Surface-Plasmon-Polaritons
for
Left Handed Maxwell Systems
In the negative refraction and sub-wavelength imaging
Part-4

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Part 4
Here the slab ENG DNG is treated in detail
Here the concept of transfer function (TF) is detailed
The cut-off due to various non-idealism in TF is discussed
Here the imaging with LHM is introduced, specifically the near
field imaging or sub-wave length imaging with LHM

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Recalling the concepts of SPP
We use refractive index term, gives a factor by which a phase velocity is decreased in material. A LHM ENG, MNG or DNG
has negative index of refraction so the phase velocity is directed against the energy flow, at particular frequency of operation
Refractive index is defined for ENG as
2
n = −ε and neglect
′
m
the losses for simple arguments. We have seen the SPP
penetrates and thus this has consequence of strong skin effect, for a slab of LHM of width d . The skin effect, of the form
−
<<
is that the field decays exponentially in the bulk of the film. We have two modes corresponding to the
ex p ( k d n )
1
0
symmetric and anti-symmetric field distribution inside the (plasmonic) metal or LHM, and thus two SPP waves given by
2
⎡
2 n
⎤
k n
where
= 2 = − ′
0
,
n
ε = 1
k
= ∞
m
is case of SPP resonance
k
= k
1 ±
ex p ( − d k n )
k
p
1, 2
p ⎢
⎥
p
4
⎣
n − 1
p
⎦
2
n − 1
It is important to note that symmetric and anti-symmetric SPP propagate on the both sides of the film. Moreover these SPP
represent eigenmodes the magnitude of the E and H fields are same on both the interfaces. The consideration holds for
arbitrary thick film, although the difference in the speed for the two SPP become exponentially small for thick films
d k n >> 1 the velocities of both the symmetric and anti-symmetric SPP are less than c . These SPP cannot be excited by
p
an external E.M. waves incident from air side, that would violate momentum conservation.
The situation changes dramatically when the film has periodically modulated material properties say n. In this case the EM
fields inside the film gets modulated. When the spatial period of the modulation coincides with the wavelength of SPP, it can
be excited by an incident EM .
The EM wave incident on the modulated film first excites the SPP
on the front of the film which then couples to the back interface, the
back side eventually converts into Transmitted EM wave.
Thus at a particular wavelength of excitation depending on the modulation period
the slab otherwise opaque can turn into extra-ordinary transmittance (transparent)
d
for EM Waves (light) . Well if the slab is having also
μ < , then the evanes
0
cent Tunneling of evanescent waves by SPP
waves inside slab can turn to be propagating one!!
coupling

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SPP modes for a slab of LHM (DNG)
This is same as was discussed for ENG material. If this slab is sufficiently thin, the modes on each sides are couples to each
other. As a consequence the number of branches in the dispersion equation of a slab will double in comparison to single
interface. Each mode splits into symmetric and anti-symmetric branches. Comparing them to the unperturbed case for single
interface show some striking feature .
Case-I
For (b), case II apart from the low
ω TM, splits
D N G
into two; two further modes, a backward and
a forward one appear that asymptotically
approach miraculous frequency at which
μ = ε = −1 . The higher is , the
k
r
r
x
closer are the two modes (one above & one
μ > 0
μ > 0
below) to the single interface curve.
Due to different character of the frequency
dependency of and
ε
, the split
μ
between the TE modes looks different from
the split between the TM modes (it is actually
Case-II
larger for TM).
D N G
D N G
μ > 0
μ > 0
μ > 0
Single interface unperturbed case

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(
)
Choosing the parameters for LHM slab
ω +
B A C K W A R D - W A V E
field
For a slab with frequency independent
μ r2 The upper branch gives the
(
)
TM
ω +
backward wave. When the frequency
μ
ε
= −
(
)
ω +
1
r 2
dependency of is put these and
(
)
ω − branches again split.
A
D
F
d
E N G
The split TE and TM of upper branch approach
μ = ε = −1 line, as
ω
k is increased.
x
o p era tio n
The choice is thus to excite SPP for DNG
where the backward wave is profuse, so that
determines the range of frequency of operation.
D N G
Tuning of thickness d so that the upper branch
(
)
μ
ω + is profuse. The choice of modulation
so that and
ω < ω
F for resonator so as to
0
p
fix the operation points for desired dispersion
μ > 0
character of LHM to work with backward wave
in DNG region. Good choice ω = 0 .6ω
0
p
This is art of design of LHM parameters.

