Alexandru Bratu - Masters Thesis - Final - 10

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METAMATERIAL LENS DESIGN FOR ULTRAHIGH RESOLUTION

FLUORESCENT MICROSCOPY

A thesis presented

by

Alexandru Bratu

to

The Department of Physics

in fulfilment of the requirements

for the degree of

MSc of NanoScience

in the subject of

BioPhysics

University College Dublin

Belfield, Dublin 4

August 2009

Thesis advisors Author

Dr. James Rice Alexandru Bratu

Dr. Brian Vohnsen



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Page 3

ABSTRACT

Metamaterials have recently shown great potential in novel lens designs

that allow imaging with resolution beyond the classical limit of diffraction. Hereits potential for ultrahigh-resolution fluorescent microscopy will be examined

with numerical analysis. Different structures based on surface-plasmon-

polariton excitation on thin metallic films have been examined with even and

uneven distributions of surface scattering elements. The results obtained show

that indeed a resolution beyond the diffraction limit can be obtained with the

novel metamaterial lens designs. This has particular importance in weak signal

fluorescent imaging of biomaterials and possibly for nano-toxicity analyses.

Significant progress has been made in the development of sub-diffraction

fluorescence microscopy methods that enable images recorded in the far field

to possess resolution down to the nanometer scale. These methods include

stimulated emission depletion and its related reversible saturable optical

fluorescent transition microscopy, stochastic optical reconstruction microscopy,

structured illumination microscopy, and photoactivated localization

microscopy. However these techniques have limitations requiring either time-

consuming point-by-point scanning or the accumulation of large data sets,

which prevents them from reaching real-time imaging applications.

The application of new lens technology made from metamaterials holds

promise for real-time sub-diffraction fluorescence imaging. A metamaterial is a

material that gains its properties from its structure rather than directly from its

composition. In the ideal case unlimited high resolution in the far field has

been found in a theoretical work [1]. Such lenses have been termed

„superlenses‟ or „hyperlenses‟ and are typically made with alternating layers of

materials with careful chosen indices of refraction and dimensions [2, 3].

Recently, they have been applied to optical microscopy and far-field

transmission based imaging with a spatial resolution of <100nm has been

demonstrated [2, 3].



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Here we outline theoretical work towards a superlens design for

fluorescence based on the proposal of Smolyaninov et al . using surface

Plasmon polaritons (SPPs) [2]. The internal structure of the magnifying

superlens consists of concentric rings of poly(methylmethacrylate) (PMMA)

deposited on a gold film surface. PMMA has a negative index of refraction

2 0n , [2]

perceived by the SPPs (the group velocity is opposite to the phase velocity).

The width of as the PMMA rings2

d is chosen so that

1 1 2 2n d n d , [2]

is satisfied, where1

d is the width of the gold-vacuum portions of the interface.

In our model we first study scattering by ordered nanostructures on a gold film

as shown schematically in Fig. 1. Ideally this can focus the incoming SPP wave

and is an essential first step towards the realization of a 2-D sub-diffraction-

limited lens. We study the SPP scattering and propagation using SPP in-plane

scattering based on a Green‟s function propagator formalism. Eventually, our

aim is to convert this theoretical proposal into a workable experimental lensfor fluorescent microscopy.



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CONTENTS

Metamaterial lens design for ultrahigh resolution fluorescent microscopy ....... 1

Abstract ................................................................................................ 3

List of FIGURES ................................................................................... 7

Chapter 1 – Plasma ............................................................................. 9

Physical properties .............................................................................. 9

SPPs (Surface Plasmon Polaritons) at smooth surfaces ............................ 11

Fundamental Properties ...................................................................... 11

Dispersion Relation ......................................................................... 11

Spatial Extension of the Surface Plasmon Fields .................................. 20

Propagation length of the Surface Plasmons ....................................... 22

Excitation of Surface Plasmons ............................................................ 23

Excitation by electrons .................................................................... 23

Excitation by Light .......................................................................... 24 Internal and Radiation Damping .......................................................... 28

Internal Damping ........................................................................... 28

Radiation Damping ......................................................................... 29



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Chapter 2 – Metamaterials ................................................................ 31

Electromagnetic metamaterials ........................................................... 31

Negative Refractive index ................................................................... 32

Cloaking devices ............................................................................... 34

Chapter 3 – Surface Plasmon Polaritons ............................................ 37

Physical Properties ............................................................................ 37

Theoretical Model .............................................................................. 38 Chapter 4 – NIM (Negative Index Media) .......................................... 47

Time reversal .................................................................................... 49

Negative index makes a perfect lens .................................................... 53

Far Field Imaging .............................................................................. 54

Chapter 5 – MATLAB Code and Simulation Results ............................. 57

Incident electric field SIMULATION ....................................................... 57

Chapter 6 – Conclusion...................................................................... 75

Acknowledgment .................................................................................. 79

Bibliography ........................................................................................ 81

Appendix ............................................................................................ 85



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LIST OF FIGURES

1: Scheme of the dispersion of plasmons in a solid

2: The charges and the electromagnetic field of the SP’s propagating on a

surface in the x direction

3: ATR coupler with its respective , , x y z coordinates and interfaces on the

plane

4: Right Handed Materials (RHM) and Left Handed Materials (LHM) and their

corresponding magnetic, electric and propagation vector

5: External SPP wave propagating under angle (with respect to the

ellipsoid x – axis) along the dielectric metal interface is scattered by an

ellipsoid particle

6: Holographic technique with a TIR reference wave

7: An NIM flat lens brings all the diverging rays from an object

8: NIM lens enhancing evanescent wave propagation such that the

amplitude of the evanescent waves are identical at the object and the

image plane

9: The dispersion of the wave in a medium with constant velocity ck

where c is the speed of light. Both positive and negative frequencies may

be viewed.



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10: Optics of a negative index slab

11: Time – reversing mirrors reproducing similar focusing effect to

subwavelength imaging in the near – field

12: Magnifying optical hyperlens with numerical simulation of imaging of sub

– diffraction limited objects

13: Program 1 diagram - dotted SPP imaging for reference point (1285,

1369)

14: Program 1 diagram - dotted SPP imaging for reference point (0, 0)

15: Program 2 diagram – gradient SPP imaging of the total E – field intensity

16: Theoretical perspective of the diagram for SPP imaging with constant

wavelength of 500nm

17: Theoretical waveguide on the quarter plane (including polarizability

factor to account for extra points)

18: STED apparatus and an intensity distribution graph

19: Microtubules (green) and clathrin – coated pits (red) [X. Zhuang

Research Labs]

20: COS – 7 cell tagged with PA – Fluorescent Protein Kaed

21: Far-field fluorescence methods which enable ultrahigh resolution imaging



Page 9

CHAPTER 1 – PLASMA

PHYSICAL PROPERTIES

This chapter stands out from much of the rest of this thesis in that it

deals with an introductory and numerical method. Each diagram has its

respective reference and listing, hence making it easy to find and understand.

The equations that follow co – work with respect to diagram reference and

those that are harder to find have been carefully addressed to with their own

respective reference.

When we think of plasma we tend to think of an ionized and

macroscopically neutral gas with sufficient amount of free charges to make

electromagnetic forces important to its macroscopic behaviour. The concept of

this fourth state of matter is treated as an electron liquid of very high density

of about 2310 3cm

, ignoring the lattice in a first approximation. From this

outlook we have that the longitudinal density fluctuations, plasma oscillations

will propagate through the metal. The quanta of these “volume plasmons” will

carry energy of the order

2

0

4 p

ne

m

, [4]

where n is the electron density of the order of 10eV. They are produced when

electrons hit the surface of the metal [4]. According to Maxwell‟s theory the

EM surface waves can propagate along a metallic surface or film with a broad



Page 10

spectrum of eigen frequencies from 0 up to 2 depending on the wave

vector k .The dispersion relation k lies to the right of light line hence the

surface plasmons have a longer wave vector than light waves of the same

energy , propagating along the surface.

Figure 1

Scheme of the dispersion of plasmons in a solid. (1) Volume plasmons,

(2) surface plasmons, (3) plasmons in a one – dimensional electron gas, (4)

plasmons in a one – dimensional system. The upper scale is valid for (1) and

(2), the scale below for (3) and (4). ll is the light line. [4]



Page 11

SPPS (SURFACE PLASMON POLARITONS) AT SMOOTH SURFACES

Surface plasmons propagate along the surface of the metal and when

coupled with a photon they create a new quasiparticle called surface plasmapolaritons.

The smooth surfaces along which they travel are not perfectly smooth,

but small enough to be negligible in calculation. The height range is a root

mean square value (r.m.s.) of 5 – 10 Å.

Surface plasmon polaritons (SPP‟s) are confined to electromagnetic

waves that travel along a metal – dielectric interface with an exponentially

decaying amplitude into the nearby media.

