7772 (1997). [14] S.–M. Kim, Y.–S. Rim, M.–J. Keum, and K.–H. Kim, "Study on the electrical and optical properties of ITO and AZO thin film by oxygen gas flow rate," J Electroceram
Abraham Vázquez–Guardado 83 INAOE 2012 Chapter 4 23, 341–345 (2008). [15] S. Franzen, "Surface Plasmon Polaritons and Screened Plasma Absorption in Indium Tin Oxide Compared to Silver and Gold," J. Phys. Chem. C 112, 6027–6032 (2008).
84 Abraham Vázquez–Guardado Appendix INAOE 2012
Appendix.
In this section we will mathematically develop some important equations and results spread throughout the previous content.
A.1–Equations of Chapter 1.
A1.1–Macroscopic Maxwell’s equations and constitutive equations.
$B ! " E = # (a.1) $t
#D ! " H = + J (a.2) #t ext
! "D = # (a.3)
! "B = 0 (a.4)
D = !0E + P (a.5)
!1 H = µ0 B ! M (a.6)
A.1.2–The wave equations.
First we apply the vectorial curl to equation (a.1), substitute equation (a.6) and finally use equation (a.2). Hence we can arrive to the general form of the wave equation for the electric field.
% $B( ! " ! " E = ! " ' # * & $t ) $ ! " ! " E = # (! " B) $t
Abraham Vázquez–Guardado 85 INAOE 2012 Appendix
$ ! " ! " E = # (! " µ0 [H + M]) $t $ ! " ! " E = #µ0 (! " H # ! " M) $t
$ % $D ( ! " ! " E = #µ0 ' + Jext # ! " M* $t & $t ) $ % $ ( ! " ! " E = #µ0 ' +0E + P + Jext # ! " M* $t & $t ( ) ) $2 E $ % $P ( ! " ! " E = #+0µ0 2 # µ0 ' + Jext # ! " M* $t $t & $t )
1 #2 E # % #P ( ! " ! " E + 2 2 = $µ0 ' + Jext $ ! " M* (a.7) c #t #t & #t )
Similarly, for deriving the wave equation for the magnetic field, we first start by taking the curl to equation (a.2), then substitute equation (a.5) and finally we use (a.1).
$ #D ' ! " ! " H = ! " & + Jext ) % #t ( # ! " ! " H = ! " * E + P + ! " J #t ( 0 ) ext # #P ! " ! " H = *0 (! " E) + ! " + ! " Jext #t #t # $ #B' #P ! " ! " H = *0 & + ) + ! " + ! " Jext #t % #t ( #t # $ # ' #P ! " ! " H = +*0 & µ0H + µ0M ) + ! " + ! " Jext #t % #t ( )( #t
1 #2 H 1 #2 M #P ! " ! " H + = $ + ! " + ! " J (a.8) c2 #t 2 c2 #t 2 #t ext
A.1.3–Electric and magnetic field components.
First of all lets consider the time–harmonic field of equation (a.1) with the apt constitutive equations. Then the expansion of the curl for the electric field leads to the determinant representation and, consequently, to the separation of each component.
86 Abraham Vázquez–Guardado Appendix INAOE 2012
! " E = iµ0µ#H xˆ yˆ zˆ $ $ $ ! " E = $x $y $z
Ex Ey Ez
% # # ( % # # ( % # # ( ! " E = Ez $ Ey xˆ $ ' Ez $ Ex * yˆ + Ey $ Ex zˆ &' #y #z )* & #x #z ) &' #x #y )* ! " E = iµ µ+ , H H H . 0 - x y z / # # iµ µ+ H = E $ E 0 x #y z #z y # # iµ µ+ H = E $ E 0 y #z x #x z # # iµ µ+ H = E $ E 0 z #x y #y x
i $ # # ' H x = ! & Ez ! Ey ) (a.9) µ0µ" % #y #z (
i $ # # ' H y = ! & Ex ! Ez ) (a.10) µ0µ" % #z #x (
i $ # # ' H z = ! & Ey ! Ex ) (a.11) µ0µ" % #x #y (
In the same way we proceed with the curl of the magnetic field.
! " H = #i$0$%E xˆ yˆ zˆ & & & ! " H = &x &y &z
Ex Ey Ez
' & & * ' & & * ' & & * ! " H = H z # H y xˆ # ) H z # H x , yˆ + H y # H x zˆ () &y &z +, ( &x &z + () &x &y +, ! " H = #i$ $% - E E E / 0 . x y z 0
Abraham Vázquez–Guardado 87 INAOE 2012 Appendix
$ $ !i" "#E = H ! H 0 x $y z $z y $ $ !i" "#E = H ! H 0 y $z x $x z $ $ !i" "#E = H ! H 0 z $x y $y x
i % # # ( Ex = ' H z $ H y * (a.12) !0!" & #y #z )
i % # # ( Ey = ' H x $ H z * (a.13) !0!" & #z #x )
i % # # ( Ez = ' H y $ H x * (a.14) !0!" & #x #y )
A.1.4–Boundary conditions.
Boundary conditions are a direct deduction of Maxwell’s equation applied to an infinite section, surface or volume, of a boundary between two media. First of all lets represent equation (a.1) and (a.2) in their integral form. Integration over a surface S and applying Stokes theorem, A !dr = ($ % A)!nda , we have !#"S #S
' &B ! " E #ns da = %) #ns da S ( ) $ (S &t ' &B E#dr = %) #ns da !$&S (S &t
As it is displayed in Figure 1.1– a, the close integral of the electric field can be decomposed in each path component.
$ "B !liEit + hi2Eni2 + hj2E jn2 + l j E jt ! hj1E jn1 ! hi1Ein1 = !& #ns da %S "t When paths perpendicular to the interface tend to zero the magnetic flux integral over the surface is then zero, consequently the only remaining field component that are left are the tangential ones l(Eit ! E jt ) = 0 . In the vectorial form this equation can be expressed as
88 Abraham Vázquez–Guardado Appendix INAOE 2012 n ! (Eit " E jt ) = 0 . (a.15) We proceed with the same workflow for the magnetic. When the two sides of the surface perpendicular boundary tend to zero, the tangential components, again, are the one that are nonzero. Additionally, while the integration of the electric displacement field tends to zero, it is not the same for the current source.
, & %D ) (! " H)#ns da = + J #ns da $S . ( + -S ' %t *
* $ "D ' H!dr = , & + J) !ns da !#"S % t ( +S " * # "D & !l H + h H + h H + l H ! h H ! h H = + J )n da i it i2 ni2 j2 jn2 j jt j1 jn1 i1 in1 , $% '( s +S "t l Hit ! H jt = J )ns da ( ) -S Finally this expression leads to the following boundary condition for the tangential component of the magnetic field. The right side of the previous equation accounts for any current density j that might be present and is perpendicular to that of the tangential component of the magnetic field.
n ! (Ht1 " Ht 2 ) = j (a.16)
Now lets integrate equations (a.3) and (a.4) over a volume V and apply Gauss theorem ! "AdV = A "nda . According to Figure 1.1–b three separate surfaces can #V !#S represent the whole surface.
! "BdV = 0 #V B"nda = 0 !#S
B"n1 da + B"n2 da + B"n3 da = 0. #S1 #S2 #S 3 When surface two tends to zero, surfaces one and three become, geometrically, equals and lead to
Abraham Vázquez–Guardado 89 INAOE 2012 Appendix
B!n1 da + B!n3 da = 0 "S1 "S 3
B!n1 da + B!n3 da = 0 "S1 "S 3 n!Bi # n!B j = 0 n!(Bi " B j ) = 0 . (a.17) The same applies for the electric displacement field.
