LUẬN văn THẠC sĩ fabrication and study of optical properties of multilayer metal insulator metal nanocups

1.4.2 Calculation transmittance model of MIM nanocups structure 19 2.1.1 Polystyrene nanoparticles fabrication 21 2.1.2 Silane coupling preparation 21 2.1.4 Fabrication of monolayer Poly

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY

FABRICATION AND STUDY OF OPTICAL PROPERTIES OF MULTILAYER METAL – INSULATOR

– METAL NANOCUPS

MASTER’S THESIS

Hanoi, 2019

VU MINH THONG

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY

VU MINH THONG

FABRICATION AND STUDY OF OPTICAL PROPERTIES OF MULTILAYER METAL – INSULATOR

DR PHAM TIEN THANH

PROF DR NGUYEN HOANG LUONG

Acknowledgements

First of all, I would like to express my sincere thank to my supervisors: Dr

Pham Tien Thanh, and Prof Nguyen Hoang Luong, for their guidance, encouragement to completed this thesis

Second of all, I would also like to thank Mr Nguyen Van Tien, and Mrs

Nghiem Ha Lien, for their support in fabricating Polystyrene nanoparticles

Third of all, I would like to express my sincere thank to my classmate Mr

Pham Dinh Dat Thank to you, I got basic knowledge of FDTD method Thank you for your willing to help

Forth of all, I would like to thank Vietnam Japan University and staff working here for their necessary supports

Last but not least, I also would like to express my sincere thank to my family, for their fully support

1.4.2 Calculation transmittance model of MIM nanocups structure 19

2.1.1 Polystyrene nanoparticles fabrication 21 2.1.2 Silane coupling preparation 21

2.1.4 Fabrication of monolayer Polystyrene nanosphere on glass substrate

3.2.1 Optical properties of MIM nanocups structure on Glass Substrate

35 3.2.1.1 Confirmation of thickness of each layer 35

3.2.1.2 Transmittance properties of samples on substrate 36 3.2.1.3 Transmittance properties of substrate after separate particle 39 3.2.2 Optical properties of MIM nanocups in water 41

List of Fig.s page Fig 1.1 Geometry for SPPs propagation at a single interface between

metal and dielectric

2

Fig 1.2 Metal – Insulator – Metal structure 6 Fig 1.3 Dispersion relation of the fundamental coupled SPP modes of

a silver/air/silver multilayer geometry for an air core of size

100 nm (broken gray curve), 50 nm (broken black curve), and

25 nm (continuous black curve) Also shown is the dispersion

of a SPP at a single silver/air interface (gray curve) and the air light line (gray line)

7

Fig 1.4 Models of shell structures (a) Concentric-type

semi-shell, (b) deposition type semi-shell without metal migration, (c) with metal migration (droplet-type semi-shell)

10

Fig 1.5 Two dipolar plasmon mode in semi – shell, the transverse

mode; b the axis mode

10

Fig 1.6 (a) Schematic drawing of metal-insulator-metal (MIM)

structure for biosensing application Layer 1 is considered to

be a detected biolayer An insulator film (layer 3) is sandwiched with two gold media (b) Optical setup for reflectivity measurement Reflectivity in the absence of layer

1, R0, is taken as reference

11

Fig 1.7 Molecular interactions (biotin-avidin) detected using (a) MIM

(0, 0), (b) MIM (10, 30), (c) MIM (10, 40), and (d) MIM (10,

55) substrates (Syahir et al)

12

Fig 1.8 Spectra of the semi-shells obtained by simulation and

experiment Optical density spectrum was obtained by experiment (solid line); extinction spectra obtained by calculation with k ¬ ez, E ¦ ez (dashed line), k ¦ ez, E ¬ ez (long dashed line), k ¦ ez, E ¦ ez (dot-dashed line), and their average

(dotted line) (R.Fujimura et al)

13

Fig 1.9 3D Yee cell showing the E and H field components 15 Fig 1.10 MIM nanocups calculation model using FDTD method 20 Fig 2.1 Preparation of Silane Coupling solution 22 Fig 2.2 Glass substrate treatment 22 Fig 2.3 Image of glass substrate after treatment by Silane Coupling 23 Fig 2.4 Fabrication of monolayer PS nanoparticle on substrate 23 Fig 2.5 Polystyrene nanoparticle attaches with the substrate through

