Optimization of Graphene-based PlasmonicSensor using Genetic Algorithm

Nosheen Younas

MS ThesisDepartment of Physics Syed Babar Ali School of Science and Engineering, LUMS

Saturday, January, 23, 2016

LAHORE UNIVERSITY OF MANAGEMENT SCIENCES

Department of Physics

CERTIFICATE

I hereby recommend that the thesis prepared under my supervision by NosheenYounas on the title Optimization of Graphene-based Plasmonic Sensor usingGenetic Algorithm be accepted in partial fulfillment of the requirements for theMS degree.

Dr. Muhammad Faryad

Signature:Date:

Recommendation of Thesis Defense Committee :Dr. Mumtaz Ali Sheikh

Signature:Date:

1

Abstract

Surface plasmon polariton (SPP) waves are the electromagnetic waves that prop-agate along the metal-dielectric interface. In these waves, the electric field decaysexponentially in the metal as well as dielectric. The polarization of these waves istransverse magnetic (TM). To excite them, p-polarized light is used in such a waythat the propagation constant of the incident light matches with that of the SPPwave along the interface. This process is very sensitive to the dielectric propertiesof the material, which can be utilized to make sensors for different applications suchas biomaterial deposition monitoring. As a first step, we need the reflection andtransmission of incident light on metal-dielectric interface because when incidentlight excites SPP waves, the reflectance hits a minimum. The change in the refrac-tive index of dielectric medium results in the change of resonance angle. This formsthe basis of the plasmonic sensor. The sensitivity of the sensor is optimized usingGenetic Algorithm. The results of this analysis and optimization produced a threelayered design for surface plasmon resonance based sensors that have sensitivitiesof 157.44 deg/RIU and 152.78 deg/RIU which exceed the published sensitivities of134.6 deg/RIU, where RIU stands for refractive index units.

Contents

1 Introduction 3

1.1 Bio-sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Exciting Surface Plasmons . . . . . . . . . . . . . . . . . . . 4

1.3 Reflection, SPR and Sensitivity . . . . . . . . . . . . . . . . . . . . 5

1.3.1 N Layers Model . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Simulations of a Graphene Based Plasmonic Sensor 10

2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Optimization Using Genetic Algorithm 15

3.1 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Electric Field within Layers . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Genetic Algorithm Optimization . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Algorithm Scheme . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 Sensor Response . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.2 Optimization Convergence . . . . . . . . . . . . . . . . . . . 23

4 Conclusion 29

1

4.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2

Chapter 1

Introduction

Surface plasmon polariton (SPP) waves are coherent electromagnetic waves thatare guided by the interface of a metal and a dielectric material. These waves canbe excited by light in a phenomenon called Surface Plasmon Resonance (SPR).It is highly sensitive to the dielectric properties of the material; therefore, it canbe utilized for making a sensor for materials that cause these changes. In thefollowing section, an overview of different optical sensors for biological materialsis discussed with a special focus on SPR based sensors. Afterwards the theoreticalbackground for surface plasmons and SPR is described. Finally an introductionto genetic algorithm is presented which will be used to optimize the sensitivity ofgraphene based SPR sensors.

1.1 Bio-sensors

A large variety of optical methods can be utilized to make bio and chemical sen-sors, for example, ellipsometry, spectroscopy, interferometery and SPR. In thesetechniques, refractive index, absorbance and fluorescence etc. are monitored tosense the desired material property [1].

The SPR based sensors can be utilized for detecting change in refractive indexcaused by a biomaterial. To achieve high sensitivity the sensor should have strongSPR as well as high absorption for bio-materials. Nobel metals such as gold andsliver have been utilized in conventional SPR sensors because they suffer low oxi-dation and at the same time have high sensitivity. The sensitivity of these sensorscan be further increased by enhancing the field intensity at the gold surface. Thisis usually achieved by coating gold with a dielectric with high refractive index, forexample silicon.

However these sensors on their own prove to be poor absorbents for bio-materials.To overcome this challenge, graphene proves to be a wonderful material. Due to its2D honey comb lattice structure it has high surface area making it an efficient ad-sorbent [2]. Furthermore, carbon based rings, present in most of the biomaterials,interact strongly with the graphene lattice. Therefore to make a highly sensitive

3

bio sensor, different layers of graphene, gold and silicon can be stacked together.

1.2 Theoretical Background

Consider a metal in contact with a dielectric material such that the metal dielectricinterface is at z = 0 plane with metal occupying the lower and dielectric occupyingthe upper half plane (Fig. 1.1). The electronic charges at the surface of metal canhave coherent charge density fluctuations known as surface plasma waves. Theyare accompanied by transverse as well as longitudinal electromagnetic field. Thisfield vanishes at a distance from the surface but at the surface it is very strongmaking it highly sensitive to surface properties [3].

