Computational study of organic dye-doped SPP
waveguides by usage of the ADE-FDTD method
Nicolas Knudde
Supervisors: Prof. dr. Zeger Hens (Ghent University), Prof. dr. Sergei Popov (KTH)
Counselor: Aleksandrs Marinins (KTH)
Master's dissertation submitted in order to obtain the academic degree of Master of Science in Engineering Physics
Optics and Photonics
Chair: Prof. dr. Sergei Popov
School of Information and Communication Technology Department of Inorganic and Physical Chemistry Chair: Prof. dr. Isabel Van Driessche
Faculty of Engineering and Architecture Academic year 2015-2016
Computational study of organic dye-doped SPP
waveguides by usage of the ADE-FDTD method
Nicolas Knudde
Supervisors: Prof. dr. Zeger Hens (Ghent University), Prof. dr. Sergei Popov (KTH)
Counselor: Aleksandrs Marinins (KTH)
Master's dissertation submitted in order to obtain the academic degree of Master of Science in Engineering Physics
Optics and Photonics
Chair: Prof. dr. Sergei Popov
School of Information and Communication Technology Department of Inorganic and Physical Chemistry Chair: Prof. dr. Isabel Van Driessche
Faculty of Engineering and Architecture Academic year 2015-2016
The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the copyright terms have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation.
Nicolas Knudde Stockholm, June 1, 2016
Preface
During my Erasmus experience in Stockholm I’ve had a lot of exciting and refreshing experiences. Writing a master dissertation however is a work of endurance and stress. There were moments that my morale suffered considerably and I thought of giving up several times. I want to thank the following people who helped me get through this period in different sorts of ways:
PhD student Aleksanders, a hard worker who shared his knowledge on the practical side of this branch of science and always presented me his own ideas for new research opportunities enthusiastically. Sometimes all you need is some glance of enthusiasm.
My parents who offered me the opportunity to study engineering and doing these exchange studies at KTH. And although they will probably not understand anything about this work, I am eternally grateful to them for having patience and always being supportive. My classmates in Belgium, who always were up for an intellectual or not so intellectual discussion. It is really them who make these sometimes tough studies enjoyable.
My friends, who visited me several times and helped me relief my stress by just being there. They always are amusing and while being in Sweden I realize that maybe I have taken them too much for granted. They support me without judgement, that is true friendship.
My housemates Carlo and Mikael, who didn’t always clean the apartment enough, however it was nice to have someone to come home to and talk about the other things in life. Politics, ethics, philosophy and economics were no rare subjects in these strange and uncertain times.
The reader, thank you for having the patience to read this work.
Nicolas Knudde Stockholm, June 1, 2016
Computational study of organic dye-doped SPP
waveguides by usage of the ADE-FDTD method
by
Nicolas Knudde
Supervisors: Prof. dr. Zeger Hens (UGent), Prof. dr. Sergei Popov (KTH) Counselors: Aleksandrs Marinins (KTH)
Master’s dissertation submitted in order to obtain the academic degree of: Master of Science in Engineering Physics
Optics and Photonics
Chair: Prof. dr. Sergei Popov
School of Information and Communication Technology Royal Institute of Technology
Department of Inorganic and Physical Chemistry Chair: Prof. dr. Isabel Van Driessche
Faculty of Engineering and Architecture Ghent University
Academic year 2015-2016
Overview
Photonic integrated circuits are becoming of interest because of their large bandwidth applicability. One of the disadvantages is that scaling is limited by the diffraction limit. This limit can be overcome by including metals which support guiding modes called surface plasmon polaritons (SPP). A limitation of using a metal is the losses that are introduced in the form of Joules losses. This loss of optical energy can be tried to be overcome by including gain material in the circuits. The goal of this work is to investigate this computationally by coupling the Maxwell equations and the semi-classical rate equations in an FDTD framework. The gain medium that is used is an organic dye called Rhodamine 800. The properties of some SPP waveguides and the influence of a gain medium is investigated. In this approach the spatial variation of inversion is taken into account.
Keywords
SPP, ADE-FDTD, gain, PIC
Computational study of organic dye-doped SPP
waveguides by usage of the ADE-FDTD method
Nicolas Knudde
Supervisors: prof. dr. Zeger Hens, prof. dr. Sergei Popov
Counselor: Aleksandrs Marinins
Abstract—Photonic integrated circuits are becoming of interest because of their large bandwidth applicability. One of the disadvantages is that scaling is limited by the diffraction limit. This limit can be overcome by including metals which support guiding modes called surface plasmon polaritons (SPP). A limitation of using a metal is the losses that are introduced in the form of Joules losses. This loss of optical energy can be tried to be overcome by including gain material in the circuits. The goal of this work is to investigate this computationally by coupling the Maxwell equations and the semi-classical rate equations in an FDTD framework. The gain medium that is used is an organic dye called Rhodamine 800. The properties of some SPP waveguides and the influence of a gain medium is investigated. In this approach the spatial variation of inversion is taken into account.
Index Terms—SPP, ADE-FDTD, gain, PIC
I. INTRODUCTION
Classically in optics the description of the wave phenomena could be approximated since the objects were much larger in comparison to the wavelength. However as the optical components got smaller and smaller there was a need to solve the Maxwell equations in its entirety. Solving these equations analytically is only possible in a few cases, so a need for numerical solutions existed. This was a problem that is already long known in classical electromagnetism and the main methods are the finite element method, the boundary integral equations, the method of moments and the finite difference time domain (FDTD) method. These all have their advantages which will not all be discussed in detail here. The FDTD method in particular had the advantage that it has a simple formulation and it is easy to implement non-linearities, for example non-linear material constants. This property will be exploited in this work to include the semi-classical rate equations.
It was already mentioned that the photonic compo-nents get smaller and smaller, which is particularly of interest in the case of the photonic integrated circuit, the optical analogue of the electric integrated circuit. One problem however with further scaling down the components and thus creating a sort of Moore law for PICs is that the confinement of optical components is subjected to the diffraction limit, which prevents
the components to become smaller. One solution is adding metalic material that supports the propagation of SPP waves, which are highly confined. The problem with a metal is that losses are introduced, which are traditionally low for regular photonic structures. Including a gain material might be a solution for this, which are discussed in this article.
II. RATE EQUATIONS
The material that is used consists of a four-level gain medium. The medium is described by using the semiclassical rate equations together with two dipole transitions. The polarization densities are described by a second order differential equation [10]:
∂2P a ∂t2 + 2Γa ∂Pa ∂t + (ω 2 a+ Γ2a)Pa (1) =−2ωa e2d2 a ~ (N3− N0)Eloc ∂2P e ∂t2 + 2Γe ∂Pe ∂t + (ω 2 e+ Γ2e)Pe (2) =−2ωe e2d2 e ~ (N2− N1)Eloc
The rate equations for a four-level system are: ∂N3 ∂t =− N3 τ32 + 1 ~ωa ∂P a ∂t + ΓaPa · Eloc (3) ∂N2 ∂t = N3 τ32− N2 τ21 + 1 ~ωe ∂P e ∂t + ΓePe · Eloc (4) ∂N1 ∂t = N2 τ21− N1 τ10 − 1 ~ωe ∂Pe ∂t + ΓePe · Eloc (5) ∂N0 ∂t = N1 τ10− 1 ~ωa ∂P a ∂t + ΓaPa · Eloc (6)
In the above edi is the dipole moment of the gain
molecule, ωi is the absorption or emission frequency
and Γi is a damping. Ni is merely the population
density of level i and the τ constants are just the relaxation time constants. The local electric field is the field that a single dye molecule experiences and is equal to +2
3 E. The gain medium that is used here is an
organic dye called Rhodamine 800 and the constants are listed in table I.