Silane Coupling

24

Fig 2.6 Procedure of making Multilayer Metal – Insulator – Metal

nanocups structure on glass substrate and in water

25

Fig 2.7 Sputtering multilayer Gold – Magnesium Fluoride – Gold on

monolayer PS nanoparticles on glass substrate

25

Fig 2.8 Dispersed particles into water by ultrasonic vibration, (a):

substrate after separated particles, (b): MIM nanocups in water

PS200MI (green straight curve), and PS200MIM (black straight curve) on glass substrate

37

Fig 3.13 Transmittance properties of PS500M (red straight curve),

PS500MI (green straight curve), and PS500MIM (black straight curve) on glass substrate

38

Fig 3.14 Schematic of near-field coupling between metallic

nanoparticles for the two different polarizations

39

Fig 3.15 Transmittance properties of sample PS500MIM: before

separate particle (red dashed curve), after separate particle (red straight curve); and MIM structure (green straight curve)

40

Fig 3.16 Transmittance properties of sample PS200MIM: before

separate particle (black dashed curve), after separate particle (black straight curve); and MIM structure (green straight curve)

40

Fig 3.17 Transmittance properties of sample PS200MIM after separate

particle (black straight curve), sample PS500MIM after separate particle (red straight curve) and MIM structure (green straight curve)

41

Fig 3.18 Transmittance property of sample PS500MIM particles in

water

42

Fig 3.19 a, Image of PS200MIM particles in water; b, Transmittance

property of sample PS200MIM particles in water

42

Fig 3.20 Simulation result of transmittance properties of PS200MIM

particle in air

43

List of abbreviations ATR: Attenuated Total Reflection CTAB Cetyl trimethyl ammonium bromide KPS: Potassium Persulfate

LSPR: Localized Surface Plasmon Resonance MIM: Metal – Insulator – Metal

PMMA: Poly(methyl methacrylate) PS: Polystyrene

PVD Physical Vapor Deposition SDS: Sodium Dodecyl Sulfate SEM: Scanning Electron Microscope SPP Surface Plasmon Polariton SPR: Surface Plasmon Resonance

ABSTRACT

The Metal – Insulator – Metal (MIM) nanocups structure on glass substrate and in water were fabricated by chemical method and sputtering, studied by transmittance properties The morphology of the sample was studied by Scanning Electron Microscope, and transmittance properties were studies by UV-Vis-NIR, and simulated by Finite Difference Time Domain (FDTD) method The result showed that MIM nanocups have surface plasmon resonance (SPR) and localize surface plasmon resonance (LSPR) phenomena, that depend on size of Polystyrene (PS) and thickness of metal and insulator layers

The purpose of this thesis is:

- Deposit monolayer PS nanoparticles on glass substrate

- Fabrication MIM nanocups structure with core of Polystyrene PS nanoparticles

- Study SPR and LSPR phenomenon of Metal – Insulator – Metal nanocups structure on substrate and in water

CHAPTER 1: INTRODUCTION

Research background

Surface plasmon resonance (SPR) phenomenon describes the oscillation of free electron cloud on the surface under electromagnetic wave of light For optical biosensors, some researchers have developed biosensor based on SPR and LSPR phenomena Surface Plasmon Resonance occurs when the frequency of the excitation light coincides with the frequency of the surface electron oscillation When SPR occurs, the spectral response at the wavelength and angle of occurrence of the SPR are sensitive to the change in refractive index at the surrounding surface To enhance the sensitivity and simplify the working condition of biosensor, Syahir group [16]

developed biosensor base on metal – insulator – metal (MIM) structure with the thickness of each size is 10 – 55nm When the incident light comes into the MIM structure, the SPR phenomenon will occur at the appropriate light The reflection and absorption of the MIM structure at the wavelength are also highly sensitive to slight changes in the refractive indices at the surface of the structure

Some researchers study plasmons resonance phenomenon in sphere and semi – shell nanoparticle, which has both SPR and LSPR resonance Fujimura group studied plasmonic properties of gold nanocup The extinction curve had two peaks at around 570nm and 750nm In this thesis, we will study SPR and LSPR phenomena

in MIM nanocups structure

1.1 Surface plasmon resonance (SPR)

1.1.1 Theory [11]

Surface Plasmon Resonance (SPR) is an optical phenomenon involving excitation of the free oscillating metal electron It occurs when incident light (p – polarized) propagate in thin film metal under total internal reflection

