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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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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
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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 Greens 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 .......................................................................... 24Internal 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 .............................................................................. 38Chapter 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 SPs 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
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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 plasmonswill
carry energy of the order
2
0
4p
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 Maxwells theory the
EM surface waves can propagate along a metallic surface or film with a broad
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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]
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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 .
Surface plasmon polaritons (SPPs) 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 1, 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
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0 exp x zE E i k x k z t , [4]
where + for z 0, for z 0, an imaginaryzk would cause an exponential
decay for zE . The equation gives that directionxk 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 SPs propagating on a
surface in the x direction are shown schematically. The exponential
dependence of the field zE is seen on the right. yH shows the magnetic field in
the y direction of this p polarized wave. [4]
With the use of Maxwells equations the retarded dispersion relation for a
plane surface semi infinite metal along with the dielectric function
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' '1 1 2 when close enough to the medium 2 as air or vacuum:
1 20
1 2
0z zk k
D
, [4]
together with
2
2 2
i x zik k
c
, [4]
or
12 2
2 , 1,2zi i xk k ic
, [4]
The wave vectorxk is continuous through the interface hence we can
rewrite0
D :
1
21 2
1 2
xkc
, [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
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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 FTmetal
kq
q , [5, 10]
where
dielectric constant of the metal, thomas fermi wavenumber, wavenumbermetal FT k q
which is derived from Lindhards formula [6] for the longitudinal dielectric
function given by
, 1k q k
q
k k q k
f fq V
i
, [6]
with
q eff ind V V q V q andcarrier distribution function,aka the Fermi - Dirac distribution
function for electrons under stable thermodynamic equilibriumkf
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We must also study the formula under the limiting case. Lindhards 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 qE 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
ki
iq
k i
q ki
k i i
q kk
q
pl
fq
kV
k q
m
V f k qq
k m
q
V fm
q NV
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 NE V
q L L m
, [6, 39]
Now consider the static limit 0i where Lindhards formula takes
the following shape:
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2 2
, ,
,0 1 1
i i
i iq q
k i k i
f fq q
k kq 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 fq q q k
k k m
, [6, 39]
where
22 2
&2
k ik
i
kk
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
ki i
q q k
k i k k
fq 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
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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
rL 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
fE n
m , [6, 39]
Hence the density n is given by
3
2
2 2
1 2
3f
mn E
, [6, 39]
At 03
T 0 K E2
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:
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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
FTmetal
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 3Nd: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:
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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
40.24 1.08 100.2209
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 smallc
, but remains
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further away than 2c
such that the surface plasmons cannot convert into
light. However at larger values ofxk or
'
1 2 , we have that
2
lim1x
p
SPk
, [4]
where thep
corresponds to the plasma frequency, given by24
p
e n
m
,
with n the electron bulk density.
Asxk gets larger we also have that
lim 0
lim 0
x
x
k
k
group velocityk
dphase 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& 0xkc
, [4]
so that the amplitude of the surface plasmons have an exponential decay
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zik ze
normal to the surface. From this information we gather that the depth value of
the field falls to 1/e, becomes 1zi
zk
or
1
' 21 2
2 2
2
'
1 21 '2
1
in the medium:2
in the metal:2
z
z
, [4]
By using 2 metals with internal absorption such as gold and silver, we
get:
2
1
2
1
390:
240600
280:
310
z nmsilver
z nmAt nm
z nmgold
z nm
, [4]
since
1lim
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 componentzE compared to the longitudinal component xE , as
'
1
z
x
E i
E , [4]
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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 metalz x
E E since
'
1
z
x
E i
E . [4]
The theory can be validated from Maxwells theory in vacuum
0 or 0yx z
EE EE
x y z
, [4]
however it suffices only outside the surface of the air/metal interface and at
largexk value the z and x component would suffice:
:
:z x
air iE 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
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22 @ 514.5iL 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
kc
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 energy 0E to the
electrons of the solid material. The wave vectorxk 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 ' sinx el elk k k with
2el
el
k
. [4]
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To observe a successful excitation the use of fast electrons is highly
recommended such as to study the dispersion relation at larger px x
k kc
,
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 ofx
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]
wherexk rises from the perturbation along the smooth surface. We observe
that the resonance can be observed as the reflected light minimum. For the
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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 componentxk
byx
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
1pr 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:
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pr sp
x x
pr prismk k
sp surface plasmon
[4, 24]
1 2
1 2
sinpr pr
c c
[4, 24]
The reflected intensity R may be given by Fresnels 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 zkk i
p pk i i k i k i k i k i k
ik kizi 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 kki ki ik
i ip 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 dp p
prp
pr ik dp p
pr
i k
p k i i k i k ik
i kk i i k
i k
r r eR r
r r e
n n n nr
n n
n n
, [4, 24]
1 1 2 2sin sin sinpr prn n n
, [4, 24]
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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 prpr
prp
pr
prpr
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 dp p
prp
pr i k dp 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 ppr 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 41 12 1 12 1z zik d ik d p p p p p p
pr pr pre t r r r t e
, [4, 24]
where the phase factor 1 1 1 1 1coszk 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 sinpr
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]
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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 .