λe λa Γe Γa de da τ32 τ10 τ21
710 nm 680 nm 1/(20 fs) 1/(20 fs) 0.09 nm 0.1 nm 100 fs 100 fs 500 ps TABLE I
CONSTANTS OF ORGANIC DYE GAIN MEDIUM[8]
III. DRUDE-LORENTZ MODEL
To describe a metal the Lorentz-Drude model is used, which assumes free electrons. There is still a damping present due to the collision of electrons with phonons. The Drude-Lorentz model states:
= ∞−
ω2 p
ω2− iγω (7)
Such that the polarization is the sum of ∞0E and
the polarization described by: ∂2P ∂t2 + γ ∂P ∂t =−ω 2 p0E (8)
The metal that is used is gold and the constants are [2]: ∞= 9.0685, ωp/2π = 2155.6THz and γ/2π =
18.36 THz.
IV. ADE-FDTD
The principle of the auxiliary differential equation FDTD method (ADE-FDTD) is that there are simply equations added to the regular FDTD algorithm. In this case these consist of dipole equations 1, 2 and 8 and the rate equations 3, 4, 5 and 6. Both can be easily discretized using a Yee FDTD scheme. The second order differential equations require the polarization to be known at timestep N − 1 in order to calculate the polarization at N + 1, which means that there is an extra value to be stored. Also a uniaxial PML (UPML) [9] is implemented in order to terminate the calculation domain.
The complicated coupling of the Maxwell equations with these auxiliary equations makes it hard to grasp in which order there equations should be solved. There-fore these are explained step by step and illustrated in figure 1.
1) First of all the D field at step N +1 is calculated from the Maxwell equation.
2) With the help of the E field and the populations Nj at step N, the Pj field at timestep N + 1 is
calculated.
3) The E field at timestep N + 1 is calculated by subtracting the calculated P field from the D field and taking into account the PML parameters 4) From this E field at step N + 1, the population Njat step N +1, as well as the polarizations Pi
at step N + 2 is calculated
5) From the E field at step N + 1, the B field at step N + 1 is calculated
6) From the B field at step N + 1, the constitutive equations and PML parameters the H field at step N + 1 is calculated D + Σ E B H Pj Ni −
Fig. 1. Calculation scheme
V. SPP
Surface plasmon polaritons are already long known and they are essentially oscillations of the electrons in the metal, only TM polarized SPP modes exist. One of their properties is that they are very well confined at the surface especially at the SPP frequency ωsp.
However the confinement always comes at a cost. The more it is tried to confine the wave the higher the losses are. Just because the field penetrate the metal more and more.
Classical structures making use of SPP propagation are a simple dielectric- metal interface, three layer structures of metal-insulator-metal or insulator-metal-insulator. More advanced structures also confine the field in the lateral directions such as the dielectric loaded SPP waveguide [6] , the metal stripe [1], gap SPPs [4] or SPPBG [3]. They all exhibit the same confinement-loss tradeoff.
The metal-insulator-metal structure and the dielec-tric loaded SPP waveguide will be investigated as well as the influence of including a gain material on the propagation in these structures.
VI. RESULTS
In the FDTD code a PML is implemented and the medium is pumped by an SPP. The main disadvantage of using an SPP to pump the medium is that there is a high loss for the pump signal. The wavelength at which the FDTD simulation is done is the absorption wavelength λa= 680nm.
0 100 200 300 400 500 2a(nm) 1000 2000 3000 4000 5000 6000 7000 8000 9000 G ai n co ef fic ie nt (c m − 1)
Fig. 2. Gain coefficient for lossless propagation
A. MIM
For three layer structures it is possible to express the dispersion relation in the case of homogeneous media as an analytical expression which is transcendent [7]. It turns out that there is a symmetrical and asymmetrical mode, but the symmetrical mode is the only mode that exists for every slab thickness 2a, and thus only this mode will be investigated.
When using gold and a dielectric medium of refrac-tive index 1.5 the gain coefficient needed for lossless propagation can be numerically calculated from this dispersion relation and is shown in figure 2 at the emission wavelength. When the gap size is very small the gain coefficient needed is very high, because the penetration of the fields in the metal increase. Here there will be worked at a gap size of 2a = 60 nm, since the gain needed is not too high and there is still a good confinement. Because of the small gap size the field distribution inside the dielectric layer will have a very small spatial variation. The gain coefficient that is needed corresponds to a concentration of dye molecules of about 7 × 1018 cm−3.
When pumping the MIM structure with an SPP there will be some gradient in population inversion inside the insulator layer. However this will be a small gradient because the mode profile inside the insulator is more or less constant because of the very small thickness. Nevertheless it is worth looking which results the ADE-FDTD method gives us. The population inversion after the pulse has passed is shown in figure 3.
From this population inversion the effective refrac-tive refracrefrac-tive index can be calculated by using FEM software, from which the propagation length can be calculated. The propagation length is defined as the length where the power of the mode has decreased by a factor e−1, which is plotted in figure 4. A positive
propagation length means that there is a net damping,
0 200 400 600 800 1000 |E| (kV/cm) 0 1 2 3 4 5 6 7 8 9 N2 − N1 (c m − 3) ×1018
Fig. 3. Population inversion for a MIM structure with gold and dielectric with refractive index 1.5
0 200 400 600 800 1000 |E| (kV/cm) −0.1 0.0 0.1 0.2 0.3 0.4 0.5 1/ Lp r o p (µ m − 1)
Fig. 4. Propagation length at 710 nm for a MIM structure with gold and dielectric with refractive index 1.5
while a negative propagation length means that there is a net amplification of the mode. It is clear from the plot that a net amplification occurs at a pump value of 400kV/cm
B. DLSPPW
The dielectric loaded SPP waveguide consists of a substrate, a layer of metal, a layer of dielectric and air, this is shown in figure 5. Even though we are still below the SPP frequency at 710 nm a good confinement is still obtained because of the large refractive index of the metal in cooperation with the regular confinement of a waveguide. The dimensions in our case are 70 nm for the metal layer and 300 nm for the dielectric layer.
Now one can wonder what the value of the population has to be for this structure in order to compensate losses in the metal. One approximation to calculate this is to approximate the population inversion profile as constant and calculate the effective mode index simply by FEM software (COMSOL vii
air metal substrate dielectric Fig. 5. DLSPPW 0.00 0.05 0.10 0.15 0.20 0.25 0.30 y(µm) 0 1 2 3 4 5 6 N2 − N1 (c m − 3) ×1018 356 kV/cm 1238 kV/cm 3334 kV/cm
Fig. 6. Inversion in the dielectric waveguide for different peak amplitudes
Multiphysics [5]) in order to compensate losses, this gives an inversion of 5.8 × 1018 cm−3. In practice
however, full inversion will be reached much earlier at the metal-dielectric interface, because of the high confinement at the metal when using an SPP. This we can investigate by using the ADE-FDTD method.
A typical behavior of the inversion is plotted in figure 6. Far from complete inversion the population inversion decreases exponentially with the distance from the metal. However, when the pump is high enough total inversion is reached at the metal, and so the population inversion does not increase anymore. This is visible in the flat shape for high amplitudes.
Now when we assume a concentration of dye molecules of about 6.8 × 1018 cm−3, we can calculate
the inversion in the waveguide, use this to calculate the spatial variation of the permittivity and eventually use FEM software to calculate the propagation length of the mode.
The results are plotted in figure 7, a positive propa-gation length means damping, a negative propapropa-gation length means an amplification. In the graph it appears that the compensation of losses occurs before full in-version is reached, the population inin-version the furthest from the metal drops as low as 0.5×1018 cm−3, more
than 10 times as low as what would be needed when assuming a homogeneous gain.