Surface Plasmon Polariton (SPP) is electromagnetic excitation propagating at the interface between a dielectric and conductor, evanescently confined in the perpendicular direction The electromagnetic surface waves arise via coupling of electromagnetic fields to the oscillation of the conductor’s electron plasma

The most simple geometry sustaining SPP is that of a single, flat interface between dielectric, non – absorbing half – space (z > 0) with positive real dielectric constant 𝜀2 and conducting half – space (z < 0), describe via dielectric function 𝜀1(𝜔) (Fig 1.1) The requirement of metallic character imply if Re[ε1] < 0

First, about transverse magnetic (TM) solution, we have:

Ez(z) = −A1 β

ωε0ε2e

iβxe−k2 z (1.1c) and for z < 0

Those function (1.1a), (1.1b), (1.1c), (1.2a), (1.2b), (1.3c) are the components

of the wave vector perpendicular to the interface in the two media Its reciprocal ẑ =1/|kz|, define the evanescent decay length of the fields perpendicular to the interface, which quantifies the confinement of wave Continuity Hy and εiEz at interface require that A1 = A2, and

k2

k1 = −

ε2

ε1 (1.3) Note that Re[ɛ1] < 0 if ɛ2 > 0, the surface wave only exists at the interface between materials with opposite signs of the real part of their dielectric permittivities, for example, between a conductor and insulator The expression of Hy further has to fulfill wave equation, yielding:

k12 = β2− k02ε1 (1.4a)

k22 = β2− k02ε2 (1.4b) Combining (1.4a), (1.4b) and (1.3), the dispersion relation of SPPs propagating at the interface between the two half - spaces are shown:

β = 𝑘0√ 𝜀1𝜀2

𝜀1+ 𝜀2 (1.5) This expression valid for both real and complex ɛ1,

For transverse electric (TE) surface mode, for z > 0, the expression for field component are:

𝐴1(𝑘1+ 𝑘2) = 0 (1.8) Since confinement to the surface require Re[k1] > 0, and Re[k2] > 0 which is only fulfil when A1 = 0, also A1 = A2 = 0, therefore no TE mode surface for TE polarization

Thus, SPPs only exist for TM polarization

The properties of SPPs will be examined by taking a closer look at their dispersion relation Radiation into metal occurs in the transparency regime ω > 𝜔𝑝

A frequency gap region with purely imaginary β prohibiting propagation exist between the regime of the bound and radiative modes

For small wave vector corresponding to low (mid – infrared or lower) frequency, the SPP propagation constant is close to k0, at the light line, and the waves extend many wavelengths into the dielectric space In this regime, SPPs therefore, acquire the natural of grazing – incidence field, and are also known as Sommerfeld – Zenneck wave

In the opposite regime of large wave vectors, the SPPs frequency approaches the characteristic surface plasmon frequency 𝜔𝑠𝑝

ωsp = ωp

√1 + ε2 (1.9)

as can be shown by inserting free – electron dielectric function into (1.5) In the limit

of negligible damping of the conduction electron oscillation (implying Im[𝜀1(𝜔)] =0), the wave vector β goes to infinity as frequency approaches surface plasmon frequency, and group velocity vg → 0 The mode thus acquires electrostatic character, and is known as surface plasmon It can indeed be obtained via straightforward solution of Laplace equation∇2φ = 0 for single interface geometry of Fig 1.1, where

φ is electric potential A solution that is wavelike in the x – direction and exponentially decaying in the z – direction is given by

φ(z) = A2eiβxe−k2 z (1.10a) for z > 0, and

φ(z) = A1eiβxek1 z (1.10b) for z < 0 The Laplace equation has solution when k1 = k2 = β: the exponentially decay lengths |ẑ| = 1/kz into insulator and into metal is equal Continuity of φ and ε𝜕𝜑

𝜕𝑧ensure continuity of the tangential field components and the normal field components

of the dielectric displacement and require that A1 = A2 and addition:

ε1(ω) + ε2 = 0 (1.11) For a metal described by a dielectric function, this condition fulfilled at 𝜔𝑠𝑝 Comparation (1.11) and (1.5), show that the surface plasmon is indeed the limiting form of a SPP as β → 0

SPP is propagating at the interface between conductor and insulator are essentially two – dimensional electromagnetic waves Confinement is achieved since