FUNDAMENTAL PROPERTIES

DISPERSION RELATION

When an electron on a metal boundary is charged energy oscillations are

given. Their existence was proven by electron energy – loss experiments by

Powell & Swan. The frequency of the fluctuations are dependent of the wave

vector by a dispersion relation. The localization of these fluctuations is done in

the z – plane with Thomas – Fermi screening length which is approximately

about 1Å, accompanied along with a mixture of the transversal and the

longitudinal EM waves which decay completely as z .By considering the

description of the electric field



Page 12

0 exp x z E E i k x k z t , [4]

where + for z 0 , – for z 0 , an imaginary zk would cause an exponential

decay for z E . The equation gives that direction xk x ;2

x

p

k

where p

is the

wavelength of the plasma oscillation [4]. An illustration of the field would show

a maximum in the surface z = 0 , which is normal for surface waves.

Figure 2

The charges and the electromagnetic field of the SP’s propagating on a

surface in the x direction are shown schematically. The exponential

dependence of the field z E is seen on the right. y

H shows the magnetic field in

the y – direction of this p – polarized wave. [4]

With the use of Maxwell‟s equations the retarded dispersion relation for a

plane surface semi – infinite metal along with the dielectric function



Page 13

' '

1 1 2 when close enough to the medium2

as air or vacuum:

1 20

1 2

0 z zk k

D

, [4]

together with

2

2 2

i x zik k

c

, [4]

or

12 2

2 , 1,2 zi i xk k ic

, [4]

The wave vector xk is continuous through the interface hence we can

rewrite0

D :

1

21 2

1 2

xk c

, [4]

However if we assume besides 2, and '' '

1 1 , we obtain a complex

' ''

x x xk k ik , [4]

with

3

' ' ''2' ''1 2 1 2 1

2' ''

1 2 1 2 1

&2

x xk k

c e c

, [4]

For '

xk one needs ' '

1 1 20& , this may be accomplished in a metal



Page 14

and also in a doped semiconductor near the eigen frequency. From this we can

also use a Thomas – Fermi wavenumber theory to deduce the relationship

between the wavelength of the incident electric field and the dielectric constant

of the metal with the presumptions that the density of the plasma oscillations

is constant throughout.

Consider a metal that has a relative absorbance, hence a complex

dielectric constant component. By using the work of Y.C. Lee and S.E. Ulloa [5]

we could find a beginning to such a theory of wavelength – dependent

dielectric constant that could later be used to calculate the incident electric

field with cumulative point – dependence:

2

21 FT

metal

k q

q , [5, 10]

where

dielectric constant of the metal, thomas fermi wavenumber, wavenumbermetal FT k q

which is derived from Lindhard‟s formula [6] for the longitudinal dielectric

function given by

, 1 k q k q

k k q k

f f q V

i

, [6]

with

q eff ind V V q V q and

carrier distribution function,aka the Fermi - Dirac distribution

function for electrons under stable thermodynamic equilibriumk f



Page 15

We must also study the formula under the limiting case. Lindhard‟ s formula

will be studied under 3D since the density applies only to 3D.

Consider the long wavelength limit 0q for the denominator and

numerator

2 2 2 2

2 22 ,2 2

...

k q k

k q k k k k k k k

k k q E E k k q q

m m m

f f f q f f q f

, [6, 39]

Substituting and letting 0 we get

2

,

0

,0 0

2

20

2

0

2 2

2 3 2

0 0

2

2

0

0, 1

1 1

1

1

41

1

k i

iq

k i

q k i

k i i

q k k

q

pl

f q

k V

k q

m

V f k qq

k m

q

V f m

q N V

m

e q N

q L m

, [6, 39]

where we have that

2 22

2 3 3

0 0

4 4, &k k q pl

e e N E V

q L L m

, [6, 39]

Now consider the static limit 0i where Lindhard‟s formula takes

the following shape:



Page 16

2 2

, ,

,0 1 1

i i

i iq q

k i k i

f f q q

k k q V V

k q k q

m m

, [6, 39]

Hence using the equation above we get that for a thermal Fermi – Dirac

thermal equilibrium

2

k k k k i i i i

i i ii i

f f f q q q k

k k m

, [6, 39]

where

22 2

&2

k ik

i

k k

m k m

, [6, 39]

Therefore

2

2

2 32, 0

2 2 2

2 3 2 2

0 0

4 1,0 1 1 1

4 41 1 1

k i i

q q k

k i k k

f q k

f emq V V f

q Lk q

m

e N e n

q L q q

, [6, 39]

is a 3D screening wave number 3D inverse screening length defined as

2

0

4 e n

, [6, 39]

But thinking in terms of the statistics of the parameter we get that the

Coulomb screening potential is given by



Page 17

2 3 2

0

2 2 3 2 2

0

2

4 1, 0

, 0

q

s

e

V q L eV q

qq L q

q

, [6, 39]

By applying a linear Fourier transform of the equation above we get

2 2

3 2 200

4 iq r r

s

q

e eV r e e

r L q

, [6, 39]

which is similar to the Yukawa potential.

Now by considering a degenerating ideal Bose – Einstein gas at which

0T 0 K= , the Fermi energy becomes

22

2 332

f E n

m , [6, 39]

Hence the density n is given by

3

2

2 2

1 2

3f

mn E

, [6, 39]

At 0 3T 0 K E

2 f

f

n n=

E

. Substituting this result into the screening

wave number equation we obtain:

2 2

0

6

f

e n e n

E

, [6, 39]

and the answer given is the 3D Thomas – Fermi screening wave number.

Proceeding further we get that for the general metal:



Page 18

2 2 2 2

22 222 220 33

00

2 2

2 2

2 2 2 23 30 0

6 1 6 1 61 1 1 1

332

2

6 2 121 1

13 3

FT metal

f

metal

k e n e n e nq

q E q qq nn

mm

e n m e mnq

q n q n

, [6, 39]

But from simple theory we have that

2

q wavenumber

, [5]

My simulation data will use a typical green laser wavelength of 500nm,

hence I had to also consider at what density such a wavelength would

propagate through a medium [7]. By considering an experimental

demonstration of Nd:YAG laser where we have Nd:YAG 1064nm , almost about

1000nm which corresponds to a density of 21 3

Nd:YAG 10 cmn . However we have

that the wavelength is inversely proportional to the density of the volume of

the ionic distribution in the medium, thus

21 3

5002 10 cm

green nmn

. [7]

Now substituting in the equation for the wavelength – dependent

dielectric constant we get that:

2 2

2 22 2 2 23 3

0 0

12 61 1

23 3

metal

e mn e mnq

n n

. [6, 39]

By inserting the values for gold we have that the inner density for the

metal may vary slightly but not much. Thus:



Page 19

2

22 2 3

0

19 31

234 2 1

212 1 21 3

61

3

6 1.602 10 9.109 10 5001

6.626 10

8.854 10 3 3.14 2 103.14

gold

e mnq

n

C kg nm

m kg s

F m cm

. [6, 39]

By converting the SI units throughout the equation we get that

0.24gold

.To verify our answer we must carry out a much simpler calculation

to verify that in the long – wavelength region the screening wave number is

correct when considering an ideal Bose – Einstein gas. Using gold as our metal

we have that by international standards its refractive index is given by [8]:

0.47gold

n . [8]

From electro – optics we know that:

22 2

0.47 0.2209gold gold n n . [8]

By comparing this with the general case in the long – wavelength range

we just evaluated we have that the local error is about

40.241.08 10

0.2209

Theoretical

Experimental

, [general]

hence almost negligible.

Now returning back to our plasma retardation we have that the

dispersion relation0

D approaches the light line 2@ xk small

c

, but remains



Page 20

further away than 2

c

such that the surface plasmons cannot convert into

light. However at larger values of xk or '

1 2 , we have that

2

lim1 x

p

SPk

, [4]

where the p

corresponds to the plasma frequency, given by24

p

e n

m

,

with n the electron bulk density.

As xk gets larger we also have that

lim 0

lim 0

x

x

k

k

group velocityk

d phase velocity

dk

, [4]

such that the surface plasmon matches a localized fluctuation of the electron

plasma.

SPATIAL EXTENSION OF THE SURFACE PLASMON FIELDS

From definitions we are aware that the wave vectors

2 1& z zk k , [4]

are imaginary due the fact that

'

1& 0 x

k c

, [4]

so that the amplitude of the surface plasmons have an exponential decay



Page 21

zik ze

normal to the surface. From this information we gather that the depth value of

the field falls to 1/e, becomes 1ˆ

zi

zk

or

1

' 21 2

2 2

2

'

1 21 '2

1

în the medium:2

în the metal:2

z

z

, [4]

By using 2 metals with internal absorption such as gold and silver, we

get:

2

1

2

1

ˆ 390:

ˆ 240600

ˆ 280:

ˆ

310

z nmsilver

z nm At nm

z nmgold

z nm

, [4]

since

1ˆlim

x

ik

x

zk

, [4]

this leads to a strong field concentration at the interface between the two

media. But for lower xk values or bigger '

1 the field has a more powerful

transverse component z E compared to the longitudinal component

x E , as

'

1

z

x

E i

E , [4]



Page 22

leading to a guided photon field from a Zenneck – Sommerfeld wave, which in

theory gives guided propagation with no radiation field to waste energy.