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Fourier transform and the transfer function
Spatial in contrast to temporal transform. Spatial information is of interest in both the actual space domain
and in ‘spatial frequency’ domain. Let us take a complex two variable function , its
g ( x , y )
F.T is
{
−
π
+
ℑ g x y }
j 2
( f x
f y )
( ,
)
= G ( f , f ) =
g ( x , y )
x
y
e
d x d y
∫
x
y
f
and are spatial
f
frequency and is amplitude.
G ( f , f )
x
y
x
y
Note that the spatial frequencies are related simply to
previously defined ‘wave numbers’ as
k = and
2π f
x
x
= π
. The problem usually arising is to find the
k
2
f
y
y
variation in x and y of some function at plane when
z 2
function at is known.
z
The space between & is
z
z
1
1
2
filled by some medium. All that can happen to a particular
pair is that it will have different amplitude and phase.
The function that tells us what happens to all the spatial frequencies
is called transfer function
T ( f .
, f ) We may denote the function (say
x
y
tangential component of E field), varying in the plane as a function
z1
of x and y by
g ( x , y )
1
z = z1 . Its spatial harmonics (which together we may call the Fourier Spectrum) are given
by its F.T
G ( f , f )
1
x
.
y
When travelling the medium each of the Fourier components will undergo some
changes corresponding to the T.F
T ( f . Hence
, f )
z
T ( f , f )G ( f , f )
x
y
the Fourier Spectra at is
2

x
y

1
x
y
.
Inverse F.T is :
−
ℑ {
j
π f x + f y
G ( f , f )
= g x y
= T f
f
G
f
f
e
d f d f
∫
x
y }
2
1
(
)
( ,
)
(
,
)
(
,
)
x
y
=
x
y
1
x
y
x
y
z
z 2

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F.T of a slab & Transfer Function example
As an example let us find the T.F. of a wave in free space propagates at an angle relative to
θ
z. Its phase varies in z & x as
1
− j [k
+
2
2
2
2
+
=
=
μ ε
z z
k x x ] where
k
k

k
ω . The period along the x-axis is determined by , it is actually
e
z
x
0
0
k x
equal to
2 π / , is sim
k
ilar to temporal period
T = 2π / . Here
ω
in analogy we may call as
k x
spatial frequency.
x
Its amplitude at
z = is equal
0
to ‘one’. When the wave propagates from to
z = 0
z , its complex
= d
amplitude will
1
−
be
j k
d
z
where
1

2 =
2
2
−
= ω
2
2
μ ε −
. In this example clearly the phase has changed. The T.F is thus
e
k
k
k
k
z
x
0
0
x
2
2
− j
ω μ ε −
0
0
k
d 1
T ( k
)
x
= e
x
x
F
4ζ e
T =
=
2
jk z 2d
2
jk z 2
A
(1 + ζ ) e
− (1 − ζ
−
)
d
e
e
e
T ( k ) = T ( k )T ( k )T ( k )
t
x
1
x
2
x
3
x
2
2
− j
ω
μ ε
−
0
0
k
d 1
T ( k
)
x
= e
1
x
d
d
d
4ζ
1
2
3
T (k )
e
=
z
2
x
2
jk z 2d2
2
jk z 2d2
(1 + ζ ) e
− (1 − ζ
−
) e
e
e
2
2
− j
ω
μ ε
−
0
0
k
d 3
T
( k
)
x
= e
3
x
What are the range of the spatial frequencies. From analogy of temporal frequency we may assert that it can take any value
from zero to infinity. Obviously the smaller details of a spatial function the higher the spatial frequencies that can reproduce
details . Note the equation
2
2
k + k
=
2
2
k
= ω m
με
ust be satisfied. When spatial frequency is high enough to satisfy this
z
x
equation the inequality then
>
is im
k
aginary (SPP) , still the expressions are valid.
k
ω εμ
x
z
Valid for propagating harmonic wave functions as well as bounded wave functions.