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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 imaginary1zk ,
21 1 1 0z xk kc
, [4]
transform the in the medium0
into a plane wave due to the real 2zk ,
since0xk
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]
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with1
p
prr 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
12p
prr 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].
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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.
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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
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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
226.15 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. Snells Law
By definition we have that 1 1 2 2sin sinN N , but since 2 0N , 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:
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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
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University of Pennsylvania by Andrea Al 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.
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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 constantp
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 - space with its center located at 0,0,p pzr . To support
SPP propagation requires that we must have the metal dielectric interface with
the conditionm r
[16]. For this scattering system we get:
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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 dzr , [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 .
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Figure 6
Schematic view of the holographic technique with a TIR reference wave
[17]
The dipole has an electric monochromatic field 0 dE 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
rE r . Further detailed study of the near field above the surface dz ,
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
3dV a a particle radius , [22]
and a dielectric permittivityd with external optical field with wavelength
bigger than the particles size hence not limiting the analysis to metamaterials.
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We now obtain a dipole moment p :
1
2
0, 0
0
I ,S
d d d
kG
p r r E r [23]
where I xx yy zz (in spherical coordinates)[22] is the unit dyadic tensor,
ok is the wave number in vacuum, 0 is the vacuum permittivity, , 'S
G r r is
the surface part of the Greens 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
1 3I
2
dd
d
V
. [23]
The dipole electric fielddE
2
00
0
, ,S
d ob ob d ob d
kG 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 Greens tensor
of free space that determines the direct electric dipole field 0 ,ob dG r r and
secondly from the scattering part of the Greens tensor accounting for the
secondary electric field which is related to reflection from the interface
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,S ob dG r r . The last Greens 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 nx x x x
x
.
, [22]
The tensor , 'SG r r also includes the excitation of SPPs for a dielectric metal
interface. Hence the electric reference field is
t obir ob t E ek r
E r , [23]
where
20 sin ,0, sin 1tk k i , [23]
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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 amplitudestE 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
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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 [email protected] 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 rE r E that propagates in the direction opposite to
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the reference wave t obir ob t E ek r
E r , with its respective wave number
' 20 sin ,0, sin 1tk k i . [23]
To evaluate the dipole momenta of the nanoparticle the full momentum
must be considered as a summation of the ith point:
2 2
00 0
0 0
, , ,N
S S
i ct i i i i i j i j j
j i
k kG G G
p E r r r p r r r r p , [23]
where , ,i i i ix y zr and , ,j j j jx 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 1 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 yy k kz yi x
kk W f x k W f x k W f x
E r
,
[12]
with , ,ob ob ob obx y zr , W is the beam waist, , ,x y z are the unit coordinate
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vectors given by the unit dyadic tensor I , 0SPPk k
is the SPP wave
number , 2 20z SPP
k k k and 22
1 obob
SPP
xf 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 1z SPPk k k k k k k
Considering our Gaussian beam propagating in free space with the
function w z with a minimum 0w 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
1oR
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
0R
wz
, [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.
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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 Snells 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
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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.
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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 phased d
v vk 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 tE 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 tis the time.
Shifting the frequency by 2 gives the shifted electric field E as:
0 0
' ei k r t
E eE , [31]
demonstrating that reversing frequency is the same as reversing time. It may
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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.
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22 2
2
0
z x yk k kc
, [31]
when zk and
22 2
2
0
z x yk i k k c
, [31]
when \zk .
Thanks to J.B. Pendrys formulae [31] we observed that the waves are
allowed to emerge from the end of the cavity. For realzk
22
0 2e ,
1
zi k r k z d t R eE E d z
R
, [32]
while for imaginary zk
2
0 22e ,
1
z
z
i k r k z t
ik d
R eE E d z
R e
. [33]
The limit of very large transmission and reflection coefficients would give
20
lim 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 dR e .
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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]
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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 weve
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.
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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 3Al 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]
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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.