β > k in dielectric, leading to evanescent decay on both sides of interface The SPPs dispersion curve thus lies to the right of light line of dielectric (given by ω = ck), and excitation by three – dimensional light beam is not able unless special techniques for phase – matching is employed

1.1.2 SPR in metal – insulator – metal (MIM) structure

MIM structure is a structure have an insulator layer sandwich between two metals layer (Fig 1.2) When incident light goes into the structure, the light energy

trap inside the insulator layer, therefore, enhance surface plasmon resonance signal

Set ε2 = ε2(ω) as dielectric function of metal and ɛ1 as dielectric constant of insulator in equation:

Glass substrateIII

I

II

Metal Insulator

Figure 1.2 Metal – Insulator – Metal structure

tanh k1a = −k2ε1

k1ε2 (1.12a)

tanh k1a = −k1ε2

k2ε1 (1.12b) Where 2a is thickness of insulator layer Function (1.12a) describe mode of odd vector parity (Ex(z) is odd, Hy(z) and Ez(z) are even function Meanwhile, function (1.12b) describe modes of even vector parity (Ex(z) is even function, Hy(z) and Ez(z) are odd) [9] From an energy confinement point of view, the most interesting mode

is the fundamental odd mode of the system, which does not exhibit a cut – off for vanishing I layer thickness [12] For example, Fig 1.3 illustrate dispersion relation

of this mode for a Silver/Air/Silver heterostructure [11] In this time, the dielectric

function ε(ω) was taken as a complex fit with data of Silver obtain by Johnson and Christy [8] Therefore, β does not go to infinite when surface plasmon frequency is

Figure 1.3 Dispersion relation of the fundamental coupled SPP modes of a silver/air/silver multilayer geometry for an air core of size 100 nm (broken gray curve), 50 nm (broken black curve), and 25 nm (continuous black curve) Also shown is the dispersion of a SPP at a single silver/air interface (gray curve) and the air light line (gray line)

approached, but folds back and eventually cross the light line, as SPPs propagating at single interface

It is apparent that large propagating constants β can be achieved even for excitation lower that 𝜔𝑠𝑝, provide width of dielectric layer is chosen sufficient small

The ability to access such large wave vectors and thus small penetration lengths 𝑧̂

into the metallic layers by adjusting the geometry indicates that localization effect that for a single interface can only be sustain at excitation near 𝜔𝑠𝑝, can for MIM structures also be attained for excitation out in the infrared

Surface Plasmon Polaritons on a flat metal/dielectric interface cannot be excited directly by light beam since β > k, where k is wave vector of light on the dielectric side of interface Therefore, to occur SPR on flat metal/dielectric interface,

a prism is needed However, in MIM structure Prism is not needed MIM structure also have another benefit, for example, easy to make, and enhance the signal of SPR which is explained as when incident light goes into MIM structure, the electromagnetic wave is trap inside insulator layer (Fig 1.5), thus increase the energy that loses in structure

1.2 Localized Surface Plasmon Resonance (LSPR)

1.2.1 Mie theory [11]

The theory of scattering and absorption of radiation by a small sphere predicts

a resonant field enhancement due to a resonance of the polarizability α = 4πa3 ε−ε m

ε+2ε m

if the Frölich condition Re[ε(ω)] = −2εm is satisfied Under these circumstances, the nanoparticle acts as an electric dipole, resonantly absorbing and scattering electromagnetic fields This theory of the dipole particle plasmon resonance is strictly valid only for vanishingly small particles However, in reality, the calculations outlined above provide a reasonably good approximation for spherical or ellipsoidal particles with dimensions below 100 nm illuminated with visible or near-infrared radiation

However, for a particle with large dimension, where quasi – approximation is not justified due to significant phase – changes of the driving field over particle volume The simplest theoretical approach available for modeling the optical properties of nanoparticles is the Mie theory estimation of the extinction of a metallic sphere in the long wavelength, electrostatic dipole limit [15] In the following equation [9]:

E(λ) =24πNAa

3εm3 2⁄λln (10) [

ε1(εr + 2εm)2+ ε2] (1.13) Where E(λ) is the extinction, which is equal to the sum of absorption and Rayleigh scattering, NA is the areal density of nanoparticles, a is radius of metallic nanosphere,