The wave vector of the surface plasmon is given by [3]

ksp =ω

c

(εmεdεm + εd

)1/2

. (1.1)

To excite surface plasmons a real ksp is needed which requires Re(εm) < 0, acondition that is easily met in metals [3].

The schematic of this phenomenon is given in Fig. 1.1.

SPP waveDielectric

Metal

Figure 1.1: A surface plasmon is being excited at a metal dielectric interface atz = 0 plane. εm represents the dielectric constants for metal and εd is dielectricconstant for the dielectric. kz =

√εd(k20 − k2x) gives the wavenumber in dielectric

and kz =√εm(k20 − k2x) gives the wavenumber in the metal.

1.2.1 Exciting Surface Plasmons

Surface plasmons can be exicte by electrons or photos [3]. In case of excitation bylight, the propagation vector of light needs to match the propagation vector of thesurface plasmon Eq. (1.1). Since kx >

ωc

therefore the wave vector of light needsto be increased to match the kx of plasmon. This is achieved by two ways

4

(a) Grating Coupler : At angle θ0, the components of an incident light with k = ω/ccan have following wave vector on the surface of a grating

ω

csin(θ0 + vg) (1.2)

where v = 0,±1,±2, ... and g = 2π/a with a as the grating constant [3]. This canthen fulfill the momentum matching condition

kx =ω

csin(θ0 + vg) =

ω

c

√ε

ε+ 1= Re(ksp)

(b)Attenuated Total Reflectance (ATR) Coupler : The light incident at a metalsurface through a dielectric (εd > 1) has momentum projection on surface givenby [3]

kx =√εdω

csin θ0. (1.3)

When this kx of light matches Re (ksp) in Eq. (1.1) for SPP, SPR takes place inwhich energy is transferred to the the surface plasmon.

1.3 Reflection, SPR and Sensitivity

In an ATR coupler the SPR condition is achieved either by changing the incidenceangel (θ0) or the frequency (ω) of incident light. These methods are known asangle interrogation and frequency interrogation method respectively. The generalsetup for both of these methods is shown in Fig. 1.2 . A metal layer is coatedon one side of a high-refractive-index prism. The light is incident on the metalfrom inside the prism at angles greater than the critical angle. More details onthis setup are given in Sec. 2.1.

Incident

light

Re�ected

Light

Prismz

x

Metal layer

Biomaterial

Figure 1.2: Prism-coupled setup of SPR through angle interrogation method.

5

Fresnel’s equations indicate that the incident light undergoes reflection, transmis-sion and adsorption. The sum of the absorbance (A), transmittance (T ) and thereflectance (R) should be equal to 1 by energy conservation. Since light is incidentat angles greater than the critical angle, therefore T = 0. Hence A + R = 1.At resonance angle (θres), where SPP waves are excited, reflectance R is nearlyzero which implies A = 1. Practically, the reflected intensity reaches a minimum(Rmin). The change in the refractive index of the metal-dielectric medium causesa change in resonance angle. This change in refractive index can be induced bythe deposition of biomaterial hence it acts like an indicator for the presence ofbio-materials. The Sensitivity (S) of SPR sensor is defined as the derivative ofthe monitored parameter with respect to the one under study. Therefore in caseof angle interrogation based SPR sensor if the change in resonance angle is 4θrescorresponding to the change in refractive index4n of the reflecting layer then [4,5]

S =4θres4n

(deg/RIU), (1.4)

where RIU stands for refractive index unit.

1.3.1 N Layers Model

Consider a stack of N layers of different materials stacked along z axis [6]. Eachlayer is characterized by a refractive index n, dielectric ε, permeability µ andthickness d. The refractive index of initial medium is n0 and the refractive indexof the final medium is given by nN+1.

If a p-polarized light is incident on this stack at an angle θ0 on the first layer, theincidence angle for subsequent layers is simply given by Snells law as follows:

n0 sin θ0 = n1 sin θ1 = ... = nN sin θN = nN+1 sin θN+1 (1.5)

where θ0 is the initial angle of incidence.

There are two waves in every medium except for the last one. One traveling inthe forward direction and the other one in the backward direction. In subsequentfigures and equations, the magnitudes of these waves are represented by right andleftward pointing arrows, respectively. In a jth layer, the x and z componentsfor forward traveling wave are given by Ej→ cos θjx and Ej→ sin θj z respectively(for components of backward traveling wave Ej→ will be replaced by Ej←). Thebackward traveling wave is a result of the reflection from the interface with thenext medium because next medium is of different refractive index. Since there isno medium after the final one, therefore there is only forward moving wave in thefinal medium. The final wave given by Ep

N+1→ gives the overall transmitted field.For the first interface the boundary conditions for the parallel components of theE and B fields lead respectively to [4]

cos θ0(Ep0→) = cos θ1(E

p1→ + Ep

1←) (1.6)

6

θ1

θn+1

θ2θn

θ0

d2d1 dn

n1

n1 n2 nn

nn+1

n0x

z

Figure 1.3: Schematic for p-polarized light incident on a stack of layers with dif-ferent refractive indices and thickness.

andn0(E

p0→ − E

p0←) = n1(E

p1→ − E

p1←) (1.7)

The arrows on the lower right corner of the E field indicate direction (right or left)of propagation of the field according to the situation presented in Fig. 1.3 andthe p on the top of E shows p-polarization. At the final interface, the boundaryconditions are

cos θN(EpN→e

ikNdN cos θN + Epj←e

−ikjdj cos θj) = cos θN+1(EpN+1→) (1.8)

nN+1(EpN→e

ikNdN cos θN ) = nN+1(Epn+1→) (1.9)

because there is no backward-traveling field in the final medium.