! "DdV = $ dV #V #V D"nda = $ dV !#S #V
D!n1 da + D!n2 da + D!n3 da = # dV "S1 "S2 "S 3 "V However, since surface two tends to zero, the right side of the last equation does not but it does become in a surface charge density ! .
D!n1 da + D!n3 da = # dS "S1 "S 3 "S Di !n $ D j !n = %
Then the final representation of this boundary condition is
n!(Di " D j ) = # . (a.18) A.1.5–Complex permittivity functions.
In order to arrive to equation (1.68) that relates the electric permittivity and the electric conductivity we do the following. First consider a linear, isotropic and nonmagnetic and spatially nondispersive medium. Then, lets use the time–harmonic representation of equation (1.6) and use (1.8), (1.12) and (1.16).
J(! ) = " (! )E(! ) = #i!P(! )
D(! ) = $0$ (! )E(! ) = $0 (! )E(! ) + P(! ) i $0$ (! )E(! ) = $0 (! )E(! ) + " (! )E(! ) !
90 Abraham Vázquez–Guardado Appendix INAOE 2012
i ! (" ) = 1+ # (" ) (a.19) !0"
In addition, we consider the complex nature of the electric permittivity ! (" ) = !#(" ) + i!##(" ) and its relationship with the complex refractive index n! ! = n ! + i" ! in the form of n! = ! , both frequency dependent. This allows us to ( ) ( ) ( ) express their component correspondence in terms of each other.
2 ! = (n + i" ) ! = n2 #" 2 + i2n" !$ + i!$$ = n2 #" 2 + i2n"
"! = n2 #$ 2 (a.20)
"!! = 2n# (a.21) Now, from equation (a.20) and doing some algebraic treatment we have the following.
"! = n2 #$ 2 n2 = "! +$ 2 "!!2 n2 = "! + 4n2 4n4 = 4n2"! + "!!2
4n4 ! 4n2#" = #""2 4n4 ! 4n2#" + #"2 = #""2 + #"2 2 (2n2 ! #") = #""2 + #"2 2n2 ! #" = #""2 + #"2 1 n2 = "! + "!!2 + "!2 (a.22) 2 ( )
A.1.6–The permittivity function of the free electron gas.
Lets consider a cloud of free electrons oscillation around a fixed positive ion core.
The equation of motion is m!x! t v + m! x! t = "eE t , without considering lattice potential ( ) ( ) ( )
Abraham Vázquez–Guardado 91 INAOE 2012 Appendix and electron–electron interactions. Time dependency will be omitted for simplicity. If we
!i"t assume that the external field oscillates harmonically, i.e. E(t) = E0e ; consequently, the
!i"t induced oscillating position will have a harmonic move too, x(t) = x0e . Then using this two expressions we have,
2 !i"t !i"t m(!i" ) x0e + m# (!i" )x0e = !eE !m" 2x ! im#"x = !eE x(m" 2 + im#" ) = eE e x = E m(" 2 + i#" )
Now, the electric polarization is defined as P = !neex , where ne is the density of electrons and e the electron charge. According to the constitutive equation linking the electric displacement and polarization and the electric field, D = !0E + P , and defining the constant
2 2 ! p = nee m"0 , the plasma frequency, we have
n e2 P = ! e E m(" 2 + i#" )
D = $0E + P n e2 D = $ E ! e E 0 m(" 2 + i#" )
% 2 ( nee D = !0 '1" * E m! # 2 + i$# & 0 ( ))
" 2 ! (" ) = 1# p (a.23) (" 2 + i$" ) A.1.7–The permittivity function of a metal with interband transitions.
We follow the same procedure as before, but with a term that keeps the electron in place at ! 0 . Then, starting from the equation of motion we arrive to the permittivity function of the metal.
92 Abraham Vázquez–Guardado Appendix INAOE 2012
2 m!x!(t) + m! x! (t) + m" 0 x(t) = eE(t) 2 2 m(#i" ) x + m! (#i" )x + m" 0 x = eE 2 2 x(#m" # im!" + m" 0 ) = eE e x = # 2 2 E m(" #" 0 + i!" )
P = #nbex 2 nbe P = 2 2 E m(" #" 0 + i!" )
D = $0E + P % "" 2 ( D 1 p E = $0 ' + 2 2 * & " #" 0 + i!" )
The term !! 2 = n e2 " m . p b 0
"! 2 ! (" ) = 1+ p (a.24) " 2 #" 2 + i$" 0 A.1.8–Dispersion of volume Plasmon.
In the limit when the product , which is true for , the permittivity ! ! " ! > ! p
2 2 function (a.23) is predominantly real and can be approximated as ! (" ) = 1#" p " . Now, for a propagating wave in this medium, the wavevector is expressed as k 2 = !" 2 c2 . Then if we combine all these equation we arrive to the volume plasmon dispersion relation expressed in equation (1.71).
" 2 k 2 = ! c2 $ 2 ' 2 2 " p " k = &1# 2 ) 2 % " ( c
2 2 " " p k 2 = # c2 c2 c2k 2 = " 2 #" 2 p
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2 2 2 2 ! = ! p + c k (a.25)
Equation (1.98) is similar to (1.73); therefor, under the same procedure we arrive to equation (a.24). Then, using equation that relates the permittivity function and the electric susceptibility, ! = 1+ " , we end up with:
! (" ) = 1+ # (" ) 2 "! p 1+ 2 2 = 1+ # (" ) " $" 0 + i%" "! 2 # (" ) = p 2 2 i " $" 0 + %" 1 ! (" ) # 2 2 . (a.26) " $" 0 + i%"
A.2–Equations of Chapter 2.
For both TE and TM polarizations, the starting point are equations (1.9) and (1.10). We consider the condition that our medium is non–magnetic, linear and charge free.
A.2.1–TE wave equation for the Two–Layer waveguide.
Starting from equation (1.9) and the aforementioned condition in the RHS of such equation the term that is non–zero is the partial derivative of the macroscopic polarization. We use the linear condition between the polarization en the electric field, P E, along = !0" with the relation between the permittivity function and the electric susceptibility ! = 1+ " . The identity for an arbitrary vectorial field ! " ! " A = ! ! # A $ !2A is used. ( )
1 #2 E # % #P ( ! " ! " E + 2 2 = $µ0 ' J + + ! " M* c #t #t & #t ) 1 #2 E #2 P ! " ! " E + = $µ c2 #t 2 0 #t 2 ! " ! " E = !(! +E) $ !2E
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2 2 2 1 $ E $ E !(! "E) # ! E + = #µ0%0 & c2 $t 2 $t 2 1 $2 E 1 $2 E #!2E + + & = 0 c2 $t 2 c2 $t 2 1 $2 E #!2E + (1+ & ) = 0 c2 $t 2
# $2 E !2E " = 0 (a.27) c2 $t 2
We introduce the harmonic propagation for the electric field, E = E(x)ei!ze"i#t , in equation (a.27).
% &2 !2 E(x)ei"ze#i$t # E(x)ei"ze#i$t = 0 ( ) c2 &t 2 ( ) 2 ' & 2 * % 2 ) 2 E(x) + 0 + (i" ) E(x), # 2 (#i$ ) E(x) = 0 (&x + c &2 $ 2 E(x) # " 2E(x) + %E(x) = 0 &x2 c2
2 ! 2 2 E(x) + k0 " # $ E(x) = 0 (a.28)! !x2 ( )
Finally, while TE polarization is considered, then the only electric field that is non–zero is y x the component. The relationship k0 = ! c is used. Omitting the dependency we continue with equation (a.28) as follows to arrive to our electric field diffusion equation.