−500 0 500 1000 1500 2000 2500 3000 3500 4000 |E| (kV/cm) −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 0.25 1/ Lpr (µ m − 1)
Fig. 7. Propagation length for a dye concentration of 6.8 × 1018cm−3
VII. CONCLUSION
An ADE-FDTD approach has been introduced to calculate the gain in SPP waveguides. From this the pump value and concentration of dye molecules can be calculated in order to eliminate losses. The structures that were studies were only planar structures, but the approach could be generalized to three dimensional structures. This could be interesting to investigate novel, more complex structures.
REFERENCES
[1] Pierre Berini. Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric struc-tures. Phys. Rev. B, 61:10484–10503, Apr 2000.
[2] Craig F. Bohren and Donald R. Huffman. Classical Theories of Optical Constants, pages 226–267. Wiley-VCH Verlag GmbH, 2007.
[3] Sergey I. Bozhevolnyi, John Erland, Kristjan Leosson, Peter M. W. Skovgaard, and Jørn M. Hvam. Waveguiding in surface plasmon polariton band gap structures. Phys. Rev. Lett., 86:3008–3011, Apr 2001.
[4] Sergey I. Bozhevolnyi and Jesper Jung. Scaling for gap plasmon based waveguides. Opt. Express, 16(4):2676–2684, Feb 2008.
[5] COMSOL AB, Stockholm, Sweden. COMSOL Multiphysics R
v. 5.2: RF Module User’s Guide, 2015.
[6] Zhanghua Han and Sergey I Bozhevolnyi. Radiation guiding with surface plasmon polaritons. Reports on Progress in Physics, 76(1):016402, 2013.
[7] S.A. Maier. Plasmonics: Fundamentals and Applications. Springer US, 2010.
[8] P. Sperber, W. Spangler, B. Meier, and A. Penzkofer. Ex-perimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers. Optical and Quantum Electronics, 20(5):395– 431, 1988.
[9] A. Taflove and S.C. Hagness. Computational Electrodynamics: The Finite-difference Time-domain Method. Artech House antennas and propagation library. Artech House, 2005. [10] Sebastian Wuestner, Andreas Pusch, Kosmas L. Tsakmakidis,
Joachim M. Hamm, and Ortwin Hess. Overcoming losses with gain in a negative refractive index metamaterial. Phys. Rev. Lett., 105:127401, Sep 2010.
Contents
Preface iii
Abstract v
Extended abstract vi
Contents x
List of Figures xii
List of Tables xiv
Nomenclature xv 1 Introduction 1 2 Maxwell’s equations 3 2.1 Maxwell’s equations . . . 3 2.2 Constitutive equations . . . 3 2.3 Wave equation . . . 4 2.4 Boundary conditions . . . 4
3 Interaction of radiation and atomic systems 6 3.1 Transitions . . . 6
3.2 The Einstein relations . . . 7
3.3 Polarization . . . 8
3.4 Modelling materials . . . 8
3.4.1 Classical electron model . . . 8
3.4.2 Drude model . . . 9
3.4.3 Interband transitions in noble metals . . . 10
3.5 Line broadening . . . 11 3.5.1 Homogeneous broadening . . . 11 3.5.2 Inhomogeneous broadening . . . 12 3.6 Amplification . . . 12 3.7 Obtaining inversion . . . 12 ix
3.8 Organic Dye . . . 14
4 Surface Plasmon Polaritons 15 4.1 SPP at metal-dielectric interface . . . 15 4.2 Multilayer SPP . . . 18 4.2.1 IMI structure . . . 18 4.2.2 MIM structure . . . 18 4.3 SPP Waveguides . . . 19 4.3.1 Metal stripe . . . 19 4.3.2 G-SPP . . . 20 4.3.3 DLSPPW . . . 20 4.3.4 SPPBG . . . 20
5 Gain assisted SPP propagation 22 5.1 SPP propagation at metal-gain medium boundary . . . 22
5.2 Current research . . . 23 6 FDTD method 25 6.1 Basic Principles . . . 25 6.1.1 Space discretization . . . 26 6.1.2 Time discretization . . . 27 6.2 Stability . . . 27 6.3 Simulation of a metal . . . 28
6.4 Simulation of four-level gain medium . . . 28
6.5 Absorbing layers . . . 31 6.5.1 Non-dispersive media . . . 31 6.5.2 Dispersive media . . . 33 7 DLSPPW 34 7.1 Structure . . . 34 7.2 No gain . . . 35
7.3 Longer propagation length . . . 36
7.4 Amplification . . . 39 8 MIM structure 41 8.1 Theoretical model . . . 42 8.2 Simulation . . . 43 9 Conclusion 45 A Publication 46 Bibliography 47 x
List of Figures
3.1 Radiative processes in a two-level system . . . 7
3.2 Refractive index for ωpτ = 100 . . . 10
3.3 Scheme of a four-level gain medium . . . 13
3.4 Jablonski diagram of an organic dye [32] . . . 14
4.1 SPP dispersion for gold for γ = 0 (black) and γ?0 (blue) . . . 17
4.2 The transversal component of the magnetic field H on a metal-dielectric interface and a IMI structure and the charge [40] . . . 17
4.3 Propagation constant of an MIM structure with gold and a material with refractive index 1.5 at a wavelength of 710nm . . . 19
4.4 Metal stripe . . . 19
4.5 Slot waveguide . . . 20
4.6 Trench waveguide . . . 20
4.7 DLSPPW . . . 21
4.8 SPPBG principle [14] . . . 21
5.1 Gain coefficient for gold and a medium with refractive index 1.5 . . . 23
6.1 Yee cell in 2D . . . 26
6.2 Calculation scheme . . . 30
6.3 Lineshape function for absorption and emission . . . 31
6.4 PML scheme . . . 32
7.1 DLSPPW slab structure . . . 35
7.2 DLSPPW mode profile at λ0 = 680nm . . . 36
7.3 Propagation length as a function of inversion for DLSPPW . . . 37
7.4 Inversion in the dielectric waveguide for different peak amplitudes . . . 38
7.5 Time dependence of the inversion in the dielectric waveguide: for different distances from the metal. The black line indicates when the peak passes. The size of the peak is |E| = 3.9 × 106 V/m at the metal boundary. . . 38
7.6 Propagation length for a dye concentration of 5.2× 1018 cm−3 . . . . 39
7.7 Propagation length for a dye concentration of 6.8× 1018 cm−3 . . . . 40
8.1 MIM mode profile at λ0 = 710nm for a slab width of 60nm . . . 41
8.2 Gain coefficient for lossless propagation atλ0 = 710nm for a MIM structure with gold and dielectric with refractive index 1.5 . . . 42 8.3 Population inversion for lossless propagation at λ0 = 710nm for a MIM
structure with gold and dielectric with refractive index 1.5 . . . 43 8.4 Population inversion for a MIM structure with gold and dielectric with
refractive index 1.5 . . . 44 8.5 Propagation length at 710 nm for a MIM structure with gold and dielectric
with refractive index 1.5 . . . 44
List of Tables
6.1 Constants of organic dye gain medium [35] . . . 30 7.1 Gold Drude model parameters [7] . . . 35
Nomenclature
β Propagation constant
δ Skin depth
Relative dielectric function 0 Vacuum permittivity
γ Damping constant in the Drude model λ0 Wavelength in vacuum
B Magnetic induction
D Electric displacement field E Electric field H Magnetic field P Polarization density µ0 Vacuum permeability ω Angular frequency ωp Plasma frequency σ Cross section
τnr Non-radiative decay lifetime τsp Spontanious emission lifetime F Photon flux
h Planck constant
k0 Wave constant in the vacuum kB Boltzmann constant
n Refractive index
Ni Population density of level i
T Temperature
DLSPPW Dielectric Loaded Surface Plasmon Polariton Waveguide PML Perfectly Matched Layer
SPP Surface Plasmon Polariton
TEM Transversal Electromagnetic Mode or TEM mode: neither electric nor mag-netic field component along the propagation direction
UPML Uniaxial Perfectly Matched Layer
Chapter 1
Introduction
Photonic integrated circuits or PICs are a large interest in the current scientific research and are the optical analogue of electric integrated circuits. One of the advantages in comparison to electric integrated circuits is that it is possible to work at optical frequencies and that a larger bandwidth can be obtained. Of course one would wish to make these PICs as small as possible, in the same way that scaling for electric integrated circuits has had a significant contribution to Moore’s law. One problem with scaling however is that the optical system is governed by the diffraction limit, which with regular dielectrics makes it impossible to make waveguides very small.