But in a metal z x

E E since

'

1

z

x

E i

E . [4]

The theory can be validated from Maxwell‟s theory in vacuum

0 or 0 y x z

E E E E

x y z

, [4]

however it suffices only outside the surface of the air/metal interface and at

large xk value the z and x – component would suffice:

:

: z x

air i E iE

metal i

.

PROPAGATION LENGTH OF THE SURFACE PLASMONS

Considering a smooth surface the intensity of the surface plasmons

decay exponentially as

''

3' ''2

2 '' 1 2 1

2' '1 2 1

@ 2

xk x

xe k c

. [4]

The length after which the intensity decays to 1/e is given by ''

1

2i

x

Lk

.

Now consider in the visible region 400 700nm nm we have that



Page 23

22 @ 514.5i L m silver nm , [4]

while at 1060 500inm L m .

Using the temporal decay iT instead of the spatial coordinate decay

along the x – axis we have that the values of the propagation and length and

temporal decay are inter – connected by:g

i i g

g

v group velocity

L T vv

k

.

By using a complex frequency ' ''i and' xk with

2

''iT

we obtain

'' '' 1 1 2

2 ''

1 21 1 2

'1 2

1 2'

'

1 2

''2

@

'

x

x

x

k c

k c

k c

. [4]

EXCITATION OF SURFACE PLASMONS

EXCITATION BY ELECTRONS

When electrons hit a solid surface they have an energy loss of E

after penetration. They transfer a momentum q and energy0 E to the

electrons of the solid material. The wave vector xk is determined from the

projection of q upon the surface of the film. The electrons are scattered at

different angles and they transfer different momenta 'sin x el elk k k with

2el

el

k

. [4]



Page 24

To observe a successful excitation the use of fast electrons is highly

recommended such as to study the dispersion relation at larger p

x xk k

c

,

since a small value irrespective of the electrons velocities will cause the

aperture of the electron beam to be sufficiently reduced. And during the

electron transmission in thin films, the dispersion relation determines the

energy of the surface plasmons, while the choice of '

elk fixes xk and

respectively.

EXCITATION BY LIGHT

To excite the surface plasmons photons are used, however they face the

difficulty of having the dispersion relation to lie right of the right line, i.e.

larger x – component wave vector. To achieve this “conversion” of photons to

SPs the photon energy must be increased by a factor of x

k . 2 suitable

methods to achieve this increase are:

1. Grating Coupler [4]

The principle of this method is that if light penetrates the grating with a

grating constant a at an angle 0 the surface component has wave vector

0 0sin sin1

x g SP x g

wk v k k with v

c c c k

, [4]

where xk rises from the perturbation along the smooth surface. We observe

that the resonance can be observed as the reflected light minimum. For the



Page 25

surface plasmons to be turned into light their propagation along the grating or

a rough surface should result in a reduction in their wave vector component xk

by x

k . Radiation is used to detect the presence of the SPs, because if the SP

is excited to a maximum value, the emitted intensity is excited as well to a

maximum.

2. ATR (attenuated total reflection) coupler [4, 24]

If light is reflected at a metal surface covered with a dielectric medium

1 pr then:sin sin

cos cos

pr

x pr pr pr pr

pr

z pr pr pr pr

k nc c

k nc c

. [4, 24]

Figure 3

ATR coupler with its respective , , x y z coordinates and interfaces on the

plane. [4, 9]

The resonance condition for the light in the prism with SP at metal (1)

|air (2) interface (Kretschmann – Raether configuration) is:



Page 26

pr sp

x x

pr prismk k

sp surface plasmon

[4, 24]

1 2

1 2

sin pr pr

c c

[4, 24]

The reflected intensity R may be given by Fresnel‟s equations of the

prism from the prism | metal | air – 3 layer system:

2 2

2 2

cos cos

cos cos

zi zk k zi i zk zi zk zi zk k i

p pk i i k i k i k i k i k

ik ki zi zk k zi i zk zi zk zi zk

k i i k k i

i k i k i k i k

k k n k c n k c k k k k n n

n n k k n n n nr r

k k n k c n k c k k k k n nn nk k n n n n

. [4, 24]

For transmission however:

1

1 1

1 1

p piik ik

k

p p pk k ki ki ik

i i p p p p

ik ki ik ik

nt r

n

n nt r r

n n

t t r r

. [4, 24]

Hence the total reflection for the 3 – layer system becomes:

1 1

1 1

22

2 1 12

12 2

1 121

cos cos

cos cos

cos coscos cos

z

z

ik d p p

pr p

pr ik d p p

pr

i k

p k i i k i k ik

i k k i i k

i k

r r e R r

r r e

n n n nr

n n

n n

, [4, 24]

1 1 2 2sin sin sin pr pr n n n

, [4, 24]



Page 27

2 2

2 2 2 2

2

2 2 2 2

2 2 2 2

2 2

1

1

12

1

sincos 1 sin 1 sin

sin sin

sin sin

sincos

sincos

pr pr

k k k k k k pr pr

k

i pr pr k pr pr

p i k

ik

i pr pr k pr pr

i k

pr pr pr

pr p

pr

pr pr

pr

nn n n n n

n

n n

r

n n

n

nr

n

n

2

1

pr

. [4, 24]

The last term can be understood better by the following expansion,

1 1

1 1

1 1 1 1 1 1

1 1 1 1 1 1 1 1

2

1 12

12 2

1 12

2 22 2 4

1 12 1 12 1 12

2 2 3 22 2 4 4

1 12 1 12 1 12 1 12

1 1

1

1 ...

...

1

z

z

z z z

z z z z

ik d p p

pr p

pr i k d p p

pr

ik d ik d ik d p p p p p p

pr pr pr

ik d ik d ik d ik d p p p p p p p p

pr pr pr pr

p

pr pr

r r er

r r e

r r e r r e r r e

r r e r r e r r e r r e

r r

1 1 1 1

1 1 1 1

1 12 1 1 12 1 12 1

2 2 22 4

12 1 1 12

2 4

1 1 12 1 1 12 1 12 1

1 1 12 1

1 ...

1 1 1 1

z z

z z

p p p p p p p p pr pr pr pr pr

ik d ik d p p p p p

pr pr

ik d ik d p p p p p p p p p

pr pr pr pr pr pr

t r t t r r r t

p p p

pr pr pr

r e r r r e

r r r r e r r r r r e

r t r t

1 1 1 12 4

1 12 1 12 1 z zik d ik d p p p p p p

pr pr pr e t r r r t e

, [4, 24]

where the phase factor 1 1 1 1 1cos z

k d k d is the optical length.

Thus the detailed phase factor becomes

2

2 2 2

1 1 1 1 1 1 1 1 12

1

cos 1 sin sin pr

z pr pr pr

nk d k d n d d n

c n c

. [4, 24]

Please note that I have expanded and shown every term to its full extent

for clearer understanding. Thanks also to the R. Adams calculus book. [22]



Page 28

INTERNAL AND RADIATION DAMPING

INTERNAL DAMPING

From the extended dispersion relations we have that the internal

absorption is given by

3

' ''2'' 1 2 1

2' '1 2 12

xk c

. [4]

The EM field of the SPs excites pairs of electron – holes at the Fermi

level, 1

1

e

kT

F

e

, [10]

chemical potential at the Fermi – Dirac Distribution function.

The de – excitation that follows produces the phonons and hence the

heating. The energy can also be lost by the emission of photoelectrons, if

photoelectric work function [11], with the heat measured with a

photo – acoustic cell. A small exception is that of silver for which the periodic

heating of the film can be registered by the periodic pressure variations in a

microphone. Experimentally the energy lacking in the reflected light is

measured with the photo – acoustic cell as the heat energy in the silver film.

The power lost by light decaying from surface plasmons can be detected

assuming we have a smooth surface with 5 Å.



Page 29

RADIATION DAMPING

If the SPs propagate along the smooth surface interface of air/metal with

a dielectric material 1o as in the ATR device, the SPs, the evanescent wave,

with an imaginary1 zk ,

2

1 1 10

z xk k c

, [4]

transform the in the medium0

into a plane wave due to the real2 zk ,

since0 xk

c

. Hence radiation damping, which is characteristic to an

asymmetric 2 – interface system.

The electrons excite the SPs which emit light into the dielectric 0 to

produce 2 intense tip – like plasma radiation peaks with an angular width anda corresponding intensity usually about 100 times stronger than the transition

radiation. The peaks arise from 2 differently valued scattering angles and due

to these angular dispersions and intense light circle around the film

perpendicular plane normal is produced.

By analyzing the phase factor for the ATR coupler we have that the

internal damping is given by 0Im xk where

0

1

3

' 221 0

1' '

1 1

2

1 1

xk d p

x pr xk e r k

c

, [4]



Page 30

with1

p

pr r given by / p zi zk zi zk ik

i k i k

k k k k r

. In the MATLAB simulation we shall

later discover that also from experiment that for a wavelength of 500nm a

silver film gives that the minimum reflectance R 2

12 p

pr r is equal to zero

corresponding to a minimum thickness of 55nm. The evaluation of the

minimum thickness depends on the dielectric function just like we showed

the dependence of the wavelength in the steps above, but it varies by at least

10%. We must also remember that the energy conservation satisfies:

1

R relative reflection

R A T A relative absorption

T relative transmission

. [4]

Hence at T = 0 and with mind we have that R = 0 A = 1, i.e. the whole

energy is absorbed in the metal film. The case of R = 0 also gives that

0Im Im xk k , i.e. where internal damping = radiation damping.