2 # ! 2 & * 2 Ey + 0 + (i" ) Ey ) ()i+ ) Ey = 0 %!x2 ( c2 $ ' !2 + 2 E ) " 2E + *E = 0 !x2 y y c2 y
!2 E + k 2" # $ 2 E = 0 (a.29) !x2 y ( 0 ) y
Abraham Vázquez–Guardado 95 INAOE 2012 Appendix A.2.2–TM wave equation for the two–layer waveguide.
In the same fashion as in the case for TE polarization and the same physical consideration we continue for the wave equation for TM polarization. However, here the linear relationship between the magnetic flux and field is used.
1 #2 H #P #M ! " ! " H + = ! " J + ! " + µ c2 #t 2 #t 0 #t 1 #2 H #P ! " ! " H + = ! " c2 #t 2 #t 1 #2 H #E ! " ! " H + = $ %! " c2 #t 2 0 #t 2 2 1 # H # !(! &H) ' ! H + = $0 % (! " E) c2 #t 2 #t 2 2 1 # H # ( #B+ '! H + 2 2 = $0 % * ' - c #t #t ) #t , 1 #2 H #2 H '!2H + = 'µ $ % c2 #t 2 0 0 #t 2 1 #2 H 1 #2 H '!2H + + % = 0 c2 #t 2 c2 #t 2 1 #2 H !2H ' (1+ % ) = 0 c2 #t 2
# $2 H !2H " = 0 (a.30) c2 $t 2
Now, we use the harmonic magnetic field propagating in the z direction, H = H(x)ei!ze"i#t .
% &2 !2 H(x)ei"ze#i$t # H(x)ei"ze#i$t = 0 ( ) c2 &t 2 ( )
2 # ! 2 & * 2 H(x) + 0 + (i" ) H(x) ) ()i+ ) H(x) = 0 %!x2 ( c2 $ ' !2 + 2 H(x) ) " 2H(x) + *H(x) = 0 !x2 c2
2 ! 2 2 H(x) + k0 " # $ H(x) = 0 (a.31) !x2 ( )
96 Abraham Vázquez–Guardado Appendix INAOE 2012 As in the previous case, for TM polarization the only non–zero component for the magnetic field is the y component. Then, the full vectorial equation (a.31) ends up in the following scalar equation.
2 # ! 2 & * 2 H y + 0 + (i" ) H y ) ()i+ ) H y = 0 %!x2 ( c2 $ ' !2 + 2 H ) " 2 H + *H = 0 !x2 y y c2 y
!2 H + k 2" # $ 2 H = 0 (a.32) !x2 y ( 0 ) y A.2.3–TE non–existence condition of SPPs.
The electric and magnetic field components for TE polarization are
! $ ! # E x = 0 Ey (x) 0 and H x = H x 0 H x . Lets propose a propagating electric ( ) "# %& ( ) " x ( ) z ( ) $
i!z kix field along the z direction and a field profile depending on x , Ey (x) = Ae e . From the set of equation (1.36) and (1.38) the other two magnetic field components can be retrieved.
i " H x (x) = Ey (x) µ µ ! "z 0 i i " H z (x) = # Ey (x) µ0µi! "x
The sub index i represents the material parameter; however, since this is the case of non– magnetic materials becomes one. According to Figure 2.1 two domains build up the µi entire structure, then the previous equation must be restated for physical meaning in both domains. In other words, the electric field solution must have a decaying behavior in the x direction. Then for domains where x < 0 and x > 0 these equations become.
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x < 0 x > 0
" i"z k1x " i"z !k2x H x (x) = ! A1e e H x (x) = ! A2e e µ µ # µ µ # 0 1 0 1 i"z k1x i"z !k2x Ey (x) = A1e e Ey (x) = A2e e
i i"z k1x i i"z !k2x H z (x) = ! k1A1e e H z (x) = k2 A2e e µ0µ1# µ0µ2#
Boundary conditions require the continuity of the tangential field at the interface, x = 0 . For the electric field we have:
E = E y1 y2 x=0 i!z k1x i!z "k2x A1e e = A2e e
A1 = A2 . (a.33) For the magnetic field component and using this last equation we have:
H = H z1 z2 x=0
i i#z k1x i i#z !k2x ! k1A1e e = k2 A2e e µ0µ1" µ0µ2"
!k1A1 = k2 A2
k1A1 + k2 A2 = 0
k1 + k2 = 0 (a.34)
The terms k depend on the material and under normal conditions they are both positive; i consequently, there is not even a remote chance that equation (a.34) fulfills. Hence the general asseveration that TE waves cannot couple to SPP modes.
A.2.4–SPP dispersion relation.
In TM polarization the components that are non–zero are E x = ! E x 0 E x # ( ) " x ( ) z ( ) $
! $ i!z kix and H x = 0 H y (x) 0 . A propagating surface mode is considered to be H x = Ae e ( ) "# %& y ( ) . Equations (1.39) and (1.41) give us the equivalence with the electric field components.
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i $ Ex (x) = ! H y (x) " " # $z 0 i i $ Ez (x) = H y (x) "0"i# $x
The terms stand for the permittivity and depend on the medium, is the permittivity !i !d of the dielectric medium and ! m that for the metal. For each domain, x < 0 and x > 0 the magnetic and electric fields are, x < 0 x > 0
! i!z kmx ! i!z $kd x Ex (x) = A1e e Ex (x) = A2e e " " # " " # 0 m 0 d i!z kmx i!z $kd x H y (x) = A1e e H y (x) = A2e e
i i!z kmx i i!z $kd x Ez (x) = km A1e e Ez (x) = $ kd A2e e "0" m# "0"d#
Applying the same boundary condition for the tangential components at x = 0 for the magnetic field component we have:
H = H y1 y2 x=0 i!z kmx i!z "kd x A1e e = A2e e
A1 = A2 . (a.35) Now, we do the same to the electric field component and use equation (a.35).
E = E z1 z2 x=0
i i#z kmx i i#z $kd x km A1e e = $ kd A2e e !0! m" !0!d" k A k A m 1 = $ d 2 ! m !d
k k m = " d (a.36) ! m !d
Finally, the propagation constant identity, k 2 2 k 2 , is used to arrive the desired i = ! " 0 #i dispersion relation.
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2 2 2 2 ! " k0 # m ! " k0 #d 2 = 2 # m #d
2 2 2 2 2 2 2 2 !d " # k0 ! m!d = ! m" # k0 ! m!d 2 2 2 2 2 2 " (!d # ! m) = k0 (! m!d # ! m!d )
2 2 " m"d (" m # "d ) ! = k 0 " + " " # " ( d m )( d m ) 2 2 " m"d ! = k0 ("d + " m )
" m"d ! = k0 (a.37) (" m + "d )
A.2.5–Dispersion relation in a three–layer system.
Since it was proved that TE waves do not couple to SPP modes then the TM will be directly used in the following analysis. In Figure 2.3 the waveguide geometry is displayed.