For any jth layer in between the initial and final layer, the mathematics can besummarized as follows[

cos θjeiβj cos θje

−iβj

njeiβj −nje−iβj

][E

(p)j→

E(p)j←

]=

[cos θj+1 cos θj+1

nj+1 −nj+1

][E

(p)j+1→

E(p)j+1←

](1.10)

where

β = 0, j = 0and β = kjdj cos θj, 1 ≤ j ≤ N

(1.11)

andE

(p)N+1← ≡ 0.

If the matrices related to the jth layer are grouped together according to

7

M(p)j =

[cos θj cos θjnj −nj

] [cos θje

iβj cos θje−iβj

njeiβj −nje−iβj

]−1(1.12)

=

[cos βj −i sin βj cos θj/nj

−inj sin βj/ cos θj cos βj

](1.13)

then we can get

[1

E(p)0←/E

(p)0→

]= A(p)

[E

(p)N+1→/E

(p)0→

1

](1.14)

where

A(p) =

[a(p)11 a

(p)12

a(p)21 a

(p)22

]=

1

2n0 cos θ0

[n0 cos θ0n0 − cos θ0

] ( N∏j=1

M(p)j

) [ cos θN+1 0nN+1 0

](1.15)

withN∏j=1

M(p)j = M

(p)1 .M

(p)2 ...M

(p)N . (1.16)

Equation (1.14) represents two equations, which must be solved simultaneously to

find the ratios E(p)← /E

(p)0→ and E

(N+1)0→ /E

(p)0→. Finally,

ttotp =E

(p)N+1→

E(p)0→

=1

a(p)11

, (1.17)

rtotp =E

(p)0←

E(p)0→

=a(p)21

a(p)11

. (1.18)

The reflection coefficient of N layer model for p-polarized light is

R =| rp |2 (1.19)

1.4 Genetic Algorithm

Genetic algorithm is a unique way to optimize a function since it gives a set (pop-ulation) of optimized solutions. It is particularly useful for optimizing functionswhich have highly non-trivial dependence on the variables. The general scheme ofgenetic algorithm consists of six main steps which are as follows:

• Initial Population; a set of randomly constructed solution is created.

• Fitness ; the function to be optimized is computed using solutions from thepopulation, this gives the fitness of the solution which tells how optimal asolution is for the given function.

8

• Selection; solutions with less than average fitness are discarded.

• Reproduction; new solutions are created form the surviving solutions by ran-dom mixing of function parameters.

• Mutation; new solutions undergo a random change in parameters.

These new solutions are then inserted in the initial population and this cyclecontinues until a solution with optimal fitness is found. This solution optimizes thefunction. In this study the quality to be optimized is the sensitivity of the sensorand the parameters are the layer thicknesses and compositions. Further detailson the theory, implementation and results of genetic algorithm based optimizationare presented in chapter 3.

9

Chapter 2

Simulations of a Graphene BasedPlasmonic Sensor

The simulations are performed in matlab. The code written for this purpose isshown in the appendix. To test the correctness and functionality of the code forreflection, transmission and absorption through multiple layers, the results of [4]are reproduced. Before moving to calculations and simulations it is importantto understand the basic experimental setup for SPR. In subsequent section theexperimental setup utilized in SPR based sensors in [4] is discussed. Afterwardsresults of [4] are reproduced using the code written in Matlab.

2.1 Setup

For exciting SPR either the angle of incidence or the wavelength of light can bechanged to fulfill the electrodynamic conditions. Experimentally it is relativelychallenging to change the wavelength of incident light. Therefore traditionally theSPR experiments are performed by angular interrogation method. In this methodthe angle of incidence is varied and the experimental setup consists of a glass prismwith high refractive index. One side of this prism is coated with multiple layers ofmetal and other elements to enhance the SPR.

The dielectric medium (bio-material) to be sensed is placed on the stack of layersdeposited on the coated side of the prism. The evanescent waves of the incidentlight penetrate the metal film and excite surface plasmon waves at metal-dielectricinterface. This results in the transfer of energy to the SPW. The reflected intensityreaches its minimum at resonance angle.