Now we can determine the function in situ using the ATR minimum.

Data shows that a detailed an accurate formula matches the experimental data

by analyzing the reflection minimum. In situ shows the variation of withrespect to the dielectric function for many absorbing metals such as silver, e.g.

25.7 81 0.4silver i [1].



Page 31

CHAPTER 2 – METAMATERIALS

ELECTROMAGNETIC METAMATERIALS

Metamaterials are materials that gain their properties from its structure

rather than its composition. They are artificially subwavelength composites

which possess extraordinary optical properties that do not exist in nature. [12]

Their importance is vastly used in the field of electromagnetism, in

particular for optics and photonics. Uses include microwave applications such

as new types of beam steerers, modulators, band – pass filters, lenses

(superlenses and hyperlenses), microwave couplers and antenna radomes.

A metamaterial has structural features smaller than the wavelength of

the electromagnetic radiation it interacts with. Instinctively for the material to

act homogenously (have size feature close to wavelength size) it must be

described by an effective refractive index. An example would be visible light

from sunlight, wavelength about 560nm for which the structures would have a

size about halfway or even smaller. For microwave structures however they

must have structures in the order of 1 decimeter.

They usually have periodic structures such as photonic crystals and

frequency selective surfaces, but they differ from metamaterials as their

features size almost matches that of the wavelength.



Page 32

NEGATIVE REFRACTIVE INDEX

Study of the exotic nature of metamaterials is studied to investigate the

possibility to create a structure with negative refractive index, since this

property is not found in any naturally occurring material. Typical optical

mediums and substances like glass or water have a positive refractive index,

with glass about 1.5 and small deviations for crown glass and clear water

about 1.33. These positive indexes of refraction have their permittivity and

permeability > 0 as well since

n . [1]

But there are metals with light absorption abilities such gold and silver

which have negative permittivity at visible wavelengths 400 700nm nm

By using the work of Vohnsen and Veselago [13, 14] we have the

following formulas to properly indicate how the sign of the refractive index

behaves with respect to the correspondent material:

2

0

0

n

n

n

&

0

0

0 ,

0

dielectrics

dielectrics

metals plasmas absorption

metamaterials

. [13, 14]

No materials with 0 are known in nature. All known transparent

materials posses positive values for and . The positive square root is taken

by convention. Some materials have both values negative and it is then



Page 33

necessary to take the negative square root. Veselago proved that such

substances can transmit light [14, 15]. For special metals such as gold and

silver we have values such as

2

26.15 1.85 800 |

5.7 81 0.4 560 |

gold

silver

i nm gold air interface

i nm silver air interface

. [14, 15]

Numerous physical properties of metamaterials have stormed the

imagination of physicists for years. The most important ones are:

1. Snell‟s Law

By definition we have that 1 1 2 2sin sin N N , but since 2 0 N , the rays of

light will be refracted on the same side of the normal (perpendicular

axis to the light plane) on entering the material.

2. Doppler Shift

The effect is reserved such that if a light source is moving towards an

observer it appears to reduce its frequency

3. Cherenkov radiation

Cherenkov radiation changes direction and thus points the other way

4. Poynting vector, S = E x H

Poynting vector is anti – parallel to the phase velocity, hence unlike a

right –handed material, the wave fronts move in the direction opposite

to the flow of energy. By making a diagram of energy flow we have that:



Page 34

Figure 4

Diagram showing the Right Handed Materials (RHM) and Left Handed

Materials (LHM) and their corresponding magnetic, electric and propagation

vector. LHM were first introduced by Victor Veselago in 1967. [14, 15]

CLOAKING DEVICES

Once a dream, now a possibility metamaterials are now being considered

to create optical illusions such as invisibility or more commonly referred to as

„cloaking‟. The mechanism of this physical concept involves surrounding the

object with a shell such as to prevent light to bounce off of it. First proposals

for such an amazing technology was given on the 14 th of February 2005 at the



Page 35

University of Pennsylvania by Andrea Alü and Nader Engheta who announced

that plasmons can cancel out the visible light on radiation coming from an

object. This „plasmonic cover‟ would in theory suppress light scattering by

resonating with illuminated light, which could render objects „nearly invisible to

an observer‟.

The first successful „cloaking‟ approach was done by a US – British team

of scientists which created a metamaterial that made an object invisible to

microwave radiation. Since this was not visible light, this proved as a stepping

stone towards more advanced nano – engineering techniques.



Page 36



Page 37

CHAPTER 3 – SURFACE PLASMON POLARITONS

PHYSICAL PROPERTIES

Surface plasmons coupled with a photon create a quasiparticle called

polaritons, thus they become surface plasmon polaritons (SPPs). They are

confined to electromagnetic waves that propagate along a dielectric – metal

interface with amplitudes which decay exponentially into both neighboring

media.

Their physical nature allows them to concentrate and guide

electromagnetic radiation using wavelength structures such as metamaterials.

Typical uses of SPPs occur in micro – optical components that may be used as

mirrors, beam splitters, micro – scatterrers and efficient SPP interferometer.

Currently SPP analysis involves the use of point – dipole system

approximation and an anisotropic free – space polarizability component that is

dependent of the of the ellipsoid major axes and hence on the particle shape.

The system involves the use of a dielectric constant p

for the particle in

the region z < 0 with dielectric constant 0r and a metal in the region z < 0

with dielectric constantm

. The particle is positioned above the metal surface

in the dielectric ½ - space with its center located at 0,0, p p zr . To support

SPP propagation requires that we must have the metal dielectric interface with

the conditionm r

[16]. For this scattering system we get:



Page 38

Figure 5

Schematic of the scattering system: an external SPP wave propagating

under angle (with respect to the ellipsoid x – axis) along the dielectric metal

interface is scattered by an ellipsoid particle. [16]

THEORETICAL MODEL

However the MATLAB model for our scattering system will involve also

contributory work given by Evlyukhin and Bozhevolnyi[18] we have that for

the point – dipole approximation using a holographic technique along with a

TIR(total internal reflection) configuration. Like in the scattering case above

we consider a point dipole located at

0,0,d d zr , [17, 18, 19]

in an air medium above a flat surface of a glass substrate(n typically 1.5

depending on the type of glass used) with dielectric constant 2 2.25n .



Page 39

Figure 6

Schematic view of the holographic technique with a TIR reference wave

[17]

The dipole has an electric monochromatic field 0 d E r with wavelength .

A holographic image of the dipole is obtained at the height due to the

interference between the induced dipole field d

E r and reference wave

r E r . Further detailed study of the near – field above the surface d z ,

gives that the coupling is not negligible, hence an increase in the magnitude of

the dipole [20, 21]. To consider these couplings we must have the polarization

of a spherical nanoparticle with volume given by

34

3d V a a particle radius , [22]

and a dielectric permittivityd with external optical field with wavelength

bigger than the particle‟s size hence not limiting the analysis to metamaterials.



Page 40

We now obtain a dipole moment p :

1

2

0, 0

0

ˆˆ ˆ Î ,S

d d d

k G

p r r E r [23]

where ˆ ˆ ˆ ˆ ˆ ˆÎ xx yy zz (in spherical coordinates)[22] is the unit dyadic tensor,

ok is the wave number in vacuum,0

is the vacuum permittivity, ˆ , 'S

G r r is

the surface part of the Green‟s tensor of the substrate – air interface system,

where we have & 'r r located in the air medium. Now considering the

polarizability tensor as the dipolar polarizability of the spherical nanoparticle

in the free space along the long – wavelength electrostatic approximation we

have

0

1ˆˆ 3I

2

d d

d

V

. [23]

The dipole electric fieldd E

2

00

0

ˆ ˆ, ,S

d ob ob d ob d

k G G

E r r r r r p , [23, 24, 25]

at the point – dipoleobr on the parallel plane to the substrate surface (i.e. P –

polarized field) now takes the contribution first of all from the Green‟s tensor

of free space that determines the direct electric dipole field 0ˆ ,ob d G r r and

secondly from the scattering part of the Green‟s tensor accounting for the

secondary electric field which is related to reflection from the interface



Page 41

ˆ ,S

ob d G r r . The last Green‟s tensor is calculated with Sommerfeld integrals

[23, 24, 25]. To calculate these integrals the use of the Lommel function,

incomplete Weber integrals and incomplete Lipschitz – Hankel integral along

with a zero – order Hankel function of the first kind. Intense numerical analysis

is required, thus a smaller substitute may be to acquire Maple and input the

functions desired and where the error function occurs use an approximation.

To do so use the first desired terms from the Taylor expansion of the error

function, remembering that the error function is an entire function(it has no

singularity except that at infinity). For completion the expansion formula would

take the following shape [22]:

2-

2 1

0

3 5 7 9

2

12

! 2 12

...3 10 42 216

xt dt

o

n n

n

erf x e

x

n n x x x x

x

.