There two flat interfaces are present, x = !a and x = a . Domains II and III correspond to a dielectric and domain I to a metal. The equations for the surface wave solutions of the wave equation (a.32) for all domains are:
x < !a !a < x < a x > a
" i"z k2 (x+a) " i"z !k1 (x+a) i"z k1 (x!a) " i"z !k3 (x!a) Ex (x) = A2e e Ex (x) = A11e e + A12e e Ex (x) = A3e e # # $ # # $ ( ) # # $ 0 2 0 1 0 3 i"z k2 (x+a) i"z !k1 (x+a) i"z k1 (x!a) i"z !k3 (x!a) H y (x) = A2e e H y (x) = A11e e + A12e e H y (x) = A3e e
i i"z k2 (x+a) i i"z !k1 (x+a) i"z k1 (x!a) i i"z !k3 (x!a) Ez (x) = k2 A2e e Ez (x) = k1 (!A11e e + A12e e ) Ez (x) = ! k3A3e e # 0# 2$ # 0#1$ # 0# 3$
Now lets apply the boundary conditions for the tangential electric and magnetic field components at each interface. For the interface at x = !a we have:
E = E z2 z1 x=!a
i i$z k2 (x+a) i i$z !k1(x+a) i$z k1(x!a) k2 A2e e = k1 (!A11e e + A12e e ) "0"2# "0"1#
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k2 k1 "2ak1 A2 = ("A11 + A12e ) (a.38) !2 !1
H = H y3 y1 x=!a i"z k2 (x+a) i"z !k1(x+a) i"z k1(x!a) A2e e = A11e e + A12e e
!2ak1 A2 = A11 + A12e . (a.39)
Now, at the interface x = a we have:
E = E z1 z2 x=a
i i$z #k1(x+a) i$z k1(x#a) i i$z #k3 (x#a) k1 (#A11e e + A12e e ) = # k3A3e e !0!1" !0! 3"
k1 "2ak1 k3 ("A11e + A12 ) = " A3. (a.40) !1 ! 3
H = H y1 y2 x=a
!2ak1 A11e + A12 = A3 . (a.41) These four equations are brought together in a matricial form to represent a system of four equations with four unknowns, M!A = 0 .
# & k2 k1 k1 "2ak1 % " e 0 ( # & ! ! ! A2 % 2 1 1 ( % ( # 0 & % "2ak1 ( % ( 1 "1 "e 0 % A11 ( 0 % () % ( = % ( k "2ak k k A % 0 ( % 0 1 e 1 " 1 " 3 ( % 12 ( % ( % 0 ( !1 !1 ! 3 % A ( $ ' % ( $ 3 ' "2ak $% 0 "e 1 "1 1 '(
Where M is the coefficient matrix, A the unknowns and 0 the zero column vector. The determinant of M has to be zero so that this system does not have a nontrivial solution. From this condition, the dispersion relation can be obtained.
Abraham Vázquez–Guardado 101 INAOE 2012 Appendix det M = 0 ) # & # & , )# & , k2 k1 k3 "2ak1 k1 "2ak1 k3 "2ak1 k1 k1 k3 "2ak1 +"1% " " ( + e % e " e ( + 0. " +% " " ( + e (0) + 0. " !2 * $ !1 ! 3 ' $ !1 ! 3 ' - !1 *$ !1 ! 3 ' - )# & , k1 "2ak1 k1 "2ak1 k3 "2ak1 e +% e " e ( + (0) " 0. " 0 = 0 !1 *$ !1 ! 3 ' -
) # & , )# & , k1 k2 k3 k2 "4ak1 k1 k2 k3 k2 k1 k1 k3 k1 "4ak1 ) k1 k1 k3 k1 , + + + e % " ( . + +% + ( . " e + " . = 0 *!1 !2 ! 3 !2 $ !1 !2 ! 3 !2 ' - *$ !1 !1 ! 3 !1 ' - *!1 !1 ! 3 !1 - )# & # & , )# & # & , "4ak1 k1 k2 k3 k2 k1 k1 k3 k1 k1 k1 k3 k1 k1 k2 k3 k2 e +% " ( " % " ( . = " +% + ( + % + ( . *$ !1 !2 ! 3 !2 ' $ !1 !1 ! 3 !1 ' - *$ !1 !1 ! 3 !1 ' $ !1 !2 ! 3 !2 ' -
!4ak ) k2 # k1 k3 & k1 # k1 k3 & , ) k1 # k1 k3 & k2 # k1 k3 & , e 1 + ! ! ! . = ! + + + + . " $% " " '( " $% " " '( " $% " " '( " $% " " '( * 2 1 3 1 1 3 - * 1 1 3 2 1 3 - # & # & # & # & !4ak1 k2 k1 k1 k3 k1 k2 k1 k3 e % ! ( % ! ( = !% + ( % + ( $ "2 "1 ' $ "1 " 3 ' $ "1 "2 ' $ "1 " 3 '
# k k & # k k & 1 + 2 1 + 3 $% " " '( $% " " '( e!4ak1 = 1 2 1 3 (a.42) # k1 k2 & # k1 k3 & % ! ( % ! ( $ "1 "2 ' $ "1 " 3 '
When the mid layer thickness tends to infinity, equation (a.42) splits into two separate dispersion equation that are equal to equation (a.37).
x ! " $ k k ' $ k k ' 1 + 2 1 + 3 %& # # () %& # # () 1 2 1 3 = 0 $ k1 k2 ' $ k1 k3 ' & * ) & * ) % #1 #2 ( % #1 # 3 (
$ k1 k2 ' $ k1 k3 ' & + ) & + ) = 0 % #1 #2 ( % #1 # 3 (
k k ! ! 1 = " 2 # $ = k 1 2 ! ! 0 ! + ! 1 2 1 2 k1 k3 !1! 3 = " # $ = k0 !1 ! 3 !1 + ! 3
102 Abraham Vázquez–Guardado Appendix INAOE 2012 Now, let’s consider that domains III and II are equal and correspond to a dielectric medium, , and domain I is a metal, , such as in symmetric waveguides, !3 = !2 = ! d !1 = ! m equation (a.42) turns to:
# k k & # k k & # k k & m + d m + d m + d $% " " '( $% " " '( $% " " '( e!4akm = m d m d e!2akm = ± m d # km kd & # km kd & # km kd & % ! ( % ! ( % ! ( $ " m "d ' $ " m "d ' $ " m "d '
This result leads to two separate solutions. Taking the positive solution we have:
# k k & # k k & m " d e"akm = m + d eakm $% ! ! '( $% ! ! '( m d m d k k m (e"akm " eakm ) = d (e"akm + eakm ) ! m !d
1 1 Using the identities sinh! = e! " e"! and cosh! = e! + e"! we obtain the first 2 ( ) 2 ( ) dispersion equation.
1 km 1 kd ! sinh(akm ) = cosh(akm ) 2 " 2 " m d sinh(akm ) kd" m = ! cosh(akm ) km"d
kd" m tanh(akm ) = ! (a.43) km"d
On the other hand, taking the negative solution and using the trigonometric identity used previously, we obtain the second dispersion equation.
# & # & km kd "2akm km kd % " ( e = "% + ( $ ! m !d ' $ ! m !d ' # & # & km kd "akm km kd akm % " ( e = "% + ( e $ ! m !d ' $ ! m !d '
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k k m (e"akm + eakm ) = d (e"akm " eakm ) ! m !d
1 km 1 kd cosh(akm ) = " sinh(akm ) 2 ! m 2 !d
sinh(akm ) km!d = " cosh(akm ) kd! m
km"d tanh(akm ) = ! (a.44) kd" m
A.3–Equations of Chapter 3.
A.3.1–The constant gain model.
The complex propagation constant of a certain mode in the z direction can be expressed as " i 1 . This parameter is related to the complex effective refractive ! = ! # 2 $ index, k = k n! , which is of the form n! = n + i! . By combining these equation and relating z 0 the real and imaginary parts the expressions equations (3.11) and (3.12) are retrieved.