ɛm is dielectric constant of the medium surrounding the metallic nanosphere (assumed

to be positive, real number and wavelength independent), λ is the wavelength of absorbing radiation, ɛi is the imaginary portion of metallic nanosphere’s dielectric function, and ɛr is the real portion of the metallic nanosphere’s dielectric function

The LSPR condition is met when the resonance term in the denominator ((ɛr+2ɛm)2) approaches 0

LSPR excitation results in wavelength selective absorption with extremely large molar extinction coefficient, resonance Rayleigh scattering with an efficiency equivalent and the enhanced local electromagnetic fields near the surface of the nanoparticle

1.2.2 LSPR in nanocups structure

Nanocups structure have hopefully application in verify field, from drug delivery to free – label sensor which is applied LSPR phenomenon [5] R Fujimura

et al studied plasmons properties of Gold nanocups through both DDASCAT [3] The

model was shown in Fig 1.4 Where Rcore is core radius, Rshell is shell radius, Rgrain is grain radius, tshell is shell thickness in concentric – type semi – shell, t’shell is shell thickness at the top of droplet – type semi – shell, hevp is evaporation thickness, dcenter

is distance between shell and core centers, and 𝜃𝑐𝑎𝑝 is capping angle Because

fabrication of semi – shell is randomly oriented in water, optical response for three different conditions was calculated (i) (k‖𝑒𝑧, 𝐸⏊𝑒𝑧), (ii) (k⏊ez, E || ex), and (iii) (k⏊ez, E⏊ez); where k is wave vector, E is electric field vector of incident wave, and

ez is rotation axis of target semi – shell Optical density and extinction were showed

semi-Figure 1.5 two dipolar plasmon mode in semi – shell, the transverse mode; b the axis mode [13]

mode Fig 1.8 In transverse mode, the electric field component of the light excites the loop current in metallic shell, leading to strong magnetic component of the plasmon resonance The hybridization and the large field confinement near the edge

of the rim lead to strong red shifts of the nanocups transverse plasmon resonance

1.3 Application of SPR and LSPR phenomena

1.3.1 Application of SPR phenomenon

SPR can be applied in many fields, such as electronic field [20][21];

biosensors [15][16][17], gas detection [12] by using sensitive characteristics of SPR

Syahir et al [16] fabricated Au – PMMA – Au by using PVD method for Au layers

and spin – coating method for PMMA layer for free – label biosensor application (Fig

1.6) By changing the thickness of insulator sandwich between two metal layers, one can control surface plasmons excited in MIM waveguides The MIM surface plasmons can be excited with propagating light without using attenuated total reflection (ATR) geometry [16] Unlike SPR phenomenon in monolayer have to have prism to occur [11], SPR phenomenon can be achieved at a normal incident without prism The resonance wave of MIM structure is independent of incident angle [14]

MIM structure also helps enhance, and make SPR signal clear, lead to easier to detect

Figure 1.6 (a) Schematic drawing of metal-insulator-metal (MIM) structure for biosensing application Layer 1 is considered to be a detected biolayer An insulator film (layer 3) is sandwiched with two gold media (b) Optical setup for reflectivity measurement Reflectivity in the absence of layer 1, R0, is taken

as reference [15]

Syahir et al using MIM structure to detect Biotin and Avidin [16] (Fig 1.7) For the

signal of sensing Biotin and Avidin using just only monolayer Gold, the signal is very small and unclear Remain thickness of Au layer on top (10nm) and changing the thickness of PMMA layer: 30nm; 40nm; 55nm As reflectance properties showed in Fig 1.2, the sensing signal increase and clearer with increasing thickness of PMMA

1.3.2 Application of LSPR phenomenon

There are also many application fields for LSPR phenomenon, such as biosensor [15] Recently, many researchers are interested in nanocups structure and that application, for example free – label sensing [5], or enhancing the charge transfer

of the counter electrode in dye – sensitized solar cells [7], and due to semi – shell

structure, it can be used in drug delivery and release R Fujimura et al studied Au

Figure 1.7 Molecular interactions (biotin-avidin) detected using (a) MIM (0, 0), (b) MIM (10, 30), (c) MIM (10, 40), and (d) MIM (10,

55) substrates (Syahir et al).