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Field quantities in the three regions and Fourier expansion.
We consider TM for simplicity. The field from source in the plane
z are
= 0
expanded in form of Fourier Series over the
propagating and evanescent components (waves) i.e. for a
k < k
k > k
ex p ( jω t )
x
0
nd
x
0 ; all of which is proportional to
− jk x
in steady state.
H ( x , z = 0 ) = ∑ A ( k )
x
e
y
x
k x
The field expansions of the full solution everywhere in x-z plane takes the form in region 1, 2 & 3 the H field as:
−
−
H
( x , z )
j k
z
j k
z
j k x
z
z
x
= ∑ A e
+ B e
e
ε
y 1
(
1
1
)
1
D ( k )
x
k
B ( k )
x
x
−
−
F ( k )
H
( x , z )
j k
z
j k
z
j k x
x
z
z
x
= ∑ C e
+ D e
e
y 2
(
2
2
)
ε
x
k
3
x
C ( k )
−
−
x
H
( x , z )
j k
z
j k x
z
x
= ∑ F e
e
ε 2
y 3
(
3
)
A ( k )
z
x
k x
Corresponding E fields are from curl H:
k
k
z1
−
−
−
−
E ( x, z )
jk
z
jk
z
jk x
x
jk z
jk z
jk x
z
z
x
= ∑
A e
− Be
e
E ( x, z)
z
z
x
= −∑
Ae
+ Be
e
z1
(
1
1
)
x1
(
1
1
)
ωε
ωε
k
k
x
1
x
1
k
k
z1
−
−
−
−
E
( x, z )
jk
z
jk
z
jk x
x
jk
z
jk
z
jk x
z
z
x
= ∑
C e
− De
e
E (x, z)
z
z
x
= −∑
Ce
+ De
e
z 2
(
2
2
)
x 2
(
2
2
)
ωε
ωε
k x
2
kx
2
k
k
z 3
−
−
−
−
E
( x, z )
jk z z
jk x x
= ∑
F e
e
E (x, z)
x
jkz z jkx x
= −∑
Fe
z 3
(
3
)
x 3
(
3
)
ωε
ωε
k
k
x
3
x
3
If the medium 1 and 3 are air with
ε = ε = 1
μ = μ = 1
1
a
2
nd
1
3
then we get the A, B, C, D, F as earlier expressed, In
particular the transmission coefficient at distance 2d is
F
− jk z 1d
T =
e
A

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Far field / Near field refraction from flat LHM slab
2
2
We shall take object in form of a Gaussian function
ex p ( − x and assume
/ τ )
to start with that object size (half width
of Gaussian function) is equal to “one-wavelength” and the width of LHM slab is large
=
.
λ The figure show
d
5 0 0
The Poynting vector stream lines taking only the propagating components into account. Note here that Poynting vector
stream lines is exactly same as the “optical ray diagram”. This is Vaselago’s (1970) method with LHM considering n = − 1
The diagram show ‘internal focus’ rather two foci.
What about evanescent waves? Surely when they travel a distance
2 5 , they are entir
0 λ
ely negligible. For a value of k x
6 8 2
just above ‘propagating spectrum’
k , (
k = 1 .0 0 0 1k
k > k
ex p ( −κ d / 2 )
1 0 −
0
x
) ;
0
x
the value of
0

1
comes to , negligible!!
Half width
1λ
Half width
5λ
When the object size is increased to five folds the Poynting vector streams are concentrated because bigger object gives
narrow beam width
Misconception:
Believed by many: The physical picture of imaging with aid of geometric optics is still approximately true in presence of
evanescent waves.
Not true, not even approximately. There is no internal focus when the object is sub-wavelength. The Poynting vector
stream lines converge upon the outer boundary of the lens.

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Near field/ sub-wave length refraction from flat LHM slab
Let us choose ‘sub-wave length’ object of half width
0 .1λ
and a slab of LHM of width
d .
= 0.5λ When we
consider ‘propagating waves’ only i.e.
0 < k < k
x

0 Then there is maxima in the middle of the slab indicating internal focus.
Let us now include the spectrum of evanescent components in range
k
< k < 1. and
2 k
k < k < 1 .3k
. The ‘gentle’
0
x
0
0
x
0
maxima is seen to be shifting towards rear surface, becoming ‘sharper’ internal maxima. When the upper limit of is
k x
chosen to be
1 .4 k then the maxima is clearly at the rear surface. The internal focus disappears, but with out influencing
0
the image. In all cases the image invariably appears in the image plane. If we include more and more evanescent spectral
component, the maxima at the rear surface rises rapidly but the position of image is fixed. This is sub wavelength imaging
with near field restoration with
n with
= −1
μ = ε = −1 is Pendry’s lens (2000).
r
r
d = 0 .5λ
2 d

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