A solution that has been proposed is the use of a combination of metals with those dielectrics. At the metal-dielectric interfaces surface plasmon polaritons or SPPs can propagate, which are collective oscillations of electrons at the surface. The small skin depth of the metal makes that these waves are heavily confined to the metal. Couplers, switches, waveguides, interferometers and so making use of this principle have been demonstrated. Various structures and SPP waveguides were constructed in literature like V-grooves, dielectric loaded SPP waveguides, slot waveguides and so on. The disadvantage of using metals is that they are lossy and only relatively low propagation lengths can be reached. It will be shown that the higher the confinement, the higher the losses and we have to make a trade-off between the two.
One method to overcome this problem is to add a gain material to the surrounding dielectric as an attempt to compensate for the losses in the metal. Some metamaterials were also based on this principle. In this work the gain in such a structure is investigated by the coupling of the semi-classical rate equations combined with Maxwells equations. By this the population inversion which causes gain can be calculated and used to calculate the losses. These calculations are done by usage of the auxiliary differential equations finite difference time domain method (ADE-FDTD) and a perfectly matched layers to simulate open boundaries.
CHAPTER 1. INTRODUCTION 2 For the calculation of the effective mode index in the presence of inversion a commercial FEM software is used (COMSOL Multiphysics).
This thesis is divided into chapters:
• Chapter 1: this short introduction and situation of the problems. • Chapter 2: a short review of the macroscopic Maxwell’s equations.
• Chapter 3: description of the interaction between radiation and atomic systems. Some models for constitutive equations are discussed.
• Chapter 4: theoretical model to surface plasmons and a few structures currently used in research.
• Chapter 5: some previous research on the topic of gain assisted SPP propagation. • Chapter 6: discussion of the ADE-FDTD method.
• Chapter 7 and 8: Two structures are being investigated by ADE-FDTD. • Chapter 9: a short conclusion of this work.
Chapter 2
Maxwell’s equations
2.1
Maxwell’s equations
The Maxwell’s equations are the equations that describe the behavior of electromagnetism and are a synthesis of the works of Coulomb, Gauss, Faraday and Ampère. The macroscopic Maxwell equations state:
∇ × E = −∂B∂t (2.1)
∇ × H = J + ∂D
∂t (2.2)
∇ · D = ρ (2.3)
∇ · B = 0 (2.4)
Or in the frequency domain (eiωt) convention:
∇ × E = −iωB (2.5)
∇ × H = J + iωD (2.6)
∇ · D = ρ (2.7)
∇ · B = 0 (2.8)
Here E is the electric field, H is the magnetic field, D is the electric induction, B is the magnetic induction, ρ is the charge density and J is the current density. There exist some additional relations:
D = 0E + P (2.9)
B = µ0(H + M) (2.10)
Here P is the polarization density wheras M is the magnetization density.
2.2
Constitutive equations
Only media without current or magnetization will be treated here such that there is only one question: how isP calculated? The equation connecting the polarization to the electric
2.3. WAVE EQUATION 4 field is called a constitutive equation and in general this is:
P = 0χ(r, ω)¯¯ · E (2.11) or thus:
D = 0( ¯¯I + ¯¯χ(r, ω))· E = ¯¯ · E (2.12) ¯¯
χ is called the susceptibility tensor whereas ¯¯ is called the permittivity tensor. These tensors are obtained phenomenologically and thus have no exact form, some models will be discussed in the following chapter. In most cases it is sufficient to assume that the medium is isotropic and homogeneous, such that the tensors are reduced to function of frequency. The relative permittivity r is defined as /0.
2.3
Wave equation
For now it is assumed that the medium is non-dispersive, isotropic and homogeneous without a current nor charges. Taking the curl of equation 2.5 and consequently substituting equation 2.6 gives us the wave equation:
∇2E + k2
0rE = 0 (2.13)
Here k0 is the wave constant in a vacuumk0 = ω√0µ0 = ω/c.
This allows us to define the refractive index n = √r which is defined by c = vn, here c is the speed of light in vacuum and v is the speed of light in the material itself. However generally the relative permittivity can be a complex number when amplification or, more commonly, absorption is present. In this case consider the dielectric function = R+ iI, which describes the polarization of a material. This is equivalent to the refractive index formulation: n = nR+ inI with n2 = . And it is easy to relate both quantities to eachother: R= n2R− n2I (2.14) I = 2nRnI (2.15) n2R= 1 2 R+ q 2 R+ 2I (2.16) nI = I 2nR (2.17)
2.4
Boundary conditions
At a boundary of two different materials (1 and 2) we need a way to connect the different fields in both materials, these are called the boundary conditions. In what follows n is the unit normal pointing inside medium 1. The boundary conditions are:
2.4. BOUNDARY CONDITIONS 5 • The normal component of the magnetic induction is continuous:
n· (B2− B1) = 0 (2.18)
• The jump in the normal component of the electric displacement is equal to the surface charge:
n· (D2− D1) = ρs (2.19) • The tangential component of the electric field is continuous:
n× (E2− E1) = 0 (2.20)
• The jump of the tangential component of the magnetic field is equal to the surface current:
Chapter 3
Interaction of radiation and atomic
systems
3.1
Transitions
In the basic action of a gain medium there are three processes that need to be taken into account, namely absorption, spontaneous emission and stimulated emission. We consider an atomic two-level system with lower energyE1 and upper energyE2as shown in figure 3.1. Absorption occurs when a photon with frequency hν0 = E2− E1 impinges on the atom, such that the atom can be excited to level 2 with a finite probability.
When the system finds itself in level 2, there is a probability for a decay by emitting a photon, which is called spontaneous emission. Photons which are produced in this process are emitted in a random direction and without any coherence. However, this decay does not have to be radiative, but can also be caused by creating kinetic energy of surrounding atoms for example, or in the case of a lattice can consist of phonons that are emitted. This kind of decay we call non-radiative.
Now, when the system finds itself in level 2 and a photon with frequency ν0 impinges on the system, it is possible that the system will decay. This is called stimulated emission and in this case the emitted photon will have the same direction, same phase (co-herence) and same frequency of propagation as the incident photon, so the flux is amplified.
3.2. THE EINSTEIN RELATIONS 7 E1 E2 (a) Absorption E1 E2 (b) Spontaneous emis-sion E1 E2 (c) Stimulated emission
Figure 3.1: Radiative processes in a two-level system
3.2
The Einstein relations
Now it is only a matter of quantifying these processes, this is what Einstein did [31]. In what follows N2 is the population of the second level while N1 is the population of the first level. We also assume a system where there are no degenerate energy levels. The rate of spontaneous emission is described by:
dN2 =−A21N2dt (3.1)
A21 is called the Einstein coefficient of spontaneous emission. The rate of absorption is described by:
dN1 =−B12ρ(ν0)N1dt (3.2) B12 is called the Einstein coefficient of absorption and ρ(ν0) is the spectral energy density of the electromagnetic field at the resonance frequencyν0. And lastly the rate of stimulated emission is described by:
dN2 =−B21ρ(ν0)N2dt (3.3) Here B21 called the Einstein coefficient of stimulated emission. The photons created by stimulated emission have the same frequency, direction, polarization, and phase as the incoming photon.