For example the setup for experiments performed in [4] consists of a glass prismwhich is coated on one side with multiple layers. First a layer of gold is depositedonto one side of the prism, afterwards layers of silicon and graphene are laid uponthe gold layer such that the top layer is graphene. On top of this stack of layers is alayer of water with some dissolved bio-material. This layers has a refractive indexwhich is a function of the properties and concentration of bio-material. Conse-

10

quently 4θres can be utilized to find 4n and hence the properties of bio-material.

The setup is shown in Fig. 2.1.

Gold

Silicon

Graphene

Biomaterial

Water

Graphene

Incident

Light

Re ected

Light

Prismz

x

Figure 2.1: Prism-coupled setup for SPR.

To enhance the sensitivity; the number, thickness and arrangement of these layerscan be changed.

For exciting surface plasmons, a monochromatic p polarized light is incident onthe multi-layer structure through one face of the prism at greater than the criticalangle. The reflected light from the multi-layer emerges from the other face of theprism as shown in Fig 2.1. The angle of incidence is changed to minimize theintensity of reflected light. This hints the SPR. This specific angle changes as thedielectric properties change due to deposition of biomaterial.

For the calculations and simulations the dispersive behaviors of all the layers fora given wavelength of the light source have to be accounted for.

To reporduce the results of Ref. 4, simulations are carried out for 633 nm light.For this wavelength the refractive index (n1) of the SF10 glass prism used in [4]is 1.7230. The refractive index of gold (n2) is 0.1726 + i3.4218 and the refractiveindex (n3) of silicon is determined from the following relation

n3 = A+ A1 exp(−λ/t1) + A2 exp(−λ/t2) (2.1)

where A = 3.44904, A1 = 2271.88813, A2 = 3.39538 µm, t1 = 0.058304 µm andt2 = 0.30384 µm; λ is the wavelength in µm. The refractive index of graphene isgiven as

n4 = 3 + iC

3λ (2.2)

where λ is the wavelength in µm and the constant C = 5.446 µ. The thickness ofa layer of graphene is dg = 0.34 nm and therefore for L number of layers the totalthickness is d4 = L × 0.34 nm. For sensing medium, the refractive index is 1.33

11

because biomaterials are typically dissolved in water. The thickness of the sensingmedium is d5 = 100 nm. Due to the adsorption of biomolecules on the graphenelayer the refractive index of the sensing medium changes by 4nbm = 0.005.

2.2 Correctness

In order to test the correctness of the matlab code, the work done in Ref. 4 isregenerated. Figure 2.2 represents the reflection as a function of angle of incidencefor a stack of layers containing 50 nm gold, 5 nm silicon and 5 layers of 0.34 nmgraphene. The wavelength of light utilized for this calculation is 633 nm. It isevident that at a certain angle the reflection completely vanishes. This angle rep-resents the resonance angle for the SPR. Figure 2.3 shows reproduced simulationsof reflection of a 633 nm light incident at the resonance angle at a prism-multilayer system with various thickness of gold layer and different number of graphenelayers. It is evident that both the thickness of gold and the number of graphenelayers gives a change in the minimum reflection at the resonance angle. A minimumreflection at resonance angle indicates a strong SPR.

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

angle of incidence θ (deg)

Re�

ectance

Figure 2.2: Reflection of light from stack containing 50 nm of gold, 5nm of silicon and five 0.34 nm thick layers of graphene for 633 nm wavelength.

The dependence of resonance angle on the deposition of bio-material is the mostcrucial factor in this experiment. To test matlab code and calculations for thisdependence, the results in [4] are reproduced where the change in resonance angle ismonitored as the refractive index of bio-material layer is changed by 0.005 becausethe deposition of bio-material changes the refractive index. The results are shownin Fig. 2.4. The results indicate that the change in the resonance angle is largerfor more number of graphene layers.

12

Re

�e

cta

nc

e

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

5

10

15

20

25

30

35

40

number of graphene layers

50nm Gold & 0nm Si40nm Gold & 0nm Si35nm Gold & 0nm Si30nm Gold & 0nm Si

Figure 2.3: Reproduction of Fig. 3b in Ref. 4 i.e. behavior of minimum reflectivityin Surface plasmon resonance curve. The multi-layer stack only has gold andgraphene. The wavelength of incident light is 633 nm.

With the addition of silicon layers, the change in resonance angle and hence sensi-tivity is enhanced as shown in Fig. 2.5. Here a stack of gold, silicon and graphenelayers is utilized with only two layers of graphene. The sensitivity is plotted as afunction of the change in refractive index of biomaterial layer. It is evident thatthis sensor is very sensitive to the biomolecule layer which is in complete agreementwith Ref. 4.

13

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.26

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

number of graphene layers

∆ θ

res

(d

eg

)

d

d

d

d

50 nm Gol

40 nm Gol

35 nm Gol

30 nm Gol

Figure 2.4: Reproduction of Fig. 4b in Ref. 4 i.e. behavior of change in resonanceangle as a function of different number of layers of graphene for various thicknessesof gold. The stack only contains gold and graphene layers. The wavelength ofincident light is 633 nm.