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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 1 since it requires the evaluation of the volume of
the nanoparticle which is impossible in 2D, i.e. as
0
1 3I
2
dielectricd
dielectric
eV
e
where
34
, radius of spherical nanoparticle, not 2D3
dV a a 3D-only
2 The electric field is P-polarized
3 The Greens 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
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2 2
2 2 2 2
2 2
2 2
, ,
Re Re cos sin cos
cosRe , Re , ,
zz
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 1z 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
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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
zjj
incident z iiij
incident
ii
k k
x yz x y
x y x y
k x y
E k y ii jjx y
E x y y
2 2
0
2 2
12209,
j
jj
x y
ii jjx y
The indexes iiandjjwere chosen such to distinguish between MATLABscomplex iandj.
Where the summation of the thj 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 eE E ez
where
2 2
2 2 , 1,2,3,..., number of points on graphi iz x x y y i N N =
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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 shortg
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 experimentswavelength_choice = questdlg('Please select the value for the wavelength in nmyou wish to choose :', ...
'Wavelength - Menu', ...'495','500','515','515');
% Handle responseswitch wavelength_choice
case'495'
lambda = 495;
case'500'
lambda = 500;
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case'515'
lambda = 515;
end
%store wavelength in nm to data filewavelength = num2str(lambda);
diary Superlens22.out
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 responseswitch number_of_graphs_choice
case'2 - 4'
%2 - 4 graphs range choicefirst_range_choice = questdlg('Please select in the range 2 - 4 :', ...'2 - 4 graphs range', ...'2','3','4','4');
% Handle responseswitch 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 choicesecond_range_choice = questdlg('Please select in the range 5 - 7 :', ...
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'5 - 7 graphs range', ...'5','6','7','7');
% Handle responseswitch 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 choicethird_range_choice = questdlg('Please select in the range 8 - 10 :', ...
'8 - 10 graphs range', ...'8','9','10','10');
% Handle responseswitch 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 filegraph_number = num2str(number_graphs);
diary Superlens22.out
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 numbersstep_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 responseswitch 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 filesize_of_step = num2str(step_size);
diary Superlens22.out
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 constantk = (200 * pi)/(lambda * sqrt(12209));
%store wave vector constant to data filekappa = num2str(k);
diary Superlens22.out
disp(['Wave vector constant is ' kappa ' nm^-1'])disp([' '])
diary off
%total number of points on graphN = 20 * number_graphs^2 + number_graphs;
diary Superlens22.out
num = num2str(N);disp(['The graph has ' num ' points in total'])disp([' '])
diary off
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%Reference point value x_2x_2 = input('Reference X - Value ---> ','s');X2 = str2num(x_2);
%Reference point value y_2y_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
diary Superlens22.out
disp(['Reference point for E - field is (' x_2 ',' y_2 ')'])disp([' '])
diary off
%storage of diagram code to data file
diary Superlens22.out
disp(['Reference graph showing evenly spread point distribution with respect tofocus reference point (X2,Y2)'])
%initial step before point - to - point uniform distributiondisp(['initial step before point - to - point uniform distribution'])ii = 0;disp([' '])
%initial value for sigma summation to used later for calculating total%electric field intensitydisp(['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 filediary 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
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%y - coordinateb = abs(0.5 * n * lambda * sin(theta));
%line - trace the point with respect to other coordinatesline([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_sumE_sum = (exp(i * k * z))/z + E_sum;
%E_total = E_sum + incident E- field with initial amplitude%E_0 = 0E_total = exp(i * k * Y2) + E_sum;
%point number,its with respective x-y coordinates and electric field%intensitydisp([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 = 1:1plot(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 ofpoints N (= 20 x number_graphs 2 + number_graphs). For N > 100,000 itcomes to within a good estimate of Sterlings 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 ofstep_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.
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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 havingx = [-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
iE z x x y y
x yz
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 tensor
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 490,500,515lambda = 500;
for p = 1:2010for q = 1:2010
E(p,q)= p*q/lambda + 1;
endend
figure(3), imagesc(abs(E)), xlabel('p(number of points)'),ylabel('q(number of
points)'),title('SPP imaging - theoretical constant wavelength of 500nm'),colorbar;
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Program 3 results
Figure 16
SPP imaging theoretical constant wavelength of 500nm
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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 Smolyaninovs 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;%
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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,
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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.
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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]
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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]
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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. Betzigs 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 structuredillumination
microscopy. There are also mathematically derivable methods such as point
accumulation for imaging in nanoscale topography (PAINT), i.e.:
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Figure 21
Far-field fluorescence methods which enable ultrahigh resolution
imaging. [38, 39]
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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 ultra- high 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 OSullivan 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.
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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)
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RefractionIndexList.html
[9] G. Steiner, V. Sablinskas, A. Hbner, Ch. Kuhne, R. Salzer, Surface
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(1996)
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[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)
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