Now rises the question if there is a relation between the different Einstein coefficients. When one assumes thermal equilibrium the population ratio is described by a Boltzmann factor. At the same time the average population densities should be constant, which means that there must be as many transitions 2→ 1 as transitions 1 → 2:
− B12ρ(ν0)N1 =−B21ρ(ν0)N2− A21N2 (3.4) From which: ρ(ν0) = A21N2 B12N1 − B21N2 (3.5) Now assume a blackbody radiation and we know that the spectral energy density can be described by Planck’s law:
ρ(ν) = 8πν 2 c3 hν ekBThν − 1 (3.6)
3.3. POLARIZATION 8 By comparing equation 3.5 and 3.6 and taking into account the Boltzmann factor, one can deduce that:
B21 = B12 (3.7)
A21 =
8πhν3
c3 B12 (3.8)
3.3
Polarization
Consider a wave passing through a dielectric, the electrons in the dielectric will be displaced under the influence of the electric field. This induces a dipole moment p for each atom. For simplicity the dielectric is assumed to be isotropic such that p is linearly proportional with the electric field by a constantα called the polarizability:
p = αE (3.9)
For media with a low density of atoms he macroscopic polarization density P is simply this microscopic dipole moment multiplied by the amount of atoms per unit volume N .
However, when the density becomes higher the local effects of nearby atoms must be taken into account and the atom experiences a local electric field Eloc. We draw a sphere about one particular atom, the inside of this sphere will be described microscopically while the outside will be described macroscopically. The macroscopic description of the electric field inside due to material outside this sphere can be reduces to a surface charge on this sphere equal to 3P
0. The atoms inside the sphere cause a so-called crystal field Ecrystal.
This crystal field is exactly 0 for cubic crystals and is negligible for many other cases such that: P = N αEloc = N α E + P 30 = 0χE (3.10)
From which the dielectric susceptibility χ can be solved: χ = 3N α
30− Nα
(3.11) And from which the Clausius-Mossotti relation can be derived
− 0 + 20 = N α 30 (3.12)
3.4
Modelling materials
3.4.1
Classical electron model
A very simple model of electromagnetic waves in a material is the classical electron model. Here the sea of electrons is considered as a plasma of particles with a certain mass m. Also some damping constant γ or analogous relaxation time τ = 1
3.4. MODELLING MATERIALS 9 This damping constant has its origin in collisions with phonons, lattice imperfections and impurities and has to be in principle be calculated from quantum mechanical considerations [19]. In general there is also a restoring spring-like force because of the boundedness to the ions. This becomes:
md 2x dt2 =−mγ dx dt − mω 2 0x (3.13)
In the presence of an electric field and working in the sinusoidal domain (E(t) = E(t)eiωt) this becomes:
x = eE
m(ω2
0− ω2+ iγω)
(3.14) And after puttingP = N ex we can calculate the dielectric susceptibility and define the plasma frequency ωp: χ = N e 2 m0 1 ω2 0 − ω2+ iγω = ω 2 p ω2 0− ω2+ iγω (3.15)
3.4.2
Drude model
Before we elaborate on the concept of plasmonics, it is important to get a clear view of how metals behave electromagnetically. As we all know, metals are non-transparent at optical frequencies, but the electromagnetic waves penetrate with a certain skin depth δ. As we go to the NIR range, this skin depth increases and finally at ultraviolet frequencies, the metal behaves like a dielectric. In this section we will try to explain this behavior.
A very simple model of electromagnetic waves in a metal is the Drude model, also called plasma model. For a metal it is assumed that there are free electrons and that there is thus no restoring force (ω0).
It is now easy to determine the dielectric function: R= 1− ω2 pτ2 1 + ω2τ2 (3.16) I =− ω2 pτ ω(1 + ω2τ2) (3.17)
ωp is called the plasma frequency and is equal to q
ne2
0m. The result for ωpτ = 100 is
plotted in figure 3.2.
Three regimes can be distinguished [24]:
1. 0 < ωτ < 1: the radiation is absorbed: nR≈ nI = q
τ ω2 p
2ω
2. 1 < ωτ < ωpτ : here the real part of the permittivity is negative. Since nI nR the metal is reflective.
3. ω > ωp: the real part of the dielectric constant becomes zero and then positive. In this regime nR nI and the metal is transparent.
3.4. MODELLING MATERIALS 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 7 8 9 10 ω/ωp n nR nI
Figure 3.2: Refractive index forωpτ = 100
The limit for small wavelengths can also be proven from a quantum mechanical point of view [24]. One has to assume however, that the skin depth is considerably larger than the electrons’ mean free path. When this is not the case a phenomenon called the Anomalous Skin Effect appears which has some interesting features that can be used to measure geometrical properties of the Fermi surface [30].
3.4.3
Interband transitions in noble metals
The most important metals used in plasmonics are gold and silver, both noble metals. In the noble metals the filled d band is close to the Fermi level. This means that at frequencies where the dielectric constant is dominated by the electrons in the s band, the dielectric constant is very high because of the highly polarized d band.
Also the interband transitions are dominated by transitions from the d band to the Fermi surface, and not higher unoccupied bands.
A solution for this is describing the electron is by adding several Lorentz resonances to the dielectric constant:
fi ω2
i − ω2+ iγiω
(3.18) When one takes all this into account the dielectric function of a metal can be described by:
(ω) = ∞+X i fiωi2 ω2 i + iγiω− ω2 (3.19)
3.5. LINE BROADENING 11
3.5
Line broadening
What does this all tell us about the transition processes? Let us first assume that we are close to the resonance frequency and assume that χ is small compared to 1:
χ = ω 2 p 2ω0(ω0− ω + iγ/2) (3.20) n = 1 +1 2χ = 1 + ω2 p 4ω0(ω0− ω + iγ/2) (3.21) The absorption coefficienta is equal to the distance it takes for the power of the optical signal to decrease with a factor 1/e.
a = 2k0nI = 4π λ ω2 pγ 8ω0((ω0− ω)2+ (γ/2)2) (3.22)
This function is not a delta distribution but has a finite width, this phenomenon is called line broadening. Line broadening causes are traditionally divided into two kinds, namely homogeneous broadening and inhomogeneous broadening. Homogeneous broadening is a characterized by a linewidth ∆ν, which is a property of all atoms, meaning that all atoms are indistinguishable. Inhomogeneous broadening is the consequence of atoms having a slightly different transition frequency, where atoms are thus distinguishable. When we normalize a we get a function g(ν) with R0∞g(ν)dν = 1, which is called the lineshape function.
3.5.1
Homogeneous broadening
The most common causes of homogeneous broadening are:
• Natural broadening: the energy levels in for example figure 3.1 are not really just lines. The Heisenberg principle tells us that ∆E∆t≥ ~/2. Since the levels have a finite lifetime, they thus have a certain energy band instead of a line. This finite lifetime is the same for every atom in the gain material and causes homogeneous broadening [36].
• Collision with a phonon: when a molecule is embedded in a crystal there may be phonons present that can be absorbed by the molecules. This the phase between the oscillations of the atoms and that of the photons present [44].
• Pressure broadening: when the concentration of atoms is high enough the atoms start to collide considerably. This causes a lifetime termination and in addition a random phase change. This results in broadening [44, 36].
3.6. AMPLIFICATION 12
3.5.2
Inhomogeneous broadening
There are two main causes of inhomogeneous broadening:
• On the one hand impurities can be present in the crystal, which alters the properties of the material ion by ion. This shifts the energy levels and causes broadening. • On the other hand a Doppler shift can be present in gaseous media. The molecules or
atoms have a finite speed and thus a doppler shift in frequency is present. However the shape of the linefunction (the normalized absorption coefficient) becomes Gaussian in that case, not Lorentzian.