, [22]

The tensor ˆ , 'SG r r also includes the excitation of SPPs for a dielectric – metal

interface. Hence the electric reference field is

t obi

r ob t E e k rE r , [23]

where

2

0 sin ,0, sin 1t k k i , [23]



Page 42

with , the incident angle being larger that the critical one for the TIR which is

given by

1 1 01 1sin sin 41.81.5

critical

glassn

. [22]

The amplitudest E are linearly connected to the incident amplitude and

can be found by using the transfer matrix of Fresnel coefficients.

The Fresnel reflection coefficient for p – polarized waves pr satisfies:

2

22

22

2

cos 1 sin1 1cos sin 1 0 0

lim lim lim 11 1

cos 1 sincos sin 1 0 0

pr

,

[26]

sincecos

lim lim 1cos

,

sinlim lim 0

sin

and

1lim 0

[22] by using

the general infinity - delta for evaluating limits.

Having a p – polarized electric field we have a component parallel and

perpendicular to the field. The parallel component disappears as the angle

reaches 90 degrees because the electric field for these waves is perpendicular

to the direction of propagation. We also notice that by inserting our theoretical

values from above we have first of all that



Page 43

21 2 2 5sin cos 1 sin 1

1.5 3 3 3 ,

then also by substituting the value for the glass dielectric constant we have

2

2

1cos sin

3 2 9@ cos ,sin ,

1 5 3 4cos sin

pr

2

2 2

3 4 9 4 3 4 65 3 6581 3 5 655 9 4 9 5 9 36 5 81

3 4 9 4 3 4 65 3 65 81 3 5 65

5 9 4 9 5 9 36 5 81

81 3 5 6581 3 5 65 81 3 5 65

81 3 5 65 81 3 5 65 81 3 5 65

19683 810 195 1625

19683 1625

21308 810 19518058

0.55

p

p

p

p

p

r

r

r

r

r

.

Hence 55% reflection from the surface. Depending on the material

surface properties the remaining 45% goes to the transmission and absorption

factors as we have seen in chapter one on plasmas.

Finally by illuminating the nanoparticle by a reconstructing plane wave

cE r starting off from the substrate surface at an incident angle , the

holographic image of the dipole source is obtained. This wave shall construct a

transmitted field 't obi

ct ob t e k r

E r E that propagates in the direction opposite to



Page 44

the reference wave t obi

r ob t E e k rE r , with its respective wave number

' 2

0 sin ,0, sin 1t k k i . [23]

To evaluate the dipole momenta of the nanoparticle the full momentum

must be considered as a summation of the i th point:

2 2

00 0

0 0

ˆ ˆ ˆˆ ˆ ˆ, , , N

S S

i ct i i i i i j i j j

j i

k k G G G

p E r r r p r r r r p , [23]

where , ,i i i i

x y zr and , , j j j j x y zr are the radius - vectors of the center of

the particles with number i and j respectively. The other parameters were

defined above and N is the total number of particles in the nanoparticle array.

The reference wave acts as an SPP Gaussian beam excited on the

surface of the metal substrate with propagation along the x – axis. By

considering the electric field at the substrate – air interface, the reference

beam propagating along the x – axis may be given by:

2

2

2

2 2 22

22 1ˆ ˆˆ 1

obSPP ob z ob

ob

yi k x k z

W f x

r ob

ob

SPP obob z z

SPPSPP ob SPP obSPP ob

e

f x

k y y k k z yi x

k k W f x k W f x k W f x

E r

,

[12]

with , ,ob ob ob ob

x y zr , W is the beam waist, ˆ ˆ ˆ, , x y z are the unit coordinate



Page 45

vectors given by the unit dyadic tensor Î ,0SPPk k

is the SPP wave

number , 2 2

0 z SPPk k k and 2

21 ob

ob

SPP

x f x i

k W

. By further detailing on

these terms we have that:

2 2 2 2

0 0 0 0 0

11

1 1 1 z SPPk k k k k k k

Considering our Gaussian beam propagating in free space with the

function w z with a minimum0

w at one place along the beam axis, known as

the beam waist. Now for a wavelength at a distance z along the beam waist,

the variation of the spot size function is given by

2

1o

R

zw z w

z

, [27]

where the origin of the z- axis is defined, without loss of generality, to coincide

with the beam waist W , and where

2

0 R

w z

, [27]

is defined as the Rayleigh range. Thus by inspecting the Gaussian distribution

on a 3D medium we can interpret the equation for the SPP reference wave in

an easier to understand manner.



Page 46



Page 47

CHAPTER 4 – NIM (NEGATIVE INDEX MEDIA)

An NIM (Negative Index Media) refers to medium which possess

negative refractive index and in time are subject to negative refraction. The

idea was originally brought out by Veselago in 1967, where as we discussed

earlier, anomalies occur such as reverse physical equivalents of Snell‟s Law,

Doppler Shift and Cerenkov radiation. The idea of negative refraction at the

interface of a negative and positive refractive index medium allows a flat slab

of NIM to focus all the diverging light from one inside and one outside of the

slab.

Figure 7

An NIM flat lens brings all the diverging rays from an object [28]

Light when emitted or scattered from an object includes not just

propagating waves but also evanescent waves, which carry the subwavelength

detail of the object, and they are sometimes referred to as the image “lost

treasures”. The evanescent waves decay exponentially in a medium with



Page 48

positive refractive index, thus their image cannot be identified in the plane by

a normal lens. But if the lens were made of an NIM placed close enough to the

object, the near – field evanescent waves can be strongly enhanced across the

lens.

Figure 8

NIM lens enhancing evanescent wave propagation such that the

amplitude of the evanescent waves are identical at the object and the image

plane [28]

After escaping the NIM lens the evanescent waves decay again till their

amplitudes reach their original level at the image plane. In the meantime the

propagating waves go through the NIM lens with a negative refraction and a

reversed phase front, ultimately leading to a phase change by the image

plane. The ultimate goal is to apply this to far – field imaging.



Page 49

TIME REVERSAL

NIM media have also established a link between time reversal and

negative refraction. The concept of negative refraction implies that as a pulse

wave moves forward, the phase evolves in the opposite direction, hence the

groups and phase velocity progress through the medium in opposite directions,

0 ,group phase

d d v v

k dk k dk

, [31]

leading eventually to phase reversal.

Considering a medium that can have both a positive and negatively

refracting state where like in graphene the pseudo – spin of an electron close

to the Fermi energy controls the phase evolution [29, 30]. Media with

properties such as these are referred to as “self – conjugate”. This self –

conjugation establishes the link between time reversal and negative refraction.

Let us have a wave of the form:

0 0e

i t E E e

k r , [31]

where E is the electric field component,0

E is a constant,0

e is a polarization

unit vector, r is a position vector and t is the time.

Shifting the frequency by 2 gives the shifted electric field E‟ as:

0 0

ˆ' ei k r t

E e E , [31]

demonstrating that reversing frequency is the same as reversing time. It may



Page 50

be viewed schematically as the following diagram:

Figure 9

The dispersion of the wave in a medium with constant velocity ck

where c is the speed of light, (c = 299792458m/s). Both positive and negative

frequencies may be viewed. Time reversal may be understood as vertical

transition between positive and negative frequencies [31].

Now think of a system having 2 time – reversing sheets separated by a

distance d . Waves are reversed by the first sheet and reversed again by the

second sheet, i.e. switching between positive and negative frequency. To

understand it better we shall make a transmission – reflection typical

coefficient analysis, but in this case we assume that the coefficients are

identical, i.e. T = R. Since the sources are in 2D, they become mirror –

symmetrical about the plane and thus give the same transmitted and reflected

fields. To solve the wave further we must first consider the wave vector

component that is perpendicular to the sheets, i.e.



Page 51

22 2

2

0

z x yk k k c

, [31]

when zk and

22 2

2

0

z x yk i k k c

, [31]

when \ zk .

Thanks to J.B. Pendry‟s formulae [31] we observed that the waves are

allowed to emerge from the end of the cavity. For real zk

22

0 2e ,

1

zi k r k z d t R e

E E d z R

, [32]

while for imaginary zk

2

0 22e ,

1

z

z

i k r k z t

ik d

R e E E d z

R e

. [33]

The limit of very large transmission and reflection coefficients would give

2

0ˆlim e zi k r k z d t

R

E e

E , [33]

given that T = R whether zk is real or imaginary. From the equation above all

Fourier components contribute respectively to the image, and in the infinite

limit the image tends to perfection, i.e. zik d R e .



Page 52

Figure 10

Optics of a negative index slab [13]

By considering this concept we thus have the idea of a superlens, which

bases itself on negative refraction. The main problem of these lenses is not the

loss since the fields are translated trough loss – less vacuum but the ability of

the time – reversing sheet to give enough resolutionmax

1

ln

d

k R .

Like in the theoretical model we discussed the idea would be to create

the following subwavelength – enhancing resolution diagram:

Figure 11

Time – reversing mirrors reproducing similar focusing effect to

subwavelength imaging in the near – field [31]



Page 53

Such a 2D hologram is challenging to build but it is possible by

considering the Fourier coefficients & the work of Bozhevolnyi and Pendry.