Au on them In shell thickness is 32 nm, optical density of the sample has two peaks

at around 570nm and 750nm (Fig 1.8) with optical density is over 0.8 and above 1.0 respectively

J Chen group studied optical fiber biosensor base on Silver Nanoparticles (AgNPs) [1] AgNPs was fabricated by using chemical method, as well as the sensor probe The results showed that when the refractive index increase due to increase of concentration of Sucrose, the reflective properties decrease, and LSPR wavelength increased with the increase of refactive index

C Li studied LSPR sensing molecular biothiols based on noncoupled gold nanorod [10] Gold nanorod was fabricated in aqueous solution, by a typical – seed mediated The seed was prepared by using NaBH4 reduced HAuCl4 with CTAB as

Figure 1.8 (Color online) Spectra of the semi-shells obtained by simulation and experiment Optical density spectrum was obtained

by experiment (solid line); extinction spectra obtained by calculation with k || ez, E ⏊ ez (dashed line), k ⏊ ez, E || ez (long dashed line), k || ez, E || ez (dot-dashed line), and their average (dotted line) [3]

surfactant, then the rod was grown by add HAuCl4 and AgNO3, using ascorbic acid

as reductant The mPEG – Au nanorod and the FITC – PEG – Au nanorod were prepared by chemical method The result showed that there is a red shift of peak of the LSPR band with addition of GSH, compared with peak of LSPR of mPEG – Au nanorod

Because in MIM structure, SPR phenomenon can be occurred without prism, and sphere structure have LSPR phenomenon, therefore, MIM nanocups structure promise a large field of application, for example biosensor [4] Therefore, the purpose

of this research is fabricated and study optical properties of MIM nanocups structure, hoping that this structure will have both SPR phenomenon in MIM structure and LSPR in sphere structure

1.4 Finite Difference Time Domain (FDTD) approach

1.4.1 Theory

The Finite Difference Time Domain (FDTD) approach is based on a direct numerical of the time – dependent Maxwell’s curl equation In 3D simulation, space and time step is defined as below:

∆x is the size in real unit of a space step along the X direction

∆y is the size in real unit of a space step along the Y direction

∆z is the size in real unit of a space step along the Z direction

∆t is the size in real unit of a time step Each field components represented for a 3D array: Ex(i, j, k); Ey(i, j, k); Ez(i, j, k); Hx(i, j, k); Hy(i, j, k); Hz(i, j,k) The field components position

is showed in Fig 1.9 This arrangement is called a Yee Cell E and H field components are interleaved at intervals of ½ cell in both space and time This way it is possible to solve sequentially all E fields and H fields in the simulation domain using a leapfrog algorithm

The time domain vectorial Maxwell’s equation are given in differential form

For H components:

Figure 1.9 A 3D Yee cell showing the E and H field components

n +12

i, j −1

2, k −

12

+∆t

μ0[

Hy|

n +12

i −1

2, j, k

= Hy|

n −12

i −1

2, j, k −

12

+∆t

μ0[

𝑛 +12

𝑛

𝑖 −1

2, 𝑗, 𝑘

+ 2∆𝑡(2𝜖 + 𝜎∆𝑡)

[

𝐻𝑧| 𝑛 +

12

𝑖 −12, 𝑗 +12, 𝑘

− 𝐻𝑧| 𝑛 +

12

𝑖 −12, 𝑗, 𝑘 +12

− 𝐻𝑦| 𝑛 +

12

𝑖 −12, 𝑗, 𝑘 −12

∆𝑧

] (1.15𝑎)

i, j −1

2, k

+ 2∆t(2ϵ + σ∆t)

[

Hx| n +

12

i +12, j −12, k

− Hz| n +

12

i −12, j −12, k

∆x

] (1.15b)

𝑛

𝑖, 𝑗, 𝑘 −1

2

+ 2∆𝑡(2𝜖 + 𝜎∆𝑡)

[

𝐻𝑦| 𝑛 +

12

𝑖 +12, 𝑗, 𝑘 −12

− 𝐻𝑦| 𝑛 +

12

𝑖, 𝑗 +12, 𝑘 −12

− 𝐻𝑥| 𝑛 +

12

𝑖, 𝑗 −12, 𝑘 −12

∆𝑦

] (1.15𝑐)

The fundamental constraint of the FDTD method is the step size for both time and space Space and time steps relate to the accuracy, numerical dispersion, and the stability of the FDTD method To keep result the results as accurate as possible, with low numerical dispersion, the mesh size usually quoted is 10 cells per wavelength,