3.6
Amplification
Assume that the radiation is monochromatic, that is the frequencies are contained within [ν, ν + dν[ where dν is very small, much smaller than the linewidth ∆ν. g(ν)dν can now be interpreted as the probability of a transition by radiation with frequency in [ν, ν + dν[. Now ignoring spontaneous emission and using the Einstein equations we get:
dN2
dt =−B21ρ(ν)N2g(ν)dν − B12ρ(ν)N1g(ν)dν (3.23) ρ(ν)dν can be interpreted as the energy density u and B21= B12:
dN2
dt =−B21ug(ν)(N2− N1) (3.24) The change of energy density is equal todu =−dN2hν and the photon density Z is equal to Zhν = u such that:
dZ
dt = hνB21g(ν)(N2− N1)Z = b21(N2− N1)Z (3.25) This means that when inversion is obtained (N2 > N1) the photon density rises exponentially and amplification is obtained. At the same time ncdt = dz such that:
dZ dz =
n
chνB21g(ν)(N2− N1)Z = γZ (3.26) Which also means an exponential rise in the spatial dimension while propagating in a medium with refractive index n.
3.7
Obtaining inversion
In previous section it was established that we can’t use a equilibrium system. So we could try to use a external beam to pump the system to inversion. However, onceN1 = N2 there would be no net absorption or emission and the material would become transparent and would be of no use. This creates a large problem and the idea of creating a gain medium
3.7. OBTAINING INVERSION 13 level 0 level 1 level 2 level 3 Rp τ10−1 τ21−1 Bφ τ32−1
Figure 3.3: Scheme of a four-level gain medium
of 2 levels must be abandoned.
Luckily there exist three or four level systems in which it is possible to create inversion under certain conditions. Consider a four level system like the one shown in figure 3.3, it is possible to pump from level 0 to level 3. Subsequently there is a fast decay from level 3 to level 2, then there is a stimulated emission which causes the transition from level 2 to 1. And then again there is a fast decay from level 1 to level 2. In this way there can be an inversion obtained between level 2 and 1.
To describe this let us assume that there is a pumpRp which is simply the amount of excitations from level 0 to level 3 per unit of time. The gain material is placed into a cavity and the lifetime of a photon in such a cavity is τc. Also the relaxation of level 3 to 2 and level 1 to 0 is infinitely fast such thatN3 = N1 = 0. The relaxation of level 2 to 1 happens with a finite time constant τ21 such that:
dN2 dt = Rp− b12ZN2− N2 τ21 (3.27) dZ dt = b12ZN2− Z τc (3.28) When we now want to build a continuous wave laser (CW), we want a steady state solution. This means that the time derivatives drop out, by which we obtain [31]:
N2 = 1 τcb12 (3.29) Z = Rpτc− N2 τ21 (3.30) This means that the pump parameter should be higher than a threshold value of N2
τ21 =
1 b21τ21τc.
3.8. ORGANIC DYE 14
3.8
Organic Dye
Dye lasers make use of organic molecules. The advantage of a dye is that the energy levels are actually not lines, but have a certain linewidth, which in the case of dyes is relatively large. This makes that the dye is working in a large range of wavelengths rather than one specific wavelength [37]. A energy or Jablonski diagram of a typical organic dye is shown in figure 3.4.
When the triple states are negligible it is usually possible to model the dye just using a 4-level system taking into account the first and second singlet state S0 and S1. The way that this triplet state T1 is populated is by a non-radiative decay called intersystem crossing. This intersystem crossing represents only losses in the system and is not preferred for most applications. However when a pump pulse is used to pump the system, that is sufficiently short, meaning much shorter than the intersystem crossing lifetimes [34], the population of T1 is negligible. Often another approach is used for this where the organic dye molecules are in a solution of a liquid such as methyl alcohol, glycerol or water and the liquid flows constantly, which will not be discussed here.
When there are too many photons impinging on the gain medium, some molecules might lose there fluorescent properties forever, this phenomenon is called photobleaching [2]. This photobleaching is due to the breaking of chemical bonds of the gain medium.
For higher concentrations fluorescence quenching starts to be increasingly important [41]. Fluorescence quenching is any process that decreases the fluorescence intensity of a sample. This may have many different reasons like excited-state reactions, molecular rearrangements, energy transfer, ground-state complex formation, and collisional quenching [21].
Chapter 4
Surface Plasmon Polaritons
4.1
SPP at metal-dielectric interface
Below the plasma frequency the refractive index of a metal has a negative real part and a positive real part. This means that waves inside bulk metal cannot propagate. However it is possible to excite a bound surface mode propagating along a metal dielectric interface, this mode is called a surface plasmon polariton or SPP. Physically this SPP can be seen as a TEM mode in the dielectric being coupled to the oscillation of the electrons in the metal, which are mostly longitudinal [16]. For this reason an SPP is never a TE mode in the dielectric.
Now we can investigate the properties of the SPP by solving the Maxwells equations, only a TM mode will be considered. The Maxwell equations in the frequency domain with the eiωt convention are reduced to:
Hz = i ωµ0 ∂Ey ∂x − ∂Ex ∂y (4.1) Ex =− i ω ∂Hz ∂y (4.2) Ey = i ω ∂Hz ∂x (4.3)
The wave equation states:
∂2H z ∂y2 + (k
2
0− β2)Hz = 0 (4.4)
When we now assume that there is a bound state along a surface between two half-infinite planes of different media (y > 0 and y < 0) propagating along the x-axis the following equations hold. For y > 0:
Hz = A1e−iβx−k1y (4.5) Ex = i k1 ω10 A1e−iβx−k1y (4.6) Ey = β ω10 A1e−iβx−k1y (4.7) 15
4.1. SPP AT METAL-DIELECTRIC INTERFACE 16 Fory < 0: Hz = A2e−iβx+k2y (4.8) Ex =−i k2 ω20 A2e−iβx+k2y (4.9) Ey = β ω20 A2e−iβx+k2y (4.10)
The boundary conditions now consist of: Hz|y=0+ = Hz|y=0− and Ex|y=0+ = Ex|y=0− [45].
This means that A1 = A2 and:
k2 k1
=−2 1
(4.11) Which indeed means that there is only a bounded mode if the real part of the dielectric constants have an opposite sign, which is the case for a metal below the plasma frequency and a dielectric.
Combining this with the wave equation the exponential tail in the y-direction can be calculated: k21 = β2− k2 01 (4.12) k22 = β2− k022 (4.13) While: β = k0 r 12 1+ 2 (4.14) and ki = ik0 s 2 i 1+ 2 (4.15) The dispersion relation for gold is plotted in figure 4.1. When there is no damping there is a frequency gap where the propagation constant is purely imaginary. This means that it is completely damped and non-propagating. When approaching this lower boundary ωsp = √1+ωp 2 the wave vector becomes infinite and an electrostatic character is obtained [22, 16]. The advantage of working at frequencies close to ωsp is that there is a large confinement to the interface, which comes in handy since we want devices and components that are as small as possible. However the eternal trade-off that has to be taken into consideration is that the imaginary part of β becomes larger. Which means that the losses are higher. When the frequency is high enough the metal becomes transparent and the regular dispersion of a non-dispersive bulk medium with dielectric constant∞ is obtained. However, when a damping is taken into account there is a finite wave vector in this gap, which represents a so-called quasibound, leaky mode [22].