0 0.005 0.01 0.0150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

∆ nbio

∆ θ

res

(d

eg

)

Figure 2.5: Sensitivity as a function of change in refractive index ofthe biomolecules layer. In this stack two layers of graphene each withthickness 0.34 nm are used. For gold and silicon the thickness is 40 nmand 7 nm respectively. For this experiment, 633 nm light is utilized.

14

Chapter 3

Optimization Using GeneticAlgorithm

One of the prime targets for this research is to achieve SPR sensors with very highsensitivities. This ultimately boils down to getting material, layer thickness andarrangement selection which provides maximum change in the resonance angle asa function of the concentration of the biomaterial. Since the concentration affectsthe refractive index therefore, in this case, the goal is to achieve a stack for whichthe resonance angle is highly sensitive to the refractive index of the biomateriallayer.

3.1 Parametric Study

A trial stack of layers is considered which contains a 40 nm thick gold layer withthree 5 nm thick layers of silicon alternating with three 0.34 nm thick layers ofgraphene. The arrangement of these layers along z-axis (as shown in Fig. 2.1) isas follows: Au, Si, C, Si, C, Si, C and Biomaterial. The biomaterial layer is 100nm thick. Using 633 nm light, the change in resonance angle as a function of therefractive index of the bio-material layers is shown in Fig. 3.1. Although this isa high sensitivity but a better sensitivity has already been reported in [4] (it hasbeen reproduced in Fig. 2.5).

As a second trial, each layer of graphene in the previous stack is replaced by adouble layer. It turns out that this results in decrease in sensitivity.

For another trial, the gold layer is removed from the stack and now only threelayers of silicon alternate with three double layers of graphene in following order:Si, C, Si, C, Si, C and Biomaterial. The sensitivity for this stack is given in Fig. 3.2which shows that removal of gold layer had very adverse effect on the sensitivity.

From these examples it is clear that in order to come up with a better design, theelectric field inside the layers needs to be calculated so that this information canbe utilized to get maximum sensitivity.

15

0 0.005 0.01 0.0150

0.2

0.4

0.6

0.8

1

1.2

1.4

∆ nbio

∆ θ

res

(de

g)

Figure 3.1: The change in resonance angle as a function of the change in refractiveindex of the 100 nm thick biomolecules layer for 40 nm gold, three 5 nm thicklayers of silicon alternating with three 0.34 nm thick layers of graphene (Au, Si,C, Si, C, Si, C, Bio). Wavelength of incident light is 633 nm.

3.2 Electric Field within Layers

So far only the z component of the total electric field has been calculated for theentire stack of layers. In order to engineering a highly sensitive SPR sensor, abetter understanding of how electric field behaves inside these layers is required.This can be helpful in deciding the correct material, number and thickness for theselayers. In order to achieve this, each layer is subdivided into numerous sub-layersof the same material. The electric field is calculated at the interface of each one ofthese sub-layers. Consequently this provides a point by point information on theelectric field inside the layers.

To find the electric field’s z component, following form of equation 1.15 has beenutilized

[E

(p)j+1→

E(p)j+1←

]=

[cos θj+1 cos θj+1

nj+1 −nj+1

]−1( j∏i

M(p)i

)−1 [cos θ0 cos θ0n0 −n0

][E

(p)0→

E(p)0←

].

(3.1)

16

0 0.005 0.01 0.0150

0.02

0.04

0.06

0.08

0.1

0.12

0.14

∆ nbio

∆ θ

res

(d

eg

)

Figure 3.2: Sensitivity as a function of refractive index of the 100 nm thickbiomolecules layer for three 5 nm thick layers of silicon alternating with threedouble (2×0.34 nm thick) layers of graphene (Si, C, Si, C, Si, C, Bio). Wavelengthof incident light is 633 nm.

3.2.1 Results

The electric field inside the stack proposed in [4] and regenerated in figure 2.5 of[4] is shown in figure 3.3. This stack consists of 40 nm gold, 7 nm thick silicon andtwo 0.34 nm thick layers of graphene. The incident light is 633 nm.

In trial stacks it was seen that the stack containing 40 nm gold, three 5 nm thicklayers of silicon which alternate with three 0.34 nm thick layers of grapene showedsome promise. Figure 3.4 shows the electric field inside this stack.

The z component of electric field inside the layers indicates a very low field in goldlayers in all the trials presented so far. This indicates that the energy in the electricfield is going into SPR in the gold layer. However the filed reaches maximum insilicon and graphene layers and then drops off slowly. It could be the slop of theelectric field drop-off that determines the sensitivity but the exact factors are stillnot obvious. Therefore the optimization of sensitivity of SPR sensors is a non-trivial function. To perform such optimization, genetic algorithms prove to be agood candidate. Further details are elaborated in next section.

17

0 50 100 1500

1

2

3

4

5

6

7

8

9

depth inside the stack (nm)

Ele

ctric

fiel

d am

plitu

de (

a.u.