! = !" # i 1 $ = k n + i% 2 0 ( )
" Re{!} = ! = k0n
" k n (a.45) ! = 0
Im{ } 1 k ! = " 2 # = 0$
2k (a.46)! ! = " 0#
The complex permittivity of the active dielectric was proposed as ! = !" + i!"" . Now we use equation (1.56) in the imaginary part of the refractive index, given by equation (1.57), we have to include the sign for the imaginary permittivity part of the dielectric. If
104 Abraham Vázquez–Guardado Appendix INAOE 2012 the real part of the permittivity is much greater than the imaginary part then we obtain equation (3.13).
"## ! = 2n "# 1 n2 = + "#2 + "##2 2 2 "# ! "## n2 " "#
#$$ ! = "2k0 2 #$
#$$ ! = "k0 (a.47) #$
A.3.2–Condition for bound solutions.
The condition for bound SPP solutions implies that the transverse complex propagation constant must fulfill such condition. In the two–layer system, propagation is in the z direction while it transversely propagates in the x direction. The complex nature of both propagation constants is accounted, " i 1 and k k! i 1 k!! . Then, the ! = ! # 2 $ xd = xd+ 2 xd propagating electric field will yield an intensity expression that shows the bound and lossless propagation condition.
i(kxd x+kzz) !i"t E(r,t) = E0e e
i #$!i 1% z i k$ +i 1 k$$ x ( 2 ) ( x 2 2 xd ) !i"t E(r,t) = E0e e e
i#$+ 1% z ik$ ! 1 k$$ x ( 2 ) ( xd 2 xd ) !i"t E(r,t) = E0e e e 2 I & E(r,t)
i#$+ 1% z ik$ ! 1 k$$ x !i#$+ 1% z !ik$ ! 1 k$$ x 2 2 ' ( 2 ) ( xd 2 xd ) !i"t * ' ( 2 ) ( xd 2 xd ) i"t * E(r,t) = E0 e e e e e e () +, () +,
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2 2 !z "kx##d x E(r,t) = E0 e e
$$ 2 ("z#kxd x) I ! E0 e (a.48)
The first exponential term accounts for amplification, as previously discussed. The second accounts for the bound condition in the dielectric medium. For the bound mode in the dielectric medium, the imaginary part must be positive, k!! 0 , so that the decaying xd > behavior in the transverse direction is assured. After this, we need to proceed on finding the mathematical expression of the transvers propagation constant. The general propagation constant is of the form k 2 2 k 2 . If we use k 2 k 2 and equation (3.9), = ! + xi = !i 0 then the transverse wave vector component is obtained.
2 2 2 kxi + ! = "ik0 2 2 2 kxi = "ik0 # !
2 2 2 !d! m 2 !i (!d + ! m ) " !d! m kxi = !ik0 " k0 = k0 !d + ! m !d + ! m
2 2 !i!d + !i! m " !d! m kxi = k0 (a.49) !d + ! m
For any value of i , equation (a.49) turns to!
2 2 2 !i kxi = k0 ! "#$%&' !d + ! m
In the dielectric medium i = d and i = m in the metal. Then, we use the full complex notation for both permittivities, " i "" and " i "" . ! d = ! d + ! d ! m = ! m + ! m
2 " "" !d + i!d 2 2 ( ) kxd = k0 !" + i!"" + !" # i!"" ( m m ) ( d d )
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2 " "" " " "" "" !d + i!d !d + ! m # i ! m + !d 2 2 ( ) ( ) ( ) kx2 = k0 !" + !" + i !"" + !"" !" + !" # i !"" + !"" ( d m) ( m d ) ( d m) ( m d ) 2 " "" $ " " "" "" ' !d + i!d !d + ! m # i ! m + !d 2 2 ( ) %&( ) ( )() kxd = k0 2 2 !" + !" + !"" + !"" ( m d ) ( m d ) 2 " "" " " !d + i!d !d + ! m $ "" "" ' 2 2 ( ) ( ) ! m + !d kxd = k0 2 2 &1# i ) " " "" "" & !" + !" ) ! m + !d + ! m + !d % m d ( ( ) ( )
2 " "" " " !d + i!d !d + ! m $ "" "" ' ( ) ( ) ! m + !d kxd = k0 &1# i ) 2 2 " " !" + !" + !"" + !"" %& ! m + !d () ( m d ) ( m d )
" " " " " " If we know that ! m ! !d then ! m + !d = # ! m + !d . Doing some mathematical work we can ( ) introduce the additional minus sign to the quadratic term in the square root.
2 2 2 ! "# + i"## = $i "# + i"## ' = !"## + i"# ( d d ) %& ( d d )() ( d d )
2 ## # # # !"d + i"d " m + "d $ ## ## ' ( ) " m + "d kxd = k0 &1! i ) 2 2 # # "# + "# + "## + "## %& " m + "d () ( m d ) ( m d )
" " ! m + !d "" "" ! m + !d "" " kxd = k0 1# i #!d + i!d 2 2 " " ( ) !" + !" + !"" + !"" ! m + !d ( m d ) ( m d )
The second square root term can be approximated using Taylor expansion. Finally the real and imaginary part of the transverse propagation constant is separated.
"## + "## "## + "## 1! i m d ! 1! i m d # # # # " m + "d 2 " m + "d ( )
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" " ! m + !d $ "" "" ' "" " ! m + !d kxd = k0 #!d + i!d &1# i ) 2 2 ( ) " " !" + !" + !"" + !"" %& ! m + !d () ( m d ) ( m d )
" " $* - * - ' ! m + !d "" "" "" "" &, " ! m + !d ""/ , " "" ! m + !d / ) kxd = k0 2 2 & !d # !d + i !d + !d ) , " " / , " " / !" + !" + !"" + !"" &, 2 ! m + !d / , 2 ! m + !d / ) ( m d ) ( m d ) %+ ( ) . + ( ). (
The previous condition states that k!! 0 then, xd >
! ! # & " m + "d !! !! % ! !! " m + "d ( kx!!2 = k0 2 2 "d + "d > 0 % ! ! ( "! + "! + "!! + "!! % 2 " m + "d ( ( m d ) ( m d ) $ ( )' !! !! ! !! " m + "d "d + "d > 0 2 "! + "! ( m d ) !! !! !! " m + "d ! "d > )"d 2 "! + "! ( m d ) 2 ! ! " m + "d 1 ( ) < ) "!! "!! + "!! "! d ( m d ) d 2"! "! + "! < )"!! "!! + "!! d ( m d ) d ( m d ) 2 "!! + "!!"!! + 2"! "! + "! < 0 ( d ) d m d ( m d )
"" From this last quadratic expression we obtain two solutions of !d , which represent the limits for bound modes,
$ 2 ' "" 1 "" "" " " " !2 < &#! m ± !2 # 8!d ! m + !d ) 2 % ( ) ( )(
$ 2 ' $ 2 ' 1 ## ## # # # ## 1 ## ## # # # &!" m ! " m ! 8"d " m + "d ) < "d < &!" m + " m ! 8"d " m + "d ) (a.51) 2 % ( ) ( )( 2 % ( ) ( )(
108 Abraham Vázquez–Guardado Appendix INAOE 2012 A.3.3–Condition for lossless SPP propagation.
The condition for lossless SPP propagation comes from the imaginary part of the complex SPP propagation constant, ! = 0 . From equation (3.14) with the full representation of the complex permittivities of both materials the complex SPP propagation constant gets the following form.