4.1. SPP AT METAL-DIELECTRIC INTERFACE 17 βc ωp 0 0.2 0.4 0.6 0.8 1 ω ωp 0 0.1 0.2 0.3 0.4 0.5 0.6
Figure 4.1: SPP dispersion for gold forγ = 0 (black) and γ?0 (blue)
453
Guided Wave Plasmonics
17.3 SPPs ON THIN METAL FILMS
17.3.1 L
ONG-R
ANGESPP M
ODESThe previous discussion pertains to the surface of a bulk metal forming an interface
with the overlaying dielectric, for example, air. Let us now consider a metal film
deposited on a dielectric substrate (dielectric constant
ɛ
3) with the film topped by
another dielectric (dielectric constant
ɛ
1). The SPP modes are expected to occur at
both the metal–dielectric interfaces, taken to be at z
= ± t/2. The bound thin-film
slab modes are obtained by requiring that the fields decay in the dielectric media
away from the two interfaces. For this, appropriate solutions to Maxwell’s equations
in the three media are matched at the two interfaces subject to the standard boundary
conditions. As before, only E
x, E
zand H
yare nonzero for TM polarization. The SPP
modes are determined by the condition [1,20]:
ε α
1 2+
ε α ε α
2 1 2 3ε α
3 2ε α
1 2ε α ε α ε α
2 1 2 3 3 22
α
20
(
)
(
+
)
+
(
−
)
(
−
)
exp
(
−
t
)
=
(17.2a)
α
j=
⎣⎢
⎡
k
x2−
ε ω ω
j( )( )
c
2 1 2⎤
⎦⎥
;
j
=
1 2 3
, ,
(17.2b)
Note that if t is much larger than the SPP decay length L
z,min metal (
=α
2)
−1, the
second term on the left-hand side of Equation 17.2 becomes negligible. (Typically,
L
z,mis about a few tens of nanometres for Ag and Au in the IR.) Equating either of the
two brackets in the first term with zero, we obtain conditions for having independent
SPP modes on each of the two metal–dielectric interfaces (compare with the
condi-tion:
α
1= −α
2(
ɛ
1/
ɛ
2) found in Section 17.2.2 for a single interface). The dispersion of
these modes is governed by Equation 17.1. As t decreases and becomes comparable
to L
z,m, the two SPP modes interact and can no longer be considered to belong to
a single interface. Figure 17.3 schematically shows the fields associated with the
modes and the corresponding charge density induced at the interfaces for the two
coupled modes on the thin-film slab. The corresponding fields and charges for the
SPP on a single interface are also shown. One of the coupled modes is an asymmetric
mode (a
b) with H
yhaving a node in the metal film and the other, a symmetric mode
Hy Hy n1 n2 + + + + + + + + + + ++ ––– ––– ––– n1 n2 n3 sb Symmetric Antisymmetric ab Metal + + + + + + ––– –––
FIGURE 17.3 The y component of the magnetic field of SPP (H
y) is shown for a single
metal–dielectric interface and for coupled SPP modes for a thin metal film, along with
cor-responding surface charges.
Figure 4.2: The transversal component of the magnetic fieldH on a metal-dielectric interface and a IMI structure and the charge [40]
4.2. MULTILAYER SPP 18
4.2
Multilayer SPP
Now the metal-insulator-metal (MIM) and isolator-metal-insulator (IMI) structures can be investigated in a similar fashion as in section 4.1. When this calculation is performed two dispersion relations are obtained [22]:
tanh k1a =− k21 k12 (4.16) tanh k1a =− k12 k21 (4.17) Here2a is the width of the inner layer, while subscript 1 stands for the medium in the inner layer and subscript 2 stands for the medium in the outer layers (a symmetric structure is assumed). It can be proven that equation 4.16 corresponds to a symmetric Hz function while 4.17 corresponds to an asymmetric Hz function. At the same time equations 4.12 and 4.13 from which the propagation constant can be calculated.
4.2.1
IMI structure
Now we consider the IMI structure in figure 4.2. A property of the odd/antisymmetric modes is that when the metal film becomes thinner, the confinement gets smaller. The mode will gradually turn into a plane wave, which means that there are lower losses and thus an increase in propagation length due to lower confinement to the metal this is called a long range SPP or LR-SPP [16]. The condition of low losses is strongly affected by the symmetry: the loss is the lowest when the dielectric at both sides is the same. On the other hand the even mode will get more confined if the metal layer gets thinner, thus higher losses and a lower propagation length are obtained this mode is also called a short range SPP or SR-SPP.
Again the trade-off between the propagation length and the confinement appears.
4.2.2
MIM structure
When the dielectric layer is thick enough, there are simply 2 possible SPP waves that can propagate. Decreasing the width of the layer couples those two SPP waves and a symmetric and antisymmetric mode appear. The only mode that propagates for all thicknesses (that is also for very small ones) is the mode that has a symmetric electric field component in the lateral direction and an antisymmetric electric field component in the direction of propagation. The propagation constant for this mode as a function of the middle layer thickness 2a is plotted in figure 4.3.
The propagation length is even higher than a regular dielectric-metal interface for certain widths of the dielectric layer, which is explained due to the fact that the mode approaches the electrostatic capacitor mode and that the fields inside the gap is very high
4.3. SPP WAVEGUIDES 19 0 10 20 30 40 50 60 2a(nm) 0 5 10 15 20 25 30 35 ℜ β k0
(a) Real part
0 10 20 30 40 50 60 2a(nm) −2.0 −1.5 −1.0 −0.5 0.0 ℑ β k0 (b) Imaginary part
Figure 4.3: Propagation constant of an MIM structure with gold and a material with refractive index 1.5 at a wavelength of 710nm
[9]. When now the metallic layers are pushed even further together the penetration of the fields in the metal start to increase and the propagation length starts decreasing again. Nevertheless a good confinement can be obtained.
4.3
SPP Waveguides
When we use a SPP for specific application, of course it is not the intention to use a field that is only confined in one direction. Hence, a way is needed to confine the light transversally also. Several structures were proposed and investigated in literature. A few of these out of the vast collection will be discussed in this section.
4.3.1
Metal stripe
A first, simple example is the metal stripe [4], it is simply an extension of the IMI structure where the metal has finite dimensions in the lateral direction and is shown in figure 4.4. A difference is that no purely TM modes exist such that the fields have components in all directions. Also there are higher order modes that can exist and their dispersion relation is discussed in [4].
metal
dielectric
4.3. SPP WAVEGUIDES 20 metal
dielectric
Figure 4.5: Slot waveguide
metal air
Figure 4.6: Trench waveguide
4.3.2
G-SPP
Gap SPP’s or G-SPP’s are structures where the wave is traveling in a gap between metal layers [11] and is a generalization of the 2D MIM structure. There are different kinds of geometries and the most important ones are slot waveguides (figure 4.5) [5] and trench waveguides (figure 4.6) [5].
4.3.3
DLSPPW
The dielectric loaded SPP waveguide or DLSPPW is simply a regular optical waveguide standing on top of a metal. The advantage of this structure shown in figure 4.7 is that an SPP only is confined in the direction perpendicular to the interface [16]. When a dielectric is added, lateral confinement is provided. However the operation frequency is still pretty far from the plasma SPP frequency as discussed in chapter 4, nevertheless still a good confinement can be obtained because of the large refractive index of the metal and the confinement by the waveguide. The structure is theoretically examined in [17] by using the finite element and effective-index method. Other structures such as bends and interferometers are investigated in [20].
4.3.4
SPPBG
This structure is introduced in [10] and is based on a regular photonic idea, namely the photonic band gap (PBG). The SPP Band Gap structure or SPPBG consists of a two dimensional crystal consisting of pillars made of a photoresist placed on a metal [14]. Due to Bragg reflection the structure can be almost perfectly reflecting for SPPs at certain wavelengths, this principle is sketched in figure 4.8. When there is now for example a line defect in the 2D crystal this can be seen as a waveguide for SPPs.
4.3. SPP WAVEGUIDES 21 air metal substrate Figure 4.7: DLSPPW Figure 4.8: SPPBG principle [14]
Chapter 5
Gain assisted SPP propagation
By now it should be clear that the largest problem with using metallic structures and SPP’s is that there are high losses. One of the solutions in literature is using a gain medium as surrounding material. Because of the high fields at the metal boundaries it is possible to pump the medium to obtain a larger propagation length. First the infinitely extended metal-dielectric interface in 4.1 will be discussed loosely based on the approach in [26].