Later I will show how such an idea may be modeled by MATLAB.

NEGATIVE INDEX MAKES A PERFECT LENS

The idea of Fourier components of a 2D image has been shown in terms

of transmission and reflection components for a p – polarized field, i.e.

11

11

lim

&

lim 0

zik d

P polarized

P polarized

T e

R

. [31]

What makes the perfect lens more “realizable” is that for a P – polarized

fields, where the electrostatics claim ownership, the dependence of

vanishes and only the dielectric constant takes effect. Hence by further

analysis the transmission coefficient of the slab in the electrostatic limit gives

that the 1 condition satisfies the existence for a surface plasmon as we‟ve

seen in chapter 1. The condition was also shown by R.H. Ritchie [34]. As

Pendry demonstrated, in silver the dielectric function finite imaginary part

prevents ideal reconstruction but a considerable focusing is achieved

throughout. The focusing idea remains valid as we have seen the use of the

quasi – electrostatic limit applied in our earlier theoretical model of a point –

dipole source.



Page 54

FAR FIELD IMAGING

Near – field imaging has shown good prospectus but to achieve the

greater goal we must also consider the far – field subwavelength imaging [12].

To do so we use a hyperlens, works like a superlens and uses cylindrical

coordinates for its analysis. The slab of material on which far – field was

proposed was a magnifying optical hyperlens which had a curved periodic

stack of gold(Au) and aluminium oxide(2 3 Al O ) both of 35nm deposited on a

half – cylindrical cavity fabricated on a quartz substrate. The material had sub

– diffraction – limited objects inscribed into a 50nm – thick chrome layer

located at the inner surface (air side).

Figure 12

Magnifying optical hyperlens with numerical simulation of imaging of sub

– diffraction limited objects. [3, 35]



Page 55

This demonstrated the capability of a hyperlens for sub – diffraction –

limited resolution imaging. The same simulation ideals will be involved in my

MATLAB coding where we have that the propagation will satisfy the condition

give by the evanescent waves, i.e.

2oblique incidence

, [13]

with the wavelength as given by several experiments such as 495, 500 and

515 nm.



Page 56



Page 57

CHAPTER 5 – MATLAB CODE AND SIMULATION

RESULTS

The coding provides detailed explanations and programming comments

to ease reading along with the following main equations for simulation of the

incident electric field.

INCIDENT ELECTRIC FIELD SIMULATION

To calculate in 2D I have made the following assumptions:

1 The polarizability factor ˆ 1 since it requires the evaluation of the volume of

the nanoparticle which is impossible in 2D, i.e. as

0

1ˆˆ 3I

2

dielectricd

dielectric

eV

e

where

34, radius of spherical nanoparticle, not 2D

3d V a a 3D-only

2 The electric field is P-polarized

3 The Green‟s function is the wave propagator exhibiting Huygens wavelet

behaviour, 2 2

1 1,ii jj

ii jj

G r r ii jjr r x y



Page 58

2 2

2 2 2 2

2 2

2 2

, ,

Re Re cos sin cos

cosRe , Re , ,

z z

z

z z

ik x yik y

incident ii ii jj jj

jj

ik y

ii z z z

zik x y ik x y

ii jj jj ii jj jj

jj jj jj

E e G r r e ii jj

e k y i k y k y

k x yG r r e G r r e ii jj

x y

with

2 2 2 2

0 0 0 0 0

11

1 1 z SPPk k k k k k k

The result obtained shows only the real part of the dielectricfunction for the metal gold since a different formula for wavelength – dielectric

function is used for semiconductors like Si and Ge, where we usually consider

a measurement of the thickness of the slab and the screening Coulomb effect

that comes into action.

Now that the SPP group and phase velocities may have opposite signs

then gold dielectric which describes the condition for exceeding the

resonant frequency where &gold dielectric

are the frequency dependent

dielectric constants of the metal and the dielectric, respectively [2].

But by using 2n where dielectric constant and n refractive index,

refractive index tables have that 2

0.47 0.47 0.2209gold gold gold

n .

Then by substituting in all the necessary values we have that the

incident electric field takes the form



Page 59

0

0 0 0

2 2

2 2

2 2

0

1 2 1 2 10000 200 1

1 1 0.2209 12209 12209

0, if and only if 0,

,0 ,

cos

cos ,

200cos

200, cos

12209

z

z jj

incident z iiij

incident

ii

k k

x y z x y

x y x y

k x y

E k y ii jj x y

E x y y

2 2

0

2 2

12209,

j

jj

x y

ii jj x y

The indexes ii and jj were chosen such to distinguish between MATLAB‟s

complex i and j .

Where the summation of the th j components involves the adding of the

respectively cumulative points along the waveguide from the surface, the

measurement will give a theoretically accurate description of the field

intensity.

To account for the ultra – high image resolution we must also include the

approximation for the SPP propagator with the complex parts. Thus the

equation for the electric field becomes:

0

0 2

0

ik z

ik y e E E e z

where

2 2

2 2 , 1,2,3,..., number of points on graphi i z x x y y i N N =



Page 60

Hence z is always positive with 2 2, x y the reference point for evaluating the

intensity of the electric field along all the points.

Now consider the actual E - field as an SPP dotted waveguide with

uniform wavelength and even distribution.

MATLAB Codes:

Program 1

% Alexandru Bratu, Bio Science Masters

% Program #1 - Generate point spread field for polarization equation, take % polarizability tensor alpha as = 1, then proceed with E- field equation% approximation to the Sommerfeld Integrals. I did so by using exponential and % arithmetic equations. At very high numbers(N > 100,000 for example) theequation may be evaluated % using Sterling's approximation to obtain a more accurate result.

%Sterling's equation --> ln N! = N * Ln(N) - N, where Ln is the natural log

close all; clear all;

format short g

diary Superlens22.out

disp(['----------------------------------- SPP Imaging with Reference Point(X2,Y2) -----------------------------------'])

diary off

%1.Wavelength

%"Wavelength" choice of 495,500,515nm; as used in observed experiments wavelength_choice = questdlg('Please select the value for the wavelength in nmyou wish to choose :', ...

'Wavelength - Menu', ... '495','500','515','515');

% Handle response switch wavelength_choice

case '495'

lambda = 495;

case '500'

lambda = 500;



Page 61

case '515'

lambda = 515;

end

%store wavelength in nm to data file wavelength = num2str(lambda);


disp(['Wavelength is ' wavelength ' nm,']) disp([' '])

diary off

%2.Number of graphs

%"Number of graph(s)" choice of 1 - 10; for accurate analysis later and %faster image processing

number_of_graphs_choice = questdlg('Please select the number of graphs you wishto choose :', ...

'Number of Graphs - Menu', ... '2 - 4','5 - 7','8 - 10','8 - 10');

% Handle response switch number_of_graphs_choice

case '2 - 4'

%2 - 4 graphs range choice first_range_choice = questdlg('Please select in the range 2 - 4 :', ... '2 - 4 graphs range', ... '2','3','4','4');

% Handle response switch first_range_choice

case '2'

number_graphs = 2;

case '3'

number_graphs = 3;

case '4'

number_graphs = 4;

end

case '5 - 7'

number_of_graphs = 2;

%5 - 7 graphs range choice second_range_choice = questdlg('Please select in the range 5 - 7 :', ...



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'5 - 7 graphs range', ... '5','6','7','7');

% Handle response switch second_range_choice

case '5'

number_graphs = 5;

case '6'

number_graphs = 6;

case '7'

number_graphs = 7;

end

case '8 - 10'

%8 - 10 graphs range choice third_range_choice = questdlg('Please select in the range 8 - 10 :', ...

'8 - 10 graphs range', ... '8','9','10','10');

% Handle response switch third_range_choice

case '8'

number_graphs = 8;

case '9'

number_graphs = 9;

case '10'

number_graphs = 10;

end

end

%store number of graphs to data file graph_number = num2str(number_graphs);


disp(['Graph has ' graph_number ' curves,']) disp([' '])

diary off

%3.Step size

%"Step size" choice 0.025,0.05,0.1; as these numbers when divided by



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%generate finite rational numbers step_size_choice = questdlg('Please select the value for the angular - step_sizeyou wish to choose :', ...