"# + i"## "# + i"## "# + i"## "# + i"## "# + "# $ i "## + "## ( m m )( d d ) ( m m )( d d ) ( m d ) ( m d ) ! = k0 k0 "# + i"## + "# + i"## "# + "# + i "## + "## "# + "# $ i "## + "## ( m m ) ( d d ) ( m d ) ( m d ) ( m d ) ( m d ) % + i& ! = k0 2 2 "# + "# + "## + "## ( m d ) ( m d )
The variables ! and ! are introduced to reduce notation.
! + i" = & #$#$ % #$$#$$ + i #$$#$ + #$#$$ )& #$ + #$ % i #$$ + #$$ ) '(( m d m d ) ( m d m d )*+'(( m d ) ( m d )*+ ! = #$#$ % #$$#$$ #$ + #$ + #$$#$ + #$#$$ #$$ + #$$ ( m d m d )( m d ) ( m d m d )( m d ) $ $ $ $ $ $ $$ $$ $ $$ $$ $ $$ $ $$ $$ $ $$ $ $$ $$ $ $$ $$ ! = # m#d# m + # m#d#d % # m#d # m % # m#d #d + # m#d# m + # m#d#d + # m#d # m + # m#d #d
# # # # # # ## # ## # ## ## ! = " m"d" m + " m"d"d + " m"d" m + " m"d "d 2 2 2 2 # $ # ## ' # $ # ## ' ! = "d & " m + " m ) + " m& "d + "d ) %( ) ( ) ( %( ) ( ) (
2 2 $ ' 2 "# + "## 2 2 & m m ) $"# " 2 ' # # ( ) ( ) # ## # d m ! = " m&"d + "d + "d ) = " m& + "d ) # ( ) ( ) & # ) & " m ) % " m ( % ( # 2 # 2 ! = "d " m + " m "d * = "##"# + "#"## "# + "# + "#"# + "##"## "## + "## ( m d m d )( m d ) ( m d m d )( m d )
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## # # ## # # # ## # # ## # # # ## # # ## ## ## ## ## ## ## ! = " m"d" m + " m"d"d + " m"d " m + " m"d "d $ " m"d" m $ " m"d"d + " m"d " m + " m"d "d ## # # # ## # ## ## ## ## ## ## ! = " m"d"d + " m"d " m + " m"d " m + " m"d "d 2 2 2 2 ## % # ## ( ## # ## ## ! = " m ' "2 + "2 * + "2 "1 + "2 "1 &( ) ( ) ) ( ) ( ) 2 2 % ( 2 "# + "## 2 2 ' d d * %"## " 2 ( ## ## ( ) ( ) # ## ## m d ! = "d '"d + " m + " m * = "d ' + " m * ## ( ) ( ) ' ## * ' "d * & "d ) & ) ## 2 ## 2 ! = " m "d + "d " m
2 2 2 2 "# " + "# " + i "## " + "## " ( d m m d ) ( m d d m ) ! = k0 2 2 "# + "# + "## + "## ( m d ) ( m d )
2 2 2 The term ! = ! " + ! "" was introduced to reduce notation. If we introduce the i ( i ) ( i ) aforementioned condition for the imaginary part, it can also apply for the imaginary part inside de square root operator, i.e. ! = 0 , which leads to a quadratic equation.
2 #!"" ! 2 & "" m d !d % + ! m ( = 0 % "" ( $ !d ' 2 2 2 !"" ! + !"" !" + !"" !"" = 0 d m m ( d ) m ( d ) 2 2 ! 2 "" m "" " !d + !d + !d = 0 ( ) "" ( ) ! m
The solutions of this equation are:
2 2 2 2 # ! !"" ± ! !"" # 4 !" m m ( m m ) ( d ) !"" = d 2 2 2 2 4 # ! !"" ± ! !"" 1# 4 !"!"" ! m m m m ( d m ) m !"" = d 2
110 Abraham Vázquez–Guardado Appendix INAOE 2012
2 2 ! $ 4 ' "" m " "" !d = ± 1# 4 !d! m ! m ) "" ( ) 2! m % (
The square root can be approximated using a Taylor expansion and trimming it to the
"" second term. Then the previous solution pair of !d is further simplified.
2 2 2 2 4 "# "## 4 "# "## ( d ) ( m ) 1 ( d ) ( m ) 1! 4 ! 1! 4 " 2 " m m
2 2 ! 4 "" m * $ " "" ' - !d = ,#1± &1# 2 !d! m ! m ) / "" % ( ) ( 2! m + . 2 2 2 2 2 2 !"!"" !" !"" ! 4 ! d m d m "" m * " "" - m ( ) ( ) !d = #1+1# 2 !d! m ! m = # = # + "" , ( ) / "" 4 2 2! m + . 2! m ! m ! m * 2 2 - * 2 2 - 2 2 2 " "" 2 " "" " "" 2 ! , !d ! m / ! , !d ! m / !d ! m ! "" m ( ) ( ) m ( ) ( ) ( ) m !d# = ,#1#1+ 4 / = ,#1+ 4 / = 2 # "" "" "" 2! m , ! m / ! m , ! m / ! m ! m + . + .
$ 2 " "" & !d !m ( ) a & # 2 ( ) & & !m !"" = (a.52) d % 2 & " "" 2 !d !m ! & ( ) m b 2 # ( ) & "" ! !m '& m
For a typical high plasmonic metal, silver for instance, solution (b) is large enough to consider it; hence the election to choose (a). Equation (3.18) can be retrieved by direct substitution of solution (a) into equation (a.47).
2 "# "## 1 ( d ) m ! = k0 2 # " m "d
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3 2 "## "# m ( d ) ! = k0 2 (a.53) " m
A.3.4–Gaussian model for the active medium.
The imaginary part of the electric susceptibility given by equation (3.31) is mathematically managed in order to apply equation (3.35) straightforwardly. First consider the optical gain function that includes the lineshape function. This last function is modeled by a Gaussian distribution g ! . ( )
! (" ) = g max g (" ) 2 "%" %( 0 ) 1 2 g (" ) = e 2$ 2#$ 2 "%" %( 0 ) e 2$ 2 ! (" ) = !" 0 g ! = max "0 2#$
Then the imaginary part of the electric susceptibility "!!(# ), using the gain function, is
n0% " g!!(# ) = $ k0 2 (#$#0 ) n c $ 2 0 e 2' " g!!(# ) = $%# 0 2&# n c "!!(# ) = $% 0 g 0 #0 2� 2 (#$#0 ) $ #0 2' 2 " g!!(# ) = " g!!(#0 ) e #
Then we use the Hilbert transform to calculate the real part of the electric susceptibility.
& 1 ' " g!!(s) " g! # = P) ds ( ) $ s %# (%&
112 Abraham Vázquez–Guardado Appendix INAOE 2012
2 s%# ' %( 0 ) A ( 1 e 2& 2 "! # = P* ds (a.54) g ( ) $ * s %# s )%'
A.3.4.1–Hilbert transform of the Gaussian.
Once we know how to relate the imaginary part of the electric susceptibility to the real part we proceed to calculate it analytically. We start with equation (a.54) and do some algebraic manipulation.