)

forward going fieldbackward going fieldtotal field

Figure 3.3: The electric field inside a stack containing 40 nm thick gold, 7 nmsilicon and two 0.34 nm thick graphene layers in following order: Au, Si, C andBiomaterial. The biomaterial layer is 100 nm thick and incident light is 633 nm.

18

0 20 40 60 80 100 120 140 1600

5

10

15

20

25

30

depth inside the stack (nm)

Ele

ctri

c !

eld

am

pli

tud

e (

a.u

.)

forward going !eld

Figure 3.4: The electric field inside a trial stack containing 40 nm thick gold withthree 5 nm silicon alternating with three 0.34 nm thick graphene layers in followingorder: Au, Si, C, Si, C, Si, C and Biomaterial. The biomaterial layer is 100 nmthick and incident light is 633 nm.

19

3.3 Genetic Algorithm Optimization

The sensitivity of the SPR sensor is a complex function of thicknesses of all thelayers. In order to optimize the sensitivity, all layer thicknesses have to be variedsimultaneously because optimal thickness of a single layer is dependent on thethicknesses of all other layers. Genetic algorithm provides a very effective methodfor such multi-variable optimization where explicit dependence of sensitivity onlayer thicknesses is unavailable [7]. This approach is derived from natural selectionand genetics [8,9].

3.3.1 Algorithm Scheme

The basic steps of the algorithm are given in the form of a flow chart in Fig. 3.5 . In

Initial population

Fitness

Selection

Reproduction

Mutation

A set of solutions is created by

randomly selecting thicknesses

for layers

Sensitivity of SPR sensor is

calculated for each solution

Solutions are ranked according

to sensitivity and those with

less sensitivity are discarded

New solutions are produced by

using layer thicknesses of old

solutions in a random mixture

Each new solution undergoes

a random change in layer

thicknesses

Figure 3.5: Flow chart containing the basic steps of genetic algorithm based opti-mization.

genetic algorithm based optimization a population of possible solutions is created.For our problem, each solution is a set of thicknesses for layers of gold, silicon andgraphene for example for a three layered SPR sensor a possible solution wouldlook like (a, b, c), where a, b and c give thicknesses of the three layers. Initially thepopulation consists of a set of randomly created solutions [9].

Each individual of the population is then given a fitness. In general fitness is thequality that needs to be maximized. For this problem the quality that needs to be

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maximized is the sensitivity of the overall sensor. Therefore fitness is synonymouswith sensitivity in this study. Sensitivity is calculated for all the solutions in thepopulation and then they are ranked in a descending order according to theirsensitivity. If the fittest solutions have sensitivity that is greater then a chosenthreshold the algorithm stops otherwise it moves to next step.

Since the solutions in population are arranged in descending order of the sensitivity,the lower half of population which contains poor solutions, is discarded. Thisprocess takes place in selection, the idea is to retain only the solutions with morethan average fitness in general. This also halves the population size. The survivorsare then selected as parents for producing new solutions. This selection processrandomly selects pairs of solutions to act as parents such that no solution is inmore than one pair.

These parent solutions then under go reproduction. Each pair produces a pair ofnew children solutions such that each child inherits some thicknesses from fatherand some from mother. This is known as crossover or mixing. For example ifparent solutions are (a, b, c) and (d, e, f) then a possible child would be (d, b, f).For the thickness of a particular layer in the child solution, the decision to use avalue from either father or mother is based on random probability. This mixingprovides a certain robustness to the algorithm. In the worst scenario, when theoptimization will not converge, it ensures that it will not diverge as well.

One final step in this algorithm is mutation. In mutation, thickness of any layerin a child solution has a probability of changing to any value within the allowedrange of thickness for that particular layer. The change in value itself is randomlyselected. The probability of mutation is kept very small because if mutation takesplace too often then the overall algorithm becomes chaotic and looses its abilityto optimize. Finally the child solutions become part of the population and thiscompletes one iteration. This population can then be utilized for next iteration andso on until the sensitivity of the best solutions in the population stops improvingor reaches a certain threshold.

Mutation is a very distinguishing and important feature of genetic algorithm be-cause without mutation a few good solutions will persist in successive iterationsand halt the improvement of sensitivity, locking us in a local maximum.

The key differences of genetic algorithm from other optimization techniques are asfollows:

• Instead of using a single point in each iteration, a population of points isused. In classic algorithms the sequence of points in successive iterationsreaches an optimal solution however in genetic algorithm the best point inpopulation reaches optimization.

• It utilizes objective function information, not derivatives or other auxiliaryknowledge to get optimized.

• It uses probabilistic transition rules instead of deterministic rules.

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Table 3.1: Results of Genetic Algorithm Optimization for layer thicknesses of SPRsensor. Optimization is performed for 633 nm p-polarized light. Thicknesses are innm and angles are in degree. Sensitivity is computed using a change in refractiveindex of δn = 0.005.