'Step size Menu', ... '0.025','0.05','0.1','0.1');

% Handle response switch step_size_choice

case '0.025'

step_size = 0.025;

case '0.05'

step_size = 0.05;

case '0.1'

step_size = 0.1;

end

%store number of graphs to data file size_of_step = num2str(step_size);


disp(['Graph has step size of ' size_of_step ]) disp([' '])

diary off

%4.Diagram of the SPP Dotted Waveguide in a P - polarized field

%wave vector constant k = (200 * pi)/(lambda * sqrt(12209));

%store wave vector constant to data file kappa = num2str(k);


disp(['Wave vector constant is ' kappa ' nm^-1']) disp([' '])

diary off

%total number of points on graph N = 20 * number_graphs^2 + number_graphs;


num = num2str(N); disp(['The graph has ' num ' points in total']) disp([' '])

diary off



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%Reference point value x_2 x_2 = input('Reference X - Value ---> ','s'); X2 = str2num(x_2);

%Reference point value y_2 y_2 = input('Reference Y - Value ---> ','s'); Y2 = str2num(y_2);

%store the reference point(X2,Y2) at which the electric field intensity is %calculated to data file


disp(['Reference point for E - field is (' x_2 ',' y_2 ')']) disp([' '])

diary off

%storage of diagram code to data file


disp(['Reference graph showing evenly spread point distribution with respect tofocus reference point (X2,Y2)'])

%initial step before point - to - point uniform distribution disp(['initial step before point - to - point uniform distribution']) ii = 0; disp([' '])

%initial value for sigma summation to used later for calculating total %electric field intensity disp(['initial value for Gaussian propagator summation used later forcalculating total electric field intensity']) E_sum(1) = 0; disp([''])

diary off

%storage of E- field calculations and respective diagrams to data file diary Superlens22.out

disp(['ii a b z E_total'])

for n = [ 1 : number_graphs ] %10 curves drawn for default research

for theta = pi * [ -1 : step_size/number_graphs : 1 ];

ii = ii + 1;

%create a radius that encompasses lambda(/2,/1,/0.5,etc...) by using %the the angle theta, -pi <= theta <= pi x = 0.5 * lambda * %cos(theta), y = 0.5 * lambda * sin(theta)

%x - coordinate

a = 0.5 * n * lambda * cos(theta);



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%y - coordinate b = abs(0.5 * n * lambda * sin(theta));

%line - trace the point with respect to other coordinates line([a X2],[b Y2])

%z - coordinate

%z = sqrt([X2 - a(N)]^2+[Y2 - b(N)]^2) z = sqrt([X2 - a]^2 + [Y2 - b]^2);

%Gaussian propagator function E_sum E_sum = (exp(i * k * z))/z + E_sum;

%E_total = E_sum + incident E- field with initial amplitude %E_0 = 0 E_total = exp(i * k * Y2) + E_sum;

%point number,its with respective x-y coordinates and electric field %intensity disp([ii a b z E_total])

plot ( a , b ,'g*')

figure(1), xlabel('a()'), ylabel('b()'),title('SPP Dotted Imagingwith 1 Reference Point (X2,Y2)')

hold on

end

end

for r = 1:1 plot(X2, Y2 , 'ro')

end

gtext('(X2,Y2)')

%intensity field value at (X2,Y2) E_field = num2str(E_total);

disp(['The value of E at (' x_2 ',' y_2 ') is ' E_field ' Joules'])

diary off

warndlg('The point spread x-y coordinate values generate a hectic gradient E -field, run the second program please.','Warning')

% N.B. DISCUSSION of Figure (1): % ------------------------------------------------------------------------- % Since the (a,b) coordinates would not give an accurate gradient % field, the E - field intensity must taken over the entire plane,i.e. from % [a_min b_min]<=>[a_max b_max] to show the variation, however the ultra -% high focusing may be viewed from the intensity of the blue lines as the % reference point reaches the semi - circular located points. The necessity % of a second program to evaluate the entire gradient E- field was therefore % a requirement. % ------------------------------------------------------------------------



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The program is very flexible and it allows for proper choice of valuesincluding number of curves(precision), wavelength values of 495,500 and 515nm(which are mentioned in [1,2,12,16]), which also determines the number of points N (= 20 x number_graphs 2 + number_graphs). For N > 100,000 itcomes to within a good estimate of Sterling‟s equation and within a few digitsaway from the exact Sommerfeld equations. In terms of wavelength Ipreferred 500nm cause it was a more exact value. And finally in terms of step_size I have chosen 0.025 and 0.05 and 0.1 because they increase from0.025 gradually and also due to the fact that when divided by they producefinite numbers.

The data that follows is taken from the external file produced fromSuperlens11.

Program 1 Results

Wavelength is 500 nm,

Graph has 10 curves,

Graph has step size of 0.1

Wave vector constant is 0.011373 nm^-1

The graph has 2010 points in total

Reference X - Value ---> 1285

Reference Y - Value ---> 1369

Reference point for E - field is (1285, 1369)

Reference graph showing evenly spread point distribution with respect to focus

reference point (X2, Y2).

The (x, y, z) coordinates and their correspondent E – field SPP

calculations are located in the appendix(end) of the paper.



Page 67

Figure 13

Program 1 diagram - dotted SPP imaging for reference point (1285, 1369)

Figure 14

Program 1 diagram - dotted SPP imaging for reference point (0, 0)



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Program 2

% Alexandru Bratu, NanoBio Science Masters

% Program #1 - Generate gradient SPP diagram %

% Warning - Due to the processsing power of my computer i have halved the % components of the y - axis coordinate!

clear all; close all;

%second program is made such to avoid the overwhelming amount of %calculations fomr the first one

%by choosing for 10 graphs we have that the range of the x,y coordinates is %given by %1. a = [-2500,2500]

%2. b = [0,5000] I_sum = 0;

k = (200 * pi)/(500 * sqrt(12209));

X2 = 1285; Y2 = 1369;

% since we have 2010 points on program 1 we now split this up to achieve % the same effect by having x = [-1005:1:1005]

for x = -1005 : 1 : 1005

for y = 0 : 1 : 2010

%Now we include indexes positive bigger than or equal to 1 since %Matlab indexing requires positive integers and not non-negative %i.e. integers including zero

%translation index for x

c = x + 1006;

%translation index for y

d = y + 1;

t = sqrt([X2 - x]^2 + [Y2 - y]^2);

I_sum = (exp(i * k * t))/t + I_sum;

I_total(c,d) = exp(i * k * Y2) + I_sum;

end

end





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2 2

2 2

1,2,...2010,1, ,

, 1285,1369i i

i E z x x y y

x y z

The high focusing could not be achieved from the first diagram since

there were too many points and on more powerful computers the field

achieved a chaotic structure in the form of pylons packed closely together.

The ultra high focusing was however observed in program 1 from the

intensity of the blue colour near (X2,Y2) but I did not include the effect of the

polarizability tensorˆ

to achieve the calculation due to more points and showthe intensity gathering even at these points. To account for this I created the

following theoretical program to illustrate my idea.

Program 3

% Alexandru Bratu, NanoBio Science Masters % program 3 - SPP intensity gradient in the 1st quarter - plane %wavelength chosen from the range 490,500,515 lambda = 500;

for p = 1:2010 for q = 1:2010

E(p,q)= p*q/lambda + 1;

end end

figure(3), imagesc(abs(E)), xlabel('p(number of points)'),ylabel('q(number of

points)'),title('SPP imaging - theoretical constant wavelength of 500nm'),colorbar;



Page 71

Program 3 results

Figure 16

SPP imaging – theoretical constant wavelength of 500nm



Page 72

Program 4

Since the first program involved the evolution of the point – focus the

following coding that was inspired with the help of my supervisors showed the

theoretical perspective of the SPP beam imaging for ultra – high resolution.

The only problem with program number 1 is that the polarization field equation

did not show the “true” gradient envelope I expected. Even after about 160

tests the expected gradient was not achieved by the equations but perhaps an

accurate solution to the Sommerfeld integrals would have achieved this. The

analysis of Smolyaninov‟s work showed that successful simulations involved

study of the beam waist0

w , a much denser and a more accurate complex

equation describing the SPP Gaussian beam. Since my computer simply did not

have such power, computations even over an square array of 1,000,000 still

took several minutes and those above it simply froze and made MATLAB crash

at times. What proved successful however was that the physical concept was

well understood with respect to coordinating the field with the dotted SPP

image. Nonetheless a theoretical model of the waveguide would something

along the following lines of code (with polarizability tensor factor of course):

% Alexandru Bratu, NanoBio Science Masters

% program 4 - theoretical image, conversion from dotted to purely gradient

% coding inspiration by Dr. Brian Vohnsen

clear all;close all;

pixels = 2010;% <--- polarizability tensor included,hence another 100 factors

for x = 1:2010, for y = 1:2010,

E(x,y)= cos(x*y*2*pi/(pixels*pixels));



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end; end;

figure(4), imagesc(abs(E)), axis square, xlabel('X(number of points'),ylabel('Y(number of points)'), title('SPP amplitude(with polarizability factorincluded)'), colorbar;

Program 4 results

Figure 17

Theoretical waveguide on the quarter plane (including polarizability

factor to account for extra points).

The diagram shows how the polarizability factor affects the overall effect

on the P – polarized plane. The desired effect was using the original equations

to create for a reference point of (0,0) the gradient field along the dotted path,



Page 74

i.e. start off with source at zero and create a constant waveguide along it with

wavelength of 500nm or 495 or 515(as discussed by Smolyaninov and his

fellow researchers). The high – resolution diagram was supposed to include a

even more complex outlook since the polarizability tensor created an

interaction between all the points such as in a numerical chain of alpha –

carbons chain – interaction. The calculations would have involved an extra

2010 points and at that stage my computer simply could not cope with 2010 *

2010 * 2010 = 8120601000 such a vast number of computations.