2 2 (s !"0 ) (s !"0 ) q = 2 ; q = 2# 2#
s = 2# q +"0 ; ds = 2# dq ' !q2 % g$$("0 )"0 ( 1 e % g$ " = 2#P* dq ( ) & )!' 2# q +"0 !" 2# q +"0 ' ( 2 %$$(" )" 1 1 e!q %$ " = g 0 0 2#P* dq g ( ) * & * 2# "0 !" "0 ) q + q + !' 2# 2# ' ( + .+ . * - 0- 0 %$$(" )" 1 1 2 g 0 0 - 0- 0 !q % g$ (" ) = P* e dq & * - "0 " 0- "0 0 * -q + ! 0-q + 0 2 2 2 )!' , # # /, # /
1 1 1 1 = ! ! ! $ ! ! ' ! $ ! ! ' ! q 0 q 0 0 0 + # + & q + # ) # & q + # ) + 2" 2" 2" % 2" 2 2" ( 2 2" % 2" 2 2" ( 2 2" ! ! ! u = 0 # & b = 2" 2 2" 2 2" 3 +**(! )! 4 - 1 0- 1 0 2 g 0 0 P e#q dq + g* (! ) = 6 / 2/ 2 , 5#3 .q + u # b 1.q + u + b 1 - 1 0- 1 0 B C / 2/ 2 = + .q + u # b 1.q + u + b 1 q + u # b q + u + b
Abraham Vázquez–Guardado 113 INAOE 2012 Appendix 1 = B(q + u + b) + C(q + u ! b) q + u = !b q + u = b 1 = !2bC 1 = 2bB
" 1 %" 1 % 1 " 1 1 % $ '$ ' = $ ! ' #q + u ! b &#q + u + b & 2b #q + u ! b q + u + b & , )(((* )* 1 - " 1 1 % 2 g 0 0 P e!q dq ) g( (* ) = / $ ! ' + 2b .!, #q + u ! b q + u + b & h1 = b ! u h2 = !(u + b) , ) g(((*0 )*0 1 - " 1 1 % !q2 ) g( * = P/ $ ! 'e dq ( ) + 2b q ! h q ! h .!, # 1 2 & " , , % ) g(((*0 )*0 1 - 1 !q2 - 1 !q2 ) g( (* ) = $P/ e dq ! P/ e dq' + 2b $ . q ! h1 . q ! h2 ' # !, !, &
The following identity gives the Hilbert transform of a Gaussian function, which is applied to the previous result.
$ 1 % 1 2 2 2 ! 2 f ! = P e#q dq = # e#! ew dw ( ) ' (0 " &#$ q #! " $ $ 1 , 1 % 1 2 1 % 1 2 / *) + = *))(+ )+ . P e#q dq # P e#q dq1 g ( ) g 0 0 2b " ' q # h " ' q # h -. &#$ 1 &#$ 2 01
2 h 2 h 1 , 2 #h 1 2 2 #h 2 2 / *) + = *))(+ )+ # e 1 en dn + e 2 em dm g ( ) g 0 0 . (0 (0 1 2b - " " 0
2 h 2 h 1 #h 1 2 #h 2 2 *) + = *))(+ )+ ,#e 1 en dn + e 2 em dm/ g ( ) g 0 0 . (0 (0 1 "b - 0
Finally, lets go back with the variables changed.
h1 = b ! u;h2 = !(u + b)
2 2 1 b!u 2 !(u+b) 2 #" $ = #""($ )$ '!e!(b!u) en dn + e!(u+b) em dm* ( ) g 0 0 ) &0 &0 , %b ( +
114 Abraham Vázquez–Guardado Appendix INAOE 2012
! ! ! u = 0 # ;b = 2" 2 2" 2 2" 2 2 ' ! ! * ' ! * . # # 0 ! !0 # 0 !0 1 2 2&" ) , # 2 ) , # 2 %$ ! = %$$(! )! 0#e ( 2" 2" + 2" 2" en dn + e ( 2" + 2" em dm3 g ( ) g 0 0 &! 0 -0 -0 3 / 2 n c g ( ) 0 ; max " g!! # 0 = $%#0 %#0 = 2&# 0 2&'
2 2 (#$#0 ) 2' 2 2 2 $#0 2' 2 gmax 2&' $(#$#0 ) 2' n $# 2' m "! # = $n c )$e e dn + e 0 e dm, g ( ) 0 2 (0 (0 2&' & # *+ -.
2 2 #$# 2& 2 2 $# 2& n c ( $ #$# 2& ( 0 ) n2 $# 2& 0 m2 + "! # = $g 0 $e ( 0 ) e dn + e 0 e dm (a.55) g ( ) max 2 * '0 '0 - % # ) ,
This is the final result of the Hilbert transform of the Gaussian function. The last result is
2 x 2 in function of the Dawson integral, F X = e! x ey dy , where the arguments are a variable ( ) "0 and a constant. If we use this notation and X = ! "! 2# and X = ! 2" then the ( 0 ) 0 0 last result is expressed as:
n c " # = g 0 &F X % F X ( (a.56) g! ( ) max 2 ' ( ) ( 0 )) $ #
This result along with the imaginary part of the electric susceptibility forms the frequency dependent complex refractive index of the active medium, equation (3.27).
"" # g" (! ) # g (! ) nd (! ) = n0 + + i 2n0 2n0
n c n c %1 # ! = g 0 &F X % F X ( = g 0 F X &F X F X %1( g" ( ) max 2 ' ( ) ( 0 )) max 2 ( 0 ) ( 0 ) ( ) $ ! $ ! '* )+ n c %1 g 0 F X & a F X # g" (!0 ) = % max 2 ( 0 ) = ( 0 ) $ !0 ! 0 & ( # g" (! ) = %# g" (!0 ) aF(X) %1 ! ' )
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2 #$# $( 0 ) # 2 "!! # = "!! # 0 e 2% g ( ) g ( 0 ) # 1 b = 2n0 n # = n + b"! # + ib"!! # d ( ) 0 g ( ) g ( )
2 !"! "( 0 ) ! ! 2 n n b 0 %aF X 1' ib 0 e 2) (a.57) d (! ) = 0 " $ g# (!0 ) & ( ) " ( + $ g##(!0 ) ! !
From here and beyond the complex refractive index will be referred as n . Once the total d refractive index is known we can use it to calculate the SPP propagation constant and then the group index.
A.3.4.2–Calculation of the group index.
The dispersion equation of the SPP in the metal/dielectric interface using equation (a.57) is
2 " n2 % 2#$ ( " n2 ! 2 = k2 m d = m d , (a.58) 0 " + n2 &' c )* " + n2 m d m d and the group index
!" c !" ng = c = . (a.59) !# 2$ !%
The group index is relates, as we already knew, to first derivative of the SPP propagation constant with respect to the frequency. Then we start with the SPP propagation constant derivative.
2 !" 2 !" ! ,% 2$# ( + n2 / 4$ 2+ ! % # 2n2 ( 2 . m d 1 m d = " = ' * 2 = 2 ' 2 * !# !# !# .& c ) + + n 1 c !# & + + n ) - m d 0 m d
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2 2 2 2 !" 4$ %m & nd ! 2 2 ! nd ) = 2 ( 2 # +# 2 + !# c ' %m + nd !# !# %m + nd * 2 2 2 2 2 !" 8$ # k0%mnd 2 ! nd = 2 2 2 + ko %m 2 !# c k0 %m + nd !# %m + nd !" 2 8$ 2 # & ! 1 1 ! ) , " 2 = k2% n2 + n2 !# c2 k2 o m ( d !# % + n2 % + n2 !# d + 0 ' m d m d *
& ) & 8 2 ) n2 1 "! % # 2 ( d " 2 " 2 + ! 2 $ 2 2 ! = ko ,m $ 2 nd + 2 nd ( + 2 ' "# c k0 * ( n "# ,m + nd "# + ' (,m + d ) * & ) & 8 2 ) k2, n2 k2, & 2 2 ) "! % # ( o m d o m + " 2 ! ! " 2 ! 2 $ 2 2 ! = $ 2 + 2 nd = $ 2 + 2 nd ( + 2 ( + ' "# c k0 * ( n ,m + nd + "# ' ,m + nd nd * "# ' (,m + d ) * & "! 8% 2 # ) ! 2 & n2 ) " ! 2 , " 2 d 1 n2 2 m n ! ( $ 2 2 !+ = 2 ( $ 2 + + d = 2 d ' "# c k0 * nd ' ,m + nd * "# nd ,m + nd "# 2 "! 4% # ! ,m " = 2 2 ! + 2 nd "# c k0 nd ,m + nd "# "! ! ! , " = + m n "# # n , + n2 "# d d m d In this result we clearly see that the SPP propagation constant derivative depends on the complex refractive index derivative. Hence we do this last operation.