Gold Silicon Graphene θ◦res (deg) θ◦res (deg) δθ◦res (deg) sensitivity(nm) (nm) Layers (without Bio.) (with Bio.) δθ◦res/δn (deg/RIU)

55 7.5 1 72.59 73.40 0.782 157.4444 8 1 74.86 75.63 0.764 152.7850 7.5 1 72.1 72.86 0.765 15354 7.5 1 72.50 73.29 0.785 157

3.4 Results and Discussion

The first layer to be deposited on prism is selected to be gold since it has verystrong plasmon resonance at 633 nm incident light. Silicon is used for the secondlayer because it provides a dielectric environment that enhances the plasmons ingold. Finally the third and top layer is made from graphene because this layerwill be in contact with the solution containing the bio-material and therefore itwill improve the adsorption of bio molecules onto the SPR sensor, which will leadto better sensitivity. The optimization is performed by assuming a 100 nm thicklayer of water containing bio-material to be tested by SPR sensor. It is assumedthat the presence of bio-material in water will change its refractive index from 1.33to 1.335 (by 0.005).

The genetic algorithm is utilized to optimize the thickness of these three layers formaximum sensitivity for the overall SPR sensor. The possible layer thicknesses forgold lie between 35 nm to 55 nm, for silicon the thickness range is 1 nm to 10 nmand the thickness of graphene is varied from 0.34 nm to 5.1 nm. The graphenethickness, in particular, is varied in steps of 0.34 nm since the thickness of onemonolayer of graphene is 0.34 nm. This results in graphene having a range of 1 to15 monolayers in this optimization.

Several combinations of layer thicknesses are that give highly sensitive SPR sensorsare obtained as a result of this optimization. The results are tabulated in table3.1.

The reflectance of these solutions is presented in Fig. 3.6. The reflectance is plottedfor two different refractive indices of the biomaterial layer. The plot shows thatboth graphs are same for most part, however θres. shows a slight change.

3.4.1 Sensor Response

Sensitivity is one of the major characteristics of the SPR sensor, however it is alsoimportant that the sensitivity does not change as a function of the concentrationof bio-material. The concentration of bio-material will fundamentally change twoaspect of bio-molecule layer i.e.

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• refractive index

• layer thickness.

Consequently the SPR sensor should show robust sensitivity for a decent rangeof change in these two parameters. Figure 3.7 shows plots of change in θres asa function of change in refractive index of the bio-material layer. The initialrefractive index is taken to be 1.33 and layer thickness is set at 100 nm. Plots reveala linear relationship between change in resonance angle and change in refractiveindex, indicating constant sensitivity.

To explore the limits of these solutions, Fig. 3.9 shows a plot of ∆θres over a verylarge range of ∆nbio. It indicates that the sensitivity levels off when the change∆nbio = 0.07, which is a very high value.

The effect of change in layer thickness is presented by Fig. 3.8. These results showthat for both the sensors presented in table 3.1, the sensitivity is maximum arounda thickness of about 115 nm for the bio-material layer. As the layer thickness in-creases the sensitivity drops since the change in resonance angle drops. However forbio-material thicknesses upto 150 nm, the change in resonance angle or sensitivityis higher than published values (for example ref. [4])

The thickness of bio-material layer in practical applications is a strong function ofdeposition time and inter-molecule interactions of the bio-material. However therefractive index of bio-molecules is a strong function of the concentration (sinceit depends on layer packing). Therefore these results for sensor configurations arestill of good practical value if used in conjugation of experimental calibration.

3.4.2 Optimization Convergence

The performance of algorithm on a typical run of optimization is shown in Fig.3.10, which represents two main features; firstly, the best solution in the populationconverges in a very few iterations. However the value to which it converges is notconsistent through different trials. Secondly, all the solutions in population startconverging towards the best solution. It can be clearly seen in Fig. 3.10 (b) wherethe blue line represents the mean value of the entire population at every iteration.

This highlights an important feature of genetic algorithm; the reproduction usingthe best solutions can trap the optimization in a local maximum if used withoutmutation.

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0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

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1

θincidence

(deg.)

Ref

lect

ivity

n = 1 (no bio)n = 1.335 (with bio)

(a) Gold 55 nm, Silicon 7.5 nm and 1 layer of Graphene

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

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0.6

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1

θincidence

(deg.)

Ref

lect

ivity

n = 1 (no bio)n = 1.335 (with bio)

(b) Gold 44 nm, Silicon 8 nm and 1 layer of Graphene

Figure 3.6: Reflectance as a function of incidence angle for the sensors presentedin table 3.1.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

x 10

0

0.2

0.4

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∆ nbio

∆θ

res

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eg

)

-3

(a) Gold 55 nm, Silicon 7.5 nm and 1 layer of Graphene

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

x 10

0

0.2

0.4

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0.8

1

∆ nbio

∆ θ

res

(d

eg

)

-3

(b) Gold 44 nm, Silicon 8 nm and 1 layer of Graphene

Figure 3.7: Change in resonance angle as a function of change in the refractiveindex of the bio-material layer for the sensors presented in Table 3.1.