Page 75

CHAPTER 6 – CONCLUSION

The coding provides theoretical evidence for creating a 2D hologram for

sub – wavelength imaging. To achieve ultra – high resolution one needs a low

density of excited fluorescent molecules. It may be achieved by:

1. De – exciting most molecules before imaging (STED): [13, 38]

Stimulated emission depletion microscopy (STED)

Excite fluorophores (here with green light just like in our

simulation, we had values of 495, 500 and 515 nm)

De – excite with STED (here red light, 625- 740 nm)

This emission [13] can be stimulated with the STED

beam(see below)

Delayed STED reduces the PSF (Point Spread Function), where the

PSF is ideally constructed by a full converging spherical

wave(4Pi)[38] . [13]



Page 76

Figure 18

STED apparatus and an intensity distribution graph [13, 36]

2. Exciting only a few molecules at a time (STORM, PALM): [13, 38]

Stochastic Optical Reconstruction Microscopy (STORM)

Multicolored probes to see multiple cellular components at the

same time, such as these microtubules (green) and small hollows

called clathrin – coated pits (red)

Figure 19

Microtubules (green) and clathrin – coated pits (red) [X. Zhuang

Research Labs]



Page 77

PhotoActivitable Localization Microscopy (PALM)

Thin section from a COS – 7 cell expressing the lysosomal trans –

membrane protein CD63 tagged with the PA – Fluorescent Protein

Kaed

Precision was set at the 50nm section by E. Betzig‟s experimental

data as seen below

Figure 20

COS – 7 cell tagged with PA – Fluorescent Protein Kaed [37]

In terms of biological purposes we have methods like Fluorescence

imaging with one nanometre accuracy (FIONA), Standing wave fluorescence

microscopy (SWFM), 4Pi microscopy, and saturated structured–illumination

microscopy. There are also mathematically derivable methods such as point

accumulation for imaging in nanoscale topography (PAINT), i.e.:



Page 78

Figure 21

Far-field fluorescence methods which enable ultrahigh resolution

imaging. [38, 39]



Page 79

ACKNOWLEDGMENT

I would like to personally thank my supervisors/thesis – advisors

Dr. Brian Vohnsen and Dr. James Rice for participating in this project.

Dr. Nicolae Viorel Buchete for inspirational ideas from molecular simulations in Unix,

Dr. Vladimir Lobaskin for notes on manipulating 2nd order Fourier transforms,

Dr. Gareth Redmond for notes on physical concept of microscopy,

Dr. Padraig Dunne especially for his last end of the year lecture notes on lasers,

Professor Nick Quirke for original methods from his papers to handle exponentials,

Dr. Dominic Zerulla for his book on Raether & surface plasmonics,

Dr. Brian Vohnsen for his papers on plasmonics and the 3 lecture notes on

BioPhotonics which helped me design my 1st, 3rd and 4th program,

Dr. James Rice for his own paper on fluorescence microscopy and for helping me to

provide accurate techniques for ultrahigh imaging and create 2nd program

Dr. Suzi.Jarvis for her papers on biomimicry and the field of lipid bilayers that are used in

fluoresce microscopy analysis

Dr. Gerry O’Sullivan for his 3rd year notes on Advanced Quantum Mechanics

Dr. Joachim Raedler for his concepts on fluorescent microscopy

& my colleagues Codrin Andrei, Andrzej S. Pitek and JiaJun Li.



Page 80



Page 81

BIBLIOGRAPHY

[1] J. B. Pendry, Physical Review Letters 85, 3966-3969 (2000).

[2] I.I. Smolyaninov, Y.J. Hung, C.C. Davis, Science 315, 1699 – 1701(2007).

[3] Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, Science 315, 1686 (2007)

[4] Raether, Surface Plasmons(Textbook)

[5] Y.C. Lee, S.E. Ulloa, J.Phys. C: Solid State Phys., 17 (1984) 2239 –

2247

[6] Book - Chow W.W., Koch S.W., Semiconductor – laser fundamentals,

page 86,129[7] Padraig Dunne, Lasers Physics (lecture notes)

[8] http://www.robinwood.com/Catalog/Technical/Gen3DTuts/Gen3DPages/

RefractionIndexList.html

[9] G. Steiner, V. Sablinskas, A. Hübner, Ch. Kuhne, R. Salzer, Surface

plasmon resonance imaging of microstructured monolayers, Journal of

Molecular Structure 509 (1999)

[10] W. Csaszar, A.L. Endrös, Fermi level dependent properties of hydrogen

in crystalline silicon, Materials Science and Engineering B36 112-115

(1996)

[11] Constantine Douketis, Tom L. Haslett, J.Todd Stuckless, Martin

Moskovits and Vladimir M. Shalaev, Direct and roughness-induced

indirect transitions in photoemission from silver films, Surface Science

297 84 (1993)

[12] Jie Yao, Zhaowei Liu, Yongmin Liu, Yuan Wang, Cheng Sun, Guy Bartal,

Angelica M. Stacy, Xiang Zhang, Science 321 930 (2008)

[13] Brian Vohnsen, Nanooptics and Biophotonics (lecture notes)

[14] Victor Veselago, Soviet Phys. Uspekhi 10 509 (1968)

[15] Victor Veselago, Usp. Fiz. Nauk 92 517 – 526 (1967)

[16] Andrey B. Evlyukhin, Sergey I. Bozhevolnyi, Science 590 173 – 180 (2005)

[17] A.B. Evlyukhin, S.I. Bozhevlonyi, Opt. Express 16, 17429 (2008)



Page 82

[18] S. I. Bozhevolnyi and B. Vohnsen,”Phase conjugation of optical near

field by a surface hologram,” Opt. Commun. 135, 19–23 (1997)

[19] B. Vohnsen and S. I. Bozhevolnyi,” Holographic approach to phase

conjugation of optical near fields,” J. Opt. Soc. Am. A 14, 1491–1499

(1997)[20] W. Lukosz and R. E. Kunz,”Light emission by magnetic and electric

dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc.

Am. 67, 1607-1615 (1977)

[21] O. Keller, M. Xiao, and S. Bozhevolnyi, ”Configurational resonances in

optical near-field microscopy: a rigorous point-dipole approach,” Surface

Science 280, 217–230 (1993)

[22] Robert A. Adams, Calculus(textbook), 5th edition

[23] L. Novotny, B. Hecht, and D. W. Pohl, ”Interference of locally excited

surface plasmons,” J. Appl. Phys. 81, 1798–1806 (1997)

[24] T. Søndergaard and S. I. Bozhevolnyi, ”Surface plasmon polariton

scattering by a small particle placed near a metal surface: An analytical

study,” Phys. Rev. B 69, 045422 (2004)

[25] M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, ”Accurate and efficient

computation of the Greens tensor for stratified media,” Phys. Rev. E 62,

5797–5807 (2000)

[26] T. Søndergaard and S. I. Bozhevolnyi, Vectorial model for multiple

scattering by surface nanoparticles via surface polariton-to-polariton

interactions, Physical Review B 67, 165405 (2003)

[27] Lee, S.J.; Lee, Y.L.; Woo, S.Y.; Hwang, W.; Lee, J.H.; Kim, J.H.; Kwak,

C.H, Simple method for determining Gaussian beam waist using lens Z-scan, Volume 1, Issue , 15-19 Dec. 2003 Page(s): 246 Vol.1 -

Digital Object Identifier 10.1109/CLEOPR.2003.1274703

[28] Liu, Z. et al. Rapid growth of evanescent wave with a silver superlens.

Appl. Phys. Lett. 83, 5184–5186 (2003)

[29] K. Kobayashi, J. Phys. Condens. Matter 18, 3703 (2006)

[30] V. V. Cheianov, V. Fal′ ko, B. L. Altshuler, Science 315, 1252 (2007)

[31] J.B. Pendry, Science 322, 71 (2008)



Page 83

[32] S. Maslovski, S. Tretyakov, J. Appl. Phys. 94, 4241 (2003)

[33] C. A. Allen, K. M. K. H. Leong, T. Itoh, IEEE Int. Microwave Theory Tech.

Symp. Dig. 3, 1875 (2003)

[34] R. H. Ritchie, Phys. Rev. 106, 874 (1957)

[35] Far – Field Optical Hyperlens Magnifying Sub – Diffraction- LimitedObjects, Zhaowei Liu, et al. Science 315, 1686 (2007);

[36] Hell & Wichmann, Opt. Lett. 19 (1994) 780

[37] E. Betzig et al, Science 313 (2006) 1642

[38] James H. Rice, Mol Biosyst. 2007 Nov; 3(11):781-93. Epub 2007 Sep 11

[39] Purchased Book, Contemporary Optical Image Processing with MATLAB

by Ting – Chung Poon & Partha P. Banerjee, Elsevier.



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APPENDIX

Results from program 1 data are displayed below for reference point

(X2, Y2) = (1285, 1369).

They include evaluation for the following:

0

0

2 2

0

1

1,2,...,2010 of points on graph

x - coordinate

y - coordinate

1285 1369

_ ,

ii

ii

ik zik Y2

ii number

a

b

z a,b a b

e z E total a b E e

z

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Alexandru Bratu - Masters Thesis - Final - 10