5.1
SPP propagation at metal-gain medium boundary
Here we assume that we have a gain medium with a certain constant, space-independent, isotropic gain. The equations 4.14 and 4.15 are still valid here. We can expand the relative permittivity in their real and imaginary parts: i = 0i+ i00i. The damping of the mode occurs because the imaginary part of the relative permittivity of the metal is non-zero and negative. However, because the gain has a positive imaginary part, the two effects might cancel out each other.
First of all we want the mode to still be confined to the metal because that is exactly the advantage of using SPP’s, which mathematically means that ki has to have a positive real part. This implies that there is a limit to the gain that the material has which is:
002−p002 2 − 801(01+ 02) 2 < 00 1 < 002+p002 2 + 801(01+ 02) 2 (5.1)
On the other hand we want to be able to calculate the gain for which the damping of the metal is exactly to investigate this we write equations 4.14 as:
β k0 2 = (|1| 20 2+|2|201) + i(|1|2002+|2|2001) (0 1+ 02)2+ (001+ 002)2 (5.2) To reverse the damping of the metal the imaginary part of this equation has to be equal to zero. This implies that:
5.2. CURRENT RESEARCH 23 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 γ0c ωp 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ω ωp
Figure 5.1: Gain coefficient for gold and a medium with refractive index 1.5
001 = |2| 2 200 2 −1 ± s 1− 4(01002)2 |2|4 ! ≈ −|2| 2 00 2 +(01) 200 2 |2|2 (5.3a) −(01) 200 2 |2|2 (5.3b) The approximation is valid since the real part of the permittivity of the metal is high. One can prove that solution 5.3a does not satisfy equation 5.1 and is thus not bounded. Now calculating the power gain coefficient γ (= −α) with solution 5.3b:
γ0 k0 = 00 2(01)3/2 |2|2 (5.4) Below this gain coefficient the mode is lossy, but of course the propagation length increases approachingγ0. When the gain coefficient is higher thanγ0the mode is amplifying and the signal gets larger. This relation is plotted in figure 5.1. The gain that is needed becomes maximum at the SPP frequency, since the propagation length of the SPP becomes minimal.
5.2
Current research
In [26] IMI structures and metal stripes were investigated theoretically by the use of a space independent gain in the dielectric and FEM software. They reported good results: low or no losses were obtained. The so-called hybrid plasmonic waveguides which are inherently three dimensional structures were investigated in [13], they also used a constant
5.2. CURRENT RESEARCH 24 gain. They concluded with the observation that one needs a higher gain with higher refractive index media which corresponds to equation 5.4. A metal-insulator-semiconductor (MIS) structure was theoretically investigated in [42] taking into account temperature effect, where they obtained a positive gain without using a bulky amplification system. A plasmonic laser consisting of a metallic nanowire was realized in [28].
Attention was drawn to the realization of nanostructres such as spherical metallic shells or metallic spheres. The approach introduced by Stockman and Bergman [3] and was called a SPASER. The SPASER consists of metallic spherical particles which have a resonance frequency. The fundamental resonant mode is for example dipolar [5]. When a resonant mode is excited the fields in the vicinity of the metal is strongly enhanced and acts as a resonator. When this is coupled to a gain medium, by using a shell made of gain material the resonator functions as the feedback mechanism and the amplification of plasmons is obtained.
Chapter 6
FDTD method
The method that is going to be used in this thesis to investigate gain in waveguides doped with a gain medium is the Finite Difference Time Domain method. It is a very classic and intuitive way to solve differential equations which relies on the approximation of derivatives by finite differentials. However some difficulties can occur which will be discussed in this chapter.
6.1
Basic Principles
The basic problem that is classically solved by the usage of the FDTD method are the macroscopic Maxwell equations, and one specifically only makes use of the curl equations:
(
∇ × E = −∂B ∂t ∇ × H = ∂D
∂t
Here the electric displacement D is equal to 0E + P and throughout this work it is assumed that we are handling non-magnetic material: B = µ0H. The polarization will be determined by the relative dielectric constant or will depend on the inversion in the doped material.
Here the discretization of the equations will be performed for a 2 dimensional problem in the (x, y) plane where the electric field only contains a field component perpendicular to this plane, thus in the z-direction.
The equations can be reduced to:
∂yEz =− ˙Bx (6.1)
∂xEz =− ˙By (6.2)
6.1. BASIC PRINCIPLES 26 y x Ex Ey Ez Bx By Bz
Figure 6.1: Yee cell in 2D
6.1.1
Space discretization
A Yee cell is used [38], which means that we discretize the fields in a grid, which is not the same for all fields. The cell that we use is pictured in figure 6.1. The Ez fields are defined on points (i dx, j dy) ( (i, j)∈ N2) , here dx and dy are the discretization steps in the x and y direction respectively. The Bx field is difined on points i dx, j + 12
dy and the By field is defined on points i +12
dx, j dy. These fields will be written in short as Ez|(i,j), Bx|(i,j+21) and By|(i+12,j).
We can now make a Taylor expansion: Ez|(i+1,j) = Ez|(i+1 2,j)+ 1 2dx(∂xEz)|(i+12,j)+ 1 8dx 2(∂2 xEz)|(i+1 2,j)+ O(dx 3) (6.3) Ez|(i,j)= Ez|(i+12,j)− 1 2dx(∂xEz)|(i+12,j)+ 1 8dx 2(∂2 xEz)|(i+12,j)+ O(dx3) (6.4) If we subtract both equations we get:
Ez|(i+1,j)− Ez|(i,j) = dx(∂xEz)|(i+12,j)+ O(dx3) (6.5) or
(∂xEz)|(i+12,j) =
Ez|(i+1,j)− Ez|(i,j)
dx + O(dx
2) (6.6)
This is exactly the derivative we need to evaluate equation 6.2, since the Bx field is defined exactly at (i + 1
2, j). This approach to calculate the derivatives is called the finite difference approach because of the finiteness of dx.
6.2. STABILITY 27 Ez|(i,j+1)− Ez|(i,j) dy =−( ˙Bx)(i,j+12) (6.7) Ez|(i+1,j)− Ez|(i,j) dx =−( ˙By)(i+12,j) (6.8) Hy|(i+12,j)− Hy|(i−12,j) dy − Hx|(i,j+12)− Hy|(i,j−12) dy = ( ˙Dz)(i,j) (6.9)
6.1.2
Time discretization
A staggered grid is also used for the time discretization which means: ˙ B|t=kdt= B|t=(k+12)dt− B|t=(k−1 2)dt dt (6.10) ˙ E|t=(k+12)dt= E|t=(k+1)dt− E|t=k dt dt (6.11)
6.2
Stability
In the FDTD method instabilities due to discretization can occur which often manifests itself as an exponential growth of the fields during the time stepping. There is no theorem that provides a sufficient condition in order to avoid this phenomenon. However, there is a necessary condition which is called after the three mathematicians that first formulated it: the Courant-Friedrich-Levy condition or CFL condition:
Theorem. If convergence is obtained using the FDTD method in a 3D problem, then the following condition is fulfilled:
C = cmaxdt r 1 dx2 + 1 dy2 + 1 dz2 ≤ 1 (6.12)
C is a dimensionless number called the Courant number and cmax is the maximum wave speed in the problem [18].
The theorem restricts the time step, which is logical. It also contains a tradeof: the spatial discretization has to be fine enough in order to describe the wave phenomena correctly: at least dx < λ4, and one can see the smaller this spatial discretization step, the smaller the time step should be. This means that a high spatial resolution forces the algorithm to use a small discretization step and will thus take longer. The theorem is easily generalized to dimensions 1 and 2.
Since it is not a sufficient condition, one still has to look out for instabilities in the solutions using the regular time stepping algorithm.