"" nd (! ) = n0 + b# g" (! ) + ib# g (! ) $ $ $ n = b #" ! + ib #"" ! $! d $! g ( ) $! g ( )
$ $ * ! - 0 &aF X 1( # g" (! ) = % +# g" (!0 ) ' ( ) % ). $! $! , ! / $ *1 $ $ 1 - aF X 1 aF X 1 # g" (! ) = %# g" (!0 )!0 + ( ( ) % ) + ( ( ) % ) . $! ,! $! $! ! / d F x = 1% 2xF x dx ( ) ( ) $ * a 1 - 1 2XF X aF X 1 # g" (! ) = %# g" (!0 )!0 + ( % ( )) % ( ( ) % ) 2 . $! ,! ! /
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! & a 2a a 1 ) XF X F X $ g# (" ) = %$ g# ("0 )"0 ' % ( ) % 2 ( ) + 2 * !" (" " " " + ! " 0 a 1 2aX a F X $ g# (" ) = %$ g# ("0 ) 2 " + % ( " + ) ( ) !" " { } 2 "%" - %( 0 ) 0 ! ! " 2 $##= / $## " 0 e 2, 2 !" g !" g ( 0 ) " ./ 12
2 "%" - %( 0 ) 0 ! ! 1 2 $##= $## " " / e 2, 2 !" g g ( 0 ) 0 !" " ./ 12
2 2 "%" "%" - %( 0 ) %( 0 ) 0 ! 1 ! 2 2 ! 1 $##= $## " " / e 2, + e 2, 2 !" g g ( 0 ) 0 " !" !" " ./ 12
2 2 "%" "%" - %( 0 ) %( 0 ) 0 ! 2 2 " %" 2 1 / e 2, ( 0 ) e 2, 2 $ g##= $ g##("0 )"0 % 2 % 2 !" " 2, " ./ 12
2 2 "%" "%" - %( 0 ) %( 0 ) 0 ! 2 2 " %" 2 1 / e 2, ( 0 ) e 2, 2 $ g##= $ g##("0 )"0 % % 2 !" ," " ./ 2, 12
2 "%" %( 0 ) ! " 2 0 2X e 2, $ g##= %$ g##("0 ) 2 " +, !" ," ( ) 2 "%" %( 0 ) ! " " 2 n b 0 a 1 2aX a F X ib 0 2X e 2, d = % $ g# ("0 ) 2 { " + % ( " + ) ( )} % $ g##("0 ) 2 " +, !" " ," ( )
Finally, the group index is
c "# n = g 2! "$
c # c # %m " ng = + 2 nd 2! $ 2! nd %m + nd "$ c # bc # % $ n m 0 a 1 2aX a F X g = & 2 ( g' ($0 ) 2 { $ + & ( $ + ) ( )} & 2! $ 2! nd %m + nd $ 2 $&$ &( 0 ) bc # % $ 2 i m 0 2X e 2) 2 ( g''($0 ) 2 $ +) 2! n % + n )$ ( ) d m d
118 Abraham Vázquez–Guardado Appendix INAOE 2012
n c n c g $1 1 g 0 F X ; 0 ; max ;a F X ;b " g! (#0 ) = $ max 2 ( 0 ) " g!!(#0 ) = $&# &# = = ( 0 ) = % # 0 2%# 0 2n 0 0 2%' 0
c " c " $ n c # %1 %1 %1 n m g 0 F X 0 F X 1 2F X X F X F X g = + 2 max 2 ( 0 ) 2 ( 0 ) # + % ( 0 ) # + ( 0 ) ( ) + 2! # 4!n n $ + n ! # # { ( ) } 0 d m d 0 2 #%# %( 0 ) c " $ n c g # 2 i m 0 max 0 2X# +& e 2& 2 2 ( ) 4!n0 nd $ + n 2!#0 2!& &# m d " c " $ 1 n g m F X 2X 1 F X g = + max 2 2 {# + ( 0 ) % ( # + ) ( )} + k0 2! ndk0 $m + nd # 2 #%# %( 0 ) c " $ 1 1 2 ig m 2X# +& e 2& max 4! n k $ + n2 2 # ( ) d 0 m d 2!&
1 3 " ! %2 c " ! %2 1 n n m g m F X 2X 1 F X g = d $ 2 ' + max 2 $ 2 ' ) + ( 0 ) * ( ) + ) ( ) + ! + n 2( ! + n ) { } # m d & # m d & (a.60) 3 2 )*) 2 *( 0 ) c " ! % 1 1 2 ig $ m ' 2X) ++ e 2+ max 4( ! + n2 2 ) ( ) # m d & 2(+
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120 Abraham Vázquez–Guardado Table of figures INAOE 2012
List of figures and tables
Figure 1.1 – Graphical representation of boundary and surface integration. 13 Figure 1.2 – Dispersion relation of the free electron gas. 18 Figure 1.3 – Volume plasmon representation. 19 Figure 1.4 – Contribution made by bound electrons to the permittivity function 22 of gold. Figure 1.5 – Graphical representation of a mathematical temporal pulse. 23 Figure 1.6 – Time slotted representation of backward pulse propagation. 27 Figure 1.7 – Slow Light induced by material resonance. 30 Figure 1.8 – Energy–level of a two– and three–level system. 32 Figure 1.9 – Energy–level model representing the stimulated Raman scattering. 33 Figure 2.1 – Two–layer system containing a flat interface. 40 Figure 2.2 – Propagation constant and group index for the two–layer system 44 SPP mode without losses in the metal. Figure 2.3 – Three–layer system sharing two flat interfaces. 45 Figure 2.4 – Mode symmetries for the electric field in the three–layer system. 47 Figure 2.5 – Dispersion curve and group index for the SPP modes supported by 49 the three–layer system without losses in the metal. Figure 2.6 – Propagation constant, group index and propagation length graphs 51 for the two–layer system SPP modes with losses in the metal. Figure 2.7 – Propagation constant, group index and propagation length graphs 52 for the three–layer system SPP modes with losses in the metal. Figure 3.1 – Active dielectric permittivity imaginary part for lossless propagation 65 and bound condition. Figure 3.2 – Propagation constant, transverse propagation constant, and 66
Abraham Vázquez–Guardado 121 INAOE 2012 Table of figures propagation length for a fixed wavelength. Figure 3.3 – Dielectric permittivity, real and imaginary parts, for the finite gain 72 model. Figure 3.4 – Group index and propagation length for the finite model gain. 73
Table 3.1 – Intensity function depending on the magnitude and sign of the 58 dielectric permittivity real and imaginary parts.
122 Abraham Vázquez–Guardado