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100 120 140 160 180 200 220 2400.55

0.6

0.65

0.7

0.75

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0.9

Thickness of Biomaterial layer (nm)

∆ θ

res

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eg

)

(a) Gold 55 nm, Silicon 7.5 nm and 1 layer of Graphene

100 120 140 160 180 200 220 2400.5

0.55

0.6

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0.85

Thickness of Biomaterial layer (nm)

∆ θ

res

(d

eg

)

(b) Gold 44 nm, Silicon 8 nm and 1 layer of Graphene

Figure 3.8: Change in resonance angle as a function of change in the thickness ofthe bio-material layer for the sensors presented in Table 3.1.

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0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

6

7

8

∆ nbio

∆ θ

res

(d

eg

)

(a) Gold 55 nm, Silicon 7.5 nm and 1 layer of Graphene

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

6

7

∆ nbio

∆ θ

res

(d

eg

)

(b) Gold 44 nm, Silicon 8 nm and 1 layer of Graphene

Figure 3.9: Convergence of the sensitivity for the sensors presented in Table 3.1.

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2 4 6 8 10 12 14 160

0.1

0.2

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Iteration

∆ θ re

s (de

g)

(a) Performance of the overall population.

1 2 3 4 5 6 7 8 9 10 110

0.1

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0.7

Iteration number

∆ θ

res.

(d

eg

)

(b) Performance of the overall population for another trial.

Figure 3.10: Convergence of the sensitivity for typical runs of genetic algorithm.The blue lines in (a) and (b) represent the mean value of ∆θres. for the entirepopulation at each iteration.

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Chapter 4

Conclusion

The analysis so far indicates that the SPR provides a very highly sensitive meansfor creating sensors which can be utilized for chemical and biological applications.The sensitivity proves to be a non-trivial function of the optical architecture of thesensor. The analysis indicates a three layer sensor to be the optimal design wherethe first layer is composed of gold since it has strong SPR, the second layer is ofsilicon as it improved the SPR in gold and the final layer is graphene due to itshigh affinity for bio-material.

Although the sensor is based on the complex phenomenon of SPR, the sensor’ssensitivity can be accurately modeled by utilizing Fresnel’s equations with appro-priate boundary conditions. However the explicit dependence of the sensitivityon layer thickness and arrangement is not obvious. Under these circumstances aparametric approach proved to be of no use.

However genetic algorithm proved to be quite adequate for this task, yieldingresults that outperform published sensors. The sensitivity for the two optimalsolutions is 157.44 (deg/RIU) and 152.78 (deg/RIU) which is considerably higherthan the results published in [4]. The sensor structures obtained from geneticalgorithm optimization show a linear response for the refractive index of the bio-molecule layer, however the response is nonlinear for the thickness of this layer.

4.1 Future Work

These studies did not take the noise into consideration. Since it can be a majorattribute of a sensor, further studies are required for its calculation. Furthermorethe robustness of sensitivity for the thickness of bio-material layer is also an impor-tant quality, sensors that show robust sensitivity as a function of the bio-materiallayer thickness need to be simulated. With the present performance of Geneticalgorithm, it can be used to achieve these optimizations as well.

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[2] S. Szunerits, N. Maalouli, E. Wijaya, J.-P. Vilcot, R. Boukherroub, “ Recent ad-vances in the development of graphene-based surface plasmon resonance (SPR)interfaces” Analytical and Bioanalytical Chemistry, Volume 405, Issue 5, pp1435-1443.

[3] H. Raether, “Surface Plasmons on Smooth and Rough Surfaces and on Grat-ings” Spriner 1986.

[4] R. Verma, B. D. Gupta, R. Jha “Sensitivity enhancement of a surface plasmonresonance based biomolecules sensor using graphene and silicon layers” Sensorsand Actuators B, 160 (2011) 623-631.

[5] M. Piliarik and J. Homola, “Surface plasmon resonance (SPR) sensors: ap-proaching their limits?,” Optics Express, Volume 17, Issue 19, pp 16505-16517.

[6] J. Peatross and M. Ware, “Physics of Light and Optics” Brigham Young Uni-versity, 2015 Edition.

[7] F. Bahrami, M. Maisonneuve, M. Meunier, J. S. Aitchison, and M. Mojahedi,“An improved refractive index sensor based on genetic optimization of plasmonwaveguide resonance,” Optics Express, Volume 21, Issue 18, pp 20863-20872.

[8] H. Awan, K. Abdullah, M. Faryad,“Implementing Smart Antenna System usingGenetic Algorithm and Artificial Immune System” IEEE, Microwaves, Radarand Wireless Communications, MIKON 2008. 17th International Conference,19-21 May 2008.

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